Review of Analysis of Tube Sheets

8/11/2019 Review of Analysis of Tube Sheets

1/19

ELSEVIER

ht. J. Pm . Ves. & Piping 61(1996) 219-297

@J 1996 Elsevier Science Limited

030&i-0161(95)00026-7

Printed in Northern Ireland. All rights reserved

030%0161/96/$15.00

Review of analysis of tube sheets

V. G. Ukadgaonker, P. A. Kale, Mrs. N. A. Agnihotri & R; Shanmuga Babu

Department of Mechanical Engineering, Indian Institute o f Technology, Powai, Bombay 400 076, Zndia

(Received 15 January 1995;accepted 9 February 1995)

Various designmethodshave been proposedby a number of researchers or

analyzing stresses nd deflections n multiperforated plates, popularly known

as tube sheets.The purpose of this paper is to show the different techniques

developed by various researchersn the analysisof tube sheets.A thorough

literature review is undertaken and the different techniquessuchas analytical,

experimental and numerical are dealt with separately. Finally the results

obtained by various researchworkers are comparedand the authors work is

also mentioned. The future scope n this field as proposed by the authors is

also discussed.

NOMENCLATURE

d

hole diameter

Eh3

D=

12(1 - Y2)

flexural rigidity of perforated plate

D* = ti*h3

12(1 - Y2)

flexural rigidity of the equivalent

solid plate

E Elastic modulus of the plate material

E* Elastic modulus of equivalent plate

h

plate thickness

P

pitch of the hole pattern

7 (D*/D) X 100 deflection efficiency

p

[(p

-

d)/p] X 100

ligament efficiency

Y Poissons ratio of the plate material

V* Poissons ratio of equivalent solid plate

Subscripts:

A triangular pitch pattern

Cl square pitch pattern

INTRODUCTION

Tube sheets, which are multiperforated plates,

Ligament eficiency: the ratio of the minimum

being important components of pressure vessels

ligament width,

(p

-

d),

to the pitch,

p,

of the

from functional, structural and cost points of

hole pattern as shown in Fig. 1, i.e. [(p - d)/p]

X

view, the optimum design is of paramount 100. Ligament efficiency varies from unity for an

importance. The stress concentration factor

isolated hole to nearly zero for as ligament width

found at the edge of the hole in a stressed plate is

becomes small relative to the distance between

of great practical importance. An exact theoreti- holes.

279

cal solution for the stresses and deformation

everywhere in the tube sheet is not possible but

approximate solutions can be obtained. During

the last decade many authors have proposed

analytical, experimental or numerical techniques

to solve this problem. Osweiller in his literature

review concerning the tube sheet has shown the

evolution of equivalent solid plate concept over

four decades. The purpose of the present paper is

to study the different concepts developed over

recent decades by various authors in the analysis

of the tube sheet.

Definitions

Hole pattern: the holes in the tube sheet can be

arranged as shown in Fig. 1 in three different

patterns viz equilateral triangular, square and

staggered square. The equilateral triangular

pattern is most widely used since it is the most

effective packaging arrangement.


2/19

280 V. G. Ukadgaonker et

al.

0000

00000

000000

0

om

0000

EQUILATERAL S Q?iARE

TRIANGULAR

(0 PITCH 1

(DPITCHI , , h , ,

STAGGERED

SQUARE

(0 PITCH 1

Fig. 1. Different types of hole patterns.

Stress concentration factor: The ratio of maxi-

mum principal stress to the nominal stress

without any discontinuity at the same section.

Literature review

The deflections and the stresses in a drilled plate

subjected to any type of loading is more

compared to a solid plate of the same dimensions

under similar loading conditions. The weakening

effect of the perforations may be described either

in terms of deflections and ligament efficiencies

or in terms of the ratios of the elastic properties

such as Youngs modulus, Poissons ratio, for the

drilled plate and the solid plate. Various

theoretical, experimental and numerical methods

have been proposed for evaluation of these

deflection efficiencies or modified elastic con-

stants. The following section discusses each of the

three techniques separately.

Analytical techniques

Gardner in 1948 was the first to introduce the

concept of equivalent solid plate for the design of

a floating tube sheet heat exchanger. He

introduced the terms called Deflection Efficiency

and Ligament Efficiency which takes into account

the weakening effect of the perforations. The

flexural rigidity D* of the equivalent solid plate is

given by

where q is the

depends upon the

and the degree of

ligament efficiency.

solid plate having

deflection efficiency which

type of penetration pattern

drilling of the plate, i.e. the

The stresses in an equivalent

the flexural rigidity D* are

calculated using

classical structural analysis

methods and then in the real plate by dividing

D*=r/D

them by p, where p is the ligament efficiency.

Due to lack of experimental results, Gardner

proposed formulae based on minimum ligament

width in circumferential direction:

n*=l-

4

- sin

[

lb

Jr

-l 1(1-

q,=1-3sine1 -

n

[;(l-P)]

which leads to values of about O-5 for typical

values of CL. Gardner later adopted the same

method for the design of fixed tube sheet heat

exchanger.3 The main drawback of Gardners

analysis is that no consideration has been given

to the interaction effect of the tube sheet with

other parts of the pressure vessel and that the

edge condition he assumed is either simply

supported or clamped, but the actual edge

condition lies somewhere in between these two.

In 1952 Miller proposed4 the use of the mean

ligament width divided by the pitch for the

calculation of 71

which leads to the values of about O-6 for typical

values of p.

In 1952 Malkin and Horvay proposed5,6 that

the perforated plates could be replaced by a

hexagonal honey comb structure as shown in Fig.

2 with parallel-sided load-carrying members. This

approximation is valid only for small ligament

efficiencies less than 20%. By equating the strain

energy for this idealized structure to that of

equivalent solid plate, they obtained curves for

the effective Youngs modulus E* and the

Poissons ratio y* of the plate and for the

corresponding stress concentration factors. Based

on these results they calculated the deflection

efficiency which was found to be

obtained by Gardner and Miller.

lower than that

Fig. 2. Circular hole approximated by hexagon.


3/19

Review of analysis of tube sheets 281

Blake and Paton proposed7 a method for

estimating stresses in a rectangular tube plate or

large circular tube plates, taking into account the

effect of a bellows or diaphragm. The theory of

Timoshenko for beams on elastic foundation was

used. Relationships between tube plate stress,

tube loads and deflection for diaphragm or

bellows-type heat exchangers were derived.

Figure 3 shows the values of deflection efficiency

obtained by mechanical tests and electrical

resistance tests based on the conducting sheet

analogy. A tube plate test rig was constructed to

test the validity of the design expressions

obtained by analytical technique. The ex-

perimental work of Blake and Paton will be

discussed later in the present paper.

