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Buckling of laminated composite cylinders: a review
Buckling of laminated composite cylinders: a review R. C. TENNYSON
A brief review of the available static buckling theory for both geometrically 'perfect' and 'imperfect' anisotropic composite circular cylinders is presented for various loading configurations. For comparison purposes, relevant experimental data are discussed, including recent combined loading test results and recommendations are made concerning the design of composite cylinders.
INTRODUCTION
The circular cylindrical shell is used extensively as a primary load carrying structure in many applications and as such is subjected to various loading configurations. From a design viewpoint, local or general instability due to the action or interaction of pressure, axial, torsional and thermal loads often represents the limiting load condition. Although a great deal of design information supported by substantial theoretical and experimental data is available for homogeneous isotropic cylinders in both the reinforced and monocoque states, this is not the case for anisotropic cylinders. Since composite materials have gained widespread usage because of the many unique advantages they offer, it is thus necessary to develop an adequate background for analysing heterogeneous*, anisotropic* structures that can result from the assembly of these material systems into various structural configurations. However, for the case of cylindrical shells, the calculation of stability criteria is a formidable task, especially if one must consider anisotropy, radial nonhomogeneity, various loading conditions, end constraints and the effects of geometric shape imperfections. Additional complexities can also arise due to differences in elastic moduli in tension and compression and nonlinear stress/strain behaviour, as will be discussed later. For the purposes of this review, only static buckling will be considered. Initially, classical linear buckling theory for geometrically perfect circular composite cylinders is examined, followed by comparisons with published experimental data. Subsequently, the effect of shape imperfec* On a macroscopic basis, an individual lamina may often be approximated as homogeneous and orthotropic (with respect to the axis of the reinforcement phase, such as fibres for example). However, for multi-ply structures composed of laminae having various orientations with respect to some arbitrary structural axis, the macroscopic structural behaviour can be considered anisotropic since the stiffness matrix is fully populated with non-zero terms and heterogeneity can occur due to the 'thickness' variation of the corresponding elastic coefficients. University of Toronto Institute for Aerospace Canada
COMPOSITES.
JANUARY
1975
Studies,Toronto,
tions on the buckling of composite cylinders is described together with appropriate experimental results. In a literature survey such as this, it is difficult to account for all references which relate to the subject matter. However, it is hoped that the absence of any pertinent reports which have been inadvertently overlooked will not seriously impair the value of the review. CLASSICAL BUCKLING THEORIES- PERFECT C YL INDERS
In general, classical buckling theories assume linear relationships between stress/strain and strain/displacements. Furthermore, a membrane prebuckled shape is often considered in order to satisfy, either partially or completely, various end boundary conditions, although an 'infinite' length shell model can also be employed. Perhaps one of the earliest stability analyses of homogeneous orthotropic cylindrical shells was that done by March et al I in 1945 in which torsional buckling of plywood cylinders was studied. Later, another report was published by March 2 for the case of plywood cylinders under axial compression based on large deflection theory. It should be noted that plywood cylinders can be regarded as nonhomogeneous anisotropic constructions since the laminae can be oriented at arbitrary angles. Other theoretical analyses limited to orthotropic shell configurations were performed by Schnell and Bruhl 3, Thielemann et al a (on combined loading) and Hess s, in which simply supported end conditions were partially satisfied. A more general treatment of buckling of heterogeneous aeolotropic (ie anisotropic) cylinders under combined loading with arbitrary boundary conditions was presented by Cheng and Ho 6,7 which requires considerable numerical analysis to obtain the critical loads. Tasi 8 investigated in more detail the effect of heterogeneous composition on the axial compressive buckling load of laminated composite cylinders, based on a generalization of the earlier work by Schnell and Bruhi. 3 In addition, Tasi also compared his results with data obtained for the same problem using Cheng and Ho's theory and found negligible differences. At this point it is worth noting that Cheng and Ho employed Flugge's linear shell theory which is considered to be more
17
refined than the linear Donnell-type stability theory adopted by Tasi and most authors. Other studies of the bending, compressive, torsional and tensile buckling of anisotropic cylinders have also been published. For the case of bending, Ugural and Cheng 9 found that coupling between in-plane stretching and bending had an important effect on the buckling loads and in Holston's analysis lo (which was based on Cheng and Ho's work) it was shown that the pure bending buckling stress was essentially equal to the uniform compression value. Investigations of the effect of the length-to-radius ratio by Holston 11 on the compressive buckling load indicated no significant effects for values greater than 1.5, neglecting coupling between shearing and extensional strain. Khot 12 also an~ysed the case of axial compression, including nonlinear strain/displacement relations which were necessary to examine postbuckling behaviour. This work served as a basis for Khot's subsequent imperfection study which will be discussed later. A torsional buckling analysis by ChehilI and Cheng 13 also utilized large deflection shell theory and included an initial shape imperfection having the form of the assumed buckling mode. However, numerical results were only presented for plywood cylinders. For tensile loading, an interesting study by Pagano et al 14 showed that buckling was possible due to the induced torque arising from the presence of end constraints. One final reference can be made to the effect of material heterogeneity which was considered by Stavsky and Friedland is for compressive buckling of laminated orthotropic cylinders. Interesting comparisons with Cheng and Ho's theory have been madeby Martin and Drew 16 using a Donnell-type analysis for the case of radial pressure. Again, the differences between the two theories were negligible for the configurations considered. Similar agreement was also found by Chao 17 who employed Timoshenko's buckling equilibrium equations and compared his results for radial pressure and torsion with different boundary conditions. However, it should be pointed out that Chao's analysis was formulated to take into account general boundary conditions, combined loading (compression, torsion and radial pressure), local panel and stiffener buckling and the optimum design of axial and circumferentially stiffened circular cylinders. Finally, a modified form of Cheng and Ho's analysis was presented by Lei and Cheng 18 who investigated the buckling of orthotropic laminated shells including the effects of simply supported (four sets) and clamped (four sets) boundary conditions corresponding to the membrane prebuckled state. Buckling results were then obtained as a function of boundary conditions and varying length-toradius ratios for axial compression, radial pressure and combined loading. Extensive studies have been carried out by several Soviet authors on the effects of variable shear modulus 19 and transverse shear strain 20 and by Rikards et al 21 on the buckling of anisotropic shells. In the latter case, comparisons were made with classical theory which indicated asymptotic agreement as the interlayer shear modulus approached an infinite value. One other analysis by Van 22 was concerned only with the effect of variable axial compressive forces on the buckling load. Stiffened composite cylinders have been analysed by Jones 2a for the condition that the principal axes of orthotropy coincide with the shell coordinate directions. He was able to demonstrate that the coupling between bending and exten18
sion had an important effect for the loading modes considered (axial compression, lateral pressure and combinations thereof). In addition to the work done by Chao 17 other studies of stiffened anisotropic shells have been performed 24,2s and these included thermal loading through the shell wall thickness. Within the last decade, computer programs have been developed by many people for analysing structural stability problems. Usi.ng such techniques as numerical integration, finite differences and finite elements, it is possible to solve almost any shell buckling problem for a variety of loading conditions and wall configurations. However, most of these programs are generally not available for proprietary reasons, and the few that are require extensive computer calculations due to the lengthy numerical procedures involved. One such program (based on a finite difference analysis) that has been reasonably well documented in the open literature and that can treat buckling of composite cylinders under axisymmetric loading was prepared by Bushnell et al. 26 Volume 2 of their report contains the necessary stiffness coefficients for describing anisotropic circular cylinders, including a nonlinear prebuckling analysis for various end constraints. Some numerical results for composite cylinders under axial compression with clamped ends are contained in Volume 1 for comparison with classical theory and experimental data. It would appear from these results that the membrane prebuckling shell model gives only slightly higher ( " 4%) buckling loads than the 'exact' solution. Before proceeding to compare available experimental data with the 'perfect' shell theories presented, there are four factors which deserve special mention insofar as they can affect this comparison: (1)
the effect of boundary conditions on the prebuckling stress state existing in the cylinder;
(2)
the assumption of linear stress/strain behaviour;
(3)
the effect of different elastic moduli in tension and compression; ~
(4)
the role of geometric shape imperfections.
