Optimal design of laminated composite plates using a global optimization technique

Composite Structures 19 (1991 ) 351-370

Optimal Design of Laminated Composite Plates Using a Global Optimization Technique T. Y. Kam* & J. A.

Snyman

Department of MechanicalEngineering,Universityof Pretoria, Pretoria 0001, Republic of South Africa ABSTRACT A multi-start global optimization technique is used to investigate the lamination arrangements of laminated composite plates designed for maximum stiffness. The multi-start global optimization technique which originated from the concept of minimizing the potential energy of a moving particle in a conservative force fieM is extended to the optimal design of laminated composite plates in which the strain energies of the plates are minimized. The optimization algorithm has been proved to be efficient and effective in producing the global optima. Numerical examples of the selection of optimal lamination arrangements of symmetrically laminated composite plates with different aspect ratios subject to different loading conditions are given. The results show that aspect ratio, loading condition and material property can affect the optimal lamination arrangement. INTRODUCTION

With the increasing use of laminated composite materials in the mechanical, aerospace, marine, and other branches of engineering, the optimal design of laminated composite structures has attracted close attention in recent years. In particular, the optimal design of laminated composite plates has been a subject of research for many years. The usual objective in the optimal design of laminated composite plates is to design layer orientations, layer thicknesses or number of layers which will give the minimum weight of the plate and satisfy the imposed constraints. A selected list of some of the literature published in this area is given in Refs 1-6. Instead of using structure weight as the objective function, some researchers have chosen other objective functions of *Correspondence address: Dept of Mechanical Engineering,National Chao Tung University, 1001 Ta Hsueh Road, Hsinchu,Taiwan. 351 Composite Structures 0263-8223/91/S03.50 O 1991 Elsevier Science Publishers Ltd, England. Printed in Great Britain

352

T. Y. Kam, J. A. Snyman

interest in their investigations. Several papers 7-11 were devoted to the design of laminated composite plates for maximum buckling strength. Tauchert and Adibhatla 12,13 investigated the lamination arrangements for plates to yield maximum bending stiffnesses or bending strengths. Although a substantial amount of effort has been devoted to this area, as indicated by the extensive literature published on the subject, a vast proportion of the published work is limited to simple plates consisting of a very few layers. For instance, the results in Refs 12 and 13 are only available for symmetrically laminated composite plates composed of four layers and subjected to concentrated loads. As is well known, laminated composite plates may be made up of many layers of different orientations and even a relatively simple composite plate may possess many design variables. The increase in the number of design variables when coupled with the highly nonlinear way in which strains and deflections vary with changes in fibre orientation can result in great difficulties in obtaining convergence to a local minimum when conventional optimization techniques, used by previous researchers, are employed. Furthermore, it appears to be extremely expensive if not intractable to find the global optimum using the conventional optimization techniques. For these reasons the aforementioned work on the optimal design of laminated composite plates were restricted to fairly simply cases and the results have therefore found only limited applications in practical design. It is also obvious that if a broader application of optimal design in laminated composite structures is desired, more efficient and reliable global optimization techniques must be used. In recent years, a few global optimization techniques have been proposed. 14-18Amongst them the multi-trajectory global minimization technique proposed by Snyman and Fatti TM has proved to be one of the most efficient and effective global minimization techniques to d a t e . 19 In this paper the multi-start global minimization technique is extended to the optimal design of laminated composite plates. The objective of the optimal design is to select lamination arrangements so as to maximize the flexural stiffness of laminated composite plates. The present objective function has been used in ref. 12 in which only two design variables were considered for plates with aspect ratios other than one and the plates were subjected to simple loadings. In this study optimal lamination arrangements of plates composed of many layers and subjected to different loading conditions are investigated. Numerical examples are given to demonstrate the efficiency and effectiveness of the present optimization technique. The results presented here should be valuable to practicing engineers as well as researchers working in this area.