In 1969 GardneP improved his floating tube

sheet method by considering the unperforated

annulus of the tube sheet periphery and proposed

a direct formula.

Figure 4 shows the comparison of deflection

efficiency obtained by various researchers includ-

ing Duncans experimental results discussed later.

The figure clearly reveals the greater disparity of

results of various authors. Solemo and Mahoney

reviewed and compared all these theories and

proposed a refinement to Horvay and Duncan

theories, which, however, does not agree with the

experimental results. In later years more

attention has been paid to the accurate

determination of effective elastic constants E*

and use of them in design of the perforated

plates.

The earlier research works, theoretical or

experimental, for the evaluation of effective

04

o-2 o-3 04 o-5

fmax -%/ fmax

01

I

I I I I I ,,,,I

01 015

02 33 OC 05 06 07 080910

LIGAMENT EFFICIENCY

Fig. 3. Deflection efficiency values obtained by Blake and Fig. 5. Sampsons effective elastic constants for bending and

Paton. plane stress.

t

'1

0 0.25 0.50 O-75 l-00

LIGAMENT EFFICIENCY

Fig. 4. Comparison of deflection efficiencies.

elastic constants was mainly focused on the

triangular penetration pattern. The effective

elastic constants given in 1971 ASME codes was

based on the expeimental results obtained by

Sampson~ which are discussed later. Sampson

found E*/E and Y* values for in-plane loading

are different than for bending loading, but they

exhibit isotropic behaviour. Figure 5 gives the

Sampson experimental elastic constants values as

a function of ligament efficiency.

- PLANE STRESS CONSTANTS


4/19

282 V. G. Ukadgaonker et al.

In 1962 ODonnell and Langer proposedl*

general effective elastic constants values based

on Sampsons results as shown in Fig. 6 which

can be used for both in-plane loading and

bending loading and for any thickness h/p > 2.

The error due to the approximation involved was

found to be 8%. They have also given the

expression for the ligament stress intensity based

on stresses averaged across the minimum

ligament section at the plate surface which is as

follows:

where

p/h

is reciprocal of the ligament efficiency

factor and crl = u, or oe whichever has largest

absolute value; and k is the stress concentration

factor whose value depends upon the biaxiality

ratio p, which can be evaluated from Fig. 7.

Similar expressions have been given for finding

ligament stress intensities averaged through the

depth of the plate and the peak stresses in

perforated plates taking into account mechanical

as well as thermal loads. Other work was based

on the application of energy principles to

idealized geometries.5*6T3-17

Methods for analysis for the plates perforated

by square penetration pattern came to be known

with the start of 1960s. In the case of square

u

z -.5

a

ii

w 0.4

z

=

it 0.3

::

0.2

1

0.10 0.15 0.20 0.30 (ZLO 0.5 0.6 030~091.0

h/R, LIGAMENT EFFICIENCY

Fig. 6. Effective elastic constants used for design.

K.

I , I I I I I 1

QfcAVERAGE STRESSINTENSITY~

IN MINIMUM LIGAME NTSEC TION

r a=6 :STRE SSES IN EQUIVALENT-

SOLID PLATE

ml=QrofQe(WHICHEVER HAS THE-

LARGEST ABSOLUTE VALUE)

0.6 Illlllll 11 IIIl11.

-1.0 - Mb 0.6- 0.4-0.2 0 l 0.2 0.4 l 0.6+0.6 et.0

/i BIAXIALITY RATIO

Fig. 7. Stress intensities in perforated plate ligaments.

penetration pattern the effective elastic constants

in pitch direction and in diagonal direction are

different, i.e. they exhibit anisotropic behaviour.

Meijers**19 has considered the most general

case of a doubly periodic pattern of equal circular

holes. The Complex Variables technique is used

i.e., normal and shear stresses are related to two

complex stress functions. Both in-plane loading

and bending loading were considered. Equations

which relate the moments per unit length

M,,M,, i& and shear forces per unit length

N,, NY to the two complex stress functions are

given. Similarly the mean values of the above

quantities are related to the two complex stress

functions. From the equivalent solid plate one

can obtain the values of the mean moments; from

this it is possible to find out the two complex

stress functions. Once the stress functions are

known it is possible to find out the moment

distribution in actual perforated plate. Meijers

considered a thin plate for the bending problem

for which Kirchotfs assumptions are valid.

Extensive numerical results were given in the

graphical form for rectangular and rhombic

penetration patterns. Results given include

effective Youngs modulus, effective Poissons

ratio and stress distribution for in-plane loads

and moment distribution for bending loads

covering the entire range of parameters given by

the ratio of hole diameter to pitch in respective

directions, required to define the doubly periodic

pattern of holes. Figures 8 and 9 show the


5/19

Review

of

analysis

of

tube sheets

283

0.2 0.L

Al 26

0.8 1.0

Fig. 8. Effective Youngs modulus for rectnagular pattern

(P. Meijers).

@6-

t

;lw 0% -

>I?

o-2 -

v,~+ EFFECTIVE POISSONS RATIO

I

02

.

1.0

Al*

Fig. 9. Effective Poissons ratio for rectangular pattern (P.

Meijers).

curves for the effective Youngs modulus and

effective Poissons ratio.

By using the Complex Variable technique

Meijers has obtained stress distribution for

square patterns

under uniaxial and shear

loading. Figure 10 shows the stress concentra-

tion in square pattern for various values of hole

diameter-to-pitch ratios. Similar curves are given

for diagonal or diamond pattern and general

rhombic pattern. He has also considered the

problem of bending of thin perforated plates.

Figure 11 shows the variation of bending

moment in the ligament of the plate and around

the hole boundary for square pattern. M,* and

44: are the bending moments per unit length

applied at the edge of the plate. He has also

given results for torsional loading on square

pitch pattern and triangular pitch pattern.

Bailey and Hicks in 1960 derived the effective

elastic constants for plates with square penetra-

tion pattern using an Airy stress function

approach which takes advantage of the symmetry

Fig. 10. Stress concentrations at point

A

for rectangular

pattern and notched strip (Meijers).

Fig. 11. Square pattern moment distribution along hole

boundaries and x-axis for M:=

My*=

1

(M,,

small)

(Meijers).

properties of the typical element. The loading

conditions were unequal uniform tension in x and

y directions and uniform applied shear. The

general problem with unequal uniform displace-

ments in the x and y directions, are split into two

cases as shown in Fig. 12 and the total solution of

the problem is obtained as superimposition of

these two cases.