As far as boundary conditions are concerned, one must be careful to distinguish at what point in the analysis the edge constraint effects were considered. In the multitude of classical theories, prebuckling deformations are not included in the analysis. Rather, initial buckling mode displacement functions are assumed which either partially or completely satisfy specific boundary conditions to describe the membrane prebuckling state of the cylinder only at the inception of instability. Considerably more complicated analysis is required to treat the prebuckled state and assess the effects of prebuckling deformations on the buckling load. However, for certain boundary conditions, the load reductions associated with the prebuckling displacements may only amount to 5% ~ 10%, such as the clamped case for example. One assumption that is common to all elastic stress analyses is that of a linear stress/strain relation. Neyertheless, it is well known that for many composite materials, linearity may not exist for all types of stress and strain. For example, material property tests were performed on 'Scotchply' Type 1002 preimpregnated epoxy/glass** (type E) tape, 36% by ** A product of Minnesota Mining and Manufacturing Co, Saint Paul, Minnesota, USA. The tubes were fabricated at the University of Toronto Institute for Aerospace Studies,
COMPOSITES. JANUARY 1975
It is of some interest to note the method used to apply pure torsion with the same machine. Fig.2 demonstrates a larger cylinder ( ~ 153 m m (6 in) diameter) mounted in such a manner that the downward motion of the lower platen results in torsion of the shell. Figs. 3, 4 and 5 present the mean stress/strain data obtained from several specimens (see Table 1) tested up to failure. Although the axial tension and compression data in the main exhibit linearity, the shear behaviour is quite nonlinear. Hence some difficulty could arise in comparing torsional buckling loads with predicted values based on a constant shear modulus, although the initial load curve can be approximated reasonably well by a linear relation.
Manufacture of laminated tubes using belt-wrapper Fig.1 apparatus
Another assumption generally made is that of a constant elastic modulus to describe both tension and compression. lones 27 has shown that variations in these material coefficients can cause significant differences in buckling loads, at least for the cases he considered. Suffice it to say that one should be aware of the potential difficulties that can arise due to moduli differences and thus complete material property characterization tests should be done. Referring again to Figs 3 and 4, it can be seen that for this material system, the elastic moduli in axial compression and tension vary by ~< + 4% from the average. The effect of geometric shape imperfections on reducing the buckling loads of anisotropic circular cylinders is a topic meriting considerable discussion. In order to provide the background for assessing the need to consider this problem, it is appropriate to compare experimental data with theoretical predictions based on geometrically 'perfect' shell models. B U C K L I N G O F COMPOSI TE C YL I N D E R S E X P E R I M E N TA L D A T A
It would appear that some of the earliest composite cylinder tests were those performed by Norris and Kuenzi 28 on ortho-
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weight resin rontent, in the form of laminated cylindrical tubes subjected to torsion, axial tension and compression in directions parallel and normal to the fibre directions. The test specimens were manufactured using a belt-wrapper apparatus (shown in Fig. 1) in which the preimpregnated tape was wrapped around a mandrel under controlled pressure supplied by a continuous 'belt' mounted on rollers. Each tube was instrumented with two strain gauges positioned diametrically opposed and loaded in a standard electrically driven, four screw, tension/compression machine.
COMPOSITES. JANUARY 1975
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Axlol stroin x IO .3 Stress/strain curves for Scotchply 1002 loaded parallel
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19
Table 1. Summary of material property tests
Test mode
Fibre angle* (degrees)
Average tube diameter mm (in)
No. of specimens tested
Average wall thickness mm (in)
Average fracMaximum +ture stress MN/m 2" (Ibf/in 2 xl04) % deviation
Tension (E 11)
0
4
50.8
(2)
0.51
(0.020)
834
(12.1)
8
Compression (Ell)
0
7
25.4
(1)
2.13
(0.084)
607
(8.8)
4
Tension (E22)
90
7
50.8
(2)
1.02
(0.040)
(0.33)
8
Compression (E22)
90
7
50.8
(2)
1.02
(0.040)
50.8
(2)
1.04
(0.041)
Torsion (G12)
90
7
22.8 938 46.9
(1.36)
5
(0.68)
8
*With respect to longitudinal axial direction
tropic plywood shells under axial compression, and later reported by Kuenzi 29 who presented a theoretical comparison with the data. Large scatter above and below the theory was found. Preliminary experimental data were also published 30 by Card and Peterson for Filament-wound glass/ epoxy circular cylinders under axial compression, followed by a more extensive investigation 31 by Card. In comparing his results with linear anisotropic theory, it was apparent that substantial disagreement occurred, although some bCackling loads were as high as 80% of the predicted values. However, the general variation in results was similar to that observed for isotropic cylinder tests. A comparable reduction factor was also reported 32 based on earlier laboratory axial buckling tests and comparisons with Hess' theory. Tasi et al 33 have also investigated the compressive buckling behaviour of fdament-wound glass/epoxy cylinders with clamped edge constraints and compared their data with solutions they obtained using Cheng and Ho's linear theory. Again, test results fell substantially below theory, although
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Fig.4 Stress/straincurves for Scotchply lar to the fibre direction 20
2
1002 loaded
perpendicu-
several of their cylinders buckled at >i 80% of the predicted loads. An interesting comparison was later made by Bushnell et al 26 with the data of References'30, 32 and 33 in which the clamped boundary was taken into account throughout the prebuckling state. The scatter was reduced considerably and agreement to within 1> 80% was obt~ned for 70% of the cylinders studied. Other experimental investigations 34,as on axially compressed glass fibrereinforced plastic shells still showed considerable disagreement with classical theory and recourse to statistical 'reduction factors' was necessary to arrive at design curves as a function of the radius.to-thickness ratio. External pressure loading of orthotropic circular cylinders has been considered by Schneider and Hofeditz.a6 Theoretical buckling pressures were compared with reported test data on 53 glass fibre orthotropic cylinder specimens in the 'long cylinder' range. Generally good agreement with the predicted values was obtained (within + 15%). An extensive experimental study of the buckling of filamentwound glass/epoxy cylinders was undertaken by Holston et al 37 for various loading modes, including axial compression, torsion, bending, combined torsion/axial compression and combined bending/axial compression. Using a linear anisotropic shell stability analysis based on the work by Cheng and Ho, the bending critical loads were found to be essentially equal to the axial compression values which agreed with theory within 67% ~ 90%. However, the experimental torsion buckling loads were considerably higher than predicted and the corresponding interaction diagram bore no resemblance to theory. This could be due to the neglect of end contraints and prebuckling deformations. Similar combined torsion/compression buckling tests on graphite/epoxy and boron/epoxy.cylinders have recently been reported by Wilkins and Love. aa Using the programme of Holston et al 37 they too observed considerable disagreement between the experimental and predicted torsion buckling loads. At the same time, the compressive buckling loads were approximately 65% of the classical values. In a torsional strength study by Wall and Card 39 a comparison with the anisotropic buckling theory by Chao 17 for those specimens which may have failed by torsional instability indicated agreement only to within ~ 67%. More recent experiments on the torsional buckling of preimpregnated tape-wound glass/epoxy cylinders have been conducted at the University of Toronto Institute for Aerospace Studies (UTIAS), the results of which are summarized in Table 2. Because of the limited number of test models available and the cost of their manufacture, it was desirable to employ a
COMPOSITES. JANUARY 1975
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8
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16
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20
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28
Sh¢or stroin x IO-3(Rod) Fig.5
Stress/strain curves for Scotchply 1002 in pure shear
non-contacting internal mandrel with a clearance of about 1.27 mm (0.05 in) from the cylinder surface to reduce the postbuckling deformations. Thus the cylinders not only buckled elastically, but could be re-used several times since no fracture of the fibres or lamina separation occurred after buckling. Consequently, the cylinders were mounted vertically and torsion loading was achieved by hydraulic pumps applying transverse twisting forces to the upper end plate while retaining the base of the cylinder fixed to a rigid platen, as shown in Fig.6. The theoretical predictions shown in Table 2 were based on the linearized Donnell equations.
