| $a_{i}$ | generalized coordinates (coefficients used to express displacement in general form) |
| $A$ | cross-sectional area |
| $\underline{B}$ | matrix relating strains to nodal displacements or relating temperature gradient to nodal temperatures |
| $c$ | specific heat of a material |
| $\underline{C}'$ | matrix relating stresses to nodal displacements |
| $C$ | direction cosine in two dimensions |
| $C_{x}, C_{y}, C_{z}$ | direction cosines in three dimensions |
| $\underline{d}$ | element and structure nodal displacement matrix, both in global coordinates |
| $\hat{\underline{d}}$ | local-coordinate element nodal displacement matrix |
| $D$ | bending rigidity of a plate |
| $\underline{D}$ | matrix relating stresses to strains |
| $\underline{D}'$ | operator matrix given by Eq. (10.3.16) |
| $e$ | exponential function |
| $E$ | modulus of elasticity |
| $\underline{f}$ | global-coordinate nodal force matrix |
| $\hat{\underline{f}}$ | local-coordinate element nodal force matrix |
| $\underline{f}_{b}$ | body force matrix |
| $\underline{f}_{h}$ | heat transfer force matrix |
| $\underline{f}_{q}$ | heat flux force matrix |
| $\underline{f}_{Q}$ | heat source force matrix |
| $\underline{f}_{s}$ | surface force matrix |
| $\underline{F}$ | global-coordinate structure force matrix |
| $\underline{E}_{c}$ | condensed force matrix |
| $\underline{F}_{i}$ | global nodal forces |
| $\underline{F}_{0}$ | equivalent force matrix |
| $\underline{g}$ | temperature gradient matrix or hydraulic gradient matrix |
| $\underline{G}$ | shear modulus |
| $h$ | heat-transfer (or convection) coefficient |
| $i, j, m$ | nodes of a triangular element |
| $I$ | principal moment of inertia |
| $\underline{J}$ | Jacobian matrix |
| $k$ | spring stiffness |
| $\underline{k}$ | global-coordinate element stiffness or conduction matrix |
| $\underline{k}_{c}$ | condensed stiffness matrix, and conduction part of the stiffness matrix in heat-transfer problems |
| $\hat{\underline{k}}$ | local-coordinate element stiffness matrix |
| $\underline{k}_{h}$ | convective part of the stiffness matrix in heat-transfer problems |
| $\underline{K}$ | global-coordinate structure stiffness matrix |
| $K_{xx}, K_{yy}$ | thermal conductivities (or permeabilities, for fluid mechanics) in the $x$ and $y$ directions, respectively |
| $L$ | length of a bar or beam element |
| $m$ | maximum difference in node numbers in an element |
| $m(x)$ | general moment expression |
| $m_{x}, m_{y}, m_{xy}$ | moments in a plate |
| $\hat{\underline{m}}$ | local mass matrix |
| $\hat{m}_{i}$ | local nodal moments |
| $\underline{M}$ | global mass matrix |
| $\underline{M}^{*}$ | matrix used to relate displacements to generalized coordinates for a linear-strain triangle formulation |
| $\underline{M}^{\prime}$ | matrix used to relate strains to generalized coordinates for a linear-strain triangle formulation |
| $n_{b}$ | bandwidth of a structure |
| $n_{d}$ | number of degrees of freedom per node |
| $\underline{N}$ | shape (interpolation or basis) function matrix |
| $N_{i}$ | shape functions |
| $p$ | surface pressure (or nodal heads in fluid mechanics) |
| $p_{r}, p_{z}$ | radial and axial (longitudinal) pressures, respectively |
| $P$ | concentrated load |
| $\hat{P}$ | concentrated local force matrix |
| q | heat flow (flux) per unit area or distributed loading on a plate |
| $\bar{q}$ | rate of heat flow |
| $q^{*}$ | heat flow per unit area on a boundary surface |
| Q | heat source generated per unit volume or internal fluid source |
| $Q^{*}$ | line or point heat source |
| $Q_{x}, Q_{y}$ | transverse shear line loads on a plate |
| r,θ,z | radial, circumferential, and axial coordinates, respectively |
| R | residual in Galerkin's integral |
| $R_{b}$ | body force in the radial direction |
| $R_{ix}, R_{iy}$ | nodal reactions in x and y directions, respectively |
| s,t,z' | natural coordinates attached to isoparametric element |
| S | surface area |
| t | thickness of a plane element or a plate element |
| $t_{i}, t_{j}, t_{m}$ | nodal temperatures of a triangular element |
| T | temperature function |
| $T_{\infty}$ | free-stream temperature |
| $\underline{T}$ | displacement, force, and stiffness transformation matrix |
| $\underline{T}_{i}$ | surface traction matrix in the i direction |
| u,v,w | displacement functions in the x, y, and z directions, respectively |
| U | strain energy |
| $\Delta U$ | change in stored energy |
| v | velocity of fluid flow |
| $\hat{V}$ | shear force in a beam |
| w | distributed loading on a beam or along an edge of a plane element |
| W | work |
| $x_{i}, y_{i}, z_{i}$ | nodal coordinates in the x, y, and z directions, respectively |
| $\hat{x}, \hat{y}, \hat{z}$ | local element coordinate axes |
| x, y, z | structure global or reference coordinate axes |
| $\underline{X}$ | body force matrix |
| $X_{b}, Y_{b}$ | body forces in the x and y directions, respectively |
| $Z_{b}$ | body force in longitudinal direction (axisymmetric case) or in the z direction (three-dimensional case) |
Greek Symbols
| $\underline{\varepsilon}_{T}$ | thermal strain matrix |
| $\kappa_{x}, \kappa_{y}, \kappa_{xy}$ | curvatures in plate bending |
| $\nu$ | Poisson’s ratio |
| $\phi_{i}$ | nodal angle of rotation or slope in a beam element |
| $\pi_{h}$ | functional for heat-transfer problem |
| $\pi_{p}$ | total potential energy |
| $\rho$ | mass density of a material |
| $\rho_{w}$ | weight density of a material |
| $\omega$ | angular velocity and natural circular frequency |
| $\Omega$ | potential energy of forces |
| $\phi$ | fluid head or potential, or rotation or slope in a beam |
| $\sigma$ | normal stress |
| $\underline{\sigma}_{T}$ | thermal stress matrix |
| $\tau$ | shear stress and period of vibration |
| $\theta$ | angle between the $x$ axis and the local $\hat{x}$ axis for two-dimensional problems |
| $\theta_{p}$ | principal angle |
| $\theta_{x}, \theta_{y}, \theta_{z}$ | angles between the global $x, y$ , and $z$ axes and the local $\hat{x}$ axis, respectively, or rotations about the $x$ and $y$ axes in a plate |
| $\underline{\Psi}$ | general displacement function matrix |
Other Symbols
| $\frac{d(\quad)}{dx}$ | derivative of a variable with respect to $x$ |