Boundary conditions are

satisfied only at discrete points at every 10

around the edges of the square element of the

plate as only a finite number of terms in the

infinite series have been taken. The error due to

this approximation was found to be only 0.01%.

Imposition of the boundary conditions as- shown

in Fig. 12 calculates the unknown arbitrary


6/19

284

V. G. Ukadgaonker et al.

CASE 1

CASE 2

5x

= 6,~ 6x z-6,,

ux =vy

Ux =-vy

Fig. 12. Symmetrical and antisymmetrica l displacements.

constants in the series. By using this approach

they have developed formulae for E*/E and Y*.

Variation of elastic constants with respect to p/d

is shown in Figs 13 and 14. Bailey and Hicks have

evaluated local stresses at certain locations in a

square penetration using a numerical method and

a digital computer. They have considered the

basic cases of isotropic tension and pure shear in

both pitch and diagonal directions. Figure 15

shows the stress distribution across the minimum

ligament section for uniaxial tension loading.

Experimental values were obtained from the

photo-elasticity tests.

Slot and ODonnell have developedl a new

theory and formulas for the thick perforated

plates for square and triangular pitch patterns

subjected to uniform in-plane loading, based on a

generalized plane strain condition. Plane stress

condition was assumed in the analysis of thin

perforated plates. The influence of Poissons ratio

,-

t-

,SY_MPTOTI C VALUE FOR c/a

la0 1.5 20 25 30 :

PITCH OF HOLES/DIA.OF

HOLES

d

1.0 1.5 2.0 2.5 3.0 3 .5

PITCH OF HOLES/DIAMETER OF HOLES

Fig. 14. Effective Poissons ratio (Bailey and Hicks).

on the effective elastic constants was also

analyzed. In this method displacement of a point

in the perforated plate is equated to its

displacement in the equivalent solid plate. Figure

16 gives the comparison of the theoretical results

of Slot and ODonnell for the perforated plates

loaded in bending with the experimental results

of Sampson9 and theoretical results of Meijers

approximate formulae. The results of ODonnell

PHOTOELASTIC MEASUREMENT OF STRESSES

ACROSS LIGAMENT

XPERIMENTAL

DISTANCE ACROSS LIGAMENT, X/2h or Y /2h

Fig. 15. Stress distribut ion across the ligament section for

uniaxial tension (Bailey and Hicks).ig. l3. Effective modulus (Bailey and Hicks),


7/19

Review of analysis

of

tube sheets 285

0 b PHOTOELASTIC TES TS (HIP=?) 1

SAMPSON

- --. THEOR ETICAL FORMULAS -

MEIJERS

-

THEORETICAL FORMULAS -

SLOT AND ODONNELL

,

OO

I I I I I I I I I

o-2 0-L 0.6 04

1.0

LIGAMENT EFFICIENCY - y1

Fig. 16. Effective elastic constants obtained by Slot and

ODonnel l for triangular pattern.

are in agreement with the design practice and

confirms that the effective elastic constants of

thick perforated plate are the same for in-plane

loading, bending or torsion.

The problem involving a number of holes in

non-symmetric arrays has been solved by Hulbert

in 1970 by using a boundary point least square

technique which is an extension of the point

matching technique.22 He used a computer

program based on this technique to calculate

stresses in plates with symmetric or non-

symmetric arrays of holes either in plane strain

or plane stress conditions. Though the numerical

work in the boundary point least squares

approach increases as the number of series

coefficients and number of boundary equations

needed increases with the number of rows of

holes, it is not too difficult.

A more rational method of analysis of heat

exchanger tube sheet stresses was presented by

Yu and Syracuse in 1955.23 Their analysis

includes the interaction effect between the tube

sheet and the connecting shell and flange. The

condition at the joint is formulated based on the

fact that the sum of the moments acting on

various parts of the joint must be equal to zero.

This condition replaces the usual one of zero

edge moment for a simply supported tube sheet

or one of zero edge rotation for a clamped tube

sheet. The edge moment condition is shown in

Fig. 17. They have found from their analysis that

in the case of an external floating head type of

Ma

Fig. 17. Balance of moments at joint (Yu and Syracuse).

K,, K,,, Kf= rotation stiffness of shell, head and flange

respectively; e,, & = edge rotations of shell.

heat exchanger, the tube stresses are not

independent of shell-side pressure, which is in

contrary to Gardners and Millers observations.

Yu and Syracuse in 1956 presented another

paper, a step further towards a more exact

analysis of tube sheet problem.24 The analysis

takes into consideration the force in the middle

plane of the tube sheet due to the motion of the

joint in a direction normal to the shell axis, the

rotation-resisting capability of the tube bundle,

and the bending moments exerted by the flanges

and shells. Figure 18 shows an element of a plate

under the various loadings as described above.

The stresses in the tube sheet calculated by this

method are bound to be lower than those

calculated by their previous method.23 However,

a direct stress is at the same time induced due to

any presence of the force N. Therefore, although

the maximum final stress in the plate is always

decreased through the additional considerations

of the effect of the tube bundle, this is not so

when iV is taken into account.

Boon and Walsh25 in 1964 took into account

the reactive bending of tubes in addition to the

interaction effects under any combination of

hydro-static differential pressure and thermal

Fig. 18. Element of plate under various loadings (Yu and

Syracuse) q and m are respectively the resisting and moment

exerted by the foundation N-force in the middle plane of the

tubesheet.


8/19


g 016 -

5

g on-

- 014 -

z

E 013-

; 012-

s Oll-

TUBE BENDING NOT

- CONSIDERED

007f

--- TUBE BOJDING

CONSIDERED

006 -

10 12 14 16 16 20

22 24 26 28 30

Nla

Fig. 19. Tube sheet deflection versus n/a (Boon and Walsh).

expansion loading,

in the analysis of fixed

tubesheet exchange.

Figure 19 shows a plot

between the tubesheet deflection at the centre of

the tube sheet and n/a, where IE s the number of

tubes and a is the inside radius of the shell. Only

a small percentage reduction in deflection,

considering tube bending, is observed which do

not justify the use of more complicated models

for practical applications.

EXPERIMENTAL TECHNIQUES

0.7

Experimental values of effective elastic constants

reported by Nuno, Fujie and Ohkumaz6 are

shown in Figs 20 and 21. Experiments conducted

on plastic plates with Poissons ratio of O-39, were

perforated in a square pattern and were tested

for uniaxial loading in both the pitch and

diagonal directions.

