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Daryl Logan's clear and easy to understand text provides a thorough treatment of the finite element method and how to apply it to solve practical physical problems in engineering. Concepts are presented simply, making it understandable for students of all levels of experience. The first edition of this book enjoyed considerable success and this new edition includes a chapter on plates and plate bending, along with additional homework exercise. All examples in this edition have been updated to Algor™ Release 12.
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 40 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.
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'  in 1941 and McHenry  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  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  developed the flexibility or force method, and in 1953 his work  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.  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 commonly known as the direct stiffness method for obtaining the total structure stiffness matrix. Along with the development of the high-speed digital computer in the early 1950s, the work of Turner et al.  prompted further development of finite element stiffness equations expressed in matrix notation. The phrase finite element was introduced by Clough  in 1960 when both triangular and rectangular elements were used for plane stress analysis.
A fiat, rectangular-plate bending-element stiffness matrix was developed by Melosh [ 10] in 1961. This was followed by development of the curved-shell bendingelement stiffness matrix for axisymmetric shells and pressure vessels by Grafton and Strome  in 1963.
Extension of the finite element method to three-dimensional problems with the development of a tetrahedral stiffness matrix was done by Martin  in 1961, by Gallagher et al.  in 1962, and by Melosh  in 1963. Additional three-dimensional elements were studied by Argyris [ 15] in 1964. The special case of axisymmetric solids was considered by Clough and Rashid  and Wilson  in 1965.
Most of the finite element work up to the early 1960s dealt with small strains and small displacements, elastic material behavior, and static loadings. However, large deflection and thermal analysis were considered by Turner et al. [ 18] in 1960 and material nonlinearities by Gallagher et al.  in 1962, whereas buckling problems were initially treated by Gallagher and Padlog [ 19] in 1963. Zienkiewicz et al.  extended the method to visco-elasticity problems in 1968.
In 1965 Archer  considered dynamic analysis in the development of the consistent-mass matrix, which is applicable to analysis of distributed-mass systems such as bars and beams in structural analysis.
With Melosh's  realization in 1963 that the finite element method could be set up in terms of a variational formulation, it began to be used to solve nonstructural applications. Field problems, such as determination of the torsion of a shaft, fluid flow, and heat conduction, were solved by Zienkiewicz and Cheung  in 1965, Martin  in 1968, and Wilson and Nickel  in 1966.
Further extension of the method was made possible by the adaptation of weighted residual methods, first by Szabo and Lee  in 1969 to derive the previously known elasticity equations used in structural analysis and then by Zienkiewicz and Parekh  in 1970 for transient field problems. It was then recognized that when direct formulations and variational formulations are difficult or not possible to use, the method of weighted residuals may at times be appropriate. For example, in 1977 Lyness et al.  applied the method of weighted residuals to the determination of magnetic field.
In 1976 Belytschko [28, 29] considered problems associated with largedisplacement nonlinear dynamic behavior, and improved numerical techniques for solving the resulting systems of equations.
A relatively new field of application of the finite element method is that of bioengineering [30, 31]. This field is still troubled by such difficulties as nonlinear materials, geometric nonlinearities, and other complexities still being discovered.
From the early 1950s to the present, enormous advances have been made in the application of the finite element method to solve complicated engineering problems. Engineers, applied mathematicians, and other scientists will undoubtedly continue to develop new applications. For an extensive bibliography on the finite element method, consult the work of Kardestuncer , Clough , or Noor ...
