AECL-6476

ATOMIC ENERGY WffR L'ÉNERGIE ATOMIQUEOF CANADA UMITED U B f DU CANADA LIMITÉE

ELASTIC STRESSES IN U-SHAPED BELLOWS

Contraintes élastiques dans les soufflets en forme de U

P. JANZEN

Chalk River Nuclear Laboratories Laboratoires nucléaires de Chalk River

Chalk River, Ontario

May 1980 mai

ATOMIC ENERGY OF CANADA LIMITED

ELASTIC STRESSES IN U-SHAPED BELLOWS

by

P. Janzen

Chalk River Nuclear LaboratoriesChalk River, Ontario KOJ 1JO

1980 May

AECL-6476

L'Energie Atomique du Canada, Limitée

Contraintes élast iques dans les sou f f l e t s en forme de II

par

P. Janzen

Résumé

Ce rapport présente des re la t ions décrivant les niveaux de contraintesélast iques méridionales et c i r con fé ren t i e l l es à l'embase et à la t ê t i è r e àcause de la pression externe et de la f l ex ion ax ia le des sou f f le ts en formede U. La dér iva t ion est basée sur l 'analyse s ta t i s t i que de donnéesthéoriques obtenues à p a r t i r d'une analyse des éléments f i n i s deconf igurat ions choisies de s o u f f l e t . Les formules mathématiques etdiverses représentations graphiques sont proposées comme aides pour laconception et l 'analyse des s o u f f l e t s .

Laboratoires nucléaires de Chalk RiverChalk River, Ontario KOJ 1J0

Mai 1980

AECL-6476

ATOMIC ENERGY OF CANADA LIMITED

ELASTIC STRESSES IN U-SHAPED BELLOWS

by

P. Janzen

ABSTRACT

This report presents relations describing the meridionaland circumferential elastic stress levels at the rootand crown due to external pressure and axial deflectionof U-shaped bellows. The derivation is based on astatistical analysis of theoretical data obtained from afinite element analysis of selected bellows configurations.The mathematical formulations and various graphicalrepresentations are proposed as aids to bellows designand analysis.

Chalk River Nuclear LaboratoriesChalk River, Ontario KOJ 1J0

1980 May

AECL-6476

iii

TABLE OF CONTENTS

page

1. INTRODUCTION 1

2. PERSPECTIVE 1

3. REFERENCE CONVOLUTION SHAPE 4

4. STRESS ANALYSIS 4

5. APPROACH TO STATISTICAL STUDY 6

6. REGRESSION ANALYSIS 7

7. RESULTS AND DISCUSSION 11

8. CONCLUSION 12

9. ACKNOWLEDGEMENT 13

10. REFERENCES 13

TABLES

FIGURES

iv

LIST OF SYMBOLS

e, exp exponential function (base of Napierian

logarithmic system)

d convolution depth

Z bellows' active (convoluted) length

nc bellows' number of convolutions

n bellows' number of material pliesp external pressure

r convolution crown torus radiuscr. bellows convolution inner radius

r = r. + d/2 bellows convolution mean radius

r convolution root torus radius

t material ply thickness

E material Young's modulus

F stress function correction factor

K bellows axial spring rate

S stress function

S. inner surface tangential stress

S outer surface tangential stress

° S Sos " SisS, = j bending stress component

S + S._ os is .S = T membrane stress componentm z

i non-dimensional configuration parameters

Y - —C ri

YF = t

~±a., i • 0,1,••',5 exponents of configuration parameters3.» i = 0,1,»••,4 exponents of non-dimensional configuration

parameters

ô axial deflection per convolutionr

À. = — radius ratiom

X- = —p- thickness ratio

y = /l2(l-v2)A1A2 shape parameter

v material Poisson's ratio

<)> angle measured in the meridional plane

A = n <5 total bellows axial deflectionc

ELASTIC STRESSES IN U-SHAPED BELLOWS

1. INTRODUCTION

Although bellows have numerous applications,frcm large,low-pressure expansion joints in air ductsand pipelines to small, high-pressure bellows stemseals in valves, little information on design oranalysis of bellows can be found in published litera-ture. The information which is available is basedon approximate solutions which frequently result inunacceptable discrepancies.

This report derives equations describing the springconstant, and the meridional and circumferentialelastic stress levels at the root and crown of U-shaped bellows due to pressurization and axial deflec-tion. The derivation is based on a statistical analysisof theoretical data obtained from a finite element analysis ofselected bellows configurations. These mathematicalformulations and various graphical representations areproposed as aids to bellows design and analysis.

2. PERSPECTIVE

The analysis of bellows to determine its response tovarious applied loads has received attention overmany years. Selected contributions will be mentionedto place this subject in perspective.

Theoretical treatments of this problem have exploitedtwo basic mathematical techniques designated "approxi-mate solutions" and "asymptotic solutions" (1) .Each approach has a limited range of validity relatedto some geometric parameter. Parameters recurring inpublished literature include a "radius ratio"\± = rc/rm, a "thickness ratio" \2 = rc/t and acombined shape parameter

/12(1-V2) A ^ = À2(l-v2)rc2/rmt.

Figure 1 defines the geometric variables and rm

+ d/2.

- 2 -

The 'approximate solution" employs a truncated infiniteseries expansion of the governing variables to solvethe exact theory equations. Its applicability islimited to small values of X^(Al<0.1) and an approxi-mate range of A 2 from 5 to 3 5

For large values of A2(^2>:1-5), the differential

equations of the exact theory can be reduced to permita closed form solution in terms of higher transcendentalfunctions. This approach yields the "asymptoticsolution".

R.A. Clark has published asymptotic solutions for an"Omega" piping expansion joint subjected to axial load,a corrugated pipe subjected to axial displacement andinternal pressure (2), and a U-shaped bellows subjectedto axial load (3) .

The same case of a complete torus slit at the inneredge and welded to a stiff cylindrical pipe subjectedto an axial load was also considered by N.C. Dahl (4).However, he developed an approximate solution utiliz-ing the principle of minimum complementary energy anda four-term series to approximate the change in themeridional tangent angles. Over a range of values ofp his solution corresponds to that of R.A. Clark.Beyond u = 30 the solutions diverged, with theasymptotic solution possessing greater accuracy.

U-shaped expansion joints have been the subject ofmore recent papers. A. Laupa and N.A. Weil (1) usedan energy method for the toroidal sections, and thetheory of symmetrical bending of circular platesaugmented by the thick-walled cylinder analysis forthe annular plate connecting the two toroidal sections.Their general solution permitted different radii tobe assigned to the inner and outer toroidal sectionsand independent changes in the average thickness ofthe toroidal sections and the interconnecting annularplate. Axial and pressure loading were both consider-ed. T. Ota, M. Hamada, T. Takezono, Y. Inove,T. Nakatani and M. Moriishi (5,6,7,8,9) extended thisanalysis slightly and included design charts toillustrate the behaviour of the maximum meridionalbending and circumferential membrane stresses asfunctions of geometric parameters.

