Journal of Engineering for Industry,
Trans. ASME, Vol. 84, August, 1962
Paper No.
61-WA-llS
W. J. O'DONNELL
Associate
Engineer, Westinghouse
Bettis
AtomicPower laboratory, Pittsburgh,Pa.
Assoc.Mem.ASME
B. F. LANGER
ConsultingEngineer,WestinghouseBettis
AtomicPower laboratory, Pittsburgh,Pa.
FellowASME
Design of Perforated Plates
1
This paper describes a method for calculating stresses and deflections in perJorated
plates with a triangular penetration pattern.
The method is based partly on theory and
partly on experiment.
Average ligament stresses are obtained from purely theoretical
considerations but e.tfective elastic constants and peak stresses are derived from strain
measurements and photoelast-ic tests. Acceptable limits for pressure stresses and thermal stresses in heat-exchanger tube sheets are also proposed.
Introduction
TIm
calculation of stresses in perforated plates is a
subject which has received considerable attention as a result of
the widespread use of flat tube sheets in heat-exchange equipment. Major contributions have been made by Horvay [1, 2],2
Malkin [3], Gardner [4, 5, 6], Duncan [7], Miller [8], Galletly
and Snow [9], and Salerno and Mahoney [10]. Most of the published work has been limited to perforations arranged in an
equilateral triangular pattern, and the present paper is no exception. The Pressure Vessel Research Committee of the Welding
Research Council is currently sponsoring work on square patterns of holes but no results are available as yet.
Most heat-exchanger tube sheets are designed to meet the
standards set by the Tubular Exchanger Manufacturers Association [11]. In these TEMA standards the thickness required to
resist shear depends on the ligament efficiency of the perforations, but the thickness required to resist bending is independent
of ligament efficiency. S This does not mean, of course, that
bending stress is not affected by ligament efficiency; it does mean,
however, that all tube sheets designed to TEMA standards are
designed to be safe with the minimum allowable ligament ef-
ficiency of 20 per cent, as specified in Par. R-2.5 of reference [11].
'When service conditions are usually severe or when the utmost is
desired in reliability and optimum design, stresses should be calculated in detail and realistic allowable stress values should be
set. It is the realization of this fact that led to the previous work
and the work described in this paper.
Most of the proposed methods for analyzing perforated plates
have involved the concept of an "equivalent" solid plate [3, 4].
In one method the equivalent solid plate has the same dimensions
as the actual plate but its flexural rigidity is reduced by a factor
called its defleetion efficiency. In another method the equivalent
plate is also the same as the solid plate, but it has fictitious elastic
constants E* and p* in place of the actual constants of the
material E and P. The latter concept is used in this paper.
General
Method
ofAnalysis
The general method of evaluating stresses and deflections in a
perforated plate having a triangular penetration pattern is:
Step 1. Calculate the nominal bending and membrane stresses
and deflections of an equivalent solid plate having the effective
modified elastic constants E* and p* and the same dimensions as
the perforated plate.
1 This work is part of a dissertation suhmitted by W. J. O'Donnell
Step 2. Calculate physically meaningful perforated-plate stress
to the University of Pittsburgh in partial fulfillment of the requirevalues from the nominal stress values in the equivalent solid plate
ments for the degree of Doctor of Philosophy.
from Step 1. Deflections of the perforated plate are the same as
2 Numbers in brackets designate References at end of paper.
the deflections of the equivalent solid plate.
3 See, for example, reference [11],paragraphs R-7.122 and R-7.123.
Contributed by the Petroleum Division for presentation at the
"Yhen the perforated plate is part of a structure, as in the case
Winter Annual Meeting, New York, N. Y., November 26-December
of a heat exchanger, Step 1 is accomplished using classical
1, 1961, of THE AMERICAN
SOCIETYOF lvIECHANICAL
ENGINEERS.
structural-analysis methods. A study of the effective elastic conManuscript received at ASME Headquarters, July 26, 1961. Paper
stants for use in Step 1 is contained herein, and values based on
No. 61-WA-115.
----Nomenclature-----------------------------Material Properties
D*
E* H3/12 (1 - p*2), effective flexural rigidity of perforated plate, lb-in.
E
elastic modulus of solid material, psi
E*
effective elastic modulus of perforated material, psi
Sm - allowable membrane stress intensity of material, psi
p
Poisson's ratio of material, dimensionless
p*
effective Poisson's ratio for perforated material, dimensionless
Pp
Poisson's ratio of plastic-model material, dimensionless
Pp *
effective elastic modulus of perforated plastic models, dimensionless
OiT
thermal expansion coefficient, in/in deg F
Co-ordinatesand Dimensions
l' = radial distance of ligament from center of circular perforated plate, in.
co-ordinates shown in Figs. 7 and 8, in.
width of plate rim, Fig. 13, in.
minimum ligament width, Fig. 6, in.
minimum ligament width for thin ligament at misdrilled holes, in.
outside radius of plate rim, Fig. 13, in.
distance between center lines of perforations, Fig. 6',
in.
radius of perforations, Fig. 6, in.
plate thickness, in.
X, Y,Z
b
2h
2hmin
p
H
Stresses
U
T'
(TO
radial and tangential stresses in equivalent solid
circular plate, psi
0' r or 0'0, whichever has largest absolute value, psi
stresses in minimum ligament section, Fig. 7, psi
(Continued
on
lIe.Tt
paoe)
Discussion on this paper will be accepted at ASME Headquarters until January 10, 1962
experimcntal results by Sampson are recommended. Methods of
evaluating average and peak ligament stresscs for Step 2 of the
analysis are developed and appropriate design limits are recommended for these values. A method of evaluating the acceptability of misplaced holes is also given.
Effective Elastic Constants for Perforated Plates
When a perforated plate is used as a part of a redundant structure, the values used for the effective elastic constants will affect
calculated stresses in the remainder of the structure, as well as in
the perforated plate itself. For example, the amount of rotation at the periphery of a steam-generator tube sheet depends on
the relative rigidity or the tube sheet with respect to the rest of
the heat exchanger. If effective elastic constants (particularly
E*) which are too low are used in the analysis, the theoretical
rotation at the periphery of the tube sheet due to pressure loads
across the tube sheet will be greater than the actual rotation. The
calculated stresses at the periphery will then be lower than the
actual stresses. This can be seen from fig. 29 of reference [12].
Correspondingly, if an effective elastie modulus which is too high
is used in the analysis, the calculated pressure stresses at the
center of the tube sheet would be low. If the tube sheet is taken
to be too rigid, the calculated stresses, due to a pressure drop
across the tube sheet, in the head and shell at their junction ",;th
the tube sheet would be lower than the actual stresses. Since
stresses in these areas are usually among the highest stresses in a
heat exchanger, it is important that they be evaluated properly.
Taking the tube sheet to be too flexible causes calculated thermal-stress values in the tube sheet and in the remainder of the
heat exchanger to be below the actual stress values.
From the foregoing discussion it may be concluded that it is not
possible to insure conservatism in heat-exchanger or tube-sheet
stress calculations by assuming effective elastic constants which
are known to be either too high or too low. The best estimates of
p* and E*, rather than the highest or lowest estimates should be
used.
