The cantilever strip plate of varying thickness and the centre of shear

A homogeneous, isotropic plate occupies the region 0 ≤ x₁ ≤ ∞, |x₂| ≤ a, |x₃| ≤ h, where the semi-thickness h = h(x₂). The ratio h(x₂)/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 x₁ = 0 is clamped while the long sides are free. This cantilever plate is now loaded at x₁ = +∞ 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 x₂. 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 M₁₁ V₁ on x₁ = 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 M₁₁, V₁. 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 → ∞, the shear centre lies at x₂ = m₁^Ba, where m₁^B is one of the weighted integrals in the bending problem.