While manufacturers have had some design formulasat their disposal for some time, either of a proprietaryor published nature (10), the proliferation of thin-ply

- 3 -

formed bellows, especially in finite life applications,has revealed a need for more exact design criteria.A number of reports have been issued in an attemptto correct this deficiency. R. A. Winborne (11) andW.F. Anderson (12) have presented design proceduresbased on asymptotic shell theory and charts for"curvature corrections". A more comprehensive reportby T.M. Trainer (13) summarized a three-yearproject whose objective was "to establish designprocedures, stress analysis methods, and other factorsessential to the successful design and fabrication ofmetallic bellows". Theoretical results of the bellowsresponse to axial and pressure loads were obtainedutilizing a multi-segment numerical-integrationtechnique. A computer program, capable of linearelastic axisymmetric and non-symmetric deformationanalysis and of non-linear axisymmetric deformationanalysis was developed. Parameters pertinent tobellows design and fabrication were identified and someparametric curves to permit convolution shapeoptimization were presented. Although the reportwas extensive, its usefulness as a practical designaid is limited. No mathematical design formulasare presented and the parametric curves are not givenin sufficient detail to aid in design.

Many of the above publications, in addition to compar-ing theoretical results with previously publishedresults, also reported some experimental results insupport of their analysis. One of the more enlighten-ing papers on experimental results was prepared byC E . Turner (14). He utilized an approximate solutionfor the range of y<5 as well as a numerical solutionfor an extended range of JJ from 0.4 to 35. Of mostsignificance, he noted a number of sources for thediscrepancies between theoretical and experimentalresults. Among these were the following:

1. The strain gage thickness is not negligiblein comparison with the ply thickness;

2. The gage grid spans an appreciable arc lengthof the convolution torus. Since the gagemeasures an average strain over the grid length,it will not indicate the maximum strain in aregion of a strain gradient;

3. Bellows generally possess a variable ply thick-ness along a convolution profile which isdifficult to anticipate analytically;

4. Nominally similar convolutions exhibited differentstrain levels at corresponding locations.

- 4 -

Even after attempting to account for all the knownexperimental errors, C.E. Turner (14) estimated potentialerrors between theory and experiment in the rangeof + 10%, in extreme cases + 20%.

Similar limitations of experimental studies have beenobserved by the author. In addition, an effect dueto ply interaction in multi-ply bellows has beennoted.

3. REFERENCE CONVOLUTION SHAPE

Although convolution shapes can be categorized, theyare far from standardized. This is mainly due tothe proprietary manner in which bellows have beendeveloped: methods of fabrication and productionmachinery itself have often been developed in-house.Because other than shape parameters can affectbellows performance, optimum convolution configura-tions may vary from one manufacturer to another.Localized changes in ply thickness and materialproperties, for example, will influence bellowsperformance. These differences in convolutioncharacteristics are especially apparent in small,high-pressure multi-ply bellows.

For the purpose of this study, an idealized,axisymmetric,U-shaped convolution with uniform material propertiesand thickness will be adopted as the referenceconfiguration. The physical parameters describingthis shape are the internal radius, r^, depth ofconvolution, d, root and crown torus radius, rr andrc, and thickness, t (Figure 1 ).

4. STRESS ANALYSIS

Stress states in bellows resulting from axialcompression and external pressure were determinedby the finite element method. Typical surface stressdistributions are depicted in Figures 2 and 3.

Thus, subjected to axial compression, a U-shapedbellows exhibits peak meridional surface stresses atroot and crown. Resultant circumferential surfacestresses are lower and more variable, not necessarilypeaking at root and crown. All surface stressesapproach zero at midspan.

- 5 -

External pressure causes peaks in all surface stressesat three locations: root, crown and midspan. Althoughof slightly different magnitudes, each principalstress has the same sign at root and crown which isopposite to that at midspan.

Theoretical analyses of stress states in thin shellsgenerally assume a linear strain variation withdistance normal to the neutral surface. In bellowsconvolutions, the ratio of torus radius to thicknesscan be small enough to yield a significantly non-linear strain distribution at root and crown. Agood approximation of the resulting stress distributionis given by the Winkler-Bach formula for curved beams.The stress distribution has a hyperbolic pattern,attaining a maximum value at the concave surface inthe case of pure bending. A study of the stressdistribution obtained from a finite element analysisof a bellows bears this out.

Due to the large circumferential radius of curvaturethe thickness ratio, r/t, is very large. The stressdistribution resulting from loading should bepractically linear and this is corroborated by thefinite element analysis of bellows.

While the maximum meridional stress in bellows due toaxial deflection frequently occurs at mid-crown andmid-root (<{>=0) , this is not always the case.N.C. Dahl (4) found that for toroidal bellows thelocation of maximum stress moved away from <f>=0 forU>5, approaching <J>=ir/2 for large y. According tothe results of the study of U-shaped bellows by T.M.Trainer (13), the location of maximum stress is at(J>=0 for y<0.75. For a transition range of y from0.75 to 2 an approximately constant maximum stresslevel exists over a portion of the torus. For valuesof v>>2 the location of the maximum stress tends to<J>=7r/2 with a significant decrease in stress occurringas

A similar situation exists for the case of bellowssubjected to external pressure with the differencethat the maximum meridional stress occurs at (j>=0 ofthe crown and root for p<5, the transition range is5<y<10, and for \i>10 the location of maximum stresstends to (j)=ir/2, again with a significant decrease instress occurring as <(>-*-0.

The distribution of circumferential stress is morecomplex. In the case of axial compression, extreme

- 6 -

levels generally occur near <J>=0 in root and crown.Depending on the bellows geometry, however, thelocation of peak stress may shift to <j>>0. Externalpressure can cause the circumferential stress stateto exhibit several locations of extreme levels.

5. APPROACH TO STATISTICAL STUDY

The study of the elastic states of stress in bellowsdue to axial compression and external pressure revealsa complex picture: the stress distribution changeswith change in any geometric parameter; the locationof the extreme meridional stress levels changes from<|)=0 for small y to 0»ir/2 for large y; externalpressure causes an extreme meridional stress level atmidspan as well as the crown and root; except forsmall y the locations of extreme meridional stresslevels due to axial deflection and external pressuredo not coincide; the circumferential stress distribu-tion is difficult to categorize. Some assumptionsmust be stated to reduce the problem to manageableproportions.

It will be assumed that the meridional and circum-ferential stress levels at <j>=0 characterize thebellows performance capability. Bellows which exhibitextreme meridional stress levels due to axialdeflection and external pressure at cj>=0 fall in therange of y<2. Such bellows have widespread applica-tion, generally at low pressure and where axialflexibility is important. Higher pressure may bereadily accommodated by increasing the number ofplies. The associated circumferential stress doesnot generally peak at <J>-0, but the resultant stressdue to the combination of axial deflection andexternal pressure will be a maximum near cf>=0.

