Preface About the author Notations General 1.1. Introduction to FEM 1.2. General problems of numerical analysis of concrete structures Truss and beam structures 2.1. Corners in frame structures - rigid regions 2.2. Beams with variable depth - inclined haunches 2.3. Beams with halving joints and openings 2.4. Soft supports - elastic bedding 2.5. Shear walls with large openings 2.6. Bracing of high-rise buildings 2.7. Design of hollow box girder bridges 2.8. Truss system - design of T-beam bridges 2.9. Support conditions 2.10. Dimensioning of reinforced beams 2.11. Material nonlinear analysis of truss and beam systems Shear walls and deep beams 3.1. Estimation of stress resultants of deep beams 3.2. Modelling the support condition 3.3. Dimensioning of deep beams 3.4. Strut-and-tie models 3.5. Singularities Slabs 4.1. General 4.2. Meshing - size of elements 4.3. Material parameters - Poisson's ratio 4.4. Support conditions for slabs 4.5. One-way slab 4.6. Slabs that can lift from the supports 4.7. Discontinuous line support 4.8. Concrete joist floors 4.9. Flat slabs 4.10. Foundation slabs 4.11. Skewed slabs 4.12. Singularities 4.13. Discretisation - generation of the element mesh 4.14. Dimensioning of spatial structures 4.15. Comparison with analytical methods and tables Shell structures 5.1. Mesh generation 5.2. T-beams 5.3. Slab-on-beam structure 5.4. Composite structures 5.5. Singularities 5.6. Material nonlinear analysis of shells and massive members Three-dimensional building models 6.1. General problems 6.2. FE modelling of a building 6.3. Design of a building 6.4. Portal frame bridge 6.5. Checking and documentation of FE analyses 6.6. The power of FE analysis 6.7. Summary and conclusions References Index
show more

<ul><li><p>Finite-element Design of ConcreteStructuresPractical problems and their solutions</p></li><li><p>Finite-elementDesign of ConcreteStructuresPractical problems and theirsolutions</p><p>Second edition</p><p>G.A. RombachUniversity of Hamburg-Harburg</p></li><li><p>Published by ICE Publishing, 40 Marsh Wall, London E14 9TP.</p><p>Full details of ICE Publishing sales representatives and distributorscan be found at: www.icevirtuallibrary.com/info/printbooksales</p><p>First published 2004Second edition 2011</p><p>Also available from ICE Publishing</p><p>Structural Dynamics for Engineers, Second edition.H.A. Buchholdt and S.E. Moossavi Nejad. ISBN 978-0-7277-4176-9Structural Analysis with Finite Elements.P. Rugarli. ISBN 978-0-7277-4093-9Designers Guide to EN 1992-1-1 Eurocode 2: Design of ConcreteStructures (common rules for buildings and civil engineeringstructures).A.W. Beeby. ISBN 978-0-7277-3105-0</p><p>Associate Commissioning Editor: Victoria ThompsonProduction Editor: Imran MirzaMarket Development Executive: Catherine de Gatacre</p><p>www.icevirtuallibrary.com</p><p>A catalogue record for this book is available from the British Library</p><p>ISBN: 978-0-7277-4189-9</p><p># Thomas Telford Limited 2011</p><p>ICE Publishing is a division of Thomas Telford Ltd, a wholly-ownedsubsidiary of the Institution of Civil Engineers (ICE).</p><p>All rights, including translation, reserved. Except as permitted by theCopyright, Designs and Patents Act 1988, no part of this publicationmay be reproduced, stored in a retrieval system or transmitted in anyform or by any means, electronic, mechanical, photocopying orotherwise, without the prior written permission of the Publisher, ICEPublishing, 40 Marsh Wall, London E14 9TP.</p><p>This book is published on the understanding that the author is solelyresponsible for the statements made and opinions expressed in itand that its publication does not necessarily imply that suchstatements and/or opinions are or reflect the views or opinions of thepublishers. Whilst every effort has been made to ensure that thestatements made and the opinions expressed in this publicationprovide a safe and accurate guide, no liability or responsibility can beaccepted in this respect by the author or publishers.</p><p>Whilst every reasonable effort has been undertaken by the authorand the publisher to acknowledge copyright on materialreproduced, if there has been an oversight please contact thepublisher and we will endeavour to correct this upon a reprint.