1. INTRODUCTION Prologue / Brief History / Introduction to Matrix Notation / Role of the Computer / General Steps of the Finite Element Method / Applications of the Finite Element Method / Advantages of the Finite Element Method / Computer Programs for the Finite Element Method / References / Problems 2. INTRODUCTION TO THE STIFFNESS (DISPLACEMENT) METHOD Introduction / Definition of the Stiffness Matrix / Derivation of the Stiffness Matrix for a Spring Element / Example of a Spring Assemblage / Assembling the Total Stiffness Matrix by Superposition (Direct Stiffness Method) / Boundary Conditions / Potential Energy Approach to Derive Spring Element Equations / References / Problems 3. DEVELOPMENT OF TRUSS EQUATIONS Introduction / Derivation of the Stiffness Matrix for a Bar Element in Local Coordinates / Selecting Approximation Functions for Displacements / Transformation of Vectors in Two Dimensions / Global Stiffness Matrix / Computation of Stress for a Bar in the x-y Plane / Solution of a Plane Truss / Transformation Matrix and Stiffness Matrix for a Bar in Three-Dimensional Space / Use of Symmetry in Structure / Inclined, or Skewed, Supports / Potential Energy Approach to Derive Bar Element Equations / Comparison of Finite Element Solution to Exact Solution for Bar / Galerkin's Residual Method and Its Application to a One-Dimensional Bar / References / Problems 4. ALGOR™ PROGRAM FOR TRUSS ANALYSIS Introduction / Overview of the Algor system and Flowcharts for the Solution of a Truss Problem Using Algor / Algor Example Solutions for Truss Analysis / References / Problems 5. DEVELOPMENT OF BEAM EQUATIONS Introduction / Beam Stiffness / Example of Assemblage of Beam Stiffness Matrices / Examples of Beam Analysis Using the Direct Stiffness Method / Distributed Loading / Comparison of Finite Element Solution to Exact Solution for Beam / Beam Element with Nodal Hinge / Potential Energy Approach to Derive Beam Element Equations / Galerkin's Method to Derive Beam Element Equations / Algor Example Solutions for Beam Analysis / References / Problems 6. FRAME AND GRID EQUATIONS Introduction / Two-Dimensional Arbitrarily Oriented Beam Element / Rigid Plane Frame Examples / Inclined or Skewed Supports—-Frame Element / Grid Equations / Beam Element Arbitrarily Oriented in Space / Concept of Substructure Analysis / Algor Example Solutions for Plane Frame, Grid, and Space Frame Analysis / References / Problems 7. DEVELOPMENT OF THE PLANE STRESS AND PLANE STRAIN STIFFNESS EQUATIONS Introduction / Basic Concepts of Plane Stress and Plane Strain / Derivation of the Constant-Strain Triangular Element Stiffness Matrix and Equations / Treatment of Body and Surface Forces / Explicit Expression for the Constant-Strain Triangle Stiffness Matrix / Finite Element Solution of a Plane Stress Problem / References / Problems 8. PRACTICAL CONSIDERATIONS IN MODELING; INTERPRETING RESULTS; AND USE OF THE ALGOR™ PROGRAM FOR PLANE STRESS/STRAIN ANALYSIS Introduction / Finite Element Modeling / Equilibrium and Compatibility of Finite Element Results / Convergence of Solution / Interpretation of Stresses / Static Condensation / Flowchart for the Solution of Plane Stress/Strain / Problems and Typical Steps Using Algor / Algor Example Solutions for Plane Stress/Strain Analysis / References / Problems 9. DEVELOPMENT OF THE LINEAR-STRAIN TRIANGLE EQUATIONS Introduction / Derivation of the Linear-Strain Triangular Element Stiffness Matrix and Equations / Example LST Stiffness Determination / Comparison of Elements / References / Problems 10. AXISYMMETRIC ELEMENTS Introduction / Derivation of the Stiffness Matrix / Solution of an Axisymmetric Pressure Vessel / Applications of Axisymmetric Elements / Algor Example Solutions for Axisymmetric Problems / References / Problems 11. ISOPARAMETRIC FORMULATION Introduction / Isoparametric Formulation of the Bar Element Stiffness Matrix / Rectangular Plane Stress Element / Isoparametric Formulation of the Plane Element Stiffness Matrix / Gaussian Quadrature (Numerical Integration) / Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadrature / Higher-Order Shape Functions / References / Problems 12. THREE-DIMENSIONAL STRESS ANALYSIS Introduction / Three-Dimensional Stress and Strain / Tetrahedral Element / Isoparametric Formulation / Algor Example Solutions of Three-Dimensional Stress Analysis / References / Problems 13. HEAT TRANSFER AND MASS TRANSPORT Introduction / Derivation of the Basic Differential Equation / Heat Transfer with Convection / Typical Units; Thermal Conductivities, K; and Heat-Transfer Coefficients, h / One-Dimensional Finite Element Formulation Using a Variational Method / Two-Dimensional Finite Element Formulation / Line or Point Sources / One-Dimensional Heat Transfer with Mass Transport / Finite Element Formulation of Heat Transfer with Mass Transport by Galerkin's Method / Flowchart of a Heat-Transfer Program / Algor Example Solutions for Heat-Transfer Problems / References / Problems 14. FLUID FLOW Introduction / Derivation of the Basic Differential Equations / One-Dimensional Finite Element Formulation / Two-Dimensional