The objective is to derive a response surface forthe stresses at root and crown in terms of bellowsgeometrical parameters. Five geometrical parameterswere considered essential to the statistical study:rc» *r> *i> d, t (Figure 1). Selected bellowsconfigurations were analysed by the finite elementmethod to generate the relevant data. A "half-replicate two level fractional factorial design"was chosen so that the number of test cases requiredwould be minimized. Then, to permit an evaluationof the quadratic terms in the response function a"star design" (15) with center point was superposedon the fractional factorial. A frequent practice

- 7 -

is to make such a design "rotatable" by a properchoice of the length of the axis arm of the stardesign, thus reducing the complexity of mathematicalcomputation. The preferred choice of the variablelevels precluded this practice. Moreover, theleast squares method was to be utilized in theregression analysis of the data and rotatabilitywas not an essential characteristic. The resultis a non-rotatable "central composite design" infive variables as shown in Table 1.

Two such designs were considered in the study: onein a range of low values of all parameters; one ina range of higher values of all parameters. Table 2gives the ranges of the parameters for each design.Taken together these arrays of tests permitted theinclusion of higher order terms in the mathematicalmodel of the response surface.

In the first design the values of y ranged from 0.09to 2.64. In the second design y ranged from 0.29to 2.28.

6. REGRESSION ANALYSIS

Conventional formulas for the maximum meridionalstress in a bellows are generally expressed as theproduct of operational (pressure or axial deflection),physical (number of plies or convolutions) andgeometrical parameters, the last raised to a power.Such an expression was adopted as the mathematicalmodel for the response surface:

a a a a a aS = e ° r l r 2 d 3 r.* t 5 ...(1)

c r i

Here S is the stress, and a.,i=0,l,...,5 are unknownexponents to be evaluated for a best fit of thefunction to the data.

In the case of axial deflection, stress is proportionalto deflection and Young's Modulus. Thus

_ (3 a a a a af • ie • r ' i ! d > r.» t ! ...(2)£i c r l

This expression is dimensionless. Then the sum of

- 8 -

oti, 1 • 1,...,5 must add to -1. A simplificationmay be introduced by dealing in the dimensionlessratios :

YB " ^ YC

Then

„ * 3 B 3 3 3S. = • ^ _ e 0 y l Y 2 V 3 y !Ci r . A i i l> &

If the bellows under consideration possessed ncconvolutions and experienced a total deflectionA = nc6, then

n r S B 6 B B 6— — - e ° Y 1 Y 2 Y 3 Y "

A E e XA YB YC Y E

Length and deflection are common design constraints.Further, the number of convolutions per length andhence the deflection per convolution, will depend onconvolution pitch. It is therefore not immediatelyobvious which convolution configuration will have themore favourable stress state.

Therefore, designers may find it advantageous to relatethe bellows axial deflection to length rather thanthe number of convolutions.

For U-shaped bellows the relation between length, i,and the number of convolutions is

% = 2 n c ( r c + r r ) . . . ( 6 )

The stress parameter can then be rewritten

s* 3 B 3 B B2 0 1 2 3 I2 s _ 0 Y 1 Y Y 3 Y

A(Y. + Y,)E e YA *B YC YEA a

- 9 -

A similar line of reasoning yields the. relation forstress resulting from external pressure!, p.

3 3 3 3 3 i6 Y A Y B Y C Y E i •••(»>

where nn is the number of plies in the bellows.P I

While the mathematical models, Eq.(7) ar>d (8) arenot linear, they can be made so by taking theirlogarithm. A multiple linear regression! analysisutilizing the least-squares method is thjen employed.This analysis minimizes the logarithm of; the errorsrather than the actual errors.These formulations were adopted for both membraneand bending stresses. Estimates of these stresscomponents were based on the surface stresses obtainedfrom the finite element method:

=

m =

S b

SOS

S o s

+2

-

2

S

S

i s

i s

...(9)

where Sos, S±s, are the outer and inner surface

stresses, Sm, Sb the membrane and bending components

of surface stresses. In view of the non-linear

stress distribution, Sm and Sb in the meridional

direction are fictitious quantities. The circumfer-

ential components however should be accurate estimates.

Design of structural components involving bellowsoften requires a knowledge of forces developed due toaxial deflection. To meet this need a responsesurface for bellows spring rate was derived based onthe mathematical model

n K oK 3 3 3 3

FT = -2— - e ° V V V

r i E r? (YA + Yn)E A B C

- 10 -

K is the spring rate.

While the response functions (7), (8) and (10) giveacceptable accuracy for many applications, animproved fit will result through a subsequentderivation of a correction factor

F = exp(f(Yi)) . . .(11)

The function f(Yi) is a polynomial of products ofthe geometric parameters. Terms of f(Yi) significantin improving the fit of the response function wereidentified by a computer program using a multiplelinear regression analysis with a forward stepwisealgorithm. Within the range of geometrical parameterlevels considered some of the circumferential stresseschanged signs. This precluded a direct applicationof Equations (7) and (8). Instead, absolute valuesof stress were first introduced into Equations (7)and (8) to evaluate the unknown exponents. Subsequent-ly a correction factor in the form of a polynomialin the geometrical parameters, F = f(Y^), was introducedto allow for the change in sign.

The response functions then take the form

- - c 3. 3. 3 33

° V cAE (YA+Y f i)AE

for axial deflection;

n_ S 3 3 3 3 3o v l v 2 v^s Y^U ...(12)

for external pressure;

,- 3 3 3 3 3F e Y Y Y

4 (YA+YB)E A B C

for the axial spring rate.

- 11 -

7. RESULTS AND DISCUSSION

The regression analysis according to the mathematicalmodels Equations (7), (8) and (10) yielded the resultsin Tables 3(a) and 3(b). These oversimplified modelsexhibit acceptable fit for many applications andindicate the effect of changes in the various geo-metrical parameters.

According to the sum of 3., i=l,...4, all meridionalstress components, S, due to axial compression andexternal pressure are practically independent ofinner radius, rj.. This is also the case forcircumferential bending stress components. The circum-ferential membrane stresses due to axial compressionare approximately inversely proportional to innerradius. Under external pressure the circumferentialstress components are approximately proportional toinner radius.

All bending stress components decrease at root andcrown for respective increases in torus radius. Achange in root torus radius has little effect onbending stresses at the crown, and vice versa. Thisis also the case for all meridional membrane stresscomponents. The circumferential membrane stresscomponents are affected by changes in either torusradius. In the case of axial deflection, theserelationships are valid for unit deflection perconvolution. If axial deflection is set per unitlength the effect is more complicated.

Convolution depth is the most influential configura-tion parameter, closely followed by ply thickness.An increase in convolution depth or ply thicknessresults in a decrease and increase, respectively, ofmost stress components due to axial compression andvice versa for external pressure. Some circumferentialmembrane stress components decrease in magnitude forincrease in both configuration parameters.

The axial spring rate is approximately proportionalto the inner radius and practically independent oftorus radii when expressed in units per convolution.Ply thickness and convolution depth are the dominantconfiguration parameters.

Tables 4(a) to 4(d) present formulas for the correctionfactor F, which includes the sign of the stresscomponents. For the range of parameter levelsconsidered, the magnitude of F is close to 1.

- 12 -

Because the regression analysis approximates themeridional non-linear stress distribution acrossthe ply thickness by a linear stress distribution,the membrane and bending stress components, separately,will be slightly in error. Should an accurateestimate of the membrane stress be desired, Table 4{e)presents appropriate formulas for the effect ofexternal pressure acting on the convoluted surfaceonly. For axial deflection, the axial spring ratemay be used.

