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Compufers & Srnrcrures Vol. 62, No. 2, pp. 231-242, 1997 Copyright 0 1996 Elsevier science Ltd

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MULTIOBJECTIVE DESIGN OF LAMINATED CYLINDRICAL SHELLS FOR MAXIMUM TORSIONAL AND AXIAL

BUCKLING LOADS

M. Walker,? T. Reisst and S. Adalit tCenter for Engineering Research, Technikon Natal, Durban, South Africa

SUniversity of Natal, Durban, South Africa

(Received 6 July 1995)

Abstract-The multiobjective design of a symmetrically laminated shell is obtained with the objectives defined as the maximization of the axial and torsional buckling loads. The ply angle is taken as the optimizing variable and the performance index is formulated as the weighted sum of individual objectives in order to obtain Pareto optimal solutions of the design problem. Single objective design results are obtained and compared with the multiobjective design. The effect of weighting factors on the optimal design is investigated. Results are given illustrating the dependence of the optimal fibre angle and performance index on the cylinder length, radius and wall thickness. Copyright 0 1996 Elsevier Science Ltd.

INTRODUCTION

When thin cylindrical shells are subjected to axial or torsional loading conditions, they may fail by buckling. Since these structures have extensive practical applications, consideration of this type of failure for their design is important.

An advantage of composite materials over conventional ones is the possibility of tailoring their properties to the specific requirement of a given application. For symmetrically laminated composite structures, the designer can select the layer fibre angle as his optimizing variable to maximize a given objective function. When these structures are subjected to multiple buckling loading conditions such as axial compressive and torsional loadings a multiobjective design approach can be formulated in order to ensure an optimally designed structure.

Various researchers have considered the design of thin lanhinated cylindrical shells. Early studies include Sherrer[ 1] who presented a theoretical elastic solution for filament wound cylinders with any number of layers and for any loading conditions; and Reuter[2], who analyzed laminated alternate-ply cylindrical shells using classical laminated shell theory. Optimal design of these structures has been considered by several authors, using analytical or numerical methods. Hu[3] investigated the influence of shell length and thickness on the optimal layer fibre angle. Onoda[4], and Tripathy and Rao[S], using the finite element method, considered optimal layups for laminated shells under axial buckling loads.

Multiobjective design studies of composite struc- tures include those by Adali et a1.[6], Sun and

Hansen[7] and Tennyson and Hanse[l], who studied the optimal design of laminated cylindrical shells under torsional, axial, external and internal pressure loadings. Kumar and Tauchert[9], and Rao et a/.(10] studied multiobjective designs of various composite structures. Shape and material optimisation was investigated by Saravanos and Chamis[ 111.

In this study, the multiobjective optimal design of simply supported symmetrically laminated cylindrical shells for a weighted combination of maximum axial and torsional buckling loads is investigated. The layer fibre angle is chosen as the optimizing variable, and the effects of cylinder length, radius, wall thickness and loading weighting are numerically investigated.

BASIC EQUATIONS

Consider a laminated circular cylindrical shell of length L, radius R, and total thickness H under torsional load T,, and axial load N, as shown in Fig. 1. The shell has a symmetric layup consisting of K layers of equal thickness t. The structure is referenced in an orthogonal coordinate system (x,y,z), where x is the longitudinal, y the circumfer- ential, and z the radial direction. The displacement components u, u and w lie in the x, y and z directions, respectively.

The fibre angle is defined as the angle between the fibre direction and the longitudinal (x) axis. The fibre orientations are symmetric with respect to the mid-plane of the shell and are given by & = (- ly + ‘8 fork<K/2and&=(-l)r0fork>K/2+l,where k=l ,...,K with k being the layer number.

231

238 M. Walker et al

L- - I I , -x \

<

Fig. I. Geometry and loading of cylindrical shell

The shell is subjected to either a compressive torsional load T,, or axial load N, leading to failure by buckling. These loads act at different times during the service life of the shell and as such should be taken into account separately. Next the loading is described explicitly.

Torsional buckling loud

For the shell, the critical torsional buckling load is given as[l2]

subject to

Axial buckling load

The equilibrium equations for the shell under the load N, are given by[12]

AHU,,, + A&,n + w,,/R) + A&,,v + u,w) = 0

A&,,, + u,,v) + AIS,,, + A&,r, + u,?,) = 0

DI, w,,.w + 2(D12 + 2D&,.\,,, + Dzzw,,~,

+[A,zu,r + A22(u,r + w/R)]/R=N,W,,,, (3)

where

D,, = ; f: @(t: - t:~ ,) (4) Ir=l

are the extensional and bending stiffnesses, repec- tively, and oi? are components of the transformed reduced stiffness matrix for the kth layer. Due to the midsurface symmetry of the shell, the bendingexten- sion coupling matrix does not appear in eqn (3). For a simply supported shell the boundary conditions are given as

w = 0, r=O at .x=O,L. (5)

The solution to the system of eqns (3) which satisfies the boundary conditions (5) is obtained by taking the displacements in the form

u = u,,,cos(l,x)sin(i,,y)

D = u,.sin(Lx)sin(l,y)

w = w,,,,cos(L,x)sin(&), (6)

where I, = rnz/L, A,, = n/R, and u,,,,, v,,, and w,,,,, are the amplitudes of the displacement components, and m and n are numbers of half waves in the buckle pattern in the axial and circumferential direction, respectively. The buckling load N, corresponding to these wave numbers is obtained as an eigenvalue of

Multiobjective design of laminated cylindrical shells 239

go _, .._. ._.___._ ._._., ,. ., ,.., ,, ,......

