Springer structural analysis with finite elements (springer)

Structural Analysis with Finite Elements Friedel Hartmann Casimir Katz Structural Analysis with Finite Elements With 408 Figures and 26 Tables 123 Friedel Hartmann University of Kassel Structural Mechanics Kurt-Wolters-Str 34109 Kassel Germany friedelhartmann@uni-kassel.de Casimir Katz SOFiSTiK AG Bruckmannring 38 85764 Oberschleissheim Germany casimir.katz@sofistik.de Library of Congress Control Number: 2006937296 ISBN-10 3-540-49698- x Springer Berlin Heidelberg New York ISBN-13 978-3-540-49698-4 Springer Berlin Heidelberg New York This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable to prosecution under the German Copyright Law Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use A X macro package Typesetting by authors using a Springer LT E Cover design: deblik, Berlin Printed on acid-free paper SPIN 11940241 62/3100/SPi Preface The finite element method has become an indispensible tool in structural analysis, and tells an unparalleled success story With success, however, came criticism, because it was noticeable that knowledge of the method among practitioners did not keep up with success Reviewing engineers complain that the method is increasingly applied without an understanding of structural behavior Often a critical evaluation of computed results is missing, and frequently a basic understanding of the limitations and possibilities of the method are nonexistent But a working knowledge of the fundamentals of the finite element method and classical structural mechanics is a prerequisite for any sound finite element analysis Only a well trained engineer will have the skills to critically examine the computed results Finite element modeling is more than preparing a mesh connecting the elements at the nodes and replacing the load by nodal forces This is a popular model but this model downgrades the complex structural reality in such a way that—instead of being helpful—it misleads an engineer who is not well acquainted with finite element techniques The object of this book is therefore to provide a foundation for the finite element method from the standpoint of structural analysis, and to discuss questions that arise in modeling structures with finite elements What encouraged us in writing this book was that—thanks to the intensive research that is still going on in the finite element community—we can explain the principles of finite element methods in a new way and from a new perspective by making ample use of influence functions This approach should appeal in particular to structural engineers, because influence functions are a genuine engineering concept and are thus deeply rooted in classical structural mechanics, so that the structural engineer can use his engineering knowledge and insight to assess the accuracy of finite element results or to discuss the modeling of structures with finite elements Just as a change in the elastic properties of a structure changes the Green’s functions or influence functions of the structure so a finite element mesh effects a shift of the Green’s functions We have tried to concentrate on ideas, because we considered these and not necessarily the technical details to be important The emphasis should VI Preface be on structural mechanics and not on programming the finite elements, and therefore we have also provided many illustrative examples Finite element technology was not developed by mathematicians, but by engineers (Argyris, Clough, Zienkiewicz) They relied on heuristics, their intuition and their engineering expertise, when in the tradition of medieval craftsmen they designed and tested elements without fully understanding the exact background The results were empirically useful and engineers were grateful because they could suddenly tackle questions which were previously unanswerable After these early achievements self-confidence grew, and a second epoch followed that could be called baroque: the elements became more and more complex (some finite element programs offered 50 or more elements) and enthusiasm prevailed In the third phase, the epoch of “enlightment” mathematicians became interested in the method and tried to analyze the method with mathematical rigor To some extent their efforts were futile or extremely difficult, because engineers employed “techniques” (reduced integration, nonconforming elements, discrete Kirchhoff elements) which had no analogy in the calculus of variations But little by little knowledge increased, the gap closed, and mathematicians felt secure enough with the method that they could provide reliable estimates about the behavior of some elements We thus recognize that mathematics is an essential ingredient of finite element technology One of the aims of this book is to teach structural engineers the theoretical