The Finite Element Method in Engineering, Sixth Edition, provides a thorough grounding in the mathematical principles behind the Finite Element Analysis technique—an analytical engineering tool originated in the 1960's by the aerospace and nuclear power industries to find usable, approximate solutions to problems with many complex variables. Rao shows how to set up finite element solutions in civil, mechanical and aerospace engineering applications. The new edition features updated real-world examples from MATLAB, Ansys and Abaqus, and a new chapter on additional FEM topics including extended FEM (X-FEM). Professional engineers will benefit from the introduction to the many useful applications of finite element analysis.
1. Part I. Introduction - p. xvii 2. Chapter 1. Overview of Finite Element Method - p. 3 3. 1.1. Basic Concept - p. 3 4. 1.2. Historical Background - p. 3 5. 1.3. General Applicability of the Method - p. 6 6. 1.4. Engineering Applications of the Finite Element Method - p. 8 7. 1.5. General Description of the Finite Element Method - p. 10 8. 1.6. One-Dimensional Problems With Linear Interpolation Model - p. 11 9. 1.7. One-Dimensional Problems With a Cubic Interpolation Model - p. 21 10. 1.8. Derivation of Finite Element Equations Using a Direct Approach - p. 25 11. 1.9. Commercial Finite Element Program Packages - p. 38 12. 1.10. Solutions Using Finite Element Software - p. 39 13. Review Questions - p. 40 14. Problems - p. 41 15. Part II. Basic Procedure - p. 52 16. Chapter 2. Discretization of the Domain - p. 55 17. Chapter 3. Interpolation Models - p. 81 18. 3.1. Introduction - p. 81 19. 3.2. Polynomial Form of Interpolation Functions - p. 82 20. 3.3. Simplex, Complex, and Multiplex Elements - p. 83 21. 3.4. Interpolation Polynomial in Terms of Nodal Degrees of Freedom - p. 84 22. 3.5. Selection of the Order of the Interpolation Polynomial - p. 86 23. 3.6. Convergence Requirements - p. 86 24. 3.7. Linear Interpolation Polynomials in Terms of Global Coordinates - p. 90 25. 3.8. Interpolation Polynomials for Vector Quantities - p. 103 26. 3.9. Linear Interpolation Polynomials in Terms of Local Coordinates - p. 106 27. 3.10. Integration of Functions of Natural Coordinates - p. 115 28. 3.11. Patch Test - p. 116 29. Review Questions - p. 118 30. Problems - p. 121 31. Chapter 4. Higher Order and Isoparametric Elements - p. 129 32. 4.1. Introduction - p. 129 33. 4.2. Higher Order One-Dimensional Elements - p. 129 34. 4.3. Higher Order Elements in Terms of Natural Coordinates - p. 131 35. 4.4. Higher Order Elements in Terms of Classical Interpolation Polynomials - p. 142 36. 4.5. One-Dimensional Elements Using Classical Interpolation Polynomials - p. 148 37. 4.6. Two-Dimensional (Rectangular) Elements Using Classical Interpolation Polynomials - p. 149 38. 4.7. Continuity Conditions - p. 151 39. 4.8. Comparative Study of Elements - p. 153 40. 4.9. Isoparametric Elements - p. 154 41. 4.10. Numerical Integration - p. 161 42. Review Questions - p. 166 43. Problems - p. 166 44. Chapter 5. Derivation of Element Matrices and Vectors - p. 173 45. 5.1. Introduction - p. 173 46. 5.2. Variational Approach - p. 174 47. 5.3. Solution of Equilibrium Problems Using Variational (Rayleigh–Ritz) Method - p. 178 48. 5.4. Solution of Eigenvalue Problems Using the Variational (Rayleigh–Ritz) Method - p. 182 49. 5.5. Solution of Propagation Problems Using the Variational (Rayleigh–Ritz) Method - p. 183 50. 5.6. Equivalence of Finite Element and Variational (Rayleigh–Ritz) Methods - p. 183 51. 