W . D esign of Perforated Plates

UntitledTrans. ASME, Vol. 84, August, 1962
P ap e r N o .
61 -W A -llS
W . J . O 'D O N N E L L A ssoc ia te E ng ineer , W estin ghouse B ett is
AtomicPower laboratory, Pittsburgh,Pa. Assoc.Mem.ASME
B . F . L A N G E R ConsultingEngineer,WestinghouseBettis AtomicPower laboratory, Pittsburgh,Pa.
FellowASME
D es ig n o f P e r fo ra te d P la te s 1
Th is paper describes a method fo r ca lcu la ting stresses and deflections in perJora ted
p la tes w ith a tr iangu la r penetra tion pa ttern . The method isbased partly on theory and
partly on experim ent. Average ligament stresses are obta ined from purely theoretica l
considera tions but e.tfective elastic constan ts and peak stresses are derived from stra in
measurements and photoelast-ic tests. Acceptab le lim its fo r p ressure stresses and ther-
m a l stresses in heat-exchanger tube sheets are a lso proposed.
----N o m en c la tu re ------------------------------
G en e ra lM e th o do fA n a ly s is
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
(C o n t in u e d o n l Ie .T t p a o e )
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 mis-
drilled 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.
U T' (TO
b
2h
2hmin
The general method of evaluating stresses and deflections in a perforated plate having a triangular penetration pattern is:
Step1. Calculate the nominal bending and membrane stresses and deflections of an equivalent solid plate having the effective modified elastic constantsE* and p* and the same dimensions as the perforated plate.
Step2. Calculate physically meaningful perforated-plate stress values from the nominal stress values in the equivalent solid plate from Step 1. Deflections of the perforated plate are the sameas the deflections of the equivalent solid plate.
"Yhen the perforated plate is part of a structure, as in the case of a heat exchanger, Step 1 is accomplished using classical structural-analysis methods. A study of the effective elastic con- stants for use in Step 1 is contained herein, and values basedon
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 shouldbe cal- culated 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.
S tresses
1 This work is part of a dissertation suhmitted by W. J. O'Donnell to the University of Pittsburgh in partial fulfillment of the require- ments for the degree of Doctor of Philosophy.
2 Numbers in brackets designate References at end of paper. 3 See, for example, reference [11],paragraphs R-7.122 and R-7.123. Contributed by the Petroleum Division for presentation at the
Winter Annual Meeting, New York, N. Y., November 26-December 1, 1961, of THE AMERICANSOCIETYOF lvIECHANICALENGINEERS. Manuscript received at ASME Headquarters, July 26, 1961. Paper No. 61-WA-115.
Material Properties
D * E* H3/12 (1 - p*2), effective flexural rigidity of per- forated 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, dimen-
sionless Pp Poisson's ratio of plastic-model material, dimensionless
Pp * effective elastic modulus of perforated plastic models, di- mensionless
O iT thermal expansion coefficient, in/in deg F
In tro d u c t io n
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 equip- ment. 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 pub- lished work has been limited to perforations arranged in an equilateral triangular pattern, and the present paper is noexcep- tion. The Pressure Vessel Research Committee of the Welding Research Council is currently sponsoring work on square pat- terns 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 Associa- tion [11]. In these TEMA standards the thickness required to resist shear depends on the ligament efficiency of the perfora- tions, 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-
Co-ordinatesand Dimensions
l' = radial distance of ligament from center of circular per- forated plate, in.
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 recom- mended for these values. A method of evaluating the accepta- bility of misplaced holes is also given.
E ffe c t iv e E la s t ic C o n s ta n ts fo r P e r fo ra te d P la te s When a perforated plate is used as a part of a redundant struc-
ture, 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 rota- tion 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 thanthe 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 atthe 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 ther- mal-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 ob- tained from theoretical considerations and two have been obtained experimentally:
1 Theoretical Horvay plane stress [1]. 2 Theoretical Horvay bending [2]. 3 Theoretical modified Horvay bending, corrected for con-
strained warping by Salerno and Mahoney [10]. 4 Theoretical Malkin bending [3].
5 Theoretical modified Malkin bending corrected forCOIl-
strained warping by Salerno and Mahoney [10]. 6 Experimental Sampson plane stress []3]. 7 Experimental Sampson bending [13].
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 forE* 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.