Plates with four different

ligament efficiencies ranging between 13% and

50% were tested. The dotted line in Fig. 21 takes

into account the influence of Poissons ratio Y*

based on the empirical relation developed by

ODonnell and Langer.*

In 1960 Sampson9

undertook experimental

tests on rectangular plastic plates with v = 0.5,

using the photo-elastic frozen stress technique for

both in-plane loading and bending. Sampsons

effective elastic constants for relatively thin plates

1-o

I

I I 11111

EFFECTIVE ELAST IC MODULII

- - BAlLEV,HlCKS AND HULBERT- BAlLEV,HlCKS AND HULBERT

THEORIESHEORIES

-- PvRC APPROXIMA TION- PVRC APPROXIMA TION ,

0.6

so.5

w

04

0.3

0.2

o-1

0

DIRECTION _IRECTION _

DIRECTION.IRECTION.

EXPERlMENtq -XPERlMENtq -

Lb1

0.15 0.2 0.3 04 0.5 0.6 04 1-O

LIGAMENT EFFICIENCY

Fig. 20. Effective modulus obtained by Nuno, Fujie and

Ohkuma.

0.9

0.6

I

o-

0.1

I I I I ' ll"'l~1

EFFECTIVE P;lSiJNS RATIO

z .

I

0

cd NUNO,FUJIEld34

0 LAWRENCE

DIAGONAL

DIRECTION

CALCULATED FOR &%iALUES

MEASURED ON MATER IALS HAVING 3 ~0.39

I I 1 I I I ,l.,IIIl&.

045 0.2 0.3

04

LIGAMENT EFFIC

O-5 0.6

IENCY

04 I.0

Fig. 21. Effective .Poissons ratio obtained by Nuno, Fujie

and Ohkuma.


9/19

Review of analysis

of

tube sheets

287

in bending differ from those of plane stress, but

as the plate gets thicker, for h/p 3 2, E* and Y*

values for bending approach those for in-plane

loads, as shown in Fig. 22. This figure shows the

variation of the elastic constants with respect to

the thickness of the plate. It appears from Fig. 22

that h/p = 2 is the transition region between thin

and thick perforated plates. Tests were also

performed on an aluminium specimen with

Y = O-327 under bending to study the effect of

the materials Poissons ratio on the effective

elastic constants. Based on the test values,

Sampson established an empirical relation as

shown in Fig. 23 to estimate the values of the

effective elastic constants for any material and for

any ligament efficiency.

Leven also conducted tests27*28on circular

plastic plates with Y = O-5. The plates were

simply supported and uniformly loaded. Plate

deflections were measured and ligament stress

variations along radial sections were obtained.

The measured deflections agreed with those

calculated using Sampsons elastic constants

thereby supporting their validity.

As mentioned earlier, Bailey and Hicks2

carried out a number of experiments to verify

h 1

-=-

O-C-R 3

h-1

o3 x-7

q o-2-

*

w

I

PLANE STRESS

0.2

0406oal

2 4 6 8 10 20 40 60 80100

H/R

1-o

. H- DEPTH OF PLATE 1

2 R - PITCH OF TRIANGULAR HOLE PATTERN

0.8

t

2h - MINIMUM LIGAME NT WIDTH

Fig. 22. Variat ion of effective elastic constants with

thickness of the plate (Sampson).

12

I I I I1 IIIIII.~

_ EMPIRICAL RELATIONSH IP

1O- v~v*p[O.L3L3(vpIV-l)(L,hIRt23026H~-~

_ WHERE V#bvp=POISSON S RATIOS FOR

_

PLASllC(\H).S)

- $6~ I POISSION S RATIOS FOR METALS

-

0.6 -

0.1 0.15

0.2 0.3 0.4 0.5 0.6 08 1.0

LIGAMENT EFFICIENCY

Fig. 23. Effect of materia l Poissons ratio Y on effective

Poissons ratio v* (Sampson).

their theoretical analysis. They conducted uni-

axial tensile tests on wide aluminium plates

perforated with circular holes having a pitch-to-

diameter ratio of 1.5. The holes were drilled such

that the load was applied in the pitch direction in

one series of tests and in a diagonal direction in a

second series. Test results are shown in Table 1.

Bailey and Hicks conducted photo-elastic tests

on an Araldite model to study the stress

distribution in a plate.

2o The model was 0.135

inch thick and was perforated with square system

of holes having a diameter of l/2 inch and a

pitch of 3/4 inch. Stresses were measured across

the ligaments and along the sides of the square

panels.

As mentioned earlier Blake and Paton

conducted tests on rolled brass strips 1 inch thick

drilled with 1 inch diameter holes on different

triangular pitches, for the evaluation of deflection

efficiency.7

Electric resistance measurements

were made in tests conducted on replicas of

these test specimens prepared in electric

resistance paper. Deflection efficiency was

Table 1. Comparison of elastic constants for square pe-

netration pattern

Loading

direction

Pitch

Pitch

Diagonal

Diagonal

Elastic

constants

E*/E

V*

E*/E

V*

Experimental

values

O-46

o-2

0.27

0.55

Analytical

values

045

0.20

O-29

o-51


10/19

288

V. G. Ukadgaonker

et al.

calculated by comparing the results with those for

undrilled specimen. The experimental results are

shown in Fig. 3 from which one can observe that

the curve does not depart greatly from the boiler

joint efficiency line especially between the

standard ligament efficiency ranges of 20% to

28.5%. The results obtained by the electrical

resistance paper tests are linear and suggests

greater stiffness than the boiler joint efficiency

line whereas the mechanical tests exhibit a dip

when diameter-to-pitch ratio approaches unity.

An apparatus (a tube plate test rig) was

constructed for measuring stresses and deflec-

tions in a tube plate for various loading

conditions. The tube sheet material was naval

brass and had dimensions 3.5 inches wide, O-5

inch thick perforated by 5/8 inch diameter holes.

The Cupro-Nickel tubes of 5 feet effective length

was expanded into the tube plate and the other

end was rigidly attached to a column. Arrange-

ment for varying the diaphragm dimensions from

0 to 8 inches was provided. The deflection was

measured by a dial gauge and stresses at any

point of the tube plate by means of strain gauges.

The experimental results are bound to be in good

agreement with theoretical design values.

Duncan in 1955 carried out experiments for the

bending of circular plates.29 The actual loading

conditions were approximated by the technique

of equivalent loading, using the concentrated

loads and simulated hydro-static loading, using

the flexible-bag technique He compared the

deflection and stress in a given test plate before

and after four-, two- and one-pan drilling and

provided specific experimental values for the

structural efficiency of each type of drilled plate.

He also investigated the dependence of this

efficiency on the ratios of hole size/pitch and

pitch/thickness. Deflection efficiency for the

three different patterns tested used were 60%

for four-pass drilling, 51% for two-pass drilling

and 41% for one-pass drilling.