5.7. Derivation of Finite Element Equations Using the Variational (Rayleigh–Ritz) Approach - p. 184 52. 5.8. Weighted Residual Approach - p. 189 53. 5.9. Solution of Eigenvalue Problems Using Weighted Residual Method - p. 197 54. 5.10. Solution of Propagation Problems Using Weighted Residual Method - p. 197 55. 5.11. Derivation of Finite Element Equations Using Weighted Residual (Galerkin) Approach - p. 198 56. 5.12. Derivation of Finite Element Equations Using Weighted Residual (Least Squares) Approach - p. 200 57. 5.13. Strong and Weak Form Formulations - p. 203 58. Review Questions - p. 205 59. Problems - p. 206 60. Chapter 6. Assembly of Element Matrices and Vectors, and Derivation of System Equations - p. 213 61. 6.1. Coordinate Transformation - p. 213 62. 6.2. Assemblage of Element Equations - p. 218 63. 6.3. Incorporation of Boundary Conditions - p. 225 64. 6.4. Penalty Method - p. 233 65. 6.5. Multipoint Constraints: Penalty Method - p. 237 66. 6.6. Symmetry Conditions: Penalty Method - p. 241 67. 6.7. Rigid Elements - p. 243 68. Review Questions - p. 247 69. Problems - p. 248 70. Chapter 7. Numerical Solution of Finite Element Equations - p. 257 71. 7.1. Introduction - p. 257 72. 7.2. Solution of Equilibrium Problems - p. 258 73. 7.3. Solution of Eigenvalue Problems - p. 267 74. 7.4. Solution of Propagation Problems - p. 278 75. 7.5. Parallel Processing in Finite Element Analysis - p. 284 76. Review Questions - p. 285 77. Problems - p. 286 78. Part III. Application to Solid Mechanics Problems - p. 292 79. Chapter 8. Basic Equations and Solution Procedure - p. 295 80. 8.1. Introduction - p. 295 81. 8.2. Basic Equations of Solid Mechanics - p. 295 82. 8.3. Formulations of Solid and Structural Mechanics - p. 313 83. 8.4. Formulation of Finite Element Equations (Static Analysis) - p. 318 84. 8.5. Nature of Finite Element Solutions - p. 322 85. Review Questions - p. 323 86. Problems - p. 324 87. Chapter 9. Analysis of Trusses, Beams, and Frames - p. 331 88. 9.1. Introduction - p. 331 89. 9.2. Space Truss Element - p. 331 90. 9.3. Beam Element - p. 344 91. 9.4. Space Frame Element - p. 349 92. 9.5. Characteristics of Stiffness Matrices - p. 360 93. Review Questions - p. 361 94. Problems - p. 362 95. Chapter 10. Analysis of Plates - p. 379 96. 10.1. Introduction - p. 379 97. 10.2. Triangular Membrane Element - p. 379 98. 10.3. Numerical Results With Membrane Element - p. 391 99. 10.4. Quadratic Triangle Element - p. 397 100. 10.5. Rectangular Plate Element (In-Plane Forces) - p. 398 101. 10.6. Bending Behavior of Plates - p. 400 102. 10.7. Finite Element Analysis of Plates in Bending - p. 403 103. 10.8. Triangular Plate Bending Element - p. 404 104. 10.9. Numerical Results With Bending Elements - p. 407 105. 10.10. Analysis of Three-Dimensional Structures Using Plate Elements - p. 410 106. Review Questions - p. 413 107. Problems - p. 414 108. Chapter 11. Analysis of Three-Dimensional Problems - p. 427 109. 11.1. Introduction - p. 427 110. 11.2. Tetrahedron Element - p. 427 111. 11.3. Hexahedron Element - p. 436 112. 11.4. Analysis of Solids of Revolution - p. 441 113. Review Questions - p. 449 114. Problems - p. 450 115. Chapter 12. Dynamic Analysis - p. 457 116. 12.1. Dynamic Equations of Motion - p. 457 117. 12.2. Consistent and Lumped Mass Matrices - p. 459 118. 12.3. Consistent Mass Matrices in a Global Coordinate System - p. 469 119. 12.4. Free Vibration Analysis - p. 470 120. 12.5. Dynamic Response Using Finite Element Method - p. 483 121. 