INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 36, 595-609 (1993)

THE EFFECT OF NON-LINEAR CURVATURE STRAINS O N THE BUCKLING O F LAMINATED

PLATES AND SHELLS

S. T. DENNIS*

Department of Engineering Mechanics, USAF Academy, Colorado Springs C o 80840, U.S.A

A. N. PALAZOTTO'

Department of Aeronautics and Astronautics, Air Force Institute of Technology, Wright Patterson AFB, OH 45433, U.S.A.

SUMMARY A versatile geometrically non-linear curved shell finite element formulation includes exact Green's strain displacement for the inplane strains and linear strain displacement for the transverse shear strains. Additional coupling results for membrane-bending and membrane-transverse shear due to non-linear curvature strain-displacement relations. Additional non-linear terms that are a function of the shell's radius of curvature are also present. The approach is applied to the bifurcation buckling of axially compressed flat plates and transversely pressure-loaded cylindrical shell panels. The method gives a more flexible result for the plates compared to von Karman non-linearity. Additionally, larger buckling pressures are found for shell panels. The effects of transverse shear flexibility are evident.

INTRODUCTION

The exact solutions to differential equations governing laminated shell behaviour often do not exist. The analyst relies on approximate numerical solutions to capture shell phenomena such as bending-membrane coupling and transverse shear deformation, especially in the presence of material orthotropy and geometric non-linearity. Furthermore, often a linear bifurcation ideal- ization can give accurate instability information instead of solving the more complex non-linear problem. Transverse shear deformation, neglected in classical theories, is significant for laminated orthotropic constructions as these are typically weak in shear. Geometric non-linearities are important in structures susceptible to instabilities. Previous investigators have combined many of these characteristics in formulating finite element approximations in analysing the stability of shell structures.

Noor' examines the stability of multilayered plates with first-order transverse shear flexibility and makes comparisons with three-dimensional (3-D) results. Further studies are found in Noor and Peters,' where symmetry conditions and delaminations are examined. Geier and Rohwer present both laminated plate and shell buckling results, comparing classical (Kirchhoff ), first- order shear deformation (Reissner-Mindlin) and parabolic transverse shear approaches. Basar and Ding4 derive an efficient family of large rotation shell finite elements based on

*Associate Professor 'Professor

0029-598 1/93/040595- 15$12.50 0 1993 by John Wiley & Sons, Ltd.

Received 30 November 1991 Revised 9 March 1992

596 S. ‘r. DENNIS AND A. N. PALAZOTTO

Kirchhoff-Love assumptions. A simplified large rotation approach by Dennis and Palazotto’ showed that additional non-linear coupling between membrane and bending triggered the non-linear response of axially compressed cylindrical shell panels without numerical geometric imperfections. Through the thickness, displacement representations have been developed by Reddy and co-workers, Barber0 and Teply, and applied to plates for deformations and inter- laminar stresses6. and buckling8 They also generalized the approach for closed cylindrical shells and solved for natural frequencies in Reference 9. Stein” assumed a von Karman non-linearity with three-dimensional flexibility and studied instability in axially compressed plates. Leissa’ ’ gives a very comprehensive review of laminated plate instability where a von Karman type non-linearity is assumed. Likewise, Bushnell12 presents an in-depth review of isotropic shell buckling. Reddyl3 derives a moderate rotation theory for laminated plates. Schmidtt4 et al. conclude that the inclusion of moderate rotation geometric non-linearity with a first-order shear deformation theory exhibits strong coupling between membrane and bending deformation. Furthermore, Schmidt and Reddy” suggest that a moderate rotation theory will indicate new buckling behaviour for plates and shells.

The present paper applies an approach for laminated composite shell analysis that combines a parabolic transverse shear distribution with a simplified large displacement and rotational geometric non-linearity to the bifurcation buckling of flat plates and cylindrical shell panels. The resulting finite element formulation is versatile, in that many relevant situations can be accurately solved. A wide range of thick to thin (shell radius to thickness greater than five) cylindrical shell and flat-plate geometries that may undergo large rotations can be represented under a variety of loading condition^.^^ Presently, buckling solutions to the theory are found via a displace- ment-based finite element approach and illustrate the influence of non-linear curvature strains. The numerical results are buckling loads for typical components of aerospace semi-monocoque constructions or thick-walled deep sea submersible structural elements.

