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VESSEL DESIGN
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Page 1: Process Equipment Design

V E S S E L

D E S I G N

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Page 2: Process Equipment Design

LLOYD E. BROWNELL

Professor of Chemical and Nuclear Engineering

University of Michigan

EDWIN H. YOUNG

Associate Professor

of Chemical and Metallurgical Engineering

Reg. pu’[email protected]

. . ACC. No..357University of Michigan

Lib. Asstt . . . . . . . . . . . . . . . r/C . . . . . . . . . .

JOHN WILEY & SONS

New York l Chichester l Brisbane l Toronto l Singapore

Page 3: Process Equipment Design

20 19 18 17 16 15 14

Copyr ight @ 1959 by John Wiley 8 Sons , Inc .

.411 rights reserved.

Reproduction or translation of any part of thi\ work beyond thatpermitted by Sections 107 or 108 of the 1976 Unlted States Copy-rtght Act without the permisbton of the copyright owner is unlaw-

ful. Requests for perm!sston or further tnformatton should beaddressed to the Permissions Department, John Wiley&Sons, Inc.

l ib rary of Congress Catalog Card Number : 5?--5882

Printed in the United States of America

ISBN 0 4 7 1 1 1 3 1 9 0

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PREFACE

This book was prepared primarily for se~io-and students in engineer-ing. The needs of design engineers and consultants as well as those of studentswere considered in selecting the topics and methods of presentation. The bookis based upon our experiences gained in industrial design offices and in 16 years ofteaching courses in equipment design at the University of Michigan. We bothhave supervised research and development of process equipment, and have actedas consultants in this field.

The book was originally prepared as class notes, which have been used forabout ten years in teaching courses in process equipment design at the senior andgraduate levels in the Chemical and Metallurgical Engineering Department ofthe University of Michigan. Typical problems have involved the design offraction@ingt o w e r s , trm vacu>m cry&all&m, condensers, heat exch-rs,high-pres?re reactors, and other types of process equipment.

The design of process equipment requires a thorough knowledge of the func-tional process, the materials involved, and the methods of fabrication. The___- ._....design factors to be considered are many and varied and, in most cases, so inter-woven that exact methods of attack are often impossible to formulate. Com-promises are necessary and the design engineer often has only experience insimilar or related fields to guide him in his choice. Thus, the engineer mustrealize that considerable engineering judgment is required in applying all recom-mended specific methods of design.

One purpose of this book is to consolidate the basic concepts, industrial prac-tices, and theoretical relationships useful in the design of processing equipment.Many of these considerations and much of this vital information are widelyscattered throughout the technical literature, industrial bulletins, appropriatecodes, and handbooks. It is not intended that this book should cover all theramifications of design problems, but it will serve as a guide to the student and thepracticing engineer for efficient and economical design of equipment for theprocessing industries.

vii

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Page 6: Process Equipment Design

. . .VIII Preface

The organization is based on the premise t)hut t,he vessel is the basic part ofmost types of processing equipment.

For example, a heat exchanger or evaporator is a vessel with tube bundles and afractionating tower is a vessel with trays. The first 12 chapters are concerned inpart with the development of fundamental relationships on which many of thecode specifications are based. Chapter 13 is concertled entirely with code prac-tice and covers selected code specifications not covered in the earlier chapters.Chapters 14 and 15 are concerned with the design of vessels beyond the scope ofthe ASME code.

The sequence of chapters was selected to permit the introduction of a brietreview of elementary theories of mechanics and strength of materials early in thebook. More advanced theory is developed as needed in subsequent chapters.The integration of theory with practice in design eliminates the necessity of aseparate section on erigineeritlg mechanics. The sequence of presentation allowsfor an orderly development of theoretical relatiotlships when the book is beingused as a textbook in teaching design. The material presented covers the raugefrom simple vessels for low-pressure service to thick-walled vessels for high-pressure applications. Tl~tf rxperierlced designer will find the book useful as areference in a design office.

In all but a few cases derivations of equations and the method of analysis havebeen given so that the etlgirleer will utlderstaltd the assumptions and limitationsinvolved. Also, example calculations and designs have been included to illustratethe use of the relationships and recommended procedures.

We wish to acknowledge the assistance given by a large number of individualsand companies in providing subject material and illustrations on process equip-ment, design and in making reviews and suggestions. We are particularlyiudebted to the following: C. E. Freese, Mechanical Consultant, and B. B. Kuist,The Fluor Corporation; W. H. Burrows, Chief Engineer, Manufacturing Depart-ment, Standard Oil Company of Indiana; A. E. Pickford, Department Head,.dpparatus Design, C. F. Braun and Company; H. B. Boardman, Director ofltesearch, L. P. Zick, Research Engineer, and E. N. Zimmerman, Chicago Bridgeand Iron Company; W. T. Gurm and Walter Samans, American PetroleurnInstitute; J. M. Evans, Chief Engineer, and F. L. Maker, Standard Oil Companyof California; R. S. Justice, Chief Engineer, Gulf Oil Corporation; F. L. Plummer,Director of Engineering, Hamrnond Iron Works; W. D. Kinsell, Manager, Con-struction, Engineering Department, The Pure Oil Company; G. E. Fratcher,Director of Engineering, A. 0. Smith Company; F. E. Wolosewick, Sargeut andLundy Engineers; P. E. Franks, Chief Engineer, Sinclair Refining Cornpany;D. W. Carswell and H. B. Peters, Chief Engineer, The Texas Company; W. T.Brown, Manager, Mechanical Division, and Harry Wearne, Construction Man-ager, Shell Oil Cornpany ; F. J. Feeley, Jr., Assistant Director, Engineering DesignDivision, Esso Hesearch aud Engineering Company; J. H. Faupel, E. I. du Pontde Nemours and Company; W. H. Funk, Lukens Steel Company; and the follow-ing additional companies and organizat3ons: Horton Steel Works, Ltd.; Blaw-Knox Company; Graver Tank and Manufacturing Company; American CyanamidCornpany; Inland Steel Company; Ryerson Steel Company; Taylor Forge andPipe Works; Aluminum Company of America; M. W. Kellogg Company; Amer-ican Standard Association, Inc.; The Girdler Company, Inc.; Baldwiu-Lima-Hamilton Corporation; Bethlehem Steel Compally, Inc.; ITnited States Depart-ment of Interior, Bureau of Miues; Great Lakes Steel Corporation; McGraw-Hill

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Preface ix

Book Company, Inc.; I‘uivrrsal-Cyclops Steel Corporatiorr: at~d the UnitedStates Steel Corporation.

We also wish to express our appreciation to the Amrricau Society of MechanicalEngineers and the American Petroleum Institute for permissiou to use selected

material from the 1956 edition of the Unfired Pressure L‘essel Code and the APISpecification for Welded Oil Storage Tanks and Production Tanks, respectivei).We are also indebted to Dr. J. McKetta, Mr. F. L. Standiford, Dr. H. H. Yang,and Dr. M. D. S. Lay, who assisted in the preparation of the course notes while

enrolled in the Graduate School of the University of Michigan, and to ProfessorDonald L. Katz, Chairman, Department of Chemical and Metallurgical Engi-neering, University of Michigan, for encouragement and advice in the preparationof this book. Many individuals have given valuable suggestions, comments, andassistance in the preparation of this book and ally omissions irr ackuowledgmentare not iutended.

Ann Arbor, Michigm!April, 1959

LLOYD E. BROWNELLEDMIK i-z. YOUNG

Page 8: Process Equipment Design

CONTENTS

. .

cChapter 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

References

Appendix A.

B.

Factors Influencing the Design of Vessels

Criteria in Vessel Design

Design of Shells for Flat-Bottomed Cylindrical Vessels

Design of Bottoms and Roofs for Flat-Bottomed Cylindrical

Vessels

Proportioning and Head Selection for Cylindrical Vessels with

Formed Closures

Stress Considerations in the Selection of Flat-Plate and Conical

Closures for Cylindrical Vessels

Stress Considerations in the Selection of Elliptical, Torispheri-

cal, and Hemispherical Dished Closures for Cylindrical Vessels

Design of Cylindrical Vessels with Formed Closures Operating

under External Pressure

Design of Tall Vertical Vessels

Design of Supports for Vertical Vessels

Design of Horizontal Vessels with Saddle Supports

De&n of Flanges

Design of Pressure Vessels to Code Specifications

High-Pressure Monobloc Vessels

Multilayer Vessels

Design Conventions

Welding Conventions

xi

1

19

36

58

76

98

120

141

155

183

203

219

249

268

296

317

323

327

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xii Contents

C .

D.

E.

F.

G.

H.

I.

J .

K.

1.

Author Index

Subject Index

Pricing of Steel Plate

Allowable Stresses

Typical Tank Sizes and Capacities

Shell Accessories

Properties of Selected Rolled Structural Members

Values of Constant C of Eq. 13.27

Charts for Determining Shell Thickness of Cylindrical and

Spherical Vessels under External Pressure

Properties of Various Sections and Beam Formulas

Properties of Pipe

Strength of Materials

330

335

346

349

353

362

364

381

386

392

395

399

Page 10: Process Equipment Design

C H A P T E R

01FACTORS INFLUENCING

THE DESIGN OF VESSELS

e

hemical engineering involves the application of thesciences to the process industries which”‘&e primarily con-cerned with the conversion of one material into another dychemical or physical means. These processes require thehandling and stor ing of large quant i t ies of mater ia ls in con-tainers of varied construction, depending upon the existingstate of the material, its physical and chemical properties,and the required operations which are to be performed.For handling such l iquids and gases a container , or “vessel ,”is used. Thzyessel is the basic part of most types of proc-essing equipment. Most process equipment units may beconsidered to be vessels with various modifications neces-sary to et ions . x

ble the units to perform certain required func-or example, an a_utoclave may be considered to be

a high-pressure vessel equipped with agitation and heatingsources; a distillation or absorption column may be consid-ered to be a vessel containing a series of vapor-liquid con-tactors; a heat exchanger may be considered to be a vesseld--containing a suitable provision for the transfer of heatthrough tube walls; and an evaporator may be consideredto be a vessel containing a heat exchanger in combinationwith a vapor-disengaging space.

Regardless of the nature of the appl icat ion of the vessel , anumber of factors usually must be considered in designingthe unit. The most important consideration often is theselection of the type of vessel that performs the requiredservice in the most sat i s fac tory manner . In developing thedesign a number of other criteria must be considered, suchas the properties of the material used, the induced stresses,the elastic stability, and the aesthetic appearance of theunit . The cost of the fabricated vessel is also important inrelat ion to i ts service and useful l i fe .

1.1 SELECTION OF THE TYPE OF VESSEL

Usually the first step in the design of any vessel is theselection of the type best suited for the particular serviceipquestio~.- The primary factors influencing this choiceare: the function and location of the vessel, the nature of thefluid, the operating temperature and’pressure, and the neces-sary volume for s torage or capaci ty for process ing. Vesselsmay be classified according to service, tempera-ture and pressure service, matkrials of construction, orgeometry of the vesse l .

The most common types of vessels may be classifiedaccording to their geometry as:

‘1. Open tanks.

2. Flat-bottomed, vertical cylindrical tanks.

3. Vert ical cyl indrical and horizontal vessels with formedends.

4 . Spherical or modif ied spherical vessels .

Vessels in each of these classifications are widely used ass~o%& vessels and as processing vessels for fluids. Therange of service for the various types of vessels overlaps,and it is difficult to make distinct classifications for allapplicat ions.

It is possible to indicate some generalities in the existinguses of lhe common types of vessels. Large volumes ofnonhazardous l iquids, such as brine and other aqueous solu-t ions, may be stored in ponds i f of very low value, or in opensteel, wooden, or concrete tanks if of greater value. If thefluid is toxic, combustible, or gaseous in the storage condi-tion, or if the pressure is greater than atmospheric, a closedsystem is required. For storage of fluids at atmosphericpressure, cylindrical tanks with flat bottoms and coni-cal roofs are commonly used. Spheres or spheroids are

1

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2 Factors Influencing the Design of Vessels

:,t

I- lo’-0”Inside diameter

I

lo’-0” UnAred Pressure Vessel

150 Ib.sq in. at 85O’F

Fig. 1.1. Example of a cylindrical vessel with formed ends designed to the original API-AWE code. (Courtesy of Amer. Pet. Inst.!__----

employed for pressure storage where the volume requiredis large. For smaller volumes under pressure, cylindricaltanks with formed heads are ,more economica l .

l.la Open Vessels. Open vessels are commonly usedas surge tanks between operations, as vats for batch opera-tions where materials may be mixed and blended, as sett l ingtanks, decanters, chemical reactors, reservoirs, and so on.Obviously, this type of vessel is cheaper than covered orclosed vessels of the same capacity and construction. Thedecision as to whether or not open vessels may be useddepends upon the fluid to be handled and the operation.

Very large quanti t ies of aqueous l iquids of low value maybe stored in ponds. I t i s doubtful i f ponds may be correct lyreferred to as vessels. They are, however, the simplest

containers made from the cheapest of materials , rol led earth.Not all types of earth can be used for storage ponds; a claywhich will form an almost watertight bottom is essential.An example of the use of ponds of rol led earth is found in theprocess whereby salt is crystallized from sea water by solarevaporation (1). When more valuable fluids are handled,more reliable but more expensive containers are required.Large c ircular tanks of s tee l (2 ) or re inforced (or prestressed)concrete (3), (4) are often used for settling ponds in whicha slowly rotating rake removes sediment from a slightlyincl ined conical bot tom. Vessels of this type, as exempli f iedby the Dorr classifier, may have diameters ranging from100 to 200 ft and a depth of several feet.

Smaller open vessels are usually of a circular shape and

Page 12: Process Equipment Design

are constructed of mild carbon steel, concrete, and some-times of wood (5). Other materials find limited use whereserious corrosion or contamination problems are encount-ered. However, in the process industries in general, themajor portion of existing vessels are constructed of steelbecause of its low initial cost and ease of fabrication. Inmany cases such vessels are lined with lead, rubber, glass,or plastic to improve resistance to corrosion. In the foodindustry fir is commonly used for pickle and kraut tanks,whereas quarter-sawed white oak is employed for wine andspirits. Redwood or Cyprus tanks are often employed forwater storage reservoirs. Wood is also used in place ofsteel for handl ing di lute solut ions of hydrochlor ic , lact ic , andacetic acids and salt solutions and is indispensable as a low-cost tank in the tanning, brewing, and pickl ing industr ies (6) .

In the food and pharmaceutical industr ies i t is often neces-sary to add materials to open vessels in the preparation ofmixtures . Small open tanks or kett les are usual ly employedfor such purposes. Glass-lined steel, copper, Monel, andstainless steel tanks are widely used in these applicat ions toresist corrosion and prevent contamination of the processmater ia ls .

1 .l b Closed Vessels. Combustible fluids, fluids emit-ting toxic or obnoxious fumes, and gases must be stored inclosed vessels (7). Dangerous chemicals, such as acid orcaustic, are less hazardous if stored in closed vessels. Thecombustible nature of petroleum and its products necessi-tates the use of closed vessels and tanks throughout thepetroleum and petrochemical industries. The extensiveuse of tanks in this field has resulted in considerable efforton the part of the American Petroleum Institute to stand-

Selection of the Type of Vessel 3

ardize design for purposes of safety and economy. Tanksused for the storage of crude oils and petroleum productsare generally designed and constructed in accordance withAPI Standard 12 C, API Specification for Welded Oil-Storage Tanks. This is the standard reference used indesigning tanks for the petroleum industry, but it is also auseful guide for other applicat ions.

CYLINDRICAL VESSELS WITH FLAT BOTTOMS AND CONICAL

OR DOMED ROOFS. The most economical design for aclosed vessel operat ing at a tmospheric pressure is the vert i -cal cylindrical tank with a conical roof and a flat bottomresting directly on the bearing soil of a foundation com-posed of sand, gravel , or crushed rock. In cases where it isdesirable to use a gravity feed, the tank is raised above theground, and the flat bottom may be supported by columnsand wooden joists or steel beams. Cylindrical, flat-bot-tomed, cone-roofed tanks are provided with “breathers” orvents which permit expansion and contraction of the fluidsas a result of temperature and volume f luctuations. Tanksup to 24 f t in diameter may be covered with a self -supportingroof ; tanks with larger diameters , up to 48 f t , usual ly requireat least one central column for support. Tanks larger than48 ft in diameter are frequently designed with multiple-column supports or with a floating or pontoon roof whichrises and falls with the level of liquid in the vessel. Ingeneral, tanks with conical roofs are limited to essentiallyatmospheric pressure . If domed roofs are used, pressuresfrom 244 to 15 lb per sq in. gage may be permitted. Thesevessels are normally smaller in diameter and of greaterheight for a given capacity than tanks with conical roofs(8, 9).

Fig. 1.2. Oil refinery installation. (Courtesy of C. F. Braun & Company.)

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4 Factors Influencing the Design of Vessels

CYLINDRICAL V ESSELS W I T H FORMED ENDS. Closedcyl indrical vessels with formed heads on both ends are usedwhere the vapor pressure of the stored liquid may dictate astronger design. Codes have been developed through theefforts of the American Petroleum Institute (10) and theAmerican Society of Mechanical Engineers (11) to governthe design of such vessels. These vessels are usu$ly lessthan 12 ft in diameter if they are to be shipped by rail.However, f ield-erected vessels may exceed 35 ft in diameterand 200 ft in length. If a large quantity of liquid is to bestored, a batt,ery of vessels may be used.

A variety of formed heads arca used for closing I hta ends ofcylindrical vessels. The formed heads include the hemi-spherical, elliptical-dished, torispherical, standard-dished,

one type or another. Figure 1.2 shows a wide variety ofsuch items in a petroleum refinery. Note that nearly allof the processing equipment shown consists of cylindricalvessels with formed ends.

SPHERWAL AND M ODIFIED SPHERICAL VESSELS. Storagecontainers for large volumes under moderate pressure areusually fabricated in the shape of a sphere or spheroid.Capacities and pressures used in this type of vessel varygreatly. Capacity ranges from 1000 t.o 25,000 bbl, andpressures range from 10 lb per sq in. gage for the largervessels to 200 lb per sq in . gage for the smal ler ones . F i g u r e1.3 shows a battery of horizont.al cylindrical vessels andspherica l vesse ls for s tor ing petroleum products a t pressuresup to 100 lb per sq in. gage.

Fig. 1.3. Spherical and horizonial storage tanks at Crown Central Petroleum Plant near Houston, Texas. (Courtesy of Hammond Iron Works.)

conical, and toriconical shapes. For special purposes flatplates are used to close a vessel opening. However, flathe;ids are rarely ustd for large vessels. For pressures uotcovered by the ASME code, the vessels are often equippedwith standard dished heads, whereas vessels that requirecode construction are usually equipped with either theASME-dished or e l l ipt ical -dished heads. The most commonshape for the closure of “pressure vessels” is the ellipticaldish. Figure 1.1 shows a drawing of a vertical cylindricalvessel with formed ends designed to the original API-ASMEcode.

Most chemical and petrochemical processing equipmentsuch as distilling columns, desorbers, absorbers, scrubbers,heat exchangers, pressure-surge tanks, and separators areessentially cylindrical closed vessels with formed ends of

Where a given mass of gas is to be stored under pressure,it is obvious that the required storage volume will beinversely proport ional to the s torage pressure . In general ,for a given mass the spherical type of tank is more economical for large-volume, low-pressure storage operation. Athigher storage pressures, the volume of gas is reduced, andtherefore the cylindrical type of storage vessel becomesmore economical. If allowance is made for the cost ofcompression and cooling of the gas, some of this apparentsaving is lost. When handling small masses of gas, thereis an advantage in the use of cylindrical storage vesselsbecause the cost of fabrication becomes the controllingfactor and small cylindrical vessels are more economicalthan smal l spherical vessels .

Further economy can sometimes be realized by using

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Page 14: Process Equipment Design

Methods of Fabrication 5

and greater re l iabi l i ty as compared with cast i ron, i t is m o r esuitable for high-pressure service where metal porosity isnot a problem. The vesse l d iameter i s s t i l l l imit ing becauseof problems in cast ing . Alloy cast -s teel vessels can be usedfor high-temperature and high-pressure instal lat ions .

Form is a method of shaping metal that is commonlyusedfor certain vessel parts such as closures, flanges, andfittings. Vessels with wall thicknesses greater than 4 in.are often forged. Ot$her special methods of shaping metal,such as pressing, spinning, and rol l ing of plates , are used forforming closures for vessel shells and are discussed later inthe text. Sheet -metal forming is s imi lar to press ing in thatmetal is shaped by means of presses and dies, but thismethod is limited to relatively thin stock. The process ofsheet-metal forming as a method of vessel fabricat ion f indsits greatest application in the field of nonferrous metalssuch as copper, Monel, and stainless steel, where cost con-siderations often preclude the use of heavier stock.

Riveting was widely used, prior to the improvement ofmodernwelding techniques, for many different kinds ofvessels , such as s torage tanks, boi lers , and a variety of pres-sure vessels (12). It is still used for fabrication of nonfer-rous vessels such as copper and aluminum. However,welding techniques have become so advanced that eventhese materials are often welded today. Because of thetrends away from riveted construction, the designs basedupon riveting as a method of fabrication will not be dis-cussed in this text.

M_achining is the only method other than cold formingthat can be used to secure exact tolerances. Close toler-ances are required for the mating parts of equipment..Flange faces, bushings, and bearing surfaces are usually--.-machined in order to provide sat is factory al ignment . Lab-oratory and pilot plant equipment for very-high-pressureservice is sometimes machined from solid stock, piercedingots, and forgings. Multilayer vessels for high-pressure

Fig. 1.4. Two multispheres for storage of nitrogen under 400 lb per sq in.

gage. (Courtesy of Chicago Bridge & Iron Company.)

modified spherical vessels such as the two multispheresshown in Fig. 1.4. These storage vessels were designed tohandle nitrogen at 400 lb per sq in. gage working pressure.Modif ied spherical vessels are a lso used for s torage of largevolumes under moderate pressures. Large e l l ipsoidal ves-sels have been built to hold 55,000 bbl at a pressure of75 lb per sq in. gage. The largest vessels for storage underpressure are the semi-ellipsoidal tanks, which have beenmade to hold as much as 120,000 bbl at a pressure of 235 lbper sq in. gage. As the capacity of an individual vessel isincreased, the pressure that the vessel can safely maintain

h (wit,hout very heavy construction) c!ecreases. A hemi-spheroid with a capacit,y of 20,000 hhl of natural gasolineat a working pressure of 2 35 lb per sq in. gage is shown inFig. 1.5.

1.2 METHGCS OF FABRICATION

,

Process equipment is fabricatel b”y a _ num$er.o&r&_____ - . . ^_-estam’methods such as fusion welding, cast ing, forging,machining, brazing and soldering, and sheet-metal forming.__. .Each method has certain advantages for particular types ofequipment. However , fusion welding is the most importantmethod. The size, shape, service, and material propertiesof t.he equipment all may influence the selection ‘of thefabricat ion method.

GLet-irqn castings have been widely used for the massproduction of small pipe f i t t ings and are used to a consider-able extent for larger items such as cast-iron pipe, heat-exchanger shells, and evaporator bodies because of thesuperior corrosion resistance of cast iron as compared withsteel. Large-diameter vessels cannot be easily cast, andthe strength of gray iron is not reliable for pressure-vesselservice . C& steel may be used for small-diameter thick-walled vessels . Furthermore, because of i ts higher s trength

Fig. 1.5. A 20,030-bbl hemispheroid gasoline-storage tank 64 ft in

diameter by 35 ft high. Designed for 2% lb per $9 in. gage woridng

pressure. (Courtesy of Chicago Bridge & Iron Company.)

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Page 15: Process Equipment Design

6 Factors Influencing the Design of Vessels

Fig. 1.6. Welding external circum-

fere ntiol seam of shell of large vessel

with automatic welder. (Courtesy of

C. F.Braun 8 Company.)

services may be fabricated by machining a series of con-centric shells and shrink fitting for producing desirable pre-stress conditions. This method of vessel fabrication is dis-cussed in a later section of the text. In general, machiningis an expensive operation and is limited to small vesselsand parts in which the cost can be justified.

1.2a Fusion Wejam. Fusion welding is the mostwidely used method of fabrication for the construction ofsteel vessels (12). This method of construction is virtuallyunlimited with regard to size and is extensively used for thefabrication and erection of large-size process equipment inthe field. Often such equipment is fabricated by the methodof subassembly. In this process, sections of the unit areshop welded and then assembled in the field. Equipmenthaving a size sufficiently small to permit transportation bytrucks, rail, or barge is usually completely shop weldedbeta

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se of the lower cost and greater control of the weldingp cedure in the shop.

There are two tvpes of fusion welding t-hat are exte.nsivelyused for the fabrrqation of vessels. These are: (1) the gaswelding ‘&ocess, in which a combustible mixture of acetyleneand oxygen supply the necessary heat for fusion, and (2)the electric-arc welding process, in which the heat of fusionis ‘supplied by an electric current (13, 14, 15, 16). Arcwelding is the preferred process because of the reduction ofheat in the material being welded, the reduction of oxidation,and better control of the deposited weld metal. A widerange of arc-welding equipment is available, from the smallportable welding units to the large automatic weldingmachines. Small arc-welding machines are widely used inwelding shops that fabricate small equipment whereas the

automatic machines are better suited for the welding ofheavy sections involving the deposition of a large quantityof weld metal. Figure 1.6 illustrates the use of an auto-matic welding machine in fabricating a large-diametervessel.

Gas welding is the preferred type of welding for lightgages of metal (20 gage or less), which are difficult to weldby the arc-welding process. Gas welding equipment isextremely useful in flame cutting either in the field or inthe shop.

One of the most recent and successful developments in thefield of arc welding of vessels is the submerged-arc weldingprocess (17). This process was virtually unknown at thebeginning of World War II. The necessity of expeditingproduction of welded equipment during the war yearsresulted in the realization of the advantages of this tech-nique. The process involves submerging of the arc beneatha blanket of granulated mineral flux. The, arc beneath theblanket generates heat to melt the electrode and depositsweld metal. A portion of the granulated flux melts, forminga protective layer on the weld metal, and solidifies with theweld metal. In addition to completely protecting the weldmetal from the atmosphere, this process makes the weldmetal virtually free of hydrogen. As the arc is covered,there is no arc flash, and also a lesser quantity of smoke andobnoxious fumes is produced as compared with the earlierwelding processes. As the weld can not be observed by theoperator, mechanical attachments are used to control thedimensions of the weld. Several inches of weld metal canbe deposited in one pass, a fact which greatly decreases thewelding time involved. However, the greatest advantage

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Methods of Fabrication 7

of the submerged-arc process is t.he e l iminat ion of the opera-tyble.

1

‘w, 1.2b Welding Standards. The success of fabricationby welding is dependent upon the control of the weldingvariables such as exFrience and training of the welder, theuse of proper materials , and welding procedures. An inex-per ienced welder or a welder us ing infer ior mater ia ls or incor-rect procedures can fabricate a vessel that has good appear-ance but has unsound joints which may fail in service.Thus it is absolutely essential that the welding variables becontrolled in order to produce sound joints in the equip-m e n t . A number of codes and standards have been estab-l ished for this purpose . Some of these standards are:

“ASME Code Welding Qualifications” (Section IX of theASME Boiler Code)

ASA Code for Pressure Piping (B 131.1, Section 6 and’Appendices I and II)

Standard Qualijication Procedures of the American Weld-< ing Soc ie ty

API Standard 12 C, API Speci$cation for Welded OilStorage Tanks (Sections 7 and 8)

The American Welding Society (AWS) established thebasic standards for qualifying operators and procedures.These standards of qualification form the basis for most ofthe standards in the various codes. For pract ical purposes ,therefore the rules for qualifying welders and weldingprocedures are essential ly the same in the various codes andstandards. Regardless of whether or not the welded vesselis intended to meet one of the codes or standards, i t is advis-able that the welding conform to one of the minimumstandards.

Each fabricat ion shop should establ ish welding proceduresbest suited to its need and its equipment. To meet thewelding standards previously mentioned, i t is not necessarythat welding procedures be the same in all shops. But itis necessary that, regardless of the procedures used, thewelded joints pass the qualification tests for welding proce-dures and that the welding operators be qualified in usingthese ,wrne procedures. To meet welding standards, weldsmade by the shop proced&es must be tested to determinetensile strength, ductility, and soundness of the weldedjoints. The req&ed tests for the welding procedures speci -fied by APmndard 12 C involve the following:

A. For groove welds :1. Reduced-sect ion-tension test ( for tensi le s trength) .2. Free-bend test (for ductility).3. Root-bend test (for soundness).

4. Face-bend test (for soundness).5. Side-bend test (for soundness).

B. For fillet welds:1. Transverse-shear test (for sbear strength).2. Free-bend test (for ductility).3. Fillet-weld-soundness test.

,,-

The minimum results required by tests such as thoselisted above are described in detai l in the various codes. Afew representat ive requirements are :

1. The tensile strength in the reduced-section-tension test

Double-welded butt joint Double-welded butt joint(V-type groove) (U-type groove)

rover Q In.Y

Single-welded butt joint T Smgle-welded butt jointwith backing strip wlthout backing strip

(may be V-or U-type groove) (may be V-or U-type groove)

Single-welded butt jointwith backing strip wlthout backing strip

Double full-fillet lap joint Single full-fillet lap jointwith plug welds

Fig. 1.7. Examples of welded joints. (Note: The two types of lap welds

shown may be used only for circumferential joints and for shell plates not

over 36 in. thick, and for attachment of nozzles and reinforcements without

thickness limitation.) (From the API-ASME code [lo].)

shall not be less than 95 y0 of the minimum tensile strengthof the material being welded.

2. The minimum permissible elongation in the free-bendtest is 20%.

3. The shearing strength of the welds in the transverse-shear test shal l not be less than 87 y0 o f the minimum tens i lestrength of the material being welded.

4. In the various soundness tests, the convex surface ofthe specimen is examined for the appearance of cracks orother defects. If any crack exceeds $6 in. in any direction,the joint is considered to have failed.

The individual welders, as well as the shop procedures,must meet certain standard qualifications. The individualwelders must qualify under the established procedureaccording to the test previously described. This is impor-tant because a welder may qualify when using one procedurebut may be unable to qualify when using another procedure.For example, an operator of an automatic welding machinemay produce satisfactory welds with that machine but maynot qualify when using manual equipment.

1.2~ Types of Welded Joints. A variety oi types ofwelded joints are used in the fabrication of vessels. Theselection of the type of joint depends upon the service, thethickness of the metal, fabrication procedures, and coderequirements . Figure 1 .7 is a diagram from the API-ASMEcode for unfired pressure vessels which illustrates some ofthe types of welded joints used in the welding of steel platesfor the fabr icat ion of pressure vesse ls . Other types of weldjoints and detai ls for the preparat ion of such joints are givenin Appendix B. Instead of drawing weld details to specifythe type of weld desired, most engineering of&es now usestandard symbols for ,welding conventions (16). Typicalwelding symbols are shown in Fig. 1 .8 .

Page 17: Process Equipment Design

8 Factors Influencing the Design of Vessels

Type of weld

GrooveBead Fillet

Plugand

Feldweld

Welda l l

aroundF l u s h

Location of welds

Arrow (or near)side of joint

1. The side of the joint to which the arrow points is the arrow side, and the

opposite side of the joint is the other side.

2. Arrow-side and other-side welds are some size unless otherwise shown.

3. Symbols apply between abrupt changes in the direction of welding, or

to the extent of hatching or dimension lines, except where the all-

around symbol is used.

4. All welds are continuous and of user’s standard proportions unless

otherwise shown.

5. Tail of arrow used for specification process or other reference. (Tail

may be omitted when reference not used.)

6. When a bevel- or J-groove weld symbol is used, the arrow shall point

with a definite break toward the member which-is to be chamfered.

(In cases where the member to be chamfered is obvious, the break in the

orrow may be omitted.)

7. Dimensions of weld sizes, increment lengths, and spacing, in inches.

8. For more detailed instruction in the use of these symbols refer to Stondord

Welding Symbols, published by American Welding Society.

I

1. I--

Fig. 1.8. Welding symbols recommended by API Standard 12 C. (Courtesy of American Petroleum Institute.)

I.3 TYPES OF CRITERIA IN VESSEL DESIGN

The select ion of the. type of vessel i s based primari ly uponthe ?&&Z&l-s&vice required of the vessel. The func-I &al requirements impose certain operating conditions inrespect to such things as temperature, pressure, dimensionall imitat ions , and various loads. I f the vessel is not designedproperly, so as t,o accommodate these requirements, the

/ve el may fail in service.

Failure may occur in one or more manners, such as byplasticformation resulting from excessive stress, by rup-ture without plastic deformation, or by elastic instabilit,y.I:aiJuz-may also result from corrosion, wear, or fat,igue.Design of the vessel to prot,ect against such fa i lures involvesthe considerat,ion of these factors and the physical proper-I ies of the mat.eriuls. Var ious types of poss ib le vesse l fa i lureand criteria in vessel design are discussed in the followingchapter .

1.4 ECONOMIC CONSIDERATIONS

.ilthough the chemical-process requirements generallyl imit the choice of mater ia ls of fabr icat ion, the f inal se lec t ionis frequently dictated by economic considerations. Fo1purposes of comparison the re lat ive costs of lO,OOO-gal tanksfabricated from various materials are tabulated in Table 1 .1(w&h steel as the unit reference) (18). An examination oft.his table indicates t,hat the cheapest const&&~materials,._.provided they can be used, are wood, concrete, and steel.These materials can frequently be lined with a thin protec-tive layer this eliminates the necessity of fablicat.ing the

vesse ls f rom more expensive meta ls or a l loys . As the sizeof the tank is increased to handle larger volumes, the rela-tive costs of using alloys and nonferrous metals increases.Prestressed or reinforced concrete may sometimes be usedto advantage for the construction of large vessels.

1.4a Steel Pricing. The bulk of chemical and petro-chemical process equipment is fabricated from plain carbons t e e l . -1 knowledge of the method of pricing steel is essen-

Table 1.1. Relative Costs of Materials of Construction

for Tanks/ Cost Relative to Steel~.~______-.

J 10,000 gal 100,000 galWOOdConcret y’(reinforced)

0.40.6 0:;

St,eel’ 1.0 1.0Lithcotr-lined steel 1.2 1.2Rubber-l ined steel 1.8 2.0Lead-lined steel 1.8 2 .ocoppe:L 2.0 2.6Aluminum 2.4 3.0Glass- l ined st.eel 2.7 3.0Xi-Clad steel 2.7 3.0Stain-clad steel 2.7 3.0Stainless steel, type 304 3 4 3.5,Mon&clad steel 3.4 3.5Incouel-clad steel 3.4 3.5Stainless steel, type 316 4.4 4.8&lone1 metal 4.4 4.8Si lver- l ined s teel 12.8 . , _

Page 18: Process Equipment Design

Economic Considerations 9

mill price to compensate for t.he handling, storage, anddelivery of the steel st.ock. Therefore, the differencebetween warehouse prices and mill prices is essentially :Iservice charge.

The relative amount of “millproduction” that was shippedto warehouses for warehouse distribution for the ten-yea1period 1944-1954 is indicat.ed in Fig. 1.9. This figure indi-cates that for the seven-year period 1945-1952 about 18ycof the total steel-mill production on the average was shippedto the warehouses. For the year 1951 the stezl millsproduced 78,928,950 tons of steel products and shipped14,399,432 tons (18.50’%) to the warehouses.

In the design of equipment for large process plants, it isnot unusual to place vessel orders with the vessel fabricatorfrom 6 to 12 months before the required shipping dates I,Oenable t)he vessel manufacturer t.o order t,he steel plate fromthe mill rather than from a warehouse.

M I L L P R I C I N G . In general, steel is purchased from themill or warehouse in the “hot-rolled” or “cold-rolled” condi-tion. The steel is further classified as sheet, strip, plates, orbars. Alloy steels and &ructural steels are classified sepa-rately. “Hot-rolled, plate steel,” or “cold-rolled stripsteel,” or “alloy steel bars,” and so on are combined classi-ficat,ions of types of available steel. Table 1.2 shows thegrade classification, by size, of flat, cold-rolled carbon steelby a typical steel mill. Table 1.3 shows t.he correspondinggrade classification by size of flat, hot-rolled carbon steel.

The steel mills and the warehouses quote “base prices”for each class of steel product. Table 1.4 shows a sectionof a typical mill base-price list as of January, 1956. Theprices are all F.O.B. cars or trucks at the mill works (IndianaHarbor, Indiana.) The prices quoted in Table 1.4 applyto an order of 10,000 lb or more of the size ordered at onetime (one thickness and one width is considered one size),of one grade or analysis, released for shipment to one destina-tion at one time. For weights of less than 10,000 lb, “item-quantity extras” apply. Item la, of Appendix C lists thequantity extras charged by a typical steel mill (InlandSt.eel Company, as of May 13, 1953) for carbon-steel plates.

:(

Table 1.2. Grade Classification by Size of Flat,

Cold-rolled Carbon Steel

(Courtesy of Great Lakes Steel Corporation, Division ofNational Steel Corporation, Detroit, Michigan)

Thickness. inches0.250 or 0.249.9 to

Width, inches t,hicker 0.0142up to 12 Bar Strip (1)Over 12 to 24 Strip (2) Strip (2)Over 12 to 24 Sheet (3) Sheet (3)Over 24 to 32 Sheet SheetOver 32 Sheet Sheet

Notes: (1) Up to 35 in. wide and less than 0.225 in. inthickness, and not to exceed 0.05 sq in. in cross section,having rolled or prepared edges is “flat-wire.” (2) Ifspecial edge, finish, or definite t,emper, as defined by ASTMSpecification A-109. (3) If no special edge, finish, ortemper is specified or required.

tial in order to arrive at economical designs for equipmenLfabricated of steel.

Steel may be purchased from t,wo sources-a steel millor a st,eel warehouse. The prices paid for the steel from thetwo sources are very different, the warehouse prices beingappreciably higher. The reason for these price differencesis found in the methods used by steel mills to obtain maxi-mum-volume production in order to minimize unit costs.It is the present practice of the steel mills to accumulateorders until they have sufficient tonnage to permit economi-cal rolling. Therefore, the steel mills usually serve cus-tomers who require material in reasonably large quantit,iesand- who can anticipate their requirements well in advance.It is apparent that this mode of operation is not conduciveto quick delivery; three or four months, or more, dependingupon the rolling schedule, may elapse before delivery.

This situation makes necessary another means of furnish-ing steel to customers who require material quickly and inquantities too small for mill production schedules. Thesteel warehouse fills this distribution need, supplying steelimmediately from large warehouse stocks. The steel ware-house secures steels from many rolling mills, in a full rangeof qualities, finishes, shapes, and sizes, and stores thesesteels. Thus fabricators using steel may purchase anyparticular product immediat,ely from stock or combineorders for various pr0durt.s and buy all at one time fromone convenient source.

Obviously the warehouse must. be paid an increase over

Table 1.3. Grade Classification of Flat, Hot-rolled

Carbon Steel

(Courtesy of Great Lakes Steel Corporation)

Width, inchesThickness, To 3% Over Over Over Over

inches incl. 355 to 6 6 to 12 12 to 48 480.2300 and thicker Bar Bar Plate Plate Plate0.2299 to 0.2031 Bar Bar Strip Sheet Plate0.2030 to 0.1800 Strip Strip Strip Sheet Plate0.1799 to 0.0568 Strip Strip Strip Sheet Sheet

Year

Fig. 1.9. Percentage of total millproduction of steal products shipped to

warehouses.

.,_ _ .-.-- _._~____~__ - -.-~- -/ I \ -i---‘ - \I 7

..__--_ _ 4.=-s _ .T-

Page 19: Process Equipment Design

1 0 Factors Influencing the Design of Vessels

Table 1.4. Mill Price List

(Courtesy of Inland Steel Company, Chicago, Illinois,January, 1956)

Base Priceper 100 lb

Hot-rolled sheets (18-gage and heavier) $4.325Hot-rolled strip 4.325Cold-rolled sheets 5.325Hot-rolled carbon-steel bars (merchant quality) 4.65Hot-rolled alloy-steel bars 5.575Reinforcing bars 4.65Carbon-steel plates 4.50Carbon-steel structural shapes 4.60

Carbon-steel plates fall into three classifications: (1)those furnished to chemical requirements, (2) those fur-nished to physical requirements, and (3) those furnished toboth chemical and physical requirements. Item lb ofAppendix C lists the “classification extras.”

Other mill-price extras of primary interest are given initem 1, Appendix C and are classified as: quality extras,length extras, width-and-thickness extras and killed-steelextras.

Circular- and sketch-plate extras are involved when itemssuch as blanks for formed heads are purchased. As theseplates are usually flame cut, gas-cutting extras also apply.Gas-cutting extras are also charged for rectangular plateswhen the thickness limits for shearing are exceeded. Item 2of Appendix C lists the gas-cutting extras per linear foot ofcutting.

The previously mentioned extras such as quality, thick-

Fig. 1 .lO. B a s e p r i c e o f s t e e l p l a t e s i n P i t t s b u r g h .

ness, and width are calculated on a dollar-per-loo-lb basis,whereas the extras for circular and sketch plates are calcu-lated on a percentage basis, as listed in item 3 of Appendix C.The percentage is calculated on the net-per-loo-lb price ofthe smallest rectangular plates from which each circular orsketch plate is obtained exclusive of freight and extras forgas cutting a quantity. The outside dimension of eachcircular or sketch plate determines the size of the smallestrectangular plate from which the circular or sketch plate isobtained.

A wide variety of other “mill extras” are quoted by thevarious steel mills. The reader is referred to companyprice lists for complete quotations on these other extras,among which are:

1. Heat-treatment extras2. Surface-finish extras3. Testing extras4. Chemical-requirements extras5. Specification extras6. Special-requirements extras7. Dimensional and workmanship extras8. Extras for special-shipment requirements9. Special-marking-of-plates extras10. Loading extras11. Bundling-of-plates extras.

Each extra is usually separate and distinct. The indi-vidual items are combined to form a “full extra” applicableto the order.

The steel-mill base prices given in Table 1.4 and thesteel-mill extras given in Appendix C are quoted as ofJanuary 31, 1956.. It must be emphasized that these pricesare representative of the prices quoted by steel mills at thattime. As economic conditions vary, prices charged formanufactured products fluctuate, and the base and extraprices are subject to change. Figure 1.10 illustrates thechanges in the base price of steel plate in Pittsburgh fromJuly, 1938 to January, 1956 (19). The horizontal line toApril, 1945 is for the period during which government con-trols were maintained on steel prices because of the nationalemergency of World War II. The curve indicates that theprice of steel plate at the mill doubled between 1945’ and1956. Reference should always be made to the most-recentavailable price lists for estimation purposes.

WA R E H O U S E P R I C I N G. Steel warehouses are strategicallylocated throughout the country to provide a convenientsource of supply for steel products. Whereas the steel millsproduce steel products of standard length and width, thewarehouse will supply steel cut to the customer’s require-ments. Typical operations in the warehouse include shear-ing, sawing, slitting, and flame cutting. Some warehouseswill supply steel plates rolled to cylindrical shapes and barshapes, bar stock rolled to rings or bent to other shapes, andplates with drilled or punched holes. Figure 1.11 showstypical stocks of steel in a warehouse.

Prices vary somewhat from warehouse to warehouse,depending upon the location of the warehouse, the distancefrom the mill, and the service performed. Item 4 of Appen-dix C gives typical warehouse prices from one warehouse(20).

_ __- _ I_- - -_ i --T - - -- - - - 7 7 -I-‘ __-_ .-- .---.--_c_- ~~

Page 20: Process Equipment Design

!Economic Considerat ions 11

Fig. 1.11. Interior view of warehouse showing typical stocks of steel. (Courtesy of Joseph T. Ryerson 8 Son, Inc.)

>’ . 1.4b Fabrication Costs. The direct costs of producing\ a piece of process equipment include the cost of materialsb and the cost of labor. Material costs consist of the shop

material used in the fabrication plus the parts purchasedfrom an outside source. The cost of steel plate, which hasbeen discussed in the previous section, usually comprises amajor portion of the material costs for vessels. The laborcosts involved in the actual fabrication of the equipment

I are often difficult to estimate accurately in advance. Howhas reported methods of short-cut estimations of welded

I process vessels (21).FABRICATION P ROCEDURE.I One of the first steps in the

fabrication of the vessel is usually the preparation of theshell for rolling. The edges of the individual plates for theshell require machining to true the edges and, in the case ofcode welding, to prepare the edge for welding. Figure 1.12shows a 40-ft planer machining a double “U” edge on alx-in. plate 29 ft long for a vessel shell. The next step is

i usually crimping the edges of the plate which will be joinedby a longitudinal weld. The crimping step is requiredbecause the rolls cannot be used to form the two ends to thedesired curvature. Figure 1.13 shows a 350-ton hydraulicpress in the foreground, crimping the edge of a plate beforerolling. In the background plates are shown being rolledinto a cylindrical shape on pyramid rolls.

MAN-HOURS AND MATERIALS. After the shell has beengiven edge preparation and rolled into shape, the vesselcomponents must be fitted and assembled by welding.

iFigure 1.14 gives curves according to How (21) for estimat-ing the man-hours involved in the various stages throughassembly of the shell and closures. The upper curve of

Fig. 1.14~ gives the cutting time in hours per linear foot forflame cutting the shell plate as a function of plate thickness.This curve may be used when the shell is cut from standardplate kept in stock, such as mill plate. If the plate is pur-chased from a warehouse, it may be obtained, cut to size,and the cutting cost included in the purchase price. Inaddition to the man-hours involved in flame cutting, amachine rate burden which includes the cost of machinetime and gas consumed is also involved in flame cutting.The curve for cutting-machine rate burden is shown in thelower part of Fig. 1.14~~.

The number of man-hours involved in edge preparationprior to crimping and rolling are given in Fig. 1.14d. Thecombined number of man-hours involved in crimping thelongitudinal seam ends and rolling the plate into a cylindricalform are given as parameters in Fig. 1.14b; the man-hoursare treated as a function of plate lengths and thicknesses.The parameters given in Fig. 1.14b are based upon therolling and crimping of a few plates; the figure thereforegives a liberal allowance for these operations when more thana few plates are rolled and crimped at one time.

The man-hours required for the fitting and assemblingof the shell and closures for the vessel are given by the solidlines in Fig. 1.14~ as a function of the plate thickness andwith three parameters for different degrees of complexityof the vessel. Also included in this figure are three curveshaving nearly the same shape as the parameters and indi-cated by the dotted lines that may be used as a rough checkon the total man-hours involved in fabrication. Theselatter three curves are intended to be used only as a checkto disclose any gross errors in the total estimation. In addi-

Page 21: Process Equipment Design

Factors Influencing the Design of Vessels

Fig. 1.12. Machining a double U edge on a plate 1 yh in. thick and 29 ft long for a vessel shell by means of o 40-ft planer. (Court+sy o f C, F. Broun g

Company.)

Fig. 1.13. A 350-ton hydraulic press crimping the end of o plate before rolling. (Courtesy of C. F. Bran & Company.)

Page 22: Process Equipment Design

4 % k i%il 1 I%12 I3 4 5 6% % 1% 1% 2%

P l a t e t h i c k n e s s , i n .

(4

3.5 I

/+I I

1 4

-9 I

I I\ I I I I I 8I

Average s tee l th ickness , in .

Cd

I

Economic Considerations 13

Pla te thickness. i n .

(b)

0.6

01111111/1((% b % % % % 1 I?$ 1% 1% 1%

Plate thickness, in.

(4

F i g . 1 . 1 4 . C u r v e s o f H o w (21) fsr e s t i m a t i n g s h o p t i m e f o r v e s s e l f a b r i c a t i o n . ( a ) C u t t i n g t i m e a n d m a c h i n e r o t e b u r d e n f o r tlgme c u t t i n g w i t h a u t o m a t i c

imachines . (b) T i m e f o r r o l l i n g plates 6 0 t o 7 2 i n . i n w i d t h a n d o f v a r i o u s l e n g t h s a n d t h i c k n e s s e s . (d F i t t i n g a n d a s s e m b l y , a n d total f a b r i c a t i o n t i m e ( f o r

rough check ) , fo r s tee l t anks and weldments. (d) W e l d i n g a n d e d g e - p r e p a r a t i o n t i m e a n d w e l d i n g - r o d w e i g h t f o r c o d e b u t t w e l d s i n c a r b o n s t e e l . (Cow-

t e s y o f M c G r a w - H i l l P u b l i s h i n g C o . )

t ion to giving the number of man-hours for edge preparation,Fig. 1.14d also gives the welding time and quantity of weld-ing rod per linear foot, involved in assembly of the vesselends and shell.

Most vessels contain two or more nozzles for chargingand discharging operations. The man-hours and welding-rod requirements for attaching different types of nozzlesare given in Fig. 1.15.

Formed closures such as dished heads can be purchasedfrom fabricators with the edges beveled for welding. Costsfor this preparation are given in a later section describingformed heads. However, if it is practical to machine theheads in the shop that fabricates the vessel, the man-hoursrequired for this operation may be estimated from Fig.1.16~ and b. Bolting flanges for nozzles may be shop fabri-cated from flat plates. The machining time for t.his opera.

1t I \ \--I--- -

Page 23: Process Equipment Design

14 Factors Influencing the Design of Vessels

1.6 -

1.4 -

1 . 2 -

B l.O-e$CLp 0.8 -

a -g 0.6 -

0.4 -

0.2 -

O-

1.611 . 4 I -

l.2t-,2 1.0 //m / /

Tii / /.E / /$ 0 . 8 I I I I/ I/ I

2 Labor. hr

LO.6I

Weld rod. I

I I I

0 I% 1 1 % 1% 2 2 % 3 3% 4

Nominal nozzle size, in.

(4

J slip-on flange!

I2 2%33%4 5 6 8 1 0 1 2 1 4 1 6 1 8

Nominal nozzle size, in.

(d

o-

0.5 -

1.0 -

2 1 . 5 -#e$ 2.0 -

12$ 2.5 -

%3 3.0 -

3.5 -

4.0 -

4.5 -

4.5

4.0

p 3.5=lox.E 3.0

f$“r 2.5&cl2 2.0

1 . 5

1.0

(1.52 2% 3 3354 5 6 8

Nominal nozzle size, in.

(b)

o-- 30

5 -

- 25

25 -

5

30 -

02 2%33%4 5 6 8 1 0 1 2 1 4 1 6 1 8

Nominal nozzle size, in.

UJ

Fig. 1.15. Curves of How (21) for estimating welding-time in hours and welding-rod requirements for nozzle attachments to vessels. (a) Welding time and

welding rod for installing XH steel couplings in unfired pressure vessels. (b) Welding time and welding rod for installing long-welding-neck forged-steel

nozzles. (c) Welding time and welding rod for fabricating 150-psi nozzles in unfired pressure vessels. (d) Welding time and welding rod for fabricating

300-psi r,ozzles in untired pressure vessels. (Courtesy of McGraw-Hill Publishing Co.)

Page 24: Process Equipment Design

Economic Considerations 15

6

5tE0E

4

2m4 3IIu-zAZk 2

1.8’E

1.6

-0 1.4

1.2

/

0.9/

0.81 5 20 25 30 35 40 50 60 70 80 100 120

2..8

..6

..4

I- 0.8 ’ ,+’ n

I

1 5 20 25 30 35 40 50Outside diameter of head, in. Outside diameter of head, in.

(4 (6)

6.56.0

5.0

8; 3.0

8i 2.52

8 2.05 1.8

0.6

k 1.6L; 1 . 4

8 1 . 2 flanges up to 6 ips inclusiv1.11.00.90.8

c

1 2 3 4 5 6 8 10 12 14 16 18 20 22 24 26 4 % k % t J6 1 1%Nominal flange size, in. Size of weld, in.

(cl (4

Fig. 1.16. Curves of How (21) for estimating man-hours of machining time and welding-rod requirements for miscellaneous operations in vessel fabrication,

(a) Machining time, flanged and dished heads, grooved face, carbon-and-nickel- or stainless-clad steel. (b) Machining time, flanged and dished heads.

beveled face, carbon-and-nickel- or stainless-clad steel. (c) Machining time for carbon-steel plate flanges, 1 to 26 ipr, >/4 to 2% in. thick. (d) Welding

time and weight of welding rod for fillet welds in carbon steel. (Courtesy of McGraw-Hill Publishing Co.)

r’ - I \7- - \I/. - -

Page 25: Process Equipment Design

1 6 Factors lnftuencing the Design of Vessels

Table 1.5. Engineering News-Record ConstructionCost Index

(Courtesy of McGraw-Hill Publishing Co.)

Year19131915192019251 9 2 61 9 3 01 9 3 219351 9 4 01 9 4 5

Index100

9 42 3 52 0 62 0 82 0 21571952 4 23 0 8

Year19461 9 4 71 9 4 81 9 4 91 9 5 01 9 5 11 9 5 21 9 5 31 9 5 41955

Index3 4 64 1 34 6 14 7 75 1 05 4 35 6 96 0 06 2 8660 (July)

i.ion is given in Fig. 1.16~. Various attachments such asskirts, saddles, and lugs may be added to the vessel, usuallyby fillet welding. The man-hours per foot and the weldingrequired for fillet welding is given in Fig. 1.16d. Addi-tional curves for some alloy and rumferrous metals are giver1Iry How (21).

1.5 ESTIMATING CURRENT COSTS

COST INDICES. Because of the constant change of costsfor material, labor, taxes, and plant overhead, available COSIdata rapidly become obsolete. Thus some method ofbringing cost data up to date is required. The procedurenormally followed is t,he application of available “cost.indices.” The cost indices are relative numbers giving thevariation in a group of costs with reference to a base year.To use a cost index the estimator simply multiplies t.heknown cost at a given date by the ratio of the current indexvalue to the index applicable a.t the date of the known cost.

Index ACost A = Cost B ___

Index B (1.1)

A number of indices are in wide use; they differ somewhatbecause of the basis used in their preparation and the refer-ence year. Three widely used indices are the Engineering‘Vews-Record (ENR) construction-cost index (22), the

Table 1.6. Marshall and Stevens Equipment-Cost Index

(Average for All Industries)(Courtesy of McGraw-Hill Publishing Co. [235])

Year lndex Year Index1913 57.9 1 9 4 7 150.61 9 1 5 55.9 1948 162.81920 153.3 1949 161.21925 105.3 1950 167.91926 100 .o 1951 180.31930 87.0 1 9 5 2 180.51 9 3 2 66.1 1953 182.51 9 3 5 78.0 1 9 5 4 184.61 9 4 0 86.1 1 9 5 5 190.61 9 4 5 103.4 1956 208 .a1 9 4 6 123.2 1957 (June) 224.1

Table 1.7. Twenty-City Average of Hourly Rates forSkilled Labor

(From Engineering News-Record)(Courtesy of McGraw-Hill Publishing Co.)

Year Rate, dollars/hr Year Rate, dollars/hr1926 1.27 1950 2.521932 1.03 1 9 5 2 2.841 9 3 9 1.44 1 9 5 3 3.011945 1.66 1 9 5 4 3.141 9 4 6 1.80 1 9 5 5 (July) 3.251949 2.41

Marshall and Stevens eyuipment-cost index (23), and theNelson refinery index (24).

The ENR construction-cost, index (22) reflects labor-wage-rate and material-price trends. The index consists of thecost of a hypothetical block of construction requiring 6 bblof cement, 1.088 M fbm of lumber, 2500 lb of steel, and200 hours of common labor. This cost was $100 in theyear 1913, which is taken as the reference year. Althoughthis index is intended to reflect average construction costsand has no particular relation to the cost of equipment, ithas proved extremely useful in estimating changes in costsfor complete plant,s. Because of its wide use it has oftenbeen the basis of estimating changes in equipment costs.Table 1.5 lists some values of this index as a function of time.

The Marshall and Stevens equipment-cost index reflectsthe comparative costs of equipment (23). It is based upont.he costs of machinery and major equipment, installationlabor, plant furniture and fixtures, tools and minor equip-ment, and oflice furniture. These costs are estimatedquarterly for 47 different industries, with a separate formulafor each industry and with the year 1926 as a reference of100. The petroleum-industry index contains the followingcomponent percentages: process machinery, 25; installationlabor, 19; power, 12; maintenance equipment, 2; and admin-istration, 6. Other process industries for which indices areprepared are: the cement industry, the chemical industry,the clay-products industry, the glass industry, the paint

Table 1.8. Average Boilermaker Wages in July,

1954, as a Function of Locale

(U. S. Bureau of Labor Statistics)dollars+

U. S. Average 3.11New England (Me., Vt.., Mass., Conn., R. I.,

N. H.) 3.00Mid-Atlantic (N. Y., Pa., N. J.) 3.44Border States (Del., Md., Ky., W. Va., Va.) 3.01Southeast (Tenn., S. C., N. C., Ala., Ga., Miss.,

Fla.) 2.90Great Lakes (Minn., Wis., Mich., Ill., Ind., Ohio) 3.13Midwest (N. Dak., S. Dak., Kans., Nebr., MO.,

Iowa) 2.96Southwest (Tex., Okla., La.) 2.90Mountain (Mont., Idaho, Wyo., Utah, Ariz.,

N. Mex., Colo.) 3.01Pacific (Wash., Oreg., Calif., Nev.) 3.05

Page 26: Process Equipment Design

industry, the paper industry, and the rubber industry. Aweighted average for the process industr ies is a lso reported,which contains, in percentages: cement, 2; chemicals, 48;clay products, 2; glass, 3; paint, 5; paper, 10; petroleum,22; and rubber, 8. Also, an average for all 47 industries ispublished, which has differed only by about 1 y0 to 2% fromthe average for the process industries alone. As this indexis based primarily upon industrial-equipment costs, it isconsidered more re l iable for est imating changes in equipmentcosts than the ENR index. Values of the Marshall andStevens index for the average of all industries are given inTable 1.6.

The Nelson refinery index (24) is a construction-costindex somewhat similar to the ENR construction-cost indexbut based upon the cost of materials and labor for the con-struct ion of petro leum ref iner ies .

Although price indices are extremely valuable in est imat-ing costs, it should be mentioned that they are based onnational averages and may be inconsistent with pricechanges for a particular locale. Also, price indices arebased on wage rates and material costs but make no allow-ance for such factors as: availability of materials, productiv-ity of labor, competitive conditions, influence of new tech-niques, business opt imism, re lat ion of demand to product ioncapacity, and other intangibles.

LABOR-COST V ARIATIONS. The cost of labor varies fromyear to year and from area to area. These costs have risenrapidly since World War II, and at the end of 1955 thehourly rates in dollars were at an all-time high. Table 1.7gives the average hourly rate for skilled labor for 20 citiessince 1926 as reported by t,he Engineering News-Record (22).

Skilled laborers such as machinists, welders, and boiler-makers are required in the fabrication of vessels. Thehourly rates, as a funct,ion of locale, that were paid boiler-makers in July, 1953, as reported by the Bureau of LaborStatistics (22), are given in Table 1.8.

S HOP OVERHEAD . In addit ion to the direct costs involvedfor materials and labor, all fabricators must add an indirectcost often termed the “shop overhead” or “burden.” Thisoverhead includes a variety of items, such as the cost ofsupervis ion, administration, engineering, sales, utilities,maintenance, depreciation, taxes, and other fixed and indi-rect costs. These costs vary from shop to shop, area toarea, and year to year, and are established by the conditionsfor a particular shop and by the accounting practice fol-lowed. This overhead usually ranges from 100 % to 200 %of the total cost for labor and materials.

PROFIT. The profit of the fabricator is estimated on thetotal cost to the fabricator, including materials, labor, andoverhead. The profit usually ranges from 5% to 205~ ofthis total cost but may be higher if the state of competit,ionpermits .

SOURCES OF P RICE I NFORMATION. In recent years a con-siderable attempt has been made to col lect , group, and corre-late price information. A series of post World W-ar IIarticles was published in Chemical and Metallurgical Engi-neering in 1946, and additions to them were made in Chemi-cal Engineering from 1947 through 1955. These articleshave been collected and reprinted in three booklets (25).Furthermore, rather recently two texts on the subject of

Typical Procurement Procedure for Vessels 1 7

chemical-engineering costs have been published (26, 27).In October, 1954, Weaver listed a bibliography of 351articles dealing with equipment costs, operating costs, andestimating methods (28).

SCALING EQUIPMENT COST WITH SIZE. Frequently apiece of process equipment having a s ize different from thatfor which the cost is known is desired. A comprehensivestudy of the cost of a variety of process equipment as a func-t,ion of size and capacity was made by Chilton (18). III Hsubsequent article (229) Chilton analyzed these data andsimilar data by Williams (230) and concluded that the“six-tenths factor” rule is useful as a short-cut method forapproximating the cost of a similar piece of equipment of adifferent s ize. This rule states that the cost of a second sizeis equal to the cost of the first size times the ratio of thesizes (or capacities) raised to the six-tenth power, or

o.6Cost A = Cost B (1.2)

The general val idity of this rule has been well establ ished,but some discre t ion should be exerc ised by l imi t ing i t to lessthan a tenfold range unless cost data are available for twoor more units over a range of s izes .

1.6 NPICAL PROCUREMENT PROCEDURE FOR VESSELS

The typical procedure followed in the procurement ofvessels for a process application will be discussed brieflyto give a perspective of the sequence of steps involved.

Normal ly a process-design group develops f low sheets forthe process involved. The f low sheets include informationrelative to the operating temperat.ure and pressure, capaci-ties, heat duties, and any particular information concerningcorrosion. The equipment-design group prepares detailedsketches of the various items of equipment, specifying thematerials of construction, shell and closure thickness, typeof closure, and code stamping. The nozzle and manholetypes and their ratings; corrosion allowances; stress reliev-ing, radiographing, and hydrostatic or air-testing require-ments are also specified by the design group. In addition,shipping limitations; the weight, of the vessel empty, withinternal attachments, and filled with water; and the operat-ing weight are usually estimated by this group. From theabove information specification sheets for each item ofequipment are prepared.

The procurement group sends copies of the detailedsketches and the specification sheets to various vessel fabri-cators for quotations on prices and delivery dates. In t bcmeantime the plant-design group are also sent copies of f.hesketches and specifications. This group prepares a layoul.of the plant design. This includes specifications for roads,utilities, sewers, fire protection, structural foundations,pumps, piping, and detai l design of the various componentsinvolved.

On the basis of the original est,imates received from thefabricators, one or more fabricat.ors are selected and finaldrawings are furnished for a rigid price quotation on thevarious vessels and other items of equipment to be pur-chased. The fabricator is then granted permission topurchase material and prepare shop drawings. Theseshop drawings are usually submitted to the purchaser for

Page 27: Process Equipment Design

18 Factors Influencing the Design of Vessels

approval. On approval of these drawings the fabricatorproceeds with construction of the vessels and other items ofequipment being supplied. Where major items of equip-ment are involved, it is customary for the purchaser to setup an inspecting-and-expediting group at the fabricator’splant. The inspectors for the purchaser normally follow

every step in fabrication, from initial inspection of plate andheads through testing and shipping.

This text covers the design problems of the engineeringoffices of both the purchaser and the fabricator. Noattempt has heen made to separate these problems, whichoften overlap.

P R O B L E M S

1. A vertical vessel designed as shown in Fig. 1.1 and 32 ft from head junction to head junc-tion is in use. If an identical vessel were to be fabricated of SA-285, Grade C steel today, whatwould be the estimated cost of:

a. The two circular blanks for the two elliptical dished heads at the warehouse (no edgepreparation) ?

b. Four shell plates cut to length only at the warehouse?(Note: see Chapter 5, section 5, Common Types of Formed Heads and Their Selection, for

diameters of head blanks.)2. Estimate the cost of labor today for the fabrication of the vessel shown in Fig. 1.1. How

much weld metal will be required?

i

Page 28: Process Equipment Design

C H A P T E R

m2CRITERIA IN VESSEL DESIGN

AAunit of process equipment may fail in service for a

variety of reasons. Consideration of the types of failurewhich may occur is one of the criteria which should be usedin equipment design. Failure may result from excessiveelastic or plastic deformation or from creep. As a result ofsuch deformation, the equipment may fail to perform itsspecified function without rupture or may fail catastroph-ically with rupture. Failure can usually be classified inone of the following catagories: excessive elastic deforma-tion, elastic instability, plastic instability, brittle rupture,creep, or corrosion.

2.1 EXCESSIVE ELASTIC DEFORMATION

2.1 a Induced Stresses. Elastic deformation is inducedby a load such that when the load is removed, the partresumes its original shape. A typical example is the steelspring in a watch. Under service conditions the variousparts of the equipment will be subjected to a variety ofinduced stresses. A stress is defined as the force per unitarea in the member under consideration. Various types ofstresses are induced, depending upon the loading condition,and are classified as: tensile, compressive, shear, bending,and torsion. These stresses may be the result simply ofthe weight of the material of construction or may be causedby loads resulting from fluid pressure, forces, wind moments,and so on. Parts under axial-compressive or tensile forceshave induced stresses which may be computed by the simplerelations:

For axial tension:

j=i (2.1)

For axial compression:

(2.2)

when j = induced axial stress, pounds per square inchP = load, poundsa = cross-sectional area, square inches

Stresses resulting from bending and torsion are morecomplex, and a large number of texts have been written onthe subject of the evaluation of such stresses (29, 30,31, 32).

Induced stresses result in corresponding induced elasticdeformations. The deformations may interfere with thefunctional operation of the equipment. A common exam-ple of this is found in the use of excessively thin flanges fora bolted closure with a gasket at the interface of the flanges.Tightening of the flange bolts in an attempt to seat thegasket in such a way that it will contain the internal pres-sure may result in excessive elastic bending of the flangebetween the bolts without transfer of the bolt load to thegasket. Another example is the excessive deflection of atray in a distillation column under the tray load, a conditionwhich produces a nonuniform liquid seal on the bubble capsand possible instability in tray operation.

2.1 b Modulus of Elasticity. In order to avoid suchsituations as described in the previous examples, sufficientrigidity must be incorporated into the design of the part torestrict the amount of deformation to a permissible value.The deformation which can be tolerated is determined bythe function of the part. Parts in simple tension or com-pression, such as exist in axial loading, deform in the elasticregion in direct proportion to the induced stress and inindirect proportion to the modulus of elasticity of thematerial of construction. Thus the proportionality con-

19

\-\ \I /- - -___-__ - - --_ - -~.---~ __-

Page 29: Process Equipment Design

20 Criteria in Vessel Design

60,000

0

-r--l I I I IGray cas t i ron (ASTM-A-276 -No . 60 )

Cold-rolled mild steel( A S T M - A - 3 7 4 )

Hot-rolled mild stee(ASTM - A - 283,

Ib io ;o 4bPercentage of elongation

Fig. 2.1. Typical stress-strain curves for various metals.

stant between stress and strain (under axial loads) is themodulus of e last ic i ty .

Typical stress-strain curves for a few selected materialsare shown in Fig. 2.1. (Note that two scales are used onthe abscissa in order to enlarge the elastic region of thecurves. )

The elastic portion of the total strain is represented bythe straight- l ine segments of the curves . The slope of thesestraight-line segments, when the strain is expressed ininches per inch, is the modulus of elasticity of the material,E, or:

f-=E (2.3)E

where j = axial stress, pounds per square inche = unit strain, inches per inch

E = modulus of elasticity, pounds per square inch

2.1~ Elastic Bending. The deflection of a part subjectedto forces which produce bending is a more complex phenome-non. In such cases the amount of deflection is inverselyproportional to the modulus of elasticity and the momentof inertia of the member. The use of relationships devel-oped in the field of theoretical mechanics are required forevaluation of the deflect ions. General procedures for suchcalculations are presented in a number of texts on the sub-ject of strength of materials (29, 30, 31, 32), and on thesubject of the theory of elasticity (33, 34, 35, 36, 37).Selected procedures for particular calculations involved invessel design are presented in later chapters of this text.The basic relationships for such calculations may be devel-oped by considering the shear and bending in a uniformly

loaded beam having uniform cross section and freely suported at the ends as indicated in Fig. 2.2.

Consider an element, dx, of a beam having a uniform cr(section, supporting a distributed load of w pounds per inof length of the beam as indicated in Fig. 2.2. The toiload acting on the element is w (dx). If w is consider’positive when the load acts downward and if dx is positi\the differential shear force, dV, must be negative. 1summation of vertical forces: Vz - VI + w (dx) = 0, or

a n d

o r

Vz - VI = -w (dx)

dV = -w (dx)

dVU’

dx(2.

Taking a summation of bending moments about pointgives :

w (dx)2 .Since ---2 is negligible,

dM = MS - Ml= Vldx

dM- = 1’d x (2.

Any beam under a load def1ect.s. A particular radiuscurvature ex is ts for the portion of the beam under considertion. Thus the loaded beam has a radius of curvature P,a dist,ance 2 from the perpendicular to the neutral axiThe bending of the beam wil l result , in a deformation Axthe f iber at any distance ?/ f rom the neutral axis , as indicatein Fig. 2.3. The corresponding strain or unit deformat.iceZ is equal to Ax/x, and by similar triangles

Ax ?/ez=--= (2.1x P

As given by Eq. 2.3, the ratio of stress to strain for elastdeformat,ion is equal to the modulus of elast,icity, E, or

ji = EEL

(b)

Fig. 2.2. Forces on an element of a uniformly loaded beam. (a) UI

formly loaded beam. (b) Detail of e lement dx.

Page 30: Process Equipment Design

Excessive Elastic Deformation 2 1

/dA

SectionA-A

Stressdiagram

Fig. 2.3. Stress and strain in on elemental strip of a curved plate or beam.

By definition the radius of curvat.ure r is defined (38) as:By substitution of Eq. 2.6

1.- = d2yidx2 --r, [l + (dy/d~)~]‘~J

( 2 . 1 2 ).f, = E YI

(2.6a)2

By summation of forces For small deflections the quaniit ?; rl?, kc is small comparedunity; thereforeof

it-at 1 d2y-N-

r- dx2(2.13)

is.of By summation of moments

Substituting Eq. 2.13 into Eq. 2.9 gives:edon M=/&

dx2(2.1‘1)

(2 .7)

(2 .9 :

(2 .10)

(2 .11)

8.6)

3tic

Also, by definition (29) the moment of inertia is:

+aI =

1.v2 dA

--c

Therefore

M = !?!?r

M2

By combination of Eq. 2.9 with Eq. 2.6a,

jpf = &!?

Gr for the outermost fiber where y = c,

Iv-here t = --) section modulus, inches3

C

By Eq. 2.5:

d M = Vak

t,herefore - = EI 9 = 1dMd x dx3

(shear force) (2.15)

.And hy Eq. 2.4

d V = -wdx! tmefore

!!lL~~d!L vu>

d x J dx4;ioad) (2.16)

Anot,her important relationship, the equation of I hedefleclioa curve, is obtained from EC{. 2.14:

These relationships for i)thiulls may be applied to platesand shells under certain corldilions, <IS described in $11 hue-quent chapters of the t.ext.

) Uni-I = moment of inertia of the cross section, inches’c = distance from neutral fiber to outermost fiber

Page 31: Process Equipment Design

2 2 Criteria in Vessel Design

2.2 ELASTIC INSTABILITY

Elastic instability is a phenomenon associated with struc-tures having limited rigidity and subjected to compression,bending, torsion, or a combination of such loading condi-tions. Elastic instability is a conditionin which the shapeof the structure is altered as a result of insufficient stiffness.It is often the controlling factor when compressive loadsare involved. A typical example of elastic instability isthe buckling of a cylindrical vessel under an external pres-sure as a result of vacuum operation. Another example isthe buckling of a horizontal vessel as a result of the bendingmoment induced by the reaction between the vessel andsaddle supports. Elastic instability in vessels is usuallyassociated with the use of thin shells.

2.20 Column Instability. The simplest type of elasticinstability occurs in the “column” action of an axial, end-loaded compression member. The mathematical relation-ship for critical loading of long, slender columns was firstdeveloped by Euler (39) over 200 years ago. The rela-tionship for such a column, pivoted at both ends and freeto rotate, may be derived as follows.

In reference to Fig. 2.4, the bending moment at distancex is equal to -Py, and by Eq. 2.14

M=EIs=-Py

Rewriting with b2 = P/El, gives:

d2y2 + b2Y = 0

The solution of this differential equation is (40) :

y = A cos br + B sin bx

where A and B are arbitrary constants.For 2 = 0, y = 0, or

y = A cos 0 + B sin 0 = 0

therefore

A=0

P

x ’r- Y

1

jI!

f i g . 2 . 4 . C o l u m n p i v o t e d a t b o t h e n d s a n d f r e e t o r o t a t e .

Table 2.1. Euler Column Formulas for Various End

Conditions

Condition Equation

1. Both ends pivoted Pcritical = T$ (2.18)

2. One end fixed, other free Pcritical = =C$’ (2.21)

3. Both ends fixed Pcritioal =‘yi (2.22)

4. One end fixed, other pivoted Pcritioal = ‘<p (2.23)

When x = 1, .y also equals 0. or

,y = B sin bl = 0

But B can not be zero if y is to have values other thanzero between 2 = 0 and x = 1. Therefore sin bl mustequal zero. If sin bl is equal to zero, the least value of theterm bl will be K. As the least value of b determines theleast value of P, bl is taken as equal to ?r for the criticalvalue of P, or

bl = Ttherefore

thereforer2EI

Periticsl = 12 (2.18)

But I may be expressed in terms of the radius of gyration,K, and the cross-sectional area, u, by (41):

I = k2a (2.19)

Substituting Eq. 2.19 into Eq. 2.18 gives:

Peritical ?r2Efcriticsl = 7 = __12/k2

(2.20)

The stress, fcriticalp is the load per unit area at whichincipient buckling occurs. This is not the maximum stressdeveloped as a very slight increase in Pcritical will result in aconsiderable amount of deflection and a rapid increase instress until failure by buckling ensues. For design, anallowable stress appreciably less than the value of fcritical isused to provide a margin of safety against buckling.

In addition to the condition of a long column pivoted atboth ends and free to rotate, column action for other typesof end loading may be developed (31). Table 2.1 listsEuler column formulas for various end conditions. ForRankine column formulas, see Chapter 4, section 4.3b.

2.2b Vessel Shells under Axial Load. In the design ofvessels the relationship for the elastic stability of a curvedplate subjected to an axial compressive load is of interestbecause this condition commonly exists in the shell of ver-tical cylindrical vessels. Timoshenko (42) has given thederivation of the following relationship :

I /-I__-

Page 32: Process Equipment Design

/n cl”\

where t = shell thickness, inchesP = shell radius, inchesp = Poisson’s ratio

Experimental tests (42) on the axial compression of thincylinders have resulted in buckling loads which are about40% of that predicted by Eq. 2.24. The safe compressivestress that can be carr ied without buckl ing was invest igatedby Wilson and Newmark (43). As a result of these testsand others (44), it was found that the safe compressivestress that can be imposed on a steel cylindrical shell with-

dout fa i lure by wrinkl ing can be expressed as fol lows:

I fallowable = 1.5 x lo6 f0

2 $ yield point (2.25)

Various applications of the criterion of elastic instabilityare discussed in subsequent chapters of this book.

2.3 PLASTIC INSTABILITY

2.3a Stress-Strain Relationships. The most widelyused criterion in the design of equipment is that of main-

1 taining the induced stresses within the elastic region of thematerial of construction in order to avoid plastic deforma-

I t ion resul t ing from exceeding the yie ld point . These stressesmust be limited to a permissible value that is accepted asbeing safe for the particular application. Usually theresults of tensile tests of standard specimens are taken asthe basis for establishing the allowable or safe workingstress . Duct i le mater ia ls such as hot-rol led mild s teel havetwo significant stress values, the yield point and the ulti-mate tensile strength. The yield point is defined as the

I tensile load at yield expressed in pounds divided by the

,

I

~-UnloadingI

J

A” Strain -

(a)

Plast ic instabi l i ty 23

original cross-sectional area of the specimen in squareinches. The tensi le load at yield is the load condit ion wherepermanent strain begins to occur. The ultimate tensilestrength is defined as the maximum tensile load divided bythe original cross-sectional area of the test specimen. Thestresses measured by the standard tensi le test are induced bya uniaxial load whereas actual loads under operating condi-t ions may induce three-dimensional s tresses . Various pro-cedures are employed to handle the problem of combinedstresses when using an allowable stress based on uniaxialtests. The procedures differ with the nature of the designproblem (see Chapters 6 through 15).

Some typical stress-strain curves for various materiaiswere presented in Fig. 2.1. Of the curves shown, only thecurve for a hot-rolled mild steel has a well-defined yieldpoint, which occurs at about 30,000 psi. The functionalservice of a member may be lost if the induced stressesexceed the yield point. For example, a machined flangeused as a closure for a vessel may no longer produce apressure-tight seal if the machined face of the flange ispermanently deformed. Thus the allowable stress in suchan application should be kept below the yield point.

Figure 2.5~ shows a s imple s tress-s tra in curve for the caseof an induced stress within the elastic region. Under load,the part may have an induced stress of A pounds per squareinch and a unit strain of A” inches per inch. On removal ofthe load the stress and strain will both return to zero.Such a stress condition is considered satisfactory for designif the induced stress at point A is kept sufficiently below theyield point to provide an adequate margin of safety.Figure 2.5~ shows a similar curve for the case in whichplastic deformation has occurred. The loading conditionhas produced a stress which has exceeded the yield point Aand has reached a stress of B pounds per square inch. I fplastic deformation had not occurred, the theoretical stress-strain condition would have been located at B’. Theactual strain resulting from an induced stress B is indicatedby B”. The permanent residual strain upon removal of

Strain --+ 0 A”

IIII1--J

C B” Strain-

(b) 03

Fig. 2.5. Stress-strain diagrams for elastic and plastic loadings. (According to Kerkhof [45]. Courtesy of the American Welding Society.)

Page 33: Process Equipment Design

24 Criteria in Vessel Design

the load is indicated by point C. If the entire cross sectionof the part undergoes plastic deformation, as shown inFig. 2.5c, there will be no residual stress upon removal of theload. Such a design is usually considered unsatisfactorybecause of the excessive permanent deformation and thepossibility of rupture.

If part of the cross section is subjected to elastic strain,as indicated in Fig. 2.5a, and the remainder of the sectionundergoes plastic deformation, as indicated in Fig. 2.5c,residual stresses and strains will remain in the cross sectionupon removal of the load condition. Figure 2.5b shows theloading and unloading condition that results when only aportion of the cross section undergoes plastic deformat,ion.It is important to note that the plastic strained conditionresulting from stress B’ is much more limited in this casethan in that shown in Fig. 2.5~ because of the restraintoffered by the adjacent elastic portion of the cross sectionwhich is undergoing elastic strain. Thus the total strainin the portion undergoing plastic strain is limited to thatpredicted by the modulus-of-elasticity line extended to cpoint B’. The actual induced stress in the plastic portioncorresponds to B pounds per square inch, and the actualstrain, to B” of Fig. 2.5b. Upon unloading, the port,ionhaving undergone plastic strain has a residual compressivestress as indicated by point C. This residual compressive-stress condition is in equilibrium with residual tensilestresses in the adjacent region that has been subjected toonly elastic loading. If the portion undergoing plasticdeformation is small in comparison with the portion under-going elastic deformation, the residual strain will be imper-ceptible. Thus the prevention of significant plastic defor-mation does not require all calculated elastic stresses to bebelow the yield point since appreciable plastic deformationcan occur only if the material yields across the entire area.

Such a loading condition as shown in Fig. 2.5b oftenexists where local stress concentrations (which are non-uniform across the section) occur, as at the junction ofvessel shell and heads. Because the major part of thecross section is in elastic strain, the small amount of plasticstrain relieves the high stress from B’ to B without seriousdeformation. Also the mean stress across the elastic-plastic zone may be sufficiently below the yield point toallow an adequate margin of safety. Thus such a conditionmay have advantages in relieving high local stresses butmay become undesirable if excessive repeated loading andunloading occur. Such a cyclic operation may result instrain hardening with corresponding loss in ductility andsubsequent failure by rupture (45, 46).

2.3b Allowable Stress. On reference to Fig. 2.5a, thepercentage of the yield strength used as the allowable stressis controlled by a number of factors, such as: the accuracywith which the loads can be estimated, the reliability ofthe stresses computed from these loads, the uniformity ofthe material, the hazard if failure occurs, and other con-siderations like local stress concentrations, impact shock,fatigue, and corrosion.

For structural steels, one half to two thirds of the yieldstrength is often used as the allowable stress for static loadsin structures. For example, the skirts used to support tallvertical vessels may be considered structures and therefore

do not need to conform to pressure-vessel code specifica-tions but should conform to local building codes. A hot-rolled mild steel with an allowable stress of 20,000 psi mightbe used for such a structure. Inspection of Fig. 2.1 showsthat this value is about two thirds of the yield point ofSA-285, Grade C steel (which is a typical hot-rolled mildsteel). If the vessel supported by this skirt is fabricatedfrom a “code steel,” such as SA-285, Grade C steel, havingthe same physical properties as the skirt steel, the ailowablestress is based on one-fourth of the ultimate tensile strengthrather than two thirds of the yield point. Thus the allow-able stress for the code vessel using this steel is 12,650 psirather than the 20,000 psi which might be used for the skirtdesign. The hazard of an exploding pressure vessel isgreat, a fact which justifies the use of a greater factor ofsafety for pressure vessels than for structures. However,the reason for the use of the ultimate strength to definethe allowable stress is not obvious and has been the subjectof considerable discussion (45, 47, 48, 49).

One reason for the use of the ultimate strength as a cri-terion for allowable stress has been the lack of a plasticzone for brittle materials, such as gray cast iron, and thelack of a well-defined yield point, as in the case of mostnonferrous materials. If the yield point is not well defined,the value of a yield point corresponding to some specifiedpermissible strain may be obtained. For example in the0.2% offset method a line is drawn parallel to the modulusline from the 0.2% elongation point, and the Litcrcept ofthe stress-strain curve with this line is taken as the yieldstrength of the material.

The curve for gray cast iron shown in Fig. 2.1 indicatesno yield point, and fracture occurs at the ultimate strength:therefore it is necessary to base the allowable stress for gra!cast iron and other brittle materials on the ultimate strength.Because of the great use made of cast iron in design duringthe period prior to World War I, the policy of basing theallowable stress on the ultimate strength was widely usedeven for materials which had well-defined yield point,s.Many engineers still use a “factor of safety” of three forstructural steel and a factor of safety of six for gray castiron, based upon the ultimate strength, when designingstructural parts. The pressure-vessel codes still use afactor of safety of four based upon the ultimate strengthfor specifying the allowable stresses for pressure vessels.This prior convention of applying a factor of safety to theultimate strength does not justify the continued use of thisprocedure. Where failure can be expected to occur as aresult of plastic deformation, the yield point should be usedas a basis for determining the allowable working stresses.However, if the vessel is designed to meet code requiremnnt.s,the procedure specified in the codes must be used. Itshould be pointed out that the maximum allowable workingstress specified by the codes is not always based on t.heultimate strength. The criteria used in establishing theallowable stresses in the’ASME eode (11) follow.

1. At temperatures below the creep range, allowablestress values were established at the lowest value of stressobtained from: (a) 25% of the specified minimum tensilestrength at room tempetature; or (b) 25y0 of the minimum

Page 34: Process Equipment Design

Material

Low-carbon nickel

25 aluminum

Copper

54s aluminum

18-8 stainless

Carbon steel

Carbon steel

Low-alloy steel

66% o f y ie ld ;

P l a s t i c I n s t a b i l i t y 2 5

ASME codeo designation

SB-162

SB-178-996A

SB-11

SB-178-GR4OA

Seamless quenched-.and-tempered steel

Navy G steel

T-l steel

SA-240 andSA-167

SA-201

SA-212

SA-362

Code case#1134

None

None

% ultimate 1 L, ultimate L62.6% of yieldPercentage of ultimate strength

LYield

Fig. 2.6. Comparison of allowable stress and yield stress as a percentage of the ultimate strength for materials within atmospheric temperature range.

(According to Zick [48]. Courtesy of the American Welding Society.)

expected tensile strength at operating temperature; or (c)I 621,s % of the minimum expected yield strength for 0.2 %

offset at operating temperature.2. For bolting material used at temperatures between

-20” F and 400” F the stress value were based on 20%of the minimum tensile strength or 25% of the minimumyield strength, whichever was lower.

Criterion lc introduces a further restriction for mate-i rials that have low yield-strength-tensile-strength ratios.

Appendix D lists allowable stresses for various materials

, as specified by selected codes and standards.The appendices of the ASME code for untired pressure

vessels (11) describe the basis for establishing values ofallowable stresses for both ferrous and nonferrous materials.Zick (48) has shown graphically (see Fig. 2.6) the allow-able stresses and yield strengths for various materials asfunctions of the percentage of the ultimate strength.

The percentages given in Fig. 2.6 are those specified byJ the code (11) except for the cases of the two high-strength

steels showr at the bottom of the figure, which as yet haveno code designation. The nonferrous materials shown in

Fig. 2.6 have allowable stresses based on two-thirds of theyield stress, which for these materials is an allowable stressless than one-fourth of the ultimate. Ferrous materialsshown have an allowable stress based on one-fourth of theultimate strength, which for these materials is an allowablestress less than 6234% of the yield stress except for 18-8stainless. The allowable stress for 18-8 stainless satisfiesboth criteria (25 % of ultimate and 6235 ‘$J of yield strength).If the criterion is a factor of safety applied to the yieldstrength, then the ratios between allowable stress andyield strength are inconsistent for the higher-strengthmaterials.

An allowable stress based on the yield point assumes thatfailure occurs by plastic deformation. If failure may beexpected to be caused by rupture rather than by excessiveplastic deformation, the use of the ultimate strength as acriterion for the allowable stress may have justification onthe basis that the fatigue limit, which controls failure byrupture, is usually proportional to the ultimate strength(48). It should be noted that failure by rupture has seldomoccurred in vessels fabricated of code-approved low-carbonsteels having high ductility.

Page 35: Process Equipment Design

26 Criteria in Vessel Design

2.4 BRITTLE RUPTURE

The current trend toward the use of higher-strengthsteels having lower ductiiity increases the possibility offailure by rupture. This has resulted in a number ofinvestigations and technical papers dealing with thisproblem (45, 46, 50, 51, 52, 106).

Stress concentrations are known to exist in a part underload where there are changes in shape or cross-sectional area.Very often these stress concentrations may be evaluated insuch parts as the j-ion of vessel closures and vesselshells. The code for the design of pressure vessels statesthat such “stress shall be considered” but does not indicatethe procedure for this consideration. The customary prac-tice has been the use of generous factors of safety and duc-tile materials. The use of high factors of safety resultsin the overdimensionlng of vessel sections. Such over-dimensioning, when used in conjunction with highly elasticmaterials, usually permits the dissipation of local stresses

strain hardening of ductile materials fn a test to rupture,and strain hardening resulting from local overstressingin repeated cyclic loading.

2.4a Notch Brittleness. Mild steels show nigh elonga-tion in the simple tensile test and are normally consideredto be ductile materials. Such materials can fail with littleor no evidence of plastic strain if the material contains acrack or notch and if the material is at a service temperaturebelow the “transition temperature” of the material. Thistype of failure is known as ‘1-f’ and hasresulted in the catastrophic fai ure of a number of weldedships and a number of storage vessels (53, 64). Severaltexts and articles discuss the phenomenon of notch brit-tleness (52-66).

The transition temperature is defined as the temperatureabove which the’dmpe of failure occurs. Below thetransition temperature a transition range may exist in whichthe material has semibrittle properties. At still lowertemperature the material becomes completely brittle.

by limited plastic deformation without failure by rupture.Currently the Pressure Vessel Research Council is support-ing research on materials having higher yield and tensilestrengths.

If no special corrosion problems are involved, failure, ifit does occur, is usually caused by either: (1) excessiveplastic strain (ductile rupture) or (2) brittle rupture (45).Figure 2.7 shows failure by excessive plastic deformation in amultilayer pressure vessel purposely tested to destruction.Such failure, discussed in the previous section, can occuronly if a high stress is distributed over a large area. Localstresses never produce great plastic deformation becausesmall plastic deformation serves to relieve these stresses.This type of failure seldom occurs in a properly designedvessel.

Below this temperature of complete embrittlement, brittlefracture may occur even though no notches or cracks existin the material. In the transition range a notch or crackmust exist for brittle fracture to occur. Above the transi-tion temperature brittle fracture will not occur even if sucha notch exists. Thus, failure below the transition tempera-ture is referred to as “notch brittleness.‘,’ Some materialshave a very narrow transition range; therefore a singletransition temperature is sufficient to define the transitionfrom the ductile to the brittle type of failure. Other mate-rials, such as plain-carbon or low-alloy-ferritic steel, havetransition ranges of hundreds of degrees.

Figure 2.8 shows the fragments of a 5000 psi monoblocvessel purposely tested to destruction. The fragmentationis typical of brittle rupture. Brittle rupture may resultfromi the use of brittle materials, “notch brittleness,”

IMPACT TESTS. The transition-temperature range isusually determined by making Charpy or Izod impact testsat various temperatures. The procedure for the Charpyimpact test for plate steels for vessel construction is describedin ASTM designation A 370-54T (67), and the minimumimpact strength permitted is given in ASTM designationA 300-54aT (67). The general procedure involved in such

I

1-

\ \ - \I I

Fig. 2.7. Ductile rupture in a multilayer vessel purposely tested to destruction. (Courtesy of A. 0. Smith Corp.)

Page 36: Process Equipment Design

Brittle Rupture 2 7

STRAIN E NERGY. When a load is applied to an elasticmaterial, the material deforms in the direction of the force,and work is done upon the material. This work is equal tothe product of the average force times the distance throughwhich the force moves. If the initial force is zero, theaverage force is equal to one half of the final force. Whenthe load is removed from an elastic body, it returns to itsoriginal shape, and in so doing it has the capacity for doingwork. Thus an elastic material under load may be said tohave “strain energy.”

Consider a cubic inch of elastic material initially underno load and apply a force sufficient to produce stress, f.The average force in terms of stress is equal to (f/Z), andthe unit deformation resulting is equal to:

Fig. 2.8. Brittle fracture in c) monobloc vessel designed for 5000 psi

and purposely tested to destruction. (Courtesy of A. 0. Smith Corp.)

tests is to prepare rectangular specimens with either a V,IJ, or keyhole notch machined across one face of the speci-men. The specimen is clamped in a vise with the notchexposed and facing a heavy pendulum. The pendulumis released in such a way that it strikes the specimen with animpact blow which causes the specimen to fracture at thenotch. The maximum swing of the pendulum after inter-action withyhe specimen is measured. From this measure-ment the foot-pounds of energy expended in causing thefracture may be determined. A brittle material fractureswith little or no plastic deformation, and the amount ofenergy required for fracture is small. A tough materialundergoes considerable plastic deformation prior to plasticfracture. This plastic deformation absorbs a good deal ofenergy. Thus the foot-pounds of energy absorbed incausing fracture is a measure of the toughness of the mate-rial. Figure 2.9 shows curves for typical data from CharpyU-notch tests for a variety of mild steels (53).

EFFECT QF &MPO~IUO_N. All of the four steels shown in. .._ -. _I..Fig. 2.9 have carbon contents between 0.20 % and 0.25%and yield strengths between 33,000 and 40,000 psi. Thecurves for the three steels shown to the right in Fig. 2.9exhibit complete brittleness at temperatures of 0 to - 10” F,whereas the ABS class C steel exhibits this phenomenon ata temperature about 30” F lower. The transition tempera-ture for all four steels covers a wide range. In general,steels with lower carbon contents exhibit greater toughnessat lower temperatures. The presence of phosphorus hasbeen shown to decrease the transition temperature. Acorrelation has been developed (60) indicating that thetransition temperature is a function of the sum of the per-centage of carbon plus 20 times the percentage of phos-phorus (C + 20P). The addition of nickel to steel cangreat!y increase its toughness and lower the transitiontemperature. Steel designated as AISI 2800 (854 % nickel)and 304 stainless steel will withstand impact loads at tem-peratures as low as - 320” F (liquid-nitrogen saturationtemperature) (61).

fe=-E

Therefore, the strain energy, IJ, is equal to:

(2.3)

For a volume of elastic material larger than 1 cu in., thetotal strain energy becomes equal.

Utota1 vol. = (U) (volume) (2.27)

GRIFFITH THEORY. A crack extending through a unit,..e.‘.... ,._.-thickness of a plate under stress deforms to produce anopening in the plate which may be considered to have theshape of an ellipse with a very large major-to-minor-axisratio. The volume of such an opening having a lengthequal to 2c is a function of c2. The strain energy released asa result of formation of the new crack volume in the ellipticalopening can be determined by use of Eqs. 2.26 and 2.21.

fut.v. = FE bc20 (2.28)

where the volume of the elliptical opening is cr&.

3 6 I , , , ,

3 2

$ 2 8

jz5, 2 4

E 2 0;it; 16

E 12

8

4

orl I -I I I I I I I I I I I I- 6 0 - 4 0 - 2 0 0 2 0 4 0 60 8 0

Temperature, degrees F

Fig. 2.9. Charpy U-notch data for some mild steels (53). (Courtesy of the

American Welding Society.)

Page 37: Process Equipment Design

28 Criteriq in Vessel Design

Fig. 2.10. Strain hardening due to cyclic load. (According to Kerkhof

[43]. Courtesy of the American Welding Society.)

Griffith (68, 69) in a rigorous derivation has shown thatfor a plate of unit. thickness this expression becomes:

RC2f2I-:t.“. = y-

The rate of release of strain energy with increase of crackdimension c as the crack propagates is obtained by dif’fer-entiation of Ey. 2.29.

The surface area of metal in the crack interface at thetime of formation of the crack is twice the product of thecrack lenghfl times the metal thickness, or 4c, for a plat,e ofunit thickness. The formation of this surface consumesenergy which is normally equal to the product of the areatimes the surface tension, .T, for brittle materials in whichfracture occurs with little plastic deformation. If someplastic strain accompanies the fracture, T will be greaterthan the surface tension and must be experimentally deter-mined. The value of T for low-carbon steel at room tem-perature is approximately equal to 2 X lo6 ergs per sq

cm (70). The energy required to form surface area ofmet.al at the crack interface is given by:

uc r a c k = 4~7’ (2.31)

The rate at which energy is consumed as the crack propa-gates is:

dUorack 4T__ =d c

(2.32)

The criterion for propagation of the crack is determined bythe ratio of the rate of strain energy released to surfaceenergy absorbed in creating the new surface at the interface,o r

(2.33)

o rf = 1/2ET/m (2.34)

It should be noted that when the ratio given by Eq. 2.33exceeds unity, a case of instability exists. For this condi-tion the crack will propagate at an increasing rate, approach-ing one third the speed of sound in the material (severalthousand feet per second for steel). Such failures occur sorapidly and are so extensive as to be catastrophic in mostinst.ances.

Substituting the approximate value of 2 X 106 ergs persq cm (70), which is equal to 11.4 in-lb per sq in., for T inEq. 2.34 and rounding off the product (2 X 11.4)~/~ to 10as an approximation gives:

__-f = dlOE/c (2.35)

If the stress f reaches the yield point, plastic deformationwill occur. Therefore the smallest crack that can init,iatebrit,tle fracture is determined by setting f in Eq. 2.35 equalto the yield point (y.p.) and solving for c.

10EGxitical = f,.,.” (2.36)

Thus a steel (SA-283, Grade C) having a yield point of30,000 psi would not propagate a crack shorter than 35 in.However, a low-alloy steel (ASTM A-242) having a yieldstrength of 50,000 psi would propagate a crack having alength of about JC in.

2.4b Repeated Cyclic loading. Brittle rupture canoccur without appreciable plastic deformation as a resultof local high stresses and repeated cyclic loading. Suchfailure may occur near limited areas of stress concentration,near defects in the plate, or near weld joints. This typeof failure does not occur during hydrostatic tests in spite ofthe fact that the stresses are higher than those induced inservice because they are not repeated so that they producefatigue. Failure by rupture usually begins by the forma-tion of a tiny crack after the vessel has been in service fora considerable period of time with cyclic loading operation.These small cracks continue to propagate with time. Thematerial surrounding the cracks becomes strain-hardenedand brittle. The extension of the cracks continues throughthe strain-hardened area and stops when ductiie materialis encountered. After continued stress cycles, the materialat the root of the cracks becomes strain-hardened and the

Page 38: Process Equipment Design

Britt le Rupture 23

The above relat ionship indicates that . in the considerat ionof failure by brittle rupture, discussed previously, theallowable stress should be based upon the yield point ratherthan upon the ultimate strength of the material. Equa-tion 2.37 does not take into account the phenomenon ofstress relaxation at elevated temperatures and was devel-oped for uniaxial s tresses (45) . Such considerations and theapplication of Eq. 2.37 are discussed in a later section ofthe text.

A number of articles have appeared reporting studies onthe problem of brittle rupture as a result of repeated cyclicloading (50, 52, 71-76).

For purposes of analysis of stress-strain relationships, thestress-strain curves can be approximated as indicated inFig. 2.11 (77). The stress-strain curves shown in Fig. 2.11a, b, and c are termed: power-function, straight-line, andideal ized curves respect ively. They can be represented byI he fol lowing equations:f = u,,o..58 (power-function, Fig. Z.lla) (2 .38)

f = f. + (E - ~0) tan (Y (st,raight-line, Fig. 2.11b)(2 .39)

f = fo ( idealized, Fig. 2 .1 lc) (2 .40)

2.4~ Other Factors Contributing to Brittle Rupture.

~YDROCEN EMBRITTLEMENT AND BLISTERING Hydro-gen will ‘diffuse into steel under certain conditions. Theaction of hydrogen at high temperatures and pressuresdiffers from that at low temperatures and pressures . W h e nsteel is exposed to hydrogen at high temperatures and pres-sures, the steel loses its tensile strength, becomes brittle,and often cracks or blisters. The mechanism of diffusionof hydrogen at high temperatures and pressures into steelis believed to result from the dissociation of hydrogenmolecules to monoatomic hydrogen. The partial pressureof monoatomic hydrogen causes the hydrogen to dif fuse intothe steel (78). As the hydrogen diffuses into the steelat high temperatures, it reacts with the carbon in the steelto form methane. The methane does not diffuse out of thesteel and accumulates to form blisters and cracks. F i g u r e2.12 shows sections through hydrogen blisters in pressure-vessel steels (78).

Stress-stra:ndiagram

_-Type

of curveStrain

hardeningconsidered

Fig. 2.11. Stress-strain curves fitting mathematical equations. (According

to Burrows et al. [77]. Courtesy of the American Society of mechanical

Engineers.)

cracks progress further. The continued strain hardeningand progression of the crack result in eventual failure bybrittle fracture.

Failure by rupture as a result of strain hardening can beexplained with reference to the stress-strain diagram givenin Fig. 2.10-according to Kerkhof (45). During theapplication of the hydrostatic test load or during the firstloading, local stress concentrations develop according tothe pattern given in Fig. 2.5b. If a small portion of thecross-sectional area of the shell thickness is under a sulb-ciently high stress intensification, removal of the load willresult in reverse yielding as shown by region CD in Fig.2.10. Reverse yielding occurs as a result of an inducedcompressive stress which exceeds the compressive yieldpoint C and produces plastic strain to point D. Such astrain can occur where bending stresses exists such as nearthe junction of vessel closures and attachments. B e n d i n gstresses may be very high on the shell surface and zero atthe center of bhe shell thickness. Upon reloading, thepath DEF is followed, and upon unloading, the path FGHi s fo l lowed. After each successive cycle the yield point ofthe material increases as a result of s train hardening. Aftera sufficient number of cycles a stress of Q is reached in theloaded condition, and of P in the unloaded condit ion withoutappreciable plastic deformation. If these stresses- reachthe fatigue limit of the material, minute cracks will formin the strain-hardened material. Continual cyclic opera-tion will result in eventual brittle rupture It should benoted that if the parallelogram OB’QP, the sides OB’ andPQ have equal length, and if Q is the tensi le stress equal andopposite to the compressive stress P, then Q = >#’ (45).Thus if the maximum value of the theoretical stress B’does not exceed twice the yield point A, the maximumvalue of Q wil l not exceed the yield point A, and fracture bystrain hardening will not occur. The maximum stressrange QP must not exceed:

fmax range 5 2fy.p. (2 .37)

where fma. rsnge = maximum local stress range not produc-ing fatigue failure, pounds per squareinch

fY.P. = initial yield point of the material at theoperating temperature, pounds per squareinch

Fig. 2.12. Section cut through hydrogen blisters in pressure-vessel steel (78).

(Courtesy of Shell Development Company and the American Welding

Society.)

Page 39: Process Equipment Design

3 0 Criteria in Vessel Design

At low temperatures and pressures the mechanism ofhydrogen diffusion is believed to be associated with theformation of hydrogen ions as a result of corrosive attack.The hydrogen ions are converted to monoatomic hydrogenby means of electron exchange. This conversion takesplace, for example, when a steel part is cathodic and deioni-zation occurs with the flow of a small amount of current.Energy is required to cause the hydrogen to penetrate thesteel. This energy is supplied by the current from thegalvanic action of corrosive attack. The driving energy ofa fraction of a volt in an electrolytic cell is equivalent tomany thousand atmospheres of hydrogen pressure (78).

The embrittlement caused in a vessel by hydrogen dif-fusion is temporary. If the equipment is shut down for aperiod of time, the hydrogen will diffuse from the metal.If the equipment is cooled slowly, the rate of hydrogendiffusion from the metal will be greatly increased. Anneal-ing for two hours at 1200” F or for one day at 225’ F willreturn the ductility to normal. The loss in ductility causedby hydrogen diffusion is not excessive. For example, ahydrogen content of 14 ppm in 1020 steel will reduce thetensile-test elongation from 40% to 22%. Many processplants operate normally without giving any particular con-sideration to hydrogen embrittlement. However, blisteringand cracking is a serious problem with equipment handlinghydrogen at high temperature and pressure.STRAIN AGING. When metal is permanently deformed

beyond the elastic limit by cold working, a precipitationmay occur because of local supersaturation along slip planesin the microstructure of the metal. Thus cold workinghas two effects. First, the hardness is increased at the timeof cold working; and this increase is followed, on the agingof the metal, by an additional increase resulting fromprecipitation. Second, the toughness decreases because of

T i m e -

Fig. 2.13. Schematic creep curve, extension plotted against elapsed time.

(A, elastic extension; 6, creep at decreasing rate; C, creep at approximately

constant rate; D, creep at increasing rate; E, elastic contration; F, perma-

nent change of shape.)

the precipitation and may be measured by impact test; asa function of aging (79, 80).TEMPER EMBRITTLEMENT. The phenomenon of temper

embrittlement occurs when hardened medium-carbon struc-tural steel is cooled slowly or held within a critical range oftemperatures below the temperature at which austenite istransformed to ferrite. This critical temperature rangeusually occurs in somewhere between 850 and 1100” F.Welded joints and service at elevated temperatures aresubject to this phenomenon. The effect is accentuated byhigh Mn, P, and Cr and is retarded by MO. The phenome-non is not completely understood but is believed to resultfrom a precipitation mechanism (80, 81, 82, 83).

2.5 CREEP

The criteria for design discussed previously have beenbased upon the premise that strain under load does not varywith time. This premise is essentially true for ferrousmaterials under load at temperatures up to about 650” F.However, beyond this temperature range the material“creeps” under load, causing an increase of strain with time.An increasing rate of creep is encountered as the servicetemperature is increased. Some materials, such as lead,creep readily at room temperature. The rate of creepdepends upon the prior history of the material and thestress as well as upon the temperature.

2.5a Creep Test. In studying the creep characteristicsof a material, a small tensile-test specimen is placed underconstant axial load while held at constant temperaturein an electric furnace. The rate at which the sampleelongates is recorded as a function of time for each tempera-ture and load. Depending upon the test conditions and t.hematerial, the duration of the test may last from a few hoursto several months and on occasion has been continued forseveral years. Figure 2.13 shows a typical creep-ratecurve obtained from such an investigation. The generalspecifications for conducting a creep test are covered byASTM specification E-22-41 (67).

Upon application of the initial load an instantaneouselastic strain occurs, resulting in an extension of the speci-men as indicated by extension A of Fig. 2.13. The initialcreep begins and continues at a decreasing rate for the timeinterval B; this region is known as the “first stage of creep.”This period is followed by a constant-rate period extendingover the time interval C; this region is known as the “secondstage of creep” and is the region which is used to limit theservice life of the equipment. The constant-rate zone isfollowed by an increasing-rate period over the time intervalD . This is known as the “third stage of creep” and endsin fracture of the specimen if the test is continued. Usuallythe test is interrupted before fracture occurs, and the speci-men undergoes an elastic contraction as indicated by exten-sion E. The amount of permanent strain is indicated byextension F.

Two typical creep curves for a high-alloy steel tested at1200” F with stresses of 20,000 and 25,000 psi are shown. inFig. 2.14 (84). It should be noted that increasing the stressfrom 20,000 to 25,000 psi greatly increases the creep rateand shortens the service life of the material. If a numberof curves such as those shown in Fig. 2.14 are obtained, the

Page 40: Process Equipment Design

Creep 31

0.060

Fig. 2.14. Time-elongation curves

a t 1200’ F , “ 1 6 - l 5-6” alloy-

solution quenched from 2 150’ F

(84). (Courtesy of Timken Roller

Bearing Co.)

g 0.050.c

.g 0.040F

B 0.030

0.020

I I I I I I I I I I I0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 7000

Time, hours

slopes of the second stage (constant rate of creep) may he de-termined and a cross-plot prepared such as the one as shownin Fig. 2.15 (85). In Fig. 2.15 the creep rate is plottedagainst stress with temperature as a parameter. T w ostandards of creep strength are commonly used for designpurposes in this country. One is the stress that will pro-duce a creep rate of 1% per 10,000 hours (approximately1% per year), and the other is the stress that will producea creep rate of 1% per 100,000 hours (approximately 1%in 11 years).

2.5b Creep-Rupture Test. Another consideration is therupture life of the part under the stress condition at theservice temperature. A test known as the stress-rupture orcreep-rupture test (67) is used to determine this rupture l i fe .It is similar to the creep test except that high loads are usedwhich produce greater creep rates so that rupture occursin a reasonable length of time. The total strain obtainedin a creep test is usually below 35% whereas in the creep-rupture test the total strain usually exceeds 50%. F i g u r e2.15 shows curves plotted from both creep-test data and

creep-rupture-test data. The time to rupture is plotted asa function of stress with parameters, as indicated in Fig.2.16 (85). The breaks in the three curves of Fig. 2.16 arethe result of a change from ductile to brittle fracture. It isessential that the rupture tests be conducted for sufficienttime to insure that the slope of the curve beyond the breakis establ ished. It is customary practice to extrapolate thisportion of the curve to 100,000 hours in order to obtain thestress to produce rupture for longer periods of time. TheASME code (11) specif ies that the al lowable stresses at hightemperatures are based on 100% of the stress to produce acreep rate of >{~a y0 per 1000 hours (0.00001 y0 per hour) oron 60 ‘% of the average stress or 80 y0 of the minimum stressto produce rupture at the end of 100,000 hours, whicheveris the lower value.

The stress to produce rupture after a specified length oftime varies with the material. Figure 2.17 compares thestress to produce rupture in 1000 hours as a function oftemperature for a variety of materials (86). The super-al loys shown in Fig . 2 .17 may be rather expensive for use in

Fig. 2.15. Correlation of creep-

test and creep-rupture-test data

for 18-8 molybdenum (type 316)

steel (05). icourtesy o f Car-

negie-!li’nois Steel Corp.)

Creep-rupture test

Minimum rate of extension, per cent per hour

b _ r \ - \ \r-T- -

Page 41: Process Equipment Design

32 Criteria in Vessel Design

1’

Fig. 2.16. Stress vs. rupture time

for 18-8 molybdenum (type 316)

steel at 1100, 1300, and 1500” F

(85) . (Courtesy of Carnegie-

Illinois Steel Corp.)

1 0 100Rupture time, hours

1000 10,000 100,000

vessel construction; they find their principal use in suchitems as turbine blades, parts for jet engines, and similarapplicat ions , but may be used for vessels for extreme service .

I t should be noted that creep- and stress-rupture-test dataare obtained under atmospheric exposure conditions inlaboratories and under uniaxial loading. The stress condi-tion existing in a vessel part under field service conditionsusually comprises stresses in three directions, a fact whichcomplicates the application of the experimental data. Inaddit ion, the vessel material may be exposed to a corrosiveatmosphere and be subject to scaling, hydrogen embrittle-ment , intergranular corrosion, and strain hardening.

2.6 CORROSION

The extent to which corros ion wi l l occur in process equip-ment depends upon the nature of the films that form on the

surface of the parts. The excellent corrosion resistance ofcopper and its alloys, for example, is the result of theirability to form thin, protective films on their surfaces.These f i lms may be the resul t of s imple oxidat ion or may becomposed of insoluble sa l ts . To be protective the coat,inpshould be thin, adherent, continuous, and relatively insolu-ble . Equipment operating under condit ions that al low theformation of a uniform protective film generally corrodesslowly and may last for a great many years.

Under severe corrosive condit ions , however , rapid corro-s ion occurs , resul t ing in cost ly delays and replacements . Byejudicious selection of material and by careful improving ofoperating conditions, corrosion can be reduced 01’ retarded,and substant,ial savings in operating and maintenance costsmay be realized. Thus an appreciat ion of the factors whichcontribute to corrosion is of value in the design of equip-

60

0400 600 800 1000 1200 1400 1600 1800 2

Temperature, degrees F

Fig. 2 .17. Stress to rupture in

1000 hours vs. temperature for

var ious mater ia ls (86) . (Cour-

tesy of Universal-Cyclops Steel

Corp. a n d Battelle M e m o r i a l

Institute.)

Page 42: Process Equipment Design

C o r r o s i o n 3 3

ment. A thorough treatment of the theory of corrosion and Table 2.2. Galvanic Series in Sea Water (236)its control is presented in the Corrosion Handbook (87). (Courtesy of American Society of Testing Materials)

2.6a Uniform Corrosion. When corrosion occurs at thesurface of equipment by the formation of soluble salts, auniform thinning of the wal l occurs . The rate of corrosiondepends upon the corroding medium, the velocity of fluidflow, the temperature, and other factors. This type ofcorros ion is encountered in ac id solut ions (part icular ly thosecontaining oxygen), in waters carrying a high oxygen orcarbon dioxide content (such as mine water), and in solu-tions havmg a solvent action on the corrosion products(such as those containing ammonium hydroxide, which dis-so lve the corros ion products f rom copper a l loys) . Attemptshave been made to reduce uniform corrosion by the applica-tion of an external electric current to provide cathodic pro-I.ertion. The limited success attained by this method hasbeen attributed more to the formation of protective filmsthan to cathodic protection.

2.6b Impingement Attack. Under normal operatingcondit ions certain local ized areas of the parts may be exposedto t,he destructive forces of a relatively high-velocity circu-la ling medium. C o r r o s i o n under such conditions isdescribed as “impingement attack.” Turbulence of thefluids causes a rapid and repeated destruction of t.he protec-tive film with subsequent corrosion of the exposed metal.This condition is considerably aggravated when the circu-la1 ing medium carr ies in suspension an abrasive mater ia l .

2.6~ Concentration-cell Attack. Corrosion may becaused by differential aeration wit,h the formation of con-centrat ion cel ls at the metal surfaces under certain operat ingconditions. Cracks, crevices, porous coatings, and breaksin protective films are sources of trouble since they trapliquid and set up differences in concentration of salts, ions,or gases in the c irculat ion medium. As a result of an elec-trochemical type of concentration-cell action, severe pit,tingof the metal surface and local ized perforation of the materialo c c u r . An example of th is type of corros ion is the “rust ing”of plain carbon steel. This type of corrosion can be mate-rially reduced by the observation of the following sugges-l i o n s ( 8 8 ) :

I. Specify butt joinls and tsttll)llasizr the necessity for com-plete penetrat ion of tlrt. weld tllat,rrial to guard against minutecrevices.

2. Avoid the use of lap j&Its, or completely seal them withweld metals , solder , or a suitable caulking compound.

:%. Avoid sharp corners and stagnant areas or other sitesfavorable to the accumulat ion of precipitates and other sol ids .

.1. Endeavor to provide uniform f low of l iquid with a mini-mum of turbulence and air entrainment.

3. Provide suitable strainers in lines to prevent local obstruc-tiou within the equipment which may start deposit at tacks orresult in impingement at tack.

2.6d Deposit Attack. When smal l part ic les se t t le out orlodge on the wall of t,he equipment, part of the metal wallbcxcomes protected by the deposit, and a special type of con-centrat ion-cel l act ion may take place. Usually the shieldedarea becomes anodic and intense pitting results. F i l t e r i n gof the circulating medium and frequent cleaning will mini-mize the occurrence of deposit a (t ack .

CORRODED END (ANODIC OR LEAST NOBLE)Magnesium, magnesium al loysZinc, galvanized steel , or galvanized wrought ironAluminum (52 SH, 4S, 3S, 2S, 53S-T in this order)Alclad, cadmiumAluminum, (A 17S-T, 17S-T, 24S-T in this order)Mild steel, wrought iron, cast ironNi-res i s t13 %-chromium stainless steel, type 410 (active)50-50 lead-tin solder18-8 stainless steel, type 304 (active), 18-8-3 stainless steei,

type 316 (active)Lead, tinMuntz metal, manganese bronze, naval brassNickel (active), Inconel (active)Yel low brass , admiral ty brass , a luminum bronze, red brass ,

copper , silicon bronze, Ambrac, 70-30 copper-nickel,camp. G-bronze, camp. M-bronze

Nickel (passive), Inconel (passive)Monel18-8 stainless steel , type 304 (passive) ; 18-8-3 stainless steel ,

type 316 (passive)

PROTECTED END (CATHODIC OR MOST NOBLE)

2.6e Galvanic-cell Attack. When dissimilar metals andalloys are in contact with each other in a conducting medium,a galvanic act ion is set up which results in the dissolut ion ofthe less-noble or anodic metal. From the electromotiveor galvanic series it is possible to predict the tendencies ofmetals and alloys to form galvanic cells and to predict theprobable direction of the galvanic action. Table 2.2 givesthe galvanic series as determined for sea water (89, 236).When use is made of such a series determined for the fluidunder considerat ion, one may be relat ively safe in choosingmetals from the same group; however, if the metals aredistant from each other in the list, the metal higher in thel is t wi l l corrode rapidly .

When one is using metals that produce a galvanic action,the relat ive areas of the two materials have a very importantbearing on the extent of corrosion. Usually the extent ofgalvanic action will be proportional to the ratio of the areaof the metal lower in the series to the area of the metalhigher in the l i s t . Thus i t is wise to avoid galvanic coupleswhere the exposed area of the cathodic metal is muchgreater than that of the anodic metal. (For example, asteel part in a copper vessel would rapidly corrode, but acopper part in a steel vessel would be relatively safe).

2.6f Stress Corrosion. As a result of the simultaneousact ion of s tresses and certa in corros ive condit ions , parts mayfai l by cracking. When the stress is applied externally, thebreak often is called a “stress-corrosion crack.” Whenresidual internal stresses are involved, the resulting breakoften is called a “season crack.” Annealing to relieveresidual stresses greatly reduces season cracking. W h e nthe part is subjected to repeated cyclic stresses duringservice, failure by “fatigue cracking” may occur, as dis-

r r__-~ .-- . . -

\ - \ \ P T---

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3 4 Criteria in Vessel Desigd

cussed previously. Such failures are characterized by theirsuddenness and by the absence of plastic deformation in thefailing section.

Stress-corrosion data on systems free of corrosion in theunstressed state indicate that there is little or no attack onmaterial that is either completely elastically deformed orcompletely plasticly deformed. But such parts subjected

to both plastic and elastic deformation in the same sectionmay undergo severe stress corrosion. Thus stress corrosionis caused by strain hardening in combination with highelastic stresses rather than by plastic deformation (45).Such strain hardening followed by corrosion has beenobserved in boilers and pressure vessels where local stressconcentrations exist without severe plastic deformation.

P R O B L E M S

1. A full-floating-head horizontal condenser has 400 >6-in-diameter No. 16 BWG admiralty-metal tubes. The floating tube sheet and head assembiy weigh 1000 lb when the head is filledwith water. If the tube-support plate is located 18 in. from the center of load of the floatinghead assembly and the overhanging tubes and head assembly are assumed to behave as a simplecantilever beam,

a. What bending stress is developed in the tubes as a result of this load?b. What is the floating-head deflection from the horizontal at the center of load?e. Is the design satisfactory if the allowable stress is 6000 psi?

3

For a simple cantilever, y = g

n4 max = Pl

For a tube, Imctanpular =s(do4 - di4)

6 4

For a g-in. No. 16 BWG tube, OD = 0.625 in.ID = 0.495 in.

Modulus of elasticity of admirality metal = 15 X lo6 psi

2. A horizontal stiffener supporting a bubble-cap tray in a fractionating column may be con-sidered to act as a uniformly loaded, simply supported beam. The deflection equation forsuch a beam is:

WZ3X- - - - -24 I

where y = dellection’at point z, inchesz = distance from end of stiffener, inches

2 = total length of stiffener, inchesE = modulus of elasticity, pounds per square inchZ = moment of inertia, inches4

ta = uniform load, pounds per linear inch

The stiffener has a moment of inertia of 1.785 in4 and a section modulus of 1.02 in.3 and isof steel (E = 30 X lo6 psi). The stiffener has a length, 1, of 100 in. and carries a uniformload, w, of 2.4 lb per linear in.

Determine:a. The maximum deflection at the center of the span, z = 50 in.b. The maximum stress at the center of the span.c. The stress 20 in. in from the support, z = 20 in.d. The shear load, pounds, at x = 20 in.

3. A copper fractionating column has vertical tray-support rods between trays. The traysare 20 in. apart and the rods are spaced so as to support 100 lb each, under column action.What is the diameter of the rods required to limit the column loading to one half of the criticacolumn load if (a) the rods are free to pivot at both ends or (b) the rods are fixed at both ends?

k = i: E = 15 X lo6 psi

Page 44: Process Equipment Design

P r o b l e m s 3 5

4. A vessel is to be fabricated of type 316 stainless steel (see Figs. 2.15 and 2.16). Thevessel is intended to be used at 1300” F for an expected service life of five years. The creepdeformation permitted during the service life of the unit is not to exceed 5%. Determine theallowable stress if the allowable stress is not to exceed either (a) two thirds of the stress to pro-duce the creep rate or (b) 50% of the stress to produce rupture. Also estimate the total creepand rupture life if the allowable design stress as determined in this manner is used in the designof the vessel.

r \ - ‘i - \I 1 e _- - -

Page 45: Process Equipment Design

C H A P T E R

DESIGN OF SHELLS

FOR FLAT-BOTTOMED CYLINDRICAL VESSELS

u hroughout the chemical and petrochemical industr ies ,gases, liquids, and solids are stored, accumulated, or proc-essed in vessels of various shapes and sizes. Such a largenumber of storage vessels or tanks are used by these indus-tries that the design, fabrication, and erection of these ves-sels have become a specialty of a number of companies. A sa result of economic considerat ions, only a few companies iuthe process industries now design storage vessels havinglarge volumetric capacity, and the usual practice is to con-tract for this equipment with companies specia!izing in t.hisfield. However, the design of this equipment involves basicprinciples which are fundamental to the design of othertypes of more special ized equipment. This and the follow-ing chapter cover these fundamentals.

Before the extensive use of welding as a means of fabrica-tion, vessels fabricated of metal were assembled either hybobing or riveting (90). At the present time most fluidsheld at atmospheric or low pressure are contained in cyl indri -cal tanks of welded construction. Because of the largequantities of petroleum and its products stored in thismanner, the American Petroleum Institute has establishedspecifications governing the design of welded, vertical,cylindrical tanks for oil storage above ground. Thisspecification code, API Standard 12 C (2), is intended toprovide the oi l industry with design speci f icat ions for tanksof adequate safety and reasonable economy in a wide varietyof capacities. It is recommended for use by the oil indus-t.ry in all sections of the United States except those areaswhich are subject to local regulat.ions that conflict with it.Although these speci f icat ions were devised primari ly for thedesign of storage vessels for petroleum and its products,they are useful guides in the design of a variety of vessels for

storage of other f luids . Figure 3.1 sholvs a typical weldedoil -s torage tank.

3.1 MATERIAL SPECIFICATIONS

The materials used in the construction of storage vesselsare usually metals, alloys, clad-metals, or materials withlinings which are suitable for containing t.he fluid. W h e r eno appreciable corrosion problem exists, the cheapest. andmost easily fabricated construction material is usually hot-rolled mild (low-carbon) steel plate. The particular typesof steel plate specified by API Standard 12 C are SA-7(open-hearth or electric-furnace processes only), SA-283,Grade C for all thicknesses greater than l>i in., or SA-283,Grade D for thicknesses less t,han I>4 in. Copper-bearingsteel, which is not specified by this code, has some advan-tages in res is t ing a tmospher ic corros ion . SA-7 is specifiedfor structural steel shapes such as angles, channels, andI-beams, and ASTM-A-27 grade 60-30 fully annealed isspecified for steel castings. The physical properties andchemical composition of these plain carbon steels are listedin Table 3.1.

Low-carbon steels are rather soft and ductile and areeasily sheared, rolled, and formed into the various shapesused in fabricating vessels. These steels are also easilywelded to give joints of uniform strength relatively freefrom localized stresses. The ultimate tensile strength isusually between 55,000 and 65,000 psi and the carboncontent between 0.15% and 0.25%. However, both theultimate strength and the carbon content cannot be speci-lied since one is a function of the other. In Table 3.1 theultimate strength is specified but not the carbon ccntent.

Low-alloy, high-strength steels are a specific class of low-carbon steels which are made stronger by the addition of a

36

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Material Specif ications 37

Fig. 3.1. Welded oil-storage tank, Humble Oil and Refining Company, Baytown, Texas. Diameter, 120 ft; height, 40 ft; nominal capacity, 80,000 bbl.

(Courtesy of Bethlehem Steel Company.)

small amount of alloying elements. Such steels have yieldpoints that are considerably higher than plain carbon steelof the same carbon content and also have higher ultimatetensile strengths. The chief disadvantage of some of thehigh-strength alloy steels of moderate alloy content is thatthey are more difficult to fabricate because they have lowductility. Some have an increasing tendency to air-harden,which may result in localized stresses in welded joints.However, these disadvantages may be avoided if the alloycontent and the carbon content are both kept relatively low,as they are in low-alloy, high-strength steels. Many steelmanufacturers do not produce these steels to standardspecifications as they do plain carbon steels but use avariety of formulas marketed under various trade names.

Most manufactures claim t,heir low-alloy, high-strengthsteels t.o be readily “weldable.” However, this is not aprecisely defined characteristic, and it should be understoodthat such “weldability” refers to conventional metal-arcwelding processes usually employed for plain carbon steel.Table 3.2 gives properties of low-alloy, high-strength steelsrecommended by API Standard 12 C, 1955.

In selecting material for large-diameter vertical vessels,consideration should be given to the types of failure thathave occurred in the past. In 1952 two unusual tank fail-ures occurred in England while the tanks were being hydro-statically tested wit,h water at 40” F (91). Figure 3.2 showsthe catastrophic nature of the failure. One tank was acrude-oil storage tank 140 ft X 54 ft with a floating roof,

0)

J?lat,es

Table 3.1. Properties of Plain Carbon Steels, Recommended by API Standard 12 C(Courtesy of American Petroleum Institute)

(2) (3) (4) (5) (6) (7) (8)Steel Phosphorus, * Sulfur, * Tensile Elongat.iou,

Specili- max, max, Strength, Yield Point, min, per centcations per cent per cent Psi min, psi In 2 in. In 8 in.

SA-7 and Acid 0.06 0.05 60,000-72,000 33,000 2 2SA-283, Basic 0.04

(

1,500,000

Grade D Tensile st.rengtht

SA-283, Acid 0.06 0.05 55,000-65,000 30,000 2 4Grade C Basic 0.04

1,500,000Tensile strength?

Structural Shapes SA-7 Requirements are same as given above for A-7 plates.- .,

Steel Castings ASTM-A-27, 0.05 0.06 60,000 min 30,000 24Grade 60-30,fully annealed

“’ From ladle analysis made by manufacturer; check analysis from finished material by purchaser may show 25 y0 more.t See exceptions listed in particular specification. Copper, when copper-steel plates or structural shapes are specilied.

miuimum percentage, 0.20.

- _ ~--- - - - - - - -I \-F--- \ I 7---------

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3 8 Design of Shells for Flat-bottomed Cylindrical Vess

and the other was a gas-oil storage tank 150 ft X 48 ft witha cone roof. A fairly large number of similar tanks havefailed in this country. These failures and those encounteredin welded “Liberty” ships and T-2 tankers (92, 93) havestimulated studies on the reason for such failures (53, 94,95, 96).

The failures of the two British tanks were traced to flawsin the welds in the shell. In the case of the crude-oil storagetank, the flaw initiating failure was incomplete penetrationin a weld-probe replacement located across the first hori-zontal welded joint. The crack progressed vertically inboth directions with brittle fracture up to the Gfth course;the fracture of the upper four courses was by shear. It issignificant that the rupture occurred entirely across theplates and not along any of the vertical welded joints in thenine courses. The gas-oil storage tank failed in a similarmanner. In this case the flaw was in a partially repairedcrack in the top 10 in. of a vertical joint in the top of thefirst course. As in the case of the crude-oil tank, this crackprogressed vertically in both directions with brittle fracturein the first four courses and with shear fracture in the remain-ing upper courses. The fracture occurred chiefly across theplates as in the other failure except for the fifth course,where it traveled through a vertical welded joint.

These two ruptures appear to be examples of failure bynotch brittleness as a result of a crack existing in the plateat a location of high stress and having a length greater thanthe critical crack length for crack propagation. See Chapter2 for discussion of notch brittleness and critical cracklengths.

A complete investigation was made of samples of steelfrom the two tanks which failed. The samples were impacttested at various temperatures and the results were com-pared with corresponding data for other steels used forvessel construction (53, 91, 94, 95). The steel used in theconstruction of these tanks was identified as BS-13, aBritish steel similar to SA-283, Grade C. Some of the

results of these tests are given in Fig. 2.9 (53), in the previouschapter. The conclusion made was that the British steelhad properties within the specification given and was asgood as similar American steels of equivalent quality. Thesteel at the time of hydrostatic test was at a temperature(40” F) within the transition-temperature range (see Fig.2.9). If the flaws had been smaller in size, or if a moreductile steel had been used, or if the hydrostatic test hadbeen carried out at a higher temperature, the vessel prob-ably would not have failed. These failures demonstratethe necessity of sound welding procedures and thoroughinspection of welded joints.

Even if cracks cannot be avoided, a high degree of protec-tion against brittle fracture can be obtained with low-carbonsteels if the steel has a minimum specification of 15 ft-lb inthe Charpy V-notch test at the lowest service temperature(47, 96). When specifying fine-grained steels or semikilledsteels of low carbon with manganese, a 30 ft-lb CharpyV-notch value would give similar protection (96). Themajority of the steels presently used for storage-tank con-struction do not meet this specification at temperaturesbelow 50’ F. It is recommended that for service tempera-tures below 50” F, steels meeting the above specificationbe used, such as ABS, class C or SA-201, Grade B (67) andthat all weld joints in the shell be radiographed.

3.2 ESTIMATED COST OF TANKS

Storage tanks require considerable capital investment.If a limited quantity of fluid is to be handled, a single tankmay suffice, in which case the magnitude of the proportionsis controlled by the volume of fluid to be stored. Where alarge number of tanks are required, it is generally true thatlarger tanks give a lower cost per unit volume of storagethan smaller tanks. This is indicated in Fig. 3.3, whichshows that the total installed cost of a l,OOO,OOO-gal cone-roofed tank is approximately $32,000 whereas the corre-sponding cost of a lO,OOO-gal tank is about $3000, a hundred-

Table 3.2. Principal Provisions of Specifications on low-alloy, High-strength Steels, Recommended by API Standard 12 C(Courtesy of American Petroleum Institute)

(1) (2) (3) (4) (5) (f-9 (7) (8) (9)Chemical Requirements, max, per cent Tensile Requirements, min

Tensile YieldM a n g a - Phos- Strength, Point, Elongation, min, per cent

Carbon nese phorus Sulfur Silicon ps i psi In 2 in. In 8 in.ASTM-A-242

Ladle Analysis 0.22 1.25 . . . 0.050 . . .Check Analysis 0.26 1.30 . . . 0.063 . . .Thicknesses :

4is to N in., incl. 70,000 50,000 1 8Over W to 135 in.,incl. 67,000 46,000 1 9Over 136 to 4 in., incl. 63,000 42,000 2 4 1 9

SAE 950 0.20 1.25 0.150 0.050 0.90Thicknesses:

0.0710 to 0.2299 in.,incl. 70,000 50,000 2 2 . . .0.2300 to $6 in., incl. 70,000 50,000 2 2Over >s to 1 in., incl. 67,000 47,000 2 2 1,500,000Over 1 to 2 in., incl. 65,000 45,000 2 2 Tensile strength

Page 48: Process Equipment Design

Optimum Tank Proportions 3 9

Fig. 3.2. Failure of British storage tank at Fawley, England (91). (Courtesy of Esso Research and Engineering Company and Petroleum Publishing Co.)

fold increase in volume for approximately 11 times the costof the smaller tank. However, the large tanks are notalways selected because of the greater flexibility permittedin storing a variety of fluids in a battery of smaller tanks.For such reasons no general rule can be made for the selec-tion of an optimum number of tanks.

The total cost of various types of large-sized storage tanksis given in Fig. 3.4 (97). The cost does not include the costof the foundation. The cost of tanks fabricated from vari-ous materials (18), field-erected steel tanks (98), and smalltanks under 1000 gal (99) has been reported in the literature.(See also 26, 27, and 28.) Figure 3.5 shows the weight ofsteel and the cost in dollars per ton for large-diametertanks (97).

J3.3 OPTIMUM TANK PROPORTIONS

Before a storage tank can be designed, the proportion ofheight to diameter must be established. The diameters ofstandard steel tanks for storage at atmospheric pressureusually range from 10 to 220 ft, and the heights vary from6 to 64 ft. Typical tank dimensions are listed in AppendixE. No general rule can be given for the selection of theheight-to-diameter ratio because t.his ratio is often a functionof the processing requirements, available land area, andheight limitations. Figure 3.6 shows a group of oil-storagetanks of various height-to-diameter ratios.

The volume of a single tank, which may be one of a

”Pz%i.E= 1035l-

1 0 2 2103 104 106

Volumetric capacity in gallons10s

Fig. 3.3. Estimated cost of small and medium-sized tlat-bottomed cone-

roofed storage tanks.

Assumption,:

Steel base cost j- extras -(- freight = 6c/lb

Total cost = total weight X (6~ + fabrication cost + erection cost)

Marshall and Stevens “Process industries Average” cost index = 180

-..

- - - _~ --~~-..-. _-_..

\ - \- \ I

7- - - _--_-- ---. - - - -

Page 49: Process Equipment Design

40 Design of Shells for Flat-bottomed Cylindrical Vessels

k

B 8052 705.G 60%8 50zP 40

30

Fig. 3.4. Estimated insblleP

costs of large-diameter tanks

197). ICourtesy o f P e t r o l e u m

Processing, McGraw-Hil l Pub-

lishing Co.)

0’. I ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ :J ’10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 i90 200Capacity in thousands of barrels

battery of tanks, may be determined by the process require- exists with tanks of small volume, in which elastic stabilityments and other considerations such as production flexihility and corrosion control the thicknesses. The upper limit forand seasonal variations in storage requirements. The the optimum ratio of D/H occurs when the shell thicknessesoptimum proportion of the tank diameter, D, to height, H, is a function of both D and H, and the unit area costs of thevaries between two limits. The lower limit for the optimum bottom and roof are independent of D and H. This condi-ratio of D/H occurs when the shell, bottom, and roof costs tion exists with tanks of large volume.per unit area are independent of D and H. This condition The optimum proportions of a mnk are influenced by the

700

650

600

550

500

450

400

350

300

250

200

150

100

50

n-0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 ZOO

Capacity in thousands of barrels

\ /,,

Fig. 3.5. Weight and cost pev

ton of large-diameter tanks (97).

(Courtesy of Petroleum Procesr-

ing, McGraw-Hill Publishing Co j

,.”

ii;

I--

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Page 50: Process Equipment Design

Optimum Tank Proportions 41

Fig. 3.6. Oil-storage tanks of various height-to-diameter ratios. Three 96,000-bbl tanks 120 ft in diameter by 48 ft high with three fire walls 172 R in

diameter by 24 ft high appear in the background. (Courtesy of Hammond Iron Works.)

cost of the foundations and the cost of the land in the tankarea that is chargeable to the tank as well as by the cost ofthe bottom, shell, and roof, if required. The value of t.heland chargeable to the tank may be expressed in terms of anannual cost per unit area. For purposes of tank proport ion-ing, it is also convenient to express the cost of the founda-tion, bottom, shell, and roof, if used, in terms of cost perunit area as fol lows.

--I

Let D = diameter of the vessel, feetH = height of t.he vessel, feet

aD’(H)V = volume of the vessel, cubic feet = ___

4or

+

“2$n-

(3.1)

When A1 = arca of the shell, square feet = TDHAs = area of the vessel bottom or the projected

area of the roof, square feetTD2=-

4c1 = annual cost of fabricated shell, dollars per

square footc2 = annual cost of fabricated bottom, dollars per

square footc3 = annual cost of the fabricated roof, dollars per

square foot of projected area

c4 = annual cost of the installed foundation underthe vessel, dollars per square foot of tank-bottom area

t h e n

cs = annual cost of the land in the tank areachargeable to the tank, dollars per squarefoot of tank-bottom area

C = total annual cost, of the vessel, dollars peryear

C = (Am) + Az(c2 + CQ + ~4 + ~5) (3.2)

Substituting for the areas -41 and 42, we obtain:

C = TDHCI + $ (c2 + c3 + ~4 + ~5)

Simplifying the equation by substituting for H in termsof D, we obtain:

c = 7 + F (c2 + c3 + c4 + c5) (3.3)

To determine the optimum tank proportions by usingEq. 3.3, it is necessary to determine which of the cost termsare var iables pr ior to di f ferent iat ion.

3.30 Tanks Having Shell Thickness Independent of Dand H. The stress in the shell is a function of both thediameter and the height of the tank, as indicated in Eqs.3.18 and 3.19. For reasons of e las t ic s tabi l i ty , the minimumshel l th ickness i s l imi ted to N s in. for 45-ft tanks and smallerand to x in. for tanks of larger diameters. Therefore,

I \ - \T’ , ..-T-- - .-_ _

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4 2 Design of Shells for Flat-bottomed Cylindrical Vessels

tanks having shell thicknesses of >/4 in. or less may be con-sidered to have a shell thickness independent of H and D.Substituting )i-in. shell thickness into Eqs. 3.18 and 3.19,results in the following:

0.25 = 0.0001456 (H - 1)DOr

andD(H - 1) = 1720 for butt-welded shells (3.4)

D(H - 1) = 1515 for lap-welded shells (3.5)

Thus, all tanks with butt-welded or lap-welded shellshaving a value of D(H - 1) equal to or less than 1720 or1515 respectively fall into this category.

Tanks of small volume falling into the category of Eq. 3.4or Eq. 3.5 have shell, bottom, and roof costs per unit areawhich may be considered independent of D and H. UsingEq. 3.3 and differentiating the total cost, C, with respect tothe diameter of the vessel, D, and considering the volume,V, to be known and the cost factors, cl, y, cz, c4, and cs, asconstant known factors for the volume considered, weobtain :

dC-=dD

- 4+ + r$ (C2 + C 3 + C4 + cs).

Actually, the individual unit costs, cl, ~2, cg, and ~4, of thevarious tank components will vary somewhat with tankproportions and with tank volume and other factors such asdesign considerations. Many correlations for estimatingtank costs have been presented in the literature. Thesecorrelations indicate that for purposes of estimating installedtank costs, the cost is primarily a function of the totalvolume and that total installed costs do not vary appre-ciably with the unit costs of component parts. Therefore,the consideration of unit-cost factors as constants is areasonable approximation.

For minimum cost the slope of the curve of cost versusthe diameter of the vessel to contain the fixed volume, V,must be equal to zero (dC/dD = 0). Therefore

- % + y (c2 + cg + c4 + c5) = 0

Solving for D, we obtain:

03 = !!! Cl

n- C2 + C3 + C 4 + C5

Substituting for the volume, V, we obtain:

C l

o r

D = 2H ClC 2 + C3 + C4 + C5

(3.6)

3.3b Tanks Having Shell Thicknesses Dependent uponD and H. Tanks having heights and diameters such thatthe quantity D(H - 1) exceeds 1720 for butt-welded shellsor 1515 for lap-welded shells have shell thicknesses whichare dependent upon D and H. The cost of the shell perunit area, cl, is a function of D and H and for purposes of

proportioning may be assumed to be directly proportionalto the shell thickness as follows:

cl = cg(H - l)D (3.7)Or

(3.8)

Substituting Eq. 3.7 for cl in Eq. 3.3 gives:

c = 4VMH - l)D] uD2D + _ (C2 + C3 + c4 f ca)4

Expanding and substituting for H by Eq. 3.1 gives:

- 4vC6 + T$ (C2 + C3 + C 4 + C5)

Differentiating and setting equal to zero gives:

dC -32csV2dD = ~03

- 0 + y cc2 + cg + c4 + es) = 0

$ (c2 + C3 + c4 + c5) = s

Substituting Eq. 3.1 for V and Eq. 3.8 for cg gives:

*D-y (Cz + C3 + C4 + c 5 ) =

((H “;;&) te)’

Or

=+ cc2 + c 3 +;c4 + c5) =((H T;u3 r+)

Since H - 1 for large tanks is approximately equal to I-r,it will be assumed that (H - 1) = H. Therefore

D(c2 + c3 + c4 + c5) = 4c1HOr

D = 4H C l

C2 >+ C3 + c4 + c5(3.9)

3.3~ Estimation of Cost Factors. In most designs thecost factors, cl, ~2, and so on, cannot be accurately evaluateduntil the design of the tank is known, a requirement whichnecessitates successive approximations in determining theoptimum proportions. The costs of the tank components,shell, bottom, and roof, are a function of the plate t.hick-ness, grade of steel, cost of forming, cost of field welding,and so forth, and include the cost of the accessories such asnozzles, manholes, pumps, ladders, platforms, and so forthwhich are attached to the various components.

These factors are all interrelated, and in order to makean estimate it is usually more convenient to express themin terms of the cost of the fabricated component parts ofthe vessel per pound of fabricated material because infor-mation concerning cost per pound is more readily available.

3.3d Simpli f ied Cases of Optimum Proport ions. Todemonstrate the use of Eq. 3.6, three simplified cases willbe presented.

For the first case, consider a small open tank and disre-gard the costs for land and foundation. For small tanks

\ \- \I/- - -- - - -

Page 52: Process Equipment Design

the shell thickness is often the same as the bottom thickness.If the cost per square foot of shell, cl, is taken as equal tothe cost per square foot of bottom, c2, then cl = c2; andif ~3, ~4, cg = 0, then, by Eq. 3.6,

D = 2H Cl = 2H (3.10)I c1+0+0+0

kHowever, the shell is generally more expensive than the

bottom partly because of the additional cost of rolling theshell plates.

For the second case, consider a small closed vessel havingthe same cost per unit area for shell, roof, and bottom anddisregard the cost of foundations and land.

Cl = cp = c3

r And if c4 and cg = 0, by Eq. 3.6,

i D = 2H- cl = HCl + Cl + 0

(3.11)

For closed vessels also the cost of the shell per unit areais generally greater than the cost of the bottoms per unitarea for the reasons previously stated. Furthermore, the

1 roof costs are generally greater than the bottom costs perunit area because of the structural steel required for theroofs of all but small vessels.

For the third case, consider a large closed tank in whichthe roof and shell costs are twice the cost of the bottomper unit area.

Cl = 2cz = c3

And if c4 and cg = 0,

D zz 4H 2c2c2 + 2c2 + 0

= #H (3.12)

Introducing the actual values of foundation and landcosts, c4 and cg, into the equations has the obvious effectof increasing the height-to-diameter ratio. It is apparent

1 that in areas where land is cheap and the tanks can beeasily supported on the soil without expensive foundations,

Shell Design of Large Storage Tanks 4 3

economical designs result in tanks having low heigbt-to-diameter ratios.

3.4 SHELL DESIGN OF SMALL AND MEDIUM-SIZEDVESSELS (PRODUCTION TANKS)

Small and medium-sized vertical vessels with flat bottoms,called production vessels in contradistinction to storagevessels) are usually fabricated from steel plate of a singlethickness. Their optimum proportions are similar to thosediscussed previously in case two (D = H). The designof such vessels is simple and has been standardized for theoil industry (100, 101) as described in Fig. 3.7 and Table3.3. The shells of such vessels are usually fabricated ofeither Ns-in. or sd-in. plate with plate widths preferablynot less than 60 in. A mild steel meeting SA-7, SA-283,Grade C or D (open-hearth or electric steel only) is speci-lied. The shell plates are usually either double-weldedbutt joints with complete penetration of weld metal orsingle-welded butt joints with backing strips with completepenetration of weld metal. The design of the bottom androof (deck) is covered in Chapter 4.

3.5 SHELL DESIGN OF LARGE STORAGE TANKS

The majority of tanks and vessels are cylindrical becausea cylinder has great structural strength and is easy tofabricate. Several types of stresses may occur in a cylin-drical shell. These may be recognized as:

1. Longitudinal stress resulting from pressure within thevessel;

2. Circumferential stress resulting from pressure withinthe vessel ;

3. Residual weld stresses resulting from localized heating:4. Stresses resulting from [email protected] such as

wind, snow, and ice, auxiliary equipment, and impact loads;5. Stresses resulting from thermal differences;6. Others, such as may be encountered in practice.

3.5a Stresses in Thin Shel ls Based on MembraneTheory. Simple equations may be derived to determine theminimum wall thickness of a thin-walled cylindrical vessel

NominalCapacity,

, bbl90

i 200210300400

H-500750

L-500II-1000

1500L-1000Y 2000

3000

ApproximateWorkingCapacity,

bbl72

166200266366479746407923

1438784

17742764

Table 3.3. Typical Dimensions for Production Tanks (100, 101)(Courtesy of American Petroleum Institute)

Height of Height ofOutside Overflow Walkway

Diameter, Height, Connection, Lugs,ft in. ft ft in. ft in.7-11, 1 0 9-6 7-7

12- 0 1 0 9-6 7-7lo- 0 1 5 14-6 12-712- 0 1 5 14-6 12-712- 0 20 19-6 17-715- 6 16 15-6 13-715- 6 24 23-6 21-721- 6 8 7-6 5-721- 6 16 15-6 13-721- 6 24 23-6 21-729- 9 8 7-6 5-729~ 9 16 15-6 13-729- 9 24 23-6 21-7

Location ofFill-Line

Connection,in.1 4141414141 41 41 41414141414

Size ofConnec-

tions,.in.

344444444

i44

---

--i- -

-

\ - \

-- I---\I-/o __ .- -____ .-- --_-.

Page 53: Process Equipment Design

44 Design of Shells for Flat-bottomed Cylindrical Vessels

D r a i n - l i n e &*24* x 36” cleanout

connection- ’

Plan

,- 20”-diam dome

I-T-=40”

I +--I

24” x 36”

I

D

.

- - & i t -

TA-

- -

ic

4Elevation

II % 4- - -----.

7Walkwaybracket

l u g s

_.

&-

1Yd

l a n k

Detail,thief-hatch outlet

]j[

Detail of dome

Detail, Iw a l k w a y b r a c k e t l u g s

‘14” min

Shell- plate

Fig. 3.7. Standard design for sm-all and medium-sized vesseh (production tanks) as recommended by API Standard 12 D. (See Table 3.3 fo. dimensions.)

’(Courtesy of American Petroleum Institute.)

L___- __... - . . - --- - - -.

\ \ \I /

Page 54: Process Equipment Design

Shell Design of large Storage Tanks 45

Fig. 3.8. Longitudinal forcer acting on thin cylinder under internalpWSS”r.3.

with an internal pressure. Figure 3.8 shows a diagramrepresenting a thin-walled cylindrical vessel in which a uni-form stress,f, may be assumed to occur in the wall as a result

I of internal pressure. The nomenclature used in Figs. 3.8and 3.9 is:

b1 = length, inches

d = inside diameter, inches

t = thickness of shell, inches

p = internal pressure, pounds per square inch gage

hngiludinat Stress. If one limits the analysis to pressurestresses only, the longitudinal force, P, resulting from aninternal pressure, p, acting on a thin cylinder of thickness t,

+1enpt.h I, and diamet,er d is:

P = force tending to rupt.ure vessel longitudinally

Inrd2z--4

anda = area of met,al resisting longit~udinal rupture

= tnd

1 l‘hert~foreP p7rd”i-l Pdj = stress = - = --- --- = 2 = induced stress, poundsa hd per square inch

ort = pd

?f(3.13)

CIRCUMFERENTIAL STRESS. If oue refers to Fig. 3.9 and-,onsiders the circumferential s~ressrs caused by internal,:ressure only, the following analysis rrriiy be developed :

I’ = force tending to rapture vcssrl circumferentially

= pdl

a = area of metal resisting force

= 2t1

P pdl pdf =I sf,ress = -. = --.. = _a 2t1 2t

1 = PC”? f

(3.14)

A comparison of Eq. 3.13 with Eq. 3.14 indicates that fora specific allowable stress, fixed diameter, and given pres-sure, the thickness required to contain the pressure for thecondition of Eq. 3.14 is twice that required for the conditionof Eq. 3.13. Therefore, the thickness as determined byEq. 3.14 is “controlling” and is the commonly used “thin-walled equation” referred to in the various codes for vessels.This equation makes no allowance for corrosion and doesnot recognize the fact that welded seams or joints may causeweakness.

J OINT EFFIC IENCIES AND CORROSION ALLOWANCE . Invessels for atmospheric storage the welded joints are seldomstress relieved or radiographed. The welded seam maynot be as strong as the adjoining rolled-steel plate of theshell. It has been found from experience that an allowancemay be made for such weakness by introducing a “jointefficiency factor,” E into Eqs. 3.13 and 3.14. This factoris always less than unity and is specified for a given type ofwelded construction in the various codes.

The thickness of metal, c, allowed for any anticipatedcorrosion is then added to the calculated required thickness,and the final thickness value rounded off to the nearestnominal plate size of equal

Equation 3.13 becomes:

t=

and Eq. 3.14 becomes:

t=

or greater thickness.

FE + c

(3.15)

(3.16)

where t = thickness of shell, inchesp = internal pressure, pounds per square inchd = inside diameter, inchesj = allowable working stress, pounds per square inchE = joint efficiency, dimensionless (see Table 13.2)c = corrosion allowance, inches

M ODIFICATION OF EQUATIONS. For storage vessels themaximum allowable working stress is considered approxi-mately one third of the ult,imate tensile strength of thesteel; that is, a safety factor of 3 is employed, which iscommon for static structural loads on steel. The stresses

F i g . 3 . 9 . Circumferential forces acting on thin cylinder under intern;s

pWS*“W.

l . _ - . - . - \ \ - _ . . - - - - . ~ I / . -_ _ - -

Page 55: Process Equipment Design

4 6 Design of Shells for Flat-bottomed Cylindrical Vessels

are computed on the assumption that the tank is tilledlevel with water at 60” F (density 62.37 lb per cubic ft)and that the tension in each ring is calculated at. a point12 in. above the center line of the lower horizontal joint ofthe horizontal row of welded plates being considered.

The hydrostat,ic pressure in cylindrical storage tanksvaries from a minimum at the top of the upper most courseto a maximum at the bottom of the lowest course. Indetermining the plate thickness for a particular course, adesign based upon the pressure at the bottom of the courseresults in overdesign for the rest of the plate and thereforedoes not represent maximum economy. A design basedupon the pressure at the top of the course would result inunderdesign, which would not be good engineering practice.However, some consideration should be given to the addi-tional restraint offered by the plates adjoining a particularcourse. In the lowest course, the plates of the vessel bot-tom offer considerable restraint to the bottom shell course.This additional restraint of the bottom edge is effective foran appreciable distance or height from the bottom of thelowest course. In an intermediate course with a course ofheavier plates below, the top of the heavier plates will beunderstressed; this will tend to cancel any overstressing ofthe bottom of the course in question. Therefore, a designbased upon the pressure at a height of 1 ft from the bottomof the course may be considered conservative. The fol-lowing equations may be derived if one assumes that thedensity of the fluid will not exceed that of water, which isused for the hydrostatic test of the tank.

p = p (ff - 1)144

(3.17)

where p = density of water at 60” F = 62.37 lb per cubic ftH = height, in feet, from the bottom of the course

under consideration to the top of the top angleor to the bottom of any overflow which limitsthe tank’s filling height

p = internal pressure, pounds per square inch

For double-welded butt-joint construction, the above,definition of p may be substituted into Eq. 3.16. Whenone uses an allowable design stress of 21,000 psi for SA-7plate and a joint efficiency of 0.85 for doubled-welded butt-.joint construction, the following substitution results:

thereforet = O.O001456(H - l)D + c

where t = shell thickness, inchesH = height as defined in Eq. 3.17, feetD = inside diameter, feetc = corrosion allowance, inches

(3.18)

If double-full-billet lap-joint construction is assumed, thecorresponding joint efficiency, E, is 0.75. Then Eq. 3.18becomes:

t = O.O001650(H - l)D + c (3.19)

If low-alloy high-strength steel is used, the maximumallowable stress is taken as 60% of the minimum yield

point. When one uses double-welded butt-joint and low-alloy high-strength steel construction Eq. 3.18 becomes:

t=-5.096(D)(W - 1) + c

.fY.P.(3.20)

For lap-welded construction,

t = 5.775(D)(H - 1)P (3.21)

.rS.P.

wheref,,,. = minimum specified yield point of steel plate,pounds per square inch

Steels of different composition should not be mixed inany course of a vessel, with the exception that it is permissi-ble to use a different steel for providing reinforcing area forshell openings. In case two different steels are used in anypart of an area of reinforcement the stress correspondingto the weaker steel should be used to determine the thick-ness of shell metal to be reinforced.

Equations 3.18 and 3.19 are given in API Standard 12Cand apply only to the steel materials approved by the code.When other materials of construction are used, the con-stants of these equations must be recalculated through theuse of the proper value for the allowable stress.

3.5b Practical Considerations in Selecting Shell-plateDimensions. It should be emphasized that Eqs. 3.15, 3.16,3.18, 3.19 are useful only for predicting the thickness ofmetal required to withstand the int,ernal pressure. Otherfactors such as structural stability, live loads, wind, iceand snow loads, and fabrication procedures must be con-sidered. Minimum-thickness specifications for tanks byAPI Standard 12 C include some of these other considera-tions; they are listed in the tables of Appendix E. Tosummarize the important points of these tables, considerfirst the thickness of the tank shell. It should not be lessthan >/4 in. for tanks 50 to 120 ft in diameter, >is in. fortanks 120 to 200 ft, and 34 in. for tanks over 200 ft; or lessthan xs in. for tanks smaller than 50 ft in diameter. Theseminima are derived from practical considerations of stiffness,corrosion allowance, wind loads, and so on. Tanks havingshell thicknesses greater than these minima may use decreas-ing thicknesses for the upper courses. The thickness forthe upper courses necessary to contain the hydrostaticpressure may be determined by substituting the appropriatedepth of liquid into Eq. 3.18 or 3.19. However, the thick-ness should not be less than the minimum.

It should be noted that these minima are expressed infractions which correspond to mill plates of standard thick-ness. In general it is more economical to fabricate thesmaller vessels from mill plate of standard thickness than toorder plate rolled to a specified thickness. However, forthe large vessels the shell must be thicker to withstand thehydrostatic pressure. With this greater shell thickness,rigidity and corrosion are no longer the controlling factors.Some reduction in cost may result from ordering these platesrolled to thickness, especially if the required plate thicknesslies about midway between standard plate sizes.

In specifying shell-plate widths a compromise must bemade between the costs of the material and the costs offield welding including plate preparation such as edge

’ -------r --_ --------- -- \ .-p - - -.--- - - -~

Page 56: Process Equipment Design

4 6 Design of Shells for Flat-bottomed Cylindrical Vessels

are computed on the assumption that the tank is tilledlevel with water at 60” F (density 62.37 lb per cubic ft)and that the tension in each ring is calculated at a point12 in. above t.he center line of the lower horizontal joint ofthe horizuntal row of welded plates being considered.

The hydrostatic pressure in cylindrical storage tanksvaries from a minimum at the top of the upper most courseto a maximum at the bottom of the lowest course. Indetermining the plate thickness for a particular course, adesign based upon the pressure at the bottom of the courseresults in overdesign for the rest of the plate and thereforedoes not represent maximum economy. A design basedupon the pressure at the top of the course would result inunderdesign, which would not be good engineering practice.However, some consideration should be given to the addi-tional restraint offered by the plates adjoining a particularcourse. In the lowest course, the plates of the vessel bot-tom offer considerable restraint to the bottom shell course.This additional restraint of the bottom edge is effective foran appreciable distance or height from the bottom of thelowest course. In an intermediate course with a course ofheavier plates below, the top of the heavier plates will beunderstressed; this will tend to cancel any overstressing ofthe bottom of the course in question. Therefore, a designbased upon the pressure at a height of 1 ft from the bottomof the course may be considered conservative. The fol-lowing equations may be derived if one assumes that thedensity of the fluid will not exceed that of water, which isused for the hydrostatic test of the tank.

p = p (H - 1)1 4 4

(3.17)

where p = density of water at 60” F = 62.37 lb per cubic ftH = height, in feet, from the bottom of the course

under consideration to the top of the top angleor to the bottom of any overflow which limitsthe tank’s filling height

p = internal pressure, pounds per square inch

For double-welded butt-joint construction, the above,detinition of p may be substituted into Eq. 3.16. Whenone uses an allowable design stress of 21,000 psi for SA-7plate and a joint efficiency of 0.85 for doubled-welded butt-joint construction, the following substitution results:

therefore

t = 62.37(H - 1)(12D) + c2(21,000)(0.85)(144)

t = O.O001456(H - l)D + c

where t = shell thickness, inchesH = height as defined in Eq. 3.17, feetD = inside diameter, feetc = corrosion allowance, inches

(3.18)

If double-full-fillet lap-joint construction is assumed, thecorresponding joint efficiency, E, is 0.75. Then Eq. 3.18becomes:

t = O.O001650(H - l)D + c (3.19)

If low-alloy high-strength steel is used, the maximumallowable stress is taken as 60% of the minimum yield

point. When one uses double-welded butt-joint and low-alloy high-strength steel construction Eq. 3.18 becomes:

t = 5z096(D)W - 1) + c (3.20)dY.P.

For lap-welded construction,

t = 5.775(D)(H - 1)E (3.21)JY.P.

wheref,.n. = minimum specified yield point of steel plate,pounds per square inch

Steels of different composition should not be mixed inany course of a vessel, with the exception that it is permissi-ble to use a different steel for providing reinforcing area forshell openings. In case two different steels are used in anypart of an area of reinforcement the stress correspondingto the weaker steel should be used to determine the thick-ness of shell metal to be reinforced.

Equations 3.18 and 3.19 are given in API Standard 12Cand apply only to the steel materials approved by the code.When other materials of construction are used, the con-stants of these equations must be recalculated through theuse of the proper value for the allowable stress.

3.5b Practical Considerations in Selecting Shell-plateDimensions. It should be emphasized that Eqs. 3.15, 3.16,3.18, 3.19 are useful only for predicting the thickness ofmetal required to withstand the int,ernal pressure. Otherfactors such as structural stability, live loads, wind, iceand snow loads, and fabrication procedures must be con-sidered. Minimum-thickness specifications for tanks byAPI Standard 12 C include some of these other considera-tions; they are listed in the tables of Appendix E. Tosummarize the important points of these tables, considerfirst the thickness of the tank shell. It should not be lessthan s/4 in. for tanks 50 to 120 ft in diameter, >i6 in. fortanks 120 to 200 ft, and 35 in. for tanks over 200 ft; or lessthan xs in. for tanks smaller than 50 ft in diameter. Theseminima are derived from practical considerations of stiffness,corrosion allowance, wind loads, and so on. Tanks havingshell thicknesses greater than these minima may use decreas-ing thicknesses for the upper courses. The thickness forthe upper courses necessary to contain the hydrostaticpressure may be determined by substituting the appropriatedepth of liquid into Eq. 3.18 or 3.19. However, the thick-ness should not be less than the minimum.

It should be noted that these minima are expressed infractions which correspond to mill plates of standard thick-ness. In general it is more economical to fabricate thesmaller vessels from mill plate of standard thickness than toorder plate rolled to a specified thickness. However, forthe large vessels the shell must be thicker to withstand thehydrostatic pressure. With this greater shell thickness,rigidity and corrosion are no longer the controlling factors.Some reduction in cost may result from ordering these platesrolled to thickness, especially if the required plate thicknesslies about midway between standard plate sizes.

In specifying shell-plate widths a compromise must bemade between the costs of the material and the costs offield welding including plate preparation such as edge

Page 57: Process Equipment Design

Shell Design of Large Storage Tanks 4 7

Square-groovedouble-welded

butt jointtpatiial penetration)

Double-beveldouble-welded

butt joint(Park4 penetration)

Single- beveldouble-welded

butt joint

Single-Vdouble-welded

butt jointSingle-U

double-weldedbutt joint

fh)

double-welded Double-Ubutt joint double-welded

butt joint

Double-weldedlap joint

Square-groovedouble-welded

butt jointDouble-welded

full-filletlap joint

Horizontal JointsVertical Joints

I Fig. 3.10. Typical shell joints recommended by API Standard 12 C. (Courtesy of American Petroleum Institute.)

working for butt welding. Plates 80 to 90 in. wide maybe purchased at base cost without the inclusion of a priceextra for width. Plate widths over 90 in. carry “widthextra” charge which increases appreciably with increaseig width. Therefore, it is advantageous to use the widest

e that does not iqvolve an excessive extra cost.plates havmg a wld$h of 96 m. are me& exten-

I siv&y M&i.In specifying plate lengths, theire are no price extras for

4 leng&s between 8 and 50 ft when miM plates are purchased.Therefore, the longest plates which can be readily handledand shipped are specified. Thus plate lengths of approxi-mately 20 to 30 ft, are selected since longer plates are diffi-cult to handle. The exact length of the plates is determinedby dividing the circumference by the number of shell plates,with proper allowance made for the vertical weld joints.

3.5~ Butt-welding versus Lap-welding. The plates ofthe shell may be butt- or lap-welded depending upon thedesign and economic considerations. However, x in. isthe maximum plate thickness for lap-welded horizontaljoints and N in. is the maximum plate thickness for lap-welded vertical joints. Butt-welded joints may be usedfor shell plates for all thicknesses up to and including 134in. for plain carbon-steel-plates and up to and including lgsin. for low-alloy high-strength steel plat,es. The plates forbutt welded joints must be squared. Squaring of the platesfor lap-welded joints is not necessary. For this reason,plates for lap-welded joints are less expensive; however,erection by butt welding is somewhat faster. Because ofthe present high labor costs most tanks are now fabricated

by butt welding. Each course of the tank must be insidethe course beneath it when the horizontal joints are lapwelded. Vertical seams should not be in alignment forany of three consecutive courses. This is a precautionagainst localized conditions of stress at welds and aids inassuring the distribution of the stresses uniformly through-out the vessel. The requirement of a minimum distanceof 2 ft between vertical joints in adjacent courses is anadditional safety measure.

In the butt welding of the shell, the joints should prefer-ably be doubled welded with complete penetration andfusion. A single-welded butt joint with backing may beused instead, with the same joint efficiency. It is particu-larly important that the vertical butt joints have completepenetration and fusion because these joints are under thefull tensile stress in the shell. The horizontal joints arenot under this tensile stress. However, for structuralstrength against wind loads, and so on, and for preventionof failure by notch brittleness, all horizontal single-beveledjoints should have complete penetration, as shown in Fig.3.10b. With squared plates (square-groove) and double-beveled butt plates for horizontal joints, as shown in Fig.3.10~ and c respectively, incomplete penetration may beused for the sake of economy. However, with partial pene-tration the thickness of the unwelded portion should notexceed one-third the thickness of the thinner plate, andthe unwelded portion should be located at approximatelythe center of the thinner plate. If a horizontal butt jointis offset because of different plate thicknesses, the insidesurfaces should be flush.

Page 58: Process Equipment Design

40 Design of Shells for Flat-bottomed Cylindrical Vessels

size is not over 3 in. nominal pipe size. For small pipesizes, screwed fittings are usually preferred because theyare cheaper than flange fittings. However, pipe withscrewed fittings having a nominal size greater than 2 in.is rather difficult to fit because such heavy pipe is so rigidthat it cannot be deflected easily to aid in aligning thethreads during fitting. Therefore, for practical reasons,it is recommended that any nozzle having a nominal sizegreater than 2 in. have flange fittings. It is usually desira-ble to locate nozzles for filling and discharging near thebottom of the tank to obtain the benefit of gravity in dis-charging and to avoid pulling a partial vacuum on fluidswhich are volat i le . However, water and sludge may sepa-rate and collect on the bottom. To avoid pumping thissludge out of the discharge line, the discharge nozzles areusually placed on the shell a short distance above the bot-tom. Another nozzle with a sump is placed at the bottomto remove material accumulating below the dischargenozzle and to completely drain the vessel . Typical nozzlesof both the screwed-f i t t ing and f lange-f i t t ing type are shownin Fig. 3.14, and standard dimensions for these nozzles aregiven in Items 1 and 2 of Appendix F.SHELL MOLES. Manholes are necessary in closed

vessels to permit inspection, cleaning, repairs, and so on.These manholes may be located on the shell or on the roaror at both locations. Manholes located on the shell havethe advantage that it is somewhat easier to use a shellopening to clean or repair a vessel. Shell manholes havethe disadvantage that they usually cannot be opened unlessthe vessel is empty and therefore are not used as often forinspect ion as roof manholes . Items 3.4, and 5 of Appendix

Fig. 3.11. Circumferential joints for tank shells. (Courtesy of Hammond

iron Works.)

Lap joints should have an overlap of at least 5 X tinches, and in no case should the overlap be less than 1 in.Vertical lap joints should have continuous full-fillet weldsboth inside and out. On horizontal lap jo ints , as shown inFig. 3 .11, the f i l let should have a s ize not less than one thirdt,he thickness of the thinner plate, and in no case should itbe less than x,j in.

Figure 3 .11 a lso shows a horizontal butt -welded shel l of as torage vesse l . Note that the heavier courses at the baseare V butt-welded, whereas the upper courses are plainbutt-welded.

3.5d Cold Forming of Shell Plates. Plain carbon-steelshell plates having a thickness of XG to N in. for tankshaving a diameter of 40 f t or more or low-al loy high-strengthsteel plates for tanks having a diameter of 50 f t or more canbe deflected on erection and therefore need not be coldformed by rolling to the radius of curvature of the shell.If the diameter is 60 ft or more for plain carbon st,eel or100 ft or more for ’ low-a l loy hi:h-strength steel , shel l plateshaving a thickness of g to 55 in. may be deflected on erec-tion without cold forming. Plain carbon-steel plates ofthicknesses from $4 to N in. and diameters over 120 ftneed not be cold formed. However, al l plain carbon-steelplates having a thickness of 36 in. and over and all low-alloy high-strength steel plates having a thickness of W in.or over must be cold formed to the shell radius regardlessof the shel l d iameter . Figure 3.12 shows the f ield weldingof horizontal butt-welded seams of the shel l of a large tank.