12.6. Nonconservative Stability and Flutter Problems - p. 490 122. 12.7. Substructures Method - p. 491 123. Review Questions - p. 491 124. Problems - p. 493 125. Part IV. Application to Heat Transfer Problems - p. 501 126. Chapter 13. Formulation and Solution Procedure - p. 505 127. Chapter 14. One-Dimensional Problems - p. 523 128. 14.1. Introduction - p. 523 129. 14.2. Straight Uniform Fin Analysis - p. 523 130. 14.3. Convection Loss From End Surface of Fin - p. 527 131. 14.4. Tapered Fin Analysis - p. 530 132. 14.5. Analysis of Uniform Fins Using Quadratic Elements - p. 533 133. 14.6. Unsteady State Problems - p. 536 134. 14.7. Heat Transfer Problems With Radiation - p. 541 135. Review Questions - p. 545 136. Problems - p. 546 137. Chapter 15. Two-Dimensional Problems - p. 553 138. Chapter 16. Three-Dimensional Problems - p. 569 139. Part V. Application to Fluid Mechanics Problems - p. 586 140. Chapter 17. Basic Equations of Fluid Mechanics - p. 589 141. Chapter 18. Inviscid and Incompressible Flows - p. 611 142. Chapter 19. Viscous and Non-Newtonian Flows - p. 631 143. Part VI. Solution and Applications of Quasi-Harmonic Equations - p. 649 144. Chapter 20. Solution of Quasi-Harmonic Equations - p. 653 145. Part VII. ABAQUS and ANSYS Software and MATLAB® Programs for Finite Element Analysis - p. 668 146. Chapter 21. Finite Element Analysis Using ABAQUS - p. 671 147. Chapter 22. Finite Element Analysis Using ANSYS - p. 703 148. Chapter 23. MATLAB Programs for Finite Element Analysis - p. 723 149. Appendix A. Comparison of the Finite Element Method With Other Methods of Analysis - p. 745 150. Appendix B. Green–Gauss Theorem (Integration by Parts in Two and Three Dimensions) - p. 757 151. Index - p. 759
The Finite Element Method in Engineering, Sixth Edition, provides a thorough grounding in the mathematical principles behind the Finite Element Analysis technique—an analytical engineering tool originated in the 1960's by the aerospace and nuclear power industries to find usable, approximate solutions to problems with many complex variables. Rao shows how to set up finite element solutions in civil, mechanical and aerospace engineering applications. The new edition features updated real-world examples from MATLAB, Ansys and Abaqus, and a new chapter on additional FEM topics including extended FEM (X-FEM). Professional engineers will benefit from the introduction to the many useful applications of finite element analysis.
1. Part I. Introduction - p. xvii 2. Chapter 1. Overview of Finite Element Method - p. 3 3. 1.1. Basic Concept - p. 3 4. 1.2. Historical Background - p. 3 5. 1.3. General Applicability of the Method - p. 6 6. 1.4. Engineering Applications of the Finite Element Method - p. 8 7. 1.5. General Description of the Finite Element Method - p. 10 8. 1.6. One-Dimensional Problems With Linear Interpolation Model - p. 11 9. 1.7. One-Dimensional Problems With a Cubic Interpolation Model - p. 21 10. 1.8. Derivation of Finite Element Equations Using a Direct Approach - p. 25 11. 1.9. Commercial Finite Element Program Packages - p. 38 12. 1.10. Solutions Using Finite Element Software - p. 39 13. Review Questions - p. 40 14. Problems - p. 41 15. Part II. Basic Procedure - p. 52 16. Chapter 2. Discretization of the Domain - p. 55 17. Chapter 3. Interpolation Models - p. 81 18. 3.1. Introduction - p. 81 19. 3.2. Polynomial Form of Interpolation Functions - p. 82 20. 3.3. Simplex, Complex, and Multiplex Elements - p. 83 21. 