Practitioners’ guide to finite element modelling of reinforced concrete structures State-of-art report prepared by Task Group 4.4 June 2008 Subject to priorities defined by the Technical Council and the Presidium, the results of fib’s work in Commissions and Task Groups are published in a continuously numbered series of technical publications called 'Bulletins'. The following categories are used: category minimum approval procedure required prior to publication Technical Report approved by a Task Group and the Chairpersons of the Commission State-of-Art Report approved by a Commission Manual, Guide (to good practice) approved by the Technical Council of fib or Recommendation Model Code approved by the General Assembly of fib Any publication not having met the above requirements will be clearly identified as preliminary draft. This Bulletin N° 45 was approved as an fib State-of-art report by Commission 4 in April 2004. This report was drafted by Task Group 4.4, Computer based modelling and design, in Commission 4, Modelling of structural behaviour and design: Koichi Maekawa 10, 1, 3, 5, 6 (Univ. of Tokyo, Japan, Co-Convener), Frank Vecchio 1, 3, 5, 6, 10 (Univ. of Toronto, Canada, Co-convener) Stephen Foster 2, 6, 3, 4, 5, 8 (Univ. of New South Wales, Australia, Editor) Oguzhan Bayrak 4, 6 (Univ. of Texas at Austin, USA), Evan Bentz 3, 4 (University of Toronto, Canada), Johan Blaauwendraad 9 (Delft Univ. of Technology, The Netherlands), Jan Cervenka 5 (Cervenka Consulting, Czech Republic), Vladimir Cervenka 5, 6, 10 (Cervenka Consulting, Czech Republic), Tetsuya Ishida 6 (Univ. of Tokyo, Japan), Milan Jirasek 6 (Czech Technical Univ. in Prague, Czech Republic), Walter Kaufmann 2 (dsp Ingenieure & Planer AG, Switzerland), Johann Kollegger 5 (Technische Univ. Wien, Austria), Daniel Kuchma 7, 8 (Univ. of Illinois, USA), Ho Jung Lee 7 (SC Solutions, Inc., USA), Giuseppe Mancini 3 (Politecnico Torino, Italy), Giorgio Monti (Sapienza Università di Roma, Italy) 4, 6, Josko O!bolt 5, 6 (Univ. Stuttgart, Germany), Clemens Preisinger 5 (Technische Univ. Wien, Austria), Enrico Spacone 4 (Univ. of Chieti-Pescara, Italy), Tjen Tjhin 8 (Buckland and Taylor Ltd. Bridge Engineering, USA) 1, 2, 3 ... Chapter number for which this member was the main preparing author 1, 2, 3 ... Chapter number for which this member provided contributions Full address details of Task Group members may be found in the fib Directory or through the online services on fib's website, www.fib-international.org. Cover image: FE modelling of high strength squat shear walls (image courtesy of S. Foster) © fédération internationale du béton (fib), 2008 Although the International Federation for Structural Concrete fib - féderation internationale du béton - does its best to ensure that any information given is accurate, no liability or responsibility of any kind (including liability for negligence) is accepted in this respect by the organisation, its members, servants or agents. All rights reserved. No part of this publication may be reproduced, modified, translated, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior written permission. First published in 2008 by the International Federation for Structural Concrete (fib) Postal address: Case Postale 88, CH-1015 Lausanne, Switzerland Street address: Federal Institute of Technology Lausanne - EPFL, Section Génie Civil Tel +41 21 693 2747 • Fax +41 21 693 6245 [email protected] • www.fib-international.org ISSN 1562-3610 ISBN 978-2-88394-085-7 Printed by Sprint-Digital-Druck, Stuttgart . Preface In September 2000, fib Commission 4 established a task group (TG4.4) with the objective of preparing a report for guiding design engineers in the safe use of computer-based analysis procedures for design of reinforced concrete structures. The working party that first met in Berlin 2001 brought together a group of highly regarded researchers from Europe, the Americas, East Asia and Australasia with the objective of producing a document for use by engineers with some background in numerical modelling. In the six years since work started on this report, advanced models have continued to be developed; however, this report is not about picking one model over another