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# Fundamental Finite Element Analysis and Applications: with Mathematica and Matlab Computations

## byM. Asghar Bhatti

### Hardcover | February 4, 2005

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**Finite Element Analysis with Mathematica and Matlab Computations and Practical Applications*is an innovative, hands-on and practical introduction to the Finite Element Method that provides a powerful tool for learning this essential analytic method.

*Support website (www.wiley.com/go/bhatti) includes complete sets of Mathematica and Matlab implementations for all examples presented in the text. Also included on the site are problems designed for self-directed labs using commercial FEA software packages ANSYS and ABAQUS.

*Offers a practical and hands-on approach while providing a solid theoretical foundation.

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The following ISBNs are associated with this title:

ISBN - 10:0471648086

ISBN - 13:9780471648086

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Table of Contents

**1. Finite Element Method: The Big Picture.**

1.1 Discretization and Element Equations.

1.1.1 Plane Truss Element.

1.1.2 Triangular Element for Two Dimensional Heat Flow.

1.1.3 General Remarks on Finite Element Discretization.

1.1.4 Triangular Element for Two Dimensional Stress Analysis.

1.2 Assembly of Element Equations.

1.3 Boundary Conditions and Nodal Solution.

1.3.1 Essential Boundary Conditions by Re-arranging Equations.

1.3.2 Essential Boundary Conditions by Modifying Equations.

1.3.3 Approximate Treatment of Essential Boundary Conditions.

1.3.4 Computation of Reactions to Verify Overall Equilibrium.

1.4 Element Solutions and Model Validity.

1.4.1 Plane Truss Element.

1.4.2 Triangular Element for Two Dimensional Heat Flow.

1.4.3 Triangular Element for Two Dimensional Stress Analysis.

1.5 Solution of Linear Equations.

1.5.1 Solution Using Choleski Decomposition.

1.5.2 Conjugate Gradient Method.

1.6 Multipoint Constraints.

1.6.1 Solution Using Lagrange multipliers.

1.6.2 Solution Using Penalty function.

1.7 Units.

**2. Mathematical Foundation of the Finite Element Method.**

2.1 Axial Deformation of Bars.

2.1.1 Differential equation for axial deformations.

2.1.2 Exact solutions of some axial deformation problems.

2.2 Axial Deformation of Bars Using Galerkin Method.

2.2.1 Weak form for axial deformations.

2.2.2 Uniform bar subjected to linearly varying axial load.

2.2.3 Tapered bar subjected to linearly varying axial load.

2.3 One Dimensional BVP Using Galerkin Method.

2.3.1 Overall solution procedure using Galerkin method.

2.3.2 Higher-Order Boundary Value Problems.

2.4 Rayleigh-Ritz Method.

2.4.1 Potential Energy for Axial Deformation of Bars.

2.4.2 Overall solution procedure using the Rayleigh-Ritz method.

2.4.3 Uniform bar subjected to linearly varying axial load.

2.4.4 Tapered bar subjected to linearly varying axial load.

2.5 Comments on the Galerkin & the Rayleigh-Ritz Methods.

2.5.1 Admissible assumed solution.

2.5.2 Solution convergence - the completeness requirement.

2.5.3 Galerkin versus Rayleigh-Ritz.

2.6 Finite Element Form of Assumed Solutions.

2.6.1 Linear interpolation functions for second-order problems.

2.6.2 Lagrange interpolation.

2.6.3 Galerkin weighting functions in the finite element form.

2.6.4 Hermite interpolation for fourth-order problems.

2.7 Finite Element Solution of Axial Deformation Problems.

2.7.1 Two Node Uniform Bar Element for Axial Deformations.

2.7.2 Numerical examples.

**3. One Dimensional Boundary Value Problem.**

3.1 Selected Applications of 1D BVP.

3.1.1 Steady state heat conduction.

3.1.2 Heat flow through thin fins.

3.1.3 Viscous fluid flow between parallel plates - Lubrication problem.

3.1.4 Slider bearing.

3.1.5 Axial deformation of bars.

3.1.6 Elastic buckling of long slender bars.

3.2 Finite Element Formulation for Second Order 1D BVP

3.2.1 Complete Solution Procedure.

3.3 Steady State Heat Conduction.

3.4 Steady State Heat Conduction and Convection.

3.5 Viscous Fluid Flow Between Parallel Plates.

3.6 Elastic Buckling of Bars.

3.7 Solution of Second Order 1D BVP.

3.8 A Closer Look at the Inter-Element Derivative Terms.

**4. Trusses, Beams, and Frames.**

4.1 Plane Trusses.

4.2 Space Trusses.

