Asymptotic Solutions of the Von Karman Equations for a Circular Plate Under a Concentrated Load

Abstract

Reissner’s form of the axisymmetric von Karman equations for a centrally, point-loaded plate are written in dimensionless differential and integral form. To concentrate on essentials, we take Poisson’s ratio to be one-third (so that the limiting Fo¨ppl membrane equations have one-term solutions) and boundary conditions of simple support. A dimensionless parameter β measures the relative bending stiffness. A nine-term perturbation solution in powers of ε = β–6, the first term of which corresponds to linear plate theory, is constructured using MACSYMA. Although the resulting deflection-load power series appears to converge only if |ε| < 1/40, successive Aitken-Shanks’ transformations yield an expression valid up to ε ≈ 1. Solutions as β → 0 are constructed using singular perturbation methods and two terms of the deflection-load curve are computed numerically, the first term corresponding to the exact nonlinear membrane solution. A graph shows that there is a region of overlap of the large and small β-approximations to the deflection-load curve.