Many different sets of effective elastic constants for perforated
materials having a triangular penetration pattern have been
proposed. Five of the best known sets of values have been obtained from theoretical considerations and two have been
obtained experimentally:
1 Theoretical
2 Theoretical
3 Theoretical
strained warping
4 Theoretical
Horvay plane stress [1].
Horvay bending [2].
modified Horvay bending, corrected for conby Salerno and Mahoney [10].
Malkin bending [3].
5 Theoretical modified Malkin bending corrected for
strained warping by Salerno and Mahoney [10].
6 Experimental Sampson plane stress []3].
7 Experimental Sampson bending [13].
COIl-
The "plane-stress" constants apply to loads in the plane of the
perforated plate; i.e., tensile or compressive loads as opposed to
bending. All of the theoretieal values for E* and p* were intended
to apply only to those perforated materials having ligaments
thinner than those usually found in tube sheets. For example,
Horvay recommends his theory only for ligament efficienciesless
than 20 per cent.
Sampson Effective Elastic Constants
The Sampson experimental values of the effective elastic constants for both plane stress and bending loads were obtained
in tests on rectangular coupons at the ~T estinghouse Research
Laboratories. The test specimens were made of plastic material,
p = 0.5. Subsequent tests were run to evaluate the effect of the
material Poisson's ratio on the values for the effective elastic constants. Plane-stress constants were obtained by applying uniaxial tensile loads, and bending constants were obtained by applying pure bending loads. These values were found to differ quite
markedly from the theoretical values.
The validity of the general method of using effective elastic
constants and stress multipliers to calculate stresses and deflections in tube sheets was checked by Leven in tests on perforated circular plates [14, ]5]. The plates were made of plastic
(p = 0.5) and were simply supported and uniformly loaded.
Plate deflections were measured and ligament stress variations
along radial sections were obtained. The results give support
to the validity of the Sampson experimental method of determining the effective elastic constants using perforated rectangular
coupons subject to uniaxial loads. The measured deflections
agreed best with those calculated using the effective elastic constants obtained experimentally by Sampson. Moreover, the
measured local stresses agreed closely "ith those calculated using
the stress-ratio factors obtained by Sampson. Hence, the Sampson effective elastic constants are considered to be the most
accurate for use in design calculations.
The Sampson effective elastic constants for relatively thin
plates in bending differ significantly from those in plane stress.
However, as a plate in bending gets thicker, the stress gradient
through the depth gets smaller and it is reasonable to expect that
a very thick plate would not be affected appreciably by the small
stress gradient in the thickness direction. Consequently, the
----N0 menclature'---------------------------i1:u 0"11' T1/X'
ur,uo
(Jrim
(jeer
U"max
Seer
Stress Multipliers
stresses averaged through depth of plate, psi
transverse shear stress averaged through depth of
plate, psi
nominal bending plus membrane stress at inside
of rim, psi
maximum principal stress basecI on average
stresses across minimum ligament section, psi
stress intensity based on stresses averaged across
minimum ligament section at plate surface, psi
maximum local stress, psi
stress intensity based on stresses averaged across
minimum ligament section and through depth
of plate, psi
(dimensionless)
K
value given in Fig. 10
Kn = value given in Fig. 14
2
Kr
K",
Ku
Y
value given in Fig. 13
value given in Fig. 15
value given in Fig. 10 for {3= 0
valuegiveninFig.]2
Others
F
11
111
P
I1P
Tp
Ts
TIl
Tc
f3
If
normal force carried uy ligament, Fig. 6, lb/in.
shear force carried by ligament, Fig. 6, lb/in.
moment carried by ligament, Fig. 6, in-lb/in.
pressure on plate surface under consirleration, psi
pressure drop across tube-sheet, psi
temperature at primary tube-sheet, surface, deg F
temperature at secondary tube-sheet surface, deg F
temperature of hot side of tube sheet, Fig. 14, deg F
temperature of cold side of tube sheet, Fig. 14, deg F
(jr/(jO
or (jO/(jr whichever gives -1
f3
1, dimensionless
angular orientation of ligament, Fig. 7, radians
<:
<:
Transactions of the AS M IE
0.6
I
I
I
I
I
I
I
I
1-
0.5
~
'"
0.4
h
E*/E
R
'\
= 1
3
"tJ"
0.3
II
h
R
0.2
PLANE
""'- .......••.
= 4.1.--,
STRESS
-=< I-
1\
0.1
2R
2h
=
=
=
PITCH OF TRIANGULAR
HOLE
MINIMUM
LIGAMENT
WIDTH
I
I
I
H
o
I
0.2
0.4
DEPTH
OF
I
PLATE
I
I
0.6 0.8 I
2
PATTERN
I
I
4
6
I
I I
I
I
8 10
20
I
I
40
60
80100
H/R
Fig.'
Variation of Sampson effective
Poisson's ratio of solid material)
10
,
I
elastic
modulus
with depth
I
I=
~EPTH
H
2R
0.8
2h
=
=
I
PITCH
OF IpLATE'
LIGAMENT
I
HOLE PATTERN
WIDTH
0.6
v*
h
R
0.4
/
0.2
,..
I
= 4"--""
,..-
~
~~
h
R
V
r-
.3
I
II
-...r- -Lr
-
/
tf
PLANE
STRESS
o
0.2
(vp = 0.5
in bending
I
I'
I
OF TRIANGULAR
MINIMUM
of a plate
li
-
I
I
0.4
0.6 0.8
I
4
2
6
8
10
20
40
60
80100
H/R
Fig. 2 Variation of Sampson effective
= Poisson's ratio of solid material)
Poisson's
values for the effective elastic constants for a plate in bending
should approach the plane stress values as the plate gets thick.
Fig. 1 shows the variation of E* with the relative thickness of a
plate in bending, and Fig. 2 shows the same variation for /1*.
Note the rather abrupt transition in the E* IE-values that occur
in the vicinity of H IR = 4. This appears to be what might be
interpreted as a transition region between "thick" and "thin"
perforated plates.
Obviously, it would be inconvenient to use one set of elastic
constants for bending loads and another set for in-plane loads.
Fortunately, this is not necessary as long as the plate is thicker
than about twice the pitch of the perforations (Hill> 4) and
this situation occurs in most heavy-duty heat-exchange equipment which requires the refined analysis described here. The
effective elastic cOllstants in bending for H IR > 4 do not differ
greatly from the plane-stress values. Fig. 3 shows the bending
constants at Hill = 7 plotted with the uniaxial plane-stress constants. Accordingly, the plane-stress constants appear to be the
Journal of Engineering
for Indllstry
ratio with a depth of a plate
in bending
(vp = 0.5
most acceptable values for plates having a relative thicknes8
4.
Notice that the uniaxial plane-stress values of effective Poisson's ratios (/1",* and /Iv *) vary with the orientation of the load
with respect to the hole pattern. The impracticality of factoring
this anisotropic behavior into the analysis is immediately evident, and values must be used which represent the approximate
Poisson's effect in all directions. This is not considered to be a
serious problem, however, partly because the principal strel:iSes
are generally not oriented in the directions resulting in the largest
differences between the effective Poisson's ratios (the x and ydirections, respectively, in Fig. 3), and partly because these differences do 1I0thave a large effeet on the calculated stresses.
