fib_Bull45_NMG.pdf

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 Technical Report State-of-Art Report Manual, Guide (to good practice) or Recommendation Model Code

minimum approval procedure required prior to publication approved by a Task Group and the Chairpersons of the Commission approved by a Commission approved by the Technical Council of fib 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 5, 6, 10 Consulting, Czech Republic), Vladimir Cervenka (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. 7, 8 7 Wien, Austria), Daniel Kuchma (Univ. of Illinois, USA), Ho Jung Lee (SC Solutions, Inc., USA), 3 Giuseppe Mancini (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

Copyright fib, all rights reserved. This PDF copy of fib Bulletin 45 is intended for use and/or distribution only within National Member Groups of fib.

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 halfcentury 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 Preamble 1.2 Notation 1.3 Sample applications (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) The question of accuracy (1.4.1 – Reasons for caution)

2

1.4 1.5 Challenges remaining 1.6 Objectives 1.7 Scope of report 1.8 References Design using linear stress analysis 2.1 Introduction 2.2 Membrane structures

1 1 2 2 20 27 29 30 30 33 33 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, fcd. – 2.2.8 Example – Design of a reinforced concrete 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)

3

2.5 References Essential nonlinear modelling concepts 3.1 Introduction 3.2 Nonlinear concrete behaviour

78 83 83 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 3.6 3.7 3.8 3.9 4

iv

Precision of nonlinear concrete FE analyses Safety and reliability Statistical analyses Concluding remarks References

Analysis and design of frame structures using non!linear models 4.1 Introduction 4.2 Notation

104 105 114 115 115 121 121 122



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 5

References

Analysis and design of surface and solid structures using non!linear models 5.1 Introduction 5.2 Notation 5.3 2D Structures with in!plane loading 5.4 Plate and shell structures (5.4.1 Layered elements) 5.5 Three dimensional solid structures

160 165 165 165 166 170 173

(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)

6

5.6 References Advanced modelling and analysis concepts 6.1 Introduction 6.2 Constitutive frameworks

190 195 195 195

(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)

7

6.5 References Benchmark tests and validation procedures 7.1 Introduction 7.2 Calibration and validation of NLFEA models

227 233 233 234

(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 7.4

Selection of global safety factor Other issues in the use and validation of NLFEA programs

239 241

(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 7.8 7.9 8

Summary and future trends in model validation Future trends in model validation References

Strut!and!tie modelling 8.1 Introduction 8.2 Notation 8.3 Overview of the STM 8.4 8.5

(8.3.1 Strut!and!tie models – 8.3.2 Components of strut!and!tie models – 8.3.3 Admissible strut!and!tie models) STM design steps (8.4.1 Complications in STM design)

Some considerations in using the STM

260 261 263 265 265 266 267 270 271

(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 8.7

Computer!based STM Modelling aspects using computer!based STM

279 280

(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)

9

8.9 References Special purpose design methods for surface structures 9.1 Introduction 9.2 Notation 9.3 Design of slabs and shear walls: perfect plastic approach

303 307 307 307 309

(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 10 Concluding remarks 10.1 Introduction 10.2 Structural performance based design in practice 10.3 Benefits of non!linear modelling and analyses 10.4 Code provisions 10.5 Specification of design loads 10.6 Maintenance 10.7 References

vi

329 331 331 331 333 335 335 336 337



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 100

Pentium 4

15 sec

Pentium III

Estimated SPECint95

10

Pentium Pro

3 min

Pentium

1

30 min

0.1

5 hours

80486 80386 80286

0.01

2 days

0.001

20 days

8088

8080

0.0001

7 months

1970

1975

CFT published

1980

MCFT published

1985

1990

Kobe Earthquake

1995

DSFM published

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

Agm

maximum aggregate size

As D !"# !$#

cross section area of rebar rebar diameter compressive strength of concrete cylinder at 28 days tensile strength of concrete

fy

yield strength of reinforcement

Mu

sectional moment capacity of beam (hand calculated)

Pu

ultimate load capacity of beam (finite element analysis)

Vu1

sectional shear capacity of beam (Simple Method of CSA A23.3)

Vu2

sectional shear capacity of beam (General Method of CSA A23.3)

!u

midspan deflection at ultimate load (finite element analysis)

"o # $ %

concrete strain at peak compressive stress 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.


! 3!


Figure 1.33: Reinforcem ment details for third storey floor f slab.

The enggineering plans p for thhe structuree specified service livve loads off 12.0 kPa for the basemennt and firstt floor, 6.0 kPa for thhe remainin ng floors, and a 1.9 kPaa for the ro oof. The specifieed concrete strength waas 20 MPa, and the allowable tenssile stress inn the reinfo orcement was 1388 MPa. Thee actual com mpressive sttrength of th he concretee, determineed from corres taken in 19788, had an average a vallue of 37.22 MPa. Fro om couponss taken at tthe same tiime, the reinforccing steel was w found to have a yield strength of 442 MPa and tensile streength of 710 MP Pa, with a modulus m of elasticity e of 210,000 MPa. The thirrd floor of the warehoouse becam me the site for the storrage of druums of nick kel pellet beginninng in mid-D December 1977. The drums, on wooden paallets, weree transporteed to the third flooor by a freiight elevatoor, and then moved to the t storage location l by forklift trucck. Each drum weighed w 2255 kg, and there were eiight drums to t a pallet. The palletss measured 915 mm square. In general, the pallets were stackked side by side, two high, h giving a live floorr load of about 43.1 kPa. Cooincidentallly, an alternnate-bay loaading patterrn evolved as one bay was left free to facilitate foork-truck acccess. Anotther perhap ps relevant detail is that the floorrs below

4

! 1 Introduction !


housed a paper-prroducts mannufacturingg plant. Wiith such opperations, thhe dangers of dust explosioons and firee are an everr-present cooncern. On Januuary 4, 19788, a large seection of thhe third floo or collapsed; see Figuree 1.4. The collapsed c area enccompassed approximattely 16 bayss of the stru ucture, withh columns, sslabs, and drums d of nickel falling f throuugh to the basement b leevel. At abo out the sam me time of thhe collapsee, a large explosioon occurredd and fire engulfed thee structure. The fire coontinued forr 48 hours before b it could bee controlledd, and two men m died in the inciden nt. Indicatedd in Figure 11.4 is the nu umber of pallets of o nickel peellet, per bayy, determineed to be preesent at the time t of colllapse.

F Figure 1.4: Looading and co ollapse patternn.

The conntention off the paper manufacturring firm was w that thee stored nicckel overloaaded the floor, resulting r inn its collappse and coonsequently y sparking an explossion and fire. fi The propriettors of the nickel opeeration arguued that an explosion and fire inn the plant was the primaryy cause of collapse; c thaat is, the inttense heat from f the firre instantly weakened the t floor above and a led to itss failure. Analysees of warehouse structture To obtaain an estim mate of the load-bearing capacity y of the slaab at the thhird floor leevel, the structurre was anallysed usingg nonlinear finite elem ment analyssis proceduures. In add dition to materiall and geoometric nonnlinearities, the analy yses considdered the membrane action, temperaature degraadation of material m strrengths, tim me-related effects andd other infl fluencing factors. The analysis approacch was to model m portions of the third floor slab, and columns above and a below, using u plane frame stripps. The mosst critically loaded porttion of the structure s was assumed to be along Coluumn Line 4.. In the mod delling, fourr bays extennding from Column Lines A through E, E one bay wide, w were considered. c To take intto account tthe influencce of the


! 5!


column capitals and reduced clear spans, beam elements representing the drop panels were used. The four stairwells, located roughly at the corners of the warehouse floor plan (Figure 1.2), likely influenced the behaviour of the floor slab as will be discussed. To estimate their lateral stiffness, the stairwells were considered to be cantilevered from the basement floor. All constitutive modelling was based on the models of the Modified Compression Field Theory (Vecchio and Collins, 1986). Vecchio and Collins (1990) provide complete details of the modelling. The nonlinear analyses indicated that the ultimate capacity of the floor in the vicinity of the collapse was approximately 45.5 kPa (superimposed load), as shown in Figure 1.5a. Including the dead load, the total capacity was 50.3 kPa. Note that the estimated floor capacity is only slightly higher than the estimated floor loading at the time of collapse, but is 7.6 times the value of the specified floor capacity. The most significant difference between linear and nonlinear analyses, in this situation, lies in the fact that the nonlinear analyses accounted for the net elongation that typically occurs in reinforced concrete elements in flexure (see Figure 1.5b). Tensile strains on the cracked face of an element are normally much larger than the compressive strains on the opposite face, particularly if the reinforcement is yielding. Hence, the tendency is for a slab to elongate as it flexes under the applied transverse loads. When the slab is prevented from elongating, as it is in this case by the columns, stairwells, elevator shafts, and floors above and below, a compressive thrust is induced in the slab. For reinforced concrete elements, axial compressive forces generally serve to increase flexural and shear capacity. This behaviour, known for many years, is commonly referred to as compression membrane action. In the case of the Kimberley-Clark Warehouse, a linear elastic analysis of the third floor subsystem predicts very little gain in the axial compressive force as the floor load is increased. The flexural capacity at the midspan of the floor is exceeded at approximately 21.5 kPa live load. Allowing for partial moment redistribution, as permitted by ACI 318 (1983), the estimated ultimate floor capacity increases to 25.4 kPa. If complete moment redistribution is considered, the calculated floor capacity increases to approximately 31.1 kPa. This remains well below the observed capacity at failure. The nonlinear analyses, on the other hand, show a much more pronounced increase in axial force with increasing floor load; see Figure 1.5c. The proper consideration of membrane action, only possible through nonlinear analyses, was instrumental in achieving an accurate estimate of the load capacity of this structure. Further analyses were conducted considering the effects of elevated temperatures if a fire were present below the heavily loaded third floor slab. A temperature of 1000!C was assumed to be acting instantaneously on the bottom surface of the slab, and standard uni-directional heat flow analyses were used to compute the progression of the thermal gradient through the slab. Reductions in materials strength and stiffness, arising from the elevated temperatures, were considered for both concrete and steel. The analyses indicated that collapse would have occurred within 2 to 3 minutes from the onset of the fire. Hence, it was concluded that the collapse could as likely have been triggered by a fire below weakening the floor as by floor overload.

6

! 1 Introduction !


Figurre 1.5: Resultss of analyses for column linne 3/4.


! 7!


1.3.2

Sleipner A offshore platform

Details of platform failure The Condeep offshore platforms, of which Sleipner A was one, serve in the hostile environment of the North Sea in water depths of up to 300 m. The construction of a typical Condeep platform starts in a large dry-dock area, and is moved to a sheltered fjord as the construction progresses. In later stages, the base structure is floated out to sea, then partially submerged and mated with the deck structure. After deck-mating, the completed structure is towed to its offshore site and lowered to a final position on the sea floor. A critical factor in the design of the base structure is the thickness of the cell walls. If the walls are too thin, they may fail under the very high water pressures to which they are subjected during deck-mating. If the walls are too thick, the structure will not float, or will not be hydrostatically stable during the towing operations. The margin between the two is narrow. The gravity base structure of the Sleipner A offshore platform, which was 110 m high, consisted of a cluster of 24 buoyancy cells. Four of these cells extended upward to form shafts connecting to the deck structure. Details are given in Figure 1.6. The exterior walls of the cells were circular, with radii of 12 m, whereas the interior walls separating the cells were straight. The intersection of the interior walls formed a small triangular void called a tricell. There were a total of 32 such tricells. Because the tricells were open at the top, they filled with water once the tops of the cells were submerged. Hence, the walls of the tricells had to resist a substantial hydrostatic pressure. On August 23, 1991, the gravity base structure for Sleipner A was slowly being submerged as part of the deck-mating operation. The intention was to lower the structure until the base was 104 m below the ocean surface. However, when a depth of 99 m was reached, a loud rumbling noise emanated from one of the drill shafts and water could be heard entering. Within minutes, the structure began to sink in an uncontrollable manner. Moments after disappearing below the surface, a series of implosions were felt as the buoyancy cells collapsed. Evidence showed that the loss of the structure was attributed to the shear failure of the wall of Tricell 23 adjacent to drill shaft D3. At failure, this 550 mm thick wall was resisting a 65 m head of seawater, resulting in a pressure of about 655 kPa. The reinforcement details in the tricell walls near the failure location, described in detail by Collins et al. (1997), are summarized in Figure 1.7. In general, a grid of horizontal and vertical bars was provided near each face of each wall of the tricell. For approximately the bottom third of the height of the cell walls, 12 mm diameter stirrups were provided at a 170 x 170 mm spacing. In the middle-third height of the walls, the stirrup spacing was increased to 170 x 340 mm spacing. Just below the location of the failure, the stirrups were terminated and the walls contained no shear reinforcement thereon up. Another detail of note is the provision of a T-headed bar placed across the throat of the tricell joint. This 25 mm diameter bar was about 1.0 m long and had steel plates welded on its ends to provide anchorage. The failure of the Sleipner base structure, which involved a total economic loss of about $700 million USD, was probably the most expensive shear failure on record.

8

! 1 Introduction !


Figure 1.6: Details off Sleipner A co oncrete base structure.


! 9!


Figgure 1.7: Reinfforcement dettails for tricelll 23.

10

! 1 Introduction !


Analysees of platforrm base stru ucture To deveelop a betteer understannding of thee factors in nfluencing thhe failure oof the tricell wall, a series of o three-dim mensional nonlinear n fiinite element analysess were unddertaken. Th he finite elementt mesh usedd for the innvestigation is shown in i Figure 1.8. Because of symmettry, only one-sixtth of the trricell needed to be moodelled. Thee one elem ment thick m mesh contaiined 342 brick ellements (244 dof) and 338 3 wedge elements e (1 18 dof). All constitutivve modelling g was in accordaance with thhe Modifiedd Compressiion Field Th heory. In the analyses, the horizon ntal axial force inn the tricell wall was heeld constantt at 5000 kN N/m while the t hydrosttatic pressurre on the inner faace of the wall w was inccreased until failure. In n addition, a constant vvertical com mpressive stress of 7 MPa waas assumed acting in thhe walls. Full details of o the finitee element modelling m are disccussed by Coollins et al. (1997).

F Figure 1.8: Fiinite element model m of tricelll.

s wo ould sustainn a brittle, sshear-criticaal failure The anaalyses indiccated that thhe as-built structure when thhe applied water presssure on thhe inner facces of the tricell reacched 620 kP Pa. This correspoonded to a head of seaawater 62 m high, and agreed welll with the 665 m head acting a at the critiical locationn at the tim me of failuree (see Figuree 1.7b). Note that this analysis waas a first run andd involved the t input of details annd material properties as then knnown, and no n ‘finetuning’ of analysiis parameteers was invvolved. Hence, the abbility of N NLFEA metthods to properlyy represent behaviour in this situaation was excellent. It is also inteeresting to note n that ACI 3118-95 proceedures leadd to a prediicted failurre pressure of 1200 kkPa while th he 1994 AASHT TO LRFD Specification S ns gives 4500 kPa; both significantlly in error. The dessigners of thhe structuree were interrested in leaarning how the strengthh of the triccell wall would have h changeed if the stirrrups, whichh had been terminated just below the failure location, l had beeen continuedd up the wall. They weere also keen n to know how h the lenggth of the T-headed T bar in thhe throat off the joint reegion influenced the faiilure load. To T answer tthese questions, two series of nonlinear finite elem ment analysees were cond ducted. In one o set the w walls were assumed a to contaain 12 mm diameter d stiirrups at 1700 x 340 mm m spacing, while w for thee other set the t walls were asssumed to contain c no stirrups. Within W each set, the lenngth of thee T-headed bar was varied from f zero (i.e., no bar)) to the maaximum posssible lengthh (1.5 m). T The results of these studies, summarizeed in Figuree 1.9, indicaate that the tricells couuld have resisted an ad dditional


! 11 !


20 m off water heaad if either the stirrupss had been continued up u the walll or if the T-headed T bars haad been aboout 500 mm m longer. The structu ure would have survivved in eith her case. Changes were madde accordinngly to the design of the replaceement struccture. On April A 29, 1993, thhe new conncrete gravvity base strructure wass successfuully mated w with the deeck, and Sleipnerr A2 was reeady to be toowed to seaa. It remainss in service today. t

Figgure 1.9: Influuence of stirruups and lengthh of T-headed bar on prediccted failure prressure for triccell.

1.3.3

F Frame corn ner

Problem m descriptioon Frame corners c are a typical deesign probleem where reinforcemennt detailingg plays an im mportant role. Thhe region of o a frame corner is subjected to t a compleex stress sttate and caannot be analysedd by beam theory. Connsequently, it is design ned by emppirical rules which typiically do not cover specific situations. The T followiing analysiss was perfoormed withiin the project of the newly built b Brixeen Universiity (structuural designeed by Berggmeister, B Brixen). Th he frame structurre is a part of the lectture auditorrium buildiing and is visually exxposed on the t back elevatioon with a laarge glass facade f creaating a signiificant aesthhetic impreession. Therrefore, a stringennt requiremeent on the crack c width limit was im mposed. Thhe girder is ppre-stressed d and the cable annchors signnificantly innfluence thee stress statee in the fraame corner. The reinfo orcement detailingg was an im mportant isssue in this project. p Thee model is reduced to a section in ncluding the sym mmetrical haalf of the giirder with the t floor slaab and partss of columnns; see Figu ures 1.10 and 1.111. Numerical analysiis based onn the work k of Cervennka (1999) was used for the reinforccement detaailing optimiisation. Numeriical model The moodel developpment was based on thhe analysis of the wholle 5-storey frame, Figu ure 1.10. At firstt the question of model complexxity was co onsidered: How H far doo we have to t go to capture the structurral behaviouur realisticaally? In this case two models m weree considered d; a two12

! 1 Introduction !


dimensional model, based on a plane stress state, and a three dimensional model with a fully 3D stress state. Obviously, the choice of model has a strong impact on calculation time and objectivity of results. While a 2D model is simpler and less time demanding, it cannot appropriately capture the confinement effect in the anchoring region of the prestressing cables. On the other hand, the 3D model is more realistic, but it requires more time for data preparation and calculation. In this case it was decided to use the 2D model for global information on the structural behaviour and for parametric studies relating to structural optimisation. The 3D model was used for a detailed analysis of selected cases. The numerical model reflected in a realistic way the geometry and reinforcement. It included three types of reinforcing models: smeared reinforcement (stirrups and slab), bars (main girder and column reinforcement), prestressing cables, see Figure 1.11.

Figure 1.10: Frame structure (left) and frame corner (right) dimensions (in m) and loading.

Figure 1.11: Reinforcing by ordinary bars (left) and pre-stressing cables (right).

Constitutive models used for concrete were of two types: For plane stress analysis the damage-based model (Cervenka, 2002), and for 3D analysis the fracture-plastic model (Cervenka and Cervenka, 1999) were used. Both models employ the same approach for tensile cracks, which is based on smeared cracks, crack band and fracture energy. Hordijk’s exponential function is applied for the crack opening law. Compressive response in plane stress is based on orthotropic damage with the failure function according to Kupfer. In 3D analysis, the plastic flow theory with a Menetre-Willam surface is utilized. The post-peak softening is employed in both tension and compression responses.


! 13 !


Serviceability analysis The prime objective of the analysis was to determine the crack width under service load in the frame corner. It was difficult to calculate these cracks with usual beam theory and therefore a FE-based simulation was employed. Crack patterns resulting from the analysis are shown in Figure 1.12 for both models. In the 2D model the maximum crack width in the frame corner was 0.28, mm and in the 3D analysis it was 0.15 mm. Both were calculated assuming a crack spacing of 100 mm. Both crack patterns are very similar in locations and directions. It is interesting that the 3D model gives smaller crack width. This is explained by the capability of the model to capture the variable crack width in the third direction and thus producing smaller crack width near the surface, where the reinforcement is located. The distribution of the principal compressive stress in the corner in 2D analysis is shown in Figure 1.13.

2D model

3D model

Figure 1.12: Crack patterns under service load.

Figure1.13: Response of frame corner.

14

! 1 Introduction !


The crack spacing must be also examined. The crack model is based on a fracture mechanics approach, and the crack width within this frame is independent of the element size. The crack spacing develops naturally as the strains localize. This works well for single discrete cracks. However, in reinforced concrete with several parallel cracks of similar thickness, one main crack need not develop due to the restraining effect of reinforcement. Then the crack width is a product of crack strain and element size, while the latter represents the assumed crack spacing. In the above case this spacing was 100 mm and represented a conservative estimate of a real spacing. Thus, another crack width could be obtained by a simple recalculation for a different spacing. It should be realized that the above crack model includes the effects of strain softening in cracks and also the so-called “tension stiffening” due to the contribution of uncracked and partially cracked concrete regions, while considering the real geometrical arrangement of the reinforcement. In the given case of design, the accuracy of the model cannot be easily examined and we have to rely on a separate validation through known experimental data, for which the tests of Hartl (1977) and Braams (1990) were used. Ultimate analysis After service load conditions were examined, the loading was further increased and the ultimate failure load and corresponding failure mode were evaluated. The full analysis was performed for two types of concrete post-peak behaviour in compression. The first type, described as “brittle”, utilizes linear softening according to van Mier (1986), which can be considered as a lower-bound behaviour. The second type was a perfect plasticity with horizontal plateau after peak, which clearly overestimates ductility. Response for both models and two types of compression softening can be seen from load-displacement diagrams in Figure 1.14.

Figure 1.14: Load-displacement diagrams.

The brittle model in 2D shows a local instability at the load of 3.2 times the service load. At this stage the compressive failure of the cracked concrete in the frame corner is initiated. Note that this region is under a strong local effect of the prestressing tendon anchor. The corner rotation due to brittle failure causes a temporary load decrease, then the load is again increased until the bending capacity in the midspan is exhausted. The ultimate load of 63.8


! 15 !


kN/m only slightly exceeded the first peak load of 58.6 kN/m before the corner fails. The redistribution of internal forces was also confirmed by changes in moment distribution in girder. Similar behaviour was observed in the case of the 3D model. However, in this case the local instability was less pronounced and the load could be significantly increased to the maximum 75 kN/m after the partial compressive failure of the frame corner at 46.6 kN/m. The higher ultimate load in the 3D model is explained by the better ability of the corner to resist moments when lateral confinement is included. In spite of the partial brittle failure of concrete in the frame corner, the structure showed relatively large ductility. The mid-span defection of 355 mm at failure in the 3D analysis was 14 times that at service load. The failure at the frame corner can be observed in Figure 1.13b, where the localized compressive strains are indicated near the tendon anchor. In order to identify the source of limited ductility, both models were also analysed by models with perfectly plastic behaviour in compression. In this case the response of the 2D and 3D models are almost identical and the ultimate load is increased to 97 kN/m with very ductile behaviour. It shows that if brittle failure of concrete in the frame corner is avoided, the bending capacity can be fully utilized. However, the brittle response is considered more realistic. Interesting is also information about the computer time required for analyses as shown in Table 1.1. Calculations were performed on a Pentium 4 (2.2 MHz). The loading was imposed in load steps with a Newton-Raphson iteration scheme up to service load. Then the solution method was changed to arc-length and loading continued to failure. The arc-length method was capable of capturing the local instability. The convergence criterion was 1% error in equilibrium, and was satisfied in all steps until the global failure.

Table 1.1: Frame corner analysis computer time.

Model 2D 3D

Number of finite elements 830 11777

Load steps 20 38

time [minutes] 15 184

Conclusions The nonlinear analysis has proven as an efficient tool for optimising reinforcement detailing. The three-dimensional analysis was more helpful for assessment of crack widths in the frame corner and of the failure mode. It better described the effects of confinement on concrete behaviour. The two-dimensional analysis could be considered as a safe estimate and, in the view of considerably lower computer time requirement, it was used for parameter study. The 2D and 3D analyses showed different results in case of brittle response. In case of ductile response the results were almost identical.

16

! 1 Introduction !


1.3.4

Base slabs in LNG storage tank

Problem description The nominal punching shear strength of RC circular slabs (3D) is larger than that of beam shear (2D) because hoop reinforcement creates confining radial compression under flexure and the shear span-to-depth ratio becomes substantially smaller compared with 2D beams and columns. As the nominal punching shear strength is size-dependent, this factor leads to higher cost performance with reduced hoop and transverse reinforcement when RC circular slabs of large scale are designed in shear. When a circular slab is supported with some deformational constraint, the hoop effect is thought to be reduced and the shear capacity is probably diminished. On the contrary, the effective shear span to depth ratio will be smaller and the capacity will be increased. Since the design formulae are based upon the experiments of simply supported slabs, they cannot be directly applied to more generic cases without any special discussion. For designing the world largest underground LNG tank, a new structural type was adopted, i.e., the base slab and sidewall shells are monolithically combined; see Figure 1.15. Here, nonlinear finite element analysis was applied for estimating the punching shear capacity of the wall confined slabs, because FEM can be freely applied to any boundary condition; see Figure 1.16.

Figure 1.15: Circular slab in punching shear and axi-symmetric 3D analysis (Akiyama et al., 1996; Maekawa et al., 2003)


! 17 !


Figure 1.16: Conventional punching shear design formulas and FEM simulation (Kawabara, 2002).

Limit state analysis First, a 3D nonlinear analysis was made of a large-scale conventional circular slab in shear and checked by the available shear strength formulas. As the hydrostatic uniform force is applied to the slab, a volume-control method or arc-length method are effective in computing the softened failure states. After checking the validity of nonlinear analysis, a parametric study was performed with different peripheral boundary conditions; see Figure 1.17. It was found that the nominal punching shear strength does not drop but increases due to more confined boundary conditions and hoop reinforcement effects expected in these cases. The most current LNG storage tank under construction has been designed with the hoop effect and successful cost savings were achieved. In this practical case, computer based modelling was utilized to extend the applicability of design formula to more general boundary conditions.

Figure 1.17: Shear capacity of different boundary conditions (Kawabara, 2002).

18

! 1 Introduction !