foundations of the finite element method, because this knowledge is invaluable in the design of safe structures This book is an extended and revised version of the original German version We have dedicated the web page http://www.winfem.de to the book From this page the programs WINFEM (finite element program with focus on influence functions and adaptive techniques), BE-SLABS (boundary element analysis of slabs) and BE-PLATES (boundary element analysis of plates) can be downloaded by readers who want to experiment with the methods Additional information can also be found on http://www.sofistik.com FriedelHartmann@uni-kassel.de Kassel Munich August 2003 Casimir.Katz@sofistik.de Friedel Hartmann Casimir Katz Acknowledgement We thank Thomas Graetsch, who wrote the program WINFEM and provided many illustrative examples for the approximation of influence functions with finite elements, and Marc Damashek and William J Gordon for their help in preparing the manuscript The permission of Oxford University Press to reprint the picture on page 145 is greatly acknowledged Preface to the second edition One of the joys of writing a book is that the authors learn more about a subject This does not stop after a book is finished So we have added additional sections to the text • • • • • • • • • The Dirac energy How to predict changes The influence of a single element Retrofitting structures Generalized finite element methods (X-FEM) Cables Hierarchical elements Sensitivity analysis Weak form of influence functions in the hope that these additional topics will also attract the readers’ interest Kassel Munich October 2006 Friedel Hartmann Casimir Katz Contents What are finite elements? 1.1 Introduction 1.2 Key points of the FE method 1.3 Potential energy 1.4 Projection 1.5 The error of an FE solution 13 1.6 A beautiful idea that does not work 15 1.7 Set theory 16 1.8 Principle of virtual displacements 23 1.9 Taut rope 29 1.10 Least squares 33 1.11 Distance inside = distance outside 37 1.12 Scalar product and weak solution 40 1.13 Equivalent nodal forces 42 1.14 Concentrated forces 44 1.15 Green’s functions 51 1.16 Practical consequences 55 1.17 Why finite element results are wrong 57 1.18 Proof 64 1.19 Influence functions 69 1.20 Accuracy 80 1.21 Why resultant stresses are more accurate 86 1.22 Why stresses at midpoints are more accurate 88 1.23 Why stresses jump 99 1.24 Why finite element support reactions are relatively accurate 99 1.25 Gauss points 104 1.26 The Dirac energy 110 1.27 How to predict changes 113 1.28 The influence of a single element 126 1.29 Retrofitting structures 130 1.30 Local errors and pollution 136 1.31 Adaptive methods 147 1.32 St Venant’s principle 172 1.33 Singularities 175 1.34 Actio = reactio? 177 X Contents 1.35 1.36 1.37 1.38 1.39 1.40 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49 1.50 The output 181 Support conditions 183 Equilibrium 184 Temperature changes and displacement of supports 187 Stability problems 193 Interpolation 197 Polynomials 199 Infinite energy 208 Conforming and nonconforming shape functions 209 Partition of unity 211 Generalized finite element methods 213 Elements 220 Stiffness matrices 221 Coupling degrees of freedom 224 Numerical details 226 Warning 235 What are boundary elements? 239 2.1 Influence functions or Betti’s theorem 240 2.2 Structural analysis with boundary elements 247 2.3 Comparison finite elements—boundary elements 262 Frames 269 3.1 Introduction 269 3.2 The FE approach 270 3.3 Finite elements and the slope deflection method 289 3.4 Stiffness matrices 292 3.5 Approximations for stiffness matrices 298 3.6 Cables 305 3.7 Hierarchical elements 309 3.8 Sensitivity analysis 313 Plane problems 327 4.1 Simple example 327 4.2 Strains and stresses 334 4.3 Shape functions 337 4.4 Plane elements 338 4.5 The patch test 344 4.6 Volume forces 346 4.7 Supports 347 4.8 Nodal stresses and element stresses 357 4.9 Truss and frame models 363 4.10 Two-bay wall 365 4.11 Multistory shear wall 365 4.12 Shear wall with suspended load 370 Contents 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 XI Shear wall and horizontal load 375 Equilibrium of resultant forces 378 Adaptive mesh refinement 383 Plane problems in soil mechanics 386 Incompressible material 393 Mixed methods 393 Influence functions for mixed formulations 399 Error analysis 401 Nonlinear problems 401 Slabs 415 5.1 Kirchhoff plates 416 5.2 The displacement model 421 5.3 Elements 422 5.4 Hybrid elements 425 5.5 Singularities of a Kirchhoff plate 429 5.6 Reissner–Mindlin plates 431 5.7 Singularities of a Reissner–Mindlin plate 436 5.8 Reissner–Mindlin elements 439 5.9 Supports 441 5.10 Columns 443 5.11 Shear forces 451 5.12 Variable thickness 452 5.13 Beam models 459 5.14 Wheel loads 460 5.15 Circular slabs 461 5.16 T beams 462 5.17 Foundation slabs 469 5.18 