To help visualize the response functions of Equation(12), Figures 4 to 9 show the relationship ofmeridional stress components at the convolution rootversus a configuration parameter for bellows subjectedto axial compression and external pressure.

Because these formulas are based on a statisticalderivation, some error is likely to be present in mostestimates. For the selected configurations of thisstudy the predicted meridional stresses agreed withcalculated stresses to within a maximum of five per cent.Errors for the circumferential stresses were higher,reaching a maximum of about seven per cent for bend-ing and more for the membrane stresses. The lattererrors were due to the low magnitude of the calculatedstresses for the selected configuration.

Results of the analysis were also compared with thoseof several sources for the example given in Â. Laupaand N.A. Weil (1). Although the convolution configurationpossessed a characteristic 11-1.47, the dimensionswere outside the range of parameter levels consideredin this study. Nevertheless, as Tables 5(a) to 5(h)show, the statistically derived response functionsyield results which approach those of the finiteelement method more closely than many design formulasproposed in literature or in current use.

8. CONCLUSION

Response functions in terms of geometrical variables*i> rr»

rc» ^ an<* *• a r e derived for U-shaped bellowswith a characteristic shape parameter y<2 for bendingand membrane stress components at root and crown inthe meridional and circumferential directions due toaxial compression and external pressure, for axial

- 13 -

spring rate and for the actual meridional membranestress due to external pressure acting on theconvoluted surface only. Suitable for design oranalysis, these formulas represent an improvementin the accuracy of response estimates and in thesimplicity of calculations over earlier designformulas.

9. ACKNOWLEDGEMENT

The finite element computer program utilized in thisstudy was developed by R. Shill of Atomic Energy ofCanada Limited, Chalk River.

10. REFERENCES

A. Laupa, N.A. Weil, Analysis of U-ShapedExpansion Joints, J. of Applied Mechanics,March 1962, pp 115-123.

R.A. Clark, On the Theory of Thin ElasticToroidal Shells, J. of Mathematics and Physics,vol. 29, 1950, pp 146-178.

R.A. Clark, An Expansion Bellows Problem, J. ofApplied Mechanics, March 1970, pp 61-69.

N.C. Dahl, Toroidal-Shell Expansion Joints,J. of Applied Mechanics, December 1953, pp 497-503.

T. Ota, M. Hamada, On the Strength of ToroidalShells, Bulletin of JSME, vol. 6, No. 24, (1963),p 638-654.

M. Hamada, S. Takezono, Strength of U-ShapedBellows, (1st. Report, Case of Axial Loading),Bulletin of JSME, vol. 8, No. 32, (1965),pp "=25-531.

M. Hamada, S. Takezono, Strength of U-ShapedBellows, (2nd Report, Case of Axial Loading -Continued), Bulletin of JSME, vol. 9, No. 35,(1966), pp 502-513.

- 14 -

8- M. Hamada, S. Takeaona, Strength of U-ShapedBellows, (3rd Report, Case of Loading ofInternal Pressure), Bulletin of JSME, Vol. 9,No. 35, (1.966), pp 513-523.

9. M. Hamada, Y. Inove, T. Nakatani, M. Moriishi,Design. Diagrams and Formulae for U-ShapedBellows, Int. J. of Près. Ves. and Piping, (4),1976, pp 315-328.

10. The M.W. Kellogg Company, Design of PipingSystems, John Wiley & Sons Inc., Revised 2ndEdition, 1955, pp 214-230.

11. R.A. Winborne, Simplified Formulas and Curvesfor Bellows Analysis, NAA-SR-9848, AtomicsInternational, California, USA, August 1964.

12. W.F. Anderson, Analysis of Stresses in Bellows,Part 1: Design Criteria and Test Results,Atomics International, NAA-SR-4527, October1964.

13. T.M. Trainer, Final Report on the Developmentof Analytical Techniques for Bellows andDiaphragm Design, T.R. Ho. AFRPL-TR-68-22,Battelle Memorial Institute, Columbus, Ohio,March 1968.

14. C E . Turner, Stress and Deflection Studies ofFlat-Plate and Toroidal Expansion BellowsSubjected to Axial, Eccentric or InternalPressure Loading, J. of Mechanical EngineeringScience, vol. 1, No. 2, 1959, pp 130-143.

15. J.S. Hunter, Determination of Optimum OperatingConditions by Experimental Methods, Part II-3,Models and Methods, Industrial Quality Control,vol. 15, No. 8, February 1959.

16. Standards of the Expansion Joint ManufacturersAssociation, Inc., Fourth Edition, 1975.

-15 -

TABtE 1 Design Matrix Used to Derive Computer -Generated Data.

*

CASE

1

2

3

4

5

6

7

8

9

1 0

1 1

1 2

1 3

1 4

1 5

1 6

17

1 8

1 9

20

21

22

23

24

25

26

27

CROWN TORUSRADIUS

rc

A

-1

+1

- 1

+1

- 1

+1

-1

+1

-1

+1

-1

+1

-1

+1

-1

+1

. 0

-1

+1

0

o0

0

o !o :0 ;

o ;

ROOT TORUSRADIUS

r r

CONVOLUTIONDEPTH,

di

B | C

t- 1

- 1

+1

+1

- 1

- 1

+1

+1

- 1

- 1

+1

+1

- 1

- 1

+1

+1

0

0

0

- 1

+1

0

0

0

0

0

0

-1

-1

! - l

- l

+i

+i

+i

+i

- l

- l

- 1

- i

+i

+i

+i

+i

0

0

0

0

o- 1

+ 1 >

0

0

0

0 :

INTERNALRADIUS,

ri

D

-1

I+ 1

+ 1

+ 1

+ 1

+ 1

+ 1

+ 1

+ 10

0

0

0

0

0

0

- 1

+ 1

0

0

PLYTHICKNESS,

t

E=ABCD

+ 1

- 1

- 1

+ 1

- 1

+ 1

+ 1

- 1

- 1

+ 1

+ 1

- 1

+ 1

- 1

- 1

+ 1

0

0

0

0

0

0

o0

0

-2

+2

2 5- l

FractionalFactorial

Center Point

ModifiedStar

Design

- 16 -

TABLE 2 : Variable Levels Corresponding toElements of the Design Matrix

VARIABLE

Crown TorusRadius,

r c

Root TorusRadius,

r r

ConvolutionDepth,

d

Inner Radius,

r i

PlyThickness ,

t

SYMBOL,UNITS

A» nun

B, mm

C> mm

D, mm

E j mm

DESIGN 1

LOW(-1)

0.762

0.762

3.175

12.70

0.203

HIGH(+1)

1.524

1.524

6.350

63.50

0.305

DESIGN 2

LOW(-1)

2.540

2.540

8.890

76.20

0.356

HIGH( + 1)

4.445

4.445

17.78

152.4

0.457

TABLE 3(a) : VALUES OF EXPONENTS IN THE RESPONSE FUNCTIONS FOR MERIDIONAL STRESSCOMPONENTS AND AXIAL SPRING RATE:

"E ri S is ^AE (Y.+YÏAEAxial Compression

External Pressure

Axial Spring Rate

n S

P

ncK

B B BF e ° Y. 1 Y 2

B *v ' v "

A 'B YC YE '

X.K2ir r± E

ri ( YA + YB ) E

STRESS

AXIALCOMPRESSION

EXTERNALPRESSURE

COMPONENT

MEMBRANE

BENDING

MEMBRANE

BENDING

AXIAL SPRING RATE

LOCATION

CROWN

ROOT

CROWN

ROOT

CROWN

ROOT

CROWN

ROOT

B0

-1.160

-1.063

-0.879

-0.818

-0.781

-1.036

-1.878

-1.760

-0.850

8i

-1.082

-0.032

-0.490

-0.029

-0.532

-0.038

-0.531

0.019

-0.028

B2

-0.006

-1.100

-0.018

-0.548

-0.020

-0.626

0.028

0.519

-0.001

B

-1.980

-1.932

-1.699

-1.655

1.462

1.636

2.255

2.353

-2.620

Bit

2.033

2.053

1.082

1.111

-0.918

-0.988

-1.901

-1.926

2.896

* F determines the sign of stress

TABLE 3(b) : VALUES OF EXPONENTS IN THE RESPONSE FUNCTIONS FOR CIRCUMFERENTIALSTRESS COMPONENTS:

Axial Compression:

External Pressure:

"c ri S

AEIS

FeB B B B

Y.1 Y * Y 3A B

Y * *C *E *

STRESS

AXIALCOMPRESSION

EXTERNALPRESSURE

COMPONENT

MEMBRANE

BENDING

MEMBRANE

BENDING

LOCATION

CROWN

ROOT

CROWN

ROOT

CROWN

ROOT

CROWN

ROOT

B0

-1.647

-0.702

-2.143

-2.096

1.566

1.538

-3.091

-2.954

1

0

-0

-0

0

0

-0

0

B

.636

.299

.496

.029

.099

.188

.534

.025

0

0

-0

-0

0

0

0

-0

B2

.452

.746

.026

.561

.509

.478

.031

.522

B3

-1.624

-1.604

-1.697

-1.655

-1.139

-0.899

2.243

2.333

3

-0.

0.

1.

1.

-0.

-0.

-1

-1

It

430

334

086

112

425

566

891

912

I

05

F determines the sign of stress.

- 19 -

TABLE 4 (a) : Formulas for Correction Factor F for MeridionalStresses due to Axial Compression, and AxialSpring Rate. Signs of Bending Stress Componentsare for Outer Surface. To obtain the InnerSurface Bending Stress Component, change the signof F.

1. MEMBRANE STRESS AT CROWN:

F - -exp[0.711 x 10"1 - 0.130 x 10<Y£/Yc) - 0.451 x 102(YA Yg2)

+ 0.297 x 1O"2(YE/YB2) + 0.344(YA

2/YE) + 0.363 x 10(YB2/Yc)

+ 0.188 x H f 1 <YC2/YE) - 0.207(YA YC/YE) - 0.340 x 10"2(YB/Y

2. MEMBRANE STRESS AT ROOT:

F « -exp[0.364 x 10"1 + 0.139 x 10"3(Yc/YA2) - 0.104

+ 0.566(YA2/YB) + 0.571(YB

2/YE) + 0.318 x lO'1^

- 0.326(YB YC/YE) - 0.256 x 10"1(YB/YB Y c ) ] .

3. BENDING STRESS AT CROWN:

F - exp[0.257-0.435 x 10"2( l /YA) + 0.820 x 10(YA Yc)

+ 0.264 x lO^CY^Yg) - 0.454(YE/YA) - 0.173 x 1 0 ' 4 ( l / Y c Y£)

- 0.753 x 10"2(YB/Yc2) + 0.596(YA

2/YB) - 0.677(YA2/YE)

+ 0.227 x 10(YB2/Yc) + 0.141 x 10" 5 ( l /YA

2 Y,,) -0 .180 x 103(YA Yc Y£)

- 0.107(YA YC/YE) + 0.141 x 10"3(Yc/YA Y B ) ] .

4 . BENDING STRESS AT ROOT:

F - -exp[0.281-0.639 x 10~2(l/Y_.) - 0.499 x 10~ 2 ( l /Y o )

+ 0.391 x ÎO^CY /Y ) - 0.211 x 10(Y_/Yn) - 0.949 x 103(Y. Y 2 )at, Ci (j A B

- 0.920 x 10(YA Yc2) - 0.703 x 104(YA Yg2) + 0.696 x Uf 4 (Yj/Y^)

+ 0.185 x 102<YB2/Yc) - 0.143 x lOttg2/^) + 0.201 x 10~5(l/YB

2 Yc)

+ 0.161 x 103(YA YB Yc) + 0.364 x 104(YA Yfl YE) - 0.623 x 10"3(YB/Yc

5. AXIAL SPRING RATE:

F - exp[-0.185 + 0.257 x 10'4(l /YB2)

+ 0.558 x 10"2(YE/YA2) + 0.417(YA

- 20 -

TABLE 4(b) : Formulas for Correction Factor F for MeridionalStresses due to External Fressure. Signs ofBending Stress Components are for Outer Surface.Inner Surface Bending Stress Components areobtained by a change of sign of F.

1 . MEMBRANE STRESS AT CROWN:

F - -exp[-0.345 + 0.870 x 10"1(Yc/YA> - 0.727 x 10~2(Yj/Yç2)

- 0.397 x 10"3(Y./Y.2) + 0.129 x 10<Y.2/Y 2 ) - 0.662 x 10"2(Y,, Y./Y 2 )C A AC B C A

+ 0 .169 x 10" 6 (Y B /Y A Y E2 ) - 0 . 2 5 8 x 10"1(Y ( , Yj /Yg 2 ) + O . 1 2 1 x l O 3 ( Y A Yg2

- 0.1A2(YB Y C2 /Y E ) + 0 .187 x 1 0 " 3 2

2. MEMBRANE STRESS AT ROOT:

F - exp[-0.782 x 10"2 - O.19O(Y-,/Y2) + 0.638 x lO'^Y^/Y-)be D &

- 0.126 x 10~5(l/Y 2 Y.) + 0.104 x 10(Y2/Y.2) - 0.998 x 10~3(Y 2/Y.2)a \i o . \J LA

+ 0.295 x 10~2(Y 2 / Y 2 ) + 0.927 x 10"10(l/Y.2 Y_2) + 0.215 x 10"2

I* D A L

•(YE/YB Yc) - 0.672 x 10~1(YB ?<?/*£+ 0.213 x 103(YA2 Yg Yc)

- 0. 338 x 10(YA2 YB/YE) - 0.170 x 10~2(YA

2/YB Y£) + 0.143 x 10"3

• ( YC2 / YB V " ° - 1 8 9 x 1 0 " 2 ( Y B YC/YA Y E ) ] -

3. BENDING STRESS AT CROWN

F - exp[0.596 x lO"1 - 0.245 x 10"4(l/YA2) + 0.167 x 102(YA Y ^

+ 0.205 x 10"4(Yc2/YE

2) + 0.190 x 10"7<l/YA2 Y^) - 0.394 x 10(Ytt YE/YA)

- 0.226 x 10(YA2 YC/YB) - 0.226 x 10(YA

2 YC/YE - 0.160 x 10(Yc2 YE/YA>],

4. BENDING STRESS AT ROOT:

F - exp[0.117 + 0.268 x 10(Y_) + 0.151 x 103(Y. Y^)

- 0.304 x 10*(Y Y 2 ) - 0.103(Y_/Y 2 ) - 0.250 x U f ^ Y ^ / Y 2 )O fit ti \f A if

- 0.371 x 10(Y. YJV) - 0.131 x 10"1(Y_/Y. Y_) + 0.118 x 103.

