fnr. 1. Engng Sci. Vol. 27, No. 9, pp. 1005-1022, 1989 00267225/89 $3.00 + 0.00 Printed in Great Britain. All rights reserved Copyright @ 1989 Pergamon Press plc ON BOUNDARY-DISCONTINUOUS DOUBLE FOURIER SERIES SOLUTION TO A SYSTEM OF COMPLETELY COUPLED P.D.E.3 REAZ A. CHAUDHURI Department of Civil Engineering, University of Utah, Salt Lake City, UT 84112, U.S.A. Abstract-A heretofore unavailable boundary-discontinuous double Fourier series based approach for solution to a system of completely coupled linear second-order partial differential equations with constant coefficients and subjected to general (completely coupled) boundary conditions is presented. This method facilitates the selection of the unknown Fourier coefficients, contributing to the complete solution and also takes into account the possible discontinuities of the assumed solution functions and/or their first derivatives at the boundaries. Dirichlet and Neumann types of boundary conditions can be treated as two special cases. The method is applied to obtain solutions of the hitherto unsolved class of problems, pertaining to arbitrarily laminated anisotropic (constant)-shear-flexible doubly- curved shells of rectangular planform, with arbitrary admissible boundary conditions and subjected to general transverse loading. Such specific cases of lamination as anti-symmetric angle-ply, symmetric angle-ply and general cross-ply, such particular case of loading as uniformly distributed transverse load and such specific case of geometry as a rectangular plate, can be obtained as special cases of the above. In addition, this method is shown to reproduce the available boundary-continuous solutions for unsymmetric cross-ply plates and doubly-curved shells with SS3 type simply-supported boundary conditions and anti-symmetric angle-ply plates with SS2 type simply-supported boundary conditions. 1. INTRODUCTION Many boundary-value problems of mathematical physics, with domains of rectangular planform, are represented by systems of highly coupled linear partial differential equations (P.D.E.) with constant coefficients, where the prescribed boundary conditions can be quite general. A subclass, represented by a system of completely coupled linear second-order P.D.E.‘s with constant coefficients, can be treated, without much loss of generality, as a representative of the above. The boundary conditions which may contain at the most first derivatives, may be Dirichlet type, Neumann type or general (mixed). The objective of the present study is to present a general method of solution to this subclass of problems, using double Fourier series, which may be continuous or discontinuous at an edge. The present study is motivated by the concern of finding exact (double Fourier series) solutions to the problems of shear-flexible arbitrarily laminated doubly-curved or cylindrical panels (open shells), with arbitrary boundary conditions. A detailed literature search reveals that the existing studies, e.g. [l-3], on obtaining exact solutions are restricted to cross-ply (for definition, see [4,5]) shells with a special type of simply-supported boundary conditions, designated SS3 under the classification of Hoff and Rehfield [6]. Reddy [3] has attempted to obtain a double Fourier series solution to an anti-symmetric angle-ply doubly-curved moderately-thick panel of rectangular planform, with the SS2 type of simply-supported boundary conditions. Sanders’ [7] kinematic relations, extended to include the first-order shear deformation theory (FSDT) based on Reissner-Mindlin hypothesis, have been used in Reddy’s [3] formulation. However, his attempt has not been crowned with success, leading to his erroneous conclusion that “unlike plates, antisymmetric angle-ply laminated shells with simply supported boundary conditions do not admit analytical solutions”. It is worthwhile to note that in the majority of the existing studies on the subject, solutions are usually assumed in the form of a double Fourier series, such that either all four or two opposite (Levy type solution) boundary conditions are satisfied a zyxwvutsrqponmlkjihgfedcbaZY priori. These assumed solutions are then substituted into the governing partial differential equations, which will yield a set of five linear algebraic equations in terms of as many unknown Fourier coefficients for each combination of m, it, where zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCB m and IZ denote the number of terms of the Fourier series. This approach has been successful only in the case of cross-ply panels of rectangular planform, with the SS3 boundary conditions. For example, in the case of an antisymmetric angle-ply