3.4. Interpolation Polynomial in Terms of Nodal Degrees of Freedom - p. 84 22. 3.5. Selection of the Order of the Interpolation Polynomial - p. 86 23. 3.6. Convergence Requirements - p. 86 24. 3.7. Linear Interpolation Polynomials in Terms of Global Coordinates - p. 90 25. 3.8. Interpolation Polynomials for Vector Quantities - p. 103 26. 3.9. Linear Interpolation Polynomials in Terms of Local Coordinates - p. 106 27. 3.10. Integration of Functions of Natural Coordinates - p. 115 28. 3.11. Patch Test - p. 116 29. Review Questions - p. 118 30. Problems - p. 121 31. Chapter 4. Higher Order and Isoparametric Elements - p. 129 32. 4.1. Introduction - p. 129 33. 4.2. Higher Order One-Dimensional Elements - p. 129 34. 4.3. Higher Order Elements in Terms of Natural Coordinates - p. 131 35. 4.4. Higher Order Elements in Terms of Classical Interpolation Polynomials - p. 142 36. 4.5. One-Dimensional Elements Using Classical Interpolation Polynomials - p. 148 37. 4.6. Two-Dimensional (Rectangular) Elements Using Classical Interpolation Polynomials - p. 149 38. 4.7. Continuity Conditions - p. 151 39. 4.8. Comparative Study of Elements - p. 153 40. 4.9. Isoparametric Elements - p. 154 41. 4.10. Numerical Integration - p. 161 42. Review Questions - p. 166 43. Problems - p. 166 44. Chapter 5. Derivation of Element Matrices and Vectors - p. 173 45. 5.1. Introduction - p. 173 46. 5.2. Variational Approach - p. 174 47. 5.3. Solution of Equilibrium Problems Using Variational (Rayleigh–Ritz) Method - p. 178 48. 5.4. Solution of Eigenvalue Problems Using the Variational (Rayleigh–Ritz) Method - p. 182 49. 5.5. Solution of Propagation Problems Using the Variational (Rayleigh–Ritz) Method - p. 183 50. 5.6. Equivalence of Finite Element and Variational (Rayleigh–Ritz) Methods - p. 183 51. 5.7. Derivation of Finite Element Equations Using the Variational (Rayleigh–Ritz) Approach - p. 184 52. 5.8. Weighted Residual Approach - p. 189 53. 5.9. Solution of Eigenvalue Problems Using Weighted Residual Method - p. 197 54. 5.10. Solution of Propagation Problems Using Weighted Residual Method - p. 197 55. 5.11. Derivation of Finite Element Equations Using Weighted Residual (Galerkin) Approach - p. 198 56. 5.12. Derivation of Finite Element Equations Using Weighted Residual (Least Squares) Approach - p. 200 57. 5.13. Strong and Weak Form Formulations - p. 203 58. Review Questions - p. 205 59. Problems - p. 206 60. Chapter 6. Assembly of Element Matrices and Vectors, and Derivation of System Equations - p. 213 61. 6.1. Coordinate Transformation - p. 213 62. 6.2. Assemblage of Element Equations - p. 218 63. 6.3. Incorporation of Boundary Conditions - p. 225 64. 6.4. Penalty Method - p. 233 65. 6.5. Multipoint Constraints: Penalty Method - p. 237 66. 6.6. Symmetry Conditions: Penalty Method - p. 241 67. 6.7. Rigid Elements - p. 243 68. Review Questions - p. 247 69. Problems - p. 248 70. Chapter 7. Numerical Solution of Finite Element Equations - p. 257 71. 7.1. Introduction - p. 257 72. 7.2. Solution of Equilibrium Problems - p. 258 73. 7.3. Solution of Eigenvalue Problems - p. 267 74. 7.4. Solution of Propagation Problems - p. 278 75. 7.5. Parallel Processing in Finite Element Analysis - p. 284 76. Review Questions - p. 285 77. Problems - p. 286 78. Part III. Application to Solid Mechanics Problems - p. 292 79. Chapter 8. Basic Equations and Solution Procedure - p. 295 80. 8.1. Introduction - p. 295 81. 8.2. Basic Equations of Solid Mechanics - p. 295 82. 8.3. Formulations of Solid and Structural Mechanics - p. 313 83. 