A First Course in the Finite Element Method

by Logan, Daryl L.

• ISBN13: 9780534552985
• ISBN10: 0534552986
• Edition: 4th
• Format: Hardcover
• Copyright: 2006-07-25
• Publisher: CL Engineering

Summary

A First Course in the Finite Element Analysis provides a simple, basic approach to the finite element method that can be understood by both undergraduate and graduate students. It does not have the usual prerequisites (such as structural analysis) required by most available texts in this area. The book is written primarily as a basic learning tool for the undergraduate student in civil and mechanical engineering whose main interest is in stress analysis and heat transfer. The text is geared toward those who want to apply the finite element method as a tool to solve practical physical problems. This revised fourth edition includes the addition of a large number of new problems (including SI problems), an appendix for mechanical and thermal properties, and more civil applications.

Table of Contents

Introduction	1	(27)
Prologue	1	(1)
Brief History	2	(2)
Introduction to Matrix Notation	4	(2)
Role of the Computer	6	(1)
General Steps of the Finite Element Method	7	(8)
Applications of the Finite Element Method	15	(4)
Advantages of the Finite Element Method	19	(4)
Computer Programs for the Finite Element Method	23	(5)
References	24	(3)
Problems	27	(1)
Introduction to the Stiffness (Displacement) Method	28	(37)
Introduction	28	(1)
Definition of the Stiffness Matrix	28	(1)
Derivation of the Stiffness Matrix for a Spring Element	29	(5)
Example of a Spring Assemblage	34	(3)
Assembling the Total Stiffness Matrix by Superposition (Direct Stiffness Method)	37	(2)
Boundary Conditions	39	(13)
Potential Energy Approach to Derive Spring Element Equations	52	(13)
References	60	(1)
Problems	61	(4)
Development of Truss Equations	65	(86)
Introduction	65	(1)
Derivation of the Stiffness Matrix for a Bar Element in Local Coordinates	66	(6)
Selecting Approximation Functions for Displacements	72	(3)
Transformation of Vectors in Two Dimensions	75	(3)
Global Stiffness Matrix	78	(4)
Computation of Stress for a Bar in the x-y Plane	82	(2)
Solution of a Plane Truss	84	(8)
Transformation, Matrix and Stiffness Matrix for a Bar in Three-Dimensional Space	92	(8)
Use of Symmetry in Structure	100	(3)
Inclined, or Skewed, Supports	103	(6)
Potential Energy Approach to Derive Bar Element Equations	109	(11)
Comparison of Finite Element Solution to Exact Solution for Bar	120	(4)
Galerkin's Residual Method and Its Use to Derive the One-Dimensional Bar Element Equations	124	(3)
Other Residual Methods and Their Application to a One-Dimensional Bar Problem	127	(24)
References	132	(1)
Problems	132	(19)
Development of Beam Equations	151	(63)
Introduction	151	(1)
Beam Stiffness	152	(9)
Example of Assemblage of Beam Stiffness Matrices	161	(2)
Examples of Beam Analysis Using the Direct Stiffness Method	163	(12)
Distributed Loading	175	(13)
Comparison of the Finite Element Solution to the Exact Solution for a Beam	188	(6)
Beam Element with Nodal Hinge	194	(5)
Potential Energy Approach to Derive Beam Element Equations	199	(2)
Galerkin's Method for Deriving Beam Element Equations	201	(13)
References	203	(1)
Problems	204	(10)
Frame and Grid Equations	214	(90)
Introduction	214	(1)
Two-Dimensional Arbitrarily Oriented Beam Element	214	(4)
Rigid Plane Frame Examples	218	(19)
Inclined or Skewed Supports---Frame Element	237	(1)
Grid Equations	238	(17)
Beam Element Arbitrarily Oriented in Space	255	(14)
Concept of Substructure Analysis	269	(35)
References	275	(1)
Problems	275	(29)
Development of the Plane Stress and Plane Strain Stiffness Equations	304	(46)
Introduction	304	(1)
Basic Concepts of Plane Stress and Plane Strain	305	(5)
Derivation of the Constant-Strain Triangular Element Stiffness Matrix and Equations	310	(14)
Treatment of Body and Surfaces Forces	324	(5)
Explicit Expression for the Constant-Strain Triangle Stiffness Matrix	329	(2)
Finite Element Solution of a Plane Stress Problem	331	(19)
References	342	(1)
Problems	343	(7)
Practical Considerations in Modeling; Interpreting Results; and Examples of Plane Stress/Strain Analysis	350	(48)
Introduction	350	(1)
Finite Element Modeling	350	(13)
Equilibrium and Compatibility of Finite Element Results	363	(4)
Convergence of Solution	367	(1)
Interpretation of Stresses	368	(1)
Static Condensation	369	(5)
Flowchart for the Solution of Plane Stress/Strain Problems	374	(1)
