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An Introduction to the Finite Element Method

ISBN: 9780072466850 | 0072466855
Edition: 3rd
Format: Hardcover
Publisher: McGraw-Hill Science/Engineering/Math
Pub. Date: 1/11/2005

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SummaryTable of Contents
J.N. Reddy's, "An Introduction to the Finite Element Method, third edition" is an update of one of the most popular FEM textbooks available. The book retains its strong conceptual approach, clearly examining the mathematical underpinnings of FEM, and providing a general approach of engineering application areas. Known for its detailed, carefully selected example problems and extensive selection of homework problems, the author has comprehensively covered a wide range of engineering areas making the book approriate for all engineering majors, an... MORE
Prefacexiv
Introduction
1(26)
General Comments
1(1)
... MORE
2(7)
Numerical Simulations
9(4)
The Finite Element Method
13(11)
The Basic Idea
13(1)
The Basic Features
13(8)
Some Remarks
21(2)
A Brief Review of History and Recent Developments
23(1)
The Present Study
24(1)
Summary
24(3)
Problems
25(1)
References for Additional Reading
26(1)
Mathematical Preliminaries, Integral Formulations, and Variational Methods
27(76)
General Introduction
27(4)
Variational Principles and Methods
27(1)
Variational Formulations
28(1)
Need for Weighted-Integral Statements
28(3)
Some Mathematical Concepts and Formulae
31(10)
Coordinate Systems and the Del Operator
31(2)
Boundary Value, Initial Value, and Eigenvalue Problems
33(3)
Integral Identities
36(3)
Linear and Bilinear Functionals
39(2)
Elements of Calculus of Variations
41(17)
Introduction
41(1)
Variational Operator and First Variation
41(3)
Fundamental Lemma of Variational Calculus
44(1)
The Euler Equations
44(3)
Natural and Essential Boundary Conditions
47(7)
Hamilton's Principle
54(4)
Integral Formulations
58(16)
Introduction
58(1)
Weighted-Integral and Weak Formulations
58(6)
Linear and Bilinear Forms and Quadratic Functionals
64(2)
Examples
66(8)
Variational Methods
74(23)
Introduction
74(1)
The Ritz Method
74(2)
Approximation Functions
76(1)
Examples
77(14)
The Method of Weighted Residuals
91(6)
Summary
97(6)
Problems
98(4)
References for Additional Reading
102(1)
Second-Order Differential Equations in One Dimension: Finite Element Models
103(52)
Background
103(2)
Basic Steps of Finite Element Analysis
105(36)
Model Boundary Value Problem
105(1)
Discretization of the Domain
106(2)
Derivation of Element Equations
108(17)
Connectivity of Elements
125(7)
Imposition of Boundary Conditions
132(1)
Solution of Equations
132(2)
Postcomputation of the Solution
134(7)
Some Remarks
141(5)
Axisymmetric Problems
146(4)
Model Equation
146(1)
Weak Form
147(1)
Finite Element Model
148(2)
Summary
150(5)
Problems
151(3)
References for Additional Reading
154(1)
Second-Order Differential Equations in One Dimension: Applications
155(78)
Preliminary Comments
155(1)
Discrete Systems
156(6)
Linear Elastic Spring
156(2)
Torsion of Circular Shafts
158(1)
Electrical Resistor Circuits
159(2)
Fluid Flow through Pipes
161(1)
Heat Transfer
162(19)
Governing Equations
162(4)
Finite Element Models
166(1)
Numerical Examples
166(15)
Fluid Mechanics
181(2)
Governing Equations
181(1)
Finite Element Model
181(2)
Solid and Structural Mechanics
183(11)
Preliminary Comments
183(1)
Finite Element Model of Bars and Cables
184(1)
Numerical Examples
185(9)
Plane Trusses
194(20)
Introduction
194(1)
Basic Truss Element
194(1)
General Truss Element
195(7)
Constraint Equations: Penalty Approach
202(9)
Constraint Equations: A Direct Approach
211(3)
Summary
214(19)
Problems
215(16)
References for Additional Reading
231(2)
Beams and Frames
233(58)
Introduction
233(1)
Euler--Bernoulli Beam Element
233(28)
Governing Equation
233(1)
Discretization of the Domain
234(1)
Derivation of Element Equations
234(9)
Assembly of Element Equations
243(2)
Imposition of Boundary Conditions
245(2)
Postprocessing of the Solution
247(1)
Numerical Examples
248(13)
Timoshenko Beam Elements
261(13)
Governing Equations
261(1)
Weak Form
262(2)
General Finite Element Model
264(2)
Consistent Interpolation Elements
266(4)
Reduced Integration Element
270(1)
Numerical Examples
271(3)
Plane Frame Elements
274(7)
Introductory Comments
274(1)
Frame Element
274(7)
Summary
281(10)
Problems
282(8)
References for Additional Reading
290(1)
Eigenvalue and Time-Dependent Problems
291(52)
Eigenvalue Problems
291(23)
Introduction
291(1)
Formulation of Eigenvalue Problems
292(3)
Finite Element Formulation
295(19)
Time-Dependent Problems
314(23)
Introduction
314(2)
Semidiscrete Finite Element Models
316(2)
Parabolic Equations
318(6)
Hyperbolic Equations
324(2)
Mass Lumping
326(2)
Applications
328(9)
Summary
337(6)
Problems
337(5)
References for Additional Reading
342(1)
Computer Implementation
343(66)
Numerical Integration
343(13)
Background
343(2)
Natural Coordinates
345(1)
Approximation of Geometry
346(1)
Isoparametric Formulations
347(1)
Numerical Integration
348(8)