THEORY

Kinematics and strains

A detailed account of the approach outlined in the following is given in Reference 16. The kinematics for a general shell geometry described by orthogonal curvilinear co-ordinates with parabolic transverse shear distribution are shown in equation (1).

u3(t* , 5 2 ) = w

where k = - 4/3h2, h is the shell thickness, u, u, w, ( y = 1,2) are functions of midsurface shell co-ordinates, <), $? are bending rotations of the midsurface normals. ( ),;, refers to differentiation with respect to &, c ly are square roots of elements of the surface metric. Cylindrical shell specialization results by letting rl = a2 = 1, R2 = R and R1 = co, and flat-plate kinematics result by letting

The strains are expressed in equation (2) using equation (1) and the Green’s strain-displace- ment relations where exact expressions are assumed for the inplane strains E ~ , E~ and & 6 , but only

= cx2 = 1 and R , = R 2 = m.

NON-LINEAR CURVATURE STRAINS 597

the linear displacement terms are retained for the transverse strains, E~ and E ~ , see Saada.20 Each of the strain components, E: , E:, and K , ~ of equation (2) are specialized for cylindrical co-ordinates and shown in Reference 16. Several representative terms (to be discussed shortly) are shown in equation (2) as well. The flat-plate straindisplacement relations can be determined from equation (2) by letting R = ax

E~ = E : + [ p ~ i p ,

E , = E: + 12xrn2, m = 4 , 5

i = 1,2,6, p = sum 1-7

where, for example, with c = 1/R, and k = - 4/3h2:

Librescuz' develops general shell equations based on varying approximations in the magni- tudes of both strain and the rotations the differential elements undergo during deformation. A consistent small strain, rnoderute-rotation approximation has non-linear (but not exact) strain-displacement relations for the inplane strains, yet linear relations for the transverse shear strains. The present approach will, therefore, be viewed as a small strain, simplified large rotation theory due to the exact inplane strain assumptionz2 together with the linear transverse strain assumption. In what follows, the cylindrical shell and flat-plate strain-displacement relations of equation (2) will be referred to as the 'simplified large rotation' (SLR) approach. The exact strain assumptions for inplane strain displacement allow theoretically arbitrary rotations for classical formulations.22 However, the linear (in displacement) assumptions for transverse shear strains will limit the accuracy for the movements associated with large rotations of differential shell elements for those cases where transverse shear is influential.

For comparison purposes, both Donnell cylindrical shell and von Karman flat-plate approaches are derived. These approaches, in the present context, are not the classical theories; but instead, also include a parabolic transverse shear distribution. The difference between the SLR and the latter pair is, therefore, only in the degree of geometric non-linearity assumed in the strain-displacement relations.

The Donnell cylindrical shell strainsz3 are defined by equation (3) and the kinematics of equation (1). This results in strain-displacement relations of the form of equation (2) but with strain components as defined by equations (4). The transverse shear strains are identical to the SLR approach. The von Karman flat-plate strain-displacement relations are represented as in

598 S. T DEhhIS AND A N PALAZOTTO

equations ( 3 ) and (4) with R = m.

E 6 = U 1 . 2 + U 2 , 1 + w . l W , 2

C ? = U . ~ + ~ W ~ ~ , ~ ~ ~ = $ ~ , 1 , K 1 3 = k ( W , 1 1 + $ 1 . 1 ) ( 4 4

t;: = u , 2 - I.v/R + 3 w f 2 , ~ 2 1 = $ 2 . 2 - u,?/R, ~ 2 3 = k(w,22 + $ 2 , 2 ) (4b)

(4C) &: = ~ , 2 + V , I + ~ , i w , 2 , ~ ~ l i = $ 1 , 2 + $ 2 , i - v,i/R, ~ 6 3 =k(2w, i2 + $ 1 , ~ + $2,1)

where k = - 4/3h2, x , J p = 2,4, 5,6, 7) = 0. Comparison of equations (4) and the individual strain terms of equation (2) shows many more

coupling terms in the latter. Not only does the SLR approach have non-linear middle plane (I:) strain-displacement relations but non-linear curvature strains ( f c i j ) as well. The K~ and K~~ terms illustrate some of the additional membrane bending non-linear coupling e.g. t,b1, u, of K~ The K 1 3 and ~ 2 3 terms show membrane-transverse shear coupling since e.g. ($1 + ~ 1 , ~ ) terms repres- ent a shear rotation, see equation (1). Many additional non-linear terms in 1/R appear in the circumferential strain as well, compared to the Donne11 relations of equations (4).