S am p so n E ffe c t iv e E la s t ic C o n s ta n ts
The Sampson experimental values of the effective elastic con- stants 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 con- stants. Plane-stress constants were obtained by applying uni- axial tensile loads, and bending constants were obtained byapply- ing 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 andde- flections in tube sheets was checked by Leven in tests on per- forated 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 determin- ing the effective elastic constants using perforated rectangular coupons subject to uniaxial loads. The measured deflections agreed best with those calculated using the effective elastic con- stants obtained experimentally by Sampson. Moreover, the measured local stresses agreed closely "ith those calculated using the stress-ratio factors obtained by Sampson. Hence, the Samp- son 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 expectthat a very thick plate would not be affected appreciably by the small stress gradient in the thickness direction. Consequently,the
----N0menclature'----------------------------
K r value given in Fig. 13 K " , value given in Fig. 15 K u value given in Fig. 10 for {3= 0
Y valuegiveninFig.]2
Others
F normal force carried uy ligament, Fig. 6, lb/in. 11 shear force carried by ligament, Fig. 6, lb/in.
111 moment carried by ligament, Fig. 6, in-lb/in. P pressure on plate surface under consirleration, psi
I1P pressure drop across tube-sheet, psi T p temperature at primary tube-sheet, surface, deg F Ts temperature at secondary tube-sheet surface, deg F T I l temperature of hot side of tube sheet, Fig. 14, deg F Tc temperature of cold side of tube sheet, Fig. 14, deg F
f3 (jr /( jO or (jO /(jr whichever gives -1 <: f3 <: 1, dimen- sionless
If angular orientation of ligament, Fig. 7, radians
stresses averaged through depth of plate, psi
(jeer
(Jrim
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
Stress Multipliers (dimensionless)
K value given in Fig. 10 K n = value given in Fig. 14
i1:u 0"11' T1/X' ur,uo
40 60 80100204 6 8 10
H/R
1-
h .1.--, ""'- .......••. 1\ R =
2R = PITCH OF TRIANGULAR HOLE PATTERN
2h = MINIMUM LIGAMENT WIDTH I I I II I I I I I I I I I I Io
0.2
0.1
0.6
0.2
0.3
0.4
0.5
Fig.' Variation of Sampson effective elastic modulus with depth of a plate in bending ( v p = 0.5 Poisson's ratio of solid material)
I , I
I H
I
2h = MINIMUM LIGAMENT WIDTH
/ ~ I I -...r- -Lr lih I PLANE STRESS -
~ R .
3 ~
H/R
6 8 10 20 40 60 80100
Fig. 2 Variation of Sampson effective Poisson's ratio with adepth of a plate in bending ( v p = 0.5 = Poisson's ratio of solid material)
values for the effective elastic constants for a plate in bending should approach the plane stress values as the plate gets thick. Fig. 1shows the variation ofE* with the relative thickness of a plate in bending, and Fig. 2 shows the same variation for/1 * .
Note the rather abrupt transition in theE* IE -va lues 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(H il l> 4) and this situation occurs in most heavy-duty heat-exchange equip- ment which requires the refined analysis described here. The effective elastic cOllstants in bending forH IR > 4 do not differ greatly from the plane-stress values. Fig. 3 shows the bending constants atH il l = 7 plotted with the uniaxial plane-stress con- stants. Accordingly, the plane-stress constants appear tobe the
most acceptable values for plates having a relative thicknes8 H IR> 4.
Notice that the uniaxial plane-stress values of effective Pois- son'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 immediatelyevi- dent, 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 (thex and y-
directions, respectively, in Fig. 3), and partly because these dif- ferences 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 ef- ficiencies under bending and plane-stress loads. He then pro- ceeded to evaluate the effect of material Poisson's ratio onthe effective elastic constants. This was accomplished by measuring
Journal of Engineering for Indllstry 3
4 Transactions of the AS !ViE
Before proceeding to the detailed calculation of stresses,it is necessary to decide which stresses are significant and, conse- quently, should be calculated and limited in order to assurean 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 loadingwill 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
P ro p o sed S tre s s L im its
0.1
1.2
0.2
1.1
1.0 p* : v t[O .4343(VpIV -11 (Lnh/R+2.3026l+lr'
0.9 WHERE: vt 6 vp = POISSON'S RATIOS FOR PLASTIC(v:O.5
v* 6v : POISSON'S RATIOS FOR METALS
0.3
LIGAMENT EFFICIENCY h/R
"-"- *,."0.4
0.7
0.8
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 creditwas taken for the tube wall in the caleulations; that is, when thehole size was taken as the ID rather than the OD of the tubes [16]. When the ligament effic.iencywas calculated on the basis ofthe 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.
