Design Handbook for RautaRuukki Structural Hollow Sections

DESIGN HANDBOOKFOR RAUTARUUKKI

STRUCTURAL HOLLOWSECTIONS

DESIGN HANDBOOK FOR RAUTARUUKKI STRUCTURAL HOLLOW SECTIONS Chapter 1

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DESIGN HANDBOOK FOR RAUTARUUKKI STRUCTURAL HOLLOW SECTIONSChapter 1

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New, revised edition 2000

ISBN 952-5010-47-3

Author Rautaruukki Oyj, Hannu Vainio (M.Sc.Tech)

Typesetting Lasjuma Oy

Translation Trantex Oy

Printers Otava Book Printing Ltd, Keuruu 2000

Orders RAUTARUUKKI OYJ

METFORM

13300 Hämeenlinna, Finland

tel. +358 3 528 60

fax +358 3 528 5873

Internet: www.rautaruukki.com/metform

FOREWORD

This volume is a completely new, revised edition of the 1/98 handbook for Rautaruukki structu-ral hollow sections, replacing all former editions. It is mainly based on the Europeanprestandard on steel structures, Eurocode 3 (ENV 1993-1-1:1992). Other parts of the Eurocodewere also used, and the design guidance for hollow section structures published by CIDECT(Comité International pour le Développement et l´Étude de la Construction Tubulaire) wereconsulted for additional reference.

The aim of this handbook is to provide design guidance for structures manufactured of Rauta-ruukki structural hollow sections. It is also intended as a textbook. The primary scope of thishandbook is building construction, but it can also be used in mechanical engineering whereapplicable.

The handbook was written by Hannu Vainio (M.Sc.Tech). On Rautaruukki's behalf, themanuscript was supervised by Reijo Ilvonen, Janne Pirttijoki and Kristian Witting. ProfessorErkki Niemi, Jouko Kouhi, Antti Helenius, Ilkka Hakola, Tiina Ala-Outinen, Arto Rokkanen,Hannu Reinikainen and Mikko Arponen also participated in the preparation of the manuscriptand revised parts of it. The handbook was translated into English by Sirpa Meriläinen atTrantex Oy. Lauri Sannikka of Lasjuma Oy prepared the lay-out. The book was printed andbound by Otava. Warmest thanks are due to all contributors.

In this document, a comma is used instead of a decimal point, and a decimal point is used ins-tead of a multiplication sign, in accordance with the usual continental practice.

Rautaruukki is happy to receive any comments and suggestions for improving the contents ofthis handbook.

Hämeenlinnassa 1.2.2000RAUTARUUKKIMETFORM

The accuracy of the contents of this manual has been carefully reviewed. Nevertheless,Rautaruukki is not responsible for any remaining errors or damage, whether direct or indirect,due to the incorrect application of the information presented in this book. The data in this bookis for reference only, and the user is responsible for verifying the accuracy of the results bycalculation. The information in this book is subject to change.


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INTRODUCTION ............................................................................................................... 81 RAUTARUUKKI METFORM STRUCTURAL HOLLOW SECTIONS ............................... 91.1 EN 10219 hollow sections ................................................................................................. 91.2 Manufacture of hollow sections ......................................................................................... 91.3 References ........................................................................................................................ 11

2. RESISTANCE OF HOLLOW SECTION STRUCTURES .................................................. 132.1 Limit state design and partial safety factors ...................................................................... 132.2 Classification of cross-sections ......................................................................................... 152.2.1 Calculating the effective cross-section .............................................................................. 172.3 Resistance of hollow sections subjected to bending moment ........................................... 182.3.1 Effect of holes on bending resistance ................................................................................ 192.3.2 Bending resistance in bi-axial bending .............................................................................. 192.3.3 Bending resistance of Class 4 circular hollow sections ..................................................... 202.3.4 Lateral-torsional buckling of hollow sections ..................................................................... 212.3.5 Examples for calculating bending resistance of various cross-sections ............................ 232.4 Resistance of hollow sections subjected to shear force .................................................... 262.4.1 Shear resistance of square and rectangular hollow sections ............................................ 272.4.1.1 Plastic shear resistance of square and rectangular hollow sections ................................. 272.4.1.2 Shear buckling resistance of square and rectangular hollow sections .............................. 272.4.2 Shear resistance of circular hollow sections ...................................................................... 282.4.2.1 Plastic shear resistance of circular hollow sections ........................................................... 282.4.2.2 Shear buckling resistance of circular hollow sections ....................................................... 282.4.3 Effect of holes on shear resistance ................................................................................... 292.5 Hollow sections subjected to torsion moment ................................................................... 252.5.1 Methods for calculating torsion resistance of hollow sections ........................................... 312.5.1.1 Plastic torsion resistance of hollow sections ..................................................................... 312.5.1.2 Torsional buckling resistance of hollow sections ............................................................... 312.6 Hollow sections subjected to axial force ............................................................................ 332.6.1 Tension resistance of hollow sections ............................................................................... 332.6.2 Compression resistance of square and rectangular hollow sections and

Class 1, 2 and 3 circular hollow sections (no buckling) ..................................................... 332.6.3 Compression resistance of Class 4 cross-sections (no buckling) ..................................... 342.7 Combined load resistance of hollow sections (no buckling) .............................................. 342.7.1 Hollow sections subjected to bending moment and axial force (no buckling) ................... 352.7.1.1 Class 1 or 2 hollow sections .............................................................................................. 352.7.1.2 Square or rectangular Class 3 or 4 hollow sections and circular Class 3

hollow sections .................................................................................................................. 362.7.1.3 Class 4 circular hollow sections ......................................................................................... 362.7.2 Hollow sections subjected to shear force and bending moment ....................................... 372.7.2.1 Square and rectangular hollow sections ............................................................................ 372.7.2.2 Class 1, 2 or 3 circular hollow sections ............................................................................. 382.7.2.3 Class 4 circular hollow sections ......................................................................................... 392.7.3 Hollow sections subjected to axial force, shear force and bending moment

(no buckling) ...................................................................................................................... 402.7.3.1 Class 1 or 2 hollow sections .............................................................................................. 402.7.3.2 Class 3 and 4 square and rectangular hollow sections and Class 3

circular hollow sections ...................................................................................................... 41.7.3.3 Class 4 circular hollow sections ......................................................................................... 412.8 Buckling resistance of hollow sections .............................................................................. 43


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CONTENTS

2.8.1 Buckling resistance of square and rectangular hollow sections and Class 1, 2 and 3circular hollow sections ...................................................................................................... 43

2.8.2 Buckling resistance of Class 4 circular hollow sections ..................................................... 442.9 Resistance of hollow sections subjected to combined loads (buckling) ............................ 462.9.1 Hollow sections subjected to bending moment and axial force (buckling) ........................ 472.9.1.1 Square and rectangular hollow sections and Class 1, 2 and 3 circular hollow sections .... 472.9.1.2 Class 4 circular hollow sections ......................................................................................... 492.10 Concentrated load resistance of hollow sections .............................................................. 542.10.1 Concentrated load acting from one side only .................................................................... 542.10.2 Concentrated load acting from both sides ......................................................................... 562.11 References ........................................................................................................................ 58

3 DESIGN OF JOINTS IN HOLLOW SECTION STRUCTURES ......................................... 593.1 Design of welded joints in lattice structures ....................................................................... 593.1.1 Joints of circular, square or rectangular brace members to square or

rectangular chords ............................................................................................................. 623.1.2 Joints of circular brace members to circular chords .......................................................... 723.1.3 Joints of circular, square and rectangular brace members to I profile chords ................... 773.2 Welded frameworks ........................................................................................................... 773.2.1 Joints of square and rectangular hollow sections subjected to bending ............................ 783.2.2 Circular hollow section joints subjected to bending ........................................................... 803.3 Welded end-to-end joints of hollow sections ..................................................................... 823.4 Bolted hollow section joints ............................................................................................... 843.4.1 End-to-end bolted hollow section joints ............................................................................. 843.4.1.1 Flange plate connections ................................................................................................... 853.4.1.2 In-line tension joint with splice plates ................................................................................ 903.4.2 Bolted beam-to-column joints ............................................................................................ 943.5 Hollow section-to-foundation joints .................................................................................... 1043.5.1 Joint between a column subjected to axial force and foundation ..................................... 1043.5.2 Joint between a column subjected to bending moment and axial force and the foundatio 1053.6 References ........................................................................................................................ 108

4. FATIGUE AND BRITTLE FRACTURE IN HOLLOW SECTION STRUCTURES ............. 1094.1 Fatigue loading .................................................................................................................. 1094.2 Stress calculation methods in fatigue design .................................................................... 1104.3 Design requirements for fatigue-loaded hollow sections (nominal stress method) ........... 1114.3.1 Conditions and necessity of fatigue design ....................................................................... 1114.3.2 Fatigue load design conditions .......................................................................................... 1124.3.3 Fatigue strength of hollow sections (nominal stress method) ............................................ 1144.3.3.1 Fatigue strength under normal and shear stress ............................................................... 1144.3.3.2 Fatigue strength of lattice structure joints (nominal stress method) .................................. 1154.4 Fatigue strength of lattice structure joints (hot spot method) ............................................. 1174.5 Design of fatigue-loaded hollow section structures ........................................................... 1224.5.1 Welded joints ..................................................................................................................... 1224.5.2 Bolted joints ....................................................................................................................... 1244.5.3 Lattice structures ............................................................................................................... 1244.6 Brittle fracture of structural hollow sections ....................................................................... 1294.6.1 Parameters affecting brittle fracture in structural hollow sections ..................................... 1294.6.2 Minimum service temperatures of Rautaruukki structural hollow sections ........................ 1324.7 References ........................................................................................................................ 132


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5 FIRE DESIGN OF HOLLOW SECTIONS ......................................................................... 1335.1 Development of temperature in fire compartments 1345.1.1 Standard time-temperature curve ...................................................................................... 1345.1.2 Development of temperature according to the parametric model ...................................... 1355.2 Development of steel temperature .................................................................................... 1355.2.1 Development of temperature in unprotected steel members ............................................. 1365.2.2 Development of temperature in fire protected steel members ........................................... 1375.3 Strength and modulus of elasticity of steel in fire situations .............................................. 1395.4 Critical temperature in hollow section structures ............................................................... 1405.5 Determining the strength of hollow section structures in fire situations ............................. 1415.5.1 Partial safety factors in fire design ..................................................................................... 1425.5.2 Determining the cross-section class in fire design ............................................................ 1435.5.3 Strength of hollow section subjected to tension in fire situations ...................................... 1445.5.4 Buckling strength of hollow sections in fire situations ........................................................ 1445.5.5 Bending strength of hollow sections in fire situations ........................................................ 1455.5.6 Shear strength of hollow sections in fire situations ............................................................ 1465.5.7 Strength of hollow sections subjected to bending moment and compressive axial

force in fire situations ......................................................................................................... 1465.6 Fire retardant methods ...................................................................................................... 1475.6.1 Fire retardation by insulation ............................................................................................. 1475.6.2 Fire retardation by increasing the heat retention capacity of structural steel .................... 1495.6.3 Structural fire retardation ................................................................................................... 1505.7 Fire design of concrete-filled columns ............................................................................... 1555.7.1 Using tables in the fire design of concrete-filled columns .................................................. 1565.8 References ........................................................................................................................ 158

6 DESIGN OF HOLLOW SECTION STRUCTURES ........................................................... 1596.1 Structural actions ............................................................................................................... 1616.1.1 Self-weight and imposed loads .......................................................................................... 1626.1.2 Snow load .......................................................................................................................... 1626.1.3 Wind load ........................................................................................................................... 1626.1.4 Additional horizontal forces ............................................................................................... 1646.1.5 Combined loads ................................................................................................................. 1656.1.6 Load determination in the model building .......................................................................... 1666.2 Designing columns ............................................................................................................ 1676.2.1 Column buckling length ..................................................................................................... 1676.2.2 Effect of joint stiffness on column buckling length ............................................................. 1676.2.3 Column-to-foundation connections .................................................................................... 1716.2.4 Column design in the model building ................................................................................. 1726.2.5 Designing the column-to-foundation joint in the model building ........................................ 1746.3 Designing the hollow section beams ................................................................................. 1776.3.1 Designing gable beam in the model building ..................................................................... 1786.3.2 Designing the door beam in the model building ................................................................ 1806.4 Design of trusses ............................................................................................................... 1836.4.1 Selection of truss type ....................................................................................................... 1856.4.2 Selection of the chord member .......................................................................................... 1876.4.3 Selection of bracing members ........................................................................................... 1896.4.4 Design of truss joints ......................................................................................................... 1896.4.5 Truss joints at the supports ............................................................................................... 1916.4.6 Estimation of the truss rigidity ............................................................................................ 1926.4.7 Designing the truss of the model building .......................................................................... 1936.5 Stiffening hollow section structures ................................................................................... 204


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6.5.1 Designing the stiffening elements in the model building .................................................... 2046.6 References ........................................................................................................................ 208

7 SHOP FABRICATION AND ERECTION .......................................................................... 2097.1 Cutting of hollow sections .................................................................................................. 2097.1.1 Cutting of circular hollow sections ..................................................................................... 2097.1.2 Cutting methods ................................................................................................................ 2117.1.3 Notching of hollow section ends ........................................................................................ 2117.2 Bending of hollow sections ................................................................................................ 2127.2.1 Bending methods for hollow sections ................................................................................ 2127.3 Bolted joints ....................................................................................................................... 2137.3.1 Thermodrilling of hollow section walls ............................................................................... 2147.3.2 Expansion bolt joints .......................................................................................................... 2147.3.3 Pilot tap joints .................................................................................................................... 2147.4 Welding of hollow sections ................................................................................................ 2157.4.1 Quality levels ..................................................................................................................... 2157.4.2 Welding methods ............................................................................................................... 2157.4.3 Welding sequence ............................................................................................................. 2167.4.4 Fillet and butt welds ........................................................................................................... 2167.4.5 Preheating ......................................................................................................................... 2187.4.6 Residual stresses .............................................................................................................. 2187.4.7 Inspection of welds ............................................................................................................ 2197.5 Tolerances ......................................................................................................................... 2207.6 Assembly of trusses .......................................................................................................... 2257.7 Fire protection .................................................................................................................... 2267.8 Transport and storage ....................................................................................................... 2267.9 Erection ............................................................................................................................. 2277.10 References ........................................................................................................................ 228

8 CORROSION PROTECTION ............................................................................................ 2298.1 Corrosivity categories ........................................................................................................ 2298.2 Surface preparation ........................................................................................................... 2308.3 Anti-corrosive painting ....................................................................................................... 2308.4 Hot-dip galvanizing ............................................................................................................ 2328.5 References ........................................................................................................................ 234

9 LIITTEET ........................................................................................................................... 235Liite 9.1 Putkipalkkien poikkileikkaus- ja kestävyysarvot teräslajille S355J2H ................................ 235Liite 9.2 Putkipalkkien nurjahduskestävyydet teräslajille S355J2H ................................................. 255Appendix Calculation tables for lattice joints 287Liite 9.4 Kehäliitosten jäykkyyden arvioiminen ................................................................................ 325Liite 9.5 Väsymisluokat ................................................................................................................... 331Liite 9.6 Poikkileikkaustekijät palomitoituksessa ............................................................................. 337Liite 9.7 Neliön ja suorakaiteen muotoisten putkipalkkien minimitaivutussäteet ............................. 341Liite 9.8 WinRAMI-ohjelma .............................................................................................................. 343


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9 APPENDIX .................................................................................................................. 235Appendix 9.1 Cross-sectional properties and resistance values for steel grade S355J2H ............... 235Appendix 9.2 Buckling tables for steel grade S355J2H .................................................................... 257Appendix 9.3 Calculation tables for truss joints ................................................................................ 289Appendix 9.4 Estimating the stiffness of moment connections ......................................................... 327Appendix 9.5 Fatigue categories ...................................................................................................... 333Appendix 9.6 Cross-section factors in fire design ............................................................................. 339Appendix 9.7 Minimum bending radii for square and rectangular hollow sections ........................... 343Appendix 9.8 WinRAMI software ...................................................................................................... 345

INTRODUCTION

The structural hollow section is a modern, adaptable element for steel structures. It is also anenvironmentally friendly element, since it is easy to recycle and re-use. The simple shape ofhollow sections and their excellent mechanical properties make them a light-weight andaffordable solution.

In a lattice structure, the high buckling resistance of hollow sections enables the use of longspans and a large spacing between diagonals. Due to the superior torsional stiffness of theclosed section, lattice structures made of hollow sections as well as individual hollow sectionshave good resistance to lateral-torsional buckling. The fabrication of standard joint details iscost-efficient, and rounded corners and easily accesible joints facilitate pre-treatment.

Hollow sections can easily be formed into light-weight and stiff frames grid structures, sincetheir torsional stiffness and bending resistance in all directions is high. The torsional stiffness ofhollow sections can be utilized also in various console structures and structures with projectingsections. In bracing members, the high stiffness of the hollow section serves to produce a lowsusceptibility to deflection.

Another application for hollow sections is in composite structures. When using a concrete-filledcomposite column, the properties of steel and concrete can be efficiently utilized, under normalloads and in fire situations.

New jointing systems, using direct attachment to the hollow section wall, enable thepreparation of flexible and simple joints in hollow section structures.

The design of a hollow section member or framework is easy and quick: the simple geometrycan be expressed with few parameters, which makes computer-aided design a feasible option.The weight, resistance and stiffness of structures can be optimized by modifying the wallthickness, without needing to change the external dimensions of the hollow section or thegeometry of the structure.

This design handbook for hollow sections includes data on materials and dimensions of hollowsections manufactured by Rautaruukki. It also provides instructions for the dimensioning anddesign of cross-sections, joints and structures. Moreover, instructions are given for shopfabrication and site installation of hollow sections, as well as for their corrosion protection andfire design.

The handbook is complemented by the WinRAMI software, designed by Rautaruukki especiallyfor the dimensioning of hollow section structures. Additional information on WinRami softwareis available in Rautaruukki sales offices.

The design guidance in this book is principally based on the European prestandard on steelstructures, Eurocode 3 (ENV 1993-1-1:1992) and other parts of the Eurocode. The guidancefor the design of hollow section structures published by CIDECT (Comité International pour leDéveloppement et l´Étude de la Construction Tubulaire) were also used as a reference. Theprimary scope of this handbook is building construction, but it can be used also in mechanicalengineering where applicable. The resistance values shown in formulae and tables areultimate design values based on the basic partial safety factors used in Eurocode 3.However, the partial safety factors used in national application documents (NADs) maydiffer from those used in Eurocode 3, and this must be taken into account in structuraldesign. The user is responsible for always verifying the information from the currently validnational application document.


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1 RAUTARUUKKI METFORM STRUCTURAL HOLLOW SECTIONS

1.1 EN 10219 hollow sections

Rautaruukki Metform manufactures cold-formed welded hollow sections conforming toEN 10219, made of steel grade S355J2H. The design guidance in this manual applies toRautaruukki Merform structural hollow sections shown in Appendix 9.1. These hollow sectionsare manufactured according to satisfy the requirements of Eurocode 3 [1] and the latestresearch [2]. According to the latest research [2], the Rautaruukki Metform structural hollowsections shown in Appendix 9.1 are suitable for welded steel structures down to the operat-ing temperature of -40°C (section 4.6).

For the longitudinally welded* structural hollow sections shown in Appendix 9.1, RautaruukkiMetform guarantees the properties required by EN 10219 standard. In addition to this, thefollowing properties are guaranteed for steel grade S355J2H:

- impact toughness value of 35 J/cm2 is guaranteed at a testing temperature of -40°C(EN 10219: requires 35 J/cm2 of a testing temperature of -20°C)

- mechanical properties meet the additional requirements for the application in plasticitytheory presented in Eurocode 3 (ENV 1993-1-1:1992, section 3.2.2.2) when tested inaccordance with standard EN 10219

- a carbon equivalent value of 0,39 maximum is guaranteed for steel grade S355J2H (EN10219: requires a carbon equivalent value of 0,45 maximum)

- chemical composition is better than required by EN 10219- tolerance of wall thickness is better than required by EN 10219- products meet the wall slenderness limitations presented in the latest research [2]:

(B+H)/T ≥ 25- corners are free of cracks- the products meet the following EN 10219 options:

• 1.2: The maximum value of the carbon equivalent is guaranteed for constructional steels.• 1.3: Alloy contents are provided on the inspection certificate• 1.7: Steel grade is suitable for hot-dip galvanizing• 1.8: No weld repairs are made to the base material of the hollow section

*The technical delivery conditions for spiral welded hollow sections are agreed separately foreach order.

1.2 Manufacture of hollow sections

Rautaruukki Metform manufactures the structural hollow sections from hot rolled steel strip bycold forming and welding. Hollow sections, with square and rectangular cross-sections andsmall circular (D ≤ 323,9 mm) hollow sections, are seam-welded longitudinally using highfrequency welding (HFW) (Figure 1.1). Large circular sizes (D ≥ 355,6 mm) are seam-weldedspirally using submerged arc welding (SAW) (Figure 1.2). The quality of hollow sections iscontrolled according to the ISO 9001 quality system, certified in all hollow section factories.


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Longitudinally welded hollow sections

The material used in longitudinally welded hollow sections is steel strip, cut accurately tocorrespond with the width of the external dimensions of the section. At the beginning of theproduction line, the steel strip is unrolled and the strip ends are welded together. The strip isthen fed into a strip accumulator to enable a continuous manufacturing process.

The steel strip is shaped with forming rolls at room temperature step by step into a circularcross secton. The edges of the strip are heated to the welding temperature with high frequencycurrent using an induction coil and pressed together. External burrs are removed from thesection. Seam quality is ensured by a continuous eddy current or ultrasonic inspection.

The diameter of a circular hollow section is calibrated to the final dimensions and the cross-section is formed to square or rectangular shape with profile rollers. A continuous marking ismade to the hollow section, and it is cut to dimensions according to customer orders. Samplesare taken for mechanical tests and flattening/expanding tests are carried out as required by thedelivery condition standard, in accordance with the factory’s quality system.

After cutting, the dimensions of the hollow sections are checked and the products are packedin stacks. Each stack is marked with a tag indicating the properties of the stacked hollowsections and their identification code. Based on the identification tag, the properties of thehollow section can be traced down to the steel manufacture.

Figure 1.1 Fabrication of longitudinally welded hollow sections

Welding

Burr removal

Forming rolling

Shaping and calibrating rolling

Weld inspection


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Spirally welded hollow sections

Rautaruukki Metform also manufactures circular hollow sections with diameters from ∅ 355,6to ∅ 1219 mm by spirally welding them from hot-rolled steel strip. At the beginning of theproduction line, separate steel strips are welded into a continuous sheet, which is thenstraightened and formed into a spiral-welded pipe using three-roll bending at roomtemperature. The spiral seam is welded both inside and outside using submerged arc welding.

Mechanical values are tested with test coupons cut from the section. Sections are cut todimensions according to customer orders, then inspected and delivered to customers.

Figure 1.2 Fabrication of spirally welded hollow sections

1.3 References

[1] ENV 1993-1-1: Eurocode 3: Teräsrakenteiden suunnittelu. Osa 1-1: Yleiset säännöt jarakennuksia koskevat säännöt, 1993(ENV 1993-1-1: Eurocode 3: Design of steel structures. Part 1.1: General rules and rulesfor buildings, 1993)

[2] CIDECT: Project 5AQ/2: Cold formed RHS in arctic steel structures, Final report 5AQ-5-96, 1996

Forming unit

Weld inspection

Feeding roll

Roll cutterWelding unit for stripcontinuity

Strip coil

Roll straighteningWelding


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2. RESISTANCE OF HOLLOW SECTION STRUCTURES

2.1 Limit state design and partial safety factors

The resistance of a member refers to the structure's ability to bear the loads it is subjected towithout failure or excessive deformation. Resistance and load vary according to time andlocation. Thus, they do not have a single absolute value, but their values are distributedaccording to statistic probability. In design, the dispersion of resistance and load must be takeninto account by using partial safety factors.

The design criterion for the ultimate limit state is:

where

Table 2.1 Partial safety factors for actions [1]

Also the design formulae may differ nationally. The examples in this book werecalculated using basic values in Eurocode 3.

When calculating the design value for load in ultimate limit state, the following formulae areapplicable when the structure is subjected to several variable actions [1]:

γ γ

γ γ

G.j k.j Q.1 k.1j

G Q (2.2 )

(2.2 )

⋅ + ⋅

⋅ + ⋅

∑

∑∑≥

a

G Q bG j k j Q i k iij

. . . .,0 91

Permanent actions Variable actions (γQ)(γG) Leading variable Accompanying

Accompanying variable action action variable action

Favourable effect 1,0 * *

Unfavourable effect 1,35 1,5 1,5

Fatigue-inducing action 1,0 1,0 1,0

Fire design ** ** **

Serviceability limit state 1,0 1,0 1,0 · ψ0

The partial safety factors presented in this table are the basic values of Eurocode 3. National values must be checked from the national application document (NAD) [9], [10], [11], [12].* Usually 0 [9]** Partial safety factors for fire design are presented in chapter 5

γγ

f

M

k

k

SR

is the partial safety factor for the load

is the partial safety factor for steel

is the characteristic value of the force or moment induced by the load

is the characteristic value of resistance

γγf k

k

MS

R⋅ ≤ (2.1)


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where

In formula (2.2a), only the primary leading variable action is taken into account. In formula(2.2b), all variable actions are taken into account. The formula to be used in the design is theone giving the most severe effect.

Alternatively, a more accurate formula (2.3) can be used for ultimate limit state calculations [1].

where

The partial safety factors of the material depend on the class of the cross-section, the loadingand the location of the designed element in the structure (Table 2.2).

Table 2.2 Partial safety factors for the material [1]

Partial safety factor

Class 1, 2 and 3 cross-sections γM0 = 1,1

Class 4 cross-sections γM1 = 1,1

Resistance to buckling γM1 = 1,1

Net section at bolt holes (area of holes subtracted γM2 = 1,25from the gross cross-section)

Resistance of bolted joints γMb = 1,25

Resistance of rivetted joints γMr = 1,25

Resistance of pin joints γMp = 1,25

Resistance of welded joints γMw = 1,25Friction joints Ultimate limit state γMs.ult = 1,25

Ultimate limit state, oversizeholes parallel to load

γMs.ult = 1,40

Serviceability limit state γMs.ser = 1,1

Resistance of joints in hollow section lattices γMj = 1,1Fatigue strength Inspection and

accessibility ofstructure

'Fail safe'components1)

Non 'fail safe'components 2)

normal γMf = 1,0 γMf = 1,25

poor γMf = 1,15 γMf = 1,351) 'Fail safe' components = failure of one structural element does not lead rapidly to the collapse of the entire structure2) Non 'fail safe' components = failure of one structural element leads rapidly to the collapse of the entire structure The partial safety factors presented in this table are the basic values used in Eurocode 3. National values must be checked from the appropriate national application document (NADs).

is the combination factor of the actionψ 0

γ γ ψ γG j k j Q k i Q i k iij

G Q Q. . . . . . .⋅ + ⋅ + ⋅ ⋅>∑∑ 1 1 0

1

3(2. )

is the partial safety factor for permanent action

is the characteristic value of the permanent action [5]

is the partial safety factor for leading variable action

is the characteristic value of the leading variable action [5]

is the partial safety factor for variable action

is the characteristic value of the variable action [5]

γ

γ

γ

G j

k j

Q

k

Q i

k i

G

Q

Q

.

.

.

.

.

.

1

1


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2.2 Classification of cross-sections

Cross-sections are divided into four classes. A single structure may contain structural elementswith cross-sections of different classes. Elements of a single cross-section (flanges and webs)may also belong to different classes. The class of a cross-section is determined by theslenderness and the stress state of the cross-section elements. The class of a single hollowsection may be different in bending and compression.

Class 1: Cross-sections which can form a plastic hinge with the rotation capacity required forplastic analysis.

Class 2: Cross-sections which can develop their plastic moment resistance, but have limitedrotation capacity.

Class 3: Cross-sections in which the calculated compression stress in the extreme fibre thecross-section can reach yield strength, but local buckling is liable to prevent the development ofthe plastic moment resistance.

Class 4: Cross-sections are those in which it is necessary to make explicit allowances for theeffects of local buckling when determining their moment recistance or compression recistance.

The classification of the cross-section is usually determined by to the highest classification ofcompression element. The forces and resistances of the structure can be calculated for allclasses using elasticity theory, if the effect of local buckling on cross-section resistance is takeninto account. Plastici theory can be applied when calculating forces for class 1 cross-sectionsand resistances for class 1 and 2 cross-sections. In practice for simplicity forces can becalculated using the method determined by the highest class.

For Class 4 square and rectangular hollow sections, the calculation of bending andcompression resistance is based on the effective cross-section. Resistances of the cross-section are thus calculated using only the effective area of elements. When calculatingresistance for class 4 circular hollow sections, the effective cross-section cannot be used, andEurocode 3 does not give instructions for calculating their resistance. In this case, resistancecan be assessed in relation to the buckling stress of the cylindrical casing. Circular hollowsections with Class 4 cross-sections are not recommended for load carrying structures.


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Table 2.3 Design methods for various cross-section classes [1]

Cross-section class Method for calculating Method for calculating Distribution of stresses when resistance actions (loads) the resistance is reached

1 Plastic analysis Plastic analysisSquare, rectangularand circular hollowsections

2 Plastic analysis Elastic analysisSquare, rectangularand circular hollowsections

3 Elastic analysis Elastic analysisSquare, rectangularand circular hollowsections

4 Effective cross-section Elastic analysisSquare, rectangularand circular hollowsections

4 Buckling stress Elastic analysisCircular hollowsections

��

��

��

��

��

fy

fy

fy

fy

0,5beff

σu < fy


16

Table 2.4 Limit values for cross-section classes [3]

2.2.1 Calculating the effective cross-section

The width reduction factor ρ for Class 4 cross-sections of square and rectangularhollow sections is calculated as follows [1]:

The slenderness of a class 4 web subjected to bending and its effective width is calculatedaccording to the guidance given in reference [1]. The slenderness of the flange or websubjected to uniform compression can be determined using the following formula (2.6):

λσ εp

y

cr

1f

bt= =

⋅56 8,(2.6)

ρ λλ

λ= − >p

ppkun

0 222

, 0,673 (2.5)when

ρ λ= ≤1 0,673 (2.4)kun p when

Cross-section class

1 2 3

Cross- fy 235 275 355 460 235 275 355 460 235 275 355 460Stress Loading section Nstate method element mm2

Compres- Web and h/t 45,0 41,6 36,6 32,2 45,0 41,6 36,6 32,2 45,0 41,6 36,6 32,2sion flange b/t

Bending Flange b/t 36,0 33,3 29,3 25,7 41,0 37,9 33,4 29,3 45,0 41,6 36,6 32,2

Bending Web h/t 75,0 69,3 61,1 53,6 86,0 79,5 70,0 61,5 127,0 117,3 103,3 90,8

Compres- Entire d/t 50,0 42,7 33,1 25,5 70,0 59,8 46,3 35,8 90,0 76,9 59,6 46,0sion and/or cross-bending section

fy is the yield strength for steelFor other steel grades, the values in column 235 N/mm2 are multiplied by correction factor ε when usingsquare and rectangular hollow sections and by correction factor ε2 when using circular hollow sections.

The cross-section class for a compression and bending web can be determined according to the instructions in reference [1], a conservative assessment is obtained when the cross-section class of the web is determined by compression only.

ε ε= =235

f

235

fy y, 2

��

��

��

��b d

h t t


17

where

Table 2.5 Effective width beff of the flat compression elements with Class 4 cross-sections in square and rectangular hollow sections [1]

2.3 Resistance of hollow sections subjected to bending moment

Hollow sections are efficient when subjected to bending about one or both principle axis. In ad-dition, the buckling resistance about the minor axis is superior to an equivalent weight I or Hprofile section and therefore lateral restraints can be placed at greater spacings.

The design criterrion for a member subject to bending about one axis is:

where

The bending resistance for hollow sections of different classes of cross-section is calculated as follows [1]:

Class 1 and 2 cross-sections

Class 3 cross-sections

Class 4 cross-sections (square and rectangular)

Class 4 cross-section (circular hollow sections)(section 2.3.3)

where:Wpl is the plastic modulus of the cross-sectionWel is the elastic modulus of the cross-sectionWeff is the effective modulus of the cross-sectionfy is the design strength

M M W f

M M W f

M M W f

c Rd pl Rd pl y M

c Rd el Rd el y M

c Rd eff Rd eff y M

. .

. .

. .

/

/

/

= = ⋅= = ⋅= = ⋅

γγ

γ

0

0

1

is the design value for bending moment

is the design value for bending resistanceMM

Sd

c Rd.

M MSd c Rd≤ . (2.7)

Stress distribution (compression positive) Effective width befff

be1 be2

b1

+ σ+ σbeff = ρb1

be1 = 0,5 beff

be2 = 0,5 beff

is the thickness of the hollow section wall

is the buckling stress

= b- 3t; flange design width or h- 3t; web design heigth

is the nominal yield strength of the material [8]

t

crσ

ε

b

fy

1

= 235

fy


18

2.3.1 Effect of holes on bending resistance

The effect of holes need not be taken into account in a flange subjected to tension when thefollowing criterion is satisfied [1]:

where

If the criterion is not satisfied, the cross-sectional area of the tension flange assumed in design,must be reduced to such an extent that the criterion is satisfied. This reduced cross-sectionalarea of the tension flange is then used to calculate the bending resistance. The effect of holesin the tension area of the web need not be considered, if the criterion (2.8) is met in the entiretension area. The tension area consists of the tension flange and the tension element of theweb. In the compression area, the effect of holes need not be considered, unless the bolt holesare oversize or slotted [1].

2.3.2 Bending resistance in bi-axial bending

The following design criteria are applied, if the hollow section is subjected to bi-axial bending[1]:

where

The effective cross-section modulus Weff.y is calculated assuming only My.Sd is present andWeff.z is calculated assuming only Mz.Sd is present. The bending resistances are thuscalculated separately.

= 2 (circular hollow sections)

=1,66 (square and rectangular hollow sections)

αα

M

M

M

My.Sd

eff.y.Rd

z.Sd

eff.z.Rd

+

≤ 1 0, (2.11)

Class 4 cross-section(square and rectangular hollow sections)

M

M

M

My.Sd

el.y.Rd

z.Sd

el.z.Rd

+

≤ 1 0, (2.10)Class 3 cross-section

M

M

M

My.Sd

pl.y.Rd

z.Sd

pl.z.Rd

+

≤

α α

1 0, (2.9)Class 1 and 2 cross-sections

is the cross-sectional area of the tension flange

is the net cross-sectional area of the tension flange

is the ultimate strength of the material

is the partial safety factor for the material (Table 2.2)

is the partial safety factor for the net effective cross-section (Table 2.2)

AAf

f

f.net

u

γγ

M

M

0

2

0,9 (2.8)⋅ ≥A

A

f

ff.net

f

y

u

M2

M0

γγ


19

2.3.3 Bending resistance of Class 4 circular hollow sections

The buckling stress must be calculated for circular hollow sections with Class 4 cross-sections.The design criterion is that the bending moment due to loading is smaller than the bendingresistance of the hollow section:

where

The buckling stress of a circular hollow section is calculated as follows, when [2]:

where

αb is the reduction factor for buckling stress, which is calculated as follows, when r/t ≤ 212[2]:

For the hollow sections for which > 212 and/or , the buckling stress should be

calculated according to guidance given in reference [2].

r

tλ > 2

α b = ++

0 18870 6734

1 0 01,

,

,r

t

(2.14)

(the radius of the wall central axis)

λα σ

σ

=⋅

=

= −

fy

b cr

cr 0,605Et

r

rd t

2

σ λu yf= − ( )( )1 0 41231 2

,,

(2.13)

is the design value for the bending moment

= σu ·Wel/γM1 (design value for bending resistance)

is the buckling stress of the hollow section

MM

Sd

c Rd

u

.

σ

M MSd c Rd≤ . (2.12)


20

λ ≤ 2

2.3.4 Lateral-torsional buckling of hollow sections

With hollow sections, lateral-torsional buckling does not normally govern. However, lateral-torsional buckling may govern in long hollow sections with a small b/h ratio. Figure 2.1 andTable 2.6 give the maximum lengths with which lateral-torsional buckling need not be takeninto account when using square and rectangular hollow sections with Class 1-3 cross-sections.Circular hollow sections are not liable to lateral-torsional buckling. The values in Table 2.6 aredetermined using the following formula [3]:

where

If condition (2.15) is not met, the lateral-torsional buckling resistance can be calculatedaccording to Eurocode 3 [1].

Table 2.6 Rectangular hollow sections: length ratios below which lateral-torsional buckling need not be taken into account [3]

fy = 235 N/mm2 fy = 275 N/mm2 fy= 355 N/mm2 fy = 460 N/mm2

0,25 27,8 23,8 18,4 14,2

0,33 41,8 35,8 27,7 21,4

0,5 73,7 63,0 48,8 37,7

0,6 93,1 79,5 61,6 47,5

0,7 112,5 96,2 74,5 57,5

0,8 132,0 112,8 87,4 67,4

0,9 151,3 129,3 100,2 77,3

1,0 170,6 145,8 112,9 87,2

These values are determined for uniform moment which is the most severe case.

b t

h t

−−

L

h t−

��

M M

L

h

b

t

is the design value for bending resistance

is the elastic critical lateral-torsional buckling moment MM

c.Rd

cr

M

Mc.Rd

cr

⋅ ≤γ M1 0 4, (2.15)


21

Figure 2.1 Limit value curves for lateral-torsional buckling (fy = 355 N/mm2)

50 75 100 125 150 175 200 225 250 275 300 325 350 375 400

0

5

10

15

20

25

30

35

40

45

0

Height (h-t) (mm)

Leng

th L

(m

)

1,0

0,9

0,8

0,7

0,6

0,5

0,33

0,25

b t

h t

−−


22

2.3.5 Examples for calculating bending resistance of various cross-sections

Example 1a: Class 1 cross-section

Consider a structural hollow section with dimensions 140 x 140 x 5. The steel grade usedis S355J2H, for which the Class 1 criteria are (Table 2.4):Flange: b / t = 28 < 29,3 (compression)Web: h / t = 28 < 61,1 (bending)

The plastic section modulus for the cross-section is determined by the following equation[6]:

where the terms induced by corner rounding are:

By inserting the hollow section dimensions 140 x 140 x 5 in the formula 2.16, we obtainthe following value for section modulus:

The bending resistance of the hollow section with dimensions 140 x 140 x 5 is thus:

Bending resistance values are presented also in tables in Appendix 9.1.

M Wf

pl Rd ply

M.

,,= = ⋅ =

γ 0

3132 10355

1 142 6 kNm

Wpl = ⋅ − − ⋅ − ⋅ − ⋅ ⋅ + ⋅ ⋅

= ⋅

140 140

4

(140 2 5)(140 2 5)

44 21, 46 67,77 4 5, 37 63,88

132 10

2 2

3 mm3

= 5 mm (internal nominal corner radius)= 10 mm (external nominal corner radius)

rri

0

A r

A r

h r

h r

z

z

i

= −

⋅

= −

⋅

= − −−

⋅

= − − −−

⋅

14

14

h

2

10 3

12 3

h 2t

2

10 3

12 3

π

π

ππ

ππ

ξ

ξ

02

12

0

Wb h b t h t

A h A hpl z z= ⋅ − − − − ⋅ + ⋅2 2

4

2 2

44 4

( )( )ξ ξ (2.16)


23

Example 1b: Class 2 cross-sections

Consider a hollow section with dimensions 160 x 160 x 5. The steel grade used isS355J2H, for which the Class 2 criteria are (Table 2.4):Flange: b / t = 32 < 33,4 (compression)Web: h / t = 32 < 70 (bending)

For Class 2 cross-sections, the plastic section modulus is determined by the followingformula (2.16):ri = 5 mmr0 = 10 mmWpl = 175·103 mm3

Thus, the following bending resistance is obtained for a hollow section with dimensions160 x 160 x 5:

Example 1c: Class 3 cross-sections

Consider a hollow section with dimensions 180 x 180 x 5. The steel grade used isS355J2H, for which the Class 3 criteria are (Table 2.4):Flange: b/t = 36 < 36,6 (compression)Web: h/t = 36 < 103,3 (bending)

For a Class 3 cross-section, the elastic section modulus is calculated as follows [6]:

where the terms, taking account of corner rounding, are:

By inserting the dimensions of the hollow section 180 x 180 x 5 in the formula (2.17), thefollowing value for elastic section modulus is obtained:

Wel = ⋅ − − ⋅ − ⋅ − + ⋅ + + ⋅

⋅

= ⋅

180 180

12

180 2 5 180 2 5

124 75 5 21 5 87 8 4 4 7 5 3 83 9

2

180

193 10

3 32 2

3

( )( )( , , , ) ( , , , )

mm3

I r

I r

r

zz

i

i

= − −−

= − −−

1

3 16

1

3 12 3

1

3 16

1

3 12 3

04

4

ππ

ππξξ

( )

( )

= 5 mm

r = 10 mm0

Wb h b t h t

I A h I A hh

el zz z z= ⋅ − − − − + ⋅ + + ⋅

⋅3 3

2 2

12

2 2

124 4

2( )( )( ) ( )ξξ ξ ξ (2.17)

M Wf

pl Rd ply

M.

,,= = ⋅ =

γ 0

3175 10355

1 156 5 kNm


24

The following bending resistance is obtained for a hollow section with dimensions180 x 180 x 5:

Example 1d: Square hollow section with Class 4 cross-section

Consider a hollow section with dimensions 200 x 200 x 5. The steel grade used isS355J2H, for which the Class 3 criteria are (Table 2.4):

Flange: b/t = 40 > 36,6 (compression) ⇒ Class 4 cross-sectionsWeb: h/t = 40 < 103,3 (bending) ⇒ Class 1 cross-sections

As the compression flange belongs to Class 4, its effective width must be determined. Theslenderness of the flange is obtained from the formula (2.6):

Now, the effective width of the flange can be calculatedfrom the formula (2.5):

The neutral axis of the effective cross-section is transferred downwards. The effectivesection modulus of the cross-section is calculated by subtracting the section modulus ofthe non-effective element from the section modulus of the entire cross-section. Theeffective section modulus for a hollow section with dimensions 200 x 200 x 5 is obtainedas follows:

where

(transfer of the neutral axis of the cross-section)

mmδ =⋅ ⋅ ⋅ − ⋅

− ⋅= ⋅ ⋅ ⋅ − ⋅

− ⋅=

b t h t

A b tnon eff

non eff

.

.

( , , ) , ( , , )

,,

0 5 0 5 17 5 5 0 5 200 0 5 5

3840 17 5 52 27

Wt

eff =− ⋅ ⋅

= ⋅ ⋅ − ⋅ ⋅ ⋅⋅

= ⋅

I + A b h - 0, 5t + )

h +

2410 10 + 3840 200 - 0, 5 5+ 2, 27)

200 + 2, 27 mm

2non.eff

2

4 2 23

δ δδ

( ,

,( . )

, , ( ,

,,

0 5

0 52 18

2 27 17 5 5 0 5

0 5227 2 103

beff = − ⋅ −( ) =0 801 0 22

0 801200 15 167 52

, ,

,, mm

λ p =

− ⋅

⋅= >

200 3 5

5

56 8235

355

0 801 0 673,

, ,

MW f

el Rdel y

.,

,=⋅

= ⋅ ⋅ =γ M0

193 10 355

1 162 3

3

kNm


25

��

��

��

��

δ

1,5t 1,5t0,5beff bnon.eff 0,5beff

bnon.eff = b – 3t – beff = 200 – 3· 5 – 167,5 = 17,5 mm (non-effective element of thecompression web)A is the area of the entire cross-section (Appendix 9.1)I is the second moment of area of the entire cross-section (Appendix 9.1)To obtain the bending resistance value, the effective section modulus is multiplied by theyield strength:

Meff.Rd = 227,2·103·355 / 1,1= 73,3 kNm

Example 1e: Circular hollow section with Class 4 cross-section

Consider a hollow section with dimensions 323,9 x 5. The steel grade used is S355J2H,for which the Class 3 criteria are (Table 2.4):

d / t = 64,78 > 59,6 ⇒ Class 4 cross-sectionFirst, determine the slenderness of the cross-section:

Buckling stress is determined by the following formula (2.13):

To obtain the bending resistance of the cross-section, the buckling stress is multiplied bythe elastic section modulus:

2.4 Resistance of hollow sections subjected to shear force

The design criterion for a hollow section subject to shear force is [1]:

where

VSd is the design value for the shear force VRd is the design value for the shear resistance

V VSd Rd ≤ ( . )2 19

MW d d t

dc Rd

u el u

M.

, , ,

, ,,= ⋅ =

⋅ − −( )[ ]⋅

=⋅ − − ⋅( )[ ]

⋅ ⋅=σ

γσ π

γπ

M1

4 4

1

4 42

32

315 0 323 9 323 9 2 5

32 323 9 1 1112 6 kNm

σ λu yf= − ⋅( ) = − ⋅( ) =1 0 4123 1 0 4123 0 339 355 315 01 2 1 2, , , ,, , N/mm2

σ

α

λα σ

cr

b

y

b cr

Et

r

f

= = ⋅ =

= ++

= ++

=

=⋅

=⋅

=

0 605 0 605 2100005

159 453984 0

0 18870 6734

1 0 010 1887

0 6734

1 0 010 775

355

0 775 3984 00 339

, ,,

,

,,

,,

,

,,

, ,,

N/mm

r

t

159, 45

5

2


26

2.4.1 Shear resistance of square and rectangular hollow sections

The method for calculating the shear resistance depends on the slenderness of the web of thecross-section as follows [1]:

Shear buckling need not be considered for square and rectangular hollow sections for whichh/t < 59,1 and the yield strength of the material fy ≤ 355 N/mm2. In practice, shear bucklinggoverns only for a very few hollow sections.

2.4.1.1 Plastic shear resistance of square and rectangular hollow sections

Plastic shear resistance is calculated using the following formula [1]:

where

(h in this case is the dimension parallel to shear force)

2.4.1.2 Shear buckling resistance of square and rectangular hollow sections

The shear buckling resistance of hollow sections is calculated using the following formula [1]:

The web shear buckling stress τba depends on the slenderness of the web λ w as follows [1]:

where

The resistance to shear buckling is calculated according to the instructions in reference [1], ifλ w ≥ 1,2.

λw =

h - 3t

t

fy86 4

235,

τ λ λba wyf= − −( )( )1 0 625 0 83

2 22, , ( . ) for 0,8 < < 1, 2w

V tba Rdba

M. ( . )= ( ) ⋅2 h - 3t

τγ 1

2 21

A Ab h

v =+h

Vpl Rd. ( . )=⋅

⋅A fv y

M032 20

γ

calculate the plastic shear resistance (section 2.4.1.1)

calculate the resistance to shear buckling (section 2.4.1.2)

h

t f

h

t f

y

y

≤ ⋅ + ⇒

> ⋅ + ⇒

69235

3

69235

3


27

��Vsd

h h

bb

Vsd

2.4.2 Shear resistance of circular hollow sections

The shear resistance of circular hollow sections can be assessed by the following methodsClass 1, 2 and 3 cross-sections ⇒ calculate plastic shear resistance (section 2.4.2.1)Class 4 cross-sections ⇒ calculate shear buckling resistance (section 2.4.2.2)

2.4.2.1 Plastic shear resistance of circular hollow sections

The plastic shear resistance is calculated by the following formula [1]:

where

2.4.2.2 Shear buckling resistance of circular hollow sections

The design value of shear buckling resistance for circular hollow sections is obtained using thefollowing formula [2]:

where

The calculation of the theoretical shear buckling stress is a complex task. A conservativeassessment is obtained by the following simplified formula [2]:

where

A more accurate formula for calculating the theoretical shear buckling stress is given inreference [2]. The difference to the results obtained by the formula (2.25) is, however, rathersmall when using normal hollow section lengths (> 1000 mm). The shear buckling stress isdetermined by the following formulae [2]:

τ τ τ

ττ

τ

ba cr

bay y

cr

or

f for

= ≤

= −

>

0 65 2 26

31 0 222 2 27

, ( . )

, ( . )

f 0, 444f

f 0, 444f

cr y

cr y

is the length of the hollow section element which is subjected to the shear VsdL

τ crt

r= ⋅

0 747 2 25

0 75

, ( . ),

Et

L

is the shear buckling stress

the central axis radius of the hollow section wallτ ba

r

V r tba Rdba

M. ( . )= ⋅ ⋅π τ

γ 12 24

A Av = 2

π

Vpl Rd. ( . )=⋅

⋅A fv y

M032 23

γ


28

2.4.3 Effect of holes on shear resistance

The effect of holes located in webs need not be considered when calculating the design valuefor plastic shear resistance, if the following criterion is satisfied [1]:

where

The cross-sectional area Av used in design is reduced to the value (fu / fy) · Av.net, if thecriterion (2.28) is not satisfied.

Example 2a:Calculate the shear resistance of a hollow section withdimensions 400 x 200 x 6. The steel grade used isS355J2H. First determine whether the buckling of theweb needs to be considered. Using the dimensionsgiven in the example, the following value is obtained:

⇒ buckling of the web must be taken into accountCalculate the slenderness of the web and the shear buckling stress τba using the formula(2.22).

The shear resistance is obtained by inserting the shear buckling stress in the formula (2.21):

Example 2b: Calculate the shear resistance of a hollow section with dimensions 400 x 200 x 8. Thesteel grade used is S355J2H.

h

t= = + = ⇒400

850

2353 59 1 < 69

f

y, buckling of the web need not be taken into

account

V tba Rdba

M.

,

,,= ( ) = ⋅( ) ⋅ ⋅ =2 h - 3t 2 400 - 3 6 kN

τγ 1

6191 4

1 1797 6

λ

τ λ

w

wyf N mm

= = =

= − ⋅ −( )( ) = − ⋅ −( )( ) =

h - 3t

t

f

400 - 18

6

355

0,906

y

ba

86 4235

86 4235

1 0 625 0 83

1 0 625 0 906 0 8355

3191 4 2

, ,

, , , , , , /

h

t= = + =400

666 67

2353 59 1, , > 69

fy

is the cross-sectional area of the web

is the net cross-sectional area of the webAA

v

v net.

A Av net v. ( . ) f

fy

u≥ 2 28


29

��yy

200

400

6

VSd

The shear resistance is obtained directly by inserting values in the formula (2.20):

Example 3a:Calculate the shear resistance of the circular hollow section inexample 1e with dimensions 323,9 X 5 and a Class 4 cross-section. The length of the hollow section is 6 m, and the steelgrade used is S355J2H. Shear force is assumed constant alongthe entire length of the hollow section. First, determine thetheoretical shear buckling stress from the formula (2.25):

The shear buckling stress is obtained from the formula (2.27), since τcr > 0,444 fy:

The resistance to shear buckling is calculated using the shear buckling stress:

Example 3b: Calculate the shear resistance for a hollow section with dimensions 323,9 x 8. The steelgrade used is S355J2H.For a Class 2 cross-section, the plastic shear resistance must be calculated.(d/t= 40,5 < 46,3):

2.5 Hollow sections subjected to torsion moment

Hollow sections are efficient when subjected to a torsional moment. Their torsional resistanceis superior to that of open sections. The design criterion for a member subjected to torsionmoment is the following:

where

is the design value for torsion moment

is the design value for torsion resistance

MM

t Sd

t Rd

.

.

M Mt Sd t Rd. . ( . ) ≤ 2 29

Vpl Rd.,

,=⋅

⋅= ⋅

⋅=

A f kNv y

M037939

2 355

3 1 1941 7

γ π

V rt

kba Rd baM

. , ,,

,= ⋅ ⋅ = ⋅ ⋅ =τ πγ

π1

157 1 159 455

1 1357 7 N

ττba

y y

cr

f f= −

= −

3

1 0 222355

31 0 222

355

337 4, ,

, = 157,1 N/mm2

τ crt

r= ⋅

= ⋅ ⋅ ⋅

=

0 747

0 7475

6000

5

159 45337 4

0 75

0 75

,

,,

,

,

,

Et

L

2,1 10 N/mm5 2

Vpl Rd.,

=⋅

⋅= ⋅

+⋅

⋅=

A f kNv y

M039124

400

200 400

355

3 1 11133

γ


30

��

yyy6000

VSd

2.5.1 Methods for calculating torsion resistance of hollow sections

As Eurocode 3 Appendix G dealing with torsion is not yet available, torsion resistance iscalculated using the same method as for shear resistance. Torsional resistance for square andrectangular hollow sections can be calculated as follows:

The torsional resistance of a circular hollow section depends on the slenderness and the lengthof the hollow section. Normally, it can be determined using the following methods:

Class 1, 2 and 3 cross-sections ⇒ (calculate the plastic torsion resistance (section 2.5.1.1)Class 4 cross-section ⇒ (calculate the resistance to torsional buckling (section 2.5.1.2)

2.5.1.1 Plastic torsion resistance of hollow sections

The plastic torsional resistance of hollow sections can be expressed by the following formula[4]:

where

2.5.1.2 Torsional buckling resistance of hollow sections

The resistance to torsional buckling can be calculated by the equation [2]:

where

is the shear buckling stress (sections 2.4.1.2 and 2.4.2.2)τ ba

MW

t b Rd bat

M. . ( . )= τ

γ 12 31

is the torsional resistance of the cross-section

is the area bounded by the central axis of the hollow section wallWA

t

t

Mf

Wf

A tt pl Rdy

Mt

y

Mt. . ( . )=

⋅≈

⋅⋅

3 32 2 30

0 0γ γ

calculate plastic torsional resistance (section 2.5.1.1)

calculate resistance to torsional buckling (section 2.5.1.2)

h

t f

h

t f

y

y

≤ ⋅ + ⇒

> ⋅ + ⇒

69235

3

69235

3


31

Example 4:Consider the hollow section in example 2a with dimensions400 x 200 x 6 subjected to torsion moment Mt.Sd = 150 kNm.The steel grade used is S355J2H.

The shear buckling stress τba, for the adjacent cross-sectionwas calculated in the example 2a. Determine the torsionalbuckling resistance of the cross-section as follows:

Example 5:Consider the circular hollow section in example 3a withdimensions 323,9 x 5 subjected to torsion moment Mt.Sd =100 kNm. (τba obtained from example 3a). The torsionalbuckling resistance is given by the following formula:

MW

M M

t b Rd bat

M

t b Rd t Sd

. .

. . .

,,

,,= = ⋅ ⋅ =

>

τγ 1

3

157 1786 6 10

1 1112 3 kNm

OK!

MW

M M

t b Rd bat

M

t b Rd t Sd

. .

. . .

,,

,,= = ⋅ ⋅ =

>

τγ 1

3

191 4877 1 10

1 1152 6 kNm

OK!


32

��yy

��

yyy6000

Mt.Sd

Mt.Sd

200

400

6

2.6 Hollow sections subjected to axial force

2.6.1 Tension resistance of hollow sections

Regardless of slenderness, the cross-section is fully effective when subjected to tension. Thus,the form of the cross-section does not affect the tension resistance. Hollow sections areefficient when used as tension members, as their joints can be made stronger and less com-plex than equivalent open sections. The design criterion for hollow section in tension is [1]:

where

The tension resistance of the cross-section is the smallest of the following [1]:

where

If ductility is required of the structure, the criterion (2.35) must be satisfied [1]:

However, it is recommended that the criterion (2.35) should always be satisfied.

2.6.2 Compression resistance of square and rectangular hollow sections and Class 1, 2 and 3 circular hollow sections (no buckling)

The design criterion for hollow sections loaded in compression is [1]:

where

for Class 1, 2 and 3 cross-sections

for Class 4 cross-sections

N

N Af

Af

A

Sd

pl Rdy

M

effy

M

eff

N

N

c.Rd

c.Rd

= =

=

. γ

γ

0

1

is the design value for compressive force

is the effective cross-section area in axial compression

NSd Nc.Rd≤ ( . )2 36

0 9 2 352

0, ( . )

A

Anet M

M

f

fy

u≥ ⋅ γ

γ

is the net area (the area of holes subtracted from the gross area)

is the ultimate strength of the material

Af

net

u

N Af

N Af

t Rdy

M

t Rd netu

M

.

.

( . )

, ( . )

=

=

γ

γ

0

2

2 33

0 9 2 34

is the design value of tensile force

is the design value of tension resistance

NN

Sd

t Rd.

N NSd t Rd ≤ . ( . )2 32


33

2.6.3 Compression resistance of Class 4 cross-sections (no buckling)

The design criterion for circular hollow sections with Class 4 cross-sections can be expressedas [2]:

where

Factor α0 is determined by the equation:

2.7 Combined load resistance of hollow sections (no buckling)

Table 2.7 presents criteria for the effect of different load combinations.

Table 2.7 Combined load criteria to be checked when no risk of buckling is present

Load combination Cross- Class 1 and 2 Class 3 Class 4 section cross-sections cross-section cross-section

Section Formula Section Formula Section Formula

Bending, Square and rectangular 2.7.1.1 (2.39) 2.7.1.2 (2.43) 2.7.1.2 (2.43)

compression or tension Circular 2.7.1.1 (2.39) 2.7.1.2 (2.43) 2.7.1.3 (2.44)

Bending Square and rectangular 2.7.2.1 (2.46) 2.7.2.1 (2.46) 2.7.2.1 (2.46)

Shear* Circular 2.7.2.2 (2.47) 2.7.2.2 (2.47) 2.7.2.3 (2.48)

Bending Square and rectangular 2.7.3.1 (2.49) 2.7.3.2 (2.50) 2.7.3.2 (2.50)

Compression or tension Circular 2.7.3.1 (2.49) 2.7.3.2 (2.50) 2.7.3.3 (2.51)

Shear*

*The shear force needs to be taken into account only if VSd > 0,5 VRdThe effect of torsion is allowed for by adding the following term in the interaction expression:

M

Mt Sd

t Rd

.

.

α 00 83

1 0 018=

+

,

,r

t

(2.3 )r

t 212≤when

when

N A

f

f

Et

r

rd t

c Rdu

M

u y

y

cr

cr

.

,,

,

=

= − ( )( )=

⋅

=

= −

σγ

σ λ

λα σ

σ

11 2

0

1 0 4123

0 605

2

λ 2≤

(the radius of the wall central axis)

NSd Nc.Rd≤ ( . )2 37


34

2.7.1 Hollow sections subjected to bending moment and axial force (no buckling)

The cross-section class of a web loaded by bending moment and axial force depends on thestress distribution. In practice, the cross-section class is more easily determined by thecompression element (web or flange).

2.7.1.1 Class 1 or 2 hollow sections

The interaction expression (2.39) can be applied for Class 1 and 2 cross-sections [1]:

The parameter α used in the calculation of bending resistance depends on the form of thehollow section:

Circular hollow sections α = 2

Square and rectangular hollow sections

where

The bending resistance is reduced by axial force and depends on the shape of the hollowsection:

M M

M M M

M M a

M M M

M M

N Rd pl Rd

N Rd pl Rd pl Rd

Ny Rd pl y Rd

Ny Rd pl y Rd pl y Rd

Nz Rd pl z Rd

. .

. . .

. . .

. . . . .

. . .

, ( . )

(

, ( . )

(

,

= −

≤

= −

≤

=−

+

1 26 1 2 40

1 33 1 2 41

1

0 5

N

N

N

N

N

N

Sd

pl.Rd

Sd

pl.Rd

Sd

pl.Rd

htht

A

≤

( . )

(. . . . .

2 41b

M M MNz Rd pl z Rd pl z Rd

, for square hollow sections

, for rectangular hollow sections

is the plastic bending resistance,section 2.3)

is the plastic bending resistance about they axis, section 2.3)

, for rectangular hollow sections

is the plastic bending resistance about the z axis, section 2.3)

however

however

however

N Af

pl Rdy

M. =

γ 0

α =

−

≤1 66

1 1 13

6,

,N

N

Sd

pl.Rd

2

M

M

M

My.Sd

Ny.Rd

z.Sd

Nz.Rd

+

≤α α

1 0 9, (2.3 )


35

��

b

h My

Mz

y

z

2.7.1.2 Square or rectangular Class 3 or 4 hollow sections and circular Class 3 hollow sections

The interaction expression (2.43) is derived according to the elasticity theory [1]:

where

Aeff is calculated for axial compression. The effective cross-section modulus Weff.y iscalculated assuming only My.Sd is present and Weff.z assuming only when Mz.Sd is present.

2.7.1.3 Class 4 circular hollow sections

The combined load criteria for Class 4 circular hollow sections can be expressed as follows:

where

M M M

N A

Sd y Sd z Sd

c Rdu

M

= +

=

. .

.

2 2

1

σγ

N

N

M

MSd

c Rd

Sd

c Rd. .( . )+ ≤ 1,0 2 44

N Af

N Af

M Wf

M Wf

M Wf

M Wf

c Rdy

M

c Rd effy

M

el y Rd el yy

Mel z Rd el z

y

M

eff y Rd eff yy

Meff z Rd eff z

y

M

.

.

. . . . . .

. . . . . .

=

=

= =

= =

γ

γ

γ γ

γ γ

0

1

0 0

1 1

ja

ja

for Class 3 cross-section


and

and

N

N

M

M

M

Ma

N

N

M

M

M

Mb

Sd

c Rd

y Sd

el y Rd

z Sd

el z Rd

Sd

c Rd

y Sd

eff y Rd

z Sd

eff z Rd

.

.

. .

.

. .

.

.

. .

.

. .

( . )

( . )

+ + ≤

+ + ≤

1,0

1,0

2 43

2 43

Class 3 cross-section

Class 4 cross-sections

M M

M M M

N Rd pl Rd

N Rd pl Rd pl Rd

. .

. . .

, ( . )

(

= ⋅ −

≤

1 04 1 2 42N

NSd

pl.Rd

1,7

circular hollow sections

however is the plastic bending resistance, section 2.3)


36

The combined effect of bending moment and axial force is allowed for by the parameter α [2]:

where

2.7.2 Hollow sections subjected to shear force and bending moment

2.7.2.1 Square and rectangular hollow sections

If the shear force is more than half of the shear resistance of the cross-section (VSd > 0,5VRd),the effect of shear force must be accounted for when calculating bending resistance. The shearresistance value VRd is either Vpl.Rd (section 2.4.1.1) or Vba.Rd (section 2.4.1.2). The bendingresistance of the cross-section is then [1]:

where

ρ = −

2V

V1Sd

Rd

2

M MV Rd c Rd. . ( . )=−

≤W

A

tf

pl

vy

M0

ρ

γ

2

82 46

σ

σ

α

α

0

00 83

1 0 01

0 18870 6734

1 0 01

=

=

=+

≤

= ++

≤

N

AM

W

Sd

bSd

el

b

r

t

r

t 212

r

t

r

t 212

,

,

,,

,

the design stress due to the axial force

the design stress due to the bending moment

when

when

α α σ α σσ σ

= ⋅ + ⋅+

0 0 b b

0 b( . )2 45

M W

f kun

f

Et

r

c Rd elu

M

u y

y

cr

cr

.

,,

,

=

= − ( )( ) ≤

=⋅

= ⋅

σγ

σ λ λ

λα σ

σ

11 2

1 0 4123

0 605

2when


37

Figure 2.2 depicts the resistance area of the combined effect of bending moment and shearforce.

Figure 2.2 The effect of shear force on plastic bending resistance. The bending resistance of the section due to the flanges is Mf.pl.

2.7.2.2 Class 1, 2 or 3 circular hollow sections

For a circular hollow section with Class 1, 2 or 3 cross-section, the bending resistance can beexpressed as follows when the shear force is more than half of the shear resistance(VSd > 0,5 Vpl.Rd) [3]:

however MV.Rd ≤ Mc.Rd

M M 1V

VV.Rd pl.Rd

Sd

pl.Rd

2

= −

( . )2 47

VRd

V

0,5VRd

Mf.pl MplM

Combined effect of shear force andbending moment

M Wf

M Wf

M Wf

c Rd ply

M

c Rd ely

M

c Rd effy

M

.

.

.

=

=

=

γ

γ

γ

0

0

1

t

Av

for Class 1 and 2 cross-sections (section 2.3)

for Class 3 cross-sections (section 2.3)


is the wall thickness of the hollow section

is the the area of the shear element formula [(2.20) or (2.21)]


38

where


For circular hollow sections with Class 4 cross-sections, the effect of shear force is accountedfor in the combined load criterion [2]:

where

Example 6Calculate the bending resistance for a hollow sectionwith dimensions 400 x 200 x 6 when subjected to ashear force equal to 600 KN. The steel grade used isS355J2H. The resistance to shear buckling for asimilar hollow section was calculated in example 2a.This value is now used. Force quantities are:

VSd = 600 kN > 0,5 Vba.Rd = 0,5 · 797,6 = 398,8 kNMSd = 290 kNm

Determine the classification of the cross-section:

Flange: 29,3 < b/ t = 200/ 6 = 33,3 < 33,4 ⇒ Class 2

Web: 61,1 < h/ t = 400/ 6 = 66,7 < 70 ⇒ Class 2

Since the cross-section of the hollow section is Class 2, the plastic bending resistancecan be used. The shear force is more than half of the resistance to shear buckling. Thus,it decreases the moment resistance.

Calculate the value for the parameter ρ :

ρ = −

= ⋅ −

=2V

V1

2

797,61Sd

Rd

2 2600

0 255,

is the design value for bending resistance (section 2.3.3)

is the design value for shear buckling resistance (section 2.4.2.2)MV

c.Rd

ba.Rd

M

M

V

VSd

c.Rd

Sd

ba.Rd 1+ ≤ ( . )2 48

M Wf

M Wf

V

V

c Rd ply

M

c Rd ely

M

Sd

pl Rd

.

.

.

=

=

= ( ) + ( )

γ

γ

0

0

V Vy.Sd2

z.Sd2

for Class 1 and 2 cross-sections (section 2.3)


is the plastic shear resistance (section 2.4.2.1)


39

�yVSd

MSd

Inserting the value for ρ in the expression (2.46):

The bending resistance of hollow section with dimensions 400 x 200 x 6 is not sufficient(MSd > MV.Rd). The wall thickness must be increased or a larger hollow section must beselected.

2.7.3 Hollow sections subjected to axial force, shear force and bending moment (no buckling)

2.7.3.1 Class 1 or 2 hollow sections

The interaction expression (2.49) can be applied to Class 1 and 2 cross-sections [1]:

where

α

α

=

−

≤

= −

≤

= −

≤

=−

+ ⋅

1 66

1 1 13

1 26 1

1 33 1

1

0 5

,

,

,

,

,

.

.

. .

. .

. .

N

N

6

= 2

MN

N

M

MN

N

M

M

N

Nh t

A

Sd

V.Rd

2

N.RdSd

V.Rd

N.Rd

Ny.RdSd

V.Rd

Ny.Rd

Nz.Rd

Sd

V.Rd

M

M

M

M

M

V Rd

V Rd

V y Rd

V y Rd

V z Rd

≤MNz.Rd MV z Rd. .

square hollow sections

square and rectangular hollow sections

rectangular hollow sections

rectangular hollow sections


however

however

however

M

M

M

M y.Sd

Ny.Rd

z.Sd

Nz.Rd

+

≤α α

1 0 2 49, ( . )

M

kNm

V Rd.

,

,

, ,

=− ⋅

=− ⋅

⋅

⋅

= ≤ =

WA

tf

kNm M

plV

y

M0

pl.Rd

ρ

γ

2 2

8906000

0 255 4584

8 6355

1 1

256 4 292 3


40

2.7.3.2 Class 3 and 4 square and rectangular hollow sections and Class 3 circular hollow sections

The interaction expression for a member subjected to compression and bending in thepresence of shear force can be expressed in a similar way to that in section 2.7.1.2. The effectof shear force is accounted for in bending resistance values [1]:


When using circular hollow sections with Class 4 cross-sections, the resistance of the cross-section must be checked using the combined load criterion [2]:

where

is the design value for bending resistance (section 2.3.3)

is the design value for compression resistance (section 2.6.3)

is the design value for shear buckling resistance (section 2.4.2.2)

NMV

c Rd

c.Rd

ba.Rd

.

N

N

M

M

V

VSd

c Rd

Sd

c.Rd

Sd

ba.Rd 1+ + ≤

.( . )2 51

N

N

M

M

M

MSd

c Rd

y Sd

V y Rd

z Sd

V z Rd.

.

. .

.

. ., ( . )+ + ≤ 1 0 2 50

MN

N

M

N.RdSd

pl.Rd

1,7

N.Rd

= −

≤

=

= − ⋅( )

1 04 1

0

0

, .

.

.

.

. .

. .

.

M

M

N Af

N A Af

M

M

M

V Rd

V Rd

pl Rdy

M

V Rd vy

M

V y Rd

V z Rd

V Rd

γ

ργ

however

is the bending resistance with the effect of the shear force taken intoaccount (by y axis) (section 2.7.2.1)

is the bending resistance with the effect of the shear force taken intoaccount (by z axis) (section 2.7.2.1)

is the bending resistance with the effect of the shear force taken intoaccount for circular hollow sections (section 2.7.2.2) or for square hollowsections (section 2.7.2.1)



41

Example 7Determine whether a hollow section with dimensions200 x 200 x 8 can carry the load shown in the adjacentfigure. The steel grade used is S355J2H, and the loadvalues are:

NSd = 1400 kNMy.Sd = 27 kNmMz.Sd = 24 kNmVz.Sd = 400 kN > 0,5 Vpl.Rd = 275,75 kNVy.Sd = 150 kN < 0,5 Vpl.Rd

Determine the classification of the cross-section (Table 2.4):

h/t = b/t = 200/8= 25 < 29,3 ⇒ Class 1

The method for calculating the shear resistance also depends on the slenderness of thecross-section:

h/t = 200/8 = 25 < 59,1 ⇒ calculate the plastic shear resistanceThe plastic shear resistance of the cross-section is the same about both axes:

The reduction in the bending resistance due to shear force depends on the parameter ρ:

Next, the resistance to axial force and bending is determined, taking into account theeffect of shear force:

M

M M

A Af

V y Rd

z

V z Rd pl z Rd

vy

M

. .

. . . .

,

,

,

,

,,

,

=− ⋅

=− ⋅

⋅

=

= =

= = ⋅ =

= − ⋅( ) = − ⋅(

WA

tf

kNm < M 135,8 kNm

kNm

N

plV2

y

M0

pl.y.Rd

V.Rd

ρ

γ

ργ

8420860

0 203 2960

8 8355

1 1

126 9

355 420 86

1 1135 8

5924 0 203 2960

2

0)) ⋅ =355

1 11718

, kN

ρ

ρ

z

y z SdV

= −

= ⋅ −

=

= ( )

2V

V1

2 400

551, 51

< 0, 5V

z.Sd

pl.Rd

2 2

pl.Rd

0 203

0

,

.for

Vpl Rd.,

,= ⋅⋅+

=⋅

⋅+

=f

3

h A

h b

355

3

200 5924

200 200 kNy

M0γ 1 1551 9


42

�yVz

Vy

N Mz

My

The effect of axial force on the bending resistance is accounted for by the followingformulae:

The parameter α for a rectangular hollow section is (section 2.7.3.1):

In this case, the interaction expression (2.49) is as follows:

2.8 Buckling resistance of hollow sections

2.8.1 Buckling resistance of square and rectangular hollow sections and Class 1, 2 and 3 circular hollow sections

Stuctural hollow sections are particularly efficient as compression members, as the hollowsection material is located equally about and at a distance from the cross-section’s mid-point.Due to high torsional stiffness, torsional buckling need not be taken into account. The designcriterion for the flexural buckling resistance of the compression member can be expressed asfollows [1]:

where

(the design value for flexural buckling resistance)Nb.Rd = ⋅ ⋅

=

=

χ βγ

χ

β

β

Ay

M

Aeff

A

eff

Af

A

A

A

A

1

1

is the reduction factor for flexural buckling

for square and rectangular Class 4 hollow sections

for Class 1, 2 and 3 cross-sections

is the effective cross-sectional area

is the total cross-sectional area

NSd Nb.Rd≤ ( . )2 52

M M

M

27 24 < 1,0 y.Sd z.Sd

Nz.RdMNy Rd. , ,,

+

=

+

=α α

29 60 31 670 765

6 6

OK!

α α=

−

=−

= ⇒1 66

1 1 13

1 66

1 1 13

6 65,

,

,

,

,N

N

1400

1718

> 6 = 6Sd

V.Rd

2 2

M M

M M

Ny Rd V y Rd

Nz Rd V z Rd

. . .

. . .

, , , ,

, , , ,

= ⋅ −

= ⋅ ⋅ −

=

= ⋅ −

= ⋅ ⋅ −

=

1 26 1 1 26 126 9 11718

29 60

1 26 1 1 26 135 8 11718

31 67

N

N

1400 kNm

N

N

1400 kNm

Sd

V.Rd

Sd

V.Rd


43

The reduction factor χ for buckling is given by the equation [1]:

where

Buckling length depends on the type of connection at the ends of the member. Usually, it isconservative to use the actual length as the buckling length for lattice structures and thetheoretical buckling length for rigid structures without allowing for joint rigidity. Thedetermination of buckling length is presented in more detail in Chapter 6.

According to Eurocode 3, there are several methods for calculating the buckling resistance ofthe cross-sections of cold formed hollow sections. A simple conservative method is to use thenominal yield strength fy and buckling curve c [7] for the hollow sections. For buckling curve c,the value of the imperfection factor α is 0,49 [1].

2.8.2 Buckling resistance of Class 4 circular hollow sections

The design criterion for the buckling resistance of Class 4 cross-sections is:

where

N A

f

Et

r

b Rdu

M

u y

cr

.

,,

,

,

,

= ⋅

= − ( )( )=

⋅

=+

=

χ σγ

σ λ

λα σ

α

σ

11 2

0

0

1 0 4123

0 83

1 0 01

0 605

f

r

t

y

cr

(design value for buckling resistance)

(buckling stress [2])

NSd Nb.Rd≤ ( . )2 56

is the buckling length

is the radius of gyration

is the factor allowing for initial deflection and residual stresses

Li

c

α

χφ φ λ

φ α λ λ

λπ

β

=+ −( )

≤

= ⋅ + −( ) + ( )[ ]=

⋅⋅

12 53

0 5 1 0 2 2 54

2 55

2 2

2

1,0

( . )

, , ( . )

( . )L

i

f

Ec y

A


44

The reduction factor χ for buckling is calculated as in section 2.8.1, except that the bucklingstress (fy = σu). is used instead of yield strength. The cross-sections of Class 4 circular hollowsections are always considered to be totally effective; thus βA= 1.

Example 8a Calculate the compression resistance of a hollowsection with dimensions 200 x 200 x 5. The steel gradeused is S355J2H, and the buckling length is 4 m. Themember is nominally pinned at both ends.Determine the classification of the cross-section of thehollow section (Table 2.4):

b / t = 200/ 5 = 40 > 36,6 ⇒ Class 4As the cross-section of the hollow section is Class 4,the effective cross-section must be determined. Theslenderness of the compression elements is calculatedusing the formula (2.6):

The dimensions of the effective and non-effective elements of the cross-section are asfollows [formula (2.5) and Table 2.5]:

Using the effective cross-section, let us determine the effective area and parameter βA:

The local buckling of the cross-section has now been taken into account. Next, considerthe buckling resistance of the hollow section. The cross-section slenderness isdetermined using the formula (2.55):

The reduction factor for buckling is calculated from the formulae (2.53) and (2.54) byinserting α = 0,49 ( for buckling curve c):

χ = 0,769

λπ

βπ

=⋅

=⋅

=L

i

f

E

4000

7

355

210000c y

A9 3

0 906 0 628,

, ,

A A b t

A

A

eff non eff

Aeff

= − ⋅ ⋅( ) = − ⋅ ⋅( ) =

= = =

4 3840 4 18 5 3480

3480

38400 906

.

,

mm

β

b h

b t b

eff eff

eff

= = − −( ) = − − ⋅( ) =

= − − = − ⋅ − =

λλ

p

p2

non.eff

b 3t 200 3 5 mm

b mm

0 22 0 801 0 22

0 801167 5

3 200 3 5 167 5 18

2, , ,

,,

,

λεp

1b

t

200 - 15

5 > 0,673=⋅

= =56 8

56 8235

355

0 801,

,,


45

��yy

200

5

2004000

NSd

The buckling resistance of the hollow section is calculated by multiplying the plasticcompression resistance of the effective cross-section by the reduction factor (2.52):

Example 8bCalculate the compression resistance of a hollow section with dimensions 200 x 200 x 8.The steel grade used is S355J2H, and the buckling length is 4 m. The member is nomi-nally pinned at both ends. Determine the classification of the cross-section (Table 2.4):

b/ t= 200/ 8 = 25 < 36,6 ⇒ Class 1 cross-section

The effective cross-section need not be calculated for Class 1, 2 and 3 cross-sections.The cross-section slenderness is calculated using the formula (2.55):

The reduction factor for buckling is calculated from formulae (2.53) and (2.54) byinserting α = 0,49 (for buckling curve c):

χ = 0,740

The buckling resistance of the hollow section is calculated by multiplying the plasticcompression resistance by the reduction factor χ (2.52):

2.9 Resistance of hollow sections subjected to combined loads (buckling)

Criteria for the effect of different load combinations when buckling is taken into account areshown in Table 2.8.

Table 2.8 Combined load criteria to be checked when buckling may be present

Loading Cross-section Class 1 and 2 cross- Class 3 cross- Class 4 cross-combination sections sections sections

Section Formula Section Formula Section Formula

Bending and Square 2.9.1.1 (2.57) 2.9.1.1 (2.57) 2.9.1.1 (2.57)

compression Rectangular 2.9.1.1 (2.57) 2.9.1.1 (2.57) 2.9.1.2 (2.60)

The effect of torsion is accounted for by adding the following term in the interaction expression:

M

Mt Sd

t Rd

.

.

N Af

b Rdy

M. ,

,,= ⋅ = ⋅ ⋅ =χ

γ 00 740 5920

355

1 11414 8 kN

λπ π

=⋅

=⋅

=L

i

f

E

4000

7

355

210000c y

7 60 675

,,

N Af

b Rd effy

M. ,

,,= ⋅ = ⋅ ⋅ =χ

γ 10 769 3480

355

1 1863 7 kN


46

2.9.1 Hollow sections subjected to bending moment and axial force (buckling)

The classification of webs subjected to bending and compression depends on the stressdistribution. In practice, the cross-section class is more easily determined by the compressionelement (web or flange).

2.9.1.1 Square and rectangular hollow sections and Class 1, 2 and 3 circular hollow sections

The interaction expression for a structure subjected to compression and bending is as follows[1]:

where

The parameters ky and kz in expression (2.57) are determined as follows [1]:

where

χχ

µ λ β

µ λ β

y

z

y

z

= −( ) +−

≤

= −( ) +−

≤

y Mypl y el y

el y

z Mzpl z el z

el z

W W

W

W W

W

2 4 0 9

2 4 0 9

. .

.

. .

.

,

,

is the reduction factor for buckling determined about the y axis

is the reduction factor for buckling determined about the z axis



kA f

kA f

yy

zy

= −⋅⋅ ⋅

≤

= − ⋅⋅ ⋅

≤

1 1 5 2 58

1 1 5 2 59

µχµχ

y Sd

y

z Sd

z

N

N

, ( . )

, ( . )

N Af

M Wf

M Wf

M Wf

M Wf

M Wf

M Wf

b Rd Ay

M

y Rd pl yy

Mz Rd pl z

y

M

y Rd el yy

Mz Rd el z

y

M

y Rd eff yy

Mz Rd eff z

y

M

. min

min

. . . .

. . . .

. . . .

= ⋅ ⋅

= =

= =

= =

χ βγ

χ

γ γ

γ γ

γ γ

1

1 1

1 1

1 1

is the minimum value of the reduction factor for buckling (about the y or z axis)

for Class 1 and 2 cross-sections


for Class 4 square and rectangularhollow sections

N

N

k

M

k

MSd

b.Rd

y

y.Rd

z

z.Rd+

⋅+ ⋅ ≤

M My Sd z Sd. . ( . )1 2 57


47

Table 2.9 The form factor of the moment plane βM [1]

Method of loading Moment diagram Equivalent uniform moment factor

End moments

βM.ψ = 1,8 - 0,7ψ

-1 ≤ ψ ≤ 1

In plane lateral uniform load

βMQ = 1,3

In plane lateral concentrated load

βMQ = 1,4

End moments and in plane lateralloads

βM is derived from the formulae

MQ = the moment with the greatestabsolute value due to lateral loading∆M = the moment with the greatestabsolute value when the sign of themoment does not change∆M = the sum of the absolute values ofthe greatest and smallest moments,when the sign of the moment changes

�y�y��yyβ β β βψ ψM M

QMQ M

MM

= + −( ). .∆

M1

ψM1

∆M

∆M

∆M

MQ

MQ

MQ

MQ

MQ

µ λ β

µ λ ββ β

λ

λ

y y My

z z Mz

My

y

z

= −( ) ≤

= −( ) ≤

=

=

2 4 0 9

2 4 0 9

,

,

Mzand



are equivalent uniform moment factors allowing for the shape of the moment diagram (Table 2.9)

is the slenderness determined by the y axis

is the slenderness determined by the z axis


48


Buckling is taken into account in the interaction expression which can be expressed as follows:

where

The parameters ky and kz in expression (2.60) are determined as follows:

where

Y and z axis are chosen in such a manner that the primary governing combination is obtainedfor bending moments.

χχ

µ λ β

µ λ ββ β

λπ

σ

λπ

σ

y

z

y y My

z z Mz

My Mz

yc y

y

u

zc z

z

u

L

i E

L

i E

= −( ) ≤

= −( ) ≤

=⋅

=⋅

2 4 0 9

2 4 0 9

ja

,

,

.

.

is the buckling reduction factor determined about the y axis

are the equivalent uniform moment factors (Table 2.9)

is the buckling reduction factor determined about the z axis

(σu is determined with parameter α0, Table 2.8.2)

(σu is determined with parameter α0, Table 2.8.2)

kN

A

kN

A

yy Sd

y u

zz Sd

z u

= −⋅

⋅ ⋅≤

= − ⋅⋅ ⋅

≤

1 1 5

1 1 5

µχ σµ

χ σ

,

,

(σu is determined with parameter α0, section 2.8.2)

(σu is determined with parameter α0, section 2.8.2)

N

f

r

t

b.Rd

y

cr

M W

f

Et

r

c Rd elu

M

u y

b

b

cr

.

,,

,,

,

,

=

= − ( )( )=

⋅

= ++

=

σγ

σ λ

λα σ

α

σ

11 2

1 0 4123

0 18870 6734

1 0 01

0 605

is the design value for buckling resistance (section 2.8.2)

(buckling stress [2])

N

N

k

M

k

MSd

b.Rd

y

c.Rd

z

c.Rd+

⋅+ ⋅ ≤

M My Sd z Sd. . , ( . )1 0 2 60


49

Example 9Calculate the resistance for the hollow section fromexample 8a with dimensions 200 x 200 x 5, when itis subjected to axial force and bending moment dueto a uniform transverse load. Only one of the axes issubjected to bending. The member is supported bynominally pinned connections at both ends.

The slenderness and compression resistance of theeffective cross-section is obtained from example 8a:

λ = 0,628

Nb.Rd = 863,7 kN (Nb.Rd = Nb.y.Rd = Nb.z.Rd, symmetrical cross-section)

The bending resistance value for a hollow section with dimensions 200 x 200 x 5 wascalculated in example 1d i.e.

My.Rd = Meff.Rd = 73,3 kNm

The equivalent uniform moment factor is given in Table 2.9:

βMQ = 1,3

Using factor βMQ the parameters µ and k in the interaction expression can be derivedfrom the formula (2.58):

Check the resistance of hollow section for the combined effect of axial force and moment(2.57):

N

N

k

M =

500

863,7 73, 3= 0,772 1,0 OK!Sd

b.Rd

y

y.Rd+

⋅+ ⋅ <

My Sd. ,1 419 10

µ λ β

µχ

y MQ

yy

kA f

= −( ) = ⋅ −( ) = −

= −⋅⋅ ⋅

= +⋅ ⋅

=

2 4 0 628 2 1 3 4 0 879

1 1 0 879500000

0 769 3840 3551 419

, , ,

,,

,y Sd

y

N


50

�y

200

5

200

4000

NSd =500 kN

MSd =10 kNm

Example 10Calculate the resistance of a hollow section withdimensions 180 x 180 x 5 to the loading shown in theadjacent figure. The buckling length of the structureis 4 m, and the member is supported by nominallypinned connections at both ends. The steel grade usedis S355J2H, and the loading values are:

NSd = 400 kNMy.Sd = 9 kNmMz.Sd = 9 kNmMt.Sd = 3 kNm (assumed constant along the

entire hollow section)

The hollow section in classified as Class 3, since33,4 < b/t= 36 < 36,6 (Table 2.4). Thus, the bendingresistance must be determined using elasticity theory.The effect of torsion must be accounted for in the inte-raction expression. The reduction factor χ for thebuckling of a hollow section subjected to compressionis calculated using buckling curve c (section 2.8.1):

Determine the compression and bending resistance of the hollow section:

The parameters µ and k depending on the shape of the moment diagram are as follows(section 2.9.1.1)

The calculation method for torsional resistance is determined by the web slenderness(section 2.5.1):

h/t = 180/ 5 = 36 < 59,1 ⇒ calculate the plastic torsion resistance

µ λ β

µχ

µ λ β

µχ

y MQ

yy

z MQ

zy

kA f

kA f

= −( ) = ⋅ ⋅ −( ) = −

⋅⋅ ⋅

= +⋅ ⋅

=

= −( ) = ⋅ ⋅ −( ) = −

⋅⋅ ⋅

2 4 0 736 2 1 3 4 1 03

1 1 03400000

0 702 3436 3551 481

2 4 0 736 2 1 4 4 0 883

, , ,

,,

,

, , ,

= 1-N

< 1, 5

= 1-N

y Sd

y

z Sd

z== +

⋅ ⋅=1 0 883

400000

0 702 3436 3551 412,

,, < 1, 5

N Af

M M Wf

b Rdy

M

y Rd z Rd ely

M

.

. .

,,

,

,,

= ⋅ = ⋅ ⋅ =

= = = ⋅ =

χγ

γ

1

1

0 702 3436355

1 1778 4

193355

1 162 28

kN

kNm

λπ π

χ

=⋅ ⋅L

i

f

E =

4000

71,1

355

210000= 0,736

= 0,702

c y


51

�yN Mz

My

Mt

4000

NN

My

y-y z-z

Mz

Plastic torsional resistance is calculated using the formula (2.30):

By adding the effect of torsion in the interaction expression (2.57), the following result isobtained:

Example 11Calculate the resistance of a circular hollow sectionwith dimensions 323,9 x 5 to the combined loadingshown in the adjacent figure. The steel grade used isS355J2H, and the hollow section length is 6 m. Themoment is assumed constant along the hollow sectionlength. The hollow section is supported by hinges atboth ends.The loading values are:

NSd = 500 kN (compression)My.Sd = 18 kNmMz.Sd = 18 kNmMt.Sd = 6 kNm

First, the buckling resistance of the hollow section is calculated. Obtain the followingvalue for the parameter α0 in compression only:

The buckling stress σu, in compression only, is calculated as follows:

The slenderness is obtained for buckling by inserting the buckling stress value into theformula (2.55):

σ

λα σ

σ λ

cr

u y

Et

r

f

= = ⋅ ⋅ =

=⋅

=⋅

=

= − ( )( ) = − ⋅( ) =

0 605 0 605 2 1 105

159 453984

0 723 3984 00 351

1 0 4123 1 0 4123 0 351 355 313 3

5

0

1 2 1 2

, , ,,

, ,,

, , , ,, ,

N/mm

f 355

N/mm

2

y

cr

2

α 00 83

1 0 010 723=

+=,

,,

r

t

N

N

k

M

k

M

62, 28 62, 28 < 1,0

Sd

b.Rd

y

y.Rd

z

z.Rd+

⋅+ ⋅ +

= + ⋅ + ⋅ + =

M M M

My Sd z Sd t Sd

t pl Rd

. . .

. .

,

, ,

,,

400

778 4

1 481 9 1 412 9 3

54 00 987 OK!

Mf W

t pl Rdy t

M. .

,

,,= ⋅ = ⋅ =

3

355

3

289 8

1 154 0

0γ kNm


52

��

yyy6000

My

Mz

Mt

N

The buckling resistance of the compression member is derived from formulae (2.52 - 2.54):

The equivalent uniform moment factor for constant moment is (Table 2.9):

βMψ = 1,1

Parametes µ and k are derived from the formula (2.58):

The bending resistance for a hollow section with dimensions 323,9 x 5 was calculated inexample 1e:

The torsional buckling resistance is taken from example 5:

The resistance values calculated can be inserted in the interaction expression (2.60).Allowing for the effect of torsion, the following result is obtained:

N

N

k

M

k

M

1074, 2

1, 498

112,6

1, 498 18

112,6 < 1,0 OK!

Sd

b.Rd

y

c.Rd

z

c.Rd+

⋅+ ⋅ +

= + ⋅ + ⋅ + =

M M M

My Sd z Sd t Sd

t b Rd

. . .

. .

,,

500 18 6

112 30 998

MW

t b Rd ba Rdt

M. . . ,= =τ

γ 1112 3 kNm

M Wc Rd elu

M. ,= =σ

γ 1112 6 kNm

µ µ λ β

µχ σ

ψy z M

y zu

k kA

= = ⋅ −( ) = ⋅ ⋅ −( ) = −

= ⋅⋅ ⋅

= +⋅ ⋅

=

2 4 0 654 2 1 1 4 1 177

1 1 177500000

0 753 5009 313 31 498

, , ,

,, ,

,= 1-N

< 1, 5Sd

y

N Ab Rdu

M. ,

,

,,= ⋅ = ⋅ ⋅ =χ σ

γ 10 753 5009

313 3

1 11074 2 kN

λπ

σπ

=⋅

=⋅

=L

i Ec u 6000

112 8

313 3

2100000 654

,

,,


53

2.10 Concentrated load resistance of hollow sections

When the hollow section is subjected to concentrated loads, the local resistance of the webmust be checked, as the walls of the hollow section are relatively thin. The larger the area intowhich the concentrated load can be distributed, the greater the resistance. The width ss of theeffective supporting surface of the area subjected to the load is directly proportional to thethickness of the material between the concentrated load and the web. Regarding the effect ofthe concentrated load, two cases can be distinguished [1]:

a) Concentrated load acting from one side onlyb) Concentrated load acting from both sides

2.10.1 Concentrated load acting from one side only

With a concentrated load acting from one side only, the resistance of the one web of hollowsection is the smallest of the following [1]:

where

s

s b tf

b

hs

h t

s

yf Ed

y

f Ed

s

= ⋅ −

−( )≤

2 1

30 2

2σ

σ

.

.

,

the width of the effective supporting surface determined by assuming that theconcentrated load is distributed in an 45° angle along continuous metal planes

is the bending stress in the flange

is the web height

is the lesser of the flange width and 25 t

R s s tf

R t

y Rd s yy

M

a RdM

.

.

