optimization of symmetric laminates for maximum buckling load including the effects of...

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OPTIMIZATION OF SYMMETRIC LAMINATES FOR

MAXIMUM BUCKLING LOAD INCLUDING THE

EFFECTS OF BENDING-TWISTING COUPLING

M. Walker,t§ S. Adali$Q: and V. Verijenkolg

+Depdrtment of Mechanical Engineering, Technikon Natal. Durban, South Africa

ZDepartment of Mechanical Engineering, University of Natal, Durban, South Africa

$ Institute of Advanced Studies, Technikon Natal, Durban. South Africa

(Received I July 1994)

Abstract-Finite element solutions are presented for the optimal design of symmetrically laminated rectangular plates subjected to a combination of simply supported, clamped and free boundary conditions. The design objective is the maximization of the biaxial buckling load by determining the fibre orientations optimally, with the effects of bending-twisting coupling taken into account. The finite element method coupled with an optimization routine is employed in analysing and optimizing the laminated plate designs. The effects of boundary conditions, the number of layers and bending-twisting coupling on the optimal ply angles and the buckling load are numerically studied.

1. INTRODUCTION

The use of laminated composite materials as structural components is becoming widespread in several branches of engineering. These structures often con- tain components which may be modelled as rectangular plates. A common type of composite plate is the symmetrically laminated angle ply configuration which avoids bending-stretching effects by virtue of mid-plane symmetry. An important failure mode for these plates is buckling under in-plane loading. The buckling resistance of the fibre composite plates can be improved by using the ply angle as a design variable and determining the optimal angles to maxi-

mize the buckling load. One phenomenon associated with symmetric angle-

ply configurations is the occurrence of bending-twisting coupling which may cause significantly different results as compared to cases in which this coupling is exactly zero [I]. The effect of bending-twisting coupling becomes even more pro- nounced for laminates with few layers. Due to this coupling, closed-form solutions cannot be obtained for any of the boundary conditions and this situation led to neglecting bending-twisting coupling in several studies involving the optimization of symmetric laminates under buckling loads[2-161. In actual fact, closed-form solutions for symmetric laminates are not available even for the simplified models where this coupling is neglected except if the boundary conditions are simply supported all around. The present study adopts a numerical approach to include the effect of bending-twisting coupling and to obtain

the optimal design solutions for a variety of boundary conditions.

Optimization of composite plates with respect to ply angles to maximize the critical buckling load is necessary to realize the full potential of fibre-re- inforced materials. Results obtained using different approaches can be found in the literature. The authors mostly dealt with either simply supported or clamped plate edges, neglecting the effect of bending-twisting coupling [2-161. These studies estab- lished that symmetric lamination yields the highest buckling load compared to other stacking sequences. However, the effects of boundary conditions and of bending-twisting coupling on the optimal designs remain mostly unknown. The influence of bending-twisting coupling on the buckling load of symmetric angle ply laminates have been investigated in Refs [17-201 with a study of the subject of Grenest- edt [21] for simply supported and clamped boundary conditions showing that the biaxial buckling load decreases when the effect is included in the analysis.

A design study of infinitely long anisotropic laminates by Nemeth [22] indicates that the twisting stiffness parameter should be increased to maximize the buckling load. A recent study by Kam and Chang [23] includes the bending-twisting effect in the design of simply supported laminates by employing a finite element approach for design analysis. A more de- tailed review of the subject can be found in Ref. [24].

As there is little reported on the optimal design of laminates with bending-twisting accounted for and with different boundary conditions, the results here aim to fill this gap. The finite element formulation

313

314 M. Walker et al.

which is used in the present study is based on Mindlin type theory for laminated composite plates. Numeri- cal results are given for various combinations of boundary conditions and optimal designs with and without bending-twisting coupling are compared.

2. BUCKLING OF SYMMETRIC LAMINATES

Consider a symmetrically laminated rectangular plate of length a, width b and thickness h which consists of n orthotropic layers with fibre angles Ok, k = 1,2, , K, as shown in Fig. I. The plate is defined in the Cartesian coordinates x, y and z with axes x and y lying on the middle surface of the plate. The plate is subjected to biaxial compressive forces N, and NJ in the x and Y directions, respectively, as shown in Fig. 1. Plates with these characteristics are commonly known as symmetric angle-ply laminates.