3.5e Shel l Par ts . In addition to the shell plates avariety of other shell parts and accessories must be consid-ered in the shell design. Figure 3.13 shows typical tankaccessories including shell nozzles, manholes, ladders, andso on.SHELL NOZZLES. Pipe l ines which bring the f luid to and

from the tank are attached to short pipe connections weldedinto the tank shel l . These connections are cal led “nozzles”arid may be fabricated of screwed pipe fittings if t,he pipe

Fig. 3.12. Field welding of shell circumferential buit jotnr. (Cow&v 04

Hammond Iron Works.)

c r \ \ - \I /

Page 59: Process Equipment Design

% gives typical dimensions for shel l manholes and manhole-cover plates designed as shown in Fig. 3.15.

REINFORCEMENT OF SHELL OPENINGS. All openings suchas nozzles and manholes, made in the shell in which t.heopening is over 2 in . in diameter should be reinforced. Thereinforcement prevents local overstressing of the shellaround the opening. The minimum cross-sectional areaof the reinforcement should not be less than the productof the vertical diameter of the hole cut in the tank shelltimes the shell plate thickness. The cross-sectional areaof the reinforcement is measured parallel to the axis of I.heshell across the center of t,he opening.

02 Conservation vent\ @ Free vent y

Shell Design of Large Storage Tanks 4 9

sidered as available for reinforcement out to a distance offour times the neck-wall thickness, measured from the out.-side of the shell. The metal in the neck lying within theshell-plate thickness may also be included. If the neckof the fitting extends both inward and outward as shownin the center and right of Fig. 3.15, credit may be taken folthe metal of the neck over a dist.ance of eight neck-wallthicknesses plus the shel l thickness .

In the case where two or more openings are located ciosctogether and the toes of t.he f i l le t welds f ix ing the reinforciupplates for these openings come within t.wice t.he shell t hi&-

manhole

@1 S h e a v e

Fig. 3.13. Usual accessories and fittings on standard cone-roof tanks. (Courtesy of Hammond Iron Works.)

Included as standard Included as extra

4. One 20” shell manhole 9. Sump 15. Connection for foam chamber

1. One 20” roof manhole 10. Swing line unit complete 16. Drain

3. One 6” gauge hatch 11. Water draw-off 17. Flame arrester

4. Roof nozzle for vent (12 or 13) 12. Conservation vent (volatile products) 18. Antifreeze valve

5. Ladder (small tanks only) 13. Free vent (nonvolatile products) 1-8 Extra units

6. Spiral stairway 14. (a) Target-type float gouge 5. (a) Inside ladder7. Two shell nozzles 14. (b) Ground-reading-type float gouge

8. Flange for water draw-off

Reinforcement metal may include any one of the fourmetal parts listed below or any combinat,ion of them:

1. The metal in the at,tachment. flange of an attachedi i t t ing ,

2. The metal of a reinforcing plate,3. The metal of any excess shell-plate thickness beyond

that required from a calculation of the minimum platethickness ,

4. The metal in the neck of a fitting. This can be con-sidered as part of the reinforcement area. If the fittinge.xtends only outward, the metal in the neck may be con-

ness of each other, a single reinforcing plate should be usc~l.This plate should be proportioned for the largest openingin the group. I f the re inforc ing plates for one or more smallopenings are of such a size that they lie entirely within I 111%area of the reinforcement plate for the largest opening. I hcbsmall openings can be included in the normally designc~tlre inforcement plate for the largest opening without incrras-ing the size of this reinforcing plate. If, however, anyopening intersects the vertical center line of another opeu-ing, the width of the reinforcement (along the vert icalcenterl ine o f eit.her opening) should not be less than the sum of lhewidths of the two plat,es that would normally be used.

Page 60: Process Equipment Design

50 Design of Shells for Flat-bottomed Cylindrical Vessels

Permissibleal ternat ivesquare cut

A circular reinforcing plate maybe substituted for the plateshown, for the 3- to l&in. sizenozzlesr inclusive, provided thediameter of the circular plateis made equal to W

Bolt holes shallradius of tank shell

straddle the flangecenter lines

Reinforcfng plate

Single flange

(4

S en o

Double flange

(4

Special flange

(a)

Fig. 3.14. Shell nozzles recommended by API Standard 12 C (see items 1 and 2 of Appendix F for typical dimensions.) (Courtesy of American Petroleum

Institute.)

Finally, if the normal reinforcing plates for the smalleropenings do not fall within the area limits of the reinforce-ment for the largest opening, the group reinforcing plateshould inc lude within i t s outer l imits the normal re inforc ingplates for a l l openings in the group.

3.5f Reinforcement of Top Course of Shell for LargeOpen Tanks. Open vessels of large diameter may not have

the necessary inherent rigidity to withstand wind loadswithout deforming and excessively straining the structure.Two methods of stiffening are available: shell plates maybe made thicker , or suitable st i f fening girders may be addedto the structure. The use of thicker shell plates usuallyis more costly than the use of stiffening girders. W i n dgirders or stiffening rings for open tanks are located at or

Page 61: Process Equipment Design

Shell Design of Large Storage Tanks 51

Gasket:20” manhole-25%” OD x 20” ID x 4” thick24” manhole-29%” OD x 24” ID x !$” thick

Long-fiber asbestos sheet

Shape manhole flange tosuit curvature of tank

I30”

Increase ifnecessary

ifor clearance

finished to provide a minimumgasket-bearing width of L in.for minimum thickness at bolt

circle, see Appendix F.

A l t e r n a t i v e d e s i g n s o f m a n h o l e s

F i g . 3 . 1 5 . Shell manholes recommended by API Standard 12 C (see items 3,4, and 5 of Appendix F for typical dimensions). (Cour tesy cf A m e r i c a n P e t r o l e u m

Institute.)

near the top of the vessel on the top course of shell plate.The st i f fening r ing is placed preferably on the outside of the

H = height of shell including any “freeboard” pro-vided above the maximum fi l l ing height , feet

I shell rather than the inside. The calculation of the available section modulus of theThe required section modulus as specified by API Stand-

ard 12 C for the stiffening ring may be computed by Eq.stiffening ring may include a portion of the tank shell

3.22:which is considered to be effective for a distance of 16 plate

z = 0.0001D2H (3.22)thicknesses from the ring, as indicated in Fig. 3.16. W h e ncurb angles are attached to the top edge of the shell ring

:vhere z = section modulus, inches3 by butt welding, this reinforcing distance sbnuld be reduced.D = nominal diameter of the tank, feet by the width of the vertical leg of the angle. Table 3.4

L-.-I-- - - - - -

_..- - - .-.._--\- - ~~ \I /

-~

Page 62: Process Equipment Design

_. .=“I-uorrotned Cylindrical Vessels

Detail B

Fig. 3.16. Typical reinforcement for top course of shells for open vessels

secommended by API Standard 12 C. (Courtesy of American Petroleum

Institute.)

lists the section moduli for the rings shown in Fig. 3.16for two shell thicknesses.

Stiffening rings may be made either of structural sect,ionsor formed-plate sections or combinations of the two. Theminimum size of the angle specified by Standard 1.2 Ceither alone or with a built-up section is 236 X 255 X j/,and the minimum plate thickness is >/4 in. If the stiffeningrings are located more than 2 ft below the top of the shell,a minimum of a 235 X 235 X Ns top curb angle is requiredfor Ns-in. shells. A minimum of a 3 X 3 X $/4 angle isrequired for M-in. shells. Other members of equivalentsection modulus may be substituted. Drain holes shouldbe provided in rings that may trap liquid. Stiffening ringsare sometimes used as walkways, in which case they shouldprovide at least 24-in. of clear walking space and shouldbe located preferably 3 ft, b in. below the top curb angle.Any such angle having a horizontal web exceeding 16 t.imesthe web thickness requires suitable vertical support.

EXAMPLE DESIGN 3.1, W IND GIRDER FOR OPEN VESSEL.h wind girder is required for an open vessel 80 ft, 0 in.in inside diameter and 40 ft high, having a top-course platethickness of f$ in.

When one uses Eq. 3.22, the minimum section modulusof Lhe wind girder is:

z = 0.0001D2H

= o.oool(so)y4o)

z = 25.6 im3

When one uses the type of construction shown in detailF of Fig. 3.16, referring to Table 3.4, he finds that the

appropriate dimension for the minimum width of theweb, b, is 12 in., which corresponds to a section modb28.1 in3 The rest of the sectional dimensions, inclwind-girder plate thickness of $6 in., are fixed byE of Fig. 3.16. A stiffening ring such as this is fabriby bending plate steel into the appropriate shape.a wide-webbed ring cannot, be easily rolled to the req

Table 3.4. Section Moduli of Various Stiffening-ril

Sections on Tank Shells, Recommended by API

Standard 12 C

(Courtesy of American Pet,roleum tnstitute)

Member Size?in.

Shell Thickness, in._--~ --.-___$16 s/4Section Moduli

Top angle: det,ail A, Fig. 3.16234 x 21.& / x 5c 0.41 0.42235 x 21,s x $iF, 0.51 0.523 x3 x 53 0.89 0.91

Curb angle: det.ail B, Fig. 3.16254 x ZS$ x 53 1.61 1.72235 x 2q x p1;la 1.89 2.043 x 3 XSk 2.32 2.483 x 3 xps 2.78 3.354 x 4 X!i 3.64 4.414 x 4 xyg k. 17 5.82

One angle: detail I:, Fig. 3.16256 x 2% x $4 1.68 1.78ajg x 255 x $f(j 1.98 2.124 x 3 XQ 3.50 3.734 x 3 x3is 4.14 4.455 x 3 XX6 5.53 5.955 x 3$$ x 3’6 6.13 6.605 x31,5x36 7.02 7.616 x4 ~9s 9.02 10.56

Two angles: detail D, Fig. 3.164 x3 x 546 11.27 11.784 x 3 x7* 13.06 13.675 x 3 xpi, 15.48 16.245 x 3 x3$z 18.17 18.895 x315x- 41 6 16.95 17.705 x335x$; 19.99 20.636 x 4 ~“6 27.?4 28.92

Formed plate: detail E, Fig 3.166, in.

10 22.31 2 28.11 4 34.31 6 40.81 8 47.72 0 54.92 2 62.42 4 70.32 6 78.52 8 87.03 0 95.93 2 105 13 1 114.73 6 124.53 8 134.74 0 145 3

Page 63: Process Equipment Design

diameter because of i ts s t i f fness . Therefore , i t i s more con-venient to weld straight sections of formed plate, making apolygonal s t i f fening member . The inside edge wil l be f lamecut and made smooth to form an arc having a radius equalto that of the shel l outs ide diameter . Usua l ly two or moresections are welded end-to-end in the shop to minimize thef ield welding required.

Figure 3.17 shows a preliminary sketch of a wind-girdersubassembly, using 20 equal sect ions with every two sect,ionswelded end-to-end for subassembly in the shop.

For determinat ion of web dimension, 5,

cos 9" = 40 ft, 0% in. + 12 in.-___-- = 0.98769

40 ft, O+ in. + s

492.25x=-- in . - 480.25 in .

0.98769

= 18.135 in. = 1 ft. 6; in.

For determinat ion of ins ide chord length of subassembly,

Chord2

= 40 ft, 0: in. (sin 18”)

Chord = 2(480.25 in.)(0.30902)

= 296.814 in. = 2.4 fl, 8:# in.

For determinat ion of out,side chord length of subassembly

Chord__ = (40 ft, 0) in.) + (1 ft, 6; in.) sin 18”

2

Chord = 2(41 ft, 63 in.)(0.30902)

= 2(498.375)(0.30902)

= 308.016 in.

= 25 ft, 8 in.

For determinat ion of outs ide chord length of sect ion,

Chord = 2(41 ft, 6g in.) sin 9”

= 2(498.375)(0.15643)

= 155.9216 in.

= 12 ft, 11 ++ in.

Figure 3 .18 shows typical detai l designs for a wind girderfor an open tank 80 ft in diameter and 40 ft high as describedin example design 3.1.

3.5g Reinforcement of Top Course of Shell for LargeClosed Tanks.

S H E L L S W I T H R O O F S H A V I N G C O L U M N S U P P O R T . If tanksare c losed with a roof , the roof provides addit ional s tructuralrigidity to the upper course of shell plates. As a result,smaller stiffening rings are used for closed vessels. For-Q&S =&\I column supports having diameters of 35 ftor fess, Zjg x .Zf$ x ji in. is the minimum-size stiffening

angle. For tanks having diameters of 35 to 60 ft; the sizeof the angle should be increased to 235 x 235 x 54s in.

Shell Design of Large Storage Tanks 5 3

continuous double-wielded but 1. joint or a continuous double-welded lap joint.SHELLS WITH SELF-SUPPOHTING ROOFS. A sel f -support -

ing roof is one which is supported only on its periphery-,without added structural support. Such roofs cause :Icompressive stress in t,he roof plates, which is transferredto t.he shell as hoop tension. A stiffening angle should beadded to the top shell course at the junction of the roof andshell to absorb the stress as a tensile load. The forcc,sacting On the ring are shown in Fig. 3.19.

The fol lowing nomenclature wil l be used in the equation*explaining Fig. 3.19:

a = cross-sectional area of stiffening ring, square inchesI) = nominal tank diameter, feet0 = angle of the cone element wit.h the horizontal, degrees

P = roof load, pounds per square foot, (live load of 25 lhper sq ft plus dead load)

7’1 = compressive force per linear inch along an element ofthe cone, pounds per linear inch

Tz = tensile force per linear inch in a circumferential diwc-tion, pounds per linear inch

TX = horizontal component of Tl, pounds per linear inchF = circumferential tensile force act.ing in stiffening ring,

pounds

W = t.otal load on roof, pounds = r$ P

f = tensile stress, pounds per square inch

F i g . 3 . 1 7 . P r e l i m i n a r y s k e t c h o f w i n d - g i r d e r s u b a s s e m b l y f o r e x a m p l e

design 3.1.

Page 64: Process Equipment Design

54 Design of Shells for Flat-bottomed Cylindrical Vessels

I4 Chord = 25’-8”

436” h 2’-8*,W.P.--

Cut back both ends. as shown,8” shop after welding sectionstogether. See secbon E-E

;owW.P. -’

Cut thts edge true andsmooth after shop weldingsections together-

Arc on edge Of PL. 25,-l%”

Section E-EChord = 24’-8’x6”

Wind-girder assembly, 10 required

Detail of wind-girder section,20 required

Fig. 3.18. Details for wind girder for example design 3.1.

In reference to Fig. 3.19a, the roof load, W, results incompressive force, Tr, in the roof plates as follows:

T1= w zzrpPDsin e(?rD) 12 48 sin 0

Compressive force, Tr, has a horizontal component, Ta, or

T3 = T1 cos fj = ED+Le

In reference to Fig. 3.19b,

2F = 12DT3

or

T3 = F‘60

- F

- F

(a)

Fig. 3.19. Loads on conical roofs.

(b)

Section A-A

alsoF = aj

TV = af = PD cot e60 48

Section B-B

To solve for a,

PD2 cot 0a =

8.f

Thereforef = 18,000 psi, allowable

PDZa = 8(18,000) tan 8

PD’a=-

if:144,000 tan B

live load = 25 lb per sq ft (for 4 in. plates)

dead load = 11 lb per sq ft

P = 36 lb per sq ft

By substituting,

36D2 Da =

144,000 tan 8 = 4000 tan e

For small angles, tan e is approximately equal to sin 8;therefore

02a =

4000 sin e

Page 65: Process Equipment Design

Example Design 3.2, Complete Shell Design for a Closed Vessel 55

The API Standard 12 C recommends the use of Eq. 3.23for determining the required reinforcing area.

02a = 3000 sin 19

In applying Eq. 3.23 credit may be taken for the cross-sectional area of the shell and roof plates within a distanceof 16 times their thickness from the stiffening angle. Inother words, the sum of these areas must be equal to orgreater than (D2/3000 sin 0) for cone-roof construction.

For dome or umbrella-roof construction a similar equa-tion may be derived.

(3.24)

where D = diameter of tank, feetR = radius of curvature of dome, feeta = reinforcing area, square inches

3.6 EXAMPLE DESIGN 3.2, COMPLETE SHELL DESIGNFOR A CLOSED VESSEL

The design calculations and drawing for the steel shellonly for a 55,000-bbl oil-storage tank having a cone roofare required. The cone roof is to be supported by internalcolumns, girders, and rafters.

3.6a Proportioning. It is estimated that the ratio ofannual cost of the shell per unit area, cl, is two times theannual cost of the bottom per unit area, ~2, and that theannual cost of the roof, cg, is 1.8 times the bottom annualcost per unit area. The annual cost of the land area, cd.and preparation, cg, together is estimated at 0.40 of thecost of the tank bottom, ~2. Corrosion allowance will benegligible.

When we substitute the above information into Eq. 3.9,

D = 4H2.002--~ = 2.5H

~2 + 1.8~2 + 0.4~2

When we substitute the same information into Eq. 3.1,

H = 4(55,000 X 42/7.48)(2SH)%\

H3 = 62,910 cu fto r

H = 39.8 ft

Since D = 2.5H,

D = 2.5 X 39.8 = 99.5 ft

The tank dimensions will be 100 ft inside diameter by40 ft high. Appendix E, item 3 indicates that such a tankwill have a volume of 55,950 bbl.

3.6b Design of Shell Courses. The thickness of thebottom shell course can be determined readily by Eq. 3.18(for butt.-welded assembly).

tl = 0.0001456(40 - l)(lOO) + c

= 0.568 in. + c

Since corrosion allowance = 0,

tl = 0.57 in.

By the use of 10 plates and with an allowance of %a in.for vertical weld joint, the center length of each plate iscalculated from the circumference.

L =nd - weld length = 3.1416(1200.57) - lO(?ia)

12n 12(10)

3770.136= ~ = 31.4178 ft

120(or 31 ft, 5 in.)

Standard mill plates of 96-in. width will be specified andbutt welding of the shell plates will be used for fabrication.Shearing of the shell plates is required to square the platesfor butt welding. Therefore, the final height of the shellplates will be slightly less than 96 in.

The thickness of the shell course above the bottom coursecan also be determined by Eq. 3.18. The proper height,however, for this calculation will be (40 ft - 8 ft) or 32 ft

t2 = 0.0001456(32 - 1)lOO + c

= 0.452 in. + c

Specify that t2 = 0.46 in.

L = 7r0200.46) - lO(O.15625) = 31 ft 4g in10 x 12

Accordingly, the thickness of the third course (H =24 ft) will be:

t3 = 0.0001456(24 - 1)lOO + c

= 0.335 in. + 0

Fig. 3.20. Elevation view of shell for example design 3.2.

A. Cable sheave

B. Winch

C. 4” steam nozzles

D. 10” shell nozzle

E. 1 s” extra-heavy couplings

F. 4” water-draw-off nozzle

G. Shell nozzle for double swing joint

H. 24” shell manhole

._- --1-- - - -- .--- \ \ \ r-7--

--

Page 66: Process Equipment Design

56 Design of Shells for Flat-bottomed Cylindrical Vessels

i-

k circumference

Section through shell

PL +4 -k ”

PL . $3-0.34

PL. %‘-0.46’

P L . Ql-0.57

Bill o f Moferioh~$$$?~ Mark D e s c r i p t i o n Lrnqth

ft. i n .

/o */ PL95~YO.57” 31 5 I O

L+- 31 ’ -5 g,“-#5 shell PL

I+-31’-5%“-+l shell PL.

Section through vertical joints

lo x2 P/. 95 Y4”x 0.46 * 31 4%; IO Pl: 96”XO.46’ 31 5Y4.

IO *3 P,! 93%*x0.34” 31 4% IO Pl 94”XO.34” 31 5%IO #4 Pl 95%“X%# .’ 31 4% I O P/. 96”X$+” 31 5?&IO “5 Pl: 9sy6/e’x 74# 31 4%” I O Pl: 94”X;/4” 31 5%”IO T A 5 L 3”X3”X+L8/B” 31 4% IO L 3”X3”X%” 31 8

Specify that tS = 0.34 in.

L = ~(1200.34) - lO(O.15625)

10 x 12= 31 ft, 48 in.

Likewise, the thickness of the fourth course (N = 15 ft)will be :

t4 = 0.0001456(15 - 1)lOO + c

= 0.204 in.

L = ~(1200.25) - lO(O.125)

10 x 120= 31 ft, 4s in.

Specify 14 and t5 = 0.25 in. since the thickness as deter-mined from the appropriate relationship results in a thick-

ness of less than >a in. This means simply that hydrostat iv

pressure stress considerations are not controlling and thrstructural stability of the thin shell is the prime consdera-lion. Thus the minimum shell thickness of >/4 in. for shellsof this size as set by API 12 C for tanks 50 ft in diameteland larger will be specified (see Appendi:: E, item 4).The required thicknesses for the various shell courses couldhave been determined from Appendix E, item 4. It will beseen that the calculated and tabulated shell-course thicknessagree.

Only the shell plates of I he bottom course need to be coldformed to the shell diameter. Frequently, however, thinnershell plates are rolled to facilitate erection. Accordingly.the shell specifications will call for rolling of the bot,t.omthree courses.

Page 67: Process Equipment Design

Example Design 3.2, Complete Shell Design for a Closed Vessel

3.6~ Design of Top Angle. The minimum-size topangle for a tank larger than 60 ft, 0 in. in diameter with a

L = a(1200.375) - lO(O.15625)

10 x 12

5 7

roof supported on columns is 3 in. X 3 in. X s in. andwill, therefore, be used. Specifications will call for buttwelding of this angle to the top course. By using 10 lengthsof top angle, the length of each angle section is calculatedas follows :

L = 31 ft, 4+$ in.

Figure 3.20 shows the elevation view of tank she!!, andFig. 3.21 shows the shell details.

P R O B L E M S

1. A cone-roof tank having a filled capacity of 100,000 bbl is to be designed. Determine theoptimum proportion of D/H from the following cost considerations. The fabricated shell, roof(including plates, rafters, girders and columns), and bottom are estimated to cost 18, 20, and14 cents per lb respectively. Foundation costs are estimated to be $4000. Fixed annualcharges including amortization, interest, and so on are estimated to be 40% per year based oninitial installed cost. The annual charge for the land allocated to the tank area is $500.

2. A wind girder is required for an open vessel 120 ft inside diameter and 48 ft high. Thetop course of the shell is fabricated from x-in. plate. If the wind girder consists of 30 identicalsections corresponding to detail E of Fig. 3.16, determine the minimum section modulus andthe girder dimensions.

3. Determine the required cross-sectional area of the stiffening ring for a self-supportingconical. roof 30 ft 0 in. in diameter having an angle 0 of 15” with the horizontal.

4. The required shell plate thicknesses for the vessel described in problem 2 may be deter-mined from item 2 of Appendix E. Using these plate thicknesses and 18 courses, prepare aplot of circumferential stress versus height for the conditions of the hydrostatic test in whichthe vessel is filled with water.

Page 68: Process Equipment Design

C H A P T E R

I’

m4A

DESIGN OF BOTTOMS AND ROOFS

FOR FLAT-BOTTOMED CYLINDRICAL VESSELS

uhe bottoms and roofs of vertical cylindrical storagevessels are usually fabricated of steel plates having thick-nesses less than those used in the shell. This is possiblefor the bottom because it is normally supported by a pre-pared base of sand or gravel resting on the soil. The roofload is usually limited to wind and snow load with a properallowance made for any anticipated additional loads.

4.1 BOTTOM DESIGN

The shape and design of the bottom for a storage vesselwill depend upon such considerations as: the foundationused to support the vessel, the method for removal of thestored material, the degree of sedimentation of suspended

n Intermittent weld

Shell-to-bottom joints

Single-weldedfull-filletlap joint

Single-welded butt jointwith backing strip

Bottom-plate joints

Fig. 4.1. Typical bottom joints recommended by API Standard 12 C.[Courtesy of American P&r&urn Institute.)

I \ \

solids, corrosion of the bottom, and the size of the storagevessel. If the considerations mentioned dictate the use ofa flat bottom and the safe bearing capacity of the soil is atleast 3000 lb per sq ft the bottom is usually placed on asand or gravel pad directly on the soil.

If the tank bottom is directly supported by the ground,flexure of the bottom is prevented, and the bottom platesare under a simple compression load. Theoretically a light-gage sheet metal, 16-gage or less would be sufficient forsuch a bottom. However, to provide greater ease in weld-ing and to allow additional metal for corrosion, plates havinga thickness of at least s/4 in. should be used. For manyyears xs-in. plate was the most common plate thicknessused for tank bottoms. Bottom plates of 72 in. or more inwidth are preferred and plates 96 in. wide are usuallyspecified. Plates of >d-in. thickness are usually lap weldedwith a lap margin of at least 1>/4 in. for all joints. Thebottom plates should extend beyond the shell-plate bottomweld at least 1 in. No more than three plate laps shouldbe located within 12 in. of each other or of the shell. Typi-cal welded joints for shell-to-bottom and bottom plates areshown in Fig. 4.1. Figure 4.2, a and b, gives alternatemethods of shaping the sketch plates under the shell ring.The sketch plates should be formed and welded in such amanner that a smooth bearing surface for the shell platesis produced.

In regard to selecting the plates for the bottom, thelargest-sized plates available that can be convenientlyhandled and that have no cost extra for size are usuallythe most economical. Plates 96 in. wide x 20 or 30 ft longare often used. If the bottom plates are laid symmetricallyin relation to the center lines of the bottom plan, t-he numberof different shaped plates will be reduced to a minimum.

5 8

Page 69: Process Equipment Design

Example Design 4.1, Bottom for a Tank 150 Ft, 0 In. in Diameter 59

h3ottom plate’

Fig. 4.2. Methods of shaping sketch plates under the shell ring recommended by API Standard 12 C. (Courtesy of American Petroleum Institute.)

This is an advantage because the plates can then be scribedand cut in groups of four, whereas if the bottom plates aresymmetrical in relation to only one center line, only twoplates can be scribed and cut at one time. A bottomasymmetrical a long both center l ines makes a large numberof plates of different sizes necessary. The simplest sym-metrical layout is to arrange the corners of four plates tointersect at the center of the tank bottom. However, thislayout should not be used with lap-welded constructionbecause four plates will lap at the joint. Also , th is layoutis sometimes wasteful in that with some groups of dimen-sions considerable scrap may result from the plates at theper imeter . In such a case the bottom plates may be rear-ranged with one plate centered on the bottom. In thislayout the center row is single, but all other rows havemates. The center row will have two perimeter plates oft.he same size, but there will be four identical perimeterplates for each succeeding row from center .

The sizes of plates and the location of cuts on perimeter

Table 4.1. Dimensions of Welded Draw-off Elbow,

Recommended by API Standard 12 C, All

Dimensions in Inches-See Fig. 4.5

(Courtesy of American Petroleum Institute)Distance

Distance f r o mDistance f r o m Diameter Diameter Center of

Nomi- f r o m Center of of Hole of Rein- Elbow tonal Center of Outlet to in Tank forcing Face ofPipe Elbow to Bottom, Bottom, Plate, OutletSize* Shell, B C DP DR Flange, E

2 7% 6 3 % 6% 123 8% 7 4% 7% 134 9 % 7’Ns 5% 9% 1 46 1 1 9% 7% 123i 168 13 1236 9 % 1655 18

* Extra-strong pipe, API Standard 5 L

plates can be readily calculated by use of Eq. 4.1 and byreference to Fig. 4.3.

A2 = B(D - B) (4.1)

A2 = 4 - ,9Means must be provided for the removal of liquid from

the bot tom of the vesse l . A sump, shown in Fig. 4.4, maybe used with a sump-pump discharge. Flat-bottomedtanks using gravity or pump discharge may discharge bymeans of a draw-off elbow, as illustrated in Fig. 4.5.

Dimensions for a draw-off e lbow are given in Table 4 .1 .

4.2 EXAMPLE DESIGN 4.1, BOTTOM FOR A TANK150 FT, 0 IN. IN DIAMETER

A bottom is required for a tank 150 ft, 0 in. in insidediameter. The minimum allowable (2) plate thickness is>/a in.; however, because of the large tank size a plate thick-ness of x6 in. will be specified to provide additional pro-tection against loss by corrosion. The bottom course ofshel l p la tes for th is vesse l i s l)is in. thick, and a XS in . f i l l e tweld will be used between the shell and the bottom plates.

4Q

------ - -

d --At

63-I II0 Q::Fig. 4.3. Relationship of bottom plate dimensions.

Page 70: Process Equipment Design

60 Design of Bottoms and Roofs for Flat-bottomed Cylindrical Vessels

The hottom plates must extend a minimum of 1 in. beyondthe shell weld, or in this case, the radius must be increasedil minimum of (1 + 1s.i~ + x~ in.) or 27; in. A radiusof 75 ft, 3 in. will be used for the bottom. The centralhottom plates will be 96 in. wide by 31 ft, 8>/4 in. in length.The bottom plates will be lap welded, and the joints willbe staggered so that no more than three plates are lappedwithin 12 in. of each other or of the shell.

Figure 4.6 shows a layout, for such a bottom. It shouldhe noted that the layout is symmetrical with respect t,o oneaxis, and thus only half of the bottom is shown. It is alsoevident that, except for the necessary staggering of theplates, it is symmetrical w-ith respect to the other axis.

To demonstrate t,he use of Eq. 4.1 in this design, dimen-sions A and C will be calculated for points 2 and y on thelayout. At point s dimension C is equal to one half theplate length of 31 ft, 812 in., and D is 150 ft,, 6 in.

3% 8% in.c= 2-

= 15 ft, lOi in.

Fig. 4.4. D r a w - o f f s u m p r e c o m m e n d e dby API Standard 12 C. (Courtesy ofA m e r i c a n P e t r o l e u m I n s t i t u t e . )

T a n k s h e l l

’Alternative ,,‘I’

m i t r e d p i p e

Alternative forbottom corner

(150 ft, 6 in.):!=k

- (15 ft, 104 in.)2

= X62.562 - 251.024

= 5411.338

l.harrfort!

A = 73 fl, 6% iu.

To obtain the width of plate S-2 at point L, 81~-in. platewidt,hs less (10 - 1) laps of 134 m.’ m u s t br subtrackedfrom dimension A. The plate width is 96 in.

S-2 plate width at z = (73 ft, 68 in.) - 8+(8 ft., 0 iu.,- 9(1+ in.) = (73 fl, 6% in.) - (67 ft, 0% in.) = 6 it, :J in.

Fig. 4.5. Welded draw-off elbowrecommended by API Standard 12 C.

(See Table 4.1 for dimensions.)(Courtesy of Amer ican P e t r o l e u mInstitute.)

Page 71: Process Equipment Design

Example Design 4.1, Bottom for a Tank 150 Ft, 0 In. in Diameter

1?7’-sy’L1?9.-‘1~“I!,,-l~%.,,~,-~

6 1

R a d i u s = 75’4” tooutside edge of bottom

Section through corner

B//l of Materiot$$'gz Mark Dercnption LWfh wt. No.Ft. h. Order Length

Ft. in.

47 SW32 P/s-96"X/2.75" 31 84 47 Ph.-96Y12.75" 31 8%IO 5124 P/r.-96'Xl2.75" 2 3 9fi 10 P/r.-96"X12.75y 2 3 9%4 5W/6 P/s. -96"Xl2.75" 15 10% 4 P/s.-96"X/2.75# I5 /OS4 S- / Pls. - SK X12.75* 2 Pls-96Vl2.75~ 16 2%4 S-2 Fls -SK X 12.75" 2 Pls -96112.75" I4 6%?4 5-3 ,&-SK X/2.7!? 2 Pk-96"X/2.75X 19 0%4 5 -4 P/s: -SK X 1275" 2 Pl~-96'X12.75x 13 9%4 S-5 P&-SK X 12.75+ 2 Pls.-96"X12.75x 22 3Q4 S-6 Plx-SK X/275* 2 Pk-96"X/2.75* 28 3%2 s-7 P/x-N x/2.75* 2 PJ~.-96'X12.75~ 15 S?Q2 S-7-A ?/s-SK Xt2.7.V'2 S-8 ,&-SK X/2.75* 2 f/s.-96'X12.75" 21 Ilk? S-8-A Pls -SK X /2.75*4 S-9 P/S.-SK X 12.75' 2 Ph.-96'Xl2.75" I3 5%4 S-IO P/s-SK X/275* 2 /'Is.-96112.75" I6 84 S-N Ph.-SK Xt2.75# 2 Pk96"X/2.75* 20 I&4 S -I2 P/s. -SK X /2.75# 2 P/s-96"X/2.75" 23 5%2 S-13 f'ls-SKXI2.75' 2 Ph.-96"Xl2.75" /2 I%

Half plan of bottomAll PLS.-12.75+- 1%” laps

2940 iin fi of 5/;.” weld

=3+1%Section through lap

Similarly, at point y dimension C is equal to 135 pla\vlengths plus 2 plate widths less 3 plate laps of l!i in., (11’

C = 1.5(31 ft, 8i in.) + 2(8 ft, 0 in.) - 3(1% in.)

= (63 ft,, 6; in.) - 32 in.

= 63 ft, 2; in.

By Eq. Lla,

A-2 = ; - (;2

= 5662.562 - (63 fl, 2#in.)*

= 5662.562 - 3996.610

= 1665.952

A = 40 ft.. 9t-i in.

Ag. 4.6. T y p i c a l b o t t o m l a y o u t f o r a 150~ft-diameter tank.

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Fig. 4.7. Weight of snow loads equaled or exceeded one year in ten years-pounds per square foot (137). (Th’IS material is reproduced from the American Sfondord Building Code Requirements forMinimum Design loads in Buildings and Other Structures, A58.1-1955, copyrighted by the American Stonderds Association.)

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R o o f D e s i g n 6 3

Therefore, for the layout shown in Fig. 4.6, the dimen-sion for S-8 at point y is:

(40 ft, 9% in.) - (34 ft, 7 in.) + la in.or

6 ft, 4& in.

4.3 ROOF DESIGN

The most comnion shape for a tank roof is a cone althoughdome or umbrella roofs are also used. In addition to theseshape classifications, tank roofs may be classified into twotypes, self-supporting and nonself-supporting.

Regardless of shape or method of support, tank roofs aredesigned to carry a minimum live load of 25 lb per sq ftin addition to the dead load. This live load is an averagefigure which allows for combined wind and snow loads andfor the weight plant personnel who may travel across a roofto inspect the vessel or to reach a manhole and so on.Figure 4.7 shows the maximum snow loads to be antici-pated in various parts of the United States (137).

4.3a Self-supporting Conical Roofs. A self-supportingroof is one which is supported only on its periphery withoutthe aid of additional support from columns. Tank diam-eters for self-supporting roofs generally do not exceed 60 ftand usually are less than 40 ft. Any greater spans requiresuch heavy rafters that it is simpler to use one or more sup-porting columns and thereby reduce the span. Such roofsusually consist of roof plates supported on rafters.

Small and medium-sized flat bottomed cylindrical tankshaving capacities of 400 and 3000 bbl or under respectivelyare extensively used in the petroleum industry (100, 101).Figure 3.7 of the previous chapter shows proportions forsuch tanks, and Table 3.3 gives typical dimensions. Theroofs of these tanks are known as “decks” and are fabricatedof mild steel having specifications meeting ASTM-A-7,ASTM-A-283 grade C or D (open-hearth or electric-furnacesteel only). The deck plates have the same thickness asthe shell plates, and a slope of 1 in. in 12 in. is used for thecone. If xs-in. plates are specified, the deck must be rein-forced with structural support if it is 15>6 ft or larger indiameter but does not require additional reinforcing if it issmaller in diameter. If >a-in. plate is used, no supportis needed for lS>&ft diameter tanks, but support is requiredfor all larger-diameter tanks. The deck may be attachedto the shell by one of the following methods. The deckmay be flanged and welded by: a double-welded butt jointwith complete penetration, a single-welded butt joint withbacking strip, or a full-fillet double-welded lap joint. Ifthe deck is not flanged, it should have full-fillet welded jointsboth inside and outside.

For larger tanks with cone roofs the equation for stressin a cone under either an internal or external pressure canbe derived as shown in Chapter 6 (see Eq. 6.139). Themaximum stress will exist at the greatest diameter of thecone and will be:

f* = -+!!L2t sin e

where p = internal or external pressure pounds per squareinch gage

d = diameter, inchest = cone shell thickness, inches0 = angle between cone element and horizontal

The stress as calculated by Eq. 4.2 will be controllingonly in the case of thick cones used with pressure vesselsof limited diameter. In the case of large-diameter conicalroofs such as those used for storage tanks, the controllingfactor is elastic instability. The theoretical critical com-pressive stress that causes failure of a curved plate bywrinkling is given by Eq. 2.24.

(2.24)

where E = modulus of elasticity of material, pounds persquare inch

t = thickness, inchesP = radius of curvature, inches (see Fig. 4.8)

fcritical = theoretical critical stress at which failure bgwrinkling occurs, pounds per square inch

The safe compressive stress that can be carried withoutwrinkling was investigated by Wilson and Newmark (43)in a series of experimental tests. As a result of these testsand others (44), it was found that the safe compressivestress that can be imposed on a steel cylindrical shell with-out failure by wrinkling is one twelfth of the theoreticalcritical stress and can be expressed for P as follows:

fallowable = 1.5(106) 4 7 1 yield pointP 3

(2.25) e?

Equation 2.25 can be modified for use with a conical roof

D = diameter of tank, feetr = radius of curvature of cone

at periphery, inches\ I‘ \

I_ 6 D

I sin B

\ I 0 = angle of cone elementwith horizontal

90-a

e B

T\i

F i g . 4 . 8 . R a d i u s o f c u r v a t u r e o f c o n i c a l roofs.

1’ \ \ I a

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64 Design of Bottoms and Roofs for Flat-bottomed Cylindrical Vessels

Fig. 4.9. Field photograph showing_

structural support for a tank roof.

(Courtesy of Aluminum Company of

America.)

by referring to Fig. 4.8 and substituting for r as follows:

fallow. = 1.5(106) l+

01

sin e = f~ll0w.D

250.000t

(4.3)

(4.4)

It is very important to recognize that the allowable com-pressive stress, fallow., is not the conventional allowablestress for the material but is the safe stress that can beapplied without danger of failure by wrinkling. The com-pressive stress induced by live and dead loads on the roofmust not exceed the allowable compressive stress, fallow..Equation 4.2 can be used to calculate the compressive stressinduced by the roof loads, or Eq. 4.2 can be substituted intoEq. 4.4 as follows:

sh e = P/144)(124 D

(2t sin @) 250,OOOt

\ \ \I /

(4.5)

If live load = 25 lb per sq ft and dead load = 7.65 lb persq ft (for ?ia-in. roof plates)

P = 32.65 lb per sq ft

If one substitutes for P in Eq. 4.5,

min sin 0 = +& d32.65/6 = ?f-4301 (4.6)

Equation 4.6 is the equation specified by API Standard12 C for self-supporting conical roofs. It should be empha-sized that the derivation of the constant is based uponselected roof loads and 3is-in. roof-plate thicknesses andshould be modified for other conditions.

4.3b Conical Roofs with Structural Support. When thedesign calls for a conical roof with structural support, a

Page 75: Process Equipment Design

slope, or pitch, of the roof of a y;i-in. rise per 12 in. is recom-mended. The roof plates may be ridged in order to decreasethe number of rafters required. Roof plates should not beattached to the rafters. Roof plates of lap design shouldhave a minimum lap of 1 in. when tack welded; moreover,if a continuous full-fillet weld is used on all seams, it isnecessary to weld only the top side of the roof. For steelconstruction a minimum thickness of yi 6 in. is recommendedfor the roof plates. Figure 4.9 is a photograph showingassembly of structural support for a tank roof before instal-lation of roof plates.

Storage tanks and’ other large vessels with conical roofsusually are designed with no attempt made to prevent theroof plates from flexing. In such a design the rafters arespaced sufficiently close to each other to prevent overstress-ing of the outer fiber of the roof plales as a result of flexure.The roof plates are assumed to ad as flat continuous beamswit,h a uniform roof load. The rafters and girders areassumed to act as uniformly loaded beams with free ends.Roof design involves the consideration of bending and shearin the roof plates, rafters, and girders. Column action inthe rafters of self-supporting roofs and in the columns ofroofs having supports must be also considered. A briefdiscussion of these relationships follows.

UNIFORMLY L OADED B EAMS WITH FREE E NDS. Referringt.o Fig. 4.10, consider any point, 2, between supports RIand Rz in the beam uniformly loaded with w pounds perlinear inch. The forces acting on the beam to the left ofpoint x produce a bending moment, M, which can be evalu-ated by a summation of the moments at x. For a uniformlyloaded beam freely supported at the ends, RI = R,;therefore

R1 ,!!!2

The force RI produces a clockwise or positive momentequal to Rlx, and the uniform load IO the left of x results ina force, wx, which produces a counterclockwise or negativemoment equal to -wx(s/2), or

(4.7)

To obtain the location of t,he maximum bending moment,let

dM - 0d x

‘I therefore

OP

therefore

;($-!$)=n

Wl- - wx = 02

1x=--

2

Roof Design

By substituting x = l/2 into Eq. 4.7, we obtain:

MWl2

max = -8(for a uniformly loaded beam freely

supported)

65

(4.8)

To determine the deflection of the beam Eq. 2.41 is sub-stituted into Eq. 4.7, and the equation is integrated.

M = E’I cy - wlxwx2dx2 2 2

therefore

To evaluate the constant of integration Cl apply theboundary condition dy/dx = 0 where x = l/2. Therefore

By substituting in Eq. 4.7 and integrating again, weobtain:

EIy = !f!$ - !$ - !$t! + C2 (4.10)

Since y = 0 wherq r = 0,

c2 = 0

By substituting and solving Eq. 4.10 for y, we obtain:

WPX- - - - -2 4

The maximum deflection occurs at t,he center of the spanwhere x = l/2; therefore

5wl”’ = 384EI

= maximum deflection (4.11)

UNIFORMLY LOADED CONTINUOUS BEAM. A uniformlyloaded continuous beam having a large number of equallyspaced supports reacts the same as a simple uniformlyloaded beam with fixed ends. Consider the uniformlyloaded beam shown in Fig. 4.11. The beam is assumed tobe a section of a continuous beam with a large number ofequally spaced supports 1 distances apart. A bendingmoment MO exists over the supports.

F i g . 4 . 1 0 . U n i f o r m l y l o a d e d b e a m w i t h f r e e end+

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46 Design of Bottoms and Roofs for Flat-bottomed Cylindrical Vessels

Fig. 4.11. U n i f o r m l y l o a d e d c o n t i n u o u s b e a m .

Taking the summation of moments at a distance z fromthe support RI, we obtain:

M = MO + Rlx - wx ;0

But

for a beam with clamped ends,and

(2.14)

Substituting for M and R1, we obtain:

M=E~+fo+~-w~ (4.12)

Integrating, we obtain:

Applying boundary conditions to evaluate the constantCl, we obtain:

“” 0-= when x = 0d x

Thereforec1 = 0

E+Mox+!$2-w~ (4.13)

Also,

therefore

dy 0\

-=d x

whens=l I. A .

MO = - $ (maximum moment for uniformly

loaded beam with fixed ends) (4.14)

Substituting into Eq. 4.12, we obtain:

M = EI ‘2 = _ w; + w+ - w$ (4.15)

The bending moment at the center of the span where2 = l/2 is:

(4.16)

Comparison of Eq. 4.14 with Eq. 4.16 indicates that the

maximum bending moment occurs over the supports, asdefined by Eq. 4.14.

The maximum deflection of the beam by inspection isobserved to occur at x = l/2 and may be obtained by sub-stituting Eq. 4.14 into Eq. 4.13, integrating, and evaluatingthe new constant of integration, Ct.

EIy = -

but

thereforey=Oatx=O

cz = 0Substituting and solving for y where x = l/2, we obtain:

WP__ =

’ = - 384EImaximum deflection (4.17)

COLUMN ACTION. Slender structural members underaxial compression tend to deflect. This deflection resultsin a bending stress superimposed on the compressive stress.Referring to Fig. 2.4 of Chapter 2, we find that the axialforce, P, causes a deflection of y in the column of length, 1,and cross-sectional area, a.

The bending moment, M (equal to the force, P, times thelever arm, y) induces a bending stress equal to MC/I, whichmust be added to the compressive stress, P/a, or

j=yc+LP~+Pa a

By definition

I = ar2 (where r = radius of gyration)

therefore

(4.18)

The column may be compared to a uniformly loadedbeam freely supported, in which by Eqs. 4.12, 4.8, and 4.11,

j+; M=!$; y=?!!&

Solving for the product yc for the uniformly loaded beam,we obtain:

= c112 (4.19)

where Cr = constant.For a column the product yc is assumed to vary as 12, as

in the case of a beam:

yc = CZP

Substituting the quantity Cs12 for yc in Eq. 4.19 andsolving for P/a, we obtain:

(4.20)

where Cs is a constant depending upon the material, themethod of loading, and the method of support. Nomethod is known for calculating from theory the value of

\ \ \I I

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R o o f D e s i g n 6 7

the constant Cs. The constant is usually determined byexperiment. Rankine experimentally evaluated the con-stants for round- and square-ended columns and found Csto be l/18,000 and l/36,000, respectively (29). Since anyslight displacement of a fixed-end or square-end columneither laterally or axially will result in an unknown eccen-tric loading, columns are usually designed as round-endedmembers.

For values of l/r between 60 and 200, the AmericanInstitute of Steel Construction (102) recommends the fol-lowing equation for steel columns:

P 18,000-=a 1 + (12/18,000r2)

(4.2i)

i;For columns having values of l/r between 0 and 60, a

column formula is not used, but a maximum value of15,000 for compressive stress, P/a, for steel columns isspecified. For values of l/r greater than 200, Euler’scolumn formula is used. (See Table 2.1, Chapter 2.)

Self-supporting roofs have rafters under combined com-pressive load and bending load. Such rafters may be con-sidered to act as beams under column loading.

The constants f and Cs of Eq. 4.20 have been specifiedas given in Eq. 4.22 by the American Institute of SteelConstruction (102) for steel beams under column action.

P 20,000-=, a 1 + (12/2000b2)

where 1 = unsupported length, inchesb = width of compression flange, inches

The application of Eq. 4.22 is limited to conditions inwhich 1 exceeds 15b but is less than 40b. If the memberis shorter than 15b, the rafter may be designed as a beam.Lateral stiffeners may be used to maintain 1 within theupper limit of 40b. The value of 20,000 specified in thenumerator of Eq. 4.22 is permitted because the bendingstress is maximum only at the outer fiber, and therefore themaximum combined compressive stress exists only at the

l outermost fiber on the top side of the rafter. The averagecompressive stress across the member will be less than20,000 psi.

RAFTER SPACING. Consider a circumferential strip ofroof plate 1 in. wide located at the outer periphery of theconical roof, and disregard the support offered by the shell.This strip is considered to be essentially a straight, flat,continuous, uniformly loaded beam. The controlling

/ bending moment is equal to w12/12 and occurs over thesupporting rafters. By Eq. 4.14,

,M

-w12 -P(1Y2 -pPmax = - = ___ =1 2 1 2 1 2

(4.23)

where 1 = length of beam (strip) between rafters, inchesP = unit load, pounds per square inch = w when

width = 1.0 in.

Introducing the stress resulting from flexure by Eq. 2.10,we obtain:

For a rectangular beam,

bt2z=-6

where b = width of beam, inchest = thickness of beam, inches

Thus, for the case where b = 1.0 in.,

t2z=-6

Substituting Eq. 4.24 and Eq. 4.20 into Eq. 2.10, weobtain:

o r1 = t d2j/p (4.25)

For an allowable stress, j, of 18,000 psi (the maximum speci-fied by API Standard 12 C for steel roofs) a roof-platethickness of Ns in., and a roof load of 32.65 lb per sq ft,that is, p = 0.227 psi, a substitution into Eq. 4.25 gives:

1 = I (2)WOW16

= 74.6 in.0.227

It is apparent from the above calculations for a %a-in.roof plate that rafter spacing in large-diameter cone-rooftanks should not exceed 6 ft unless heavier roof plates areused. API Standard 12 C specifies a maximum rafterspacing of 27r feet or 75% in. on the outer perimeter of aring of rafters and a maximum of 535 ft on the inner perim-eter (2).

The minimum number of rafters adjacent to the shellis determined by dividing the shell circumference by themaximum rafter spacing. The actual number of raftersto be specified should be a multiple of the number of sidesof the polygon of girders supporting the other end of therafters to provide a symmetrical layout; this is a furtherrestriction.

The minimum number of rafters to be used between twoadjacent inner girders should be based on the perimeter ofthe outer polygon of girders. The length of one side of apolygon having sides of equal length is:

where L = polygon-side length, feetN = number of sides of polygonR = radius of circle circumscribing polygon, feet

The minimum number of rafters, n, required will then beequal to (12NL/I) or,

n = 24NR sin 360

1 2N(4.27)

where n = minimum number of rafters1 = maximum rafter spacing, inches

The actual number of rafters to be specified should be a

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6 8 Design of Bottoms and Roofs for Fiat-bottomed Cylindrical Vessels

multiple of the number of sides of the polygon, N, to main-lain a symmetrical layout.

SELECTION O F RAFTERS AND G IRDERS . Rafters aredesigned as uniformly loaded beams with free ends. Eachrafter is considered to support the roof plates and roof loadover an area extending on either side of the rafter andbounded by the center line to the adjacent rafter. F r o mEq. L8 it is seen that the maximum bending momeut forsuch a beam is equal to w12/8 and occurs at the center of thr~bi111. The maximum fiber stress is directly proportionalto ~hc square of the length of the beam, 12. Therefore, toavoid use of excessively heavy rafters the length of the rafteris usually limited to from 20 t.o 24 ft or less.

adequate lateral support to the compression flanges of theraf ters .

SELECTION 0F COLUMNS. The design of beams andrafters for roofs i s based on a safety factor of approximately3, that is, one third the ultimate strength. For rol led-s tee lstructural shapes of 55,000 psi ultimate tensile strength,an allowable tensile stress of 18,000 psi is recommended.The same value may be used for the maximum allowablecompress ive st,ress of ro l led-s tee l sec t ions i f la tera l def lec t ionis prevented. In the case of columns, the lateral deflectionshould be considered, and the maximum al lowable compres-sive sl.ress should not exceed 15,000 psi. This stress canbe computed by Eq. 4.21.

This may be demonstrated by considering 20 ft, 0 in.rafttxrs spaced 6 ft. 0 in. apart at the outer side and t ft.0 in. apart at the inner side with a roof-design Io;~d of0.5 psi. The maximum bending moment is, by K’:(I. 1.8:

M = Ef _ (0.25)(5 x 12)(20 x 12)”8 8

= 108,000 in-lb

The allowable compressive stress so calculated is basedon the gross sect,ion of the column (including area of weld)if the column consists of two or more sections weldedtogether. For main compression members the l/r ratioshould not exceed 180, and the ratio for bracing and second-ary compression members should be l imited to 200.

Rewriting Eq. 2.10 as,M

z=-

f

Supporting columns for roofs may be either of standardstructural shapes or of pipe, depending upon preference in

design. In the installation of columns, clip angles shouldbe used on the tank bottoms to prevent any possible lateralmovement of the column bases.

for f = 18,000 psi (assumed allowable value) we obtain:

108,000 in.-lb2=

18,000 lb per in.2= 6 o in 3

’ ’

From item 1 of Appendix G a 7 x 2Jg in. channel weigh-ing 9.8 lb per ft in which z = 6.0 in.3 may be select.ed toful f i l l the requirements .

4.3~ Dome and Umbrella Roofs. A dome roof is oneformed to a spherical surface. At the beginning of thecentury, tanks with dome roofs were used for a great varietyof services. Today they are seldom used for atmosphericstorage as the more simple cone-roof tank is cheaper. ’Dome roofs are still used extensively for cylindrical flat-bot tom storage vessels des igned for low-pressure service .

For vessels of large diameter in which the rafter spru isreduced by the use of girders, column supports must beused for each ring of girders and also at the center of thrtank. Usually five or more straight girders art’ joinedend-to-end to form a polygonal support for the ends of theraf ters . The girders are designed in the same manner asthe rafters. The girder load is considered to be a uniformload equal to the roof load plus the weight of the rafters.The roof area contributing this girder load is the length ofthe girder times the distance on either side halfway to t.hrnext rafter support. The rafters form a series of concen-trated loads on the girders, but for practical considerat.ionsthe load may be treated as uniform whenever four or morerafters are supported on one girder .

The umbrel la roof is formed so that any horizontal sect ionthrough the roof is a regular polygon wit,h as many sidesas there are plates. Umbrella roofs are a compromisebetween cone roofs and dome roofs. Umbrel la roofs haveapproximately the strengt,h of dome roofs but are easierto install because the roof plates are curved in only onedirect ion.

The equation for stress in a spherical thin-walled vesselcan be developed in a manner similar to that used in devel-oping Eq. 3.13 with t.he following result:

t=p_d+c4f

Self -support ing roofs have raf ters under a combined com-pressive load and bending load. In such designs, if theunsupported length, 1, exceeds 15b where b equals the widthof the compression flange, the stress in pounds per squareinch should not exceed the value calculated from Eq. 4.22.The laterally unsupported length of beams and girdersshould not exceed 40 times b (the width of the compressionflange).

By comparing Eq. 4.28 with Ey. 3.14 it is apparent thatfor the same radius of curvature and the same shell thick-ness the spherical shape is twice as strong as the cylindricalshape.

The above restr ic t ions , l imit ing beams to lengths with an1 ‘b ratio not greater than 40 and to stress not greater thanpermitted by the formula for Z/b ratios greater than 15, donot apply to rafters which are in contact with the steel roofplating. It is assumed that under full-load conditions,fr ic t ion between the roof sheets and the raf ters wil l provide

Thus, for equal strength, the radius of curvature of asphere should be twice that of the cylinder. Therefore, itis customary to make the radius of curvature of a dome roofequal to twice the radius-of the shell. The API Standard12 C recommends this proportion and permits a 20 7, varia-t ion in e i ther d i rec t ion , or

o rR=D (4.29)

Rmir, = 0.8D (4.30)o r

Rmax = 1.20 /

Page 79: Process Equipment Design

Example Design 4.2, Design of Roof and Structural Supports for a 122-Ft Tank 6 9

where R = radius of curvatjure of dome, feetD = radius of tank shell, feet

ELASTIC S TABILITY OF U MRRELLA R OOFS. The specifica-tions of an umbrella roof are also determined by the roofloads and the elastic stability of the roof under load. Byfollowing the same method of derivation as that used forEq. 4.6 for self-supported conical roofs, Eq. 4.32 can bederived for umbrella roofs as follows, start.ing with Eq. 2.25:

but

R=;

t,herefore1.5 X 1OV = pr2

p = 32.65 lb per sq ft. = .227 psia (as for conical roofs)

1.5 x 10” t2 = r20.227

&iXXlo”=rt

2.58 X 1000 = ; = y

R = 215t (4.32)

Equation 4.32 is recommended by API Standard 12 C forhoth umbrella roofs and dome roofs. When one appliesthe same fact.or of safety, 12, dome-shaped roofs havegreater elastic stahilit,y t.han umbrella roofs. This may beshown as follows.

ELASTIC STARILIT~ OF DOME HOOFS. For elastic sta-hility of a thin sphere under external pressure, (42)

2Et”Peril iwl = _--

r2 d3(1 - p2)( 4 . 3 3 )

For steel, p = 0.3, and E = 30 X 10” psi; therefore

2 x 30 x 10”t2

Introducing an elastic-stability fact,or of safety of 12.0,the same safety factor that was included in Eq. 2.25, usedin derivations of cone- and umbrella-roof operations, weohtain:

; = d(36.3 X 106)/12p = 103 d3.03/p

. Assuming p = 0.227, the same as for conical and umbrellaroofs, we oht.ain:

P-=t

10” 4X03/.227

r~- = 3650t

R 3 6 5 0~- - 30-1

t 12o r

R = 304t (4.34)

The constant in Eq. 4.34 differs from the constant inEq. 4.32 by 4. Th’1s is to be expected since the constantin Eq. 4.28 differs from the constant in Eq. 3.14 by a factorof 2.0; moreover, t and r occur in the Pcriticar equation to thesfxwid power.

4.4 EXAMPLE DESIGN 4.2, DESIGN OF ROOF ANDSTRUCTURAL SUPPORTS FOR A 122 FT., 0 IN.DIAMETER STORAGE TANK 48 FT, 0 IN. HIGH

SELWTION OF ROOF PLATES. A tank of t,his diameterrequires column supports and rafters and girders. If therafter lengths are to be limited to about 20-ft spans, twopolygonal rings of girders plus one central column will benecessary. All roof plates will be cut from two st.andardsizes of plates carried in stock: plate size A, 72 in. wide x25 ft, 3$/, in. long, and plate size B, 72 in. wide x 22 ft,l\$ in. long. All plates will be g{s in. thick (7.65 lb persq ft).

A study of various combinations of the above platestogether with sketch-plate requirements indicates that aneconomical roof-plate layout will be like the one shown inFig. 4.12. Such a layout results in a small amount of scrapin cutting out the sketch plates.

RAFTER AND G IRDER SPARING. A suitable rafter-spacinglayout based on the use of two polygonal groups of girdersis next determined. Girders of approximately 26-ft inlength will be used because spans greater than 30 ft requireexcessively- heavy structural sections. Using a radius of22 ft, 0 in. t.o circumscribe the inner polygon of girdersrequires a polygon of five sides. To maintain symmetry adecagon will be used for the outer polygon, as shown inFig. .I. 13.

To determine t,he rafter spacing. Eq. -t.25 may be used.The design load is 25 lb per sq ft live load plus 7.65 lb persq ft dead load @ifi, in. plate). The allowable design st,ressfor the roof plat.es will be taken as 18,000 lb per sq ft.Therefore, by Eq. 1.25,

31=&2flp=

16

= 74.6 in., maximum rafter spacing

Also, maximum rafter spacing = 2x ft = 75.4 in.The minimum number of rafters in the outer ring can be

determined by dividing the circumference of the shell bythe maximum rafter spacing and equals (2xr/t) or

2 x 3.14 x 61 x 12&in =

74.6

= 61.5 rafters, minimum number

Page 80: Process Equipment Design

70 Design of Bottoms and Roofs for Flat-bottomed Cylindrical Vessels

W-IO” diameter

B” 1/5’-9k”13,-Z%”

6’~,q m

C u t 2 P L S . 7 2 ” I 10 .2# x 2 5 ’ ~ X 5 ” ‘- C u t 2 P L S . 7 2 ” x lO.f# I 25’4%” Description MarkPls 7Z"Xl~?2"XZ5-3~" AP/s. 72"XlO.2*X22'/~2" BP/s. 72*X/0.2+X lP'7?6" C

P/s 72"X/O.2+X II-074. 0Pls. IO2 *X Skefch 2P/s lO2"XSketch I2P/s IO2 *X Sketch 7P/s IO2 x X Sketch II

P/s I02 x X Sketch 8P/s IO2 #X Sketch I4

P/s 102 * X Sketch 5

P/s 102 * X Sketch 4

Pls IO2 #X Sketch 6Pfs IO2 *X Sketch 9

P/s IO2 # X Sketch I OP/s IO2 x X Sketch I

Pls 102 e X Sketch lAP/s /02# X Sketch 3P/s IO2 +# X Sketch I3

Make From30 Pf~.72"xlO.2~X25~3~'36 P/s. 72"X10.2*X22~l~'-2P/s 72"XIO2"X25'3%'2Pls 72"X102xX22'I%'

2Pls 72"%/02 "X25-374'

NO .30

364

444

44444

44442

244

2Pls 72"X10.txX25'-3%"

2 P/s 72"X102xX2S'-3%"

2Pls 72"XlO..2'X25'-3%"

2 P/s 72"X102cX25'3%"

2Pls 72"XlO2*X22-I%'

2P/s 72"X/O2*XZ5'3%'

2Pls 72"X/OZ*X25'-3tim

3’$!+r5’-9%;p’-‘!q 9’4%”

C u t 2 P L S . 7 2 ” I I O . 2 9 I ( 2 5 ’ - 3 % ”

(12’ 7 % ”

C u t 2 P L S . 7 2 ” x 1 0 . 2 # x 2 5 ’ - 3 % ”

W-8%” , ,_ l3’-6%”

C u t 2 P L S . 7 2 ” x I O 2 # x 2 5 ’ - 3 % ”

ppl11’ 0%”C u t 2 P L S . 7 2 ” x 10 2 # x 22’-1%” C u t 2 P L S . 7 2 ” x I O 2 # x 22’ - 1%’

1 . 10,-O” _I!_ II’-11%” 4

C u t 2 P L S . 7 2 ” x 10.2+ I 22’-1%”

Shipping Weights30 "A"PlS. 46,41036 B"Pls. 48,744/2;1"Sketches 18,564

BB'Skefches 10,832

C u t 2 P L S . 7 2 ” I 10.2$ I 22’-1%

Fig. 4.12. Roof plates for a 122~ft-diameter tank.

\I I

Page 81: Process Equipment Design

Example Design 4.2, Design of Roof and Structural Supports for a 122-Ft Tank 71

As indicated in Fig. 4.13, a lo-sided polygon, or decagon,will be used to support one end of these rafters. Therefore,the minimum number of rafters required must be a multipleof 10. Thus it may be seen that the minimum number ofrafters between the outer shell and the decagon of girdersis equal to 6.15 rafters per girder. An integral number of7 rafters per decagon girder will be used, giving a total of70 rafters.

To check the spacing of these rafters on the girders Eq.4.26 may be used.

L = 2R sin 3602N

360= (2)(39.75) sin (2)(10)

= 79.5 sin 18”

= (79.5)(0.309) = 24.6 ft

Average rafter spacing = 24.6 ft/7 rafters = 3.51 ft

This spacing is less than the 5jd ft maximum allowablerafter spacing on the inner ring.

The minimum number of rafters required between thedecagon and the pentagon can be determined by usingEq. 4.27, or

24NR . 360n = ~ sm -

1 2N

360(24)(10)(39.75) sin (2)(10)

n =74.6

= 127.8 sin 18’

= (127.8)(0.309) = 39.5

The actual number of rafters to be used must be a mul-tiple of the number of sides in a pentagon and the numberof sides in a decagon. Therefore, 40 rafters will be specified.

The minimum number of rafters within the pentagon mayalso be determined by use of Eq. 4.27, or

360

n =

(24)(5)(22.0) sin o(5)

74.6

= 34.7 sin 36”

= (34.7)(0.588) = 20.4

The actual number to be specified should be a multiple offive. This means that 20 rafters would be a most con-venient number if its use can be justified. An examinationof Fig. 4.12 indicates that the maximum span betweenrafters is less than that determined by Eq. 4.27, which isconservative. Therefore, 20 rafters will be specified.SELECTION OF RAFTER SIZE. The rafter having the

greatest span controls the size of the structural sectionrequired. The maximum rafter span is indicated as rafterR4 in Fig. 4.13 and has a length of 22 ft, 9xi6 in. Thespacing between the rafters at the shell periphery is approxi-

N o t e s :$” inside of shell

All holes I%.” I#Z unless other&e notedAll caps to be welded to columns in shop

N O . Descrrption

5 G i r d e r s

4JIO G i r d e r s

L uqs

I O R a f t e r sI O

4 07 0

I C o l u m nC h a n n e l

C h a n n e l

C l i p

Cop5 C o l u m n s

C h a n n e lC h a n n e l

CIIP

CapI O C o l u m n s

C h a n n e lC h a n n e lC l i p

Cap5 S p l i c e s

IO S p l i c e s

I6 C o l u m n b a s e sC h a n n e l s

G u s s e t s70 lugs

B o l t l i s t

Bill o f Materio/sM a r k

Cl

A to F incl

62G r0 L incl

RIR2

R3R 4

C l

M a k e f r o m

5 6 l5’@33.9~X25’9%’lBar S”XWX33’-9%”

IO& 15’833.9XXZ44”6~

/Bar 5”X%‘X46’-9Y2’2’

IO t b%llS~ x II’ 7%’/0&8%‘Il.5*X/9’6%6=do& 8’@f/.5x x2/‘- 70/l”70688ll.5*X24’7~e”

CIC

c2

lu /2”@20.7*XS4’7~e/UlO’@ 15.30xs3’-ll~4/r’

/L 6”X6’XJ/B’XO’-272”IPL 32”( X94”

c2cc3

56/2*@ 20.7xX50’10%k”

5131 9*@ 13.4eX5050”3?aa5b 6”Xb”X %“XO’2%”

5Pls. /2”X20.4xXl’5”

N C

G/SG2S

Cl9

l0&2”@ 20.7*X43 ‘7%~’IO& 9 ° C M4~X47’N’%s”

IO& 6”X6”Xs/8B”XO’-2W/OP/s. /2”X2~7.4~X/‘2”

5Pls 4”xj/ss”xl’o~u’8”fOPIs. 4UxJ/6 ” x l’O3/4/r”

11

486 8”@ll.S*X2’6”32Pls 10”@10.2xXl”10”706 6”X4”X%“XO”4”

4 0 Rafters t o c a p c o u n t e r s i n k Y--n x 2%”6 0 Girders to cofumns 74’X2”6 0 G i r d e r s p l i c e s v4’X2”

340 R a f t e r s t o girdem Y4/** x I%’

140 R o f t e r s t o l u g s 74. x /J/4*

6 4 C o l u m n s t o b a s e s o/4’ x /Y2#3 2 C o l u m n s t o b a s e s 3/4=x /Y4#

F i g . 4 . 1 3 . R o o f - s u p p o r t a s s e m b l y f o r a 122~ft-diameter t a n k

Page 82: Process Equipment Design

7 2 Design of Bottoms and Roofs for Flat-bottomed Cylindrical Vessels

Inner-row girdersmake 5, mark Gl

m m

_ _ _______---------------- - - - - - - - - - - - - - - - - - - - - - - - _-1 2%”

2” -a.2%“+ A+

- -2”--F---C” L” *

e 24’-6%” *

Outer-row girdersmake 10, mark G2

10 PLS 5” Y ;L”x 531” Required mark

10 PLS 5” x %“x 6&” Required mark @

10 PLS 5” x 3j”x 7%” Required mark @10 PLS 5” x %j”x 8y,” Reqwed mark @

Girder lugs 10 PLS 5” x %,“x 8Q” Required mark @

2OPLS 5”X%“X4” Required mark @

20 PLS 5”~ 4”~ 4hM Required mark @

POPLS 5”X%“X5%” Required mark @

20 PLS 5” x %“x 5%” Required mark @20 PLS 5” x s” x 51%,” Required mark @

1OPLS 5-x %“x 6%” Required mark @

Fig. 4.14. G i r d e r d e t a i l s f o r a 1 2 2 - f t - d i a m e t e r t a n k .

mately 5p; ft., and the rafter spacing at lhe decagon end of! he R4 rafters is approximately 31,~ ft.

The design of the rafter is based upon Lhe roof load plusthe weight of the rafter. Since the rafter size and weightis unknown, a preliminary design based on the roof loadonly wil l be made and later wil l be checked with the weightof the rafter selected.

For the preliminary calculation the raft.er load will be

Inner-girder splicesmake 5, mark GlS

assumed to be uniform and wil l be taken as t.he load inducedby a roof plate having an average width of 4j4 ft. Thelive load is taken as 25 lb per sq ft, and the x6-in. roof-plate weight is 7.65 lb per sq ft; this gives a total designload of 32.65 lb per sq ft. This corresponds to a load of0.227 psi.

Fol lowing the procedure presented in the sect ion enti t led“Selection of Rafters and Girders,” we find that the uni-

\ \ \I /-.---

Page 83: Process Equipment Design

formly loaded beam with this t,hin type of end support hasa maximum bending moment. as given hy Eq. 4.8.

MWP

mitx = -8

Example Design 4.2, Design of Roof and Structural Supports for a 122-Ft Tank 73

(see i tem 1 of Appendix G) is a 15 x 3$6 in. channel weighing33.9 lb per ft and having a section modulus of ‘41.7 in.R.Checking the girder by including the weight of t,he girder,w e find:

w = 0.227 psi X (4.5 X 12) = 12.25 lh per in.;roof l(jad1 = 22.67 X 12 = 272 in .

M(12.25)(272)”

ma.x = 8

57.6 + 33.9 = co.~w=~12-- '

.1r,,,,* =(60.S)(2Y5)2

= 656.000 in-lb8

= 113,500 in-lb

M 1 1 3 , 5 0 02=-= __ = 6.28 in.3f 18,000

Referr ing to i tem 1 of Appendix G, we lirld I hat the l ightest ,American Standard channel sect ion that can he specifiedis an 8-in. x 21i in. 11.5-lb-per-ft h(Ailnl having a sectionmodulus of 8.1 in.3.

The weight of rafters should he included in I he r:lftertoad. This added load amount s I 0:

11.5 lb per flw = - ~~~- ~~

1 2= 0.938 It) prr in.

The total toad is 12.25 + .96 or 13.21 1h prc in.Recalculating M,,,,, we oblain:

M 656,000 _ . 32 = - = ~-~~~ = ,y(j.,, 111f 18,000

Therefore, the girder selected is sat,isfartory.SELECTION OF COLUMN SIZE. A total of 16 columns wilt

be required: 10 for the decagon, 5 for the pentagon plus Isupport,ing the apex of the cone as shown in Fig. 4.13.The roof area and i ts corresponding lo;ld increase per columnas the distance from the tank cent,er increases as a resultof the roof-support layout .

The total roof load supported by each column (C3 ofFigs. k.13 and 4.15) of the decagon is equal to the toad pegdecagon girder plus the weight of the girder itself or is60.4 lh per tin in. of girder length from t.he previous girder-design calculation. Therefore,

P = 60.4 X 295 = 17,800 Ih

M,,,,, = (!“:&!!:‘“) ’8

= 122,000 in-lb

122,000z = ~ ~ = 6.78 in.”1 8 , 0 0 0

The minimum radius of gyratjion of the column sectionis a function of the length of the column under considera-t ion. If the ratio of (I/r is not to exceed 180 a11Cl the 1enpt.hof the column is 48 ft, Ox, in. (576.5 in.) as shown in Fig.4.15. then the minimum radius of gg.raIion is:

Therefore, the rafter selected is satisfactory.SELECTION OF GIRDER SIZE. Consider first the girders

(G2 of Figs. 4.13 and 4.14) of the decagon, which have aspan of 24 ft, 6% in. The girders wilt he assumed to art.as a uniformly loaded beams carrying the raft,er loads.Each of these girders supports one end of 11 rafters. Themaximum rafter loading is 13.21 lb per t in in. over an averagerafter span of (61 - 22)/2 or 19.5 ft. Assuming that halfof the total load carried on each raft,er is supported by thegirder , we can calculate the roof-plus-raft .er load as fol lows:

I= 180r

(13.21)(19.5)(12)(11)Roof-plus-rafter load = --~ ~---

(29% (2)

= 57.6 lb per lin in.

MWP

max = -8

= (57.6)(295)28

1 5 7 6 . 5r = - = -~ = 3.21 in.

180 180

Referring to item 9 of Appendix G under the heading,“properties of sections consisting of t.wo channels,” oneobserves that the lightest channel combination which willprovide a radius of gyration of 3.21 in. or more about boththe X-Z and y-y axis is the combination of a 9-in., 13.4-lbchannel and a 12-in., 20.7-lb channel. This combinationhas a value of 3.41 in. in relation to the 2-z axis and a valueof 3.62 in. in relation to the y-y axis. This combinationprovides a cross-sectional area of 9.92 sq in. and has aweight of 34.1 lb per ft of combined sect.ion.

The allowable compressive st.ress for t,he column may hecalculated by use of Ey. 4.21 as follows:

= 627,000 in-lb

z = _M = 627,000-= 34.9 in.3f 1 8 , 0 0 0

The l ightest channel which wil l provide this sect ion rrlohh~

f=- 18,000 18,0001 + (L2/18,000r2) = (600)2

l + (Is,000)(3.41)2

1 8 , 0 0 0= ___ = 6620 psi‘7 -7.7LI.‘L

\ \ \ I 7 -

c

.

Page 84: Process Equipment Design

Inner-row raftersMake 10, mark RlMake 10. mark R2

Columns-Make 1, mark ClMake 5, mark C2Make 10, mark C3

19’-6%”20'- 1 1’+1/,,”

Middle-row raftersMake 40. mark R3

22"7'&"-22’-93;,” w

Outer-row raftersMake 70. mark R4

6"x4"x%"L xO>4" i!,-3" 8”@11.5* x2’-6”2 PLS. [email protected]# x l’-1w

- C o l u m n b a s e s

R

Make 16. mark CL%

Rafter lugsMake 70. mark L1

j+- 2’-8”+

Inner-cob cap Center-col. capMake 5, mark CZC Make 1. mark ClC

Fig. 4.15. Rafter and column detoilr for o 122-ft-diameter tank.

Outer-cob capMake 10, mark C3C

Page 85: Process Equipment Design

Example Design 4.2, Design of Roof and Structural Supports for a 122-Ft Tank 75

The actual induced stress is:

j=;

.f=17,800 + (50 X 34.1)

9.92

19,505= ~ = 2270 psi

9.92

Therefore, the column combination is satisfactory, andthe radius of gyration is controlling.

The size of the pentagon column supports (C2 of Figs.4.13 and 4.15) can be readily determined because it is recog-nized that here also the radius of gyration is controlling.These columns have a length of 50 ft, 4 in., or 604 in., asshown in Fig. 4.15. Therefore,

1 6 0 4P = - = - = 3.36 in.

1 8 0 1 8 0

Thus the same column combination called for at the decagonsupports will also be specified here.

The central column (Cl of Figs. 4.13 and 4.15) has agreater length because of the roof pitch and the fact that

it does not support girders. Its length is 54 ft, 0t.d in. or648 in. Therefore,

1 6 4 8r = - - - = 3.61 in.

1 8 0 1 8 0

Consider a combination of a lo-in. and a 12-in. channel.This combination has a radius of gyration of 3.83 in. in rela-tion to the x-x axis and a radius of gyration of 3.52 in.in relation to the y-y axis; therefore the average radius ofgyration is 3.68 in.

In view of the low induced compressive stress on thiscolumn and the average radius of gyration of 3.68 in. ascompared with the 3.61 in. required, its use in this applica-tion can be justified.

It is customary for many tank fabricators to use built-upstructural sections for columns as is done in this example. .However, it is apparent that an appreciable saving in columnmaterial could be realized by using pipe in which the inducedstress would be higher. For example, lo-in. schedule-10pipe having a radius of gyration of 3.74 in. and a cross-sec-tional area of 5.49 sq in. would be satisfactory. This areacompares with the area of 10.50 sq in. for the lo-in., 15.3-lband 12-in., 20.7-lb channel combination for the centercolumn. Thus a material saving of 48’% can be realized.This gain may be partially offset by the greater cost of pipe,

P R O B L E M S

1. Using the dimensions given in Fig. 4.16, determine the required section modulus, Z, forrafters RA, RB, RC, and RD.

2. Using the dimensions given in Fig. 4.16, determine the required section modulus, Z, forgirders GA, GB, and GC.

3. Using the dimensions given in Fig. 4.16, determine the required radius of gyration forcolumns Cl, C2, C3, and c4.

4. Derive a relation comparable to Eq. 4.6 for use with aluminum roof plates.

C- 1 I1 reauiredl

Fig. 4.16. Rafter and girder layout for a 150-ft-diameter cone-rooi

tank.

Page 86: Process Equipment Design

C H A P T E R

PROPORTIONING AND HEAD SELECYION

FOR CYLINDRICAL VESSELS

WITH FORMED CLOSURES

u he real need for the use of formed c!osurrs on cyliutl ri-cal vessels arose w-ii h the development of the power steamboiler early in the nineteenth century. As a result of thefrequent occurrence of boiler explosions, the British Houseof Commons in 1817 made the recommendation that theheads of cylindrical boilers be hemispherical (12). Sincethen a wide variety of formed closures termed “heads” havebeen developed, standardized, and extensively used in thefabrication of process pressure vessels. The developmentof the thermal cracking process in the petroleum industryduring the period from 1915 to 1930 resulted in the con-struct ion of thousands of pressure vessels with formed headsoperating in the range of from 100 to 400 psi. The headsof these early vessels usually were of the torispherical-dishtype with a small knuckle radius.

The first formed heads were of a small size and werehand-forged by “bumping out” a flat plate. One of t.heearly American steel producers, Lukens Steel Company,in 1885 formed a 5-ft-diameter dished head by digging ahole in the ground to the approximate radius of the dishand bumping the heated plate into the depression by the useof mauls . Since then metShods of forming heads have beenhighly developed by the use of dies and forging and spinningtechniques. Figure 5.1 shows a photograph of the world’slargest flanging machine spinning a head with a 20 ft, 6 in.diameter .

5.1 GENERAL CONSICERATIONS

5.la Development of Welded Ccnstruction. The earlythermal-cracking plarts of the prtrcleum industry usedpressure vessels in which the formed htads were rivrtrd to

t.he shell. These vrsst~ls had the fau11 of frequent leakagearound the rivet. heads. AI tempts to correct this diffrcul! ywere made by means of fillet welding the plate edges andseal welding the r ivet heads. These vessels often were notsatisfactory unless the fillet welds were made so large thatthe loads were carried by the fillet. welds rather than hythe r ivets . When it was realized that the welds were carry -ing the loads rather than the rivets, a large number ofvessels for low-pressure service (walls less than 1 in. thick)were fabricated entirely by oxyacetylene we!ding. Thelimitations of the welding art, at this time, in particular thebrit.tleness of the bare electrode welds, made the construc-tion of heavy-walled vessels impracticable. With thedevelopment of flux-coated electrodes ductile welds werepossible. This development resulted in practical obsolrs-cence of riveted-fabrication techniques for pressure-vesselservice .

5.1 b Use of Formed Heads. Cylindrical vessels withformed heads are used for a wide variety of applications inwhich cylindrical tanks with flat bottoms cannot be used.These applications can be grouped into three Gasses: (1)funct.ional use, (2) pressure consideration, and (3) sizel imi ta t ions .

Processing equipment such as distillation columns,desorption units, packed i owers, evaporators, crystallizers,and heat exchangers are essentially cylindrical vessels hav-ing formed heads plus other required functional parts.If the working pressure of the process vessel is to be 01 herthan atmospheric pressure, formed heads are usually usedto c lose the vesse l .

In general, all cylindrical vessels requiring a workingpressure in the vapor space of abo& 5 lb pep sq in. gage ormore are fabricated with formed heads. Large-diameter

76

Page 87: Process Equipment Design

Mater ia l Speci f icat ions 77

Fig. 5.1. World’s largest flanging machine spinning a head 20 ft, 6 in. in outside diameter. (Courtesy of Lukens Steel Company.)

flat-bottomed, cone-roofed storage vessels are limited to .If -working pressure in the vapor space of only a few ounces.

Xowever, cylindrical vessels with flat bottoms and con-I( siderably smaller diameters may operate under allowable

working pressure of several pounds per square inch if adomed or umbrella roof is used. Equipment designed tooperate under less than atmospheric pressure will alsorequire the use of formed heads. Smal l hor izonta l s toragevessels supported off the ground are usually fabricated withformed heads although flat ends of heavy plate are some-I imes used.

5.1~ Vertical versus Horizontal Vessels. In general,! he functional requirements of the vessel determine whetherthe vesse l shal l be ver t ica l or hor izonta l . For example, dis-tilling columns and packed towers, which utilize the force:*f gravi ty for phase separat ion, require vert ica l insta l lat ion.Heat exchangers and storage vessels may be either verticalor hor izonta l . In the case of heat exchangers, the selectionis often controlled by the routing of the fluids and heat-transfer considerat ions . In the case of storage vessels, theinstallation location is important. If the vessel is to beinstalled outdoors, the wind loads on vertical vessels may

7 impose the necessity of heavy foundations to prevent over-t.urning. For this reason, horizontal storage vessels are

usually more economical. However, ot.her important con-siderations such as available floor space or ground area,head room, and maintenance requirlmcnts may be deter-mining fac tors .

5.2 MATERIAL SPECIFICATIONS

Vessels with formed heads are most commonly fabricatedfrom low-carbon s teel wherever corrosion and temperatureconsiderations will permit its use because of the low cost,high strength, ease of fabrication, and general availabilityof mild steel. Low- and high-alloy steels and nonferrousmetals are used for special services .

The steels commonly used fal l into two general classifice-tions: (1) the steels specified by the ASME code for unfiredpressure vessels (ll), often referred to as “boiler-platesteels , ” of flange or firebox quality; and (2) structural-grade steels, some of which are permitted by the abovecode in certain applications and which arc widely used fcrthe construction of storage vessels under specificationsgiven in API Standard 12 C (2). The design of vessels il;accordance with the ASME code for unfired pressurevessels is treated in Chapter 13, which includes a descriptionof the materials and specifications. The discussion in thischapter will be restricted to those st,eels used in the fabrica-

Page 88: Process Equipment Design

tion of vessels with formed ends not requiring fabricationin accordance with these codes.

5.20 Comparison of Specifications for Structural- andBoiler-quality Steel Plates. Structural-quality steel ratherthan boiler-plate-quality steel is used in the fabrication ofmany vessels with formed heads because of economic con-siderations and its availability. Both types of steel areavailable in the “killed ” and the “semikilled ” or rimmedquality. A “killed ” steel is one completely deoxidized bythe addition of aluminum, silicon, or manganese at the timeof the casting of the ingot. The purpose of killing is tominimize the interaction of carbon and oxygen and toreduce the formation of blow holes. A completely killedsteel requires “hot capping,” more time in the soaking pit,and more time for the ingot heating. “Hot capping ” is,the use of an insulated mold on top of the ingot mold tohold a molten reservoir of metal for feeding the ingot as itshrinks on solidifying. A partially killed or rimmed steelis a partially deoxidized steel. An ingot of rimmed steelhas a high-purity, low-carbon steel rim from which it obtainsits name. Fully killed structural steels have no advantages.over boiler-plate steel because of their high cost and limitedavailability.

One of the major differences between boiler-plate steeland structural-plate steel is the “ quality ” control dictatedby the number and severity of test requirements. As faras chemical requirements are concerned, the principal differ-ence expressed by ladle analysis is the more restrictive-limit placed on phosphorus and sulfur for boiler-plate steels.

The thickness tolerances are the same for boiler-platesteels and structural steels when plates are ordered to agiven thickness. The physical tests are the same for bothsteels except for the number of tests and the stipulated loca-tion for test specimens. Structural-quality plate steelsrequire only two tension and two bend tests from each heat,of metal which may contain over 100 tons. Flange-qual i tyboiler-plate steel requires one tension and one bend testfrom each plate rolled. Firebox-quality boiler-plate steelrequires two tension tests and one bend and one homo-geneity test from each plate as rolled. There are also minordifferences in the methods permitted for repairing surfacedefects in the slabs prior to rolling.

Boiler-plate steel such as SA-285 flange quality and lire-box quality had mill quality extras of $0.40 and $0.50 per100 lb respectively as of January 1956 (see Appendix C).Other boiler-plate steels such as SA-212 and SA-201 hadmill quality extras of from $1.20 to $1.55 per 100 lb, depend-ing upon thickness and grade. Killed steels had mill extrasof $0.65 per 100 lb. The use of structural-grade steelsresults in the minimum of quality-extra charges, and theuse of these steels is justified whenever permissible. Inselecting steels for pressure-vessel fabrication to satisfy coderequirements, Chapter 13 should be consulted.

5.2b Types of Structural-steel Plates. The most widelyavailable types of plain-carbon structural-steel plates arelisted (67) in ASTM-A6-54T. Those most suitable forvessel construction are A-7, A-113, A-131 and A-283.Specification ASTM-A6-54T gives the general requirementssuch as permissible variations in dimensions and weight,methods of testing, correcting of defects, and rejection (67).

78 Proportioning and Head Selection for Cylindrical Vessels with Formed Closures

ASTM-A-7, A-283, Grade C, and A-283, Grade D are themost commonly used plain carbon steels in the constructionof storage vessels and are widely used for vessels withformed heads, especially the steel designated as ASTM-A-283, Grade C. Steel A-283-54 is of the structural qualityintended for general applications. It is available in fourgrades, A, B, C, and D, having minimum tensile strengthsof 45,000, 50,000, 55,000 and 60,000 psi respectively, asgiven in Table 5.1. This steel is available in thicknessesup to and including 2 in., but its use in vessels designed tocode specification is limited to thicknesses up to andincluding K in. Grades A and B are primarily used insevere cold-forming applications where high ductility is ofprime importance and tensile strength is a minor considera-tion. On the other hand, Grade D does not have sufficientductility for easy shell and head forming and is not as easilywelded as Grade C. As a result, Grade C is the most widelyused structural-quality plate steel for vessel construction.The major. portion of all oil-storage tanks, elevated tanks,water standpipes, and other varieties of tanks of all descrip-tions, involving both dishing and rolling, are constructedof ASTM-A-283, Grade C.

Steel A-7 is intended for use in the construction of bridgesand buildings and for general structural purposes. It hasphysical properties identical to A-283, Grade D. Thesesteels are the same whether made by the open-hearth orelectric-furnace processes. However, steel A-7 is also madeby the acid-Bessemer process, and steel made by this processis not recommended for vessel construction. Steel A-7 isavailable in all standard thicknesses, and its use is permitted .in vessels designed to present code specifications and havingshell thicknesses up to and including s/4 in., providing thesteel has properties equivalent to A-283, Grade D.

Steel ASTM-A-113-55 is a structural steel used for theconstruction of locomotives and railroad cars except wherefirebox boiler plate is required. It is made by either theopen-hearth or the electric-furnace process and is availablein nearly all standard thicknesses. This steel is made in3 grades, A, B, and C. Steel A-113-55, Grade B has prop-erties approximately midway between those of steels A-283-54, Grades C and B, as shown in Table 5.1. Note that thegrade specifications for tensile strength for the A-113 steelsrun in the reverse order of the grade specifications for A-283steels. There is no particular advantage to using this steelin preference to A-283 steels except when it is more readilyavailable. It may be used for vessels designed to presentcode specifications with the same limitations as for A-283grade steel.

Steel ASTM-A-131-55 is an improved structural steelintended primarily for use in ship-construction. Formerly,the specifications for this steel were essentially the same asfor A-7 and A-283, Grade D. To improve the quality ofship-hull steels, the specification was changed in 1950 inorder to include an increase in quality specificationswith increasing thicknesses. This logical requirement of .increased quality with increased thickness warrants con-sideration of this steel as a material of construction forheavy-vessel fabrication. For this steel there is a limitationon the maximum percentage of carbon and a range of from0.60 % to 0.90 y0 manganese for all plates thicker than G in.

---T-- -\- - - - - - - - - - l - - - r - - - - - --_- -- _ ~__ ~-~- - I -II

---- -\ \ \I I

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.i

SteelA-283-54

Grade AGrade BGrade CGrade D

A-7-55TA-131-55*

Grade AGrade BGrade C

A-l 13-54Grade AGrade BGrade C

TensileStrength,

ps i

Proportioning of Vessels with Formed Heads 79

Table 5.1. 1955 ASTM Steel Specifications (67)

Max Max % Max %Min Yield Thickness Min % Min To Max % S

Point, Available, Elone.. in.. Elone.. in.. C (1at;le) (ladle)ps i in. 8X. ’ 2in. ’ (ladle) (basicj (basicj

45,000 to 55,000 24,000 250,000 to 60,000 27,000 255,000 to 60,000 30,000 260,000 to 72,000 33,000 260,000 to 72,000 33,000 15

58,000 to 71,00058,000 to 71,00058,000 to 71,000

60,000 to 72,000 33,00050,000 to 62,000 27,00048,000 to 58,000 26,000

* See text for limitations.

32,000 $5 and less32,000 35 to 132,000 1 and over

Also, for plates having a thickness of 1 in. or more, a require-ment of 0.15% to 0.30% silicon is specified. In addition,it is stipulated that this steel be manufactured to have aninherent fine-grained structure. This steel is available in awide range of thicknesses and is of higher quality than A-7but presently is not permitted in the construction of vesselsdesigned to meet unfired-pressure-vessel codes. The addi-tional quality requirements for heavier plates of this steelwill increase its cost and may thereby eliminate any savingsfrom using it instead of boiler-plate steels.

Other structural-quality steels listed in ASTM designa-tion A-6-54T are A-8, A-94, A-284, and A-242. Steel A-8is a 3.0% to 4.0~~ nickel steel containing a maximum of0.43~~ carbon and having a tensile strength of from 90,000to 115,000 psi. It is intended for use in main stress-carryingstructural members. The nickel addition results in liner,

t stronger, and tougher pearlite than is found in plain carbonsteel and appreciably increases the yield point, fatigue limit,and impact strength. The difficulties of welding this steel

I plus the cost extras for nickel addition precludes its use forvessel construction. Steel A-94 is a structural silicon steelcontaining a maximum of 0.40% carbon and a minimumof 0.20 y. silicon and having a tensile strength of from 80,000to 95,000 psi and a minimum yield point of 45,000 psi.This steel may be eliminated from consideration for vesselconstruction on the basis of welding difficulties and the costextras for fully killed steel. Steel A-284 is a low- andintermediate-strength carbon-silicon steel containing from0.10% to 0.300/, silicon and having tensile strengths of

i from 50,000 to 60,000 psi depending upon the grade. Thesteel is coarse-grained and requires heat treating for grainrefinement. The presence of the silicon tends to dissociatecarbides to form soft graphite thereby weakening thewelded joints. For these reasons and because this steel isa fully killed steel and therefore involves cost extras, it isnot economical to use it for vessel construction.

Steel A-242 is a low-alloy structural steel intended pri-T marily for use as a stress-carrying material of structural

members when saving in weight and atmospheric-corrosion

2 7 3 0 no spec. 0.04 0.052 5 2 8 no spec. 0.04 0.052 3 2 7 no spec. 0.04 0.052 1 2 4 no spec. 0.04 0.052 1 2 4 no spec. 0.04 0.05

2 1 2 4 no spec. 0.04 0.05(see Ref. 67 ) 0.23 0.04 0.05(see Ref. 67) 0.25 0.04 0.05

2 1 2 4 no spec. 0.04 0.05 .2 4 3 8 no spec. 0.04 0.052 6 no spec. no spec. 0.04 0.05

resistance are important. Thicknesses are limited to notunder sis in. and not over 2 in. It contains a maximumof 1.25Cy manganese and a maximum of 0.20yo carbon.This steel has a yield point of 50,000 psi for thicknesses offrom xi6 to N in., 45,000 for thicknesses of K to 134 in.and 40,000 for thicknesses of l>s to 2-in. in comparison toa yield point of 30,000 psi for A-283, Grade C. For theplates l>s-in. thick and less this represents an increase of50% or more in yield strength. Using the same designfactor of safety based on yield point results in a proportionaldecrease in metal thickness required to resist a given load.In designs in which stress rather than elastic stability orbrittle fracture is controlling, the use of t,his steel ratherthan a plain carbon steel such as A-283, Grade C may resultin a saving. See Table 3.2 for specifications for this steeland Chapter 3 for further discussion of its use.

5.3 PROPORTIONING OF VESSELS WITH FORMEDHEADS

In general, the cost of a vessel may be considered to beproportional to the weight of the steel used in its construc-tion. It would therefore appear that for storing a fluidunder uniform pressure a vessel having the minimum surfacearea and thickness per unit volume would be the most eco-nomical. A spherical vessel has the minimum surface areaper unit volume and the minimum shell thickness for a givenpressure and volume. If the cost of fabrication were not aprime consideration, the most economical shape for a vesselwould therefore appear to be a sphere. However, thefabrication costs of spherical vessels are so great that theiruse is limited to special applications. Cylindrical vesselsare more easily fabricated, in the majority of cases are con-siderably simpler to erect, are readily shipped, and aretherefore more widely used in the process industries.

For a simple cylindrical vessel with formed heads, the .optimum ratio of length to diameter, L/D, is a function ofthe cost per unit area of the shell and the formed heads.More complex vessels such as distillation columns, heatexchangers, and evaporators have additional parts such as

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*od . Proportioning and Head Selection for Cylindrical Vessels with Formed Closures

D i m e n s i o n s f o r a 2 : 1 e l l i p s o i d a l d i s h e d h e a d .

trays in distillation columns and tube bundles in heatexchangers which must also be considered in determining theoptimum proportions.

The proportioning of a simple vessel may be based eitheron the cost per pound of the material or the cost per unitarea of the material. In Chapter 3 the proportioning offlat-bottomed, cylindrical, cone-roofed tanks was basedon the cost per unit area because land and foundation costs,which are important for such vessels, can best be consideredon a unit-area basis. In addition, the cost of coned roofsand flat bottoms are relatively constant on a unit-areabasis for large-diameter tanks. However, cylindrical tankswith formed ends for various pressure services have widevariations in thickness and therefore vary in cost per unitarea. The cost of land area and foundations is usually aminor consideration for such vessels. Therefore, ii is moreadvantageous to consider the cost of shell and heads interms of unit weight rather than in terms of unit area.

5.30 Equations for Optimum Proportions of Vesselswith Elliptical Dished Heads.

VOLUME RELATIONSHIPS. A cylindrical vessel closed at_---__ -l,ot’i;-~ds-~wi~~-~llipt,ical dished heads has a volume equalto the volume of the cylindrical section plus twice the vol-ume contained in one of the heads. The volume containedin a head can be expressed in terms of a cylinder of equiva-lent volume having the same inside diameter as the cylin-drical section of the head. Figure 5.2 is a cross section ofc -,___._. :.. -.._ . . . . . . .an ellipsoidal head having a 2: 1 malor-to-mmor-axis ratio.~~qi&ions for the volume relationships for a 2:l

ellipsoidal head ( 103) are as follows.

The equation of au ellipse is:

(5.1)

For a 2: 1 ellipsoidal dished head

a = 2b

Substituting we obtain:

-$+$=IExpanding we obtain:

x2 + 4y2 = 4b2

Solving for x2 we obtain:

z2 = 4b” - ty2 = 4(b2 - s”;

Differential volume,

dV = A dy = rx2 dy

Integrating we obtain:

The v-olume of an equivalent cylinder is:

1. = vra2H

where H = length of cylinder

Equating we obtain:

3Ta2H = !!!.

3

Thus the volume of IWO ellipsoidal heads having a major-t,o-minor-axis ratio of 2.0 is:

I-n=(3(y),=Tg .

Therefore, the t,o a1 vohmte coutwiueti in the vessel is:

~~;‘_, = [($)I* +gj

where I, = length of the vessel, t.angent line t.o tangent line,between heads, feet.

Solving for L, we obtain:

L+p]COST RELATIONSHIPS. The diameter of a circular plate

required for forming an ellipsoidal head is approximately22% greater than the internal diameter of the finished ves-sel (103). Also, the cost of the formed heads is approxi-mately 50% greater than the cost of the steel from whichthey are formed. This increase in cost results from costextras for circular plates and the cost of forming andmachining. Let

.

c, = cost of fabricated shell, dollars per pound1.5 c, = cost of fabricated head, dollars per pound

1 = thickness of head and shell, inchesp = density of steel, pounds per cubic foot

The cost of the shell section of t,he fabricat.ed vessel is:

’ i* t

‘, A ; ”

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and the cost of two elliptical dished heads is

2 x 1.&p[; (1.22D)Z ;1

or the total cost of the vessel is:

t 4VG=c,pr- - -

[

D2

12 uD T %(1.22D)2 1+

But according to Eq. 3.14,

. t = I!!! = POw 24f

Substituting we obtain:

C = c,pu !@2885

1.275 ; + 0.782D2 1= c,k[1.275V + 0.782D31 (5.3)

where k = f!?288j

PROPORTIONING . The cost of the shell is not a constantbut is a function of the weight of the vessel, which in turnis a function of the pressure and diameter. For vesselshaving a shell plate thickness of up to 2 in., the cost of thevessel may be estimated as varying approximately inverselywith D% (103). Or

C8’C8 = Dys

Substituting in Eq. 5.3, we obtain:

G = c,‘k F + 0.7820% 1Holding V constant, differentiating, and equating to zero

in order to obtain the minimum, we find that

dC 1.275V-=dD

-g D,i + fi(0.782)05* = 0

6.25D3 = 1.275V

D3 = 0.204V

. .Substltutmg for V we find that

1D = 0.16OL + 0.0530

L = 0.947~ D = 5.930 rr 600.160

(5.4)

Or use L/D = 6 for vessels with plate thickness up to 2 in.

Selection of Optimum Plate Dimensions 81

For vessels fabricated from plates from 2 in. to 6 in. inthickness, the thickness extra will modify the cost per unit.weight. In this range of thickness the cost of the vesselmay be estimated as varying approximately inversely withD46 (103). Or

cs ,Ics = gr’l

Substituting into Eq. 5.3, we obtain:

C = c,“k F + 0.782D2.75 1Differentiating and equating to zero to obtain the mini-

mum, we find that

dG 1 1.275V&j = - z D5/4 ,~ + 9-(0.782)0~.~~ = 0

8.60D3 = 1.275V

D3 = 0.148V

Substituting for V, we obtain:

1D = 0.116L + 0.039D

0.961L=-0.116

= 8.280 z 80

Or use L/D = 8 for vessels with plate thickness of from 2 in.to 6 in.

DIAMETER AND LENGTH LIMITATIONS. The selection ofthe proportions of a vessel may be influenced by other fac-tors such as the maximum diameter or length that can beshipped by railroad flatcar. In general, the maximumdiameter that can be shipped on most railroad lines is13 ft, 6 in. Larger diameters may be shipped by rail butrequire special routing of the shipment. If water trans-portation is available between the fabrication shop and theerection site, large-diameter equipment may be shipped bybarge or floated to the site. Two other alternatives are:(I) shop forming and partial fabrication by welding in sec-tions with final fabrication in the field or (2) field assemblyof plates cut and formed in the shop.

The length of a vessel is not as critical as the diameter withrespect to railroad shipping limitations because more thanone flatcar may be used. Figure 5.3 shows an oil-refineryfractionation column loaded on three flatcars, supportedon the two end flatcars with no load supported on themiddle car. This permits the cars to negotiate a curve withthe vessel pivoting on the end cars.

Other considerations such as the selection of plate widthsand plate lengths to minimize the number of welded jointsmay influence the proportions of the vessel. (See the fol-lowing section.)

5.4 SELECTION OF OPTIMUM PLATE DIMENSIONS

P?ATE W IDTH. The cylindrical shells of vessels withformed heads may be fabricated by rolling and welding oneor more plates together. A choice exists as to the plate

I ---3?+ .---iv-\- - \I /- - -- - --

- - - -.-+ . . . --..-.

Page 92: Process Equipment Design

82 Proportioning and Head Selection for Cylindrical Vessels with Formed Closures

“Ts*

Fig. 5.3. O i l - re f ine ry f rac t iona t ing tower ready fo r sh ipment on th ree f l a t ca rs . (Courtesy of C. F. Braun & Co.)

widths and number of plates to be used. Usually a cir-cumferential weld and sometimes a longitudinal weld maybe avoided by using a larger plate width. Plates havingwidths in excess of 90 in. bear a cost extra which increaseswith increasing width. The most economical design is oftenone in which a wider plate is used; providing that a weldedjoint is thereby eliminated and the cost saved by eliminatingsuch a joint exceeds the extra cost of wider plates. Anexample of the reduction in cost that may be realized by theselection of a plate size that will eliminate a welded jointis given by W. G. Theisinger (104) in regard to a purchaseorder involving 20 vessels 48 in. in diameter and 20 vessels

.=: 60g 50

: 4oaJ$ 30

f-a 20

i10

0.1 0.2 0.3 0.4 0.6 0.8 1.0 2 3 4C, = cost extra, dollars per 100 lb

F i g . 5 . 4 . Width extras for carbon-steel plates os of 1953.

54 in. in diameter. The shells for these vessels might havebeen ordered as follows:

1. Two-plate shellsa. 48-in.-diameter vessels

20 plates 15’735 X 87 X 141 in.20 plates 15735 X 85 X 134 in.

b. 54-in.-diameter vessels20 plates 1763s X 91 X llT<e m.20 plates 17634 X 93 X llp{,j in.

2. Single-plate shellsa. 48-in.-diameter vessels

20 plates 171 X 15735 X 13/4 in.b. 54-in.-diameter vessels

20 plates 183 X 1763d X 119is in.

For two-plate shell construction the extra fabrication costswere estimated to average $436.00 per shell or a total of$17,440.00 for the 40 vessels. With single-plate shell con-struction the extra for the 40 wide plates at width extrasof $1.25 per 100 lb for the 157>4-in.-wide plate and $1.50per 100 lb for the 17634-in.-wide plate and the overweightallowance totaled $9,853.00; therefore, a net saving of$7587.00 was realized by the purchaser by using single-plate shell construction. In addition, the fabricating timewas reduced by 5800 man-hours, and this resulted in quickerdelivery. It should be pointed out that these figures arefor prices existing in 1944 and are not representative ofcurrent prices.

When a vessel shell may be fabricated by one- or two-piece construction, the selection may be made by simplyestimating the costs for each design and selecting the designgiving the lesser cost. However, for larger vessels in whichthe shell must be fabricated from many plates, the above

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.

i

‘E

i

Table 5.2. Average Extra Fabrication Cost, C, (104)Dollars per Foot as of 1944

(Based on 133% Shop Burden)Code Welded Unclassified

3.75 2.504.50 3.005.40 3.606.38 4.25’

6.98 4.657.50 5.008.10 5.408.74 5.83

9.30 6.269.83 6.55

10.43 6.9511.10 7.40

11.78 7.8512.60 8.4013.43 8.9514.25 9.50

15.15 10.1016.13 10.7517.25 11.5018.53 12.35

19.50 13.0020.55 13.7021.53 14.3522.80 15.20

23.85 15.9024.90 16.6025.88 17.2527.00 18.00

28.28 18.8529.10 19.4030.15 20.1031.35 20.90

procedure is not so simple since a number of designs maybe possible. To determine thepEt~.~~-~.ofplates,the plate width resulting in the minimum cost for the fabri-cated shell can be evaluated mathematically.

The width extras for plain-carbon-steel plates as of 1953are shown in Fig. 5.4, in which the cost extra in dollars per100 lb is plotted against w - 90 where w is the plate widthin inches. The equation of the line given in Fig. 5.4 is:

ce = $ (w - 90)1.23

where C, = dollars per 100 lbw = plate width, inches

The cost of circumferential welding, Cw, including thecost of preparation of the joint, is usually expressed interms of dollars per foot of weld and is given in Table 5.2.The fabrication cost per circumferential weld will be

Selection of Optimum Plate Dimensions 83

+WL. The total cost of all of the circumferential shellwelds for N number of plates (excluding the head welds)will be:

(N - l)wDC, = (5.7)s

where 1 = length of shell, inchesD = shell diameter, feet

The additional cost of using plates wider than 90 in. isgiven by the equation:

t$) (2) = (l;f;;o) [A] (w - 9O)‘.23 (5.8)

The total extras for using plate widths wider than 90 in.plus the costs for all the circumferential joints exclusive ofhead joints is given by the sum of Eqs. 5.7 and 5.8 as follows:

C=?rD ’K >

- - 1 C, + O.O0023511(w - 9O)‘.23I

G-9)W

Differentiating the cost, C, with respect to plate width, w,and equating to zero to obtain the minimum, we find that

dC - TD -GEdur-

7 + 0.000235t1.23(~ - 90)“.23 = 0

c, w2- = 3460t - (w - g(Qo.23 (5.10)

Solving Eq. 5.10 for w gives the optimum width of plateto give minimum fabrication cost for the shell as a functionof joint fabrication cost, C,, and shell thickness, 1. Thisequation is plotted in Fig. 5.5 for convenience.

Since 1953 the width extra has been combined with thethickness extra (see Appendix C). Therefore Fig. 5.5 isuseful only for first approximations.

PLATE THICKNESS. Plates having thicknesses of from--- -.w$6 &.‘to~&~areXvailable from mills at base cost with nothickness extras. To avoid extras for plates thicker than1 in., a higher-strength steel often may be used to advan-tage. This is of particular importance in connection with^-

65

4

390 100 110 120 130 140 150 160 170 180 190 200

w = optimum plate width, inches

Fig. 5.5. Optimum plate widths for vessel-shell construction bared onwidth extras as of 1953.

Page 94: Process Equipment Design

84 Proportioning and Head Selection for Cylindrical Vessels with Formed Closures

Fig. 5.6. F o r m i n g dished heads b y “ d r a w i n g ” i n o p r e s s . D e t a i l o: Toking c i r c u l a r b l a n k p l a t e f r o m f u r n a c e f o r f o r m i n g . Detail b: Forming of head 1 x 6 in.t h i c k f o r o v e s s e l 6 0 i n . i n i n s i d e d i a m e t e r i n o 1000~ton press. ( C o u r t e s y o f C . F . Broun & C o . )

vessels designed to meet code requirements and is con- Although the cost of heads formed from flat platessidered in detail in Chapter 13. The 1957 practice in involves the additional cost of forming, the use of forrne!lsteel pricing combined thickness extras with width extras heads as closures is usually more economical than the use(see Appendix C). of flat plates as closures except for closures of small diam-PLATE LENGTH.._.._ _.-- - Plates having lengths between 8 and eter. This can be shown by comparing the thickness

50 ft are available from mills at no length extra. If pos-$1

required for closures of flat plates with that of various typessible, the plate lengths selected for a vessel should be of formed heads.within these limits. Warehouses usually do not stock Figure 5.7 illustrates various types of the more commonplates longer than 40 ft, and this length is usually carried formed heads whereonly in plate thicknesses of $Q in. or less and plate widthsof 72 in. or less. The plates of heavier gage (up to and “\ f = head thickness, inchesincluding 3 in.) and greater width are usually carried in20-ft lengths at the warehouses. The maximum plate ;/-.i icr = inside-corner radius, inches

length, thickness, and width that can be handled by thec

sf = straight flange, inchesshop fabricating the vessel may impose a limitation on thesize of plate that can be handled. P = radius of dish, inches

5.5 COMMON TYPES OF FORMED HEADS ANDTHEIR SELECTION

OD = outside diameter, inches

b = depth of dish (inside), inchesNearly-all formed heads are fabricated from a single

circular flat plate by spinning, as shown in Fig. 5.1, or by“drawing”i+itb dies in a press, as shown in Fig. 5.6.Detail a of Fig. 5.6 shows a single-blank plate being removedfrom a furnace for forming to a head, and detail b shows theplate in a press during the forming operation.

a = ID/2 = inside radius, inches

s = slope of cone, degrees

OA = overall dimension, inches

H = diameter of flat spot, inches

___ ._ -..--

Page 95: Process Equipment Design

The inside depth of dish and overall dimension, OA, maybe determined by use of the dimensional relationships forflanged and dished heads given in Fig. 5.8.

For purposes of welding heads to the shells of vessels,various styles of machined edges can be supplied on theformed head by the manufacturer. Standard machiningstyles for heads supplied by one manufacturer are shownin Fig. 5.9. It should be noted that in styles C6 and C7the dimension t ( the head thickness) must exceed dimensions (the shell thickness) by at least & in. and in styles D8,D9 and DlO the dimension t must exceed dimension s byk in.’ “Table 5.3 gives the cost extras for the variousstandard machining styles and applies to al l types of formedheads. Quantity differentials (1955) must be applied tot.he cost extras given in Table 5.3 as follows for types Aand B only: list plus 90 *h for 1 to 4 heads, list plus 50 CA for11 to 50 heads. All other styles, diameters, and gages arelist plus 90%.

5% Flanged-only Heads. The formed head mosteconomical to fabr icate is that produced by s imply forminga flange with a radius on a f lat plate. This head is identif iedas a “flanged-only head” and is illustrated in detail a of

Fig. 5.1. Various types of more com-

mon formed head,: (a) flanged only,

(b) Aangad and shal low d ished, (c)

tlanged and s tandard d ished, (d)

AWE and API-AWE code flanged and

dished (torispherical), (e)‘Jilliptical

dished (ellipsoidal), (f) hemispherical,

(g) fiunged and conical dished

[toriconical) (105). (Courtesy of Lukenr

Steel Co.)

Common Types of Formed Heads and Their Selection85 1/’

Fig. 5.7. The radius of a flanged-only head decreasessomewhat the abruptness in change of shape at the junctionof the flat head and the cylinder. The resulting gradualchange in shape reduces local stresses .

The flanged-only head finds its widest application inclosing the ends of horizontal cylindrical storage vessels a~atmospheric pressure. These vessels typically store I’urloil, kerosene, and miscellaneous liquids having low vab)()tpressures . Flanged-only heads may be used for the bot IWIIheads of vertical cylindrical vessels that rest on concrr~c?

slabs and do not have diamet,ers in excess of 20 ft. Table5.4 gives the straight-flange length and inside-corner radiusfor such heads as functions of head thickness. These hwtlsare fabricated on the basis of using the outside diameteras the nominal diameter. Head diameters based on theoutside diameter are available in increments of 2 in. from12 to 42 in., in increments of 6 in. from 42 to 144 in., andin increments of 12 in. from 144 to 240 in. A head wit.h a246-in. outside diameter is also avai lable .$\During @@i&&of the heads, thinning out of the plate I%curs at the corner radius. Therefore, for heads havingan outside diameter of under 150 in . , plate thicknesses must

rt

‘;T Inside depth of dishI r t

wOD----- ’(b)

+nside depth of dish /

cc-----+OD--------l

(d)

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86 Proportioning and Head Selection for Cylindrical Vessels with Formed Closures

Table 5.3. Cost Extras for Standard Machining Styles for Heads (105)(Courtesy of Lukens Steel Company)

Outside Diameter GageStyle A >p” g’ 1” 134” 1>4” 136” 2”

24” and under $2.00 $3.50 $4.00 $5.00 $5.5036” $4.50 $5.00 $5.50 $7.50 $8.00 &YOO &L5048” $5.50 $6.50 $7.00 $9.00 $9.50 $10.00 $10.5060” $7.00 $8.50 $9.00 $11.50 $12.00 $12.50 $13.00

72” $8.50 $9.50 $10.00 $13.00 $13.50 $14.00 $14.5084” $10.00 $11.50 $12.00 $15.00 $15.50 $16.00 $16.5096” $11.00 $12.50 $13.00 $17.00 $17.50 $18.00 $18.50

108” $12.00 $13.50 $15.00 $18.00 $19.00 $19.50 $20.00120” $14.00 $15.00 $17.00 $20.00 $20.50 $21.00 $22.00

-256” 3 I,

. . .. . .

$ii.ho$14.00

$12.00$15.00

$15.50 $16.50$17.50 $19.00$19.50 $21.00$21.00 $22.50$23.50 $25.00

$25.00$28.00$31.00$43.00$56.00

$26.50$29.50$33.00$46.00$60.00

$18.00$19.00$21.00$28 00$35.00

$19.50$20.50$22.50$30.00$37,00

$23.50$25.50$28 50$41.00$54.00

132”144”160”176”192”

Style B or C24” and under

36”48”60”

$21.50 $22.00 $23.00$23.00 $24.00 $24.50$25.00 $25.50 $27.00$35.00 $37.00 $39.00$45.00 $48.00 $51.00

$20.50$22.50$24 50$32.00$39.00

-

$4.00$6.00$7.50

$10.00

$2.50$5.00$6.50$8.50

$4.50$6.50$8.00

$10.50

$6.00 $6.50$8.50 $9.00 $ii.bO

$10.50 $11.00 $11.50$13.50 $14.00 $15.00

sii.50$12.50$16.50

$ii.bo$20.50

$i+.‘oo$23.00

72” $9.50 $11.00 $11.50 $14.50 $15.00 $16.50 $18.0084” $11.50 $12.50 $13.50 $17.50 $18.00 $19.50 $20.5096” $12.50 $14.00 $15.00 $19.00 $20.00 $21.50 $23.00

108” $13.50 $15.00 $17.00 $20.50 $21.50 $23.50 $25.50120” $15.00 $17.00 $19.00 $22.50 $23 00 $25 00 $27.00

$22.00$25.50$28.00$30.00$33.00

$25.50$29.00$33.00$37.00$39.50

$20.00$21.00$22.50$30.00$39.00

$21.00$22.50$24 50$32.00$41.00

$23 00$25.00$27.00$35.00$43.00

$24.00$25.50$28.00$38.00$48 00

$24.50$26.00$28.50$41.00$53.00

$26.50$29 00$32.00$45.00$58.00

$28.50$32.00$37.00$50.00$63.00-___

$36.00$39.00$44.00$57.00$70.00

132”144”160”176”192”

- - - -Style D

24” and under36”48”60”

$42.00$44.00$50.00$662.00$75.00

$7.50$11.00$13.50$17.50

$4.00$6.50$8.00

$10.50

$5.00$7.50$9.50

$12.00

$5.50$8.00

$10.00$13.00

$7.00$10.50$13.00$17.00

$ii. 50$14.00$19.00

$15’.bO$16.00$21.00

. .

$1;. bo$26.00

$21.50$29.50

72” $12.00 $13.00 $14.00 $18.00 $19 00 $21.00 $22.5084” $13.50 $15.00 0 $16.00 $20.50 $21.50 $23.50 $25.5096” $14.50 $17.00 $17.50 $22.50 $23.00 $25.50 $28.00

108” $15.50 $18.00 $19.00 $24.00 $24.50 $27.00 $30.00120” $17.00 $20.00 $21.00 $25.00 $26 00 $29.00 $32.00

132” $21.50 $25.00 $26.00 $27.00 $27.50 $31 .oo $34.50144” $22.50 $26.50 $27.50 $28.50 $29 00 $33.00 $37.00160” \ $24.00 $28.00 $29.00 $30.00 $31.00 $35.50 $40.00176” $35.00 $40.00 $45.00 $47.00 $50.00 $55.00 $60.00192” $46.00 $52.00 $61.00 $65.00 $70.00 $75.00 $80.00

$29.50$34.00$37.00$40.00$43.00

$35.00$38.00$41.50$45.00$49.00

$46.50$49.00$52.00$68.00$85.00

$52.00$55.50$61.00$77.00$93.00

be increased by >is in. for plates up to 1 in. in thickness and34 in. for plates 1 to 2 in. in thickness if the minimum platethickness is to be maintained throughout the corners. 3 I

The manufacturer’s catalog should be consulted forgreater thicknesses and diameters and for blank weight andforming costs.

5.5b Flanged Standard Dished and Flanged ShallowDished Heads. The pressure rating of a flanged-only headcan be increased if the flat portion is dished. Such heads,not designed to code specifications, are formed from a flatplate into a dished shape consisting of two radii: the “crown”radius or radius of dish and the inside-corner radius, some-

..--.-- - ---\I I\ \

Page 97: Process Equipment Design

Table 5.4. Dimensions of Standard Flanged-only

Heads for All Diameters

(Courtesy of Lukens Steel Company)Gage Standard Straight Inside-corner

(Thickness) Flange (in.) Radius (in.)t s.f icr

Ns l>h-2 3iss/a 1>+2>5 4i3i6 135-3 l5.;63 4 135-3 1%5i6 135-3 $5 15’/lG

!ci lf+-3j$ 1%34 lj$-3>5 1%39 13+3jg 2%74 1>84 2%

1 135-4 31% 1X-4% 344l?L 1>6-4j& 3%144 19j4ijs 4%1% 1>+4js 4%1% 1%4X 5%2 1 jq--4>5 6

times referred to as the “knuckle” radius. If the radius ofdish is greater than the shell outside diameter, the head&known as a “flanged and shallow dished head.” If the radiusof dish is equai to or less than theoutside diameter, the headis known as a “flanged and standard dished head.” Aflanged and shallow dished head is shown in Fig.-5.7, detailb, and a flanged and standard dished head is shown in Fig.5.7, detail c. These heads are fabricated on the basis ofusing the outside diameter as the nominal diameter. Headdiameters based on the outside diameter are avalable inincrements of 2 in. from 12 to 42 in., in increments of 6 in.from 42 to 144 in., and in increments of 12 in. from 144 to240 in. A 246-in.-outside-diameter head is also available.

It should be emphasized that because of the high localizedstresses due to the small inside-corner radius, the use offlanged and shallow dished heads and flanged and standarddished heads is not permitted in vessels which must meetpressure-code requirements.

Typical applications of these heads occur in the construc-

Common Types of Formed Heads and Their Selection 87

Fig. 5.0. Dimensional relationships for flanged and dished heads.

IDa=--

2

b = r - d(BC)2 - (Al+r

IDA0 = - - (kr)

2

BC = r - (icr)

AC = d(BC)2 - (ASIO A = t + b + s f

tion of vertical process vessels for low pressures, of hori-zontal cylindrical storage tanks for volatile fluids such asnaphtha, gasoline, and kerosene, and of large-diameterstorage tanks in which the vapor pressure and hydrostaticpressure is too great for the practical use of flanged-oniyheads. Vessels with flanged and shallow dished heads areprimarily used for horizontal storage tanks.

Table 5.5 gives the dimensions of flanged and shallowdished heads. Table 5.6 gives the dimensions of flangedand standard dished heads except for the radius of dish.The radius of dish varies with thickness and diameter, and

1

Style A

2 3 4 5 6 7 8 9 10I-.

Style I3 Style C Style 0 .’- .~ ,’I ,.

‘ I.Fig. 5.9. Standard machining styles for heads (105). (Courtesy Op Lukens Steel Company.)

I’

I ----~ / \ \ \I /

Page 98: Process Equipment Design

88 Proportioning and Head Selection for Cylindrical Vessels with Formed Closures

Table 5.5. Dimensions of Flanged and Shallow

Dished Heads in Inches(See Fig. 5.7.)

OD(Courtesy of Buffalo Tank Company)

6 6 72 76 84 9 0 9 6 1 0 2.__Gage icr - sf P r r r r r r

?4 35 3% 1 2 0 120 120 120 1 9 7 1 9 7 1 9 7?iS 54 338 1 2 0 120 120 120 1 9 7 1 9 7 1 9 734 N 4% 1 2 0 120 120 120 1 9 7 1 9 7 1 9 7%6 74 6 1 2 0 120 120 120 1 9 7 1 9 7 1 9 735 1 6 120 120 120 120 197 197 197

/f--lOD 108 114 120 126 132 138 144~~.... ~-- _ _ _Gage icr - sf r r r r r r r

$4 !i 355 1 9 7 197 3001 9 7

946 36 3% 1 9 7 197 300 300 3 0 0 3 0 0 3 0 0197 197

34 94 4.34 1 9 7 197 300 300 3 0 0 3 0 0 3 0 0197 197

746 74 6 1 9 7 197 300 300 3 0 0 3 0 0 3 0 0197 197

$5 1 6 197 197 300 300 300 300 300197 197

46 1% 3 0 0 3 0 0 3 0 0

the manufacturer’s catalog should be consulted for thisdimension, blank weight, and forming costs.

5.5~ Flanged and Dished Heads (Torispherical) toASME Code. The pressure rating of flanged and dishedheads can be increased by decreasing the local stresseswhich occur in the inside corner of the head. This may beaccomplished by forming the head so that the inside-cornerradius is made at least equal to three times the metalthickness; for code construction, the radius should in nocase be less than 6% of the inside diameter. Also, theradius of dish may be made equal to or less than the diam-eter of the head. Figure 5.7, detail d shows a sketch of across section of a flanged and dished head meeting theASME Code, in which is identified as a “torispherical”head. These heads are fabricated on the basis of using theoutside diameter as the nominal diameter. Head diametersbased on the outside diameter are available in incrementsof 2 in. from 12 to 42 in., in increments of 6 in. from 42 toI44 in., and in increments of 12 in. from 144 to 240 in.Heads having outside diameter of 210 in. and 246 in. arealso available.

The volume in cubic feet of heads having icr equal to6% of the outside diameter (not including the straight-flange portion)‘is approximately equal to:

V = 0.000049d;3 \,I (5.11)

where di = inside diameter of vessel, inches1’ = volume of torispherical dished head to straight

flange, cubic feet

Heads of this type are used for pressure vessels in thegeneral range of from 15 to about 200 lb per sq in. gage.These heads may be used for higher pressures; however,for pressures over 200 lb per sq in. gage it may be more

economical to use an elliptical hanged and dished head.These heads are used principally for vessels designed t(J

meet the ASME codes for unfired pressure vessels. Ingeneral, these heads are used either for horizontal or ver-tical vessels for a great variety of process equipment withinthe pressure ranges specified above. For pressures in therange of 150 lb per sq in. gage and for higher pressures, acost comparison should be made between the code flangedand dished heads and the code elliptical dished heads.The optimum choice based on total cost varies with pres-sure, diameter, thickness, and material of construction.

Table 5.7 gives the inside-corner radius and radius of dishfor code flanged and dished heads. Table 5.8 gives .thestraight-flange length for different head thicknesses offlanged and dished heads.

For purposes of cost estimation it is necessary to know theblank weight in order to obtain the cost of the steel usedand the cost of forming the head at the fabrication plant.The approximate blank .diam;fter may, be determined byuse of the following relatronshrps:

.-.__ _.. . .diameter = OD + s + 2sf + Qicr

. . ~~(for gages under 1 in.) (5.12)

diameter = OD + g + 2sj + $icr + t

(for gages 1 in. and over) (5.13)

where OD = outside diameter of dish, inchessf = straight-flange length, inches

icr = inside-corner radius, inchesf = gage thickness, inches

Table 5.6. Dimensions of Flanged and Standard

Dished Heads(Courtesy of Lukens Steel Company)

Thickness(in.)

t%6

?4

Inside-cornerRadius (in.)

icr

956%

l?is1%1x61%.1742%2%3-3js3 %4%4354165w5%6

Page 99: Process Equipment Design

Gco

Z- I I I I I I I I

2 ‘+llli1~xi&~ I+ I I I l ll$$

?ll’hI!! ’- - l I : I ’ ! I I I I+I-I

I 1 I 1 ; ! I I 1 I I ’ I

I

/

Page 100: Process Equipment Design

9 0 Proportioning and Head Selection for Cylindrical Vessels with Formed Closures

Table 5.7. Dimensions of ASME Code Flanged and Dished Heads (Continued) -T

-40 42 48 54 60 66

i c r

*r

I

I

+4

c4%i- -- - -- -- -- -- -- -- -

9

72

i c r

4%A

II

/

i4 %-4%3%-5%-5%-6-6%-7%-Wi9

i c r

3 %a

I

JI3 %

c3%- -- -- -- -_ - -- -- -- -- -9

i c r i c ri c r

iI

I

I

I

i33%- -

- - ---_- -- - .- -. - -- -- -

Wi-9

r r r rr

40k

I

f4036

t

--I-- t --1. --I-- -1_

!-+-:36

rr

60k

I

JI6054I

I

I-t-- - .I -- -

--t-- -

T -54

2%k

III1

2%33%

- - .- -- - ---.- -. - -- - -_--- - -

8%

4240I

III

G4042

_- --+-- I -- - -- -

- tIt:

i- -42

48

t4842

"iIII- -

-$I--4%481-- --

iI+- --I--+ -48

3%

k

IIi3%3 %- - ..--- -- -- - -- - -- -- -- - -. - -9

YI95448

"i

IA.-t-- t -

7_- _.4’4h-A --i-54

66

1v6660

kII

I-r--I-- t-t

z60

724

I

i7266

!IIIIIIt66

84,

r

11478 ! l(s

i c r

5%AIIIIIII

i5%

~6- - -- - -- - -9

i c r i c r i c r i c ri c r r r r r r r

5%

3IIIIII15%5%- -,--- - .--..--9

84A

I

Ii8478kI

I3--I---o-78

9696906

III

$I9084

fI

.I_-+--!4-

96

t

IIi

9690

1I

II

- l -- v-

90

5 %k

I

It5%

-5%- -- - -- - -- -9

78787872

1

I

/I

1-1- ---L_- --472

909084

tIIII

II

z.1---l--

6%aIIII

I

It

6%6%----A9

108 ’6

IIIIIt

108102

k

II

0LO2

102

k

I$

1029 6

6

II

-t- - -9 6

\ ,-~-~ _ _ -. _ - -.- -..-

Page 101: Process Equipment Design

WcQ-4xx+-----------g-.8

sz5 z-E+--c 0

--------*=0

WOCO-----

x

------oa

WC0wf-----~-----~ 4:x A 15’

sir-0--------->~

0

WC0

x--------v----d

i?w’- --------E

5-e ------*=0

+-----------,G YEl

p-----------g-g- -----+zx

&----+g0

+---------- --,E5

p------------z k-5x

b’3

Page 102: Process Equipment Design

9 2 Proportioning and Head Selection for Cylindrical Vessels with Formed Closures

m 2505=B

I

Note: Multiply costs by 1 :factor gwn in table 5.10.

II

0 , I , / I

1 0 20 30 40 50 60 7 0 80 90 1 0 0 1 1 0 1 2 0 1 3 0 1 4 0Head, outside diameter, inches

The weight of material may be calculated from the blankdiameter, gage thickness, and density of steel (490 lb percu ft). The cost of the steel may be obtained from Appen-dix C. Table 5.9 gives the cost of forming per head forheads having an outside diameter of up to 48 in. F i g u r e5.10 presents the cost of forming per head for larger-diam-eter heads. Table 5.9 and Fig. 5.10 are based on 1955forming costs and include the quantity differential thatapplies when two to four heads are purchased. Table5.10 gives the quantity-differential pricing factors for thepurchase of other quantities of heads.

If the straight flanges are machined for welding by thefabricator, Table 5.3 with appropriate quantity differen-Gals may be used for estimating the machining cost’ extras.If heads having straight f langes in excess of s tandard lengths

15

Fig. 5.10. Approximate 1955 cost of

forming per head for ASME code

flanged and dished heads if two to

four heads are purchased.

'0

are desired, an extra equal to 5y6 of list price fcr eachadditional !6-in. flange length is charged.

To determine the total cost of the head at the mill, thecost for the steel blank is first determined. To this isadded the forming cost, the machining cost, quantity dif-ferentials, and any extras for ordering flanges longer thanstandard.

5.5d Elliptical Dished Heads Meeting ASME and API-

ASME Code Saecifications. The elliptical dished (ellip-soidal) heads are used in preference to code flanged anddished heads formed with two radii for many vesselsdesigned for pressures in the range of 100 psi and for mostvessels designed for pressures over 200 psi. The ellipticaldished heads are formed on dies in which the diametricalcross section is an ellipse. If the ratio of major to minor

\I /-.----,_ - .---_-._

Page 103: Process Equipment Design

Common Typesof Formed Heods and Their Selection 93

Table 5.9. Typical Forming Costs as of 1955 for ASME

Code Flonged and Dished Heads Hoving Diameters up

to 48 Inches

Prices per Head for 2-4 Heads*

Table 5.8. Typical Stondord Straight Flange for ASME

Code Flanged and Dished Heads

RecommendedMax

StraightFlange,

in.233 %4 %6

66888888888888888

Notes on MaxStraight Flange

3” for 60” diam3” for 60” diam3” for 96” + 10” diam3” for 126” diam4” for 132” - 144” diam335” for 156” diam5” for 168” diam3” for 180” diam and above3” for 180” diam and above3” for 180” diam and above3” for 180” diam and above3” for 180” diam and above3” for 180” diam and above3” for 180” diam and above3” for 180” diam and above3” for 180” diam and above3” for 180” diam and above3” for 180” diam and above3” for 180” diam and above3” for 180” diam and above3” for 180” diam and above3” for 180” diam and above3” for 180” diam and above

Head, ODThickness 12-18 20-24 26-30 32-36s6- >i 9.61 10.54 11.63 12.40x6- 46 10.23 10.85 12.40 13.95x6- $5 11.16 12.40 13.95 16.2836 12.40 13.18 15.50 17.0538: 15.50 17.05 20.15 22.48

I 23.40 30.23 33.15 37.05136 -1’4 35.10 40.95 46.80 52.65l$h - l!& 54.60 60.45 66.30 72.15Ia, --134 87.75 92.63 97.5017/x - 2 117.00 117.00 121.882q 136.50 136.50 146.252’; 165.75 177.45;? 3 i 210.60 220.353

* Zlultiply by factors in Table 5.10.

36-42 48.~1 5 . 5 0 1 7 . 8 21 7 . 0 5 20.1519.38 23.2521.70 24.802 6 . 3 5 3 1 . 0 04 2 . 9 0 4 8 . 7 55 8 . 5 0 6 4 . 3 57 8 . 0 0 8 3 . 8 5

1 0 2 . 3 8 1 0 7 . 2 5126.75 132.60154.05 161.85189.15 200.852 3 0 . 1 0 2 3 9 . 8 5

278.85

axis is 2: 1, the strength of the head is approximately equalto the strength of a seamless cylindrical shell having thecorresponding inside and outside diameters. For this rea*son most manufacturers have standa d’ ed on ellipticaldished heads having a 2 : 1 ratio of axis.&‘he inside depth

-

i

Fig. 5.11. Horizontal storage tanks with elliptical closures on the right and a hemispherical closure on a vessel in the left foreground.

Brow a CO.)iCmriesy of C. F.

Page 104: Process Equipment Design

94 Proportioning and Head Selection for Cylindrical Vessels with Formed Closures

Table 5.11. Standard Straight Flanges for Available

ASME Code Elliptical Dished Heads(Courtesy of Lukens Steel Company)

Table 5.10. Typical Quantity-differential PricingFactors for Formed Heads

Based on 1954 and 1956 Prices and on Fig. 5.10 and Table 5.9

Heads up to48” OD and48” ID for

Gages up to76” InclusiveMultiply by1 9 5 4 1 9 5 6

1.32 2.201.00 1.650.92 1.500.775 1.300.71 1.200.645 1.050.58 0.95

Heads up to48” OD and48” ID forGages over

?4“, andHeads over48” OD and48” ID forAll Gages

Quantity1 head of a size2 to 4 heads of a size5 to 10 heads of a size

11 to 20 heads of a size2 1 to 50 heads of a size5 1 to 60 heads of a sizeOver 60 heads of a size

Multiply by1954 1 9 5 6

1.05 2.201.00 2.100.897 1.850.795 1 650.720 1.500.615 1.300.565 1.20

of dish is half of the minor axis and is equal to one fourthof the inside diameter of the head.p

The right side of Fig. 5.11 shows elliptical heads 1 We in.thick on the ends of butane-storage tanks 144 in. in diam-eter by 120 ft long. These tanks are rated at 100 lb persq in. gage at 400’ F.

Figure 5.7, detail e, shows a sketch of a cross section of anelliptical dished head. These heads are fabricated on thebasis of using the inside diameter as the nominal diameter.

Blank diameter, inches

DiameterAvailable Standard Gage

546

5 %?‘iS

wNN%

11%1%1%1%1%1%1%2

(in.) sf (in.) -(in,)

12- 36 2-2x 2%1% 42 2-3 2%12- 66 2 - 3 %i

12-120 2-33512-120 2-3x bi12-126 2-3x 3 %12-138 2 4 3 %12-156 2-4 412-180 2 - 4 4%12-192 2-4s 4 %12-204 2-434 4%12-216 241,-4x 512-216 2%-4x 5%12-216 2x6-4% 5%12-216 2%4% 5%12-216 2is6-4% 612-216 34%

DiameterAvailable Standard

(in.) sf (in.)

20-204 2x4~26-192 3%-p%26-186 4%4%34-174 4%34-168 4x-534-156 5X-5%34-138 5%-648-132 654-126 6%-6M54-120 6N-760-120 7?+-7%60-114 7 %72- 96 7%-872- 96 w-8%‘i’2- 96 8%-972- 96 9

Head diameters based on the inside diameter are availablein increments of 2 in. from 12 to 42 in. and in incrementsof 6 in. from 42 to 216 in. Table 5.11 gives the standardstraight flanges for various elliptical dished heads. Themanufacturer’s catalog should he consulted for the maxi-mum straight flange available at extra host. The volume

2 0 3 0 40 50 60 70 8 0

1

110

100

90

80

70

60

I0 70 80 90 100 110 120 130 140 150 160 170 180 190 20:’

Fig. 5.12. Blank dlamltc.: versus

head inside diameter for various

thicknesses of ASME ell iptical

heads.

Blank diameter. inches

Page 105: Process Equipment Design

Common Types of Formed Heads and Their Selection 95

600

Fig. 5.13. Approximate 1955cost of forming par head forASME coda elliptical heads iftwo to four heads are pur-c h a s e d .

Thickness 3’ /l-

I

I INote: Multiply costs byfactors in table 5.1~. 1 / 1 I

in cubic feet contained within the head, not including thestraight-flange portion, is approximately equal to:

-----T

--\ ___- - .- -., .._V = 0.000076 di3 (5.14)_ - -..-. --..-.. -_I __ I

where di = inside diameter of vessel, inchesV = volume of elliptical dished head to straight

flange, cubic feet

For cost-estimating purposes the procedure presented inthe previous section for code flanged and dished heads isfollowed; Fig. 5.12 is used for the blank diameter and Table5.12 and Fig. 5.13 are used for the cost of forming per headfor two to four heads. Table 5.10 may be used for deter-mining other quantity diflerentials.

5.5e Hemispherical Heads. For a given thickness,hemispherical heads are the strongest of the formed heads.These heads can be used to resist approximately twice thepressure rating of an elliptical dished head or cylindricalshell of the same thickness and diameter. The degree offorming and accompanying costs are greater than for anyof the heads previously described; also, the available sizes

i0Head, inside diameter, inches

Table 5.12. Typical Forming Costs as of 1955 for

ASME Code Elliptical Heads Having Diameters

up to 48 Inches

Prices per Head for 24 Head Lots*

Thickness 12-18 20-24 26-30 32-36 3842 48

x 12.40 13.95 17.83 20.15N-- x6 13.95 14.73 18.60 22.48 26.35 31.00KG-j6 13.95 17.05 20.93 24.80 29.45 34.10

x 15.50 19.38 23.25 27.90 33.33 38.75x 20.15 24.80 31.00 35.65 41.85 48.05

1 30.23 35.10 44.85 50.70 58.50 66.30l>+lK 37.05 46.80 52.65 58.50 66.30 75.081%--l% 46.80 50.70 60.45 70.20 79.95 88.731%-1x 66.30 70.20 79.95 91.65 101.40 109.201x-2 85.80 91.65 101.40 111.15 120.90 132.602% 115.05 124.80 138.45 148.20 159.902% 150.15 173.55 183.30 195.002% 189.15 212.55 224.25 235.953 247.65 259.35 271.05* Multiply by factors in Table 5.10.

Page 106: Process Equipment Design

PC Proportioning and Head Selection for Cylindrical Vessels with Formed Closures

Fig. 5.14. Flanged hemispherical head 168 in. in inside diameter with

3-tn. straight flange weighing 16,500 lb and constructed from seven

formed sections. (Courtesy of Lukens Steel Company.)

formed from single plates are more limited. One-piecespun hemispherical heads available from one fabricator aregiven in Table 5.13. A great variety of hemispherical-head diameters and gages are available in segmental formand can be field or shop welded. Figure 5.14 shows ahemispherical head of a 168-in. inside diameter fabricatedfrom six plates dished to shape over ‘an 84-in. radius formerand then welded together with a dished head at the end tocomplete the hemisphere.

5.5f Conical and Toriconical Heads. Conical heads arewidely used as bottom heads for a variety of process equip-ment such as evaporators, spray driers, crystallizers, andsettling tanks. The particular advantage of the use ofconical bottoms lies in the accumulation and removal ofsolids from such equipment. Cones having an angle atthe apex of 60” are commonly used for the removal ofsolids. Greater angles may cause accumulation of solidsas a result of the frictional resistance between the solidsand the inside cone surface. Another common applicationof conical sections is for changing the diameter of cylindricalshells; this is often necessary in some fractionating-columndesigns. Figure 5.3 illustrates such an application.

Conical sections can be formed on the rolling equipmentused to roll cylindrical shells. This is done by spreading

the rolls on one end according to the angle of the cone.The smaller opening of the conical section must be largeenough to accommodate the diameter of the bending roll,which usually is 4, 8, 10, or more inches in diameter. Thegreater the apex angle of the cone, the larger the smalleropening must be in order to accommodate the roll.

Toriconical heads differ from simple conical heads in thatthey have a radius at the flanged end, as illustrated indetail g of Fig. 5.7. Toriconical heads may be formed fromflat plates in the same manner as flanged and dished heads.They are more expensive t,han simple conical heads butare better suited for pressure-vessel applications because thelocalized stresses near the junction of the cone and theshell are more uniformly distributed in the toriconical sec-tion. Thus the concentrated localized stresses which wouldexist without the knuckle radius are greatly reduced.

Spun toriconical heads are available from manufacturersin a variety of sizes ranging from 30 to 198 in. outsidediameter (or inside diameter) in 6-in. increments for includedangles of 90” and 120°, and in gages from M in. through2 in. Included angles of 114’ can also be obtained indiameters of from 42 to 216 in. and, included angles of140’ in diameters of from 66 to 240 in. Diameters of240 in. are available in each of the above angles. Pressedsegmental cones having included angles of 60”, 75“, and90” are also available in a variety of sizes.

5.5g Some Oiher Types of Formed Heads. Three typesof formed heads not previously described are shown inFig. 5.15. The flanged and reverse dished head shown

Table 5.13. Dimensions of Available One-piece SpunHemispherical Heads

(Courtesy of Lukens Steel Company)ID (in.)

1 2Wi614%1534l@%t517%18352324.W%28%303435363841%4247%485034546064%7294

ThicknessLightest to Heaviest

% 1%w 1%36 1%96 1%N 1%N 1%N 1%N 1%N 1%N 1%M 1%; 2

21,s34 35% 3% 3; 3 3

94 336 3% 334 3%94 3 ;,4% 3%% 3%952 31,g

sj (in.)

o-2o-2o-2o-2o-2o-2o-2o-2O-2%04%o-2 350-254@-wio-3o-3o-3o-3o-3o-3o-3o-3o-3O-3%O-3%0-3x0-3x

Page 107: Process Equipment Design

in detail a of Fig. 5.15 is often used on the ends of vesselsintended to face against a plane surface such as the bottomof steel vessels resting on concrete slabs. In this respectthis type of head has the advantage of a bearing surface inone plane and also the additional strength resulting fromdishing. The radius of dish, R, is equal to .the outsidediameter of the vessel, and the heads are available in Z-in.increments from 18 through 24 in. outside diameter and in6-in. increments from 24 through 132 in. outside diameter.A head 144 in. in diameter also is available. The gagesvary from x~ in. through 1 in.

Dished-only heads such as shown in detail b of Fig.5.15 can be used as the center piece of built-up hemi-spherical heads of large sizes andmay also beusedin speciallydesigned equipment where flanges are undesirable. Suchheads are available in a large number of sizes, from 12through 24 in. in increments of 2 in. on the outside diam-eter, from 24 through 144 in. in increments of 6 in., andfrom 144 through 180 in. in increments of 12 in. Gagesrange from xs through 3 in.

Flared and dished heads, shown in detail c of Fig. 5.15,may be used for cover plates of kettles and hoppers andin specially designed equipment. Such heads are availablein sizes from 18 through 132 in. inside diameter in incre-ments of 6 in. The corresponding outside diameter is equalto the inside diameter plus 6 in. The radius of dish is

Common Types of Formed Heads

equal to the inside diameter.x6 in. through 2 in.

and Their Selection 9 7

Gages are available from

Fig. 5.15. Some other types of formed heads: (a) flanged and reverse

dished, (b) dished only, (cl flared and dished (105). (Courtesy of Lukens

Steel Co.)

d P R O B L E M S

1. A horizontal vessel 10 ft in diameter and 60 ft long is to be fabricated from ASTM-A-283,Grade C plate 54 in. thick by code welding. Determine the optimum number of plates for theshell and the plate width. (See Appendix C, item 4, section g.)

2. Heads for six vessels approximately 48 in. in diameter for 190 psi, service are required.Determine the cost of 12 elliptical dished heads 48 in. in inside diameter and 44 in. thick, andcompare it with the cost of 12 torispherical heads 48 in. in outside diameter and s in. thick.(Roth heads are suitable for 190 psi service.) Steel is to be ASTM-A-285, Grade C, and heads

I are to be shipped with standard straight flanges.

Page 108: Process Equipment Design

C H A P T E R

STRESS CONSIDERATIONS IN THE SELECTION

OF FLAT-PLATE AND CONICAL CLOSURES

FOR CYLINDRICAL VESSELS

0 n the fabrication of process equipment flat plates or conesmay he used as closures for cylindrical vessels because suchclosures are easily formed with conventional shop equip-ment, but their use usually is limited to low-pressure serviceor to closures for small-diameter vessels. Flat plates areoften used as closures for hand holes, manholes, andso on.

The sharp discontinuity in shape existing at the junctionof a cylindrical vessel and either a flat plate or a conicalclosure results in localized stress concentrations at thejunction. In low-pressure service, where the magnitudeof these stresses is low, they often may be disregarded.However, for adequate evaluation of a design, a knowledgeof the magnitude of these stresses is essential.

The nature of the stress concentrations is complex in thatbending moments, shear, and stress reversals must be con-sidered in addition to the membrane stresses resulting frominternal pressure.

The use of a flat plate as a closure often results in theplate’s being considerably thicker than the cylinder towhich it is attached. The difference between the flexi-bility of the flat plate and of the cylinder results in the twoparts of the vessel attempting to deform radially and angu-larly at different rates under the influence of internalpressure. This movement is prevented by the rigid junc-ture of the two elements and results in shear and flexuralstresses which may be quite severe. Similar but lesssevere localized stresses result when conical heads are usedas closures. In addition to the juncture stresses, bendingstresses in the central portion of the flat closure may alsobe critical.

9 8

6.1 RELATIONSHIPS BASED ON THE THEORY OFELASTICITY

6.1 a Stress-Strain Relationships. By definition themodulus of elasticity, E, is the slope of the straight line orelastic portion of the “stress-strain” diagram (see Fig. 2.1)or by Eq. 2.3:

EJ (2.3)e

where j = stress, pounds per square inche = unit strain, inches per inch

E = modulus of elasticity, pounds per square inch

Therefore, if the elastic limit is not exceeded, elasticdeformation occurs under induced stresses. The amountof deformation or “strain” is simply related to the inducedstress by the above relation. Thus

When a specified segment of metal is loaded in one direc-tion only, with resulting induced stress and correspondingstrain, strain is also induced in a direction or directions at.right angles to the induced stress. For example, a tensile-test specimen elongates under tensile load and reduces itsdiameter by lateral contraction.

Experiments have proved that such axial elongation isrelated to the corresponding lateral contraction. The ratioof these two deformations is a constant within the elasticlimit and is known as Poisson’s ratio. This ratio may be

Page 109: Process Equipment Design

Relationships Based on the Theory of Elasticity 99

And the corresponding net unit elongation, ~2, in the ydirect ion wi l l be :

expressed as fol lows:

EC-=p (6.2)c

wh& eC = unit lateral contractionc3 = unit axial elongation~1 = Poisson’s ratio, a constant depending upon the

material (0.3 for structural steel)

This relat ionship may also be used to calculate the lateralexpansion resulting from axial compression of a material.

In the case of a closed cylindrical vessel containing apressure, the shell may be considered to be subjected tothree forces . One force results from the pressure pushingagainst the heads. This load on the heads is transmittedto the shell in an axial direction and therefore results in alongitudinal tensile force and tensile stress being set up inthe shell. A circular or “hoop“ stress is also induced inthe shell by the contained pressure acting against the cir-cular shell. A third stress exists in the radial direction,which may be disregarded in thin-walled vessels . The twoprincipal tensile stresses act at right angles to each otherproducing a two-dimensional s tress condit ion.

The resulting elongation in one of these directions willdepend not only upon the tensile stress in this directionbut also upon the stress in the perpendicular direction.If one refers to one direction as z and the other as y, theunit elongation by Eq. 6.1 in the x axis direction due to thetens i le s t ress , ji, wi l l be :

E fi2=-E

(6.la)

The tensile stress in the y direction, jr,, will produce anelongation, e2/, in the y direction and a lateral contraction,6, in the x direction, as indicated previously.

E fEl= 2

E(6.lb)

The accompanying contraction, eC, in the x direction willequal PQ, (because t, is equal to the ey resulting from j2/) or

Therefore

E, = e2/ (for f J

6 = Pqj (from Eq. 6.2)

.f!lcc=/.-E

If both stresses ji and j, are acting simultaneously, thenet unit elongation, ~~2, in the x direction will be (the sub-script 2 refers to the net stress or strain for the biaxiall o a d i n g c o n d i t i o n ) :

c2AL&!z E Eo r

(6.41

fi2 = EE,Z = fi - pjg (6.4a)

.f, fze2/2 = - - p -E E (6.5)

o r.f,z = Eq,2 = .f, - ~jz (6.5aj

Equation 6.4 can be combined with Eq. 6.5 to give a pairof useful equations in which the stresses ji: and jr, are func-

tions of the strains e22 and ~2 as follows. f,Noting that -E

in Eq. 6.4 may be substituted for by its equivalent from

Eq. 6.5, we find that from Eq. 6.5 f fi2 = Ey2 + p -.E E

Substituting in Eq. 6.4, we obtain:

Expanding we obtain:

o r

*fcz2 = I2 - pry2 - /.p -?

E E

cz2E = f.z - 13,2E - cc2fz

Factoring we obtain:

cz2E.= .fi(l - p2) - peZ12E

S o l v i n g f o r ji, ,we obtain:

o r

ez2.E + w,zEfz= I-

P2

jx = (%2 + wu2w

1 - p2 (6.6)

In like manner it may be shown that

jY

= (~2 + wzz)E1 - /.Ls (6.7)

Equations 6.6 and 6.7 are convenient relat ionships givingthe stresses in two direct,ions perpendicular to each otherin terms of the respective strains resulting from thestresses, Poisson’s ratio, and the modulus of elasticity ofthe material in question. It is assumed that. the materialis homogeneous and the deformations are all within theelas t i c l imi t .

6.lb General Bending Relationships. It has previ-ously been shown by Fig. 2.3 and by Eq. 2.6 that the unitelongation in a deflected beam is equal to:

Substituting for l/r as given by Eq. 2.13, we obtain:

d2y%=Ys (6.8)

An elemental strip of a plate under deflection can becompared to a simple beam, and Eq. 6.8 can be used to

Page 110: Process Equipment Design

100 Selection of Flat-Flate and Conical Closures for Cylindrical Vessels

express the unit strain in terms of the radius of eur~aturrof the deflected plate. In reference to Fig. 2.3, the unitstrain of an elemental strip in the z direction is assumed tobe negligible. Rewriting Eqs. 6.4, 6.5, and 6.6 for theelement shown in Fig. 2.3, we obtain:

fi fiE,2 = - - /.-E E

(6.9)

(6.11)

hut ifE,2 = 0

then

jz+P2

(6.12)

Substituting Eq. 6.8 for E, in Eq. 6.12, we obtain:

(6.13)

For the strip of the plate in the J: direction having thirk-ness, t, and unit width in the z direction, Eq. 2.7 ma! bemodified to:

M =/+;f j-g dA (6.14)

Substituting forji by use of Eq. 6.13, we obt.ai!::

But by Eq. 2.8

I

+t/2I 5= y2 dA :;. 8’

-tra

and for a unit skin

t’; s -12

Therefore

Let

M = _ E t3 d2y~--1 - p2 12 dx2

D = Et312(1 - p2)

where D = flexural rigidity; then

(6.15)

A comparison of Eq. 6.16 and Eq. 2.14 shows that D, theflexural rigidity of a plate, is equivalent to the quantityEZ for beams. It therefore follows that the Eqs. 2.15and 2.16 for beams may be modified for plates by the sub-

and

(6.18)

6.1~ Bending Relat ionships in a Circular F lat P late.

Referring to Fig. 2.3 and considering it to represent a stripin a circular plate, we find that the dishing of a circularflat plate under uniform pressure will result, in curvaturesin both the z and z directions. In reference to Eq. 2.6the unit strains map he written:

Substituting for t.he straius in Eqs. 6.6 autl 6.i, we find that

jz

=: (w + wz2W = -3.

1 - /.l2(6.19)

(6.20)

Figure 2.3 also shows that the strain and corras~on&ngstress is zero at the midplane and is at a maximum at theouter fibers. The effect of the stress on either side of themidplane is to produce a couple which may be expressedas a bending moment. Equation 2.7 may be applieci knoth the x and z directions to determine the bendingmoments from the combined stresses as follows (see pa@41 of Peferonce 107) :

I Il.st,_ o.st~ jzy dy dx = M, d3: (6.22)

Substituting Eqs. 6.19 and 6.20 for jJ and j, in Eqs. 5.21and 6.22, respectively, integrating, and substituting l”;;q.6.15 (107) we obtain:

h uniformly loaded circular plate will dish in a sphericalmanner; t,herefore r, = r, = rzz, and M, = M, = MS+Therefore, Eqs. 6.23 and 6.24 reduce to:

1 MZZ-= ___-rxz W + PCL)

(6.25)

Referring to Eq. 2.13 and letting the radial distanre r.equal to x, we obtain:

1_ = 3rx

(6.26)

Page 111: Process Equipment Design

Relationships Based on the Theory of Elasticity 101

Referring 1.0 Pig. 6.1, by similar triangles, we obtain:

01rz dl

1 djP dr

(6.27)rz

Substituting Eqs. 6.26 and 6.27 into Eqs. 6.23 and 6.21,we obtain:

(6.28)

(6.29)

/‘wIltIre Mz is the bending moment prr unit lengt,h along the? circumferential section, and M, is the bending moment pe1

iunit, length along the diametral section of circular plate.In reference to Fig. 6.2, a circ*umferential section of a flat

rover plate having a uniformly distributed load, p, isdesignated as element ah-d. A summation of the forcesresulting from the bending moment may be taken aboutthe sides ad and bc. Arc length acl = r d+, and arc lengthhc = (r + dr) d+

ad couple = M,r d+ (6.30)

M, + f$ d r (r + dr) dr$I

(6.31)

The sides ab and cd have couples which are each equal toMz dr, and they have a resultant in the z-y plane equal to

M, dr d+ (6.32)

The symmetry of the element results in no shear on thesides ab and dc. The shear per unit length, &, times thelength of the arc ad gives tbe tot.al shear on the arc and isequal to &r d+. The tobl shear on the side bc is correspond-ingly equal to

(6.33)

Fig. 6.1. Deflection in a dished circular plate.

If t.he small differences in shear on these two sides aredisregarded, t,hese two forces result in a couple in the x-yplane equal to

- Qr d+ d r (6.34)

Summing up the couples in the x-y plane wit.11 properregard for sign, we find that (107)

M, + $< d rI

(r + dr) d+ - M,r d+ - &r d+ d r

-M, d r d+ = 0 (6.35)

Disregarding small high-order quantities, we find that

M, + $f r - M, - Qr = 0 (6.36)

Subst.itut.ing Eqs. 6.28 and 6.29 into Ey. 6.36, we obtain:

(6.37)

Fig. 6.2. Bending moments in a

circular flat plate.

IY

L,I Y

View A - A

Page 112: Process Equipment Design

1 0 2 Selection of Flat-plate and Conical Closures for Cylindrical Vessels

Equation 6.37 may be written as (107) : Combining terms and factoring gives:

(6.38) (6.48)

The shear per unit length of circumference, &, at anyradial distance, P, in a uniformly loaded circular plate is:

force pm-2 prQ=-===-circumference 2ar 2

(6.39)

Substituting Eq. 6.39 into Eq. 6.38 gives:

The maximum defection will occur at the center of theplate, where r = 0.

(6.49)

To determine the stresses in a flat cover plate, Eqs. 6.28and 6.29 are used with the substitution of Eq. 6.48 for y,by which(6.40)

f (1 + P) - r2(3 + P)I

(6.50)Integrating once gives:

Id dy--(r-)=g+Cor dr dr

(6.41) and

$ (1 + j.~) - r2(1 + 3~)I

(6.51)Multiplying both sides of the equation by r and integratingagain gives: At the edge of the plate, where r = d/2,

(6.42) (6.52;

Dividing by r and integrating again gives:(6.53)

y = f$ + 7 + Cl log r + c2 (6.43)At the center of the plate, where r = 0,

UNIFORMLY LOADED FLAT PLATE WITH THE EDGES

CLAMPED. The constants of integration, CO, Cr, and Csmust be evaluated from the boundary conditions at theedge of the plate. For a circular plate with the edgeclamped, the slope of the plate at the center and at theperiphery is zero. Therefore,

(6.54)M, = M, = (1 +6;)Pd2

The maximum moment is at the edge, as indicated by thecomparison of Eqs. 6.52, 6.53, and 6.54. The maximumstress may be determined by substituting Eq. 6.52 intoEq. 2.10.($)i=o=o= [f&+T+tqrzo (6.44) f=!$!f

andwhere z for a unit-width rectangular strip of the flat platelocated on the center line is equal to th2/6. Therefore,1 (6.45)

r=(d/2)

From Eq. 6.44, Cr = 0 and by substituting for Cl inEq. 6.45, CO is evaluated. Therefore,

c =-pd20320

Substituting for CO and Cr in Eq. 6.43, gives:

or

th = d db%)(P/f) (655a)

UNIFORMLY LOADED FLAT CIRCULAR PLATE SIMPLY SUP-PORTED AT THE EDGE. The condition of the circular platewith a clamped edge differs from the condition of a circularplate freely supported by the bending moment at the edgeof the plate as given by Eq. 6.52. If this bending momentis removed, spherical dishing will result.

The radius of curvature, rZ, of a spherical dished platemay be determined by reference to Fig. 4.3 and Eq. 4.1.

A2 = BD - B2 (4.i;

For small values of B, the term B2 may be disregarded.Therefore A2 = BD (approximately). Using the notation

for circular plates, A = r, B = y, D = 2r,, gives:

r2 S 2yr,

y _ pr4P d2r2640 1280+c2

(6.46)

Applying the condition that y = 0 at r = d/2 (for aclamped plate) and solving for C2 gives:

(6.47)

Substituting for C’s in Eq. 6.46 gives:

y _ pr4P d2r2640 1 . 2 8 0

+ pd410240

Page 113: Process Equipment Design

Relationships Based on the Theory of Elasticity 1 0 3

CONCENTRIC L OAD ON A FLAT C IRCULAR PLATE. A con-centric load on a flat circular plate creates a bendingmoment, Ml, and a deflection, y, which dishes the plate.

Equation 6.27 gives the radius of curvature of a spher-ically dished cylindrical plate in terms of r.

1_= dyr dr

Rewriting we obtain:

idI dy = /;2 d rr 1

s

d/2-=- r dr

rz rz r

therefore

therefore

d2Y - -

max - 8r,(6.56)

To determine the deflection at any point, r,

Substituting for l/r, by Eq. 6.25, we obtain:

(6.57)

By Eq. 6.52

(6.58)

Equation 6.58 is the deflection equation for pure bendingof a circular plate with a bending moment at the edge equaland opposite to the bending moment of a uniformly loadedcircular plate with clamped edges. If the deflections forthese two conditions are combined by superposition, thedeflection of a uniformly loaded circular plate simply sup-ported at the edges will be obtained. Thus, Eq. 6.48 may

* be combined with Eq. 6.58 to give (107) :

,I‘--p E-g

Y= (A, ) [(z) (t? - r2] (6.59)

The maximum deflection occurs at r = 0; therefore

Ymax = ~ (6.59a)

The bending moments for a uniformly loaded flat circularplate simply supported may be obtained by adding theedge bending moment, pd2/32, to Eqs. 6.50 and 6.51 togive :

f (1 + p) - r2(3 + p) 1 + ‘g

therefore

Mz=~[C3+pl~-r2)] (6.60)

and

Mz = 5 f (1 + IL) - r2(1 + 3~)[ 1 + ‘g

therefore

: (3 + p) - r2(1 + 3~) 1 (6.61)

’ * The maximum bending moment occurs at the center ofthe circular flat plate when r = 0; therefore

M, = M, = ; $ (3 + /.I)0

(6.61a)

.,.The corresponding maximum stress is:

f6 M

max(r=O) = 7 =6pd2(3 + P>

th (16)(4)th2

(6.62)

But by Eq. 6.25,

1 MZ-=rz Wl + PI

Substituting gives:

(6.63)

(6.64)

where y = deflection due to concentric load on flat circularplate

MI = moment caused by concentric load on flat cir-cular plate

6.ld Bending Relationships for a Beam on an ElasticFoundation. The deflection of an elemental longitudinalstrip of the shell or radial strip of the flat-plate closure issimilar to the deflection of a beam on an elastic foundation.Therefore, the relationships for such a beam will be dis-cussed. The deflection of such a beam at any point in thebeam will be proportional to the reacting force on the beamat the point in question.

Such a beam under a point load will deflect immediatelyunder the load because the supporting foundation is elastic.The stiffness of the beam will transfer a portion of the loadto either side of the force; this will resuldeflection which is a function of the k

in a smaller elasticresis ante of the founda-

tion and the distance from the point of load application.Let

5 = horizontal distance along the beamy = deflection of the beamw = resistance of the foundation per unit length at any

pointcl = constant depending upon the stiffness of the beam and

the resistance of the foundationE = modulus of elasticity of the beamZ = moment of inertia at the beam

M = bending moment at point z

For a beam on an elastic foundation, the deflections of theunloaded portion of the beam at any distance, 2, from thepoint of load application is given by:

yz = Cl% (6.65)

The resistance w of the foundation per unit length at any

Page 114: Process Equipment Design

Selection of Flat-plate and Conical Cfosures for Cylindrical Vessels

Qo Qo the load approaches zero as the distance from the pain‘.of load application increases. This reyuires that the con-stants A and B be equal to zero. Therefore

y = ewPx(C cos /3x + F sin 82) (6.69)

The constants C and F must be evaluated for the par-ticular conditions of the beam under consideration. In the

’ --b-=4following section these constants are evaluated for a cylin-

d drical shell with a flat-plate closure.

6.2 STRESSES IN CYLINDRICAL VESSELS WITH FLAT-PLATE.CLOSURES

Fig. 6.3. Forces and moment at junction of fiat cover plate and cylindrical

shell (108, 109).

where p = internal pressure, pounds per square inch gage

lh = thickness of flat cover plate, inches

1, = thickness of cylindrical shell, inches

Q. = shear force at junction per unit length of circumference,

pounds per inch

No = tensile force at junction per unit length of circumference,

pounds per inch

A&, = bending moment at junction per unit length of circum-terence, inch pounds per inch

d = diameter of cover plate and shell, inches

x = longitudinal distance along shell from junction, inches

yl = deflection of shell at junction, inches

ys = deflection of flat plate at junction, inches

I = radial distance along flat plate, inches

NI = head radial tensile force per unit length of circumfer-

ence, pounds per inch

MI = axial head bending moment per unit length of cir-

c u m f e r e n c e

poillt, zr, can be expressed in I erms of (1~ (Ir, bhe slope ofthis deflection curve.

(2.16)

In reference to Fig. 6.3, the internal pressure, p, actsupon the flat cover of t.hickness, lh, and upon the cylindricalshell of thickness, t,, to produce the forces QO and No andthe moment MO. The force Qo is the shear force per unitlength of circumference, which acts to restrain the shellfrom expanding and separating circumferentially from theflat cover plate. The force ‘lie is the axial tensile force perunit length of circumference resulting from the pressureload on the flat cover plate, which acts to separate thecover plate axially from the shell. The result of theseforces is the bending moment MO. The relations, basedupon the theory of elasticity, between QO and MO can bederived. A longitudinal strip of the shell in the neighbor-hood of the junction which is bent inward is selected foranalysis. The force causing this deformation can be con-sidered to be an inward radial shear force acting on theend of the strip. This force is resisted by the bendingforces set up in the strip and by the compressive hoop stressopposing a tendency for the shell circumference to decrease.The total resistance to this tendency to deform inwardresults in radial shear forces, longitudinal bending stress,and circumferential compressive stresses.

6.20 Bending in the Shell. The deflection curve of anelemental longitudinal strip of a shell resulting from theforces and moments in Fig. 6.3 is comparable to that of abeam on an elastic foundation. The general solution of theeyuation for the deflection curve for such a beam is givenby Eq. 6.69.

Substituting for u, in Eq. 2.16 by means of Eq. 6.65 andreii rranging gives :

d;y _ Ydx” - - clEI

(6.66)

lett.ing the yuantitb- l;‘Elc, = -f/Y’ gives:

(6.67)

y = eppZ(C cos /3x + F sin /3x) (6.69)

The two constants C and F can be determined from theconditions at the loaded end of the elemental strip. FromEy. 6.16

M” = (Mz)x:,o = --I~1 (6.70)ZCO

:~nd from Ey. 6.17Equation 6.67 hds the following solution (107):

.y = c@(A cos px + B sin PC) + Cpz(C cos /3r + F sin @r)(6.68)

The arbitrary constants must be evaluated from theknown conditions at certain points along the element. Thedeflection,’ y, of the beam at points greatly removed from

Qo = (Qr1.m = ($$)xzo = -Dl (>z=o (6.71)

where D1 =El,3

12(1 - /.42)

By subst,ituting Eq. 6.69 for .v in Eqs. 6.70 and 6.71 and

Page 115: Process Equipment Design

differentiat,ing, the constants C and F are determined.

(6 .73)The slope of the deflection curve at C C = 0 resulting from

the combined effects of bending and pressure stresses isobtained by differentiation of Eq. 6.80. This equation is

Therefore identical with Eq. 6.76.To eva1uat.e t,he constant /3 in Eq. 6.80 reference is made

,y = 2g [pMo(sin /%z - (‘OS /3x) - Qo (‘OS pz] to Fig. 6.4. Detail a of t,his figure shows a cross-sectional

(6 .74)view of a longitudinal elemental strip of a shell of radius r.The width of the element is h inches. The curvat.ure of the

The maximum deflection due to bending nccurs at the shell results in the forces .f.J, on each side of t,he element

, . end of the shell at the junction with the head. Therefore react,ing l ess t,han 180” apart,. This results in a component ,w, radially inward, which is normal t,o the surface. 1(See

(y)z=o = - & W~fo + Qo) (6 .75) detail h of Fig. 6.4.)1

The slope of this deflection curve at. this point is equal to:

&0z2 x=0 = L @PM0 + Qo)

2P2D1(6 .76)

The relationships as given in Eqs. 6.75 and 6.76 arelimited to the bending resulting from the reaction at t,hejunction of the shell and the closure. The shell will alsodeform as a result of the effect. of longit,udinal and circum-ferential stresses from internal pressures. These twostresses as determined by Eqs. 3.13 and 3.14 may be com-bined by the use Eq. 6.5 to give the radial deformationas fo l lows:

Stresses in Cylindrical Vessels with Flat-plate Closures 1 0 5

the posi t ive s direct ion taken away- from the junction. Thestrain due to bending (Eq. 6.75) and the strain due t,o pres-sure-stress considerations (Eq. 6.79) are in the same direc-Con and must have the same sign.

For small auples sin (0 7) = h/21+, or

For R Iongitudin~l *trip of unit widt.h b = 1; t.herefore

where u’ = normal force. pounds per linear inch

By- Eqs. h.la anti 6.78

Substituting Eqs. 3.14 and 3.13 into Eq. 6.5 gives: Therefore ’pd pd

Ey2 = 2t,E - Cc St,E (6 .77) (6 .82)

To convert from the unit radial deformation as given byEq. 6.77 to total deformat,ion it is necessary t.o multiply

: Eq. 6.77 by P.(6 .83)

(6 .78)

pd2Yp = ~

/.vd”F---b---l

4&E 8&Eo r : t,

t(6 .79) 912

where yP = total strain due to pressure

The deformation as given by Eqs. 6.75 and 6.79 aredirectly additive (in accordance with the principle ofsuperposi t ion) . Therefnre

?‘oombined = (6.80)

The second term is negative because the location of thefd (b)

1 .r-y axis was originally taken at the center line of theshel l wal l with the posi t ive .y direr&n radially outward and

Fig. 6.4. Circumferential stresses in cylindrical shell under internal

pressure.