( . )

, ( . )

= +( )

= ⋅+

γ

γ

1

2

1

2 61

0 51

3

2 62E f

s

h - 3ty

s


54

FSd

FSd

FSd

a)

b)

Figure 2.3 Width of effective supporting surface

The width of the effective supporting surface is calculated by the following formula, when thecorner rounding of the uppermost hollow section is filled by the weld (Figure 2.3, Detail 2):

where

Alternatively, when load is transmitted from the I section (Figure 2.3, Detail 1):

where

Additionally, when bending moment is present the interaction expression must be checked [1]:

F

R

M

M

F

R

M

M

Sd

a.Rd

Sd

c.Rd

Sd

a.Rd

Sd

c.Rd

+ ≤

≤

≤

1 5 2 66

1 0 2 67

1 0 2 68

, ( . )

, ( . )

, ( . )

is the web thickness of the I section

is the flange thickness of the I section

is the internal corner radius of the hot rolled I section

is the throat thickness of a welded I-section

tt

ra

w

f

i

b

s t t r

s t t a

s w f i

s w f b

= + + −( ) ⋅

= + + ⋅

2 2 2 2 2 64

2 2 2 2 65

( . )

( . )

(hot rolled I section)

(welded I section)

is the wall thickness of the uppermost hollow section (Figure 2.3, Detail 2)

is the internal corner radius of the uppermost hollow section (Figure 2.3, Detail 2)tri

s t rs i= + −( ) ⋅2 2 2 2 63( . )

��yy �

�yy1:11:

11:1

Detail 1 Detail 2

ss

FSdtwFSd

t

ri

Fillet weldss

FSdFSd

FSd

t f

t

ri

load from I profile load from hollowsection


55

2.10.2 Concentrated load acting from both sides

With a concentrated load acting from both sides, the resistance of one web of the hollowsection for one web is the smallest of the following [1]:

where

Formula (2.70) determines the compression resistance of the hollow section web by treatingthe web as a compression member whose width is beff and height is h - 3t. The buckling lengthis the web height h - 3t. The buckling load is calculated using the formulae (2.52) - (2.56)presented in section 2.8.1.Depending on the location of the concentrated load, the effective width of the web is calculatedas follows [1]:

a) concentrated load on the hollow section span

b) concentrated load close to the end of the hollow section

Figure 2.4 Effective web width

a)

beff

beff

Ss

beff

beff

b)

Ssa

hh

b h s a s h seff s s s= ⋅ + + + ≤ +0 5 0 52 2 2 2, ,

b h seff s= +2 2

is the buckling resistance of the compression member formed by the web

is the buckling reduction factor in buckling class c

is the effective width of the web (Figure 2.4)

Rb.Rd

χbeff

R

R

y.Rd

b.Rd

= +( )

= ⋅ ⋅

s s tf

b tf

s yy

M

effy

M

γ

χγ

1

1

2 69

2 70

( . )

( . )


56

Example 12Calculate the concentrated load resistance of the member shown in the adjacent figureusing hollow sections with dimensions 200 x 200 x 10 and 100 x 100 x 5. The steel gradeused is S355J2H.The resistance is derived from the formulae (2.61) and (2.62), as the concentrated loadin the joint acts from one side only. Flange plates (t = 10 mm) increase the width of theeffective supporting surface in proportion to their thickness.

For a hollow section with dimensions 200 x 200 x 10, the following result is obtained(concentrated load acting on both webs):

The resistance of the hollow section with dimensions 200 x 200 x 10 is the smallest of thevalues calculated above, that is, RRd = 918,9 kN.

For the hollow section with dimensions 100 x 100 x 5, the following result is obtained(concentrated load is distributed on webs):

R s s tf

kN

R t E f

s

h t

y Rd s yy

M

a Rd y

s

M

.

.

,,

,,

,

= ⋅ +( )

= ⋅ ⋅ + −( ) + ⋅ + ⋅ + ⋅( ) ⋅ ⋅ =

= ⋅ ⋅ ⋅+

−

= ⋅ ⋅ ⋅ + ⋅( ) =

2

2 2 10 2 2 15 2 10 2 10 2 100 5 5355

1 1366 3

2 0 51

3

3 5 210000 3551 3 0 2

1 1314

1

2

1

2

γ

γ

,,0 kN

R s s tf

kN

R t E f

s

h t

y Rd s yy

M

a Rd y

s

M

.

.

,,

,

,

,

= +( )

= ⋅ ⋅ + −( ) + ⋅ + ⋅ + ⋅( ) ⋅ ⋅ =

= ⋅ ⋅ ⋅+

−

= ⋅ ⋅ ⋅ + ⋅( ) =

2

2 2 5 2 2 5 2 10 2 10 2 200 10 10355

1 1918 9

2 0 51

3

3

1 10 210000 3551 3 0 2

1 11255

1

2

1

2

γ

γ

,, ,9 0 2 kN s

h - 3ts

( )≤

��yy

��

yyy

FSd

Ss

Ss 100x100x5

200x200x10

Ss

FSd 0,5FSd0,5FSd

Ss


57

The resistance of the hollow section with dimensions 100 x 100 x 5 is the smallest of thevalues calculated above, that is, RRd = 314 kN. The concentrated load resistance of theentire joint is determined by the buckling of the web of the hollow section withdimensions 100 x 100 x 5. Thus, the greatest allowed concentrated load value affectingthe joint is FSd = 314,0 kN.

2.11 References

[1] ENV 1993-1-1:Eurocode 3: Teräsrakenteiden suunnittelu. Osa 1-1: Yleiset säännöt jarakennuksia koskevat säännöt, 1993(ENV 1993-1-1: Eurocode 3: Design of steel structures. Part 1.1: General rules and rulesfor buildings, 1993)

[2] ECCS: Technical Committee 8- Structural stability- Technical working group 8.4- Stabilityof shells: Buckling of steel shells, European recommendations, 4th Edition, 1988

[3] CIDECT: Structural stability of hollow sections, Verlag TÜV Rheinland GmbH, Köln 1992

[4] CIDECT: Design guide for structural hollow sections in mechanical applications, VerlagTÜV Rheinland GmbH, Köln 1995

[5] ENV 1991-2-1:Eurocode 1: Suunnitteluperusteet ja rakenteiden kuormat. Osa 2-1:Rakenteiden kuormat: Tiheydet, oma paino ja hyötykuormat, 1995(ENV 1991-2-1:Eurocode 1: Basis of design and actions on structures. Part 2-1: Actionson structures. Densities, self-weight and imposed loads, 1995)

[6] EN 10219-2: Kylmämuovatut hitsatut seostamattomat rakenne- ja hienoraerakenne-teräsputkipalkit. Osa 2: Toleranssit, mitat ja poikkileikkaussuureet, 1997(EN 10219-2: Cold formed welded structural hollow sections of non-alloy and fine grainsteels. Part 2: Tolerances, dimensions and sectional properties, 1997)

[7] CIDECT: Research project No 2R-2-16: Buckling behaviour of a new generation of coldformed hollow sections, Draft final report-2R-2-16 final, Aachen 1996

[8] EN 10219-1: Kylmämuovatut hitsatut seostamattomat rakenne- ja hienoraerakenne-teräsputkipalkit. Osa 1: Tekniset toimitusehdot, 1997(EN 10219-2: Cold formed welded structural hollow sections of non-alloy and fine grainsteels. Part 1: Technical delivery requirements, 1997)


58

3 DESIGN OF JOINTS IN HOLLOW SECTION STRUCTURES

3.1 Design of welded joints in lattice structures

Joints in lattice structure are usually assumed to be nominally pinned, and brace members aredesigned for axial load only. Depending on the dimensions of the chord and brace members,the effect of joint rigidity can be accounted for, by reducing the buckling length of the bracemember. The transverse loads on the chord span between the braces introduce bending mo-ments, and the chord must therefore be designed for compression and bending. In terms ofcompression resistance, a hollow section with thin walls is the most practical solution.However, when considering the resistance of the joint , a thin-walled, wide chord is not as goodas a thick-walled, narrow chord. The design formulae for lattice structure joints are partiallybased on test results. When using the formulae, it must be ascertained that the hollow sectionsmeet the validity conditions given in the tables. Appendix 9.3 includes formulae for assessinglattice structure joints for different hollow section types.

Table 3.1 shows different types of uniplaner lattice structure joints. Joint in multiplaner framesare dealt with in references [2] and [3]. The figures also feature the following parametersessential for the joint design:

Eccentricity value is taken as positive when the neutral axes of the brace members intersectthe chord below the centre of gravity (Table 3.1). Eccentricity is negative when the intersectionis located above the chord’s centre of gravity (Table 3.1). The joint gap refers to the spacebetween the brace members. The joint is overlapped when the brace members are partially orcompletely overlaid by each other. The overlap can also be expressed as a negative-value.Eccentricity and gap are interrelated in the following manner [2]:

where

θi

ihh0

is the smaller of the angles between the brace member and the chord

is the height of the brace member

is the height of the chord

g eh h h

eh h

gh

= +

+ − −

= + +

+

−

0 1 2

1 2

1

1

2

2

1

1

2

2

1 2

1 2

0

2 2 2

2 2 23 2

sin( )

sin sin sin sin

sin sin

sin sin

sin( )( . )

θ θθ θ θ θ

θ θθ θθ θ

(3.1)

is eccentricity

is gap

is overlap

g

e

g

q


59

q

e

-e

g

Table 3.1 Joint types in lattice structures

Table 3.2 presents different failure modes of hollow section lattice structures. The governingfailure mode depends on the dimensions of the chord and brace members, and on the jointgeometry.

Joint type Gap joint Overlap joint

N

K

KT

T

X

Y


60

gθ1 θ2

e

θ1 θ2

-e q

θ1 θ2

g

e

θ1 θ2

-e

q

g1 g2

θ1 θ2θ3

e

θ3θ1 θ2

θ1

θ1

θ1

q2q1e=0

Table 3.2 Failure modes of lattice structure joints

Failure mode Structure in which the failuremode is possible

Flexural failure of the chordface

Thin-walled chord, bracemember narrower than chord

Punching shear failure of thechord face

Thin-walled chord with greatb0, brace member slightlynarrower than chord

Tension failure of the bracingmember or weld failure

Thick-walled brace memberand thin-walled chord

Local buckling of the bracingmember

Thin-walled brace memberwith a great bi or hi

Overall shear failure of thechord

Thin-walled chord with small h0

Local buckling of the chordwalls

Thin-walled high chord ofequal width as the bracemember

Local buckling of the chordface

Thin-walled chord with great b0


61

3.1.1 Joints of circular, square or rectangular brace members to square or rectangular chords

Before calculating the resistance of the joint, members must be designed according to theirloads (chapter 2). Joints are usually assumed pinned, so the brace members are designed foraxial force only. When calculating the resistance of the joint, the moments due to theeccentricity of the joint need not be taken into account, if the eccentricity is between

where

However, when designing the chord, the moments due to the joint eccentricity must be takeninto account.

The chord face resistance is affected by the axial force and the bending moment. This functionis determined by using the parameter n [1]:

where

In examples 13-17, the resistance of the joint is determined by using Tables 9.3.1, 9.3.2 and9.3.3 in Appendix 9.3. When using the tables, it must be ascertained that the lattice membersand the joint geometry meet the validity conditions presented in the tables. The principle is tocalculate the resistance of the joint for different failure modes and select the smallest value asthe final resistance of the joint. The steel designation used in all examples is S355J2H. Whencalculating the resistance of the hollow section lattice structure joints, the partial safety factorγMj, is used with the value of 1.1.

The validity condition for the joint are met in examples 13-27, but their checking is notpresented in the examples.

In the tables in Appendix 9.3 and examples 13-27, it is assumed that the value of thepartial safety factor γγγγM0 of the material is 1.1. The tables in Appendix 9.3 apply for steelgrades with the yield strength value of 355 N/mm2 or smaller.

σ 0

0

0

0

.

.

.

Ed

Sd

Sd

y

NMf

is the greatest compression stress in the flange on the side of the joint

is the axial force of the chord

is the bending moment of the chord

is the yield strength of the chord

nf

N

A f

M

W fM Mj Ed

y

M Mj Sd

y

Sd

el y=

⋅=

⋅⋅

+⋅

γ γ σ γ γ0 0

0

0 0

0 0

0

01 1 1 1, ,. . . (3.4)

is the height of the chordh0

− ≤ ≤0 55 0 250 0, ,h e h (3.3)


62

Example 13

A Y joint with a tension brace member(Table 9.3.1).

The joint geometry and loading are asfollows:

Brace member: 100 x 100 x 5, A1 = 1836 mm2

⇒ NRd = 1836 · 355/ 1,1 = 592,5 kNChord: 200 x 200 x 8, A0 = 5924 mm2

N0.Sd = 936,4 kN (compression)N1.Sd = 590 kN (tension)θ = 45° β = b1/ b0 = 100/ 200 = 0,5 η = h1/ b0 = 100/ 200 = 0,5

The chord axial force N0.Sd influences the resistance of the joint in the form of the term kn:

Chord face yieldSince β = 0,5 < 0,85 the chord face resistance must be checked:

Resistance of the jointThe resistance of the joint is thus N1.Rd = 225,1 kN, which is remarkably less than thebrace member axial force N1.Sd = 590 kN. A larger hollow section must be selected as thebrace member or the chord face must be reinforced to obtain a sufficient resistance ofthe joint.

N k

kN

Rd nMj M

10

2

0

2

1

24 1

1 1

55 8

1 0 5 45

2 0 5

454 1 0 5 0 908

1

1 1225 1

.( ) sin sin

,

( , ) sin

,

sin, ,

,,

=⋅

−+ −

⋅

= ⋅−

⋅ + −

=

f t

3 < N

y

Sd

β θηθ

βγ γ

nN

A f

kn

M Mj Sd

y

n

=⋅

⋅=

⋅=

= − = − ⋅ =

γ γ

β

0 0

01 1

1 1

1

936400

5924 3550 490

1 30 4

1 30 4 0 490

0 50 908

,

,,

,,

,, ,

,,

.


63

��

��h 0

h1

t1

b1

b0

t0N0.Sd

θ

N1.Sd

Example 14 A T joint with a compression bracemember (Table 9.3.1).The joint geometry and loading are thefollowing:


⇒ NRd = 1836 · 355/1,1 = 592,4 kNChord: 100 x 100 x 6, A0 = 2163 mm2

N0.Sd = 400 kN (compression)N1.Sd = 350 kN (compression)θ = 90° β = b1/ b0 = 100/ 100= 1,0

Note that the term kn is not relevant in this example, since the chord face yield is not thegoverning failure mode (β > 0,85).

Chord web bucklingThe chord web buckling may be a relevant failure mode for the design, since β is 1,0.First, determine the buckling stress using buckling curve c:

Now we can calculate the chord web resistance:

Brace member failureNext, determine the effective width of the brace member:

Resistance of the jointThe chord web resistance determines the resistance of the entire joint, which then is:N1.Rd = 376,1 kN > N1.Sd OK !

bb t

b tb

N f t h t b

eff

Rd y effMj M

= ⋅⋅

= ⋅ ⋅⋅

= ≤

= ⋅ − +( ) ⋅= ⋅ ⋅ − ⋅ + ⋅( ) =

10 10 100 6

100 572

2 4 21

355 5 2 100 4 5 2 721

1 1522 8

1 02

0 1

2

1

1 1 1 10

kN.,

,γ γ

N RdMj M

10 1

00

210

1 265 2 6

90

2 100

9010 6

1

1 1376 1.

sin sin

,

sin sin ,,= ⋅ +

⋅

= ⋅ ⋅ + ⋅

=f t h

t kNb

θ θ γ γ

λθ π π

χ

= −

= −

=

=

= ⋅ =

3 46 21

3 46 29

10 664

0 747

0 747 355 265 2

,)

,)

,

,

, ,

h

t

f

E(sin

100

6

355

210000(sin 0

N

mm

0

0

y

2fb

nN

A f

kn

k

M Mj Sd

y

n n

=⋅

⋅=

⋅=

= − = − ⋅ = > ⇒ =

γ γ

β

0 0

01 1

1 1

1

400000

2163 3550 573

1 30 4

1 30 4 0 573

1 01 07 1 0 1

,

,,

,,

,, ,

,, ,

.


64

��

��h 0

h1 t1

b1

b0

t0N0.Sd

θ

N1.Sd

Example 15 An X joint with compression brace members(Table 9.3.1).The joint geometry and loading are asfollows:

Brace members: 180 x 180 x 6, A1 = 4083 mm2

⇒ NRd = 4083 · 355/ 1,1 = 1318 kNChord: 200 x 200 x 8, A0 = 5924 mm2

N0.Sd = 620 kN (tension) ⇒ kn =1N1.Sd = 1000 kN (compression)θ = 30°β = b1 / b0 = 180/ 200 = 0,90η = h1 / b0 = 180/ 200 = 0,90γ = 0,5b0 / t0 = 0,5 · 200/ 8 = 12,5

Chord face punching shearThe chord face punching shear must be checked, since 0,85 < β < 1-(1/γ)= 0,92:

Chord face yield and chord web bucklingThe chord resistance must be determined for both chord face and web, since0,85 < β < 1,0. The resistance of the joint is calculated for the chord face when β = 0,85and for the chord web when β = 1,0. Then, the resistance is determined by linearinterpolation when β = 0,9:

a) β = 0,85

Nt

N

RdMj M

Rd

102

0

2

1

1

24 1

1 1

355 8

1 0 85 30

2 0 9

304 1 0 85

1

1 11418 1 1318

.

.

( ) sin sin

,

( , ) sin

,

sin,

,,

=⋅

−+ −

⋅

= ⋅−

⋅ + −

= ⇒ =

fk

kN > N kN

yn

Rd

β θηθ

βγ γ

bt b

bb

N

ep

RdMj M

= ⋅ ≤ = ⋅ ⋅ = ≤

=⋅

+

⋅

= ⋅ ⋅ +

=

10 10 8 180

20072 180

3

2 1

3 30

2 180

3080

1

1 12897 8

0 1

01

11

0

f t hb + b

355 81 +72 kNy 0

1 ep.sin sin sin sin ,

,θ θ γ γ


65

��

�

h 0

h1

t1

b1

b0

t0N0.Sd

θ

N1.Sd

h1

b) β = 1,0

Now, determine the chord resistance by interpolation using the values calculated above:

N1.Rd = 533,8+ (1318 – 533,8)(1– 0,9)/ 0,15 = 1056,6 kN

Chord shearNext, check the chord shear resistance:

Brace member failureAlso the effective width of the brace member must be checked:

Resistance of jointThe resistance of the joint is the smallest of the above values, that is, N1.Rd = 1022,4 kN > N1.Sd OK !

bb t

b tmm b

N f t h t b

eff

Rd y effMj M

= ⋅⋅

= ⋅ ⋅⋅

= <

= ⋅ − +( ) ⋅= ⋅ ⋅ − ⋅ + ⋅( ) =

10 10 180 8

200 696

2 4 21

355 6 2 180 4 6 2 961

1 11022 4

1 02

0 1

2

1

1 1 1 10

kN.,

,γ γ

N RdMj M

10

1

30

1

1 11192 5.

sin sin ,,=

⋅⋅

= ⋅ =f A

3

355 3200

3 kNy v

θ γ γ

λθ π π

χ

θ θ γ γ

= −

= −

=

=

= ⋅ ⋅ ⋅ =

= ⋅ +

⋅

= ⋅

3 46 21

3 46 23

11 473

0 323

0 323 0 8 30 355 45 87

210

1 45 87 8

301

0 10

0

,)

,)

,

,

, , sin ,

sin sin

,

sin.

h

t

f

E(sin

200

8

355

210000(sin 0

N

mmf t h

t

0

0

y

2

b

f

N

b

RdMj M

22 180

3010 8

1

1 1533 8

⋅ + ⋅

=

sin ,, kN


66

Example 16 A gapped K joint (Table 9.3.2).The joint geometry and loading are asfollows:

Brace members:150 x 150 x 6, A1 = 3363 mm2

⇒ NRd = 3363 · 355/ 1,1 = 1085 kNChord: 200 x 200 x 8, A0 = 5924 mm2

θ1 =θ2 = 45°N0.Sd = 1363,6 kN (compression)N1.Sd = 600 kN (compression)N2.Rd = 600 kN (tension)

β = 150 / 200= 0,75 < 1 – (1/γ)= 0,92, so the chord punching shear must be checked.

The joint gap presented by the joint geometry is [2]:

e = 20 mm, e < 0,25h0 = 50 mm (eccentricity is within the limits allowed in Appendix9.3)Determine the resistance of the joint by brace member 1 only, since the brace membersare of equal size and carrying equal loads.

Chord face yieldFirst, calculate the resistance by chord face yield:

N RdMj M

10

2

0 0

2

8 91 1

8 9355 8

45

150 150920 12 5

1

1 1634 2

. ,sin

,

,sin

, ,,

,

=⋅

+

⋅

⋅

= ⋅ +⋅

=

= =∑ ∑f t

b h

2m bk

2 2000 kN

yi

i 1

m

ii 1

m

nθγ

γ γ

g eh h h= +

+ − − =0 1 2

1 2

1

1

2

22 2 227 9

sin( )

sin sin sin sin,

θ θθ θ θ θ

mm

nN

A f

kn

M Mj Sd

y

n

=⋅

⋅=

⋅=

= − = − ⋅ =

= ⋅ =

γ γ

β

γ

0 0

01 1

1 1

1

1363600

5924 3550 713

1 30 4

1 30 4 0 713

0 750 920

0 5200

812 5

,

,,

,,

,, ,

,,

, ,

.


67

��

�b1,2

t1,2

b0

t0

θ 2

N0.Sd

h 0

θ 1g

h1 h2N2.Sd

e

N1.Sd

Chord shearObtain the following value for the shear resistance of the entire chord:Av = (2h0 + α · b0)t0 = (2 · 200+ 0,241 · 200)8 = 3586 mm2

Brace member failureThe effective width of the brace member is:

Chord punching shearIn this case, the chord punching shear resistance must also be taken into account:bep = 10t0 · b1 / b0 = 10 · 8 · 150/ 200 = 60 ≤ 150

Resistance of the jointThe chord face yield determines the resistance of the joint, which then is:N1.Rd = 634,2 kN > N1.Sd OK!

N RdMj M

11

03

2 1 1

3 45

2 150

4550

1

1 11337 1

.sin sin

,

sin sin ,,

=⋅

+

⋅

= ⋅ ⋅ +

=

f t hb + b

355 81 + 60 kN

y 01 epθ θ γ γ

bb t

b t

N f t h t b b

eff

Rd y effMj M

= ⋅⋅

= ⋅ ⋅⋅

=

= ⋅ − + +( ) ⋅

= ⋅ ⋅ − ⋅ + +( ) =

10 10 150 8

200 680

2 41 1

355 6 2 150 4 6 150 801

1 1979 8

1 02

0 1

2

1 1 1 1 10

mm < b

kN

1

.,

,,

γ γ

α

θ γ γ

=+

=+ ⋅

⋅

=

=⋅

⋅= ⋅ =

1

14

3

1

14 27 9

3

0 241

1 1

45

1

1 1944 9

2

10

g

t 8

f A

3

355 3586

3 kN

2

02

2

y v

,,

sin

,

sin ,,.N Rd

Mj M


68

In a gapped KT joint, it must also be checkedthat the sum of the brace member verticalforce components is less than the resistanceof the joint. The vertical rigidity in the chordface is poor, so this condition is relevant. Fourdifferent load modes can be suggested.These are presented in Figure 3.1. In 3.1c,the lattice joint is subjected to a down-pullingpoint load. In 3.1d, an intermediate support isplaced at the corner. In the cases presentedin the figure, the conditions for the resistanceof the joint are the following [2]:

a N N N

b N N N

c N N N N

d N N

Rd Sd Sd

Rd Sd Sd

Rd Sd Sd Sd

Rd Sd

) sin sin sin ( . )

) sin sin sin ( . )

) sin sin sin sin ( . )

) sin sin

. . .

. . .

. . . .

. .

2 2 1 1 3 3

1 1 2 2 3 3

2 2 1 1 3 3 4 4

1 1 2 2

3 5

3 6

3 7

θ θ θθ θ θθ θ θ θθ θ

≥ +≥ +≥ + +≥ ++ +N NSd Sd3 3 4 4 3 8. .sin sin ( . )θ θ


69

Figure 3.1 A gapped KT joint

θ2

θ3

1 23a) b)

c) d)

θ1 θ1

θ1 θ1

θ2 θ2

θ2

θ3 θ3

θ3

1

1 1

2

2 2

3

3 3

4 4θ4θ4

Example 17a A gapped K joint (Table 9.3.3).The joint geometry and loading are asfollows:

Brace members: 140 x 140 x 6, A1 = 3123mm2

⇒ NRd = 3123 · 355/1,1 = 1008 kN Chord: 180 x 180 x 8, A0 = 5284 mm2

θ1 =θ2 = 60°N0.Sd = 1500 kN (tension)N1.Sd = 800 kN (compression)N2.Sd = 800 kN (tension)

The joint overlap expressed by the joint geometry is as follows [2]:

e = –30 mm > –0,55h0 = –99 mm (joint eccentricity is within the limits allowed inAppendix 9.3)

The relative value of the overlap λov is:

λov = q· sin(θ1)/ h1 = 92,4 sin(60)/ 140 = 0,57

Brace member failureNow 0,5 ≤ λov < 0,8, so the following value is obtained for the effective width:

Resistance of the jointThe resistance of the joint is not sufficient, since N1.Rd = 772,5 kN is less than N1.Sd.With an overlap (q) of 130 mm, a sufficient strength of the joint is obtained.

bb t

b tmm b

bb t

b tmm b

N f t h t b b

eff

e ov

Rd y e e ovMj M

= ⋅⋅

= ⋅ ⋅⋅

= ≤

= ⋅⋅

= ⋅ ⋅⋅

= ≤

= ⋅ − + +( ) ⋅

= ⋅ ⋅ − ⋅

( )

( )

10 10 140 8

180 683

10 10 140 6

140 660

2 41 1

355 6 2 140 4

1 02

0 1

2

1

1 22

2 1

2

1

1 1 1 10

.,

γ γ

66 83 601

1 1772 5+ +( ) =

,, kN

q eh h h= − +

+⋅

− −

=0 1 2

1 2

1

1

2

22 2 292 4

sin( )

sin sin sin sin,

θ θθ θ θ θ

mm (overlap scale)


70

��

�h1 h

2

b1,2

t1,2

b0

t0

θ 2

N0.Sd

h 0

θ 1

N2.Sd

-e

N1.Sd

q

Example 17bA lower corner joint in a lattice structure(Table 9.3.3).The resistance of the lattice’s lower corner canbe determined using the formulae foroverlapped joints which consider the lowerchord as continuous (Figure 3.2) [2]. Theresistance of the joint can be determined forthe lower corner with the following jointmembers:


⇒ NRd = 1836 · 355/ 1,1 = 592,4 kNChord: 150 x 150 x 6, A0 = 3360 mm2

θ1 = 90°θ2 = 45°N1.Sd = 500 kN (compression)The joint must be designed in such a manner that eccentricity e = 0.

The following overlap value is obtained:

Brace member failureNow λov > 0,8, so the following value forthe effective width is obtained:

Resistance of the jointThus, the resistance of the joint is N1.Rd = 529,3 > N1.Sd OK !

bb t

b tmm b

N f t h t b b

e ov

Rd y e e ovMj M

( )

( )

= ⋅⋅

= ⋅ ⋅⋅

= ≤

= ⋅ − + +( ) ⋅

= ⋅ ⋅ − ⋅ + +( ) =

10 10 100 6

150 548

2 41 1

355 5 2 100 4 5 100 481

1 1529 3

1 22

2 1

2

1

1 1 1 10

kN

.,

,,

γ γ

q eh h h

qh

= − +

+⋅

− −

=

= ( ) = ( ) =

0 1 2

1 2

1

1

2

2

1

1

2 2 281 1

81 190

1000 81

sin( )

sin sin sin sin,

sin,

sin,

θ θθ θ θ θ

λ θ

mm

ov


71

Figure 3.2a A lower cornerjoint in a lattice structure

θ2θ1

N1.Sd

N1.Sd

θ2θ1

Figure 3.2b The calculationmodel

q

3.1.2 Joints of circular brace members to circular chords

The joint of a circular brace member to a circular chord is designed according to Tables 9.3.4,9.3.5 and 9.3.6 presented in Appendix 9.3. Otherwise, lattice structures constructed of circularhollow sections are designed by the same principles as square and rectangular hollowsections, presented in section 3.1.1. When determining the resistance of the joint, the momentsdue to joint eccentricity need not be taken into account, if the eccentricity is between:

where

d0 is the diameter of the chord

However, when designing the chord, the moments due to joint eccentricity need to be takeninto account.

The chord axial force and bending moment influence the chord face resistance. This function isdetermined by parameter np.

where

σ

θ

θ

p Ed

p Sd Sd i Sd i

Sd

i Sd

i

Sd

y

N N N

NN

Mf

.

. . .

.

.

.

cos= − ( )0

0

0

0

Σ

is the chord compression stress due to force Np.Sd and bending moment M0.Sd

is the axial force of the chord

is the axial force of the brace member

is the angle between the brace member and the chord

is the bending moment of the chord

is the yield resistance of the chord

nf

N

A f

M

W fp

M Mj p Ed

y

M Mj p Sd

y

Sd

el y=

⋅=

⋅⋅

+⋅

γ γ σ γ γ0

0

0

0 0

0

01 1 1 10

, ,. . . (3.1 )

− ≤ ≤0 55 0 25 90 0, ,d e d (3. )


72

Example 18 A T or a Y joint (Table 9.3.4).The joint geometry and loading are as follows:

Brace member: 168,3 x 5, A1 = 2565 mm2

⇒ NRd = 2565 · 355/ 1,1 = 827,8 kN Chord: 219,1 x 10, A0 = 6569 mm2

θ = 90°N1.Sd = 450 kN (compression)Np.Sd = 1018,2 kN (compression)

The axial force of the chord Np.Sd influences theresistance of the joint in the form of the term kp.

Chord face yieldThe resistance of the joint determined by the chord face yield is:

Chord punching shearThe chord punching shear resistance is given by:

Resistance of the jointThe resistance of the joint is the smallest of the above values N1.Rd = 462,1 kN > N1.Sd OK !

Nd

RdMj M

10 1

20

23

1

2

1 1 355 10 168 3

3

1 90

2 90

1

1 1985 2.

sin

sin

, , sin

sin ,,=

⋅ ⋅ ⋅ +

⋅

= ⋅ ⋅ ⋅ +

=

f t kNy π θ

θ γ γπ

N RdMj M

12

0

2

2 8 14 21 1

902 8 14 2 0 77 11 0 0 79

1

1 1462 1

.sin

, ,,

sin, , , , ,

,,

=⋅

+( ) ⋅

= ⋅ + ⋅( ) ⋅ =

f tk

355 10 kN

y 02

0.2p

20,2

θβ γ

γ γ

nN

f A

k n n

pM Mj p Sd

y

p p p

=⋅

⋅=

⋅=

= − +( ) = − +( ) =

= =

= =

γ γ

β

γ

0

0

2 2

1 1

1 1

1

1018200

355 65690 48

1 0 0 3 1 0 0 3 0 48 0 48 0 79

168 3

219 10 77

219 1

2011 0

,

,,

, , , , , , ,

,

,,

,,

.


73

��

��

��

�

d1

d 0

Np.Sdt0

t1

N1.Sd

θ

Example 19An X joint (Table 9.3.4).The joint geometry and loading are as follows:

Brace members: 193,7 x 6, A1 = 3538 mm2

⇒ NRd = 3538 · 355/ 1,1 = 1142 kN

Chord: 219,1 x 10, A0 = 6569 mm2

θ = 90°N1.Sd = 450 kNNp.Sd = 1018,2 kN (compression)

kp = 1,0 – 0,3(np+np2) = 1,0– 0,3(0,48+ 0,482) = 0,79β = 193,7/ 219,1 = 0,884

Chord face yieldThe resistance of the joint determined by the chord face yield is:

Chord punching shearThe chord punching shear, resistance is given by:

Resistance of the jointThe resistance of the joint is the smallest of the above values N1.Rd = 466,9 kN > N1.Sd OK !

Nd

RdMj M

10 1

20

23

1

2

1 1 355 10 193 7

3

1 90

2 90

1

1 11133 8.

sin

sin

, , sin

sin ,,=

⋅ ⋅ ⋅ +

⋅

= ⋅ ⋅ ⋅ +

=

f t kNy π θ

θ γ γπ

N RdMj M

10

5 2

1 0 81

1 1

90

5 2

1 0 81 0 8840 79

1

1 1466 9.

sin

,

,

,

sin

,

, ,,

,,=

⋅−

⋅

= ⋅− ⋅

⋅ =f t

k355 10

kNy 02

p

2

θ β γ γ

⇒ =⋅

=np1 1

1

1018200

355 65690 48

,,


74

��

��

��

��

d1

d 0

Np.Sdt0

t1

N1.Sd

θ

d1

N1.Sd

Example 20 An overlapped K joint (Table 9.3.5).The joint geometry and loading are asfollows:

Chord: 219,1 x 10, A0 = 6569 mm2

Brace members: 168,3 x 5 (compressionmember)⇒ N1.Rd = 2565 · 355/ 1,1 = 827,8 kN 139,7 x 5 (tension member)⇒ N2.Rd = 2116 · 355/ 1,1 = 682,9 kN θ1 =50°θ2 = 60°N1.Sd = 600kN (compression)N2.Sd = 530,7 kN (tension)Np.Sd = 636,4 kN (compression)

kp = 1,0 – 0,3(np + np 2) = 1,0 – 0,3(0,30+ 0,302) = 0,88β = (168,3+139,7)/ (2 · 219,1) = 0,703γ = 219,1/ 20 = 11.0g = 25 mme = 42,6 mm [formula (3.2)]e < 0,25d0 = 54,8 mm OK !

Chord face yieldThe chord face yield resistance for the compression member is given by:

For the tension member, the corresponding resistance is:

N NRd Rd2 11

2656 5

50

60580 7. .

sin

sin,

sin

sin,= ( )

( )

= ( )

( )

=

θθ

kN

k

e e

N k

g g

t

Rd p

= + ⋅

+

= + ⋅

+

=

=⋅

+( ) ⋅

−

⋅

−

γ γ

θβ

0 21 2

21 33

0 21 2

25

2 101 33

10

2

10 024

1

11 10 024 11

1

1 974

1 8 10 21

0

,,

,

,,

,

.

, ,,

sin, ,

,f tky

g11

355 10

501 8 10 2 0 703 1 974 0 88

1

1 1656 5

0

2

γ γMj M⋅

= ⋅ + ⋅( ) ⋅ ⋅ =sin

, , , , ,,

, kN

⇒ =⋅

=np1 1

1

636400

355 65690 30

,,


75

��

��

��

�

�N1.Sd N2.Sd

d 0

Np.Sd

θ1 θ2t1,2

t0

d1d2

g

Chord punching shearThe chord punching shear resistance for the compression member is given by:


Resistance of the jointThe resistance of the joint is the smallest of the above values:

Compression member: N1.Rd = 656,7 kN > N1.Sd OK !Tension member: N2.Rd = 580,7 kN > N2.Sd OK !

Example 21 An overlapped K joint (Table 9.3.6).The joint geometry and loading are as follows:

Brace members: 139,7 x 5, A1 = 2116 mm2

⇒ NRd = 2116 · 355 / 1,1 = 682,9 kNChord: 219,1 x 10, A0 = 6569 mm2

θ1 = 40°θ2 = 50° N1.Sd = 600 kN (compression)N2.Sd = 503,5 kN (tension)Np.Sd = 636,4 kN (compression)

kp = 1,3 – 0,3(np+np2) = 1,0 – 0,3(0,30+ 0,302) = 0,88β = 139,7/ 219,1 = 0,64γ = 219,1/ 20 = 11,0q = 85 mm (overlap)e = –53,0 mm [formula (3.2)]e > – 0,55d0 = –120,5 mm OK !

⇒ =⋅

=np1 1

1

636400

355 65690 30

,,

Nd

RdMj M

20 2 2

22 0

23

1

2

1 1 355 10 139 7

3

1 60

2 60

1

1 11017 3.

sin

sin

, , sin

sin ,,=

⋅ ⋅ ⋅ +

⋅

= ⋅ ⋅ ⋅ +

=

f t kNy π θ

θ γ γπ

Nd

RdMj M

10 1 1

21 0

23

1

2

1 1 355 10 168 3

3

1 50

2 50

1

1 11482 4.

sin

sin

, , sin

sin ,,=

⋅ ⋅ ⋅ +

⋅

= ⋅ ⋅ ⋅ +

=

f t kNy π θ

θ γ γπ


76

��

�

��

��

N1.Sd N2.Sd

d 0

Np.Sd

θ1 θ2t1,2

t0

d1 d2

q

d1,2

Chord face yieldThe chord face yield for the compression member is given by:


Resistance of the jointThe resistance of the joint expressed by the brace members is:Compression member: N1.Rd = 846,3 kN > N1.Sd OK !Tension member: N2.Rd = 710,1 kN > N2.Sd OK !

3.1.3 Joints of circular, square and rectangular brace members to I profile chords

The joint is designed according to the principles presented in section 3.1.1, except that nowTables 9.3.7 and 9.3.8 in Appendix 9.3 are used.

3.2 Welded frameworks

Members in welded frameworks are subjected to both bending and axial loads. Since the jointrigidity influences the scale of the joint moment and the joint moment influences the jointrigidity, the final distribution of forces and moments should be determined by iteration. Aconservative method is to determine the joint moments assuming complete rigidity of the joints,and the span moments by assuming the jointsare pinned. Appendix 9.4 contains instructions forestimating joint rigidity when using square andrectangular hollow sections.

Figure 3.3 presents the moment-rotation curvefor welded joints. The curve slope represents thejoint rigidity. This depends on the relationbetween the hollow section width and the columnwidth, and on the wall thickness of joint elements.The greater the b1/b0 relation and the wallthickness, the stiffer the joint.

N NRd Rd2 11

2846 3

40

50710 1. .

sin

sin,

sin

sin,= = =θ

θ kN

k

e e

N k

g q

t

Rd p

= + ⋅

+

= + ⋅

+

=

=⋅

+( ) ⋅

− −

−⋅

−

γ γ

θβ

0 21 2

21 33

0 21 2

85

2 101 33

10

2

10 024

1

11 10 024 11

1

2 30

1 8 10 2

0

,,

,

,,

,

.

, ,,

sin, ,

f tky

g11 1

355 10

401 8 10 2 0 64 2 30 0 88

1

1 1846 3

0

2

,

sin, , , , ,

,,

γ γMj M⋅

= ⋅ + ⋅( ) ⋅ ⋅ = kN


77 ��

Figure 3.3 Rigidity of frame joints

b1/b0= 1

b1/b0< 0,85

θ

M

b1

b0

θ

M

3.2.1 Joints of square and rectangular hollow sections subjected to bending

Table 9.3.9 in Appendix 9.3 includes the formulae for determining the bending resistance ofsquare and rectangular hollow sections for loads both parallel and perpendicular to the chordaxis. The combined effect of the axial force and bending moment are accounted for by the in-teraction expression [1]:

where

Moreover, the resistance of the joint to axial force (section 3.1) and to bending moment(section 3.2) must be checked separately.

Example 22A compression and bending T joint (Table9.3.9).The joint members and loading are the following:Chord: 150 x 150 x 6 NRd = 3363 · 355/ 1,1 = 1085 kNMRd = 180 · 355/ 1,1 = 58,1 kNmBrace member: 150 x 150 x 6NRd = 3363 · 355/ 1,1 = 1085 kNMRd = 180 · 355/ 1,1 = 58,1 kNmLoads:Mip.1.Sd = 15 kNmN1.Sd = 150 kN β = b1/ b0 = 150/ 150 = 1,0

Brace member failureSince β =1, the effective width of the brace member is calculated first (Table 9.3.9):

bb t

b tmm b

M f Wb

bb h t

eff

ip Rd y pleff

Mj

= ⋅⋅

= ⋅ ⋅⋅

= ≤

= − −

⋅ ⋅

= − −

⋅ ⋅

=

10 10 150 6

150 660

11

355 180000 160

150150 150 6

1

1 132

1 02

0 1

2

1

1 11

1 1 1

kNm

. . .

,

γ

is the axial force of the brace member

is the bending moment parallel to the plane of the frame

is the bending moment perpendicular to the plane of the frame

NM

M

Sd

ip Sd

op Sd

1

1

1

.

. .

. .

N

N

M

M

M

MSd

Rd

ip Sd

ip Rd

op Sd

op Rd

1

1

1

1

1

11 1.

.

. .

. .

. .

. .+ + ≤ (3.1 )


78

��

��

N1.Sd

Mip.1.Sd

h 0

θ t1

t0

b1

h1

b0

Chord web yieldThe bending resistance determined by the chord web yield is (Table 9.3.9):

T joint, fyk = fy

Bending resistance of the jointThe bending resistance of the joint is the smallest of the above values.Mip.1.Rd = 31,4 kNm > MSd

The brace member resistance to compression is determined according to the guidance insection 3.1.1. Since the brace member and chord are of equal width, the brace memberfailure and the chord face buckling due to axial force need to be checked:

Brace member failure (Table 9.3.1)

Chord face buckling (Table 9.3.1)

Compression resistance of the jointThe compression resistance of the joint is the smallest of the above values.N1.Rd = 359,7 kN

Combined load condition of the jointNow, the calculated resistance values are substituted in the interaction expression (3.11).

N

N

M

MSd

Rd

ip Sd

ip Rd

1

1

1

1

150

359 7

15

31 40 895.

.

. .

. . , ,,+ = + = < 1,0 OK!

λθ π π

χ

γ γ

= −

( )

= −

( )

=

=

= ⋅ =

= ⋅ +( )⋅

= ⋅ ⋅ ⋅ + ⋅

3 46 21

3 46150

62

355

210000 90

11 041

0 516

0 516 355 183 3

2 101 1

183 3 6 2 150 10 6

0

0

1 0 1 00

,sin

,sin

,

,

, ,

,,.

h

t

f

E

f

N f t h t

y

b

Rd bMj M

N

mm2

(( ) =1

1 1359 7

,, kN

bb t

b tmm b

N f t h t b

eff

Rd y effMj M

= ⋅⋅

= ⋅ ⋅⋅

= ≤

= ⋅ − +( ) ⋅= ⋅ ⋅ ⋅ − ⋅ + ⋅( ) =

10 10 150 6

150 660

2 4 21 1

355 6 2 150 4 6 2 601

1 1766 8

1 02

0 1

2

1

1 1 1 10

kN.,

,,

γ γ

M f t h tip Rd ykMj M

. . ,,

,,

,1 0 1 02

0

20 5 51 1

0 5 355 6 150 5 61

1 131 4= ⋅ +( )

⋅= ⋅ ⋅ ⋅ + ⋅( ) =

γ γ kNm


79

3.2.2 Circular hollow section joints subjected to bending

Table 9.3.10 in Appendix 9.3 presents the formulae for determining the moment resistance ofcircular tubes for loads both parallel and perpendicular to the plane of the frame. The combinedeffect of axial force and bending moment is allowed for by using the following interactionexpression [1]:

Moreover, the resistance of the joint for the axial force (section 3.1) and the bending momentmust be checked separately (section 3.2).

Example 23A compression and tension T joint.The joint members and loading are asfollows:

Chord: 219,1 x 5 NRd = 3363 · 355/ 1,1 = 1085 kNMRd = 229 · 355/ 1,1 = 73,9 kNm

Brace member: 219,1 x 5 NRd = 3363 · 355/ 1,1 = 1085 kNMRd = 229 · 355/ 1,1 = 73,9 kNm

Loads:Mip.1.Sd = 30 kNmNp.Sd = 272,7 kN (chord)N1.Sd = 70 kN (brace member)β = d1 / d0 = 219,1/ 219,1 = 1,0γ = 219,1/ (2 · 5)= 21,91

First, we calculate the effect of the chord axial force:

kp = 1,0– 0,3(np + np 2) = 1,0– 0,3(0,251+ 0,2512) = 0,906

Chord face yieldThe bending resistance calculated by the chord face yield is (Table 9.3.10):

Bending resistance of the jointSince d1 = 219,1 mm > d0 – 2t0 = 219,1– 2 · 5 = 209,1 mm, the punching shear resistanceof the chord need not be taken into account in calculating the bending resistance. Thebending resistance of the joint is thus Mip.1.Rd= 36,4 kNm.

M f t dk

ip Rd yp

Mj M. .

,

,

,sin

,

, , , ,,

sin ,,

1 02 0 5

10

2 0 5

4 851 1

4 85 355 5 21 91 1 0 219 10 906

90

1

1 136 4

= ⋅ ⋅ ⋅ ⋅( ) ⋅

= ⋅ ⋅ ⋅ ⋅ ⋅ ⋅( )

=

γ βθ γ γ

kNm

N np Sd p. ,,

,= ⇒ =⋅

=272 71 1

1

272700

355 33630 251 kN (puristusta) (compression)

N

N

M

M

M

MSd

Rd

ip Sd

ip Rd

op Sd

op Rd

1

1

1

1

21

11 2.

.

. .

. .

. .

. .+

+ ≤ (3.1 )


80

��

��

�

�

��

�t0

N1.Sd

d 0

Np.Sd

t1d1

Mip.1.Sd

Chord face yieldThe compression resistance for the brace member is calculated according to theguidance presented in section 3.1.1. By the chord face yield, we obtain (Table 9.3.4):

Chord punching shearThe chord punching shear need not be calculated, since d1 = 219,1 mm > d0 – 2t0 =209,1 mm.

Compression resistance of the jointThe compression resistance of the joint calculated by the chord face is N1.Rd = 230,4 kN.

Now, the calculated resistance values are substituted in the interaction expression(3.12):

N

N

M

MSd

Rd

ip Sd

ip Rd

1

1

1

1

2 270

230 4

30

36 40 983.

.

. .

. . , ,,+

= +

= < 1,0 OK!

N kRd pMj M

10

22 2

0

22 0 2

2 8 14 21 1

355 5

902 8 14 2 1 0 21 91 0 906

1

1 1230 4

.

,

sin, ,

,

sin, , , , ,

,,

=⋅

+( ) ⋅⋅

= ⋅ + ⋅( ) ⋅ ⋅ =

f t

kN

y

θβ γ

γ γ


81

3.3 Welded end-to-end joints of hollow sections

The resistance of hollow section end-to-end joints should be designed to be at least equal tothe hollow section plastic resistance in order to utilize the entire plastic resistance of the crosssection. The resistance of the welded joint is at least equal to the hollow section resistance, ifthe weld is a full penetration weld and the conect weld material for the parent metal is selected.

The weldability of structural steels is good with all welding procedures. Preheating is necessaryonly if the external temperature is below 5 °C or if the hollow sections are damp. Instructionsfor selecting the form of the backing and grooves are given in Tables 3.3 and 3.4.

Table 3.3 Welded end-to-end joints with equal wall thicknesses [4]

Groove type Wall α b c Backingthickness thickness

Without backing platet ≤ 3 mm 0° t - -

3 ≤ t ≤ 20 mm 60° 0 ≤ b ≤ 3 mm - -

t ≤ 20 mm 60° 0 ≤ b ≤ 4 mm 1,5 ≤ c ≤ 4 mm -

With backing platet0 = 3 mm 0° 3 ≤ b ≤ 5 mm - t1 =3 mm

t0 = 5 mm 0° 5 ≤ b ≤ 6 mm - 3 < t1 ≤ 5 mm

t0 = 6 mm 0° 6 ≤ b ≤ 8 mm - 3 < t1 ≤ 6 mm

t0 < 20 mm > 60° 5 ≤ b ≤ 8 mm 1 ≤ c ≤ 2,5 mm 3 < t1 ≤ 6 mm

t

b

α

α

α

tt

b

b

b

b

t 1t 1

t 0t 0

cc


82

Table 3.4 Welded end-to-end joints with different wall thicknesses [4]

Groove type Difference in thicknesses α βt2 - t1 ≤ 0,5 t2 ≤ 3 mm 60°- 80° -

Unlimited 60°- 80° ≤ 30°

t2 - t1 ≤ 1,5 mm 60°- 80° -

1,5 < t2 - t1 < 3 mm 60°- 80° -

t2 - t1 ≥ 3 mm 60°- 80° ≤ 14,036°

Root gap b is selected as in Table 3.3


83

α

b

t 1 t 2

α

α

α

α

t 1t 1

t 1t 1

t 2t 2

t 2t 2

b

b

b

b β

β

3.4 Bolted hollow section joints

3.4.1 End-to-end bolted hollow section joints

In structures utilizig hollow section structures, it is normal practice to assemble structureelements in the workshop by welding and then connect the elements on site with bolted joints.Bolted joints are quicker and easier to prepare on site than welded joints. Alternatives for end-to-end bolted joints are shown in Figure 3.4.

Figure 3.4 Alternatives for end-to-end connections

When designing a joint, it is essential to ensure that the load is as concentric as possible inrelation to the cross section and that the rigidity of the joint components is uniform. In thisrespect, a tension joint is best constructed using the alternatives b, c, f or g. In these jointtypes, the tension load is transmitted more directly to the hollow section than in joints a, d or e,which also include the risk of lamellar tearing. In the flange joints d and e, a sufficient flangethickness must be selected to minimize the bolt prying forces due to flange elasticity.

a) b) c)

d) e) f)

g)


84

3.4.1.1 Flange plate connections

Flange plate connections for square and rectangular hollow sections

A flange plate connection can be designedassuming it a bi-dimensional joint of Telements, in which the bolts are placed atopposite sides of the hollow section (Figure3.5). Due to the tension load, a plastic hinge isformed in the flanges at the bolt rows and thehollow section walls. The tension resistance ofthe flange can be determined using the plasticflange moment. When determining the loadingon bolts, the prying force due to flangebending must be accounted for.

When calculating the flange plate jointresistance, the first thing to determine is factorδ for the relative net area of the bolt row [2]:

where

Parameter αh accounts for the effect of holes on the flange plastic moment values at the boltlocation, when the tensile force of the bolts is assumed equal to their tension resistance [2]:

where

Kb

f p

b b d tbadtB

f

red M

y

red

t Rd

y

= ⋅⋅

− +

4

0 9

0 5

0

0

0

γ,

,

.

(the bolt row lever arm by the plastic hinge)

is the distance of the bolts from the hollow section edge

is the distance of bolts from the flange edge

is the diameter of the bolt

is the thickness of the hollow section wall

is the tensile resistance of bolts or the punching resistance of the flange

[Eurocode 3:6.5.5.4] [1] (select the smaller value)

is the yield strength of the flange

αδh

t Rd

p

K B

t

a d

a b t= ⋅ −

++ +( )

. ,2

01

0 54(3.1 )

is the diameter of bolt holes

is the spacing between the bolt centresdp

0

δ = −

1 30d

p(3.1 )


85

��

��NSdNSd

Figure 3.5 Flange plate connection

ab

a red

b red

t0

d0

p p p/2p/2

For the flange thickness, the following minimum and maximum values are obtained [2]:

where

The resistance of the joint can be determined by expressing the work done in the plastic hingesequal to the work done by the external load [2]:

where

n is the number of bolts

Due to the prying effect, the axial force introduced to the bolt is greater than Nt.Sd. This axialforce in the bolt is expressed with the symbol Np.Sd [2]:

where

The axial force Np.Sd must be smaller than the tension resistance of the bolt.

a a d b d

K N

t

red

pt Sd

p

= + ≤ +

= ⋅ −

0 5 1 25 0 5

11

2

, , ,

.

αδ

N Nb

ap Sd t Sd

red

red

p

p. .= +

⋅+ ⋅

1

17

δ αδ α

(3.1 )

Nt n

KRd

p h=+ ⋅( )2 1

6δ α

(3.1 )

is the axial force in one bolt

is the thickness of the flange

Nt

t Sd

p

.

K Nt K Nt Sd

p t Sd⋅+

≤ ≤ ⋅..

15

δ (3.1 )


86

Example 24Calculate the tension resistance of the adjacentflange plate joint which is subjected to axial forceNSd = 1086 kN. The hollow section dimensions are120 x 120 x 8, and the steel grade used is S355J2H.The steel grade used in the flange is S355J2.

The strength grade of the M 27 bolts is 10.9. Theflange hole position parameters are:

d0 = 30 mmp = 90 mm ⇒ δ = 1– (d0/ p) = 1– (30/ 90)= 0,67b = 45 mmd = 27 mm t0 = 8 mm ⇒ bred = b– 0,5d + t0 = 45– 0,5 · 27+ 8 = 39,5 mmK = 4bred ·γM0 /(0,9fy · p) = 4· 39,5· 1,1/ (0,9 · 345 · 90) = 6,22 mm2/ kN(tp > 16 mm ⇒ fy = 345 N/mm2)a = 45 mm ⇒ ared = a+ 0,5d = 45+ 0,5 · 27 = 58,5 mm

The bolt force in one bolt is:Nt.Sd = NSd / 6 = 1086/ 6 = 181 kN

The tension resistance of the bolt is [1]:Bt.Rd = 0,9fub · As /γMb = 0,9 · 1000 · 459/ 1,25 = 330 kN

where

As is the tension cross section of the bolt fub is the ultimate strength of the bolt γMb is the partial safety factor for bolt joints (chapter 2)

Flange resistanceThe minimum and maximum values for the flange thickness are obtained by the formula(3.15):

Select a flange thickness of tp = 28 mm.

K Nt K N

t

t

t Sdp t Sd

p

p

⋅+

≤ ≤ ⋅

⋅+

≤ ≤ ⋅

≤ ≤

..

,

,,

1

6 22 181

1 0 676 22 181

26

δ

mm 33,6 mm


87

��

��d 0

a b bred ared

p/2

p/2

pp

t0

NSd

NSd

The joint resistance is determined using the formula (3.16):

Resistance of boltsThe tensile resistance of the bolts is checked, with the prying effect taken into account bythe formula (3.17):

Resistance of weldsThe resistance of a fillet weld is calculated as shown in Eurocode 3, subsection 6.6.5.3

where

In the example, the hollow section is welded to the flange from all edges. In such cases,the required throat thickness is:

The joint resistance is sufficient for an axial force of 1086 kN, which is also the plastictension resistance of the 120 x 120 x 8 hollow section.

aN

f Lw Mw Sd

u w= ⋅ ⋅ ⋅

⋅= ⋅ ⋅ ⋅

⋅=3 3 0 9 1 25 1086

490 4809 0

β γ , ,, mm

faL

u

w

w

Mw

βγ

is the ultimate strength of the weaker joint componentis the thickness of the weld throatis the length of the weldis the strength factor (S355 ⇒ βw = 0,9) [1] is the partial safety factor of the welded joints (chapter 2)

Ff a L

w Rdu w

w Mw. = ⋅ ⋅

⋅ ⋅38

β γ(3.1 )

αδ

δ αδ α

pt Sd

p

p Sd t Sdred

red

p

p

K N

t

N Nb

a

= ⋅ −

= ⋅ −

=

= +

⋅+ ⋅

= +

⋅+

.

. .

,

,,

,

,

, ,

2 211 6 22 181

281

1

0 670 651

11

181 139 5

58 5

0 67 0 651

1 00 67 0 651

218 1

, ,

,

⋅

= kN < 330 kN OK!

Nt n

K

K B

t

a d

a b t

Rdp h

ht Rd

p

=+ ⋅( )

= + ⋅( ) =

= ⋅ −

++ +

= ⋅ −

+ ⋅+ +

2 2

20

2

1 28 1 0 67 1 442 6

6 221487

10 5 6 22 330

281

45 0 5 27

0 67 45 45 8

δ α

αδ

, ,

,

,

( )

, ,

, ( ).

kN > N OK!Sd

= 1 442,


88

Flange plate joint in circular hollow sections

Figure 3.6 Shape coefficient of the flange f3 [3]

The required flange thickness for a circular hollow section is calculated from the followingformula [3]:

where

For the number of bolts, we obtain the following equation [3]:

where

r d er d e1 0 1

2 0 1

0 5 20 5

= += +

,,

n

Nf

fr

r

B

Sd

t Rd≥

− +⋅

⋅

11 1

0 6720

33

1

2ln

, .(3. )

fNf

Sd

y

3 is the shape coefficient of the flange (Figure 3.5)is the design value for the tensile force of the jointis the yield strength of the flange

tN

f fp

Sd M

y≥ ⋅ ⋅

⋅ ⋅2

90

3

γπ

(3.1 )


89

10

8

6

4

2

00 0,2 0,4 0,6 0,8 1,0

d t

d t e0 0

0 0 12

−− +

f3

�

��

t0

t p

d0e1 e2

3.4.1.2 In-line tension joint with splice plates

An in-line tension joint with splice plates issuitable as a splice joint for the lower chord ina lattice structure. Since the load is parallel tothe plates, there is no risk of lamellar tearingof the splices (Figure 3.7).

The resistance of the joint is determinedseparately for bolts and splices. The boltstransfer the force affecting the joint by theirshear resistance. The bolt’s shear resistanceper shear plane is determined from thefollowing formula, assuming the shear planedoes not pass through the threaded portionof the bolt [1]:

where

The tension resistance of the splice plates is calculated by taking into account both the netcross section and the bearing resistance. The resistance of the net cross section can becalculated by the same principle as that of a hollow section in tension (chapter 2). The bearingresistance of a splice plate depends on the positioning of the holes and the strength of thebolts. This relationship is illustrated by the parameter α, obtained as the minimum value fromthe following equation [1]:

however, α ≤ 1,0

where

edpffub

u

1

0

1

is the distance of the bolt from the edge parallel to force

is the diameter of the bolt hole

is the distance between bolts parallel to force

is the ultimate strength of the bolt

is the ultimate strength of the splice

α = −

min ( . )

e

d

p

d

f

fub

u

1

0

1

0

3

3

1

43 22

fA

ub

Mbγ

is the ultimate strength of the boltis the cross section of the boltis the partial safety factor of the bolt joints (chapter 2)

Ff A

v Rdub

Mb.

,= ⋅0 621

γ(3. )


90

NSd

NSd

NSd

NSd

Figure 3.7 In-line joint with splice plates

a1

a 2

Lv

a1

Lv

t

The bearing resistance of a double shear joint per one bolt is determined from the equation [1]:

where

Addihonally, the resistance for block shear failure of the splice in the middle must be checked.The design value for block shear failure is determined from the formula [1]:

In this case, the effective shear area Av.eff can be calculated from the following formula [1]:

where

ff

kL a a a

u

y

v

= 2 5

1 2

,, nd

is the ultimate strength of the hollow section

is the yield strength of the hollow section

are defined in Figure 3.7

A t L t L a a k df

f

L L d a k df

f

v eff v eff vu

y

v eff vu

y

. .

.

= ⋅ = +( ) + − ⋅( )

≤ +( ) + − ⋅( )

2 25

2 5

1 2 0

2 0

(3. )

however

Vf A

eff Rdy v eff

M.

.=⋅3

240γ

(3. )

dt

Mbγ

is the diameter of the boltis the thickness of the spliceis the partial safety factor of the bolt joints

Ff d t

b Rdu

Mb.

,= ⋅ ⋅ ⋅ ⋅2 523

αγ

(3. )


91

Example 25Calculate the tension resistance of theadjacent joint. The dimensions of the hollowsection are 150 x 150 x 6,3, and the steeldesignation used is S355J2H.

The steel grade used in splices is S355J2.The strength grade of the M24 bolts is 8.8.The parameters of the joint geometry are:

t1 = 20 mmt2 = 10 mmd = 24 mmd0 = 26 mma1 = e1= e2 = 40 mmLv = a2 = p1 = p2 = 80 mmhp = 160 mm

Resistance of the splice plate net cross-sectionThe splice plates can be taken as tension cross sections. Thus, the resistance of a crosssection containing holes can be obtained from formulae (2.33) and (2.34) [1]:

Thus, the tension resistance of the net cross section is Ft.Rd = 762,0 kN.

Bearing resistance of splice platesWhen the holes are situated as in the example, the bearing resistance of splice plates isas follows [1]:

Now there are 4 bolts per plate, so the bearing resistance is:

Fb.Rd = 4 · 241,3 = 965 kN

α

αγ

= −

= [ ] = ≤

= ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ =

min ; ;

min , ; , ,

, , ,

,,.

e

d

p

d

f

f

Ff d t

ub

u

b Rdu

Mb

1

0

1

03 3

1

4

0 513 0 513 1 0

2 5 2 5 0 513 490 24 20

1 25241 3

0,776; 1,632

kN

A mm

A mm

FA f

FA f

net

v

t Rdv y

M

t Rdnet u

M

= − ⋅( ) =

= ⋅ =

=⋅

= ⋅ = > ⇒ =

= ⋅ = ⋅ ⋅ =

20 160 2 26 2160

20 160 3200

3200 345

1 11003 6 16 345

0 9 0 9 2160 490

1 25762 0

2

2

1

2

kN t mm f N

mm

kN

p y 2.

.

,,

, ,

,,

γ

γ


92

e1

e 2

p1

NSd

NSd

NSd

NSd

d0

t 1t 2h pp 2a 2

Lw

Lv a1

Block shear failure resistance of splice platesFirst, calculate the effective shear area from the formula (3.25):

Obtain the block shear failure resistance by substituting in formula (3.24):

Shear resistance of boltsThe shear resistance of bolts is determined by assuming that the shear plane does notpass through the threaded portion of the bolt [1]:

There are four bolts and the joint has two shear planes, so the bolt resistance is:

Fv.Rd = 8 · 173,6 = 1389 kN

Resistance of weldsDesign the fillet welds with a throat thickness of 5 mm applying formula (3.18). The axialforce introduced into the weld is assumed equal to the hollow section plastic tensionresistance. The plastic tension resistance of a hollow section with dimensions 150 x 150x 6,3 is [1]:

Npl.Rd = NSd = 3485 · 355/ 1,1 = 1125 kN

The following value for the required weld length is obtained:

Resistance of jointsThe entire resistance of the joint is then determined by the resistance of the net crosssection:

Ft.Rd = 762 kN

LN

f aw

w Mw Sd

u≥ ⋅ ⋅ ⋅

⋅= ⋅ ⋅ ⋅

⋅ ⋅=3

4

3 0 9 1 25 1125

4 490 5224

β γ , , mm

Ff A

v Rdub

Mb.

, ,

,,= ⋅ ⋅ = ⋅ ⋅ =0 6 0 6 800 452

1 25173 6

γ kN

Vf A

eff Rdy v eff

M.

.

,,=

⋅⋅

= ⋅⋅

=3

2345 5226

3 1 1946 3

0γ kN

A t L t L a a k df

f

L L d a k df

f

v eff v eff vu

y

v eff vu

. .

.

,

,

= ⋅ = +( ) + − ⋅( )

= +( ) + − ⋅( )

=

= +( ) + − ⋅( )

1 1 1 2 0

2 0

2

20 2 80 40 80 2 5 26490

3455226

261 3 5

mm

mm < 2

2

yy

= 421 3, mm


93

3.4.2 Bolted beam-to-column joints

A hollow section or an I profile can be joined to a hollow section column by several differentmethods, as shown in Figures 3.8-3.16. Introducing rigidity to the joint requires the use of endplates, which means that the tolerance on length must be more rigorous. In structures withmultiple bays, the variation of length may accumulate, and the length deviation must be evenedout with intermediate plates. More flexible joints in which bolts transmit the shear forces allowfor greater adjustment. However, even in flexible joints, it is important to take into account themoments due to the eccentricity of shear force in the design of the column.

Figure 3.8 A beam-to-column connection between I sec-tion and hollow section subjected to shear, bending andnormal loads. The joint resistance is usually limited by thebuckling or plastification of the column web.

Figure 3.9 A typical beam-to-column connection betweenI section and hollow section. The end of the beam must bestiffened with a plate, which then transfers the shear force,through contact in bearing, to the stiff portion of thesupport component. The joint is suitable mainly for beamswith minor shear force.

Figure 3.10 Semi-rigidity of the joint is obtained by usingvery stiff end plates. However, the most practical way is toassume the joint pinned and take the bending moment ofthe extension into account in the column design. Due to itssimplicity, this type of joint is frequently used in latticestructures. Complex joint details are made by welding, andsimple straight members are connected to the outstands,starting from the corner point, by flange plate joints. Therigidity between the column and the outstand can beestimated as shown in Appendix 9.4.

Figure 3.11 The joints shown in Figures 11a and 11bbehave in a similar way. When the bolted joint is made asan ordinary joint, carrying the load by bolt shear, theclearances between the bolts and the holes makes thejoint indeterminate with regard to transmission of bendingmoment. With a friction grip bolted joint, the moment canbe transferred from the beam to the support plate,preserving the rigidity.

If the joint is subjected to shear force only, the supportplate can be connected directly to the column flange. In ajoint carrying the bending moment or axial force, thecolumn flange usually must be stiffened with a reinforcingplate.


94

� ��

��Figure 3.8��

Figure 3.9

��

Figure 3.11a

Figure 3.10

��

Figure 3.11b

Figure 3.12 In the adjacent figure, a support plate withthreaded holes is welded to the column flange. Since thebolts are short and the threaded part is subjected to shearload, the plastic deformation capacity of the joint is low.The best way to design the bolted joint is to ensure that it'sresistance is equal to that of the incoming hollow section.When the flange is made from ordinary structural steel, itis advisable to select such plate thicknesses and boltdimensions that the thread of the bolt reaches up thethreaded hole at least by the length of the bolt’s nominaldiameter, but the end of the bolt does not touch thecolumn wall. The column wall thickness should not beincluded in the effective thread length, if the holes aredrilled and threaded after the plate is welded.

Figure 3.13 The joint shown in figure 3.13 can be made bythreading the holes that are made in the column wall bythermodrilling (Figure 3.13 a). In thermodrilling, the hollowsection wall is thickened next to the hole, so a sufficientlength for the thread is obtained. An alternative is to useexpansion bolts with the bolt holes drilled as usual (Figure3.13 b). The expansion bolts press against the tube wallwhen the bolt is tightened.

Figure 3.14 Figures a and b show examples of connectinga bracing to the column.

Figure 3.15 This figure illustrates an end-to-end joint of achord of a lattice structure. Diagonals should not bewelded to the end flanges, but to the chord. The actualgap (ga) of the joint is the distance between diagonal weldand the chord flange weld. The gap must meet therequirements given in the tables in Appendix 9.3.

Figure 3.16 Figures a and b show beam-to-column jointsin which the beam is continuous. For the properfunctioning of this kind of joint, it is essential that the loadsduring erection and use of the structure are close tosymmetrical. When bending moments and shear forcesare unequal, the column must be sufficiently strong toresist bending. The column flange plate joint is taken as ahinge in relation to buckling, unless the rigidity of the jointis increased with specific methods.


95

��Figure 3.12

�

��

��

��

Figure 3.13

Figure 3.14

��

Figure 3.15

��Figure 3.16b ��

Figure 3.16a

Detail 1

Detail 1ga

a) b)

a) b)

Hollow section flange plate joint subjected to bending moment

The flange plate joint is capable of transferring boththe bending moment and the shear force. Whencalculating the bending resistance, the joint is dividedinto components. - bending resistance of the column webshear resistance of the column web- tension resistance of flanges and bolts

The lowest of the component resistances determinesthe bending resistance of the entire joint.

Bending resistance of the column web

The column web bending resistance can be estimated using the formulae for welded latticejoints (Table 9.3.9). When the column and the beam are of equal width, the following bendingresistance is obtained [1]:

where

Shear resistance of the column web The shear resistance of the column must also bechecked, since the moment load is transferred frombeam flanges to the column as a shear force. It isassumed that the column has no external shear load,and the column shear force then consists of the jointload only. The bending resistance of the joint whengoverned by the shear is obtained by multiplying theshear resistance of the column web by the columnheight [1]:

where

Vf A

ht

pl Rdy v

M. =

⋅⋅3 0

1

1

γis height of the column

is the thickness of the beam flange

(design value of the column shear resistance, section 2.4)

M V h tRd pl Rd= −( ). 1 1 27(3. )

is the yield strength of the columnfy

M f t h tip Rd yMj M

. . ,,

1 0 1 02

00 5 5

1 126= ⋅ +( )

⋅γ γ(3. )


96

��

��

MSd

MRd

h1

t1

Vpl.Rd

Vpl.Rd

t0

Tension resistance of flanges and bolts

The resistance of flanges and bolts can be estimated by calculating the resistance of the jointbetween the flange and hollow section using equivalent T models. Equivalent T modelsconsist of a column and a flange, and a hollow section and a flange. There are three potentialfailure modes for a T model. According to these modes, the tension resistance values of thebolt row are as follows. (Figure 3.17) [1]:

Figure 3.17 Failure modes for an equivalent T model. Ft.Rd is the force of the bolt row in the tension area of the joint.

a) flange yield at the location of the bolt row and at the hollow section webs

b) bolt failure as the flanges yield at the hollow section webs

c) bolt or flange failure

where

F

ML f t

L

t

meB

t Rd

pl Rdeff y p

M

eff

p

t Rd

.

.

.

,=

⋅ ⋅0 25 2

0γ

is the tension resistance of the bolt row

is the effective length of the bolt row

is the thickness of the flange

is the bolt’s distance from the outer edge of the hollow section

is the bolt’s distance from the edge of the flange, e ≤ 1,25 m

is the tension resistance of the bolt or the punching resistance of the flange

[Eurocode 3: 6.5.5 (4)] [1] (select the smaller value)

F Bt Rd t Rd. .= 2 30(3. )

FM e B

m et Rd

pl Rd t Rd.

. .=+ ⋅ ⋅+

2 229(3. )

FM

mt Rd

pl Rd.

.=4

28(3. )


97

��

��

��

��a) b) c)

Ft.Rd Ft.Rd Ft.Rd

Ft.RdFt.RdFt.Rd

e m em

The effective length of the bolt row depends on the shape of the flange’s yield line pattern.From the following equations, select the one giving the smallest result [1]:

where

p is the vertical distance between the horizontal bolt rows

The bending resistance of the joint is obtained by multiplying the tension resistance values ofthe horizontal bolt rows by the distance of the bolt rows from the centre of compression. Onlythe bolt rows in the tension zone are considered. The tension zone of the joint is located abovethe neutral axis of the hollow section. The following value for bending resistance is thereforeobtained [1]:

where

Fh

t Rd

r

. is the design value for the bolt row’s tension resistanceis the distance of the bolt row distance from the compression centre

M F hRd t Rd ii

r i= ( ) ( )∑ . (3. )34

L m

L m e

L p m e a

L p b

eff

eff

eff

eff

= ⋅

= +

= + +

=

2 31

4 1 25 32

0 5 2 0 625 33

33

π (3. )

(3. )

(3. )

(3. )

,

, ,

(all bolts)

(all bolts)(the uppermost and lowermost bolt row, if there areseveral rows)

(other bolt rows, if there are several rows)


98

Example 26Calculate the bending resistance of the flangeplate joint. The column dimensions are 200 x 200x 6,3 and those of the hollow section are 300 x200 x 8. The steel grade used is S355J2H. Theflange thickness is 20 mm. The steel grade usedin the flanges is S355J2. The strength grade ofthe M22 bolts is 8.8.The bending resistance values of the jointcomponents are:300 x 200 x 8: M1.Rd = 244 kNm200 x 200 x 6,3: M0.Rd = 110 kNmUsually, the joint also includes shear force,which must be taken into account in the jointdesign.

Bending resistance of the column webThe resistance of the column web is determined from the formula (3.26):

Shear resistance of the column webShear resistance of the column web is determined as shown in chapter 2:

By multiplying the shear resistance by the height of the hollow section, the momentresistance for the column web is obtained [formula(3.27)]:

MRd = Vpl.Rd(h1– t1) = 442(0,3 – 0,008) = 129,1 kNm

Resistance of flanges and boltsFirst, calculate the effective length of the bolt row [formulae (3.31)- (3.33)]:

Since the latter formula gave the smallest value, it is used as the effective length of thebolt row Leff = 262,5 mm.

Then, substitute the effective length of the bolt row in the failure mode equations for theT stub [formulae (3.28) - (3.30)]:

ML f t

pl Rdeff y p

M.

, , ,

,,=

⋅ ⋅= ⋅ ⋅ ⋅ =

0 25 0 25 262 5 345 20

1 18 233

2

0

2

γ kNm

L m

L m e

eff

eff

= ⋅ = ⋅ =

= + = ⋅ + ⋅ =

2 2 50 314

4 1 25 4 50 1 25 50 262 5

π π mm

mm, , ,

Vf A

pl Rdy v

M,

,=

⋅⋅

= ⋅⋅

=3

355 2372

3 1 1442

0γ kN


. . ,,

, , ,,

,1 0 1 02

0

20 5 51 1

0 5 355 6 3 300 5 6 31

1 1111 7= ⋅ +( )

⋅= ⋅ ⋅ + ⋅( ) =

γ γ kNm


99

��

��

MSd

200 x 200 x 6,3

300 x 200 x 8

300

5050

200

50

h r

a) flange yield at the location of the bolt row and at the hollow section web

b) bolt failure as the flanges yield at the hollow section webs

c) bolt or flange failure

We see that the failure mode to be used in the design is therefore bolt failure with flangeyielding Ft.Rd = 339,2 kN. It is normally recommended to design the flange joint so thatthe flanges yield before the bolts fail. The failure mode is then ductile. For flanges andbolts, the joint bending resistance is [formula (3.34)]:

MRd = Ft.Rd · hr = 339,2 · (0,25– 0,004) = 83,44 kNm

Design of weldsThe welds must transfer the tensile force due to bending moment into the hollowsection’s upper flange

NSd = MRd / 0,3 = 278,1 kN. The fillet weld is provided across the width of the entirecolumn (200 mm), which gives a required throat thickness of:

Bending resistance of the jointThe smallest bending resistance value is that involving failure of the bolts and flangeyielding, so it is selected as the bending resistance of the entire joint:

MRd = 83,44 kNm

aN

f Lw Mw Sd

u w≥ ⋅ ⋅ ⋅

⋅= ⋅ ⋅ ⋅

⋅=3 3 0 9 1 25 278 1

490 2005 5

β γ , , ,, mm

F Bt Rd t Rd. . ,= ⋅ = ⋅ =2 2 174 5 349 kN

FM e B

m et Rd

pl Rd t Rd.

. . , ,,=

+ ⋅ ⋅+

= ⋅ + ⋅ ⋅+

=2 2 2 8 233 50 2 174 5

50 50339 2 kN

FM

mt Rd

pl Rd.

. ,,= = ⋅ =

4 4 8 233

50658 6 kN


100

Example 27Calculate the resistance of the adjacent shearjoint.The dimensions of the hollow sections are 200 x200 x 6,3, and the steel grade used S355J2H. Atthe end of the hollow section, the joint issubjected to a force FSd = 150 kN. The thicknessof the splice plate is 15 mm, and the strengthgrade of the M20 bolts is 8.8.

The vertical load introduces a bending momentinto the column:MSd = 150 · 0,04 = 6 kNmThe axial force of the column is:NSd = 300 kN (compression)

Bearing resistance of the splice platesThe bearing resistance of the splices is calculated as shown in example 25 [1]:

2 bolts ⇒ Fb.Rd = 2 · 133,7= 267,4 kN > Fsd OK !

Block shear failure resistance of splice platesFirst, calculate the effective cross-section [1]:

The resistance to block shear failure is obtained by substituting in the formula (3.24):

Vf A

eff Rdy v eff

M.

.

,,=

⋅⋅

= ⋅⋅

=3

355 3000

3 1 1559 0

0γ kN > F OK!Sd

A t L t L a a k df

f

L L a a n df

f

v eff v eff vu

y

v eff vu

y

. .

.

,

,

= ⋅ = + + − ⋅( )

= + + − ⋅( )

=

= ≤ + + − ⋅( )

=

1 2 0

1 3 0

15 120 40 40 0 5 22490

3553000

200 215 3

mm

mm mm

2

LL L a av eff v. = ≤ + +( ) =

= ≤

200 200

40

1 3 mm mm

a mm 5d = 100 mm1

α

αγ

= −

≤

= [ ] = ≤

= ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ =

min ; ;

min , ; ,

, , ,

,,.

e

d

p

d

f

f

Ff d t

ub

u

b Rdu

Mb

1

0

1

03 3

1

4

0 606 0 606

2 5 2 5 0 606 490 20 15

1 25178 2

1,0

1, 568; 1,632 1,0

kN


101

�

200 x 200 x 6,3

200 x 200 x 6,3FSd

40

120

NSd

40

4040

Shear resistance of boltsIn the equation for determining the shear resistance of bolts, it is assumed that the shearplane passes through the threaded portion of the bolt [1]:

Two bolts and a single-lap joint ⇒ Fv.Rd = 2 · 94,1= 188,2 kN > Fsd OK !

Resistance of the column wallThe resistance of the column wall is calculated as shown in Table 9.3.11 in Appendix9.3:

Resistance of weldsThe weld between the column and the splice must transfer the force FSd vertically and thetensile force due to moment MSd horizontally. Therefore determine the stress componentsof the weld:

where

ττσ

ll

⊥

⊥

is the shear stress parallel to the weld axisis the shear stress perpendicular to the weld axisis the axial force perpendicular to the weld design surface

τ

τ σ

ll =⋅

= = ⋅ ⋅⊥ ⊥

F

a LM

W

t

a

Sd

Sd

el

(3. )

(3. )

35

1

236

nN

A f

M

W f

k n

M kf t h

t

b

h

b

t

b

M Mj Sd

y

Sd

el y

m

Rd my

p

p

=⋅

⋅+

⋅

=

⋅+

⋅=

= −( ) = −( ) =

=⋅ ⋅

−+ −

γ γ0

102

1

0

1

0 0

1 1

1 1

1

300000

4745 355

6000

292 2 3550 260

1 3 1 1 3 1 0 260 0 962

0 51

24 1

,

,

,,

, , , ,

,.

⋅

= ⋅ ⋅ ⋅

−

⋅ + −

=

1 1

0 5 0 962355 6 3 200

115

200

2 200

2004 1

15

200

1

1 17 79

0

2

,

, ,,

,,

γ γMj M

kNm > M OK!Sd

Ff A

v Rdub s

Mb.

, ,

,,= ⋅ ⋅ = ⋅ ⋅ =0 6 0 6 800 245

1 2594 1

γ kN


102

The weld strength is checked with formulae in Eurocode 3 Appendix M [1]:

Try a throat thickness of 3 mm, which gives the following weld stresses (fillet welds onboth faces of the plate):

Checking the conditions (3.37) and (3.38):

The same throat thickness can be used in the weld between the hollow section end plateand the splice.

Resistance of the jointCompare the calculated resistance values to the force quantities:

VRd = 188 kN > VSd = 150 kN OK !MRd = 7,79 kNm > MSd = 6 kNm OK !

σ τ τβ γ

σγ

⊥ ⊥

⊥

+ +( ) = ≤⋅

=⋅

=

= ≤ = =

2 2 23 303 2490

0 9 1 25435 6

106 1490

1 25392

ll ,, ,

,

,,

N

mm

N

mm

N

mm

N

mm

2 2

2 2

f

f

u

w Mw

u

Mw

OK!

OK!

τ

τ σ

ll =⋅

=⋅ ⋅

=

= = ⋅ ⋅ = ⋅⋅

⋅ =⊥ ⊥

F

a LM

W

t

a

t

Sd

Sd

el

2

150000

2 3 200125

2

1

2

6000

100 2 3

1

2106 1

N

mm

N

mm

2

2,

σ τ τβ γ

σγ

⊥ ⊥

⊥

+ +( ) ≤⋅

≤

2 2 23 37

38

llf

f

u

w Mw

u

Mw

(3. )

(3. )


103

3.5 Hollow section-to-foundation joints

The joint between the column and the foundation is generally assumed to be either rigid orpinned. In practice, all joints are semi-rigid. However, in practical design, it is seldom necessaryto take the semi-rigidity into account.

It is usual to provide a nominally pinned joint. It is seldom necessary to design a freely rotatingjoint (Figure 3.18a) however as the base plate (Figure 3.18b) can be made sufficiently flexible.The moment at the base of the column need not be taken into account in the design when thejoint is sufficiently flexible.

Reinforcing a rigid joint with stiffeners is seldom an economical alternative. It is usually moreefficient to increase the thickness of the base plate.

Figure 3.18 Joint of a hollow section to foundation

3.5.1 Joint between a column subjected to axial force and foundation

The joint is subjected to axial force only, so the concrete surface pressure is obtained directlyby dividing the load by the base plate area:

The area is calcutated for axial force concentric with base plate. When the axial force iseccentric, the base plate area must be reduced. Only the portion of the base plate concentric tothe axial force is included in the reduced area. The base pressure must be less than theconcrete compression resistance.

pN

ASd= (3. )39

��

��

a) b)


104

The pressure under the baseplate induces a bending moment into the base plate. The baseplate can be treated as a cantilever whose length is the distance between the hollow sectionwall and the plate edge. When calculating the elastic plate resistance, we obtain the followingcondition for the base plate thickness [1]:

where

In addition, the load during erection should be checked, which is when the holding down boltstransfer the forces to the foundation.

3.5.2 Joint between a column subjected to bending moment and axial force and the foundation

In a rigid joint, the bending moment in the column must betaken into account. At the ultimate limit state, thecompression stress is limited by the concrete compressionresistance design value of fcd. The tensile force istransferred into the foundation through the holding downbolts on the tension face. The following conditions for thebalance of forces are then obtained:

where

Af

fb

s

yb

cd

eff

is the stress cross-section of the foundation bolts (on the tension face)is the dimensioning valve for the foundation boltsis the design value for the compression strength of concreteis the effective width of the bottom plate on the compression side

N N N b y f A f

M N a a d N d y

Sd c s eff cd s yb

Sd Sd c

= − = ⋅ ⋅ − ⋅

+ − −( )[ ] = −( )(3. )

(3. )

41

0 5 0 5 42, ,

Mb p a

ba

f

Sd

y

M

= ⋅ ⋅ 12

1

0

2

tp

γ

is the width of the base plate

is the length of the cantilever

is the thickness of the base plate

is the yield strength of the base plate

is the safety factor of the material

t 6M

pSd≥ ⋅

⋅= ⋅γ γM

y

M

yb fa

p

f0

103

40(3. )


105

MSd

NSd

Ns

dy

f cd

Nc

��

a1

b

a

Using this pair of equations, we can determine the length of the concrete compression area yand the required stress cross-section of the holding down bolts As:

The holding down bolts have no tensile force if the formula (3.41) yields a negative value forNs. The bending load of the base plate is created from the distribution of stresses in thecompression area.

The base plate thickness is determined similarly as in section 3.5.1, except the tension load forthe holding down bolts most also be checked. The holding down bolts may be subjected tosome tension, if the moment is high and the axial force is low. In addition, the load during erec-tion must also be checked.

Example 28Calculate the joint resistance of a hollow sectionwith dimensions 200 x 200 x 8. The steel gradeused is S355J2H, and the design value for thecompression resistance of concrete is 14 N/mm2.The base plate dimensions are a x b = 400 x 400.The steel grade used in the holding down bolts isS355. The column is subjected to the followingloads:

NSd = 1500 kNMSd = 35 kNmVSd = 100 kN

First, determine whether the holding down boltsare subjected to tension at the ultimate limitstate:

y = 178 mm (or 522 mm)Nc = beff · y · fcd = 400·178·14 = 996,8 kNNs = Nc – NSd = 996,8 kN – 1500 kN = –503,2 kN ⇒ holding down bolts are not subjected to tension

yb f d b f d b f M N d a

b f

y

eff cd eff cd eff cd Sd Sd

eff cd=

⋅ ⋅ ± − ⋅ ⋅( ) − ⋅ + −( )[ ]⋅

=⋅ ⋅ ± − ⋅ ⋅( ) − ⋅ ⋅ ⋅ + ⋅ − ⋅( )[ ]

⋅

2

2 6 3

2 0 5

400 14 350 400 14 350 2 400 14 35 10 1500 10 350 0 5 400

400 14

,

,

yb f d b f d b f M N d a

b f

eff cd eff cd eff cd Sd Sd

eff cd=

⋅ ⋅ ± − ⋅ ⋅( ) − ⋅ + −( )[ ]⋅

22 0 5

43,

(3. )


106

��

MSd

NSd VSd

a1

p 1p 2b

a

Resistance of the holding down boltsSince the holding down bolts are not subjected to tension, the holding down bolts needbe designed for shear only:

4 bolts Ø 24 ⇒ AS = 1412 mm2 > 552 mm2 OK !

Base plate resistanceThe value of the bending moment in the base plate at the column edge is as follows:

The thickness of the base plate is obtained by substituting the bending moment MSd intothe formula (3.40):

⇒ select tp = 35 mmtM

b fp

Sd M

y≥ ⋅

⋅= ⋅ ⋅

⋅=6 6 28 1 1

400 34536 60γ ,

, mm

Mb a f

kNmSdeff cd=

⋅ ⋅= ⋅ ⋅ =1

2 2

2

400 100 14

228

AV

fmms

Sd M

y> ⋅ = ⋅ =3 100 3 1 1

3455520 2γ ,


107

3.6 References

[1] ENV 1993-1-1:Eurocode 3: Teräsrakenteiden suunnittelu, Osa 1-1: Yleiset säännöt jarakennuksia koskevat säännöt, 1993 (Sisältää myös liitteen K: ENV 1993-1-1:1992/A1:1994)(ENV 1993-1-1: Eurocode 3: Design of steel structures. Part 1.1: General rules and rulesfor buildings, 1993)(Include also annex K: ENV 1993-1-1:1992/ A1:1994)

[2] CIDECT: Design guide for rectangular hollow section joints under predominantly staticloading, Verlag TÜV Rheinland GmbH, Köln 1992

[3] CIDECT: Design guide for circular hollow section joints under predominantly static loading,Verlag TÜV Rheinland GmbH, Köln 1991



108

4. FATIGUE AND BRITTLE FRACTURE IN HOLLOW SECTION STRUCTURES

4.1 Fatigue loading

Fatigue refers to failure of an element due torepeated loading. Normally, failure resultsfrom stress concentrations created atstructural discontinuities. Local stressconcentrations cause the propagation ofminor initial defects in the weld and at theweld toe. Fatigue loading can cause failure atlower stress values than static loading.

Fatigue loading varies in magnitude, directionand position. This type of loading occurs, forinstance, in crane gantries, bridges andmachinery foundations. Fatigue loading generates cracks in theelement and propagates existing initialdefects. In welded structures, the mostsusceptible position for crack propagation isnormally the line between weld and parentmaterial. Crack propagation is initiated in theweld or in its proximity, since the weld alwayscontains minor defects. Careful welding isthus especially important for structuressubjected to fatigue loading. Welded jointshave a decisive role in the fatigue design ofthe entire member, since the fatigue strengthof a structural hollow section is rarely lowerthan that of the joint.

The fatigue strength of a welded detail depends on the following factors:

- stress range (load amplitude)- stress frequency (number of stress cycles)- shaping of structural discontinuity locations- weld geometry- size of the initial crack- residual stress state- toughness of the material

In dynamically loaded members, the effect of vibration on stress values must be accounted for.The increase of stress is significant if the natural frequency of the members is close to thevibration frequency of the load. In practice, members are usually designed in such a mannerthat the lowest natural frequency is higher than the frequency of the dynamic load. In this way,the stress concentrations due to resonance can be prevented. The frequency of the dynamicload can also be higher than the natural frequency, if the resonance frequency is passedthrough rapidly (e.g. in machinery foundations).


109

��Figure 4.1 Structural discontinuities

Figure 4.2 Stress concentration at the weldundercut

4.2 Stress calculation methods in fatigue design

Stress values can be calculated using four methods that represent different levels:- nominal stress method- hot spot stress method- notch stress method- fracture mechanics

In the nominal stress method, structuralstress values are calculated, usuallyaccording to the elasticity theory, withoutaccounting for the effect of structuraldiscontinuities. This method is simple andwell-suited for manual calculations. Thefatigue design values presented in Eurocode3 are based mainly on the nominal stressmethod.

Hot spot stress is the stress present at thecritical point of a structural discontinuity(Figure 4.4). It is at this location that thefatigue crack growth is assumed to start.Structural discontinuities, which appear forinstance where the cross-section changes orthe attachment ends, are taken into accountin the hot spot method. However, this methoddoes not account for the effect of weldgeometry. Due to non-uniform stressdistribution, hot spot stress values are usuallyhigher than nominal stress values. The stressvalues(σa, σb) are calculated for at least twopoints at the proximity of the weld, and thesevalues are used to extrapolate the hot spotstress in the edge of the weld (σhot spot). Thestructure must then be modeled using asuitable calculation program to determine thestresses. The stress calculation points can beselected at a distance of 0,4t and 1,0t (seeFigure 4.4) from the edge of the weld [3].Linear extrapolation is used if twoextrapolation points are selected. With moreextrapolation points, parabolic extrapolation isused. Hot spot stress values can also bemeasured from the prototype or calculatedusing the concentration factors (Ks) takenfrom reference manuals.


110

+

+ +

=

=

��

��

Figure 4.3 Stress calculation methods

a) Nominal stress method

b) Hot spot stress method

c) Notch stress method

Figure 4.4 Determining hot spot stressvalues

σm

σm

σm

σb σhot spot

σb σnlp σln

σm is the nominal stressσb is the bending stress depending on the joint

geometryσhot spot is the hot spot stressσnlp is the non-linear stress concentration due to

the notchσln is the notch stress

σaσb

σhot spot

t

0,6t

0,4t (at least 4 mm)

Notch stress refers to the actual stress at the bottom of the notch. The notch is usuallygenerated next to the weld or at some other structural discontinuity. The area affected by thenotch stress concentration is approximately 0,3 times the plate thickness [3]. To determine thenotch stress, the structural calculations must be performed using the FE method whichaccurately takes into account the geometry of the structure, including the actual weld geometryand corner radii. The design must not include non-rounded corners, since FE calculationgenerates an infinite stress in them as the element size is decreased using shell elements (notapplicable to solid-elements).

The size and shape of the initial crack is also taken into account in the model based onfracture mechanics. With this model, the rate of crack propagation can be calculated usingthe geometry and the properties of the material.

