Optimization of laminated cylinders for buckling

June·

1987

OPTIMIZATION OF LAMINATED CYLINDERS FOR BUCKLING

by

Guojun Sun

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DELFT

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4 NOV. 1987

UTlAS Report No. 317

eN ISSN 0082-5255

OPTIMIZATION OF LAMINATEO CYLINOERS FOR BUCKLING

Submi tted May 1987

June 1987

by

Guoj un Sun

© Guoj un Sun 1987

UTIAS Report No. 317 CN ISSN 0082-5255

'

.

Acknowledgement

The author wishes to express his gratitude to his supervisor, Prof. J.S. Hansen for providing the opportunity to pursue the graduate program, suggesting the research topic, encouragements and continued i nterest in the present study. Thanks are al so due to the other two members of his Ph. O. committee, Prof. R.C. Tennyson, who has a profound i nterest in the stabil ity of compos ite she 11 s and suggested the second topic of this study, and ' Prof. P.A. Sullivan for their valuable discussions and advice throughout the investigation.

I would like to thank my friend Mr. Heng Zhang for helpful discussion on optimization methods and my colleague t~r. G. Hharram for his help on the experiments.

This work was supported by the National Science and Engineering Research Council of Canada under Grant A3663.

Abstract

Optimization of the buck1 ing load of a 1aminated-composite circu1ar cy1indrica1 she11 subjected to axia1 compression, externa1 pressure, torsion, or a combination thereof, is undertaken. In the optimization procedure, the buck1 ing load is taken as the objective function to be maximized while the fiber orientation of each 1amina represents the optimizing parameter. Powell's method is used for the optimization, with the initia1 point obtained by a random search technique. For the shell ana1ysis, the von-K~rm~n-Donnell thin shell equations are solved using the finite difference technique. C1amped boundary conditions and non1 inear prebuck1 ing effects are inc1 uded; _{Koiter ' s} perturbation approach is used to yie1d both the buck1 ing load and the post-buck1 ing character of the shell. The procedure deve10ped is demonstrated for eight 1 oad i ng confi gurati ons and compari sons are prov ided for some reference 1aminate configurations. It is found that in each 10ading case the optimized configuration is reproducib1e and the improvement to the buck1ing strength due to optimization is significant. In addition, se1ected laminates were chosen for an experimenta1 pro~ramme involving a series of four-ply graphite/epoxy shells. The predicted analytical and the measured experimental buckling loads are in good agreement.

In the second part of this study, the analysis is extended to the case of laminated noncircular cy1indrical shells subjected to axial compression. The buckl ing and initial post-buckl ing behaviour are investigated by two analytical approaches; both are based on Koiter's asymptotic methode In the first, the effects of boundary constraints and

prebuckling deformations are ignored; this simplified analysis offers a first approximation to the prob1em. In the second, the inf1uence of clamped boundary condition associated with nonlinear prebuckling deformations is included; this development yields an accurate but much more complicated analysis. Calcu1ations are carried out for several example configurations and the results obtained from these two quite different approaches are compared. It is found that the buckling load and the post-buck1ing character of laminated noncircular cylinders, like their circul ar counterparts, are affected si gni fi cantl y by the 1 aminate configuration.

1. Ac knowl edgement Abstract Table of Contents Li st of Tabl es Li st of Figures · Nomencl ature Introduction Table of Contents 1.1 Postbuc kl i ng Theory Page ii iii iv vii viii xi 1 1 1.2 OptimlJll Design of Composite Cyl indrical Shells Subjected

to Combined Loads 4

1.3 Buckling and Postbuckling Behaviour of Noncircular

Cyl inders 7

PART I 11

OPTIMIZATION OF LAMINATED CIRCULAR CYLINDRICAL SHELLS SUBJECTED TO COMBINED LOADS

2. Optiml.ll1 Design of Composite Circular Cylindrical Shells

Subjected to Combined Loads 11

11 11 15 2.1 Theoretical Anal ysi s

2.1.1 Governing Equations 2.1. 2 Prebuc kl i ng Sta te • • 2.1.3 Wo and F 0 Probl em 2.1.4 Buckling Equations 22 24 2.1.5 Iteration Procedure to Solve the Buckling Equations 26 2.1.6 The Second Order Fieldand Imperfection Sensitivity .

Co~fficient Ibl ₂₈

2.1.7 Boundary Conditions 2.2 Optimization Procedure

32 33

3.

4.

2.3 Experiments

2.3.1 Anisotropic Cylinders

2.3.2 Combined Loading Buckling Tests

2.3.3 Compari son Between Theory and Experiment 2.4 Di scuss ion

2.4.1 A Physical Insight into the High Buckling of the Optimized Cyl inders

2.4.2 Convergence in Search of the Optimum 2.4.3 Accuracy in the Buckling Load Calculation 2. 5 Co n c 1 u s ion

PART II

A STUDY ON THE STABILITY OF

LAMINATED NONCIRCULAR CYLINDRICAL SHELLS Composite Noncircular Cyl inder Under Axial Compression (A Simpl ified Approach)

3.1 Theoret ical Anal ysi s

3.1.1 Geometryof Noncircular Cross Sections 3.1.2 Governing Equations

3.1.3 Buckling and Postbuckling Fields 3.2 Results of Analysis and Discussion 3.3 Concl usions

Composite Noncircular Cyl inder Under Axial Compression (An Accurate Approach)

4.1 Theoretic~ Analysis

4.1.1 Governing Equations 4.1.2 Prebuc kl ; ng Sta te

4.1.3 The Determination of ~l(Y) and ~2(Y)

• •

4.1.4 Wo and Fo Problem

4.1.5 Solution of the Buckling Equations

Streng th 34 34 36 37 40 40 42 43 44 45 45 45 46 50 51 55 58 59 59 61 64 70 73 74

4.1.6 Postbuckling Field 4.1. 7 Boundary Conditions 4. 2 Di sc u s ion 4.3 Concl us ion 5. Summary References PART III SUMMARY

Appendix A Laminate Constitutive Equations Appendix B Ex panded Equa t ions for the Anal ysi s Appendix C Ex panded Equa t ions for the Anal ysi s

Cyl inders (A Simpl ified Approach) Appendix D Expanded Equations for the Analysis

Cyl inders (An Accurate Approach)

of Ci rc ul ar Cyl i nder of No nc i rc u 1 ar of No nc i rc ul ar 76 77 77 80 82 83 85 Al

BI

Cl Dl

List of Tables

Table Title

1. Optimal Configurations for Example Loading Cases 2. Composite Cyl indrical Shell Data for Experiments 3. Compari son of Experiments wi th Theory