Direct design method 476 5.19 Point supports 477 5.20 Study 480 5.21 Sensitivity analysis 480 Shells 485 6.1 Shell equations 485 6.2 Shells of revolution 488 6.3 Volume elements and degenerate shell elements 490 6.4 Circular arches 491 6.5 Flat elements 493 6.6 Membranes 498 XII Contents Theoretical details 503 7.1 Scalar product 503 7.2 Green’s identities 508 7.3 Green’s functions 516 7.4 Generalized Green’s functions 519 7.5 Nonlinear problems 526 7.6 The derivation of influence functions 529 7.7 Weak form of influence functions 535 7.8 Influence functions for other quantities 539 7.9 Shifted Green’s functions 541 7.10 The dual space 552 7.11 Some concepts of error analysis 560 7.12 Important equations and inequalities 568 References 579 Index 593 582 References 63 Ciarlet PG (1978) The Finite Element Method for Elliptic Problems NorthHolland Amsterdam 64 Ciarlet PG, Lions JL Eds (1991) Handbook of Numerical Analysis, Volume II: Finite Element Methods North-Holland, Amsterdam 65 Ciarlet PG, Lions JL Eds (1996) Handbook of Numerical Analysis, Volume IV: Finite Element Methods (Part 2), Numerical Methods for Solids (Part 2) Elsevier, Amsterdam 66 Ciarlet PG (2005) An Introduction to Differential Geometry with Applications to Elasticity Springer Verlag 67 C ¸ irak F, Ramm E (2000) “A posteriori error estimation and adaptivity for elastoplasticity using the reciprocal theorem”, Int J Num Methods in Eng 47: 379–393 68 Clough RW, Tocher JL (1965) “Finite element stiffness matrices for analysis of plate bending”, Proc (1st) Conf On Matrix 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33: 1331–1364 263 Zienkiewicz OC, Zhu JZ (1992) “The superconvergent patch recovery and a posteriori error estimates, Part 2: Error estimates and adaptivity”, Int J Num Methods in Eng 33: 1365–1382 Index ||u||E 507 ||f ||0 505 ||u||m 506 (p, u) 505 (p, ϕi ) 507 (p, w) 505 C -elements 338 C -elements 424 Gj 81 H m (Ω) 506 L2 scalar product 505 Ph 28 S 22 V 22 559 V Vh Vh 559 Vh+ 185 δ0 69 δi 69 div 503 div2 M 426 tr, trace 395 ∇ 503 ⊗ 504 ∂ i w 536 C[ ], elasticity tensor 395 E 503 S 503 uG 134 ˆ ) 504 a(u, u a(w, w) ˆ 35 a(w, w) 10 p(ϕi ) 507 a posteriori error indicators accuracy 80 actio = reactio? 177 adaptive mesh refinement 383 adaptive methods 147 adaptivity, dimension 154 adaptivity, model 154 adding a member 323 approximations for stiffness matrices 298 asymptotic error estimates 560 Aubin–Nitsche trick 540, 564 Babuˇska–Brezzi condition 399 backward error analysis 201 Bathe–Dvorkin element 439 beam girders 462 beam models 459 beam-like elements 356 Bessel’s inequality 10 bilinear elements 340 bilinear form 504 boolean matrix 329 boundary layer effect 436 Boussinesq 256 brachistochrone 175 bulk modulus 393 147 cables 305 capacity 56 Castigliano’s Theorem 50 Cauchy-Schwarz inequality 506 change in an elastic support 316 changes in a structure 313 circular arches 491 circular slabs 461 columns 443 composition 22 concentrated forces 44 condition number 230 continuity 221 594 Index corner force 421 coupling degrees of freedom 224 cracks in a beam 316 critical angles, plane problem 336 critical load 302 CST element 338 cubic elements 338 cycloid 175 d-adaptivity 154 deformation space degenerate shell elements 490 dichotomy 149 dimension argument 45 dipole 97 Dirac delta 69 Dirac energy 110, 111 direct design method 476 direct product 504 discontinuities 389 displaced point supports 354 displacement methods 393 displacement model of slabs 421 displacements 40 distance 37 DKT element 440 drilling degrees of freedom 344 drop panels and column capitals 451 DST element 441 dual load 69 dual quantities 70 dual weighted residual error estimate 525 duality 40 duality techniques 156, 160, 520 effective length factor 302 effective width 467 efficiency of estimator 567 eigenvalue 230 eigenvector 230 eigenwork 17 Element Free Galerkin method element stresses 357 elements 220 elements, plane problems 338 elements, slabs 422 ellipticity 21 energy method 215 energy norm 151 equilibrium 86, 184, 420 equilibrium of resultant forces 378 equilibrium point 20 error estimators 566 error in energy 151 error indicator 566 essential boundary conditions 570 Euler equation 25, 40 Euler-beam I 193 explicit error estimators 567 finite element 197 first fundamental form 486 flat elements 493 flexibility matrix 198 folded plates 463 forces 40 foundation slabs 469 fundamental solution 241 Galerkin method 193 Galerkin orthogonality 11 Gateaux derivative 402 Gaussian curvature 488 Generalized Finite Element method 213 geometric matrix 301 global equilibrium 185 goal-oriented recovery 160, 520 goal-oriented refinement 412 Green’s first identity 508 Green’s function 51, 57, 516 Green’s identities 508 Green-Lagrangian strain tensor 401 h-methods 154 half-space model 256, 470 hard support 436 hat functions 197 