Optimal design of laminated compositeplates

3 53

PROBLEM FORMULATION The objective in the optimal design of a laminated composite plate is the selection of the lamination arrangement which gives the maximum stiffness of the plate. The optimality criterion used here is that of minimum strain energy which is equivalent to the criterion of maximum stiffness.12 In mathematical form the optimization problem is:

mininfrze

U(O)

(1)

subject to

0°< Oi~ 180 °

(2)

NL

Z ti=t

(3)

i=1

where U is the strain energy of the plate, 0 = (01, 02,..., 0NL)Tthe vector of ply orientations, ti the thickness of the ith ply, t the total thickness of the plate and NL the number of layers. For simplicity only the optimal design of symmetrically laminated composite plates is considered here. The moment-curvature relations of the plate in Fig. 1 may be expressed in the matrix form:

M--D2

(4)

M= My

(5)

where the vector

contains the bending and twisting moments per unit length, and

0y 2

2

(6)

2 azw Ox i)y

gives the corresponding curvatures for the classical plate theory. Here w denotes the transverse deflection of the plate middle surface. The

354

T.Y. Kam, J. A. Snyman z

TQ

vrlqq

x

Fig. 1.

Geometry of symmetrically laminated-plate.

coefficients of the bending stiffness matrix D in eqn (4) can be written as: 20

Dij=k=lZ(Qij)k -

-2

tkZk + 121

(i,j = 1,2,6)

(7)

where t k is the thickness and Zk =½(Zk+Zk_ l ) is the distance to the centroid of the kth ply; (Qij)k is the transformed reduced stiffness, which can be defined in terms of the ply angle Ok and the elastic constants El, E2, v12, VZl and G12 of the orthotropic layer. The transverse displacement can generally be approximated by using the Rayleigh-Ritz method. For this purpose w(x, y) is represented in the series form

w( x ,y ) = ¢rc

(8)

where ~ is a vector of functions which satisfy the kinematic boundary conditions, and c is a vector of arbitrary coefficients which are to be determined through the applications of the principle of minimum potential energy. For the case of a simply-supported rectangular plate, a suitable form for the elements of the shape-function vector ~ is ~lm_l)+, = sin

m arx . a

nary

sin--

b

m= 1,2,...,M n= 1,2,...,N

(9)

The choice M = N = 7 gives solutions based upon a total of 49 terms and were found to be in good agreement with exact solutions in various

Optimaldesignof laminatedcompositeplates

3 55

special cases involving isotropic and specially orthotropic plates. Differentiation of eqn (8) according to eqn (6) leads to the curvature expression (10)

=BC

where the 3 x 49 matrix B is given by (11)

B = [bxbyb,q]r

and the components of the vectors bx, by and b~ are

bxi= -02q~-----jl byi =-02qj-----j h .= 02~i OX 2 ' Oy2' vxY' OxOy'

i=1,2,...,49

(12)

The strain energy associated with bending of the plate is given by:2° (13) Substitution of eqn (10) into eqn (13) gives U=½crKc

(14)

where K=

BrDBdxdy

(15)

It can be shown that the derivatives of strain energy with respect to ply orientatons are OU 1 rOK OOk= - ~ c -~k c(k= 1, 2,...,NL)

(16)

The results evaluated from eqns (13) and (16) for strain energy and its gradient will be used in the global minimization technique to search for the global optimum. OPTIMIZATION PROCEDURE The optimization problem stated in eqns (1)-(3) will be solved by the multi-start global minimization technique proposed by Snyman and Fatti. 18 The search trajectories used in the minimization algorithm are

T.Y. K a m , J. A . S n y m a n

356

derived from the equation of motion of a particle in a conservative force field, where the strain energy to be minimized represents the potential energy. The trajectories are modified to ensure convergence to a local minimum, but in such a way as to increase the probability of convergence to a comparatively local minimum. In this manner, the regions of convergence of the lower minima, including in particular that of the global minimum, are increased. A Bayesian argument is adopted by which, under mild assumptions, the confidence level that the global minimum has been attained may be computed. The version of the algorithm used here is essentially the same as that described by Snyman and Fatti. 18 Trajectories are initialized at random starting points within the domain of interest defined by the constraints in eqn (2). Each individual trajectory converges to a local minimum and if the minimum value of the energy corresponds to the lowest value achieved to date the probability of it being the global minimum is computed. If a trajectory leaves the domain of interest at the point 0p where one or more of the components Opitake on values such that either Opi> zt or Opi< 0 then the constraints are imposed by continuing the trajectory at the point O'pwith components identical to Op except for the components corresponding to the violated constraints. These components are replaced as follows !