4. Computed Results for (90,0,0,90) Oval Cyl inder (Simpl ified Theory) as Shown in Figure 27

5. Buckl ing Load for (90,45,-45,90) Oval Cyl inder (Simpl ified Theory) as Shown in Figure 28

6. I bi Coefficient for (90,45,-45,90) Oval Cyl inder (Simpl ified Theory) as Shown in Figure 28

7. Buckling Load for 90° Oval Cylinder (Simplified Theory) as Shown in Figure 29

8. I bi Coefficient for 90° Oval Cylinder (Simpl ified Theory) as Shown in Figure 29

9. Computed Resul ts for 0° Oval Cyl inde'r (Simpl ified Theory) as Shown in Figure 30

10. Computed Resul ts for 0° E11 i pt i cal Cylinder (Simpl ified Theory) as Shown in Figure 31

11. Computed Results for (90,0,0,90) E11iptical Cylinder (Simplified Theory) as Shown in Figure 32

12. Computed Results for (45,0,-45,90,90,45,0,-45) Oval Cylinder (Simpl i fied Theory) as Shown in Figure 33

13. Computed Results for (90,0,0,90) Oval Cylinder (Accurate Theory) as Shown in Fi gure 37

14. Computed Resul ts for 0° Oval Cyl inder (Accurate Theory) as Shown in Figure 38

15. Computed Results for (45,0,-45,90,90,45,0,-45) Oval Cylinder (Accurate Theory) as Shown in Figure 39

List of Figures

Figure Title

1. Geometry and Loading of the Ci rcul ar Cyl indrical Shell 2. Laminate Coordinate System

3. Circular Cylinder Containing an Axisymmetric Geometric Imperfection

4. Axisymmetric Prebuckl ing Deflections at Various Compressive Load Level s

5. Infl uence of Buckl ing t-tlde Shape Imperfections

6. Iteration Procedure in the Solution of Buckling Problem 7. Buckling f.'bde of 0° Unidirectional Circular Cylinder 8. Buckling Mode of (90,45,-45,90) C,rcular Cylinder

9. Buckling Load for (90,8,-8,90) Circular Cylinder

10. The Ibl

Coefficient for (90,8,-8,90) Circular Cylinder 11. Multi-modal Objective Function

12. Load Versus Axial Strain for Shell 1 Subjected to Axial Compress ion

13. Load Versus Axial Strain for Shell 7 Subjected to Axial Compress ion

14. Load Versus Axial Strain for Shell 9 Subjected to Axial Compression

15. Experimental Set-up for Buckl ing Tests

16. Postbuckling Configuration of Shell 1 (26,-42,76,-03) Under Axial Compress ion

17. Postbuckling Configuration ofShell 11 (0,90,90,0) Under Axial Compress ion

18. Postbuckling Configuration of Shell 4 (-59,05,51,-59) Under Combined Axial Compression and Torsion R_x:R_xy=3:1

19. Postbuckling Configuration of Shell 11 (0,90,90,0) Under Hyd rosta tic Pressure

20. Postbuckl ing Configuration of Shell 11 (0,90,90,0) Under Combined Torsion and External Hydrostatic Pressure R :R =6:1

xy y

21a. Critical Combinations of Axial Compressive and Torsional Loads (optimized shells versus reference shell 9 (90,0,0,90))

21b. Critical Combinations of Axial Compressive and Torsional Loads {optimized shells versus reference shell 10 (0,45,-45,90))

21c. Critical Combinations of Axial Compressive and Torsional Loads (optimized shells versus reference shell 11 (0,90,90,0))

22a. Critical Combinations of Torsional and External Pressure Loads (optimized shells versus reference shel1 9 (90,0,0,90))

22b. Critical Combinations of Torsional and External Pressure Loads (optimized shells versus reference shell 10 (0,45,-45,90))

22c. Critical Combinations of Torsional and External Pressure Loads (optimized shells versus reference shel1 11 (0,90,90,0))

23a. Cr itic al Combinations of Axial Compressive and External Pressure Loads (optimized shells versus reference shel1 9 (90,0,0,90)) 23b. Critical Combinations of Axial Compressive and External Pressure

Loads (optimized shells versus reference shel1 10 (0,45,-45,90)) 23c. Critical Combinations of Axial Compressive and External Pressure

Loads (optimized shells versus reference shel1 11 (0,90,90,0))

*

24. Distributions of Flexural Rigidity 0 ($)

25. Geometry of Ell iptical and Oval Cross Sections

27. Buckling Load and 'bi Coefficient for (90,0,0,90) Oval Cylinder 28. Buckl ing Load and 'b' Coefficient for (90,45,-45,90) Oval

Cyl inder

29. Buckl ing Load and 'b' Coefficient for 900

Oval Cyl inder 30. Buc kl i ng Load and 'b' Coeffic i ent for 00

Oval Cyl i nder 31. Buckling Load and 'bi Coefficient for 00

Elliptical Cylinder 32. Buckling Load and 'bi Coefficient for (90,0,0,90) Elliptical

Cyl i nder

33. Buckling Load and 'bi Coefficient for (45,0,-45,90,90,45,0,-45) Oval Cyl i nder

34. Buckl ing Load of 00 Laminated E11 iptical Cyl inders Versus Isotropic Ell iptical Cyl inders

35. Buckl ing Load of (90,0,0,90) Laminated E11 iptical Cyl inders Versus Isotropic Elliptical Cylinders

36. Buckl ing Load of (45,0,-45,90,90,45,0,-45) Laminated E11 iptical Cyl i nders Versus Isotropic Ell i ptical Cyl i nders

37. Comparison of Resul ts From Accurate Approach and Simpl ified Approach for (90,0,0,90) Oval Cylinder

38. Compar;son of Resul ts From Accurate Approach and Simpl ified Approach for 00

Oval Cyl i nder

39. Comparison of Resul ts From Accurate Approach and Simpl ified Approach for (45,0,-45,90,90,45,0,-45) Oval Cylinder

A .. lJ B .. lJ Dij A .. lJ B .. lJ D .. lJ [ A*] [B*] [0 *] -* A .. lJ -* B .. lJ -* Dij A, B b 0 Ot e Nomenclature 1 --2- ~ (Qij) k(h~-h~_l)

+

~ (Qij)k{h~-h~_l)

[Ar1 -[Ar1[B] [D]-[B][Ar1[B] 1

*

E ₁₁ t A _lJ ..

*

tB .. lJ

major and minor semi-axes of noncircular cross-section Koiter's postbuckl ing parameter

Et3/12{1-v2 )

discriminant defined in Eq. 2.25, Eq. 4.21

eccentricity parameter of oval cylinder

EU' E

22, G12 elastic constants of lamina

F Ai ry stress function

prebuc~ing state of

F

F

₁

F

_II

F

L Le m Mx' My' Mx' My' N_{x '} N_{y '} N_x' Ny' P P qo t~ xy Mxy N_xy N_xy

bue kl ing sta te of F seeond order sta te of F

Ellt 3F

eyl inder 1 ength

ei reumferenee 1 ength of the eyl i nder

number of hal f waves in the axial direetion of a no ne i reul ar eyl i nder

moment resul tants

non-dimensional fonn of r~x' _{My' Mxy defined in Eq.} 2.1 and Eq. 3.14

stress resul ta n ts

non-dimensionalform of N_x' N_{y '} N_{xy defined in Eq.} 2.1 and Eq. 3.14

axial eompressive load external pressure

a quarter of eireumferential length of noncireular eyl i nder

G₁₂

Q11 C'+ +2 (Q 12 +2Q 66) C 2S 2+Q22 S4 Qll S'+ +2 (Q 12 +2Q 66 ) S2C 2 +Q22 C4 (Qu +Q22 -4Q66 ) S2C2+Q 12 (S4+C 4 )

