THE CANTILEVER STRIP PLATE UNDER TORSION, BENDING OR FLEXURE AT INFINITY

A homogeneous, isotropic plate occupies the region 0 less than or equa l to x(1) 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 ratio h/a is sufficien tly small so that the classical theory of thin plate bending 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. Despite the fundamental nature of these problems, and the long history of thi n plate theory, no solutions are to be found in the existing literatur e that will determine (for instance) the important unknown resultants V-1, M(11) at the clamped end x(1) = 0. The main reason for this is th at this combination of boundary conditions leads to severe oscillating singularities of the field in the corners (0, +/-a). The fact that su ch singularities must exist is widely known, but we present here for t he first time a method of solution that takes these singularities full y into account. Our numerical results show that the values of M(11), V -1 on x(1) = 0 bear little resemblance to those of the corresponding S aint-Venant 'solutions', which do not fully satisfy the boundary condi tions at the clamped end. Indeed, significantly large values of these resultants were found at points far enough from the corners so as to b e relevant in actual engineering applications. Also of interest are ce rtain weighted integrals of M(11), V-1 which we calculate. These const ants determine the effect of the clamping at 'large' distances (greate r than 4a, say) from the clamped end. At such distances, the effect of the clamping is merely to impose an additional rigid body deflection on the plate. Finally, we consider the plate of finite length. Provide d that the aspect ratio is 2 or more, we give accurate approximate sol utions for the torsion, bending or flexure of a finite plate clamped a t both ends.