O pi ~" Opi -- m : r

if Opi ~> J[

and (17)

Opi--Opi+ mT[ if Opi
m

Here the value of m is chosen in such a way that constraints.

Opi satisfies

the

NUMERICAL EXAMPLES The application of the multi-trajectory global optimization technique is illustrated by means of several numerical examples. Consider the cases of rectangular plates constructed of equal thickness layers (t k = t / N L ) , symmetrically laminated with respect to the middle surfaces and simply supported on all edges of the plates. In the numerical examples graphite/ epoxy is used for which the material constants a r e 21 E1 = 181 E0 G12 = 7"17 E 0

E 0 = 1 '0 GPa

E 2 = 10"3 E o v12 = 0"28

Optimal design of laminated composite plates

3 57

The optimal ply orientations of plates composed of four to 16 layers are computed by the above global optimization method for various different aspect ratios. The maximum displacements of the plates are also evaluated in order to determine the optimal number of layers for the plates.

Example 1: Optimum ply orientations of centrally loaded plates The optimum ply orientations of centrally loaded plates of various aspect ratios made up of various number of layers are determined. The results of optimum ply orientations and nondimensionalized central displacements, (wc.lO3)/(QaS/Eobt3), are given in Table 1. Plots of nondimensionalized central displacement versus number of layers for plates of selected aspect ratios are given in Fig. 2. It is noted that the optimal number of layers for the plates is approximately 8.

Example 2: Optimum ply orientations of uniformly loaded plates The optimum ply orientations of uniformly loaded plates are evaluated and given in Table 2. Plots of nondimensionalized central displacement, ( wc . 103)/(qa4/E ° t 3), v e r s u s number of layers for plates of selected aspect ratios are given in Fig. 3. The optimal numbers of layers for b/a ~ 1.4 and b/a > 1.4 are, respectively, 8 and 4. It is interesting to note that irrespective of loading condition and aspect ratio we obtain multiple global optima of ply orientations as the number of plies increases. Table 3 shows some results for uniformly loaded plates of aspect ratios b/ a = 1.8 and 2.0.

Example 3: Optimum ply orientations of plates subjected to combined loads The optimum ply orientations of plates subjected to a centrally applied load and a uniformly distributed load are determined. The ratio of the uniform load to the point load, qaE/Q, is 1.0. Results for b/a-- 1.4 and 2.0 are given in Table 4. As is expected, the optimum results are bounded by those of centrally loaded plates and uniformly loaded plates, respectively.

Example 4: Reduction of displacements by using optimum design Displacements of symmetric 16-layer plates of c o m m o n l a y - u p s 22 [45, 0, - 4 5 , 902, - 4 5 , 0, 45]s and [0, - 4 5 , 0, - 4 5 , 0, 45, 0, 45]~, subject to

358

T.Y. K a m , J. A. Snyman

vI vI r:.. e ~ m

I

I

I

I

I

I

I

m

"d e~

< 0

"r2, ~'~ ~

•

"~I"

0

I

I

I

I

~

~

I

,,~

',~I" I ........

,~

'~

e~ m

.

.

.

.

.

I

----.- "-~. •~-,,,~-

.

',~-',,~-

',~,~"

I

~

I ~. ~

"'"

I

',O

I

i

u

E E

' Io oo

r~

@ .=

oo ,--~

I oO

I

"~ ~•

E E e~

0

I

.~

I

Optimal design of laminated composite plates

~,-~

~

~

I

~

I

3 59

•

e~l"~'"~

r~

0

360

T. Y. Kam, J. A. Snyman _.b

a

1.2 w

:

~

~"

103

Qa3/Eot3

1.4

--

1.a

--O-- ---4- - ~

2.0

-e- - - e - - 4

-"

~

-

JI,

6.0

5.0 4.0

a...... ~

,~

" " 0.-.-... ~ . - 0 - - - .