In the present study, a first-order shear deformable theory is employed to analyse the problem and the following displacement field is assumed:

u = u,(x, Y) + zti/,(x, Y)

u = &Ax, Y) + z$, (x3 Y)

w = w(x, YX (1)

where rlO, vO and wO are the displacements of the reference surface in the x, Y and z direction, respectively, and $,, $, are the rotations of the transverse normal about the x and Y axes.

The in-plane strain components can be written as a sum of the extensional and flexural parts and they are given as

where

and

[S]=a 2 5 hh lcl(k, dz k=l hk ,

[D] = f rhr [&z2dz. Fig. 1. Geometry and loading of the laminated plate. k=IJhk ~I

Here, a subscript after the comma denotes differen- tiation with respect to the variable following the comma.

The transverse shear strains are obtained from

(4)

The equations for in-plane stresses of the kth layer under a plane stress state may be written as

and similarly for the transverse shear stresses as

where a is a shear correction factor [25], and Qij are the transformed stiffnesses.

Equations (5) and (6) may be written in compact form as

ok = Qkt, (7)

where Qk refers to the full matrix with elements

(0, )k, 7 and at and e represent in-plane and transverse stresses and strains, respectively. The resulting shear forces and moments acting on the plate are obtained by integrating the stresses through the laminate thickness, viz.

s h/2

{q’=(v,, V,)= (7.x;, t,.,) dz -hi2

hi2 {MJT = (4, MT, KY) = s (a,, u,., u,.)z dz. -h/2

(8)

The relations between V and M, and the strains are given by

where the stiffness matrices [S] and [D] are computed from

(10)

Optimization of symmetric laminates for maximum buckling load 315

From the condition that the potential energy of the plate is stationary at equilibrium, and neglecting the pre-buckling effects, the equations governing the biaxial buckling of the shear deformable laminate are obtained as

M,.,. + My,,., - Y, = 0, (11)

where N, and N? are the pre-buckling stress components which are shown in Fig. 1. As no simplifica- tions are assumed on the elements of the [D] matrix, eqn (11) includes the bending-twisting coupling as exhibited by virtue of D,, # 0, D,, # 0.

3. FINITE ELEMENT FORMULATION

We now consider the finite element formulation of the problem. Let the region S of the plate be divided into n sub-regions S, (S,E S; r = 1,2,. . . , n) such that

l-I(u) = E II”(u), (12) r=l

where II and IIs’ are potential energies of the plate and the element, respectively, and II is the displacement vector. Using the same shape functions associated with node I’ (i = 1, 2, . . , n), S,(x, y), for interpolating the variables in each element, we can write

” = li w%Yh, (13) i=l

where ui is the value of the displacement vector corresponding to node i and is given by

u = {I@, o{‘, wb”, +I”, +‘,!I}‘. (14)

The static buckling problem reduces to a general- ized eigenvalue problem of the conventional form, viz.

WI + 4&l){uI = 0, (15)

where [K] is the stiffness matrix and [KG] is the initial stress matrix. The lowest eigenvalue of the homo- geneous system [eqn (15)] yields the buckling load.

4. OPTIMAL DESIGN PROBLEM

The objective of the design problem is to maximize the buckling loads N, and h$ for a given thickness h by optimally determining the fiber orientations given by &=(-1)‘+‘0 for k<K/2 and Ok=(-l)kO for

k>K/2+1. Let N,=N and N!=A-N where 0 Q ), < 1 is the proportionality constant. The buckling load N(B) is given by

N(B) = min,.,[N,,(m, n, 611, (16)

where Nmn is the buckling load corresponding to the half-wave numbers m and n in the x and y directions, respectively. The design objective is to maximise N(B) with respect to 8, viz.

N,,,,, & max [N(e)], 0” < 8 < 90”, (17) 0

where N(B) is determined from the finite element solution of the eigenvalue problem given by eqn (16). The optimization procedure involves the stages of evaluating the buckling load N(B) for a given 0 and improving the fibre orientation to maximize N. Thus, the computational solution consists of successive stages of analysis and optimization until a conver- gence is obtained and the optimal angle eopt is determined within a specified accuracy. In the optimization stage, the Golden Section [26] method is employed.