F ig .4 E ffec t o f m a ter ia l P o isson 's ra t io II on effec tive Po isson 's ra t io v*
E*/E
LIGAMENT EFFICIENCY, h/R
~2h<l•... V >
U
0.2
0.1
the effective elastic constants of an aluminum specimen( I I =
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(H IR > 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 ofthe effective elastic constants for any material and for any ligament efficiency. This relation is given in Fig. 4. The maximum devia- tion of any of the aluminum-bar test points from. this empirical relation is 7 per cent. The corresponding relation betweenp*
for steels ( I I = 0.3) and I Ip * for plastic ( l ip = 0.5) was used to modify the Sampson plane-stress II*-values obtained in tests on plastic specimens in order to obtain corresponding values applica- ble to metal plates. The resulting values of1 1 * for I I = 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 ratiosE* I E were found to be unaffected by changes in the Poisson's ratio of the material. Hence, the Sampson plane-stress values ofE* 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 larg- est error in the maximum local stresses or in the maximum average ligament stresses that are limited by the design criteria recom- mended herein for any type or direction of loading and any plate thickness(H IR > 4) was found to be 8 per cent.
W ffe n in g E ffe c t o f T u b e s
When tubes are rolled or welded into a tube sheet, the question
Fig. 3 Comporison of Sompson effective elostic constants for bending and plane stress
0.2 0.3 0.4 0.5 0.6 0.7 0.80.9 1.0
h/R, LIGAMENT EFFICIENCY
z 0 0.6u u I- en 0.5 <t .-J W
W > 0.4 I- u W lJ.. lJ.. 0.3w
D*/D E*/E 0.2
0.9
1.0
Fig. 5 Effective elastic constants for perforated plates
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 alge- braic 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 Typical Ligament in a Uniform Pattern
(a ) llfechan ica l Loads (i.e., pressure loads but not thermal loads):
(i) The stress intensity based on stresses averaged across the minimum ligament sectionand 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 apres- sure 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
the yield strength of a nonferrous material.) Let us call thi~ basic st.ress intensity allowanceSm .
(ii) The stress intensity based on stresses averaged acrossthe minimum ligament section butnot through the thickness of the plate should be limited to prevent excessive deflection. This stress is the sum of membrane plus bending effects and, sincethe limit-design factors for flat plates are greater t.han 1.5,it can safely be allowed to reach a value of 1.5Sm .
(b) Combined ilfechan i-ea l 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, asde- scribed 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 re- laxed. For combined pressure and thermal loads, the stress in- tensity based on average stresses in the ligament cross seet-ion should be limited to 3S " , and peak stresses must. still, of course, be subject to fatigue evaluation.
Journal of Engineering tor Industry 5
F = 2 ( r rT cos I / ; )R cos I/; + 2 [ lIo cos(I/; - 7 l" /2 ) ]R cos(I/; - 7l"/2)
(1)
A n a ly s is o f A v e ra g e L ig am en t S tre s s In te n s it ie s a t S u r fa c e s o f P la te
(4)
(3) 1 flL R
- rr d x = - [rr cos2 I/; + IIO sin2 1/;] 2h - IL y h T
1 flL R (Tyz).Vg = - Tyz d x = - [ ( r rT - r ro ) sin I/; cos1/;]
2h - IL h
and
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-dimen- sional 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 stressrr x, i.e.,
Hence, the average stresses in a ligament at any arbitrary angle I/; ,,·;th the principal directions of the equivalent solid plate stresses rrT and r ro (as shown in Fig . .7)are given by
v = 2(rr T sin I / ; )R cos I/;
+ 2 [r ro sin (I/; - 7 l" /2 ) ]R cos(I/; - 7l"/2) (2)
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 momentsI I I 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 ten- sile stresses in the minimum ligament section, is analyzed in order to evaluate the average stress intensities which are limited by the proposed design criterion.
