A homogeneous, isotropic plate occupies the region 0 less than or equal to x(1) less than or equal to infinity, \x(2)\ less than or equal to a, \x(3)\ less than or equal to h, where the semi-thickness h = h(x(2)). The ratio h(x(2))/a is supposed to be everywhere sufficiently small so that the classical theory of bending of thin plates (of non-uniform thickness) applies. The short end of the plate at x(1) = 0 is clamped while the long sides are free. This cantilever plate is now loaded at x(1) = +infinity by an applied twisting moment, by a bending moment or by flexure. We solve these problems for the case in which h varies exponentially with x(2). We use the projection method which overcomes the difficulty that the boundary conditions lead to severe oscillating singularities in the corners (0, +/-a).Our numerical results show that the values of M-11, V-1 on x(1) = 0 bear little resemblance to those of the corresponding Saint-Venant 'solutions', which do not fully satisfy the boundary conditions at the clamped end. Indeed, very large values of these resultants are found at points near the 'thick' corner which could affect the integrity of the plate in actual engineering applications. We also determine the values of certain weighted integrals of M-11, V-1. These constants determine the effect of the clamping at 'large' distances (greater than 4a, say) from the clamped end.As a further application, we consider the corresponding plate of finite length 2L. Provided that the aspect ratio L/a is 2 or more, we give accurate approximate solutions for the torsion and flexure of a finite plate clamped at both ends. The flexure problem for the finite plate enables us to calculate the position of the 'centre of shear' according to Reissner's definition. This has not previously been possible due to the complicated nature of the underlying boundary-value problem. In the limit as L/a --> infinity, the shear centre ties at x(2) = m(1)(B)a, where m(1)(B) is one of the weighted integrals in the bending problem.

In a plate-theoretical formulation of the shear center problem, the relevant boundary-value problem is for a cantilevered rectangular plate of variable thickness with two free opposite edges and with the edge opposite to the clamped end subject to a rigid vertical displacement and free of bending moment. For plates with Poisson's ratiov equal to zero, there is an exact elementary solution for this boundary-value problem from which the exact location of the shear center can be calculated. When Poisson's ratio is not zero, an approximate elementary solution may be obtained within the framework of a Saint-Venant flexure solution for plates by satisfying the displacement boundary conditions at the clamped edge approximately. Different forms of this approximation are discussed in [7], some with rather marked Poisson's ratio effects. Among these, the minimum complementary energy approach of [6] gives a shear center location identical to the exact solution forv=O.