LX = 1 (Maximum T,)

60 -

70 -

30 -

20 -

5

lo L(m) l5

Fig. 2. &., vs L with R = lm (single objective design).

the linear system of equations obtained by substitut- ing eqn (6) and eqn (3) and is given by

Cl, Cl2 Cl,

I I

CZI c22 cn

CM C32 C33

Cl, Cl2 ’ (7)

I I c2, c22

where

CI, = A,& + Ad.’

C22 = A221.2 + A&,

c,, = D,,I::+2(Dn + 20&~~,2

-+- 0221: + AZ/R’

C12 = C2, = (A12 + A&m~~

CP = C,, = AdmlR

C23 = C32 = A22lJR. (8)

L Cm>

go _

a-1 .!a-

20.““““““““’ 1 2 3 4 5 6 7 6 9 10

R cm>

Fig. 4. O,,, vs R with L = 15m (single objective design).

The critical buckling load N,,(e) is calculated from

NC, = minN&n,n;0). (9) J%”

OPTIMAL DESIGN PROBLEM

The present study is concerned with the multiobjec- tive optimization of the laminated shell by determin- ing the Pareto optimal value of the design variable 8. The objectives of the design involve the maximization of the buckling loads T,, and N,. In general, these objectives conflict with each other necessitating a multiobjective formulation. With this situation in mind, the performance index J(cQI;e) of the design problem is specified as

where a, /I > 0, a + fi > 0 are the weighting factors, and the To and No denote the values of T,, and NC, at 0 = 0”. Single objective designs correspond to ~1 = 1, /I = 0 for maximum T,, and a = 0, /I = 1 for maximum N,. For a, /I > 0, the fibre orientation 0 maximising J(aJ?$) gives the Pareto optimal 8.

L/L/T.(a-0

J

1.6 -

1.7 -

1.6 ” ” a ” s. ” ” I. 1 2 3 4 5 6 7 6 9

R cm>

Fig. 3. J vs L with R = Im (single objective design). Fig. 5. J vs R with L = 15m (single objective design).

240 M. Walker et al.

o.,t 0 10 20 30 40 50 50 70 80 90

0

Fig. 6. J vs 0 for various values of n with R = Im.

Thus the design objective can be stated as

subject to

0” d H d 90

The optimization procedure involves the stages of evaluating the buckling loads r,,(0) and N,,(B) for a given 0 and improving the fibre orientation to maximize J(a,p;@ at a given set of weighting factors. Thus, the computational solutions consists of successive stages of analysis and optimization until a convergence is obtained and the Pareto optimal angle eupt is determined within a specified accuracy. In the optimization stage, the go/den section method[ 131 is employed.

NUMERICAL RESULTS

The results reported in this section are for eight-layered symmetrically laminated cylinders with simply supported ends. The material properties used in the design are those of T300/5208 graphite epoxy

““I 60

,‘.. _‘\\

a=2/3 \__* ..,

__-_ ' ‘\>.,'

_---\<___-------

70 <,

k, ,\ , '._ f -\,

\a _\

60 \; ., a=1/2

"\

50 “\

‘,

'_ :..., ; \ I.

40 - '. '_ ; a=1/3

'\ I . .

30 r ,\:

5 10 15 20

L Cd

Fig. 7. &,t vs L with R = lm (multiobjective design).

a =2/3 ------'--___.__.__

L (ml Fig. 8. J vs L with R = lm (multiobjective design).

given by El = 181 GPa, E, = 10.34 GPa, El2 = 7.17 GPa and V, = 0.28. In the numerical results, the weighting factors are constrained as a + /I = 1 so that [j = I -x. 0 < a d 1. In Figs 2-l 3, the plate thickness is taken as H = 0.05m and the accuracy is specified as 0.1

Single objective designs

Results for the single objective designs involving the maximisation of either r,, or NC, are given first. Figure 2 shows the effect of cylinder length on the optimal fibre angles for NC, (a = 0) and maximum T,, (1 = I) with R = lm. For the axial loading case, &, generally decreases with increasing length. The optimal fibre angle for torsional loading is 90” for all L. Figure 3 shows the corresponding performance indices plotted against length L. The curves for the optimal fibre angles plotted against cylinder radius R

are shown in Fig. 4 where f. = 15m The curve is stepped due to changes in the buckling mode. Figure 5 shows the corresponding curves for the design objectives. It is observed that NC,/NO decreases as the length L or radius R of the shell increases.

Multiobjectiw designs

The dependence of the performance index J on the fibre angle is investigated in Fig. 6 for four weighting

go/

201 “‘I”’ / ” 1 ” 1.