Fig.7 View of cylinder with thrust bearing under combined loading of axial compression and torsion
Table 2. Comparison of torsional buckling loads with theory for glass/epoxy (Scotchply 1 XP250) preimpregnated tape-wound cylinders 2
Shell No
Fibre orientation 3 (degrees) Inner Mid Outer
Experimental torque buckling load N m (Ibf in)
Theoretical torque buckling load N m (Ibf in)
Experiment Theory
la
-70
70
0
1136.1 (10055)
1291.9 (11434)
0.879
4b
90
-45
45
1595.9 (14125)
1636
(14480)
0.976
6b
45
0
-45
1361.5 (12050)
1164.7 (10308)
1.169
7b
-45
45
90
1145.1 (10135)
1375.4 (12173)
0.833
9b
0
-45
45
1118.0 (9895)
1121.4 (9925)
0.997
11a
30
90
30
1118.0 (9895)
1231.2 (10897)
0.908
11b
20
90
30
1145.1 (10135)
1222.8 (10823)
0.936
12a
30
90
-30
1027.8 (9097)
1301.9 (11523)
0.790
12b
30
90
-30
1118.0 (9895)
1325.4 (11731)
0.844
I A product of Minnesota Mining and Manufacturing Co, Saint Paul, Minnesota, USA 2 Nominal geometric properties as follows: radius = 159 mm (6.26 in); length = 317.5 mm (12.5 in); total thickness = 0.686 mm (0.027 in). Nominal material properties as follows: E l l ~-- 37.92 GN/m 2 (5.5 x 106 Ibf/in2); E22 = 17.93 GN/m 2 (2.6 x 106 Ibf/in2); G12 ~- 4.83 GN/m 2 ~0.7 x 106 Ibf/in2); P 12 = 0.37 With respect to longitudinal cylinder axis
COMPOSITES . J A N U A R Y 1975
21
Torque (Ibf in)
1995 !
39,90
9097 (T¢,I
7980
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23"71 (53301(Pcr}
21.53 (4840)
BUCKL ING OF GEOMETRICAL L Y IMPERFECT COMPOSITE CYL INDERS
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16.86 (37901
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The effect of geometric shape imperfections on the reduction in buckling strength for isotropic (stiffened and
+~
10.28 3101
E 0 U
t' 225
A31 b76 Torque (Nm) Axial strain
902
1(~28 (Tot}
Fig.8 Variation of axial strain with compressive load for varying values~of applied torque: the figures at the 'top' of each curve represent the buckling loads, measured in kilonewton$ (Ibf given in parentheses}
neglecting prebuckling displacements. Surprisingly good agreement with classical theory was obtained. Again the assumption of linear elastic behaviour was made and, for the magnitude of thelaminae shear stresses occurring at buckling, this was a reasonable approximation.t It should be noted that the sign of the applied torque (or shear stress) must be taken into account since significant differences in the torsional buckling loads can result. Combined loading tests involving axial compression and torsion have also been done at UTIAS with the shell models described in Table 2, again using an internal mandrel. However, to allow rotation of the cylinder during axial compressive loading, a thrust bearing was inserted between the top plate and load platen, as shown in Fig.7, and four pairs of strain gauges were mounted on the outer shell wall surface at equal spacings around the circumference. Thus for various values of fixed torque, it was possible to determine the variation in average axial strain with applied compressive load as shown in Fig.8. The main purpose of this plot was to check on the linearity of the structural response. The actual critical loads were obtained from the compression machine, although the strain gauge signals can be used for confirmation. From the combined load buckling results, an interaction diagram was constructed (Fig.9) for the anisotropic shells considered. The combined axial compression (P) and torsional (7+) buckling loads have been made nondimensional by the experimental values obtained when T = 0 (Per) and P = 0 (Ter), respectively. It can readily be seen that the interaction curves are nonlinear and vary substantially with laminate construction. This same degree of nonlinearity, however, was not observed in Reference 38 for graphite/ epoxy and boron/epoxy cylinders. No theoretical comparisons were made in Fig.9 because of the large discre+ The range of linear shear behaviour for Scotchply XP250 and "lO02jis not Significantly different f r o m that shown in Fig.5.
22
pancies between the observed and predicted compressive buckling loads. These differences were attributed primarily to the presence of geometric shape imperfections in the test cylinders. Thus it is appropriate to consider next the role played by imperfections on the buckling behaviour of composite cylinders.
monocoque) cylinders and spherical shells has been well known for some time. An excellent review of the theory and experimental evidence relevant to this subject has been published by Hutchinson and Koiter. 4° However, by comparison, relatively few studies have been unc[ertaken to investigate shape imperfection effects on the stability of composite shells. Theoretical analyses have been published based on anisotropic shell theory for the loading cases of pure torsion 13 and axial compression. 41 In both treatments, a Donnelltype of analysis including nonlinear strain/displacement relations and an initial shape imperfection having the form of the assumed buckling mode were employed and the total potential energy was minimized to yield the buckling loads. Load/deflection curves were tl3.en obtained as a function of the imperfection amplitude. For the case of torsion, results were presented only for plywood cylinders. On the other hand, Khot demonstrated the sensitivity of the compressive buckling loads not only for various imperfection amplitudes (ie using a ratio of the sum of the imperfection amplitudes/shell wall thickness) but also as a function of ply orientation for three-layer composite cylinders fabricated from glass/epoxy and boron/epoxy. Similar analyses were later done by Khot et al 42 including the effect of internal pressure and examining in some detail the postbuckling loads of composite cylinders.43 It would appear that the first application of Koiter's imperfection shell theory to composite cylinders was pub-
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C r i t i c a l t o r q u e ratio (T/Ter) F~g.9 Interaction diagram for combined axial compression and torsion loading for laminated glass/ep'bxy circular cylinders
COMPOSITES. JANUARY
1975
lished by Card. 44 In this analysis, Card employed Koiter's theory to examine the imperfection sensitivity of laminated orthotropic cylinders based on a perturbation technique applied to a 'perfect' shell at the inception of instability. Furthermore, the prebuckling deformations associated with simple support and clamped edge constraints were considered and buckling loads computed for three structural configurations (glass/epoxy, boron/epoxy and boron/aluminium) as a function of helix angle. Experiments with helical filament-wound, glass/epoxy cylinders were performed to determine their axial compressive buckling loads, and the results compared with 'perfect' shell predictions. The agreement of the data with consistent theory varied from 63% "" 91%, the average being 79%, thus indicating substantial shape imperfection effects. Calculations of the imperfection sensitivity were made for qualitative comparison with the experiments but since no measurements of actual imperfection distributions were done, a quantitative buckling load comparison was not possible. Using Koiter's theory, Khot and Venkayya 45 considered an initial shape imperfection having the form of the classical asymmetric 'perfect' shell buckling mode to calculate the initial postbuckling coefficients for a cylinder under axial compression. The effects of prebuckling deformations were neglected and an asymptotic formula was employed to estimate buckling loads for small values of the imperfection amplitude. Compressive buckling load calculations were then made for glass/epoxy and boron/ epoxy cylinders as a function of imperfection amplitude and ply angle. Later, an axisymmetric shape imperfection analysis was performed by Tennyson et a146 using Koiter's theory (neglecting boundary conditions) tO determine the effect of imperfection amplitudes and wavelengths on the compressive buckling load of anisotropic circular cylinders as a function of fibre orientation. Specific results were presented only for three-ply, glass/epoxy cylinders. One of the interesting factors that emerged was that critical imperfection wavelengths can occur which lead to minimum buckling loads and these values do not differ substantially for varying fibre angles. This was also observed for graphite• epoxy and boron/epoxy cylinders. Subsequently, axial compression buckling tests were performed by Tennyson and Muggeridge 47 on three-ply, glass/epoxy cylinders of the type described in Table~2. Each cylinder was surveyed to obtain profiles of the geometric shape imperfection distributions along several generators. Since the distributions were random in nature, it was necessary to formulate a buckling criterion in order to apply axisymmetric imperfection theory to calculate buckling loads for comparison with the experimental values. The approach used considered the maximum measured root mean square (rms) imperfection amplitude to represent an 'equivalent' axisymmetric shape imperfection concentrated at the critical wavelength. Thus, based on the analysis in Reference 46, buckling loads were computed and the results compared with the experiments. It was found that agreement ranged from 80% ~ 97%, with a mean of 87%. It should also be noted that the average shell buckling load was about 60% of the classical 'perfect' shell predicted value with a corresponding mean rms imperfection amplitude of approximately 40% of the shell wall thickness. Hence it would appear from these results that for the case of axial compression, this approach provides reasonably accurate estimates of the buckling loads. However, the test results apply only over a small range of imperfection amplitudes and for one material
COMPOSITES. JANUARY 1975
configuration. A more detailed discussion of this work and a comparison with independent calculations by Khot can be found in Reference 48. In a recent publication by Wilkins 49 a similar method was employed to predict the buckling loads of laminated graphite/epoxy curved panels under axial compression. Although the imperfection measurements were based only on thickness variations, reasonable correlation was obtained with the experimental values.