•(YA YB Yc2) - 0.337 x 103(YA YE

2/YC) - 0.260 x 10"5(Yc/YA

2 YB)

+ 0.106 x 10~7(l/YA YB2 Yc) - 0.278 x 10(Yg

2 Yp/Yg)].

- 21 -

TABLE 4(c) : Formulas for Correction Factor F for CircumferentialStresses due to Axial Compression. Signs of BendingStress Components are for Outer Surface. InnerSurface Bending Stress Components are obtained bya change of sign of F.

1. MEMBRANE STRESS AT CROWN:

F - [0.139 + 0.280 x lO^d/Y.) - 0.765 x 10"1(Y,,/Y_)U IS Ci

- 0.182 x 102(Yn Y r) - 0.570 x 10"4(l/Y,. Y_) + 0.104 x 102(Y. Y 2 )

+ 0.375 x 10(YB2/YA) - 0.278 x 10"

5(l/YA2 YB) - 0.916 x 10~J

•<YE/YA YB) + 0.991 x 10'6(l/YA Yg Yg)].

2. MEMBRANE STRESS AT ROOT:

F - - 1.00

3. BENDING STRESS AT CROWN:

F - exp[0.318 - 0.679 x lo"2(l/YA) + 0.673 x 10"X(YA/YB)

- 0.176 x 10(Y£/Yc) - 0.465 i^^/\) + 0.240 x 10(YB2/Yc>

- 0.918 x 10~2(Yc2/YE) + 0.135 x 10"

5(l/YA2 Yç) + 0.428 x 10~*

•WE»'

4. BENDING STRESS AT ROOT:

F - -exp[-0.628 x 10"1 - 0.320 x 10"1(l/Y(;) + 0.182 x 102 (YA Y )

+ 0.101 (Y_/Y,J - 0.925 x 103(Y. Y_2) - 0.100 x 105(Y. Y_2)a t A D At

+ 0.951 x 10~2(YA/Yc2) - 0.806 x 10"4(YB/YE

2) - 0.226 x 10"?

.(YC/YB2) + 0.407 x 10"7(l/Yc Y£

2) - 0.453 x 103(YB2 Y£)

+ 0.142 x 10(Y2/Y.) + 0.176 x 102(Y2/Y,,) - 0.201 x 10<Y 2/X )DA O Li D IS

+ 0.112 x 103(YA YB Yc) + 0.436 x 104(YA YB Y£) + 0.199 x 10"l

(YE/YB Yc) + 0.131 x 10~5(l/YA YB Yc)].

- 22 -

TABLE 4(d) : Formulas for Correction Factor F for CircumferentialStress due to External Pressure. Signs of BendingStress Components are for Outer Surface.Inner SurfaceBending Stress Components are obtained by a changeof sign of F.

1. MEMBRANE STRESS AT CROWN:

F - - [ - 0 . 1 7 5 x 10 + 0.521 x 102(YA) + 0 .128( l /Y c ) - 0.160 x U f 1 (1/Yg)

+ 0.145 x 10(Y c2) + 0.331(Yc/YE) + 0.448 x 10(YA

2/YE)

- 0.393 x 10"?(Y c2 /YE

2) + 0.354 x 103(YA YR Yc) - 0.534 x 10(YA Y c /

- 0.111 x 102(YA YE/YC2) + 0.936 x 10"2 (Yfi Yç/Yg2) +0.236 x 10"?

.(YC/YB YE2) + 0.374 x 10"3(YB/Yc

2 Y£) - 0.230 x l o V / Yg/Yç)

- 0.296 (YB2/YC Y£) - 0.543 x 10"2(Yc

2/YA Y^,)].

2 . MEMBRANE STRESS AT ROOT:

F - - [ - 0 . 4 7 1 X 10 + 0.113 x 102(Yo) + 0.922 x 10~ 2 ( l /Y o )

+ 0.191(Yc/YE) + 0.575 x 102(YE/Yc) + 0.200 x 10"2(Yc /YB2)

- 0.345(Yc2 /YA) - O.729(YC

2/YE) - 0.134(YA2/YB

2)

- 0.228 x 10"2(Y 2 /Y_2) + 0.757 x 10"1<Y_/Y. Y.) - 0.449 x 10"2.Vi CM B A I»

•(YB/YA Y c2) + 0.183 x 10"5(YB/YA Y£

2) + 0.361 x l O ' V g / Y g Y,,2)

- 0.26 x 10~V A/Y B2 YE) - 0.519 x 10'

2(YE/YB2 Yc> + 0.551 x 10~*

•(YA YC/YB YE) - 0.346 x 103(YE

2/Yc2)l.

3. BENDING STRESS AT CROWN:

F - exp[0.461 x 10"2 - 0.105 x 10"*(l/YA2) + 0.650 x 102(YA Y,,)

- 0.249 x 103(Y. Y_) - 0.943(Y_2/Y.)+ 0.104(Y. Y_/Y_ )U £• V A A t> b

+ 0.674(YB YE/YC2) - 0.869 x 10~6(Yc/YA Yj

2) + 0.234 x 10 •

.(Y Y 2/Y ) + 0.325 x 103(Y Y_2/Y ) - 0.664 x 10 (Y 2 Y /Y )].A D L U b D A U £

4 . BENDING STRESS AT ROOT:

F - exp[- 0.124 x 10"1 + 0.674 x 10(YB> + 0.153 x 10~1(Y(;/YB)

- 0.468 x H f V / Z Y j 2 ) + 0.151 x 103(YB2 Y c

2) - 0.574 x 10*

•<YB Yc Y£2) - 0.390 x 10(YB Yg/Yç2) - 0.559 x 10"1(Y(, Yg/YA

2)

- 0.341 x 10(YB2 YC/YE)J.

- 23 -

TABLE 4(e) : Actual Meridional Membrane Stress due toExternal Pressure Acting on the Convolu-ted Surface Only.

1. ROOT

exp(-0.826)Y - ° -0 6 8 Y R ° - W Y,1'112 Y -1'050

A -b L £•

2. CROWN

PpSmc / np

- 24 -

TABLE 5 : Bellows Response to Loading in the Exampleof Laupa and Weil(l):

r = r = 1 3 . 7 mm,c r

d = 58.2 mm,

y = 1.47

r = 304.6 mm,

t = 1.2 7 mm

v = 0.3

(a) Bellows Axial Spring Rate and Meridional StressComponents at Root due to Axial Compression.