5. NUMERICAL RESULTS AND DISCUSSION

5.1. VeriJication

In order to verify the finite element formulation described above, some solutions are compared with those available in the literature. A single-layered simply supported square plate was modelled with 0 = 30”, 2 = 0 (uniaxial compression) and material properties E, = 60.7 GPa, El = 24.8 GPa, G,, = 12 GPa and v,* = 0.23. The analytical solution for this problem is available in Ref. [27]. Table 1 illustrates the effect of the number of finite elements on the non-dimensionalized buckling load N,, where

N,a2 Nb = - E,h3

D, and D, =

12(1 - h2V21)

The plate thickness ratio is specified as h/b = 0.01 m. The use of 256 elements for a square plate resulted in an error of less than 0.08% as compared to the analytical solutions. This mesh density was accepted as providing sufficient accuracy. Consequently, in the present study, a square plate is meshed with 256 elements. Plates of aspect ratios

Table 1. Effect of the number of elements of the buckling load

Number of elements N,

10 x 10 25.57 13 x 13 25.33 16 x 16 25.22 20 x 20 25.14

Exact (Ref. [27]) 25.20


- 55.85 ---_.-- FSM ----- F_s.C.S --- c.s.cs -- -,- - c.c.c.c

Fig. 2. Buckling load plotted against the ply angle for Fig. 5. Maximum buckling load plotted against the aspect rectangular laminates with a/b = 0.5. ratio with E. = 0 (uniaxial load).

Fig. 3. Buckling load plotted against the ply angle for square plates.

other than one are meshed with a corresponding proportion of 256 elements.

5.2. Numerical results

Numerical results are given for a typical T300/5208

graphite/epoxy material with E, = 181 GPa,

El= 10.3 GPa, G,: = 7.17GPa and viz= 0.28. The symmetric plate is constructed of four equal thickness layers with 0, = - O2 = - 8, = 0, = 0 and as before, the thickness ratio is specified as h/b = 0.01. Different

combinations of free (F), simply supported (S) and clamped (C) boundary conditions are implemented at

go(

I

8’ Is’ !

, 1 .50 0.75 1.00 1.25 1.50 1.75 2.00

4

Fig. 4. Optimal ply angle plotted against the aspect ratio with ,I =0 (uniaxial load).

Fig. 6. Optimal ply angle plotted against the aspect ratio with i. = 0.5 (biaxial load).

the four edges of the plate. In particuIar, five different combinations are studied, namely (F, S, F, S),

(F, s, c, S), (& s, s, S), (C, X C, S) and

(C, C, C, C), where the first letter refers to the first plate edge and the others follow in an anti-clockwise direction as shown in Fig. 1.

For each of the boundary conditions, three in- plane load cases are considered, namely, uniaxial compression (N,. = 0), biaxial compression with N,./N, = 0.5 and biaxial compression with N,./N, = 1.

The results presented in this section are obtained

for rectangular plates with aspect ratios varying

1500 I \

a/b

Fig. 7. Maximum buckling load plotted against the aspect ratio with A = 0.5 (biaxial load).


15

8.h I

0.75 1.00 1.25 1.50 1 .75 2.00

o/b Fig. 8. Optimal ply angle plotted against the aspect ratio

with E. = I (biaxial load).

between 0.5 and 2. The non-dimensionalized buckling parameter Nh is defined as

Nb=$ 0

(18)

where N is the critical buckling load and E, is a reference value having the dimension of Young’s modulus and is taken as E, = 1 GPa.

The dependence of the buckling load Nb on the fibre angle is investigated for the five cases of boundary conditions in Fig. 2 for r = 0.5, and in Fig. 3 for r = 1. With r = 0.5 the maximum buckling load occurs at 0” for all the boundary cases, but this is not

so with Y = 1. It is clear that the maximum buckling load for a given boundary condition and aspect ratio occurs at a specific value of the fibre angle (referred to as the optimal fibre angle) and this value can be several times higher than the buckling load at other fibre angles. This fact emphasizes the importance of carrying out optimization in design work of this nature to obtain the best performance of fibre composite plates.

