6 - Two–Dimensional Problems of Elasticity

Summary

Two-dimensional problems of plane stress and plane strain are considered. The plane stress problems are the problems of thin plates loaded over their lateral boundary by tractions which are uniform across the thickness of the plate, while its flat faces are traction free. The plane strain problems involve long cylindrical bodies, loaded by tractions which are orthogonal to the longitudinal axis of the body and which do not vary along this axis. The tractions over the bounding curve of each cross section are self-equilibrating. Two rigid smooth constraints at the ends of the body prevent its axial deformation. The stress components are expressed in terms of the Airy stress function such that the equilibrium equations are automatically satisfied. The Beltrami–Michell compatibility equations require that the Airy stress function is a biharmonic function. The Airy theory is applied to analyze pure bending of a thin beam, bending of a cantilever beam by a concentrated force, and bending of a simply supported beam by a distributed load. The approximate character of the plane stress solution is discussed, as well as the transition from the plane stress to the plane strain solution.