Q66 Q16 Q26 S, C R R(y) Ra Rx' Ry' Rxy T t x, y, z x, y, z U,

V, W

U,

V, W

Q' \~o

W

1

WIl

Z ö nns E: 0 E:_{x '} E: 0 y' Yxy 0 À Àc IJ. v_{12 '} v₂₁ (Qll +Q22- 2Q 12- 2Q 66 )S2C2+Q66 (S4+C4) (QII-Q12-2Q66)SC3+(Q12-Q22+2Q66)S3C (QII-Q12-2Q66)S3C+(Q12-Q22+2Q66)SC3 sin9 k, COS9k respectively radius of circul ar cyl inder

radius of curvature of noncircular cylinder

an equivalent radius of the circle with the same circumfe-rence as the perimeter of the noncircular cross section axial, circumferential and torsional stress ratios torque

total thickness of shell

axial, circumferential and nonnal coordinates

non-d imens ional fonn of x, y, z defi ned in Eqs. 2.1, 3.14

axial, circumferential and nonnal displacements

non-dimensional fonn of U, V, W defined in Eqs. 2.1, 3.14

Axisymmetric imperfection defined in Eq. 2.2

prebuckl ing state of W

buckl ing state of W

second order state of

W

L2/(Rt)

root mean square ofaxisymmetric imperfection ampl itude buc kl i ng mode ampl itude

mid- surface stra ins

non-d imens ional load parameter critical value of load parameter

axisymmetric imperfection ampl itude

w axisymmetric wave length parameter defined in Eq.(2.2)

1. INTRODUCTION

1.1 Postbuckling Theory

The stability of thin elastic shells remains one of the most cha 11 eng ing probl ems of the theory of el asti c ity. Extensi ve research has been conducted in order to reduce the disparity between theoretical and experimental results, especia11y for the case of cylindrical shells subjected to compression. Because of the importance of various shell structures as the load carrying members in aircraft, spacecraft, ships, submarines, as well as civil and mechanical structures, this topic still receives a great deal of attention. In recent years, high strength fiber reinforced composites have been widely appl ied in structures in which low weight i s of the vital concern. Thus, theoretical and experimental

investigations of the stability of elastic thin shells have been extended to consider anisotropic material effects.

Classical linear buckling theory of thin cylinders, based upon the assumptions of perfect shell geometry and smal 1 deflections, predicts much higher buckl ing loads than those reveal ed by various tests. For instance, the experimental buckl ing strengths of thin cyl inders under ax ial compression gave only twenty to sixty per cent of the theoretical val ues [1]. In 1934 Donnell [2] introduced into the probl em both the consideration of nonlinear terms due to large deflections and the concept of initial shape deviations. This work had a great influence on von ~rmiin and Tsien [3] in 1941 when they developed a large deflection theory for shells under compression. Further refinement of the theory by considering the effects of imperfections were made by Donnell and Wan [1], and Loo [4].

In 1945, Koiter [5] developed a general theory of elastic stabil ity. His theory was not available in English until the 1960's when the basic ideas of thi s theory sta rted to become wi del y known. By anal yzi ng up to fourth order variations of the incremental potential energy fr om the fundamental equilibrium state to an adjacent equilibrium state, Koiter's general theory not only specifies the stabil ity behaviour of the equilibrium in the vicinity of the critical bifurcation point, but also prov ides a quantita ti ve asymptoti c desc ri pt i on of sensiti v ity of the structure to buckling mode type imperfections. In this pioneering work, Ko iter demonstrated tha t for an ax i symmetr ic imperfec ti on wi th an ampl itude of only one tenth of the wall thickness the buckl ing strength could be reduced to as 1 ittl e as 60% of the corresponding perfect shell. An equival ent variation of Koiter' sapproach, written in a version of the principle of virtual work, was developed by Budiansky and Hutchinson [6-10]. Koiter al so developed a spec ial theory [11] in which the effect of small but finite imperfections in the shape of the classical axisymmetric buckl ing mode of a perfect cyl indrical shell was accounted for. The results of this less restricted analysis fully confirmed the predictions of his general theory.

It is now generally accepted that geometric imperfections are the dominant factor in reducing the buckling load of thin elastic shells. The literature in this field is rich as revealed in survey papers [12-14]. Further develoJlllent of the theory was made to incl ude the influence of edge restraints and the effects of nonl inear prebuckl ing deformations [15-18]. Almroth [19] extended the study of the effects of axisymmetric imperfections of Koiter's special theory [11] by taking into account the exact boundary restraint and nonl inear prebuckl ing deformations and found that the resul ts were in reasonably good agreement

with Koiter' s approximate sol ution. Tennyson and Muggeridge [20] made

exact model calculations and tested accurately manufactured shells

containing axisymmetric imperfections \'1ith various values of amplitude

and wavel ength, demonstrating the serious degrading effect of the

axisymmetrical imperfections to the buckl ing strength. Their work also

indicated that a critical axisymmetric wavelength existed which yielded a minimum buckling load, consistent with the prediction of Koiter's special

theory. Budiansky and Hutchinson' s version of Koiter' s initial

postbuckl ing theory was general ized by Fitch [21-22], Cohen [23], and Hutchi nson and Frauenthal [24] to incl ude the infl uences of nonl inear prebuckl ing behaviour.

I n t her e a 1 wo rl d, s h e 11 st r u c t ure s a r e l i k e 1 y t 0 con t a i n

imperfections of a random nature. Arnazigo and Budiansky [25-26] derived asymptotic expression for the buckl ing of long cyl inders wi th homogeneous

random axisymmetric imperfections under axial compression. Tennyson,

t1uggeridge, and Caswell [27-28] conducted a combined analytical and

experimental study of the effects of various shape imperfections on the

buckling of circular as well as noncircular cylinders. They proposed

design criteria for predicting buckl ing loads of cyl inders in terms of the measured root-mean-square value ofaxial profiles of the actual wa"

mid-surface deviations. For imperfections more general than axisymmetric

in shape, Hutchinson [29] investigated the buckling of a long pressurized shell under axial compression by considering two types of imperfections , one in the shape of classical axis}111metric mode and the other in the shape of a classical asymmetric mode, indicating that axis}111metric imperfections of a certain wave length are particularly detrimental. Arbocz and Babcock [30] generalized Hutchinson's analysis to include more modal imperfection terms and concluded that certain modal components of

the imperfection were more degrading than others. As an extension of

Koiter's pioneering work [5] in which only the influence ofaxisymmetric imperfection was analysed, the effect of completely general imperfections within the context of an asymptotic approach was analysed by Hansen [31], who demonstrated that both axisymmetric and non-axisymmetric imperfections could pl ay important rol es. Later the same author [32] published an analogous probabilistic development to the problem to provide a sound theoretical basi s for the des i gn of randomly imperfect cyl ind rical shell s. Yamaki [33] compared the resul ts obta ined by applying the perturbation procedure to Donnell' s nonl inear equations with those of the fully nonl inear analysis to examine the range of val idity of the Koiter type asymptotic approach on the initial postbuckl ing behavior and the effects of imperfections in the shape of a non-axisymmetric buc kl i ng mode.