Sampson evaluated the effective elastic constants for perforated
plastic materials (/lp = 0.5) over a wide range of ligament efficiencies under bending and plane-stress loads. He then proceeded to evaluate the effect of material Poisson's ratio on the
effective elastic constants. This was accomplished by measuring
HIR>
3
1.0
1.2
1.1
0.9
1.0
olYo
o 0 o-x
0.8
V>
•...
Z
0.7
<l
•...
u
p*:
WHERE:
(Lnh/R+2.3026l+lr'
vt[O.4343(VpIV-11
vt
6 vp
v* 6v
=
POISSON'S
: POISSON'S
RATIOS FOR PLASTIC(v:O.5
RATIOS
FOR METALS
0.8
0.7
~2h
V>
Z
0
0.9
2R
0.6
~ 0.6
I
*
U
IV>
0.5
0.5
"-"*,."0.4
<l
...J
w
0.4
0.3
LoJ
~
0.2
I-
u 0.3
w
0.1
ll..
ll..
LoJ
E*/E
0.2
0.15
0.1
0
0.1
Fig.4
0.15
0.2
LIGAMENT
0.3
0.4
EFFICIENCY,
.!!I'I
0.5 0.6 0.70.50.91.0
h/R
Fig. 3 Comporison of Sompson effective elostic constants for bending
and plane stress
the effective elastic constants of an aluminum specimen (II =
0.327) in pure bending. The specimen had a relative thickness
in the range of "thick plates" (H IR = 7). Hence, the test values
obtained from this specimen are felt to be applicable in the entire
range of parameters (HIR > 4), and for plane-stress loads as
well as bending loads. Based on these test values, correlations
were established on an empirical basis to estimate values of the
effective elastic constants for any material and for any ligament
efficiency. This relation is given in Fig. 4. The maximum deviation of any of the aluminum-bar test points from. this empirical
relation is 7 per cent. The corresponding relation between p*
for steels (II = 0.3) and IIp* for plastic (lip = 0.5) was used to
modify the Sampson plane-stress II*-values obtained in tests on
plastic specimens in order to obtain corresponding values applicable to metal plates. The resulting values of 11* for II = 0.3 are
recommended for use in design calculations. These values are
given in Fig. 5. They can be used for both plane stress and
bending loads in the plate, as discussed previously.
The effective elastic-madulus ratios E* I E were found to be
unaffected by changes in the Poisson's ratio of the material.
Hence, the Sampson plane-stress values of E* IE, taken from Fig.
3, are recommended for use in design calculations. These values
are also given in Fig. 5.
The smallest ligament efficiency of the coupons tested by
Sampson was 15 per cent. Hence, the values given in Fig. 5
~hould not be extrapolated much below this value.
The error in stress values calculated using the general effective
elastic constants given in Fig. 5 instead of the constants measured
by Sampson (which depend on the type of loading, direction of
loading, and the thickness of the plate) was evaluated. The largest error in the maximum local stresses or in the maximum average
ligament stresses that are limited by the design criteria recommended herein for any type or direction of loading and any
plate thickness (HIR > 4) was found to be 8 per cent.
Wffening Effect of Tubes
When tubes are rolled or welded into a tube sheet, the question
4
0.2
0.3
0.4
0.5 0.60.70.80.91.0
LIGAMENT EFFICIENCY h/R
Effect of material Poisson's
ratio
II
on effective Poisson's
ratio
v*
always arises regarding the degree to which the tubes increase
the stiffness of the plate. As mentioned previously, it is not
always conservative to assume either a maximum or a minimum
value for the stiffness. In some strain-gage tests by A. Lohmeier,
of the 'Vestinghouse Steam Division, on a steam generator which
had seen considerable service, very good correlation was obtained
between calculated and measured sti'esses when full credit was
taken for the tube wall in the caleulations; that is, when the hole
size was taken as the ID rather than the OD of the tubes [16].
When the ligament effic.iencywas calculated on the basis of the
OD of the tubes, the measured stresses due to pressure loading
averaged about 75 per cent lower than the calculated values.
While this one test cannot be considered as conclusive evidence,
the authors believe that it is a strong indication. Furthermore,
it can be shown that sinee the membrane stresses in the tube
sheet are usually low, very little residual compression is required
in the tube wall to make it follow the strains in the drilled hole.
Therefore the authors tentatively recommend that fuJI credit be
taken for the tube-wall thickness. Further confirmatory tests
are planned.
Proposed Stress Limits
Before proceeding to the detailed calculation of stresses, it is
necessary to decide which stresses are significant and, consequently, should be calculated and limited in order to assure an
adequate design. The peak stress in a perforated plate is not
necessarily the most significant one. Primary stresses, those which
are required to satisfy the simple laws of equilibrium of internal
and external forces, and are consequently not self-limiting, should
be the ones most severely limited. Secondary stresscs, those
which are only required to accommodate to an imposed strain
pattern (e.g., thermal expansion) can be allowed to go higher than
primary stresses. If the latter are kept lower than twice the
yield strength, loadings subsequent to the initial loading will
produce strains within the elastic limit. Peak stresses in localized
regions are of interest only if they are repeated often enough to
produce fatigue. For tube sheets, consideration must also be
given to distortion of the holes which may cause leakage around
the tube.
The use of the maximum-shear theory of failure rather than the
m(lximum-st,ress theory of failure is recommended. In order to
Transactions of the AS !ViE
1.0
0.9
0.8
H
(/)
=
H /R
z 0.7
j:!
(/)
z
0
u
OF
THICKNESS
I-
>
4
0.6
u
I-
en 0.5
<t
.-J
W
W
> 0.4
I-
u
W
lJ..
lJ..
w
0.3
E*/E
D*/D
0.2
0.1
o
0.15
0./
0.2
h/R,
Fig. 5
0.3
Typical Ligament
in a Uniform
Pattern
(a) llfechanical Loads (i.e., pressure loads but not thermal
loads):
(i) The stress intensity based on stresses averaged across the
minimum ligament section and through the thickness of the plate
should be limited to prevent stretching of the plate. This stress
is analogous to the average stress intensity in the shell of a pressure vessel under internal pressure and, consequently, should be
limited to a value about t.he same as the allowable stress values in
t.he ASME Boiler Code. (The quest.ion of whether or not the
values in t.he 1959 edition of t.he Code are t.oo conservative for
vessels which are analyzed carefully for high stress is beyond the
scope of this paper. In t.he 1959 Code, the allowable stresses do
not exceed 5/8 of the yield strength of a ferrous material or 2/3 of
Journal of Engineering tor Industry
0.5 0.6 0.7 0.80.9 1.0
LIGAMENT EFFICIENCY
Effective elastic constants
make allowable shear-stress values comparable to the more
familiar tensile values, calculated stresses are expressed in terms
of two times the maximum slWar stress; which is the largest algebraic difference between any"two of the three principal stresses.