Figure 4 shows the effect of the plate aspect ratio r = a/b on eopt for the five cases of boundary conditions for plates under uniaxial loading. In the case of (F, S, F, S), the optimal fibre angle is 0’ for 0.5 <r ,< 2. The case (F, S, C, S) is interesting

500 / Y, I

a/b Fig. 9. Maximum buckling load plotted against the aspect

ratio with 1 = I (biaxial load).

“-.

a/b

..

0 0.50 0.75 1.00 1.25 1.50 1.75 2.00

Fig. 10. EKecl of bending twisting coupling on the optimal ply angle.

because the optimal angle remains 0’ between r = 0.5

and 1.79 at which point eupf jumps to 47.5 For the case (S, S, S. S), the jump in the optimal ply angle occurs at r = 0.8. With (C, S. C, S), Oop, displays several jumps which occur at r = 0.67, 1.27 and 1.88. Finally, for (C, C, C, C), cop, is non-zero for r > 1.2

after which the optimal ply angle fluctuates between 29.5’ and 47

It is noted that the discontinuities that occur in O,,,

as the aspect ratio increases from 0.5 to 2 are due to changes in the buckling modes.

The values of the maximum buckling load N,

corresponding to the optimal ply angles given in Fig. 4 are shown in Fig. 5. As expected, the clamped plate gives the highest buckling loads.

The results for biaxial loading with N,,/N, = 0.5 are

given in Fig. 6. In this case, an interesting situation occurs with discontinuities for all the boundary conditions. For (F, S, F, S), the relationship between I and U,,, is worth noting. At r = 1.33. a,,, displays a discontinuity jump to 23 , whereupon it remains flat up to r = 2. The trends for the cases (C, S. C. S) and

(C. C, C, C) show similarities to those shown in Fig. 4, although the number of jumps for the case (C, C, C, C) increases. Figure 7 shows the values of N, corresponding to flop, shown in Fig. 6.

The results for the second biaxial loading case with

N,, N, = I are presented in Fig. 8. The case (F, S, F, S) does not display any discontinuity and 0 0pc remains between 20 and 31 The cases (C. S, C, S) and (C. C. C, C) show similar trends to those illustrated in Figs 4 and 6. The plates with

(F, S, C, S) have an approximate O,,,, value of 27 between r = 0.5 and r = 1.43 at which point the optimal ply angle jumps to 46‘ and then increases to Bop, = 75’. Figure 9 gives the values of N, corresponding to 8,,, shown in Fig. 8. The trends are similar to those of the other loading cases shown in Figs 5 and 7, but as expected, the values for N,, are less than before due to increased loading.

Finally, Fig. 10 compares the optimal ply angles of simply supported laminates with and without bending-twisting coupling. In essence, neglecting the effect of bending-twisting coupling corresponds to having


an infinite number of layers. Figure 10 indicates that this effect is substantial (particularly around r = 0.75 and r = I .30) in the case of four-layered plates and its neglect may lead to incorrect optimal fibre orientations, resulting in substantially reduced buckling loads.

6. CONCLUSION

A finite element solution for the optimal design of laminated composite plates for maximum buckling load was presented. This formulation is based on Mindlin-type laminated plate theory. The numerical approach employed in the present study is necessi- tated by the fact that the inclusion of the bending-twisting coupling effect and the consideration of various combinations of free, clamped and simply supported boundary conditions rule out an analytical approach.

The effect of optimization on the buckling load was investigated by plotting the buckling load against the design variable (Figs 2 and 3). The results show that the difference in the buckling loads of optimal and non-optimal plates could be quite substantial, em- phasizing the importance of optimization for fibre composite structures.

It is observed that optimal fibre angles display several jump discontinuities when plotted against the aspect ratio. The present study shows that the number and location of these discontinuties caused by the changes in the buckling mode, depend on the specific boundary conditions and the biaxial loading ratio NJ/N,. In most cases the optimal fibre angle is quite sensitive to the value of the aspect ratio with (F, S, F, S) plates exhibiting the least sensitivity and (C, S, C, S) plates exhibiting the most sensitivity.

A comparison of optimal fibre angles with and without bending-twisting coupling (Fig. 10) showed that this effect cannot be neglected at certain aspect ratios. It is noted that the effect can be minimized by using a large number of layers in the laminate construction.

Acknowledgemenf-Mark Walker would like to thank Peter Metcalf for his help in writing some of the software used to produce the results presented in this paper.

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