# Structural Analysis with Finite Elements

Springer Science & Business Media, Jan 30, 2007 - Technology & Engineering - 598 pages

Structural Analysis with Finite Elements, 2nd Edition provides a solid introduction to the foundation and the application of the finite element method in structural analysis. It offers new theoretical insight and practical advice on 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. This second edition contains additional sections on sensitivity analysis, on retrofitting structures, on the Generalized FEM (X-FEM) and on model adaptivity. An additional chapter treats the boundary element method, and related software is available at www.winfem.de.

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### Contents

 What are ﬁnite elements? 1 13 Potential energy 6 14 Projection 8 15 The error of an FE solution 13 16 A beautiful idea that does not work 15 17 Set theory 16 18 Principle of virtual displacements 23 19 Taut rope 29
 Plane problems 327 42 Strains and stresses 334 43 Shape functions 337 44 Plane elements 338 45 The patch test 344 46 Volume forces 346 47 Supports 347 48 Nodal stresses and element stresses 357

 110 Least squares 33 111 Distance inside distance outside 37 112 Scalar product and weak solution 40 113 Equivalent nodal forces 42 114 Concentrated forces 44 115 Greens functions 51 116 Practical consequences 55 117 Why ﬁnite element results are wrong 57 118 Proof 64 119 Inﬂuence functions 69 120 Accuracy 80 121 Why resultant stresses are more accurate 86 122 Why stresses at midpoints are more accurate 88 123 Why stresses jump 99 125 Gauss points 104 126 The Dirac energy 110 127 How to predict changes 113 128 The inﬂuence of a single element 126 129 Retroﬁtting structures 130 130 Local errors and pollution 136 131 Adaptive methods 147 132 St Venants principle 172 133 Singularities 175 134 Actio reactio? 177 135 The output 181 136 Support conditions 183 137 Equilibrium 184 138 Temperature changes and displacement of supports 187 139 Stability problems 193 140 Interpolation 197 141 Polynomials 199 142 Inﬁnite energy 208 143 Conforming and nonconforming shape functions 209 144 Partition of unity 211 145 Generalized ﬁnite element methods 213 146 Elements 220 147 Stiffness matrices 221 148 Coupling degrees of freedom 224 149 Numerical details 226 150 Warning 235 What are boundary elements? 239 21 Inﬂuence functions or Bettis theorem 240 22 Structural analysis with boundary elements 247 23 Comparison ﬁnite elementsboundary elements 262 Frames 269 32 The FE approach 270 33 Finite elements and the slope deﬂection method 289 34 Stiffness matrices 292 35 Approximations for stiffness matrices 298 36 Cables 305 37 Hierarchical elements 309 38 Sensitivity analysis 313
 49 Truss and frame models 363 410 Twobay wall 365 412 Shear wall with suspended load 370 413 Shear wall and horizontal load 375 414 Equilibrium of resultant forces 378 415 Adaptive mesh refinement 383 416 Plane problems in soil mechanics 386 417 Incompressible material 393 419 Inﬂuence functions for mixed formulations 399 420 Error analysis 401 Slabs 415 51 Kirchhoff plates 416 52 The displacement model 421 53 Elements 422 54 Hybrid elements 425 55 Singularities of a Kirchhoff plate 429 56 ReissnerMindlin plates 431 57 Singularities of a ReissnerMindlin plate 436 58 ReissnerMindlin elements 439 59 Supports 441 510 Columns 443 511 Shear forces 451 512 Variable thickness 452 513 Beam models 459 514 Wheel loads 460 515 Circular slabs 461 516 T beams 462 517 Foundation slabs 469 518 Direct design method 476 519 Point supports 477 520 Study 480 Shells 485 62 Shells of revolution 488 63 Volume elements and degenerate shell elements 490 64 Circular arches 491 65 Flat elements 493 66 Membranes 498 Theoretical details 503 72 Greens identities 508 73 Greens functions 516 74 Generalized Greens functions 519 75 Nonlinear problems 526 76 The derivation of inﬂuence functions 529 77 Weak form of influence functions 535 78 Influence functions for other quantities 539 79 Shifted Greens functions 541 710 The dual space 552 711 Some concepts of error analysis 560 712 Important equations and inequalities 568 References 578 Index 593 Copyright