4.3 Design requirements for fatigue-loaded hollow sections (nominal stress method)

4.3.1 Conditions and necessity of fatigue design

The guidance in Eurocode 3 can be used in fatigue design, provided that the followingconditions apply [1]:- the normal stress range is lower than 1,5 fy- the shear stress range is lower than 0,866 fy- the structure is corrosion-protected so that pit depth is less than 1,0 mm- the temperature of the structure is below 150 °C

Fatigue strength need not be calculated if the stress range ∆∆∆∆σσσσ fulfills the followingcondition [1]:

where

Fatigue need not be taken into account, either, if the number of stress cycles fulfills thefollowing condition [1]:

where

is equivalent constant-amplitude fatigue stress range at 2 ·106 stress cycles

[N/ mm2] (Figure 4.5)

N is the number of stress cycles

∆σ E

N

2

NMf Ff E

≤ ⋅⋅ ⋅

2 10

366

2

3

γ γ σ∆(4.2)

γγ

σ

Ff

Mf

∆

is the partial safety factor for fatigue loading (Table 2.1)

is the partial safety factor for fatigue strength (Table 2.2)

is the greatest nominal stress range [N/mm2]

γ σγFf

Mf⋅ ≤∆ 26

(4.1)


111

4.3.2 Fatigue load design conditions

Constant-amplitude load

For constant amplitude loading, the fatigue load design condition is presented in the followingform [1]:

where

Variable-amplitude load

In variable-amplitude loading, the principle is to calculate the cumulative effect of differentstress ranges on the structure. The number of stress cycles featuring different stress rangesare thus compared with the numbers of corresponding fatigue strength stress cycles and theobtained quotients are added. The sum must meet the following condition (Palmgren-Minerrule) [1]:

where

Ni can be determined for normal stresses as follows [1]:

N

N

N

iD

Mf Ff iFf

Mf

iD

Mf Ff i MfFf i

Mf

i FfMf

= ⋅⋅ ⋅

⋅ ≥

= ⋅⋅ ⋅

> ⋅ ≥

= ∞ ⋅ <

5 10

5 10

63

65

∆∆

∆ ∆

∆∆

∆ ∆ ∆

∆ ∆

σγ γ σ

γ σ σγ

σγ γ σ

σγ

γ σ σγ

γ σ σγ

when

when

when

iD

D L

iL

(4.5)

(4.6)

(4.7)

is the number of stress ranges with the magnitude of ∆σi (load)

is the number of the failure-inducing stress ranges with the magnitude of ∆σi

for the relevant detail category

nN

i

i

Σi

i

i

n

N≤ 1 (4.4)

is the fatigue strength which is dependent on the fatigue category and the

number of stress cycles

∆σ R

γ σ σγFf

R

Mf⋅ ≤∆ ∆

(4.3)


112

where

In the case of shear stress, the corresponding value for Ni is [1]:

where

The combined load condition for normal stress and shear stress is [1]:

Σ ∆∆

Σ ∆∆i

i i

i i i

i i

i i

n

N

n

N

σσ

ττ

( )( ) + ( )

( ) ≤ 1 0(4.1 )

is the shear stress range caused by the load

is the shear stress range at 2⋅106 stress cycles for the relevant fatigue detail

(Figure 4.6)

is the shear stress cut off limit (= 1 ·108 stress cycles) for the relevant fatigue

detail (Figure 4.6)

∆∆

∆

ττ

τ

i

C

L

N

N

ic

Mf Ff iFf i

L

Mf

i Ff iL

Mf

= ⋅⋅ ⋅

⋅ ≥

= ∞ ⋅ <

2 1065

∆∆

∆ ∆

∆ ∆

τγ γ τ

γ τ τγ

γ τ τγ

when

when

(4.8)

(4.9)

is the normal stress range caused by the load

is the normal stress range of the constant-amplitude fatigue limit

(= 5 ·106 stress cycles) for the relevant fatigue detail (Figure 4.5)

is the cut off limit (= 1 ·108 stress cycles) for the relevant fatigue detail

(Figure 4.5)

∆∆

∆

σσ

σ

i

D

L


113

4.3.3 Fatigue strength of hollow sections (nominal stress method)

When calculating the fatigue strength with the nominal stress method, the fatigue category ofthe member or element must first be determined. The fatigue category number indicates thefatigue strength of the detail [N/ mm2] at 2 ⋅106 stress cycles.

4.3.3.1 Fatigue strength under normal and shear stress

The fatigue strength curves for hollow section details (4.5) and (4.6) represent the stressranges for details in different fatigue categories. The determination of a detail's fatigue categoryis explained in Appendix 9.5.

Figure 4.5 Fatigue strength curves for various normal stress ranges [1]

Number of stress cycles N

104 105 106 2·106 5·106 107 108 109

1000

500

400

300

200

100

50

40

30

20

10

Nor

mal

str

ess

rang

e (N

/mm

2 ) ∆σ

Constant-amplitudefatigue limit (∆σD)

Cut-off limit (∆σL)

140125112100908071635650454036

Detail category


114

Figure 4.6 Fatigue strength curves for various shear stress ranges [1]

4.3.3.2 Fatigue strength of lattice structure joints (nominal stress method)

For the design of hollow section lattice structures, the fatigue curves presented in Figure (4.7)are used. Due to secondary bending moments, lattice structure members feature local stressconcentrations. To account for them, the forces in the brace members and chords aremultiplied by the factors presented in Table 4.1. The stresses obtained are compared with thefatigue strength ∆σR. The values in Table 4.1 are approximate empirical values or valuesbased on testing. The nominal stress method yields only a rough estimate of the fatiguestrength of the structure. For instance, the combined effect of chord and brace memberstresses on the fatigue strength is difficult to account for in the nominal stress method. Chapter4.4 presents the more accurate hot spot calculation method for the hollow section lattice joints.


104 105 106 2·106 107 108 109

1000

500400

300

200

100

5040

30

20

10

She

ar s

tres

s ra

nge

(N/m

m2

) ∆τ

100

80

Detail category

Cut-off limit (∆τL)(∆τC)


115

Figure 4.7 Fatigue strength curves for lattice structure joints determined by nominal stress method [1]


104 105 106 2·106 107 108 109

1000

500

400

300

200

100

50

40

30

20

10

Str

ess

rang

e (N

/mm

2 ) ∆σ

90

71

565045

36

Detail category

Cut-off limit (∆σL)


116

Table 4.1 Stress correction factors for lattice structure joints [1]

4.4 Fatigue strength of lattice structure joints (hot spot method)

When using the hot spot method for fatigue design, the nominal stress values of latticemembers are multiplied by the concentration factors Ks. The hot spot stress range obtained isused as in the nominal stress method, except that the fatigue stress curves are selected fromFigure 4.8 according to the wall thickness. The design condition for the hot spot method is:

For variable-amplitude loading, the number of stress cycles causing failure Ni is determinedfrom the following formulae:

N n

N n

it

i

it

i

= ≤ ⋅

= ⋅ < ≤ ⋅

− ( )−

− ( )+

10 5 10 4 12

10 5 10 1 10 4 13

12 476 3

1 0 1816

6

16 327 5 2 0116

6 8

, log

, log

, log , log

( . )

( . )

∆

∆

σ

σ

γ σ σγFf s

R

MfK⋅ ⋅ ≤∆ ∆

( . )4 11

Joint type Chords Verticals Diagonals

Circular hollow sections

Gap K joint 1,5 - 1,3

N joint 1,5 1,8 1,4

Overlap K joint 1,5 - 1,2

N joint 1,5 1,65 1,25

Square and rectangular hollow sections

Gap K joint 1,5 - 1,5

N joint 1,5 2,2 1,6

Overlap K joint 1,5 - 1,3

N joint 1,5 2,0 1,4


117

Figure 4.9 Hot spot fatigue strength curves for lattice structure joints of square and rectangular hollow sections [8]. With wall thicknesses 2-4 mm, the 4 mm curve can be used.

The calculation of stress concentration factors for hollow section lattice structure joints is dealtwith in source [7]. Concentration factors are expressed as functions of chord and bracemember dimensions and joint dimensions (gap or overlap and joint angle). Formulae fordetermining stress concentration factors (Ks) for T, X and K jointed square hollow sections arepresented in Tables 4.2 and 4.3.


104 105 106 5·106 107 108 109

1000

500

400

300

200

100

50

40

30

20

10

Hot

spo

t str

ess

rang

e (N

/mm

2 ) ∆σ

45681012,5

Wall thickness mm

Cut-off limit (∆σL)Constant-amplitudefatigue limit (∆σD)


118

In the case of T or X joints, the concentration factors must be calculated for several points. Thegoverning point is the one with the greatest stress range. The stress range of the members in Tand X joints can be expressed as follows:

Chord:

where

The quantities marked with the subscript i are for the brace member, respectively.

The stresses in chord and brace members do not necessarily vary at the same phase. Thestress range of the chord is determined by using the maximum values for the chord’s stressrange. The brace member stress ranges are calculated from the forces acting at the same timeas the chord maximum forces and are added to the chord stress ranges. This is done similarlyfor the brace members, except that the maximum values are attributed to the brace memberstress ranges.

With K joints, the maximum concentration factor values need not be calculated at severalpoints, since Table 4.3 gives the formulae needed for calculating the maximum concentrationfactor for chords and brace members. An axially loaded K joint also generates secondarybending moments, the effect of which is accounted for by multiplying the nominal stresses bythe correction factors given in Table 4.4 [7]. In other cases, the formulae (4.14) and (4.15) areapplied.

∆

∆∆∆

σ 0

0

0

0

0

0

0

0

0

0

KK

K

NM

M

AW

W

s N

s M ip

s M op

ip

op

ip

op

.

.

.

.

.

.

.

( )

( )

is the normal stress range of the chord

is the stress range of the chord bending moment parallel to the lattice plane

is the stress range of the chord bending moment perpendicular to the lattice plane

is the area of the chord

is the chord section modulus parallel to the lattice plane

is the chord section modulus perpendicular to the lattice plane

is the concentration factor for the stress due to the chord bending momentperpendicular to the lattice plane

is the concentration factor for the stress due to the chord bending momentparallel to the lattice plane

is the stress range of the chordis the concentration factor for the stress due to the axial force of the chord

∆ ∆ ∆ ∆

∆ ∆ ∆

∆ ∆

σ

σ

00 0

0

0 0

0

0 0

0

0

4

= ⋅ +⋅

+⋅

+

⋅ +⋅

+⋅

= ⋅

( ) ( )

( ) ( )

K N

A

K M

W

K M

W

K N

A

K M

W

K M

W

K

s N s M ip ip

ip

s M op op

op

si N i

i

si M ip ip i

ip

si M op op i

op

is N

. .max . . .max

.

. . .max

.

. . . . .

.

(4.1 )

NN

A

K M

W

K M

W

K N

A

K M

W

K M

W

s M ip ip

ip

s M op op

op

si N i

i

si M ip ip i

ip

si M op op i

op

0

0

0 0

0

0 0

0

5

+⋅

+⋅

+

⋅ +⋅

+⋅

( ) ( )

( ) ( )

. .

.

. .

.

. .max . . .max . . .max

∆ ∆

∆ ∆ ∆(4.1 )

Brace member:


119

Table 4.2 Concentration factors for T and X joints with square hollow sections [7]

T joint X joint

Load Concentration factor SCFIn-plane bendingmoment in bracingmember

Chord:

Axial force in bracemember

Chord:

Normal stress causedby axial force andbending moment inchord

Chord:

Minimum concentration factor value is 2

X joints for which β = 1,0 ⇒ Ks0.N.C and Ks0.M(ip).C multiply by 0,65

X joints for which β = 1,0 ⇒ Ks0.N.D and Ks0.M(ip).D multiply by 0,50

Fillet welds: ⇒ Ksi.M(ip)A.E and Ksi.N.A.E multiply by 1,40

Parameters: Conditions:τ = t1 / t0 0,25 ≤ τ ≤ 1,0γ = b0 / (2t0) 12,5 ≤ 2γ ≤ 25,0β = b1 / b0 0,35 ≤ β ≤ 1,0; θ = 90°

K

K

K

s M ip B

s M ip C

s M

02 1 722 1 151 0 697 0 75

02 0 690 5 817 4 685 0 75

0

0 011 0 085 0 073 2

0 952 3 062 2 382 0 0228 2 2

2

2

. ( ).( , , , ) ,

. ( ).( , , , ) ,

. (

, , ,

, , , ,

= − + −( )( )= − + + ⋅( )( )

+ −

− + −

β β γ τ

β β γ γ τ

β β

β β

ipip D

si M ip A E

).( , , , ) ,

. ( ). ,( , , , )

, , ,

, , ,

= − + −( )( )

= − +( )( )

− +

− + −

0 054 0 332 0 258 2

0 390 1 054 1 115 2

2 2 084 1 602 0 527 0 75

2 0 154 4 555 3 809

2

2

β β γ τ

β β γ

β β

β βK

Brace member:

K

K

K

s N B

s N C

s N D

02 1 377 1 715 1 103 0 75

02 1 565 1 874 1 082 0 75

0

0 143 0 204 0 064 2

0 077 0 129 0 061 0 0003 2 2

0 208

2

2

. .( , , , ) ,

. .( , , , ) ,

. .

, , ,

, , , ,

,

= − +( )( )= − + − ⋅( )( )= −

+ −

+ −

β β γ τ

β β γ γ τ

β β

β β

00 387 0 209 2

0 013 0 693 0 278 2

2 0 925 2 389 1 881 0 75

2 0 790 1 898 2 109

2

2

, ,

, , ,

( , , , ) ,

. . ,( , , , )

β β γ τ

β β γ

β β

β β

+( )( )

= + −( )( )

+ −

+ −Ksi N A E

Brace member:

K K

K K

s M ip C s N C

s M ip D s N D

0 00 248 0 19

0 00 205 0 24

0 725 2

1 373 2

. . . .( , ) ,

. . . .( , ) ,

, ( )

, ( )

( )

( )

= =

= =

γ τ

γ τ

β

β

��

��

��

M1.ip M1.ip

N1 N1t1 t1

t0 t0

b0 b0

AEB

CD

AEB

CD

b1 b1

M0.ip N0M0.ip N0


120

Table 4.3 Concentration factors for K joints with square hollow sections [7]

Table 4.4 Nominal stress correction factor for K joints (hot spot stress method) [7]

Source [7] includes parametric concentration factor formulae also for circular hollow sectionjoints.

Joint type Chord Brace member

Gapped 1,5 1,5

Overlapped 1,3 1,3

Gap joint Overlap joint

Joint type Member Concentration factor SCFGap Brace member

Gap Chord

Overlap Brace member

Overlap Chord

The minimum concentration factor value is 2

Parameters: Conditions for Conditions forτ = ti / t0 gapped joints: overlapped joints:

ξ = g / bi gapped 0,25 ≤ τ ≤ 1,0 0,4 ≤ τ ≤ 1,0

ξ = -q / bi overlapped 0,25 ≤ ξ ≤ 0,75 -1,0 ≤ ξ ≤ -0,4γ = b0 / (2t0) 12,5 ≤ 2γ ≤ 25,0 12,5 ≤ 2γ ≤ 25,0β = bi / b0 0,35 ≤ β ≤ 1,0 0,35 ≤ β ≤ 0,7

1,5 ≤ ≤ 7,0 2,5 ≤ ≤ 17,0

35° ≤ θ ≤ 55° 35° ≤ θ ≤ 55°

3,62 2-τ τ ξ γ ξ γ

γ β γ β γ

( ) + ⋅ − ⋅( ) +

⋅ − ⋅( ) − ⋅

−

0 336 0 3 0 01

0 044 6 38 4 18100

2 2

2

2

0

2

, , ,

, , , ,g

t

1 1 0 00288 5 73 1 0 178

0 166 1 73

3

0

2

0

3

0

2

, , , ,

, ,

τ γ ξ ξ

β

+

+ −

−

−

g

t

g

t

g

t

0 144 1 0 813 1 84

3 23 1 94 1 910

0 26

2 2

2 2 33

, , ,

, , , ,

β γ β β τ

ξ τ τ γ

⋅ −( ) + ⋅ +

− −

−

− ⋅ −( ) + +( ) −

⋅ − ⋅ + −

40 22 1 0 59 0 028 8 9

5 41 0 008 2 109 4 24

2 2 2

3 2 3 6

, , , ,

, , , ,

ξ β ξ γ β τ

γ β ξ γ ξ

gt0

qt0

��

��

�h 1 h

2

N1 N1N2 N2

t1,2t1,2

b0 b0

b1,2b1,2

t0 t0

h 0 h 0

g

q

h1 h2

N0

θ2θ1 θ1 θ2


121

4.5 Design of fatigue-loaded hollow section structures4.5.1 Welded joints

Structural welded joints should be placed in locations subjected to the lowest load, wheneverthis is feasible (Table 4.5). In welded joints, the fatigue strength does not depend significantlyon the yield strength of steel, which means that the use of high strength steel does notnecessarily bring the same advantages as it does in static design. The level of stress can bedecreased by expanding the cross-sectional area, but this increases the weight of thestructure. The optimal dimensions of a structure must therefore be determined on case by casebasis, taking into account the effects of weight and service life of the structure.

From the fatigue point of view, it is more advantageous toconstruct the welded hollow section end-to-end jointswithout using intermediate plates, since a better fatiguecategory can be obtained. It is essential for the weldstrength that the weld is free of defects especially at theroot side, which is the side more difficult to inspect. If thereis a possibility of defects remaining at the root side (e.g.when using a high material thicknesses), it is advisable touse backing plates. However, it should be kept in mind thatthe use of backing plates generates a stress concentrationwhich may decrease the fatigue strength of the structure.

A high weld convexity increases the stress concentration at the weld toe, so a concave weldprovides a better fatigue strength. Other factors improving the fatigue strength of the weldinclude a low-gradient weld joint angle and a great corner radius of the weld toe.

Post-weld heat treatment

Post weld heat treatment of the weld toe can be used toimprove the fatigue strength of the joint. In the posttreatment, the weld toe is re-melted with, for instance, aTIG torch or plasma torch. Another possibility is to removeminor initial cracks by grinding the weld toe. At the sametime, the corner radius of the weld’s joining location isincreased (Table 4.5). Grinding depth is typically at themost 10% of the plate thickness. A recommended grindingcorner radius for plates with thickness less than 20 mm is10 mm [5].

It is also possible to generate, in the welded detail, a compression stress that counteracts thecrack initiation and decelerates the propagation of the crack. A compression stress can begenerated, for instance, by shot blasting. Stress relieving or annealing is a method intended forreducing the internal residual tension stresses due to the weld. In annealing, the yield strengthof the material is temporarily lowered by increasing the temperature, which relieves the internalstresses of the element. To prevent the generation of new stresses, heating and cooling shouldbe performed as slowly as possible.

Post weld treatment can be profitable when the treated area is small or when the treatment canbe automated in the shop. This treatment can also be used for improving the fatigue resistanceof old structures. The effect of post weld treatment has not been accounted for in the designguidance of Eurocode 3 [1]. An empirical observation is that grinding increases fatigue strengthby 30-100% (2⋅106 stress cycles) and re-melting by 10-170% (2⋅106 stress cycles). Shotblasting can yield a fatigue strength improvement of 30-170% (2⋅106 stress cycles) [5].


122

Grinding

Table 4.5 Methods for improving fatigue strength of details

Method Original detail Improved solution

Stress reduction

Reduction of bending moment inflange plates

Connection placement in a locationsubjected to lower load

Smoother attachment plategeometry

Smoother splice plate geometry

Overlapping of a truss connection

L joint reinforcement with anintermediate plate

Welding method selection Manual welding Machine weldingWeld end rounding by grinding

��

��

��

σ1 σ1σ2 < σ1 σ2 < σ1

h 1 h2 < h1

connection connection

M M

Det 1 Det 1

Det 1Det 1

grinding

grindingt 1

t 2 >

t1


123

4.5.2 Bolted joints

In bolted joints subject to tension, the use of prestressed bolts increases the fatigue strength ofthe joint significantly. The fatigue-inducing stress decreases, if the prestressing force is greaterthan the tension load on the joint. Long and elastic bolts increase the fatigue strength, sincethe elasticity of flange plates does not significantly reduce the prestressing force of long bolts.In flange joints subject to tension, it is advisable to place the bolts as close to the weld aspossible in order to reduce additional stress caused by eccentricity.

The prestressing of bolts is also a useful practice in bolted joints that are subjected to shearforce. Prestressing reduces the stress in the material at the edge of the hole, since part of theforce is transferred through in friction between the splices. However, it is important to ensurethat the friction between the splices is sufficient to prevent the bolt from slipping towards theedge of the hole, which would result in the loss of the prestressing benefits.

4.5.3 Lattice structures

In lattice structure joints, increasing the chord wall thickness and decreasing the brace memberwidth improves the fatigue strength of the structure, as the bending stresses on the chord wallare then reduced. The proportion of brace member width to chord width is expressed by theparameter β. For fatigue strength, the optimal solution would be to have chords and bracemembers of equal width, which would yield a β value of 1.0. In that case, the load istransmitted from the brace member directly to the chord web. With a thick-walled chord,however, the great corner radius may make it difficult to weld a brace member with equal widthto the chord. The increase in the β value improves fatigue strength in cases in which β isgreater than 0.5-0.7 [6].

Welds in lattice structures must be made so that the initiation and termination points do notcoincide with the brace member corners. To obtain the required weld throat thickness when thejoint angle is smaller than 60°, the brace member end must be tapered (chapter 7). The throatthickness of lattice structure joints must be sufficient to prevent the weld root side fromgoverning the fatigue strength. The effect of joint eccentricities is more significant in fatiguedesign than in static design, since the fatigue design of lattice structure joints must alsoaccount for the secondary bending moments due to eccentricities. The need for reinforcingplates in lattice structure joints must be judged on a case by case basis. Reinforcing platesincrease the static strength of the joint; on the other hand, they also generate discontinuities atwhich stress concentrations are generated.

Example 29Calculate the fatigue strength of a hollowsection with dimensions 200 x 200 x 8, subjectedto a fatigue load ∆NSd. A non-loaded plate iswelded to the side of the hollow section. Theload on the member fluctuates as follows:

∆NSd(kN) Number of stress cycles(ni)

1 220 1,5 ·107

2 350 1 ·106

3 50 1 ·107


124 ��

75

NSd

When determining the fatigue strength, stresses are calculated using elastic theory.Calculate the nominal stress range caused by the first load fluctuation for the hollowsection:

The fatigue strength of the structure is calculated using the Palmgren-Miner rule,[formula (4.4)], since the load is not of constant-amplitude. The fatigue category for thenon-load bearing joined element is 71, when the plate length is 75 mm (Appendix 9.5,Table 9.5.2). In fatigue category 71, the normal stress ranges of fatigue limits are thefollowing (Figure 4.5):

∆σD = 52 N/ mm2 (stress range of the constant-amplitude fatigue limit)∆σL = 29 N/ mm2 (stress range of the cut off limit)

Let us assume a fail-safe structure and normal accessibility. A value of 1,0 (Table 2.2) isobtained for the safety factor of the material in fatigue design γMf The number of stresscycles resulting in failure is obtained by substituting in the formula (4.6):

The strength of the hollow section is calculated from the summation equation (4.4):

Σi

i

i

n

N= ⋅

⋅+ ⋅

⋅+ ⋅

∞= <1 5 10

2 69 10

1 10

3 41 10

1 100 85 1 0

7

7

6

6

7,

, ,, , OK !

N

N

D

Mf Ff i MfFf i

Mf

D

Mf Ff

16

5

16

1

56

57

2

5 10

5 10 5 1052

1 0 1 0 37 132 69 10

350000

592459 08

= ⋅⋅ ⋅

> ⋅ ≥

= ⋅⋅ ⋅

⋅

⋅ ⋅

= ⋅

= =

∆∆

∆ ∆ ∆

∆∆

∆

σγ γ σ

σγ

γ σ σγ

σγ γ σ

σ

when

=

D L,

, , ,,

,NN

mm

N

N

mm

N

D

Mf FfD

L

2

26

2

36

36

3 2

3

5 10 5 1052

1 0 1 0 59 083 41 10

50000

59248 44

= ⋅⋅ ⋅

⋅

⋅ ⋅

= ⋅ >( )

= =

= ∞ <( )

∆∆

∆ ∆

∆

∆ ∆

σγ γ σ

σ σ

σ

σ σ

= 2

3

, , ,,

,

Load fluctuations 2 and 3 are calculated in a similar manner:

∆ ∆σ 11

2220000

592437 13= = =N

A

N

mmSd. ,


125

Example 30Check the fatigue strength of a gapped Kjoint using the nominal stress method.Dimensions of the chord are 200 x 200 x10 and those of the brace members are140 x 140 x 5. The axial force range forbrace members ∆N1.Sd is 190 kN and forthe chord ∆N0.Sd is 590 kN. The axialforces in the chord and the bracemember fluctuate, the chord axial forcebeing 100 kN when the axial force in thebrace members is 190 kN. Similarly, theaxial force in the brace members is49 kN when the axial force in the chordmembers is 590 kN.

The dimensions of the joint are asfollows:

θ1 = 37°g = 35 mme = 0,8 mm (≈ 0 mm)

When determining the stresses on lattice structure joints, the effect of secondary bendingmoments must be taken into account. The corrected stress range value is obtained bymultiplying the uniform stress range by the correction factor presented in Table 4.1:

The fatigue category of the joint is determined by the ratio of the thickness of the chordwalls to brace member walls (Table 9.6.3):

t0 / t1 = 2,0 ⇒ fatigue category 71

Assume a fail-safe structure and a normal accessibility, which yields a material factorfor fatigue designl (γMf) of 1,0 (Table 2.2)

The fatigue strength at 5 ·104 stress cycles, for fatigue category 71, is as follows (Figure4.7):

∆σR = 148 N/ mm2

∆ ∆

∆ ∆

σ

σ

11

12

00

02

1 5 1 5 190000

2636108

1 5 1 5 590000

7260122

= = ⋅ =

= = ⋅ =

, ,

, ,

.

.

N

A

N

mm

N

A

N

mm

Sd

Sd

Brace member:

Chord:


126

��

�b1

t1

b0

t0

θ 1

N0.Sd

h 0

θ 1g

h1 h1N1.Sd

e

N1.Sd

Chord

Brace member

Time

N (kN)

590

190

10049

Compare the fatigue strength with chord and brace member stress ranges (4.3):

Example 31Calculate the fatigue strength of the joint shown in example 30 using the hot spot stressmethod. First determine the values of the parameters required in Table 4.3:

The stress concentration factor (Ks) values for the brace member and the chord areobtained separately by substituting in the formulae given in Table 4.3:

Brace member:

Chord:

Kg

t

g

t

g

ts N0

3

0

2

0

3

0

2

3 2

1 1 0 00288 5 73 1 0 178 0 166 1 73

1 1 0 5 0 00288 1035

105 73 0 25 1 0 178 0 25

35

10

0 166

. , , , , , ,

, , , , , , ,

,

= +

+ −

−

−

= ⋅ ⋅ +

+ ⋅ − ⋅

−

τ γ ξ ξ β

ββ 3235

101 73 2 46

− =, ,

K

g

t

si N. , , , ,

, , , ,

, , , , , , , ,

,

= −( ) + ⋅ − ⋅( ) +

⋅ − ⋅( ) − ⋅

−

= ⋅ −( ) + ⋅ ⋅ − ⋅ ⋅( ) +

3 62 2 0 336 0 3 0 01

0 044 6 38 4 18100

2 2

3 62 0 5 2 0 5 0 336 0 25 10 0 3 0 01 0 25 10

0

2

2

0

2

2

τ τ ξ γ ξ γ

γ β γ β γ

044044 10 0 7 6 38 10 0 7 4 1810 35

100 102 2 2 772

2

⋅ ⋅ − ⋅( ) − ⋅⋅

− =, , , , , ,

τ

ξ

γ

β

= = =

= = =

= = =

= = =

t

t

g

b

b

t

b

b

i

i

i

0

0

0

0

5

100 5

35

1400 25

2

200

2010

140

2000 7

,

,

,

γ σ σγ

γ σ σγ

FfR

Mf

FfR

Mf

N

mm

N

mm

N

mm

N

mm

∆ ∆

∆ ∆

0 2 2

1 2 2

1 0 122 0 122148

1 0148

1 0 108 108148

1 0148

= ⋅ = < = =

= ⋅ = < = =

, ,,

,,

OK !

OK !


127

Now determine the hot spot stresses, multiplying nominal stress values by theconcentration factor and by the correction factors given in Table 4.4 (in this example,1.5):

Brace member:

Considering the wall thickness, obtain the following fatigue strength values at 5 ·104 stress cycles (Figure 4.9):

Brace member:∆σR.1= 543 N/ mm2

Chord:∆σR.0= 447 N/ mm2

Now compare the fatigue strength with the chord and brace member stress ranges.

Brace member:

γ σ σγ

γ σ σγ

FfR

Mf

FfR

Mf

N

mm

N

mm

N

mm

N

mm

⋅ = < = =

⋅ = < = =

∆ ∆

∆ ∆

1 21

2

0 20

2

350543

1 0543

377447

1 0447

.

.

,

,

OK !

OK !

Chord:

γ γ

γ γ

Ffi

isi N Ff s N

Ffi

isi N Ff s N

N

AK

N

AK

N

mm

N

AK

N

AK

.max. .

..max

.

, , , , , ,

, , , , ,

+

= ⋅

⋅ + ⋅

⋅ =

+

= ⋅

⋅ + ⋅

0

00

2

0

00

1 0 1 5190000

26362 77 1 0 1 5

100000

72602 46 350

1 0 1 549000

26362 77 1 0 1

55590000

72602 46 377 2

⋅ =,

N

mm

Chord:


128

4.6 Brittle fracture of structural hollow sections

Brittle fracture is the rapid failure of an element with no clearly distinguishable plasticdeformation. Brittle fracture initiates from a small crack initiated by fatigue or a weld defect andpropagates rapidly even in a defect-free structure. The probability of brittle fracture depends onthe following factors:

- steel strength grade- thickness of the material- loading rate- service temperature- steel toughness

A hollow section with high strength and thick walls is more sensitive to brittle fracture than onewith low strength and thin walls. A high loading rate increases the risk of brittle fracture, and sodoes a low service temperature. A tougher steel grade is, however, better in low temperatures.Vulnerability to brittle fracture is indicated by the parameters of transition temperature andimpact toughness.

The methods for calculating the minimum service temperature presented in different versionsof Eurocode 3 are contradictory and thus not recommended. This manual presents a simplifiedmethod for calculating the minimum service temperature, which is based on tests made to coldformed hollow sections at low temperatures [2].

4.6.1 Parameters affecting brittle fracture in structural hollow sections

The probability of brittle fracture in cold-formed hollow sections depends on both the materialand the dimensions of cross-sections and joints. This section presents conditions that must bemet to prevent brittle fracture. The conditions are valid for the Eurocode 3 service conditioncategory C, defined as follows:

C2: Fracture of critical members or joints, where local failure would cause complete structuralcollapse with serious consequences to life or very high costs.

Mechanical properties of structural hollow sections

The ultimate strength and yield strength of structural steel must meet the following condition[2]:

where

is the nominal ultimate strength of steel

is the nominal yield strength of steelffu

y

f

fu

y≥ 1 2 6, (4.1 )


129

The elongations, measured on coupons cut longitudinally from the hollow section flange, mustfulfill the following conditions [2]:

where

For the steel grade S355J2H, the required uniform elongation is Ag ≥ 3,38%.The impact toughness KV measured on coupons cut longitudinally from the hollow sectionflange must fulfill the following condition [2]:

The test is carried out at the minimum service temperature of the structure:

Dimensions of the cross-section

Table 4.6 Minimum values for the corner radius of hollow sections

Furthermore, the manufacturer must show that the manufacturing method used is feasible forthe constant production of hollow sections whose internal corners do not have cracksexceeding the allowed values. The maximum depth allowed for a flaw with a blunt notch is 0.2mm. For flaws with a sharp crack-like tip, the maximum depth allowed is 0,05 mm [2].

The slenderness of the walls of the hollow section must fulfill the following condition [2]:

where

is the width of the hollow section

is the height of the hollow section

is the wall thickness of the hollow section

bht

b h

t

+ ≥ 25 20(4. )

Wall thickness (mm)

Minimum internal corner radius Minimum external corner radius

t ≤ 6 mm 0,6 t 1,6 t

6 < t ≤ 10 mm 1,0 t 2,0 t

t > 10 mm 1,4 t 2,4 t

KV ≥ 35 9 J

cm2 (4.1 )

is the ultimate elongation when the measured length of the test piece is 5,65

is the cross-sectional area of the test piece

is the uniform elongation corresponding to the ultimate tensile strength (%)

is the elongation at yield strain (%)

ASAg

y

5

0

ε

S0

A

Ag y

5 15 7

20 8

≥≥

% (4.1 )

(4.1 )ε


130

Minimum gap value

In gapped N, K and KT joints, the actual gap ga (Figure 4.9) must fulfill the following condition[2]:

where

The gap g between brace members (Figure 4.9) must fulfill the following conditions [4]:

where

Figure 4.9 Actual gap

��

�Detail 1 (θ > 60°)Detail 1 (θ ≤ 60°)

ga ga

g g

t 0g

θ θ

t0

Detail 1

t 0

t1 and tbb

2

i

0

are the wall thicknesses of the brace members

is the width of the brace member

is the width of the chord

g t t

g

b

g

b

≥ +

≥

≤

1 2

0

0

22

23

24

(4. )

(4. )

(4. )

0, 5 1-b

b

1, 5 1-b

b

i

0

i

0

is the wall thickness of the chordt0

g ta ≥ 1 5 210, (4. )


131

4.6.2 Minimum service temperatures of Rautaruukki structural hollow sections

The Rautaruukki Metform hollow sections shown in Appendix 9.1 (steel grade S355J2H) fulfillall the requirements for hollow section properties (section 1.1) stated in section 4.6.1.According to the latest research [2], they can be used in welded structures down to atemperature of - 40 °C.

4.7 References

[1] ENV 1993-1-1:Eurocode 3: Teräsrakenteiden suunnittelu. Osa 1-1: Yleiset säännöt jarakennuksia koskevat säännöt, 1993(ENV 1993-1-1: Eurocode 3: Design of steel structures. Part 1.1: General rules and rulesfor buildings, 1993)

[2] CIDECT: Project 5AQ/2: Cold formed RHS in arctic steel structures, Final report 5AQ-5-96,1996

[3] Niemi, E.:Stress determination for fatigue analysis of welded components, IIW/ 115-1991-93, 1995


[5] Tarjavuori, P.: Hitsin väsymislujuuden parantaminen jälkikäsittelyllä, Lappeenrannanteknillinen korkeakoulu, Konetekniikan osasto, 1995


[7] CIDECT: Research Project 7M: Working draft: Design guide for hollow section structuresunder fatigue loading, Aachen 1996

[8] IIW International conference on Performance of dynamically loaded welded structures:Scale effects on the fatigue behaviour of tubular structures, San Francisco, 1997


132

5 FIRE DESIGN OF STRUCTURAL HOLLOW SECTIONS

In fire situations, the temperature of the steel increases together with the temperature of thegases in fire compartment. As the temperature of the steel increases, its strength anddeformation properties are transformed. According to their use, structures have different fireresistance requirements (e.g. requirements for bearing capacity and compartmentation).

Often it is necessary to protect steel components in order to slow down the increase intemperature during fire. Several fire retardant methods are applicable for use with hollowsections, for instance, the use of protective materials such as mineral wool or fire-retardantpaint. The heat retention capacity of hollow sections can be improved for instance with aconcrete infill. Hollow sections are efficient in fire design, since their section factor (the ratio offire-exposed area to unit mass) is smaller than that of open sections. In addition, hollowsections with their rounded corners are well-suited for fire-retardant painting. Fire retardantmethods are described in more detail in Section 5.6.

The strength of a hollow section in a fire situation can be calculated by two different methods:either by the properties of the material (yield strength and modulus of elasticity) in a firesituation or by determining the critical temperature of structural steel as a function of degree ofutilization. The non-uniform temperature distribution of the steel component can be taken intoaccount when carrying out fire design by the properties of the material. When using the criticaltemperature method, the temperature distribution of structural steel is assumed uniform. Themethods are illustrated in Figures 5.1 and 5.2.

Kuva 5.1 Fire design based on properties of material

Figure 5.2 Fire design based on critical temperature

Required fireresistanceperiod

Calculate maxsteel tempera-ture during fireresistanceperiod θ a.max.

Determine util-ization ratioµ0

Calculate criti-cal structuraltemperature θa.cr

Improve structural fire retardationor select larger hollow sectionsize

OKθa.cr ≤ θa.max

no

yes

Required fireresistanceperiod

Calculate maxsteel tempera-ture during fireresistanceperiod θ a.max.

Determineproperties ofmaterialfy.θ and Eθ

Calculatestrength in firesituation Rfi.d

Improve structural fire retardationor select larger hollow sectionsize

OKEfi.d ≤ Rfi.d

no

yes


133

5.1 Development of temperature in fire compartments

Factors affecting the development of a real fire include the mode of combustion, the shape offire compartment, the magnitude and type of fire load, the supply of air needed for combustionand the fire extinguishing system. However, the models used in practical design are simpler.Fire design can be based on either the standard time-temperature curve [1] common to all firesituations or on parametric temperature-time curves.

5.1.1 Standard time-temperature curve

The temperature of fire compartment varies with time. In ISO-834, this is expressed with thefollowing formula [1]:

where

Figure 5.3 Standard time-temperature curve according to ISO-834

0 10 20 30 40 50 60 70 80 90 100 110 1200

0

100

200

300

400

500

600

700

800

900

1000

1100

0

Time (min)

Tem

pera

ture

(°C

)

is the temperature of the gases in the fire compartment (°C)is the time (min)

θgt

θg = + ( )20 345 log 8t + 1 (5.1)

DESIGN HANDBOOK FOR RAUTARUUKKI HOLLOW SECTIONSChapter 5

134

5.1.2 Development of temperature according to the parametric model

An alternative method for calculating the evolution of temperature in fire compartment is theparametric model presented in Eurocode 1, Section 2.2 [1]. This model also accounts foropenings in the fire compartment, the thermal properties of the wall materials and themagnitude of the fire load in determining the development of the temperature. The parametricmodel can be used if the fire compartment area is less than 100 m2, there are no openings inthe fire compartment ceiling, and the height of the fire compartment does not exceed 4 m [1].The temperature of the fire compartment increases as long as there is flammable material.Finally, the temperature reaches a maximum value of Θmax (Figure 5.4), after which the firecompartment temperature starts decreasing. The parametric model is not explained moreextensively in this manual, since many buildings do not meet the conditions for using it.

Figure 5.4 Temperature-time curves in standard and parametric fire models (sketch)

5.2 Development of steel temperature

In fire situations, the temperature of steel members increases slower than that of the firecompartment. The development and distribution of steel temperature depends on the shape ofthe steel member and its thermal properties. It is always necessary to calculate the steeltemperature up to the required fire resistance period, since a steel member may reach itsmaximum temperature during fire at a point where the fire compartment temperature startsdecreasing according to the parametric fire model. By using a fire retardant material, theevolution of steel temperature can be slowed down, which lengthens the fire resistance period.

Standard time-temperature curve

Parametric fire curve

Cooling phaseHeating phase t

Θg

Θmax

DESIGN HANDBOOK FOR RAUTARUUKKI HOLLOW SECTIONS Chapter 5

135

5.2.1 Development of temperature in unprotected steel members

The increase of temperature in unprotected steel members can be determined from the formula(5.2), when the temperature distribution in the cross-section is assumed uniform [2]:

where

The net heat flux consisting of radiation and convection can be expressed as follows [1]:

where

The convective net heat flux is obtained from the following formula [1]:

where

αθθ

c

g

m

is the convective heat transfer coefficient (Eurocode 1 default value αc= 25 W/m2K

is the ambient gas temperature (°C)

is the steel surface temperature (°C)

˙.h

W

mnet c c g m= −( )

α θ θ 2 (5.4)

is a factor allowing for the differences in national testing

(Eurocode 1 default value γn.c = 1,0)

is a factor allowing for the differences in national testing

(Eurocode 1 default value γn.r = 1,0)

is the convective net heat flux (W/ m2)

is the radiative net heat flux (W/ m2)

γ

γ

n c

n r

net c

net r

h

h

.

.

.

.

˙

˙

˙ ˙ ˙. . . . .h h h

W

mnet d n c net c n r net r= ⋅ + ⋅

γ γ 2 (5.3)

is the section factor of an unprotected steel member (m-1), at least 10 m-1

(Appendices 9.1 and 9.6)

is the exposed surface area of the member per unit length (m2)

is the volume of the member per unit length (m3)

is the specific heat of steel (J/ kgK)

is the net heat flux per unit area (W/ m2)

is the time interval (s), maximum 5 s

is the unit mass of steel (7850 kg/m3)

A

VAVc

ht

m

m

a

net d

a

˙.

∆ρ

∆ ∆θρa t

m

a anet d

A

Vc

h t. .˙=

⋅⋅ (5.2)


136

Respectively, the radiative net heat flux is determined from the following equation [1]:

where

The constant ca= 600 J/ kgK can be used for the specific heat of steel. Alternatively, specificheat can be determined from the following formulae [2]:

where

5.2.2 Development of temperature in fire protected steel members

The increase of temperature in fire protected steel members is calculated from the formula,when the temperature distribution in the cross section is considered uniform [2]:

where

is the section factor of the fire protected steel member (m-1),

(Appendices 9.1 and 9.6)

is the interior area of the fire retardant material in the member per unit lengt (m2)

is the specific heat of steel (J/kgK)

is the volume of the member per unit length (m3)

is the thickness of the fire retardant material (m)

A

VA

Vcd

p

p

a

p

∆ ∆ ∆θλ θ θ

ρ φ θφ

a t

pp

g t a t

p a a

g t

A

V

d ct e.

. .

.=−( )

⋅ ⋅ +

− −

≥

13

1 0 010 (5.1 )

θa is the steel temperature

c when

c when

c when

c when

a a a a

aa

aa

a

= + − ⋅ + ⋅ ≤ < °

= +−

≤ < °

= +−

≤ < °

= ≤ < °

− −425 0 773 1 69 10 2 22 10 600

66613002

738735

54517820

731900

650 1200

3 2 6 3, , ,θ θ θ θ

θθ

θθ

θ

20 C

600 C

735 C

900 C

a

a

a

a

(5.6)

(5.7)

(5.8)

(5.9)

Φεθ

res

r

is the configuration factor (Eurocode 1 default value Φ = 1,0)

is the resultant emission factor (Eurocode 1 default value εres = 0,50)

is the ambient radiation temperature

˙ ,.hW

mnet r res r m= ⋅ ⋅ +( ) − +( )[ ]

−Φ ε θ θ5 67 10 273 2738 4 4

2 (5.5)


137

The parameter φ is determined as follows [2]:

where

The delay in the increase in temperature of the steel member due to moisture evaporation canbe taken into account if the moisture content of the fire retardant material is great. Duringmoisture evaporation, the steel member temperature is constant (= 100°C). The delay time canbe expressed as follows [3]:

where

Figure 5.5 Effect of the moisture content of the fire retardant material to increase in temperature

Fire compartment temperature

Steel member temperature

Tem

pera

ture

(°C

)

t (min)

100tv

p

d

p

p

p

p

ρ

λ

is the moisture content of the fire retardant material (%)

is the unit mass of the fire retardant material (kg/m3)

is the thickness of the fire retardant material (m)

is the thermal conductivity of the fire retardant material (W/mK)

tp d

vp p p

p=

⋅ ⋅ ( )ρλ

2

52min (5.1 )

cp

pρis the specific heat of the fire retardant material (J/kgK)

is the unit mass of the fire retardant material (kg/m3)

φρρ

=⋅⋅

c

cd

A

Vp p

a pp

p (5.1 )1

is the time interval (s), maximum 30 sis the temperature of the steel member (°C)is the temperature of the fire compartment (°C)is the increase in the fire compartment temperature within interval ∆t (°C)is the thermal conductivity of the fire retardant material (W/mK) depending onthe temperature of the fire retardant material

is the unit mass of steel (kg/ m3)

∆

∆

t

a t

g t

g t

p

a

θθ

θλ

ρ

.

.

.


138

5.3 Strength and modulus of elasticity of steel in fire situations

As the temperature increases, the strength and the modulus of elasticity of the steel changes.However, the room temperature values for yield strength can be used up to 400 °C. The yieldstrength correponds to a total elongation of 2%. The modulus of elasticity is constant up to100 °C. The dependence of strength and modulus of elasticity on temperature is given in Table5.1 and Figure 5.6.

Table 5.1 Effect of temperature on the strength and modulus of elasticity of steel [2]

Figure 5.6 Relative strength of steel and modulus of elasticity as a function of temperature

0 100 200 300 400 500 600 700 800 900 1000 1100 12000

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0

Temperature (°C)

kE.θ ky.θ

kk

yE

..

,θ

θ

Temperature θa (°C) Reduction factor for yield strengthky.θ

(fy.θ/ fy)

Reduction factor for modulus ofelasticity kE.θ

(Ea.θ/ Ea)

20 1,0 1,0

100 1,0 1,0

200 1,0 0,9

300 1,0 0,8

400 1,0 0,7

500 0,78 0,6

600 0,47 0,31

700 0,23 0,13

800 0,11 0,09

900 0,06 0,0675

1000 0,04 0,045

1100 0,02 0,0225

1200 0,0 0,0

Intermediate values can be determined by linear interpolation


139

5.4 Critical temperature in hollow section structures

The critical temperature of hollow sections can be determined as a function of the degree ofutilization for Class 1, 2 and 3 cross-sections and for hollow sections with Class 4 cross-sections loaded in tension [2]:

In critical temperature calculations, it is assumed that the temperature of the steel member isdistributed uniformly through the entire cross-section. However, the method gives aconservative value even if the steel temperature distribution is non-uniform. For Class 4 cross-sections of other than hollow sections in tension, the strength in fire situations is sufficient if thesteel temperature during fire is lower than 350 °C. The degree of utilization µ0 is determinedfrom the following formula:

where

The critical temperature for different degrees of utilization is presented in Table 5.2 and Figure5.7.

Table 5.2 Critical temperature of steel θa.cr as a function of the degree of utilization µ0 [2]

µ0 θa.cr µ0 θa.cr µ0 θa.cr µ0 θa.cr

0,10 829 0,34 645 0,58 560 0,82 490

0,12 802 0,36 636 0,60 554 0,84 483

0,14 779 0,38 628 0,62 549 0,86 475

0,16 759 0,40 620 0,64 543 0,88 467

0,18 741 0,42 612 0,66 537 0,90 458

0,20 725 0,44 605 0,68 531 0,92 448

0,22 711 0,46 598 0,70 526 0,94 436

0,24 698 0,48 591 0,72 520 0,96 421

0,26 685 0,50 585 0,74 514 0,98 398

0,28 674 0,52 578 0,76 508 1,00 349

0,30 664 0,54 572 0,78 502

0,32 654 0,56 566 0,80 496

E

Rfi d

fi d

.

. .0

is the design value for the loads in fire situations (Section 5.5.1)

is the design value of the strength at room temperature

µ00

4=E

Rfi d

fi d

.

. .(5.1 )

θµa cr. ,= −

+39 19 1 482 3 ln1

0,9674 03,833 (5.1 )


140

Figure 5.7 Critical temperature of steel

5.5 Determining the strength of hollow section structures in fire situations

The fire design criterion is expressed as follows [2]:

where

E

Rfi d

fi d

.

. .

is the design value for the effect of loads in a fire situation

is the design value for hollow section strength in a fire situation (varies

according to time and temperature)

E Rfi d fi d. .≤ (5.1 )5

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

0

100

200

300

400

500

600

700

800

900

0

Crit

ical

tem

pera

ture

(°C

)

Degree of utilization µ0

1,0


141

5.5.1 Partial safety factors in fire design

The design value of loads in a fire situation is affected by the expansion and deformation of thematerial due to temperature. The simpler method is to calculate the fire situation loads bymultiplying the design value of loads in normal temperature by the fire situation reduction factorηfi, so the effect of the structural heat expansion need not be taken into account. The designvalue for the fire situation load is as follows [2]:

where

The fire design reduction factor ηfi is determined from the following formula [2]:

where

Figure 5.8 shows various curves of the fire design reduction factor ηfi with different values ofcombination factor ψ1.1 for γGA = 1,0, γG = 1,35 and γQ= 1,5 (Eurocode 3 basic values).

The values of partial safety factors may vary country by country. The partial safetyfactors must be checked from national application documents (NADs).

is the partial safety factor for permanent loads in an accident situation [7]

is the partial safety factor for permanent load [7]

is the partial safety factor for variable load [7]

is the combination factor for variable loads [7]

is the permanent load

is the principal variable load

γγγψ

GA

G

Q

k

k

GQ

1 1

1

.

.

η γ ψγ γfiGA k k

G k Q k

G Q

G Q= ⋅ + ⋅

⋅ + ⋅1 1 1

17. .

.(5.1 )

η fi

dE

is the design load reduction factor in a fire situation

is the design value for force or moment at room temperature

E Efi d fi d. = ⋅η (5.1 )6


142

Figure 5.8 Various reduction factor ηfi curves in fire design

5.5.2 Determining the cross-section class in fire design

For compression members, the cross-section classification is calculated as in section 2.2(Table 2.4). For other structural members, the correction factor ε is used when determining thecross-section limit values. The correction factor is calculated from the following formula [2]:

The formulae given in Sections 5.5.3 - 5.5.7 for the determination of strength are valid only forhollow sections with Class 1, 2 and 3 cross-sections, and for hollow sections with Class 4cross-sections in tension. Class 4 cross-sections of other than hollow sections in tension mustbe fire protected so that the hollow section temperature does not exceed 350 °C during the fire.

ε θ

θ= 235

8f

k

ky

E

y

.

.(5.1 )

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 50

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0

Q

Gk

k

ψ1,1 = 0,2

ηfi ψ1,1 = 0,5

ψ1,1 = 0,7

ψ1,1 = 0,9


143

5.5.3 Strength of hollow section subjected to tension in fire situations

The strength of a hollow section in tension is given by the following formula [2]:

where

5.5.4 Buckling strength of hollow sections in fire situations

The buckling strength for a hollow section in compression is given by the following formula [2]:

where

The modified strength properties of steel are taken into account when calculating theslenderness of a hollow section in a fire situation [2]:

where

L

ik

k

fi

y

E

.

.

θ

θ

is the buckling length in a fire situation

is the moment of inertia

is the reduction factor for the yield strength of steel in temperature θa

is the reduction factor for the modulus of elasticity of steel in temperature θa

λπθ

θ

θ=

⋅⋅⋅

L

i

k f

k Efi y y

E

.

.(5. )21

ky

fi

M fi

.

.

θ

χ

γ


is the reduction factor for buckling in a fire situation (always calculated by thebuckling curve c)

is the partial safety factor of the material in a fire situation (Eurocode 3 defaultvalue γM.fi = 1,0)

N A kf

fi Rdfi

yy

M fi. . .

.,θ θ

χγ

=

⋅ ⋅1 2

20(5. )

k

Ny

Rd

M

M fi

.

.

θ

γγ

1


is the tension strength in normal temperature (Appendix 9.1)

is the partial safety factor of the material

is the partial safety factor of the material in a fire situation (Eurocode 3 default

value γM.fi= 1,0)

N k Nfi Rd y RdM

M fi. . .

.θ θ

γγ

= ⋅ 1 19(5. )


144

The buckling length in a fire situation is calculated as at room temperatures, excluding thefollowing cases [2]:

- The buckling length of a column in a non-sway frame can be determined by assuming asemi-rigid column having rigid supports at both ends above or below the fire compartment. Apre-requisite for this is that the fire resistance of the structural members susceptible tobuckling, which limit the fire compartment, is at least the same as the fire resistance of thecolumns.

- In a non-sway building in which each floor constitutes a separate fire compartment, thebuckling length of the column can be determined as follows:

a) columns in intermediate floors Lfi = 0,5 Lb) columns in the top floor Lfi = 0,7 L

The value of these buckling length may vary country by country. The values should bechecked from the national application documents (NAD's)

5.5.5 Bending strength of hollow sections in fire situations

The bending strength of a hollow section is determined by the following formula [2]:

where

With a non-uniform temperature distribution of the hollow section cross-section, the values ofthe adaptation factor κ1 are the following:

- hollow section exposed to fire on all sides κ1 = 1,0- hollow section exposed to fire on three sides with

a concrete or a composite plate on the fourth side κ1 = 0,7

With a non-uniform longitudinal temperature distribution of the hollow section, the values of theadaptation factor κ2 are as follows:

- supports of a statically undetermined hollow section κ2 = 0,85- other cases κ2 = 1,0

k

My

Rd

M

M fi

.

.

θ

γγ

1

is the reduction factor of the yield strength of steel in temperature θa

is the bending strength at room temperature (Appendix 9.1)



valueM.fi = 1,0)

M k Mfi Rd y RdM

M fi. . .

.θ θ

γγ κ κ

= ⋅⋅ ⋅

1

1 222(5. )


145

5.5.6 Shear strength of hollow sections in fire situations

The shear strength of a hollow section is given by the following formula [2]:

where

The adaptation factors κ1 and κ2 are given as in the case of bending strength.

5.5.7 Strength of hollow sections subjected to bending moment and compressive axial force in fire situations

The interaction expression for hollow sections subjected to bending moment and compressiveaxial force is as follows [2]:

where

λ λ λ λ χχ

χχ

θ θy z yf

zf y. z.

y. i z. i⇒ ⇒ ⇒ ⇒1 2 1 2, ,

is the compression strength of a hollow section in a fire situation intemperature θa (5.18)

is the bending strength of a hollow section in a fire situation in temperatureθa (by y axis)

is the bending strength of a hollow section in a fire situation in temperatureθa (by z axis)

is the compressive axial load in a fire situation

is the bending load in a fire situation (by y axis)

is the bending load in a fire situation (by z axis)

are calculated as in Section 2.9.1.1, with the following modifications:

N

M

M

N

M

M

k

fi Rd

y fi Rd

z fi Rd

fi Ed

y fi Ed

z fi Ed

y

. .

. . .

. . .

.

. .

. .

θ

θ

θ

and kz

N

N

k M

M

k M

Mfi Ed

fi Rd

y y fi Ed

y fi Rd

z z fi Ed

z fi Rd

.

. .

. .

. . .

. .

. . .θ θ θ+

⋅+

⋅≤ 1 24(5. )

k

Vy

Rd

M

M fi

.

.

θ

γγ

1


is the shear strength in normal temperature



value γM.fi = 1,0)

M k Vfi Rd y RdM

M fi. . .

.θ θ

γγ κ κ

= ⋅⋅ ⋅

1

1 223(5. )


146

5.6 Fire retardant methods

Steel structures can be protected against fire by insulating them or increasing their heatretention capacity. Structural solutions can also be used to increase the fire resistance period.In addition to the cost of materials, installation and maintenance costs should be consideredwhen selecting the fire retardant method.

Table 5.3 Fire retardant methods

5.6.1 Fire retardation by insulation

As compared to unprotected structures, insulated structures are slower to heat and slower toreach the critical temperature. Structures can be insulated with boards or sprayed materials.The thermal conductivity λp of the fire retardant material depends on the temperature of thematerial, which must be taken into account when calculating the temperature of the steelmember. The thermal conductivity characteristics of the fire retardant material are usuallyshown in manufacturers' brochures. The following is a description of the properties and use ofthe most common fire retardant materials.

Mineral wool boards

The fire retardant properties of mineral wool are based on its good thermal conductivity. Thesintering temperature of fire protective mineral wool, that is, the temperature in which the fibresmelt, must be sufficiently high. Depending on the fire resistance period, a sintering temperatureof 800-1000 °C is required. The density of the boards varies between 100-400 kg/ m3, and theirthickness varies between 10-100 mm.

Mineral wool boards can be fixed mechanically or with glue. In mechanical fixing, nails andbolts are used. Steel spikes and lock plates are also an alternative. Nails are fixed in place byshooting or by resistance butt welding. Steel spikes must be fixed before installing theinsulation material. The insulation material is attached to the steel member by lock plates.When using glue, the steel surface must be dry and clean from any dust or oil.

Vermiculite boards

The basic material in vermiculite boards is exfoliated mica. The moisture content of the boardsis high, as the binding agent is a mineral containing silicate. Vermiculite boards have a goodthermal insulation capacity, and the evaporation of moisture in fire situations increases the fireresistance period. The density of boards varies between 350-500 kg/m3 and their thicknessbetween 16-80 mm.

Principle MethodsHeat insulation - insulation boards

- fire retardant paints- sprayed insulation materials

Improvement of heat retention capacity - concrete infill- water infill- sprinkler systems

Structural fire retardation - ceiling screens- placing the columns outside the fire compartment- placing the columns inside the wall


147

Vermiculite boards can be fixed with glue or nails. Normally, boards are fixed to form a casingaround the tube. When using nails, an air slot of approximately 3 mm must be left between thecasing and the tube. When using glue, the work temperature must be over 0 °C. The surface ofa vermiculite board is smooth and fit for painting.

Calcium silicate boards

The fire retardant properties of calcium silicate boards are almost as good as those ofvermiculite boards. The density of calcium silicate boards varies between 430 - 950 kg/ m3 andtheir thickness between 6 - 65 mm. Calcium silicate boards are fixed to form a casing aroundthe tube, normally with bolts.

Plaster board and profiled elements

The use of plaster board as a fire retardant is based on the high content of absorbed water inplaster. After the evaporation of absorbed water, non-reinforced board fails and loses itsthermal conductivity. The strength of the board can be improved by reinforcing it with glassfibre, which secures the insulation capacity of plaster board even after the evaporation ofabsorbed water. The density of boards varies between 770 - 980 kg/m3. The profiled thicknessis normally 13 mm, and it can be installed in several layers.

Plaster board is fixed to the hollow sections with bolts. One to four board layers can beinstalled. A mixture of plaster, perlite and glass fibre can be used to pre-form profiled elements,to the shape of the hollow section, which are then fixed around the tube on-site. The profiledelements are fixed with glue or with separate cover plates.

Wood fibre plaster boards

Wood fibre plaster boards are made by pressing a mixture of wood fibres and plaster to form ahard-surface board. The density of wood fibre boards is approximately 1200 kg/m3. Boards arefixed in place with bolts or nails.

Cement cellulose boards

The material of cellulose cement boards is cellulose, and binding agents include variousmaterials which contain silicate. The density of the board is approximately 1100 kg/m3. Boardsare used primarily in light-weight fire-resistant partition walls. Cellulose cement boards areusually fixed directly to the frame with bolts.

Sprayed mineral fibre

In sprayed mineral fibre, mineral wool fibres and cement are sprayed together with water onthe hollow section surface. The density of the sprayed layer is 220 - 500 kg/ m3, and thethickness of the layer is 10 - 60 mm. If the thickness of the layer is more than 35 mm, areinforcement is installed around the hollow section.

The finished surface of the sprayed layer is porous. The sprayed surface can be painted, plas-tered or clad with boards. The surface of the mineral fibre layer must be protected from oil and,when used outdoors, from moisture.

Sprayed vermiculite

The sprayed mass consists of vermiculite aggregate and cement, lime or plaster binding agent.The mass is sprayed in a similar manner to mineral fibres. The thickness of the insulation layeris normally 10 - 60 mm, and it can be sprayed in one or more layers of 10 - 15 mm. The densityof the sprayed layer is 300 - 800 kg/m3. Sprayed masses with greater density form a strongersurface.


148

Fire retardant paints

The protective effect of fire retardant paints is based on the heat-insulating foam generated asthe temperature increases to 250 - 300 °C. However, the foam layer cannot resist long periodsof fire, but is peeled off as the fire advances. Fire retardation by paint is thus suited formaximum fire resistance periods of one hour.

The thickness of a paint layer is 0,2 - 3 mm, and several layers can be applied. Fire retardantpaint can be applied to the tube surface in the same way as normal anti-corrosive paint.Normally, an anti-corrosive undercoat must be applied to the hollow section before theapplication of fire retardant paint. During on-site work and transport, it is important to keep inmind that the fire retardant coat must not be subjected to mechanical stress or moisture. Thecompatibility of undercoat and top coats must be checked with the paint manufacturer. Theadvantage of a fire retardant paint is the thin coat and a finished surface which retains theoriginal shape.

5.6.2 Fire retardation by increasing the heat retention capacity of structural steel

Concrete infill of hollow sections

A concrete-filled column is a simple and effective fire retardant method which retains theappearance and the dimensions of the hollow section. The use of reinforcement significantlyimproves the fire resistance period of the hollow section. The amount of reinforcement can beadjusted to regulate the strength of the column at normal temperature and in fire situations.This way, the same column size can be used in multistory buildings from the ground floor to thetop.

Since the concrete infill is usually carried out on-site, the light weight of hollow sections can befully utilized during erection. In normal temperature, a concrete-filled hollow section functionsas a composite structure, and in a fire situation, the majority of loads is transferred by theconcrete filling and the reinforcement.

For fire situations, the tube must be provided with steam exhaust openings. During the fire, thesteam pressure is then dissapated through the openings without damaging the section. Whenplacing the concrete infill, sufficiently thin layers must be used and consolidation performedwith great care. Fire design tables for concrete-filled columns are shown in reference [4].

Water infill of hollow sections

Water infill in a hollow section functions as a cooling agent. The thermal energy generated byfire is consumed by heating and vaporizing the water contained in the hollow section. Theeffect of water cooling can be enhanced by connecting the hollow sections with an overheadwater tank. In a fire situation, the vaporized water ascends to the tank and returns to the hollowsections cooled. To prevent the water from freezing, an agent such as calcium carbonate orcalcium nitrate must be added.

Water cooling is an effective fire retardant method. By arranging a water circulation, thetemperature of hollow sections normally stays at 200 - 250 °C for the entire duration of the fire.Water cooling can be applied only when protecting columns. To prevent leakage, specialattention must be paid to sealing the tube joints of the tubes trougt which water circulates.


149

Sprinkler systems

A sprinkler is an automatic fire extinguishing system which starts operating as the temperatureincreases in a fire situation. The fire compartment temperature does not increase after thesprinkler system has started operating. National regulations include instructions on allowing forsprinkler systems in the fire design. The profitability of installing a sprinkler system depends onthe ratio of its installation costs to the cost of other fire retardation methods.

5.6.3 Structural fire retardation

With appropriate structural solutions, separate fire retardation of hollow section structures canbe reduced or completely omitted. The use of structural solutions to improve the fire resistanceof structural elements and joints reduces the need for fire retardant materials which increasematerial and installation costs. Structural fire retardation must be applied individually for eachcase, and it should be taken into account at the planning stage.

Fire proof ceilings

To obtain the space required for Heating, Ventilating and Air Conditioning installations, theroom height can be reduced with a false ceiling. False ceilings are also used to cover pipeinstallations and other services, and a fire proof ceiling can be utilized for the fire protection ofstructural components (e.g. floor joists) in the intermediate space. In such a case, the fire proofceiling must be designed and dimensioned appropriately. Also the fixtures that connect the thefire proof ceiling to the floor above must be sufficiently strong to bear the loads during the fire.In practice, the fixtures often constitute governing elements in a fire situation.

Placing columns outside the fire compartment

When placing columns outside the external walls, the increase in fire compartment temperatureneed not be taken into account in column design. A prerequisite for this is that the column isplaced sufficiently far from window openings. In a fire situation, the hot gases and flamesexiting through window openings increasing the temperature of steel columns that are close tothe openings. Window openings are usually placed so close to one another that a flameretardant must be used in the columns. Sheet steel is an example of a flame retardant material.

Placing columns inside the wall

The size of the column cross-section exposed to fire is reduced if the column can be placedpartially or completely inside the fire retardant material used in the wall structure. The materialsused in the wall structure at column locations must be fire-resistant in order to be able to taketheir protective effect into account in the fire design. A problem may be the connection of thebracing members to the column placed inside the wall.


150

Example 32Calculate the resistance of columns (180 x180 x 5) to axial loads in the building shownin the adjacent figure. In a fire situation, thecompressive load on a hollow section is Efi.d = Nfi.Ed = 550 kN.

The fire resistance period required for thebuilding is 15 min. The steel grade used isS355J2H and the buckling length of thecolumn is Lfi 4,0 m. The temperatureevolution in the fire compartment isdetermined with the standard time-temperature curve [Formula (5.1)].

Development of temperature in an unprotected hollow sectionThe temperature increase of unprotected steel is obtained from the formula (5.2):

The net heat flux per area consists of convection and radiation:

By replacing the material constants and the section factor for steel in the formula (5.2),we obtain:

A

Vm

cJ

kgK

t s

m

a

a

=

=

=

=

−205

600

7850

5

1

ρ kg

m3

∆

(Appendix 9.6)

˙ ˙ ˙

˙

˙ ,

, , ,

. . . . .

.

.

h h h

h

h

net d n c net c n r net r

net c c g m g m

net r res g m

g m

= ⋅ + ⋅

= −( ) = −( )= ⋅ ⋅ +( ) − +( )[ ]= ⋅ ⋅ ⋅ +( ) − +( )

−

−

γ γ

α θ θ θ θ

ε θ θ

θ θ

25

5 67 10 273 273

1 0 0 5 5 67 10 273 273

8 4 4

8 4 4

Φ

[[ ]

∆ ∆θρa t

m

a anet d

A

Vc

h t. .˙=

⋅⋅


151

15000

5000

4000

600

Figure 5.9 presents the evolution of an unprotected 180 x 180 x 5 hollow section in astandard fire. The curve is calculated using the formula above with time steps of 5seconds. The maximum temperature conforming to the required fire resistance period(15 min) is:

Development of temperature in a protected hollow sectionA column of 180 x 180 x 5 is protected with 15 mm mineral wool boards. Thetemperature increase of the fire protected steel structure conforms with the formula(5.10):

Calculate the properties of the fire retardant material and steel using the formula above.

A

V

cJ

kgK

d

W

mK

kg

m

c

cd

A

V

c

p

a

p

a

p p

a ap

p

p

=

=

=

=

=

=⋅⋅

= ⋅⋅

⋅ =

=

210

600

15

0 25

7850

1000 150

600 78500 015 210 0 1003

1000

3

m

mm

t = 5 s

J/kgK

= 150 kg/m

-1

p

3

∆

λ

ρ

φρρ

ρ

,

, ,

The parameter φ is determined with the formula (5.11):

(for simplicity, heat conductivity is assumed constant)

(appendix 9.6)

∆ ∆ ∆θλ θ θ

ρ φ θφ

a t

pp

g t a t

p a a

g t

A

V

d ct e.

. .

.=−( )

⋅ ⋅ +

− −

≥

13

1 010

θa C.max = °673

∆ ∆

∆ ∆

∆

θρ

θ θ θ θ

θ θ θ θ θ

a t

m

a anet d

g m g m

a t g m g m

A

Vc

h t

t t

. .

.

˙

, ,

, ,

=⋅

⋅

= ⋅ −( ) + ⋅ +( ) − +( )[ ]= ⋅ −( ) + ⋅ +( ) − +( )[ ]

− −

− −

1 088 10 1 2339 10 273 273

5 4406 10 6 1696 10 273 273

3 12 4 4

3 12 4 4


152

By replacing the values of material properties and the parameter φ in the formula(5.10), the following expression is obtained:

Figure 5.9 shows the increase in temperature of a 180 x 180 x 5 hollow section protectedwith 15-mm mineral wool boards during a standard fire. The curve is calculated usingthe formula above with time steps of 5 seconds. The maximum temperature conformingto the required fire resistance period (15 min) is:

θa.max = 301 °C

Figure 5.9 Increase in temperature of unprotected hollow sections and hollow sections protected with mineral wool boards (t=15 mm) of dimensions 180 x 180 x 5

Calculating structural strengthWhen calculating the compression resistance, the changes in the steel strength andmodulus of elasticity caused by temperature must be taken into account. The strength isdetermined by the maximum temperature during fire resistance period. The adaptationfactors for strength and the modulus of elasticity ky.θ and kE.θ are determined by linearinterpolation from Table 5.1. The results are listed in Table 5.4.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

0

100

200

300

400

500

600

700

800

0

t (min)

Tem

pera

ture

(°C

)

Fire compartment temperature

Unprotected hollow sectiontemperature

Fire protected hollow sectiontemperature

∆ ∆θ θ θ θa t g t a t g t. . . ., ,= −( ) −0 003595 0 01008


153

Table 5.4 Compression strength of columns in a fire situation

The column can resist a 15 minute fire if a 15 mm layer of mineral wool is used, becausethe compression resistance at 301°C is:

Critical temperature by the degree of utilizationThe fire resistance period for the above structure can also be determined by the degreeof utilization. Substituting values in formula (5.14), we obtain:

Using the critical temperature, we obtain the same result as above:

Unprotected hollow section: θa.max = 673 °C > θa.cr = 505 °CFire protected hollow section: θa.max = 301 °C < θa.cr = 505 °C OK !

N Af

N

N

fi Rd fiy

fi Ed

fi Rd

a cr

. . .

.

. .

. ,

,,

,,

,,

, ,

0 0

0

03 833

1 20 702 3436

355

1 2713 6

550

713 60 771

39 19 1 482 39 190

1 482

505

= ⋅ = ⋅ =

= = =

= −

+ =⋅

−

+

= °

χ

µ

θµ

kN

ln1

0,9674 ln

1

0,9674 ,771

C

0

3,833

The critical temperature is obtained from formula (5.13)

N A kf

N

fi Rdfi

yy

M fifi Ed

fi Rd fi Ed

. . ..

.

. . .

,

,

,,

,,

,

θ θ

θ

χγ

=

⋅ =

⋅ = =

= =

1 2

0 647

1 23436 1 0

355

1 0657 7 550

171 4 550

kN > N kN

kN < N kN

An unprotected column does not meet the fire resistance requirements, because, becau-se the compression resistance at 673°C is:

Fire situation θa.max ky.θ kE.θ λfi χfi Nfi.θ.Rd (kN)°C Formula(5.21) Formula(5.20)

Unprotected 673 0,2948 0,1786 0,946 0,572 171,4hollow section

Fire protected 301 1,0 0,7990 0,824 0,647 657,7hollow section


154

5.7 Fire design of concrete-filled columnsThe strength of a concrete-filled column in a fire situation is based on the slower heating of theconcrete filling and the reinforcement. The hollow sections walls and the external casing ofconcrete filling heats rapidly as the fire compartment temperature increases, but thereinforcement and the inner part of the concrete filling retain the normal temperature for alonger period (Figure 5.10).

Figure 5.10 Decrease in yield strength of a concrete-filled column in a fire situation

When designing concrete-filled columns, it is recommendable that hollow sections with largeexternal dimensions and thin walls are used. In this way, the portion of the column's concrete-filledinterior area is large. In a fire situation, the concrete and the reinforcement bear more load in alarger column, since the interior temperature is lower and the strength is higher (Figure 5.11).

Figure 5.11 Temperature distribution in cross-sections of concrete-filled hollow sections of various sizes

��

R = 100 mm

R = 150 mm

90 min

60 min

30 min

y/R

tem

pera

ture

0 0,25 0,50 0,75 1,0

R

y

reinforcement

concrete

steel

time

Rel

ativ

e yi

eld

stre

ngth


155

5.7.1 Using tables in the fire design of concrete-filled columns

A simple conservative design method is presented in Table 5.5 which lists the minimumdimensions for concrete-filled columns. Minimum dimensions depend on the degree ofutilization ηfi.t, calculated from the formula [5]:

where

In addition, reference [5] presents theoretical formulae for the fire design of concrete-filledcolumns.

An alternative design method for concrete-filled columns is to use pre-calculated design tables.Reference [4] lists the buckling strength values in a fire situation for hollow sections withvarious reinforcement ratios and strength values of concrete. The tables include bucklingstrength values for fire resistance periods of 60, 90 and 120 minutes.

N

Nfi Ed

b Rd

.

.

is the design value for axial force in a fire situation

is the buckling strength of a concrete-filled column in normal temperature [6]

η fi tfi Ed

b Rd

N

N.

.

.= ≤ 1 25(5. )


156

Table 5.5 Minimum dimensions for columns in a fire situation [5]

Fire resistance period

As is the area of reinforcementAc is the area of concrete filling

Degree of Minimum dimension R30 R60 R90 R120 R180

utilizationηfi.t

0,3 b or d (mm) 160 200 220 260 400

Reinforcement ratio As / (Ac+ As) (%) 0 1,5 3 6 6

Reinforcement position us (mm) - 30 40 50 60

0,5 b or d 260 260 400 450 500

Reinforcement ratio As / (Ac+ As) (%) 0 3 6 6 6

Reinforcement position us (mm) - 30 40 50 60

0,7 b or d 260 450 500 - -

Reinforcement ratio As / (Ac+ As) (%) 3 6 6 - -

Reinforcement position us (mm) 25 30 40 - -

- when calculating the degree of utilization, the assumed yield strength value of steel is 235 N/mm2

- the steel grade used as reinforcement is S500- when calculating the buckling resistance Nb.Rd in normal temperature, greater reinforcement ratios than 3% are not taken into account- the following conditions must be met: b/ t ≥ 25 and d/ t ≥ 25

��

��

us

us

t

As

Acb d

h


157

5.8 References

[1] ENV 1991-2-2: Eurocode 1. Suunnitteluperusteet ja rakenteiden kuormat: Osa 2-2 Palollealtistettujen rakenteiden kuormat, 1995(ENV 1991-2-2: Eurocode 1. Basis of design and actions on structures. Part 2-2: Actionson structures. Actions on structures exposed to fire, 1995)

[2] ENV 1993-1-2: Eurocode 3: Teräsrakenteiden suunnittelu: Osa 1-2: Rakenteellinenpalomitoitus, 1996(ENV 1993-1-2: Eurocode 3: Design of steel structures. Part 1-2: General rules. Structuralfire design, 1996)

[3] ECCS: Tecnical Committee 3- Fire safety of steel stability: Design manual on the Europeanrecommendations for the fire safety of steel structures, First edition, 1985

[4] CIDECT: Design guide for structural hollow section columns exposed to fire, Verlag TÜVRheinland GmbH, Köln 1994

[5] ENV 1994-1-2: Design of composite steel and concrete structures: Part 1.2: Structural fire design, 1994

[6] ENV 1994-1-1: Design of composite steel and concrete structures:Part 1-1: General rules and rules for buildings, 1994

[7] ENV 1991-1: Eurocode 1: Suunnitteluperusteet ja rakenteiden kuormat:Osa 1: Suunnitteluperusteet, 1995(ENV 1991-1: Eurocode 1: Basis of design and actions on structures. Part 1: Basis ofdesign, 1995)


158

6 DESIGN OF HOLLOW SECTION STRUCTURES

A structural hollow section is a versatile structural element suitable for various parts of abuilding. Hollow sections show their best characteristics in columns and lattice structures. Thedesign of structures made of hollow sections is uncomplicated since, due to their good torsionalstiffness, lateral-torsional buckling and torsional buckling are usually not governing factors. Thisalso enables the efficient utilization of design software. In this chapter, the design procedure ofhollow section structures is handled in its entirety. First, the central issues affecting the designsolutions are handled in detail, after which a design solution for a model building is presented.

Figure 6.1 The model building

The model building is an exhibition hall shown in Figure 6.1. The building is used for arrangingfairs and meetings.

The frame of the building consists of hollow section columns and of primary and secondarylattices. The building is stiffened with a horizontal lattice located in the roof and by wind bracingin the walls.

Hollow sections were chosen as columns, because the wall structure is constructed of light-weight wool elements which do not support the columns about the minor axis. Hollow sectionshave high torsional and bending stiffness about the minor axis, which makes them a goodsolution in this case.

10000048000

1450

0

1050

0

door beam

primary frame column

eaves beam

gable beam


159

Usually, it is advantageous to select a normal K, N or KT type lattice as the roof-supportingstructure. In these structures, the chords are hollow sections and the brace member joints areuncomplicated. Usually the full depth of the roof height would be used for the lattice girder. Inthis case the height between the eaves and the apex is 4,0 m, which would provide an efficientlattice girder or truss. Due to transport requirements however the maximum depth of prefabri-cated unit must not exceed 2,5 m; therefore a reduced depth parallel sides N girder has beenused, which follows the roof profile. A tie is provided at eaves level across the widtt of the build-ing. See figure 6.1

A frame spacing of 10 m is chosen, as it produces an efficient solution for the model building.Between the primary frames, there are purlin trusses with a spacing of 4 m, which enables theuse of a shallow profile for the roof. In addition, the purlin trusses are an easy way to providelateral restraint to the upper and lower chords of the primary lattice.

The building is braced using horizontal lattices in the roof plane (Figure 6.2). This solutionproduces smaller foundations and external column dimensions than using a rigid portal solu-tion. Hollow sections are used as bracing members due to their excellent compressionresistance.

Figure 6.2 Horizontal stiffening in the model building

On the external walls, there are bracing elements constructed from hollow sections, functioningas tension members. Here, too, hollow sections are an efficient solution, as their stiffness facili-tates installation and the elements retain their shape well. However, regarding the foundations,the best solution would be to dimension the bracing elements both as tension and compressionmembers, as the load on the foundations can then be divided into two (Figure 6.3).

Static model of the column

bracing lattice


160

Figure 6.3 Stiffeners at the end wall of the building (load in the rafter is transferred to the tension diagonal)

The wind columns on the side walls of the building, carrying wind load, are supported at theupper end by roof bracing to make them non-sway. The supporting force at the upper end ofthe wind columns is transferred to primary frames through the roof profile.

6.1 Structural actions

The actions to which the structure is subjected are divided into permanent, variable andaccidental actions. Permanent actions include:

- self-weight of the structure- fixed equipment

Variable actions include:

- imposed loads- snow loads- wind loads

Accidental actions include:

- fire loads- seismic loads

The design values of the loads are used in structural design. A design value is obtained bymultiplying the characteristic value of the load by the partial safety factor.

For load calculation, only the formulae applicable to the model building are shown. Loadcalculation is handled in more detail in references [1], [2], [3] and [4].


161

compression membertension member

apex

rafter

6.1.1 Self-weight and imposed loads

Design values for weights of materials and for imposed loads are given in reference [1]. Theweight of the partition walls can be distributed to generate a uniform load. When designing thestructural floor elements of a single storey building, the load must be taken into account in theweakest area. The effect of concentrated load must be considered separately. In some cases,the imposed loads on the, structural floor in a single storey building, can be reduced. [1].

When designing columns, the loads on storeys are assumed uniformly distributed. Also in thecase of multi-storey buildings, imposed loads can sometimes be reduced [1].

6.1.2 Snow load

The characteristic value of the snow load depends on the geographical location and shape ofthe building. The following equation is obtained for the characteristic value of snow load on roofstructure [2]:

where

National application documents (NADs) may present different methods for calculatingthe characteristic value of the snow load.

Reference [2] gives characteristic values for a snow load on the ground and snow load formfactor values.

6.1.3 Wind load

As in the case of snow load, the magnitude of wind load depends on the geographical locationof the building and on the shape of the structure. The total force due to wind can be expressedas follows [3]:

where

is the exposure coefficient (Figure 6.4) dependent on ct and cr

is the topography coefficient [3]

is the roughness coefficient [3]

is the dynamic pressure coefficient [3]

is the pressure coefficient [3]

is the area perpendicular to the wind

ccccc

A

e

t

r

d

p

ref

F q c c c c c Aw ref e t r d p ref= ⋅ ( ) ⋅ ⋅ ⋅, (6.2)

is the snow load shape coefficient

is the characteristic value of the snow load on the ground (kN/ m2) depending on

the geographical location of the building [2].

µi

ks

s si k= ⋅µ (6.1)


162

Table 6.1 Terrain categories

Figure 6.4 Exposure coefficient ce when ct = 1,0 (building stands on level ground)

The formula for calculating the exposure coefficient ce when ct is not equal to 1,0 is given inreference [3].

The values of the pressure coefficient cp for wall structures with wall area Aref greater than 10m2 are shown in Table 6.3. The values for cp with Aref smaller than 10 m2 are given inreference [3].

1,4 1,6 1,8 2 2,2 2,4 2,6 2,8 3 3,2 3,4 3,6 3,8 4

0

10

20

30

40

50

60

70

0

Hei

ght o

f bui

ldin

g (m

)

ce

Terra

in ca

tego

ry IV

Terra

in ca

tego

ry II

I

Terra

in c

ateg

ory

II

Terra

in c

ateg

ory

I

Terrain category Description

I Rough open sea, lakes with at least 5 km fetch upwind and smooth flat country withoutobstacles

II Farmland with boundary hedges, occasional small farm structures, houses or treesIII Suburban or industrial areas and permanent forestsIV Urban areas in which at least 15% of the surface is covered with buildings and their

average height exceeds 15 m

ρ = 1,25 kg/mm3 (air density)

vref is the average wind velocity in a 10-minute period measured at the distanceof 10 m above the ground in terrain category II (Table 6.1). The annual probabilityfor exceeding the wind velocity vref is 2% [3]. National application documents(NADs) may include different methods for calculating the characteristicvalue of wind load.

q vref ref= ⋅0 5 2, ρ


163

Table 6.2 Pressure coefficient cp in vertical walls (Aref > 10 m2)

Figure 6.5 Pressure coefficients for the wall

6.1.4 Additional horizontal forces

In addition to wind load, horizontal forces in the structure are generated by eccentricities andinstallation tolerances. A further factor to be taken into account in the design are the horizontalforces transmitted from structural member in compression to the members proving restraint.

Transverse force due to a compression structural element

The load caused by a compression structural member in compression (e.g. the upper chord ofthe lattice) on restraining members (e.g. horizontal diagonal members) is determined as follows[5]:

qN

Lwhen

L

qN

Lwhen

L

= ≤

= +( ) >

50 2500

1

60 2500

q

q

δ

α δ

(6.3)

(6.4)

A

b

d 0,8e

0,2e

d-e

d-0,2e

0,2e

1. d > e

dB

C

A

B

C

D

E

2. d < e

A

B

D

Eb

wind wind

A

B

e is min (b, 2h)h is the height of the buildingb is the width of the building

Wall A B C D E

d/h ≤ 1 -1,0 -0,8 -0,5 0,8 -0,3

d/h ≥ 4 -1,0 -0,8 -0,5 0,6 -0,3

The intermediate values can be determined by linear interpolation


164

where

The use of the formula (6.4) leads to iteration. Formulae (6.3) and (6.4) are applicable forstiffening system that support one member only. When there are several supported members,the horizontal load is determined as follows [5]:

where

6.1.5 Combined loads

Chapter 2 presents combined loads in the ultimate limit state [4] [formulae (2.2a, 2.2b and2.3)]. In the serviceability limit state, the equation for combined loads is the following [4]:

ψ0, ψ1 and ψ2 are load combination factors obtained from Eurocode 1 [4] or from nationalapplication documents (NADs).

Permitted deflections for various structural elements are presented in Table 6.3.

G Q Q

G Q Q

G Q

k j kj

ii

k i

k j kj

ii

k j

k j ii

k ij

. . . .

. . . . .

. . .

+ + ⋅

+ ⋅ + ⋅

+ ⋅

∑ ∑

∑ ∑

∑∑

>

>

≥

1 01

1 1 1 21

21

ψ

ψ ψ

ψ

(6.7)

(6.8)

(6.9)

(rare combination)

(normal combination)

(long-time combination)

ΣN

kn

n

rr

r

= +0 21

,

is the sum of the compressive forces of members

is the number of members

q Nk

Lwhen

L

q Nk

Lwhen

L

r

r

= + ≤

= + >

Σ

Σ

0 2

60 2500

60 2500

,

q

q

δ

α δ

(6.5)

(6.6)

NLδ

αδ

q

q= ≥500

0 2L

,

is the the axial compression

is the length of the compression member

is the horizontal deflection in the stiffening system caused by the force q and

external loads


165

Table 6.3 Permitted deflections [5]

6.1.6 Load determination in the model building

Self-weightThe self-weight of the purlin trusses and roof is estimated at Gk = 0,5 kN/ m2.

Snow load

The characteristic value of thesnow load on the ground in thebuilding area is 1,5 kN/ m2. Theroof angle is 1:6 (α = 9,5°). Fora pitched roof, the shape coeffi-cients µ1 = µ2 = 0,8, when 0° ≤ α1

= α2 ≤ 15°. Of the load combi-nations in Figure 6.6, the mostonerous one is chosen. In thiscase, the snow load is determinedby the combination of a) and c),since the structure is symmetrical.

s = µ · sk = 0,8 · 1,5 = 1,2 kN/m2

Structure Recommended limits for deflection

Horizontal deflection

Frame structure without crane gantry rails h/ 150

Other one-story building h/ 300

One story in a multi-story building h/ 300 (h is the height of one story)

Total height of a multi-story building h0/ 500 (h0 is the height of the building)

Vertical deflection

δmax δ2

Roofs in general L/ 200 L/ 250

Roofs with person load L/ 250 L/ 300

Intermediate floors in general L/ 250 L/ 300

Intermediate floors supporting brittle structures L/ 250 L/ 350

Intermediate floors supporting columns L/ 400 L/ 500

Deflection which can affect the appearance of the building L/ 250

δmax = end deflection

δ0= pre-camber

δ1 = deflection due to permanent loads

δ2 = deflection due to variable loads

δ 1δ 2

δ 0δ m

ax


166

Figure 6.6 Snow loads on a pitched roof

a)

b)

c)

d)

α1 α2

µ1, α2

µ2, α2

0,5 µ1, α2

µ2, α2

0,5 µ1, α1

µ1, α1

Wind load

In the building area, the wind velocity vref = 23 m/s. The reference mean velocitypressure qref is determined from the formula:

In terrain category III, the exposure coefficient ce has the following value with a buildingheight of 10,5 m measured at the eaves level (Figure 6.4):ce = 1,816

Thus the basic value for the wind load is obtained using the formula (6.2):

The following pressure coefficient values are obtained for the model building (Figure 6.5):

Wind parallel to the side wall (d/h = 9,52 > 4) A = – 1,0, B = – 0,8, C = – 0,5, D = +0,6, E = – 0,3

Wind parallel to the end wall (d/h = 4,57 > 4) A = – 1,0, B = – 0,8, C = – 0,5, D = +0,6, E = – 0,3

Reference [3] also shows separate pressure coefficients for roof structures.

6.2 Designing columns

As was shown in Chapter 2, a hollow section is an efficient cross-section for a column. Themass of a hollow section is located far from the centre, so the radius of gyration of the hollowsection is relatively large in all directions. In column design, the essential factors are thebuckling length, the effect of the joint stiffness and the column-to-foundation joint.

6.2.1 Column buckling length

The buckling length of the column is influenced by the actual length, the fixing method of theends and the lateral support to the member. Theoretical buckling lengths for columns arepresented in Table 6.4.

q q c c c c cwk ref e d p p p= ⋅ ⋅ ⋅ = ⋅ ⋅ =0 33 1 816 1 0 0 6, , , ,

q vref ref= ⋅ = ⋅ ⋅ =0 5 0 5 1 25 23 0 332 3, , , ,ρ kN

m2


167

Table 6.4 Theoretical buckling length Lc of compression members

In frame structures with rigid joints, the benefits of structural hollow sections can be utilizedwhen determining the column buckling length values. Another factor influencing the bucklinglength in frames is the lateral support of the frame. A non-sway structure can be stiffened eitherwith lattices or by supporting it with a rigid structural element (a lift shaft or a stair well).Generally speaking, a frame structure can be classified non-sway if the following condition ismet [5]:

where

The stiffening of a sway structure is based on columns functioning as cantilevers and fixed tofoundations with a rigid joint, or on the rigidity of the joints.

In the case of a continuous column, the buckling length can be determined using Figures 6.7and 6.8. The distribution factors η1 and η2 in the figures are determined as follows [5]:

η

η

11

1 11 12

22

2 21 22

1

2

= ++ + +

= ++ + +

K K

K K K K

K K

K K K K

c

c

c

c

(6.1 )

(6.1 )

(upper assembly point)

(lower assembly point)

is the design value of the vertical total load

is the buckling load according to the frame elasticity theory in case of sway

buckling mode

VV

Sd

cr

V

VSd

cr≤ 0 1 0, (6.1 )

Pinned at both ends

Fixed at one end Fixed at both ends Fixed at both ends, one sway joint

Fixed at one end,pinned at the other

Lc = 1,0 L Lc = 2,0 L Lc = 0,5 L Lc = 1,0 L Lc = 0,7 L

L L L L L


168

where

Table 6.5 Effective stiffness coefficients K for hollow sections [5]

In Table 6.5, the moments in the hollow section are assumed to be elastic (MSd < Wel · fy /γM0).The hollow section is assumed pinned if its moment exceeds the elastic moment [5].

The buckling length of columns in rigid jointed structures is obtained from Figure 6.7 for non-sway frames and from Figure 6.8 for sway frames. The curve values represent the relation ofbuckling length to the actual column length.

Conditions of rotational restraint at the far end of the hollow section Effective stiffness coefficient

Fixed 1,0 Ib / Lb (1-0,4 N / Ne)

Pinned 0,75 Ib / Lb (1-1,0 N / Ne)

Rotation as at near end (double curvature) 1,5 Ib / Lb (1-0,2 N / Ne)

Rotation equal and opposite to that of near end (single curvature) 0,5 Ib / Lb (1-1,0 N / Ne)

Ib is the hollow section’s moment of inertia parallel to frame

Lb is the length of the hollow section

N is the compressive force of the hollow section

Ne = π2 · E · Ib / Lb2

KI

L

KI

L

KI

LI I

L LK K K K

c =

=

=

11

1

22

2

1

1

11 12 12 22

,

,,

and I

and L, and

2

2

are the values of inertia forcorresponding columns parallel to frame

are the effective stiffnesscoefficients for corresponding hollow sections (Table 6.5)

are the values of height for corresponding columns


169

K1

K2

K11 K12

K21 K22

Kc

η1

η2

column in question

Figure 6.7 Column buckling length values of non-sway frames Lc /L [5]

Figure 6.8 Column buckling length values of sway frames Lc /L [5]

5,0

= ∞

4,03,02,82,62,42,2

2,01,91,81,71,6

1,51,4

1,31,251,2

1,151,1

1,051,0

LcL

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0rigid

rigid

pinn

edη1

η2

pinned

1,0

0,9

0,8

0,7

0,6

0,5

0,4

0,3

0,2

0,1

0

= 1,00,95

0,9

0,85

0,8

0,75

0,70,6750,65

0,6250,6

0,5750,55

0,5

0,525

LcL

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0rigid

rigid

pinn

edη1

η2

pinned

1,0

0,9

0,8

0,7

0,6

0,5

0,4

0,3

0,2

0,1

0


170

6.2.2 Effect of joint stiffness on column buckling length

The joints of a frame structure can be considered rigid if the joints are stiffened as shown inFigure 6.9. A non-stiffened joint must be considered semi-rigid when determining the columnbuckling length. The calculation of the rigidity for non-stiffened hollow section frame structuresis dealt with in Appendix 9.5. The effect of a semi-rigid joint on the column buckling length(formulae 6.11 and 6.12) is accounted for in the value of effective inertia of the hollow section.This is determined by the following formula [6]:

where

The stiffness of the joint varies according to the applied moment, as the increasing momentcauses the plastification of the joint components that are subjected to the greatest loads. Thetotal moment-rotation curve should thus be known in order to utilize the effect of semi-rigidjoints on the calculation of the column buckling length. In case this is not known, theassumption that the support of the column is pinned at the location of the joint is conservative.

Figure 6.9 Stiffened hollow section frame structures

� ��

h0,75b0

2,5h

3h

≥0,75b0

M M

M M

0,85h

1,4t

t

h

t

t

h

≥0,75b0

b00,7h

h

is the stiffness of the joint (Nm/rad)

is the inertia of the hollow section parallel to frame

is the length of the hollow section

SIL

j

b

b

I E I

S L

Ib effb

j b

b. =+ ⋅

⋅

1

13 3(6.1 )


171

6.2.3 Column-to-foundation connections

Connections between the column and the foundation were dealt with in Section 3.5. If a rigidcolumn-to-foundation connection is assumed in the structural model the moments transferredinto the foundation by the column must be accounted for when designing the holding downbolts and the base plate. If a pinned joint is assumed in the model, moments need not be takeninto account.

The holding down bolts must be designed such that they are able to carry the constructionloads the column is subjected to. The thickness of the second stage concrete layer is taken intoaccount when calculating the buckling length for the design of the holding down bolts.

6.2.4 Column design in the model building

Design the columns in a primary frames. The column-to-foundation connection is in thiscase assumed rigid. The moment transferred from the column to the foundation must thusbe taken into account when designing the connection. The horizontal loads on thebuilding are carried by the bracing lattices, so the columns are supported by hinges atthe upper end. The buckling length of the columns can thus be obtained directly fromTable 6.4, giving Lc.y = 0,7· 10,3 = 7,21 m.

Force quantities

The forces on the columns are determined simply by the area of load carried. Theresistance must be checked separately for two different load combinations, since at thisstage it is not known whether the dominant load is the snow load or the wind load.

In case the wind load is dominant:

(wind load)

(snow load)

where

is the frame spacing

is the width of the building

L

Bf

G GkN

m

N G s L B kN

d G k

Sd d d f

= ⋅ = ⋅ =

= +( ) ⋅ = +( ) ⋅ =

γ . . , , ,

, , , ,

1 1 21 35 0 5 0 68

0 5 0 5 0 68 1 08 10 48 422

(self-weight)

M kNm

V kN

s QkN

m

sd

sd

d Q k

= =

= ⋅ ⋅ =

= ⋅ ⋅ = ⋅ ⋅ =

3 610 3

847 7

5

83 6 10 3 23 2

0 6 1 5 1 2 1 08

2

0 2 2 2 2

,,

,

, , ,

, , , ,. . .