SOURCE

Finite ElementAnalysis

StatisticalModel

Laupa & Weil(1)

Salzmann (1)

Hamada

(7)

Kellogg (10)

Anderson (12)

EJMA (16)

SpringK

10

r

2

2

2

3

2

2

2

2

3

* Assumed linear

ncK

ri E

nc

i E

.88

.99

.90

.06

.64

.92

.21

.62

.05

Rate,

K

*

*

MembraneStress,Sm

102nc r., Si mAE

-0

-0(-0

-0

-0

-0

-0

-0

-0

-0.(-0.

.11

.35*11) t

11

12

10

10

08 t

10 t

0912) t

stress distribution.

= "c ri Sm . 2*tAE ri

Surface BendingStress, Sb

10 2n c riSfe

AE

Inner

8

6

8

10

9

8

12

11.

10.

69

82*

80

51

91

91

48

73

24

Outer

-9

-8

-8

-10

17

82*

80

51

-9.91

-8

-12.

-11.

-10.

t Derived from K

; F --= KA

91

48

73

24

Resultant SurfaceStress, SR

10 2 n c r i SRAE

Inner

8

8

8

10

9

8

12

-11

10

.58

.48

.6<)

.40'

.81

81

40

63

15

4 Derived from

Outer

-9

-9.

-8.

-10.

-10.

-9.

-12.

-11.

-10.

V

28

17

92

63

01

02

57

83

33

N * Sjjt, Membrane force per unit length.F * Axial Force.

_' Bending moment per unit length.

- 25 -

TABLE 5 (Cont'd)

(b) Bellows Axial Spring Rate and Meridional StressComponents at Crown due to Axial Compression.

SOURCE

Finite ElementAnalysis

StatisticalModel

Laupa & Weil(1)

Salzmann (1)

Hamada Y'

Kellogg (10)

Anderson (12)

EJMA (16)

Spring Rate,K

105n Kc

r i E

2.88 +

2.99

2.90

3.06

2.64 *2.92

2.21

2.62

3.05

MembraneStress, Sm

1 0Vi SmAE

-0,09

-0.34*(-0.10)t

-0.09

-0.10

-0.09-0.09

-0.07t

-0.08t

-0.09(-0.10)t

Surface BendingStress, Sb

1 0 2 nc ri SbAE

Inner

-9.00

-8.49*

-8.68

-10.51

-9.91-8.71

-12.48

-11.73

-10.24

Outer

8.51

8.49*

8.68

10.51

9.918.71

12.48

11.73

10.24

Resultant SurfaceStress, SR

102 nc r± SR

AE

Inner

-9.09

-8.83

-8.77

-10.61

-10.00-8.81

-12.56

-11.81

-10.33

Outer

8.42

8.15

8.59

10.41

9.838.62

12.41

11.64

10.15

*

t*

Assumed

Derived

Derived

linear stress distribution

from

from

n c K

r i E

N -

M -

F -

K

Sm

n„ c

Sbt2

6Axial

rJ Si m .AE

Membrane

, Bending

Force

Zirt (d + 'ri ri

Force per

Moment per

*- ï

Unit

Unit

F -

Length

Lengf

- 26 -

TABLE 5 (Cont'd)

(c) Bellows Meridional Stress Components at Root dueto External Pressure

SOURCE

Fin i te Element Analysis

S t a t i s t i c a l Model

Laupa & Weil (1)

Hamada (9)

Kellogg (10)

Anderson (12)

EJMA (16)

MembraneStress ,S m

% S mP

23.4

(39.6)*23.6 t

23.9

136.9

-

120.5

22.9

Surface BendingSt ress , Sj,

n S

P

Inner

-588.4

(-600.4)*

-582.?

-625.4

-1050

-758.3

-608.3

Outer

620.8

(600.4)*

582.9

625.4

1050

758.3

608.3

Resul tant SurfaceS t r e s s , SR

"pSR

P

Inner

-565.0

-560.8

-559.0

-488.5

-1050

-637.8

-585.4

Outer

644.2

640.0

606.8

762.3

1050

878.8

631.2

* Assumed Linear Stress Distribution

t Obtained from Axial Force Equilibrivan

Root Axial Force F = S r . tr mi

_ 27 -

TABLE 5 (Cont'd)

(d) Bellows Meridional Stress Components atCrown due to External Pressure

SOURCE

Finite Element Analysis

Statistical Model

Laupa & Hell (1)

Hamada (9)

Kellogg (10)

Anderson (12)

EJMA (16)

MembraneStress,Sm

p p Sm

P

-22.5

(-39.0)*-22.3 t

-22.1

-115.0

-

-120.5

-22.9

Surface BendingStress, S],

P

Inner

-588.2

(-571.0)*

-591.3

-625.4

-1050

-758.3

-608.3

Outer

555.4

(571.0)*

591.3

625.4

1050

758.3

608.3

Resultant SurfaceStress, SR

"pSRP

Inner

-610.7

-610.0

-613.4

-740.4

-1050

-878.8

-631.2

Outer

532.9

532.0

569.2

510.4

1050

637.8

585.4

* Assumed Linear Stress Distribution

t Obtained from Axial Force Equilibrium

Crown Axial Force F Sm ( ri + d ) t

- 28 -

TABLE 5 (Cont'd)

(e) Bellows Circumferential Stress Componentsat Root due to Axial Compression

Finite

SOURCE

Element Analysis

Statistical Model

Laupa &

Hamada

Weil (1)

(7)

MembraneStress,Sm

1Q2 ViSmAE

-4.59

-4.43

-4.35

-4.56

Surface BendingStress, S^

102 nc r i S bAE

Inner

2

2

2

2

.67

.56

.64

.73

Outer

-2.67

-2.56

-2.64

-2.73

ResultantStress,

102 n r. SC 1AE

Inner

-1.92

-1.87

-1.71

-1.83

SurfaceSR

R

Outer

-7.26

-6.99

-6.99

-7.28

(f) Bellows Circumferential Stress Componentsat Crown due to Axial Compression

SOURCE

Finite Element Analysis

Statistical Model

Laupa & Weil (1)

Hamada (7)

MembraneStress,Sm

102n r.Sc l mAE

3.84

3.75

4.04

3.92

SurfaceStress

Bending

' Sb

102 n r. S,c l bAE

Inner

-2.62

-2.47

-2.60

-2.67

Outer

2.62

2.47

2.60

2.67

ResultantStress

10 2n c

Surface

' SR

r i S RAE

Inner

1.22

1.28

1.44

1.25

Outer

6.46

6.21

6.65

6.58

- 29 -

TABLE 5 (Cont'd)

(g) Bellows Circumferential Stress Componentsat Root due to External Pressure

SOURCE

Finite Element Analysis

Statistical Model

Laupa & Weil (1)

EJMA (16)

MembraneStress,S

' m

np SmP

87.2

86.7

66.6

97.6

SurfaceStress

"p S

P

Inner

-181.1

-175.1

-174.9

1

Bending

• Sb

b_

Outer

181.1

175.1

174.9

-

Resultant SurfaceStress, S.,

K

"pSRP

Inner

-93.9

-88.4

-108.3

-

Outer

268.3

261.8

241.5

-

(h) Bellows Circumferential Stress Components atCrown due to External Pressure

f

Finite

SOURCE

tt

Element Analysis .