SUMMARY
It is worthwhile to consider to what extent design criteria ca~ be formulated for calculating buckling loads of laminated composite cylinders. Although numerous analyses of anisotropic cylinders exist for a variety-of loading conditions, it is evident that yery little experimental verification has been obtained. Thus it is necessary to treat any stability calculations with some caution due to the lack of experimental data. From the work reported, it would appear that classical theory is acceptable for predicting torsional and external pressure buckling loads for practical end constraint conditions. Since bending critical loads can be accurately estimated based on axial compression theory, one must only consider an appropriate analysis for evaluating the compressive behaviour of cylinders. It has been well established that shape imperfections can drastically reduce the compression strength, although they apparently have only a mirior influence on the torsion buckling loads. However, the major difficulty with incorporating imperfection effects is the requirement for statistical profile surveys. It is absolutely essential to have some measure of the magnitude of the imperfections present in the shell structure in order to estimate the probable load reductions. Otherwise, recourse to empirical correction factors is necessary. Because of the presence of shape imperfections, varying ply orientations to optimize the compressive buckling loads amounts to only about a 10% increase. Finally, limited experimental data does exist for estimating the combined buckling loads for cylinders under compression/ bending and compression/torsion. In conclusion, it should be emphasized that extensive experimental test programmes must yet be undertaken to study the buckling behaviour of both stiffened and unstiffened laminated composite cylinders under all loading conditions. Only in this way will sufficient data be available to support the existing theories and provide the necessary background for reliable stability load calculations. A CKNOWL EDGEMEN TS
The author wishes to acknowledge the contributions made to this review by Mr J. Young, Mr M. Booton, Mr D. Band and Mr R. Carducci who provided experimental and analytical results which have not been previously published. The research conducted at the University of Toronto Institute for Aerospace Studies in composite materials and structures was sponsored by grants to the author from the National Research Council of Canada (Grant No A-2783) and the National Aeronautics and Space Administration (Grant No NGR 52-026-039). The shell test models were supplied by Dr N. S. Khot of the US Air Force Flight Dynamics Laboratory (Stfiactures Division), Wright-Patterson Air Force Base, Ohio.
23
REFERENCES
25
1
26
2 3 4
5 6 7 8 9 10 11 12
13 14 15 16
17 18 19
20
21 22 23 24
24
March,H. W. et al. 'Buckling of thin-walled plywood cylinders in torsion', Forest Products Lab, Madison, Wisc Report 1529 (June 1945) Match, H.W. 'Buckling of long, thin plywood cylinders in axial compression', Forest Products Lab Report 1322-A (March 1956) Schneil, W. and Bruhl, C. 'Die Langsgedruckte Orthotrope Kreiszylinderschale bei Innendruck', Zeitschrift Flugwiss 7 (1959) Thielemann, W. F.~ Schnell, W. and Fischer, G. 'Buckling and postbuckling behaviour of orthotropic circular cylindrical shells subjected to combined axial and internal pressure', Zeitschrift Flugwiss 8 Heft 10/11 (1960) Hess, T.E. 'Stability of orthotropic cylindrical shells under combined loading', JAmerican Rocket Society 31 No 2 (Feb 1961) Cheng, S. and Ho, B. P.C. 'Stability of heterogeneous aeolotropic cylindrical shells under combined loading', AIAA Journal 1 No 4 (April 1963) Ho, B. P. C. and Cheng, S. 'Some problems in stability of heterogeneous aeolotropic cylindrical shells under combined loading', AIAA Journal 1 No 7 (July 1963) Tasi, J. 'Effect of heterogeneity on thestability of composite cylindrical shells under axial compression', AIAA Journal 4 No 6 (June 1966) Ugural, A. C. and Cheng, S. 'Buckling of composite cylindrical shells under pure bending', AIAA Journal 6 No 2 (Feb 1968) Holston, A. Jr. 'Buckling of inhomogeneous anisotropic cylindrical shells by bending',AIAA Journal 6 No 10 (Oct. 1968) Holstdn, A. Jr. 'Buckling of filament-wound cylinders by axial compression',AIAA Journal 6 No 5 (May 1968) Khot, N. S. 'On the effects of fiber orientation and nonhomogeneity on buckling and postbuckling equilibrium behaviour of fiber-reinforced cylindrical shells under axial compression', USAF, ICPAFB, Tech Report AFFDL-TR-68-19 (May 1968) Chehill, D. S. and Cheng, S. 'Elastic buckling of composite cylindrical shells under torsion', AIAA Journal, Spacecraft and Rockets 5 No 8 (August 1968) Pagano, N. J., Halpin, J. C. and Whitney, J.M. 'Tension buckling of anisotropic cylinders', J Comp Mat 2 No 2 (April 1968) Stavsky, Y. and Ftiedland, S. 'Stability of heterogeneous orthotropic cylindrical shells in axial compression', Israel J Technology 7 No 1-2 (March 1969) Martin, R. E. and Drew, D. D. 'A Batdorf type modified equation for the stability analysis of anisotropic cylindrical shells', Texas A & M University, Tech Report No 10 (March 1969) Chao, T. L. 'Minimum weight design of stiffened fiber composite cylinders, USAF, ICPAFB Tech Report AFMLTR-69-251 (Sept 1969) Lei, M. M. and Cheng, S. 'Buckling of composite and homogeneous isotropic cylindrical shells under axial and radial loading', J AppliedMech, Trans ASME (Dec 1969) Kravehuk, V. S. 'Influence of the shear modulus on the critical stresses of fiberglass-reinforced plastic cylindrical shells subjected to an axial compressive force', Prikladnaia Mekhanika 5 (Sept 1969) Pelekh, B. L., Rikards, R. B. and Teters, O. A. 'Stability of anisotropic shells weakly resistant to shear in a geometrically nonlinear formulation', Trans 7th All-Union Conference on Theory o f Plates and Shells (Dnepropetrovsk, Ukranian SSR, Sept 1969) Rikards, R. B. and Teters, G. A. 'Stability analysis of fiberglass-reinforced plastic shells with allowance for shear stresses', Mekhanika Polimerov 5 (Sept-Oct 1969) Van Fo Fy, G. A. 'Stability of orthotropic cylindrical shells under variable pressure', Mekhanika Polimerov 6 (Sept-Oct 1970) Jones, R. M. 'Buckling of circular cylindrical shells with multiple orthotr0pic layers and eccentric stiffeners', AIAA Journal 6 No 12 (Dec 1968) Terebushko, O. I. 'Stability of stiffened and anisotropic shells', Trans of 7th All-Union Conference on Theory of Plates and Shells (Dnepropetrovsk, Ukranian SSR, Sept 1969)
27 28 29 30 31 32 33 34 35 36
37 38
39 40 41
42
43 44 45
46
47 48 49