Page 116: Process Equipment Design

1 0 6 Selection of Flat-plate and Conical Closures for Cylindrical Vessels

Substituting Eq. 6.83 into Eq. 6.81 gives:

-4t,EyW=d2

Substituting for w in Eq. 6.18 gives:

(6.84)

Substituting for D by Eq. 6.15 and rearranging gives:

$q12(yp2)] +y = 0 (6.85)

Or

~+4[!zL&)]y=o

by Eq. 6.67

$+4/34y = 0

therefore

(6.86)

6.2b Bending in a Flat-plate Closure. The flat-plateclosure is considered to deflect as a uniformly loaded cir-cular plate simply supported at its edge with a superimposedconcentric load from the reaction at the junction with theshell. Superimposing the deflections by means of Eqs. 6.59and 6.64 gives Eq. 6.93

-p d” - g

y2 = (:,,, )[G::)(:) -r2]+ (6.93)

where

D2 = Eth312(1 - $)

Differentiating Eq. 6.93 with respect to P and evaluatingat r = d/2 gives:

(3r=d,2 = - [ ,,,a:‘: p) - 2D2;‘: ,)] (““)

A stress intensification factor for the stress caused by thedeflection y is obtained by dividing Eq. 6.82 by Eq. 3.14.

(6.87)

Equation 6.87 can be expressed in terms of the momentand shear by substituting Eq. 6.80 for y.

(6.88)Since by Eqs. 6.15 and 6.86

fl4& = $? (6.89)

substituting Eq. 6.89 into Eq. 6.88 results in a dimensionlessrelationship.

25 - 4-aYl -2MoP2 2QoPfh o o p d2p P P

(6.90)

Another dimensionless equation in terms of the bendingmoment, MO, and the shear, Qs, can be derived from Eq.6.76. Multiplying Eq. 6.76 by Eq. 6.89 gives:

= P4D1 hl WMo + Qo) 1Therefore

Et, dyl-(->

= 2PMo + Qop2d2 d x z=. 2

Dividing Eq. 6.91 by (-pd/2t,) gives:

(6.91)

-PMo Qo=---pd 2pd

(6.92)

Inspection of Fig. 6.3 shows that (dyl/dx) = (dyz/dr).Substituting for (dyz/dr) in Eq. 6.94 and multiplyingthrough by Eq. 6.89 gives:

. . . -pd 2t,3p2Drvidmg through Eq. 6.95 by 21, and by ~

th2gives :

The shear force at the junction QO results in a radialtensile force, Nr, in the cover plate. If Ni is taken at themidplane of the cover plate, a lever arm equal to &/2exists between these forces. This results in a bendingmoment, Ml, which may be evaluated by the summationof moments.

NI = -Qo (6.97)and

Ml = ‘A - M2 0 (6.98)

(6.96)

Substituting Eq. 6.98 for Ml in Eq. 6.96 gives:

p2 d2t, B24Qoz=o = - 640 + P)th + 4p(l + EL)

B2tsMo

2P(l + P)th(6.99)

Substituting Eq. 6.86 for /3 and rearranging gives:

(6.100)

(6.101)

Page 117: Process Equipment Design

Stresses in Cylindrical Vessels with Flat-plate Closures 107

where br, bs, and bB are the respective coefficients in theparenthesis of Eq. 6.100. (See Table 6.1 for tabulatedvalues.)

Inspection of Fig. 6.3 shows that the radial stress in theflat plate is equal to Nl/th and the unit radial strain is equalto 6/r. By elastic theory and Eq. 6.4,

6-=e,= &&

r E E

Substituting dy/dr and6.93 gives:

(6 .4)

(See Reference 32, p. 121.)

d2y/dr2 by differentiating Eq.

But

and

therefore

Substituting for jr and jC in Eq. 6.4 gives:

a+-p)i$P (6.102)

The strain 6 is measured at the mid-plane of the plate,and the displacement at the junction will be:

(6.103)

At the junction the displacement of the shell must equalthe displacement of the head. Therefore

y1= -61 = -6 (6.104)

By using Eq. 6.82 and substituting for yl, we obtain:

,J

64DI;;3+e)2

+ -Mid-

2D2U + P))ISubstituting Eq. 6.98 for M1 and Eq. 6.15 for

dividing through by (pd/2t,) gives:

Eta1 --1dl - p)Qor 3tsd(l - cc)- =d2p thpd2 - 32tf&2

+ x3(1 - P)QO _ x4 - PL)MO2P dth P dth2

(6.105)

where bq, bg, and bs are the respective coefficients given inEq. 6.111. (Note that be = 0.) (See Tables 6.2 through6.9 for tabulated values.)

Equation 6.110 may be equated to Eq. 6.108 to give:

Dz andal $ + u2 p% + u3 = u4 p% + a 5 p$ + a6 (6.113)

Equation 6.101 may be equated to Eq. 6.112 to give:

(6.106) b1$+b2$+b3 = bdp%+ba$+be (6 .114)

Multiplying through Eq. 6.106 by (th/&) and rearranginggives:

+ [ 3d(;2; “1 (6.107)

Ethyl MO Qo- = a1 - + u2 - + a 3d2p pd2 pd

(6.108)

where al, ~22, and u3 are the respective coefficients given inthe brackets of Eq. 6.107. (See Table 6.1 for tabulatedvalues.)

6.2C Combination of Relationships in DimensionlessGroups. Watts and Lang (108) combined the relationshipsfor bending and shear in the shell and cover plate in theform of dimensionless groups as given by Eqs. 6.101 and6.108. The advantage of this procedure is to give genera1relationships independent of the system of units. Thecoefficients or “influence numbers” group the variables.describing the geometry of the vessel.

The equations for the shell may be put in the same formas Eqs. 6.101 and 6.108. Multiplying 6.90 by (th/4ts) andrearranging gives:

~=[-yy$)]!$

+[?(?)I$+ [2(q)] (6 .109)

Ethyl MO Qo-=ur-++5-+ufjpd2 pd2 pd

(6.110)

where ~4, ~5, and a6 are the respective coefficients given inthe brackets of Eq. 6.109. (See Tables 6.2 through 6.9for tabulated values.)

Equation 6.92 is multiplied by the ratio z0

2

and isL

rearranged to give:

~(~)z=o = [-(W ($I$+ [ +“]$+O (6.111)

(6.112)

Page 118: Process Equipment Design

108 Selection of Fiat-plate and Conical Closures for Cylindrical Vessels

Or

M,, = pd2(a3 - ad@5 - h2) - (a5 - a2)h3----- 1(a4 - al)@5 - h2) - (a5 - u2)vh - h)

and(6.115)

Q. = pd(~4 - adba - (~3 - ad(ba - bd

~____~-~

(~4 - ad& - bd - (a5 - az)(h - h) 1(6.116)

Table 6.1. Coefficients for Flat Head (108)

(Extracted from Transactions of the ASME with Permissionof the Publisher, the American Society of Mechanical

Engineers, 29 West 39th St., New York, N. Y.)

Wlh (1 I ,I 2 02 61 hr hr

b.0 - 8 . 4 0 0 0 +I 4000 +0.2625 + 5 . 0 8 3 9 - 0 . 6 3 5 5 -K158Y10 n o - 2 1 0 0 0 0 +I 4000 +0.6563 C12.7098 - 0 . 6 3 5 5 - 0 . 3 9 7 2t o . 0 0 - 8 4 . 0 0 0 0 +I.4000 f2.6250 f50.83Yl - 0 . 6 3 5 5 - 1 . 5 8 8 7

RO.00 - 168.0000 +I.4000 +5 2 5 0 0 +101.6782 - 0 . 6 3 5 5 - 3 . 1 7 7 41 0 0 . 0 0 -2lo.nnnn +1,4Onn +6 5 6 2 5 f12i.0978 - 0 . 6 3 5 5 - 3 . 9 7 1 83 0 0 . 0 0 -630,OOOO f1.4000 i-lY.6875 f381 2 9 3 3 - 0 . 6 3 5 5 - 1 1 . 9 1 5 45 0 0 . 0 0 -1n50.0000 +1.4nnO +32.815 +635.4889 - 0 . 6 3 5 5 - 1 9 . 8 5 9 0

Table 6.2. Coefficients for Cylinder, /h/l,Y = 0.5

(Extracted from Transactions of the ASMlS with Permissionof the Publisher, the American Society of Mechanical

Engineers, 29 West 39th St., New York, N. Y.)

d/t. a4 as a 6 h hs___-.-- -.---.

4 -3.30513 -0 .9090 -0.10625 -0 .9090 -0 .125010 -8 .2630 - 1 .4373 -0.10625 -1 .4373 -0 .125020 -16 .5240 -2 .0325 -0.10625 -2 .0325 -0 .125030 -24 7885 -2 .4894 -0.10625 -2.4894, -0 125040 -33.0510 -2 .8745 -0 .10625 -2 .8745 -0 .1250

bg = 0

Table 6.3. Coefficients for Cylinder, Ih/& = 0.8

(Extracted from Transactions of the ASME with Permissionof the Publisher, the American Society of MechanicalEngineers, 29 West 39th St., New York, N. Y.) (108)

d/t. a4 a6 a6 b4 hs

3.20 -4 .2298 -1 .3007 -0 .1700 -2 .0812 -0 .32008.00 - 10.5745 -2 .0567 -0 .1700 -3 .2906 -0 .3200

16.00 -21.1491 -2 .9085 -0 .1700 -4 .6537 -0 .320024.00 -31.7236 -3 .5622 -0 .1700 -5 .6996 -0 .320032.00 -42.2981 -4 .1133 -0 .1700 -6 .5813 -0 .320064.00 -84.5963 -5 .8171 -0 .1700 -9 .3073 -0 .320080.00 - 105.7454 -6 .5037 -0 .1700 .- 10.4059 -0.3200

740.00 -317.2361 -11 .2647 -0 .1700 -18 .0236 -0 .3200400.00 -528.7268 -14 .5427 -0 .1700 -23 .2684 -0 .3200

bG = 0

Table 6.4. Coefficients for Cylinder, /h/l, = 1.0

(Extracted from Transactions of the ASMK with Permissilmof the Publisher, the Amrrican Society of MechanicalEngineers, 29 West 39th St., New York, N. Y.) (108)

d/t. a4 a 5 a6 64 h:,

4.00 -6 .6091 -1 .8178 -0 .2125 -3 .6357 -0 .500010.00 -16.5227 -7.87-13 -0 .2125 -5 .7485 -0 .500020.00 -33.0454 -4 .0618 -0 .2125 -8 .1296 -0 .500030.00 -49.5681 -4 .9784 -0 .2125 -9 .9567 -0 .500040.00 -66.0908 --5.7185 -0 .2125 -11 .4970 -0 .500080.00 -132.1817 -8 .1296 -0 .2125 -16 .2592 -0 .5000

100 .00 -165.2271 -9 .0892 -0 .2125 -18 .1784 -0 .5000300.00 -495.6813 -15.7229 -0..2125 -31 .4859 -0 .5000500.00 -826.1356 -?0.3”41 -0 .2125 -40 .6481 -0 .5000

bs = 0

Table 6.5. Coefficients for Cylinder, /h/Is = 1.2

(Extracted from ‘I’ransac%ions I,f the ASMK with Permis:;ilbllof the Publisher. the Ameri(~an Society of MechanicalEngineers, 29 \\‘est 39th Si... New York, N. Y.) (108)

d/t, a4 (I b a6 b4 b5-___.

4.80 -9 .5171 -2 .3896 -0 .2550 -5 .7351 -0 .720012.00 -23 .7927 -3 .7783 -0 .2550 -9 .0679 -0 .720024.00 -47.5854 -.5.3433 -0 .2550 -12 .8240 -0 .720036.00 -71.3781 -6 .5442 -0 .2550 -15.7061 -0 .720048.00 -95 .1708 -7 .5566 -0 .2550 -18 .1359 -0 .720096.00 -190.3416 -10 .6867 -0 .2550 -25 .6480 -0 .7200

120 .00 -237.9271 -11.91,81 -0 .2550 -28 .6754 -0 .7200360 .00 -713.7812 -20.69,l: -0 .2550 -49 .6672 -0 .7200600.00 -1189.6352 -26 .7167 -0 .2550 -64 .1200 -0 .7200

be = 0

Table 6.6. Coefficients for Cylinder, th/& = 1.6

(Extracted from Transactions of the ASME with Permissicnlof the Publisher, the American Society of MechanicalEngineers, 29 West 39th St,.. New York, N. Y.) (108)

d/t, a4 (15 a6 br bs

6.40 -16 .9193 -3 .6790 -0 .3400 -11 .7730 -1 .280016.00 -42 .2981 -5 .8171 -0 .3400 -18 .6147 -1 .280032.00 -84 .5963 -8 .2266 -0 .3400 -26 .3251 -1 .280048.00 -126.8944 -10.0755 -0.3400 -32 .2416 -1 .280064.00 -169.1926 -11.6342 -0.3400 -37.2294 - 1.2800

128.00 -338.3851 -16.4532 -0.3400 -52 .6503 -1 .2800160.00 -422.9814 -18.3952 -0.3400 -58 .8648 -1 .2800480 00 -1268.9443 -31.8615 -0.3400 -101 .9568 -1 .2800800.00 -2114.9071 --41.1330 -0.3400 -131 .6257 -1 .2800

Page 119: Process Equipment Design

,Stresses in Cylindrical Vessels with Flat-plate Closures 1 0 9

Table 6.7. Coefficients for Cylinder, G/t, = 2.0

(Extracted from Transactions of the ASME with Permissionof the Publisher, the American Society of MechanicalEngineers, 29 West 39th St., New York, N. Y.) (108)

d/L a4 a 6 a 6 br ba

At xl.

~1 and x2. The bending moment at x1 is Ml and at 22 isMs. The induced bending stresses vary from zero at thex axis to a maximum at the outer fiber. The bending stressin the strip at any distance y from the x axis is given byEq. 2.10.

8.00 -26.4363 -5.1416 -0.4250 -20.5665 -2.000020.00 -66.0908 -8.1296 -0.4250 -32.5185 -2.000040.00 -132.1817 -11.4970 -0.4250 -45.9881 -2.000060.00 -198.2725 -14.0809 -0.4250 -56.3237 -2.000080.00 -264.3634 -16.2593 -0.4250 -65.0370 -2.0000160.00 -528.7268 -22.9941 -0 4250 -91.9762 -2.0000200.00 -660.9085 -25.7081 -0.4250 -102.8325 -2.0000600.00 -1982.7254 -44.5278 -0.4250 -178.1112 -2.00001000.00 -3304.5424 -57.4851 -0.4250 -229.9406 -2.0000

- -

bg = 0

Table 6.8. Coefficients for Cylinder, th/ls = 6.0

IHxtracted from Transactions of the ASME with Permissionof the Publisher, the American Society of Mechanical

Engineers, 29 West 39th St., New York, N. Y.)

tl/l. a4 a 6 a6 ba 6s

Since the element is in equilibrium, the summation of theforces must be equal t,o zero. Therefore

4 -39.6612 -10.908 -1.275 -130.8960 -18.001 0 -99.1560 -17.2476 -1.275 -206.9640 -18.0010 -198.288 -24.2820 -1.275 -292.6800 -18.0030 -297.4614 -29.8728 -1.275 -358.4736 -18.0040 -396.6120 -34.4940 -1.275 -413.928 -18.00X0 -793.1640 -48.7800 -1.275 -585.360 -18.00

100 -991.5372 -54.5400 -1 275 -654.480 -18.00300 -2973.916 -94.4550 -1.275 -1133.460 -18.00M O -4957.268 --121.9500 -1.275 -1463.400 -18.00

Rearranging gives:

66 = 0 fCM2 - MI)Y

shear = Zb dx JcydA (6.117a)

21

MAY '_ _ _Jz 2/

y dA - M+s

ey dA - fshesrb dx = 0 (6.117)

1/

Table 6.9. Coefficients for Cylinder, &/Is = 10

rlxtracted from Transactions of tht= ASME with Permissionof the Publisher, the American Society of Mechanical

Engineers, 29 West 39th St., New York, N. Y.)

4 -66.1025 -18.18 -2.1250 -363.60 -50.0010 -165.260 -28.745 -2.1250 -574.90 -50.0020 -330.480 -40.65 -2.1250 -813.00 -50.0030 -495.769 -49.788 -2.1250 -995.76 -50.0040 -661.020 -57.49 -2.1250 -1149.80 -50.0080 -1321.940 -81.30 -2.1250 -1626.00 -50.00100 -1652.562 -90.90 -2.1250 -1818.00 -50.00300 -4956.526 -157.40 -2.1250 -314'8.00 -50.00,500 -8262.113 -203.25 -2.1250 -4065.00 -50.00

be = 0

6.2d Stresses in the Shell.

SHEAR STRESSES AT THE JUNCTION. Figure 6.5 shows ahmgitudinal strip of the shell of width b under bendingit&on. Consider an elemental length, &c, between points

At x2.

The corresponding forces acting upon the-element dA are

f,.fdA.

fgA,, = FJ

‘y dA1/

At x2,

j2/AYz = 7s

‘y dAY

Also, a shear force exists at y equal to:

But (M2 - Ml) = dM, and by Eq. 2.5 dM/dx = V.Therefore

In a rectangular section of width b and depth t,

(6.118)

j ’shear = zsY

‘by dy = 3 (c2 - Y2) (6.119)

The value of fshear = 0 when y = c and is maximum at

H JLl i2 Section H - H

Fig. 6.5. Bending in an element of a plate.

Page 120: Process Equipment Design

1 1 0 Selection of Flat-plate and Conical Closures for Cylindrical Vessels

the vertical axis where y = 0. Therefore the maximum This stress exists at xa, the location of the maximum axialshear stress is: stress in other places than at the junction.

fvc2 v(t2/4) 3 v

shear = ?? = 2(bt3/12) = 2bt(6.120)

If the shear at the junction is expressed as shear per unitlength of circumference, Q(bo = l.O),

f -2 Qoshear - 2 -

4(6.121)

AXIAL SrnEssEs. The combined axial stresses in theshell at the junction will consist of the axial stress frominternal pressure and the axial bending stress, or by Eqs.3.13 and 2.10:

faxial oombined = $ + F = $ + (6.122)s 8

CIRCUMFERENTIAL STRESSES AT THE JUNCTION. Thecombined circumferential stresses will consist of the hoopstress from internal pressure and the circumferential bendingstress, plus the component of the axial bending stress.The circumferential bending stress (circum. bending) maybe obtained by substituting Eq. 6.75 for y in Eq. 6.82, or

fcircum. bending = F = $ “Mi2i Q”) (6.123)1

but

/34& = $ (See Eq. 6.89.)

Substituting in Eq. 6.123 gives:

f&cum. bending = pd(‘M: + Q”)s

(6.124)

fcircum. combined = f + iBd@Mlo + Q”) + F8 s II I.s

(6.125)

STRESSES IN THE SHELL IN OTHER PLACES THAN AT THE

JUNCTION. Watts and Lang (108) have given the followingrelationships for stresses in the shell in other places thanat the junction:

flshesr = 3Qo d1 + 2(i3;;s(~) + 2@Mo/Qd2 (6.126)

where x, = location of maximum shear stress in the shell inother places than at the junction,

(6.127)

combined = longitudinal pressure stress + axial

f)axisl = E + 3.41 fi (flshear) (6.128)s

6.2e Stresses in the Flat-plate Closure.

SHEAR STRESSES IN THE PLATE AT THE JUNCTION. FromEq. 6.39

Q=$!

The shear force per unit length in the plane at 90’ is ofequal magnitude (29). Therefore by Eq. 6.121

(6.130)

AXIAL S TRESSES AT THE J UNCTION. The combined axialstress in the plate at the junction is composed of the axialshear stress plus the bending stress from the action of theshell on the plate, or

f&al combined = (6.131)

Substituting Eq. 6.98 for MI in Eq. 6.131

faxial combined = l$l+!$-~~ ( 6 . 1 3 2 )

CIRCUMFERENTIAL STRESSES AT THE JUNCTION. Thecombined circumferential stresses in the plate at the junc-tion are composed of the circumferential shear stress, thecircumferential bending stress from the action of the shellon the plate, and the circumferential bending stress fromthe action of pressure on the plate (see Eq. 6.61). Thebending moments causing these stresses have opposite signs.

fcireum. combined = !$I I

+ c2 1K - M,I (6.133)

Substituting Eq. 6.98 for MI and Eq. 6.61 for Mz evalu-ated at P = d/2 in Eq. 6.133 gives:

fcircum. combined = tI 1

+ ?-3$-6p th0

d 2 (1 1 (6.134)

STRESS IN CENTER OF FLAT PLATE. At the center of theflat-plate closure the axial stress is equal to the circum-ferential stress. The combined maximum stress is equalto the sum of the shear stress and the bending stress fromthe action of pressure on the plate (see Eq. 6.62a).

fmax combined = I ia0th + ; 1% - Mzzl (6.135)

Substituting Eq. 6.98 for Ml and Eq. 6.62a for M,, inEq. 6.135 gives:

fmax combined =

(6.136)

Page 121: Process Equipment Design

Stresses in Cylindrical Vessels with Flat-plate Closures 111

Solution of the previous equations shows that when thethickness of the flat plate is greater than the thickness ofthe shell, the maximum stress is the combined axial stresslocated in the shell at the junction. If the flat plate has athickness equal to or less than the shell, the maximumstress is the combined axial stress located in the flat plate atthe junction. Thus, there is a ratio between the thicknessof the cover plate and the thickness of the shell, th/tsrwhich will result in equal stress in the plate and shell.This ratio will vary from 1:0 to 1:2 depending upon theratio of diameter to thickness of the cover plate. However,it is usually advantageous to use a ratio of th/ts greater than1:2 in order to reduce the magnitude of the maximumstress concentration. Therefore, for such designs themaximum stress concentration is located in the cylinder atthe junction with the flat cover plate and may be deter-mined by Eq. 6.122.

1 For purposes of comparison it is convenient to expressthe stress concentration at the junctionf,,,, in terms of thehoop stress in the shell for the same conditions.

Max stress ratio, I = ‘s = L%?!-ffhoop pd/%

(6.137)

Figure 6.6 is a plot of I, the ratio of the combined axialstress in the shell at the junction to the hoop stress in theshell. Watts and Lang (108) have evaluated and tabulatedthe stress-intensification factors for the major stress in theshell and flat-plate closure.

Figure 6.6 indicates the stress concentrations that wouldexist if the theory of elasticity held up to the stress levelsindicated. These stresses usually are not reached becausethe yield point of the material used in the construction ofthe vessel usually is exceeded. Above the yield point,plastic deformation occurs and the theory of elasticity nolonger holds. For analysis of the plastic stresses, thetheory of plasticity must be applied. The present stateof this theory is such that it is not possible to evaluate these

I stresses. Strain gages applied at anticipated points ofstress concentrations make it possible to evaluate the actual

!stress concentration.

Figure 6.6 shows that the stress concentration at thejunction between the shell and the head may result in alocalized bending stress in the shell many times greaterthan the hoop stress in the shell even when relatively thickcover plates are used. This is quite often ignored in manydesigns. Fortunately, plastic deformation at the point ofstress concentration tends to relieve an excessive localizedbending stress and thereby to prevent failure. This illus-trates the definite advantage of using formed heads whichtend to reduce localized stresses at the junction.

6.2f Example Design 6 .1 . Determine the varioustheoretical stresses in the shell and in the flat-plate closureof a steel condensate collector operating at 100 lb per sqin. gage. The shell of the collector is fabricated of rolledsteel plate s/4 in. thick, and the flat-plate closures have athickness of 134 in. The vessel is 6 ft long and 3 ft indiameter.

Solution:

d 36 = I44-=-i-4 ;i

4_h=si=6tp: T

(6.86)~= 0.606

a1 = -3(1 - p) $ = -50.4 b1 = 6(1 - PIP2tht.s

= 30.5

a2 = 2(1 - p) 1.4 b 23(I - cl)= = _ ___ =

P2d4-0.635

a3 = 3; (1 - p) ; = 1.575 b 3 =-?l-- ~=16 ,B2thts

-0.953

u4 = -(@d)2 th = = =2 4 - 1 4 2 8 b4 -/3d -785.4

as= -pd$ = -@j&j b5 = - f = - 1 8s

a6= - !!!k! =4 8

-1.275 b6 co

Substituting into Eqs. 6.115 and 6.116 gives:

MO = pd2 (a3 - a.s)(bs - bz) - (a5 - a2)bs(a4 - al)(bs - b2) - (a5 - a2)(b4 - bl) = 478

Qo = pd(a4 - alhi - (a3 - as)(b4 - bl)

(a4 - al)(bs - bz) - ( a5 - a2)(b4 - bl) = -426

Stresses in the shell :By Eq. 6.121,

f 21shear = 3_ (-426) = -2,550 psiz

By Eq. 6.122,

faxial c o m b i n e d = ‘“1” ,“2” + p = 49,300 psi4

8

6

Ratio of shell diameter to shell thickness, d/t,

F i g . 6 . 6 . M a x i m u m s t r e s s r a t i o i n t h e s h e l l a t t h e j u n c t i o n f o r f l a t - p l a t eclosures.

.I _ - -\ - \ \I 7

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.

112 Selection of Flat-plate and Conical Closures for Cylindrical Vessels

By Eq. 6.125,

fcircum. combined = ‘F

The stress in the center of a flat plate by Eq. 6.136 is:

- 2 5 5 6 4 0 . 5 6 6 (-2556)(+0.752)= +1288psi=- - . ~___- - -.e2.i 14.9

- .

Location of this stress is given by Eq. 6.127.

i!1

: i/

i

!/ ‘:

--___---I

+ I y.py(36NO.6061a

+ (0.3 x 6)(478) ~

T-5

= 7,200 + 11,900 + 13,750 = 32,850 psi

x 478) - 42611-__-__-

Stresses in the shell in other places than at the junction:By Eq. 6.126,

f(3/2)(-426) l/i.+-(2)(--0.68)-+ (2)(0.463)

.shear = -~~-_.~~~._~

W4)e0.606x4.45

1= -hn--’ -426--~~~0.606 (0.606)(478) + 1

= 4.45 in. from junction

1~~ Eq. 6.128,

flaxial = !yg + 3.41 &i/a k128.8

= 3600 + (3.41)(+12)(+128.8)

= 3600 _+ 3260 = +8860 psior -1660 psi

I.ocation of t.his stress is given by Eq. 6.129.

7rlT,, = 2* - - - = 4.45

4P-;

1x __- = I.45 - I .3

0.606

= 3. I5 in. from junction

Stresses in thr ,llal-plalr closure:By Eq. 6.130.

By Eq. 6.132,

j&, eomk,inrr, = q!? j + j !y! - wp2,2 H

= 284 + 1275 + 852 = 2441 psi

By Eq. 6.134,

= j-2841 + 1127s + 852 - 5980'= 1137 psi

jmax combined = / -2841 + 11275 + 852 - 14,080\= 12,137 psi

If this vessel had been fabricated from SA-283, Grade Csteel, two of the stresses in the shell, faxis combined andfcircumferential oombinedv would be above the minimum yieldpoint of 30,000 psi for this material. This indicates thatsome plastic deformation will occur and that these com-puted theoretical stresses will not be reached since theywill be relieved by plastic deformation. If this vessel wasoperated under cyclic loading conditions, failure might haveoccurred by.brittle fracture if the stress range had exceededtwice the yield point. (See Chapt.er 2.)

It should be pointed out that vessels of such design areseldom used for pressures higher than 25 psi because theyare impractical. Vessels of such size operating at pressuresabove 25 psi usually have formed closures, termed dishedheads.

6.29 Practical Design of Cylindrical Vessels with Flat-plate Closures. As a first approximation, the thickness of acylindrical shell under internal pressure may be determinedby the membrane equation (Eq. 3.14). The thickness of theflat-plate closure may be determined by considering the plateas a circular plate with clamped edges (Eq. 6.55a) or

t =Ei8w

(See Eq. 3.14)

t,, = d d3p/16j (See Eq. 6.55a)

th-= d 43pD6.f W) _t

- 0.866 u/j/p (6.138)x pd

The thickness ratio obtained by the use of Eq. 6.138results in ratios in which the thickness of the flat-plateclosure is several times the thickness of the shell for usualvalues of allowable stress and operating pressure. Someimprovement in design proportions may be made possibleby increasing the thickness of the shell to reduce the maxi-mum theoretical combined axial stress in the shell at thejunction. However, such an increase in shell thicknessdoes not result in a corresponding decrease in the requiredthickness of the flat-plate closure. (The maximum theo-retical combined axial stress in the shell at the junctionalso can be decreased by increasing the thickness of theflat cover plate.)

The maximum theoretical local combined stress mayexceed the yield point appreciably with imperceptiblepermanent strain since only the outer fibers undergo plasticdeformation. (See Fig. 2.5, detail b.) If the vessel isstressed under cyclic conditions, the maximum theoreticallocal stress range can approach a value equal to twice theyield point without danger of causing failure by brittlefracture. (See Eq. 2.37). If the vessel is not to be stressedin a cyclic manner, a higher theoretical (elastic) local stressmight be tolerated. Under such conditions the theoreticalstress loses its significance, and it is suggested that strainbe used as a design criterion.

\ \ \I /.- ~-..~ _ - - ~--

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Stresses in Cylindrical Vessels with Conical Closures 1 1 3

By small-angle relationships, we obtain (see Fig. 7.6b and c):6.3 STRESSES IN CYLINDRICAL VESSELS WITHCONICAL CLOSURES

A flat plate may be cut and rolled to the shape of a coneand used as a closure for a cylindrical vessel. Shops havingfacilities for rolling cylindrical shells usually can roll conicalshapes; this makes this type of head convenient for manytypes of process vessels.

Figure 6.7 shows an element in a conical closure definedby length dl and angle dqi For very small angles the sideopposite the angle is numerically equal to the angle inradians times the length of a long side. Therefore thearea, A, of the element is:

A = di (P d+)

The force, FP, on the element and uormal to the surface isequal to the internal pressure times t.he area, or

F, = pA = p dl (P dq5)

This force is resisted by the normal components of theforces of the stresses induced in the element. The meridio-nal forces, F,, are 180” apart on either side of the elementand therefore have no normal component. However, thehoop forces, Fh, are not 180” apart and have a normal com-ponent, Fhrn. The normal component, FhTn, may be deter-

, mined from the radial component, Fhr, as follows:

Fh = jhj dl

F hrn = Fhv COS Ct = j,,l dl tk$ (‘OR CY

By a summation of forces in the normal direction

Fhrn - P, = 0

jhtdld+cosa - ptllrd+ = 0

therefore

(6.130)

(For angle 0, see Fig. 4.8 and Eq. 1.2.)At the junction of the conical head and the cylindrical

shell, a compressive force is exert.ed by the cone on thecylinder. The shell under the influence of internal pressureattempts to expand radially outward against this inwardforce; this results in a bending moment and shear at thejunction. The inward compressive force produced by theconical head can be evaluated as follows, by referring toFig. 6.8.

Axial tension in the shell = I’ pounds per linear inch ofshell circumference.

p = pd4

(6.140)

F i g . 6 . 7 . H o o p s t r e s s i n a c o n i c a l h e a d .

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114 Selection of Flat-plate and Conical Closures for Cylindrical Vessels

Fig. 6.8. Compressive force at junction of cone and cylinder.

The component of axial tension in the cone = T poundsper linear inch.

P pdT=--.--cos a 4 cos ff

(6.141)

The component of axial tension in the cone expressed ascompression at the junction = C pounds per linear inch.

C = P tan LY = p_d tan ix4

(6.142)

As a result of this compressive force (Eq. 6.142) it isimpossible to design a conical head to eliminate momentand shear at the junction since the cone always tends todeflect inward, and the shell outward under the influenceof internal pressure.

6.30 Bending in the Shell and Conical Closure at theJunction. It was shown previously, in the analysis of a shellhaving a flat-plate closure, that two dimensionless equations(Eqs. 6.109 and 6.111) could be derived for the section ofthe shell at the junction. For convenience these wererewritten to give Eqs. 6.110 and 6.112. These equations,including the coefficients, are valid for the shell at thejunction of the cone. Relationships for the conical closurecomparable to Eqs. 6.101 and 6.108 for the flat-plate closuremay be developed as reported by Watts and Lang (110).

Eta 1~ = a,$ + as ;; + aspd2

(6.143)

= b,p%+b.F;+bg (6.144)

where y1 = radial deflection of shell at junction, inchestc = thickness of conical section, inches

Fig. 6.9. Forces and moment at junction of cylindrical shell and conical

head (1 101.

b&o2 tan a! sin crb7 =

2G,$02 tan a~a7 =

2 C + 2pG4

a8 = (Eo2B/4) - ~‘G sin cr A/2C + .bG

bg = ~C + 2pG

2-P 3ba(l + P)a9 = -8

set ff - y tan cr +F02

set a

b b8 kUI ff9

= 60 + PCC)3 set ff c s c

E04

b7 csc2 (y4 %02

where

A = fo(berz’ [O beiz &J - be&’ (0 berz to)

B = (berz’ 50)’ + (beiz’ [o)2

C = Eo(berz .fo berz’ [o + beiz to beiz’ 50)

G = (berz toI2 + (bei (012

and

.$!o = cone parameter at the base

= /3d 1/2(t,/t,) cot (Y csc (Y

The ber and bei Bessel functions are given by MacLachlan (111).

The calculations for the coefficients are based on therelations:

Figure 6.9 shows diagrammatically the location of theseforces and moments. The nomenclature used in Fig. 6.9is the same as for Fig. 6.3 in the previous section on flatcover plates.

Q o

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L

Equation 6.143 may be equated to Eq. 6.110 to give:

and Eq. 6.144 may be equated to Eq. 6.112 to give:

b,~+bg$+b9=h4~+b5!$+bi (6 .146)

Or

M,, = pd2 (a - as)(bs - ba) - (a5 - as)bs(a4 - a7)(bs - b8) - (a5 - us)(br - b7) 1

(6.147)

Qo = pd(~4 - a7)bg - (us - &j)(b4 - b7)

(a4 - a716 - b8) - (a5 - us)(br - bi) 1(6.148)

The coefficients u4 through a9 and bd through bg are tabu-lated in Tables 6.2 through 6.13.

Table 6.10. Coefficients for Cone, a = 15” (110)(Extracted from Transactions of the ASME with Permission

of the Publisher, the American Society of MechanicalEngineers, 29 West 39th St., New York, N. Y.)

d/lb 0 1 ( 1 9 ao 87 ba be

3.9231 -6.3685 +I.6824 +0.1020 +3.6189 -0.4913 f0.01999.8076 -16.0642 +2.7141 +0.0360 +5.7543 -0.4957 +0.0276

29.4229 -48.4194 +4.7646 -0.0999 +9.9995 -0.4980 +0.031449.0382 -80.7915 +6.1753 -0.1941 +12.9205 -0.4986 +0.0322

104.9320 -173.0657 +9.0713 -0.3879 +18.9165 -0.4991 +0.0329194.0819 -320.2775 +12.3660 -0.6085 +25.7384 -0.4994 +0.0331310.4304 -512.4285 +15.6604 -0.8291 +32.5598 -0.4995 +0.0333498.6781 -823.3524 +19.8700 -1.1111 +41.2758 -0.4996 +0.0333

Table 6.11. Coefficients for Cone, a = 30” (110)(Extracted from Transactions of the ASME with Permission

of the Publisher, the American Society of MechanicalEngineers, 29 West 39th St., New York, N. Y.)

Utn ai as ao 87 68 b9

4.0825 -6.3676 f1.5220 +0.0030 +3.8148 -0.4720 +0.044716.3299 -26.5094 f3.2502 -0.2297 +7.7977 -0.4913 +0.063440.8248 -66.8682 +5.2433 -0.5138 +12.3988 -0.4957 to.068377.0489 -126.6027 +7.2665 -0.8047 +17.0707 -0.4972 +0.0700

122.4743 -201.5488 +9.2045 -1.0840 +21.5459 -0.4980 +0.0707204.1239 -336.2996 f11.9299 -1.4770 +27.8398 -0.4986 f0.0713308.1955 -508.0946 +14.6953 -1.6760 +34.2261 -0.4989 f0.0715500.0751 -824.8978 +18.7620 -2.4629 +43.6180 -0.4992 +0.0718

Table 6.12. Coefficients for Cone, a = 45” (110)(Extracted from Transactions of the ASME with Permission

of the Publisher, the American Society of MechanicalEngineers, 29 West 39th St., New York, N. Y.)

d/h a7 as ao b7 bs bv

4.0000 -5.7796 +1.2340 -0.0725 +4.0325 -0.4372 +0.074110.0000 -15.5974 +2.1524 -0.2654 +6.6074 -0.4720 +0.097740.0000 -64.9345 +4.5965 -0.8558 +13.5059 -0.4913 +0.116380.0000 -130.8284 +6.6064 -1.3547 +19.1883 -0.4949 to.1202

100.0000 -163.7929 +7.4152 -1.5562 +21.4754 -0.4957 +0.1211300.0000 -493.6916 +13.0171 -2.9547 +37.3186 -0.4980 +0.1235500.0000 -823.7623 +16.8714 -3.9179 +49.2200 -0.4986 +0.1240

Stresses in Cylindrical Vessels with Conical Closures 1 1 5

Table 6.13. Coefficients for Cone, a = 60” (110)(Extracted from Transactions of the ASME with Permission

of the Publisher, the American Society of MechanicalEngineers, 29 West 39th St., New York, N. Y.)

d/h a7 as a9 br ba 69

8.485321.213258.101884.8528

169.7056212.1320294.1406400.3580636.3959

-12.2604 +1.5114 -0.3207 +6.9846 -0.4372 +0.1486-33.0871 +2.6361 -0.7559 +11.4443 -0.4720 -CO.1809-93.7150 +4.6013 -1.5823 +19.2842 -0.4881 +0.2009-137.7469 +5.6295 -2.0229 +23.3930 -0.4913 +0.2052-277.5290 +8.0911 -3.0837 +33.2351 -0.4949 +0.2103-347.4572 +9.0817 -3.5114 +37.1965 -0.4957 +0.2114-482.6728 +10.7465 -4.2314 +43.8550 -0.4966 +0.2126-657.8465 +12.5860 -5.0271 +51.2120 -0.4972 +0.2135-1047.2780 +15.9426 -6.4797 +64.6377 -0.4980 +0.2145

6.3b Stresses in the Shell.

Stresses at the Junction. The maximum shear stress atthe junction is given by Eq. 6.121. The maximum com-bined axial stress is given by Eq. 6.122. The maximumcombined circumferential stress (circum. combined) is givenby Eq. 6.125 with the following modifications:

f&cum. combined = , $ - Bd(pM; + Q")8 s

Note that the sign of the second term in Eq. 6.149 is nega-tive whereas that of the second term in Eq. 6.125 is positivebecause the direction of bending in the shell with a conicalclosure is in opposite direction to the bending in the shellwith a flat-plate closure.

Stresses in the Shell in Other Places than at the Junction.Watts and Lang (110) have derived relationships for themaximum stresses in the shell in other places than at thejunction as follows:

fshear max =3Qol/l + 2 (PMolQo) + WMO/QO)~ (6 150)

2t, tan-’(&+1) .

The location of this maximum shear stress in the shell isgiven by Eq. 6.127. The maximum axial stress is given byEq. 6.128, and the location of this stress is given by Eq.6.129.

6.3~ Stresses in the Conical Closure. Relationships forevaluating the stresses in the cone at the junction havebeen presented by Watts and Lang (110).

(6.151)

faxialoombined = f/ y + yc

+6;s 7c I !

(6.152)

p cos ff-I- 2a3 + ~

4 II ( 6 . 1 5 3 )

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116

14

Selection of Flat-plate and Conical Closures for Cylindrical Vessels

v I 0 = s apex angle of coneI I

I1 0 0

I200

dlts

I I300. 400

Fig. 6.10. Maximum stress ratio in the shell at the junction for conical

chnursr.

6.3d Location of Maximum Stresses. If the thicknessof the cone is equal to or greater than the thickness of theshel l , the maximum stress concentrat ion usual ly is the axialbending stress in the shel l at the junct ion. This stress mayhe calculated by use of Eq. 6.122, which assumes elasticbehavior to the stress level indicated. For purposes ofcomparison it is convenient to express the stress concentra-tion as a stress ratio, as indicated in Eq. 6.138. Figure 6.10graphically indicates the stress concentration in the shellat the junction with conical heads as a function of angle a!for head-to-shell thickness ratios (110) of 1 and 2.

Conditions under which the axial stress in the shell maynor.jbe maximum are as fo l lows:

1. When thickness of the cone is less than the thicknessof the she l l .

2. When the thickness of the cone is equal to the thick-ness of the shell, but the ci/& ratio is small.

3. When (Y = 45” or less, and the ratio of d/t, is small.

These exceptions apply to designs of unusual proport ionsthat rarely occur in process equipment. For exceptions 1and 2 the maximum stress concentration will be the axialstress in the cone. For exception 3 the maximum stressconcentration will be the circumferential stress in the coneat the junction. For convenience in locat ing the maximum

stress concentration and determining its value, Watts ant1Lang (110) have presented suitable tables.

For cones with LY equal to or less than 45’, it is desirableto make the thickness of the cone equal to the shell thirk-ness. When the cone is thinner than the shell there is alarge axial stress in the cone at the junction. Similarly.when the cone is thicker than the shell, there exists a 1;rrpe

axial stress in the shell at the junction. If it is not practic-able to make the shell and cone the same thickness in isdesirable to make the cone t,hirker than the shell. When ais greater than 45”, the cone should be made thicker thanthe she l l .

All the stress c,oncentrations except the circumferent irl Istresses increase in magnitude as the angle (Y is increased.This is indicated for the shell axial-stress ratio in Fig. 6.10.At one limit, where (Y is O’, the cone becomes a cylinderhaving the same mean diameter as the shell, and no stress

intensifications will occur if the two cylinders have thesame thickness. In the ot,her limit, when QI = 90’. thecone becomes a flat cover plate, and the analysis of I heprevious section on flat plates holds (see the dashed line inFig. 6.10). This limit is shown in Fig. 6.6. Comparingthe ordinate of Fig. 6.6 to that of Fig. 6.10 indicates thatan approximately tenfold reduction in stress concentrationexists when conical heads are used rat,her than flat coverplates.

The high stress ratios shown in Figs. 6.6 and 6.10 are adirect result of sharp-corner construction. The use of :Iformed “toriconical” head rather than a s imple conical headwill greatly reduce the stress concentrations at the shell-and-cone junction.

6.3e Exper imental ly Determined Stresses. To verif\the theoretical relationships, experimental tests were con-ducted by O’Brien et al. on conical vessels by the use (Jf

strain gages (112).The experimental vessels were 48 in. in diameter wi~tr

conical heads having a = 45’. The cones were slightl)thicker than the shells, the cones having a thickness of0.755 in. and the shells a thickness of 0.633 in. Figure6.11 shows the profiles of the junction of the cones with theshells. Vessel A-2 (Fig. 6.11) approximates a toriconicalh e a d .

Strain gages were suitably placed both inside and out,sideof the vessel in the vicinity of the shell-cone junction tomeasure the c ircumferent ial and axial s trains . The stresses

Radius 0.00 assumed0.80 actual

Profile at cone-shelljunction, vessel A-2

Profile at cone-shelljunction, vessel A-l

Fig. 6.11. Profiles at cone-shell junctions of experimental vessels (112).

(Courtesy of American Welding Society.)

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Stresses in Cylindrical Vessels with Conical Closures 117

Cylinder

Outer circumferential stress curves

Fig. 6.12. Measured and computed stresses in experimental vessel (1 12). (Courtesy of American Welding Society.)

were computed from the strains determined experimental lyby means of Eqs. 6.6 and 6.7.

where jr, f, = longitudinal and circumferential st.ress. re-spectivelg , pounds per square inch

p = Poisson’s ratio

.f, = 1”-E = modulus of elast.icity, pounds per square inch

P2(%a? + EIq/z) (6.6) E,Z, ~~2 = longitudinal and circumferential strain, re

spectively, inches per inch

(6.7) The inner and outer 1on;ritudinal stresses computed fromstrain measuremenls are compared in Fig. 6.12 wit,h the

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1 1 8 Selection of Flat-plate and Conical Closures for Cylindrical Vessels

stresses as determined from theoretical considerations.The solid curves represent the theoretical stresses for thevessel A-l having the sharp junction, and the circles are theexperimentally determined stresses for the same vessel.The dashed curves represent the theoretical stresses forvessel A-2 having a toriconical junction, and the solidcircles are the corresponding experimentally determinedstresses.

The hoop stress in the cylindrical shell is:

The inner longitudinal (or axial) stress is a tensile stresswhereas the corresponding outer stress is compressive. InEq. 6.122 the two terms are additive for the inner surfaceand are of opposite sign for the outer surface because thebending moment Ma is positive in one case and negativein the other. Thus, the greatest stress occurs at the innersurface, where the stress due to bending and the longitudinalpressure stress are additive. Examination of Fig. 6.12indicates that, except for a few local discrepancies, thegeneral trends of the stress distribution for the head-to-shelljunction in vessel A-l are the same whether determinedexperimentally or calculated from theory. The agreementis not quite so good for vessel A-2, however. The maximumstress in this vessel is considerably lower than that invessel A-l; this indicates the reduction in maximum stressobtained by the use of a toriconical head rather than aconical head. Figure 6.12 indicates that the theoreticalrelationships satisfactorily predict the stress distributionwithin the elastic range.

A comparison may be made between Figs. 6.10 and 6.12by calculation of the maximum stress at the junction asfollows:

Diameter of vessel A-l = 48 in.

Shell thickness, t, = 0.633 in.

Cone thickness, t, = 0.755 in.

d 4 8ts - 75’70.633

2 t 0 .7554

_ ----El20.633 .

Referring to Fig. 6.10 and using the line for a! = 45” andt,/t, = 1.2, we find that the maximum stress ratio is 3.8.

P R O B L E M S

pd ~48j = ?i = (2)(0.625) = 38’4p

The maximum stress is:

fnlsx = (3.8 X 38.4~) = 146~Ifp = 1 psi

fmax = 146

This compares favorably with the maximum inner longi-tudinal stress indicated in Fig. 6.12, which is also plottedfor 1.0 psi.

Although the stress concentrations shown in Fig. 6.12which exceed the yield point are seldom reached because ofplastic deformation, the results of the elastic analysis indi-cate the location of the stress concentration and the zone inwhich plastic deformation can be expected to occur.

Most commercial pressure vessels, including those havingconical heads, are given a hydrostatic test at one and one-half times the working pressure. The vessel deforms elas-tically during the early part of the test. At some pressurelevel the yield point of the material is exceeded at zones oflocal stress concentration. Some plastic deformation fol-lows which permanently deforms the zone of stress con-centration, giving the vessel a new shape. This permanentdeformation may not be very great and may not be appar-ent. However, a system of residual stresses is set up in theplastic zone that remains after the pressure is removed.In service this vessel may deform elastically at the workingpressure, but the stresses cannot be calculated because theshape and the stress-intensity pattern are not known.Thus, in many practical cases plastic flow permits a vesselto resist ccncentrated stresses without damage, providedthat the material of construction has sufficient ductility.

The general discussion in the last paragraph of section6.2g on flat-plate closures also applies to conical closures.The ASME code for unfired pressure vessels (11) uses thefollowing modified form of Eq. 6.139 for conical heads thathave a half-apex angle, cy, not greater than 30’:

t = pd2 cos oc(jE - 0.6~)

(6.154)

Additional information on the design of conical closures isgiven in Chapter 13.

1. A cylindrical vessel has a shell fi in. thick and 20 in. in inside diameter. The vessel isciosed at both ends with a flat cover plate welded to the shell and has a thickness of 135 in. I fthe internal pressure is 200 psi, calculate: MO, 00, and the maximum local fiber stress in thevessel.

2. For the vessel described in problem 1, calculate:a. shear stress in the plate at the junction,b. maximum combined circumferential stress in the shell,c. maximum combined axial stress in the plate at the junction,

I - - - - - - - -__- ~ - ~ - - - T - - - - I - - - - -

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Problems 119

d. maximum combined circumferential stress in the plate at the junction,e. maximum combined stress in the center of the plate.

3. For the vessel in problem 1, determine the maximum shear and maximum axial combinedstresses in the shell in obher places than at the junction and the location of these stresses.

4. If one of the flat cover plates in problem 1 is replaced with a 90”cone (01 = 45’) having athickness of >/4 in., calculate: Me, Qe, and the maximum local fiber stress at the cone end.

5. For the vessel in problem 4, calculate the maximum shear stress in the shell at the coneend in other places than at the junction, and determine its location.

Page 130: Process Equipment Design

C H A P T E R

STRESS CONSIDERATIONS IN THE SELECTION

OF ELLIPTICAL, TORISPHERICAL,

AND HEMISPHERICAL DISHED CLOSURES

FOR CYLINDRICAL VESSELS

lliptical, torispherical, and hemispherical dishedclosures, shown previously in Fig. 5.7, are considerablystronger than conical or flat-plate closures for cylindricalvessels. The great majority of cylindrical vessels haveeither elliptical or torispherical dished heads. Figure 7.1shows the welding of an elliptical dished head to a lOZ-in.-diameter shell. Although the hemispherical dished headis stronger than either the elliptical or torispherical dishedhead, it is not so widely used because of the excessiveforming required in its fabrication. In general, this resultsin a higher fabricating cost and a more limited range ofavailable sizes. However, hemispherical heads in limitedsizes are now extensively used as closures for propane andbutane horizont.al storage vessels and for this service are*more economical than elliptical dished heads. See Fig. 5.11.

The principal advantage of dished heads over flat coverplates or cones as closures is the large reduction in the dis-continuity in shape at the junction between the cylindricalvessel and the closure, resulting in a reduction of discon--_- -tinuity stresses at or near the junction.

The hoop and meridional stresses in the central portionof these three types of dished heads are relatively easy toevaluate. For a given vessel these stresses will be at amaximum in the torispherical head and at a minimum inthe hemispherical head. However, the discontinuitystresses at or near the junction are difficult to evaluate.As a result of the difficulty of evaluating these stresses, anextensive literature has developed in this field (1133125).A history of the design of heads for pressure vessels isincluded in a recent article entitled “Report on the Designof Pressure Vessel Heads” prepared by the Design Divisionof the Pressure Vessel Research Committee (12).

7.1 ELLIPTICAL DISHED CLOSURES

The excessively high stress concentrations existing in thesharp-corner junctions of vessels with flat cover plates andwith conical heads have been discussed in the previouschapter. The use of a knuckle radius on a cone to form atoriconical head reduces the stress concentration at thejunction. An additional improvement in design is obtainedif the torus ring is retained and the conical element isreplaced with a spherical dished element. Such a head isreferred to as a torispherical head and is widely used in t,hefabrication of process equipment. A further improvementin design resulting in a greater reduction of stresses in aformed head is obtained from the use of elliptica! dishedheads. The stresses in such heads have been analyzed bytwo methods: (1) by strain measurements on heads ofvessels subjected to internal pressure and then by m;t t he-matical computation of the stresses from the strain measure-ments, and (2) by mathematical analysis ui the headgeometry.

7.2 STRESS ANALYSIS FROM STRAIN MEASUREMENTS

An early study of strain measurements on ellipticaldished heads was reported by E. Hijhn (113) a translatiouof which appeared in Mechanical Engineering (114). Inthis study an elliptical head having a 1.97 to 1.00 major-to-minor-axis ratio and having an outside diameter of 31.7 in.,a head-plate thickness of 0.47 in., and a shell plate thicknessof 0.313 in. was subjected to internal pressures of 8, 16, and24 atm.

Before and after application of the pressure to the vessel,HShn made tomplates of the contour of t,be elliptical dishedhead and also templates of torispherical dished heads thatwere included in the investigation. He found that the

\- \ \I I

120

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torispherical dished heads deformed considerably, approach-ing the shape of an ellipse with a major-to-minor-axis ratioof about 2.0, whereas the elliptical head retained its origi-nal contour . This tends to indicate that the natural shapefor a disllzc: head on a cylindrical vessel under internalpressure is an e l l ipse having a 2 : 1 major- to-minor-axis rat io .

Strain gages were placed about the vessel in such a man-ner that strains were measured in the radial and longitudinaldirections at 18 different positions along the shell andaround the head.

The strains measured were the result of elastic deforma-tion in three directions. In thin-walled vessels radialstresses are of minor importance, and the two principalstresses in the head act at right angles to each other andare identified as hoop stress, fiL, and meridional stress fm.At the junction of the head with the shell, these stressesbecome the hoop (or circumferential) stress of the shell andthe longitudinal stress of the shell, respectively. If bothof these stresses are acting simultaneously, they may becalculated from strain measurements by use of Eqs. 6.6and 6.7.

fh = (~h2 + wm2)E

1 - $a n d

fm

= (~n2 + /.e2)E

1 - /ls(6.7)

If these stresses are divided by the shell hoop stress, stress-intensification factors are obtained which are convenientfor comparing the two principal head stresses with the cor-responding shell hoop stress existing outside of the dis-continuity at the junction. Figures 7.2 and 7.3 are plots

Fig. 7.2. Hoop-rtrers-intenriflco-

tion factor for vessels with ellipti-

cal closures-from Hijhn’s experi-

mental data (114).

Stress Analysis from Strain Measurements

Fig. 7.1. Welding elliptical head % in. thick to shell with automatic

welder--vessel is 102 in. in diameter and designed for 50 lb per sq in.

gage, 200” F service. (Courtesy of C. F. Braun & Co.)

1.6, , , , , , , , , ,

s 0 .6.-52 0.4‘ci5 0.2.EID 0sco

8 - 0 . 222 - 0 . 4II

z - 0 . 6

- 0 . 8

-10~J I I I I I I I I0 0.1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1

!_ Head radial d i s t a n c e t o m e r i d i a n -Shell radius

5 0.1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 . 8 0.0 i

- Shell axial distance from junction -.-Shell radius

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1 2 2 Selection of Elliptical, Torispherical, and Hemispherical Dished Closures

of these stress-intensification factors as calculated from the central section of the head thinned out in forming thusHijhn’s data. accounting for the peculiar hump in the head-stress curve

Figure 7.2 shows the hoop-stress-intensification factor, near the center of the head. Figure 7.3 shows the corre-Ih = f&h shell, in the head and in part of the shell of the sponding curve for the meridional-stress-intensification fac-vessel. Note that Ih in the head from the center of the tor in the head and the longitudinal-stress-intensificationhead to the point of tangency of shell and head is plotted factor in the shell. Note that similar reversals also occur.on the left-hand portion of the figure whdpeas Ih in the shellis shown in the right-hand portion of the figure. Therefore,

@he horizontal axis to the left of the point of tangency of‘.head and shell represents fractional radial distances from

the axis of the vessel to the point under consideration. Thehorizontal axis to the right of the tangency represents dis-tances along the longitudinal axis of the shell measuredfrom the point of tangency in terms of fractions of theradius. The vertical-a% intensification factor representstensile stress above the reference line of zero and compres-sive stress below the reference line. Figure 7.2 shows that

7.3 HUGGENBERGER’S THEORETICAL ANALYSIS FORMEMBRANE STRESSES

The work of Hijhn previously described in which stresseswere determined from strain measurements indicated thattwo critical stress groups exist : (1) the stresses in an elementof an elliptical dished head resulting from internal pressureand the geometry of the head and (2) the stress concentra-tions in the neighborhood of the junction of t,he head andthe shell. Huggenberger (115, 116) developed an analysisof the stresses in an element of an elliptical dished head

L,O 0 . 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1,0 0 . 1 0.2 0.3 0.4 0.5 0.6

F i g . 7 . 3 . Meridional-~hass-intensi-

fication factor for vessels with ellip-

tical closure&rom Hiihn’s experi-

mental data (114).

L Head radial distance to meridian _ Shell axial distance from junctionShell radius Shell radius

the central portion of the head is under tensile stress andthat a reversal to a compressive stress occurs at 0.86 radiusunits from the axis. A maximum compressive stress isreached at about 0.95 radius units from the axis, after whichpoint the compressive stress decreases to zero at the pointof tangency. The curve further indicates the influence ofthe head on the hoop stress in the shell. The hoop-stressintensification factor in the shell increases from zero at thepoint of tangency, reaching a maximum at about 0.3 radiusunits from the point of tangency; it then drops to a lesservalue, after which drop it levels off at about 0.7 radius unitsfrom the tangent line to a constant value of 1.0, the normalhoop-stress-intensification factor in the shell.

The dotted line on Fig. 7.2 from the center of the head tothe experimental curve indicates the probable curve thatwould have resulted if the elliptical dished head used byHShn had been of uniform thickness throughout. Actually,

from mathematical considerations. In Huggenberger’sanalysis the bending moments at the junction of the headand shell were disregarded. The equations of Huggen-berger are developed in the following section.

Figure 7.4 is a three-dimensional sketch of a differentialelement of an elliptical dished head. Arc m-n is a meridianformed by passing a plane through point 0 and through thew-w axis. Arc O-O’ is formed by passing a plane throughpoint 0 perpendicular to the w-w axis. The stress jm isthe stress on the element in the meridional direction and iscalled the meridional stress. Stress fh is the stress actingcircumferentially on the element and is termed the hoopstress. These are principal stresses, and no shearing stressesexist on the sides of the element.

Figure 7.5 is a view of the meridional plane through theelement. Because of the symmetry of the head about thew-w axis, the location of this meridian is not significant.

Page 133: Process Equipment Design

Fig.7.4. Stresses acting on differential element in an elliptical dished head

Huggenberger’s Theoretical Analysis for Membrane Stresses 1 2 3

Similarly, the circumferential or hoop force acting onopposi te s ides i s :

Fh = fhtrl d4

The location of the hoop plane is defined by angle do madeby the normal to the surface of the element and the W--Waxis . The radius, ~1 defines the curvature of the elementin the plane of the meridian; the radius r2 defines the curva-ture of the element about the W-W axis, about which axisit generates a cone; and PO is the radial distance of the ele-ment from the W-W axis.

The length of both s ides of the element in the meridionalplane is equal to rr d+, and the length of the upper side ofthe element in the circumferential plane through point 0 isrg d0. This fo l lows from the fact that for very smal l anglesexpressed in radians, the side opposite the angle is numer-ically equal to the angle in radians times the length of along s ide. The surface area of the element is equal to theproduct of the two sides, or

A = rlro d+ de (7.1)

The internal pressure acting on this area produces anormal force having the value of:

FP = pA = prlro dq5 de (7.2)

where p = internal pressure, pounds per square inch

This force is resisted by the components of the principalforces, the components being taken in the direction of thenormal to the plane of the element. The principal forcein the meridional direct ion act ing on opposi te s ides is :

F, = f&o de (7.3)

Figure 7.6, detail a shows that this force has a componentin the direction normal to the plane of the element becausethe forces on opposite faces are not 180’ apart. For afini te e lement ,

Fmn = F, d+ = f&o de d+

This fo l lows because of smal l angle re lat ionships .

As shown in Fig. 7.6, detail b, this force has a resultant,force in the radial direction which is perpendicular to thew-w axis and is:

deF h r = 2Fh - = f&l dc# de

2

As shown in Fig. 7.6, detail c, this resultant force has acomponent in the direction normal to the plane of the ele-ment, which is:

Fh,.,, = Fh de sin $I = fhtrl sin 4 d4 d0

Summing the forces normal to the plane of the elementresul ts in :

o rFmn i- Fhrn - FP = 0

f&o de d4 -I- fhtrl sin 4 d4 de - prlro d$ de = 0

C o m b i n i n g g i v e s :

f&0 + fhtrl sin 4 = prm

Dividing by rlrot gives:

f”+-=-fh sin 4 Prl r0 t

From Fig. 7.5 it follows that rg = r2 sin 4; therefore,

fm fh p,+,=i

d

r1 = meridional radius ofcurvature at point P.inches

r2 = radius of curvature of thesection perpendicular to themeridian at point I? inches

Fig. 7.5. M e r i d i o n a l s e c t i o n o f e l e m e n t o f a n e l l i p t i c a l d i s h e d h e a d .

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124 Selection of Elliptical, Torispherical, and Hemispherical Dished Closures

(a) WJ

Fig. 7.6. Components of forces in meridional and circumferential planes. (a) Components of forces in a meridional plane. (b) Components of forces in a

circumferential plane. (c) N ormol component of forces from o circumferential plane.

Tire axial force (along the w-w axis of Fig. 7.5) equalsmeridional force, F,,,, times sin 4.

Axial force = m”“p = P,, sin #J

Force = (SI ress) (area)

F,, = .fN1 (2mt)

7rrg’p = f,,L(2ar,,l) sin 4i,r

The equat ion of an e l l ipse is :

(7.6)

where rg, 2, a, and b have the significance of 5, y. a, and b,respectively, as indicated in Fig. 5.2.

Figure 7.5 indicates that fill is perpendicular to rp and istangent 1.0 the ellipse at the point P. The slope of jm isc&&ted as fo l lows:

luslopr = - -dro

Dilfereutiating Eq. 7.6 gives:

dZ b” rg- = - -.dr,, a2 z

Therefore the distance I in Fig. 7.5 is given by.

b2 ro2

a2 2

Let k = u/b. Therefore

ro2z = k2z

From Fig. 7.5 and by Ey. 7.7,

(7.7)

Substituting for Z2 its equivalent, as given by Eq. 7.6and making t: = a/b, we obtain:

Z 2 = -$ (a2 - ro2) (7.8)

Therefore

r0

sin C#I- [(ak)” + ro2(1 - k2)]‘y

\lultiplyinp both sides of Eq. 7.8 by (p/2t) gives:

(7.9)

[(ak)2 + ro”(l - k )I2 51

\ \ \I /- -~--- -

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Huggenberger’s Theoretical Analysis for Membrane Stresses 1 2 5

lht

jm = _‘op.Lsin 4 2t

(from Eq. 7.5)

Therefore

Pjm = - [(ak)Z + r&l - P)]”2t

( 7 . 1 0 )

By mathematical definition t,he radius of curvature, rl, is:

From Eq. 7.8,

Z = i (a2 - ro2)E

dZ- = Z’ = -; (a2 -dro

ro2)-%0

a2d2Z _ zjt = 2

dro2 k (U2”T

kbstitufing in F-q. 7.11, we find that

[

PO2 1 32l+ gp-Ir”ijr1 = -

- ; (a2 _ ro2)%

Simplifying, we find that

[(ak)2 + ro2(1 - k2)]“”r1 = __a2k2

Rearranging Eq. 7.4, we find that

P fm.fh=r2 --;[ 1t

I Substituting from Ecp. 7.10 and 7.12 gives:

jh = p2 P- - -t

jh = 1’2 1 --[

i [(ak)2 + ro(l - k2)]ts

[(ak)’ + ro2(1 - k2)]“z_____-____ ..~_a2k2

a2k22[(ak)2 + ro2(1 -

FIYJIII Fig. 7.j and Eq. 7.9,

t-0r2 = mFsm $I

= [(ak)2 + ro2(1 - k2)]s’

Therefore

(7.13)

(7.14)

3a2k2- -.1

E2[(ak)’ + ro2(l - k’i t

(7.11)

(7.12)

(7.15)

Equations 7.10 and 7.15 describe t.he two principalstresses, jm and .fh, in terms of the geomeky of the elliptical

dished head. These equations were first given by Hug-genherger (115) in 1925.

Examining these equations for the extreme conditionswhen r = a and b = m, we find that the stresses at thejunction of the head and the attached cylinder for thiscondition reduce to the longitudinal and circumferenlialstresses in a cylinder, .or

and

fm = z

.fh = T

At the crown center, when rg = 0, the two st,resses fhand jnL hecome equal and have the value:

(7.16)

For a hemispherical head, when k = 1.0, Eq. 7.16 reducesIO the st.ress equation for a hemi,;pherical head.

rp.fh = .I,,, = zl

For HII elliptical dished head whrre a = 2b, Eq. 7.16heconies:

jh = j,,, = !f

This indicates that an elliptical dished head ,,aving a major-to-minor-axis ratio of 2: 1 should be expected to have thesame stress in the crown as the hoop stress in the attachedshell if bending moments at the junction are not considered.

For any elliptical dished head having a value of k greaterthan 1.0 and less than 2.0, the maximum stress will occurin the center of the crown and will lie between t.he valuesof rp/t and rp/2t, as given by Eq. 7.16.

Huggenberger’s equations, 7.10 and 7.15, are cumbersomebecause of their length. The principal advantage in t,heiruse lies in the expression of the stresses in terms of the radialdistance rg. These equations may be modified so as toexpress the stresses in terms of the radius of curvature dthe closure.

From Fig. 7.5 it is evident that

r2

Substituting into Eq. 7.5 gives:

.rm = pzThis relationship ran be transferred t.o stress-intensifica-

tion form (press. is pressure) by dividing by fhOop(st&) togive :

I ( 7 . 2 7 )

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1 2 6 Selection of Elliptical, Torispherical, and Hemispherical Dished Closures

value at the junction with the shell (if bending :nc,mentsat the junction are ignored). For all elliptical dished heads

Head radial distance to meridianShell radius

Fig. 7.7. Elliptical-headstress-intensi~cation factors based on membrane

theory of Huggsnberger (115, 116).

If lh = t*,

A comparison of Eqs. 7.15 and 7.14 indicates that Eq.7.15 can be written in the following form:

(7.19)

Converting Eq. 7.19 to stress-intensification form gives:

.fh k2t ’Ihbress.) =

fhoop(shell)= 21 m(press.) -

4zm(preL%)th2(7.20)

where Zm(pre,,.) is given by Eq. 7.17.

If k = a/‘b = 2, and 1, = Zh,

1Zh(press.) = 21 m(press.) - ~Z

(7.21)WL(Pr’28S.)

The values of Zh(press.) and Zm(press.) given by Eqs. 7.17and 7.20 are plotted in Fig. 7.7.

As shown in Fig. 7.7, the hoop stress, jh, has a maximumvalue in tension at the center of the crown where PO = 0.As ~0 approaches a, the hoop stress in tension decreases,passes through zero, and reaches a maximum negative

in which k is greater than 2.0, the maximum stress will bethis compressive stress (bending moments being ignored)occurring at or near the junction with the shell.

Although Huggenberger’s relationships do not take intoaccount the bending-stress concentrations in the head nearthe junction with the shell, they. are useful in analyzingthe stresses in the remainder of an elliptical dished headfor any major-to-minor-axis ratio.

7.4 COATES’S THEORETICAL ANALYSIS FOR BENDINGSTRESSES AT THE HEAD-SHELL JUNCTION

The relationships developed by Huggenberger are limitedbecause no allowance was made for bending moments thatexist in the head at and near the junction. The existenceof this bending moment is graphically indicated by thecurves determined from Hijhn’s data as shown in Figs. 7.2and 7.3 and is further indicated in Fig. 7.7, in which hoopstresses calculated from Eqs. 7.17 and 7.20 are plotted.Comparing the hoop pressure stress for an elliptical dishedhead in which k = 2.0 as predicted by the appropriatecurve of Fig. 7.7 with the combined hoop stress in Pig. 7.2.we find that a sharp discontinuity is indicated at the junc-tion of the head and the shell. This discontinuity existsbecause the head and shell have been treated as separate,disconnected membranes with no restraints at the edges.Figure 7.8 shows the deformation of an ellipsoid or revolu-tion having a major-to-minor-axis ratio of 2.0 when underinternal pressure. A vessel of such a shape when underinternal pressure will deform to become more spherical,the shell deforming outward along the minor axis andinward along the major axis.

Figure 7.9 illustrates the effect of the deformation of anelliptical dished head when attached to a cylindrical vessel.Under internal pressure the elliptical dished head shown inFig. 7.9 will tend to deform inward as was shown in Fig.7.8. The cylindrical shell will tend to deform radially out-ward under the influence of internal pressure. The joiningof the head to the shell restrains both of these deformations.This results in the introduction of bending moments in thehead and in the shell. The effect of the junction is to bendthe shell inward and to bend the head outward with respectto their unrestrained positions. As a result a compressivestress is induced in the outermost fiber of the knuckle of the

Fig. 7.8. Deformation of an ellipsoid of revolution under internal pressure.

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Coates’s Theoretical Analysis for Bending Stresses at the Head-Shell Junction 1 2 7

with zero pressure

Fig. 7.9. Deformation of an elliptical dished head and shell in the junction

zone. (Courtesy of F. 1. Maker.)

head whereas a tensile stress is induced in the innermostfiber at the same location. Similarly, the bending of theshell in the opposite direction tends to increase the tensilestress in the outermost fiber of the shell in the neighborhoodof the junction.

7.4a Coates’s Relotionships for local Bending Stressesat the Junction when Head and Shell Are Not Joined.W. M. Coates (117) mathematically investigated the bend-ing at the junction of a dished head and cylindrial shell.A longitudinal strip of the shell in the neighborhood of thejunction which is bent inward under the influence of thehead is selected for analysis. The force causing thisdeformation can be considered to be an inward radial shearforce acting on the end of the strip. This force is resistedby the bending forces set up in the strip and by the com-pressive hoop stress opposing a tendency for the shell cir-cumference to decrease. The total resistance to this ten-dency to deform inward results in radial shear forces, longi-tudinal bending stresses, and circumferential compressivestresses.

,Since the shell behaves in an elastic manner and is rela-

tively thin in comparison to the diameter. the radial forceon the strip under consideration can be considered to beproportional to the radial deflection. Therefore, the stripmay be considered to act like a beam on an elastic founda-tion. Such a beam under a point load will deflect immedi-ately under the load because the supporting foundation iselastic. The stiffness of the beam will transfer a portion ofthe load to either side of the force, and this will resultin a smaller elastic deflection, which is a function of theresistance of the foundation and the distance from the pointof load application. The theory for such a beam wasdeveloped in Chapter 6. The equation of the deflectioncurve for such a beam is given by Eq. 6.74. By use of thisrelationship Coates developed an analysis of the discon-tinuity stresses.

Figure 7.10 illustrates an element of the shell underconsideration where

P = radius of shell& = thickness of shell

x1 = longitudinal distance from junction

y1 = deflection of shell at junctionT2 = normal hoop forceQ = meridional shearing force

MI = meridional bending momentMz = hoop bending moment

The following relationships summarize Coates’s deriva-tions for the shell:

1” = - 2~3D~ e-“+~ [Qo cos /3gcl + /31Mr,(cos /31x,

- s i n /3rzr)] (6.74)

where D = Ets312(1 - $+)

and

p = Poisson’s ratio

- sin &zr)] (7.22)

Q = D $ = -e-B1z1 [QO(COS /3lzl - sin /3rzr)1

- 2/?lM0 sin /31x1] (7.23)

day1 1MI = - D - = - e-flP1 [Q. sin plzlda2 81

+ ,&MO(COS PIZI + sin 8141 (7.24)

M2 = lrMt (7.25)

The corresponding relationships of Coates for the headare :

(7.26)

*I92 =

-\i.

3(1 - UP)rz2th2

Fig. 7.10. Forces and moments acting on element of shdl

(7.27)

Page 138: Process Equipment Design

- - - \ \ \I I

128 Selection of Elliptical, Torispherical, and Hemispherical Dished Closures

a n d MO = 0

0 1 ’ ’ ! ’ !\-0.1 , , , , / , /-0.21 11 11 11

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0ax

Fig. 7.11. Functions for computing stresses based on the theory of beams

on elastic foundations (29).

where x2 = distance measured along meridian from planeof junction, inches

r2 = radius of curvature of hoop section of head,inches (See Fig. 7.5.)

1 (e-~SZI)[Qo c o s (32X.23% = 2p13D

- /3&fo(cos pzx2 - sin /32s2)] (7.28)

T2 = 3 yz = 2Plrr-2

2x2 - sin @2x2)] (7.29)

(7.30)

Ml = D __ = p- (e-@2”2)[ -Qo sin p2x2d%dxz2 r2Pl

+ P~Mo(cos P2x2 + sin P2x2)l (7.31)

M2 = /.LM,

Substituting these values in the equations developed forthe cyl inder and for the head, gives the fol lowing equations.

For the she l l ,

6Mlt,2 = fm(bending) = f7+1 sin plxl (7.33)

6M2t,2 = f Qbending) = ?--41’r1 sin PlrCi (7.34)

For an elliptical dished head (r2 is a variable)

6Ml - 3pk2a2-=th2

.f m(bending) = -_____

4 d33(1 - p2)thr2I

-=th2

f h(bending) =cos (/-‘I P2dsz)

0

GM2

22 p2 dx2 j (7.35)

(7.36)

7.4~ Solution of Equations for the Shell. Equations7.33 and 7.34 predict the major bending stresses developedin the shell as a result of the discontinuity at the junction.The terms in the right-hand side of these equations areusually constant for a given vessel and pressure with theonly variable, x1, the linear axial distance (inches) from thejunct ion measured along the shel l . Use of Fig. 7.11 reducesthe solution of these equations to a simple calculation.The constant @I is evaluat,ed for the shell, and suitableincrements of x1 are selected (such as Ax1 = 0.05a). Valuesof the product plzl are tabulated, and the correspondingfunctions of e-alZl sin ,81x1 and e-Slzl cos plzl are deter-mined by use of Fig. 7.11. These functions are multipliedrespectively by 3pk2/4@12t,2 and by -pk2a/4t, to give thetW0 major discontinuity stresses,fm(hendiIlr) andfh(bending), inthe shell as a function of x1.

The StreSS-intenSifkatiOn factors, zm(bendinr) and zh(bending),

may be computed by the use of the fol lowing relat ionships: )tan (Y dy2

M2 = /.LM~ - D(l - /L’) ~ .-r2 dm

(7.32)

where LY is the angle between the normal to the head andthe plane of the junction. Figure 7.11 is shown for con-venience in Pvaluating the functions of ,Blxl or &x2.

7.4b Analysis of Edge Loads and Local Stresses whenHead and Shell Are Joined. The joining of the head to theshel l imposes the fo l lowing restr ic t ions i f the shel l and headare of the same thickhess:

1. The radial displacement of the shell and of the head atthe junction must be equal.

2. The slope of the total-displacement curves for thehead and for the shell at the junction must also be equal.

Therefore

Q. 2k”fvl

I fm(bending)m(bending) =

fe-@+l sin /lIzI (7.37)

h(preas.)

Z f h(bending)h(bending) = ___ =

fe--Bl+ sin /31x1 (7.38)

hbreas.)

If k = 2, Eqs. 7.37 and 7.38 reduce to:

Z (7.39)

Z h(bending) =

The functions e-sz sin px and eYflz cos @x have maximumvalues at bx = a/4 and /3x = s/2, respectively. As aresult the maximum meridional bending stress in the shelloccurs at a distance of ,81x1 = ?r/4, and the maximum hoopbending stress in the shell occurs at /?lxl = 0. This cor-responds to the fol lowing.

Page 139: Process Equipment Design

Coates’s Theoretical Analysis for Bending Stresses at the Head-Shell Junction 129

The location of maximum Zm(bending in sheu) is:

x1 = 0.61 dat,

The location of maximum Zh\l,,.nding in sherr) is:

x1 = 0

The distance to damp out hending stresses is:

x = 2.44 dat,

7.4d Solution of Equations for an Elliptical Dished Head.Equations 7.35 and 7.36 predict the major bending stressin an elliptical dished head as a result of the discontinuityat the junction. The solution of these equations is moredifficult than the solution of those for the shell since theradius of curvature of a hoop section of the head, r2 (seeFig. 7.5) is a variable. Therefore f12, which is dependentupon r2 (see Eq. 7.27) is also a variable. A further com-plication arises from the fact that ~2 is the linear distancealong the meridian of the head (as measured from thejunction). This distance is not a simple function of r-0(the head radial distance to the meridian) and can only beevaluated graphically or by the use of elliptic integrals.This prevents the formal integration of the term p2 dx2 inboth Eq. 7.35 and 7.36.

To show a method of determining the distance 22 as afunction of ro, reference is made to Fig. 7.12, which illus-trates an ellipse of axis u-b circumscribed by a circle ofradius a (126).

The distances ro and Z may be expressed as:

r0 = a sin cy (7.41)And by Eq. 7.6

.~-z = P d,2 - ro2

a

Z = b cos 01 (7.42)

Measuring the arc length, s, from point B, the top of theminor axis , we obtain :

BP = / ds = 1 l/dro2 + dZ2 (7.43)

Rut

dro = a cos (Y dor

I hereforedZ = -bsinardor

S= I ca da + b2 sin2 (Y dol (7.44)

where the upper limit of cx corresponds to the value of a! atpoint P.

Substituting 1 - sin” (Y for cos2 (Y gives:

s = /u l/iie- (a2 - b2) sin2 ar dar0

=a /oa di - K2 sin2 a! dcu (7.45)

where K i= 1/a2? = eccentricity of the ellipse (7.46)a

(7.46a)

Mathematicians have proved that Eq. 7.45 can not heevaluated in f ini te form in terms of the e lementary funct ionsof CY. Equation 7.45 defines a function denoted by E(K, (Y)which because of its origin is termed an “elliptic integral,”o r

E(K, a) = lo” msin’ (Y dol (0 < K < 1) (7.47)

Values of the function E in terms of’K and cy have beendetermined by the use of inf ini te-ser ies calculat ions . Tablesof values of the function are given in the literature (127).

To determine ~2 as a function of r-0, the value of s at thepoint in question is subtracted from the value of s fora = 90”. As the function E is equal to s/a (see Eqs. 7.45and 7.46) and as it is convenient to plot stress-intensihca-tion factors versus the dimensionless ratio ro/a, a table maybe prepared for an el l ipt ical dished head giving correspond-ing values of CY, ro/a, s/a, x2/a, and rz/a. Such a tabula-tion is shown in Table 7.1 for an ellipse with a major-to-minor-axis ratio (k) equal to 2.0 to 1. In preparing thistable even angles of a were selected rather than even incre-ments of ro/a in order to avoid the necess i ty of interpolat ionof tables of elliptic integrals (127) used to determine valuesof E. Selecting o( fixes ro/a because by Eq. 7.41 ro/a equalssin CL The value of rz/a (radius of curvature divided byshell radius) is determined by use of Eq. 7.14.

Equations 7.35 and 7.36 may be solved by tabulating rz,fi2, and z2 as a function of rg. A summation of the incre-ments of pz Ax2 is used to determine the integral of p2 dxz.The values of this integral may then be used to determinethe corresponding functions of e-‘zz2 sin &x2 and e%% cos&xz by means of Fig. 7.11. Substitution of the values ofthese functions and the corresponding value of r2 into Eqs.7.35 and 7.36 permits the determination of the major

h’g. 7.12. T r i g o n o m e t r i c v a r i a b l e s f o r a n e l l i p s e .

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1 3 9 Selection of Elliptical, Torispherical, and Hemispherical Dished Closures

Table 7.1. Dimensionless Ratios for an Ellipse in

Which a/b = 2:OIf k = 2 and t, = th,

(See Fig. 7.12 [127].) I m(bending) =

ro-=a

a sin (Y

90” 1.000089” 0.9998588” 0.9993987” 0.9986386” 0.9975685’ 0.9961980” 0.9848175” 0.9659370” 0.9396965=’ 0.9063160” 0.8660355O 0.819155o” 0.7660445O 0.707114o” 0.6427935” 0.57358

S-=a

E.2

1.21111.20231.19361.18481.17611.16731.12251.07591.02660.97430.91840.85880.79540.72820.65750.5833

5 2-=a

(Ego - Ed

00.00880.01750.02630.03500.04380.08860.13520.18450.23680.29270.35230.41570.48290.55360.6278

f-2-a

1.0001.00151.0021.0041.0081.0121.0441.0961.1641.2391.3221.4101.4961.5811.6601.734

(7.47a)

bending stresses, fm(bendinp) and fh(bendins), in an ellipticaldished head.

Equations 7.35 and 7.36 may be converted into stress-intensification-factor form by dividing by @d/2&), or

I- 3t,k2a

m(bending) =‘ithr2 d3(1 - P2)

(sin [ /3s &s) (7.48)

Ih(bending) (7.49)

Ih(bending)

7.4e Combined Stress- intensif icat ion Factors. Thebending stresses can be combined with the pressure stressesto give the maximan combined stresses.

For the shell,

I n+(combined) = Im(bending) + Zm(press.)

= I m(bendina) + 0.5 (See Eq. 7.37.) (7.52)

Ih(combined) = I h(bending) + Z h(press.)

= zh(bending) + 1-O (See Eq. 7.38.) (7.53)

For the elliptical closure,

Z m(combined) = In(bending) + Zm(press.) (7.54)

where Im(ben&ns) is given by Eq. 7.50, and Im(press.) is givenby Eq. 7.17.

Z h(combined) = Ih(bending) + Ih(press.) (7.55)

where Ih(bendins) iS given by Eq. 7.51, and Ih(preSS.) iS given

by Eq. 7.20.

7.5 EXAMPLE CALCULATIONS USING COATES’SMETHOD OF ANALYSIS

A cylindrical steel vessel has elliptical dished closureswith a major-to-minor-axis ratio of 2.0, The diameter ofthe vessel is 80 in., and the vessel is to operate under aninternal pressure of 100 psi. The shell and the heads havethe same thickness, 1>/4 in. Determine the maximum

Table 7.2. Solution of Stress-intensification Factors in the Shell

(3) (4) (5) (6) (7) (8) (9)

XlXl

0a01x1 F,(Pl& FZ(@lxl) Im(bending) Ih(bending) Im(combined) zh(combined)

--_~~- ~..~-__0 0 0 1.0 0 0 0 0.50 1.001 0.025 0.182 0.82 0.15 0.272 0.082 0.772 1.082 0.05 0.364 0.66 0.255 0.463 0.139 0.963 1.143 0.075 0.546 0.50 0.30 0.544 0.163 1.044 1.164 O.JO 0.728 0.37 0.325 0.590 0.177 1.090 1.186 0.15 1.092 0.15 0.30 0.545 0.164 1.045 1.168 0.20 1.456 0.03 0.23 0.418 0.125 0.918 1.13

10 0.25 1.820 -0.04 0.155 0.281 0.084 0.781 1.0812 0.30 2.185 -0.06 0.100 0.182 0.055 0.682 1.061 4 0.35 2.558 -0.06 0.025 0.045 0.014 0.545 1.011 6 0.40 2.912 -0.04 0.010 0.018 0.005 0.518 1.011 8 0.45 3.276 -0.01 0 0 0 0.500 1.002 0 0.50 3.640 0 0 0 0 0.500 1.00

where Fl(xr) = e-p1z1 cos plzlF2(zl) = e&r1 sin /3lzl

Page 141: Process Equipment Design

Example Calculations Using Coates's Method of Analysis 131

stress-intensification factors in the shell and closures in the To solve the above equations, Table 7.2 was prepared.neighborhood of the junction. One-inch increments of ~1 were selected up to 4 in. from the

For the shell, a = 40 in., b = 20 in., k = a/b = 2.0. junction, and 2-in. increments from 4 to 20 in. TheseBy Eq. 6.86 and with p = 0.3, values were divided by a (40 in.) and tabulated in column

42, and also were multiplied by PI (0.182) and tabulated in

Pl =d4 3(1 - P2) =

y’3(1--0.32)

r2t 2= o 182 column 3. The values of eVP121 cos @rzr and e-@lzl sin @rzr

8 (40)2(1.25)2 ’ were determined from Fig. 7.11 and tabulated as Fr(zr)

Thereforeand Fs(sr), respectively, in columns 4 and 5. Substitutionof values from COhmn 4 into Eq. 7.40 gives Ih(bending) tabu-

3k2

( )___ = 1.815

lated in column 7. Substitution of values from column 5

4812t,a into Eq. 7.39 gives Im(ben&ns) tabulated in column 6. Sub-stitution of values from column 6 into Eq. 7.52 and of

By Eq. 7.39, zm(ben&ng) = 1.815eCsl”l S i n @iZi COhnn 7 into Eq. 7.53 gives zm(combined) and zh(oombined)By Eq. 7.40, zh(bending) = -e-@?l c o s plxl tabulated in columns 8 and 9, respectively.BY Eq. 7.52, ~m(combined) = zm(bending) + 0.5 The example calculation for the head is as follows.BY Eq- 7.53, ~h(combined) = zh(bending) + 1.0 Since the head has a major-to-minor axis ratio of 2.0 tol,

(1) (2)

Table 7.3. Solution of Stress-intensification Factors in the Head

(3) (4) (5) (f-9 (7) (8)

a r2 no/a P2 x2 ( i n . ) Ax2 B2 Ax2 Wz Axz

90898887868580757065605550454035

40.00 1.000 0.1818 0.000 0.000 0.0000 0.000040.06 0.999 0.1817 0.352 0.352 0.0640 0.064040.08 0.999 0.1815 0.700 0.348 0.0632 0.127240.16 0.998 0.1813 1.052 0.352 0.0638 0.191040.32 0.997 0.1811 1.400 0.348 0.0630 0.254040.48 0.996 0.1808 1.752 0.352 0.0636 0.317641.76 0.985 0.1780 3.544 1.792 0.3190 0.636643.84 0.966 0.1738 5.408 1.864 0.3240 0.960646.56 0.940 0.1688 7.380 1.972 0.3329 1.293549.56 0.906 0.1634 9.472 2.092 0.3418 1.635352.88 0.866 0.1580 11.708 2.236 0.3533 1.988656.40 0.819 0.1531 14.092 2.384 0.3650 2.353659.84 0.766 0.1488 16.628 2.536 0.3774 2.731063.24 0.707 0.1447 19.316 2.688 0.3890 3.120066.40 0.643 0.1405 22.144 2.828 0.3973 3.517369.30 0.573 0.1382 25.112 2.968 0.4102 3.9275

(9) (10)

FliW2 Ad F2W2~d hbend.) zm(bend.) zm(press.) zh(press.) kcomb.) &comb.)

1.000 0 1.000 0.000 0.500 -1.000 0.500 0.0000.936 0.058 0.934 -0.105 0.501 -0.993 0.396 -0.0590.874 0.110 0.872 -0.198 0.502 -0.990 0.304 -0.1180.811 0.156 0.809 -0.282 0.503 -0.986 0.221 -0.1770.751 0.193 0.745 -0.347 0.505 -0.975 0.158 -0.2300.226 0.226 0.684 -0.405 0.506 -0.966 0.101 -0.2820.426 0.314 0.408 -0.544 0.523 -0.868 -0.021 -0.4600.220 0.313 0.201 -0.518 0.548 -0.729 0.030 -0.5280.075 0.264 0.065 -0.411 0.583 -0.545 0.172 -0.480

-0.012 0.194 -0.010 -0.284 0.620 -0.375 0.336 -0.385-0.055 0.125 -0.042 -0.172 0.661 -0.193 0.490 -0.235-0.070 0.068 -0.047 -0.087 0.705 -0.010 0.618 -0.057-0.060 0.026 -0.040 -0.032 0.748 0.158 0.716 0.118-0.044 0.001 -0.028 -0.001 0.790 0.314 0.789 0.286-0.028 -0.011 -0.017 0.012 0.830 0.455 0.842 0.438-0.014 -0.014 -0.008 0.015 0.865 0.572 0.880 0.564

f--__r_ ~-I . ..-- .-

-\--\- - \ I 7- ----- --- -.

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132 Selection of Elliptical, Torispherical, and Hemispherical Dished Closures

1.2

1.(1 I -

= 0.82

5: 0.6

g 0.42.$ 0.2.L!5 IIca.E& -0.2250.48* -0.6

/ -

-0.8 \I

-1.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.1 0.2 0.3 0.4 0.5 0.6Head radial distance to meridian =(2)-L_

Shell axial distance from junction

Shell radius Shell radius= (2)-

Fig. 7.13. Hoop-stress intensifi-c a t i o n b a s e d o n c o m b i n a t i o no f H u g g e n b e r g e r ’ s membrane

s t r e s s e s a n d Gates’s b e n d i n gstresses.

Table 7 .1 may be correctly used in the analysis of I hr st rcssesin the head. By Eq. 7.27,

The value of r2 may be determined from colrmm 6 ofTable 7.1 (a = 40 in.) for the values of angle a selected incolumn 1 of Table 7.3. The coefiicien( for Eq. 7.50 is:

- 3 a (--3)(JO) -72.5,~~~~~~ =

P2 d3(1 - P2) r2 d~~~33) = pp

By Eq. 7.50,

By Eq. 7.51,

As pz varies with ~2, the above equations can not beformal ly integrated. Integration of /32 dx2 is performed bysteps. By use of column 5 of Table 7.1, values of x2 aredetermined for corresponding values of LY and r2. Thesevalues of x2 are tabulated in column 4 of Table 7.3. Di f fer -ences are taken to give the incremental values of Ax2 listedin column 5. Values of Ax2 are multiplied by /32 and tabu-lated in column 6. Accumulative summations of /32 Ax2are tabulated in column 7 and correspond to the integralfrom 0 to 52 of /32 dxz. Then values of Fl(x2) and Ft(x2)are determined by the use of Fig. 7.11 in the same manner aspreviously indicated to give Zh(hendinp) and Z,,,(l,en~~i~~e) listedin columns 10 and 11.

To det,ermine the combined stress-intensification factors,the pressure-stress-intensification factors must be added tothe bending-stress-intensification factors. The meridionaland hoop pressure-stress-intensification factors may bedetermined by Eqs. 7.17 and 7.20, respectively, and aretabulated in columns 12 and 13, respectively, of Table 7.3.The combined meridional and hoop stress-intensificationfactors determined by use of Eqs. 7.54 and 7.55 are tabu-lated in columns 14 and 15, respectively.

The bending-stress-intensification factors for this vesselat and Ilear the junction, zh(bending) and zm(bending), areplotted in Figs. 7.13 and 7.14. It should be pointed OUIthat these curves are specific for a vessel having the dimen-sions given in the problem and are not general .

Referring to Fig. 7.13, we find that the computed hoopstress-intensification factor at or near the junction by useCoates’s method of computation is shown as a dashed curvelabeled B. Inspection of the curve indicates that thebending stresses in both the head and the shell are rapidlydamped out within a short distance from the junc*tion.Huggenberger’s theoretical curve for t.he hoop stress in anelliptical dished head having a major-to-minor-axis ratioof 2 to 1 and the theoretical hoop stress in the shell areshown in Fig. 7.13 by the dotted curves labeled A, Addi-tion of curve A to curve B results in the combined curveshown as a solid line labeled C. This combined curve, C,gives the predicted stress-intensification factor in both theelliptical dished head and shell for the vessel described.Figure 7.14 shows the corresponding meridional stress-intensification factors for the same vessel.

Although the combined curves of Figs. 7.13 and 7.11 arespeci f ic for the vesse l descr ibed, i t i s in teres t ing to comparethese curves with those of Figs. 7.2 and 7.3, which werecomputed from strain measurements of E. Hiihn. In gen-

Page 143: Process Equipment Design

era1 it may be seen that the meridional and hoop stress-intensification factors have the same shapes in the corre-sponding rurves.

7.6 EFFECT OF MAJOR-TO-MINOR-AXIS RATIO

The American Pet.roleum Institute formed the API Com-mittee on Untired Pressure Vessels in 1930 to formulate acode for the petroleum induslrj. An unpublished memo-randum entitled “Streses in Heads of Pressure Vessels”was prepared for this committee in 1932 by F. L. Maker,then of the Standard Oil Company of California.* In thismemorandum curves were presented for computed maxi-mu111 stresses and maximum strains for elliptical heads ofvarious depth ratios; the computations were made by usingGates’s method of calculation and a ratio of shell radius,P, lo head thickness, t, of 32 (r/t = 32). Figure 7.15 showsthe ratio of maximum stress or strain in the head to thehoop stress or strain in the shell as a function of the ratioof major to minor axis k of the elliptical head. The follow-ing is a quotation from Maker’s original memorandum inreference to Fig. 7.15: “It was found that as the ratio of theaxes of the ellipse varied there was a change in the locationof the point at which the maximum stress occurred. Forheads flatter than k = 2.5, hoop compression on the outsideface of the knuckle is the governing point. For headshaving the value of k between 2.5 and 1.2, the meridionaltension on the inside face of the knuckle governs. Forheads having the value of k between 1.2 and 1.0 (the lastrepresenting a hemispherical he’dd) hoop tension at thejuncture is the governing st,ress. For the usual shape ofelliptical head having a depth ratio of 2.0, the discontinuityPtresses added to the membrane stresses give a maximum

* Private communiratiou.

Effect of Major-to-minor-axis Ratio 1 3 3

of 1.09 times the stress in the cylindrical shell of the samethickness.”

The upper dashed line in Fig. 7.15 shows the maximumstrain ratios calculated by Coates’s relationships. Thesolid line from k = 3.5 to k = 2.5 and the dotdash linefrom k = 2.5 to k = 1.0 are also maximum stress ratioscalculated from Coates’s relationships. The maximum ofstrain ratios is greater than the maximum of stress ratiosover the entire range for k greater than 1.25. However,the maximum-stress theory is generally used as the basisof pressure-vessel design since experimental tests have comeout in agreement with calculations based on this theory.Experimental tests have shown that elliptical heads havingk = 2.0 are as strong or stronger than the shell. (One.point from a test by T. M. Jasper is shown in Fig. 7.15 atk = 2.0.) For this reason the proposed curve was loweredslightly to pass through 1.0 at k = 2.0. Calculations basedon the maximum-strain theory failure have indicated thata hemispherical head should have a thickness of only 41 y0of the shell thickness. Simple membrane theory withoutallowance for the discontinuity stresses would indicate ahead thickness of 50% of the shell thickness. Tests ofvessels with hemispherical heads having thicknesses of 50 y0of the shell thickness have shown such heads to be as strongor stronger than the shell. For these reasons the proposedcurve was drawn to pass through 0.5 at k = 1.0.

For many years the proposed curve shown as a solid linewas used as the stress-intensification factor, referred to as v,by the ASME code (11). In recent years this curve hasbeen replaced by Eq. 7.56.

l’ = $(2 + k2) (7.56)

t= ..@--+e2fE - 0.2~

(7.57)

Fig. 7.14. Meridional-stress intan-1 siflcation based on combination of

‘hggenberger’s membrane stresses

and Coat&s bending stresses.

ii 0 . 8s-” 0 . 6EE 0.4s‘Cyj 0.2z‘Z5 0.E:,f$ - 0 . 2z1 - 0 . 4s.-02, - 0 . 6E

- 0 . 8

-1.00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 liO 0.1 0.2 0.3 0.4 0.5 ( iLH e a d radial d i s t a n c e t o m e r i d i a n

= (;)A Shell axial distance from junctionShell radius Shell radius I (2)-

Page 144: Process Equipment Design

134 Selection of Elliptical, Torispherical, and Hemispherical Dished Closures

f;\

I-

,-

Hoop strain-outsideface of knuckle

Y . I Proposed;J minimum depth

for eUiptica\ heads

Meridional tension-inside face of knuckle

II I I I I I I I I I I I I I I I I I I1 I I I

3.0 2.5 2.0 1.5 1.0k, ratio of major to minor axis of elliptical head

Fig. 7.15. Computed maximumstresses and strains in ellipticalheads as ratios of stresses andstrains in attached cylindricalshell. (Courtesy of F. 1. Maker.)

where V = stress-intensification factor from Eq. 7.56E = welded-joint efficiency (See Chapter 13.)c = corrosion allowance, inches

a

The most widely used el l iptical dished head is that havinga major-to-minor-axis ratio of 2.0 and a thickness approxi-mately equal to the thickness of the shell to which it isattached. The discussion here will be limited primarilyto vessels with heads of this shape.

Inspection of the hoop and meridional stress-intensifica-tion curves computed from strain measurements (see Figs.7.2 and 7.3) and from elastic theory (see Figs. 7.13 and 7.14)indicates that in the neighborhood of the junction, themaximum tensi le s tress i s the combined meridional bendingand longitudinal s tresses in the outer f ibers of the outer sur-face of the shell near the junction. The combination ofhoop discont inuity and shel l -hoop stresses in the same regionproduces a tensile stress nearly as great. The maximumcompressive stress is the combined membrane and meridionalbending stresses located in the outer fiber on the outer sur-face of the head near the junction.

The meridional bending stress produces a tensile stresson the outer surface of the shell near the junction and acompressive stress of about equal magnitude on the innersurface. The net effect is to produce a nonuniform stressdistribution across the thickness of the head and of theshell. A very small amount of plastic deformation willrelieve these high longitudinal stresses and cause a redis-tribution of the stresses which cannot be evaluated by thetheory of e las t ic i ty .

The tendency of the elliptical dished head to deform

inwardly at the junction under internal pressure results ina radial shear force inward on the shel l . This force opposesthe internal pressure acting in the opposite direction on theshell at the junction. Thus the head may be considered to

act as a reinforcing ring on the end of the shell reducingthe average hoop stress on the shell near the junction.Although the outer-fiber stresses resulting from bendingmay exceed the hoop stress in the shell proper, the netresult of the bending stress, membrane stress, and shearwil l be a reduct ion in average s tress in the shel l cross sect ionin the longitudinal direction at and near the junction.

The shear between the head and the shell acts radiallyoutward on the head opposing the tendency of the head tobend inward at the junction. The net result is to reducethe compressive stress in the head near the junction. Thusthe shell serves to reinforce the head.

At the center of an elliptical dished head having a major-to-minor-axis ratio of 2.0, both the meridional and cir-cumferential hoop stresses are equal to the hoop stress inthe shell proper. However, this is a point condition, andthe average stress over a finite area at the center of thehead is less than the hoop stress in the shell proper.

The statement is sometimes made that an ellipticaldished head having a major-to-minor-axis ratio of 2.0 is asstrong as the cylindrical shell to which it is attached. O nthe basis of average stresses over any f inite area of the heador the shell near the junction, such a head and the junctioncan be considered to be slightly stronger than the cylinderproper . This has been verified by years of experience anda multitude of tests. In 1931, T. M. Jasper, then Directorof Research of A. 0. Smith Company, made the comment(124) “Elliptical heads of 2 to 1 ratio will fail a cylinderwall of the same thickness if properly made and attached.”

---

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Relationships of Rhys-Stresses at Junction of Knuckle and Crown in Torispherical Closures 135

7.7 TORISPHERICAL DISHED CLOSURES

The torispherical head, often referred to as the flanged-and-dished head, has previously been shown in Fig. 5.7(details c and d). This head is formed from a flat platewith two radii, the larger being the crown radius or radiusof dish and the smaller being the knuckle radius, sometimesreferred to as the inside-corner radius. Heads of this typewere widely used before the development of dies for theelliptical dished head. In the early years the knuckle didnot have any particular radius, and the radius was fre-quently only 1 or 2 in. The radius used was whatever theboilermaker or flangesmith could hammer out on hot form-ing blocks. The crown radius is always made equalto the diameter of the shell on the theory that this providesa stress in the spherical portion of a head equal to the stressin the cylindrical shell. Fai ures in the field and a number1of tests indicated the weakness of the knuckle of this typeof dished closure. Because of these failures, larger knuckleradii were used, and the ASME Boiler Construction Codeadded the restriction that the knuckle radius be not lessthan 6% of the vessel diameter. This limitation is animprovement but does not define an optimum dished head.

Figure 7.16 illustrates some of the various shapes of tori-spherical dished heads possible for a given depth of dish,b, less than the diameter of the vessel. One extreme caseis shown by the uppermost curve where the crown radiusis infinity and the knuckle radius, rb, is a maximum andis equal to the depth of dish. The other extreme case isshown by the lowermost curve where the knuckle radiusis equal to zero and the crown radius, ~3, is a minimum.An infinite number of knuckle and crown radii combinationsexists between these two extremes, one of which is illus-trated by ~1 and rz.

An examination of Fig. 7.16 indicates that as the crownradius increases from its minimum value of rg to infinity,the ratio of the knuckle radius to the crown radius (r1 to rc)goes from zero to a maximum value and back to zero.Hiihn (114) and Boardman (128) showed that when thisratio is maximum, the resulting torispherical dished headmost closely approximates an elliptical dished head.

HShn showed that the ratio of rl/r, of a torispherical headis maximum when

r1 m-krcmrtx d(k2 + 1) - 1

where k = ”b

(7.58)

7.8 APPLICATION OF COATES’S RELATIONSHIPS TOTORISPHERICAL CLOSURES FOR STRESSES AT THEJUNCTION OF THE SHELL AND KNUCKLE

The same procedures used for calculating hoop andmeridional bending stresses in elliptical closures may beused for torispherical closures in the knuckle zone. Theradius of curvature, r2, of a torispherical knuckle is differentfrom that in an ellipse and may be calculated as follows:

r2 = rl + (a - r1) set (90 - 4) (7.59)

where r1 = knuckle radius, inchesa = radius of shell, inches

(90 - 4) = angle between rl and diameter

In determining /3z (Eq. 7.27) Eq. 7.59 is used to evaluater2 rather than Table 7.1, which was used for the ellipticalclosure.

For the knuckle zone,

Equation 7.60 may be used with Eq. 7.58 to determinethe values of /3szZ for the solution of Eqs. 7.50 and 7.51 forthe knuckle.

The limit of the knuckle in terms of the angle $I is givenby:

a - r1sin 4 = ___( )rc - rl

where r, = radius of crown, inches

The value of the angle r$ determined by the aboveequation may be substituted into Eq. 7.60 to determinethe maximum value of 2s in the knuckle zone.

Tables similar to Tables 7.2 and 7.3 may be establishedfor the torispherical head to determine values of bothI m(bending) and Ih(bendinz) in the shell and head, respectively,at and near the junction of the head and shell.

7.9 RELATIONSHIPS OF RHYS FOR STRESSES AT THEJUNCTION OF THE KNUCKLE AND CROWN INTORISPHERICAL CLOSURES

The change in the meridional radius of curvature at thejunction of the knuckle and crown produces an additionaldiscontinuity. C. 0. Rhys (124) has suggested the follow-ing equations for calculating the bending stresses at andnear the junction of the knuckle and the crown.

For the knuckle,

.fh(press.) = ‘sh 2 - ;( )

(7.61)

Fig. 7.16. Possible variations of radii in torispherical heads. (Courtesyof F. 1. Maker.)

\-- -

TFT- ~--- -~ - -

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1 3 6 Selection of Elliptical, Torispherical, and Hemispherical Dished Closures

(4 (b)

Fig. 7.17. Deformation of o torispherical head under internal pressure

(after Hiihn [l 141). (a) Elastic deformation. (b) Plastic deformation.

(7.62)

fh(bending)

(7.63)

(7.64)

Z 4sm(press.) = - -

2ath(7.66)

zh(txnding) = [ 3;2!! (2 - l)] e-J”os=f” COS r/j* &

h r1 0

(7.67)

%or the crown,

r2tyZh(press.) = ______2ath

Jo(7.68)

(7.69)

(7.70)

(7.71)

(7.72)

(7.73)

Zm(bendinr)

7.10 DISCUSSION OF EQUATIONS OF COATESAND RHYS

In using the equations of both Coates and Rhys for tori-spherical closures, the different radii that are used shouldnot be confused. The knuckle radius, rl, is a radius in ameridional plane, is a constant for a given head, and isidentical to icr in Table 5.7. The knuckle has anotherradius of curvature, r2, in the plane perpendicular to themeridian, as shown for the case of the ellipse in Fig. 7.5.This radius is a variable, as given by Ey. 7.59, is equal tothe shell radius at the junction of the knuckle and shell,and is equal to the radius of the crown, r,, at the junctionof the crown and knuckle. Because pt and r2 are variablesin the knuckle, integration of the functions of p dx arerequired. In the crown, r2 is a constant equal to r,. There-fore 62 is also constant, and &x can be determined withoutintegration.

In using the relationships of Rhys, the distance z is thepositive distance from the junction of the knuckle andcrown in the direction of the knuckle for the knuckle stressequations and in the direction of the crown for the crownstress equations.

According to the relationships of Rhys, the maximumbending stress is the meridional bending stress in the knuckledue to the discontinuity at the crown-knuckle junction.This stress is equal to:

fm(bending) max =ath

(7.75)

The relationships of Coates and Rhys for bending stressesin torispherical heads are discontinuous when combinedwith the pressure stresses in the head. Also, these calcu-lated combined stresses fail to damp out in the same manneras those determined by Coates’s relationships for an ellip-tical dished head. A further complication exists in thefact that prior to failure a torispherical head plasticallydeforms. This fact invalidates the use of elastic-theoryrelationships.

7.11 DEFORMATION OF TORISPHERICAL HEADSUNDER INTERNAL PRESSURE

Although a torispherical head having approximately theshape of an ellipse is considered to be the strongest head ofthis type, it is rarely used because of the availability ofdies for elliptical dished heads when heads of such strengthare desired. However, for reasons of economy torisphericalheads having less than the maximum ratio of rl/rc areextensively used.

Torispherical heads having less than the maximum ratioof rl/r, t,end to deform to the shape of an ellipse under theinfluence of internal pressure, as was shown by Hijhn (114).Figure 7.17a (from Hiihn) illustrates the change in shapethat occurs within the elastic limit of the material, andFig. 7.17b indicates the permanent set produced when sucha head is plasticly deformed. The crown expands out-

-- - - - - ---

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Development of Stress-intensification Factor for Code-designed Torispherical Closures 137

ward, the knuckle flattens, and the profile becomes moreelliptical. As a result of this plastic deformation, thehead becomes s tronger with a redistr ibut ion of the s tresses .Elastic t.heory does not permit the calculation of suchstresses after plastic set has occurred. Therefore, it hasbeen necessary to use experimental tests of vessels underpressure supplemented with observations of vessel failuresto establish an empirical correlation for the design of tori-spherical heads.

Hiihn reported 344 cases of failures of dished heads inEurope, most of which had ring cracks on the inside ofthe knuckle discovered in the course of inspections and infifteen of which violent ruptures occurred. Hiihn pointedout that as the meridional and hoop stresses are of oppositesigns at the inner surface of the knuckle, large shearingstrains exist at an angle of 45” with the meridians butobserved that no case of failure had occurred which couldbe attributed to these strains. The cracks that wereobserved always occurred along a r ing in the knuckle . Thebending stresses in the knuckle are large because of thesharp radius of the knuckle, and the bending stresses donot follow the linear rule in passing from compression totension throughout the thickness of the knuckle. Thestress-distribution curve has a hyperbolic shape, and thestress is much greater at the inner surface than it would bein the bending of a straight beam. It has not been feasibleto determine the stresses on the interior surface of a tori-spherical dished head by strain measurements on t.he exte-r ior sur face . However, such external st.rain measurements

have been useful in developing empirical correlations fort,he design of tor ispherical heads.

7.12 DEVELOPMENT OF STRESS-INTENSIFICATIONFACTOR FOR CODE-DESIGNED TORISPHERICALCLOSURES

Hiihn analyzed all of the reliable test data then availableon torispherical heads and performed many tests of his own.He noted that the yield point of the head material was firs1reached in the knuckle, and measured the pressure requiredto produce such yielding. He then computed the stress inthe center of the crown at. this pressure by use of the equa-tions for spherical shells based upon membrane theory.Next he determined the ratio of the yield point to the stressin t,he center of the crown and used this s tress- intensi f icat ionfactor for correlation purposes. After trying a number ofmethods of correlation, he concluded that the results couldbest be correlated by comparing the computed stress ratioswith the ratio of the knuckle radius to the crown radiush/r,.

Hiihn plotted 20 points with rl/rC varying from 0.025 to0.19 and drew a curve through these points for which hefitted an empirical equation. Hijhn’s data and equationare shown in Fig. 7.18. The equation or curve gives afactor by which the head thickness computed by means oft,he equation for spherical shells may be multiplied to givea t,orispherical head thickness having suficient strength.

Figure 7 .18 was original ly part of a memorandum of F. L.

3.5

A Elastic tests by Siebel and Koerber0 Elastic tests by Hahn+ Test of A.O. Smith Co.

\ n

\ I-NW a

1 q-1-I

A\-

- - - - - -J1. T o 1 . 0+ 0

b-----~at r/R=

- - .- 1% . -,-, _ -

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20

1.0

Ratio of minimum knuckle radius to crown radius, r/R

Pig. 7.18. Computed maximum stresses and strains in elliptical heads as ratios of stresses and strains in attached cylindrical shell (12). (Courtesy of

American Welding Society.)

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138 Selection of Elliptical, Torispherical, and Hemispherical Dished Closures

Maker* prepared for the API-ASRIE committee and waslater published (12). The data and curves of Hiihn werereproduced, and additional data from tests in this countrywere added. A curve was also included based on computa-tions made by using Coates’s method for determiningstresses in elliptical dished heads of various depth ratios.A comparison of this curve with the empirical curve ofHijhn indicated that the curve for torispherical dished headsshould turn upward more abruptly than was specified byHiihn. Therefore, the “proposed” curve was bent upwardto the form given in Fig. 7.18 in order to give better agree-ment with some of the data for the low ratios of ri/rc.It should be emphasized that the curves for torisphericalheads in Fig. 7.18 are based on tests in which stress-intensi-fication factors were determined by observation of thepressure at which yielding occurred on the outside of theknuckle of the head, and are empirical. The “proposed”curve was included in the first edition of the API-ASMEcode, published in 1934, and has withstood the test of20-years’ use. The 1956 edition of the ASME code (11)substituted Eq. 7.76 for evaluating the stress-intensi-fication factor, W.

al,, = -Mlbz

w = $3 + dim (7.76)

t = preW2fE - 0.2~ +c (7.77)

where W = stress-intensification factor for torisphericaldished heads

c = corrosion allowance, inches

7.13 HEMISPHERICAL DISHED CLOSURES

A thin-walled spherical vessel theoretically requires onlyhalf the shell thickness of a cylindrical vessel of the samediameter to contain the same internal pressure. This isindicated by a comparison of Eq. 3.14 and 4.25, whichdefine the theoretical hoop stress based on membranetheory. If large access openings for manways or nozzlesfor piping are to be cut into formed closures, the hemispheri-cal head may be used to advantage because its greaterstrength will make less reinforcement required. Someconsideration must be made of the thickness discontinuitythat exists when a hemispherical dished head having lesserthickness than the shell is attached to a cylinder. Theextent of bending and shear at the junction should also beconsidered.

G, W. Watts and H. A. Lang (129) analyzed the stressesin a pressure vessel with a hemispherical head. Themethod used was the same used by these authors for flat-plate closures and conical closures, described in Chapter 6.Dimensionless equations having the form of Eqs. 6.113 and6.114 were used. However, for a hemispherical headattached to a cylindrical shell, the coefficient brs = bs = 0.For the cylinder ~4, us, u& b4, bg, and b6 are the same asfor the cylinder in Eqs. 6.109, 6.110, 6.111, and 6.112 andare tabulated in Tables 6.2 through 6.6. The constantsalo, ail, ar2, and blo, bll, blz for the hemispherical headwere defined by Watts and Lang as follows:

* Private communication.

M2c-ill = + 7

j,2 + jg-.p(J2J1 - J2i1) + Mz(J& + Jzi)1

a12 = - 0.0875

MlMzbl0 = + ~2 [

J12 + J2’- -r(JzJ1 - Jzth) -1+ Mz(Jk + Jzlbz)

MIbll = - -JZJI - JZJI-

4 p(jzJ1 - Jzi) + M’L(J& + J2j2)I

bl2 = 0

Ml = d12(1 - p2) ;0 n

The coefficients (influence numbers) ala, all, ais, andblo, bll are given in Table 7.4. The quantities J1 and J2are, respectively, the values at the junction of the real andimaginary parts of the hypergeometric function resultingfrom the theory of spherical shells. The quantities jr andis are the derivatives of J1 and J2, respectively (109).Because of the difference between the coefficients for ahemispherical head and for a flat plate, Eqs. 6.113 and 6.114become Eqs. 7.78 and 7.79.

MO Qoa10 - + a11 - + a12 = a4

pd2 pdMo +

pd2u5 :; + u6 (7 .78)

blo p2 + bll$ + be = b4 p$ + bs: + bs (7.79)

but bs = 0, b12 = 0; therefore

Mo = pd2 (a12 - ad(bs - bll)(a4 - alo)& - bd - ( a s - w)(br - blo)

(7 .80)

Qo = pd -(a12 - as)@4 - ho)

(a4 - ulo)(bs - bll) - (us - m)(br - blo)(7.81)

7.14 STRESSES IN THE SHELL AT ITS JUNCTION WITHHEMISPHERICAL DISHED CLOSURES

The maximum shear stress in the shell at the junction isgiven by Eq. 6.121. The maximum combined axial stress

Table 7.4. Coefficients for Hemispherical Head (129)

(Extracted from Transactions of the ASME with Permissionof the Publisher, the American Society of Mechanical

Engineers, 29 West 39th St., New York, N. Y.)

d/h a10 a11 a12 ho bll

4.00 -6.9690 f1.8200 -0.0875 +3.6608 -0.527210.00 -16 .7131 +2.8631 -0 .0875 +5.7319 -0 .505820.00 -33.2457 +4.0574 -0.0875 +8.1196 -0.503030.00 -49.7670 +4.9729 -0.0875 +9.9470 -0.502040.00 -66.2900 +4.7438 -0.0875 +11.4886 -0.5015

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Location of the Maximum Stress 1 3 9

(7.84)!n the shell is given by Eq. 6.122. The maximum combinedcircumferential stress is given by Eq. 6.125.

7.15 STRESSES IN THE SHELL IN OTHER PLACES THANAT THE JUNCTION

The maximum shear stress in the shell in other placesthan at the junction is given by Eq. 6.126, and its locationis defined by Eq. 6.127. The maximum axial stress in theshell in other places than at the junction is given by Eq.6.128, and its locat.ion is defined by Eq. 6.129. The maxi-mum circumferential stress in the shell in other places thanat the junction is given by:

f&cum. = $ + 0.733 a (fshear? (7.82)s

wherefshear’ = Eq. 6.126

The location of this stress is given by:

1.072ccc = za + __

P(7.83)

where xa is from Eq. 6.129, and /3 is from Eq. 6.86.

7.16 STRESSES IN THE HEMISPHERICAL HEAD AT THEJUNCTION

The maximum shear stress in the hemispherical head isgiven by Eq. 6.121 with the modification that th, the thick-ness of the head, is used rather than t,, the thickness ofthe shell. or

The maximum combined axial stress is given by Eq. 6.125with the same substitution for wall thickness, or

(7.85j

The maximum combined circumferential stress is givenby Eq. 7.86.

fcircum. = 5: + 1'9 / + z r* + UllQO ( 7 . 8 6 ))

7.17 STRESSES IN THE HEMISPHERICAL HEAD INOTHER PLACES THAN AT THE JUNCTION

For a hemispherical head the axial and meridional stressesare identical and are given by Eq. 7.87.

7.18 LOCATION OF THE MAXIMUM STRESS

The maximum stress in a vessel having a hemisphericalclosure is dependent upon the ratio of the thickness of thehead to the thickness of the shell, as indicated in Fig.7.19, in which the stresses are plotted as stress-intensifica-tion factors versus the ratio of th/ts. For ratios of head toshell thickness of from 0.6 to 2.0, the maximum stress is

1.4

0 L0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

Ratio of head thickness to shell thickness, t,,/t,

Fig. 7.19. Stress-intensification factors for vessels with hemispherical closures.

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1 4 0 Selection of Elliptical, Torispherical, and Hemispherical Dished Closures

located in the shell and is the circumferential stress asdefined by Eq. 7.82; it is equal to 1.037 times the hoopstress in the cylinder. For head-to-shell ratios of 0.6 andunder the maximum stress is the circumferential combinedstress in the head at or near the junction as defined byEq. 7.86.

The other stress-intensification factors plotted in Fig.7.19 may be identified as follows:

I aj(shell) =

~rj(shell) =

zaj(head) =

I am(head) =

maximum combined axial stress intensificationin the shell at the junction, Eq. 6.122maximum combined circumferential stressintensification in the shell at the junct,ion, Eq.6.125maximum combined axial stress intensificationin the head at the junction, Eq. 7.85meridional hoop stress intensification in thehead in other places than at the junction, Eq.7.87

7.19 PRACTICAL CONSIDERATIONS

Figure 7.19 shows that the bending st.resses at or near thejunction in vessels with hemispherical closures are much lesssignificant than in vessels with flat-plate, conical, elliptical,or torispherical closures.

If one judges on the basis of membrane theory, the hemi-spherical head has twice the strength of a cylindrical shellof the same diameter, as indicated by Eqs. 3.13, 3.14 (notethat the hoop stress in a hemisphere is the same as the axialstress in a cylinder). This is shown in Fig. 7.19 where thecurve I,, is equal to 1.0 when th/ts = 0.5. The maximum

combined stresses in the head and shell are equal at th/tp =0.6. This is indicat,ed in Fig. 7.19 where the curvesIem(shel1) and Z,j(l,e*d) intersect at Zh/Zs = 0.6. This, inturn, indicates that the optimum ratio of th/ts is 0.6. Forsuch a ratio, the shell thickness may be evaluated by thethin-wall equation, and the maximum stress in the shellwill be only 1.7% greater than the theoretical circum-ferential stress. If the head thickness is equal to 0.6 timesthe shell thickness, the maximum shell stress will be only3.7% greater than the theoretical circumferential shellstress. For Zh/Zs ratios of less than 0.6, both the axial andthe circumferential stresses in the head at the junction areincreased rapidly as this ratio decreases. If the head ismade with half the l.hickness of the shell, the combinedcircumferential head st.ress at the junction is 11.5 y$, greaterthan the theoretical hoop stress in the shell, and the com-bined axial stress in the head is 7.9 ‘% greater. This increasein stress is a result of t.he bending moments at and near thejunction. If either of t,hese stresses exceeds the yield pointof the material, plast.ir deformation will result, which willrelieve the excessive st,ress. If some plastic deformationcan be tolerated in high-stress conditions or in hydro-static testing, a thickness ratio of 0.5 can be consideredsatisfactory.

The ASME Code (I 1) gives the following relationship fc;:the thickness of hemispherical heads:

Ih = pdi..~___ + cIjE - 0.4p

where E = joint efKrienc:y

P R O B L E M S

1. A natural-gasoline stabiliziug colurnu is to be constructed of ASTM-A 285, Grade C steel.The column is to operate at 200 psi. The vapor leaves the top of the column at 160” F. Thetower has a nominal diameter of 6 ft. Select a torispherical closure and au elliptical dishedclosure for this application if the allowable design stress is 13,750 psi and the minimum corrosionallowance is Sic in. (The om e iciency for a single-piece closure is loo%.)j . t ff

For the case of the elliptical closure, assuming a joint efficiency, E, of 85 y0 in the shell, drtrr-mine the combined hoop stress-intensification factor curves in the shell and in the closure.

t ehe” Prinaide + c= fE - OJjp (See Reference 11.)

2. For the conditions giver1 in problem 1, determine the combined meridional stress-intrusifi-cation factor curves in the shell and closure.

3. For the case of the torispherical closure in problem 1, assuming a joint efficiency of 8SyGin the shell, determine the meridional bending stresses in the knuckle resulting from the changein the meridional radius of curvature at the junction of the knuckle and crown.

1. For the conditions in problem 3, calculate the meridional bending stresses in the crown

resulting from the change in the meridional radius of curvature at the junction of the knuckleaud crown.

5. For purposes of heat transfer it is desired to attach a steel hemispherical head !i iu. thickto a steel cylindrical vessel 1 in. thick and 10 in. in inside diameter. If the vessel is under apressure of 1000 psi, calculate the maximum stress in the head.

6. Calculate the plot the maximum-shear-stress-intensification factor in the shell at thejunction for various hemispherical closures as a function of th/ts when d/th = .40.

(7.88)

Page 151: Process Equipment Design

C H A P T E R

DESIGN OF CYLINDRICAL VESSELS

WITH FORMED CLOSURES

OPERATING UNDER EXTERNAL PRESSURE

AAwide variety of chemical and petrochemical processes

require equipment operating under part ial vacuum. Exam-ples are: vacuum condensers for evaporators and distilla-lion columns, vacuum columns such as lube oil columns,vacuum crystallizers, and so on. Such vessels are underexternal pressure from the atmosphere. Vessels are some-times jacketed and heated by means of steam, Dowtherm,or other condensing vapors under pressure in the jacket;this a lso produces an external pressure on the vessel . Thesevessels are usual ly cyl indrical with formed heads as c losures .

A cyl indrical vessel under external pressure has an inducedcircumferential compressive stress equal to twice the longi-t udinal compressive stress because of external-pressureeffects alone. Under such a condition the vessel is apt tocollapse because of elastic instability caused by the cir-cumferential compressive stress. The collapsing strengthof such vessels may be increased by the use of uniformly:;paced, in ternal or external c i rcumferent ia l s t i f fening r ings .From the standpoint of elastic stability such stiffeners havethe effect of subdividing the length of the shell into sub-sections equal in length to the center-to-center spacing ofthe s t i f feners .

Long, thin cylinders without stiffeners or with stiffenersspaced beyond a “critical length” will buckle at stressesbelow the yield point of the material. The correspondingcritical pressure at which buckling occurs is a function onlyof the t/d ratio and the modulus of elasticity, E, of themater ia l . If the length of the shell with closures, I, or thedistance between circumferential stiffeners, 1, as the casemay be, is less than the critical length, the critical pressureat which collapse occurs is a function of the l/d ratio aswell as of the t/d ratio and the modulus of elasticity, E.

The relationships for the conditions beyond the criticallength wil l be developed f irst . (See Timoshenko [42] .)

8.1 ELASTIC STABILITY OF LONG, THIN CYLINDERSUNDER EXTERNAL PRESSURE

A cylindrical shell under external pressure tends todeform inward as a result of the external radial pressure.The relationship between the radius of curvature, the prod-uct EI, and the bending moment producing curvature isgiven by Eq. 2.9.

Consider a cylindrical shell having an original radius ofcurvature of ro under no load. A local section of this shellunder external radial pressure will have a new radius ofcurvature of P. The relationship between the bendingmoment, the two radii, and the product EI is given by:

M=EI L-1( )PO r

Figure 8.1 is a diagram of a shell section before and afterdeformation under external pressure. The shell has anoriginal radius of curvature of ro and at the point underconsideration has a radius of curvature of P after deformingthrough a radial distance, w. Points a and b represent thel imits of an e lemental s t r ip , ds, in the shel l pr ior to deforma-t ion. The points a’ and b’ are the corresponding limits ofthe same elemental s tr ip after deformation. The elementalstrip subtends an angle of d0 before deformation and anangle of de -l- A0 after deformation.

141

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1 4 2 Design of Cylindrical Vessels with Formed Closures Operating under External Pressure

Fig. 8.1. Deformation of a cylindrical-shell section under external pressure.

From Fig. 8.1 and small-angle relationships,

ab = dS = ro de

therefore

(8.2)

1 de-=-q, dS

(8.2a)

After bending1 de + Ad0-=P dS + Ad&’

where dS + Ads = length of element a’b’

(8.3)

If the small angle dw/dS is disregarded, for small-anglefunctions where the tangent of the angle equals the angleexpressed in radians,

de = dS + Ads

PO - w (8.4)

de (r. - w) = dS + Ads

Subtracting Eq. 8.2 from Eq. 8.4 gives:

A& = -w&j = -wdsr0

(8.5)

Inspection of Fig. 8.1 indicates that the difference between

the angles ($$ and ($$+sdS) is the same as the

,difference between the angles de and (de + Ado), or

($+$dS)-($)=(dO+AdO)-de (8.6)

therefore

(8.7)

Substituting Eqs. 8.5 and 8.7 into Eq. 8.3 for AdS andAdo, respectively, gives:

1de + $$ dS

-=r

Substituting l/r0 for de/d& by Eq. 8.2a, gives:

1 1-=-r r0 ( >

1+w +$r

Substituting Eq. 8.9 into Eq. 8.1 gives:

1 1( >- - -r r0

= R) + @!f2 = - E;

Assuming rro = ro2, we find that

!k+E2= -z

W)

(8.9)

(8.10)

Multiplying through by ro2 and substituting for Eq. 8.2egives, therefore (42) :

(8.11)

Figure 8.2 shows a quadrant section of a cylindrical vesselunder external pressure. The dotted lines show a possibledeformation of this shell under the influence of external

Fig. 8.2. Bending moments in o shell deformed by external pressure.

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Elastic Stability of long, Thin Cylinders under External Pressure 1 4 3

pressure, p. In the deformed condition the bendingmoment Ms and the force F will exist at point c. Consid-ering a circumferential element of unit longitudinal width,we find that the compressive force F will be equal to thepressure times the projected area, or

F = p(uc) = p(ro - wo) (8.12)

Taking a summation of moments about any arbitrarypoint, e, in the deformed shell gives:

M = MO + F(bc) - p(ce)(he) (8.13)

Substituting for F in Eq. 8.13, by Eq. 8.12, we obtain:

M = MO + p[(ac)(bc) - tee’] (8.14)

Considering the two triangles abe and cbe, we find that

and(ae)2 = (~b)~ + ( b e ) ’

(ce)’ = (bc)2 + (be)2

Substituting gives:

(ae)’ = (~b)~ + (ce)2 - (bc)2

= (ce)2 + (UC - b~)~ - (bc)2

(ae)2 = (ce)2 + (a~)~ - 2(bc)(ac)or

+(ae2) = 8(ce)2 + fan - (bc)(ac)

therefore

(ac)(bc) - i(ce)2 = ~[(uc)~ - (ae)21

Substituting Eq. 8.15 into 8.14 gives:

M = MO + &~[(uc)~ - (ue)2]

(8.15)

(8.16)

l3ut UC = r. - ~0, and ue = PO + (-w). Therefore,substituting into Eq. 8.16 gives:

M = MO + +p-P[(Po2 - 2row0 + wo2) - (ro2 - 2r0w + w2)]

[

2 - wo= MO + p row - row0 + S-r 1Furthermore, the small quantity $(wo’ - w2) may be dis-,*egarded. Therefore

M = M O + pr0(w - w0) (8.17)

Substituting Eq. 8.17 into Eq. 8.11 gives:

d2wz+

M0r02WC -__EI

~+w(1+~)~pro3w0~~0r02

Let

Therefore (42)

(8.18)

(8.19)

$+q2w = pro3wo - Moro2EI

(8.20)

The solution to this differential equation is:

w = A sin q0 + B cos q0 + pro2+pE;rc” (8.21)

Introducing the conditions at c and g (Fig. 8.2) where thedeformed shell is perpendicular to the axes, we find that

0

andd w0de e=*/2 =

0

Equation 8.21 may be differentiated and set equal to zerofor the conditions of 0 = 0:

d w0de *=o= A(cos qe) - B sin q0 = 0

thereforeA=0

For the condition of 0 = s/2 we have:

= -B sin q0 = 0

therefore

-Bsinq:=O

and therefore

sin q: = 0 (8.22)

These unique values of q define corresponding uniquevalues of p in Eq. 8.19. The lowest of these values isq = 2 and defines p critical. Equation 8.22 is satisfied whenq is equal to 2 or multiples of 2. Substituting this value of qin Eq. 8.19 and solving for p gives (42)

3EI 24EIPtheoretioal = -=------

ro3 do3(8.23)

Equation 8.23 expresses the theoretical or critical loadper unit circumferential length of unit width of circum-ference. For a strip of unit width the critical load is thepressure at which buckling theoretically occurs. If the ringis a part of a long cylindrical shell, the adjacent metal oreither side of the ring will offer restraint to the longitudinaldeformation of the strip. To allow for this restraint Eq.8.23 may be divided by (1 - ~1~). (See Eqs. 6.la and 6.12.)To express the critical stress in terms of the shell thickness,t,, a substitution for I may be made for a rectangular strip.

I=!!?12

where b = 1 for a strip of unit width. Making these sub-stitutions in Eq. 8.23 gives:

(8.242

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1 4 4 Design of Cylindrical Vessels with Formed Closures Operating under External Pressure

where 1, = critical length, inchesd = diameter of shell, inchest = shell thickness, inches

(+I = (dtstance between stiffeners

d i a m e t e r o f v e s s e l >

Fig. 8.3. Collapse coefficients for cylindrical shells under external pressure

(134).

Substituting for Poisson’s ratio, p = 0.3, gives:

Equation 8.24 gives the theoretical “critical” pressure(external) at which a long cylindrical vessel will buckle.This equation is the generally accepted theoretical formulaof Bresse (130) and Bryan (131) for long, thin tubesunder external pressure.

Stewart in a number of tests using commercial tubing andpipe investigated the applicability of Ey. 8.25 and foundwhat collapse occurred at a critical pressure of 27 y0 less-I han the theoretically predicted pressure. For design oflong, thin cyl indrical vessels operat ing under external pres-sure, a factor of safet.y of 4 may be applied to Eq. 8.25.giving (1,%2) :

t 3Pallo\\nlIle = 0.55E 2

0(8.26)

8.2 CRITICAL LENGTH BETWEEN STIFFENERS

Equations 8.25 and 8.26 apply to long, thin cylindersunder external pressure without circumferential stiffeningrings or with the stiffening rings spaced at or beyond the“critical length.” To make allowance for the addedrestraint offered by stiffeners spaced at less than the criticallength, the critical length may first be evaluated.

The expression for the critical length was first developedby Southwell (133) . Southwel l ’s analysis involves a 15-rowdeterminant solution and is beyond the scope of this text.The relationship resulting from Southwell’s analysis isgiven by:

4~ 4%1, = - - - - (i/i-T)(d l/dlt, (8.27)

Substituting, p = 0.3, gives:

8.3 COLLAPSING PRESSURE OF VESSEL SHELLS W!TbiCIRCUMFERENTIAL STIFFENERS

For vessels in which circumferential stiffeners are spacedat less than the critical length, the coefficient of Eq. 8.25must be modif ied according to the proport ions of the vessel(158, 159), or

Ptheoretical = KE f0

3

Applying a factor of safety of 4 gives:

(8.30)

where fi = roefiicient accordin g IO the proportions of thevessel, as indicated in Fig. 8.3 (134). (Noiethat the minimum value of K is 2.2 as by Eq.8.25.)

Substituting Ey. 8.29 into Eq. 3.14, (disregarding cor-rosion) we find that the circumferential compressive stressfrom external pressure at which collapse occurs is:

(8.31)

Equations 8.25 and 8.31 may he plotted for conveniencaof solution as shown in Fig. 8.4 (IL). The inf lect ions in theparameters occur at the critical lengths, which correspondto the critical lengths determined by Eq. 8.28. The ver-tical parameters of d/t above the inflections represent theregion where the spacing between stiffeners exceeds thecritical length and the collapsing pressure is independentof the l/d ratio. Equation 8.25 applies in this region.The inclined parameters below the inflection represent theregion where stiffeners have an effect and the collapsingpressure is a function of the l/d ratio as expressed by thecoefficient K in Eq. 8.29. It is significant to note that Fig.8.4 is general and is independent of the material of construc-t ion. To use the chart to predict the ratio of l/d at whichcollapse occurs, it is necessary to know the value of (f/E)for the material at the temperature under consideration.

Figure 8.5 shows a group of stress-strain curves for se>,-era1 materials and indicates that for mild steel at room tem-perature the stress-strain curve can be approximated bytwo straight lines (135). Unfortunately, this approxima-tion is the exception rather than the rule and is limited tocarbon and low-alloy steels at temperatures below 500” F.Carbon steels at temperatures above 500” F and othermaterials such as high-alloy steels and nonferrous metalshave nonlinear stress-strain curves with a variable modulusof elasticity and no definite yield point. Figure 8.6 showsthe variation of the modulus of elasticity, E, for plain car-bon steel and austenite steel as a function of temperature.

The error caused by using a constant modulus of e last ic i ty1, = l.lId m (8.28)

Page 155: Process Equipment Design

Collapsing Pressure of Vessel Shells with Circumferential Stiffeners 145

i

Fig. 8.4. General chart for cd-

lapse of vessels under external

pressure showing relationship

between the dimensional ratios

,d/t and I/d and the physical

property f /E (135).

Fig. 8.5. Typical stress strain

curves for several materials.

(Extracted from Transactions of

the ASME 11351 with permission

of the publisher, the American

Society of Mechanical Engineers,

29 West 39th St., New York,

N.Y.)

6

43

(l/d) *

1.00.80.60.40.30.2

0.08 I /llll/ I llllll0.06 I IllIll I l/llll

2 3 4 5 6 8 2 3 4 5 6 8 ;! 3 4 5 6 8 3 456 8 2 3 4 5 6 82

0.000001 0.00001 0.0001 0.001 0.01 0 . 1t = (f/E)

80,000 p

:70,000 I

:I Structural silicon steel

60,000 -V- I

I27 ST aluminum alloy

aluminum-annealed

03 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016

Strain, inches per inchdetermined on 8-in. gage length

r-- - -- --T==-‘ ._-__- - -\ \ \IT --~- - -

Page 156: Process Equipment Design

1 4 6 Design of Cylindrical Vessels with Formed Closures Operating under External Pressure

32

3 0

‘g 28

2.o 26

2 24

q 22zgj 20

g, 1 8

1 6

140 200 400 600 800 loo0

Temperature, deg, F

Fig. 8.6. Modulus of elasticity of plain carbon and oustenitic steels as o

function of temperature. (Extracted from the 1956 edition of the ASME

Boiler and Pressure Vessel Code, Unfired Pressure Vessels [l 11, with per-

mission of the publisher, the American Society of Mechanical Engineers,

29 West 39th St., New York, N.Y.)

for materials having nonlinear stress-strain curves may beavoided by using a “tangent” modulus of elasticity, that is,the slope of the stress-strain curve at the stress and tem-perature under consideration. For convenience, to avoidthe necessity of measuring the tangent modulus, the simul-taneous value of (j/E) can be plotted versus stress in termsof pressure and dimensions of the vessel. In designing avessel for a given value of (j/E) based upon the material ofconstruction and operating temperatures, one bases hisdesign upon the strain at which collapse occurs rather thanupon an al lowable stress .

Using a design factor of safety of 4, in which the allow-able pressure is considered to be one fourth of the theoret icalpressure at which collapse occurs, we obtain:

wPtheoretical = - -d - 4Psllow. (8.32)

where Ptheoretical = theoretical external pressure at whichcollapse occurs, pounds per square inch

~~11~~. = allowable external pressure, pounds persquare inch

Rearranging Eq. 8.32, we obtain:

0d ‘B= Ptheoretical W = 2 =

o r

(8.33)

Using Eq. 8.33 and appropriate stress-strain diagrams,we can determine simultaneous values of (j/E) and (j/2)and can plot them as shown in Fig. 8.7. The 100” curvein Fig. 8.7 has an inclined line of constant slope below theyield point because at 100” F and below the modulus ofelast ic i ty does not vary with s tress . Above the yield pointin the plastic region the stress-strain curve is nearly a

horizontal line and corresponds to the horizontal line ofFig. 8.5. The 650” F line shows a break at a lower valueof (j/2) and levels off thereafter.

Figures 8.4 and 8.7 can be used to determine the safeexternal working pressure of an existing vessel underexternal pressure . The dimensional ratios l/d and d/t arefirst computed and the corresponding value of j/E is deter-mined from Fig. 8.4. This value is used with Fig. 8.7 todetermine the value of the quantity (panow.) (d / t ) f rom whichthe value of psnoW. is directly computed. In designing avessel the dimensional ratio Z/d is usually known, but thevalue of d/t is unknown as t is to be determined. Thevalue of t must first be assumed and the calculated safeallowable working pressure checked with the desired work-ing pressure as indicated above.

As both of the curves of Figs. 8.4 and 8.7 have a commonabscissa of j/E, they may be conveniently superimposedas indicated in Figs. 8.8 and 8.9 (for plain carbon steel upto 900“ F).

8.4 EXAMPLE DESIGN OF A SHELL

Given: A fractionating tower 14 ft in inside diameter by21 ft in length from tangent line to tangent line of theclosures . The tower contains removable trays on a 39-in.tray spacing and is to operate under vacuum at 750” F.The material of construction is SA-283, Grade B plaincarbon steel, which has a yield strength of 27,000 psi (seeTable 5.1.).

The required thickness of the shell will be determinedboth without stiffeners and with stiffeners located at thetray pos i t ions .

20,000

10,000

8WJcq

I IN”3 JO00

I Iz2 3,000

%3 2,000

1,000

800

Strain, e = (f/E)

Fig. 8.7. Chart for plain carbon steel showing allowable pressure under

external loading at 100’ F to 900’ F. (Extracted from Transactions of the

ASME [135] with permission of the publisher, the American Society of

Mechanical Engineers, 29 West 39th St., New York, N.Y.)

Page 157: Process Equipment Design

Fig. 0.0.

from the

Mechanic

1

:C

2.01.81.61.4

0.90I III I\1

0.80 I III I

0.60

0.50

0.400.35

0.06 I I I I -r--1 I II

0.05 ’ ! ’ ! ! 1 ! ! !‘I

Example Design of a Shell

MCtOr A = f/E = B

Combined chart for determining thickness for carbon-steel shells under external pressure-for yield strengths of 24,000 to 30,000 psi.

956 edition of the ASME Boiler and Pressure Vessel Code, Unfired Pressure Vessels [1 11, with permission of the publisher, the American

II Engineers, 29 West 39th St., New York, N.Y.)

(Extracted

Society of

Page 158: Process Equipment Design

14% Design of Cylindrical Vessels with Formed Closures Operating under External Pressure

40,00035,OilO30,000

25,000

,-7OOOF - 14,0007- /-8OO’F - 12,000

.--900°FI I , I I I 10.000

6.0 I I1.111111 I I I I I ,,,,Ir , I I\ I I I I I I I I ! I I I I I I I 6.000

I I I11111I I I I I I I I II 5,000

m 2,500 5

2 5.00835 4.0

11 7 3.5gwE p 3.0g g 2.5

-2 g 2.0

.f : :f22 1.4*:!. @ 1.2Es 1.0g; 0.900 $ 0.80B z 0.70$ 0.60

o--0.50

kt+i# 800

500

25

0.400.350.300.25

400350300250

0.200.180.160.140.120.100.090.080.070.060.05

2 3 4 5678 2 3 4 5678 2 3 4 5 6 7 8 10.00001 0 . 0 0 0 1 0 . 0 0 1 0.01

Factor A = fJE = e

Kg. 8.9. Combined chart for determining thickness for carbon-steel shells under external pressure-for yield strengths of 30,000 to 38,000 psi. (Extl-acteL

from the 1956 edition of the ASME Boiler and Pressure Vessel Code, Unfired Pressure Vessels [l I], with permission of the publisher, the American Society of

Mechanical Engineers, 29 West 39th St., New York, NY./

Page 159: Process Equipment Design

8.4~ Required Shell Thickness without Stiffeners. Thedetermination of the shell thickness is a successive-approxi-mation calculat,ion. Assume a shell thickness of x in.

1 2 1 X 12 2 5 2

do = (14 x 12) + 1.25= ~ = 1.49

1 6 9 . 2 5

d o 169.25-= ~- = 271t 0 . 6 2 5

Enter Fig. 8.8 with l/d = 1.49 and move horizontally tointersect the diagonal line for d/t = 271; this gives: E =0.0002 in. per in. Move vertically to the material line for750’ F (interpolating between the 700” F and 800” F mate-rial lines) and then move horizontally to the right-handside of the chart and read B = 2300. The maximum allow-able external pressure for the assumed shell thickness of$4 in. is:

Therefore

BPallow. = - -

dolt(8.33)

Pallow. = 2&$c = 8.48 psi

Since this pressure is considerably lower than the desiredexternal pressure of 15 psi for full vacuum, the calculationmust be repeated with a greater thickness assumed. There-fore assume a shell thickness of lKs in.

1 2 5 2-YE ~ = 1.49do 1 6 9 . 6 3

do 1 6 9 . 6 3-= __ = 208.5t 13/16

Entering Fig. 8.8 with Z/d = 1.49 and moving to do/t =208.5 gives E = 0.00029 in. per in. Moving vertically tothe ‘i50” F material line and horizontally to the right givesB = 3400; therefore

P3 4 0 0

a l l o w . = 208.5~ = 16.3 psi

Therefore a 1Ws in. plate, if available, is adequate.A shell plate of this thickness weighs 33.15 lb per sq ft.

The shell weight is:

wt = vrdlp

wt = 3.14 x 14 x 21 x 33.15

wt = 30,700 lb

8.4b Required Shell Thickness with Stiffeners. Therequired shell thickness of 1%~ in. far the condition of nointernal stiffeners can be materially reduced by the inclusionof stiffening rings at the tray locations. Assume a shellthickness of 7/i6 in.

1 = 39 in.

1 3 9 3 9-=do (14 x 12) + 0.875 = i%%% = 0231

1‘ 0 1 6 8 . 8 7 5~ - 386

t 0.4375

Design of Circumferential Stiffeners 149

Enter Fig. 8.8 with an l/do of 0.231 and move to a do tof 386; this gives: E = 0.0012 in. per in. Move verticall\to the 750” F line and to the right to give B = 6200.The maximum allowable pressure for t,he assumed shellthickness is:

pallow. = ‘&&$ = 16.1 psi (which is adequat.e)

The weight of the shell is 17.85 lb per sq ft. Therefore

the shell weight = 3.14 X 14 X 21 X 17.85

= 16,450 lb

This represents a saving in shell steel of (30,700 - 16,450)= 14,250 lb. However, this is offset in part by the weight.of the stiffening rings. This point is covered in the exam-ple design in the section following the stiffening-ring sectioaThe weight of the rings of a satisfactory design was foundto be 2700 lb. Therefore a net saving of (14,250 - 2700) or13,100 lb of steel is realized.

8.5 DESIGN OF CIRCUMFERENTIAL STIFFENERS

In designing circumferential stiffening rings for VA&Sunder external pressure each stiffener is considered to resistthe external load for a (1/2 distance on either side of I hering (where E is the spacing between rings). Thus the loadper unit length on the ring at collapse is equal t,o Z(ptheorrtical).We may rewrite Eq. 8.23 noting that in t,his equation ihoterm 1 is taken as unity. Therefore

p = ~theoretical(~) = ‘9’

where P = load on combined shell and stiffener in pounusper inch of circumferential length.

Or

I = Pth.o;;;c

Multiplying by t/t and rearranging gives:

Substituting Eq. 3.14, f = pd/2t, and Eq. 6.1, E = *f/E,gives:

(8.3.5)

The moments of inertia of the stiffening ring and the shellact together to resist collapse of the vessel under extcl,rl;t:pressure. Timoshenko (42) has shown that the combiurdmoment of inertia of the shell and stiffener may be coti-sidered as equivalent to that of a thicker shell, or

t,=~+$Lt++ (II.:iC!)

u

where t, = equivalent thickness of +ll, irlchcsA, = cross-sectional area c:! , ,I,? ~it,c,rlniferrnli:ll

stiffener, square inches1 = tl, = distance betwecu ( il,(‘lllilfei.(~lltii?; stiii’erle:,a,

inches

r - \ \I / _ -__-.-.-.T_~.~I \

Page 160: Process Equipment Design

150 Design of Cylindrical Vessels with Formed Closures Operating under External Pressure

Substituting Eq. 8.36 into Eq. 8.35 gives:

(8.37)

where Z = required moment of inertia of stiffening ring,inches’

Equation 8.37 is the same as that specified by the 1956ASME code (11) for stiffening rings for vessels underexternal pressure except that the coefficient in the denomi-nator is 14 in the code equation rather than 12 as in Eq.8.37. The value of 14 in the code equation may be approxi-mated empirically. In general, the combined moment ofinertia of the stiffener and the shell together varies from30% to 7Oa/, greater than the moment of inertia of thestiffener alone (10). Using a conservative allowance of a30y0 increase in the Z of the stiffener when combined withthe shell and introducing additional safety in design byincreasing the load by lo%, we obtain:

Therefore

1.1 d21I=-

(1.3)(12)

(8.38)

where

E = unit strain (see Figs. 8.8 and 8.9)

Equations 8.37 and 8.38 give the moment of inertia requiredfor the stiffening for the same collapsing pressure as that ofthe vessel designed by use of Fig. 8.8. The allowableoperating pressure is one-fourth the pressure at whichcollapse theoretically occurs.

In order to utilize Fig. 8.8 in the design of stiffening rings,it is necessary to proceed in the opposite direction, enteringthe figure with the value of B. If B is expressed in termsof the equivalent shell thickness, leq., (including the con-tribution of the stiffening ring), then by Eq. 8.33,

B = J = ~eoretic~ldo = mlo\v.do

2 ux&) t + (.AIllil(8.39)

If gaps are placed in the stiffening rings, they should bestaggered between alternate rings. In this case the dis-tance between stiffening rings should be taken as twice thering spacing to allow for the lack of continuous support ofthe shell at the gaps in the stiffening rings. Permissiblegaps in the stiffening rings of external-pressure vessels arespecified by the ASME code (11). (See Chapter 13 oncode vessels.)

8.6 EXAMPLE DESIGN OF CIRCUMFERENTIALSTIFFENERS

The design of the circumferential stiffening rings forthe 14-ft-diameter fractionating tower given previously(Required Shell Thickness with Stiffeners) in this chapterwill be illustrated. The required shell thickness was foundto be xG in., and the stiffeners spaced at 39 in. Thetower operates under full vacuum. The calculation of therequired moment of inertia of the stiffening rings requiressuccessive approximations.

Assume a 7 in. channel weighing 12.25 lb per ft. 1 =

24.1 in.4 (see Appendix G) and A, = 3.58 sq LB Sub-stituting into Eq. 8.39 gives:

B = pallow.do 15 X 168.875

t + (4,/O = 0.4375 + (3.58/39)= 4739

Enter the right side of Fig. 8.8 with B = 4790 and movehorizontally to the material line for 750” F; then movevertically to the bottom of the chart where E = c.00045.Substituting into Eq. 8.38 gives:

(169)2 X 39 0.4375 + $8 0.00045z = >

1 4

Z = 18.95 in.4

As the required moment of inertia is less than tnat pro-vided by the assumed 7-in. channel, the design is satis-factory. The weight of five such stiffening rings is:

Wt of rings = 5 X 3.14 X 14 X 12.25

= 2,700 lb

The weight of the shell is:

Wt of shell = 16,450 lb + 2700 lb

= 19,150

The total weight of the shell with stiffeners is 17,600 lb,compared to a total weight of 30,700 lb if a shell withoutstiffeners is specified; this represents a saving of l&100 lbsteel.

8.7 OUT-OF-ROUNDNESS OF SHELLS

Any out-of-roundness after fabrication of a vessel designedfor external pressure will reduce the strength of the vessel.The out-of-roundness results in increased stress concentra-tions, and the effect of external pressure is to aggravate thecondition. Thus a shell of elliptical shape or a circularshell, either dented or with flat spots, is less strong underexternal pressure than a vessel having a true cylindricalshape. The following procedure may be used to determinethe additional stress from elliptical out-of-roundness.

In reference to Fig. 8.2, the dashed section may be COG-sidered to be a deformed cylinder in which zag is the maxi-mum eccentricity radially inward. Timoshenko (136) hasshown that the initial radial displacement, w’, at any pointmay be assumed to follow the relationship:

WI = wo cos 28 (8.40)

If a uniform external pressure, p, is superimposed uponthis initially deformed cylinder, an additional radlai dis-

placement, w, will result,. This displacement w was pre-viously defined by Eq. 8.11.

d2ws+w=+ (8.11)

Noting that D, the flexural rigidity of a plate as given byEq. 6.15, may be substituted for EZ, we obtain:

-12(1 - &M?Ets3

(8.41)

Page 161: Process Equipment Design

To determine the bending moment M in the shell, con-sider a strip in the circumferential direction of unit width.By Eq. 8.17 for the region A to g (see Fig. 8.2)

A4 = pro(w - wo)

and for t.he region c to A (see Fig. 8.2)

(8.42)

M = pro(w + wo) (8.43)

For a vessel having an ini t ia l maximum out-of-roundnessradially inward of w. and a radial out-of-roundness at anypoint of w’, as given by Eq. 8.40, Eqs. 8.42 and 8.43 reduceto:

M = pro(w + w’)

Substituting into Eq. 8.41 gives:

= pro(w + wg cos 20) (8.44)

-12(1 - g>$+w= Et3 pro3(w + wo CoS 28)

3

Rewri t ing gives :

12(1 - $)Et3 pro3

8 1= -12Et 3 p2) pro3wo cos 20 (8.45)

8

However, by Eq. 8.24,

E t 3Ptheoretioal = 4(1 _ p2I ;

0

Substituting into Eq. 8.45 gives:

d’w- -d02 I

= -32~~ cos 28 (8.46)

Timoshenko (42) has shown that by substituting Eq.8.24 into Eq. 8.45 and applying the condition at A of Fig.8.2 (45’ position), the following solution to the differentialequation Eq. 8.46 is obtained:

WOPw = cos 28 (8.47)Ptheoretical - P

where Ptheoreticsl = theoretical col lapsing pressure (Eq. 8 .24)p = external pressure acting on shell, pounds

per square inch

The bending moment at point A of Fig. 8.2 is zero,whereas the maximum bending moment occurs at 0 = 0and at 0 = g, where

123 WOPlnax = pro wo +Ptheoretical - P >

Elastic Stability of Hemispherical and Torispherical Dished Closures 151

For a longitudinal strip of the shell of unit width to = 1)

@mwo (8.50)P

Ptheoreticsl

The total stress in an out-of-round shell unde; externalpressure is the sum of the stress given by Eq. 8.50 pius theexternal -pressure s t ress , or

Wowo

P(8.51)

Ptheoretical

Substituting panow. for P and letting PAIN. = ><ptheoreticalin accordance with Eq. 8.33 gives:

fmax ( 8 . 5 2 )

The maximum stress as given by Eq. 8.52 occurs in theouter f iber only and may be permitted to exceed the averagecompressive stress in the shell but should not exceed theyield point . The ASME code (11) gives a table of permis-sible values for out-of-roundness. (See Chapter 13, oncode vessels . )

8.8 ELASTIC STABILITY OF HEMISPHERICAL ANDTORISPHERICAL DISHED CLOSURES

Formed closures under external pressure are subject tofailure by elastic instability as are shells. Equation 4.33applies in the case of hemispherical or torispherical headsand gives the theoretical pressure at which collapse wouldoccur because of elastic instability.

2 E(t)2Ptheoretical =

r2 d3(1 - p2)(4.33)

o rt theoretical = r 43(1 - ~‘1 dp/2E (8.53)

Applying an approximate design factor of safety of 4.4,that is, using a thickness 4.4 times as great as that at whichbuckl ing theoret ical ly occurs , we obtain :

th = hittheoretical = 4.4r dp/2E g3(1-- + c (8.54)

For steel construction where p = 0.3,

+ C = 4r dp/E + C (8.55)

where th = thickness of head, inchesp = maximum external pressure, pounds per square

inch

(8.48)

This bending moment resulting from external pressureacting on an out-of-round shell produces a stress, f,,,:

c = corrosion allowance, inchesE = modulus of elasticity at operating temperature,

pounds per square inchr = radius of dish for hemispherical and tor ispherical

dished heads, equivalent head radius for el l ipti -cal dished heads, inches

Page 162: Process Equipment Design

152 Design of Cylindrical Vessels with Formed Closures Operating under Externol Pressure

As Ey. 8.55 contains the modulus of elasticity, E, whichmay decrease aL elevat,ed temperatures as a function ofstress, it is convenient to use Fig. 8.8 in a manner similarto that for cylindrical shells. The dashed line labeled“Sphere l ine” of Fig. 8 .8 is shown for this purpose. To usethe same chart, the scale for the sphere line is modified inthat, the vertical axis is now equal to r/100& where P is theradius of curvature (outside of head) and th is the headthickness (both in inches) . Figure 8 .8 is used to determineB, the same procedure being used as for shel l design. Themaximum allowable pressure, psllou,., is then determined byEq. 8.56.

BPallow. = -

r/th(8.56)

8.9 EXAMPLE DESIGN OF A HEMISPHERICAL DISHEDCLOSURE

A hemispherical closure will be designed for the vesseldescribed in the section entitled “Example Design of aShel l .” The design is a successive approximation becausethe tangent modulus of elasticity at this temperature(750” F) is also a function of the stress, f.

Radius of curvat,ure = 9 = 84.5 in.

Assume a head thickness of ?x in.

rc 84.5~~~~ = 2.7100th = 100x 0.3125

Enter the left-hand side of Fig. 8.8 at a value of 2.7 andmove horizontally to intersect the sphere line at f/E =0.00044. Move vertically to the material line for 750” F(interpolating between 700 and 800” F) and then movehorizontally to read B = 4800. The maximum allowableexternal pressure for the assumed shell thickness of x~ in.i s :

BPallow. = -

r/th

4800Pallow. = --- cc 17.8

2.7 X 100

As the vessel is to be designed for 1 atm. (14.7 psia) theassumed head thickness is satisfactory.

8.10 EXAMPLE DESIGN OF A TORISPHERICAL DISHEDCLOSURE

A torispherical closure will be designed for the vesseldescribed in the section entitled “Example Design of aShel l .” This design is a lso a success ive approximation.

Radius of dish (radius of curvature) = 169 in.

Assurlle a head thickness of $6 in.

PC~ = 2.7100th

Proceeding as before, B = -1800. (See example designof shel l . )

4800Pallow. = -~2.7 x 100

= 17.8 psi;1 (which is greater rhan

14.7 psia)

Therefore the thickness assumed is satisfactory.

8.11 ELASTIC STABILITY OF ELLIPTICAL DISHECCLOSURES UNDER EXTERNAL PRESSURE

The radius of curvature of an elliptical dished closurc~changes aboul the meridian of lhe head. To use the prr-vious relationships for elliptical closures, an eqnivaienlradius of curvature must be used. The radius of curvat ur’eof an elliptical dished head is maximum at the center ofthe head and at this point is equal to twice the radius of t hc>shell for a head having a major-to-minor-axis ratio of 2.0.Design of the head based on this maximum radius of curva-ture would result in considerable overdesign be-ause thrkradius of curvature decreases as the point under considera-tion is moved away from the center toward the junrt.ionwith the shell. This decrease in the radius results in anincrease in rigidity and greater elastic stability. Thus, aI1elliptical dished head has greater elastic stability than atorispherical dished head having the same diameter, thick-ness, and radius of curvature at the center of the head. Asthe radius of curvature of an elliptical dished head variesalong the meridian, an average radius may be used. How--

ever, the average must not be taken too far from the centr[of the head, which is the least stable point on the head.Table 8.1 lists the equivalent radius of curvature as a funr-tion of the major-to-minor-axis ratio for heads of vesselsunder external pressure (11). (Not same as Code.)

8.12 EXAMPLE DESIGN OF ELLIPTICAL DISHEDCLOSURE

An elliptical dished closure, a/b = 2.0, will be designedfor the vessel described in the section entitlerl “ExampleDesign of a Shell.” This design also involves successiveapproximation.

From Table 8.1,

rc- = 0.90dr, = (0.90)(169) = 152.1 in.

Table 8.1. Equivolent Radius of Curvature to Be Used

for Design of Elliptical Dished Heads under External

Pressure (11)(Kxtracted from the 1956 Edition of the ASME Boiler a11t1

Pressure Vessel Code l~nfirrd Pressure Vessels , withPermiss ion of the Publ isher , the American Society

3f Mechanical Engineers, “9 \\‘est 39th St.,New York, N.Y.)

,\laJor-to-lninor-axIs ratio,u/h 3.0 2.8 2.6 2.4 2 2 2 0 1 II

hvg radms of curvature P<-’b~essel diametw d 1.36 1.27 1.18 1.08 0.99 0.90 0.81

Major-to-minor-axis ratio.u/b 1.6 1.4 1.2 1.0

Avg radius of curvature rC_ Vessrl d i a m e t e r ‘7 0.73 0 65 0.57 0.W

Page 163: Process Equipment Design

Pipes and Tubing under External Pressum 1538.15 PIPES AND TUBING UNDER EXTERNAL PRESSURE

The re lat ionships presented ear l ier for long, thin cyl indersunder external pressure are conservative. Tubing andpipe usually have (t/do) ratios greater than 0.02 and as aresult are subject to failure by elastic-plastic bucklingrather than by elastic failure. Because of this and thegeneral uniformity of commercial tubing and pipes, moreliberal values of the allowable external working pressuremay be permitted than for the case of vessel shells. TheASME Special Research Committee on Vessels underExternal Pressure reviewed this problem with the objectof increasing the value of a l lowable external working pres-sures (213).

Stewart (132) developed the fol lowing empirical relat ion-ship for the collapsing pressure of a steel pipe having ayield strength of 37,000 psi at room temperature:

Assume a head thickness of gj6 in.

-~ = 152.1/(100)(0.5625) = 2.7100th

Proceeding as before, B = 4800. (See example designof shel l . )

Thereiore the assumed thickness of x/i6 in. is satisfactory.

8.13 ELASTIC STABILITY OF CONICAL CLOSURESUNDER EXTERNAL PRESSURE

Conical closures under external pressure can be classedin three groups. If the apex angle is small (45O or less) theconical closure is considered to behave as a cylindricalshell having the same diameter as the large end of the coneand a length equal to the axial length of the cone, providedthe cone has no st i f fening r ings . If circumferential stiffen-ing rings are used, the metal thickness may be decreasedin each successive section as the apex is approached. Inthis case, each section is designed by using the greatestdiameter of the section as the equivalent shell diameter, D,and t.he axial length between stiffeners (center to center)as the equivalent shel l length between st i f feners , L.

For conical heads having an intermediate apex angle(45’ to 120’) the same procedure is followed except that thediameter at the large end of the cone is taken as the lengthof the equivalent cylinder if no circumferential stiffenersare used. If circumferential stiffeners are used, the pro-cedure is the same as for stiffened cones with apex anglesof less than 45”, described above.

For flat cones having apex angles greater than 120”, theconical head is designed as a flat plate having a diameterequal tc the largest diameter of the cone.

8.14 EXAMPLE DESIGN OF A CONICAL CLOSURE

A conical closure having an apex angle of 45” and withoutstiffeners will be designed for the vessel described in thesection entitled “Example Design of a Shell.” This designis a lso a success ive approximation.

&sume a thickness of I’{6 in.

l=A!.c =-~__.84.5 = 204 in.tan o( 0.4142

do 169 - 246th 0.6875

Entering Fig. 8.8 with l/do = 1.21, move horizontallyto do/& = 246 intersecting curve at f/E = 0.00030. M o v evertically to 750” F material line to read B = 3700.

B _ s’100.P,lhv. = G - 2 46 = 15.05 psia

Therefore. the thickness of 1 ?.I6 in. is satisfactory.

p = 86,670 G0

- 1386d o

(8.57)

The ASME code committee revised Stewart’s formula(213) to include the effect of material having a yield pointother than 37,000 psi as follows:

p = 2.344+) - 1.064.f~ f;r (8.58)

10,0008,0006,000

- UJUUE j

,000800600

; 500F 400i 300BF 2003E

1008060504030

0 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32t/D ratlo

F i g . 8 . 1 0 . A l l o w a b l e w o r k i n g p r e s s u r e s f o r t u b i n g a n d p i p e u n d e r e x t e r n a lpressure (213) .

Page 164: Process Equipment Design

1 5 4 Design of Cylindrical Vessels with Formed Closures Operating under External Pressure

where p = collapsing pressure, pounds per square incht = tube thickness, inches

d = tube outside diameter, inchesfy = yield strength at operating temperature, pounds

per square inchE = modulus of elasticity at operating temperature,

pounds per square inch

The code committee considered that the modified Stewartformula, given by Eq. 8.58, was conservative when usedfor (t/do) ratios up to 0.14.

No extensive data and no theoretical relationships existfor the collapsing pressure of very thick tubes having at/do ratio of 0.30 or greater. However the committee wasof the opinion that such thick tubes fail by plastic yieldingrather than by collapse. On this basis the allowable pres-sure for tubes having a t/do ratio of 0.30 or greater wascalculated upon a modified hoop-stress relation as given byEq. 8.59.

4.fY tp=1.05 d,0

P R O B L E M S

(8.59)

Equation 8.59 is based upon thin-wall theory (see Eq. 3.14)and the assumption that the yield strength is one half ofthe ultimate strength. A 5% reduction factor is includedin the denominator on the basis of both theoretical con-siderations and experience.

For the intermediate range of t/do ratios lying between0.14 and 0.30, a gradual transition was selected by thecommittee (213). By selecting allowable working stresses(w.s.) equal to 40% of the yield strength for steel andallowable working pressure equal to one fifth of the col-lapsing pressure as determined by Eq. 8.58, Fig. 8.10 wasconstructed.

In reference to Fig. 8.10, the transition zone was drawnas a straight line from t/do = 0.14 to t/do = 0.30 for eachof the curves. A cross plot of this curve with numerousparameters of t/d is presented in Fig. 13.5 of Chapter 13for convenient use in design applications. Although Fig.8.10 and Fig. 13.5 were prepared for mild-steel tubes, thesefigures can be used for other ferrous materials when a factorof safety of either 4 or 5 is employed and for nonferrousmaterials when a factor of safety of 5 is employed.

1. Determine the maximum allowable vacuum that can be applied to a low-carbon-steelcylindrical vessel 10 ft in outside diameter with elliptical dished heads (k = 2.0). The beadand shell thickness are both 36 in., and the vessel is 30 ft long (tangent line to tangent line).

2. Determine the minimum number of equally spaced circumferential shell stiffeners requiredto permit use of full vacuum with the vessel described in problem 1.

3 . (a) Determine the minimum required moment of inertia for the stiffeners required in prob-lem 2. (6) Select the minimum-weight channel suitable (see Appendix G) for this purpose,

4. The shell for a vacuum crude tower 30 ft in diameter is constructed of low-carbon-steelplate 1x6 in. thick. Circumferential shell stiffeners are to be located 6 ft apart. Determinethe required moment of inertia for the stiffeners, and suggest a possible design for the stiffeners.A corrosion allowance of 41s in. is required.

5. A vacuum crystallizer 12 ft in diameter is to be fabricated of mild steel and is to have a60’ cone (apex angle) at the base and a torispherical closure at the top. The distance fromthe junction of the cone with the shell to the point of tangency of the top closure is 18 ft.Specify the design for: (a) the torispherical head. Cb) the shell, and (c) the cone.

I \ \ \I /

Page 165: Process Equipment Design

C H A P T E R

I-190DESIGN OF TALL VERTICAL VESSELS

Tu a!1 vertical vessels may or may not be designed to be

self-supporting. The design of self-supporting vertical ves-sels is a relatively recent concept in equipment design;high structures were formerly stabilized by the use of guywires. The self-supporting type of tower is widely usedtoday since it has been found uneconomical to allocatevaluable space for the wires of guyed towers. Further-more, the esthetic appearance of a clean-looking plant hasbeen recognized as having commercial value. Self-sup-porting columns 200 or more feet high that possess attrac-tiveness, safety, utility, and economy of construction are inuse today. The conditions under which vertical pressurevessels operate are often severe, and since the contents arequite often inflammable, structural failure is a seriousmatter. Simple membrane stress relationships are insufi-cient to predict the stresses induced by the action of windand seismic forces.

9.1 INDIVIDUAL STRESSES IN THE SHELL

The stresses in the shell of vertical vessels are essentially:(1) the axial and circumferential stresses resulting frominternal pressure or vacuum in the vessel; (2) the compres-sive stresses resulting from dead loads including the weightof the vessel itself plus its contents and the weight of insula-tion and attached equipment; (3) stresses resulting frombending moments caused by wind loads acting on the vesseland its attachments; (4) stresses caused by any eccentricityresulting from irregular load distribution; (5) stresses result-ing from seismic (earthquake) forces.