Hydrostatic loading experiments were per-

formed to study the effect of deflection values

when hole size of given pattern was enlarged

keeping the thickness of the model constant and

when thickness was varied for a fixed drilling

pattern. Table 2 gives the results for hydrostatic

loading experiments for two-pass drilling

patterns.

Duncan and Upfold in 1963 conducted30

flexural and tension tests on a series of

rectangular sections of steel, gun metal and

Table 2. Dellection efficieocy values obtained by hydros-

tatic load ing experiments for two-pass dri lli ig

Deflection

efficiency

Holes

S/32 inch

dia.,

2 inch

thick

Holes

l/8 inch

dia.,

2 inch

thick

Holes

3132 inch

dia.,

2 inch

thick

7

54% 68% 81%

77

44% 56% 66%

Where n = deflection of undrilled plate (theoretical)/

deflection of drilled plate obtained by experiments X 100;

17 = deflection of undrilled plate obtained by experiments/

deflection of drilled plate obtained by experiments X 100.

perspex perforated by triangular, square and

square/diagonal layout. The holes were progres-

sively jig drilled on a fixed pitch covering a range

of ligament efficiencies from 100 to 0 for all three

drilling layouts. An interferometric technique

was used to observe flexural behavior for mild

steel specimen and the Salet-Ikeda technique

was used for other materials. Tensile tests were

carried out for all the three type of drilling

patterns. Figures 24-27 summarize the results of

bending and tension tests carried out on 45

specimens. The authors concluded from their

experiments that different materials with

different modulus of elasticity and Poissons

ratio exhibit similar equivalent physical pro-

perties when perforated in a geometrically similar

manner. The experimental results also support

the theory of Bailey and Hicks.

ODonnell in 1972 conducted31 bending tests

on a series of aluminium beam specimens

perforated in triangular or square array for

various ligament efficiencies of 50%, 20% and

10%. The thickness-to-pitch ratio ranged from

3.5 for the thickest specimen to O-25 for the

thinnest. For the case of plate perforated in a

square penetration pattern, effective elastic

constants are evaluated for loading in pitch

direction as well as in diagonal direction.

ODonnells solution21 for thick perforated plate,

Meijers solutions18,19 for thin perforated plates

and Sampsons11 experimental results are also

included in his graphs.

For the case of a triangular pattern,

theoretical values of effective Youngs modulus

for thin plates are significantly higher than those

for thick plates for the same ligament efficiency.

ODonnells experimental results also follow the

same trend, i.e. an increase in effective Youngs

modulus with decrease in

h/p.

Theoretical values


11/19

Review

of

analysis

of

tube sheets

289

0.6-

I

TOFVALIDITY ,sO.Z

BOILER EFFld-.. _ . ,

0 0.2 0.k 0.6 0*6 1.0

LIGAMENT EFFICIENCY

Fig. 24. Effective Youngs modulus for triangular pattern obtained by Duncan and Upfold.

I

I

4

--A--A- DUNCAN

(TENSILE 1

.-&.-&-.-A-.- DUNCAN

+ TURLEY

TRIANGULAR PATTERN

0

0.2

04 0.6

O-8

P-d/P

Fig. 25. Effective Poissons ratio for triangular pattern

obtained by Duncan and Upfold.

of effective Youngs modulus for the thick

perforated plate appear to be valid for the entire

range of thickness h/p

3 2. Experimental results

of ODonnell also confirm this behaviour. But

ODonnells experimental vaues of E*/E in the

thin plate region is slightly higher than

theoretical results, which implies that thin plate

theoretical results of E*/E are applicable to only

much thinner plates. Again the experimental

results of Y* obtained by ODonnell follows the

same trend but slightly higher than the

theoretical results which implies that theoretical

GUN

METAL

3

SQUARE PATTERN

LAWRENCE (FLEXURE 1

A DUNCAN 1 FLEXURE 1

DIAGONAL PATTERN

LAWRENCE ( FL EXUREI-

DUNCAN (FLEXURE)

BATT (PHOTOELASTIC

PLANE STRESS 1

THEORETICAL CURVES

FROM BAILEY AND

HICKS

i

1.5 2 2.5 3 3

P/d

L

I

L I

0 O-2 04 0.5 O-6 0.7

( P-d 1 /P

Fig. 26. Effective Youngs modulus for square pattern

obtained by Duncan and Upfold.

thin plate Y* values are applicable to much

thinner plates.

For the case of square pitch pattern,

experimental results of E*/E obtained by

ODonnell are in good agreement with the


12/19


L

0

1

I

4 DUNCAN (F\EXRk)

0 BATT(PH~TO-ELASTIC),

0

LAWRENCE (FLEXURE I

GUN METAL

THEORETICAL

CURVES FROM

BAILEY AND

SQUARE PATTERN

I

I I I

1.5

I

' p/d2;' 1

0.33 o-5

0.6 0.66

(p-d)/P

1

Fig. 27. effective Poissonsatio for squarepattern obtained

by Duncan and Upfold.

theoretical results in the thick plate region. In the

thin plate region, the measured values of Y* are

slightly higher than theoretical values. Measured

values of Y* in the pitch direction differ from the

theoretical values even in thick plate region

especially for a ligament efficiency of 10%.

ODonnell has suggested one should use the

theoretical values for Y*.

NUMERICAL TECHNIQUES

Jones has determined3* the elastic stress distribu-

tion in the perforated plate with triangular

penetration pattern for in-plane loads and

bending loads. 3D analysis was made for a plate

with 5% ligament efficiency and 2D analysis for a

plate with 10% ligament efficiency. Only the

shaded portion of the plate as shown in Fig. 28 is

required to be modelled. Figure 29 shows the 3D

model. Boundary conditions, in terms of

displacements for the shaded region are obtained

from the concept of equivalent solid plate and

type of loading. Results are given in the form of

contours of stress intensity as shown in Fig. 30. In

addition to this Jones has also considered the

problem of determining the stress distribution in

EA OF FINITE

EMENT STUDY

Fig. 28. Shaded area used for finite element modelling

(Jones).

1 LCO ELEMENTS

I

925 NODES

t =c*o

t/p-2*0

L

/b r0.05

_-----

X

w

(a) TOP VIEW

(bl OVERALL VIEW OF THE

THREE DIMENSIONAL

MODEL

Fig. 29. Three-dimensional inite element model (Jones).

a circular plate with a centrally placed circular

hole which is subjected to step change of

temperature on the surface.