This quantity is called the "equivalent intensity of combined
stress," or more briefly, the "stress intensity."
The following stress limits are proposed:
1
0.4
for perforated
plates
the yield strength of a nonferrous material.) Let us call thi~ basic
st.ress intensity allowance Sm.
(ii) The stress intensity based on stresses averaged across the
minimum ligament section but not through the thickness of the
plate should be limited to prevent excessive deflection. This
stress is the sum of membrane plus bending effects and, since the
limit-design factors for flat plates are greater t.han 1.5, it can
safely be allowed to reach a value of 1.5 Sm.
(b) Combined ilfechani-eal and Thermal Loads:
(i) The stress intensity based on stresses averaged across the
minimum ligament section but not through the depth should be
limited to 3 Sm.
(ii) The peak stress intensity at any point due to any loading
should be limited by cumuintivc fatigue considerations, as described in [17].
2 Isolated or Thin Ligament.
If a high stress occurs in a single
ligament due to a misdrilled hole, the foregoing limits may be relaxed. For combined pressure and thermal loads, the stress intensity based on average stresses in the ligament cross seet-ion
should be limited to 3 S", and peak stresses must. still, of course, be
subject to fatigue evaluation.
5
From the foregoing we see that three stress intensities should be
calculated:
(1) Average in ligament cross section, called Serr
(2) Average across ligament width at plate surface, called rrerr
(3) Peak, called rrmax
Analysis of Ligament Stress Intensities
Expressions for the average ligament stress intensities, limited
by the design criteria suggested in the foregoing, are derived in
this section from purely theoretical considerations. The analysis
is quite general and can be used for any biall.;ality condition of the
stress field in the equivalent solid plate, and for any ligament
orientation in the stress field. The accuracy of simplifying assumptions used in the analysis is examined using photoelastic
test results. The analytical results are simplified and presented
in a form suitable for design calculations.
In the concept of an equivalent solid plate, as considered herein,
stresses and deflections of a solid plate having the effective elastic
properties of the perforated material are evaluated. There is a
unique state of stress within a body having a given set of elastic
properties and subject to a particular load. Therefore, the stress
field in an equivalent solid plate is the same as the stress field in
the perforated plate on the same macroscopic scale for which the
effective elastic constants were evaluated. Hence, the resultant
loads carried by ligaments (at any arbitrary depth in the tube
sheet) at any particular location must be equal to the resultant of
the load carried by the equivalent solid plate. This is the basis
of the analytical approach presented herein.
In perforated plates such as tube sheets, the perforations and
ligaments are quite small relative to the over-all dimensions of the
plate itself. As a result, the rate of change of the tangential and
radial stresses with radial position in the equivalent solid plate
(given by classical circular-plate theory) is small relative to the
perforations. Hence, one can assume that there exists only a
negligible variation of load from any ligament to its adjacent
parallel ligaments. Under these conditions, there are no sidesway
bending moments in the minimum ligament sections. This can
be seen by considering the equilibrium of an arbitrary cut at the
surface, or at any arbitrary depth of the plate, as shown in Fig. 6.
The stress field in the equivalent solid plate is given by rrT and rro
where the radial and tangential directions are principal directions
Fig. 6
6
Loads aeling on a typical seelion
in the equivalent solid plate. This stress field must be carried by
the minimum ligament sections. Since there is no variation of
stress from hole to hole, no net moment is supported by the cut
section. Hence, the moments in the minimum ligament sections
M· must be zero. Since the orientation of the cut is arbitrary, it
is apparent that the sidesway moments III are zero in all minimum
ligament sections.
Yielding would tend to produce a uniform distribution of stress
across the minimum ligament sections. Hence, in this analysis a
three-dimensional element, subject to the average shear and tensile stresses in the minimum ligament section, is analyzed in
order to evaluate the average stress intensities which are limited
by the proposed design criterion.
Analysis of Average Ligament Stress Intensities at Surfaces
of Plate
Having the principal stresses lIT and rro at either surface of
the equivalent solid plate, the problem of evaluating loads in the
minimum ligament sections becomes statically determinate. The
resultant load carried by the ligaments must be equal to the resultant load carried by the equivalent solid plate. The loads
carried by the ligaments, as shown in Fig. 6, are then given by
+
F = 2(rrT cos I/;)R cos I/;
2[lIo
cos (I/; - 7l"/2)]R
cos (I/; - 7l"/2)
(1)
v
= 2(rrT sin I/;)R cos
I/;
+ 2[rro
sin (I/; - 7l"/2)]R
cos (I/; - 7l"/2) (2)
Hence, the average stresses in a ligament at any arbitrary
angle I/; ,,·;th the principal directions of the equivalent solid plate
stresses rrT and rro (as shown in Fig . .7)are given by
-
flL
1
2h
R
rr dx = - [rr cos2 I/;
-IL
h
y
+
IIO
sin2 1/;]
(3)
T
and
(Tyz).Vg
flL
1
= -
2h
-IL
Tyz
dx
R
= -
[(rrT -
rro) sin I/; cos 1/;]
(4)
h
In order to specify completely the state of stress in a minimum
ligament section and to evaluate the ligament stress intensities
(maximum-shear stresses) that are limited by the design criterion,
something must be known about the stresses transverse to th"
ligament at the minimum ligament section rrx' A three-dimensional view of a ligament is shown in Fig. 8(a). The average
stresses acting on an element at a surface of the plate are shown
in Fig. 8(b). The three-dimensional Mohr circle based on these
average stresses, given by equations (3) and (4), is shown in
Fig. 9. The Mohr circle, assuming zero transverSe stress rrx, i.e.,
Fig. 7
Stresses in a typical ligament
Transactions of the AS M [
z
I
x
I
\
(b)
AVERAGE
I
A
/
/
STRESSES
OR SECONDARY
I
AT PRIMARY
SURFACE
",
\
//
\
,,/
"
(0)
3- DIMENSIONAL
VIEW
OF
LIGAMENT
(e) AVERAGE
STRESSES
THROUGH
Fig. 8
Three-dimensional
AVERAGED
DEPTH
stresses
The comparable expression for the stress intensity (twice the
maximum shear stress) in the minimum ligament section is given
by
ACTUAL STRESSES IN PLANE OF
TUBE SHEET
ACTUAL STRESSES IN PRINCIPAL
TRANSVERSE PLANES
CALCULATED STRESSES IN PLANE OF
TUBE SHEET ASSUMING PLANE STRESS
..
tT
\
\
\
( O,'t"yx)
"-
---
:::--
(CTX,"yX)-Fig. 9 Three-dimensional
Mohr circle for stresses averaged
minimum ligament section at su;:face of perforated plate
across
plane stress, is also shown for the plane of maximum shear. For
purposes of this analysis, the transverse stresses 0" x will be taken
equal to zero. The significance of this important assumption will
be explained subsequently. The corresponding maximum principal stress, based on the average value of the stresses across the
minimum ligament section, is given by
~ {O"r
h
cos21/;
+ 0"0 sin
2
if;
2
+
(O"T
-
0"0)2
Journal of Engineering for Industry
COS2
if; sin2 if;
J'h}
(5)
where (J"cff is the stress intensity limited by the design criterion.
