Closed form stress distribution in 2D elasticity for all boundary conditions

Abstract

This paper applies a Hamiltonian method to study analytically the stress distributions of orthotropic two-dimensional elasticity in (x, z) plane for arbitrary boundary conditions without beam assumptions. It is a method of separable variables for partial differential equations using displacements and their conjugate stresses as unknowns. Since coordinates (x, z) can not be easily separated, an alternative symplectic expansion is used. Similar to the Hamiltonian formulation in classical dynamics, we treat the x coordinate as time variable so that z becomes the only independent coordinate in the Hamiltonian matrix differential operator. The exponential of the Hamiltonian matrix is symplectic. There are homogenous solutions with constants to be determined by the boundary conditions and particular integrals satisfying the loading conditions. The homogenous solutions consist of the eigen-solutions of the derogatory zero eigenvalues (zero eigen-solutions) and that of the well-behaved nonzero eigenvalues (nonzero eigen-solutions). The Jordan chains at zero eigenvalues give the classical Saint-Venant solutions associated with averaged global behaviors such as rigid-body translation, rigid-body rotation or bending. On the other hand, the nonzero eigen-solutions describe the exponentially decaying localized solutions usually ignored by Saint-Venant’s principle. Completed numerical examples are newly given to compare with established results.