Meijers has given33 the refined theory of

bending and torsion of a thin perforated plate.

The classical solution requires refinement as the

ratio h/R tends to zero, where h is the thickness

of the plate and

R

is the hole radius. For very

thick perforated plates, i.e. for h/R+ 03,

solutions are available which are approximations

of plane stress or generalized plane strain

conditions. Meijers has given interpolation of

results for the intermediate values of h/R. The

accuracy of the interpolated results was checked

by using finite-element solution. Figure 31 shows

the element type and the element distributions in

the ligament. The element used is a prism


13/19


291

3 0 * 5019

4 O-5675

5 O-63&1

6 0.7006

7 O-7672

10

8 0*8339

9 0 -9005

10 l 987

Fig. 30. Contour plot of stress intensity for plane stress,

isotropic loading (Jones).

LIGAMENT EFFICIENCY

Fig.

32. Comparison of results.

results are in good agreement with the theoretical

results of Bailey and Hicks,* Meijers and

Hulbert** in respective cases of loading.

Ukadgaonker and Kale have considered35 the

problem of plates with square penetration

patterns with ligament efficiencies of 17*14%,

28*57%, 37.14% and 48.57% subjected to

in-plane loads in pitch and diagonal directions.

Finite element analysis was done by using

ANSYS version 5.0 program. 8-noded quad-

rilateral elements are used for analysis. Figure 32

shows the graph of the stress concentration factor

versus ligament efficiency for pitch and diagonal

directions.

COMPARISON OF RESULTS

Fig. 31.

Element type and element distribution (Meijers).

Triangular penetration pattern

element with 18 nodal points. He stated that he

found that the finite element solution agreed well

with interpolated results. No numerical results

were given in his paper.

Kushwaha et al. have carried out finite element

analysis of thin perforated plates with square

penetration pattern.34 Plates with thickness-to-

pitch ratio from 0.17 to O-28 were considered

with ligament efficiencies varying from 15% to

50%. Stress concentration factors were found out

for in-plane as well as bending loads. The finite

element program COSMOS was used. Their

Thick perforated plates. As pointed out by

Sampson in Fig. 22, a thick perforated plate is

one having the thickness-to-pitch ratio greater

than 2. For thick plates the effective elastic

constants for bending loads approach their

respective values for in-plane loads. Hence for

comparison purposes the results have been

shown for in-plane loads only. Figures 34 and 35

show the values of effective Youngs modulus

and effective Poissons ratio respectively, ob-

tained by various researchers. ASME Code36


14/19

292

V. G. Ukadgaonker et

al.

Fig.

33. Infin ite plate with square pattern of holes.

prescribes the values obtained by ODonnell and

Lange?* to be used in design equations.

Therefore their results have been shown by solid

curves in these figures. Figures 34 and 35 show

that the results obtained by other researchers are

in close agreement with that of ODonnell and

Langer.

Thin perforated plates. These are ones having

ratio of thickness-to-pitch less than 2. For thin

plates effective elastic constants are different for

Inn

2 0.50 -.50 -

2

0

L

. 0.40 -.40 -

z

2

a--

3 ; OJO-; OJO-

/

0

/

/

/

2

2

//

2 02020-

/

w

>

10

F

y OJO-

w

0.00

+f ,

0 SL OT ANDLOT AND ODONNELL[211DONNELL[Zll -

A

MEIJERS [191EIJERS [191

+ ODONNELL AND LANGEROD:ODONNELL AND LANGERO D:

Q HORVAY [61HORVAY [61

I

y 0.10

-x- SAMPSON [lo]x- SAMPSON [lo]

I

. DUNCAN 1301UNCAN 1301

0.60

0.00 I m I m

0.00.00 0.20.20 OLOLO 060 04060 040

1.00.00

LIGAMENT EFFICIENCYIGAMENT EFFICIENCY

Fig. 34. Comparison of results for thick plate/triangularig. 34. Comparison of results for thick plate/triangular

pattern/in-plane loading.attern/in-plane loading.

o-70111~~1

0 - SLOT

AND ODONNELL I211

AA - MEIJER S Cl91

0.60

o o -ODONNELL AND LANGER Cl21

1.00

0.20 *

0.60 0.00 l-00

LIGAMENT EFFICIENCY

fig. 35. Comparisonsof results in plane loading (thick

plate/triangular pattern).

in-plane loads and bending loads. Figures 36 and

37 show the results obtained by various

researchers. In these figures a solid curve is

shown for the results obtained by Meijers.

Square penetration pattern

Thick perforated plates. Figures 38 and 39 show

the values of effective Youngs modulus in pitch

and diagonal directions obtained by various

z 0*60-

Ga

z

= 0*50 -

.P

z

2 040 -

*

.

ul

2 0.30-

2

0

=0.20-

?

L

k O.lO-

L

w

-MEIJERS[19]

- DDONNELLB

+-+ SAMPSON

o.ootm 11

0.00

.20 O-40 0.60

O-60 l-00

LIGAMENT EFFICIENCY

Fig. 36. Comparison of results (thin plate/triangular

pattern/bending).


15/19


293

0.60

1

&e-a MEIJERS [191

A AA ODONNELL [311

o o o SAMPS ON 1101

Q BAILEY 6 HICKS 1201BAILEY 6 HICKS 1201

A MEIJERS [191MEIJERS [191

-t- SLOT 6 ODONNEL[211t- SLOT 6 ODONNEL[211

o NUNO, FUJIE 6 OHKUMANUNO, FUJIE 6 OHKUMA

LIGAMENT EFFICIENCY

Fig. 39. Comparison of results for thick plate/square

pattern/in-plane loading.

LIGAMENT EFFICIENCY

Fig. 37. Comparison of results (thin plate/triangular

pattern/bending).

035

g 0.30

C

0.

i 025

a

a i

investigators for plane stress loading. Bailey and

Hicks results are based on accurate theoretical

method hence these values are represented by a

solid curve in Figure 40 and Figure 41 which

show the values of effective Poissons ratio for

plane stress loading. Since the effective elastic

constants in bending are the same as those for

in-plane loading, they have not been shown

separately.

(: 0.20

=:

i 0.15

t

0

-o-BAILEY 6 HlCKSt201

A MEIJERS[lSI

-.-SLOT 6ODONNELLt211 :

o NUNO,FUJIE 6 OHKUMA t261:

0.001

0.00

0.20 0.40

0.60 0.80 1.00

LIGAMENT EFFICIENCY


plate/in-plane loading.

Thin perforated plates. Figures 42 and 43 show

the values of effective Youngs modulus for

bending loads applied in pitch and diagonal

directions. Experimental results of ODonnell

and theoretical results of Meijers have also been

shown. Figures 44 and 45 show the variation of

effective Poissons ratio for bending loads.