Equation (6) gives the stress intensity based on the average
stress across any particular minimum ligament section for any
ligament orientation if; at either surface of the plate.
Consider the significance of assuming a zero transverse stress
at the minimum ligament section. Obviously, the transverse
stress must be zero at the edges of the minimum ligament section.
Moreover, this stress is usually small, even at the center of the
ligament. Photoelastic tests [18] have shown that the average
transverse stress usually has the same sign as the average longitudinal stress, as shown in Fig. 9. When these stresses have the
same sign, the calculated value of the stress intensity in the plane
of the plate, based on stresses averaged across the minimum
ligament section, will always be equal to or greater than the
correct value of the stress intensity in that plane. This is illustrated in Fig. 9.
There are conditions for which the maximum shear does not
occur in the plane of the plate. This happens when the minimum
principal stress in the plane of the plate has the same sign as the
maximum principal stress in that plane (the transverse shear
stresses being zero at the surfaces). The maximum shear can then
be found by rotat.ing the element in the principal plane perpendicular t.o the plat.e because t.he difference bet.ween the maximum
principal st.ress and the zero Z-direction stress' is great.er t.han
t.he difference between any other principal stresses. However,
the maximum shear st.resses in t.he plane of t.he plate calculated by
, Thc Z-dircction stress due to pressure acting at the surface of a
plate is attenuated a short distance from the surface in the manner of
a bearing stress. Hence, although this stress should be considered in
the fatigue analysis of local peak stresses, it need not be considered in
the average stress-intensity limitations because the latter are only
intended to prevent excessiveyielding and deformation.
7
assuming plane stresl'; are always equal to, or greater than, the
actual maximum shear stresses in any other plane. This can be
seen by again considering the aet-ual three-dimensionrL! Mohr
circle, as shown in Fig. 9. Hence, it is not necessary to write
equations for the shear stresses in planes other than the plane of
the plate, provided that zero transverse stress (T" is assumed at
the minimum ligament sections.
At the center of a circular perforated plate the stress field in the
equivalent solid p1a.te is isotropic. Hence, as indicated by equation (4), there are no shear stresses Ty" acting at the minimum
ligament section. The maximum shear stress in this ('.ase (found
by rotating the element as previously described) acts on a plane
at 45 deg to the plane of the plate.
The theoretical expression
for the maximum shear stress assuming plane stress in the
minimum ligament section then gives the correct value for the
actual maximum shear stress, even though the latter does not
occur in the plane of the p1a.te. Hence, the theoretical approach
used herein gives the exact values of average stress intensities in
ligaments near the center of a circular perforated plate regardless of the magnitude of the transverse stresses in the minimum
ligament sections.
At the edge of a circular plate, however, high stresses may
exist under any biaxiality conditions. For many of these conditions, the maximum shear stress occurs in thc plane of the plate,
as illustrated in Fig. 9. The equation for the average stress intensity across the minimum ligament section, equation (6), then
gives values which are higher than the actual values for many
ligament orientations because of the assumption of zero transverse stress in the ligaments. The significance of this error was
evaluated by making use of measured values of the transverse
stress CJ'" obtained photoelastically by Sampson [18].
The error for a perforated plate under tensile loading having a
ligament efficiency of 25 per cent and a minimwn ligament width
of 0.25 in., was evaluated.
The maximum error for any biaxiality condition and any orientation of the ligament in the stress
field was fOlIDdto be less than 3 per cent. This error increases
with increasing ligament efficiencies. For a plate having a
ligament efficiency of 50 per cent and a minimum ligament width
of 0.5 in., the maximum error was found to be 5 per cent. These
errors might tend to be greater for bending loads on relatively
thin plates than for the tensile loads used in the photoelastic
tests. However, epoxy resin having a Poisson's ratio of 0.5 was
used in the photoelastic tests and the resulting transverse stresses
were probably higher than they would be for metals. Hence, the
maximum error in the calculated stress-intensity values is probably no greater in a metal plate than the error evaluated herein
from photo elastic tests on plastic models.
The equation for the average stress intensity in the minimum
ligament section, equation (6), Inay be simplified further for design calculations by consiaering the symmetry of the hexagonal
array of neighboring holes surrounding the typical hole. It is
apparent that the same stress distributions would result if the
orientation of the ligaments were shifted ±60 deg in the equivalent solid-plate stress field, the actual stress distribution in the
ligaments also being shifted ±60 deg. Consequently, at least
two of the ligaments surrounding the typical hole pattern will be
at most 30 deg rotated from that orientation which would produce
the maximum stress intensity in the minimum ligament section.
Near the cent.er of a symmetriC<'1.lly
loaded circular plate, the stress
field is very nearly isotropic and the orientation of a particular
ligament does not affect the stresses in that ligament appreciably.
Near the periphery of a plate such as a tube sheet which contains
a large number of holes, the angular orientation of the hole patterns "ith respect to the radii of the plate varies gradually around
the periphery, encompassing the entire range of possible orientations. From these considerations, it is apparent that the expression for tbe stress intensity, equation (6), can be maximized
8
with respect to if; for tube-sheet design calculations without introducing undue conservatism. The resulting expression should
be used to obtain stress inteusities for typical ligaments in a uniform pattern, !":ither than for isolated ligaments. The expression
for the orientation which gives the maximum skess intensity is
given by
-
(Tr'
cos3
+
if; sin if;
002
+ (0,2
-
sin3
if; cos if;
j-(T,(TO
+
(T02)
sin
2if;
cos
2l/J
=
0
(7)
From equations (6) and (7) it is possible to evaluate ligament
stress intensities, maximized \\ith respect, to angular orientation
in the stress field, for any ligament effieiency and any biaxiality
condition. These equations can be written as functions of the
biaxiality ratio fJ = (T,/ (To or (To/a" whichever gives -1 ~ fJ ~ l.
1 for isotropic loads
0 for uniaxial loads
= -1 for pure shear
=
fJ
1
=
Equation (7), written in terms of fJ, was used to find the orientation if; which gives the maximum average ligament stress intensity in a stress field of biaxiality fJ. This orientation was then
used in equation (6) to evaluate the corresponding value of the
average stress intensity. The resulting values are given by:
(8)
where
(T off
K
R/h
=
ligament stress intensity based on stresses averaged
across minimum ligament section at either plate
surface
value given in Fig. 10
reciprocal of ligament efficiency, Fig. 6
a, or (To, whichever has the largest absolute value.
(For example, if (T, = -3000 psi and (To = 2500 psi,
then (T, = -3000 psi and /(T,j = 3000 psi)
stresses at either surface of equivalent solid plate obtained from Step 1 of analysis
To calculate ligament stress intensities based on stresses
averaged across the \\idth of the ligament but not through the
depth of the plate, substitute the values of (T, and (TOat the surface
of the plate into equation (8). The K-values for equation (8),
given in Fig. 10, depend on the biaxiality of the stress field and
vary with radial location in the plate. The resulting stress intensities will, of course, vary from one side of the plate to the
other and will depend on the radial location in the plate.