Detailing Another application of axisymmetric 3D nonlinear analysis is addressed to slab-to-wall joints in combined flexure and shear. Linear finite element analysis was used for deciding reinforcement ratio and orientation in the slab-wall connection. Its size was so large that no way of arranging reinforcing bars was found, because available bar diameters are inescapably much smaller than the size of the structure. Furthermore, the back-check by nonlinear finite element analysis revealed that the connection may fail in shear before the failure of the slab and the sidewall due to size effect of shear failure. In general, elastic analysis based arrangement of reinforcement is safe and possible for construction when the size of structures is small. But it is a different situation with large-scale structural design. Thus, several other design details were considered and the nonlinear finite element analysis was conducted for verification of its safety performance under static design loads (underground water and soil pressure); see Figure 1.18. Practitioners in both design office and construction site could have the modified dimensioning and steel arrangement which satisfies the design requirement of safety and waterproofing serviceability.

Figure 1.18:- RC slab-sidewall connection: linear analysis based arrangement of rebars (left) and nonlinear analysis aided design (right) (Nakano, 2002).

Discussion In the four case studies presented, the use of advanced analytical techniques served to greatly aid in the understanding of the factors and mechanisms influencing behaviour. With the warehouse structure, the puzzle from an analysis perspective was not what event finally triggered collapse but rather how could the floor sustain such high overloads in the first place. In the case of the offshore structure, the challenge was to rigorously and accurately capture


! 19 !


the influence of local reinforcement details in an analysis of predicted response. In both cases, as was shown, conventional analysis/design calculations were largely inadequate in providing an accurate assessment of load capacity. Advanced nonlinear analysis procedures, on the other hand, provided much improved modelling capability. The four structures examined, while in themselves insufficient to form the basis for a definitive statement, also point to another finding. That is, as design procedures become more refined, and as economic and technologic pressures increase, the margin for error diminishes. In the bulky concrete structures constructed a half-century ago, there was often much redundancy in the structural system and a large margin of safety inherent in the design and construction procedures. Hence, a structure such as the Kimberley-Clark warehouse was able to withstand service floor loads almost an order of magnitude higher than design-specified. On the other hand, with modern precision-engineered structures of today, based on more exact methods of analysis and design, the margin for error is small and the penalty for miscalculation is high. In such an environment, the value of sophisticated analysis procedures is great provided they have been adequately verified and are properly applied.

1.4

The question of accuracy

Despite the increasing sophistication of NLFEA tools, users must be ever mindful of the question of how accurately and reliably can they represent the behaviour of reinforced concrete structures. In this regard, it is useful to examine the results of three ‘prediction competitions’. At the 1981 IABSE Symposium in Delft, a ‘blind’ competition was organized involving four panels tested in a comprehensive research program then underway at the University of Toronto. The test panels were orthogonally reinforced, and subjected to uniform, proportional and monotonically increasing stress conditions; seemingly a very simple problem to model and analyse. The results of these panel tests were not disclosed prior to analysts submitting their predictions of strength and load-deformation response. Approximately thirty entries were received, many from the leading researchers in the field at the time. Shown in Figure 1.19 is the range of responses for Panel C, one of the better predicted of the four panels. The analysis results submitted showed a wide variation in predictions of the panel’s shear strength, and an even wider divergence in computed load-deformation responses (Collins, Vecchio and Mehlhorn, 1985). Clearly the collective ability to model nonlinear behaviour of reinforced concrete, particularly in shear-critical conditions, was not well advanced. More recently, in 1995, the Nuclear Power Engineering Corporation of Japan (NUPEC) staged a prediction competition involving a large-scale 3D shear wall subjected to dynamic cyclic loading (NUPEC, 1996). The flanged shear wall exhibited highly nonlinear behaviour before sustaining a sliding shear failure along the base of the web. One facet of the competition called for estimates of the ultimate strength, and corresponding displacement, of the wall as determined from static push-over analyses. Again, over thirty sets of predictions were received; the results are summarized in Figure 1.20. The predictions of strength, as a group, showed better correlation than was seen with the Toronto panels; however, the deformation estimates still showed large scatter. Nevertheless, it could be concluded that the ability of NLFEA to accurately capture the behaviour of reinforced concrete had measurably advanced. It should be noted, however, that this was not a completely blind competition since some of the test results had been disclosed to analysts prior to the competition.

20

! 1 Introduction !


Most recently, ASCE-ACI Committee 447 organized an informal competition centred on results from a series of large-scale columns tested at the University of California at San Diego. Many of the analyses undertaken pursuantly are documented in papers contained within an ACI Special Publication (William and Tanake, 2001). From these, it can be noted that: i) a number of significantly different analysis approaches were taken; ii) predictions of strength and pre-peak response generally correlated well with the experimental results; and iii) predictions of post-peak response were generally not as accurate and still require further attention. Bear in mind, once again, that this was not a blind competition. Analysts had the opportunity to calibrate parameters, optimize material models, and refine analyses. One should also bear in mind that experimental results themselves are subject to scatter and error. Repeating a test, particularly if conducted at different laboratories, may yield differing results. Nevertheless, it is an inevitable conclusion that our ability to accurately model the behaviour of reinforced concrete structures has seen significant improvement over the past three decades. It has approached a stage of development where we may be inclined to proceed with a certain degree of confidence.

Figure 1.19: Delft Panel competition results (Panel C): a) Variations in predicted shear strength; b) Variations in predicted load-deformation response.


! 21 !


1.4.1

R Reasons foor caution

Despite these sign nificant adv vancements in our abillity to accu urately moddel the resp ponse of reinforcced concretee, the users of NLFEA A proceduress need to bee mindful of several isssues and potentiaal dangers:

Diversitty of theoreetical approaches A numb ber of ratheer diverse approaches a exist for NLFEA N mo odelling of rreinforced concrete structurres. Among those availlable are mo odels built on o nonlineaar elasticity, plasticity, fracture mechan nics, damagee continuum m mechanics, endochro onic theory, and other hhybrid formu ulations. Crackin ng can be modelled m disscretely, or using u smearred crack ap pproaches; the latter caan range from fu ully rotating g crack mo odels, to fix xed crack models, m to multiple m noon-orthogon nal crack models,, to hybrid d crack mo odels. Som me approach hes place heavy empphasis on classical mechan nics formulaations, others draw mo ore heavily on empiriccal data andd phenomen nological models.. It can geenerally be said of an ny approach h that it will w be morre suited to o certain structurre/loading situations s an nd less so to t others. No N one app proach perfo forms well over the entire raange of stru uctural detaiils and loadiing conditio ons encounttered in pracctice.

Figure 1.20:- NUPEC Wall predictiion results: a)) Variations in n predicted laateral load cappacity; b) Varriations in predictedd lateral displlacement at ulltimate load.

22

! 1 Introduction !


Diversity of behaviour models Reinforced concrete structures, particularly in their cracked states, are dominated in their behaviour by a number of second-order mechanisms and influencing factors. Depending on the particular details and conditions prevailing, a structure’s strength, deflection, ductility and failure mode may be significantly affected by mechanisms such as: compression softening due to transverse cracking, confinement effects, tension stiffening, tension softening, aggregate interlock and crack shear slip, rebar bond slip, rebar dowel action, rebar compression buckling, scale effects, and creep and shrinkage, to name a few. For each of these, a number of diverse formulations can exist. In the case of tension stiffening, for example, the stiffening effect can be ascribed to a post-cracking average tensile stress in the concrete or, quite differently, to the load-deformation response of the reinforcement. The user of a NLFEA software must be aware of what mechanisms are likely to be significant in the problem at hand, be certain that it is included in the analysis model, and have some confidence that the model being used is reasonably accurate. Incompatibility of models and approaches The formulation and calibration of a concrete behaviour model, as it is being developed, is often dependent on the particular analysis methodology being used. As a consequence, some models cannot be randomly transplanted from one analysis approach to another, or freely combined with other models. Often, they are developed in combination with other complementary material models, or analysis approaches, and should not be separated. As a case in point, consider the observed and predicted behaviour of Panel PV19 tested by Vecchio and Collins (1986); this was, in fact, Panel C from the 1981 Delft Competition, represented in Figure 1.19. An important mechanism influencing the shear strength and deformation response of this element was the softening of the concrete in compression due to transverse cracking, with the panel eventually sustaining a concrete shear failure. Shown in Figure 1.21 are the predicted responses obtained using the compression softening model of Vecchio and Collins (1986) implemented in a rotating crack formulation, and that of Okamura and Maekawa (1991) implemented in a fixed crack formulation (i.e., each correctly matched with the crack model for which it was first developed). Both provide equally good simulations of response. The Vecchio-Collins formulation slightly over-estimates strength and slightly under-estimates ductility. Conversely, the Maekawa formulation slightly under-estimates the strength and slightly over-estimates the ductility. Either one, however, is certainly well within the margins of accuracy we can hope to achieve with NLFEA. But consider the consequences if one implements the Vecchio-Collins model into a fixed crack formulation, or if one uses the Okamura-Maekawa model in a rotating crack formulation. In both cases, the results are much less satisfactory; strength, ductility and failure mode are subject to significant miscalculation. Experience required Use of NLFEA for modelling and analysis of reinforced concrete structures requires a certain amount of experience and expertise. Unlike, say, the use of plane section analysis techniques to calculate the flexural strength of a beam cross section, the application is rarely straightforward. Decisions made with respect to modelling of the structure and selection of material behaviour models will have significant impact on the results obtained. Again, unlike sectional analysis techniques, two analysts may obtain widely diverging results when modelling the same structure using the same analytical model and the same software. Decisions made regarding mesh layout, type of element used, representation of reinforcement details, support


! 23 !


Figure 1.21: Influence of compression softening model on computed load-deformation response of Panel PV19.

conditions, method of loading, convergence criteria, and selection of material behaviour model will produce a divergence of results. Here lies the explanation for the significant difference in accuracy of results obtained in ‘blind’ competitions as opposed to those obtained when the desired results are known in advance. Add to this the increased likelihood of errors in input due to the relative complexity of NLFEA. As an illustration, consider the two simple-supported beams shown in Figure 1.22, tested by Podgorniak-Stanik (1998); one with shear reinforcement and one without. Ten analysts, all using the same software program and all previously experienced in its use, where asked to independently provide predictions of the expected load capacity (Pu) and corresponding midspan deflection (!u) for these shear-critical beams. Also requested were: the theoretical section moment capacity (Mu) determined using hand calculations based on the common rectangular stress block approach; the sectional shear capacity (Vu1) determined using the Simple Method of the Canadian code specifications, which is essentially the standard 45degree truss model; and the sectional shear capacity (Vu2) determined using the General Method of the Canadian code specifications, ostensively a more accurate calculation involving the consideration of compatibility conditions, inclination of the stress field, and reduction in concrete strength due to transverse cracking. The results are summarized in Table 1.2. The calculations of moment capacity, using simple hand methods, were consistent amongst analysts for both beams, showing a coefficient of variation (COV) of less than 3.5%. Calculations of the shear capacity using code specifications produced larger variations. It is interesting to note that the calculation of Vu1 (Simple Method) wasn't any less scattered than

24

! 1 Introduction !


Figuree 1.22: Detailss of Podgornia ak-Stanik test beams. Table 1.22: Results of analyses of Pod dgorniak-Stannik beams.


! 25 !


that of Vu2 (General Method), due to some ambiguity present in the current code formulations making interpretation of the code provisions subjective in certain situations. Also note that the results were significantly more scattered for the beam without shear reinforcement (Beam No. 2), as one might expect. Finally, let us examine the results obtained from the finite element analyses. The estimates of load capacity (Pu) were relatively close to the codecalculated shear capacities, but showed significantly more scatter with COVs of about 17% for the beam containing shear reinforcement, and 28% for the beam containing no web steel. The estimates of deflection at ultimate load (!u) were widely divergent, in part because some analysts were predicting a brittle shear failure and others a ductile flexural failure. It is worth repeating that the same finite element program was used by the analysts, all experienced in its use. Differences arose primarily from the selection of material behaviour models available within the program library, and from the modelling of details such as reinforcement (smeared or discrete) and support/loading conditions. Incidentally, the experimentally observed peak load and corresponding deflection for Beam No. 1 were 672 kN and 20.7 mm; for Beam No. 2, 370 kN and 5.9 mm. Too much information NLFEA investigations invariably produce large quantities of data, typically spawning output files of several megabytes in size or larger for each load stage. Information may be provided on: stresses and strains at each integration point of each element, both with respect to local and principal axes, both for the element and for the concrete component; nodal displacements; sectional forces per unit width at each integration point; reactions; reinforcement stresses and strains; stiffness matrix coefficients, and more. Considering that typical problems can involve tens of thousands of degrees of freedom, the total amount of data quickly escalates to the point where the use of post-processors is virtually essential. Even then, the analyst must have an awareness of what to look for and how to interpret it. Despite the use of sophisticated postprocessors and graphics capabilities, there remains the possibility for misinterpretation. Incomplete knowledge We must accept that we still do not understand well, let alone have accurate models for, all aspects of reinforced concrete behaviour (see Section 1.6 Challenges Remaining). Applications of NLFEA should be done with a healthy degree of caution and scepticism. Wherever possible, analysis software and models should be validated or calibrated against benchmark tests involving specimens of similar construction and loading details, dependent on mechanisms anticipated to be significant in the analysis problem at hand (as far as this can be done). Wherever possible, results should be supported by analyses based on different models or approaches. Research philosophy Lastly, it may be said that the research community, and associated technical committees, may have failed to meet the needs of the profession in some respects. Many working in the area have directed their efforts to developing sophisticated models and methods of analysis, in many cases basing their work on esoteric models or rigorous application of classical mechanics approaches not directly suited to reinforced concrete. Consequently, advancements have been made in developing the NLFEA concepts and methodologies, but often with reinforced concrete being merely the application. Unfortunately, reinforced concrete is a complex and stubborn material that sometimes refuses to act according to accepted rules of

26

! 1 Introduction !


mechanics. Researchers might do well to refocus some efforts towards better understanding and modelling reinforced concrete behaviour, with finite element analyses being merely the tool. Certainly, however, there is room and need to advance on both fronts.

1.5

Challenges remaining

To allay the notion that we now fully understand the behaviour of reinforced concrete, and can accurately model it in all situations, consider two series of beams tested by Angelakos et.al. (2001). The first series involved five beams; all were 6000 mm in length, 1000 mm in depth, 300 mm wide, reinforced with approximately 1.0 percent longitudinal reinforcement, containing no transverse reinforcement, and subjected to a monotonically increasing load applied at the midspan (Beams DB120, DB130, DB140, DB165 and DB180; see Figure 1.23 for beam details). The only variable was the compressive strength of the concrete, ranging from 20 MPa to 80 MPa. These beams failed in a shear-critical manner upon the formation of the first web shear crack. Current design code formulations would say that the shear strength of these beams is directly proportional to a ‘concrete contribution’ related to the tensile strength of the concrete, which in turn is normally related back to the compressive strength (typically, the tensile strength is taken as proportional to the square root of the compressive strength). Hence, the 80 MPa beam would be expected to have a shear strength of close to double that of the 20 MPa beam. Finite element analyses could also be expected to produce similar trends in predicted strength, since the concrete tensile strength is the over-riding parameter in most analyses of such cases. Shown in Figure 1.24 are the load-deformation responses measured In the experiments. Note that there is little difference in the strengths and pre-peak deflection responses observed; certainly nothing approaching the doubling of strength anticipated. In fact, the 80 MPa beam exhibits a shear strength lower than the 20 MPa beam. At work are mechanisms related to smoothness of the fracture plane, aggregate interlock mechanisms, and crack slip mechanisms. A second series of beams from the same test program involved four beams similar in dimensions and loading to the first (beams DB120M, DB140M, DB164M and DB 180M; see Figure 1.23). The principal difference was that these beams contained the near-minimum amount of shear reinforcement (0.08 %). Again, the compressive strength of the concrete ranged from 20 MPa to 80 MPa. Shown in Figure 1.25 are the load-deformation responses recorded. Note that here, a small amount of shear reinforcement had a substantial influence on the strength and failure mechanisms observed. Although the higher strength concrete beams did exhibit a higher shear strength, there was still a good deal of perplexing behaviour observed. See Angelakos et al. (2001) for a more complete description of the test program, and a more thorough discussion of results and significance. These two series of test results provide a stringent test of any NLFEA model. Would-be analysts are encouraged to formulate all structural models, and select all constitutive models and analysis parameters in advance of any ‘preliminary’ computations, and to conduct the analyses in a group and only once (i.e., forego any “calibration” or “fine-tuning’ work).


! 27 !


Figgure 1.23: Deetails of Angellakos test beam ms.

Figure 1.24: Observedd load-deform mation responnse of Angela akos Series I beams (beam ms containing g no shear reinforceement).

28

! 1 Introduction !


Figure 1.25: Observedd load-deform mation response of Angellakos Series II beams (beeams containing shear reinforceement).

1.6

Objectivees

In addittion to ongo oing work in n the develo opment of im mproved co onstitutive m models and analysis procedu ures, researcch effort iss also requiired to mak ke NLFEA procedures more ameenable to practicaal applicatio on and to rem move somee of the conccerns regard ding accuraacy and conssistency. Specificc short-term m objectives should incllude the following:

"

Providee design en ngineers and d non-experrts guidancee in the appplication of NLFEA procedu ures to com mmon and prractical prob blems.

"

Establish databank ks and ben nchmark prroblems, fo or various sstructure ty ypes and loading g conditionss, to facilitaate the valid dating or callibrating off material beehaviour modelss and analyssis procedurres.

"

Recogn nize that div versity in th he analysis procedures p a available is itself valuaable, and encouraaged it. At the same tiime, work towards t thee harmonizaation of con nstitutive modelss and anaalysis apprroaches must m proceeed where appropriatte. The identifiication and d description n of relevaant behavio our mechannisms, and suitable modelss for their reepresentatio on, should be accentuateed.

"

Providee guidance on modelliing issues relating r to assessmentt, rehabilitation and forensic engineering; areas where w NLFE EA procedu ures are findding signifiicant use today.

"

Work towards t dev veloping acccurate, con nsistent and easy-to-usee automated d design softwarre, with NL LFEA being g one of thee viable means of perfo forming bacckground calculaations.

Task Grroup 4.4 (Computer-Baased Modellling and Deesign), of fib b Commission 4 (Mod delling of Structurral Behavio our and Deesign) was established to work towards theese objectiv ves. This report constitutes c a initial efffort towardss achieving them. an


! 29 !


1.7

Scope of report

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. Chapter 2 discusses design procedures based on linear elastic modelling, the level at which most code-based design specifications are formulated. Chapter 3 examines basic material properties and behaviour models for concrete and reinforcement. Nonlinear modelling concepts are explored in their application to plane frames and to two- and three-dimensional structures in Chapters 4 and 5, respectively. Chapter 6 provides a more detailed and comprehensive discussion of nonlinear constitutive models and behaviour mechanisms, and their application to modelling. The use of benchmark tests and calibration procedures, a necessary step in the use of any new modelling tool, is discussed in Chapter 7. Strut-and-tie modelling concepts are reviewed in Chapter 8, special models for design of planar elements in Chapter 9 and conclusions are discussed in Chapter 10.

1.8

References

ACI Committee 318. (1983), “Building Code Requirements for Reinforced Concrete”, American Concrete Institute, Detroit, 111 pp. Angelakos, D., Bentz, E. C., and Collins, M. P. (2001), “The Effect of Concrete Strength and Minimum Stirrups on the Shear Strength of Large Members”, ACI Structural Journal, (in press). ASCE (1982), “Finite Element Analysis of Reinforced Concrete”, State-of-the-Art Report”, ISBN 0-87262-307-6, 545 pp. Akiyama, H., Goto, S. and Nakazawa, T. (1996), “Shear strength of Large Reinforced Concrete Circular Slabs Under Uniformly Distributed Load”, Proc. of JCI, Vol.18, No.2, pp.1097-1102. Bentz, E.C. (2000), Presentation at ACI Annual Convention, Toronto, October. Braam C.R. (1990), “Control of Crack Width in Deep Reinforced Beams”, Heron, Vol. 4, No. 35. Bresler, B., and Scordelis, A.C. (1963), “Shear Strength of Reinforced Concrete Beams”, American Concrete Institute J., Vol. 60, No. 1, pp. 51-74. Cervenka, J. and Cervenka, V. (1999), “Three Dimensional Combined Fracture-Plastic Material Model for Concrete”, Proceedings of the 5th U.S. National Congress on Computational Mechanics, Boulder, Colorado, USA, August. Cervenka, V. (2002), “Computer Simulation of Failure of Concrete Structures for Practice”, First fib Congress in Osaka, October 13-19, in Japan, keynote lecture in Session 13.

30

! 1 Introduction !


Collins, M.P., Vecchio, F.J., and Mehlhorn, G. (1985), “An International Competition to Predict the Response of Reinforced Concrete Panels”, Canadian Journal of Civil Engineering, 12, No.3, pp. 624-644. Collins, M.P., Vecchio, F.J., Selby, R.G., and Gupta, P.R., (1997). “The Failure of an Offshore Platform”, Concrete International, Vol. 19, No. 8, pp 28-35. Hartl, G. (1977), “Die Arbeitslinie Eingebetete Stäbe bei Erst- und Kurzzeitbelastung”, Dissretation, Universität Innsbruck. Kawabara, Y. (2002), “Shear Capacity of Rigid Base-slab Connection for LNG Storage Tanks”, JSCE annual review. Maekawa, K., Pimanmas, A. and Okamura, H. (2003), “Nonlinear Mechanics of Reinforced Concrete:, Spon Press. Nakano, M. (2002), “In-ground LNG Storage Tanks – Technological Trends and Latest Technological Developments”, Concrete Library of JSCE, No.39, June. Nuclear Power Engineering Corporation (NUPEC) (1996), Comparison Report, “Seismic Shear Wall ISP”, NUPEC’s Seismic Ultimate Dynamic Response Test, Report No. NUSSWISP-D014, 407 pp. Okamura, H., and Maekawa, K. (1991), “Nonlinear Analysis and Constitutive Models of Reinforced Concrete”, University of Toyko, ISBN 7655-1506-0, pp 182. Podgorniak-Stanik, B. (1998), “The Influence of Concrete Strength, Distribution of Longitudinal Reinforcement, Amount of Transverse Reinforcement and Member Size on Shear Strength of Reinforced Concrete Members”,. M.A.Sc. Thesis, Dept. of Civil Engineering, University of Toronto, pp 711. van Mier J.G.M (1986), “Multiaxial Strain-softening of Concrete, Part I: Fracture”, Materials and Structures, RILEM, Vol. 19, No.111. Vecchio, F.J., and Collins, M.P. (1986), “The Modified Compression Field Theory for Reinforced Concrete Elements Subjected to Shear”, Journal of the American Concrete Institute, 83, No.2, pp. 219-231. Vecchio, F.J., and Collins, M.P. (1990). “Investigating the Collapse of a Warehouse”, Concrete International, Vol. 12, No.3, pp 72-78. Vecchio, F.J. (2000), “Disturbed Stress Field Model for Reinforced Concrete: Formulation”, ASCE Journal of Structural Engineering, 126, No. 9, pp. 1070-1077.


! 31 !


[Blank Page]

! !


2

Design using linear stress analysis

2.1

Introduction

Elastic based stress analysis using finite elements provides engineers with a valuable tool in their design armory. Today, most (if not all) structural engineering companies have access to a finite element package although of varying degrees of sophistication. In fact a number of public domain and shareware programs are available via the web. Dimensioning of structures based on linear analyses of frames, as one example, is commonplace. Less common is detailing of concrete plates, shells and membrane type structures. Yet it is in some of these structures where designers can benefit most by undertaking stress analyses, remembering always that concrete has limited ductility. Some of the advantages of detailing based on linear FE modelling include: "

Linear FE modelling is well established and relatively easy to apply.

"

Multiple load cases can be accommodated quickly and with a minimal change in the input data.

"

The greatest quantity of reinforcement is placed in the high-tension regions. That is in regions corresponding to the initial crack locations thus helping to control crack propagation.

The main drawbacks in using the method are: "

No information is attained as to the collapse load of the structure (provided that ductility demands are met, elastic analyses give a lower bound to the strength limit design load)

"

Detailing guidelines need to be established and followed to ensure ductility and serviceability demands are met.

"

No information is provided on inelastic phenomena such as crack widths, crack spacings or deflections.

The above aside, however, it is not intuitively obvious how to interpret output from a finite element analysis, in particular a linear analysis and how to detail the reinforcement. For example, how do we detail the results of a shell element with 8 stress resultants into 4 layers of longitudinal reinforcement plus transverse reinforcement? It is the objective of this chapter to address this issue and to give guidance to the practicing engineer on some of the strengths and on the limitations of using linear finite element modelling for reinforcement detailing. Models for load-deformation analyses are treated later in this report with the focus of this section on classical yield conditions for membrane, slab and shell elements and structures derived from plasticity theory. This allows for a straightforward reinforcement design based on the applied loads. Furthermore, a consistent method to account for transverse shear forces in membranes and slabs (Marti, 1990) will be outlined.


! 33 !


2.2

Membrane structures

2.2.1

Notation

As

area of reinforcing steel

Es

elastic modulus of steel

f cd

design compression strength of concrete

f c#

characteristic cylinder compression strength of concrete

fy

yield stress of reinforcing steel

f yd , f ydt

design yield stress of reinforcing steel ' f y & s

f ycd .x , f ycd . y

design yield stress of compression reinforcing steel placed parallel to the

$

%

global X and Y directions, respectively.

f ytd .x , f ytd . y

design yield stress of tension reinforcing steel placed parallel to the global

t (

X and Y directions, respectively. thickness of membrane disturbance factor

)1

tension strain in the major principal direction

&c, &s

partial safety factors for concrete and reinforcing steel (taken as & c ' 1.5 , & s ' 1.15 )

* a , *b

reinforcement ratios in the directions of an arbitrary set of orthogonal axes

*x , * y

reinforcement ratios in the global X and Y directions, respectively.

+o

normal stress at the centre of Mohr’s circle of concrete stresses

+s

stress in reinforcing steel

+ sa , + sb

stress in the reinforcing steel placed in the directions of a and b, respectively

+ sx , + sy

stress in reinforcing steel placed parallel to the global X and Y directions, respectively

+ xc , + yc

stress in the concrete in the global X and Y directions, respectively

+ x , + y , , xy

normal stresses in the X and Y directions and the shear stress, respectively

+1, +3

major and minor principal stress resulting from the applied loads, respectively major and minor principal stress in the concrete, respectively

+ 1c , + 3c

-F

34

angle between the global X axis and the major principal stress in the concrete (measured in the anti-clockwise direction). angle between the global X axis and the major principal stress due to the applied loads (measured in the anti-clockwise direction).

