2次元高次理論による厚板の解析

Abstract

Based on the expansion of displacement components of a plate into the power series of a thickness coordinate, a set of the fundamental plate equations of two-dimensional higher order theory is derived through the principle of virtual work in terms of the expanded strain tensors and the higher order stress resultants. For elastic plates of isotropic materials, a set of linear constitutive equations is also derived in terms of the expanded displacement components. Through the equilibrium equations of a three-dimensional continuum, stress components are determined in terms of the expanded displacement components with satisfying the stress boundary conditions on the upper and lower surfaces of a plate. The governing set of equations is completely decoupled into the sets of equations governing the in-plane and the out-of-plane deformation modes. There are stress resultants of higher order than bending moments and shear forces involved in the governing equations. These higher order stress resultants or the corresponding expanded displacement components may be necessarily prescribed in the specification of boundary conditions on the middle surfasce of a plate. Truncated approximate equations of the present theory are solved for an infinite plate subjected to sinusoidally (or uniformly) distributed pressure on the upper surface. All the components of displacement and stress are obtained in power series of the thickness coordinate for each level of several approximate theories. The results are compared with those obtaind from exact solution available from three-dimensional elasticity theory. The relative accuracy of the solutions of the present approximate equation is then determined by direct comparison of the approximate solutions with the exact solution.