ψ γ

(at the lower end of the column)

q q LkN

mwd Q wk f= ⋅ ⋅ = ⋅ ⋅ ⋅ =γ . , , , ,2 1 5 0 8 0 6 5 3 6


172

10,3

m

qwd

NSd

In case the snow load is dominant:

For the column in this example, the equivalent uniform moment factors shown in Table 2.9 are obtained as follows:

The final value for the equivalent uniform moment factors is obtained from the following formula (Table 2.9):

The column is designed using the formula shown in Section 2 (2.57). Try a hollowsection with dimensions 300 x 200 x 6 and steel grade S355J2H. The cross-section of thehollow section is Class 4, as h/ t= 300/ 6 = 50 > 36,6. The calculation of the effectivecross-section and the determination of column compression and bending resistance aredetailed in Chapter 2, so only the results are presented here (Appendices 9.1 and 9.2):

N N kN

N kN

M kNm

b Rd b z Rd

b y Rd

y Rd

. . .

. .

.

,

,

= =

=

=

815 3

1090

189 7

(about the z axis)

(about the y axis)

β β β ψM yQ

MQ MM

M. ,

, , , ,

= +

⋅ −( )

= +⋅

⋅ −( ) =

1 8

1 8128

8 251 3 1 8 1 48

∆

M q L

M qL

Q

MQ

M

= ⋅

= +

⋅ = ⋅

=

=

1

81

8

9

128

25

128

1 3

1 8

2

2∆ q L2

β

β ψ

,

,

(uniform transverse load)

(restraint moment at the lower end of the column)

q q LkN

m

M kNm

V kN

s QkN

m

G

wd Q wk f

Sd

Sd

d Q k

d G

= ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ =

= ⋅ =

= ⋅ ⋅ =

= ⋅ = ⋅ =

= ⋅

ψ γ

γ

γ

0 2 2

2

1 1 2

1

0 6 1 5 0 8 0 6 5 2 16

2 1610 3

828 6

5

82 16 10 3 13 9

1 5 1 2 1 8

. .

. .

.

, , , , ,

,,

,

, , ,

, , ,

GGkN

m

N G s L B kN

k

sd d d f

. , , ,

, , , ,

1 21 35 0 5 0 68

0 5 0 5 0 68 1 8 10 48 595

= ⋅ =

= +( ) ⋅ = +( ) ⋅ ⋅ =

(wind load)

(snow load)

(self-weight)


173

��

∆M

M0

z

y

Next, determine the value for parameter ky. The wind load and the snow load arecalculated separately, since NSd affects the value of parameter ky .

In case the wind load is dominant, [formula (2.58)]:

In case the snow load is dominant, [formula (2.58)]:

Now we have determined the necessary parameters. The column resistance can bechecked using the following condition (2.57):

Regarding shear resistance, the wind load is dominant. We need to calculate the plasticshear resistance, since h / t = 47,6 < 59,1 (Section 2.4.1 and Appendix 9.1):

Thus, the resistance of a 300 x 200 x 6 hollow section is sufficient.

6.2.5 Designing the column-to-foundation joint in the model building

The column-to-foundation joint is made as shown in Figure 6.10. Since the joint wasassumed rigid, it is designed according to the guidance in Section 3.5.2.

Force quantities

The dominant forces and moments are:

M kNm

N kN

N kN

V kN

Sd

Sd

Sd

Sd

===

=

47 7

162

422

23 2

,

,

.min

.max

(construction loads)

V V kNpl Rd Sd. , ,= > =644 3 23 2 OK!

N

N

k M

M

N

N

k M

M

Sd

b Rd

y y Sd

y Rd

Sd

b Rd

y y Sd

y Rd

.

.

.

.

.

.

,

, ,

,,

,

, ,

,,

+⋅

= + ⋅ = <

+⋅

= + ⋅ = <

422

815 3

1 246 47 7

189 70 831 1

595

815 3

1 347 28 6

189 70 933 1

OK ! (if wind load is dominant)

OK ! (if snow load is dominant)

µ λ β

µχ

y y M y

y ySd

y yk

N

A f

= −( ) = ⋅ −( ) = −

= −⋅ ⋅

= +⋅ ⋅

= <

2 4 0 776 2 1 48 4 0 807

1 1 0 807595000

0 677 5763 3551 347 1 5

. , , ,

,,

, ,

µ λ β

µχ

y y M y

y ySd

y yk

N

A f

= −( ) = ⋅ −( ) = −

= −⋅ ⋅

= +⋅ ⋅

= <

2 4 0 776 2 1 48 4 0 807

1 1 0 807422000

0 677 5763 3551 246 1 5

. , , ,

,,

, ,


174

The column location is eccentric to the base plate (e1=75 mm), so the moment due toaxial force must be taken into account:

(This moment direction is dominant in the design of the base plate.)

Designing the holding down bolts

The design value for the compressive resistance ofconcrete in the foundation is fcd = 14 N/ mm2. Thewidth of the compression area in the concrete isdetermined by the following formula (example 28):

The axial force in the bolts is determined from thevertical equilibrium condition (example 28):

Ns = by · fcd – NSd = 400· 8,29· 14 – 422000 = –395,6 kN ⇒ no tension in holding down bolts

During installation, however, the holding downbolts carry the total load the column is subjectedto. The compressive force of a holding down boltdue to construction loads is:

(The direction of the dominant moment is now reversed as compared to the base platedesign. e2 is the eccentricity of the column to the group of holding down bolts.)

Design the holding down bolts of steel grade S355J2 for shear force and axial force(example 28):

Considering Ø 30 ⇒ As = 561 mm2:

FA f

kNV

OK

ØN

mm

v Rds y

M

Sd.

,, !=

⋅⋅

= ⋅⋅

= >

⇒ =

3

561 345

3 1 1101 6

4

345

0

2

γ

> 16 mm f y

(shear resistance per bolt)

FN M N e

kNc SdSd Sd Sd

..min .min,

,,

, ,

,,= + + ⋅ = + + ⋅ =

40 5

0 3

162

40 5

47 7 162 0 1

0 3147 02

yM

b f dd

mm

Sd tot

cd

= − −⋅ ⋅

= − − ⋅⋅ ⋅

=

1 12

1 12 16 05

400 14 0 35350 8 29

2

2

.

,

,,

M M N e

kNm

Sd tot Sd Sd.

, , ,

= − ⋅

= − ⋅ =1

47 7 422 0 075 16 05


175��

MSd

225150 e1=75

450

300

200

30040

0

300100 100

Figure 6.10 Base plate

e2= 100

NSd

t p

300

The buckling length of the holding down bolts is equal to the thickness of the secondstage concrete layer, that is, 70 mm:

The size of holding down bolts is usually between Ø 24-36 mm. The combined loadcriterion for a holding down bolt subjected to shear force and axial force is expressed asfollows:

Designing the base plateThe resistance of the base plate is determined using the value of contact pressure(example 28):

Using the previously calculated stresses, obtain the value for the base plate bendingmoment at the column edge as follows:

Thus, the resistance of the base plate tp = 30 mm and that of the holding down bolts ofØ 30 are sufficient.

M pa

p pa

b kNm

tM

b fmm p mm

t

Sd

pSd MO

y

p

= + −( )

= + −( )

=

≥ ⋅ ⋅⋅

= ⋅ ⋅⋅

= ⇒ =

> ⇒ =

21

2

1 212 2 2

2 33 53

150

23 53 2 74

150

3400 18 26

6 6 18 26 1 1

400 34529 6 30

16 345

, , , ,

, ,,

rovide t

mm f

p

y

γ

NN

mm2

The thickness of the base plate is obtained by substituting the bending moment MSd

in the formula (3.40):

pN

a b

M

a b mm

p pM

a b

a

a

N

mm

Sd Sd tot

Sd

1 2 2 2

2 1 21

2 2

6 422000

450 400

6 16050

450 4003 53

123 53

12 16050

450 400

150

4502 74

=⋅

+⋅

=⋅

+ ⋅⋅

=

= −⋅

= − ⋅⋅

=

. ,

, ,

N

(at the base plate edge)

(at the column edge)

F

F

F

Fv Sd

v Rd

c Sd

b Rd

.

.

.

.

, ,

,

,

,,+ = ⋅ + = <0 25 23 2

101 6

147 0

176 00 612 1 OK!

L

A fkN F

c

s y

MOc Sd

=

⇒ ⇒

= =⋅

= ⋅ = >

70

561345

1 1176 0

mm

i = 7, 5 mm = 0,12 < 0, 2 = 1,0

F F b.Rd t.Rd

λ χ

γ ,, . OK!


176

6.3 Designing the hollow section beams

When designing a beam, the use of plastic theory is recommended whenever feasible.Plasticity theory can be used when calculating force quantities for Class 1 cross-sections andwhen calculating resistance values for Class 1 and 2 cross-sections.

Rectangular hollow sections are a more efficient alternative when the axial force is smallcompared to the bending moment and the bending moment is uniaxial. Even with hollow sec-tions with a high h/b ratio, the resistance for lateral-torsional buckling is rarely governing.

Allowing for the continuity of the hollow sections reduces span moments, which often makes itpossible to select a smaller hollow section size. It is thus recommended that continuous hollowsections as long as possible are used. However, the effect of shop manufacture, transport andsite installation on the length of structural elements must be taken into account. By placing thesplices appropriately, the forces affecting the joints can be kept to a minimum, which makes itpossible to also select a pinned joint.

It is recommended that the end of the hollowsection is stiffened with an end plate so thatcross-sectional deformation is prevented intransferring the reaction from the webs to thesupporting plate (Figure 6.11a). The cornerrounding in the hollow section increases the riskof buckling in bending, even when intermediatesupports are used, if the notch below the cornerrounding is not filled with the weld (Figure6.11b). The cross-section of a hollow sectiontends to become distorted if it is subjected totorsion moment. The cross-sectional distortioncan be prevented if the torsional moment istransferred to the hollow section as shown inFigure 6.11c.

The semi-rigidity of the joints can be taken intoaccount when designing hollow sections in aframe structure. However, in such a case, themoment-rotation curve of the joint must beknown, since the rigidity of the joint variesaccording to the joint moment. Reference [6]and Appendix 9.4 include equations fordetermining the moments in a hollow sectionwith semi-rigid joints at both ends and uniformload distribution. Appendix 9.4 also deals withthe estimation of rigidity in hollow section joints.


177

��

��

Figure 6.11 Preventing the distortion ofa hollow section

FF

WeldFF

F

a)

b)

c)

6.3.1 Designing gable beam in the model building

Now design the gable beam of the model building. The width of the building is 48 m andthe column spacing at the end of the building is 6 m. Divide the gable beam into fourparts which are joined to the end columns on-site using bolted joints. The gable beam isvertically loaded by the support reactions of the purlin trusses. The gable beam transfersthe transverse wind load of the hall to the bracing lattices, which means it is alsosubjected to axial force. Consider a load combination with dominant snow load. Thewind load is multiplied by the combination factor ψ0= 0,6.

Loads

The compressive force on the gable beam is assumed constant along the entire length ofthe section:

where

L is the length of the building

The total portion of the roof is included inthe value of the horizontal load, which isconservative. A more accurate result can beobtained by taking into account the effect ofthe pressure coefficients of the roof’s windload.

With dominant snow load, the following restraining force for the purlin truss is obtained:

(self-weight and snow load)

where

Lp is the purlin spacing Lf is the frame spacing

The self-weight Gk.1 can be assumed smaller (0,4 kN/ m2), since the weight of the primarylattice need not be taken into account.

F G Q L L kNy Sd G k Q k p f. . . . ., , , , , , ,= ⋅ + ⋅( ) ⋅ = ⋅ + ⋅( ) ⋅ ⋅ =0 5 0 5 1 35 0 4 1 5 1 2 4 10 46 81 1 1 1γ γ

q c c q ckN

m

N q H HL

kN

d Q p p ref e

Sd d

= ⋅ +( ) ⋅ = ⋅ ⋅ +( ) ⋅ =

= +( ) = ⋅ ⋅ +( ) =

ψ γ0 2 2 1 2 2

1 2

0 6 1 5 0 6 0 3 0 6 0 486

0 3752

0 486 0 375 10 5 4100

2193

. . . . , , , , , ,

, , , ,

(wind load)


178

H1

H2

Cp.1 Cp.2

Calculate the forces and moments using plastic theory. For vertical loading, thefollowing static model is obtained:

By using equalizing the internal and external work, the plastic moment can be determined:

Resistances at the ultimate limit stateThe resistance of the hollow section is determined from the formula (2.39), since thehollow section is also subjected to compression load. Consider a hollow section withdimensions 150 x 100 x 8 and steel grade S355J2H. The cross-section of the hollowsection is Class 1. The resistance values are as follows (Appendix 9.1):

The bending resistance, reduced by the axial force, is (Section 2.7.1.1):

In the interaction expression, there is now only one bending load, so the effect of theparameter α is omitted:

M

MSd

N y Rd. .,

= 0 686 < 1,0 OK!

M MN

N

M M

N y Rd pl y RdSd

pl Rd

N y Rd pl y Rd

. . . ..

. . . .

, , , ,

,

= −

= ⋅ −

=

⇒ = =

1 33 1 1 33 54 59 1193

113760 28

54 59

kNm > M

kNm

pl.y.Rd

M kNm

V kN V

N kN

pl y Rd

pl z Rd Sd

pl Rd

. .

. .

.

,

,

=

= >

==

54 59

394 0

1137

OK!

M F L M FL

kNm

VML kN

Sd Sd Sd Sd

SdSd

3 22

32

1537 44

2

3

37 44

θ θ θ+( ) = ⋅

⇒ = ⋅ =

= =

,

,

4000 4000 4000FSd FSd

L = 6000 L = 6000

FSd FSd

θθ 2θ 2θ

3θ 3θ


179

Check that the hollow section will not buckle before the mechanism is generated. Thebuckling length is 6 m. The buckling resistance is as follows (Appendix 9.2):

The gable beam is restrained laterally supported by the roof profile, so lateral bucklingneed not be checked. In this case, the effect of bending moment on buckling need not betaken into account, since vertical buckling leads to the expected failure mechanism.

Stresses and deflection at the serviceability limit stateCheck that the stresses do not exceed the yield strength of the material withserviceability limit state loads. The partial safety factors for loads are given by theformula (6.7):

We determine the deflection of the gable beam using elasticity theory:

Thus, the resistance of a 150 x 100 x 8 hollow section is sufficient.

6.3.2 Designing the door beam in the model building Design the side wall door beam which isjoined to the intermediate column and to thecolumn of the primary frame. The height of thedoor is 5 m and its self-weight is 0,75 kN/ m2.This is a sliding door, and the assembly rail isplaced at a distance of 200 mm from thehollow section’s z axis and at a distance of100 mm from its y axis. The door beam issubjected to the self-weight of the door and tothe wind load. It is assumed that the loweredge of the door is supported by the floor, inwhich case only half of the wind load istransferred to the door beam.

δ

δ

= ⋅ −( )⋅ +

= ⋅ −( )⋅ ⋅ ⋅ ⋅ +

=

<

−1

6 2

1

6

32000 4 6 4

2 1 10 1008 10

4

2 6 420 2

200

2 2

11 8F a L a

E I

a

L a

L

Sd

,, mm =

L

297

OK! (Table 6.3)

F G Q L L kN

M F kNm

N Q

Sd k k p f

el y Sd Sd

Sd k

= +( ) ⋅ ⋅ = +( ) ⋅ ⋅ =

=

=

=

= ⋅ = +( ) ⋅ ⋅ +( )

0 5 0 5 0 4 1 2 4 10 32

10

9

10

932 35 6

0 6 0 6 0 3 0 6 0 375 10 5 4100

2

1

0 2 2

, , , ,

,

, , , , , ,

.

. .

.

ψ

=

= + = + = < =

128 6

35 6

134 4

128 6

3524301 4 3552 2

,

,

,

,,max

. .

.

kN

M

W

N

A

N

mmf

N

mmel y Sd

el y

Sdyσ

(self-weight and snow load)

(wind load)

Nb y Rd. . ,= 369 9 kN > N OK!Sd


180

�doors Vy

5000

5000

qwdVz

Mz

My

200

100

Mt

a

a

a-adoor beam

LoadsThe hollow section is subjected to biaxial bending and to torsional moment. No axialforce is present. The force quantities are determined using elasticity theory. Consider ahollow section with dimensions of 180 x 100 x 6 and steel grade S355J2H.

(torsional load due to the weight of the door and the wind load)

where

Resistances at the ultimate limit stateSince the hollow section has a Class 2 cross-section, the resistance values aredetermined using plasticity theory (Appendix 9.1).

Mt.Rd = 33,3 kNm Vpl.z.Rd = 374,1 kNm > Vz.Sd OK! Mpl.y.Rd = 58,4 kNm Vpl.y.Rd = 207,8 kNm > Vy.Sd OK! Mpl.z.Rd = 38,8 kNm

The interaction expression based on plasticity theory can be used, since there is no axialforce present and the hollow section has a Class 2 cross-section [formula (2.9)]. Theportion of torsion is checked separately, since the maximum value of the torsion momentis at a different location than that of the bending moment:

M

M

M

M

M

M

y Sd

pl y Rd

z Sd

pl z Rd

t Sd

t Rd

.

. .

,

.

. .

, , ,

.

.

,

,

,

,, ,

,

,, ,

+

=

+

= <

= = <

1 66 1 66 1 66 1 6615 8

58 4

4 22

38 80 139 1 0

2 88

33 30 086 1 0

OK!

OK!

eeH

Mq G H

q H

d

t Sd

d G k d

Q wk d

1

2

1 1

1

2 88

1 35 0 75 5 5 0615 8

12 70 50 5 1 5 0 6 0 6 5 1 354 223 38

.

. .

.

,

, , ,,

,,, , , , ,,,

== ⋅ ⋅= ⋅ ⋅ ==== ⋅ ⋅= ⋅ ⋅ ⋅ ⋅ ===

kNm

kN/mM kNm

V kNq

kN/mM kNmV kN

y.Sd

z.Sd

wd

z.Sd

y.Sd

γ

γ

is the eccentricity of the vertical loadis the eccentricity of the wind loadis the height of the door

(torsional moment at the support)

(load causing bending due to the weight of the door)(vertical bending moment at the centre of the span)

(vertical shear force at the support)

(load causing bending due to wind)(horizontal bending moment at the centre of the field)

(horizontal shear force at the support)

q G e q H e

kNm

m

td G k Q wk d= ⋅ ⋅ + ⋅ ⋅ ⋅

= ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ ⋅ =

γ γ. . .,

, , , , , , , , ,

1 1 1 1 20 5

1 35 0 75 5 0 2 0 5 1 5 0 6 0 6 5 0 1 1 15


181

Stresses and deflection at the serviceability limit stateCheck that the stresses do not exceed the yield strength of the material withserviceability limit state loads. The load values are given by the formula (6.7):

Determine the deflection:

The resistance of a 180 x 100 x 6 hollow section is thus sufficient for a door beam. It isoften necessary to restrict the deflection of the door beam to ensure smooth functioningof the door mechanism.

δ

δ

yk

z

q L

E I

L

L

= ⋅⋅

= ⋅⋅ ⋅ ⋅

=

<

= <

−5

384

5

384

3750 5

2 1 10 1310 1011 1

250

6 7250

4 4

11 8,,

,

mm

=L

424 = OK!

mm =L

751 OK!

(Table 6.3)

G

M kNm

QkN

mM kNm

M

W

M

Wf

mm

k

el y Sd

k

el z Sd

el y Sd

el y

el z Sd

el zy

=

=

=

=

= + = + = < =

3 75

11 7

0 90

2 81

11 7

145 5

2 81

104 8107 2 355 2

,

,

,

,

,

,

,

,,

. .

. .

max. .

.

. .

.

kN

m

N

mm

N OK!2σ

(self-weight)

(wind)

(vertical bending moment at the centre of the field)

(horizontal bending moment at the centre of the field)


182

6.4 Design of trusses

When designing trusses, the joints between brace members and chords are usually assumedpinned, so the brace members are subjected to axial force only. Bending moments need not betaken into account in the design of joints if the conditions shown in the tables of Appendix 9.3are met. However, the chords as continuous members are also subjected to bending stresses.The chord moment can be reduced by directing the load to the joints of the lattice. Hollowsections function efficiently as compression and tension members, which makes the lattice alight-weight structure in relation to its load-bearing capacity.

Table 6.6 Steps in truss design

Task Illustration1. Determine the loads in the structure. Determine the

most severe load combination. The direction of theload is an important consideration, as the tensionand buckling resistance values of a hollow sectiondiffer.

2. Determine the height of the lattice. Often, thisdepends on the need for space and on the functionalrequirements of the building, as well as on therequirements of transport and installation. Select thetype and the purlin spacing of the lattice. With thesedata, generate a static model of the lattic

3. After calculating the lattice moment by treating thelattice as a section, make a preliminary selection ofthe members. Divide the maximum moment value bythe lattice height, which gives the initial chord forcevalue (N0 ≈ Mmax/h). Calculate the initial value forthe brace member load using the shear force valueof the section (V0 ≈ Vmax ). Check the resistance ofthe joint subjected to the largest load. Adapt thedimensions of the brace members and chords sothat the relation of their widths is approximately 0,7-0,8.

12

hh

Mm

axV

max

N0 = Mmax /h = q · L2 /(8h)

V0 = Vmax = (q · L/2)

M

V

2 2

3


183

4. Determine the actual member forces using anappropriate design software. Calculate the memberforces for all load combinations. Use a safety factorof 1,0 for the self-weight in cases where the effect ofthe self-weight is advantageous. In the calculationmodel, assume that the chord is continuous and thatthe brace members is pinned. Check the resistanceof selected members to actual member forces.Calculate the resistance of brace members in termsof either tension or buckling resistance and checkthe resistance of chords from the interaction expres-sion for moment and axial force. If the member sizesneed to be adjusted, recalculate the member forcesusing the new dimensions. Check that thedimensions of the members meet the validity criteriashown in the tables of Appendix 9.3.

5. Calculate the local resistances of joints. Decidewhether to reinforce the joints subjected to thegreatest loads or to select stronger members. Thereinforcement of joints is efficient if the reinforcementcosts are smaller than the increased material costsof the members.

6. Calculate the deflection and compare it with thepermitted value.

7. Design the transverse support of the lattice and thepurlin-to-lattice joints. Determine the location ofassembly joints of the lattice, taking transport intoaccount

��

682 kN 682 kN

1166 kN

-1431 kN -1431 kN

-480

kN -480 kN

��

N2

b0

t0

t1, 2

b1, 2

h 0 e

N0

h2h1

θ1 θ2

N1

δ max

Primary lattice

detail 5

detail 5

Purlin


184

In lattice design, it is recommended that design software package are used which includeresistance data for structural hollow sections (e.g. WinRAMI, Appendix 9.8).

The costs of a lattice does not only consist of the weight of the steel, but also of shopfabrication and on-site installation. A lattice with gapped joints and few members may thus beless costly than a lighter weight lattice with several members and overlapped joints. The mostadvantageous type of lattice and joint shape must be decided on a case by case basis.

6.4.1 Selection of truss type

The most commonly used lattice types are K, KT and N trusses.

Figure 6.12 Various truss types

A K truss is suitable for long-spanned structures where loads can be transferred directly tolattice joint locations. In K type trusses, the number of members is small and joints are simple.Wide spacings between members also leaves room for tube lead-throughs. However, thebuckling length of the upper chord is large, which may result in a heavier chord than in theother lattice types. In general, a K truss is simple and very affordable in terms of fabricationcosts.

In a KT truss, the spacing of the upper chord supports is more dense, so the resistance of thechord is better than that of a K truss. The joints in a KT truss are, however, more complex toprepare. The joints in the lower chord must often be made overlapped, which increasesfabrication costs.

In an N truss, the number of members is larger as compared to that of a K truss. In deep andshort lattices, the brace member forces are great in comparison with the chord forces. In sucha case, an N truss is efficient, since the compressed brace members are shorter than the onesin KT trusses. The joints must usually be made overlapped to avoid high values of eccentricity.

N truss

KT truss

K truss


185

In long-spanned structures, there are big differences in the brace member loads. Close to thesupport, the loads are greater than in the central area of the lattice. To reduce the weight of thestructure, the brace members subjected to smaller loads can be made of lighter-weight hollowsections. However, to facilitate shop fabrication, it is not advisable to use more than 2 to 4different brace member sizes. With hollow sections having the same external diameter, onlyone wall thickness should be used in one lattice to avoid confusing them during construction.For simplicity's sake, the chord is usually made of one hollow section size, although axial forcevaries according to the length of the chord. The chord section can be constructed of hollowsections of different sizes if the lattice is divided into assembly blocks. In such a case, the sizeof the chord changes at the assembly joint.

At the intermediate support of a continuous lattice girder, the supporting force and the momentare at their maximum values. The buckling length of the lower compression chord can bereduced by placing a vertical brace member above the support. It is an advantage if thesupporting force is received by a vertical member, since the longer diagonal members aresubjected to tension. In subsequent diagonal compression members, axial force is reducedfrom the value at the support (Figure 6.13).

Figure 6.13 Intermediate support of a continuous lattice girder

Laterally joined trusses

Supports for pipes, ducts and working platforms are usually shaped as a bridge by joining theprimary lattices with horizontal wind lattices (Figure 6.14). The lattices can also be joined withplates whose joints are designed according to the horizontal loads. It is advisable to brace astructure constructed of two lattices with lacing perpendicular to the lattice plane if the latticesare also subjected to torsion load. (Figure 6.14a). The distance between the lattice nodesperpendicular to the lattice plane can be used as the buckling length of a laterallynonsupported upper chord. In such a case, the transverse force generated by the upper chordmust be taken into account in the design. The transverse force can be estimated using theformulae shown in Section 6.1.4. A laterally nonsupported upper chord is presented, forinstance, in Figure 6.14b.


186

Figure 6.14 Laterally joined trusses (Fn is the transverse force preventing buckling)

Due to services, the lattice must sometimes be perforated. To guarantee a sufficient shearresistance, the lattice must be reinforced at the openings. A lower lattice can be constructedbetween the upper chord and the opening if the height of the opening is smaller than that of thelattice. If the opening and the lattice are of equal height, the opening must be reinforced with aframe. The bending moment due to shear force must be taken into account in the design of theframe and chords. Usually, it is advisable to place the openings in areas with the least possibleshear force.

Figure 6.15 Reinforcing openings in trusses

6.4.2 Selection of the chord member

A decisive factor in the selection of chord section are the buckling lengths about different axes.When the buckling lengths are close to equal in both directions, square hollow sections are themost advantageous. The use of rectangular hollow sections can be efficient if the bucklinglength values differ significantly. However, a broad and shallow chord is not a good solutionbecause of the local strength of the chord face and the shear resistance. It is advisable toselect a deep chord when the chord bears significant bending load between the lattice nodes.

��

�� Lc

Mt

a) b)FN FN

FN FN


187

The in-plane buckling length of the chord equals the distance between nodes. When thedirection of buckling is out of the plane, the buckling length equals the distance betweenlaterally supported points. The above buckling length values can be multiplied by 0,9 if thejoints are welded all round and the brace members are not flattened [5]. Lateral stiffeningelements are designed for transverse loads and transverse forces due to compression chords(Section 6.1.4). The bending moments generated by joint eccentrities must be taken intoaccount in the chord design. The joints of square and rectangular hollow sections are simplerthan the joints of circular hollow sections. Exceptional cases in which the use of circular hollowsections is advisable are triangular lattice sections and space frames.

A thick chord wall is efficient in terms of joint resistance, but in terms of the compressionresistance of the chord the situation is quite the contrary. A feasible compromise must bereached in the design, or the chord face must be reinforced.

Lattices that are subjected to heavy loads and that have a large spacing between the lateralrestraints of the compression chord can be constructed using a double chord. In a doublechord lattice, the chords are joined to each other directly (Figure 6.16b) or through diaphragmmembers (Figures 6.16a and c). The horizontal inertia of the chord increases significantly if thechords are joined by diaphragm members. Regarding the resistance of the joint, chord faceyield is not possible, since the forces from brace members are transferred directly to chordwebs. In the design of a joint shown in Figure 6.16c, forces due to three-dimensionality must betaken into account. When designing the joint shown in Figure 6.16b, the same formulae as withan I profile chord can be used if the space between corner roundings is welded at jointlocations. The web thickness value of the I profile tw is replaced with the combined thicknessvalue of both chord webs (2 t0), and the corner radius is the inner corner radius of the chord.The dimensions of the joint shown in Figure 6.16a must be selected so that the brace memberwelds are accessible. Regarding the shear resistance of the chords in Figure 16a, the followingsectional areas are used [7]:

where

Figure 6.16 Double chord

��

��

��

a) b)

c)

hitsi

is the height of the chord

is the width of the chordhb

o

o

A t hen b

A t when bv o o

v o o

= ⋅ ≥= ⋅ <

2 6 4

2 0 5

, /

, /

h w h 1

h h 1 o o

o o

(6.1 )

(6.1 )


188

6.4.3 Selection of bracing members

The selection of brace members is less complex, since a brace member with thin walls andlarge outer dimensions is better in terms of both joint strength and member resistance.However, the slenderness of a brace member should be kept within the limits specified in thetables in Appendix 9.3. Special attention must be paid to the weld between the chord and thebrace member when the chord and brace member are of almost equal width (Section 7.4.4).Usually, it is advisable to select such dimensions for the brace member so that the ratiobetween its width and the chord width falls between 0,7 and 0,8.

It is always conservative to take the actual length of the brace member as its buckling length.However, joints welded at all sides have rigidity, and thus a buckling length of 0,75 times theactual length of the member can be used. The buckling length of the brace members can alsobe calculated with the formulae presented in reference [5], if the b1/b0 ratio is less than 0,6.However, in joints of completely overlapped (λov = 100 %) and flattened brace members, thebuckling length is always the actual length of the member.

With large values of the joint angle θi , it is advisable to use rectangular brace members to keepthe joint eccentrities to a minimum (Figure 6.17a). With small joint angles or near the support ofa single span lattice square brace members can be used. (Figure 6.17b). Near the support of asingle span lattice, the axial forces in the chord are small, so the chord can more easily carrythe bending moments due to joint eccentrities.

Figure 6.17 The effect of brace member shape

6.4.4 Design of truss joints

Truss joints can be divided into two main groups: gapped and overlapped. Gapped joints areeasier to make, since the brace members can be cut to the conect angle in one go. There isalso some tolerance when assembling the lattice. In a gapped joint made of square bracemembers, the eccentricity is usually large when the brace members are of the same width asthe chord. The eccentricity increases the bending load on the chord. The shear resistance ofthe joint may also govern in a gapped joint. An overlapped joint is more complex to prepare,because the overlapping member must be cut to two different angles. The tolerances of theelements are more restricted than in gapped joints. Respectively, the resistance of the joint isgreater, and eccentricity can be removed completely if an appropriate overlap is used.

��

��

θθθθa) b)


189

The smallest angle permitted for brace members in the tables is 30°. In practice, small jointangles should be avoided, as they make the welding of the acute angle side quite difficult. Withsmall angles, even minor flaws in cutting the hollow section can result in large root gaps in thejoints. If the joint angle θI is smaller than 60°, the ends of brace members must be chamfered.According to tables in Eurocode 3 Appendix K, the minimum gap for gap joints is (t1+ t2).It is also advisable to check that the gap meets the condition given in reference [8], that is,ga ≥ 1,5 t0, to obtain a sufficient plastic deformation capacity of the chord (ga is the distancebetween the weld toes, see Figure 6.18).

Figure 6.18 Minimum gap values

In the joints of hollow section lattices, local stress concentrations are generated at the joint.However, the stress concentrations are somewhat evened out by the yielding of the bracemembers and chords. Due to this, the welds at the joints must be designed to have the samestrength as the members. The welds that qualify are full penetration single-V butt welds or filletwelds with a throat thickness meeting the following conditions [7]:

In overlapped joints, the lower brace member need not usually be welded at the side whichremains hidden. However, in totally overlapped joints (λov = 100 %) even the hidden side mustalways be welded. This is also the case when the components of the forces in the bracemember, parallel to the chord, differ from each other by more than 20 % [7].

Reinforcing truss joints

Reinforcing the joint is advantageous if the number of joints to be reinforced is small comparedto that of members in the lattice. On one hand, the reinforcement of joints increases fabricationcosts; on the other, it reduces the weight of the structure and removes the need to use toomany hollow section sizes.

a

a

a

≥ =( )≥ =( )≥ =( )

0 95 235 16

1 00 275 7

1 07 355 8

,

,

,

t f N mm

t f N mm

t f N mm

y2

y2

y2

(6. )

(6.1 )

(6.1 )

g > t1 + t2

ga >1,5 t0 ga >1,5 t0

θ1 > 60° θ2 > 60° θ1 ≤ 60° θ2 ≤ 60°

t 0 t 0

t1 t1t2 t2

g > t1 + t2


190

The chord face can be reinforced with plates (Figure 6.19a). This is an effective method forstructures with brace members distinctly narrower than the chords. When determining theresistance of joints, the thickness of the chord face t0 is replaced with the thickness of thebracing tp, and the width of the chord is replaced with the width of the bracing bp. Theresistance of joints reinforced on the chord face is shown in Tables 9.3.13 and 9.3.15. Thebracing is prone to lamellar tearing, which must be taken into account when selecting thebracing material.

The shear resistance of the chord can be improved with plates welded to the chord side (Figure6.19b). The height of the plates is equal to that of the chord. When calculating the sectionalarea, the thickness of the chord web t0 is replaced with the sum t0+ tp. The resistance of jointsreinforced by the chord webs is given in Tables 9.3.14 and 9.3.16. The shear resistance of thejoint often governs when the brace member and the chord are of equal width.

Overlapping of brace members increases both the resistance of the chord face and the shearresistance of the chord. Brace members of different widths can be overlapped withreinforcement (Figure 6.19c). The thickness of the reinforcing plate must be at least twice thethickness of the brace member wall (tp ≥ 2t1). A further advantage gained by the use ofreinforcement is a symmetrical joint. Preparing the joint shown in Figure 6.19c withoutreinforcement is not advisable, as the resistance of the joint is smaller than the results shown inthe tables in Appendix 9.3. The resistance of reinforced overlapped joints is shown in Table9.3.17.

Figure 6.19 Reinforcement of truss joints

6.4.5 Truss joints at the supports

The transmission of the lattice shear force to the column must be examined carefully. If thecentre of gravity of diagonal member, chord and column do not intersect at the same point, thereaction is transferred as a bending moment to the column. In practice, it is often advantageousto allow a small eccentricity if this facilitates the preparation of the joint. The end of the chord isusually sealed with a plate to achieve sufficient resistance to concentrated load. The jointlocation of the chord must be designed taking into account the combined effect of axial force,shear force and bending moment. Especially at intermediate support location in continuouslattice sections, the effect of axial force is significant. Various methods are presented in Figure6.20.

� ��

��

a) b)

c)


191

Figure 6.20 Truss joints at the support

6.4.6 Estimation of the truss rigidity

Usually, the deflection of the lattice is obtained directly from the output of the lattice designsoftware. In the software, the lattices can be modelled using continuous chords and pinnedbrace members. In gapped joints, due to the flexibility of the joints the actual deflection can bemore than 12-15% greater than the calculated deflection [7]. In the preliminary planning stageit may be necessary to estimate the lattice deflection by manual calculations. The stiffness ofthe lattice can be calculated taking into account the effect of the chords only:

where

are the cross-sectional areas of chords

is the distance between the centre of gravity

axes of chords

A AH

1 2 and

A2

H

I A H

A

AA

A

A H A

A

= ⋅+

+ ⋅+

12

2

1

2

1

22

2

11

1

1(6.19)

2 2


192

��

A1

6.4.7 Designing the truss of the model building

Loads

Now design the primary trusses in the model building. Loads on the lattice consist of theself-weight of the structure and the snow load. The purlin spacing is 4 m, so it is an ad-vantage to use the same spacing in the brace member joints. With the 10 m spacing ofprimary lattices, the following load on the lattice is obtained:

where

Truss shape

In a roof lattice, the lower chord can be made either straight or bent. Axial forces aregreater in a bent lower chord. With a straight lower chord, the height of the lattice isgreater in the centre of the span, which is also the location with the greatest moment.Thus, the axial force is smaller in a straight lower chord. With to a straight lower chord,the length of brace members is greater, which may increase the weight of the lattice ascompared to a lattice with a bent lower chord. In the model building, the lower chord isbent, but the axial force of the lower chord is resisted by a tension rod. The use of thetension rod reduces the amount of steel in the brace members, but increases the numberof joints in the lattice.

The span is long, which makes the shear force small compared to the bending moment.An N lattice is thus too heavy to be used in the model building. For a better upper chordresistance, a KT lattice is selected, since the spacing of the purlins is large. The heightof the lattice is estimated by the span. In practice, the optimal height varies between L/9-L/12. In the model building, a lattice height of 6,5 m is selected.

Figure 6.21 Primary truss

lower chord

upper chord

vertical

diagonal

tension rodL

H α =

9,5

°

N1.Sd = -480 kNN1.Sd = 682 kNN0.Sd = 1166 kN

N0.Sd = -1431 kN

N0.Sd = -462 kN

is the frame spacingis the purlin spacing

L

Lf

p

q G Q LkN

mF q L kN

d G k Q k f

Sd d p

= ⋅ + ⋅( ) = ⋅ + ⋅( ) =

= ⋅ = ⋅ =

γ γ, , , , , , , , ,

, ,

1 1 1 1 1 35 0 5 1 5 1 2 10 24 8

24 8 4 99 2

(load on the node)


193

Determining the member forces

The initial load on chords and brace members can be determined by calculating thelattice forces with the formulae of a simply supported section:

Based on these values, select an upper chord of 200 x 200 x 8 and brace members of 120x 120 x 6. For the buckling length of the upper chord, take 90% of the horizontaldistance between purlins and, 90% of the vertical distance between the lattice nodes.Thus the following buckling length for the upper chord are obtained:

The size of the lower chord is more complex to define manually in this particular case, soby considering the brace member joint, select a lower chord of dimensions 140 x 140 x 5.The axial force on the tension rod can be assumed equal to that on the upper chord.Select a tension rod of Ø80 mm (fy = 345 N/mm2). The output from the design softwaregives the following maximum values for the forces:

Resistance of the upper chord

First, consider the resistance of the upper chord with a 200 x 200 x 8 hollow section andthe steel designation S355J2H. The hollow section has a Class 1 cross-section and its re-sistances are as follows (Appendix 9.2):

N N kN

N kN

M kNm

N

N

k M

M

b Rd b z Rd

b y Rd

pl y Rd

Sd

b Rd

y y Sd

y Rd

. . .

. .

. .

.

.

.

,

, ,

,, ,

= ==

=

+⋅

= + ⋅ = <

1484

1806

135 8

1431

1484

1 182 4 0

135 80 999 1 0

(buckling resistance, Lc= 3,65 m)

(buckling resistance, Lc= 1,83 m)

Substituting the resistances in the interaction expression (2.57):

OK !

Upper chord:

Lower chord:Tension rod :Brace members:

N kNM kNmN kNN kNN kNN kN

Sd

Sd

Sd

Sd

Sd

Sd

0

0

0

0

1

1

14314 0

4621166

480682

.

.

.

.

.

.

,= −== −== −=

(compression)

(compression)(tension)(compression)(tension)

L m

L m

c y

c z

.

.

,cos ,

,

,cos ,

,

= =

= =

0 92

9 51 83

0 94

9 53 65

Nq L

HkN

N q L kN

Sdd

Sd d

0

2 2

1

8

24 8 48

8 6 51099

0 5 2 0 5 24 8 48 2 841 7

.

.

,

,

, , , ,

≈ ⋅( )

= ⋅⋅( )

=

≈ ⋅ = ⋅ ⋅ =

(load on upper chord)

(load on brace member)


194

Designing the lower chord and the tensionrod

The use of the tension rod generates com-pression in the lower chord. Compression ispresent only in the first diagonal spacing,after which the lower chord is subjected totension. Support the compression element ofthe lower chord laterally to the latticepurlin with stays. This way, the bucklinglength of the lower chord of dimensions 140x 140 x 5 is equal in both directions:

Resistance of the brace membersThe resistance of a tension brace member of dimensions 120 x 120 x 6 is determined in a similar way to that of the tension rod:

In brace members subjected to smaller loads, a smaller hollow section can be used.

NA f

A fkN

t Rdy

M

y

M

.,

, , ,

,

,

=⋅

= =

= ⋅ =

=⋅ ⋅

= ⋅ ⋅ = >

γ

χγ

1

1

2643355

1 1853

0 75 3 54 2 66

0 690 2643 355

1 1589 480

kN > 682 kN

L m

N kN

c

b.Rd

This buckling length is used to determine the buckling resistance:

The buckling length of a 120 x 120 x 6 compression brace member is 0,75L:

L m

NA f

kN kN

NA f

kN kN

c

b Rdy

M

t Rdy

M

=

=

=⋅ ⋅

= ⋅

= >

=⋅

= ⋅

=

0 94

9 53 65

0 617 2636355

1 1

525 1 462

40345

1 1

1576

1

0

2

,cos ,

,

,,

,

,

.

.

> 1166

χγ

γπ

The resistance of the tension rod (Ø80 mm) is determined by the area and the yieldstrength:

OK!

( Ø > 16 mm ⇒ fy = 345 N/mm2 )

OK!


195

��a

a

a

a

a - a

Resistance values of joints

To simplify shop fabrication, the joints of the upper chord in the model building aredesigned gapped and those of the lower chord overlapped. Thus, the eccentricities canbe kept small. In a simply supported lattice section, the shear force is at its greatest atthe supports, which is also where the brace member forces are at their greatest. Due tothis, the resistance values of joints at the two outermost purlins (corners 1 and 5 in thecalculation model) must be checked. In practical design, the resistance values of alljoints must be checked.

Figure 6.22 Joint at corner 1

The joint at corner 1 is a Y joint with a tension brace member. The formulae for this typeof joint are shown in Table 9.3.1 The geometry and forces at the joint are as follows:

Chord: 200 x 200 x 8 (A0 = 5924 mm2)Brace member: 120 x 120 x 6

N kN puristusta k k

N kN

b

b

h

b

Sd n n

Sd

0

1

1

0

1

0

479 1 30 4

0 6

479000

5924 3551 13 1 0

682

54

120

2000 6

120

2000 6

.

.

,,

,, ,

,

,

= − ( ) ⇒ = −

⋅

= ⇒ =

=

=

= = =

= = =

θ

β

η

o

(compression)

(Figure 6.21)

�Joint of corner 1

detail 1

detail 1

h 0

t0

t1

b1

b0

h1

θ

N0

N1


196

Chord face yield

Since β = 0,6 < 0,85, the governing failure mode is the yielding of the chord face:

The joint resistance is not sufficient, since N1.Sd = 682 kN > N1.Rd. When the chord facevalue governs, reinforcement welded to the chord face helps improve the resistance ofthe joint. In the model building, reinforcement is an affordable solution, since only theoutermost joints must be reinforced.

Figure 6.23 Reinforcing the joint at corner 1

Yield of the reinforced chord faceDetermine the resistance of the joint which has a 275 x 185 x 15 reinforcing platewelded to the chord face (Table 9.3.13):

N kN kN

L mmh

b b b mm

L mm

Rd

p p p

p

1

2

1

11

355 15

1 0 65 54

2 0 65

544 1 0 65 1 0

1

1 11019 682

275120

54185 185 120 257

275 1 5120

54

., sin

,

sin, ,

,

sin sin

,sin

= ⋅−( )

⋅ + −

= >

= ≥

+ −( ) =

+ −( ) =

= ≥

OK!θ

== 222 mm

Check the length of the bracing:

��

b0

t1

t0

b1

Lph 0

t p

h1

N0bp

θ

N1

Nf t

k

kN

Rdy

nMj M

10

2

0

2

1

24 1

1 1

355 8

1 0 6 54

2 0 6

544 1 0 6 1 0

1

1 1256 1

.sin sin

,

, sin

,

sin, ,

,,

=⋅

−( ) + −

⋅

= ⋅−( )

⋅ + −

=

β θηθ

βγ γ


197

The dimensions of the chord and brace members must fall within the validity area givenin Table 9.3.13:

Joint at corner 5

Figure 6.24 Joint at corner 5

The joint at corner 5 is a gapped K joint. The formulae for this type of joint are given inTable 9.3.2. The geometry and forces at the joint are as follows:

Chord: 200 x 200 x 8, A0 = 5924 mm2

Brace members: 120 x 120 x 6

N kN puristusta k

N kN

N kN

Sd n

Sd

Sd

0

1

2

1 2

859 1 1 30 4

0 6

859100 1 1

5924 3551 00

480

480

54

120 120

2 2000 6 1

10 92

.

.

.

, ,,

,

,,

, , , ,

= − ( ) ⇒ = −

⋅⋅

=

= −=

= =

= +⋅

= < −

=

θ θ

β βγ

γ

o

== = =b

t0

02

200

1612 5,

(compression)

so the punching shear of the chord mustbe checked

(θ < 60°, so the brace member must be welded to the chord with a Vgroove)

�det 5

det 5h 0 t0

t1, 2

b1, 2

b0

ga θ2

N0

N2N1

θ1

g

h1 h2

e

b

b

h

b

h

b

h

b

b

t

h

t

1

0

1

0

1

1

0

0

0

0

0

0

120

2000 6 0 25

0 5 1 2

0 5 1 2

200

825 35

= = = >

< = <

< = <

= = = <

, ,

,

,

b h

t

b

t

h

t

b h

t

b mm b t mmp

0 0

0

1

1

1

1

1 1

1

0 0

400

850 25

120

620 35

240

640 25

185 2 184

+ = = >

= = = <

+ = = >

= > − =

OK!

OK!

OK!

OK!

OK!

OK!

OK!

OK!


198

The eccentricity at the joint created by the joint gap is as follows:

Yield of the chord faceFirst, determine the resistance by the yield of the chord face:

Chord shearObtain the value for the shear resistance of the entire chord as follows:

In the determination of the compression resistance of the chord, the effect of shear forcemust be taken into account, since:

V V

N A AV

V

f

N N kN

Sd Rd

Rd vSd

pl Rd

y

M

Rd Sd

>

= − −

= − ⋅ −

=

> =

0 5

21

5934 34192 388 3

637 11

355

1 1

859 1

0 0

2

0

2

0 0

,

,

, ,

,

..

. .

γ

1862 kN

OK!

A h b t

g

t

Nf A

kN

Vf A

v

Rdy v

Mj M

pl Rdy v

M

= + ⋅( ) = ⋅ + ⋅( ) =

=+

=+ ⋅

⋅

=

=⋅

⋅= ⋅ =

=⋅⋅

=

2 2 200 0 137 200 8 3419

1

14

3

1

14 50

3 8

0 137

3

1 1 355 3419

3 54

1

1 1787 4

3

0 0 0

2

02

2

2

10

0

α

α

θ γ γ

γ

,

,

sin

,

sin ,,.

.

mm

2

637637 1

480 54 388 3

,

sin ,

kN

kN

VSd = ( ) =

Nf t

b h

m bkRd

yi

i

m

ii

m

nMj M

10

21 1

0 0

2

8 92

1 1

8 9355 8

54

120 120 120 120

4 2001 0 12 5

1

1 1482 0

. ,sin

,

,sin

, ,,

,

=⋅

+

⋅

⋅

= ⋅ + + +⋅

=

= =∑ ∑

θγ

γ γ

kN OK!

g

hg

h

=

+ +

⋅+( ) − =

50

2 236 52

2

1 2

1 2

0

mm

e =h

2sin mm1

1θ θθ θθ θsin

sin sin

sin, (eccentricity at the joint)


199

Failure of the brace memberThe effective width of the brace member is:

Shear failure of the chord faceIn this case, the shear failure resistance of the chord must also be taken into account:

Resistance of the jointThus, the chord face yield was the most important governing failure mode N1.Rd = 482,0 kN > N1.Sd = 480 kN OK !Check the validity criteria in the joint table:

b

b

b

t

h

t

b

b

b

t

b h

t

h

b

h

b

b

t

h

t

b h

t

1

0

1

1

1

1

1

0

0

0

1 1

1

1

1

0

0

0

0

0

0

0 0

0

120

2000 6 0 35

120

620 35

0 6 0 10 01

0 35240

640 25

0 5 1 2

0 5 1 2

200

825 35

= = > = = = <

= > + = + = = >

< = <

< = <

= = = <

+

, ,

, ,,

,

,

,

== = >400

850 25

OK!

OK!

OK!

OK!

OK!

OK!

OK!

OK!

bt b

b

Nf t h

b b

kN

ep

Rdy

epMj M

= ⋅ = ⋅ ⋅ = ≤

=⋅

+ +

⋅

= ⋅ ⋅ + +

=

10 10 8 120

20048 120

3

2 1 1

355 8

3 54

2 120

54120 48

1

1 1856 1

0 1

0

10 1

10

mm

.sin sin

,

sin sin ,,

θ θ γ γ

bb t

b t

N f t h t b b

kN

eff

Rd y effMj M

= ⋅⋅

= ⋅ ⋅⋅

=

= ⋅ − + +( ) ⋅

= ⋅ ⋅ − ⋅ + +( ) =

10 10 120 8

200 664

2 41 1

355 6 2 120 4 6 120 641

1 1774 5

1 02

0 1

2

1 1 1 1 10

mm < 120 mm

.,

,,

γ γ


200

Lattice joint at the support

Figure 6.25 Lattice joint at the support

The lattice-to-column joint is made using an end plate. To facilitate the fabrication of thejoint, eccentricity is allowed about the neutral axis of the column. As a result, the upperchord must transfer the shear force to the column. The eccentricity also causes bendingmoment in the upper chord. When the eccentricity is 200 mm, the forces loading the endof the upper chord at the diagonal/chord intersection are as follows:

V kN

MSd

Sd

== ⋅ ==

595

595 0 2 119

78

kNm

N kNSd

,

(joint at corner 1)

�e

a

a

tp

200 x 200 x 8

120 x 120 x 6

300

x 20

0 x

6

a - a2 x

15

b

t

h

t

E

f

e

h

g

b

g

b

g g

y

a

1

1

1

1

0

0

0

20 1 25 30 4

0 5536 5

2000 18 0 25

50

2000 25 0 5 1 0 5 1 0 6 0 2

50

2000 25 1 5 1 1 5 1 0 6 0 6

= = < =

− < = = <

= = > −( ) = −( ) =

= = < −( ) = −( ) =

= −

, ,

,,

, ,

, , , , ,

, , , , ,

β

β

22 2 90 50 2 90 54 6

41 3 1 5 1 5 8 121

0

L g t

g mm ta

= − −( ) = − −( )= > = ⋅ =

tan tan

, , ,

θ mm

OK!

OK!

OK!

OK!

OK!

OK!


201

L

θ2

t2

L

g

ga

t1

t 0 L

θ1

L

Resistance of the end of the chord to the combined loadThe chord needs to be reinforced, because the shear resistance of a 200 x 200 x 8 hollowsection is not sufficient (Vpl.Rd = 552 kN < VSd). Reinforce the chord by plates welded tothe side of the chord. The thickness of the plates is tp= 5 mm. Reinforcing plates extendover the joint of the diagonal member. The plates welded to the sides of the chord aretaken into account when determining the resistance of the chord end, and the platewelded to the lower flange of the chord is taken into account when determining theresistance of the joint (joint at corner 1). The resistance values of the reinforced upperchord are:

The shear force now exceeds half of the shear resistance; thus, its effect must be takeninto account in the interaction expression. The effect of shear force is taken into accountwhen calculating the resistance to bending and axial force [formula (2.46)]:

The effect of axial force is taken into account in the interaction expression (2.49).Bending moment now occurs only in the other direction, so the latter term in thecondition is omitted. Buckling need not be taken into account, as it is the local resistanceof the chord which is considered here. The following value for bending resistance withthe presence of axial force is obtained:

Resistance of the chord end to concentrated loadNext, calculate the resistance of the upper chord webs to the reaction using the formulaeshown in Section 2.10. The end of the upper chord is sealed with a plate, and the gapbetween the corner rounding and the splice is filled with weld. The deformation of thehollow section and the resulting secondary bending loads therefore need not be takeninto account.

M MN

NkNm M

M M kNm M M kNm

N Rd V RdSd

V Rdpl Rd

N Rd V Rd N Rd Sd

. ..

,

. . .

, , , ,

,

= −

= ⋅ −

= >

⇒ = = ⇒ > =

1 26 1 1 26 158 5 178

2356193 1

158 5 119

OK!

M

WA

tf

kNm

N A Af

kN

V Rd

plv

y

M

V Rd vy

M

.

.

,

,,=

− ⋅

=− ⋅

⋅

=

= − ⋅( ) =

ρ

γ

ργ

2

0

2

0

8520900

0 126 4962

8 13355

1 1158 5

2356

V Ah

b ht h

fkN

M W t hf

pl Rd py

M

pl Rd pl py

M

.

.

,,

, ,,

,

=+

+ ⋅( )

⋅

=+

+ ⋅( )

⋅

=

= + ⋅( )[ ] = + ⋅ ⋅( )[ ] =

23

5924200

200 2002 5 200

355

3 1 1924 6

2 0 25 420900 2 0 25 5 200355

1 1168 1

0

2

0

2

γ

γ

kNm

Npl.Rd == + ⋅( )[ ] = + ⋅( )[ ] =A t hf

kNpy

M2 5924 2 5 200

355

1 12557

0γ ,


202

The magnitude of the concentrated load to each web is:

In the previous resistance calculations, the web reinforecement was included in thethickness of the webs. The web thickness used in calculations is then t = 8 + 5 = 13 mm.

Deflection of the latticeThe second moment of area of the lattice is estimated with the formula (6.19):

where

A1 is the area of the upper chord A2 is the area of the tension rod

At the serviceability limit state, loads are calculated using the characteristic values (6.7),which gives the following load on the lattice:

where

Lf is the frame spacing

The deflection is calculated by assuming the lattice section is simply supported:

δ = ⋅⋅

= ⋅ ⋅⋅ ⋅ ⋅

= = <5

384

5 17000 48

384 2 1 10 0 114947

1032 200

4 4

11q L

E Imm

L Lk

, , OK ! (Table 6.3)

q G Q LkN

mk k k f= +( ) = +( ) =0 5 1 2 10 17, ,

I A H

A

AA

A

A H A

A

= ⋅+

+ ⋅+

= ⋅+

+ ⋅+

=

12

2

1

2

1

2

22

2

1

2

2

2

2

2

1

1

1

5924 6500

5027

5924

15027

5924

5027 65001

15027

5924

0 1149, m4

FV

kN

R s s t tf

kN F

R t t E f

SdSd

y Rd s y yy

MSd

a Rd p yM

= =

= +( ) +( ) = ⋅ + ⋅ + ⋅( ) +( ) = >

= +( ) ⋅ + ⋅( ) = +( ) ⋅ +

2297 5

2 6 4 15 2 200 13 8 5355

1 1730 0

0 51 3 0 2

0 5 5 8 210000 3551

01

02

1

2

,

,,

,,

,

.

.

γ

γ33 0 2

1 1

1060 1

⋅( )

=>

,

,

,

.

kN

R Fa Rd Sd

The resistance to concentrated load is determined by formulae (2.61) and (2.62), whichgive the following results:

OK!

OK!


203

6.5 Stiffening hollow section structures

To keep the displacements generated by the horizontal loads within permissible limits, thestructure must be stiffened. The stiffening methods can be divided into five groups:- mast stiffening- frame stiffening- tower stiffening- plate stiffeninglattice stiffening

In mast stiffening, the columns function as masts fixed to the foundation with a rigid joint,receiving the horizontal loads. When the height of the building increases, the foundations aresubjected to larger moments, and the buckling length factor of the columns is great. Thismakes mast stiffening suitable only for low buildings. The advantage of this method is thesimplicity of the installation, as no separate supports are needed during installation.

Frame stiffening is a natural method to use when frame joints are rigid and lattice stiffening isnot feasible due to, for instance, openings in the walls. In frame stiffening, horizontal loads aretransferred as moments to the corners of the frame. Rigid joints are, however, more costly tomake than pinned ones.

In tower stiffening, horizontal loads are transferred through rigid elements such as lift shaftsor stairwells to the foundation. This method is well-suited for buildings in which stiffeningelements are made using slip casting. The problem is to produce a sufficiently firm joint whenjoining the building frame to the stiffening elements.

In plate stiffening, horizontal forces are transferred to the ground through plates joined to theframe. Plates can be vertical (e.g. walls cast on-site) or horizontal (e.g. profile). Plate stiffeningis an affordable method: since the plate already forms a part of the building, no separatestiffening elements are needed. Horizontal loads must also be taken into account whendimensioning the joining of the stiffening plates, as this makes the joining firmer than it wouldbe without taking the stiffening effect into account.

Lattice stiffening is an effective method in high buildings. Stiffening lattices transfer thehorizontal loads to the members as axial forces. Hollow sections make excellent stiffeninglattice members, since their radius of gyration is large about both axes. A disadvantage of thelattice stiffening is the space needed for lacings at doors, windows and other openings made towalls.

Several stiffening methods can be used in the same building. The buckling length values of thestructure can be determined using the model for non-sway frames (Figure 6.7) when usingtower, plate or lattice stiffening. In mast and frame stiffening, the buckling length values mustbe determined using the sway-frame model (Figure 6.8).

6.5.1 Designing the stiffening elements in the model building

In the model building, the lattice stiffening method is used. This manual deals with thedesign of transverse stiffening elements only for the model building. The longitudinalstiffening elements are designed according to the same principles.


204

Lateral stiffeningThe wind loads in the side wall of the building are transferred to the stiffening lattices inthe end wall through the horizontal lattice in the roof parallel to the side walls. However,the horizontal force of the side wall wind columns is transferred to the primary columnsthrough the roof profile. This way, the eaves section of the side wall can be made lighter.The joints of the stiffening profile must be checked for the loads created by the horizontalforces.

End wall bracingDesign the bracing elements of the end wall. The horizontal load is divided evenly in theend walls, as the building is symmetrical and the bracing in the end walls is similar. Inthe pressure coefficient of the wind load, the effect of negative pressure must also betaken into account, so cp= 0,6 + 0,3= 0,9. Thus the following value for the horizontalforce of the end wall is obtained:

where

H1 is the height of the eaves H2 is the height of the roof structure L is the length of the building

The bracing is made with two members, one subjected to compression and the other totension. The total horizontal force is transferred by the tension lacing. The tensile forceis as follows:

NSd = 321,5 / cos 48 = 480 kN

Consider a hollow section of dimensions 100 x 100 x 4:

Nt.Sd = 1495⋅355 / 1,1 = 482,5 kN

Figure 6.26 Connecting the bracing

��

��

FSd

Det 1

Det 1

250

x 15

0 x

12,5

100 x 100 x 4

F c q c H HL

kNSd Q p ref e= ⋅ ⋅ ⋅ ⋅ +( ) = ⋅ ⋅ ⋅ +( ) =γ . , , , , , , ,1 1 20 3752

1 5 0 9 0 6 0 375 10 5 4100

2321 5


205

The bracing-to-column connection is made as a single-lap joint through a plate. Thethickness of plates is 20 mm, and their width is 230 mm. The strength grade of the M30bolts is 8.8. The shear resistance of bolts is calculated assuming that the shear planedoes not pass through the threaded portion of the bolt (example 25):

The bearing resistance of the plate is limited by the edge distance. In the joint shown inthe example, the edge distance is 50 mm. The following bearing resistance is obtained:

The resistance of the column-to-end plate connection is calculated from the formulae inTable 9.3.11:

Column flange shear failure: N1.Rd = 628,9 kNSplice failure: N1.Rd = 463,9 kNColumn web yield: N1.Rd = 665,6 kN

The resistance of the joint is sufficient, since the horizontal component of the lacing’saxial force is

NSd = 321,5 kN < N1.Rd

Roof Bracing

Figure 6.27 Roof bracing

main column

wind column

purlin section

eaves section

qd.2

qd.1

D2

D1

F kN Nb Rd Sd. , ,,

,= ⋅ ⋅ ⋅ ⋅ = >2 2 5 0 521 490 3020

1 25612 7 OK!

F kN Nv Rd Sd. ,,

= ⋅ ⋅ ⋅ = >2 707 0 6800

1 25543 OK!


206

The roof bracing is subjected to transverse wind load. The loads on the wind columnsare transferred as reactions, so part of the wind load is transferred directly to thefoundation. The total wind load on the roof is transferred to the roof bracing. Whendetermining the wind load, it is important to distinguish between the effects of negativeand positive pressure due to the difference between the compression and tensionresistance values of hollow sections. The following wind load is obtained on the sidewall:

where

H1 is the height of the eaves H2 is the height of the building at the apex

The axial force values in diagonal members D1 and D2 (Figure 6.27) is obtained fromthe reaction at end of the roof bracing:

The resistance of the upper chord of the purlin truss and that of the eaves beam, to theaxial force due to wind load, must also be checked, as well as the deflection of the roofbracing.

D qL

kN

D qL

kN

L L

d

d

c y c z

10 5

404 28

0 5 100

40279 4

20 5

402 14

0 5 100

40139 7

20 24

65 2

2236355

1 1721 6

0 328 2236355

1 1

1

2

2 2

:,

cos,

,

cos,

:,

cos,

,

cos,

,

,,

,,

. .

N

N

m

N kN > N

N

Sd.1

Sd.2

t.Rd Sd.1

b.Rd

= = ⋅ =

= = ⋅ =

= = + =

= =

= ⋅ = 236236 4, kN > NSd.2

The diagonal is supported at purlin trusses, so the buckling length is:

Consider a hollow section of dimensions 120 x 120 x 5. The following values fortension and buckling resistance is obtained:

(tension)

(compression)

OK!

OK!

q c q c H H

kN

m

q c q c H H

kN

m

d Q p ref e

d Q p ref e

1 1 1 2

2 1 1 2

0 375

1 5 0 6 0 6 0 375 10 5 4 4 28

0 375

1 5 0 3 0 6 0 375 10 5 4 2 14

= ⋅ ⋅ ⋅ ⋅ +( )

= ⋅ ⋅ ⋅ +( ) =

= ⋅ ⋅ ⋅ ⋅ +( )

= ⋅ ⋅ ⋅ +( ) =

γ

γ

.

.

,

, , , , , ,

,

, , , , , ,

(pressure)

(negative pressure)


207

6.6 References

[1] ENV 1991-2-1: Eurocode 1: Suunnitteluperusteet ja rakenteiden kuormat, osa 2-1:Rakenteiden kuormat, tiheydet, oma paino ja hyötykuorma, 1995(ENV 1991-2-1:Eurocode 1: Basis of design and actions on structures. Part 2-1: Actionson structures. densities, self-weight and imposed loads, 1995)

[2] ENV 1991-2-3: Eurocode 1: Suunnitteluperusteet ja rakenteiden kuormat, osa 2-3:Rakenteiden kuormat, lumikuormat, 1995(ENV 1991-2-3: Eurocode 1: Basis of design and actions on structures. Part 2-3: Actionson structures, Snow loads, 1995)

[3] ENV 1991-2-4: Eurocode 1: Suunnitteluperusteet ja rakenteiden kuormat, osa 2-4:Rakenteiden kuormat, tuulikuormat, 1995ENV 1991-2-4: Eurocode 1: Basis of design and actions on structures. Part 2-4: Actions onstructures, Wind loads, 1995)

[4] SFS-ENV 1991-1: Eurocode 1: Suunnitteluperusteet ja rakenteiden kuormat, osa 1Suunnitteluperusteet, 1995(ENV 1991-1: Eurocode 1: Basis of design and actions on structures. Part 1: Basis ofdesign, 1995)

[5] SFS-ENV-1993-1-1: Eurocode 3 Teräsrakenteiden suunnittelu, Osa 1-1: Yleiset säännöt jarakennuksia koskevat säännöt, 1993 (Sisältää myös liitteen K: ENV 1993-1-1:1992/A1:1994)(ENV 1993-1-1: Eurocode 3: Design of steel structures. Part 1.1: General rules and rulesfor buildings, 1993)(Include also annex K: ENV 1993-1-1:1992/ A1:1994)

[6] ECCS: Technical Committee 8- Structural stability- Technical working group 8.1/ 8.2Skeletal structures: Analysis and design of steel frames with semi-rigid joints, First edition1992




208

7 SHOP FABRICATION AND ERECTION

In addition to structural design, the cost of a hollow section structure includes shop fabricationand erection. Design aims at minimizing the weight of the structure, since the price of a hollowsection is almost directly proportional to its weight. However, the lightest structure is notnecessarily the most economical solution with regard to the whole of the construction project. Amaximally optimized structure may be expensive to manufacture and install, which may resultin losing the savings gained in material costs to increased manufacturing and erection costs.To reach an optimal result, it is important that the designer, the shop and the site all work inclose cooperation and that all parties have sufficient knowledge about on all aspects of theconstruction project.

The general principle is to perform the most demanding and complex phases in the workshopto make the erection quick and cost-efficient. In practice this means that all welded joints aremade at the shop, and erection then consists of joining of preassembled units with bolts.

7.1 Cutting of hollow sections

Hollow sections can be ordered either standard-length ( 6, 12 or 18 m) or cut to size (even 24m). No waste pieces are produced when ordering cut-to-size sections. The cutting is oftenmade at the fabricators if the structure requires slanted cutting surfaces (e.g. lattices). Squareand rectangular hollow sections are easy to cut, since the cut can usually be made in oneplane. However, various cutting planes are needed in overlapped joints. Circular hollowsections are more complex to cut particularly for joints between several circular sections. Insuch a case, the end of the circular hollow section must often be profiled.

7.1.1 Cutting of circular hollow sections

A circular hollow section can sometimes be joined to the edge of another circular section bycutting the hollow section in one plane. A prerequisite for this is that the external dimensions ofthe hollow sections differ distinctly. This way, the root gap remaining at the edges of thesmaller hollow section is sufficiently small with regard to welding. A circular hollow section canbe cut in one plane if the following conditions are met [1]:

where

In practice, most structures made from circular hollow sections do not meet the conditions of formulae (7.1) and (7.2).

is the root gap at the external edge of the smaller hollow sectionis the root gap at the internal edge of the smaller hollow sectionis the wall thickness of the larger hollow section

is the wall thickness of the smaller hollow section

g

g

t

t

1

2

0

1

t1

g 2

g 1

g t t

g1 0 1

2 3 2

≤ ≤≤

and g

mm1 (7.1)

(7 ).


209 ��

d1

d0 t0

The size of the root gap in circular holllow sections can also be decreased by cutting the hollowsections at different angles. Using the expressions in Figure 7.1, the following equations forcutting angles αg and αd are obtained [1]:

where

Figure 7.1 Cutting of circular hollow sections

�

��

��

��

�

L L

h

αd

t1

d 0

αg

d1

θ

L r r t

hd d

r t

rd t

ddt

= − −( )

= − − −( )

= −

12

1 12

0 02

1 12

11 1

0

1

1

2 42

2is the diameter of the larger hollow section

is the diameter of the smaller hollow section

is the wall thickness of the smaller hollow section

α θ θθ θ

α θ θθ θ

g

d

h

r h L

h

r h L

= − + ⋅+ ⋅ − ⋅

= − + +− ⋅ − ⋅

90

90 4

1

1

o

o

arctansin

cos sin

arctansin

cos sin.

(7.3)

(7 )


210

7.1.2 Cutting methods

Sawing

Sawing is the most commonly used method for cutting hollow sections. Usually, a disc saw or aband saw is used. An increase in the sawing speed usually decreases the accuracy in cuttingand generates burrs which then need to be removed. In addition to the sawing speed, the easytransport of sections to and from the sawing site is a factor worth considering. It is also possibleto cut both ends of a hollow section simultaneously to save time. Saw blades must be changedoften, as the decreased sharpness of the saw blade increases the dimensional deviation andthe quality of the cut seam deteriorates.

Other cutting methods

a) Thermal cutting

Thermal cutting, when made free-hand, is a less accurate cutting method than sawing. It issuitable, for instance, for shaping the ends of circular hollow sections in lattice joints,particularly if a profile cutting machine is used.

b) Cutting by punching

This method is feasible only with thin-walled hollow sections. Its advantage is the possibility ofgenerating the most complex of cutting surfaces. In circular hollow section joints, the ends canbe shaped in one go with punching.

c) Laser cutting

Laser cutting is an accurate method. A further advantage is the reduced area of temperaturechange in the area of the cut. A disadvantage of laser cutting is the expensive equipmentneeded.

7.1.3 Notching of hollow section ends

The use of splices in bolted joints may require notches in the hollow section wall (Figure 7.2a).In joints subjected to small loads however it is more better cut the plate rather than the hollowsection (Figure 7.2b). In order to transfer forces in joints subjected to heavy loads, it is, con-versly, better to cut the hollow section and keep the splice intact, since it is difficult to make asufficiently long weld inside the hollow section. Also in joints using small hollow sections, it isnecessary to cut the section, since welding the interior is practically impossible.

Notches can be made using the cutting methods described above.

Figure 7.2 Joining splice plates to hollow sections

a) b)


211

7.2 Bending of hollow sections

A hollow section can be bent either by cold or hot bending. Cold bending is more cost-efficientand simpler, and thus more commonly used. Circular hollow sections are easier to bend thansquare and rectangular hollow sections, since in the latter case the shape of the cross-sectiontends to get distorted.

Factors affecting the success of bending include:

- the ultimate strength and yield strength of the material- the chemical composition and ultimate elongation of the material- the relation of the wall thickness to the height and width of the hollow section (t/h, t/b, t/d)- the relation of the bending radius to the height and width of the hollow section (r/h, r/d, r/b)

When using square and rectangular hollow sections, the effect of the distortion of the cross-section shape to structural appearance must be evaluated on a case by case basis. The morecritical the appearance of the structure, the greater the bending radius must be. Hollowsections with lower ultimate strength and yield strength are easier to bend. In addition, agreater wall thickness in relation with the height and width of the hollow section facilitatesbending. A small radius is always more complicated to produce than a greater one.

The cross-sectional properties of square and rectangular hollow sections decrease duringbending. Reduced values for second moment of area are presented in Appendix 9.7. Thedistortion of a hollow section’s wall during bending must also be taken into account whendetermining its compression resistance.

More demanding bending procedures usually require practical expertise, a workshop thatspecializes in bending and is equipped with appropriate machinery. For successful bending,the cooperation and expertise of the designer and the manufacturer is important.

7.2.1 Bending methods for hollow sections

Roller bending In roller bending, the hollow section is directed throughthree or four rollers. The size of the rollers is determinedby the size of hollow sections. The middle rollersdetermine the magnitude of the bending radius. Normally,one of the rollers is freely rotating. Minimum bending radiifor square and rectangular hollow sections are shown inAppendix 9.7.

Induction bending

In induction bending, the hollow section is heated during bending with an induction coil. A smallportion at a time is simultaneously heated and bent. This is repeated until the entire hollowsection is processed. Compared to roller bending, induction bending is a more expensivemethod, but it has the advantage of producing smaller minimum bending radii.


212

Curved lattice structures can be made from straight elements. The curved shape is producedby joining the straight chord member elements together in an angle corresponding to thebending radius (Figure 7.3).

Figure 7.3 Curved hollow section lattice

7.3 Bolted joints

The bolts used in load-bearing structures are normally of strength grade 8.8. Therecommended clearances of bolt holes are presented in Table 7.1.

Table 7.1 Recommended clearances of bolt holes [2]

Bolts are tightened such that the normal stress (or shank tension) Fp generated in the bolt isequal to (7.5) [2]:

where

It is advisable to prevent the loosening of bolts by locking the nuts with an appropriate method(e.g. torque type nuts, special washers or glue).

Holes can be made either by drilling or punching. However, in tension members the holes forbolts must always be made by drilling. In joints subjected to lighter loads, self tapping bolts ordrill bolts can be used. With these types of bolts, the most commonly used sizes are 5,5 and6,3 mm. The bolt is always fixed at the side of the thinner element side of the joint. The corehole for a self tapping bolt must be slightly smaller than the bolt itself. The use of drill bolts issimpler and faster, since no core hole needs to be drilled. When using self tapping bolts anddrill bolts, the instructions provided by the bolt manufacturer must be followed.

is the ultimate strength of the bolt

is the tension cross-section area of the boltfA

ub

s

F f Ap ub s= ⋅0 7 5, .(7 )

Bolt diameter Clearance Bolt diameter Clearance Bolt diameter Clearance (mm) (mm) (mm) (mm) (mm) (mm)

12 1 18 2 24 214 1 20 2 27 316 2 22 2 30 3


213

In Chapter 3, we looked at various joint details for bolted joints. The openings were usuallylocated in the structural element, usually a plate, joined to the hollow section. The simplest waywould be to drill the hole directly in the hollow section, but in practice, the slender walls ofhollow sections cannot bear much load. However, by using special drilling methods or specialbolts, the hollow section joints can be made directly via the hollow section wall.

7.3.1 Friction Drilling in the Wall of the Hollow Section Tube

When using friction drilling a special hard metal tool is used, which heats the wall of the hollowtube due to friction. The heated material is first pressed to the outside of the tube surface andafter the penetration to the inside. Thus the basematerial itself forms a bushing, which can bethreaded. The threads are made in a separate workphase, either with a traditional thread cut-ting tool or with a thread rolling peg. Due to the bushing, the thread length is almost doubledcompared to the wall thickness. The method has been tested up to material thicknesses of 12,5 mm. With increasing material thicknesses and bolt diameters special friction drilling settingsare required from the drilling equipment. For building purposes sizes M16 - M20 are suitable.

Kuva 7.4 Friction drilling

7.3.2 Expansion bolt joints

In expansion bolt joints, the opening to the section wall is drilled using normalmethods. The opening is then equipped with an expansion bolt whichfunctions in a similar way those used in connections to concrete structures.Inside the expansion bolt is a threaded hole in which the bolt is placed. Whenthe bolt is tightened, the conical end of the expansion bolt clamps against thesection wall. Simultaneously, the cone spreads the blanket of the expansionbolt, which generates the necessary tightening torque in the bolt. Themethod is suited for M8-M20 bolts and for all wall thicknesses [4].

7.3.3 Pilot tap joints

Joints subjected to light loads only can also be made using tapped studswhich are welded to the section wall. The opening on the joined elementmust be enlargened from the section wall side, so that the weld does notbear against it after the bolts are tightened. Another alternative is the use ofwashers between the element to be joined and the section. Special attentionshould be paid to the protection of tapped studs during transport andinstallation.

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yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy

��

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��

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��


��

yyy

Phase 1 Phase 2


214

��

��

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7.4 Welding of hollow sections

The structural steel used in Rautaruukki hollow sections has good weldability with all weldingmethods. Weldability depends on the welding method and the chemical composition of thematerial. The chemical composition of steel with regard to welding is best described by thecarbon equivalent value (CEV):

With normal wall thickness values of Rautaruukki longitudinally welded hollow sections (< 16mm) and with steel grade S355J2H, no special welding methods are needed (CEV ≤ 0,39).Special methods are necessary only when the carbon equivalent value is higher than 0,4. If thecarbon equivalent value is greater than 0,45, preheating is required. When welding twodifferent materials together, the parameters of the welding process (temperature, weldingmethod and filler material) must be selected according to the material which has higherstrength. The welded elements must be dry and free from grease and oil.

7.4.1 Quality levels

Table 7.2 Quality levels for arc welding according to the European standard

The quality level is determined by the designer responsible, who must consider structuralsafety requirements and the ease of inspection and manufacture.

When determining the weld quality level, factors to be considered include post-weld surfacefinishing, type of structural loading (static or dynamic), operating conditions (temperature andenvironment) and the consequences of potential failure. In addition to affecting the weldingcosts, the choice of quality level also has an effect on the weld inspection and testing costs.

7.4.2 Welding methods

Two principal methods used in the welding of hollow sections are manual metal arc weldingwith covered electrode and gas shielded arc welding. Metal arc welding is used principally inon-site installations. Its advantage is the light-weight and easily transportable equipmentrequired.

Gas shielded arc welding is the most common method used in shop fabrication. Advantagesassociated with gas shielded arc welding are better productivity and the possibility for theautomation of the welding procedure.

In the welding of hollow section members, the requirements set for the welder’s performanceby the standard EN-287-1 must be considered [9].

Level symbol Quality levelD ModerateC IntermediateB Stringent

CEV CMn Cr Mo V Ni Cu= + + + + + +6 5 15

6(7 ).


215

7.4.3 Welding sequence

Due to the stresses generated by welding and to the deformations of the joined elements, thecorrect welding sequence is important. The welding of hollow sections should not be started orfinished in a comer. The maximum throat thickness generated in one pass is 5 mm. Thewelding sequence used depends on the accessability of the weld and the ability to turn thesection.

In Figure 7.5a, the section can be rotated horizontally. The welding point thus remains thesame and the element is rotated a full circle. In Figure 7.5b, the element is positionedhorizontally and can also be rotated about its axis. The welding direction is at first from thebottom upwards. The lower side is welded after rotating the section. In Figure 7.5c, the sectionis fixed in a vertical position. The welding is performed continuously over the entire section. InFigure 7.5d, the section is fixed in a horizontal position. Now, the lower seam must be weldedfrom below.

Figure 7.5 Welding sequence

7.4.4 Fillet and butt welds

With hollow sections, the aim is to design joints in such a manner that fillet welds can be used.Fillet weld are the simplest and most cost-efficient weld type, since no weld preparation isneeded. However, depending on the joint geometry, a groove must, in some cases, be made toensure a sufficient throat thickness of the weld. It is advisable to make the weld as symmetricalas possible to minimize the consumption of the weld metal. Weld preparations for end-to-endconnections are treated in section 3.3. Fillet and butt welds for lattice joints made of square andrectangular hollow sections are shown in Table 7.3.

��

��

��

��

a) b)

c) d)

360°

1

1

1

1

1

1 2

2

2

2

2

4

3

3

3

4

4

180°

180°


216

Table 7.3 Fillet and butt welds in joints made of square and rectangular hollow sections

X Z

θ ≤ 60° End chamfered after cutting

t1 < 8 mm

θ > 60°

t1 < 8 mm

t1 ≥ 8 mm

θ > 60°

t1 ≥ 8 mm

Y

b0-b1 ≥ 2r0

r0 is the external corner radius

��

r0

t1

t0

b0

θ

X YZ

b1

∗ = − −( ) −( ) − −( ) + ⋅ − − < −( )root r r t b b b b t r t w b r t gap hen b00 0 1 0 1 0 12

1 0 12

1 0 10 25 2 2, ,

< 2

< 2

< 2

t1

t 1 1-2,5

< 2

< 2

t1

> t 1

1-2,5

r0< 2

< 2 < 2

>60°

>60°>60°

>60°

0 ≤ b0- b1 < 2r0 and

maximum root gap * 2 mmminimum groove angle 60°

0 ≤ b0- b1 < 2r0 and root gap * more than 2 mmThe brace member end must be shaped


217

7.4.5 Preheating

The structural steels used in hollow sections do not require preheating when the maximumnominell yield strength of the steel is 355 N/ mm2 and the wall thickness is less than 13 mm.Hollow sections must, however, be preheated to the temperature of 40 °C for a minimum of 75mm away from the joint, if the surface of the hollow section is damp or the temperature of thestructure to be welded is below 5 °C [1]. In thermal cutting, preheating is usually not needed ifthe carbon equivalent value of steel is less than 0,45 [5].

7.4.6 Residual stresses

During welding, residual stresses due to the temperature are always generated in the steel.This may result in permanent deformations of the member.

Deformations are generated both parallel and perpendicular to the weld direction. Themagnitude of the deformation depends on the welded material, the number of passes and therigidity of the structure. In addition to deformation, there is also angular distortion, in which theangle between welded elements changes due to welding stresses.

Figure 7.6 shows the angular distortions due to welding in their unrestricted form.

Figure 7.6 Angular distortions due to welding

In practice, the elements are supported during welding in such a manner that the deformationsare restricted, which generates stresses in the material. The deformation is significantlyreduced if a fillet weld is made with a both side groove, or a double V groove is used instead ofa single V groove. In this case, welding is made from both sides in turns. A single side Vgroove can also be made by placing the welded elements in an angle which eliminates thedeformation due to welding. The estimation of the angle magnitude is based on empiricalknow-how.

When welding an entire structural element such as a lattice, it is recommendable to first tackweld the joints, then make the fillet welds and finally the full penetration single-V butt welds [1].

Stress relief by heat-treatment is usually needed only in special cases. The normal stress relieftemperature is 550-650 °C [5].


218

For instance the following factors increase the residual stresses due to welding:

- large throat thickness of weld- large distance of the weld from the neutral axis of the hollow section- supporting the hollow sections during welding- high rigidity of the hollow sections- wrong weld sequence- weld methods with high heat input

7.4.7 Inspection of welds

The determination of the weld quality depends on the required quality level, the structuralloading and the consequences of a potential failure. Non-destructive weld inspection methodsare presented in EN 10246. In fatigue loaded structures, welds are usually the weakest links,and the improvement of the weld quality therefore increases the service life of the entirestructure. The weld quality can be ensured by various methods, which are selected on a caseby case basis.

Visual inspection

Visual inspection is the least expensive and least accurate inspection method. The inspectormust have sufficient experience in welding and the necessary expertise for the task. Visualinspection can detect errors in the throat thickness and the largest flaws which reach thesurface. Visual inspection is not sufficient for detecting minor defects such as hairline cracks.

Magnetic particle testing

Magnetic particle testing is a method for detecting surface flaws. The inspected surface iscovered with a fine-grained ferrite powder, and a magnetic field is generated in the material byan electric coil. The powder particles form a line at crack locations. Cracks as small as 0,1 µmwide can be detected in the test.

Penetrant testing

Penetrant testing is another method for locating surface defects in welds. First, the inspectedsurfaces are carefully cleaned, after which dye penetrant is applied on the surface. The dyehas low surface tension and high capillarity and can thus penetrate even the narrowest ofcracks. The dye is left on the surface for 5-10 minutes, after which time excess dye is wipedoff. The surface is dried and covered with developer which reacts with the dye penetrant,bringing it to the surface. The pattern generated by the developing agent indicates the defecttypes and their severity.


219

Ultrasonic inspection

Ultrasonic inspection also serves to detect internal defects. Ultrasonic inspection is a very rapidmethod but it requires an experienced and qualified inspector. The test consists of sendinghigh-frequency waves to the point being inspected. The spectrum of the waves is examinedwith an oscilloscope which shows the weld defects as intensity peaks. The minimum materialthickness feasible for ultrasonic inspection is approximately 6 mm, but with current equipment,welds connecting plate thicknesses as low as 4 mm can be inspected provided that the weldmaterial itself is thicker than 6 mm. The disadvantage of the method is that defects parallel tothe sound waves are difficult to detect. Due to this, ultrasonic inspection must always beperformed in various angles.

Radiographic inspection

Radiographic inspection can also detect internal defects in the weld. In this method, the weld isradiated with x-rays or gamma rays, and the passage of rays through the weld location isphotographed. Radiographic inspection is not feasible for inspecting joints in structures withvarying wall thickness or lattice joints. X-rays are hazardous to health, and the inspection mustbe conducted in a controlled environment.

7.5 Tolerances

Tolerances in the industrial manufacture of hollow sections are dealt with in Chapter 1. Tables7.4, 7.5 and 7.6 give the tolerances for the geometry of fabricated elements. Erection toler-ances are shown in Table 7.7. The tables concentrate on the most important tolerancesconcerning hollow sections. Quality requirements regarding fabrication procedures anderection are discussed in more detail in references [2], [6] and [7].


220

Table 7.4 Tolerances for singular members in workshop fabrication [2]

Tolerance Parameter Permitted deviationHollow section length measured atflange centre line

Cutting length in general:∆ = ± (2+ L / 5000) (mm)

Cutting length when force istransferred by contact in bearing: ∆ = ± 1 mm

Hollow section straightness byhorizontal and vertical axes

∆ = L / 1000 (mm) ∆ = 3 mm Select whichever is greater

Deviation of curvature in a benthollow section measured at thecentre point of span

∆ = L / 1000 (mm) ∆ = 6 mm Select whichever is greater

Squareness of hollow section end Squareness in general:∆ = ± h / 300

Squareness when force istransmitted by contact in bearing:∆ = ± h / 1000


221

L + ∆

L∆

L

∆

h

∆

Table 7.5 Tolerances for openings and cuttings in workshop fabrication [2]

Tolerance Parameter Permitted deviationDeviation of a single bolt openingfrom the designed location

∆ = ± 2 mm

Deviation of a bolt group from thedesigned location dimension a: ∆ =

*

dimension b: ∆ = ± 2 mm

dimension c: ∆ = ± 5 mm

dimension d: ∆ = ± 2 mm, when h ≤ 1000 mm

dimension d: ∆ = ± 4 mm.when h > 1000 mm

Deviations in the dimensions of a cutelement

dimension d: ∆ =

dimension L: ∆ =

Squareness of the cutting of hollowsection wall ∆ = ± 0,1t

* In reference [2] tolerance is ∆ = ± 5 mm, but a negative tolerance may cause a bearingresistance remarkably smaller than the one determined by calculations


222

∆

a

b

c

d

h

L

d

∆

t

+2 mm-0 mm

+2 mm-0 mm

+5 mm–0 mm

Table 7.6 Tolerances for lattices in shop fabrication [2], [6]

Tolerance Parameter Permitted deviationStraightness of brace member ∆ = L / 500 (mm)

∆ = 6 mmL is the length of the brace memberSelect whichever is greater

Distance between corner points ∆p = ±5 mm

∑∆p = ±10 mm

Gap tolerance g ≥ t1 + t2 and ga ≥ 1,5 t0*

∆g = and ∆ga =

t1 and t2 are the thickness values ofbrace member walls tt0 is the wall thickness of the chord

Lattice height, width and diagonal ∆D = ±3 mm, when D ≤ 300 mm∆D = ±5 mm, when 300 < D < 1000 mm∆D = ±10 mm, when D ≥ 1000 mm

∆W = ±3 mm, when W ≤ 300 mm∆W = ±5 mm, when 300 < W < 1000 mm∆W = ±10 mm, when W ≥ 1000 mm

∆X = ±3 mm, when X ≤ 300 mm∆X = ±5 mm, when 300 < X < 1000 mm∆X = ±10 mm, when X ≥ 1000 mm

Straightness parallel to the latticeplane

∆max = L / 500∆max = 12 mmSelect whichever is greater

* Deviates from tolerances in reference [6]


223

��

L

+5 mm*–0 mm

+5 mm*–0 mm

p

∆

∆

∆g, ∆ga

ga

g

W + ∆W

D +

∆D

X + ∆X

∆1∆2 ∆3 ∆4 ∆5

∆6

Table 7.7 Installation tolerances of steel structures [2], [7]. The smallest tolerance value is selected from the ones presented in references.

Tolerance Structure Permitted deviationDeviation of the distance betweenadjacent columns

e = ± 5 mm

Slope of columns in a multi-storybuilding between successiveintermediate floors

e = ± 0,002h

h is the height of a floor

Transverse deviation of a column ina multi-story building at all levels ofintermediate floors

Σh is the height in questionmeasured from the base level n is the number of floors from thelevel in question to the base level

Slope of a column in a one-storybuilding when the column is notsupporting a crane gantry

h is the height of the column

Slope of a column in a portal framewhen the column is not supporting acrane gantry (average)

e = ± 0,010h (singular value)

Slope of a column in one-storybuilding or in portal frame when thecolumn is supporting a crane gantry

h < 5 m: e = ± 5 mm

5 ≤ h ≤ 25 m: e = ± 0,001h

h > 25 m: e = ± 25 mm


224

��

eh

n= ± Σ

300

L + e

e

h

e

Σh

e

h

h 1h 2

h 3

eh= ±

300

e eh1 2

20 002

+ = ± ,

e

h

h

e1 e2

7.6 Assembly of trusses

In lattice assembly, an important factor is the number of lattices to be fabricated. With largeseries, it is worthwhile speding more time in the design and preparation of the assembly frame(or jig). Also the accuracy requirements must be taken into account when designing the jig.Regarding fabrication costs, it is important to be able to store the hollow sections close to theassembly site, since this saves time during transport.

The jig must be made of sufficiently strong elements so that thermal and mechanical stressesgenerated during the assembly do not distort it. The distances between lattice membersupports must be sufficiently short to avoid deformations in the lattice due to welding. In the jig,it is advisable to join the flange plates of the lattice to the plates welded to the assembly framewith bolts (Figure 7.7).

Figure 7.7 Assembly frame

To speed up the shop fabrication phase, the members in the jig can be tack welded together,and the final welding can be done at another worksite. In such a case, attention must be paid tothe firmness of tack welding to avoid the generation of high deformations during the finalwelding.

In gapped joints, attention must be paid to retaining a sufficiently large gap. Even small gapsmust meet the minimum gap tolerances shown in Table 7.6. In overlapped joints, the largerbrace member that is overlapped must always be welded first.


225

7.7 Fire protection

It is not always an advantage to prepare the fire protection in the workshop. Most fire retardantmaterials will be damaged during transport and must then be repaired on-site. The fireprotection method best suited for shop fabrication is fire retardant (intumescent) painting. Eventhen, sufficient care must be taken during transport to keep the damage to the painted surfaceto a minimum. The damaged spots must be repaired on-site, and it may, in extreme cases benecessary to apply a new coat of paint to ensure the required fire resistance period. Concretefilling can be carried out in the shop, but this increases the weight of hollow sections, thuscomplicating their transport and installation. Pouring of concrete is usually done on site, sincemost intermediate floor slabs require a surface-cast layer in any case.

7.8 Transport and storage

During the structural design phase, the transport of structural elements must also be taken intoaccount. Large pre-assembled units make the erection quicker but may increase tranport costs,especially if special transport is needed. When planning special transport, it is necessary tokeep in mind the limits to the size of units set by bridges and roads at the proximity of the site.In international projects, it must be considered that regulations concerning maximum weightand dimensions of the transport vary country by country. Road transport is the simplest way ofmoving the material when shop and site are located relatively close to each other.

The following issues should be considered in the transport of hollow sections [3]:

- the tarpaulins used to cover the load must be dry, clean, undamaged and sufficiently large- hollow sections must be placed in the platform of the transport vehicle so that they are not

exposed to bumps, abrasion or any other type of damage- when loading hollow sections one over the other, the sections with the thickest walls and the

greatest length and weight must be placed the lowermost- heavy products which might cause damage to hollow sections must not be piled on them- the platform must be clean, dry and even- the load must be secured to prevent shifting during transport- the load is tied with straps so that it does not touch the side or end columns of the platform- straps or chains must not be attached so tightly that they cause dents on the hollow

sections; if necessary, the points of contact must be covered- suitable props must be used in the transport

The following issues must be kept in mind in the storage of hollow sections [3]:

- the depot must be clean, dry and properly ventilated- the entry of water from condensation into the sections must be prevented- a sufficient amount of props below and within the stack must be used


226

7.9 Erection

The erection of hollow section structures is similar to the erection of other steel structures.However, the greater torsional stiffness of hollow sections and the greater bending stiffnessabout both axes make them easier to lift and erect and the need for temporary lateral supportsis less than with conventional sections. Structures utilizing hollow sections are also less proneto the effect of wind during erection compared to open sections.

A hollow section structure is erected according to an erection plan made by the designer orfabricator. The plan must take into account the routes of vehicles and the craneage. Connec-tions on site are usually bolted to allow speedy erection.

it is usual to commence erection from one of the braced bays, usually at one end. The firstframe may require temporary guys, but the usual practice is to provide 4 holding down bolts inthe baseplate to give temporary stability. Erection then proceeds witt the next frame are thepermanent bracing is erected between the frames. Purlins are then connected to provide rafterstability. Having ensured that the end two frames are lined and levelled, erection of the otherframes can proceed. The purlins and bracing being erected to provide stability to each frame inturn.