Computer Implementation
356(14)
Introductory Comments
356(1)
General Outline
357(2)
Preprocessor
359(1)
Calculation of Element Matrices (Processor)
360(3)
Assembly of Element Equations (Processor)
363(2)
Imposition of Boundary Conditions (Processor)
365(2)
Solving Equations and Postprocessing
367(3)
Applications of Program FEMID
370(31)
General Comments
370(1)
Illustrative Examples
370(31)
Summary
401(8)
Problems
401(5)
References for Additional Reading
406(3)
Single-Variable Problems in Two Dimensions
409(116)
Introduction
409(1)
Boundary Value Problems
410(32)
The Model Equation
410(1)
Finite Element Discretization
411(1)
Weak Form
412(3)
Finite Element Model
415(2)
Derivation of Interpolation Functions
417(8)
Evaluation of Element Matrices and Vectors
425(11)
Assembly of Element Equations
436(4)
Postcomputations
440(1)
Axisymmetric Problems
441(1)
A Numerical Example
442(11)
Some Comments on Mesh Generation and Imposition of Boundary Conditions
453(5)
Discretization of a Domain
453(2)
Generation of Finite Element Data
455(1)
Imposition of Boundary Conditions
456(2)
Applications
458(32)
Conduction and Convection Heat Transfer
458(14)
Fluid Mechanics
472(13)
Solid Mechanics
485(5)
Eigenvalue and Time-Dependent Problems
490(14)
Introduction
490(1)
Parabolic Equations
491(8)
Hyperbolic Equations
499(5)
Summary
504(21)
Problems
504(18)
References for Additional Reading
522(3)
Interpolation Functions, Numerical Integration, and Modeling Considerations
525(52)
Introduction
525(1)
Element Library
525(15)
Triangular Elements
525(7)
Rectangular Elements
532(5)
The Serendipity Elements
537(2)
Hermite Cubic Interpolation Functions
539(1)
Numerical Integration
540(21)
Preliminary Comments
540(3)
Coordinate Transformations
543(6)
Integration over a Master Rectangular Element
549(8)
Integration over a Master Triangular Element
557(4)
Modeling Considerations
561(8)
Preliminary Comments
561(1)
Element Geometries
562(1)
Mesh Generation
563(4)
Load Representation
567(2)
Summary
569(8)
Problems
570(5)
References for Additional Reading
575(2)
Flows of Viscous Incompressible Fluids
577(30)
Preliminary Comments
577(1)
Governing Equations
577(2)
Velocity-Pressure Formulation
579(4)
Weak Formulation
579(2)
Finite Element Model
581(2)
Penalty Function Formulation
583(5)
Preliminary Comments
583(1)
Formulation of the Flow Problem as a Constrained Problem
583(1)
Lagrange Multiplier Model
584(1)
Penalty Model
585(3)
Time Approximation
588(1)
Computational Aspects
588(3)
Properties of the Matrix Equations
588(1)
Choice of Elements
589(1)
Evaluation of Element Matrices in the Penalty Model
590(1)
Postcomputation of Stresses
591(1)
Numerical Examples
591(11)
Summary
602(5)
Problems
603(2)
References for Additional Reading
605(2)
Plane Elasticity
607(28)
Introduction
607(1)
Governing Equations
607(5)
Plane Strain
607(1)
Plane Stress
608(2)
Summary of Equations
610(2)
Weak Formulations
612(2)
Preliminary Comments
612(1)
Principle of Virtual Displacements in Vector Form
612(1)
Weak Form of the Governing Differential Equations
613(1)
Finite Element Model
614(3)
General Model
614(3)
Eigenvalue and Transient Problems
617(1)
Evaluation of Integrals
617(3)
Assembly of Finite Element Equations
620(2)
Examples
622(7)
Summary
629(6)
Problems
629(4)
References for Additional Reading
633(2)
Bending of Elastic Plates
635(32)
Introduction
635(2)
Classical Plate Theory
637(9)
Displacement Field
637(1)
Virtual Work Statement
638(4)
Finite Element Model
642(1)
Plate Bending Elements
643(3)
Shear Deformation Plate Theory
646(7)
Displacement Field
646(2)
Virtual Work Statement
648(2)
Finite Element Model
650(2)
Shear Locking and Reduced Integration
652(1)
Eigenvalue and Time-Dependent Problems
653(2)
Examples
655(8)
Summary
663(4)
Problems
663(2)
References for Additional Reading
665(2)
Computer Implementation of Two-Dimensional Problems
667(44)
Introduction
667(2)
Preprocessor
669(1)
Element Computations (Processor)
669(6)
Applications of the Computer Program FEM2D
675(28)
Introduction
675(6)
Description of Mesh Generators
681(5)
Applications (Illustrative Examples)
686(17)
Summary
703(8)
Problems
705(4)
References for Additional Reading
709(2)
Prelude to Advanced Topics
711(46)
Introduction
711(1)
Alternative Finite Element Models
711(14)
Introductory Comments
711(1)
Weighted Residual Finite Element Models
712(10)
Mixed Formulations
722(3)
Three-Dimensional Problems
725(11)
Heat Transfer
726(1)
Flows of Viscous Incompressible Fluids
727(1)
Elasticity
728(3)
Three-Dimensional Finite Elements
731(4)
A Numerical Example
735(1)
Nonlinear Problems
736(7)
General Comments
736(1)
Bending of Euler--Bernoulli Beams
736(2)
The Navier--Stokes Equations in Two Dimensions
738(1)
Solution Methods for Nonlinear Algebraic Equations
739(1)
Numerical Examples
740(3)
Errors in Finite Element Analysis
743(7)
Types of Errors
743(1)
Measures of Errors
744(1)
Convergence and Accuracy of Solutions
745(5)
Summary
750(7)
Problems
751(2)
References for Additional Reading
753(4)
Index757

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