but, rather, how designers can use existing and future models as tools in design practice, in benchmarking of their models against established and reliable test data and in selecting an appropriate safety factor as well as recognising various pitfalls. Non-linear computer analysis methods have seen remarkable advancement in the last half- century with much research activity in the manner of constitutive modelling of reinforced concrete behaviour and in the development of sophisticated analysis algorithms. These advancements are well documented in various state-of-the-art reports and remain the subject of intensive research today. Linear and non-linear analysis methods, combined with plasticity design processes, and with local detailing methods such as strut-and-tie modelling, can form the basis of design of new, complex, structures that are not easily dimensioned using other rational design methods. The state-of-the-art in non-linear finite element analysis of reinforced concrete has progressed to the point where such procedures are close to being practical, every-day tools for design office engineers. No longer solely within the domain of researchers, they are finding use in various applications; many relating to our aging infrastructure. Non-linear computer analysis procedures can be used to provide reliable assessments of the strength and integrity of damaged or deteriorated structures, or of structures built to previous codes, standards or practices deemed to be deficient today. They can serve as valuable tools in assessing the expected behaviour from retrofitted structures, or in investigating and rationally selecting amongst various repair alternatives. Non-linear finite element analysis procedures are also proving particularly valuable in forensic analyses. In the near future, they will likely form the main engine in computer-based automated design software, although in a form likely invisible to the user. This report provides an overview of concepts and techniques relating to computer-based modelling of structural concrete. It attempts to provide a diverse and balanced portrayal of the current technical knowledge, recognizing that there are often competing and conflicting viewpoints. The report is written primarily for the benefit of the practicing engineer, rather than as a state-of-the-art for researchers, concentrating more on practical application and less on subtleties in constitutive modelling. To the members of the working party, our sincere thanks for the extensive and voluntary work undertaken over an extended period to get this report competed. Stephen Foster, Editor, chair of fib Commission 4 Koichi Maekawa, co-convener of TG 4.4 Frank Vecchio, deputy chair of fib Commission 4 and co-covener of TG 4.4 14 November 2007 fib Bulletin 45: Practitioners’ guide to finite element modelling of reinforced concrete structures iii . Contents 1 Introduction 1 1.1 Preamble 1 1.2 Notation 2 1.3 Sample applications 2 (1.3.1 Kimberley!Clark warehouse – 1.3.2 Sleipner A offshore platform – 1.3.3 Frame corner – 1.3.4 Base slabs in LNG storage tank) 1.4 The question of accuracy (1.4.1 – Reasons for caution) 20 1.5 Challenges remaining 27 1.6 Objectives 29 1.7 Scope of report 30 1.8 References 30 2 Design using linear stress analysis 33 2.1 Introduction 33 2.2 Membrane structures 34 (2.2.1 Notation – 2.2.2 General – 2.2.3 Reinforcement in one direction – 2.2.4 Isotropically reinforced panels – 2.2.5 The general solution – 2.2.6 Some comments on the angle ! – 2.2.7 The design concrete compression strength, f . – 2.2.8 Example – Design of a reinforced concrete cd squat shear wall) 2.3 Slabs and shells 52 (2.3.1 General – 2.3.2 Stress resultants – 2.3.3 Equilibrium, stress transformation and boundary conditions for slabs – 2.3.4 Normal moment yield criterion for slabs – 2.3.5 Sandwich model for the dimensioning of shell elements – 2.3.6 Dimensioning of slab and shell elements in design practice – 2.3.7 Example 1 – 2.3.8 Example 2) 2.4 3D solid modelling 70 (2.4.1 Introduction – 2.4.2 Background – 2.4.3 Application to reinforced concrete – 2.4.4 Reinforcement dimensioning for 3D stresses ! example 1 – 2.4.5 Reinforcement dimensioning for 3D stresses ! example 2) 