~. ~ 4 ~ -

_A '-0-'--

_,

.~

• --4-'-.

3.0 -.o-----o--

2.0

.-o---o----o

1.0

0

Fig. 2.

i

i

i

6

6

8

t

10

t

12

t

t

14

16

HL ( N o .

of

Plies)

Effects of number of plies upon the central displacements of centrally loaded plates of various aspect ratios.

central loads or uniform loads, are determined. The results are tabulated in Table 5 and compared with the optimum solutions obtained in previous examples. Notably reductions in central displacements of 9.4-42.5% can be achieved by using the optimum designs.

Example 5: Effect of Young's ratio upon optimum ply orientation The optimum ply orientations of plates of various Young's modulus ratios ( E l / E 2 = 5, 10, 17-573, 30 and 40) subject to different loading conditions are determined and given in Tables 6 and 7. The results demonstrate the effects of Young's modulus ratio on optimum ply orientations of plates. In particular, the differences between the optimum ply orientations for low and high Young's modulus ratios are evident.

361

Optimal design of laminated composite plates

.=. ~1~ "

oo

oo

~-

~-

r-

~.

~.

< ¢q

O0

v~

¢q

v~

tt~

tt~

I

I

°~

"5

<

I

I

I

e'~

"r~ Oa~ o% ¢q

r-

¢~1 OX Oa

~

t",l

I

O~

¢q

I "~

I ''a

¢¢~ t ~

¢¢5¢e

O~ o~

6

6

~

6

6

~

O0 tt~ 0

_=

©

I

tt~

.6

362

T.Y. Kam, J. A. Snyman

0

0

.oo

0

.~

.oo

0

~

0

•

,~

t"-

.op

oo

~

~

I

m

~

~

0

I I

I

I

I

I

I

~

I

q~"

~

O0

O0

O0

O0

O0

oo c~

oo 6

oo 6

oo c~

oo c~

oo o

.~ o

Optimal design of laminated composite plates

36 3

a Cla4/Eot3

1.0 1.2

---e---S--

- 4 -

1.1 1.6

"

v

1.0

-

0

.--e--

2.0 0.9

•

0.8

•

,,

A

~

• . . . .

•

em

A

e 0.7 "--o

. . . .

-

-

"e

0.6 0.5 0.4 0.3 0.2 0.1

t

0 Fig. 3.

4

t

I

6

8

i

I

I

12

16

16

I

~'-

In, ( N o . o f P l i e s )

Effects of number of plies upon the central displacementsof uniformlyloaded plates of various aspect ratios.

In the above numerical examples, optimum solutions have been obtained with the use of only a few starting points and for each resulting trajectory the local optimum was evaluated very efficiently. The multitrajectory global minimization technique has therefore proved itself to be a very promising method for the optimal design of laminated composite structures. It is the intention to extend the application of the current global optimization technique to the design of laminated composite structures where other different objective functions and constraints, such as minimization of structure weight subject to stress, displacement, buckling and frequency constraints, are of importance.

CONCLUSION Optimal lamination arrangements of plates designed for maximum stiffness have been investigated via a multi-trajectory global minimization technique. Results for multi-layer plates of various aspect ratios sub-

364

T.Y.

Kam, J. A. Snyman

¢",1

r~

~

°

°

II

.<

¢,,q

..d

tm

¢,q ee~

',"~ ....~.. t"'.-

I

I

i eq~

©

eq~

C'q

"~"

~e~

I e~

I

Optimal design of laminated composite plates

36 5

he%

tr~

tt~

6`6`~_ .d

0

L)

-2

I


q-

0')

. ,~. a e~

©

qM

I

i

i

366

T.Y. K a m , .I.A. S n y m a n

ooo

,¢eq

~. ~.

~

~D

k.

tt~

r, < 0

0

k..

,d

eD

0

~t

~t

Optimal design of laminated composite plates

•

..~

.

I

I

I

°~

~

I

I

367

~.

o.

.

~.

I

I

I

I

~.

.

'0 0

•

-... ~.

