A First Course in the Finite Element Method

by Logan, Daryl L.

• ISBN13: 9780534385170
• ISBN10: 0534385176
• Edition: 3rd
• Format: Hardcover
• Copyright: 2001-04-05
• Publisher: CL Engineering

Summary

This third edition provides a simple, basic approach to the finite element method that can be understood by readers. It does not have the usual prerequisites (such as structural analysis) required by most available books in this area. The book is written primarily as a basic learning tool for civil and mechanical engineering readers whose main interest is in stress analysis and heat transfer. The book is geared toward those who want to apply the finite element method as a tool to solve practical physical problems.

Table of Contents

Introduction	1	(25)
Prologue	1	(1)
Brief History	2	(1)
Introduction to Matrix Notation	3	(3)
Role of the Computer	6	(1)
General Steps of the Finite Element Method	6	(7)
Applications of the Finite Element Method	13	(5)
Advantages of the Finite Element Method	18	(1)
Computer Programs for the Finite Element Method	19	(7)
References	22	(3)
Problems	25	(1)
Introduction to the Stiffness (Displacement) Method	26	(37)
Introduction	26	(1)
Definition of the Stiffness Matrix	26	(1)
Derivation of the Stiffness Matrix for a Spring Element	27	(5)
Example of a Spring Assemblage	32	(3)
Assembling the Total Stiffness Matrix by Superposition (Direct Stiffness Method)	35	(2)
Boundary Conditions	37	(13)
Potential Energy Approach to Derive Spring Element Equations	50	(13)
References	58	(1)
Problems	59	(4)
Development of Truss Equations	63	(74)
Introduction	63	(1)
Derivation of the Stiffness Matrix for a Bar Element in Local Coordinates	63	(6)
Selecting Approximation Functions for Displacements	69	(2)
Transformation of Vectors in Two Dimensions	71	(3)
Global Stiffness Matrix	74	(4)
Computation of Stress for a Bar in the x-y Plane	78	(2)
Solution of a Plane Truss	80	(7)
Transformation Matrix and Stiffness Matrix for a Bar in Three-Dimensional Space	87	(5)
Use of Symmetry in Structure	92	(3)
Inclined, or Skewed, Supports	95	(6)
Potential Energy Approach to Derive Bar Element Equations	101	(11)
Comparison of Finite Element Solution to Exact Solution for Bar	112	(4)
Galerkin's Residual Method and Its Application to a One-Dimensional Bar	116	(21)
References	119	(1)
Problems	120	(17)
Development of Beam Equations	137	(51)
Introduction	137	(1)
Beam Stiffness	138	(5)
Example of Assemblage of Beam Stiffness Matrices	143	(2)
Examples of Beam Analysis Using the Direct Stiffness Method	145	(9)
Distributed Loading	154	(11)
Comparision of the Finite Element Solution to the Exact Solution for a Beam	165	(6)
Beam Element with Nodal Hinge	171	(5)
Potential Energy Approach to Derive Beam Element Equations	176	(3)
Galerkin's Method for Deriving Beam Element Equations	179	(9)
References	181	(1)
Problems	181	(7)
Frame and Grid Equations	188	(76)
Introduction	188	(1)
Two-Dimensional Arbitrarily Oriented Beam Element	188	(4)
Rigid Plane Frame Examples	192	(19)
Inclined or Skewed Supports-Frame Element	211	(1)
Grid Equations	212	(17)
Beam Element Arbitrarily Oriented in Space	229	(5)
Concept of Substructure Analysis	234	(30)
References	240	(1)
Problems	240	(24)
Development of the Plane Stress and Plane Strain Stiffness Equations	264	(43)
Introduction	264	(1)
Basic Concepts of Plane Stress and Plane Strain	265	(5)
Derivation of the Constant-Strain Triangular Element Stiffness Matrix and Equations	270	(14)
Treatment of Body and Surface Forces	284	(5)
Explicit Expression for the Constant-Strain Triangle Stiffness Matrix	289	(2)
Finite Element Solution of a Plane Stress Problem	291	(16)
References	301	(1)
Problems	301	(6)
Practical Considerations in Modeling; Interpreting Results; and Examples of Plane Stress/Strain Analysis	307	(37)
Introduction	307	(1)
Finite Element Modeling	308	(10)
Equilibrium and Compatibility of Finite Element Results	318	(2)
Convergence of Solution	320	(1)
Interpretation of Stresses	321	(2)
Static Condensation	323	(4)
Flowchart for the Solution of Plane Stress/Strain Problems	327	(1)
Computer Program Results for Some Plane Stress/Strain Problems	328	(16)
References	331	(1)
Problems	332	(12)
Development of the Linear-Strain Triangle Equations	344	(14)