0.00 0.20 040 0.60

0.80 l-00

LIGAMENT EFFICIENCY

Fig. 38. Comparison

of results (thick plate/square

pattern/in-plane loading)

Comparison of stress concentration factors

Figure 46 shows the ratio of the maximum local

stress to the nominal stress in the equivalent solid


16/19


g 0.30

. .

. NUNO,FUJlE AND

::

OHKUMA1261

o M BAILEY AND HICKSI -- BAILEY AND HICKSI

A A A MEIJERS1191A A MEIJERS1191

0

- SLOT AND ODONNEL[Zl I-SLOT AND ODONNELI21 I

. .

. NUNO,FUJlE AND

OHKUMA1261

5

g 0.250.25 -

\?

?

f

s 0.200.20 -

a

v)

)

g 0.150.15 -

w

>

u 0.10 0.10 -

L

E

\

w 0.05 -

\

\

-

0 DONNELL [31]

MElJERS c 91

O-00-00

OBOBO o-co-co 0.60.60 O-80-80

TOOOO

o-00 0.20 040 060

080 I-00

LIGAMENT EFFICIENCYIGAMENT EFFICIENCY

LIGAMENT EFFICIENCY


pattern/in-plane loading.

1:

0.70 1,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,1,,,,,,,,~

/

+1

0 0 ODONNELL131J :

-MC MEIJERS[lS]

- 0.00 0.20 040 0.60 O-80 1.00

LIGAMENT EFFICIENCY

Rg. 42. Comparison of results for thin-plate/square

pattern/bending.

plate or the stress concentration factor for

isotropic nominal stress field. Figures 47 and 48

show the same ratio for uniaxial and pure shear

nominal stress fields respectively. Theoretical

solutions obtained by Bailey and Hicks* and

Hulbert** are plotted in these figures.

Nuno, Fujie and Ohkuma have conducted26

photo-elastic tests using uniaxial tensile loads in

both pitch and diagonal directions. Specimens

having four different ligament efficiencies ranging

Fig. 43. Comparison of results for thin plate/square

pattern/bending.

9

f

0 .l 0

Y

k

w 0.05

0 0 0 ODONNELLt311

- MEIJERS [ 19 I

0

1./

0

0

0.00 0.20 0.10 0.60 0.80 1.00

LIGAMENT EFFICIENCY


pattern/bending.

between 15% and 50% were tested. The

two-dimensional photo-elastic method using thin

models at room temperature was used.

Satio has published3 some series solutions for

infinite rows of holes in an isotropic stress field.

These results are included in Figure 46.

Stepanek has reported results of some

photoelastic work with uniaxial tension in pitch

direction and with isotropic tension.38 These

results are included in Figure 46 and 47.


17/19

Review

of

analysis

of

tube sheets

295

L

;; 0.70 -

H

$ o*so -

9

2 o-50 -

m

-21 040 -

In

: 0.30 -

w

>

5 0.20 -

W

IL

:: 0.10 -

HZ1

P

a

0 ODONNELL(30

- MEIJERS (19)

o*oot~

0.00 0.20

040

0.60 0-30 l-00

LIGAMENT EFFICIENCY


pattern/bending.

-BAILEY HICKS & -

\ HUBERT THEORIES _

- . NUNO,FUGlEit OHKUMA

\

1 - & SAlfO

_ e STEPANEK

0

I I I I .IlIIIII.lL

0.1

0.15 0.2

0.3 0.4 a546 08 10

$ LIGAMENT EFFICIENCY

Fsg. 415.Maximum stress multip liers for equal biaxial

(isotropic) tension.

SCOPE FOR FUTURE WORK

Future work is proposed to be undertaken in the

following three areas: Analytical, Numerical and

Experimental.

Analytical approach

The method of solution proposed here is for a

plate with square penetration pattern. The

---EiAILEV,HICKSANDHULBE

- -PVRCAPPROXlYAlION

NUN0 FUJIELOHK UMA _

EXPERIMENTS

0 DIAGONAL DIRECTION -

l SQUARE DIRECTION

o L EVEN EXPERIMENTAL -

0.1 0.15 0.2 0.3 0.1 0.50.6 0.6 180

LIGAMENT EFFICIENCY

Fig. 47. Maximum stress mult iplier for uniaxial tension.

NUNO, FUJIE 6. OHKUMA

EXPERIMENTS

l DIAGONAL DIRECTION

0 SQUARE DIRECTION

36

I 1 II11111

SHEAR LOAD

- BAILEY h HICKS AND

HULBERT THEORIES

--- PVRC APPROXIMA TION

LIGAMENT EFFICIENCY

Fig. 48. Maximum stress mult iplier for pure shear stress

field conditions.

solution is based on the Complex Variable

technique and the results obtained from finite

element analysis. Consider the infinite perforated

plate as shown in Fig. 33. The dotted square

portion in Fig. 33 shows the general square

element with a hole. Stress boundary conditions

are determined first at the boundaries of this

square boundary with stress-free hole boundary

by finite-element analysis for uniaxial tension in


18/19

296

V. G. Ukadgaonker

et al.

the pitch direction. For finite-element analysis

the plate with a finite number of holes is taken.

Then the unknown complex stress functions are

found out which satisfy the edge boundary

condition on this square boundary. Similarly the

two complex stress functions can be found out for

complex loading conditions such as biaxial

tension, hydrostatic tension, bending, etc.

Numerical approach

10.

The entire tube sheet could be modelled with

actual geometry details in three dimensions. The

11.

actual boundary conditions can be simulated.

Combined action of the in-plane loads and the

bending due to the lateral fluid pressures can be

considered. It is also possible to include the

stiffening effect of the tubes and the temperatures

at various locations to account for the thermal

stresses. Solution may be found by finite-element

or finite-difference method. Due to the

enormous data in three dimensions a high speed

super computer will have to be used.

Experimental approach

A loading frame may be devised to test a

photo-elastic model in biaxial hydrostatic

tension. The model may be tested for combined

loading, i.e. in-plane and bending loads. Effects

of pressurized holes may be simulated by

introducing fluid at very high pressures in tubes

by some hydraulic loading arrangement. All

these tests may be carried out by stress freezing

technique used for 3D photo-elasticity.

REFERENCES

1. Osweiller, F., Evolution and synthesisof the effective

elastic concept for the design of tubesheets,

ASME

Transaction of Pressure Vessel Technology, 111, (1989)

209-217.

