Most problems involving complex-shaped deformable bodies require a numerical solution of the governing equations and boundary conditions. However, there are some important simple problems that can be solved analytically and that reveal the nature of the stresses and deformations in geometries of practical significance. In this chapter we will examine a number of such problems, including thick-wall pressure vessels, shrink-fits, the stress concentration at a hole in a plate, the bending of a curved beam, and a concentrated force acting on a wedge. Both displacement-based approaches as well as stress-based approaches using the Airy stress function will be considered.

Previous chapters examined the topics of equilibrium, compatibility, strain–displacement relations, and stress–strain relations. When these elements are combined, we can form up different complete sets of governing differential and algebraic equations. In order to solve those sets of equations we must also specify the conditions that arise from having known loads or geometric constraints. These are called the boundary conditions for the problem.In this chapter we will examine some of the choices we have for formulating complete sets of the governing equations and how those governing equations can be combined with appropriate boundary conditions to solve for the stresses and deformations. We will also discuss the principle of Saint-Venant, which gives us some flexibility in how we specify the boundary conditions. Finally, we will also show how structural analysis problems can be expressed in terms of algebraic matrix–vector equations, which are the counterparts of the governing differential/algebraic equations. A classical deformable body problem, Navier's table problem, will be used as an example of these purely algebraic methods.

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.