Having the principal stressesl IT and r ro at either surface of the equivalent solid plate, the problem of evaluating loadsin the minimum ligament sections becomes statically determinate. The resultant load carried by the ligaments must be equal to the re- sultant load carried by the equivalent solid plate. The loads carried by the ligaments, as shown in Fig. 6, are then given by
From the foregoing we see that three stress intensities should be calculated:
(1) Average in ligament cross section, calledSerr
(2) Average across ligament width at plate surface, calledrrerr
(3) Peak, called rrmax
A n a ly s is o f L ig am en t S tre s s In te n s it ie s 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 as- sumptions 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. Thereis a unique state of stress within a body having a given set of elastic properties and subject to a particular load. Therefore, thestress field in an equivalent solid plate is the same as the stress field in the perforated plate on the same macroscopic scale for whichthe 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 atthe surface, or at any arbitrary depth of the plate, as shown in Fig. 6. The stress field in the equivalent solid plate is given byr rT and r ro
where the radial and tangential directions are principal directions
Fig. 6 Loads aeling on a typical seelion Fig. 7 Stresses in a typical ligament
6 Transactions of the AS M [
I x I
OR SECONDARY SURFACE
(0) 3- DIMENSIONAL VIEW OF LIGAMENT (e) AVERAGE STRESSES AVERAGED
THROUGH DEPTH
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 averagedacross m in im um ligam en t sec tion a t su ;:face o f p e r fo ra te d p la te
plane stress, is also shown for the plane of maximum shear. For purposes of this analysis, the transverse stresses0 " x will be taken equal to zero. The significance of this important assumption will be explained subsequently. The corresponding maximum princi- pal stress, based on the average value of the stresses acrossthe minimum ligament section, is given by
~ {O "r cos21/; + 0"0 sin2 if; h 2
The comparable expression for the stress intensity (twice the maximum shear stress) in the minimum ligament section is given by
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 ofthe ligament. Photoelastic tests [18] have shown that the average transverse stress usually has the same sign as the average longi- tudinal 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 il- lustrated 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 asthe 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 perpen- dicular t.o the plat.e because t.he difference bet.ween themaximum 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
J'h}+ (O "T - 0 "0 )2 COS2 if; sin2 if;
Journal of Engineering for Industry
, 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
(5) the average stress-intensity limitations because the latter are only intended to prevent excessiveyielding and deformation.
7
where
A n a ly s is o f l ig am en t S tre s s In te n s it ie s A ve ra g ed T h ro u g h
D ep th o f P la te The value of (T , averaged through the depth of a plate at any
location is equal to the value of00 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
(8)
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 (T o , whichever has the largest absolute value.
(For example, if(T , = -3000 psi and(T o = 2500 psi, then (T , = -3000 psi and / (T , j = 3000 psi)
stresses at either surface of equivalent solid plate ob- tained from Step1of 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 (T O at 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 stressin- tensities 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 in- tensity 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 stressin a ligament having an arbitrary angular orientationif;.
with respect to if; for tube-sheet design calculations without in- troducing undue conservatism. The resulting expression should be used to obtain stress inteusities for typical ligaments in a uni- form pattern, !":ither than for isolated ligaments. The expression for the orientation which gives the maximum skess intensityis given by
- (T r ' cos3 if; sin if; + 002 sin3 if; cos if;
+ (0 ,2 - j- (T , (T O + (T 0 2 ) sin 2if; cos2l/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 , / (T o or (T o /a " whichever gives-1 ~ fJ ~ l.
1 = 1 for isotropic loads
fJ = 0 for uniaxial loads = -1 for pure shear
Equation (7), written in terms offJ , was used to find the orien- tation if; which gives the maximum average ligament stress in- tensity in a stress field of biaxialityfJ . This orientation was then used in equation (6) to evaluate the corresponding value of the average stress intensity. The resulting values are given by:
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 fieldin the equivalent solid p1a.te is isotropic. Hence, as indicated by equa- tion (4), there are no shear stressesTy" 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 regard- less 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 con- ditions, the maximum shear stress occurs in thc plane of the plate, as illustrated in Fig. 9. The equation for the average stressin- tensity 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 trans- verse 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 bi- axiality 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 transversestresses were probably higher than they would be for metals. Hence, the maximum error in the calculated stress-intensity values isproba- bly 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 furtherfor de- sign calculations by consiaering the symmetry of the hexagonal array of neighboring holes surrounding the typical hole. Itis apparent that the same stress distributions would result ifthe orientation of the ligaments were shifted ±60 deg in the equiva- lent 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.llyloaded 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 pat- terns "ith respect to the radii of the plate varies graduallyaround the periphery, encompassing the entire range of possible orienta- tions. From these considerations,it is apparent that the expres- sion for tbe stress intensity, equation (6), can be maximized
8 T ra n s a c t io n s o f the A S M E
K
2.0
1.9
>- !:: 1.6 (J)
~ 1.2
1.1
1.0
CTr8CT8=STRESSES IN EQUIVALENT SOLID PLATE
CT,= CTror CT8(WHICHEVER HAS THE
LARGEST ABSOLUTE VALUE)
8 CT
r
0.8 - 1.0 - 0.8 -0.6 -0.4 - 0.2 0+ 0.2 +0.4 +0.6 + 0.8+ 1.0
(3, BIAXIALITY RATIO
Fig. 10 Stress intensities in perforated-plate ligaments
loads T liZ is zero at the mllllmum ligament sections. The transverse shear stress averaged through the depthT lI' varies linearly with radial locationr in 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-dimen- sionai Mohr circle based on these stress values is shown in Fig. 11. Since the transverse stress in the ligamentiT z has the same sign as the longitudinal stressiT ; (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 sub- ject to the transverse shearT il'. The average stress intensity in a ligament at any radial distancer from the center of the plate is given by
R [(AP r)' J '/' (ma,.'I:with r = radius Self = - -- + ( iT r )2 . (9)
h H to outermost lIgament)
where
AP pressure drop across plate r = radial distance of ligament from center of plate
iT r = iT o stresses averaged through depth of equivalent solid plate
H thickness of plate
Peak Stresses in Perforated Plates Maximwn local stresses due to all loads (mechanicltl and
Journal of Engineering for Industry
- STRESSES IN PRINCIPAL PLANE OF MAXIMUM SHEAR STRESS
- STRESSES IN OTHER PRINCIPAL PLANES
Fig. 11 Three-dimensional Mohr circle for stresses averaged across
minimum ligament section and averaged through depth of plate
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
30
28
26
24
22
~20 )(
0
10
8
6
4
2
CTmax = Y CTI
\ SOLID PLATE
CT 1 ~
\ fir I
'\. \ q~'\. 1\ '\. \ 2h,
\ I I I '"
'" ~UNI~XI~L ,S~R~S~(f3= ~ )
"""-. "" .-/ "
..•.•..••.. " /'
..•••..•... ~ ..•.. "-....•.•..•.../ " ....... ~ .............. .....•.. •••• .•..~ ~ ...-
0""'-- -- ---- o 0.1 0.15 0.2 0.3 0.4 0.5 0.6 0.70.80.9 1.0
h/R, LIGAMENT EFFICIENCY
Fig. 12 Maximum local stresses in perforated plotes
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-platefJ = ( J r / ( J o or ( J o / ( J r (whichever gives -1 ~ fJ ~ ). This ratio varies, of course, with radialloca- tion in the plate. The maximum stress for any particular thermal or pressure load is then given by the relation:
(10)
where
( J 1 (J r or ( J o (whichever has the largest absolute value) Y value given in Fig. 12 P pressure acting on surface
All thermally induced maximum local stresses, as well as pres- sure stresses, must be considered in the cumulative fatiguelimita- tions on the values of( J m ,, ' 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
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 approxi- mated by the expression
(11 )
where
(Jrim nominal bending plus membrane stress at inside of rim
[ (r 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 dimen- sions.
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
O'"mox =K rC"'nm
, I I I I I I I II CTMAX = KOO"NOM 1- j-D-j-
I I I (!)p ~o-
\ EaT(TH-Tc) 0 50_
Tc o O"MAX ~o-
\ o ~o 0 ~o_
ti 1.8
§2.6 b~
~2.4
b
Evaluation of Special Cases of Thermally Induced Stresses in Tube Sheets
.02 .04 .06.08 .10 .12 .14 .1618 .20.22 .24 .26 .28.30
P o
Fig. 14 Peak thermal stresses at perforations adjacent to a diametrallane
where
K " uniaxial (/3 = 0) stress multiplier from Fig. 12 E* effective elastic modulus for tube sheet
The stresses at the edges of the holes adjacent to the unper-
a T thermal expansion coefficient, in/in/deg F T p primary temperature, deg F T / metal telnperature, at secondary tube-sheet surface,
deg F 0
(l4) KDEaT(TH - 7 '.)
2(1 - II) ([max
where
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 ligamentsin 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 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 as- sumed that there is no redistribution of load to nearby ligaments. The limited stress intensity in a particular out-of-tolerance liga- ment can then be evaluat-ed by substituting the smallest ligament
G Reference [191. figs. 20, 21, and 32.