Chang, L. K. and Card, M. F. 'Thermal buckling analysis for stiffened orthotropic cylindrical shells', NASA TN D-6332 (1971) Bushnell, D., Almroth, B. O. and Sobel, L. H. 'Buckling of shells of revolution with various wall constructions', NASA CR-1050 1, 2, 3, (May 1968) Jones, R. M. 'Buckling of circular cylindrical shells with different moduli in tension and compression', USAF, Air Force Report No SAMSO-TR-70-S5(Dec 1969) Norris, C. B. and Kuenzi, E. W. 'Buckling of long, thin plywood cylinders in axial compression - experimental treatment' US Forest Products Lab Report 1322-B (1943) Kuenzi, E. W. 'Buckling of layered orthotropic and sandwich cylindrical shells in axial compression', NASA TN D-1510 (Dec 1962) Card, M. F. and Peterson, J. P. 'On the instability of orthotropic cylinders', NASA TN D-1510 ('Collected Papers on Instability of Shell Structures', Dec 1962) Card,M. F. 'Experiments to determine the strength of filament-wound cylinders loaded in axial compression', NASA TN D-3522 (Aug 1966) Ravenhall, R. 'Stiffness and buckling in filament-wound motors', AIAA Journal, Spacecraft and Rockets 1 No 3 (1964) Tasi, J., Feldman, A. and Stang, D.A. 'The buckling strength of filament-wound cylinders under axial compression', NASA CR-266 (July 1965) Gumeniuk, V. S. and Kravchuk, V.S. 'Study of the stability of cylindrical fiberglass reinforced plastic shells', Mechanika Polimerov 5 (Sept-Oct 1969) Schulz, U. 'Buckling stability of anisotropic cylindrical grp shells', der Bauingenieur 47 No 5 (May 1972) Schneider, M. H. and Hofeditz, J. T. 'Buckling of fiberglass cylinders under external pressure', Proceedings of the Winter Annual Meeting, American Soc of Mech Eng (New York, 1964) Paper No 64-WA/UNT-12 Holston, A., Jr, Feldman, A. and Stang, D.'A. 'Stability of filament-wound cylinders under combined loading', USAF, ICPAFB, Tech Report AFFDL-TR-67-55 (May 1967) Wilkins, D. J. and Love, T. S. 'Combined compression-torsion buckling tests of laminated composite cylindrical shells', presented at the 15 th AIAA Structures, Structural Dynamics and Materials Conference (Los Vegas, April 1974) Wall,L. D., Jr, and Card, M. F. 'Torsional shear strength of filament-wound glass-epoxy tubes', NASA TN D-6140 (Aug 1971) Hutchinson, J. W. and Koiter, W. T. 'Post-buckling theory', Applied Mechanics Reviews 23 No 12 (Dec 1970) Khot, N. S. 'On the influence of initial geometric imperfections on the buckling and postbuckling behavior of fiberreinforced cylindrical shells under uniform axial compression', USAF, WPAFB, Tech Report AFFDL-TR-68-136 (Oct 1968) Khot, N. S., Venkayya, V. B. and Berke, L, 'Buckling and post-buckling behavior of initially imperfect orthotropic cylindrical shells under axial compression,and internal pressure', presented at 1UTAM Conference 'Instability of continuous systems' (Herrenalb, West Germany, Sept 1969) Khot, N. S. 'Buckling and postbuckling behavior of composite cylindrical shells under axial compression', AIAA Journal 8 No 2 (Feb 1970) Card, M. F. The sensitivity of buckling of axially compressed fiber-reinforced cylindrical shells to small geometric imperfections', NASA TMX-61914 (June 1969) Khot, N. S. and Venkayya, V. B. 'Effect of fiber orientation on initial postbuckling behavior and imperfection sensitivity of composite cylindrical shells', USAF, WPAFB Tech Report AFFDL-TR-70-125, (Dec 1970) Tennyson, R. C., Chan, K. H. and Muggeridge, D. B. 'The effect of axisymetric shape imperfections on the buckling of laminated anisotropic circular cylinders', Trans Canadian Aeronautics and Space Inst 4 No 2 (Sept 1971) Tennyson, R. C. and Muggeddge, D. B. 'Buckling of laminated anisotropic imperfect circular cylinders under axial compression', AIAA JSpacecraft and Rockers 10 No 2 (Feb 1973) Tennyson, R. C. et al. 'Buckling of fiber-reinforced circular cylinders under axial compression', USAF, WPAFB, Tech Report AFFDL-TR-72-102 (Aug 1972) Wilkins, D. J. 'Compression buckling tests of laminated graphite-epoxy curved panels', presented at the 12th AIAA Aerospace Sciences Meeting (Washington, DC, Jan 1974)
COMPOSITES . J A N U A R Y 1975
A brief review of the available static buckling theory for both geometrically 'perfect' and 'imperfect' anisotropic composite circular cylinders is presented for various loading configurations. For comparison purposes, relevant experimental data are discussed, including recent combined loading test results and recommendations are made concerning the design of composite cylinders.
INTRODUCTION
The circular cylindrical shell is used extensively as a primary load carrying structure in many applications and as such is subjected to various loading configurations. From a design viewpoint, local or general instability due to the action or interaction of pressure, axial, torsional and thermal loads often represents the limiting load condition. Although a great deal of design information supported by substantial theoretical and experimental data is available for homogeneous isotropic cylinders in both the reinforced and monocoque states, this is not the case for anisotropic cylinders. Since composite materials have gained widespread usage because of the many unique advantages they offer, it is thus necessary to develop an adequate background for analysing heterogeneous*, anisotropic* structures that can result from the assembly of these material systems into various structural configurations. However, for the case of cylindrical shells, the calculation of stability criteria is a formidable task, especially if one must consider anisotropy, radial nonhomogeneity, various loading conditions, end constraints and the effects of geometric shape imperfections. Additional complexities can also arise due to differences in elastic moduli in tension and compression and nonlinear stress/strain behaviour, as will be discussed later. For the purposes of this review, only static buckling will be considered. Initially, classical linear buckling theory for geometrically perfect circular composite cylinders is examined, followed by comparisons with published experimental data. Subsequently, the effect of shape imperfec* On a macroscopic basis, an individual lamina may often be approximated as homogeneous and orthotropic (with respect to the axis of the reinforcement phase, such as fibres for example). However, for multi-ply structures composed of laminae having various orientations with respect to some arbitrary structural axis, the macroscopic structural behaviour can be considered anisotropic since the stiffness matrix is fully populated with non-zero terms and heterogeneity can occur due to the 'thickness' variation of the corresponding elastic coefficients. University of Toronto Institute for Aerospace Canada
COMPOSITES.