1.2 Optimum Design of Laminated Circular Cylindrical Shells

For the 1 ast two decades or so, the introduction of high perfonnance fiber reinforced composites to 1 ight weight structures has resulted in a considerable amount of research. One of the advantages of these materials lies in the flexibility for the designer to tailor the property of the laminate to the appl ication through fiber orientation and stacking sequence. Buckl ing is one of the criteria which is of ten crucial in the design of composite structures. There is no doubt that to optimize the buckl ing strength of laminated cyl indrical shells presents additional difficul ties. The effect of fiber orientation on the buckl ing streng th

has been recognized by many authors. Earl ier, Khot [34-35] investigated the effect of fiber orientation on buckling, postbuckling and imperfection sensitivity of composite cyl inders. Tennyson et al. [36]

\0

extended Koiter' s special theory [11] to 1aminated circu1 ar cy1 indrica1 shells containing axisymmetric imperfections. They found that drastic buckling 10ad reductions occurred, comparab1e with isotropic shells, for very sma11 va1ues of the imperfection arnp1itude and fiber orientation p1ayed a vita1 ro1e on the buck1ing 10ad. The first part of the present study was prompted by Tennyson and Hansen' s recent paper [37] in which they demonstrated that the buck1 ing strength of composite cy1 inders cou1d be increased significant1y through a judicious choice of 1aminate configuration. The optimum design of 1aminated p1ates under axia1 compression and torsion has been studied by Hi rano [38-39]. However, 1itt1e information is avai1ab1e on the optimum design of 1aminated composite shell s for buck1 ing with the exception of Nshanian and Pappst 's paper [40]. In that paper, it was assumed that the 1aminate was symmetric and ba1anced; this work presented the optima1 design of simp1y supported composite cy1 inders for free vibration and buck1 ing. Thus the optimal design of 1aminated composite shells for buck1ing strength is still 1arge1y unexp10red.

In the first part of the present study, the optimization of the buck1 ing 10ad of a 1aminated-composite circu1ar cy1 indrica1 shell subjected to axia1 compression, externa1 pressure, torsion or a combination thereof is undertaken using Powell's method [41] with the initia1 point obtained by a random search technique. The objective function was composed of an accurate numerical sol ution of the eigenprob1em of the von Kitrmiln-Donnell thin shell equations with c1amped end conditions inc1 uding the non1 inear prebuck1 ing deformations. During the optimi za ti on procedure, the cal cu1 ati on of buc k1 ing 10ads were based on the perfect cy1 inder since the imperfection amp1 itudes were unknown before she11 s were manufactured. For the purpose of comparing wi th

experimental data, the cal cul ation of the buckl ing loads for the

resulting optimal configurations, as well as some reference

configurations, were based on cylinders containing an axisymmetric shape imperfection of which the ampl itudes were obta ined from the actual shell s

made to verify the analysis. In addition, the imperfection sensitivity

parameter Ibl _{was also evaluated to indicate the sensitivity to}

asymmetric imperfections in the shape of the buckl ing mode. The lowest

buckling load associated with the nonlinear prebuckling path was

determined by an iteration procedure in conjunction with the inverse power methode Using this procedure, the least eigenvalue can of ten be determined in several iterations without the knowledge of the possible

lower bound of the eigenvalues. Thi s procedure proved to be very

efficient especially in the evaluation of buckling loads of randomly produced configurations in which no information for an initial estimate

of the buckling load was available. Comparing with the determinant

plotting method, which is also enployed in the liter'ature to solve the buckl ing problem along a nonl inear prebuckl ing path [42J, this approach can reduce the computer time substantially. The CPU time expended on a Perkin-Elmer mini computer is about one hal f of an hour for each run to

search for the optimal configuration. Computations are carried out on

four-ply graphite/epoxy cylinders for eight loading cases: (1) pure

axial compression, (2) pure external pressure, (3) pure torsion, (4) axial compression and torsion with load ratio R_{x:Rxy=3:1, (5,6) axial} compression and external pressure with load ratios R

x:Ry=3:1 and R

x:Ry=8:1, (7,8) torsion and external pressure with load ratios

R :R =1:1 and R :R =6:1. It is found that in each loading case, with

xy y xy y

configurations, the optimized configuration is practically reproducible. The results of the analysis show that the buckling load of an optimized struc ture can be more than three times as hi gh as tha t of a reference configuration. Contrary to the assumptions of ten taken in the 1 iterature dealing with the optimum design of composites, the optimized laminations are neither mid-plane symmetric nor balanced (i.e., an equal number of plies in the directions of +e and -e). A physical insight into the high buckling resistance of the optimized lamimate is then given based on the

*

plot of the distribution of the flexural rigidity 0 (4)) as a function of off-axis angle 4>. In addition,

a

series of experiments for optimal, four ply, graphite/epoxy circular cylindrical shells were performed. The correlation between theory and experiment was found to be good.

1.3 The Stability of Composite Noncircular Cylindrical Shells

Basically two topics are involved in this study, both of them util izing Koiter ' s initial postbuckl ing theory. In the first part the emphasis is on the optimization of the laminate configuration for the buckl ing of circul ar cyl inders. In the second part, the probl em of buckling and initial postbuckling behaviour of composite noncircular cylindrical shells under compression is analyzed. For practical purposes the noncircular cylinders are at least as important as their circular counterparts. The noncircularity is either deliberately introduced into the desi gn of cyl indrical shell structures for various appl ications or due to the resul t of manufacturing imperfections in supposedly circul ar cyl indrical shell s. Al though considerabl e attention has 'been given to the stabil ity probl em and the effect of shape imperfections on the buckling of circular cylinders, only a few investigations of the buckling and imperfection sensitivity of noncircular cylinders exist for isotropic

materials; none are avai1ab1e for composites. Earlier studies made by Kempner and Chen [43-44

J

ind icated tha t, for a suffi c ient1 y out-of- round cy1 inder, the far postbuck1 ing region exhibited a maximum 10ad which cou1d exceed the c1assica1 buck1ing 10ad. This suggested that ova1 cy1inders with sufficient1y eccentric cross sections might be 1ess sensitive to initia1 imperfections than circu1ar cy1inders. Hutchinson [45], on the other hand, app1 ied Koiter' s general theory to investigate the sensitivity of the ova1 cylinder to initia1 imperfections, showing that cy1 inders with ratio of major to minor axes B/A>0.2 were drastically imperfection sensitive. Subsequent1y, Kempner and Chen [46J confirmed Hutchinson' s results for the region immediate1y adjacent to the c1assica1 buck1 ing point but indicated that the snap-through behaviour for an ova1 cy1 inder of moderate to large eccentricity wou1d be substantially 1ess drastic than that for a circu1ar cylinder. They still suggested that for ova1 cy1 inders with moderate to high eccentricity the collapse 10ad, which was of the order of, or higher than ·the c1assica1 buck1ing 10ad, served as a more meaningfu1 stabil ity parameter. Tennyson et al. [47J conducted experiments on near-perfect and imperfect ova1 she11 s and demonstrated significant imperfection sensitivity of ova1 she11s. Feinstein et al. [48J investigated the effect of c1amped boundary conditions associated with a rather accurate non1 inear prebuck1 ing state on the buck1ing of finite ova1 shells under compression, finding noticeab1 e dev ia ti ons between the resu1 ts of the accurate ana1 ys is and those estab1ished for infinite1y long shells. Kempner and Chen [49J a1so studied the stabil ity of ova1 cy1 inders subjected to cornbined axia1 compression and bend ing. Recent1 y, a moda1 expansi on method was ernp10yed by Vo1pe et al. [50] to solve the buck1ing prob1em of orthogonally stiffened finite ova1 cy1inders.