Constitutive equations

Consider transversely isotropic materials, i.e. those characterized by unidirectional fibres embedded in a matrix. Assuming plane stress, (03 = 0), the transverse normal strain, E ~ , is eliminated and the constitutive relations in material axes are given by

( 5 )

where the oN are elements of the second Piola-Kirchhoff stress tensor. For transverse isotropy

A = 1 - ~~~v~~

Now consider a shell that is constructed of layers of the material described above. Generally, the fibres of the kth individual layer or ply are oriented at an arbitrary angle and, therefore, the constitutive relations of equation (5) for that ply must be transformed into shell co-ordinates resulting in equation (7).

(7)

NON-LINEAR CURVATURE STRAINS 599

where the Q i j ( i , j = 1,2,6) and Qm, (m, n = 4, 5) are elements of symmetric arrays of transformed stiffnesses for the kth ply and C T . ~ and c N are functions of shell midsurface co-ordinates 5 , and shell transverse co-ordinate i.

Potential energy

The total potential energy. lip, of the cylindrical shell (where a flat plate is a special case) based upon SLR and Donnell assumptions can be expressed as a function of the strain components as shown in equation @),I6 where many of the terms vanish for a von Karman plate or Donnell shell since many of the curvature terms ( K ~ ~ ) of equation (8) are totally non-linear functions of displacement and the simpler non-linear theories consist of only linear curvature terms. The terms of equation (8) are shown explicitly in Reference 16. The elements of the elasticity arrays, A,, B i j , . . ., K j are defined in equation (9). Additionally, for symmetrically arranged laminates, the elements of the elasticity arrays associated with odd powers of [ are identically zero.

Carrying out the first variation of equation (8) results in five coupled non-linear partial differential equations in the displacement variables. For symmetric laminates, the Donnell and von Karman strain assumptions give non-linearity only in the membrane stiffness terms, i.e. those containing elements of the array, A i j . In contrast, the SLR strain assumptions give non-linear terms that contain elements from all of the elasticity arrays: i s . non-linearity exists in the curvature as well as the midsurface strains.

{c? E P Ai,j + 2 & . 7 ( ~ i I B i j + t i i 2 D i j + t i i 3 E i j

+ ~ i 4 F i j + t i i s G i j + t i i 6 H i j + t i i 7 Z i j )

+ K j ~ K i ' D i j + 2 t i j l K i z E i j + ( 2 1 i j l K i 3 + K j z K i z ) F i , ,

+ z ( k . j l K i 4 + t i j Z ~ i 3 ) G i j + ( 2 K j l t i i 5 + 2 K j z K i 4 f K i 3 K i 3 ) H i j

+ 2 ( K j i K i 6 -k K j z K i s + t i j 3 K i 4 ) l i . j + ( 2 ti.jl t i i 7 + 2 K j z K i 6

+ 2 K j s K i 5 + K j 4 K i 4 ) J i j + 2 ( K j z t i i 7 f K j 3 K i 6

+ K j 4 K i ~ ) K i j + ( 2 K j s K i 7 + 2 t i j 4 K i 6 + K j 5 t i i 5 ) L i j

FINITE ELEMENT SOLUTION

The shell domain is discretized such that the continuum displacements are approximated by nodal values and interpolation functions. The potential energy is given for the discretized domain

600 S . T. DENNIS AND A. N. PALAZOTTO

in equation (10). Its first variation of the energy, bn,, gives the equilibrium equations, F(q) , also shown in equation (10). The symmetric arrays K , NI, and N , were defined from the potential energy expression (8) using a generalization of the approach described by Rajasekaran and Murray:24

where q is a column array of the nodal displacements, R is a column array of the nodal loads, K is an array of constant stiEness coefficients, N , is an array of stiffness coefficients that are linear in displacement, and N2 is an array of stiffness coefficients that are quadratic in displacement.

Rectangular four- and eight-noded flat and curved elements have been developed for flat plate and cylindrical shells, respectively. The flat element has 28 degrees of freedom and assumes linear distributions for the inplane displacements u and v, and the bending rotations, The remaining nodal degrees of freedom, w and its two first derivatives, are interpolated by non-conforming cubic shape functions. The curved element has 36 degrees of freedom and is similar to the flat element except that the inplane displacements are quadratically distributed, see Figure 1. A curved element with 28 degrees of freedom is also used in the following.