Statistical Model

i

Laupa

EJMA

& Weil

I

(1) j

(16) ,

MembraneStress,Sm

np Sm

P

47.7

46.6

69.7

97.6

SurfaceStress

n_JLP

Inner

-171.2

-166.6

-177.4

Bending, Sb

Outer

171.2

166.6

177.4

i

ResultantStress

Surface

• SR

"p SRP

Inner

-123.5

-120.0

-107.7

_ !

Outer

218.9

213.2

|247.1

-

t : PLY

CUFF

OD -

ID -

THICKNES:

1

OUTER D)

INNER D

U

I/SPAN

ROOTWIDTH

AMETER

AMETER

CONVOLUTI 9N

HRnWNTORUSRADIUS CROWN

GAPCROWNWIDTH

1

I I \\

)ROOT

GAP

/ V / V

1_r r : ROOT

TORUSRADIUS

CONVOLUTIONDEPTH, d_ ( O D - I D )

2

ID1 2

00ro= T

i

1

FIGURE 1 Bellows Nomenclature

OUTSIDE MERIDIONAL

INSIDE CIRCUMFERENCE

OUTSSDE CIRCUMFERENCE

— INSIDE MERIDIONAL

0.5 1.0 1.5 2.0 2.5 3.0 3.5CENTRELINE DISTANCE (mm)

4.0 4.5

FIGURE 2 Typical Bellows Stress Distribution Due to Axial Compression of1 mm per Convolution.

I

200h

-150

OUTSIDE MERIDIONAL

INSIDE CIRCUMFERENCE

OUTSIDE CIRCUMFERENCE

INSIDE MERIDIONAL

0.0 05 1.0 1.5 2.0 2.5 3.0 3.5 4.0

CENTRELINE DISTANCE (mm)

i

FIGURE 3 Typical Bellows Stress Distribution Due to External Pressure of 1 MPa.

a) Membrane Stress: Y, LB 0.02 b) Bending Stress: Y = YB

0.02

FIGURE 4 Compressive Meridional Membrane and Outer Surface Bending Stress Componentsat Convolution Roots Due to Axial Compression as a Function of ThicknessParameter, Y, t/ri"

10 . -

o.ooo .rot .1

10

9 .0» .004

c) Membrane S t r e s s : Y. = YB 0.04 d) Bending Stress = YB = 0.04

FIGURE 4 cont 'd

I

10

0.000

e) Membrane Stress : YA

0.06 f) Bending Stress: 0.06

FIGURE 4 cont'd

10",

o.ooo -m .a» .ou .016 .020 .021 .02s

1

0.000

g) Membrane S t r e s s : Y. = Y_ = 0 . 0 8n a

h) Bending Stress: Y 0.08

FIGURE 4 cont 'd

0.000 .032

10

0.000

1) Membrane Stress: Y 0.10 j) Bending Stress: Y,

FIGURE 4 cont'd

o.ooo

1LOGOI

0.000 .004 .008 .012 .016 .028 .03?

k) Membrane Stress : Y = 0.12 1) Bending Stress : Y. = Y_. = 0.12A J5

FIGURE 4 cont 'd

10*

10*

10" 1

It"0.000 .004 .00* .012 .OU .020

a) Membrane S t r e s s : Y, = YA B

0.02 b ) B e n d i n g S t r e s s : Y = Y = 0 . 0 2A 1J

FIGURE 5 Tensile Meridional Membrane and Outer Surface Bending Stress Componentsat Convolution Roots due to External Pressure as a Function of ThicknessParmeter, Yw =

- 4 0 -

IIIIIII i i—I» — I — h n r r n

- S S X S S S R R K s e s s s s

-a-oo11

60

a•Hcai

IttUJ.U J .'J J IUUJJJ J UiUU J

I11111 1 1 1 him 1 1 1 tur U

o

o

(0toaiM

0)GM

•8

10

0.000 .00410"'.

-C-

e) Membrane S t r e s s : YA « YR = 0 .06 f ) Bending S t r e s s ; Y = Y = 0 .06A a

F i g u r e 5 c o n t ' d

.E-

0.000 .O0< .01» .012 .0» .0» 0.000 .004 .001 .012 .01* .020 .OU .ait .032

g) Membrane S t r e s s : Y = Y = 0.08A B

h) Bending S t r e s s : YA ~ Y B » ° ' 0 8

Figure 5 cont'd

«.0» .OU

i ) Membrane S t r e s s : Y. * Y_ = 0.10A D

j) Bending Stress: YA = Y - 0.10

Figure 5 cont'd

10'. .

10

I

k) Membrane Stress: t = "Ï = 0.12 1) Bending Stress: YA =

YB = 0.12

Figure 5 cont'd

I

O.D .1

a) Membrane Stress: 0.008 b) Bending Stress: Yt 0.008

FIGURE 6 Compressive Meridional Membrane and Outer Surface Bending Stress Componentsat Convolution Roots Due to Axial Compression as a Function of ConvolutionDepth Parameter Y_ = d/r..

10*.

1

i•

'"'I

10'

10"

: 1 1—

/

~ i —

1

1 1

0-

1

^^-Jo.02

-^*^O--0.04

Y I

:

1 1

ONI

a) Membrane Stress: Yn = 0.008 Bending Stress: Yv = 0.008

FIGURE 7 Tensile Meridional Membrane and Outer Surface Bending Stress Components atConvolution Roots due to External Pressure as a Function of Convolution DepthParameter Yn = d/r., .

10'

îo-l

"r i LO.OO .02 .0*

i) Membrane Stress: Yw = 0.008Eb) Bending Stress: ••= 0.008

FIGURE 8 Compressive Meridional Membrane and Outer Surface Bending Stress Componentsat Convolution Roots due to Axial Compression as a Function of Root TorusRadius YB = rr/r±.

I

00

0.0 .02 .01

a) Membrane Stress Y_ = 0.008ft

b) Bending Stress: = 0.008

FIGURE 9 Tensile Meridional Membrane and Outer Surface Bending Stress Components atConvolution Roots due to External Pressure as a Function of Root TorusRadius YB = rr/r..

ISSN 0067 - 0367

To identify individual documents in the serieswe have assigned an AECL- number to each.

Please refer to the AECL- number when re-questing additional copies of this document

from

Scientific Document Distribution OfficeAtomic Energy of Canada Limited

Chalk River, Ontario, CanadaKOJ 1J0

ISSN 0067 - 0367

Pour identifier les rapports individuels faisantpartie de cette série nous avons assignéun numéro AECL- à chacun.

Veuillez faire mention du numéro AECL- sivous demandez d'autres exemplaires de cerapport

au

Service de Distribution des Documents OfficielsL'Energie Atomique du Canada Limitée

Chalk River, Ontario, CanadaKOJ 1J0

Price $4.00 per copy Prix $4.00 par exemplaire

1112-80