The purpose of the present study is to invest i gate the buc k1 ing and initia1 postbuckl ing behaviour of noncircu1 ar composite cy1 inders under axia1 compression by app1ying Koiter's perturbation methode Two ana1ytica1 approaches were deve10ped to solve the prob1em. First1y, a simp1ified approach was deve10ped in which the inf1uence of edge restraints and prebuck1ing deformations were neg1ected. Considering the many 1aminate configurations which may be invo1ved in the ana1ysis of composite shel1s, this approach serves as a first approximation to the

prob1em with low cost. Second1y, taking into account the effect of c1amped boundary conditions in conjunction with an accurate representation of the non1inear prebuck1ing state, an accurate but much more complicated ana1ysis was deve1oped. It shou1d be noted that

Koiter ' s method can be used to advantage on1y for bifurcation buck1 ing prob1ems where the prebuck1ing state is relative1y simple. (e.g., the prebuckling state is neg1ected, 1inear, or described by ordinary, non1inear differential equations.) However, in the second approach, the prebuck1 ing state of the noncircu1 ar cy1 inder is basicall y governed by non1inear partia1 differentia1 equations, which is exact1y the c1ass of prob1em that is circumvented by using an asymptotic methode The prebuckling equations, however, need to be simp1ified in order that the bifurcation buck1ing ana1ysis can be conducted. The limitation to the second approach, within the scope of the Koiter type asymptotic method, is then discussed in this context. Comparing the resu1ts obtained from these two different analyses for ova1 cy1 inders of severa1 laminate configurations, it is found that the simplified analysis provides a quantitatively good approximation for the buckling load and a qua1 itative1y good approximation for the initial postbuckl ing behavior to the accurate analysis; the buckling and initial post-buck1ing character

of laminated noncircular cylinders, like their circular counterparts, are affected significantly by the laminate configuration.

PI\RT I

OPTIMIZI\TION OF LAMINI\TED CIRCULI\R CYLINDRICI\L SHELLS SUBJECTED TO COHBINED LOI\DS

2. OPTIMUM DESIGN OF COMPOSITE CIRCULI\R

CYLINDRICI\L SHELLS SUBJECTED TO COMBINED LOADS

2.1 Theoretical I\nalysis

2.1.1 Governing Eguations

Let the mid-surface of a perfect circular cylindrical shell be the reference surface and the origin of the coordinates be located at the middle length of the cylinder (Fig. 1). The coordinates x, y, z are measured in the axial, circumferential and normal directions respectively. The fiber orientation of the k-th lamina (staking from the inner surface of the shell) is shown in Fig. 2. The cylinder is supposed to have length L, radius Rand wa" thickness t. 1\ parameter Z is defi ned as Z=L2 /( Rt) • The nond imensi onal parameters are obta ined by the following relations, with the agreement that the same notation with or without an overbar I _ I represents dimensional or nondimensional fonus of

-* 1 * Aij = Eu t Aij -* * B .. = tB .. lJ lJ ( i, j=1, 2, 6) -* * D .. = 1 J Eut3Dij N = EU t 2 tl x R x x=Mx (2.1) N_y = ElI t 2 _R N Y

y

=

IRt y Nxy= EU t 2 R Nxy

W

= t W

MX

= EU t 3 _R Mx EU t 3 _{t 2} My = _R My V = - V IRt

M

= EU t 3 _Mxy

;(1

= .-L e:0 xy R x R x (2.1) EU t3F e:0 t F = _y = -R- e:~ Eu t 2 _{-0 _ t 0} P = P _Yxy-

R

_Yxy R2 * * *

where A .. , B .. and 0 .. are campl iance coefficients of the laminate [51],

lJ lJ lJ

N , N and Nare the axial, circumferential and torsional stress

x y xy

resul tants, M

x' My and Mxy are moment resul tants, e:~, e:~ and Y~y are mid-plane stra;ns, F ;s the Airy stress function, p is the external

pressure, E₁₁ is the longitudinal elastic modulus of the lamina, U, V, W are the displacementsalong x, y, z coordinates respectively. Following

the method of [27] and [52], a small but finite initial axisymmetric midsurface deviation is assumed to be (Fig. 3)

W{x) = - !.I. cos{wx) (2. 2)

where

According to [27], ö

rms is the measured root-mean-square value of the actual wall mid-surface profil es and !.I. is the equival ent ampl itude of the mid-surface imperfection in the shape of cl assical axisymmetric buckl ing mode. Also, w is the normalized wave length parameter which corresponds to the classical axisymmetric buckling mode. As shown theoretically and experimentally, this imperfection yields a minimum buckling load under

axial compression [11,20]. The nonlinear von ~rm~n-Donnell strain

displacement relations containin';J the axisymmetric imperfection term tI can be expressed in the form

e:Q=v +W+1.{W)2

Y 'y 2 ' y (2. 3)

*

*

*

*

*

*

e:o

X All A12 A16 Bll B12 B16 Nx

*

*

*

*

*

*

e:O _{A12 A22 A26 B2l B22 B26} N

Y Y

*

*

*

*

*

*

°

A16 A26 A66 B61 B62 B66 N_xy Yxy

=

(2.4)

*

*

*

*

*

*

M _{X -B 11 -B 21 -B 61 0 11 012 016} -W,XX

*

*

*

*

*

*

M _{-B 12 -B 22 -B 62 012 °22 °26} -W Y 'yy

*

*

*

*

*

*

M xy -B 16 -B 26 -B 66 0 16 D26 D66

-2W

'xy

The development of the constitutive equations can be found in Appendix A. For aanisotropie eomposite eireular eylindrieal shell with an initial axisymmetrie shape imperfection

W,

the equilibrium equations are

N + N

=

0

x 'x xy'y

N + N

=

0

xy'x y'y

(2.5)

The Airy stress funetion F is introdueed to satisfy the first two equations of

(2.5)

identieally

F,yy(X,y)

=

NX(x,y) F,XX(X'y)

=

Ny(X,y)

F,xy(X,y)

= -

Nxy(x,y)

(2.6)

Oifferentiating the first two equations of (2.3) and substituting them into the third one to eliminate U and V, making use of

(2.4)

to replaee the strain eomponents, a eompatibility equation in terms of F and W is

obtained. _{Also substituting Mx ' My'} t~xy fr om (2.4) into the third equation of (2.5) yields another form of the equilibrium equation. As a result, in terms of the Airy stress function F and normal displacement W, the von Kitrmttn-Donnell type compatibil ity and equil ibrium equations can be written for the composite circular cylindrica1 shell containing an axisymmetric imperfection

W

as

(2.7)

where the operators are

(2.8)

2.1.2 Prebuckling State

The prebuc kl ing di sp1 acements Uo, Vo, Wo are assumed to be a function ofaxia1 coordinate x on1y, since the external loads and shell geometry are axisymmetric [52],

Uo

=

Uo(x) Vo = Vo(x)

Wo = Wo(x)

'Then according to constitutive relation (2.4), the stress resultants and moment resultants N, M, etc. are also functions of x only. The strain

x x

displ acement relations become

E~O=

Uo'x +

t

(WO'x)2 + W'xWo'x

EO = W

yo 0

Y~yo

= VO'x

whereas the equilibrium equations (2.5) reduce to

N~y'x

=

0

o NO 0 ("'" ) 0

r~x ' xx - y + _{Nx W, xx +W} 0' xx + p

=

(2.10)

(2.11 )

Here the subsc ri pts '0' on U; V, W, _EX' 0 _{Ey' Y~y and the super sc ri pts} 0 ' 0'

on N , Ny' Nxy ' Mx' My' Mxy are used x to ind icate tha t they are pararoeters in the prebuckling state. Considering the fi rst two equations of (2.11),

the stress res ul tants can be written as

FO' yy= N~ = constant

Fo,xx= N~(X) (2.12 )

FO'xy= - NO = _{xy constant}

Then the prebuckl ing stress function can be expressed as

1

where

(2.14)

The prebuckl ing equations are identical to the governing equations (2.7) except that Wand F are replaced by Wo and Fo. Recalling that Wo is a function of x only and substituting (2.13) into (2.7) yields the compatibility and equilibrill1l equations for the prebuckling state