The stability of the roof structure under its own dead weight should be checked to ensure that itcan safetly be lifted into place. If necessary frames can be lifted in braced pairs to ensure sta-bility of individual frames.

Hollow section structures are light-weight, and even large units can be easily lifted and installedon the site. The bolts are tightened only after the positions of structural elements have beenchecked. After this, the bolts are tightened. (Section 7.4). The assembly of large latticestructures, especially that of space frames, is usually most easily carried out at ground level,after which the finished lattice is erected to its final position.


227

7.10 References

[1] CIDECT: Design guide for fabrication, assembly and erection of hollow section structures,1996

[2] ENV 1090-1: Teräsrakenteiden valmistus ja asennus- Osa 1: Yleiset säännöt ja raken-nuksia koskevat säännöt, 1996(ENV 1090-1: Execution of steel structures- Part 1: General rules and rules for buildings,1996)

[3] Rautaruukinterästuotteiden käsittelyohje (The Handling Manual of Rautaruukki´s SteelProducts, in finnish), 1995

[4] CIDECT: Report 6G-14(A)/96: Hollofast and hollobolt system for hollow sectionconnections, 1996

[5] Rautaruukin teräkset: Hitsaajan opas (Rautaruukki´s Steels. The Welder Manual, infinnish), 1995

[6] prENV 1090-4 Execution of steel structures: Part 4: Supplementary rules for hollow sectionlattice structures, 1997

[7] ENV-1993-1-1: Eurocode 3: Teräsrakenteiden suunnittelu. Osa 1-1: Yleiset säännöt jarakennuksia koskevat säännöt, 1993(ENV 1993-1-1: Eurocode 3: Design of steel structures. Part 1.1: General rules and rulesfor buildings, 1993)

[8] EN-25817: Terästen kaarihitsaus: Hitsiluokat, 1993(EN-25817: Arc-welded joints in steel. Guidance on quality levels for imperfections, 1993)

[9] EN-287-1: Hitsaajan pätevyyskoe: Sulahitsaus: Osa 1: Teräkset, 1992(EN-287-1: Approval testing of welders. Fusion welding. Part 1: Steels, 1992)


228

8 CORROSION PROTECTION

Corrosion of steel surfaces is caused by oxygen and moisture in the air. The corrosion rate ofsteel depends on air temperature and moisture as well as air pollution. The pollution factorhaving the greatest effect on corrosion is the chloride and sulphur content of the air. Thedecrease of temperature below zero decelerates corrosion or may prevent it completely. Theshape of a hollow section is advantageous regarding corrosion protection, since the arearequiring protection is small compared to the weight of the section.

8.1 Corrosivity categories

Based on the corrosivity of the climate, environments can be divided into categories. Thecorrosivity categories presented in reference [1] are shown in Table 8.1. The temperature andmoisture content of the building also influences corrosion. The propagation of corrosion is likelywhen relative humidity exceeds 80 % and temperature exceeds 0 °C. However, the likelihoodof corrosion is great even at lower humidity levels if the amount of air pollution is great orsalinity is high.

The effect of strong chemical agents (acids, alkalis, salts, organic solvents) must be taken intoaccount when designing the corrosion protection. The corrosion rate increases substantially ifthe structure is exposed to simultaneous mechanical and chemical stress.

Table 8.1 Corrosivity categories [1]

Corrosivitycategory

Thickness loss oflow-carbon steel(µm/ year) *

Thickness loss ofzinc (µm/ year) *

Examples

C1 ≤ 1,3 ≤ 0,1 Heated buildings with clean atmospheres, e.g. offices,shops, schools, hotels.

C2 1,3- 25 0,1- 0,7 Rural areas with low level of air pollution. Unheatedbuildings where condensation may occur, e.g. depots,sports halls.

C3 25- 50 0,7- 2,1 Urban areas with moderate amount of air pollution.Coastal areas with low salinity. Production rooms withhigh humidity and some air pollution, e.g. food-processing plants, laundries, breweries, dairies.

C4 50- 80 2,1- 4,2 Industrial and coastal areas with moderate salinity.Chemical plants, swimming pools, coastal ship- andboatyards.

C5-I 80- 200 4,2- 8,4 Industrial and coastal areas with high humidity andaggressive climate. Buildings with almost permanentcondensation and with high level of pollution.

C5-M 80- 200 4,2- 8,4 Coastal and sea areas with high salinity. Buildingswith almost permanent condensation and with highlevel of pollution.

* after first year of exposure


229

8.2 Surface preparation

Hollow sections are normally delivered unprotected or with a light protective oil coating.Cleaning the surface before applying paint is thus essential for successful corrosion protection.The cleaning method selected depends on the amount and quality of the impurity and theshape and size of the member. Cleaning methods are summarized in Table 8.2. In workshopand paint shop conditions, the most commonly used cleaning method is shot blasting(preparation grades Sa 21/2 and Sa 3). Grease and salt must be removed from the surfacesbefore cleaning if acid pickling or shot blasting is used. When renovating old structures, wirebrushing can be used in some special cases (St 2 and St 3).

Table 8.2 Cleaning methods for steel surfaces [3]

8.3 Anti-corrosive painting

Paint is applied on a dry and clean surface according to the paint manufacturer’s instructions.The best results are obtained in controlled paint shop conditions. During site painting, therelative humidity of air should be less than 80 % and the temperature should exceed +5 °C andbe at least 3 °C above dew point. With some paint types (e.g. epoxy paints), a higher paintingtemperature is required. Recommended paint combinations for hollow sections are presentedin Table 8.3. Paint selection also depends on the special characteristics required of the paintsurface. Table 8.4 presents the applicability of different paints in cases where specialcharacteristics are required of the painted surface.

Prefabrication primer is a rapidly drying paint applied on the section surface in a thin layer ofapproximately 15 µm. The purpose of the primer is to protect hollow sections during storageand transport, and it must always be removed before applying the final paint.

In normal conditions, internal corrosion of hollow sections is rare. The insides of hollowsections usually need not be protected against corrosion. However, the seepage of rain waterinto the sections must be prevented. Hollow section structures must be provided with openingsfor the removal of water from condensation, especially if there is a possibility of freezing.

Preparationgrade

Cleaning method Surface quality

Sa 2 1/2 Shot blasting Mill scale, rust, paint coatings and foreign matter are removed. Anyremaining traces of contamination shall show only as slight stains inthe form of spots or stripes.

Sa 3 Shot blasting Mill scale, rust, paint coatings and foreign matter are removed. Thesurface shall be metallic clean.

St 2 Wire brushing Poorly adhering mill scale, rust, paint coatings and foreign matter areremoved.

St 3 Wire brushing Poorly adhering mill scale, rust, paint coatings and foreign matter areremoved. The surface shall have a metallic sheen.

Be Acid pickling Mill scale, rust and residues from paint coatings are removedcompletely. Paint coatings shall be removed prior to acid pickling.


230

In designing to prevent corrosion, one of the central issues is to avoid recesses and pocketswhere water might be trapped. For instance, reinforcing the lattice corner joints with plateswelded to brace member side generates a space where water and debris may accumulate. Thejoining of two metals with different electrochemical potential (e.g. steel and copper) must beavoided, or the metals must be insulated from each other, since corrosion will take place at thejoint in the less noble of metals.

The metal surfaces to be painted must be as smooth and rounded at corners as possible. Dueto their rounded corners, hollow sections are also suitable for blast-cleaning. Weld splatter andother irregularities must be removed before painting. The welds must also be as smooth-surfaced as possible to avoid spots remaining inadvertently unpainted. Joints must bedesigned so that the structure can easily be painted from all sides. When painting with a brush,the space between splices must be at least as wide as the brush. The corrosion resistance ofbolts and nuts must be at least as high as that of structural materials.

Table 8.3 Recommended paint combinations in various corrosivity categories [4]

Corrosivitycategory

Surfacepreparationgrade

Primingcoats

Top coats

Paint type Numberof coats

Nominal DryFilmThick-ness(µm)

Paint type Number ofcoats

Nominal DryFilmThick-ness(µm)

C1, C2 Sa 2 1/2 AK 1- 2 80 AK 1- 2 80C1, C2 St 2 AK 2 80 AK 2- 3 120C1, C2 Sa 2 1/2 AY, CR, PVC 1- 2 80 AY, CR, PVC 1- 2 80C1, C2 Sa 2 1/2 EP 1- 2 80 EP, PUR 1- 2 80C3 Sa 2 1/2 AK 1- 2 80 AK 2- 3 120C3 Sa 2 1/2 EP 1 160 AY 1 40C3 Sa 2 1/2 AY, CR, PVC 1- 2 80 AY, CR, PVC 2- 3 120C3 Sa 2 1/2 EP 1- 2 80 EP, PUR 2- 3 120C4 Sa 2 1/2 EP, PUR, Zn(R) 1 40 EP, PUR 2- 3 200C4 Sa 2 1/2 ESI. Zn(R) 1 80 EP, PUR 2- 3 160C4 Sa 2 1/2 EP 1- 2 80 EP, PUR 2- 3 200C5-I Sa 2 1/2 EP, PUR 4 160 AY, CR, PVC 1 40C5-I Sa 2 1/2 EP, PUR 1- 2 80 EP, PUR 3- 4 240C5-I Sa 2 1/2 ESI, Zn(R) 1 80 EP, PUR 3 200C5-M Sa 2 1/2 EP, PUR, Zn(R) 1 40 EP, PUR 2- 3 280C5-M Sa 2 1/2 EP, PUR 1- 2 80 EP, PUR 3- 4 240AK = Alkyd PVC = Polyvinyl chlorideCR = Chlorinated rubber PUR = PolyurethaneAY = Acrylic Zn(R) = Zinc rich primerEP = EpoxyESI = Ethyl silicate


231

Table 8.4 Applicability of different paint types for conditions requiring special characteristics [4]

8.4 Hot-dip galvanizing

Protecting hollow section structures with galvanization is an efficient anti-corrosive method, andin many climates it can be sufficient on its own. The protective effect of zinc is based on itsoxidation. Due to the cathodic protective effect of zinc, minor surface flaws in the zinc layer donot lead to the corrosion of the steel. The durability of the protective effect of zinc is directlyproportional to the thickness of the zinc layer. In severe environmental conditions (e.g.corrosivity categories C5-I and C5-M), even a galvanized structure requires an anti-corrosivepainting.

Proper oxidation of zinc is prevented if hollow sections are placed in poorly ventilated andhumid premises directly after galvanizing. This may be the case, for instance, in tightly packedstacks of sections where rain or water from condensation can accumulate. In such conditions,white rust is generated on the galvanized surface. White rust can be removed by brushing orusing an appropriate detergent. To avoid white rust, hollow sections must always be stored onprops. It is also important to allow enough space between hollow sections to ensure sufficientcirculation of air and evaporation of moisture. If the zinc layer has already been oxidized, whiterust is no longer generated.

Required characteristic Paint typePVC CR AY B AK PURp PURa EP ZnS PURc CTV

Gloss retention ++ ++ +++ + ++ + +++ + - + +Colour retention ++ ++ +++ + ++ + +++ + - - -Resistance to:- water immersion ++ ++ + +++ + + ++ +++ ++ +++ +++- condensate/rain +++ +++ +++ +++ ++ +++ +++ +++ +++ +++ +++- solvents + + + + + ++ + ++ +++ + +- solvents (splash) + + + + ++ +++ +++ +++ +++ + +- acids + + + + + ++ + + + + +- acids (splash) ++ ++ ++ + + ++ ++ ++ + ++ +- alkalis + + + + + + + +++ + ++ +- alkalis (splash) ++ ++ ++ ++ + + +++ +++ + +++ ++Resistance to dry heat:60- 70 °C +++ +++ +++ ++ +++ +++ +++ +++ +++ +++ +++70- 120 °C + + ++ + ++ +++ +++ +++ +++ ++ +120- 150 °C + + + + + ++ ++ ++ +++ + +>150 °C + + + + + + + + +++ + +Physical properties:- abrasion resistance + + + + ++ +++ ++ +++ +++ ++ +- impact resistance ++ ++ ++ ++ + +++ +++ ++ + +++ ++- flexibility ++ ++ ++ ++ + ++ +++ +++ + ++ ++- hardness ++ ++ ++ ++ +++ +++ ++ +++ +++ ++ +Application by:- brushing ++ ++ ++ +++ +++ ++ ++ +++ + ++ ++- rollercoating + + + +++ +++ ++ ++ ++ + ++ ++- spraying +++ +++ +++ +++ +++ +++ +++ +++ +++ +++ +++PVC = Polyvinyl chloride PURp = Polyurethane, Zns = Zinc silicate - = not relevantCR = Chlorinated rubber polyester type PURc = Coal tar + = poorAY = Acrylic PURa = Polyurethane, polyurethane ++ = goodB = Bitumen acryl type CTV = Coal tar vinyl +++ = excellentAK = Alkyd EP = Epoxy


232

Hot-dip galvanized steel surfaces can be painted, for instance, using the paint combinationsshown in Table 8.5. However, the galvanized surface must be treated before painting by, forinstance, sand blasting (= light shot blasting). Sand blasting removes the impurities on thesurface and makes the surface rougher, which improves the adhesion of the paint coat. In caseshot blasting cannot be used, the galvanized surface must be cleaned with an alkalinedegreasant. After the cleaning, the surface is rinsed and dried and is then ready for painting.

Table 8.5 Recommended paint combinations for hot-dip galvanized steel surfaces [4]

Hot-dip galvanized structures must be made as open as possible to produce a smooth zinclayer. Galvanized hollow section structures must be designed so that zinc can flow freely intothe hollow section and out of it. If necessary, the structure must be provided with openings toensure a sufficient flow of zinc. Site connections of hot-dip galvanized hollow sections shouldpreferably be bolted. Welded joints made on-site must be protected by applying zinc-rich paintor by spraying them with zinc. Where galvanized sections are bolted the male thread must bemade slightly smaller than the female thread of the joined element to allow sufficient space forthe zinc layer. The female thread is usually made to the hot-dip galvanized element only aftergalvanizing, since the galvanizing of the joined element protects the inner thread fromcorrosion.

Corrosivity category Priming coat(s) Top coat(s)Paint type Number of

coatsNominal DryFilm Thicknessµm

Paint type Number ofcoats

Nominal DryFilm Thicknessµm

C1, C2 - - - PVC 1 80C1- C3 PVC 1 80 PVC 1 80C1- C4 PVC 1 80 PVC 2 160C1- C4 EP, PUR 1 80 EP, PUR 1 80C1- C4, C5-I, C5-M EP, PUR 1 80 EP, PUR 2 160


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8.5 References

[1] ISO/ FDIS 12944-2: Paints and varnishes- Corrosion protection of steel structures by pro-tective painting systems. Part 2: Classification of enviroments, 1997

[2] ISO/ FDIS 12944-3: Paints and varnishes- Corrosion protection of steel structures by pro-tective painting systems. Part 3: Design considerations, 1997

[3] ISO/ FDIS 12944-4: Paints and varnishes- Corrosion protection of steel structures by pro-tective painting systems. Part 4: Types of surfaces and surface preparation, 1997

[4] ISO/ FDIS 12944-5: Paints and varnishes- Corrosion protection of steel structures by pro-tective painting systems. Part 5: Protective paint systems, 1997


234

9 APPENDIX

Appendix 9.1 Cross-sectional properties and resistancevalues for steel grade S355J2H

DESIGN HANDBOOK FOR RAUTARUUKKI STRUCTURAL HOLLOW SECTIONS Appendix 9.1

235

M = weight Wt = torsional section modulus PL = cross-section class in concentric compressionA = cross-section area I = moment of inertia Nc.Rd = compression resistance without bucklingAu = external area Wel = elastic section modulus Mc.Rd = bending resistanceAm/V = cross-section factor in fire design Wpl = plastic section modulus Vpl.Rd = shear resistanceIt = torsional modulus i = radius of gyration

r0 = 2,0 x t when t ≤ 6,0 mm1) = recommended r0 = 2,5 x t when 6,0 mm < t ≤ 10,0 mm

series r0 = 3,0 x t when t > 10,0 mm

z

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Table 9.1.1 Cross-sectional properties and resistance values for square hollow sections of steel grade S355J2H. (fy = 355 N/mm2).

The calculated resistance values are design values (section 2.1) based on the default value (1.1) of the partialsafety factor of material γγγγM0 used in Eurocode 3. The partial safety factor values may differ in each country.National values must be checked from the NAD (National Application Document) of the country in question.

1) h b t M A Au Am/V It Wt I Wel Wpl i PL Nc.Rd Mc.Rd Vpl.Rdmm mm mm kg/m mm2 m2/m 1/m mm4 mm3 mm4 mm3 mm3 mm kN kNm kN

x x 102 x 104 x 103 x 104 x 103 x 103 x 10x 40 40 2 2,31 2,94 0,153 521 11,28 5,23 6,94 3,47 4,13 1,54 1 94,78 1,33 27,36x 40 40 2,5 2,82 3,59 0,151 422 13,61 6,21 8,22 4,11 4,97 1,51 1 115,8 1,60 33,44x 50 50 2 2,93 3,74 0,193 517 22,63 8,51 14,15 5,66 6,66 1,95 1 120,6 2,15 34,81x 50 50 2,5 3,60 4,59 0,191 417 27,53 10,22 16,94 6,78 8,07 1,92 1 148,1 2,61 42,75x 50 50 3 4,25 5,41 0,190 351 32,13 11,76 19,47 7,79 9,39 1,90 1 174,5 3,03 50,38x 60 60 2,5 4,39 5,59 0,231 414 48,66 15,22 30,34 10,11 11,93 2,33 1 180,4 3,85 52,07x 60 60 3 5,19 6,61 0,230 348 57,09 17,65 35,13 11,71 13,95 2,31 1 213,3 4,50 61,56x 60 60 4 6,71 8,55 0,226 265 72,64 21,97 43,55 14,52 17,64 2,26 1 275,9 5,69 79,64x 70 70 2 4,19 5,34 0,273 512 63,96 17,48 40,73 11,64 13,52 2,76 3 172,2 3,76 49,72x 70 70 2,5 5,17 6,59 0,271 412 78,49 21,22 49,41 14,12 16,54 2,74 1 212,7 5,34 61,39x 70 70 3 6,13 7,81 0,270 345 92,42 24,74 57,53 16,44 19,42 2,71 1 252,0 6,27 72,74x 70 70 4 7,97 10,15 0,266 262 118,5 31,11 72,12 20,61 24,76 2,67 1 327,5 7,99 94,54

80 80 2 4,82 6,14 0,313 510 96,34 23,16 61,70 15,42 17,85 3,17 4 180,1 4,70 57,17x 80 80 2,5 5,96 7,59 0,311 410 118,5 28,22 75,15 18,79 21,90 3,15 2 244,9 7,07 70,70x 80 80 3 7,07 9,01 0,310 344 139,9 33,02 87,84 21,96 25,78 3,12 1 290,7 8,32 83,92x 80 80 4 9,22 11,75 0,306 261 180,4 41,84 111,0 27,76 33,07 3,07 1 379,1 10,67 109,5x 80 80 5 11,3 14,36 0,303 211 217,8 49,68 131,4 32,86 39,74 3,03 1 463,3 12,82 133,8

90 90 2 5,45 6,94 0,353 509 138,1 29,64 88,86 19,75 22,78 3,58 4 187,9 5,73 64,63x 90 90 2,5 6,74 8,59 0,351 409 170,3 36,23 108,6 24,12 28,00 3,56 3 277,2 7,79 80,02x 90 90 3 8,01 10,21 0,350 343 201,4 42,51 127,3 28,29 33,04 3,53 2 329,5 10,66 95,10x 90 90 4 10,5 13,35 0,346 259 260,8 54,17 161,9 35,98 42,58 3,48 1 430,8 13,74 124,4x 90 90 5 12,8 16,36 0,343 210 316,3 64,70 192,9 42,87 51,41 3,43 1 527,9 16,59 152,4x 90 90 6 15,1 19,23 0,339 176 367,8 74,16 220,5 49,00 59,54 3,39 1 620,7 19,22 179,2

90 90 6,3 15,5 19,73 0,333 169 382,3 76,21 221,1 49,14 60,30 3,35 1 636,7 19,46 183,8100 100 2 6,07 7,74 0,393 508 190,5 36,92 123,0 24,60 28,30 3,99 4 194,0 6,84 72,08

x 100 100 2,5 7,53 9,59 0,391 408 235,2 45,23 150,6 30,13 34,86 3,96 4 281,4 9,17 89,33x 100 100 3 8,96 11,41 0,390 342 278,7 53,19 177,0 35,41 41,21 3,94 2 368,2 13,30 106,3x 100 100 4 11,7 14,95 0,386 258 362,0 68,10 226,4 45,27 53,30 3,89 1 482,4 17,20 139,3x 100 100 5 14,4 18,36 0,383 209 440,5 81,72 271,1 54,22 64,59 3,84 1 592,4 20,85 171,0x 100 100 6 17,0 21,63 0,379 175 514,2 94,12 311,5 62,29 75,10 3,79 1 698,2 24,24 201,5

100 100 6,3 17,5 22,25 0,373 168 536,0 97,02 314,2 62,83 76,38 3,76 1 718,1 24,65 207,3100 100 7,1 19,4 24,65 0,370 150 589,2 105,6 340,1 68,03 83,59 3,71 1 795,6 26,98 229,7

x 100 100 8 21,4 27,24 0,366 134 644,5 114,2 365,9 73,19 91,05 3,67 1 879,2 29,38 253,8

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110 110 2,5 8,31 10,59 0,431 407 314,9 55,23 202,4 36,80 42,47 4,37 4 291,3 10,78 98,65x 110 110 3 9,90 12,61 0,430 341 373,5 65,07 238,3 43,33 50,27 4,35 3 406,9 13,98 117,5x 110 110 4 13,0 16,55 0,426 258 486,5 83,63 305,9 55,62 65,21 4,30 1 534,1 21,05 154,2x 110 110 5 16,0 20,36 0,423 208 593,6 100,7 367,9 66,90 79,27 4,25 1 657,0 25,58 189,6x 110 110 6 18,9 24,03 0,419 175 694,9 116,5 424,6 77,19 92,46 4,20 1 775,6 29,84 223,9

110 110 6,3 19,4 24,77 0,413 167 725,8 120,4 430,1 78,21 94,36 4,17 1 799,4 30,45 230,8x 120 120 3 10,8 13,81 0,470 340 487,7 78,15 312,3 52,06 60,24 4,76 4 405,1 15,85 128,6x 120 120 4 14,3 18,15 0,466 257 636,6 100,8 402,3 67,05 78,33 4,71 2 585,7 25,28 169,1x 120 120 5 17,6 22,36 0,463 207 778,5 121,8 485,5 80,91 95,45 4,66 1 721,5 30,80 208,3x 120 120 5,6 19,5 24,82 0,461 186 860,3 133,6 532,3 88,71 105,3 4,63 1 800,9 33,97 231,2x 120 120 6 20,8 26,43 0,459 174 913,5 141,2 562,2 93,69 111,6 4,61 1 853,1 36,02 246,3

120 120 6,3 21,4 27,29 0,453 166 955,5 146,2 571,6 95,26 114,2 4,58 1 880,7 36,86 254,2120 120 7,1 23,8 30,33 0,449 148 1056 160,1 623,3 103,9 125,7 4,53 1 978,7 40,55 282,5

x 120 120 8 26,4 33,64 0,446 132 1163 174,6 676,9 112,8 137,8 4,49 1 1086 44,48 313,4120 120 8,8 28,6 36,48 0,442 121 1252 186,5 719,9 120,0 147,9 4,44 1 1177 47,73 339,9

x 140 140 4 16,8 21,35 0,546 256 1023 139,8 651,6 93,09 108,2 5,52 3 689,0 30,04 198,9x 140 140 5 20,7 26,36 0,543 206 1256 169,8 790,6 112,9 132,3 5,48 1 850,6 42,70 245,5x 140 140 5,6 23,0 29,30 0,541 185 1391 186,9 869,6 124,2 146,3 5,45 1 945,5 47,20 273,0x 140 140 6 24,5 31,23 0,539 173 1479 197,9 920,4 131,5 155,3 5,43 1 1008 50,13 291,0

140 140 6,3 25,4 32,33 0,533 165 1550 205,4 940,8 134,4 159,6 5,39 1 1043 51,51 301,2x 140 140 7,1 28,3 36,01 0,529 147 1719 226,0 1031 147,3 176,3 5,35 1 1162 56,89 335,5x 140 140 8 31,4 40,04 0,526 131 1901 247,7 1127 161,0 194,2 5,30 1 1292 62,67 373,1x 140 140 8,8 34,2 43,52 0,522 120 2055 265,8 1205 172,1 209,2 5,26 1 1405 67,52 405,5x 140 140 10 38,1 48,57 0,517 106 2274 290,9 1312 187,4 230,4 5,20 1 1567 74,35 452,5x 150 150 4 18,0 22,95 0,586 255 1265 161,7 808,0 107,7 124,9 5,93 4 701,2 33,61 213,8x 150 150 5 22,3 28,36 0,583 206 1554 196,8 982,0 130,9 153,0 5,89 2 915,1 49,37 264,2x 150 150 6 26,4 33,63 0,579 172 1833 229,8 1146 152,8 179,9 5,84 1 1085 58,05 313,3

150 150 6,3 27,4 34,85 0,573 164 1922 238,8 1174 156,5 185,2 5,80 1 1125 59,75 324,7150 150 7,1 30,5 38,85 0,569 147 2134 263,1 1289 171,9 204,8 5,76 1 1254 66,09 361,9

x 150 150 8 34,0 43,24 0,566 131 2364 289,0 1412 188,2 226,0 5,71 1 1396 72,92 402,9150 150 8,8 36,9 47,04 0,562 120 2560 310,7 1513 201,7 243,9 5,67 1 1518 78,70 438,3

x 150 150 10 41,3 52,57 0,557 106 2839 341,0 1653 220,3 269,2 5,61 1 1696 86,87 489,7160 160 4 19,3 24,55 0,626 255 1541 185,3 987,2 123,4 142,8 6,34 4 720,2 37,57 228,7

x 160 160 5 23,8 30,36 0,623 205 1896 225,8 1202 150,3 175,2 6,29 2 979,7 56,53 282,8x 160 160 6 28,3 36,03 0,619 172 2239 264,2 1405 175,7 206,2 6,25 1 1163 66,56 335,7

160 160 6,3 29,3 37,37 0,613 164 2349 274,7 1442 180,3 212,6 6,21 1 1206 68,60 348,2160 160 7,1 32,7 41,69 0,609 146 2611 303,2 1587 198,4 235,4 6,17 1 1345 75,97 388,4

x 160 160 8 36,5 46,44 0,606 130 2897 333,6 1741 217,7 260,1 6,12 1 1499 83,95 432,7160 160 8,8 39,7 50,56 0,602 119 3141 359,2 1870 233,7 281,1 6,08 1 1632 90,73 471,1

x 160 160 10 44,4 56,57 0,597 106 3490 395,1 2048 256,0 311,0 6,02 1 1826 100,4 527,0160 160 12 50,9 64,86 0,578 89 3997 443,1 2224 278,0 346,1 5,86 1 2093 111,7 604,3160 160 12,5 52,6 67,04 0,576 86 4114 454,6 2275 284,4 355,7 5,83 1 2164 114,8 624,6


x x 102 x 104 x 103 x 104 x 103 x 103 x 10

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M = weight Wt = torsional section modulus PL = cross-section class in concentric compressionA = cross-section area I = moment of inertia Nc.Rd = compression resistance without bucklingAu = external area Wel = elastic section modulus Mc.Rd = bending resistanceAm/V = cross-section factor in fire design Wpl = plastic section modulus Vpl.Rd = shear resistanceIt = torsional modulus i = radius of gyration

r0 = 2,0 x t when t ≤ 6,0 mm1) = recommended r0 = 2,5 x t when 6,0 mm < t ≤ 10,0 mm

series r0 = 3,0 x t when t > 10,0 mm

z

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h

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Table 9.1.1 Cross-sectional properties and resistance values for square hollow sections of steel grade S355J2H (fy = 355 N/mm2), continued.



x x 102 x 104 x 103 x 104 x 103 x 103 x 10x 180 180 5 27,0 34,36 0,703 205 2724 289,8 1737 193,0 224,0 7,11 3 1109 62,28 320,1x 180 180 6 32,1 40,83 0,699 171 3223 340,1 2037 226,3 264,4 7,06 2 1318 85,31 380,4

180 180 6,3 33,3 42,41 0,693 163 3383 354,1 2096 232,8 273,1 7,03 1 1369 88,13 395,1180 180 7,1 37,2 47,37 0,689 146 3768 391,7 2313 257,0 303,1 6,99 1 1529 97,81 441,3

x 180 180 8 41,5 52,84 0,686 130 4189 432,2 2546 282,9 335,7 6,94 1 1705 108,3 492,3180 180 8,8 45,2 57,60 0,682 118 4551 466,6 2742 304,6 363,6 6,90 1 1859 117,3 536,7

x 180 180 10 50,7 64,57 0,677 105 5074 515,3 3017 335,2 403,5 6,84 1 2084 130,2 601,5180 180 12 58,5 74,46 0,658 88 5865 583,7 3322 369,1 453,6 6,68 1 2403 146,4 693,7

x 180 180 12,5 60,5 77,04 0,656 85 6050 600,1 3406 378,5 467,1 6,65 1 2486 150,7 717,8x 200 200 5 30,1 38,36 0,783 204 3763 361,8 2410 241,0 278,9 7,93 4 1125 73,38 357,3x 200 200 6 35,8 45,63 0,779 171 4459 425,5 2833 283,3 329,7 7,88 2 1473 106,4 425,1

200 200 6,3 37,3 47,45 0,773 163 4682 443,5 2922 292,2 341,2 7,85 2 1531 110,1 442,1200 200 7,1 41,6 53,05 0,769 145 5223 491,6 3232 323,2 379,3 7,81 1 1712 122,4 494,2

x 200 200 8 46,5 59,24 0,766 129 5815 543,6 3566 356,6 420,9 7,76 1 1912 135,8 551,9200 200 8,8 50,8 64,64 0,762 118 6328 588,1 3850 385,0 456,6 7,72 1 2086 147,4 602,2

x 200 200 10 57,0 72,57 0,757 104 7072 651,5 4251 425,1 508,1 7,65 1 2342 164,0 676,1200 200 12 66,0 84,06 0,738 88 8230 743,4 4730 473,0 575,6 7,50 1 2713 185,8 783,1

x 200 200 12,5 68,3 87,04 0,736 85 8502 765,5 4859 485,9 593,5 7,47 1 2809 191,5 810,9x 220 220 6 39,6 50,43 0,859 170 5976 520,6 3813 346,7 402,2 8,70 3 1628 111,9 469,9

220 220 6,3 41,2 52,49 0,853 163 6277 543,0 3940 358,2 416,8 8,66 3 1694 115,6 489,0220 220 7,1 46,1 58,73 0,849 145 7010 602,9 4366 396,9 464,0 8,62 2 1895 149,7 547,1

x 220 220 8 51,5 65,64 0,846 129 7815 667,9 4828 438,9 515,6 8,58 1 2118 166,4 611,6220 220 8,8 56,3 71,68 0,842 117 8514 723,6 5221 474,7 560,2 8,53 1 2313 180,8 667,8

x 220 220 10 63,2 80,57 0,837 104 9533 803,6 5782 525,7 624,7 8,47 1 2600 201,6 750,6220 220 12 73,5 93,66 0,818 87 11149 922,3 6487 589,7 712,0 8,32 1 3023 229,8 872,6220 220 12,5 76,2 97,04 0,816 84 11530 950,8 6674 606,7 734,9 8,29 1 3132 237,2 904,1

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x 250 250 6 45,2 57,63 0,979 170 8843 681,2 5672 453,8 524,5 9,92 4 1646 136,0 536,9250 250 6,3 47,1 60,05 0,973 162 9290 711,2 5873 469,8 544,4 9,89 4 1770 143,4 559,4250 250 7,1 52,8 67,25 0,969 144 10388 791,0 6522 521,7 607,0 9,85 3 2170 168,4 626,5

x 250 250 8 59,1 75,24 0,966 128 11598 878,2 7229 578,3 675,8 9,80 2 2428 218,1 701,0250 250 8,8 64,6 82,24 0,962 117 12653 953,3 7835 626,8 735,3 9,76 1 2654 237,3 766,2

x 250 250 10 72,7 92,57 0,957 103 14197 1062 8707 696,5 822,0 9,70 1 2987 265,3 862,4250 250 12 84,8 108,1 0,938 87 16691 1226 9859 788,8 943,6 9,55 1 3487 304,5 1007

x 250 250 12,5 88,0 112,0 0,936 84 17283 1266 10161 812,9 975,2 9,52 1 3616 314,7 1044260 260 6 47,1 60,03 1,019 170 9970 739,5 6405 492,7 568,8 10,33 4 1669 145,3 559,3260 260 6,3 49,1 62,57 1,013 162 10475 772,3 6635 510,4 590,8 10,30 4 1797 153,4 582,9260 260 7,1 55,0 70,09 1,009 144 11717 859,4 7373 567,1 658,9 10,26 3 2262 183,0 653,0260 260 8 61,6 78,44 1,006 128 13087 954,7 8178 629,1 734,0 10,21 2 2532 236,9 730,8260 260 8,8 67,3 85,76 1,002 117 14283 1037 8869 682,2 799,0 10,17 1 2768 257,9 799,0260 260 10 75,8 96,57 0,997 103 16035 1156 9865 758,8 893,8 10,11 1 3116 288,5 899,6260 260 11 81,9 104,4 0,983 94 17498 1247 10479 805,8 956,5 10,02 1 3368 308,7 972,3260 260 12,5 91,9 117,0 0,976 83 19553 1381 11548 888,3 1063 9,93 1 3777 343,0 1090

x 300 300 6 54,7 69,63 1,179 169 15434 996,8 9964 664,2 764,2 11,96 4 1746 184,6 648,7300 300 6,3 57,0 72,65 1,173 161 16218 1042 10342 689,5 794,9 11,93 4 1887 195,4 676,8300 300 7,1 63,9 81,45 1,169 144 18161 1161 11514 767,6 887,9 11,89 4 2303 228,6 758,8

x 300 300 8 71,6 91,24 1,166 128 20312 1293 12801 853,4 990,7 11,84 4 2787 266,2 850,0300 300 8,8 78,4 99,84 1,162 116 22195 1406 13902 927,4 1080 11,80 3 3222 299,3 930,2

x 300 300 10 88,4 112,6 1,157 103 24966 1572 15519 1035 1211 11,74 2 3633 390,8 1049300 300 12 104 132,1 1,138 86 29514 1829 17767 1184 1402 11,60 1 4262 452,3 1230

x 300 300 12,5 108 137,0 1,136 83 30601 1892 18348 1223 1451 11,57 1 4423 468,2 12771) h b t M A Au Am/V It Wt I Wel Wpl i PL Nc.Rd Mc.Rd Vpl.Rd

mm mm mm kg/m mm2 m2/m 1/m mm4 mm3 mm4 mm3 mm3 mm kN kNm kNx x 102 x 104 x 103 x 104 x 103 x 103 x 10

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M = weight Wt = torsional section modulus PL = cross-section class in concentric compressionA = cross-section area I = moment of inertia Nc.Rd = compression resistance without bucklingAu = external area Wel = elastic section modulus Mc.y.Rd = bending resistance by y axisAm/V = cross-section factor in fire design Wpl = plastic section modulus Mc.z.Rd = bending resistance by z axisIt = torsional modulus i = radius of gyration Vpl.y.Rd = shear resistance in the direction of y axis

Vpl.z.Rd = shear resistance in the direction of z axisr0 = 2,0 x t when t ≤ 6,0 mm

1) = recommended r0 = 2,5 x t when 6,0 mm < t ≤ 10,0 mm series r0 = 3,0 x t when t > 10,0 mm

Table 9.1.2 Cross-sectional properties and resistance values for rectangular hollow sections of steel grade S355J2H (fy = 355 N/mm2).


1) h b t M A Au Am/V It Wt Iy Wel.y Wpl.y iy Iz Wel.z Wpl.z iz PL Nc.Rd Mc.y.Rd Mc.z.Rd Vpl.y.Rd Vpl.z.Rd

mm mm mm kg/m mm2 m2/m 1/m mm4 mm3 mm4 mm3 mm3 mm mm4 mm3 mm3 mm h b kN kNm kNm kN kNx x 102 x 104 x 103 x 104 x 103 x 103 x 10 x 104 x 103 x 103 x 10

40 30 2 1,99 2,54 0,133 524 7,07 3,79 5,49 2,75 3,37 1,47 3,51 2,34 2,77 1,18 1 1 81,88 1,09 0,89 20,26 27,01x 50 30 2 2,31 2,94 0,153 520 9,77 4,84 9,54 3,81 4,74 1,80 4,29 2,86 3,33 1,21 1 1 94,78 1,53 1,07 20,52 34,20x 60 40 2 2,93 3,74 0,193 516 20,70 8,12 18,41 6,14 7,47 2,22 9,83 4,92 5,65 1,62 2 1 120,6 2,41 1,82 27,85 41,78x 60 40 2,5 3,60 4,59 0,191 416 25,14 9,72 22,07 7,36 9,06 2,19 11,74 5,87 6,84 1,60 1 1 148,1 2,92 2,21 34,20 51,30x 70 50 2 3,56 4,54 0,233 513 37,45 12,20 31,48 8,99 10,80 2,63 18,76 7,50 8,58 2,03 3 1 146,4 3,49 2,42 35,22 49,31x 70 50 2,5 4,39 5,59 0,231 413 45,75 14,72 38,01 10,86 13,16 2,61 22,59 9,04 10,45 2,01 1 1 180,4 4,25 3,37 43,39 60,75x 70 50 3 5,19 6,61 0,230 348 53,62 17,06 44,05 12,59 15,40 2,58 26,10 10,44 12,21 1,99 1 1 213,3 4,97 3,94 51,30 71,83x 80 40 2,5 4,39 5,59 0,231 413 37,58 13,24 45,11 11,28 14,15 2,84 15,26 7,63 8,72 1,65 2 1 180,4 4,56 2,81 34,71 69,43x 80 60 2 4,19 5,34 0,273 511 61,22 17,08 49,53 12,38 14,73 3,05 31,87 10,62 12,11 2,44 4 2 163,2 4,75 3,22 42,62 56,82x 80 60 2,5 5,17 6,59 0,271 411 75,07 20,73 60,13 15,03 18,02 3,02 38,61 12,87 14,81 2,42 2 1 212,7 5,82 4,78 52,62 70,16x 80 60 3 6,13 7,81 0,270 346 88,35 24,14 70,05 17,51 21,16 3,00 44,89 14,96 17,37 2,40 1 1 252,0 6,83 5,61 62,35 83,14x 80 60 4 7,97 10,15 0,266 262 113,1 30,32 87,92 21,98 26,99 2,94 56,12 18,71 22,12 2,35 1 1 327,5 8,71 7,14 81,04 108,1

80 70 2,5 5,56 7,09 0,291 410 96,21 24,47 67,64 16,91 19,96 3,09 55,11 15,75 18,23 2,79 2 1 228,8 6,44 5,88 61,64 70,4580 70 3 6,60 8,41 0,290 345 113,4 28,58 78,94 19,74 23,47 3,06 64,26 18,36 21,43 2,76 1 1 271,4 7,57 6,91 73,11 83,5680 70 4 8,59 10,95 0,286 261 145,9 36,08 99,48 24,87 30,03 3,01 80,84 23,10 27,40 2,72 1 1 353,3 9,69 8,84 95,19 108,880 70 5 10,5 13,36 0,283 212 175,5 42,67 117,4 29,34 35,99 2,96 95,21 27,20 32,81 2,67 1 1 431,0 11,61 10,59 116,1 132,790 50 2 4,19 5,34 0,273 511 53,37 15,88 57,88 12,86 15,74 3,29 23,37 9,35 10,50 2,09 4 1 154,2 5,08 2,66 35,52 63,93

x 90 50 2,5 5,17 6,59 0,271 411 65,3 19,24 70,26 15,61 19,25 3,27 28,24 11,29 12,82 2,07 3 1 212,7 6,21 3,64 43,85 78,92x 90 50 3 6,13 7,81 0,270 346 76,67 22,36 81,85 18,19 22,60 3,24 32,74 13,10 15,03 2,05 2 1 252,0 7,29 4,85 51,96 93,53

90 60 2,5 5,56 7,09 0,291 410 88,99 23,48 79,84 17,74 21,44 3,36 42,75 14,25 16,24 2,46 3 1 228,8 6,92 4,60 52,84 79,2590 60 3 6,60 8,41 0,290 345 104,8 27,39 93,21 20,71 25,21 3,33 49,77 16,59 19,08 2,43 2 1 271,4 8,14 6,16 62,67 94,0090 60 4 8,59 10,95 0,286 261 134,4 34,50 117,5 26,11 32,26 3,28 62,40 20,80 24,36 2,39 1 1 353,3 10,41 7,86 81,60 122,490 70 2 4,82 6,14 0,313 510 93,2 22,76 73,37 16,30 19,26 3,46 49,98 14,28 16,24 2,85 4 3 178,8 5,26 4,11 50,03 64,3290 70 2,5 5,96 7,59 0,311 410 114,6 27,73 89,41 19,87 23,63 3,43 60,81 17,37 19,91 2,83 3 1 244,9 7,63 5,61 61,86 79,5490 70 3 7,07 9,01 0,310 344 135,3 32,43 104,6 23,24 27,82 3,41 71,00 20,29 23,44 2,81 2 1 290,7 8,98 7,56 73,43 94,4190 70 4 9,22 11,75 0,306 260 174,2 41,05 132,3 29,40 35,70 3,36 89,57 25,59 30,04 2,76 1 1 379,1 11,52 9,69 95,77 123,190 70 5 11,3 14,36 0,303 211 210,1 48,70 156,8 34,84 42,91 3,30 105,8 30,23 36,06 2,71 1 1 463,3 13,85 11,64 117,0 150,5

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90 80 2 5,13 6,54 0,333 509 115,2 26,20 81,11 18,03 21,02 3,52 67,78 16,95 19,41 3,22 4 4 184,0 5,50 4,90 57,32 64,4890 80 2,5 6,35 8,09 0,331 409 141,8 31,98 98,98 22,00 25,81 3,50 82,66 20,66 23,83 3,20 3 2 261,1 8,33 6,67 70,93 79,7990 80 3 7,54 9,61 0,330 343 167,6 37,47 115,9 25,76 30,43 3,47 96,74 24,19 28,09 3,17 2 1 310,1 9,82 9,07 84,25 94,7890 80 4 9,85 12,55 0,326 260 216,5 47,61 147,1 32,69 39,14 3,42 122,6 30,65 36,11 3,13 1 1 405,0 12,63 11,65 110,0 123,890 80 5 12,1 15,36 0,323 210 261,9 56,70 174,9 38,86 47,16 3,37 145,5 36,38 43,49 3,08 1 1 495,6 15,22 14,03 134,7 151,590 80 6 14,2 18,03 0,319 177 303,8 64,78 199,3 44,28 54,50 3,32 165,6 41,41 50,23 3,03 1 1 582,0 17,59 16,21 158,1 177,990 80 6,3 14,5 18,47 0,313 169 315,2 66,43 199,0 44,23 55,03 3,28 165,7 41,41 50,75 2,99 1 1 596,1 17,76 16,38 162,0 182,2

100 40 2 4,19 5,34 0,273 511 41,47 13,89 65,38 13,08 16,54 3,50 15,61 7,81 8,69 1,71 4 1 144,4 5,34 2,07 28,41 71,03x 100 40 2,5 5,17 6,59 0,271 411 50,52 16,76 79,32 15,86 20,23 3,47 18,78 9,39 10,59 1,69 4 1 198,6 6,53 2,81 35,08 87,69

100 50 2 4,50 5,74 0,293 510 61,59 17,73 74,98 15,00 18,50 3,62 25,67 10,27 11,46 2,12 4 1 157,3 5,97 2,75 35,63 71,26x 100 50 2,5 5,56 7,09 0,291 410 75,39 21,49 91,2 18,24 22,67 3,59 31,06 12,42 14,01 2,09 4 1 214,7 7,32 3,73 44,03 88,06x 100 50 3 6,60 8,41 0,290 345 88,56 25,01 106,5 21,29 26,66 3,56 36,06 14,42 16,44 2,07 2 1 271,4 8,60 5,30 52,22 104,4x 100 60 2 4,82 6,14 0,313 510 84,08 21,56 84,59 16,92 20,46 3,71 38,60 12,87 14,43 2,51 4 2 170,2 6,60 3,48 42,88 71,47x 100 60 2,5 5,96 7,59 0,311 410 103,3 26,23 103,1 20,62 25,11 3,69 46,88 15,63 17,68 2,49 4 1 230,9 8,10 4,71 53,03 88,38x 100 60 3 7,07 9,01 0,310 344 121,7 30,64 120,6 24,11 29,57 3,66 54,65 18,22 20,79 2,46 2 1 290,7 9,54 6,71 62,94 104,9x 100 60 4 9,22 11,75 0,306 260 156,3 38,68 152,6 30,52 37,94 3,60 68,68 22,89 26,60 2,42 1 1 379,1 12,24 8,59 82,09 136,8

100 70 2 5,13 6,54 0,333 509 108,5 25,40 94,19 18,84 22,42 3,80 54,60 15,60 17,60 2,89 4 3 181,9 6,08 4,25 50,15 71,65100 70 2,5 6,35 8,09 0,331 409 133,5 30,98 115,0 22,99 27,55 3,77 66,50 19,00 21,60 2,87 4 1 247,0 8,89 5,74 62,06 88,66100 70 3 7,54 9,61 0,330 343 157,7 36,27 134,7 26,94 32,48 3,74 77,74 22,21 25,45 2,84 2 1 310,1 10,48 8,21 73,72 105,3100 70 4 9,85 12,55 0,326 260 203,4 46,03 171,0 34,20 41,78 3,69 98,29 28,08 32,68 2,80 1 1 405,0 13,48 10,55 96,27 137,5100 70 5 12,1 15,36 0,323 210 245,7 54,73 203,4 40,67 50,34 3,64 116,4 33,25 39,31 2,75 1 1 495,6 16,25 12,69 117,8 168,3100 80 2 5,45 6,94 0,353 509 134,6 29,24 103,8 20,76 24,38 3,87 73,87 18,47 20,97 3,26 4 4 187,0 6,35 5,07 57,45 71,81100 80 2,5 6,74 8,59 0,351 409 165,8 35,73 126,9 25,37 29,98 3,84 90,17 22,54 25,77 3,24 4 2 263,1 9,68 6,83 71,13 88,91

x 100 80 3 8,01 10,21 0,350 343 196,1 41,91 148,8 29,76 35,39 3,82 105,6 26,41 30,40 3,22 2 1 329,5 11,42 9,81 84,54 105,7x 100 80 4 10,5 13,35 0,346 259 253,8 53,38 189,5 37,89 45,62 3,77 134,2 33,54 39,15 3,17 1 1 430,8 14,72 12,64 110,5 138,2x 100 80 5 12,8 16,36 0,343 210 307,6 63,72 225,9 45,19 55,09 3,72 159,6 39,90 47,24 3,12 1 1 527,9 17,78 15,24 135,5 169,3x 100 80 6 15,1 19,23 0,339 176 357,4 72,98 258,4 51,68 63,82 3,67 182,1 45,53 54,67 3,08 1 1 620,7 20,60 17,64 159,3 199,1

100 80 6,3 15,5 19,73 0,333 169 371,4 74,97 258,8 51,75 64,58 3,62 182,8 45,70 55,39 3,04 1 1 636,7 20,84 17,88 163,4 204,2110 40 2 4,50 5,74 0,293 510 46,87 15,34 83,29 15,14 19,31 3,81 17,06 8,53 9,45 1,72 4 1 146,9 6,23 2,13 28,51 78,39110 40 2,5 5,56 7,09 0,291 410 57,12 18,52 101,2 18,41 23,65 3,78 20,54 10,27 11,53 1,70 4 1 203,6 7,63 2,92 35,22 96,86110 40 3 6,60 8,41 0,290 345 66,77 21,45 118,1 21,47 27,80 3,75 23,73 11,86 13,49 1,68 3 1 271,4 8,97 3,83 41,78 114,9110 50 2 4,82 6,14 0,313 510 69,94 19,57 94,95 17,26 21,47 3,93 27,98 11,19 12,42 2,14 4 1 159,8 6,93 2,83 35,73 78,61110 50 2,5 5,96 7,59 0,311 410 85,65 23,75 115,7 21,03 26,34 3,90 33,88 13,55 15,20 2,11 4 1 219,7 8,50 3,87 44,19 97,22110 50 3 7,07 9,01 0,310 344 100,6 27,66 135,3 24,59 31,01 3,88 39,38 15,75 17,85 2,09 3 1 290,7 10,01 5,08 52,45 115,4110 60 2 5,13 6,54 0,333 509 95,89 23,80 106,6 19,39 23,63 4,04 41,97 13,99 15,59 2,53 4 2 172,7 7,63 3,58 42,99 78,81110 60 2,5 6,35 8,09 0,331 409 117,8 28,99 130,1 23,66 29,03 4,01 51,02 17,01 19,12 2,51 4 1 235,9 9,37 4,89 53,20 97,52110 60 3 7,54 9,61 0,330 343 138,9 33,89 152,5 27,72 34,22 3,98 59,52 19,84 22,50 2,49 3 1 310,1 11,04 6,40 63,19 115,8110 60 4 9,85 12,55 0,326 260 178,5 42,86 193,5 35,19 44,01 3,93 74,96 24,99 28,84 2,44 1 1 405,0 14,20 9,31 82,52 151,3110 70 2 5,45 6,94 0,353 509 124,2 28,04 118,3 21,51 25,79 4,13 59,23 16,92 18,96 2,92 4 3 184,4 6,94 4,37 50,27 78,99110 70 2,5 6,74 8,59 0,351 409 152,9 34,23 144,6 26,29 31,72 4,10 72,20 20,63 23,29 2,90 4 1 252,0 10,24 5,95 62,24 97,8110 70 3 8,01 10,21 0,350 343 180,7 40,12 169,6 30,84 37,43 4,08 84,48 24,14 27,46 2,88 3 1 329,5 12,08 7,79 73,97 116,2110 70 4 10,5 13,35 0,346 259 233,3 51,00 216,0 39,27 48,25 4,02 107,0 30,57 35,32 2,83 1 1 430,8 15,57 11,40 96,72 152,0110 70 5 12,8 16,36 0,343 210 282,1 60,76 257,6 46,84 58,27 3,97 127,0 36,28 42,56 2,79 1 1 527,9 18,81 13,73 118,5 186,2



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dix 9.1




Table 9.1.2 Cross-sectional properties and resistance values for rectangular hollow sections of steel grade S355J2H (fy = 355 N/mm2), continued.




110 90 2 6,07 7,74 0,393 508 186,6 36,52 141,6 25,75 30,11 4,28 104,4 23,19 26,30 3,67 4 4 193,4 7,54 6,10 64,87 79,29110 90 2,5 7,53 9,59 0,391 408 230,3 44,73 173,5 31,54 37,09 4,25 127,7 28,38 32,38 3,65 4 3 280,1 10,18 8,26 80,40 98,27110 90 3 8,96 11,41 0,390 342 272,8 52,59 204,0 37,09 43,85 4,23 150,0 33,33 38,26 3,63 3 2 368,2 14,15 10,76 95,65 116,9110 90 4 11,7 14,95 0,386 258 354,2 67,31 261,0 47,45 56,73 4,18 191,5 42,56 49,46 3,58 1 1 482,4 18,31 15,96 125,3 153,2110 90 5 14,4 18,36 0,383 209 430,8 80,74 312,8 56,87 68,77 4,13 229,1 50,91 59,91 3,53 1 1 592,4 22,19 19,34 153,9 188,1110 90 6 17,0 21,63 0,379 175 502,6 92,94 359,6 65,38 79,98 4,08 262,9 58,42 69,62 3,49 1 1 698,2 25,81 22,47 181,4 221,7110 90 6,3 17,5 22,25 0,373 168 523,8 95,78 362,3 65,87 81,29 4,04 265,4 58,97 70,85 3,45 1 1 718,1 26,23 22,86 186,6 228,0110 100 2 6,39 8,14 0,413 507 220,1 40,76 153,3 27,87 32,27 4,34 132,6 26,52 30,26 4,04 4 4 196,5 7,79 7,03 72,20 79,42110 100 2,5 7,92 10,09 0,411 407 271,8 49,98 187,9 34,17 39,78 4,32 162,5 32,50 37,30 4,01 4 4 286,3 10,42 9,49 89,52 98,47110 100 3 9,43 12,01 0,410 341 322,2 58,83 221,2 40,21 47,06 4,29 191,2 38,23 44,12 3,99 3 2 387,5 15,19 12,34 106,6 117,2110 100 4 12,4 15,75 0,406 258 419,1 75,47 283,5 51,54 60,97 4,24 244,8 48,96 57,14 3,94 1 1 508,2 19,68 18,44 139,7 153,7110 100 5 15,2 19,36 0,403 208 510,7 90,74 340,4 61,88 74,02 4,19 293,7 58,74 69,34 3,90 1 1 624,7 23,89 22,38 171,7 188,9110 100 6 17,9 22,83 0,399 175 596,9 104,7 392,1 71,29 86,22 4,14 338,0 67,60 80,74 3,85 1 1 736,9 27,82 26,06 202,6 222,9110 100 6,3 18,5 23,51 0,393 167 622,9 108,1 396,2 72,04 87,82 4,11 341,9 68,37 82,29 3,81 1 1 758,7 28,34 26,56 208,6 229,5120 40 2 4,82 6,14 0,313 510 52,32 16,78 104,1 17,34 22,28 4,12 18,50 9,25 10,21 1,74 4 1 148,9 7,19 2,19 28,59 85,76120 40 2,5 5,96 7,59 0,311 410 63,77 20,27 126,7 21,12 27,32 4,09 22,30 11,15 12,47 1,71 4 1 207,7 8,82 3,01 35,35 106,1120 40 3 7,07 9,01 0,310 344 74,56 23,51 148,0 24,67 32,16 4,05 25,79 12,89 14,60 1,69 4 1 270,5 10,38 3,85 41,96 125,9120 50 2 5,13 6,54 0,333 509 78,39 21,41 118,0 19,67 24,64 4,25 30,28 12,11 13,38 2,15 4 1 161,8 7,95 2,90 35,82 85,98120 50 2,5 6,35 8,09 0,331 409 96,03 26,01 144,0 23,99 30,26 4,22 36,70 14,68 16,39 2,13 4 1 223,8 9,77 3,99 44,33 106,4120 50 3 7,54 9,61 0,330 343 112,9 30,32 168,6 28,10 35,67 4,19 42,69 17,08 19,26 2,11 4 1 289,8 11,51 5,12 52,65 126,4120 60 2 5,45 6,94 0,353 509 107,9 26,05 131,9 21,99 27,00 4,36 45,33 15,11 16,75 2,56 4 2 174,7 8,71 3,67 43,08 86,17120 60 2,5 6,74 8,59 0,351 409 132,6 31,75 161,2 26,87 33,20 4,33 55,15 18,38 20,56 2,53 4 1 240,0 10,71 5,04 53,35 106,7

x 120 60 3 8,01 10,21 0,350 343 156,3 37,14 189,1 31,52 39,18 4,30 64,40 21,47 24,21 2,51 4 1 309,2 12,64 6,45 63,40 126,8x 120 60 4 10,5 13,35 0,346 259 201,1 47,05 240,7 40,12 50,49 4,25 81,25 27,08 31,08 2,47 2 1 430,8 16,29 10,03 82,90 165,8

120 80 2 6,07 7,74 0,393 508 175,0 35,32 159,8 26,63 31,72 4,54 86,04 21,51 24,09 3,33 4 4 191,6 8,18 5,34 57,66 86,50120 80 2,5 7,53 9,59 0,391 408 215,8 43,23 195,8 32,63 39,07 4,52 105,2 26,30 29,65 3,31 4 2 272,2 12,61 7,29 71,47 107,2

x 120 80 3 8,96 11,41 0,390 342 255,5 50,80 230,2 38,37 46,20 4,49 123,4 30,86 35,02 3,29 4 1 347,9 14,91 9,32 85,03 127,5x 120 80 4 11,7 14,95 0,386 258 331,2 64,93 294,6 49,10 59,77 4,44 157,3 39,32 45,23 3,24 2 1 482,4 19,29 14,60 111,4 167,1x 120 80 5 14,4 18,36 0,383 209 402,3 77,77 353,1 58,86 72,45 4,39 187,8 46,94 54,74 3,20 1 1 592,4 23,38 17,66 136,8 205,2x 120 80 6 17,0 21,63 0,379 175 468,5 89,40 406,1 67,68 84,25 4,33 215,0 53,76 63,55 3,15 1 1 698,2 27,19 20,51 161,2 241,9

120 80 6,3 17,5 22,25 0,373 168 487,8 92,07 408,5 68,08 85,57 4,28 217,1 54,28 64,68 3,12 1 1 718,1 27,61 20,87 165,8 248,7

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120 90 2 6,39 8,14 0,413 507 211,8 39,96 173,7 28,95 34,08 4,62 112,1 24,91 28,06 3,71 4 4 195,5 8,51 6,24 64,98 86,64120 90 2,5 7,92 10,09 0,411 407 261,5 48,98 213,0 35,50 42,01 4,59 137,3 30,51 34,56 3,69 4 3 284,2 11,46 8,50 80,57 107,4120 90 3 9,43 12,01 0,410 341 309,9 57,64 250,7 41,79 49,71 4,57 161,4 35,86 40,87 3,67 4 2 367,3 16,04 10,85 95,89 127,9120 90 4 12,4 15,75 0,406 258 402,8 73,88 321,5 53,58 64,41 4,52 206,3 45,85 52,90 3,62 2 1 508,2 20,79 17,07 125,8 167,7120 90 5 15,2 19,36 0,403 208 490,4 88,76 386,2 64,37 78,20 4,47 247,2 54,93 64,16 3,57 1 1 624,7 25,24 20,71 154,6 206,1120 90 6 17,9 22,83 0,399 175 572,7 102,3 445,1 74,18 91,09 4,42 284,1 63,13 74,66 3,53 1 1 736,9 29,40 24,10 182,3 243,1120 90 6,3 18,5 23,51 0,393 167 597,3 105,6 449,3 74,88 92,73 4,37 287,5 63,88 76,12 3,50 1 1 758,7 29,93 24,57 187,7 250,3120 100 2,5 8,31 10,59 0,431 407 309,4 54,73 230,3 38,38 44,95 4,66 174,4 34,88 39,73 4,06 4 4 290,5 11,73 9,77 89,68 107,6120 100 3 9,90 12,61 0,430 341 367,0 64,47 271,3 45,21 53,22 4,64 205,3 41,06 47,03 4,04 4 2 386,7 17,17 12,45 106,8 128,1120 100 4 13,0 16,55 0,426 257 477,8 82,83 348,4 58,07 69,05 4,59 263,2 52,65 60,98 3,99 2 1 534,1 22,28 19,68 140,2 168,2120 100 5 16,0 20,36 0,423 208 582,9 99,75 419,3 69,88 83,95 4,54 316,3 63,25 74,09 3,94 1 1 657,00 27,09 23,91 172,4 206,9120 100 6 18,9 24,03 0,419 174 682,0 115,3 484,1 80,68 97,93 4,49 364,6 72,91 86,38 3,89 1 1 775,6 31,61 27,88 203,5 244,3120 100 6,3 19,4 24,77 0,413 167 712,3 119,1 490,00 81,67 99,89 4,45 369,6 73,91 88,19 3,86 1 1 799,4 32,24 28,46 209,8 251,7140 40 2,5 6,74 8,59 0,351 409 77,20 23,79 189,3 27,04 35,41 4,69 25,82 12,91 14,34 1,73 4 1 214,1 11,43 3,16 35,56 124,5140 40 3 8,01 10,21 0,350 343 90,30 27,62 221,8 31,68 41,76 4,66 29,90 14,95 16,82 1,71 4 1 281,7 13,48 4,09 42,27 147,9140 60 2,5 7,53 9,59 0,391 408 162,7 37,26 236,6 33,79 42,29 4,97 63,43 21,14 23,43 2,57 4 1 246,3 13,65 5,28 53,60 125,1140 60 3 8,96 11,41 0,390 342 191,9 43,64 278,1 39,73 49,98 4,94 74,16 24,72 27,63 2,55 4 1 320,5 16,13 6,84 63,77 148,8140 60 4 11,7 14,95 0,386 258 247,1 55,42 355,6 50,80 64,63 4,88 93,81 31,27 35,56 2,51 3 1 482,4 20,86 10,09 83,56 195,00140 70 2,5 7,92 10,09 0,411 407 213,1 44,00 260,2 37,17 45,72 5,08 89,3 25,51 28,35 2,98 4 1 262,5 14,76 6,43 62,66 125,3140 70 3 9,43 12,01 0,410 341 252,0 51,66 306,2 43,75 54,09 5,05 104,7 29,91 33,49 2,95 4 1 339,8 17,46 8,32 74,58 149,2

x 140 70 4 12,4 15,75 0,406 258 326,0 65,94 392,6 56,09 70,07 4,99 133,2 38,05 43,24 2,91 3 1 508,2 22,62 12,28 97,81 195,6x 140 70 5 15,2 19,36 0,403 208 395,1 78,88 471,5 67,35 85,05 4,94 158,7 45,35 52,31 2,86 1 1 624,7 27,45 16,88 120,2 240,4x 140 80 3 9,90 12,61 0,430 341 317,1 59,69 334,4 47,77 58,20 5,15 141,2 35,31 39,64 3,35 4 1 359,2 18,78 9,86 85,43 149,5x 140 80 4 13,0 16,55 0,426 257 411,6 76,48 429,6 61,37 75,51 5,10 180,4 45,10 51,31 3,30 3 1 534,1 24,37 14,56 112,1 196,2x 140 80 5 16,0 20,36 0,423 208 500,5 91,83 517,1 73,87 91,80 5,04 215,9 53,99 62,24 3,26 1 1 657,0 29,63 20,09 137,9 241,4x 140 80 6 18,9 24,03 0,419 174 583,8 105,8 597,00 85,29 107,1 4,98 248,0 61,99 72,43 3,21 1 1 775,6 34,56 23,37 162,8 285,00x 140 80 6,3 19,4 24,77 0,413 167 608,5 109,2 602,7 86,10 109,1 4,93 251,4 62,85 73,97 3,19 1 1 799,4 35,20 23,87 167,8 293,7

140 100 2,5 9,10 11,59 0,471 406 387,3 64,23 331,1 47,30 56,04 5,35 198,2 39,63 44,61 4,14 4 4 296,8 14,51 10,23 89,97 126,00140 100 3 10,8 13,81 0,470 340 459,6 75,76 390,7 55,82 66,42 5,32 233,5 46,70 52,85 4,11 4 2 397,9 21,44 13,16 107,2 150,1140 100 4 14,3 18,15 0,466 257 599,3 97,57 503,6 71,94 86,39 5,27 300,1 60,02 68,66 4,07 3 1 585,7 27,88 19,37 140,9 197,3140 100 5 17,6 22,36 0,463 207 732,1 117,8 608,2 86,89 105,3 5,22 361,4 72,29 83,59 4,02 1 1 721,5 33,98 26,98 173,6 243,00140 100 6 20,8 26,43 0,459 174 858,0 136,5 704,8 100,7 123,2 5,16 417,7 83,53 97,66 3,97 1 1 853,1 39,75 31,52 205,2 287,3140 100 6,3 21,4 27,29 0,453 166 896,9 141,2 715,4 102,2 125,9 5,12 425,0 84,99 100,0 3,95 1 1 880,7 40,64 32,27 211,9 296,6140 110 2,5 9,49 12,09 0,491 406 451,3 70,98 354,7 50,67 59,47 5,42 245,7 44,68 50,53 4,51 4 4 301,8 14,98 11,61 99,11 126,1140 110 3 11,3 14,41 0,490 340 536,1 83,80 418,9 59,84 70,53 5,39 289,9 52,70 59,90 4,49 4 3 409,1 19,31 14,92 118,1 150,3140 110 4 14,9 18,95 0,486 256 700,0 108,1 540,6 77,23 91,83 5,34 373,4 67,89 77,93 4,44 3 1 611,5 29,64 21,91 155,3 197,7140 110 5 18,3 23,36 0,483 207 856,4 130,8 653,8 93,4 112,1 5,29 450,7 81,95 95,02 4,39 1 1 753,8 36,16 30,67 191,5 243,7140 110 6 21,7 27,63 0,479 173 1005 151,8 758,7 108,4 131,2 5,24 522,0 94,91 111,2 4,35 1 1 891,8 42,34 35,88 226,5 288,3140 110 6,3 22,4 28,55 0,473 166 1052 157,3 771,8 110,3 134,4 5,20 531,9 96,71 114,0 4,32 1 1 921,4 43,36 36,78 234,1 297,9140 120 3 11,8 15,01 0,510 340 615,4 91,84 447,00 63,86 74,64 5,46 353,4 58,90 67,26 4,85 4 4 416,4 19,51 16,73 129,1 150,6140 120 4 15,5 19,75 0,506 256 804,5 118,7 577,6 82,52 97,27 5,41 456,1 76,02 87,61 4,81 3 2 637,3 31,39 24,53 169,8 198,1140 120 5 19,1 24,36 0,503 206 985,5 143,8 699,4 99,91 118,8 5,36 551,6 91,94 107,0 4,76 1 1 786,00 38,34 34,51 209,5 244,4140 120 6 22,6 28,83 0,499 173 1158 167,2 812,6 116,1 139,3 5,31 640,2 106,7 125,3 4,71 1 1 930,5 44,94 40,44 248,0 289,3140 120 6,3 23,4 29,81 0,493 165 1213 173,3 828,1 118,3 142,8 5,27 653,1 108,9 128,6 4,68 1 1 962,00 46,08 41,49 256,4 299,1



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150 50 2,5 7,53 9,59 0,391 408 127,7 32,78 254,1 33,88 43,52 5,15 45,17 18,07 19,95 2,17 4 1 232,7 14,04 4,27 44,67 134,0150 50 3 8,96 11,41 0,390 342 150,2 38,28 298,6 39,81 51,43 5,12 52,65 21,06 23,49 2,15 4 1 305,5 16,60 5,55 53,14 159,4150 60 2,5 7,92 10,09 0,411 407 177,9 40,01 281,3 37,50 47,21 5,28 67,56 22,52 24,87 2,59 4 1 248,8 15,23 5,38 53,71 134,3150 60 3 9,43 12,01 0,410 341 210,0 46,89 331,0 44,13 55,84 5,25 79,04 26,35 29,34 2,57 4 1 324,9 18,02 6,99 63,93 159,8150 60 4 12,4 15,75 0,406 258 270,5 59,60 424,0 56,54 72,31 5,19 100,1 33,37 37,80 2,52 4 1 488,6 23,34 10,32 83,84 209,6150 70 2,5 8,31 10,59 0,431 407 233,7 47,25 308,5 41,13 50,89 5,40 94,99 27,14 30,04 3,00 4 1 265,0 16,42 6,55 62,78 134,5150 70 3 9,90 12,61 0,430 341 276,4 55,51 363,4 48,45 60,25 5,37 111,4 31,84 35,50 2,97 4 1 344,3 19,44 8,51 74,75 160,2150 70 4 13,0 16,55 0,426 257 357,7 70,92 466,7 62,22 78,15 5,31 141,9 40,54 45,88 2,93 4 1 514,4 25,22 12,55 98,11 210,2150 70 5 16,0 20,36 0,423 208 433,6 84,92 561,5 74,86 94,98 5,25 169,3 48,37 55,56 2,88 2 1 657,0 30,65 17,93 120,7 258,6150 90 2,5 9,10 11,59 0,471 406 358,9 61,74 362,9 48,38 58,27 5,60 166,0 36,89 41,13 3,78 4 3 293,1 15,61 9,08 80,98 135,0150 90 3 10,8 13,81 0,470 340 425,7 72,77 428,2 57,10 69,07 5,57 195,4 43,43 48,70 3,76 4 2 383,0 22,29 11,74 96,48 160,8150 90 4 14,3 18,15 0,466 257 554,2 93,60 552,0 73,59 89,83 5,51 250,7 55,72 63,22 3,72 4 1 566,0 28,99 17,29 126,8 211,3150 90 5 17,6 22,36 0,463 207 676,0 112,8 666,6 88,88 109,5 5,46 301,4 66,99 76,91 3,67 2 1 721,5 35,33 24,82 156,2 260,4150 90 6 20,8 26,43 0,459 174 791,1 130,6 772,4 103,0 128,0 5,41 347,7 77,27 89,78 3,63 1 1 853,1 41,32 28,98 184,7 307,8150 90 6,3 21,4 27,29 0,453 166 826,3 135,0 783,2 104,4 130,8 5,36 353,8 78,62 91,94 3,60 1 1 880,7 42,22 29,67 190,7 317,8

x 150 100 3 11,3 14,41 0,490 340 507,2 81,40 460,6 61,42 73,48 5,65 247,6 49,53 55,76 4,15 4 2 402,4 23,71 13,46 107,4 161,1x 150 100 4 14,9 18,95 0,486 256 661,6 104,9 594,6 79,28 95,67 5,60 318,6 63,71 72,50 4,10 4 1 591,8 30,87 19,79 141,2 211,8x 150 100 5 18,3 23,36 0,483 207 808,7 126,8 719,2 95,89 116,7 5,55 384,0 76,80 88,34 4,05 2 1 753,8 37,67 28,51 174,1 261,1x 150 100 6 21,7 27,63 0,479 173 948,3 147,1 834,7 111,3 136,7 5,50 444,2 88,84 103,3 4,01 1 1 891,8 44,11 33,34 206,0 308,9x 150 100 6,3 22,4 28,55 0,473 166 991,6 152,3 848,3 113,1 139,9 5,45 452,7 90,53 105,9 3,98 1 1 921,4 45,14 34,18 212,8 319,2

150 100 7,1 24,9 31,75 0,469 148 1096 166,7 926,6 123,6 154,1 5,40 493,5 98,69 116,5 3,94 1 1 1024,6 49,72 37,61 236,6 354,9x 150 100 8 27,7 35,24 0,466 132 1206 181,9 1008 134,4 169,2 5,35 535,7 107,1 127,9 3,90 1 1 1137,4 54,59 41,26 262,7 394,0

150 110 2,5 9,88 12,59 0,511 406 498,8 76,23 417,3 55,64 65,64 5,76 260,2 47,30 53,22 4,55 4 4 304,3 16,49 11,83 99,24 135,3150 110 3 11,8 15,01 0,510 340 592,6 90,04 493,1 65,74 77,89 5,73 307,1 55,83 63,11 4,52 4 3 413,5 21,22 15,25 118,3 161,3150 110 4 15,5 19,75 0,506 256 774,2 116,3 637,3 84,97 101,5 5,68 395,9 71,98 82,17 4,48 4 1 617,6 32,76 22,38 155,7 212,3150 110 5 19,1 24,36 0,503 206 947,7 140,8 771,8 102,9 124,0 5,63 478,3 86,96 100,3 4,43 2 1 786,0 40,01 32,36 192,0 261,8150 110 6 22,6 28,83 0,499 173 1113 163,6 896,9 119,6 145,3 5,58 554,5 100,8 117,4 4,39 1 1 930,5 46,90 37,89 227,3 309,9150 110 6,3 23,4 29,81 0,493 165 1165 169,6 913,4 121,8 148,9 5,54 565,8 102,9 120,5 4,36 1 1 962,0 48,07 38,88 235,0 320,4160 40 2,5 7,53 9,59 0,391 408 90,78 27,31 269,0 33,63 44,5 5,30 29,34 14,67 16,22 1,75 4 1 218,7 14,36 3,27 35,73 142,9160 40 3 8,96 11,41 0,390 342 106,2 31,74 315,9 39,49 52,57 5,26 34,02 17,01 19,04 1,73 4 1 290,0 16,97 4,27 42,51 170,1160 50 3 9,43 12,01 0,410 341 162,8 40,93 352,9 44,11 57,28 5,42 55,97 22,39 24,90 2,16 4 1 309,4 18,49 5,67 53,27 170,5

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160 60 3 9,90 12,61 0,430 341 228,2 50,14 389,9 48,73 61,99 5,56 83,91 27,97 31,05 2,58 4 1 328,7 20,01 7,14 64,07 170,9160 60 4 13,0 16,55 0,426 257 294,0 63,79 500,4 62,54 80,38 5,50 106,4 35,46 40,04 2,54 4 1 498,1 25,94 10,61 84,09 224,2160 70 3 10,4 13,21 0,450 341 301,0 59,36 426,8 53,35 66,70 5,68 118,2 33,76 37,51 2,99 4 1 348,1 21,53 8,68 74,9 171,2160 70 4 13,6 17,35 0,446 257 389,7 75,90 549,0 68,63 86,62 5,63 150,6 43,04 48,52 2,95 4 1 523,9 27,96 12,91 98,38 224,9160 70 5 16,8 21,36 0,443 207 472,5 90,96 661,6 82,70 105,4 5,57 179,9 51,39 58,81 2,90 2 1 689,2 34,02 18,98 121,1 276,8160 80 3 10,8 13,81 0,470 340 380,3 68,59 463,8 57,98 71,41 5,80 159,0 39,76 44,26 3,39 4 1 367,5 23,05 10,29 85,76 171,5

x 160 80 4 14,3 18,15 0,466 257 494,1 88,03 597,7 74,71 92,86 5,74 203,5 50,89 57,39 3,35 4 1 549,7 29,97 15,29 112,7 225,4x 160 80 5 17,6 22,36 0,463 207 601,3 105,9 721,7 90,21 113,2 5,68 244,1 61,03 69,74 3,30 2 1 721,5 36,52 22,51 138,9 277,7x 160 80 6 20,8 26,43 0,459 174 702,1 122,3 836,0 104,5 132,3 5,62 280,9 70,22 81,31 3,26 1 1 853,1 42,70 26,24 164,2 328,3

160 80 6,3 21,4 27,29 0,453 166 732,3 126,3 846,5 105,8 135,1 5,57 285,7 71,43 83,25 3,24 1 1 880,7 43,60 26,87 169,5 339,0160 90 3 11,3 14,41 0,490 340 465,4 77,82 500,8 62,60 76,12 5,90 206,8 45,95 51,31 3,79 4 2 386,8 24,57 11,97 96,65 171,8160 90 4 14,9 18,95 0,486 256 606,2 100,2 646,4 80,80 99,1 5,84 265,5 59,01 66,66 3,74 4 1 575,5 31,98 17,77 127,1 226,0160 90 5 18,3 23,36 0,483 207 739,7 120,9 781,8 97,72 120,9 5,79 319,5 71,00 81,16 3,70 2 1 753,8 39,02 26,19 156,7 278,5160 90 6 21,7 27,63 0,479 173 866,0 140,0 907,2 113,4 141,6 5,73 368,9 81,98 94,82 3,65 1 1 891,8 45,69 30,60 185,4 329,5160 90 6,3 22,4 28,55 0,473 166 904,7 144,8 920,9 115,1 144,8 5,68 375,9 83,53 97,21 3,63 1 1 921,4 46,73 31,37 191,5 340,5

x 160 90 7,1 24,9 31,75 0,469 148 997,9 158,4 1006 125,7 159,4 5,63 409,2 90,93 106,9 3,59 1 1 1025 51,45 34,50 213,0 378,6160 100 3 11,8 15,01 0,510 340 555,5 87,05 537,8 67,22 80,83 5,99 261,8 52,35 58,67 4,18 4 2 406,2 26,09 13,73 107,6 172,1160 100 4 15,5 19,75 0,506 256 724,9 112,3 695,1 86,88 105,3 5,93 337,0 67,40 76,34 4,13 4 1 601,3 34,00 20,33 141,5 226,4160 100 5 19,1 24,36 0,503 206 886,4 135,8 841,9 105,2 128,7 5,88 406,6 81,32 93,09 4,09 2 1 786,0 41,52 30,04 174,6 279,3160 100 6 22,6 28,83 0,499 173 1040 157,7 978,4 122,3 150,8 5,83 470,7 94,15 108,9 4,04 1 1 930,5 48,67 35,16 206,6 330,6160 100 6,3 23,4 29,81 0,493 165 1088 163,3 995,4 124,4 154,5 5,78 480,4 96,07 111,8 4,01 1 1 962,0 49,85 36,08 213,6 341,8160 120 4 16,8 21,35 0,546 256 979,5 136,6 792,4 99,06 117,8 6,09 510,0 84,99 96,89 4,89 4 2 653,0 38,02 25,73 170,5 227,3160 120 5 20,7 26,36 0,543 206 1201 165,8 962,0 120,3 144,2 6,04 617,8 103,0 118,5 4,84 2 1 850,6 46,52 38,23 210,5 280,6160 120 6 24,5 31,23 0,539 173 1414 193,2 1121 140,1 169,3 5,99 718,3 119,7 139,0 4,80 1 1 1008 54,63 44,85 249,4 332,5160 120 6,3 25,4 32,33 0,533 165 1481 200,4 1144 143,0 173,8 5,95 734,6 122,4 142,9 4,77 1 1 1043 56,10 46,11 258,2 344,2160 120 7,1 28,3 36,01 0,529 147 1641 220,3 1255 156,9 192,0 5,90 804,5 134,1 157,7 4,73 1 1 1162 61,96 50,90 287,5 383,4160 120 8 31,4 40,04 0,526 131 1814 241,4 1371 171,4 211,5 5,85 877,9 146,3 173,7 4,68 1 1 1292 68,26 56,04 319,8 426,3160 120 8,8 34,2 43,52 0,522 120 1960 258,9 1467 183,4 227,9 5,81 938,0 156,3 187,0 4,64 1 1 1405 73,55 60,36 347,6 463,4160 120 10 38,1 48,57 0,517 106 2166 283,0 1597 199,6 251,0 5,73 1019 169,9 205,8 4,58 1 1 1567 80,99 66,42 387,8 517,1180 100 4 16,8 21,35 0,546 256 853,9 127,1 926,0 102,9 125,9 6,59 373,9 74,78 84,02 4,18 4 1 616,9 40,63 21,26 142,1 255,7

x 180 100 5 20,7 26,36 0,543 206 1045 153,9 1124 124,9 154,0 6,53 451,8 90,35 102,6 4,14 3 1 850,6 49,71 29,16 175,4 315,7180 100 5,6 23,0 29,30 0,541 185 1155 169,1 1237 137,4 170,3 6,50 495,7 99,14 113,3 4,11 2 1 945,5 54,95 36,56 195,0 350,9

x 180 100 6 24,5 31,23 0,539 173 1227 178,9 1310 145,5 180,8 6,48 523,8 104,8 120,2 4,10 2 1 1008 58,36 38,80 207,8 374,1180 100 6,3 25,4 32,33 0,533 165 1283 185,5 1335 148,3 185,5 6,43 535,8 107,2 123,6 4,07 1 1 1043 59,88 39,89 215,1 387,3

x 180 100 7,1 28,3 36,01 0,529 147 1420 203,5 1463 162,6 204,9 6,38 585,6 117,1 136,3 4,03 1 1 1162 66,11 44,00 239,6 431,3x 180 100 8 31,4 40,04 0,526 131 1565 222,5 1598 177,6 225,6 6,32 637,5 127,5 149,9 3,99 1 1 1292 72,81 48,39 266,5 479,6

180 120 4 18,0 22,95 0,586 255 1160 154,6 1050 116,7 140,0 6,76 563,8 93,97 106,2 4,96 4 2 668,6 45,17 26,88 171,0 256,6180 120 5 22,3 28,36 0,583 206 1424 187,8 1277 141,9 171,5 6,71 684,0 114,0 130,0 4,91 3 1 915,1 55,35 36,79 211,3 317,0180 120 6 26,4 33,63 0,579 172 1677 219,1 1491 165,7 201,7 6,66 796,3 132,7 152,7 4,87 2 1 1085 65,10 49,27 250,7 376,0180 120 6,3 27,4 34,85 0,573 164 1757 227,6 1525 169,5 207,4 6,62 816,1 136,0 157,2 4,84 1 1 1125 66,94 50,73 259,7 389,6180 120 7,1 30,5 38,85 0,569 146 1949 250,5 1676 186,2 229,4 6,57 895,2 149,2 173,8 4,80 1 1 1254 74,04 56,07 289,5 434,3180 120 8 34,0 43,24 0,566 131 2156 274,8 1835 203,9 253,1 6,51 978,4 163,1 191,6 4,76 1 1 1396 81,70 61,83 322,3 483,4180 120 8,8 36,9 47,04 0,562 119 2332 295,1 1967 218,6 273,2 6,47 1047 174,5 206,6 4,72 1 1 1518 88,16 66,68 350,6 525,9180 120 10 41,3 52,57 0,557 106 2582 323,3 2149 238,8 301,5 6,39 1141 190,1 227,8 4,66 1 1 1696 97,31 73,52 391,8 587,7



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200 80 4 16,8 21,35 0,546 256 663,6 111,1 1046 104,6 132,4 7,00 249,8 62,45 69,55 3,42 4 1 577,6 42,72 16,58 113,7 284,1200 80 5 20,7 26,36 0,543 206 808,4 134,1 1269 126,9 161,9 6,94 300,4 75,11 84,74 3,38 4 1 794,4 52,24 22,50 140,3 350,8

x 200 80 6 24,5 31,23 0,539 173 944,8 155,2 1477 147,7 190,0 6,88 346,7 86,69 99,07 3,33 2 1 1008 61,31 31,97 166,3 415,7200 80 6,3 25,4 32,33 0,533 165 986 160,6 1503 150,3 194,7 6,82 354,3 88,58 101,8 3,31 2 1 1043 62,84 32,86 172,1 430,3

x 200 100 5 22,3 28,36 0,583 206 1206 171,9 1459 145,9 181,4 7,17 496,9 99,39 112,1 4,19 4 1 858,9 58,53 29,87 176,1 352,2x 200 100 6 26,4 33,63 0,579 172 1417 200,1 1703 170,3 213,3 7,12 576,9 115,4 131,5 4,14 2 1 1085 68,83 42,44 208,9 417,8

200 100 6,3 27,4 34,85 0,573 164 1483 207,6 1739 173,9 219,1 7,06 591,2 118,2 135,4 4,12 2 1 1125 70,72 43,70 216,5 432,9200 100 7,1 30,5 38,85 0,569 146 1641 228,0 1910 191,0 242,3 7,01 647,0 129,4 149,5 4,08 1 1 1254 78,19 48,25 241,3 482,5

x 200 100 8 34,0 43,24 0,566 131 1811 249,6 2091 209,1 267,3 6,95 705,4 141,1 164,7 4,04 1 1 1396 86,25 53,14 268,6 537,2x 200 120 5 23,8 30,36 0,623 205 1652 209,9 1649 164,9 200,9 7,37 750,1 125,0 141,5 4,97 4 1 923,4 64,83 37,69 212,1 353,5x 200 120 6 28,3 36,03 0,619 172 1947 245,1 1929 192,9 236,6 7,32 874,4 145,7 166,3 4,93 2 1 1163 76,34 53,68 251,8 419,6

200 120 6,3 29,3 37,37 0,613 164 2040 254,7 1976 197,6 243,5 7,27 897,7 149,6 171,5 4,90 2 1 1206 78,60 55,36 261,1 435,2200 120 7,1 32,7 41,69 0,609 146 2265 280,7 2174 217,4 269,7 7,22 985,8 164,3 189,8 4,86 1 1 1345 87,03 61,25 291,3 485,5

x 200 120 8 36,5 46,44 0,606 130 2507 308,3 2386 238,6 298,0 7,17 1079 179,8 209,5 4,82 1 1 1499 96,17 67,61 324,5 540,8200 120 8,8 39,7 50,56 0,602 119 2714 331,4 2562 256,2 322,0 7,12 1156 192,7 226,2 4,78 1 1 1632 103,9 72,99 353,3 588,8

x 200 120 10 44,4 56,57 0,597 106 3007 363,7 2806 280,6 356,1 7,04 1262 210,4 249,8 4,72 1 1 1826 114,9 80,62 395,2 658,7220 120 5 25,4 32,36 0,663 205 1885 231,9 2082 189,3 232,2 8,02 816,3 136,1 153,0 5,02 4 1 943,4 74,95 39,09 212,8 390,1

x 220 120 6 30,2 38,43 0,659 171 2222 271,1 2439 221,7 273,8 7,97 952,4 158,7 180,0 4,98 3 1 1240 88,36 51,23 252,7 463,4220 120 6,3 31,3 39,89 0,653 164 2329 281,9 2501 227,4 282,2 7,92 979,2 163,2 185,9 4,95 3 1 1287 91,06 52,67 262,3 480,9220 120 7,1 35,0 44,53 0,649 146 2586 310,8 2756 250,6 312,8 7,87 1076 179,4 205,8 4,92 2 1 1437 101,0 66,42 292,8 536,8

x 220 120 8 39,0 49,64 0,646 130 2864 341,7 3029 275,4 346,0 7,81 1179 196,6 227,4 4,87 1 1 1602 111,7 73,39 326,5 598,5220 120 8,8 42,5 54,08 0,642 119 3102 367,7 3258 296,1 374,3 7,76 1265 210,9 245,8 4,84 1 1 1745 120,8 79,31 355,7 652,1

x 220 120 10 47,5 60,57 0,637 105 3440 404,1 3576 325,1 414,7 7,68 1383 230,6 271,8 4,78 1 1 1955 133,8 87,72 398,3 730,2x 250 150 5 30,1 38,36 0,783 204 3285 336,9 3304 264,3 319,8 9,28 1508 201,1 225,5 6,27 4 2 1064 103,2 54,35 268,0 446,7x 250 150 6 35,8 45,63 0,779 171 3886 395,7 3886 310,8 378,1 9,23 1768 235,8 266,3 6,23 4 1 1366 122,0 69,73 318,9 531,4

250 150 6,3 37,3 47,45 0,773 163 4078 412,2 4001 320,1 390,9 9,18 1825 243,3 275,7 6,20 4 1 1447 126,2 73,55 331,5 552,6250 150 7,1 41,6 53,05 0,769 145 4543 456,3 4427 354,1 434,5 9,13 2015 268,7 306,2 6,16 3 1 1712 140,2 86,70 370,7 617,8

x 250 150 8 46,5 59,24 0,766 129 5050 504,0 4886 390,9 482,2 9,08 2219 295,9 339,6 6,12 2 1 1912 155,6 109,6 413,9 689,9250 150 8,8 50,8 64,64 0,762 118 5488 544,5 5274 422,0 523,1 9,03 2392 318,9 368,1 6,08 1 1 2086 168,8 118,8 451,7 752,8

x 250 150 10 57,0 72,57 0,757 104 6121 602,1 5825 466,0 582,0 8,96 2634 351,2 409,2 6,02 1 1 2342 187,8 132,1 507,0 845,1250 150 12 66,0 84,06 0,738 88 7088 684,4 6458 516,6 658,0 8,77 2925 390,0 463,3 5,90 1 1 2713 212,3 149,5 587,3 978,9

x 250 150 12,5 68,3 87,04 0,736 85 7315 704,1 6633 530,6 678,3 8,73 3002 400,3 477,5 5,87 1 1 2809 218,9 154,1 608,2 1014

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x 260 140 6 35,8 45,63 0,779 171 3646 382,5 4082 314,0 385,9 9,46 1567 223,9 251,8 5,86 4 1 1339 124,6 64,79 297,6 552,7260 140 6,3 37,3 47,45 0,773 163 3825 398,4 4202 323,2 399,0 9,41 1617 231,0 260,7 5,84 4 1 1420 128,8 68,41 309,4 574,7260 140 7,1 41,6 53,05 0,769 145 4259 440,8 4647 357,5 443,4 9,36 1785 254,9 289,5 5,80 3 1 1712 143,1 82,27 345,9 642,5

x 260 140 8 46,5 59,24 0,766 129 4731 486,5 5129 394,5 492,0 9,30 1964 280,6 320,9 5,76 2 1 1912 158,8 103,6 386,3 717,5260 140 8,8 50,8 64,64 0,762 118 5138 525,3 5536 425,9 533,7 9,25 2115 302,2 347,8 5,72 1 1 2086 172,3 112,2 421,6 782,9

x 260 140 10 57,0 72,57 0,757 104 5724 580,4 6113 470,2 593,8 9,18 2328 332,5 386,4 5,66 1 1 2342 191,6 124,7 473,2 878,9x 260 180 6 39,6 50,43 0,859 170 5566 501,4 4856 373,5 446,9 9,81 2763 307,1 347,9 7,40 4 2 1494 144,2 89,50 384,4 555,3

260 180 6,3 41,2 52,49 0,853 163 5844 523,0 5013 385,6 462,9 9,77 2856 317,4 360,6 7,38 4 1 1583 149,4 94,52 400,1 577,9260 180 7,1 46,1 58,73 0,849 145 6523 580,3 5556 427,4 515,3 9,73 3162 351,4 401,3 7,34 3 1 1895 166,3 113,4 447,6 646,6

x 260 180 8 51,5 65,64 0,846 129 7267 642,4 6145 472,7 572,7 9,68 3493 388,1 445,8 7,29 2 1 2118 184,8 143,9 500,4 722,7260 180 8,8 56,3 71,68 0,842 117 7912 695,6 6647 511,3 622,2 9,63 3774 419,4 484,1 7,26 1 1 2313 200,8 156,2 546,4 789,3

x 260 180 10 63,2 80,57 0,837 104 8850 771,9 7363 566,4 693,8 9,56 4174 463,8 539,5 7,20 1 1 2600 223,9 174,1 614,1 887,1260 180 12 73,5 93,66 0,818 87 10328 884,4 8245 634,2 789,9 9,38 4679 519,9 614,9 7,07 1 1 3023 254,9 198,5 713,9 1031

x 300 100 5 30,1 38,36 0,783 204 2044 262,2 4065 271,0 348,2 10,29 722,8 144,6 159,6 4,34 4 1 930,8 112,4 34,14 178,7 527,4x 300 100 6 35,8 45,63 0,779 171 2403 306,2 4777 318,5 411,4 10,23 842,4 168,5 187,9 4,30 4 1 1222 132,8 44,43 212,6 637,7

300 100 6,3 37,3 47,45 0,773 163 2515 318,3 4907 327,1 424,9 10,17 868,1 173,6 194,5 4,28 4 1 1302 137,1 47,03 221,0 663,1300 100 7,1 41,6 53,05 0,769 145 2787 350,7 5422 361,5 472,0 10,11 953,9 190,8 215,5 4,24 4 1 1549 152,3 55,30 247,1 741,3

x 300 100 8 46,5 59,24 0,766 129 3080 385,2 5978 398,5 523,5 10,05 1045 209,0 238,3 4,20 4 1 1833 168,9 64,48 276,0 827,9300 150 6 40,5 51,63 0,879 170 4988 478,6 6074 404,9 499,6 10,85 2080 277,3 309,5 6,35 4 1 1416 161,2 74,32 320,7 641,4300 150 6,3 42,2 53,74 0,873 162 5235 498,9 6264 417,6 517,3 10,80 2150 286,6 320,9 6,32 4 1 1506 167,0 78,73 333,8 667,6300 150 7,1 47,2 60,15 0,869 144 5834 553,0 6946 463,0 576,0 10,75 2378 317,0 357,0 6,29 4 1 1778 185,9 92,65 373,6 747,1300 150 8,8 57,7 73,44 0,862 117 7058 661,5 8312 554,1 695,7 10,64 2831 377,4 430,2 6,21 3 1 2370 224,5 121,8 456,2 912,3300 150 10 64,8 82,57 0,857 104 7879 732,8 9209 614,0 775,9 10,56 3125 416,7 479,2 6,15 2 1 2665 250,4 154,6 512,8 1026

x 300 200 6 45,2 57,63 0,979 170 8115 651,2 7370 491,4 587,8 11,31 3962 396,2 446,1 8,29 4 2 1609 189,7 107,7 429,5 644,3300 200 6,3 47,1 60,05 0,973 162 8524 679,8 7624 508,3 609,9 11,27 4104 410,4 463,2 8,27 4 2 1709 196,8 114,1 447,6 671,3300 200 7,1 52,8 67,25 0,969 144 9524 755,7 8469 564,6 680,0 11,22 4553 455,3 516,2 8,23 4 1 2007 219,5 134,0 501,2 751,8

x 300 200 8 59,1 75,24 0,966 128 10627 838,4 9389 626,0 757,1 11,17 5042 504,2 574,5 8,19 4 1 2350 244,3 156,5 560,8 841,2300 200 8,8 64,6 82,24 0,962 117 11586 909,5 10178 678,6 823,8 11,12 5459 545,9 624,9 8,15 3 1 2654 265,9 176,2 613,0 919,5

x 300 200 10 72,7 92,57 0,957 103 12987 1012 11313 754,2 920,9 11,05 6058 605,8 698,1 8,09 2 1 2987 297,2 225,3 689,9 1035300 200 12 84,8 108,06 0,938 87 15236 1167 12788 852,5 1056 10,88 6854 685,4 801,2 7,96 1 1 3487 340,8 258,6 805,4 1208

x 300 200 12,5 88,0 112,04 0,936 84 15768 1204 13179 878,6 1091 10,85 7060 706,0 827,9 7,94 1 1 3616 352,2 267,2 835,1 1253400 100 6 45,2 57,63 0,979 170 3421 412,4 10132 506,6 669,6 13,26 1108 221,6 244,3 4,38 4 1 1283 216,1 48,10 214,8 797,7400 100 6,3 47,1 60,05 0,973 162 3579 429,1 10447 522,3 693,6 13,19 1145 229,0 253,5 4,37 4 1 1373 223,9 51,20 223,8 860,9400 100 7,1 52,8 67,25 0,969 144 3968 473,5 11587 579,4 772,8 13,13 1261 252,2 281,4 4,33 4 1 1652 249,4 60,98 250,6 1002400 120 6 47,1 60,03 1,019 170 4831 505,2 11063 553,2 716,9 13,58 1655 275,8 303,1 5,25 4 1 1360 231,4 60,57 258,1 797,7400 120 6,3 49,1 62,57 1,013 162 5062 526,3 11423 571,2 743,2 13,51 1713 285,5 314,8 5,23 4 1 1455 239,9 64,50 269,0 860,9400 120 7,1 55,0 70,09 1,009 144 5628 582,5 12683 634,2 828,6 13,45 1892 315,3 350,1 5,20 4 1 1743 267,4 76,84 301,4 1005400 120 8 61,6 78,44 1,006 128 6245 643,1 14056 702,8 922,4 13,39 2084 347,4 388,7 5,15 4 1 2086 297,7 91,12 337,3 1124400 120 8,8 67,3 85,76 1,002 117 6774 694,6 15231 761,6 1004 13,33 2246 374,4 421,9 5,12 4 1 2403 323,9 103,9 368,8 1229

x 400 200 6 54,7 69,63 1,179 169 12069 877,1 14789 739,5 906,0 14,57 5092 509,2 562,5 8,55 4 2 1670 292,4 116,5 432,5 797,7400 200 6,3 57,0 72,65 1,173 161 12673 916,2 15330 766,5 941,7 14,53 5286 528,6 585,2 8,53 4 2 1780 303,9 124,0 451,2 860,9400 200 7,1 63,9 81,45 1,169 144 14169 1020 17068 853,4 1052 14,48 5875 587,5 653,2 8,49 4 1 2110 339,4 147,3 505,9 1012

x 400 200 8 71,6 91,24 1,166 128 15820 1133 18974 948,7 1173 14,42 6517 651,7 728,1 8,45 4 1 2499 378,7 174,4 566,7 1133400 200 8,8 78,4 99,84 1,162 116 17260 1231 20619 1031 1279 14,37 7069 706,9 793,1 8,41 4 1 2857 412,8 199,0 620,1 1240

x 400 200 10 88,4 112,6 1,157 103 19368 1373 23003 1150 1434 14,30 7864 786,4 888,1 8,36 4 1 3408 462,7 236,2 699,1 1398400 200 12 104 132,1 1,138 86 22782 1591 26248 1312 1656 14,10 8977 897,7 1027 8,24 2 1 4262 534,5 331,4 820,2 1640

x 400 200 12,5 108 137,0 1,136 83 23594 1644 27101 1355 1714 14,06 9260 926,1 1062 8,22 2 1 4423 553,1 342,8 851,2 1702



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M = weight Wt = torsional section modulus PL = cross-section class in concentric compressionA = cross-section area I = moment of inertia Nc.Rd = compression resistance without bucklingAu = external area Wel = elastic section modulus Mc.Rd = bending resistanceAm/V = cross-section factor in fire design Wpl = plastic section modulus The effect of shear buckling has not been accounted forIt = torsional modulus i = radius of gyration hollow sections of cross-section class 4 (section 2.4.2.2)

1) = recommended series

Table 9.1.3 Cross-sectional properties and resistance values for circular longitudinally welded hollow sections of steel grade S355J2H (fy = 355 N/mm2).

