The previous chapter focused on the behavior of the stress vector and stresses at any fixed point in a body. However, stresses will also vary from point to point within a deformable body so that we need to describe those spatial variations. As we will see in this chapter, local equations of equilibrium involving the stresses must be satisfied everywhere within a body. We will examine if the elementary theories of axial loads, bending, and torsion considered in Chapter 1 satisfy these equations of equilibrium and relate these equilibrium equations for the stresses to the force and moment equilibrium equations normally used in elementary strength of materialsdiscussions.

Partial differential equations whose solution specifies the elastic response of a loaded body are summarized. If all boundary conditions are given in terms of tractions, the boundary-value problem can be specified entirely in terms of stresses. The governing differential equations are then the Cauchy equations of equilibrium and the Beltrami–Michell compatibility equations. If some of the boundary conditions are given in terms of the displacements, the boundary-value problem is formulated in terms of the displacement components through the Navier equations of equilibrium. The boundary conditions can be expressed in terms of displacements, or in terms of displacement gradients. Due to the linearity of all equations and boundary conditions, the principle of superposition applies in linear elasticity. The semi-inverse method of solution and the Saint-Venant principle are introduced and discussed. The solution procedure is illustrated in the analysis of the stretching of a prismatic bar by its own weight, thermal expansion of a compressed prismatic bar, pure bending of a prismatic bar, and torsion of a prismatic rod with a circular cross section.