*

*

A22 f o ,xxxx- B21 WO'xxxx: Wo'xx

(2.15)

An expression fot' fo'xx is obtained by eliminating e:~ from (2.10) and the second equation of constitutive relation (2.4)

(2. 16)

Substituting (2.16) into (2.15) and el iminating fo 'xxxx results in a fourth order ordinary differential equation for Wo

* * *2 * *

(A22Dll+B21)WO'xxxx+(2B21-A22N~)Wo'xx+Wo :

(2.17)

WoUZ/2)

=

0

W_o'x(iZ/2) = 0

(2. 18)

where Wo is an even function of x since the right hand side of (2.17) is an even function. Therefore, the boundary conditions at the mid-length

(x=O) of the cyl inder are assumed to be

(2.19)

In order that the external loading can be characterized by a single load parameter À, the loads "are assumed to increase proportionally by the ratios Rx' Ry' and Rxy which are related to N~, p, and N~y such that

(2.20)

Then the external loads (axial compression force P, external pressure p and torqure T) are rel ated to À as

(2. 21 )

Here R I is used to represent the part of the axial stress caused by net

x

axial load 1tR2

p

should be added to P which yields the relation

(2.22 )

Rewriting the prebuckling equation (2.17) in a compact form,

(2.23) wh ere

* *

*

2 Tl

=

A22Dll + 821

*

*

T2

=

_{28 21} + _A22 RxÀ _(2.24) T₃

=

1

The solution to (2.23) can be written in different foms, depending on the sign of the di scriminant,

(2.25) or,

(2.26)

*

*

*

Note tha t the val ue of Dt depends on 1 _{aminate parameters A22 , Du, 821} and the axial stress resultant N~.

(1) When Dt<O, Wo takes the fom

where and I a 0 = } ( 21l

31

Tl -

~

)

2

1 T 1 b 0 =

2" (

2/T 31 Tl +

r )

'2

1 (2.28 ) (2.29)

The coefficients Cl and C₂ are detennined from the boundary conditions,

Cl = - Ao{[Ca+Cbcos(wO)][aosin(bl)ch(al)+bocoS(bl)sh(al)] +Cbw sh(al)sin(bl)sin(wo)} C₂

=

AO{[Ca+Cbcos(wO)][aocos(bl)sh(al)-bosin(bl)ch(al)] where al = ao/ÏJ2 bI

= b

_o

/Z/2 Wo

=

w /Z/2 Ao = [aosin(2bl) + 2bosh(al)ch(al)]-1

(2) When Dt>O, Wo takes the fonn

(2.30)

(2.31 )

where T 2-/Dt I ao =

[

]2 2T I (2.33) T 2+/Dt I bo =

[

_2T I

]2

Note that usually the quantities in the brackets for ao and bo of (2.33) are positive. Using the following notation

w_o=W/Z/2;

(2.34 )

the coefficients C₃ and C,+ can be determined from the boundary conditions as

The expressions for Ca and C

b are the same as found in (2.29). (3) When Dt=O, Wo ta kes the fonn

where Denoting al = ao/Z/2; Wo = W/Z/2 Ao = [al+sin(al)cos(al)]-l (2.35) (2.36 ) (2.37) (2.38 )

the coefficients Cs and ,C₆ are

(2.39 )

It should be mentioned that based on the analysis of [23], 0t=O leads to an expression for N~

* * *

2 !

*

2(A₂₂0₁₁+B_{21 )2 -} 2B₂₁

*

A₂₂ (2.40 )

which is the critical load corresponding to the classical axisymmetric buckl ing mode for an anisotropic cyl inder under axial compression. In Fig. 4, solutions of the prebuckling equation (2.17) are depicted for a

(90,45,-45,90) laminated circular cylindrical shel1. (Note, the sequence

of numbers separated by commas are the orientations of layers in degrees, starting from the inner surface of the cyl inder. The notation for degree

101 _{has been omitted. The çyl inder confi guration will be denoted in thi s}

way throughout the rest of thi s work.) The fi rst three curves are

prebuckling displacements at equal load intervals. They are nearl y

uniform and small in ampl itude. When the value of the discriminant Ot a pproaches zero and subsequentl y turns positi ve, the ampl itude of Wo i nc reases rapi d ly, as shown by the 1 ast curve, and the sha pe of Wo becomes trigonometrie in nature.

• •

2.1.3 Wo and fa problem

load parameter À are defi ned as

(2.41 )

These quantities are required at various prebuckl ing load level s in the solution of the buckling problem and the evaluation of postbuckling coefficient Ibl

• The equations for

Wo

and

fo

can be derived directly by

differentiating (2.17) and (2.16) with respect to À. Considering (2.20), the fourth-order ord inary d ifferenti al equa ti on for

Wo

is

(2.42 )

and the expression for

fo

is

(2.43 )

The corresponding boundary conditions at the clamped end are

(2.44 )

The boundary conditions at the mid-length of the cylinder are

Equation (2.42) is solved numerically using the finite difference method.

2.1.4 Buckling Eguations

The classical buckling load of the cylinder with an initial

axisymmetric imperfection Q' is denoted by Àc. Following the method

outlined in [24], an asymptotic perturbation expansion of the solution, val id in the neighbourhood of the bifurcation point, is taken in the

form

+ •••••• (2.46)

where Wo' Fo is the prebuckling field, W!, Fr is the buckling field and

Wrr' F_II is the second order field, 'E' is the normalized amplitude of

the buckl ing mode W

r and serves as the perturbation parameter. The load À

in the vicinity of the buckling load Àc is expressed in the form

{2.47}

where 'a' and 'bi are postbuckling coefficients. In every case

considered in this study the first coefficient 'a' is identically zero

due to the sinusoidal nature assumed for the buckl ing mode. Thus the

initial postbuckl ing behaviour of the structure hinges on the sign and

ampl itude of 'b'. A negative 'b' means that the load carrying abil ity

sensitive, while a positive Ibl

means that some ability is retained in the shell to stand increased load af ter buckl ing. The dependence of load À on the ampl i tude of the buckl ing mode shape imperfection and the I bi coefficient is depicted in Fig. 5.

Substituting (2.46) into the governing equations (2.7) and setting the coefficients of E: to zero yields the following linear buckling equa t ions

_______

=~~:~~~~:~~~_~l~tt~~l~~~

________ _

(W~

'xx +Q"xx) F I 'yy +Rx ÀW I 'xx +2R xyÀW I 'xy

(2.48 )

+Fo,~xWI'yy-FI'XX

where the supersc ri pt I c I on Wo and Fa i nd icates tha t they are eval ua ted at the critical load Àc. The solution of these equations is assumed in the separable form

(2.49)

where the nond imensi onal wave number Nis rel ated to the integer circumferential wave number n as

N = It/R n (2.50)

Substituting Eq. 2.49 into Eq. 2.48 and equating the coefficients of cos(Ny) and sin(Ny) to zero respectively lead to four coupled fourth

order ordinary differential equations for wl(x), w₂(x), f1(x), and f₂(x). They are 1 isted in Appendix B. These equations in conjunction with the appropriate boundary conditions are then solved using a central finite difference scheme. Providing that there is regularity of geometry and of the loading system, the finite difference method is more strai9ht forward

to implement than the finite element method; thus it has been used here. Most of the numerical sol utions for the shell buckl ing probl ems are carried out us i ng thi s method. The resul ti ng set of simul taneous 1 inear equations are then solved using a highly efficient FORTRAN subroutine package called 'LINPACK ' [53]. The subroutine SGBFA is used to perfonn lower and upper decomposition of the banded matrix and the subroutine

SGBSL is called to perfonn the backward substitution. Considering the

large number of runs involved in the optimization routine, single

precision versions of the subroutines are employed to reduce the

computing expense. Details ab out the utilization of the LINPACK equation solver can be found in [54].