Heaviside function 218 Hellinger–Reissner principle 396 Hermite Polynomials 201 hierarchical elements 309 higher-order elements 344 homogeneous deflection curve 289 hourglass modes 230 How to predict changes 113 hp-method 154 Index 595 HSM element 425 hybrid elements 425 hyperelastic material 402 missing member 325 mixed methods 393 Mohr’s integral 240, 535 implicit error estimators 567 incompressible material 393 infinite energy 208 influence functions 42, 69 influence functions for mixed formulations 399 influence of a single element 126 integral operators 82 integration by parts 508 interpolation 197 isotropy 221 natural boundary conditions 570 Nitsche trick 540 nodal influence functions 77 nodal stresses 357 nonlinear problems 123, 161, 167, 401, 526 normal equations 34 kernel of a matrix 231 kernel of an influence function kinematic constraints 347 Kirchhoff plates 416 Kirchhoff shear 420 Kronecker delta 199 Lagrange element 200, 343 Lagrange multipliers 226 Lagrange polynomials 200 least action 20 least squares 33 linear elements 338 linear strain triangle 341 lintels 356 load case δ 17 load case p 17 loads 255 local equilibrium 185 local solution 136 locking 437 loss of a frame member 320 loss of a support 317 LST element 341 m-adaptivity 154 mapped polynomials 203 master degree of freedom 226 Maxwell’s theorem 550 membrane locking 491, 492 membrane stresses 487 membranes 498 Mindlin shell elements 490 58 order of the singularity 46 order of the strain energy 46 output 181 p-methods 154 particular deflection curve 289 partition of unity 211 Partition of Unity method 213 Pascal’s triangle 200 patch test 344 peaks 102 piecewise polynomials 197 piled raft foundations 258 plane elements 338 plane strain 222, 334 plane stress 222, 334 point supports 353, 477 Poisson’s ratio 416, 419 polynomials 199 potential energy primal problem 157 primary load case 386 principal stresses 336 principle of minimum potential energy principle of virtual displacements 25 principle of virtual forces 240 projection 515 projection method pulley 111 Pythagorean theorem 573 Q4 340 Q4 + 341 Q8 343 quadratic elements 338 596 Index r-adaptivity 154 Rayleigh quotient 195 recovery based error estimators 567 reduced global stiffness matrix 228 reduced integration 228 reduced stiffness matrix 185, 224 Reissner–Mindlin plates 431 remeshing 154 residual 236 residual forces 15 restriction retrofitting structures 130 Riesz’ representation theorem 521, 559 rigid-body motions 221 rope rotational invariance 221, 338 rounding errors 230 safety of structures 326 scalar product 33, 40, 503 scalar product of matrices 504 second fundamental form 486 second Piola-Kirchhoff stress tensor 401 self-equilibrated stresses 386 sensitivity analysis 313 serendipity elements 200, 343 set theory 16 settlement of supports 187 settlements 387 shape functions 337 shear forces 451 shear locking 437 shear modulus 336, 393 shearing strain 432 shell equations 485 shells 485 shells of revolution 488 shift 82 Shifted Green’s Function Theorem 547 shifted Green’s functions 541 singular matrices 227 singularities 175 skew projection 13, 22 slabs, Reissner–Mindlin 431 slave degree of freedom 226 smoothing process 362 Sobolev norms 12 Sobolev’s Embedding Theorem 46 soft support 436 spurious modes 211 St Venant’s principle 172 stability problems 193 stiffness matrices 221 stiffness matrices for 1-D elements 292 stiffness matrices, slabs 424 stiffness matrices, three properties 223 stiffness matrix, 2-D 339 strain energy product 31, 35, 221, 339, 395 stream model 330 substitute load case superconvergence 206 support conditions 183 support reactions 99, 353 supports 441 symmetries 354 T beams 226 tangential supports 354 Taylor series 199 temperature 187 tension tension stiffening 288 test range 36 test space 22 three-moment equation 198 Timoshenko beam 296 Timoshenko beam element 301 Tottenham’s equation 64 translations 212 trial space 6, 22 truss models 363 values at a point 62 variable thickness 452 virtual displacements 24 volume elements 490 weak convergence 40 weak equal sign weak form of influence functions weak solution 40 weighted least squares 35 weighted residual method 194 wheel loads 460 535 Index Wilson’s Element 341 Winkler model 255, 469 X-FEM 215 yielding of a rigid support zero-energy modes 230 319 597 ...Friedel Hartmann Casimir Katz Structural Analysis with Finite Elements With 408 Figures and 26 Tables 123 Friedel Hartmann University of Kassel Structural Mechanics Kurt-Wolters-Str 34109... always be obtained from Springer Violations are liable to prosecution under the German Copyright Law Springer is a part of Springer Science+Business Media springer. com © Springer- Verlag Berlin... 235 What are boundary elements? 239 2.1 Influence functions or Betti’s theorem 240 2.2 Structural analysis with boundary elements