In addition, stresses may result from fabrication proce-dures such as cold forming, and welding. Figure 9.1 showsthe cold forming of a cylindrical shell from flat plates, and

155

Fig. 9.2 shows the welding of an inside longitudinal seamof a vessel shell. Figure 9.3 shows a completed weldedvessel entering the oven for stress relieving.

9.2 AXIAL AND CIRCUMFERENTIAL PRESSURE STRESSES

A cylindrical vessel under internal pressure tends toretain its shape in that any out-of-roundness or dents result-ing from shop fabrication or erection tend to be removedwhen the vessel is placed under internal pressure. Thus,any deformation resulting from internal pressure tends tomake an imperfect cylinder more cylindrical. However,the opposite is true for imperfect cylindrical vessels underexternal pressure, and any imperfection will tend to beaggravated with the result of possible collapse of the vessel.For this reason, a given vessel under external pressure ingeneral has a pressure rating only about 60% as high as itwould have under internal pressure. This is only anapproximation and other considerations must be taken intoaccount in determining the rating of vessels under externalpressure.

9.20 Tensile Stresses Resulting from Internal Pressure.The axial and circumferential stresses due to internal pres-sure in the shell of a closed vessel were developed in Chapter3 and are given by Eq. 3.13 and Eq. 3.14, respectively.

The axial tensile stress is:

Rdfw = 4(t, (3.13)

The circumferential tensile stress is:

fcp = -EL..2(& - c)

( 3 . 1 4 )

Page 166: Process Equipment Design

Design of Tall Vertical Vessels

Fig. 9.1. Rolling vessel shell from 3-in. plate. (Courtesy of C. F. Brow &

Co)

9.2b Compress ive Skesses Result ing from ExternalPressure. External pressure act ing upon a cyl indrical shel land its heads may result in failure of a vessel either byJ-ielding or by buckling. If the vessel has a relatively thinw-all, the stress at which wrinkling or buckling begins tooccur is usu3Uy below the yield strength of the material.If the vessel has a relatively thick wall, the stress at whichbuckling occurs is the yield point of the material underconsiderat ion at the temperature of service .

Since vessels operating under external pressure must bedesigned in accordance with elastic-stability criteria, thedesign is based upon the cr i t ical pressure at which buckl ingoccurs rather than upon an al lowable s tress for the mater ia l .The design procedure to be fol lowed is given in the previouschapter. After designing the vessel for external pressureservice (using procedures given in Chapter 8) we may useEys. 3.13 and 3.14, referred to above, to evaluate theinduced compressive axial and c ircumferent ia l s tresses .

9.3 COMPRESSIVE STRESSES CAUSED BY DEAD LOADS

The dead load acting on the vessel is determined by theweight and location of all the exterior and interior attach-ments such as: trays, overhead condensers, platforms,insulation, and so on. Those loads which act eccentr ical lymay be reduced to vertical forces and moments acting atthe central axis of the tower. This section will consideronly the vertical compressive forces acting on the vessel,and a later sect ion wil l cover the summation of the momentsproduced by eccentric loads.

Stresses caused by dead loads may be considered in threegroups for convenience: (1) Stress induced by shell andinsulation (2) Stress induced by liquid in the vessel (3)Stress induced by attached equipment.

STRESS INDUCED BY SHELL AND INSULATION. At anydistance, X feet, from the top of a vessel having a constantshel l th ickness ,

where W = weight of shell above point, T, poundsDo = outside diameter of shell, feelDi = inside diameter of sheli, feetX = distance from top to point under consideration,

feetps = density of shell material, pounds per cubic foot

= 490 lb per cu ft for steel constructionA n d

Wiw = $ Dins.Xtins.Pim. (9.2)

where Dins, = mean diameter of insulat ion, feetW i n s . = weight of insulat ion

Pins. = insulation density, pounds per cubic foot= 40 lb per cu ft for most insulation

tin.3, = insulat ion thickness , inches

Since compressive stress is force per unit area, disregard-ing corros ion a l lowance, c , g ives :

fdead wt s h e l l =ir/4(Do2 - Di2)Xp, Xp,a/4(Do2 - Di2)144 = z

= 3.4x (if py = 490 lb per cu fL) (9.3a)

The stress due to the dead weight of insulation is:

fdead wt ins. = __144mD,(t, - c)

where D, = mean diameter of the shell, feetDim. x DO = diameter of insulated vessel, feet

t, = shell thickness, inches

(9.4)

Wshell = ; (Do2 - Di2)p,X (9.1) Fig. 9.2. Welding inside longitudinal seam of 3-Lthick shell section.

(Courtesy of C. F. Brow 8 Co.)

- - - - ---r- -. -.--

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Tensile and Compressive Stresses 157

F i g . 9 . 3 . W e l d e d c o l u m n e n t e r i n g o v e n f o r s t r e s s r e l i e v i n g . ( C o u r t e s y o f C . F . Bran & C o . )

Therefore,

fPins.Xfins.

dead WC ins. = 144Ct, _ el (9.4a)

STRESS INDUCED RY SUPPORTED LIQUID.

I 1: liquid wt(9.5)

STRESS INDUCED BY ATTACHMENTS SUCH AS TRAYS,OVERHEAD CO N D E N S E R S. TOP H EAD , FLATFORMS, A N D

LADDERS.

., f2 weight of attachments

dead wt attach. = -__-~--~~---__-lzaD,(t, - c)

(9.6)

The weight of steel platforms m;ly be estimated at 35 lbper sq ft of area, and the weight of steel ladders at 25 lbper lin ft for cagec! ladders and 10 lb per lin ft for plainladders !i Z). ‘l‘rays in distilling columns, including liquidhold-up on trays, IXIY be estimated to have a weight of 25 lbper sq ft of tray area.

The total dead-load stress, fd7, ncticg along the longi-tudinal axis of the shell is then the sum of the above daad-weight stresses

If&z = .fdead w t she’1 f .filrrtd wt ina. -b fdead w t lir!

+ fdasd st attach. (9.7)

where .h, = lhr total dead-load stress acting along thelongitudinal axis at point X, pounds per squareinch

If the vessel does not contain internal attachments, suchas trays which support liquid, but consists only of the shellinsulation, the heads, and minor attachments such as man-holes, nozzles, and so on, the additional load may be esti-mated as approximately equal to 18yo of the weight of asteel shell, or as shown by Nelson (139),

j& = (1.18)(3.4)X = 4.0X (9.8)

9.4 TENSILE AND COMPRESSIVE STRESSES CAUSEDBY WIND LOADS IN SELF-SUPPORTIKG VESSELS

The stresses produced in a self-supporting vertical veseeiby the action of the wind are calculated by considering thevessel to be a verticle, uniformly loaded cantilever beam.The wind loading is a function of the wind velocity, airdensity, and the shape of the tower. The United StatesWeather Bureau (137) has correlated the above factors inthe following relation:

(9.9)

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158 Design of Toll Vertical Vessels

Fig. 9.4. Minimum allowable resultant wind pressures (137). (Th is material is reproduced from the American Standard Building Code Requiremenfr for

Minimum Design Loads in Buildings and Other Strucfures, A58.1-1955, copyrighted by the American Standards Association.)

where B = barometric pressure, inches, mercuryP, = wind pressure on a flat surface, pounds per

square footVW = wind velocity, miles per hourF, = shape factor = 1.0 for flat plate at 90” to the

wind

For a barometric pressure of 30 in. of mercury Eq. 9.9becomes:

P, = 0.004Vw2F, (9.10)

The shape factor, F,, for a smooth cylinder has been foundto be 0.60 (137). Thus the resistance of a smooth cylinderis 60% of that of a flat surface normal to the wind and hav-ing the same projected areas as the cylinder. Projectionsof auxiliary equipment loaded on the tower will causeturbulence, and the use of a value of F, based on smoothcylinders is questionable. Therefore, the value of F, usedby designers varies from 0.60 to 0.85, depending on theamount and shape of the projections on the vessel. If avalue of 0.60 is used for the shape factor then Eq. 9.10becomes

P, = 0.0025Vw2 (9.11)

The appropriate wind velocity that should be used inEq. 9.11 is dependent upon the location in which the equip-

ment is to be erected. In the Gulf Coast area winds up to125 mph are experienced. Most other regions experienceintermediate maximum wind velocities; therefore a figureof 100 mph is often used. Figure 9.4, published by theAmerican Standards Association (137) is a map of theUnited States indicating minimum allowable resultant windpressures at 30 ft of elevation.

To obtain the design force, P,, the wind-velocity pres-sures should be multiplied by a shape factor and a heightfactor. In the use of Fig. 9.4 a shape factor of 0.6 is recom-mended for chimneys and clean circular towers, and a shapefactor of 1.0 for rectangular buildings and structures.The height factor is 1.0 for structures having heights from30 to 49 ft. For higher structures the height factor variesdirectly as the (height/30) raised to the >f power (138).

In using Table 9.1 reference is made to Fig. 9.4 to deter-mine the wind pressure at an elevation of 30 ft for thelocality in question. The design pressure for the tower isobtained from Table 9.1 after one knows the height of thetower. The value obtained from Table 9.1 should bemultiplied by the appropriate shape factor, F, for cylindricaltowers. These design values are recommended as minimumand do not provide allowance for tornadoes.

As pointed out by Bergman (140) the relationships givenby the ASA (137) presented here for use with Fig. 9.4 and

Page 169: Process Equipment Design

Tensile and Compressive Stresses 559

where M,, = bending moment due to wind at X distancefrom the top, inch-pounds

deff. = effective diameter of vessel, inches

This equation is subjected to the limitation that the windacts over the total distance, X.

The stress in the extreme fiber of the shell. due to thewind, is obtained by use of Eq. 2.10:

,Re

Fig. 9.5. Drag coefficients for circular cyliiders (141). (Courtesy of

M c G r a w - H i l l B o o k C o . )

Table 9.1, do not consider the effect of velocity on the dragcoeflicient. The drag coefficient is similar to a frictionfactor and varies with the Reynolds number, Re, as shownin Fig. 9.5.

Figure 9.5 shows that between Re = 500 and Re =500,000, the drag coefficient is fairly constant, with a valueof about 1.1 for cylinders with L/D = A. However, at avalue of Re equal to 6 X 106, the drag coeflicient dropsabruptly to 0.7 for rough-surface cylinders and to 0.3 forsmooth cylinders. The wind pressure determined by useof Table 9.1 is based upon the higher drag coefficient andvalues of Re between 5 X lo2 and 5 X 106. The value ofRe is equal to 9100 DV where D is the vessel diameterin feet, and V is the wind velocity in miles per hour (140).Thus a vessel having a diameter of 8 ft in a wind having avelocity of only 10 mph would have a Reynolds number of7.28 X 105, which is above the transition value of about6 x 105. Therefore the use of Fig. 9.4 and Table 9.1results in a wind-pressure safety factor of about 2 to 3,depending upon the smoothness of the vessel.

The force, P,, acts over the projected area of the column,and some designers compensate for the turbulence causedby the projections by using an “effective” diameter, d,a., ofthe vessel and the allied equipment. This effective diam-eter is the diameter of the vessel plus twice the thickness ofthe insulation plus an allowance for the projected area ofpiping and attached equipment. For open-framed struc-tures the effective area is taken as twice the projectedarea, and an allowance of 17 in. is made for caged ladders(139).

Figure 9.6 shows a group of self-supported vertical vesselswith caged ladders and platforms. Note also the externalpiping, which increases the effective diameter (d& towind loads.

After determining the values of the wind loading and theprojected area upon which it acts, the bending moment anydistance X from the top of the tower can be expressed as:

Mm, = P,X ($) (y) = ~PwX2d,,. (9.12)

At the base of the tower,

fwb = W&o)

I

(9.13)

where PO = outside radius of shell, inchesI = rectangular moment of inertia perpendicular to

and through the longitudinal axis, inches 4

fWZ = stress at extreme fiber due to wind load, poundsper square inch (compressive stress on down-wind side, tensile stress on upwind side)

In design calculations it is assumed that auxiliary equip-ment will add load to the vessel but will not aid in its sup-port; therefore, the extreme fiber is at the outside surfaceof the shell.

For any values of t/r that would be encountered in vesseldesign, this relationship can be simplified as follows.

The equation of a circle is:

x2 + y2 = P2and by Eq. 2.8:

I, = j-z” dA

If the integration is performed in the first quadrant, then

I, = 4J

or x2 dA

Assume that the area of a thin shell is 2mt and that dA =tds . Introducing the derivative of an arc length (38) gives:

ds = 41 + (dy/dx)2 dx =

Therefore

dA = d&2 dx

TI, = 4ts

XP dx

0 2/772(9.15)

Table 9.1. ASA Recommended Wind Pressures for

Various Height Zones above Ground (137)(Courtesy of American Standards Association)

Wind pressure-map areasHeight zone (lb per sq ft)

et) 2 0 2 5 3 0 3 5 4 0 4 5 5 0

Less than 30 1 5 20 25 25 30 35 4030 to 49 2 0 2 5 3 0 3 5 4 0 4 5 5 050 to 99 2 5 3 0 4 0 4 5 5 0 5 5 6 0100 to 499 30 40’ 45 55 60 7 0 75

- _ _~~... - ---\--- \ - - \ i--f-----L----‘ -. w..- -

Page 170: Process Equipment Design

169 Design of Tall Vertical Vessels

Fig. 9.6. S&f-supporiing verfkol vessels with caged ladders, platforms, and external piping. (Courfe~y of C. F. drain g Co.)

Page 171: Process Equipment Design

Guyed Vessels 161

Substituting x = P sin 0 9.5 GUYED VESSELS

r/2I, = 4tr3 sin’ 0 dt9

= 4tr” _ 1[

e T/2sin 0 cos e

2 2 1 0

= 4lr3 $[I

thereforeI, = 7dt (9.16)

Or, including an allowanc~e for corrosion and using Ihemean radius, r,, we obtain:

Z = 7rrm3(f, - c) (9. I6a)

\vliere

Substituting E(4. 9.16 hit.0 E(4. 9.13 gives (lwwuse

The chief advantage of using guy wires for restraininga tall vertical vessel is the reduction in the size of the founda-tion and in the size and number of foundation bolts. Thisis offset in part by the anchorages for the guy wires and bythe nuisance created by the wires. The design of guyedvessels has been discussed by Marshall (142, 143).

The wires used for guying purposes are usually of wirerope fabricated of high-strength-steel strands around ahemp core saturated with a preservative and a lubricant.The wire rope is generally specified by two numbers, ofwhich the first gives the number of strands per cable, andthe second gives the number of wires per strand. Thus, a6 x 7 wire rope, which is commonly used for guying pur-poses, contains 6 strands with 7 wires per strand.

/ Commercial wire rope is available in several grades:\? extra-high strength, plow steel, and cast steel. The approxi-

mate breaking strengths in tons for these grades are givenby the following equations for 6 x 7 cable:

P?lL = ro): T = 41d2 (for extra-high strength) (9.22).I(9.17) T = 36d2 (for plow steel) (9.23)

By substitut.ing Eqs. 9.12 and 9.15 into E(4. 9.13 thegeneral equation for the bending stress on the vessel shelldue to a given wind load, P,, may be obtained.

f,,,x - PwX2&vo27rrm3(fs - c)

(9.18)

But r, = ro and doi2 = r,n; therefore, Eq. 9.18 rz111 be

T = 30d2 (for cast steel) (9.24)

where T = approximate breaking strength, tonsd = wire-rope diameter, inches

The approximate costs of 6 x 7 cable is (144) :

Dollars per foot = 0.4.4d2 (for extra-high strength)(9.25)

simplified to:

(9.19)Dollars per foot = 0:EOd’

Dollars per foot = 0.28d2

(for plow steel)

(for cast steel)

(9.26)

(9.27)

Simplifying Eq. 9.19 for the case where P,,; is 25 lb persq ft gives I he following two equations.

For insulated towers,

15.89deE.X2J”,,, =

do2& - c)(9.20)

For noninsulated t,owers,

(9.21)

Equations 9.20 and 9.21 are limited by the followingrtasl rictions:

. For design purposes a factor of safety of 4 is usuallyapplied to Eqs. 9.22, 9.23, and 9.24 to obtain the allowablecable loads. Greater factors of safety are used wheresudden loads are anticipated.

Usually three or four sets of guy wires are equally spacedaround the vessel. For vessels up to 50 ft in height onecable is used in each position; for vessels 75 ft in heighttwo cables are often used in each position; for vessels over75 ft in height three or more cables are commonly used ineach position. These wires are attached to a rigid collarusually located at two thirds and sometimes three quartersthe height of the vessel.

I. The wind pressure is 25 lb per sq ft.2. The wind acts at the above intensity over the entire

length of the column.3. There are no external attachments on the tower.4. The moment of inertia of the shell about its transverse

axis is:Z = 7rrm3(t, - c)

5. The mean radius of the shell is approximately equalto the outside radius.

The general form of Eq. 9.19 incorporates assumptions 2,4, and 5 above.

9.50 Tension in Guy Wires. To avoid the danger ofhaving a slack guy wire, an initial tension of one fourth ofthe allowable cable load is applied by tightening the turn-buckles on the cable.

The guy wires are used to counteract the bending momentcaused by the wind load, given by Eq. 9.12. Using thisequation with X equal to the total height, H, and with theguy ring located at y<H, we find that the horizontal force,F,, to be absorbed by each set of wires in pounds is:

(9.28)

In Eq. 9.28 the lever arm of the total wind load is assumed

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162 Design of Tall Vertical Vessels

to be j<H whereas the lever arm of the horizontal componentof the force on the guy wires is y<H.

If for% sets of Anumber of guy wires are used on the vesseland if allowance is made for a wind load from any direction,each set should be designed to carry this load in additionto the initial tension load. The guy wires are usually con-nected to the vessel so as to make an angle, 8, with thevertical of from 30’ to 70°. Therefore, the total tensileforce, F,,, in each guy wire on the upwind side will be:

Fgw = B Pto Hdef i .16n sin 0

(9.29)

9.5b Compressive Stress in Vessel Induced by Guy-wire Tension. The tension in the guy wires will have a ver-tical component downward which will add to the compres-sive stresses in the shell. This component Fcg is equal to thetotal tension in each cable times the number of cables timesthe cos 8. For four sets of n number of guy wires with Fwdownwind and both neutral sides equal to F&5 upwind,

(9.30)

Therefore, since do x d,:

fcg = f)Q = PwH&.

s 8 tan fhrdo(t, - c)

fee =O.O4P,H deffL

tan 0(t, - C) do(9.31)

wheref,, = compressive stress in shell from vertical com-ponent of four sets of guy wires, pounds persquare inch

The compressive stress feg induced in the shell by thevertical component of the guy wires must be added to thecompressive stresses from dead weight, and so on in analyz-ing the combined stresses in the vessel.

9.5 Bending Moment. From the top of the vessel tothe guy ring, the vessel is designed like an unguyed vessel,and .I;y?le bending moment in this region is given by Eq.9.12. From the guy ring to the base of the vessel, thebending moment caused by a wind load on a guyed toweris considered to be that of a beam overhanging one supportwith a uniformly distributed load. If the guy ring islocated at 3$H, the bending moment, Mwsg, between thering and the base is given as:

Mwrg = ‘+ (3Hx _ H2 - 2x2) (9.32)

The maximum value of the moment MWZg is given byEq. 9.33 and is located at a distance of H/4 feet above thebase.

M P,detr H2wzg(msx) =L

3 2

By substituting by means of Eq. 9.13bending stress, fWZr,, may be determined.

(9.33)

the corresponding

fwza = Mwzg7rro2(t, - c)

(9.34)

where iVWZO = bending moment between guy ring and base,inch-pounds

P, = wind pressure, pounds per square footdee. = effective diameter of shell for wind load,

inchesH = vessel height, feetX = distance from top of vessel, feetPO = i&de radius of shell, inchest, = shell thickness, inchesc = corrosion allowance, inches

9.6 EXAMPLE CALCULATION 9.1

Compare the maximum bending moments from windloads and stresses induced by such bending in a self-sup-ported vessel with those in a guyed vessel. The vessel is100 ft, 0 in. high, 7 ft, 0 in. in outside shell diameter, andhas ‘a shell thickness of N in. Corrosion allowance is $4 in. ;magnesia insulation 3 in. thick is used. No other attach-ments are considered in this problem. The maximumanticipated wind pressure is 25 lb per sq ft. For the guyedcondition four cables equally spaced making an angle of45” with the vertical will be used with the guy ring at66 ft, 8 in.

In the self-supported tower,

Pw = 25 lb per sq ft

The bending moment in the unguyed vessel will be amaximum at the base.

By Eq. 9.12, for an unguyed vessel where X = 100 ft,

Mm = iP,X2dee.

= +(25)(100)2(84 + 6)

= 11.25 X lo6 inch pounds

By Eq. 9.17

jwz = Mw, 11.25 x 1067rro2(t, - c) = 7r(3.5 x 12)2(0.5)

= 4070 psi

Or by Eq. 9.20

fwz =15.89detr.X2do(4 - c)

= W.WW’W’O~” = 4050 psi842(0.5)

In a guyed tower, by Eq. 9.33,

M Pwdetr.H2 (25)(90)(100)2wzg(max) = ___ =

3 2 3 2= 0.703 X lo6 inch pounds

By Eq. 9.34

By Eq. 9.31

fco =O.O4P,H detr. (0.04)(25)(100) (84 + 6)

tan e(t, - C) do = (1.00)(0.5) (84)= 213 psi

Thus the maximum bending stress due to wind load is4070 psi in the self-supported tower and oniy 254 psi in

Page 173: Process Equipment Design

the guyed tower. However, an additional compressivestress of 213 psi is induced in the guyed tower as a result ofthe downward vert ical component of the guy-wire react ion.

To calculate the s ize of guy wire required when four cablesare used, by Eq. 9.29,

Fgw =(5) (WUW W’)P H&rz > _ = 6200 lb or 3.1 tons

1 6 n s i n 0 (4)(16)(4)(0.707)

Using a factor of safety of 4, and a cast-steel cable, wefind that by Eq. 9.24

Breaking strength = T = 30d2

Tallowable = 30d2 = 3 1 tons4 *

S o l v i n g f o r d we obtain:

d = (4)(3.1)___ = 0.642 in.3 0

Therefore, use four cables 31 in. in diameter (or a largernumber of smal ler-s ized cables) .

9.7 STRESSES RESULTING FROM SEISMIC FORCES

Earthquake phenomena in certain geographic locationsresult in the production of vibrational loads. The seismicprobability in various localities of the United States asprepared by the U. S. Coast and Geodetic Survey is shownin Fig. 9.7. The chart indicates a division of the UnitedStates into seismic zones according to the amount of damagecaused by earthquakes. Maximum hazard of damage toequipment occurs in seismic zone 3 .

Current practice in designing for these seismic forces isempirical and is based upon the theory of vibration. Indeveloping the re lat ionships for v ibrat ional loads , i t i s con-venient to use the theories of strain energy and simpleharmonic mot ion.

9.7a Strain-energy Relationships. Consider the gen-eral condition of a uniformly loaded beam having a dis-tributed load of w lb per lin in. Each elemental load willbe equal to w dx (see Fig. 2.3). If the load is graduallyapplied, the average elemental force wil l be equal to (w/2) dx.The beam will deflect a distance of y (see Eq. 2.17) and thework done by each elemental force wil l be equal to (w/Z)y dx.The work done by the load on the beam is equal to theintegra l .

U=; y d xJ

(9.35)

In order to solve this relationship, the deflection equationfor the beam must first be evaluated and then substitutedfor y in Eq. 9.35.

The strain energy may also be evaluated in terms of theinternal resisting stress in the beam. Equation 2.26 givesthe unit strain energy at any point in the beam in terms ofthe s tress at the sect ion in quest ion. The s t ress var ies f roma maximum tensi le s t ress to a maximum compress ive s t ress ,passing through zero at the neutral axis, as defined by Eq.2.10.

Stresses Resulting from Seismic Forces 1 6 3

Note: The term y in Eq. 9.35 is the deflection of a beam,whereas the term yc in Eq. 2.10 is the distance from theneutral axis of a beam to the fiber in question. The maxi-mum value of ye is equal to c.

The bending moment M varies along the length of thebeam; therefore, the stress f varies both with the distancefrom the neutral axis and along the beam. Subst i tut ingEq. 2.10 into Eq. 2.26, gives:

&fz2 E

(2.26)

(9.36)

Consider a differential volume in the beam dA dx (seeFig. 6.5). The dif ferent ial s train energy in this di f ferent ialvolume is :

Integrating we obtain:

UL-2 E

$’ yc2 dA dx

From Eq. 2.10

sye2 dA = I

Substituting into Eq. 9.38, we obtain

u 1Total = z

sF dx

(9.38)

(9.39)

Substituting for M in Eq. 9.39, by Eq. 2.14, we obtain:

&EI2

(9.40)

“r”’ To apply Eq. 9.40 to the vibration of a vertical vesselunder seismic load, the deflection curve for the vessel must

be known. A vert ical vessel bolted to a foundation behavesas a cantilever beam. .,

‘- 9.7b Deflect ion of a Uniformly Loaded Canti leverBeam. Figure 9.8 is a sketch of a beam of uniform crosssect ion def lect ing under uniform load.

The moment at any point, x is:

l - xM = w(Z - 2) y- = ; (1 - x)2 (9.41)

But

(2.14)

Therefore

(9.43)

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t/I

-I

41

25

125. IM‘ 115. 110' IOE'_- _.. tw 95 90'I 85.I I 80.I , 15. 70"\ 65'\

I !-- l / /MISS

Statute mksLI I I I 1 I

100 50 0 1 0 0 2ca 3 0 0 4 0 0KllOm&rS

III I I I I I I 1100 50 0 2 0 0 4 0 0 600 a00I

Zone O-no damageZone 1- nvnor damageZone 2 -moderate damageZone 3 - maior damaee

F i g , 9 . 7 . S e i s m i c p r o b o b i l i t y m o p o f t h e U n i t e d S t a t e s ( f r o m t h e U . S . C o a s t a n d G e o d e t i c S u r v e y ) .

Page 175: Process Equipment Design

Fig. 9.8. Cantilever beam deflecting under uniform load.

Integrating Eq. 9..22 ,oivrs:

I’.,* - lx” + $ + (:,I

(9 .44)

H u t dy/dx = 0 a~ r = 0; t herrforc C, = 0. IntCgralingagain gives:

But. y = 0 when L = 0; therefore (Jz = 0. ‘I’ht!refore

9.7~ Stra in Energy of a Def lected Vert ical Vessel .If a vessel is deflected elastically fro~rl its vertical axis byseismic forces or wind load, elastic (strain) energy is storedin the deflected vessel. The same relationships hold asin the case of the elastic deformation of a beam; the strainenergy is defined by Eq. 9.40, and the second derivative ofthe deflection by Eq. 9.43. By combining the IWTo equa-tions, with h, the vessel height, substitut.ed for 1, the beamlength, the fol lowing re lat ionship is obtained:

h(h - x)a dx

Integrating we ohlain:

If a vertical tower is vibrating, the maximum velocit)and therefore the maximum kinetic energ)- occurs at zerodisplacement. As the t.ower displacement approaches amaximum, the kinetic energg approaches zero and theebstic energy of strain approaches a maximum. Themaximum kinetic energ) (at zero displacement) must eyualthe maximum strain energy (at maximum displacement) ifI he energy in t,he system is considered to be constant . Thesolution of this equality involves the evaluation of kineticenergy due to harmonic vibrat ion.

9.7d H a r m o n i c V i b r a t i o n . A displacemen i n t h eearth’s crust produced by seismic forces results in a suddenshift in the foundation of a vessel relative to its center ofgravi ty . The inertia of the vessel restrains the vessel frommoving s imultaneously with the foundation and this resultsin an e last ic def lect ion of the vessel . This e last ic def lect ion Fig. 9.9. Harmonic vibration components.

Stresses Resulting from Seismic Forces 1 6 5

initiates a harmonic vibration in a vertical vssel, similar tothat. in a reed or a tuning fork clamped at the base and setin motion by a sudden force.

The relationships for such a simple harmonic vibrationcan be derived by considering the motion of a weight sus-pended on the end of a completely elast ic spring. Considerthis system first in its equilibrium position. If a force, F,is applied to the weight parallel to the axis of the system,the weight will be displaced along the axis. The displace-ment, .y, will be proportional to the force, F, and to thespring constant, k. Considering the force , F, to be posi t ivedownward ( increasing spring tension) and negative upward(decreasing spring tension) , we obtain:

Fd o w n = -F,, = ky (9 .47)

Using Newton’s re lat ionship ( force is equal to mass t imesacceleration), expressing mass as W/g and acreleration ast,he second derivative of y with respect to time, t, we obtain:

I (9 .48)

Combining Eqs. 9.47 and 9.48 for a force upward, we obtain:

Using the notalion that p2 = kg/W, we obtain:

(9.50)

The generalis :

solution to this differential eyuation (9.50)

.y = Cl cos pt + C2 sin pl (9.51)

The term p has the significance of angular velocity andthe term pl is the angular displacement in radians at an)time, 1. The functions cos pt and sin pt are periodic func-tions that repeat when the angular displacement reaches2 7 r . The interval of t ime between such repet i t ions is cal ledthe period T, and the reciprocal of the period l/Z’ is thefrequency of vibration. Therefore

pT = 2~ (9 .52)

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1 6 6 Design of Tall Vertical Vessels

h

Fig. 9.10. V i b r a t i o n a l d i s p l a c e m e n t o f a v e r t i c a l t o w e r .

Combining Eqs. 9.52 and 9.49 gives:

T=Z=2*MgP

To evaluate the constants of Eq. 9.51 consider the condi-tion when the weight, W, has a displacement of yo from itsequilibrium condition and is moving at a velocity of(dyo/dt). If this condition is taken as zero time (t = 0),by Eq. 9.51.

yo = Cl cos 0 + C2 sin 0therefore

Cl = yo (9 .54)

The first differential of Eq. 9.51 with respect to t is:

dy-=dt

-pCl sin pt + pC2 cos pt (9 .55)

Therefore at t = 0 the following is obtained:

= -pC1 sin 0 + pC2 cos 0 = pC2

therefore

(9 .56)

Substituting these values of Cl and Cz into Eq. 9.51 gives

sin pt (9 .57)

Inspection of Eq. 9.57 shows that the vibration consists oftwo parts. The major part is a vibration which is propor-tional to cos pt and depends upon the initial dispiacement,yo, as is shown in Fig. 9.9. The minor part is a vibrationwhich is proportional to sin pt and depends upon the initialvelocity dyo/dt.

If the minor vibration is disregarded,

y s yo cos pt (9 .58)

and

Using the fundamental relationship that kinetic energyis equal to one half the mass times the square of the velocitydy/dt gives:

where h = tower height measured from base

But the maximum kinetic energy exists at zero deflection,y = 0, and when pt = ?r/2. Substituting this conditioninto Eq. 9.59 gives:

dy-=dt

-pyo sin 5 = --pyo

Therefore

(9.61)

d?/ ’0ii- = P2Y02 (9 .62)

However, in a vibrating tower the velocity of sway varieswith the height, increasing to a maximum at the top, a3shown in Fig. 9.10. The total kinetic energy of the swayis equal to the integral from 0 to h, or

KE=!f h dy 2/( >_ &29 o dt

Substituting Eq. 9.62 into Eq. 9.63 gives:

(9.63 j

(9.64)

Substituting for yo by Eq. 9.45 gives:

df- 2(x4 -

2 4 E I4hx3 + 6h2x2)

1d;c

w3p2= 1152gE212 0s

h (x4 - 4hr3 + 6h2x2)2 dz

Therefore

(9 .65)

Equating Eq. 9.65 to Eq. 9.46 gives:

Solving for p gives:

thereforep = 3.53 1/(EIg/wh4)

The period of vibration, T, is equal to p/27r by Eq. 9.55,therefore

(9 .67)

Page 177: Process Equipment Design

Stresses Resulting from Seismic Forces 167

Table9.2. Data on Some Disastrous Earthquakes (145)

(Courtesy of American Society of Civil Engineers)

ApproximateHorizontal

Main Period, AccelerationLocation Date seconds (fraction of g) Comment

Japan June 1894 1.3 0.09 Serious damageSan Francisco April 1906 ? 0.25* DisastrousTokyo September 1923 1.35 0.10 DisastrousLong Beach, Calif. March 1933 0.30 0.23 Damage of $40-70,000,OOO

1.50 0.111.80 0.02

Helena, Mont. October 1935 0.13 0.13 Damage of $3-4,000,OOO0.31.0

* Estimated.

For a vertical steel cylindrical tower Eq. 9.67 may besimplified as follows. Let

E = 30,000,OOO psi, modulus of elasticityI = moment of inertia of shell given by Eq. 9.16g = 32.2 ft per sec2, gravitational constant

Substituting gives:

T = 2.65 x 1O-5 (9.68)

where T = period of vibration, secondsH = total tower height, feet (vessel plus skirt)D = tower diameter, feetw = tower weight, pounds per foot of height

t = tower shell thickness, inches

9.7e Seismic Load Coefficients. Design for earth-quake loads is primarily based upon empirical analysis ofstructures that have withstood severe earthquakes. Earth-quake motions are quite complex and not simple harmonicmotions as are the vibrations induced in vessels. However,earthquakes have had certain marked periods as shown inTable 9.2.

The fourth column of Table 9.2 gives the seismic coeffr-cient, which is the approximate horizontal accleration interms of fractions of g, the acceleration due to gravity.From Newton’s relationship that force equals mass timesacceleration we obtain:

F=!!WzCW (9.69)9

where a/g = C = seismic coefficient.Although Table 9.2 gives only the horizontal component

of seismic acceleration, it should be noted that a verticalcomponent also exists. However, the vertical seismic forcesare usually far less damaging to vertical structures andusually are disregarded in the designing of vessels.

In the empirical application of seismic coefficients tobuilding design it has been customary to allow for a hori-zontal force equal to CW where W is the weight of thebuilding and C is a seismic coefficient. Coefficients usedtoday for such purposes were based primarily upon the

data obtained from the disastrous Japanese earthquake of1923 known as the “Great Kwanto Disaster.” Surveyswere made of all buildings left standing with minor damageafter this disaster, and it was found that buildings designedwith a seismic coefficient of 0.10 or greater successfullywithstood the seismic forces. It has also been observedthat flexible structures can more readily absorb seismicforces without damage than rigid structures.

The Uniform Building Code (Pacific Coast) of 1952 (146)specifies a seismic coefficient of 0.10 for free-standing stacksand similar structures such as vessels in zone 3 along withan increase in allowable stress of $4. For buildings theeffect of flexibility of structure in reducing the coefficientis recognized by the specification of a variable coefficient.Although the Uniform Building Code does not specify sucha variable coefficient for free-standing stacks and relatedequipment such as vessels, it is believed that such a con-sideration is reasonable.

Vessels having short periods of vibration of 0.4 set or lessmay be considered to be rigid structures, and a seismiccoefficient of 0.20 for zone 3 is recommended for suchdesigns. This. coefbcient is twice the value of the coefficientfor free-standing stacks recommended JQL-& UniformBuilding Code. Vessels having a period of 1.0 set or greatermay be considered to be flexible and are therefore morecapable of absorbing seismic forces. For such vessels acoefficient of 0.08 for zone 3 is recommended. For vesselsin zone 3 having a period in the transition region between0.4 and 1.0 set, it is recommended that a seismic coefficientof 0.08/T be used where T is the period in seconds as deter-mined from Eqs. 9.68. A summary of recommended coeffr-cients for various seismic zones and various vessel periodsis given in Table 9.3.

Table 9.3. Recommended Coefficients for Various

Sesmic Zones (145)

Seismic Coefficient, CSeismic Period Period Period

Zone < 0.4 set 0.4-l set > 1.0 set1 0.05 0.02/T 0.022 0.10 0.04/T 0.043 0.20 0,08/T 0.08

Page 178: Process Equipment Design

168 Design of Toll Vertical Vessels

~

cw

H r

Fig. 9.11. Seismic forces on a vertical vessel.

9.7f Shear and Bending Moment Resulting from SeismicForces in Unguyed Vessels. The seismic forces act to pro-duce horizontal shear in vertical unguyed vessels. Thisshear force in turn produces a bending moment about thebase of the vessel. The shear loading will be triangularwith the apex at the base, as shown in Fig. 9.11. Thecenter of action for such a triangular loading is located at,2.SH. The shear force at the base resulting from seismicforces is given by Eq. 9.69.

The shear force, V,, (pounds) at any horizontal plane inthe tower X feet down from the top is given by:

v82

= CWX@H - X)

HZ(9.70)

where C = seismic coefficient from previous sectionW = total weight of tower, poundsH = total height of tower, feet

The bending moment MS, (inch pounds) at plane Xresulting from the shear forces above plane X is given by:

The corresponding heading stress may be determined byEq. 9.17. Therefore

(9.72)

The maximum shear and bending moment are located atthe base of the tower and may be found by substitutingX = H in Eqs. 9.70 and 9.71, respectively, or

vsb = cw (9.73)and

2CWH(12 in. per ft) = 8(,w~~A&t, = ~~- ~~ 1

3(9.74)

Substituting Eq. 9.74 into Eq. 9.72 gives the seismichending stress at the base of the skirt of the vessel.

(9.75)

where r = tower radius, inches1, = skirt thickness, inches

9.8 EXAMPLE CALCULATION 9.2

For the unguyed vertical tower described in ExampleCalculation 9.1, calculate the seismic bending moment andresultant stress at 25 ft and 50 ft above the base of thetower. The tower weighs 1800 lb per vertical foot of height(180,000 lb total) and is to be erected in southern California.

Calculation of vessel period:By Eq. 9.68

T = 2.65 X IO-” (;>‘(,>,

therefore

T = 0.856 set

Determinuiion of seismic coeficient:In reference to Fig. 9.7, southern California is in seismic

zone 3, and by Table 9.3 the period of vibration 0.856 setlies between 0.4 and 1.0 sec. Therefore t,he seismic coeffi-cient, C, is given hy

c - 0.08

T

c = 0.08 = 0.09350.856

Determination of seismic bending moments at 25 and 50 ft:By Eq. 9.71

Msz

= 4cWX2(3H - X )

H”

TV = 180,000 lb, at 25 ft (X = 75 ft), therefore

,~~, = ~f~0.0935)(180,000)(75)2(300 - 7 5 )h.r (IO@2 ___--~--

= 8.50 X lo6 in-lbAt X = 50 ft

M,, = 4.20 X lo6 in-lb

Determination of seismic stresses at the 25 arid 50 ft levels:By Eq. 9.72

At 25 ftM,, = 8.50 X lo6 in-lb

Therefore

fsz =8.50 X lo6~(42)~(0.5)

= 3080 psi

At 50 ft

ThereforeM,, = 4.20 X lo6 in-lb

fw = ;q2=; = 1514 psi

9.9 STRESS CAUSED BY ECCENTRIC LOADINGS

In vessels such as bubble-cap columns the shell and traysare placed symmetrical about the longitudinal axis, butexternal attached equipment usually acts as an eccentric

Page 179: Process Equipment Design

Combined Stresses in the Shell 1 6 9

Fig. 9.12. Erection ofdioxide absorption towerby 7 ft. 6 in. (CourteGirdler Company and theC h e m i c a l C o r p . )

a carbon84 ft. 0 in.

sy of TheMississippi

load and should be considered as such. Most externalattached equipment produces a negligible moment, andengineering judgment must be used in the calculation ofstresses. Equipment such as small ladders, pipes, arltl

manholes may usually be disregarded, but the total com-bined moment of heavier equipment such as overhead orside condensers is important. The eccentricity is calculatedby:

ZM,e=Zw,

(9.7C)

where e = eccentricity, the distance from the column axis tocenter of reaction, inches

Zkf, = summation of moments of eccentric loads, inchpounds

ZW, = summation of all eccentric loads, pounds

Eccentric loads produces a bending moment equal tozW,,(e). The additional bending stress at plane X causedhy this n~omen~ is:

Substituting ftn I hy Eq. 9.16 gives:

(9.77)

9.10 COMBINED STi?ESSES IN THE SHELL

A controlling combined tensile or compressive stressoccurs as a resu!t of combinations of stresses. It is import-ant to consider the intended construction, erection, andtest schedule which is to be followed in erecting the vessel

r-- -- I .“* --\ -~ \ -\r 7-_ ?$

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Design of Tall Vertical Vessels

fWz (from wind load)

[J (from seismic load)

fh (from total developed load)

fW (axial stress from internalpressure)

Fig. 9.13. Stress conditions in the ihell of a vertical vessel.

and placing it on stream. With this in mind, the condi-tions must he determined which establish the controllingstresses. Figure 9.12 shows the erection of a tall self-supporting tower on its foundation.

An evaluation must be made of the effect of the combinedstresses induced in the shell of a vertical pressure acted upon :by wind loads. These stresses are additive at specificpoints in the shell of the vessel. During the constructionand subsequent use of the vessel, the total combined stresseswill vary according to the forces acting on the vessel atany given time. The stressed condition of the vessel maybe divided into the following possible cases:

Case 1: vessel under constructiona. empty shell erectedb. shell and auxiliary equipment such as trays or pack-

ing but no insulationCase 2: vessel completed but shut downCase 3: vessel under test conditions

a. hydrostatic testb. air test

Case 4: vessel in operation

In the consideration of wind and earthquake loads it isassumed that the possibility that the most adverse windand earthquake load will occur simultaneously is remote andthe possibility that these lateral forces will occur in thesame direction is even more remote. Therefore, the result-ing stresses for wind loads and earthquake loads are com-puted separately, and the most adverse loading conditionused in the design.

In analyzing the combined stresses calculations are usu-ally made beginning at the top of the vessel. The mini-

mum shell thickness in the upper portion of the tower isusually controlled by the circumferential stress resultingfrom internal pressure or vacuum. The shell plate thick-ness at the top of the column can usually be specified on thisbasis by using a standard plate thickness slightly largerthan the minimum. At lower sections of the vessel, wherecompressive dead loads and wind loads become significant,the shell thickness must be increased in order to resist theseadditional loads.

The distance down the vessel for which the initial shellplate thickness may be used without exceeding the allow-able stress is determined by calculating the combinedstresses again, remembering that the specified distanceshould be reduced to some multiple of standard plate widthsto avoid cutting a plate unnecessarily.

In designing the shell it isnot necessary to allow for thecompressive stress resulting from the weight of the liquidin the hydrostatic test since the bottom head of the shelltransfers this load directly to the skirt. It is essential tocheck the lower sections for wrinkling failure before thetinal specifications are fixed.

Therefore, in the case where thermal, eccentric, and live-load stresses are negligible, and for positive pressures, thefollowing equations for the maximum combined stress canbe applied in the calculation of plate thickness. It must benoted, however, that if the stress due to eccentricity isappreciable, it should be included in the dead-weight-stressand bending-stress terms.

Figure 9.13 is a diagram indicating the stress conditionsin a vertical vessel resulting from the loads indicated. Byreference to this figure it is apparent that the maximumtensile stress exists on the upwind side of the vessel and themaximum compressive stress exists on the downwind sideof the vessel.

Maximum tensile stress (upwind side) at point X withan unguyed vessel under internal pressure and in the absenceof eccentric loads is:

flmnx = UWZ or f& + fap - fd~ (9.78)

For external pressure the equation becomes:

ftmax = (fwz or fsz) - fap - .fdz (9.79)

Maximum compressive stress (downwind side) at point Xwith an unguyed vessel under internal pressure and in theabsence of eccentGc loads is:

fcmax = (fwz or fsz) + fdz - fap

For external pressure the equation becomes:

(9.80)

fcmax = (fwz or .fsz) + fdz + fap (9.81)

9.11 DETERMINATION OF THICKNESSES OF SHELLPLATES FROM TENSILE STRESSES

The diameter and height of the vertical vessel are deter-mined by the process requirements. In the case of a dis-tillation column the diameter is determined by the maximumvapor load and allowable vapor velocity. The height ofthe tower is determined by the number of trays requiredto effect the desired separation and the selected tray spacing.The material of construction is determined by corrosion

Page 181: Process Equipment Design

Checking Shell Compressive Stresses for Elastic Stability 171

requirements, temperature and pressure levels, economic in thickness in order to withstand the increase in tensileconsiderat ions, and avai labi l i tv . stress resulting from wind load. In this case the calcula-

The minimum required shell thickness usually occurs in tion using Eq. 9.83 is repeated, with the larger plate thick-the top course of the vessel where cumulative stresses from ness substituted. The bending moment due to wind loadswind loads and dead weight are small . A t r ia l th ickness for increases with X2. Therefore, the plate thickness requiredthe top course can be selected on the basis of pressure con- wil l increase more rapidly with respect to X near the bottom

Isiderations by using Eq. 13.1. However, seismic loads may of the tower than in the upper region.be s ignif icant in the upper port ion of the vessel i f the vessel In checking the tensile load conditions the various casesis to be erected in seismic zone 3, in which case the selected of stress conditions should be investigated individually tothickness must be checked for the combination of pressure determine the controlling condition. If the vessel isstresses and stresses from seismic forces by using Eq. 9.72 designed for high-pressure service, the limiting conditionin Eq. 9.78 or 9.81, whichever is appropriate. I f the selected wil l usual ly exis t when the vessel i s operat ing under pressurethickness is satisfactory, it is next necessary to determine and under a wind load. As the vessel would not be testedhow many courses of this thickness may be used in the with air pressure under high-wind conditions, this case isvesse l . The limiting value of X for the initial plate thick- not considered to be controlling. Vessels designed forness selected may be determined by substituting into Eq. low-pressure service may encounter maximum-stress con-9.78, using Eq. 9.7 forjdz, Eq. 9.17 forjwz, Eq. 9.72 forjSz, ditions when the erected empty vessel is exposed to a highand Eq. 3.13 for jap. This may be done by trial and error wind load.bv assuming the number of courses down from the top and Any additional compressive loads such as those inducedchecking the combined stresses at this level. by trays with l iquid ins ide a dis t i l la t ion column or overhead

An alternative procedure is to solve directly for the dis- external condensers wil l re l ieve the tensi le load.tance X down from the ton of the vessel at which the maxi-mum induced stress is equal to the allowable stress bymeans of a quadratic relationship. For unguyed vessels

9.12 CHECKING SHELL COMPRESSIVE STRESSES FOR

under internal pressure fabricated of steel and for the case ELASTIC STABILITY

in which the maximum lateral loads are determined by 9.12a Critical Compressive Stress. After determiningwind loads rather than seismic forces for operating condi- the shell plate thicknesses in order to satisfy tensile stresstions, the fol lowing.Cquation may be used: requirements , the design should be checked for compressive

f2P,X2

+ pd awft)xstresses on the downwind side. This analysis is more com-

t*X = ?rd(t, - c) 4(t, - d(t,,- c)plex because elastic stability must be considered. Thin-walled columns stressed along their longitudinal axis may

where P, is given by Eq. 9.11. fail in two ways: by Euler’s buckling or by wrinkling.This equation may be modified to lit erection and other Fai lure due to Euler ’s buckl ing involves bending of the shell

condit ions not covered by the l imitat ions above. To deter- as a whole and is se ldom control l ing in vert ica l vessels withmine X, the distance at which the induced tensile stress cyl indr ica l she l l s . Wrinkling is local in nature and dependsis equal to the allowable stress, jrtllOw. may be substituted upon the combined compressive stresses at the point under

for ft max and the equation arranged in the form of a considerat ion. I t is necessary to determine the stress underquadratic. which this phenomenon occurs. The allowable critical

compressive s tress at which wrinkl ing does not occur whena steel cylinder is under axial compression was given inChapter 2 by Eq. 2.25.

+ pd~~ - fdhv.4(k T c) 1 = 0 (9.82)

fcallow. = 1.5 x 10s c 5 Q y.p. (2.25)r

Equation 9.82 has the following form:

aX2+bX+C=0 9.12b The Influence of Stiffeners. Vertical vessels maybe st i f fened by the addit ion of internal or external members

,/ where the coefllcients a, b, and c are defined by the quantit ies attached to the shell. The members may be attached inin the respective brackets. The solution of the binomial either the longitudinal or circumferential direction and in

, equation is : some cases in both direct ions . Timoshenko has shown how

x= -b + 1/b2 - 4acal lowance may be made for the st i f fening effect i f the mem-bers are uniformly spaced (42).

2 a Equation 2.25 may be modif ied as fol lows to al low for the

After determining the value of X by means of Eq. 9.83,st i f fening ef fect :

it may be desirable to adjust the plate thickness, t,, for thei

‘4‘

top courses so that the height of the section, X, will be a fcallow. =

multiple of the plate width used. Usually the plate thick-1.5 x lo6 f %qt,t,/t2)

ness originally selected is satisfactory for a number of 1.5 x 10s=courses. Plates below distance X must have an increase

fi 6 35 y.p. (9.84)P

Page 182: Process Equipment Design

1 7 2 Design of Tall Vertical Vessels

where

t,=t+$(the equivalent thickness of the (9.86~)shell in the circumferent.ial direc-Yl i o n )

t,=t+$(lhe equivalent thickness of the (9.84h)

z she l l in t.he longitudinal direr t ion)

where .12/ = cross-sectional area of one ~ircumferenlial slif-fener, square inches

d, = distance hetween circm~lferential stiffeners,inches

A, = cross-sectional area of one longitudinal sbif-fener, square inches

(1, = distance between longitudinal stiffeners, inches

It is significant that any additional area added to t.heshell as stiffeners increases the buckling strength of thevessel in proportion to the square root whereas the addi-t ional metal area added uniformly to the shel l by increasingits thickness will increase its strength in direct proportion.The obvious conclusion is that it is more economical tostiffen a shell by increasing the shell thickness t.han hyadding st i f feners .

9.13 EXAMPLE DESIGN, SHELL CALCULATIONS FORA TALL VERTICAL VESSEL

A close fract ionat ion tower is to he fabricated and insMledin the west-central area of Texas. The vessel has the fol-lowing speci f icat ions :

Shell outside diameter = 7 ft, 0 in.Shell length, tangent line to tangent line = 150 ft., 0 in.Operating pressure = 40 lb per sq in. gageOperating temperature = 300” FShell material = SA-283, Grade CShell, double welded butt joints stress relieved but not

radiographedSkirt height = 10 ft, 0 in.Tray spacing = 24 in. (71 trays)Top disengaging space = 4 ft, 0 in.Bottom separator space = 6 ft, 0 in.Tray loading including liquid = 25 lb per sq ftTray-support rings = 236 in. x 2>$ in. x N in. anglesCorrosion allowance, c = $6 in.Overhead vapor line = 12 in., outside diameterInsulation (ins.) = 3 in. on column and vapor linesAccessories = one caged ladder

Allowable stress (see Chapter 13 on code vessels) :SA-283, Grade C stress rel ieved but not radiographed has

an allowable stress of 12,650 psi (see Table 13.1), and awelded-joint efficiency of 0.85 is specified by the ASMEcode (11) (see Table 13.2).

Calculation of minimum shell thickness:

t = -J-QSE - 0.4p

+ c (13.1)

(40)(42)_(12,650) (0.85) - (0.4) (40)

+-fi

= 0.156 + 0.125 = 0.281

Therefore use ?is in.SeWion of head:A preliminary calculation indicates that the required

ellipt,ical-head thickness will be 516 in. Since ellipticaldished heads 7 ft, 0 in. in diameter are not made this t,hin,a t.orispherical dished head will he used. Also , tor ispher ica lheads are nominal on the out.side diameter.

Calculation:By Eq. 13.12

t = !):885pr,.fE - O.lp

= -----___,12,6;;;;;::“~~&oj + c = 0.275 + 0.125

= 0. &OO in. = &-in. t.hick head

By Eq. 3.12

Diameter = 011 + tz + 2sf + Qicr (5. L2j

From Table 5.7, icr = 514 in. , and from Table 5.8,sj= 3l$in.

Diameter = 84 + $$ + 2(1+) + g(5;)

= 84 + 3.5 -+ 3 + 3.4

= 93.9 in.

rrd2t pWeight. of head = --- _~

4 1 7 2 8

7r(93.9)?&) #&- ~-.___-~4

= 860 lh

Calculation of axial sfrrss in shel l :d< z do; therefore 11s~ = 84 in.By Eq. 3.13

jap = .A?-. = (-to) CS4)4(& - c) (4)(0.1875)

= .W80 psi

Calculation of dead weights:

j-dead wt shel l = 3.4x (by Eq. 9.3a)

Note: Eq. 9.3a applies only t,o shells of constant thick-nesses and may he used for t.he top section where the thick-ness is constant.

I) Pins.Xtins.Jdead wt ins. = 1~4(t, _ ej (by Eq. 9.4a)

40x3= (144)(0.1875)

= 4.44x

\ \ \I / -.

Page 183: Process Equipment Design

\\ I of top head = 860 lb

\\ t of ladder =WI. of 12-in. schedule

Appendix K) =\\‘ 1, of pipe insulation

; (1.52 - 1.02)40

Example Design, Shell Calculations for a Tall Vertical Vessel 173

25 lb per ft30 pipe (from

43.8 lb per ftzz

= 39.3 lb per fl

Total (860 + 108.1X)lh

W = 860 + 108.1X

#‘deed wt attachments =ZW

(not including trays) nd(t, - e)

860 + 108.tA= (3.14)(84)(0.1875)

= 17.4 + 2.19X

The weight of trays plus liquid (below X = 4) is calcu-lated as follows.

.fdrad wt (liquid + trays) =

= (9.73X - 19.45)

jclw = 3.4X + 4.44X + 2.19X + 17.4 + 9.73X - 19.45

= (19.76X - 2.05)

Calculation of stress due to wiud loads:The wind pressure HS obt.ained from Fig. 9.4 and Table 9.1

is:

From Pig. 9.4, wind pressure at 30 ft = 25 psfFrom Table 9.1, the corresponding wind pressure above

100 ft = 40 psf

If a shape factor of 0.65 is applied, the effective wind pres-sure above 100 ft will be 26 psf. Table 9.1 shows thatwind pressure corrected for shape will be about 20 psf below100 ft elevation. Therefore, a design wind pressure of25 psf will be used in the design calculations. This permitsthe use of Eq. 9.20.

(To minimize wind load place ladder 90” to vapor line.)

deff. = (insulated tower + vapor line)= (84 + 6) + (12 + 6) = 108 in.

By Eq. 9.20

15.89deR.X”jTOZ = ____ = 15.89(108X2)

d&t, - c) (8-4)2(0.1875)

4 Thereforefwt = 1.297X2

Calculation of combined stresses under operating condit ions:Upwind side: By Eq. 9.78

.f t (max) = .fwz + .fap - .fdr

= 1.297X2 + 4480 - 19.76X + 2.05

= 1.297X2 - 19.67X + 4.178

For an allowable stress of 12,650 psi and a joint efficiencyof 0.85.

(12,650)(0.85) psi = 1.297X2 - 19.76X + 447801

X2 - 15.25X - 4830 = 0

Solving for X gives:

Lhwnwind side:By Eq. 9.80

fc(nrax) = fw, - jap + .fdz

= 1.297X2 - 4480 + 19.76X - 2.05

= 1.297X2 + 19.76X - 4478

From elastic stability, by Eq. 2.25,

jc = 1.5 x 106 4 (= +y.p.0 P

Therefore

= 6690 psi

1.297X2 + 19.76X - 4478 = 6690

x2 + 15.3x - 8640 = 0

x = --lJ5 4 d(15.25)2 + (4)(1)(8630)2

= 85.5 ft

If credit is taken for the stiffening effect of tray supportrings, a higher allowable compressive stress will result.Therefore

t, = t, + 9 (see Eq. 9.8.t;l)II

whew I, = equivalent thickness of sheli, inchesill, = cross-sectional area of one rircun!ferential

stiffnerd, = distance between circumferrntial stiffeners,

inchest, = t, (since no longitudinal stiffener are used)

\ - \ \I / - v-

Page 184: Process Equipment Design

1 7 4 Design of Tall Vertical Vessels

The tray-support rmgs are 235 x ,s21’ x s in. angles.Therefore

A2/ = 1.73 sq. in.

dg = 24 in. (tray spacing)

t, = 0.1875 + g = 0.1875 + 0.072

t, = 0.26 in.

By Eq. 9.84

.fc = 1.5 x 10sdg 5 6 Y.P.

P

= =-$) 4(0.26)(0.1875)

(see Table 5.1) = 125 ft

30,0005 g y.p. 5 3

Downwind side:(No credit is taken for stiffening rings.)

fc(msx) = fwz + fdz = 6690

= 0.727X2 + 4.234X + 10.42Therefore

= 7880 < 10,000

1.297X’ + 19.76X - 4478 = 7880

X2 + 15.25X - 9.540 = 0Therefore

x= -15.25 + 2/(15.25)2 + (4)(9540)2

= 90.0 ft

This credit for stiffness by the tray-support rings resultsin a X value of 90.0 compared to the previous computedvalue of 85.5 ft. The next step is to check the shell forempty condition, no trays, no insulation, no pressure, vaporline in place, only wind load acting. Credit may be takenfor corrosion allowance under erection conditions.

Calculation of stresses :Upwind side :Calculation of dead weight:

fdead wt shell = 3.4X

Other dead weights are:

Wt of top head = 860 lb

Wt of ladder = 25 lb per ftWt of vapor line = 43.8 lb per ft

Total

ZWfdoad wt a t t a c h m e n t s = __

zrdts

68.8 lb per ft + 860

68.8X + 860= (3.14)(84)(0.3125)

= 0.834X + 10.42

jdw = 3.4X + 0.834X + 10.42

= 4.234X + 10.42

Wind-load stress:(Note that deff. is increased by 17 in. for caged ladders.)

d eff. = 84 + 17 = 101

fwz =1=W’WX2 = o 727x2

(84)2(0.3125) ’

Calculation of combined stresses for condition of partialerection:

Upwind side :

jtcmax) = 0.727X2 - 4.234X - 10.42

= (12,650)(0.85) = 10,750

X2 - 5.81X - 14,800 = 0Therefore

x = +5.81 + d(5.81)2 + (4)(1)(14,800)2

ThereforeX2 + 5.81X - 9,200 = 0

x =-5.81 + l/(5.81)2 + (4)(1)(9,200)

2

= 93.1 ft

Thus the controlling stress conditions exist under oper-ating load with a superimposed wind. The 72-ft distancewill be acceptable since the upwind condition under oper-ating load is controlling. For this reason, specify ninecourses of 8-ft-wide ><s-in. plate.

Calculation of second (lower) tower section consisting ojxs-in. plate:

(A value of us in. is assumed for 1, in the next lowersection.)

Operating conditions:Calculation of axial stresses :

Calculation of dead weights:Equation 9.3a can no longer be used directly because of

the change in shell thickness. Equation 9.3a can be modi-fied by treating the top zone as a constant load.

fdead wt shell =r Dm 0 - W(@W + 3.4(x _ 72)

a Dm (t2 - c)144

= 3.4x - 97.5

f 40X(3)desd at in’. = (144)(0.3125)

= 2.66X

W = 860 + 108.1X

f860 + 108.1X

dw attachments = Tc84j co.3125j = 10.4 + 1.31x

25(7)

fdead wt (trays + liquid) = = 11.65(48) (0.3125)

; - 1)/= 5.83X - 11.65

Page 185: Process Equipment Design

Example Design, Shell Calculations for a Tall Vertical Vessel i75

Therefore Erection conditions:

f& = 3.4X - 97.5 + 2.66X + 10.4 + 1.31X+ 5.83X - 11.65

= 13.2X - 98.75

Calculation of stress due to wind loads:

The previous calculation for the top section indicatedthat x6-in. plate would not be overstressed under erectionconditions with wind load on the upwind side for a distanceof X = 124.9 ft and on the downwind side for a distance ofX = 93 ft. Therefore, only the downwind side of the x s-in.plate section will be checked.

Calculation of dead-weight stress:15.89(108)X2fwz = (84)2(0.3125) = o*776x2

Calculations of combined stresses under operating conditions:Upwind side :

ftcmsx) = fwz: +fap -f&z

= 0.776X2 + 2,680 - 13.2X + 98.75

For an allowable stress of 10,750 psi (12,650 X 0.85)

10,750 = 0.776X2 - 13.2X + 2778.8

X2 - 17.00X - 10,250 = 0

Therefore

x = +17 00 +_ 1/(17.00)2 + (4)(10,250)2

= 110.1 ft

The length of x6 in. plate = 72 ft; therefore

length of WC-in. plate = (110.1 - 72)

= 38.1 ftDownwind side :

fc(msx) = fwz - fap +fdz

= 0.776X2 - 2,680 + 13.2X - 98.75

The equivalent thickness of the shell with stiffness creditfor tray-support rings is as follows.

By Eq. 9.84a

t, = 1, + 2Y

By Eq. 9.84

= 0.3125 + 0.072

= 0.383 in.

f1.5 x 10”

c(allow.) = 42 d(O.383)(0.3125) 5 10,000

= 12,350 > 10,000

Therefore

andfc = 10,000

0.776X2 + 13.2X - 2779 = 10,000

X2 + 17.00X - 16,450 = 0Therefore

x= -1700 + 1/(17.00)2 + (4)(16,450)2

= 120.0 ft

fdesd wt shell =?rDt1(72) (490)

aDts(144)+3.4(X - 72)

= 3.4x - 70.5

f68.8X + 860

dead wt attachments = x(84j (o.4375j = 0.595x + 7.44

f&,, = 3.4X - 70.5 + 0.595X + 7.44

= 4.0X - 63.06

Calculation of wind-load stress:(A total of 17 in. is added to deE for a caged ladder.)

detr. = 84 + 17 = 101

15.89(101)X2fwz = (84)2(0.4375) = o’519x2

Calculation of allowable compressive stress:

= 0.4375 + 0.72

= 0.5095 in.

1.5f

x 106C(SllOW.) = r

d(O.51)(0.4375) 6 gy.,,

30,000= 1.5 X lo6 dO.472/42 $ _^3

Therefore16,900 > 10,000

fc(allow.) = 10,000

Calculation of combined compressive stresses:

fc(msx) = fwz + fdw

= 10,000

0.519X2 + 4.0X - 63.06 = 10,000

x2 + 7.7x - 19,400 = 0

x= 7.7 f 2/(7.7)2 + (4)(19,400)2

= 135.5 ft

Therefore the erection condition is not controlling, andfour courses of plate 8 ft, 0 in. wide and xs in. thick willbe used for the second section. At the end of the Ws-in.plate section X = 72 + 32 = 104 ft.

Calculation of third section:It is apparent that since the wind load stress varies with

5\ ~\- .-- ------T-r I

Page 186: Process Equipment Design

176 Design of Tall Vertical Vessels

X”, the plate thickness must increase at a more rapid rate.Therefore ?s-in. plate will he used for the third section.

Calculation oj third (lower) tower section consisting ofsi-in. plate:

Operat ing cortdilions:Calculation of axial slress:

pdfw = ;&I (J) (40) (84)= ~~~ = 1680 psi(4)(0.501

(Ialcuiatiorr of dead wrights:

.f40X(3)

d e a d wt i n s . = (144)(o.50j = 1.67x

f

860 + 108.1dw attaohmmts = a(84)(0.50)

= 6.52 + 0.82X

fdead wt (trays + liquid) =(X/2 - l)W)U)

(48)(0.50)

= 3.64X - 7.28

.f,jx = 3.4.x - l94.2 + 1.67X + 6.52 + 0.82X+ 3.64X

= 9.53x - 195

Calculation of stress due to wind loads:

15.89(108)X2fw.c = ~84)2(o.50) = 0.486X2

7.28

Calculation of combined stresses under operating condit ions:Upwind side :

ft(max) = .fw.r +.fa, - fdz

= 0.,486X" + 1680 - 9.53X + 195

For an allowable sl rta.++ 0 f 1’3,750 psi

10,750 = 0.186X” - 9.53X + 1680 + 195

X2 - 19.6X18,250 = 0Therefore

x = fl9.6 I!Z fi.6)’ + (4)(18,250)2

= I 1.5.3 ft

Downwind side:

?CilllHXl = .f,l,., - .f,r ,, + .fd,

= 0.486X" - 1680 + 9.53.Y - I95

.fcka,\\-.) = 4 y.p.

= IO,000

therefore

0.486X2 + 9.53 - 1680 - 195 = 10,000

X" + 19.6X - 24,400 = 0

x = -J(J,6 + d(19.6)2 + (4)(24,400)0

= 146.7 ft

Use five courses of plates 8 ft, 0 in. wide and s in. thick.X at the end of the 36-in.-plate section = 72 + 32 + 40 =1’41 ft.

The bottom course will be made of one plate 6 ft, 0 in.wide and 3/4 in. thick. (This s-in. course must be speci-fied as ASTM-A-285, Grade C to meet code requirements-see Table 13.1, footnote.) The calculation of stress at thebottom tangent line is X = 150 ft for yd-in. plate.

Calculation of axial s t r e s s :

Calculation of dead weights :

5.fdlrr(shell) = [(3.4)(150) - 194.2]& = 283.4 psi

TiT

(40)(150)(3).fdw(ins.) = (r44)(o.5625) = 222 Psi

.fdr+h-llments) =860 + (108.1)(150) 17,060

(3.14)(84)(0.5625)= -__ = 115 pi

149

.tkc(trayti + liquid) = --____(48)(o.5625)

f&,,(t&,) = 283.4 + 222 + 115 + 460 = 1080 psi

Calculation of stress due to wind loads:

jUIT

= (~5.WUWWW2(84)2(0.5625)

= 9700 psi

Upwind side :

ft(max) = .fwT + .fa,> - .fd.,

= 9700 + 1495 - 1080

= 10,105 psi

Downwind side :

jcknax) = .fwz - .fu, + fd3.

= 9700 - 1495 + 1080

= 9285

Therefore the design is satisfactory with regard to loadingconditions in which the wind load rather t,han the seismicload is controlling.

Check of s tresses due to se ismic loads:

Page 187: Process Equipment Design

Referring to Fig. 9.7, we find that West. Central Texasis located in seismic zone 2 .

The weight of the tower plus attachments, liquids, andso on at the bottom tangent line may be calculated t)\mult iplying the total compressive stress due to dead weightsby the cross-sectional area of the tower at this position, or

or

~W(z=150) = .fdw(total)~4

= (1080)1r(84)(0.6875)

W = 196,000 lb

W196,000

(avn) =~~- = 1305 lb per ft150

The period of vibration may be calculated by use of Eq.9.68.

T = 2.65 x LO-” ($2 (T!y

The determinat ion of the period of vibrat ion of a columnhaving thickness variations is somewhat involved. It issuff ic ient for design purposes to approximate the period byusing a single shell thickness. The most conservativedesign will result when the greatest thickness is used.Therefore

T = 2.65 x lo-” (~)‘((~~~:~))11 = 1.59 set

Referr ing to Table 9 .3 , we f ind that the seismic coeff ic ientC, for zone 2 and for a period greater than 1.0 set is equal to :

c = 0.04

The moment due to seismic forces may be calculated byEq. 9.71.

M 82= sWX2(3H - x)

H2

The seismic moment at the bottom tangent line is:

Ms(x=160) = (4)(0.04)(196,000)(150)2[(3)(160) - 1501(160)2

= 9,090,OOO in-lb

The corresponding stress is given by Ey. 9.72.

9,090,000= ~(42)~(0.5626)

= 2910 psi

h comparison of t.he seismic stress of 2910 psi with thewind load stress of 9700 psi indicates that the wind stressesare controlling at the bottom tangent line.

Is the next upper course occurs only 6 ft above t,hebottom tangent line, no calculation of the seismic load willbe made at lhis point, but it will be evaluated at X = 104ft where t, = x6 in.

Oiher Methods of Analysis 177

ZW(x404) = fd.d(ts - e)

= (13.2X - 98.8)(7r)(84)(0.3l25)

= 105,500 lblherefore

M,,(u=lo?, = (~1~)(.04)(103,OO.i)(lO &)'(A80 - 104), I

-(160)'- -

= 2,783,OOO in-Il)

The corresponding s tress i s :

= 1550 psi

The stress due to wind load at this pcGul is 0.776X2.

fu.(X=,04 = (0.776)(10!)" = 8380

Therefore the wind load is controlling itI this level of I.hvcolumn.

The evaluation of seismic stresses at X = 72 ft where theshell thickness is 5~~ in. is:

w(X=72) = f&?rd(& - e )

= (19.76X - 2.0.')(~)(8~)(0.1875)

= 70,300 lb

h&(X=72) =(4)(0.04)(70,300)(72)2(C80 - 72)

~-(160)"

= 930,000 in-lh

930,000fs(x=72) = ----~~--~(42),~(0.1875)

fs(x=72) = 893 psi

The stress due to wind load at this point is 1.297x”.

fw(x=72) = (1.297)(72)' = 6700 psi

Therefore t.he wind stresses rather than seismic stresses wecontrol l ing over the column height . . This is due to a longperiod of v ibrat ion and corresponding low seismic coefflcien~.However, i f this vessel had been located on the lower WestCoast in seismic zone 3 rather t,han in zone 2, the seismic*stresses would have been doubled, and the wind stressc~sreduced.

The design of the skirt, bolt.ing riug, and foundations f’o~this tower is covered in Chapter 10. Figure 9.14 is a sketchof the tower designed in this example.

9.14 OTHER METHODS OF ANALYSIS

In the preceding sections of this chapter essentially onemethod of combining t,he s t resses has been employed. Thismethod is known as the maxirliulll-i)riIlrip;ll-stress t,heorb-and is the method most widely used. Other theories maybe used to det,ermine the t,hickness of lhe shell required.T h e m:~xiIriurn-prirlcipal-strrss I heorg autl t h r e e other

Page 188: Process Equipment Design

178 Design of Tall Vertical Vessels

12’ OD vapor linewith 3’ insulati=

Tor ispher ica l headKC’ ihick, 7’-0”

9 courses 8’4’ ea.-of %C plate

-3” insulation on shell

-71 trays (24” spacing)

-4 courses 8’-0” ea.of %.C plate

5 courses V-0” ea.-of Y plate

The tensile stress, jt, in the axial direction measuredover the area in the m-n plane is:

1 course 6’-0”-of %s” plate

Fig. 9.14. Sketch of a 160-ft, O-in. tower for the example design.

methods of analysis are listed below and compared in thefollowing sections.

1. Maximum-stress theory2. Maximum-shear theory3. Maximum-strain theory4. Modified-strain-energy theory

9.140 Maximum-stress Theory. The maximum-stresstheory, sometimes known as Robinson’s theory, is the oldestand simplest theory for designing any section subjected tostresses in three directions. It simply assumes that themaximum of the three stresses ji, j,, and jz controls thedesign because yielding is taken to occur when the maximumstress reaches the yield point of the material. This theorymakes no allowance for the effect of the components of thetwo minor &resses on the principal stress. This simpletheory may’result in over or underdesign of the sectionunder copsideration.

The theory is most applicable for brittle materials.However, vessels with designs based on relationship3 devel-oped from this theory and fabricated from structural steelsare widely used and have proved satisfactory in service.It is assumed that ji > j, > ji.

Some structural materials have physical properties suchthat the maximum-principal-stress theory is not the bestmethod of analysis. High-tensile-strength materials inwhich the tensile strength to shear strength ratio is higherthan usual are more likely to fail in shear than in tension.Other evidence of failure by shear has been observed insimple tension tests when there is slipping of planes inclinedto the tensile-load axis. This would indicate the need ofconsidering more than just the principal stress.

9.14b Maximum-shear Theory. The maximum-sheartheory, developed by J. J. Guest (147) assumes that yieldingstarts when the maximum shearing stress becomes equal tothe maximum shearing stress at the yield point in a simpletensile test.

The relationships involved can be developed with refer-ence to Fig. 9.15, which shows the stress condition in anelement under uniaxial tensile load. Consider any plane,m-n, at an angle, 0, to the axis of the element. The cross-sectional area at right angles to the axis of the element istaken as a. The area of the element lying in the planem-n is:

a%7&n = -cos e

P Pjt = - = - (COS e) = j*ri*l CO9 e

Gn a

The component of jt normal to the plane m-n is:

jn = jt COS 0 = j*xial COS* 0

The tangential component called the shearing stress, j,, is:

j8 = jt sin u = faxiar sin 0 cos e v

j, will be maximum when 0 = 45”, or

ff axis1

e(mar) = __2

Fig. 9.15. Stresses in an element under uniaxial tensile lu&

Page 189: Process Equipment Design

Other Methods of Analysis 1 7 9

In the case of two-dimensional stress such as exists in theshell of a thin-wall cylindrical vessel, the two principaltensile stresses at right angles to each other are:

1. The longitudinal stress, jr (y-axis direction).

fz = g2. The hoop stress, fh (x-axis direction).

fh = $

(3.13)

(3.14)

Consider the plane m-n with its normal at an angle 0 tothe axis of the vessel as shown in Fig. 9.16 noting that jh =Zjr. The shearing component of jr in the plane m-n willbe obtained by letting ji in the y axis = jaxiar.

j81 = 6ji sin 28

The shearing component of fh in the plane m-n will beobtained by letting fh in the x axis = faxisr.

The components fsh and jsl are in opposite directions;therefore the resul t ing combined shear s tress in two-dimen-sional analysis (f& is:

fs2 = +(fi - fh) Sin 20

The value of j8s will be maximum when 8 is 90”. There-f o r e

fs2max = 8(fZ - fh)

In a similar manner it can be shown (29) that in the caseof three-dimensional-stress analysis the maximum shearstress is def ined by one half the algebraic dif ference betweenthe maximum and minimum of these s tresses , or

fs3 = kfi - fi) = +fy.p. (9.85)

This theory is in good agreement with experimentalresults obtained on ductile materials. For purposes ofcomparison with other theories the maximum actual shearstress and the allowable shear stress will be doubled byremoving the 35 from each side of the equation. Therefore

fi - fi = fmax (9.86)

It should be noted that in Eqs. 9.85 and 9.86 both stressesji and ji are assumed to have t,he same sign, that is, bothare compress ive s t resses or tens i le s t resses . Designs basedon relationships developed from this theory are more con-servative than designs based on the principal-stress theory.

9.14~ Maximum-strain Theory. The maximum-straintheory developed by Saint-Venant (148) assumes that yield-ing of a ductile material begins when the maximum strainbecomes equal to the strain at the tensile-test yield point.This theory permits the combining of stress-strain relation-ship in three dimensions .

Equations 6.4 and 6.5 give the strains, and Eqs. 6.4aand 6.5a give the stresses from two-dimensional stress-s tra in analys is . Equations 6.4 and 6.5 can be modified forthree-dimensional analysis as fo l lows:

f fw23-=--=E E cz3 = -$fz - PL(fu +fJ1 (9.87)

f2/3 = fyP = Ey3 = 1 [ji - p( j, + j*)]E E E

(9.88)

f- = fz = cz3 = 1 [fz - P(fi + &)I23

E E E(9.89)

where the subscripts x3, y3, and 23 refer to the 2, y, and zdirect ions in a three-dimensional system.

Equations 9.87, 9.88, and 9.89 can he transformed toexpress the stresses in terms of strains as follows:

fi3 = pE(1 + Pco(1 - 2Pcr)

(%3 + q/3 + ez3) + E- CT3l+P

(9.90)

P*E Efv3 = (l+co(l -2p) (%3 + eY3+ tzr) + (l+p) %3 (9.91)

PE Efi3 = (1 +~)(l -2r) (c,3+ ‘U3+ e,3> + (1 +~j Ez3 (9.92’

In the case of thin-walled vessels in which the radialstresses can be disregarded, ji may be taken as equal to zero.This simplification results in Eqs. 6.4a and 6.5a. There isl imited evidence against the use of this theory. In the caseof a simple plate subjected to tension in two perpendiculardirections, the elongation in each direction will be reducedby the tension in the perpendicular direction. This wouldindicate that a plate loaded in this manner would have agreater yield point than a plate loaded in simple tension inone direct ion. This conclusion is not supported by limitedexperimental tests .

9.14d Modif ied-strain-energy Theory. There are anumber of modifications of the theory based upon thepremise that yie lding begins when a given quanti ty of s trainenergy is accumulated in a given volume of material .

Equation 9.35, which gives the strain energy in a deflected

Fig. 9.16. S t r e s s e s i n a n e l e m e n t o f a v e s s e l s h e l l u n d e r t w o - d i m e n s i o n a lstress.

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180 Design of Tall Vertical Vessels

Fig. 9.17. Graphical comparison of four theories of failure.

beam, can be modified for the strain energy of an elementstrained in IWO directions as follows:

UY = ~ft,h (9.93)

ux = Jo’=fz de, (9.94)

The total strain energy stored in the element is equal tothe sum:

uxy = uy + ux (9.95)

Equations 6.4 and 6.5 give the strains in the x and ydirections resulting from the stresses fi and f,

fi Pfl/&c2. = - - -E E

f, Pfz%2 = - - -E E

Substituting into Eq. 9.95 gives:

therefore

uz, = f (fg2 +fz2 - 2Pfz/fz) (9.96)

For an element strained in three directions it can be shownin a similar manner that

crz,z = & Kfi” +fg2 +fz2) - 2PLfzfz/ +fifi +fufi)l(9.97)

Equation 9.97 is due to Haigh (149) and has been called t,hestrain-energy function.

Yielditig is assumed to occur when the total strain energyis equal to that obtained in simple tension at the yield point:therefore

Substituting into Eq. 9.94 for the condition at the yieldpoint gives:

1 hUZ(Y.P.) = jj s

fi dfi0

therefore

(9.99)

Setting Eq. 9.97 equal to Eq. 9.99 gives:

f.r2 + f,’ + fz2 - a4fifU + fifi + f,fi) = fy.,." (9.JW

SeLting p equal to the theoretical maximum value of 0.3gives a simplified equation

4Kf.z - f,j2 + Cfi/ - fd2 + (f* - fz121 = fY.P.2 (9.101)

Equation 9.101 is identical to that developed by Hubel(150), Hencky (151), and Von Mises (152) to bring thetheory into agreement with the fact that materials canundergo large hydrostatic pressures without yielding.

Equation 9.100 for two-dimensional strain reduces to:

fi2 +fg” - %-Jfzf, = fy.,." (9.102)

9.15 GRAPHICAL COMPARISON OF THE FOURTHEORIES

Figure 9.17 graphically shows a comparison of the foulmethods of analysis for a two-dimensional stress system(fi = 0). The upper right quadrant represents tension folboth fi and f,. The upper left quadrant represents fi, intension and fi in compression, and the lower right quadrantrepresents fi in tension and f2, in compression. The solidlines in the figure represent the locus of the conditions atwhich yield is assumed to begin according to the fourtheories. The square u-b--c-d represents the maximumstress theory. Point a represents equal tension in both thex and y perpendicular directions, both of which are con-sidered to be equal to the yield-point stress obtained froma simple tensile test.

According to the maximum-stress theory, yielding doesnot occur inside the square. As f, is decreased and fz isheld constant, fi is controlling from point 1 to point 2, at.which points fyy.p. in compression is taken as equal to f,u,.,.in tension.

The irregular hexagonal figure l-a-2-3-& representsthe maximum-shear theory and is the same as the maximum-stress theory in the upper right quadrant, l-u-2, and thelower left quadrant, 3-c+ but is more conservative in theother two quadrants where the stresses are of opposite signs.This can be explained by Eq. 9.85, which is for the threedimensional theory. If f2/ and fi are both positive and fi isequal to zero, the maximum minus the minimum stress willbe equal to either fi minus zero or f, minus zero. However,iffi: is negative and f, is positive and f2 is zero, the maximumminus the minimum will be f, - fi, which will give thediagonal line l-4.

The rhombus A-B-C-D represents the maximum-straintheory. By referring to Eqs. 9.87 and 9.88, in which fi istaken as equal to zero for two-dimensional stress, it may hcseen that if the stresses fi and f, have the same sign, then

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Example of a Vessel in Which the Four Theories are Compared 181

either jZ or j, can appreciably exceed the yield point of themater ia l . This is true because the strain in the y directionreduces the strain in the z direction and vice versa when thestresses have the same sign, as is the case at point A orpoint C.

iThe strain-energy theory is represented by the ellipse

and can be evaluated by means of Eq. 9.102.

I 9.16 EXAMPLE OF A VESSEL IN WHICH THE FOURTHEORIES ARE COMPARED

Because a tall fractionating tower has inherent stressesresulting from dead weight, pressure stresses, and super-imposed loads such as wind or seismic forces, it is ideal forcomparing the four theories discussed in the previous sec-tions. Brummerstedt (153) in 1943 presented an exampledesign of a tower comparing the four theories . His problem

i concerned an analysis of stresses in a 70-tray fractionatingtower which had a 10 ft inside diameter and a height of

! 160 ft. The tower was to withstand an internal pressureof 200 psi and a superimposed seismic force equal to 20%of i ts operat ing weight . This force was assumed to applyat the center of gravity of the tower.

The tower was to be designed to a maximum allowablestress of 13,750 psi for the material of construction. Thewelded-joint eff iciency was to be 82 c&. This resulted in anallowable stress is 13,750 X 0.82 or 11,300 psi. (These

1 values came from the API-ASME code res tr ic t ions in e f fec t1 in 1943). The operating weight including steel plate,I attachments such as piping platforms, ladders, and operat-

ing l iquid was estimated to be 620,000 lb.The seismic force of 20”/0 resulted in a horizontal force of

124,000 lb. This force induced a moment, of 9,600,OOO f t - lbon the column.

The circumferential- and longitudinal-pressure stressesresulting from internal pressure were computed by Brum-merstedt (153) and combined with the dead weight andseismic stresses under the assumption that the radial stressin the tower shel l was zero. These stresses were combined/ by the four theories, and the results tabulated for compari-

: son. Tables 9.4 and 9.5 summarize the results.The example tower design is such that the percentage

difference in total weights as indicated in Table 9.5 is notvery great . This is due to the comparatively high designpressure of 200 psi. A s imilar comparat ive analysis madeof such a tower operating under a low pressure of under50 psi of ten results in a considerably greater variat ion. 111such a case Lhe method of analysis becomes of greater

PROELEMS

Table 9.4. Summation of Combined Stresses in a Tall

Tower (According to Brummerstedt) (153 i

Resulting Stress, psi RatioT h e o r y (based on l>+in. shell) (basis of max stress)

Maximum-stress 9 , 6 2 0 1 .OOO:lMaximum-stra in 9,204 0.957:1Maximum-shear 1 1 , 7 8 0 1.225:1Stra in-energy 1 0 , 4 5 0 1.088:1

importance. However, as indicat.ed in an earlier section. :It.all tower operating under a low desigu pressure may failbecause of elastic instability. To design for such a condi-Lion t,he appropriate relat ionships presented must be applied.

Shell and head thicknesses based only upon membrane-stress equations provide no allowance for superimpc#setlloads on vertical vessels. However, the shell thicknessesobtained by such equations provide a convenient stari.ingpoint for evaluating the thicknesses for vertical vesselssince the thicknesses thus obtained may be modified tosat,isfy structural requirements. In the case of vesselsoperat ing under internal pressures of 30 lb per sq in . gage ormore it is usually convenient to first check the cumulaG\-etensile stresses from pressure, wind bending moments,and/or seismic moments. The design of tall vessels foloperation at low internal pressures or the design of ‘wn!-

vessel under external pressure is controlled by the cumul:~-t ive compress ive forces . The design of such vessels can bedetermined most rapidly by beginning t.he calculat ions withthe cumulative compressive stresses rat.her than with thet,ensile stresses.

Table 9.5. Summation of Thicknesses and WeightsRequired by the Four Theories (153)

Prr-Weight crnt-

of avShel l Ol

Bottom Next Next Top Plates MHXTheory 20 ft 20 ft 20 ft 100 ft (lb) st>rrss

Maximum-stress 1316 in. 1M in. 136 in. 136 in. SO,000 100.0

Maxirnurn-strain 1>/4 in. I ss in. 136 in. llg in. 285,000 98.2 .

Maxilnum-shear 1x6 in. 136 in. I%/4 in. 1’6 in. 300,000 103. 5

Sl.rain-rwrgy 1756 in. 154s in. I:j’16 in. 1'6 in. 295,000 IQ! .I!

1. An insulated steel fract ionating column located at Oakland, Cal i fornia , is 6 f t , 0 in . ininside diameter and 160 ft, 0 in. from tangent to tangent between heads. The heads project

I 1 f t , 6 in. beyond the point of tangency. The skirt is 8 ft, 0 in. from the base to the shell junc-tion at the point of tangency with the bottom head. The vessel is designed to operate at

t 100 lb per sq in. gage. It is const,ructed of SA 285 grade C steel (13,750 psi maximum allowabletensile stress). The effective wind area of external attachments i s e&mated to be 10% of the

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182 Design of Tall Vertical Vessels

area of the uninsulated column. The insulation is 3 in. thick and weighs 40 lb per cu ft. Thetray spacing is 18 in., and there are 102 trays with an estimated weight of 25 lb per sq ft ofcolumn cross section. The top tray is 4 ft, 0 in. below the top tangent line. Calculate theminium shell thickness at the bottom tangent line resulting from wind moment.

2. For the vessel in problem 1, calculate the maximum stress at the bottom tangent lineresulting from the seismic moment.

3. If the shell of the vessel in problem 1 is fabricated from 20 plates 96 in. wide, specify thethickness for each course allowing a minimum of xs in. for corrosion.

4. Redesign the vessel in the example design for full vacuum operation.5. Redesign the vessel in the example design for 160 lb per sq in. gage operating pressure.

6. Redesign the vessel in the example design for the same conditions but on the basis of themaximum-shear theory.

7. Redesign the vessel in the example design for the same conditions but on the basis of themaximum-strain theory.

8. Redesign the vessel in the example design for the same conditions but on the basis of themaximum-strain-energy theory.

9. A fractionating tower is required to separate styrene from a dilute feed. Preliminarycalculations indicate that 70 trays will be required with the feed entering on tray 32 (from thebottom). The reboiler will be separate from the column. Ninety-five per cent recovery ofthe styrene in the feed is desired. Annual production of styrene from the columns is to be8000 tons.

StyreneEthylbenzeneTolueneBenzene

Stream Comnosition. Weight PercentagesFeed

Mol. Wt (saturated liquid) Bottoms

104.14 3 7 . 0 9 9 . 7106.16 6 1 . 1 0 . 392 .13 1 . 178.11 0 . 8

Temperatures Top of column 54” cBottom of column 90” c

Pressures Top of column 30 mm HgBottom of column 310 mm Hg

Reflux ratio, L/D = 7 (mol ratio)

The tower is self-sustaining (no guy wires) and is to have a IO-ft skirt extending from the topof the foundation to the tangent line of the bottom dished head. The erected tower is to belocated in the Houston, Texas, area. The overhead condenser is to rest on the ground, and thereflux is to be pumped back. The client specifies that the bubble caps are not to be larger than5 in. but may be smaller if desired. The tower is to be designed for full-vacuum service.

A tray layout and tower design excluding skirt, foundation bolts, nozzles, and bubble-capdetails are required.

REFERENCES FOR PROBLEM 9

Bolles, W. L., “Optimum Bubble Cap Tray Design,” Part I, Petroleum Processing, Vol. 11, No.2 (1956); Part II, No. 3; Part III, No. 4; Part IV, No. 5.

Boundy, Ray, Styrene ACS Monograph No. 115, Reinhold Publishing Company, New York,1952.

Davies, J. A., “Bubble Trays-Design and Layout-Part I,” Petroleum Refiner, Vol. 29, No. 8(1950); Part II, ibid., No. 9.

Page 193: Process Equipment Design

C H A P T E R

DESIGN OF SUPPORTS

FOR VERTICAL VESSELS

ertical vessels are normally supported by means of asuitable structure resting on a reinforced-concrete founda-tion. This support structure between the vessel and thefoundation may consist of a cylindrical steel shell termed a“skirt.” An alternate design may involve the use of lugsor brackets at tached to the vessel and rest ing on columns orbeams. These more common designs for support ing vert icalvessels wi l l be descr ibed.

10.1 SKIRT SUPPORTS FOR VERTICAL VESSELS

10.1 Skirt Thickness. Tall vertical vessels are usuallysupported by skir ts . Because cyl indrical shel ls have al l themetal area located at the maximum distance (for a givendiameter) from the neutral axis, the section modulus, 2, ismaximum, and the induced stress minimum for the metalinvolved. Thus the cylindrical skirt is an economicaldesign for a support for a tall vertical vessel. The skirt isusual ly welded direct ly to the vessel . Because the skirt isnot required to withstand the pressure in the vessel, theselect ion of mater ia l i s not l imited to the s tee ls permit ted bythe pressure-vessel codes, and structural steels with cor-responding allowable stresses may be used with someeconomy. (The steels used in the design of flat-bottomedcylindrical storage tanks (se&hapter 3) are suitable for theskirts of vertical vessels9 For structural loads a factor ofsafety of 3 based on the ultimate tensile strength is usuallyused, whereas@ factor of safety of 4 is used with pressurevessels.) Thus t&e, allowable stress in the skirt is usually3334 ‘% higher than that in the shell of a pressure vesselwhen the steels in each case have the same ultimate tensilestrength.

The skirt may be welded directly to the bottom dished

head, flush with the shell, or to the outside of the shell. Ibthe skirt is welded flush with the shell, the weight of thevessel in the absence of wind and seismic loads places theweld in compression. On the other hand, if the skirt iswelded to the outside of the vessel , the weld joint is in shear ;therefore this method is not so satisfactory, but it is an easymethod of erect ion and is of ten used for smal l vessels .

There will be no stress from internal or external pressurefor the skir t , unl ike for the shel l of the vesse l , but the s tressesfrom dead weight and from the wind or seismic bendingmoments wil l be a maximum. The same procedure may beused for designing the skir t as for designing the shel l , whichwas described in Chapter 9. Note: Subscript b refers tothebase of the skirt.

Wind-load stress = fwt, =15.89deR.H2

d 2t (9.20)0

(9.17)

8CWHSeismic-load stress = fsb = 7 (9.75)

Dead-weight stress = f& = ET dt (9.6)

Max permiss ib le compress ive s t ress = fcattoW.1.5 x 106= l/t,t, s 6 y.p. (9.841

P

Max tensile stress = ftmax = (fwb orf&) - fdb (9.78)

Max compressive stress = foaX = (f& or fsb) i- fdb (9.80)

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1 8 4 Design of Supports for Vertical Vessels

and

Substituting gives:

I-,”

(b)

F i g . 1 0 . 1 . S k e t c h o f l o a d i n g o f a n c h o r b o l t s .

After the skirt and bearing plat,e have been designed, theskirt design should be checked for the reaction of the boltingchairs or ring. (See section lO.lg.)

10.1 b Skirt -bearing-plate and Anchor-bolt Design.The bottom of the skirt of the vessel must be securelyanchored to the concrete foundation by means of anchor

bolts embedded in the concrete to prevent overturning fromthe bending moments induced by wind or seismic loads.

The concrete foundation is poured with adequate rein-forcing steel to carry tensile loads (143, 154, 155). Theanchor bolts may be formed from steel rounds threaded atone end and usually with a curved or hooked end embeddedin the concrete. The bolting material should be clean andfree of oil so that the cement in the concrete will bond to theembedded surface of the steel.

When either a compressive or tensile load is applied to theanchor bolts, the load is transferred from the steel throughthe bond to the concrete. Surface irregularities, bends, andhooks aid in transferring loads from steel to concrete. Asthe steel and concrete are bonded, the resulting strain is thesame for both the,steel and concrete at the bond. Themodulus of elasticity of steel, E,, is about 30 X lo6 psiwhile that of concrete, EC, varies from about 2 X lo6 to4 X lo6 psi depending upon the mix employed. The ratioof these moduli is:

ESn=-

FJCrewrit.ing gives:

ButEcn = E,

(10.1)

But because of the bond, E, = e,s,

fs induced = %fc induced ‘10.2)

Table 10.1 gives the value of n as a function of tY,e com-pressive strength of the concrete, which in turn is a fmctionof the mix used for the concrete.

The bending moment and weight of the vertical vesselresult in a loading condition on the concrete foundationsomewhat similar to that in a reinforced-concrete beam.Figure 10.1 is a sketch representing the loading condition ofthe anchor bolts in the concrete foundation.

Figure 10.1, detail a is a sketch showing the bearing plateat the base of a skirt for a vertical vessel. In the calcula-tions it is assumed that the bolt circle is in the center of thebearing plate. Sometimes the bolt circle is made largerthan the mean diameter of the bearing plate but should betaken equal to it for simplicity of calculation since the erroris small and is on the safe side. “,The wind l.oad and thedead-weight load of the vessel result in a tensile load on theupwind anchor bolts and a compressive load on the down-wind anchor bolts.’ If je is the compressive stress in theconcrete, the induced compressive stress in the steel boltsin the concrete is given by Eq. 10.2. Thus njc !s the inducedcompressive stress in the steel bolts on the downwind side,and js is the maximum tensile stress on the upwind side.As the stress is directly proport,ional to the distance fromthe neutral axis, a straight line may be drawn from js tonfc, as shown in detail b of Fig. 10.1. The neutral axis islocated a distance kd from the downwind side of the bearingplate and a distance (d - kd) from the upwind side.

By similar triangles, we obtain:

fs 7&~-(d - kd) = kd

therefore

k = nfc = ~_ 1

nfc + fs 1 + U../n.fJ(10.3)

Table 10.1. Average Values of Properties nf Three

Concrete Mixes

Water Content fc’ n .“cU.S. Gallons 28-day Ultimate 3. x IO6 Allowable

per 94-lb Sack Compressive Compressiveof Cement Strength, psi Ec St.rength, psi

7% 2 0 0 0 15 8 0 06% 2 5 0 0 12 1 0 0 06 3 0 0 0 10 1 2 0 05 3750 n 1 4 0 0

Page 195: Process Equipment Design

Skirt Supports for Vertical Vessels 1 8 5

where Ct is the term in the brackets and is a constant fora given value of k.

To determine the distance 11 consider the element whichis located a distance of r(cos a + cos 0) from the neutralaxis. The moment of the force on this element times thislever arm is:

dMt = dFt r(cos a + cos 13)

where ,‘; = maximum induced tensile stress in steel atbolt-circle center line on upwind side, poundsper square inch.

fe = maximum induced compressive stress in con-crete at bolt-circle center line on downwind side,pounds per square inch /E‘n=2x7

If the maximum induced tensile stress in the volts, f8, andt.he maximum induced compressive stress in the concrete,

fc, at the center line of the bolt circle-are known, k may bedetermined by use of Eq. 10.3.

Taylor, Thompson, and. Smulski (156) have expressedthe area of bolting steel% terms of an equivalent shell ofsteel of thickness tl having the same total cross-sectionaltIrea of steel as shown in Fig. 10.2.

Referring to Fig. 10.2, we find that the location of theneutral axis may be defined in terms of angle (Y (156).

d/2 - kd‘OS a = ~~

--cl--k42

(10.4)

In the same figure consider an element of the bolting steelmeasured by angle d0. The area of this element is given by:

dA, = tlr d0 (10.5)

The distance of this element from the neutfal axis is

r(cos a + co9 e>

The maximum distance from the neutral axis for suchan element is:

r(1 + cos (Y)

The stress in the element, fs’, is directly proportional tothe distance from the neutral axis, and if the maximumstress is fs,

p tcos a + cos e)A =.A; (1 +

cos a)(10.6)

Multiplying the stress by the elemental area gives theelemental force in tension, dF,.

dFt = fstlr(co9 a -I- cam e) de

(1 + cos a)(10.7)

The summation of the elemental forces on the bolting steelin tension can be represented by tensile force Ft located atthe center of tension and distance 11 from the neutral axis.Similarly the summation of the compressive forces on theconcrete in compression can be represented by a compressiveforce F, located at distance l2 from the neutral axis.

RELATIONSHIPS FOR THE TENSION SIDE. By integrationof Eq. 10.7 for the upper and lower halves on both sides ofthe center line, we obtain:

Ft = f,tlr2(co9 a! + toss) de

(1 + cos cy)

= f..rlr [ L- ((T - n) cos cx + sin a)] (10.8)1 + cos a

J = jstlrct (10.9)

= fd1r[

ccos a + cos e)(1 + cos a)

, r(cos e + cos CX) 1 de

= fdlr [ r(cos 8 + cos a!)2(1 + cos a) 1 de

By integration,

Mf = j,tlr22J

* (COS a + cos epdea (1 + cos a)

= 2fstlr2[

(7r - a) cos2 a + Q(sin (Y cos (11) + &(7r -- a~)1 + cos (11 I

(10.10)

Dividing Mt by FL gives 11.

11 =

[

(T - a) cos2 a + &sin (Y cos (Y) + +(7r-). r(7r - a) cos a + sin LY I

(10.11)

(Note that 11 is a constant for a given value of k.)RELATIONSHIPS FOR THE COMPRESSION S IDE . On the

compression side a similar procedure is used. A differentialelement of concrete and steel is considered having an areaof:

dA, = tzr de (10.12)

where t2 = concrete width (exclusive of bolting steel, tl)under the bearing plate, inches.

The distance of this element from the neutral axis is:

r(cos e - cos a)

Fig. 10.2. Plan view of loading on bolting steel and bearing plate.

r - - l - - - - -_ ‘..-

\ \ -\I T

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1 8 6 Design of Supports for Vertical Vessels

Table 10.2. Values of Constants C,, C,, Z, and j as aFunction of k (156)

k C C Ct z j

0.050 0.600 3.008 0.490 0.7600.100 0.852 2.887 0.480 0.7660.150 1.049 2.772 0.469 0.7710.200 1.218 2.661 0.459 0.7760.250 1.370 2.551 0.448 0.7790.300 1.510 2.442 0.438 0.7810.350 1.640 2.333 0.427 0.7830.400 1.765 2.224 0.416 0.7840.450 1.884 2.113 0.404 0.7850.500 2.000 2.000 0.393 0.7860.550 2.113 1.884 0.381 0.7850.600 2.224 1.765 0.369 0.784

The maximum distance from the neutral axis for suchan element is:

r(1 - cos f-I)

The stress fc’ in the element is directly proportional tothe distance from the neutral axis, and if the maximuminduced stress is fc,

fc’ = fcr(cos e - cos a)

r(1 - cos a)J10.13)

The corresponding compressive (camp.) stress, Sg(comp.),in the steel on the compression side (see Eq. 10.2) is:

Ss(com*.) = nfc ‘y y;oy;)a) (10.14)

The corresponding compressive forces in the element areobtained by multiplying the elemental stresses by the ele-menlal areas.

dFc (concrete) = fc’ dA, = f&r

dFe(steel) = nfc’ dA, = nf,tlr

The total compressive force on the element is equal tothe sum of the above two equations, or

dFc(totau = (t2 + nh)rfc “O;’ I3 - cos a de- cos C Y IBy integration,

F, = (t2 + ntl)rfc2/0 p ““1” I3 - cos a de- cos a

F, = (tz + ntdrfc2(sincu - (YCOS(Y)

1 - cos (Y I(10.17)

F, = 02 + ntl)rfJA (10.18)

where C, = the term in the bracket and is a constant for agiven value of k.

To determine the distance 12 the same procedure is usedas for the tension side. The moment of the force on the

element times the lever arm is:

dM, = dFc r(cos tl - cos a)

= (t2 + ntl)r2fc “: ~mc~~ap)2 de

By integration,

MC = (tz + ntdfcr22s

p (COS e - cos a]2de

0 1 - cos LI1

= (t2 + ntdfcr22cos2 a - ?&sin a! cos a) + &

-1 - cos 4 1(10.19)

Dividing MC by F, gives 12.

12 =[

a cos2 a - +(sin (Y cos (Y) + +asin LY - (Y cos Q ! 1 (r) (10 .20 )

(Note that 12 is a constant for a given value of k.)The total distance between the forces Ft and F,: is equal

to II + 12. This distance divided by d gives the dimension-less ratio, j. -

j=I1 + 12

ld=2

[

(T - a) cos2 a + +(7r - a) + $ sin a cos a I(7r - a) cm a + sin a

++ [ +cu-$sinarcoscu+a~os~,isin (Y - ff cos cr i

(10.21)

Referring to Fig. 10.2, we find that the distance from theneutral axis to the center line of the vessel is (d/2)(cos a)and distance zd is equal to:

dzd = l2 + - cos a

2(10.22)

2 = ; cos a +[ (

+a - 3 sin a cos LY + a! cos2 LYsin 01 - (Y cos (Y >I

(10.23)

The quantities Ct, C,, j, and z are given in Table 10.2as a function of k.

BOLTING AREA AND BEARING-PLATE W IDTH. Taking as&&&n of moments about F, (see Fig. 10.2) we obtain:

Mw i n d - W&,zd - Ftjd = 0

thereforeFt = Mwind - Wdwzd

jd'(10.24)

Substituting for Ft by Eq. 10.9 we obt.a!‘a:

(10.25)

And A, = 2mtl; therefore

A, E zT Mwind - Wd,,,.zdGfdd I

(10.26)

Page 197: Process Equipment Design

Referring to Fig. 10.2 and taking a summation of verticalforce? we obtain:

i Ft + wdw - F, = 0 (10.27)

Substituting for Ft by Eq. 10.9 and F, by Eq. 10.18, weobtain:

jstlrct + wdw - ct.2 + ntl)rfccc = 0

Solving for tr, we obtain:

12

= wdw + (GA - CJ,n)rli

WJ(10.28)

The total width of the bearing plate will be tl + tz (Eq.10.25 plus Eq. 10.28). Therefore

Width of bearing plate, t3 = tl + t2 (10.29)

Nomographs for the solution of anchor-bolt problems bythe method of Taylor, Thompson, and Smulski have beenpresented by Gartner (233). An alternate procedure hasbeen presented by Jorgensen (234).

DETERMINATION OF BEARING-PLATE THICKNESS. Thethickness of the bearing plate is determined by the com-pression load on the downwind side of the vertical vessel.The minimum required width of the bearing plate was pre-viously determined by use of Eq. 10.29. The maximumcompressive stress between the bearing plate and the COIT-Crete occurs at the outer periphery of the bearing plate.The induced compressive stress at the bolt-circle center linewas determined by successive approximation in calculatingthe required width of bearing plate (see Eq. 10.29). Equa-tion 10.30 gives the relationship between the maximuminduced compressive stress at the outer periphery and thecorresponding stress at the bolt circle.

f2kd + t3

c(max induced) = (fc(bolt circle induced)) ~( >2kd

(10.30)

Although the compressive stress varies from the maximumgiven in Eq. 10.30 to a lesser value at the junction of theskirt and bearing plate, the value at the bolt circle may beused for simplicity of calculation in determining the requiredthickness of the bearing plate.

BEARING PLATES WITHOUT GUSSETS. A bearing platewithout gussets may be assumed to be a uniformly loadedcantilever beam with je(max induced) the uniform load. Themaximum bending moment for such a beam occurs at thejunction of the skirt and bearing plate for unit circumfer-ential length (b = 1 in.) and is equal to:

MC,,,a=, = jo ,,,a= bl f0

= -ff (for b = 1) (10.31)

where 1 = outer radius of bearing plate minus outer radiusof skirt, inches

The maximum stress in an elemental strip of unit widthis given by:

.f(6M(maw 3.flY In** l2

max) = ~ =btd2 t42

(for b = :)

where t4 = bearing-plate thickness, inches

Skirt Supports for Vertical Vessels

Letting jcmax) = f@no&,le) and solving for t4 givesI

187

us:

tr = 11/ 3.fc msx/f(allow.) (10.32a)

The thickness of the bearing plate, t4, as calculated byEq. 10.32~ is usually rounded off to the next larger standardthickness of plate. ‘I

BEARING P LATES WITH GUSSETS. If gussets are used tostiffen the bearing plates, the loading condition on the sec-tion of the plate between two gussets may be consideredto act similarly to that of a rectangular uniformly loadedplate with two opposite edges simply supported by thegussets, the third edge joined to the shell, and the fourthand outer edge free. Timoshenko (107) has tabulated ,thedeflections and bending moments for this case as shown inTable 10.3.

Note in Table 10.3 that for the case where l/b = 0 (nogussets or gusset spacing, b = 00) the bending momentreduces to Eq. 10.31, and the thickness of the flange isdetermined by Eq. 10.32. Also note that when l/b is equalto or less than %, the maximum bending moment occurs atthe junction with the shell because of cantilever action.If l/b is greater than 4, the maximum bending momentoccurs at the middle of the free edge.

To determine the bearing-plate thickness from the bend-ing moments, Eq. 10.33 may be used.

(10.32b)I

DESIGN PROCEDURE FOR BOLTING CALCULATIONS AND

SIZING OF BEARL~G PLATE. The location& the neutralaxiyiae&ined by&e Rio of induced stresses, as indi-cated by Eq. 10.3. Thus the determination of minimumbolting and minimum width of bearing plate requires suc-cessive-approximation calculations. The value of k deter-mines the constants Ct, C,, j, and z, which in turn determinethe values of Ft and F, and their locations.

As a first approximation in the determination of k, js maybe taken as the maximum allowable stress in the boltingsteel, but je should not be taken as the maximum allowablecompressive stress in the concrete since the maximum com-

Table 10.3. Maximum Bending Moments in a Bearing

Plate with Gussets (107)(Courtesy of McGraw-Hill Book Co.)

0 8. - 0.5OOjJ245 0078jcb2 - 0 428jJ2w 0 0293j,b2 -0.319j~PI% 0. 0558f,b2 - 0 . 227jJ21 0. 0972f,b2 -0.119j012% 0. 123f,b2 - 0 . 124f,122 0 131f,b2 - 0 . 125f,123 0. 133f,b2 - 0. 125f,1200 0. 133f,b2 - 0 . 125f,12

b = gusset spacing (5 direction) inches.1 = bearing-plate outside radius minus skirt outside

radius (y direction) inches.

Page 198: Process Equipment Design

1 8 8 Design of Supports for Vertical Vessels

Table 10.4. Bolt Data (157)

(Courtesy of Taylor Forge & Pipe Works)

Bolt Standard Thread 8-thread Series~.-~Size No. of Root No. of Root

d Threads Area Threads Area13 0.126 INO. 811 0.202 thread10 0.302 series9 0.419 below 1”8 0.551 8 0.551

5% 1.515 8 1.6805 1.744 8 1.9805 2.049 8 2.304

4 % 2.300 8 2.652

4 % 3.020 8 3.4234 3.715 8 4.2924 4.618 8 5.2594 5.621 8 6.324

0.693 8 0.7280.890 8 0.9291.054 8 1.1551.294 8 1.405

Bolt Spacing*Minimum Pre-

B, ferred1 Q” 3 I,

11~5 31% 32’16 32% 3

MinimumRadial

DistanceR

‘x6”

‘3461?,41%1%

EdgeDistance

E5/‘8”

N‘%S

‘546

1’4 6

1%2

2%2%2%2%

MaximumNl.lt Fillet

Dimem:i on Radius(across flats) P

w M )’1!16 %6

l?G w

13iS N1% 746

I1946 x6

2 7462x6 %6

2% 36

2%6

2% i

w6 36

3 % ‘Y6

3% ‘416

3 % ‘946

4% N4 % ‘546

* B, = center-t.o-cent,er distance het.ween bolts, inches.

pressive stress in the concrete is developed at the outerperiphery of the bearing plate rather than at the center lineof the bolt circle.

After evaluating k by Eq. 10.3, the minimum area ofbolting steel required may be determined by Ey. 10.26and the preceding relationships. This permits the selectionof the number and size of bolts having sufficient root areato equal or slightly exceed the minimum required boltingarea. Table 10.4 gives the necessary information for t.hisselection. Usually the number of bolts selected is a mul-tiple of four to permit greater ease in bolt layout.

A s(act.) = ~VAH 2 ,Js(min) (10.33)

where N = number of boltsAe = root area of bolt, square inches (Table 10.3)

A s(min) = minimum area of bolting steel, square inches(Eq. 10.26)

A a(act.) = actual area of bolting steel used, square inches

The width of the bearing plate may be evaluated by useof Eqs. 10.25, 10.28, and 10.29.

In the first trial the area of bolting steel is select.ed andthe width of the bearing plate determined by use of t,heoriginal evaluation of k. The values of the induced tensile

and compressive stresses in the stee! and concrete based onthis area and this width are next determined. An inducedtensile stress in the steel based upon the first calculationof k may be determined by Eq. 10.34.

.fs (induced) = .fs(first estimate) es(act.)

(10.34)

An induced maximum compressive stress in the concretealso based upon the first calculation of k may be determinedby use of Eq. 10.30. These values of f8 and fc may be usedto obtain a more correct value of k by Eq. 10.3. If the newvalue of k differs appreciably from that originally calculated,a complete recalculation should be made using the constantsCt, C,, j, and z based upon the new value of k. After con-sistent values of fs, fe, and k have been determined, thebolting design and bearing-plate width are established.The thickness of the bearing plate may then be establishedby use of Eq. 10.31. To complete the design of -the skirtand bolting ring the compression ring or bolting “chairs”should also be considered.

10.1~ Example Calculation 10.1, Bearing-plate Design.A proposed fractionation column is 10 ft in diameter and150 ft high and rests on a foundation of 3000-psi-strength

Page 199: Process Equipment Design

concre$e. The proposed bearing plate under the skirt hasa 9 ft, 8 in. inside diameter and an outside diameter of11 ft, 8 in. The bolt circle is 11 ft, 0 in. in diameter andcontains 24 steel bolts 215 in. in diameter (from Table 10.4,the area per bolt = 3.72 sq in). Assume that under oper-ating conditions the dead weight of the tower is 600.000 lb.A high wind velocity develops a wind moment of 8,OOO.OOOft-lb. A continuous compression ring is used. FromTable 10.1, n = Es/EC = 10; js(atlon.) for the structural-steel skirt is 20,000 psi. Determine the maximum inducedstress in tension in the bolts and the maximum inducedstress in compression in the bolts. Also determine I hemaximum compression stress in the concrete at the outer-most edge of the bearing plate on the downwind side and thewidth and thickness of the bearing plate.

For first trial assume js E 20,000.From Table 10.1, jccmax) = 1200.

t(pro

posed) = (11 ft, 8 in.) - (9 ft. 8 in.) = 12 in.2

By Eq. 10.3, estimating jr(t)olt circle) = 1000,

1 1k (spprox.) = - = = 0.3331+f”

de

1+-- 20,000

(10)(1000)

By rearranging Eq. 10.30 and solving for jc+,l+ rirclp) we*obtain :

fc(bolt circle)(2)(0.333)(11)(12)

1(2)(0.333)(11)(12) + 12_

= 1055 psi

To evaluate the induced stresses the constants are readfrom ‘I’able 10.2. For k = 0.333

Cc = 1.588

Ct = 2.376

t = 0.431

j = 0.782

The tensile load may be calculated by Eq. 10.24.

~~ = Mwind - Wdw.zd 8 X lo6 - 6 X 10S(0.431)(11)=-.id (0.782)(11)

= 8 ' lo6 - 2'85 ' lo" = 598 000 lb8.6

The induced stress in t,he steel js based upon k = 0.333may be evaluated by Eg. 10.9.

,c tl = A = (24)(3*72) = 0.215 in.c-w-. ._ ,& 411) (12)

598,000 = j,(O.215)(5.5)(12)(2.376)

fs = 17,700 psi

Skirt Supports for Vertical Vessels 189

The compressive load may be cakulated by a summationof vertical forces using Eq. 10.27.

Ft + Wa,,. - F, = 0

600,000 + 598.000 = F, = 1,198.OOO lb

The induced stress in the concrete al the bolt circle. jCbased upon k = 0.333 may he evaluated by Eq. 10.18.

F, = (tz + rrt,)rj-J:,But

IS = tB - t, = 12 - 0.215 = 11.785 in.

1.198.000 = [(11.785) + (10 X 0.21~)](5.5)(12)(1.588)j~

jc = 818 psi

Rechecking k by Eq. 10.3 gives:

k =1 1

= ~~ = 0.317l + 17,700 1 + 2.16

(10)(X18)

Rechecking const,ant,s and stresses gives the following.From Table 10.2 for k = 0.31i,

c, = 1.554

Cf = 2.405

i = 0.434

j = 0.782

F

2= 8 X lo6 - 10”(6)(0.-4,3~4)(11) = 396000

(0.782)(11)

596,000.f* = co-q-( 12)(2. iOj) = 17AjO Pi

F,. = 600,000 + 396.000 = I, 196,000 lb

.fc =1,196,OOO

-- = 835 psi(11.785 + 2.15)(5.5)(12)(1.55.4)

k = 1 I-- = 0.321

17,400

’ + (10)(835)1 + 2.08

k = 0.32, approximalely , by interpolation

The rechecks (Jf fS, .f,, and k are in sufbcienl agreement fordesign purposes.

To check maximum compressive stress in bolts and con-crete, E(t. 10.2 is used.

fs(comp.) = gafc = 10jc = 8350 psi

By Eq. 10.30

fr(msx induced) = (.fr(tmlt circle induced))

= 835 (2)(0.3w)(w + 12(2)(0.32)(11)(12) >

= 835(-$) = 965psi

Page 200: Process Equipment Design

1 9 0 Design of Supports for Vertical Vessels

Ski,rt-, -Ful l - f i l le t weld

F i g . 1 0 . 3 . R o l l e d - a n g l e b e a r i n g p l a t e .

Determination of bearing-plate thickness by Eq. 10.32 isas follows:

1 11 8 in. - 10ft, ft, 0 in.= = 1 0 in.

2

t4 = 1 0(‘)(“5)-=. *3 81 in

20,000(without gussets)

As this thickness is considered to be excessive, the bearingplate will be stiffened with 24 gussets equally spaced andstraddling the bolts.

The gusset spacing, b, is

b = dll)W = 17.3 in.2 4

1 1 0-z--z 0.58b 11.3

Interpolating between l/b = 34 and l/b = 45 from Table10.3 gives:

M max = M, = -0.26fJ2 = (-0.26)(965)(100)

= -25,100 in-lb

By Eq. 10.32b

t4 = 1/6(25,100/20,000) = 2.75 in.

Further reduction in bearing-plate thickness could berealized if the gusset spacing were decreased by using 4 8gussets.

For 48 gussets, gusset spacing, b is:

b = 7411lW) = 8.65 in.4 8

1 1 0-=-=1.255b 8.65

Interpolating again from Table 10.3 gives:

M msx = M, = -0.121fJ2

= -11,700 in-lb

By Eq. 10.32b

t4 = ~6(11,700/20,000) = 1.875

Therefore, use lid-in. plate.

10.1 d Practical Considerations in Designing Bearing

Plates.

ROLLED-ANGLE BEARING PLATE. If the vertical vesselis not very high and a skirt is used to support the vesselrather than legs, lugs, or columns, a simple design maysuffice for the bearing plate. If the calculated thickness ofthe bearing plate is 36 in. or less, a steel angle rolled tti fitthe outside of the skirt may be lap welded as shown in Fig.10.3.SINGLE-RING BEARING PLATE. If the required bearing-

plate thickness is 35 in. to N in., a design using a single-ringbearing plate may be employed, as shown in Fig. 10.4.If the bearing-plate width is less than 5 in. and the thicknessless than $4 in., the rolled-angle design (Fig: 10.3) willprobably be more economical.CENTERED CHAIRS. If the required bearing-plate thick-

ness is N in. or greater for the design shown in Fig. 10.4, abolting “chair” can be used to advantage. Figure 10.5shows a typical design for a centered anchor-bolt chair.Although the number and size of bolts required should bechecked for each design, Table 10.5 gives some typicalvalues of the maximum number of chairs usually insertedin a vessel or skirt of a given diameter.

In checking the bearing-plate thickness for a -enteredchair the plate inside the stiffeners may be considered toact as a concentrated loaded beam with fixed ends.

The concentrated load, P, is produced by the bolt andis equal to maximum bolting stress times the bolting area,or

P = fax&, (10.35)

wheref, = maximum induced stress in bolting steelAb = root area of anchor bolt, square inches (The

values of f8 and & are those determined in anearlier section.)

P = maximum bolt load, pounds

The maximum bending moment in the bearing plateinside the chair occurs at upwind dead center and is located

Skir+ I

j-- 5” minimum _I_t(

F i g . 1 0 . 4 . S i n g l e r i n g b e a m , p l a t e with gusrats.

I \ - \ \I /

Page 201: Process Equipment Design

Plan View B-B

W a s h e r

Bolt size + 1

Section A-A

LWasher (thickness equalto bolt diameter)

E l e v a t i o n

Fig. 10.5. Centered anchor-bolt chair.

at or near tl bolt where the cross-sectional area is mini-m u m . T .roment is given by Eq. 10.36.

where M,,, = maximum bending moment, inch-poundsb = spacing inside chairs, inches (usually 8 in.)

The hole in the bearing plate reduces the effective beam

Skirt Supports for Vertical Vessels 191

Table 10.5. Maximum Number of Centered Chairs in

Various-sized Vessel Skirts

Skirt diameter, ft No. of Chairs3 44 85 86 1 27 1 68 1 69 2 0

10 2 4

width of the plate. With this consideration the requiredbearing-plate thickness inside the chair may be calculatedby Eq. 10.37.

t4 =J

f5Mmx (10.37)-(t3 - bWfan,w.

where t3 = bearing-plate width, inchest4 = bearing-plate thickness, inches

bhd = bolt-hole diameter in bearing plate, inchesf&llow. = allowable stress, pounds per square inch

The bending moment in the bearing plate outside thestiffeners (between chairs) may be controlling and can bedetermined by use of Table 10.3. The thickness can bedetermined by use of Eq. 10.32b.

EMPIRICAL DIMENSIONS FOR EXTERNAL CHAIRS. If then&be;-of bolts required exceeds the number given inTable 10.5, external bolting chairs may be used, as shownin Fig. 10.6. The proportions for the chair may be deter-mined empiricaliy by the relationships given in detail b ofFig. 10.6. Note that thehole in the bearing plate is made- - - - - _ _ _ _ --.- -- -

Bolt size + 9”Bolt size + %’ , , .T

Bolt size + y” min

Fig. 10.6. External bolting chair.

Page 202: Process Equipment Design

192 Design of Supports for Vertical Vessels

plate

Fig. 10.7. V e s s e l s k i r t w i t h e x t e r n a l b o l t i n g c h a i r s .

larger than the hole in the__top plate for ease in erection of- -the vessel.

C A L C U L A T I O N OF COMPRESSION-PLATE THICKNESS. Thezximum load on the compression plate at the top of anexternal chair occurs on the upwind side of the verticalvessel where the reaction of the bolts produces a compressionload. The compression plate may be considered to act asa rectangular plate bounded by the two gusset plates, t,heskirt, and the outside of the plates. The bolt load may beconsidered to be a uniformly distributed load acting over acircufar area cquaf to the bolt area. The fact that the

compression plate is welded to the skirt and gusset platesas indicated in Fig. 10.7 provides additional rigidity onthese sides, which tends to compensate for the lack of sup-port on the fourth side. As an approximation the platewill be considered to act as a plate freely supported on foursides. Timoshenko (107) has developed the relationshipsfor a rectangular plate freely supported on four sides with aconcentrated load acting as a uniformly distributed loadover a circular area of radius e. In reference to the “Plan”view of Fig. 10.6 with y in the radial direction and r in thecircumferential direction, the maximum bending momentsM, and M, are given by Eqs. 10.38 and 10.39, respectively.

(10.38)

where M, =

M, =

P =

maximum bending moment along radisi axis,inch-poundsmaximum bending moment along circum-ferential axis, inch-poundsmaximum bolt load on upwind side (see Eq.10.35) pounds

cc=In =a =

Poisson’s ratio (0.30 for steel)natural logarithm

I=

b =e =

sz

radial distance from outside of skirt to boltcircle, inchesradial distance from outside of skirt to outeredge of compression plate, inchesgusset spacing, inchesradius, of action of concentrated load, inchesone-half distance across flats of bolting nut,inches

71, Y2 = constant,s from Table 10.6

A comparison of Eqs. 10.38 and 10.39 using the constantsin Table 10.6 indicates that for (b/Z) = unity, M, = M,,and that for all cases in which (b/l) is greater than unity,M, is greater than M, and M, is therefore controlling.

After the determination of the size of the bolt and thewidth of the bearing plate and after the selection of thebolt-circle diameter and gusset spacing, the dimensionsa,b,e, and 1 are fixed. The constants yi and ys may beevaluated by use of Table 10.6, and the maximum bendingmoments in the radial and circumferential directions maybe computed by use of Eqs. 10.38 and 10.39.

For the case in which a is selected to be l/2 and M, iscontrolling, Eq. 10.38 reduces to:

My = 0 (1 + p) 111: + (1 - -YI) (1O.N~)

To determine the maximum stress in the compressionring a strip of unit width is considered. For this case,

f6Mll

max = 2t5

Table 10.6. Constants for Moment Calculation in

Compression Ring (107)

(Courtesy of McGraw-Hill Book Co.)b/l 1.0 1.2 1.4 1.6 1.8 2.0 =

.-

Yl 0.565 0.350 0.211 0.125 0.073 0.042 0Yz 0.135 0.115 0.085 0.057 0.037 0.023 0

Note: for a b/l less than 1.0 invert b/l and rotate axes 90”.

Page 203: Process Equipment Design

Skirt Supports for Vertical Vessels 1 9 3

as the upper plate of the bolting ring. Such a continuousring is preferred when the spacing of external chairs becomesso small that the compression plates approach a continuousring. As in the case of the compression plate the maximumload on a continuous compression ring occurs on the upwindside of the vertical vessel where the reaction of the boltsproduces a compression load on the ring. This load pro-duces a bending stress in the compresson ring. As in thecase of external chairs the vertical gusset plates transfer thiscompression load to the bearing plate.

In determining the thickness of the continuous compres-

Or. if fi is assumed to be follow.

i5 = dWK,/f,n0w.) (10.41)

where t5 = thickness of compression plate, inches.fsuoW. = allowable working stress, pounds per square inch

lO.le Example Calculation 10.2, External-chair Design.An external chair will be designed for a column 8 ft, 0 in.in diameter having 12 bolts 114 in. in diameter with a cal-culated induced stress of 17,500 psi. The bolt-circle diam-eter is 8 ft, 6 in., and the outside diameter of the bearingp1at.e is 9 ft, 0 in. The gusset height, h, is 12 in.

By Fig. 10.6,

t6 = (Sg)(lys In.) = 0.515 in. (Use 34-in. plate.)

A = 9 in. + (lj$ in.) = 1014 in.

b = 8 m. + (l?,< m.) = 936 m.

By Table 10.4,

root, area of bolt, Ab = 1.405 sq in.

The bolt load by Eq. 10.35 is:

P = fs&, = (17,500)( 1.405) = 24,600 lb

By Fig. 10.6,

(8 ft, 6 in.) - (8 ft. 0 in.) = 3 iIa=---2

1 = (9 ft, 0 in.) - (8 ft, 0 in.)- = 6 in.2

From Table 10.4,

nut’dimension across flats 2.375e = --~~~~ = ~ = 1.188 in.2 2

The ccmpression-plate thickness is:

b 9.5 1 -8-=-= .J1 6

Interpczlating from Table 10.6 gives:

‘, y1 = 0.134

Substituting in Eq. 10.40 gives:

, (I +2 1

p)h-~ + 1 -en- y1 1( o ( 6 ) + 1 - 0.134

7r1.188)

I

= +8200 in-lb

Substituting into Eq. 10.41 withf,llow. = 17,500 psi gives:

i,=g;.=dT=1.672

Therefore use lx-in. plate for compression plate.10.1 f Continuous-compression-ring Thickness. Figure

10.8 shows a sketch of a continuous compression ring used

sion ring the assumption is made that each section of thering between gussets acts as a rectangular plate bounded bythe two gusset plates, the shell, and the outer ring. Thebolt load will be considered to be a uniformly distributedload acting over the area of the bolt.

Therefore the method used in determining the thicknessof the compression plates for external bolting chairs isapplicable. This method involves the use of Eq. 10.38,10.39, or 10.40 and of Table 10.6 plus Eq. 10.41.CALCULATION OF GUSSET-PLATE THICKNESS FOR COM-

PRESSION RINGS. If the gussets are evenly spaced alter-nately between bolts, the gusset plate may be considered toreact as a vertical column. Normally the gusset is weldedto the shell, but no credit is taken for the stiffening effectproduced by the shell. The moment of inertia of the gussetabout the axis having the least radius of gyration is givenin Appendix J, item 1 as:

.

I! I I ICompression/ ring

Fig. 10.8. Skirt with continuous compression ring and strap.

Page 204: Process Equipment Design

1 9 4

OF

Design of Supports for Vertical Vessels

ta2r2 = -12

(10.42)

force, Qc (see (Fig. 6.3) produced in shells with closures,and the calculation may be treated accordingly.

In Chapter 6 the following relationships were derived:

(YLO = & @MO + Qo>1

(6.76)

where a = area of cross section, square inchesis = radius of gyration, inches

t6 = gusset-plate thickness, inches1 = width of gusset, inches

Equation 4.21 may be used to express the relationshipfor steel columns in which the value of (h/r) is from 60 to200.

In detail a of Fig. 10.6 y is the horizontal deflection of theskirt corresponding to yr in Fig. 6.3 and varies with distanceabove the compression ring in accordance with Eq. 6.65.Applying the boundary condition that the slope of thedeflection curve dy/dx is equal to zero at the top of the ringwhere x = 0, we obtain:

(4.21)18,0001 + (h2/18,000r2)

where h = height of gusset, inches

Substituting Eq. 10.42 into this relationship gives.

dy

0dz z=o=(I= L @PM0 + 00)

W2D1(10.48)

Noting that y1 of Fig. 6.3 is taken as equal to -y in Fig.10.6, detail a, we obtain:(10.43)f

18,000anow~ = 1 + (h2/15001sZ)

(YLO = & @MO + Qo> (10.49)1The allowable stress, fsuoW., in Eq. 10.43 must be:

Solving Eqs. 10.48 and 10.49 for MO and Qo gives:

Qo = 4P3&y (10.50)

MO = -2#S2Dly (10.51)

(10.44),fBolt load P

allow. =0s =a

Substituting in Eq. IO.43 gives:

whereBolt load 18,dbO

its = 1 + (hz/1500teqP= J

4 3(1 - p2)r2t2 (6.86)

or

(6.15)18,000Z1r,3 ‘- (bolt load)ts2 -h2(bolt load)

1 5 0 0= 0 (10.45)

By Eq. 6.84

W tE-=-Y r2

Examination of Eq. 10.45 indicates that when the gussetheight, h, is small, the third term in the equation may bedisregarded. In this case Eq. 10.45 reduces to the rela-tionship for straight compression without column action or

(10.52)

where w = load, pounds per linear incht = skirt thickness, inchesP = radius of curvature, inches

Substituting Eqs. 6.15 and 10.52 into Eqs. 10.50 andinto Eq. 10.51 gives:

(10.53)

(10.46)

Equations 10.45 and 10.46 are based on the asumptionthat the compression plate is sufficiently thick for the boltload to be transferred to the gusset plates without intro-duction of eccentric action. The stiffening resulting fromthe welding of the compression plates and gussets to theshell introduces a margin of safety which justifies the aboveassumption.

If the gussets are not evenly spaced, an eccentric loadingwill result in an induced bending moment. The thicknessef such gussets may be proportioned empirically, as in thexase of gussets for external chairs.

ts = gts (10.47)

lO.lg Reaction of External Bolting Chairs and Compres-sion Rings. The use of external bolting chairs or a compres-sion ring results in a loading condition that produces a reac-tion in the skirt. This reaction, R, is similar to the shear

(10.54)MO = -p2t2wr2 _ -w6(1 ,- p2) 2f12

Taking a summation of moments about the junction ofthe skirt with the bearing plate gives:

P(a) = Q&m

where Qs is the force per linear inch on the skirt and isassumed to act over an arc distance of m, or

(10.55)

Page 205: Process Equipment Design

Skirt Supports for Vertical Vessels 1 9 5

This thickness can’ be reduced by increasing the gussetheight. Assuming a gusset height of 18 in. rather than 12in. will reduce the skirt thickness to:

1 = 1.214(+$)1a = 0.926 in. or 1.0 in.

lO.li Thermal Stresses in the Skirt. For the case inwhich the vessel is operated at a temperature considerablydifferent from atmospheric temperature, a thermal stressmay be induced in the skirt as a result of the temperaturegradient near the junction of the skirt and the vessel.

TEMPERATURE GRADIENT IN SKIRT. To minimize thetemperature gradient in the skirt, the skirt may be insulatedboth inside and out. Skirts of vessels are insulated insideand out for fire protection when manways are cut into theskirt. The modulus of elasticity decreases rapidly withincreasing temperature above 600” F with resulting loss inelastic stability. F. E. Wolosewick (160) has given anapproximate equation for the skirts of vessels with 2 to435 in. of insulation both inside and out.

T, = (TV - 50°) - 6.037x - 0.2892’+ 0.009~~ - 0.00007~~ (10.60)

Differentiating with respect to z gives:

dTZda:=

-6.037 - 0.578~ + 0.027~~ - 0.00028~~ (10.61)

where T, = temperature of skirt at z distance below junc-tion of skirt and shell, degrees Fahrenheit

TV = temperature of fluid in vessel bottom, degreesFahrenheit

5 = distance below junction of skirt and shell,inches

Substltutine Eq. IO.53 into 10.55 gives:

WWC-m h

Substituting Eq. 10.56 into Eq. 10.54 for u) gives:

Mo = _ P3t2Par26(1 - p2)mh

(10.57)

For a strip of unit width under flexure

f6Mo p3Par2

allow. =-cm

t2 (1 - p2)mh(10.58)

Substituting for /3 by Eq. 6.86 and solving for t gives:

t = [3(I - P2)P [ 1p” % $6( 1 - p2)?’ m h j

For steel in which p = 0.3,

t = l.76(&yrts (10.59)

where t = skirt thickness required to resist reaction ofexternal chairs or compression ring, inches

P = radius of skirt, inchesm = 2A (see Fig. 10.6) or bs (bolt spacing)P = maximum bolt load, poundsa = radial distance from outside of skirt to bolt circle,

inchesh = gusset height, inches

lO.lh Example Calculation 10.3, Reaction of a BoltingRing. The tower described in Example Calculation 10.1 isto be modified so that it has a bearing plate extending635 in. out from the skirt with a bolt circle 3>/4 in. outside theskirt. Twenty-four bolts 235 in. in diameter are to be usedwith n continuous compression ring, and the gusset heightis to be 12 in. Determine the required thickness of theskirt to resist the reaction of the bolting ring. The maxi-mum induced stress in the bolts is 17,450 psi, and the maxi-mum allowable stress in the skirt is 20,000 psi.

a = 3>/,-in.

~(126.5)m = ~ = 16.6 in.

24

fallow. = 2%000 Psi,h = 12 in.

p = 17,450 X 3.72 sq in. per bolt

P = 60 in.

By Eq. 10.59

Substituting gives:

(17,450)(3.72)WW(16.6)(12)(20,000) 1 j6 (60)s

= 1.214 in. or 134 in.

THERMAL EXPANSION. As an insulated vessel is brought‘up to operating temperature, it will undergo thermalexpansion. If there is no restraint to this expansion, nostress will be induced. The metal both in the skirt and inthe shell at the junction will have the same temperature.From the junction to the foundation a temperature gradientwill exist, which will tend to produce a varying thermalexpansion. At any given point in the skirt the radialthermal expansion, y, is proportional to the coefficient ofthermal expansion, LY, the radius of the skirt, F, and thetemperature difference T’, or

y = cw(Tv - Tz) = MT’ (10.62)

where y = radial thermal expansion, inches(Y = coeflicient of thermal expansion inches per inch

per degree Fahrenheitr = radius of skirt, inches

T’ = (TV - T,) = temperature of vessel bottomminus skirt temperature, degrees Fahrenheit

Differentiating Eq. 10.62 with respect to 2, the distancealong the skirt from its junction with the vessel, gives:

dy ar dT’-=-dx dx

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1 9 6 Design of Supports for Vertical Vessels

but dT’ = d(T, - T,) = -dT,; therefore

dy ar dT,-= - -d x d x

(10.63)

STRESS FROM BENDING MOMENTR AND SHEAR. Theterm dyjdz represent the slope of the skirt.from t.he verticalas a result of thermal deformation. This deflection can becompared to the deflection dyl/dx for a cylindrical shelljoined to a flat-plate closure, shown in Fig. 6.3. ht thejunction, where 2 = 0,

(YLO = - 2i&l (@MO + Qo) (6.75)

dy0iii z=o

= $n, (Wfo + Qo) (6.76)

Substituting Eq. 10.62 into Eq. 6.75 and Eq. IO.63 inloEq, 6.76 gives:

crrT’ = yz,o = - -!-2P3D,

(@MO + Qo)

(YP dT’ dr_- = -2d x 0 =

ij;D GVMo + Qo).dx s=o 1

Clearing fractions on I he right side of these equationsgives:

arT’(2P3D,) = -@MO - Qo

a=$ (2/3*&) = 2PMo + Qo (10.61)

Adding the two equations gives:

arT1(2P3D1) + zd$: (2p2D1) = PM,

therefore

MO = 2c@D~(T’/3 +g) (10.63)

Substituting into Eq.

=$ (2/3201) =

therefore

10.64 gives:

4M2h(T’B+~)+Qo

Q o = 2&12D1 (ZT’/l - z) (10.66)

At the junction of the skirt and bottom dished head ,T’ =TV - T,, and T,, = T,; therefore T’ = 0. And, t.herefore,

dT’Mo==ad@D,-

d x(10.67)

Qo = -ad/3zD,dg (10.68)

(10.69)

and

Q. = a dp2 D1 ‘2 (10.70)

A comparison of Eqs. 10.69 and 10.70 indicates that.

PMo = -Qo (10.71),YAXIAL IHEAMAL STRESS, f,t, AT JUNCTION WITH SHELL.

(See Eq. 6.122.)

~IRW~FERENTIAL THERMAL STRESS, fCt, AT JUNCTIONWITH SHELL. (See Eq. 6.125.)

frt = /#(PMo + Qo)

1

-~ +‘$y (10.73)I

But bj Eq. 10.71 /3Mo = -Qo; therefore

fct = I !!$p (10.74)

SHEAR THERMAL STRESS,~~~, AT JUNCTION WITH SHELL.(See Eq. 6.121.)

(10.75)

lO.li Example Calculat ion 10.4, Thermal Stresses.Consider a vessel having a diameter of 130 in. and a skirtthickness of, 3% in. insulated inside and out with the skirtsupporting a shell in which the bottom temperature is700” F. Assume that the temperature distribution in theskirt is given by Eq. 10.60. Calculate the thermal stressesa1 the junction.

a = 7.6 X 10e6 deg F

E = 25.5 X 106 psi (from Fig. 8.6)

/.L = 0.27

By Eq. 6.86

P= - P2) = 4r2t2 J

By Eq. 6.15

D, = --t”_- = ______12(1 - jL2)

25.5 x 10”(0.5)3 = 28-7 )’ 1o4

’ ’12(1 - 0.272)

The temperature gradient at the junction, 1’ = 0, byEq. 10.61 is:

= -6.037x=0

Axial thermal stress:Substituting into Eq. 10.69 gives:

= -(7.6 x 10-6)(130)(0.227)(28.7 X lo’)\ -6.037)

= 388 in-lb per in.

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Lug Supports for Vertical Vessels 1 9 7

are set up on the windward side when the vessel is emptybecause in this case the dead load is subtracted from thewind load. Therefore the stresses on the leeward side arethe determining factor for design of the supports. Themaximum total compression load in P pounds in the mostremote column is (164):

By Eq. 10.72

= 9320 psi

~‘ircumjerential thermal stress:By Eq. 10.70

= (7.6 X 10-6)(130)(0.227)2(28.i X 104)(-6.037)

= -88.2 lb per in.

By Eq. 10.7.4

jet = / !!!y! = [(0.27)(9320)] = 2115 psi

Shear thermal slress:By Eq. 10.75

10.2 LUG SUPPORTS FOR VERTICAL VESSELS

The choice of the type of supports for a vertical pressurevessel depends on the available f loor space, the convenienceof location of the vessel according to operating variables,the size of the vessel, the operating temperature and pres-sure, and the materials of construction.(Brackets or lugs offer many advantages over other types

of supports .4

They are inexpensive, can absorb diametricalexpansions y sliding over greased or bronze plates, are

easily attached to the vessel by minimum amounts ofwelding, and are easi ly leveled or shimmed in the f ield.

As a result of the eccentricity of this type of support,compressive, tensile, and shear stresses are induced in thewal l of the vesse l . The tensi le and compressive forces causeindeterminate flexural stresses which must be combinedwith pressure s tresses c i rcumferent ia l ly and longi tudinal ly .The shear forces act in a direct ion paral le l to the longitudinalaxis of the vessel, and the shear stress induced by theseforces is relatively so small that they are often disregarded.

Lug supports are ideal for thick-walled vessels since thethick wal l has a considerable moment of inert ia and is there-fore capable of absorbing the flexural stresses due to theeccentricity of the loads. In thin-walled vessels , however,this type of support is not convenient unless the properreinforcements are used or many lugs are welded to thevesse l .

/ If a vessel with lug supports is located out of doors thew-ind load, as well as the dead-weight load should be con-sidered in the calculation of P. However, as lug-supportedvessels are usually of much smaller height than skirt-sup-ported vessels , the wind loads may be a minor considerat ion.The wind load tends to overturn the vessel, particularlywhen the vessel is empty. The weight of the vessel whenfilled with liquid tends to stabilize it.

The highest compressive s tresses in the supports occur onthe leeward side when the vessel is full because dead loadand wind load are additive. The highest tensile st.resses

p = 4P,(H - L) I ZWn&e n

(10.76)

where P,, = total wind load on exposed surface, poundsH = height of vessel above foundation, feetI, = vessel clearance from foundation to vessel bot-

tom, feetDbC = diameter of anchor-bolt circle, feet

n = number of supportsZB’ = weight of empty vessel plus weight of liquid

and other dead load, pounds

10.20 Lugs with Horizontal Plates. Figure 10.9 shows asketch of a vessel supported on four lugs, each lug havingtwo horizontal -plate s t i f feners . Such lugs are of essent ia l lythe same design as that shown in Fig. 10.6 for external chairs ,and the same design procedure may be used. This typeof lug uses to advantage the axial stiffness and strength ofthe cyl indrical shel l to absorb the bending stresses producedby the concentrated loads of the supports. Both the topand bottom plates should have continuous weids as themaximum compressive and tensi le s tress occurs in these twoplates, respectively. These welds and the intermit.tentwelds of the vertical gussets to the shell carry the verticalshear load. The load, P, on the column has a lever arm, a,measured to the center l ine of the shel l plate . This moment

-

Fig. 10.9. Sketch of vessel on four-lug supports with horizontal-plate

stiffeners.

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1 9 8 Design of Supports for Vertical Vessels

uP

Front v iew Side v iew

f

7

E

R

1 51

1 R

F ig . 10 .10 . S k e t c h o f l u g i n w h i c h o n l y t h e v e r t i c a l g u s s e t s o r e w e l d e d t o

s h e l l . ( C o u r t e s y o f P e t r o l e u m R e f i n e r . 1

is resisted by the couple, Qah, acting at the center lines oft,he top and bottom horizontal stiffening plates. As in thecase of bolting loads on external chairs, discussed previously,a reaction is established in the shell producing axial and cir-cumferential bending stresses which should be considered.The axial stress is defined by:

6Mo - 03Par2faxial = 12 = t1 _ p2jmh (10.58)

where m = effective arc width, taken as equal to 2A.

This method of analysis is recognized as being approxi-mate because it is based upon an assumed effective arcwidth and an assumed uniformly distributed moment overthis effective arc width. This method does not providefor the calculation of the bending moments and stresses inthe circumferential direction between lug supports. Ifthe horizontal plates are not used or if the bottom plate isnot welded to the shell, this method of analysis is notapplicable.

10.2b Lugs without Horizontal Plates. Wolosewick(161) has considered the case of lugs fabricated of twovertical gussets welded to the shell without horizontalplates welded to the shell. This analysis is based upon thestresses resulting from bending moments induced in a shellring section. No allowance is made for the axial stiffeningeffect of the shell. This method of analysis was developedin the Boulder Canyon Project and was presented in a re-port of the project, Bulletin No. 5, Penstock Analysis andStiflener Design (162).

Figure 10.10 shows a sketch of a lug without a top stiffen-ing plate in which the two vertical gussets but not the bot-tom plate are welded to the shell.

DE R I V A T I O N O F E Q U A T I O N S. In Wolosewick’s analysis ofa lug of the type shown in Fig. 10.10, the stiffening effectof the heads of the vessel as well as the restraint of longi-tudinal fibers of the shell is ignored. Therefore the calcu-lated stresses based on the assumed ring action of the shellwill be larger than the actual stresses induced by the loads;this places this approximate analysis on the conservativeside.

The following assumptions are made in the developmentof the bending-moment equations:

1. The stress in the ring is a linear function across thethickness of the shell.

2. The modulus of elasticity in tension and the modulusof elasticity in compression are the same.

3. Elementary circumferential fibers behave freely andare not restrained by longitudinal fibers of the vessel. (Thisassumption may be modified by multiplying the stresses by(1 - P2)).

4. There is no warping of the vessel cross section, and asection that is plane section before bending remains a planesection after bending.

For a ring on which a set of loads symmetrically locatedare acting, such as the one shown in Fig. 10.11, the only Iunknown to be determined is the moment MO acting on thecenter line of the cut section.

Shear forces for this model of symmetrically arrangedforces are considered to act as follows: The shear force at Awill be opposed by the shear force at B, which acts in theopposite direction; and, therefore, the net shear force on thering above is zero since the magnitude of the shear force atA is the same as that of the shear force at B.

Figure 10.12 is a diagram showing the angles involvedin the derivation of the relationships. The angle 6 is theangle between the y axis and R and is equal to one half theangle bounded by the two vertical gusset plates of a lug.The angle $I is the angle from the y axis to any point s.The moment MO is the internal moment in the shell at thecenter line, and the moment M, is the internal moment inthe shell at any point, s, between axes.

The strain energy stored in a structural element resultingfrom the action of bending moments is given by:

lJ=L M2ds2EI .I

(9.39)

For small-angle relationships in reference to Fig. 10.12,

ds = rd4Therefore

lJ = &s

M2r d+ (10.77)

R R

4-l- yMo+--Ru ‘A

R e

R &4--e

\\ OB-N$ --R

M O

F ig . 10 .11 . L o a d i n g c o n d i t i o n o n s h e l l r i n g r e s u l t i n g f r o m a c t i o n o f fourlugs (161). (Courtesy of Petroleum Refiner.)

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Lug Supports for Vertical Vessels 1 9 9

In an analogous manner, the following generalized equa-tion can be obtained:

Table 10.7. Bending Moments for Four Uniformly

Spaced Double-Gusset Lugs with Various Angle

Spacing between Gussets (161)

Ma/M

8” r#l=o +=tJ f$ = 45O

0 0.137 0.137 -0.0715 0.091 0.095 -0.068

1 0 0.055 0.065 -0.0601 5 0.027 0.045 -0.05020 0.002 0.033 -0.0332 5 -0.018 0.030 -0.0133 0 -0.033 0.035 +0.0103 5 -0.043 0.048 0.0384 0 -0.049 0.068 0.06745 -0.050 0.095 0.095

Castigliano’s first theorem states that the partial deriva-tive of the total strain energy with respect to the load isequal to the momement of the load in the line of action ofthe load. In the case of couples the partial derivative withrespect to the moment is equal to the rotation of the beamat the axis of the couple (30).

au I-=-aM,, EZ J

&,f aws--r@

aM0(10.78)

Equation 10.78 represents the value of the slope of thering at the center line, where MO is acting and is equal tozero since no rotations are possible. Therefore

au---COaM0

(10.79)

From 4 = 0 to 4 = 13

MS = MO + Rr(1 - cos 4) (10.80)

From & = 6 to 4 = (w/2 - 0)

M, = MO + Rr(1 - cos 4) - Rr(sin 4 - sin 0) (10.81)

From + = (w/2 - 0) to 4 = s/2

MS = MO + Rr(1 - cos 4) - Rr(sin 4 - cos 4) (10.82)

Therefore,

[MO + RrU - cos +)I dd

t /

(*/Q--B+ [MO + Rr(1 - cos 4) - Rr(sin 4 - sin 0)] d+

9

.I s/2+ [MO + Rr(1 - cos 4)(r/2) - 8- Rr(sin 9 - cos $J)] d4 = 0 (10.83)1

Formal integration of Eq. 10.83 gives:

M0

sin e1 (COS e + 0 sin e)-__ -

2 7r 2+ cot;1 (10.85)

where rz = an even number of lug supports symmetricallyspaced

Using the above equations Wolosewick (161) has pre-sented: graphical solutions for the cases of two, four, andeight-lug supports; graphical solutions for the conditionsof two and four uniform loads; equations for two, four, andeight lugs with shell-reinforcing plates.

The value of the moments along the circumference can bedetermined from Eqs. 10.80, 10.81, and 10.82 if the valueof MO from Eq. 10.85 and the direct forces (represented byR in Fig. 10.12) are available. Table 10.7 gives values ofthe dimensionless group (M,/Rd) at 4 = 0, 4 = 0, and4 = 45” for various values of 0. (Note: at 9 = 0, M, =MO).

As shown by Table 10.7, a decrease in B will increase M,.The reduction in moment by an increase in the angle 0of the gussets is useful in design. If the stresses induced inthe vessel by a lug exceed the allowable value, the angle ofspread of the gussets can be increased to reduce the stresses.EFFECTIVE WIDTH OF SHELL. In the developmentofthe

bending-moment equations it is assumed that only a por-tion of the shell withstands the flexural stresses caused bythe reaction between the lugs and the vessel. This assump-tion disregards the reinforcement effect of the remainderof the shell as well as of the heads of the vessel. Thus thecomputed bending moments for the shell are conservative.

A rigorous determination of the effective width of shellthat resists the flexural stresses produced by the eccentricloading imposed by the lugs requires a laborious mathe-matical analysis. Wolosewick assumed that for this typeof loading the effective width of shell equals the width ofshell adjacent to the upper end of the hypothetical column

f’ R

MO

2(cos 8 + fl sin 0)

sin 0-7r 2

-i cot i1 (10.84)

F ig . 10 .12 . O n e q u a d r a n t o f a s h e l l w i t h f o u r lugs as suppor ts .

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2oC Design of Supports for Vertical Vessels

Fig. 10.13. Lug with single vertical gusset (continuous weld all around).

cf the gusset plus 0.78 V% inches on each side, OI

b = h + (2)(0.78) k’?t

= h + 1.56 V5i (10.86)

utlere b = effective width of shell, inchesh = width of shell adjacent t.o hypothetical colu~~lr~ of

gusset, inches (See Fig. 10.10.)r = mean radius of shell, inchest = thickness of shell, inches

The effective width extends beyond the value given byEq. 10.86. Therefore, the value of the computed stresswill be higher than the value of the actual localized stress,and the design will be conservative. However, the valueof the stress as computed is not exceedingly high, and thesize of the designed lugs is practical.

COMPUTATION OF B ENDING STRESSES I N SHELL . Thecalculation of the bending stress in the shell follows oncethe values of the moments are obtained. If the dimensionsof t,he lugs are assumed, the value of b, the effective widthof shell, can be computed, and the stress calculated fromEq. 2.10.

or for a plate Z = (bt”/6)

(2.10)

(10.87)

wheref = bending stress, pounds per square inchM = bending moment, inch-pounds

t = thickness of shell, inchesb = effective width of shell, inches

= h + 1.56 V% (See Fig. 10.10.)

Several methods are used for the design of lugs for verticalpressure vessels. The assumptions involved in any methodare diversified, and the procedure used often depends on thejudgme-,; and experience of the design engineer.

One method frequently used is to consider the gussets ascolumns fixed by the bearing plate at one end and by thewall at the other. The contact plane with the remainderof the lug area adds additional restraint to the column.Figure lO.l(! illustrates the strip of the rib that is assumedto resist the load.

The effective width, b, of the column is controlled by Ihe

width of t,he base plate mult,iplied by the cosine of the angie9.

III reference t.o Fig. 10. IO, the bending moment Pa isresisted by a couple in the shell, R(3@), for each gusset orfor the case of two gussets as shown in Fig. 10.10.

DESIGN OF W ELDED LUGS. If the two vertical gussetsshown in Fig. 10.10 are welded to the shell by four verticalwelds, the welds will be subjected to both a vertical shearand tensile and compressive stresses produced by the bend..ing moment.

The vertical shear in t,he welds expressed in pounds perlinear inch will be equal to the vertical force P divided bythe length of the welds, or

V=;The maximum tensile force produced in the welds

expressed in pounds per linear inch will exist in the lower-most fiber of the vertical welds and will be equal to:

T - MU2 6Pa13/12 = 12

(10.90)

The resultant combined force (shear plus tension) in thewelds is:

F=+f2+T2 (10.91)

The usual allowable stress for weld metal in shear is13,600 psi (102). If .15” fillets are used, the weld will havea minimum cross-sec:tional area of 0.707w sq. in. per in. ofweld length. Therefore Fallo,r. is:

F anow, = (13,600)(0.707)~ = 9,600~ lb per lin in. (10.92)

where w = width of one leg of fillet, inches

A single vertical gusset plate with a horizontal base plateand with a continuous weld all around is sometimes used aea lug, as shown in Fig. 10.13. In such a case the weldsare no longer symmetrical with respect to the neutral axis.The location of the neutral axis (center of gravity orcentroid) and the moment of inertia of the welds about thisaxis must be evaluated. This may be accomplished bydetermining the distance, b, from the base to the neutralaxis. The welds are treated as lines, and the summation ofthe moments (the length of the line times the distance fromthe centroid of the line to the reference axis) is divided bythe I otal length of the lines, or (164)

11 + l2 + 2dhb=2(1$l+2d)

(10.93)

The moment of inertia I, of the welds about the neutralaxis at the center of gravit,y, e.g., is given by:

I I’.“. = z[sy2 dA + Az2] (10.94)2

+ s + 2 b 2”0

’ + & + tc2

+ (2d + t)b”

= b3&!f + f + tc2 + lb2 + ; + 2d(g2 + b2) (10.95’

‘f \ \I /

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Lug Supports for Vertical Vessels 201

as a beam fixed at one end but guided al the ot.her end wit.ha concentrated load af the guided end (30).

c

The shear force on the weld expressed in pounds per linearinch is

v = ----!I!--2 + 4d + 2t

(10.96)

The maximum tensile force on the weld (in pounds perlinear inch) produced by t.he bending moment occurs a I Ihebottom horizontal weld and is given by:

(10.97)

The resultant force is given hl Eq. 10.91, and the allow-able force by 10.92.

10.2~ Column Supports and Column Bearing Plates.

COLUMN SUPPORTS FOR LUGS. If the column is attachedin such a manner that it can be considered as a columnunder concentric axial load, the allowable fiber st,ress isgiven by Eq. 4.18.

fc =18,000

1 + (1’/18,000r2)(4.21)

where 1 = unbraced length of column inchesP = least radius of gyration of column, inches

fc = allowable compressive st.ress in column, poundsper square inch

The maximum permissible l/r ratio is 120, and t,he maxi-mum fiber stress = 15,000 lb per sy in.

The required cross-sectional area, A, of the column foraxial compression is, by Eq. 2.2:

“=Pccw

where p = maximum compression load per column, poundsfc = allowable compression stress from Eq. 4.21 above

Usually the columns are attached to the vessel with a dis-tance, a, between the center line of the column and thecenter line of the shell, as shown in Figs. 10.9 and 10.10.This produces an eccentric loading and an additional stressin the column supports. This stress is given as:

fee = 4 (10.98)

where 2 = section modulus of columrl, inches3

The column supports may also be subjected to bendingas a result of a wind load. The moment produced bythe wind load was considered by Siemon (164) to be equalto PL/2; this is comparable to considering the column

fbend. =(PW/ fl)l, 2

2(10.99)

where Pw = wind load on vessel, poundsR = number of colu~nn supports

I = column height, inchesZ = column se&ion modulus, cubic inches

When columns are subjected both to direct loads and tobending produced by eccentric loads, the American lnsti-tute of Steel Construction specification (102) states that, thesum of the axial compressive stresses divided by the allon-able column stress, plus the bending stress divided by theallowable flexural stress shall not exceed unity, or

y! + ~--~ ” .-..pm’-m 5 1(Pa Z) + (Puh3nZ)(10.100)

fc .ftCOLUMN BEARING PLATE . The procedure used for

designing bearing plates for column supports is similar IOthat used for designing a bearing ring for skirts of verticaltowers discussed in the earlier part of this chapter. The col-umn load is usually transferred t.o a concrete foundat.ion.The allowable compressive stress for concrete foundationsmay be selected from Table 10.1. The maximum columnload, P, divided by t.he allowable compressive stress, fc,gives the minimum required area for t,he bearing plate.The column is usually located in the center of the plate, andthe base plate is usually proportioned to give approximatelyequal overhang on all sides. If the load is assumed to beuniformly distributed, a uniform load, fc, is applied to theunderside of the overhang of the p1at.e. This overhang maybe considered to act as a uniformly loaded cantilever beamwhich has a maximum bending moment of (fcP/2) (see Eq.10.31) at the junction with the column. The requiredthickness is given by Eq. 10.32 where 1 is t.he greatest over-hang distance.

A comprehensive analysis of stresses from local loadingsin cylindrical pressure vessels was reported by Bijlaard(163). The method of analysis involves the developmentof loads and displacements into double Fourier series.This method may be used for: (1) a load uniformly dis-tributed within a rectangle, (2) a point load, (3) a momentin the longitudinal direction uniformly distributed over ashort distance in the circumferential direction, and (4) amoment in the circumferential direction uniformly dis-tribut,ed over a short distance in the longitudinal direction.For the t,angentiti loading condition eyuations for displace-ment, bending moment, and membrane forces are covered.

P R O B L E M S

1. An external chair is to be designed for a column 6 ft, 0 in. in diameter with eight anchorbolts 1 in. in diameter and with a calculated induced stress of 18,000 psi. The bolt-circle diam-

\ \I /~____ -~--I -

\. .-m

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.2 0 2 Design of Supports for Vertical Vessels -I

eter is 6 ft, 6 in., and the outside diameter of the bearing’plate is 7 ft, 0 in. The gusset heightiq 8 in. Determine the required thickness of the compression ring.

2. The concrete foundation supporting the tower base shown in Figure 10.14 is a 2500-psi28-day-ultimate-strength concrete. Assuming a loo-mph wind velocity, determine the requiredsize of anchor bolts if the allowable stress is 18,000 psi for the bolting steel.

~24 holes equa l ly spaced

L IGussets s t r a d d l e b o l t s

T o w e r h e i g h t : 1 5 0 f t ( f r o m t o p o f c o n c r e t e s l o b t o t o p o f t o w e r )U n i n s u l a t e d t o w e r d i a m e t e r : 8 f t , 0 i n .

Insulation: 3 in.E f f e c t i v e p r o j e c t e d d i a m e t e r f o r w i n d a c t i o n : IO f tEstimated tower weight, empty = 175,000 lb

Number of trays = 85 at 18-in. spacing (top vapor-disengaging.zone =

3 ft)Normal operating load = 30 lb par sq ft of Aoor troy orea (includes shell

a n d i n s u l a t i o n )S k i r t h e i g h t = 1 1 f t , 0 i n . ( f r o m t o p o f c o n c r e t e s l a b t o b o t t o m t a n g e n t l i n e )The 24 gussets are equal ly spaced.

F ig . 10 .14 . S k e t c h o f d i s t i l l a t i o n - t o w e r b a s e f o r p r o b l e m s .

3. For the tower base shown in Fig. 10.14 and the conditions given in problem 2, determinethe required thickness, z, of the compression ring if the allowable stress in the ring is 18,000 psi.

4. If the concrete foundation supporting the tower base shown in Figure 10.14 is a 3000-psi28-day-ultimate-strength concrete, the compression-ring thickness, z, is 1% in., the 24 boltsare 2 in. in diameter (eight-thread series), and the maximum wind pressure is 30 lb per sq ft,determine the stress in the skirt resulting from the interaction between the compression-ringassembly and the skirt.

5. For the tower base shown in Fig. 10.14 and the conditions given in problem 4, determinethe maximum stress in the bearing plate.

6. If the tower described in problem 2 were located in seismic zone 3, determine the size ofbolts required to withstand the seismic loads, assuming the wind moment is not controlling.

Page 213: Process Equipment Design

/C H A P T E R

DESIGN OF HORIZONTAL VESSELS

WITH SADDLE SUPPORTS

u he selection of the type of support for a pressure vessel1s dependent on several variables, such as: the size of thevessel, its wall thickness, the floor space available, the ele-vation of the vessel in relation to the ground or floor, thematerials of construction, and the operating temperature.Horizontal cylindrical pressure vessels are commonly sup-ported by saddle supports or cradles. If the underside ofthe vessel is to be located only a short distance above thegrade line, steel saddles resting on the top of concrete piersmay be used. When vessels are elevated, a structural-steel-~

. - __II) ,?> I-frame may be usea to sup@~&?l~s?~ccradles. If~~~znpport~t?YGd~ the‘ load resultingfr%iiiie welght‘Pifflb’~ei;essle.~~~~~~.contei;ts will be equallyd&ded even though one support may settle more than the_ .._ - --_other.

. ..- ____._._ .., - ‘~~71r”l- -.. . -.Smce the loads may not be equally drvrded after.- .,_ ..-”

tG supports settle -if m--~;-~~~~~~~r~~~~~~~~~,,---_. _“. ..-..“-x.,-- ,. ith~two-sul$ort system has an advantage over a system,.----- --- -.-----employing a larger number o~sup~oir&$

Figure 11.1 shows a group of horizontal butane- andgasoline-storage tanks each 12 ft in diameter by 120 ft longsupported on two saddles.

Horizontal vessels when resting on saddle supports suchas shown in Fig. 11.2 behave as beams. An analysis ofthe stresses induced in the shell by the supports was reportedby Zick (165) who developed equations for the stresses.Zick’s relationships contain empirical constants determinedexperimentally.* By using this method of analysis fol-lowing stresses can be evaluated:

1. The maximum longitudinal stress.2. The tangential shear stress.3. The circumferential stress at the horn of the saddle.4. The additional stress in the head used as a stiffener.* For steel vessels.

The maximum unstiffened length of the vessel between theheads, the ring compression in the shell over the saddle, thestresses on the ring stiffeners, and the total horizontalforce acting against the horns of the saddle may also bedetermined.

advantage of the stiffening effect of the head.A is often selected so that A = 0.4R.

Dimension

=-.-“.---- _Ic_____, * __.. . ‘~-~- ..F Dimension A shouldnever exceed 20% of dlmensron L; otherwise the stressesresulting from cantilever action will be excessive.

A cylindrical vessel with dished closures at the endsmay be treated as an equivalent cylinder having a lengthequal to (L + $iH) where L is the distance between thetangent lines of the vessel and H is the depth of a dishedclosure. *his approximation assumes that the weight ofthe head and the fluid contained in the head is equal totwo-thirds of the weight of a cylinder of length,H and thefluid contained in it. This approximation ,is valid forhemispherical heads and elliptical dished heads and can bedemonstrated by use of Eq. 5.14 for an elliptical closure for alOO-in.-diameter vessel.

V = (0.000076)(100)3 = 76 cu ft

The depth of dish, from Fig. 5.7, is ID/4 = 25 in. Thevolume of a cylinder 100 in. in diameter and 25 in. deepis 114 cu ft. The ratio of the volume of the head to thevolumhe cylinder is 76/114 or N.

The weight of the fluid and the vessel may be consideredto be a uniform load equal to the total weight divided by theequivalent length, or

J” 2QW=LH

where w = uniform load, lb per ft

Page 214: Process Equipment Design

\/204 Design of Horizontal Vessels with Saddle Supports

I

!

I

Fig. 11.1. Butane and gasoline horizontal storage tanks 12 ft in diameter by 120 ft long supported on two saddles. (Courtesy of C. F. Braun & Co.)

In the loaded condition the shell, over the distance L, 11.1 LONGITUDINAL BENDING STRESSESbehaves as a uniformly loaded beam. The load of the headsintroduces a shear load at the junction of the heads and thecylinder equal to ~~HuJ. This load produces a verticalcouple acting at a distance of 3gjH from t,he point oTtsg:gency and a horizontal couple acting with a lever arm ofR/4 where R is the radius of the vessel in feet.

As in the case of an overhanging beam with two supports ,two maximum bending moments exist in the longitudinaldirection of the vessel. One maximum occurs over thesaddle supports , and the other maximum occurs in the centerof the vessel span. The shell acts as a beam over the twosupports under the uniform load of the vessel and its con-

-

L -H-l I

tents, as shown in Fig. 11.3.aThe maximum moment over the supports, M,, may bedetermined by referring to Fig. 11.3 and by taking bendingmoments about the center of reaction, Q, over the distanceH + A :

Fig. 11.2. Sketch of horizontal vessel with two saddle supports.

A = distance from tangent line to saddle, feet

’ L = length of vessel, tangep! to tangent, feet

H = depth of heod,%et

Q = total load per saddle, pounds

= total weight divided by two

R = radius of vessel, feet

b = width of saddle (or width of concrete for formed concrete

saddles, inches

r = radius of vessel, inches

t = shell thickness, inches

v B = total included angle, degrees

w = load per unit length, pounds/ft

Vertical shear moment = sHw(A) counterc lockwise

Vertical couple = $Hw($H) counterclockwise

AOverhanging-shell moment = WA z

0counterc lockwise

c lockwise

Therefore

Substituting Eq. 11.1 for w, we obtain:

..-. 1.I (11.2)

Page 215: Process Equipment Design

Longitudinal Bending Stresses 205

The ceptroid of the shell included in angle 24 is located adistance of r&n A/A) from the ~-2 axis (where A is meas-ured in radians).

Therefore the moment of inertia of the arc of the shellabout its own centroid (cent.) is:

I c e n t . = I, - Ad2

B e n d i n g - m o m e n t d i a g r a m i n ft-lb

F i g . 1 1 . 3 . Cylindrical shell acting os beam over supports, according

to Zick (165). ( C o u r t e s y o f A m e r i c a n W e l d i n g S o c i e t y . )

The maximum bending moment at the center of the spanis determined by taking the summation of the bendingmoments about the saddle over t.he distance H + L/2.In addition to the moments over the distance H + A is themoment :

W

Taking a summation of these moments with due regardfor signs giyesghe moment MC at the center of the span, or

-Mc=w(L - 2A)2

8 1

To det.ermine the stress the moment of inertia of the shell- _..must be evaluated. Above ,each saddle support circum-ferential bending moments are produced which permit theunstiffened upper portion of the shell to deform. Thisdeformation makes this portion of the shell ineffective as abeam and reduces the effective cross section in the samemanner as if a horizontal section were cut from the vesselsome distance above the saddle. The arc A measured fromboth sides of the center line of the saddle up to thesefictitious “cuts” defines the effective cross section of thevessel, shown in Fig. 11.4.

By Eq. 9.15 the moment of inertia, I,, of the arc of theshell in the lower tw~@iZIi%its &luded by angle 2A is:

13. = 2tr3 /,,’ sin2 A dA

= 2tr31-I

= 2tr3Aii+

sinA?A] _ (~s~A)‘~~~~(~)

= tr3sin2 A

A + sin A cos A - 2 ~A 1 (11.4)

The section modulus, 2, for the side in tension at thesaddle is:

z = J = r3t[A + sin A cos A - 2(sin2 A/A)]- -C r(sin A/A) - r cos A

o rz = r2t A-+--sin A cos A - 2(sin2 A/A) .’

(sin A/A) - cos A(11.5)

The stress f, at t.he saddle will be (from Eqs. 11.2 and11.5) :

fl =

=

R2 - H2-

12QA 1 -l-;+-

2A

1+g

Asin2 A

+ sin A cos A - 2 ~A )I

o r

f1= k3S (11.6)

Section A-A

F i g . 1 1 . 4 . Sketch of effective area of shell under boom action (165).

