Essentials of the Finite Element Method, Second Edition aims to simplify search by offering a comprehensive yet concise resource suitable for newcomers to FEM or those in need of a refresher. This edition begins by explaining the basics of FEM and gradually introduces advanced topics, while also illustrating the practical applications of the theory in engineering. The book covers various specific subjects, including linear spring elements, bar elements, trusses, beams and frames, plates, heat transfer, structural dynamics, and buckling. Throughout the text, readers are provided with step-by-step detailed analyses for the development of finite element equations.
Moreover, the book demonstrates the programming aspect of FEM, offering examples in MATLAB, EXCEL, CALFEM, and ANSYS. This allows readers to gain insights into developing their own computer code. Designed for a wide range of readers, from first-time BSc/MSc students to experienced researchers and practicing mechanical/structural engineers, the book serves as a comprehensive reference text suitable for modern engineers.
1. Title of Book - p. i 2. Cover image - p. i 3. Title page - p. ii 4. Table of Contents - p. iii 5. Copyright - p. iv 6. Preface - p. ix 7. Chapter 1. An overview of the finite element method - p. 1 8. Chapter 2. Mathematical background - p. 21 9. 2.1 Vectors - p. 21 10. 2.2 Coordinate systems - p. 26 11. 2.3 Elements of matrix algebra - p. 31 12. 2.4 Variational formulation of elasticity problems - p. 38 13. Chapter 3. Linear spring elements - p. 47 14. Chapter 4. Bar elements and hydraulic networks - p. 63 15. Chapter 5. Trusses - p. 87 16. 5.1 The element equation for plane truss members - p. 87 17. 5.2 The element equation for three dimensional trusses - p. 89 18. 5.3 Calculation of the bar's axial forces (internal forces) - p. 91 19. 5.3.1 Step 1: Derivation of the local stiffness matrices - p. 92 20. 5.3.2 Step 2: Expansion of the local element equations to the degrees of freedom of the structure - p. 95 21. 5.3.3 Step 3: Assembly of the local element equations to compose the structure equation - p. 98 22. 5.3.4 Step 4: Boundary conditions - p. 98 23. 5.3.5 Step 5: Internal forces - p. 100 24. 5.3.6 Step 1: Calculation of the direction cosines - p. 104 25. 5.3.7 Step 2: Derivation of the local stiffness matrices - p. 106 26. 5.3.8 Step 3: Expansion of the local element equations to the degrees of freedom of the structure - p. 108 27. 5.3.9 Step 3: Assembly of the local element equations to compose the structure equation - p. 109 28. 5.3.10 Step 4: Boundary conditions - p. 110 29. 5.3.11 Step 1: Degrees of freedom - p. 130 30. 5.3.12 Step 2: Load vector {f} - p. 130 31. 5.3.13 Step 3: Boundary conditions {bc} - p. 131 32. 5.3.14 Step 4: Derivation of global stiffness matrix [K] - p. 131 33. Chapter 6. Beams - p. 149 34. 6.1 Element equation of a 2D beam subjected to nodal forces - p. 150 35. 6.2 2-D element equation of a beam subjected to a uniform loading - p. 166 36. 6.3 2-D element equation of a beam subjected to an arbitrary varying loading - p. 170 37. 6.4 2-D element equation of a beam on elastic foundation subjected to uniform loading - p. 197 38. 6.5 Engineering applications of the element equation of the beam on elastic foundation - p. 203 39. 6.6 Element equation for a beam subjected to torsion - p. 213 40. 6.7 2-D element equation for a beam subjected to nodal axial forces, shear forces, bending moments and torsional moments - p. 216 41. 6.8 3-D element equation for a beam subjected to nodal axial forces, shear forces, bending moments and torsional moments - p. 217 42. Chapter 7. Frames - p. 237 43. 7.1 Framed structures - p. 237 44. 7.2 2-D frame element equation subjected to nodal forces - p. 237 45. 7.3 2-D frame element equation subjected to arbitrary varying loading - p. 237 46. 7.4 3-D beam element equation subjected to nodal forces - p. 237 47. 7.5 Distribution of bending moments, shear forces, axial forces, torsional moments of each element - p. 237 48. Chapter 8. The principle of minimum potential energy for 1-D elements - p. 307 49. Chapter 9. From “isotropic” to “orthotropic” plane elements: Elasticity equations for 2-D solids - p. 317 50. 9.1 The generalized Hooke's law - p. 317 51. 