2.5 References 78 3 Essential nonlinear modelling concepts 83 3.1 Introduction 83 3.2 Nonlinear concrete behaviour 84 (3.2.1 Concrete in compression – 3.2.2 Concrete in tension – 3.2.3 Modelling of tension stiffening – 3.2.4 Modelling of concrete cracks – 3.2.5 Modelling of reinforcement) 3.3 Nonlinear concrete modelling framework 98 (3.3.1 Elasticity – 3.3.2 Plasticity – 3.3.3 Damage – 3.3.4 Mixed models – 3.3.5 Discrete modelling frameworks) 3.4 Solution methods 102 (3.4.1 Newton!Raphson method – 3.4.2 Modified Newton!Raphson method) 3.5 Precision of nonlinear concrete FE analyses 104 3.6 Safety and reliability 105 3.7 Statistical analyses 114 3.8 Concluding remarks 115 3.9 References 115 4 Analysis and design of frame structures using non!linear models 121 4.1 Introduction 121 4.2 Notation 122 iv fib Bulletin 45: Practitioners’ guide to finite element modelling of reinforced concrete structures . 4.3 Nonlinear models of frame elements 123 (4.3.1 Lumped versus distributed plasticity – 4.3.2 Distributed models – 4.3.3 Section models: fibre elements vs. strut!and!tie – 4.3.4 Modelling of shear – 4.3.5 Modelling Bond Slip in Beams – 4.3.6 Analysis of a section) 4.4 Interpretation of results 148 (4.4.1 Localisation problems – 4.4.2 Physical characteristics of localised failure in concrete – 4.4.3 Regularisation techniques for force!based frame elements – 4.4.4 Practical considerations) 4.5 References 160 5 Analysis and design of surface and solid structures using non!linear models 165 5.1 Introduction 165 5.2 Notation 165 5.3 2D Structures with in!plane loading 166 5.4 Plate and shell structures (5.4.1 Layered elements) 170 5.5 Three dimensional solid structures 173 (5.5.1 Introduction – 5.5.2 Models based on non!linear elasticity – 5.5.3 Fracture!plasticity modelling – 5.5.4 Microplane model – 5.5.5 Examples of the application of 3D FE modeling) 5.6 References 190 6 Advanced modelling and analysis concepts 195 6.1 Introduction 195 6.2 Constitutive frameworks 195 (6.2.1 Non!linear elasticity – 6.2.2 Plasticity – 6.2.3 Continuum damage mechanics – 6.2.4 Smeared crack models – 6.2.5 Microplane models) 6.3 Solution strategies 214 (6.3.1 Introduction – 6.3.2 Newton!Raphson method – 6.3.3 Modified Newton!Raphson method – 6.3.4 Incremental displacement method – 6.3.5 The constant arc length method – 6.3.6 Line searches – 6.3.7 Convergence criteria – 6.3.8 Load!displacement incrementation) 6.4 Other issues 223 (6.4.1 Post peak response of compression elements – 6.4.2 Effects of ageing and distress in concrete – 6.4.3 Effects of ageing and distress in reinforcing steel – 6.4.4 Second order effects) 6.5 References 227 7 Benchmark tests and validation procedures 233 7.1 Introduction 233 7.2 Calibration and validation of NLFEA models 234 (7.2.1 Overview of model calibration and validation process – 7.2.2 Level 1: model calibration with material properties – 7.2.3 Level 2: validation and calibration with systematically arranged element–level benchmark tests – 7.2.4 Level 3: validation and calibration at structural level) 7.3 Selection of global safety factor 239 7.4 Other issues in the use and validation of NLFEA programs 241 (7.4.1 Problem definition and model selection – 7.4.2 Working within the domain of the program’s capability) 7.5 Case 1: Design of a shear wall with openings 244 (7.5.1 Objective – 7.5.2 Level 1 calibration – 7.5.3 Level 2 and 3 validation – 7.5.4 Evaluation of global safety) 7.6 Case study II: design of simply supported deep beam 250 (7.6.1 Objective – 7.6.2 Calibration and validation of NLFEAP!1 – 7.6.3 Calibration and validation of NLFEAP!2 – 7.6.4 Analysis of deep beam) fib Bulletin 45: Practitioners’ guide to finite element modelling of reinforced concrete structures v . 