(Courtesy of American Welding Society.)

Page 216: Process Equipment Design

206 Design of Horizontal Vessels with Saddle Supports

Values of A/L when R = H

// / / / / / / /I I I I /I ’ 1.6

0.8

0.6

H/L=0 --!? I loH/L = 0.05 \\\\\\\\\\\\

H/L=O.lO0

1.0

K, and Kg0.8

0.6

Values of A/L when R = 2H

Fig. 11.5. Plot of longitudinal bending moment constants KI and KZ (165). (Courtesy of American Welding Society.)

where

( 1 1 . 7 )

+ sin A cos A - 2 ~

In a similar fashion, using Eqs. 11.2 and 11.5, we find thatthe stress at the mid-span, ji, will be:

(11.8)

where

Values of K1 and K2 for different design proportions canbe obtained from Fig. 11.5 (165). In Fig. 11.5, K1 is

plotted for the condition of H = R when jr governs, andKg for the condition of H = 0 when js governs. Theseapproximations simplify the calculations and give conserva-tive designs.

It should be noted that Eq. 11.9 was obtained by dividingthe maximum bending moment by the corresponding sectionmodulus. The stress SO obtained will be the maximumaxial stress in pounds per square inch in the shell due tobending as a beam. This maximum bending stress may beeither tension or compression.

The tensile stress as obtained by Eq. 11.6 or 11.8 whencombined with the axial stress due to internal pressureshould not exceed the allowable tensile stress of the materialtimes the efficiency of the girth joints.

According to Zick (165) the compressive stress as deter-mined by Eq. 11.6 or 11.8 when combined with axial pressurestress should not exceed one half of the compression yieldpoint of the material or the value given by:

fallow. = XY (11.10)

where Y = 1 for 4 7 60r

Y =21,600

18,000 + (L/r)2for 4 > 60

r

X=(l,OO0,OOO~)(2-~lOO~)forf~O.015

X = 15,000 fort 2 0.015r

Equation 11.10 is applicable when t 2 j/4 in. (166).

T-7----------

Page 217: Process Equipment Design

Fig. 11.6 Shear diagram for shell stiffened with ring (165) (Cour-

tesy of American Welding Society.)

It should he noted that the reduction in compression stressas a result of elastic instability is not a factor in a vesselwhich is designed for pressure or in which t/r 2 0.005.

Consideration must be given to the stress due to bendingmoment before adding the stress due to internal or externalpressure. This is especially important when the combinedstress is less than the bending stress before internal orexternal pressure is applied.

11.2 TANGENTIAL SHEAR STRESS

d1.2a Shel l St i f fened by Ring in Plane of Saddle.When the shell is held to a cylindrical shape, the tangentialshear stress varies as the sine of the central angle, 4, meas-ured from the vertical. The maximum shear stress occurs atthe equator. In this case the analytical solution is simple.Let V = shear force as shown in Fig. 11.6. Then betweensupports

V = Q - w(A + H + m)

where w = 2Q/(L + $H) lb per ft, or

V=Q- 2Q(3L + 4H)/3

(A + ff + m)

At the saddle, where m = 0,

V=Q-6QsH

then

Tangential Shear Stress 207

v = Q (L - 2.4 - 0.7H) ~ Q L - 2A - H

L + 1.3H L + H >(11.11)

Consider a section of shell of length dx, as shown in Fig.11.7. From Eq. 2.10

By Eq. 2.5

(11.12)

z = 7rr3t (see Eq. 9.16) (11.13)

y = r cos f#J (11.14)

dA = tdl = trd4 (11.15)

dP = df dA (11.16)

On section ABDC the moment at AB is M, and at CD is(M + dM). If the element WA0 on the ring from -4 to+$J is isolated, flexural forces will exist on the ends, andlongitudinal shear forces on the radial planes at Wand 0,as shown in Fig. 11.8.

By a static balance of forces, shown in Fig. 11.8,

ZF, = 0or

zfdA+zdfdA-zfdA+2u=O

where u is the total longitudinal shear force on the section Wand 0.

Substituting Eq. 11.16 into the above equation and can-celing terms, we have:

IdP = j’df dA = -2a (11.17)

Substituting Eqs. 11.12, 11.13, and 11.14 into Eq. 11.16gives :

A P = s sd P =++ Vcos+d4

-* 7rP= =$ (11 .18 )

Fig. 11.7. Shear and moment dia-

gram for shel l st i f fened by r ing

( 1 6 5 ) . ( C o u r t e s y o f American

Weldi;.g Society.)

dM

Page 218: Process Equipment Design

2 0 8 Design of Horizontal Vessels with Saddle Supports

fa + dfdA/’

Fig. 11.8. Shear on side of element shown in Fig. 11.7.

where AP is the change in the longitudinal force on the por-tion WA0 per unit length of the ring.

4P is balanced by the longitudinal shear on a unit lengthof the radial sections W and 0. Subst.it,uting Eq. 11.18into Eq. 11.17 gives:

2V sin 4-2~ = (2t) (unit shear) = Bp

OL‘

Unit shear = -I’ sin ??rrt (11.19)

If a shearing stress occurs at a given point on a plane in astressed body, there must exist a shearing stress of equalmagnitude at that point on a second plane at right angles tothe first plane (231).

Since the shear has the same intensity on adjacent edgesof the rectangular element, the unit shear on the ends ofthe free body WA0 at the points W and 0 also equals(V sin +/d), and it,s direction is normal to the radialplanes and is, therefore, tangent to the shell, or

V sin C#Jor=-n-r

(11.20)

where rt = transverse tangential shear per unit length of arc

The shear force CL is tangent to the shell at all points andvaries from zero at the top to a maximum at mid-point, andback to zero at the bottom.

The summation of the vertical components of the trans-verse tangential shears on both sides of the stiffener gives Q.For this case the term C in Eq. 11.20 is replaced by Qand the vertical component. is P sin I$ times the shear.

2sin $I cos 4 1 * =

2Q

0(11.21)

The tangential transverse shear stress at any point on asection on both sides of the stiffener is:

V sin C#Jj3 = srt = St [ !*$!I sin #I

o r

(11.22)

where

KS = YiL4 (11.23)*

For the maximum value of js, sin #J = 1, and Ka = l,/?r =

0.319. The value for K3 is independent of 13, the angle c-fcontact with the support saddle.

For design purposes the value of j3 should not exceed theallowable tensile stress of the material times 0.8, or

j3 = 0.8 X allowable tensile stress of material

11.2b Unstiffened Shell with Saddles Awoy from Head.When the shell of the vessel is free to deform above thesaddle, the tangential shear stresses act on a reduced effec-tive cross section, and the maximum stress occurs at thehorn of the saddle. Here the shears are proportional tosin $I but act only on twice the arc given by (e/2) + (b/20)or * - cy. This angle is the assumed position at whichmaximum tangential shears occur on a shell which is freeto deform above the saddle and beyond the influence ofthe head. Zick reported (165) that this assumption wasverified very closely by strain-gauge experiments.

Fig. 11.9 represents a section taken in the plane of thesaddle for a shell with supports away from the head.

If a portion of the shell is noneffective, as shown in Fig.11.4, the shear (it is increased in the effective portion.Since the summation of the vertical components must stillequal the vertical load Q the shears will be increased ininverse proportion to the integral of the function, or

flt(unstiffened)

‘JtWffened)

~t(unstiffened) =V sin I$

r(7r - a + sin a! cos a)

The shear stress j4 will then be:

o r

f4 =Q sin 4 L-H-2A]

rt(7r - a sin (Y cos (Y) LfH j

j4 _ Q&L - H - ~4rt L + fZ 1

where

K4 =sin $I

7r - ff + sin a! cos a!

(11.24)

(11.25)

Location of - -assumed pomt ofmaxtmum shear

Fig. 11.9. Location of assumed point of maximum shear in unstiffened shell

(165). (Courtesy of American Welding Society.)

Page 219: Process Equipment Design

For design, f4 $ 0.8 X the allowable tensile stress of thematerial.

The maximum shear stress occurs at the point of maxi-mum shear, or where $I = cy,

sin (YK4 = ___.-.--7r - a + sin (Y cos (Y

Values of Ka are given in Fig. 11.10.11.2~ Shell Stiffened by Head. When the saddle sup-

ports are located near the head, the tangential shear stressesare first carried from the saddle to the head. Then theload is transferred back to the head side of the saddle by:aniential shear stresses which act on an arc of angle largerthan the angle of contact of the saddle. Here the shearsvary as the sine of 42. The angle 42 varies from (r - a)t.0 n-.

Above angle (Y these shears are directed downward andvary as the sine of & from 0 to (Y. Below angle (Y they aredirected upward, on the head side of t,he saddle. This canbe represented as shown in Fig. 11.11.

In order to have static balance at the left of section A-Aof Fig. 11.11, the downward forces must balance the upwardforces.

cforces down = 2

.I

a Q sin2 41___ I- d+,

0 TP

2Q 41=- - -[

sin 41 cos 41 an- 2 2 -0I

sin (Y cos a)

forces up = 2 /ou sin’ 42 d&

JT sin’ +J d& 1 r dh

a=,~Qs~~& [a-sinpcosa T]d42

?r - a + sm (Y cos a!

2Q a - sin OL cos a!=-lr 7r - a + sin (Y cos ff I

[42

I

*

- - + sin +2 co9 $I~2 a+- sin ~11 cos cy) (11.26)

170

160

1 5 0

g 140

$4 1 3 0.E= 1 2 0

1 1 0 \ I

iO00.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6

K, and KS

Circumferential Stress at Horn of Saddle 209

A-7 I

Shell stiffened by head ,

I

ki A - AwShear diagram whensaddle is near head

Fig. 11.11. Shear in shell stiffened by head (165). (Courtesy of American

Welding Society.)

The shear stress in the head is:

f5 =2Q sin $12 a - sin a cos (Y ’

2th(rr) 7r - a + s i n (Y c o s (Y1or

-f5A!c!rth

Similarly,

where

Qh-6fs = shear stress in shell = T8

K = sin 42[

a - sin a! cos a!5 __ --__-

n - 7r - a + sin (Y cos 01 1The maximum stress occurs at 42 = cr. Then

K5 =Sma* F

a - sin (Y cos ff- -~ --___-a 7r - a + sin (Y cos (YI

(11.27)

(11.28)

(11.29)

Values of Kg as a function of 13 are given in Fig. 11.10.

11.3 CIRCUMFERENTIAL STRESS AT HORN OF SADDLE

The theoretical analysis leading to t,he determination ofthe circumferential stress at the horn of the saddle has notbeen carried out successfully. The maximum stress occursat the point of maximum bending moment due to tangent.ialshear. When a stiffening ring is used to restrain the shellfrom deforming above the saddle, the mathematical analysisof the bending moment due to tangential shear can be solved.

The one-half arc of Fig. 11.12 is in equilibrium under theaction of the forces shown. From symmetry the verticalshear is zero both at point A and at point C. At any point,U, the shear ut is:

Q sin I,!Jut = ~

m

per unit length of arc, or for a length of arc dl, is:

Fig. 11.; 0. Values of Kd and KS as a function of saddle angle 0. 7rr

Page 220: Process Equipment Design

,210 Design of Horizontal Vessels with Saddle Supports

= maximum shear on I FY-A!Mboth sides of ring perunit length of arc

Gravi ty

I Q

Fig. 11.12. Forces acting on one-half arc of shell stiffened by ring in plane

of saddle (165). (Courtesy of American Welding Society.)

But dl = P d#; therefore

= g sin $J d#a

The 2 component of this shear = Q= = (Q/r) sin Ic,cos J/ d#. The moment arm of uZ with respect to N =P cos # - r cos 4. The y component of this shear = uU =(Q/r) sin2 # d$. The moment arm of uy with respect toN = r sin $I - r sin #. Therefore the moment of the tan-gential shear about N is:

ji&,=/dM=[; sin # cos #(rxos # - r cos 4) dfi

- ___/

+ Q sin2 #(r sin $J - r sin #) d#

0 n-

Or by integration

M, = p[

.L - cos f.j - z sin 4 1 (11.30)7r

As stated before, from symmetry, the vertical shear iszero at both A and C. There are, therefore, only threeunknowns acting on the free body, Pt, MA. and MC.

Timoshenko has shown (see Reference 29, Part II, p. 68)that for a thin curved beam the small angle of rotation,A dq5. between two neighboring cross sections may beexpressed as follows:

M, d2A d + = -

EI

From symmetry about the vertical axis of the shell, therotations of A and C are both zero, or

c C

c

Mg dl-=Oor

EI cM.+dl=O

A A

Roark has shown (see Reference 166, p. 147) that thehorizontal deflection of an element in a curved beam may beexpressed as follows:

x=I

$dl

Likewise, from symmetry about the vertical axis of thashell, the horizontal movements of A and C are both zero.or

C c

c

Mbmd15=OorEI c

M+mdl= 0A A

But dl = r d+; thereforec C

cM+mr dqb = 0 or

cM4m d$ = 0

A A

Likewise, m dl = r d+ (r cos + -- r cos 0). ThereforeC

cM# d$ (cos qi - cos ,8)] = 0

A

c C

c

M+ cos I$ d4 -cos p c

M+d4 = 0A A

c

But2

M+ d4 = 0; therefore (11.31)A

c

cMmcos+d+=O (11.32)

A

From a static balance the total moment is:

ZM=O

M$ = -Ptr(l - cos 4) + MA

+ $ (2 - 2 cos Cp - C$ sin 4) (11.33)

Substituting Eq. 11.33 into Eq. 11.31 and integratingfrom 0 to p gives:

-P&P - sin /3) + /~MA = $ [3 sin /3 - /3 cos /3 - 2/3]

(11.34)

Subst,ituting Eq. 11.33 into Eq. 11.32 and integratingfrom 0 to p gives:

-Ptr[sin @ - +@ - t sin 2p] + MA sin ,f3

= 2 [Q sin 2/3 - k/3 cos 2/3 + 0 - 2 sin 01 (11.35)

Simultaneous solution of Eqs. 11.34 and 11.35 gives:

(11.36)

Page 221: Process Equipment Design

0.24

0.20 0.10

q 0.16 0.08$&-$ 0.12 $0.06 b

z 0.08 0.04 tYf

0.04 0.02

00 20 40 60 80 100 120 140 160 lsoo

Values of fi in degrees

Fig. 11 .13. Constants Klo and K1l as a function of angle 8.

MA = SK [sin2 /?(l - + cos /3 + ip sin /3 - Q2)

- $0 sin /3 + $@ cos j3(2/3 + sin 2/3- 5 sin /3 + /3 cos fi)] (11.37)

Substituting Eqs. 11.36 and 11.37 into Eq. 11.33 gives:

M+ = & [cos +(sin2 /3 - Q@ sin Z/3 + +B2 cos 2fi)

+ 4 sin q5(+fi2 + + sin 2/3 - sin2 /3)+ $3 cos PGV + sin 33)

where

- sin /3(&b + 4 sin 2@ + %p cos 2p)] (11.38)

K = sin2 p - $32 - $3 sin 2/3 (11.39)

It should be noted that K and the quantities in paren-theses in Eqs. 11.36, 11.37, and 11.38 are functions of p andwill have the same value for all values of 4 for a given saddlesupport.

Values of P,/Q and MA/@ computed by the use of Eqs.11.36 and 11.37 are given by the diagrams of Fig. 11.13,and values of M,/Qr for various values of 4 and p computedby the use of Eq. 11.38 are given in Fig. 11.14.

As seen in Fig. 11.14(

note that p = 180 - i>

the maxi-

mum moment occurs at 4 = /3. Therefore

MB = KeQr (11.40)

where MB = maximum circumferential bending moment ininch-pounds

Values of KG were determined from Fig. 11.14 and plottedin Fig. 11.15. The use of Ks in the design of shells withring stiffeners in the plane of the saddle is treated in section11.6.

J11.30 Unstiffened Shell. When the saddles are locatedaway from the head so that the shell is free to deform, theshears tend to accumulate near the horn of the saddle sothat the actual maximum circumferential moment in theshell is less than the value obtained for M from Eq. 11.40.Zick reports (165) that this has been confirmed by strain-gauge measurements, which show that the effective lengthon top of the saddle that resists the moment is about four

Circumferential Stress at Horn of Saddle 211

times the radius of the shell or one half the length of thevessel, whichever is smaller. Therefore, the use of thevalue-of’the hypothetical moment MB given by Eq. 11.40wi

JI”render calculated stresses in accord with actual stresses.

11.3b Shel l St i f fened by Head. When the shell isstiffened by the head, the shear stresses are carried acrossthe saddle to the head, and then the load is transferredback to the saddle, as previously shown. As in the case ofthe unstiffened shell, the shears tend to concentrate nearthe horn of the saddle. Since the stiff members are rela-tively short, this transfer reduces the circumferential bend-ing moment still more; that is, the circumferential bendingmoment is smaller in the shell stiffened by the head than inthe unstiffened shell. This effect is introduced when thecircumferential bending moment is defined as:

MB = KTQr (11.41)

where K, = KG for values of A/R greater than 1. Forvalues of A/R less than 0.5, K7 = >/4K,+ ’

For design purposes the following equations are recom-mended:

f7 = -Q 3K7Q-__

4t(b + 1.56 6)ifLh 8R

2t2(11.42)

f7 Q 1 2 K7QR= - -___4t(b + 1.56 6) Lt2

if L < 8R ,1 (11.43)

where f, =

t=t=

b =

maximum circumferential combined compressivestress at the horn of the saddlevessel-shell thickness for unattached wear platecombined thickness of shell and wear plate whenwear plate extends r/10 inches above horn ofsaddle and saddle is located near head (A/R 50.5). (Otherwise, t equals thickness of shellonly).width of saddle, inches

0.080

0.06 ~fl

0.04 I I I II- i%t

It’= 180’2 *

I

0 20 40 60 80 100 120Value of 4 in degrees

Fig. 11 .14. V a r i a t i o n o f c i r c u m f e r e n t i a l m o m e n t a r o u n d s h e l l .

Page 222: Process Equipment Design

2 1 2 Design of Horizontal Vessels with Saddle Supports

= 120

110

100

901 ’ ’ ’ ’ ’ ’0 0.01 0.02 0.03 0.04 0.05 0.060.07 0.08 0.09 0.1 0.2

Ka and K,,

F ig . 11 .15 . Values of Ka and K13 as Al function of the saddle angle 0.

Zick suggests t.hat for multiple supports one should useL equal to twice the length of the load carried by the saddle.If L 2 8R, use the first formula.

Equation 11.42 takes into account the assumed value ofthe effective length of shell that resists the bending moment,as outlined previously. It also takes into considerationthe fact that the change in shear distribution reduces thedirect load at the horns of the saddle. This reduced directload is assumed to be equal to Q/4 for shells without ringstiffeners. Equation 11.42 also takes into account the factthat the effective length of the shell resisting this reducedload is limited by that portion which is stiffened by thecontact saddle. The assumed value for this effective length-is 0.72 drt on each side of the saddle plus the portiondirectly above the saddle.

Values of K, can be obtained from Fig. 11.16; they werederived from Fig. 11.15 through use of the assumptionslisted above.

For design purposes Zick recommends:

f7 s 1.25 X allowable tensile stress of material

LNote: when rings are placed in the plane of the saddle, alongitudinal bending stress occurs at the edge of the ring.This local stress would be 1.8 times the design ring stress.

11.4 ADDITIONAL STRESS IN HEAD USEDAS STIFFENER

The stiffness of the head is often utilized by locating thesaddles near the heads.

In the derivation of Eq. 11.20 it was shown that shearshave both tangential and horizontal components, as illus-trated in Fig. 11.8. When the saddle is close to the head,the horizontal components will cause tension across theentire height of the head as if the head were a flat disk.

The following analysis is based on the assumption thatthe head is a flat disk and that the maximum stress fsinduced in the head by the horizontal components of thetangential shears is 1.5 times the average.

The total load due to the horizontal shears will be (seeFig. 11.11):

/“0 sin 41 cos +lr d+l

0 rr

- s “0 s i n C#~Z ~0s & ~ ~L1! In- [ a - sin cy cos a!7r - ff -1 s i n cy cos a! 1 r &a

=p([sq!!!]y[ ?r ~.~~F25~%!%] [ ?q!!q:)

Q sin’ o(=-2 [

-43r--cu+sinacosa:

The maximum cross-sectional area of the disk will be 2rtb..

fs = g x 1.5

(sin’ a)/(?r - a + sin cy cos a)2 [--

~~-_ 1 . 52rth 1

01

fs=QKsrth

(11.44)

where

Ks = 8 sin’ o(~~7r - a + sin a! cos a 1 (11.45)

The value of Ks as a function of saddle angle 0 are givenin Fig. 11.17.

The stressfs is a tensile stress in the head and should be.combined with the stress induced by internal pressure.

0' I I I I J0 0 . 5 1 . 0 1 . 5 2 . 0 2.5

R a t i o o f A / R

F ig . 11 .16 . P l o t o f c i r c u m f e r e n t i a l b e n d i n g m o m e n t c o n s t a n t K:.

Page 223: Process Equipment Design

I I I nl I I I I I I I I i

1 1 0

1 0 0

0 0.25 0.50 0.75 1 . 0 0 1 . 2 5KS and K,

fig. 11.17. Valves of KS and Kp as o function of the saddle angle 0.

For design purposes the combined stress on the headshould he permitted to he 257, great,er than the allowabletensile stress of the material.

For cases involving negative pressures the head stressis compressive and the combined stress equals (+fs - headstress). This stress can usually be disregarded.

11.5 WEAR PLATES-RING COMPRESSION IN SHELL.OVER SADDLE

There are forces acting on the shell hand directly overthe saddle causing ring compression in the shell hand. Thetangential shear forces act over the arc from a! to ?r and aredirected toward the center, 0, because the saddle reactionsare radial. Fig. 11.18 shows these reactions with t,heassumption that there is frictionless contact between thesurfaces of the shell and the saddle. Taking momentsabout point 0 indicates that the ring compression at anypoint, A, is given by the summation of the tangential shearsbetween cr and 0, or

“(2 sin dwd41 = _ 4 Q sin 4; d&TP (T - LY + sin O( cos 01)

This summation of shears w-ill become a maximum whencp = n-. The above expression w-ill then become:

lOad = Q - ’ +:!I!=7r - ct + sin Q ! cos Q !

The width of shell that will resist. this load will be 1.56 42

Design of Ring Stiffeners 213

on each side of the saddle plus the width of the wear plate,if used.

l + cOs a!fg - o-~._pt(b + 1.56 dz) (

_ _7r - a + sin (Y cos a! )

QKY ~_l(b + 1.56 h)

(11.46)

where b = wear-plate widt,h, inches

Ky = ~1 + cos a____--~-

7r - a + sin LY cos cy(ii.47i

The values of Kg as a function of saddle angle 8 are pi\-cbnin Fig. 11.17.

The stress fs is especially important when concretnsaddles are used. It should also be checked for large-diam-eter vessels. For design purposes fs = 0.5 X compressionyield point of material.

The ring compression stress may be reduced by attachinga wear plate somewhat larger than the surface of the saddleto the shell directly over the saddle. The thickness, t, ma>be taken as the combined thickness of the shell and I hewear plate in the formula for fg provided the width of lheplate equals (b + 1.56 &).

Note: when the wear-plate thickness is added to the shellthickness as stated above, the thickness, t, in Eq. 11.42 canalso be taken as the combined thickness of the shell and thewear plate if the wear-plate width equals (b + 1.56 fi)and if the plate extends r/10 inches above the horn of asaddle near the head.

11.6 DESIGN OF RING STIFFENERS

In the case of thin-walled vessels or the case of saddleslocated away from the head (A/R > 34) the shell alonemay not resist the circumferential bending moment. Ringstiffeners are then attached to the shell to alleviate the ioaron the shell. The length, I, of the shell that will act witheach stiffener can be assumed (162) to be equal to 0.78 fiFig. 11.19 shows two recommended types of internal ring

$(

Fig. 11.18. loads and reactions on saddles (I 65). (Courtesy of American

Welding Society.)

Page 224: Process Equipment Design

214 Design of Horizontal Vessels with Saddle Supports

l--l ‘lxc=+-

Vassel s h e l l -. -7

Composite section A-A

P-I‘ t+-l~C

-7

Alternate compositesection A-A

Fig. 11 .19. E x a m p l e o f i n t e r n a l s t i f f e n i n g r i n g s .

stiffeners, and Fig. 11.20 shows corresponding types ofexternal ring stiffeners. An inside ring stiffener is mostdesirable from the strength standpoint because the maxi-mum stress is compression in the shell, which is reduced bythe internal operating or test pressure.

An external ring stiffener is not very desirable from theappearance standpoint and is even less desirable from thestrength standpoint because the maximum stress may beeither compression in the outer flange or tension in the vesselshell due to load Q.

The value of load Pt on the top of the ring can be devel-oped by a procedure identical to the one followed to obtainEq. 11.36. If the radius of the ring is taken equal as to r,then

Pt = $K [3 sin2 B - zp sin 2/l - B2 + */3’ cos 261n-

(See Eq. 11.36.)o r

where

Pt = KloQ (11.43)

Klo = GK (3 sin2 a - -# sin 2b - p2 + &3’ cos 2@)

(11.49)in which

K = sin2 /3 - ;/3’ - i sin 29 (11.39)

Values of P,/Q (equal to Klo) are plotted in Fig. 11.13for several values of the angle p.

Likewise, the circumferential bending moment at the topwill be:

MA = SK [sin2 /I(1 - t cos 0 + +3 sin /? - $12)

- $fi sin p + &l cos /3(2/3 + sin 2@ - 5 sin p + /? cos /3)]

o r

MA = KllQr (11.50)

where

K11 = bK [sin2 /3(1 - : cos p + +p sin p - $3’)

- &? sin p + $B cos 8(28 + sin 2p - 5 sin /3 + fi cos fl)]

Values of MJQr (equal to Krr) are plotted in Fig. 11.13for several values of the angle /3.

The maximum circumferential bending moment occurs atthe point at which #I = /l, as shown previously. Then byEq. 11.40

MB = KsQr

The value of KG is plotted in Fig. 11.15 as a fun&on ofthe angle of contact of the saddle, 8.

The moment due to the tangential shear at any point isgiven by Eq. 11.30, or

M, = 9 1n-

- cos 4 - 2” sin 4>

The moment due to the tangential shear at the horn ofthe saddle will be:

(Mb)~ = 2 (1 - cos /3 - $/3 sin /3)n-

orU%)B = K12Qr (11.51)

Table 11.1 gives values of Kl2 for different values of p.Consider now the section of ring from the vertical to the

horn of the saddle (at which the circumferential bendingmoment is a maximum) under the action of the forcesshown in Fig. 11.21.

The load on the ring at the horn of the saddle can bedetermined by taking moments about the center, 0.

Pt + Pdr + MB = MA + (M,)Bo r

PB = 1 [ M A + (MC)0 - MB] - Pt (11.52)r

Substituting Eqs. 11.50, 11.51, 11.40, and 11.48 intoEq. 11.52 gives:

PB = (Ku + K12 - Kc - KIO)Qo r

PB = KnQ (11.53)

Table 11.1 gives values for K13 for several values of fi.The stress on the ring will be the sum of the stresses due

to the load Pp plus the stress due to the circumferential

Bcl

Composite section B-B

Alternate composite section B-B

F i g . 1 1 . 2 0 . E x a m p l e o f e x t e r n a l s t i f f e n i n g r i n g s .

Page 225: Process Equipment Design

Table 11.1. Approximate Values of Constants for theEvaluation of Ring Stresses

P 0 Kll KlZ Km Ks K13135 9 0 0.03 0.286 0.132 0.082 0.102120 120 0.02 0.189 0.100 0.0528 0.056105 1 5 0 0.012 0.119 0.078 0.0316 0.021

9 0 180 0.006 0.0684 0.057 0.017 0.0004

moment MB, or

(11.54)

where A, = cross-sectional area of the ring stiffener, squareinches

Z- = section modulus of the ringC

When n rings are used,*

(11.55)

When the ring is attached to the outside surface of theshell adjacent to the saddle or to the inside surface of theshell directly over the saddle, the maximum combined stressis a compressive stress in the shell, fro being negative.

For design Zick recommends that the maximum combinedcompressive stress resulting from liquid load and pressureshould not exceed one half the compression yield point ofthe material (165). The maximum combined tensile stressresulting from liquid load and pressure should not exceedt k allowable tensile stress of the material.

$11.7 DESIGN OF SADDLES

The saddle must be capable of resisting the loads imposedby the vessel. Fig. 11.18 indicates the radial-load conditionacting on a saddle. To resist the horizontal componentsof these radial loads, the saddle must be designed to preventseparation of the horns of the saddle when the vessel iscarrying a full liquid load. Therefore, at the lowest pointof either a steel or concrete saddle a minimum cross-sectionalarea must exist sufficient to resist the horizontal componentsof the reactions. A summation of the horizontal compo-nents on one half of the saddle is given by:

F =- cos C j J + cos p

7r - /3 + sin fl cos flP sin (z - 4) d+

- cos #I + cos p= sin C#J dqt7r - p + sin fl cos /3

= Q[

-+ sin2 4. - cos 4 cos p?r - p + sin /3 cos /3 1

Tp

Q[

1 + cos p - & sin2 /3=7r - /3 + sin ,f3 cos p 1

= K14Q (11.56)

* When two circumferential stiffening rings per saddle areattached to the shell (one on each side of a saddle) the minimum

spacing between the rings should be 1.56 & inches, and themaximum spacing, R feet.

where

Zick’s Nomograph 215

K14 =1 + cos p - + sin2 p7r - p + sin /3 cos @

(11.57)

According to Zick the effective section of the saddle resist-ing this horizontal force should be limited to a distance ofr/3 below the shell at the lowest point of the saddle. Thissame restriction should also apply to the reinforcing steelcross section in a concrete saddle. The average designstress should be limited to two thirds of the allowable tensilestress of the saddle material.

For a saddle where 0 = 120”, /3 = 120”, and

K14 =1 + cos 120” - i sin2 120”

A - (120/180) + sin 120” cos 120’

Similarly, for a saddle where 0 = 150”,

K14 = 0.260

11.8 ZICK’S NOMOGRAPH FOR AID IN THE DESIGNOF VESSEL SUPPORTS

As an aid in the design of supports for horizontal vessels,Zick (165) has presented the nomograph shown in Fig. 11.22,which indicates the most economical locations and types ofsupports for vessels on two supports. The nomograph isbased on a liquid density of 42 lb per cu ft. If liquids ofdifferent densities are involved or different materials ofconstruction are to be used, a preliminary design may beobtained by use of the figure.

Large-diameter vessels constructed of thin-wall materialshould be supported near the closures provided that theshell can support the load between the saddles. Theclosures must be stiff enough to transfer the load to thesaddles. Thick-walled horizontal vessels are sometimes toolong to act as simple beams. According to Zick such vesselsshould be supported where the maximum longitudinal bend-ing stress in the shell at the saddles is about equal to themaximum longitudinal bending stress at the mid-span.The shell must be stiff enough to resist this bending andto transfer the load to the saddles. If the shell is unable

Fig. 11.21. F o r c e s o n r i n g s t i f f e n e r ,

Page 226: Process Equipment Design

1.

216 Design of Horizontal Vessels with Saddle Supports

Shell thickness, t, inches

AHI+-----L------~HH

Basis of designA-285 Grade C carbbn steelliquid wt = 42 lb per cu tf

Example shown by arrowsUse 120” saddles

Fig. 11.22. Location and type of support for horizontal pressure vessels on two supports by L. P. Zick (165). (Courtesy of American Welding Society.)

to provide the necessary stiffness, ring stiffeners should beadded near the saddles.

11.9 EXAMPLE CALCULATION OF STRESSES

Estimate the stresses induced by the supports in a vesseldesigned for storing lube oil and having the following designdata:

Lube-oil API gravity 16.5Working pressure 75 psiDesign pressure 9$ psiDesign temperature 500’ FMaterial SA-285, Grade CAllowable working stress 13,750 psiJoint efficiency 80%Corrosion allowance 36 in.Shell diameter (ID) 10 ftShell thickness (including corrosion allow-

ance) N in.Head thickness (including corrosion al-

J

lowance) K in.Tangent length 68 ft.“\Bearing-plate width, b 10 in.Heads elliptical dished,

2:l ratioemployed

To analyze the type of support to be employed for sup-porting the vessel, use is made of Fig. 11.22. By enteringthe figure with a shell-thickness value of s in. (with allow-ance for corrosion) and with a tangent length of 68 ft, it isfound that the resulting zone indicates that A/R S 0.5with 0 equal to 120” and that the head-plate thickness shouldhe checked. For this vessel R = 5 ft; therefore A will betaken as 234 ft (W R) in order to take advantage of thestiffening.effect of the head.

A sketch of the vessel with 120” saddle supports is shownin Fig. 11.23.

Following is the calculation of the weight of one head ofthe vessel. Fig. 5.12 the required blank diameter for3ia20-in-diameter elliptical dished head -t-i52 iQThe weight of the plate is:

~(152)‘(0.75)(490;-.-(4) (1728)~

= 3850 lb per headY,. 1 $I;

R q -. ’

’The weight of two heads is 7700 lb. $1The shell weight is:

410) (68) (0-W (490’) = 65-~I,5oo lb

\12

/:

Page 227: Process Equipment Design

The volume of one head is given by Eq. 5.14

‘-G.= 0.000076 Di3 \\_ -_

= 0.000076(120)3J

= 131 cu ft per head

The total volume of the two heads is 262 cu ft.The volume of the shell is:

n-y (68) = 5340 cu ft

The total volume is:

262 + 5340 = 5600 cu ft

The densit’y of the fluid is 59.7 lb per cu ft.

The total weight of the fluid (vessel full) = (59.7)(5600)

= 334,000 lb

The weight of the vessel and its contents= 334,000 + 65,500 + 7700

Therefore= 407,200 lb

andQ (load per saddle) = 204,000 lb

H A 2.5 = 0 0368-=-L L=68 .

Maximum longitudinal bending stress :The saddles are located close to the heads of the vessel.

The maximum longitudinal bending stress exists at thecenter of the span between the saddles and is given byEq. 11.8:

vf2 = +=zQL

7rr2(t - c)From Fig. 11.5

K2 = 0.82therefore .

= +4920 psi

Since t/r = 0.625/60 = 0.0104 > 0.005, the compressionstress is not a f&&r in the design.

f, = longitudinal pressure stress

By Eq. 3.15

(4) (0.80) (0.625) 5400 psi

4 fz +f, = 10,320 < 0.8faaoW. = 11,000 psi

Tangent ial shear s tress :the tangential

shear stress in the head is given by Eq. 11.127.

fs _ Q!

4th - c)

ji _

Example Calculation of Stresses

From Fig. 11.10 and with 0 equal to 120”

K5 = 0.88therefore

217

fs = (204,000)(0.88) = 4780 psi(60)(0.625)

The tangential shear stress in the shell is given by Eq.11.28; and as &hell equals thead, the shear stress in the headequals the shear stress in the shell.

fs = fi = 4780 psiP

Circumferential s tress at horn of saddle :Since the shell is stiffened by the head and since L > 8R,

Eq. 11.42 gives the circumferential stress at t.he horn ofthe saddle.

From Fig. 11.16, A/R = 0.5, and 0 = 120°; therefore

K, = 0.013

For the condition in which no credit for t is taken for thewear-plate thickness,

f, =-204,000

(4)(0.625)[10 + 1.56 d(60)(0.625)]_ (3)(0.013)(204,000)

(2) (0.625) 2= -4180 - 10,180 = -14,360 psi

The maximum permissible stress equals:

12,650 X 1.25 = 15,800 psi

As the stress f7 is less than the allowable stress, it is notnecessary to take credit for t,he wear plate.

L - - - - z- - - - - - -I rT-r

Fig. 11.23. Sketch of vessel in example calculation.

Page 228: Process Equipment Design

218 ‘.’ Design of Horizontal Vessels wtth Saddle Supports

Additional stress in head used as sti f fener :The additional stress induced in the head when it is used

as a stiffener is given by Eq. 11.44.

js zz --QL

4th - c)

From Fig. 11.17 and with ~9 equal to 120’

thereforeKg = 0.40

js

= (204,000)(0.40)(60)(0.625) = 2180 psi

For an elliptical dished head (K = 2.0) the maximumpressure stress may be taken as equal to the circumferentialhoop stress in the shell (see Chapter 7 and Eq. 7.57). FromEq. 7.57

fP =p[Vd + 0.2(t - c)]

2E(t - c)By Eq. 7.56

v = i(2 + k2) = 1.0

Using a one-piece head, we find that E = 1.0.

The maximum combined stress in the head equals:

2180 + 8650 = 10,830 psi

The maximum allowable stress in the head is:

(12,650)(1.25) = 15,800 psi

Ring compression in shell over saddle :The stress in the shell band directly over the saddle is

given by Eq. 11.46.

jg = QKgt(b + 1.56 &)

From Fig. 11.17 and with 8 equal to 120”

K9 = 0.76(204,000) (0.76)

jg = (0.625)[10 + 1.56 1/(60) (0.625)]= 12,700 psi

The allowable stress equals the yield point divided by two.According to reference 67 the yield point of SA-285, Grade Csteel equals 30,000 psi. Therefore

allowable stress =30,000__ = 15,000 psi

2j

P= 90[(1)(120) + (0.2)(0.625)1

(2)(l) (0.625)= 8650 psi

P R O B L E M S

1. Recalculate the stresses in the vessel described in the example calculation (see Fig. 11.23)if the shell and head thickness is r+{a in. rather than K in.

3. In reference to Fig. 11.1, the horizontal storage tank is 12 ft, 0 in. in inside diameter x 120ft, 0 in. long. The vessel is used to store butane at 100 psi and 400” F and has elliptical dishedheads. The heads and shell have a thickness of rgs in. with a >Q-in. corrosion allowance.Assume the saddles are located 80 ft apart, 8 equals 120”, wear-plate width equals 10 in., wear-plate thickness equals N in., and the wear-plate extends 6 in. above the horn of the saddle.Assume the vessel is fabricated of ASTMA-285, Grade C steel with a joint efficiency of 80%,and calculate the stresses in the shell and head for the case in which internal stiffeners are notused.

3. Redesign the storage vessel described in problem 2 for gasoline storage at 50 psi and400” F using internal stiffeners in the plane of the saddle (see Fig. 11.19), ASTMA-283, Grade Csteel, a joint efficiency of 80%, and no corrosion allowance.

4. Design the concrete saddles for the vessel described in problem 2, and specify the area ofreinforcing steel if a 1:2>4:3js concrete mix is used (see Table 10.1).

-- -’ - --. - \ \ \I /

Page 229: Process Equipment Design

C H A P T E R

DESIGN OF FLANGES

AA “”variety of attachments and accessories are essential

to vessels. These items include flanges for closures, nozzles,manholes, and handholes; and flanges for two-piece vessels.Supports, platforms, and ladders are examples of othertypical accessories. Flanges may be used on the shell of avessel to permit disassembly and removal or cleaning ofinternal parts. Flanges are also used for making connec-tions for piping and for nozzle attachments of openingslarger than lj+in. nominal pipe size. Threaded-pipe con-nections such as couplings and half couplings are used foropenings smaller than l$d-in. pipe size. Figure 12.1 showsa sectional view of an autoclave with a shell flange on thevertical vessel and nozzle flanges on the discharge and feedattachments.

12.1 SELECTION OF STANDARD FLANGES

A great variety of types and sizes of “standard” flangesare available for various pressure services. The flangesdesignated as “American Standards Association (ASA)B16.5-1953” are used for most steel pipe lines over l>S-in.nominal pipe size; therefore these flanges are extensivelyused for nozzles and other attachments to vessels (187).These flanges are normally forged from ASTM A-181 andASTM A-105 carbon steels. Forged-alloy-steel flanges arealso available. These flanges are called “companionflanges” because they are almost always used in pairs.Although they are usually manufactured from forged steel,cast-iron companion flanges may be used for low-pressureservice. Forged-steel flanges are manufactured in thefollowing standard types for all pressure ratings:

J1. Welding-neck.2. Slip-on.3. Screwed.4. Lap-joint.5. Blind.

Other standard types such as reducing flanges, socket-welding flanges, orifice flanges, and nonstandard flanges arealso available for certain ratings.

12.la Welding-neck Flanges. Figure 12.2 shows asectional view of one and lists the dimensions of severalstandard X0-lb welding-neck flanges from 55 to 24-in.nominal pipe size. Welding-neck flanges differ from othertypes in that they have a long, tapered hub between theflange ring and the weld joint. This hub provides a moregradual transition from the flange-ring thickness to thepipe-wall thickness, thereby decreasing the discontinuitystresses and consequently increasing the strength of theflange. This type of flange is preferred for extreme serviceconditions such as: repeated bending from line expansion orother forces, wide fluctuations in pressure or temperature,high pressure, high temperature, and subzero temperature.These flanges are recommended for the handling of costly,flammable, or explosive fluids, where failure or leakage of aflange joint might bring disastrous consequences.

12.1 b Slip-on Flanges. Figure 12.3 shows a sketch ofa standard 150-lb slip-on flange and gives dimensions forsuch flanges of $4 to 24-in. nominal pipe sizes. The slip-ontype of flange is widely used because of its greater ease ofalignment in welding assembly and because of its low initialcost. The strength of this flange as calculated from internal-pressure considerations is approximately two-thirds that ofa corresponding welding-neck type of flange. The use of

219

_-._-_ _____ ~--, -- --...- --;-.: __. ‘-\- __.. r-- -- - n-7

Page 230: Process Equipment Design

220 Design of Flanges

BAdapter block with

1 gage 0 to 18 00 lb0 1 rupture-disc

assembly for1150 lb at 300°F

1 gas-inlet valve.

*

Thermometer well

-Coil

;P If<ig.$---- Insulation

b o x

Fig. 12.1. Sectional view of electrically heated stainless steel autoclave for service at 900 psi and 375” F (167). (Courtesy of Blaw-Knox Co.)

this type of flange should be limited to moderate serviceswhere pressure fluctuations, temperature fluctuations, vibra-Cons, and shock are not expected to be severe. Thefatigue life of this flange is‘approximately one-third thatof a welding-neck flange.

12.1~ Lap-joint Flanges. Figure 12.4 shows a sectionalview of a steel lap-joint flange and lap-joint stub. Lap-joint flanges are usually used with a lap-joint stub. Thecombined cost of the two parts is about 30% greater thanI he cost of a welding-neck flange of the same size and rating.These flanges have about the same ability to withstandpressure without leaking as the slip-on flange, which is less

than that of the welding-neck flange. Also, these flangeshave the disadvantage of having only about 10% of thefatigue life of welding-neck flanges. For these reasons,these flanges should not be used for connections wheresevere bending stresses exist. The principal advantage ofthese flanges is that the bolt holes are easily aligned, andthis simplifies the erection of vessels of large diameter andunusually stiff piping. These flanges are also useful in caseswhere frequent dismantling for cleaning or inspection isrequired, or where it is necessary to rotate the pipe byswiveling the flange.

Page 231: Process Equipment Design

Rating of Standard Flanges 221

are also used to block off the ends of piping and valves.In this application a valve followed by a blind flange isfrequently used at the end of a line to permit additions tothe line while it is “on st,ream.” Blind flanges absorb highbending stresses but do not have to absorb stresses causedby t,hermal expansion or by the weight of the piping system.

12.2 RATING OF STANDARD FLANGES

Standard flanges are rated as 150, 300, 400, 600, 900,1500, and 2500-lb flanges. These ratings correspond toservice pressures at specified service temperatures as givenin Table 12.1. For example, a 150-lb flange has a ratingof 150-psi service pressure at 500” F and a rating of 230 psiat 100” F. A 2500-lb flange has a rating of 2500 psi at800” F and a rating of 5000 psi at 100” F.

12.ld Screwed Flanges. Figure 12.5 shows a sectionalview of a screwed flange. This flange can be rapidly con-nected to threaded pipe without welding. The threadedconnection is susceptible to leakage under almost any typeof cyclic operation. Applications which involve bendingor thermal cycles should not employ this type of flange.These flanges may be used to advantage in extremely-high- ,/pressure service with alloy steel, which has the requiredstrength for the pressure service but which is not easilywelded.

12.le Blind Flanges. Figure 12.6 shows a sectionalview of a standard 150-lb blind flange and gives the dimen-sions for such flanges of $5 to 24-in nominal pipe size. Theseflanges are used extensively to blank off pressure-vesselopenings such as handholes and inspection ports. They

AMERICAN STANDAHU-kki B16~-1939 FORCED AND ROLLEUSTEEL--ASI'M A 181Out.side Diameter Diameter Inside

Outside Thickness Diameter of of Hub Diameter of Drilling TemplateNominal Diameter o f o f Hub at Point Length Standard

- Approx.Yo. Diam.

Pipe of Flange, Raised at o f t,hrough Wall ..o fSize Flange Minimum Face Base Welding Hub Pipe Holes--__l_____-

A T R E K L B

WeightEach,

PoundsInches

l/i%i

1

1%1'35622%2%

0.841.051.321.661.90

0.62 40.82 41.05 41.38 41.61 4

, 1%1%

2wi33%4

568

1 01 2

14 2116 231/i18 252a 27%24 32

677%8369

101 113yi1619

7%68%

10%12%15

WAWi212327%

2.382.883.504.004.50

5.566.638.63

10.7512.75

14.0016.0018.0020.0024.00

2.07 42.47 43.07 43.55 84.03 8

5.05 86.07 87.98 8

10.02 1212.00 12

13.25 1215.25 1617.25 1619.25 2023.25 20

68

101215

1924395280

102127140170260

Fig. 12.2. Standard 150~lb steel welding-neck flanges (168). (Courtesy of Taylor Forge and Pipe Works.)

Page 232: Process Equipment Design

2 2 2 Design of Flanges

AMERICAN H~aao~m -ASA D16s-1939out-

Out- Thick- Side IXam-side r.ess Diam- eta

Fonae~ AND ROLLED STEEL-ASTM A 181

5 1 0 ‘946 75,in 6:<a 17,lfJ 5.66 8 74, 3,i 8Yz 1 56 11 1 855 78//1‘S l%O 6.72 8 ‘/b ?‘a 941 198 13jh 1% 10% S’>(a 1 % 8.72 8 ?/d $‘r 11% 30

10 1 5 l%o 1236 1 2 l’>/la 10 88 1 2 1 74 14$1 431 2 1 3 lf’r 1 5 1438 2% B 12.88 1 2 1 7/6 1 7 6 4

14 2 1 1 % 16% 1 5 % 2% 14.19 12 1% 1 18% 8516 23% l?,io 18ji 18 2 4 1 16.19 16 1% 1 21y1 9 31 8 2 5 l%a 2 1 1 9 % 2’4ie 18.19 1 6 1 % 1% 2 3 % 1 2 02 0 27Si l’$(e 23 22 2% 20.19 20 1H 1% 25 1552 4 3 2 1% 27ti 26411 3 % 24.19 20 1% 1% 2955 210

Fig. 12.3. Standard 150-lb forged sl ip-on f langes (168) . (Courtesy of

Taylor Forge and Pipe Works.)

Fig. 12.4. Sectional view of steel lop-joint flange ond lap-joint stub.

52.3 STANDARD FLANGE FACINGS

Standard f langes are available with a variety of machinedfaces, as shown in Fig. 12.7 (corresponding dimensions aregiven in Table 12.2).

Steel flanges with a raised face are extensively usedbecause of the simplicity of the design and because theyhave been proved adequate for average service conditions.For severe service involving high pressure , high temperature ,thermal shock, or cycl ic operat ions , this type of f lange fac ingmay not be satisfactory. Flanges with ratings of 150 and300 lb have a >is-in.-high raised face, and flanges havinghigher ratings have >i-in.-high raised faces. The raisedface is machined with spiral or concentric grooves approxi-mately 364 in. deep with about >s2-in. spacing. The edgesof these grooves serve to deform and hold the gasket.Flat-ring composition gaskets normally are used having a

width equal to the width of the raised face whereas flatmetal gaskets may be used having a width equal to thatused with the large tongue-and-groove type of face.

The male-and-female facings have the advantage of con-fining the gaskets thereby minimizing the possibility ofblowout of the gaskets. They have the disadvantage thatthe two mating flanges are not identical. For this reasonthese f langes are not as widely used on pipe-l ine connectionsas are the raised-face flanges. They are used extensivelyon heat exchangers, and sometimes for manholes and as endflanges. Male-and-female facings have another disadvan-tage compared to tongue-and-groove flanges in that theyoffer no protect ion against forc ing the gasket into the vessel .The male-and-female facings are standardized with ax6-in-deep recess on the female face and a j/4.-in.-highraised face on the male part. The gasket surfaces areusual ly smooth f inished as the outer diameter of the femaleface serves to locate and retain the gasket. The width ofthe face is so large in the case of large male-and-femalegasket-contact surfaces that full-face metal gaskets cannot

Fig. 12.5. Sectional view of a screwed flange.

6’

AMERICAN STANDARD~ASA B16~-1939FORGED AND ROLLED STEEL- ASTM A 181

out-Out- Thick- side

Diam- of et.er __D r i l l i n g T e m p l a t e Approx.

Diam. WeightB o l t E&l,

C i r c l e Pounds

568

1012

10 15;s 7X611 1 asi13% 1% lO>i;16 1x5 123;19 13; 15

14 21 1% 163;I 6 2336 IN6 18%18 25 1916 212 0 27% l’Me 232 4 32 1% 27%

N o . Diam.

88a

1212

1216162 02 0

479

1317

2 02 64 570

110

3111702092 7 2411

Fig. 12.6. Standard 150-lb bl ind f lange (168) . (Courtesy of Taylor

Forge and Pipe Works.)

Page 233: Process Equipment Design

Table 12.1. Pressure Ratings for

Carbon-steel Flanges (168)

With Standard Facings (Other than Ring Joints)for Water, Steam, and Oil Service

Primary Service-p r e s s u r e R a t i n g s(lb per sq in.) 150 300 4 0 0 6 0 0 9 0 0 1500 2506

Maximum Hydro-s t a t i c - s h e l l - t e s tPressures*( l b per sq in.) 350 900 1200 1800 2700 4500 7500

Serv i ceT e m p e r a t u r e s Maximum Nonshock Service-pressure

Fluid (dee F) Ratings (lb per sq in.)100 2 3 0 600 800 1200 1800 3000 5000150 2 2 0 590 785 1180 1770 2950 49152 0 0 2 1 0 580 770 1160 1740 2900 4830250 2 0 0 570 760 1140 1710 2850 4750

300 190 560 740 1120 1680 2800 4660

1 350 180 550 725 1095 1645 2740 4565400 170 5 4 0 710 1075 1615 2 6 9 0 4475

Wd‘3-. 4 5 0 160 525 700 1050 1580 2630 4380St-;o r O i l 15ot 500 665 1000 1500 2500 4165

550i 500

140 475 630 950 1420 2370 3950600 130 445 590 890 1330 2220 37006 5 0 120 415 550 830 1240 2070 3450

700 110 380 500 760 1140 1900 3160750 100 340 450 6 8 0 1020 1700 2830800 92 300t 4oot 600t 9 0 0 t 15OOt 25OOt850 82 245 330 4 9 0 740 1230 2050

O i l 900{ 950

70 2 1 0 2 8 0 4 2 0 630 1050 1750OdY 55 165 2 2 0 330 495 8 2 5 1375

1000 40 120 160 240 3 6 0 6 0 0 1000* T e m p e r a t u r e o f t e s t w a t e r s h a l l n o t e x c e e d 125O F .t P r i m a r y s e r v i c e - p r e s s u r e r a t i n g s .

be used because of the excessive tightening loads requiredto seat the gasket.

The tongue-and-groove type of gasket face has advantagesand disadvantages similar to those of the male-and-femaletype of gasket face. The presence of retaining metal oneither s ide of the gasket gives protect ion against deformingsoft gaskets into the interior of the vessel; this is an advan-tage over the male-and-female type of face. Also, thegasket i s less sub ject to eros ive or corros ive contact wi th thef luid in the vesse l . In service the tongue is more likely tobe damaged than the groove; therefore the tongue should beplaced on the part that can be most ‘easily removed fromthe vessel. This type of facing is standardized for bothsmall and large f langes. The small area of the tongue-and-groove surface provides the minimum area that it is advis-able to use with f lat gaskets . Therefore, this type of facingprovides the minimum bolting load for compressing a flatgasket .

One advantage of the ring-joint type of facing is that itoffers the greatest protect ion under severe service condit ionsor with the use of hazardous fluids. This type of flange iswidely used in petroleum, petrochemical , and high-pressureservice . Close tolerances and high standards of machiningare required, and as a result, this type of flange is seldomused in nominal s izes larger than 36 in. Another advantagelies in the fact that the internal pressure acts on the ring toincrease the sealing force on the joint. The fact thatmating f langes are ident ical reduces the problems of s tockingand of assembly. Also, because the gasket-contact surfaces

Nonstandard Flanges 2 2 3

are below the flange face, the gasket-contact faces are pro-tected from damage. The main disadvantage of this typeof facing is the high cost of manufacturing it; this is themost expensive gasket face.

12.4 NONSTANDARD FLANGES

Nonstandard flanges are available in sizes from 26 in.to 96 in. These flanges are fabricated by rolling a hotannular blank, as shown in Fig. 12.8. Nonstandard flangesare available in a variety of ratings. Typical ratings forflanges supplied by one manufacturer are 50, 125, and 250psi. Figure 12.9 shows a sectional view of a large-diameterwelding-neck flange rated at 50 psi (at 100” F service tem-perature) and, gives the dimensions for such f langes.

Raised face

Large male and female

Large tongue and groove

Lap joint

Small male and female

Small tongue and groove

Ring-type joint

Fig. 12.7. American Standard Range facings (168). (Courtesy of TaylorF o r g e a n d P i p e W o r k s . )

Page 234: Process Equipment Design

224 Design of Flanges

Table 12.2. American Standard Flange Facings (168jFor 150, 300, 400, 600, 900, 1500, and 2500-lb Flanges

Outside Diameter3

NominalPipeSize,

Inches?4N

11%

RaisedFace,Lap

Joint,LargeMale,and

LargeTongue5

R1%l’.YiS2255

1%22%3

8 10%1 0 12%1 2 1514 16>/4

16 18%1 8 212 0 2 32 4 27%

HeightInside Inside Raised Face,

Diame- Outside Diameter3Diame- Large and

ter of Large ter of Kaised Small MaleLarge Female Large Face, and Tongue,and and and 250 and 400, 600, Depth of

Small Small Small Large Small Small Small 300-lb 900, 1500, GrooveMalea.5 Tongue5 Tongue3*5 G r o o v e 5 Female4f5 Groove5 Groove3v5 Stand- and 2500-lb or

s i;2x2 1%'X6 11!46

1416 1%

1% 2%

1% 2%2% 3%21xfi 3%3x6 4%

wi6 5%4x6 wi65% 619466% 8

8% 1 01ojg 1 212% 14?413% 15%

15% 17%17% 203419% 2 223% 26>i

iJ11x61%1%

9%11>/413%14%

16%19%2 125M

W X1x16 I2%21%

2x6 1%2%6 1x6

w6 I'%6%S 2x64%6 2%5x6 3%

5x6 3%6% 4%7% 5x6

8946 me

101$i6 8x612%6 10x6

w6 w616546 13%6

18756 wi6%iS 17lg,j23x6 19%627x6 29x6

Y 21x6 '%61% 1%I'%6 17i62x6 113,ic

ardsl Standards2$4s/454,ki

2%G 23/16 l/i 6 ?43x6 21936 %6 %i3%6 3x6 %6 K41x6 4Ke '46 M

411d,j5%6%7x6

14 6'% 6

f4 6'4 6

lo?46 9x6 '/i' 612x6 11x6 %s14x6 13x6 %6l5%6 14l146 %6

17l3i6 1@?46 %620x6 19Ns x6=?iS wi6 ?4626x6 25x6 X6

Ns946ws

%6

?'i6

946%6%6

1 Regular facing for 150 and 300-lb steel flanged fittings and flange standards is a x s-in. raised face included in the minimumflange thickness. A x6-in. raised face is also permitted on the 400, 600, 900, 1500, and 2500-lb flange standards, but it mustbe added to the minimum flange thickness.

* Regular facing for 400, 600, 900, 1500, and 2500-lb flange standards is a x-in. raised face not included in minimum-flanpe-thickness dimensions.

3 A tolerance of plus or minus 0.016 in. (364 in.) is allowed on the inside and outside diameters of all facings.4 Care should be taken in the use of joints of these dimensions (they apply particularly on lines where the joint is made

on the end of pipe) to insure that pipe used is thick enough to permit sufficient bearing surface to prevent crushing the gasket.Threaded companion flanges are furnished with plain face and are threaded with American Standard Locknut Thread.

6 Gaskets for male-female and tongue-groove joints shall cover the bottom of the recess with minimum clearances takingnto account the tolerances prescribed in note 3.

12.5 GASKETS AND THEIR SELECTION

Leakproof metal-to-metal surfaces in which gaskets arenot used are difficult to fabricate even by use of very accur-ate machined surfaces. Irregularities in clearances of onlya few millionths of an inch will permit the escape of a fluidunder pressure. The function of a gasket is to interposea semiplastic material between the flange facings, whichmaterial through deformation under load seals the minutesurface irregularities to prevent leakage of the fluid. Theamount of flow of the gasket material that is required toproduce a tight seal is dependent upon the roughness of thesurface. The amount of force that must be applied to the

gasket to cause the gasket to flow and seal the surfaceirregularities is known as the “yield” or “seating” force.This force is usually expressed as a unit stress in pounds persquare inch and is independent of the pressure in the vessel.Thus, this yield stress represents the minimum load thatmust be applied to the gasket to seat it even though verylow pressures are used in the vessel. Usually the gasket isseated by tightening the bolt load on the flanges prior tothe application of the internal pressure in the vessel.

Upon the application of the internal pressure in the vessel,an end force tends to separate the flanges and to decrease theunit stress on the gasket. Figure. 12.10 shows the threemajor forces acting on the gasket.

Page 235: Process Equipment Design

Gaskets and Their Selection 225

the contact surface area increases. Serrated gaskets areuseful where soft gaskets or laminated gaskets are unsatis-factory and the bolt load is excessive with a flat-ring metalgasket. Smooth-finished flange faces should be used withserrated gaskets.

Corrugated gaskets with asbestos filling are similar tolaminated gaskets except that the surface is rigid with con-centric rings as in the case of serrated gaskets. Corrugatedgaskets require less seating force than laminated or serratedgaskets and are extensively used in low-pressure liquid andgas service. Corrugated metal gaskets without asbestosmay be used to higher temperatures than those with asbestosfilling and are extensively used in sealing water, steam, gas,oil, and acid and other chemicals.

Two standard types of ring-joint gaskets are available forhigh-pressure service. One type has an oval cross section,and the other has an octagonal cross section. These ringsare fabricated of solid metal, usually soft iron, soft steel,monel, 4-6% chrome, and stainless steels. The alloy-steelrings should be heat treated to soften them. For low-temperature service plastic rings may be used for corrosionresistance and as a means of electrically insulating the flangejoint.

There is a considerable possible choice of gasket materialin many applications. The decision as to which gasketmaterial is to be selected is often based upon the requiredgasket width. If the gasket is made too narrow, the unitpressure on it may be excessive. If the gasket is made toowide, the bolt load will be unnecessarily increased. A rela-tionship for making a preliminary estimate of the propor-tions of the gasket may be derived as follows:

(Gasket seating force) - (Hydrostatic pressure force)= (Residual gasket force)

The residual gasket force can not be less than that requiredto prevent leakage of the internal fluid under operatingpressure. Therefore

Leakage will occur under pressure if the hydrostatic endforce is sufficiently great that the difference between it andthe bolt-load force reduces the gasket load below a criticalvalue. Also, it may be possible with too low a contactpressure on the gasket for the gasket to be blown out bythe internal pressure. The ratio of the gasket stress, whenthe vessel is under pressure, to the internal pressure istermed the “gasket factor.” The gasket factor is a propertyof the gasket rnxd the construction and is independ-ent of the internal pressure over a wide range of pressures.In selecting the proper gasket for an existing closure, one ofthe first steps should involve the determination of the totalamount of force necessary to make the gasket yield and tomaintain a tight seal under operating conditions.

Figure 12.11 shows sectional views of some common typesof gaskets and lists the gasket factor, m, and the minimumdesign seating stress, y, for each type of gasket (11). Figure12.11 also indicates the recommended facings for the varioustypes of gaskets. The effective width of the gasket, b, forvarious types of facings is shown in Fig. 12.12 (11).

Flat-ring gaskets are widely used wherever service condi-tions permit because of the ease with which they may becut from flat sheets and installed. They are commonlyfabricated from such materials as rubber, paper, cloth,asbestos, plastics, copper, lead, aluminum, nickel, monel,and soft iron. The gaskets are usually made in thicknessesof from 5d4 to $4 in. Paper, cloth, and rubber gaskets arenot recommended for use above 250” F. Asbestos-com-_..., . ..-.cposition gaskets may be used up to 650” F or slightly higher,and ferrous and nickel-alloy metal gaskets may be used upto the maximum temperature rating of the flanges.

Laminated gaskets are fabricated with a metal jacket anda soft filler, usually of asbestos. Such gaskets can be usedup to temperatures of about 750” F to 850” F and requireless bolt load to seat and keep tight than solid-metal flat-ring gaskets.

Serrated metal gaskets are fabric;lted of solid metal andhave concentric grooves machined into the faces. Thisgreatly reduces the contact area on initial tightening,thereby reducing the bolt load. As the gasket is deformed,

F i g . 1 2 . 8 . Rolling of a large flange (168). ( C o u r t e s y o f T a y l o r F o r g e a n d P i p e Works.)

J-m--~\, r - -

Page 236: Process Equipment Design

226 Design of Flanges

FORGED AND ROLLED STEEL ASTM A 18150 Lb Pressure at 100” F-Welding-neck Type

Outside OutsideDiameter Thickness Diameter Compressed-

of ofLength Diameter Drilling Template Approx.

Taper-Hub through of No. Diam.Size Flange Flange

of~~~~d Asbestos- WeightGasket Diameters Hub Bore of

Inches A T R Size E K L B Holes 12 6 3 1 % 1% 2846 273C x 28% 2734 26% 3 26

3 3 % 13-i 3036 29% x 30% 2934 2 8 % 3 28283032

35563 8 %

32% 31% x 32% 31% 30% 3 3035 33% x 35 33% 32% 3% 32

32363636

Each,Pounds

98105112140

34 Wi36 42x42 4948 55

37 35% x 37 35% 34% 3 % 3439 37% x 39 37% 36% 3% \ 364 5 % 44%x45% 43% 42% 335 425 1 % 50% x 5154 49vh 483/ 4 3% 48

40404852

54 61?460 67%66 7472 80

2 6 332 8 3530 3732 39%

1%1%

1%1%1%1%

1%1%1%2%

1%1%1%1%

15-i1%1%1%

5 7 % 56% ~57% 56 54% 4 54634i 6234 x 63% 62 60~$ 4% 607036 68Xx 7036 68 67 4% 667636 74%~ 76Ji 74 73 5% 72

50 Lb Pressure at 100’ F-Slip-on Type30 29% x 30 28% 28 2% %l?da32 31% x 32 30% 30 2% 28%34 33% x 34 32% 32 2% 3w636% 3534 x 36% 34% 3444 2% 3wiS

-64727280

32363640

34 4 1 %36 43%42 5048 56

54 6235 1%60 68% 1%66 7 5 % 1%7 2 8134 2

3 8 % 3736 x 38% 36% 36% 2% 34l%640% 39%x40% 38% 38% 2% 361X,54 6 % 45% ~46% 44% 44% 3-i 42l9i652% 51%x52% 50% 50% 2% 48%

59 57% x 59 57% 56% 3% 55%665 63% x 65 6 3 % 6 2 % 3 % 613167 1 % 69Xx71x 69% 68% 4 67%7 7 % 75Xx77X 75% 74% 4 % 734iS

404 44856

%--7280

of BoltHoles Circle

1 29%1 3 1 %1 33941% 3 6 %

1% 3 8 %1% 4 0 %1% 4 6 %1% 52%

1% 591M 651% 7 1 %1% 77%

1 311 331 351% 3 7 %

1% 3 9 %1% 4 1 %1% 4 7 %1% 5 3 %

1% 60%1% 661,/,1% 7 31% 7 9

149157209241

312398556705

122140148171

181191234269

335451591728

Fig. 12.9. Nonstandard large-diameter flanges (168). (Courtesy of Taylor Forge and Pipe Works.)

where y = yield stress, pounds per square inch(see Fig. 12.11)

m = gasket factor (see Fig. 12.11)p = internal pressure, pounds per square inch

d,, = outside diameter of gasket, inches4 = inside diameter of gasket, inches

In Eq. 12.1 it is assumed that the hydrostatic force extendsto the outer diameter, 4, of the gasket and that all thehydrostatic force is utilized in relieving the gasket load thatexisted prior to application of the internal pressure. Theseassumptions disregard elastic deformation of the bolts,gasket, and flanges, but the relationship is a useful one forthe initial proportioning of the gasket. For convenience

Eq. 12.1 may be rewritten as follows:

-_--do Y-pm&= - P(m + 19

(12.2)

In the case where it is desirable to retain the gasketmaterial selected and to decrease the gasket width, a gasketseating stress greater than y may be used with certainreservations. If the seating stress greatly exceeds y. thegasket may be crushed, or a ductile, unrestrained gasketmay be squeezed out between the &urge faces. In generalthe use of seating stresses exceeding y should be limited tosolid-metal gaskets in tongue-and-groove joints.

Page 237: Process Equipment Design

Design of Special Flanges 2 2 7

12.6 OPTIMUM SELECTION OF BOLTS FORSPECIAL FLANGES

4

The maximum bolt load will be the greater of the two fol-lowing forces: the force required to seat the gasket and theforce required to withstand the internal pressure and main-tain the gasket-factor pressure (mp) at the same time.After the greater of these two forces has been determined,the required bolting area of bolting steel may be determinedby dividing the maximum force by the allowable bolting

,’

.-

stress. A number of combinations are possible in providingthe required bolting area. In general a larger number ofsmaller-sized bolts will provide the same bolting area as alesser number of larger-sized bolts. The minimum boltspacing based on wrench clearances limits the number ofbolts that can be placed in a given bolt circle. The maxi-mum bolt spacing is limited by the permissible deflectionthat would exist between flanges. If this deflection isexcessive, the gasket joint will leak. Taylor Forge recom-

1 mends (188) the following empirical relationship for maxi-mum bolt spacing:

\

’ \Bs(max) = 2d + 5 (12.3) Fig. 12.10. The three major forces acting on a gasket (169).

where Ba(max) = maximum bolt spacing for a tight joint,inches. d = bolt diameter, inches

r t = flange thickness, inchesm = gasket factor (see Fig. 12.11)

1,

Before the bolting calculations can be completed, thediameter of the vessel, B, and the value of gr (hub thickness)must be known.

1 I,

I

: ‘,

The wrench clearances limit the minimum bolt distancefrom the hub. ~%n general it is desirable to use a minimum-diameter bolt circle and an even number of bolts, preferablya multiple of four. The minimum diameter of a bolt circlemay be determined by setting up a table in which the num-ber of bolts required, the root area, the preferred bolt spac-ing, B,, and the radial spacing, R, are tabulated as functionsof bolt size (see Table 10.4 for root area, B,, and R). Theminimum bolt-circle diameter will be either the diameter nec-essary to satisfy the radial clearances [d = B + s(gl + R)]or the diameter necessary to satisfy the bolt-spacing require-ment [d = (NBS/r)], whichever is greater. The optimumdesign is usually obtained when these two controlling diam-eters are approximately equal. The following exampledemonstrates the procedure recommended.

Table 12.3. Selection of Optimum Bolt SizeMin

Eolt. Root No. of Actual NB,Size Area Bolts No. (N) B, R n- B+%l+R)

N 0.302 73.7 76 3 l>Q 72.5 36%‘A 0.419 53.3 56 3 1% 53.4 37

1 0.551 40.4 44 3 1% 42.0 3 7 %1% 0.728 30.6 32 3 1% 30.5 37%lx 0.929 24.0 24 3 1% 22.9 38

Example of selection of optimum size and number of bolts.Given :

Inside diameter, B = 32 in.

Hub thickness, g1 = 12 in. (see Fig. 12.14)

Allowable bolt stress = 20,000 psi .:

Maximum bolt load = 446.000 lb

The minimum bolting area, AB(min) is given by:

WA~(min) = - =

446,000- = 22.3 sq in. .

fallow. 20,000

Root area, B, preferred, and radial distance, R, areobtained from Table 12.3.

Inspection of Table 12.3 indicates that the minimum boltcircle will result from the use of 32 bolts 136 in. in diameter.The size of the bolt circle is 3734 in.

12.7 DESIGN OF SPECIAL FLANGES

Process vessels are often of such large size that standardpipe flanges are not available in the sizes required. In suchcases special flanges must be designed. Large-size flangesmay be rolled from an annular ring (see Fig. 12.8) or maybe rolled from bar stock and welded. If a slip-on flangewithout a hub is to be used, the ring for the flange may beflame cut from flat steel plate.

The earliest method of designing flanges was the so-called“locomotive method” of Risteen (170). Cracker andStanford (171) developed a method of flange design inwhich the flange was considered to behave as a beam.Den Hartog (172) compared the “locomotive” and CrackerStanford methods by vector analysis and showed them tobe the same although the deriGations were different.Waters and Taylor (171) developed a method of analysis

Page 238: Process Equipment Design

2 2 0 Design of Flanges

Gasket Factors (m) for Operating Condit,ions and JIinimum Design Seating Stress (y)Note: This table gives a list of many commonly used gasket materials and contact facings with sug

gested design values of m and y that have generAy pro~rd satisfactory in actual service when usingelfective gasket, seating w-idth? h, given in Fig. 12.13.

Refer toFig. 12.12

Gaskel ~u:~trrial

-

5

hlindesignseatingstress.

y : ~-_- ~. -

Rubber without f;rbric or ;I high ~wrcc~l~~rge uf ushtos lilwr:Relow 75, Show I Wometer75 or higher, Short, I )uromet w

0 50

I 00

2 0 0

2 . i33 *so

0‘00

16003700fl300

-

Ilubber with iIst)f~st~I~-f;~l)r~i(~ insertion. wilh or willlout ::-[‘I!

wire reiriforcemfv~ 1 2-plyI-d\

., ‘ ? --. -3

2 4- 0

7.i.i-

I .;5 1 LOO-

2900I300

-2.7-i 3700

Corrugated nlr*t ;11, asbt‘st.os inst~rtt~tlor

Corrugated melal. j:~cl,c~~~d, nsl~r~osfilled

sot’{ il~llIIIiJlUIlI

Ydt CO~~pt~J o r tbl?lS?;Tron 01‘ s o f t stwl

MOlld 01’ ,I-hC;) chlY~lllf

Slainless slr&

2 3 0- -

2 ,a

3 0 0

3.253.30

290037004500.X00h.iOO-

S o f t alufiiiiluiti 2.73 3700Soft CC,,,I”“’ or I,rans 3.00 ‘1500Iron or soft slwl 3 2 5 5500\lo~irl o r l-h’,‘; (*liroliit 3.50 6500Stainloss steels 3.i.5 7600

Flat metal, jacketed, asbestos filled

Soft :iluriiiriuriiSoft ccbr)wr or l)razslron or’sht stcv~lAlone1

3 "3 53003.50 65003.75 76003.50 80003.75 90003.75 9000

3.25 55003.50 65003.75 76004 06 8800I 23 10100

I.00 8800‘k.i5 13000.i 50 180000.00 "1800h .50 26000

Iron or ~ofl. slrrl 3.50 18000Yvlorlf~l 0,’ I - h ’ ;, f~hrolllc 6.00 21800St:iinlea$ stfds 6.50 26000

-

Sketche.< ~IIICInotes

Facingimitations

Usecol.

Lisp 1 4 69 ,OIll\-

Sl)irill-wOlJIlfl Ill~‘lill. :ls~~f~stoc; lilltd

Use laonly

Use lit, 2*only-

Use I. 2, 3011ly

h-me

Corrugated mrtal

Grooved iron )Imetal jackel,

Solifl fkdt metal

Kin:: joint,

* The surface of a gasket. llil\miIlg 21 lap should 1~1: ag:hst ltic s1~1001l1 snrtiw of the hciri,u and JIOI ap;linst the nubbin.

Fig. 12.1’, Gasket materials and contact facings. ( E x t r a c t e d f r o m t h e 1 9 5 6 e d i t i o n o f t h e ASME B o i l e r a n d P r e s s u r e V e s s e l C o d e , U n f i r e d P r e s s u r e Versel~,with permission of the publisher, the American Society of Mechanical Engineers [I 11.)

Page 239: Process Equipment Design

Design of Special Flanges 2 2 9

combining the theory for a beam on an elastic foundationwith the theory for a flat plate which made possible thecalculat ion of the stresses in the radial , tangential , and axialdirect ions . The Taylor-Waters method was extended by\Vaters, Rossheim, Wesstrom, and Williams (173, 174).This method of flange design has been the basis of theMIME-code (11) procedures for flange design. A com-pdrison of the theoretical st,ressea with stresses determinedfrom strain measurements has been reported (175) forflanged joints of vessels and piping in low-pressure service .

Loose ring flange Loose hubbed flange

Riveted flange Fusionlap-weldedring flange

Fusionlap-welded

hubbed flangeltin W i d t h . b,jCoIumu I I

Facing Sketch Basic Gasket SExaggerated Column ,I

W+3N8

Forgedintegral flange

Fusionbutt-welded

hubbed flange

Fusionthru-welded

ring flange

Fig. 12.13. Various types of flanges subject to the method of analysis of

Waters, et al. (Extracted from Transocfions of the ASME with permission

of the publisher, the American Society of Mechanical Engineers [174].)!!+!Y; (y ruin)

The fol lowing sect ions describe a method of f lange designbased upon the procedures developed by Waters, Rossheim,IVesstrom, and Williams. The method is general andapplies to circular flanges of bolted joints under pressureand free to deflect under the action of the bolt load. Thisincludes all types of flange facing in which the gasket orcont.acting f lange surfaces are ent i re ly within the bol t c i rc leand excludes all types in which there is any contact outsidethe bolt. circle. Various types of flanges to which thismethod applies are shown in Fig. 12.13.

3 Na

7N16

N4

3 NT-

W-I-‘-Jl IIT-Ith, b

b = b, w h e n b. 5 >;”

VT”b = 2 w h e n b, > fL” I

rz (radius)

r, (radius)

Location of Gasket-load i ?actiou

Note: The gasket factorslisted only apply to tlangedjoints in which the gasketis contained entirely withinthe inner edges of the holth o l e s .

The design values and other details given we suggested only end ure notmandatory.

Shell Hub Ring -

Fig. 12.12. Effective gasket width and location of gasket load reaction.

(Extracted from the 1956 edition of the ASME Boiler and Pressure Vessel

Code, Unfired Pressure Vessels, with permission of the publisher, the Ameri-

con Society of Mechanical Engineers [l 11.)

.’

Fig. 12.14. Analysis of forces ond moments in o tapered hub flange.

(Extracted from Tronsocfions of fhe ASME with permission of the publisher,

the American Society of Mechanical Engineers, [174].)

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230 Design of Flanges

F ig . 12 .15 . T y p e - l l o a d i n g o f t a p e r e d h u b R a n g e (173).

The assumptions made in the derivation are:

1. Creep and plastic yielding do not occur.2. The bolt load has been determined.3. The lever arm of the bolt load has been computed.4. The effect of the external moment applied to the flange,

equal to the product of the bolt load and the lever arm, isindependent of the location of the bolt-loading circle andof the forces balancing the bolt load.

The tapered hub flange is analyzed by dividing the flangeinto three parts, as shown in Fig. 12.14, and consideringeach part as an independent unit. The loading is assumedto consist of: (1) a moment acting on the ring, so distributedthat it may be replaced by an equivalent couple producedby the force Wr at the inside diameter and outside diameterof the ring, as shown in Fig. 12.15; (2) internal hydrostaticpressure acting radially on the base of the flange and axiallythrough an assumed closure, as shown in Fig. 12.16. Theeffects of each loading on the flange are analyzed independ-ently and are assumed to be linearly related in such a waythat the complete solution may be obtained by superposi-tion, as shown in Fig. 12.17.

12.7a Analysis of a Shell Connected to a Flange. Ashell connected to a flange is considered to act as a beam onan elastic foundation. The deflection equation for thiscondition is given by Eq. 6.69, which may be written asfollows:

y = gz(cl sin &r + c2 cos @z) (12.4)

Equation 12.4 has the opposite sign on the exponent /3xfrom that of Eq. 6.69 because of a difference in sign conven-tion used by Waters, Rossheim, Wesstrom, and Williams(173,174). In Eq. 6.69 the distance x was taken as positiveaway from the junction along the shell whereas Waters et al.take x as positive in the direction of the ring. The deflec-tion y is taken as positive in the radial direction outwardin Eq. 12.4 whereas in Eq. 6.69 y was taken as positive inthe radial direction inward. . The constant p has the samevalue as it had in Eq. 6.86. The constants cl and c2 maybe evaluated by considering two of the boundary conditionsat the junction.

F ig . 12 .16 . T y p e - I I l o a d i n g o f t a p e r e d h u b flange (173).

The bending moment is given by Eq. 6.16.

M=DSwhere D is given by Eq. 6.15.

The shear is given by Eq. 6.17.

Q=D$

The load is given by Eq. 6.18.

d4Yu,=D-dx4

(6.16‘,

(6.17)

(6.18)

Successively differentiating Eq. 12.4 gives:

2 = Pt+[cr(sin px + cos /3x) + cz(cos px - sin /3x)]

(12.5)

2 = 2P2$“[c1 co9 px - c2 sin Px]

2 = 2P3t@[cr(sin /3x - cos @x) + cs(sin /?x + cos Bs)]

$ = 4/34eSz[cr sin 82 + cs cos /3x]

Therefore M, Q, and w are:

M = Et3p22z[cl cos 82 - c2 sin Pxl6(1 - ~‘1

(12.6)

Q = Et3B3eBZ[cr(sin px - cos /3x) + cs(sin ox + cos &)I

6(1 - p2)(12.7)

Et3/342z[cl sin /3x + cs cos @xlw =

3(1 - P2)(12.8)

In Eq. 12.8, p4 is given by Eq. 6.86.

Substituting Eqs. 12.4 and 6.86 into Eq. 12.8 in terms ofthe shell radius, rsr gives:

Et3yWC-P 2t 2s *

(12.9)

27rr1 p 27r-r# 27r+

F ig . 12 .17 . C o m b i n e d l o a d i n g o f t a p e r e d h u b f l a n g e (173).

Page 241: Process Equipment Design

To evaluate the constants cl and c2 the boundary condi-tions at the junction of the shell with the hub are applied.

At z = 0 (by Eq. 12.4)

cz = Yo (12.10)

At z = 0 (by Eq. 12.6)M = Et=P%

06U - ~‘1

cl = MO [6(;;B:1)] (12.11)

The above equations for the shell cannot be solved untilthe corresponding relationships have been developed forthe ring and the hub.

12.7b Analysis of the Ring of a Flange. A relationshipgiving the shape of the deflection curve of a flat plate havingradial symmetry in terms of the shear, Q, and the flexuralrigidity, D, is given by Eq. 6.38 with z as the axial directioninstead of y.

(6.38)

or for Q = 0 (see Reference 107, p. 63)

rf[3$(r$)] = 0

Differentiating with respect to r gives:

z {r$$(r$)] = 0 (12.12)

By dividing Eq. 12.12 by P and rearranging it, we canwrite it as:

(~+~~)($+~~)I=0 (12.13)

Waters, Rossheim, Wesstrom, and Williams (173, 174)have shown that Eq. 12.12, derived for the specific case ofQ = 0, can also be derived for the general case of Q f 0.(See Appendix A of References 173.)

By four successive integrations of Eq. 12.12, we obtain:2

z=Klnr-fr2+4 8

(12.14)

Letting f = ~5, and @co - 2c)8 = ~6, we obtain:

2 = cgr2 In r + c+jr2 + c3 In r + c4 (12.15),

By successive differentiation we obtain:dz- = 2cgr In r + (c5 + 2cg)rd r

d2z- = 2C5 ln r + 3C5 + 2~6 -dr2

&2 2~5 2c3/ -=

dr3 -p + 3,

(12.16)

c3

2(12.17)

(12.18)

and shear relationships for a circular flate given by Eq. 6.28, 6.29 and 6.36. A comparison

Design of Special Flanges 231

(Q+dQ)(r+dr)dB

+dM,)(r+dr)dB

Fig. 12.18. Shear and moments on on element of ring (173).

of Fig. 6.2 with Fig. 12.18 indicates that these relationshipsmust be written in the following forms:

M,=D($+;$)

Mt= D(;$+$)

Q=-!%+“;”

(12.19)

(12.20)

(12.21)

Substituting Eqs. 12.16 and 12.17 into Eqs. 12.19 and12.20 gives:

M, = D 2c5(1 + P) ln r + (3 + dc5 + 2(1 + P)C6

- (1 - /J) $ 1 (12.22)

Mt = D 2c5(l + cc) h r + (1 + 3P)cs + 2(1 + Phi

+ (1 - /J) s 1 (12.23)

Differentiating Eq. 12.22 with respect to r gives:

dMr-=d r

2c5(l + cc) +x(1 - c0c3

r3 1 (12.24)r

Substituting Eqs. 12.22, 12.23, and 12.24 into Eq. 12.21gives:

Q+! (12.25)

But

therefore

Q = (total load)W _ - Wcircumference 27rr

Wlc5 = go (12.26)

where WI = equivalent bolt load or total force applied atthe outside diameter of the ring, and (oppo-sitely) at the inside diameter of the ring, whichmultiplied by the radial breadth of the ringequals the total moment loading on the ring,pounds

Page 242: Process Equipment Design

Fig. 12.19. Segment of a tapered hub (173).

Q = shear on o unit sector of hub, at any point, pounds per inch. Sub-

scripts 0 and 1 some os for M,,.

T = Hoop tension, pounds per linear inch of axial hub length.

Mn = moment on o unit sector of hub, at ony point, inch pounds per inch.

Subscripts 0 and 1 refer to this moment at the small and large ends

of the hub, respectively.

In reference to Fig. 12.18, where t.he angle B is the slopeof the middle surface at the inner edge of the ring, the deflec-tion, slope, and moment at any point may be expressedin terms of ~9, the loading, the ring dimensions, and theelastic constants. For a small angle of rotation

(9 = dzdr

(12.27)

Substitution of Eqs. 12.26 and 12.16 int,o Eq. 12.27 gives:

~=~(2lnr~+l)+2ryr+~ ( 1 2 . 2 8 )

Solving for cg gives:

Examination of Fig. 12.14 indicates that the monlrut onthe ring at ~2 is equal to zero. Substituting Eq. 12.26 for(‘5 and Eq. 12.29 for c6 in Eq. 12.22 for the condition ofP = r2 gives:

MC9 = 0 = iyil [2(1 + p) In rz + (3 + p)] + 2(1 - p)7i-

Regrouping gives:

0 = iyh 2(1 + p) In ‘? + 2n- [ rl 1

r-2’I[--~ ---~(1 + PW’ + (1 + PI 1 (12.30)

The constant c4 is obtained from Eq. 12.15 by notingthat z = 0 at r = r1, and therefore

c4 = -c5r12 In rl - c6q2 - c3 In r1

Substituting the values of ~3, ~5, and cg given by Eqs.12.30, 12.26, and 12.29, respectively, into Eqs. 12.22 and12.23 for the case r = r1 gives:

MT, = -K2 - 1

( 1 2 . 3 1 )

Mt, = K-2 + I

(12.32)

12.7~ Analysis of the Hub of a Flange. In the analysisof the hub the assumption is made that the stresses anddeformations are the same as those for a beam with a varyingsection and on an elastic foundation. In this case the beamis represented by a longitudinal strip of the hub of unitwidth. The unit dimesion is taken at the inner surfacewhere r = rl.

Consider a segment of a transverse section of the hub likethe one in Fig. 12.19.

A summation of the forces in the radial plane must equalzero for equilibrium to exist; therefore

orrldQ - sdx = 0 (12.33)

Force r d2.\rea = ~-g dx

= hoop stress = E 2 (see Eq. 6.82)rl

dV 7 41-=.- =-dx r1 r12

Y

Taking a summation of moments in the radial plane gives:

(.ZIf, + t/,Vh)(r1) dr#~ - Mh(rl) d4 + Qrl(dz) d4 = 0

(12.35)

For a beam of infinite breadth (see Eq. 6.16)

(6.16)

Difl’erentiatiug Eq. 6.16 once and substituting Eq. 12.35,differentiating again and substituting Eq. 12.34 gives:

where K = ‘2r1

d2>j (12.36)

Page 243: Process Equipment Design

Design of Special Flanges 233

The moments at either end and the shear at the shellend are given by:

Equation 12.36 can be written in dimensionless form.

d2dj2

(1 + CYj)3 $ + $(l + Cxj)w = 01 (12.37)

where j =

CY=

w =

9=

go = shell thickness, inches91 = maximum hub thickness, inches9= intermediate hub thickness, inchesh = hub length, inches

2- = dimensionless axial dist.ance along hubh

taper factor for hub = @-ILqdgo

dimensionless radial displacement of hub or shell

at any point, rrl

Equation 12.37 may be solved in one of three ways: (1)by an exact solution of the differential equation, with g as avariable, which will give a solution in terms of Bessel func-tions; (2) by writing the total energy of the system as afunction of the deflections and minimizing the total energy;(3) by writing the strain energy as a function of the loadsand determining the deflection at any point. Waters,Rossheim, Wesstrom, and Williams (173) using the strain-energy method derived an approximate solution.

In the strain-energy method three parameters, al, ~2,and us, are selected and so related that if al is used alone, afirst approximation is obtained with all boundary conditionssatisfied. Similarly, if al and us are used together, or al,us, and us are used together, second and third approxima-tions, respectively, are obtained with all boundary condi-tions satisfied. There are four boundary conditions to besatisfied; therefore the solution involves a fourth-orderequation.

The four boundary conditions for the hub that are to bespecified are :

1. The radial displacement at the large end.2. The moment at the large end.3. The moment at the small end.4. The shear at the small end.

Here only the first boundary condition is known (zero)and the rest are unknown.

Let

A define the curvature factor at the smalld2w0- dj2 1 a end of the hub when x = 0 and j = 0

(12.38)

d2wA1 = _

dj2 I

define the curvature factor at the large1 end of the hub when z = h and j = 1

(12.39)

d3wB. = 7

dJ3 I

define the shear factor at the small endo of the hub when x = 0 and j = 0

(12.40)

ZIh = Ego3rlA o0 12(1 - p2)h2

,Mh = &0~(1 + a13rlAlI 12(1 - p2)h2

(12.41)

(12.42)

Q o = -Erlgo3@aAo + Bo)

12(1 - /.@I3(12.43)

or in terms of the huh modulus, $, by:

M = Egoh2Aoh, ~--

4

hfh = (1 + d3-Qoh2A1- .--__L rlfi

(12.41a)

(12.42a)

Q. = _ .%oWaAo + Bo)rl+

(12.43a)

Waters et al. (173) have shown that w can be written asa polynomial in powers of j with al, us, and us, and Ao, AI,and B. appearing in the coefficients as follows:

w = (1 - j)ur + (j - $j4 + $j5)u2 + (j” - Sj’ + +j6)a3- (&j - ij-j” + &j4)Ao - (hzj - &$)A1

- (&j - ijj” + &j4)B0 (12 .44 )

The known boundary condition is applied to Eq. 12.44.If j = 1, we obtain: w = 0. This satisfies this condition.Also, if Eq. 12.44 is differentiated and j = 0 and j = 1 aresubstituted, the three boundary conditions given by Eqs.12.38, 12.39, and 12.40 are satisfied. The parametersu2andu3 represent successive approximations and may be droppedwithout affecting the validity of Eq. 12.44 with regard toboundary conditions.

In solving Eq. 12.44 use was made of the total energy ofthe hub, that is, the sum of the energy of bending, Ur;the energy of stretching, U2; the external energy of rotation,Us; and the external energy of translation, Ud. The sumof these energies after deformation, lJtotsl = U1 + lJ2 + U3+ U4, must be a minimum at equilibrium or (dUtot,l)/(du,)= 0. This condition approximately s&&es Eq. 12.37.

This step permits a solution of Eq. 12.37 in terms of threeunknown constants of integration, the fourth being zero(wr = 0). Three equations result.

~11~1 + ~12~2 + ~13~3 = ~14.40 + c15A1 + c16Bo

c21al + c22a2 + c23u3 = c24AO + C25Al + c26BO

C31a1 + c32a2 + C33a3 = c34Ao + c35A1 + C36BO

Waters et al. (173) have tabulated the solutions for the con-stants in the above equations and have presented curves forthe determination of al, us, and us in terms of Ao, Al,and Bo.

With the boundary conditions fixed, the values for al, us,and us may now be computed, and the quantities Ao, A:,and Bo determined. These determinations in turn permitthe determination of the deflection, slope, and moment of

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234 Design of Flanges

any point of the hub. If the hub is free at the small end(loose flange) A0 = Bo = 0.

12.7d Relationships for the Hub, Shell, and Ring WhenCombined. The relationships for each part of the flange(shell, ring, and hub) have been developed independentlywith undetermined boundary conditions. With the partsassembled, the adjacent parts have common boundaryconditions, and in addition certain conditions are known atthe free boundaries.

According to the theory used in developing the hub rela-tionships, two of the constants of integration disappear forthe shell condition of constant thickness, and Eq. 12.4 maybe solved with:

(See Eq. 12.37 for definitions.) (12.45)

As stated above, the displacements, slopes, moments, andshears are identical on either side of the shell-hub interface,and four equations exist by which cl, ~2, Ao, and Bo may beexpressed in terms of AI. With these relationships theslope 0, the moment Mh,, and the shear Qr at the large endof the hub can be expressed in terms of Al. For the shell-and-hub junction, from Eq. 12.44 and its derivatives: atx = 0 and j = 0, the deflection is:

the slope is:

yo = c2 = wrl = alrl (12.46)

dy 1 (- Boz z=o =

- - al + a2 - T%AO - &AI Pl-

12 h

= P(Cl + c2)

the shear is:

(12.47)

- F (3aAo + B,,) = 6(fg$) (cl - ~2) (12.48).

the moment is:

-Go3P2c1W - p2)

(12.49)

The parameters al, a~, and a3 and constants ~5, cg, Ao,and Bo can all be expressed in terms of A 1 at the junction ofthe hub and ring. On either side of this junction the slopesand moments are equal, respectively, and the displacementis assumed to be negligible. The ring acts on the hub,producing a moment Mh,, with the additional moment of->$Qrt resulting from the shear Qr. Therefore the deflec-tion and the moment with respect to the intersection of theinner surface of the ring and its midpoint are:

Slope = 0 = (--al - &22 - 5~13 + 4 A0

+ +A1 + &BO) ; (12.50)

Moment = Mr, = Mh, - fQlt

>Al - +Q,t (12.51)

An expression for Qr may be obtained by integrating Eq.12.34:

w dj + QO

Qo is given by Eq. 12.43a. Therefore

Q1 z !&ohr1

' (1 + aj)w dj _ 3aAo++ BO 1Substituting Eq. 12.44 for w and integrating with sub-

stitution of the limits as indicated gives:

QI =~[(;+++(;+$2+($j+&)as

-(&+;+y)“o-(;+;)A1

- & + ; + ; > Bo (12.52)

There now exist four equations, 12.31, 12.50, 12.51, and12.52, containing the four unknowns 0, Mr,, Al, and Qr.The solution of these four equations is the key to the designof the flange. Inspection of Eq. 12.52 for Qr and Eq.12.50 for 0 indicates that both Qr and 0 are defined in termsof known dimensions, the loading factors A,,, Bs, and Al,and the parameters al, ~2, and u3. The loading factorsand the parameters can be expressed in terms of cr and #.Therefore, if Eq. 12.52 and 12.50 are divided by Al andsubstitutions are made for parameters, the loading factorsare obtained.

The two factors F and V of Eqs. 12.53 and 12.54 arefunctions solely of cx and #. By definition

#=12(1 - p2)h4

r12g02

therefore

ti____ =3(1 - pZ)

(12.55)

Therefore F and V of Eq. 12.53 and 12.54 may be plottedwith the group 2h/&go as a parameter. Also, F and Vare functions of CZ’, by definition cy = (gr - ge)/go, which isa function of gr/go. Figure 12.20 shows plots of F and Vas functions of the groups indicated above.

The next step in the solution is the determination of thevalue of Al and the evaluation of the three remaining vari-ables by substitution. To derive the expression for Al,Eqs. 12.51 and 12.31 are divided by Al, and M,,/Al iseliminated from the resulting equations. Substituting for

Page 245: Process Equipment Design

Design of Special Flanges

0.6

0.51 . 1 D 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

g, kl

I I /IIIIII hGl

0.6

0.5

0.4

0.3Y

0.2

0.1

01.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

g1’go

Fig. 12.20. Values of F and V for integral flanges (188). (Courtesy of Taylor Forge and Pipe Works.)

Page 246: Process Equipment Design

236 Design of Flanges

4

D and Jti30 - P2)

their respective equivalents,Et3

12(1 - $)

and __-G&, gives :

(12.56)

This equation may be written as:

where M = !%B

(12.58)

x= 1f

(12.59)

U= (12.61)

The above equations are too cumbersome for use indesign. Therefore they were reduced by Waters, Rossheim,Wesstrom, and Williams to those for the three criticalstresses (173).

A study of the stress distribution shows that for thetapered hub flange (Fig. 12.14) the critical stresses are:(1) the radial and hoop stresses at the inside diameter of thering and (2) the axial hub stress at the outer surface of thehub and at the large end of the hub for a hub with littletaper or at the small end of the hub for a hub with a largetaper. In the second case the critical stress may be dis-placed into the shell for a very rigid hub.

12.7e locating the Critical Section in the Hub. For ahub whose section modulus is variable,

jh=(~)(;)=(~)@) (12.62)

from Eq. 6.1.6

This equation may- be rewritten in t,he following dimen-sionless form :

Also

9 = 90 + E (sl - 90) = .90U + 4 (12.63)

From Eqs. 12.62, 6.16, and 12.63:

7

E900 d2wjh = -. L--- (1 + a.;) -2(1 - p2)h2 dj2

(12.64,

To obtain the maximum stress, set djh/dj equal to 0.Therefore

d3w d2w(1 + Cxj) -- + Ly - = 0

,lja dj2(12.65)

By substituting the second and third derivatives of Eq.12.44 into Eq. 12.65. a fourt.h-degree equation is obtained.

&Aj“ + (A + &B)j3 + (B + &C)j2+ (C + 2aD)j + (D + aE) = 0 (12.66)

where A = 96 3AI

B = 90Tl - 1082

C = -60:+24?-2$+2-223

D = B.

ET2

Here again all the constants can be evaluated in terms otAl, which is in turn a function of the hub quantities a! and#. The roots of Eq. 12.66, therefore, represent values of jor X at which the axial stress reaches a maximum in thehub. The corresponding maximum stress could be evalu-ated by substituting the roots into Eq. 12.64; obviously~roots outside of the region 0 _< j 5 1 are meaningless andshould be disregarded. From the above operation it isfound that the maximum axial st,ress usually occurs eit,herat, one end of the hub or at the other.

In solving Eq. 12.42 for Mh,, Eq. 12.57 may be substitut.etlfor A1 to give:

M _ gh, - 6

(12.6i)

Therefore the axial hub stress, jh,, at the inner surface ofthe large end is given by

(12.68)

d2y Mdl - p2)d;c2 = EI

The maximum axial stress, Jo at the outer surface of thehub or shell can be related to the corresponding stress,

Page 247: Process Equipment Design

Design of Special Flanges 237

J Fig. 12.21. Values of stress con

tion factor, f ’ (188). (Courtesy

I Taylor Forg e and Pipe Works.)

25

3

2.5

I fr‘,, at the inner surface by the factory, or

f’ = 1 (minimum)

f’ = 1 for hubs of uniform thickness (gl/go = 1)

f’ = 1 for loose hubbed flanges

fH = j’? (12.69)

fH fhj(max)where f’ = - or __fh, fh,

(whichever is greater) (12.70)

In the case wheref’ = 1, the maximum stress isfH and iskated at the junction of hub and ring. Values off’ were

/ determined by Waters, Rossheim, Wesstrom, and Williams1 (173) and are shown in Fig. 12.21 as functions of the hub

quantities (h/ho) and (gl/go).The maximum radial stress occurs at the inside diameter

:,f the ring. The moment MT1 is determined by substitutionof Eqs. 12.57 and 12.53 into Eq. 12.51. The extreme fiber41 ress is calculated from the moment M,, and is added to theradial stress due to Q1, giving:

1 (12.71)

Similarly, the maximum tangential stress at the insidediameter of the ring is determined from Ey. 12.57, 12.53,12.32, and 12.32, giving:

fT = F - ZjR (12.72)

where X = X in Eq. 12.59M = M in Eq. 12.58F = F in Fig. 12.20

(1 - p) + 6 (1 + F)$!~~~ In Ka 1

(12.73)

y=KZ+lJ

K2 - 1(12.74)

Values of T, U, Y, and Z are presented in Fig. 12.22 forrelationships in which p = 0.3.

12.7f Reduction to the Other Types of Flanges. Theprevious relationships apply to the general case of an integral

Page 248: Process Equipment Design

238 Design of Flanges

10987

6

2

1 K=$

1.02 1.03 1.04 1.06 1.08 1.10 1.20 1.30 1.80 2.00 3.GO 4.00 5.00

F ig . 12 .22 . Valuer of T, U, Y, and Z when p = 0.3 (188). ( C o u r t e s y o f T a y l o r F o r g e a n d P i p e W o r k s . )

T=K2(1 + 8.55246 loglo K) - 1

(1.04720 + 1.9448K2)(K - 1)

U=K*(l + 8.55246 log lo K) - 1

1.36136(K2 - l)(K - 1)

1Y = __

K2 log,0 K

K - 10.66845 + 5.71690 ~-

K2 - I 1

z=K2fK2 - 1

P o i s s o n ’ s r a t i o a s s u m e d = 0 . 3

flange composed of a ring, a tapered hub, and a shell ofuniform thickness extending indefinitely beyond the hub.These relationships can be modified for special cases, suchas the case of loose flanges.

The ring and hub of a loose flange (see Fig. 12.13) areindependent of the shell and are not subject to internalhydrostatic pressure.

The curvature factor, Ao, and the shear factor, Be (seeEqs. 12.38 and 12.40) are both equal to zero, and a different

set of F and V values must be used. These values, denotedas FL and VL, are shown in Fig. 12.23. Strain measure-ments on hubs of loose flanges have indicated that themaximum bending stress always occurs at the large endof the hub. ThereforeS’ = 1.0 for loose flanges.

The critical-stress equations for loose flanges (based onFL and VL) are:

fH = x$ (12.75)

Page 249: Process Equipment Design

j~=xM

1 + ,.,,*a>

t2(12.76)

jT = F - ZjR (12.77)

For the extreme case of a loose flange without a hub, thestress equations become:

fff = 0 (12.78)

fR = 0 (12.79)

fT = y (12.80)

12.7g Flange-Stress Equations Used in the Pressure-vessel Code. In the flange equations previously devel-oped, the quantity X or its reciprocal, L, may be expressedin terms of other factors. (See Eq. 12.59.) This may beperformed by noting that he = d&o and by letting theterm d equal (U/V)hOge2 and the term e equal F/ho. Bycombining these factors the quantity L may be defined asthe reciprocal of X, or

L+!+Pd

(12.81)

6

5FL

4

0.:0.80.70.60.5

Fig. 12.23. FL, VL, valuer for loose Aanges (1881 I. (Courtesy of Taylor Forge and Pipe Works.)

Design of Special Flanges 239

By substituting Eq. 12.81 and the relationships M =MO/B and e = F/ho into Eqs. 12.69, 12.71, and 12.72, andinto Eqs. 12.75 and 12.76 for the loose flange, the followingequations giving the desired stresses are obtained.

For integral-type jlanges and all hubbed jlanges:

Longitudinal hub stress, jH = =$$ (12.82)

Radial flange stress, jn = csteL;2;‘Mo (12.83)

YMoTangential flange stress, jr = t26 - ZjR (12.84)

For ring jlanges of the loose type:

fT = z

.fR = 0 (12.863

.fH = 0 (12.87)

Typical flanges permitted by the ASME code (11) areshown in Fig. 12.24.

Table 12.4 lists the nomenclature for Fig. 12.24 and thecode equations for flange design.

10080604030

108643

v, 2

0.: 1 l7-

0.40.30.2

0 %0.060.040.030.02

1.0 1.5 2.0 3.0 4.0 5.0

Page 250: Process Equipment Design

240 Design of Flanges

Loose-type flangesl o a d i n g s a n d d i m e n s i o n s n o t s h o w n a r e t h e s a m e a s i n s k e t c h 2

v

To b e t a k e n a t m i d p o i n t o fc o n t a c t b e t w e e n f l a n g e a n d l a p ,independent 01 gasket location.

(1)

fhveted or s c r e w e d f l a n g ewith or without hub.

(2)

Integral-type flanges

W h e r e h u b slope adjacantt o f l a n g e excetdr 1 : 3 .usa d e t a i l 6a o r 6 b .

(6)

I p h 1 ~l.%(min)

go

Optional-type flangesThese may be calculated as either loose-or Integral-type.

L o a d i n g s a n d d i m e n s i o n s n o t s h o w n a r e t h e s a m e a s i n 2 f o r l o o s e - t y p e f l a n g e s o r in 7 for i n t e g r a l - t y p e

t I

O.lt,(min)

f o r h u b t a p e r s 6O o rless. use g, = g,.

(4)

0.25&r, b u t n o t less t h a n % i n . , t h e m i n i m u mfor e i t h e r l e g . T h i s weld m a y b e m a c h i n e d t oa c o r n e r r a d i u s a s p e r m i t t e d i n s k e t c h 5.in which case g, = go.

(7)

:,+ k” (mf u l l p e n e t r a t i o n &d backchtp

(8) @a) @b) (9)

Fig. 12.24. T y p e s o f f l a n g e s p e r m i t t e d b y t h e ASME C o d e . (Extracted from the 1956 edition of the ASME Boiler ond Pressure Vessel Code, Unfired Pressure

Vessels, with permission of the publisher, the American Society of Mechanical Engineers [l 11.)

Code rules .for desigr~ ing flanges:Bolt loads:Two bolt loads exist: that developed by tightening up the

bolts, Wm2, and that which exists under the operating con-ditions, W,l. The bolt load for the tightening-up condi-tion must exert sufficient force, H,, on the gasket to causeyielding of the gasket in order to produce a tight joint.This load is equal to the effective area of the gasket timesthe gasket y ie ld s tress , or

Wm2 = H, = rbG<v (12.88)

The bolt load under the operating condition cor1sist.s ofthe force necessary to resist the internal pressure and tokeep the gasket t ight during operation. The internal pres-sure produces an end force, H, given by:

H = ;G2p (12.89)

The force required to keep the gasket from leaking. H,, is

given by:

Therefore

Bolt areas:

H, = 2hGmp (12.90)

Wml = H + HP (12.91)

I f the operat ing temperature is suff ic ient ly high to reducethe allowable stress of the bolting steel in comparison withthe allowable stress at room temperature, different bolt-ing areas will be required for the two conditions. For theoperating condition the minimum bolting area is given by:

AWml

ml = --~fb

(12.92)

For the bol t ing-up condit ion (no internal pressure)

AWm2nLs = -~

.fo.(12.93)

Note: see Chapter 13 for allowable stresses.

Page 251: Process Equipment Design

A

&

A,

A Wkl

A m2

B

Bl

BI

b

26

bo

Cd

e

F

7

G

I7091H

/ HD

Table 12.4. NomenclatureFrom the 1956 ASME Unfired-Pressure-Vessel Code

with Permiss ion of the American Socie tyof Mechanical Engineers

(See Fig. 12.24)outside diameter of flange or, where slotted holesare used with swing bolts, diameter to bottom ofslots , inches .actual total cross-sectional’area of bolts at root ofthread or section of least diameter under stress,square inches.total required cross-sectional area of bolts, takenas the greater of A,,,1 and Am2, square inches.total required cross-sectional area of bolti at rootof thread or section of least diameter under stressfor operat ing or working condit ions , square inches .total required cross-sectional area of bolts at rootof thread or section of least diameter under stressfor atmospheric-temperature conditions wit,houtinternal pressure, square inches.inside diameter of f lange, inches. When B is lessthan ZOgl, it will be optional fo? the designer tosubstitute B1 for B in the formula for longitudinalhub stress, f~.B + g1 for loose-type hubbed flanges and forintegral-type f langes when f is less than 1 .B + go for integral-type flanges when f is equal toOI greater than 1.effective gasket or joint-contact-surface sea t.ingwidth, inches (see Figs. 12.11 and 12.12).effective gasket or joint-contact-surface pressurewidth, inches.basic gasket seating width, inches (see Figs. 12.11and 12.12).bolt-circle diameter, inches (see Fig. 12.24).

= factor:

for integral-type flanges d = i hogo

for loose-type flanges d = $ hogo

= factor:

for integral-type flanges e = f

for loose-t,ype flanges e = 2

= factor for integral-type flanges; obtain from Fig.12.20.

= factor for loose-type f langes; obtain from Fig. 12 .23.= hub stress-correction factor (for int,egral flanges);

obtain from Fig. 12.21. For values below chartusef’ = 1.

= diameter at location of gasket-load reaction.Except as noted in sketch 1 of Fig. 12.2*, G isdefined as follows (see Fig. 12.12):

When 60 s $C in., G = mean diameter of gasketcontact face, inches.When bo > 34 in., G = outside diameter ofgasket contact face minus 2b, inches.

= thickness of hub at small end, inches.= thickness of hub at back of flange, inches.= total hydrostatic end force, pounds, = 0.785G’p.= hydrostatic end force on area inside of flange =

0.785B2p.

Ho =

H.v =

HT =

H =h =ho =

ha =

ho =hT =

K =

L =

MD =

MG =zz

MO =MT =

M, =

Mmax =

mi v

==

component of moment due to Ho, inch-pounds,= Hoho.component of moment due t,o HG, inch-pounds,H&c.total moment acting upon the flange, inch-pounds.component of moment due 1.0 HT, inch-pounds =H&T.moment under bolting-up conditions, inch-pounds= WhG.maximum moment, greater of MO or Maj~o/j~a,inchlpounds.gasket factor; obtain from Fig. 12.11.possible contact width of gasket, inches (see Fig.12.12).

P

R

fb

ff

“fH

fR

.fTTtuV

VLW

= maximum al lowable working pressure, pounds persquare inch.

= radial distance from bolt circle to point of inter-section of hub and back of flange, inches (integraland hubbed flanges) (see Table 10.4).

= maximum allowable bolt stress at atmospherictemperature, pounds per square inch.

= maximum allowable bolt stress at operating tem-perature, pounds per square inch.

= maximum al lowable design stress for f lange materialor nozzle neck, pounds per square inch.

= longitudinal stress in hub, pounds per square inch.= radial stress in flange, pounds per square inch.= tangential stress in flange, pounds per square inch.= factor involving K; obtain from Fig. 12.22.= flange thickness, inches.= factor involving K; obtain from Fig. 12.22.= factor for integral-type flanges; obtain from Fig.

1 2 . 2 0 .= factor for loose-type f langes; obtain from Fig. 12 .23.= flange-design bolt load, pounds.

Wrnl = required bol t load for maxirnuru operat ing or work-ing conditions, pounds (see Eq. 12.91).

w,, = required initial bolt load at, atmospheric-tempt+ature conditions without. internal pressure, po~r~tds(see Eq. 12.88).

Y = fact.or involving Zi; &lain from Fig. 12.22.Y = gasket or joint-contact-surface unit seating l(jiid.

pounds per square inch (see Fig. 12.11).2 = factor involving K; obtain l’rom Fig. 12.22.

Design of Special Flanges 241

difference between flange-design bolt load and totalhydrostatic end force, pounds, = 1Y - H.total joint-contact-surface compression load,pounds.difference between total hydrostatic end force andhydrostatic end force on area inside of flange,pounds, = H - HO.total joint-contact-surface sea tinp load, pounds.hub length, inches.radial distance from bolt cir& to circle on whichHD acts.radial distance from gasket-load react.ion to t,otIcircle, inches, = (C - G)/Z.factor = z/Bgo, inches. cradial distance from bolt circle to circle on whichHT acts.rat io of outside diameter of f lange to inside diameterof flange = A/B.

factorJe;l f ;. ” 5 tq+ +$ ’

Page 252: Process Equipment Design

242 Design of Flanges

The procedure for determining the actual bolt size waspresented in section 12.6. The actual bolt area providedusually exceeds the minimum required bolt area because anintegral number, usually a multiple of four, is used. Theexcess bolting area may result in overstressing of the flangein the bolting-up operation. To provide a margin of safetyagainst such overstressing, the code specifies that the designload, W, for the bolting-up condition be based on the averageof the minimum and the actual bolting areas, or

For the operating condition

w = W,l (See Eq. 12.91.) (12.95)

Flange moments:The various axial forces on the flange produce bending

moments. The summation of moments is taken about thebolting axis. For flanges classified as the integral type,the total moment must be at least equal to the sum of themoments acting upon the flange, or:

Flange Loads X Lever Arms = Moments

Ho = 0.785B2p hD=R+e2

MD = HD X ho

(12.96)

HT = H - HDh =R+g,+h,

T2

MT = HT X hT

(12.97)

Ho=W-H G -,, =C-G2

Mo = Ho X ho

(12.98)

and total moment, MO = MD + MT + Mo (12.99)

In the case of loose-type flanges in which the flange bearsdirectly on the gasket, the force HD is considered to act onthe inside diameter of the flange and on the gasket load atthe center line of the gasket face. The lever arms for themomeuts are:

(12.100)

h =C-GG-2

(12.101) therefore

(12.102)

The lever arms given in Eqs. 12.100, 12.101, and 12.102apply to the flanges shown in sketches 2, 3, and 4 of Fig.12.24 and also to those in sketches 8 through 9 when theseflanges are calculated as loose-type flanges. In the caseof the lap-joint flange of sketch 1 of Fig. 12.24, the leverarm ha is given by Eq. 12.100, and lever arms ho and hT are,identical and are given by Eq. 12.101.

’ 12.7h Design Procedure for Ring Flanges (LooseType).Ring flanges are widely used because of their simplicity andtheir ease of fabrication. Some typical ring flanges areshown in sketches 7 through 9 of Fig. 12.24. Flanges cor-

responding to these types are designed as integral flangesif any of the following values are exceeded:

g0 > Q in.

B-. > 30090

p > 300 psi

Operating temperature > 700” F

If the operating conditions and proportions are such thatnone of the above limits are exceeded, flanges correspondingto sketches 8 through 9 of Fig. 12.24 may be designed asloose-type flanges. In this case the minimum flange thick-ness may be calculated by use of Eq. 12.85 rewritten inthe following form:

t = 2/O’M,ax)/UB) (12.85)

To illustrate the design procedure, a ring-type flangewith a plain face for a heat-exchanger shell will be designedto the following specifications:

Design pressure = 150 psiDesign temperature = 300” FFlange material = ASTM A-201, grade BBolting steel = ASTM A-193, grade B-7Gasket material = asbestos compositionShell outside diameter = 31 in. = BShell thickness = x in.Shell inside diameter = 30>/4 in.Allowable stress of flange

material = 15,000Allowable stress of bolting

material = 20,000

Flange type See sketch 8 of Fig. 12.24.

Calculation of gasket width-by Eq. 12.2:

do y-pmz=

Y - P(m + 1)

Assuming a gasket thickness of ~1~ in., from Fig. 12.11 wefind :

y = 3700

m = 2.75

d oz=

3 7 0 0 - (150)(2.75) = 1 021

.3700 - (150)(3.75)

Assume that di of the gasket equals 32% in.; then

do = (1.021)(32.75) = 33.5 in.

Minimum gasket width = 33.5 ; 32.75) = !

/ 8

Therefore, use a ><-in. gasket width.

JThe mean gasket diameter G = 32.75 + 0.5 = 33.25 in.Calculation of bolt loads:Load to seat gasket-by Eq. 12.88:

Wm2 = Hv = brGy

Page 253: Process Equipment Design

From Fig. 12.12,

Thereforeb = bo if bc Z 0.25 in.

b = 0.25

Substituting, we obtain :

Hy = 0.25(?r)(33.25)(3700)

= 93,600 lb

Load to keep joint tight under operation-by Eq. 12.90:

HP = 2bGmp

= 2(0.25)(?r)(33.25)(2.75)(150)

= 21,600 lb

Load from internal pressure-by Eq. 12.89:

*G2H=T-p=

a(33f5)2(150)

= 130,500 lb

Total operating load-by Eq. 12.91:

W,l = H + HP = 130,500 + 21,600 = 152,100 lb

IV,,,1 is greater than Wm2.The controlling load is 152,100 lb.Calculation of minimum bolting area, A,,,l, as given by

Eq. 12.92:

A !+ 152,100 lbml =

20,000 psi= 7.6 sq. in.

Calculation of optimum bolt size (see Table 10.4):

Min ActualBolt Root No. of No. of NB.-Size AW% Bolts Bolts* R B, E r ID + Z(1.415go + Rlt

$6 0.202 37.6 40 ‘Hs 3 N 38.2 34.031 0.302 25.2 2 8 136 3 ‘%a 26.8 34.3M 0.419 18.15 2 0 lfi 3 1%~ 19.1 34.6

*’ Use multiples of 4.t Allow (l/0.707)90 = 1.41500 for weld.

From the above table the minimum bolt circle is 34.3 in.when you are using N-in. bolts. For simplicity in dimen-sioning specify 28 bolts of K-in. diameter on a 35-in. boltcircle.

Bolt-circle diameter, C = 35 in.

Calculation of flange outside diameter:

Flange OD = bolt-circle diameter + 2E

= 35 + 2(H) = 36g = A

Check of gasket width:

Abactual = (28)(0.302) = 8.45 sq. in.

Minimum gasket width = A fb actual a l l o w .

2Yn-G(8.45)(20,000)

= (2)(3700)(~)(33.25)

= 0.219 in. (compared with 4 in. specified)

Design of Special Flanges 243

Moment computations:For bolting-up condition (no internal pressure):The design load is given by Eq. 12.94.

w = +(Ab + Am)fa

= $(7.6 + 8.45)(20,000)

= 160,500 lb.

The corresponding lever arm is given by Eq. 12.101:

ho =;(C-G)

= +(35 - 33.25)

= 0.875 in.

The flange moment is as follows (see Table 12.4):

Flange moment,“kr = Who

= (160,500)(0.875)

= 140,500 in-lb

For operating condition (W = W,,,,-see Eq. 12.95):For Hn see Eq. 12.96.

Hn = 0.785B2p

= (0.785)(31)2(150)

= 113,000 lb

The lever arm, hn (from Eq. 12.100) is:

,, =C-BD-2

35 - 31=-2

= 2.0 in.

The moment, Mn (from Eq. 12.96) is:

M D = HD Xho

= (113,000)(2.0)

= 226,000 in-lb

Ho by Eq. 12.98 is:

Ho=W-H=W,l-H

= 152,100 - 130,500 = 21,600 lb

The corresponding lever arm by Eq. 12.101 is:

hG

= C - G 3 5 - 3 2 . 2 5~ =2

~__ = 1.375 in.2

The moment by Eq. 12.98 is:

MO = HG X ho

= 21,600 x 0.875 = 18,900 in-lb

HT by Eq. 12.97 is:

HT = H - Hn

= 130,500 - 113,000 = 17,5@0 lb

Page 254: Process Equipment Design

244 Design of Flanges

The corresponding lever arm by Eq. 12.102 is:

hr, + hG 2.0 + 0.875 = 1 44inhT = _~__ = ~.~-. . .2 2

The moment is given by Eq. 12.97.

MT = HT X her= 17,500 x 1.44 = 25,200 in-lb

The summation of moment for the operating condition,MO, by Eq. 12.99 is:

MO = MD + -'G + MT

= 226,000 + 18,900 + 25,200

= 270,100 in-lb

Therefore the operating moment is controlling; and

Mmex = 270,100 in-lb

Calculation of flange thickness by Eq. 12.85:

t = d(YMm,x)/(fB)

K = A - 36%B 31

= 1.211

.From Fig. 12.22 with K equal to 1.211,

Y = 10.25

Therefore

t = ~~~~~~~~~~~~~~~~ = 2/5 = 2 44 in

* . .(15,000)(31)

Therefore specify 2>$-in. plate.This flange design corresponds to TEMA (176) class C,

150 lb, 31-in-size flange but is >/a” thinner.12.7i Design Procedure for Integral Flanges. For pro-

portioning of the hub in order to comply with the ASMEcode (ll), integral hubbed flanges should have a hub taperequal to or less than 1: 3 unless a double-taper hub slope isused. If a double t,aper is used, the hub slope-adjacent tothe weld must be equal to or less than 1: 3 for a minimumlength equal to 1.5 go measured from the center of the weld.Also, a single taper hub may be used with a slope greaterthan 1: 3 if a straight section is used which also has a mini-mum length of 1.5gs.

In proportioning the hub as a first trial, the hub thickness,gi, may be taken as equal to two times go for values of goup to 1.5 in. and 1.5 times go for values exceeding 1.5 in.The flange loads, lever arms, and bending moments can thenbe evaluated, and t,he constants K, T, 2, Y, and U deter-mined. Taylor Forge (188) suggests that at this stage ofthe calculation a value of g1 be calculated from the followingequation.

g1 = 0.7 dMo/jB (12.103)

where j = allowable stress, pounds per square inch

If g1 as calculated by Eq. 12.103 is less than go, the cal-culation indicates that a tapered hub flange is not required,and gi can be made equal to go with h equal to the minimumrequired for welding. If gi as calculated by Eq. 12.103 isgreater than go, a value for h may be determined from Fig.12.21 by letting the factor j’ of Fig. 12.21 equal 1.0. Wit,11these values of gi and h the stresses in the flange and hubmay be calculated for a selected flange thickness, t.

ALLOWABLE STRESSES. Equations 12.82 through 12.84give the induced stresses in integral-type and all hubbed-type flanges. For purposes of design it is important thatthese individual stresses do not exceed the correspondingallowable stress. In selecting the allowable stress themoment distribution between the hub and the flange shouldbe considered. In this case a small amount of yielding ofthe weaker of these t,wo parts will shift part of the momentto the stronger part. If any yielding is to occur, it ispreferable that the yielding take place in the hub rather thanin the flange, which has a machined gasket face and whichmight leak under limited deformation. For this reason ithas been recommended that a higher allowable stress beused in the hub than in the flange, and it is customary touse one and one-half t.imes the allowable tensile stress ofthe material as the permissible longitudinal hub stress.However, if this overstressing of the hub is permitted, anallowance must also be made in the ring for absorbing anyload shifted to the ring from the hub. This may be accom-plished by limiting the averages of the induced stresses, theaverage of Jo + f~ and the average of j’~ + jr, to not morethan the allowable stress in tension. For example, if fallow.for the flange is 15,000 psi, then the maximum induced hubstress is permitted to be 22,500 psi. If Jo inducedis 22,500 psi, then neither fr nor Jo can exceed 7500 psi.However, if the induced stress Jo is less than 22,500 psi,then both Jo and jr may exceed 7500 psi as long as theaverage of Jo -l- fz or of Jo + fT does not exceed 15,000 psi.These limitations may be summarized as follows.

The maximum induced longitudinal hub st,ress* is:

fH 5 ~.5fallow. (12.104)

The maximum induced radial flange stress* is:

fR 5 faliow. (12.105)

The maximum induced tangential flange stress* is:

fT 5 f~lhv. (12.106)

* Provided that

fH + fT < j ,, w~~__2 = &O. (12.107)

and

fH+fR<jIIw.___

2 zz E&o. (12.108)

The steps involved in designing an integral flange aremore tedious than those involved in designing a ring flange

Page 255: Process Equipment Design

.Design Conditions

Design of Special Flanges 245- -

Gasket and Boliinp Calculations -l

Ii,, + H = 82,800 + 363.000 = 445,800.

Allowable Oper. temp. jFo 12,950 psi A4 = 23.3Ilangestress Atm. temp. jFli 13,000 ps i 1Y = OS(A, + Aw)j, = 456,(;il:)

All dimensions are shownW,,,l=H,+H= ~.lS.800

in the corroded condition Gasket widthcheck

N ABX a= ---~ ~- =,,,,n2~7~3

0 3c),)

Flange Loads (Operating Condition) Lever Arms Flange Moments (Operating Condiiion)

Ho = ;B2p = 322,000. ho = R + O.jq, = 2 . 1 2 5 hlr,=Hl,xhD = 684,,000.!

Ho = W,l - H = H, = 82,800. hc = 0.5(C - G) = 1 . 7 5 0 MG=HGxhc = 145,000.HT = H - Ho = 41,000. hT = 0.5(R + ql + hc) = 2.250 MT = Hr x hT zz 92,200.

M, = lzl~ + l-kfc + AtIT z= 921,200.

Flange Load (Bolting-up Condition) 1 Lever Arm ( Flange Moment (Bolting-up Condition) )

HG=W= 456,000. 1 hc = 0.5(C - G) = 1.75 IMa=Hc:xhc=(Equivalent to checking for AZ, at.

M = the greater of M, or Ma X :z = 921,200 allowable flange stress of ~FA andmax

798,000.

separately for M, at allowableM = .*?--? = 28,800.

flange stress of fill<1 ~. 1., I J~~-. ~

If bolt spacing exceeds 2d + 1, apply corrrc*t.ion factor. C.F. = Bolt, Spacing_ --~- -..- =2d + t

&f _ Mm xC.F. -B

Allowable Stress Calculation

Shape Const an IS

K=A/B = 1.24 ho = 1/&g; = 4.00

T = 1.82 h/ho 0.75

From Z= 4.72 From FZ 0.760

Fig. 12.22 y = 9.15 Fig. 12.20 1’ = 0.135[- = 10.05 FromFig. 12.21 j = 1.00

!Ili’SO = 2.50 e = F,‘ho = 0.190Z’

d = hogo =?; 7-1,. 50

1.5 fro Longitudinal hub stress, j,r = f’M/Xgr’ 18,100. IfF0 Hadial flange stress, jn = flM/Xt” 7,380.

1

Tangential flange stress, jr = (AZl’,t?) - Zjr---

.fFO 7,300. -i

fF0 Greater of O.j(j~ + fir) or O.j(fll + jr) 12,iO0. - 1

Fig. 12.25. Calculation sheet with example calculation for integral flanges (188). (Courtesy of Taylor Forge and Pipe Works.)

Page 256: Process Equipment Design

Design of Flanges246

Design Conditions Caskel and Bolting Calculations

psi 1 Gasket details“F .“F. Facing deta

Operating pressure, pOperating temp.Atmospheric t emp,Flange material I IBolting material I ICorrosion allowance I I

Allowableflangestress

II, = b&y =

H, = 2baCmD =

Allowablebolt

stress

Oper . temp. jbti’7Tp

ps i H = r =

Atm. temp. ja p s i A, =H H +Hthe greater of -? or pm-- =.fa fb

Oper . temp. j~0 ps i AD =

Atm. temp. jt,n ps i W = 0.5(A, + A~l)&t,~ =

All dimensions are shownin the corroded condition

W,+=H,+H=AR XfaGa$le=fdth IfT,in = _ _ _ =

2y?rG

Flange Loads (Operating Condition)I I

Lever Arms Flange Moments (Operating Condition)

&, e.;Bfp =

Ho = W,,,l - H = H, =HT = H - Ho =

ho = R + 0.5gl =

hr: = 0.5(C - G) =hr = 0.5CR + .rl~ + hc) =

MD = HD x ho =

Mc = HG X hc =MT = HT x hr =Mo = MD + ML’ + MT =

Flange Load (Bolting-up Condition) Lever ArmI

Flange Moment (Bolting-up Condition)

Hc=W= 1 hG = 0.5(C - C) = 1 M, = HG x hc =

&I,,,,, = the greater of AZ,, or M, X ‘2(Equivalent to checking for M, at allowableflange stress of jpo and separately for hi, at hl = Mm:,,__ =

fFA allowable flange stress of jF.1) B

If bolt spacing exceeds 2d + t, apply correction factor. C.F. = Bolt Spacing-- ------ =2tf + 1

l,f = hl,n;,, x CF. _B

1, = .-I :B =

Shape Constants

ho = dBgo =

t--8,=

:I-e-R = tB=

VWHD

t- hn -2

-c C=

IT= ( h,./ho z IFrom z =

Fig. 12.22 1. =c- =

!I 1 / g 0 =c:

tf = 7 hog,,? =T

From F =Fig. 12.20 tr =

FromFig. 12.21 f=

e=F/ho =

F ig . 12 .26 . B l a n k c a l c u l a t i o n s h e e t f o r i n t e g r a l - t y p e f l a n g e s ( 1 BE). ( C o u r t e s y o f T a y l o r F o r g e a n d P i p e W o r k s . )

Page 257: Process Equipment Design

Design of Special Flanges 247

ho = R + g1 = MD = HD X hD =I

Hc: = W,,,l - H = HG~ = hc = OS(C - C) = Mc = Hc x hc =HT = H - HD = hT = 0.5(R + CJI + hc) = MT = HT X hT

M,, = MD + MC + MT -r

Flange Load (Bolting-up Condition) Lever Arm Flange Moment (Bolting-up Condition)

HG=W= hc = 0.5(C - G) = M = HG x hc: = ~..

M fF0(Equivalent to checking for M, at allow-

= the greater of M, or M, X - able flange stress of fFA and separately for M,l,,Xm a r M = __ =

.~FA M, at allowable flange StreSS of f.VA) B___~

If bolt spacing exceeds 2d + t, apply correction factor. C.F; =Bolt Spacing-=

2d + tM = Mm,, x C.F. =

B

Shape Constants

I h-= A / B = hn=2/% =

t.,r=h= /+E= +--RI

iI(assumed)cr=te+l

6 = t3/dAllowable Stress calculation X=-y+6

l.j.fFO Longitudinal hub stress, jff = M/X,q12fF3 Radial flange stress, f~ = flM/XP-_...fFCl Tangential flange stress, jr = MY/P - Zf,.

.- .-_i..I

fF3 Greater of O.S(frr + jr?) or 0.5(f,r + f~)

f.ig. 12.27. Bla:k calc~lalion sheet far loose-type fiangitj [188). ( C o u r t e s y o f T a y l o r F o r g e a n d P i p e Wcdrs.)

Page 258: Process Equipment Design

2 4 8 Design of Flonges

(loose type) previously illustrated. Considerable time maybe saved by using a work sheet of the type developed byTaylor Forge Company (188). Such a work sheet with anexample calculation is illustrated in Fig. 12.25. Figures12.26 and 12.27 illustrate blank calculation sheets forwelding-neck flange designs and lap-joint flange designs,respectively.

It should be noted that the determination of the finalthickness, t, normally requires calculation by successiveapproximation. For estimating the initial thickness Taylor

P R O B L E M S

Forge suggests the use of the following equation (188):

t = o.72 dGGw@%l,,w.) (12.109)

Equation 12.109 does not give a precise determination ofthe thickness. If the hub stress is excessive, adjustmentcan be made by increasing the hub thickness, 91, by the

quantity 4f~/1.5fan~~.. If the flange is overstressed, thethickness may be increased by the factor dfxiy or

z/frlfallow. (188).

1. Figure 12.28 shows the partial design of a pressure vessel. Determine the maximumoperating pressure for the vessel based upon the shell dimensions and material (see Chapter 13for allowable stresses and code equations). For this pressure select a suitable gasket materialand determine the optimum number and size of the bolts assuming a hub thickness, 91, equalto 3 in. The service temperature is 400” F.

SA-212Grade B=2”

F ig . 12 .28 . Pressure vesse l for problems 1 , 2 , and 3 .

2. For the same vessel described in problem 1, design a suitable flange of the integral-hub

type.3. For the vessel described in problem 1, design a ring-plate flange without a hub.

4. Calculate the highest service pressure which can be used at 300” F for a 24-in., 150-psiwelding-neck flange fabricated of SA-105, Grade IL steel (see Fig. 12.2). The gasket is ofasbestos composition 31s in. thick and covers the full width of the raised face. The bolts areof SA-19-B7, eight-thread series (see Table 10.4).

Page 259: Process Equipment Design

C H A P T E R1 m1DESIGN’ OF PRESSURE VESSELS

TO CODE SPECIFICATIONS

arious codes governing the procedures for the design,fabrication, inspection, testing, and operation of pressurevessels have been developed, partly as a safety measure.These procedures furnish standards by which any state canbe assured of the safety of pressure vessels installed withinits boundaries. The specifications in these codes wereoriginally based upon the specifications developed for steamboilers. The code used for unfired pressure vessels is sec-tion VIII of the ASME Boiler and Pressure Vessel Code,1956 (11). Many states require that pressure vessels bedesigned and fabricated according to these code specifica-tions. Although all states do not have such regulations,it is usually necessary that pressure-vessel equipment bedesigned to a specified code in order to obtain insurance onthe plant in which the vessel is to be used. The NationalBureau of Casualty Underwriters publishes a Synopsis ofBoiler and Pressure Laws, Rules, and Regulations for thevarious states and provinces and for selected cities in theUnited States and Canada (227). Regardless of the methodof design, pressure vessels within the limits of the ASMEcode specifications are usually checked against these speci-fications. Before discussing the code it is of interest tobriefly review the antecedents of the code and its develop-ment. A. M. Greene, Jr. has published an extensive historyof the ASME boiler code (177-184).

13.1 ANTECEDENTS OF THE ASME CODE

Although steam devices were used to some extent priorto 1800, the pressures involved were low. Significant steampressures were not used until the development of the Wattsteam engine in the early part of the nineteenth century.

Many boiler explosions occurred during this period bothin the United States and in Europe, causing public concern.The Joint Committee of the Councils of the City of Phila-delphia reported in 1817 on the subject of steam boilerexplosions in boats and recommended the passing of a statelaw requiring testing the strength of boilers, the proper useof safety valves, and monthly inspections. A very dis-astrous boiler explosion occurred in London in 1815, andthe cause of this disaster was investigated by a committeeof the British House of Commons. This committee con-cluded that improper construction or improper material andan excessive though gradual pressure rise caused the explo-sion. The committee recommended that hemispherical orsegmental heads be used for cylindrical boilers, that wroughtiron be used as a material of construction, and that twosafety valves be used.

In 1830 the Franklin Institute appointed a committeeto investigate the cause of boiler explosions. As a result ofthe efforts of this committee, a government grant was pro-vided in 1831 for conducting experiments to test many ofthe causes of, and preventives of, the explosion of steamboilers. The nature of these experiments and the conclu-sions drawn from them are described by Greene (177).

As a result of the Franklin Institute studies two actswere passed by Congress. The first act, passed in 1833,provided for the inspection of hulls and boilers. Thesecond act, termed the “Steamboat Act of 1852,” establishedsteamboat inspection service under the Secretary of theTreasury. This act contained the original rules and repu-lations pertaining to the design and construction of boilersand provided for evaluation of the cause of boiler failures.The act was amended a number of times.

The American Boiler Manufacturers’ Association was249

Page 260: Process Equipment Design

250 Design of Pressure Vessels to Code Specifications

crganized in 1889. This group established a committeefor the preparation of uniform specifications and laws to bebased on the best American boiler-shop practice. Thecommittee presented a report to the association in 1898which covered such items as the specification of materials,riveting, factors of safety, specifications for bumped anddished heads, flanging, and the hydrostatic pressure test.

The first company for insurance against loss by boilerexplosions was arganized in 1855 in Great Britain. A num-ber of additional insuring associations were organized inBritain and Continental Europe during the next decade.The first such company to be organized in the United Stateswas the Hartford Steam Boiler Inspection and InsuranceCompany, founded in 1866. This company trained inspec-tors and established procedures; it has continued the writingof insurance on boilers and the practice of inspection untilthe present time.

At the beginning of the twentieth century about 350to 400 boiler explosions were occurring yearly in theUnited States with a tremendous loss of life and property.Although inspection and licensing of firemen and boilertenders were customary practice and although rules forboiler construction had been formulated by government andby insurance agencies, there was as yet no legal boiler codein any state in the Union. The first such code was passedin Massachusetts in 1907 and was known as the “Massa-chusetts Rules.” These rules specified a factor of safetyof 434, a minimum tensile strength of 55,000 psi for steeland 45,000 psi for wrought iron, and in addition containedmany specifications covering nozzles, heads, safety valves,and so on. In 1911 the state of Ohio adopted a set ofboiler rules similar to the Massachusetts rules. Althoughthese rules for boilers represented an important step, con-siderable differences existed between the various specifica-tions in the Massachusetts rules and those in the Ohio rules.There existed a need for a uniform code acceptable to allstates.

13.2 DEVELOPMENT OF THE ASME CODE

In 1911 the American Society of Mechanical Engineersestablished a committee to formulate standard specificationsfor the construction of steam boilers and other pressurevessels. This committee reviewed the Massachusetts andOhio rules and conducted an extensive survey amongsuperintendents of inspection departments, engineers,fabricators, and boiler operators. A number of preliminaryreports were issued and revised. A final draft was preparedin 1914 (178) and was approved as a code and copyrightedin 1915.

The introduction to the code stated that public hearingson the code should be held every two years. The first ofthese hearings was held in December of 1916. A largerepresentation of manufacturers, builders, designers, andusers was present as a result of the issuance of 1500 invita-tions. Mcst of the discussions were concerned with thecode requirements dealing with boiler construction.

In 1916 it was recognized that there was a need for asafety code for unfired pressure vessels (181), but becausethe revisions for the 1918 code were we11 under way, no

attempt was made to include a separate section for unfiredpressure vessels at that time.

In 1918 a revised edition of the ASME code was issued.This edition contained a number of new section:; and there-fore constituted an expansion of the code. In 1924 thecode was revised with the addition of a new section, VIII,which represented a new code for untired pressure vessels.

In the late twenties a considerable development of weld-ing techniques occurred. The code committees attemptedto develop rules for safe welded vessels but could not agreeon the rules required to produce reliable structures. Thefinal approval of fusion welding of the shells and parts ofpressure vessels occurred during the period 1928 to 1931(182). Also during this period the rules for the design ofdished heads were revised.

The first rules for the fusion welding of pressure vesselswere revised in the 1932 edition of the code. Also, revisionswere made in section VIII, for unfired pressure vessels.

13.3 THE API-ASME CODE

In 1931 a joint API-ASME Committee on Unfired Pres-sure Vessels was appointed to prepare a code for safe prac-tice in the design, construction, inspection, and repair ofunfired pressure vessels for petroleum liquids and gases.The API-ASME code was first published in September,1934, and was revised in 1936, 1938, 1943, and 1951. Theearly API-ASME code was considerably more lenient thansection VIII of the then-existing ASME code. Thisresulted in the reduction of fabrication costs for vesselsdesigned to the joint code. In recent years the ASME codehas been “broadened and improved so that it more com-pletely covers the petroleum industry’s pressure vessel needsand is in some respects more advanced” (186). In May,1956, the API-ASME code was officially discontinued, thediscontinuation to become effective December 31, 1956,(186) and was supplanted by the 1956 edition of sectionVIII of the ASME Boiler and Pressure Vessel Code (11).

13.4 SCOPE OF SECTION VIII OF THE ASME CODE

On April 26, 1956, the revised section VIII of the 1956edition of the ASME code for unfired pressure vessels wasapproved by the council of the ASME. Pressure vesselsdesigned and constructed in accordance with these speciflca-tions may be marked with the code symbol. Vessels builtafter January 1, 1957, in accordance with the 1949 editionand earlier editions of the code are not to be marked withthe code symbol.

This code is intended to cover the design and constructionpractices for unfired pressure vessels operating at pressuresof up to 3000 psi. In general vessels which do not exceed5 cu ft in volume and 250 psi in design pressure or whichdo not exceed 135 cu ft in volume for any design pressuremay be exempted from inspection, provided that theycomply in all other respects with the requirements of thecode. The jurisdiction of the code with regard to externalpiping of the vessel is limited to: the first circumferentialjoint in welded-end connections; the face of the first flangein bolted-flange connections; and the first threaded jointin threaded-pipe connections.

The code does not cover all details of design and construc-

Page 261: Process Equipment Design

Table 13.1. Maximum Allowable Stress Values in Tension for Carbon and Low-alloy Steels, Pounds perSquare Inch (11)

From the 1956 ASME Unfired-Pressure-Vessel Code with Permission of the American Society of Mechanical EngineersMaterial S&XC

and&xi- P- Mio For Metal Temperatures Not Exceeding Deg Ffication Nominal Num- Ten- -20 toNumber Grade Composition her sile Notes. 650 700 750 800 850 900 950 1000 1050 1100 1150 1200

PLateSieeeelaCarbon Steels

SA-7 . . . 60,000 (l)(3) 12.6505 5 . 0 0 0 13.750 13,250

13,25011.650.

.

12,05012.05010.700

10,20010,2009300

835083507900

8350865089509250. .. . .. . ..7750805083509550. .

95509900895092509550

14,40015,00015,900

. . .65006500

6500650065006500

, . . . . . .. . . . . .. .. . . .. . . .. . .

. . .2500 . . .2500 . . .2500 . . .2500 . . .

. . .

. . .. . . . .. . . . . .

. . .

. . .. . . .2500 . . .

. . .

2500 r...2500 . . .2500 . . .2500 . . .2500 . . .6250 . . .6250 . . .6250 . . .

. . .. . . .

. . .

.

. .

.

.

.

. .

I .

. . .

,

.

.

. .

.

. . .

. .

. .

. . .

. . .

. .

. . .

. .

. . .

. .

. . .

. . .

. . .

. . .

.

.

6250 .._ _..7500 5000 28006250 ,.. . . .6250 ._, .._

7300 5200 3300

.........

.........

.........2500 ......2500 ......2500 ......2500 ....... . . . . . . . .. . . . . . . . .

6250 ......6250 ......

... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... .........1550 1000.. . ...... ...... ...2200 1500

.. . ...

. . . ...

. . . ...

. . . ...

. . . . . .. . . .... . . . . .. . . .... . . . . .

. . . . . .... . . .

6250 . . . . . . . . .

SA-30 mangeSA-30 Firebox A ::I s5;ooo (4) 13;750

48.000 (4) 12,00048,000 ( L ) ( 3 ) 11.05040,000 . . . 10,00044,000 . . . 11,00042.000 . . . 10,500

4500450045004500

10,00010.00010.00011,00010,0007000

10.80011,00011,0009000

10,00010,000

SA-30 Fir&or BSA-113 CSA-129 ASA-129 BSA-129 CSA-201 ASA-201 B

yg "'

SA-212 A c-s:SA-212 B C-SiSA-283 ASA-283 BSA-283 CSA-283 D . . .SA-285 A .

:::22:: :SA-299 . . . C-Mu-SiSA-300 .

Low-alloy Steels

:t:g iCr-Mn-SiCr-Mn-Si

SA-203 A,D 2% and 355 NiSA-203 B, ESA-203 C

;E Eid 3% Ni

:;:::a ;:C-$4 MOc-$4 MO

SA-204 C] C-HMOSA-225 A Mu-VI,812-225 B Mu-VSA-301 A $4 Cr-$4 MOSA-301 B lCr--HMOSA-302 A Mn-$4 MOSA-302 B Mn--$5MoSA-353 . . . 9NiSA-357 . . . 5 Cr->4 MO

Pipes and TubesSeamlese Carbon steelaSA-53 A . . .SA-53 B . . .SA-83 A . . .SA-106 A . . .SA-106 B . . .SA-192 . . . . . .SA-210 . . . . . .SA-333 C .SA-334 C

12.05012.95013,85014,750

.13,25014.35015;soo16,600

10,20010,80011,40012,000.

. . .. . .

. . .. . .

. . .11,000 10.25012.100 11,15013,250 12,05017.700 15.650

.

. . .

. . .

.90009600

10.20012,600.

17,700 15.650 12.60019,800 17,700 12,800

55;ooo . . . 13,75060.000 . . . 15.0006S;OOO . . . La;25070,000 17,50045,000 (L)(3) 10,35050,000 (L)(3) 11.50055,000 (L)(3) 12,65060.000 (L)(3) 12.65045;ooo csj(4j 11,25050.000 (21(41 12.500

6500650065006500

i

55;ooo izji4j 13:75075.000 18,750

(13) . .

75,000 . . . 18,75085.000 . . . 21.25065.000 . . . 16,25070,000 . . . 17,50075.000 . . . 18,75065,000 . 16,25070,000 . . . 17,50075,000 . . . 18,75070,000 ( 1 2 ) 17,50075,000 ( 1 2 ) 18,75065,000 . . . 16,25060.000 .., 15,00075,000 . . . 18.75080.000 20,00090,000 (17) 22.50060,000 ( 1 4 ) . .

65006500650065006500

12,50012,75013.000

. . .

15,50016.600

13;85014.750

11,40012.000

17;70016.25017,50018,75017,50018,750

15:65016.25017.50018;75017.500

12:60015,65016.90018.000.

18:75016,25015,00018,75020,000

.15,65014.750

14,400 12,50014.200 13.100

18;OO0 L$9QO 13;ooo19,100 16.800 13,250

. . .13,100 12.800 12,400 11.500

16,25015.00018;75020,000

13,400

48,000 (4)(6) 12,00060,000 (4)(6) 15,000

(4)(6) 11.750

11,65014,35011,45011.65014,35011,45014,350

10.70012.950

930010,80092009300

10.800

7900 65008650 6500

10:55010,74012,95010,55012,950

. . .

7850 65007900 65008650 65007850 65008650 6500. .

. .

48.000 :.: 12.00060.000 _.. 15.000

. . 11:75060,000 15,00055.000 (4)(6) 13,75055,000 (4)(6) 13,750

d20010.800

1Seamlees Low-allov steeLn

SA-209 TlSA-209 TlaSA-209 TlbSA-213 T3SA-213 T5SA-213 T7SA-213 T9SA-213 TllSA-213 T12SA-213 T5bSA-213 T3bSA-213 T5cSA-213 TL7SA-213 T21SA-213 T22SA-333 3SA-333 5SA-334 3SA-334 5SA-335 PlSA-335 P2SA-335 P3SA-335 P3bSA-335 P5SA-335 P5bSA-335 P5c

C-+GMoC+6MoC--WMoL4g Cr-0.7 MO5Cr-$5 MO7 Cr--$4 Ma9Cr-1 MOl)i (k-35 Mo-Si1 Cr--45 MO5Cr--)4 Mo-Si2 Cr-54 MO50-36 Mo-TilCr-v

3 55,0003 60,0003 53.000

13.75015,00013.25015.000

(14)( 1 4 )

13,75015,00013.250

13,75015,00013.25015,00013,10013.10013,10015,00015,00013,10015,00013.100

13,45014.400

13,150 12.50013,750 12,50012,750 12.50014,400 13.10012,400 11.50011,500 950012.500 12.00014.400 13,10014.200 13.10012.400 10,90014,000 12.50012,400 11,500

15;ooo15.00012,80012,50012,80015.000

60;OO060,00060,00060,00060.000

15,00013.400

780073005000850078007500550062007300

5500 40005200 33003500 25005500 33005500 4000

25002200

12001500120013;400

13.40015,00015,00013,40015,00013,400

1800220025001550180017501800

(1s ,... 15,000

15,000(14) . .

15.000

15001200

60:00060,00060,00060,00060,000

14,75012.800

5000 28003500 2500

1000120012001200

14;70012.800

4200 27504800 2800(14) :.

(5) 15,000 .7000 5500 4000 2700 15007800 5800 4200 3000 2000

3 Cr-0.9 MO2$i Cr-L MO;S&Ni

;S&Ni

C-?JMo34 G--W MO1% Cr-0.7 Mo2 CT-j.4 MOSCr--54 MO50-g Mo-Si5 Cr--)d Mo-Ti

or CbL>i 0-36 Mo-SiLCr+B MO134 Si-$6 MO7 Cr--fh MO9Cr-LMo3 0-0.9 MO

60.00060,00065,00065,00065,00065.000

. .15.000 14,800 14,500 13,90015.000 15,000 15,000 15,00016.250 . . . . . .16,250 . . . . . . . . .16.250 . . . . . . . . .16.250 _..

13.20014,400

.

. .

13,15013,15014,40014,00012,40012,40012,400

14,40014,20013.75011,50012.50013,200

12,000 900013,100 11,000

. . . . . .

.

. . . . . .

. . .12,500 10.00012.500 10,00013.100 11,00012,500 10,00011,500 10,00010,900 900011,500 10.000

13,100 11,00013.100 11.000

6250625078006200730055007300

. .. . . . . , . .

. . . . .. . , . .. . .

.5500 4000 25004200 2750 17505200 3300 22003500 2500 18004800 2800 1800

.

. . I

. . .

. . .

.

12001200150012001200

7800 5500 4000 2500 12007500 5000 2800 1550 1000

999

.

15;ooo 15;000 L5;ooo 15;00015,000 15,000 15,000 14.700. 13,400 13.100 12,800

13,400 13.100 12.80013,400 13,100 12,800

SA-335 PLLSA-335 P12SA-335 PL5SA-335 P7SA-335 P9SA-335 P2L

4 60,0005 60.000

60;OO060,000

60.00060,00060,000

15.000 15,000 15,000 15.00015.000 15.000 15.000 14.75015;000 L5;ooo 15;ooo 14;400

13,400 13,100 12,50013.400 13,100 12.800

15.000 14,800 14,500 13,900251

12;500 10;0009500 7000

6250 . . . ,,_ ,..5000 3500 2500 1800 12008500 5500 3300 2200 15007000 5500 4000 2700 1500

60,00060.000

(14)(14) Li.000 10,800

12,000 90006O;OOO

Page 262: Process Equipment Design

Table 13.1. (Continued)F r o m t h e 1 9 5 6 ASME U n f i r e d - P r e s s u r e - V e s s e l C o d e w i t h P e r m i s s i o n o f t h e A m e r i c a n S o c i e t y o f M e c h a n i c a l Engineera

Mate r i a l Specend Spa% P- Min For Metal Temperatures Not Exceeding Deg F

tkation NOIIliId Num- Ten- -20t0Number G r a d e C o m p o s i t i o n her ‘de Notes 6 5 0 700 750 800 8 5 0 9 0 0 9 5 0 1000 1050 1100 1150 1200

SA-335 P22 2$i C r - 1 MO 5i

mnnn--.--- 15.00055.000 .:. -~’

15.000 15.000 15.000 14.400 13.100 11.0001 3 . 7 5 0 i3:750 1 3 . 7 5 0 i3:450 13:150 121500 1O:OOO

7800 5800 4200 3000 200%SA-369 FPl C-44 MO

SA-369 FP2 ja G-w MO

SA-369 FP3b 2 CT--Ji MOSA-369 FPll lj/, Cr-36 Mo-SiSA-369 FP12 1 Cr--)d MO

SA-369 FP21 3 Cr-1 MOSA-369 FP22 2% Cr-1 MO

5 5 ; o o o 13;75060,000 . . . 15,0006 0 . 0 0 0 . 15.000

13,750 13;750 13,45015,000 15,000 14,70015,000 15,000 15,00015,000 15,000 14,75014.800 14.500 13.900

13,150 12,50014.000 12,50014,400 13,10014,200 13,10013.200 12.000

10,00010.000

62506200 4200

5500500055005800520035005500

2750400011;000

11,0009000

11,00010,000

700010,800

7800750070007800730050008500

2500155027003000220018002 2 0 0

60,000 _.. 15,00060,000 _.. 15,00060,000 15,ono60,000 ( 1 4 )60,000 ( 1 4 ) _..60.000 (14)

2800400042003300

15;ooo 15;ooo 15;ooo13,400 13,100 12.80013,400 13,100 12,50013,400 13,100 12,800

14,400 13;10012.408 11,58011.500 950012.500 12,000

SA-369 FP5SA-369 FP7SA 369 FPY

F0rghgsC a r b o n S t e e l s

SA-105 I

5.cr-$( MO7 Cr-35 MO

9 Cr-1 MO

25003300

60.000 (4) 15,00070,000 (4) 17,50060,000 ( 4 ) 15.00070,000 (4) 17.50060,000 15,00070,000 17,50075,000

.:.18,750

60.000 15,000

70,000 17,50070,000 (14) . . .YO,OOO ( 1 4 ) . . .70,000 ( 1 4 ) .

100,000 ( 1 4 )70,000 17,50070,000 17,50070,000 ( 1 4 )80,000 ( 1 4 )80,000 .._ 20,00070,000 17,500

14.350 12,950 10,80016.600 14.750 12.000

8650 65009250 65008650 65009250 65008650 65009250 65009 5 5 0 6 5 0 0

25002500SA-105 II

SA-181 I 14;350 12:950 lo;80016,600 14,750 12,00014,350 12,950 10,80016,600 14,750 12,00017.700 15,650 12,600

. . . . . . . .

17,500 17,500 16.90016,150 15,500 14,85017,500 16,000 14,50016,150 15,500 15.00021,200 20,000 17,70017,500 17,500 17,50017,500 17,500 16.90016,150 15,500 14,85016,500 15,500 14,50020,000 20,000 18,000

2500250025002 5 0 02500

6250750073007800850078006250750073007800

SA-181 IISA-266 ISA-266 IISA-266 IIISA-350 LFl

Low-alloy SteelsSA-182 FlSA-182 F12

. . .

. . .

C-x MO1 Cr-$6 MO5 Cr-$6 MO

l)C Cr--35 MO9 Cr-1 MO2>/4 C r - 1 MO

C-X MO1 Cr.-$6 MO5 Cr-x MO

Zjl, C r - 1 MO3% Ni

15,000 12,75014.200 13.100 5000

5200550055005800

2800 1559SA-182 F5SA-182 FllSA-182 F9SA-182 F22SA-336 FlSA-336 F2

13,000 11,50014.400 13.100

33004000

2 2 0 02500

15001200

15;400 13;10016,000 14,00015,000 12,75014,200 13,10013,000 11,50016,000 i4,,oon

33004200

280033004200

22003000

155022003000

15002000

1000500052005800

SA-336 F5aSA-336 F22

15002000

SA-350 LF3 9Cas t i ng s

C a r b o n S t e e l sSA-95SA-216 WCASA-216 WCB

70,000 (7) (18) 17,5006n.000 (7) 15.000

16,600 14,750 12,00014,350 12,950 10,80016,600 14,750 12,000

. . . . . .

16,250 16,250 15,65017,500 17,500 17,00017,500 17,500 17,00017,500 17,500 17,00017,500 17,500 17,00021,600 20,400 19,00022,000 21,000 19.400

.

9 2 5 0 65008650 65009250 6500. . .

14,400 12.50015,800 14,00015,800 14,00015,800 14,00015,800 14,00017,000 13.60017,300 15,000

.

450045004500

10,00010,00011,00011,00011,00010,00011,750

25002500

.11 7o;ooo (7j 17:500

65,000 (7) (16) 16,250

65,000 (7)(8) 16,25070,000 (7)(8) 17,50070,000 (7)(8) 17,50070,000 (V(8) 17,50070,000 (7)(8) 17,50090,000 (7)(a) 22,50090.000 (7)(8) 22,50065,000 (7)(16) 16,25065,000 (7)(16) 16,25065,000 (7)(16) 16.250

2500

6250625078007800780073008500

SA-352 LCBL o w - a l l o y S t e e l s

SA-217 WC1SA-217 WC4SA-217 WC5SA-217 WC6SA-217 WC9SA-217 C5SA-217 Cl2SA-352 LClSA-352 LC2SA-352 LC3

B o l t i n gC a r b o n S t e e l s

SA-261 BOSA-307 BS A - 3 2 5

Low-ailoy SteelsSA-193 B5SA-193 B7SA-193 B7aSA-193 B14SA-193 B16SA-320 L7. LY

. . .

C--j6 MONi-Cr-% MONi-Cr--1 MOl$i Cr-$6 MO

2j/, C r - 1 MO5 Cr-$4 MO9 Cr-1 MO

. . .

. . .

3

5500580052005500

4000420033003300

... ... ...

... ... ...

... ... .... . .

100,000 (Y)(lO) 16,25055,000 (11)

(9)(10) 18,750

14,950 12,500 10,000

17,200 15,650 .

20,000 20,000 20,00020,000 20,000 20,00020,000 20,000 20,00020,000 20,000 20,00020,000 20,000 20,000

. . . . .

17,200 15,650 . . .18,400 16,750 .18,400 16,750 . . .

6900.

. . . . . .

17,250 13,75016,250 12,50017,250 13,75018,750 16,65018,750 16,650

. . .

. . . . . ..

4800 2750 . . .

iii0 i;io : : :6250 27506250’ 2750 : .:

. . .

5 Cr--fh MO1 Cr-0.2 MO

1 CT-0.6 MO1 Cr-0.3 MO-VI c-15 MO-V

(9)(10) 20,000(9)(10) 20,000(9)(10) 20,000(9) (10) 20,000

. (9) (10) 20,OOb(9)(15)

. . (9)(10) 18,750(9)(10) 20,000(9)(10) 20,000

10,3008500

10,30014,25014,250

ilo,SA-354 BBSA-354 BC

'L43. .

. . .

. . ..

.

. .... ... ...... ... ...... ... ...SA-354 BD

BarsC a r b o n S t e e l s

111

50,000 . 12,50055,000 13,75060,000 _.. 15,000

SA-306 50SA-306 55

. . .

. . .. . ... ...

... .... .

SA-306 60

F r o m t h e 1 9 5 6 ASMEPntired-PressureVessel CZde w i t h P e r m i s s i o n o f t h e A m e r i c a n S o c i e t y o f M e c h a n i c a l E n g i n e e r sNotes: The stress virlues i n t h i s table m a y h e i n t e r p o l a t e d t o d e t e r m i n e v a l u e s f o r i n t e r m e d i a t e t e m p e r a t u r e s . .A l l s t r e s s v a l u e s i n s h e a r a r e 0 . 8 0 t i m e s t h e v a l u e s i n t h e a b o v e t a b l e .A l l s t r e s s valwes i n b e a r i n g a r e 1 . 6 0 t i m e s t h e v a l u e s i n t h e above tab le .( 1 ) S e e s e c t i o n 1 3 . 5 , f i r s t p a r a g r a p h .(2) Flange quality in this specification not permitted over 850° F.(3) These stress values are one fourth the specified minimum tensile strength multiplied by a quality factor of 0.92, except for SA-283, Grade D, and SA-7.(4) F o r s e r v i c e t e m p e r a t u r e s a b o v e 850° F i t i s r e c o m m e n d e d t h a t k i l l e d s t e e l s c o n t a i n i n g n o t l e s s t h a n 0 . 1 0 % r e s i d u a l s i l i c o n b e u s e d . K i l l e d s t e e l s w b i c b

have b e e n d e o x i d i z e d w i t h l a r g e a m o u n t s o f a l u m i n u m a n d r i m m e d s t e e l s m a y h a v e c r e e p a n d s t r e s s - r u p t u r e p r o p e r t i e s i n t h e t e m p e r a t u r e r a n g e a b o v e 850’ Fw h i c h a r e s o m e w h a t l e s s t h a n t h o s e o n w h i c h t h e va lues in the above t ab l e a r e based .

(5) Between temperatures of 650 and 1000” F, inclusive. the stress values for specification SA-201, Grade B. may be used until high-temperature test databecome ava i l ab le .

(6) Only killed ateel (silicon) shall be used above 900’ F.252

Page 263: Process Equipment Design

Welded-Joint EtSciencies 253

Table 13.1. (Continued)(7) T o t h e s e stress v a l u e s a q u a l i t y f a c t o r s h a l l b e a p p l i e d , a s s p e c i f i e d i n t h e c o d e .(8) These stress values apply to normalized and drawn material only.(9) These stresss values are established from a consideration of strength only and will be satinfactory for average service. F o r b o l t e d j o i n t s , w h e r e f r e e d o m

from leakage over a long period of time without retightening is required, lower stress vabms may be necaasary, determined from the rnllttive flexibility of t.heflange a n d b o l t s a n d c o r r e s p o n d i n g r e l a x a t i o n p r o p e r t i e s .

(10) Between temperatures of -20 and 400’ F, st,resa values equal to the lower of the following will be pnrmit.tnd: 20 ‘7 of the specified tensile strength, or25 % of the specified yield strength.

(11) Not permitted above 450~ F; allowable stress value, 7000 psi.(12) Between t,emperatures of 750 and 1000° F, inclusive, the stress values for specification S4-212, Grade R, may he used until high-twnperature test datit

become ava i l ab le .(13) The stress values to be used for tempnraturtis helow -20” F when steels ate made to conform with specification SA-300 shall be those that are given

in the column for -20 to 650” F.(14) Maximum allowable stress values for kmperetures below 700° F are giver1 in the following table:

For Metal Temperatures For Metal TemoeratureaNot Exceeding Deg F Not Exceeding Deg F

S p e c i f i c a t i o n P- -20 to Specificatioo P- -20 toNumber Grade Number 400 500 6 0 0 650 Numhnr t i r a d e Number 4 0 0 500 6 0 0 6 5 0

~~___.~SA-213 T5 5 15,000 14,500 14,000 13,700 SA-355 . 15,000SA-213 T7 5 15,000 14,500 14,000 13,700 SA-369 FP5 5 15,000SA-213 T 9 5 15,000 14.500 14,000 13,700 SA-369 FP7 5 15,000SA-213 T5b 5 15.000 14,500 14.000 13,700 SA-369 FP9 15,000SA-213 T5c 5 15,000 14.500 14.000 13,700 SA-335 P7

z15,000

SA-335 P5 5 15.000 14.500 14,000 13,700 SA-335 P9 5 15,000SA-335 P5b 5 15,000 14.500 14.000 13,700 SA-182 F12 4 17.500SA-335 P5c 5 15,000 14;500 14,000 13,700 SA-182 F5a 5 22;500SA-336 F 2 4 17.500 17.500 17.500 16.800 SA-182 Fll 4 17.500SA-336 F5 5 20;ooo 19;200 18;OOO 17;300 SA-182 F9 5 25;OO0

(15) For temperatures below 400’ F, stress values equal to 20 % of the specified minimum I.ensile strengt,h will be permitted.(16) See par. UCS-67(d) of Reference 11.( 1 7 ) S e e s p e c i a l r u l i n g i n r e g a r d t o w e l d i n g w i t h e l e c t r o d e s h a v i n g a t e n s i l e s t r e n g t h less t h a n t.he ba se meta l .( 1 8 ) T h i s m a t e r i a l n o t s u i t a b l e f o r w e l d i n g .

14,50014,50014,50014.560la;sbo14,50017,50021,60017,500

24.000

14,000 13,70014,000 13,70014,000 13,70014,000 13,70014,000 13,70014,000 13,70017,500 16,80020,100 19,00017.500 16$00

22.700 22,nno

tion but does list a number of mandatory requirements anda number of nonmandatory suggestions for practice indesign. It is expected that the designer and the fabricatorswill use accepted engineering practices and good judgmentwhen dealing with those parts of the vessel not covered bycode specifications.

13.5 MATERIAL SPECIFICATIONS

Plain-carbon- and low-alloy-steel plates are usually usedwhere service conditions permit because of the lesser costsand greater availability of these steels. Such steels may befabricated by fusion welding and oxygen cutting if the car-bon content does not exceed 0.35 %. Vessels may be fabri-cated of plate steels meeting the specifications of SA-7,SA-113, Grade C, and SA-283, Grades A, B, C, and D pro-vided that (1) the vessel does not contain lethal liquids orgases, (2) the operating temperature is between --20 and650” F, (3) the plate thickness does not exceed 46 in., (4)the steel is manufactured by the electric furnace or open-hearth furnace, and (5) the material is not used for unfiredsteam boilers.

The allowable stresses for these and other plate steelstogether with those for steels used for pipes, forgings,castings, and boltings are given in Table 13.1.

One of the most widely used steels for general purposes inthe const,ruction of pressure vessels is SA-283, Grade C.This steel has good ductility and forms, welds, and machineseasily. It is also one of the most economical steels suitablefor pressure vessels. However, its use is limited to vesselswith plate thicknesses not exceeding 46 in. For vesselshaving shells of a greater thickness, SA-285, Grade C ismost widely used in moderate-pressure applications. Inthe case of high pressures or large-diamet.cr vessels a higher-strength steel may be used t.o advantage to reduce the wallthickness. SA-212, Grade B is well suited for such applica-

tions and requires a shell thickness of only 79’~ of thatrequired by SA-285, Grade C. This steel also is easilyfabricated but is more expensive than the other steels.

The SA-283 steels cannot be used in applications withtemperat.ures over 650” F; the SA-285 steels cannot be usedfor services with temperatures exceeding 900” F; and theSA-212 steels cannot be used at temperatures over 1000” F.However, both the SA-285 and the SA-212 steels have verylow allowable stresses at the higher temperatures. There-fore, for temperatures between 650 and 1000” F, steelSA-204, which contains 0.4 to 0.6% molybdenum, is satis-factory and has good creep qualities. For low-temperatureservice ( - 50 to - 150” F) a nickel steel such as SA-203 maybe used. The allowable stress for this steel is not specifiedfor temperatures below -20” F. Normally, the fabricatormust run impact tests to determine the applicability of thesteel and its freedom from brittle fracture for low-temper-ature service.

13.6 WELDED-JOINT EFFICIENCIES

The use of a welded joint may result. in a reduction inthe strength of the part at or near the weld. This may bethe result of metallurgical discontinuities and residualstresses. The code rules make allowance for these factorsby specifying joint efficiencies for various types of weldswith and without stress relief and radiographing. Thedesigner is permitted some option in the selection of thekind of welded joint to be used and in whether or not thevessel and its parts must be stress relieved and whether ornot the welded joints must be radiographed. The generalthickness limitations for various types of joints are givenin Table 13.2. Additional t,hicltness limitat.ions for varioustypes of st.eels are given in the previous sect,ion, entitledMat,erial Specifications.

All vessel shells having a thickness greater than l$/, in.,_~

Page 264: Process Equipment Design

254 Design of Pressure Vessels to Code Specifications

Table 13.2. Maximum Allowable Efficiencies for

Arc- and Gas-welded Joints (11)

From the 1956 ASME Unfired-Pressure-Vessel Codewith Permission of the American Society of

Mechanical EngineersMaxi-

Basic lIl”lIlJoint The- JointEffi- mally Effi-

ciency, Stress ciency,TYPO Per Radio - Re- Per

of Joint Limitations cent graphed lieved cent

D o u b l e - w e l d e d NOIll3 No No 80butt joint

S i n g l e - w e l d e d Longitudinal joints not 80 No Yes 85 ,butt joint over 1M in. thick. No Yes No 90with backing thickness limitation on Yes Ye8 95s t r i p

Sindeweldedcircumferential joints.

Circumferential iointa 70 No No 70butt jointwithout back-

only, not 0~~~56 in.thick.

N O Yes 75

ing stripDouble full-fillet Longitudinal joint. not 65 No No 65

lap joint over Q$ in. thick. Cir- N O Yes 70cumferential joints not

S i n g l e f u l l - f i l l e tlap joint withplug welds

over J$ in. thick.Circumferential joints 60 No No 60

only. not over FQ in. No Yea 65thick; attachment ofheads not over 24 in.in outaide diameter toshells not over 56 in.thick.