Since equation (8) was developed by maximizing the stress intensity with respect to the angular orientation of the ligament, it
may be overly conservative for plates having a small number of
holes. As previously pointed out, the stresses near the center of
the plate do not depend on the angular orientation of the ligament
because the stress field is isotropic. However, it may be worth
while to evaluate ligament stresses individually when the limiting
value given by equation (8) occurs at the periphery of a plate
having a small number of holes. Equation (6) gives the stress in
a ligament having an arbitrary angular orientation if;.
Analysis of ligament Stress Intensities Averaged Through
Depth of Plate
The value of (T, averaged through the depth of a plate at any
location is equal to the value of 00 averaged through the depth at
that location. Moreover, these average values do not vary with
location in a symmetrically loaded circular plate because they
are produced by membrane-type loads. From equation (4), the
average shear stress in the plane of the plate due to membrane
Transactions of the AS M E
K
2.0
CTeff = AVERAGE STRESS INTENSITY
MINIMUM LIGAMENT SECTION
1.9
a:::
0
CTr8CT8=STRESSES
IN EQUIVALENT
SOLID PLATE
1.8
CT,= CTror CT8(WHICHEVER
HAS THE
LARGEST
ABSOLUTE
VALUE)
r-
u
~ 1.7
{3 =
>-
!:: 1.6
(J)
z
w
I-
z
IN
CTr or CT8
CT
CT
r
8
WHERE -I ~ {3:s I
1.5
1.4
w
a::: 1.3
I00
00
00
~
1.2
1.1
1.0
0.8
- 1.0 - 0.8 -0.6 -0.4 - 0.2
(3,
Fig. 10
0
BIAXIALITY
Stress intensities
R [(APr)'
--
H
+
(iTr)2
J'/'
r
(ma,.'I:with = radius
.
to outermost lIgament)
RATIO
in perforated-plate
loads TliZ is zero at the mllllmum ligament sections. The
transverse shear stress averaged through the depth T lI' varies
linearly with radial location rin a circular plate.
Fig. 8(c) shows an element subject to the shear and tensile
stresses averaged across the minimum ligament section and
averaged through the depth of the plate. The three-dimensionai Mohr circle based on these stress values is shown in Fig.
11. Since the transverse stress in the ligament iTz has the same
sign as the longitudinal stress iT; (as previously discussed), it is
apparent that the maximum shear stress -due to membrane loads
can be found by rotating an '~lement in the principal plane subject to the transverse shear Til'. The average stress intensity in a
ligament at any radial distance r from the center of the plate is
given by
Self = h
+ 0.2 +0.4 +0.6 + 0.8 + 1.0
ligaments
-
STRESSES
IN PRINCIPAL
-
STRESSES
IN OTHER
PLANE
PRINCIPAL
OF
MAXIMUM
SHEAR
STRESS
PLANES
(9)
where
AP
iTr
pressure drop across plate
r = radial distance of ligament from center of plate
= iTo
stresses averaged through depth of equivalent solid
H
plate
thickness of plate
Peak Stresses in Perforated Plates
Maximwn local stresses due to all loads (mechanicltl and
Journal of Engineering for Industry
Fig. 11 Three-dimensional
Mohr circle for stresses averaged
minimum ligament section and averaged through depth of plate
across
thermal) are also limited by the suggested design criteria of this
paper. These stresses can be evaluated from the known stresses
in the equivalent solid plate using the stress multipliers obtained
photoelastically by Sampson. A minor correction was made on
these multipliers to account for the nonlinearity of the stress dis-
9
I
30
1\
28
\
\
26
\
22
)(
\
b
0
~
0::
en
en
w
0::
1
f3
16
CTr
~
= -0"8 ORCT
14
\
-
\
\
\
1\
,
12
\
\
'"
2h
I
I
I
"- '""" "
"
I
"""-.
..•.•..••..
/'
..•••..•...
6
....•.•..•.. ./
.............
4
~
I
"~
0""'--
2
o
0.1
0.15
Fig. 12
0.2
h/R,
where
P
I I I
I
,S~R~S~(f3=~)
I
I
f
-,
T
".
.....•.. ~
.......
••••
-- ----
.•.. ... -
toelastic tests on tube-sheet models have revealed the existence
of high local stresses at the perforations adjacent to the rim (15].
These peak stresses appear to be due to the influence of the rim
and cannot be calculated by equation (10), but may be approximated by the expression
(11 )
where
(Jrim
or (Jo (whichever has the largest absolute value)
value given in Fig. 12
pressure acting on surface
(Jr
All thermally induced maximum local stresses, as well as pressure stresses, must be considered in the cumulative fatigue limitations on the values of (Jm,,' The values given by equation (10)
are the peak stresses throughout the perforated portion of the
plate.
Most perforated circular plates have unperforated rims. Pho-
10
I
I
Maximum local stresses in perforated plotes
(10)
Y
I
0.3
0.4
0.5 0.6 0.70.80.9 1.0
LIGAMENT EFFICIENCY
tribution through the thickness of the coupons used by Sampson.
The multipliers Yare functions of the biaxiality of the stress
field in the equivalent solid-plate fJ = (Jr/(Jo or (Jo/(Jr (whichever
gives -1 ~ fJ ~).
This ratio varies, of course, with radiallocation in the plate. The maximum stress for any particular thermal
or pressure load is then given by the relation:
(J1
I
,<SO~ROPIC STRESS({3=1)
..•..
~
:
I
~UNI~XI~L
.-/
8
T
S,H~A~(/3 = -,')
\~ /\URf
'\,
10
-
I
etC)
000
q~
i
'\.
I
WHERE-I < f3 < I
fir
1\
'\.
'\.
I
= smESSES IN EQUIVALENT
SOLID PLATE
~
= CTr OR CTe (WHICHEVER HAS THE
~
LARGEST ABSOLUTE VALUE)
I 2R
,
I
CT8
r
t;
>0-
a
\
~20
E 18
CT
\
\
\
24
0
CTr
II I I
= Y CTI
CTmax
[(r
nominal bending plus membrane stress at inside of
rim
value given in Fig. 13
(Jrim is evaluated in Step 1 of the general analytical approach,
the rim being treated as a plate or ring depending on its dimensions.
The Kr-values in Fig. 13 were derived from known values of
stress concentration in a bar with a semicircular notch5 and were
checked against the photoelastic results of Sampson and Leven.
5
Reference [191. figs. 15.35.85.
and 86.
Transactions of the UME
3.0
2.9
I
2.8
, I
§2.6
:; 2.5
o
z 2.4
b~
•.....•..
~x
~2.4
<l
2.3
\\
CTNOM
I
I
I
<l:
a:
'"
2.1
g
2.0
0::
1.9
f-
~
w
1.8
~
1.7
w
a:
(J)
ti 1.8
Tc
\.
"-
O"MAX
~
-
..........
-
...•......