FULLY LINEARIZED BIFURCATION ALGORITHM

The fully linearized bifurcation approach based upon the N-notationZs was adopted for buckling solutions. In this approach we seek the load that just gives a non-positive-definite second variation of the potential energy. In linear bifurcation theory, the structure behaves linearly up to the point of instability and, therefore, a reference load, RrcEr can be scaled by a positive constant,

I "

P R \

\ Figure 1. A 36-degree-of-freedom rectangular shell element of thickness h. Corner nodes have seven degrees of freedom,

midside nodes have two degrees of freedom

NON-I.TNEAR CURVATURE STRAINS 60 1

A, where the critical load, R , , , is given by equation (1 1). Carrying out an additional variation on equation (10) and using equation (11) gives equation (12):

where gl =f(qrer) and E2 = f(qrZer), I?, Critical - loads are found by choosing a reference load, Rref, and then solving the linear problem

of Kqrer = Rret for qrcf . After substituting qrcr into the arrays f l , and flZ, A is incrementally increased until the determinant of equation (12) equals zero. The A that satisfies equation (12) is the bifurcation load, Acr. In practice, equation (1 2 ) is simplified by neglecting the higher-order terms of the array, f 1 2 . This approach is called the fully linearized approach and results in a standard generalized cigenvalue problem.26

and ,Q2 are global stiffness arrays.

NUMERICAL RESULTS

In what follows, the fully linearized bifurcation algorithm of the previous section with the SLR and Donne11 shell approaches is illustrated by several flat plate and cylindrical shell panel cases. These structural elemcnts and loadings are typical of semi-monocoque constructions and applications.

Flult plures

Consider an axially compressed, simply supported square isotropic flat plate as shown in Figure 2, where CT is the axial pressure. The boundary conditions when biaxial symmetry is valid27 are shown in equation (1 3). As can be seen, the symmetry conditions include constraints on the inplane displacements, u and t i , as well as the transverse displacements:

along .x = 4 2 : M, = IJ~ = 0

x = 0: symmetry, u = w . ~ = = 0

along s = b/2: w = = 0

s = 0: v = w.2 = *2 = 0

Convergence to the classical buckling load {Acr = 4.00) is shown for the thin plate case in Table I, where critical loads calculated for four mesh arrangements that represent one quarter of the plate are shown, Next, using the 12 x 12 mesh of 28-degree-of-freedom elements, the thickness of the plate is varied and results are generated for both simplified large rotation (E,,,,) and the von Karman ( A v K ) elements, see Tables I1 and 111, for plate aspect ratios a,h = 1 and 3, 0.2, respectively. Both can be compared to the published results of Reddy and Phan2* (ARp), who used a Navier series classical buckling solution to the parabolic transverse shear theory and also to Stein" whose results are based on a series solution to the 3-D elasticity equations. Both ;IKP and L - , , are obtained through the intermediate von Karman non-linearity.

As can be seen, the ,IVK compare very well with the Reddy and Phan results. Additionally, note that even for the very thick plate where the length of a side of the plate is only 5 times the thickness, the present A V K solution (a/b = 1) compares very well with the 3-D results. The SLR solution compares very closely with the von Karman results except for in the thicker plates where i.,,, reflects a gradually more flexible structure for each of the three aspect ratios, most significantly for a/h = 0-2. Indeed, these results are more flexible than the 3-D elasticity results

602 S. T. DENNIS AND A. N. PALAZOTTO

l x

a

b

Figure 2. Axially compressed flat plate, aspect ratio = ajh, thickness, h. One quadrant discretized when biaxial symmetry is valid

(a/h = l), thus indicating that the addition of the inplane non-linear displacement terms used in defining the strains for the SLR element have a greater influence at these thicknesses than does the inclusion of 3-D flexibility. The additional non-linear coupling between the membrane strains and transverse degrees of freedom apparently precipitates instability at a smaller load for the SLR plates. This effect is most pronounced for a/b = 0.2 and S = 5 where lsLR is 24% less than LVK.