</p><p>Typeset by Academic Technical, BristolPrinted and bound in Great Britain by CPI Antony Rowe Limited,Chippenham and Eastbourne</p></li><li><p>Contents Preface viiAbout the author ixNotations xi</p><p>01 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General 11.1. Introduction to FEM 5</p><p>1.2. General problems of numerical analysis of concrete</p><p>structures 7</p><p>References 12</p><p>02 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Truss and beam structures 132.1. Corners in frame structures rigid regions 14</p><p>2.2. Beams with variable depth inclined haunches 23</p><p>2.3. Beams with halving joints and openings 29</p><p>2.4. Soft supports elastic bedding 37</p><p>2.5. Shear walls with large openings 56</p><p>2.6. Bracing of high-rise buildings 61</p><p>2.7. Design of hollow box girder bridges 80</p><p>2.8. Truss system design of T-beam bridges 84</p><p>2.9. Support conditions 105</p><p>2.10. Dimensioning of reinforced beams 110</p><p>2.11. Material nonlinear analysis of truss and beam systems 114</p><p>References 136</p><p>03 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shear walls and deep beams 1393.1. Estimation of stress resultants of deep beams 141</p><p>3.2. Modelling the support condition 145</p><p>3.3. Dimensioning of deep beams 158</p><p>3.4. Strut-and-tie models 166</p><p>3.5. Singularities 169</p><p>References 171</p><p>04 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Slabs 1734.1. General 173</p><p>4.2. Meshing size of elements 180</p><p>4.3. Material parameters Poissons ratio 181</p><p>4.4. Support conditions for slabs 183</p><p>4.5. One-way slab 185</p><p>4.6. Slabs that can lift from the supports 197</p><p>4.7. Discontinuous line support 199</p><p>4.8. Concrete joist floors 205</p><p>4.9. Flat slabs 206</p><p>4.10. Foundation slabs 228</p><p>4.11. Skewed slabs 234</p><p>4.12. Singularities 237</p><p>4.13. Discretisation generation of the element mesh 246</p><p>4.14. Dimensioning of spatial structures 254</p><p>4.15. Comparison with analytical methods and tables 261</p><p>References 269</p><p>05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shell structures 2715.1. Mesh generation 271</p><p>5.2. T-beams 283</p><p>v</p></li><li><p>5.3. Slab-on-beam structure 296</p><p>5.4. Composite structures 310</p><p>5.5. Singularities 310</p><p>5.6. Material nonlinear analysis of shells and massive</p><p>members 312</p><p>References 318</p><p>06 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Three-dimensional building models 3216.1. General problems 321</p><p>6.2. FE modelling of a building 323</p><p>6.3. Design of a building 329</p><p>6.4. Portal frame bridge 335</p><p>6.5. Checking and documentation of FE analyses 344</p><p>6.6. The power of FE analysis 348</p><p>6.7. Summary and conclusions 350</p><p>References 350</p><p>Index 351</p><p>vi</p></li><li><p>Preface Over the last few years, electronic data processing has changedthe work of engineers throughout all elds of construction. Thisis particularly true for design of structures, doing which is</p><p>impossible to imagine today without the help of computer</p><p>software. Even simple structures such as, for example, a simply</p><p>supported reinforced concrete beam under uniform loading, are</p><p>designed using the help now easily available from computers.</p><p>One must admire this. In many cases, these computer</p><p>calculations are faster, less costly and thus more protable than</p><p>using manual calculations.</p><p>The developments over the last year or so have involved using</p><p>yet more complex numerical models, as can be seen from the</p><p>various contributions to conferences and journals on the subject.</p><p>Today, it seems that modelling arbitrary complex reinforced</p><p>structures with almost unlimited accuracy is only a question of</p><p>adequate computer capacity, the size of the element mesh and</p><p>the accurate modelling of the nonlinear materials behaviour.</p><p>However, this involves the great danger that one only believes</p><p>the results from the computer, and the engineer loses his or her</p><p>feelings for the real behaviour of the structure. Thus, in this</p><p>book, the author is critical of blind belief in computer-based</p><p>results. The author advocates that one should not have totally</p><p>blind condence in the output of computer calculations, but</p><p>rather take the numerical model used, and hence the results</p><p>achieved, with a pinch of salt.</p><p>With the increasing complexity of numerical models, it becomes</p><p>likely that important details may be overlooked, primarily due to</p><p>the ood of information produced by computers. The collapse of</p><p>the Sleipner platform (see Chapter 5), resulting from an</p><p>erroneous nite-element (FE) calculation, impressively</p><p>demonstrates this danger.</p><p>A complex numerical calculation should not be used to</p><p>compensate for the engineers lack of knowledge of the</p><p>structural behaviour of a structure. An engineer should be able</p><p>to simplify any real structure into well-dened, known,</p><p>understandable and designable equivalent structural systems.