The calculated resistance values are design values (section 2.1) based on the default value (1.1) of the partial safety factor of material γγγγM0 used in Eurocode 3.The partial safety factor values may differ in each country. National values must be checked from the NAD (National Application Document) of the country in question.

1) d t M A Au Am/V It Wt I Wel Wpl i PL Nc.Rd Mc.Rd Vpl.Rd

mm mm kg/m mm2 m2/m 1/m mm4 mm3 mm4 mm3 mm3 mm kN kNm kNx x 102 x 104 x 103 x 104 x 103 x 103 x 10x 33,7 2 1,56 1,99 0,106 533 5,02 2,98 2,51 1,49 2,01 1,12 1 64,28 0,65 23,63x 42,4 2 1,99 2,54 0,133 524 10,38 4,90 5,19 2,45 3,27 1,43 1 81,92 1,05 30,11x 42,4 2,5 2,46 3,13 0,133 425 12,52 5,91 6,26 2,95 3,99 1,41 1 101,1 1,29 37,17

42,4 2,6 2,55 3,25 0,133 409 12,93 6,10 6,46 3,05 4,12 1,41 1 104,9 1,33 38,5642,4 2,9 2,82 3,60 0,133 369 14,11 6,66 7,06 3,33 4,53 1,40 1 116,1 1,46 42,69

x 42,4 3 2,91 3,71 0,133 358 14,49 6,84 7,25 3,42 4,67 1,40 1 119,8 1,51 44,05x 48,3 2 2,28 2,91 0,152 522 15,62 6,47 7,81 3,23 4,29 1,64 1 93,89 1,38 34,51x 48,3 2,5 2,82 3,60 0,152 422 18,92 7,83 9,46 3,92 5,25 1,62 1 116,1 1,69 42,67

48,3 2,6 2,93 3,73 0,152 408 19,55 8,10 9,78 4,05 5,44 1,62 1 120,5 1,75 44,2848,3 2,9 3,25 4,14 0,152 367 21,40 8,86 10,70 4,43 5,99 1,61 1 133,5 1,93 49,06

x 48,3 3 3,35 4,27 0,152 356 22,00 9,11 11,00 4,55 6,17 1,61 1 137,8 1,99 50,6448,3 3,2 3,56 4,53 0,152 336 23,17 9,59 11,59 4,80 6,52 1,60 1 146,3 2,10 53,78

x 60,3 2 2,88 3,66 0,189 516 31,16 10,34 15,58 5,17 6,80 2,06 1 118,2 2,19 43,45x 60,3 2,5 3,56 4,54 0,189 416 37,99 12,60 18,99 6,30 8,36 2,05 1 146,5 2,70 53,85

60,3 2,6 3,70 4,71 0,189 401 39,31 13,04 19,65 6,52 8,66 2,04 1 152,1 2,80 55,9160,3 2,9 4,11 5,23 0,189 361 43,18 14,32 21,59 7,16 9,56 2,03 1 168,8 3,09 62,03

x 60,3 3 4,24 5,40 0,189 350 44,45 14,74 22,22 7,37 9,86 2,03 1 174,3 3,18 64,0660,3 3,2 4,51 5,74 0,189 329 46,94 15,57 23,47 7,78 10,44 2,02 1 185,3 3,37 68,09

x 60,3 4 5,55 7,07 0,189 267 56,35 18,69 28,17 9,34 12,70 2,00 1 228,3 4,10 83,9276,1 2 3,65 4,66 0,239 513 63,96 16,81 31,98 8,40 10,98 2,62 2 150,3 3,54 55,23

x 76,1 2,5 4,54 5,78 0,239 413 78,37 20,60 39,19 10,30 13,55 2,60 1 186,6 4,37 68,5776,1 2,6 4,71 6,00 0,239 398 81,18 21,34 40,59 10,67 14,05 2,60 1 193,8 4,53 71,2176,1 2,9 5,24 6,67 0,239 358 89,48 23,52 44,74 11,76 15,55 2,59 1 215,2 5,02 79,11

x 76,1 3 5,41 6,89 0,239 347 92,19 24,23 46,10 12,11 16,04 2,59 1 222,3 5,18 81,7276,1 3,2 5,75 7,33 0,239 326 97,56 25,64 48,78 12,82 17,02 2,58 1 236,5 5,49 86,93

x 76,1 4 7,11 9,06 0,239 264 118,1 31,04 59,06 15,52 20,81 2,55 1 292,4 6,72 107,5x 76,1 5 8,77 11,17 0,239 214 141,8 37,28 70,92 18,64 25,32 2,52 1 360,4 8,17 132,5

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88,9 2 4,29 5,46 0,279 511 103,1 23,20 51,57 11,60 15,11 3,07 2 176,2 4,88 64,77x 88,9 2,5 5,33 6,79 0,279 411 126,8 28,51 63,37 14,26 18,67 3,06 2 219,0 6,02 80,49

88,9 2,6 5,53 7,05 0,279 396 131,4 29,55 65,68 14,78 19,37 3,05 2 227,5 6,25 83,6288,9 2,9 6,15 7,84 0,279 356 145,0 32,63 72,52 16,31 21,46 3,04 1 252,9 6,92 92,94

x 88,9 3 6,36 8,10 0,279 344 149,5 33,64 74,76 16,82 22,15 3,04 1 261,3 7,15 96,0388,9 3,2 6,76 8,62 0,279 324 158,4 35,64 79,21 17,82 23,51 3,03 1 278,1 7,59 102,2

x 88,9 4 8,38 10,67 0,279 261 192,7 43,35 96,34 21,67 28,85 3,00 1 344,3 9,31 126,6x 88,9 5 10,4 13,18 0,279 212 232,8 52,36 116,4 26,18 35,24 2,97 1 425,3 11,37 156,3

88,9 6 12,3 15,63 0,279 179 269,9 60,72 134,9 30,36 41,31 2,94 1 504,3 13,33 185,488,9 6,3 12,8 16,35 0,279 171 280,5 63,10 140,2 31,55 43,07 2,93 1 527,6 13,90 193,9

101,6 2 4,91 6,26 0,319 510 155,3 30,56 77,63 15,28 19,84 3,52 3 202,0 4,93 74,23x 101,6 2,5 6,11 7,78 0,319 410 191,2 37,64 95,61 18,82 24,56 3,50 2 251,2 7,93 92,32

101,6 2,6 6,35 8,09 0,319 394 198,3 39,03 99,14 19,52 25,49 3,50 2 261,0 8,23 95,92101,6 2,9 7,06 8,99 0,319 355 219,2 43,15 109,6 21,57 28,26 3,49 2 290,2 9,12 106,7

x 101,6 3 7,29 9,29 0,319 343 226,1 44,50 113,0 22,25 29,17 3,49 2 299,9 9,42 110,2101,6 3,2 7,77 9,89 0,319 323 239,7 47,19 119,9 23,59 31,00 3,48 1 319,3 10,00 117,3

x 101,6 4 9,63 12,26 0,319 260 292,6 57,59 146,3 28,80 38,12 3,45 1 395,8 12,30 145,5x 101,6 5 11,9 15,17 0,319 210 354,9 69,87 177,5 34,93 46,70 3,42 1 489,7 15,07 180,0

101,6 6 14,2 18,02 0,319 177 413,4 81,37 206,7 40,68 54,91 3,39 1 581,6 17,72 213,8101,6 6,3 14,8 18,86 0,319 169 430,1 84,67 215,1 42,34 57,30 3,38 1 608,7 18,49 223,7108 2 5,23 6,66 0,339 509 187,2 34,66 93,58 17,33 22,47 3,75 3 214,9 5,59 79,00

x 108 2,5 6,50 8,29 0,339 409 230,7 42,72 115,4 21,36 27,83 3,73 2 267,4 8,98 98,29108 2,6 6,76 8,61 0,339 394 239,3 44,31 119,6 22,15 28,89 3,73 2 277,8 9,32 102,1108 2,9 7,52 9,58 0,339 354 264,6 49,00 132,3 24,50 32,04 3,72 2 309,0 10,34 113,6

x 108 3 7,77 9,90 0,339 342 273,0 50,55 136,5 25,28 33,08 3,71 2 319,4 10,68 117,4108 3,2 8,27 10,54 0,339 322 289,6 53,62 144,8 26,81 35,16 3,71 2 340,0 11,35 125,0

x 108 4 10,3 13,07 0,339 259 353,9 65,54 177,0 32,77 43,29 3,68 1 421,8 13,97 155,0108 5 12,7 16,18 0,339 210 430,1 79,65 215,1 39,83 53,09 3,65 1 522,2 17,13 191,9108 6 15,1 19,23 0,339 176 501,8 92,93 250,9 46,46 62,50 3,61 1 620,5 20,17 228,1108 6,3 15,8 20,13 0,339 168 522,5 96,75 261,2 48,38 65,24 3,60 1 649,6 21,06 238,8114,3 2 5,54 7,06 0,359 508 222,5 38,94 111,3 19,47 25,23 3,97 3 227,7 6,28 83,70

x 114,3 2,5 6,89 8,78 0,359 409 274,5 48,03 137,3 24,02 31,25 3,95 2 283,4 10,09 104,2114,3 2,6 7,16 9,12 0,359 394 284,8 49,82 142,4 24,91 32,45 3,95 2 294,5 10,47 108,2114,3 2,9 7,97 10,15 0,359 354 315,1 55,13 157,6 27,57 36,00 3,94 2 327,5 11,62 120,4

x 114,3 3 8,23 10,49 0,359 342 325,1 56,88 162,6 28,44 37,17 3,94 2 338,5 12,00 124,4114,3 3,2 8,77 11,17 0,359 321 344,9 60,36 172,5 30,18 39,51 3,93 2 360,5 12,75 132,5114,3 4 10,9 13,86 0,359 259 422,1 73,86 211,1 36,93 48,69 3,90 1 447,3 15,71 164,4

x 114,3 5 13,5 17,17 0,359 209 513,8 89,91 256,9 44,96 59,77 3,87 1 554,1 19,29 203,7x 114,3 6 16,0 20,41 0,359 176 600,4 105,1 300,2 52,53 70,45 3,83 1 658,8 22,73 242,2

114,3 6,3 16,8 21,38 0,359 168 625,4 109,4 312,7 54,72 73,57 3,82 1 689,8 23,74 253,61) d t M A Au Am/V It Wt I Wel Wpl i PL Nc.Rd Mc.Rd Vpl.Rd

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Table 9.1.3 Cross-sectional properties and resistance values for circular longitudinally welded hollow sections of steel grade S355J2H (fy = 355 N/mm2) continued.






127 2 6,17 7,85 0,399 508 306,9 48,33 153,4 24,16 31,25 4,42 4 224,1 6,93 93,16x 127 2,5 7,68 9,78 0,399 408 379,1 59,70 189,5 29,85 38,76 4,40 3 315,6 9,63 116,0

127 2,6 7,98 10,16 0,399 393 393,3 61,94 196,7 30,97 40,24 4,40 3 327,9 9,99 120,5127 2,9 8,88 11,31 0,399 353 435,6 68,59 217,8 34,30 44,67 4,39 2 364,9 14,42 134,1

x 127 3 9,17 11,69 0,399 341 449,5 70,79 224,8 35,39 46,14 4,39 2 377,2 14,89 138,6127 3,2 9,77 12,45 0,399 320 477,2 75,15 238,6 37,57 49,06 4,38 2 401,7 15,83 147,6

x 127 4 12,1 15,46 0,399 258 585,2 92,16 292,6 46,08 60,54 4,35 1 498,8 19,54 183,4127 5 15,0 19,16 0,399 208 714,3 112,5 357,1 56,24 74,46 4,32 1 618,5 24,03 227,3127 6 17,9 22,81 0,399 175 836,9 131,8 418,4 65,90 87,92 4,28 1 736,1 28,37 270,6127 6,3 18,8 23,89 0,399 167 872,4 137,4 436,2 68,70 91,86 4,27 1 771,0 29,65 283,4133 2 6,46 8,23 0,418 508 353,2 53,11 176,6 26,56 34,32 4,63 4 233,9 7,59 97,64

x 133 2,5 8,05 10,25 0,418 408 436,5 65,64 218,3 32,82 42,58 4,61 3 330,8 10,59 121,6133 2,6 8,36 10,65 0,418 392 453,0 68,12 226,5 34,06 44,22 4,61 3 343,7 10,99 126,3133 2,9 9,30 11,85 0,418 353 501,8 75,46 250,9 37,73 49,09 4,60 2 382,5 15,84 140,6133 3 9,62 12,25 0,418 341 517,9 77,88 259,0 38,94 50,71 4,60 2 395,4 16,37 145,3133 3,2 10,2 13,05 0,418 320 550,0 82,70 275,0 41,35 53,92 4,59 2 421,1 17,40 154,8

x 133 4 12,7 16,21 0,418 258 675,1 101,5 337,5 50,76 66,59 4,56 2 523,2 21,49 192,3133 5 15,8 20,11 0,418 208 824,8 124,0 412,4 62,02 81,96 4,53 1 648,9 26,45 238,5133 6 18,8 23,94 0,418 175 967,4 145,5 483,7 72,74 96,85 4,50 1 772,6 31,25 284,0133 6,3 19,7 25,08 0,418 167 1008,9 151,7 504,4 75,85 101,2 4,49 1 809,3 32,67 297,5139,7 2,9 9,78 12,46 0,439 352 583,4 83,52 291,7 41,76 54,28 4,84 3 402,2 13,48 147,8

x 139,7 3 10,1 12,88 0,439 341 602,2 86,21 301,1 43,11 56,07 4,83 3 415,8 13,91 152,8139,7 3,2 10,8 13,72 0,439 320 639,6 91,56 319,8 45,78 59,63 4,83 2 442,9 19,25 162,8

x 139,7 4 13,4 17,05 0,439 257 785,7 112,5 392,9 56,24 73,68 4,80 2 550,3 23,78 202,3x 139,7 5 16,6 21,16 0,439 207 961,1 137,6 480,5 68,80 90,76 4,77 1 682,9 29,29 251,0x 139,7 6 19,8 25,20 0,439 174 1129 161,6 564,3 80,78 107,3 4,73 1 813,3 34,64 298,9

139,7 6,3 20,7 26,40 0,439 166 1177 168,5 588,6 84,27 112,2 4,72 1 852,1 36,21 313,2x 139,7 8 26,0 33,10 0,439 133 1441 206,2 720,3 103,1 138,9 4,66 1 1068 44,84 392,6x 139,7 10 32,0 40,75 0,439 108 1724 246,8 861,9 123,4 168,6 4,60 1 1315 54,40 483,3

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152 2,9 10,7 13,58 0,478 352 755,2 99,37 377,6 49,69 64,48 5,27 3 438,4 16,04 161,1152 3 11,0 14,04 0,478 340 779,7 102,6 389,9 51,30 66,61 5,27 3 453,2 16,56 166,6152 3,2 11,7 14,96 0,478 320 828,4 109,0 414,2 54,50 70,86 5,26 3 482,8 17,59 177,4152 4 14,6 18,60 0,478 257 1019 134,1 509,6 67,05 87,64 5,23 2 600,2 28,28 220,6152 5 18,1 23,09 0,478 207 1249 164,3 624,4 82,16 108,1 5,20 1 745,2 34,88 273,9152 6 21,6 27,52 0,478 174 1469 193,3 734,5 96,65 128,0 5,17 1 888,2 41,30 326,4152 6,3 22,6 28,84 0,478 166 1533 201,8 766,6 100,9 133,8 5,16 1 930,7 43,19 342,1159 2,9 11,2 14,22 0,500 352 866,7 109,0 433,3 54,51 70,67 5,52 3 459,0 17,59 168,7159 3 11,5 14,70 0,500 340 894,8 112,6 447,4 56,28 73,02 5,52 3 474,5 18,16 174,4159 3,2 12,3 15,66 0,500 319 950,9 119,6 475,4 59,8 77,69 5,51 3 505,5 19,30 185,8

x 159 4 15,3 19,48 0,500 257 1171 147,3 585,3 73,63 96,12 5,48 2 628,6 31,02 231,1159 5 19,0 24,19 0,500 207 1436 180,6 717,9 90,30 118,6 5,45 1 780,7 38,28 286,9159 6 22,6 28,84 0,500 173 1690 212,6 845,2 106,3 140,5 5,41 1 930,7 45,35 342,1159 6,3 23,7 30,22 0,500 165 1765 222,0 882,4 111,0 147,0 5,40 1 975,4 47,44 358,5168,3 2,9 11,8 15,07 0,529 351 1031 122,5 515,5 61,26 79,34 5,85 3 486,3 19,77 178,8168,3 3 12,2 15,58 0,529 340 1065 126,5 532,3 63,25 81,98 5,85 3 502,8 20,41 184,8168,3 3,2 13,0 16,60 0,529 319 1131 134,5 565,7 67,23 87,24 5,84 3 535,7 21,70 196,9168,3 4 16,2 20,65 0,529 256 1394 165,7 697,1 82,84 108,0 5,81 2 666,3 34,85 244,9

x 168,3 5 20,1 25,65 0,529 206 1712 203,4 855,9 101,7 133,4 5,78 2 827,8 43,04 304,3x 168,3 6 24,0 30,59 0,529 173 2017 239,7 1009 119,9 158,1 5,74 1 987,3 51,03 362,9

168,3 6,3 25,2 32,06 0,529 165 2107 250,4 1053 125,2 165,4 5,73 1 1035 53,39 380,3x 168,3 8 31,6 40,29 0,529 131 2595 308,3 1297 154,2 205,7 5,67 1 1300 66,40 477,9x 168,3 10 39,0 49,73 0,529 106 3128 371,7 1564 185,9 250,9 5,61 1 1605 80,98 589,9

193,7 4 18,7 23,84 0,609 255 2146 221,5 1073 110,8 144,0 6,71 3 769,3 35,75 282,8193,7 5 23,3 29,64 0,609 205 2640 272,6 1320 136,3 178,1 6,67 2 956,6 57,47 351,6

x 193,7 6 27,8 35,38 0,609 172 3119 322,1 1560 161,1 211,5 6,64 1 1142 68,24 419,7193,7 6,3 29,1 37,09 0,609 164 3260 336,6 1630 168,3 221,3 6,63 1 1197 71,43 440,0

x 193,7 8 36,6 46,67 0,609 130 4031 416,2 2016 208,1 276,1 6,57 1 1506 89,09 553,6x 193,7 10 45,3 57,71 0,609 106 4883 504,2 2442 252,1 337,8 6,50 1 1862 109,0 684,6

193,7 12 53,8 68,50 0,609 89 5678 586,3 2839 293,2 396,8 6,44 1 2211 128,0 812,5193,7 12,5 55,9 71,16 0,609 86 5869 606,0 2934 303,0 411,1 6,42 1 2296 132,7 844,1219,1 4 21,2 27,03 0,688 255 3128 285,5 1564 142,8 185,1 7,61 3 872,3 46,07 320,6219,1 5 26,4 33,63 0,688 205 3856 352,0 1928 176,0 229,2 7,57 2 1085 73,98 398,9

x 219,1 6 31,5 40,17 0,688 171 4564 416,6 2282 208,3 272,5 7,54 2 1296 87,96 476,5219,1 6,3 33,1 42,12 0,688 163 4772 435,6 2386 217,8 285,4 7,53 2 1359 92,10 499,6

x 219,1 8 41,7 53,06 0,688 130 5919 540,3 2960 270,2 356,7 7,47 1 1712 115,1 629,3x 219,1 10 51,6 65,69 0,688 105 7197 657,0 3598 328,5 437,6 7,40 1 2120 141,2 779,2

219,1 12 61,3 78,07 0,688 88 8400 766,8 4200 383,4 515,3 7,33 1 2520 166,3 926,1x 219,1 12,5 63,7 81,13 0,688 85 8689 793,2 4345 396,6 534,2 7,32 1 2618 172,4 962,41) d t M A Au Am/V It Wt I Wel Wpl i PL Nc.Rd Mc.Rd Vpl.Rd


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Table 9.1.3 Cross-sectional properties and resistance values for circular longitudinally welded hollow sections of steel grade S355J2H (fy = 355 N/mm2), continued.






273 5 33,1 42,10 0,858 204 7562 554,0 3781 277,0 359,2 9,48 3 1359 89,39 499,4x 273 6 39,5 50,33 0,858 170 8974 657,5 4487 328,7 427,8 9,44 2 1624 138,1 597,0

273 6,3 41,4 52,79 0,858 163 9392 688,0 4696 344,0 448,2 9,43 2 1704 144,6 626,1x 273 8 52,3 66,60 0,858 129 11703 857,4 5852 428,7 562,0 9,37 2 2149 181,4 790,0x 273 10 64,9 82,62 0,858 104 14308 1048 7154 524,1 692,0 9,31 1 2667 223,3 980,1

273 12 77,2 98,39 0,858 87 16792 1230 8396 615,1 818,0 9,24 1 3175 264,0 1167x 273 12,5 80,3 102,3 0,858 84 17395 1274 8697 637,2 848,9 9,22 1 3301 274,0 1213

323,9 5 39,3 50,09 1,018 203 12739 786,6 6369 393,3 508,5 11,28 4 1427 112,6 594,2x 323,9 6 47,0 59,92 1,018 170 15145 935,2 7572 467,6 606,4 11,24 3 1934 150,9 710,8

323,9 6,3 49,3 62,86 1,018 162 15858 979,2 7929 489,6 635,6 11,23 3 2029 158,0 745,6x 323,9 8 62,3 79,39 1,018 128 19820 1224 9910 611,9 798,5 11,17 2 2562 257,7 941,8x 323,9 10 77,4 98,61 1,018 103 24317 1501 12158 750,8 985,7 11,10 1 3183 318,1 1170

323,9 12 92,3 117,6 1,018 87 28639 1768 14320 884,2 1168 11,04 1 3795 376,9 1395x 323,9 12,5 96,0 122,3 1,018 83 29693 1833 14847 916,7 1213 11,02 1 3947 391,4 1451

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M = weight I = moment of inertia PL = cross-section class in concentric compressionA = cross-section area Wel = elastic section modulus Nc.Rd = compression resistance without bucklingAu = external area Wpl = plastic section modulus Mc.Rd = bending resistanceAm/V = cross-section factor in fire design i = radius of gyration Vpl.Rd = shear resistanceIt = torsional modulus The effect of shear buckling has not been accounted forWt = torsional section modulus hollow sections of cross-section class 4 (section 2.4.2.2)

Table 9.1.4 Cross-sectional properties and resistance values for circular spirally welded hollow sections of steel grade S355J2H (fy = 355 N/mm2).(Technical delivery conditions to be agreed when ordering)


d t M A Au Am/V It Wt I Wel Wpl i PL Nc.Rd Mc.Rd Vpl.Rd

mm mm kg/m mm2 m2/m 1/m mm4 mm3 mm4 mm3 mm3 mm kN kNm kNx 102 x 104 x 103 x 104 x 103 x 103 x 10

355,6 5,6 48,3 61,58 1,117 181 18862 1061 9431 530,4 686,1 12,38 4 1757 152,2 730,4355,6 6 51,7 65,90 1,117 170 20141 1133 10071 566,4 733,4 12,36 3 2127 182,8 781,7355,6 6,3 54,3 69,13 1,117 162 21094 1186 10547 593,2 768,8 12,35 3 2231 191,4 820,1355,6 8 68,6 87,36 1,117 128 26403 1485 13201 742,5 966,8 12,29 2 2819 312,0 1036355,6 10 85,2 108,6 1,117 103 32447 1825 16224 912,5 1195 12,22 2 3504 385,6 1288355,6 12 102 129,5 1,117 86 38279 2153 19139 1076,0 1417 12,16 1 4180 457,4 1537355,6 12,5 106 134,7 1,117 83 39704 2233 19852 1117 1472 12,14 1 4348 475,1 1598406,4 6 59,3 75,47 1,277 169 30257 1489 15128 744,5 962 14,16 4 2141 212,4 895,3406,4 6,3 62,2 79,19 1,277 161 31699 1560 15849 780,0 1009 14,15 4 2256 223,5 939,3406,4 8 78,6 100,1 1,277 128 39748 1956 19874 978,1 1270 14,09 3 3231 315,6 1188406,4 10 97,8 124,5 1,277 103 48952 2409 24476 1205 1572 14,02 2 4019 507,2 1477406,4 12 117 148,7 1,277 86 57874 2848 28937 1424 1867 13,95 2 4798 602,6 1764406,4 12,5 121 154,7 1,277 83 60061 2956 30031 1478 1940 13,93 1 4992 626,1 1835457 6 66,7 85,01 1,436 169 43236 1892 21618 946,1 1220 15,95 4 2383 267,0 1008457 6,3 70,0 89,20 1,436 161 45308 1983 22654 991,4 1280 15,94 4 2513 281,1 1058457 8 88,6 112,9 1,436 127 56893 2490 28446 1245 1613 15,88 3 3642 401,8 1339457 10 110 140,4 1,436 102 70183 3071 35091 1536 1998 15,81 2 4532 645,0 1666457 12 132 167,8 1,436 86 83113 3637 41556 1819 2377 15,74 2 5414 767,1 1990457 12,5 137 174,6 1,436 82 86290 3776 43145 1888 2470 15,72 2 5633 797,3 2071508 6 74,3 94,62 1,596 169 59623 2347 29812 1174 1512 17,75 4 2622 327,8 1122508 6,3 78,0 99,30 1,596 161 62493 2460 31246 1230 1586 17,74 4 2767 345,3 1178508 8 98,7 125,7 1,596 127 78560 3093 39280 1546 2000 17,68 4 3586 443,6 1491508 10 123 156,5 1,596 102 97040 3820 48520 1910 2480 17,61 3 5049 616,5 1856508 12 147 187,0 1,596 85 115072 4530 57536 2265 2953 17,54 2 6035 952,9 2218508 12,5 153 194,6 1,596 82 119511 4705 59755 2353 3070 17,52 2 6280 990,7 2308559 6 81,8 104,2 1,756 168 79702 2852 39851 1426 1835 19,55 4 2856 394,0 1236559 6,3 85,9 109,4 1,756 161 83552 2989 41776 1495 1925 19,54 4 3015 415,3 1298559 8 109 138,5 1,756 127 105130 3761 52565 1881 2429 19,48 4 3916 535,1 1643559 10 135 172,5 1,756 102 130002 4651 65001 2326 3014 19,41 3 5566 750,5 2046559 12 162 206,2 1,756 85 154327 5522 77164 2761 3591 19,34 3 6655 891,0 2446559 12,5 169 214,6 1,756 82 160324 5736 80162 2868 3734 19,33 2 6926 1205 2546

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610 8 119 151,3 1,916 127 137103 4495 68551 2248 2899 21,29 4 4242 634,3 1795610 10 148 188,5 1,916 102 169693 5564 84847 2782 3600 21,22 4 5398 800,6 2236610 12 177 225,4 1,916 85 201627 6611 100814 3305 4292 21,15 3 7276 1067 2674610 12,5 184 234,6 1,916 82 209509 6869 104755 3435 4463 21,13 3 7572 1108 2783610 14,2 209 265,8 1,916 72 236008 7738 118004 3869 5042 21,07 2 8578 1627 3153660 8 129 163,9 2,073 127 174176 5278 87088 2639 3401 23,05 4 4555 738,9 1944660 10 160 204,2 2,073 102 215741 6538 107870 3269 4225 22,98 4 5806 934,7 2422660 12 192 244,3 2,073 85 256534 7774 128267 3887 5039 22,91 3 7884 1254 2898660 12,5 200 254,3 2,073 82 266613 8079 133306 4040 5241 22,90 3 8206 1304 3016660 14,2 226 288,1 2,073 72 300526 9107 150263 4553 5923 22,84 3 9298 1470 3417711 8 139 176,7 2,234 126 218324 6141 109162 3071 3954 24,86 4 4869 853,0 2096711 10 173 220,2 2,234 101 270603 7612 135301 3806 4914 24,79 4 6218 1081 2612711 12 207 263,5 2,234 85 321981 9057 160991 4529 5864 24,72 3 8504 1461 3126711 12,5 215 274,3 2,234 81 334686 9415 167343 4707 6099 24,70 3 8852 1519 3254711 14,2 244 310,9 2,234 72 377470 10618 188735 5309 6895 24,64 3 10032 1713 3687762 8 149 189,5 2,394 126 269366 7070 134683 3535 4548 26,66 4 5178 974,4 2248762 10 186 236,3 2,394 101 334057 8768 167028 4384 5655 26,59 4 6623 1237 2802762 12 222 282,7 2,394 85 397710 10439 198855 5219 6751 26,52 4 8068 1497 3354762 12,5 231 294,3 2,394 81 413462 10852 206731 5426 7023 26,50 4 8429 1562 3491762 14,2 262 333,6 2,394 72 466542 12245 233271 6123 7942 26,44 3 10766 1976 3957813 8 159 202,3 2,554 126 327801 8064 163901 4032 5184 28,46 4 5481 1103 2400813 10 198 252,3 2,554 101 406728 10006 203364 5003 6448 28,39 4 7024 1403 2992813 12 237 302,0 2,554 85 484469 11918 242235 5959 7700 28,32 4 8565 1700 3582813 12,5 247 314,4 2,554 81 503721 12392 251860 6196 8011 28,31 4 8951 1774 3729813 14,2 280 356,4 2,554 72 568630 13988 284315 6994 9062 28,25 3 11500 2257 4227813 16 315 400,6 2,554 64 636443 15657 318222 7828 10165 28,18 3 12929 2526 4752914 10 223 284,0 2,871 101 580294 12698 290147 6349 8172 31,96 4 7800 1758 3369914 12 267 340,1 2,871 84 691779 15137 345890 7569 9764 31,89 4 9534 2136 4034914 12,5 278 354,0 2,871 81 719417 15742 359708 7871 10159 31,88 4 9967 2230 4199914 14,2 315 401,4 2,871 72 812689 17783 406344 8892 11498 31,82 4 11440 2548 4761914 16 354 451,4 2,871 64 910283 19919 455142 9959 12904 31,75 3 14567 3214 5354

1016 10 248 316,0 3,192 101 799699 15742 399850 7871 10121 35,57 4 8562 2153 37491016 12 297 378,5 3,192 84 953969 18779 476984 9389 12097 35,50 4 10490 2622 44901016 12,5 309 394,1 3,192 81 992246 19532 496123 9766 12588 35,48 4 10972 2739 46741016 14,2 351 446,9 3,192 71 1121524 22077 560762 11039 14252 35,42 4 12610 3134 53011016 16 395 502,7 3,192 64 1256959 24743 628479 12372 16001 35,36 4 14343 3549 59621219 10 298 379,8 3,830 101 1388029 22773 694014 11387 14617 42,75 4 10017 3040 45051219 12 357 455,0 3,830 84 1657432 27193 828716 13597 17483 42,68 4 12328 3719 53981219 12,5 372 473,8 3,830 81 1724362 28291 862181 14146 18196 42,66 4 12906 3888 56201219 14,2 422 537,5 3,830 71 1950668 32004 975334 16002 20613 42,60 4 14872 4462 63751219 16 475 604,7 3,830 63 2188182 35901 1094091 17951 23157 42,54 4 16953 5066 7173

d t M A Au Am/V It Wt I Wel Wpl i PL Nc.Rd Mc.Rd Vpl.Rd

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DESIGN HANDBOOK FOR RAUTARUUKKI STRUCTURAL HOLLOW SECTIONSAppendix 9.1

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Appendix 9.2 Buckling tables for steel grade S355J2H


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Table 9.2.1 Buckling resistance values for square hollow sections of steel grade S355J2H (fy = 355 N/mm2) in buckling category c.

1) = recommended seriesh = heightb = widtht = wall thicknessLc = buckling lengthNb.Rd = buckling resistancez

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1) h b t Nb.Rd (kN) h b tmm mm mm Lc (m) mm mm mm

x 0 0,5 1,0 1,5 2 2,5 3 3,5 4 4,5 5 6 7 8 9 10x 40 40 2 94,78 83,79 59,80 37,88 24,44 16,75 12,12 9,156 40 40 2x 40 40 2,5 115,8 101,9 71,86 45,03 28,91 19,78 14,30 10,79 40 40 2,5x 50 50 2 120,6 112,3 89,54 64,65 44,80 31,75 23,39 17,86 14,05 11,33 50 50 2x 50 50 2,5 148,1 137,5 109,0 78,05 53,78 38,02 27,96 21,33 16,77 13,52 50 50 2,5x 50 50 3 174,5 161,7 127,7 90,92 62,42 44,04 32,36 24,67 19,39 15,63 50 50 3x 60 60 2,5 180,4 173,0 145,7 114,6 85,13 62,71 47,16 36,44 28,90 23,43 19,35 60 60 2,5x 60 60 3 213,3 204,2 171,7 134,6 99,58 73,20 54,98 42,46 33,65 27,27 22,53 60 60 3x 60 60 4 275,9 263,3 220,1 170,8 125,3 91,61 68,61 52,89 41,87 33,91 28,00 60 60 4x 70 70 2 172,2 169,0 147,7 123,6 98,25 75,92 58,73 46,16 36,99 30,21 25,08 18,05 70 70 2x 70 70 2,5 212,7 208,4 181,9 151,9 120,4 92,84 71,72 56,32 45,11 36,82 30,57 21,99 70 70 2,5x 70 70 3 252,0 246,7 214,8 178,8 141,1 108,4 83,59 65,56 52,46 42,80 35,52 25,54 70 70 3x 70 70 4 327,5 320,0 277,9 230,2 180,6 138,1 106,2 83,15 66,48 54,20 44,96 32,30 70 70 4

80 80 2 180,1 180,1 162,2 142,4 120,7 98,89 79,76 64,39 52,49 43,37 36,31 26,40 20,00 80 80 2x 80 80 2,5 244,9 244,0 217,8 189,0 157,4 126,8 101,0 80,79 65,49 53,89 45,00 32,61 24,65 80 80 2,5x 80 80 3 290,7 289,3 257,9 223,3 185,4 148,9 118,3 94,51 76,54 62,95 52,54 38,05 28,76 80 80 3x 80 80 4 379,1 376,6 334,9 288,8 238,6 190,6 150,8 120,2 97,23 79,87 66,62 48,20 36,41 80 80 4x 80 80 5 463,3 459,5 407,9 350,6 288,3 229,4 181,0 144,0 116,3 95,45 79,56 57,52 43,43 80 80 5

90 90 2 187,9 187,9 175,0 158,1 139,6 120,0 100,9 84,03 69,99 58,67 49,63 36,56 27,92 21,97 90 90 2x 90 90 2,5 277,2 277,2 253,4 225,6 194,8 163,0 133,8 109,3 89,9 74,67 62,77 45,87 34,86 27,34 90 90 2,5x 90 90 3 329,5 329,5 300,7 267,2 230,2 192,2 157,4 128,4 105,5 87,54 73,54 53,71 40,80 31,99 90 90 3x 90 90 4 430,8 430,8 392,0 347,4 298,2 247,9 202,2 164,6 134,9 111,8 93,88 68,49 51,99 40,75 90 90 4x 90 90 5 527,9 527,9 478,8 423,2 361,8 299,4 243,4 197,5 161,6 133,8 112,3 81,81 62,06 48,62 90 90 5x 90 90 6 620,7 620,7 561,5 495,1 421,9 347,8 281,9 228,4 186,6 154,4 129,4 94,22 71,44 55,95 90 90 6

90 90 6,3 636,7 636,7 574,5 505,3 429,1 352,5 284,8 230,2 187,9 155,3 130,1 94,63 71,72 56,15 90 90 6,3100 100 2 194,0 194,0 185,2 170,6 154,9 137,8 120,3 103,4 88,28 75,35 64,57 48,37 37,29 29,52 23,91 100 100 2

x 100 100 2,5 281,4 281,4 264,9 241,4 215,9 188,5 161,0 135,8 114,2 96,39 81,94 60,75 46,55 36,71 29,66 100 100 2,5x 100 100 3 368,2 368,2 343,4 310,7 274,9 236,8 199,5 166,4 138,8 116,4 98,55 72,67 55,51 43,69 35,24 100 100 3x 100 100 4 482,4 482,4 448,9 405,4 357,7 307,0 257,9 214,5 178,5 149,6 126,5 93,14 71,10 55,93 45,10 100 100 4x 100 100 5 592,4 592,4 549,9 495,7 436,0 372,9 312,2 259,0 215,1 180,0 152,1 111,9 85,33 67,09 54,08 100 100 5x 100 100 6 698,2 698,2 646,4 581,5 510,0 434,6 362,6 300,0 248,7 207,8 175,4 128,9 98,23 77,20 62,21 100 100 6

100 100 6,3 718,1 718,1 663,8 596,4 522,1 443,9 369,6 305,3 252,8 211,1 178,1 130,7 99,61 78,26 63,05 100 100 6,3100 100 7,1 795,6 795,6 733,6 657,6 573,8 486,1 403,3 332,2 274,6 229,0 193,0 141,5 107,8 84,62 68,15 100 100 7,1

x 100 100 8 879,2 879,2 808,9 723,8 629,9 532,0 440,1 361,7 298,6 248,7 209,5 153,5 116,8 91,68 73,82 100 100 8

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110 110 2,5 291,3 291,3 280,0 259,1 236,8 212,7 187,5 162,8 140,1 120,4 103,6 78,12 60,45 47,97 38,91 32,16 110 110 2,5x 110 110 3 406,9 406,9 386,0 353,9 319,2 281,7 243,4 207,4 175,8 149,3 127,4 94,97 73,01 57,69 46,66 38,49 110 110 3x 110 110 4 534,1 534,1 505,7 463,0 416,7 366,7 316,1 268,6 227,3 192,6 164,3 122,3 93,91 74,17 59,97 49,45 110 110 4x 110 110 5 657,0 657,0 620,9 567,6 509,7 447,4 384,5 326,0 275,2 232,9 198,4 147,5 113,2 89,35 72,22 59,54 110 110 5x 110 110 6 775,6 775,6 731,6 667,8 598,4 523,8 448,8 379,5 319,7 270,2 229,9 170,7 130,9 103,3 83,44 68,77 110 110 6

110 110 6,3 799,4 799,4 753,1 686,8 614,6 537,0 459,3 387,8 326,3 275,5 234,3 173,8 133,2 105,1 84,88 69,95 110 110 6,3x 120 120 3 405,1 405,1 392,3 365,0 336,0 304,7 271,7 238,7 207,5 179,7 155,7 118,3 91,99 73,22 59,52 49,27 120 120 3x 120 120 4 585,7 585,7 562,5 520,3 475,2 426,4 375,5 325,7 280,1 240,4 206,9 155,8 120,5 95,61 77,54 64,08 120 120 4x 120 120 5 721,5 721,5 691,9 639,3 582,9 521,9 458,5 396,7 340,5 291,8 250,8 188,6 145,8 115,6 93,71 77,42 120 120 5x 120 120 5,6 800,9 800,9 767,3 708,5 645,3 577,1 506,3 437,4 375,0 321,1 275,8 207,2 160,1 126,9 102,8 84,94 120 120 5,6x 120 120 6 853,1 853,1 816,7 753,7 686,1 613,1 537,2 463,8 397,2 339,9 291,8 219,1 169,2 134,1 108,7 89,75 120 120 6

120 120 6,3 880,7 880,7 842,3 776,9 706,5 630,4 551,6 475,5 406,8 347,7 298,4 223,8 172,7 136,8 110,9 91,56 120 120 6,3120 120 7,1 978,7 978,7 934,5 860,8 781,5 695,7 607,2 522,1 445,8 380,5 326,1 244,3 188,4 149,1 120,8 99,73 120 120 7,1

x 120 120 8 1086 1086 1035 952,7 863,6 767,5 668,4 573,6 489,0 416,9 356,9 267,1 205,8 162,9 131,9 108,9 120 120 8120 120 8,8 1177 1177 1121 1030 932,0 826,2 717,7 614,4 522,6 444,9 380,5 284,3 218,9 173,1 140,1 115,6 120 120 8,8

x 140 140 4 689,0 689,0 676,0 634,2 590,7 544,1 494,5 443,3 393,0 345,9 303,7 234,9 184,6 148,0 120,8 100,3 140 140 4x 140 140 5 850,6 850,6 833,8 781,9 727,6 669,5 607,7 544,1 481,7 423,5 371,4 286,9 225,3 180,4 147,3 122,3 140 140 5x 140 140 5,6 945,5 945,5 926,2 868,2 807,4 742,4 673,2 602,1 532,5 467,7 409,8 316,3 248,2 198,7 162,2 134,6 140 140 5,6x 140 140 6 1008 1008 986,9 924,8 859,8 790,1 716,0 639,9 565,5 496,4 434,8 335,3 263,0 210,5 171,8 142,6 140 140 6

140 140 6,3 1043 1043 1021 955,8 887,9 815,1 737,7 658,3 580,9 509,3 445,6 343,2 269,0 215,2 175,5 145,6 140 140 6,3x 140 140 7,1 1162 1162 1136 1063 986,5 904,7 817,6 728,6 642,0 562,2 491,4 377,9 295,9 236,6 192,9 160,0 140 140 7,1x 140 140 8 1292 1292 1261 1180 1094 1002 903,7 803,7 707,0 618,1 539,6 414,3 324,1 259,0 211,0 175,0 140 140 8x 140 140 8,8 1405 1405 1370 1280 1186 1085 977,3 867,8 762,3 665,7 580,5 445,1 347,9 277,8 226,3 187,7 140 140 8,8x 140 140 10 1567 1567 1526 1425 1318 1204 1082 958,6 840,2 732,3 637,7 487,9 380,9 304,0 247,5 205,1 140 140 10x 150 150 4 701,2 701,2 696 657,6 618,2 576,5 532,1 485,6 438,4 392,5 349,5 276,1 219,8 177,6 145,9 121,6 150 150 4x 150 150 5 915,1 915,1 904,8 853 799,4 742,6 682,0 618,8 555,2 494,2 437,8 343,3 272,0 219,1 179,6 149,5 150 150 5x 150 150 6 1085 1085 1072 1010 946,0 877,8 805,1 729,4 653,5 580,8 513,9 402,2 318,3 256,3 209,9 174,7 150 150 6

150 150 6,3 1125 1125 1110 1045 978,0 907,0 831,0 752,0 672,8 597,3 528,0 412,6 326,3 262,5 214,9 178,8 150 150 6,3150 150 7,1 1254 1254 1236 1164 1088 1008 922,7 833,9 745,2 660,7 583,5 455,3 359,7 289,2 236,7 196,9 150 150 7,1

x 150 150 8 1396 1396 1375 1293 1208 1118 1022 922,0 822,7 728,4 642,4 500,4 394,9 317,3 259,6 215,8 150 150 8150 150 8,8 1518 1518 1494 1405 1312 1213 1107 998,0 889,1 786,2 692,7 538,8 424,8 341,1 279,0 231,9 150 150 8,8

x 150 150 10 1696 1696 1668 1567 1461 1349 1230 1106 983,0 867,9 763,6 592,6 466,7 374,4 306,0 254,3 150 150 10160 160 4 720,2 720,2 720,2 685,4 648,7 610,3 569,7 526,9 482,7 438,4 395,6 319,0 257,5 210,0 173,5 145,2 160 160 4

x 160 160 5 979,7 979,7 975,6 923,8 870,7 814,7 755,3 692,8 628,9 566,0 506,4 403,0 322,4 261,3 215,0 179,5 160 160 5x 160 160 6 1163 1163 1157 1095 1032 964,9 893,7 818,9 742,6 667,6 596,7 474,2 379,0 307,0 252,5 210,7 160 160 6

160 160 6,3 1206 1206 1199 1135 1068 998,4 923,9 845,8 766,1 688,0 614,4 487,4 389,2 315,0 259,0 216,1 160 160 6,3160 160 7,1 1345 1345 1337 1264 1190 1111 1027 939,4 850,0 762,5 680,2 538,9 429,9 347,8 285,8 238,4 160 160 7,1

x 160 160 8 1499 1499 1488 1407 1323 1234 1140 1041 940,5 842,5 750,7 593,6 472,9 382,3 314,0 261,9 160 160 8160 160 8,8 1632 1632 1619 1530 1438 1340 1237 1128 1018 911,1 811,0 640,4 509,7 411,8 338,1 281,8 160 160 8,8

x 160 160 10 1826 1826 1809 1708 1604 1494 1377 1254 1129 1009 896,7 706,4 561,5 453,2 371,9 309,9 160 160 10160 160 12 2093 2093 2068 1949 1826 1695 1556 1410 1264 1124 995,2 779,4 617,2 497,0 407,1 338,9 160 160 12160 160 12,5 2164 2164 2137 2013 1885 1749 1603 1452 1301 1156 1022 799,7 632,8 509,4 417,1 347,1 160 160 12,5


x 0 0,5 1,0 1,5 2 2,5 3 3,5 4 4,5 5 6 7 8 9 10

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Table 9.2.1 Buckling resistance values for square hollow sections of steel grade S355J2H (fy = 355 N/mm2) in buckling category c, continued.

1) = recommended seriesh = heightb = widtht = wall thicknessLc = buckling lengthNb.Rd = buckling resistance


x 0 0,5 1,0 1,5 2 2,5 3 3,5 4 4,5 5 6 7 8 9 10x 180 180 5 1109 1109 1109 1066 1013 959,1 901,8 841,4 778,5 714,5 651,3 534,3 436,5 358,8 298,0 250,5 180 180 5x 180 180 6 1318 1318 1318 1266 1203 1138 1069 996,4 920,9 844,3 768,8 629,6 513,7 421,8 350,2 294,2 180 180 6

180 180 6,3 1369 1369 1369 1314 1248 1180 1108 1033 953,8 873,9 795,2 650,5 530,3 435,3 361,2 303,4 180 180 6,3180 180 7,1 1529 1529 1529 1466 1392 1316 1235 1150 1061 971,4 883,2 721,5 587,6 481,9 399,7 335,6 180 180 7,1

x 180 180 8 1705 1705 1705 1634 1551 1465 1374 1278 1178 1077 978,4 797,7 648,8 531,7 440,7 369,8 180 180 8180 180 8,8 1859 1859 1859 1779 1688 1594 1494 1389 1279 1169 1060 863,3 701,5 574,4 475,9 399,2 180 180 8,8

x 180 180 10 2084 2084 2084 1992 1889 1782 1668 1549 1425 1300 1178 957,1 776,4 635,1 525,8 440,9 180 180 10180 180 12 2403 2403 2403 2288 2167 2039 1905 1763 1616 1469 1326 1071 865,1 705,7 583,2 488,4 180 180 12

x 180 180 12,5 2486 2486 2486 2366 2240 2107 1967 1819 1666 1513 1366 1102 889,2 725,0 598,9 501,5 180 180 12,5x 200 200 5 1125 1125 1125 1105 1060 1014 966,0 916,3 864,1 809,9 754,5 644,8 544,1 457,6 386,3 328,4 200 200 5x 200 200 6 1473 1473 1473 1436 1374 1310 1243 1173 1100 1024 947,0 798,2 665,7 555,1 465,7 394,2 200 200 6

200 200 6,3 1531 1531 1531 1492 1427 1360 1291 1218 1141 1062 981,7 826,5 688,8 574,1 481,4 407,4 200 200 6,3200 200 7,1 1712 1712 1712 1667 1594 1519 1441 1358 1272 1183 1093 919,0 765,0 637,1 534,0 451,7 200 200 7,1

x 200 200 8 1912 1912 1912 1860 1778 1694 1605 1513 1415 1315 1214 1019 847,3 704,9 590,4 499,2 200 200 8200 200 8,8 2086 2086 2086 2029 1938 1846 1749 1647 1540 1430 1319 1106 918,3 763,4 639,1 540,1 200 200 8,8

x 200 200 10 2342 2342 2342 2275 2172 2067 1957 1841 1720 1595 1469 1229 1019 845,6 707,3 597,4 200 200 10200 200 12 2713 2713 2713 2628 2507 2382 2251 2113 1969 1821 1673 1392 1149 951,4 794,2 669,8 200 200 12

x 200 200 12,5 2809 2809 2809 2719 2594 2464 2328 2184 2034 1881 1727 1435 1184 979,3 817,2 689,0 200 200 12,5x 220 220 6 1628 1628 1628 1606 1544 1481 1416 1348 1277 1203 1127 974,0 829,8 703,3 597,1 509,7 220 220 6

220 220 6,3 1694 1694 1694 1671 1606 1540 1472 1400 1326 1248 1169 1009 859,0 727,6 617,3 526,7 220 220 6,3220 220 7,1 1895 1895 1895 1869 1795 1721 1644 1564 1480 1393 1304 1124 956,0 809,0 686,0 585,1 220 220 7,1

x 220 220 8 2118 2118 2118 2087 2005 1922 1835 1745 1651 1553 1452 1251 1063 898,7 761,5 649,2 220 220 8220 220 8,8 2313 2313 2313 2278 2188 2096 2001 1902 1798 1690 1580 1359 1153 973,7 824,5 702,5 220 220 8,8

x 220 220 10 2600 2600 2600 2558 2456 2352 2244 2132 2014 1892 1766 1517 1285 1084 917,1 780,9 220 220 10220 220 12 3023 3023 3023 2967 2846 2723 2595 2461 2321 2176 2027 1733 1462 1230 1038 882,6 220 220 12220 220 12,5 3132 3132 3132 3073 2947 2819 2686 2547 2401 2249 2095 1789 1508 1268 1070 909,3 220 220 12,5

x 250 250 6 1646 1646 1646 1646 1606 1554 1501 1447 1391 1333 1273 1147 1019 895,6 782,3 682,5 250 250 6250 250 6,3 1770 1770 1770 1770 1722 1665 1607 1548 1486 1422 1356 1218 1078 943,7 821,9 715,2 250 250 6,3250 250 7,1 2170 2170 2170 2170 2098 2024 1949 1871 1791 1707 1620 1441 1262 1094 945,7 818,0 250 250 7,1

x 250 250 8 2428 2428 2428 2428 2345 2263 2178 2091 2000 1906 1808 1606 1405 1217 1051 908,6 250 250 8250 250 8,8 2654 2654 2654 2653 2562 2471 2378 2283 2183 2079 1972 1750 1530 1324 1143 987,3 250 250 8,8

x 250 250 10 2987 2987 2987 2984 2881 2778 2673 2564 2451 2334 2212 1960 1711 1480 1276 1101 250 250 10250 250 12 3487 3487 3487 3477 3356 3234 3109 2979 2845 2704 2559 2260 1967 1696 1458 1257 250 250 12

x 250 250 12,5 3616 3616 3616 3605 3478 3351 3221 3086 2946 2800 2649 2338 2033 1751 1506 1297 250 250 12,5

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260 260 6 1669 1669 1669 1669 1639 1590 1539 1488 1435 1380 1323 1204 1081 960,0 846,0 743,5 260 260 6260 260 6,3 1797 1797 1797 1797 1761 1706 1651 1594 1536 1475 1412 1281 1146 1014 890,7 780,7 260 260 6,3260 260 7,1 2262 2262 2262 2262 2199 2125 2050 1973 1894 1811 1725 1547 1367 1195 1039 903,2 260 260 7,1260 260 8 2532 2532 2532 2532 2459 2376 2292 2206 2116 2023 1926 1725 1523 1330 1156 1004 260 260 8260 260 8,8 2768 2768 2768 2768 2687 2596 2504 2409 2310 2208 2102 1881 1659 1448 1257 1092 260 260 8,8260 260 10 3116 3116 3116 3116 3023 2920 2816 2708 2596 2480 2359 2109 1858 1619 1405 1219 260 260 10260 260 11 3368 3368 3368 3368 3263 3151 3037 2919 2797 2670 2538 2265 1991 1732 1501 1301 260 260 11260 260 12,5 3777 3777 3777 3777 3655 3529 3398 3265 3126 2982 2832 2523 2214 1923 1664 1440 260 260 12,5

x 300 300 6 1746 1746 1746 1746 1746 1710 1667 1624 1580 1535 1489 1393 1292 1187 1082 979,0 300 300 6300 300 6,3 1887 1887 1887 1887 1887 1843 1795 1748 1699 1650 1599 1492 1380 1264 1148 1036 300 300 6,3300 300 7,1 2303 2303 2303 2303 2295 2235 2175 2114 2051 1987 1921 1784 1638 1490 1343 1203 300 300 7,1

x 300 300 8 2787 2787 2787 2787 2766 2690 2613 2536 2456 2375 2290 2113 1928 1740 1557 1387 300 300 8300 300 8,8 3222 3222 3222 3222 3186 3096 3004 2911 2816 2718 2616 2404 2181 1958 1743 1545 300 300 8,8

x 300 300 10 3633 3633 3633 3633 3590 3488 3384 3279 3171 3059 2944 2702 2450 2197 1955 1731 300 300 10300 300 12 4262 4262 4262 4262 4206 4084 3961 3836 3707 3574 3437 3149 2849 2550 2263 2001 300 300 12

x 300 300 12,5 4422 4422 4422 4422 4363 4236 4108 3978 3844 3706 3563 3262 2951 2639 2342 2069 300 300 12,51) h b t Nb.Rd (kN) h b t

mm mm mm Lc (m) mm mm mmx 0 0,5 1,0 1,5 2 2,5 3 3,5 4 4,5 5 6 7 8 9 10

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Table 9.2.2 Buckling resistance values for rectangular hollow sections of steel grade S355J2H (fy = 355 N/mm2) in buckling category c.


1) h b t Nb.Rd (kN) h b t mm mm mm Lc (m) mm mm mm

x 0 0,5 1,0 1,5 2 2,5 3 3,5 4 4,5 5 6 7 8 9 1040 30 2 y-y 81,88 71,51 49,63 30,66 19,57 13,35 9,636 7,268 40 30 2

z-z 81,88 66,50 39,28 21,88 13,43 9,003x 50 30 2 y-y 94,78 86,87 67,11 46,40 31,29 21,89 16,01 12,17 9,550 7,687 50 30 2

z-z 94,78 77,73 46,86 26,37 16,24 10,91 7,81x 60 40 2 y-y 120,6 114,8 95,48 73,54 53,56 39,00 29,14 22,44 17,75 14,36 11,85 60 40 2

z-z 120,6 108,0 79,29 51,67 33,77 23,28 16,90 12,79 10,01x 60 40 2,5 y-y 148,1 140,7 116,5 89,17 64,59 46,88 34,97 26,90 21,26 17,20 14,19 60 40 2,5

z-z 148,1 132,2 96,40 62,38 40,63 27,97 20,29 15,35 12,00x 70 50 2 y-y 146,4 142,8 123,6 101,9 79,49 60,54 46,42 36,28 28,98 23,61 19,57 14,05 70 50 2

z-z 146,4 137,3 111,1 81,92 57,62 41,15 30,44 23,29 18,35 14,82 12,20x 70 50 2,5 y-y 180,4 175,7 151,9 124,9 97,10 73,78 56,49 44,12 35,22 28,68 23,77 17,06 70 50 2,5

z-z 180,4 168,8 136,1 99,90 69,97 49,88 36,85 28,18 22,20 17,91 14,75x 70 50 3 y-y 213,3 207,5 178,9 146,5 113,4 85,86 65,61 51,18 40,82 33,23 27,53 19,75 70 50 3

z-z 213,3 199,3 160,1 116,8 81,57 58,04 42,84 32,74 25,78 20,80x 80 40 2,5 y-y 180,4 177,6 156,0 131,7 105,8 82,49 64,18 50,62 40,66 33,25 27,64 19,92 15,01 80 40 2,5

z-z 180,4 162,2 120,2 79,14 51,98 35,92 26,10 19,77 15,47x 80 60 2 y-y 163,2 162,5 145,0 125,6 104,5 84,04 66,82 53,43 43,29 35,61 29,73 21,54 16,28 80 60 2

z-z 163,2 158,2 135,6 109,8 84,04 63,12 48,00 37,34 29,73 24,17 20,01 14,34x 80 60 2,5 y-y 212,7 210,8 187,0 160,6 131,9 104,9 82,68 65,75 53,09 43,57 36,31 26,25 19,81 80 60 2,5

z-z 212,7 205,0 174,4 139,3 105,1 78,17 59,12 45,83 36,42 29,56 24,45 17,50x 80 60 3 y-y 252,0 249,7 221,2 189,7 155,4 123,3 97,07 77,12 62,23 51,05 42,53 30,74 23,20 80 60 3

z-z 252,0 242,7 206,0 164,0 123,3 91,50 69,13 53,56 42,53 34,52 28,54x 80 60 4 y-y 327,5 323,7 286,0 243,9 198,4 156,4 122,6 97,10 78,21 64,09 53,35 38,51 29,05 80 60 4

z-z 327,5 314,4 265,5 209,6 156,2 115,3 86,84 67,16 53,27 43,20 35,7080 70 2,5 y-y 228,8 227,4 202,4 174,9 144,7 115,9 91,85 73,29 59,30 48,74 40,66 29,43 22,24 80 70 2,5

z-z 228,8 224,8 196,8 165,3 131,9 102,3 79,29 62,40 50,05 40,89 33,97 24,4680 70 3 y-y 271,4 269,4 239,5 206,4 170,3 135,9 107,5 85,64 69,23 56,86 47,42 34,30 25,91 80 70 3

z-z 271,4 266,2 232,6 194,8 154,8 119,6 92,5 72,73 58,29 47,60 39,53 28,4480 70 4 y-y 353,3 350,2 310,5 266,4 218,6 173,6 136,7 108,7 87,73 71,98 59,98 43,35 32,72 80 70 4

z-z 353,3 346,0 301,6 251,3 198,6 152,8 117,9 92,48 74,03 60,41 50,14 36,0680 70 5 y-y 431,0 426,4 377,1 322,1 262,8 207,5 162,9 129,2 104,1 85,34 71,07 51,32 38,71 80 70 5

z-z 431,0 421,2 365,7 302,9 237,7 181,9 139,8 109,5 87,51 71,35 59,18 42,52

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90 50 2 y-y 154,2 154,2 140,3 124,4 106,7 88,68 72,35 58,87 48,25 40,00 33,58 24,49 18,59 14,57 90 50 2z-z 154,2 146,7 121,8 93,60 68,03 49,48 36,95 28,43 22,49 18,20 15,01

x 90 50 2,5 y-y 212,7 212,6 190,8 167,0 140,8 114,8 92,18 74,21 60,39 49,83 41,68 30,28 22,92 17,94 90 50 2,5z-z 212,7 200,1 162,9 121,3 85,95 61,63 45,67 35,00 27,59 22,29 18,36

x 90 50 3 y-y 252,0 251,7 225,6 197,0 165,6 134,6 107,9 86,74 70,52 58,15 48,62 35,30 26,71 20,90 90 50 3z-z 252,0 236,7 192,1 142,4 100,5 71,92 53,24 40,77 32,14 25,95 21,38

90 60 2,5 y-y 228,8 228,8 206,6 181,8 154,5 127,1 102,7 83,09 67,83 56,08 46,98 34,19 25,91 20,29 90 60 2,5z-z 228,8 221,1 188,7 151,8 115,3 86,12 65,29 50,70 40,32 32,75 27,10 19,41

90 60 3 y-y 271,4 271,4 244,5 214,8 182,1 149,3 120,5 97,28 79,33 65,54 54,88 39,91 30,24 23,67 90 60 3z-z 271,4 261,8 222,8 178,4 134,8 100,4 75,94 58,9 46,81 38,01 31,44 22,50

90 60 4 y-y 353,3 353,3 317,2 277,8 234,4 191,3 153,8 123,9 100,8 83,21 69,62 50,58 38,30 29,97 90 60 4z-z 353,3 340,1 288,3 229,2 172,0 127,6 96,30 74,56 59,20 48,04 39,72

90 70 2 y-y 178,8 178,8 164,3 146,8 127,5 107,5 88,72 72,82 60,04 49,98 42,07 30,80 23,43 18,39 90 70 2z-z 178,8 177,2 157,0 134,6 110,3 87,50 68,88 54,72 44,15 36,22 30,18 21,81

90 70 2,5 y-y 244,9 244,9 222,2 196,3 167,9 138,9 112,9 91,65 75,00 62,11 52,09 37,96 28,80 22,56 90 70 2,5z-z 244,9 241,0 211,6 178,5 143,2 111,5 86,70 68,35 54,88 44,87 37,30 26,87

90 70 3 y-y 290,7 290,7 263,4 232,5 198,4 163,9 133,1 107,9 88,22 73,02 61,22 44,60 33,82 26,50 90 70 3z-z 290,7 285,9 250,6 210,9 168,8 131,2 101,8 80,21 64,37 52,61 43,72 31,49

90 70 4 y-y 379,1 379,1 342,3 301,3 256,1 210,6 170,3 137,7 112,4 92,94 77,86 56,66 42,94 33,62 90 70 4z-z 379,1 372,0 325,1 272,1 216,3 167,1 129,3 101,6 81,44 66,50 55,23

90 70 5 y-y 463,3 463,3 416,6 365,3 308,8 252,5 203,3 163,9 133,5 110,2 92,26 67,06 50,79 39,75 90 70 5z-z 463,3 453,5 395,0 328,8 259,5 199,4 153,7 120,6 96,50 78,71 65,33

90 80 2 y-y 184,0 184,0 170,2 152,9 134,0 114,0 94,99 78,50 65,04 54,32 45,83 33,65 25,65 20,15 90 80 2z-z 184,0 184,0 167,1 147,8 126,6 104,9 85,40 69,39 56,81 47,07 39,49 28,79 21,85 17,12

90 80 2,5 y-y 261,1 261,1 237,8 211,0 181,4 151,1 123,4 100,6 82,48 68,42 57,45 41,93 31,84 24,96 90 80 2,5z-z 261,1 260,5 233,0 202,9 169,9 137,6 109,9 88,20 71,61 58,99 49,30 35,76 27,05

90 80 3 y-y 310,1 310,1 282,0 249,8 214,2 177,9 145,1 118,0 96,68 80,14 67,26 49,06 37,24 29,18 90 80 3z-z 310,1 309,1 276,1 240,0 200,3 161,7 128,9 103,3 83,78 68,98 57,61 41,77 31,58

90 80 4 y-y 405,0 405,0 367,1 324,3 277,0 229,0 186,0 150,9 123,4 102,2 85,70 62,44 47,37 37,10 90 80 4z-z 405,0 403,1 359,5 311,5 259,0 208,2 165,5 132,3 107,2 88,17 73,60 53,31 40,30

90 80 5 y-y 495,6 495,6 447,8 394,4 335,4 276,1 223,4 180,8 147,6 122,1 102,3 74,46 56,44 44,20 90 80 5z-z 495,6 492,4 438,1 378,2 312,7 250,1 198,1 158,0 127,8 105,0 87,59 63,39 47,88

90 80 6 y-y 582,0 582,0 524,0 460,0 389,6 319,1 257,3 207,6 169,3 139,8 117,1 85,11 64,48 50,47 90 80 6z-z 582,0 577,2 512,3 440,3 362,1 288,1 227,3 180,8 146,0 119,9 99,92 72,25 54,54

90 80 6,3 y-y 596,1 596,1 535,2 468,7 395,4 322,7 259,4 208,9 170,1 140,4 117,4 85,32 64,61 50,55 90 80 6,3z-z 596,1 590,3 522,9 447,9 366,6 290,5 228,5 181,4 146,4 120,0 100,0 72,25 54,52

100 40 2 y-y 144,4 144,4 133,9 120,6 106,0 90,58 75,74 62,78 52,11 43,59 36,81 27,06 20,64 16,23 100 40 2z-z 144,4 133,3 104,6 73,79 50,37 35,44 26,00 19,80 15,56 12,53

x 100 40 2,5 y-y 198,6 198,6 181,9 162,2 140,3 117,8 96,87 79,29 65,26 54,25 45,62 33,37 25,37 19,90 100 40 2,5z-z 198,6 180,9 137,9 93,8 62,68 43,65 31,86 24,19 18,96 15,25


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Table 9.2.2 Buckling resistance values for rectangular hollow sections of steel grade S355J2H (fy = 355 N/mm2) in buckling category c, continued.



x 0 0,5 1,0 1,5 2 2,5 3 3,5 4 4,5 5 6 7 8 9 10100 50 2 y-y 157,3 157,3 146,6 132,6 117,3 100,9 84,99 70,84 59,04 49,52 41,91 30,89 23,59 18,57 14,97 100 50 2

z-z 157,3 150,6 126,4 98,9 73,08 53,66 40,28 31,10 24,64 19,97 16,49x 100 50 2,5 y-y 214,7 214,7 197,9 177,4 154,8 131,1 108,7 89,53 73,99 61,69 51,99 38,12 29,02 22,79 100 50 2,5

z-z 214,7 203,4 167,8 127,5 91,80 66,4 49,44 37,99 30,01 24,27 20,01x 100 50 3 y-y 271,4 271,4 248,1 220,8 190,7 159,6 131,0 107,1 88,01 73,11 61,46 44,91 34,13 26,77 100 50 3

z-z 271,4 255,3 207,9 154,8 109,7 78,65 58,29 44,66 35,22 28,44 23,44x 100 60 2 y-y 170,2 170,2 159,2 144,4 128,2 110,9 93,81 78,50 65,63 55,16 46,74 34,52 26,39 20,78 16,77 100 60 2

z-z 170,2 166,6 145,1 120,7 95,30 73,18 56,41 44,23 35,4 28,88 23,97 17,23x 100 60 2,5 y-y 230,9 230,9 213,9 192,5 168,9 144,1 120,3 99,58 82,59 69,04 58,28 42,83 32,65 25,66 20,68 100 60 2,5

z-z 230,9 224,4 193,3 157,9 121,9 92,20 70,36 54,85 43,73 35,59 29,48 21,15x 100 60 3 y-y 290,7 290,7 267,3 239,1 208,0 175,5 145,1 119,2 98,34 81,90 68,96 50,51 38,43 30,17 24,29 100 60 3

z-z 290,7 281,0 239,8 192,9 146,5 109,4 82,97 64,42 51,23 41,62 34,44 24,66x 100 60 4 y-y 379,1 379,1 347,5 309,9 268,4 225,4 185,5 151,9 125,1 104,0 87,48 63,99 48,65 38,17 30,72 100 60 4

z-z 379,1 365,6 310,9 248,4 187,4 139,4 105,4 81,72 64,93 52,71 43,60 31,20100 70 2 y-y 181,9 181,9 170,8 155,4 138,6 120,5 102,6 86,26 72,37 60,98 51,77 38,32 29,34 23,12 18,67 100 70 2

z-z 181,9 181,0 161,3 139,5 115,8 92,90 73,78 58,93 47,72 39,24 32,75 23,71 17,92100 70 2,5 y-y 247,0 247,0 229,7 207,3 182,8 156,7 131,5 109,3 90,93 76,16 64,38 47,40 36,17 28,45 22,94 100 70 2,5

z-z 247,0 244,3 215,9 184,3 150,1 118,5 92,9 73,64 59,34 48,63 40,49 29,23 22,05100 70 3 y-y 310,1 310,1 286,4 257,1 224,8 190,8 158,7 130,9 108,3 90,41 76,23 55,94 42,61 33,47 26,97 100 70 3

z-z 310,1 305,3 268,1 226,4 181,9 141,8 110,3 87,02 69,89 57,16 47,52 34,24 25,80100 70 4 y-y 405,0 405,0 373,0 334,1 291,1 246,2 204,0 167,9 138,7 115,6 97,36 71,36 54,32 42,66 34,35 100 70 4

z-z 405,0 398,0 348,7 293,2 234,3 181,9 141,1 111,1 89,12 72,83 60,51 43,57 32,81100 70 5 y-y 495,6 495,6 455,2 406,8 353,3 297,7 245,7 201,7 166,3 138,4 116,5 85,29 64,88 50,92 40,99 100 70 5

z-z 495,6 486,0 424,4 354,9 281,7 217,4 168,1 132,1 105,8 86,38 71,73 51,60

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100 80 2 y-y 187,0 187,0 176,7 161,4 144,9 127,1 109,1 92,42 77,97 65,97 56,17 41,74 32,02 25,28 20,43 100 80 2z-z 187,0 187,0 171,1 152,3 131,6 110,2 90,51 74,00 60,85 50,56 42,50 31,06 23,61 18,52

100 80 2,5 y-y 263,1 263,1 245,4 222,1 196,5 169,3 142,7 119,0 99,24 83,26 70,48 51,97 39,70 31,24 25,20 100 80 2,5z-z 263,1 263,1 237,0 208,2 176,4 144,6 116,6 94,10 76,75 63,40 53,09 38,60 29,25 22,89

x 100 80 3 y-y 329,5 329,5 305,5 275,2 241,8 206,5 172,6 143,1 118,7 99,31 83,86 61,66 47,02 36,96 29,79 100 80 3z-z 329,5 328,9 294,5 256,9 215,5 174,8 139,9 112,4 91,29 75,23 62,89 45,64 34,53 27,01

x 100 80 4 y-y 430,8 430,8 398,5 358,1 313,7 267,0 222,4 183,8 152,3 127,2 107,3 78,80 60,05 47,18 38,02 100 80 4z-z 430,8 429,4 383,6 333,4 278,3 224,7 179,1 143,5 116,4 95,80 80,03 58,02 43,88

x 100 80 5 y-y 527,9 527,9 487,0 436,8 381,4 323,3 268,4 221,3 183,0 152,6 128,6 94,35 71,85 56,43 45,45 100 80 5z-z 527,9 525,2 468,2 405,4 336,7 270,4 214,8 171,6 139,0 114,3 95,40 69,10 52,22

x 100 80 6 y-y 620,7 620,7 571,1 511,0 444,7 375,5 310,7 255,4 210,8 175,6 147,9 108,3 82,43 64,72 52,11 100 80 6z-z 620,7 616,8 548,7 473,6 391,6 313,2 248,0 197,8 160,0 131,5 109,7 79,36 59,95

100 80 6,3 y-y 636,7 636,7 584,2 521,5 452,3 380,4 313,6 257,1 211,8 176,2 148,3 108,5 82,52 64,75 52,12 100 80 6,3z-z 636,7 631,8 561,0 482,6 397,3 316,5 249,9 198,9 160,7 131,9 110,0 79,53 60,05

110 40 2 y-y 146,9 146,9 139,0 127,2 114,3 100,5 86,52 73,46 62,09 52,60 44,83 33,35 25,61 20,22 16,35 110 40 2z-z 146,9 136,5 108,5 77,93 53,81 38,08 28,02 21,38 16,82

110 40 2,5 y-y 203,6 203,6 190,5 172,8 153,4 132,7 112,3 94,02 78,61 66,08 56,01 41,37 31,63 24,91 20,10 110 40 2,5z-z 203,6 186,6 144,2 99,75 67,28 47,07 34,43 26,18 20,54

110 40 3 y-y 271,4 271,4 250,7 225,2 197,0 167,4 139,3 115,0 95,18 79,45 67,01 49,18 37,47 29,44 23,71 110 40 3z-z 271,4 245,0 183,3 121,9 80,46 55,73 40,55 30,74 24,07

110 50 2 y-y 159,8 159,8 151,7 139,2 125,7 111,1 96,13 82,03 69,60 59,14 50,51 37,67 28,97 22,90 18,53 110 50 2z-z 159,8 153,7 130,2 103,4 77,52 57,43 43,33 33,55 26,63 21,61 17,86

110 50 2,5 y-y 219,7 219,7 206,5 188,1 167,9 146,2 124,7 105,0 88,14 74,32 63,13 46,76 35,81 28,23 22,80 110 50 2,5z-z 219,7 209,3 174,2 134,3 98,00 71,39 53,37 41,10 32,51 26,32 21,72

110 50 3 y-y 290,7 290,7 270,4 244,1 215,2 184,6 155,0 128,8 107,2 89,79 75,92 55,89 42,66 33,56 27,05 110 50 3z-z 290,7 274,0 223,8 167,4 119,1 85,55 63,47 48,66 38,39 31,01 25,56

110 60 2 y-y 172,7 172,7 164,5 151,3 137,1 121,7 105,8 90,71 77,25 65,81 56,32 42,11 32,44 25,66 20,78 17,15 110 60 2z-z 172,7 169,7 148,7 125,0 99,80 77,44 60,07 47,29 37,93 31,00 25,75 18,54

110 60 2,5 y-y 235,9 235,9 222,6 203,4 182,4 159,8 137,1 116,0 97,83 82,73 70,42 52,30 40,12 31,66 25,58 21,09 110 60 2,5z-z 235,9 230,1 199,4 164,6 128,6 98,02 75,21 58,81 46,98 38,28 31,75 22,80

110 60 3 y-y 310,1 310,1 289,7 262,6 232,8 201,0 169,9 142,0 118,6 99,61 84,38 62,28 47,60 37,48 30,24 110 60 3z-z 310,1 300,2 256,9 207,6 158,4 118,8 90,21 70,13 55,81 45,36 37,55 26,90

110 60 4 y-y 405,0 405,0 377,5 341,5 302,0 259,9 218,9 182,5 152,1 127,6 108,0 79,59 60,79 47,84 38,58 110 60 4z-z 405,0 390,9 333,1 267,0 202,1 150,7 114,1 88,51 70,35 57,14 47,27 33,83


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x 0 0,5 1,0 1,5 2 2,5 3 3,5 4 4,5 5 6 7 8 9 10110 70 2 y-y 184,4 184,4 176,2 162,4 147,5 131,5 114,9 98,89 84,50 72,18 61,89 46,39 35,78 28,34 22,96 18,95 110 70 2

z-z 184,4 184,1 164,8 143,7 120,6 97,80 78,24 62,82 51,04 42,06 35,16 25,51 19,30110 70 2,5 y-y 252,0 252,0 238,6 218,5 196,6 173,1 149,1 126,8 107,2 90,91 77,52 57,70 44,32 35,00 28,30 23,34 110 70 2,5

z-z 252,0 250,0 222,1 191,0 157,3 125,3 98,96 78,78 63,65 52,26 43,57 31,51 23,79110 70 3 y-y 329,5 329,5 309,2 281,2 250,6 217,8 185,2 155,6 130,5 109,9 93,29 69,03 52,84 41,64 33,62 27,69 110 70 3

z-z 329,5 324,9 286,0 242,5 195,8 153,3 119,7 94,50 76,02 62,22 51,75 37,32 28,13110 70 4 y-y 430,8 430,8 403,2 366,0 325,1 281,5 238,5 199,8 167,1 140,5 119,1 88,01 67,31 53,01 42,78 35,23 110 70 4

z-z 430,8 423,9 372,1 313,9 251,9 196,1 152,5 120,2 96,50 78,92 65,60 47,26 35,60110 70 6,3 y-y 636,7 636,7 591,6 533,7 470,0 402,6 337,5 280,3 233,0 195,1 164,9 121,3 92,56 72,79 58,68 110 70 6,3

z-z 636,7 623,3 542,8 451,8 356,6 274,0 211,2 165,6 132,6 108,2 89,76 64,53110 90 2 y-y 193,4 193,4 186,6 173,2 158,9 143,4 127,2 111,1 96,13 82,91 71,61 54,19 42,03 33,40 27,13 22,44 110 90 2

z-z 193,4 193,4 182,2 166,2 148,7 130,0 111,1 93,80 78,97 66,69 56,71 42,06 32,24 25,43 20,54110 90 2,5 y-y 280,1 280,1 266,9 245,5 222,3 197,4 171,7 147,2 125,3 106,8 91,38 68,33 52,63 41,64 33,71 27,82 110 90 2,5

z-z 280,1 280,1 260,0 234,3 206,1 176,2 147,5 122,3 101,6 85,00 71,81 52,81 40,28 31,67 25,53110 90 3 y-y 368,2 368,2 347,7 317,7 285,0 249,9 214,5 181,7 153,3 129,6 110,4 82,00 62,92 49,66 40,13 33,08 110 90 3

z-z 368,2 368,2 338,0 301,9 262,0 220,6 181,9 149,2 123,0 102,4 86,14 63,05 47,96 37,64 30,30110 90 4 y-y 482,4 482,4 454,7 414,8 371,3 324,7 277,8 234,7 197,6 166,9 142,0 105,3 80,74 63,69 51,46 42,41 110 90 4

z-z 482,4 482,4 441,6 393,4 340,3 285,3 234,5 191,8 157,8 131,2 110,3 80,63 61,29 48,07110 90 5 y-y 592,4 592,4 557,2 507,5 453,3 395,2 337,1 284,1 238,7 201,3 171,0 126,7 97,08 76,55 61,82 50,93 110 90 5

z-z 592,4 592,4 540,7 480,5 414,0 345,6 283,1 231,0 189,7 157,4 132,3 96,6 73,38 57,54110 90 6 y-y 698,2 698,2 655,2 595,8 530,9 461,4 392,4 329,7 276,5 232,9 197,6 146,2 112,0 88,23 71,23 58,67 110 90 6

z-z 698,2 698,2 635,6 563,7 484,2 402,8 328,9 267,8 219,6 182,1 152,9 111,5 84,69 66,38110 90 6,3 y-y 718,1 718,1 672,7 610,9 543,3 471,0 399,6 335,0 280,5 236,0 200,1 147,9 113,2 89,15 71,96 59,26 110 90 6,3

z-z 718,1 718,1 652,1 577,0 494,1 409,6 333,5 270,9 221,9 183,8 154,2 112,4 85,32 66,85

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110 100 2 y-y 196,5 196,5 190,4 177,2 163,3 148,2 132,3 116,3 101,2 87,75 76,07 57,85 45,01 35,83 29,13 24,12 110 100 2z-z 196,5 196,5 188,5 174,2 158,9 142,3 125,1 108,3 93,00 79,74 68,56 51,58 39,87 31,61 25,63 21,18

110 100 2,5 y-y 286,3 286,3 274,1 253,0 230,2 205,7 180,2 155,6 133,2 114,0 97,86 73,48 56,73 44,95 36,43 30,09 110 100 2,5z-z 286,3 286,3 270,9 247,9 222,8 195,9 168,6 143,1 120,9 102,5 87,32 64,95 49,87 39,38 31,83 26,25

110 100 3 y-y 387,5 387,5 366,8 335,8 302,0 265,7 228,9 194,4 164,4 139,3 118,8 88,38 67,88 53,61 43,34 35,74 110 100 3z-z 387,5 387,5 362,3 328,4 291,3 251,8 212,9 178,0 148,7 125,0 105,9 78,17 59,76 47,05 37,96

110 100 4 y-y 508,2 508,2 480,1 438,8 393,9 345,6 296,8 251,5 212,2 179,6 152,9 113,6 87,21 68,84 55,64 45,87 110 100 4z-z 508,2 508,2 474,0 428,9 379,5 326,8 275,5 229,7 191,6 160,7 136,1 100,3 76,63 60,31 48,65

110 100 5 y-y 624,7 624,7 589,0 537,5 481,4 421,2 360,6 304,8 256,7 216,9 184,5 136,9 105,0 82,84 66,93 55,16 110 100 5z-z 624,7 624,7 581,5 525,4 463,8 398,4 334,9 278,7 232,1 194,5 164,5 121,2 92,51 72,78 58,68

110 100 6 y-y 736,9 736,9 693,3 631,7 564,5 492,4 420,3 354,3 297,8 251,2 213,5 158,2 121,2 95,60 77,21 63,62 110 100 6z-z 736,9 736,9 684,3 617,1 543,1 464,9 389,4 323,2 268,6 224,8 190,0 139,7 106,6 83,84 67,58

110 100 6,3 y-y 758,7 758,7 713,0 649,0 579,2 504,3 429,7 361,6 303,6 255,9 217,3 160,9 123,3 97,17 78,46 64,64 110 100 6,3z-z 758,7 758,7 703,2 633,1 555,9 474,5 396,4 328,3 272,4 227,7 192,3 141,3 107,8 84,71 68,27

120 40 2 y-y 148,9 148,9 143,2 132,6 121,3 109,0 96,17 83,58 72,00 61,88 53,30 40,20 31,12 24,70 20,04 16,56 120 40 2z-z 148,9 139,4 112,3 82,32 57,65 41,08 30,35 23,21 18,28

120 40 2,5 y-y 207,7 207,7 197,7 181,7 164,4 145,8 126,6 108,4 92,24 78,52 67,16 50,19 38,64 30,56 24,74 20,41 120 40 2,5z-z 207,7 191,5 149,9 105,4 71,80 50,47 37,01 28,18 22,14

120 40 3 y-y 270,5 270,5 255,1 232,9 208,7 182,6 156,5 132,3 111,5 94,21 80,16 59,50 45,63 36,00 29,09 23,98 120 40 3z-z 270,5 246,5 188,1 128,1 85,62 59,65 43,54 33,06 25,92

120 50 2 y-y 161,8 161,8 156,1 144,8 132,8 119,9 106,3 92,77 80,23 69,17 59,73 45,19 35,04 27,85 22,61 18,70 120 50 2z-z 161,8 156,4 133,4 107,2 81,34 60,75 46,04 35,74 28,42 23,09 19,10

120 50 2,5 y-y 223,8 223,8 213,9 197,2 179,2 159,7 139,6 120,2 102,7 87,77 75,27 56,43 43,54 34,48 27,93 23,06 120 50 2,5z-z 223,8 214,2 179,9 140,7 104,0 76,35 57,31 44,25 35,06 28,41 23,46

120 50 3 y-y 289,8 289,8 274,8 251,9 227,0 200,1 172,8 147,1 124,6 105,8 90,25 67,24 51,68 40,83 33,02 27,23 120 50 3z-z 289,8 275,1 227,7 174,0 125,8 91,26 68,04 52,32 41,35 33,45 27,59

120 60 2 y-y 174,7 174,7 169,0 157,0 144,3 130,6 116,2 101,8 88,36 76,37 66,08 50,12 38,93 30,97 25,16 20,82 120 60 2z-z 174,7 172,4 152,0 129,1 104,5 81,97 64,05 50,65 40,74 33,36 27,75 20,02

120 60 2,5 y-y 240,0 240,0 230,1 212,5 193,8 173,5 152,3 131,8 113,1 96,88 83,27 62,61 48,38 38,36 31,09 25,69 120 60 2,5z-z 240,0 235,0 204,8 170,6 134,9 103,7 80,02 62,78 50,25 41,00 34,04 24,47

x 120 60 3 y-y 309,2 309,2 294,3 270,5 244,7 216,9 188,4 161,3 137,2 116,8 99,89 74,64 57,46 45,45 36,79 30,36 120 60 3z-z 309,2 300,9 259,8 213,0 165,1 125,2 95,70 74,72 59,61 48,53 40,22 28,86

x 120 60 4 y-y 430,8 430,8 407,1 372,2 334,3 293,4 252,1 213,8 180,5 152,8 130,1 96,71 74,23 58,60 47,36 39,05 120 60 4z-z 430,8 416,6 355,9 286,7 218,0 163,1 123,7 96,10 76,45 62,12 51,4 36,81


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x 0 0,5 1,0 1,5 2 2,5 3 3,5 4 4,5 5 6 7 8 9 10120 80 2 y-y 191,6 191,6 186,5 174,0 161,0 146,9 132,0 116,8 102,3 89,16 77,61 59,35 46,32 36,96 30,09 24,94 120 80 2

z-z 191,6 191,6 177,5 159,7 140,2 119,6 99,9 82,67 68,58 57,32 48,39 35,56 27,11 21,31120 80 2,5 y-y 272,2 272,2 262,3 243,2 222,8 200,8 177,8 155,0 133,8 115,3 99,45 75,16 58,25 46,27 37,56 31,06 120 80 2,5

z-z 272,2 272,2 248,5 220,8 190,2 158,8 130,0 106,1 87,12 72,31 60,75 44,37 33,70 26,43x 120 80 3 y-y 347,9 347,9 333,2 307,6 280,1 250,3 219,5 189,6 162,4 139,0 119,4 89,67 69,25 54,88 44,48 36,74 120 80 3

z-z 347,9 347,9 314,6 277,4 236,2 194,7 157,7 127,7 104,3 86,31 72,33 52,66 39,93 31,27x 120 80 4 y-y 482,4 482,4 459,2 422,0 381,9 338,5 294,1 251,7 214,2 182,3 155,9 116,5 89,70 70,95 57,42 47,39 120 80 4

z-z 482,4 481,9 431,9 377,2 317,1 257,7 206,6 166,0 135,0 111,3 93,07 67,57 51,14 40,00x 120 80 5 y-y 592,4 592,4 562,9 516,6 466,6 412,6 357,4 305,2 259,1 220,3 188,2 140,4 108,0 85,39 69,09 57,00 120 80 5

z-z 592,4 591,0 528,8 460,5 385,6 312,2 249,5 200,2 162,5 133,9 111,9 81,15 61,39 48,00x 120 80 6 y-y 698,2 698,2 661,8 606,5 546,4 481,7 415,9 354,0 299,8 254,4 217,0 161,7 124,2 98,16 79,39 65,47 120 80 6

z-z 698,2 695,4 620,7 538,6 448,7 361,5 287,8 230,3 186,7 153,6 128,3 92,95 70,27120 80 6,3 y-y 718,1 718,1 679,4 621,7 559,0 491,5 423,1 359,3 303,7 257,2 219,3 163,1 125,2 98,90 79,96 65,93 120 80 6,3

z-z 718,1 714,5 636,9 551,4 457,9 367,8 292,1 233,4 189,1 155,5 129,8 93,98 71,03120 90 2 y-y 195,5 195,5 191,1 178,8 166,0 152,3 137,6 122,7 108,1 94,70 82,78 63,68 49,87 39,88 32,52 26,98 120 90 2

z-z 195,5 195,5 185,1 169,5 152,5 134,3 115,7 98,40 83,20 70,53 60,14 44,76 34,39 27,16 21,96120 90 2,5 y-y 284,2 284,2 274,6 255,1 234,4 211,9 188,4 164,9 142,9 123,5 106,8 80,92 62,82 49,96 40,58 33,58 120 90 2,5

z-z 284,2 284,2 265,2 240,1 212,5 183,1 154,4 128,9 107,5 90,22 76,38 56,33 43,04 33,88 27,32120 90 3 y-y 367,3 367,3 352,6 326,0 297,6 266,9 234,8 203,5 174,9 150,1 129,1 97,19 75,16 59,61 48,34 39,95 120 90 3

z-z 367,3 367,3 339,7 305,3 267,5 227,6 189,6 156,7 129,8 108,4 91,50 67,18 51,19 40,23 32,41120 90 4 y-y 508,2 508,2 485,1 446,7 405,4 360,8 314,7 270,5 230,9 197,0 168,8 126,4 97,46 77,15 62,48 51,58 120 90 4

z-z 508,2 508,2 466,3 416,3 361,0 303,7 250,3 205,2 169,1 140,7 118,4 86,61 65,87 51,69 41,61120 90 5 y-y 624,7 624,7 595,2 547,5 496,0 440,3 383,1 328,4 279,8 238,3 204,0 152,6 117,5 92,98 75,27 62,13 120 90 5

z-z 624,7 624,7 571,5 508,9 439,8 368,4 302,6 247,4 203,5 169,1 142,1 103,9 78,96 61,94 49,84120 90 6 y-y 736,9 736,9 700,9 643,8 582,1 515,5 447,3 382,5 325,2 276,6 236,5 176,6 135,9 107,5 86,98 71,77 120 90 6

z-z 736,9 736,9 672,5 597,7 515,0 429,9 352,0 287,2 235,9 195,8 164,5 120,1 91,25 71,55 57,56120 90 6,3 y-y 758,7 758,7 720,3 660,8 596,4 526,8 455,8 388,8 329,9 280,2 239,3 178,4 137,2 108,5 87,74 72,38 120 90 6,3

z-z 758,7 758,7 691,2 613,3 527,2 439,0 358,7 292,2 239,7 198,8 167,0 121,9 92,52 72,53 58,34