! 2 Design using linear stress analysis !


2.2.2

General

In the output of linear elastic FE modelling of membranes the analyst is generally given stresses and strains in a Cartesian coordinate system, which, for the analysis that follows are chosen as collinear with the envisaged reinforcement directions. Whilst principal stresses are of interest as they give the elastic load path, it is usually preferable to detail the reinforcement along axes orthogonal to the axes of the structure or structural element being designed. For example, Figure 2.1a shows normal and shear stresses on a 200 mm thick plane stress membrane element, stress output that is typical for a linear FE analysis. From the normal stresses it is not directly evident what quantity of steel reinforcement is needed to carry tension stress or in what direction the steel should be placed. Plotting Mohr’s circle the principal stresses and their directions are calculated and given in Figures 2.1b and Figure 2.1c. Assuming that the strength of the concrete is sufficient to carry the full compression stress but carries no tension, steel reinforcement is needed such that *1 f y ' 8.5 MPa and is placed at

- ' 70.7 0 degrees to the global X axis. This requirement is satisfied with f ytd ' 435 MPa and with 2-layers of 20-mm diameter bars at 160-mm centres. ,

5 MPa

8.5

Y 10 MPa

(-20,10)

+n

20 MPa

23 . 5

o .-'/0/12

M Pa

MPa

8.5 MPa -23.5 MPa

X

(a)

3413 o

(5,10)

(b)

Figure 2.1: a) Example output from FE modelling of a membrane;

(c) b) and c) calculation of principal stresses.

Whilst with this solution equilibrium is satisfied, it is not usually convenient to place reinforcement aligned with the principal directions. Rather it is normally preferred to place the steel orthogonal to the global axes. In this case the concrete and reinforcement stress resultants must sum to the global stresses on the membrane, as depicted in Figure 2.2. In the compression field approaches (such as for example the diagonal compression field theory of Mitchell and Collins, 1974) the concrete is taken to carry no tension, that is + 1c ' 0 . Figure 2.3 shows the relationship between the applied stresses, the concrete stresses and the reinforcement. Using this approach it is seen that there lies an infinite number of solutions to the stress field. Some of the more interesting solutions are discussed in brief, below. 2.2.3

Reinforcement in one direction

Looking closely at the example presented in Figure 2.1, the designer may choose a stress regime such that * x+ s ' 0 , that is to provide reinforcement solely in the global Y direction. In this case the Mohr’s circles are as shown in Figure 2.4. From the resulting geometry it can be shown by equating the radii of the Mohr’s circle of concrete stresses calculated from two points on the circle (see Figure 2.4) that


! 35 !


1

+y

*55+ y sy

+3

1

+

Y

, xy

=

+

3

+x

*55+ x sx

-

X

Figure 2.2: Concrete and steel reinforcement stress components.

*55+ x sx , $+556,555% x xy .-

+3c

.-F

+1c 555'4

+n

$+5567,555% y xy

concrete stresses

applied stresses

*55+ y sy Figure 2.3: Compression field approach.

Y

,

+y (+556,555) x xy

,xy +x ,xy

+3c

R

+o

R

+1c

X

+n (+5567,555) y xy

*55+ y sy

Figure 2.4: Mohr’s circles for reinforcing steel placed solely in the Y-direction

36



+o '

2 + x2 8 , xy

2+ x

(2.1)

where + o is the normal stress at the centre of the Mohr’s circle of concrete stresses and + x and , xy are the X-normal and shear stresses, respectively (shown in Figure 2.2). Given that + 1c ' 0 , the principal compression stress in the concrete may be obtained from + 3c ' 2+ o . The geometry of the circles also shows that the stress in the Y-direction reinforcement required for equilibrium is

* y+ sy ' + x 8 + y 7 2+ o

(2.2)

Further examination of Figure 2.4 shows that Eqs 2.1 and 2.2 only give a solution provided that the inequality + x 9 0 holds true. Similarly it can be shown that for the case where reinforcement is placed only in the global X direction (that is * y+ s ' 0 ), then provided that + y 9 0

+o '

2 + 2y 8 , xy 2+ y

* x+ sx ' + x 8 + y 7 2+ o

(2.3) (2.4)

For the example presented in Figure 2.1, if it is desired to reinforce the element solely in the Y-direction then the resulting reinforcement requirement (by Eqs. 2.1 and 2.2) is * y+ sy ' 10 MPa . This requirement is satisfied with f ytd ' 435 MPa and with 2-layers of 20-mm diameter bars at 130-mm centres. The minor principal stress + 3c ' 2+ o ' 725 MPa . Thus, in the example presented, the solution of providing reinforcing steel in the Y-direction only is less efficient than when reinforcing in the major principal stress direction. This is to be expected but likely to be more practical. 2.2.4

Isotropically reinforced panels

One of the more interesting solutions is for the case of an isotropically reinforced panel. The Mohr’s circle for this case is shown in Figure 2.5. It is seen that the circle representing the concrete stresses is a translation of the global stresses and that * x+ sx ' * y+ sy ' + 1 . The area of reinforcement in each of the global X and Y directions is equal to that for the case where the steel is placed in the direction corresponding to the major principal stress. Equilibrium can be maintained by replacing the force in the major principal direction $F1 % with any two concurrent forces $Fa and Fb % . Letting F1 act over a unit length and taking the forces Fa and Fb as orthogonal, as shown in Figure 2.6, then Fa ' F1 sin :

Fb ' F1 cos :


(2.5)

! 37 !


*55+555'5*55+ x sx z sy , $+556,555% x xy

-'-F .-

+3c

+1c

.-F + 1

+3

+n

$+5567,555% y xy *55+555'5*55+ y sy x sx Figure 2.5: Mohr’s circle for isotropically reinforced panels.

F1

:

Fb 1

1. c o

s:

:

+b 1. sin

+1

:

+a Fa

Figure 2.6: Equivalent concurrent orthogonal forces and resulting stresses.

+a '

F1 sin : ' +1 t. sin :

+b '

F1 cos : ' +1 t. cos :

(2.6)

where the angle : is as shown in Figure 2.6 and t is the thickness of the element. That is + a ' + b ' + 1 . As the angle : is arbitrary, all orthogonal reinforcement sets are valid solutions with the steel stress in each direction equal the major principal applied stress $+ 1 % . Looking again at the example of Figure 2.1, assuming that the strength of the concrete is sufficient to carry the compression but carries no tension then steel reinforcement may be placed in any two orthogonal directions (such as, for example, in the global X and Y axis directions) such that * a+ sa ' *b+ sb ' 8.5 MPa . This requirement is satisfied with

f ydt ' 435 MPa and with 2-layers of 20-mm diameter bars at 160-mm centres placed in each of the X and Y directions. The example of the isotropically reinforced plate may appear to contradict the solution obtained when the steel is placed in the major principal stress direction without an equivalent quantity of orthogonal reinforcement. Examination of Figure 2.6 shows that placing the steel

38



in the major principal direction corresponds to the particular solution : ' 0o giving Fa ' 0 , sin : ' 0 and leads to a breakdown of Eq. 2.6. This is the only condition for which this occurs. 2.2.5

The general solution

Using the theory of plasticity, Nielsen (1964, 1971) derived a set of general solutions for isotropically and orthotropically reinforced plates. Clarke (1976), Clyde (1977), Müller (1978), Marti (1979, 1980), Morley (1979) and many others have also made significant contributions to the field. The cases examined above are particular solutions of a general failure criterion. For the most general case the strength of a membrane is limited by the yield surface expressed by the following criteria (Kaufmann and Marti, 1998):

$

%$

%

2 Y1 ' , xy 7 * x f ytd .x 7 + x * y f ytd . y 7 + y ' 0

(2.7a)

$

%$

%

(2.7b)

2 Y3 ' , xy 7 f cd 7 * y f ytd . y 8 + y * y f ytd . y 7 + y ' 0

$

%$

%

(2.7c)

2 Y4 ' , xy 7 f cd 4 ' 0

(2.7d)

2 Y2 ' , xy 7 f cd 7 * x f ytd .x 8 + x * x f ytd .x 7 + x ' 0

$

%$

$

%$

$

%$

%

2 Y5 ' , xy 8 f cd 8 * x f ycd .x 8 + x * x f ycd .x 8 + x ' 0

(2.7e)

%

2 Y6 ' , xy 8 f cd 8 * y f ycd . y 8 + y * y f ycd . y 8 + y ' 0

(2.7f)

%

2 Y7 ' , xy 7 f cd 8 * x f ycd .x 8 + x f cd 8 * y f ycd . y 8 + y ' 0

(2.7g)

where f cd is the design value of the concrete compression strength f ytd .x and f ytd . y are the design strengths of the steel reinforcement in tension in the global X and Y directions, respectively, and f ycd .x and f ycd . y are the design strengths of the steel reinforcement in compression in the global X and Y directions, respectively. The yield surface given by Eqs 2.7a to 2.7g is plotted in the three-dimensional stress space of + x , + y ,, xy in Figure 2.7 for a

$

%

panel with * x ' 0.01 , * y ' 0.005 and f yd ' 500 MPa . In Eqs. 2.7a-g, Y1 corresponds to under reinforced elements where the steel in both the X and Y directions yields before failure of the concrete, Y2 and Y3 correspond to web crushing failure and Y4 represents the no-yield condition of crushing in pure shear. The criteria Y5 , Y6 and Y7 are for plates in biaxial compression. Whilst the application of Eq. 2.7 is relatively straightforward it is not as easy to picture the failure surface in the 3-dimensional stress space. As the yield criterion is based on the principal stresses we again revert to Mohr’s circles of stress for an alternative graphical solution. The case of an isotropically reinforced plate is used to demonstrate the relationship between Mohr’s circle and the general failure criterion given by Eq. 2.7. In Figure 2.8a-d


! 39 !


10

10

5

, xy (MPa) 5

0 -25 -20 -15 -10 -5 + x (MPa)

0 0

-5 -10

+ y (MPa)

, xy (MPa)

-15 0

-20 -25

(a)

5

+ y (MPa) + x (MPa)

-30

2

Regime 1

10

5 Regime 4

3

6

Regime 7

-25

(b)

Figure 2.7:

40

Yield surface for a reinforced concrete membrane with *x =0.01, *y = 0.005 and fyd = 500 MPa; a) 3D View; b) plan view identifying yield regimes.



;,;

;,;

regime 1

+3

+n

*5f yd

+n

collapsing Mohr's circle

;,;

*5f yd

fcd

(a)

(b)

regime 4

regimes

5&6

5&6

fcd /2

+1

+3

fcd

regimes

regimes 2&3

sliding Mohr's circle

fcd /2

expanding Mohr's circle

fcd /2

regime 4

regimes 2 & 3

regime 7

;,;

+3

fcd /2

sliding Mohr's circle

+1

*5f yd

+n

fcd

fcd

(c)

+1

*5f yd

+n

(d)

.5*5f yd R = fcd /2

R

Region I

,

R

+n

Region II

fcd + *5fyd

*5f yd

(e)

Figure 2.8:

Isotropically reinforced plate: a-d) Mohr’s circles defining the failure surface; e) admissible stress envelope.


! 41 !


Mohr’s circles are plotted for an isotropically reinforced plate. Starting from the case of equibiaxial tension (that is +1 ' + 3 ' *f yd ) the circle expands with reducing + 3 until the point where the radius reaches the limiting shear strength of f cd 2 (Figure 2.8a) with the corresponding minor principal stress being +3 ' 7 fcd 7 * f yd . This phase represents the

$

%

regime 1 shown in Figure 2.7. In the second phase the circle shifts towards the negative, whilst maintaining a constant radius, until + 1 ' 0 and + 3 ' 7 f cd (Figure 2.8b). This represents regimes 2, 3 and 4 in Figure 2.7. In the third phase, corresponding to regimes 4, 5 and 6 in Figure 2.7, the circle slides further to the negative and the reinforcing steel moves into compression (Figure 2.8c). Finally, the circle collapses with the minor principal stress remaining constant until the equi-biaxial compression point is reached, that is the point +1 ' +3 ' 7 fcd 8 * f yd , as shown in Figure 2.8d.

$

%

From the Mohr’s circles shown in Figures 2.8a-d the admissible stress field is plotted (Figure 2.8e). For any solution to be admissible two criteria must be met. Firstly, the Mohr’s circle of the applied stresses must lie wholly within the circular region I, where region I is centered on the normal stress axis and has a diameter of f cd . Secondly, region I lies wholly within region II (refer Figure 2.8e). If these two criteria are met then the membrane is safe. Note that in practice it is sufficient to show that any two diagonally opposed points on the Mohr’s circle of stress meet these criteria to demonstrate admissibility of the solution (for example the stress pairs ( + x ,, xy ) and ( + y ,7, xy )). For membranes subject to tension and/or shear the zone of greatest interest is that of regime 1 in Figures 2.7 and 2.8a and the design equation given by Eq. 2.7a. Plotted in Figure 2.9 is the Mohr’s circle of stress for the applied stresses and the stresses in the concrete for a membrane in bi-axial tension. From the upper shaded triangular region it is seen that the concrete stress in the X direction is + xc ' + x 7 *x f ytd.x and therefore steel reinforcement is required such that

* x f ytd . x < + x 8 , xy tan -

(2.8)

where 7 90 o 9 - 9 90 o and - = 0 .

Similarly, from the lower shaded triangular region + yc ' + y 7 * y f ytd. y and therefore to provide sufficient capacity the Y direction reinforcement is required such that

* y f ytd . y < + y 8 , xy cot -

(2.9)

In Figure 2.9b the feasible domain of solutions to Eq. 2.7a is plotted. In this form the relationship between Eq. 2.7a and Eqs. 2.8 and 2.9 is clear.

42



*555f x ytd.x , $+5556,5555% xc xy +3c

$+556,555% x xy -

.-F

+1c

+n

-

concrete stresses

$+55567,5555% yc xy

$+5567,555% y xy

applied stresses

*555f y ytd.y

*y fyd +y585;,xy|5cot|-| feasible domain

+x585;,xy| tan|-| +y 0 0 Figure 2.9:

+x

*x fyd

a) Mohr’s circle of stress for an under-reinforced concrete plate; b) solution domain for underreinforced element.

Equations 2.8 and 2.9 are identical to those derived by Nielsen (1964) using the theory of plasticity 1. For Eqs. 2.8 and 2.9 to govern the behaviour (that is for the panel to be under reinforced) the strength of the concrete must be such that f cd < 7+ 3c , that is the concrete must not crush. Applying the first invariant of stresses the compression failure criterion can be written as

+3c 8 +1c ' $+ x 7 *x f ytd.x % 8 $+ y 7 * y f ytd. y %

(2.10)

Taking + 1c ' 0 (as per the compression field theory) and with f cd < 7+ 3c crushing failure is not critical provided that

fcd < *x f ytd.x 8 * y f ytd. y 7+ x 7+ y

(2.11)

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 1

Orthotropic membrane theory can be traced to Nielsen (1920) and Eqs. 2.8 and 2.9 to Leitz (1923) for tan- ' 1 [according to Nielsen (1964, 1971)].


! 43 !


For strength limit design of reinforced concrete membranes the strength criteria are fully satisfied by Eqs. 2.8, 2.9 and 2.11 and are identical to the equations presented by Kaufmann (1999) based on the yield surface defined by Eq 2.7. Taking once more the example presented in Figure 2.1 and with - ' 65o , Eq. 2.8 gives *x f ytd.x ' 1.45 MPa and * y f ytd. y ' 9.67 MPa. This requirement is satisfied with

f ytd ' 435MPa and with 2-layers of 12-mm diameter bars at 330-mm centres placed in the X-direction and 2-layers of 20-mm diameter bars at 140-mm centres placed in the Y-direction. The required design concrete strength is given by Eq. 2.11 and is f cd < 26.1 MPa . 2.2.6

Some comments on the angle -.

In developing solutions using linear FE modelling it is important that the designer “respect” the limitations of the concrete material. In a panel subject to a constant ratio of normal and shear stresses (with at least one tensile principal stress) before cracking the stress field in the concrete remains relatively elastic and the stresses in the steel reinforcement are negligible. After cracking the tension stresses in the concrete rapidly reduce while those in the reinforcing steel increase. Assuming that the concrete does not fail in compression then the crack directions will remain relatively stable until yield of the steel in one-direction. After yield in one direction the forces in the structural element are continuously redistributed to balance the applied tractions until yield in all directions has occurred. Concrete panels, however, have a limit on the amount of redistribution that is capable of being achieved. This is referred to as ductility demand. As a rule concrete elements such as membranes and panels should not be pushed far beyond that which is “natural”. Designers must critically examine the load path being assumed to satisfy themselves that the element has a sufficient level of ductility to meet the demands of the structure. To this end plots of principal stress directions from linear-elastic modelling can give the designer valuable insight. Care and “engineering common sense” needs to be applied when designing any structural element but even more so for the case of non-flexural membranes. For a membrane element with its principal stresses in biaxial tension, steel reinforcement is required in at least two non-concurrent directions to carry the tension stresses with the concrete taking the shear stresses. For the case of orthogonal reinforcement - ' 45o leads to the lowest concrete strength requirement. Many structural elements fall into the category of tension-compression elements. Again

- ' 45o provides the lowest concrete strength requirement and in many cases provides for a good solution. In some cases, however, - ' 45o may not be desirable. Figure 2.10a shows an element in tension-compression where taking - ' 45o leads to tension reinforcement in the X direction and compression reinforcement in the Y direction. This, in some circumstances, can lead to a significant requirement for reorientation of the cracks requiring a high ductility demand. Figure 2.10b shows that there are limits on - if compression in the steel is not wanted. Adding to Eq. 2.8 the condition *x f yd < 0 we get the limits

44



*55+ x sx , .-5'5A4!

*55+ x sx ,

tension steel

$+556,555% x xy

$+556,555% x xy +3c

+1c 555'4

+n

tension steel

+3c

.-

+1c 555'4

+n

$+5567,555% y xy

$+5567,555% y xy

applied stresses

applied stresses

*55+ y sy compression steel

(b)

(a)

o

Figure 2.10: a) Example showing the requirement of compression reinforcement for - = 45 ; b) no compression steel solution by limiting -.

for+ x 9 0 :

7 + x , xy ? tan - 9 >

for+ x < 0 :

0 9 tan - 9 >

(2.12a)

Similarly applying the requirement that * y f ytd. y < 0 to Eq. 2.9 gives

for+ y 9 0 :

0 9 tan - ? 7 , xy + y

for+ y < 0 :

0 9 tan - 9 >

(2.12b)

Choosing a value of - to meet the requirements of Eq. 2.12 ensures that the stresses in the reinforcing steel do not go into compression. It is also observed for Eq. 2.12 that the lower limit on Eq. 2.12a for + x 9 0 and the upper limit on Eq. 2.12b for + y 9 0 correspond to the angles for unidirectional reinforcement in the x and y directions, respectively. In Figure 2.11 the solutions to Eq. 2.7a are plotted for the stresses of the example shown in Figure 2.1. The figure shows that for - 9 63.43o the X direction steel is in compression and for - @ 63.43o the steel is in tension. The angle - ' 63.43o corresponds to the solution of reinforcement only in the Y direction. Lastly, for the case of elements subject to biaxial compression, the angle - should be matched to that of the principal stress directions due to the applied loads. In this case reinforcing steel may be used, if desired, to carry a proportion of the compression forces.


! 45 !


40

+x = -20 MPa +y = 5 MPa , xy= 10 MPa

*y fyd (MPa)

30 Eqn. 2.7a

20

- = 63.43o

10

0 -20

X dirn. reo. in compression - < 63.43o

-10

X dirn. reo. in tenision - > 63.43o

0

10

20

*x fyd (MPa) Figure 2.11: Solution domain for element subject to plane stress.

2.2.7

The design concrete compression strength, fcd.

It has been shown by a number of researchers (Robinson and Demorieux, 1977, Vecchio and Collins, 1982, 1986, Miyakawa et al., 1987, Belarbi and Hsu, 1991, Pang and Hsu, 1992, amongst others) that the disturbing effect of passing tension reinforcement through concrete in compression weakens the concrete. In addition, as concrete strength is increased the concrete becomes more brittle. To account for the imperfect assumption that concrete is a rigid plastic material and that the ductility demands can be met an efficiency factor is introduced to ensure that the concrete is not overstressed. There is considerable debate on the efficiency factor for concrete in compression in D-regions and much has been written (see for example the quite contradictory articles of section 4.6 Nielsen, 1999, and Foster and Malik, 2001). For a history of various efficiency factors methodologies the reader is referred to the report on recent approaches to shear design by ASCE-ACI Committee 445 (1998). In recent approaches to quantifying of the efficiency of concrete in uniaxial compression and subject to transverse disturbance the overall efficiency factor is separated into two components (MacGregor, 1997, Foster and Malik, 2001). The first factor accounts for the variation of the in-situ strength of concrete relative to that of the standard cylinder test. The second factor accounts for the disturbance due to the transverse stress fields and the brittleness of unconfined concrete. In planned changes to the ACI code (Cagley, 2001, p129) the strength of struts subject to transverse tensions in strut-and-tie models is f cd ' 0.85 ( f c# & c

(2.13)

where the 0.85 factor accounts for the variation between the in-situ and cylinder strengths and ( is a disturbance factor. Equation 2.13 is of a generally suitable form for the design of concrete plates subject to in-plane stresses. Whilst there are a number of variants of the efficiency factor relationship, the model by Collins and Mitchell (1986) has generally withstood the test of time and has been incorporated into the Canadian code of practice (CSA84, 1984). Based on the panel tests of Vecchio and Collins (1982), Collins and Mitchell proposed that

46



('

1 ? 1.0 0.8 8 170)1

(2.14)

where ) 1 is the major principal strain normal to the direction of the compression field. The transverse strain is required to be sufficiently large for the ductility demand to be met. Adopting a minimum transverse strain of twice the yield strain of the steel reinforcement gives

('

1 0.8 8 340 f y Es

(2.15)

where f y and Es are the yield strength and the elastic moduli of the steel reinforcement, respectively. For 500 MPa grade reinforcing steel ( ' 0.6 and f cd ' 0.5 f c# . Not all members or sub-elements of members are subject to significant transverse strains and a reduction to the degree indicated by Eq. 2.15 is unjustified. For the case where transverse strains are small and the concrete is essentially in uniaxial compression (transverse tensions can be carried without cracking of the concrete) and for the case of biaxial compression then the concrete is undisturbed by crossing tension stress fields. As a guide, if the major principal stress due to the applied loads is such that +1 9 0.33 fc# then the disturbance factor maybe taken as

( ' 1.0

(2.16)

Examples of members and regions of members where Eq. 2.16 applies include beams and columns outside the shear zones and at edges of members where principal stresses and strains are aligned with the boundary of the element. 2.2.8

Example – Design of a reinforced concrete squat shear wall.

Shear walls are common elements in buildings that are subject to in-plane stresses. In this example the squat shear wall shown in Figure 2.12 is designed to carry of factored design load

H * ' 4000 kN . The load may be applied in either direction as indicated in the figure. The wall is 200 mm thick and is bounded by 500-mm square stiff elements on both sides and a 1200-mm by 400-mm beam at the top of the wall. A linear-elastic finite element model of the wall was set up with the mesh shown in Figure 2.13. The mesh consists of 30 by 8-node isoparametric elements defined by 113 nodes. The material properties were taken as E ' 25 GPa and B ' 0.18 for all elements. The model was analysed with integration on a 3x3 Gaussian quadrature with the stress output on a 1 gauss point quadrature (that is at the centre of the element). The resulting normal stresses for load case 1 are shown in Figure 2.13. For load case 2, the element stresses are a reflection of the results of load case 1 about the Y-Y axis (refer Figure 2.12). For the reasons of ductility demand, discussed above, it is important to make an assessment of the flow of forces in the wall and adjacent boundary elements. In the design that follows, for the wall elements the minor principal stress direction for the concrete is taken as aligned with the diagonal of the wall (refer Figure 2.12) giving - ' 58o . For the side face boundary elements the principal stresses are taken as aligned with the global stress directions as shown fib Bulletin 45: Practitioners’ guide to finite element modelling of reinforced concrete structures

! 47 !


in Figure 2.12. The resulting stresses in the X and Y direction reinforcing steel are shown in Figure 2.14. For detailing of the reinforcement, the wall is divided into vertical and horizontal strips, as shown in Figure 2.15. Reviewing the steel stresses in the horizontal (x) direction it is seen that the maximum stress in each of wall strips 1X, 2X and 3X are of similar magnitude (5.45, 5.92 and 5.54 MPa, respectively). It is deemed therefore appropriate to maintain a single steel stress in the X direction of 5.92 MPa for the full height of the wall. The area of steel reinforcement per unit height of wall is given by As '

* f yd t h

(2.17)

f yt & s

where t is the wall thickness, h is a unit height, f yt is the yield strength of the steel and & s is the partial safety factor for the steel reinforcement (taken as & s ' 1.15 ). For 500 MPa grade reinforcing steel and *x f yd ' 5.92 MPa, Eq. 2.17 gives As ' 2720 mm2 m . This is satisfied with 2 layers of 16-mm diameter bars at 150-mm centres. In the vertical (Y) direction the greatest stresses occur in design strips 1Y and 4Y (noting that in strip 4Y the greatest stress occurs for load case 2). For 500 MPa grade reinforcing steel and * y f yd ' 4.22 MPa, Eq. 2.17 (with h replaced by the unit width, b) gives an area of reinforcing steel of As ' 1940 mm2 m . This is satisfied with 2 layers of 16-mm diameter bars at 200-mm centres. In wall strips 2Y and 3Y the requirement is for * y f yd ' 2.22 MPa giving

As ' 1020 mm2 m and leading to 2 layers of 12-mm diameter bars at 200-mm centres. For the boundary strips 1Y and 2Y the concrete stress angle is taken as 60 degrees giving a maximum stress in the longitudinal (Y-direction) steel of * y f yd ' 7.5 MPa and with

f ytd ' 435MPa this gives As ' 4310 mm2 . This is satisfied with 8C28-mm bars. The stress in the x direction of *x f yd ' 1.81MPa is carried three legged ties at 160 mm centres. The reinforcement details for the shear wall are shown in Figure 2.16. The minor principal stresses in the concrete $+ 3c % are obtained from Eq. 2.11 and given in Figure 2.14. In this case the + 1c axis is taken as coinciding with the major principal axis and thus varies through the length of the element. In the compression boundary element the transverse tension strains are low and the disturbance factor is taken as that given by Eq. 2.16. The maximum compression stress is 5.66 MPa and when the effect of the axial reinforcement in the section is considered it is seen that the compression strength of the boundaries does not control the design. In the web the compression stresses are reasonably uniform, indicating a degree of efficiency in the design as the whole of the wall is utilised. The minimum minor principal stress in the concrete wall is + 3c ' 9.0 MPa and as the transverse tensions are significant the disturbance factor is taken as that given by Eq.2.15. The minimum specified concrete strength is obtained from Eq. 2.13 and is f c# < 26.5 MPa . A concrete strength of f c# ' 30 MPa is adopted.