Finite Element Formulation / Flowchart of a Fluid-Flow Program / Algor Example Solutions for Two-Dimensional Steady-State Fluid Flow / References / Problems 15. THERMAL STRESS Introduction / Formulation of the Thermal Stress Problem and Examples / Algor Example Solutions for Thermal Stress Problems / References / Problems 16. STRUCTURAL DYNAMICS AND TIME-DEPENDENT HEAT TRANSFER Introduction / Dynamics of a Spring-Mass System / Direct Derivation of the Bar Element Equations / Numerical Integration in Time / Natural Frequencies of a One-Dimensional Bar / Time-Dependent One-Dimensional Bar Analysis / Beam Element Mass Matrices and Natural Frequencies / Truss, Plane Frame, Plane Stress/Strain, Axisymmetric, and Solid Element Mass Matrices / Time-Dependent Heat Transfer / Algor Example Solutions for Structural Dynamics and Transient Heat Transfer / References / Problems 17. PLATE BENDING ELEMENT Introduction / Basic Concepts of Plate Bending / Derivation of a Plate Bending Element Stiffness Matrix and Equations / Some Plate Element Numerical Comparisons / Algor Example Solutions for Plate Bending Problems / References / Problems / APPENDIX A: MATRIX ALGEBRA / Introduction / Definition of a Matrix / Matrix Operations / Cofactor or Adjoint Method to Determine the Inverse of a Matrix / Inverse of a Matrix by Row Reduction / References / Problems / APPENDIX B: METHODS FOR SOLUTION OF SIMULTANEOUS LINEAR EQUATIONS / Introduction / General Form of the Equations / Uniqueness, Nonuniqueness, and Nonexistence of Solution / Methods for Solving Linear Algebraic Equations / Banded-Symmetric Matrices, Bandwidth, Skyline, and Wavefront Methods / References / Problems / APPENDIX C: EQUATIONS FROM ELASTICITY THEORY / Introduction / Differential Equations of Equilibrium / Strain/Displacement and Compatibility Equations / Stress/Strain Relationships / Reference / APPENDIX D: EQUIVALENT NODAL FORCES / Problems / APPENDIX E: PRINCIPLE OF VIRTUAL WORK / References / APPENDIX F: BASICS OF ALGOR™ / Introduction / Hardware Requirements for Windows Installation / Conventions / Getting Around the Menu System / Function Keys / Algor Processor Names / File Extensions Generated by the Algor System / Checking Model for Defects by Using Superview / ANSWERS TO SELECTED PROBLEMS / INDEX
General principles are presented for each topic, followed by traditional applications of these principles, which are in turn followed by computer applications where possible. This approach is taken to illustrate concepts used for computer analysis of large-scale problems. The book proceeds from basic to advanced topics and can be suitably used in a two-course sequence. Topics include basic treatments of (1) simple springs and bars, leading to two- and three-dimensional truss analysis; (2) beam bending, leading to plane frame and grid analysis, and space frame analysis; (3) elementary plane stress/strain elements, leading to more advanced plane stress/strain elements; (4) axisymetric stress; (5) isoparametric formulation of the finite element method; (6) three-dimensional stress; (7) heat transfer and fluid mass transport; (8) basic fluid mechanics; (9) thermal stress; and (10) time-dependent stress and heat transfer.
New topics/features include: how to handle inclined or skewed supports, beam element with nodal hinge, beam element arbitrarily located in space, the concept of substructure analysis, a completely new chapter on fluid mechanics, and a diskette including the source codes of six basic programs used in the text.
The direct approach, the principle of minimum potential energy, and Galerkin's residual method are introduced at various stages, as required, to develop the equations needed for analysis. Appendices include: (1) basic matrix algebra used throughout the text, (2) solution methods for simultaneous equations, (3) basic theory of elasticity, and (4) the principle of virtual work. More than 60 solved problems appear throughout the text. Most of the examples are solved "longhand" to illustrate the concepts; many of them are solved by digital computer to illustrate the use of the computer programs provided on the diskette enclosed in the back of the book. More than 300 end-of-chapter problems, including a number of new ones, are provided to reinforce concepts. Answers to many problems are included in the back of the book. Those end-of-chapter problems to be solved using a computer program are marked with a computer symbol. Computer programs are incorporated directly into relevant places in the text to create a natural extension from basic principles to longhand examples and then to computer program examples. The programs are written specifically for instructional purposes and source codes written in FORTRAN language are included on the accompanying diskette. Each program solves a specific class or type of problem. These programs arc easy for students to use. A single lecture is sufficient to explain how to use most of them.
To run any of the six special-purpose programs, you must create a ".EXE" file of the program. This is done using a FORTRAN compiler program such as the MS-DOS FORTRAN compiler.