Mechanics of materials (also called strength of materials) is an area of engineering that describes the deformation of bodies and the distribution of forces within those bodies. Introductory courses in this discipline normally consider three types of simple deformations – the stretching or compression of axially loaded bars, the bending (flexure) of laterally loaded beams, and the torsion (twisting) of circular shafts. Many complex structures are composed of members that have these simple types of behavior. This chapter will outline the analysis of these three types of problems and review the basic assumptions made so that those assumptions can be later examined and, in some cases, relaxed.

The bending of unsymmetrical beams was considered in Chapter 10, where we showed that, for thin sections, we could obtain explicit expressions for both the flexure stresses and the shear flows (shear stresses). In Chapter 11 similar explicit expressions were found for the shear stresses or shear flows for the torsion of thin, open and closed cross-sections. In this chapter we will examine cases where a thin member is in a combination of bending and torsion. We will also consider the case where axial loads acting on a thin section can induce torsional deformations. A new quantity called a bimoment will be shown to be a resultant of the axial stress distribution that produces torsion.

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.

Just as the concept of stress gives us a measure of force distributions in a deformable body, the concept of strain describes the distribution of deformations locally at every point within the body. In this chapter we will define strains and describe how strains change with directions and with the choice of coordinates, as was done with stresses. Strains will also be related to the displacements of the deformable body. It will be shown that strains must satisfy a set of compatibility equations at every point in a body to ensure that they represent a well-behaved deformation. Since the strains often found in practice are quite small, this book will only consider problems for small strains.

Work–energy concepts are important for two reasons. First, they provide an alternative way to guarantee equilibrium and compatibility, which are two key elements in all stress analyses. Second, energy methods have become the basis of formulating numerical methods so they are at the heart of the field of computational mechanics, which will be discussed in the next chapter.In this chapter we will discuss two types of internal energy in deformable bodies – strain energy and complementary strain energy. Although we will show that these internal energies are equal for linear elastic bodies, we will see that they play distinct roles in terms of work–energy relations. A number of important general principles and theorems will be described including the principle of virtual work, the principle of complementary virtual work, the principle of minimum potential energy, the theorem of minimum complementary potential energy, and the reciprocity theorem.The classical theorems of Castigliano and Engesser and the principle of least work will be used to solve problems with discrete forces and moments.

Structures can fail in different ways so that one needs to examine a variety of failure modes. In this chapter we will consider (1) a number of the commonly used static failure theories, (2) fatigue failure under alternating loads, and (3) fracture theory. We will also briefly discuss how nondestructive inspections can be used in conjunction with crack growth laws to keep structures safe while in use. Another way that structures can fail is through a loss of stability. The sudden buckling of columns, also called a bifurcation type of instability, will be described as well as other types of instabilities such as limit-load instabilities and snap-through buckling instabilities.

The distribution of the external forces acting on a body affects both the internal and external deformation of the body. The internal deformations in particular depend on how the forces are distributed throughout the body. Stress is a key concept that gives us a way to characterize those internal force distributions. This chapter will discuss in depth the stress concept, including stress transformations, principal stresses, states of stress, and Mohr's circle. MATLAB® will be used as the principal tool for calculations.

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.

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.

The engineering beam bending theory summarized in Chapter 1 assumed that the beam cross-sectional area has a plane of symmetry and that bending moments were acting along a single axis. In this chapter we want to remove those restrictions and to examine the multiaxis bending of beams with nonsymmetrical cross-sections. This will lead to a generalization of the flexure formula for the normal stress in the beam.In Chapter 1 we also obtained an expression for the shear stresses induced in symmetrical beams. It is difficult to obtain similar analytical shear-stress forms for beams with general unsymmetrical cross-sections. However, we will show that when the cross-section is thin one can obtain explicit expressions for the shear stresses. Analysis of the bending of thin beams will demonstrate that the shear force in the beam must pass through a specific point, called the shear center, if the beam is to bend without twisting. A new cross-sectional area property, called the principal sectorial area function, will be shown to play a key role in locating the shear center.

Most complex deformable body problems commonly found in practice can only be solved with numerical methods. The stiffness-based finite element method is today the numerical method of choice for analyzing deformable bodies as well as many other engineering problems. In this chapter we will describe both stiffness-based and force-based finite elements. It will be shown that while the stiffness-based finite element method is based on solving equilibrium equations for the displacements, the force-based method relies on appropriately combining equilibrium and compatibility to solve directly for the forces (or stresses). We will examine both stiffness-based and force-based finite elements for axial load and bending problems. Simple examples will be used that allow us to work through the application of these finite element methods in detail and to compare results with analytical solutions. Finally, we will outline an important alternative to a finite element approach - the boundary element method – for solving stress problems numerically.