Single full-fillet Only for attachment of 50 No No 50lap joint with- heads convex to pres- N O Yes 55out plug welds sure to shells not over

5$ in. thick, and forattachment of headsConCeYe to pressure notover 24 in. in outsidediameter to shells notover x in. thick.

WeGhan (d + 50)/120 (where d = inside diameteror 20 in., whichever is greater) must be thermally stressrelieved. Vessels of any thickness fabricated from thefollowing low-alloy steels must be stress relieved; SA-301,Grade B; SA-302; SA-217, Grades WC4 and WC5; SA-357;SA-387, Grades B, C, D, and E; and chrome-molybdenumsteel having a chrome content greater than 0.7%. Also,vessels having a shell thickness greater than 0.58 in. mustbe thermally stress relieved if they are fabricated of thefollowing steels: SA-202, SA-203, SA-204, SA-225, SA-299,SA-301, Grade A, SA-387, Grade A, and any steel having aspecified molybdenum content of 0.4 to 0.65 y0 and a chromecontent not greater than 0.7a/ Also, steels greater than1 in. in thickness must be stress relieved if they meet thespecifications of the following: SA-212, SA-105, Grade II,SA-181, Grade II, SA-266, Grade II, SA-95, and SA-216,Grade WCB.

If high-alloy steels are used, stress relieving is not requiredin the case of austenitic chromium-nickel stainless steels.The increase in joint efficiency may be used if these steelsare heat treated at over 900” F. If the vessels are con-structed of ferritic chromium stainless steels, stress relievingis required in all vessel thicknesses except in the case oftype 405 welded with electrodes, a process producing theaustenitic weld. The code gives the temperatures anddescribes the procedures to be used in thermal stress reliev-ing. Allowable stresses as specified by the code for high-alloy steels are given in Appendix D.

Radiographic examination is required for double-weldedbutt joints if the plate thickness is greater than 134 in. Ifthe plate thickness is greater than 1 in., complete radio-graphing of each welded joint is required if the vessel isfabricated of SA-202, SA-203, SA-212, SA-225, SA-294,SA-299, SA-301, or SA-302. Vessels of all thicknesses thatare fabricated of SA-353, SA-357, or SA-387 must be radio-graphed. Also, vessels constructed of high-alloy steels suchas type 405 welded with straight chromium electrodes andtypes 410 and 430 welded with any electrodes must beradiographed in all thicknesses except when the carboncontent does not exceed 0.08 %, the plate thickness does notexceed 155 in., and austenitic welds are used.

13.7 DESIGN OF CYLINDRICAL SHELLS UNDERINTERNAL PRESSURE

The equations for determining the thickness of cylindricalshells of vessels under internal pressure are based upon amodified membrane-theory equation. The development ofthis equation is described in the following chapter (seeEq. 14.34). The modification empirically shifts the thin-wall equation (see Eq. 3.14) to approximate the “Lame”equation for thick-walled vessels (see Fig. 14.5). The equa-tion may be written in either of the following forms:

t=fE 2.6~1 = fE ::.4p

(13.1)

o r

p,*tdf!Li * t-0 - 0.4t

(13.2)

where t = minimum required thickness of the shell exclu-sive of corrosion allowance, inches

p = design pressure, or maximum allowable workingpressure, pounds per square inch

E = welded-joint efficiency (see Table 13.2)f = maximum allowable stress, pounds per square

inch (see Table 13.1 or Appendix D).ri = inside radius of the shell, inchesPO = outside radius of the shell, inches

If the thickness of the shell exceeds 50% of the insideradius, or when the pressure exceeds 0.385fE, the Lame equa-tion should be used to calculate the vessel-shell thickness(see Chapter 14). The following forms of the Lame equa-tion are given by the code (11).

With the pressure p known,

23 - 1t = r@!:! - 1 ) = p. __( )zw (13.3)

where&fE+p

fE - P(13.4)

When t is known,

where

z=(?~)‘+y$!LJ (13.6)

Page 265: Process Equipment Design

Design of Cylindrical Shells Under External Pressure 255

13.8 DESIGN OF CYCLINDRICAL SHELLS UNDEREXTERNAL PRESSURE

The design of cylindrical vessels under external pressureis based upon consideration of the elastic stability of theshell, as described in Chapter 8. The calculation is madeby successive approximation by using the following equa-tion (see Eq. 8.33):

Bp = do/t

(13.7)

B-here p = allowable working pressure, pounds per squareinch

da = external diameter of shell, inchest = minimum thickness of shell exclusive of corrosion

allowance, inchesB = factor from Fig. 8.8 for carbon steel (see Appen-

dix I for charts for other steels and alloys)

Considerable reduction in vessel-shell thickness is oftenobtained by the use of stiffening rings, in which case Eq. 13.7may be modified as follows (see Eq. 8.39):

B = Not + (ML)

(13.8)

where A, = cross-sectional area of the stiffening ring,square inches

L = design length of a vessel section, as shown inFig. 13.1

The required moment of inertia of the stiffening ring maybe calculated by use of Eq. 13.9 (see Eq. 8.38).

I , =1 4

(13.9)

where A = the factor given in Fig. 8.8

Stiffening rings should extend completely around thecircumference of the vessel. If it is necessary to includejoints between the ends of sections of such rings, as shownin details C, D, F, and G of Fig. 13.2, the moment of inertiaof the ring must be maintained bv the addition of metal.

The internal stiffening ring may be replaced in part bythe external stiffener, as shown in detail H, so that themoment of inertia of the ring is maintained. In designingsuch stiffeners the moment of inertia of each section is takenabout its own neutral axis. Gaps in the stiffening ring suchas those shown in details A and E should not exceed thepermissible length of arc given in Fig. 13.3 unless the addi-tional reinforcement shown in detail H is provided. Someexceptions to this limitation are permitted by the code ifthe arcs of the stiffening rings are staggered 180”.

Stiffening rings are attached to the shell by either con-tinuous or intermittent welding. In the case where inter-mittent welding is used on each side of the stiffening ring,the total length of weld should be at least equal to one-halfof the outside circumference of the vessel for external rings,and at least equal to one-third of the circumference forinternal rings. Intermittent welds may be spaced a maxi-mum distance of 8t in. apart.

Any out-of-roundness of the shells of vessels subjected toexternal pressure reduces the strength of the vessel. Thisproblem was discussed in section 8.7 of Chapter 8. Themaximum permissible deviation, e, from a circular formpermitted by the code for vessels under external pressure isgiven by Fig. 13.4.

Moment axis of ring

h = depth of head

G-...h = depth

of head

Fig. 13.1. Design length of vessel section L for Eq. 13.8. (Extracted from the 1956 edition of the ASME Boiler and Pressure Vessel Code, Unfirad Pressure

Vesw!s, with permission of the publisher, the American Society of Mechanical Engineers [l 11.)

Page 266: Process Equipment Design

2 5 6 Design of Pressure Vessels to Code Specifications

moment of ikn.. -required for ring Length of any gap in

u n s u p p o r t e d s h e l l n o tto exceed l e n g t h of arc

Unstiffened cylinder

t h a n

This section &all have momentof inertia required for ring

H

Fig. 13.2. Various arrangements of stiffening rings for unfired cylindrical

vessels subjected to external pressure. (Extracted from the 1956 edition

of the ASME Boiler and Pressure Vessel Code, Unfired Pressure Vessels, with

permission of the publisher, the American Society of Mechanical Engineers

[ill.)

In determining the maximum out-of-roundness, e, asegmental c ircular template is used having the inside or theoutside radius specified by the design (depending uponwhether the measurements are made inside or outside).The length of the chord of the template should equal twicethe maximum permissible unsupported arc length as deter-mined by Fig. 13.3. In the case of vessels with butt joints,the value oft used in Fig. 13.4 is the nominal plate thicknessless the corros ion a l lowance . If longitudinal lap joints areused, t is equal to the nominal plate thickness, and thepermissible deviation is equal to (t + e).EXAMPLE CALCULATION. The vessel considered in sec-

tion 8 .4 of Chapter 8 wil l be used to i l lustrate the method ofdetermining the permiss ible out-of -roundness of cyl indricalshells. The shell is 14 ft in diameter and r3<s in. thick.

The difference between the maximum diameter and theminimum diameter cannot exceed -1 y0 of the nominaldiameter, or 0.01 .< (14 ft X 12) = 1.68 in.

d o- = 208.5tT

” = 1.49

From Fig. 13.4, e = 0.80t = 0.80(1x6) = 0.65 in. E i o mFig. 13.3 , the maximum permissible unsupported arc length

is found to be 0.138dc. The chord length for the templateis twice this value, or 2(0.138)(169.3) = 46.7 in. There-fore, in a chord length of 46.7 in. the maximum plus orminus deviat ion from circular form must not exceed 0 .65 in .

13.9 DESIGN OF PIPES AND TUBES UNDEREXTERNAL PRESSURE

When tubes or pipes are subjected to external pressure,an increase in allowable pressure is permitted over thatdetermined for shells by use of Fig. 8.8 (which is basedupon elastic-stability considerations). Figure 13.5, fromthe 1956 code, gives the al lowable design pressure for pipesor tubes subjected to external pressure as a function of theallowable stress of the material of construction and theratio of t/do. When corrosion or eros ion is expected, addi-tional metal must be supplied. If the pipe or tubes arethreaded, additional metal equal to (0.8/n) inches, where nis the number of threads per inch, must be provided.

13.10 FORMED CLOSURES UNDER INTERNAL PRESSURE

The most common types of closures for vessels underinternal pressure are the elliptical dished head (ellipsoidalhead) with a major-to-minor-axis rat io equal to 2.0 : 1 .0 andthe torispherical head in which the knuckle radius is equalto 6% or more of the inside crown radius (ASME standarddished head).

13.10a El l ipt ica l Dished Heads. In the case of thetwo-to-one el l ipt ical dished head the fol lowing relat ionshipsapply (see Eqs. 7.56 and 7.57):

(13.10)

(13.11)

where d = inside diameter of the head skirt

For elliptical dished heads which have a major-to-minor-axis ratio other than 2.0: 1.0, Eq. 7.56 is used with Eq. 7.57for determining the shel l th ickness .EXAMPLE-DESIGN CALCULATIONS. The thickness exclu-

sive of corrosion allowance for a seam& elliptical dishedhead is to be determined for a vessel having an insidediameter of 40 in. which is to be operated at an internalpressure of 200 psi . The inside depth of dish is to be 9 in.and the al lowable stress is 13,750 psi . The joint efficiency,E, is equal to 1.0 for seamless heads.

In reference to Eq. 7.56

v = 3(2 + k2) = +[2 -t- ($$s”-!“j

= 1.15

By Eq. 7.57

t = PdV 200(4O)(l.L5)2fE - 0.2~ = 2(13,750) - 0.2(200)

= 0.33 in.

13.10b Torispherical Dished Heads. For torisphericaldished heads in which the knuckle radius is 6 $& of the inside

Page 267: Process Equipment Design

Formed Closures under Internal Pressure 2 5 7

0.1 0.15 0.2 0.3 0.4 0.50.6 0.8 1.0 1 . 5 2 3 4 6 6 78 10 15 20 $5Length between heads or stiffening rings i outside diameter, L/do

Fig. 13.3. Maximum arc of shell left unsupported because of gap in stiffening ring of cylindrical shell under external pressure. (Extracted from the 1956

edition of the ASME Boiler and Pressure Vessel Code, Unfired Pressure Vessels, with permission of the publisher, the American Society of Mechanical Engineers

Ll 1 I.)

800700600500400

40

30250.10 0.15 0.2 0.3 0.4 0.5 0.6 0.8 1.0 1.5 2 3 4 5678910 15 20 25

Lengths between heads or stiffening rings i outside diameter, L/do

Fig. 13.4. Maximum deviation from o circular form, e, for vessels under external pressure. (Extracted from the 1956 edition of the ASME Boiler and Pressure

Vessel1 Code, Unfired Pressure Vessels, with permission of the publisher, the American Society of Mechanical Engineers [l I!.)

Page 268: Process Equipment Design

258 Design of Pressure Vessels to Code Specifications

10,ooo9 , 0 0 08,ooo7,0006,0005,ooo

4,000

1 , 0 0 0‘B 900Q 800g 7003 600

; 500‘ii5Em

Design stress, psi

Fig. 13.5. Chart for determining wall thickness of tuber under external pressure. (Extracted from the 1956 edition of the ASME Boiler and Pressure Vessel

Code, Un&d Pressure Vessels, with permission of the publisher, the American Society of Mechanical Engineers [l 11.)

crown radius, t,he following relationships hold (see Eqs. 7.76and 7.77) :

t = 0.88W, (13.12)fE - O.lp

OT

stress is 13,750 and the joint efficiency is 1.0. Determinethe thickness exclusive of corrosion allowance.

By Eq. 7.76

W = %(3 + 1/pJrl) = t[3 + 3.161

fEt’ ‘s 0.885r, + 0.U

(13.13)

where r, = inside spherical or crown radius, inchesFor torispherical dished heads having greater than 6%

knuckle radii. Eas. 7.76 and 7.77 are used.

= 1.54

By Eq. 7.77

pr,W’ = 2fE - 0.2P =

200(40)(1.54)2(13,750) - 0.2(2OG::

EXAMPLE-&&N CALCULATIONS. A vessel having aninside diameter of 40 in. with a seamless torispherical dishedhead having a radius of dish of 40 in. and a knuckle radiussf 4 in. is to be operated at 200 psi pressure. The allowable

= 0.45 in,

13.10~ Hemispherical Dished Heads. When the thick-ness of a hemispherical head does not exceed 0.356r,, where

Page 269: Process Equipment Design

Formed Closures undes Internal Pressure

In reference to Table 13.3

259

r, is the inside spherical radius, or when p does not exceed0.665fE. Eq. 7.88 applies. For thick hemispherical headsexceeding these lim.its, the following formulas apply (11) :

YS - 1t = I’i(YSh - 1) = ro ~( >Y45

(13.14)

where

or

y = 2w + P)2fE - P

(13.15)

where

13.10d Conical Dished Closures. For conical closuresor conical shell sections in which half the apex angle, a! (seeFig. 6.8) is not greater than 30”, Eq. 6.154 is used. Thediscontinuity stresses at the junction of the conical closurewith the shell, described in Chapter 6, may cause excessivedeformation. This may be prevented by the addition of acompression ring at the junction. When LY exceeds A,determined from Table 13.3, conical heads without a knucklewill require a compression ring at the section where thecone joins the shell.

Table 13.3. Value of A for Conical Closures (11)

; 0.001 0.002 0.003 0.004 0.005 0.006

A, deg 13 18 22 25 28 31

The required cross-sectional area of the compression ringin square inches is given by (11) :

.4~~~+)(lzg) (13.16)

where A = c&ical value from Table 13.3

When the thickness of the head or the shell exceeds therequired thickness, exclusive of corrosion allowance, for thedesign pressure, credit may be taken for the excess thick-ness. The area included in a distance of eight plate thick-nesses on each side of the joint times the excess thicknessmay be credited towards the required compression-ringarea, A.EXAMPLE-DESIGN CALCULATION. A vessel having an

inside diameter of 200 in. is to operate under an internalpressure of 50 psi. A conical closure is to be used with anapex angle of 30”. The material has an allowable stress of13,750 psi and a joint efficiency of 80%.

By Eq. 6.154

t= pd2 cos cu(fE - 0.6~)

;= 50(200)2(0.866)(13,750 x 0.80 - 0.6 x 50)

= 0.526 in.

50,$ = 13,750(0.80) = o’0045

A = 26.5”

As (o = 30’) exceeds a A of 26.5’, a compression ring isrequired. By Eq. 13.16

A = 0.0045(200)2;.577)] [ 1 23

= 1.52 sq in.

13.10e Toriconical Closures. Conical closures or coni-cal shell sections in which half the apex angle, cr is greaterthan 30’ must be connected to the shell by means of a torusring section at the junction to reduce junction stresses.Also, a toriconical head may be used when the angle a! isless than 30” but greater than A (see Table 13.3) in orderto avoid the use of compression rings. The minimumknuckle radius must be equal to the greater of either 6%of the outside diameter of the head skirt or three times theknuckle thickness. The required thickness of the knuckleis determined by use of modified forms of Eqs. 7.76 and 7.77with L substituted for r, where

and

L=A.!-2 CO8 a

(13.17)

dl = inside diameter of the conical portion of the toriconicalhead at the point of tangency with the knuckle, meas-ured perpendicular to the axis of the cone, inches (seeFig. 13.6)

The thickness of the cone is determined by use of Eq.6.154, in which dl is substituted for d.EXAMPLE-DESIGN CALCULATION. It is desired to use a

toriconical closure having a knuckle radius of 20 in. for thevessel in the previous section in order to avoid the use of acompression ring. Determine the thickness of ihe knuckleand the cone.

Fig. 13.6. Toriconical closure (11).

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2 6 0 Design of Pressure Vessels to Code Specifications

Loose-flange type Integral-flange type(4

Ringgasket -I ‘, ’

shown \‘\

Ringgasket -shown

Fig. 13.7. Spherically dished-steal-plate coven with bolting flanges. (Extracted from the 1956 edition of the ASME Boiler and Pressure Vessel Code, Unfired

Pressure Vessels, with permission of the publisher, the American Society of Mechanical Engineers [ 111.)

The inside diameter of the cone, di, at the point oftangency to the knuckle is:

dl = 200 - 2(2O)(J - 0.866) = 194.64 in.

The thickness of the knuckle is de&mined by means ofEqs. 7.76 and 7.77 modified by use of L in place of P,. where I,.is given by Eq. 13.17.

I = 194.64___- 1 ti2 in.i 2(0.866)

By Eq. 7.76_ _

w = P(3 + dJr1) = t(3 + &12/20) = 1.3-L

By Eq. 7.77

t(knuckle) =50(112)(1.34)

2(13,750)(0.8) - 0.2(50)= 0.342 in.

The thickness of the cone by Eq. 6.154 with di substitutedfor d is:

t = ~~~ (50)(194.64)2(0.866)[(13,750)(0.8) -i miiijj = o'512 ln'

13.11 FORMED CLOSURES UNDER EXTERNAL PRESSURE

In designing elliptical or torispherical dished heads forexternal pressure (pressure on the convex side) the thick-ness must be at least equal to that computed by use ofEq. 7.57 for elliptical closures and Eq. 7.77 for torisphericalclosures. In using either of these equations the pressure onthe concave side is taken as equal to 1.67 times the external

design pressure and a joint efficiency of 1.0 is used. Theminimum thickness as determined by the above methodmust be compared with the thickness computed for thesame closure by means of the procedure for external pressurevessels given in Chapter 8, section 8.8 or 8.11 as the casemay be.

Hemispherical and conical closures are designed in accord-ance with the procedures outlined in Chapter 8.

13.12 SPHERICAL DISHED COVERS

A torispherical dished closure or a spherical dished flatplate may be combined with a bolting ring to produce adished cover, as shown in Fig. 13.7.

COVERS WITH PRESSURE ON THE CONCAVE SIDE. Thethickness of the covers shown in detail a of Fig. 13.7 isdetermined by the method used for torispherical covers.The thickness of the spherical dished covers shown indetails b, c, and d of Fig. 13.7 is determined by use of themembrane equation for spherical shells under internal pres-sure with an empirical factor of N to allow for the dis-continuities at the junction with the ring. The minimumthickness of the dished covers in details b, c, and d ofFig. 13.7 is given by (11):

(13.18)

where p = internal design pressure, pounds per square in.chL = radius of crown, inchesf = allowable stress

If the bolting rings are within the range of the American

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Nozzles, Openings, and Reinforcements 261

COVERS WITH PRESSURE ON THE CONVEX S IDE. Sphericaldished covers may be used with the pressure on the convexside. The dimensions of the cover and the ring are deter-mined by using the same procedure as was used in the caseof pressure on the concave side. However, the thicknessof the cover plate must be checked for elastic instability byusing the procedures for the design of closures underexternal pressure given in Chapter 8.

Slandards (ASA) B16.5.1953. the flange dimensions andfacing details should conform to these standards. Boltingrings larger than these standards allow and correspondingto detail a of Fig. 13.7 are designed as ring-plate flanges,as described in Chapter 12. If the bolting ring correspondsto that shown in detail b of Fig. 13.7, the following equa-tions may be used to determine the flange thickness (11).

For a ring gasket, the flange thickness is:

For a full-face gasket, the flange thickness is:

(13.19)

t = 0.6 B(A + B)(C - B).4 - B I

(13.20)

\vhrre A = outside diameter of bolting ring, inchesB = inside diameter of bolting ring, inchesC = diameter of bolt circle, inches

MO = total moment as determined in Chapter 12ring-plate flanges

f = allowable stress, pounds per square inchp = internal pressure, pounds per square inch

f o r

If the cover plate corresponds to detail c of Fig. 13.7, thefollowing equations may be used to determine the flangethickness (11).

For a ring gasket, the flange thickness is:

t=Q[l+Jl+~] (13.21)

For a full-face gasket, t.he flange thickness is:

where

If the bolting ring corresponds to detail d of Fig. 13.7,the following relat.ionship applies for determining the flanget,hickness (11) :

Flange thickness, t = F[l + 1/l + (J/F2)] (13.24)

M here F = PB<4.L’~~!-!WA - B)

(13.25)

(13.26)

In Eq. 13.26, MO is calculated as before except that amoment equal to H,h, must be added or subtracted, as thecase may be, from the moment H&D where

H,. = radial component of membrane load in sphericalsegment = Ho cot B1, pounds

h, = lever arm of force H, about centroid of flange ring,inches

$1~ = axial component of membrane load in sphericalsegment, pounds

ho = lever arm of force HD about centroid of flangering, inches

13.13 FLAT-PLATE CLOSURES

Flat plates are often used as closures or as cover platesfor the openings in vessels. Blind flanges may be used forthis purpose and should conform to ASA-B16.5-1953. Forunstayed flat heads, blind flanges, and cover plates, such asthose shown in Fig. 13.8, the minimum thickness may bedetermined by a modification of Eq. 6.55a with the constantof Ws in this equation replaced by a constant of C, givenin Appendix H, to give:

t = d dC(p/f) (13.27)

where t = minimum thickness of plate, inchesC = constant from Appendix Hp = design pressure, pounds per square inchf = maximum allowable stress

13.14 NOZZLES, OPENINGS, AND REINFORCEMENTS

Nozzles and openings are necessary components of pres-sure vessels for the process industries. Openings in acylindrical shell, conical section, or closure may producestress concentrations adjacent to the opening and weakenthat portion of the vessel. In order to minimize such stressconcentrations, it is preferable that. the openings be circularin shape. As a second choice the openings may be madeelliptical, or as a third choice they may be made obround.An obround opening has two parallel sides and two semi-circular ends. Openings of other shapes are permissibleif the vessel is tested hydrostatically.

If the opening in a closure of a cylindrical vessel exceedsone half the inside diameter of the shell, the opening andclosure should be fabricated by one of the methods inFig. 13.9.

In reference to Fig. 13.9, the design of the opening includ-ing the knuckle radii and the angle (Y must meet the require-ments for conical or toriconical closures given in section13.10.

If the opening is less than one half the inside diameter ofthe shell, the opening may correspond to one of the designsshown in Fig. 13.10.

Openings in shells and closures other than those of thetype shown in Fig. 13.9 may require reinforcement. Smallsizes of openings welded or brazed to a vessel do not requirereinforcement. The particular sizes IWL requiring reinforce-ment are: 3-in. nominal pipe size in a shell or closure witha thickness of 96 in. or less, and 2-in. nominal pipe sizein a shell or closure with a thickness greater than N in.Threaded, studded, or expanded connections having anominal pipe size of 2 in. or less also do not requirereinforcement.

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2 6 2 Design of Pressure Vessels to Code Specifications

-dd

(4 (‘4

c= 0.25

W)

C-=0.30+ q.p

fs)

C- 0.25

(4

c=o.30+ qp

00

breadedring

c = 0.30

04

I

C= 0.30

w

tl E 2 times required thickness ofa seamless shell, but new lessthan 1.

c = 0.50

(0

ectionatring

Fig. 13.8. Some acceptable types of flat heads and covers. (Extracted from the 1956 edition of the ASME Boiler and Pressure Vessel Code, Unfired Preaum

Vessels, with permission of the publisher, the American Society of Mechanical Engineers [l 11.) \

TT - -

Page 273: Process Equipment Design

Nozzles, Openings, and Reinforcements 263

A 12-in., extra-heavy, lap-joint stub-end nozzle, con-forming to specification SA-105, Grade I, is attached bywelding to a vessel that has an inside diameter of 60 in.and a shell thickness of K in. The material of the shelland reinforcement plate conforms to specification SA-201,Grade B. The vessel operates at 250 psi and 700” F.There is no allowance for corrosion, and the vessel is notstress-relieved. Check the adequacy of the reinforcementplate and the attachment welds shown in Fig. 13.11.

Wall thicknesses required:

‘0 = 0 12R, mmimum

Fig. 13.9. Large head openings-reverse-curve and conical shell-reducer

’ sections., (Extracted from the 1956 edition of the AWE Boiler ad Pressure

Vessel Code, Unfired Pressure Verse/s, with permission of the publisher, the

American Society of Mechanical Engineers [l I].)

J13.140 Reinforcement for Openings in Shel ls and

Formed Closures. In the case of a shell opening requiring* reinforcement in a vessel under internal pressure, the metal

removed in the axial plane of the shell must be replacedby the metal of reinforcement. The required reinforcingarea for an opening in a vessel under internal pressure isgiven by Eq. 13.28. If the vessel is under external pres-

! sure, only 50 % of the area specified by Eq. 13.28 is required.

a=dXt (13.28)

where a = cross-sectional area of reinforcement, squareinches

d = diameter of opening (in corroded condition),inches

t = thickness of shell, inches

In addition to providing the area of reinforcement, ade-quate welds must be provided to attach the metal of rein-forcement, and the induced stresses must be evaluated.The following example from the code illustrates the pro-cedure (11).

EXAMPLE OF A NALYSIS OF REINFORCING A REA FOR O PEN-ING IN SHELL. The following example is quoted from the1956 edition of the ASME code for unfired pressure vessels(11). This example illustrates the recommended procedurefor calculating the required reinforcement for an openingin a shell.

Shell250 x 30

t, = = 0.528 in.14,350 X 1.0 - 0.6 X 250

,

Nozzle250 x 6

t,., = = 0.106 in.14,350 X 1.0 - 0.6 X 250

Size, of welh required:

Inner fillet weld = 1.41 X 0.7 X 0.375 = 0.375 in.

Reinforcement-plate tfillet weld =. 1.41 X 0.5 X 0.375 = 0.264 in.

The weld sizes used are satisfactory.

Area of reinforcement required:

A = 11.75 X 0.528 = 6.20 sq in.

Area of reinforcement provided:

A1 = 11.75(0.75 - 0.528) = 2.61

Aa = 2 x 1.625(0.563 - 0.106) = 1.29

A3 = 2 X &[(T\)’ + (+)2] = 0.24

Area provided by A1 to A3 = 4.14 sq in.Area required in A4 = 6.20 -- 4.14 = 2.06 sq in.use Aq = 2 X 3.0 X 0.375 = 2.25 sq in.Total area provided = 6.39 sq in.

Load to be carried by welds:

W = (6.20 - 2.61)14,350 = 51,500 lb

Unit stresses:

Shear in fillet weld = 0.46 X 14,350 = 6600 psi

Tension in groove weld = 0.70 X 14,350 = 10,050 psi

Shear in nozzle wall = 0.70 X 14,350 = 10,050 psi

Strength of connection elements:

A, inner fillet weld = 1.57 X 12.75 X 0.375x 6600 = 49,500 lb

B, nozzle neck in shear = 1.57 X 12.25 X 0.50x 10,050 = 96,600 lb

C, groove weld in tension = 1.57 X 12.75 X 0.75x 10,050 = 150,700 lb

D, outer fillet weld = 1.57 X 18.75 X 0.312x 6600 = 60,600 lb

These elements provide sufficient strength so that there

Page 274: Process Equipment Design

264 Design of Pressure Vessels to Code Specifications /- Backingstrlp. d used, may be removed af ter welding. - , L’

(b) (CJ

(e-1) k-2) (fJ1,+ t*= l%l,i”

Typical flush-type nozzles

fmJ (nJ

Fig. 13-10. Some acceptable types of welded nozzles for vessels. (Extracted from the 1956 edition of the ASME Boiler and Pressure Vessel Code, Unfired

Page 275: Process Equipment Design

Nozzles, Openings, and Reinforcements 265

(P)

(9-2)

@J

Either method of attachinent is satisfactory

-t,+ ts= lW,i” \t, or ts not less than ?gmin or L”

(u-2)* h-1) (u-2)’ (w-1) ( w - 2 ) ’ w-3) @J

‘For 3” pipe size and smaller, see exemptions

than L”

(.V)Section I-I

(2)

\ Typical tube connections(When used for other than square, round, or oval headers, round off corners)

Pressure Vessels, with permission of the publisher, the American Society of Mechanical Engineers [l l!;;

Page 276: Process Equipment Design

264 Design of Pressure Vessels to Code Specifications

Fig. 13.11. Example of reinforced opening. (Extracted from the 1956

edition of the ASME Boiler and Pressure Vessel Code, Unfired Pressure

Vessels, with permission of the publisher, the American Society of Mechanical

Engineers [l 11.)

is no need of specifying a minimum width of fusion betweenthe groove weld and the reinforcement plate.

Possible paths of failure are (1) through B and D witha design strength of 157,200 lb, (2) through A and C with adesign strength of 200,200 lb, (3) through C and D witha design strength of 211,300 lb. All of these paths arestronger than the required strength of 51,500 lb.

The design strength of the outer fillet weld attachingthe reinforcement plate to the shell is 60,600 lb, or greaterthan the reinforcement plate strength of 2.25 X 14,350 =32,300 lb .

In the case of an opening which is located entirely withinthe spherical portion of a torispherical dished head, thethickness to be used in Eq. 13.28 is calculated by Eq. 7.77with a joint efficiency of 1.00 and a shape factor of 1.00.

If an opening requiring reinforcement is located in aconical closure or section, the thickness to be used inEq. 13.28 is that required for a seamless cone having adiameter, d, measured where the nozzle axis pierces theinside wall of the cone.

13.14b Reinforcement for Openings in Flat-plate Clo-sures. Openings in flat-plate closures in which the size ofthe opening does not exceed one half of the head diameteror the shorter span, as indicated in Fig. 13.8, require onehalf the reinforcing area specified by Eq. 13.28. If theopening exceeds one half the head diameter or the shorterspan, the flat-plate closure should be designed as a flange(see Chapter 12).

13.14~ Limits of Reinforcement. In general the l imits ofreinforcement measured normal to the vessel wall shouldnot exceed the lesser of the fol lowing:

.m, - c)

2Gz - c) + thickness of added reinforcementexclusive of weld metal (13.30)

where tS = shell or head thickness, inchest, = thickness of nozzle wall, inchesc = corrosion allowance, inches

Credit may be taken for any excess thickness present inthe shell or nozzle over that required on the basis of thedesign pressure and corrosion allowance. The area avail-

1 I \I /

able for credi t i s the larger of the fo l lowing:

a = (Et - t,)d (13.31)

a = 2(Et - tJ(t + t,) (13.32)

where a = available area contributing to reinforcement,square inches

d = diameter of opening (in corroded condition),inches

E = joint efficiency= 1.0 i f opening does not pass through welded joint

Credit may also be taken for the cross-sectional area ofthe attachment welds. The required cross-sectional areaof the reinforcing plate may be determined by subtractingthe area credited from excess shell and/or nozzle thicknessand the area credited from the attachment welds from thetotal required area as determined by Eq. 13.28. In caseof multiple openings or threaded connections the codeshould be consulted.

13.15 PRESSURE TESTING OF CODE VESSELS

All pressure vessels designed to code speci f icat ions exceptthose exempted because of small size must be tested eitherhydrostatically, pneumatically, or by means of the “proof-test” (11).

In the case of the hydrostatic test the vessel must be sub-jected to a hydrostatic test pressure at least equal to oneand a half times the maximum allowable pressure at thetest temperature. Following the application of the testpressure, all joints and connections must be inspected withthe vessel under a pressure not less than two thirds of thetest pressure. Although water is normally used in thistest , any nonhazardous l iquid may be used below its boi l ingtemperature.

If the vessels are designed so that they can not safely befilled with water (as in the case of tall vertical towersdesigned to handle vapors) , pneumatic testing may be used.The pneumatic test pressure should be at least one anda half times the maximum allowable pressure at the testtemperature. In conducting the pneumatic test the pres-sure in the vessel should be gradual ly increased to not morethan one half the test pressure. Thereafter the pressureshould be raised in increments of one tenth of the testpressure until the test pressure is reached. Fol lowing th isthe pressure should be reduced to the maximum allowablepressure and held there for a sufficient length of time topermit inspect ion of the vesse l .

The “proof test” can be used to establish the allowableworking pressure in vessels that have parts in which thestresses can not be computed with satisfactory accuracy.In one procedure of this test all areas of probable highstress-concentration are painted with a wash of lime oranother brittle coating. The pressure is raised and thevessel i s inspected for s igns of y ie lding indicated by f lakingor strain lines in the wash. The vessel is rated at the testtemperature at one half the test pressure at which yieldingis f i r s t observed.

Strain gage measurements may be used in nondestructivetest ing. In this case the pressure is increased in increments

Page 277: Process Equipment Design

of one tenth the test pressure, each increment followed byrelaxation of the pressure, until a permanent strain of 0.2 y0is reached. The vessel rating at the test temperature isequal to one half the pressure producing this permanentstrain. A modification of the strain-gage-measurementprocedure known as the displacement-measurement pro-cedure is also permitted by the code. This method involvesthe use of measuring gages at diametrically opposed refer-ence points in a symmetrical structure.

In another version of the proof test a sample vessel istested to destruction and identical vessels are rated at thetest temperature at one fifth the pressure at which thetested vessel failed.

The maximum operating (op.) pressure for a vessel at anelevated temperature is determined by the following

P R O B L E M S

relationship:

Pressure Testing of Code Vessels 267

fPop = Pstm Op

fatm(13.33)

where pop = maximum operating pressure at operating.temperature

fop = allowable working stress at operating temper--ature

fatm = allowable working stress at atmospheric tem-perature (test temperature)

This chapter represents only a condensation of selectedsections of the code. The code should be consulted for-more detailed information on the design of pressure vesselsaccording to code specifications (11).

1. A horizontal storage vessel 16 ft, 0 in. in inside diameter and 32 ft, 0 in. from tangent lineto tangent line is required to process petroleum hydrocarbons at 200 psi and 200” F. Ellipticaldished closures and single-welded butt joints with backing strip are to be used without radio-graphing or thermal stress relieving. Corrosion allowance is 31s in. Determine the totalweight of the vessel shell and closures exclusive of attachments if the steel used is (1) SA-283.Grade C, (2) SA-285, Grade C, (3) SA-212, Grade B. Do all designs meet code specifications 3

2. Estimate the total cost of each of the vessels in problem 1 if four courses are used for theshell.

I 3. Design a vertical vessel 15 ft, 0 in. in outside diameter and 24 ft, 0 in. long from the pointof tangency at the upper torispherical dished head to the junction with the 60”-apex-angleconical closure at the bottom to operate at full vacuum (1) without stiffeners, (2) with stiffeners.The material is to be SA-285, Grade C with a joint efficiency equal to 0.8.

4. Design a spherical dished cover as shown in detail d of Fig. 13.7 for a vessel in whichdimension B is equal to 2434 in. that is to operate at 150 psi and 500” F. The gasket is Ks-in.-thick asbestos composition and has an inside diameter of 24% in. and an outside diameter of2555 in. The bolt-circle diameter, C, is equal to 26% in., and 24 bolts s in. in diameter areto be used. The steel used meets SA-285, Grade C specifications.

5. Determine the thickness of a flat cover plate such as the one shown in detail g of Fig. 13.8which might be used as a substitute for the dished cover described in problem 4, using thesame dimensions and specifications for gasket, bolts, and materials.

6. Specify the reinforcement required for the case of SA-285, Grade C steel if a circularmanway fabricated of 18-in. nominal-size pipe, schedule 20, is used in the top of the vesseldescribed in problem 1. The manway does not intersect a weld seam in the shell.

Page 278: Process Equipment Design

C H A P T E Rm14A

HIGH-PRESSURE MONOBLOC VESSELS

Tu he appl icat ion of high pressure to the chemical processindustries opened a new f ield to the design engineer. Thisrelatively new technique originated about 1913 with theindustr ia l synthesis of ammonia from i ts e lements and withthe Burton process for the cracking of oil. Numerousapplications have been made since then, and the range ofpressures used today extends up to about 50,000 psi . Earlyinvest igators quickly real ized that the factor l imit ing at ta in-able pressures was the mechanical propert ies of the materialsused. As a result t.he behavior of construction materialsunder these working conditions became the object of anextensive study. Rapid progress has been made, and theinformation obtained, as well as the new methods of con-struction, constitutes what is usually called high-pressuretechnique.

Vessels under high-pressure service may fai l as a result of-various types of rupture depending upon the material ofconstruction, method of fabrication, and operating con-dit ions . A simple type of construction consists of a singlethick-walled cylinder with suitable end closures, which istermed monobloc construction. Such a vessel may befabricated without residual stresses or may be deliberatelyprestressed. Some advantages of prestressing are thelowering of the maximum stress level under operatingcondit ions and the reduct ion of required wall thickness .

Several cr i ter ia for the fa i lure of nonprestressed monoblocvessels are presented and are compared both theoreticallyand with rupture data; this is followed by a discussion ofthe behavior of such vessels at elevat.ed temperatures. Theadvantages of prestressing and rnet,hods of design usingautofrettage are then considered.

Designing high-pressure equipmen presents many factorsand condit ions to be taken into considerat ion. To mentiononly a few, the following may be listed:

1. Dimensions-diameters, length, and so on, and theirl imi ta t ions .

2. Operating conditions-pressure and temperature.3. Temperature gradient,s in the walls.4. Methods of transferring heat.5. Corrosive nature of reactants and products.6. Type of operation-batch or continuous.7. Number and size of openings and closures in vessels.8. Vertical or horizontal installations.9. Available materials, their physical properties and

cost .10. Types of const,rurt.ion, that is, forged, welded, cast.

Among these the operat ing condit ions are the most impor-tant since almost all the other factors are dependent vari-ables of the working pressure and temperature. A treat-ment of the pressure-stress relationships in the material ispresented here , exclusive of temperature-stress re lat ionships .

14.1 THEORIES OF ELASTIC FAILURE

Elastic failure of a given material may be considered tooccur when the elastic limit of the material is reached.Beyond this limit the specimen is permanently deformed 01ruptured. Of the various theories developed to accountfor elastic failure, four are of special interest.

According to Lamk and Clapeyron (189), it was sufficientto find the greatest principal stresses in order to determinewhen the elastic limit was reached. Saint-Venant (148)considered that the elastic limit was reached when the

Page 279: Process Equipment Design

Lam6 Theory of Stress Analysis for Thick-walled Cylinders 269

sum of the downward forces is equal to (p07rd,2/4) +fa(7r/4)(do2 - h2). Then, by a summation of forces, thefollowing relation must hold:

Fig. 14.1. Stresses in a thick-walled vessel under internal pressure.

maximum strain attained a certain magnitude. Coulomband Tresca (190) maintained that the elastic limit wasreached when the shear stress attained a certain value.Perry and Duguet considered that shear stress togetherwith friction determined when the elastic limit was reached.

In other words, according to the four theories failures willoI:cur when:

1. The maximum principal stress equals the stress at theerastic limit under simple tension.

2. The maximum strain equals the strain at the elasticlimit under simple tension.

3. The maximum shear stress reaches a critical value.t. The strain energy per unit volume reaches a critical

value.

14.2 LAME THEORY OF STRESS ANALYSIS FORTHICK-WALLED CYLINDERS

Consider a thick-walled cylinder with closed ends sub-mitted to an internal and an external pressure, respectivelydenoted by pi and po, with inside and outside diametersof di and-d,. Stresses are induced in the cylinder wall tooppose the pressure effects. Stresses, like forces, possess amagnitude or intensity, a direction, and an orientation, andmay be resolved into three components, each componentperpendicular to the other two components. These com-ponents are the stress projections on the reference axes.The axes used are those illustrated in Fig. 14.1. The threestress components are also indicat.ed on the same figure.They are:

fa, the axial stressfr, the radial stressft, the hoop or tangential stress

14.2a Determination of the Axial Stress. The force ex-erted by the internal pressure in the axial direction is givenby the quantity (pgrdc2)/4. This force is opposed by actionof the external pressure and also by an axial stress set in thecylinder wall. Visualizing forces acting on one half ofthe cylinder, we find that static equilibrium requires that thesum of the forces in the upward direction must equal thesum of the forces in the downward direction. For the upperhalf of the cylinder the upward force is (pgrdi2)/4, and the

andfa = piWi”/4) - p&do”/4

b/‘4)(do2 - di2)

or

fa = pidi2 - p,do2do2 - di2

(14.1)

In this analysis uniform distribution of the axial stress onthe cylinder cross section has been assumed. The axialstress is then a function of the geometry of the cylinder andthe applied pressures. For any given condition the axialstress is constant.

14.2b Determination of the Tangential and RadialStresses. Consider a cross section of the cylinder in adirection perpendicular to the 1 axis, as shown in Fig. 14.2.The radii P and (P + dr) indicate the projection of a thin-walled cylinder of thickness dr in the r-t plane.

Figure 14.2 illustrates a small element of this cylinderand the applied stresses under static equilibrium conditions.

Considering the element shown in Fig. 14.2 to behaveas a thin-walled shell, we find that the following summationof forces can be made about the diametrical plane:

j,(2rl) - (jr + dj,.)2(r + dr)l = 2ft drl (l4.2a)

Eliminating 21 and rearranging gives:

-fr dr - r djT - dr dfr = ft dr

Dividing through by dr gives:

ft = -jr - r $ - df7

Disregarding the small quantity djv gives:

jt = -jr-r$

As the differential thickness is reduced indefinitely, thelimit is reached:

jt = -jr - r z = - $ ( jTr) (14.3)

Fig. 14.2. Relations for o circumferential element of a thick-walled vessel.

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270 High-pressure Monobloc Vessels

According to the theory of elasticity, for one directionalstrain where the stress is below the yield point, Eqs. 6.laand 6.lb apply. Therefore

faEa = -E

where jh = stress in axial direction, pounds per square inchE, = strain in axial direction, inches per inchE = modulus of elasticity, pounds per square inch

With positive internal pressure, the stress in the axialdirection, fat and the stress in the tangential direction willbe tensile stresses whereas the stress in the radial direction,jr, will be a compressive stress.

By Eqs. 9.87 through 9.89, for three-dimensional strainwith proper regard for signs (axial and tangential stress intension, radial stress in compression) following relationshipis obtained:

For a given material stressed within the elastic limitunder a given internal pressure, the quantities eas, ja, and Eare all constants, or

where a is a constant.Substituting for jt by means of Eq. 14.3 gives:

-2jv-r$=2aT

Rearranging gives:

dfr - 2d,.-=-.fr+a r

Integrating gives:

In (jr + a) = -In r2 + Cl

where Cr is a constant of integration.Taking the antilogarithms where antilogarithm Cr = b

gives:b

.&+a=-r2o r

&.=$-a (14.6a)

where b and a are constantsIf the convention of positive values for tensile stresses

and negative values for compressive stresses is adopted, theequation for jr becomes (radial stresses are compressiveunder the influence of internal pressure) :

By Eq. 14.5 with substitution forj,. and with solving forjr,

f,=$+a (14.7)

Since tangential stresses under the influence of internalpressure are positive, Eq. 14.7 gives the tangential stresswith proper sign. To evaluate the constants a and b, theboundary conditions of the shell are applied; these are:

at r = r,, p = p. = fro

at r = r;, p = pi = friSubstituting in Eq. 14.6b for r = r, and for r = r;,

respectively, gives:

-(.frLo = - a - -$ = p.[ 1- (.fr)r=ri = - a - $ = pi

[ I2

Subtracting Eq. 14.8 from Eq. 14.9 gives:

pi-p*,b2ri2 roz

b = ro2ri2h - po)

r. 2 - Pi2o r

b = do2di2(pi - PA

4(do2 - di2)

For the usual condition, in which p. = 0,

b =d 2d.2

4(&l -” di2) pi (14.10bj

Substituting Eq. 14.10a into Eq. 14.8 and solving for agives :

(14.8)

(14.0>

(14.lOa)

[

ro2ri2(pi - PO)

(ro2 - ri2)ro2 1 -a=p,

therefore

a = [ri~o~~r~)] -po

By expanding, Eq. 14.1 is obtained as follows:

a = Piri2 - poro2 = p&i2 - podo2r. 2 - ri2 do2 - di2

= fa (14.11a)

For the case where p. = 0,

(14.11b)

Substituting the constants into Eqs. 14.6b and 14.7 gives:

The mathematical stress relationships-given above were

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Lam& Theory of Stress Analysis for Thick-walled Cylinders

The constant b is given by Eq. 14.10b.

271

\ i-0---

3 r.-‘-

Y3!-

\

.--,

4fb)

Fig. 14.3. Tangential stress distribution in thick-walled cylinders.

originally stated by Lamk (189), and the results are applica-L ble within the elastic region. From the expressions obtained

for the axial, radial, and tangential stresses, it is seen thatfor given conditions fa is constant and ft and f,. are inverselyproportional to the square of the radius. It is then possibleto give a graphical illustration of the stress distributionwithin the cylinder wall.

i14.2~ Stress Distribution in a Cylindrical Shell.

TANGENTIAL STRESS DISTRIBUTION. To obtain a graphi-

I cal presentation of the variation of the tangential stresswith the radius, the limits of Eq. 14.7 may be considered.As r approaches zero, the tangential stress approaches plusinfinity; and as r approaches infinity, the tangential stressapproaches a, as indicated in detail a of Fig. 14.3.

Detail b of Fig. 14.3 is a representation of the tangentialstress variations in the wall of a thick-walled vessel. Asthe internal pressure is usually greater than the externalpressure, the tangential stress is positive and has a maxi-

, mum value at the inside surface of the cylinder.RADIAL STRESS DISTRIBUTION. To obtain a graphical

presentation of the variation of the radial stress with theradius, the limits of Eq. 14.6b may be considered. As rapproaches zero, fr approaches minus infinity; and fr isequal to zero where r = &&, as indicated in detail a ofFig. 14.4, for the general case where p. # 0. If p. = 0,then r, = a. See detail b of Fig. 14.4.

Examination of Fig. 14.4 indicates that the radial stressis compressive and has a maximum value at the insidesurface of the cylinder.

14.2d Example Calculat ion 14.1 , Based upon LamiTheory. A thick-walled alloy-steel vessel of monobloc con-struction having an inside diameter of 12 in. and an outsidediameter of 23xs in. is subjected to an internal pressureof 20,000 psi. The tangential and radial stress variationsin the wall are desired.

The Lam& relationship for tangential stresses in a thick-walled vessel is given by Eq. 14.7:

d 2d.2b = z Pa

4(d12 - di2)

Substituting gives:

do = 23+78 in.

di = 12 in.

thereforepi = 20,000 psi

b = 975,800

The constant a is given by Eq. 14.1Ib.

pidi2a _do2 - di2

Substituting gives:

pi = 20,000 psi

di = 12 in.

do = 23~‘~ in.therefore

a = 7106

Therefore, Lamb’s equation for the tangential stress vari-ations in the wall of this vessel becomes:

ft = ‘F + 7106

The tangential stresses at various points in the shell are:

ftcr=s in.) = 34,212 psi (inside surface)

ft(r =7.5 in.) = 24,454 psi

ftcr=9xis in.) = 18,666 psi

ft+11.719 i n . ) = 14,212 psi (outer surface)

It is apparent that in this vessel the hoop stress at theouter surface is only 41.5% of the maximum stress whichexists at the inner surface.

r0----- --

_/

ri.-- ----

ir

‘0 1’-m-s--

ri- -

&-CT

/k

,’/

--)

:

i0

r

ft=s+aFig. 14.4. Radial stress distribution in thick-walled cylinders.

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2 7 2 High-pressure Monobloc Vessels

The Lam& relationship for the radial stresses in a thick-walled vessel is given by Eq. 14.6h.

&=a-;

Therefore, for d = do,

and for d = di

This relat.ionship for the vessel under considerationreduces to:

fr = 7106 - y

The radial stresses at various points in the shell are:

f+.=~j~.) = -20,000 psi

frcT = 7.5 in,) = - 10,240 psi

fr(T=g4ie in,) = -4450 psi

.f+= 11.719 i n . ) = 0 psi

14.3 CRITERIA FOR SHELL FAILURE BASED ON THEORYOF ELASTICITY

14.30 Maximum-principal -stress Theory. Accordingto the maximum-principal-stress theory, failure is con-sidered to occur when any one of the three principal stresses,Jo, jf, or f,., reaches the st,ress equal to the elastic limit takenas the yield point (jY.p.) of the material.

By Eq. 14.1

IBy Eq. 14.12

By Eq. 14.13

A comparison of the above three equations indicatesthatft is the largest and therefore is the limiting factor to betaken into considerat,ion for design. At failure the followingequality is assumed to hold:

fi(max) = f,.,.

Inspection of Eq. 14.12 shows that. the maximum valueofft is obtained at the inside wall, that is, where d = di andp0 = 0 gage (atmospheric pressure).

Rewriting Eq. 14.12 for p. = 0 gives:

Pidi2ft -do2 - di2

Let

theu

do K-=4

fi(d=d.) = AK2 + l)* K2 - 1Therefore

.ft(*nax) = fy.p. = P i g[ I

(14.14a)

Equation 14.14a gives the stress at failure according tothis theory. For design purposes a factor of safety, X, maybe introduced so that the induced stress will be less thanthe elastic limit, jy.r., of the material in question. Fordesign purposes Eq. 14.14a would be written:

(14.14h)

where X = factor of safety

Equation 14.14b may be solved for K to give:

K =J

(f,.dx&) + ’(fy.,.hPi) - 1

(14.14c)

Equation 14.14a is the so-called maximum-principal-stress equation and is also known as the Lame equation.It may be noted that Poisson’s ratio does not appear in thisequation. In many cases it has been found that the equa-tions resulting from the application of the maximum-strainand maximum-strain-energy theories are in much betteragreement with the experimental results than the so-calledLame equation. These equations contain Poisson’s ratio,and many authors have come to the conclusion that theLame equation is not theoretically correct because it doesnot include this ratio. On the contrary, it has been clearlyshown that this equation follows from a rigorous mathe-matical stress analysis in which the lateral contraction, ofwhich Poisson’s ratio is a measure, has been taken intoaccount.

14.3b Maximum-shear-stress Theory. This second cri-terion postulates that failure should occur when the maxi-mum shear stress equals the shear stress set up in thematerial at the elastic limit (taken as the yield point, fY.r.).Disregarding the effect of the axial stress, ja, we lind thatthe radial and tangential stresses form a two-dimensionalsystem. Knowing that the maximum shear stress is equalto the algebraic difference between the stresses considered,we find that the corresponding maximum shear stress isgiven by Eq. 9.85.

fs(n,ax) = at - “fr)n,ax = &fy.*.

The shear stress at radius r is:

fw = at - .&jr

Substituting for ft by Eq. 14.7 and for j,. by Eq. 14.6bgives :

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! Criteria for Shell Failure Based on Theory of Elasticity 273

therefore again with the proper regard for sign convention, gives:

f Ji87 = -

Substituting for b by Eq. U.lOb for the (‘ase in which the thereforeexternal pressure is zero, and noting that js is a maximumwhen r = di/2, we find by Eq. 9.85 that.

fdo2Pi

Max) = do2 _ dig = ify.,,. (14.16)

5 = I(4

Let

then

fy.,. = i&y 1 Pi = ?fd*nax) (14.17a)

The term js(max) in Eq. 14.17a is the induced shear stressthat exists at failure in accordance with this theory and isnumerically equal to one half of the tensile stress in simpletension at the elastic limit. This theory may be used fordesign if a factor of safety, X, is included.

Therefore, for design purposes Eq. 14.17a may be writtenas:

(14.17b)

In discussing the strength of cylinders under high pres-sure, Manning (237) states: “Overstrain of a cylindricalwall occurs when the maximum shear stress reaches avalue l/d times the tensile upper yield stress. . . . Themost satisfactory basis of design is arranging for the maxi-mum shear stress . . . to equal (when raised if required byan appropriate safety factor) the tensile yield stressdivided by &.” Equation 14.16 may be modified(equated to &./fi instead of jY.P./2) to give:

(14.17c)

Fo: purposes of design Eq. 14.17~ may be solved for Kas follows:

Examination of Eq. 14.18 indicates that the st.rain ismaximum when P is equal to di/2. Therefore

Substituting for u by Eq. 14.1lb and for b by Eq. 14. IObgives:

pi%(rn%X) = -

(1 - /.+A2 + (1 + ddo2

E 1 (14.19)

To satisfy the criterion that. failure occurs when tl,(mnx)equals the st,rain at the elastic limit we let

thereforeet2 = ey.p.

fy.,. = E+.p. = &n(n,nx,

Subst.ituting for d,/di = K gives:

Equation 14.20a states the conditions at. which failure isassumed to occur according to this theory. For designpurposes a safety factor, X, is introduced into this equat.ionfor proportioning the vessel. Thus, for design purposesEq. 14.20a can be written as:

Equation 14.20b may be solved for K to give:...~___

K = (fy.p.lxPij t (l - Cc)(fY.P./~Pi) - (1 + 4

(14.20~)

K =J_ fY.P. -_

- XPi v5(14.17d)

fy.1,.

Equations 14.17~ and 14.17d are based on the shear-strain-energy hypothesis (237).

14.3~ Maximum-strain Theory. In this third case,rupture is considered to occur whenever the strain set upin the material reaches the strain at, the elastic limit. Inthe discussion of the maximum-principal-stress theory itwas shown that the tangential stress, jt, was the maximumor limit,ing stress. Consequently, within the elastic range,et is the limiting strain to consider in design.

By Eq. 6.4

14.3d Maximum-strain-energy Theory. St.rain energyrefers to the mechanical energy absorbed by a body stressedwithin the elastic range. According t,o this fourth criterion,the strain energy accumulated in the material when it isstressed to its elastic limit is the det.ermining factor forrupture. Tn Chapter 2, section 2.4a, the strain energywas shown to be equal to the work done on the material(see Eq. 2.26). The work done in a one-dimensionalstress system is given by:

(2.26)

For a two-dimensional stress system it has been shoWnthat the strain in either direction is the algebraic sum of 11letwo components, as given by Eq. 6.4.

Substituting for jt by Eq. 14.7 and for jr by Eq. 14.6b, E.z2 = f [.f, - /4-t/1 (6.A)

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2 7 4 High-pressure Monobloc Vessels

Substituting (l/E)[f, - pfV] for (f,/E) in Eq. 2.26 gives: criterion, we let

u f 2. .Ir(max) = $!j- (14.28:

ua! = & u-z2 - kf&/l (14.21)

It then follows thatLikewise. for the y direction,

f,.,. =wu - PM4 + (1 + /4d04

(d,c-- di2)2(14.29)

therefore

uz, = & LL2 +.f,” - w.&1 (14.23)Letting do/di = K, substituting p = 0.25 for steel, and

multiplying the numerator and denominator by 2 gives:In an analogous manner it can be shown that the strain

energy for a three-dimensional system is given by:

U q/z = & lfz2 +.h2 +.fz2 - P(fi/fi +.fi.f, + fifi)l(14.24)

This last expression has been called the strain-energyfunction.

If Eq. 14.23 is applied to the two-dimensional stresssystem formed by ft and fv in the case of the cylinder wall,the strain energy is given by:

f46 + 10K4

Y.P. = pi 2(K2 - 1)(14.30aj

Equation 14.30a gives the conditions at which failure isassumed to occur by the maximum-strain-energy theory.For design purposes a factor of safety, X, is included tcproportion a vessel. Equation 14.30a can then be writtenas:

fAPi 46 + 10K4

Y.P. = 2(K2 - 1)(14.30b)

1ut, = - [A2 +A2 - a-dtfrl2E

14.3e Comparison of the Four Theories of Failure withExperimental-test Results. Newitt (191) reported a num-ber of experimental tests on mild steel in which the elasticstrain of a cylinder under pressure was measured andplotted as a function of internal pressure. He found thatthe strain was proportional to the pressure until the elasticlimit was reached. This procedure gave values of stress,strain, and internal pressure at the elastic limit. Thesetests were made on a number of vessels having different Kratios (from K = 1:35 to K = 3:65). The experimentalvalue of p/fy.p. was then compared with the values pre-dicted by the four different theories. Table 14.1 sum-marizes the results obtained with mild steel.

By making the substitutions for ft and f,., Ut, may beexpressed as a function of the radius of the cylinder, signconvention being used,

ut~=~[(u+~)2+(u-~)-++$)(a--$)] (14.26)

Carrying out the algebraic operations and simplifying,we obtain:

ut.=g2+;-p(u2-;)]

= ;[

(1 - p)a2 + (1 + cc) 91Table 14.1. Results of Tests on Mild-steel

Cylinders (191)

streas atRatio of Yield in Yield Experi-Externa l S imple Pressure menta lt o In ter - Tens i on in Crlin- Va lue

Calculated Valuea of P/&..~.According to

Max- MU- Max- M&X-The strain energy is a function of the fourth power of theradius and has its maximum value at the inside wall whereP = dJ2. Therefore

prin.- prin.- shear- strein-stress strain stress energy

Theory Theory Theory Theory

0.291 0.295 0.225 0.2620.402 0.393 0.287 0.3440.430 0.415 0.300 0.3630.430 0.415 0.300 0.363

nal Di-ameter

1:351:531:581:581:741:771:791:791:791:791:861:972:192~192:452~662:883:053~263:65

(fy.p.)* der -(p) , o fIh/sq in. lb/w in. p/fy.p.35,300 9,700 0.275

U Ir(Irax) = -;[

(1 - /da2 + (1 + cc) $z 1 35,300 12,000 0.340

35,300 12,500 0.35435,300 12,500 0.35435,300 14,700 0.41635.300 14.400 0.407

0.506 0.475 0.336 0.41:0.515 0.483 0.340 0.417By making the substitutions for the constants a and b by

Eqs. 14.11b and 14.10b, the expression becomes: 35;300 15;400 0.43615,200 0.43015.400 0.436

0.525 0.490 0.344 0.4220.525 0.490 0.344 0.4220.525 0.490 0.344 0.4220.525 0.490 0.344 0.4220.554 0.511 0.356 0.4490.590 0.539 0.372 0.4600.655 0.583 0.395 0.4940.655 0.583 0.395 0.4940.713 0.625 0.416 0.522

35,30035.300

lilr(max, = JE (1 - /.tL) (d ry;,2)2 + (1 + cl)[

pi2d040 z (do2 - di2)2 1

U (1 - PL)di4 + (1 f PIdotr(max) (do” di2)2 1 (14.27)

35;30034,00034,00036,86036,86036,860

14;600 0.41313,600 0.40014.100 0.41518,090 0.49018,090 0.49018,740 0.508

36,86036,86036,86036,86036,860

20,150 0.546 0.752 0.649 0.429 0.53920.300 0.550 0.784 0.672 0.439 0.553

The strain energy of the material at the elastic limit isgiven by (f,.,. 2/2E) (see Eq. 2.26). To satisfy the fourth

20:200 0.547 0.806 0.684 0.446 0.56221,700 0.588 0.827 0.697 0.452 0.57121,800 0.591 0.860 0.718 0.467 0.583

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Criteria for Shell Failure Based on Theory of Elasticity 2 7 5

Table 14.2. Results of Tests on High-tensile-steel Cylinders (192)

Tens i leElasticLimit ,(fs.,.),

K tons/sq in.

Yield Pressure inCylinder (p),

tons/sq in.11.4811.509.00

12.1011.9314.1012.6812.10

p/j,,,,,. Calculated According toMax- Max- Max- Max-nrin.- nrin.- shear- strain-

Nickel steel 2.15 28.08Nickel steel 2.15 28.94Nickel steel 2.50 21.31Nickel steel 2.50 28.80Nickel-chromium steel 2.00 29.57Nickel-chromium-molybdenum steel 2.00 37.62Nickel-chromium-molybdenum steel 2.00 33.44Nickel-chromium-molybdenum steel 2.00 32.45

1

s t r e s s s t r a i nTheory TheoryP/f,.,,.

0.4090.3970.4220.4200.4040.3740.3790.373

0.644 0.5770.644 0.5770.724 0.6310.724 0.6310.600 0.5460.600 0.5460.600 0.5460.600 0.546

s t r e s s energyT h e o r y T h e o r y0.392 0.4880.392 0.4880.420 0.5270.420 0.5270.375 0.4660.375 0.4660.375 0.4660.375 0.466

Class of Steel

A comparison of the experimental values of p/fy.*. andthe theoret ical predict ions shows that the c losest agreement ,between theory and experimental value was obtained withthe strain-energy equation.

Similar tests were also made on a variety of high-tensile-steel cylinders by Macrae (192). The results of these testsare given in Table 14.2.

A comparison of the experimental values of ~/fy.~. andthose predicted by the various theories shows that in thecase of high-tensile steels the best agreement is obtainedby use of the maximum-shear-stress equation. This might

I ‘nave been anticipated because these materials have shearstrengths which are quite low in comparison with their

i tens i le s t rengths .Cook and Robertson (193) reported similar information

on cast-iron cylinders. However, in their study the pres-sure was increased until the cylinders ruptured becausecast iron does not have a well-defined yield point . Resultscf these tests are given in Table 14.3.

In Table 14.3 only the comparison between the experi-mental value of p/j and that predicted by the maximum-principal-stress theory is given. The fact that the agree-ment i s good indicates that th is mater ia l fo l lows this theory.

14.3f Comparison of the Lam6 Theory with the Mem-brane Theory. The membrane equation for hoop stress isgiven in by Eq. 3.14 as:

Rewriting in terms of jt with t = (rO - PJ gives:

ft = & (14.31)

I where r,, = outside radius of shell, inchesri = inside radius of shell, inches

If the ratio of r,,/‘ri equals K, then Eq. 14.31 indicatest.hat the hoop stress determined by the membrane equationbecomes:

Dft = Pi L

CK - 1)(14.32)

and Eq. 14.14 indicates that the hoop stress determined by

the Lame equation becomes:

ft = pi (K2 + 1)(K2 - 1)

(See Eq. 1.4.14.)

The ratio of jt/pi may be conveniently plotted against Kas shown by Maccary and Fey (194) and as indicated inFig. 14.5. The determination of shell thickness using theLame equation involves calculation by successive approxi-mation. The same calculation using the membrane equa-tion is more convenient, being a direct calculation, but islimited in its application to vessels in which t/o$ is equalto or less than 0.10.

The range of the membrane equation has been extendedby the empirical modification of adding the constant 0.6.This new equation is known as the ASME modified mem-brane equation and is in much closer agreement with theLam& equation. At a t/d+ value of 0.25 the ASME modifiedmembrane equation agrees with the Lame equation within1% (194). The ASME modified membrane equation is:

.ft- = & + 0.6 (14.33)Pi

or if welded-joint efficiency and corrosion allowance areincluded (ll), it is:

t=jtE yO.Sp’+ ’

(14.34)

14.39 Graphical Comparisons of the Various Theories.A graphical comparison of the membrane theory, the ASMEmodified membrane theory, the principal-stress thenry, the

Table 14.3.

Exter- Inter- calcu-n a l ml T e n s i l e Bursting lated

Dinm- D i a m - Strength Pressure Bursting Calcu-e t e r , eter, (f), (PI 9 Pressure, Observed latedin. in. K Ib/aqin. Ib/sqio. lb/aqin. P/f P/f

1.133 0.873 1.30 1 8 , 6 0 0 5 , 0 6 0 4 , 7 6 0 0.272 0.2561.420 0.923 1.54 2 4 , 5 0 0 9 , 5 2 0 9 , 9 5 0 0.388 I l . 4 0 61.390 0.755 1.83 2 3 , 5 5 0 1 3 , 0 0 0 1 2 , 7 1 0 0.552 0.5401.710 0.922 1.85 2 6 , 9 0 0 1 4 , 5 5 0 1 4 , 8 0 0 0.540 0.55n1.561 0.793 1.97 2 4 , 2 0 0 1 5 , 1 0 0 1 4 , 3 0 0 0.623 0.5901.475 0.750 1.97 2 4 , 7 5 0 1 6 , 4 6 0 1 4 , 6 0 0 0.665 0.5901.516 0.635 2.40 2 6 , 7 0 0 1 9 , 2 5 0 1 8 , 8 0 0 0.720 0.7041.870 0.630 2.96 2 1 , 7 0 0 1 7 , 4 1 0 1 7 , 3 0 0 0.802 0.796

Results of Tests on Cast-iron

Cylinders (193)

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276 High-pressure Monobloc Vessels

C-IQ.II

\ I I I I I I I I I

1 0 \;\ I

;\ \

6 \?, I

4 I\\ I I I I I I I

0.6 1 I I I I I I I I 1

0.4

0.31 . 0 1 . 2 1 . 4 1 . 6 1.8 2.0 2.2 2.4 2.6 2.8

K = Outside diameterInside diameter

maximum-shear-stress theory, the maximum-strain theory,and the maximum-strain-energy theory is shown in Fig. 14.6.

14.3h Example Calculation 14.2, Comparing the FourTheories of Failure. A vessel is to be designed to with-stand an internal pressure of 20,000 psi. An internaldiameter of 12 in. is specified, and a steel having a yieldpoint of 70,000 psi has been selected. Calculate the wallthickness required by the various theories with a factor ofsafety of X = 1.5.

Maximum-principal-stress Theory:By Eq. 14.14~

thereforeK = 1.58

a!,, = (1.58)(12) = 19 in.

thereforet = 3.5 in.

Using Fig. 14.6 for Ap/f = (1.5 X 20,000)/70,000 = 0.428and reading from the principal-stress curve, we obtain:

L-34t .therefore

t = +4 = g = 3.53 in.

Maximum-shear-dress Theory:By Eq. 14.17d

Fig. 14.5 . A compar ison of the

Lam&tangential stress with mem-

brane-theory tangential stress (194).

(Courtesy of McGraw-Hill Publishing

Co.)

K =d

fY.P.

f,.,. - APi fi

J---

70,000=70,000 - 1.5 x 20,000 x 1.732

= 1.97

d, = (1.97)(12) = 23.6 in.

thereforet = 5.8 in.

Using Fig. 14.6 for X~i/fy,~. = 0.428 and reading from ~hrmaximum-shear-stress curve (fsmex = j,.,./ti), we obi.ain:

T&L20t .

therefore

t = g = 6.0 in.

Maximum-strain Theory:By Eq. 14.20~

K = (fy.p./APi) + (1 - PI>

(fy.,,./~Pi) - (1 + Pcl;

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Criteria for Shell Failure Based on Theory of Elasticity 2 7 7

thereforeK = 1.69

t = g = 4.14

d, = (1.69)(12) = 20.3 in.therefore Maximum-strain-energy Theory:

, t = 4.15 in. By Eq. 14.30b

I Referring to Fig. 14.6 for X&f,,,,, = 0.428, we obtain:

di = 2.9I

1.00 I--

F ig. 14.6. Compar ison of var i-

ous theories for shell design.

0.950.90

0.850.800.75

0.70

0.65

0.60

0.55

0.50

0.45

0.40

0.35xl!f

0.30

0.25

0.20

0.15

0.10

-

==G

--

tlm-tI I I IIll. I

1 Thin-wall equation2 ASME modified equation3 Maximum-principal-stress equation (Lam&)4 Maximum-strain equation (M = 0.25)5 Maximum-strain-energy equation6 Maximum-shear-stress

equation Cfs max = fY,P, I437 Maximum-shear-stress

equation Cfs milx = fY,P. /2)

> 0.3 0.4 0.5 0.6 0.70.8 1 . 0 1 . 5 2 3 4 5 6 7 8910 15dit

--

---

-

-

-

-

-

-

-

-

-

-

-

-

~\

.?(

Page 288: Process Equipment Design

278 High-pressure Monobloc Vessels

K

1.1681.1671.5002.0003.0004.000

Table 14.4. Stress Conditions at Onset ofOverstrain (237)

Pi Max Tangential Stress Max Shear Stressh-9 (psi) (psi)7,560 49,280 28,6707,820 51,070 29,610

15,680 40,760 28,22021,950 36,470 29,28026,620 33,350 29,97028,220 32,000 30,000

Substituting with pi = 20,000 psi and again usingfv.r, = 70,000, we obtain:

70,000 46 +lOK*= =1.5 x 20,000 2(K2 - 1)

2.33

6 + lOK* = (4.66)2(K2 - 1)2

6 + lOK* = 21.7K4 - 43.4K2 + 21.7

0 = 11.7K4 - 43.4K2 = 15.7Let K2 = CT.

11.722 - 43.42 + 15.7 = 0

-b& db2-4ac 43.4 f 33.95=

2a = (2)(11.7)

For a maximum value z = 3.3; therefore

K = dg = 1.82

d, = (12)(1.82) = 21.8 in.

t = 4.9 in.

From Fig. 14 .6 , reading from the maximum-strain-energycurve at Xp/f,.,, = 0.428, we find that

di = 2.4

t = 12 = SOin2.4 ' '

Discussion:The required thicknesses of the vessel shell under con-

sideration according t,o the equations for the four theoriesare as fol lows:

T h e o r yMaximum-shearMaximum-strain-energyMaximum-stra inMaximum-principal -s tress

Thickness (in.)5.84.94.153.5

I t i s obvious that the var ious theories give widely di f ferentanswers for the required thickness. Experimental resultshave agreed with one theory when work was being donewith one material and with another theory when work wasbeing done with another material. When working with anew material, it is important to make a preliminary testwith a vessel constructed of the mater ia l . The experimentalvessel may be hydrostatically tested until a permanent set

is obtained. The pressure measured at the beginning ofpermanent set can be compared with that predicted by thevarious theories , and the most appropriate equation selected.

In the case of the example calculations presented above asteel with a tensile yield point of 70,000 psi was specified.This would be classified as a high-strength tensile steel andwould be expected to fail by shear as indicated by Macrae(192) (also see Table 14.2). Manning (237) concludes thatsteels with high tensile strength would be expected to failby shear. Table 14.4 lists the maximum shear stress andthe maximum tensi le s tress at which overstrain was observedto occur for vessels of various K ratios fabricated of high-tens i le -s t rength s tee l .

Table 14.4 shows that onset of overstrain occurred whenthe maximum shear stress reached a mean value of 29,300 psi(+ about 3 %), whereas the maximum tensile stress varieswidely. Furthermore, Manning states that judging on thecriterion of onset of overstrain when the shear stress equalsthe tensile yield stress divided by 43, one would predictonset of overstrain when the shear stress reached 30,200 psi ,which is less than 235 ‘% higher than the mean experimentalvalue.

In the case of a vessel of monobloc construction the termK2/(K2 - 1) in Eq. 14.17~ approaches unity at high valuesof K, and the maximum internal pressure, pi, becomesequal to jY.JX & as a limiting condition. For theprevious problem, in which j,.,. was 70,000, this limitingpressure would be 27,000 psi. This is a serious restrictionbut may be circumvented by other procedures such asprestress ing and mult i layer construct ion.

14.4 CRITERIA FOR SHELL FAILURE BASED ON THETHEORY OF PLASTICITY

The theory of the plastic behavior of materials is ingeneral beyond the scope of this text. A number of excel-lent texts are available on this subject (195-200). How-ever, some of the relationships developed from the theoryof plast ic i ty appear to explain the yield and burst ing charac-teristics of thick-walled cylinders better than any of therelationships based upon the theory of elasticity. There-fore, a limited discussion on the plastic failure of vessels isconsidered appropriate.

On subject ing a thick-walled vessel to increasing internalpressure, stresses are induced in the shell which are maxi-mum at the bore, as predicted by the various criteria forfailure based upon the theory of elasticity. Contrary towhat might be expected, failure of a shell of ducti metalusually does not begin at the fibers along the bore but atthe fibers along the outside surface of the shell (217). O nstressing beyond the yield point, most metals pass througha region of plast ic f low in which e longat ion progresses with-out an increase in resisting stress. This condition is firstreached in the inner part of the cylinder. However, thest,rain of the inner zone is limited by the outer zone, whichis not strained beyond the yield point; thus the inner fibersare incapable of rupture. The inner fibers of an over-stressed vessel often show evidence of slip where failurebegan but halted because of the restraint offered by theouter f ibers . The inner f ibers therefore are prevented fromfailing, provided that the outer fibers offer suflicient

Page 289: Process Equipment Design

Criteria for Shell Failure Based on the Theory of Plasticity 279

restraint. There is no such protection of the outer fibersby the inner fibers.

Manning has discussed the rupture of thick-walled cylin-ders of ductile metal (203). The pressure necessary for theyield point for the metal fibers in the bore to be reached isknown as the “elastic-breakdown” pressure. At this pres-sure the maximum fiber stress is the tangential stress atthe inner surface. The radial stress also has its maximumvalue at the bore, and this stress is equal to the internalpressure. As the pressure is raised, the region of plasticflow, termed “overstram, ” moves radially outward andcauses the tangential stress to decrease in the inner layersand to increase rapidly in the outer layers. Progressiveincrease in pressure moves the elastic-plastic interfaceradially outward until the interface reaches the outer radiusand no elastic zone remains. In this situation the maxi-mum hoop stress is at the outside surface. Manning hasreported (203) that for the beginning of overstrain for avessel in which the outside diameter to inside diameterratio was 2: 1 and the pressure was 12,750 psi, the tangentialstress at the inner radius was 21,000 psi, and at the outerradius 8000 psi. For the same vessel with 100 y0 overstrain(plastic-elastic interface at rO) at a pressure of 27,620 psi,the tangential stress at the bore was 16,000 psi, and at theouter surface 34,000 psi. Thus it is apparent that thetangential-stress distribution is totally different in the 100 y0plastic state than in the completely elastic state. On theother hand, the radial-stress patterns have similar shapesfor the completely elastic and completely plastic states.

When failure occurs in the shell of a ductile metal asthe result of progressive increase in stress, it usually followsthe path of a continuous helix from the outer surfaceinward, as shown in Fig. 14.7.

Prager and Hodge (195) have defined the internal pres-sure in a cylindrical vessel that is required to place theelastic-plastic interface on the outside surface of the vessel.This is the pressure required to place all of the vessel wallbeyond the yield point. In deriving this relationship it isnecessary to establish the condition under which plasticflow is initiated. A widely used yield criterion is that ofVon Mises (205). This criterion can be expressed by thefollowing relationship:

fy.,. = f&Y. Ah (14.35)

where f,.,. = yield-point stress of the material in simpletension, pounds per square inch

fs.v. = yield limit in simple shear, pounds per squareinch

It was shown previously, by Eq. 9.85, that the maximumshear stress in a three-component system is:

Combining the above relationships with Eq. 14.35 gives:

ft - fr

( >

_ fY.P. (14.36)2 Ah

By making allowance for sign convention with the com-pressive stress, fr, negative, the Eq. 14.2b may be written as:

ft = f?. + I’: (14.37)

Substituting for ft in Eq. 14.37 by Eq. 14.36 gives:

2fy.p. drdfr = r -v3 r

Integration of Eq. 14.38 gives:

P= 2fy.p. In 24 Pi

(14.39)

where p = internal pressure required to stress the outersurface to the yield point, pounds per squareinch

fY.P. = yield point of the shell material in single tension,pounds per square inch

r, = outside radius of the vesselri = inside radius of the vessel

Equation 14.39 was derived for an ideal plastic solid,that is, a material that has a stress-strain diagram illus-trated by the “idealized” curve of Fig. 2.11. For thisideal condition the yield strength and tensile strength (t.s.)have the same value; therefore, the bursting strength wouldhe that predicted by either Eq. 14.39 or Eq. 14.40.

pS.%ln~ (14.40)

where ft.*. = ultimate tensile strength, pounds per squareinch

Fig. 14.7. Typical failure as a result of overstraining of thick-walled cylin-

der of ductile metal. (Courtesy of J. H. Faupel [201].) (Extracted from

Transactions of fhe ASMF with permission of the publisher, the American

Society of Mechanical Engineers.)

Page 290: Process Equipment Design

280 High-pressure Monobloc Vessels

30

Y (

E.B. = Elastic breakdown at bore

E.B. O.S. = Overstrain through the wall

10 - 2o E.B. B. = Bursting (x = bursting for experimental curve)- Calculated

10 - - - - E x p e r i m e n t a l

0, 00Jt4000L2000~

0External hoop strain, microin./in.

Fig. 14.8. P r e s s u r e - v e r s u s - s t r a i n c u r v e s f o r t h i c k - w a l l e d c y l i n d e r s u n d e r i n t e r n a l p r e s s u r e ( 2 1 5 ) . ( E x t r a c t e d f r o m T r a n s a c t i o n s o f the ASME w i t h p e r m i s s i o no t t h e p u b l i s h e r , t h e A m e r i c a n S o c i e t y o f M e c h a n i c a l E n g i n e e r s . )

T r e a t m e n t a n d P r o p e r t i e s o f C y l i n d e r s

C y l i n d e rN o .

39

1 01 1

M a t e r i a l a n d T r e a t m e n tQuenched-and-tempered

and quenched SAE 3320Annealed SAE 1035Annealed SAE 3320Annealed Cr-Ni-MO-V

WallR a t i o

2.742.752.752.75

In the actual case for ordinary metals the ultimate tensilestrength is appreciably higher than the yield strength, andthe stress at bursting will lie between the yield and ultimatestrengths. Faupel (201) has proposed that Eqs. 14.39 and14.40 be modified as follows:

p=~(22$$31n~)+(I-~)($$1n~) (14.41)

Equation 14.41 reduces to:

p=%[ln??][2-k] (14.42)

Equations 14.41 and 14.42 proportionally weight thestress values to their ratios; thus, when the ratiofY.,./ft.s. isequal to 0.25, Eq. 14.39 contributes 75% and Eq. 14.40contributes 25y0 to the bursting pressure (201). Faupelreported the tests on the rupture of nearly 100 thick-walledcylinders fabricated from a variety of high-strength steeIsand showed that Eq. 14.42 was reliable within k 15 y0 forpredicting the observed rupture pressure on a 90 O/,-certainty

P r o p e r t i e s i n T r a n s v e r s e D i r e c t i o n a t Bore o f C y l i n d e r -Y i e l d S t r e n g t h , Elongo- Reduc-

psi U l t i m a t e , tion in tion in0.01 % 0.2% S t r e n g t h , 1 in., area,o f f s e t o f f s e t psi p e r c e n t p e r c e n t

74,300 79,500 105,400 2 4 5039,950 38,600 73,100 2 5 3 258,100 68,500 112,600 20 4145,100 45,300 83,300 25 37

basis (201). The vessels were tested at temperatures fromambient to 660” F. Figure 14.8 shows some of the datareported by Faupel and Furbeck (215) on elastic-breakdownpressures and observed and calculated bursting pressures.

Table 14.5 presents selected test results of Faupel (201)and of Crossland and Bones (202) and includes a comparisonof five theoretical methods of predicting bursting pressure,and the observed data.

1

I

a

4

(K - 1)P = ?L (K + 1)In K

5 Manning method (see below and section 14.11j

Page 291: Process Equipment Design

Monobloc Vessels at Elevated Temperatures 281

prior to rupture is dependent upon the time-stress historyof the vessel.

Voorhees, Sliepcevich, and Freeman (204) have presenteda procedure for calculating the time of rupture from creepand stress-rupture data normally available to a designer.Prior to the work of Voorhees the design of thick-walledvessels at high pressures and elevated temperatures wasusually based upon the maximum principal stress and anallowable stress determined from creep and stress-rupturetest data. This is the current method recommended bythe ASME code (11) for vessels operating at pressures upto 3000 psi.

14.50 Equivalent Stress (Shear-stress Invariant).Voorhees’ analysis assumes that the creep-rupture life of avessel under complex stressing is controlled by an equivalentstress, 7, termed the “shear-stress invariant.” This averagestress is also known as the octahedral shear stress, theeffective stress, the intensity of stress, and the quadraticinvariant. The theory for the biaxial-stress condition wasdeveloped by Von Mises (205), and this theory was furtherdeveloped to apply to the triaxial-stress condition inde-pendently by Hencky (206, 207, 208) and by Huber (209).A derivation of the relationship between the equivalentstress, j, and the three principal stresses, fl, fi, and frc wherefl > fi > f3 was given by Eichinger (210). The relation-ship between these stresses is:

7” = +rc.f1 - f2) 2 + (fl - j-d2 + u2 - f3)2! (14.43)

I comparison of the observed rupture pressures inTable 14.5 with those obtained by the five methods of pre-dicting the rupture pressure from theory shows that thebest agreement is obtained in the case of Faupel’s datawith either the Faupel or the Manning method. TheFaupel met,hod makes use of Eq. 14.42, and the Manningmethod makes use of torsion-test data. The Manningmrt,hod involves the graphical integration of the stress-strain curve in a torsion test and cannot be expressed in a

I

single equation (see section 14.11) (203). In the absenceof torsion-test data Eq. 14.42 is recommended for the pre-diction of the bursting strength of thick-walled cylinders.

14 .5 MONOBLOC VESSELS ATELEVATED TEMPERATURES

\I’hen pressure vessels are used at elevated temperatureswith induced stresses within the creep range, the phenome-non of creep must be taken into consideration. A discussionof creep and the use of stress-rupture curves was presentedin Chapter 2, section 2.5.. Thick-walled vessels for high-pressure service have steep

stress gradients. These stress gradients change under theinfluence of creep at elevated temperatures. This redis-tribution of the stress gradients under the influence of creepis known as “stress leveling.” The vessel geometry andthe creep- and rupture-strength properties of the materialof construction influence the degree of redistribution ofstress under the action of creep. The life of the vessel

Table 14.5. Bursting Pressure of Cylinders (201)

Calculated pb (psi)c D RA B-

Mean-Dev. diameter2(%) Method-17 77,500-15 57,000-22 77,000

++; 81,000 75,000-15 74,200-19 97,500-9 57,500

-26 128,500-45 121,000-49 138,500

Dev.(o/o)+9

r;

+:;7

+Yf

+f i:-10-4

Manning5 D e v .Method (% )85,00059,500 ::

89,00070,000 +Z69,000 f2570,000 f4

105,000 +:64,500 +9

150,000191,000 $1157,000 -18

Observed Lamb1psi Pb (Psi) Method

90,480 79,000 65,30068,350 57,000 48,50091,550 83,000 65,00089,700 63,000 68,70080,100 55,000 61,50074,900 67,500 57,300

104,650 98,500 79,750105,650 59,000 53,600137,850 143,000 105,500105,500 168,000 91,200106,700 192,500 97,600

Dev. Classical3 Dev. Faupeld(%) Method (%o) Method-2 95,000 +20 85,700

0 70,500 +24 54,000-7 94,500 +14 84,700

+29 115,000 +83 67,400f36 106,500 +94 63,000+10 96,000 +42 66,500-10 131,000 +43 111,300-3 68,300 +16 63,000

-10 178,000 +24 158,500-28 160,000 -5 152,000-28 191,000 -1 184,000

FaupelTest No.

71314304011$244454647

Crosslandand

Hones (202)‘rest No.

R2.492.432.442.752.762.752.741.752.753.694.71

G_ _ _ _ _-10 29,300-2 18,700

-12 43,700-17 52,000-18 60,000-11 37,000-20 64,500-8 40,400

-17 56,100-23 68,700-12 47,700-26 74,500-28 76,000

J KF27,80018,20039,40045,00049,50034,20052,00036,80047,50054,000

H-6 34,400

0 21,700-2 52,500-4 63,200

0 74,600-4 44,000-1 81,000

0 48,200-2 69,500-2 88,5000 57,800

-2 93,200-4 100,400

-1.571.331.992.292.661.782.901.882.483.182.133.603.72

66,000 31,00066,000 18,64,066,000 44,60066,000 54,00066,000 60,10066,000 38,40066,000 65,30066,000 40,20066,000 57,40066,000 70,00066,000 47,80066,000 76,00066,000 79,000

+11+16+I8+17f24+15

25,800 -17 29,000 -616,300 -13 18,000 039,400 -12 43,500 -247,400 -12 53,000 -256,000 -7 62,000 +332,700 -15 38,000 -160,600 -7 67,000 +336,000 -10 41,000 f252,000 -9 57,000 -166,100 -6 72,000 +343,200 -10 47,000 -273,000 -4 78,000 +374,800 -5 81,000 3-3

2345

I67 $24

+20+21f26+17+23+27

I 84 9101112

42,00056.500

/ 13I

57;oooII.- --I-- ----. - ~..-~-

\ Ye--- \ r 7------ ---~-

Page 292: Process Equipment Design

2 8 2 High-pressure Monobloc Vessels

For a cylindrical pressure vessel under internal pressurethe maximum principal stress, fl, is equal to ft as given byEq. 14.7. The intermediate principal stress, fi, is equal tola as given by Ey. 14.11b. The minimum of the threeprincipal stresses, fs, will be the radial stress, fr, as givenby Eq. 14.6b. By comparing these equations it may beobserved that fa is equal to the arithmetic average of ft andf,., orfa = (f,. + ft)/2. Substituting into Eq. 14.43 gives:

3” = Brut --.a2 + (ft -.M2 + cfa -fr121but

4

therefore

3 = d5 (ft - fa)

At the inside surface of a thick-walled vessel

(14.44)

Substituting for b by Eq. 14.10b gives:

J(r=TI) = di do2pi(do2 - di2)

(14.45)

Or in terms of the outside diameter, do, and thickness, t,

3cr=rij = d (do/02Pi4(4/O - 4

(14.46)

Voorhees analyzed the experimental data obtained onnotched-bar samples tested at elevated temperatures (211,214) and on pressure vessels tested at elevated temperaturesand high pressures (204, 211). He concluded that theequivalent stress, 3 (shear-stress invariant) was more usefulin correlating these experimental data than the maximumprincipal stress or the maximum shear stress. Voorheesalso reviewed the work of other investigators in this fieldand concluded that these studies also indicated the useful-ness of the equivalent stress (211).

14.5b Effect of Creep in High-pressure Vessels. Theinitial stress distribution at the time of first application ofload may be determined by means of Lamk’s analysis.Under the influence of elevated temperature and high-pressure stress gradients the shell material creeps causing aredistribution of the stresses, as mentioned previously.The most rapid stress redistribution will occur in the regionof greatest stress. This region is located near the innersurface of the shell, and the maximum principal stress is thetangential stress, ft. For a given material, given dimen-sions, and given operating conditions, a vessel at elevatedtemperature will have a definite life prior to rupture termedthe “rupture life.” Creep-rate curves and stress-rupturecurves for various materials of construction are availablefor design purposes. Typical examples of such curves areshown in Fig. 2.15 and Fig. 2.16, respectively.

The rupture life of a vessel at elevated temperatures isdependent upon the history of the stress conditions andcreep phenomena. Thus, if a vessel is held at a giventemperature under a high stress, it will have a shorter lifethan if it is held at the same stress level at a lower temper-ature. Voorhees verified Robinson’s theory (212) that the

fraction of the total vessel life dissipated at any stress isequal to the ratio:

(

Actual Time at Given Stress LevelRupture Life at That Stress in a Conventional

Constant-load Test)

To test the validity of this relationship Voorhees raisedand lowered the stress levels on 18 conventional unnotchedbars. On averaging the results for the 18 tests, he foundthat the additions of the fractions of rupture life checkedwith the experimental observations within 1 y0 (204).Voorhees points out that this rule of addibility of rupture-lifefractions can not reasonably be expected to hold true ifappreciable structural alterations occur.

To analyze the effects of stress leveling in the shell of athick-walled vessel, Voorhees arbitrarily subdivided thecross section of the shell into a number of concentric ringsor shells in such a way that the conditions at the centroidof a particular ring were representative of that ring. Hisstudy indicated that a subdivision into a minimum of sixconcentric shells each with twice the circular cross sectionof the adjacent inner shell gave a satisfactory coverage ofthe total range of stresses across the wall. In applyingthis analysis the procedure consists of replacing the con-tinuous change in stress pattern with an equivalent seriesof time intervals over each of which the creep rate andstress in a given ring may be considered nearly constant.The fraction of rupture life expended during each intervalis calculated for each ring; and when the accumulativefraction for any ring reaches unity, rupture should occurat that location, and failure of the entire vessel is imminent.

14.5~ Stress Redistr ibution by Creep Relaxation.Consider two adjacent rings in the shell of a thick-walledvessel with the inner ring having a principal stress of f2 anda creep rate of C2 and the outer ring having a principalstress of fl and a creep of Cr. If f2 > fr, then C2 > Crwith both rings at the same temperature. After some creephas occurred, the plastic strain in the inner ring will exceedthe plastic strain in the outer ring, but as both rings arejoined by the fibers of the material, this difference in plasticstrain must be absorbed by elastic strains in each of thetwo rings. These elastic strains result in an increase in thestress in the outer ring and a decrease in the stress in theinner ring. On the basis of the assumption of plasticincompressibility the changes in principal stress are equalto 2G times the corresponding principal elastic strains. Asimilar relationship holds for the elastic changes involvingthe equivalent stress and strain (204). The relationshipbetween the modulus of elasticity, E, Poisson’s ratio, p, andthe factor 2G is as follows (29) :

E2G = __1+/J

(14.47)

where E = modulus of elasticity in tension, pounds persquare inch

G = modulus of elasticity in shear, pounds persquare inch

p = Poisson’s ratio at operating temperature= 0.32 for most high-strength steels at lOOO-

1300” F

Page 293: Process Equipment Design

In the case of plain carbon steels and alloys with highcreep rates, the effect of stress redistribution by creep is astress equalization in a short period of time with 1% creepor l ess . This is followed by a progressive thinning of theshell, an increase in the stress, and a reduction of therupture life of the vessel. Steels with slow creep ratesbehave differently. In this case the creep rate at theoperating condition may not be rapid enough to producestress equalization before an appreciable fraction of therupture life of the vessel is consumed. Thus, the steelswith iow creep rates may be used at higher temperaturesand greater pressures, and the phenomenon of stress equali-zation may not occur.

14.5d Example Calculation 14.3, I l lustrat ion of theVoorhees Method. To illustrate this procedure, a vesseloperatmg at 1050” F and 5500-psi internal pressure will beconsidered. The vessel is fabricated of annealed carbonsteel iraving a modulus of elasticity of 24,000,OOO psi at1050” F. The vessel has an outside diameter of 12 in. andan inside diameter of 6 in. Creep-rate data for this steelare given in Fig. 14.9, and the stress-rupture curve is givenin Fig. 14.10 (211).

For purposes of calculation the cross section of the shellis arbitrarily divided into six rings each having an areaequal to twice the area of the adjacent inner ring. Thusthe area ratios will be 1, 2, 4, 8, 16, and 32; and the inner-most ring will contain j&a of the total area, the secondring 34a, and so on. Table 14.6 presents a summary ofthe pre l iminary ca lculat ions for these s ix r ings .

In reference to Table 14.6, the six ring divisions arenumbered from the outside diameter inwards, as indicatedin column 1. Column 2 gives the fraction of the totalcross-sectional area contained in each ring. Column 3gives the ratio of the centroid radius of each ring to theoutside radius of the vessel. Columns 4, 5, 6, and 7 givethe ratios of various calculated stresses to the internalpressure that exists at the centroid of each ring (and alsoat the inner and outer surfaces). The tangential stress, ft,in column 4 was calculated by use of Eq. 14.12. The

Monobloc Vessels at Elevated Temperatures 283axial stress, fa, in column 5 was calculated by use of Eq.14.1; and the radial stress, fr, in column 6, by Eq. 14.13.The equivalent s tress (shear-stress invariant) , f, o f c o l u m n 7was determined by use of Eq. 14.43.

The next step in the calculation involves the determi-nation of the “rate of redistribution of the initial stressgradients,” given in Table 14.7, as controlled by creeprelaxat ion. Data on creep rate versus stress at 1050” F forthe annealed carbon steel under consideration are requiredin this step. Such data are given in Fig. 14.9.

In reference to Fig. 14.9, the dashed line is taken as theaverage of the test data. The creep rates corresponding tothe values of stress f, used for column 7 of Table 14.6, aretaken from the dashed line of Fig. 14.9 and tabulated inTable 14.7.

The differential strain rate at each interface between suc-cess ive pairs of r ings i s determined by subtract ion. This isi l lustrated for the innermost three shel ls as fo l lows:

Shel l No. Effective Creep Rate6 0.0405 -0.025

6 - 5 0,015 differential rate

5 0.0254 -0.011

5 - 4 0.014 differential rate

For a short time interval the creep rate may be con-sidered constant, and the creep-rate differential may beconverted to a stress change by multiplying the differentialcreep rate by the assumed time interval and by the shearmodulus of elasticity. In the case of the interactionbetween rings number six and five and between rings num-ber live and four, the stress changes are 546 psi and 509 psi,respect ively, calculated as fol lows.

For rings 5 and 6,

0.015In.. I(in.) (hr)(0.002 hr) (24 X lo6 psi) = 546 psi

6

Creep rate, in.lin.lhr

. - - 0 to 1o- 10 to 45 (minimum rate petlod). - - - - so

I I I I I I0.01 0.1

Fig. 14.9. Creep rate versus stress at 1050’ F. for annealed carbon steei (204). (Courtesy of the American Chemical Society.)

Page 294: Process Equipment Design

284 High-pressure Monobloc Vessels

tSampling Specimen I I I

Code direction diameter I Illlll0 Longitudinal 0 . 1 6 0

5 - 0 Longitudinal 0.350. Longitudinal 0.505

- 0 Tangential 0.160L l Radial 0.160

I I I I I1 1 0

Rupture life, hr1 0 0 1000

Fig. 14.10. Stress verse rupture life for three conditions (204). (Courtesy of the American Chemical Society.)

For rings 4 and 5,

(0.002 hr) (24 X IO” psi) &(. )

= 509 psi

The calculated results are summarized in Table 14.8.The stress interaction results in the relaxation of the

stress in the inner shell and a transfer of the stress to theadjacent outer shell. The transfer of stress is distributedinversely as the respective areas are distributed. As theouter shell of a pair has twice the area of the inner adjacentshell because of the selection of the area sequence, the innershell will receive two thirds of the total stress change.

By following the procedure illustrated in Table 14.8, thestress changes for all six rings are determined, and the netchange for each ring is established. After the initial timeinterval of 0.002 hours under load, creep will reduce theequivalent stress in ring number 6 to 12,430 - 364 = 12,066psi. In ring number 5 the net change results in an equiva-

Table 14.6. Position and Stress Pattern at Centroids

of Each of Six Rings in Vessel with OD = 21D (204)

1 2 3 4 5 6 7Fraction of PIP”

Shell Total Cross (at cen-N O . Section__. ..-ID

tro’d, j&e fo/P f/P l/P1.67 0.33 -1.00 2.30

6 lax 0.506 1.64 0.33 -0.97 2.265 “6 3 0.523 I.55 0.33 -0.88 2.104 -%i a 0.555 1.41 0.33 -0.74 1.863 96 3 0.617 1.21 0.33 -0.54 1.522 I943 3 0.723 0.97 0.33 -0.30 I .lO

O’D 3%3 . . 0 , 9 0 0 0 . 7 4 0 . 6 7 0 . 3 3 0 . 3 3 -0.08 0.00 0 . 7 1 0 . 5 8

lent stress at the end 0.002 hours of 11,570 + 182 - 339= 11,413 psi.

The fraction of the rupture life expended in each ringduring the time interval of 0.002 hours may he determinedby use of the rupture-life data shown in Fig. 14.10.

For ring number 6 the rupture life of the vessel at theinitial stress of 12,430 psi at 1050” F is estimated fromFig. 14.10 to be 12 hours. The reduction in stress causedby creep during 0.002 hours has increased the life of thevessel to 13 hours. Using an average life of 12!5 hours, wefind that the fraction of the life used up during the intervalof 0.002 hours is equal to (O.O02/12>Q(lOO%) = 0.016%.The calculation sequence is repeated, starting with thestress distribution existing at the end of the first timeinterval. The fraction of the rupture life consumed in thissecond interval is then determined. When the summationof the life fractions equals unity, failure is imminent andthe total of the accumulated time intervals gives the antici-pated rupture life.

As successive creep occurs with relaxation of the higherstresses, the stress variations across the wall tend to levelout with a decrease in creep rate. Consequently, longertime intervals may be used in each successive calculation.In the above example the stresses are essentially uniformacross the wall with a value of 6000 psi at the end of1.5 hours. This corresponds to an expenditure of about,only 0.8% of the rupture life of ring number 6. The reduc-tion of the maximum stress from 12,430 psi to 5800 psiappears to increase the rupture life of the vessel from12 hours to about 840 hours, as indicated by Fig. 14.10.However, the continuation of creep tends to increase thestress level and accelerate the creep rate, and this shortenst)he life of the vessel.

The stress rise is proportional to the rate of change of the

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ratio of the diameter to the wall thickness. In the absenceof localized bulging the st,ress rise at the end of a period ofcreep will equal the initial stress at the beginning of theperiod t imes the rat io :

(

1 + Creep Strain during the Interval_---.-1 - Creep Strain during the Interval )

The creep rate at a stress of 6000 psi is 0.00048 in. per in.per hr (see Fig. 14.9). A period of 8.5 hours at this con-dition would result in a total creep of (8.5 hr)(0.00048) =0.0041 in. per in. Therefore, at the end of a total elapsedtime of 10 hours the uniform equivalent stress should be:

1 . 0 0 4 16000 psi ox = 6045 psi(. >

Although the stress rise is only 45 psi, it r,esults in adecrease in the rupture l i fe of the shel l from 840 to 780 hours.

, The first 1.5 hours consume 0.8%, and the next 8.5 hours1.5% of the rupture life of ring number 6. A continuat ionof these calculations show that the rupt,ure life of the vesselwould be reached after 300 hours if no local deformationor out-of-roundness developed. However, small irregu-larities in geometry are augmented by creep, and thisshortens the rupture life of the vessel. Voorhees statesthat an experimental vessel with s imilar loading fai led after55 hours (204).

14.6 PRACTICAL CONSIDERATIONS

In the case of plain carbon steels and alloys with highcreep rates the amount of creep required to cause incipientfailure as a result of out-of-roundness, eccentricity, orextraneous s t resses i s very smal l . After s tress equal izat ionoccurs, the vessel continues to deform plastically, and anyirregularity or out-of-roundness is accentuated; thus insta-bility is produced, followed by rupture. Voorhees, Sliep-cevich, and Freeman (204) recommend that for the caseof plain carbon steels and alloys with high creep rates,the selected design stress be equal to 0.8 times the valueof the stress producing a 2% creep during the anticipatedli fe o f the vesse l . As the initial stress is rapidly equalized,the mean integrated stress will be reached in short timeand may be calculated by formal integration rather thanby the stepwise procedure previously described. Thisintegrat ion i s :

(14.48)

Table 14.7. Initial Conditions for an

4nnealed-carbon-steel Vessel at 5500 psi (204)

Effective CreepShel l No. 7, psi Rate, in./in./hr

6 1 2 , 4 3 0 0.0405 1 1 , 5 7 0 0.0254 10,240 0.0113 8 , 3 4 0 0.00272 6 , 0 5 0 0.005i 3 , 9 1 0 0.0008

Prestressed Monobloc Vessels 285

Table 14.8. Stress Changes Resulting from Creep (211)

Total Stress Stress Change inShel l No. Interaction, psi Each Ring, psi

6 5 4 6 -g (546) = -36k5 +$$ (546) = +1825 5 0 9 - g (509) = -3394 +g (509) = +170

But bs Eqs. 14.44, 14.7, and 14.11b,

Substituting the above equation into Eq. t-L.47 forjgives:

(14.49)

Equation 14.49 may be set. equal to 0.8 of the st.ressgiving 2 o/o creep during the anticipated life, and the prc )-portions of the vessel determined. This proposed designprocedure is not recognized by the code but should besatisfact,ory for most of the more ductile high-temperaturealloys which show a marked degree of strengthening in anotched-bar rupture test.

In t.he case of alloys for high-temperature service havinghigh creep strength, the design should be based on stress-rupture data rather than creep rate. The recommendedstress for such a design is 0.8 of the stress for rupture in t,heanticipated service life of the vessel (204). In order forthis design criterion to be used, the operating conditions oftemperature and pressure must be carefully controlled, andreliable data on the alloy employed must be available.Normally the maximum equivalent, stress, J, will be theinitial equivalent stress at the inner radius, pi.

For ideal conditions the shell material at the inner wallshould have a high creep rate in order to quickly relax t.heh igh in i t ia l s t resses . On the other hand, a low creep rateis desirable at the outer surface in order to restrain theshell of the vessel from thinning and to maintain stability-.These two criteria are incompatible for a single material,but both can he realized if two shell constructions areused (204). The inner shell material should be chosen tohave moderate creep strength and to be capable of with-standing extended creep without rupture. The materialfor the outer shell should have a low creep rate and a highcreep strength. Such a design may be used with loose fitssince the inner shell will creep until contact is made wit.hthe outer she l l .

14.7 PRESTRESSED MONOBLOC VESSELS

14.7a The Advantage of Prestressing. The advantageof prestressing a vessel is either the reduction of the maxi-mum stress existing under operating conditions or thereduction of the required shell thickness when a specifiedmaximum allowable design stress is used. Regardless ofthe theory employed to calculate the maximum stress in the

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2 8 6 High-pressure Monobloc Vessels

t ” p 5.P

G‘i

‘04 3

ET-

1 2

tp.s.P

64

‘i

‘08 7

#5 6

(a) (b)

Fig. 14.11. S t r e s s d i s t r i b u t i o n s i n s h e l l s u n d e r l o a d .

wall of a thick-walled vessel, a nonuniform stress distribu-tion under pressure will be found to exist in a monoblocvessel that has not been prestressed. The principle ofprestressing is to induce a permanent residual compressivestress existing under zero pressure at the point in the shellwhere the maximum tensi le s tress is induced under pressure.

14.7b Ideal Stress Distribution. The ideal stress dis-tribution in a prestressed shell was discussed by Maccaryand Fey (194) and was shown to depend upon the theorybeing used for design. When the maximum-principal -s tresstheory is the theory used, Fig. 14.11, detail a, shows thestress distribution under load for a nonprestressed shellwhereas detail b shows the stress distribution under loadfor an ideal prestressed shel l .

In reference to detail b, by summation of forces

CF= Zpiril= 2i:‘fdrlo r

piri = /: f dr = (fayg n.p,s.)(tn.p.s.) = area l-2-3-4 (14.50)

where n.p.s. = nonprestressed

favg n.p.s. = & =1-PiK-l (14.51)

fallow. n.p.8. = Pi $$

In reference to detail b, by use of the same maximumstress (p.s. stands for prestressed)

Also , by summation of forces

piri = (favn ,.s.)(~I,.s.) = area 5-6-7-8 (14.52)

Equating Eq. 14.50 to Ey. 14.52 (for the same pressureand inside diameter) gives:

Pi(z) tw. = Pi ($q) tn.,,.,.

Shel l th ickness , prestressed t,.,.Shell thickness, nonprestressed = c.

= ($2) (K+) = $$ (14.53)

Thus, employing the maximum-principal-stress theory,in which the maximum stress is determined by ILamBanalysis , and employing a given maximum al lowable stress ,one finds that the use of an ideal-prestressed vessel insteadof a nonprestressed vessel will result in a reduction in shellthickness of (K + 1)/(K2 + 1). To calculate this reduc-tion in required thickness the design pressure and maximumallowable working stress must be known. Equation 14.14bcan be rearranged to solve for K.

K = difauow. + p)/(fallow. - P) (14.54)

The concept of ideal prestressing is useful for realizingthe possibilities of prestressing. However, the ideal-prestress condit ion can only be approximated by the methodof autofrettage of monobloc vessels.

14.8 THERMAL PRESTRESSING

The work of Voorhees (211) on the creep of vessels sub-jected simultaneously to high-temperature- and high-pressure-service conditions suggests the use of creep tothermal ly prestress a th ick-wal led vesse l . Voorhees reportsthat a creep of 1 y0 i s suff ic ient to produce s tress equal izat ionunder the operating pressure. If a vessel has the pressureon it raised to the operating pressure and then is heatedslowly and uniformly until a 1% strain from creep hasresul ted, the s t ress dis tr ibut ion across the vesse l wal l shouldbecome nearly uniform, approaching the ideal-prestressedcondit ion shown in detai l b of Fig. 14.11. If the pressure isreleased and the vessel cooled with sufficient rapidity toprevent additional creep, residual stresses will result in thes h e l l . Upon placing the vessel in service at the same pres-sure as that used in prestressing but at a temperature belowthat producing creep, the ideal-prestressed condit ion wil l beapproached for the loaded condition. Although thismethod of prestressing appears to be very promising formonobloc vessels, there is no known report of the use ofthis method. The most widely used method of prestressingmonobloc vessels is that of “autofrettage.”

14.9 AUTOFRETTAGE PRESTRESSING

The process of “autofrettage” is the oldest method ofprestressing monobloc cylinders. Toward the end of thenineteenth century some monobloc cy l inders for gun barre lswere prestressed by a special casting technique using achilled core, but this process has not been used in modernpractice. The word “autofrettage” comes from the Frenchlanguage, and a literal translation is “self-hooping” fromthe similarity of prestressing by means of shrink fittingsuccessive shells to the prestressing of barrels by hooping.The process cons is ts s imply of s t ress ing a l l , or more usual ly ,part , of the monobloc shel l beyond the yield point by meansof hydrostat ic pressure . This produces a greater unit s tra inin the inner portion of the shell than in the outer portion.On release of the overstress ing pressure the di f ference in unitelongation results in a residual compressive stress in theinner and a res idual tensi le s tress in the outer port ion of thes h e l l . Usually the permanent set as measured on the insidediameter is limited to between 2.5 and 6.07& Although itmight appear that the maximum strengthening be producedby overstressing the entire shell beyond the elastic limit,

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Autofrettage Prestressing 2 8 7

where p* = autofrettage pressure that will piace the elastic-plastic interface at ri, pounds per square inch

P ** = autofrettage pressure that will place the elastic-plastic interface at r,, pounds per square inch

The elastic-plastic interface located at radius P, dividesthe shell into two zones, a plastic zone and an elastic zone.Separate relationships must be used for determining theautofrettage-pressure stresses for each zone.

For the plastic zone, where ri 5 r < rc,

this is not. the case. It is usually desirable to limit theoverstressing to keep an outer layer of the shell within theelastic region. However, Manning (203) suggests thatthere are advantages to 100% overstrain and states that“it is doubtful whether they (other workers) have obtainedthe fullest possible advantage from the peculiar redistribu-tion of stress which overstrain brings about.”

14.9a Stress Relationships for Autofrettage Pressure.The theory of the autofrettage process has been studied bya number of investigators. One of the original investi-gators was Macrae (192). His work was later reviewed byNewitt (191). Other reviews and discussions of the subjecthave been presented by Hill (197), Comings (218), Pragerand Hodge (195), Faupel (201, 216), and Faupel and Fur-beck (215). Hill considers the autofrettaging of sphericalshells as well as cylindrical shells. Faupel and Furbeckreport experimental data on autofrettaged vessels tested tobursting and compare the theory with experimental results.

The procedure used in the analysis of autofrettaged shells, consists of the following:

1. The determination of the autofrettage pressure whichwill locate the elastic-plastic interface at the desired positionin the wall of a vessel having a specified inside and outsidediameter.

2 . The determination of the autofrettage-pressure stresses.3. The determination of the changes in pressure stresses

resulting from the unloading process.4. The combination of residual stresses with operating-

pressure stresses by the method of superposition.5. The calculation of elastic breakdown and bursting

pressures.

The derivation of the relationships used in the procedureoutlined above is presented by Prager and Hodge (195) andwill be only summarized here.

Prager and Hodge (195) have shown that integration ofEq. 14.38 leads to a relationship for the autofrettage pres-sure required to locate the elastic-plastic interface at aradial distance of r, (between ri and ro), or

P = .fs.y.(1-k~-22ni

r. PC >(14.55)

where js,Y. = yield limit in simple shear, pounds per squareinch (see Eg. 14.35)

r, = radial distance of elastic-plastic interface,inches

ri = inside radius of shell, inchesr,, = outside radius of shell, inchesp = autofrettage pressure, pounds per square inch

The two limits p* and p ** of Eq. 14.55 are reached whenrc = ri or when rc = r,,, respectively.

p* =f*.Y. [I - ($1

P ** = 2ja.Y. In 5ri

(14.56)

jr = -js.y. 1 - $Y - 2 In i>

(14.583

ft = jg.y.(

1 + $ + 2 In i>

.fa =6.,.(!$+2lnk)

(14.59)

where jY = autofrettage radial stress, pounds per squareinch

jt = autofrettage tangential stress, pounds per squareinch

ja = autofrettage longitudinal stress, pounds persquare inch

For the elastic zone, where r, 5 r 5 r,,

jr = p” 1 - 5( >

jt = p” 1 + $( >

P,, _ .fs.y.rcz

r.2

14.9b Changes in Stresses as a Result of Unloading theAutofrettage Pressure.

VESSELS WITH UNLOADING STRESSES WITHIN THE ELASTIC

REGION. Prager and Hodge (195) have shown that themethod of computing the changes in stresses resulting fromunloading the autofrettage pressure will depend uponwhether or not the unloading stresses exceed the compres-sive yield strength of the material. Vessels will haveunloading stresses within the elastic region if

K = 2 5 2.22 (14.64)pi

andp* _< Ap < p** (14.65)

or ifK > 2.22

andAP 5 ZP* (.?4.66)

where Ap = the unloading pressure, pounds per square inch

For vessels satisfying the above conditions, the followingrelationships give the changes in autofrettage-pressure,

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2 8 8 High-pressure Monobloc Vessels

stresses for both the elastic and plastic regions:

Afa = -Ap’ (14.69)

(14.70)

where Ap = (pl - pz)pl = autofrettage pressure, pounds per square inchp2 = unloading pressure, pounds per square inch

(normally zero)

VESSELS WITH UNLOADING STRESSES BEYOND THE ELASTIC

REGION. Vessels will have unloading stresses beyond thecompressive yield strength if

(14.71)

and2p* 5 Ap < p** (14.72)

For the conditions above, Eq. 14.55 must be modified asfollows to give the elastic-plast,ic interface after unloading:

Ap = 2js.y. 1 - :: - 2 ln z>

(14.73)3

where rj = radial distance of elastic-plastic interface, inches

The changes in autofrettage-pressure stresses uponunloading for the plastic zone are:

i-i 5 r 5 rj

(14.74).

(14.76)

For the elastic zone,

(14.77)

(14.78)

The residual autofrettage stresses are determined asfollows:

(fv) = fr + Ah (14.80)

where (jr) = residual radial stress, pounds per square inchjr = j,. from Eqs. 14.58 and 14.61

Aj,. = Aj,. from Eq. 14.67 or from Eqs. 14.74 and14.77

Also,U-t) = ft + Aft (14.81 j I

where (ji) = residual tangential stress. pounds per squareinch

jt = jt from Eqs. 14.59 and 14.62Aft = Aft from Eq. 14.68 or from Eqs. 14.75 and

14.78And

Ma) = fa + 4-a (14.82)

where (ja) = residual axial stress, pounds per square inchja = ja from Eqs. 14.60 and 14.63

Aja = Aja from Eq. 14.69 or from Eqs. 14.76 and14.79

14.9~ Combined Stresses under Operating Pressure.The combined stresses under operating pressure are deter-mined by the method of superposition. Lame’s equationsfor j,., jt, and ja (Eqs. 14.6b, 14.7, and 14.1, respectively) arecombined with the appropriate residual stresses of Eqs.14.80, 14.81, and 14.82.

The unit radial strain in both t,he elastic and plasticregions under autofrettage pressure is given by:

(14.83)I

Ewhere G = ___

20 + d(See Eq. 14.47.)

andE = modulus of elasticity of material, pounds per

square inchp = Poisson’s ratio

Upon unloading of the autofrettage pressure a change inunit radial strain, A+, occurs.

For conditions satisfying Eqs. 14.64, 14.65, and 14.66

(14.84)I

where Ap’ is given by Eq. 14.70.

For conditions satisfying Eqs. 14.71, 14.72, and 14.73i

(14.85) /

14.9d Example Calculation 14.4, Determination of Opti-mum Autofrettage Pressure. An autofrettage vessel is tobe designed to contain a fluid at 20,000 psi internal pressure.An internal diameter of 12 in. is specified. A high-strengthsteel having a 70,000-psi yield point is to be used.

Using a factor of safety, X equal to 2.0, the required shellthickness, according to the maximum-principal-stress theory,is 5.5 in. (for a similar vessel shell). Therefore, K = do, d;= 1.916. The shell thickness for the ideal-prestressed con-dition is, by Eq. 14.53:

K + 1 2.916 o 623tp.s. _---=-=tn.p.L% K2 + 1 4.67 ’

thereforetp.s. = (5.5)(0.623) = 3.43 in.

Page 299: Process Equipment Design

Autofrettage Prestressing 289

As the ideal-prestressed condit,ion cannot be obtained byautofrettage, an intermediate value between the ideal-prestressed and the nonprestressed value will be selected.

Determine the autofrettage pressure which will give theminimum combined hoop stress if a shell thickness of41s in. is used.

Solution:Data:

By Eq. 14.35

/ti = 6.0 in.

r, = 10.5 in.

f,.,. = 70,000 psi

therefore

js.&= %I = 40,300 psi

To determine the optimum autofrettage pressure a num-ber of values of r, must be selected, and the correspondingstresses evaluated.

The upper and lower limits of the autofrettage pressureare determined by Eqs. 14.56 and 14.57.

The lower limit is:

P * = f8.y. 1 - <( >ro

= 40,300(, -i&)

= 27,150 psi

The upper limit is:

p** = 2fs.y. In 2ri

10.5= (2)(40,300) I11 __

6

= 45,200 psi

The stresses for r, = 6, 7, 8, 9, and 10.5 in. will be deter-mined. Only the calculations for rC = 9 in. will be pre-sented here.

The autofrettage pressure required to locate the elastic-plastic interface at r, = 9 in. is determined by Eq. 14.55.

p = f*,,,(

1 - $1 - 2 In 2>

= 40,300(

I - & - 2 In +>

= 43,500 psi

Under autofrettage pressure the radial, tangential, andlongitudinal stresses for r, = 9 in. are as follows.

For the plast ic region:By Eq. 14.58

f7 = -.f8.y.(

1 - $ - 2 In k)

therefore

jr = -40,300(

1 - & - 2 In $>

For r = 6,7, 8, and 9 in., jr = -43,500, -31,000, -19,400.and - 10,700 psi, respectively.

By Eq. 14.59

(2

ft = f8.Y.r

1 + ” + 2 111 -r. PC >

therefore

ft = 40,300 1 + &(

+21n$>

For r = 6, 7, 8, and 9 in.,ft = +37,400, +49,800, +61,100,and +69,000 psi, respectively.

By Eq. 14.60

therefore

fa = 40,300( .6.2111;

>

For r = 6, 7, 8, and 9 in.,fa = -3,030, +9,300, +20,900,and +29,700 psi, respectively.

For the e last ic region:By Eq. 14.63

P,, _ .fs.y.rcz--= (40,300) (81)

2 = 29,700 psir. 110.3

By Eq. 14.61

fT = p” 1 - r!I( >

therefore

J.=29,700(1-y)

For r = 9, 9x, and 1035 in., jr = -10,750, -4760, and0 psi, respectively.

By Eq. 14.62

ft = p” 1 + 5( >

therefore

jt = 29,700(1 + y)

For r = 9, 9x, and 1056 in., ft = +69,000, +64,000, and+59,400 psi, respectively.

By Eq. 14.63

fa = p" = +29,700 psi

P I \ \ \I /- - -

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290 High-pressure Monobloc Vessels

6 0

5 0 t--CR44 0

8 0

7 0

-20

-30 /

-40 ,’

-506 7 8 9 10 1 1

r (inches)

Fig. 14.12. Induced stresses under autofrattaga pressure, 43,500 psi.

Figure 14.12 is a plot of the induced autofrettage-pressurestresses for the case in which r, = 9 in.

Determination of changes in autojrettage-pressure stressesresult ing from unloading:

Check of unloading conditaons:

K = 2 = y = 1.75pi

Comparing this value with Eq. 14.64 indicates thatEqs. 14.67, 14.68, and 14.69 are applicable. First, Eq. 14.70must be evaluated.

AP = PI - ~2

p2 = 0therefore

Ap = autofrettage pressure

and= 43,500 psi (for r’, = 9 in.)

AP’ = 43,500 (,,,.;‘” ,,)

= 21,200 psi

Substituting into Eq. 14.67 gives:

Aj,. = -Ap’ 1 - ‘$( >

therefore

Afr = -21,200(1 -y)

For r = 6, 7, 8, 9, 9x, and IO>5 in., Ajr = 43,700, 26,500,15,400, 7640, 2390, and 0 psi, respectively.

Substituting into Eq. 14.68 gives:

Ajt = - Ap’

therefore

Aft = -21,200(1 + y)

For r = 6, 7, 8, 9, 95<, and lO$$ in., Aft = --86,000,-69,000, -58,000, -50,000, -48,000, and -42,400 psi,respectively.

Substituting into Eq. 14.69 gives:

Aja = -Ap’ = -21,200 psi

Calculation of residual s tresses :Substituting the above values into Eqs. 14.81, 14.82, and

14.83 gives the following residual stresses (for rC = 9 in.):

ifT f t fa0 -48,600 - 24,230

7 -4500 - 19,200 - 11,9008 - 4000 +3,000 -3009 -3060 +19,000 +8,5009% -2370 +18,200 +8,500

1ojg 0 +17,000 +8,500

Figure 14.13 graphically presents the above residualstresses.

Figure 14.14 shows the residual hoop (tangential) stressesresulting from unloading the autofrettage pressure. The

I /f/I I I I IL I0

8 -5sb -10I:g -15

E -20

-25

-30

-35

-40

-45

-508 9 1 0 11

r (inches)

Fig. 14.13. Residual stresses for rC = 9 in.

Page 301: Process Equipment Design

Results from Experimental Studies of Autofrettoge 2 9 1

6 0

2 5

- 5

-10

-15

-20

-25

-30

-35

-406 7 8 9 1 0 1 1

r (inches)

operating hoop stress calculated by Lam&‘s relationship isalso shown. This operating-pressure stress is combinedwith the residual stresses to give a series of combinedstresses for various values of rc. The curve at the topof the figure shows the locus of the maximum combinedstress and indicates a minimum value where r, is about7 in. If the vessel had not been autofrettaged, the maxi-mum stress would have been 39,600 psi. If the vessel hadbeen autofrettaged so that r, was equal to 7 in., this stresswould have been reduced to 35,000 psi. This amounts toabout an 11 y0 reduction in the maximum stress at operatingpressure. Examination of Fig. 14.14 indicates that thevessel could be overautofrettaged with the result that thecombined stress would be greater than that in the samevessel without autofrettage.

14.10 RESULTS FROM EXPERIMENTAL STUDIESOF AUTOFRETTAGE

Faupel and Furbeck (215) experimentally studied residualstresses in autofrettaged vessels constructed of plain carbonsteel and various alloys. The theoretical residual stresseswere computed from relationships which may be derivedfrom equations presented earlier in this chapter.

In these derivations Eq. 14.70 is substituted into Eqs.14.67, 14.68, and 14.69, and these last three equations addedto Eqs. 14.58, 14.59, and 14.60, respectively, to give theresidual stresses:

(14.86)

(14.87)

(14.88)

Faupel and Furbeck (215) determined experimental valuesof residual stresses in autofrettaged vessels by means of themethod developed by Sachs (219). In this procedure,prestressed thick-walled cylinders were aligned in a lathe,and concentric layers of metal machined from the bore insuccessive steps. The axial and circumferential strainswere recorded for each successive cut (usually about 0.05 in.measured on the diameter). Also, after each cut the borediameter was measured to within 0.0001 in. by use offlush-cooling and temperature-compensating gages. Thesedata on the strains and bore diameters were substituted intothe following equations developed by Sachs (219) to givevalues of residual stresses based upon these measurements.

(14.89)

ft’ = + [P2 (Ao-An)-$-(+&ye] (14.90)

fa’ = i-“- [EL2

( A o - A ) $ - ct 1 (14.91)

Fig. 14.14. Plot of residual hoop stresses and combined operating hoop-

pressure stress for various values of rC.

where A0 = initial cross-sectional area of cylinder includingthe bore, square inches

A=

CC=

O=

E, =EC =E =/J=

cross-sectional area of the bore following amachine cut, square inches(E, + pE,), inches per inch(EC + pE,), inches per inchaxial strain, inches per inchcircumferential strain, inches per inchmodulus of elasticity, pounds per square inchPoisson’s ratio

In one test a cylinder with an outside diameter to insidediameter ratio of 2.74 fabricated of Cr-Ni-MO-V steel washeat-treated to a hardness of 42 Rockwell C to removeresidual stresses. It was then autofrettaged at a pressureof 200,000 psi for 4.5 hours. This procedure was followedby a stabilizing heat-treatment at 650” F for 3 hours. Thecylinder was checked for stability and elastic behavior upto the autofrettage pressure by repeating the loading to200,000 psi through four cycles of operation, and then theresidual stresses were determined by the Sachs method.The results are shown in Fig. 14.15.

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292 High-pressure Monobloc Vessels

P 2.757’ -_I

40

- 2 0 0

- 2 4 0

- 2 8 0

6 4 2 0I I I ,

2 42 4 66Cross-sectional area, square inches

Fig. 14.15. Theoretical and experimental residual-stress distributions in an

outofrettaged cylinder (215). (Extracted from Transactions of the ASMEwith permission of the publisher, the American Society of Mechanical

Engineers.)

In Fig. 14.15 the theoretical stresses predicted by Eys.14.86, 14.87, and 14.88 show good agreement with theexperimental values.

14.11 MANNING’S METHOD

Table 14.5 compares experimental values of bursting withvarious theories of failure and shows good correlationbetween the experimental values and those predicted bythe method of Manning (203). Manning’s hypothesis isthat since vessels for high-pressure service are usually fabri-cated of high-tensile steel, which usually fails by shear, anexperimentally determined shear stress should be used inthe design of such vessels rather than tensile-test data.He proposes that a simple torsion test be used to determinethe shear stress-strain relationships and has presentedmethods of using these data in the design of high-pressurevessels (203, 237-240).

According to Manning, two assumptions are involved:(1) the cross-sectional area remains constant under strain,(2) the relation between maximum shear stress and maxi-mum shear strain is the same in a cylinder as in a torsiontest. The first assumption permits the calculation of theshear strain at any point in the wall of the vessel and thesecond permits the calculation of shear stress from torsiondata. Equation 14.2b (with proper allowance for signconvention) is a statement of the first assumption, or:

ff -jr = P $

The corresponding maximum shear stress at radius P isgiven by Eq. 9.85:

fs = at - .fr) (9.85)

If Eq. 9.85 is substituted into Ey. 14.2b we have:

In Ey. 14.92 the value of P must be that of the strainedcondition. This can be expressed as follows for the int.e-grated form :

.f7 = - p i + 2 ~~~~ d(r + Ar) (14.93)

where Ar = shift of point at radius r as a result of straininduced by pressure pi

Ari = shift of inside radius, r-i, as a result of straininduced by pressure pi

pi = internal autofrettage pressure

In order to integrate Eq. 14.93 the shear st.ress, fS, as afunction of strain must be known. For this purposeManning uses the results from a torsion test and the follow-ing eyuation given by Nadai (241):

(14.91)

where T = toryue8 = unit twistd = diameter of shaft used in torque test

If the cross-sectional area is constant, the following rela-tion is true:

m2 - vi2 = r(r + Ar)” - r(ri + Ari)’or

2r Ar + Ar’ = 2ri Ari + Ari’ (14.95)

Using Eqs. 14.94 and 14.95 and experimental data fortorsion tests, Manning integrates the integral group ofEy. 14.93 and plots twice the integral as a function ofradius ratio r/pi. If twice the integral is combined with-pi as indicated in Eq. 14.93, the radial stress is obtainedunder autofrettage conditions.

This stress curve is plotted in Fig. 14.16 and is indicatedby the line labeled “Manning’s value.” The curve isplotted over a range of r/pi from 1.0 to 10.0 but the inte-gration has been made for a vessel of infinite external radius.

I --\-- - ~~-

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Manning’s Method 293

240,000

180,000

'2160,000

3'3 140,0002E8

120,000

tg 100,000Ixe 80,000

60,000

01 2 3 4 5 6 7 8 9

R a d i u s r a t i o

F ig . 14 .16 . C o m p a r i s o n o f M a n n i n g ’ s m e t h o d w i t h E q . 1 4 . 5 8 .

For a vessel of infinite external radius the maximum valueof the integral for the example given (see page 511 ofReference 239) is 185,800 psi. In the example quoted theelastic-plastic interface was placed at a radius ratio of 3.0.

The dashed line was obtained by use of Eq. 14.58, usingvalues of ri = 1 in., r, = 3 in., r, = 00, and js.Y. = 57,800psi. The value of js.Y. = 57,800 was obtained from Man-ning’s plot on page 511 of Reference 239. Note that theformal integration given by Eq. 14.58 gives essentiaily thesame curve as obtained by Manning, who used the rathertedious procedure of stepwise integration. The problemlies more in the determination of the correct shear yieldstress than in the method of integration.

The reason for using the ratio of r/ri on a logarithmics&e is for convenience in vessel design. Manning pointsout that for an isotropic material the same stress and straincurves versus radius ratio (r/rJ will apply regardless of theabsolute dimensions of the vessel . Jasper (242,243)) on theother hand, states that for “very thick-walled vessels theratio of thickness to diameter is not as good a criterion asdelinite thickness.” It must be recognized that vesselsmay not behave ideally and their characteristics may beinfluenced by the nonisotropic nature of the material ofconstruct ion. Such deviations can only be determined bylrsts with vessels of full size. In the absence of such datacorrelations based on dimensional similarity are the onlymethod of analysis .

The abscissa of Fig. 14.16 is labeled “Radius Ratio.”I‘ltk ratio is equal to r,, ‘P; for the case in which r, = m.

For use in the analysis of actual vessels in which r, is lessthan infinity this ratio becomes 3r/r, and a correction mustbe made to the values of stress determined from the figuresince the l imits of the integrat ion are no longer f rom one toinf in i ty . The use of the figure in this manner assumes thecriterion of dimensional similarity mentioned earlier. Toil lustrate the use of the s tress curve examples wi l l be given.

Example 1. ri = 3 in., PC = 9 in., and r, = 30 in.Therefore, for r = ri = 3 in.,

3r-= 3(3) _ 1rc 9

From Fig. 14.16, j,.(uncorrected) = - 185,800 psi.For r = r,,

3’=3i-c

From Pig. 1.1.16, jrc ,,,, corlected, = -57,800 psi.For r = ro,

3r (3)(30) = 1o-= -~rc 9

From Fig. 14.16, fr(nncorrected) = -5200 psi.

Since the radial pressure at I-,, is zero (for a vessel withzero external pressure) and since the shape of the radial

-- -. .

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2 9 4 High-pressure Monobloc Vessels

X

Change in outside diameter

Fig. 14.17. Hysteresis in autofrettaged vessel (237). (Courtesy of

American Chemical Society.)

stress curve depends upon the radius ratio (assumingdimensional similarity), the curve has only to be shifteddownward until the radial stress is zero at 3r/re = 10. Thesame result may be obtained by subtracting the right-handintercept of 5200 psi from the uncorrected stress valuesobtained from the f igure. Therefore the corrected stressesbecome:

fri = - 185,800 - (-5200) = -180,600 psi

= autofret tage pressure

frc = -57,800 - (-5200) = -52,600 psi

a n d

fr. = 0

The radial stress at any other point in the wall of the vesselcan be determined in a s imilar manner .

Example 2 . Pi = 8 in. , r, = 1 2 in. , r, = 2 0 in.For P = ri,

3r-zr (3)(8) _ 2PC 1 2

From Fig. 14.16, fr(uncOrrected) = - 104,600 psi.For r = rc,

3r 3-=rc

From the figure, fr(Unc,,rrected) = - 57,800 psi.For r = r,,

3r (3)(20) 5-= __ =PC 1 2

From the figure, fr(UncOrre&d) = -20,800 psi.To correct these values so that jr, = 0, -20,800 psi is

subtracted from the values given above to give:

f,.< = -104,600 - (-20,800) = -83,800

= autofret tage pressure

fr, = -57,800 - (-20,800) = 37,000 psi

a n d

It should be mentioned that Fig. 14.16 could have beenprepared using r,/ri = 4 or any other convenient quantityrather than 3. If such a plot were prepared with rJri = 4,then the radius rat io would become 4r/r, and would changecorrespondingly for any other value.

Manning recommends that the elastic-plastic interface rcbe located approximately at r, = 1/z for optimum pre-stress ing with autofret tage.

14.12 PRACTICAL CONSIDERATIONSIN AUTOFRETTAGE

In the process of autofrettage it is the usual practice toprovide additional metal on both the inside and outsidesurface of the wall. The excess metal on the inner surfaceis necessary in order to allow for permanent radial strainand yet maintain the desired internal diameter . Also , somemetal must be provided both on the inner and the outersurfaces for machining off the scale and for truing subse-quent to the heat-treating operation which should followautofrettage.

The desirability of heat treatment after autofrettage canbe explained by reference to Fig. 14.17.

Figure 14.17 was used by Manning (237) to illustrate theproblem of hysteresis in an autofrettaged vessel. If in theoriginal overstraining the internal pressure is carried topoint B in Fig. 14.17 and released, the outside diameterwill have a permanent strain as indicated by point C.Decrease in diameter to C’ may result i f the vessel is a l lowedto remain in the unloaded state. On reloading the vesselto a pressure of B’ the strain of the outside diameter willfollow the path between C’ and B’, producing a hysteresisloop. However, if the vessel is given a low-temperatureanneal (575”-600” F for mild steel), the hysteresis loop willbe replaced by a straight line characteristic of elastic strain.Furthermore, the yield point will rise from B’ to E. Thisphenomenon was first investigated by Macrae (192), whotermed the rise in yield from B’ to E the “elastic gain.”Thus a low-temperature heat treatment has the advantageof producing greater dimensional s tabi l i ty in the f inal vesse l .

One suggested procedure of design consists of determiningthe thickness of a shell that would burst at twice the servicepressure by means of Eq. 14.42. This vessel can then beautofrettaged with a pressure sufficient to place the elastic-plast ic interface at the geometric-mean radius. This shouldbe followed by a low-temperature heat treatment andmachining to desired dimensions. The vessel will have afactor of safety of 2 based on the working pressure.

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Practical Considerations in Autofrettagn 295

P!?OBLEMS

1. A thick-walled pressure vessel having an inside diameter of 8 in. and an outside diameterof 16 in. is subjected to an internal pressure of 15,000 psi. Determine the maximum inducedstress according to the maximum-principal-stress theory, the maximum-shear-stress theory,the maximum-strain theory, and the maximum-strain-energy theory.

2. Prepare a plot of the tangential stress, ft, and the radial stress, f,., for the vessel in prob-lem 1.

3. Determine the bursting strength of the vessel described in problem 1 if the vessel isfabricated of SA-302 grade B steel (see Chapter 12 for ft.,. and Fig. 2.6 for f,.,.).

4. Calculate the stress distribution across the vessel shell for the vessel described in section14.5d (Illustration of the Voorhees Method) after 1% strain from creep has occurred in theshell.

5. A vessel with an inside diameter of 20 in. and an outside diameter of 32 in. is to be operatedat 1100” F. What is the maximum allowable working pressure if the vessel is fabricated of18-8 type 316 steel (see Fig. 2.16) and its anticipated life is 20,000 hours? (See Eq. 14.49).

6. For the vessel described in problem 1 calculate the optimum autofrettage pressure thatwill give the minimum peak stress under operating conditions.

7. A reactor is needed in our plant for service at 400” F with an internal pressure of 25,000 lbper sq in. gage. We have elected to use an inside diameter of 16 in.

The thickness of the vessel is to be selected so that the vessel will rupture at twice the servicepressure by use of the equation of Faupel, which is based on experimental-test data.

The vessel is to be autofrettaged with a pressure sufficient to place the elastic-plastic inter-

face at the geometric-mean radius, that is, at l/r&.The physical properties of the steel are:

Tensile strength = 105,000 psi

Yield strength = 90,000 psi

Using the above suggested design procedure, determine:u. The required thickness of the shell.b. The autofrettage gage pressure required for locating the elastic-plastic interface at the

geometric-mean radius.e. The radial compressive stress under autofrettage pressure at:

1. the inside surface2. the elastic-plastic interface3. the outer surface.

Page 306: Process Equipment Design

C H A P T E R

m a15MULTILAYER VESSELS

II n the previous chapter the Lamk relationships basedupon the theory of elasticity were presented for tangential,axia l , and radial s tress dis tr ibut ion in a thick-wal led vessel .Also, the relationships of Voorhees and Faupel based uponthe theory of plasticity were discussed. These relation-ships make possible the determinat ion of the required thick-uess of thick-walled vessels for high-pressure applicationsand the determination of the stress variations according tothe theory of “plane strain.” The possibility of reducingthe required wal l th ickness by using mult iwal l construct ionin which the concentric shells are “shrink-fitted” togetherwil l now be considered.

The Lame hoop-stress equation indicates that the maxi-mum stress occurs at the inner surface of the vessel. Byshrink-fitting concentric shells together the inner shells areplaced in res idual compress ion so that the ini t ia l compress iveloop s tress must be re l ieved by the internal pressure beforehoop tensile stresses are developed. Therefore the maxi-mum hoop tensi le s tress as determined by- Lame’s re la t ion-ship is appreciably reduced with the result that t,here is areduct ion of the total wal l thickness required to contain thepressure when the vessel-wall thirkness is designed with asprcifred a l lowable s tress .

15.1 MULTILAYER VESSELS WITHSHRINK-FITTED SHELLS

Tbe relationships which follow were first presented byH. I,. Cox in 1936 (220). The t.beory as developed is basedupon the assumption that the maximum combined stresses(hoop-pressure stress plus hoop-shrinkage stress) cxistiug:rt the inner surface of each of the several shells will attain

a certain identical value. It is assumed that both theinternal and external diameters are known and that thenumber of shells is to be a minimum. It is also assumedthat the combined cylinder is fabricated by shrinking-oneach successive shell from the inside outwards and thatafter each shell is shrunk-on, the outside diameter ismachined to size before the next cylinder is shrunk-on tothe inner she l l or she l l s . Therefore the designer must deter-mine the number of shells and their radii plus the inter-ference (the amount by which the outside diameter exceedsthe inside diameter of the next shell). The relationshipsfor designing such a vessel may be derived as fol lows:

Consider a cylinder fabricated from n shells like the onein Fig. 13.1 where:

4, dl, . . . d,-l = diamet,er of successive intershell sur-faces, inches

pi = internal pressure, pounds per squareinch

p,, = external pressure, pounds per squareinch

PI, 1J.L. ’ . p,,-1 = successive interface pressures with piand p. acting

f, = hoop stress set up at the inside of eachshel l wi th pi and p0 acting, pounds persquare inch (to be the same for allshe l l s )

PI’, pr’, . * pn’ = interface shell pressure that existswhen pi = p,,

According to the Lamk thick-walled-vessel theory, con-sidering the rth + 1 shell and using the sign convention asgiver1 by Ey. 14.6b gives:

296

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Multilayer Vessels with Shrink-fitted Shells

vo

297

hit

thereforef7 = -Pr

(15.1)

andb 46

pr+l = -jr+1 = - a + __ = - a - i - -~rv+12 dr+12

(15.2)

And by Eq. 14.7

f*=h=-$+u=f2+u (15.3)

where u and b are defined by Eqs. 14.llb and 14.lOa,respectively. Therefore - _

n

,

pr -j, = --2a

Subtracting Eq. 15.3 from Eq. 15.2 gives:

pr+l &+I - f&r2 = --a(dr+1’ + dr2)

Dividing through by dr2 gives:

Pr+l dr+12 _ j = _d,.2

* u(“$ + I )

Letdr+l~ = K4

rtl

Substitutiug Eq. 15.6 into Eq. 15.5 gives:

pr+l Kr+12 -f, = -a(K,+12 f 1)

Dividing Eq. 15.7 by Eq. L5./4 gives:

.P~+I &+I’ ---Lq Kr+12 + 1Pr - fq = - 2

Let!

C %+I~‘+I = 1 + K,+12

It then follows from Eq. 15.8 that

c+1 Pr+1 - Pr = (1 - C+df*and similarly

c TPT - P r - 1 = (1 - Gl.fqNowifr = 1,

ClPl - p i = (1 - CllfnI If r = 2,

czpz - p1 = (1 - C,)jq, If r = 3,

C3P3 - p2 = (1 - (:3).f9

If r = n,CnP0 - P n - 1 = (1 - Wfp

This series can be represented by:

(C1C2C.( * * * C,)p, - p i = ( 1 - ClC2 . . Glfp

L e t

CIC&3 * - - C, = F

(15.4)

(15.5)Fig. 15.1. Diagram of Q multilayer shell showing notation used in dcri-vations.

(15.6j Equation 15.12 then becomes:

FP, - pi = (1 - Of,

Solving for j, gives:

(15.14)

(15.7) (pi - PO) - VP, - PJF - l

therefore

(15.8) f, = -p. + p*( )

(15.15)

(15.9)

(15.10)

(15.11)

(15.12)

The smallest value of j, for a given pressure difference(pi - po) obviously exists when F has a maximum value.The method of Lagrange multipliers may be used fordetermining such a constrained maximum (34). Thismethod indicates that the maximum value of F exists when

K1 cz K2 = K3 , . . = K, = K (15.16)

Substituting Eq. 15.6 into Eq. 15.16 gives:

K= $k!+$+ . !,d,-1

(15.17)12 z 1

A comparison of Eq. 15.16 with Eq. 15.9 indicates that

Cl zz c2 = c3 = . . . c, = c (15.18)

SubstituCng Eq. 15.18 into Eq. 15.13 gives:

F = C1C3C3 * . . c, = C” (15.19j

SubstituGng Eq. 15.19 into Eq. 15.15 gives:

(15.13) f, = -po + p*( >

(15.20)

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2 9 8 Multilayer Vessels

where p. = external pressure on vessel, pounds per squareinch

pi = internal pressure in vessel, pounds per squareinch

n = total number of shells in vessel wall with thick-nesses satisfying Eq. 15.17

f, = combined stress at interface of each shell (pres-sure stress superimposed on shrinkage stress)

C = constant as defined by Eqs. 15.9 and 15.6

A comparison of Eqs. 15.16 and 15.18 indicates thatEq. 15.9 may be written as:

where

C=2K21 + K2

(15.21)

K=(2)=@)=($)= . ..(-+) (15.22)

Therefore it follows that.-In o2n

‘” = (l;‘;i2)”(15.23)

Substituting Eq. 15.23 into Eq. 15.20 gives:

fQ (1 + K2)“h -ho) _ p2nK2" - (1 + K2)n I ’

( 1 5 . 2 4 )

Equation 15.24 is the general relationship for determiningthe maximum combined stresses at the interfaces of theconcentric shells and at the inside surface of the innermostshell. The combined stresses (hoop) at these lor.ations allhave the same numerical value. This equation for theusual case where the external pressure is zero (gage),p0 = 0, reduces to:

.f, =pdl + K2jn

2nK2n - (1 + K2)%(15.25)

15.la Example Calculation 15.1. To illustrate theapplication of Eq. 15.25, consider a multilayer vesselhaving an inside diameter of 12 in. and an outside diameterof 23x6 in. which has been formed by shrink-fitting. Thevessel is to operate under an internal pressure of 20,000 psiand is constructed of three shells. The interface diametersare 15 in. and 183i in., respectively. Determinef,.

K=Kl+K2+= 23.4375K3 = _____ = 1.2518.250

Because K1 = Kz = KS, this vessel satisfies Eq. 15.16;therefore Eq. 15.25 is applicable and may be used to calcu-late f,. Substituting in Eq. 15.25, we find that:

pi = 20,000

K = 1.25, K2 = 1.5625

n=3therefore

“67 =20,000(1 + 1.5625)3

23(1.25)6 - (1 + 1.5625)3

fq = 24,582 psi (compressive stress)

If this vessel had been fabricated of two shells insteadof three, then in order to satisfy Eq. 15.16 the interfacediameter would have had to be 16.8 in. For this vessei

n=2and

K - ‘1”;” - ‘“;i3i5 _ 1.4, K2 = 1.96

Substituting into Eq. 15.25 gives:

f-2 =20,000(1 + 1.96)2

22(1.4)4 - (1 + 1.96)2

= 26,530 psi

If the shell had been of monobloc construction, the maxi-mum stress would have been 34,212 psi (see ExampleCalculation 14.1). A reduction of 2235 y0 was achieved bytwo-shell construction whereas a reduction of only 28>/4%was realized by three-shell construction. It is thereforeapparent that from the economic consideration of shrink-litfabrication the use of more than two shells may not bejustified.

15.1 b Determination of Interface Pressures. To deter-mine the variations in hoop stress throughout the wall ofany shell by using the Lam& relationships it is necessaryto determine the combined pressure at any interface (pres-sure resulting from internal pressure plus shrinkage stresses).

Eq. 15.15 may be rewritten as follows:

fq(F - 1) = -Fpo + p i

At the fth shell

pi = pr and F = C”-’ (by Eq. 15.19)

thereforef,(c- - 1) = - c”-‘po + pr

Substituting for f, by Eq. 15.20 and rearranging gives:

Pr = [‘:I:] (Pi-P01 +Po (15.263

15.lc Example Calculation 15.2. Determine the inter-face pressures of Example Calculation 15.1. The combinedpressure existing at the two interfaces (shrinkage stress witha superimposed 20,000-psi pressure stress) is given byEq. 15.26.

but

and

P r = [c~~~-ll] ( P i -PO) +Po

PO = 0

C=2K2 2(1.5625)~ = 1.2221 + K2 = 1.5626

therefore

P r =[

1.222cn+ - 1 (20 ooo)1.222n - 1 1 ’

Therefore when n = 3 and P = 2 (second interface),substituting for (n - r) = 3 - 2 = 1.0 gives:

(pr)r=2 = Cl.222 - 1)w4000)1.2223 - 1

= 5350 psi

: I

I+--, ~~i. -~~ ----- -- --\ \ \I /

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Multilayer Vessels with Shrink-fitted Shells 2 9 9

and(ft)d=d, = 17,820 psi

For the outer shell

di = 18.75 in.

do = 23.4375 in.

pi = 5350 psi

PO = 0

Substituting into Eq. 14.12 gives:

(ft)d=di = 24,400 (in comparison to 24,582 psiobtained by use of Eq. 15.25)

and(ft)d=do = 19,040 psi

To summarize the stresses existing in the three shells(combined shrinkage and pressure stress),

1. For the inner shell

ji = 24,970 psi

j. = 16,870 psi

(24,582)

2. For the middle shell

ji = 24,270 psi

j. = 17,820 psi

3. For the outer shell

(24,582)

ji = 24,400 psi

j. = 19,040

(24,582)

15.1 f Determination of Shrinkage Stresses. Theshrinkage stresses may be readily determined by the methodof superposition. The stress variation, assuming the vesselis a monobloc shell having the same inside diameter andoutside diameter as the multilayer vessel, may be deter-mined in each by use of LamB’s equations.

The stress variation in each of the shells can be determinedby the methods presented in the previous section. Theshrinkage stresses are obtained by subtraction with properallowance for signs.

15.1 g Example Calculation 15.4. The vessel under con-sideration in Example Calculations 15.1, 15.2, and 15.3 willbe used for demonstrating the method of obtaining shrinkagestresses.

In Example Calculation 14.1 a monobloc vessel wasconsidered which had the same proportions as the multi-layer vessel in Example Calculations 15.1, 15.2, and 15.3.The shrinkage (shrink.) stresses are obtained by subtractingthe individual combined stresses in each of the shells of themultilayer vessel from the monobloc-vessel stresses asfollows.

Inner surface of inner shell (P = 6):

jt = 34,212 psi (for monobloc)

jt = 24,970 psi (for multilayer)

34,212 + ft(shrink.) = 24,979

Therefore fr(.&ink,) = - 9242 psi (compression).

Also, for n = 3 and P = 1 (the first interface),

(p ) = = Iu.22a2 - 11(2o~owTT 1 1.2223 - 1

= 11,920 psi

To summarize the pressure-stress considerations:

1. The pressure at the interface of the inner shell is equalto the operating pressure of 20,000 psi.

2. The pressure at the interface of the inner shell andthe middle shell is 11,920 psi.

3. The pressure at the interface of the middle shell andthe outer shell is 5350 psi.

4. The pressure at the outer surface of the outer shell is0 lb per sq in. gage.

15.1 d Determination of the Hoop Stresses at the OuterSurfaces of Multilayer Vessels. The individual shells maybe treated in accordance with Lame’s theory if the pressuresat the interfaces are computed by use of Eq. 15.26. Thehoop stresses may then be computed by use of Eq. 14.12 asan alternative to the use of Eq. 15.25.

15.le Example Calculation 15.3. By using the vesselunder consideration in Example Calculation 15.1 and theresults obtained in Example Calculation 15.2, the hoop-stress variation in each of the three shells may be readilydetermined.

The Lame relationship for hoop stresses as given inEq. 14.12 is:

jt = Pdi’ - do2 d.2d 2 Pi - POdo2 - di2

+” ___d2 [ 1do2 - di2

Each of the individual shells can be treated as a monoblocvessel as follows.

For the inner shell

di = 12 in.

do = 15 in.

pi = 20,000 psi

p. = 11,920 psi

Substituting into Eq. 14.12 gives:

(jt)d=di = 24,970 psi

The value of 24,970 psi is within slide-rule agreement of thevalue of 24,582 psi obtained by use of Eq. 15.25 in ExampleCalculation 15.1.

(ft)d=d, = 16,870 psi

For the middle shell

di = 15 in.

do = 18.75 in.

pi = 11,920 psi

p. = 5350 psi

Substituting into Eq. 14.12 gives:

(j&& = 24,270 psi (in comparison to 24,582 psiobtained by use of Eq. 15.25)

Page 310: Process Equipment Design

300 Multilayer Vessels

40,000 1 I I I I I I I I I I I I I I

Lamb monobloc

O--------._-------- - _______-- ---_-- nQ-184 -846 .g E

gz

6 7 8 9 1 0 11Radial distance, inches

Outer surface of inner shell (P = 7.5):

jt = 24,454 psi ( for monobloc )

jt = 16,870 psi ( for mul t i layer )

24,454 + ft(shrink.) = 16,870

Therefore shrinkage jt = -7584 psi (compression).inner surface of middle shell (r = 7.5):

jt = 24,454 psi ( for monobloc )

jt = 24,270 psi ( for mul t i layer )

24,454 + ft(shrink.) = 24,270

Therefore ft(8hrink.j = - 184 psi (CompreSSiOn).

Outer surface of middle shell (r = 93ie):

jt = 1 8 , 6 6 6 psi ( for monobloc )

jt = 17,820 psi ( for mul t i layer )

18,666 + ft(shrink.) = 17,820

Therefore fi(shrink.) = -846 psi (compression).Inner surface of outer shell (r = 9x6):

jt = 18,666 psi ( for monobloc )

jt = 24,400 psi ( for mul t i layer )18,666 + ft(shrink.) = 24,400

Therefore ft(&i&.) = +5734 psi (tension).Outer surface of outer shell (r = 11.718):

jt = 14,212 psi ( for monobloc )

jt = 17,820 psi ( for mul t i layer )

Therefore ft(shrink,) = +5608 psi (tension).

Figure 15.2 is a graphical representation of the super-imposed Lamk pressure-stress curve for a monobloc vessel

1 2 1 3

Fig. 15.2. Graphical representationof the stress distribution in vessel ofE x a m p l e C a l c u l a t i o n 1 5 . 4 .

having the same dimensions as the multilayer vessel, theshrinkage stresses of Ihe multilayer vessel, and the com-bined operating-pressure stresses of the multilayer vessel.A reduction of 9630 psi in the maximum hoop stress at t,heinner surface is obtained as a result of shrink-fittingfabricat ion.

15.1 h Determinat ion of Interferences Required inShrink-fitted Vessels. The necessary total difference indiameters (interference) may he calculated by the relation-ships developed by Cox (220). The general re lat ionship is :

EU, 1-=4

[c’r--r(K2r + 1 ) - 2P](& - PO)(C” - l)(P’ - 11

(15.27)

where U, = difference in diameters, inchesd, = outside diameter of the rlh shell, inchesE = modulus of elasticity of shell material, pounds

per square inch

C2!!51 + K2

(see Eq. 15.21)

n = total number of shellsr = interface number, numbering outward

K = E ratioID

(see Eq. 15.22)

pi = internal pressure, pounds per square inch gagepO = external pressure, pounds per square inch gage

Substituting for C in Eq. 15.27 by means of Eq. l5.21gives :

EU, 2n--rK2n-2r[(h-2 + ,)r(K2’ + 1) - 2’+1K2’](pi - pO)-=4 (K2r - 1)[2nK2” - (K2 + l)“]

(15.28)

For a multilayer shrink-fit,ted vessel fabricated from two

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Multilayer Vessels with Shrink-fitted Shells 301

For the second interface, where n = 3 and r = 2, Eq.15.31 applies.

shells. n = 2 and r = 1. Therefore

EUldl

(15.29)

For a multilayer shrink-fitted vessel fabricated fromthree shells, n = 3, and r = 1 and r = 2.

For r = 1

EUl 4K4-= --’dl 1K4++1 1 (Pi - PO) (15.30)

For P = 2

2K2(K4 + 4K2 + 1)~-

(K2 + 1)(7K4 + 4K2 + I) I(Pi - PO) (15.31)

For a multilayer vessel fabricated from four shells R = 4,ztud r = 1, 2, and 3.

For r = 1

EU1 = .-m-t!!?--15K6 + llK4 + 5K2 + 1

(pi - PA 05.32)

For r = 2

Kc-2 4K4(K4 + 4K2 + 1)d2

- ~~- (Pi - PO)(K2 $ 1)(15K6 + 11K” + 5K-2 + 1)

For r = 3(15.33)

EU, 2~2(#8 + X6 + UK4 + 5K2 -t l)(~i - &)_- =ic3 (5~4 + Ks + 1)(15K6 jZ1K455K2 + 1)

(15.34)

15.li Example Calculation 15.5. The interferences for1 he multilayer vessel described in Example Calculation 15.1will be determined. The nominal dimensions of the three-layer vessel from Example Calculation 15.1 iIre:

1. Inner-shell inside diameter, 12 in.; outside diameter,13 in.

2. Middle-shell inside diameter, 15 in.; outside diameter,183% in.

3. Outer-shell inside diameter, 18% in.; outside diameter,23x6 in.

Also, K = 1.25, pi = 20,000 psi, and p. = 0.The modulus of elasticity will be taken as 30,000,OOO psi.For the first interface, n = 3 and r = 1, Eq. 15.30

:rpplies.

EUI _dl

- 7K4 +4f2+ i (Pi - PO)

Substituting and solving for U1 gives:

(4)(1.25)4(7)(1.25)4 + (4)(1.25)2 + 1>

(20,000)

therefore

ZJ1 = 0.0040 in,

The outside diameter of the inner shell must exceed theSde diameter of the middle shell by 0.0040 in. beforeshrink-Wing.

EU2 2K2(h’ + AK’ + 1)-=4 (K2 + l)(7h-4 + 4sK2 + 1) (” - “)

Substituting and solving for ZJ2 gives:

u2=(&g

(2)(1.25)2[(1.25)4 + (4)(1.25)2 + l][(1.25)2 + 1][(7)(1.25)4 + (4)(1.25)2 + l] 1 (20’ooo)

therefore

U, = 0.0061

The outside diameter of the subassembly consisting ofthe inner and middle shells must be machined to provide aninterference on the diameter of 0.0061 in. with the outershell before shrink-fitting of the outer shell.

15.1 i Simplified Relationships. The general relation-ships for multilayer-vessel construction have been presentedin the previous sections according to the method developedby Cox (220). The use of these relationships is somewhatinvolved; also, for practical reasons most shrink-fitted multi-layer vessels consist of two shells. Therefore it is desirableto work with simplified relationships for the case of two-shell construction in which the external pressure is zero.For this condition the combined shrinkage stress and pres-sure stress at the inner surface of each shell (f,) is givenby Eq. 15.25.

pi(l + K2jnfq = ___2”K2n - (1 + K2)n

This equation may be rewritten as:

/l + K2\”

(15.35)

Eq. 15.21 C is defined as:

Substituting into Eq. 15.35 gives:

(15.36)

Solving this equation for C gives:

C=Wi (15.37)

This relationship gives the value of C for II number ofshells with pi (internal working pressure) and f, (allowablestress) specified. The corresponding value of K is givenby Eq. 15.21 rewritten as follows:

K = 2/C/(2 - C) (15.38)

An examination of Eqs. 15.37 and 15.38 indicates thatwhen n = 2, pi = 3fq, C = 2, and K = m. Thus, when

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3 0 2 Multilayer Vessels

the working pressure is three times the allowable stress, thetheoretical required thickness approaches infinity. Thissituation may be compared with that of the Lam& equationfor a monobloc shell, in which the theoretical required shellthickness approaches infinity when the working pressureapproaches the allowable stress of the material. Thus, theuse of multilayer-vessel theory theoretically permits athreefold extension of the design range, two-shell construc-tion being assumed. A greater extension may be obtainedby using more than two shells. The practical limits oftwo-shell construction occur in the range in which theworking pressure is from one half to twice the allowableworking stress.

15.lk Example Calculation 15.6, a Two-shell Shrink-fitted Vessel. A multilayer vessel having an ID of 12 in.and consisting of two shells which operate under a workingpressure of 20,000 psi with an allowable working stress of24,582 psi (same as the maximum stress in the three-shellvessel of Example Calculation 15.1) will be designed.

By Eq. 15.37

C= .;/(pi/jq) + 1 = 1/(20,000/24,582) + 1 = 1.348

By Eq. 15.38

K = X4(2 - C) = 41.348/(2 - 1.348) = 1.44

dl = (12)(1.44) = 17.28

dz = (17.28)(1.44) = 24.85 in.

Thus, for the same maximum-stress limitations, the two-shell vessel has an outside diameter of 24.85 in. and a three-shell vessel has an outside diameter of 23% 6 in. (see ‘ExampleCalculation 15.1).

If this vessel had been of monobloc construction and themaximum stress had been limited to 24,582 psi, the follow-ing would have been true.

By Eq. 14.14a

j K2 + 1 24,582 K2 + 1 1,229-=-=-- --=P K2 - 1 20,000 K2 - 1

therefore

K = 3.12

OD = (12)(3.12) = 37.4 in.

Thus, the monobloc shell would be 37.4/23.44 = 1.59times as thick as the vessel of three-shell construction. Thevessel of two-shell construction would be 24.85/23.44 = 1.06times as thick as the vessel of three-shell construction.

15.11 Thermal Expansion for Shrink-fitting. In orderto shrink-fit successive shells upon one another, the outershell must be expanded by heating. It may then be slippedover the inner shell or shells and allowed to cool. Theexpansion must be sufficient to overcome the required inter-ference afiar cooling and also to provide some clearance foreasy assembly. The diametral enlargement of the outershell resulting from heating is:

Ad = a(At)d (15.39)

where Ad = increase in shell diameter, inchesd = inside diameter of shell, inches

At = temperature of heating minus room tempers-ture, degrees Fahrenheit

(Y = coefficient of thermal expansicn= 0.0000067 in. per in. per “F (for steel)

15.lm Example Calculation 15.7, Determination of Pre-heat Temperature for Shrink-fitting. In reference to Exam-ple Calculation 15.5, the required interferences for a three-shell shrink-fitted vessel were found to be 0.0040 in. and0.0061 in., for the inner and outer interfaces, respectively.The preheat temperature necessary to provide the requiredthermal expansions without clearances and also with anadditional clearance of 0.050 in. is to be determined.

The interface diameters are 15 in. and 18% in. Theinside diameter is 12 in. and the outside diameter is 23Tf6 in.

By use of Eq. 15.39 the temperature to which the she&must be preheated may be determined.

Ad = a(At)d

where LY = 0.0000067” FAt = (1 - 70” F)Ad = interference + clearance

For the case of no clearance:For the middle shell

therefore

Ad = 0.0040 in.

d = 15 in.

0.0040’ - 7o = (15)(0.0000067)

t = 110°F

For the out,er shell

Ad = 0.0061 in.

d = 18% in.

therefore

39.9” F

t - 7 0 = 0.0061= 48.5” F

(18.75)(0.0000067)

t = 118.5” F

For the practical case in which a clearance of 0.050 in. i;provided:

For the middle shell

Ad = 0.0040 + 0.050 = 0.054 in.

d = 15 in.

therefore

0.0540’ - 7o = (15)(0.0000067)

= 538” i?

t = 608” F

Page 313: Process Equipment Design

For the outer shell

Ad = 0.0061 + 0.050 = 0.561 in.

d = lSyd in.

lherefore

0.0651

t - 7o = (1a75)@.000a.@47) = 4470 F

t = 517’ F

Thus, a shrinkage resulting from a tempera&& differenceof only 40” to 50’ is sulllcient, but to insure erraapf assemblythe two shells age heated to about 500 to B&Y F. (Notethat after the middle shell is shrunk on the inner shell, thispartial assembly is cooled and machined to accurate sizebefore the third shell is added.)

15.2 MULTILAYER CONSTRUCTION USINGWELD SHRINKAGE

The previous section on the thermal expansion for shrink-fitting demonstrated that the necessary interference forprestressing could be obtained with a very small increase inthe temperature of the outer ring (40 to 50” F), uniformheating of the ring and no clearance for assembly beingassumed. It follows that if a narrow longitudinal handof from one tenth to one twentieth of the circumference ofthe outer shell were heated to 10 to 20 times this temper-ature difference, the same effect would be produced bycooling. In the cooling of a longitudinal welded joint suchan effect can be produced. Therefore a possible convenient

Multilayer Construction Using Weld Shrinkage 3 0 3

method of prestressing would take advantage of the shrink-age s&ess,es in the longitudinal welded joints of shells.

The A. 0. Smith Company has developed this technique(221) and haa successfully fabricated a considerable numberof law pressure vessels operating at a pressure above5000 psi.

The shrinkages of longikieinal we&& have been correlatedby Spraragen and Ettinger (222). For@ansv@se shrinkageof butt welds the following relations i

s = 0.1716; + 4&H&+ (15.40)

where s = transserse~sh :&Al%Su = cr.--sectional area, &weld, square inchest = plate thickness,, &&es

w = average width of weld, inches

Substituting into Eq. 15.40 the dimensions for a single-Vbutt weld of a J/4-in. plate gives shrinkage values in theorder of magnitude of >Q in. This shrinkage is greater thannecessary to prestress successive shells to the desiredamount. Therefore some of the shrinkage must be absorbedby the provision of a clearance between successive shells atthe time of welding.

As this procedure depends upon shop technique, it is notpossible to compute the final stresses that exist uponcompletion of the fabrication. Peening after welding mayalso be used to reduce the shrinkage stresses.

In the technique used by the A. 0. Smith Company aninner shell having a thickness usually greater than 3/4 in.and often $5 in. is first fabricated. This shell is not perfo-

F i g . 1 5 . 3 . A r c - w e l d i n g o f m u l t i l a y e r v e s s e l s . ( C o u r t e s y o f A . 0 . S m i t h C o r p o r a t i o n . )

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Multilayer Vessels

Fig. 15.4. Multilayer vessel tested to destruction, d/t = 9.3 (245). (Courtesy of A. 0. Smith Corporation.)

rated and serves to contain the fluid to be held underpressure in the vessel. Subsequent shells usually >/4 in. inthickness are progressively wrapped around the inner shell,tightened mechanically, and welded longitudinally. Thesesubsequent shells are perforated with small holes for venting.Cylindrical rings are inserted at both ends of the inner shellduring these operations to maintain a true cylindrical shape.The welds are staggered around the circumference tominimize localization of any excessive stresses at or nearthe welded joints and are ground flush prior to the addingof subsequent layers. After the vessel shell has been builtup to the desired thickness with successive layers, the endsof the built-up shell are machined for the welding groovefor the attachment of a formed head. In the final vessel

0 4 8 12 16 20 24 28 32Number of layers welded

Fig. 15.5. Increase in compressive stress in inner layer or o fraction of

number of wrapped layers (di = 48 in., ti = 35 in., t, = 34 in. (245).

fCowtesy of A. 0. Smith Corporation.)

12 or more successive layers are often used. Figure 15.3shows the arc welding of multilayer shells.

A number of these vessels were tested to destruction.The test data obtained have provided some valuable infor-mation concerning this method of fabrication. Whereasmonobloc vessels which fail under high-pressure serviceoften fragment, the multilayer vessels do not fail in sucha manner. Considerable deformation occurs prior to failure,and when a leak develops in the inner shell, fluid escapesthrough the perforations in the outer shells. This providesa system of venting which gives warning of possible rupture.In the tests to destruction some of the vessels were stressrelieved and others were not. One vessel which was notstress relieved withstood 8y0 greater stress than the corre-sponding vessel which was stress relieved. This indicatesthe desirability of not stress relieving, in order to retain thecompressive stresses developed in fabrication. Figure 15.4shows a multilayer vessel tested to destruction.

15.2a Analysis of Test Data. Figure 15.5 shows a plotof the induced compressive hoop stresses at the insidesurface of the inner shell as determined by circumferentialstrain measurements during the progressive wrapping ofsuccessive shells. The vessel used in this test had an insidediameter of 48 in. The vessel wall consisted of an innershell 34 in. thick wrapped with 32 concentric layers each34 in. in thickness. The curve gives an indication of thecumulative effect of the shrinkage of successive layers onthe inner shell.

One method of analyzing these data involves the pre-diction of the experimental curve by use of simplifiedtheoretical relationships. Consider the inner shell and thefirst wrapped layer as an inner shell under external pressureand an outer shell under internal pressure, respectively.The interface pressure is common to both shells. A sum-mation of forces about a diametrical plane can be made

Page 315: Process Equipment Design

according to the membrane t,heory. The forces in the t.woshel ls are equal and opposite , or

-Fi = --f&l ( for inner core )

+Fi = +fdd (for f i rs t wrapped layer)

Equating and solving for fi gives:

t1(fi)n=l = -fl -- = incremental stress induced in innert;

b u tcore by f irst w-rapped layer

(fi)n=O = O

there fore

AA = (.fi)n=~ - (.fi)n=o = -.fi!1

Assuming that there is uniform stress addition across theshell and considering that the inner core and the first layerare a unit, we may treat the second wrapped layer in likemanner . The incremental stress induced in the inner coreplus the first layer by a second wrapped layer is:

there fore

A(.fi)n=3 = .f3 (c~+~mi;) (15.41)

For uniform t,hickness of successive shells, tl = tz = t3 = t,.Therefore

A(.fi)n = -fn ~---l)& (15.51a)

For the vessel data in Fig. 15.5, where ti = !,i in. andt1, h . . . t32 = +/4 in .,

AWn = -.fn (15.42)

The stress in the rzth wrapped shell is produced by inducingstrain resulting from weld shrinkage. Weld shrinkage isdescribed by Eq. 15.40, where

s = 0 . 1 7 1 6 ; + 0.0121W0

(15.43)

For $i-in. plate with -15” bevel welds and fi6-in. clearance,

a = 0.06 sq in.

t = 0.25

w = 0.30

s = 0.041 + 0.004 = 0.045 in. (c i rcumferent ia l s t ra in)

A calculation from the data given in Fig. 15.5 shows thatall of this shrinkage was not used to develop induced stress .In reference to Fig. 15.5, the point given for the firstwrapped layer which is most consistent with the other dataoccurs at fi = 2000 psi.