48



120 00 mm x 400 mm bou undary element

Y

Loadd Case 2

1c

400

Load Case 1

2c

500 mm squaare boundary eleement 2c

1c

3500

58°

2c

t = 200 mm m

60°

1c varies

Y 500

5500

500

F Figure 2.12: Finite F element analysis and d design of squuat shear walll.


! 49 !


+555'5 x 7F1/E5MPa +555'5 y 741435MPa ,5555'55 xy 41.E5MPa

+555'5 x 721A35MPa +555'5 y 7414/5MPa ,5555'55 xy 41.A5MPa

+555'5 x 7.1425MPa +555'5 y 7414/5MPa ,5555'55 xy 41.05MPa

+555'5 x 741DA5MPa +555'5 y 541445MPa ,5555' xy 5541/F5MPa

+555'5 x 7/10.5MPa +555'5 y 5/1A/5MPa ,5555'55 xy 41E05MPa

+555'5 x 7.1325MPa +555'5 y 7414.5MPa ,5555'55 xy 213.5MPa

+555'5 x 7.1DE5MPa +555'5 y 741.05MPa ,5555'55 xy 21A05MPa

+555'5 x 7/13D5MPa +555'5 y 7412E5MPa ,5555'55 xy 21205MPa

+555'5 x 7413F5MPa +555'5 y 741345MPa ,5555' xy 55.1./5MPa

+555'55 41445MPa x +555'5 y 741F.5MPa ,5555' xy 5541225MPa

+555' x 7414/5MPa +555'55 21E.5MPa y ,5555'5 xy 541045MPa

+555'5 x 741235MPa +555'55 41AF5MPa y ,5555'5 xy 521.F5MPa

+555'5 x 741A25MPa +555'5 y 7412A5MPa ,5555'55 xy 014D5MPa

+555'5 x 7/14/5MPa +555'5 y 741EF5MPa ,5555'55 xy 21345MPa

+555'5 x 741D.5MPa +555'5 y 7/1F/5MPa ,5555'5 xy 5.13A5MPa

+555'5 x 7414.5MPa +555'5 y 7/1EE5MPa ,5555'5 xy 5410.5MPa

+555'55 41425MPa x +555'5 y 5D12.5MPa ,5555'5 xy 5410F5MPa

+555'55 410.5MPa x +555'5 y 5/1AD5MPa ,5555'5 xy 5214.5MPa

+555'5 x 741425MPa +555'5 y 741205MPa ,5555'55 xy 213.5MPa

+555'5 x 741DE5MPa +555'5 y 7/1/25MPa ,5555'55 xy 21FD5MPa

+555'5 x 741F45MPa +555'5 y 7.1./5MPa ,5555'55 xy 21/05MPa

+555'5 x 7414.5MPa +555'5 y 721D35MPa ,5555'55 xy 41DD5MPa

+555'55 41205MPa x +555'55 31445MPa y ,5555'55 xy 41ED5MPa

+555'55 41AF5MPa x +555'55 .10F5MPa y ,5555'55 xy .1E/5MPa

+555'55 41/D5MPa x +555'5 y 741/F5MPa ,5555'55 xy 21245MPa

+555'5 x 7412/5MPa +555'5 y 7/1.E5MPa ,5555'55 xy 21235MPa

+555'5 x 741AE5MPa +555'5 y 7.1D45MPa ,5555'55 xy 214A5MPa

+555'5 x 741235MPa +555'5 y 7D10A5MPa ,5555'55 xy 41A05MPa

4000 kN

Figure 2.13: Finite element mesh for squat shear wall example and stress output.

4000 kN *55f x yd '5/12/5MPa *55f y yd '5.12A5MPa

*55f x yd '5213.5MPa x yd '521..5MPa *55f *55f y yd '5.1..5MPa y yd '5.12/5MPa *55f +555 '7E1.E5MPa +555 '7E13F5MPa 3c 3c

*55f x yd '521DA5MPa *55f x yd '5.1335MPa *55f y yd '5/1345MPa *55f y yd '541FE5MPa +555 '7310.5MPa +555 '701A.5MPa 3c 3c

+3c '5741335MPa

*55f x yd '541FE5MPa *55f y yd '5014D5MPa

*55f x yd '501E05MPa *55f x yd '5D1D05MPa *55f y yd '521445MPa *55f y yd '5.1/05MPa +555 '731.05MPa +555 '7A1445MPa 3c 3c

*55f x yd '501A.5MPa *55f x yd '521AD5MPa *55f y yd '5/10F5MPa *55f y yd '541/05MPa +555 '7E1.25MPa +555 '7F1./5MPa 3c 3c

+555'5 3 c 7/1A35MPa

*55f x yd '541E25MPa *55f y yd '5D1DA5MPa

*55f x yd '5D1.D5MPa *55f x yd '5D1A.5MPa *55f y yd '521E25MPa *55f y yd '5/1AA5MPa +555 '7F13/5MPa +555 '7E1.35MPa 3c 3c

*55f x yd '5D1.F5MPa *55f x yd '5501025MPa *55f y yd '5/1/D5MPa +555 '7E1/.5MPa +555 '7F1AA5MPa 3c 3c

+555'5 3 c 721FD5MPa

*55f x yd '5/1E/5MPa *55f y yd '5310A5MPa

*55f x yd '5D10D5MPa *55f x yd '5D10.5MPa *55f y yd '501..5MPa *55f y yd '5/1A45MPa +555 '7F1.D5MPa +555 '731225MPa 3c 3c

*55f x yd '5D14E5MPa *55f x yd '5521AF5MPa *55f y yd '541E25MPa +555 '7310A5MPa +555 '7F1EF5MPa 3c 3c

+555'5 3 c 7D1FF5MPa

Figure 2.14: Reinforcing steel stresses for shear wall orthogonally reinforced in XY.

50



Wall Strip 4X

Wall Strip 3X

Wall Strip 2X

Boundary Strip 2Y

Wall Strip 4Y

Wall Strip 3Y

Wall Strip 2Y

Wall Strip 1Y

Boundary Strip 1Y

Wall Strip 1X

Figure 2.15: Design strips in the X and Y directions for detailing of the reinforcement. 2750

1375 400

1375

2 x C12@200 2 x C16@150

8 C.E

2 x C16@200

3 x C12@160 stirrups

500

5500

8 C.E

3500

2 x C16@200

500

Figure 2.16: Reinforcement details for squat shear wall example.


! 51 !


2.3

Slabs and shells

2.3.1

General

Similar to that for membranes, the dimensioning of plate, slab and shell elements for stresses determined from a FE-analysis is typically based on yield conditions derived from plasticity theory. While careful consideration of the limited ductility of concrete is important in the dimensioning of membrane elements, there is generally much less concern with this aspect in slabs and shells as such structures are typically under-reinforced. That is, failure is governed by yielding of the reinforcement while the concrete does not crush. Important exceptions to this observation are concentrated transverse forces, which may result in brittle punching failures in slabs and shells without transverse reinforcement. Over the past decades, several investigations have been carried out on the load-deformation analysis of shell elements, taking into account general non-linear material behaviour (see for example Khalifa, 1986, Kirschner and Collins 1986). The resulting approaches, which typically use a subdivision of the shell elements in finite layers, are suitable for general loaddeformation analyses of single elements, however, they do not allow for a straightforward dimensioning. Thus, non-linear FE modelling is less suitable for practical design purposes where many elements must be dimensioned for different load-cases and the reinforcement quantity and arrangement is yet to be determined. The required amount of reinforcement could be minimized by aligning the reinforcement with the principal (tensile) stress directions. This type of reinforcement has been applied to numerous slab and shell structures in the past, and even today, some designers prefer to provide this so-called “trajectory reinforcement”, particularly in shells. However, reinforcement layouts following the principal stress trajectories are usually impractical, requiring a high amount of complex work on site. Moreover, if several different load-cases must be considered, principal stress directions will vary, making the provision of reinforcement aligned with them – strictly speaking – impossible. For these reasons, orthogonal reinforcement is provided in almost all slab and shell structures, and the following considerations will thus focus on this type of reinforcement. 2.3.2

Stress resultants

Integrating the stresses acting along the boundaries of a shell element, Figure 2.17 (a), one obtains the element’s stress resultants shown in Figure 2.17(b). In the general case of a plane shell element, there are eight independent stress resultants, they are, the bending and twisting moments h2

mx '

h2

h2

G + x z dz , m y ' G + y z dz , mxy ' m yx ' G , xy z dz

7h 2

7h 2

(in kNm/m = kN) (2.18)

7h 2

the transverse shear forces h2

vx '

G , zx z dz , v y ' G , zy z dz

7h 2

52

h2

(in kN/m)

(2.19)

7h 2



and the membrane forces h2

n x ' G + x dz , n y ' 7h 2

h2

G + y dz , nxy ' n yx '

7h 2

(a)

h2

G , xy dz

(in kN/m)

(2.20)

7h 2

(b) h/2 y

x

z h/2 dz

z 1

z

x nx

1 +x dz +y dz

,xy dz

,yx dz

,zx dz

y

ny

my mxy

,zy dz

nxy

nyx

vx

mx

myx

vy

Figure 2.17: Plane shell element: (a) Stresses; (b) Stress resultants in the general case of combined membrane and bending action.

The stress resultants of Eqs. 2.18 to 2.20 can be subdivided into two groups: the in-plane axial forces nx , ny , nxy (membrane action) and the out-of-plane forces bending and shear forces

H

I

Hmx , my , mxyI Hvx , vyI (bending action). For plane elements, membrane and bending action

are independent, that is, two complete sets of equilibrium conditions involving only in-plane or out-of-plane forces can be formulated. Curvature of the shell middle plane will result in a coupling of membrane and bending action, such that equilibrium conditions of a curved element will generally involve all eight stress resultants. The definition of stress resultants is also affected by curvature; in the case of thin curved shell elements a good approximation can be obtained by multyplying the integrands in Eqs. 2.18 to 2.20 by a factor of 1 7 z rx (stress resultants with last subscript y) or 17 z ry

(stress resultants with last subscript x). Thus, strictly speaking, for rx = ry one obtains that

mxy = myx and nxy = nyx (although , xy ' , yx still applies). However, for thin shells z rx J 0 and z ry J 0 and thus, Eqs. 2.18 to 2.20 can usually be applied without modification. In the following, we will denote elements subjected to pure membrane action, mx ' my ' mxy ' 0 and vx ' vy ' 0 , as membranes, and elements subjected to pure bending action, nx ' ny ' nxy ' 0 , as plates or slabs. Elements subjected to combined membrane and bending forces are called shells. While shell elements can be part of either plane or curved structures, it is evident by these definitions that membranes and slabs must essentially be plane structures. Slabs are probably the most important application of structural concrete. Thus, while this


! 53 !


chapter treats the dimensioning of shells, special attention is given to slabs, including equilibrium and boundary conditions, stress transformation and reinforcement design. For stresses and stress resultants, the sign convention illustrated in Figure 2.17 will be used throughout this chapter. Accordingly, positive stresses are acting in the direction of positive coordinates on boundaries with positive outer normal; with regard to normal stresses, this means that tensile stresses are positive. Positive membrane and transverse shear forces are those corresponding to positive stresses, and positive bending and twisting moments correspond to positive stresses, according to the above definition, for positive values of coordinate z. With regards to the double subscript indices, the first index stands for the direction of the stress, while the second denotes the normal direction of the boundary on which the stress or stress is acting. 2.3.3

Equilibrium, stress transformation and boundary conditions for slabs

Equilibrium conditions Formulating equilibrium of the stress resultants acting on a slab element, Figure 2.18, one gets

vx, x 8 vy, y 8 q ' 0

(2.21a)

mx, x 8 mxy, y 7 vx ' 0

(2.21b)

my, y 8 myx, x 7 vy ' 0

(2.21c)

where subscripts after a comma denote partial derivatives with respect to the corresponding variable. Evaluating Eqs. 2.21b-c for v x , x and vy, y and substituting into Eq. 2.21a gives the elastic plate equation

mx, xx 8 2mxy, xy 8 my, yy 8 q ' 0

(2.22)

which consists of the sum of equilibrium conditions for pure bending of unit strips in the xand y-direction plus an additional coupling term consisting of the twisting moments.

(a) !

dx

vy dx

vx dy

dy

mx5dy my5dx

myx dy

x

z

(my+my,y dy) dx

(mx+mx,xdx) dy

y

mxy dx

q dxdy

(myx+myx,xdx) dy (vx+v x,xdx) dy

(mxy+mxy,ydy) dx

(vy+v y,y dy) dx

Figure 2.18: Equilibrium conditions for slab element.

54



Transformation of bending and twisting moments Moment equilibrium of the slab element shown in Figure 2.19(a) yields mn ' mx cos 2 K 8 m y sin 2 K 8 mxy sin 2K

(2.23a)

mt ' mx sin 2 K 8 m y cos 2 K 7 mxy sin 2K

(2.23b)

$

%

mtn ' my 7 mx sinK cosK 8 mxy cos2K

(a)

mxysinK

my sinK

K

mxcosK

mn

my cosK

x

mtn

1

t

1

T

n myxsinK mxsinK

mnt

mtn X

x

K

mt n

myxcosK

(b)

mxycosK

2K

K1 K

Y

y

(c)

(d)

vn

vxcosK

x

K vxsinK

vt 1

mn

v0

n vx

t

vy v

t

v0 vy K0 L 2

y

Q (Pol)

vx K0 K vn

vn

vy cosK

n 1

t

1 N

t

x

2K1

2

y

vy sinK K

(2.23c)

L

3L 2

2L

K

y

Figure 2.19: Stress transformation: (a), (b) Bending and twisting moments; (c), (d) Transverse shear forces.

Equations 2.23a-c can be interpreted as equations for the transformation of bending and twisting moments acting on any boundary perpendicular to the direction n, where the orientation is determined by the angle K . This may be represented graphically using Mohr’s circle, Figure 2.19(b); in this case, twisting moments must be taken as positive if the corresponding positive moment arrow is pointing towards the boundary under consideration. The principal directions, for which twisting moments are zero, mtn ' 0 , are determined by the condition tan 2K1 '

2m xy mx 7 m y

(2.24)

and the principal moments m1 and m2 in the corresponding directions are


! 55 !


m1,2 '

mx 8 m y 2

M

$mx 7 m y % 2 8 4mxy2 2

(2.25)

Transformation of transverse shear forces Equilibrium of the forces acting on the slab elements shown in Figure 2.19(c) yields vn ' v x cos K 8 v y sin K vt ' 7v x sin K 8 v y cos K

(2.26)

These relationships can be interpreted as equations for the transformation of transverse shear forces acting on any boundary perpendicular to the direction n, whose orientation is determined by the angle K . The trigonometric functions can be represented graphically using Thales’ circle, Figure 2.19(d). At any point of the slab, the principal transverse shear force vo ' v x2 8 v 2y

(2.27)

is transferred in direction Ko , where

tan K o '

vy vx

(2.28)

In the direction perpendicular to Ko , no transverse shear is transferred. Apart from some special cases, the principal directions of bending moments and transverse shear forces are not coincident, Ko = K . Boundary conditions for slabs Along the boundary of a slab, there may exist, generally, a bending moment mn , a twisting moment mnt and a transverse shear force vn (Figure 2.20a). However, the fundamental equation for thin elastic slabs with small deflections (Kirchhoff, 1850) cannot satisfy more than two boundary conditions. Thus, one may introduce a further condition for simply supported and free edges as follows: replace the twisting moments mtn by a continuous distribution of vertical forces, Figure 2.20(b), which, at the boundaries of infinitesimal elements of length dt , compensate each other with the exception of the increase mtn,t dt . The increase per unit distance, mtn ,t , is now combined with the transverse shear force vn , giving a resultant edge force of vn 8 mtn,t ' mn, n 8 2mnt ,t

(2.29)

as shown in Figure 2.20(c). At a slab corner, the twisting moments mtn of the two joining edges result, according to Figure 2.20(d), in a corner force of

2mtn

56

(2.30) ! 2 Design using linear stress analysis !


(a)

(b)

(c)

dt

dt

(d) dt

dt

mtn dt

mn dt

5

5

n

(vn5+5mtn,t)5dt

mtn + mtn,t dt 5

5

5

(e)

mtn

mnt

mtn

vn5dt

t

dt

dn

(f)

(g) dt vn–5dt

mn– dt

Vt

5

t

– mtn dt

mtn

n

5

m+n dt

+ dt mtn 5

5

1 -Vt5=5mtn 25mnt

t

vn+5dt

n

Vt +Vt,t dt

Figure 2.20: Static boundary conditions and discontinuities in slabs: (a) Stress resultants at slab edge; (b) Forces corresponding to twisting moment; (c) Resultant edge orce; (d) Corner force; (e) Edge shear force; (f) Force flow at corner; (g) Discontinuity.

The treatment of twisting moments along slab edges outlined above, first proposed by Thomson and Tait (1883), can be justified by Saint Venant’s principle. However, it is not entirely satisfactory due to its foundation on Kirchhoff’s theory and the restrictive conditions for the application of the latter. Figure 2.20(e) illustrates an alternative explanation of the behaviour near slab edges, based only on equilibrium considerations. In order to satisfy equilibrium of the edge strip, there must exist a transverse shear force transferred along the edge such that, provided the slab boundary is stress-free and stresses + t within the edge strip do not vary along the direction t (Clyde, 1979)

Vt ' 7mtn

(2.31)

Note that the edge shear force (Eq. 2.31) is a force (for example in kN), while the resultant edge force (Eq. 2.29) is a force per unit width (kN/m). The condition (Eq. 2.30) for corner forces and the contribution mtn ,t of the twisting moments to the resultant edge force (Eq. 2.29) are directly obtained from the existence of the edge shear forces (Eq. 2.31). The corresponding boundary conditions can be summarised as follows: "

clamped edge: any values of mn , mtn and vn are possible

"

simply supported edge: mn ' 0 , resultant edge force vn 8 mtn,t ' mn, n 8 2mnt ,t

"

free edge: mn ' 0 , zero resultant edge force vn 8 mtn,t ' mn, n 8 2mnt ,t ' 0 .


! 57 !


These boundary conditions follow directly from equilibrium considerations and are thus valid for any material behaviour. For thin elastic slabs more restrictive conditions can be formulated (Timoshenko, 1959); however, these are not relevant if detailing and dimensioning are carried out according to plasticity theory. The edge shear forces (Eq. 2.31) must be taken into account when dimensioning free edges of concrete slabs and shells with transverse reinforcement provided capable of resisting Vt . This is illustrated using Figure 2.20(f), which visualises the force flow at the corner of a slab under pure torsion. Two sets of inclined concrete compression struts, perpendicular to each other, form at the top and bottom faces of the slab with their components normal to the slab edges. These forces are resisted by corresponding reinforcement. The components parallel to the slab edges are transferred to inclined concrete compressive struts along the edge strips. The vertical components of the struts correspond to the edge shear forces (Eq. 2.31) and transverse (vertical) reinforcement must be provided to resist these forces. This can be achieved by providing hairpins or corresponding bends in the in-plane reinforcement (providing full anchorage of the bent-back bar ends is achievable). Statical discontinuities in slabs By joining two slabs along their free edges it follows, from the equivalence of twisting moments along slab edges and edge shear forces according to Eq. 2.31, that while bending moments mn must be continuous (Figure 2.20g), twisting moments mnt and transverse shear forces vn may jump across statical discontinuities within a slab. At a statical discontinuity, 8 7 7 mnt along which a shear force Vt is transferred, the conditions mn7 ' mn8 , Vt ' mnt and

Vt ,t ' vn7 7 vn8 must be satisfied (see Figure 2.20g).

General remarks on yield conditions The ultimate load of concrete slabs can be investigated on the basis of plasticity theory by considering local stresses and strains inside the slab and corresponding yield conditions and flow rules for concrete and reinforcement. This general procedure gives correct results in all possible cases but its application is cumbersome and can only occasionally be justified. In most cases it is sufficient to express statical and kinematical conditions as well as yield conditions and flow rules in terms of generalized stresses and strains. That is, to consider the stress resultants Eq. 2.18, Eq. 2.19 and Eq. 2.20 and corresponding generalised deformations, neglecting local distributions of stresses and strains. Yield conditions and flow rules in terms of generalised stresses and strains can be derived from both the kinematic or upper bound method as well as the static or lower bound method of plasticity theory. According to the kinematic method, deformations are restricted, by means of kinematic assumptions, to a limited class of theoretically possible deformations that can be described by a finite set of kinematic parameters. In the next section, the normal moment yield criterion for slabs is derived on this basis. According to the lower bound method, statically admissible stress distributions inside an element which do not violate the yield conditions for concrete nor reinforcement are investigated. Strictly speaking, however, the relationships obtained in this way (such as for the sandwich model presented later in this chapter) are no yield conditions since, generally, no compatible mechanism can be found.

58



2.3.4

Normal moment yield criterion for slabs

Yield lines Consider the shell element given in Figure 2.21(a), with reinforcement in directions x and y, loaded by bending and twisting moments as well as membrane and transverse shear forces. Figure 2.21(b) illustrates an element of a yield line in arbitrary direction t where, N! n and O!n are the relative rotation rate of the rigid slab parts and the relative extension rate at the level of the middle plane. For this type of kinematically admissible mechanism, only the bending moments mn and the normal forces nn contribute to the energy dissipation, D ' mnN! n 8 nnO!n

(2.32)

For any given value of nn , the depth of the compressive zone c ' h 2 7 O!n N! n can be determined from equilibrium of stresses in direction n, and one obtains an expression for the ultimate moment mnu in direction n and, consequently, for the energy dissipation. (a)

(b)

1

ny my mxy

h /2

x

z

y

x

K

1 nx

nxy

nyx

vx

mn

c

n

t

n

h /2

myx

nn

.

On

mx

.

Nn

vy y

myusinK

(c) fc mxu

cx

' fy asx

fc myu

' fy asy

z

nn

y

asx fy

z

asy fy

x

K

mxucosK

x z

cy

ntn mn

t

n mtn 1

y

(d)

(e)

mxy

myu

k= 1

m'yu

mx myu

mxu

my

m'xu

|mxy| my mxu mx

|mxy|

Figure 2.21: Normal moment yield criterion: (a) Shell element; (b) Yield line; (c)Superposition of ultimate moments in directions x and y; (d) Yield condition; (e) Dimensioning.


! 59 !


Consideration of yield line mechanisms according to Figure 2.21(b), result in non-zero reactions for mtn and ntn (Figure 2.21c). These do not contribute to the energy dissipation since the corresponding kinematic parameters are zero, that is, mtn and ntn are generalized reactions within the domain of plasticity theory. By introducing a third kinematic parameter O!tn , representing a relative shear deformation along the yield line, a generalized yield line is obtained in which the membrane shear forces ntn contribute to the energy dissipation according to D ' mnN! n 8 nnO!n 8 ntnO!tn but the twisting moments mtn remain as generalized reactions. In the following, only yield lines with O!tn ' 0 are treated, and membrane forces are neglected, that is nn ' 0 . Thus, only the bending moments mn contribute to the energy dissipation. Deduction of the normal moment yield criterion By superimposing the ultimate moments mxu and myu in the reinforcement directions while setting mxy ' nx ' ny ' 0 , a statically admissible state of stress is obtained (Figure 2.21c). For an arbitrary direction n it follows that nn ' ntn ' 0 , mn ' mxu cos 2 K 8 m yu sin 2 K and

$

%

mtn ' myu 7 mxu sinK cosK (see Eq. 2.23). Generally, the depths of the compression zones in the two reinforcement directions do not coincide, that is cx = cy and there is no compatible mechanism according to Figure 2.21(b). Thus, the value of mn determined in this way is a lower bound for the ultimate moment mnu in direction n, mxu cos 2 K 8 m yu sin 2 K . The differences between mn and mnu obtained for cx = cy are generally negligible and the inequality can be suppressed, leading to the conditions mnu ' mxu cos 2 K 8 m yu sin 2 K

(2.33a)

# ' m#xu cos 2 K 8 m#yu sin 2 K mnu

(2.33b)

Equation 2.33b, which is valid for negative moments, follows from the same considerations as for Eq. 2.33a. A state of stress mx , my and mxy corresponds, according to Eq. 2.23, to bending and twisting moments in the direction n of mn ' mx cos 2 K 8 m y sin 2 K 8 mxy sin $2K %

(2.34)

and from the condition that

# ? mn ? mnu 7 mnu

(2.35)

one obtains, by considering all possible directions K , the yield conditions

60



$ % 2 Y # ' mxy 7 $m#xu 8 mx %$m#yu 8 m y % ' 0 2 7 $mxu 7 mx % m yu 7 m y ' 0 Y ' mxy

(2.36)

where mxu 7 mx < 0 , myu 7 my < 0 , m#xu 8 mx < 0 and m#yu 8 my < 0 . The conditions Y ' 0

$

%

and Y # ' 0 can be represented by two elliptical cones in the mx , my , mxy -space, as shown in Figure 2.21(d). Similarly to the yield conditions for orthogonally reinforced membrane elements, it is possible to represent the yield conditions defined by Eq. 2.36 in parametric form:

mxu < mx 8 k mxy

m yu < m y 8 k 71 mxy

m#xu < 7mx 8 k mxy

m#yu < 7m y 8 k 71 mxy

(2.37)

Equation 2.37 is suitable for straightforward dimensioning of the four reinforcement layers, Figure 2.21(d) where, typically, k = 1 is used. Skew reinforcement Yielding reinforcement layers in arbitrary, skew, directions can always be substituted by an equivalent orthogonal reinforcement if it is assumed that all reinforcing bars are located, approximately, in the same plane. The effect of several reinforcement layers oriented in directions differing by angles : i from

$

%

the x-axis and with resistances nis ' as f sy per unit width corresponds to a fictitious i reinforcement in directions x and y with resistances

nxs ' P nis cos 2 : i , i

n ys ' P nis sin 2 : i , i

nxys ' P nis sin : i cos : i ,

(2.38)

i

The resistances nxs , nys and nxys can be transformed using Eqs. 2.23a-c (setting ns instead of m) just like bending and twisting moments and there exist two perpendicular principal directions $R ,Q % , differing by an angle ( from the x- and y-axis, for which nRQs ' 0 . The ultimate moments mRu and mQu corresponding to the resistances nRs and nQs can be substituted in the yield condition Eq. 2.36 together with the bending and twisting moments mR , mQ and mRQ acting in the same direction, giving

$

%$

%

2 Y ' mRQ 7 mRu 7 mR mQu 7 mQ ' 0

(2.39)

where mRu 7 mR < 0 and mQu 7 mQ < 0 . An alternative procedure consists of transforming the ultimate moments mRu and mQu into the directions x and y using Eq. 2.23, leading to m xu ' mRu cos 2 ( 8 mQu sin 2 (


(2.40a) ! 61 !


m yu ' mRu sin 2 ( 8 mQu cos 2 (

$

(2.40b)

%

mxyu ' mQu 7 mRu sin ( cos (

(2.40c)

Comparing Eqs. 2.40a-c with the bending and twisting moments mx , my and mxy leads to

$

%

2 Y ' mxy 7 $mxu 7 mx % m yu 7 m y ' 0

(2.41)

where mxu 7 mx < 0 and myu 7 my < 0 . The conditions Y ' 0 and Y # ' 0 (which has not been given but follows from the same considerations as outlined for positive moments) can be graphically represented as elliptical cones in the mx , my , mxy -space, similar to the yield

$

%

conditions of Eq. 2.36 but with the vertices no longer lying in the plane mxy ' 0 . Assuming that the effective depth is constant in all directions, it is possible to evaluate mxu ,

myu and mxyu directly from Eq. 2.38 without having to determine the principal directions

$R ,Q % and substitute them into Eq. 2.41.