7.7 Summary and future trends in model validation 260 7.8 Future trends in model validation 261 7.9 References 263 8 Strut!and!tie modelling 265 8.1 Introduction 265 8.2 Notation 266 8.3 Overview of the STM 267 (8.3.1 Strut!and!tie models – 8.3.2 Components of strut!and!tie models – 8.3.3 Admissible strut!and!tie models) 8.4 STM design steps (8.4.1 Complications in STM design) 270 8.5 Some considerations in using the STM 271 (8.5.1 Rules in defining D!regions – 8.5.2 Two! and three!dimensional D!regions – 8.5.3 Capacity of struts – 8.5.4 Uniqueness of strut!and!tie models – 8.5.5 Strain incompatibility of struts and ties – 8.5.6 Tension stiffening in ties – 8.5.7 Influence of tie anchorages – 8.5.8 Size, geometry, and strength of nodal zones – 8.5.9 Load redistribution and ductility requirements) 8.6 Computer!based STM 279 8.7 Modelling aspects using computer!based STM 280 (8.7.1 Identifying strut!and!tie models – 8.7.2 Refining strut!and!tie models – 8.7.3 Other considerations – 8.7.4 Static indeterminacy of strut!and!tie models – 8.7.5 Procedures to solve statically indeterminate strut!and!tie models – 8.7.6 Dimensioning nodal regions) 8.8 Design example using computer!based tools 298 (8.8.1 Problem statement – 8.8.2 Solution) 8.9 References 303 9 Special purpose design methods for surface structures 307 9.1 Introduction 307 9.2 Notation 307 9.3 Design of slabs and shear walls: perfect plastic approach 309 (9.3.1 Slabs subjected to bending loads – 9.3.2 Ultimate load determination – 9.3.3 Failure mode determination – 9.3.4 Material optimization – 9.3.5 Plates subjected to in!plane loads) 9.4 Design of slabs using the reinforcement field approach 318 (9.4.1 Linear yield conditions for element nodal forces – 9.4.2 Material optimisation through stress redistribution – 9.4.3 Slab subjected to bending loads – 9.4.4 Dimensioning procedure) 9.5 Design of shear!walls: the stringer!panel approach 321 (9.5.1 Linear!elastic version – 9.5.2 Non!linear version – 9.5.3 A three!step design procedure – 9.5.4 Example) 9.6 References 329 10 Concluding remarks 331 10.1 Introduction 331 10.2 Structural performance based design in practice 331 10.3 Benefits of non!linear modelling and analyses 333 10.4 Code provisions 335 10.5 Specification of design loads 335 10.6 Maintenance 336 10.7 References 337 vi fib Bulletin 45: Practitioners’ guide to finite element modelling of reinforced concrete structures . 1 Introduction 1.1 Preamble Computer-based analysis procedures for reinforced concrete structures, and finite element analysis procedures in particular, have seen tremendous advancement in the last half-century. Much research activity has occurred in the manner of constitutive modelling of reinforced concrete behaviour and in the development of sophisticated analysis algorithms. These advancements are well documented in various state-of-the-art reports, and still remain the subject of intensive research. Occurring at the same time, and no less significant, has been the accelerated development of computing technology and hardware. Data compiled by Bentz (2006), shown in Figure 1.1, provide a clear measure of the exponential growth in computing power in recent years. Shown is the time required to conduct a nonlinear shear analysis of a prestressed T-beam using a layered beam element algorithm. It is seen from the graph that, in 25 years, computing speed has increased by five orders of magnitude. Analyses that required several days of CPU time on supercomputers two decades ago run in minutes on personal desktop computers today. The advent of powerful and relatively inexpensive computers has greatly expanded the size and complexity of problems that can be analysed, and has greatly reduced the computer time required for their solution. Computer Performance Pentium 4 100 15 sec Pentium III Pentium Pro 10 3 min 5 Pentium 9 t Cin 1 30 min 80486 E P 80386 S d 0.1 5 hours e t 80286 a m ti 0.01 2 days 8088 s E 0.001 20 days 8080 CFT MCFT Kobe DSFM published published Earthquake published 7 months 0.0001 1970 1975 1980 1985 1990 1995 2000 2005 Year of CPU introduction Figure 1.1: Increase in computing power in recent years (Bentz, 2006). The state-of-the-art in nonlinear finite element analysis (NLFEA) of reinforced concrete has thus progressed to the point where such procedures are close to being practical, every-day tools for design office engineers. No longer solely within the domain of researchers, they are finding use in various applications; many relating to our aging infrastructure. NLFEA procedures can be used to provide reliable assessments of the strength and integrity of damaged or deteriorated structures, or of structures built to previous codes, standards