2.1.5 Iteration Procedure to Solve the Buckling Eguations

The inverse power method is one of the most efficient to be used to fi nd the 1 east eigenval ue of the eigenprobl em

Ax

=

~Bx (2. 51 )

where the matrices A and 8 are real but, in general ,

non-symmetrie. As the convergence rate of the inverse power method is

proportional to the ratio of the smallest to the second sm all est

eigenvalues, the number of iterations required for convergence may be consi derabl y reduced i f all the eigenval ues are shi fted by the same

amount. However, the direct use of the shi fted inverse power method is impeded since the present eigenprob1em with a non1inear prebuck1ing path does not have the simp1 e form of (2.51). Some of the el ements of the B matrix are functions of the prebuck1 ing deformation Wo which depends on the 10ad parameter À non1 inear1y. One of the techniques of ten used to solve the eigenprob1em (2.48) associated with a non1 inear prebuck1ing path is the determinant p10tting method in which the determinant of the coefficient matrix of the buck1ing equations is eva1uated at incrementa1 10ad steps until the va1ue of the determinant first changes sign [42]. The prerequisite for using this method is a good initia1 guess of the 10wer bound of all eigenva1ues, otherwise the computing expense cou1d be extreme1y high if starting from a 10ad level much 10wer than the rea1 buck1ing 10ad. For the purpose of determining the optima1 1 amination, many cal cu1 ations of the buck1 ing 10ad are required for random1y produced configurations with no forehand know1edge of the buck1ing 10ad. Thus a Newton-type iteration procedure associated wi th the inverse power raethod, which converges in severa1 iterations without this initia1 estimate for the 10west eigenva1ue, is very effective. As illustrated in Fig. 6, the a1gorithm for this iteration

procedure is:

(1) Take an initia1 guess of eigenva1ue À • If no know1edge of the 10wer 9

bound of the eigenva1ues is avai1ab1e, set Àg=O.

(2) Assume the prebuck1ing displacement Wo in the form

(2. 52 )

respectively.

(3) Substituting Eq.(2.52) into the buckl ing equation (2.48), the eigenprobl em can be expressed in the matrix form

where

A

x =

B

x + À

B

x 9 A is independent of À, while • (2.53 )

and

B

are matrices dependent on terms WOg and WOg respectively. The inverse power method is then applied to the linearized eigensystem (2.53) to obtain Àl' as shown in Fig. 6.

(4) A new initial guess À is obtained as gnew

À = À + rÀ₁

gnew 9 (2.54 )

where r is a reduction ratio used to keep the new guess Àgnew lower than Àc' The process is repeated from step (1) until the resul ti ng À~s of two successive iterations are close enough to be taken as the critical load Àc' The eigenval ues corresponding to a number of circumferential wave numbers are evaluated from which the least is taken as the critical load. The buckling modes for 0° and (90,45,-45,90) cylinders, for

instance, are shown in Figs. 7-8.

2.1. 6 The Second Order Field and the Imperfection Sensitivity Goeffi ci ent I bi

Substituting (2.46) into the governing equations (2.7) and equating the coefficients of e:2 to zero yields the equations for the second order field

-(~'XX+W~'XX)

__________________ tt ____________ t __________ tt ___ _

WIl'

+WIl'xx+\~I'x

2-W_{I 'xx}WI'

-(W~'XX~,XX)FII'Yy+Rx~eWII'Xx+2RXy~eWII'Xy-FII'XX

+F 0

,~X

WIl' yy +F I ' yyW I 'xx -2F I ' xyW I 'xy +F I 'xx W I 'yy

(2.55)

The sol ution to the postbuekl ing equations is assumed to be

WIl = w3(x) + w₄(x)eos(2Ny) + w_S(x)sin(2Ny) F

II = f3(x) + f4(x)eos(2Ny) + f s (x)sin{2Ny)

(2.56)

Substituting (2.56) into (2.55) and eolleeting terms involving eos(2Ny), sin(2Ny) and non-trigonometrie terms respeetively, yield two eoupled ordinary differential equations for w3(x) and f 3{x), and four eoupled fourth order ordinary differential equations for w₄(x), ws(x), f₄(x), fs(x). They are listed in Appendix B. These equations in eonjunetion with the eorresponding boundary eonditions are again solved using a

finite di fferenee seheme.

With the solution of the above sets of equations, it is possible to eval uate the post-buekl ing eoefficient I bi [24]

b

=

(2.57)

where

(2. 58)

and the foll owi ng shorthand nota tion has been used,

(2.59)

Expanding the coefficients of (2.58), the following expressions are obta i ned,

b 1 =

f f

{F I I ' x x W I

,~+F

I I ' Y Y W I

,~-

2 F I I 'x y W I 'x W I ' Y +2 F I ' x x W I ' y W I I ' Y

-FI'XyWO,~WI'yJdxdy}2

b4

=

Àl{

f

J[

~~

'xx W I ,§-RxW I

,~+2RxyW

I 'x W I ,y+2F I ,yywg 'x WI,x

• c ] 2

-2FI'xyWO'xWI'y dxdy}

(2.60 )

where W

I and FI are the critical buckling mode with the amplitude of WI

• •

normalized to unity, the superscript 'Cl on Wo' Fo, Wo and Fo indicates

that they are evaluated at the critical point Àc. The integration is over the entire midsurface of the cylinder. As shown in Fig. 5, Às denotes the maximum value of À of an imperfect structure in the case of b<O, while Àc is the bifurcation point of a reference 'perfect'

structure. If an initial buckl ing mode shape imperfection is expressed as

(2.61 )

where ö is the normal ized imperfection ampl itude, then a measure of the imperfection sensitivity is given by the asymptotic formula [8J

(2.62 )

The relations between \/À

c and ö for various values of bare plotted in the lower figure of Fig. 5. It should be mentioned that unlike the literature dealing with initial postbuckling problem using Koiter's perturbation method where the fundamental path is based on the prebuckl ing deformation of a perfect shell (i.e., ~=O in the governing equation (2.7)), here the fundamental path is the deformation of the cyl inder wi th an initial midsurface axisj1llmetric deviation

W.

Thi s means that the buckl ing load Àc thus obtained incorporates the effect of an axisj1llmetric imperfection. The evaluation of the postbuckling coefficient

tb' is also based on the buckling mode of the cylinder containing an initial axisymmetric imperfection ~. In order to demonstrate the effect ofaxisymmetric imperfection ampl itude on the buckl ing load as well as the 'b' coeffi c ient, these quanti ties are plotted for cyl inders of (90,9,-9,90) laminate constructions in Figs. 9-10 for the range of -90<9<90. It is found that the maximum buckling strength is reached at 9=60° for a perfect cyl inder but the peak dies out when axisymmetric imperfections with ampl itude of only a few tenth of wall thickness are included. The value of the tb' coefficient also varies accordingly.

