In this chapter we introduce the concept of an arbitrary virtual displacement which may also be considered as an arbitrary weight function. This will be used to express the equilibrium equation for 3D solids and structures in a scalar integral form. Subsequently, the divergence theorem is applied to transform the scalar integral into another form to which the force boundary condition can be introduced. This results in the statement for the principle of virtual work involving internal virtual work and external virtual work. Internal virtual work and external virtual work will then be expressed in matrix form so that they can be used for the finite element formulation in later chapters. We then consider plane stress and plane strain problems in which the principle of virtual work can be expressed in 2D domains in accordance with simplifying conditions. In the last section, the Lagrange equation is derived within the context of deformable solid bodies, starting from the principle of virtual work.

In this chapter we first describe how to construct the element stiffness matrix and load vector in 2D domains. The mapping and shape functions derived in the previous chapter are introduced to express strain components in terms of nodal DOF. Extension of the finite element formulation to 3D domains is demonstrated using the eight-node hexahedron as an example. For dynamic problems, the element mass matrix can be formed by treating the inertia effect as a body force applied to the element. The global mass matrix is then assembled to construct the equation of motion for analyses of free vibration and forced vibration. In the last section, we briefly discuss important aspects of finite element modeling and analysis that often arise in 2D and 3D problems where the number of DOF can be large. We discuss issues, such as sparse matrices and mesh generation, which early students of the finite element method may find helpful for future reference.

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.

Stresses describe the local distributions of forces within a deformable body and strains describe the local deformations. In this chapter we want to describe the relations between stresses and strains as these are the key relationships that allow us to connect the loads applied to a body to its changes in shape. We will only consider linear elastic materials in this book where the stresses are proportional to the small strains present. Both isotropic and anisotropic linear elastic materials will be discussed. How the elastic constants appearing in the general stress–strain relations for an anisotropic material change with choice of orientation of the coordinate system being used will be given explicitly. The use of strain gages and stress–strain relations to determine the state of stress on the surface of a body will be discussed.

Two-dimensional problems of plane stress and plane strain in polar coordinates, both axisymmetric and non-axisymmetric, are considered. Among axisymmetric problems, the bending of a curved beam by two end couples and the problem of a pressurized hollow disk or cylinder are analyzed. Among non-axisymmetric problems, solutions are derived for problems of bending of a curved cantilever beam by a vertical force, loading of a circular hole in an infinite medium,concentrated vertical and tangential forces at the boundary of a half-plane, and a semi-elliptical pressure distribution over the boundary of a half-space. The problems of diametral compression of a circular disk (Michell problem), stretching of a large plate weakened by a small circular hole (Kirsch problem), stretching of a large plate strengthened by a small circular inhomogeneity, and spinning of a circular disk are also analyzed and discussed. The chapter ends with an analysis of the stress field near a crack tip under symmetric and antisymmetric remote loadings, the stress and displacement fields around an edge dislocation in an infinite medium, and around a concentrated force in an infinite plate.

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.

In this chapter, the equations developed in Chapter 9 are used to solve for the stress and velocity distribution in an idealized “glacier” consisting of a slab of ice of infinite extent on a uniform slope. Solutions are first obtained for ice with a perfectly plastic rheology.Both the shear stress and the surface-normal stress increase linearly with depth. The shear stress equals the plastic yield stress at the bed. Longitudinal normal stresses vary non-linearly, and are compressive in the ablation area and extending in the accumulation area.The surface-normal velocity also varies linearly with depth. The surface-parallel velocity varies non-linearly, with a high gradient near the bed. The stress solutions and the solution for the surface-normal velocity are essentially the same in a non-linear material, except that the shear stress does not reach a limiting value at the bed.However, the surface-parallel velocity, while varying with depth in a similar way, is now dependent on the longitudinal strain rate, and the solution is much more complicated. Interestingly, however, it does not matter whether the longitudinal strain rate is compressive or extending.