JANUARY
1975
Studies,Toronto,
tions on the buckling of composite cylinders is described together with appropriate experimental results. In a literature survey such as this, it is difficult to account for all references which relate to the subject matter. However, it is hoped that the absence of any pertinent reports which have been inadvertently overlooked will not seriously impair the value of the review. CLASSICAL BUCKLING THEORIES- PERFECT C YL INDERS
In general, classical buckling theories assume linear relationships between stress/strain and strain/displacements. Furthermore, a membrane prebuckled shape is often considered in order to satisfy, either partially or completely, various end boundary conditions, although an 'infinite' length shell model can also be employed. Perhaps one of the earliest stability analyses of homogeneous orthotropic cylindrical shells was that done by March et al I in 1945 in which torsional buckling of plywood cylinders was studied. Later, another report was published by March 2 for the case of plywood cylinders under axial compression based on large deflection theory. It should be noted that plywood cylinders can be regarded as nonhomogeneous anisotropic constructions since the laminae can be oriented at arbitrary angles. Other theoretical analyses limited to orthotropic shell configurations were performed by Schnell and Bruhl 3, Thielemann et al a (on combined loading) and Hess s, in which simply supported end conditions were partially satisfied. A more general treatment of buckling of heterogeneous aeolotropic (ie anisotropic) cylinders under combined loading with arbitrary boundary conditions was presented by Cheng and Ho 6,7 which requires considerable numerical analysis to obtain the critical loads. Tasi 8 investigated in more detail the effect of heterogeneous composition on the axial compressive buckling load of laminated composite cylinders, based on a generalization of the earlier work by Schnell and Bruhi. 3 In addition, Tasi also compared his results with data obtained for the same problem using Cheng and Ho's theory and found negligible differences. At this point it is worth noting that Cheng and Ho employed Flugge's linear shell theory which is considered to be more
17
refined than the linear Donnell-type stability theory adopted by Tasi and most authors. Other studies of the bending, compressive, torsional and tensile buckling of anisotropic cylinders have also been published. For the case of bending, Ugural and Cheng 9 found that coupling between in-plane stretching and bending had an important effect on the buckling loads and in Holston's analysis lo (which was based on Cheng and Ho's work) it was shown that the pure bending buckling stress was essentially equal to the uniform compression value. Investigations of the effect of the length-to-radius ratio by Holston 11 on the compressive buckling load indicated no significant effects for values greater than 1.5, neglecting coupling between shearing and extensional strain. Khot 12 also an~ysed the case of axial compression, including nonlinear strain/displacement relations which were necessary to examine postbuckling behaviour. This work served as a basis for Khot's subsequent imperfection study which will be discussed later. A torsional buckling analysis by ChehilI and Cheng 13 also utilized large deflection shell theory and included an initial shape imperfection having the form of the assumed buckling mode. However, numerical results were only presented for plywood cylinders. For tensile loading, an interesting study by Pagano et al 14 showed that buckling was possible due to the induced torque arising from the presence of end constraints. One final reference can be made to the effect of material heterogeneity which was considered by Stavsky and Friedland is for compressive buckling of laminated orthotropic cylinders. Interesting comparisons with Cheng and Ho's theory have been madeby Martin and Drew 16 using a Donnell-type analysis for the case of radial pressure. Again, the differences between the two theories were negligible for the configurations considered. Similar agreement was also found by Chao 17 who employed Timoshenko's buckling equilibrium equations and compared his results for radial pressure and torsion with different boundary conditions. However, it should be pointed out that Chao's analysis was formulated to take into account general boundary conditions, combined loading (compression, torsion and radial pressure), local panel and stiffener buckling and the optimum design of axial and circumferentially stiffened circular cylinders. Finally, a modified form of Cheng and Ho's analysis was presented by Lei and Cheng 18 who investigated the buckling of orthotropic laminated shells including the effects of simply supported (four sets) and clamped (four sets) boundary conditions corresponding to the membrane prebuckled state. Buckling results were then obtained as a function of boundary conditions and varying length-toradius ratios for axial compression, radial pressure and combined loading. Extensive studies have been carried out by several Soviet authors on the effects of variable shear modulus 19 and transverse shear strain 20 and by Rikards et al 21 on the buckling of anisotropic shells. In the latter case, comparisons were made with classical theory which indicated asymptotic agreement as the interlayer shear modulus approached an infinite value. One other analysis by Van 22 was concerned only with the effect of variable axial compressive forces on the buckling load. Stiffened composite cylinders have been analysed by Jones 2a for the condition that the principal axes of orthotropy coincide with the shell coordinate directions. He was able to demonstrate that the coupling between bending and exten18
sion had an important effect for the loading modes considered (axial compression, lateral pressure and combinations thereof). In addition to the work done by Chao 17 other studies of stiffened anisotropic shells have been performed 24,2s and these included thermal loading through the shell wall thickness. Within the last decade, computer programs have been developed by many people for analysing structural stability problems. Usi.ng such techniques as numerical integration, finite differences and finite elements, it is possible to solve almost any shell buckling problem for a variety of loading conditions and wall configurations. However, most of these programs are generally not available for proprietary reasons, and the few that are require extensive computer calculations due to the lengthy numerical procedures involved. One such program (based on a finite difference analysis) that has been reasonably well documented in the open literature and that can treat buckling of composite cylinders under axisymmetric loading was prepared by Bushnell et al. 26 Volume 2 of their report contains the necessary stiffness coefficients for describing anisotropic circular cylinders, including a nonlinear prebuckling analysis for various end constraints. Some numerical results for composite cylinders under axial compression with clamped ends are contained in Volume 1 for comparison with classical theory and experimental data. It would appear from these results that the membrane prebuckling shell model gives only slightly higher ( " 4%) buckling loads than the 'exact' solution. Before proceeding to compare available experimental data with the 'perfect' shell theories presented, there are four factors which deserve special mention insofar as they can affect this comparison: (1)
the effect of boundary conditions on the prebuckling stress state existing in the cylinder;
(2)
the assumption of linear stress/strain behaviour;
(3)
the effect of different elastic moduli in tension and compression; ~
(4)
the role of geometric shape imperfections.
As far as boundary conditions are concerned, one must be careful to distinguish at what point in the analysis the edge constraint effects were considered. In the multitude of classical theories, prebuckling deformations are not included in the analysis. Rather, initial buckling mode displacement functions are assumed which either partially or completely satisfy specific boundary conditions to describe the membrane prebuckling state of the cylinder only at the inception of instability. Considerably more complicated analysis is required to treat the prebuckled state and assess the effects of prebuckling deformations on the buckling load. However, for certain boundary conditions, the load reductions associated with the prebuckling displacements may only amount to 5% ~ 10%, such as the clamped case for example. One assumption that is common to all elastic stress analyses is that of a linear stress/strain relation. Neyertheless, it is well known that for many composite materials, linearity may not exist for all types of stress and strain. For example, material property tests were performed on 'Scotchply' Type 1002 preimpregnated epoxy/glass** (type E) tape, 36% by ** A product of Minnesota Mining and Manufacturing Co, Saint Paul, Minnesota, USA. The tubes were fabricated at the University of Toronto Institute for Aerospace Studies,
COMPOSITES. JANUARY 1975
It is of some interest to note the method used to apply pure torsion with the same machine. Fig.2 demonstrates a larger cylinder ( ~ 153 m m (6 in) diameter) mounted in such a manner that the downward motion of the lower platen results in torsion of the shell. Figs. 3, 4 and 5 present the mean stress/strain data obtained from several specimens (see Table 1) tested up to failure. Although the axial tension and compression data in the main exhibit linearity, the shear behaviour is quite nonlinear. Hence some difficulty could arise in comparing torsional buckling loads with predicted values based on a constant shear modulus, although the initial load curve can be approximated reasonably well by a linear relation.
Manufacture of laminated tubes using belt-wrapper Fig.1 apparatus
Another assumption generally made is that of a constant elastic modulus to describe both tension and compression. lones 27 has shown that variations in these material coefficients can cause significant differences in buckling loads, at least for the cases he considered. Suffice it to say that one should be aware of the potential difficulties that can arise due to moduli differences and thus complete material property characterization tests should be done. Referring again to Figs 3 and 4, it can be seen that for this material system, the elastic moduli in axial compression and tension vary by ~< + 4% from the average. The effect of geometric shape imperfections on reducing the buckling loads of anisotropic circular cylinders is a topic meriting considerable discussion. In order to provide the background for assessing the need to consider this problem, it is appropriate to compare experimental data with theoretical predictions based on geometrically 'perfect' shell models. B U C K L I N G O F COMPOSI TE C YL I N D E R S E X P E R I M E N TA L D A T A
It would appear that some of the earliest composite cylinder tests were those performed by Norris and Kuenzi 28 on ortho-
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weight resin rontent, in the form of laminated cylindrical tubes subjected to torsion, axial tension and compression in directions parallel and normal to the fibre directions. The test specimens were manufactured using a belt-wrapper apparatus (shown in Fig. 1) in which the preimpregnated tape was wrapped around a mandrel under controlled pressure supplied by a continuous 'belt' mounted on rollers. Each tube was instrumented with two strain gauges positioned diametrically opposed and loaded in a standard electrically driven, four screw, tension/compression machine.
