Chapter 2 - Two-dimensional problems in elasticity

Abstract

In most structural problems the object is usually to find the distribution of stress in an elastic body produced by an external loading system. It is, therefore, more convenient in this case to determine the six stresses before calculating any required strains or displacements. The solution of problems in elasticity presents difficulties, but the procedure may be simplified by the introduction of a stress function. For a particular two-dimensional case, the stresses are related to a single function of x and y such that substitution for the stresses in terms of this function automatically satisfies the equations of equilibrium no matter what form the function may take. The task of finding a stress function satisfying the preceding conditions is extremely difficult in the majority of elasticity problems, although some important classical solutions have been obtained in this way. This chapter discusses an alternative approach, known as the inverse method, and the principle of St. Venant, which may be summarized as stating that “while statically equivalent systems of forces acting on a body produce substantially different local effects, the stresses at sections distant from the surface of loading are essentially the same”. The chapter ends with the discussion on bending of an end-loaded cantilever.