1 2 3 4 5 6 7 8 9

R(m)

Fig. 9. U,,,, vs R with L = 15m (multiobjective design).

Multiobjective design of laminated cylindrical shells 241

1.90 1 I

.._..‘-‘.,_.-.-‘-.__,_._,

J 1.82 -

1.74 - 1.72

1.70

1 2 3 4 5 6 7 a 9 IO

R (4

Fig. 10. J vs R with L = 15m (multiobjective design).

2.1 ,

1.6'. ‘. ” ” ” ” ” ” ” ” J 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

cx

Fig. 12. J vs a (multiobjective design).

cases with a = 0 and u = 1 indicating the dependence of NC, and T,, on the fibre orientation. For this figure, the cylinder length was specified as L = 15m and the radius as R = lm for a = 1.0, the graph rises monotonically with a maximum at 90” and this is the optimal II for maximum T,,. As a decreases the graphs become less monotonic and the maximum is found at decreasing fibre angles. For a = 2/3, the optimal angle is 77.21”, for a = l/3, the optimal angle is 74.73”, and for u = 0, cop, is 24.22”, which is the optimal 8 for maximum NC,. It is interesting to note that at 54.78” the four curves intersect.

Figure 7 illustrates the effect of cylinder length on the optimal fibre angle for a cylinder of radius R = lm and h = O.OSm, for three different weight- ings. For a = l/3, the value of the optimal fibre angle lies between 68” and 83”, whereas for a = l/2, 0+ varies between 47” and 78”. The optimal fibre angles for a = 2/3 again are correspondingly less, and vary across a greater range. All three of these curves show discontinuities at various values of L, and this is due to mode changes.

Figure 8 shows the performance index J(a,~;0) corresponding to the optimal fibre angles shown in Fig. 6. For a = l/3 and a = l/2, the performance index J generally decreases with increasing cylinder length L. For cx = 2/3, the index remains fairly constant.

a

The curves for tlopl vs cylinder radius R are shown in Fig. 9 for the three weightings with L = 15m. As a decreases, the optimal angle is generally lower for all values of R. The discontinuities in the graphs of (I,,, vs R are once again due to mode changes. The corresponding curves for J (Fig. 10) show that the performance index fluctuates as a decreases. The performance index J generally decreases with increasing R for all three values of a.

Figure 11 shows the dependence of 0,, on the value of the weighting factor a for three different cases of L and R. As the proportion of the torsional load increases, viz. as a increases, so &, correspondingly approaches 90”, and for all three cases of L and R, &,, = 90” at a = 1, corresponding to maximum T,,. O,,, values for a = 0 are different for different values of L and R indicating that NC, reaches its maximum at different values of tk,,,, depending on L and R.

Figure 12 shows the corresponding values of the performance index J. In all three cases, the trends show a minimum between a = 0.65 and a = 0.8. At 0: = 1, the loading is purely torsional, and thus J = Tc,/To, which is independent of R and L [see eqn (l)]. Therefore in all three cases the performance index converges to the same value J = 2.05.

Table 1 shows the values of the performance index J and kpl for various values of H, with L = 15m and R = lm. The performance index increases with

2.05

1.85

U-0.6-24' \

5 1.55 1.75 1

Ncr/N,

Fig. 11. CL,, vs a (multiobjective design). Fig. 13. Trade-off curves of T./TO vs N-/NO.

242 M. Walker et al

Table 1. The design index J vs H with L = 15m and R = lm for different values of a

H (m) J (a = l/3) &,, J (a = 213) O,,, (a = 2/3)

0.001 1.37 88.73 1.71 0.005 1.54 78.44 1.78 0.01 1.62 76.33 1.81 84.86 0.05 1.75 74.73 1.85 83.88’ 0.1 1.88 63.68 1.84 70.83

It is observed that at single-objective designs, the other objective becomes quite low. This drawback is overcome by choosing a suitable intermediate value of the weighting c( in order to achieve the required compromise design. Studies of this type are thus essential if an optimal design for a certain shell geometry is to be obtained.

increasing H. Also, cop, generally decreases with increasing H for both values of t(.

Figure 13 shows the trade-off curve for K, and Ncr with L = 15m and R = lm. The individual loadings

N,, and T,, are evaluated separately at the optimal fibre angles t&,, corresponding to the range of weightings 0 < a < 1. The nonweighted contri- butions of these loads are plotted against each other to give the trade-off between the weightings c( = 0 andcc= I.

CONCLWION

A multiobjective design is given for simply supported laminated cylindrical shells subject to a combination of axial and torsional buckling loads. The objective is defined as the maximization of the performance index specified by the sum of the nondimensionalized weighted loadings. Results for single objective and multiobjective designs are presented. The effect of cylinder length, radius, wall thickness and weighting on the optimal fibre angle is investigated.

The mode changes which result due to the nature of the loading lead to designs whose trends are not unimodal with regard to parameters such as the length and radius. Further, it is noted that the optimal fibre angle has a large effect on the maximum buckling load and varies markedly with geometry

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REFERENCES

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