COMPOSITES. JANUARY 1975
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Axlol stroin x IO .3 Stress/strain curves for Scotchply 1002 loaded parallel
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19
Table 1. Summary of material property tests
Test mode
Fibre angle* (degrees)
Average tube diameter mm (in)
No. of specimens tested
Average wall thickness mm (in)
Average fracMaximum +ture stress MN/m 2" (Ibf/in 2 xl04) % deviation
Tension (E 11)
0
4
50.8
(2)
0.51
(0.020)
834
(12.1)
8
Compression (Ell)
0
7
25.4
(1)
2.13
(0.084)
607
(8.8)
4
Tension (E22)
90
7
50.8
(2)
1.02
(0.040)
(0.33)
8
Compression (E22)
90
7
50.8
(2)
1.02
(0.040)
50.8
(2)
1.04
(0.041)
Torsion (G12)
90
7
22.8 938 46.9
(1.36)
5
(0.68)
8
*With respect to longitudinal axial direction
tropic plywood shells under axial compression, and later reported by Kuenzi 29 who presented a theoretical comparison with the data. Large scatter above and below the theory was found. Preliminary experimental data were also published 30 by Card and Peterson for Filament-wound glass/ epoxy circular cylinders under axial compression, followed by a more extensive investigation 31 by Card. In comparing his results with linear anisotropic theory, it was apparent that substantial disagreement occurred, although some bCackling loads were as high as 80% of the predicted values. However, the general variation in results was similar to that observed for isotropic cylinder tests. A comparable reduction factor was also reported 32 based on earlier laboratory axial buckling tests and comparisons with Hess' theory. Tasi et al 33 have also investigated the compressive buckling behaviour of fdament-wound glass/epoxy cylinders with clamped edge constraints and compared their data with solutions they obtained using Cheng and Ho's linear theory. Again, test results fell substantially below theory, although
Frocture
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I 2 3 4 5 b 7 8 9 IO II 12 13 14 15 Axiol stroin x IO "3
Fig.4 Stress/straincurves for Scotchply lar to the fibre direction 20
2
1002 loaded
perpendicu-
several of their cylinders buckled at >i 80% of the predicted loads. An interesting comparison was later made by Bushnell et al 26 with the data of References'30, 32 and 33 in which the clamped boundary was taken into account throughout the prebuckling state. The scatter was reduced considerably and agreement to within 1> 80% was obt~ned for 70% of the cylinders studied. Other experimental investigations 34,as on axially compressed glass fibrereinforced plastic shells still showed considerable disagreement with classical theory and recourse to statistical 'reduction factors' was necessary to arrive at design curves as a function of the radius.to-thickness ratio. External pressure loading of orthotropic circular cylinders has been considered by Schneider and Hofeditz.a6 Theoretical buckling pressures were compared with reported test data on 53 glass fibre orthotropic cylinder specimens in the 'long cylinder' range. Generally good agreement with the predicted values was obtained (within + 15%). An extensive experimental study of the buckling of filamentwound glass/epoxy cylinders was undertaken by Holston et al 37 for various loading modes, including axial compression, torsion, bending, combined torsion/axial compression and combined bending/axial compression. Using a linear anisotropic shell stability analysis based on the work by Cheng and Ho, the bending critical loads were found to be essentially equal to the axial compression values which agreed with theory within 67% ~ 90%. However, the experimental torsion buckling loads were considerably higher than predicted and the corresponding interaction diagram bore no resemblance to theory. This could be due to the neglect of end contraints and prebuckling deformations. Similar combined torsion/compression buckling tests on graphite/epoxy and boron/epoxy.cylinders have recently been reported by Wilkins and Love. aa Using the programme of Holston et al 37 they too observed considerable disagreement between the experimental and predicted torsion buckling loads. At the same time, the compressive buckling loads were approximately 65% of the classical values. In a torsional strength study by Wall and Card 39 a comparison with the anisotropic buckling theory by Chao 17 for those specimens which may have failed by torsional instability indicated agreement only to within ~ 67%. More recent experiments on the torsional buckling of preimpregnated tape-wound glass/epoxy cylinders have been conducted at the University of Toronto Institute for Aerospace Studies (UTIAS), the results of which are summarized in Table 2. Because of the limited number of test models available and the cost of their manufacture, it was desirable to employ a
COMPOSITES. JANUARY 1975
IO 9
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8
50
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6
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IO I ~ { Li{ Lineor portion ) GI2:2.9x IO3MN/m2(4'2 x IO5 Ib flirt2 ) I
I
4
1
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1
8
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12
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16
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I
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20
1
24
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28
Sh¢or stroin x IO-3(Rod) Fig.5
Stress/strain curves for Scotchply 1002 in pure shear
non-contacting internal mandrel with a clearance of about 1.27 mm (0.05 in) from the cylinder surface to reduce the postbuckling deformations. Thus the cylinders not only buckled elastically, but could be re-used several times since no fracture of the fibres or lamina separation occurred after buckling. Consequently, the cylinders were mounted vertically and torsion loading was achieved by hydraulic pumps applying transverse twisting forces to the upper end plate while retaining the base of the cylinder fixed to a rigid platen, as shown in Fig.6. The theoretical predictions shown in Table 2 were based on the linearized Donnell equations.
Fig.7 View of cylinder with thrust bearing under combined loading of axial compression and torsion
Table 2. Comparison of torsional buckling loads with theory for glass/epoxy (Scotchply 1 XP250) preimpregnated tape-wound cylinders 2
Shell No
Fibre orientation 3 (degrees) Inner Mid Outer
Experimental torque buckling load N m (Ibf in)
Theoretical torque buckling load N m (Ibf in)
Experiment Theory
la
-70
70
0
1136.1 (10055)
1291.9 (11434)
0.879
4b
90
-45
45
1595.9 (14125)
1636
(14480)
0.976
6b
45
0
-45
1361.5 (12050)
1164.7 (10308)
1.169
7b
-45
45
90
1145.1 (10135)
1375.4 (12173)
0.833
9b
0
-45
45
1118.0 (9895)
1121.4 (9925)
0.997
11a
30
90
30
1118.0 (9895)
1231.2 (10897)
0.908
11b
20
90
30
1145.1 (10135)
1222.8 (10823)
0.936
12a
30
90
-30
1027.8 (9097)
1301.9 (11523)
0.790
12b
30
90
-30
1118.0 (9895)
1325.4 (11731)
0.844
I A product of Minnesota Mining and Manufacturing Co, Saint Paul, Minnesota, USA 2 Nominal geometric properties as follows: radius = 159 mm (6.26 in); length = 317.5 mm (12.5 in); total thickness = 0.686 mm (0.027 in). Nominal material properties as follows: E l l ~-- 37.92 GN/m 2 (5.5 x 106 Ibf/in2); E22 = 17.93 GN/m 2 (2.6 x 106 Ibf/in2); G12 ~- 4.83 GN/m 2 ~0.7 x 106 Ibf/in2); P 12 = 0.37 With respect to longitudinal cylinder axis
COMPOSITES . J A N U A R Y 1975
21
Torque (Ibf in)
1995 !
39,90
9097 (T¢,I
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BUCKL ING OF GEOMETRICAL L Y IMPERFECT COMPOSITE CYL INDERS
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The effect of geometric shape imperfections on the reduction in buckling strength for isotropic (stiffened and
+~
10.28 3101
E 0 U
t' 225
A31 b76 Torque (Nm) Axial strain
902
1(~28 (Tot}
Fig.8 Variation of axial strain with compressive load for varying values~of applied torque: the figures at the 'top' of each curve represent the buckling loads, measured in kilonewton$ (Ibf given in parentheses}
neglecting prebuckling displacements. Surprisingly good agreement with classical theory was obtained. Again the assumption of linear elastic behaviour was made and, for the magnitude of thelaminae shear stresses occurring at buckling, this was a reasonable approximation.t It should be noted that the sign of the applied torque (or shear stress) must be taken into account since significant differences in the torsional buckling loads can result. Combined loading tests involving axial compression and torsion have also been done at UTIAS with the shell models described in Table 2, again using an internal mandrel. However, to allow rotation of the cylinder during axial compressive loading, a thrust bearing was inserted between the top plate and load platen, as shown in Fig.7, and four pairs of strain gauges were mounted on the outer shell wall surface at equal spacings around the circumference. Thus for various values of fixed torque, it was possible to determine the variation in average axial strain with applied compressive load as shown in Fig.8. The main purpose of this plot was to check on the linearity of the structural response. The actual critical loads were obtained from the compression machine, although the strain gauge signals can be used for confirmation. From the combined load buckling results, an interaction diagram was constructed (Fig.9) for the anisotropic shells considered. The combined axial compression (P) and torsional (7+) buckling loads have been made nondimensional by the experimental values obtained when T = 0 (Per) and P = 0 (Ter), respectively. It can readily be seen that the interaction curves are nonlinear and vary substantially with laminate construction. This same degree of nonlinearity, however, was not observed in Reference 38 for graphite/ epoxy and boron/epoxy cylinders. No theoretical comparisons were made in Fig.9 because of the large discre+ The range of linear shear behaviour for Scotchply XP250 and "lO02jis not Significantly different f r o m that shown in Fig.5.
