On the theory of plane stress

Abstract

The classical theory of plane stress is concerned with a thin elastic plate subjected to edge forces which deform it so that there is no normal displacement of its middle plane. This theory may be obtained from the exact three dimensional linear theory of elasticity for homogeneous and isotropic materials, by neglecting some of the compatibility equations and assuming that the shear and normal stresses, transverse to the middle plane of the plate, vanish and the remaining stresses are independent of z, the coordinate normal to this mid-plane. Hence, the relationship between the two theories is shown by systematically deriving from the exact theory the differential equations and boundary conditions of plane stress. Each stress component is expanded in a power sertes in the plate thickness, h, and equated to determine the expansion coefficients. The plane stress theory appears as the zeroth order interior problem, the solutions of which supply a first approximation to the three- dimensional stress distribution. The expansion procedure also provides three- dimensional corrections to the plane stress theory. (N.W.R.)