Computer Program Assisted Step-by-Step Solution, Other Models, and Results for Plane Stress/Strain Problems	374	(24)
References	381	(1)
Problems	382	(16)
Development of the Linear-Strain Triangle Equations	398	(14)
Introduction	398	(1)
Derivation of the Linear-Strain Triangular Element Stiffness Matrix and Equations	398	(5)
Example LST Stiffness Determination	403	(3)
Comparison of Elements	406	(6)
References	409	(1)
Problems	409	(3)
Axisymmetric Elements	412	(31)
Introduction	412	(1)
Derivation of the Stiffness Matrix	412	(10)
Solution of an Axisymmetric Pressure Vessel	422	(6)
Applications of Axisymmetric Elements	428	(15)
References	433	(1)
Problems	434	(9)
Isoparametric Formulation	443	(47)
Introduction	443	(1)
Isoparametric Formulation of the Bar Element Stiffness Matrix	444	(5)
Rectangular Plane Stress Element	449	(3)
Isoparametric Formulation of the Plane Element Stiffness Matrix	452	(11)
Gaussian and Newton-Cotes Quadrature (Numerical Integration)	463	(6)
Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadrature	469	(6)
Higher-Order Shape Functions	475	(15)
References	484	(1)
Problems	484	(6)
Three-Dimensional Stress Analysis	490	(24)
Introduction	490	(1)
Three-Dimensional Stress and Strain	490	(3)
Tetrahedral Element	493	(8)
Isoparametric Formulation	501	(13)
References	508	(1)
Problems	509	(5)
Plate Bending Element	514	(20)
Introduction	514	(1)
Basic Concepts of Plate Bending	514	(5)
Derivation of a Plate Bending Element Stiffness Matrix and Equations	519	(4)
Some Plate Element Numerical Comparisons	523	(1)
Computer Solution for a Plate Bending Problem	524	(10)
References	528	(1)
Problems	529	(5)
Heat Transfer and Mass Transport	534	(59)
Introduction	534	(1)
Derivation of the Basic Differential Equation	535	(3)
Heat Transfer with Convection	538	(1)
Typical Units; Thermal Conductivities, K; and Heat-Transfer Coefficients, h	539	(1)
One-Dimensional Finite Element Formulation Using a Variational Method	540	(15)
Two-Dimensional Finite Element Formulation	555	(9)
Line or Point Sources	564	(2)
Three-Dimensional Heat Transfer Finite Element Formulation	566	(3)
One-Dimensional Heat Transfer with Mass Transport	569	(1)
Finite Element Formulation of Heat Transfer with Mass Transport by Galerkin's Method	569	(5)
Flowchart and Examples of a Heat-Transfer Program	574	(19)
References	577	(1)
Problems	577	(16)
Fluid Flow	593	(24)
Introduction	593	(1)
Derivation of the Basic Differential Equations	594	(4)
One-Dimensional Finite Element Formulation	598	(8)
Two-Dimensional Finite Element Formulation	606	(5)
Flowchart and Example of a Fluid-Flow Program	611	(6)
References	612	(1)
Problems	613	(4)
Thermal Stress	617	(30)
Introduction	617	(1)
Formulation of the Thermal Stress Problem and Examples	617	(30)
Reference	640	(1)
Problems	641	(6)
Structural Dynamics and Time-Dependent Heat Transfer	647	(61)
Introduction	647	(1)
Dynamics of a Spring-Mass System	647	(2)
Direct Derivation of the Bar Element Equations	649	(4)
Numerical Integration in Time	653	(12)
Natural Frequencies of a One-Dimensional Bar	665	(4)
Time-Dependent One-Dimensional Bar Analysis	669	(5)
Beam Element Mass Matrices and Natural Frequencies	674	(7)
Trusts, Plane Frame, Plane Stress/Strain Axisymmetric, and Solid Element Mass Matrices	681	(5)
Time-Dependent Heat Transfer	686	(7)
Computer Program Example Solutions for Structural Dynamics	693	(15)
References	702	(1)
Problems	702	(6)
Appendix A Matrix Algebra	708	(14)
Introduction	708	(1)
Definition of a Matrix	708	(1)
Matrix Operations	709	(7)
Cofactor or Adjoint Method to Determine the Inverse of a Matrix	716	(2)
Inverse of a Matrix by Row Reduction	718	(4)
References	720	(1)
Problems	720	(2)
Appendix B Methods for Solution of Simultaneous Linear Equations	722	(22)
Introduction	722	(1)
General Form of the Equations	722	(1)
Uniqueness, Nonuniqueness, and Nonexistence of Solution	723	(1)
Methods for Solving Linear Algebraic Equations	724	(11)
Banded-Symmetric Matrices, Bandwidth, Skyline, and Wavefront Methods	735	(9)
References	741	(1)
Problems	742	(2)
Appendix C Equations from Elasticity Theory	744	(8)
Introduction	744	(1)
Differential Equations of Equilibrium	744	(2)
Strain/Displacement and Compatibility Equations	746	(2)
Stress/Strain Relationships	748	(4)
Reference	751	(1)
Appendix D Equivalent Nodal Forces	752	(3)
Problems	752	(3)
Appendix E Principle of Virtual Work	755	(4)
References	758	(1)
Appendix F Properties of Structural Steel and Aluminum Shapes	759	(14)
Answers to Selected Problems	773	(26)
Index	799