22
pancies between the observed and predicted compressive buckling loads. These differences were attributed primarily to the presence of geometric shape imperfections in the test cylinders. Thus it is appropriate to consider next the role played by imperfections on the buckling behaviour of composite cylinders.
monocoque) cylinders and spherical shells has been well known for some time. An excellent review of the theory and experimental evidence relevant to this subject has been published by Hutchinson and Koiter. 4° However, by comparison, relatively few studies have been unc[ertaken to investigate shape imperfection effects on the stability of composite shells. Theoretical analyses have been published based on anisotropic shell theory for the loading cases of pure torsion 13 and axial compression. 41 In both treatments, a Donnelltype of analysis including nonlinear strain/displacement relations and an initial shape imperfection having the form of the assumed buckling mode were employed and the total potential energy was minimized to yield the buckling loads. Load/deflection curves were tl3.en obtained as a function of the imperfection amplitude. For the case of torsion, results were presented only for plywood cylinders. On the other hand, Khot demonstrated the sensitivity of the compressive buckling loads not only for various imperfection amplitudes (ie using a ratio of the sum of the imperfection amplitudes/shell wall thickness) but also as a function of ply orientation for three-layer composite cylinders fabricated from glass/epoxy and boron/epoxy. Similar analyses were later done by Khot et al 42 including the effect of internal pressure and examining in some detail the postbuckling loads of composite cylinders.43 It would appear that the first application of Koiter's imperfection shell theory to composite cylinders was pub-
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COMPOSITES. JANUARY
1975
lished by Card. 44 In this analysis, Card employed Koiter's theory to examine the imperfection sensitivity of laminated orthotropic cylinders based on a perturbation technique applied to a 'perfect' shell at the inception of instability. Furthermore, the prebuckling deformations associated with simple support and clamped edge constraints were considered and buckling loads computed for three structural configurations (glass/epoxy, boron/epoxy and boron/aluminium) as a function of helix angle. Experiments with helical filament-wound, glass/epoxy cylinders were performed to determine their axial compressive buckling loads, and the results compared with 'perfect' shell predictions. The agreement of the data with consistent theory varied from 63% "" 91%, the average being 79%, thus indicating substantial shape imperfection effects. Calculations of the imperfection sensitivity were made for qualitative comparison with the experiments but since no measurements of actual imperfection distributions were done, a quantitative buckling load comparison was not possible. Using Koiter's theory, Khot and Venkayya 45 considered an initial shape imperfection having the form of the classical asymmetric 'perfect' shell buckling mode to calculate the initial postbuckling coefficients for a cylinder under axial compression. The effects of prebuckling deformations were neglected and an asymptotic formula was employed to estimate buckling loads for small values of the imperfection amplitude. Compressive buckling load calculations were then made for glass/epoxy and boron/ epoxy cylinders as a function of imperfection amplitude and ply angle. Later, an axisymmetric shape imperfection analysis was performed by Tennyson et a146 using Koiter's theory (neglecting boundary conditions) tO determine the effect of imperfection amplitudes and wavelengths on the compressive buckling load of anisotropic circular cylinders as a function of fibre orientation. Specific results were presented only for three-ply, glass/epoxy cylinders. One of the interesting factors that emerged was that critical imperfection wavelengths can occur which lead to minimum buckling loads and these values do not differ substantially for varying fibre angles. This was also observed for graphite• epoxy and boron/epoxy cylinders. Subsequently, axial compression buckling tests were performed by Tennyson and Muggeridge 47 on three-ply, glass/epoxy cylinders of the type described in Table~2. Each cylinder was surveyed to obtain profiles of the geometric shape imperfection distributions along several generators. Since the distributions were random in nature, it was necessary to formulate a buckling criterion in order to apply axisymmetric imperfection theory to calculate buckling loads for comparison with the experimental values. The approach used considered the maximum measured root mean square (rms) imperfection amplitude to represent an 'equivalent' axisymmetric shape imperfection concentrated at the critical wavelength. Thus, based on the analysis in Reference 46, buckling loads were computed and the results compared with the experiments. It was found that agreement ranged from 80% ~ 97%, with a mean of 87%. It should also be noted that the average shell buckling load was about 60% of the classical 'perfect' shell predicted value with a corresponding mean rms imperfection amplitude of approximately 40% of the shell wall thickness. Hence it would appear from these results that for the case of axial compression, this approach provides reasonably accurate estimates of the buckling loads. However, the test results apply only over a small range of imperfection amplitudes and for one material
COMPOSITES. JANUARY 1975
configuration. A more detailed discussion of this work and a comparison with independent calculations by Khot can be found in Reference 48. In a recent publication by Wilkins 49 a similar method was employed to predict the buckling loads of laminated graphite/epoxy curved panels under axial compression. Although the imperfection measurements were based only on thickness variations, reasonable correlation was obtained with the experimental values.
SUMMARY
It is worthwhile to consider to what extent design criteria ca~ be formulated for calculating buckling loads of laminated composite cylinders. Although numerous analyses of anisotropic cylinders exist for a variety-of loading conditions, it is evident that yery little experimental verification has been obtained. Thus it is necessary to treat any stability calculations with some caution due to the lack of experimental data. From the work reported, it would appear that classical theory is acceptable for predicting torsional and external pressure buckling loads for practical end constraint conditions. Since bending critical loads can be accurately estimated based on axial compression theory, one must only consider an appropriate analysis for evaluating the compressive behaviour of cylinders. It has been well established that shape imperfections can drastically reduce the compression strength, although they apparently have only a mirior influence on the torsion buckling loads. However, the major difficulty with incorporating imperfection effects is the requirement for statistical profile surveys. It is absolutely essential to have some measure of the magnitude of the imperfections present in the shell structure in order to estimate the probable load reductions. Otherwise, recourse to empirical correction factors is necessary. Because of the presence of shape imperfections, varying ply orientations to optimize the compressive buckling loads amounts to only about a 10% increase. Finally, limited experimental data does exist for estimating the combined buckling loads for cylinders under compression/ bending and compression/torsion. In conclusion, it should be emphasized that extensive experimental test programmes must yet be undertaken to study the buckling behaviour of both stiffened and unstiffened laminated composite cylinders under all loading conditions. Only in this way will sufficient data be available to support the existing theories and provide the necessary background for reliable stability load calculations. A CKNOWL EDGEMEN TS
The author wishes to acknowledge the contributions made to this review by Mr J. Young, Mr M. Booton, Mr D. Band and Mr R. Carducci who provided experimental and analytical results which have not been previously published. The research conducted at the University of Toronto Institute for Aerospace Studies in composite materials and structures was sponsored by grants to the author from the National Research Council of Canada (Grant No A-2783) and the National Aeronautics and Space Administration (Grant No NGR 52-026-039). The shell test models were supplied by Dr N. S. Khot of the US Air Force Flight Dynamics Laboratory (Stfiactures Division), Wright-Patterson Air Force Base, Ohio.
23
REFERENCES
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5 6 7 8 9 10 11 12
13 14 15 16
17 18 19
20
21 22 23 24
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COMPOSITES . J A N U A R Y 1975