</p><p>Unimportant details should be neglected. It should always be</p><p>kept in mind that even very complex structures, such as the</p><p>chapel of St. Peters Church in Rome or the temples in Luxor</p><p>and Karnak, have been built without the help of computers, and</p><p>possibly even without knowledge of mechanics.</p><p>This book has been written for both the practicing structural</p><p>engineer and for students who use computer software for</p><p>designing concrete structures. The problems of FE calculations</p><p>are illustrated, not just by theoretical systems, but also by</p><p>relating to real structures, mostly those on which the author has</p><p>actually worked. They concern systems from all elds of</p><p>engineering. Furthermore, this book should help those people</p><p>vii</p></li><li><p>who develop software for structural design to understand the</p><p>difference between theory and the daily problems of designing</p><p>reinforced concrete structures.</p><p>This book could not have been written without abundant help</p><p>and support from friends and colleagues in practice and</p><p>research. I am much indebted to Peter Whiting LL.B (Hons),</p><p>BSc, FICE, for his thorough reviewing of the manuscript and for</p><p>support of my work.</p><p>Guenter Axel Rombach</p><p>Hamburg</p><p>April 2011</p><p>viii</p></li><li><p>About theauthor</p><p>Prof. Dr.-Ing. Guenter Rombach graduated in 1983 as a civil and</p><p>structural engineer from the University of Karlsruhe, Germany,</p><p>where he also obtained his doctorate in 1991 in the eld of</p><p>numerical simulations of granular ow in silos. In 1990, he</p><p>joined a large construction company, where he worked as design</p><p>and project manager for various projects, mainly big bridges,</p><p>both in Europe and further aeld. In 1995, he became technical</p><p>manager in the construction project of one of the biggest</p><p>segmental bridges in the world, the Second Stage Expressway</p><p>System in Bangkok, Thailand. Since 1996, he has been full</p><p>professor in the eld of design of concrete structures at the</p><p>University of Hamburg-Harburg, where he is in charge of</p><p>undergraduate and graduate courses in the design of reinforced</p><p>and prestressed concrete structures. In 1999, he became a</p><p>certied checking engineer. Prof. Rombach is a member of</p><p>several national and international professional committees that</p><p>deal with design of concrete structures, for example, bridges. He</p><p>has published several books about nite-element design of</p><p>concrete structures and the design of prestressed structures.</p><p>ix</p></li><li><p>Notations In general, the symbols of Eurocode 2 are used. These are listedhere, together with the additional abbreviations used in this book.1 Latin upper case letter</p><p>A Accidental action; cross-sectional area</p><p>Ac Cross-sectional area of concrete</p><p>Ap Area of a prestressing tendon or tendons</p><p>As Cross-sectional area of reinforcement</p><p>As,min Minimum cross-sectional area of reinforcement</p><p>As,prov Area steel provided</p><p>As,req Area steel required</p><p>Asw Cross-sectional area of shear reinforcement</p><p>bc Width of column</p><p>C Symbol for grade of normal concrete</p><p>CM Wrapping torsional stiffness</p><p>D Diameter of mandrel</p><p>E Effect of action (member force)</p><p>Ec Tangent modulus of elasticity of normal weight concrete</p><p>at a stress of c 0 and at 28 daysEc,eff Effective modulus of elasticity of concrete</p><p>Ecd Design value of modulus of elasticity of concrete</p><p>Ecm Secant modulus of elasticity of concrete</p><p>Es Design value of modulus of elasticity of reinforcing steel</p><p>EI Bending stiffness</p><p>F Force; action</p><p>Fd Design value of an action</p><p>FE Finite element</p><p>Gk Characteristic permanent action</p><p>H Horizontal force</p><p>I Second moment of area</p><p>L Length</p><p>M Bending moment</p><p>MEd Design value of the applied internal bending moment</p><p>MT Torsional moment</p><p>N Axial force</p><p>NEd Design value of the applied axial force (tension or</p><p>compression)</p><p>P Prestressing force</p><p>P0 Initial force at the active end of the tendon immediately</p><p>after stressing</p><p>Pmt Mean value of the prestressing force at time t, at any</p><p>point distance x along the member</p><p>Qk Characteristic variable action</p><p>R Resistance</p><p>Rd Nominal value of resistance</p><p>S Internal forces and moments</p><p>S First moment of area</p><p>SM Centre of torsion of a cross-section</p><p>SS Centre of gravity of a cross-section</p><p>SLS Serviceability limit state</p><p>xi</p></li><li><p>T Torsional