Ap

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120 100 2,5 y-y 290,5 290,5 281,8 262,4 241,9 219,9 196,6 173,1 150,9 130,9 113,6 86,51 67,34 53,64 43,63 36,13 120 100 2,5z-z 290,5 290,5 276,1 253,5 229,1 202,7 175,7 150,2 127,6 108,5 92,70 69,20 53,24 42,10 34,06 28,11

120 100 3 y-y 386,7 386,7 371,9 344,4 315,0 283,2 250,0 217,4 187,3 161,0 138,7 104,6 81,01 64,31 52,17 43,13 120 100 3z-z 386,7 386,7 363,9 331,6 296,4 258,6 220,8 186,2 156,5 132,1 112,2 83,2 63,75 50,27 40,60 33,46

120 100 4 y-y 534,1 534,1 510,9 471,3 428,8 382,8 335,1 289,0 247,4 211,5 181,5 136,2 105,2 83,31 67,50 55,75 120 100 4z-z 534,1 534,1 499,2 452,6 401,5 346,9 293,4 245,3 205,0 172,2 145,9 107,7 82,35 64,84 52,31

120 100 5 y-y 657,0 657,0 627,5 578,2 525,1 467,7 408,4 351,4 300,1 256,2 219,7 164,6 126,9 100,5 81,42 67,22 120 100 5z-z 657,0 657,0 612,7 554,5 490,5 422,5 356,1 297,0 247,7 207,8 175,9 129,7 99,06 77,96 62,88

120 100 6 y-y 775,6 775,6 739,5 680,5 616,9 548,2 477,5 409,7 349,3 297,8 255,0 190,8 147,0 116,3 94,19 77,75 120 100 6z-z 775,6 775,6 721,7 651,8 575,0 493,6 414,6 344,8 287,0 240,5 203,3 149,7 114,3 89,91 72,50

120 100 6,3 y-y 799,4 799,4 761,1 699,7 633,4 561,8 488,2 418,2 355,9 303,0 259,2 193,8 149,2 118,0 95,53 78,84 120 100 6,3z-z 799,4 799,4 742,7 670,0 590,1 505,5 423,7 351,9 292,5 244,9 207,0 152,3 116,2 91,39 73,67

140 40 2,5 y-y 214,1 214,1 209,1 195,7 181,6 166,5 150,4 133,9 118,0 103,3 90,23 69,36 54,30 43,41 35,39 29,36 140 40 2,5z-z 214,1 199,6 159,8 115,9 80,61 57,24 42,20 32,24 25,37

140 40 3 y-y 281,7 281,7 273,2 254,4 234,4 212,9 190,3 167,5 145,9 126,5 109,7 83,51 64,99 51,76 42,09 34,85 140 40 3z-z 281,7 259,6 202,8 142,3 96,80 67,99 49,84 37,95 29,8

140 60 2,5 y-y 246,3 246,3 241,9 227,2 211,8 195,3 177,8 159,7 141,8 125,0 109,9 85,16 67,00 53,73 43,90 36,47 140 60 2,5z-z 246,3 242,9 213,9 181,3 146,5 114,7 89,49 70,71 56,85 46,53 38,71 27,91

140 60 3 y-y 320,5 320,5 312,8 292,5 271,2 248,3 224,0 199,2 175,2 153,1 133,7 102,6 80,25 64,12 52,25 43,33 140 60 3z-z 320,5 314,0 273,9 228,6 181,0 139,5 107,7 84,55 67,70 55,26 45,88 33,00

140 60 4 y-y 482,4 482,4 465,7 432,3 396,8 358,4 318,1 278,0 240,7 207,7 179,5 135,9 105,4 83,81 68,06 56,30 140 60 4z-z 482,4 467,5 400,9 324,9 248,7 186,9 142,1 110,6 88,04 71,58 59,26 42,47

140 70 2,5 y-y 262,5 262,5 258,3 242,8 226,7 209,5 191,2 172,2 153,4 135,6 119,4 92,86 73,20 58,77 48,05 39,94 140 70 2,5z-z 262,5 262,5 236,3 207,5 175,7 143,9 116,0 93,60 76,32 63,04 52,78 38,37 29,07

140 70 3 y-y 339,8 339,8 332,5 311,4 289,2 265,6 240,4 214,5 189,3 166,0 145,3 111,9 87,70 70,16 57,22 47,48 140 70 3z-z 339,8 338,5 302,2 262,2 218,5 176,0 140,1 112,1 90,90 74,80 62,46 45,26 34,22

x 140 70 4 y-y 508,2 508,2 492,1 457,8 421,4 382,2 340,8 299,3 260,2 225,3 195,2 148,3 115,3 91,79 74,61 61,76 140 70 4z-z 508,2 501,8 442,5 376,3 305,0 239,6 187,4 148,3 119,3 97,71 81,30 58,66 44,23

x 140 70 5 y-y 624,7 624,7 604,1 561,4 516,1 467,2 415,7 364,3 316,2 273,3 236,5 179,4 139,4 110,9 90,08 74,54 140 70 5z-z 624,7 615,5 541,2 458,0 368,9 288,2 224,6 177,3 142,5 116,6 96,9 69,87 52,65

x 140 80 3 y-y 359,2 359,2 352,1 330,2 307,3 282,8 256,7 229,8 203,5 178,9 156,9 121,2 95,18 76,24 62,23 51,67 140 80 3z-z 359,2 359,2 328,5 292,5 252,7 211,6 173,7 142,0 116,8 97,02 81,56 59,61 45,30 35,53

x 140 80 4 y-y 534,1 534,1 518,7 483,5 446,2 406,2 363,8 321,0 280,3 243,5 211,5 161,3 125,7 100,2 81,50 67,50 140 80 4z-z 534,1 534,1 480,2 421,1 355,9 291,0 234,3 188,9 153,9 127,1 106,3 77,29 58,54 45,81

x 140 80 5 y-y 657,0 657,0 637,0 593,1 546,7 496,6 443,7 390,5 340,2 295,1 255,9 194,8 151,6 120,7 98,17 81,29 140 80 5z-z 657,0 656,7 589,0 515,1 433,8 353,4 283,6 228,2 185,7 153,2 128,1 93,03 70,43 55,1

x 140 80 6 y-y 775,6 775,6 750,8 698,3 642,6 582,5 519,2 455,8 396,2 342,9 297,0 225,6 175,4 139,6 113,4 93,89 140 80 6z-z 775,6 774,1 692,8 603,8 506,0 410,1 328,0 263,2 213,8 176,2 147,2 106,8 80,81 63,2

x 140 80 6,3 y-y 799,4 799,4 772,8 718,1 659,9 597,2 531,1 465,3 403,6 348,8 301,8 228,8 177,7 141,4 114,8 95,03 140 80 6,3z-z 799,4 797,3 713,0 620,5 519 419,8 335,2 268,8 218,2 179,7 150,1 108,9 82,35


x 0 0,5 1,0 1,5 2 2,5 3 3,5 4 4,5 5 6 7 8 9 10

269

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x 0 0,5 1,0 1,5 2 2,5 3 3,5 4 4,5 5 6 7 8 9 10140 100 2,5 y-y 296,8 296,8 294,1 277,6 260,7 242,7 223,5 203,5 183,2 163,6 145,4 114,5 90,96 73,4 60,22 50,17 140 100 2,5

z-z 296,8 296,8 284,5 262,8 239,5 214,4 188,2 162,8 139,6 119,6 102,8 77,26 59,70 47,32 38,36 31,69140 100 3 y-y 397,9 397,9 391,3 367,7 343,2 317,0 289,0 260,1 231,4 204,4 179,9 139,7 110,1 88,36 72,23 60,03 140 100 3

z-z 397,9 397,9 377,5 346,1 312,1 275,4 238,0 202,8 171,9 146,0 124,6 92,86 71,39 56,41 45,63 37,64140 100 4 y-y 585,7 585,7 571,3 534,0 494,8 452,7 408,0 362,5 318,5 278,2 242,7 186,1 145,5 116,2 94,71 78,53 140 100 4

z-z 585,7 585,7 549,4 499,5 444,9 386,4 328,4 275,8 231,2 194,7 165,2 122,2 93,53 73,7 59,50 49,01140 100 5 y-y 721,5 721,5 702,9 656,5 607,7 555,1 499,4 442,8 388,4 338,7 295,1 226,0 176,5 140,9 114,7 95,10 140 100 5

z-z 721,5 721,5 675,4 612,9 544,5 471,5 399,4 334,6 279,9 235,3 199,5 147,4 112,7 88,79 71,66 59,01140 100 6 y-y 853,1 853,1 829,8 774,2 715,6 652,6 585,8 518,1 453,4 394,7 343,4 262,4 204,7 163,2 132,9 110,1 140 100 6

z-z 853,1 853,1 796,7 721,7 639,5 551,9 466,1 389,3 325,0 272,9 231,1 170,6 130,3 102,6 82,78140 100 6,3 y-y 880,7 880,7 855,8 798,0 736,9 671,1 601,5 531,2 464,2 403,5 350,7 267,6 208,6 166,3 135,3 112,1 140 100 6,3

z-z 880,7 880,7 821,8 743,9 658,5 567,5 478,6 399,4 333,2 279,6 236,7 174,6 133,4 105,0 84,68140 110 2,5 y-y 301,8 301,8 299,9 283,6 266,8 249,2 230,3 210,6 190,5 170,8 152,4 120,7 96,25 77,86 63,98 53,37 140 110 2,5

z-z 301,8 301,8 293,3 273,5 252,7 230,2 206,4 182,3 159,3 138,6 120,4 91,93 71,67 57,14 46,50 38,52140 110 3 y-y 409,1 409,1 403,3 379,6 354,9 328,7 300,7 271,6 242,6 215,1 189,9 148,1 117,0 94,02 76,94 63,99 140 110 3

z-z 409,1 409,1 393,8 364,9 334,0 300,6 265,7 231,3 199,5 171,7 148,0 111,7 86,54 68,72 55,76 46,11140 110 4 y-y 611,5 611,5 597,5 559,1 518,9 475,7 429,8 382,8 337,2 295,2 258,0 198,3 155,3 124,1 101,2 83,95 140 110 4

z-z 611,5 611,5 582,0 534,9 484,1 429,1 372,7 319,1 271,5 231,1 197,6 147,7 113,7 89,93 72,79 60,07140 110 5 y-y 753,8 753,8 735,6 687,8 637,6 583,7 526,5 468,1 411,6 359,8 314,0 241,0 188,5 150,6 122,7 101,8 140 110 5

z-z 753,8 753,8 716,2 657,4 593,7 525,0 454,8 388,4 329,7 280,2 239,4 178,7 137,4 108,6 87,90 72,52140 110 6 y-y 891,8 891,8 869,2 812,1 752,0 687,4 618,9 549,1 482,0 420,6 366,6 280,9 219,5 175,2 142,7 118,3 140 110 6

z-z 891,8 891,8 846,0 775,7 699,5 617,3 533,5 454,6 385,3 327,1 279,2 208,1 160,0 126,4 102,3 84,35140 110 6,3 y-y 921,4 921,4 897,2 837,7 775,0 707,6 636,1 563,5 493,9 430,5 374,9 286,8 223,9 178,7 145,5 120,6 140 110 6,3

z-z 921,4 921,4 873,1 799,9 720,4 634,8 547,7 466,0 394,5 334,6 285,4 212,6 163,3 129,0 104,3 86,05

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140 120 3 y-y 416,4 416,4 411,7 388,1 363,7 337,9 310,3 281,5 252,6 224,8 199,2 156,2 123,7 99,68 81,68 68,00 140 120 3z-z 416,4 416,4 405,8 379,1 350,9 320,7 288,6 256,0 224,7 196,0 170,8 130,8 102,2 81,59 66,45 55,09

140 120 4 y-y 637,3 637,3 623,7 584,3 543,0 498,7 451,7 403,4 356,2 312,5 273,6 210,8 165,3 132,3 107,9 89,55 140 120 4z-z 637,3 637,3 614,0 569,1 521,3 469,7 415,5 362,1 312,6 269,2 232,2 175,4 135,9 108,0 87,62 72,46

140 120 5 y-y 786,0 786,0 768,4 719,3 667,7 612,5 553,8 493,6 435,2 381,2 333,3 256,4 200,8 160,6 130,9 108,6 140 120 5z-z 786,0 786,0 756,1 700,2 640,4 575,8 508,3 441,9 380,8 327,3 282,0 212,7 164,7 130,7 106,1 87,69

140 120 6 y-y 930,5 930,5 908,5 849,8 788,1 721,8 651,5 579,6 510,1 446,1 389,5 299,1 234,1 187,0 152,5 126,4 140 120 6z-z 930,5 930,5 893,7 826,7 754,9 677,4 596,5 517,4 444,9 381,9 328,6 247,5 191,5 151,9 123,2 101,8

140 120 6,3 y-y 962,0 962,0 938,4 877,2 812,8 743,6 670,2 595,4 523,2 457,0 398,6 305,7 239,0 190,9 155,5 129,0 140 120 6,3z-z 962,0 962,0 923,1 853,3 778,5 697,7 613,5 531,4 456,4 391,4 336,6 253,3 195,8 155,3 125,9 104,0

150 50 2,5 y-y 232,7 232,7 230,3 217,2 203,7 189,4 174,2 158,3 142,2 126,8 112,5 88,34 70,08 56,50 46,32 38,58 150 50 2,5z-z 232,7 225,4 193,2 156,4 119,6 89,76 68,23 53,07 42,25 34,34 28,43

150 50 3 y-y 305,5 305,5 300,4 282,3 263,4 243,2 221,7 199,4 177,4 156,7 137,9 107,0 84,31 67,65 55,30 45,95 150 50 3z-z 305,5 293,5 248,1 196,2 146,5 108,3 81,58 63,11 50,07 40,61 33,57

150 60 2,5 y-y 248,8 248,8 246,7 233,0 218,9 203,9 188,0 171,3 154,4 138,0 122,7 96,74 76,93 62,11 50,98 42,48 150 60 2,5z-z 248,8 246,2 217,7 186,1 151,8 119,9 94,2 74,67 60,18 49,34 41,08 29,67

150 60 3 y-y 324,9 324,9 320,2 301,3 281,7 260,8 238,5 215,4 192,3 170,4 150,4 117,2 92,56 74,40 60,87 50,63 150 60 3z-z 324,9 319,4 280,0 235,6 188,4 146,4 113,6 89,48 71,80 58,68 48,76 35,11

150 60 4 y-y 488,6 488,6 476,8 445,9 413,4 378,4 341,4 303,5 267,0 233,4 203,7 156,3 122,3 97,70 79,62 66,03 150 60 4z-z 488,6 475,0 409,3 334,6 258,6 195,6 149,4 116,5 92,90 75,61 62,64 44,93

150 70 2,5 y-y 265,0 265,0 263,2 248,8 234 218,4 201,7 184,3 166,6 149,3 133,0 105,2 83,88 67,82 55,71 46,46 150 70 2,5z-z 265,0 265,0 240,0 211,8 180,7 149,3 121,1 98,2 80,29 66,45 55,71 40,58 30,78

150 70 3 y-y 344,3 344,3 340,0 320,4 300,0 278,4 255,3 231,3 207,2 184,1 162,9 127,5 100,9 81,24 66,54 55,38 150 70 3z-z 344,3 343,8 307,9 268,8 225,7 183,3 146,7 117,9 95,80 78,98 66,03 47,92 36,26

150 70 4 y-y 514,4 514,4 503,4 471,6 438,3 402,6 364,6 325,6 287,5 252,3 220,8 170,2 133,4 106,8 87,11 72,29 150 70 4z-z 514,4 509,3 451,0 386,2 315,9 250,2 196,8 156,2 126,0 103,3 86,08 62,19 46,93

150 70 5 y-y 657,0 657,0 640,5 598,5 554,3 506,9 456,5 405,3 355,8 310,6 270,8 207,6 162,2 129,5 105,5 87,49 150 70 5z-z 657,0 647,9 570,3 483,6 390,5 305,8 238,6 188,5 151,6 124,1 103,2 74,41 56,08

150 90 2,5 y-y 293,1 293,1 292,1 276,7 260,9 244,3 226,6 208,0 189,0 170,3 152,5 121,5 97,29 78,90 64,95 54,24 150 90 2,5z-z 293,1 293,1 277,2 253,6 227,9 200,3 172,3 146,2 123,6 104,6 89,18 66,32 50,92 40,20 32,50

150 90 3 y-y 383,0 383,0 379,5 358,3 336,4 313,2 288,5 262,7 236,5 211,2 187,7 147,8 117,5 94,80 77,76 64,80 150 90 3z-z 383,0 383,0 359,1 326,3 290,4 252,0 214,0 179,6 150,4 126,6 107,4 79,44 60,79 47,89 38,66

150 90 4 y-y 566,0 566,0 556,4 522,6 487,5 450,0 410,0 368,7 327,8 289,3 254,5 197,5 155,5 124,7 101,9 84,71 150 90 4z-z 566,0 566,0 523,9 471,2 413,1 351,9 293,5 242,7 201,2 168,1 141,9 104,2 79,43 62,42 50,29

150 90 5 y-y 721,5 721,5 706,9 662,7 616,5 567,0 514,3 460,2 407,1 357,7 313,5 242,0 190,0 152,1 124,2 103,1 150 90 5z-z 721,5 721,5 663,8 594,0 517,0 436,6 361,2 296,9 245,1 204,1 171,9 125,9 95,8 75,25 60,59

150 90 6 y-y 853,1 853,1 834,9 782,1 726,8 667,5 604,5 539,9 476,8 418,3 366,1 282,2 221,2 177,0 144,4 119,8 150 90 6z-z 853,1 853,1 783,1 699,4 607,0 511,0 421,5 345,7 284,9 237,1 199,5 146,1 111,1 87,18 70,18

150 90 6,3 y-y 880,7 880,7 860,9 805,9 748,2 686,2 620,4 553,1 487,6 427,1 373,4 287,2 225,0 179,9 146,7 121,7 150 90 6,3z-z 880,7 880,7 807,1 719,8 623,4 523,5 430,9 352,9 290,5 241,6 203,2 148,6 113,0 88,66 71,35


x 0 0,5 1,0 1,5 2 2,5 3 3,5 4 4,5 5 6 7 8 9 10

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x 0 0,5 1,0 1,5 2 2,5 3 3,5 4 4,5 5 6 7 8 9 10x 150 100 3 y-y 402,4 402,4 399,2 377,1 354,4 330,4 304,8 278,0 250,8 224,4 199,8 157,7 125,5 101,4 83,22 69,38 150 100 3

z-z 402,4 402,4 383,3 352,4 319,2 283,2 246,3 211,1 179,7 153,1 131,0 97,90 75,43 59,68 48,31 39,87x 150 100 4 y-y 591,8 591,8 582,8 548,1 512,0 473,6 432,7 390,1 347,9 307,8 271,4 211,3 166,7 133,9 109,5 91,04 150 100 4

z-z 591,8 591,8 557,5 508,4 454,9 397,4 339,9 286,9 241,4 203,8 173,3 128,5 98,50 77,72 62,78 51,74x 150 100 5 y-y 753,8 753,8 740,0 694,6 647,3 596,7 542,8 487,2 432,3 380,9 334,6 259,2 203,8 163,4 133,5 110,9 150 100 5

z-z 753,8 753,8 706,5 641,8 571,1 495,4 420,5 352,8 295,5 248,7 210,9 156,0 119,3 94,02 75,89 62,50x 150 100 6 y-y 891,8 891,8 874,6 820,3 763,7 703,1 638,6 572,1 506,8 445,9 391,2 302,4 237,6 190,3 155,4 129,0 150 100 6

z-z 891,8 891,8 834,4 757,0 672,1 581,6 492,4 412,2 344,7 289,7 245,6 181,4 138,7 109,2 88,15 72,58x 150 100 6,3 y-y 921,4 921,4 902,6 846,0 786,8 723,4 656,0 586,7 518,9 455,8 399,4 308,2 241,9 193,6 158,0 131,2 150 100 6,3

z-z 921,4 921,4 860,9 780,2 691,7 597,3 504,7 421,9 352,3 295,9 250,7 185,0 141,4 111,3 89,83150 100 7,1 y-y 1025 1025 1002 939,0 872,4 801,1 725,3 647,5 571,6 501,3 438,7 338,0 265,0 212,0 172,9 143,5 150 100 7,1

z-z 1025 1025 955,6 864,7 765,0 658,9 555,3 463,1 386,2 324,0 274,3 202,2 154,5 121,6 98,1x 150 100 8 y-y 1137 1137 1112 1040 965,6 885,4 800,3 713,1 628,4 550,3 481,0 369,9 289,6 231,6 188,8 156,6 150 100 8

z-z 1137 1137 1059 957,0 844,4 725,3 609,7 507,4 422,4 354,0 299,4 220,6 168,4 132,5 106,8150 110 2,5 y-y 304,3 304,3 304,3 289,6 274,2 258,0 240,8 222,8 204,1 185,4 167,3 135,0 109,0 88,85 73,41 61,46 150 110 2,5

z-z 304,3 304,3 296,7 277,3 256,9 235,0 211,7 187,9 165,1 144,1 125,7 96,4 75,31 60,14 49,00 40,62150 110 3 y-y 413,5 413,5 411,2 389,0 366,2 342,2 316,6 289,7 262,4 235,6 210,3 166,8 133,2 107,8 88,61 73,93 150 110 3

z-z 413,5 413,5 399,3 370,8 340,4 307,6 273,2 238,9 206,9 178,6 154,4 116,9 90,75 72,14 58,60 48,48150 110 4 y-y 617,6 617,6 609,2 573,5 536,5 497,0 455,1 411,4 367,7 326,1 288,1 224,9 177,7 142,9 117,0 97,31 150 110 4

z-z 617,6 617,6 590,1 543,9 494,1 440,3 384,7 331,2 283,0 241,7 207,2 155,4 119,8 94,90 76,87 63,47150 110 5 y-y 786,0 786,0 773,1 726,4 677,9 626,1 570,9 513,8 457,2 403,8 355,5 276,1 217,5 174,6 142,7 118,6 150 110 5

z-z 786,0 786,0 747,9 687,2 621,6 550,8 478,2 409,1 347,9 296,1 253,2 189,1 145,6 115,1 93,19 76,90150 110 6 y-y 930,5 930,5 914,2 858,4 800,3 738,3 672,2 603,9 536,5 473,1 415,9 322,4 253,7 203,5 166,3 138,1 150 110 6

z-z 930,5 930,5 884,1 811,5 732,9 648,1 561,4 479,3 407,0 345,9 295,5 220,5 169,6 134,1 108,5 89,51150 110 6,3 y-y 962,0 962,0 944,3 886,2 825,7 760,9 692,0 620,8 550,8 485,1 426,0 329,8 259,3 207,9 169,8 141,0 150 110 6,3

z-z 962,0 962,0 913,0 837,4 755,4 667,0 576,8 491,8 417,0 354,1 302,3 225,4 173,3 137,0 110,8 91,39160 40 2,5 y-y 218,7 218,7 217,9 206,3 194,5 182,1 168,8 154,9 140,7 126,6 113,4 90,25 72,23 58,56 48,19 40,24 160 40 2,5

z-z 218,7 206 168,0 125,5 89,09 63,95 47,42 36,35 28,67160 40 3 y-y 290,0 290,0 286,9 270,6 253,8 235,9 216,9 197,0 177,0 157,7 139,8 109,8 87,06 70,17 57,52 47,90 160 40 3

z-z 290,0 270,0 215,3 155,4 107,7 76,32 56,21 42,92 33,76160 50 3 y-y 309,4 309,4 306,9 289,9 272,4 253,9 234,2 213,6 192,7 172,3 153,4 121,0 96,30 77,79 63,86 53,23 160 50 3

z-z 309,4 298,3 253,6 202,5 152,7 113,6 85,87 66,57 52,89 42,93 35,51

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160 60 3 y-y 328,7 328,7 326,8 309,1 290,9 271,8 251,4 230,0 208,2 186,8 166,7 132,2 105,5 85,34 70,14 58,52 160 60 3z-z 328,7 324,1 285,1 241,5 194,8 152,4 118,8 93,84 75,43 61,72 51,33 37,01

160 60 4 y-y 498,1 498,1 490,5 461,3 431,0 398,7 364,2 328,5 292,9 259,2 228,5 177,9 140,4 112,8 92,23 76,68 160 60 4z-z 498,1 485,7 420,5 346,6 270,3 205,9 157,8 123,4 98,5 80,28 66,56 47,79

160 70 3 y-y 348,1 348,1 346,7 328,2 309,3 289,4 268,3 246,1 223,3 201,0 179,8 143,1 114,4 92,75 76,31 63,71 160 70 3z-z 348,1 348,1 313,2 274,8 232,5 190,2 153,2 123,6 100,7 83,18 69,63 50,61 38,34

160 70 4 y-y 523,9 523,9 517,2 487,2 456,1 423,1 387,9 351,2 314,4 279,3 247,0 193,2 152,8 123,0 100,7 83,81 160 70 4z-z 523,9 520,0 462,0 397,7 327,8 261,4 206,5 164,5 132,9 109,1 90,99 65,81 49,70

160 70 5 y-y 689,2 689,2 677,0 635,6 592,5 546,5 497,4 446,7 396,7 349,8 307,4 238,2 187,4 150,3 122,8 102,0 160 70 5z-z 689,2 680,2 599,5 509,3 412,4 323,6 252,9 200,0 160,9 131,7 109,6 79,05 59,6

160 80 3 y-y 367,5 367,5 366,6 347,4 327,8 307,3 285,4 262,4 238,8 215,5 193,2 154,3 123,8 100,5 82,76 69,15 160 80 3z-z 367,5 367,5 339,2 304,4 266,1 225,8 187,6 154,8 128,0 106,8 90,07 66,08 50,33 39,54

x 160 80 4 y-y 549,7 549,7 543,8 512,9 481,0 447,1 411,0 373,3 335,3 298,7 264,9 208,0 164,9 132,9 109,0 90,75 160 80 4z-z 549,7 549,7 499,5 442,2 378,9 314,3 256,0 208,1 170,5 141,3 118,5 86,43 65,59 51,4

x 160 80 5 y-y 721,5 721,5 710,3 667,9 623,9 576,9 526,8 474,9 423,3 374,4 330,0 256,7 202,5 162,6 133,0 110,6 160 80 5z-z 721,5 721,5 648,7 568,9 480,9 393,2 316,5 255,2 207,9 171,7 143,7 104,4 79,08 61,89

x 160 80 6 y-y 853,1 853,1 838,8 788,1 735,3 678,9 618,9 556,8 495,3 437,3 384,8 298,8 235,3 188,8 154,3 128,3 160 80 6z-z 853,1 852,7 764,8 668,9 563,3 458,8 368,2 296,3 241,0 198,8 166,3 120,8 91,43 71,53

160 80 6,3 y-y 880,7 880,7 865,0 812,2 757,1 698,3 635,6 570,8 506,9 446,9 392,8 304,4 239,5 192,1 156,9 130,3 160 80 6,3z-z 880,7 879,8 788,4 688,6 578,8 470,5 377,1 303,1 246,4 203,2 169,9 123,3 93,35 73,02

160 90 3 y-y 386,8 386,8 386,4 366,5 346,2 324,8 302,2 278,4 253,9 229,6 206,3 165,3 132,8 107,9 88,99 74,41 160 90 3z-z 386,8 386,8 364,2 332,0 296,9 259,2 221,5 186,9 157,1 132,6 112,8 83,61 64,07 50,53 40,82

160 90 4 y-y 575,5 575,5 570,4 538,6 505,7 471,0 434,0 395,2 356,0 318,1 282,7 222,7 177,0 142,9 117,2 97,69 160 90 4z-z 575,5 575,5 534,6 482,2 424,6 363,5 304,6 252,9 210,2 176,0 148,7 109,4 83,48 65,65 52,92

160 90 5 y-y 753,8 753,8 743,8 700,4 655,4 607,5 556,5 503,3 450,3 399,6 353,1 275,9 218,1 175,4 143,6 119,5 160 90 5z-z 753,8 753,8 694,6 622,4 542,8 459,5 380,9 313,6 259,2 216,0 182,0 133,5 101,6 79,79 64,26

160 90 6 y-y 891,8 891,8 878,9 826,9 773,0 715,6 654,4 590,8 527,5 467,2 412,3 321,4 253,7 203,9 166,9 138,8 160 90 6z-z 891,8 891,8 819,6 732,7 636,8 536,9 443,5 364,2 300,3 250,1 210,5 154,2 117,3 92,06 74,11

160 90 6,3 y-y 921,4 921,4 907,1 852,9 796,7 736,6 672,7 606,4 540,5 478,1 421,3 327,8 258,5 207,6 169,8 141,2 160 90 6,3z-z 921,4 921,4 845,8 755,4 655,7 551,9 455,3 373,4 307,8 256,1 215,5 157,8 120,0 94,17 75,81

x 160 90 7,1 y-y 1025 1025 1008 946,8 883,6 816,1 744,2 669,7 595,9 526,3 463,3 359,8 283,5 227,5 186,0 154,6 160 90 7,1z-z 1025 1025 938,4 836,5 724,0 607,4 499,6 409,0 336,6 279,8 235,3 172,1 130,8 102,6

160 100 3 y-y 406,2 406,2 406,2 385,5 364,5 342,4 319,0 294,3 268,9 243,6 219,2 176,1 141,8 115,4 95,20 79,64 160 100 3z-z 406,2 406,2 388,3 358,0 325,4 290,1 253,7 218,5 186,8 159,6 136,9 102,7 79,22 62,74 50,83 41,97

160 100 4 y-y 601,3 601,3 596,9 564,1 530,4 494,7 456,7 416,9 376,5 337,2 300,3 237,3 189,0 152,8 125,5 104,6 160 100 4z-z 601,3 601,3 568,4 519,7 466,7 409,7 352,1 298,5 252,1 213,4 181,7 135,1 103,7 81,85 66,16 54,54

160 100 5 y-y 786,0 786,0 777,0 732,4 686,4 637,4 585,3 530,9 476,2 423,7 375,3 294,2 233,0 187,7 153,8 128,0 160 100 5z-z 786,0 786,0 738,0 671,4 598,6 520,5 443,0 372,4 312,4 263,2 223,5 165,4 126,6 99,80 80,58 66,37

160 100 6 y-y 930,5 930,5 918,9 865,7 810,6 752,0 689,5 624,5 559,3 496,9 439,6 343,9 272,1 219,0 179,4 149,3 160 100 6z-z 930,5 930,5 871,8 791,7 704,1 610,4 517,8 434,1 363,4 305,8 259,3 191,7 146,6 115,5 93,24 76,78

160 100 6,3 y-y 962,0 962,0 949,1 893,6 836,0 774,7 709,5 641,6 573,7 509,0 449,7 351,2 277,6 223,2 182,7 152,0 160 100 6,3z-z 962,0 962,0 900,1 816,6 725,1 627,4 531,2 444,7 371,8 312,6 264,9 195,7 149,6 117,8 95,10 78,31


x 0 0,5 1,0 1,5 2 2,5 3 3,5 4 4,5 5 6 7 8 9 10

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x 0 0,5 1,0 1,5 2 2,5 3 3,5 4 4,5 5 6 7 8 9 10160 120 4 y-y 653,0 653,0 649,9 615,2 579,5 542 502,1 460,1 417,3 375,2 335,4 266,6 213,1 172,6 142,0 118,5 160 120 4

z-z 653,0 653,0 632,9 589,1 542,7 492,7 439,9 386,9 336,8 292,0 253,1 192,6 149,8 119,3 97,00 80,30160 120 5 y-y 850,6 850,6 843,4 796,5 748,1 697,0 642,5 585,5 527,7 471,7 419,5 330,7 263,0 212,3 174,3 145,2 160 120 5

z-z 850,6 850,6 820,2 760,7 697,4 629,0 557,3 486,2 420,3 362,2 312,7 236,5 183,3 145,7 118,3 97,80160 120 6 y-y 1008 1008 998,5 942,4 884,6 823,3 758,0 689,7 620,8 554,1 492,1 387,3 307,6 248,2 203,6 169,6 160 120 6

z-z 1008 1008 970,7 899,7 823,8 741,9 656,0 571,4 493,1 424,4 366,0 276,5 214,2 170,1 138,0 114,1160 120 6,3 y-y 1043 1043 1033 974,3 914,0 850,0 781,9 710,6 638,8 569,6 505,4 397,1 315,1 254,1 208,3 173,5 160 120 6,3

z-z 1043 1043 1004 929,8 850,7 765,2 675,7 587,7 506,7 435,7 375,5 283,3 219,4 174,2 141,3 116,8160 120 7,1 y-y 1162 1162 1149 1083 1016 943,6 866,8 786,7 706,2 628,7 557,1 437,0 346,4 279,1 228,7 190,4 160 120 7,1

z-z 1162 1162 1117 1033 944,4 848,1 747,5 649,0 558,6 479,7 413,0 311,3 240,9 191,1 155,0 128,1160 120 8 y-y 1292 1292 1277 1203 1127 1046 959,5 869,5 779,2 692,7 613,1 480,0 380,0 305,9 250,6 208,6 160 120 8

z-z 1292 1292 1240 1146 1046 937,0 824,1 713,8 613,1 525,7 452,1 340,2 263,0 208,6 169,1 139,7160 120 8,8 y-y 1405 1405 1387 1306 1222 1134 1039 940,3 841,6 747,3 660,8 516,5 408,6 328,8 269,2 224,0 160 120 8,8

z-z 1405 1405 1346 1243 1133 1013 889,4 768,9 659,4 564,7 485,2 364,7 281,7 223,3 181,0 149,5160 120 10 y-y 1567 1567 1545 1453 1359 1258 1150 1038 927,1 821,3 724,7 564,9 446,0 358,5 293,3 243,9 160 120 10

z-z 1567 1567 1499 1383 1257 1122 982,0 846,2 724,0 618,9 531,0 398,4 307,4 243,5 197,3 163,0180 100 4 y-y 616,9 616,9 616,9 591,3 561,4 530,4 497,7 463,1 427,2 390,9 355,2 289,8 235,9 193,4 160,3 134,6 180 100 4

z-z 616,9 616,9 586,7 538,9 487,3 431,5 374,3 320,1 272,0 231,4 197,8 147,7 113,7 89,90 72,75 60,03x 180 100 5 y-y 850,6 850,6 850,4 807,1 762,9 716,6 667,4 615,7 562,4 509,3 458,3 368,0 296,1 241,0 198,8 166,3 180 100 5

z-z 850,6 850,6 800,3 729,2 651,7 568,4 485,2 409,0 343,8 290,1 246,5 182,7 140,0 110,4 89,15 73,45180 100 5,6 y-y 945,5 945,5 944,9 896,5 847,1 795,3 740,4 682,5 622,9 563,7 506,9 406,5 326,9 265,9 219,3 183,4 180 100 5,6

z-z 945,5 945,5 888,5 808,8 721,8 628,4 535,5 450,6 378,3 318,9 270,9 200,6 153,6 121,1 97,78 80,56x 180 100 6 y-y 1008 1008 1007 955,2 902,4 846,9 788,1 726,1 662,4 599,1 538,5 431,6 346,9 282,0 232,5 194,4 180 100 6

z-z 1008 1008 946,8 861,6 768,4 668,7 569,4 478,9 401,9 338,7 287,6 212,9 163,0 128,5 103,8 85,48180 100 6,3 y-y 1043 1043 1041 987,5 932,3 874,4 812,9 748,1 681,6 615,7 552,7 442,1 355,0 288,3 237,6 198,6 180 100 6,3

z-z 1043 1043 978,8 889,8 792,5 688,3 585,0 491,3 411,8 346,7 294,3 217,7 166,6 131,3 106,0 87,29x 180 100 7,1 y-y 1162 1162 1159 1098 1036 971,2 902,0 829,2 754,5 680,6 610,3 487,3 390,7 317,2 261,2 218,2 180 100 7,1

z-z 1162 1162 1088 987,9 878,1 760,8 645,0 540,5 452,3 380,5 322,6 238,4 182,4 143,6 115,9 95,47x 180 100 8 y-y 1292 1292 1288 1220 1150 1077 998,6 916,6 832,7 750,0 671,5 535,0 428,3 347,3 285,9 238,7 180 100 8

z-z 1292 1292 1208 1095 971,4 839,4 709,8 593,6 495,9 416,7 353,0 260,6 199,2 156,9 126,6

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180 120 4 y-y 668,6 668,6 668,6 642,8 611,2 578,4 543,9 507,5 469,6 431,1 393,0 322,5 263,5 216,6 179,9 151,2 180 120 4z-z 668,6 668,6 651,3 608,4 563,1 514,4 462,8 410,4 359,9 313,9 273,5 209,4 163,5 130,5 106,3 88,13

180 120 5 y-y 915,1 915,1 915,1 872,1 826,0 777,8 726,8 673,0 617,4 561,6 507,4 410,2 331,7 270,7 223,8 187,4 180 120 5z-z 915,1 915,1 884,2 821,3 754,4 682,2 606,2 530,6 459,9 397,2 343,5 260,3 202,1 160,7 130,5 108,0

180 120 6 y-y 1085 1085 1085 1033 978,0 920,3 859,2 794,8 728,2 661,6 597,1 481,8 389,0 317,2 262,1 219,5 180 120 6z-z 1085 1085 1047 972,2 892,0 805,4 714,5 624,2 540,2 466,0 402,5 304,7 236,4 187,9 152,6 126,2

180 120 6,3 y-y 1125 1125 1125 1069 1012 951,7 887,9 820,7 751,2 681,8 614,8 495,3 399,6 325,6 268,9 225,1 180 120 6,3z-z 1125 1125 1084 1006 922,1 831,7 736,8 642,9 555,7 478,9 413,4 312,6 242,4 192,6 156,3 129,3

180 120 7,1 y-y 1254 1254 1254 1191 1126 1058 986,4 910,7 832,7 754,8 679,9 546,7 440,5 358,6 296,0 247,7 180 120 7,1z-z 1254 1254 1207 1119 1025 922,8 816,0 710,7 613,4 528,0 455,3 343,9 266,4 211,6 171,7 142,0

180 120 8 y-y 1396 1396 1395 1323 1251 1174 1093 1008 920,4 833,1 749,3 601,2 483,6 393,4 324,5 271,4 180 120 8z-z 1396 1396 1342 1243 1137 1022 902,3 784,4 675,9 581,1 500,6 377,6 292,4 232,1 188,3 155,6

180 120 8,8 y-y 1518 1518 1516 1438 1359 1275 1186 1093 996,6 901,1 809,7 648,7 521,3 423,7 349,3 292,1 180 120 8,8z-z 1518 1518 1459 1349 1233 1107 975,0 846,0 727,8 624,9 537,9 405,3 313,5 248,7 201,8 166,7

180 120 10 y-y 1696 1696 1692 1604 1514 1419 1318 1212 1103 995,2 892,6 712,9 571,8 464,2 382,3 319,5 180 120 10z-z 1696 1696 1627 1503 1371 1227 1078 932,9 800,7 686,2 589,8 443,6 342,8 271,8 220,3 182,1

200 80 4 y-y 577,6 577,6 577,6 560,9 535,7 509,7 482,5 453,9 424,0 393,2 362,2 302,9 251,0 208,4 174,3 147,2 200 80 4z-z 577,6 577,6 533,3 478,6 418,3 355,0 295,1 243,4 201,4 168,0 141,7 104,0 79,19 62,20

200 80 5 y-y 794,4 794,4 794,4 764,7 727,6 689,1 648,6 605,9 561,4 516,0 471,1 387,5 317,2 261,0 217,0 182,5 200 80 5z-z 794,4 794,4 723,6 641,9 551,8 459,3 375,3 305,7 250,7 208,0 174,6 127,4 96,80 75,85

x 200 80 6 y-y 1008 1008 1008 964,2 914,8 863,4 809,0 751,7 692,2 632,0 573,3 466,4 378,7 310,0 256,8 215,4 200 80 6z-z 1008 1008 908,2 797,9 676,3 554,5 447,4 361,3 294,6 243,4 203,8 148,2 112,3 87,90

200 80 6,3 y-y 1043 1043 1043 996,8 945,1 891,3 834,4 774,3 712,1 649,4 588,2 477,5 387,1 316,6 262,0 219,7 200 80 6,3z-z 1043 1043 938,8 823,7 696,9 570,4 459,5 370,7 302,1 249,4 208,8 151,8 115,0 89,99

x 200 100 5 y-y 858,9 858,9 858,9 830,4 791,5 751,4 709,3 664,9 618,6 571,0 523,6 434,1 357,4 295,4 246,2 207,5 200 100 5z-z 858,9 858,9 814,1 745,9 671,9 592,2 511,1 435,0 368,4 312,5 266,6 198,6 152,6 120,5 97,50 80,40

x 200 100 6 y-y 1085 1085 1085 1044 992,4 939,3 883,3 824,3 762,8 700,2 638,4 523,9 428,2 352,0 292,4 245,8 200 100 6z-z 1085 1085 1021 930,5 831,5 725,3 619,1 521,9 438,7 370,1 314,5 233,1 178,6 140,8 113,7 93,7

200 100 6,3 y-y 1125 1125 1125 1080 1026 970,9 912,2 850,4 786,0 720,6 656,2 537,4 438,4 360,0 298,9 251,1 200 100 6,3z-z 1125 1125 1057 962,8 859,6 748,9 638,5 537,6 451,5 380,7 323,4 239,6 183,5 144,7 116,8 96,20

200 100 7,1 y-y 1254 1254 1254 1203 1143 1080 1014 944,4 872,0 798,6 726,4 593,8 483,9 397,0 329,3 276,6 200 100 7,1z-z 1254 1254 1177 1070 953,5 828,6 704,7 592,2 496,5 418,2 355,0 262,7 201,1 158,5 127,9 105,4

x 200 100 8 y-y 1396 1396 1396 1337 1269 1199 1125 1046 965,0 882,6 801,8 653,9 532,0 436,0 361,5 303,4 200 100 8z-z 1396 1396 1307 1187 1056 915,4 776,5 651,1 545,1 458,6 388,9 287,5 219,9 173,3 139,8 115,2


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x 0 0,5 1,0 1,5 2 2,5 3 3,5 4 4,5 5 6 7 8 9 10x 200 120 5 y-y 923,4 923,4 923,4 895,9 855,3 813,4 769,5 723,4 675,2 625,5 575,6 480,4 397,6 329,7 275,6 232,6 200 120 5

z-z 923,4 923,4 897,3 836,7 772,6 703,6 630,6 556,9 486,6 423,1 367,7 280,6 218,7 174,3 141,9 117,5x 200 120 6 y-y 1163 1163 1163 1123 1069 1014 956,4 895,4 831,8 766,7 701,8 580,2 476,7 393,3 327,5 275,8 200 120 6

z-z 1163 1163 1124 1045 960,0 868,7 772,6 676,8 587,1 507,4 438,9 332,9 258,5 205,6 167,0 138,2200 120 6,3 y-y 1206 1206 1206 1163 1107 1050 989,3 925,5 859,0 790,9 723,3 596,9 489,7 403,7 336,0 282,8 200 120 6,3

z-z 1206 1206 1165 1082 993,4 898,0 797,6 697,8 604,6 522,0 451,3 341,9 265,4 211,0 171,4 141,8200 120 7,1 y-y 1345 1345 1345 1296 1234 1169 1101 1029 954,0 877,6 801,8 660,4 541,2 445,7 370,7 311,9 200 120 7,1

z-z 1345 1345 1298 1204 1105 997,2 884,2 772,2 668,0 576,1 497,5 376,5 292,0 232,0 188,4 155,8x 200 120 8 y-y 1499 1499 1499 1443 1372 1300 1223 1142 1058 972,3 887,4 729,6 597,1 491,3 408,3 343,4 200 120 8

z-z 1499 1499 1444 1339 1227 1106 979,0 853,2 736,9 634,7 547,6 413,9 320,7 254,8 206,8 171,0200 120 8,8 y-y 1632 1632 1632 1569 1492 1412 1328 1239 1147 1053 959,7 787,7 643,7 529,2 439,6 369,5 200 120 8,8

z-z 1632 1632 1571 1455 1332 1198 1059 921,1 794,4 683,3 589,0 444,6 344,3 273,4 221,8 183,4x 200 120 10 y-y 1826 1826 1826 1752 1665 1575 1479 1378 1273 1167 1062 869,2 708,8 581,9 482,9 405,6 200 120 10

z-z 1826 1826 1754 1623 1482 1331 1172 1017 875,3 751,5 646,8 487,3 377,0 299,1 242,6 200,5220 120 5 y-y 943,4 943,4 943,4 927,8 890,5 852,7 813,4 772,4 729,5 684,9 639,2 548,3 464,1 391,3 330,9 281,7 220 120 5

z-z 943,4 943,4 920,5 860,6 797,6 730,0 658,2 585,0 514,3 449,4 392,1 300,8 235,3 187,9 153,1 127,0x 220 120 6 y-y 1240 1240 1240 1211 1159 1106 1051 992,6 931,7 868,8 804,9 680,4 568,8 475,1 399,1 338,1 220 120 6

z-z 1240 1240 1201 1117 1028 932 830,3 728,9 633,6 548,4 475,0 360,8 280,5 223,2 181,4 150,2220 120 6,3 y-y 1287 1287 1287 1256 1202 1146 1088 1027 963,8 898,0 831,2 701,5 585,7 488,8 410,3 347,4 220 120 6,3

z-z 1287 1287 1245 1157 1064 964 857,9 752,2 653,1 564,7 488,8 370,9 288,2 229,2 186,3 154,2220 120 7,1 y-y 1437 1437 1437 1401 1340 1277 1212 1144 1072 998,1 923,2 777,8 648,5 540,7 453,6 383,9 220 120 7,1

z-z 1437 1437 1389 1290 1185 1072 953,0 834,8 723,9 625,4 540,9 410,1 318,4 253,2 205,7 170,2x 220 120 8 y-y 1602 1602 1602 1560 1492 1421 1348 1271 1190 1107 1023 860,0 715,9 596,2 499,7 422,7 220 120 8

z-z 1602 1602 1546 1435 1317 1189 1055 921,0 797,4 687,8 594,2 449,7 348,9 277,3 225,2 186,3220 120 8,8 y-y 1745 1745 1745 1698 1623 1546 1466 1381 1292 1201 1108 930,4 773,4 643,5 539,0 455,7 220 120 8,8

z-z 1745 1745 1683 1561 1431 1291 1143 998,0 862,4 743,2 641,6 485,2 376,1 298,8 242,6 200,7x 220 120 10 y-y 1955 1955 1955 1899 1814 1727 1635 1539 1439 1335 1231 1030 854,6 709,9 594,0 501,8 220 120 10

z-z 1955 1955 1881 1743 1595 1435 1268 1103 952,0 818,5 705,6 532,6 412,5 327,5 265,7 219,7

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x 250 150 5 y-y 1064 1064 1064 1064 1031 995,1 959 921,7 883,1 842,9 801,2 714,9 628,3 546,6 473,5 410,4 250 150 5z-z 1064 1064 1064 1015 961,9 906,4 847,8 785,9 721,8 657,4 594,7 481,7 390,0 318,6 263,5 220,9

x 250 150 6 y-y 1366 1366 1366 1362 1315 1267 1218 1168 1116 1061 1004 887,8 772,9 666,9 573,9 494,8 250 150 6z-z 1366 1366 1364 1294 1222 1147 1067 982,9 896,4 810,5 728,3 583,5 468,9 381,1 314,2 262,7

250 150 6,3 y-y 1447 1447 1447 1442 1391 1339 1287 1232 1175 1116 1055 929,7 806,8 694,1 596,0 513,0 250 150 6,3z-z 1447 1447 1444 1368 1291 1210 1123 1032 939,4 847,4 759,8 606,6 486,3 394,7 325,1 271,6

250 150 7,1 y-y 1712 1712 1712 1699 1637 1574 1509 1442 1371 1298 1223 1069 921,0 787,5 672,9 577,1 250 150 7,1z-z 1712 1712 1701 1609 1513 1413 1306 1194 1080 968,7 864,0 684,2 545,7 441,4 362,7 302,5

x 250 150 8 y-y 1912 1912 1912 1896 1826 1755 1683 1607 1528 1446 1361 1188 1022 873,3 745,6 639,2 250 150 8z-z 1912 1912 1898 1794 1687 1574 1454 1328 1200 1075 957,6 757,2 603,3 487,7 400,5 334,0

250 150 8,8 y-y 2086 2086 2086 2068 1991 1913 1833 1750 1663 1573 1480 1291 1109 946,2 807,3 691,6 250 150 8,8z-z 2086 2086 2070 1956 1838 1714 1581 1442 1302 1165 1037 818,6 651,6 526,4 432,2 360,3

x 250 150 10 y-y 2342 2342 2342 2319 2232 2144 2054 1959 1861 1759 1653 1439 1234 1052 896,3 767,3 250 150 10z-z 2342 2342 2321 2192 2058 1917 1766 1608 1449 1294 1150 906,3 720,3 581,4 477,1 397,5

250 150 12 y-y 2713 2713 2713 2680 2577 2473 2365 2253 2136 2014 1888 1635 1396 1185 1007 860,1 250 150 12z-z 2713 2713 2683 2529 2371 2203 2024 1837 1649 1468 1301 1020 808,5 651,4 533,9 444,5

x 250 150 12,5 y-y 2809 2809 2809 2774 2667 2558 2446 2329 2207 2080 1949 1686 1438 1219 1036 884,3 250 150 12,5z-z 2809 2809 2776 2617 2452 2276 2090 1895 1699 1511 1338 1049 830,5 668,8 548,0 456,2

x 260 140 6 y-y 1339 1339 1339 1339 1295 1250 1204 1157 1108 1056 1003 893,6 783,9 680,8 589,0 509,9 260 140 6z-z 1339 1339 1330 1257 1183 1104 1020 932,4 843,0 755,7 673,8 533,3 425,2 343,8 282,5 235,6

260 140 6,3 y-y 1420 1420 1420 1420 1371 1323 1273 1222 1169 1113 1056 937,2 819,4 709,6 612,4 529,2 260 140 6,3z-z 1420 1420 1409 1331 1251 1166 1075 980 884,2 790,9 703,7 555,3 441,8 356,9 292,9 244,2

260 140 7,1 y-y 1712 1712 1712 1703 1643 1581 1518 1453 1385 1315 1242 1092 945,6 812,3 696,5 599,0 260 140 7,1z-z 1712 1712 1690 1591 1489 1381 1265 1145 1024 909,0 803,7 628,1 496,7 399,6 327,2 272,2

x 260 140 8 y-y 1912 1912 1912 1901 1833 1764 1693 1620 1543 1464 1381 1213 1049 899,9 771,0 662,5 260 140 8z-z 1912 1912 1885 1775 1660 1537 1407 1272 1136 1008 889,8 694,3 548,6 441,1 361,0 300,3

260 140 8,8 y-y 2086 2086 2086 2073 1998 1923 1845 1764 1680 1593 1502 1317 1138 975,0 835,0 717,1 260 140 8,8z-z 2086 2086 2056 1934 1807 1673 1529 1380 1232 1091 962,5 749,9 592,0 475,7 389,2 323,6

x 260 140 10 y-y 2342 2342 2342 2325 2241 2155 2067 1976 1880 1781 1679 1469 1267 1085 928,0 796,0 260 140 10z-z 2342 2342 2305 2167 2023 1870 1706 1537 1369 1210 1066 828,9 653,5 524,7 429,0 356,6


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x 0 0,5 1,0 1,5 2 2,5 3 3,5 4 4,5 5 6 7 8 9 10x 260 180 6 y-y 1494 1494 1494 1494 1451 1403 1353 1302 1250 1195 1138 1020 900,8 787,2 684,4 594,8 260 180 6

z-z 1494 1494 1494 1452 1388 1321 1252 1179 1103 1024 944,9 792,2 658,0 547,0 458,0 387,1260 180 6,3 y-y 1583 1583 1583 1583 1535 1483 1430 1375 1318 1259 1198 1071 943,3 822,2 713,3 618,9 260 180 6,3

z-z 1583 1583 1583 1537 1468 1396 1322 1243 1161 1077 991,5 828,8 686,7 569,9 476,5 402,4260 180 7,1 y-y 1895 1895 1895 1894 1829 1764 1697 1628 1557 1483 1406 1247 1089 942,2 812,7 701,9 260 180 7,1

z-z 1895 1895 1895 1830 1744 1654 1560 1461 1358 1252 1147 948,7 779,8 643,7 536,2 451,5x 260 180 8 y-y 2118 2118 2118 2115 2043 1969 1895 1817 1737 1653 1567 1388 1211 1047 902,1 778,7 260 180 8

z-z 2118 2118 2118 2044 1946 1846 1740 1628 1511 1392 1274 1052 863,5 712,0 592,8 499,0260 180 8,8 y-y 2313 2313 2313 2309 2229 2149 2066 1981 1893 1801 1706 1509 1316 1136 978,3 844,0 260 180 8,8

z-z 2313 2313 2313 2231 2124 2013 1897 1774 1646 1516 1386 1143 937,6 772,7 643,0 541,2x 260 180 10 y-y 2600 2600 2600 2593 2503 2412 2318 2222 2122 2017 1909 1687 1468 1266 1089 939,0 260 180 10

z-z 2600 2600 2600 2504 2383 2257 2125 1986 1841 1692 1546 1272 1042 857,7 713,2 599,9260 180 12 y-y 3023 3023 3023 3008 2901 2793 2682 2568 2448 2324 2195 1931 1673 1438 1233 1061 260 180 12

z-z 3023 3023 3023 2903 2760 2610 2453 2287 2114 1939 1766 1447 1181 969,7 805,0 676,4x 300 100 5 y-y 930,8 930,8 930,8 930,8 921,0 895,0 868,8 842,1 814,8 786,6 757,5 696,5 632,7 568,5 506,6 449,3 300 100 5

z-z 930,8 930,8 901,7 839,0 772,6 701,0 625,4 549,7 478,2 414,3 359,0 272,9 212,3 169,0 137,4 113,7x 300 100 6 y-y 1222 1222 1222 1222 1202 1165 1129 1092 1053 1013 972,3 886,2 797,1 708,9 625,8 550,7 300 100 6

z-z 1222 1222 1174 1086 992 890,8 784,9 681,1 585,9 503,1 433,0 326,3 252,4 200,3 162,4 134,2300 100 6,3 y-y 1302 1302 1302 1302 1278 1238 1199 1158 1116 1073 1028 934,1 837,3 742,1 653,1 573,2 300 100 6,3

z-z 1302 1302 1248 1153 1050 940,0 824,8 713,1 611,6 523,8 450,0 338,3 261,3 207,1 167,9 138,7300 100 7,1 y-y 1549 1549 1549 1549 1513 1464 1415 1364 1312 1258 1202 1085 965,6 849,9 743,4 649,2 300 100 7,1

z-z 1549 1549 1475 1356 1228 1090 947 811,8 691,1 588,6 503,6 376,5 290,0 229,4 185,7 153,2x 300 100 8 y-y 1833 1833 1833 1833 1782 1722 1662 1600 1535 1468 1399 1254 1108 968,6 842,4 732,3 300 100 8

z-z 1833 1833 1735 1588 1429 1257 1082 919 777,5 658,8 561,6 417,9 320,9 253,5 204,9 169,0

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300 150 6 y-y 1416 1416 1416 1416 1400 1360 1319 1279 1237 1193 1148 1055 956,8 858,4 763,9 676,7 300 150 6z-z 1416 1416 1416 1355 1285 1213 1137 1057 973,0 888,8 806,3 656,2 533,0 436,3 361,5 303,2

300 150 6,3 y-y 1506 1506 1506 1506 1486 1443 1399 1355 1309 1262 1214 1112 1006 900,1 798,9 706,1 300 150 6,3z-z 1506 1506 1506 1437 1362 1284 1201 1114 1024 933,0 844,1 684,3 554,3 452,9 374,7 314,1

300 150 7,1 y-y 1778 1778 1778 1778 1748 1696 1642 1588 1532 1475 1415 1290 1160 1032 910,8 801,4 300 150 7,1z-z 1778 1778 1778 1689 1597 1501 1399 1292 1181 1071 965,0 775,7 625,0 508,9 420,0 351,4

300 150 8,8 y-y 2370 2370 2370 2370 2315 2241 2165 2088 2009 1926 1840 1662 1479 1302 1139 994,7 300 150 8,8z-z 2370 2370 2357 2230 2100 1962 1816 1662 1506 1352 1207 958,0 764,8 619,1 509,0 424,7

300 150 10 y-y 2665 2665 2665 2665 2600 2516 2431 2343 2253 2159 2062 1859 1652 1452 1269 1107 300 150 10z-z 2665 2665 2647 2503 2355 2198 2032 1857 1679 1505 1342 1063 847,2 685,2 562,9 469,5

x 300 200 6 y-y 1609 1609 1609 1609 1597 1553 1509 1464 1418 1371 1322 1220 1113 1004 898,5 799,8 300 200 6z-z 1609 1609 1609 1593 1533 1472 1409 1344 1276 1205 1132 984 843,1 717,7 611,2 523,0

300 200 6,3 y-y 1709 1709 1709 1709 1693 1646 1598 1550 1501 1450 1397 1286 1171 1054 941,3 836,3 300 200 6,3z-z 1709 1709 1709 1689 1625 1559 1492 1421 1348 1271 1193 1034 883,5 750,5 638,1 545,3

300 200 7,1 y-y 2007 2007 2007 2007 1982 1925 1868 1809 1749 1687 1622 1488 1348 1207 1073 949,0 300 200 7,1z-z 2007 2007 2007 1977 1899 1820 1738 1652 1562 1469 1374 1182 1004 848,5 718,8 612,7

x 300 200 8 y-y 2350 2350 2350 2350 2313 2244 2175 2104 2031 1956 1878 1715 1546 1378 1219 1074 300 200 8z-z 2350 2350 2350 2307 2213 2117 2018 1914 1805 1692 1577 1348 1137 956,9 807,8 686,9

300 200 8,8 y-y 2654 2654 2654 2654 2606 2527 2447 2365 2280 2193 2103 1913 1718 1525 1344 1181 300 200 8,8z-z 2654 2654 2654 2599 2491 2380 2264 2144 2017 1886 1753 1490 1252 1049 883,7 750,0

x 300 200 10 y-y 2987 2987 2987 2987 2931 2841 2751 2658 2562 2463 2360 2145 1924 1706 1502 1318 300 200 10z-z 2987 2987 2987 2923 2799 2674 2543 2406 2262 2113 1962 1664 1396 1169 983,6 834,3

300 200 12 y-y 3487 3487 3487 3487 3415 3309 3201 3090 2976 2858 2736 2480 2217 1960 1721 1508 300 200 12z-z 3487 3487 3487 3405 3259 3109 2953 2789 2618 2441 2261 1910 1597 1334 1120 948,9

x 300 200 12,5 y-y 3616 3616 3616 3616 3540 3429 3317 3202 3083 2961 2833 2567 2294 2027 1779 1558 300 200 12,5z-z 3616 3616 3616 3529 3377 3222 3059 2889 2711 2526 2339 1976 1650 1378 1157 980,0

400 100 6 y-y 1283 1283 1283 1283 1283 1280 1253 1226 1199 1172 1144 1087 1027 965,0 899,6 833,6 400 100 6z-z 1283 1283 1252 1170 1084 992,0 894,2 794,5 698,2 609,9 532,1 408,1 319,1 254,9 207,7 172,2

400 100 6,3 y-y 1373 1373 1373 1373 1373 1367 1338 1309 1279 1250 1219 1157 1091 1022 950,9 878,9 400 100 6,3z-z 1373 1373 1337 1248 1154 1054 947,0 838,6 734,8 640,3 557,4 426,4 332,8 265,6 216,2 179,2

400 100 7,1 y-y 1652 1652 1652 1652 1652 1637 1601 1564 1527 1490 1452 1372 1289 1201 1112 1022 400 100 7,1z-z 1652 1652 1598 1486 1367 1238 1103 967,7 841 727,2 629,6 478 371,4 295,5 240,2 198,8


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x 0 0,5 1,0 1,5 2 2,5 3 3,5 4 4,5 5 6 7 8 9 10400 120 6 y-y 1360 1360 1360 1360 1360 1359 1331 1303 1275 1247 1218 1159 1096 1031 963,0 894,3 400 120 6

z-z 1360 1360 1354 1282 1208 1130 1047 959,5 870,4 782,8 700,0 556,5 445 360,5 296,6 247,6400 120 6,3 y-y 1455 1455 1455 1455 1455 1450 1420 1390 1359 1328 1297 1232 1163 1092 1018 943,0 400 120 6,3

z-z 1455 1455 1445 1367 1286 1201 1110 1015 918,0 823,3 734,4 581,6 463,8 375,2 308,3 257,2400 120 7,1 y-y 1743 1743 1743 1743 1743 1731 1694 1656 1617 1579 1539 1457 1371 1281 1189 1095 400 120 7,1

z-z 1743 1743 1725 1627 1525 1418 1303 1184 1063 947,0 839,8 659,2 522,8 421,4 345,4 287,6400 120 8 y-y 2086 2086 2086 2086 2086 2063 2016 1969 1921 1873 1823 1720 1612 1499 1383 1268 400 120 8

z-z 2086 2086 2054 1931 1803 1667 1522 1372 1222 1081 952,5 740,9 584,2 469,2 383,7 319,0400 120 8,8 y-y 2403 2403 2403 2403 2403 2367 2312 2256 2199 2141 2082 1959 1829 1694 1556 1419 400 120 8,8

z-z 2403 2403 2356 2210 2057 1893 1720 1540 1364 1200 1053 813,7 639,2 512,1 418,1 347,1x 400 200 6 y-y 1670 1670 1670 1670 1670 1670 1643 1610 1577 1544 1510 1441 1368 1292 1214 1133 400 200 6

z-z 1670 1670 1670 1670 1616 1560 1502 1443 1382 1318 1252 1115 978,0 849,0 734,4 635,7400 200 6,3 y-y 1780 1780 1780 1780 1780 1780 1748 1712 1677 1641 1604 1529 1451 1368 1283 1196 400 200 6,3

z-z 1780 1780 1780 1779 1719 1658 1596 1532 1465 1396 1324 1176 1029 891,3 769,4 665,0400 200 7,1 y-y 2110 2110 2110 2110 2110 2107 2064 2021 1977 1933 1888 1795 1698 1597 1491 1384 400 200 7,1

z-z 2110 2110 2110 2102 2028 1954 1877 1798 1716 1630 1541 1359 1180 1016 873,1 751,8x 400 200 8 y-y 2499 2499 2499 2499 2499 2488 2435 2381 2328 2274 2219 2104 1985 1859 1729 1598 400 200 8

z-z 2499 2499 2499 2481 2391 2299 2205 2108 2006 1900 1790 1567 1351 1156 988,6 848,3400 200 8,8 y-y 2857 2857 2857 2857 2857 2836 2774 2711 2649 2585 2520 2385 2243 2095 1942 1789 400 200 8,8

z-z 2857 2857 2857 2828 2722 2614 2503 2388 2267 2142 2012 1750 1500 1278 1088 931,5x 400 200 10 y-y 3408 3408 3408 3408 3408 3370 3294 3217 3139 3060 2979 2812 2635 2450 2261 2073 400 200 10

z-z 3408 3408 3408 3360 3229 3096 2958 2814 2663 2507 2346 2024 1722 1457 1236 1054400 200 12 y-y 4262 4262 4262 4262 4262 4192 4092 3991 3889 3784 3677 3455 3220 2976 2728 2483 400 200 12

z-z 4262 4262 4262 4179 4007 3831 3648 3457 3257 3050 2838 2419 2037 1711 1443 1226x 400 200 12,5 y-y 4423 4423 4423 4423 4423 4349 4245 4139 4033 3924 3813 3581 3337 3082 2824 2570 400 200 12,5

z-z 4423 4423 4423 4335 4156 3973 3783 3584 3376 3160 2939 2505 2108 1770 1492 1267

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Table 9.2.3 Buckling resistance values for circular longitudinally welded hollow sections of steel grade S355J2H (fy = 355 N/mm2) in buckling category c.

1) = recommended seriesd = external diametert = wall thicknessLc = buckling lengthNb.Rd = buckling resistance

1) d t Nb.Rd (kN) d tmm mm Lc (m) mm mm

x 0 0,5 1,0 1,5 2 2,5 3 3,5 4 4,5 5 6 7 8 9 10x 33,7 2 64,28 51,07 28,85 15,74 9,597 6,413 33,7 2x 42,4 2 81,92 70,98 48,39 29,46 18,69 12,72 9,166 6,907 42,4 2x 42,4 2,5 101,1 87,25 58,89 35,58 22,50 15,29 11,01 8,293 42,4 2,5

42,4 2,6 104,9 90,53 61,12 36,93 23,36 15,87 11,43 8,611 42,4 2,642,4 2,9 116,1 100,0 67,21 40,47 25,56 17,36 12,50 9,413 42,4 2,9

x 42,4 3 119,8 103,2 69,31 41,72 26,35 17,89 12,88 9,701 42,4 3x 48,3 2 93,89 84,3 62,29 40,87 26,80 18,50 13,44 10,18 7,965 48,3 2x 48,3 2,5 116,1 103,9 76,32 49,73 32,50 22,41 16,27 12,31 9,631 48,3 2,5

48,3 2,6 120,5 107,8 79,15 51,56 33,69 23,22 16,86 12,76 9,980 48,3 2,648,3 2,9 133,5 119,3 87,34 56,72 37,01 25,50 18,50 14,00 10,95 48,3 2,9

x 48,3 3 137,8 123,2 90,13 58,52 38,18 26,30 19,09 14,44 11,29 48,3 3x 60,3 2 118,2 111,1 90,33 67,08 47,43 33,97 25,16 19,27 15,19 12,27 10,11 60,3 2x 60,3 2,5 146,5 137,6 111,7 82,77 58,42 41,81 30,95 23,70 18,68 15,08 12,43 60,3 2,5

60,3 2,6 152,1 142,7 115,7 85,47 60,21 43,04 31,85 24,38 19,21 15,51 12,78 60,3 2,660,3 2,9 168,8 158,3 128,0 94,39 66,39 47,41 35,07 26,83 21,14 17,07 14,06 60,3 2,9

x 60,3 3 174,3 163,4 132,2 97,47 68,55 48,96 36,21 27,71 21,83 17,62 14,51 60,3 360,3 3,2 185,3 173,6 140,2 103,1 72,36 51,63 38,16 29,20 23,00 18,56 15,29 60,3 3,2

x 60,3 4 228,3 213,5 171,8 125,7 87,90 62,59 46,22 35,34 27,83 22,45 18,49 60,3 476,1 2 150,3 146,5 126,7 104,3 81,25 61,82 47,36 37,01 29,55 24,07 19,95 14,33 76,1 2

x 76,1 2,5 186,6 181,6 156,9 128,8 100,0 75,89 58,07 45,33 36,18 29,46 24,41 17,52 76,1 2,576,1 2,6 193,8 188,7 162,9 133,7 103,8 78,79 60,28 47,06 37,56 30,58 25,34 18,19 76,1 2,676,1 2,9 215,2 209,5 180,8 148,2 114,9 87,11 66,61 51,98 41,47 33,76 27,98 20,08 76,1 2,9

x 76,1 3 222,3 216,4 186,8 153,1 118,7 89,98 68,80 53,69 42,84 34,88 28,90 20,74 76,1 376,1 3,2 236,5 230,1 198,4 162,4 125,7 95,22 72,76 56,76 45,27 36,85 30,53 21,91 76,1 3,2

x 76,1 4 292,4 284,0 244,3 199,1 153,4 115,8 88,31 68,80 54,84 44,61 36,95 26,50 76,1 4x 76,1 5 360,4 349,5 299,9 243,4 186,7 140,4 106,9 83,17 66,23 53,86 44,60 31,96 76,1 5

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The calculated resistance values are design values (section 2.1) based on the default value (1.1) of thepartial safety factor of material γγγγM1 used in Eurocode 3. The partial safety factor values may differ ineach country. National values must be checked from the NAD (National Application Document) of thecountry in question.

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88,9 2 176,2 175,0 155,6 134,2 110,9 88,59 70,1 55,88 45,19 37,12 30,96 22,40 16,92 88,9 2x 88,9 2,5 219,0 217,5 193,3 166,6 137,4 109,7 86,77 69,13 55,89 45,90 38,28 27,69 20,92 88,9 2,5

88,9 2,6 227,5 225,8 200,6 172,7 142,4 113,5 89,69 71,43 57,72 47,40 39,52 28,59 21,59 88,9 2,688,9 2,9 252,9 250,9 222,8 191,7 157,8 125,7 99,27 79,02 63,84 52,41 43,70 31,60 23,86 88,9 2,9

x 88,9 3 261,3 259,3 230,2 198,1 163,1 129,9 102,6 81,64 65,96 54,15 45,15 32,65 24,65 88,9 388,9 3,2 278,1 275,8 244,8 210,4 173,0 137,7 108,6 86,44 69,81 57,31 47,77 34,54 26,08 88,9 3,2

x 88,9 4 344,3 341,1 302,3 259,2 212,4 168,5 132,6 105,4 85,02 69,75 58,11 41,99 31,69 88,9 4x 88,9 5 425,3 420,9 372,4 318,4 260,0 205,6 161,5 128,1 103,3 84,68 70,53 50,94 38,43 88,9 5

88,9 6 504,3 498,5 440,4 375,5 305,5 240,8 188,7 149,5 120,4 98,69 82,16 59,30 44,73 88,9 688,9 6,3 527,6 521,3 460,3 392,1 318,6 250,9 196,5 155,6 125,3 102,6 85,43 61,65 46,49 88,9 6,3

101,6 2 202,0 202,0 184,2 163,6 140,9 117,5 96,18 78,43 64,38 53,43 44,88 32,77 24,89 19,52 101,6 2x 101,6 2,5 251,2 251,2 228,8 203,0 174,5 145,3 118,7 96,72 79,33 65,80 55,25 40,33 30,62 24,00 101,6 2,5

101,6 2,6 261,0 261,0 237,8 211,0 181,4 151,0 123,4 100,5 82,48 68,42 57,45 41,93 31,84 24,96 101,6 2,6101,6 2,9 290,2 290,2 264,2 234,3 201,2 167,4 136,7 111,3 91,26 75,68 63,54 46,36 35,20 27,59 101,6 2,9

x 101,6 3 299,9 299,9 273,0 242,1 208,0 173,0 141,3 115,0 94,31 78,21 65,66 47,91 36,37 28,51 101,6 3101,6 3,2 319,3 319,3 290,5 257,4 221,0 183,7 149,8 121,9 100,0 82,87 69,56 50,75 38,52 30,19 101,6 3,2

x 101,6 4 395,8 395,8 359,5 318,1 272,3 225,7 183,8 149,3 122,3 101,3 84,98 61,95 47,01 36,84 101,6 4x 101,6 5 489,7 489,7 443,9 392,1 334,9 276,9 224,9 182,4 149,2 123,5 103,6 75,48 57,26 44,85 101,6 5

101,6 6 581,6 581,6 526,1 463,9 395,3 325,9 264,2 214,0 174,8 144,6 121,2 88,27 66,93 52,42 101,6 6101,6 6,3 608,7 608,7 550,3 485,0 412,8 340,1 275,4 223,0 182,1 150,6 126,2 91,91 69,68 54,57 101,6 6,3108 2 214,9 214,9 198,6 178,3 156,0 132,6 110,3 91,06 75,38 62,92 53,07 38,95 29,67 23,31 18,78 108 2

x 108 2,5 267,4 267,4 246,8 221,5 193,5 164,2 136,4 112,5 93,08 77,65 65,47 48,03 36,58 28,74 23,15 108 2,5108 2,6 277,8 277,8 256,5 230,1 201,1 170,6 141,7 116,9 96,68 80,66 68,00 49,89 38,00 29,85 24,04 108 2,6108 2,9 309,0 309,0 285,1 255,7 223,3 189,3 157,2 129,5 107,1 89,36 75,32 55,25 42,07 33,04 26,62 108 2,9

x 108 3 319,4 319,4 294,5 264,0 230,4 195,2 161,9 133,4 110,3 91,95 77,49 56,83 43,27 33,98 27,37 108 3108 3,2 340,0 340,0 313,5 281,1 245,3 207,8 172,4 142,0 117,4 97,89 82,50 60,50 46,07 36,18 29,14 108 3,2

x 108 4 421,8 421,8 388,3 347,6 302,7 255,8 211,8 174,2 143,8 119,8 100,9 73,97 56,30 44,20 35,59 108 4108 5 522,2 522,2 479,9 429,0 372,9 314,4 259,7 213,2 175,9 146,4 123,3 90,27 68,68 53,91 43,40 108 5108 6 620,5 620,5 569,0 507,7 440,0 369,8 304,6 249,6 205,6 171,0 143,8 105,2 80,02 62,79 50,54 108 6108 6,3 649,6 649,6 595,3 530,9 459,8 386,1 317,8 260,3 214,3 178,2 149,9 109,6 83,35 65,40 52,63 108 6,3114,3 2 227,7 227,7 212,7 192,7 170,7 147,4 124,5 104,0 86,81 72,89 61,74 45,55 34,82 27,41 22,11 114,3 2

x 114,3 2,5 283,4 283,4 264,4 239,4 211,9 182,6 154,0 128,5 107,2 89,96 76,16 56,17 42,91 33,78 27,25 114,3 2,5114,3 2,6 294,5 294,5 274,7 248,7 220,1 189,7 160,0 133,5 111,4 93,45 79,12 58,35 44,58 35,09 28,30 114,3 2,6114,3 2,9 327,5 327,5 305,5 276,4 244,6 210,6 177,5 148,1 123,5 103,6 87,68 64,65 49,39 38,87 31,35 114,3 2,9

x 114,3 3 338,5 338,5 315,7 285,7 252,8 217,7 183,5 153,0 127,6 107,1 90,62 66,82 51,04 40,17 32,40 114,3 3114,3 3,2 360,5 360,5 336,0 304,0 268,8 231,3 194,8 162,4 135,4 113,6 96,10 70,84 54,11 42,58 34,34 114,3 3,2114,3 4 447,3 447,3 416,4 376,2 332,1 285,3 239,8 199,5 166,1 139,2 117,8 86,75 66,23 52,10 42,01 114,3 4

x 114,3 5 554,1 554,1 515,1 464,8 409,6 351,1 294,5 244,7 203,5 170,4 144,1 106,0 80,92 63,65 51,31 114,3 5x 114,3 6 658,8 658,8 611,2 550,7 484,2 413,8 346,2 287,0 238,3 199,3 168,4 123,8 94,43 74,24 59,84 114,3 6

114,3 6,3 689,8 689,8 639,7 576,2 506,3 432,4 361,5 299,6 248,7 207,9 175,6 129,1 98,46 77,40 62,38 114,3 6,3x d t Nb.Rd (kN) d t

mm mm Lc (m) mm mm0 0,5 1,0 1,5 2 2,5 3 3,5 4 4,5 5 6 7 8 9 10

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Table 9.2.3 Buckling resistance values for circular longitudinally welded hollow sections of steel grade S355J2H (fy = 355 N/mm2) in buckling category c, continued.


x d t Nb.Rd (kN) d tmm mm Lc (m) mm mm

0 0,5 1,0 1,5 2 2,5 3 3,5 4 4,5 5 6 7 8 9 10127 2 224,1 224,1 215,2 199,0 181,7 162,9 143,4 124,3 106,9 91,70 78,89 59,40 45,94 36,44 29,55 24,42 127 2

x 127 2,5 315,6 315,6 299,9 275,4 248,8 220,1 190,8 163,0 138,5 117,7 100,6 75,08 57,76 45,67 36,95 30,49 127 2,5127 2,6 327,9 327,9 311,7 286,2 258,5 228,7 198,2 169,4 143,8 122,3 104,5 78,00 60,01 47,45 38,39 31,67 127 2,6127 2,9 364,9 364,9 346,7 318,2 287,4 254,2 220,2 188,0 159,6 135,7 115,9 86,49 66,54 52,60 42,56 35,11 127 2,9

x 127 3 377,2 377,2 358,4 328,9 297,1 262,7 227,6 194,3 165,0 140,2 119,8 89,40 68,77 54,37 43,99 36,29 127 3127 3,2 401,7 401,7 381,5 350,1 316,1 279,4 241,9 206,4 175,2 148,9 127,2 94,85 72,95 57,67 46,66 38,49 127 3,2

x 127 4 498,8 498,8 473,3 433,9 391,3 345,3 298,5 254,3 215,6 183,0 156,2 116,4 89,52 70,74 57,22 47,19 127 4127 5 618,5 618,5 586,1 536,9 483,5 426,0 367,6 312,8 264,8 224,6 191,6 142,7 109,6 86,60 70,03 57,75 127 5127 6 736,1 736,1 696,5 637,3 573,1 503,8 433,7 368,3 311,3 263,7 224,8 167,2 128,4 101,4 81,97 67,58 127 6127 6,3 771,0 771,0 729,2 667,1 599,5 526,8 453,3 384,7 325,0 275,3 234,6 174,4 133,9 105,7 85,48 70,48 127 6,3133 2 233,9 233,9 226,1 210,1 193,1 174,8 155,5 136,2 118,2 102,2 88,39 67,04 52,07 41,42 33,65 27,84 133 2

x 133 2,5 330,8 330,8 316,7 292,3 266,1 237,7 208,3 179,8 154,0 131,8 113,2 84,98 65,61 51,99 42,14 34,80 133 2,5133 2,6 343,7 343,7 329,1 303,7 276,5 247,0 216,5 186,9 160,1 137,0 117,6 88,30 68,18 54,02 43,78 36,16 133 2,6133 2,9 382,5 382,5 366,1 337,8 307,4 274,5 240,4 207,5 177,6 151,9 130,4 97,90 75,58 59,88 48,52 40,08 133 2,9133 3 395,4 395,4 378,4 349,2 317,7 283,8 248,5 214,4 183,6 157,1 134,8 101,2 78,13 61,90 50,16 41,43 133 3133 3,2 421,1 421,1 402,9 371,7 338,1 301,8 264,2 227,9 195,0 166,8 143,1 107,4 82,92 65,69 53,23 43,96 133 3,2

x 133 4 523,2 523,2 500,0 460,9 418,9 373,4 326,4 281,1 240,3 205,3 176,1 132,0 101,8 80,66 65,34 53,95 133 4133 5 648,9 648,9 619,6 570,7 518,1 461,3 402,6 346,2 295,6 252,3 216,2 162,0 124,9 98,90 80,10 66,13 133 5133 6 772,6 772,6 736,9 678,3 615,1 546,9 476,5 409,2 348,9 297,6 254,8 190,8 147,0 116,4 94,22 77,77 133 6133 6,3 809,3 809,3 771,7 710,1 643,8 572,1 498,2 427,6 364,5 310,7 266,1 199,1 153,4 121,4 98,31 81,15 133 6,3139,7 2,9 402,2 402,2 387,8 359,7 329,8 297,4 263,5 229,9 198,7 171,2 147,8 111,8 86,67 68,86 55,90 46,23 139,7 2,9

x 139,7 3 415,8 415,8 400,8 371,7 340,6 307,1 271,9 237,1 204,9 176,5 152,3 115,2 89,30 70,92 57,57 47,61 139,7 3139,7 3,2 442,9 442,9 426,9 395,9 362,8 327,1 289,6 252,6 218,3 188,0 162,3 122,7 95,09 75,54 61,33 50,72 139,7 3,2

x 139,7 4 550,3 550,3 530,0 491,2 449,8 405,0 358,2 312,0 269,2 231,7 199,8 150,9 116,9 92,86 75,37 62,32 139,7 4x 139,7 5 682,9 682,9 657,0 608,6 556,8 500,8 442,3 384,7 331,6 285,1 245,7 185,4 143,6 114,0 92,49 76,46 139,7 5x 139,7 6 813,3 813,3 781,6 723,3 661,0 593,6 523,2 454,2 390,9 335,7 289,1 217,8 168,6 133,8 108,5 89,67 139,7 6

139,7 6,3 852,1 852,1 818,6 757,4 691,9 621,1 547,2 474,8 408,5 350,7 301,9 227,4 176,0 139,6 113,2 93,58 139,7 6,3x 139,7 8 1068 1068 1024 946,5 863,0 772,7 678,8 587,4 504,2 432,0 371,4 279,3 215,8 171,1 138,7 114,6 139,7 8x 139,7 10 1315 1315 1259 1161 1057 943,8 826,7 713,3 610,7 522,4 448,4 336,6 259,9 205,9 166,9 137,8 139,7 10

152 2,9 438,4 438,4 427,6 399,7 370,4 338,8 305,4 271,3 238,4 208,2 181,6 139,3 108,9 86,97 70,86 58,76 152 2,9152 3 453,2 453,2 442,1 413,2 382,9 350,3 315,7 280,5 246,4 215,2 187,8 144,0 112,6 89,92 73,26 60,75 152 3152 3,2 482,8 482,8 470,8 440,0 407,6 372,8 335,9 298,3 262,0 228,8 199,5 153,0 119,6 95,5 77,80 64,50 152 3,2152 4 600,2 600,2 584,9 546,4 505,8 462,2 416,0 369,0 323,8 282,5 246,1 188,5 147,3 117,6 95,76 79,38 152 4152 5 745,2 745,2 725,6 677,5 626,8 572,3 514,5 455,8 399,5 348,2 303,2 232,0 181,1 144,5 117,7 97,53 152 5152 6 888,2 888,2 864,2 806,4 745,6 680,1 610,7 540,4 473,1 412,0 358,5 274,0 213,8 170,5 138,8 115,0 152 6152 6,3 930,7 930,7 905,3 844,7 780,8 712,0 639,1 565,3 494,7 430,7 374,6 286,3 223,3 178,1 145,0 120,1 152 6,3

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159 2,9 459,0 459,0 450,3 422,5 393,5 362,4 329,4 295,3 261,8 230,4 202,3 156,5 123,0 98,56 80,48 66,84 159 2,9159 3 474,5 474,5 465,5 436,8 406,8 374,7 340,5 305,3 270,6 238,2 209,1 161,8 127,1 101,9 83,20 69,09 159 3159 3,2 505,5 505,5 495,8 465,1 433,1 398,8 362,4 324,7 287,8 253,3 222,2 171,9 135,0 108,2 88,35 73,37 159 3,2

x 159 4 628,6 628,6 616,2 577,8 537,7 494,8 449,1 402,1 356,0 313,0 274,5 212,0 166,5 133,4 108,9 90,38 159 4159 5 780,7 780,7 764,7 716,8 666,7 613,0 555,9 497,1 439,6 386,2 338,4 261,1 204,9 164,1 133,9 111,1 159 5159 6 930,7 930,7 910,9 853,3 793,0 728,4 659,6 589,1 520,2 456,4 399,5 307,9 241,4 193,2 157,6 130,8 159 6159 6,3 975,4 975,4 954,3 893,9 830,5 762,6 690,4 616,3 544,1 477,2 417,6 321,7 252,2 201,8 164,6 136,6 159 6,3168,3 2,9 486,3 486,3 480,4 452,7 424,0 393,6 361,1 327,2 293,3 260,7 230,7 180,6 143,0 115,1 94,32 78,50 168,3 2,9168,3 3 502,8 502,8 496,7 468,0 438,4 406,9 373,3 338,3 303,2 269,5 238,6 186,8 147,9 119,0 97,51 81,16 168,3 3168,3 3,2 535,7 535,7 529,1 498,5 466,8 433,2 397,3 360,0 322,5 286,6 253,6 198,5 157,1 126,5 103,6 86,21 168,3 3,2168,3 4 666,3 666,3 657,8 619,5 579,9 537,7 492,9 446,1 399,3 354,6 313,5 245,1 193,9 156,0 127,7 106,3 168,3 4

x 168,3 5 827,8 827,8 816,7 768,9 719,4 666,7 610,5 552,0 493,7 437,9 386,9 302,2 238,8 192,1 157,2 130,8 168,3 5x 168,3 6 987,3 987,3 973,2 915,8 856,3 792,8 725,2 655,0 584,9 518,3 457,5 356,7 281,7 226,4 185,3 154,1 168,3 6

168,3 6,3 1035 1035 1020 959,5 897,0 830,3 759,3 685,5 612,0 542,2 478,4 372,9 294,4 236,6 193,6 161,0 168,3 6,3x 168,3 8 1300 1300 1280 1203 1124 1039 948,3 854,5 761,4 673,3 593,3 461,4 363,8 292,2 238,9 198,6 168,3 8x 168,3 10 1605 1605 1578 1482 1383 1276 1163 1046 930,2 821,1 722,4 560,6 441,5 354,2 289,5 240,6 168,3 10

193,7 4 769,3 769,3 769,3 733,1 694,4 653,9 611 565,8 519,0 472,1 426,6 344,8 278,8 227,5 188,1 157,6 193,7 4193,7 5 956,6 956,6 956,6 910,7 862,2 811,4 757,7 701,1 642,5 583,8 527,1 425,4 343,6 280,3 231,6 193,9 193,7 5

x 193,7 6 1142 1142 1142 1086 1028 967,2 902,6 834,6 764,4 694,1 626,2 504,9 407,5 332,1 274,4 229,7 193,7 6193,7 6,3 1197 1197 1197 1139 1077 1013 945,6 874,2 800,4 726,6 655,4 528,2 426,2 347,4 286,9 240,2 193,7 6,3

x 193,7 8 1506 1506 1506 1431 1353 1271 1185 1094 1000 906,8 816,8 656,8 529,1 430,8 355,6 297,5 193,7 8x 193,7 10 1862 1862 1861 1766 1669 1567 1458 1344 1227 1110 998,4 800,7 644,0 523,7 431,9 361,2 193,7 10

193,7 12 2211 2211 2207 2093 1976 1854 1724 1587 1446 1306 1173 938,8 753,8 612,5 504,8 421,9 193,7 12193,7 12,5 2296 2296 2292 2173 2051 1923 1788 1645 1498 1353 1214 971,2 779,4 633,1 521,6 436,0 193,7 12,5219,1 4 872,3 872,3 872,3 846,6 808,3 768,9 727,6 684,1 638,7 591,9 544,9 455,1 376,8 312,6 261,3 220,6 219,1 4219,1 5 1085 1085 1085 1053 1005 955,3 903,6 849,1 792,2 733,7 674,9 562,9 465,5 385,9 322,4 272,1 219,1 5

x 219,1 6 1296 1296 1296 1257 1199 1140 1078 1012 944 873,8 803,4 669,3 553,1 458,2 382,7 322,9 219,1 6219,1 6,3 1359 1359 1359 1317 1257 1195 1129 1061 989,1 915,3 841,4 700,7 578,9 479,5 400,4 337,8 219,1 6,3

x 219,1 8 1712 1712 1712 1658 1581 1502 1419 1331 1240 1146 1052 874,7 721,5 597,0 498,1 420,0 219,1 8x 219,1 10 2120 2120 2120 2050 1954 1855 1751 1641 1527 1409 1292 1071 882,0 728,8 607,6 511,9 219,1 10

219,1 12 2520 2520 2520 2433 2318 2198 2073 1941 1804 1663 1523 1259 1035 853,9 711,2 598,9 219,1 12x 219,1 12,5 2618 2618 2618 2528 2408 2284 2153 2016 1873 1726 1580 1306 1073 885,5 737,5 621,0 219,1 12,5

273 5 1359 1359 1359 1354 1306 1258 1209 1158 1105 1050 992,8 875,5 760,4 654,7 562,5 484,4 273 5x 273 6 1624 1624 1624 1618 1560 1503 1444 1383 1319 1253 1184 1043 905,1 778,7 668,6 575,5 273 6

273 6,3 1704 1704 1704 1696 1636 1576 1514 1450 1383 1313 1241 1093 948,3 815,7 700,2 602,7 273 6,3x 273 8 2149 2149 2149 2139 2063 1986 1907 1825 1740 1651 1560 1372 1189 1021 875,8 753,2 273 8x 273 10 2667 2667 2667 2652 2557 2460 2362 2260 2153 2042 1928 1693 1465 1257 1077 925,5 273 10

273 12 3175 3175 3175 3155 3041 2926 2807 2684 2556 2423 2285 2003 1730 1483 1269 1090 273 12x 273 12,5 3301 3301 3301 3280 3161 3040 2917 2789 2655 2516 2373 2079 1795 1537 1315 1129 273 12,5

323,9 5 1427 1427 1427 1427 1414 1374 1335 1294 1253 1210 1166 1074 977,9 880,5 786,2 698,5 323,9 5x 323,9 6 1934 1934 1934 1934 1902 1844 1787 1728 1667 1604 1539 1403 1262 1123 991,4 872,4 323,9 6

323,9 6,3 2029 2029 2029 2029 1994 1934 1874 1812 1748 1682 1614 1471 1323 1177 1039 914,1 323,9 6,3x 323,9 8 2562 2562 2562 2562 2517 2441 2364 2285 2204 2121 2034 1852 1664 1479 1304 1147 323,9 8x 323,9 10 3183 3183 3183 3183 3125 3029 2933 2834 2733 2628 2519 2292 2057 1825 1608 1413 323,9 10

323,9 12 3795 3795 3795 3795 3723 3609 3493 3375 3253 3127 2997 2723 2442 2165 1905 1673 323,9 12x 323,9 12,5 3947 3947 3947 3947 3871 3752 3632 3508 3382 3250 3114 2829 2536 2247 1978 1735 323,9 12,5

1) d t Nb.Rd (kN) d tmm mm Lc (m) mm mm

x 0 0,5 1,0 1,5 2 2,5 3 3,5 4 4,5 5 6 7 8 9 10

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Table 9.2.4 Buckling resistance values for circular spirally welded hollow sections of steel grade S355J2H (fy = 355 N/mm2) in buckling category c, continued.(Technical delivery conditions to be agreed when ordering)


d t Nb.Rd (kN) d tmm mm Lc (m) mm mm

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16355,6 5,6 1757 1757 1757 1669 1579 1484 1384 1278 1169 1060 955 857 768 689 619 558 504 355,6 5,6355,6 6 2127 2127 2114 1999 1882 1758 1625 1487 1345 1207 1077 959 854 762 681 612 551 355,6 6355,6 6,3 2231 2231 2218 2097 1974 1843 1704 1559 1411 1265 1129 1005 895 798 714 641 578 355,6 6,3355,6 8 2819 2819 2801 2648 2491 2325 2149 1963 1775 1591 1419 1262 1123 1001 895 804 724 355,6 8355,6 10 3504 3504 3479 3288 3091 2884 2663 2431 2196 1966 1752 1557 1385 1234 1103 990 891 355,6 10355,6 12 4180 4180 4148 3919 3683 3434 3168 2890 2608 2334 2078 1846 1640 1461 1306 1171 1055 355,6 12355,6 12,5 4348 4348 4314 4075 3829 3570 3293 3003 2709 2424 2157 1916 1702 1516 1355 1215 1094 355,6 12,5406,4 6 2141 2141 2141 2076 1981 1883 1781 1673 1560 1445 1328 1215 1107 1007 915 833 759 406,4 6406,4 6,3 2256 2256 2256 2187 2086 1983 1875 1761 1642 1519 1397 1277 1163 1057 961 874 796 406,4 6,3406,4 8 3231 3231 3231 3102 2948 2788 2619 2441 2255 2067 1881 1704 1540 1390 1256 1137 1031 406,4 8406,4 10 4019 4019 4019 3855 3663 3462 3250 3027 2795 2560 2329 2108 1903 1718 1551 1403 1273 406,4 10406,4 12 4798 4798 4798 4600 4368 4127 3873 3605 3326 3044 2767 2503 2259 2037 1839 1663 1508 406,4 12406,4 12,5 4992 4992 4992 4784 4543 4292 4027 3748 3457 3163 2874 2600 2346 2115 1909 1726 1565 406,4 12,5457 6 2383 2383 2383 2348 2255 2161 2063 1962 1855 1744 1631 1517 1404 1295 1192 1096 1007 457 6457 6,3 2513 2513 2513 2475 2377 2277 2174 2066 1953 1836 1716 1595 1476 1361 1252 1151 1058 457 6,3457 8 3642 3642 3642 3554 3401 3245 3081 2910 2730 2545 2356 2170 1990 1820 1662 1518 1388 457 8457 10 4532 4532 4532 4421 4229 4033 3829 3614 3390 3157 2922 2689 2465 2253 2057 1878 1716 457 10457 12 5414 5414 5414 5278 5048 4813 4567 4310 4040 3761 3478 3199 2930 2677 2443 2230 2037 457 12457 12,5 5633 5633 5633 5491 5252 5006 4750 4482 4200 3909 3615 3325 3045 2781 2538 2316 2116 457 12,5508 6 2622 2622 2622 2616 2525 2433 2339 2242 2141 2036 1928 1816 1704 1592 1483 1378 1279 508 6508 6,3 2767 2767 2767 2759 2663 2566 2466 2364 2257 2145 2030 1912 1793 1675 1560 1449 1345 508 6,3508 8 3586 3586 3586 3570 3443 3315 3184 3048 2907 2760 2607 2452 2295 2140 1990 1846 1710 508 8508 10 5049 5049 5049 4990 4799 4606 4407 4199 3982 3756 3523 3288 3054 2827 2610 2406 2217 508 10508 12 6035 6035 6035 5961 5732 5500 5261 5011 4751 4479 4200 3917 3637 3365 3105 2861 2635 508 12508 12,5 6280 6280 6280 6203 5964 5722 5473 5213 4941 4658 4367 4073 3781 3497 3227 2973 2738 508 12,5559 6 2856 2856 2856 2856 2788 2699 2608 2514 2418 2318 2214 2108 1998 1888 1778 1669 1564 559 6559 6,3 3015 3015 3015 3015 2942 2847 2751 2652 2550 2444 2334 2221 2105 1988 1871 1756 1645 559 6,3559 8 3916 3916 3916 3916 3814 3688 3561 3430 3295 3154 3008 2858 2705 2550 2396 2246 2100 559 8559 10 5566 5566 5566 5560 5369 5177 4981 4779 4568 4349 4122 3889 3654 3419 3190 2969 2759 559 10559 12 6655 6655 6655 6645 6416 6186 5951 5707 5454 5191 4918 4639 4356 4075 3800 3536 3284 559 12559 12,5 6926 6926 6926 6915 6677 6437 6192 5939 5675 5401 5117 4826 4532 4239 3953 3677 3416 559 12,5

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The calculated resistance values are design values (section 2.1) based on the default value (1.1) of thepartial safety factor of material γγγγM1 used in Eurocode 3. The partial safety factor values may differ in eachcountry. National values must be checked from the NAD (National Application Document) of the country inquestion.