Also shown in Table I1 are very thin plate geometries, S = a/h = 200 and 400. These results are given as an indication that at least for ranges of plate thicknesses studied herein, shear locking is at most an extremely minimal influence.

NON-LINEAR CURVATURE STRAINS 603

Table I. Convergence of buckling load for flat square isotropic plate, 28-degree-of-freedom elements, quarter plate discretization. Simple supports, w and $ parallel to plate edge re-

strained

2 x 2 4 x 4 8 x 8

12x 12

7.545 4.184 4.0 12 4.002

Notes: E = 1Q x lo6 psi, Y = 0.3, h = 0.08 in, a = b = Sin

u h a z

i72D *Classical A,, = = 4.00

Table 11. Flat square isotropic plate buckling for various plate thickness ratios, S

400 4.032 4.032 200 44x39 4009 4.00 100 4.002 4~000 3.9977 490 50 3.993 3986 3.9909 20 3.945 3.905 3.9443 10 3.786 3.649 3.7865 3.75

5 3.265 2.926 3.2653 3.20

Table 111. Isotropic flat plate, aspect ratios, ulb = 3 and 0.2

a/b = 3 alb = 0.2

100 26.87 26.69 26.843 50 4.020 4.014 3.991 26.27 25.61 26.270 20 3.954 3.914 3.944 10 3.789 3.652 3.787 15.64 12.38 15.658

5 3.263 2.924 3.265 7.043 5-32 7.053

Similar solutions are presented for simply supported square laminated flat plates where the published results of Phan and ReddyZ9 are used for comparison. They present a displacement- based finite element solution for buckling loads based on a ‘classical’ bifurcation approach that includes parabolic transverse shear distribution and von Karman non-linearity. Chang and ChenZ6 explain the differences between classical buckling and the fully linearized approach used here. In the first case, a [0/ 45/90], laminate, the two sets of boundary conditions shown in equation (14) were applied. The material properties are typical of a high stiffness advanced

604 S. T. DENNIS AND A. N. PALAZOTTO

Table 1V. Buckling parameters for [0/9Ols laminate for various El / E 2 , 8 x 8 mesh of 28-degree-of-freedom elements, R = oa2/E, h2 , simply sup- ported. E, varied, E , = 1.5 x 10' psi, G,, = G I , = 0.9 x lo6 psi,

G 2 3 = 075 x lo6 psi, v I 2 = 0.25

3 5.404 5153 5-1 143 5.2944 5,7538 10 9.935 9.522 9.7740 9.7621 11.492 20 15.27 1 14.137 15.298 15,019 19.712 30 19.628 19.037 19.957 19.304 27.936 40 23.278 22662 23-340 22.881 36.160

composites, see caption of Table IV with E 1 / E z = 40. The two sets of boundary conditions are identical except in the second, the normal inplane displacement along s = b/2 (see Figure 2) is additionally restrained. As a result of the boundary conditions used for the simply supported isotropic plate in the previous case and the subsequent close correlation of those results with established results based upon classical buckling, those same boundary conditions were used to represent the simple supports for this case as well. The additional boundary condition places a constraint on the normal inplane displacement u along s = k b/2 and, therefore, will cause a compressive stress field throughout the plate in the s direction since free movement has been restrained at the boundary. The classical buckling approach assumes a constant compressive load only in the x direction for this case and, generally, does not consider non-uniform prebuckled stress fields:

along x = u/2, - 4 2 : w = $2 = 0

along s = b/2, -6/2: w = = 0 or z ' = w = $ 1 - - 0

The classical thin plate solution is reproduced well (Phan and Reddy result,29 ,IPR = 42.8 versus the present, = 43.2) using the first boundary condition set. as was applied in the isotropic case. However, the biaxial stress field due to the additional compressive stresses in the s direction from the v boundary condition and the compressive applied loading in the x direction gives a buckling load for the present approach that is significantly less than the classical (AvK = 32.7). The additional compressive stresses due to the t' boundary condition result in a more flexible structure a$ would be expected and have a major influence on the buckling load.

Results for [0/90], laminates are given in Table IV, where S = a/h = 10 and various degrees of orthotropy, i.e. E l / E z , are examined. Also shown are solutions due to Phanz9 (ApR) and Noor' (j.j-D). The classical laminated plate theory (ACLPT) results are taken from Reference 29.

