In this chapter we first consider finite element modeling of slender bodies undergoing bending deformation. This will be followed by a discussion on frame structures which can be modeled as an assemblage of slender bodies rigidly connected. First, we will introduce the Bernoulli--Euler theory of beam bending as a review and extension of what is typically covered in an undergraduate sophomore-level course on mechanics of materials. We will then introduce the frame element which can be used to model frame structures deforming in the 2D plane and 3D space.

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 present the finite element formulation of heat transfer problems which can be used to determine temperature distributions in solid bodies, starting with heat conduction in the 1D domain. Similar to the notion of virtual displacement in earlier chapters, a virtual temperature or an arbitrary weight function is introduced to derive an integral equivalent of the governing equation to which the finite element formulation is applied. Methods for heat conduction and convection, in 1D, 2D, and 3D domains, including time-dependent effects, will be covered. Mathematical equivalence with other scalar field problems is also discussed.

The finite element method is a powerful technique that can be used to transform any continuous body into a set of governing equations with a finite number of unknowns called degrees of freedom (DOF). In this chapter, we will introduce the fundamentals of the finite element method using a system of linear springs and a slender linear elastic body undergoing axial deformation as examples. These simple problems are chosen to describe the essential features of the finite element method which are common to analysis of more complicated structural systems such as 3D bodies.

In the finite element formulation, the body is divided into elements of various types. This chapter describes mapping functions for the description of element geometry in the undeformed configuration and shape functions for the description of displacement and thus deformed geometry in the 2D and 3D domains. We introduce the "isoparametric" formulation in which mapping functions and shape functions are identical. This is followed by discussions on integration in the mapped domains and numerical integration.

This chapter deals with the finite element formulation for thin plate and shell structures. We will review the assumptions on the kinematics of deformation from classical plate bending theories, introduce them into the finite element formulation for plates, and then extend the formulation to curved shell structures within the isoparametric formulation. For 3D solid elements that can be used for plates and shell analysis, we will first look at solid elements with three nodes through the thickness. We will then show how solid elements with two nodes through the thickness can be constructed for analysis of plate and shell structures.

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.

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.