Discussion of the normal moment yield criterion

The normal moment yield criterion has been established and confirmed by various researchers (for example Johansen, 1962, Nielsen, 1984, Park and Gamble, 1980, Wolfensberger, 1964, Wood, 1961) and corroborated with a wide range of experimental data. Due to its simplicity, it is widely used for the dimensioning of slabs in design practice. While its application is generally unproblematic due to the typically ductile behaviour of slabs, it does have some weaknesses of which the designer should be aware. Supposing that sections perpendicular to the element middle plane remain straight and perpendicular to the deformed middle plane, the deformations of thin slabs with small deflections are fully determined by six kinematic parameters ) x 0 , ) y 0 , & xy0 , S x , S y , S xy . These six parameters come from elongation in x and y, distortion of middle plane, bending in directions x and y and twist (see Figure 2.22). The consideration of yield lines is kinematically more restrictive and the yield conditions given by Eqs. 2.36 and 2.37 are thus generally – in spite of the slight inequality suppressed in the deduction of Eq. 2.33 – upper bounds for the ultimate load. This observation holds particularly true for elements subjected to high twisting moments with respect to the reinforcement directions (reinforcement directions deviating significantly from the directions of the principal moments) and high reinforcement ratios (Marti, 1980, Nielsen, 1984). Special care is required in the dimensioning of elements and excessive reinforcement ratios should be avoided unless a more sophisticated analysis is undertaken, for example using to the sandwich model outlined below. Furthermore, the normal moment yield criterion does not take into account transverse shear forces which are always present in slabs. In the case of potentially shear-critical slabs (high shear forces, thick slabs or slabs with horizontal construction joints) it is advisable to use a model which allows for a consistent treatment of bending and twisting moments as well as transverse shear forces. The sandwich model approach outlined below is particularly suitable for such situations.

62



(a)

y

z

(b)

(c)

(e)

(f)

x

(d)

Figure 2.22: Kinematics of a thin slab element: (a)-(c) elongations )x0, )y0, and distortion &xy0 of element middle plane; (d)-(f) curvatures Sx, Sy and twist Sxy .

2.3.5

Sandwich model for the dimensioning of shell elements

The eight stress resultants acting on a shell element, Figure 2.23(a), can be resisted by a sandwich model. In this model the bending and twisting moments mx , my , mxy as well as

H

H

I

I

the in-plane forces nx , ny , nxy are attributed to the sandwich covers, while the sandwich

H

core resists the transverse shear forces vx , vy

I (illustrated in Figure 2.23b). Thus, the

sandwich model represents a lower bound solution with its basis in plasticity theory.

(a)

(b)

(c) y

dv z

ny my mxy

nxy

nx nyx

vx

-

my ny + dv 2

mxy nxy + dv 2

mx nx + dv 2

v0

dv

dv 1

vy

my ny + dv 2

mxy nxy + dv 2

x

z

vx

vy

mx myx

-

K0

mx nx + dv 2

dvcot-

v0cot2 v0cot-

v0

v0cot2

Figure 2.23: Sandwich model: (a) Shell element; (b) Layer forces; (c) Transfer of transverse shear force in uncracked and cracked core.

The sandwich core transfers the principal transverse shear force v0 ' v x2 8 v 2y in the

$

%

direction K 0 ' tan 71 v x v y . If the transverse shear forces are small, that is, nominal shear stresses v0 dv are below the nominal shear cracking resistance , c, red , the core will remain uncracked and the forces in the bottom and top sandwich covers follow directly as


! 63 !


m n n x inf,sup ' M x 8 x dv 2 n y inf,sup ' M n xy inf,sup ' M

my

8

dv m xy

ny

8

dv

(uncracked core, v0 d v 9 , c, red )

2

(2.42)

n xy 2

If the principal transverse shear force v0 ' v x2 8 v 2y is high enough to produce cracking of the sandwich core, the latter can be treated like the web of a girder of flanged cross-section running in direction K o (Figure 2.23c). The corresponding tensile forces in the element plane must be resisted by the sandwich covers in addition to the forces given by Eq. 2.42, resulting in the following total forces in the sandwich covers [Marti (1990)] mx nx v x2 8 8 n x inf,sup ' M 2 2v0 tan dv n y inf,sup ' M n xy inf,sup ' M

my dv

8

m xy

ny

8

dv

8

2

n xy 2

v 2y

(cracked core, v0 d v < , c, red ) (2.43)

2v0 tan 8

vxv y 2v0 tan -

According to Eqs. 2.42 and 2.43, a state of plane stress is obtained in the sandwich covers, and they can be dimensioned as membrane elements. For the case of failure governed by yielding of the reinforcement with the concrete remaining elastic in both sandwich covers (defined as Regime 1 and given by Eq. 2.7a), the force per unit width in the reinforcement in the x and y directions are determined with asx f y < asy f y <

mx nx v x2 8 8 8 2 2v0 tan dv my dv

8

ny 2

8

v 2y 2v0 tan -

k k 71

8

v x2

m n 8 a#sx f y < 7 x 8 x 8 2 2v0 tan dv a#sy f y < 7

my dv

8

ny 2

8

v 2y 2v0 tan -

mxy dv mxy dv

k# 7

8 k #71 7

8 8

m xy dv m xy dv

n xy

8

2 n xy 2 8 8

8

n xy 2 n xy 2

vxv y 2v0 tan vxv y 2v0 tan 8 8

vxv y

(2.44)

2v0 tan vxv y 2v0 tan -

where as and a#s are the bottom and top reinforcement areas per unit width of the slab, respectively. For an uncracked core, that is v0 d v 9 , c, red , the terms in Eq. 2.44 containing vx or vy can be omitted and no transverse shear reinforcement is required. That is * z ' 0 . If the core is cracked, v0 d v < , c, red , transverse reinforcement of

64



*z '

v0 tandv f y

(2.45)

is required for the core to carry transverse shear force. In Eqs. 2.42 to 2.45, dv is the effective depth of the membrane forces in the sandwich covers, k and k # are arbitrary positive factors and - is the inclination of the diagonal compression field in the sandwich core. Different values of k and k # can be adopted for each element of a shell or slab, avoiding abrupt changes or providing sufficient development length in order to anchor the differential forces in the reinforcement while in choosing the inclination angle, - . In this regard similar considerations apply as in the design of membranes. For simplicity in design practice using k ' k # ' 1 and

- ' 45o often provides for a good solution. Equation 2.44 is valid only if the concrete in the sandwich covers does not crush. In order to satisfy this condition, the thickness of the sandwich covers, and consequently the effective depth dv , should be chosen such that

$ $

%

f c tinf < a sx f y 8 a sy f y 7 n x inf 8 n y inf f c t sup < a #sx f y 8 a #sy f y 7 n x sup 8 n y sup

%

(2.46)

where tinf , tsup = thickness of bottom and top sandwich cover and f c = effective concrete compressive strength, respectively. Substituting Eqs. 2.43 and 2.44 in Eq. 2.46, we write

$

f c tinf < k 8 k 71

$

f c t sup < k 8 k

71

%

mxy dv

%7 d

mxy v

8 8

nxy 2 nxy 2

8 8

vxv y 2v0 tan vxv y

(2.47)

2v0 tan -

As a first and often sufficient approximation, dv can be chosen such that the middle planes of each sandwich cover coincides with the centre between its two reinforcement layers. The threshold value , c, red for considering cracked or uncracked behaviour of the sandwich core should be carefully selected. Basically it could be taken as the nominal shear strength of the slab or shell without transverse reinforcement, which, due to size effect, varies with the element thickness; for example , c , red ' f ctd for membrane shear according to [CEB-FIP (1990)]. Since the provision of transverse reinforcement is quite time-consuming, it is advisable to choose the element thickness such that, apart from zones where concentrated forces are introduced, no transverse reinforcement is required. In cases with high axial compression nx , ny or high in-plane shear forces nxy , it is possible to attribute part of these forces to the sandwich core and then, strictly speaking, the concrete compression in the sandwich core must be checked taking into account the combined action of nx , ny , nxy and v0 . The case of high axial compression is likely to occur if prestressing is

$

%

treated as forces acting on the structure. However, this is beyond the scope of this report.


! 65 !


2.3.6

Dimensioning of slab and shell elements in design practice

The design equations derived from the sandwich model (Eqs. 2.44 and 2.45) are simple and transparent, allowing for a straightforward dimensioning of in-plane as well as transverse reinforcement in the general case of a shell element subjected to eight independent stress resultants. In the special case of slabs and neglecting transverse shear forces, Eq. 2.44 is equivalent to the normal moment yield criterion (Eq. 2.37) if exact values are taken for dv . The incorporation of Eqs. 2.44 and 2.45 into a finite element program for shells does not any present more difficulties than the implementation of the normal moment yield criterion (Eq. 2.37) for the dimensioning of reinforcement, a post-processing option which is already offered by a number of commercial programs today (often called “Wood-Armer” moments). Usually, no explicit check of the assumed failure mode is made in slabs when applying Eq. 2.37, assuming implicitly that the reinforcement ratios obtained are small enough to avoid concrete crushing. While this condition is usually satisfied in slabs, a check of the assumed failure mode should always be carried out for shells, particularly for elements subjected to significant axial compression or cases with high reinforcement ratios. Thus, it is advisable to implement the corresponding check of concrete resistance according to Eq. 2.46 together with Eqs. 2.44 and 2.45. Transverse shear forces obtained from a finite element calculation will generally be less accurate than bending and twisting moments, since, as can be seen from Eqs. 2.21b-c, which can be rewritten as vx ' mx, x 8 mxy, y and v y ' my, y 8 myx, x , transverse shear forces are derivatives of the bending and twisting moments. Thus, in order to get reasonable results, a finer element mesh is required when using Eq. 2.44 and 2.45 in their general form than in cases where transverse shear forces can be neglected a priori. 2.3.7

Example 1

Consider the slab element shown in Figure 2.24(a); according to the normal moment yield criterion, Eq. 2.37, using a value of k = 1, the required bending resistances in the reinforcement directions are m xu < m x 8 m xy ' $96 8 133% T kN ' 229 kN J asx,inf < 1520 mm 2 m J C20 @ 200 mm m yu < m y 8 m xy ' $7 117 8 133% T kN ' 16 kN

(2.48b)

2

J asy ,inf < 100 mm m J * min m#xu < 7 m x 8 m xy ' $7 96 8 133% T kN ' 37 kN

(2.48c)

J asx, sup < 240 mm 2 m J * min m#yu < 7 m y 8 m xy ' $117 8 133% T kN ' 250 kN 2

(2.48a)

J asy , sup < 1670 mm m J C20 @ 170 mm

(2.48d)

Note that the required reinforcement areas have been determined for uniaxial bending according to CEB-FIP (1993).

66



(a)

(b)

mx ' x

z

y

ny my mxy

nx nyx

nxy

vx

myx

mx

(c)

96 kNm/m

nx ' 7 127 kN/m

m y ' 7117 kNm/m

nx ' 119 kN/m

mxy ' 133 kNm/m

nx ' 7 138 kN/m

vy

$

%

mx '

96 kNm/m

m y ' 7117 kNm/m

f yd ' 435MPa f yk ' 500 MPa

mxy ' 133 kNm/m

f cd ' 24.0 MPa $ f ck ' 50 MPa %

v x ' 358 kN/m

, c, red ' 1.87 MPa $, c, red ' f ctd %

v y ' 624 kN/m

h ' 0.40 m, tinf,sup ' 0.08 m dv ' 0.32 m Figure 2.24: Examples: (a) element and materials; (b) slab element actions; (c) shell element actions. Note that actions and strengths given are design values according to CEB-FIP (1993).

In many practical cases, one or more of the reinforcement areas resulting from the application of Eq. 2.37 will be negative. This means that no reinforcement is required in the corresponding direction. Also, in such cases it is basically possible to reduce the reinforcement area in the other direction by changing the value of k such that the reinforcement area which was negative for k = 1 equals zero or corresponds to the minimum reinforcement to be provided. However, this is rarely done in design practice. Consider next the shell element shown in Figure 2.24(b). With a principal transverse shear force of vo ' v x2 8 v 2y ' 719 kN , see Eq. 2.27, cracking of the core must be expected since

vo dv ' 2.25 MPa @ , ctd . The resulting forces in the bottom and top layers follow thus from Eq. 2.42 as nx,inf ' 7274 kN m , ny,inf ' 736 kN m , nxy,inf ' 502 kN m (bottom layer) and

nx,sup ' 7274 kN m ,

ny,sup ' 696kN m ,

nxy,sup ' 7329 kN m

(top

layer).

Substituting these membrane forces into Eqs. 2.8 and 2.9, or directly using Eq. 2.44, gives asx,inf < 1900 mm 2 m J C20 @ 150 mm

(2.49a)

asy ,inf < 1070 mm 2 m J C20 @ 250 mm

(2.49b)

asx, sup < 130 mm 2 m J * min

(2.49c)

asy , sup < 2360 mm 2 m J C20 @120 mm

(2.49d) 2

2

The transverse reinforcement resulting from Eq. 2.45 is * z ' 0.52% ' 5170 mm m , which can be provided, for example, using 26 stirrups Ø16mm per square meter (Ø16@150 x 250). Finally, concrete compression must be checked. Rearranging Eq. 2.45 gives


! 67 !


$

asx,inf f y 8 asy ,inf f y 7 nx,inf 8 n y ,inf

%

tinf

$

asx, sup f y 8 asy , sup f y 7 nx, sup 8 n y , sup t sup

%

' 12.6 MPa ? f cd

(2.50a)

' 8.2 MPa ? f cd

(2.50a)

and thus, the reinforcement dimensioning is valid. If the reinforcement of the slab element of Figure 2.24(a) is dimensioned as if it were a shell element using the sandwich model, one obtains reinforcement areas of asx,inf ' 1650,

asy,inf ' 120, asx,sup ' 270 and asy,sup ' 1800 mm2/m as opposed to that given by Eqs. 2.48a-d obtained using the normal moment yield criterion, Eq. 2.37. It is seen that reinforcement dimensioning according to the sandwich model is slightly on the safe side. The differences are due to the larger lever arms used in the calculation of the bending resistances for the normal moment yield criterion as compared with the averaged value of dv used in the sandwich model. 2.3.8

Example 2

While a comprehensive treatment of the design process and its results are beyond the scope of this report, some basic considerations will be outlined in order to illustrate the potential of the sandwich model. The example selected is the shell roof of the fish market at the Port of Santander, Spain, designed by the engineer Prof. Juan José Arenas and built 2001-2002. The roof is shown in Figure 2.25. The roof consists of 23 modules measuring approximately 5.1 x 45 m each, with a typical thickness of 0.12 m. The geometry of each module is defined by a circular arch of 30 m span complemented by two cantilevers of 7.5 m on either side in the transverse direction, and by 5.10 m wide parabolas of variable height in the longitudinal direction. Thus, globally, each module is acting as an arch, tied together by a post-tensioned, suspended floor or by posttensioned concrete ties depending on the module. Construction was carried out using two complete scaffolds, each for an entire module, starting at the two extremes of the roof. The repeated use of the scaffolds reduced the impact of their construction costs on the overall economy of the structure to a reasonable level. Due to the high live loads on the suspended floors, a reinforcement following the principal stress trajectories was not possible since these directions shift considerably depending on the load-case. Therefore, and because it was judged to be more practical, an orthogonal reinforcement layout (in plan), aligned with the longitudinal and transverse axes of the building, was adopted. The reinforcement was dimensioned using the sandwich model outlined in 2.3.6, above. In a first step, based on the different load combinations considered according to the pertinent building codes, envelopes of all eight stress resultants were computed with a conventional linear finite element program using eight-node shell elements (six degrees of freedom per node). Each of the envelopes calculated consisted of the maximum and minimum values of the stress resultant under consideration along with the concomitant values of the remaining seven stress resultants. The results obtained in this way were then analysed according to

68



Figure 2.25: Fish market Port of Santander, Spain (designer Prof. Juan José Arenas).

Eqs. 2.44 to 2.47, with dv such that the middle planes of each sandwich cover coincided with the centre between its two reinforcement layers for an assumed maximum bar diameter of 12 mm in each direction. Using Eq. 2.45, the envelopes of the transverse shear forces were analysed to determine whether or not the core would crack. In this case, as a result of the double curvature, loads are transferred primarily through in-plane actions, such that transverse shear forces remain small and the provision of transverse shear reinforcement was only required in the edge modules. Next, the required cross-sections of the four reinforcement layers were determined using Eq. 2.44 for a bar spacing of 150 mm. Finally, the assumed thickness of the sandwich covers was checked according to Eq. 2.47.


! 69 !


Each of the steps outlined above was carried out for all of the sixteen envelopes (maximum and minimum of each stress-resultant) at each of the roughly 1100 nodes per module. Thus, it was decided to implement the entire design procedure in a specially written program which was then run as a post-processor to the finite element program used in the global analysis. The post-processor output consisted of numerical and graphical results, containing a CAD file with all the data required for drafting of the reinforcement plans. Based on these results, reinforcement plans such as the one shown in Figure 2.26 were produced and constructed (Figure 2.27).

2.4

3D solid modelling

2.4.1

Introduction

As for two-dimensional modelling, linear stress analysis can be employed as an effective design tool for the design of three-dimensional structures. Three dimensional frame analysis programs, for example, are used in common practice with details given in numerous textbooks. Three-dimensional solid modelling is less commonly utilised. As for twodimensional elements and shell elements it is not always intuitive on how to dimension the reinforcing steel in 3D solids to meet the stress demands of the applied tractions. One method is to place adequate reinforcing steel in the direction of any principal tension stress and to ensure that the concrete has sufficient strength to meet all principal compressive stress demands. However, placing reinforcement in principal directions is not always convenient and placement in the local structural or global directions is preferred. The question is then how to dimension reinforcement for any set of orthogonal axes for the 6 components of any applied stress tensor that defines the stresses at a point. The answer to this question is addressed below. For further reading on design using 3D stress analysis the reader is referred to Foster and Marti (2003) where the concepts are discussed in further detail, including a general procedure for the dimensioning of reinforcement and the determination of “optimum” reinforcement. 2.4.2

Background

In 3D space the stresses at a point are completely defined by the tensor (using von Karman’s notation)

Z+ x , xy , xz W U X + ij ' X, xy + y , yz U X, xz , yz + z U V Y

(2.51)

where x, y, z are any set of orthogonal axes and the stresses are defined as shown in Figure 2.28a. For every point in a body there exist three stresses + x# , + y # and + z # on the local x# y # z # axis system such that , x#y# ' , x#z# ' , y#z# ' 0 . These stresses are known as the principal stresses and x# , y # , and z # the principal axes. It is well established that the principal stresses are equal to the eigenvalues of the stress tensor. Perhaps less well recognised, however, is that the direction cosines to the principal axes are given by the norms of the eigenvectors of the stress tensor.

70


Figure 2.26: Reinforcement details for one module of the fish market roof Port of Santander, Spain.

! !



71


Figure 2.27: Construction of the fish market roof at the Port of Santander, Spain.

H

I

For any oblique plane (refer Figure 2.28b) having a unit normal n ' nx , ny , nz passing

through a point P, the stresses at P can be resolved into a component normal to the plane $+ n %

and a shear component parallel to the plane $Sn % . For a stress to be principal S n ' 0 which, from Eq. 2.51, implies that

+y ,yz

,xy ,yz ,xz

+z

,xy

+n sn

+x

,xz

(b)

(a)

Figure 2.28: a) 3D stresses at a point defined in the orthogonal xyz axis system; b) normal and shear stress for an arbitrary plane passing through a point.

72

!

2 Design using linear stress analysis


+ x nx 8 , xy n y 8 , xz nz ' + n nx , xy nx 8 + y n y 8 , yz nz ' + n n y

(2.52)

, xz nx 8 , yz n y 8 + z nz ' + n nz Rewriting Eq. 2.52 of the form +.n = 0 it is seen that the equations are homogeneous. As all three components of n cannot be zero the solution is non-trivial only if the determinant of the coefficients + ' 0 , that is

+ x 7+ n , xy , xz , xy + y 7+ n , yz ' 0 , xz , yz + z 7+ n

(2.53)

Expansion of Eq. 2.53 leads to the characteristic equation

+ n 7 I1+ n 8 I 2+ n 7 I3 ' 0

(2.54)

where I1 , I 2 and I 3 are the invariants of the stress tensor and are given by

I1 ' + x 8 + y 8 + z ' +1 8 + 2 8 +3

(2.55a)

$ % 2 2 % ' +1+ 2+ 3 I 3 ' + x+ y+ z 8 2, xy, xz, yz 7 $+ x, 2yz 8 + y, xz 8 + z, xy

2 2 I 2 ' + x+ y 8 + x+ z 8 + y+ z 7 , xy 8 , xz 8 , 2yz ' + 1+ 2 8 + 1+ 3 8 + 2+ 3

(2.55b) (2.55c)

and where + 1 , + 2 and + 3 are the principal stresses. By common definition the principal stresses are ordered such that + 3 ? + 2 ? +1 . As for two-dimensions, stresses at a point in 3D can be plotted in the form of Mohr’s circles (shown in Figure 2.29) where the normal stress is plotted on the horizontal axis and the shear stress plotted on the vertical axis. Three principal circles are possible between the principal stress pairs + 1 7 + 2 , + 2 7 + 3 and +1 7 + 3 . In failure theorems the principal stress pair 1-3 is regarded as the most important and the circle generated through this stress pair is referred to as the major principal stress circle.

+55?5+55?5+ 3 2 1

+3

Sn

+2

+1

+n

Figure 2.29: Mohr’s circles for stresses at a point in 3D.

! fib Bulletin 45: Practitioners’ guide to finite element modelling of reinforced concrete structures

73


In xyz space + x , + y and + z are, by definition, normal to the YZ, XZ and XY planes, respectively. The shear stresses on these planes is then given by either the positive or negative roots of 2 2 2 2 S x ' , xy 8 , xz ; S y ' , xy 8 , 2yz ; S z ' , xz 8 , 2yz

(2.56)

As the orientation of the xyz axis system is arbitrary, its interpretation represents all possible planes. It can be then shown that all points of $+ n , Sn % must line on or between the principal stress circles and, thus, the feasible domain of solutions lies with the hatched region of Figure 2.29. 2.4.3

Application to reinforced concrete

In the applications that follow the xyz axes are taken to correspond with reinforcing directions. As for two dimensions, normal stresses applied at a point in a reinforced concrete solid element are carried by reinforcing steel and/or the concrete whilst shear stresses are carried by the concrete alone. Given that the applied stress tensor has been determined, for example by 3D finite element solid modelling, the Mohr’s circles of applied stress can be plotted, as shown in Figure 2.30. Within the circles the stress points $+ i , Si % are also plotted where i ' x, y, z . As the reinforcing steel cannot carry shear stress it follows that the points $+ ci , Sci % must fall within the hatched region of the concrete stress circles where

+ ci ' + i 7 *i+ s and Sci ' Si and where *i is the volumetric ratio of reinforcement in the ith direction and + s is the stress in the steel.

Sn

+555'5+557 ci i 5*5+ i s

*5+ i s

$+555, ci Sci)

+c3

+c2

+c1 +3

Concrete Stresses

$+55, i S i)

+2

+1

+n

Applied Stresses

Figure 2.30: Compression field for 3D stress at a point.

Applying Eq. 2.51 to the stresses defined in Figure 2.30, the tensor of the concrete stresses is written as

$

Z + x 7 * x f yd .x X + cij ' X , xy X , xz Y

74

%

W , xy , xz U $+ y 7 * y f yd. y % , yz U , yz $+ z 7 * z f yd.z %UV

!