moment</p><p>ULS Ultimate limit state</p><p>V Shear force</p><p>VEd Design value of the applied shear force</p><p>2 Latin lower case letters</p><p>a Distance; geometrical data</p><p>a Deviation of geometrical data</p><p>al Shift of moment curve</p><p>asup Breadth of the support</p><p>b Overall width of a cross-section, or actual ange width</p><p>in a T- or L-beam</p><p>bw Width of web on T-, I- or L-beams</p><p>c Concrete cover</p><p>d Diameter; depth</p><p>d Effective depth of a cross-section</p><p>dg Largest nominal maximum aggregate size</p><p>e Eccentricity</p><p>f Strength (of a material)</p><p>fc Compressive strength of concrete</p><p>fcd Design value of compressive strength of concrete</p><p>fck Characteristic compressive cylinder strength of concrete</p><p>at 28 days</p><p>ft Tensile strength of reinforcement</p><p>fy Yield strength of reinforcement</p><p>h Height</p><p>h Overall depth of a cross-section</p><p>i Radius of gyration</p><p>k Coefcient; factor</p><p>l Length; span</p><p>lb Anchorage length</p><p>lcol Height of a column</p><p>leff Effective span of beams and slabs</p><p>ln Clear distance from the faces of the supports</p><p>m Moment per unit length; mass</p><p>n Number of vertical continuous members</p><p>r Radius</p><p>1/r Curvature at a particular section</p><p>s Distance; spacing of stirrups</p><p>p Mean transverse pressure over the anchorage length</p><p>t Time being considered; thickness</p><p>t0 The age of concrete at the time of loading</p><p>u Perimeter of concrete cross-section, having area Acu, v, w Components of the displacement of a point</p><p>v shear force per unit length</p><p>v Coefcient relating the average design compressive</p><p>stress in struts to the design value of the concrete</p><p>compressive strength ( fcd)</p><p>v Angle of inclination of a structure, assumed in assessing</p><p>effects of imperfections</p><p>xii</p></li><li><p>x Neutral axis depth</p><p>x, y, z Coordinates</p><p>z Lever arm of internal forces</p><p>3 Greek lower letters</p><p> Angle; ratio</p><p> Angle; ratio; coefcient</p><p> Partial safety factor</p><p>C Partial factor for concrete</p><p>G Partial factor for permanent action, G</p><p>M Partial factor for a material property</p><p>Q Partial factor for variable action, Q</p><p> Increment; redistribution ratio</p><p> Reduction factor; distribution coefcient</p><p>' Strain</p><p>'c Compressive strain in concrete</p><p>'c1 Compressive strain in the concrete at the peak stress fc'cu Ultimate compressive strain in the concrete</p><p>'u Strain of reinforcement or prestressing steel at</p><p>maximum load</p><p>'u Characteristic strain of reinforcement or prestressing</p><p>steel at maximum load</p><p> Angle; rotation</p><p>F Angle between the x-axis and the major principal stress</p><p>in the concrete (measured in the anti-clockwise</p><p>direction)</p><p> Slenderness ratio</p><p> Coefcient of friction between tendons and their ducts</p><p> Moment coefcient</p><p> Poissons ratio</p><p> Strength reduction factor for concrete cracked in shear</p><p> Longitudinal force coefcient for an element</p><p> Ratio of bond strength of prestressing and reinforcing</p><p>steel</p><p> Over-dry density of concrete in kg/m3</p><p>l Reinforcement ratio for longitudinal reinforcement</p><p>w Reinforcement ratio for shear reinforcement</p><p> Normal stress</p><p>c Compressive stress in the concrete</p><p>s Tensile stresses in reinforcement</p><p> Torsional shear stress</p><p>; Diameter of a reinforcing bar or of a prestressing duct;n Equivalent diameter of a bundle of reinforcing bars(t, t0) Creep coefcient, dening creep between times t and t0,</p><p>related to elastic deformation at 28 days</p><p>(1, t0) Final value of creep coefcient Factors dening representative values of variable actions</p><p>0 for combination values</p><p>1 for frequent values</p><p>2 for quasi-permanent values</p><p>xiii</p></li><li><p>4 Subscripts</p><p>c Concrete; compression; creep</p><p>b Bond</p><p>d Design</p><p>e Eccentricity</p><p>eff Effective</p><p>f Flange</p><p>fat Fatigue</p><p>fav Favourable</p><p>freq Frequent</p><p>g Permanent action</p><p>i Indices; notional</p><p>inf Inferior; lower</p><p>j Indices</p><p>k Characteristic</p><p>l Low; lower</p><p>m Mean; material; bending</p><p>max Maximum</p><p>min Minimum</p><p>nom Nominal</p><p>p Prestressing force</p><p>perm Permanent</p><p>pl Plastic</p><p>q Variable action</p><p>rep Representative</p><p>s Reinforcing steel; shrinkage</p><p>sup Superior; upper</p><p>t Torsion; time being considered; tension</p><p>unf Unfavourable</p><p>w Web</p><p>y Yield</p><p>xiv</p></li><li><p>Finite-element Design of Concrete Structures</p><p>ISBN: 978-0..</p></li></ul>