~

0

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

0

° 0~ 0

°~

0

368

T.Y. K a m , J. A . S n y m a n

°~

0

o

t~ 0

~S

t~

,.d

"~

0"3

~

¢'q

~'~

I

I

I

I

° 0~

o

©

o'~

~

~

Optimal design of laminated composite plates

369

ject to different loading conditions are obtained. It has been shown that the present optimization technique can yield the global optima in a very reliable and efficient way. The multi-start global minimization algorithm is of promise for further applications to the optimal design of more complex laminated composite structures.

REFERENCES 1. Stroud, W. J. & Agranoff, N., Minimum-mass design of filamentary composite panels under combined loads: Design procedure based on simplified buckling equations. NASA TN D-8257, NASA Langley Research Center, Hampton, VA, 1976. 2. Khot, N. S., Computer program (OPTCOMP) for optimization of composite structures for minimum weight design. AFFDL TR-76-149, Wright-Patterson Air Force Base, Dayton, OH, 1977. 3. Schmit, L. A. & Farshi, B., Optimum design of laminated composite plates. Int. J Numer. M eths E ngng, 11 ( 1977 ) 623 -40. 4. Massard, T. N., Computer sizing of composite laminates for strength. J. Reinf. Plastic Composites, 3 (1984) 300-45. 5. Kam, T. Y. & Lai, M. D., Multilevel optimal design of laminated composite plate structures. Comput. & Struct., 31 (1989) 197-202. 6. Kam, T. Y. & Chang, R. R., Optimal design of laminated composite plates with dynamic and static considerations. Comput. & Struct., 32 (1989) 387-93. 7. Hirano, Y., Optimum design of laminated plates under shear. J. Comp. Mater., 13 (1979) 329-34. 8. Hirano, Y., Optimum design of laminated plates under axial compression. AIAA Journal, 17 (1979) 1017-19. 9. Muc, A., Optimal fibre orientation for simply supported, angle-ply plates under biaxial compression. Comp. Struct., 9 ( 1988) 161-72. 10. Chen, T. L. C. & Bert, C. W., Design of composite material plates for maximum uniaxial compressive buckling. Proc. Oklahoma Academy of Science, 56 (1976) 104-7. 11. Adali, S. & Duffy, K. J., Design of antisymmetric hybrid laminates for maximum buckling load: I. Optimal fibre orientation. Comp. Struct., 14 (1990) 49-60. 12. Tauchert, T. R. & Adibhatla, S., Design of laminated plates for maximum stiffness. J. Comp. Mater., 15 (1984) 58-69. 13. Tauchert, T. R. & Adibhatla, S., Design of laminated plates for maximum bending strength. Engng Optim., 8 (1985) 253-63. 14. Griewank, A. O., Generalized descent for global optimization. J. Optim. Theory & Appl., 34 (1981) 11-39. 15. Dixon, L. C. W., Gomulka, J. & Szeg6, G. E, Toward a global optimization. In Toward Global Optimization, ed. L. C. W. Dixon & G. E Szeg6. NorthHolland Co., Amsterdam, 1975, pp. 29-54.

370

T.Y. Kam, J. A. Snyman

16. Rinnooy Kan, A. H. G. & Timmer, G. T., Stochastic methods for global optimization, Amer. J. Math, Managmt Sci., 4 (1984) 7-40. 17. Price, W. L. Global optimization by controlled random search. J. Optim. Theory & Appl., 40 (1983) 333-48. 18. Snyman, J. A. & Fatti, L. P., A multi-start global minimization algorithm with dynamic search trajectories. J. Optim. Theory & Appl., 54 (1987) 121-41. 19. T6rn, A. & Zilinkas, A., Global Optimization. Springer-Verlag, Heidelberg, FRG, 1988. 20. Jones, R. M., Mechanics of Composite Materials. Scripta, Washington DC, USA, 1975. 21. Tsai, S. W. & Hahn, H. T., Introduction to Composite Materials. Technomic, Westport, CT, USA, 1980. 22. Mottram, J. T. & Selby, A. R., Bending of thin laminated plates. Computers & Structures, 25 (1987) 271-80.