1.6
STRESS
50_
-,0-
o~
o ~oo ~o
~o_
0
UNOM
"\
If)
1.6
~o-
(!)p
v)
I
\
<l
~2.0
I
j-D-j0
, EaT(TH-Tc)
2 (1-
I
1-
\
b 2.2
a
g22
I
\
:;
b
I
KOO"NOM
O'"mox =K rC"'nm
26
I
=
27
2.8
I
CTMAX
.....•...
-
1.5
r-- I--
14
-
1.3
1.4
.05
.15
.10
.20
PI
.25
.30
.35
.40
1.2
b
1.1
Fig. 13
Peak stresses
at perforations
adjacent
,
to rim
1.0
,
,
,
.02 .04 .06.08 .10 .12 .14 .1618
Evaluation of Special Cases of Thermally Induced Stresses in
Tube Sheets
In heat exchangers, the major part of the
tube-sheet thickness is at the primary temperature by virtue of
the perforations through which the primary fluid passes. The
difference in temperature hetween the primary and secondary
sides of the tube sheet occurs very near the secondary surface,
resulting in what is commonly called a thermal skin effect. Because of the thermal film drop, the entire difference between the
primary and secondary fluid bulk temperatures does not contribute to the skin effect. Credit may be taken for the temperature drop in the thermal boundary layer at the secondary side of
the tube sheet when this drop can be evaluated. Stresses due to
this effect are given by
.20.22 .24 .26 .28.30
P
o
Fig. 14
Peak thermal stresses at perforations
adjacent
to a diametrallane
Thermal "Skin Effect."
(12)
where
aT
Tp
T/
Stresses
thermal expansion coefficient, in/in/deg F
primary temperature, deg F
metal telnperature, at secondary tube-sheet surface,
deg F
0
for Temperature
Drop Across
Diametral
Lane of U-Tube Type
In the case of a U-tube type steam
generator, the unperforated diametral lane separates the inlet
and outlet sides of the tube sheet, and large thermal stresses
may arise because of a temperature difference between these
sides. The resulting maximum local stresses in the ligaments
of the tube sheet can be approximated by
Steam-Generator
Tube Sheet.
K"E*aT(TH
O"max
-
TJ
2
uniaxial (/3 = 0) stress multiplier from Fig. 12
effective elastic modulus for tube sheet
The stresses at the edges of the holes adjacent to the unper-
Journal of Engineering for Industry
KDEaT(TH([max
2(1 -
7'.)
II)
(l4)
where
KD = stress-concentration factor from Fig. 14
E, II = material properties of tube sheet
The KD-values given in Fig. 14 were derived from known
values of stress concentration in a bar with a row of semicircular
notches.6 These values apply over the entire range of ligament
efficiencies :::;60 per cent.
Evaluation of Acceptability of Improperly Drilled Holes
The presence of a particular out-of-tolerance thin ligament will
result in increased peak stresses and increased ligament stress
intensities. A method which can be used to determine how far a
hole can be drilled from its normal position in a hole pattern
without exceeding the proposed stress limits is developed in this
section. Since these increases occur only at thin ligaments in
nominally uniform patterns, the stresses in these ligaments are
limited by the less restrictive criteria previously described.
Transverse shear loads as well as loads in the plane of the tube
sheet contribute to the stress intensity based on stresses averaged
across the width of the ligament and through the depth of the
(13)
where
K"
E*
forated diametrallane, Fig. 14, can be approximated by assuming a linear temperature drop across the diametrallane:
plate. Hence, it would be extremely difficult to evaluate the
effect of load redistribution caused by the existence of a particular
ligament being thinner than average. To be safe, it must be assumed that there is no redistribution of load to nearby ligaments.
The limited stress intensity in a particular out-of-tolerance ligament can then be evaluat-ed by substituting the smallest ligament
G
Reference
[191. figs. 20, 21, and 32.
11
3.4
I
'.
Q
3.0
11"0\
2.8
IJJ,uJ
2.4 -
a:z
-
-!
\
1.8
1\
1.2
0..0..
1.0
-
"<
-
\~
11\
VlVl
~~
«<l:
!
-I
'1\
II"-I~
wW
a: a: 1.4
1-1-
\
.\
\ ,~
i'-.. ."'"""'"
ww
;;e
I
\
f"\
1.6
-_.I
"I\~
\
2.2
-lI
~
ZZ
VlVl
VlVl
I
,,\,\
'\
11"1
u:< 2.0
::>a:
fi:lo
I
\
2.6
<.:lICl
~-J
O-J
IU<l:
I
I
1/ Kmh'~"1
3.2
I-IIzz
--':;;:
<<r
""1-
I
I
-
(15)
-
~
~ ~
where
~
-
<TI
K",
P
0.8
Y
0.6
<Tr or <To (whiehever has t.he largest absolute value)
value given in Fig. 15
pressure acting on surraee
value given in Fig. 12
The K",-v:tlue givcn in Fig. 15 can be used for any ligament
efficieney.
0.4
Summary and Conclusions
0.2
o
width at the misplaced hole into elJuation (9). The resulting
stress value is limited to 3 S""
The ma),i/lllJlll local stresses tlue (,0 all loatls (mcchanical alltl
thermal) ill tm iso!:J.tedor thin ligament in a nominally uniform
pattcrn are limitlxl by fatigue considerations in the same manner
as the peak stresses ill a typical ligament in a uniform pattern.
The increase in the Joeal stresses caused by the presenee of a particular out-of-tolerance thin ligamen(, was evaluated in photoelastic tests. The inerease in peak stresses was found to be a
function of the biaxiality of the stress field, and the direction of
the displacement of the misdrilled hole with respect to the hole
pattern, as expected. The variation of the increasc in peak
stresses \dth Jig:uneut efTiciclll'Ywas foulld (,0 be small. The
maximum increase in local stresses occurred when the hole wa~
displaced at 30 deg to the line of hole centers. Using the results
for this ease, the maximum loeal st.ress in a.thin ligament is given
by
I
o
I
I
Q2
I
I
Q4
I
I
Q6
I
I
Q8
I
I
ID
hmln REDUCED LIGAMENT WIDTH
-h-"
NORMAL LIGAMENT WIDTH
fig. 15
Increase of peak slress due 10 misplaced
Load
Stress intensity
( Average across ligament at either surface of plate
Pressure
Coml.:ined pressure
thermal
Cyclic
pressure
thermal
hole
1 Effeetive elastic constants for both plane stress and bending
loads for any plate thickness (H/R > 4) are given in Fig. 5.
2 A complete structural-design criterion for perforated plates
is proposed. The limited stress values are summarized in the
following table for a nominal ligament in a uniform pattern.
and
and
I
Equation
1.58",
(9)
8m
Average across ligament and through
thickness
Average across ligament at either surface of plate
Peak in ligaments
(10)
{ Peak at perforations adjacent to rim
(11)
Cyclic thermal skin effect
Peak at surface
1
';'
( Peak ill ligaments
Cyclic thermal (temperature
difference across diametral Peak at holes adj:tcent to diametral
lane)
lane
Limit
(8)"
(8)"
(12)
(13)
(14)
38",
Cumulative
fatigue
Cumulative
fatigue
Cumulative
fatigue
Cumulative
fatigue
Cumulative
fatigue
o Equation (8) was obtain cd by maximizing thc stress intensity with respect to the angular orientation of the ligament. If the plate contains only a small number of holes and if the limiting stresses
occur at the periphery of the plate, a more accurate evaluation of this stress intensity. which takes into
account the angular orientation of the ligament may be justified. Equation (6) gives the corresponding
stress intensity in a ligament with any angular orientation f. The maximnm value of this stres>;irltensity for all ligaments should be limited as indicated in the table.