For all E 1 / E z ratios, the comparisons between either of the present results, ivK or IVSLR, with either IupR or A 3 are very close. Comparing the results from any of the values of the first four columns, i.e. those buckling loads resulting from theories that include transverse effects, versus the results from the classical laminated plate theory (LLPT), we see that for increasing E1/EZ, the buckling parameters diverge. Therefore, for laminates constructed of highly orthotropic mater- ials. i.e. high ratios of E 1 / E 2 , the influence of transverse shear deformation becomes increasingly more pronounced as evidenced by the increasingly smaller buckling loads compared to the thin plate solution, A,,,,. A comparison of %VK versus shows that the latter are, once again, indicative of a slightly more flexible structure. Comparing i,,, and ASLR versus the 3-D elasticity result, i3+, the former pair enclose the 3-D values for all values of El/&, This also occurred in

NON-LINEAR CURVATURE STRAINS 605

the isotropic case, see TableII. As was also seen in the isotropic flat plate, the inclusion of coupling in the non-linear curvature strain-displacement terms that define the SLR element apparently has a greater influence on the buckling load than including 3-D flexibility to the intermediate non-linearity that defines a von Karman plate.

Next, non-dimensionalize the buckling load as shown in equation (15) and allow the plate to become thick. These results are shown in Table V for a square [O/90ls laminate using the material properties of Table IV with E , / E 2 = 40. The classical thin-plate result (ACLT) is due to Reddy and Phan,” who use an alternative non-dimensionalization. The simplified large rotation approach again gives increasingly more flexible buckling loads for thicker geometries. For S = u/h = 5, i,,,, is 19% less than AVK:

Cylinclricul .shell punel

Consider an isotropic clamped cylindrical shell panel of square planform subjected to uniform radial pressure, see Figure 3. The critical buckling pressure, q, for several mesh refinements are shown in Table VI for the SLR equations of equation (2). Both the 28 degree-of-freedom element (L) and the 36-degree-of-freedom element (Q) were tested. The results show very good correlation

Table V. [0/90], square simply supported plate, iCLT = 3-028

200 100 50 20 10 5

3.054 3.051 3.038 3.037 2.986 2,979 2,613 2.643 1.971 1.919 1.013 0,8249

Figure 3. Cylindrical shell panel of thickness h, subjected to uniform transverse pressure, qo. One quadrant discretized due to biaxial symmetry. All geometries are square planform

606 S. T. DENNIS AND A. N. PALAZOTTO

Table VI. Clamped cylindrical shell buckling, in psi. From Reference 26, q,, = 0285 psi

4 x 4 Q 4 x S Q 8 x S Q 4X4L 8X8L 12 x 12L

0-396 0-323 0.290 0423 0.298 0.2856

Notes: E = 4.5 x lo5 psi, v = 0.3, L = 20 in, R = 100 in, h = 0,125 in

with the published result26 and, therefore, validate the cylindrical shell fully linearized buckling algorithm. The Donnell element based on equations (3) and (4) gives nearly identical results compared to the SLR element and these are not shown.

Again, transverse shear deformation can be examined in this geometry by assuming the non-dimensionalization given by equation (16). This non-dimensionalization is that used for a completely closed cylinder subjected to external pre~sure.~’ If the shell geometrical ‘Batdorf’ parameter, Z, of equation (17) is held constant, then equation (16) represents a valid non- dimensionalization. Sobel et al.jo suggest a similar parameter for buckling of isotropic shell panels that are subjected to axial compression. Therefore, thicker shells can be examined by holding Rh and L constant. For a value of Z, a thickness ratio R/h is chosen. This will f ix the ratio L/h. Similarly, if a ratio Llh is chosen, again for a given Z , then R / h is fixed.

Tables VIT and VIlI give isotropic clamped panel critical buckling loads for two values of Z based on the Donnell (A,) and SLR (ASLR) strain-displacement relations. Table VTI clearly demonstrates the non-dimensionalization as 2, and ASLR remain nearly constant for the wide range of thickness ratios R/h or L / h that represent thin shell geometries shown. As these ratios indicate a thick shell, the critical buckling load decreases where for Z = 100, R/h = 5, AD is 23% less than the thin shell case. The increased flexibility is probably due to transverse shear deformation. (Note: non-linear analyses for Z = 100, both thin and thick geometries, indicate that compressive hoop stress near the critical buckling pressure-normalized by h/Rq-are similar, perhaps ruling out differences in bending-membrane coupling as a counterargument.) The A D and As,, are nearly identical for curvature Z = 30.5 of Table VII. However, for the deeper panel Z = 100 of Table VTII, AsLR indicates a stiffer response for the thicker panels.

