In this chapter we consider the finite element formulation for bending of slender bodies under a tensile or compressive axial force. In order to capture the effect of axial force we look at the force and moment equilibrium in the deformed configuration, but still assuming small translational displacement and rotation of the cross-section. In the finite element formulation, it is shown that the effect of axial force on bending manifests as an effective bending stiffness. It will be shown that the finite element formulation of a slender body under compressive axial force results in a matrix equation for eigenvalue analysis from which we can determine the static buckling load and the buckling mode. Subsequently, we consider the finite element formulation for vibration analysis of slender bodies to investigate the effect of axial force on the natural frequencies and modes. Finally, we introduce the finite element formulation of slender bodies subjected to a compressive follower force in which the direction of the applied force is always parallel to the body axis in the deformed configuration.

Under certain conditions, a finite element may lose its ability to deform and become excessively stiff. This phenomenon is called "element locking." In this chapter we will consider three forms of locking, including transverse shear locking, membrane locking, and incompressibility locking. Approaches for alleviating or avoiding locking will also be described.

Truss structures are built up from individual slender-body members connected at common joints. The members are connected through hinge joints which are free to rotate and thus cannot transmit moment. Individual members carry only axial tensile or compressive force. In this chapter, the truss is comprised of uniaxial elements introduced in the previous chapter. However, in order to construct the global stiffness matrix of a truss structure in 3D space, it is necessary to construct the element stiffness matrices with 3 DOF at each node, corresponding to three displacement components in the Cartesian coordinate system. After developing the finite element formulation for 3D truss structures, the effects of thermal expansion and uniaxial members subject to torsional deformation are treated.

In this chapter, we consider time-dependent problems of discrete systems with N DOF. We will show how the finite element formulation is used to construct the element mass matrices, which are assembled into the global mass matrix. We will consider free vibration, to determine natural frequencies and natural modes of a finite element model through eigenvalue analysis, and numerical methods for integrating the equation of motion in time which can be used to determine dynamic response under applied loads and given initial conditions. The Lagrange equation will be shown to demonstrate how it can be applied to construct equations of motion. Once again we will consider slender bodies undergoing uniaxial vibration, torsional vibration, and bending vibration. A formal derivation of the Lagrange equation will be considered in a later chapter.

In this chapter we extend Cauchy’s Theorem to cases where the integrand is not analytic, for example, when the integrand possesses isolated singular points. Each isolated singular point contributes a term proportional to what is called the residue of the singularity. This extension, called the residue theorem, is very useful in applications such as the evaluation of definite integrals of various types. The residue theorem provides a straightforward and sometimes the only method to compute these integrals. We also show how to use contour integration to compute the solutions of certain partial differential equations by the techniques of Fourier and Laplace transforms.

This chapter introduces complex numbers, elementary complex functions, and their basic properties. It will be seen that complex numbers have a simple two-dimensional character which submits to a straightforward geometric description. While many results of real variable calculus carry over, some very important novel and useful notions appear in the calculus of complex functions. Applications to differential equations are briefly discussed as well.

The representation of complex functions frequently requires the use of infinite series expansions. The best known are Taylor and Laurent series, which represent analytic functions in appropriate domains. Applications often require that we manipulate series by termwise differentiation and integration. These operations may be substantiated by employing the notion of uniform convergence. Series expansions break down at points or curves where the represented function is not analytic. Such locations are termed singular points or singularities of the function. The study of the singularities of analytic functions is vitally important in many applications including contour integration, differential equations in the complex plane, and conformal mappings.

A large number of problems arising in fluid mechanics, electrostatics, heat conduction, and many other physical situations, can be mathematically formulated in terms of Laplace’s equation (see also the discussion in Section 2.1).