## Structural Analysis with Finite ElementsStructural Analysis with Finite Elements develops the foundations and applications of the finite element method in structural analysis in a language which is familiar to structural engineers. At the same time, it uncovers the structural mechanics behind the finite element method. This innovative text explores and explains issues such as: why finite element results are "wrong", why support reactions are relatively accurate, why stresses at midpoints are more reliable, why averaging the stresses sometimes may not help or why the equilibrium conditions are violated. An additional chapter treats the boundary element method and related software is available at www.winfem.de. Structural Analysis with Finite Elements provides a new foundation for the finite element method that enables structural engineers to address key questions that arise in computer modelling of structures with finite elements. |

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### Contents

1 What are finite elements? | 1 |

13 Potential energy | 5 |

14 Projection | 8 |

15 The error of an FE solution | 12 |

16 A beautiful idea that does not work | 15 |

17 Set theory | 16 |

18 Principle of virtual displacements | 23 |

19 Taut rope | 28 |

44 Plane elements | 252 |

45 The patch test | 258 |

46 Volume forces | 260 |

47 Supports | 261 |

48 Nodal stresses and element stresses | 271 |

49 Truss models | 277 |

410 Twobay wall | 278 |

411 Multistory shear wall | 280 |

110 Least squares | 33 |

111 Distance inside distance outside | 36 |

112 Scalar product and weak solution | 39 |

113 Equivalent nodal forces | 41 |

114 Concentrated forces | 43 |

115 Greens functions | 50 |

116 Practical consequences | 52 |

117 Why finite element results are wrong | 55 |

118 Proof | 62 |

119 Influence functions | 67 |

120 Accuracy | 75 |

121 Why resultant stresses are more accurate | 80 |

122 Why stresses at midpoints are more accurate | 84 |

123 Why stresses jump | 93 |

124 Why finite element support reactions are relatively accurate | 94 |

125 Gauss points | 99 |

126 Local errors and pollution | 105 |

127 Adaptive methods | 112 |

128 St Venants principle | 127 |

129 Singularities | 129 |

130 Actio reactio? | 132 |

131 The output | 135 |

132 Support conditions | 137 |

133 Equilibrium | 138 |

134 Changes in the temperature and displacement of supports | 141 |

135 Stability problems | 144 |

136 Interpolation | 148 |

137 Polynomials | 151 |

138 Infinite energy | 158 |

139 Conforming and nonconforming shape functions | 160 |

140 Partition of unity | 161 |

141 Elements | 163 |

142 Stiffness matrices | 164 |

143 Coupling degrees of freedom | 167 |

144 Numerical details | 170 |

145 Warning | 178 |

2 What are boundary elements? | 181 |

21 Influence functions or Bettis theorem | 182 |

22 Structural analysis with boundary elements | 189 |

23 Comparison finite elementsboundary elements | 204 |

3 Frames | 211 |

32 The FE approach | 212 |

33 Finite elements and the slope deflection method | 227 |

34 Stiffness matrices | 231 |

35 Approximations for stiffness matrices | 237 |

4 Plane problems | 241 |

42 Strains and stresses | 248 |

43 Shape functions | 251 |

412 Shear wall with suspended load | 287 |

413 Shear wall and horizontal load | 289 |

414 Equilibrium of resultant forces | 292 |

415 Adaptive mesh refinement | 296 |

416 Plane problems in soil mechanics | 300 |

417 Incompressible material | 306 |

418 Mixed methods | 307 |

419 Influence functions | 312 |

420 Error analysis | 313 |

421 Nonlinear problems | 314 |

5 Slabs | 325 |

51 Kirchhoff plates | 326 |

52 The displacement model | 331 |

53 Elements | 332 |

54 Hybrid elements | 335 |

55 Singularities of a Kirchhoff plate | 339 |

56 ReissnerMindlin plates | 341 |

57 Singularities of a ReissnerMindlin plate | 346 |

58 ReissnerMindlin elements | 349 |

59 Supports | 351 |

510 Columns | 353 |

511 Shear forces | 361 |

512 Variable thickness | 362 |

513 Beam models | 364 |

514 Wheel loads | 369 |

516 T beams | 372 |

517 Foundation slabs | 378 |

518 Direct design method | 384 |

519 Point supports | 386 |

6 Shells | 391 |

62 Shells of revolution | 394 |

63 Volume elements and degenerate shell elements | 396 |

64 Circular arches | 397 |

65 Flat elements | 399 |

66 Membranes | 404 |

7 Theoretical details | 409 |

72 Greens identities Integration by parts | 414 |

73 Greens functions | 419 |

74 Generalized Greens functions | 421 |

75 Nonlinear problems | 428 |

76 The derivation of influence functions | 432 |

77 Shifted Greens functions | 437 |

78 The dual space | 447 |

79 Some concepts of error analysis Asymptotic error estimates | 453 |

710 Important equations and inequalities | 461 |

471 | |

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### Common terms and phrases

3-D elasticity applied approximate beam bending moments Betti's theorem bilinear element boundary conditions calculated column corner points cross section deflection curve deformations degrees of freedom derivatives differential equation Dirac delta distributed load edge elastic equilibrium equivalent nodal forces error exact solution FE analysis FE method FE program FE solution finite element Gateaux derivative Gauss points gravity load Green's first identity Green's function Hence horizontal infinite influence function integral interpolation Jn Jn Kirchhoff plate kN/m linear load case ph mesh nodal displacements nodal unit displacements nodes normal orthogonal point load point supports Poisson's ratio polynomials potential energy problem Reissner-Mindlin plate rigid-body motions rotations scalar product shape functions shear force shear stresses shear wall single force singular slab Sobolev space stiffness matrix strain energy product support reaction tensor uh(x vector vertical virtual displacements volume forces wh(x zero