BOUNDARY-CONDITIONS AT THE EDGE OF A THIN OR THICK PLATE BONDED TO ANELASTIC SUPPORT

At the clamped edge of a thin plate, the interior transverse deflectio n w(x1, x2) of the mid-plane x3 = 0 is required to satisfy the boundar y conditions w = partial-derivativew/partial-derivative n = 0. But sup pose that the plate is not held fixed at the edge but is supported by being bonded to another elastic body; what now are the boundary condit ions which should be applied to the interior solution in the plate? Fo r the case in which the plate and its support are in two-dimensional p lane strain, we show that the correct boundary conditions for w must a lways have the form [GRAPHICS] with exponentially small error as L/h - -> infinity, where 2h is the plate thickness and L is the length scale of w in the x1-direction. The four coefficients W(B), W(F), THETA(B), THETA(F) are computable constants which depend upon the geometry of t he support and the elastic properties of the support and the plate, bu t are independent of the length of the plate and the loading applied t o it. The leading terms in these boundary conditions as L/h --> infini ty (with all elastic moduli remaining fixed) are the same as those for a thin plate with a clamped edge. However by obtaining asymptotic for mulae and general inequalities for THETA(B), W(F), we prove that these constants take large values when the support is 'soft' and so may sti ll have a strong influence even when h/L is small. The coefficient W(F ) is also shown to become large as the size of the support becomes lar ge but this effect is unlikely to be significant except for very thick plates. When h/L is small, the first order corrected boundary conditi ons are w = 0, dw/dx1 - 4THETA(B)/3(1 - nu) h d2w/dx1(2) = 0, which co rrespond to a hinged edge with a restoring couple proportional to the angular deflection of the plate at the edge.