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.

The torsion of a solid or hollow bar with a circular cross-section is one of the important problems considered in elementary strength of material texts. In this chapter we consider the torsion of bars having more general cross-sections, where the axial warping deformations produced requires that one develop a much more complex solution procedure. We will first consider the idealized case of uniform torsion where the bar is completely free to warp. Solutions of uniform torsion problems are obtained using both a warping function and a Prandtl stress function approach. The case of nonuniform torsion, where the rate of twist varies along the length of the bar and the warping of the bar is restrained, is also considered using a warping function approach. Uniform and nonuniform torsion problems for general cross-sections typically require a numerical solution. However, in the case of thin members, one can obtain more direct solutions for both open and closed cross-sections. It will be shown that the sectorial area function plays a major role in the solution of both uniform and nonuniform torsion problems for thin cross-sections.

In this chapter, we discuss several topics of numerical analysis. The topics include numerical interpolation, differentiation, integration, and solution of linear systems of equations. These numerical techniques are essential in the finite element method. MATLAB functions and subroutines performing these numerical calculations in the implementation of FE codes for engineering analysis are presented. While numerical analysis is a broad area, this chapter focuses on the methods and techniques that are used in the development and solution of finite element formulations and models. It should be noted that, while MATLAB has many built-in functions for performing numerical analysis tasks, this book is intended to reveal the underlying theories and techniques of the numerical methods. Therefore, we explain various numerical methods from the basics and create the MATLAB functions from scratch.

In this chapter, we discuss the finite element analysis procedure for two types of time-dependent problems: transient heat transfer and structural dynamics (or elastodynamics) problems. In the transient finite element analysis, similar to the discretization of the spatial domain, the continuous time axis is discretized into time steps for time-dependent solutions. Time integration schemes are used to calculate the solution at a given time step by using information already known from previous time steps. The time integration represents the major difference in the solution steps of FEA for time-dependent problems. This chapter contains two sections. The first section introduces the FEA procedure for transient heat transfer analysis by using a 2-D example problem. The second section discusses elastodynamic analysis by demonstrating the procedure of calculating the dynamic response of a 2-D cantilever beam, as well as finding its natural frequencies and vibrational modes. MATLAB codes for solving these problems are presented.

In this chapter, we discuss the general finite element analysis procedure for linear vector field problems. A vector field problem is a problem whose primary unknown physical quantity is a vector quantity at any spatial location in the computational domain. As solid mechanics and fluid dynamics are representatives of vector field problems, this chapter demonstrates the solutions of a set of solid mechanics and fluid dynamics problems. The chapter contains four sections. The first section briefly reviews the theory of linear elasticity. The second section introduces the FEA procedure for structural analysis of a 2-D elasticity problem. The third section discusses a 3-D elasticity problem and illustrates the FEA steps. The fourth section discusses the FEA procedure for 2-D steady state incompressible viscous flow problems. At the end of each section, MATLAB codes for solving these problems are presented.

In this chapter, we discuss the finite element analysis of several special types of structures: trusses, beams, frames, and plates. These types of structures are defined based on their geometric characteristics and mechanical behavior. From a geometric point of view, these structures can be viewed as objects with lower dimensions: line/curve segments and thin surfaces in the 3-D space. In finite element analysis, special types of elements were created to model the low-dimensional structures on both geometric and mechanics aspects. In this regard, these structural elements can be considered as reduced-order 3-D elasticity elements. The mathematical models and finite element formulations are derived based on such model reductions or simplifications. In this chapter, we demonstrate the finite element analysis procedures for these types of structures by solving three example problems. MATLAB codes for solving the example problems are also presented.

In this chapter, we start the illustration of general procedure of FEA for linear static analysis through a step-by-step solution of a 1-D elasticity problem. We introduce the important concepts and numerical techniques that make up the finite element method as we go through the steps. Then, computer implementation of the procedure is discussed, and a MATLAB code for solving the 1-D elasticity problem is provided. By completing both the hand and computer calculations for the same problem, we demonstrate how the step-by-step solution can be automated by a computer code. The chapter then goes on to demonstrate that other types of physical problems (heat transfer and advection?diffusion transport) can be solved by using the same FEA procedure.

In this chapter, we discuss basic mathematical concepts and methods that will be used as tools in the development of finite element formulations and solution of finite element models. Basic knowledge of linear algebra, calculus of vectors and matrices, variational calculus, and integral equations is necessary in the derivation of finite element formulations. These fundamental mathematical concepts and methods are reviewed as the building blocks for the following content of this book. Numerical methods for numerical approximation, differentiation, integration, discretization, and solution of linear systems will be discussed in Chapter 3. The mathematical tools and numerical methods are then utilized, along with relevant physical principles, in the illustration of the FEA procedure for different types of physical problems in later chapters.

Meshing is a process of discretizing a computational domain into a set of discrete elements with simple geometries. The non-overlapping elements combined represent the geometry of a computational domain. In addition, the elements are the small volumes where the physical quantities are approximated using simple mathematical functions, and the mesh of the elements ensures that the functions are stitched together piece by piece. Depending on the characteristics of the geometry and the type of the physical problem, the computational domains and elements can be categorized into 1-D, 2-D, and 3-D types. In this chapter, we first introduce some of the basic modeling and meshing concepts and techniques for different types of computational domains. Next, we focus on 2-D domains and describe in detail the modeling method of planar straight- line graphs and the meshing approach of Delaunay triangulation, and refinement for generating 2-D meshes of triangular elements.

In this chapter, we discuss the general finite element analysis procedure for 2-D and 3-D linear scalar field problems. A scalar field problem is a problem whose primary unknown physical quantity is a scalar at any spatial location in the computational domain. We demonstrate the finite element analysis procedure by solving 2-D and 3-D steady state heat transfer problems. The steady state heat transfer problems are solved step by step in the same fashion as solving 1-D problems. Strong and weak forms of the governing equations are derived from the law of energy conservation and the method of weighted residuals. 2-D and 3-D finite element approximations and elements are described in detail. Numerical integration over multi-dimensional elements is also described in detail. Convergence considerations are discussed. MATLAB codes for solving these problems are presented.