12
TransacHons of the AS M E
3 A method of evaluating the acceptability
of misdrilled
holes is given.
The relevant stresses and their proposed limits
are given in the following table.
Load
Combined pressure
and thermal
Stress intensity
Avera-ged across ligamen t
and through depth
Cyclic pressure
thermal
Peak in ligaments
and
Equation
Equation
(9) with
hmiH
substituted
for h
(15)
4 The effective elastie COlJstalJbi and peak stress multipliers
recommended herein are based on those obtained experimentally
by Sampson.
Stresses and ddlec:tiolls caleulated
using these
values showed better agreement with the test results obtained
by Leven (on uniformly loaded, simply supported,
circulnr perforated pl:Ltes) thalJ any of the other appro:Lehl~s mentioned
herein!
For most conventional
steam generators,
the design basis
recommended herein allows a slightly thinner tIl be sheet than does
TEMA
[11].
For example, in a typiclll high-pressure
design
where TE:MA requires It minimum tube-sheet thid:ness of 10 in.,
the design methods descrihed herein require a minimum thickness
of 91/, in. if S", is taken as 5/s of the yield strength of the material
and full credit is taken for the tubes.
On the other hand, where
severe thermal loads are antieipated, it may be necessary to make
design modifications in order to meet the criteria recommended
herein, whereas TEMA does not account for thermal loads.
Acknowledgments
The design methods proposed on this paper are the culmination
of a program sponsored by the Bureau of Ships and co-ordinated
by the vVestinghouse Bettis Atomic 1'o"'er Laboratory.
The
stress limits proposed, however, represent only the opinions of the
authors.
The experimental
work used as a basis for the proposed design methods was performed by Messrs. IVr. .M. Leven
and R. C. Sampson at the \Vestinghouse Research Laboratories
and by Mr. A. Lohmeier at the vVestinghouse Steam Division.
To avoid duplication of effort, the program was co-ordinated by
the authors with a somewhat broader program on stresses in
ligaments
being sponsored by the Pressure
Vessel Research
Committee of the vVelding Research Couneil.
References
1 G. Horvay,
JO'll1"1talof Applied
pp. 355-360.
2 G. Horvay,
Journal of A1Jplied
pp. 122-123.
3 1. Malkin, "Notes on a Theoretical Basis for Design of Tube
Shects of Triangular Layout," TRANS. ASME, vol. 74, 1952, pp.
387-396.
"The Plane-Stress Problem of Perforated Plates,"
111echanies, vol. 19, TRANS.AS ME, vol. 74, 1952,
"Bending of Honeycombs and Perforated Plates,"
111eehanics,"'ol. 19, TRANS. ASME, vol. 74, 1952,
7 See reference [14], figs. 7-11, and reference [15], figs. 18, 19,20,24,
and 25
Limit
3S",
Cumulative
fatigue
4 K. A. Gardner, "Heat Exchanger Tube Sheet Design,"
,Journal of Applied 111echanics, vol. 15, TRANS.ASME, vol. 70, 1948,
pp. 377-385.
5 K. A. Gardncr, "I-Ieat E.'xchanger Tube Sheet Design-2, Fixed
Tube Sheets," Jottrual of Applied Jllechanics, vol. HI, TUANS.ASME,
vol. 74, 1952, pp. 159-lG6.
6 K. A. Gardner, "Heat Exchanger Tuhe Sheet Design-3,
U-Tube and Bayonet Tube Sheets," Journal of AP1Jlied 111eehanics,
vol. 27, TRANS.ASME, Series E, vol. 82, 1960, pp. 25-33.
7 J. P. Duncan, "The Structural Efficiency of Tube Plates for
Heat Exchangers," Proc:eerlings, I. 111eeh. E., vol. 169, 1955, pp.
789-810.
8 K. A. G. i\liJler, "The Design of Tube Plates in Heat Exchangers," Proceedings, I. ilIech. E., vol. 1 13, 1952, pp. 215-231.
9 G. D. Gallet!y and D. It. Snow, "Some Results on Continuously Drilled Fixed Tube Plates," IJresented at the ASME
Petroleulll Conference, Nmv Orleans, La., September, 1960, Paper
No. 60-Pet-16.
10 V. L. Salerllo and J.B. i\fal1OlIcy, "A Heview, Comparison and
i\ Codification of Presen t Deflection Theory for Flat Perforated
Plates," Welding Research Council Bulletin No. 52, July, 1959.
11 "Standards of Tubular Exchanger Manufacturers
Association," fourth edition, Tubular Exc:hanger Manufacturers Association,
New York, N. Y., 1959.
12 S. Timoshenko and S. Woinowsky-Krieger,
"Theory of
Plates aud Shells," second edition, McGraw-Hill Book Company.
Inc., New York, N. Y., 1959, p. 5(;.
13 R. C. Sampson, "Photoe1asti<.: Frozen Stress Study of the
Effective Elastic Constants of Perforated Materials, A Progress
Report," WAPD-DLE-3f9,
May, 1959; available from Office of
Teehni<.:al Service, Department of Commerce, \Vashington 25, D. C.
14 i\if. i\L Levell, "Preliminary Report on Deflection of Tube
Shel,ts," vVAPD-DLE-320, i\Jay, 1959; available from Offiee of
Technical Serviees, Departmcnt of eOlTlIlIer<.:e,
vVashington 25, D. C.
15 i\f. lVI, Leven, "Photoelastic Determinat.iolT of Stresses in
Tube Sheets and COlllparison "Tith Calculated Value'S," Bettis Technical Review, vVAPD-BT-1S, April, 1960; available from Office of
Technical Services, Department of Commerce, \Vashington 25, D. C.
I6 W. J. O'Donnell, "The Effect of the Tubes on Stresses and
Deflections in U-Tube Steam Generator Tube Steets," Bettis Technical Review, \VAPD-BT-21,
Novembe'r, 1960; available from
Office of Teehnieal Services, Department of Commerce, vVashington
25, D. C.
17 B. F. Langer, "Design Values fol' Thermal Stress in Ductile
iVlaterials,', ASi\IE Paper No. 58-Met-l,
The Weldina Joumal, Research Supplement, September, 1958, pp. 411s-417s.
18 R. C. Sampson, "Photoelastic Analysis of Stresses in Perforated Material Subject to Teusion 01' Beuding," Bettis Technical
Review, WAPD-BT-18. April, 1960; available from Office of Technical Services, Department of Commerce. \Vashington 25, D. C.
19 R. E. Peterson, "Stress-Concentration
Design Factors," John
Wiley & Sons, New York, N. Y, 1953.
l"'inted in U. S. A.
Journal oi Engineering for Industry
13