Multilayer Construction Using Weld Shrinkage

By Eq. 15.42

3 0 5

there foref2 = 6600 psi

f2E2 = - = 0.0002 in. per in.

E

Total c~ircurrlf~relltial strain = ~ntl (15.4$)

= 0.0002(x) 19.25 = 0.031 ic.

Apparently 0.03 I /0.045 or about 70’ ; of Ihe weld shrink-age is converted to induced stress.

The total “free” weld shrinkage is a funct,ion of wald-joint dimensions. If the same plate thickness and weldjoint is used in each successive weld, the free weld shrinkagewill be a constant. Part of this free weld shrinkage is usedin overcoming clearance between layers, and part in com-press ing the inner layers e las t ica l ly . The major portion ofthe free weld shrinkage develops elast ic strain and resultants t ress in the layer i t se l f . Assuming for purposes o f s impl i f i -cation that each clearance in the welding operation absorbsthe same amount of the free weld shrinkage, we find thatthe total circumferential strain contributing to stressdevelopment will remain the same. The unit strain willthen be proportional to the diameter. or by Eq. 15.44

total strain 0 . 0 3 1E n=

4,= ~ = 0.0098'1 in. per in.

4

f, = & = 10.00987)r30 x 106) _ 296,000

d< + 1 + 4 - 0.25

(15.45)

Substituting Eq. 15.45 into Eq. 15.42 for jn, we obtain:

O F

A(& = -(-296,000)(2)

122 + 100.5n + 99..5

Or as an approximation,

Nfi), =592,000

112 + lOOn + 100(15.46)

Equation 15.46 represents the incremental increase in theinduced compressive stress in the inner core. Since 1 heincrements are small, s/4 in., and the number of shells islarge, Eq. P5.46 will be written in differential form and willbe formal ly integrated, or

dfi _ 5.92 X 10”4, n2 + 1OOn + 100

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3 0 6 Mu l t i l aye r Vesse ls

2 2 . 0 0 0

20,000

18,000

1 6 , 0 0 0

I I0 Values f r o m F i g . 1 5 . 5

I I

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 20 22 24 26 28 30 32

Fig. 15.6. Comparison of predicted

stresses with experimental data of Fig.

15.5.

N u m b e r o f l a y e r s , n

therefore Substituting for K = do/di gives:

j;: -5.92 x 105 ln

(

2n + 1 0 0 - 41002 - 4(100)=

2/(100)2 - 4(100) 2n + 100 + d1002 -4(100) >

-5.92 X lo5 -f i = In 2n + 100 9 898 2n + 100 + 9 8

fi = - 6 0 3 0 l n 2n+22n + 198

(15.47)

Equation 15.47 gives the change in the inside-wall hoopcompressive stress resulting from n number of shells. Theanalysis of the stresses was based on thin-wall theory. Asthe number of shells increases, the wall becomes progres-sively thicker, and the thin-wall analysis should be adjustedfor this factor. This can be accomplished by applicationof Lame’s analysis for thick-wall vessels. The wall is underan external pressure because of the weld shrinkage. ByEq. 14.12, and for --pi = 0 and d = di,

f- %,do2

1(La1nb) = do2 - di2(15.48)

Let. K = do/di, then

(15.49)

The thin-wall hoop-stress equation, 3.14, can be writtenfor external pressure as:

fl(membrane) = __2t

(15.50)pod,

Substituting for t = (do - di)/Z gives:

f ---Pod,t(membrane) =-

do - di(15.51)

f-poK

t(membrane) =-K - l

(15.52)

The ratio of the maximum fiber stress at the insidesurface as given by Lamb’s analysis, by Eq. 15.49, to thecorresponding stress based on membrane theory, by Eq.15.52, is:

fi(Lam9 = (-2poK2)/(K2 - 1)

f~(membrane) (-poK)/(K - 1)= +& (15.53)

Rewriting Eq. 15.53 in terms of n number of wrappedlayers for the particular vessel under consideration, weobtain:

or

K _ 48 + 1 + 5% - 1,: 97 + nz-48 96

ft(LamB) 2(97 + n)96= - - (15.54).tl(membrane) 96(97 + n + 96) = 193 + n

Multiplying Eq. 15.47 by the ratio given in Eq. 15.54corrects the stress based on membrane theory to the stressbased on Lam& theory for thick walls, or

.fi(corrected)= [y;3'+;][ - 6030h;$8] -27,200

(15.55)

Table 15.1 lists calculations for the vessel under con-sideration.

Inspection of Table 15.1 indicates that the rate of increasein the compressive hoop stress in the inner core decreasesrapidly as the number of outer shells is increased.

Figure 15.6 shows a plot of the experimental data fromFig. 15.5 compared with the predicted values based on thh-

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Ribbon- and Wire-wound Vessels 307

Table 15.1. Calculation of Stresses in a Multilayer

VesselC& = 48 in., ti = 35 in., t, = x in.)

illn(g&) -66030 ln $&$ ~~3~~ (:rrected)

0 -4.501 27,2001 -3.922 23,700 -3,500 1.01 -3,5302 -3.527 21,300 -5,900 1.02 -6,0104 -3.034 18,300 -8,900 1.03 -9,150

1 0 -2.293 13,850 -13,350 1.06 -14,10020 -1.734 10,480 -16,720 1.10 -18,40032 -1.378 8,320 -18,880 1.15 -21,70050 -1.075 6,480 -20,520 1.21 -24,800

wall analysis (column 4 in Table 15.1) shown in the dashedline and the predicted values based on thick-wall analysis(column 6 in Table 15.1) shown by the solid line.

15.2b Discussion. The stress distribution across theshell of the completed vessel may be predicted by use of therelationships given. To do this the initial stress in thegiven layer as it is wrapped on is determined by Eq. 15.45.The contribution to this value of each subsequent layermust be determined by integration with the correct con-stants for the layer under consideration.

The relationships given apply only to the vessel describedand are limited by the assumptions made in the analysis.Variations in manufacturing methods will alter the analysis.It should also be pointed out that certain assumptionswere made concerning the shrinkage of the layers. Theseassumptions were approximations and may or may not haveaccurately represented the shrinkages that were used in thefabrication of the vessel. The final relationship given byEq. 15.45 fits the experimental data in a satisfactory manner(see Fig. 15.6). Seely and Smith (231) have presented ananalysis of the stresses in a hollow cylinder made of thin-walled shells having various wrapping pressures.

15.3 RIBBON- AND WIRE-WOUND VESSELS

The technique of wire winding cylindrical shells subjectedto high internal pressure is old and has long been used forreinforcing gun barrels. More recently this technique hasbeen extended to include the use of flat and interlockingribbons for prestressing shells.

15.3a Wire and Flat-ribbon Windings. Wire and flat-ribbon windings are used only for absorbing hoop and

Fig. 15.7. T y p i c a l s e c t i o n o f a prestressad v e s s e l w i t h w i r e o r f l a t - r i b b o nw i n d i n g s ( 1 9 4 ) .

W e l d s 4

TCore tube

‘LProfile roller

Fig. 15.8. Detail of Wickelofen wrapping (225). (Courtesy AmericanS o c i e t y o f M e c h a n i c a l E n g i n e e r s . )

radial stresses and offer no restraint to axial load. An innermonobloc shell must be used having a minimum thicknesssufficient to absorb the axial internal-pressure load. In theprocess of winding, the inner shell may be considered tobehave as a vessel under an external pressure induced bythe winding. This pressure at the interface between theshell and wire windings also acts as an internal pressureon the wire windings.

An internal pressure in the vessel induces hoop-tensionstresses in both the inner monobloc shell and the outerwindings. Thus, under the operating conditions the innermonobloc shell may be considered to have both internal andexternal pressures, and the windings to have inducedstresses resulting from winding tension and internal pressure.

A typical section of a flat-ribbon- or wire-wound vessel isshown in Fig. 15.7.

15.3b Interlocking-ribbon Winding. A method of con-structing prestressed vessels with spiral-wound interlockingribbons of steel was developed in Germany by Schierenbeck(223) and described by Holroyd (224) and by Donovan,Josenhans, and Markovits (225). The principle involvedin this design consists of interlocking the winding by meansof grooved profiles to permit the winding to carry a portionof the axial load.

Figures 15.8, 15.9, and 15.10 illustrate the method ofinterlocking-ribbon construction, which is also known as“Wickelofen” wrapping.

In prestressing Wickelofen-wound vessels, shrinkage, isobtained by preheating the ribbon to from 1100 to 1550” F(prior to winding), as indicated in Fig. 15.9. The wound

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308 Multilayer Vessels

,-Wickelofen drum

Profile-r o l l e r

Coupl ing- /

Fig. 15.9. Wickelofen-wrapping lathe (225). (Courtesy of American

Society of Mechanical Engineers.)

ribbon is cooled first by air jets and then by water jets.After each layer of tape is spirally wound around the lengthof the vessel, the end is welded to the previous wrappedlayer. Winding speeds can be as high as 15 fpm (224).The ribbon may vary in thickness from s/4 in. to N in., andthe width is usually about 10 times the thickness.

Special lathes are required for wrapping. These lathescan be of the same type used in machining monobloc-vessel

Double-cone\ gasket

Shrunk flange

Wickelofin flange

Fig. :5.10. Wickelofen flanges (225). (Courtesy of American Society of

Mechanical Engineers.)

Fig. 15.1 1. Flanged end of a Wickelofen vessel (225) . (Courtesy of

Bureau of Mines, United States Deportment of the Interior and American

Society of Mechanical Engineers.)

shells except that the tool carriage must be modified toaccommodate the reel of tape, the tape heating and coolingequipment, and a profile back-up roll.

Figure 15.11 shows the flanged end of a Wickelofen-wound vessel. Figure 15.12 shows a sectional drawing of adesign for a high-pressure vessel for coal hydrogenationreported by Donovan, Josenhans, and Markovits (225).This vessel was designed to be fabricated of a low-chromevanadium steel similar to SAE 6115 suitable for being heattreated during wrapping. This steel has a yield strengthof about 90,000 psi and an ultimate strength of about130,000 psi. Donovan recommends that the yield strengthbe between 60 y0 and 75 ‘% of the ultimate strength and thatthe elongation not be less than 17%. In the designing ofthe vessel the strip was stressed to 70,000 psi on the outerlayer; this resulted in a compressive prestress of 50,000 psiin the core. Twenty-two layers 5is in. thick were usedover a core 135 in. thick (see Fig. 15.12).

15.4 THEORY OF RIBBON AND WIRE WINDING

Ribbon- and wire-wound vessels may be fabricated witheither a thin or a thick shell (inner core plus windings).Thick shells with windings are used obviously because ofthe greater strength resulting from wound construction incomparison with monobloc construction. The reason forthe use of wound thin shells is not so apparent. Oneexample of the use of such shells is the case in which theinner core must be fabricated of a noncorrosive or ductilematerial that does not have sufficient strength to resist thetensile (hoop) loads produced by the internal pressure.

15.4a Thin Shells Wound at Constant Tension. Thesimplest application of ribbon or wire winding involves theuse of a thin shell onto which is wound flat ribbon or wireof the same material of construction as the shell with con-stant tension in the wire during winding. HS the ribbonor wire is wound onto the shell, it causes a cncumferentialcompressive stress to develop in the shell.

In such an application an internal shell must be designed

Page 319: Process Equipment Design

of suf f ic ient th ickness to res is t the ax ia l load resul t ing f rominternal pressure . The axial stress is only one half of thehoop stress according to membrane theory; therefore theinner shell needs to he about half as thick as the corre-sponding monobloc shell. The necessary additional shellmaterial required to absorb the hoop stress is made up ofthe wire winding. The inner core may be considered tobehave as a thin-walled vessel if the operating pressure ismoderate.

The compressive stress induced in the inner shell may bedetermined by taking a summation of forces about adiametral plane with no internal pressure in the vessel.At this plane the total tensile force in the wires is equal toand opposi te in s ign to the compress ive forces in the shel l , or

t?l(Afhz = -fn ji + (n _ l)& (See Eq. 15.41a.)

The following equations are for a ribbon-wound vesselwhere

n = number of layers of windingt, = thickness of ribbon, inches

W ,, = width of ribbon, inchesfn = stress in ribbon, pounds per square inch

ti = thickness of inner shell, inchesfi = induced compressive stress in inner shell, pounds

per square inch(Afi& = induced-stress increment in inner shell from the

winding on of n layers of r ibbon, pounds per squareinch

T = tension in ribbon, pounds

a n d

.fn = $ (15.56)R 18

Substituting Eq. 15.56 into Eq. 15.41a gives:

1 (15.57)n

For the case where the ribbon thickness is small com-pared to the shell thickness,

dji = LT s n 1d n

W7L 0 4 + (n - I)&2

Integrating and substituting limits gives the followingequations.

For r ibbon windings ,

(15.58) Shrink ring

For square-wire windings where wn = t,,

(fi)% = L$l ln lncnt ,-ml; + ti (15.59)It 2 R

For circular-wire windings where wn = 2r, = t,, and wirearea = nrW2, Eq. 15.57 may be modified to:

Theory of Ribbon and Wire Winding 3 0 9

Carrying out the operations indicated for the derivationfor the r ibbon-wound condit ion gives :

(fi)n = s2 ln

7rru(rz - 1) + 2ti

2ti - ?rr, I(15.60)

w

Under the operating condition of internal pressure thecombined-stress, fi, total in the shell will be (f& plus thepressure stress, (fi)p. Assuming uni form dis t r ibut ion of thepressure stress and using the method of superposition, weobtain:

(fi) total = mn + WPBut for r ibbon and square-wire windings,

(fi)p = !g = 2t, p;tntz a

and for round wire,

(.fi)p =Pd

2ti + mm,

8’1lD, liquid in-

8’ ID, liquid out

Vickers-Anderson

J-sections

12%-chromecore tube

Threaded neck flange

iinsulation

Shrink ring

joint

SAE 611522 layer

Vickers-Andersojoint

(15.61)

(15.62)

(15.63)

Detail B

Fig. 15.12. Bureau of Miner converter design based on the Wickelofea

principle (225). (Courtesy of Bureau of Mines, United States Department

of the Interior and American Society of Mechanical Engineers.)

Page 320: Process Equipment Design

310 Multilayer Vessels

Combining Eq. 15.62 with Eq. 15.58 and Eq. 15.60 withEq. 15.63 gives the following equations.

For ribbon windings,

- T(fihd = tw ln

t,(n - 1) + ii + Pi4ti - t, 2ti + 2nt,

(15.64)n n

For round wire,

pi42ti + mr,

( 1 5 . 6 5 )

Equations 15.58, 15.59, and 15.60 become indeterminateas t, approaches ti. This is also true in Eqs. 15.60 and15.65 when rW approaches (2/r)&. This case is seldomencountered except when n = 1. If n = 1, the equationsmentioned above may be modified as follows.

For ribbon,

Wn=l = ,;nz

For square wire,

(fi)n=l = fi”nz

For round wire,

(15.5833)

(15.59b)

(15.60b)

For ribbon,

(15.64b)

For round wire,

LfJtotal = 2, + 2ti y:,, (15.6513)wz w

For calculations in which n > 1 but t, or 2r, approaches0, successive calculations using the above equations maybe made for each value of n, and the stresses combined bythe method superposition.

The stress in the winding will be a maximum at the outerlayer when the vessel is under internal pressure and willequal the sum of the tensile stress from the tension duringwinding plus the pressure stress.

For ribbon winding,

(15.66)12

For round wire,

(f?Anax = s2 + Pi42ti + mr,

(15.67)

T5.4b Thin Shells Wound at Constant Tension withWinding and Shell of Dissimilar Metals. Corrosive fluidsunder pressure may be contained in vessels having an innershell of corrosion-resistant material dissimilar to the wind-ing. The relationships derived for the induced compressivestress in the shell (ji) for n layers of winding, including

Eqs. 15.58, 15.59, and 15.60, apply. However, in the caseof a shell that has a modulus of elasticity which differs fromthat of the windings, Eqs. 15.62, 15.63, 15.64, 15.65, 15.66,and 15.67 must be modified. A summation of forces maybe taken at a diametral plane as follows:

2~’ = Shell + Winding - Fprescwre = 0

= ji2til + j,2nt,l - pidil = 0

= fiti + j,nt, = ‘+ (15.68)

Under the influence of internal pressure, the shell andthe winding expand together. Clark (31) assumes that theunit strain in the shell and the winding is proportional tothe radial distance of the point under consideration; or

(G)p ri-=-(6dp rn

But

(fi,,&

= (4p

and

Jl?.dE = (QJpE,

therefore

op riEi

(.fJp r,E,

Substituting for the mean value of ri and r,, for ribbonwinding, gives:

+(di + ti)Ei= &(di + 2ti + nt,)E,

(15.69)

Equations 15.68 and 15.69 contain an unknown ji and anunknown j,, which may be determined by simultaneoussolution.

(.fn)p =Pi4

(4 + ti)Ei 1(15.70)

2ntn + 2ti(4 + 24 + nt,)E,

and

(.fi)p =Pi4

2nt12

(4 + 24 + &)E, + 2t,

(4 + ti)Ei I

(15.71)

To calculate the stress in the shell under the influenceof internal pressure, Eq. 15.71 may be substituted for thesecond term in Eq. 15.65. To calculate the maximumstress in the winding under the influence of pressure, Eq.15.70 may be substituted for the second term in Eq. 15.66.

Equations for square- and round-wire windings on a shellof dissimilar metal corresponding to Eqs. 15.70 and 15.71may be derived.

15.4~ Example Calculation 15.8, Wound Thin Shell withWinding and Shell of Dissimilar Metals. A copper vesselhaving an internal diameter of 6 in. and a wall thicknessof ~5 in. is to be used under a 2000-psi internal pressure.The copper cylinder is to be wire-wound with two layers

Page 321: Process Equipment Design

of square steel wire 0.05 x 0.05 in. Calculate (1) the com-pressive stress in the copper cylinder before application ofinternal pressure, (2) the tension to be used in winding thewire if the maximum tensile stress in the copper is to be6000 psi, and (3) the maximum tensile stress in the windingunder operating pressure.

Ecopper = 16 X lo6 psi, &tee, = 30 X lo6 psi

Solution:1. Determination of compressive skess. Solving Eq.

15.71 for (jJP, we obtain:

(fi)P =ZOOO(6)

4(. 05)

. [

(65 1 + 0.10)30 x lo66.5(16 X 106) I

+ 2(0.5)

1 2 , 0 0 0=--0.20(2.05) + 1.0

= 8510 psi

But since the maximum tensile stress in the copper is6000 psi, the copper must be prestressed with a compressivestress of 8310 - 6000 = 2510 psi.

2. Detcrmmation of winding tension. Substituting ( ji)Pfor the second term of Eq. 15.64, we obtain the followingequations.

- T-2510 = - ln 0.55

0.0025 0 . 4 5

T = 2510(0.0025)0.203

= 30.9 lb ( tension in wire during winding)

26.3jrn = (0.05)2

= 12,360 psi

3. Determination of maximum wire stress under oper-ating pressure. Substituting into Eq. 15.70, we obtain:

(f&l =(2000) (6)

4(0.05) + 2(0.5)6.5(16 X 106)

(6 + 1 + 0.10)30 X lo6 112,000 1 2 , 0 0 0~~ = 17,800 psi

= 0.20 + 0.474 = G/4

Therefore

Maximum combined &tress in outerwinding layer under pressure = 10,500 + 17,800

= 28,300 psi

15.4d Thick Shells with Winding Applied under Con-stant Tension. Windings of wire or r ibbon may be appl iedunder various degrees of tension. The simplest method offabrication is to apply the winding under constant tension.

Theory of Ribbon and Wire Winding 311

To design such a vessel it is necessary to be able to predictthe residual-stress distribution throughout t.he inner mono-bloc core and throughout the windings prior to applicationof internal pressure. The pressure stresses may be super-imposed upon the residual stresses to obtain the combinedstresses under operat ing condit ions . To obtain the residual-stress distribution in the inner core (when pi = 0) it isnecessary to predict the radial pressure at the junction(interface) between the core and the windings. The coremay then be treated as a shell under external pressure, andthe stress distribution computed. It is also necessary t,opredict the residual stresses in the windings after all thewindings have been added.

The induced hoop tensi le stresses and radial compressionstresses resulting from an external pressure can be relatedto each other by use of the relationships based on Lamb’sanalysis. These relat,ionships, given by Eqs. 14.12 and14.13 in terms of diameter, can be rewritt,en in terms ofradi i for the case of no internal pressure as fo l lows.

For c i rcumferent ia l s t resses ,

PJO 2

ftc = __~r2 + ri2

pi2 - r02 l-----lr2

For radia l s t resses ,

The above two equations can be c*omt)ined to give ;,re la t ionship for no int,ernal pressure.

(15.72)

Equation 15.72 gives the hoop st,ress, jtr, in terms of theradial stress, jic, and the inside radius of the vessel, ri, atany radius, r. This relationship is useful in predicting thestresses in the wire windings and in the core which resultfrom the compression of subsequent windings. It shouldbe noted that both jtc and jVc of Eq. 15.72 are compressionstresses resul t ing f rom external pressure .

Let

fw = stress in wire during winding of layer underconsiderat ion (initial stress in wire)

After n layers have been wound, the stress fw* in I hpinner layer becomes :

f*=(f -f)=f -f LT+ri2w w tc w rc ___[ Ir2 - ri2

(15.73)

where fw * = total stress in the winding under considera tiolkfor pi = 0

jte = stress induced in the winding under considera-tion by the external windings for pi = 0

The total hoop stress in an inner winding may also beexpressed in terms of the radial stress, jr, and the radius, r,in differential form by means of Eq. 14.3; or

(15.74)

Page 322: Process Equipment Design

3 1 2 Multilayer Vessels

Equations 15.73 and 15.74 both define the stress distribu-tion in the windings. Equating 15.73 to 15.74 gives:

--p 4-mdr

Equation 15.75 is written in differential form and musthe integrated to give the cumulative radial stress, jrc, atany radius, r, when a given tensile winding stress, fW, isused. The equation can be rearranged as fol lows to permitintegrat ion:

--p dfrcdr

there fore

fw-= dfre I 2freri2r dr r(r2 - ri2)

Multiplying through by -r2/(r2 -

-fur2 r2 &r(r2 - ri2) = dr(r2 - ri2) -

The right-hand side of Eq. 15.76equal to the fo l lowing di f ferent ia l :

ri2) g i v e s :

2jrcri2r2r(r2 - ri2)2

(15.76)

can be shown to be

Different iat ing the above express ion gives :

(r2 - ri2) (r2 ‘2 + 2&r) - 2r3fTc

(r2 - ri2)2

Expanding the above and regrouping results in the right-hand side of Eq. 15.76. Substituting the differential forthe right-hand side of Eq. 15.76 gives:

-fur(r2 - ri2)

o r r2TfTc~ = -jw s r drr2 - Pi2 (r2 - ri2)

there fore

r2fTcr2 - ri2

= II) ln (r2 - ri2) + C (15.77)

To evaluate the constant of integration, it is noted thatf,.c = 0 at the outer surface of the windings, where r = r,.

0 = 2 In (ro2 - ri2) + C

there fore

C = $ In (ro2 - ri2)

Substituting for C in Eq. 15.77 gives.

r2fTcr2 - ri2

= II) ln (r2 - ri2) + ‘f In (ro2 - ri2)

.L= P-

(ro2 - ri2)(r2 - ri2)

OP

Substituting Eq. 15.78 for jrc in Eq. 15.73 gives:

o r

fw* =fw [l - f*)ln$$)!j (15.79)

It should be noted that the variable r under considerationlies between rj and r, where rj is the radius at the junction(interface) .

Equation 15.78 gives the radial stress at the outer surfaceof the monobloc core due to the windings (prior to theapplication of internal pressure) if rj is substituted for r.This radial stress is numerically equal to the external pres-sure on the core and, together with the dimensions of themonobloc, permits the computation of the residual-stressdis t r ibut ion .

Equation 15.79 gives the hoop-stress distribution in thewindings (without internal pressure, pi = 0). The com-bined-stress distribution, including the effect of internalpressure, may be determined by superposition of theinternal -pressure s tresses upon the res idual s tresses .

The application and design conditions normally fix theinside diameter of the vessel, the operating pressure, theoperating temperature, and the material of construction.The axial load induced by the operating pressure fixes theminimum wall thickness of the inner core. The materialof construct ion and operat ing temperature f ix the a l lowablestress of the inner core. The remaining variables are thetotal thickness of the windings, the winding tension, andthe maximum stress induced in the windings under oper-ating conditions. The maximum hoop stress in the corefor the operating conditions, the winding tension.. and themaximum combined stress in the windings are a functionof the thickness of the windings. As all of the variablesfor the windings cannot be fixed, one may be selected, andthe other two calculated. In the example which follows,r, will be fixed, and the tension end combined stresscalculated.

15.4e Example Calculation 15.9, Thick Shell Wound atConstant Tension. Consider a vessel having the same insidediameter (12 in.) and operating at the same internal pres-sure (20,000 psi) as the one in Example Calculation 14.1but fabricated by wire winding with high-tensile-strengthwire with the result that its outside diameter is 20 in. Thestress in the inner core is not to exceed 25,000 psi. Deter-mine the inner-core thickness , the winding tension required,

Page 323: Process Equipment Design

Theory of Ribbon and Wire Winding 313

and the stress distributions with and without internalpressure.

Calculation of inner-core thickness:By Eq. 14.1

fn= Pdi’ - Pjdj2

dj2 - di2= 25,000 psi

pj at vessel ends = 0. Therefore

or

Therefore

25 ooo = 2wJw12)2dj2 - (12)2

dj2 = s (144) + 144 5 259

dj = 16.1

t = 2.05 in.

rj = 6 in. + 2.05 in. = 8.05 in.

r, = 10 in. (given)

Pi = 6 in.

Induced hoop stresses in the core and winding due to internalpressure. .

By Eq. 14.12 with p0 = 0 and pi = 20,000 psi, r, = 10 in.,and ri = 6 in.,

= 20,000(36) + (100)(36)100 - 36 7 [l::'ooo36]

therefore

ft = 11,250 + 1'12f;ooo

For r = r, = 10 in.,ft = 11,250 + 11,250 = 22,500 psi

For r = 9 in.,ft = 11,250 + 13,900 = 25,150 psi

For r = 8.5 in.,ft = 11,250 + 15,600 = 26,850 psi

For r = 8.05 in.ft = 11,250 + 17,500 = 28,750 psi

Forr = ?in.,ft = 11,250 + 22,950 = 34,200 psi

For r = 6 in.,ft = 11,250 + 31,250 = 42,500 psi

The necessary residual hoop stress at the inside surfaceof the inner core is obtained by subtracting the 42,500-psihoop pressure stress from the allowable value of 25,000 psito give:

.ft(residual) = 25,000 - 42,500 = -17,500 psi

The required external pressure on the inner core (withpi = 0) may be calculated by use of Eq. 14.12 with p0 = pi,P = ri, and r,, = r+

ft = -17,500 - -Pipi I $2Pj2 - ri2

I ZPj 2 1

-64.8pj-17,500 = ___-64.8 - 36 + 64.8 ik",6]

therefore

17,500Pj = ___ = -3890 psi

-4.50

The residual-stress distribution in the inner-core wdmay now be calculated by using pi = 3890; or

f = -64.8(3890) + (64.8)(36) -3890t

28.8 r2 [ 128.8

therefore

ft = -8760 25!!!!

For r = 8.05 in.,ft = -8760 - 4870 = -13,630 psi

For r = 7 in.,ft = -8760 - 6440 = -15,200 psi

For r = 6.5 in.,ft = -8760 - 7460 = -16,220 psi

Calculation of required winding stress:By Eq. 15.78 with r = rj, and frc = pj at rj,

fw = 29,100 psi

Calculation of residual hoop stresses in windings:Equation 15.79 gives the residual hoop stress in the

windings at any radial distance.

When r = r,,fw* = fw = 29,100 psi (by inspectionj

When r = 9 in.,

= 21,700 psiWhen r = 8.5 in.,

ft0* = 29,100 [l - (z)ln(-$J]

= 16,650 psi.-

When r = ri = 8.05 in.,

fw* = 29,100 [l - (s)In($),I

= 11,080 psi

Page 324: Process Equipment Design

314 Multilayer Vessels

2 5

6 76

i Wire winding ‘0‘_I

Fig. 15.13. Hoop-stress distribution in wire-wound vessel (ExampleCalcu la t ion 15 .9 ) .

Combined stresses in the windings and core:

For P = 6 in.,jt = 42,500 - 17,500 = 25,000 psi

For P = 7 in.,jt = 34,200 - 15,200 = 19,000 psi

For r = 8.05 in. (in core),ft = 28,750 - 13,630 = 15,120 psi

For r = 8.05 in. (in winding),jtm = 28,750 + 11,080 = 39,830

For r = 8.5 in.,ftw = 26,850 + 16,650 = 43,500 psi

For r = 9.0 in.,jt, = 25,150 + 21,700 = 46,850 psi

For r = 10 in.,jt,,, = 22,500 + 29,100 = 51,600 psi

The stress distributions are plotted in Fig. 15.13. Themaximum induced hoop stress from internal pressure of42,500 psi has been reduced to 25,000 psi by the residualstress from the outer windings. The maximum stress inthe vessel occurs under operating conditions, exists in theoutermost winding, and has a value of 51,600 psi. Althoughthis appears to be a high stress, it is not excessive since

high-strength colddrawn steel wire is available that hasultimate strengths in the range of 200,000 psi.

15.4f Thick Shells with Windings Applied under Vari-able Tension to Produce Constant Tension under Oper-ating Conditions. It is apparent from the previousexample, Example Calculation 15.9, that the maximum com-bined stresses in the windings can be reduced if the windingsare applied under variable tension so that an ideal pre-stressed condition results. Derivations of relationships forthis condition have been presented by Comstock (226).If the windings are to possess constant combined stressunder operating conditions, the thin-wall equation is appli-cable for determining the pressure at the junction, rj; or byEq. 3.14,

where jtU, = tension in wire under operating conditions,pounds per square inch

The value of pj required to reduce the induced pressurestress at the inner surface of the inner core to the allowablelevel may be calculated by use of Eq. 14.12 after the thick-ness of the inner core has been established. This thicknessis established by use of Eq. 14.1 and is equal to (rj - r-i).After the value of pj has been determined, a value foreither r, or jte may be selected, and the value for the otherdetermined from Eq. 15.80.

If Eq. 15.80 is substituted for p,, in Eq. 14.12, rj substi-tuted for r,,, and the equation solved for the conditionr = ri, Eq. 15.81 results.

(jt)r=,, = Pi(rj2 + ri2) - 2rjftdr, - rj)E

Pi2 - ri2(15.81)

.If (jt),.=,.i IS taken as ihe maximum allowable stress inthe core, Eq. 15.81 may be rearranged to give jtw.

jtw = Pi(rj2 + ri2 1 - .ft(rtli0w.)(rj2 - ri2)

2rj(r, - Pj)--~ ( 1 5 . 8 2 )

If it is desirable to fix both the combined stress in thewire, ftw, and jt(altow.), then r, can be calculated directly byeither of the above equations.

When the internal pressure becomes zero, the residualstress in the wire is equal to the combined stress, jtlu, minusthe pressure stress, jt; or

ft(residua1) = ftw - & + r2c~~~2r~~,2j]r. - ri2 - 2

piri2

-ftul - 2 (15.83)r. - ri2

The hoop stress corresponding to a radial st,ress is givenby Eq. 15.72.

Comstock has shown (226) that the radial stress, jr, pro-duced by the wire windings beyond radius r (withoutinternal pressure) is :

fr = - !cY2s jtw _ (15.84)r

Page 325: Process Equipment Design

Theory of Ribbon ond Wire Winding 3 1 5

existing at r (without internal pressure). Thus the requiredwinding tension is obtained by subtracting Eq. 15.85 fromEq. 15.83 to give:

Substituting Eq. 15.84 forf,. in Eq. 15.72 gives the changein hoop stress, Aft, in the wire at radius r as a result of thewindings beyond r (with no internal pressure).

Aft = ( 1 +q (1+$)

(15.85)

The required winding tension, fW, at radius r is obtainedby subtracting the change in stress, Aft, due to windingsbeyond radius I from the residual hoop stress, ft(residud)

fw = ftw ro(r2 + r12) - 2rri2 1 2piri2r(r2 - n2)

- ~ (15.86)r2 - ri2

Comstock (226) has presented an illustrative designdemonstrating the use of Eq. 15.86 in a wire-wound vesselto operate at 15,000 psi with a constant tension of 40,000 psiin the wire winding under operating conditions.

P R O B L E M S

I. A new high-tensile-strength, high-yield-strength, low-alloy plate steel with a minimumultimate strength of 105,000 psi and a minimum yield strength of 70,000 psi and with a mini-mum elongation in 2 in. of 22% has been described in the literature (228). (The chemicalcomposition is given as: a maximum of 0.25% carbon, 1.50% manganese, and 0.35~~ siliconas well as a minimum of 0.10% vanadium and a nickel range of from 0.40 to 0.70%.)

Design the shell of a two-shell shrink-assembled multilayer vessel using this steel. Thevessel is to have an inside diameter of 24 in. and is to operate at a pressure of 20,000 psi. Theservice temperature is to be under 600” F, and an allowable tensile stress of 30,000 psi is tobe used.

2. Redesign the vessel described in problem 1 as a three-shell shrink-assembled multilayervessel.

3. What are the interface pressures for the vessel described in problem Z?

4. What metal interferences are required for the vessel described in problem 2 3

5. A 12-in.-inside-diameter pressure vessel is fabricated of an inner shell of copper 1 in.thick and an outer shell of steel 35 in. thick in such a manner that the interface pressure is zeroand the two shells are in contact with each other. The reactor is 4 ft long from tangent line totangent line with ellipsoidal heads (also of double layer).

If end effects are ignored, determine the hoop stress in both of the shells if the vessel containsan internal pressure of 2500 psi and if E of copper = 15 X lo6 psi and E of steel = 30 X lo6 psi.(Suggestion: assume that thin-wall theory is valid.)

6. A copper vessel with an internal diameter of 10 in. and a wall thickness of f$ in. is to beoperated at a lOOO-psi internal pressure. The copper shell is to be wire-wound with squaresteel wire having a cross section of 0.1 in. by 0.1 in. If the wire is wound at a constant tensionof 20,000 psi, what is the minimum number of layers of winding required to keep the hooptension in the copper core below 6000 psi under operating conditions? What is the axial stressin the copper shell under operating conditions? What is the residual compressive hoop stressirr the copper shell when the internal pressure is removed?

7 . Redesign the shell described in problem 6 using the same material and same inside diameterbut using wire winding at a constant tension of 25,000 psi.

8. Redesign the shell described in problem 6 using the same material and same inside diameterbut using a variable-tension winding to produce a constant tension of 35,000 psi in the windingunder operating pressure.

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Page 327: Process Equipment Design

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Page 328: Process Equipment Design

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210. Eichinger, A., Statement in: Proceedings, Second Inter-national Congress of Applied Mechanics, Ziirich, 1926,p. 325.

211. Voorhees, H. R., “The Creep-Rupture Life of EngineeringStructures with an Initial Stress Gradient,” Ph.D. Thesis,University of Michigan, 1956.

212. Robinson, E. L., “Effect of Temperature Variation on theLong-Time Rupture Strength of Steels,” Trans. Am. Sot.Mech. Engrs., 74 (1952), pp. 777-781.

213. Higgins, M. B., “Allowable Working Pressure for LongTubes Subject to External Pressure,” Paper No. 4,8-A-123,ASME Meeting, December, 1948 (abstracted in Me&Eng., 71 (1949), p. 169).

214. Freeman, J. W., and Voorhees, H. R., “Selection ofAlloys for Service Requirements,” Znd. Eng. Chem., 48,No. 5, p. 861.

215. Faupel, J. H., and Furbeck, A. R., “Influence of ResidualStress on Behavior of Thick-walled Closed-end Cylinders,”Trans. Am. Sot. Mech. Engrs., 75 (1953), pp. 345-354,.

216. Faupel, J. H., “Residual Stresses in Heavy-wall Cylin-ders,” J. Franklin Inst., 259 (January-June, 1955), pp-405-419.

217. Bridgman, P. W., The Physics of High Pressure, G. Belland Sons, London, 1949.

218. Comings, E. W., High Pressure Technology, McGraw-Hill,New York, 1956.

219. Sachs, G., “Residual Stresses, Their Measurement andTheir Effects on Structural Parts,” Symposium on theFailure of Metals by Fatigue, Melbourne University Press,Melbourne, Australia, 1947, pp. 237-247.

220. Cox, H. L., “The Design of Built-up Cylinders,” Engineer,162 (1936), p. 179.

221. Jasper, T. M., and Scudder, C. M., “Multilayer Con-struction of Thick-wall Pressure Vessels,” Trans. Am.Inst. Chem. Engrs., 37 (1941), p. 885.

222. Spraragen, W., and Ettinger, W. G., “Shrinkage Dis-tortion in Welding,” Welding J. (N. Y.), Res. Suppl., 29(1950), p. 323-S.

223. Schierenbeck, J., Jr., U. S. Patent 2,326,176, August, 1943.224. Holroyd, R., “Report of Investigations by Fuels and

Lubricants Teams,” U. S. Bur. Mines Inform. Circ. 7375,1946.

225. Donovan, J. T., Josenhans, M., and Markovits, J. A.,“High Pressure Vessels in Coal Hydrogenation Service,”Trans. Am. Sot. Mech. Engrs., 72 (1950), p. 357.

226. Cornstock, C. W., “Some Considerations in the Design ofThick-wall Pressure Cylinders,” Trans. Am. Inst. Chem.Engrs., 30 (1943), pp. 299-318.

227. “Synopsis of Boiler and Pressure Vessel Laws, Rules andRegulations: by States, Provinces and Cities (UnitedStates and Canada),” National Bureau of Casualty Under-writers, 1955.

228. Fratcher, G. E., “New Alloys for Multi-layer Vessels,”Petroleum Refiner, 33, No. 11 (1954), pp. 137-141.

229. Chilton, C. H., “Six-tenths Factor Applies to CompletePlant Costs,” Chem. Eng., 57, April (1950). p. 112.

230. Williams, Rodger, Jr., “Six-tenths Factor Aids in Approxi-mating Costs,” Chem. Eng., 54, December (1947), p.124.

231. Seely, F. B., and Smith, J. O., Advanced Mechanics ofMaterials, Wiley, New York, 2nd ed., 1952.

232. Taylor, C. P., Glenday, C., and Faber, O., Engineering,85 (1908), p. 325.

233. Gartner, Abraham I., “Nomograms for the Solution ofAnchor Bolt Problems,” Petroleum Rejtner, 36, No. 7(1951), pp. 101-106.

--

Page 332: Process Equipment Design

322 References

234. Jorgensen, S. M., “Anchor Bolt Calculations,” PetroleumRefiner, 25, No. 5 (1946), pp. 211-213.

2 3 5 . “Equipment Cost Rise Accelerates,” Chem. Eng., 64(March, 1957), pp. 266-267.

236. La Que, F. L., and Cox, G. L., “Some Observations of thePotentials of Metals and Alloys in Sea Water,” Am. Sot.Testing Materials, Proc. 40 (1940), pp. 670-687.

237. Manning, W. R. D., “Strength of Cylinders,” Ind. Eng.Chem., 49, No. 12 (1957), p. 1969.

238. Manning, W. R. D., “The Design of Compound Cylindersfor High Pressure Service,” Engineering, 163 (1947).pp. 349-352.

239. Manning, W. R. D., “The Design of Cylinders by Auto-frettage,” Engineering, 169 (1950), pp. 479, 509, 56?

240. Manning, W. R. D., “Residual Contact Stresses inBuilt-Up Cylinders,” Engineering, 170 (1950), p. 464.

241. Nadai and Wahl, Plasticity, McGraw-Hill Book Company,1931.

242. Jasper, T. M., letter to the editor of Engineering, Engi-neering, 160 (1947), p. 16.

243. Jasper, T. M., letter to the editor of Engiaeering, Engi-neering, 160 (1947), p. 160.

244. Langenberg, F. G., “Effect of Cold Working on theStrength of Hollow Cylinders,” Trams. Am. Sot. SteelTreating, 8 (1925), p. 447.

245. “Multi Layer Engineering for Safety,” A. 0. Smith Corp.Bulletin No. V-53, Milwaukee, Wisconsin, 1950.

Page 333: Process Equipment Design

A P P E N D I X

DRAWING CONVENTIONS

Item 1. DRAWING SIZES AND SCALES

All drawmgs should be made on vellum tracmg paperunless otherw,se speclfled.

The drawng paper or cloth to be used should be thestandard trimmed sheets of such wes that they wll fold to theletter size of E+“x 11’:

Sheet SIX

A 8f”x 11”B 11” x 17”C 17” x 22”cl 22” x 34”E All over D sue

The foltowng drawing scales are to be used:

1:l (full size)1:2 (half size)1:41:s1:121:161:32

The scale should be selected to show a clear picture of thepart being drawn. Detail drawings are usually made with scalesOf l:l, 1:2, or 1:4, whereas assembly drawings are more oftenmade wth scales of 1:4, 1:8, or 1:16.

The scale used should always be indicated in the titleblock. If more than one scale is used on the same drawing sheet,each Scale is to be indicated under the title of that respectivesection or wew.

3 2 3

I t e m 2 . TITLE BLOCK AND BILL OF MATERIALS

The Title Block identifies the drawing and should contain thefollowiilg information:

I. Name of manufacturer an-address2. Name of equipment or part drawn3. Name of purchaser and address4. Date of completion of drawing.5. Scale6. Names of draftsman. checker. and tracer7. Drawng number

Tracing paper is usually purchased wth the border. titleblock, manufacturer’s name and address,etc. printed on thestandard sizes of tracing paper. The remainder of the title block islettered free hand to supply the necessary addItIonat information.When drawings are made on blank tracing paper. the enbre btleblock and border must be drawn.

The title block should be placed on the lower right hand cornerof the paper. The revision block should always accompany the titleblock in the manner shown.

A bill 01 materials. listing all the material required in the formin which it is purchased or cut for fabrication should be includedabove the title block with two exceptions. In the case of simple detailscompising only one or two parts. the material, size, etc. may beindicated by notes on!he drawing. In the case of large assembliescomprising many parts.the materials are listed on separate sheetsaccompanying the drawings.

;:’ 1 PART NAME ) SIZE O F S T O C K /MATERIALI QUANTITY I NOTEs

CHEMCO. INC.

Page 334: Process Equipment Design

324 Drawing Conventions

I t e m 3 . DRAWING CONVENTIONS

Alphabet of Lines

The line is the basis of the englneenng drawng. The welghtof an indiwdual line is used to slgnlfy the function of that lone.Three weights of linesare used,namely: heavy, medium. and light,and are used in the following manner.

HeavyOutline of parts

- - - - - - - Cutting plane lmesShort break lines

Madfom - - - - - - - - - - - - - - - - H i d d e n e d g e l i n e s

- - - - -Cross-section lines

I” “I Long break lines

Material Symbols

Cast won Bronze. brass,and copper and babbitt

Aluminum Electricinsulation

Sound or heatinsulation

Fire brickrefractories

Electric Concrete Common Woodwindings brick

The material of which a part IS constructed IS designated, in asection view. by crosshatching. Accepted symbols of the most commonengmeermg materials are shown above.

Light parallel lines should be used for most cross-sectlomng.All slant hnes are 45”. All adlacent parts are cross-sectioned in oppositedirectnons and in the case of three or more adlacent parts. an angleOf 60’ or 3O’may be used on the addItIonat parts.

I t e m 4 . LETTERING I

Lettermg SUppIleS mformatlon whxh cannot be gwen by lanesalOne. and therefore It 1s One of the most Important elements of agood drawng.

Lettering Standards

1 All lettermg must be I” the smgle stroke commercial gothic style.

2 . Lettering may be either verbcal or inchned (between 60 and 70degrees) but must be cowstent on each drawmg.

Example: VERTICAL OR /NCLfNED /60’- 70’)3 Only upper case letters are to be used.

4 All letters shall be of umform sue wth the exception that theftrst letter Of each word Or group of words may be approxtmately50 percent larger for emphasis. but the form of lettermg must becofwstent.

Example: FINISH ALL CONTACT SURFACES

5. Spacmg between knes of lettermg should be 2/3 of the heightof the letter.

6 A permanent gude hne may be used l/16” below the letters ofa word or group Of words. These gwde lines may be extended toform a leader. Double hnes may be used for tdles and foremphasis.

Example: BUT1 WEfD BUTT WELODETAIL O F ARM

7 The following letter heights should be used %s’

TITLES AND DRAWING NUMBERS~3"HEADINGS AND PROMINENT NOTES-++

BILL OF MATERIALS, DIMENSIONS, GENERALNOTES, ETC.

Page 335: Process Equipment Design

km 5. cont.

Drawing Conventions 325

D I M E N S I O N S IItem 5. D I M E N S I O N SI

Dlmensmns are placed on a drawmg so that a part may be madeby reading the drawmg The dlmensmns must be so complete thatthe necessary lnformabon as to the we and locabon of parts 1sobwous wthout scaling the drawng or maklng computations.

DEFINITIONS

I D~mensmn lines are full. hght knes. broke” only where thedtmenslon 1s Inserted and are parallel to the object or hne bemgdlmensKwed.

Dlmenslon knes are used for two purposes: for Speclfylng IocabOnand for showng size.

Example:--g&J%& rIro”

Dlmenslon lanes are termmated by arrows at the surface of thepart or at the extension knes

Example: +------ 2’I6”-----+

2. Extenwn lmes or wetness lmes tndwate the distance measuredwhen the dlmensmn tine 1s placed outsidethe object. Theyare full. hght lanes startmg l/16” f&n the oblect and extendmgl/8” beyond the dimension hne.

Example:

-

3 Leaders are hght. stratght knes which lead from a note or dlmensmnand whtch are terminated by an arrowhead touching the part towhich attenbon 1s directed. If two or more are used in one drawmg.they should be kept parallel If powble.

Example: ‘7 DRILL $20 N.C.TAP

4 Center tmes are fine lines composed 01 alternate long and shortdashes whxh are used to represent axes of symmetrical partsand which serve as extension hnes I” the location of holes or otherSlmilac features. Important dtimensions should not be referred to acenter kne that has no finished hole on it or has no finishedsurface comcidmg with it

Dimension Standards

1. Dimensions for structural steel work. welded parts, castings,assembly drawmgs. etc, where tolerances are greater than+1/16”.ishould be given in inches and fractions of inches up to.but not includmg. 72”. Any distance of 72” or more should begiven in feet and inches.

2. Dimensions for machine shop work. where tolerances are lessthan,tO.Ol”,should be given in inches and decimals only. Longshafts may be dlmensioned in feet and Inches.

Example: 4 . 5 0 3 ‘$88’03. Feet are designated by a single quotabon mark 0. Inches are

designated by a double quotation mark (“). Feet and Inches areseparated by a dash.

Example: 7’85”

4. In all cases where feet are given, the inches must also beindicated. In case of even feet, a (-0”) must be added.

Example: 8’01’ 1 0 ’ 0 ”

5. Dimensions for location should be made from a reference lmesuch as a center line or a base lme and not from an edge. Thisis especially true if the edge is to be machined or 8s the edgeof a casting.

6. Horizontal and sloping hnes should read from left to right whilevertical lines should read from bottom to top.

Example:

7. Dimensions should not be added or repeated unnecessarily.

Page 336: Process Equipment Design

326 Drawing Conventions

Item 5. cont. D I M E N S I O N SI

8 Dlmenslons are placed OutwJe of a wew or oblect unless it wlladd clearness or slmplwty to the dra’wlng 11 placed nnsldeThey should “ever be placed I” cut (?&owed) surfacesunlessabsolutely necessary. at which bme.ttw.secbo,nlng IS omlttedaround the numbers and hnes

9 Overall dlmenslons should be given on all views

IO Dlmenslons should not be crowded If space IS small.one of

I1 When dlmenstons are placed on an angle, they should beI

placed horuontally on the arc as on a dlmenston lhne Forlarge angles, place the dlmenslon un hne with the arc. I

12 For equally spaced holes in a circular flange or disc, give thedram&r of the bolt hole wth the number and sue of the holes I

ExampleS’DRILL 6 HOLESEVENLY SPACEDON I t ’ 6.C

I 13. Circular sections should always be dlmensloned from centerto center and never from the edge of a part. I

14. An outhne of a drawng preferably should not be used as anextenston Ilne. and a hldden edge hne should never be usedas an extension lhne

I - - - -L

Page 337: Process Equipment Design

A P P E N D I X

WELDING CONVENTIONS

I t e m 1 . WELDING SPECIFICATIONS ttem 2. WELDING INSTRUCTIONS

20 Gage and Less, Use acetylene welding or seam weldmg. Arcweldang of light gages 1s very dlfflcult.

18 Gage and Heawer: Use arc welding or seam-weldmg. Arc weldingIS cheaper than gas welding and seam-welding IS cheaper than arcwelding

SIX of Welds

Butt and plug welds, and all welds wth grooves (V and Llf do notteqwre dlmenslons of bead The gap between the sheets or grooveshas to be at feast completely talled wth weld Other welds aremeasured as follows

r* =lb b’iiir”il

Fillet Weld Corner Weld Edge Weld

Welds made on the rotary seam welder (called seam welds) also donot reqwre any cross-sectional dunenslon as no matellal 1s deposwd

Edges to be Welded. Sheared edges f5/8” manmum) are sufficienttar ordinary welds, heawer plates tlame cut and scale removed.

Code Weldmg Is requmng edges machined or flame cut and ground.

Beveltng 30°bevel can be sheared-5/S” maximum-all other anglesand larger wed plates are flame cut and ground over or machined.If both plates are beveled 30’ IS suffuent. If one plate only IS beveled55’bevelmg should be used. Up to l/4” plates no bevel is requiredfor ordinary weld-up to 3/4” plates smgle bevel is sufficient-heavierplates reqwre double bevel or single U weld if one side only isaccessible for welding and double U welds 11 both sides are accessible.“U” grooves can be made only by machining.

Jf&$“j$j>&~

SmgleTV//////////

‘-rs bevelSingle be=% Double bevel

Contlnulty

A conhnuous weld 1s one which IS continued over the lull lengthof seam as shown III the followng figure

C Double U.\\\\\\\\\\\\\\

Double J

Spectfy on dIdwIng C W e l d

Corner Welds: No beveling IS requwd, weld outside.Size of fillet isnot to be specdled-fillet a ltttle heavier than plate is understood. Forheavy strams weld one bead iwde in addition to outside weld. Seefollowing examples:

7 ===I!

Corner welds

327

Page 338: Process Equipment Design

328 Welding Conventions

km 2. ml. WELDING INSTRUCTIONS

T Welds II beveled no speclfacataon of size of weld requred If notbeveled gw sue of Illlet-specify smgle or double fillet-continuous.lntermlttent or staggered-sue of bevel and grind asrequlred.

lntermlttent and conbnuous smgle IIlkt welds are used for veryhght loads only.

For medwm hght loads a staggered weld hawng the ttrst and lastweldadouble weld should be used. For all heavier loads a doublemtermittent or a double contmuous weld IS to be preferred as nobendlng I” the weld IS present

Fillet weld normally l-l/Z bmes the plate thickness. For kght loadsthe Idlet IS same as the plate thtckness and for heawer loadstwcetfw plate thwkness.

Heavy loads use smgle or double V weld wth 55Oangle and 1/16”lorgap and toe. Single V up to 5/8” plate and double up to 1” plate.

Heawer welds and all code welds use single or double U welds. For all Uwelds gap and toe 3/32”. Use the dlmenslons gwen under code weldmg.

Examples

zzIziL&ASmgle IlM Smgle v Smgk u

T weld T weld

zzii%n 72JiBL

Oouble fillet Double V Double U

Plug Weld: Use only where pant of weldmg IS not otherwiSe accessible.Ommeter 01 hole at least l-l/Z bmes the thxkness of the plate andcountersunk. Holes three bmes the thickness of plate but not lessthan l/2” teqwres no countersmkmg. Specaly as required: Holediameter. Csk. locabon and distance of welds, and grind.

Examples:

-e=DIA. =TfE

Plug weld Plug weldGrand flnlsh

tern 2. cont. WELDING INSTRUCTIONS

Butt Welds. 100 percent penetrabon IS understood-no size of beadSpeClllCatlOn 1s requred-speclly single or double butt weld. Grmd,flushand bevel as reqwred.

Smgle and double plarn butt welds are for kght loads only, single up tol/8” plate and double to l/4”-gap for smgle equal to plate thickness.for double l/2 plate thickness.

Heawer loads “se single and double V welds-up to 5/g” single. Up to1” double-gap and toe l/16” for all welds.

Extra heavy loads and all code welds use single or double U-welds,For all U welds gap and toe 3/32”. use groove dimensions g,venunder code welding.

Examples:

mm-

Single butt Smgle V butt Single U buttNo bevel 30° bevel Groove code sto.

=a==--Double butt Double V butt Double U butt

No bevel 30” bevel Groove code std.

m-Single V butt Smgle U butt

Welded both sides Welded both sides

Lap Welds: Single lap welds use very hght loads only-for all heaviecloads use double lap weld and make Illlets l-l/Z ttmes the plate.thickness to avad concentrabon of stress in the weld.Specify: single or double-we of flllet-contlnuous-staggered-mtermlttent-double IntermIttent.

Examples:I% TIMES

PLATE THICKNESS

dm7

Single lap weld Double lap weld

Page 339: Process Equipment Design

Welding Conventions 329

1j_ 1 WELDING INSTRUCTIONS 1 ,,

Edge Welds: Do not use for loads Edge welds are only used to maketight. I” many cases seam weld can be used Instead Up to 3/16plates no bevel 1s reqwred. l/4” to 7/16” plates “se a 3O’bevelOver full thickness of the plates. l/Z” and heavier 30”bevel IS usedover 3,‘4 of plate thickness Specify: SIX of weld (the SIX IS equal tothe wdth of the weld)-conttnuous-IntermIttent-grind as required

Examples.

I

Edge weld Edge weld Edge weld‘Up to 3/16” plates l/4” to 7,‘16” plates l/Z” plate and up

em 2. cont. WELDING INSTRUCTIONSI

Strength of Welds (non code): Permwble umt stress for fillet weldsmade wth coated welding rod IS 14,000 Ibs per square Inch. tak,“gInto conslderabon that practlcallyeveryf~llet weld is subfect to shear.

The folIowang table gwes Safe workmg values for fillet welds:

Sue Of fillet Load I” Ibs. per Itneal Inch

l/S” . . . . . . .._____................................ 12503/16” . . . .._._.___................................. 1 8 7 5l/4” ..,......._._................................ 2 5 0 05/16” . . . . . . . . . ..___............................... 3 1 2 53/6” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3750l/2” ..___........................................ 5 0 0 05/8” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62503/4” _____........................................ 7 5 0 0

Calculated length of weld should be ancreased l/4” for startmg andstoppmg of the arc.

For welds with no shear present the above values can be increased to:

16,000 Ibs. per square inch for tension, and18.700 Ibs. per square Inch for compression.

Butt welds and all welds employing a groove and having 100 percentPenetrattOn are ConsIdered 01 the same size as the plate even if weldIS built up higher smce the addtbonal material is not adding to thestrength of the weld.

Item 3. I CODE WELDING GROOVES I

Single butt weld Single tee weld

Double butt weld Double tee weld

b-.7

I \ \ \I I A

Page 340: Process Equipment Design

A P P E N D I X

PRICING OF STEEL PLATE

1. Mil l Carbon-steel-plate Extras ( in Dollars per 100

Pounds)

(Courtesy Inland Steel Company-Jan. 1956)

a. Item Quantity Extras

10,000 lb or overUnder 10,000 lb to 6000 lb, incl.Under 6000 lb to 4000 lbUnder 4000 lb to 2000 lbUnder 2000 lb ta 1000 lbUnder 1000 lb

b. Classification Extras

$/lo0 lbNone0.100.200.501.001.50

$/loo lb

d.

(1) Carbon-steel plates subject to chemical-composition limits or ranges None

(2) Carbon-steel plates subject tu physicalrequirements (melt test only) 0.05

(3) Carbon-steel plates subject to ladle chem-istry (carbon and/or manganese, phos-phorus, and sulfur) and to physical re-quiremenls in qualities lower than “flangegrade” 0.10

c. Quality ExtrasThe following quality extras contain the applicable

classification extras listed in item 6 above.$1100 lb

Hot pressing steel (not boiler llange steel) 0.20Cold pressing or cold flanging steel (not

boiler flange steel) 0.25Flange steel, ASTM h-285 or equivalent 0.40Ordinary firebox steel, ASTM A-285 or

equivalent o.soLocomotive flange steel 0.55Locomotive firebox steel 0.65

330

Drawing quality 0.35*Specified grain size (McQuaid-Ehn test) 0.90*Forging quality 0.80

* Includes extras for killed steel or for any minimumspecified silicon up to 0.15 %, inclusive, or any maxi-mum specified silicon over 0.10 to 0.30’%, inclusive.Specification Extras

The following extras, applicable to specificationslisted under this caption or to equivalent specifications,include the classification extras Quality and ChemicalRequirement, but no other extras, unless otherwisespecified.

ASTM Specification A-201, ASME SA-201,or Equivalent

Grade A Grade B-Fire- Fire-.

Thickness Flange box Flange box1%” and under 1.20 1.25 1.20 1.25Over 1%” to 2”, incl. 1.35 1.40 1.35 1.40*Over 2” to 3” 9 incl . 1.45 . . . 1.45

ASTM Specification A-212, ASME SA-212,or Equivalent

Grade A Grade BFire- Fire-

Thickness Flange box Flange box1 s” and under 1 . 2 0 1 . 2 5 1 . 2 0 1 . 2 5Over 1%” to 2”, incl. 1 . 3 5 1 . 4 0 1 . 3 5 1 . 4 0*Over 2” to 3”, incl. . 1 . 4 5 1.55

* Includes extra for heat treating test specimens.

When material is required to be normalized or annealed,the extra shown under heat-treatment and surface-

Page 341: Process Equipment Design

e.

finish extras shall apply in addition to the specificationextra, which in the case of plates over 2” in thicknessshall be reduced by $0.05 per 100 lb.Length Extras-All Plates, Rectangular or Otherwise

$/lo0 lb8’0” or over up to published limit of length,

but not over 50’0” NoneUnder 8’0” to 5’0” incl.Under 5’0” to 3’0”’ incl.

0.10

Under 3’0” to 2’0”’ incl.0.20

Under 2’0” to 1’0”’ incl0.30

Under 1’0” to 6” ’ *0.502.00

Over 50’0” to 60’0”, incl. 0.10Over 60’0” to 80’0” incl. 0.30

f. Width and Thickneis Extras, Dollars per 100 PoundsThe size extras in Tables I and II also apply to

plates ordered to weight per square foot on the basisof equivalent thicknesses in the ranges listed below.

When plates are specified with the greater dimensionat right angles to the direction of rolling, this dimen-sion shall be considered as the width, and the extrafigured accordingly.

Table I. Sheared, Gas-cut, and Universal Mill

Edge PlatesThickness, Inches

1OYPT? , /4 to J/i 5 46 to y; It0 1:;

Under 3/i6, to 44, 3’2, to 1, 134, to3,Width, Inches !/a Excl. Excl. Excl. Excl. Incl. Incl.

%Over 8to12,incl. 1 . 5 5 1 . 3 5 1 . 2 0 1 . 1 0 0 . 9 5 0 . 9 5 1 . 0 0‘Over12 to24,incl. 1.50 1.30 1.15 1.05 0.90 0.90 0.95Over24to30.incl. 1.40 1.20 1.05 0.90 0.75 0.75 0.80Over 30to36;incl. 1.25 1.05 0.90 0.75 0.60 0.65 0.70Over36ta48,incl. 1.20 1.00 0.80 0.65 0.50 0.55 0.60Over48 to60.incl. 1.05 0.85 0.65 0.50 0.30 0.40 0.45Over 60 to 80; incl. 1.00 0.65 0.45 0.30 0.10 0.25 0.30Over80to90,incl. 1.00 0.60 0.40 0.25 None 0.15 0.20Over 90 to 96, incl. 0.75 0.50 0.35 0.15 0.25 0.30

1 Add 0.65 per 100 lb to all extras shown above for plates OVRP 1 ?.$I”thick unless killed steel is specified or implied.

2 Maximum leneth limits for aide shearine in these widths:O v e r 8” to 12”, &cl., but not over 45” thLk, incl.--120”Over 12” to 24”, excl., hut not over )-i“ thick, incI.-240”Over 12” to 24”. excl., and over x” to %” thick, incl.-180”

All sheared-edge plates longer than the foregoingshearing limits or over x” thick must be gas rut to

size at the listed gas-cutting extras.

Table I I . Mil l-edge Plates

Thickness, Inches- -+/a to %6 %to

Under xs, to 5% PiWidth, InchesOver 3 0 to 3 6 , incl. 1 ‘i. ;x,c; ;y; ;x,c; oT;5

Over 3 6 to 4 8 , incl. 1.10 0.90 0.70 0.55 0.40Over 4 8 to 6 0 , incl. 1.05 0.85 0.65 0.50 0.30Over 6 0 to 7 2 , incl. 1.00 0.65 0.45 0.30 0.20

g. Killed-steel Extra*Silicon-killed steel, aluminum-killedsteel, or steel killed by any deoxidiz-ing agent, specified or implied $0.65 per 100 lb

* Extra does not apply to forging quality or speci-fied-grain-size quality (McQuaid-Ehn test).

Pricing of Steel Plate 331

Mill-Gas-cutting Extras per Linear Foot of Cutting

(Court.esy Inland Steel Comp;\nl--Xfa! 1953)

Thickness,Inches

1 or under1%1%1%1%1%1%1%22%wi23/2>-;2%2%2%3356

Extras,$ per LinearFoot of Cut

0.370.380.390. .400:410.420.430.440.450.460.170.480.490.510.530.550.570.59

Extras,r% per LinealFoot of Cut

0.610.630.650.670.690.710.730.750.7;0.790.810.850.800.930.970.991.031.0;

Plates ordered to a maximum of carbon exceeding0.39% or a minimum of 0.25% carbon together with amanganese content of over 1.00% require edge tra;rl-ment. The extra for such edge treatment is $1.50 per100 lb to be assessed in addition to the foregoing gas-cutting rates.Mill Circular- and Sketch-plate Extras

(Courtesy Inland Steel Company-May 1953)Circular plates 35’j&Semicircular plates 35Y,Sketch plates furnished to a radius 35 76Regular sketch plates with not more t,han four

straight edges 25 ‘i;iIrregular sketch plat.es wit,h more than four straight

edges 40 ‘;,Warehouse Base Prices and Extras

a .

b.

(Courtesy J. T. Ryerson and Son, Inc.,Chicago, Ill.-April 14, 1955)

Base Price of Hot-rolled Carbon Steels$/lOO ItI

Structural shapes 5.99Junior beams 6.64Stair channels 6.64Bars and bar shapes 5.81Hot-rolled strip 5.92Plates, hot-rolled 5.82Sheets, hot-rolled 5.68Base Price of Cold-finished Carbon Bars (Rounds,Squares, Flats, Hexagons)

$/lOO lbChicago 7.25

Page 342: Process Equipment Design

332

C.

cl.

e.

Pricing of Steel Plate

Hot-rolled Carbon SteelQuantity Extras $/IO0 lb

30,000 lb and over Base20,000 to 29,999 lb 0.2010,000 to 19,999 lb 0.405,000 to 9,999 lb 0.602,000 to 4,999 lb 0.701,000 to 1,999 lb 1.00

400 to 999 lb 1.95100 to 399 lb 3.70

Under 100 lb 6.70

Cold-finished Carbon BarsQuantity Extras $/loo lb

2000 lb and over Base1000 lb to 1999 lb 0.50500 lb to 999 lb 1.50300 lb to 499 lb 3.35

Item under 300 lb 4.00299 to 150 lb (total order) 7.00149 to 75 lb (total order) 12.00Under 75 lb (total order) 17.00

(Total order applies when the total weight of cold-finished carbon bars purchased in one day for shipmentat one time to one destination falls within one of thequantity ranges indicated.)Cutting Extras in Dollars per 100 Pounds

Rectangles

To Length OnlyTo 8’ 5’ to

Length and under Under Under Under UnderOnly over 8’ 5’to3’ 3’to2’ 2’tol’ 1 ’

S6 tol” 0 . 0 0 0 . 2 0 0 . 3 0 0 . 5 0 0 . 7 5 1.55thick*

To Width and Length,xs” to 1” Thick*

To Width 6” Wide Over 6” Over 10”and and to lo”, to 24”, Over 24”

Length+ under Incl. Incl. WideOver 12’0” 1.50 0.75 0.60 0.455'0" to 12'0" 0.60 0.40 0.30 0.25Under 5’ to

3’, incl. 0.70 0.45 0.35 0.30Under 3’ to

2’, incl. 0.85 0.55 0.50 0.50Under 2’ to

l’, incl. 1.20 0.80 0.75 0.75Under 1’0” 2.80 1.85 1.80 1.75

* Over 1” thick-see flame-cutting extras, but onhigh-carbon and abrasion-resisting use flame-cuttingextras on items over 35” thick.

t This schedule also covers stock-length platessheared to width only.

Sketch PlatesFor simple sketches with straight sides and nore-entrant cuts, add 0.10 cwt to above width

and length extras.(A simple sketch is a rectangle modified by only

one additional cut.)

Universal Mill Plates

1” thick and lighter Use shearing schedule above11”’ thick and heavier,18

cut under 5’0” 1ong See structural schedule136” thick and heavier,

cut 5’0” and longer No charge for cutting

f. Plate Flame-cutting Charges

Thickness,Inches

346%.%6N746JSX6w34Td

11%1%1%1%1%1%22%2%2%33?d3 %3 %44 %55 %66%77 %89

10

Extras per Linear FootAddl. Footage.

First 100 Ftof Any Item

$0.190.190.200.210.220.230.240.250.260.350.370.380.390.400.410.420.430.450.470.490.530.570.610.650.690.730.810.890.991.071.161.241.331.461.732.01

over 100 Ftof Any Item

$0.100.100.110.120.130.140.150.160.170.260.280.290.300.310.320.330.340.360.380.400.440.480.520.560.600.640.720.800.900.981.071.151.241.371.641.92

Charges are based on linear feet for each item.Fractions of an inch are charged at the next full inch.The minimum charge is $1.50 net per item.

Note: All other mill extras apply to warehouse stocks.

Page 343: Process Equipment Design

Pricing of Steel Plate 333

d. Flange-quality and Firebox-quality Steel Plates

(Courtesy J. T. Ryerson and Sons, Inc., 1954-55 stock list and reference book)

Size,In.

1/4x3 0364 24 85 46 07 2849 6

1 2 05/16x

3 03 642485 46 07 28 496

1 2 03/8x

3 03 64 2485 46 07 28 49 6

1 2 0?/16x

3 03 64 24 85 46 07 28 49 6

1 2 01/2x

3 03 64 24 85 46 07 28 49 6

1 2 0

Wtper Ft

Lb

Size and Quality(S/100 lb)Extra

Flange Firebox

StockLengths,

Ft

25.50 1.35 1.60 S-3030.60 1.20 1.45 S-3035.70 1.20 1.45 S-3040.80 1.20 1.45 S-3045.90 1.05 1.30 S-3051.00 1.05 1.30 S-3061.20 0.90 1.15 S-3071.40 0.90 1.15 S-3081.60 0.95 1.20 S-30

102.00 1.30 1.55 S-30

31.88 1.20 1.45 S-3038.25 1.05 1.30 S-3044.63 1.05 1.30 S-3051.00 1.05 1.30 S-3057.38 0.90 1.15 S-3063.75 0.90 1.15 S-3076.50 0.75 1.00 S-3089.25 0.75 1.00 S-30

102.00 0.80 1.05 S-30127.50 1.15 1.40 S-30

38.25 1.10 1.35 S-3045.90 0.95 1.20 S-3053.55 0.95 1.20 S-3061.20 0.95 1.20 S-3068.85 0.80 1.05 S-3076.50 0.80 1.05 S-3091.80 0.65 0.90 S-30

107.10 0.65 0.90 S-30122.40 0.70 0.95 S-30153.00 1.05 1.30 S-30

44.63 1.10 1.35 S-3053.55 0.95 1.20 -’ S-3062.48 0.95 1.20 S-3071.40 0.95 1.20 S-30so.33 0.80 1.05 S-3089.25 0.80 1.05 S-30

107.10 0.65 0.90 S-30124.95 0.65 0.90 S-30 1142 SO 0.70 0.95 S-30178.50 1.05 1.30 S-30

51.00 0.95 1.20 S-3061.20 0.80 1.05 S-3071.40 0.80 1.05 S-3081.60 0.80 1.05 S-3091.80 0.65 0.90 S-30

102.00 0.65 0.90 S-30122.40 0.50 0.75 S-301442. SO 0.50 0.75 S-30163.20 0.55 0.80 S-30204.00 0.90 1.15 S-30

Hot Rolled-Open HearthTensile Strength 55,000 to 65,000 psiConforms to ASME SA-285, grade C

ASTM A-285 (latest) grade C

Size,In.

9/16x3 03 64 24 85 46 07 28 49 6

5/8x3 03 64 24 86 07 28 49 6

1203/4x

3 03 64 24 86 07 28 49 6

1 2 07/8x

3 03 64 86 07 2

120l x

3 03 64s6 07 28 49 6

1 2 011/8x

7 211/4x

7 211/2x

7 213/4x

7 2

Wtper Ft,

Lb

Size and Quality(S/l00 lb)Extra

Flange

57.38 . . *68.85 . . .SO.33 . . .91.80 . . .

103.30 . . .114.80 . . .137.70 . . .160.65 . . .183.60 . . .

63.75 0.9576.50 0.8089.25 0.80

102.00 0.80127.50 0.65153.00 0.50178 50 0.50204.00 0.55255.00 .

76.50 0.9591.80 0.80

107.10 0.80122.40 0.80153.00 0.65183.60 0.50214.20 0.50244. SO 0.55306.00

89.25107.10142 SO178.50214.20357.00

0.950.80

d.650.50

102,oo122.40163 20204.00244, SO285 60326.40408.00

275.40

306.00

367.20

428.40

0.950.80

0’. 650.500.500.55

. . .

0.50

0.50

0.50

1.50

Firebox

StockLengths,

Ft

1.20 S-301.05 S-301.05 S-301.05 S-300.90 S-300.90 S-300.75 S-300.75 S-300.80 S-30

1.201.051.051.050.900.750.750.801.15

1.201.051.051.050.900.750.75

l.15

1.201.051.050.900.751.15

1.201.051.050.900.75

1.‘15

. . .

. . .

. . .

. . .

S-30S-30S-30S-30S-30S-30S-30S-30S-30

S-30S-30S-30S-30S-30S-30S-30S-30S-30

S-30S-30S-30S-30S-30S-30

S-30S-30S-30S-30S-30S-30S-30S-30

3 0

3 0

3 0

3 0

Page 344: Process Equipment Design

A P P E N D I X

ALLOWABLE STRESSES

Item 1. Maximum Allowable Stress Values in Tension for Carbon and low-alloy Pipe and Tubes of Welded Manufucture,

in Pounds per Square Inch

(Extracted from the 1956 Edition of the ASME Boiler and Pressure Vessel Code, Unfired Pressure Vessels,with Permission of the Publisher, the American Society of Mechanical Engineers)

(Joint efficiencies used for preparing this table are: electric-resistance welded-85 %, lap welded-80 70, but,t welded-60s.)

Speci-Specifi- fied For Met,al Temperatures Not Exceeding Deg F-~__--.cation Nominal Min -2oto

Number Grade Composition Weld Notes Tensile 650 700 750 800 850 900 950 1000..~______ ..-__- .--SA-53 Carbon steel

‘A’ Carbon steelLap (1) 45,000 9000 8800 8200

SA-53 Resist. (l)(2) 48,000 10,200 9000 9100 i9bo...bo

5500 . . .:::

SA-53 B Carbon steel Resist. (l)(2) 60,000 12,750 12,200 11,000 9200 7350 5500 . . . . . .SA-72 . . Wrought iron Lap . 40,000 8000 7800 7300 . . . ,.. . . . . . . .SA-72 Wrought iron

*A‘ Carbon steelButt 40,000 6000 5850 5500

SA-135 Resist. (i)‘(i) 48,000 10,200 9900 9100 i4bo 6jbo ” “’ : 1:5500 . .SA-135 B Carbon steel Resist. (2)(3) 60,000 12,750 12,200 11,000 9200 7350 5500 . . .SA-178 A Low-carbon steel Resist. (2)(3) . . 10,000 9700 8950 7800 6650 5500 3800 iii0SA-178 B 0. H. iron Resist. 8500 8300 7750SA-178 C Medium-carbon steel Resist. (i)‘(i) 6d,bbO 12,750 12,200 11,000 bib0 i350 Go i&i0 2idoSA-226 Low-carbon steel Resist. (2) (3) 10,000 9700 8950 7800 6650 5500 3800 2100SA-250 Tl Carbon-+molybdenum R e s i s t . 5i,bbO 11,700 11,700 11,700 11,450 11,200 10,650 8500 5300SA-250 Tla Carbon-timmolybdenum Resist . 60,000 12,750 12,750 12,750 12,250 11,700 10,650 8500 5300SA-250 Tlb Carbon-54 molybdenum Resist. 53,000 11,250 11,250 11,250 11,050 10,850 10,650 8500 5300SA-333 C Carbon steel R e s i s t . 55,000 11,700 _.. . . . . . . ,.. _., .‘. . . .SA-333 3 316 nickel R e s i s t . 65,000 23,800 _.. . . . . . . . . ,., . . . . . .SA-333 5 3 nickel R e s i s t . 65,000 13,800 . . . . . . . . . . . . . . . . . _..SA-334 C Carbon steel Besist. . 55,000 11,700 . . . . . . . . . .,. . . .s-334 3 335 nickel R e s i s t . 65,000 13,800 . . . . . . . . . . . . . . . .,. . . .SA-334 5 5 nickel R e s i s t . 65,000 13,800 ,__ . . . . . .

rliotes: The st.ress values in this table may be interpolated to determine values for intermediate tempera&es. ’ ’ ’ . .(1) These stress values permitted for open-hearth and electric-furnace steels only.(2) For service temperatures above 850 F it is recommended that killed steels containing not less than 0.10% residual

silicon be used. Killed steels which have been deoxidized with large amounts of aluminum and rimmed steels may have creepand stress-rupture properties in the temperature range above 850 F, which are somewhat less than those on which the valuesin the above table are based.

(3) Only (silicon) killed steel shall be used above 900 F.335

Page 345: Process Equipment Design

336 Allowable Stresses

Item 2. Maximum Allowable Stress Values in Tension for Nonferrous Metals, in Pounds per Square Inch

Aluminum and Aluminum-alloy Products(Extracted from the 1956 Edition of the ASME Boiler and Pressure Vessel Code, Unfired Pressure Vessels,

with Permission of the Publisher. the American Society of Mechanical Engineers)

Specifi-cation

Number Alloy

Sheet and Plate

SpecifiedTensile

Strength,

MinimumYield

Strength,For Metal Temperatures Not Exceeding Deg F

psi Notes 100 150 2 0 0 2 5 0 3 0 0 3 5 0 4800

SB-178

SB-178

SB-178

SB-178

SB-178

SB-178

SB-178

SB-178

SB-178

SB-178

996A

990A

MlA

Clad MlA

MGllA

Clad MGllA

GlA

GRZOA

GSllA

Clad GSllA

Bars, Rods, and Shapes

SB-211 GSllASB-211 CG42ASB-211 CS41ASB-273 GSllA

SB-273 CG42A

Bolting Materials

SB-211 GSllA T6T6 weldedT4

42,00024,000*62,000

/SB-211 CG42A

I S!B-211 CS41A T6 65,000

Temper pi

0H112H12H140H112H12H140H112H12H140H112H12H140H112H32H340H112H32H340H112H32H340H112 1H32H34T4T6T6 weldedT4T6T6 welded

9 5 0 0 250010,000 400011,000 900012,000 10,00011,000 3 5 0 012,000 5 0 0 014,000 11,00016,000 14,00014,000 500014,500 6 0 0 017,000 12,00020,000 17,00013,000 450014,500 600016,000 11,00019,000 16,00023,000 8 5 0 023,000 9 0 0 028,000 21,00032,000 25,00022,000 800022,000 850027,000 20,00031,000 24,00018,000 600020,000 800022,000 16,00025,000 20,000

25,000

31,00034,00030,00042,00024,00027,00038,00024,000

T6 42,000T4 62,000T6 65,000T6 38,000T6 welded 24,000*T4 60,000

9 5 0 0

23,00026,00016,00035,000

14,bbo32,000

. . .

35,00040,00055,00035,000

4u,ouo

35,000

4d,bbo

(l&h

(1)

$p

(1)

(ii&(1)(1)

i($v

(1)

iij

(1)(1)

iii(1)(1)ii;(1). .(1)is,’(5)

I$

. . .

(5)

i7j8400 8 2 0 0 7 9 0 0 7 5 0 04800 4 7 0 0 4600 4 4 0 0

10,000 9700 9 4 0 0 9 0 0 055,000 (7) 13,000 12,200 11,600 10,400

\ \ \I /

1 6 5 0 1 6 5 0 1 6 0 0 1 4 5 0 1250 1200 10502 5 0 0 2150 1950 1 7 0 0 1500 1300 11002750 2 5 5 0 2350 2100 1900 1600 14003 0 0 0 3 0 0 0 2 9 0 0 2700 2350 2000 16002 3 5 0 2 3 5 0 2 3 0 0 2 1 0 0 1850 1600 13003 0 0 0 2 8 0 0 2 5 5 0 2 2 5 0 2000 1700 14003 5 0 0 3 4 0 0 3 1 5 0 2900 2650 2400 21004000 3900 3650 3300 3000 2700 22003350 3150 2 9 0 0 2 7 0 0 2400 2100 18003 6 0 0 3 2 5 0 3 0 0 0 2 8 0 0 2500 2200 19004250 4000 3 8 0 0 3 6 0 0 3300 3000 26505 0 0 0 4 8 5 0 4700 4400 4000 3500 31003 0 0 0 2 9 0 0 2 7 0 0 2500 2200 2000 17003 6 0 0 3 2 0 0 3 0 0 0 2 8 0 0 2500 2200 19004000 3 8 0 0 3600 3 4 0 0 3100 2800 25004800 4600 4400 4 2 0 0 3800 3400 29005 6 5 0 5 6 5 0 5 6 5 0 5500 4650 3850 31505 7 5 0 5750 5 7 5 0 5500 4650 3850 31507 0 0 0 7 0 0 0 7 0 0 0 6 5 5 0 5800 5050 43008000 8 0 0 0 8 0 0 0 7400 6550 5600 47005 3 0 0 5 3 0 0 5 3 0 0 5200 4400 3700 30005 5 0 0 5 5 0 0 5 5 0 0 5 2 0 0 4400 3700 30006 8 0 0 6 8 0 0 6 8 0 0 6 3 0 0 5600 4900 41007 8 0 0 7 8 0 0 7 7 0 0 7 2 0 0 6300 5400 46004 0 0 0 4000 4000 4000 4000 3350 28005 0 0 0 5 0 0 0 5 0 0 0 4 9 0 0 4500 3700 28005 5 0 0 5 5 0 0 5 5 0 0 5 3 5 0 4800 3800 28006 2 5 0 6250 6 2 0 0 6 0 5 0 5400 3950 2800

6 2 5 0 6 2 5 0 6 2 0 0 6 0 0 0 5400 4650 3900

7 7 5 0 7 7 5 0 7 6 5 0 7 1 0 0 6400 5600 48008 5 0 0 8 5 0 0 8400 7700 6900 6100 53007 5 0 0 7 2 0 0 7 0 0 0 6700 6400 5600 4000

10,500 10,200 9900 9400 7900 6200 44006000 5 9 0 0 5 7 0 0 5 4 0 0 5000 4200 32006 8 0 0 6 5 0 0 6 2 0 0 6 0 0 0 5800 5100 36009 5 0 0 9200 9000 8500 7200 5600 40006 0 0 0 5 9 0 0 5 7 0 0 5 4 0 0 5000 4200 3200

8 4 0 0 8100 7 7 0 0 7 1 0 0 6000 4800 340010,000 9 7 0 0 9 4 0 0 9 0 0 0 7800 6200 460013,000 12,200 11,600 10,400 7200 4400 3000

9 5 0 0 9 2 0 0 9 0 0 0 8 5 0 0 7200 5600 40006 0 0 0 5 9 0 0 5700 5 4 0 0 5000 4200 3200

15,000 14,300 13,700 12,000 9100 5700 3950

6300 4900 33004000 3400 26007800 6200 46007200 4400 .300G

Page 346: Process Equipment Design

Allowable Stresses 337

Item 2. Maximum Allowable Stress Values in Tension for Nonferrous Metals, in Pounds per Square Inch (Continued)

Aluminum and Aluminum-alloy Products (Continued)SpecifiedTensile

Strength,Temper ps i

MinimumYield

Strength,For Metal Temperatures Not Exceeding Deg F

psi Notes 100 150 200

Specifi-cation

Number Alloy

Pipe and TubeSB-274SB-274SB-234SB-274SB-274SB-274 GSlOA

SB-274 GSllASB-234SB-274 >

ForgingsSB-247 MlASB-247 CS41A

SB-247 GSllA

SB-247 GSllB

0 14,000H112 14,500

H14 20,000H18 27,000T42 17,000T5 22,000T6 32,000T4 26,000T6 38,000T6 welded 24,000*

F 14,000T4 55,000T6 65,000T6 38,000T6 welded 24,000*T6 36,000

250 300 350 400

50006000 iij

17,000 (1)24,000 (1)10,000 (5)1500025,000 ii;16,000 (5)35,000 (5)

. . . . . .

500030,000 iij55,000 (5)35,000 (5)

sd,bbo iij

3350 31503600 32505 0 0 0 48506750 64004250 42005500 51008000 76006500 62009500 92006000 5900

3350 3150 2900 2700 2400 2100 180013,800 12,800 12,000 11,000 10,200 5750 390016,200 15,200 14,400 14,000 11,300 5750 39009500 9200 9000 8500 7200 5600 40006000 5900 5700 5400 5000 4200 32009000 8400 7900 7300 6100 4700 3200

2900 2700 2400 2100 18003000 2800 2500 2200 19004700 4400 4000 3500 31006050 5700 5250 4400 35004200 4150 4050 3300 21004900 4600 4150 3300 21007200 6550 4800 3300 21006000 5800 5600 4900 35009000 8500 7200 5600 40005700 5400 5000 4200 3200

* Strength of full-section tensile specimen required to qualify welding procedures.Notes :(1) For welded construction, stress values for 0 material shall be used.(2) For nominal thicknesses not greater than 0.500 in., the stress values for H14 material may be used; for nominal thick-

nesses of 0.501 to 1.000 in., the values for H12 material may be used; for thicker material the values listed shall be used.(3) For nominal thicknesses not greater than 2.000 in.; for thicker material the stress values for 0 material shall be used.(4) For nominal thicknesses not greater than 0.500 in., the stress values for H12 material may be used; for thicker material

the values listed shall be used.(5) The stress values given for this material are not applicable when either welding or thermal cutting are employed.(6) For nominal thicknesses not less than 0.25 in.

Page 347: Process Equipment Design

Item 2. Maximum Allowable Stress Values in Tension for Nonferrous Metals, in Pounds per Square Inch (Continued)

Copper and Copper AlloysSpecified Minimum

Tens i l e Yield For Metal Temperatures Not Exceeding Deg FStrength, Strength. ---____- ~~ ____..

n.; .?a: 100 150 200 250 300 350 400 450

30,ooo 10,000* 6700 6700 b500 6300 5000 3800 250030,000 I o,ooo* 6700 6700 6500 6300 5000 3800 250030,000* 9000* 6000 6000 5900 5800 5000 38OIJ 250030,000* 9000* 6000 6000 5900 5800 5000 3800 250036,000* :lo~ooo* 9000 9000 8700 8300 8000 5000 25004 5 , 0 0 0 ’ 40,000* 11,300 11,300 11,000 10,500 8000 5000 250030,000* 9000* 6000 6000 5900 5800 5000 3800 250036.000* 30;000* 9000 9000 8700 8300 8000 5000 250045.000* 40,000* 11,300 11.300 11,000 10,500 8000 5000 250036,000* 30,000* 9000 9000 8700 8300 8000 5000 250045,000* 40,000* 11,300 11.300 11,000 10,500 8000 5000 250030,000 in,ooo* 6700 6700 6500 6300 5000 3800 2500

. . .

. . .

. .

. .

4o,olJo*4n.nno*

60006000

20002000

45,000*45.000

12,000* 8000 8000 8000 8000 800012,non* 8000 8000 8000 9000 8000

15,000* 10,000 10,000 10,000 10,000 10,00015,000 10.000 10.000 10,000 10,000 10,000

18,000 12.000 12,000 12,000 12,000 12,000

80008000

50,000* 7500

7500

75007500

30003000

50005000

3000

2000

20002000

30003000

2000

50,000 20,000 1 2 . 5 0 0 1 2 , 5 0 0 1 2 , 0 0 0 1 1 , 2 0 0 10,500

50.000*50.000

2O,UOO* 1 2 , 5 0 0 1 2 , 5 0 0 1 2 , 0 0 0 11,200 10,50020,000 1 2 , 5 0 0 1 2 , 5 0 0 1 2 , 0 0 0 1 1 . 2 0 0 1 0 . 5 0 0

52,000 18.000 12.000 11,600 11,300 11,000 10,800 10,600 10,30050,000 20.000 12,500 12,500 12,500 12,500 12,200 12,000 11,700

45.000 16,000 10,700 10,600 10,500 10,400 10,300 10,100 9900

40,000* 15.000* 10,000 10,000 9800 9500 9300 9000 8700

10,10011,300

9600

8 3 0 0

50,000* 19,000 12,500 12,400 12.200 11,900 11.600 10,000 6 0 0 0 400090,000 36.000 22,500 2 2 , 5 0 0 2 1 , 0 0 0 19,500 18.000 1 6 . 5 0 0 1 5 . 0 0 0 1 3 . 5 0 0

70,000 30,000 17,500 17.500 16.800 16,000 15,500 1 5 . 0 0 0 1 4 . 5 0 0 1 2 , 0 0 0

50,000 18,000 12,000 12.000 11.900 11,700 10,000 5000 . . . . . .52,000 15,000 10,000 10,000 10,000 10,000 10,000 5000 . . . . . .70,000 38,000 14,000 14,000 14,000 14,000 14,000 10,000 . . . . . .

40,000 12,000 8000 8000 8000 8000 700055,000 20,000 11,000 11,000 11,000 11,000 10.000

5000 . . . . . .8000 . . . . . .

500

.

.

.

. . .

. . .

. . .

. . .

.

.

. . .

990011,000

9300

7500

200012,000

10.000

. . .

. . .

. . .

. . .

M a t e r i a l a n d S p e c i f i c a t i o nNUUlhW Coudit,iorl Size. iu. 550 600

. . .

. . .

. . .

. . .

. .

. .

.

.

.

980010,500

8900

6700

9600 9500 940010,000 9500 9000

8400 7700 7000

6000 . . . . .

.10.500

. .

. . .

. .

. . .. . .

. . . .9 0 0 0 7 5 0 0 6 0 0 0

... ... ...

... ... ...

... ... ...

... ... ...

... ... .. .

... ... ...

...... : .

650

. .

. .

.. . .

. . .

. . .

. .

. .

. . .

. . .

. . .. . .

.

. . .

. . .

. . .

. .

. . .. . .

. . .

C o p p e rS B - 1 1SB-12SB-13SB-4288-42SB-42SB-75SB-75SB-75S B - 1 1 158-111SB-152

Red b ra s sSB-43SB-111

P l a t e s Annea ledRods Annea ledS e a m l e s s b o i l e r t u b e s Annea ledP i p e Annea ledP i p e L i g h t d r a w nP i p e H a r d d r a w nS e a m l e s s t u b e s Annea ledS e a m l e s s t u b e s L i g h t d r a w nS e a m l e s s tubes H a r d d r a w nSeamless condenser tubes Light drawnSeamless condenser tubes Hard drawnPlate steel, at,rip and har Annea led

phosphorus,deox id i zed

P i p e Annea ledSeamless condenser tubes Annealed

Admiralty, A. B. C. DSB-111 Seamless condenser tubesSB-171 Tube plates

Aluminum brass. B. C. DSB-111 Seamless condenser t.uhea

Naval brassSB-171 Tube platea

M u n t x m e t a lS&l11 Seamless condeuser tubesSB-171 Tube plates

C o p p e r - n i c k e l , 7 0 - 3 0SB-111 Seamless condenser tubesSB-171 Tube plates

C o p p e r - n i c k e l , 8 0 - 2 0SB-111 Seamless condenser tubes

Annea ledAnnea led

Annea led

Anuesled

Annea ledAnnea led

Annea ledAnnea led

A nun&d

C o p p e r - n i c k e l , 9 0 - 1 0SB-111 Seamless condenser tubes Anuealed

A l u m i n u m b r o n z eSB-111 Seamless condenser t,ubes AnnealedSB-171 Tube plates Annealed

Aluminum bronze DSB-169 Plate. sheet A nnealnd

Copper-silicon, A, CSB-96 P l a t e , s h e e t ( 1 ) Annea ledSB-98 Rods (1) S o f tSB-98 Rods (1) H a l f h a r d

C o p p e r - s i l i c o n BSB-98 Rods (1) S o f tSB-98 Rods (1) Helf hard

BOLTING MATERIALS (3)Copper

SB-12 Rod Soft 30,000 10,000* 2500 2500 2500 2 4 0 0 2200 2 1 0 0 2000 . . . . , . .

Page 348: Process Equipment Design

Item 2. Maximum Allowable Stress Values in Tension for Nonferrous Metals, in Pounds per Square Inch (Continued)

Copper and Copper Alloys (Continued)Specified MinimumTens i l e Yield For Metal Temperatures Not Exceeding Deg F

M a t e r i a l a n d S p e c i f i c a t i o nNumber

Copper-silicon, A, DSB-98 Rod (1)

Copper-silicom BSB-98 Rod (1)

Condition

S o f tQ u a r t e r b a r dH a l f h a r d

S o f tB o l t t e m p e r

Size, in.Strength, Strength,

p s i osi 100 150 200 2 5 0 300 350 400 4 5 0 500 550 600 6 5 0 700

52,000 15,000 3800 3800 3800 3800 3800 350055,000 24.000 6000 6000 5900 5800 5600 550070,000 38,000 9500 9500 9300 9000 8800 8500 . . . . . . . . .

. . . . .

. . . . . . . .

. . . . . . . . .40,000 12,00085,000 55,00075,000 45,00075,000 40,000

13,80011,20010,000

300013,80011,20010.000

300013,10010,800

9600

290012,70010,500

9400

280012,30010.000

9000

. . . . . . . .. . . . . .

. . . .. .

.80.000 40,000 10,000 10,000 10,000 9900 9800 98oO 9 7 0 0 95ou 9400 9000 7 5 0 0 6 0 0 0 4 5 0 075,000 37,500 9400 9400 9300 9200 9200 9100 9000 8900 8800 8600 7 5 0 0 6 0 0 0 4 5 0 072,000 35.000 8800 8800 8700 8600 8600 8500 8400 8300 8200 8000 7 5 0 0 6 0 0 0 4 5 0 0

100,00090,00085,000

90,00075,0007P,OOO

50,000 12,500 12,50045,000 11,200 11,20042,500 10,600 10,600

12,500 12,500 12,400 12,400 12,30011,200 11.200 11,200 11,200 11,10010,600 10,600 10,600 10.500 10,500

12,20011.000

12,00010.800

10,500 9 0 0 0 7 5 0 0 6 0 0 010,500 9 0 0 0 7 5 0 0 6 0 0 010,000 9 0 0 0 7 5 0 0 6 0 0 0

to,000 10,000 10,000 10,000 9900 9800 9700 960035,000 8800 8800 8800 8700 8600 8600 840032,000 8000 soon 7900 7800 7800 7600 7600

10;20010;400

Y50083007400

700070007000

.

. . .

. . .

34,000 6800 6800 6800 6800 6500 6000 550030,000 6000 6000 5800 5500 5000 4500 3500

33004000 . . . . . . . . .. . . , . . . . .

. . .. . .

A l lUp to )i, incl.Over $5 to 1. incl.Over 1 to 19;. incl.

Aluminum bronzeSB-150 Alloy No. 1 U p t o 36, id.

Over 56 to 1. incl.Over 1

36 to 1, incl.Over 1 to 2, incl.O v e r 2 to 4, incl.

Up to 56, incl.Over 36 to 1, incl.Over 1 to 2. iocl.

SB-150 Alloy No. 2 . . .

m-150 \llcry N o . 3 . . .

CASTING MATERIALSB-61(2) . . .SB-62(2) . . .

* Expec ted va lue , not i n c l u d e d i n s p e c i f i c a t i o n s ._.

(1) C o p p e r - s i l i c o n a l l o y s a r e n o t always s u i t a b l e w h e n e x p o s e d t o c e r t a i n m e d i a a n d h i g h t e m p e r a t u r e s , p a r t i c u l a r l y s t e a m a b o v e 2 1 2 F .for the service for which it is to be used.

The user should satisfy himself that the alloy selected is satisfactory

(2) In the absence of evidence that the casting ia of high quality throughout, valuea not in excew of 80 x of those given in the table shall be used..,o r e c o g n i z e d s t a n d a r d s .

This is not intended to apply to valves and fittingE made

Page 349: Process Equipment Design

1

I

Item 2. Maximum Al lowable Stress Values in Tension for Nonferrous Metals, in Pounds per Square Inch (Continued)

Nickel and High-nickel Alloys,

Material Form, andSpecific&n Number Condition

Y i e l dSpecified Strength

T e n s i l e (0.2 %Strength. Offset),

For Metal Temperaturea Not Exceedin* Deg F

p s i N o t e s 100 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 9 0 0 1 0 0 0 1 1 0 0 12002

elrs, rods, and shapea

NicksB a

SB-160SB-160

BoltingSB-160SB-160

Pipe or tubingS B - 1 6 1

::::::Co;&..;~ tubing

SR.163Pla #a, sheet, or strip

SB-162SB-162 (plate only)

Low-carbon NickelBagti-y6$ and shapes

SB-160Bolting

SB-160Pipe or tubing

E???,u-I”I

odenser tubing3B-1633B-163ke, sheet. or strip3B-162jB-162 (plate only)

,p lx‘0 Bs, and shapes

PI:II

N i c k e l - c cBars, r

SB-164 (classSB-164 (class

BoltingSB-164 (class!y;;s6”A (class

Class BPipe or tubing

SB-165SB-165-am ILC

A and B)A only)

>“-I”.3

odenser tubing3B-1633B-163te, sheet, or strip3B-127jB-127 (plate only)

-- ___PiPSegyi6ybing

Co;cl~f;~ tubing

Pl;tt&$et, or strip

SB-168 (plate only)N o t e s :

wg

Hot or cold worked-annealedHot rolled or forged-hot finished

5 5 . 0 0 0 1 5 , 0 0 06 0 . 0 0 0 1 5 . 0 0 0

Hot or cold worked--annealedHot rolled or forged-hot finished (3)

5 5 , 0 0 06 0 , 0 0 0

1 5 , 0 0 01 5 , 0 0 0

S~*IId~SS--a~Il~*ledSenmless-hard. stress relievedSeamless--stress equalized

5 5 , 0 0 06 5 , 0 0 0

1 5 , 0 0 04 0 , 0 0 0

7 0 , 0 0 0 5 0 , 0 0 0

S e a m l e s s - - a n n e a l e dSeamless-stress relieved

Sl,OOO6 5 , 0 0 0

1 5 . 0 0 04 0 . 0 0 0

Hot or cold rolled-annealedHot rolled-as rolled

5 5 , 0 0 05 5 . 0 0 0

1 5 , 0 0 02%M)O

Hot or cold worked-annealedHot rolled or forged-hot finished

5 0 , 0 0 05 0 , 0 0 0

Hot worked or annealed 5 0 , 0 0 0

1 0 . 0 0 01 0 , 0 0 0

1 0 . 0 0 0

S e a m l e s s - - a n n e a l e dSeamless-hard, stress relieved

5 0 , 0 0 06 0 , 0 0 0

1 2 , 0 0 03 0 , 0 0 0

S e a m l e s s - - a n n e a l e dSeamless-stress relieved

5 0 , 0 0 0 1 2 , 0 0 06 0 , 0 0 0 3 0 , 0 0 0

Hot or cold rolled--annealedHot rolled--as rolled

5 0 , 0 0 05 0 . 0 0 0

Hot or cold worked-annealedHot rolled or forged-hot finished

7 0 . 0 0 0 2 5 , 0 0 08 0 . 0 0 0 4 0 , 0 0 0

1 6 . 6 0 0 1 4 , 6 0 0 1 3 . 6 0 0 1 3 , 2 0 0 1 3 . 1 0 02 0 , 0 0 0 1 8 , 9 0 0 1 8 , 4 0 0 1 8 . 2 0 0 32 , 1 8 , 2 0 0

Hot or cold worked--annealedHot rolled or forged-hot finished (3)

7 0 , 0 0 0

Cold drawn--as drawn (4) (5)8 0 . 0 0 0

Cold drawn--as draw, (4) (5)9 0 , 0 0 08 5 , 0 0 0

2 5 , 0 0 04 0 , 0 0 07 0 , 0 0 05 0 , 0 0 0

Seamless-annealedSeamless-hard. strew relievedSeamless-stress equalized

6 1 0 0 5 7 0 0 5 2 0 0 5 0 0 0 4 9 0 01 0 . 0 0 0 9 6 0 0 9 4 0 0 9 0 0 0 i%: 8 5 0 01 7 . 4 0 0 1 6 , 9 0 0 1 6 , 2 0 0 1 5 , 5 0 0 1 5 , 4 0 01 2 . 4 0 0 1 2 , 0 0 0 1 1 , 5 0 0 1 1 , 1 0 0 1 1 , 1 0 0

1 7 , 5 0 0(1)

1 6 , 5 0 0 1 5 , 5 0 0 1 4 . 7 0 0 1 4 , 7 0 02 1 , 2 0 0

(1)2 0 , 2 0 0

1 4 , 8 0 0

2 1 . 2 0 0 2 0 . 2 0 01 9 , 5 0 0 1 9 . 2 0 0 1 9 , 2 0 01 9 , 5 0 0 1 9 , 2 0 0 1 9 , 2 0 0

S e a m l e s s - - a n n e a l e dSeamless-stress relieved

7 0 , 0 0 08 5 . 0 0 08 5 , 0 0 0

7 0 , 0 0 08 5 , 0 0 0

2 8 , 0 0 05 5 , 0 0 06 5 , 0 0 0

2 8 , 0 0 05 5 . 0 0 0

2 8 , 0 0 04 0 , 0 0 0

3 0 , 0 0 03 5 , 0 0 0

3 0 , 0 0 03 5 , 0 0 0

3 0 , 0 0 0

3 0 , 0 0 0

3 0 , 0 0 03 5 , 0 0 0

1 7 , 5 0 0 1 6 , 5 0 0 1 5 . 5 0 02 1 , 2 0 0 2 0 , 2 0 0

1 4 , 8 0 0 1 4 . 7 0 0 1 4 , 7 0 01 9 , 5 0 0 1 9 . 2 0 0 1 9 , 2 0 0 1 9 , 2 0 0

Hot or cold rolled-annealedHot rolled--as rolled

7 0 , 0 0 07 5 . 0 0 0

Hot or cold worked--annealedHot rolled or forged-hot finished

Hot or cold worked--annealedHot rolled or forged-hot 8ni&ed

Seamless-annealed

S e a m l e s s - - a n n e a l e d

Hot or cold rolled-annealedHot rolled--as rolled

8 0 , 0 0 08 5 . 0 0 0

8 0 , 0 0 08 5 . 0 0 0

8 0 , 0 0 0

8 0 . 0 0 0

8 0 , 0 0 08 5 , 0 0 0

1 2 , 0 0 01 2 , 0 0 0

1 0 , 0 0 0 1 0 , 0 0 0 1 0 , 0 0 0 1 0 , 0 0 0 1 0 . 0 0 0 1 0 , 0 0 0 . . .1 0 . 0 0 0 1 0 . 0 0 0 1 0 , 0 0 0 1 0 , 0 0 0 9 5 0 0 8 3 0 0 . . .

. .

3 7 0 0 3 7 0 0 3 7 0 0 3 7 0 0 3 7 0 0 3 7 0 03 7 0 0 3 7 0 0 3 7 0 0 3 7 0 0 2 7 0 0 3 4 0 0 .

.

1 0 , 0 0 0 1 0 , 0 0 0(1) 1 6 . 2 0 0

1 0 . 0 0 0:s”,:i 2%

1 0 , 0 0 0

(1) 16.200;?;Wz

,;;,o”W;

, 15 :ooo 14:500 : ::

.

...

. . ..

1 0 , 0 0 0 1 0 . 0 0 0 1 0 , 0 0 0 1 0 , 0 0 0 1 0 . 0 0 01 6 , 2 0 0 1 5 , 3 0 0

1 0 , 0 0 01 5 , 0 0 0 1 5 . 0 0 0 1 4 , 5 0 0

. . . ..

. . .

1 0 , 0 0 0 1 0 , 0 0 0 1 0 , 0 0 0 1 0 , 0 0 0 1 0 , 0 0 0 1 0 , 0 0 0 . .1 3 , 3 0 0 1 3 , 3 0 0 1 3 . 3 0 0 1 3 , 3 0 0 1 2 , 5 0 0 1 1 . 5 0 0

.. . . .

... ...

... ...

... ...... ...... ...... ...... ...... ...... ...... ...... ...

4 5 0 04 5 0 0

(1)

6 7 0 0 6 4 0 0 6 3 0 0 6 2 0 0 6 2 0 0 6 2 0 06 7 0 0 6 4 0 0 6 3 0 0 6 2 0 0 6200 6 2 0 0

2 5 0 0 2408 2 3 0 0 2 3 0 0 2 3 0 0 2 3 0 0

8 0 0 0 7 7 0 0 7 5 0 0 7 5 0 0 7500 7 5 0 01 5 , 0 0 0 1 4 , 2 0 0 1 3 . 8 0 0 1 3 . 5 0 0 1 3 , 5 0 0 .*.

8 0 0 0 7 7 0 0 7 5 0 0 7 5 0 0 7500 7 5 0 01 5 . 0 0 0 1 4 . 2 0 0 1 3 . 8 0 0 1 3 . 5 0 0 1 3 . 5 0 0 1 3 , 0 0 0

8 0 0 0 7 7 0 8 7 5 0 0 7 5 0 08 0 0 0 7 7 0 0 7 5 0 0

;i::7 5 0 0

6 2 0 06 2 0 0

2 3 0 0

7 4 0 0

3 0 0 0 2 0 0 03 0 0 0 2 0 0 0

7 4 0 01 2 , 0 0 0

7 4 0 07 4 0 0

5 9 0 05 9 0 0

2 2 0 0

7 2 0 0

7 2 0 01 1 , 0 0 0

7 2 0 07 2 0 0

2 1 0 0

4 5 0 0

4 5 0 01 0 , 0 0 0

4 5 0 04 5 0 0

2 0 0 0 1 8 0 0

3 0 0 0 2 0 0 0. . . .

3 0 0 0 2 0 0 0. . .

3 0 0 0 2 0 0 03 0 0 0 2 0 0 0

1 3 , 1 0 01 7 , 6 0 0

4 9 0 08 5 0 0

::::z

4 9 0 08 3 0 0

1 4 , 5 0 0

1 4 , 5 0 01 5 , 0 0 0

1 4 . 5 0 01 4 , 5 0 0

1 7 , 0 0 02 0 , 0 0 0

%:

1 7 . 0 0 0

1 7 , 0 0 0

1 7 , 0 0 02 0 , 0 0 0

8 0 0 04 0 0 0

4 7 0 04 0 0 0

1 7 , 5 0 01 8 . 7 0 0 :z~

1 5 . 5 0 0 1 4 , 8 0 0 1 4 , 7 0 0 1 4 . 7 0 0( 1 7 . 0 0 0 1 7 , 0 0 0 1 7 . 0 0 0 1 7 . 0 0 0

1 4 , 7 0 0

1 4 , 7 0 01 8 , 5 0 0

1 4 , 7 0 01 6 , 5 0 0

8 0 0 0

.

8 0 0 0(2)80004 0 0 0

2 0 , 0 0 0 1 8 . 6 0 0 1 8 , 0 0 0 1 8 , 0 0 02 1 , 2 0 0

1 8 , 0 0 0 1 8 , 0 0 02 0 , 2 0 0 2 0 , 0 0 0 2 0 , 0 0 0 2 0 , 0 0 0 2 0 , 0 0 0

7 3 0 0 6 9 0 0 6 8 0 08 7 0 0 8 5 0 0 8 2 0 0 2%

6800 6 8 0 07900 7 9 0 0

2 0 , 0 0 0 1 8 , 6 0 0 1 8 , 0 0 0 1 8 . 0 0 0 1 8 , 0 0 0 1 8 , 0 0 0

2 0 , 0 0 0 1 8 , 6 0 0 1 8 , 0 0 0 1 8 , 0 0 0 1 8 , 0 0 0 1 8 . 0 0 0

2 0 , 0 0 0 1 8 , 6 0 0 1 8 . 0 0 0 1 8 , 0 0 0 1 8 , 0 0 0 1 8 , 0 0 02 1 . 2 0 0 2 0 , 2 0 0 2 0 , 0 0 0 2 0 . 0 0 0 2 0 , 0 0 0 2 0 , 0 0 0

1 7 , 5 0 02 0 , 0 0 0

%8

1 7 , 5 0 0

1 7 . 5 0 0

1 7 . 5 0 02 0 , 0 0 0

1 6 . 0 0 01 9 . 5 0 0

... ...

... ...

... ...

... ...

... ...

... ...... ...... ...... ...... ...... ...... ...... ...

7 0 0 0 3 0 0 01 4 , 5 0 0 7 2 0 0

6 3 0 0 6 0 0 0 3 0 0 07 4 0 0 7 3 0 0 7 2 0 0

1 6 , 0 0 0

1 6 , 0 0 0

1 6 , 0 0 01 9 . 5 0 0

7 0 0 0 3 0 0 0

7000 3 0 0 0

7 0 0 0 3 0 0 01 4 , 5 0 0 7 2 0 0

i5-z.?%f

. . .

. . ..

..

12001 2 0 0

1 2 0 0

1 2 0 0

1 2 0 0

1 2 0 01 2 0 0

.

.

2 0 0 05 5 0 0

2 0 0 05 5 0 0

2 0 0 0

2 0 0 0

2 0 0 05 5 0 0

Allowable working stress for this material is based upon Se-163, stress-relieved condenser tubing. The

on stresses for the hot-rolled or forged hot-finished temper 0dess other specific data are available. Tbr

established at 500 %.

Page 350: Process Equipment Design

Allowable Stresses 341

Item 3. Typical Physical Properties of Materials(Extracted from the 1.956 Edition of the ASME Boiler and Pressure Vessel Code, Unfired Pressure Vessels, with

Permission of the Publisher, the American Society of Mechanical Engineers)

Weight,ASME lb per

Material Spec. No. cu in.Aluminum alloy 996A SB-178 0.098

990A SB-178 0.098MlA SB-178 0.099MGllA SB-178 0.098GR20A SB-178 0.097GSllA SB-178 0.098

Copper, deoxidized SB-11 and 111 0.323Red brass SB-111 and 43 0.316Admiralty SB-111 and 171 0.308Aluminum brass SB-111 0.301Naval brass SB-171 0.304Muntz metal SB-171 0.30330 y0 Cupronickel SB-111 and 171 0.32320 y0 Cupronickel SB-111 and 171 0.32310 y0 Cupronickel SB-111 0.323Copper-silicon (A, C, D) SB-96 and 98 0.308Copper-silicon (B) SB-98 0.316Aluminum bronze (D) SB-171 0.281Aluminum bronze (E) SB-171 0.274Nickel and low-carbon nickel SB-162 0.321Nickel-copper SB-127 0.319Nickel-chromium-iron SB-168 0.300Steel SA-30 0.279

(a) Thermal expansion per degree F from 68 F to 212 F.(b) Thermal expansion per degree F from 68 F to 572 F.(c) Thermal expansion per degree F from 32 F to 212 F.

Approx.MeltingRange,

“F1195-12201190-12151190-12101165-12051100-12001080-1205

19801810-18801650-17201710-17801630-16501650-16602140-22602100-22002020-21001780-18801890-19401850-19001900-19302615-26352370-24602540-2600

ThermalConductivity, Thermal Specific

32 F-212 F, Expansion, Heat, Btu/Btu/sq ft/hr, 1OV in./in., lb/OF

“Fjin.16601 5 4 01 3 4 01 1 3 0

9 6 01 1 9 02 3 5 21 1 0 4

7 6 86 9 68 0 48 5 22 0 42 4 03 2 42 5 23 9 65 5 22644 2 01801 0 44 6 0

“F13 2(a)13.1(a)12.9(a)13.3(a)13,2(a)13.1(a)9.8(b)

10 4(b)11.2(b)10,3(b)11,8(b)11.6(b)9.1(b)9.3(b)9.5(b)

10.0(b)9’. 9(b)9.0

7.2(c)7.8(c)6.4(c)6.7(c)

at 212F0.230.230.230.230.230.230.090.090.090.090.090.090.090.090.090.090.090.090.090.130.138.11

Page 351: Process Equipment Design

J 3 4 2 Allowable Stresses

Item 4. Maximum Allowable Stress Values in Tension(Extracted from the 1956 Edition of the ASME Boiler and Pressure Vessel Code, Urlfired Pres-

For Metal TemperaturesMaterial andSpecificatiofl

Namber Grade TypeNominal kin -20 to

Composition Tensile Notes 100 200 300 400 500 600 650

Plate SteelsSA-167 3SA-167 3SA-167 5SA-167 6SA-167 a

304 18 Cr-8 Ni304 18 Cr-8 Ni321 18 Cr-8 Ni-Ti347 1 8 Cr-8 Ni-Cb309 25 Cr--12 Ni310 25 Cr-20 Ni310 25 Cr-20 Ni316 1 8 Cr-IO N i - 2 MO410 1 3 Cr

1 5 Cr34; 18 Cr-8 Ni-Cb430 17 Cr

75,000 (1) 18,750 17,000 16,000 15,450 15,100 14,900 14,85015,000 13,650 12,500 11,600 11,20017,000 15,800 15,200 14,900 14,85017,000 15,800 15,200 14,900 14,85017,300 16,700 16,600 16,500 16,45018,500 18,200 17,700 17,200 16,90018,500 18,200 17,700 17,200 16,90017,900 17,500 17,200 17,100 17,05015,100 14,600 14,150 13,850 13,70016,300 15,650 15,100 14,600 14,30017,000 15,800 15,200 14,900 14,85016,300 15,650 15,100 14,600 14,300

75,000 :. :

75;ooo75,000

75.000

.75,000

.

75,000 I:;75,000 :.65.000

18;750 18,750

18,750 16,650

18,750 18,75018,750 18,75018,750 18,750

18.750 18,750

18,750 18,75016,250 15,60017,500 17,50018,750 18,75017,500 17,500

SA-167 10SA-167 10SA-167 11SA-240 ASA-240 B 70;ooo

75,00070,000 ibj

..75,QO& .’ :60,00075,000 iij75,00075,000

SA-240 cSA-240 DSA-?9?. fSA-240SA-240 sSA-240 sSA-240 T

E’ipes and TubesSeamless

SA-213 TP304SA-213 TP304SA-213 TP321SA-213 TP347SA-213 TP316SA-213 TP310SA-213 TP310SA-268 TP405SA-268 TP410

304 18 Cr-8 Ni304 18 Cr-8 Ni321 18 Cr-8 Ni-Ti l

1 8 , 7 5 0 18,750. 17,9(?0 1 7 , 5 0 0 l.7,200 1 7 , 1 0 0 .17,95015.000 15.000 i4:yioo 14.400 13.950 13.400 13.000.---is;750 171000 161000 15:450 15;100 141900 14;85018,750 16,650 15,000 13,650 12,500 11,600 11,20018,750 18,750 17,000 15,800 15,200 14,900 14,850

18 Cr-8 Ni. , , 18 Cr-8 Ni

. 18 Cr-10 Ni-Ti1 8 C r - 1 0 N i - C b. 16 Cr-13 Ni-3 MO

. 25 G-20 Ni2 5 C r - 2 0 N i1 2 C r - A l1 3 C r1 6 C r. 18 G-8 Ni

. 18 Cr-8 Ni. 25 G-12 Ni. 25 Cr-20 Ni. 25 Cr-20 Ni. . 18 Cr-10 Ni-Ti. . 18 Cr-10 Ni-Cb

. . 16 Cr -13 Ni -3 MO18 Cr-13 Ni-t MO

. 18 Cr-8 Ni

75,000 (1)75,00075,00075,00075,00075,000 iij75,000 (3)60,00060,00060,000 Gj75,000 (1)75,00075,00075,000 iit j75,000 (3)75,00075,00075,090 .75,00075,000 iij75,000 . . .75,000 . .75,000 .75;ooo .

75,00075,000 y)

75,000 (2)(4)75,000 (3)(4)

75,00075,000 ii;75,000 (4)

18,750 17,00018,750 16,65018,750 18,75018,750 18,75018,750 18,75018,750 18,75018,750 18,75015,000 15,00015,000 14,45015,000 15,00018,750 17,00018,750 16,65018,750 18,75018,750 18,75018,750 18,75018,750 18,75018,750 18,75018,750 18,75018.750181750

18,75017,000

18,750 16,65018.75018;750

18,75018,750

18,750 18,750

14,45014,15016,00016,00016,00016,00016,000

16,00015,00017,00017,00017,90018,50018,50014,70014,00014,10016,00015,00017,30018,50018,50017,00017,00017,90017,90016,00015,00017,00017,00017,900

13,60012,75015,75015,750

15,450 15,10013,650 12,50015,800 15,20015,800 15,20017,500 17,20018,200 17,70018,200 17,70014,400 13,95013,500 13,10013,400 13,00015,450 15,10013,650 12,50016,700 16,60018,200 17,70018,200 17,70015,800 15,20015,800 15,20017,500 17,20017,500 17,20015,450 15,10013,650 12,50015.800 15.20015;SOO IS;20017,500 17,200

14,900 14,85011,600 11,20014,900 14,85014,900 14,85017,100 17,05017,200 16,90017,200 16,900‘13,400 13,00012,850 12,70012,500 12,20014,900 14,85011,600 11,20016,500 16,45017,200 16,90017,200 16,90014,900 14,85014,900 14,85017,100 17,05017,100 17,05014,900 14,85011,600 11,20014,900 14,85014,900 14,85017,100 17,050

13,150 12,800 12,700 12,65011,600 10,600 9 8 5 0 9 5 0 015,500 15,050 14,600 14,40015,500 15,050 14,600 14,40013,400 12,900 12,700 12,65013,400 12,900 12,700 12,65014,900 14,600 14,550 14 ,50014,900 14,600 14,550 14,50012,250 11,900 11,400 11,05011,500 11,150 10,900 10,80011,400 11,050 10,650 10,40013,150- 12,800 12,700 12,65011,600 10,600 9 8 5 0 9 5 0 014,200 14,100 14,050 l$,OOO15,500 15,050 14,600 14,40015,500 15,050 14,600 14,40013,400 12,900 12,700 12,65013,400 12,900 12,700 12,hSO14,900 14,600 14,550 14,50014,900 14,600 14,550 14,500

SA-268 TP430SA-312 TP304SA-312 TP304SA-213 TP309SA-312 TP310SA-312SA-312SA-312SA-312SA-312SA-376SA-376SA-376SA-376SA-376

WeldedSA-249

TP310TP321TP347TP316TP317TP304TP304TP321TP347TP316

TP304

18 Cr-8 Ni18 Cr-10 Ni-Ti

. 18 Cr-10 Ni-Cb

. . 1 6 G-13 N i - 3 MO

. . 18 Cr-8 Ni 16,00016,00016.00016;OO016,00016,00016,00016,00012,75012,75012,75016.00016;OO016,00016,00016,00016,00016.00016100016,000

SA-249 TP304SA-249 TP310SA-249 TP310SA-249 TP321SA-249 TP347SA-249 TP316SA-249 TP317SA-268 TP405SA-268 TP410SA-268 TP430SA-312 TP304SA-312 TP304SA-312 TP309SA-312 TP310SA-312 TP310SA-312 TP321SA-312 TP347SA-312 TP316SA-312 TP317

18 Cr-8 Ni. . . 25 G-20 Ni. . . 25 Cr-20 Ni. . . 18 G-10 Ni-Ti. . . 18 G-10 Ni-Cb

14,45014.450

. . . 16 Cr -13 Ni -3 MO

. . . 18 Cr -13 Ni -4 MO

. . 12 G-Al

. . . 13 Cr

. 16 Cr

. . . 18 Cr--8 Ni

. . . 18 G-8 Ni

15;20075,000 (4)60.000 (4)

16,000 15,20012.750 12.500

60;OO0 id60,000 (4) (9)75,000 (1) (4)

75,00075,000 ii;75,000 (2) (4)75,000 (3) (4)

12;30012,75014,45014,150

11;90012,00013.60012;750

. 25 Cr-12 Ni25 G-20 Ni25 Cr-20 Ni

16,000 14,75016.000 15.7501 6 ; O O O 15;75016,000 14,45016,000 14,45016,000 15,20016,000 15,200

18 Cr-10 Ni-Ti 75,000 (418 Cr-10 Ni-Cb 75.000 (4)

. 16 G--i3 Ni-3 MO

. . . 18 Cr-13 Ni-4 Mo75;ooo75,000

f;j

Page 352: Process Equipment Design

Allowable Stresses

Mechan [ical Elngineerrs)

343

for High-alloy Steel, in Pounds per Square Inchsure Vessels, with Permission of the Publisher, the American Society ofNot Exceeding Dee F

1200 1500700 750 800 850 900 950 1000 1050 1100 1150 1250 1300 1350 1400 1450

14,800 14,700 14,550 14,300 14,000 13,400 12,50010,800 10,400 10,000 9700 9400 9100 880014,800 14,700 14,550 14,300 14,100 13,850 13,50014,800 14,700 14,550 14,300 14,100 13,850 13,50016,400 16,200 15,700 14,900 13,800 12,500 10,50016,600 16,250 15,700 14,900 13,800 12,500 11,00016,600 16,250 15,700 14,900 13,800 12,500 11,00017,000 16,900 16,750 16,500 16,000 15,100 14,00013,400 13,100 12,750 12,100 11,000 8800 640013,900 13,500 13,100 12,500 11,700 9200 650014,800 14,700 14,550 14,300 14,100 13,850 13,50013,900 13,500 13,100 12,500 11,700 9200 650017,000 16,900 16,750 16,500 16,000 15,100 14,00012,450 11,800 11,000 10,100 9100 8000 400014,800 14,700 14,550 14,300 14,000 13,400 12,50010,800 10,400 10,000 9700 9400 9100 880014,800 14,700 14,500 14,300 14,100 13,850 13,500

10,000 7500 57508500 7500 5750

13,100 12,500 800013,100 12,500 80008500 6500 50009750 8500 72507100 5000 3600

12,200 10,400 85004400 2900 17504500 3200 2400

13,100 12,500 80004500 3200 2400

12,200 10,400 8500

ld,bbO %I0 5?sb8500 7500 5750

13,100 12,500 8000

45004500

3250 2450 1800 1400 10003250 2450 1800 1400 10003600 2700 2000 1550 12003600 2700 2000 1550 12002900 2300 1750 1300 9004750 3500 2350 1600 11001450 750 450 350 2505300 4000 3000 2350 1850

. . . . .

3600 2;oo

5300 4bo.o

3250 i4503250 24503600 2700

.

2b00

3iOfl

. . .

i5sb

2350

i&o 1.4.601800 14002000 1550

. .

1200

1850

i&i010001200

750750

10001000750750200

1500. .

1bbo

lso0.,. .750750

1000

50003800

25006800

14,800 14,700 14,550 14,300 14,000 13,400 12,50010,800 10,400 10,000 9700 9400 9100 880014,800 14,700 14,550 14,300 14,100 13,850 13,50014,800 14,700 14,550 14,300 14,100 13,850 13,50017,000 16,900 16,750 16,500 16,000 15,100 14,00016,600 16,250 15,700 14,900 13,800 12,500 11,00016,600 16,250 15,700 14,900 13,800 12,500 11,00012,450 11,800 11,000 10,100 9100 8000 400012,500 12,250 11,950 11,600 11,000 8800 640011,850 11,500 11,100 10,600 10,000 9200 650014,800 14,700 14,550 14,300 14,000 13,400 12,50010,800 10,400 10,000 9700 9400 9100 880016,400 16,200 15,700 14,900 13,800 12,500 10,50016,600 16,250 15,700 14,900 13,800 12,500 11,00016,600 16,250 15,700 14,900 13,800 12,500 11,00014,800 14,700 14,550 14,300 14,100 13,850 13,50014,800 14,700 14,550 14,300 14,100 13,850 13,50017,000 16,900 16,750 16,500 16,000 15,100 14,00017,000 16,900 16,750 16,500 16,000 15,100 14,00014,800 14,700 14,550 14,300 14,000 13,400 12,50010,800 10,400 10,000 9700 9400 9100 880014,800 14,700 14,550 14,300 14,100 13,850 13,50014,800 14,700 14,550 14,300 14,100 13,850 13,50017,000 16,900 16,750 16,500 16,000 15,100 14,000

12,600 12,500 12,400 12,150 11,900 11,400 10,6009200 8850 8500 8250 8000 7750 7500

14,100 13,800 13,350 12,700 11,700 10,600 935014,100 13.800 13,350 12,iO0 11,700 10,600 935012,600 12,500 12,350 12,150 12,000 11,800 11,50012,600 12,500 12,350 12,150 12,000 11,800 11,50014,450 14,350 14,250 14,000 13,600 12,800 11,90014,450 14,350 14,250 14,000 13,600 12,800 11,90010,600 10,000 9350 8600 7750 6800 340010,650 10,400 10,150 9850 9350 7500 545010,100 9800 9450 9000 8500 7900 550012,600 12,500 12,400 12,150 11,900 11,400 10,6009200 8850 8500 8250 8000 7750 7500

13,950 13,800 13,350 12,700 11,700 10,600 890014,100 13,800 13,350 12,700 11,700 10,600 935014,100 13,800 13,350 12,700 11,700 10,600 935012,600 12,500 12,350 12,150 12,000 11,800 11,50012,600 12,500 12,350 12,150 12,000 11,800 .l1,50014,450 14,350 14,250 14,000 13,600 12,800 11,900

10,000 7500 5750 4500 3250 2450 1800 1400 1000 7508500 7500 5750 4500 3250 2450 1800 1400 1000 750

13,100 12,500 8000 5000 3600 2700 2000 1550 1200 100013,100 12,500 8000 5000 3600 2700 2000 1550 1200 100012,200 10,400 8500 6800 5300 4000 3000 2350 1850 15009750 8500 7250 6000 4750 3500 2350 1600 1100 7507100 5000 3600 2500 1450 750 450 350 250 200

iibo 29bO 17504500 3200 2400

10,000 7500 57508500 7500 57508500 6500 50009750 8500 72507100 5000 3600

13,100 12,500 800013,100 12,500 800012,200 10,400 850012,200 10,400 850010,000 7500 57508500 7500 5750

13,100 12,500 800013,100 12,500 800012,200 10,400 8500

loo017504500450038006000250050005000

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

680045004500500050006800

3250 2.45b 1800 i4tio iGo 7503250 2450 1800 1400 1000 7502900 2300 1750 1300 900 7504750 3500 2350 1600 1100 7501450 750 450 350 250 2003600 2700 2000 1550 1200 10003600 2700 2000 1550 1200 10005300 4000 3000 2350 1850 15005300 4000 3000 2350 1850 15003250 2450 1800 1400 1000 7503250 2450 1800 1400 1000 7503600 2700 2000 1550 1200 10003600 2700 2000 1550 1200 10005300 4000 3000 2350 1850 1500

8500 6400 4900 3800 2750 2100 15507200 6400 4900 3800 2750 2100 1.5508300 7200 6150 5100 4050 3000 20006000 4250 3050 2100 1250 650 400

11,100 10,600 6800 4250 3050 2300 170011,100 10,600 6800 4250 3050 2300 170010,400 8850 7200 5800 4500 3400 255010,400 8850 7200 5800 4500 3400 2550

ii%0 iii0 l&lb bib3800 2700 2050 15008500 6400 4900 38007200 6400 4I)oo 38007200 5500 4250 32508300 7200 6150 51006000 4250 3050 2100

11,100 10,600 6800 425011,100 10,600 6800 425010,400 8850 7200 5800

. . . . . . .

. . . . . . .

2750 2ibb lssb2750 2100 15502450 1950 15004050 3000 20001250 650 4003050 2300 17003050 2300 17004500 3400 2550

120012001350300

1300130020002000

. . .. . .

Go120011001350300130013002000

850 650850 650950 650200 150

1000 8501000 8501550 13001550 1300

. . . . .. . . . . .

ii0 '850850 650750 650950 650200 150

1000 8501000 8501550 1300

14,450 14,350 14,250 14,000 13,600 12,800 11,900 10,400 8850 7200 5800 4500 3400 2550 2000 1550 1300

Page 353: Process Equipment Design

344 Allowable Stresses

Item 4. Maximum Allowable Stress Valutis in Tension

Spec. For Meta! Temperatures

Min - 2 0 toTensile Notes 100 200 300 400 500 600 650

Material andSpecification

Number

ForgingsSA-182SA-182SA-182SA-182SA-182SA-182SA-182SA-336SA-336SA-336SA-336SA-336SA-336SA-336

CastingsSA-351SA-351SA-351SA-351SA-351SA-351SA-351SA-35 1SA-351

BoltingsSA-193SA-193SA-193SA-193SA-193SA-320

Grade ‘beNominal

Composition

F6F304F 3 0 4F321F347F 3 1 6F 3 1 0F 6F8F8FatF8cF8mF 2 5

410 13 Cr304 18 C r - 8 Ni304 18 C r - 8 Ni321 18 C r - 8 Ni-Ti347 18 C r - 8 Ni-Cb316 18 Cr-8 Ni-3 MO310 25 Cr-20 Ni410 13 Cr304 18 Cr-8 Ni304 18 Cr-8 Ni321 18 Cr-8 Ni-Ti347 18 Cr-8 Ni-Cb316 18 C r - 8 N i - 3 MO310 25 Cr-20 Ni

CA15CF8CF8CF8MCF8MCF8CCF8CCH20CK20

B6BatB8cB8B8F

(8 grades)

iii21,25018,750

. . . 18,750

. . . 18,750

. . . 18,750

20,400 19,750 19,000 18,500 18,100 17 ,90017,000 16,000 15,450 15,100 14,900 14,85016,650 15,000 13,650 12,500 11,600 11 ,20018,750 17,000 15,800 15,200 14,900 14 ,85018,750 17,000 15,800 15,200 14,900 14 ,85018,750 17,900 17,500 17,200 17,100 17,05023,750 23 ,750 23 ,200 22,400 21 ,500 20 ,85018,100 17,500 16,900 16,400 16,000 15,70017,000 16,000 15,450 15,100 14,900 14 ,85016,650 15,000 13,650 12,500 11,600 11 ,20018,750 17,000 15,800 15,200 14,900 14 ,85018,750 17,000 15,800 15,200 14,900 14,85018,750 17,900 17,500 17,200 17,100 17 ,05023,750 23 ,750 23,200 22,400 21,500 20 ,850

22,500 22,500 22,500 22,500 22,000 21 ,60016,500 15,600 15,000 14,600 14,350 14 ,200

. . . 18,750

. . . 23,750

. . . 13 Cr--fs M O

. . . 18 Cr-8 Ni

. . . 181750(1) 18,750. . . 18,750. . . 18,750. . . 18,750. . . 18,750. . . 23,750

(6) 22,500(1) (6) 17,500

. . . 18 Cr-8 Ni

. . . 18 Cr-9 Ni-2)s MO. . . 18 Cr-9 Ni--235 MO. . . 18 Cr-9 Ni-Cb. . . 18 Cr-9 Ni-Cb

70,00070,000

17,50017,50017,50017,50017,500

15,700 14,25016.900 16,500

13,10016,40016,30016,10014.200

12,20016,350

11,70016,30015,35014,70012.200

11 ,50016 ,25015,00014 ,20011 ,90014 ,40014 ,400

70;ooo70,00070,00070,00065,000

16;900 16;500 15;90017,100 16,60017.000 15.600

15,50013.000

. . . 25 Cr-13 Ni

. . . 25 Cr-20 Ni

416 13 Cr321 18 Cr-8 Ni-Ti347 18 Cr-8 Ni-Cb v304 18 Cr-8 Ni303 18 Cr-8 Ni

17;500 16;lOO 15;150 141600 14;550 14;45016,250 15,300 14,900 14,600 14,550 14,450

20,000 19,300 18,700 18,300 17,850 17,000 16 ,50015,000 13,600 12,650 12,200 11,900 11 ,85015,000 13,600 12,650 12,200 11,900 11 ,85013,300 12,000 10,900 10,000 9300 8 9 5 0

. .

75,bbo75,00075,00075,000

15,00015.00015;ooo15,000

Notes: The stress values in this table may be interpolated to determine values for intermediate temperatures.All stress values in shear are 0.80 times the values in the above table.All stress values in bearing. are+60 times the values in the above table.‘(1) At temperatures of from 200 F through 1050 F these stress values meet all criteria specified for establishing stress values,

except that they exceed 62;s ‘$ZO but do not exceed 90 y0 of the yield strength at temprature. They may be used where slightly greaterdeformation is not objectionable.

(2) These stress values at temperatures of 1050 F and above should be used only when assurance is provided that the steel has apredominant grain size not finer than ASTM No. 6.

(3) These stress values shall be considered basic values to be used when no effort is made to control or check the grain size of thesteel

Page 354: Process Equipment Design

Allowable Stresses 345

for High-alloy Steel, in Pounds per Square Inch (Continued)

Not Exceeding Deg F L

700 750 800 850 900 950 1000 1050 1100 1150 1200 1250 1300 1350 l-E00 1450 1500 ,

17,500 17,050 16,300 14,000 11,000 8800 6400 4400 2900 1750 100014,800 14,700 14,550 14,300 14,000 13,400 12,500 10,000 7500 5750 4500 3250 2450 1800 lb00 1000 75010,800 10,400 10,000 9700 9400 9100 8800 8500 7500 5750 4500 3250 2450 1800 1400 1000 75014,800 14,700 14,550 14,300 14,100 13,850 13,500 13,100 12,500 8000 5000 3600 2700 2000 1550 1200 100014,800 14,700 14,550 14,300 14,100 13,850 13,500 13,100 12,500 8000 5000 3600 2700 2000 1550 1200 10001 7 , 0 0 0 1 6 , 9 0 0 1 6 , 7 5 0 1 6 , 5 0 0 1 6 , 0 0 0 1 5 , 1 0 0 1 4 , 0 0 0 1 2 , 2 0 0 1 0 , 4 0 0 8 5 0 0 6 8 0 0 5 3 0 0 .4000 3 0 0 0 2 3 5 0 1 8 5 0 1 5 0 020,000 18,500 17,000 15,500 14,000 12,500 11,000 9750 8500 7250 6000 4750 3500 2350 1600 1100 75015,400 15,100 14,650 14,000 11,000 8800 6400 4400 2900 1750 100014,800 14,700 14,550 14,300 14,000 13,400 12,500 10,000 7500 5750 4500 3250 2450 1800 1400 1000 ‘75010,800 10,400 10,000 9700 9400 9100 8800 8500 7500 5750 4500 3250 2450 1800 1400 1000 75014,800 14,700 14,500 14,300 14,100 13,850 13,500 13,100 12,500 8000 5000 3600 2700 2000 1550 1200 100014,800 14,700 14,550 14,300 14,100 13,850 13,500 13,100 12,500 8000 5000 4600 2700 2000 1550 1200 10001 7 , 0 0 0 1 6 , 9 0 0 1 6 , 7 5 0 1 6 , 5 0 0 1 6 , 0 0 0 1 5 , 1 0 0 14,,000 1 2 , 2 0 0 1 0 , 4 0 0 8 5 0 0 6 8 0 0 5 3 0 0 4 0 0 0 3 0 0 0 2 3 5 0 1 8 5 0 1 5 0 020,000 18,500 17,000 15,500 14,000 12,500 11,000 9750 8500 7250 6000 4750 3500 2350 1600 1100 750

20,700 19,600 18,300 16,000 11,000 7600 5000 3300 2200 1500 100014,050 13,850 13,600 13,350 13,000 12,600 12,100 9600 7500 5750 4500 3250 2450 18bb 14;O 1000 75011,300 11,100 10,900 10,650 10,400 10,100 9850 9600 7500 5750 4500 3250 2450 1800 1400 1000 7.5016,200 16,100 15,900 15,500 15,000 13,500 12,000 10,600 9400 8000 6800 5300 4000 3000 2350 1850 150014#,700 14,350 14,000 13,500 13,000 12,350 11,700 10,600 9400 8000 6800 5300 4000 3000 2350 1850 150013,700 13,300 12,900 12,600 12,300 11,900 11,600 11,200 10,800 8000 5000 3600 2700 2000 1550 1200 100011,700 11,600 11,500 11,350 11,200 11,100 11,000 10,900 10,800 8000 5000 3600 2700 2000 1550 1200 100014,350 14:300 14,150 13,900 13,500 12,500 10,500 8500 6500 5000 3800 2900 2300 1750 1300 900 75014,350 14,300 14,150 13,900 13,500 12,500 11,000 9750 8500 7250 6000 4750 3500 2350 1600 1100 7.50

15,750 14,900 13,800 12,500 11,0001 1 , 8 0 0 1 1 , 7 5 0 1 1 , 6 5 0 1 1 , 4 5 0 1 1 , 3 0 0 l;,ibO 10,k‘OO 1 0 , 5 0 0 1 0 , 0 0 0 8000 SO00 3 6 0 0 2700 sion 1550 1 2 0 0 1 0 0 011,800 11,750 11,650 11,450 11,300 11,100 10,800 10,500 10,000 8000 5000 3600 2700 2000 1550 1200 1000

8650 8300 8000 7750 7500 7250 7050 6800 6300 5750 4500 3250 24,50 1800 1100 1000 i50. .

. . . . .‘(4) These stress values are the basic values multiplied by a joint-efficiency factor of 0.85.(5) These stress values are established from a consideration of strength only and will be satisfactory for average service. For bolted

joints where freedom from leakage over a long period of time without retightening is required, lower stress values may be necessary asdetermined from the flexibility of the flange and bolts and corresponding relaxation properties.

(6) To these stress values a quality factor shall be applied (see ASME code).(7) These stress values permitted for material that has been carbide-solution treated.(8) For temperatures below 100 F, stress values equal to 20% of the specified minimum tensile strength will be permitted.(9) This steel may be expected to develop embcittlement at room temperature after service at temperatures above 800 F: conse

quently, its use at higher temperatures is not recommended unless due caution is observed.

Page 355: Process Equipment Design

A P P E N D I X

mE‘, TYPICAL TANK SIZES AND CAPACITIES

Item 1. Typical Sizes and Corresponding Approximate Capacities for TanksRecommended by API Standard 12 C

1 2Approx.Capacityper Footof Height

(bbl)14.031.556.087.4

126

(Courtesy of American Petroleum Institute)

3 4 5 6 7 8

Tank Height (ft)

9 10 1 1

TankDiameter

(ft)10152 02 53 0

12 18

2 n

2 4-

3 0 3 6 4 2 4 8 5 4 6 0

s

2 5 05 6 5

1,0101,5702,270

Number of Courses in Comnleted Tankr

1703 8 06 7 0

1,0501,510

4 3

335 4 2 07 5 5 9 4 5

1,340 1,6802,100 2,6203,020 3,780

65 0 5

1,1302,0103,1504,530

9

21350 216903,670 4,2005 , 2 9 0 6,040

4j206,800

10

512507,550

3 5 171 2,060 3,080 4,110 5,140 6,170 7,200 8,230 9 , 2 5 0 10,2804 0 2 2 4 2 , 6 9 0 4,030 5,370 6,710 8,060 9 , 4 0 0 10,740 12,090 13,4304 5 2 8 3 3,400 5,100 6,800 8,500 10,200 11,900 13,600 15,300 17,0005 0 3 5 0 4,200 6 , 2 9 0 8 , 3 9 0 10,490 12,590 14,690 16,790 18,880 20,9806 0 5 0 4 6,030 9,060 12,909 15,110 18,130 21,150 24,170 27,190 30,220

7 0 6 8 5 8,230 12,340 16,450 20,5608 0 8 9 5 10,740 16,120 21,490 26,8609 0 1 1 3 3 13,600 20,390 27,190 33,990

1 0 0 1 3 9 9 16,790 25,180 33,570 41,970120 2 0 1 4 . . 36,260 48,340 60,430

24,680 28,790 32,900 37,010 41,13032,230 37,600 42,970 48,350 53,72040,790 47,590 54,380 61,180 67,98050,360 58,750 67,140 75,540 83,93072,510 84,600 9 6 , 6 9 0 108,800 120,900

1 4 0 2742 . . .160 3 5 8 1 . .180 4 5 3 2 . . .2 0 0 5 5 9 5 . . .2 2 0 6 7 7 0 * . .

49,350 65,800. . . .

. . . . . .

. . . . . .

. . . . . .

82,250 98,700 115,100107,400 128,900 150,400136,000 163,200 190,400167,900 201,400 235,000203,100 243,700 284,400

131,600171,900217,500268,600322,300D = 219

148,000193,400244,800284,500D = 194

164,500214,900254,300D = 174

with 72-in. Butt-welded Courses

The approximate capacities shown are based on the formula:Capacity (42-gal bbl) = 0.14D2H, where D = listed tank diameter and H = listed tank height.

Capacities and diameters below the heavy lines (~01s. 9-11) are maximum for the tank heights shown, on the basis of thelx-in. maximum permissible thickness of shell plates and the maximum allowable design stresses.

346

Page 356: Process Equipment Design

Typical Tank Sizes and Capacities 347

1

TankDiam(ft)

1015202530

3540455060

708090

100120

140160180200220

Item 2. Shell Plate Thicknesses for Typical Sizes of Tanks with 72-in. Butt-welded Courses

Recommended by API Standard 12 C

(Courtesy of American Petroleum Institute)

9 10 1 1

48 5- E 60

2 3 4 5 6 7 8Tank Height (ft)

6 12 18 24 30 36 42Number of Courses in Completed Tank- . n ” 4 5 6 7

Shell Plate Thickness (In.)8 9 10z .s

346 946 %S %6

x6 346 336 %6

946 % 6 %6 336

%S 356 %6 x6

x6 346 346 946

3/i 6 94 63’i6 0.190.19 0.21

0'. 20 0'.220.24 0.26

12MaximumAllowableHeight forDiametersListed (ft)

. . .

3’i6 %6 %6 0.19 0.21 0.24 0.27 0.30 .346 %6 0.19 0.21 0.24 0.28 0.31 0.35346 946 0.19 0.23 0.27 0.31 0.35 0.39!i >/a s/a 0.26 0.30 0.35 0.39 0.43 .+/a ?4 0.26 0.31 0.36 0.41 0.47 0.52 . .

si 0.25 0.30 0.36 0.42 0.48 0.54 0.61 .>I 0.27 0.34 0.41 0.48 0.55 0.62 0.69 . .% 0.31 0.38 0.46 0.54 0.62 0.70 0.78 .0.25 0.34 0.43 0.51 0.60 0.69 0.78 0.86 .0.30 0.41 0.51 0.62 0.72 0.83 0.93 1.03 .

0.350.400.450.500.55

0.470.540.610.670.74

0.600.680.760.850.93

0.720.820.921.021.13

0.840.961.081.201.32

0.961.101.241.37

1.081.241.39

1.211.38..

6i.358.252.547.8

Plate thicknesses shown in item 2 in fractions are thicker than those required for hydrostatic loading but for practical reasonshave been fixed at the values given; therefore, plates for these courses may he ordered on a weight basis. Plate thicknessesshown in item 2 in decimals are based on maximum allowable stresses, and therefore plates for these courses must be orderedon a thickness basis.

In deriving the plate-thickness values shown, it was assumed, on the basis of average mill practice, that the edge thicknessof plates 72-in. wide and ordered on the weight basis would underrun the nominal thickness by 0.03 in. The actual thicknessmay underrun a calculated or specified thickness by 0.01 in.; consequently, fractional thickness values are shown only when thefractional value exceeds the calculated thickness of the course in question by more than 0.02 in.

The maximum allowable height for diameters listed in feet is based on the I!,$-in. maximum permissible thickness of shellplates and the maximum allowable design stresses.

Page 357: Process Equipment Design

348 Typical Tank Sizes and Capacities

Item 3. Typical Sizes and Corresponding Approximate

Capacities for Tanks with 96-in. Butt-welded Courses

Recommended by API Standard 12 C

(Courtesy of American Petroleum Institute)

1 2 3 4 5 6 7 8 9APPrOX.d&30-ity per ~~~ Tank Height (ft)

Tank Foot of 1 6 2 4 3 2 4 0 48 5 6 6 4Diam Height Number of Courses in Completed Tankw Wl) 2 3 T- 5 6 7 a

1 0 14.0 2 2 5 335 450 . . .15 3 1 . 5 505 755 1 , 0 1 0 1 , 2 6 02 0 5 6 . 0 900 1,340 1,790 2,240 2,690 : : : : : :25 87.4 1 , 4 0 0 2 , 1 0 0 2 , 8 0 0 3,500 4,200 4,900 5,6003 0 126 2,020 3.020 4,030 5,040 6,040 7,050 8 , 0 6 0

35 171 2,740 4,110 5 , 4 8 0 6 , 8 5 0 8 , 2 3 0 9,600 10,96040 2 2 4 3 , 5 8 0 5 , 3 7 0 7,160 a.950 10,740 12,530 14,32045 2 8 3 4,530 6 , 8 0 0 9,060 11,330 13,600 1 5 , 8 6 0 1 8 , 1 3 050 350 5,600 a .390 1 1 , 1 9 0 1 3 . 9 9 0 1 6 , 7 9 0 1 9 , 5 8 0 2 2 , 3 8 060 504 8,060 12,090 16,120 20,140 24,170 28,200 32,230

70 6 8 5 10,960 16,450 21,930 27,420 32,900 38,380 43,87080 8 9 5 1 4 , 3 2 0 2 1 , 4 9 0 2 8 . 6 5 0 3 5 . 8 1 0 4 2 , 9 7 0 5 0 . 1 3 0 5 7 . 3 0 09 0

1 0 01 2 0

1 4 01 6 01802002 2 0

1133 1 8 , 1 3 0 2 7 , 1 9 0 36;260 45;320 5 4 , 3 9 0 63;450 72;5201399 22,380 3 3 , 5 7 0 4 4 , 7 6 0 5 5 , 9 5 0 6 7 , 1 4 0 7 8 , 3 4 0 8 9 , 5 3 02 0 1 4 . . . 58,340 64,460 80,580 96,690 112 ,800 128 ,900

2742 . . . 65,800 87,740 109,700 131,600 153,500 175,5003581 . . . . . 114 ,600 143 ,200 171 ,900 200 ,500 229 ,2004532 145,000 181.300 217 ,500 253 ,800 238 ,100

5595 179,100 223,800 268,600 274,200 D = 1 6 36770 . . . . . 216,700 270,800 322,300 D = l a 7

D = 219

The capacities in item 3 are based on the formula: Capacity (42-gal bbl) =0.14DzH. where D = listed tank diameter and H = listed tank height,.

Capacities and diameters below the heavy lines (~01s. 7-9) are maximumfor the tank heights shown, on the basis of the l$&in. maximum permissiblethickness of shell plates and the maximum allowable design stresses.

Item 4. Shell Plate Thicknesses for Typical Sizes of

Tanks with 96-in. Butt-welded Courses

Recommended by API Standard 12 C

(Courtesy of American Petroleum Institute)

1 2 3 4 5 6 7 a 9 10Tank Height (ft)_ _ _ ___~~ -. __ Maximum

a 16 24 32 40 48 56 6 4 AllowRblnTank Number of Courses in Completed Tank Height forD i a m 1 2 3 4 5 6 7 a D i a m e t e r s

(W Shell Plate Thickwas (in.) Linted (h)..- -___10 4is 3f6

2;; K63;s .

1 5 Ksk6 34 6

vi6 Hi/is .20 ?‘f6 "46 s/is 3fe2 5 346 He se Hn 3i I 3 0.19 0 . 2 0 0 . 2 33 0 %a %6 348 ?f6 n.19 0 . 2 1 0 . 2 4 0.2a _..

3 5 %e %a %a 0.19 u.20 0 . 2 4 0.28 0 . 3 34 0 Ne vi.5 346 0 . 1 9 0 . 2 3 0 . 2 8 0.32 0 . 3 7 .,,4 5 X6 416 0.19 0.21 0 . 2 6 0 . 3 1 0 . 3 6 0 . 4 250 % % % 0 . 2 5 0 . 2 9 0 . 3 5 0 . 4 0 0 . 4 6 .,_60% % 5’4 0 . 2 7 0 . 3 4 0 . 4 1 0.48 0 . 5 5

70 % s/a 0.25 0 . 3 2 0 . 4 0 0 . 4 8 0 . 5 6 0 . 6 5 . .80 5-i 3i 0.2i 0.3i 0 . 4 6 0 . 5 5 0 . 6 4 0 . 7 4 . . .90 % % 0.31 0 . 4 1 0 . 5 2 0 . 6 2 0 . 7 2 0.83 _..

100 4/a 0 . 2 5 0 . 3 4 0 . 4 6 0 . 5 7 0 . 6 9 0 . 8 0 0 . 9 2 .1 2 0 )/+ 0 . 2 7 0 . 4 1 0.5s 0 . 6 9 0 . 8 3 0.97 1.10 ..,

1 4 0 % 0 . 3 1 0 . 4 7 U . 6 4 0 . 8 0 0 . 9 6 1 . 1 3 1 . 2 9160 Pi 0 . 3 5 0 . 5 4 0 . 7 3 0 . 9 1 1 . 1 0 !.29 1 . 4 7 6 5 3ia0 )/a 0 . 4 0 0 . 6 1 0.82 1 . 0 3 1 . 2 4 1.45 . . . 58.22 0 0 >i 0 . 4 4 0 . 6 7 0 . 9 1 1 . 1 4 1 . 3 7 . . . . . . 5 2 . 5220 0 . 2 5 0 . 4 8 O.i4 1.00 1.25 . . . . . . . . 47.8

Page 358: Process Equipment Design

A P P E N D I X

SHELL ACCESSORIES

Item 1. Shell Nozzle Dimensions in Inches as a Function of Nozzle Size, Recommended by API Standard 12 C-

Use with Item 2 and Fig. 3.14

(Courtesy of American Petroleum Instit.ute)

4 3 6 7 8 9 10Diameter Length of Side Distance, Distance, Distance fromof Hole in of Width of Shell to Shell to Rottom of Tank toReinforcing Heinforcine Reinforcing Flange Flange Center of Nozzle

Plate .DR

20361836163614341x410%8%634G43 %2%2

PlateI.

43393 53 12819241,~2O’a16’i1 210

PlateW

52jg4 7 %4mi383 53ojg2 520>/4153612%

LowType C

213519%1734153514%12>/410368%65

Face,Outside, ./

IO10101 01010

886666

Face,Inside, h-

888888666666

RegularType H

2422201 8171 51 31 1

9876

3 %3

1 2 3Flanged Nozzle.

MinimumSizeof

.A ozzle2 0181 61 412IO

8613‘7 *1J “i*

ODof

Pipe2 01 81 614w410%8%6%4%3 %-wfl1.90

Pipe-wallThickness1

I1See item 2, WI. 2

0.500.500.500.4320.3370.3000.2180.200

St37161

N

4.0002.8752.2001.5761.313

CouplingCouplingCouplingCouplingCoupling

1 0 12%. . .. . .. . .

. . .

. . .

. . .

. . .

. . .

. . .

* Flanged nozzles 2 m. and 1 j ,\n. in diameter do not require reinforcing plates.*\shell plate, and weld A will be&given in Appendix E, item 3, col. 7.

DR will be the diameter of the hole in theReinforcing plates may be used if desired.

t Screwed nozzles 3 in. in diameter require a reinforcing plate, the details for which shall be the same as shown for 3-in.flanged nozzles. Reinforcing plates may be used on the smaller fittings if desired.

$ Extra-strong pipe, API Std. 5L., made from formed plate, electrically butt welded, may be substituted.349

Page 359: Process Equipment Design

350 Shell Accessories

Item 2. Shell Nozzle Dimensions in Inches as a Function of Shell Plate Thickness, Recommended by API Standard

12C-Item 1 and Fig. 3.14

(Courtesy of American Petroleum Institute)

Shell Thickness (t)and Reinhcing-plate Thickness

CT)*

‘346

1 ‘i 61%13461 r/41x6

1%1x6

1%

20” 18” 16” and14” Flanged Nozzle,Minimum Pipe-wall

Thickness (n) t35?d?4

‘%6N5%

Diameter of Hole in ShellPlate, D,, Equals OD of Pipe

nlus the Following ValuesFor Max D, For Min D,Add to OD Add to OD(Weld A in (Weld A in

Shop) Field)3.6 94I% N.% NNNK‘3’i6

‘546

‘3’i6

l?‘i6,

lx6

l3i6

1%l?iIsi1%1%1%1%1%1%1%

746

% 6

%6

%6

?‘i6

K ‘346

N 1N 13’i6

K 136N 1x6

I%6 1%‘946 1%6

‘?‘f6 1%‘?‘iS 1x6

‘%6 1%

Size of FilletWeld B

946

Ji%6

347’i6

w%6

‘>iS

N‘946

?4

Size of FilletWeld A for

Nozzles Largerthan 2”

Size of FilletFVeld A forNozzles 2”,

l>h”, l”,and x”

* If a thicker shell plate is used than is required for the hydrostatic loading, the excess shell-plate thickness may be con-sidered as reinforcement, and the thickness of reinforcing plate (T) decreased accordingly.

t Based on API Std. 5L for pipe of >s-in. wall thickness; for pipe of over >$in. wall thickness, use ASTM A-53, A-135, orA-139 of latest issue. Pipe made from formed plate, electrically butt welded, may be substituted for any of the above-mentioned pipe sections.

Item 3. Shell-manhole Cover-plate Thickness and BoltiAg-

flange Thickness, Recommended by API Standard 12 C-

See Item 4 and Fig. 3.15

(Courtesy of American Petroleum Institute)

MaxTank

Height,(ft)2 03 55 47 9

Equiva- Cover-platelent Thickness,

Pres- Min (in.)sure* 20-in. 24-in.(psi) Manhole Manhole8.7 946 N

15.2 M w23.4 w %i34.2 36 s/a

* Based on water loading.

Bolting-flangeThickness after

Finishing,Min (in.)

ZO-in. 24-in.Manhole Manhole

34 siPi N

wN

Page 360: Process Equipment Design

Shell Accessories 351

Item 4. Dimensions in Inches for 20-in. Shell Manhole, Recommended by API Standard 12 C-See Item 3 and Fig. 3.15

(Courtesy of American Petroleum Institute)Shell

Thicknessand ManholeAttachment

FlangeThickness

t, TX6%i316

NX6wNsN‘416

94‘346

76‘546

1l5i6

1%1946

1%1516

1%1x6

1%

Approx.Radius Length

(Applies to o fSize of Fillet Formed Type Side

Weld A Weld B Only) R L453445%45%45 344545454544444444433543 3443 3443434235423542 3442%4235

Width ofReinforc-ing Plate

W54jq54%54545335533453%53%5251%51%51355 159%50%50504949494949

MaxDiam

of Holein Shell*

DP2455243424%243524%24%2525jq25>/,253525 34262626>i26352635263526%27272727

Inside Diam of Man-hole Frame

Min ID Max ID20 w420 223520 22%20 22ji20 223620 2220 217620 21%20 219420 213520 21%20 21>/420 21>620 2020 .w620 w420 20%20 20)s20 2%20 20%20 203620 20

* Hole in shell may be oval, with horizontal major diameter of 29 in., where necessary for removal of rigid scaffold brackets.

Page 361: Process Equipment Design

3 5 2 Shell Accessories

Item 5. Dimensions in Inches for 24-in. Shell Manhole, Recommended by API Standard 12 C-See Item 3 and Fig. 3.15

(Courtesy of American Petroleum Inst,itute)Shell

Thicknessand Manhole A p p r o x .Attachment Radius Lengt.h

FlangeThickness S ize o f F i l l e t

(Applies to of.__ Formed Type Side

t, T Weld A Weld B Only) R 1,

I

MaxWidth of D i a m Diam of Diam of

ing PlateW

Reinforc- of Hole Inside Diirrn of Mau-in She l l

D,28pi283528;s28 Js283428%29>i2 9 X/4293429N29%3 0 5i30’130f’430>$305430 “53o,a/,30%30%30%3 1

hole FrameMin ID Max ID-

64$i6 46 46 46 36 36 3629,62 ?,$62$,,6 261%6 16 16 06 06 058yd58%5 85 85 8

2 4 26%2 4 26352 4 26%2 4 26$/,2 4 26162 4 2 62 4 25%2 4 25%2 4 25%2 4 25362 4 25%2 4 25>c2 4 2 5 $62 4 2 52 4 24742 4 24N2 4 24%24 24%2 4 24362 4 245.i2 4 24562 4 2 4

CoverPlateD C

32%32%32%32%32%32%32%32%32%32%32%32%32%32%32%32%32%32%32%32%32%32%

The dimensions shown in col. 1 of items 4 and 5, are based on heaving the thickness of the attachment flange, T, and alsothat of the neck for a distance of at least 4T extending outward from the connecting face of the attachment f lange, equal to thethickness of the tank shell, t. If the manhole att,aches to a thicker shel l plate than is required for the hydrostat ic loading, theexcess shell-plate thickness 111ay he considered as reinforcement, and the thickness, T, of the manhole attachment flange maybe decreased accordingly. The entire neck or port ions of the neck may he thinner than the attachment f lange, provided thereis sufficient reinforcement furnished otherwise; however, the neck should not be thinner than the thickness of the shell plateor the a l lowable minimum f inished thickness of the bol t ing f lange, whichever is the smal ler . The finished manhole holting-flange thickness shall not be less than that of the cover plate less 36 in., with a minimum of s/4 in.

Page 362: Process Equipment Design

A P P E N D I X

I PROPERTIES OF

SELECTED ROLLED STRUCTURAL MEMBERSI

Item 1. Channels, American Standard, Properties of Sections

7 .

t

l- t-1

-4;

Axis 2-2s P 2’

(in.3) (in.) (in.)56 1.04 0.885.3 1.06 0.8;5.1 1.09 0.801.9 1.10 0.90

AVg

Area D e p t h Widt,h F l a n g e W e bof of of

Section Channel FlangeT h i c k - T h i c k - ~~~ Axisi-1ness ness I .”

SectionIndex andNominal

Size

C 6018 x 4

R = 0.625

Weightper

Foot(lb)

58.051.945.842.7

(in.2)16.9815.1813.3812.48

C l 50.0 14.6415x3% 40.0 11.70R = 0.50 33.9 9.90

c 2013 x 4

R = 0.48

50.0 14.6640.0 11.7135.0 10.2431.8 9.30

c 2 30.0 8.7912 x 3 25.0 7.32

R = 0.38 20.7 6.03

c 310x2%R = 0.34

30.0 8.8025.0 7.3320.0 5.8615.3 4.47

c 4 20.0 5.869~2% 15.0 4.39

R = 0.33 13.4 3.83

c 5 18.75 5.498 x 2>/4 13.75 4.02

R = 0.32 11.50 3.36

-I

(in.)4.2004.1004.0003.950

(in.) (in.) (in.4) (in. 3, (in.) (in.4)0.625 0.700 670.7 74.5 6.29 18.50.625 0.600 622.1 69.1 6.40 Ii.10.625 0.500 573.5 63.7 6.55 15.80.625 0.450 549.2 61.0 6.64 15.0

3.716 0.650 0.716 401.4 53.6 5.24 11.2 3 83.520 0.650 0.520 346.3 46.2 5.44 9.3 3.43.400 0.650 0.400 312.6 41.7 5.62 8.2 3.2

4.412 0.610 0.787 312.9 48.1 4.62 16.i4.185 0.610 0.560 271.4 41.7 4.82 13.94.072 0.610 0.447 250.7 38.6 4.95 12.5,I. 000 0.610 0.375 237.5 36.5 5.05 11.6

3.170 0.501 0.510 161.2 26.9 4.28 3.23.047 0.501 0.387 143.5 23.9 4.43 4 52.940 0.501 0.280 128.1 21.4 4.61 3.9

3.033 0.436 0.673 103.0 20.6 3.42 C.02.886 0.436 0.526 90.7 18.1 3.52 3 .42.739 0.436 0.379 78.5 15.7 3.66 2.82.600 0.436 0.240 66.9 13.4 3.87 2.3

2.648 0.413 0.448 60.6 13.5 3.22 2.42.485 0.413 0.285 50.7 11.3 3.40 1.92.430 0.413 0.230 47.3 10.5 3.49 1 .8

2.527 0.390 0.487 43.7 10.9 2.82 2.002.343 0.390 0.303 35.8 9.0 2.99 1.502.260 0.390 0.220 32.3 8.1 3.10 1.30

4.94.34.03.9

2.11.91.7

1.71.51.31.2

1.21.00.97

I .oo0.860.79

0.87 0.800.89 0.780.91 0 79

1.07 0.981.09 0.971.10 0.991.11 1.01

0.77 0 680.79 0.680.81 0.70

0.67 0.650.68 0.620.70 0.610.72 0.64

0.65 ti.590.67 0.590.67 0.61

0.60 0.570.62 0.560.63 0.58

(in.)

18

15

13

12

10

9

8

y,.- __YV__)__YYU__Y__IN

353

Page 363: Process Equipment Design

354 Properties of Selected Rolled Structural Members

Item 1. Channels, American Standard, Properties of Sections (Continued)

Area Depth Width Flange Webof of of Thick- Thick-

Section Channel Flange ness nessAxis l-l Axis 2-2

I(in. 2, (in.)

4.323.58 72.85

(in.)

2.2992.1942.090

(in.) (in.) (in. 4,

0.366 0.419 27.10.366 0.314 24.10.366 0.2.0 21.1

SectionIndex andNominal

Size

C 67 x 2JQ

R = 0.31

Weightper

Foot(lb)

14.7512.259.80

13.0010.508.20

9.006.70

s r(in. “) (in .)

7.7 2.516.9 2.596.0 2.72

5.8 2.135 . 0 2 . 2 24.3 2.34

3.5 1.833.0 1.95

I s I-(in.4) (in.3) (in.)

Y(in.)

1.40 0.79 0.57 0.531.20 0.71 0.58 0.530.98 0.63 0.59 0.55

c 76 x 2

R = 0.30

3.81 2.157 0.343 0.437 17.33.07 6 2.034 0.343 0.314 15.12.39 1.920 0.343 0.200 13.0

1.10 0.65 0.53 0.520.87 0.57 0.53 0.500.70 0.50 0.54 0.52

C85X1X

R = 0.292.63 1.885 0.320 0.325 8.81.95 5 1.750 0.320 0.190 7.4

0.64 0.45 0.49 0.480.48 0.38 0.50 0.49

c 94x1s

R = 0.287.255.40

2.12 41.56

1.720 0.296 0.320 4.5 2.3 1.47 0.44 0.35 0.46 0.461.580 0.296 0.180 3.8 1.9 1.56 0.32 0.29 0.45 0.46

c 10 6.00 1.75 1.596 0.2733x1s 5.00 1.46 3 1.498 0.273

R = 0.27 4.10 1.19 1.410 0.273

0.3560.258

2.1 1.4 1.08 0.31 0.27 0.42 0.461.8 1.2 1.12 0.25 0.24 0.41 0.441.6 1.1 1.17 0.20 0.21 0.41 0.44

Item 2. Beams, American Standard, Properties of Sections

2

l- , -1

12

Sectionl’ndex andNominal

Size24” I

WeightperFootWI

Area Depth Widthof of of

Section Beam Flange

AvgFlangeThick-

ness(in.)

WebThick-

ness(in.)

Axis l-l Axis 2-2I Z S

(in. 2, (in.) (in.) (in. 4, (inT3)

250.9234.3

r(in.) (in. 4, (in. 3,

84.9 21.178.9 20.0

P

(in.)

/ B 18/ 24, x 716

R = 0.60

120.0105.9

35.1330.98

2 4 8.0487.875

1,102 0.798 3010.81.102 0.625 2811.5

9.269.53

1.561.60

24” IBl

24 x 7R = 0.60

IIi’ 20” I

B 220 x 7

R = 0.7011 20” I

B320 x 639R = 0.60

100.0 29.25 7.247 0.871 0.747 2371.8 197.6 9.05 58.4 13.4 1.2990.0 26.30 2 4 7.124 0.871 0.624 2230.1 185.8 9.21 45.5 12.8 1.3279.9 23.33 7.000 0.871 0.500 2087.2 173.9 9.46 42.9 12.2 1.36

95.0 27.7485.0 24.80 2 0 7.200

7.0530.916 0.800 1599.7 160.0 7.59 50.5 14.0 1.350.916 0.653 1501.7 150.2 7.78 47.Q 13.3 1.38

75.0 21.9065.4 19.08 2 0 6.391

6.2500.789 0.641 1263.5 126.3 7.60 30.1 9.4 1.170.779 0.500 1169.5 116.9 7.83 27.9 8.9 1.21

18” IB4

18 x 6R = 0.56

70.0 20.4654.7 15.94 18 6.251

6.0000.691 0.711 917.5 101.9 6.70 24.50.691 0.460 795.5 88.4 7.07 21.2

1.091.15

15” IB 7

15 x 535R = 0.51

50.0 14.5942.9 12.49 1 5 5.640

5.5000.622 0.550 481.1 64.2 5.74 16.00.622 0.410 441.8 58.9 5.95 14.6

1.051.08

Page 364: Process Equipment Design

I

SectionIndex andNominal

Size12” IB 8

12 x 5jqR = 0.56

12” IB 9

12 x 5T = 0.45

10” IB 10

.-I +/ 10x495R = 0.41

# -> 8" I--T B 1 2

8x4R = 0.37

7" IB 137x356

R = 0.35

6" IB 146x346

R = 0.33

5" IB 155x3

R = 0.31

4" IB 164x238

R = 0.29

3" IB 173x296

R = 0.27

AwWeight Area

per o fFoot Section(Ills) (in. 2,

50.0 14.5740.8 11.84

Depth Widtho f o f

Beam Flange(in.) (in.)

Axis l-l Axis 2-2r

1

12 5.4775.250

Flange WebThick- Thick-

ness ness(in.) (in.)