(2.57)



where f yd.x , f yd. y and f yd.z are the design yield strengths of the reinforcing steels in the x y and z directions, respectively. The invariants of the concrete stress tensor are given by Eq. 2.55 with the appropriate substitutions for + x , + y and + z . Comparing Eqs. 2.57 with Figure 2.30 it is seen that there are 5 unknowns + c 2 , + c 3 , * y f yd. y , *x f yd.x and *z f yd.z (with + c1 ' 0 or another prescribed limit such as + c1 ' fct ). As the solution to Eq. 2.53 provides for a maximum of three real roots an infinity of solutions exist to Eq. 2.57. The designer then has the freedom to apply two constraint equations with the three invariant equations making up the five equations required for a solution. The design process outlined above is demonstrated in the examples below. As the three resulting stress points $+ ci , Sci % are not constrained to the boundary of the stress circles, as is the case for two dimensions, graphical solutions for three-dimensional stresses tend to be somewhat more complex than for 2D. For this reason the examples presented below are limited to analytical solutions. 2.4.4

Reinforcement dimensioning for 3D stresses - example 1

The results of a stress analysis on a concrete structural element give the stress tensor in the xyz dimensions as 6 7 4W Z2 X + ij ' X 6 7 2 2 UU MPa XY 7 4 2 5 UV

(2.58)

It is desired to reinforce the element in the orthogonal directions of xyz. For the stresses defined by Eq. 2.58, the magnitudes of the shear stresses are S x ' 2 13 [ 7.21 MPa , S y ' 2 10 [ 6.32 MPa

S z ' 2 5 [ 4.47 MPa

and

and the principal stresses are

+ 1 ' 8.28 MPa , + 2 ' 4.32 MPa and + 3 ' 77.60 MPa . The Mohr’s circle of stress for the tensor of Eq. 2.58 is shown in Figure 2.31a. Viewing the stress plot (Figure 2.31a) it is decided to seek the solution that gives the lowest demand on the concrete strength. This is established by selecting the smallest diameter for the major principal stress circle. As only the concrete carries shear stress the radius of the major stress circle $R173 % is constrained such

$

%

and thus for our example R173 ' Sx = 2 13 MPa. Therefore, for the absolute minimum compression stress in the concrete the constraint equations are given by

that R173 < max S x , S y , S z

+ c3 ' 72 S x ' 74 13 MPa , and

(2.59a)

* x f yd.x ' + x 8 S x ' 2 8 2 13 MPa

(2.59b)


75


|Sn | (MPa) $+565; x S x |)

$+565; y S y |)

$+565; z S z |)

+n (MPa) +355= -7.60

+255= 4.32

+155= 8.28

(a) Applied Stresses

*x +sx = 9.21 $+5556; cx S x |)

|Sn | (MPa)

*y +sy = 3.88 $+556; x S x|)

7.21 MPa

$+556; y S y|)

$+5556; cy S y |)

$+556; z S z |)

*z +sz = 9.21

$+5556; cz S z |)

+n (MPa)

+555 c2 = -2.88

+555= c3 -14.42

(b) Concrete Stresses Figure 2.31: Mohr’s stress circles for example 1: a) applied stresses and b) concrete stresses.

Substituting Eqs. 2.59a and b into the stress invariant equations given by Eqs. 2.55a-c we write I1 \

(2.60a)

+ c 2 7 4 13 ' 3 7 2 13 7 * y f yd . y 7 * z f yd .z

$

% $

%$

%

I 2 \ 74 13+ c 2 ' 72 13 3 7 * y f yd . y 7 * z f yd . z 7 2 8 * y f yd . y 5 7 * z f yd .z 7 56 I3 \

$

%$

%

0 ' 16 * y f yd . y 8 36 * z f yd .z 7 2 13 7 2 7 * y f yd . y 5 7 * z f yd .z 7 244 8 8 13

Solving Eqs. 2.60a-c gives + c 2 ' 72.88 MPa , * y f yd. y ' 3.88 MPa and 9.21 MPa. The final solution is plotted in Figure 2.31b. 2.4.5

(2.60b) (2.60c)

*z f yd.z '

Reinforcement dimensioning for 3D stresses - example 2

In our second example we are given the stress tensor Z 7 3 6 7 4W + ij ' XX 6 7 7 2 UU MPa XY 7 4 2 0 UV

76

!

(2.61)



and it is required to dimension the reinforcing steel. For the tensor of Eq. 2.61 the magnitude of the shears are Sx ' 7.21MPa, S y ' 6.33 MPa and Sz ' 4.47 MPa and the principal

stresses are + 1 ' 3.28 MPa , + 2 ' 70.68 MPa and + 3 ' 712.60 MPa . The Mohr’s circle of stress for the applied tractions is plotted in Figure 2.32a. After reviewing the stress circles it is decided to seek a solution such that no reinforcing steel is required in the y-direction, that is * y f yd. y ' 0 . Substituting this constraint into Eqs. 2.55a-c gives

I1 \ + 2c 8 +3c ' 7107 *x f yd.x 7 *z f yd.z

(2.62a)

I2 \ + 2c+3c ' 7358 7*x f yd.x 810*z f yd.z 8 *x f yd.x *z f yd.z

(2.62b)

I3 \ 0 ' 288 4*x f yd.x 815*z f yd.z 7 7*x f yd.x *z f yd.z

(2.62c)

$+565;S x |) x

$+565;S y |) y

|Sn | (MPa) $+565;S z |) z

+n (MPa) +255= -0.68

+355= -12.60

+155= 3.28

(a) Applied Stresses

$+565; x Sx |) $+5565; cx Sx |)

|Sn | (MPa)

*x +sx = 3.77 *z +sz = 3.77

$+5565; cy Sy |)

$+565; z Sz |)

$+5565; cz Sz |) +555 c3 = -14.57

+555 c2 = -2.98

+n (MPa)

(b) Concrete Stresses Figure 2.32: Mohr’s stress circles for example 2: a) applied stresses and b) concrete stresses.


77


Solutions to Eqs. 2.62a-c are plotted in Figures 2.33a and 2.33b for the intermediate principal concrete stress + c 2 versus the stress in the x and z reinforcement and for + c 2 versus + c 2 , respectively. From the first stress invariant it is seen that the minimum volume of reinforcement for a unit element (for f yd.x ' f yd. y ' f yd.z ' f yd ) occurs at the point where I1 is a minimum. In general, the optimal solutions lie at the upper end of + c 2 as shown in Figures 2.33a and 2.33b for the example at hand. After consideration a solution is chosen such that *x f yd.x ' *z f yd.z = 3.77 MPa, + c 2 ' 72.98 MPa and + c3 ' 714.57 MPa . The stress

circles for the chosen solution are shown in Figure 2.32b.

20

-13

*x fyd + *z fyd

-14 (MPa)

10

*z fyd

+ c3

* fyd

-20

-15

*x fyd

5

-16

-25

-17 -18

+c3

Ic1 (MPa)

(MPa)

15

-15

Ic1

-30

-19 0 -10

-9

-8

-7

-6 -5 +c2 (MPa)

-4

-3

-20 -10

-2

(a)

-9

-8

-7

-6 -5 +c2 (MPa)

-4

-3

-35 -2

(b)

Figure 2.33: Solutions to Eqs. 2.62a-c for +c2 versus a) reinforcement ratios and b) principal concrete stresses.

2.5

References

ASCE-ACI Committee 445 on Shear and Torsion, (1998), “Recent approaches to shear design of structural concrete”, Journal of Structural Engineering, ASCE, Vol. 124, No. 12, December, pp. 1375-1417. Belarbi, A., and Hsu, T.T.C. (1991), “Constitutive Laws of Reinforced Concrete in Biaxial Tension Compression”, Research Report UHCEE 91-2, Department of Civil Engineering, University of Houston, Houston, Texas. Cagley, J.R. (2001), “Changing from ACI 318-99 to ACI 318-02 What’s New?”, Concrete International, Vol 23, No. 6, June, pp. 69-183. CEB-FIP (1993), CEB-FIP Model Code 1990 for Concrete Structures. Comité EuroInternational du Béton, Bulletin d’Information No. 213/214, Lausanne, May 1993, 437 pp. Clark, L.A. (1976), “The Provision of Tension and Compression Reinforcement to Resist In-Plane Forces”, Magazine of Concrete research, Vol. 28, No. 94, March, pp. 3-12. Clyde, D.H. (1979), “Nodal Forces as Real Forces,” Final Report, IABSE Colloquium on Plasticity in Reinforced Concrete, Copenhagen, International Association for Bridge and Structural Engineering, IABSE Vol. 29, pp. 159-166.

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Clyde, D.H., (1977), “A General Theory for Reinforced Concrete Elements”, Australasian Conference on the Mechanics of Structures and Materials, Christchurch, New Zealand, August. Collins, M.P., and Mitchell, D., (1986), “Rational Approach to shear Design - The 1984 Canadian code Provisions”, ACI Structural Journal, Vol. 83, No. 6, Nov-Dec, 925-933. CSA84 (1984), “Design of Concrete Structures for Buildings”, CAN3-A23.3-M84, Canadian Standards Association, Rexdale, Onterio, 281 pp. Foster, S.J., and Malik, A.R., (2002), “Evaluation of Efficiency Factor Models used in Strut and Tie Modelling of Non-Flexural Members”, ASCE, Journal of Structural Engineering, Vol. 128, No. 5, May, pp. 569-577. Foster, S.J., Marti, P. and Mojsilovi!, N. (2003), “Design of Reinforced Concrete Solids Using Stress Analysis”, ACI Structural Journal, V100, N6, Nov-Dec, pp. 758-764. Johansen, K.W. (1962), Yield Line Theory, Cement and Concrete Association, London, 181 pp. Kaufmann, W., and Marti, P. (1998), “Structural Concrete: Cracked Membrane Model”, Journal of Structural Engineering, ASCE, Vol. 124, No. 12, pp. 1467-1475. Khalifa, J. (1986), “Limit Analysis and Design of Reinforced Concrete Shell Elements,” Ph.D. Thesis, University of Toronto, Toronto, 314 pp. Kirchhoff, G. R. (1850), “Über das Gleichgewicht und die Bewegung einer elastischen Scheibe,” A. L. Crelle’s Journal für die reine und angewandte Mathematik, Berlin, Vol. 40, No. 1, pp. 51-58. Kirschner, U., and Collins, M.P. (1986), “Investigating the Behaviour of Reinforced Concrete Shell Elements,” University of Toronto, Department of Civil Engineering, Publication No. 86-09, Toronto, 210 pp. Leitz, H., (1923) “Eisenbewehrte Platten bei allgemeinem Biegungszustande, Die Bautechnik, Vol. 1, pp.155-157, 163-167. MacGregor, J.G., (1997) “Reinforced Concrete - Mechanics and Design”, 3rd Edition, Prentice Hall, New Jersey. Marti, P. (1979), “Plastic Analysis of Reinforced Concrete Shear Walls”, Plasticity in Reinforced Concrete, IABSE Colloquium, Kopenhagen, 21-23 May, Introductory Report, Reports of the Working Commission, Vol. 28, pp. 51-69. Marti, P. (1980), "Zur Plastischen Berechnung von Stahlbeton”, Institut für Baustatik und Konstruktion, ETH, Zürich, Bericht Nr. 104, Birkhäuser Verlag Basel, 176 pp. Marti, P. (1980), “Zur Plastischen Berechnung von Stahlbeton,” Institut für Baustatik und Konstruktion, ETH Zürich, IBK Bericht Nr. 104, 176 pp. Marti, P. (1990), “Design of Concrete Slabs for Transverse Shear,” ACI Structural Journal, Vol. 87, No. 2, pp. 180-190.


79


Mitchell, D., and Collins, M.P. (1974), “Diagonal Compression Field Theory – A Rational Model for Structural Concrete in Pure Torsion”, ACI Journal, Vol. 71, August, pp. 396-408. Miyakawa, T., Kawakami, T., and Maekawa, K. (1987), Nonlinear Behavior of Cracked Reinforced Concrete Plate Element Under Uniaxial Compression”, Proceedings of the JSCE No. 378, August, pp. 249-258. Morley, C.T. (1979), “Yield Criteria for Elements of Reinforced Concrete Slabs”, Plasticity in Reinforced Concrete, IABSE Colloquium, Kopenhagen, 21-23 May, Introductory Report, Reports of the Working Commission, Vol. 28, pp. 35-47. Müller, P., (1978), “Plastische Berechnung von Stahlbetonscheiben und Balken”, Dissertation Nr. 83, Birkäuser Verlag Basel und Stuttgart. Nielsen, M.P. (1963), “Yield Conditions for Reinforced Concrete Shells in the Membrane State”, Non-Classical Shell Problems, Procedings of the I.A.S.S. Symposium, Warsaw, 2-5 September, pp.1030-1040. Nielsen, M.P. (1971), “On the Strength of Reinforced Concrete Discs”, Acta Polytechnica Scandinavica, Civil Engineering and Building Construction Series No. 70, Copenhagen, 261 pp. Nielsen, M.P. (1984), Limit Analysis and Concrete Plasticity, Prentice-Hall, Englewood Cliffs, 420 pp. Nielsen, M.P., (1999), “Limit Analysis and Concrete Plasticity”, 2nd Edition, CRC Press LLC, 908 pp. Nielsen, N.J., (1920), Beregning af Spaendinger i Plader (Calculation of Stresses in Slabs), Copenhagen. Pang, X.B., and Hsu, T.T.C. (1992), “Constitutive Laws of Reinforced Concrete in Shear”, Research report UHCEE92-1, Department of Civil Engineering, University of Houston, Houston, Texas, 188 pp. Park, R., and Gamble, W.L. (1980), Reinforced Concrete Slabs, John Wiley & Sons, New York, 618 pp. Robinson, J.R., and Demorieux J-M. (1977), “Essais de Modeles d’ame de Poutre en Double Tè, Annales de l’Institut Technique du Bâtiment et des Traveaux Publics, No. 354, October, Serie: Beton No. 172, pp. 77-95. Thomson, W., and Tait, P.G. (1883), “Treatise on Natural Philosophy,” Vol. 1, Part 2, Cambridge University Press. Timoshenko, S.P., and Woinowsky-Krieger, S. (1959), Theory of Plates and Shells, Mc Graw-Hill, International Student Edition, 580 pp. Vecchio F.J., and Collins M.P (1986), “The Modified Compression Field Theory for Reinforced Concrete Elements Subjected to Shear”, ACI Journal Proceedings, Vol. 83, No. 22, Mar.-Apr., 219-231.

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Vecchio, F., and Collins, M.P. (1982), “The response of Reinforced Concrete to In-plane Shear and Normal Stresses”, The Department of Civil Engineering, University of Toronto, Canada, March, 332 pp. Wolfensberger, R. (1964), “Traglast und optimale Bemessung von Platten,” Institut für Baustatik und Konstruktion, ETH Zürich, IBK Bericht Nr. 2, 119 pp. Wood, R.H. (1961), Plastic and Elastic Design of Slabs and Plates, Thames and Hudson, London, 344 pp.


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3

Essential nonlinear modelling concepts

3.1

Introduction

In Chapter 2 it was demonstrated that, for new structures, linear elastic finite element analysis provides a practical way of dimensioning structures, or structural elements, for determination of strength demands and reinforcement arrangements. These designs will often require internal stress redistribution after cracking or yielding and reinforcement detailing needs to allow for this. In general, a design based on linear elastic analysis will provide safe solutions to the problem of satisfying the strength limit state. For some design and analysis problems, however, a linear analysis may not be sufficient. As a simple example, consider the requirement of satisfying a serviceability limit states such as calculating deflections and crack widths. For such a design-check, the local extent of cracking is important and a linear analysis may not be sufficient to confirm that this limit state has been satisfied. While it is practical to design structures with linear analyses, it is sometimes helpful to check additional limit states by nonlinear analysis methods. For new structures a non-linear analysis may be performed on the structure (or its elements) after initial proportioning using a plasticity-based design procedure based on a linear elastic analysis (Chapter 2). In addition, nonlinear analysis can also assist in the evaluation of complex geometry or poorly detailed structures where the effects of localized cracking, for example, may be poorly modelled by linear analysis. The following list provides some general cases where a nonlinear analysis may prove useful: 1) Confirmation of safety for complex design details: Design codes offer methods to check safety for standard members such as beams and columns but are often not sufficient for more complex design geometries such as complicated walls systems with openings. While the inclusion of strut-and-tie methods into modern design codes has assisted greatly in expanding the generality of design codes, nonlinear analysis can be helpful to allow an independent check of whether the development of cracking, for example, will be consistent with the details of the reinforcement placement. 2) Assessment of safety of existing structures: Some structures, such as those built to older design standards, may contain reinforcement quantities and details that are not consistent with modern standards. In such situations there may be very significant costs associated with strength upgrades of the structure and a nonlinear analysis can assist in providing a better estimate of the safety factor of the as-built structure against collapse. This can be particularly important in seismic regions where design earthquake forces have tended to increase in severity through the years as knowledge of seismic hazards have improved. 3) Pushover analyses for structural capacity computation: Many codes of practice require a pushover analysis of structural strength to estimate failure modes and to determine if the collapse mode conforms to capacity design principles. That is, it is important to be able to predict how a structure is expected to collapse so that it may be designed to do so in a ductile mode such as flexure, rather than a brittle mode such as shear. 4) Explanation of observed distress in structures: As mentioned in Chapter 1, structural distress seen in the field may be best investigated with the assistance of nonlinear structural analysis. In particular, the results of nonlinear methods capable of estimating critical events and the extent and patterns of cracking at the actual applied loads can be directly compared to the structure and used to confirm the assessed mode of structural


83


behaviour. This allows a higher level of confidence to be placed in the calculated safety factor than that obtained from a “blind” analysis. 5) Estimation of P-O effects: The influence of geometric nonlinearity, in the form of the P-O effect, can be important in the moment and axial load design of frames. As concrete material science continues to improve, it can be expected that columns of higher slenderness will become more common and these will be more sensitive to the effect. The combination of material nonlinearity, in the form of cracking, and geometric nonlinearity, is a complex phenomenon that can require nonlinear finite element analysis. 6) Resistance of structures to extreme events: Design against events such as earthquakes or sabotage must include nonlinear effects to be realistic. Ignoring the ability of structures to redistribute stresses can result in cost prohibitive, overly conservative solutions. 7) Resistance to fire: Structures that must be designed to resist significant fire can show major effects resulting from the thermal expansion of parts of the structure that may not be captured by linear analysis. Considering the effects of fire and structural collapse is a nonlinear problem that requires complex analysis. As reinforced concrete is a nonlinear material, the salient nonlinear effects that concrete shows are described next. Afterwards, the various frameworks with which these nonlinear effects can be incorporated into finite element solutions are described followed by a short discussion concerning the expectation of precision in concrete nonlinear analyses. Finally, a short discussion of safety reliability calculations is presented. In each case, the description of behaviour or implementation is not written to be exhaustive but simply to demonstrate to the reader some of the more important aspects that are required for the engineer to use nonlinear finite element tools intelligently.

3.2

Nonlinear concrete behaviour

Structural concrete is an inherently nonlinear material both at strength limit states and service loads. Table 3.1 summarizes some of the important nonlinear effects observed in structural concrete. These are organized in terms of effects that are primarily observed in concrete, steel or a combination of the two. While not all nonlinear finite element analyses will need to consider all the listed effects, it is important that engineers are aware of the different nonlinear phenomena so that they can judge whether or not a particular set of analysis assumptions are appropriate. When considering nonlinear behaviour, an important aspect is whether or not strain localization will occur as “size effects” are expected in such cases (see Table 3.1). Strain localizations occur when a material stress-strain behaviour shows a decreasing stress for an increasing strain such as concrete in tension after cracking or in compression after reaching its peak compressive capacity. Nonlinear effects with localization can show a size effect whereby larger elements fail at lower stresses than geometrically similar smaller ones. In addition, material behaviour showing significant localization generally cannot be easily defined in terms of strain but rather must be defined in terms of an absolute displacement. Thus the direct tensile, or cohesive, stress that can be transmitted across a crack is a function of the width of the crack and not on a strain term. This effect of localization is of particular importance to finite element analysis as the mesh size itself introduces a fixed length scale that can bias the results of localization sensitive problems. As an example, an analysis on

84

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3 Essential nonlinear modelling concepts


Table 3.1: An overview of nonlinear behaviour in reinforced concrete structures Types of Behavioural Effects

Tension

Compression

Macrocracking Tension softening * Cyclic response Creep Crack closing effect Shrinkage

X X

Crushing Nonlinearity at high strains Post-peak unloading Cyclic response Creep Rate of loading Bi or triaxial confinement Poisson's ratio Thermal effects

X X

X

X X X X X X

Examples of behaviour

X X X X X X

X X X X X X X X X

Size dependent

Plastic strains

Bifurcation

Plain Concrete Behaviour

Stiffness

Energy balance may need to be considered to fully model listed effect Elements must include variable stiffnesses to fully model the listed effect Indicates a discrete change in behaviour rather than a smooth transition Indicates that simple nonlinear elasticity may not provide a full solution Behaviour cannot be fully modelled with strain terms alone. Modelling also requires an absolute distance relationship such as a crack width

Energy

Energy : Stiffness: Bifurcation: Plastic strains: Size dependent:

X X X

X X X X

(Reinhardt and Xu, 1999)

X

X X (Sakai and Kawashima, 2006)

Reinforcement Behaviour Tension

Compression Shear

Yielding Strain hardening Thermal effects Rate of loading Rupture Buckling Dowel action

X

X

X X X X X

X X X

X X X

X X (Ozcebe, Ersoy, Tankut, 1999)

Combined Concrete & Reinforcement Behaviour Tension

Compression Shear

Bond Tension stiffening * Tension splitting Compression Softening Aggregate interlock

X

X X X X X

X X X

X X

Damage Effects

(Harajli, 2004)

Material damage Fatigue

X

X

e.g. alkali silica reaction

X

* Tension softening is distinct from tension stiffening behaviour. See text sections 3.2.2, 3.2.3.


85


plain concrete in tension will produce different load-deformation curves after cracking if meshes with different element sizes are used and special measures are not taken (Bažant and Oh, 1983, Bažant and Planas, 1998). While plain concrete can be subjected to localization effects under increasing deformations, these effects are often not of significant importance in structures reinforced with a minimum quantity of ductile reinforcement. With such structures, the ductile properties of the reinforcing steel dominate the global behaviour. Examples of behaviour where localization effects can have a significant influence on behaviour include punching shear, shear in members without transverse reinforcement, over-reinforced columns and beams and large unreinforced concrete structures such as dams. In each of these cases the behaviour at failure is more controlled by that of plain concrete than that of the reinforcement. In using Table 3.1, and nonlinear finite element procedures in general, engineers must check their finite element discretization and modelling assumptions used in their analyses. Note that many of the attributes in the table are contentious in the research community and, as such, are open to debate. Responsible engineers must use experience, judgment and intuition as to whether, for example, localization issues will be important. While it may seem that nonlinear finite element analyses could allow non-engineers to produce reasonable and accurate results, this is not the case; the experience and judgment of engineers is more important in nonlinear finite element analysis than it is with linear analysis due to the significantly higher complexity level. The following sections expand on a subset of the important behaviour in Table 3.1 explaining the particularly important experimentally observed characteristic nonlinear concrete behaviours. 3.2.1

Concrete in compression

Concrete subjected to compression acts in a nonlinear way as has been recognized for many years (e.g. Turneaure and Maurer, 1908). Three important aspects of concrete compressive behaviour are described in more detail here: localization in compression, confinement of concrete, and compression softening. Influence of localization on compression behaviour of concrete

As concrete shows a decrease in stress for increasing strains beyond the strain associated with the compressive strength, localization of concrete in compression can be expected. That is, a damage band forms where all the deformation associated with the post-peak region occurs while the remainder of the specimen unloads with decreasing strain. Two primary effects will result from this; firstly larger specimens failing primarily in a compressive mode can be expected to show less ductility in the post-peak region than smaller specimens and, secondly, a size effect is predicted for some member types whereby larger elements in compression will be weaker in terms of stress than smaller elements. Both of these effects may be of particular importance in practice as most laboratory testing is done on small-scale models and, thus, may not be representative of the real safety of large structures in the field. It is important to note, however, that there remains much debate on the importance of localization effects for practical structures that contain realistic quantities of reinforcement even within the laboratory testing field. For example, some tests on specialized structural components confirm the presence of a size effect in unreinforced flexural compression zones (Kim, et al. 2000). In contrast, other carefully performed tests on actual laboratory beams 86

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subjected to constant moment have not always shown the decrease in ductility expected with size (Alca et al. 1997). Confinement and compression softening

It is well known that concrete can carry higher compressive stresses with larger deformations when it is laterally confined as illustrated in Figures 3.1 and 3.2. Finite element models can require either biaxial or triaxial confinement relationships. Early experimental investigations on biaxially loaded RC elements focused on the strength. Kupfer et al. (1969) reported on the strength, deformational characteristics and micro-cracking of concrete under biaxial stress conditions. Figure 3.1 illustrates the stress-strain relationships under biaxial compression. As can be seen in both Figure 3.1a and Figure 3.1b, the maximum compressive strength of concrete increases for the biaxial compression state. Kupfer et al.’s (1969) results suggest that a maximum strength increase of about 25% is achieved at a stress-ratio of +2/5+1 = 0.5. In addition, concrete ductility increases under biaxial compression.

(a) Figure 3.1:

(b)

Concrete in biaxial compression: (a) stress versus strain; and (b) strength envelope (Kupfer et al., 1969)

It is interesting to note that during the latter stages of the loading, an increase in overall volume occurs (dilatancy); this is typically attributed to the progressive growth of micro cracking in concrete. For triaxial compression, Richart et al. (1928) conducted tests at low and moderate confining stresses with typical results shown in Figure 3.2a. Balmer (1949) conducted triaxial tests at very high confining stress levels (Figure 3.2b). As can be observed from the figures, depending on the confining stress, concrete may act as a quasi-brittle, plastic-softening or plastic hardening material. This is mainly due to the fact that under higher confining stresses, the possibility of bond failure between cement paste and aggregates is significantly reduced, and the failure takes place through the crushing of the cement paste. As can also be seen, pronounced increases in strength with increasing confining pressure were observed by both Richart et al. and Balmer. In fact, experiments show that concrete has a fairly consistent failure surface that is a function of the principal stresses under triaxial loading.


87


(a) Figure 3.2:

(b)

Triaxial compression stress versus axial strain: (a) low confinement (Richart et al., 1928); and (b) high confinement (Balmer, 1949).

