Structural Optimization 15, 114-123 Q Springer-Verlag 1998 Minimum stress design of transversely isotropic sandwich plates based on higher-order theory V.E. Verijenko, E.B. Summers, S. Adali Department of Mechanical Engineering, University of Natal, Durban 4001, Republic of South Africa M. Walker Department of Mechanical Engineering, Technikon Natal, Durban 4001, Republic of South Africa Abstract The minimum stress design of thick laminated sand- wich plates is given based on a higher-order theory of plates which includes the effects of the normal and shear deformation. The sur- face layers are made of a transversely isotropie composite material and the results are given for isotropie and transversely isotropic core layers which can model a variety of materials including hon- eycomb. The theory is implemented using dedicated symbolic computation routines developed in the C programming language. The analysis is incorporated into an optimization algorithm to determine the optimal thicknesses of the surface layers for the minimum stress deisgn of three-layered sandwich plates. Numer- ical results are given for plates under sinusoidal loading and the effects of various input parameters are investigated. The stress behaviour which cannot be studied using a classical theory or a shear deformable theory only is indicated. 1 Introduction Accurate analysis of the stress and strain behaviour of lami- nated plates is essential for the optimal design of such struc- tures. This is particularly true for thick plates made of ad- vanced composite materials. It is known that the classical theory yields, in general, inaccurate results for composite structures as a result of neglecting the contributions of shear and normal deformation in the analysis and in the case of thick structures, this simplification leads to intolerable errors. This situation necessitates the use of a higher-order theory to analyse the stress behaviour of thick composite plates. The objective of the present study is to obtain a minimum stress design of sandwich plates under flexural loads based on a higher-order theory which includes the effects of both shear and normal deformation. In thick sandwich plates with lay- ers those mechanical properties differ substantially, the effect of normal deformation is considerable and the stress distri- bution through the thickness is no longer symmetrical even for symmetrical structures. This is due to the fact that the load is applied on the top surface and it is physically clear that due to the nonsymmetry of loading, the resulting stress distribution cannot be symmetrical as predicted by theories which fail to take normal deformation into account. Based on this observation, the minimum stress problem involves the computation of relative thicknesses of layers such that the re- sulting design will reflect the stress pattern in a more realistic fashion and thereby will minimize the maximum stress. In this regard, the present study departs from conventional de- signs which automatically assume a symmetrical lamination. In the case of thick structures for which the effect of normal deformation cannot be neglected, such conventional designs cease to be optimal as shown in this paper. Moreover, it is shown that design optimization leads to up to 40% reduction in the maximum stress as compared to symmetrical designs. A number of refined theories was developed for sandwich plates to include the effect of shear deformation in the surface and core layers by Pandya and Kant (1988), Gordaninejad and Bert (1989) and Bert and Cho (1989). However, opti- mum designs of sandwich plates and shells were mostly based on classical sandwich theory. Various optimization studies for sandwich structures include minimum weight beams by Huang and Alspaugh (1974), plates under compressive loads by Vinson (1986) and Ding (1989) and bending loads by Mar- tin (1987), and acoustic sandwich panels by Makris el al. (1988). Design of sandwich shells with fibre composite sur- face layers was given by Min and de Charentenay (1986) and Evans (1991). Design of sandwich plates under uncertain bending loads was given by Adali et al. (1994). Design of thick sandwich structures does not seem to be studied using a higher-order theory which is capable of including normal as well as the shear deformation. In fact previous studies on the optimal design of thick laminated structures seems to be based on shear deformable theories only. In this regard stud- ies include maximum frequency design by Adali (1984) and Kam and Chang (1993), maximum buckling load by Kam and Chang (1992a,b, 1993), and maximum stiffness design by Kam and Chang (1992a,b). In the present study, the design analysis is based on the higher-order theory developed by Piskunov et al. (1993). This theory can accurately model the stress/strain behaviour of thick composite laminates which may have layers with signif- icantly different mechanical properties as is the case with a sandwich structure. However, the computational implemen- tation of the theory poses special problems due to the need to evaluate multiple integrals through the thickness of the lami- nate to compute stiffnesses. The expressions to be integrated are power series whose coefficients depend on the current lam- ina and the coordinate of integration. For the general case, the integrated stiffnesses may not be derived in a form suit- able for direct numerical implementation. The calculation of