4.3 Temperature Changes and Initial Strains in Trusses.

4.4 Spring Elements.

4.5 Transverse Deformation of Beams.

4.5.1 Differential equation for beam bending.

4.5.2 Boundary conditions for beams.

4.5.3 Shear stresses beams.

4.5.4 Potential energy for beam bending.

4.5.5 Transverse deformation of a uniform beam.

4.5.6 Transverse deformation of a tapered beam fixed at both ends.

4.6 Two Node Beam Element.

4.6.1 Cubic assumed solution.

4.6.2 Element equations using Rayleigh-Ritz method.

4.7 Uniform Beams Subjected to Distributed Loads.

4.8 Plane Frames.

Contents

4.9 Space Frames.

4.9.1 Element equations in local coordinate system.

4.9.2 Local to global transformation.

4.9.3 Element Solution.

4.10 Frames in Multistory Buildings.

**5. Two Dimensional Elements.**

5.1 Selected Applications of the 2D BVP.

5.1.1 Two dimensional potential flow.

5.1.2 Steady-state heat flow.

5.1.3 Bars subjected to torsion.

5.1.4 Waveguides in Electromagnetics.

5.2 Integration by Parts in Higher Dimensions.

5.3 Finite Element Equations Using the Galerkin Method.

5.4 Rectangular Finite Elements.

5.4.1 Four node rectangular element.

5.4.2 Eight node rectangular element.

5.4.3 Lagrange interpolation for rectangular elements.

5.5 Triangular Finite Elements.

5.5.1 Three node triangular element.

5.5.2 Higher-order triangular elements.

**6. Mapped Elements.**

6.1 Integration Using Change of Variables.

6.1.1 One dimensional integrals.

6.1.2 Two dimensional area integrals.

6.1.3 Three dimensional volume integrals.

6.2 Mapping Quadrilaterals Using Interpolation Functions.

6.2.1 Mapping lines.

6.2.2 Mapping quadrilateral areas.

6.2.3 Mapped mesh generation.

6.3 Numerical Integration Using Gauss Quadrature.

6.3.1 Gauss quadrature for one dimensional integrals.

6.3.2 Gauss quadrature for area integrals.

6.3.3 Gauss quadrature for volume integrals.

6.4 Finite Element Computations Involving Mapped Elements.

6.4.1 Assumed solution.

6.4.2 Derivatives of the assumed solution.

6.4.3 Evaluation of area integrals.

6.4.4 Evaluation of boundary integrals.

Fundamental Finite Element Theory and Applications.

6.5 Complete Mathematica and Matlab Based Solutions of 2DBVP Involving Mapped.

Elements.

6.6 Triangular Elements by Collapsing Quadrilaterals.

6.7 Infinite Elements.

6.7.1 One dimensional BVP.

6.7.2 Two dimensional BVP.

**7. Analysis of Elastic Solids.**

7.1 Fundamental Concepts in Elasticity.

7.1.1 Stresses.

7.1.2 Stress failure criteria.

7.1.3 Strains.

7.1.4 Constitutive equations.

7.1.5 Temperature effects and initial strains.

7.2 Governing Differential Equations.

7.2.1 Stress equilibrium equations.

7.2.2 Governing differential equations in terms of displacements.

7.3 General Form of Finite Element Equations.

7.3.1 Potential energy functional.

7.3.2 Weak form.

7.3.3 Finite Element Equations.

7.3.4 Finite Element Equations in the Presence of Initial Strains.

7.4 Plane Stress and Plane Strain.

7.4.1 Plane stress problem.

7.4.2 Plane strain problem.

7.4.3 Finite element equations.

7.4.4 Three node triangular element.

7.4.5 Mapped quadrilateral elements.

7.5 Planar Finite Element Models.

7.5.1 Pressure Vessels.

7.5.2 Rotating Disks and Flywheels.

7.5.3 Residual Stresses due to Welding.

7.5.4 Crack-Tip Singularity.

**8. Transient Problems.**

8.1 Transient Field Problems.

8.1.1 Finite element equations.

8.1.2 Triangular element.

8.1.3 Transient heat flow.

8.2 Elastic Solids Subjected to Dynamic Loads.

8.2.1 Finite Element Equations.

8.2.2 Mass matrices for common structural elements.

Contents

8.2.3 Free Vibration Analysis.

8.2.4 Transient Response Examples.

**9. p-Formulation.**

9.1 *p-*Form*p*ulation for Second-Order 1D BVP.

9.2 *p*-Form*p*ulation for Second-Order 2D BVP.

Appendix A: Use of Commercial FEA Software.

A.1 Ansys Applications.

A.1.1 General Steps.

A.1.2 Truss Analysis.

A.1.3 Steady-State Heat Flow.

A.1.4 Plane Stress Analysis.

A.2 Optimizing Design Using Ansys.

A.2.1 General Steps.

A.2.2 Heat Flow example.

A.3 Abaqus Applications.

A.3.1 Execution procedure.

A.3.2 Truss Analysis.

A.3.3 Steady State Heat Flow.

A.3.4 Plane Stress Analysis.

Appendix B: Variational Form for Boundary Value Problems.

B.1 Basic concept of variation of a function.

B.2 Derivation of Equivalent Variational Form.

B.3 Boundary Value Problem Corresponding to a Given Functional.

Bibliography.

Index.