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d

t

610 8 4242 4242 4242 4242 4179 4055 3931 3804 3673 3538 3399 3255 3106 2955 2803 2651 2501 610 8610 10 5398 5398 5398 5398 5309 5149 4989 4824 4655 4481 4300 4113 3921 3726 3529 3333 3140 610 10610 12 7276 7276 7276 7276 7100 6871 6638 6400 6153 5898 5633 5359 5080 4797 4516 4239 3970 610 12610 12,5 7572 7572 7572 7572 7389 7150 6908 6659 6403 6136 5860 5575 5284 4989 4696 4408 4128 610 12,5610 14,2 8578 8578 8578 8578 8366 8095 7820 7538 7246 6943 6629 6305 5974 5640 5306 4979 4662 610 14,2660 8 4555 4555 4555 4555 4530 4408 4286 4162 4035 3905 3771 3632 3489 3343 3194 3043 2893 660 8660 10 5806 5806 5806 5806 5765 5608 5450 5289 5125 4956 4781 4601 4416 4226 4033 3838 3644 660 10660 12 7884 7884 7884 7884 7770 7541 7310 7075 6833 6584 6325 6058 5784 5504 5222 4940 4661 660 12660 12,5 8206 8206 8206 8206 8087 7848 7608 7363 7112 6851 6582 6304 6019 5727 5433 5139 4850 660 12,5660 14,2 9298 9298 9298 9298 9159 8889 8616 8338 8051 7756 7450 7134 6809 6478 6144 5810 5481 660 14,2711 8 4869 4869 4869 4869 4869 4762 4641 4520 4397 4271 4141 4008 3871 3730 3585 3439 3290 711 8711 10 6218 6218 6218 6218 6218 6069 5913 5756 5596 5432 5263 5089 4911 4727 4539 4348 4156 711 10711 12 8504 8504 8504 8504 8453 8225 7995 7763 7525 7280 7028 6768 6500 6225 5945 5663 5380 711 12711 12,5 8852 8852 8852 8852 8799 8560 8321 8079 7831 7576 7313 7042 6763 6476 6185 5891 5597 711 12,5711 14,2 10032 10032 10032 10032 9968 9697 9426 9150 8869 8579 8280 7972 7655 7329 6998 6663 6329 711 14,2762 8 5178 5178 5178 5178 5178 5110 4991 4872 4751 4628 4503 4374 4242 4106 3966 3824 3679 762 8762 10 6623 6623 6623 6623 6623 6524 6370 6215 6058 5898 5734 5566 5394 5216 5034 4849 4660 762 10762 12 8068 8068 8068 8068 8068 7936 7746 7555 7361 7164 6962 6754 6540 6321 6096 5866 5633 762 12762 12,5 8429 8429 8429 8429 8429 8289 8090 7889 7686 7479 7267 7050 6826 6596 6360 6120 5875 762 12,5762 14,2 10766 10766 10766 10766 10766 10506 10235 9962 9683 9399 9106 8804 8494 8175 7849 7517 7183 762 14,2813 8 5481 5481 5481 5481 5481 5451 5334 5217 5098 4978 4856 4731 4603 4472 4338 4200 4060 813 8813 10 7024 7024 7024 7024 7024 6973 6820 6667 6513 6357 6197 6034 5866 5695 5519 5338 5155 813 10813 12 8565 8565 8565 8565 8565 8493 8304 8115 7925 7731 7534 7332 7125 6912 6694 6471 6243 813 12813 12,5 8951 8951 8951 8951 8951 8872 8675 8477 8278 8075 7868 7656 7439 7216 6988 6754 6515 813 12,5813 14,2 11500 11500 11500 11500 11500 11315 11045 10773 10497 10216 9928 9633 9329 9017 8697 8370 8039 813 14,2813 16 12929 12929 12929 12929 12929 12717 12412 12105 11795 11478 11153 10820 10478 10125 9765 9396 9023 813 16914 10 7800 7800 7800 7800 7800 7800 7694 7544 7394 7243 7090 6935 6776 6614 6448 6279 6105 914 10914 12 9534 9534 9534 9534 9534 9534 9391 9206 9020 8833 8643 8450 8253 8052 7846 7635 7420 914 12914 12,5 9967 9967 9967 9967 9967 9967 9815 9621 9426 9230 9031 8828 8622 8411 8195 7974 7748 914 12,5914 14,2 11440 11440 11440 11440 11440 11440 11254 11030 10804 10577 10347 10112 9873 9628 9378 9121 8859 914 14,2914 16 14567 14567 14567 14567 14567 14522 14217 13912 13605 13294 12977 12654 12324 11985 11638 11282 10919 914 16

1016 10 8562 8562 8562 8562 8562 8562 8552 8406 8259 8112 7964 7814 7663 7508 7351 7191 7027 1016 101016 12 10490 10490 10490 10490 10490 10490 10463 10281 10099 9916 9732 9546 9357 9165 8969 8770 8566 1016 121016 12,5 10972 10972 10972 10972 10972 10972 10941 10749 10559 10367 10174 9979 9780 9579 9373 9163 8949 1016 12,51016 14,2 12610 12610 12610 12610 12610 12610 12563 12342 12120 11898 11674 11448 11218 10983 10745 10501 10252 1016 14,21016 16 14343 14343 14343 14343 14343 14343 14278 14025 13771 13517 13260 13000 12736 12467 12193 11913 11627 1016 161219 10 10017 10017 10017 10017 10017 10017 10017 10017 9908 9767 9626 9485 9343 9200 9055 8908 8759 1219 101219 12 12328 12328 12328 12328 12328 12328 12328 12328 12172 11997 11821 11644 11466 11287 11105 10922 10735 1219 121219 12,5 12906 12906 12906 12906 12906 12906 12906 12906 12739 12554 12369 12184 11997 11808 11618 11425 11228 1219 12,51219 14,2 14872 14872 14872 14872 14872 14872 14872 14872 14663 14448 14233 14018 13800 13581 13359 13133 12905 1219 14,21219 16 16953 16953 16953 16953 16953 16953 16953 16947 16700 16453 16206 15958 15708 15455 15200 14940 14677 1219 16

d t Nb.Rd (kN) d tmm mm Lc (m) mm mm

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

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Appendix 9.3 Calculation tables for truss joints

This appendix includes formulae for calculating uniplanar lattice structure joints, based onEurocode 3 (ENV-1993-1-1:1992) annex K. The tables also include values from references [2]and [3].


289

Table Joint type Chord Brace member

9.3.1 T-, Y- and X-joint Square RectangularRectangular Square

Circular9.3.2 Gap Square Rectangular

N-, K and KT-joint Rectangular SquareCircular

9.3.3 Overlap Square RectangularN-, K and KT-joint Rectangular Square

Circular9.3.4 T-, Y- and X-joint Circular Circular

9.3.5 Gap Circular CircularN-, K and KT-joint

9.3.6 Overlap Circular CircularN-, K and KT-joint

9.3.7 Gap I-profile RectangularN- and K-joints and SquareT-, Y- and X-joint Circular

9.3.8 Overlap I-profile RectangularN- and K-joint Square

Circular9.3.9 T and X-joint Square Rectangular

bending resistance Rectangular Square9.3.10 T-, Y- and X-joint Circular Circular

bending resistance9.3.11 Plate joint Square Plate

Rectangular9.3.12 Plate joint Circular Plate

9.3.13 T-, Y- and X-joint Square Rectangularwith reinforced chord face Rectangular Square

Circular9.3.14 T-, Y- and X-joint Square Rectangular

with reinforced chord web Rectangular SquareCircular

9.3.15 Gap Square RectangularN-, K and KT-joint Rectangular Squarewith reinforced chord face Circular

9.3.16 Gap Square RectangularN-, K and KT-joint, Rectangular Squarewith reinforced chord web Circular

9.3.17 Overlap Square RectangularN- and K-joint, reinforced Rectangular Squarewith intermediate plate Circular

Symbols

Ai is the cross-sectional area of the chord memberAv is the shear area of the chordE is the modulus of elasticity for steelMip.i.Rd is the design value of the joint bending resistance parallel to the plane of

the joint (table 9.3.9)Mip.i.Sd is the design value of the bending resistance perpendicular to the plane of

the joint (table 9.3.9)Mop.i.Rd is the design value of the bending resistance perpendicular to the plane of

the joint (table 9.3.9)Mop.i.Sd is the design value of the bending moment perpendicular to the plane of

the joint (table 9.3.9)Ni.Rd is the design value of the joint’s resistance to axial forceNi.Sd is the design value of the brace member axial forceWel.i is the elastic section modulus of the brace memberWpl.i is the plastic section modulus of the brace memberbi is the width of the brace memberb0 is the width of the chordbeff is the effective width of the brace member in calculating the brace member

resistancebe(ov) is the effective width of the overlapping brace member in overlapped jointbe.p is the effective width of the brace member in calculating the shear failure

resistance of the chordbp is the width of the bracingbw is the effective width of the chord web with an I profile chorddi is the diameter of the circular brace memberd0 is the diameter of the circular chorddw is the height of the chord when the chord is an I profilee is the eccentricity of the jointfb is the buckling length of the chord webfyi is the design value of the brace member yield strength

(In Eurocode 3 Annex K, fyi is the design value of yield resistance)fy0 is the design value of the chord yield strength

(In Eurocode 3 Annex K, fy0 is the design value of yield resistance)g is the theoretical gap of the joint (table 9.3.2)ga is the actual gap of the joint (table 9.3.2)hi is the height of the brace memberh0 is the height of the chordi is the number of the brace member (1, 2, 3)kg is the reduction factor for the resistance of gap and overlap joints with circular

hollow sectionskm is the reduction factor for the resistance of joints between plates and square or

rectangular hollow sectionskn is the reduction factor for the resistance of joints with square or rectangular chordskp is the reduction factor for the resistance of joints with circular hollow

sectionsM0.Sd is the design value for the chord bending momentm is the number of brace members


290

n is the compression stress to yield resistance with square and rectangular chords

np is the compression stress due to Np.Sd and M0.Sd to the yield resistance with circular hollow sections

N0.Sd is the design value with the greatest absolute value of the chord axial forceNp.Sd is the normal force passing through the chord jointr is the rounding radius of the I profiletf is the thickness of the I profile flangeti is the wall thickness of the brace membert0 is the wall thickness of the chordtp is the thickness of the bracingtw is the thickness of the I profileβ is the brace member diameter or width to the chord diameter or width.

The width of the brace member is taken as an average if the joint contains several brace members.

βp is the brace member width to the bracing width

βp = bi/ bp

γ is the chord width to chord wall thickness divided by twoγ = 0,5d0/ t0,

= 0,5b0/ t0 or= 0,5b0/ tf

γMj is the partial safety factor for lattice structuresη is the brace member height to the chord width

η = hi/ b0 ηp is the brace member height to the bracing width

ηp = hi/ bpλov is the relative magnitude of the overlap

λov = q · sin(θi)/ hiθi is the smaller angle between brace member and chord

In all tables of this appendix, the following limitations apply:

• fy ≤ 355 N/ mm2

• t0 ≥ 2,5 mm• ti ≥ 2,5 mm• θi ≥ 30° (also for the angle between brace members)• -0,55 ≤ e/h0 ≤ 0,25 or -0,55 ≤ e/d0 ≤ 0,25• the cross-sections of compression elements of members must belong to Class 1 or 2 when subjected to bending only (section 2.2)

• the brace member ends must not be flattened

Depending on the joint type, the tables may include additional limitations.


291

β β=⋅

=⋅

= =∑ ∑b

m b

d

m d

ii

m

ii

m

1

0

1

0 tai

Table 9.3.1 Resistance of T, Y and X joints. Chords are square or rectangular hollow sections. Brace members are square, rectangular or circular hollow sections[1], [3].


292

��

N1

M0

N0N0

h 0M0

h1

h1

N1

t1

b1

t0

b0

θ1


293

Resistance Parametersβ ≤ 0,85, chord face yield

β = bi / b0

η = hi / b0

Tension chord:kn = 1Compression chord:kn = 1,3- (0,4 / β)n ≤ 1

0,85 < β < 1,0Use the values β = 0,85 and β =1,0 when calculatingthe resistance. Resistance is defined by linearinterpolation based on the original value of β.β = 1,0, chord web buckling or yield

Tension chord: fb = fy0

Compression chord:fb = χ· fyo (T and Y joints)fb = 0,8χ(sinθi)fy0 (X joints)χ = reduction factor for bucklingusing buckling curve when theslenderness is:

0,85 ≤ β ≤ 1 - (1/γ), punching shear failure of the chord face

β > 0,85, brace member failure

β = 1,0, X joints when θ < 90°, chord shear Av = 2h0 · t0

Validity areaSquare and rectangularbrace members:Brace members ingeneral:

bi / b0, hi / b0 ≥ 0,25 3)

0,5 ≤ hi / bi ≤ 2Tension brace member:bi / ti, hi / ti ≤ 35Compression bracemember:bi / ti, hi / ti ≤ 35

Circular bracemembers:Brace members in general:0,4 ≤ di / b0 ≤ 0,8Tension bracemember:10 ≤ di / ti ≤ 50 2)

Compression bracemember:10 ≤ di / ti ≤ 50 2)

Chords:

NiRdi i Mj M

. ( )sin sin,=

⋅ ⋅−

+ −

⋅

k f tn y0 02

012

4 111

β θηθ

βγ γ

nNA f

MW f

M Mj Sd

y

Sd

y=

⋅⋅

+⋅

γ γ0 0

0 0

0

0 011,. .

NiRdi

i

i Mj M. sin sin

,= ⋅ +

⋅

f t htb 00

0

210

11θ θ γ γ

λθ π

= −

3 46 21

,)

ht

f

E(sin0

0

y0 3)

i

b ht

i i

i

+ ≥ )25 1

bt

ht

i

i

i

i, ,≤ 1 25

Efyi

3)

dti

i≤ 1 5,

Efyi

3)

b ht

hb

bt

ht

0 0

0

0

0

0

0

0

0

25

0 5 2

35

+ ≥

≤ ≤

≤

,

,

NiRdi

i

i Mj M.

sin sin,=

⋅+

⋅

f t hby0 0

ep3

22

11

0θ θ γ γ

N f t h t biRd yi i i i effMj M

.,= ⋅ − +( )⋅

2 4 211

0γ γ

NiRdi Mj M

.sin

,=⋅

⋅f A

3y0 v

θ γ γ11

0

With circular brace members, the resistance values are multiplied by π / 4, and bi and hi are replaced with thediameter di.fy ≤ 355 N/ mm2, 30° ≤ θi ≤ 90°, t0 ≥ 2,5 mm, ti ≥ 2,5 mm1 These limit values are defined in reference [2].2) Eurocode 3 does not define a lower limit for this parameter.3) These limit values are defined in reference [3].

bb t f

b t fbeff

y

y=

⋅ ⋅⋅ ⋅

≤10 1 0

20

0 1 11

bt bb

b

bt

ep = ⋅ ≤

=

10

2

0 1

01

0

0γ

Table 9.3.2 Resistance of gap N, K and KT lattice joints. Chords are square or rectangular hollow sections. Brace members are square, rectangular or circular hollow sections [1], [3].


294

��

��

h2

h3

h 1

h2

h 1

N1

M0N0

h 0

θ2

N2

θ1

M0

eg

b1, 2

t1, 2

t0

b0

ga

g

ga

g

t 0

N1

N3N2

θ2θ1θ3

M0M0N0

e

g1 g2

Det 1

Det 1

Det 1

θ ≤ 60° θ > 60°


295

Resistance Parameters

m is the number of brace membersγ = b0 /(2t0)Tension chord:kn = 1Compression chord:kn = 1,3- (0,4/ β)n ≤ 1

Chord shear

When VSd > 0,5 Vpl.Rd

When VSd ≤ 0,5 Vpl.Rd

Av = (2h0 + α · b0)t0 (square andrectangular brace members) Av = 2h0 · t0 (circular brace members)

VSd is the chord shear failure atjoint

Brace member failure

In a KT joint, also check the following conditions:N1.Rd · sinθ1 ≥ N1.Sd · sinθ1 + N3.Sd · sinθ3 N1.Rd · sinθ1 ≥ N2.Sd · sinθ2

β ≤ 1- (1/ γ), chord shear failure

Validity areaSquare and rectangularbrace members:Brace members ingeneral:

β ≥ 0,350,5 ≤ hi / bi ≤ 2

Tension brace member:bi / ti, hi / ti ≤ 35

Compression bracemember:bi / ti, hi / ti ≤ 35

Circular brace members:Brace members in general:0,4 ≤ di / b0 ≤ 0,8Tension Brace member:10 ≤ di / ti ≤ 50 2)


Chords:0,5 ≤ h0 / b0 ≤ 2b0 / t0, h0 / t0 ≤ 35

Gap:g / b0 ≥ 0,5(1-β) g / b0 ≤ 1,5(1-β)g ≥ t1+ t2ga / t0 ≥ 1,51)

NiRdi Mj M

. ,sin

,=⋅

+

⋅

⋅= =∑ ∑

8 9110

2

0 0

f tb h

2m bky0

ii 1

m

ii 1

m

nθγ

γ γβ η=

⋅==

∑b

m b b

ii 1

m

0 0,

hi

NiRdi Mj M

.sin

,=⋅

⋅f A

3y0 v

θ γ γ11

0

αγ

=+

=⋅⋅

1

14

30g

t

Vf A

32

02

pl.Rdy0 v,

M

η β, , b

100t 0

0

3)≥ +0 1

bt

ht

Ef

i

i

i

i yi, ,≤ 1 25

3)

dt

Ef

i

i yi≤ 1 5,

3)

b ht

0 0

0

125+ ≥ )

N f t h t b biRd yi i i i i effMj M

.,= ⋅ − + +( )⋅

2 411

0γ γ

NiRdi

i

i Mj M.

sin sin,=

⋅+

⋅

f t hb + by0 0

i ep3

2 11

0θ θ γ γ

bb t f

b t fbeff

i y

i yii=

⋅ ⋅⋅ ⋅

≤10 0

20

0

Chord face yield (in KT joints, select the distance between the brace memberssubjected to the greatest load as the gap)

nNA f

MW f

M Mj Sd

y

Sd

y=

⋅⋅

+⋅

γ γ0 0

0 0

0

0 011,. .

With circular brace members, the resistance values are multiplied by π/ 4, and bi and hi are replaced with thediameter di.If g / b0 > 1,5(1-β), the joint is treated as two separate T or Y joints in calculations.fy ≤ 355 N/ mm2 , 30° ≤ θi ≤ 90°, t0 ≥ 2,5 mm, ti ≥ 2,5 mm1) These limit values are defined in reference [2].2) Eurocode 3 does not define a lower limit for this value3) These limit values are defined in reference [3].

bt bib

b

bt

ep = ⋅ ≤

=

10

2

0

01

0

0γ

N A Af

Rd vy

M0 0

0

01. = − −

2VV

Sd

pl.Rd

2

γ

N Af

Rdy

M0 0

0

0. =

γ

b ht

i i

i

+ ≥ 25

Table 9.3.3 Resistance of overlap N, K and KT lattice joints. Chords are square orrectangular hollow sections. Brace members are square, rectangular orcircular hollow sections [1], [3]. With in the range of validity presented in thistable, only brace member failure is considered a failure mode. Theresistance needs to be checked for the overlapping brace member only.


296

��

��h

2h 1

h2

h 1

h3

N1 N2

θ2θ1

h 0

M0 M0N0

b1, 2

t1, 2

t0

b0

N1 N2

N3

θ2θ1θ3

q1 q2

M0N0M0

-e

q

-e


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N and K joints

KT joints

(i = 1,2)

0,5 ≤ λov < 0,8, brace member failureN and K joints

KT joints

(i = 1,2)

λov ≥ 0,8, brace member failureN and K joints

KT joints

(i = 1,2)

Validity areaSquare and rectangularbrace members:Brace members in general:

0,5 ≤ hi / bi ≤ 2

N and K joints:t1 / t2 ≤ 1,0b1 / b2 ≥ 0,75KT joints:ti / t3 ≤ 1,0 (i = 1,2)bi / b3 ≥ 0,75 (i = 1,2)Tension brace member:bi / ti, hi / ti ≤ 35Compression brace mem-ber:bi / ti, hi / ti ≤ 35

Circular brace members:Brace members in general:0,4 ≤ di / b0 ≤ 0,8Tension brace member: 10 ≤ di / ti ≤ 50 2)

Compression brace mem-ber:10 ≤ di / ti ≤ 50 2)

Chords:0,5 ≤ h0 / b0 ≤ 2b0 / t0, h0 / t0 ≤ 35

Overlap:

0,25 ≤ λov ≤ 1,0

Overlap:

0,25 ≤ λov ≤ 1,0

N f t h t b bRd y ov eff e ovMj M

1 1 1 1 10

2 2 411

.,= ⋅ −( ) + +[ ] ⋅( )λ

γ γ

N f t h t b bRd y eff e ovMj M

1 1 1 1 10

2 411

.,= ⋅ − + +[ ] ⋅( ) γ γ

bb

hb

i i

0 00 25, ,≥ 3)

bt

ht

i

i

i

i, ,≤ 11

Efyi

3)

dti

i≤ 1 5,

Efyi

3)

b ht

0 0

0

125+ ≥ )

N f t h t b bRd y e ovMj M

1 1 1 1 1 10

2 411

.,= ⋅ − + +( ) ⋅( ) γ γ

0,25 ≤ λov < 0,5, brace member failure

b ht

i i

i

+ ≥ )25 1

With circular brace members, the resistance values are multiplied by π/ 4, and bi and hi are replaced with thediameter di. fy ≤ 355 N/ mm2 , 30° ≤ θi ≤ 90°, t0 ≥ 2,5 mm, ti ≥ 2,5 mm1) These limit values are defined in reference [2].2) Eurocode 3 does not define a lower limit for this value.3) These limit values are defined in reference [3].

λθ

λθ

ov

effy

y

e ovy

y

ovi

i

effi y

i yii

e ov

q

h

bb t f

b t fb

bb t f

b t fb

q

h

bb t f

b t fb

b

=⋅ ( )

=⋅ ⋅⋅ ⋅

≤

=⋅ ⋅⋅ ⋅

≤

=⋅ ( )

=⋅ ⋅⋅ ⋅

≤

( )

sin

sin(i = 1,2)

(i = 1,2)

1

i

1

1 02

0

0 1 11

1 22

2

2 1 11

02

0

0

10

10

10

(( ) =⋅ ⋅⋅ ⋅

≤10 3

23

31

b t f

b t fbi y

i yi(i = 1,2)

N f t h t b biRd yi i ov i i eff e ovMj M

.,= ⋅ −( ) + +[ ] ⋅( )2 2 4

11

0λ

γ γ

N f t h t b biRd yi i i i eff e ovMj M

.,= ⋅ − + +[ ] ⋅( )2 4

11

0γ γ

N f t h t b biRd yi i i i i e ovMj M

.,= ⋅ − + +( ) ⋅( )2 4

11

0γ γ


N and K joint:

KT joint:

Table 9.3.4 Resistance of T, Y, and X joints. Chords and brace members are circular hollow sections [1].


298

��

��

��

��

��

��

��

N1

d1 θ1

d0

t1

t0NpM0 M0

Np


299


β = di / d0

γ = d0 / (2 t0)Tension chord:kp = 1Compression chord:

kp = 1,0- 0,3(np + np2) ≤ 1

X joints, chord face yieldβ = di / d0

γ = d0 / (2 t0)Tension chordekp = 1Compression chord:

di ≤ d0 - 2t0, T, Y and X joints, punching shear failure of the chord face

Validity areaT, Y and X joints:

0,2 ≤ di / d0 ≤ 1,0

10 ≤ di / ti ≤ 50

T and Y joints:

10 ≤ d0 / t0 ≤ 50

X joints:

10 ≤ d0 / t0 ≤ 40

NiRdi Mj M

. sin, ,

,=⋅

+( ) ⋅f t

ky0 0,2p

02

2

02 8 14 2

11θ

β γγ γ

NiRdi Mj M

. sin,,

,=⋅

−

⋅

f tky0

p0

2

0

5 21 0 81

11θ β γ γ

nN

A fM

W fpM Mj p Sd

y

Sd

y=

⋅⋅

+⋅

γ γ0

0 0

0

0 011,. .

k n n

nN

A fM

W f

p p p

pM Mj p Sd

y

Sd

y

= − +( ) ≤

=⋅

⋅+

⋅

10 0 3 1

11

2

0

0

0

0 0

, ,

,. .γ γ

T and Y joints, chord face yield

Nd

iRdi

Mj M.

sin

sin

,=⋅ ⋅ ⋅ +

⋅

f ty0 02

03

1

2

11π θθ γ γ

fy ≤ 355 N/ mm2 , 30° ≤ θi ≤ 90°, t0 ≥ 2,5 mm, ti ≥ 2,5 mm

Table 9.3.5 Resistance of gap N, K and KT joints. Chords and brace members are circular hollow sections [1].


300

��

��

��

��

�

��

�

��

��

��

��

��

��

d 1

d2

d3

d2

d1

N1 N2 d1, 2

t1, 2

t0

d0

M0Np

Det 1

g

θ2θ1

M0

Det 1

θ2θ1

θ3

e

M0M0Np

g1 g2

ga

g

ga

g

t 0

Det 1

N1

N3N2

e

θ ≤ 60° θ > 60°


301


Tension member:N2.Rd = N1.Rd [sin(θ1)/ sin(θ2)]


m is the number of brace mem-bers

di ≤ d0 - 2t0, punching shear failure of the chord face

Validity area0,2 ≤ di / d0 ≤ 1,0

10 ≤ di / ti ≤ 50

10 ≤ d0 / t0 ≤ 50

Gap:

g ≥ t1+ t2

ga / t0 ≥ 1,5 1)

N kiRd pMj M

. sin, ,

,=⋅

+( ) ⋅⋅

f tky0

g0

2

1 01 8 10 2

11θ

βγ γ

k n n

nN

A fM

W f

k

e

d

m d

p p p

pM Mj p Sd

y

Sd

y

g gt

ii

m

= − +( ) ≤

=⋅

⋅+

⋅

= + ⋅

+

=⋅

−

=∑

10 0 3 1

11

10 024

1

2

0

0 0

0

0 0

0 212

2133

1

0

, ,

,

,

. .

,,

,

γ γ

γ γ

β00

Chord face yield

Nd

iRdi

Mj M.

sin

sin

,=⋅ ⋅ ⋅ +

⋅

f ty0 02

03

1

2

11π θθ γ γ

fy ≤ 355 N/ mm2 , 30° ≤ θi ≤ 90°, t0 ≥ 2,5 mm, ti ≥ 2,5 mm

1) These limit values are defined in reference [2].


Table 9.3.6 Resistance of overlap N, K and KT joints. Chords and brace members are circular hollow sections [1].


302

��

��

��

��

��

��

��

��

��

��

��

�

��

��

d3d2

d 1

d 1

d2

d1, 2

t1, 2

t0

d0

N1 N2

N3

M0 M0

q1 q2

-e

θ2θ1

Np

θ3

-eN1 N2

q

θ2θ1

M0NpM0


303


Compression member:

Tension member:m is the number of brace membersγ = d0 / (2 t0)Tension chord:kp = 1Compression chord:

fy ≤ 355 N/ mm2, 30° ≤ θi ≤ 90°, t0 ≥ 2,5 mm, ti ≥ 2,5 mm

Validity area0,2 ≤ di / d0 ≤ 1,0

10 ≤ d0 / t0 ≤ 50

10 ≤ d1 / t1 ≤ 50

Overlap:λov ≥ 0,25

N kRd pMj M

10

2

1 01 8 10 2

11. sin

, ,,=

⋅+( ) ⋅

⋅f t

ky0gθ

βγ γ

k n n

nN

A fM

W f

k

e

p p p

pM Mj p Sd

y

Sd

y

gqt

= − +( ) ≤

=⋅

⋅+

⋅

= + ⋅

+

− −

10 0 3 1

11

10 024

1

2

0

0 0

0

0 0

0 212

2133

0

, ,

,

,

. .

,,

,

γ γ

γ γ

=⋅ ( )λ

θov

i

q

d

sin i

Chord face yield

N NRd Rd2 11

2. .

sin

sin= ( )

( )θθ

β =⋅

=∑d

m d

ii

m

1

0


Table 9.3.7 Resistance of T, Y and X joints and gap N and K joints. Chords are Iprofiles. Brace members are square, rectangular or circular hollow sections[1].


304

��

��

��

�

�

�

h 1

h2

h1

b1, 2

t1, 2

b0

r

r

tw

t f

b1, 2

t1

b0

tw

t f

d wd w

N1

θ1

N1 N2

ge

θ2θ1

ga

g

ga

g

t f

Det 1

Det 1

θ ≤ 60° θ > 60°


305

Resistance ParametersT, X and Y joints, chord web yield

Square and rectangular bracemembers:bw = (hi / sinθi)+ 5(tf + r)bw ≤ 2ti + 10(tf + r)Circular brace members:bw = (di / sinθi)+ 5(tf + r)bw ≤ 2ti + 10(tf + r)

T, X and Y joints, brace member failure

N and K joints, chord web yield

N and K joints, brace member failure

m is the number of brace membersChecking is not necessary if thefollowing conditions are true:β ≤ 1,0 - (0,03γ) g/ tf ≥ 20 - (28β), Square and rectangular bracemembers:0,75 ≤ b1/ b2 ≤ 1,33Circular brace members:0,75 ≤ d1/ d2 ≤ 1,33

N and K joints, chord shearAv = A0 - (2- α)b0 · tf + (tw + 2r)tf

Square and rectangular bracemembers:

Circular brace members: α = 0

Validity areaAll joint types in the table:Brace members in general:

Compression brace member:

Tension brace member:hi / ti ≤ 35bi / ti ≤ 35

10 ≤ di / ti ≤ 50 2)

Chords:

T and Y joints andN and K joints:Brace members:hi / bi = 1Chords:

X joints:Brace members:0 ,5 < hi / bi ≤ 2,0Chords:

Gap:

ga ≥ 1,5 tf 1)

Nf t b

iRdy w w

i Mj M. sin

,=⋅ ⋅

⋅0

0

11θ γ γ

N f t biRd yi i effMj M

.,= ⋅ ⋅⋅

211

0γ γ b t rf

ft beff w

yo

yif i= + +

≤2 7

b ht

i i

i

+ ≥ )25 1

bt

ht

Ef

dt

Ef

i

i

i

i yi

i

i yi

, ,

,

≤

≤ ≤

11

10 1 5

bt

Ef

d mm

f y

w

0

00 75

400

≤

≤

,

Nf t b

iRdy w w

i Mj M. sin

,=⋅ ⋅

⋅0

0

11θ γ γ

Nf A

iRdy v

i Mj M.

sin

,=⋅

⋅0

03

11

θ γ γ

b t rf

ft b

b

m btai

d

m bbt

eff wyo

yif i

ii

m

ii

m

f

= + +

≤

=⋅ ⋅

== =∑ ∑

2 7

21

0

1

0

0β γ ,

fy ≤ 355 N/ mm2 , 30° ≤ θi ≤ 90°, ti ≥ 2,5 mm 1) This limit value is defined in reference [2]2) Eurocode 3 does not define a lower limit for this value.

N f t biRd yi i effMj M

.,= ⋅ ⋅⋅

211

0γ γ

dt

Ef

w

w y≤ 1 5

0,

dt

Ef

w

w y≤ 1 2

0,

α =+

1

14

3

2

2g

tf

0r di

or di

Table 9.3.8 Resistance of overlap N, K and KT joints. Chords are I profiles. Bracemembers are square, rectangular or circular hollow sections [1]. Theresistance needs to be checked for the overlapping brace member only.


306

��

��

h 1

h2

b1, 2

t1, 2

b0

twt f

r

d w

N1 N2

θ2θ1

-e

q


307




λov ≥ 0,8, brace member failure

Validity areaBrace members in general:0,5 < hi / bi ≤ 2,0b1/ b2 ≥ 0,75

Compression brace mem-ber:

Tension brace member:hi / ti ≤ 35bi / ti ≤ 35

10 ≤ di / ti ≤ 50 2)

Chords:

Overlap:

0,25 ≤ λov ≤ 1,0

N f t h t b bRd y ov eff e ovMj M

1 1 1 1 10

2 2 411

.,= ⋅ −( ) + +[ ] ⋅( )λ

γ γ

N f t h t b bRd y e ovMj M

1 1 1 1 1 10

2 411

.,= ⋅ − + +[ ] ⋅( ) γ γ

N f t h t b bRd y eff e ovMj M

1 1 1 1 10

2 411

.,= ⋅ − + +[ ] ⋅( ) γ γ

dt

bt

w

w

f

≤

≤

≤

1 2

400

0 750

,

,

Ef

d mm

Ef

y0

w

y0

b ht

i i

i

+ ≥ )25 1

bt

ht

dt

i

i

i

i

i

i

, ,

,

≤

≤ ≤

11

10 1 5

Ef

Ef

yi

yi

λ θov

eff wy

yif

e ovy

y

qh

b t rf

ft b

bb t f

b t fb

= ⋅

= + + ≤

=⋅ ⋅⋅ ⋅

≤

sin

( )

1

1

01

1 22

2

2 1 11

2 7

10

λ θov

eff wy

yif

e ovy

y

qh

b t rf

ft b

bb t f

b t fb

= ⋅

= + + ≤

=⋅ ⋅⋅ ⋅

≤

sin

( )

1

1

01

1 22

2

2 1 11

2 7

10

λ θov

e ovy

y

qh

bb t f

b t fb

= ⋅

=⋅ ⋅⋅ ⋅

≤

sin

( )

1

1

1 22

2

2 1 11

10

With circular brace members, the resistance values are multiplied by π/ 4, and b1 and h1 are replaced with thesection diameter d1.fy ≤ 355 N/ mm2 , 30° ≤ θi ≤ 90°, ti ≥ 2,5 mmMember 1 = overlapping memberMember 2 = overlapped member1) This limit value is defined in reference [2].2) Eurocode 3 does not define a lower limit for this value.

Table 9.3.9 Bending resistance of T and X joints. Chords and brace members are square or rectangular hollow sections [1], [3].


308

��

��

b1

t1

t0

b0

h 0

Mip.1 Mop.1

M0N0M0

h1

N0

θ1


309

Resistance ParametersIn-plane bending moment, β ≤ 0,85, chord face yield

β = bi / b0

Tension chord:kn = 1Compression brace member:

In-plane bending moment, 0,85 < β ≤ 1,0, brace member failure

In-plane bending moment, 0,85 < β ≤ 1,0, chord web yieldT joints:fyk = fy0

X joints:fyk = 0,8fy0

T joints:fyk = fy0

X joints:fyk = 0,8fy0

Validity areaBrace members ingeneral:bi / b0 ≥ 0,250,5 ≤ h1 / b1 ≤ 2

Tension brace member:b1 / t1, h1 / t1 ≤ 35

Compression bracemember:b1 / t1, h1 / t1 ≤ 35

Chords:

M

f t hb

hh

b

ip Rd

yMj M

. .

( )( )

,

1

0 02

10 1

0 0

12 1

2

1 111

=

⋅ ⋅ − +−

+−

⋅

ββ β γ γ

kn

M f Wbb

b h tip Rd y pleff

Mj M. . .

,1 1 1

11 1 1

01

11= − −

⋅ ⋅

⋅γ γ

bb t f

b t fbeff

i y

i yii=

⋅ ⋅⋅ ⋅

≤10 0

20

0

b ht

1 1

1

125+ ≥ )

bt

ht

Efyi

1

1

1

11 25, ,≤

2)


. . ,,

1 0 1 02

00 5 5

11= ⋅ +( )⋅γ γ

M f W t bbbop Rd y pleff

Mj M. . . ,

,1 1 1 1 1

2

1

2

00 5 1

11= − ⋅ −

⋅γ γ

M f t h t b h t b hop Rd yMj M

. .,

1 0 0 1 0 0 0 0 0 00

211= ⋅ ⋅ + ⋅ ⋅ +( )[ ] ⋅γ γ

fy ≤ 355 N/ mm2, θ ≈ 90°, ti ≥ 2,5 mm, t0 ≥ 2,5 mm1) These limit values are defined in reference [2].2) These limit values are defined in reference [3].

M f th b b

kop Rd y nMj M

. .,

1 0 02 1 0 1

0

1

2 1

2 1

111= ⋅

+( )−( ) +

⋅ +( )−

⋅

ββ

ββ γ γ

M f t h t b top Rd ykMj M

. . ,,

1 0 1 0 0 00

0 5 511= ⋅ +( ) −( )⋅γ γ

k n

nNA f

MW f

n

M Mj Sd

y

Sd

y

= −

≤

=⋅

⋅+

⋅

1 30 4

1

110 0

0 0

0

0 0

,,

,. .

β

γ γ

b ht

hb

bt

ht

0 0

0

1

0

0

0

0

0

0

25

0 5 2

35

+ ≥

≤ ≤

≤

)

,

,Out-of-plane bending moment, β ≤ 0,85, chord face yield [3]

Out-of-plane bending moment, 0,85 < β ≤ 1,0, brace member failure

Out-of-plane bending moment, 0,85 < β ≤ 1,0, chord web yield

Out-of-plane bending moment,T joints, distortion of the chord section

bb t f

b t fbeff

i y

i yii=

⋅ ⋅⋅ ⋅

≤10 0

20

0

Table 9.3.10 Bending resistance of T, Y and X joints. Chords and brace members are circular hollow sections [1].


310

��

��

��

��

��

��

��

d1

t1

t0

d0

Mip.1 Mop.1

d1

M0

NpNp

M0

θ1


311


In-plane bending moment, chord face yieldβ = d1/ d0


Validity areaT ,Y and X joints:

0,2 ≤ d1 / d0 ≤ 1,0

10 ≤ d1 / t1 ≤ 50

T and Y joints:

10 ≤ d0 / t0 ≤ 50

X joints:

10 ≤ d0 / t0 ≤ 40

M f t dk

ip Rd yp

Mj M. .

,,sin

,1 0 0

2 0 51

1 04 85

11= ⋅ ⋅ ⋅ ⋅ ⋅ ( ) ⋅γ β

θ γ γ

M f t dk

op Rd y ip

i Mj M. . sin

,,

,1 0 0

2

0

2 71 0 81

11= ⋅ ⋅ ⋅ ( ) − ⋅θ β γ γ

k n n

nN

A fM

W f

p p p

pM Mj p Sd

y

Sd

y

= − +( ) ≤

=⋅

⋅+

⋅

10 0 3 1

11

2

0

0 0

0

0 0

, ,

,. .γ γ

Mf

t d

Mf

t d

ip Rdyo

Mj M

op Rdyo

Mj M

. .

. .

sin

sin

,

sin

sin

,

11

21

0 12

0

11

21

0 12

0

3

1 3

4

11

3

3

4

11

=+ ( )

( )⋅

⋅

=+ ( )

( )⋅

⋅

θθ γ γ

θθ γ γ

fy ≤ 355 N/ mm2, 30° ≤ θi ≤ 90°, ti ≥ 2,5 mm, t0 ≥ 2,5 mm

Out-of-plane bending moment,chord face yield

di ≤ d0 - 2 t0, In-plane or out-of-plane bending moment, punching shear failure of the chord

Table 9.3.11 Joints between plates and square or rectangular hollow sections [1].


312

��

��

��

t1

t0

b0

h 0

t1

t0

b0

h 0

b1

N1

N1

Mip.1

h1

N0 N0M0M0


313


η = h1 / b0

Tension chord:km = 1Compression chord:km = 1,3(1- n) ≤ 1

Validity areaLongitudinal plates:

Transverse plates:0,5 ≤ β ≤ 1,0

Chords:

Nf t

tb

tb

k

M N h

Rdy

mMj M

ip Rd Rd

10 0

2

1

0

1

0 0

1 1 1

12 4 1

11

0 5

.

. . .

,

,

=⋅

−+ −

⋅

⋅

= ⋅

ηγ γ

N f t t tRd yMj M

1 0 0 1 00

2 1011

.,= ⋅ +( )⋅γ γ

nNA f

MW f

M Mj Sd

y

Sd

y=

⋅⋅

+⋅

γ γ0 0

0 0

0

0 011,. .

Nf t

t bRdy

epMj M

10 0

103

2 211

.,=

⋅+( ) ⋅γ γ

Plate transverse to hollow sectionb1 ≥ b0 - 2t0, chord web yield

Plate transverse to hollow sectionbi ≤ b0 - 2t0, punching shear failure of the chord

Plate longitudinal to hollow sectionβ ≤ 0,85, chord face yield

0 5 2

35

30

25

0

0

0

0

0

0

0 0

0

, ≤ ≤

≤

≤

+ ≥

hb

ht

bt

b ht

1)

tb

1

00 2≤ ,

β = bb

1

0

b tbb

bep = ≤10 01

01

Plate transverse to hollow section0,5 ≤ β ≤ 1, failure of the plate

N f t bRd y effMj M

1 1 10

11.

,= ⋅ ⋅⋅γ γ b

t b f

b t fbeff

yo

y=

⋅ ⋅⋅ ⋅

≤10 0

21

0 1 11

fy ≤ 355 N/ mm2, t0 ≥ 2,5 mm

1) This limit value is defined in reference [2].

Table 9.3.12 Joints between plates and circular hollow sections [1].


314

��

��

��

�

��

��

��

��

�

��

t1

t0

d0

d0

t0

b1t1

N1

N1

Mip.1

Mop.1

M0M0

M0Np

Np Np

h1

M0 Np


315


η = h1 / d0

Tension chord:kp = 1Compression chord:

β = b1/ d0

fy ≤ 355 N/ mm2, t0 ≥ 2,5 mm

Validity areaLongitudinal plates:

Transverse plates:β ≤ 0,4

Chords:

N k f t

M N h

M

Rd p yMj M

ip Rd Rd

op Rd

1 0 02

0

1 1 1

1

5 1 0 2511

0 5

0

.

. . .

. .

,,

,

= ⋅ ⋅ +( )⋅

= ⋅

=

ηγ γ

N k f t

M N h

M

Rd p yMj M

ip Rd Rd

op Rd

1 0 02

0

1 1 1

1

5 1 0 2511

0 5

0

.

. . .

. .

,,

,

= ⋅ ⋅ +( )⋅

= ⋅

=

ηγ γ

k n n

nNA f

MW f

p p p

pM Mj Sd

y

Sd

y

= − +( ) ≤

=⋅

⋅+

⋅

10 0 3 1

11

2

0 0

0 0

0

0 0

, ,

,. .γ γ

N k f t

M N b

M

Rd p yMj M

op Rd Rd

ip Rd

1 0 02 2

0

1 1 1

1

4 2011

0 5

0

.

. . .

. .

,

,

= ⋅ ⋅ +( ) ⋅

= ⋅

=

βγ γ

Longitudinal plate on both sides of the hollow section, chord face yield

Transverse plate on one side of the hollow section, chord face yield

Longitudinal plate on one side of the hollow section,chord face yield

10 500

0≤ ≤d

t

td

1

00 2

4

≤

≤

,

η

Transverse plate on both sides of the hollow section, chord face yield

Nk f t

M N b

M

Rdp y

Mj M

op Rd Rd

ip Rd

10 0

2

0

1 1 1

1

5

1 0 8111

0 5

0

.

. . .

. .

,,

,

=⋅ ⋅

− ⋅

= ⋅

=

β γ γ

Longitudinal or transverse plate,punching shear failure of the chord

NA

MW

tt fSd Sd

el

yo

Mj M

1

1

1

11

0

0

2

3

11. .

.

,+

≤⋅

⋅γ γ

Table 9.3.13 Resistance of T, Y and X joints with chord flange plate reinforcement. Chords are square or rectangular hollow sections. Brace members are square, rectangular or circular hollow sections [1], [3].


316

��

��

b1

t1

t0

b0

bp

h1

h 0

θ1

LpM0M0

N0N0

t p

N1


317

Resistance ParametersYield in the tie beam surface 4)

Buckling or yield in the tie beam web 4)

Tension chord:fb = fy0

compression chord:fb = χ · fy0 (T and Y joints)fb = 0,8χ(sinθi)fy0 (X joints)χ = reduction factor for bucklingin buckling class C whenslenderness is:

Punching shear failure in the tie beam 4)

Failure in the strut

X-joints when θ < 90°, tie beam shear yield 4)

Av = 2h0 · t0

Validity areaSquare and rectangularbrace members:Brace members in general:hi / b0, bi / b0 ≥ 0,250,5 ≤ hi / bi ≤ 2

Tension brace member:bi / ti, hi / ti ≤ 35Compression bracemember:bi / ti, hi / ti ≤ 35

Circular brace members:Brace members in general:0,4 ≤ di / b0 ≤ 0,8Tension brace member:10 ≤ di / ti ≤ 50 2)


Chords:0,5 ≤ h0 / b0 ≤ 2b0 / t0, h0 / t0 ≤ 35

Plate:

Lp ≥ 1,5 hi / sinθi

bp ≥ b0 - 2t0

Nf t

iRdyp p

p i

p

ip

Mj M.

sin sin,=

⋅−( ) + −

⋅

2

01

24 1

11

β θη

θβ

γ γ

Nf t h

tiRdb

i

i

i Mj M. sin sin

,= ⋅ +

⋅

00

0

210

11θ θ γ γ

b ht

i i

i

+ ≥ )25 1

bt

ht

Ef

i

i

i

i yi, ,≤ 1 25

3)


.,= ⋅ − +( )⋅

2 4 211

0γ γ

λθ π

= −

( )3 46 2

10

0

0,sin

ht

f

Ey

i

3)

With circular brace members, resistance values are multiplied by π/ 4, and bi and hi are replaced with thediameter di.fy ≤ 355 N/ mm2 , 30° ≤ θi ≤ 90°, t0 ≥ 2,5 mm, ti ≥ 2,5 mm1) These limit values are defined in reference [2].2) Eurocode 3 does not define a lower limit for this value.3) These limit values are defined in reference [3].4) The values in the table are valid, when βp ≤ 0,85. The smallest value calculated from the different failure modes is selected to be the joints strength.

Nf t h

biRdyp p

i

i

iep

Mj M.

sin sin,=

⋅+

⋅3

22

11

0θ θ γ γ

b ht

0 0

0

125+ ≥ )

Lh

b b bpi

ip p i≥ + −( )

sinθ

β

η

pi

p

pi

p

bb

hb

=

=

dt

Ef

i

i yi≤ 1 5,

3)

Nf A

iRdy v

i Mj M.

sin

,=⋅

⋅0

03

11

θ γ γ

bt b

bb

b

t

epp i

pi

p

p

=⋅

≤

=

10

2γ

bb t f

b t fbeff

i p yp

p i yii=

⋅ ⋅⋅ ⋅

≤10 2

Table 9.3.14 Resistance of T, Y and X joints with chord side plate reinforcement. Chords are square or rectangular hollow sections. Brace members are square, rectangular or circular hollow sections [1], [3].


318

��

��

b1

t1

t0

b0

h1

h 0

θ1

M0M0N0N0

tp

N1

Lp

h p


319

Resistance Parametersβ ≤ 0,85, chord face yield

β = bi/ b0

η = hi/ b0

Tension chord:kn = 1Compression chord:kn = 1,3- (0,4/ β)n ≤ 1

0,85 < β ≤1,0Calculate the resistance using the values β = 0,85and β = 1,0. Use the original value of β in linearinterpolation.

β = 1,0, chord web buckling Tension chord:fb = fy0

compression chord: fb = χ · fy0 (T and Y joints)fb = 0,8χ(sinθi)fy0 (X joints)χ = reduction factor for buckling inbuckling class C when slendernessis:

0,85 ≤ βp ≤ 1- (1/ γ), punching shear failure of the chord

βp > 0,85, brace member failure

β = 1,0, X-joints, when θ < 90°, chord shear

Av0 = 2h0 · t0Avp = 2hp · tp

Validity areaSquare and rectangularbrace members:Brace members in general:hi / b0, bi / b0 ≥ 0,25 3)

0,5 ≤ hi / bi ≤ 2

Tension brace member:bi / ti, hi / ti ≤ 35Compression bracemember:bi / ti, hi / ti ≤ 35

circular brace members:brace members in general:0,4 ≤ di / b0 ≤ 0,8Tension brace member:10 ≤ di / ti ≤ 50 2)


Chords:0,5 ≤ h0 / b0 ≤ 2b0 / t0, h0 / t0 ≤ 35

Plate:Lp ≥ 1,5 hi / sinθi

Nk f t

iRdn y

i i Mj M. sin sin

,=⋅ ⋅

−( ) + −

⋅

0 02

012

4 111

β θηθ

βγ γ

Nf t t h

t tiRdb p

i

i

ip

Mj M. sin sin

,=+( )

+ +( )

⋅

00

0

210

11θ θ γ γ

b ht

i i

i

+ ≥ )25 1

bt

ht

Ef

i

i

i

i yi, ,≥ 1 25

3)


.,= ⋅ − +( )⋅

2 4 211

0γ γ

λθ π

=+

−

( )3 46 210

0

0,sin

ht t

f

Ep

y

i

3)

With circular brace members, resistance values are multiplied by π/ 4, and bi and hi are replaced with thediameter di.fy ≤ 355 N/ mm2 , 30° ≤ θi ≤ 90°, t0 ≥ 2,5 mm, ti ≥ 2,5 mm1) These limit values are defined in reference [2].2) Eurocode 3 does not define a lower limit for this value.3) These limit values are defined in reference [3].

Nf t h

biRdy

i

i

iep

Mj M.

sin sin,=

⋅+

⋅

0 0

03

22

11

θ θ γ γ

b ht

0 0

0

125+ ≥ )

nNA f

MW f

M Mj Sd

y

Sd

y=

⋅⋅

+

γ γ0 0

0 0

0

0 011,. .

dt

Ef

i

i yi≤ 1 5,

3)

Nf A f A

iRdyo v yp vp

i Mj M.

sin

,=⋅ + ⋅

⋅0

03

11

θ γ γ

bb t f

b t fbeff

i y

i yii=

⋅ ⋅⋅ ⋅

≤10 0

20

0

bt bb

b

bt

ep = ⋅ ≤

=

10

2

0 1

01

0

0γ

Table 9.3.15 Resistance of gap K, N and KT joints with chord flange plate reinforcement.Chords are square or rectangular hollow sections. Brace members aresquare, rectangular or circular hollow sections [1].


320

��

��h 1

h 1

h2

h2

h3

b1, 2

t1, 2

t0

b0

bp

h 0

θ1

Lp

M0M0 N0

N1

θ2

N2

ge

N1 N2

N3

Lp

e

M0N0M0

t p

t p

g1 g2

θ2θ1θ3

Det 1

Det 1

Det 1

ga

g

ga

g

t 0t p

θ ≤ 60° θ > 60°


321


m is the number of brace membersγ = b0/(2t0)Tension chord:kn = 1Compression chord:kn = 1,3- (0,4/ β)n ≤ 1

Chord shearAv = (2h0+ α · b0)t0 (square andrectangular brace members)Av = 2h0 · t0 (circular brace mem-bers)

VSd is the chord shear force atjoint


βp ≤ 1- (1/ γ), punching shear failure of the chord


β ≥ 0,350,5 ≤ hi / bi ≤ 2

Tension brace member:bi / ti, hi / ti ≤ 35Compression brace member:bi / ti, hi / ti ≤ 35

Circular brace members:Brace members in general:0,4 ≤ di / b0 ≤ 0,8Tension brace member:10 ≤ di / ti ≤ 502)

Compression brace member:10 ≤ di / ti ≤ 50 2)

Chords:0,5 ≤ h0 / b0 ≤ 2

b0 / t0, h0 / t0 ≤ 35

Plate:

bp ≥ b0- 2t0, tp ≥ 2ti

Gap:g / b0 ≥ 0,5(1-β)g / b0 ≤ 1,5(1-β)g ≥ t1 + t2ga ≥ 1,5tp

1)

Nf t

b h

m bkiRd

yp p

i

i ii

m

i

m

pn

Mj M. ,

sin,=

⋅+

⋅

⋅==∑∑

8 92

11211

0θγ

γ γ

Nf A

iRdy v

i Mj M.

sin

,=⋅

⋅0

03

11

θ γ γ

b ht

i i

i

+ ≥ )25 1

bt

ht

Ef

i

i

i

i yi, ,≤ 1 25

3)

αγ

=+

=⋅⋅

1

14

3

32

02

0g

t

Vf A

plRdyo v

M, .

With circular brace members, resistance values are multiplied by π/ 4, and bi and hi are replaced with thediameter di . If g/ b0 > 1,5(1-β), the joint is treated as two separate T or Y joints in the calculation.fy ≤ 355 N/ mm2 , 30° ≤ θi ≤ 90°, t0 ≥ 2,5 mm, ti ≥ 2,5 mm1) These limit values are defined in reference [2].2) Eurocode 3 does not define a lower limit for this value.3) These limit values are defined in reference [3].


.,= ⋅ − + +( )⋅

2 411

0γ γ

b ht

0 0

0

125+ ≥ )

nNA f

MW f

M Mj Sd

y

Sd

y=

⋅⋅

+⋅

γ γ0 0

0 0

0

0 011,. .

dt

Ef

i

i yi≤ 1 5,

3)

Nf t h

b biRdyp p

i

i

ii ep

Mj M.

sin sin,=

⋅+ +

⋅3

2 11

0θ θ γ γ


Lh

gpi

ii

i

m

i

m

≥ +

=

−

=∑∑1 5

1

1

1

,sinθ

bb t f

b t fbeff

i p yp

p i yii=

⋅ ⋅⋅ ⋅

≤10 2

bt b

bb

b

t

epp i

pi

p

p

=⋅

≤

=

10

2γ

β η=⋅

=∑b

m b

ii

m

1

0, =

hb

i

0

W V

N A AV

V

f

plRd

Rd vSd

plRd

y

M

hen VSd >

= − −

0 5

210 0

20

0

, .

.. γ

W V Af

plRdy

Mhen V NSd 0.Rd≤ =0 5 0

0

0, ,. γ


η β, 0,1+b

100t 0

0

3)≥

Table 9.3.16 Resistance of gap K, N and KT joints with chord side plate reinforcement. Chords are square or rectangular hollow sections. Brace members are square, rectangular or circular hollow sections [1], [3].


322

��

��

��

h 1

h2

h3

h 1h2

b1, 2

t1, 2

t0

b0

h 0

θ1

Lp

M0M0N0

N1

θ2

N2

g

e

tp

N1 N2N3

M0N0M0

g1 g2

θ2θ1

Lp

θ3

e

Det 1

Det 1

Det 1

h p

ga

g

ga

g

t 0

θ ≤ 60° θ > 60°


323


m is the number of bracemembersTension chord:kn = 1Compression chord:kn = 1,3- (0,4/ β)n ≤ 1

Chord shearAv0 = (2h0+ α · b0)t0 Avp = 2hp · tp(square and rectangular bracemembers)Av0 = 2h0 · t0 Avp = 2hp · tp(circular brace members)

VSd is the chord shear force atjoint


βp ≤ 1- (1/ γ), punching shear of chord

Validity areaSquare and rectangular brace members:Brace members in general:

β ≥ 0,350,5 ≤ hi / bi ≤ 2

Tension brace member:bi / ti, hi / ti ≤ 35Compression brace member:bi / ti, hi / ti ≤ 35



Chords:0,5 ≤ h0 / b0 ≤ 2

b0 / t0, h0 / t0 ≤ 35

Plate:

Gap:g/ b0 ≥ 0,5(1-β)g/ b0 ≤ 1,5(1-β)g ≥ t1+ t2ga ≥ 1,5t0

1)

Nf t

b h

m bkiRd

y

i

i ii

m

i

m

nMj M

. ,sin

,=⋅

+

⋅

⋅==∑∑

8 92

110 02

11

0 0θγ

γ γ

Nf A f A

iRdy v yp vp

i Mj M.

sin

,=⋅ + ⋅

⋅0 0

03

11

θ γ γ

η β, , 3)≥ +0 1100

0

0

bt

b ht

i i

i

+ ≥ )25 1

α

γ

=+

=⋅ + ⋅

⋅

1

14

3

3

2

02

0

0

g

t

Vf A f A

plRdy vo yp vp

M.

With circular brace members, the resistance values are multiplied by π / 4, and bi and hi are replaced with thediameter di. If g/ b0 > 1,5(1-β), the joint is treated as two separate T or Y joints in the calculation.fy ≤ 355 N/ mm2 , 30° ≤ θi ≤ 90°, t0 ≥ 2,5 mm, ti ≥ 2,5 mm1) These limit values are defined in referencereference [2].2) Eurocode 3 does not define a lower limit for this value.3) These limit values are defined in referencereference [3].


.,= ⋅ − + +( )⋅

2 411

0γ γ

b ht

0 0

0

125+ ≥ )

nNA f

MW f

M Mj Sd

y

Sd

y=

⋅⋅

+⋅

γ γ0 0

0 0

0

0 011,. .

dt

Ef

i

i yi≤ 1 5,

3)

Nf t h

b biRdy

i

i

ii ep

Mj M.

sin sin,=

⋅+ +

⋅

0 0

03

2 11

θ θ γ γ


Lh

gpi

ii

i

m

i

m

≥ +

=

−

=∑∑1 5

1

1

1

,sinθ

bt

ht

Ef

i

i

i

i yi, ,≤ 1 25

3)

bb t f

b t fbeff

i y

i yii=

⋅ ⋅⋅ ⋅

≤10 0

20

0

bt bb

b

bt

epi

i= ⋅ ≤

=

10

2

0

0

0

0γ

W V

N A AV

V

f

h t AV

V

f

Rd

Rd vSd

plRd

y

M

p p vpSd

plRd

yp

M

hen V Sd >

= − −

+

⋅ − −

0 5

21

22

1

0 0 0

20

0

2

0

,

..

.

γ

γ

W V

N Af

h tf

Rd

Rdy

Mp p

yp

M

hen VSd ≤

= + ⋅

0 5

20 00

0 0

,

. γ γ


β η=⋅

=∑b

m b

ii

m

1

0, =

hb

i

0

Table 9.3.17 Resistance of reinforced overlap N and K joints. Chords are square orrectangular hollow sections. Brace members are square, rectangular orcircular hollow sections [1]. In the validity area presented in this table, onlythe brace member failure is a governing failure mode.


324

��

��

b1, 2

t1, 2

t0

b0

h1

h 0

θ1

M0M0 N0

N1

θ2

h2N2

q

-e

tp

bp


325





0,5 < hi / bi ≤ 2,0

b1 / bp ≥ 0,75Tension brace member:bi / ti, hi / ti ≤ 35Compression brace member:bi / ti, hi / ti ≤ 35



Chords:0,5 ≤ h0 / b0 ≤ 2b0 / t0, h0 / t0 ≤ 40

Plate:tp ≥ 2ti

Overlap:0,25 ≤ λov < 0,8

N f t h t b biRd yi i ov i i eff e ovMj M

.,= ⋅ −( ) + +[ ] ⋅( )2 2 4

11

0λ

γ γ

With circular brace members, the resistance values are multiplied by π/ 4, and bi

and hi are replaced with the diameter di. fy ≤ 355 N/ mm2 , 30° ≤ θi ≤ 90°, ti ≥ 2,5 mm t0 ≥ 2,5 mm1) These limit values are defined in reference [2].2) Eurocode 3 does not define a lower limit for this value.3) These limit values are defined in reference [3].

bb

hb

i i

0 00 25, ,≥ 3)

N f t h t b biRd yi i i i eff e ovMj M

.,= ⋅ − + +[ ] ⋅( )2 4

11

0γ γ

b ht

i i

i

+ ≥ )25 1

dt

Ef

i

i yi≤ 1 5,

3)

bt

ht

Ef

i

i

i

i yi, ,≥ 11

3)

b ht

0 0

0

125+ ≥ )

λ θov

i

i

effi y

i yii

e ovi p yp

p i yii

qh

bb t f

b t fb

bb t f

b t fb

= ⋅

=⋅ ⋅⋅ ⋅

≤

=⋅ ⋅⋅ ⋅

≤

sin

( )

10

10

02

0

0

2

λ θov

i

i

effi y

i yii

e ovi p yp

p i yii

qh

bb t f

b t fb

bb t f

b t fb

= ⋅

=⋅ ⋅⋅ ⋅

≤

=⋅ ⋅⋅ ⋅

≤

sin

( )

10

10

02

0

0

2

References

[1] ENV 1993-1-1: Eurocode 3: Teräsrakenteiden suunnittelu: Liite K: Putkipalkeistavalmistettujen tasoristikoiden liitokset, 1994(ENV 1993-1-1: Eurocode 3: Design of steel structures: Annex K: Hollow section latticegirder connections, 1994)


[3] CIDECT: CIDECT: Design guide for rectangular hollow section joints under predominantlystatic loading, Verlag TÜV Rheinland GmbH, Köln 1992


326

Appendix 9.4 Estimating the stiffness of moment connections

In framed structures, the stiffness of the joints can be taken into account, and the bendingmoments transferred across the joint. This reduces the span moments and therefore producesan efficient design. The following formulae are obtained for the bending moments of a beamsubjected to uniform load and with a semi-rigid joint at both ends, when the supports areassumed non-deflecting [1]:

where

Figure 9.4.1 A beam with semi-rigid joints.

The rotational stiffness of a welded T joint in square and rectangular hollow sections can bedetermined according to the guidance in reference [2]:

where

Sj is the rotational stiffness of the joint (Nm/Rad)t0 is the wall thickness of the hollow section (mm)

C* is the constant obtained from figures 9.4.3- 9.4.7 (N/ mm2)

SC t

j =⋅ ⋅1000

529 4 30

3*

( . . )

cS L

E I

S

L

I

j

j

=⋅⋅

is the rotational stiffness of the jointis the length of the sectionis the second moment of area of the section

Mc

cq L

Mcc

q L

1

2

0

2

2 129 4 1

62 24

9 4 2

=+

⋅

= ++

⋅

( . . )

( . . )

(restraint moment)

(field moment)


327

Sj Sj

L

M1 M1

M0

q

With square hollow sections, the constant C* is taken from figure 9.4.3 when b1/ b0 is less thanor equal to 0.7. In other cases, it is taken from figures 9.4.4- 9.4.7.

With rectangular hollow sections, the constant C* is calculated as for square hollow sections.This result is multiplied by the correction factor shown in figure 9.4.7.

Formula (9.4.3) yields for rotational stiffness an approximation which best corresponds with thebending moment values of the joint, up to the yield moment (Mel.c) of the joint.

Figure 9.4.2 The moment-rotation curve of a semi-rigid joint.

Mc

Sj

Mel.c

Ø


328

Figure 9.4.3 Values of constant 1000/ C* for T joints in square hollow sections, when b1/ b0 ≤ 0,7 and b0/ t0 ≥ 10 [2].

t / t = 2,0

01

1,751,51,251,0

0,75

0,5

1000/ C*

25

20

15

10

5

0,4 0,45 0,50 0,55 0,60 0,65 0,70 b1/b0

b1

h 1

b0

h 0

t0

t1


329

Figure 9.4.4 Values of constant C* for T joints in square hollow sections,when b1/ b0 ≥ 0,7 and b0/ t0 = 15 or b0/ t0 = 20 [2].


2,5

2,0

1,5

1,0

0,5

2,5

2,0

1,5

1,0

0,5

0,5

0,75

1,0

1,5

2,03,0

0,5

0,75

1,0

1,5

2,03,0

b1/b0 b1/b0

1000· C*

0,7 0,75 0,8 0,85 0,9 0,7 0,75 0,8 0,85 0,9

1000· C*

b0/t0 = 25

b1/b0 ≥ 0,7

b0/t0 = 30

b1/b0 ≥ 0,7

t0 / t 1t0 / t 1

2,0

1,5

1,0

0,5

2,0

1,5

1,0

0,5

0,5

0,75

1,01,25

2,03,0

1,5

0,5

0,75

1,0

1,52,03,0

1000· C*

b1/b00,7 0,75 0,8 0,85 0,9 0,7 0,75 0,8 0,85 0,9 b1/b0

1000· C*b0/t0 = 15

b1/b0 ≥ 0,7b0/t0 = 20

b1/b0 ≥ 0,7

t0 / t 1t0 / t 1


330


Figure 9.4.7 Values of constant C* for rectangular hollow sections to those for square,hollow sections [2].

0 0,25 0,5 0,75 1,0 1,25 1,5 1,75

5

4

3

2

1

h 1

h 1b1 b1

b1/h1

C* / C*

0,5

0,75

1,0

1,5

2,0

3,0

0,5

0,75

1,0

1,5

2,03,0

4,0

3,0

2,0

1,0

4,0

3,0

2,0

1,0

b1/b00,7 0,75 0,8 0,85 0,9 b1/b00,7 0,75 0,8 0,85 0,9

1000· C* 1000· C*b0/t 0 = 35

b1/b0 ≥ 0,7

b0/t 0 = 40

b1/b0 ≥ 0,7

t0 / t 1

t0 / t 1


331

References

[1] ECCS: Technical Committee 8- Structural stability- Technical working group 8.1/ 8.2 Skele-tal structures: Analysis and design of steel frames with semi-rigid joints, First edition 1992

[2] Mang, F et al: The development of recommendations for the design of welded joints bet-ween steel structural hollow sections (T- and X- type joints). Final Report No. 5 AD. CI-DECT Düsseldorf, 1983


332

Appendix 9.5 Fatigue categories

Tables 9.5.1-9.5.4 present the fatigue categories for hollow sections which are valid whenusing the nominal stress method in fatigue design.

Table 9.5.1 Fatigue categories for hollow sections and splice joints [3].


333

Fatigue category Structure Description140 Cold-formed machine-welded hollow

sections.

Weld must be free from defects [1] with nostop/start positions.

71 End-to-end joint with circular hollowsections.

Weld must be free from defects [1] with nodiscontinuities.

The height of the weld convexity must notbe greater than 10% of the weld width, withsmooth transition to section surface.

A fatigue category of 90 can be selected ifthe wall thickness is greater than 8 mm.

56 End-to-end joint with square andrectangular hollow sections. Weld must be free from defects [1] with nodiscontinuities.

The height of the weld convexity must notbe greater than 10% of the weld width, withsmooth transition to section surface.


50 End-to-end joint with intermediate plate withcircular hollow sections.



��

��

��

��

��

��

��


334

45 End-to-end joint with intermediate plate withsquare and rectangular hollow sections.



40 End-to-end joint with intermediate plate(fillet weld) with circular hollow sections.

Wall thickness is less than 8 mm.

36 End-to-end joint with intermediate plate (fil-let weld) with square and rectangular hollowsections.

Wall thickness is less than 8 mm.

71 Hollow section welded directly to anothercross-section.

Weld is non-load-carrying.

The cross-section width parallel to stress isless than 100 mm.

��

��

��

��

��

��

��≤ 100 mm

Table 9.5.2 Fatigue categories for hollow section-to-plate joints [3].

Fatigue category Structure Description80

71

50

L ≤ 50 mm

50 < L ≤ 100mm

L > 100 mm

Longitudinal non-load-carrying plateswelded to hollow section.

80

71

t ≤ 12 mm

t > 12 mm

Transverse non-load-carrying plates weldedto hollow section.

Welds end at a minimum distance of 10 mmfrom the hollow section edge.

80 Shear connectors welded to hollow sectionflange.


335

��

�

��

�

��

�

t

> 10 mm> 10 mm

L

Table 9.5.3 Fatigue categories for joints of hollow section lattice structures [3].


336

Fatigue category Structure Description90

45

t0 / ti ≥ 2,0

t0 / ti = 1

Intermediatevaluesdetermined bylinearinterpolation

Gap K or N joints. Members arecircular hollow sections.

ga ≥ 1,5t0 1)

71

36

t0 / ti ≥ 2,0

t0 / ti = 1


Gap K or N joints. Members aresquare and rectangular hollowsections. The gap must meet the followingconditions:0,5(b0- b1) ≤ g ≤ 1,1(b0- b1) g ≥ 2 t0

ga ≥ 1,5t0 1)

71

56

t0 / ti ≥ 1,4

t0 / ti = 1


Overlap K joints with overlapbetween0,3 ≤ λov ≤ 1,0

Square and rectangular hollowsections:

λov = q / sin(θ1)h1

Circular hollow sections:

λov =

��

��

��

��

�

�d 0

θ1 θ2t1,2

t0

d1 d2

g

��

�b1,2

t1,2

b0

t0

θ 2

h 0

θ 1

ga

h1 h2

e

��

�b1,2

t1,2

b0

t0

θ 2

h 0

θ 1

-e

��

�

��

��

d 0

N0

θ 1 θ 2t1,2

t0

d1 d2

q

ga

d1,2

e

g

q

h1 h2

-e

d1,2

N1 N2

N1 N2

N0

N0

N2N1

N0

N1 N2

q · sin(θ1)

h1

q · sin(θ1)

d1


337

71

50

t0 / ti ≥ 1,4

t0 / ti = 1


Overlap N joints

Validity area of this table:0,4 ≤ bi / b0 ≤ 1,0, 0,25 ≤ di / d0 ≤ 1,0

b0 ≤ 200 mm, d0 ≤ 200 mm

-0,5 h0 ≤ e ≤ 0,25 h0

-0,5 d0 ≤ e ≤ 0,25 d0

t0 ≤ 12,5 mm, ti ≤ 12,5 mm

35° ≤ θ ≤ 50°, b0 / t0 ≤ 25

Eccentricity perpendicular to lattice plane: eop < 0,02b0 and eop < 0,02d0

Fillet welds allowed when ti ≤ 8 mm1) These limit values are determined in reference [2].

��

�

��

��

��

�

��

b ht

b ht

i i

i

0 0

025

+ + ≥, 1)

h 0

θ1

h2

-e

θ2

h1b1,2

t1,2

t0

b0

q

d2

d 0

t1,2

d1,2

-e

q

θ2θ1

d1

t0

N0

N1 N2

N0

N1 N2

Table 9.5.4 Fatigue categories for bolted joints [3].

References

[1] ENV 1090-1: Teräsrakenteiden valmistus ja asennus. Osa 1: Yleiset säännöt ja raken-nuksia koskevat säännöt, 1996(ENV 1090-1: Execution of steel structures- Part 1: General rules and rules for buildings,1996)

[2] CIDECT: Project 5AQ/2: Cold formed RHS in arctic steel structures, Final report 5AQ-5-96, 1996

[3] SFS-ENV 1993-1-1:Eurocode 3: Teräsrakenteiden suunnittelu. Osa 1-1: Yleiset säännötja rakennuksia koskevat säännöt, 1993(ENV 1993-1-1: Eurocode 3: Design of steel structures. Part 1.1: General rules and rulesfor buildings, 1993)

Fatigue category Structure Description112 Bolted joint with prestressed or non-

prestressed bolts transferring shear force.Resistance for prestressed bolts isdetermined by gross cross-section and fornon-prestressed bolts by tension area.

100 Bolted joint with bearing type adjusting boltstransferring shear force. Bolt resistance is determined by gross cross-section.

36 Bolted joint transferring tensile force.

Bolt resistance is determined by tensionarea.��

�Adjusting bolt


338

Appendix 9.6 Cross-section factors in fire design

The calculation of cross-section factors is presented in this appendix as general formulae. Thecross-section factors for hollow sections exposed to fire on all sides are given in the tables inAppendix 9.1.

9.6.1 Hollow section exposed to fire on all sides

Hollow section Am / V Ap / V

Square or

rectangular

Circular


339

��

��

��

Fire protection follows the surface Am / V

Fire protection does not followthe surface Am / V

t

h h

t

r0 r0

b b

ri ri

t

d

2 4

2 2 4

0 0

02 2

b h r r

t b h t r ri

+ − +( )+ −( ) − −( ) −( )

π

π

2

2 2 4 02 2

b h

t b h t r ri

+( )+ −( ) − −( ) −( )π

d

d t t⋅ − 2

9.6.2 Hollow section exposed to fire on three sides

Hollow section Am / V Ap / VSquare

Rectangular with

non-exposed short side

Rectangular with

non-exposed long side


340

��

��

h

t

r0

b

ri

h

t

r0

b

ri

b

t

r0

h

rib

t

r0

h

ri



b h r r

t b h t r ri

+ − +( )+ −( ) − −( ) −( )

2 6 2

2 2 4

0 0

02 2

π

π

b h r r

t b h t r ri

+ − +( )+ −( ) − −( ) −( )

2 6 2

2 2 4

0 0

02 2

π

π

2 6 2

2 2 4

0 0

02 2

b h r r

t b h t r ri

+ − +( )+ −( ) − −( ) −( )

π

π

b h

t b h t r ri

++ −( ) − −( ) −( )

2

2 2 4 02 2π

b h

t b h t r ri

++ −( ) − −( ) −( )

2

2 2 4 02 2π

2

2 2 4 02 2

b h

t b h t r ri

++ −( ) − −( ) −( )π

9.6.3 Hollow section exposed to fire on two opposite sides


341

��

��

h

t

r0

b

ri

b

t

r0

h

ri

h

t

r0

b

rib

t

r0

h

ri



Hollow section Am / V Ap / VSquare

Rectangular with

non-exposed short sides

Rectangular with

non-exposed long sides

2 4 2

2 2 4

0 0

02 2

h r r

t b h t r ri

− + ⋅( )+ −( ) − −( ) −( )

π

π

2 4 2

2 2 4

0 0

02 2

h r r

t b h t r ri

− + ⋅( )+ −( ) − −( ) −( )

π

π

2 4 2

2 2 4

0 0

02 2

b r r

t b h t r ri

− + ⋅( )+ −( ) − −( ) −( )

π

π

2

2 2 4 02 2

h

t b h t r ri+ −( ) − −( ) −( )π

2

2 2 4 02 2

h

t b h t r ri+ −( ) − −( ) −( )π

2

2 2 4 02 2

b

t b h t r ri+ −( ) − −( ) −( )π

9.6.4 Hollow section exposed to fire on two adjacent sides

Hollow section Am / V Ap / V

Square orrectangular


342

�� h

t

r0

b

ri

ht

r0

b

ri



b h r r

t b h t r ri

+ − + ⋅( )+ −( ) − −( ) −( )

4 1 5

2 2 4

0 0

02 2

, π

π

b h

t b h t r ri

++ −( ) − −( ) −( )2 2 4 0

2 2π

Appendix 9.7 Minimum bending radii for square and rectangular hollow sections

Tables 9.7.1 and 9.7.2 present the minimum bending radii for square and rectangular hollowsections when bending is made in room temperature with three-roller cold bending. The valuesin these tables are guideline minimum values that can be obtained with good equipmentand careful workmanship.

Table 9.7.1 Guideline values for minimum bending radii of square hollow sections in three-roller cold bending [1]

Pb is the flow of cross-section [%]

Pe is the inverted deflection of the compression flange [%]

Rt is internal bending radius [m]

Ired is the reduction of the second moment of inertia

Iy due to cross-sectional distortion [%]

Pb = 1 % Pb = 2,5 % Pb = 5 % Pb = 7,5 % Pe = 0,5 % Pe = 1 % Pe = 2,5 % Pe = 5 %

h b t Rt Ired Rt Ired Rt Ired Rt Ired Rt Ired Rt Ired Rt Ired Rt Ired

mm mm mm m % m % m % m % m % m % m % m %40 40 4,0 1,44 1 0,31 1 0,22 1 0,22 1 0,22 1 0,22 2 0,22 4 0,22 850 50 4,0 2,88 1 0,61 2 0,22 3 0,22 3 0,79 1 0,29 2 0,22 4 0,22 750 50 5,0 2,72 1 0,58 1 0,22 2 0,22 2 0,43 1 0,22 2 0,22 4 0,22 860 60 4,0 5,07 1 1,08 2 0,34 3 0,22 5 2,55 1 0,95 2 0,26 4 0,22 760 60 5,0 4,79 1 1,02 2 0,32 3 0,22 3 1,37 1 0,51 2 0,22 4 0,22 770 70 4,0 8,17 1 1,74 3 0,54 5 0,27 7 6,85 1 2,55 2 0,69 4 0,26 770 70 5,0 7,72 1 1,65 2 0,51 3 0,26 6 3,68 1 1,37 2 0,37 4 0,22 780 80 4,0 12,36 2 2,64 3 0,82 6 0,41 8 16,12 1 5,99 2 1,62 4 0,60 680 80 5,0 11,68 1 2,49 3 0,77 5 0,39 7 8,66 1 3,22 2 0,87 4 0,32 780 80 6,3 11,02 1 2,35 2 0,73 3 0,73 5 4,55 1 1,69 2 0,46 4 0,22 790 90 4,0 17,80 2 3,80 3 1,18 7 0,60 10 34,29 1 12,75 2 3,45 3 1,28 690 90 5,0 16,83 2 3,59 3 1,11 6 0,56 8 18,42 1 6,85 2 1,85 4 0,69 790 90 6,3 15,87 1 3,38 3 1,05 4 0,53 6 9,68 1 3,60 2 0,97 4 0,36 7

100 100 4,0 24,67 2 5,26 4 1,63 8 0,82 12 67,38 1 25,05 2 6,77 3 2,52 6100 100 5,0 23,32 2 4,97 4 1,54 6 0,78 9 36,20 1 13,46 2 3,64 4 1,35 6100 100 6,3 22,00 1 4,69 3 1,46 6 0,74 7 19,03 1 7,07 2 1,91 4 0,71 7100 100 8,0 20,71 1 4,42 2 1,37 3 0,69 6 9,78 1 3,64 2 0,98 4 0,37 7120 120 4,0 43,41 3 9,26 6 2,88 12 1,45 18 216,84 1 80,60 2 21,79 3 8,10 6120 120 5,0 41,03 3 8,75 5 2,72 8 1,37 13 116,51 1 43,31 2 11,71 3 4,35 6120 120 6,3 38,70 2 8,25 4 2,56 7 1,29 11 61,23 1 22,76 2 6,15 4 2,29 7120 120 8,0 36,44 1 7,77 3 2,41 6 1,22 7 31,49 1 11,70 2 3,16 4 1,18 7150 150 5,0 81,91 3 17,47 7 5,43 13 2,74 20 487,11 1 181,06 2 48,94 3 18,19 6150 150 6,3 77,27 3 16,48 5 5,12 9 2,58 14 255,98 1 95,15 2 25,72 3 9,56 6150 150 8,0 72,75 2 15,51 4 4,82 7 2,43 11 131,64 1 48,93 2 13,23 4 4,92 7150 150 10,0 68,76 2 14,66 3 4,55 6 2,30 8 70,73 1 26,29 2 7,11 4 2,64 7180 180 6,3 123,65 3 20,04 6 5,06 11 2,26 17 831,60 1 251,71 2 51,86 3 15,70 6180 180 8,0 81,47 3 13,21 6 3,33 10 1,49 15 416,40 1 126,03 2 25,96 4 7,86 6180 180 10,0 55,18 3 8,94 6 2,26 10 1,01 14 218,22 1 66,05 2 13,61 4 4,12 7200 200 6,3 176,76 3 28,66 6 7,24 10 3,23 16 1046,76 1 316,83 2 65,27 3 19,76 6200 200 8,0 116,47 3 18,88 5 4,77 9 2,13 14 524,13 1 158,64 2 32,68 3 9,89 6200 200 10,0 78,88 3 12,79 5 3,23 9 1,44 13 274,69 1 83,14 2 17,13 4 5,18 7250 250 6,3 376,81 3 61,08 5 15,42 9 6,89 14 1704,11 1 515,80 2 106,26 3 32,16 6250 250 8,0 248,27 3 40,25 4 10,16 8 4,54 12 853,27 1 258,27 2 53,21 3 16,10 6250 250 10,0 168,15 2 27,26 4 6,88 7 3,08 10 447,18 1 135,35 2 27,88 3 8,44 6300 300 8,0 460,71 2 74,69 4 18,86 7 8,43 11 1270,17 1 384,45 2 79,20 3 23,97 6300 300 10,0 312,03 2 50,58 3 12,77 6 5,71 9 665,71 1 201,50 2 41,51 3 12,56 6

��b b1

h h 1

h 2

yyy

t Pb b

b

Peh

b

e

= −

=

1 100

100

e

a) before bending b) after bending

y


343

Table 9.7.2 Guideline values for minimum bending radii of rectangular hollow sections in three-roller cold bending [1]

References

[1] Kennedy John B: Minimum bending radii for square & rectangular hollow sections (3-rollercold bending). CIDECT report 11C-88/14-E.

Pb is the flow of cross-section [%]

Pe is the inverted deflection of the compression flange [%]

Rt is internal bending radius [m]

Ired is the reduction of the second moment of inertia

Iy due to cross-sectional distortion [%]

��b b1

h h 1

h 2

yyy

tP

b bb

Peh

b

e

= −

=

1 100

100e

a) before bending b) after bending

Pb = 1 % Pb = 2,5 % Pb = 5 % Pb = 7,5 % Pe = 0,5 % Pe = 1 % Pe = 2,5 % Pe = 5 %

H B T Rt Ired Rt Ired Rt Ired Rt Ired Rt Ired Rt Ired Rt Ired Rt Ired

mm mm mm m % m % m % m % m % m % m % m %50 30 4,0 2,18 1 0,47 1 0,22 2 0,22 2 0,26 1 0,22 2 0,22 4 0,22 760 40 4,0 4,07 1 0,87 2 0,27 3 0,22 3 1,06 1 0,39 2 0,22 3 0,22 680 40 4,0 8,48 1 1,81 2 0,56 3 0,28 4 3,59 1 1,34 2 0,36 3 0,22 580 40 5,0 8,01 1 1,71 2 0,53 3 0,27 3 1,93 1 0,72 2 0,22 3 0,22 690 50 4,0 12,93 1 2,76 3 0,86 4 0,43 5 9,61 1 3,57 2 0,97 3 0,36 590 50 5,0 12,22 1 2,61 2 0,81 3 0,41 4 5,16 1 1,92 2 0,52 3 0,22 6

100 50 4,0 16,92 1 3,61 2 1,12 5 0,57 5 15,02 1 5,58 1 1,51 3 0,56 5100 50 5,0 15,99 1 3,41 2 1,06 3 0,53 5 8,07 1 3,00 2 0,81 3 0,30 5100 60 4,0 18,69 2 3,98 3 1,24 5 0,62 7 22,30 1 8,29 2 2,24 3 0,83 5100 60 5,0 17,66 1 3,77 3 1,17 4 0,59 6 11,98 1 4,45 2 1,20 3 0,45 6100 60 6,3 16,66 1 3,55 2 1,10 3 0,56 4 6,30 1 2,34 2 0,63 3 0,24 6120 60 4,0 29,77 1 6,35 3 1,97 5 1,00 8 48,35 1 17,97 1 4,86 3 1,81 5120 60 5,0 28,14 1 6,00 3 1,86 5 0,94 6 25,98 1 9,66 1 2,61 3 0,97 5120 60 6,3 26,54 1 5,66 2 1,76 3 0,89 6 13,65 1 5,07 2 1,37 3 0,51 6120 80 4,0 34,81 2 7,42 4 2,31 7 1,16 11 90,14 1 33,50 2 9,06 3 3,37 5120 80 5,0 32,90 2 7,02 3 2,18 6 1,10 9 48,43 1 18,00 2 4,87 3 1,81 6120 80 6,3 31,04 1 6,62 3 2,06 4 1,04 6 25,45 1 9,46 2 2,56 3 0,95 6150 100 4,0 69,50 3 14,82 5 4,60 11 2,32 18 376,84 1 140,08 2 37,86 3 14,07 5150 100 5,0 65,69 3 14,01 5 4,35 8 2,20 12 202,48 1 75,26 2 20,34 3 7,56 5150 100 6,3 61,97 2 13,21 3 4,10 6 2,07 9 106,41 1 39,55 2 10,69 3 3,97 6150 100 8,0 58,34 1 12,44 3 3,86 6 1,95 7 54,72 1 20,34 2 5,50 3 2,04 6160 80 4,0 176,59 1 28,63 2 7,23 5 3,23 7 244,29 1 73,94 1 15,23 3 4,61 5160 80 5,0 119,60 1 19,39 2 4,90 5 2,19 6 128,03 1 38,75 1 7,98 3 2,42 5160 80 6,3 79,88 1 12,95 2 3,27 4 1,46 5 65,56 1 19,85 1 4,09 3 1,24 5160 80 8,0 52,63 1 8,53 3 2,15 3 0,96 5 32,83 1 9,94 2 2,05 3 0,62 5200 100 5,0 254,95 1 41,33 3 10,44 4 4,66 5 208,42 1 63,09 1 13,00 3 3,93 5200 100 6,3 170,28 1 27,60 3 6,97 3 3,12 5 106,74 1 32,31 1 6,66 3 2,01 5200 100 8,0 112,19 1 18,19 2 4,59 3 2,05 5 53,45 1 16,18 2 3,33 3 1,01 5200 100 10,0 75,98 1 12,32 2 3,11 3 1,39 4 28,01 1 8,48 2 1,75 3 0,53 6250 150 6,3 366,57 1 59,42 3 15,00 4 6,71 5 316,79 1 95,89 1 19,75 3 5,98 5250 150 8,0 241,53 1 39,15 3 9,89 3 4,42 5 158,62 1 48,01 2 9,89 3 2,99 5250 150 10,0 163,58 1 26,52 2 6,70 3 2,99 5 83,13 1 25,16 2 5,18 3 1,57 6300 200 6,3 684,25 1 110,92 3 28,01 5 12,52 6 667,45 1 202,02 1 41,62 3 12,60 5300 200 8,0 450,84 1 73,09 3 18,45 4 8,25 5 334,20 1 101,16 2 20,84 3 6,31 5300 200 10,0 305,34 1 49,50 3 12,50 3 5,59 5 175,15 1 53,01 2 10,92 3 3,31 6400 200 8,0 1177,85 1 190,94 1 48,21 2 21,55 3 242,86 1 73,51 1 15,14 3 4,58 5400 200 10,0 797,71 1 129,32 1 32,65 3 14,60 3 127,28 1 38,52 1 7,94 3 2,40 5

y


344

Appendix 9.8 WinRAMI softwareThe WinRAMI software by Rautaruukki is an easy-to-use tool for the design of hollow sectionstructures. The program is intended for calculating uniplanar frame structures. WinRAMI canbe used for calculating the actions of the structure and for designing members and their jointswith structures made of hollow sections. For other types of structures, WinRAMI can be usedfor calculating the force quantities only.

The user interface of WinRAMI utilizes the latest Windows technology. You can create thestructural model by drawing the image of the structure with the mouse. The profiles you want touse are defined after drawing the model in the following way.

• Select the parts of the structural model you want to place the profile in.• Select the appropriate profile from the Hollow section module.• Move the selected tube profile on the structural model by using the mouse.• The program creates automatically a link (OLE2) to the Hollow section module.

Also the loads, edge conditions and joints of the structure can be defined similarly, by simplypointing with the mouse.

WinRAMI calculation methods

The structural model is determined using the element method. WinRAMI uses an element with7 degrees of freedom. With the element, the displacement of the centre of gravity axis in cross-section class 4 can be accounted for. The default axis is the total cross-section centre ofgravity axis.

Hardware requirements

The program functions well with Pentium 100Mhz processor and 16MB RAM. Operatingsystem can be either Windows 3.xx, 95 or NT.

Figure 9.8.1 Structure of WinRAMI

For additional information on WinRAMI, please contact Rautaruukki technical customer service(see page 351).

OLE 2

link

WinRAMI

Structural model, loads, actionsand displacements

Static processor: Structural element modules:

Hollow sectionHollow section cross-sectional

properties and resistance values

Moncont

moment-rigid jointsLiicont

lattice chord and bracemember joints

Joint modules:


345

Design Handbook for RautaRuukki Structural Hollow Sections

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