9.2 From “isotropic” to “orthotropic” plane elements - p. 326 52. 9.3 Hooke's law of an orthotropic 2-D element, with respect to the global coordinate system - p. 329 53. 9.4 Transformation of engineering properties - p. 330 54. 9.5 Elasticity equations for isotropic solids - p. 336 55. Chapter 10. The principle of minimum potential energy for 2-D and 3-D elements - p. 343 56. 10.1 Interpolation and shape functions - p. 343 57. 10.2 Isoparametric elements - p. 368 58. 10.3 Derivation of stiffness matrices - p. 375 59. Chapter 11. Plates and shells - p. 413 60. 11.1 Bending of flat rectangular plates - p. 413 61. 11.2 Interpolation function for displacements - p. 417 62. 11.3 Stiffness matrix of a rectangular plate element - p. 419 63. 11.4 Stiffness matrix of rectangular plate element under uniform pressure distribution - p. 425 64. 11.5 Interpolation function for displacements - p. 429 65. 11.6 Stiffness matrix of a triangular plate element under uniform pressure - p. 434 66. 11.7 Plate on elastic foundation - p. 435 67. 11.8 Shell elements - p. 437 68. 11.9 Stiffness matrix of triangular shell elements in the global coordinate system - p. 442 69. 11.10 Stress computation of plates and shells - p. 446 70. Chapter 12. Structural dynamics - p. 481 71. 12.1 The dynamic equation - p. 481 72. 12.2 Mass matrix - p. 482 73. 12.2.1 Bar element - p. 482 74. 12.2.2 2-D truss element - p. 484 75. 12.2.3 3-D truss element - p. 489 76. 12.2.4 2-D beam element - p. 493 77. 12.2.5 3-D beam element - p. 495 78. 12.2.6 Inclined 2-D beam element (2-D frame element) - p. 498 79. 12.2.7 Linear triangular element (CST element) - p. 500 80. 12.3 Solution methodology for the dynamic equation - p. 501 81. 12.4 Free vibration—Natural frequencies - p. 504 82. Chapter 13. Heat transfer - p. 531 83. 13.1 Conduction heat transfer - p. 531 84. 13.2 Convection heat transfer - p. 538 85. 13.3 Finite element formulation - p. 539 86. 13.3.1 1D heat transfer modeling using a variational method - p. 539 87. 13.3.2 2D and 3D heat transfer modeling using a variational method - p. 557 88. Appendix - p. 603 89. Index - p. 625
Essentials of the Finite Element Method, Second Edition aims to simplify search by offering a comprehensive yet concise resource suitable for newcomers to FEM or those in need of a refresher. This edition begins by explaining the basics of FEM and gradually introduces advanced topics, while also illustrating the practical applications of the theory in engineering. The book covers various specific subjects, including linear spring elements, bar elements, trusses, beams and frames, plates, heat transfer, structural dynamics, and buckling. Throughout the text, readers are provided with step-by-step detailed analyses for the development of finite element equations.
Moreover, the book demonstrates the programming aspect of FEM, offering examples in MATLAB, EXCEL, CALFEM, and ANSYS. This allows readers to gain insights into developing their own computer code. Designed for a wide range of readers, from first-time BSc/MSc students to experienced researchers and practicing mechanical/structural engineers, the book serves as a comprehensive reference text suitable for modern engineers.
1. Title of Book - p. i 2. Cover image - p. i 3. Title page - p. ii 4. Table of Contents - p. iii 5. Copyright - p. iv 6. Preface - p. ix 7. Chapter 1. An overview of the finite element method - p. 1 8. Chapter 2. Mathematical background - p. 21 9. 2.1 Vectors - p. 21 10. 2.2 Coordinate systems - p. 26 11. 2.3 Elements of matrix algebra - p. 31 12. 2.4 Variational formulation of elasticity problems - p. 38 13. Chapter 3. Linear spring elements - p. 47 14. Chapter 4. Bar elements and hydraulic networks - p. 63 15. Chapter 5. Trusses - p. 87 16. 5.1 The element equation for plane truss members - p. 87 17. 5.2 The element equation for three dimensional trusses - p. 89 18. 5.3 Calculation of the bar's axial forces (internal forces) - p. 91 19. 5.3.1 Step 1: Derivation of the local stiffness matrices - p. 92 20. 5.3.2 Step 2: Expansion of the local element equations to the degrees of freedom of the structure - p. 95 21. 5.3.3 Step 3: Assembly of the local element equations to compose the structure equation - p. 98 22. 