or ! fib Bulletin 45: Practitioners’ guide to finite element modelling of reinforced concrete structures 1! . practices deemed to be deficient today. They can serve as valuable tools in assessing the expected behaviour from retrofitted structures, or in investigating and rationally selecting amongst various repair alternatives. In situations that have not turned out well, NLFEA procedures are finding applications to forensic analyses and litigations that follow. In the near future, they will likely form the main engine in computer-based automated design software, although in a form likely invisible to the user. 1.2 Notation A maximum aggregate size gm A cross section area of rebar s D rebar diameter !# compressive strength of concrete cylinder at 28 days " !# tensile strength of concrete $ f yield strength of reinforcement y M sectional moment capacity of beam (hand calculated) u P ultimate load capacity of beam (finite element analysis) u V sectional shear capacity of beam (Simple Method of CSA A23.3) u1 V sectional shear capacity of beam (General Method of CSA A23.3) u2 ! midspan deflection at ultimate load (finite element analysis) u " concrete strain at peak compressive stress o # shear strain $ reinforcement ratio % shear stress 1.3 Sample applications The failure of two reinforced concrete structures is recounted below; one involving a warehouse structure and the other an offshore platform base-structure. The structures were subsequently analysed using nonlinear finite element analysis procedures, taking into account relevant second-order behaviour models. The analyses were found to provide an accurate assessment of the load capacities and failure modes observed; in addition, they provided meaningful insights into the underlying behaviour mechanisms and factors leading to the failures. Hence, these two sample applications serve to show that nonlinear analysis techniques can be useful everyday tools for design office applications, particularly in forensic work. As well, they provide evidence that errors made in the design of modern structures can be potentially more catastrophic, and that advanced assessment techniques will assume increased importance as a result. Also discussed below are two additional examples in which nonlinear finite element analysis procedures were used to aid in the design of structures. The first involves a prestressed concrete frame; the second, the base slab in an underground liquid natural gas (LNG) storage tank. Again, both examples serve to illustrate the usefulness of advanced analysis procedures in solving difficult design problems. ! 2 1 Introduction! . 1.3.1 Kimberley-Clark warehouse Details of warehouse failure The Kimberley-Clark building was built in 1944, in Niagara Falls, Canada, in accordance with then-current building codes. The building was a simple four-storey structure with basement, having plan dimensions of approximately 38 x 36 m (see Figure 1.2). The structural system employed was primarily a reinforced concrete flat slab system with six bays in each direction. The centre-to-centre column spacing was approximately 6.25 m in the N-S direction, and 5.85 m in the E-W direction. Exterior columns were rectangular with haunches, while interior columns were circular with capitals. Column and capital diameters decreased with increasing elevation; the columns supporting the third floor were 450 mm in diameter with 1.5 m diameter capitals. The floor-to-floor height ranged from 3.35 m to 3.65 m. Exterior walls were constructed of brick masonry, and stair/elevator shafts were located at various points around the perimeter of the structure. Figure 1.2: Floor plan of Kimberley-Clark warehouse. The floor slabs were typically 200 mm thick. At the third floor level, the slab was thickened by 150 mm around the perimeter, over a width of 1.3 m. The floor slabs were reinforced with No.4 (13 mm dia.) and No. 5 (16 mm dia.) deformed bars. The reinforcement details were consistent with a column-strip/middle-strip design method, as shown in Figure 1.3. Similar reinforcement patterns and amounts were used in both directions. ! fib Bulletin 45: Practitioners’ guide to finite element modelling of reinforced concrete structures 3!