K D = 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.
forated diametrallane, Fig. 14, can be approximated by assum- ing a linear temperature drop across the diametrallane:
(13)
(12)
O"max
where
Thermal "Skin Effect." 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. Be- cause of the thermal film drop, the entire difference between the primary and secondary fluid bulk temperatures does not con- tribute to the skin effect. Credit may be taken for the tempera- ture drop in the thermal boundary layer at the secondary sideof the tube sheet when this drop can be evaluated. Stresses due to this effect are given by
Stresses for Temperature Drop Across Diametral Lane of U-Tube Type
Steam-Generator Tube Sheet. 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
Journal of Engineering for Industry 11
I '. I I I I I I
1/ Kmh'~"1 Q
"""'"
~ ~ ~ -
I I I I I I I I I I I
3.4
3.2
3.0
2.8
~-J 2.2
ZZ
~~ 1.2 «<l: ww 0..0.. 1.0 ;;e
0.8
0.6
0.4
0.2
hmln REDUCED LIGAMENT WIDTH -h-" NORMAL LIGAMENT WIDTH
Increase of peak slress due10 misplaced hole
width at the misplaced hole into elJuation (9). The resulting stress value is limited to 3S " "
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 apar- ticular out-of-tolerance thin ligamen(, was evaluated in photo- elastic 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
(15)
where
<TI < T r or < T o (whiehever has t.he largest absolute value) K", value given in Fig. 15
P pressure acting on surraee Y value given in Fig. 12
The K",-v:tlue givcn in Fig. 15 can be used for any ligament efficieney.
S u m m a ry an d C o n c lu s io n s
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.
';' ( Peak ill ligaments Cyclic thermal (temperature 1
difference across diametral Peak at holes adj:tcent lane) lane
Load
Pressure
I face of plate
Average across ligament at either sur- face of plate
{
Peak at surface
12
o Equation (8) was obtain cd by maximizing thc stress intensity with respect to the angular orienta- tion 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 orientationf . The maximnm value of this stres>;irl- tensity for all ligaments should be limited as indicated in the table.
TransacHons of the AS M E
J o u rn a l o i E n g in e e r in g fo r In d u s try
Equation
(15)
holes is given. The relevant stresses and their proposed limits
are given in the following table.
Limit
3S",
Cumulative fatigue
3 1. Malkin, "Notes on a Theoretical Basis for Design of Tube Shects of Triangular Layout," TRANS. ASME, vol. 74, 1952, pp. 387-396.
13
l"'inted in U. S. A.
4 K. A. Gardner, "Heat Exchanger Tube Sheet Design," ,Journa l o f App lied 111echan ics,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,"Jo ttrua l o f App lied J llechan ics, 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,"Journa l o f AP1Jlied 111eehan ics,
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," P roc:eerlings, I. 111eeh. E ., vol. 169, 1955, pp. 789-810.
8 K. A. G. i\liJler, "The Design of Tube Plates in Heat Ex- changers," P roceed ings, I. ilIech . E ., vol. 1 13, 1952, pp. 215-231.
9 G. D. Gallet!y and D. It. Snow, "Some Results on Con- tinuously 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 Associa- tion," 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, \Vashington25, 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 Tech- nical 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," BettisTech- nical 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 W eld ina Joumal, Re- search Supplement, September, 1958, pp. 411s-417s.
18 R. C. Sampson, "Photoelastic Analysis of Stresses in Per- forated Material Subject to Teusion01' Beuding," Bettis Technical Review, WAPD-BT-18. April, 1960; available from Office of Tech- nical Services, Department of Commerce. \Vashington 25, D.C.
19 R. E. Peterson, "Stress-Concentration Design Factors,"John Wiley & Sons, New York, N.Y, 1953.
Stress intensity
Peak in ligaments
Load Combined pressure
Cyclic pressure and thermal
R e fe re n c e s 1 G. Horvay, "The Plane-Stress Problem of Perforated Plates,"
JO 'll1 "1 ta lo f App lied 111echan ies,vol. 19, TRANS.AS ME, vol. 74, 1952,
pp. 355-360. 2 G. Horvay, "Bending of Honeycombs and Perforated Plates,"
Journa l o f A1Jp lied 111eehan ics," 'o l. 19, TRANS.ASME, vol. 74, 1952, pp. 122-123.
7 See reference [14], figs. 7-11, and reference [15], figs. 18, 19,20,24,
and 25
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 per-
forated 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 as5/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 tomake
design modifications in order to meet the criteria recommended
herein, whereas TEMA does not account for thermal loads.
A ckn o w le d g m en ts 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 pro-
posed 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.