Laminated [0/90], and [90/0], cross-ply clamped panel critical pressures are given in Table IX. The material properties of Table IV with E , / E 2 = 40 are again used. The results show that the fibres parallel to the shell’s curvature, i.e. the 90” fibres, resist buckling for this geometry as DZ2 is larger for the layup with the larger critical pressures. The transverse shear effect is much more pronounced for the laminated shell than for the isotropic. Additionally, for this case, the SLR

NON-LINEAR CURVATURE STRAINS 607

Table VII. Isotropic non-dimensionalized clamped cylindrical shell buckling, Donnell equations, Batdorf parameter, Z = 305,

12 x 12L mesh

Z = 30.5

800 160 14.39 14.39 450 120 14.34 14.34 200 80 14.27 14.26 112.5 60 14.17 14.16 28.13 30 13.60 13.58 12.5 20 12.76 12.76 7.03 15 11.70 11.80

Table VIII. Isotropic non-dimensionalized clamped cylindrical shell buckling, Donnell equations, Batdorf parameter, Z = 100,

4 x 16Q mesh

z = 1 0 0

160 130 22.40 22.60 80 91.6 2 1 -40 2 I .82 20 45.8 19.63 2 1.05 10 32.4 18.15 20.84 5 22.9 15-68 20.60

Table IX. Laminated non-dimensionalized clamped cylindrical shell buckling, Donnell and SLR equations, Batdorf parameter, Z = 32,

4 x 16Q mesh

800 160 23.28 23.28 7,349 7.352 450 120 22.89 22.89 7.185 7.192 200 80 21.79 21.79 6.956 6.971 112.5 60 20.49 2 0 5 1 6.772 6.793 50 40 17.46 17.55 6247 6.305 28 30 14.35 14.56 5.626 5.728

7 15 6.045 5.994 3.421 3.729

critical pressures are increasingly more stiff for thicker panels. This contrasts with that shown for the axially compressed flat plates and could be a result of the tensile field developed longitud- inally. Lastly, Table X show cross-ply and unidirectional laminated panels for 2 = 100. As was seen for the isotropic panel, a stiffer response is seen for the SLR results,

608 S. T. DENNIS AND A. N. PALAZOTTO

Table X. Laminated non-dimensionalized clamped cylindrical shell buckling, Donnell and SLR equations, Batdorf parameter, Z = 100,

4 x 16Q mesh

160 126 30.94 3 1.27 6.797 6-824 80 89.4 29.03 2967 6.554 6602 20 44.7 21.20 23.37 5,937 6.1 11 10 31.6 15.21 18.70 5.365 5.689 5 22.4 9.151 13.8 1 4.498 5.064

CONCLUSIONS

Simplified large rotation assumptions for the shell strain-displacement relations lead to many additional coupling terms between membrane and transverse shear strains and give isotropic and laminated flat-plate buckling loads less than that from von Karman plate equations for very thick geometries ( S < 10). Increased flexibility from transverse shear deformation is seen for a cylin- drical shell panel. The SLR assumptions give higher buckling loads for the panel compared to Donnell assumptions, most noticeably for deeper panels.

REFERENCES

I. A. K. Noor, ‘Stability of multilayered composite plates’, Fibre Sci. Tech., 8, 81-89 (1975). 2. A. K. Noor and J. M. Peters, ‘Buckling and postbuckling analyses of laminated anisotropic structures’, Int . j . numer.

3. B. Geier and K. Rohwer. ‘On the analysis of the buckling behaviour of laminated composite plates and shells’, Int.

4. Y. Basar and Y. Ding, ‘Finite rotation elements for the non!inear analysis of thin shell structures’, Int. J . Solids Struct.,

5. S. T. Dennis and A. N. Palazotto. ‘Static response of a cylindrical composite panel with cutouts using a geometrically

6. 3. N. Reddy, E. J. Barbero and J. L. Teply, ‘A plate bending element based on a generalized laminate plate theory’, Inr.

7. E. J. Barbero, J. N. Reddy and J. Teply, ‘An accuratc determination of stresses in thick laminates using a generalized

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