Introduction	344	(1)
Derivation of the Linear-Strain Triangular Element Stiffness Matrix and Equations	344	(5)
Example LST Stiffness Determination	349	(3)
Comparision of Element	352	(6)
References	354	(1)
Problems	355	(3)
Axisymmetric Elements	358	(28)
Introduction	358	(1)
Derivation of the Stiffness Matrix	358	(10)
Solution of an Axisymmetric Pressure Vessel	368	(8)
Applications of Axisymmetric Elements	376	(10)
References	380	(1)
Problems	381	(5)
Isoparametric Formulation	386	(35)
Introduction	386	(1)
Isoparametric Formulation of the Bar Element Stiffness Matrix	386	(6)
Rectangular Plane Stress Element	392	(3)
Isoparametric Formulation of the Plane Element Stiffness Matrix	395	(9)
Gaussian Quadrature (Numerical Integration)	404	(3)
Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadrature	407	(6)
Higher-Order Shape Functions	413	(8)
References	417	(1)
Problems	417	(4)
Three-Dimensional Stress Analysis	421	(20)
Introduction	421	(1)
Three-Dimensional Stress and Strain	421	(2)
Tetrahedral Element	423	(7)
Isoparametric Formulation	430	(11)
References	436	(1)
Problems	436	(5)
Plate Bending Element	441	(17)
Introduction	441	(1)
Basic Concepts of Plate Bending	441	(4)
Derivation of a Plate Bending Element Stiffness Matrix and Equations	445	(5)
Some Plate Element Numerical Comparisions	450	(2)
Computer Solution for a Plate Bending Problem	452	(6)
References	454	(1)
Problems	455	(3)
Heat Transfer and Mass Transport	458	(50)
Introduction	458	(1)
Derivation of the Basic Differential Equation	459	(3)
Heat Transfer with Convection	462	(1)
Typical Units: Thermal, Conductivities, K; and Heat-Transfer Coefficients, h	463	(1)
One-Dimensional Finite Element Formulation Using a Variational Method	464	(14)
Two-Dimensional Finite Element Formulation	478	(9)
Line or Point Sources	487	(3)
One-Dimensional Heat Transfer with Mass Transport	490	(1)
Finite Element Formulation of Heat Transfer with Mass Transport by Galerkin's Method	491	(4)
Flowchart and Examples of a Heat-Transfer Program	495	(13)
References	499	(1)
Problems	499	(9)
Fluid Flow	508	(24)
Introduction	508	(1)
Derivation of the Basic Differential Equations	508	(5)
One-Dimensional Finite Element Formulation	513	(8)
Two-Dimensional Finite Element Formulation	521	(5)
Flowchart and Example of a Fluid-Flow Program	526	(6)
References	527	(1)
Problems	528	(4)
Thermal Stress	532	(27)
Introduction	532	(1)
Formulation of the Thermal Stress Problem and Examples	532	(27)
References	553	(1)
Problems	554	(5)
Structural Dynamics and Time-Dependent Heat Transfer	559	(57)
Introduction	559	(1)
Dynamics of a Spring-Mass System	559	(2)
Direct Derivation of the Bar Element Equations	561	(4)
Numerical Integration in Time	565	(12)
Natural Frequencies of a One-Dimensional Bar	577	(4)
Time-Dependent One-Dimensional Bar Analysis	581	(5)
Beam Element Mass Matrices and Natural Frequencies	586	(5)
Truss, Plane Frame, Plane Stress/Strain, Axisymmetric, and Solid Element Mass Matrices	591	(4)
Time-Dependent Heat Transfer	595	(7)
Computer Program Example Solutions for Structural Dynamics	602	(14)
References	609	(1)
Problems	610	(6)
Appendix A Matrix Algebra	616	(14)
Introduction	616	(12)
A.1 Definition of a Matrix	616	(1)
A.2 Matrix Operations	617	(7)
A.3 Cofactor or Adjoint Method to Determine the Inverse of a Matrix	624	(2)
A.4 Inverse of a Matrix by Row Reduction	626	(2)
References	628	(1)
Problems	628	(2)
Appendix B Methods for Solution of Simultaneous Linear Equations	630	(22)
Introduction	630	(19)
B.1 General Form of the Equations	630	(1)
B.2 Uniqueness, Nonuniqueness, and Nonexistence of Solution	631	(1)
B.3 Methods for Solving Linear Algebraic Equations	632	(11)
B.4 Banded-Symmetric Matrices, Bandwidth, Skyline, and Wavefront Methods	643	(6)
References	649	(1)
Problems	650	(2)
Appendix C Equations from Elasticity Theory	652	(8)
Introduction	652	(7)
C.1 Differential Equations of Equilibrium	652	(2)
C.2 Strain/Displacement and Compatibility Equations	654	(2)
C.3 Stress/Strain Relationships	656	(3)
Reference	659	(1)
Appendix D Equivalent Nodal Forces	660	(3)
Problems	660	(3)
Appendix E Principle of Virtual Work	663	(4)
References	666	(1)
Answers to Selected Problems	667	(22)
Index	689