2.1.7 Boundary Conditions

In order to provide a valid comparison of predictions with

experiments on cylinders of finite length, the fully clamped boundary condition at the end x=/Z/2 is imposed by the following equations

W = 0 W,

=

0 x

*

*

*

*

*

A12F'yy+A22F'xx-A26F'Xy-B21W'xx+2B26W,XY= 0 (2.63 )

*

*

*

*

*

A 16 F , y y y +2 A 2 6 F , x x y - A 2 2 F , x x x - ( A66 +A 1 2 ) F , x y y

*

*

*

*

*

+(2B26-B61)W'XXy+(B22-2B66)W'Xyy+B21W,xxx= 0

The third equation is equivalent to e:~=O (or V'y=O) which means that the circumferential strain is zero at the ends. It implies that an unifonn circumferential displacement V=constant (a free rotation) is allowed. It should be mentioned that this boundary condition (V'y=O) is different

,

from the more rigorous condition of V=O for non-orthotropic cyl inders. In the case of pure axial compression, the fully fixed end constraint V=O will cause additional torsional stress for cyl inders having non-zero

*

*

stretching-torsion coupling coefficients A16 , A26 • The fourth equation is equivalent to unifonn axial displacement condition U=constant, or, written in tenns of strain components

(2.64 )

The boundary conditions for each of the sequence of boundary val ue problems are obtained by substituting the perturbation expression (2.46)

into the above equations, as listed in Appendix B. The boundary conditions at the mid-l ength of the cyl inder are obta ined from symmetry and continuity requirements of Wand F. These equations are also listed

in Append ix B.

2.2 Optimization Procedure

Given the lamina elastic constants E_{11 ,} E_{22 , G12 , v12' shell} geometry (wa11 thickness t, radius to thickness ratio Rit, she11 length L, number of layers) and load ratios Rx' Ry' Rxy' the problem that is posed is to find the orientation of each layer of the laminate which will yield the highest buckl ing strength. Cyl inders composed of four-layers of Hercul es AS4/3501-6 graphi tel epoxy with material property Eu =21.1x106

psi., E₂₂=1.56x106 _{psi., G12 =O.84xl0}6 _{psi., v12=O.29 were selected for}

investigation. The buckl ing load of the perfect cyl inder with the

nominal geometry R/t=165, Z=L 2/(Rt)=500 was taken as the objective

function while the orientation of the constituent laminae were taken as

the design parameters. One of the best unconstrained optimization

techniques - Powell's method (without using the derivatives) [41J was

employed. The computer code for calculating buckling loads, which is

based on the analysis of the previous section, was merged with a code developed by Powell [55J to conduct a systematic search. Powell ' s method requires the objective function to be unimodal. However, in most cases the objective function for the buckl ing load is not unimodal. Thus, the initial point for the optimization of Po we 11 I s method was obtained using a random search technique. As depicted in Fig. 11, let f be the the objective function and x be the vector of design variabl es. A set of

uniform random, laminate configurations is produced to Isprayl the entire space Ll defined by the lower and upper bounds of orientation of each ply

ranging from -90° to 90°. The buckling loads for this set of uniformly distributed random laminate configurations were calculated among which the one with highest buckling load was then chosen as the initial point

for the systematic search using Powell's methode The number of random

configurations required was determined by trial since the behaviour of the objective function was unknown. The optimal laminate configurations

were searched for eight example loading cases. They were: (1) pure

axial compression, (2) pure external pressure, (3) pure torsion, (4) axial compression and torsion with a load ratio R

x:Rxy=3:1, (5,6) axial

compression and external pressure with load ratios R_x:R_{y =3:1 and}

R

x:Ry =8:1, (7,8) torsion and external pressure with load ratios

R :R =1:1 and R :R =6:1. Several optimization runs were conducted for

xy y xy y

each loading case. The initial points obtained from the random search

are listed in the third column of Table 1 while the optimal configurations obtained from the systematic search .are listed in the fourth column of Table 1. It is found that the initial points have quite a large scatter, but the optimal configurations, which · have almost the same buckl ing load À-c' are practically reproducible for each of these loading cases.

2.3 Experiments

2.3.1 Anisotropic Cylinders

In total 11 four-layer cyl inders were manufactured of Hercul es

AS/3501-6 graphite/epoxy. Shell s No .1-8 were made accord ing to the

optimized configurations of the eight loading cases of Table 1. Shells

No.9-11 have laminate configurations (90,0,0,90); (0,45, -45,90) ;

(0,90,90,0) to serve as reference tests. The geometry of shells were 6 inches long, 6.56 inches in diameter and approximately 0.02 inches

thick. Detailed data for these shells is listed in Table 2. They were manufactured from preimpregnated tape in the structures 1 aboratory at

UTIAS. Graphite/epoxy preimpregnated tape wound cyl inders were enclosed

in a vacuum bag. They were cured in an autoclave by raising the

temperature to 2400F under 85 psi. pressure and holding for one hour.

Then the vacuum bag was vented. The temperature was raised to 350°F

while the pressure was raised to 100 psi. and held for two hours before

cool ing down to roOm temperature. The cyl inders were then bonded to

aluminum end plates using epoxy to provide the clamped boundary

condition. The bonding epoxy were prepared by mixing 100 parts by weight

of Hysol 6175 epoxy resin with 29 parts of Hysol 3561 hardener.

Considering the relatively low Youngls modulus E=4xl05 _{psi. of this}

bonding material compared with the high modulus of the cured

graphite/epoxy cyl inder material , the real boundary constraints may be weaker than the assumed fully clamped end condition. But this effect was

not accounted for in the analysis. Cured cyl inders were pl aced over

internal mandrels to prevent excessive postbuckling deformations and subsequent fracture. As such, some cyl inders could be tested repeatedly without obvious damage. However, when a high ratio of torsional load was applied it was of ten found that the cylinders failed. This is probably due to the crudeness of the torsional loading facilities.

A profil e measuring apparatus was used to obta in an average wall

thickness as well as axial imperfection profil es. These data were

obtained for several axial generators at various circumferential

positions. The root mean square value of the initial radial imperfection was cal cul ated for each of the generators. The largest root mean square

value ö was then used to calculate an equivalent axisymmetric

imperfection ampl itude !J.=/f örms/t [52]. When the experiments were conducted, four foil-type strain gauges were bonded in the axial direction at various circumferential locations at the cyl inder mid-length to check for uniformity of appl ied axial compressive load. Shown in Figs. 12-14 are the load versus axial strain curves obtained from axial compress ion tests for shell s I , 7 and 9.

2.3.2 Combined Loading Buckling Tests

The equipnent for combined loading buckling tests is shown in Fig. 15 and some of the post-buckling configurations for shells 1,4 and

11 are shown in Figs. 16-20. Two calibrated Scheffer hydraulic pistons controlled by means of a hand oil pump were used to apply torsional loading. Each of the pi stons was attached to opposi te sides of a circular plate which in turn was bolted to the top end plate of the shell, so that a torsional coupl e could be appl ied about the cyl inder ax is (Fi g. 18). In combined load ing tests, external pressure was appl ied by partially evacuating the inside of the shell which was sealed at two ends by the end pl ates. An empty pressure gas cyl inder was fi rst evacua ted to about 25 mm Hg to serve as a vacuum reservoir. Then the shell was evacuated through the reservoir to the des i red val ue of external pressure. The highest external pressure supplied by this technique was about 25 mm Hg. Therefore, shells No.5 (optimized for loading ratio R :R =3:1) and No.7 (optimized for loading ratio

x y

R _xy :R =1:1) could not be tested at the corresponding loading ratio _{y .} because the external pressure required exceeded the range of the testing device. The pure hydrostatic pressure test, which was obviously beyond the capacity of the evacuating technique, was carried out in a sealed