One of the earlier 3D failure surface models that incorporated this strength enhancement was that of Ottosen (1977); shown in Figure 3.3. While there are many models that have been published, the important point is not which of the models is used but whether or not the model produces a fair representation of the failure surface. Many commercial packages will include, for example, a Mohr-Coulomb failure model. While the Mohr-Coulomb failure criteria can be made to represent concrete strength at very high levels of confinement, as at these levels the failure surface for the concrete approaches circular, the model does not reproduce well the failure surface for concrete under the levels of confining pressure typical of that in common reinforced concrete structures. If the Mohr-Coulomb approach is to be used it must be tuned for the problem being investigated. The better software packages for concrete analysis will include a specific failure model for concrete but even then some care is needed as some failure surface models require different calibration for biaxial compression than for other triaxial compressive stress conditions.

Figure 3.3:

88

Ottosen (1977) failure surface for triaxial (+1, +2 +3) stresses (plotted in the Haigh-Westergaard coordinate space).

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Many different stress-strain models exist to account for either the biaxial or triaxial behaviour of concrete in compression. These depend on many parameters and are generally specific to continuum modelling of triaxially loaded materials or are specific to the behaviour of columns, for example. One model which is often used in non-linear elasticity is the Kent and Park (1971) model later modified by Scott et al. (1982) to include the strength and ductility enhancement due to confinement effects and the effect of strain rate (Figure 3.4). Such models, however, ignore size and compression localization effects and can lead to erroneous (but usually conservative) results when considering post peak behaviour. Such models are usually sufficient, however, in adequately determining the response of a member for increasing loading up to the maximum load. Nevertheless, as for all non-linear modelling, the results should be scrutinized with due caution. In analyses of cracked concrete in compression, confinement may often be ignored with little penalty as the confining stress is almost completely released normal to the cracks.

Kfc#

)50h

0.5Kfc# 0.2Kfc#

Unconfined concrete ) 0=0.002 ) 50u )50c

Confined concrete

) 20c

)c

Figure 3.4: Modified Kent and Park (1971) model (Scott et al., 1982).

A related but not identical concept to confinement is that of weakening and softening in compression due to the effects of transverse tensile strains beyond cracking (Collins 1978). Figure 3.5 shows the results from many shear tests as reported by Vecchio and Collins (1993) to quantify this effect. The specimens in this figure were subjected to combinations of shear and axial loads and were reinforced in such a way that they were able to resist large tensile strains transverse to the direction of the applied compression. As can be seen in the figure, some of these specimens failed by crushing of concrete in compression at only 20-40% of the strength of concrete control specimens. As strains perpendicular to the direction of applied compression increase, this strength reduction effect, often called compression softening, becomes more severe. This effect can be thought of as an extension of the confinement effect into the tensile domain and is particularly important in understanding of the behaviour of concrete in shear. 3.2.2

Concrete in tension

The one type of nonlinearity that is expected in all concrete structures is cracking. This phenomenon is difficult to account for in a simple fashion, yet is vital in making realistic estimates of structural stiffness.


89


Figure 3.5:

Peak principal compressive stresses versus the principal tensile strain ratios (Vecchio and Collins, 1993)

When plain concrete cracks, it is able to resist residual, direct, tensile stresses across the crack for small crack widths. These stresses, sometimes called bridging stresses or cohesive stresses are sufficiently small that they are often ignored in traditional beam design. The decrease in these bridging stresses as the crack width increases is called tension softening and models based on the energy of fracture (Gf) are usually used to quantify the effect. Overall the behaviour of cracking in plain concrete is reasonably well modelled by methods such as those based on fracture mechanics. In concrete that is reinforced, the process of cracking is more complex. Shown in Figure 3.6 are the multiple cracks that form in reinforced concrete around the bars (bond cracking) as well as macro-cracking that extends to the surface of the member (Goto 1971). Each of the cracks shown in Figure 3.6 can be expected to behave consistently with the tension softening behaviour mentioned above for plain concrete. When the effects of all these cracks are integrated together, including the bond cracks, and the entire concrete section is considered as a whole, behaviour consistent with that shown in Figure 3.7 is obtained and the concrete component is termed tension stiffening. This tension stiffening is the difference in behaviour between the response of a bare-bar and the observed response of the composite reinforced specimen. This tension stiffening can be thought of as a macroscopic property applied on a large-scale compared to tension softening which is a property on a smaller-scale. As Figure 3.8 shows, an analysis with very small elements that considers the bond cracking and a tension-softening model does produce the same macroscopic tension stiffening behaviour as that observed in experiments. Macroscopic tension stiffening behaviour can be considered as being caused by an average tensile stress in the concrete between the cracks. Figure 3.9 shows a plot of the stresses in the reinforcement and in the concrete for a tensile element like that in Figure 3.8. It shows the average concrete tension stiffening stresses varying along the member length are almost zero at the location of the macro crack and are greater than zero between. The lower plot shows the reinforcement stresses varying from a maximum at the crack, where there is no concrete tension, to a minimum between the cracks. This is a helpful way to think of macroscopic tensile stresses in concrete as it makes it clear that reinforcement will yield at a crack before it does between the cracks and also that equilibrium at a crack itself must be able to be

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Figure 3.6: Internal cracking in reinforced concrete.

Tensile force

Y

Member response c

Tensile force

Bare bar

R

b

a b c d

S

a

Reinforcement (unembedded)

Average strain

Uncracked Crack formation Stabilized cracking Post-yielding

R First crack S Last crack Y Yielding

Elongation

(a) Figure 3.7:

d

(b)

Schematic of: (a) the tension stiffening effect; (b) the stress- strain relationship of embedded reinforcing bar by the CEB-FIP Model Code 1990 (1993).

& 6-1)7-8+/01-//+49+ 31)3:-8+34.31-0-

%"# %

Macro -tension stiffness model $"# $ !"# !

Plain concrete! softening

Computed from micro analysis #!!

$!!!

$#!!

%!!!

& '()*+, -).+/01)(.+2, (3145 bond crack

penetrating crack bond deterioration zone

Figure 3.8: Modelling of tension stiffening with tension softening


91


maintained ignoring all concrete stresses beyond those expected by tension softening. This equilibrium criterion is why the macroscopic concrete stresses are called tension stiffening and not tension strengthening: they cannot increase the strength of the tension specimen in Figure 3.9, only increase its stiffness before yield at a crack.

Figure 3.9: Average stresses in tension stiffening analysis

Considering the differences noted in the previous section between tension stiffening and tension softening, it is perhaps not surprising that there is more than one way to include tension in finite element analyses of concrete structures. It is helpful to divide up the level of analysis complexity into the following three levels: Model Small-scale Medium-scale Large-scale

concrete tension softening stiffening / softening stiffening

reinforcing bond perfect bond bond law perfect bond

Element size very small medium large

With small-scale tension modelling, as shown in Figure 3.8, it is necessary to use extremely small elements to capture the extensive cracking that occurs due to bond and slip effects and the analysis should be fully three-dimensional. These models may use the assumption of “perfect bond” whereby the nodal displacements of the concrete adjacent to the reinforcement are assumed to be the same as that of the reinforcement itself. Due to the number of elements, this type of modelling is rarely practical for realistically sized structures at the present time. With medium-scale tension modelling, the bond forces are accounted for with a macroscopic bond model that provides a relationship between the relative displacements of the concrete compared to the reinforcement. While this simplification means that more nodal degrees of freedom are required, it has the advantage that much larger elements are allowable and twodimensional analysis becomes possible. Larger elements are possible as it is no longer necessary to be able to resolve the extensive bond cracking that the small-scale tension modelling requires. If a concrete tension softening relationship is used, however, it will still be necessary to use small elements to capture the change in concrete stresses between the cracks. If, instead, a concrete tension stiffening effect is included, then still larger elements may be used, though with the cost that special routines may be needed (depending on the formulation) to ensure that the stress in the reinforcement at a crack does not exceed the yield stress.

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Large-scale tension modelling is perhaps the most practical method of including concrete tension in nonlinear analyses currently. Perfect bond between the concrete and reinforcement again becomes appropriate and much larger elements may be used in two or three dimensions. Care is required to ensure that the tension stiffening relationship used is consistent with the bond properties of the reinforcement and that local conditions at a crack are checked for equilibrium. Due to the use of perfect bond, effects caused by insufficient development of reinforcement must also be explicitly considered. A reasonable way to account for this effect with perfect bond programs is to progressively increase the cross sectional area of the reinforcement over the development length of the bar to properly model the total force that can be transferred to the concrete at every location. Finally, again, it is necessary with all analyses based on tension stiffening to ensure that the stresses in the reinforcement at cracks are not allowed to exceed the yield stress with some sort of “crack check”. For further discussion, see Vecchio and Collins (1986), Petrangeli and Ožbolt (1996), Maekawa and An (2000) and Foster and Marti (2002, 2003). 3.2.3

Modelling of tension stiffening

Modelling of tension stiffening can be undertaken in one of two ways: the first is to modify the stiffness of reinforcing bars; the second is to modify the concrete stiffness to carry the tension force after generation of cracks. In the first model, the stress transfer carried by both steel and concrete due to bond action can be expressed by changing the stiffness of reinforcing bars, as per Figure 3.7b (Gilbert and Warner, 1978). The CEB-FIP Model Code (1993) gives the following phases: uncracked concrete, crack formation, stabilized cracking where only crack opening occurs and post-yielding. While such a treatment has an advantage in terms of simple 1D calculation of response, this is not generally applicable to 2D or 3D problems since the stiffness is unchanged even if the direction of cracks relative to the axis of the bar changes (Okamura and Maekawa, 1991). The second approach of modifying the concrete properties can be sub-classified as i) methods based on a macroscopic smeared crack model, which describes spatially averaged behaviour of RC containing multiple cracks in a defined control volume, or ii) methods based on mesoscopic discrete crack models. Tension stiffening models based on smeared crack concepts use a spatially averaged constitutive relationship, which involves multiple cracks and non-uniform local stress between the cracks in a control volume. Figure 3.10 gives three examples of models representing average stress-strain relationships of concrete in tension. Since this type of modelling is generally assumed independent of the spacing of cracks, direction of reinforcing bars and reinforcement ratio, the stabilization and distribution of propagating cracks is assured in a control volume. For this reason, it is one of the most versatile and applicable constitutive relationships in executing structural analyses in cracked reinforced concrete. When the stress-strain relationship of concrete is introduced in an averaged manner, the relationship of reinforcing bars embedded in concrete should also be determined on an average basis. Here, it has to be noted that the average stress-strain relationship of the reinforcement is different from the corresponding relationship of bare bar after yielding (Shima et al., 1987), since the local stress and strain of reinforcing bars in concrete is disturbed due to the effect of bond. An example of one method of calculating an equivalent average stress for a given average strain is presented in Figure 3.11 (Maekawa and An, 2000). If steel remains elastic, the constitutive model can be set equal to that for the bare bars. However, as soon as the bar yields at a crack, the elastic relationship cannot be maintained


93


Figure 3.10: Examples of stress-strain models of concrete in tension (from Bentz 2005).

+'

1 G +( x, y )dv VV

+'

Local stress

1 G +( x, y )dv VV

Local stress

Local strain

)'

Stress distribution

Stress distribution

Local strain

1 )(x,y)dv V

G v

Yield plateau jump!

)'

1 )(x,y)dv V

G v

Strain distribution

Strain distribution

(b)

(a) +'

1 G +( x, y )dv VV

+'

Local stress

1 G +( x, y )dv VV

Local stress

Local strain

Strain distribution

)'

Strain distribution

Stress distribution

Stress distribution

Local strain

1 )(x, y)dv V

G

Plastic zone

v

)'

Plastic zone

1 )(x, y)dv V

G v

(c)

(d)

Figure 3.11: Average stress-strain relationship of a steel bar in concrete: a) at crack initiation; b) at beginning of yield; c)developing yield; d) full yield (after Maekawa and An, 2000).

94

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even though the parts of the bar away from the cracks are still linear-elastic (Figure 3.11a). As the strain increases, the yielded portion of the bar is extended (Figure 3.11b and c). Then,the average yield stress becomes lower than that of bare bar. For accurate prediction of deformational behaviour of RC members after yielding, the constitutive relationship capable of considering this phenomena is needed. Usually a bilinear or tri-linear relationship is assumed for the average response of reinforcing bars. For normal concrete combined with two-way reinforcement with a ratio of 0.1 to 2 percent in each direction, the constitutive law based on a smeared crack approach is independent of the control volume (Maekawa et al., 2003). In other words an average stress can be evaluated uniquely for an average strain history regardless of density of macroscopic cracks within the control volume and, therefore, there is no size effect in the constitutive law for reinforced concrete in a uniform strain field. Similarly it has been shown that mean stress-strain relationships of reinforcement in RC plates are also not influenced by the number of cracks within an element, including in the post-yielding regime. This observation is convenient for many structural analyses as it is not necessary to define individual macroscopic cracks within the control volume for members that are provided with minimum, or greater, reinforcement. This is not the case, however, where minimum reinforcement is not provided and cracks pass through an unreinforced (or under-reinforced) concrete section. The case where the distributed cracking assumption is not appropriate and cracks are localized is discussed in detail in the following section. A relatively new approach to the modelling of tension stiffening has been developed by Marti et al. (1998) known as the “tension chord model” (Figure 3.12). The significance of this approach is that an equivalent plastic bond stress-slip model (Figure 3.12c) is explicitly included in the formulation allowing for calculations of crack spacing, crack widths and tension stiffening directly from the model. The model was further developed into a 2D formulation (the cracked membrane model) by Kaufmann and Marti (1998), shown in Figure 3.13, and included in a FE model by Foster and Marti (2002, 2003). With adoption of the stepped rigid-perfectly plastic bond-slip relationship (Figure 3.12c), the stresses in the steel and in the concrete can be determined for any point within the differential element between cracks (Figure 3.12b). The stresses in the concrete and in the reinforcing steel at and between cracks are then calculated directly from equilibrium. (a)

srm ]

(b)

N

N

Ac

x (c)

dx dx

,b

+c+ d+c

, b0

,b

, b1 Oy

+c +s+ d+s

,b

+s

O

Figure 3.12: Tension chord model: a) reinforced tension chord; b) differential element; c) bond shear stressslip relationship (Marti et al., 1998).


95


(b) Y

^555f y ct

srmy

-r srmx

^555f x ct

X

srmx

srmx

s rmy s rmy s rmy

sr m

sr

cft

m

sr

m

(a)

Tension Stiffening Stresses

Figure 3.13: 2D tension stiffening stresses in: a) material axis directions; b) equivalent orthogonal tension chords (Kaufmann and Marti, 1998, Foster and Marti, 2003).

If a discrete modelling approach is preferred over that of a smeared crack approach, for the modelling of reinforced concrete members, it is necessary to explicitly deal with: each individual crack that propagates around the reinforcing bars; the distribution of local stress and strain in steel bars; and the interaction at the interface between concrete and bar (that is, stress transfer through bond, bond deterioration, and so on). The crack propagation around a steel bar is caused as a result of three-dimensional strain fields and bond transfer mechanisms and, since the fracture process in 3D stress field differs from that in 2D, a 3D analysis is appropriate for modelling of reinforced concrete elements using discrete crack models (Morita and Kaku, 1979, Maekawa et al. 1999). In adopting this approach, fine meshes are needed with many degrees of freedom and, thus, the use of discrete crack models for the analysis of reinforced concrete members has generally not gained favour over that of the more general distributed smeared crack approach. For problems that involve tensile fracture of the concrete, however, discrete, crack band or non-local models are needed to capture localized behaviour and is discussed in more detail in the next section. 3.2.4

Modelling of concrete cracks

An important issue with the modelling of tension in finite element models is the way that cracks are considered. In general they can be taken either as smeared throughout the element or only present at finite element boundaries which is also called discrete cracking. Modelling of reinforced concrete with dispersed cracking as a quasi-continuous material or smeared cracking was firstly introduced by Rashid (1968) and Cervenka and Gerstle (1971,1972) and has been widely applied for mechanics of cracked reinforced concrete. It requires that the cracks within the element be accounted for in terms of their effect on stiffness, strength, and energy characteristics. Generally, this is accomplished via smearing the crack within the element (see Figure 3.14a) based upon a space averaging process in stress and strain. Smeared crack tension stiffening models can be solely formulated based on experimentally extracted behaviour and will, generally, “lump” all nonlinear processes such as bond, aggregate interlock, tension softening and others into simple constitutive relationships in one-dimension.

96

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The second option is to model the cracks discretely by allowing gaps to form between different elements. This is the option taken by discrete element and lattice models discussed below and these models are generally based on tension softening relationships. Just as care is required when considering the stress-strain relationship between tension stiffening and tension softening, it is also important to consider that localization problems can occur for smeared cracking problems as well. Consider the finite element mesh shown in Figure 3.14b. The structural cracks are shown within the elements rather than between them indicating that this is a smeared crack analysis. However, the cracks are only within one line of elements indicating that virtually all the displacement is localized within this region of the member. As such, this extent of cracking in the model is consistent with a tension softening type of formulation which is designed to account for localization. In contrast, the mesh in Figure 3.14a shows cracking smeared throughout the tension region and thus no localization problems are present, consistent with a tension stiffening relationship.

Element to Model

a) no localization

b) localization

Figure 3.14: Localization in Reinforced Concrete FE Analysis.

A final issue to consider with the modelling of cracking in concrete relates to the effects of a change in principal average stress directions that can often results from redistribution. Initial cracks are usually assumed to form at the angle of principal tensile stress in the continuous material before cracking, but after cracking the average stress directions may no longer align with this axis. In general there are two approaches for dealing with this: fixed angle cracks and rotating angle cracks. Both are approximations. Experiments on large elements subjected to pure shear have shown that the angle of principal strains, averaged over a sufficient gauge length to include several cracks, can rotate significantly between first cracking and ultimate limit states. In practice, new cracks form at different angles to the previous cracks as the loading increases. This behaviour is also observed in beams with stirrups as new shear cracks form at different angles to that of initial shear cracks. Rotating crack models, therefore, allow the angle of cracking to change as the analysis progresses to capture this effect. By contrast, a fixed crack approach forces the angle of the cracks to remain constant after initial cracking with the shear stresses calculated on that crack surface usually checked against aggregate interlock relationships.


97


3.2.5

Modelling of reinforcement

Reinforcing bars in structural concrete are generally assumed to be one-dimensional line elements without transverse shear stiffness nor flexural rigidity. The point-wise stress-strain relationship can be formulated by the elasto-plastic modelling as shown graphically in Table 3.1. The plastic evolution characteristics, yield and fracture strengths and plastic elongation, are normally dependent on the micro-properties and some high strength tendons, used in prestressed concrete, and hard-drawn wires, used in mesh, have lower fracture to yield strength ratios and strains at the limit of uniform elongation. Reinforcement in a nonlinear concrete analysis can generally be treated as either discrete or smeared. Discrete reinforcement involves the inclusion of individual axial or axial-flexural elements into the finite element mesh that model each layer of reinforcement explicitly. This has the advantage of providing a one-to-one correspondence between the real structure and the model and a corresponding lower likelihood of input errors. Smeared reinforcement involves calculating an average stress-strain relationship that applies to the entire element area and is included directly as part of the overall concrete element stiffness matrix. Introduction of steel into a reinforced concrete analysis, particularly one that includes smeared concrete tension stiffening stresses, should consider the effects of those concrete stresses on the overall behaviour. In general, this can be done by adjusting the concrete tensile behaviour at first yield at a crack or by adjusting the steel behaviour at first yield. Figure 3.15a) shows the option of using bare-bar stress-strain relationships for the steel summed with a smeared concrete tension stiffening stress. At first yield at the crack, the concrete tensile forces are explicitly reduced to zero to ensure that the steel stress at a crack does not exceed yield. In this example, it is conceptually assumed that strain hardening at a crack will not occur. Fig 3.15b considers the opposite option of leaving the concrete stress-strain relationship unchanged and deriving new equations for the embedded steel behaviour. Both of these methods produce the same overall answer in this example. In any case, it can be seen that simply adding the average concrete tension stiffening stresses to the bare bar steel stresses would violate equilibrium at a crack after first yield at a crack. Finally, special care must be taken in analyses that involve non-ductile reinforcement such as glass or carbon fibre or lowductility steel reinforcement. In such situations, the increase in reinforcement stresses that occur at the cracks will result in rupture when the average strain is still well below the rupture strain of the material. Figure 3.16 shows a comparison of the observed components contributing to the tension force for a steel bar embedded in concrete for three tests of Shima et al. (1987).

3.3

Nonlinear concrete modelling framework

3.3.1

Elasticity

The most familiar continuum model is the elasticity model where loading and unloading are path-independent and follow Hooke’s law (Figure 3.17). Structural steel before yielding and structural concrete before cracking can be modelled well by such a linear elastic model. Purely elastic models can be of arbitrary complexity, and need not require isotropic behaviour. Orthotropic modelling of concrete has been found to work well when consideration is given to the observed tension behaviour being dramatically different from compressive behaviour in principal directions. The most general anisotropic modelling allows all 21 independent elastic moduli in three dimensions to be defined.

98

!



fc not reduced

fc lowered to zero by crack check

*fy

*fy 1st yield at crack stress

stress

1st yield at crack fc fcr

fc

average steel stress less than yield

fcr

fs

fs strain

strain

(a) Bare bar

(b) Embedded steel

Figure 3.15: Two approaches for checking local yield at a crack.

Load (MN)

0.3

Specimen No.6

Specimen no.5

Specimen no.4

Bare bar

0.2

Bare bar

Bare bar

0.1

Tensile force

Tensile force

Tensile force

0

Stress (GPa)

0.8

Bare bar

Bare bar

0.4

0

Stress (MPa)

Bare bar

2

Average stress of reinforcement



Average stress of concrete



1

0

1

Average strain (10-2)

0

1


0

1


2

Figure 3.16: Contributions to the tension force in a reinforced concrete element and corresponding stresses (Shima et al., 1987).

One direct result from the analysis of elastic bodies is that the stress distributions around the tips of cracks appear as real values in the output. Simple elastic theory for a fine crack predicts a stress singularity at the tip of the crack, which is clearly impossible. Concrete responds to this by the formation of a fracture process zone just ahead of the crack and cohesive bridging stresses after cracking. Thus, even linear elastic models contain approximations to reality after cracking. The accuracy of deformation by linear analysis is far from the reality for cracked concrete structures but in many cases of ultimate limit state verification, computed section forces by linear analysis provide reasonably close approximations to the exact nonlinear solution. This is because the accuracy of section forces does not primarily depend on the absolute values of section stiffness but on the relative stiffness profile. This is the reason why the linear analysis is often adopted as a computational tool for design so to obtain the section forces even for ultimate limit state examination.


99

stress

stress

stress


linear elastic strain (recoverable strain)

elastic strain recoverable strain but irrecoverable stiffness

plastic hardening strain (irrecoverable)

combination of basic concepts RC modeling +Y

Average shear stress

,YX Y

+X ,XY

Average shear stress

7

C

6

LX

X

, (MPa )

7

5

4

4

3

3

2

2

1

1 0

2

4

Average shear strain

6

C

6

5

0

, (MPa )

8

) ( 10 -3 )

0

Y

0

2

4

Average shear strain

6

8

) ( 10 -3 )

examples of computed stress-strain

Figure 3.17: Basic concepts of elasticity, plasticity, damage and their combination.

3.3.2

Plasticity

Plasticity allows for overall nonlinear behaviour by accumulation of irrecoverable strains. It is generally defined as an elastic stiffness component summed with an accumulating plastic strain component. The criterion of plasticity and elasticity is generally specified in terms of stresses and the plastic evolution generally brings about hardening of metallic materials (Figure 3.17). This is defined as plastic strain hardening. It is often used today with a flow theory of plasticity, which defines the direction of plastic flow or a ratio of plastic strain increment. Perfect plasticity with no hardening is useful when solely determining a maximum capacity for a given structural element. This capacity is derived by assuming that all the steel yields and the solution is therefore relatively easy to obtain. In general, the ultimate strengths obtained from the nonlinear analysis methods in this practitioners’ guide will be below or at most equal to the strengths obtained from an assumption of perfect plasticity. Note that due to the general inclusion of an elastic component coupled with a plastic component, the concepts of stresses at crack tips, perhaps modelled with fracture mechanics of concrete, fit equally well with plasticity models as with purely elastic methods. 3.3.3

Damage

Methods based on damage mechanics allow the elastic stiffness of the material to be reduced as a result of accumulation of damage from strain excursions or repeated loadings. This

100

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nonlinearity may represent the loss of mechanical volume or degraded capacity to absorb the elastic energy, which is thermodynamically reversible in nature. Concrete shows an irreversible reduction of elastic unloading/reloading stiffness when small lateral confinement is provided. When higher lateral confinement is applied to concrete in compression, damage of the elasticity properties is observed in softening of the unloading and reloading paths. Damage is also seen in concrete tension after cracking as described above. In many practical cases, concrete nonlinearity in compression can be simply characterized by plasticity although plasticity modelling cannot cover all observed nonlinear behaviours of the continuum. Elasto-plasticity is the basis of limit state design and capacity computation with yield-lines and/or reinforced concrete strips. When it is needed to simulate the post-peak behaviour for verification of some limit states, a damage approach is useful for collapse simulation and assessment of the remaining performance of a damaged structure. 3.3.4

Mixed models

Models based partly on elasticity with some basic concepts from plasticity and damage mechanics can provide simple models that can capture many important nonlinear effects in reinforced concrete materials and reinforced concrete structural elements. One of the simplest examples is a no-tension linear elastic concrete model coupled with an elasto-plastic steel model. This is a well known simple approach, which has been popularly used for flexural behavioural computations of reinforced concrete sections based on Euler-Kirchoff’s in-plane assumption. For both confined and cracked concrete in compression, combinations of elasticity, plasticity and damage can be consistently applied in structural analysis. Smeared-crack in-plane modelling of reinforced concrete is a further example of combining complex combinations of compression, tension and shear across cracks in a multi-dimensional model and can include nonlinear elastic orthotropic behaviour of concrete coupled with an elasto-plastic hardening behaviour of steel. Such mixed models often provide relatively simple and reasonably expressive nonlinear concrete models. 3.3.5

Discrete modelling frameworks

Rather than treating concrete as a quasi-continuous material based upon space-averaged constitutive models, it is also possible to treat it as a series of interacting discrete bodies for structural analysis and design (e.g., Rots and Blaauwendraad 1989). This can be interpreted in two ways. In the first way, a framed structure can be made from elastic elements with lumped plasticity at their ends. With this type of discrete model, nonlinear behaviour is often defined by predefined moment rotation models, or moment curvature relationships coupled with an assumed plastic hinge length which has much to do with the post-buckling behaviour of the reinforcing steel. The second way of utilizing a discrete element model is for a continuous body where the locations of the discontinuities (cracks) are not known a priori when the mesh is created. One option is then to explicitly re-mesh the analysis space at each load step to follow crack propagation in a more explicit way. For fracture simulation, Ingraffea and Saoumu (1984) first introduced the discrete approach to concrete structures. The discontinuities of the displacement field resulting from the failure processes are introduced directly into the numerical model. New elements are added at the crack surface to explicitly account for tension softening and aggregate interlock. In this approach, the crack location is instantly recognizable but it can be a lengthy procedure especially for three-dimensional analyses.