0.659 0.6870.659 0.460

(in. 4,

301.6268.9

35.0 10.2031.8 9.26 12 5.078

5.0000.544 0.428 227.0 37.8 4.72 10.0 3.9 0.990.544 0.350 215.8 36.0 4.83 9.5 3.8 1.01

35.0 10.2225.4 7.38 1 0 4.944

4.6600.491 0.594 145.8 29.2 3.78 8.5 3.4 0.910.491 0.310 122.1 24.4 4.07 6.9 3.3 0.97

23.0 6.71 8 4.171 0.425 0.44118.4 5.34 4.000 0.425 0.270

64.2 16.0 3.09 4.4 2.1 0.8156.9 14.2 3.26 3.8 1.9 0.84

20.0 5.83 7 3.860 0.392 0.45015.3 4.43 3.660 0.392 0.250

y.9 12.0 2.68 3.1 1.6 0.7436.2 10.4 2.86 2.7 1.5 0.78

17.25 5.02 3.565 0.359 0.465 26.012.5 3.61 6 3.330 0.359 0.230 21.8

14.7510.0

9.57.7

7.55.7

4.292.87 5 3.284

3.0000.326 0.494 15.0 6.0 1.87 1.7 1.0 0.630.326 0.210 12.1 4.8 2.05 1.2 0.82 0.65

2.762.21 4 2.796

2.6600.293 0.326 6.70.293 0.190 6.0

2.171.64 3 2.509

2.3300.260 0.349 2.90.260 0.170 2.5

Properties of Selected Rolled Structural Members 355

Item 2. Beams, American Standard, Properties of Sections (Continued)

s(in. 3,

50.344.8

P

(in.)

4.554.77

I(in. 4,

16.013.8

s(in.3)

_A/--5.85.3

P

(in.)

1.051.08

8.7 2.28 2.3 1.3 0.687.3 2.46 1.8 1.1 0.72

3.33.0

1.91.7

1.56 0.91 0.65 0.581.64 0.77 0.58 0.59

1.15 0.591.23 0.46

0.470.40

2

0.520.53

Item 3. Wide-flange Light Beams, Stanchions, and Joists, Properties of Sections

I - -1

IlIE

SectionIndex Weight Area Depth Flange Weband per o f o f Thick- Thick- Axis l-l Axis 2-2

Nominal Foot Section Section Width ness ness I s P I sSize (lbs) (in.2) (in.) (in.) (in.) (in.) (in.4) (in.3) (in.) (in.“) (in. 3, (ii.,

Light Beams

CBL 12 22.0 6.47 12.31 4.030 0.424 0.260 155.7 25.3 4.91 4.55 2.26 0.8412 x 4 19.0 5.62 12.16 4.010 0.349 0.240 130.1 21.4 4.81 3.67 1.83 0.81

R = 0.30 16.5 4.86 12.00 4.000 0.269 0.230 105.3 17.5 4.65 2.79 1.39 0.76

Page 365: Process Equipment Design

356 Properties of Selected Rolled Structural Members

Item 3. Wide-flonge light Beams, Stanchions, and Joists, Properties of Sections (Continued)

and perNominal Foot

Size (lbs)CBL 10 19.010 x 4 17.0

R = 0.30 15.0

CBL 88 x 4 15.0

R = 0.30 13.0

SectionIndex Weight Area

o fSection

(im2)

5.614.984.40

Depth FlangeThick-o f

Section Width(in.) (in.)

10.25 4.02010.12 1.01010.00 1.000

Web

ness(in.)

0.3940.3290.269

Thick-ness(in.)

0.2500.2400.230

Axis l-lI s P

(in.)

4.144.053.95

Axis 2-2I s

(in. 4, (in. 3,96.2 18.881.8 16.268.8 13.8

(in.4) (in.“)4.19 2.083.45 1.722.79 1.39

CBL 66 x 4 16.0

R = 0.30 12.0

CBS 6 25.0 7.35 6.37 6.080 0.4566 x 6 20.0 5.88 6.20 6.018 0.367

R = 0.25 15.5 4.59 6.00 6.000 0.269

CB 515 x 5 18.5

R = 0.3 16.C

CBJ 1212 x 4 14.0

R = 0.30

CBJ 1010 x 4 11.5

R = 0.30

CBJ 88 x 4 10.0

R = 0.30

CBJ 66 x 4 8.5

R = 0.25

P

(in.)

0.860.830.80

4.43 8.12 4.015 0.314 0.245 48.0 11.8 3 29 3.30 1.65 0.863.83 8.00 4.000 0.254 0.230 39.5 9.88 3 21 2 62 1.31 0.83

4.72 6.25 4.030 0.404 0.260 31.7 10.1 2.59 4.32 2 .14 0.963.53 6.00 4.000 0.279 0.230 21.7 7.24 2.48 2.89 1.44 0.90

Stanchions

0.3200.2580.240

53.5 16.8 2 69 17.1 5.6 1.5241.7 13.4 2.66 13.3 4.4 1.5030.3 10.1 2.56 9.69 3.2 1.43

5.45 5.12 5.025 0.420 0.265 25.4 9.94 2.16 8.89 3.54 1.284.70 5.00 3.000 0.360 0.240 21.3 8.53 2 13 7.51 3.00 I.26

Joists

4.14 11.91 3.970 0.224 0.200 88.2 .4 61 2.25 1.13 0.74

3.39 9.87 3.950 0.204 0.180 51.9 3.92 2.01 1.02 0.77

2.95 7.90 3.940 0.204

2.50 5.83 3.940 0.194

0.170

0.170

30.8

14.8

10.5

7.79

5.07

3.23 1.99 1.01 0.82

14.8 2 .43 1 89 0.96 0.87

Item 4. Equal Angles, Properties of Sections

Section SizeIndex (in.)

WeightThickness per Foot

(in.) (lb),‘lf$ 56.9

7. 51.0A l xi 45.0

R-H 8 x 8 94 38.9% 32.7Qi6 29.634 26.4

Area ofSection

(in.2)16.7315.0013.2311.449.618.687.75

I(in.4)98.089.079.669.759.454.148.6

Axis l-l and Axis 2-2.- __.-s P

(in.3) (in.)17.5 2.4215.8 2.4414.0 2.4512.2 2.4710.3 2.499.3 2.508.4 2.51

2

(in.)2.412.372.322.282.232.212.19

Axis 3-3r min(in.)I.55J.561.561.571.581.581.58

Page 366: Process Equipment Design

Properties of Selected Rolled Structural Members 3 5 7

Item 4. Equal Angles, Properties of Sections (Continued)

Section SizeIndex (in.)

A 2R=ffl

A 3R=M

A 4R=g4

A 5R=s

6 x 6

5 x 5 ,

4 x 4

3% x 3%

Weight Area ofper Foot Section

(lb) (in. 2,37.4 11.0033.1 9.7328.7 8.4424.2 7.1121.9 6.4319.6 5.7517.2 5.0614.9 4.3612.6 3.66

Axis l-l and Axis 2-2Z s

(in. “) (in. 3,35.5 8.631.9 7.628.2 6.724.2 5.722.1 5.119.9 4.617.7 4.115.4 3.513.0 3.0

f(in.)1.801.811.831.841.851.861.871.881.89

X(in.)1.861.821.781.731.711.681.661.641.61

Axis 3-3r min(in.)1.161.171.171.171.181.181.191.191.19

27.2 7.98 17.8 5.2 1.49 1.57 0.9623.6 ‘6.94 15.7 4.5 1.50 1.52 0.9720.0 5.86 13.6 3.9 1.52 1.48 0.9716.2 4.75 11.3 3.2 1.54 1.43 0.9814.3 4.18 10.0 2.8 1.55 1.41 0.9812.3 3.61 8.7 2.4 1.56 1.39 0.9910.3 3.03 7.4 2.0 1.56 1.36 0.99

18.5 5.44 7.7 2.8 1.19 1.27 0.7715.7 4.61 6.7 2.4 1.20 1.23 0.7712.8 3.75 5.6 2.0 1.22 1.18 0.7811.3 3.31 5.0 1.8 1.23 1.16 0.789.8 2.86 4.4 1.5 1.23 1.14 0.798.2 2.40 3.7 1.3 1.24 1.12 0.796.6 1.94 3.0 1.0 1.25 1.09 0.79

11.1 3.25 3.6 1.5 1.06 1.06 0.689.8 2.87 3.3 1.3 1.07 1.04 0.688.5 2.48 2.9 1.2 1.07 1.01 0.697.2 2.09 2.5 0.98 1.08 0.99 0.695.8 1.69 2.0 0.79 1.09 0.97 0.69

* Special gage.

Item 5. Equal Angles, Properties of Sections

2

3;rlrj+

14+

’ -j-l

T2 ‘3

Set tion SizeIndex (in.)

A 7R = x6 3 x 3

tA 9R=?4 wix2?4

tA 11R=?4

2 x 2

Thickness(in.)

55756

N

516

*E6

Weight Area ofper Foot Section

(lb) (in.2)9.4 2.758.3 2.437.2 2.116.1 1.784.9 1.443.71 1.09

Z(in.4)2.22.01.81.51.20.96

% 7.7 2.25 1.2N 5.9 1.73 0.98%6 5.0 1.47 0.85‘/i 4.1 1.19 0.70946 3.07 0.90 0.55

w 4.7 1.36 0.48%6 3.92 1.15 0.42Y4 3.19 0.94 0.35%6 2.44 0.71 0.28w 1.65 0.48 0.19

Axis l-l and Axis 2-2S r

(in.“) (in.)1.1 0.900.95 0.910.83 0.910.71 0.920.58 0.930.44 0.94

0.73 0.740.57 0.750.48 0.760.39 0.770.30 0.78

0.35 0.590.30 0.600.25 0.610.19 0.620.13 0.63

2(in.)0.930.910.890.870.840.82

Axis 3-3r min(in.)0.580.580.580.590.590.59

0.81 0.470.76 0.480.74 0.490.72 0.490.69 0.49

0.64 0.390.61 0.390.59 0.390.57 0.400.55 0.40

Page 367: Process Equipment Design

3 5 8 Properties of Selected Rolled Structural Members

Item 5. Equal Angles, Properties of Sections (Continued)

Weight Area of Axis l-l and Axis 2-2Section Size Thickness ner Foot Section I sIndex

tA 12R=M

(in.)

1% x 1%

(in.) (lb) (in. 2, (in. “) (in. 3,% 3.99 1.17 0.31 0.2654.5 3.39 1.00 0.27 0.2334 2.77 0.81 0.23 0.19946 2.12 0.62 0.18 0.1434 1.44 0.42 0.13 0.10

iA 15R = x6 1fC x l>i

tA 16 %R = 34 1 x 1 Ns

M* Special gage.t Bar size.

2.34 0.69 0.14 0.131.80 0.53 0.11 0.101.23 0.36 0.08 0.07

1.92 0.56 0.08 0.091.48 0.43 0.06 0.071.01 0.30 0.04 0.05

1.49 0.44 0.04 0.061.16 0.34 0.03 0.040.80 0.23 0.02 0.03

r X

(in.) (in.)0.51 0.570.52 0.550.53 0.530.54 0.510.55 0.48

0.45 0.470.46 0.440.46 0.42

0.37 0.400.38 0.380.38 0.35

0.29 0.340.30 0.320.31 0.30

Thick-Section Size nessIndex (in.) (in.)

N

Weight Arean f Axis l-1

Pi-A 26 x.5

R=H 4 x 3JSN

%A 27

R=s 4x3 ;”

.wAZ8 ?46

R=N 335 x 3 N%6

Axisper Axis 2-2-_ 3-3

Foot Section Z s P X Z s P Y r min(in.) (in.2) (in.“) (in.3) (in.) (in.) (in.4) (in. 3, (in.) (in.) (in.)14.7 4.30 6.4 2.4 1.22 1.29 4.5 1.8 1.03 1.04 0.7211.9 3.50 5.3 1.9 1.23 1.25 3.8 1.5 1.04 1.00 0.7210.6 3.09 4.8 1.7 1.24 1.23 3.4 1.4 1.05 0.98 0.729.1 2.67 4.2 1.5 1.25 1.21 3.0 1.2 1.06 0.96 0.737.7 2.25 3.6 1.3 1.26 1.18 2.6 1.0 1.07 0.93 0.736.2 1.81 2.9 1.0 1.27 1.16 2.1 0.81 1.07 0.91 0.73

13.6 3.98 6.0 2.3 1.23 1.37 2.9 1.4 0.85 0.87 0.6411.1 3.25 5.1 1.9 1.25 1.33 2.4 1.1 0.86 0.83 0.649.8 2.87 4.5 1.7 1.25 1.30 2.2 1.0 0.87 0.80 0.648.5 2.48 4.0 1.5 1.26 1.28 1.9 0.87 0.88 0.78 0.64

546 7.2 2.09 3.4 1.2 1.27 1.26 1.7 0.73 0.89 0.76 0.65*>i 5.8 1.69 2.8 1.0 1.28 1.24 1.4 0.60 0.90 0.74 0.65

10.2 3.00 3.5 1.5 1.07 1.13 2.3 1.1 0.88 0.88 0.629.1 2.65 3.1 1.3 1.08 1.10 2.1 0.98 0.89 0.85 0.627.9 2.30 2.7 1.1 1.09 1.08 1.9 0.85 0.90 0.83 0.626.6 1.93 2.3 0.95 1.10 1.06 1.6 0.72 0.90 0.81 0.635.4 1.56 1.9 0.78 1.11 1.04 1.3 0.59 0.91 0.79 0.63

9.4 2.75 3.2 1.4 1.09 1.20 1.4 0.76 0.70 0.70 0.538.3 2.43 2.9 1.3 1.09 1.18 1.2 0.68 0.71 0.68 0.547.2 2.11 2.6 1.1 1.10 1.16 1.1 0.59 0.72 0.66 0.546.1 1.78 2.2 0.93 1.11 1.14 0.94 0.50 0.73 0.64 0.544.9 1.44 1.8 0.75 1.12 1.11 0.78 0.41 0.74 0.61 0.54

8.5 2.50 2.1 1.0 0.91 1.00 1.3 0.74 0.72 0.75 0.527.6 2.21 1.9 0.93 0.92 0.98 1.2 0.66 0.73 0.73 0.526.6 1.92 1.7 0.81 0.93 0.96 1.0 0.58 0.74 0.71 0.525.6 1.62 1.4 0.69 0.94 0.93 0.90 0.49 0.74 0.68 0.534.5 1.31 1.2 0.56 0.95 0.91 0.74 0.40 0.75 0.66 0.53

* Special gage.

RAc2i6 3% x 2%?‘f6

N

746

A 32 %S

R = 956 3x2~ %946

M

Item 6. Unequal Angles, Properties of Sections

Axis 3-3r min

(in.)0.340.340.340.350.35

0.290.290.30

0.240.240.25

Page 368: Process Equipment Design

Section SizeIndex (in.)

A 33R = ss 3 x 2

Rt”=“i 2% x2

tA 37R=j/ 2x134

tA 645R=% 2 x 1%

tA 39

R=j/, 1% x 1M

* Special gage.t Bar size.

Properties of Selected Rolled Structural Members 359

Item 7. Unequal Angles, Properties of Sections

2

Weight AreanfT h i c k - per

ness Foot(in.) (lb)

3-5 7.7l/i6 6.85% 5.9x6 5.0?4 4.1

*%6 3.07

Section Z(in.2) (in.4)2.25 1.92.00 1.71.73 1.51.47 1.31.19 1.10.90 0.84

N 5.3N6 4.5?4 3.62%6 2.75

%6 3.92% 3.19%6 2.44

%i 2.7794.5 2.12%i 1.44

M 2.55%6 1.96

Y 4 2.34%6 1.8034 il.23

Axis l-1Axis

Axis 2-2 2-2

1.55 0.911.31 0.791.06 0.650.81 0.51

1.15 0.710.94 0.590.72 0.46

0.81 0.320.62 0.250.42 0.17

0.75 0.300.57 0.23

0.69 0.200.53 0.160.36 0.11

s(in.3)1.00.890.780.660.540.41

P

(in.)0.920.930.940.950.950.97

X

(in.)1.081.061.041.020.990.97

1 S(in. 4, (in.3)0.67 0.470.61 0.420.54 0.370.47 0.320.39 0.260.31 0.20

P

(in.)0.550.550.560.570.570.58

Y(in.)0.580.560.540.520.490.47

P min(in.)0.430.430.430.430.430.44

0.55 0.77 0.83 0.51 0.36 0.58 0.58 0.420.47 0.78 0.81 0.45 0.31 0.58 0.56 0.420.38 0.78 0.79 0.37 0.25 0.59 0.54 0.420.29 0.79 0.76 0.29 0.20 0.60 0.51 0.43

0.44 0.79 0.90 0.19 0.17 0.41 0.40 0.320.36 0.79 0.88 0.16 0.14 0.41 0.38 0.320.28 0.80 0.85 0.13 0.11 0.42 0.35 0.33

0.24 0.62 0.660.18 0.63 0.640.13 0.64 0.62

0.320.320.33

0.23 0.63 0.710.18 0.64 0.69

0.150.120.09

0.090.07

0.090.070.05

0.14 0.430.11 0.440.08 0.45

0.10 0.340.08 0.35

0.18 0.54 0.600.14 0.55 0.580.09 0.56 0.56

0.410.390.37

0.330.31

0.350.330.31

0.27 ’0.27

0.10 0.350.08 0.360.05 0.37

0.270.270.27

Item 8. Tees, Equal and Unequal, Properties and Dimensions of Sections

SizeRadius

Weight Area Thickness ofper of Toe Root Fillet Axis l-l Axis 2-2 -

Section Foot Section Flange Stem a bIndex (lb) (in.2) (in.) (in.) (in.) (in.) (iff.) (irf.‘) (iz3) (if;.) (ii.) (ir!.‘) (iz3) (i:.)

Equal Tees

Tl 13.5 3.97 4 4 w %6. % 5.7 2.0 1.20 1.18 2.8 1.4 0.84T 8 7.8 2.27 3 3 N 7i6 %6 1.8 0.86 0.90 0.88 0.90 0.60 0.63

t: ;o 6 . 4 6 . 7 1 . 8 7 1 . 9 5 3 234 3 2% %6 w 5% x6 x6 Pi 1 . 6 1 . 0 0 . 5 9 0 . 7 4 0 . 9 0 0 . 7 4 0 . 8 6 0 . 7 6 0 . 5 2 0 . 7 5 0 . 4 2 0 . 5 0 0 . 6 2 0 . 5 3tT 11 5.5 1.60 2Js 2% x6 94 hi 0.88 0.50 0.74 0.74 0.44 0.35 0.52tT 13 4.1 1.19 .wi 2% % 316 M 0.52 0.32 0.66 0.65 0.25 0.22 0.46tT 14 4.3 1.26 2 2 %6 96 Pi 0.44 0.31 0.59 0.61 0.23 0.23 0.43tT 15 3.62 1.05 2 2 % ?I6 ki 0.37 0.26 0.59 0.59 0.18 0.18 0.42

P- I \ \ \I / -_ -

Page 369: Process Equipment Design

3 6 0 Properties of Selected Rolled Structural Members

Item 8. Tees, Equal ond Unequal, Properties and Dimensions of Sections (Continued)

Size -~Radius

Weight Area Thickness o fper of Toe Root . Fillet Axis l-l Axis 2-2--. -~-~. -

Section Foot Sect ion Flange Stem a b7

Index (lb) (in.2) (in.) (in.) (in.) (in.) (if.) (ilf.‘) (ifs) (ii.) (i:.) (inl.‘) (ii”) (ii.)

Unequal Tees

* *T 50 13.6 4.00 5 3 % $6, '332 346, % % 2.7 1.1 0.82 0.76 5.2 2.1 1.14

11.5 3.37 5 3 94, 1%2 7<6, 36 % 2.4 1.1 0.84 0.76 3.9 1.6 1.10

T 60 11.2 3.29 4 4 % w ws 3 % 6.3 2.0 1.39 1.31 2.1 1.1 0.80T 6 1 9.2 2.68 4 3 w 746 N 2.0 0.90 0.86 0.78 2.1 1.1 0.89T 62 8.5 Z.-E8 4 2pi %I Ks 94 1.2 0.62 0.69 0.62 1.2 1.0 0.92T 79 6.1 1 .x 3 2% %6 94 546 0.94 0.52 0.73 0.68 0.75 0.50 0.65

* Where two dimensions are shown, the first is for t.he flange, the second for the stem.t Bar size.

Item 9. Two Channels, Properties of Sections

Vertical HorizonlalChannel Channel

Size Sizeand and

Weight Weight3"- 4.1 4" 5.4

4"- 5.44"- 3.45"- 6.7

5"- 6.75"- 6.76"- 8.27"- 9.8

6"- 8.2

5"- 6.76"- 8.27"- 9.88"-11.59 ” - 1 3 . 4

10”-15.3

7"- 9.8

6"- 8.27"-- 9.88"-11.59"-13.410"-15.3

8"-11.5

6"- 8.27”- 9.88"-11.59"-13.410"-15.312'-20.7

TotalArea(in.2)2.75

3.123.51

3.904.344.80

4.344.785.245.756.286.86

5.245.706.216.747.32

5.756.216.727.257.839.39

Weightpel

Foot Z

(lb) (ima)9.5 4.5

LO.8 8.812.1 9.7

13.4 16.614.9 17.916.5 19.1

14.9 26.616.4 28.518.0 30.419.7 32.221.6 34.023.5 36.0

18.0 42.819.6 45.521.3 48.023.2 30.725.1 53.2

19.7 61.521.3 65.223.0 68.824.9 72.526.8 75.932.2 83.8

r Yl(in.) (in.)1.28 2.61

1.691.66

3.233.38

2.062.031.99

3.994.164.31

2.472.442.412.372.332.29

4.574.764.935.095.245.37

2.862.822.782.742.70

5.335.525.715.876.03

3.273.243.203.163.112.99

5.886.096.296.476.657.02

Axis X-X~~r TAs1 =.;

1(in. “)1.7

2.72.9

4.24.34.4

3.86.06.26.36.56.7

8.08.28.48.68.8

10.510.710.911.211.411.9

(in. 3,2.3

3.84.1

6.06.56.9

8.49.09.6

10.210.711.1

11.912.813.514.214.9

15.216.317.318.319.221.4

Axis Y-YZ

(in. “)4.0

4.17.7

7.913.521.6

8.113.721.833.048.067.6

14.022.133.34a.367.9

14.322.433.648.668.2

'129 4

s(in.3)2.0

2.13.1

3.24.56.2

3.24.66.28.2

10.713.5

4.76.38.3

10.7.L3.6

,E 86.48.4

10.813.621.6

P

(in.)1.21

1.151.48

1.421.762.12

1.371.692.042.402.763.14

1.691.972.312.683.05

1.581.902.242.592.953.71

Page 370: Process Equipment Design

Properties of Selected Rolled Structural Members 361

VerticalChannel

Sizeancl

\Veight

9”-13.4

I O”-15.3

12”---20 7

15,‘--33.9

18”42.7

HorizontalChannel

Sizeand

Weight7”- 9.88”-11.59”-13.4

lO”-15.312”-20.7

TotalArea(in.2)6.747.257.788.369.92

Weightper

Foot(lb)

I s, =;(in. 3,

r

13.613.914.114.415.0

Axis X-7

& = LY2

(in. 3,20.321.622.824.026.9

r Yl

23.224.926.828.734.1

(in.4)90.295.299.8

104.5115.2

(in.) (in.)3.66 6.643.62 6.853.58 7.063.54 7.253.41 7.66

I(in.4)22.934.149.168.7

229.9

Axis Y-YS

(in. “)6.58.5

10.913.721.6

8”-11.5 7.83 26.8 127.8 17.3 26.3 4.04 7.40 34.6 8.69”-13.4 8.36 28.7 134.2 17.6 27.8 4.01 7.61 49.6 11.0

lO”-15.3 8.94 30.6 140.3 17.9 29.4 3.96 7.82 69.2 13.812”-20.7 10.50 36.0 154.3 18.7 33.0 3.83 8.27 130.4 21.715”-33.9 14.37 49.2 178.3 19.8 40.4 3.52 8.99 314.9 42.0

9”-13.4 9.92 34.1 233.2 27.2 39.9 4.35 8.59 51.2 11.4lo”-15.3 10.50 36.0 243.5 27.6 42.2 .4.82 8.83 70.8 14.212”-20.7 12.06 41.4 267.3 28.6 47.8 4.71 9.35 132.0 22.015”-33.9 15.93 54.6 309.0 30.2 59.7 4.40 10.22 316.5 42.2

lO”-15.3 14.37 49.2 519.0 51.8 68.6 6.01 10.03 75.1 15.012”-20.7 15.93 54.6 568.7 53.6 77.5 5.97 10.60 136.3 22.715”-33.9 19.80 67.8 661.4 56.8 97.8 5.78 11.64 320.8 42.818”-42.7 22.38 76.6 717.6 58.9 106.0 5.66 12.18 557.2 61.9

12”-20.7 18.51 63.4 935.4 76.9 106.5 7.11 12.16 143.1 23.815”-33.9 22.38 76.6 1086.4 81.5 134.6 6.97 13.33 327.6 43.718”-42.7 24.96 85.4 1175.6 84.3 147.0 6.86 13.95 564.0 62.7

Item 9. Two Channels, Properties of Sections (Continued)

Centers of gravity of both channels are in the same vertical line.

r(in.)

1.842.172.512.873.62

2.102.442.783.524.68

2.272.603.314 .46

2.292.934.024.99

2.783.834.75

Page 371: Process Equipment Design

A P P E N D I X

VALUES OF CONSTANT C OF EQ. 13.27*

C = 0.162 for plates rigidly riveted or bolted to shells,flanges, or side plates, as shown in Fig. 13.8~.

C = 0.162 for integral flat heads, as shown in Fig. 13.8b,where dimension d does not exceed 24 in., and the ratioof thickness of the head to dimension d is at least equalto or greater than 0.05.

C = 0.30 for flanged plates attached to vessels, as shown inFig. 13.8c, by means of circumferential lap jointsriveted, welded, or brazed, and meeting all the require-ments therefore, where the corner radius on the insideis not less than three times the thickness of the flangeimmediately adjacent thereto; and for flanged plateswith the same inside-corner radius screwed over theends of vessels, in which the design of the threadedjoint against failure by shear, tension or compression,resulting from the end force due to pressure, is based ona factor of safety of at least 4, and the threaded partsare at least as strong as the threads for standard pipingof the same diameter. Seal welding may be used, ifdesired.

C = 0.25 for heads forged integral with or butt welded tovessels, as shown in Fig. 13.8d and Fig. 13.8e, where thecorner radius on the inside is not less than three timesthe thickness of the flange immediately adjacentthereto, and where the welding meets all the require-ments for circumferential joints given in the code,including those for stress relieving and radiographicexamination; d = G as defined in Chapter 12, inches.

C = 0.50 for plates having a diameter, d, not exceeding 18 in.inserted into vessels and welded thereto, as shown inFig. 13.8i, and otherwise meeting the requirements forthe respective types of welded vessels, including stress

* Extracted from the 1956 edition of the ASME Boiler andPressure Vessel Code, Unfwed Pressure Vessels, with permissionof the publisher, the American Society of Mechanical Engineers.

362

relieving when required for the vessel but omittingradiographic examination. The end of the vesselshall be crimped over not less than 30” nor more than45”. The crimping shall be done cold only when thisoperation will not injure the metal. The throat of theweld shall be not less than the thickness of the flathead.

C = 0.75 for plates screwed into the end of a vessel havingan inside diameter, d, not exceeding 12 in., as shown inFig. 13.8j, or for heads having an integral flangescrewed over the end of a vessel having an insidediameter, d, not exceeding 12 in., where the design ofthe threaded joint against failure by shear, tension, orcompression, resulting from the end force due to pres-sure, is based on a factor of safety of at least 4, and thethreaded parts are at least as strong as the threads forstandard piping of the same diameter. Seal weldingmay be used, if desired.

C = 0.30 for plates inserted into the ends of vessels and heldin place by some suitable positive mechanical-lockingarrangements, such as those shown in Fig. 13.8k andFig. 13.81, where all possible means of failure, either byshear, tension, or compression, due to the hydrostaticend force, are resisted with a factor of safety of 4.Seal welding may be used, if desired.

C = 0.50 for plates welded to the inside of a vessel, asshown in Fig. 13.8f, and otherwise meeting the require-ments for the respective types of welded vessels,including stress relieving when required for the vesselbut omitting radiographic examination. The size ofweld shall be not less than two times the requiredthickness of a seamless shell or less than 1.25 times thenominal shell thickness and shall not be greater thanthe head thickness. The weld shall be deposited in awelding groove with the root of the weld at the innerface of the head, as shown in Fig. 13.8f.

Page 372: Process Equipment Design

Values of Constant C of Eq. 13.27 363

W = bolt load under operating conditions, pounds (seeChapter 12)

ho = radial distance from the bolt-circle diameter tothe diameter d, inches

H = total hydrostatic end force as defined in Chap-ter 12, pounds

C = 0.30 for plates held by set bolts in line with the gasket,as shown in Fig. 13.8m, provided the design of allholding parts against failure by shear, tension, or com-pression, resulting from the end force due to pressure,is based on a factor of safety of at least 4, and threadedjoints, if any, are at least as strong as those for standardpiping of the same diameter.

C = 0.050 for beveled carbon and low-alloy steei plateshaving a diameter, d, not exceeding 18 in. insertedinto shells, pipes, or headers, the ends of which arecrimped over the bevel with the limitations shown inFig. 13.8n. The crimping shall be done when theentire circumference of the cylinder is uniformly heatedto a proper forging temperature for the material used.For this construction the ratio tz/d shall be not lessthan the ratio P/S nor less than 0.05.

C = 0.30 + (1.4OW&/Hd) for plates bolted to shells,flanges, or side plates in such a manner that the settingof the bolts tends to dish the plate, and where thepressure is on the same side of the plate as the boltingflange, as shown in Fig. 13.8g and Fig. 13.8h, where:

Page 373: Process Equipment Design

A P P E N D I X

RmI

CHARTS FOR DETERMINING SHELL THICKNESS

OF CYLINDRICAL AND SPHERICAL VESSELS

UNDER EXTERNAL PRESSURE

364

Page 374: Process Equipment Design

40 EI

35 :i

3025

20181614

\12

24

8.0 <

7.06.0

3 5.0 \

8g< 4.0;$ 3.5 \

L In 3.0fmEs 2.5g$

$2 :.;gs 1:fjz's; 1.45: 1.222. . - 1.0$yg 0.90g': 0.80sz 0.7052~1. g 0.602 0.50

0.400.350.300.25

0.200.180.16 f

0.14 +

0.120.100.090.080.070.060.05

Charts for Cylindrical and Spherical Vessels Under External Pressure 365

25,000

20,00018,00016,00014,00012,00010,0009,0008,0007,0006,000

5,000

4,0003,5003,000 bl #n

$$!2,500 5 =a

3"2,000 2;1,800 2%1,600 %g1,400 -1,200 4

1,000 $900800 3

700600500

400350300

250

200180160140120100

it706050

0.001Factor A = f/E = e

Itom 1. Chart for determining shell thickness of cylindrical and spherical vessels under external pressure when constructed of low-carbon nickel. (From tine

1956 ASME Unflred Pressure Vessel Code with permission of the Am. Sot. Mech. Engrs.)

Page 375: Process Equipment Design

366 Charts for Cylindrical and Spherical Vessels Under External Pressure

IlIIIIIII I III III I I I 111111 I I I I I lII11150~000

Item 2. Chart for determining shell thickness of cylindrical and spherical vessels under external pressure when constructed of aluminum alloy MIA-O or

MIA-H’ .lZ. (From the 1956 ASME Unfired Pressure Vessel Code with permission of the Am. Sot. Mech. Engrs.)

40 40,00035 35,00030 30,000

25 25,000

20 20,00018 18,00016 16,00014 14,00012 12,000

10 10,ooo9.0 9,0008.0 8,0007.0 7,0006.0 6,000

5.0 5,000

4.0 4,0003.5 3,5003.0 3,000 tl IJa

SE2.5 2,500 $a

L8.2

2.01.8

a330 321,800 ,o>

1.6 1.600 xx1.4 \ \\ \

1.2 \it i.\.\,1,400 LJ

\\ \ 1,200 41 . 0 1,000 8

900 2800700600500

-.-0.900.800.700.60

0.50

0.400.350.300.25

0.26

400350300250

2000.18 1800.16 1600.14 1400.12 1200.10 1000.09 900.08 800.07 700.06 600.05 50

2 3 4 5 6 7 8 2 3 4 5678 2 3 4 5678 2 3 4 56780.00001 0 . 0 0 0 1 0.001 - 0.01 - - -0.1

Factor A = f/E = c

\I / --.-_.--- \

Page 376: Process Equipment Design

109.08.07.06.0

.,. f 1.61 . 4

?-?I$ 12!jje .3-j 1.0$j .g 0.90s ;I; 0.803 E 0.70b2IA p 0.60

2 0.50

0.400.350.300.25

0.200.180.160.140.120.100.090.080.070.060.05

Charts for Cylindrical and Spherical Vessels Under External Pressure

25,000

20,00018,00016,00014,000

10,0009,0008,000

H 7,0006,000

J-K-1 5,000

4,000ttttttttl 3,500

3,000 $ g2,500 2 %

c:2,000 ci1,800 2'1,600 2%

II\1 I Ill1, I I I I I II 1,400 -

0.001Factor A = f/E = c

1,200 41,000 $900800 a700600500

400350300250

200180160140120

100

ii706050

367

Item 3. Chart for determining shell thickness of cylindrical and spherical vessels under external pressure when constructed of aluminum alloy MIA-HI4,

(From the 1956 ASME Unfired Pressure Vessel Code with permission of the Am. Sot. Mech. Engrs.)

Page 377: Process Equipment Design

368 Charts for Cylindrical and Spherical Vessels Under External Pressure

501 I IIIYII IIIIiIlll I I I III llIIIIIII I I I I I I I IIIIIIIll I I I I I lIIIr150.ooo

201816-1412

9':8.07.06.0

! 5.0

2.01.81.61 . 41 . 21 . 0

0.900.800.70

0.60

0.50

0.400.350.300.25

0.20 I-

0.090.080.070,0060.05

0.001Factor A = f/E = c

40353025

4.03.5

3.02.5

I II I I I II

I II I IIll

40,00035,00030,00025,000

20,00018,00016,00014,00012,000

8;OO07,000

400350300250

200180

hem 4 . Chart for determining shell thickness of cylindrical and spherical vessels under external pressure when constructed of aluminum alloy MGl l A-0 01

MG; :A-HI 12. (From the 1956 ASME Unfired Pressure Vessel Code with permission of the Am. Sot. Mech. Engrs.)

Page 378: Process Equipment Design

Charts for Cylindrical and Spherical Vessels Under External Pressure 3 6 9

I I I I I III11150.oQo

40,00035,00030,000

25,000

20,00018,00016,00014,000

12,000

10,000

-0.001

Factor A = f/E = c

Item 5 . Chart for determining shell thickness of cylindrical and spherical vessels under external pressure when constructed of aluminum alloy MGI l A-H34.

(From the 1956 ASME Unfired Pressure Vessel Code with permission of the Am. Sot. Mech. Engrs.)

Page 379: Process Equipment Design

3 7 0 Charts for Cylindrical and Spherical Vessels Under External Pressure

403 530

2 5

201 81 61 4

1 2

2:8.07.0

6.0

,e 5.0

2 5 0.60

2 0.50

0.400.35

0.30

0.25

0.200.180.160.14

0.12

0.100.090.080.070.06

25,000

20,00018,00016,00014,00012,000

10,0009,0008,0007,0006,000

5,000

4,0003,5003,000 e

E2,500 8%

Lzp

2,000 :;--,1,800 5%1,600 %%1,400 -

1,200 4

1,000 8900 2800700

600

500

400350300

250

200180160140

120

100908 070

605 0

0.001Factor A = f/E = c

Item 6. Chart for determining shell thickness of cylindrical and spherical vessels under external pressure when constructed of nickel. (From the 1956 ASMEUnfired Pressure Vessel Code with permission of the Am. Sot. Mech. Engrs.)

Page 380: Process Equipment Design

Charts for Cylindrical and Spherical Vessels Under External Pressure 371

25

201 81 61 4

1 2

1 09.08.07.0

6.0

2.5

2.01.81.61.41.2

1.00.900.800.70

0.60

0.25

0.200.180.160.140.12

0.100.090.080.07

0.06

0.052 3 4 5678 2 3 4 5678 2 3 4 5678 2 3 4 5678

0.00001 o.ooo1 0 . 0 0 1Factor A = f/E = 6

0 . 0 1 0 . 1

25,000

20,00018,00016,00014,00012,000

10,0009,0008,0007,0006,000

5,000

4,0003,500

3,000@J

2,500 .E S3”

2,000 5 51,800 s$1,600 gz1,400-

1,200 ;

1,000 E900800

2

700

600

500

400350

300

250

200180160140

120

100908070

60

50

Item 7. Chart for determining shell thicknest of cylindrical and spherical vessels under external pressure when constructed of annealed nickel-copper alloy.

(From the 1956 ASME Unfired Pressure Vessel Code with permission of the Am. Sot. Mech. Engrs.)

Page 381: Process Equipment Design

3 7 2 Charts for Cylindrical and Spherical Vessels Under External Pressure

25

201 81 61 4

1 2

9!008.07.0

6.0

5 . 0

4 . 03.5

3.0

2.5

2.01.81.61.41.2

1.00.900.800.700.60

0.50

0.400.35

0.30

0.25

0.200.180.160.14

0.12

0.100.090.080.07

0.06

0.052 3 4 5678 2 3 4 5678 2 3 4 5 6 7 8

0.00001 0.0001 0.001Factor A = ffE = c

Item 8 Chart for determining shell thickness of cylindricbl and spherical vessels under external pressure when constructed of annealed nickel-chromium-irorl

alloy. (From the 1956 ASME Unfired Pressure Vessel Code with permission of the Am. Sot. Mach. Engrs.)

Page 382: Process Equipment Design

Charts for Cylindrical and Spherical Vessels Under Eiternal Pressure 3 7 3

5 0 50.000

40 40,0003 5 35,0003 0 30.000

2 5 25.000

20 20,0001 8 18.0001 6 16.00014 14.000

12 12.000

10 )O.OOO9.0 9,0008.0 8.0007.0 7,000

6.0 6.000

5.0 5.000

4.0 4.0003.5 3.5003.0

2.01.81.61.41.2

2.53qooo g f2,500 s f

E :2,000 2 51,800 2s1,600 $ g1,400 _cj1,200 4

900 $1,000

800 ’700

600

1.00.900.800.70

0.60

0.50 500

0.400.35

0.30

0.25 250

0.20 2000.180.16 ’ ’ ’ ’ ’ ”

180160

0.14 I I I I\ I Y I I\II\ h I \ I \I RI I I\1 \ I \I l\l l\l \slll400.12

0.100.090.080.070.06

2 3 4 56780.00001 0.000 1 0.001

Factor A =flE = e

ltem 9. Chart for determining shell thickness of cylindrical and spherical vessels under external pressure when constructed of annealed copper, type DHP-

(From the 1956 ASME Unftred Pressure Vessel Code with permission of the Am. Sot. Mech. Engn.)

I/b I \ \ \I /

Page 383: Process Equipment Design

374 Charts for Cylindrical and Spherical Vessels Under External Presusre

Item 10. Chart for determining shell thickness of cylindrical and spherical vessels under external pressure when constructed of copper-silicon alloys

(From the 1956 ASME Unfired Pressure Vessel Code with permission of the Am. Sot. Mech. Engrs.)

30,00025,000

20,00018,000

9,0008,0007,0006,0005,000

4,0003,500

700600500

400350300250

0.090.080.07

0.06

0.052 3 4 5678

0.00001 0.0001 2 3 4 5 678 2 3 4 5678 2 30.001 0.01Factor A = f/E = E

4 5 6 7 80 . 1

Aand C.

Page 384: Process Equipment Design

Charts for Cylindrical and Spherical Vessels Under External Pressure 375

50 50,000

40 40,00035 35,OOu30 30,000

25 25,000

20 20,0001 8 18,0001 6 16,0001 4 14,0001 2 12,000

52 ;ogoo

8.0 8:OOO7.0 7,000

6.0 6,000

5.0 5,000

4.0 4,0003.5 3,500

3.0 3,000

2.5 2,500

2.0 zoo01.8 WO1.6 1,6001.4 1,400 u1.2 II

m1.0

0.900.800.70

0.60I

I q\ I\

2 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3 4 5 6 7 8

800700

600

0.50

0.400.350.30

0.25

0.200.180.160.14

0.12

0.100.090.08 I I II I111111 I III IllllYl

0.07 * P *‘o=# o-

0.06

0.05

500

400350

300

250

200180160140

120

100908070

6050

0.00001 0 . 0 0 0 1 0 . 0 0 1

Factor A = f/E = e

0.01 0.1

:lem 11. Char t fo r de te rmin ing she l l th ickness o f cy l indr ica l and spher ica l vesse ls under ex te rna l p ressure when cons t ruc ted of annea led 90-T 3 copper -n icke la l loy . (From the 1956 ASME U n f i r e d P r e s s u r e V e s s e l C o d e w i t h p e r m i s s i o n o f t h e A m . Sot. Mech. Engrs.)

L. _ I \ \ \I /- -

Page 385: Process Equipment Design

376 Charts for Cylindrical and Spherical Vessels Under External Pressure

403530

25

201 81 61 41 2

SC8.07.0

6.0

5.0

4.03.5

3.0

2.5

2.01.81.61.41.2

1.00.900.800.70

0.60

0.50

0.400.35

0.30

0.25

0.20 I /I II I ill’0.18 I/ I I I I I II,0.160.14

0.09 I I

0.080.07 l-

0 . 0 0 1Factor A = f/E = e

Item 12. Chart for determining shell thickness of cylindrical and spherical vessels under external pressure when constructed of annealed 70-30 copper-nickel

alloy. (From the 1956 ASME Untired Pressure Vessel Code with permission of the Am. Sot. Mech. Engrs.)

\ ---i- ---- \I /---

Page 386: Process Equipment Design

Charts for Cylindrical and Spherical Vessels Under External Pressure 377

50,000

40,00035,000

30,000

25,000

20,00018,00016,OOt I14,00012,000

10,0009,0008,0007.000

6,000

5,000

4,0003,500

3,000

7 5nn-,-v.,

7 nnn2.01.81.61.4

-,-_-1,800 ”1,600 2 x1,400-

1.200 4

1,000900800

0.60

0.50

0.400.35

0.30

‘d

400350

300

2500.25

0.200.180.160.14

0.12

0.100.09

200180

100908070

605.

3 4 56780.1

0.06 -

0.05, 4 ,b,L1 2 3 4 5 6 7 8

0 . 0 0 0 0 1 0.’ 0001 0.001

Page 387: Process Equipment Design

378 Charts for Cylindrical and Spherical Vessels Under External Pressure

AlI l,‘l~IlAlllI’I ‘II ‘II ‘II lll1_11 I I I I I Illlll ̂ . ̂ ̂ .- 50,OOd

16s

1.8s I \ Y \ l\hl ‘, Y ‘, \ \ ‘, .\ \ \-zoo0 $2

1.61 ‘11 , 8 0 0 Q‘,

\ \ \ 1,600 .%,

lml I& / 40,000

a” a” 35,00030.000

IIIIIll\lI i II

18jOO0

16,00014,000

400” F 12,000

10,0009,0008,000

0.800.70

0.60

0.50

0.100.090.080.07

0.06

0.05 t

0.000012 3 4 5678

0.0001Factor A = f/E = E

1,000 2i900 2800700

600

400350

300

250

2001801 6 01 4 0

1 2 0

Item 14. Chart for determining shell thickness of cylindrical and spherical vessels under external pressure when constructed of austenitic steel ii 6 Cr-8 Ni,

type 304). (From the 1956 ASME Unfired Pressure Vessel Code with permission of the Am. Sot. Mech. Engrr.)

Page 388: Process Equipment Design

Charts for Cylindrical and Spherical Vessels Under External Pressure 329

‘IIIIIWI I I I I

40,00035,00030,000

\ I 2,500

1,600

1 , 4 0 0

1,200

o.ocoo1 - - - -o.ooo1 0.001 0 . 0 1 0 . 1

Factor A = f/E = E

Item 15. Chart for determining shell thickness of cylindrical and spherical vessels under external pressure when constructed of oustenitic steels (18 Cr-8 Ni j-

MO, type 316; 18 G-8 Ni + Ti, type 321; 18 G-8 Ni f Cb, type 347). (From the 1956 ASME Unfired Pressure Vessel Code with permission of the Am.Sot. Mech. Engrs.)

Page 389: Process Equipment Design

380 Chari ts for Cylindrical and Spherical Vessels Under External Pressure

40,ooo35,00030,ooo

25,000

(- ,---

I12,m

1.41.2

1.00.900.80

0.50

0.400.35

Factor A = f/E = e

Item 16. Chart for determining shell thickness of cylindrical and spherical vessels under external pressure when constructed of cast iron.

AWE Unfired Pressure vessel Code with permission of the Am. Sot. Mech. Engrs.)

r . t \ \ \I /

Page 390: Process Equipment Design

A P P E N D I X

PROPERTIES OF VARIOUS SECTIONS

AND BEAM FORMULAS

f

381

Page 391: Process Equipment Design

3 8 2 Properties of Various Sections and Beam Formulas

Item 1. Properties of Various Sections

Area of Section‘4

(12

a2 - 012

b h

b h

b h

b h

bh - bth,

b(h - hl)

: h* tan 30” = 0.866h*2 r/

c

; h2 tan 30’ = 0.866h2

Z/L* tan 22;O = 0.828h*

h(b + bd2

h(b + b,)2

Courtesy of theDistance from Axisto Extremities of

Sectiong and II

II=0

=-$ = 0.707n

g=t2

*=0

y = s = 0.707a

hup-

2

u-A

b h

u’=m 1

y=hcosa+bsince2

y=n2

#,h2

u=”2

hy = ~ = 0.5i7h

2 cm 30’

u=l2

h(br + 2b)Y=3(bl

Ub + 2bdy’ = 3(b

y=h

lldwin Locomotive Works)

Moment of InertiaI

a ’ii

ad - a,’--1 2

d - (1143

a4 - a,’1 2

bha12

bh33

b3hJ

6(b* + AZ)

E (hz COG P + b* sin* a)

bh3 - b,h,’- -1 2

b(h3 - h,J)1 2

A h*(l +2cos2300)12 [ 4 cd 30’ I

= 0.06h’

A A*(1 + 2 cos* 30”)ii 4 CD9 30” 1

= 0.06hI

A k2(1 + 2 co.32 223”)12 [ 4 cm* 22:” 1

= 0.055h~

h3(bz + lbb, + br*)36(b + bd

h3(b + 3b)n1 2

Section Modulus

-6 = 0.118d6&

a4 - 111’6 a

a( -a,’3n

bh’

-ii-

bh=

-27

b2h=

6db2+h2

b h A2 cd a + b1 sin* P

s h cos LI + b sin a >

bh3 - blhls- -6h

b(h3 - h13)6 h

A h(l + 2 cos* 30’)

6 C 4 cd 30” 1= 0.12h3

.4 C h(1 + 2 cm* 30”)6 4 cos 30’ I

= 0.104h”

A h(l + 2 cm* 22:“)

6 4 02s 221 I= O.lOSh~

hz(b2 + 4661 + bl*)

12(bl + 2b)

h*(b + 3b)l12

Radius of Gyration

Id-

1I=;i

-& = 0.2890

-$ = 0.577a

a*+m*\I-- = 0.289 da2 + at’

1 2

&2 = 0.289A

= 0.577h

b h

d6(bZ + h’9

h2 co82 P + b2 sin2 a1 2

bh3 - b,h,a

lZ(bh - bihd

h3 - ht3

li(h -hi)

I + 2 COSZ 30’3

= 0.264h

h 1 + 2 COSZ 30”4 cos 300 3

= 0.264h

h 1 + 2 cos2 225”_.~4 cos 22;” 3

= 0.257h

& d\/l(b” + 4bbl + blz)

1

-.

,

Page 392: Process Equipment Design

Properties of Various Sections and Beam Formulas 383

Item 1. Properties of Various Sections (Continued)

Sections

.

isi

Area of SectionA

bh5

bhz

r(d* - dl*)4

= 0.7354(d* - d?)

rd’-=0.3927dz8

r(d* - dl’)8

= 0.3927(d* - dl*)

T = 0.735468

r(bh - blhl)4

= 0.7854(bh - blhl)

(b = h = I)n-r)It- -

4= 0.2146r’

bs + 2ec+ blat

l g2

rK

Distance from Axisto Extremities of

Sectiony and Yl

y=A

UC!!2

$ = 0.049d’

U’d2

d(3r - 4)y = - = 0.283d

en.7

11 , = 2 = 0.212d

2(d3 - dls)Y=3*(d2

u, = 3rdfd* - dl*) - 4(dJ - d?)Br(d’ - dl’)

A;i

A;i

I6 1-T( >= 0.7767r

2ch* + (bl - 2c)s1* + (b - 2c)(2h - 81)’I=

2Aul=h-u

rK1

Angle 4 K

51015202530354050

ti8090

0.000440.003520.011800.027670.053310.090580.141020.205730.380260.614180.900341.225251.57080

Moment of InertiaI

bhs38

bhai i -

r(d’ = O.O49(d’ - dl’)64

d’(w - 64) - o,oo69d~1152~

.l(d4 _ &“)(dz - dl*) - 64(d3 - dlJ)P

-1152*(d* - dl*)

rbA3K = 0.049bAJ

r(bAJ - b,h,‘)64

= O.O49(bha - bth13)

rI !-.TL(3 16 36 - 9s >

= 0.0075r’

bw3 + b1/13 - (bl - 2c)(r - 81)’3

_ (b - 2c)(m - s)~3

Kl

0.99860.99080.97960.96400.94390.92000.89210.86070.78310.70510.61440.51970.4244

K2

O.OOOOO0.00000O.OOOOO0.000010.000030.000100.000330.000760.003370.010640.027110.058480.10976

Section Modulus

s=fY

bh224

bh ’rz

rd332 = 0.09&f*

r(d’ - dl”)32d

= 0.098 d’d

dJ(9a-* - 64)- -192(3r - 4)= 0.0238d3

F = 0.093bh’

r(bh3 - btA13)3 2 h

0.093(bh3 - blh13)=A

I

L= 0.0097rJ

I

;

I

U

I

y--rCDS4

adius of Gyration

It= 4Ii

-!L = 0.236hI.6

& - 0.408h

di

d/d’4

d 49rz - 6412r

= 0.132d

hi

I bAJ - blAl*i bh - bdt

~o.o349c*= 0.187r

d ili

di;5

* 0 in radian measure (1 radian = 180/r darees, and I” = 0.017453 radians).4 About the XX axis.L\ About neatral axis of shaded area.

Page 393: Process Equipment Design

384 Properties of Various Sections and Beam Formulas

Item 2. Beam Formulas

(Courtesy of the Baldwin Locomotive Works) -.--

MomentFsw,

Moment

It? = w

M- = g (32 - 2)

2WlM“an= = --ii-

WZQc = T (21 - I)

,,m1 = wQw 1113

/ = F, $.J blU.9

Rr = w

M, = W(z -a), [z > a]

MIILOZ = Wb

w 1L = jg .z

RI =R,=;

r = ;, f haz)

R,=f[W(l-a)+W,b] R,=!%1 (

&.+ f&z- R, = RI = %’

W(2a + b)Rx = 21 Ma=+-z)

MI = RIZ. [z < a or = a]W(z - a)* M

M, = RIZ 2b v

[z ; (a + b) and > a] Qz = ; - wzMz = Rdl - d. [z > (a + b)lQc = RI, [z < a or = a] Q

II = dc(l +a)/31 Qz = !!!p _ f (= - =), wl 50/=g.gg(maZ).

Iz < (a + b) and > alQz = Rz, [z > (a + b) and < 4

Moment/~+

R, = R2 =;

M,= W+f;)

M WIIPI”1 = 7’ [t = JiJ]

Q,.= W(;-F+T) M,=w~(~~~~;;~~“)~ Qr=+z

W Qw = ; W, [z = I] w(L. - d. 12 > Cl 5701Qm.. - 2. [z = 01 Q !>,,,1_ Q. = R I, [z < cl

Q",,,.. = T

Ww 313

Qz = RI, [z > cl/=rI.GJ20(ma2)

I = o.01304EIlJ (mm)W f = 2; . ~5(maz).[z = 0.42211

[z = 0.51911 / = 0.0098 jj P haz)

[z = c = 0.41411

_ ___ .-~._ __ .._ . ..- - - -- III \ \ \I /

Page 394: Process Equipment Design

Properties of Various Sections and Beam Formulas 385

Wb - W,aR, = ___

r1 = Wl L b) + Wda + 01

bfs = Ita, 1% < (I - b)lbf=, = Itan - W(b + z, - I)

[ZI > (I - b) and <I1Mz, = Wl(O - zz)Qz = RI, Qz, = RI - W&?I, = WI

Beam Formulas(Con,

yfTzg2Gq

RI+H,- Shear

I 1-R;

w3c + CdRI=W(T)

w3c1+ 0RI=W(I;-)

M e&z--W=s[z<C].c 1=

- wcz - C), 12 > cQ. = RI. fz < ClQz = (RI - WA [z > Cl

RI = w[(c + f)* - c1*]/21Rr = w[(c, + 1)” - c*]/21bfz = >bur(c - 212M., = $Pw(c + .a)* - Rmhf.” = yzwh - zrP

tin ued)

RI, Rz, M,, Mm.,, Q., QXI, and/w. rWC the mne aa in Cal. 8. above

W a b cbn = 6E11 (1 + b), Izr = cl

Wabdra = 6E11 0 +a). ICI = 4

R = reactionM = momentw = concentrated load clr total bad30 = unit load

Weight of beam neglected in each ease.Upward forces are positive. (+j; downward forces, negative (-).Shear (at B given section) is the algebraic *urn of all forces on either side of the section.

Moment (at a given section) is the dgebraic turn of all momenta produced by the forces on one aide Of the &ion.A moment bending a barn convex downward is positive (+), and one bending a beam eonvex Uprud b nega-

Q = shear/ 7 deflectionE = modulus of elebtricityI = moment of inertia

tive (-).No account of the signs of shears and moments haa been taken in the formulas above, but the applicStim Of

these orinciul~ will at once determine them. Formulas will give values numerically eorreet. and for the wmlcase this is ill that is needed.

Deflections in those ea~ea where no formulaa we given may be handled by the methods Of APPfa EfWli&vTimoshenko and LesselIs, 1st ed.. pp. 82-94. Or see Marka Handbwk. 3rd & P. 452.

-.

Page 395: Process Equipment Design

A P P E N D I X

mKPROPERTIES OF PIPE (Courtesy of Tube Turns, Louisville Kentucky.)

I

Item 1. Dimensions of Seamless and Welded Steel Pipe

Nominal Nominal Wall Thickness for

Pipe Outside Sched. Sched. Sched. Sched. Stand- Sched. Sched. Extra Sched. Sched. Sched. Sched. Sched. XXSize Diam. 5* lot 20 30 ard$ 40 60 Strongt 80 100 120 140 160 Strong34 0.405 .,. 0 .049 . . . . . . 0.068 0.068 . . . 0.095 0.095 . . . . . . . . . . . . . . .J/a 0.540 . . 0.065 . . . . . . 0.088 0.088 0.119 0.119 . . . . . . . . . . . . . . .

2 0.675 0.840 . . .0:065

0.065 0.083 . . . . . . . . . . . . 0.109 0.092 0.091 0.109 ::: . . . 0.147 0.126 0.126 0.147 . . . . . . . . . oIii)7 o&4.% 1.050 0.083 . . . . . . 0.113 0.113 0.154 0.154 ::: ::: ::: 0.218 0.308

1 1.315 (I.065 0.109 . . . . . . 0.133 0.133 ::: 0.179 0.179 . . . . . . . . . 0.250 0.358u/a 1.660 0.065 0.109 . . . . . . 0.140 0.140 . . . 0.291 0.191 . . . . . . . . . 0.250 0.3821% 1.900 0.065 0.209 . . . . . . 0.145 0.145 . . . 0.200 0.200 . . . . . . . . . 0.281 0.4002 2.375 0.065 0.109 . . . . . 0.154 0.154 . . . 0.228 0.218 . . . . . . . . . 0.343 0.4362?4 2.875 0.083 0.120 . . . . . . 0.203 0.203 . . . 0.276 0.276 . . . . . . . . . 0.375 0.552is* 4.0 3.5 0.083 0.083 0.120 0.120 . . . . . . . . . . . . 0.216 0.226 0.226 0.216 . . . . . . 0.300 0.328 0.300 0.318 . . . . . . . . . . . . 0.438 0.600

4 4.5 0.083 0.120 . . . . . 0.237 0.237 . . . 0.337 0.337 0:&8 ::: o:.k 0%45 5.563 0.109 0.134 . . . . 0.258 0.258 0.375 0.375 ::: 0.500 . . . 0.625 0.7506 6.625 0.109 0.134

0:&O O&70.280 0.280 ::: 0.432 0.432

8 8.625 0.109 0.148 0.322 0.322 0.406 0.500 0.500 O&30.562

0:8i20.718 0.864

0.718 0.906 0.87510 10.75 0.134 0.165 0.250 0.307 0.365 0.365 0.500 0.500 0.593 0.718 0.843 1.000 1.125 . . .12 12.75 0.165 0.180 0.250 0.330 0.375 0.406 0.562 0.500 0.687 0.843 1.000 1.125 1.312 . . .14 OD 14.0 . . . 0.250 0.312 0.375 0.375 0.438 0.593 0.500 0.750 0.937 1.093 1.250 1.406 . . .16 OD 16.0 . . . 0.250 0.312 0.375 0.375 0.500 0.656 0.500 0.843 1.031 1.218 1.438 1.593 . . .18 OD 18.0 . . . 0.250 0.312 0.438 0.375 0.562 0.750 0.500 0.937 1.156 1.375 1.562 1.781 . . .20 OD 20.0 . 0.250 0.375 0.500 0.375 0.593 0.812 0.500 1.031 1.281 ,1.500 1.750 1.968 . . .22 OD 22.0 . . . 0.250240D 24.0 . . . 0.250 O&5 0:;62

0.375 0.5000.375 0:687 0:968 0.500 1:2i8 1:&l 1:8i2 2:i)62 21343 :::

26 OD 26.0 . . .ok2 0:&o 0:6i5

0.375 . . . 0.500 . . . . . . . . . . . . . . . . . .30 OD 30.0 . . . 0.375 . . . . . . 0.500 . . . . . . . . . . . . . . . . .34 OD 34.0 . . . . . . . . . 0.375 . . . . . . 0.500 . . . . . . . . . . . . . . . . . .36 OD 36.0 . . . . . . . . . 0.375 . . . . . . 0.50042 OD 42.0 . . . . . . . . . 0.375 . 0.500 : : : : : : : : : : : : : : : : ::

Commercial Pipe Sizes and Wall Thicknesses. This table lists the pipe sizes and wall thicknesses currently established as standard,or specifically:

1. The traditional standard weight, extra strong, and double extra strong pipe.2. The pipe wall-thickness schedules listed in American Standard B36.10, which are applicable to carbon steel and alloys other than

stainless steels.3. The pipe wall-thickness schedules listed in American Standard B36.19, which are applicable only to stainless steels.All dimensions are given in inches.The decimal thicknesses listed for the respective pipe sizes represent their nominal or average wall dimensions. The actual thick-

nesses may be as much as 12.5% under the nominal thickness because of mill tolerance. Thicknesses shown in light face for Sched-ule 60 and heavier pipe are not currently supplied by the mills, unless a certain minimum tonnage is ordered.

* Thicknesses shown in italics are for Schedules 5S and lOS, which are available in stainless steel only.t Thicknesses shown in italics are available also in stainless steel, under the designation Schedule 40s.$ Thicknesses shown in italics are available also in stainless steel, under the designation Schedule 80s.

306

Page 396: Process Equipment Design

Properties of Pipe 387

Item 2. Design Properties of Pipe

NominalPipe Size

andOutside

Diameterinches

$4D = 0.405

5iD = 0.540

5%D = 0.675

$5D = 0.84C

$3D - 1.050

D -11.315

1%D = 1.680

134D=1.900

2D = 2.375

2%D = 2.875

D =:%I

3%D=4.000

ScheduleNumberand/orWeight

10s40 ST 40s80 xs 805

WallThick-

ne88inches

t0.049,068.095

InsideDiameter

FifthPowerof IDin.5

dh0.00273

.0014100046

Surface AreaInsideOutside

sq ftper It

A.0.106

.106

.106

Areas and WeightsCrca-Sectional

Metal FlowWeight of

W&S

d0.307

.269

.215

sq ft Ali All33per ft sq in. sq in.

Ai A Al0.080 0.055 0 074

,070 .072 ,057.056 ,092 ,036

Pipelb

per ftw

0.186,245,314

lbper ft

to,0.032

,025,016

Radiuso f

Gfla-tion

inches70

0.1271.1215.1146

Momentof

Inertiain.4

I0.0009

.cQll.0012

SectionMOdiJlU#

in.’2

0.0043.0052.0060

10s ,065 ,410 .01159 ,141 ,107 ,097 ,132 ,330 ,057 .1694 .0028 .010340 ST 40s ,088 ,364 .00639 ,141 ,095 ,125 .104 ,425 ,045 .1628 .0033 .012380 xs 80s ,119 ,302 .00251 .141 ,079 ,157 .072 .535 .031 .1547 .0038 .0140

10s ,065 ,545 .04808 ,177 ,143 ,124 ,233 ,423 ,101 .2169 .0059 .017440 ST 40s ,091 ,493 .02912 .177 .129 ,167 .191 ,568 ,083 .2090 .0073 .021680 xs 80s .126 423 .01354 ,177 ,111 ,217 ,140 ,739 ,061 .1991 .0086 .0255

10s ,083 .674 .13909 .220 .176 .197 .357 ,671 ,154 ,269 .0143 .034140 ST 405 .109 ,622 .09310 ,220 ,163 ,250 .304 ,851 ,132 ,261 .0171 .040780 xs SOS .147 ,546 ,04852 .220 ,143 .320 ,234 1 088 ,101 250 .0201 .0478

160 .187 .466 .02198 .220 ,122 .384 .171 1.304 ,074 240 .0221 .0527x x ,294 ,252 .00102 ,220 .066 .504 ,050 1.715 ,022 ,219 .0243 .0577

5s10s

40 ST 40s80 xs 80s

160x x

,065 ,920 .6591 275 ,241 ,201 ,664 .683 .288 .349 .0245 .0467,083 .884 .5398 ,275 ,231 ,252 ,614 ,857 .266 ,343 .0297 .0566,113 ,824 .3799 ,275 ,216 ,333 ,533 1.131 ,231 ,334 .0370 .D706.154 ,742 .2249 .275 ,194 .434 .432 1.474 ,187 ,321 .0448 .8053,188 ,675 .1401 ,275 ,177 ,503 ,358 1.728 ,155 ,312 .0495 .0943,218 ,614 .0873 ,275 161 ,570 ,296 1.937 ,128 ,304 .0527 .1004.308 ,434 .0154 ,275 .114 .718 ,148 2.441 ,064 ,284 .0579 .1104

5s10s

40 ST 40s80 xs 805

160x x

.065 1.185 2.337 ,344 ,310 255 1.103 .867 .478 .443 .0500 .0760,109 1.097 1.589 ,344 ,287 ,413 ,945 1.404 .409 ,428 .0757 .1151,133 1.049 1.270 ,344 ,275 ,494 ,864 1.679 ,374 ,420 .0874 .1329,179 ,957 ,303 ,344 ,250 .639 ,719 2.172 ,311 .407 .1056 .1606,219 ,877 .519 ,344 ,230 ,754 ,604 2.564 ,262 .395 .1178 .1791.250 ,815 ,360 ,344 ,213 .836 522 2.844 .226 ,387 .1252 .1903,358 ,599 ,077 ,344 ,157 1.076 ,282 3.659 ,122 .361 .1405 .2137

5s ,065 1.530 8.384 .435 ,401 .326 1.839 1.108 ,796 ,564 .1037 .125310s ,163 1.442 6.235 ,434 ,378 .531 1.633 1.305 ,707 ,550 .1605 .1934

40 ST 40s ,140 1.380 5.005 ,434 .361 ,668 1.496 2.273 ,643 ,540 .1948 .23468OXS 80s ,191 1.278 3.409 ,434 .334 ,881 1.283 2.997 ,555 ,524 .2418 .2914

160 .250 1.160 2.100 ,434 .304 1.107 1.057 3.765 ,458 ,506 .2839 .3421x x ,382 .896 ,577 .434 .234 1.534 ,630 5.215 .273 .472 .3412 .4111

5s ,065 1.770 17.37 .497 ,463 .375 2.461 1.275 1.066 ,649 ,158 .16610s ,109 1.682 13.46 .497 ,440 .613 2.222 2.085 .962 ,634 ,247 ,260

40 ST 40s ,145 1.610 10.82 .497 ,421 .799 2.036 2.718 .882 ,623 ,310 ,32680 XS 80s .200 l.M)o 7.59 ,497 ,393 1.068 1.767 3.632 .765 .605 .391 ,412

160 ,281 1.337 4.27 ,497 ,350 1.431 1.404 4.866 .608 .581 .483 308x x ,400 1.100 1.61 .497 ,288 1.885 ,954 6.409 ,411 .549 .568 .598

5s10s

40 ST 40s

80 XS 80s

160x x

5s10s

40 ST 40s

80 xs 80s160

x x

5s10s

,065 2.245 57.03 ,622 ,588 .47i 3.958 1.605 1.714 ,817 ,315 .265,109 2.157 46.69 ,622 ,565 .776 3.654 2.638 1.582 ,302 .499 .420,154 2.067 37.73 ,622 ,541 1.074 3.356 3.653 1.453 ,787 ,666 ,561,167 2.041 35.42 ,622 .534 1.158 3.272 3.938 1.417 ,783 ,710 ,598,188 2.OaO 32.00 ,622 524 1.283 3.142 4.381 1.360 ,776 ,777 ,654,218 1.939 27.41 ,622 ,508 1.477 2.953 5.022 1.278 ,766 ,868 ,731,250 1.875 23.17 ,622 ,491 1.669 2.761 5.674 1.196 ,756 ,955 ,804,312 1.750 16.41 ,622 ,458 2.025 2.405 6.884 1.041 .738 1 102 ,928,343 1.689 13.74 ,622 ,442 2.190 2.240 7.445 ,970 ,728 1.163 ,979,436 1.503 7.67 ,622 ,393 2.656 1.774 9.030 ,768 ,703 1 312 1.104

,083 2.709 145.9 ,753 ,709 728 5.76 2.475 2.496 ,988 ,711 ,495,120 2.635 127.0 ,753 ,690 1.039 5.45 3.531 2.361 ,975 .988 ,687,203 2.469 91.8 ,753 ,646 1.704 4.79 5.794 2.073 ,947 1.530 1.064,217 2.441 86.7 ,753 ,639 1.812 4.68 6.180 2.026 ,943 1.611 1.121,276 2.323 67.6 .753 ,608 2.254 4.24 7.662 1.835 .924 1.925 1.339.375 2.125 43.3 .753 ,556 2.945 3.55 10.01 1.536 ,894 2.353 1.637,552 1.771 17.4 .753 ,464 4.028 2.46 13.70 1.067 .844 2 872 1.998

40 ST 405

80 xs SOS

160x x

5s10s

.083 3.334 411.9 ,916 ,873 ,891 8.73 3.03 3.78 1.208 1.300 ,743

.120 3.260 368.2 .916 ,853 1.274 8.35 4.33 3.61 1.196 1.822 1.041,125 3.250 362.6 .916 ,851 1.325 8.30 4.51 3.59 1.194 1.890 1.080148 3.204 337.6 ,916 .839 1.558 8.06 5.30 3.49 1.186 2.194 1.253.188 3.124 297.6 ,916 ,818 1.956 7.66 6 65 3 32 1.173 2.692 1.538,216 3.068 271.8 ,916 ,803 2.228 7.39 7.58 3.20 1.164 3.018 1.724,241 3.018 250 ,916 ,790 2.467 7.15 8.39 3.10 1.155 3.29 1.883,254 2.992 240 ,916 ,783 2.590 7.03 8.81 3.04 1.151 3.43 1.962,289 2.922 213 ,916 ,765 2.915 6.71 9 91 2.90 1.140 3.79 2.165,300 2.900 205 ,916 ,759 3.016 6.60 10.25 2 86 1.136 3.90 2.226,312 2.875 196 ,916 ,753 3.129 6.49 10.64 2 81 1.132 4.01 2.294,406 2.687 140 .916 ,703 3.950 5.67 13.43 2.46 1 103 4.81 2.748,438 2.624 124 ,916 ,687 4.213 5.41 14.33 2.34 1.094 5.04 2.879,600 2.300 64 ,916 ,602 5.466 4.15 18.58 1.80 1.047 5.99 3.425

.083 3.834 828 1.047 1.004 1.021 11.55 3.47 5.00 1.385 1.96 ,979

.120 3.760 752 1.047 ,984 1.463 11.10 4.97 4.81 1.372 2.76 1.378

.128 3.744 736 1.047 .980 1 557 11.01 5.29 4.77 1.370 2.92 1.461

.134 3.732 724 1.047 .977 1.623 10.94 5 53 4.74 1.368 3.04 1.522

.148 3.704 697 1.047 ,970 1.791 10.78 6.09 4.67 1.363 3.33 1.664

Page 397: Process Equipment Design

3 8 8 Properties of Pipe

SominalPipe Size

adOutside

1)iameterIrwhes

33;D = 4.MKl

4D=4.500

5D = 5.563

6D = 6.525

8D = 8.625

ScheduleNumberand/orWeight

40 ST 40s

80 xs 80s

5sIOS

40 ST 40s

80 xs 805

120

I60XX

5sIOS

40 ST 405

80 XS ROY

120160

XX

5s10s

40 ST 40s

80 X8 80s

120160

x x

5s10s

203040 ST 40s

60

80 X6 80.3100

120140

x x160

5s10s

D2750 ao.30

WallThick-

nessinches

t

FifthPOWVof ID

in.5d6

Surface AreaArw and WeightsCross-Yxtional Wei:ht of

InsideDiameter

inchesd

3.6243.5483.4383.3643.3123.0622.728

OUtSi& Inside Metal Flow Pipe w a t e rsq ft sq rt Arei% Area lb lbper ft per ft sq in. sq in. per ft per ft

A, A& .4 Al w wla1.047 0.949 2.251 10.31 7.65 4.4i1.047 .929 2.680 9.89 9.11 4 281.047 ,900 3.283 9.28 11 16 4.021.047 ,881 3.678 8.89 12.51 3 851.047 .867 3.951 8.62 13.43 3 731 047 ,802 5.203 7.36 17 6.l 3.191.047 ,714 6.721 5.84 22 8.5 2 53

Radiuso f

GyCr-tion

inches70

1.3491.3371.3191 3071.2981.2591.210

oiInertia

in.’I

0.188,226,281,318,344,469,636

625562480431399269151

4.104.795.716.286.668.259.85

SectionModulus

in.3z

2.0502.3942.8553.1413.3314.1274 925

,083 4.334 1529 1.178 1.135 1.151 14.75 3.91 6.39 1.562 2.81 1.248,120 4.260 I-103 1.178 1.115 1.651 14.25 5.61 6.17 1.549 3.96 1 762,128 4 244 137; 1.178 I 111 1.758 14.15 5.98 6.13 1.546 4.21 1.869,134 4.232 1358 1.178 1.108 1.838 14.07 6.25 6 09 1.544 4.38 1.949,142 4.216 1332 1.178 1.104 1.944 13.96 6 61 6.04 1.542 4.62 2.054.165 4.170 1261 1.178 1.092 2.247 13.66 7.64 5.91 1.534 5.29 2.350,188 4.124 1193 1.178 1.080 2.55 13.36 8.66 5.78 1.526 5.93 2.64.205 4.090 1144 1.178 1.071 2.77 13.14 9.40 5.69 1.520 6.39 2.84.237 4.026 1058 1.178 1.054 3.17 12.73 10.79 5.51 1.510 7.23 3.22.250 4.000 1024 1.178 1.047 3.34 12.57 11.35 5.44 1.505 7.56 3.36,271 3.95R 971 1.178 1.036 R.60 12 30 12.24 5.33 1.498 8.08 3.59,281 3.93x 947 1.178 1.031 3.74 12 1x 12.72 5.27 1.495 8.33 3.70,300 3 900 902 1.178 1.021 3 96 11.95 13.46 5.17 I 489 8.78 3.90312 3.876 875 1.178 1 015 4.10 11.80 13.96 5.11 I 485 9.05 4.02337 3.8X 820 1.178 1.002 4.41 11.50 14.99 4.98 1.477 9.61 4.273i5 3.i50 742 1.178 ,982 4.U6 11.04 16.52 4.78 1 464 10.42 4.6343x 3.624 G25 1.178 ,949 5.59 10.31 19.00 4.4; 1.444 11.66 5.18,500 3 500 525 1.178 ,916 6.2R 9 62 21 36 4 Ii 1.425 12.7i 5.67,531 3.438 480 1.178 ,900 6 62 9.28 22.51 4.02 1.416 13.27 5.90,674 3.152 311 1.178 .825 8.10 7.80 27.54 3 38 1.374 15.29 6.79

,109 5.345 4363 1.456 1.399 1.88 22 43 6.38 9 71 1 928 6.97 2.51,134 5.295 4162 1.4j6 1.386 2.29 22.02 7.77 9.53 1.920 8.43 3.03.258 5.04i 3275 1.456 1 321 4 30 20.01 14.62 8 66 1 878 15.17 5.45,352 4.859 2708 1.456 1.272 5.76 18.54 19.59 8.03 I.847 19.65 7.07375 4.813 2583 1.456 I 260 6.11 18 19 20.78 7 RX 1.839 u).68 7.43438 4 688 2264 1.456 1.227 7.04 17.26 23.95 7.47 1.819 23.31 8.38,500 4.563 1978 1.456 1.194 7.95 16.35 27.04 7 06 1.799 25.74 9.25,625 4.313 1492 1.456 1.129 9.70 14.61 32.97 fi 33 1 760 30 03 10.80,750 4.063 1lOi 1.456 1.064 11.34 12.97 38.55 5.61 1.722 33.64 12.10

,109 6.407 10 80 1.734 1.677 2.23 32.2 7.58 13.96 2.304 11.84 3.58.134 6.357 10.38 1.734 1.664 2.73 31.7 9.29 13.74 2.295 14.40 4.35,169 6.287 9.82 1.734 1.646 3.43 31.0 11.66 13.44 2.283 17.87 5.40,180 6.265 9.65 1.734 1.640 3.64 30.8 12.39 13 35 2.280 18.94 5.72,188 6.249 9.53 1.734 1.636 3.80 30.7 12.93 13.28 2 277 19.71 5.95,219 6.187 9.07 1.734 1.620 4.41 30.1 14.99 13.02 2.266 22.64 6.83,250 6.125 8.62 1.734 1.604 5.01 29.5 17.02 12.75 2.256 25.5 7.69,277 6.071 8.25 1.734 1.589 5.52 28.9 18.78 12.53 2.246 27.9 8.42,280 6.065 8.21 1.734 1.588 5.58 28.9 18 98 12.51 2.246 28.1 8.50,375 5.875 7.00 1.734 1.538 7.36 27.1 25.04 11.73 2.214 36.1 10.90,432 5.761 6.35 1.734 1.508 8.40 26.1 28.58 11.29 2.195 40.5 12.23,500 5.625 5.63 1.734 1.473 9.62 24.9 32.71 10.76 2.173 45.4 13.71562 5.501 5.04 1.734 1.440 10.70 23.8 36.40 10.29 2.153 49.6 14.9,,718 5.189 3.76 1.734 1.358 13.32 21.1 45.30 9.16 2.104 59.0 17.81,864 4.897 2.82 1.734 1.282 15.64 18.8 53.17 8.16 2.060 66.3 20.03

,109 8.407 42.0 2.258 2.201 2.92 55.5 9.91 24.04 3.61 26.4 6.13,148 8.329 40.1 2.258 2.180 3.94 54.5 13.30 23.59 3.00 35.4 8.21,158 8.309 39.6 2.258 2.175 4.20 54.2 14.29 23.48 2.99 37.7 8.74,165 8.295 39.3 2.258 2.172 4.39 54.0 14.91 23.40 2.99 39.3 9.10,188 8.249 38.2 2.258 2.160 4.98 53.4 16.94 23.14 2.98 44.4 10.29,203 8.219 37.5 2.258 2.152 5.37 53.1 18.26 22.97 2.98 47.7 11.05,219 8.187 36.8 2.258 2.143 5.78 52.6 19.66 22.94 2.97 51.1 11.86,238 8.149 35.9 2.258 2.133 6.27 52.2 21.32 22.58 2.97 55.2 12.80,250 8.125 35.4 2.258 2.127 6.58 51.8 22.37 22.45 2.96 57.7 13.39,277 8.071 34.2 2.258 2.113 7.26 51.2 24.70 22.15 2.95 63.4 14.6Y,322 7.981 32.4 2.258 2.089 8.40 50.0 28.56 21.68 2.94 72.5 16.81,344 7.937 31.5 2.258 2.078 8.95 49.5 30.43 21.42 2.93 76.9 17.82,352 7.921 31.2 2.258 2.074 9.15 49.3 31.1 21.3 2.93 78.4 18.19375 7.875 30.3 2.258 2.062 9.72 48.7 33.0 21.1 2.92 82.9 19.22,406 7.813 29.1 2.258 2.045 10.48 47.9 35.6 20.8 2.91 88.8 20.58,469 7.687 26.8 2.258 2.012 12.02 46.4 40.9 20.1 2.89 100.3 23.25,500 7.625 25.8 2.258 1.996 12.76 45.7 43.4 19.8 2.88 105.7 24.52,593 7.439 22.8 2.258 1.948 14.96 43.5 50.9 18.8 2.85 121.4 28.14,625 7.375 21.8 2.258 1.931 15.71 42.7 53.4 18.5 2.84 126.5 29.32,718 7.189 19.2 2.258 1.882 17.84 40.6 60.6 17.6 2.81 140.6 32.60,812 7.001 16.8 2.258 1.833 19.93 38.5 67.8 16.7 2.78 153.7 35.63875 6.875 15.4 2.258 1.800 21.30 37.1 72.4 16.1 2.76 162.0 37.57,906 6.813 14.7 2.258 1.784 21.97 36.5 74.7 15.8 2.75 165.9 38.48

.134 10.482,165 10.420,188 10 374,203 10.344,219 10.316.250 10.250,279 10.192,307 10.136,348 10.054

127 2.81 2.74 4.47 86.3 15.2 37.4 3.75 63.0 11.72123 2.81 2.73 5.49 35.3 18.7 36.9 3.74 76.9 14.30120 2.81 2.72 6.24 84.5 21.2 36.6 3.73 87.0 16.19118 2.81 2.71 6.73 84.0 22.9 36.4 3.73 93.6 17.41I16 2.81 2.70 7.28 83.5 24.7 36.1 3.72 100.9 18.78113 2.81 2.68 8.25 82.5 28.0 35.7 3.71 113.7 21.16110 2.81 2.67 9.18 81.6 31.2 35.3 3.70 125.9 23.42107 2.81 2.65 10.07 30.7 34.2 34.9 3.69 137.5 25.57103 2.81 2.63 11.37 79.4 38.7 34.4 3.68 154.0 28.66

Item 2. Design Properties of Pipe (Continued)

r- ----l--- \ \ - \ I - ->.

Page 398: Process Equipment Design

Properties of Pipe 389

1 0

20

a0 ST

D = :&hM 4oxs

60

801001 2 01 4 0

Pipe Sizeand Schedule

OutsideDiameter

inches

Numberand/orWeight

40ST 40s

60 X8 80s

10 80

D = 10.750 100

120140

160

5s1 0 s

20

30

D&750 4’

ST 405

xs 80s60

80loo

1201 4 0

160

102 030 ST

4 0

14D -14.006 6om

8 0loo1 2 01 4 0

1020

S T3 0

18 X8D = 18.000 40

6 0

WallThick-ness

inrhent

0 365,395,500.531,593,718,750.843

1.0001.0621.125

Surface AreaOutside Inside

CrawSectional Weizht of

InsideDiameter

FifthPOWUofID

in.6d’

10198.088.185.380.0i O . l6 7 . 76 1 . 25 1 . 34 7 . 74 4 4

Metal Flow Pip+ - water

d10.0909 9609.7309.6Bi9.5649.3149.2309.0648 . 7 5 08.6258 . 5 0 0

sq It sq ft. ARti AI& l b lbper ft per ft sq in. sq in. per ft per ft

A0 Ai A A, w WC2 . 8 1 2 . 6 2 11.91 7 8 . 9 4 0 . 5 3 4 . 12 . 8 1 2 . 6 1 12.85 77.9 43.7 33.72 . 8 1 2 . 5 5 16 10 7 4 . 7 54.i 3 2 . 32.81 2 . 5 4 17.06 7 3 . 7 58 0 31.92 . 8 1 2 . 5 0 18.92 71 8 64 3 31.12 . 8 1 2 . 4 4 2 2 . 6 3 6 8 . 1 76.9 29.52 81 2.42 23.56 6 7 . 2 80 I 29 I2 . 8 1 2 . 3 7 26 24 64.5 89 2 27 92 . 8 1 2.29 30 63 6 0 . 1 104.1 26.02 . 8 1 2.26 3 2 . 3 3 5 8 4 109 9 2 5 . 32 . 8 1 2 . 2 3 34.02 5 6 . 7 1 1 5 . 7 2 4 . 6

Radiusof

Gyra-tion

inches7”

3 . 6 73.663.633.623.603.563.553.523 . 4 73 . 4 53.43

Momentof

Inertiain.’I

160.8172.5212.02 2 3 . 42 4 4 . 8286.2296 332 4 . 33 6 7 . 9384.0399.4

165:tao

1 2 . 4 2 012.390

,203 12.344,219 12 312.238 12.274.250 12.250.279 12.192.300 12.150.330 12.090,344 12.062.375 12.000,406 11.938,438 11.874,500 11.750,562 11.626,625 11.500,687 11.376.x4:( 11.064,872 11.000

1.000 10.7501.125 1 0 . 5 0 01.219 1 0 . 3 1 31.312 1 0 . 1 2 6

296 3 . 3 4 3 . 2 5 6 . 5 2 1 2 1 . 2 2 2 . 2 52.5 4.45 129.2292 3.34 3 . 2 4 7 . 1 1 1 2 0 . 6 2 4 . 2 52.2 4 . 4 4 1 4 0 . 52 8 7 3 . 3 4 3 . 2 3 8 . 0 0 119.7 2 7 . 2 5 1 . 8 4 . 4 4 1 5 7 . 52 8 3 3 . 3 4 3 . 2 2 8 . 6 2 119.1 29.3 5 1 . 6 4 . 4 3 1 6 9 . 3279 3 . 3 4 3 . 2 1 9.36 118.3 31.8 5 1 . 2 4 . 4 2 1 8 3 . 22 7 6 3 . 3 4 3 . 2 1 9 . 8 2 117.9 33.4 5 1 . 0 4 . 4 2 191.9269 3 . 3 4 3 . 1 9 1 0 . 9 3 116.7 3 7 . 2 5 0 . 6 4.41 212.72 6 5 3.34 3 . 1 8 11.73 115.9 39.9 5 0 . 2 4 40 227.52 5 8 3 . 3 4 3 . 1 7 1 2 . 8 8 114.8 4 3 . 8 49.7 4.39 2 4 6 . 52 5 5 3 . 3 4 3 . 1 6 13.41 114.3 45.6 4 9 . 5 4.39 2 5 3249 3 . 3 4 3 . 1 4 1 4 . 5 8 113.1 49.6 49.0 4 . 3 8 2792 4 2 3 . 3 4 3 . 1 3 1 5 . 7 4 111 9 53.5 4 8 . 5 4 . 3 7 3 0 0236 3 . 3 4 3 . 1 1 16.94 110.7 5 7 . 6 47.9 4.36 3 2 12 2 4 3.34 3 . 0 8 19.24 108.4 6 5 . 4 4 7 . 0 4 . 3 3 3622 1 2 3 . 3 4 3 . 0 4 21.52 106.2 7 3 . 2 46.0 4 . 3 1 4 0 1201 3 . 3 4 3 . 0 1 23.81 103.9 80.9 4 5 . 0 4 . 2 9 439191 3.34 2.98 2 6 . 0 4 101.6 88.5 4 4 . 0 4 . 2 7 4 7 51 6 6 3 . 3 4 2.90 3 1 . 5 3 96.1 1 0 7 . 2 4 1 . 6 4 . 2 2 5 6 2161 3 . 3 4 2 . 8 8 3 2 . 6 4 95.0 111.0 41.1 4.21 5 7 9144 3 . 3 4 2.31 36.91 9 0 . 8 125.5 39.3 4 . 1 7 6 4 2128 3 . 3 4 2 . 7 5 41.09 86.6 139.7 3 7 . 5 4.13 701117 3 . 3 4 2 . 7 0 44.14 8 3 . 5 150 1 3 6 . 2 4.10 7 4 2106 3 . 3 4 2 . 6 5 4 7 . 1 4 8 0 . 5 160.3 3 4 . 9 4.07 781

,188 1 3 . 6 2 4 469 3 . 6 7 3 . 5 7 8 . 1 6 1 4 5 . 8 2 7 . 7 6 3 . 1 4.88 195,220 1 3 . 5 6 0 45E 3 6i 3 . 5 5 9 . 5 2 1 4 4 . 4 3 2 . 4 6 2 . 5 4 . 8 7 2 2 6,238 13 524 4 5 2 3 . 6 7 3 . 5 4 1 0 . 2 9 143 6 3 5 . 0 6 2 . 2 4 . 8 7 2 4 4.250 1 3 . 5 0 0 4 4 3 3 . 6 7 3 53 10.80 1 4 3 . 1 3 6 . 7 6 2 . 0 4.86 2 5 5,312 13.375 4 2 8 3 . 6 7 3 . 5 0 13.44 140.5 4 5 . 7 6 0 . 8 4 . 8 4 3 1 5,375 1 3 . 2 5 0 408 3 . 6 7 3 . 4 7 16.05 137.9 54.6 59.7 4 . 8 2 3 7 3.4ofi 1 3 . 1 8 8 399 3 . 6 7 3 . 4 5 17.34 136.6 59.0 59.1 4 . 8 1 401.43x 1 3 . 1 2 5 389 3 . 6 7 3 . 4 4 18 66 135.3 6 3 . 4 5 8 . 6 4 . 8 0 429.469 1 3 . 0 6 2 380 3.67 3 . 4 2 19.94 134.0 6 7 . 8 58 0 4.79 4 5 7.500 13 000 371 3 . 6 7 3 . 4 0 21.21 1 3 2 . 7 7 2 . 1 5 7 . 5 4 . 7 8 4 8 4,593 12 814 3 4 5 3 . 6 7 3 . 3 5 24.98 1 2 9 . 0 84.9 5 5 . 8 4 . 7 4 5 6 2,625 1 2 . 7 5 0 3 3 7 3 . 6 7 3 . 3 4 26.26 1 2 7 . 7 8 9 . 3 5 5 . 3 4 . 7 3 5 8 9,656 1 2 . 6 8 8 329 3 . 6 7 3 . 3 2 27.50 1 2 6 . 4 93.5 54.8 4 . 7 2 6 1 4.75u 1 2 . 5 0 0 3 0 5 3 . 6 7 3 . 2 7 3 1 . 2 2 1 2 2 . 7 106.1 5 3 . 1 4 . 6 9 6 8 7,937 1 2 . 1 2 5 2 6 2 3 . 6 7 3 . 1 7 3 8 . 4 7 1 1 5 . 5 1 3 0 . 8 50.0 4 . 6 3 825

1 . 9 9 3 1 1 . 8 1 4 230 3 . 6 7 3 . 0 9 4 4 . 3 2 1 0 9 . 6 150.7 4 7 . 5 4 . 5 8 9301 . 2 5 0 11.500 201 3 . 6 7 3 . 0 1 5 0 . 0 7 1 0 3 . 9 170.2 45 0 4.53 1 0 2 71 . 3 4 4 11.313 1 8 5 3 . 6 7 2.96 5 3 . 4 2 100.5 181.6 43.5 4.50 1 0 8 21 . 4 0 6 11.183 1 7 5 3 . 6 7 2 . 9 3 55.63 9 8 . 3 189.1 4 2 . 6 4 . 4 8 1117

.188 15.624 931

.238 1 5 . 5 2 4 9 0 2

.250 15.500 895.281 1 5 . 4 3 8 8 7 7.312 1 5 . 3 7 5 8 5 9,344' 1 5 . 3 1 2 8 4 2,375 1 5 . 2 5 0 8 2 5.406 1 5 . 1 8 8 808,438 1 5 . 1 2 4 791,469 1 5 . 0 6 2 7 7 5.500 1 5 . 0 0 0 7 5 9.531 14.938 7 4 4.656 1 4 . 6 8 8 6 8 4.68X 1 4 . 6 2 5 6 6 9,750 1 4 . 5 0 0 6 4 1,843 1 4 . 3 1 4 601

1.031 1 3 . 9 3 8 5 2 6I.218 1 3 . 5 6 4 4 5 91.43x 1 3 . 1 2 4 3 8 9I.506 13.o!lo 3711.593 1 2 . 8 1 4 3 4 5

4.09 9 . 3 4 1 9 1 . 7 31.8 83.0 5 . 5 9 2 9 24 . 0 6 1 1 . 7 8 1 8 9 . 3 40.1 8 2 . 0 5 . 5 7 3 6 64.06 12 37 1 8 8 . 7 4 2 . 1 8 1 . 7 5 . 5 7 3 8 44 . 0 4 1 3 . 8 8 1 8 7 . 2 4 7 . 2 81.1 5 . 5 6 4 2 94.02 1 5 . 4 0 1 8 5 . 7 5 2 . 4 8 0 . 4 5 . 5 5 4 7 44.01 1 6 . 9 2 1 8 4 . 1 5 7 . 5 79.7 5 . 5 4 5 1 93.99 18.41 1 8 2 . 7 62.6 7 9 . 1 5 . 5 3 5623.98 1 9 . 8 9 1 8 1 . 2 07.8 7 8 . 4 5 . 5 2 6 0 53.96 2 1 . 4 1 179.6 7 2 . 8 77.8 b.bO 6 4 93.94 2 2 . 8 8 1 7 8 . 2 7 7 . 8 77.2 5 . 4 9 6913.93 2 4 . 3 5 1 7 6 . 7 8 2 . 8 7 6 . 5 5.48 7 3 23.91 25.81 1 7 5 . 3 87.7 75.9 5 . 4 7 7 7 33 . 8 5 3 1 . 6 2 169.4 1 0 7 . 6 73.4 5 . 4 3 9 3 33 . 8 3 3 3 . 0 7 168.0 1 1 2 . 4 72.7 5.42 9 7 23.80 3 5 . 9 0 165.1 1 2 7 . 5 7 1 . 5 5.40 1 0 4 73 . 7 5 4 0 . 1 4 160.9 1 3 6 . 5 69.7 5 . 3 7 1 1 5 73 . 6 5 4 8 . 4 8 152.6 1 6 4 . 8 6 6 . 1 5.29 1 3 6 53 . 5 5 5 6 . 5 6 144.5 1 9 2 . 3 62.6 5 . 2 3 1 5 5 63 . 4 4 6 5 . 7 9 135.3 2 2 3 . 7 5 8 . 6 5 . 1 7 1 7 6 13 . 4 0 6 8 . 3 3 132.J 2 3 2 . 3 57.5 5 . 1 5 1 8 1 53 35 7 2 . 1 0 1 2 9 . 0 2 4 5 . 1 5 5 . 8 5 . 1 2 1894

,250 1 7 . 5 0 0 1641,312 1 7 . 3 7 5 1 5 8 4,375 1 7 . 2 5 0 1527.43x 1 7 . 1 2 4 1472.500 1 7 . 0 0 0 1420.562 16.876 1369.594 1 6 . 8 1 3 1344,625 1 6 . 7 5 0 1 3 1 8,719 1 6 . 5 6 2 1247,750 1 6 . 5 0 0 1 2 2 3

4 . 1 94.194.194 . 1 94 . 1 94 . 1 94.194.194.194.194.194 . 1 94 . 1 94 . 1 94 . 1 94.194 . 1 94 . 1 94.194.194.19

4.714 . 7 14.714 . 7 14 . 7 14 . 7 14.714 . 7 14 . 7 14 . 7 1

4 . 5 8 13.94 2 4 0 . 5 4 7 . 4 104.1 6 . 2 8 5 4 94 . 5 5 17.36 237.1 5 9 . 0 1 0 2 . 7 6 . 2 5 6794 . 5 2 2 0 . 7 6 2 3 3 . 7 7 0 . 6 1 0 1 . 2 6.23 8 0 74 . 4 8 24.li 2 3 0 . 3 8 2 2 99.7 6 . 2 1 9324 . 4 5 27.49 227.0 93.5 98 3 6.19 1 0 5 34 . 4 2 30.79 22a.i 101. i 96.9 6.17 11714 . 4 0 32 46 222 0 110 4 96 I 6 16 12314 . 3 9 3 4 . 1 2 220 4 116 0 95.4 6 . 1 5 12894 . 3 4 3 8 . 9 8 2 1 5 1 5 132 5 93 3 6 . 1 2 14584.32 4 0 . 6 4 2 1 3 . 8 138.2 92.6 6.10 1515

Item 2. Design Properties of Pipe (Continued)Areas and Weizhia

SectionMOdUlllE

in.3Z

29.913 2 . 139.441.64 5 . 553.255.16 0 . 36 8 . 47 1 . 47 4 . 3

2 0 . 322.02 4 . 726.628.73 0 . 133.43 5 . 739.04 0 . 543.84 7 . 150.456.762.86 8 . 87 4 . 58 8 . 190.8

100.7109.9116.4122.6

2 7 . 83 2 . 334.836.545.053.357.361 465 369 186.384.187.798.2

1 1 7 . 91 3 2 . 81 4 6 . 81 5 4 . 6159.6

36.545.848.053.659.364.870.375.681.186.391.596.6

1 1 6 . 6121 4130 91 4 4 . 6170.61 9 4 . 52 2 0 . 1226.9236.7

61.075.589.6

103.6117.0130.2136.8143.3162.0168.3

TI \

- - -\I -

-1 I ----- -

Page 399: Process Equipment Design

3 9 0 Properties of Pipe

NomrnalPipe Size

andOutside

Diameterinches

sdleduleNU?ObZand/orWeight

80

I8100

D = !a.000 120140

160

20 60D = 20.000

10 ,250 5.24 5.11 15.51 298.6 129.3 75.7.312 5.24 5.07 19 36 294.8 127.6 93.8

20 ST .375 5.24 5 04 23.12 291 .o 126.0 111.4.438 5.24 5.01 26 9 287.2 124.4 128.9

30 xs ,500 5.24 4 97 30.6 283.5 122.8 145.7,562 5 24 4 94 34.3 279.8 121.2 162.4

40 ,593 5 24 4 93 36.2 278.0 120.4,625 5.24 4 91 38 0 276.1 119.6,812 5.24 4 81 48 9 265.2 114.8,875 5 24 4 78 52 6 261.6 113.3,906 5 24 4 76 54 3 239.8 112.5

80 1.031 5.24 4.70 61 4 252.7 109.41.250 5 24 4.58 73 6 240.5 104.1 325.1

100 1 281 5 24 4 57 75.3 238 8 103.4 331.6120 1.500 5 24 4 45 87.2 227.0 98.3 375.5140 1.750 5 24 4 32 100.3 213 8 92.6 421.7

1.844 5.24 4.27 105.2 209.0 90.5 437.9160 1.968 5.24 4.21 111.5 202.7 87.8 458.6

19.50019.37519.25019.12419.00018 87518.81418 75018.37618.25018 18817 93817 50017.43817.00016 50016 31316.064

2 822.732.642.562 482.402 362 322 102 021.991 861 641 611 421 221.161.07

52.765.878.691.5

104.1116 8122.9129 3166 4178.7184 8208.9250.3256.1296.4341.1357.5379.1

6.986 966.946 926.906 886.866 85F 796 776.766.726.646.636 566 486.456.41

757938

1114128914571624170417872257240924832 7 7 2325133163755421743794586

170.4178.7225.7240.9 8248.3 ?277.2 I

22 10

D = 22.000 STx s

21.50021 25021.000

4 594.334 08

58.186.6

114.8

7 697.657 60

101014901953

,250 5.76 5.63 17 1 363 157.2 91.8,375 5 76 5.56 25.5 355 153 6 135.4,500 5 76 5.50 33 8 346 150.0 177.5

10 ,250 6.28 6.16 18.7 434 187.8 109.6,312 6.28 6.12 23.2 429 185.8 135 8

20 ST ,375 6.28 6.09 27.8 425 183.8 161.9.438 6.28 6.05 32.4 420 181.9 187.4

xs ,500 6 28 6.02 36.9 415 179.9 212.530 ,562 6.28 5.99 41.4 411 178.0 237 0

625 6.28 5.96 45.9 406 176.0 261,687 6.28 5 92 50 3 402 174.1 285,750 6 28 5.89 54.8 398 172.2 309,968 6.28 5.78 70.0 382 165.6 388

1.031 6.28 5.74 74.4 378 163.7 41080 1.218 6.28 5.65 87.2 365 158.1 473

100 1.531 6.28 5.48 108.1 344 149.1 571120 1.812 6.28 5.33 126.3 326 141.2 652140 2.062 6.28 5.20 142.1 310 134 3 719

2 188 6.28 5.14 149.9 302 131.0 751160 2.343 6.28 5.06 159.4 293 126.9 788

24 40

D = 24.000 6.

23 50023.37623.25023.12523 00022.87622.75022.62622 50022.06421.93821.56420.93820.37619.87619.62519.314

7.176.986 796 616 446 266.095.935.775.235.084 664 023.513.102.912.69

63.478.994 6110.1125.5140 7156.0171.1186.3238.1252.9296.4367.4429.4483.2509.7542.0

8.408.388.358.338.318.298.278.258.228.158.138.077.967.877.797.757.70

13161629194322492550284031403420371046534920567068527824863090109455

~PLMSTxs

30D = 30.000 20 xs

29.37629.25029.12529.00028.87528.750

21.921.421.020.520.119.6

98.9118.7138.0157.6176.8196.1

10.5010.4810.4510.4310.4110.39

321938334434504056356230

34 STD = 34.000 X5

33.25033.000

40.639.1

134.7178.9

11.8911.85

55997383

36 STD = 36.000 X S

35.25035.ow

54.452.5

142.7189.6

12.6012.55

66598786

42 STB = 42.%0% X 6

10 ,312 7.85 7.69 29.1 678 293.5 214ST ,375 7.85 7.66 34.9 672 291.0 255

,438 7.85 7.62 40.6 666 288.4 296,500 7.85 7.59 46.3 661 286.0 336,562 7.85 7.56 52.0 655 283.6 37%

30 .625 7.85 7.53 57.7 649 281.1 415

,375 8.90 8.70 39.6 868 376.0 329,540 8.90 8.64 52.6 855 370.3 434

,375 9.44 9.23 42.0 976 422.6 370.wo 9.44 9.16 55.8 962 416.6 488

,375 11.0 10.80 49.0 1336 578.7 506,500 11.0 IO.23 65.2 1320 571.7 668

Thii table ia believed to be the mast nearly complete tabulation of the dimensional properties of commercially ‘available sizes of steel pipe ever published. It includes: the older weights ofpipe (ST = standard weight, XS = extra strong, XX = double extra strong), the schedules given id ASA Standard B36.10 (10, 20. etc.). and those given in ASA Standard U3G.19 (5S, IOS,4OS, 80s). the latter being applicaable to stainless steel only.

Piping designers will find herein-all the dimensional data they may need to determine:Pipe wall thickneasea required to resist internal pressure.Bending stresses resulting from line expansion.Bending streesa caused by weight loadings.Pipe column sizes required to sustain given axial loads.Flow areas and fifth powers of the diameter, useful in pressur?edrop calculations.Surface weas for use in evaluating heat losses and insulation and coating requirements.Definition of Properties Listed in Item 2

41.25041.000

119.4115.9

166.7221.6

14.7214.67

1062114037

WallTbick-

inchesf

0.812,937

1.1561.3751.5621.6881.781

InsideDiameter

inchesd

16.37516.12615.68815.25014.87614.62514.438

FifthPowerof ID

in.’d6

11771090950825728669627

sq It sq ft AWlI Amper ft per ft sq in. sq in.

A . Ai A AI4.71 4.29 43.87 210.64.71 4.22 50.23 204.24.71 4.11 61.17 193.34.71 3.99 71.81 182.74.71 3.89 80.66 173.84.71 3.83 86.48 168.04.71 3.78 90.75 163.7

Pipelb

per ftto

149.2170.8208.0244.2274.3294.0308.5

Ibper R

WUJ91.288.483.779.175.372.770.9

R a d i io f

GYiWtion

inchesrP

6.086.045.975.905.845.805.77

Momeoto f

Inertiain.’

I1624183421802498275029083020

180.5203.8242.2277.6305.6323.1335.6

.375 6.81 6.61 30.2 501 216.8 191,500 6.81 6.54 40.1 491 212.5 250

25.25025.000

10 269.77

102.6136.2

9.069.02

24793257

Item 2. Design Properties of Pipe (Continued)

Surface AreaOUtside Inside

Areas and WeightsCraneSeetional

M&l FlowWeight of

D - outside diameter of pipe, in.d = inside diameter of pipe, in.I = nominal wall thicknm of pipe, in.

A, = ff = outside pipe surface, 8q ft per ft length

Ai = % = inside pipe surface, sq It per ft length

(D* - d*)rA = ~ = crawsectional metal area, aq in.

4

A, = $ = crws-wtional flow area. sq in.

w = 3.4.4 = weight of pipe. lb per ft length10~ = 0.433Ay = weight of w#ter filling, lb per It lengtn

r* = I d/Dz+lr

- = ~ = radius of gyration, in.A 4

I = Ary* = 0.0491(0’ - d’) = moment of inertia in.’

z = ; = o,og*2 DE++= section modulus. in.3

i

Page 400: Process Equipment Design

A P P E N D I X L

Page 401: Process Equipment Design

I

A P P E N D I X

~

STRENGTH OF MATERIALS*

c Stress in Thousands of Pounds per Square Inch I/

(Cour~~esy of the Baldwin Locomot.ive Works)

Metals and AlloysAluminum, castAluminum, bars, sheetsAluminum, wire, hardAluminum, wire, annealedAluminum, Z-70/, Ni, Cu, Fe, etc.Aluminum Bronze, 5% to 7% v0 AlAluminum Bronze, 10% Al

Brass, 17% ZnBrass, 23% ZnBrass, 30% ZnBrass, 39% ZnBrass, 50% ZnBrass, cast, commonBrass, wire, hardBrass, wire, annealed

Bronze, 8% SnBronze, 13% SnBronze, 20% SnBronze, 24% SnBronze. 30% SnBronze, gun metal, 9% Cu, 1% SnBronze, Manganese, cast

110% Sir

Br.onze, Manganese, rolled 2% M nBronze, Phosphorus, cast 9% Sn

IBronze. Phosphorus, wire 1% PBronze, Silicon, cast, 3% SiBronze, Silicon, cast, 5% SiBronze, Silicon, wireBronze, Tobin, cast

I

38% ZnBronze, Tobin. rolled 135% suBronze, Tobin, cold-rolled >i% Pb

Copper, castCopper, plates, rods, boltsCopper, wire, hardCopper, wire, annealed

Delta Metal, castDelta Metal, platesDelta Metal, barsDelta Metal, wire I

55-60s CuW-40% Zn

2- 4% Fel- 2 % SK1

Tension, ElasticIJltimate Limit

*Courtesy of the Baldwin Loconroti\e \l’orks.

15 6.524-28 12-1430-65 16-3020-35 14-1-o-.50 25i5 40

85-100 60

32.6

ii:141.13 1

1 a-24HO30

a.27.68.6

17.117.9

6

1 6 ’

28.529.43322

5.625-55

6010050

1005575

1086680

1 0 0

1920

2 25.6

10308024.

.

4u

‘ 5X-3535-65

36

610

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Page 402: Process Equipment Design

Strength of Materials 393

Stress in Thousands of Pounds per Square Inch (Continued)

Elasticf.imit18.8

1 Tension,Metals and Alloys I Jltirtmtr

German Silver, 17.2% Zn, 21.1% Ni 40 9

Gold, cast ‘0Gold, wire 30Gold, Copper, 5% Au, 1% CII 50

Iron, cast, common 15-18Iron, cast, gray 18-24Iron, cast, malleahl~~ 27-3.5

lron, wrought, shapes 48Iron, wrought, bars .50Iron, wrought, unannealed 8 0Iron, wrought, wire, anneal14 6 0

I,t:ad, cast 1.8I,ead, pipe, wire 2.2-2.5I,ead, rolled, sheets 3 3

Platinum, wire, unanuealrd 3 3Platinum, wire, annealed 32

Silver, rolled 40

Steel, boiler plates,* fire-box 55-6.5Steel, boiler plates, flange plates 52-6'Steel, castings,* soft 60 -Steel, castings, medium 70Steel, castings, hard 8 0Steel, reinforcing bars,* plain, structural grade 55-70Steel, reinforcing bars, plain, intermediate 70-85Steel, reinforcing bars, plain, bard 8 0Steel, reinforcing bars, deformed, structural grade 55-70Steel, reinforcing bars, deformed, intermediate 70-85Steel, reinforcing bars, deformed, hard 8 0Steel, reinforcing bars, cold-twistedSteel, rivets,* boilers 43-XSteel, rivets, bridges *46-56Steel, rivets, buildings .46-56Steel, rivets, cars .48-58Steel, rivets, ships 55-65

Steel Shapes, bridges S5-6SSteel Shapes, buildings 55-6.5Steel Shapes, cars SO-65Steel Shapes, locomotives 55-65Steel Shapes, ships S8-68

Steel Alloys, Nickel Steel,* 3.25% NSteel Alloys, Nickel Steel, Shapes, plates, hars 85-100Steel Alloys, Nickel, Steel, rivets 70-80Steel Alloys, Nickel Steel, eye hars, unannealed 95-110Steel Alloys, Nickel Steel, eye bars, annealed 90-10s

s_ Steel Alloys, Copper Steel, 0.50% Cu 60-68

e- Steel Springs. untempered 65-110

Steel Wire, unannealed 120Steel Wire, annealed 8 0Steel Wire, bridge cable "00

Tin, cast 3 s- 1.6‘I’in, antimony, 10% Sn, 1% Sh 1 1

Zinc castZinc: rolled sheets

I,-6T-16

,* See Specifications of the American Society for Tcstiug Materials

I

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Page 403: Process Equipment Design

AUTHOR INDEX

Aries, Ft. S. (27):* 17, 317Armstrong, T. N. (56, 61): 26, 27, 318

Bagsar, A. B. (65): 26, 318Bergman, E. 0. (135, 140): 145, 146, 158, 319, 320Bibber, L. C. (49): 24, 318Bijlaard, P. P. (163): 201, 320Boardman, H. C. (12, 128): 5, 6, 76, 120, 135, 13i, 143, 31i, 319Bones, J. A. (202): 280, 281, 321Bowman, C. E. (51, 72, 76): 26, 29, 318Bresse, M. (130): 144, 319Bridgman, P. W. (217): 278, 321Brucker, W. H. (83): 30, 318Brummerstedt, E. F. (103, 153): 80, 81, 181, 319, 320Bryan, G. H. (131): 144, 319Burington, Il. S. (38): 21, 159, 318Burrows, W. R. (77, 109): 29, 104, 138, 318, 319

Campbell, H. C. (81): 30, 318Castigliano, A. : 199Chilton, C. H. (18, 229): 8, 17, 317, 321Clapeyron, B. P. E. (189): 268, 271, 321Clark, D. A. Il. (31): 19, 20, 22, 310, 317Coates, W. M. (117,123): 120, 126,127, 132, 133,135,136, 319Coe, E. H. (99): 39, 319Cohen, A. (40): 22, 318Comings, E. W. (218): 280, 287, 321Comstock, C. W. (226): 314, 315, 321Cook, G. (159, 193): 144, 275, 320, 321Coulomb, C. A. (190): 269, 321Cox, G. L. (236): 33, 322Cox, H. L. (220): 220, 300, 301, 321Crossland, B. (202): 280, 281, 321

Den Hartog, J. P. (32): 19, 20, 107, 317Dolan, T. J. (51, 72, 75, 76): 26, 29, 318Donnell, L. H. (44): 23, 63, 318Donovan, J. T. (225): 307, 308, 309, 321Dykhuizen, M. G. (112): 116, 117, 319

EIlinger, R. T. (78): 29, 30, 318Eichinger, A. (210): 281, 321Ellis, W. E. (83): 30, 318Ettinger, W. G. (222): 303, 321Euler, Leonard (39): 22, 171, 318

Faber, 0. (232): 185, 186, 321Faupel, J. H. (201, 215, 216): 279, 280, 287, 291, 292, 321Feely, F. J. (53, 91, 94, 95, 96): 26, 27, 38, 39, 318, 319Fey, R. F. (194): 275, 276, 286, 307, 321Fox, Lyle E. (98): 39, 319Franks, R. (57): 26, 318Fratcher, G. E. (228): 315, 321Freeman, J. W. (204, 214): 204, 282, 283, 284, 285, 321Freudenthal, A. M. (199): 278, 321Furbeck, A. R. (215): 280, 287, 291, 292, 321

Gagnebin, A. B. (56): 26, 318Gartner, Abraham I. (233): 187, 321Gensamer, M. (58, 96): 26, 38, 318, 319Glenday, C. (232): 185, 186, 321Goodier, J. N. (35): 20, 318Greene, Arthur M., Jr. (177, 178, 179, 180, 181, 182, 183, 184):

249, 250, 320Greene, T. W. (120, 122): 120, 319Griffith, A. A. (68, 69): 28, 318Gross, J. H. (50, 52, 73): 26, 29, 318

* Numbers in parentheses are reference numbers. See References section starting on page 317.395

Page 404: Process Equipment Design

396 author Index

Grossman, N. (62): 26, 318Gucer, D. E. (73): 29, 318Guest, J. J. (147): 178, 320

Haigh, B. P. (149): 180, 320Harris, L. A.. (71): 29, 318Hencky, H. (151, 206, 207, 208): 180, ‘81; 320, 321Hesse, H. C. (144): 161, 320Higgins, M. B. (213): 153, 154, 321Hill, R. (197): 278, 287, 321Hodge, P. G. (195): 278, 279, 287, 321Hoffman, 0. (198): 278, 321HShn, E. (113, 114): 120, 121, 122, 125, 126, 132, 135, 136,

137, 138, 319IIolloman, J. H. (54, 55): 26, 318Holroyd, R. (224,): 307, 308, 321How, H. (21): 11, 13, 14, 15, 16, 317Hrtko, D. (53): 26, 27, 38, 318lluber, M. T. (150, 209): 180, 281. 320, 321IIuggenberger, A. (115, 116): 120, 122, 126, 132. 133. 319

Jaffe, L. D. (54, 55, 82): 26, 30, RI8Jasper, T. M. (221, 242, 243): 133, 134, 293, 303, 421, 322Jefferson, T. B. (13, 14): 6, 317Johnston, B. G. (66): 26, 318Jorgensen, S . M. (234) : 187 , 322Josenhans, M. (225): 307, 308, 309, 321

Kerkhof, W. P. (45): 23, 24, 26, 28, 29. 3 I, 318Kinzel, A. B. (63, 121): 26, 120, 318, 319Kleppe, S. R. (53, 96): 26, 27, 38, 318, 319

Lam& G. (189): 254, 268, 269, 2i1, 272, 27.5, 288, 321Lang, H. A. (108, 110, 129): 104, 107, 108, 109, 110, 114. 115,

116, 138, 319Langenberg, F. G. (244): 322I,a Que, F. L. (236): 33, 322Larson, D. E. (7): 3, 317 ’Lavine, I. (26): 17, 317lazzair, C. C. (154): II%%, 32ULove, A. E. H. (33): 20, 317Lynn, C. V. (97): 39, 40, 319

Maccary, R. R. (194): 275, 276, 286, 307, 321MacCutcheon, E. M. (93): 38, 319MacGregor, C. W. (62): 26, 318MacLachlan, N. W. (111): 114, 319Macrae, A. E. (192): 27.5, 278, 287, 29 I. 321Maker, F. L.: 127, 133, 134, 135, 138Manning, W. R. D. (203,237, 238. 239. 240,: 273,278, 2’79, 281,

292, 293, 294, 321, 322Markovits, J. A. (225): 307, 308, 309, 321Marshall, V. 0. (142, 143): 161, 18.L, 320Mason, J. L. (4): 2, 317McCarthy, D. E. (55): 26, 318McGeady, L. J. (59, 60): 26, 27. 318Michel, R. (77): 29, 318Miller, F. H. (41, 126): 22, 129, 318, 319Miller, S. W. (118, 119): 120, 319Morley, A. (30): 19, 20, 201, 317Morris, J. L. (1s): 6, 317

Nadai, A. I,. (196, 24.1): 278, 292, Xl, 322Nehl, F. (74): 29, 318Yclwn, G. A. 178): 29. 30, 157. 318

Nelson, J. G. (139): 159, 320Newitt, D. M. (191): 273, 287, 321Newmark, N. M. (43, 71): 23, 29, 63, 318Newton, R. D. (2i): 17, 317Nordmark, G. E. (71): 29, 318Northrup, M. .S. (53, 91, 94, 95, 96): 26, ‘7. 38, 39, 318, 319Norton, M. R. (55) : 26, 318

O’Brien, H. L. (112, 134): 116, 1 I;, 1 I I, 319Orowan, E. (4’6, 70): 26, 28, 318Osborn, C. J. (66): ‘6, 318

Pagan. W. W. (138): 158, 320Pearson, K. (14,8) : 179, 268, 320Peirce, B. 0. (127): 129, 130, 319Plummer, F. I,. (8, 9): 3, 317Prager, W. (195): 278. 279. 287, 321

Rankiu, A. W. (77): 29, ,318Fiankiue, W. J. M.: 67Reddick, H. W. (126): 129, 319Hhys, C. 0. (1248): 120, 135, 136, 319Roark, R. J. (166): 210, .320Robertson, A. (193): 275, 321Robertson, T. S. (106): 26, 275, 319Robinson, E. L. (212): 282, 321Rossheim, D. B. (173, 174): 229, 230, 231, 232, 233, 236, 320Rouse, H. (141): 1.59, 320Rushton, J. H. (144): 161, 320

Sachs, G. (198, 219): 278, 291, 321Saint -Venant: 268Sanderson, C. F. (154): 184, 320Schierenbeck, J . , Jr . (223): 307, 321Schrier, E. (1): 2, 317Scotchbrook, A. F. (66): 26, 318&udder, C. M. (221): 303, 321Sechler, E. (37): 20, 318Sdy , F _ B . (231) I 2’38.307 .32XSiemon, K. 0. G. (164): 201, 320Sliepcevich, C. M. (204): 204, 282, 283, 284, 285, 321Smith, J. 0. (231): 208, 307, 321Smulski, E. (156): 185, 186, 320Sokolnikoff, I. S. (34): 20, 297, 318Sout,hwell, R. V. (36, 133): 20, 144,, 318, 319Spraragen, W. (222): 303 , 321StevenS, R. W. (23): 16, 311Stewart, R. T. (132): 144, 153, 319Stout, R. D. (50, 52, 59, 60, 66, 73): 26, 27. 29, 318Strum, R. G. (112, 134): 116, 117, 144, 319Summer, W. B. (155): 184, 320

Taylor, C. P. (232): 185, 186, 321Taylor, F. W. (156): 185, 186, 320Taylor, J. H. (1’71, 172): 227, 229, 320Theisinger, W. G. (104): 82, 83, 319Thielsch, H. (79): 30, 318Thompson, S. E. (156): 185, 186, 320Timoshenko, S. (29, 35, 42, 107, 136): 19, 20, 22, 23, 67, 69, 104,

11% 128, 149, 150, 151, 171, 187, 192, 210, 282, 3.17, 318, 319.320

‘l’odhuntcr, I. (148): 179, 268, 320Tresca, II. (190): 269, 321Trilling, C. (158): 144, 320‘I’saug, S. (50): 26. 29, 318

-

Page 405: Process Equipment Design

Author Index 397

Wesstrom. 1). f%. (173. li4). 229, 230, 231, 232, 233, 236,320

Williams, F. S. (;. (173. 17 f, 175) : 229, 230, 231, 232, 233, 236,320

Williams, f<oger, Jr. (230): 17, 321Wilson, UT. M. (43): 23, 63, 318Windrnburg, D. F. (158): 144, 320Witterstrom, E. (112): 116, 117, 319Wolosewick, F. E. (160, 161): 195, 198, 199, 320

Vanderheck, R. W. (S8): 26, 318van Iterson, F. K. T. (200): 278, 321Van Miser;. R. (152, 205): 180, 279, 281. 321, 320Voorhers, fl. K. (704, 211, 214): 204, 281. 282, 283, 284, 285,

286, 321

Wahl (241): 292, 322Waters, E. 0. (171, 172, 173. 171, 175): “27. 229, 230, 231. 232,

233,236,320Watts, Cr. W. (108, 109, 110, 129): 104. 107. 108, 109, 110, 114,

115, 116, 138, 319Weaver, J. B. (28): 17, 317Weher, W. W. (17): 6, 317Wentworth, R. P. (81): 30, 328

Zick, I,. P. (48, 165): 24, 25, 203, 205, 206, 207, 208, 209, 211,212, 213, 215, 216, 318, 320

Zimmerman, E. N. (90): 36. 319Zirnmerrnau, 0. ‘r. (26): 17, 317

Page 406: Process Equipment Design

SUBJECT INDEX

Allowable pressure, for shells under external pressure, 146on long, thin cylinders under external pressure, 144

Allowable stress, for nonferrous metals in tension, 336-340Allowable stresses, factors influencing, 24

in flange design, 244see also Stress, allowable

Allowable working pressure, chart for pipes and tubing underexternal pressure, 153

Aluminum alloys, chart for external-pressure vessels, 366-369Aluminum and aluminum alloys, ASME allowable stress in

tension, 336, 337Aluminum-bronze (alpha) chart for external pressure vessels,

377Aluminum Company of America, 64American Boiler Manufacturer’s Association, organization, 250American Institute of Steel Construction, 67, 319American Petroleum Institute, 3API accessories for tank shells, 349API-ASME code, 4, 7, 85, 92, 133, 138, 250

on Unfired Pressure Vessels, 250API dimensions for shell manholes, 350, 351API Standard 12 C, 3, 7, 8, 36, 37, 38, 46, 47, 50, 51, 52, 55, 56,

58, 59, 60, 68, 69, 77use of in vessel design, 36

API Standard 12 D, 44API tank-shell nozzle dimensions, 350API thicknesses for shell-manhole cover plates, 350API typical tank sizes and capacities, 34,6-348American Society of Mechanical Engineers, 4ASME allowable stress, for carbon and low-alloy pipes and

tubes, 335for nonferrous metals in tension, 336-340in tension for aluminum and aluminum alloys, 336, 337

+r tension for copper and copper alloys, 336, 337in tension for nickel and high-nickel alloys, 340

ASME allowable tensile stress for high-alloy steels, 34,2-345

ASME boiler code, development of, 250ASME code, antecedents of, 249

flange design, 240section VIII, scope of, 250vessels, testing of, 266welding, qualifications, 7

ASME code for unfired pressure vessels, 4, 24, 25, 31, 77. 85.88, 92, 93, 94, 95, 118, 133, 135, 146, 147, 148, 150, 151,152, 228, 229, 239, 240, 241, 250, 251, 252, 254, 255, 256,257, 258, 260, 263, 264, 266, 267

ASME stress-intensification factors for torispherical closures,138

American Society of Testing Materials, 33American Standards Association, 62, 158, 159

Code for Pressure Piping, 7specification for standard flanges, 219

American Welding Society, 7, 23, 25, 28, 29, 116, 117, 137, 205,207, 208, 209, 210, 213, 216

Anchor-bolt chair, centered, 191Anchor-bolt loading, for skirt of vertical vessel, 184Arc, unsupported maximum in ASME external-pressure vessels,

257Autofrettage prestressing of vessel shells, 286Autofrettaged shells, procedure in analysis, 287Axial stress in Lame analysis of thick-walled vessels, 269

Battelle Memorial Institute, 32Beam, uniformly loaded, 20

uniformly loaded continuous, derivation of equations forbending moment and deflection, 66

uniformly loaded with free ends, derivation of equations forbending moment and deflection, 65

Beam on elastic foundation, applied to shell connected to aflange, 230

bending relationships, 103chart of functions, 128

399

Page 407: Process Equipment Design

400 Subject Index

Beam formulas, 384Bearing plate, rolled-angle, 190

single-ring, 190sketch showing loading for vertica, vessel?, 185thickness, 187width, for skirts of vertical vessels 187

Bearing plates, for column supports, 201relationships for compression side for skirts of vertical \ rssrls,

185’relationships for tension side for skirts of vertical vessels,

185with centered chairs, 190

Bending, diagram of, in an element of a plate. 109in flat-plate closure, 106in shell of cylindrical vessel with flat-plate closure, IO C

Bending moment, diagram for horizontal vrssels, 205equation for cylindrical vessel with flat cover plate, 108equation for various types of beams, 381for wind load on vertical vessel, 159from seismic forces on unguyed vrrt,icaal \ rssrls. 168in plates, 100maximum longitudinal in horizontal L rss&. 204

Bending moments, in bearing plate with gussets, 187in guyed vessels, 162in shell under external pressure, 142table for bearing plates with gussets, 187table for four double-gusset lugs on vrrtical vessels. 199

Bending strength, of various metals and alloys, 392Bending stresses, in elliptical dished head near junction with

shell, 128in shell near jrmction with elliptical head, 128

Bethlehem Steel Company, 37Binomial equation, used in design of shells of vertical vessels, 171Blaw-Knox Company, 220Bolt data, table, 188Bolt size, optimum, 227Bolting area, for skirt supports of vertical vessels, 188Bolting calculations for bearing plates for vertical vessels, 187Bolting chairs, maximum number of centered chairs in vessel

skirts, 191Bolting steel, area for vertical vessels, 186Bottom design, for flat-bottomed cylindrical vessels, 58Bottom joints, typical, as recommended by API Standard It C,

58Bottom plate layout, drawing showing dimensions for large

flat-bottomed tank, 61Bottom plates, relationship of plate dimensions, 59Braun, C. F., and Company, 3, 6, 12. 82, 84. 93, 121. 156, 157,

160, 204Bresse, M., equation for external pressure on long. thin cylin-

ders, 144British tanks, failure of, 38Brittle fracture, 38

as function of size of crack, 28as influenced by yield point, 28effect of composition on failure, 27photograph showing in monobloc vessel, 27

Brittle rupture, 26Bryan, G. H., equation for external pressure on long, thin cylin-

ders, 144Buckling, critical stress in column, 22Buffalo Tank Company, 88Bursting pressure of thick-walled vessels, 281Butt joints, double-welded, 7

I single-welded, 7Butt welding, versus lap welding in vessel shells, 47

Ca,culation sheet for flange design, 246Ca,culations, for design of shell of a tall vertical vessel, 172Cantilever&earn formulas, 384Capacities and sizes of typical API storage tanks, 346-348Carbon steel, grade classification of flat, hot-rolled, 9Carbon steels, properties of. as recommended hy API Standtin!

12 c, 37Carnegie-Illinois Steel Corporation, 31, 32Cast iron, chart for external-pressure vessels. 380Centered chairs, used for bolting skirts of vertical vessels, 190Channels, properties of rolled-steel members, 353Charpy test, 26

figure showing U-not,CI data for some mild steels, 27Chart for thickness of \ternal-pressure vessels fabricated of

carbon steel, yield point 24,000 to 30.000 psi. 247yield point 30,000 to 38.000 psi, 148

Chemical engineering, deiiuition of, 1Chicago Bridge and Iron Company, 5Circular plate, dished, diagram showing detlection in, 101

flat, diagram showing b~~ndiug moments in, 101Circumferential moment around shell of horizontal vessel, 211Circumferential stress, at horn of saddle in she-11 of horizontal

vessel stiffened by head, 211at horn of saddle in unstiffened shell of horizontal vessel, 21 Idiagram showing in cylindrical shell under internal pressure,

105in cylindrical shell, derivation of equation for, .ESin horizontal vessel at horn of saddle support, 203, 209

Coates, W. M., theoretical analysis for local bending strrssrsat junction of head and shell, 126, 132, 133, 135, 136

Coates and Rhys, discussion of equations, 136Coates’s relationships, applied to torispherical closures, 135Coefficients, for cone, for calculation of bending moment and

shear, 115for cylinder, for calculation of bending moment and shear,

108for flat head, for calculat.iou of bending moment and shear,

108for hemispherical brads. 138

Cold-rolled steel. 9Collapse, chart for coef1icirnt.s for shells under external prrs-

SUIY. l/E4of shell by external pressurr. 143

Collapsing pressure, for pipes and tubing undrr external prrs-sure, 153

of vessel shells with circumferential ntifl’eners and undt.1external pressure, 144

Column, action, in support,ed roofs of tanks. 66instability, 22size, selection of, for Example Design 4.2, 73supports, for vertical vessels, 201

Combined stresses, ‘in shell of vertical vessel. 170under operating pressure in autofrettaged vessels, 288

Components of forces in elliptical dished head, 124Compression ring, calculation of thickness, 192

gusset-plate thickness, 193Compressive force on supports for vertical vessels, 186Compressive strength of various metals and alloys, 392Compressive stress, allowable axial in cylindrical shells, 23

caused by dead loads in vertical vessels. 156induced by guy-wire tension in vertical vessels, 162

Comstock, theory of wire-wound multilayer vessels, 314Concentrated-loaded-beam formulas, 384, 385Concentration cell, as cause of corrosion, 34Concrete, allowable compressive stress. 18t

mixes, table of average properties, 184

\ \ \ I -5_.

Page 408: Process Equipment Design

Cyclic loading, 24

Subject Index 401

strain hardening from, 28Cylindrical shells, under external pressure, ASME design, 255

udder internal pressure, ASME code design, 254with flat-plate closure, hending in the shell, 104

Dead loads, in vertical vessels, 156Dead-weight stress, in skirt of vertical vessel, 183Deflection, derivation of equation for uniformly loaded beam, 21

equation for beam on elastic foundation, 104equations for various beams, 384

Deformation, of cylindrical-shell section under external pres-sure, 142

Density, of various metals and alloys, 341Derivation, of equation for moment of inertia of cylindrical

shell, 159Design of saddles for horizontal vessels, 215Diameter limitations for vessels with fornled heads, 81Dimensions of seamless and welded steel pipe, 386Discontinuity stresses at junction of head and shell, 126Dome and umbrella roofs, for tanks, 68Dorr classifier, 2Drag coefXcients, for circular cylinders, 159Drawiugs, alphabet of lines, 324

conventions, 323dimension standards, 325, 326dimensioning, 325lettering, 324material symbols, 324sizes and scales, 323title block and bill of materials, 323

Draw-off elbow, figure showing for flat-bottomed tank, 60welded, dimensions for, as recommended by API Standard

12 c, 59Draw-off sump, figure showing for flat-bot,tomed tank, 60Ductile rupture, 26

Earthquakes, data on some disastrous, 167Eccentric loading, on vertical vessels, 168Eccentricity, ASME maximum for external-pressure vesscis,

25;h’, for au ellipse, 129

Effective area for shell of horizontal vessel. 205Efliciencies, for arc- and gas-welded joints, 254Elastic bending, 20Elastic deformation, as criterion in design, 19Elastic gain in heat-treated autofrettaged vessel.3, 2~4Elastic instability, 22

in vessel shells under axial load, 22Elastic limit, of various metals and alloys, 392Elastic-plastic interface, in autofrettage, 287

optimum location in autofrettage, 29 IElastic-plastic loadings, 23Elastic stability, equation for curved plate, 151

in design of shells of vertical vessels, 171of conical closures under external pressure. 153of dome roofs for tanks, 69of elliptical closures under exterual pressure, 152of hemispherical and torispherical closures under axternai

pressure, 151of long, thin cylinders under external pressure, 141of umbrella roofs for tanks, 69

Ellipse, equation for, 124figure showing trigonometric variables for, 129

Elliptic integrals, 129Elliptical closures, elastic stability under exter,nal pressure, 152

Ccmical closure, diagram showing compressive force at junctionwith shell, 114

experimentally determined stresses, 116Lonical closures. bending at the junction, 114

dished, for ASMF: code design, 259elastic stability uuder external pressure, 153location of maximum stresses, 116stresses in the cone, 115stresses in the shell, 115

Conical head, diagram of hoop stress, 113diagram of moment and shear at junction with shell. 114

Conical heads, 96Conical roof loads, equations for, .55

sketch showing, 54Conical roofs, radius of curvature, 63

self-supporting, 63Constant for Eq. 13.27, 362Constants, table for loads on bearing plate for vertical vessel,

186Coutinuous-beam formrdas, :384Continuous compression ring, 193Copper and copper alloys, ASMF: allowahlr stress in tension,

338, 339Copper, chart for external pressure \ essels, 373,opper-nickel alloys, charts for external pressure vessels. 375,

376Copper-silicon alloys, A and C. chart for external pressure

vessels, 374Lorrosion, concentration-cell attack, 33

deposit attack, 33galvanic-cell attack, 33impingement attack, 33jtress, 33types of, 32uniform, 33

Corrosion Handbook, 33, 319Cost, estimated, for flat-bottomed cylindrical vessels, 38

for large-diameter tanks as installed, 10for small and medium-sized flat-bottomed cone-roofed

storage tanks, 39per ton for large-diameter tanks, 40

factors, in proportioning tanks. 22iudices, 16of forming for ASME dished heads, 92relationships, in proportioning vessels with elliptical dished

heads, 80Costs, fabrication, 11Cox, equations for rnultilayrr vessels, 296Crack propagation, rate of, in brit,tle fracture, 28Cracks, in welds, 7Creep, 30

iu high-pressure vessels, 282-rupture test, 31test, 30

Crimping, photograph of press for, 12Criteria for failure of thick-walled shells hased on theory of

elasticity, 272Critical compressive stress, in shell of vertical 1 essels, 171Critical length between stifl’enerv for extrrual-pressure vessels,

144Critical load in shell under external pressure, 143Critical stress, in columns, 22

iu shell under external pressure. 12.3iu vessel shells, 22

Cutting, machine rate burden, 11Zutting time, for flame-cut steel plate, 11

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4 0 2 Subject Index

Elliptical dished heads, chart of blank dimensions, 94force components in, 124for ASME code design, 256

Elliptical head, chart of Huggenberger stress-intensificationfactors, 126

figure of deformation in the junction zone, 127figure of deformation of ellipsoid under internal pressure, 126photograph showing welding of, 121

Elliptical heads, cost of forming, 95equation for volume of, 95table of forming costs, 95

Elongation, minimum permissible in welds, 7percentage of, for various metals and alloys, 392

Embrittlement, 29Engineering News-Record, construction-cost index, 16, 17Equal angles, properties of rolled-steel members, 356Equations for various types of beams, 384Equipment cost, scaling of, 17Equipment-design group, in vessel procurement, 17Equivalent stress, 281Erection of vertical tower, photograph, 169Esso Research and Engineering Company, 39Euler column formulas, 22Euler formula, derivation of, 22Example calculation, comparing four theories of elastic failure

for thick-walled vessels, 276design of bearing-plate for vertical vessel, 188design of external bolting chairs for vessel skirt, 193for monobloc vessel at elevated temperature, 283guyed vertical vessel, 162of a two-shell shrink-fitted vessel, 302of combined stress at interface of shrink-fitted shells, 298of combined stresses in shrink-fitted multilayer vessels, 299of interface pressure in multilayer vessel, 298of interferences required in shrink-fitted vessel, 301of maximum out-of-roundness for ASME external-pressure

vessel, 256 ’of optimum autofrettage pressure, 288of preheat temperature for shrink fitting, 302of reaction of bolting ring with vessel skirt, 195of reinforcing area for opening in ASME vessel, 263of seismic forces on unguyed vertical vessel, 168of shell thickness and compression-ring area for ASME conical

closures, 259of shrinkage stresses in multilayer vessels, 299of stress distribution in thick-wall cylinder based on Lame

analysis, 271of stresses in a horizontal vessel, 216of stresses near junction of shell and elliptical dished head

using Coates’s method, 130of thermal stresses in skirt of vertical vessel, 196of thick shell wound at constant tension, 312of thickness for ASME toriconical closure, 259of thickness of ASME elliptical dished head, 256of thickness of ASME torispherical head, 258of wound thin shell with winding of dissimilar metal, 310

Example design, bottom for large flat-bottomed tank, 59complete shell design for a closed vessel (storage tank), 55design of vessel with flat-plate closure, 111for ring flange, 242of a torispherical closure for vessel under external pressure,

152of circumferential stiffeners for externai-pressure vessels, 150of conical closure for vessel under external pressure, 153of cylindrical shell under external pressure, 146of elliptical closure for vessel under external pressure, 152

Example design, of hemispherical closure for vessel under exter-nal pressure, 152

of integral flange, 245of roof and structural supports for large storage tank, 69of wind girder for open vessel, 52shell calculations for a tall vertical vessel, X2

External chair, bolting, for skirts of vertical vessels, 191External chairs, empirical dimensions, 191External pressure, effect on elastic stability of long, thin

cylinders, 141External-pressure vessels, chart for alpha-aluminum bronze, 277

chart for aluminum alloys, 366-369chart for annealed copper, 373chart for cast iron, 380chart for copper-silicon alloys A and C, 374chart for low-carbon nickel steel, 365chart for nickel, 370chart for nickel-chromium-iron alloy, 372chart for nickel-copper alloy, 371charts for austenitic stainless steel, 378, 379charts for copper-nickel alloys, 375, 376critical stress from external pressure, 143design of circumfcrcntial stiffeners, IJO

External stiffening rings for horizontal vessels, 214Extras, circular- and sketch-plate for steel, 10

classification for steel, 10for mill carbon-steel plate, 330for shearing and gas-cutting mill plate, 331for warehouse steel, 332gas-cutting for steel, 10length for steel, 10mill for steel, 10quality for steel, 10width and thickness for steel, 10

Fabrication-cost extras, table listing, 83Fabrication costs, 11Facings, for gaskets, 229

for standard flanges, 222Factor of safety, in determining allowable stress, 24Failure, of British storage tanks, photograph of, 39

types of, 19Failures, of dished heads, 137Faupel, equations for bursting pressure of thick-walled vessels,

280Faupel and Furbeck, elastic-breakdown pressures, 280

experimental values of residual stresses in autofrettagedvessels, 291

Field welding, photograph of shell circumferential butt joint,4 8

Firebox-quality steel plates, 333Firebox-quality steels, 78Flange, analysis for hub, 232

analysis for ring, 230analysis for shell connection, 230maximum axial hub stress, 236maximum radial ring stress, 237maximum tangential ring stress, 237-quality steel plates, 333-quality steels, 78relationships for hub, shell, and ring combined, 234stress equations used in ASME code, 239stress in loose or slip-on type, 238stresses, location of critical, 236thickness, for ASME spherical dished covers, 261with tapered hub, analysis of forces and moments, 229

Page 410: Process Equipment Design

Subject Index 403

Hammond Iron Works, 4, 41, 48, 49Harmonic vibration, componeuts, 165

in vertical vessels, 165Hartford Steam Boiler Inspection aud insurance Company, 250Heads, ASME dished, 88

cost of forming, 92table of dimensions, 89

cost extras for standard machining styles, 86dished only, 97elliptical dished, 92flanged and reverse dished, 97flared and dished, 97torispherical, 88

Heat treating, vessel shell, 157Heat treatment, for autofrettaged vessels, 274Height-to-diameter ratio for tanks, 41Hemispherical closure, dished, 138

elastic stability under external pressure, I51stress in head at junction, 139stress in head other than at junction, 139stress in shell other than at junction, I39

Hemispherical head, photograph showing, 96Hemispherical heads, 95

dished, for ASME code design, 258table of dimensions of available sizes, 96

Hemispheroid, photograph of, 5High-alloy steel, ASME allowable stress in tension, 342-345High-temperature service, selection of vessel material, 285HGhn, E., stress analysis by, 120, 121, 122, 126, 135, 136, 137Hoop stress, in elliptical closure, 121

in multilayer vessels, 299intensification factor, chart with combination of Huggen-

berger’s and Coates’s relationships, 132for vessel with elliptical closure from HShn data, 121

Horizontal storage tank, photograph showing, 93Horizontal vessel, example calculation of stresses, 216

supports for, 203with two saddle supports, 204

Hot-rolled steel, 9Hub of flange, proportioning, 244Hriggenberger, analysis of membrane stresses in elliptical

closures, 122, 126, 132, 133Hydrogen blisters, photograph of, in pressure-vessel steel, 29Hydrogen embrittlement, 29Hysteresis, in autofrettaged vessel, 294

I beams, properties of rolled steel members, 354Ideal stress distribution in thick-walled vessel, 285Impact tests, 26Impingement, as cause of corrosion, 33Influence numbers, for calculation of bending moment and

shear at junction of shell and conical closure, 115for calculation of bending moment and shear at junction of

shell and flat cover plate, 107for hemispherical closure, 138

Inland Steel Company, 10Inspecting-and-expediting group, in procurement of vessels,

18Instability, in shells under external pressure, 143Integral flange, design procedure, 244

example design, 245Interface pressures, in shrink-fitted shells, 298Interferences, required in shrink-fitted vessels, 300Interlocking-ribbon-wound vessels, 307Internal stiffening rings for horizontal vessels, 214Izod test, 26

Flanged and dished heads, ASME, 88figure showing dimensional relationships, 87

Flanged-only heads, 85standard, table of dimensions, 87

Flanged shallow-dished heads, 86Flanged standard-dished heads, 86Flanges, blind, 221

code rules for designing, 240lap-joint, 220nonstandard, 223screwed, 221slip-on, 219welding-neck, 219

Flanging machine, photograph of world’s largest, 77Flat circular plate, uniformly loaded and simply supported, 102

uniformly loaded with edges clamped, 102with concentric load, 103

Flat cover plate and shell, diagram of forces and moment atjunction, 104

Flat cover pl&s, value of constant for Eq. 13.2i, 362Flat heads and covers, ASME acceptable types, 262Flat-plate closures, for ASME design, 261

practical design of, 112stresses in, 110

Flat-ribbon-wound vessel, 307Flexural rigidity of plates, 100Flexure, in uniformly loaded beam, 20Forces and moments in tapered hub flange, 229Forgings, 5Formed closures, early uses, 76

under external pressure for ASME design, 260under internal pressure for ASME code design, 256

Formed head, drawing showing various types, 85Formed heads, hemispherical, elliptical-dished, torispherical,

standard-dished, conical, and toriconical, 4other types, 96pricing quantity differentials, 94types and their selection, 84types of applications, 76

Forming costs, for ASME flanged and dished heads, 93Franklin Institute, committee on cause of boiler explosions, 249Functions, elliptic integral, 130Fusion welding, 6

Galvanic series, table of, for metals in sea water, 33Gas welding, 6Gasket, equation for proportioning, 226

factors, table for, 228residual force, 225seating force, 225seating width, 229

Gaskets, and their selection, 224flat-ring, laminated, serrated, corrugated, and ring-joint, 225

Girder, details, drawing for large tank, 72size, selection of, for Example Design 4.2, 73

Girdler Corporation, 169Graphical comparison, of four theories of failure for vessel

shells, 180of various theories for design of vessel shells, X7

Gray-iron castings, 5Great Lakes Steel Corporation, 9Greene, A. M., Jr., history of ASME boiler code, 249Griffith theory of brittle fracture, 27Gusset plates, for compression rings of vessel skirts, 193Guy wires, 161Guyed vessels, 161

k I \ \ \I F--.--- - -

Page 411: Process Equipment Design

404 Subject index

k, for reinforced concrete, 184K, eccentricity of ellipse, 129

Labor, cost, variations of, 17hourly rates, for skilled, 16

Ladders, platforms and external piping, shown iu photograph ofvertical vessels, 160

Lam8 theory, compared with membrane theory, 275of stress analysis for thick-walled cylinders, 269

Lap-joint flanges, 220Lap joints, 7Length limitations, for vessels with formed heads, 81Load, derivation of equation based on deflectiorl of uniformly

loaded beam, 21Loading conditions, for vertical vessel, 170Location of saddle support, 203l~nng, thin cylinders, under external pressure. 141Longitudinal bending, moment constants RI and KT for hori-

zontal vessels, 206Longitudinal stress, in cylindrical shell. d&vation of equation

for, 45in horizontal vessels, 203

Low-alloy, high-strength steels, properties of, as recommendedby API Standard 12 C, 38

I,ug supports for vertical vessels, 197Lugs, with double vertical gussets for vessel supports. 198

with horizontal plates, for vertical vessels. 197with single vertical gusset for vessel support, 200without horizontal plates, 198

Lukens Steel Company, 77, 86, 87, 88, 89, 94, 96, 97

;llaccary and Fey, ideal prestressing of vessel shells, 286Machining, 5Machining styles, for flanges of formed heads, 87,Machining time, charts for, 15Major-to-minor-axis ratio, effect on maximum streHs and maxi-

mum strain in elliptical dished heads, 133Manhole cover-plate thickness for API tank shells, 350Manholes, charts for machining time, 1.5

for shells of storage tanks, 48Man-hours, required in fabrication, 11Manning, method of vessel design, 292

rupture of thick-walled vessels, 279Manning’s shear-stress theory, 273Marshall and Steveus equipmttut-cost index, 16“Massachusetts H&s ” for boilers, 250Material specifications, for code vessels, 2%

for shells of flat-bottomed cylindrical vessels, 36for vessels with formed heads, 77

Materials, typical physical properties, 341Materials of construction, relative costs of, 8Maximum compressive stress, in skirt of vertical vessel, 183Maximum moment, in center of span for horizontal vessel. 205

over supports for horizontal vessel, 204Maximum operating pressure for ASME vessels, 267Maximum permissible compressive stress, in skirt of rrrtical

vessel, 183Maximum-principal-stress theory, 272Maximum-shear-stress theory, 272Maximum-shear theory, in design of vertical vessels. 178Maximum-strain-energy theory, 273.Maximum-strain theory, 27311Zaximum-stress theory, in design of vertical vessels, 178Maximum tensile stress, in skirt of vertical vessel, 183Melting point of various metals and alloys, 3 JIMembrane circumferential stress, in elliptical dished head. 125

Membrane hoop stress, in elliptical dished head, 125Membrane theory. equations for cylindrical shell, 45Meridional section, for element of an elliptical dished head, 123Meridioual-stress-intensification factor, chart with combination

of Huggenberger’s and Coates’s relationship, 133for vessel with elliptical closure from Hiihn’s data, 122

Methods of analysis, other than maximum-principal-stresstheory for vertical vessels, 177

Methods of fabrication, 5Mill carbon-steel-plate extras, 330Mill price list for steel, 10Mill prices for steel, 9Modified spherical vessels, .iModulus of elasticity, 19, 98

chart for plain carbon and austmitic steels versus temper-ature, 146

of various metals and alloys, 392Moment, constants for calculation in compression ring of vessel

skirt, 192Moment of inertia, for cylindrical shell. derivation of approxi-

mate equation, 159of arc of shell for horizontal venal. 205of rolled-steel channels, 353of rolled-steel equal angles, 356of rolled-steel I beams, 354of rolled-steel tees, 359of rolled-steel unequal angles, 358of rolled-steel wide-flange beams, steel, 355of two channels of rolled steel, 360of various geometrical sections, 382required for circumferent.ial stiffeners of external-pressure

vessels, 150Monobloc vessels, at elevated temperatures, 281

bursting pressure, 280description of failure, 278experimental studies of autofrettage, 291test results” on cast-iron cylinders, 275test results on high-tensile-steel cylinders, 275test results on mild-steel cylinders, 274theories of elastic failure, 268

Multilayer vessel, photograph showing ductile rupture, 26Multilayer vessels, determination of interface pressures, 298

interferences required in shrink fitting, 300ribbon-wound, 307using weld shrinkage, 303with shell and winding of dissimilar metals, 310with shrink-fitted shells, 296

Multispheres, photograph of, 5

National Bureau of Casualty Underwriters. Svr1opsi.s of Boilernnd Pressure Laws, Rub and Rrgulations. 249

Nelson refinery index, 16Newitt’s comparisou of experimental-test results with theories

of failure, 274Nickel, chart for extrrual-pressure vessels. 370Nickel and high-nickel alloys, ASMR allowable stress in tension,

3.40Nickel-chromium-iroll alloy, chart for external-pressure vessels,

372Nickel-copper alloy, chart for external-pressure vessels, 371Nickel steel (low-carbon), chart for external-pressure vessels,

365Nomenclature for flange design, 241Nonferrous metals, ASME allowahle stress in tension, 336-340Nonstandard flanges, 223Notch brittleness, 26

Page 412: Process Equipment Design

Nozzle dimensions for API tank shells, 349, 350Nozzles, acceptable types for ASME \ rssrls. 264

for shells of storage tanks, ,t8openings, and reinforcements for ASMK vessels, 263

Oil refinery installation, photograph of. 3Oil-storage tank (welded), photograph of. 37Optimum bolt size, 227Optimum length-to-diameter ratio for vessels with formed

heads, 81Optimum plate dimensions for vessels, 81

width for vessel-shell construction (19.53), 83Optimum proportions, for vessels with elliptical dished heads. 80Optimum tank proportions, 39Out-of+oundness, ASME maximum for external-pressure ces-

sels, 256in external-pressure vessels, 150

Overhead, shop, 17Overstraining of thick-walled vessels. 279

Period of vibration in vertical vessels. 166Physical properties, of materials, 341

of various metals and alloys, 392Pipe, welded and seamless steel, 386Pipes and tubes, ASME allowahle stresses, 335

under external pressure, 256Pipes under external pressure, 153Plant-design group, in vessel procurement., I7Plastic instability, 23Plate, length, selection of, for vessel construction, 84

width, optimum for vessel construction, 81Plates, equations for bending moment, shear. and load, 100Poisson’s ratio, 98

in Lame analysis, 272Ponds, settling and storage, 2Prager and Hodge, equations for analysis of autofrettage, 287

plastic theory of failure, 279, 287Pressure, allowable on long, thin cylinders under exterilal pres-

sure, 144limits in autofrettage, 287ratings for carbon-steel flanges, 223testing of ASME code vessels, 266

Pressure Vessel Research Council, 26, 120Prestressed monobloc vessels, 285Prestressing of monohloc vessels, advantages, 285Price, base for steel, IO

information, sources of, 17Pricing, steel plate, 330Process-design group, in vessel procurement, 17Procurement group, 17Production tanks, shell design of, 43

standard design for, 44typical dimensions for, 43

Profit, in vessel fabrication, 16“Proof test” of ASME code vessels, 266Properties, of rolled structural members, 353

of steel pipe, 386of various geometrical sections, 382

Proportioning, of vessels with formed heads, 79

Radial stress in Lame analysis of thick-walled vessels, 269Radius of curvature, equivalent for design of elliptical heads

under external pressure, 152mathematical definition of, 21

Radius of gyration, of rolled-steel channels, 353of rolled-steel equal angles, 356

r

Subject Index

Radius of gyration, of rolled-steel 1 beams, 354of rolled-steel tees, 359

405

of rolled-steel unequal angles, 358of rolled-steel wide-flange beams, 35.5of two channels of rolled steel, 360of various geometrical sections, 382

“ Radius ratio,” use of, in vessel design, 293Rafter and column details, for Example Design 4.2, 74Rafter and girder selection, or roof supports in tanks, 68Rafter spacing for roof supports in tanks, 67Rating of standard flanges, 221Ratio of minimum knuckle radius to crown radius, effect on

stress in torispherical closures, 137Reaction with skirt, of compression riugs and external bn,lting

chairs, 194References, 317Reinforced concrete design, 184Reinforcement. for openings in ASME flat-plat,e closures, 266

for shell openings in storage tanks, .t9of openings in ASME vessels, 263of top course of shell for large closed tanks, 33typical for top course of shells for open vessels. 52

Reynolds’ number, versus drag coefficients, 159Rhys, C. O., relationships for stresses at the junction of the

knuckle and crown in torispherical clnsures. 135, 136Ribbon- and wire-winding theory. 308Ribbon- and wire-wound vessels, 307Ring flanges, design procedure, 2 k2Ring stiffener design for horizontal vessels. 213Riveting, 5Rolled angle as hearing plate, 190Rolled-structural-member properties. 353Rolling of vessel shell. photograph. 156Roof design, for large cylindrical flat-bottomed tanks, 63Roof plates, drawing showing dimensions for large tank, 70Roof support, drawing showing assemhly for large tank, 71

photograph showing for large st,orage t,ank. 6 1Rupture, brittle, 26Rupture time, as furlction of stress at elevated temperatures,

32Ryerson, J. T., and Son, Inc., 11

Sachs, method of determining residual stresses, 291Saddle design for horizontal vessels. 215Saddle supports for horizontal vessels, 203Screwed flanges, 221Season cracking, 33Section mod&, table of, for stiffening rings of open tanks, 52Section modulus, derivation of equation for that required in

uniformly loaded beam, 21of rolled-steel channels, 353of rolled-steel equal angles, 356of rolled-steel I beams, 354of rolled-steel tees, 359of rolled-steel unequal angles. 358of rolled-steel wide-flange beams. 355of two channels of rolled steel, 360of various geometrical sections, 382

Seismic forces, sketch for vertical vessel, 168Seismic load, coefficients, 167Seismic-load stress, in skirt of vertical vessel. 183Seismic probability, chart for United States. 164Selection of Rafter Size, Example Design 1.2. 71Semi-ellipsoidal tanks, 5Shape factor, for vessels exposed to wind loads. 1.58Shape factors, ASME, for torispherical closures. 138

Page 413: Process Equipment Design

4 0 6 Subject Index

Shear, from seismic forces on unguyed vertical vessel, 168in horizontal vessel shell stiffened by head, 209in plates, 100

Shear diagram, for horizontal vessel stiffened with ring, 207Shear equation, for cylindrical vessel with flat cover plate, 108Shear equations, for various types of beams, 384Shear force, derivation of equation based on deflection of uni-

formly loaded beam, 21Shear strength, of various metals and alloys, 392Shear stress, derivation of equation for vessel shells, 178

invariant, 281Shearing strength, minimum in welds, 7Shell, accessories for storage tanks, 3 19

bending moments under external pressure, 142construction, photographs showing rolling and w-elding, 156deformation under external pressure, 1,&2design for large storage tanks, 43design of shell courses in Example Design 3.2, 55design of top angle in Example Design 3.2, 57drawing of details of, for Example Design 3.2, 56drawing of elevation view for Example Design 3.2, 55proportioning of, for Example Design 3.2, 55reinforcement of top course for large open tanks, SO

Shell Development Company, 29Shell joints, typical aa.J;ecommended by API Standard 12 C, 47Shell manholes, 48

as recommended by API Standard 12 C, 51Shell nozzles, 48

as recommended by API Standard 12 C, 50Shell parts, 48Shell-plate dimensions, selection of, 46Shell plates, cold forming of, 48Shell stress, critical, produced by external pressure, 143Shell thickness, based on tensile stress in vertical vessel, 170Shipment of long vessels, photograph of oil-refinery fraction-

ating tower, 82Shop time, for cutting, 13

for fitting and assembly, 13for welding and edge preparation, 13, 14

Shrink fitting of multilayer shells, 296Shrinkage stresses, in multilayer vessels, 299“Six-tenths factor,” 17Sizes and capacities of typical API storagcb tanks, 34,6-348Sizing of bearing plates for vertical vessels, 187Sketch-plate shaping under shell ring, as recommended by API

Standard 12 C, 59Skirt, reaction of compression rings and external bolting chairs,

194Skirt-anchor-bolt design, 184Skirt-bearing-plate design, 184,Skirt supports, for vertical vessels, 183Skirt thickness, for vertical vessels, 183Slip-on flange, maximum stress, 238Slip-on flanges, 219Smith, A. O., construction of multilayer vessels using weld

shrinkage, 303Corporation, 26, 134vessels, method of fabrication, 303

Snow loads, chart for United States, 62Southwell, R. V., equation for critical length between stiffeners

for vessels under external pressure, 14,4Special flanges, design of, 227Specific heat, of various materials, 311Specifications, for material in ASME code vessels, 253

of low-alloy, high-strength steels, 38of plain carbon steels, 37

Spherical and modified spherical vessels, 4Spherical dished covers with pressure on concave side for

ASMF: design, 260Spraragen and Ettinger, equation for weld shrinkage, 303Stainless steel (austenitic), chart for externai-pressure vesseis,

378, 379Standard flange facings, 222Standard flanges, ratings of, 22

selection of, 219Standard Qualification Procedures of American Welding So-

ciety, 7Steel, A-7, properties of, 78

A-8, properties of, 79A-113, properties of, 78A-131, properties of, 78A-242, properties of, 79A-283, properties of, 78A-285, properties of, 79ASTM-A6-54T, specifications, 78ASTM-A-27, specifications of, 37ASTM-A-242, specifications of, 38comparison of specifications for structural- and boiler-quality

plates, 78cutting extras for mill plate, 331extras, 10for pressure vessels, 253mill plate extras, 330pipe, properties of, 386plates, flange- and firebox-quality, 333pricing, 8purchased from mill, 9purchased from warehouse, 9SA-i, specifications of, 37SA-283, Grades C and D, specifications of, 37SAE-950, specifications of, 38specifications, table for 1955 ASTM, 79structural members, properties of, 353warehouse base prices and extras, 331

Stewart, R. T., tests on cylinders under external pressure, 144Stiffeners, circumferential, for shells of vessels under external

pressure, 144design for external-pressure vessels, 149influence of, in design of shells of vertical vessels, 171

Stiffening rings for external-pressure ASMl< design, 255Straight-flange dimensions for ASME dished heads, 93Strain, two-dimensional, 99

unit, 98Strain aging, 30Strain energy, in a deflected vertical vessel, 165

in brittle fracture, 27relationships, derivation of equations, 163

Strain-energy theory (modified) for vertical vessels, 179Strain hardening, from cyclic loading, 28Strain measurements, 120Strength of materials, 392Stress, ASME allowable for aluminum and aluminum alloys,

336, 337ASME allowable for copper and copper alloys, 338, 339ASME allowable for high-alloy steels in tension, 34,2-345ASME allowable for nickel and high-nickel alloys, 34,OASME allowable for nonferrous metals in tension, 336-340allowable, figure showing comparison with yield point and

ultimate strength for various metals, 25allowable in flange design, 24,4allowable in reinforced concrete, 184analysis, from strain measurements, 120

Page 414: Process Equipment Design

I !i

I’Subject Cndex 407

Stress, during autofrettage, 287from unloading autofrettage pressure, 287in a thick-walled vessel under internal pressure by Lame

analysis, 269in bolting steel for skirt supports of vertical vessels, 185in cylindrical vessels with conical closures, 113in shell near junction with hemispherical closure, 138in skirt of vertical vessels, 183in the crown of torispherical dished head, 136in the knuckle of torispherical dished head, 13.5induced, 19induced by attachments to vertical vessels, 157induced by supported liquid in tall vertical columns, 1.57induced by weight of shell and insulation in vertical vessels,

156seismic forces, 163

Structural-quality steels, 78Superalloys, for high-temperature service, 32

Stress, at mid-span in horizontal vessel, 206at the saddle for horizontal vessel, 205axial ai junction of shell and flat cover plate, 110axial in flat-plate closure at junction, 110caused by eccentric loading on vertical vessels, 169caused by seismic forces on vertical vessels, 168circumferential at junction of shell and flat cover plate, 110circumferential in flat-plate closure at junction, 110circumferential in shell at junction with conical closure, 115combined at interface of shrink-fitted shells, 298combined in flat-plate closure, 110compressive increase from weld shrinkage in multilayer

vessels, 304conditions in shell of vertical vessel, 170corrosion cracking, 33derivation of equation for bending stress based on deflection

of uniformly loaded beam, 20distribution in shrink-fitted multilayer vessel, 300distribution in thick-walled cylinders, 271distribution in wire-wound vessel, 314from ring compression over saddle support for horizontal

vessel, 213in center of flat-plate closure, 110in cylindrical shell with flat-plate closure, 104in external-pressure vessels produced by out-of-roundness, 151in head used as stiffener for horizontal vessels, 203, 212in shell for vessel with flat cover plate, 109in shell other than at junction for vessel with flat cover plate,

110-intensification factors for vessels with hemispherical closures,

139leveling in shell of a thick-walled vessel, 282location of maximum in vessel with hemispherical closure, 139maximum axial in hub of tapered flange, 236maximum in loose-type flange, 238maximum radial in ring of flange, 237maximum tangential in ring of flange, 237range, permissible to avoid brittle fracture failure, 29ratio, chart of maximum for vessel with conical closure, 116ratio, chart of maximum in vessel with flat-plate closure,

111ratio, in vessel with flat-plate closure, 111redistribution by creep relaxation, 282relieving, photograph showing annealing of large vessel, I57shear at junction of shell and flat cover plate, IO9shear in shell other than at junction of vessel with couical

closure, 115-strain curves, idealized to mathematical equations, 29-strain curves, typical for several materials, 145-strain curves, typical for various metals, 20-strain diagram showing in elemental strip of curved plate or

beam, 21-strain relationships in elastic-plastic loadings, 23-strain relationships in vessel shells, 98two-dimensionla, 99

%resses, ASME allowable for carbon and low-alloy pipes andtubes, 335

ASME allowable for carbon and low-alloy steels, 251nt and near junction of shell with hemispherical head, I39calculated for multilayer vessel fabricated by weld shrinkage,

307caused by dead loads in vertical vessels, 156caused by wind loads on vertical vessels, 157, 161combined, in shell of vessel with flat-plate closure, 110diagram for differential element in an elliptical dished head,

123

Tangential shear stress, in horizontal vessel shell stiffened byring in plane of saddle, 207

in horizontal vessels, 203in shell of horizontal vessel stiffened by head, 209in unstiffened shell of horizontal vessel having saddles away

from head, 208Tangential stress in Lamb analysis of thick-wal!ed vessels, 269Tank, breathers, 3

costs, 8sizes and capacities, 346-348

Tanks, cylindrical, flat-bottomed, 3designed to API Standard 12 C, 3for crude oil and petroleum products, 3redwood or Cyprus, 3with domed roofs, 3with self-supporting roofs, 3with supported roofs, 3

Tapered-hub-flange loadings, 230Taylor Forge, calculation sheet for integral-type flange, 246

calculation sheet for loose-type flange, 247Taylor Forge and Pipe Works, 188, 221, 222, 223, 225, 226, 235,

237, 238, 239, 245, 246, 247, 248Taylor, Thompson, and Smulski, relationships for bearing plates

for vertical vessels, 185Tees, properties of rolled-steel members, 359Temper embrittlement, 30Temperature, effect on brittle fracture, 27

effect on creep, 31Tensile force, on supports for vertical vessels, I86Tension, constant, in wound thick shells, 3iI

in wound vessels, 308in guy wires, 161test, in welds, 7variable in winding to produce constant iension under load,

314 \Theories of failure, example for a vertical vessel, 181Theory of elasticity, applied to thick-walled vessels, 272Theory of plasticity, as criteria for failure of thick-walled

vessels, 278Thermal conductivity, of various materiais, 341Thermal expansion, for shrink fitting, 302

in skirt of vertical vessel, 195of various materials, 341

Thermal prestressing of vessel shells, 286Thermal stresses, in skirt of vertical vessel, 195

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Page 415: Process Equipment Design

408 Subject index

Timken Roller Bearing Company, 31Top course of shell, reinforcement of, for large open tanks, SOToriconical closures for ASME design, 259Toriconical heads, 96Torispherical closures, development of ASME stress-intrusifi-

cation factor, 137dished, 135elastic stability under external pressure, 151

Torispherical heads, 88deformation under internal pressure, 136dished, for ASME code design, 256possible variations of radii, 135

Transition temperature, effect on brittleness, 26Tubes under external pressure, ASME chart for wall thickness,

258Tubing under external pressure, 153Two channels, properties of rolled-steel members, 360Two-shell construction for monobloc vessel at elevated temper-

ature, 285Two-shell vessels, simplified relationships, 301

U edge, photograph of machining of, 12Ultimate strength, as basis for allowable stress, 24, 25

of various metals and alloys, 392Unequal angles, properties of rolled-steel members, 358Uniform Building Code, of Pacific Coast, 167Uniform-loaded-beam formulas, 384Uniformly loaded beam, 20Uniformly loaded cantilever beam, derivation of deflection

equations, 163U. S. Bureau of Labor Statistics, 16Universal-Cyclops Steel Corporation, 32

Vessel, closed, 3design, criteria in, 8designed to original API-ASME code, 2types, selection of, 1

Vessels, cast-steel, 5 .cylindrical with formed ends, 4for cmihz8tib~e &ids, 3for dangerous chemicals, 3forged, 5gray-iron cast, 5machined, 5multilayer, 5open, 2procurement, procedure for, 17riveted, 5spherical and modified spherical, 4vertical, individual stresses in shell, 155vertical versus horizontal, 77

Vibrational displacement in vertical vessels, 166Volume, of elliptical heads, equation for, 95

of torispherical heads, equation for, 88Volume relationships, in proportioning vessels with elliptical

heads, 80Von Mises, criterion for plastic failure, 279Voorhees, calculation of time of rupture for monobloc vessels at

elevated temperatures, 281

Voorhees, Sliepcevich, and Freeman, integrated equation fordesign of monobloc vessel at elevated temperatures, 285

Wages, average, boilermaker’s as a function of locale, I6Warehouse, prices for steel, 9

pricing, 10steel pricing, 331

Waters et al., analysis of flanges, 229, 230, 233, 236Watts and Lang, combined relat,iouships for hending and &t-al

in shell and conical closurr, 11,scombined relationships fee hending and shear in shell aud

flat cover plate, 107Wear plates, for saddle suppori~s of horizontal vessels, 213Weight of fluid in horizontal vrssels, 203Weld shrinkage used in multilayer construction, 303Welded construction, development of, 76Welded-joint efficiencies, 2%Welded joints, figure showing rxarnplrs of, 7

types of, 7Welding, automatic machine, 7

code grooves, 329conventions, 327electric-arc, 6gas, 6instructions, 327, 328, 329operators, 7photograph of automatic welding of external circumferential

seam of shell of large vessel, 6procedures, 7specifications, 327standards, 7strength of welds, 329submerged-arc, 6symbols, 7

figure showing, 8tests, 7time and rod requirements, 13types, 327, 328, 329vessel shell, photograph showing inside hn~gibudinal seam, 156

Welds, mipimum tensile strength, 7Wickelofen lathe for ribbon winding, 308Wickelofen-wound vessels, 307Wide-flange beams, properties of rolled-steel members, 355Wind girder, drawing showing details, 54

for open vessel, example design for, .52preliminary sketch of subassembly of, 53

Wind-load stress, in skirt of vertical vessel, 183Wind pressure, equation for, 157

on vertical vessels, 157Wind pressures, as function of height above ground, 159

chart for various areas in the llnited States, 158Wire or ribbon winding vessels at constant tension, 308Wire rope, used for guys, I61

Yield point, as basis for allowablr stress, 2 1, 25

Zick’s nomograph for design ol’ supporl.s for horizontal vessels,216

Zick’s relationships for saddle supports, 20:3