8.4. Formulation of Finite Element Equations (Static Analysis) - p. 318 84. 8.5. Nature of Finite Element Solutions - p. 322 85. Review Questions - p. 323 86. Problems - p. 324 87. Chapter 9. Analysis of Trusses, Beams, and Frames - p. 331 88. 9.1. Introduction - p. 331 89. 9.2. Space Truss Element - p. 331 90. 9.3. Beam Element - p. 344 91. 9.4. Space Frame Element - p. 349 92. 9.5. Characteristics of Stiffness Matrices - p. 360 93. Review Questions - p. 361 94. Problems - p. 362 95. Chapter 10. Analysis of Plates - p. 379 96. 10.1. Introduction - p. 379 97. 10.2. Triangular Membrane Element - p. 379 98. 10.3. Numerical Results With Membrane Element - p. 391 99. 10.4. Quadratic Triangle Element - p. 397 100. 10.5. Rectangular Plate Element (In-Plane Forces) - p. 398 101. 10.6. Bending Behavior of Plates - p. 400 102. 10.7. Finite Element Analysis of Plates in Bending - p. 403 103. 10.8. Triangular Plate Bending Element - p. 404 104. 10.9. Numerical Results With Bending Elements - p. 407 105. 10.10. Analysis of Three-Dimensional Structures Using Plate Elements - p. 410 106. Review Questions - p. 413 107. Problems - p. 414 108. Chapter 11. Analysis of Three-Dimensional Problems - p. 427 109. 11.1. Introduction - p. 427 110. 11.2. Tetrahedron Element - p. 427 111. 11.3. Hexahedron Element - p. 436 112. 11.4. Analysis of Solids of Revolution - p. 441 113. Review Questions - p. 449 114. Problems - p. 450 115. Chapter 12. Dynamic Analysis - p. 457 116. 12.1. Dynamic Equations of Motion - p. 457 117. 12.2. Consistent and Lumped Mass Matrices - p. 459 118. 12.3. Consistent Mass Matrices in a Global Coordinate System - p. 469 119. 12.4. Free Vibration Analysis - p. 470 120. 12.5. Dynamic Response Using Finite Element Method - p. 483 121. 12.6. Nonconservative Stability and Flutter Problems - p. 490 122. 12.7. Substructures Method - p. 491 123. Review Questions - p. 491 124. Problems - p. 493 125. Part IV. Application to Heat Transfer Problems - p. 501 126. Chapter 13. Formulation and Solution Procedure - p. 505 127. Chapter 14. One-Dimensional Problems - p. 523 128. 14.1. Introduction - p. 523 129. 14.2. Straight Uniform Fin Analysis - p. 523 130. 14.3. Convection Loss From End Surface of Fin - p. 527 131. 14.4. Tapered Fin Analysis - p. 530 132. 14.5. Analysis of Uniform Fins Using Quadratic Elements - p. 533 133. 14.6. Unsteady State Problems - p. 536 134. 14.7. Heat Transfer Problems With Radiation - p. 541 135. Review Questions - p. 545 136. Problems - p. 546 137. Chapter 15. Two-Dimensional Problems - p. 553 138. Chapter 16. Three-Dimensional Problems - p. 569 139. Part V. Application to Fluid Mechanics Problems - p. 586 140. Chapter 17. Basic Equations of Fluid Mechanics - p. 589 141. Chapter 18. Inviscid and Incompressible Flows - p. 611 142. Chapter 19. Viscous and Non-Newtonian Flows - p. 631 143. Part VI. Solution and Applications of Quasi-Harmonic Equations - p. 649 144. Chapter 20. Solution of Quasi-Harmonic Equations - p. 653 145. Part VII. ABAQUS and ANSYS Software and MATLAB® Programs for Finite Element Analysis - p. 668 146. Chapter 21. Finite Element Analysis Using ABAQUS - p. 671 147. Chapter 22. Finite Element Analysis Using ANSYS - p. 703 148. Chapter 23. MATLAB Programs for Finite Element Analysis - p. 723 149. Appendix A. Comparison of the Finite Element Method With Other Methods of Analysis - p. 745 150. Appendix B. Green–Gauss Theorem (Integration by Parts in Two and Three Dimensions) - p. 757 151. Index - p. 759