| $dt$ | time differential |
| $(^{\cdot})$ | the dot over a variable denotes that the variable is being differentiated with respect to time |
| $[ \quad ]$ | denotes a rectangular or a square matrix |
| $\{ \quad \}$ | denotes a column matrix |
| $( \quad )$ | the underline of a variable denotes a matrix |
| $(^{\cdot})$ | the hat over a variable denotes that the variable is being described in a local coordinate system |
| $[ \quad ]^{-1}$ | denotes the inverse of a matrix |
| $[ \quad ]^{T}$ | denotes the transpose of a matrix |
| $\frac{\partial(\quad)}{\partial x}$ | partial derivative with respect to $x$ |
| $\frac{\partial(\quad)}{\partial\{d\}}$ | partial derivative with respect to each variable in $\{d\}$ |
| ■ | denotes the end of the solution of an example problem |
# Prologue
The finite element method is a numerical method for solving problems of engineering and mathematical physics. Typical problem areas of interest in engineering and mathematical physics that are solvable by use of the finite element method include structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential.
For problems involving complicated geometries, loadings, and material properties, it is generally not possible to obtain analytical mathematical solutions. Analytical solutions are those given by a mathematical expression that yields the values of the desired unknown quantities at any location in a body (here total structure or physical system of interest) and are thus valid for an infinite number of locations in the body. These analytical solutions generally require the solution of ordinary or partial differential equations, which, because of the complicated geometries, loadings, and material properties, are not usually obtainable. Hence we need to rely on numerical methods, such as the finite element method, for acceptable solutions. The finite element formulation of the problem results in a system of simultaneous algebraic equations for solution, rather than requiring the solution of differential equations. These numerical methods yield approximate values of the unknowns at discrete numbers of points in the continuum. Hence this process of modeling a body by dividing it into an equivalent system of smaller bodies or units (finite elements) interconnected at points common to two or more elements (nodal points or nodes) and/or boundary lines and/or surfaces is called discretization. In the finite element method, instead of solving the problem for the entire body in one operation, we formulate the equations for each finite element and combine them to obtain the solution of the whole body.
Briefly, the solution for structural problems typically refers to determining the displacements at each node and the stresses within each element making up the structure that is subjected to applied loads. In nonstructural problems, the nodal unknowns may, for instance, be temperatures or fluid pressures due to thermal or fluid fluxes.
This chapter first presents a brief history of the development of the finite element method. You will see from this historical account that the method has become a practical one for solving engineering problems only in the past 50 years (paralleling the developments associated with the modern high-speed electronic digital computer). This historical account is followed by an introduction to matrix notation; then we describe the need for matrix methods (as made practical by the development of the modern digital computer) in formulating the equations for solution. This section discusses both the role of the digital computer in solving the large systems of simultaneous algebraic equations associated with complex problems and the development of numerous computer programs based on the finite element method. Next, a general description of the steps involved in obtaining a solution to a problem is provided. This description includes discussion of the types of elements available for a finite element method solution. Various representative applications are then presented to illustrate the capacity of the method to solve problems, such as those involving complicated geometries, several different materials, and irregular loadings. Chapter 1 also lists some of the advantages of the finite element method in solving problems of engineering and mathematical physics. Finally, we present numerous features of computer programs based on the finite element method.
# d 1.1 Brief History
This section presents a brief history of the finite element method as applied to both structural and nonstructural areas of engineering and to mathematical physics. References cited here are intended to augment this short introduction to the historical background.
The modern development of the finite element method began in the 1940s in the field of structural engineering with the work by Hrennikoff [1] in 1941 and McHenry [2] in 1943, who used a lattice of line (one-dimensional) elements (bars and beams) for the solution of stresses in continuous solids. In a paper published in 1943 but not widely recognized for many years, Courant [3] proposed setting up the solution of stresses in a variational form. Then he introduced piecewise interpolation (or shape) functions over triangular subregions making up the whole region as a method to obtain approximate numerical solutions. In 1947 Levy [4] developed the flexibility or force method, and in 1953 his work [5] suggested that another method (the stiffness or displacement method) could be a promising alternative for use in analyzing statically redundant aircraft structures. However, his equations were cumbersome to solve by hand, and thus the method became popular only with the advent of the high-speed digital computer.
In 1954 Argyris and Kelsey [6, 7] developed matrix structural analysis methods using energy principles. This development illustrated the important role that energy principles would play in the finite element method.
The first treatment of two-dimensional elements was by Turner et al. [8] in 1956. They derived stiffness matrices for truss elements, beam elements, and two-dimensional triangular and rectangular elements in plane stress and outlined the procedure