5.3.4 Step 4: Boundary conditions - p. 98 23. 5.3.5 Step 5: Internal forces - p. 100 24. 5.3.6 Step 1: Calculation of the direction cosines - p. 104 25. 5.3.7 Step 2: Derivation of the local stiffness matrices - p. 106 26. 5.3.8 Step 3: Expansion of the local element equations to the degrees of freedom of the structure - p. 108 27. 5.3.9 Step 3: Assembly of the local element equations to compose the structure equation - p. 109 28. 5.3.10 Step 4: Boundary conditions - p. 110 29. 5.3.11 Step 1: Degrees of freedom - p. 130 30. 5.3.12 Step 2: Load vector {f} - p. 130 31. 5.3.13 Step 3: Boundary conditions {bc} - p. 131 32. 5.3.14 Step 4: Derivation of global stiffness matrix [K] - p. 131 33. Chapter 6. Beams - p. 149 34. 6.1 Element equation of a 2D beam subjected to nodal forces - p. 150 35. 6.2 2-D element equation of a beam subjected to a uniform loading - p. 166 36. 6.3 2-D element equation of a beam subjected to an arbitrary varying loading - p. 170 37. 6.4 2-D element equation of a beam on elastic foundation subjected to uniform loading - p. 197 38. 6.5 Engineering applications of the element equation of the beam on elastic foundation - p. 203 39. 6.6 Element equation for a beam subjected to torsion - p. 213 40. 6.7 2-D element equation for a beam subjected to nodal axial forces, shear forces, bending moments and torsional moments - p. 216 41. 6.8 3-D element equation for a beam subjected to nodal axial forces, shear forces, bending moments and torsional moments - p. 217 42. Chapter 7. Frames - p. 237 43. 7.1 Framed structures - p. 237 44. 7.2 2-D frame element equation subjected to nodal forces - p. 237 45. 7.3 2-D frame element equation subjected to arbitrary varying loading - p. 237 46. 7.4 3-D beam element equation subjected to nodal forces - p. 237 47. 7.5 Distribution of bending moments, shear forces, axial forces, torsional moments of each element - p. 237 48. Chapter 8. The principle of minimum potential energy for 1-D elements - p. 307 49. Chapter 9. From “isotropic” to “orthotropic” plane elements: Elasticity equations for 2-D solids - p. 317 50. 9.1 The generalized Hooke's law - p. 317 51. 9.2 From “isotropic” to “orthotropic” plane elements - p. 326 52. 9.3 Hooke's law of an orthotropic 2-D element, with respect to the global coordinate system - p. 329 53. 9.4 Transformation of engineering properties - p. 330 54. 9.5 Elasticity equations for isotropic solids - p. 336 55. Chapter 10. The principle of minimum potential energy for 2-D and 3-D elements - p. 343 56. 10.1 Interpolation and shape functions - p. 343 57. 10.2 Isoparametric elements - p. 368 58. 10.3 Derivation of stiffness matrices - p. 375 59. Chapter 11. Plates and shells - p. 413 60. 11.1 Bending of flat rectangular plates - p. 413 61. 11.2 Interpolation function for displacements - p. 417 62. 11.3 Stiffness matrix of a rectangular plate element - p. 419 63. 11.4 Stiffness matrix of rectangular plate element under uniform pressure distribution - p. 425 64. 11.5 Interpolation function for displacements - p. 429 65. 11.6 Stiffness matrix of a triangular plate element under uniform pressure - p. 434 66. 11.7 Plate on elastic foundation - p. 435 67. 11.8 Shell elements - p. 437 68. 11.9 Stiffness matrix of triangular shell elements in the global coordinate system - p. 442 69. 11.10 Stress computation of plates and shells - p. 446 70. Chapter 12. Structural dynamics - p. 481 71. 12.1 The dynamic equation - p. 481 72. 12.2 Mass matrix - p. 482 73. 12.2.1 Bar element - p. 482 74. 12.2.2 2-D truss element - p. 484 75. 12.2.3 3-D truss element - p. 489 76. 12.2.4 2-D beam element - p. 493 77. 12.2.5 3-D beam element - p. 495 78. 12.2.6 Inclined 2-D beam element (2-D frame element) - p. 498 79. 12.2.7 Linear triangular element (CST element) - p. 500 80. 12.3 Solution methodology for the dynamic equation - p. 501 81. 12.4 Free vibration—Natural frequencies - p. 504 82. Chapter 13. Heat transfer - p. 531 83. 13.1 Conduction heat transfer - p. 531 84. 13.2 Convection heat transfer - p. 538 85. 13.3 Finite element formulation - p. 539 86. 13.3.1 1D heat transfer modeling using a variational method - p. 539 87. 13.3.2 2D and 3D heat transfer modeling using a variational method - p. 557 88. Appendix - p. 603 89. Index - p. 625
