TABLE OF CONTENTS Solid mechanics is based on general principles that lead to differential equations of stress equilibrium and, through stress-strain relationships, to differential equations for displacement components. Solving these is in general quite complicated and beyond the scope of this text. Some idealized problems of fundamental character which can be relatively simple are discussed in this chapter. These are axisymmetric compression or expansion and rotation of thick-walled cylinders and torsion of noncircular prisms. We also solve one more complicated problem, namely, bending of a high beam, using the finite differences method (see also Chapter 6).
The equilibrium equations derived in Chapter 2 for a three-dimensional solid have the form
Since only three equations in six unknowns are available, we evidently need additional equations relating stress components to displacement components. Replacing the unknown stress components ? xx, in Eqs. (9-1) by the unknown displacement components u, v, w, we make the number of unknowns in the resulting system equal to the number of equations. The nature of the resulting system of equations depends on the relationship between stress and strain, that is, on the constitutive equations (see Chapter 4). The simplest example is a linearly elastic isotropic solid, which we shall consider here. Substituting the relations between strain and displacement components [see Eq. (3-11)]
into Hooke's law [Eq. (4-111)],
we find the following relations between stress components and displacement components:
Substituting Eqs. (9-4) into the equations of equilibrium (9-1) and simplifying, we have
Equations (9-5)...
