# Structural Analysis with Finite Elements

Springer Science & Business Media, 2004 - Mathematics - 484 pages
Structural 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 References 471 Copyright