2. Gardner, K. A., Heat-exchanger tubesheet design,

ASME Journal of Applied Mechanics, 70,

(1948)

377-385.

3. Gardner, K. A., Heat-exchanger tubesheet design-2,


(1952)

159-166.

4. Miller, K. A. G., The design of tube plates in

heat-exchangers, roceedings of Institution of Mechani-

cal Engineers, 18, (1952)

215-231.

5. Malkin, T., Notes on a theoretical basis or the designof

tubesheets of triangular layout,

ASME Journal of

Applied Mechanics, 74,

(1952)389-396.

6. Horvay, G., The plane stress problem of perforated

7.

8.

9.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

plates,ASME Journal of Applied Mechanics, 74, (1952)

355-360.

Blake, C. S. Paton, A. D., Designof rectangular tube

plates for large heat exchangers,ournal of Mechanical

Engineering Sciences, 5, (1963)

211-226.

Gardner, K. A., Tubesheet design: a basis for

standardization, First International Conference on

Pressure Vessel Technology,

Delft, (1969) 621-647.

Salerno, V. L. Mahoney, J. B., A review, comparison

and modification of the presentdeflection theory for flat

perforated plates, Welding Research Council Bulletin,

52, July 1959.

Sampson,R.

C., Photoelastic Frozen Stress

Study

of the

Effective Elastic Constants

of

Perforated Plates,

WAPD-

DLE-319, May 1959.

Sampson, R. C., Photoelastic analysis in perforated

material subjected to tension or bending,

Bettis

Technical Review,

WAPD BT

18,

April 1960.

ODonnell, W. J. Langer, B. F., Design of perforated

plates,

ASME Journal o f Engineering for Indust ry,

(1962) 1047-1060.

Horvay, G., Bending of honeycombs and perforated

panels,


(1952)

122-123.

Mahoney, J. B. Salerno, V. L., Stress analysis of

circular plate containing a rectangular array of holes,

Welding Research Council Bulletin, 106, 1965.

Mahoney, J. B., Salerno, V. L. Goldberg, J. E.,

Analysis of perforated circular plate containing

rectangular array of holes,

Welding Research Council

Bulletin, 80, August 1962.

Mahoney, J. B. Salerno, V. L., Stiffnesscoefficients

for a circular plate with rectangular array of holes,ATA

Research Report

No. ZOO,April 1964.

Mahoney, J. B., Salerno, V. L. Goldberg, J. E.,

Analysis of Perforated Plates Containing a Rectangular

Array of Holes, RE-145, Grumman Aircraft Engineer-

ing Corporation ResearchDepartment, April 1961.

Meijers, P., Doubly periodic stress distribution in

perforated plates, thesis, University of Technology of

Delft, 1967.

Meijers, P., Plates with doubly periodic pattern of

circular holes loaded in plane stress or in bending,

Proceedings of the First International Conference on

Pressure Vessel technology, Part Z, Design and

Analysis,

Delft, ASME, (1969) 551-570.

Bailey, R. Hicks, R., Behaviour of perforated plates

under plane stress, ournal o f Mechanical Engineering

Science, 2, (2), (1960) 143-161.

Slot, T. ODonnell, W. J., Effective elastic constants

for thick perforated plates with square and triangular

penetration pattern,

ASME Journal o f Engineering for

Industry, 93, (1971) 1081-1088.

Hulbert, L. E. Niedenfuhr, F. W., Accurate

calculations of stressdistributions in multiholed plate,

ASME Journal o f Engineering for Indust ry, 87,

(1965)

331-336.

Yi-Yuan, Yu Syracuse, N. Y., Rational analysis of

heat exchanger-tube sheet stresses, SME Journal of

Applied Mechanics, 23, (Sept. 1956)468-473.

Yi-Yuan Yu Syracuse,N. Y., Axisymmetric bending

of circular plates under simultaneousaction of lateral

load, force in the middle plane and elastic foundation,

ASME Journal of Applied Mechanics,

79, (March 1957)

141-143.

Boon, G. B. Walsh, R. A., Fixed tubesheet heat

exchangers,

ASME Journal o f Applied Mechanics, 86,

(June 1964) 175-180.


19/19

Review of analysis of

tube sheets 297

26. Nuno, H., Fujie, T. Ohkuma, IL, Experimental Study

of the Elastic Properties of Perforated Plates with

Circular Holes in Square Pattern,

MAP1 Laboratory

Research Report No. 74, Mitsubishi Atomic Power

Industries Laboratory, March 1964.

27.

Leven, M. M.,

Preliminary Report on Deflection of Tube

Sheet, WAPD-DLE320, May 1959.

28. Leven, M. M., Photo-elastic deformations of stress n

tubesheets nd comparisonwith calculatedvalues,Bettis

Technical Review, WAPD-BT-18, (April 1960).

29. Duncan, J. P., The structural efficiency of tubesheets or

heat exchangers,

Proceedings

of

Institute of Mechanical

Engineers, 169,(1955) 789-802.

30.

Duncan, J. P. Upfold, R. W., Equivalent elastic

properties of perforated bars and plates,

Journal of

Mechanical Engineering Science, 15, (l), (1963) 53-65.

31. ODonnell, W. J., Effective elastic constants for the

bending of thin perforated plates with triangular and

square penetration patterns, ASME Journal of Engi-

neering for Industry,

95, (1973) 121-128.

32. Jones, D. P., FEA for perforated plates containing

triangular penetration pattern of 5 and 10 percent

ligament efficiency, ASME Journal of Pressure Vessel

Technology, 97, (August 1975)199-205.

33. Meijers, P., Refined theory for bending and torsion of

perforated plates,

ASME Journal of Pressure Vessel

Technology,

loS, (November 1986)423-429.

34. Bhatacharya, A., Kushwaha, H. S. Mahajan, S. C.,

Determination of Stress Multipliers for Thin Perforated

Plates with Square Penetration Pattern, Bhabha Atomic

ResearchCenter Report, 1991.

35. Ukadgaonker, V. G. Kale, P. A., Stress analysisof

perforated plates by finite-element and photo-elasticity

methods,

39th Congress of the Indian Society for

Theoretical and Applied Mechanics,

Visakhapattanam,

India, December 1994.

36. ASME Boiler and Pressure Vessel Code, Section II I,

Article A-8000,

1971.

37. Saito, H., Stress n a plate containing infinite parallel

rows of holes,Math. Mech., 37, (April 1957) 111-115.

38. Stepanek, S., Skoda Works, Pilsen, Czechoslovakia,

transmitted these results to James L. Mershon, Vice

Chairman, PVRC DesignDivision.

Review of Analysis of Tube Sheets

Documents