101


An alternate approach is to implicitly assume that all elements have crack-like gaps between them from the start that may or may not be active. Methods such as lattice models fall into this category (e.g., van Mier et al., 1995, Bolander and Saito, 1998). The discrete crack gap can be numerically embedded in finite element volumes and the connectivity of these embedded crack gaps can be satisfied on the element boundaries based upon hybrid stress and strain hypotheses. Thus, differences between smeared and discrete approaches is becoming less distinguishable for geometrically continuous domains and, while the methods were first applied to the analysis of material level problems, large structural problems can also be modelled as has been demonstrated by Bolander et al. (2000) and Bolander and Hong (2002).

3.4

Solution methods

As analysis of concrete structures usually involves non-linear modelling of materials and possibly geometry, the analysis requires at least some basic knowledge of non-linear solution strategies. The most commonly used solvers use a Newton or modified-Newton method to solve for the system on non-linear equations obtained. The basis of Newton solution methods are discussed below with the reader referred to Chapter 6 for some more advanced approaches. Let the quantities (qo , & p) denote the displacement and current load vector at a given point on the equilibrium path, as shown in Figure 3.18, where & is a load magnification parameter and p is a reference vector of applied loads. The equation governing equilibrium can then be written as

^ p 7 f (q) ' 0

(3.1)

or

K( q )q 7 f (q) ' 0

(3.2)

where f is a vector of internal forces and is a function of the current displacement state. Eqs. 3.1 and 3.2 describe the equilibrium state of the discretized structure. While the solution of the linear equation system Kq – f = 0

(3.3)

can be calculated directly, this is not possible for non-linear systems. Solution techniques for non-linear systems typically require the solution of linear systems repeatedly until convergence is obtained with the most frequent schemes used based on a Newton method. 3.4.1

Newton-Raphson method

The most frequently used iteration scheme for the solution of non-linear equations is some form of the Newton-Raphson procedure. In the case where Eq. 3.1 cannot be solved exactly, it can be shown that (refer Chapter 6) the solution may be progressively calculated by adding the change in displacements O qi to the current displacement state calculated from 71

Oqi ' K Ti r ( qi )

102

!

(3.4)



i where KT is the tangent stiffness determined for the current state and r (qi ) are the out of balance forces. The Newton-Raphson solution process is illustrated in Figure 3.18, noting that for every step the current stiffness matrix is formed and the linearized equations solved for O qi .

Figure 3.18: Schematic representation of Newton-Raphson method.

3.4.2

Modified Newton-Raphson method

From the above calculations it is seen that the formulation of a new tangent stiffness for each iterative cycle, and the solution of a new system of equations, must be undertaken. In computing terms, this can be time consuming and costly. To overcome this cost the approximation K Ti ' K To is often made. This modifies Eq. 3.4 to 71

O qi ' K To r ( qi )

(3.5)

and resolution of the same equation set is repeatedly used. The solution at each iteration is sped up; however, more iterations to convergence are required, as can be seen in Figure 3.19. The overall economy of the solution procedure is dependent on the problem size and nonlinear behaviour. An updated mNR approach may be adopted where the tangent stiffness matrix, K Ti , is updated if convergence is not obtained after a pre-determined number of iterations, n, and continues to be updated every n iterations following.


103


Figure 3.19: Schematic representation of modified N-R method.

3.5

Precision of nonlinear concrete FE analyses

Upon seeing the results of analysis tools that take into consideration some or all of the complexities outlined above in modelling of reinforced concrete structures, one may be tempted to conclude that such a computer program will be able to produce a very precise solution to a given problem. This is, of course, an unwise conclusion. As an example of the variability that may be expected for some analyses, consider the data shown in Figure 3.20 of measured stiffness values of concrete as a function of compressive strength. Note the wide scatter in the observed values. Consider that at a concrete compressive strength of 25 MPa, limestone aggregates have can produce concretes with stiffnesses varying from 20,000 MPa up to 35,000 MPa. Calculations sensitive to the uncracked stiffness of concrete, such as the displacements of a beam before cracking, can be expected to be similarly variable. No structural analysis should be expected to provide extreme precision in results whatever the complexity of the modelling. Engineers should perform sensitivity analyses to ensure that a given set of results is not simply a result of mesh sensitivity or unduly controlled by a parameter in which the engineer has low confidence. For concrete, one also should take into account the difference of strength between control specimens and structural concrete realized in the structures concerned. The difference is caused by size effects, concrete placing work, shrinkage, temperature hysteresis and so on. When seeing a figure like Figure 3.20, proponents of high precision in nonlinear finite element analyses may counter that such scatter simply indicates that the analysis was not properly calibrated to the experiment that was to be modelled. Of course practicing engineers 104

!



know that for most design or analysis problems, there is no experiment to calibrate to. Claims that extremely precise fits between experiment and analysis are guaranteed should be treated with caution and not used to justify dramatically lower safety factors, for example. Lastly, it is important that the model being adopted be verified against benchmark tests of the problem type being considered. Benchmarking testing is discussed in detail in Chapter 7.

Figure 3.20: Modulus of elasticity as a function of compressive strength (Rashid et al. 2002).

3.6

Safety and reliability

To ensure that safety and reliability criteria are met when using numerical analyses, the model must first be validated by experiments and benchmark tests (see Chapter 7). The verification process should include validation using: "

basic material tests,

"

structural tests, and

"

mesh sensitivity tests.

The validation of material models are generally done by modelling of selected standardized benchmark or control tests (for example, tests on reinforced panels or fracture of control sized specimens). Alternatively, validation of the material models can be performed as a part of a standardized numerical simulation task. An assessment of the capability of the material models to perform, as required to simulate the behaviour of a structure, is normally undertaken when selecting and assessing the appropriateness of the software package. A validation of structural model is undertaken for a particular model and software package in a special-purpose study undertaken by means of benchmark calculations. In this process, specific types of structural and material behaviour are tested. Such studies form a rational basis for choosing appropriate material models and software for modelling of a particular structure, structural member or element loaded under similar conditions. For example, if a


105


shear wall is to be simulated under a particular loading regime for a given set of boundary constraints, a validation of the finite element software against shear wall experiments should be performed using test data identified as being reliable. In this process of verification, it is vital that all the potential modes of failure are identified and the model verified to show that these failure modes are captured with accuracy. For performance assessment, some indicators are necessary to suitably characterize the structural responses. In general, there can be three types of indicators; force/stress related indicators, deformation/strain related indicators and energy related indicators (force multiplied by displacement or stress by strain). The first two indicators are practically useful for static design and the energy-based indicator for seismic design or damage control (Akiyama, 1985). If our problem is a perfectly linear one, the same safety allowance can be expressed no matter what sorts of indicators are used for the safety check, because force and deformation are linearly correlated. The safety checking scheme in most modern codes is based on limit states and associated partial safety factors. In this scheme, the resistance or response is based on design (extreme) parameters, which are derived from nominal ones using a reduction by the partial safety factor #p. By this way, uncertainty of materials, geometry and other properties is included in input data. Similarly, the design loads or actions are considered by extreme characteristic values, where the load factors #load are considered. In a typical static situation, material characteristic parameters are reduced, while loads are increased, both to design values. Safety of design is checked locally at particular points, for example in beam sections, by the limit state condition as R(&p ) < E(&L )

(3.6)

where R(&p) is the extreme local resistance or capacity of the structural response and E(&L) is the extreme local action of load or response indicator. This condition assures the safety of each local point. It does not indicate a direct measure of global safety of the structure but it generally results in conservative and safer assessment under static loads. This method is popular in practice, since it is easy to apply and current partial safety factors based on linear analysis are available. However, when we work with nonlinear problems, linear correlation of force and displacement does not hold in nature. Some codes such as Eurocode 2 (2002) explicitly mention that the partial safety factor method is not applicable to nonlinear analysis. The problem of applying the method of partial safety factors to non-linear modelling is demonstrated using two extreme cases. Nonlinear forces versus deformation relationships (left side of Figure 3.21) are common in reinforced concrete mechanics. For example, we can imagine relationship between the transverse shear force and the strain in the stirrup reinforcement. If the strength of concrete and amount of web reinforcement are changed, we have different static force versus deformation relationships, as shown in Figure 3.21. Let us consider the steel yield as a limit state safety check and an allowable safety margin in terms of strain (i.e., a deformation related indicator). Here, the corresponding safety allowance in terms of shear (force related indicator) differs greatly. Since a force indicator is roughly proportional to external static loads, it is thought reasonable to specify the safety allowance (safety factor) by the force-based index for expressing reliability of nonlinear analysis to identify the limit state concerned. In fact, limit state design methods do not adopt stirrup strain but apply sectional shear force as the limit state indicator for the static safety assessment.

106

!


_R

Rcap

static force, stress : R



_R small allowance ratio

strain, displacement based index

RC/PC nonlinear response toward external force/stress

Rcap strain, displacement based index




_R

_R


Elasto-plastic response toward external force/stress

Figure 3.21: Safety factor for nonlinear structural problems.

For the case of elasto-plastic problems like steel structures, however, any indicator identifies the same safety allowance (see Figure 3.21) when the initiation of plasticity is selected as the limit state because of the proportionality of internal/external forces and deformations. Then, in the discussion of this chapter, the safety allowance or variation is depicted by force related indicators. There exists similar feature on fatigue limit state design for service life. Nonlinear analysis is typically a simulation of a loading test for an entire load history up to failure, including the serviceability and ultimate limit states. In current codes there are different partial safety factors for different load stages and failure modes. It is difficult to apply current safety factors according to their definitions in nonlinear analysis. A material model based on the partial safety factors represents an imaginary, not real, material. It does not represent an extreme behaviour with a certain probability of failure. Automatic reduction of all parameters may cause much weaker material than required by the safety concept. In some cases, it can be an unsafe model. Thus, considering the design values of material parameters in nonlinear analysis does not guarantee that the target safety values, as prescribed and calibrated by standards, are achieved. Let us first consider a simply supported beam under uniform loading, where the bending moment in the mid-section is completely independent of material behaviour. If we use a beam finite element with plane section hypothesis, the nonlinear analysis is identical with the current design methods (based on elastic assessment of internal forces) and the partial safety factors can be applied. If we consider a statically indeterminate structure, the actions in sections or material points depend on material behaviour and need not be proportional to the magnitude of the external load. In general, a redistribution of internal forces due to nonlinear behaviour can produce either positive or negative effects on local failure. An example of such situation is concrete under a confining pressure where higher lateral pressures provide increases in strength. In such cases, the application of partial safety factors is not justified. Since the strong nonlinear behaviour can be expected in most of reinforced concrete structures, it is concluded that the partial safety factor approach should be used with care in statically indeterminate structures, especially when local post-yield states are allowed in the design. ! fib Bulletin 45: Practitioners’ guide to finite element modelling of reinforced concrete structures

107


Recently, static pushover analysis has been carried out in use of nonlinear FE analysis; not only for seismic design but also for static safety design of statically indeterminate structures composed of several members. The pushover analysis may present the whole structural capacity under static loads and the scenario of collapse after the capacity limit state. Figure 3.22 shows a schematic of pushover analysis and the design value of loads. Up to the peak capacity of the global structural system, some component members may fail or locally exceed their capacity. Thus, the computed characteristic capacity of the global structural system should be factorized by a global safety factor with consideration to the accuracy of the nonlinear structural analysis. This factorized design value of the global system capacity has to be greater than the overall design static load (vector). The performance assessment in terms of the global system safety is advantageous rather than the conventional methods where all constituent elements are required not to exceed their ultimate limit state. This means that failure of any member is not accepted, even if the global structural stiffness remains positive. When section failure of structural members is itemized as the limit state for structural safety, some indicators to judge the occurrence of flexural and shear failures are needed. The combined axial and flexural failure can be detected by monitoring the longitudinal compressive strains of the most extreme fibre of the section, as shown in Figure 3.23. When 2D or 3D solid elements are used, point-wise local strains tend to be dependent on the sizes of the finite elements and detailed patterns of mesh discretization (Chapters 3 and 6). Then, a space averaged mean strain or stress indicator (sectional forces) is preferable as a design index because the space-averaged indicator is not sensitive to the detail of the mesh discretization. Another way is to calculate sectional forces by integrating computed local 2D and/or 3D stresses. This is an averaging of stress field and is compared with the sectional capacity computed in advance by design formula or by the nonlinear analysis. Discussed is the shear deformation based indicator to recognize the shear mode of failure accompanying diagonal cracks. Shear deformational intensity along cracking is supposed to be a candidate but magnitude of limit value is not uniquely decided because the shear slip/deformation response depends on the axial compression force, dimensioning and size, etc. The axial mean deformation of members is also expected to be an indicator for shear failure. Regarding the restorability of structures after extreme loads, material damage based indicators are also proposed, especially in seismic designs for practice (JSCE, 2002). Damage induced in structural concrete has a close correlation with repair/strengthening costs. Figure 3.24 shows an example of design modelling for concrete used in fibre modelling (Chapter 4). The local damage indicator denoted by F is assumed to be a function of maximum experienced compressive strain in a load- or time-step run of a nonlinear structural analysis with nonlinear material models. The sectional averaged F value is used as a damage indicator in regards to restorability after the application of extreme loads. The transverse displacement of the columns corresponding to F = 0.5 is almost the same as the allowable ductility level specified in the past design codes based on empirical macroscopic models of members. Confinement (Chapters 3, 4 and 6) by lateral ties and steel is taken into consideration by modifying fibre stress-strain relationships. As the lateral confinement moderates the damage intensity, the damage evolution indicator F can be modified as shown in Figure 3.24(3). The damage indicator was formulated to represent reduction ratio of the elastic stiffness on unloading paths and is a measure of how much reversible strain energy is stored in the damaged continuum.

108

!


force (vector)


computed characteristic structural capacity

design value of loads

design value of capacity

global safety factored in terms of analysis model used

partial safety margin factored in terms of failure/collapse modes importance of structures, etc.

force (vector)

displacement

computed characteristic structural capacity

design value of loads

A member reaches the limit state in terms of stress/strain/section forces design value of capacity

partial safety factored in terms of failure/collapse modes, etc.

global safety factored in terms of analysis model used

displacement

Figure 3.22: Safety factor and the whole structural limit state by push-over analysis.

Axial-Flexural failure of members (safety assessment)

moment

index: bending capacity of the section (fiber/shell), extreme fiber strain (2D-3D, fiber/shell), reinforcement strain/stress

moment

curvature

curvature

Transverse shear failure of members (safety assessment) index: out-of-plane shear capacity (fiber/shell) shear strain along cracking (2D-3D), strain of web reinforcement (2D-3D) cracking pattern (2D-3D), comp. mean strain along member axis (2D-3D) axial mean strain of column

shear failure occurs

drift angle of column

In-plane shear failure of members (safety assessment) index: stress of reinforcement compressive concrete strain along cracking, or principal axis (shell, 2D-3D solids), in-plane shear capacity

Figure 3.23: Flexural and shear failure indicator of beam/column members.


109


F > 0.5

Design damage indicator K

(2)

(1)

confined cases

(3)

(4)

(5)

(6)

(7)

Figure 3.24: Structural concrete damage based indicator (Tsuchiya and Maekawa, 2006).

110

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In case of shell structures subjected to coupled in-plane and out-of-plane actions, cracking of concrete and yielding of reinforcement are likely to develop over a wide domain of the structures because the reinforcement is uniformly distributed and flow of stress tends to expand over the structure. Then, the structural response level at the start of reinforcement yield is well below the capacity. In these types of structures, compressive localization can be a design limit state for safety and serviceability performance (Chapter 6). Figure 3.25 shows a transient process of compressive localization after cracking of the concrete and yielding of the reinforcement in shells. When the principal compressive strain averaged within the control volume reaches the limit value (nearly 5000B)), some elements exhibit high strain rate while the strain rate of neighbouring elements decreases. In case of circular shells subjected to shear actions, a localized band accompanying shear-compression kinematics can be seen when the principal compressive strain reaches this limit state. Then, for underground tank design, the averaged principal compressive strain is now used as the indicator for damage control with regard to compressive localization. This limit state almost corresponds to the structural capacity as shown in Figure 3.25. The rationale of this criterion was checked with reinforcement volumes from 0.4% to 2.0% (Harada et al., 2001). The global safety condition can be written with nonlinear-based indicators as Rm / & R < E(&L )

(3.7)

where Rm is the resistance or the capacity limit obtained by nonlinear analysis, as stated above, based on the mean material characteristic parameters, &R is the global safety factor of the corresponding resistance and E(&L) is the factorized external action or the response indicator of the structural system concerned. The global safety factor describes the safety of the system on a global level. Thus, it should cover the uncertainties of all components of the structural system and analysis. Since safety is related to the average resistance, it can be described by a global central safety factor. This represents a generally accepted safety margin of usual structures produced according to general standards. This covers wide and rather unspecified range of uncertainties with the safety not related to specific random properties of a given structure or product. The practical determination of global safety factor is more difficult comparing to partial safety specification, where it is related to known random variations of dimensions and material properties. Therefore, the values of global safety factors are not yet proposed in the majority of codes. For example, Eurocode 2 (2002) states that the partial safety factor concept is not applicable to the nonlinear analysis, and does not propose a format for safety check. DIN 1045 (1998) proposes the global safety factor for the “system resistance” &R = 1.3 valid only for ductile modes of failure. Higher values should be considered in case of brittle models of failure such as concrete shear and diagonal tension, or compression. The nonlinear analysis considered in practical codes is mostly limited to the models based on beam-column systems. The mean material parameters are based on nominal values that, typically, represent 5% probability. e qud c & gl d

& Rd E $& G G 8 & QQ % ? Rc

b e ` or & Sd & Rd E & g G 8 & q Q ? Rc qud ` c & gl a d

$

%

b ` ` a

(3.8)

where, &G, &Q are the global safety coefficients respectively for permanent and variable loads,


111

Principal compressive strain (B)


270 °

Element No.45

-10000

Element No.57 -7500

180 °

0 °

-5000 Maximum load

-2500

90° 90°

0

-1.0

; << 0.0

1.0

2.0

3.0

4.0

=

Deformation angle (%) @!!!

Maximum Load 2871kN

Rebar Yielding 1637kN

?!!! "

C4)8+2:D 5

%!!!

Diagonal Cracking 931kN

E F-):

!"

E F-):

Collapse Load 2352kN

$!!!

Flexural Cracking 735kN

>%"!

!

>$"!

!"!

$"!

%"!

?"!

@"!

#"!

>$!!! Test

>%!!!

Analysis COM3 >?!!! A -941, )0(4.+&.7*-+2B5

crack pattern and deformation localization in the web of RC shell tank subjected to lateral force

Figure 3.25: Structural concrete damage based indicator (Harada et al., 2001).

&Rd is the model uncertainty coefficient on the resistance side (suggested value gRd = 1.06), &Sd is the model uncertainties coefficient on the action side (suggested value gSd=1.15), &gl is the

global structural safety factor (suggested value gg = 1.20, but gRd ggl = 1.27 = gGl), qud is the ultimate level of the internal actions path, reached in the incremental process of nonlinear analysis. For the case where no model uncertainties are considered, the inequalities in Eq. 3.8 are modified to

eq E & G G 8 & QQ ? Rcc ud d & Gl

$

%

b `` a

(3.9)

The JSCE (1999) LNG-Tank design code presents full 3D nonlinear analysis of soil-RC shell interacting systems and proposes a factor of 1.3 for the global safety factor in regard to the global deformational indicator, as shown in Figure 3.25. Here, we input the most probable

112

!



values for material characteristic parameters of structural concrete and soil foundation. According to current practice, at serviceability limit states, we can use directly for the mean response &R = 1.0, and unfactored loads or actions. At ultimate limit states, the global safety factor should be &R < 1.3, depending on the ductility of the response. Concerning the ultimate limit state, Mancini (2002) and Bretagnoli et al. (2004) proposed for nonlinear static analyses that for the evaluation of qud, the analysis should be stopped when the ultimate strength and the corresponding deformation are reached within the most critical region and where the whole structure is incapable of supporting any further load increments. An example of deep beam capacity is shown in Figure 3.26. During the analysis, the first critical element initially crushed but the structure was able to carry further load increments up to the crushing of a second element. At this point the model was unable to reach the equilibrium for any further load increments. This last step has been considered as the final point of the internal actions path. Figure 3.26 shows the resisting interaction surface for the critical second element, and the internal action path in the same element, up to the intersection with the resisting surface. Figure 3.26 also illustrates the procedure for the application of safety format in a vectorial combination of internal actions and, by means of definition of a safety interaction surface, derived by the limit one by linear transformation referred to the axes origin. If the critical element and its internal paths are not clearly identified, the external loads magnified by the global safety factor can be applied to the structural concrete. If the magnified loads would not cause failure of the structure, it implies the design satisfies the safety requirement.

Critical Element 1

450 400

P [kN] Numerical Experimental

350 300 250 200

Reinforcement

x A

150 Critical Element 2

100 50 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 x [m]

Figure 3.26: Application of safety format of strength limit state of internal actions of a deep beam (Mancini 2002, Bretagnoli et al., 2004).


113


In the draft Australian concrete structures standard (DR 05252, 2005), for new designs and/or assessment of existing structures, a system safety approach is used to ensure that the structure (or structural element) meets expected safety demands. The draft adopts the following stress check procedure for use with non-linear frame and stress analysis: It shall be confirmed that the design resistance of the structure or the component member is equal to or greater than the design action -

Rd < Ed where Rd is the design capacity of the structure based on mean strengths of materials and Ed is the critical combination of factored actions (for example, for dead and live load combinations the worse case of 1.25G + 1.5Q, 0.8G + 1.5Q and 1.35G, where G = dead load and Q = live load). The design capacity of the structure is limited by

Rd ' C sys Ru.sys where Ru.sys is the capacity of the structure and is determined for the same combination of actions adopted for Ed with mean values of material properties, and Csys is a system reduction factor and is dependent on the mode of failure.

The values for Csys adopted by the draft standard are dependent on the type of failure. For structural systems in which the deflections and local deformations at high overload are an order of magnitude greater than those for service conditions; and yielding of the reinforcement occurs well before the peak load is reached, Csys = 0.7. Otherwise, Csys = 0.5 (although larger values than 0.5 may be used if it can be shown that, at high overload, adequate warning is given of impending collapse).

3.7

Statistical analyses

Statistical or risk analysis is the most conceptually rational method of safety assessment in the present state-of-the-art. The analysis includes deterministic and probabilistic domains. The basic structural model is generated as a deterministic one with mean (central) parameters. Certain parameters of this model are assumed to be random variables. They can be material parameters, dimensions, etc. Based on these input data, the structural response can be obtained in a statistical form in which the state variables of the response, such as ultimate load, deflection or stress state in a point, are described by a random distribution (with mean, standard deviation and other parameters). The safety margin can be formulated by comparing the response R and actions E. The probability of failure pf is defined as a probability of Z < 0:

Z = R - E, pf = p(Z < 0)

(3.10)

The method is illustrated in Figure 3.27.

114

!



!

Figure 3.27: Statistical analysis of probability of failure.

This method allows an assessment of the response under given loading conditions and with consideration of random nature of the input parameters and can be used in a rationally based assessment of global safety. Solution of this problem can be performed numerically combining the structural and statistical analyses. The numerical model of structure is based on a deterministic nonlinear analysis using the finite element method. The probability of distribution of the response can be obtained by numerical methods based on random sampling. In these methods, the random variables of samples are generated by statistical methods and sample response is realized by a nonlinear solver. Finally, the statistical parameters of the response are analyzed again by statistical methods. An example of this approach, applied to bridges, is presented in Bergmeister (2002). The advantage of the above approach is that the reliability can be rationally evaluated by failure probability or by a safety index. Both safety measures are well justified in reliability engineering and prescribed by standards (eg., Eurocode 1, 2002). The resulting safety margin (global safety factor) is based on actual input parameters, their random variation and their mechanical relevance.

3.8

Concluding remarks

While non-linear finite element modelling can be an extremely useful and powerful approach in determining the behavioural response of complex concrete structures, extreme care is needed in the setting up of the models, in the verification of the model, in assessing the models capability to correctly identify critical behaviour and in the interpretation of results. For this, experience is needed in both computational modelling and in design and construction of concrete structures. The output of FE models should never be considered in isolation of the problem being investigated and a prudent engineer will always have predicted the results using simple calculations and experience before the model is run. Finally, equilibrium checks on the input/output are, of course, essential.

3.9

References

Akiyama, H. (1985), Earthquake-Resistant Limit-State Design for Buildings, UT-Press, Tokyo. Alca, N., Alexander, S.D.B., MacGregor, J.G., (1997) “Effect of size on flexural behavior of high-strength concrete beams,” ACI Structural Journal, 94(1), pp. 59-67.


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Bergmeister, K., et. al. (2002), “Structural analysis and safety assessment of existing concrete structures”, Proceedings of fib Congress - Concrete Structures in 21 Century -, Osaka, Japan, Session 11, 47-54. Bretagnoli, G., Carbone, V. I., Giordano, L. and Mancini, G. (2004), “Safety format for nonlinear analysis”, Proceeding of the fib Symposium, Avignon, France. DR 05252 (2005), Concrete Structures, Draft for Public Comment Australian Standard, Standards Australia, Revision of AS3600-2001. Balmer, G.G. (1949), “Shearing strength of concrete under high triaxial stress-computation of Mohr's envelope as a curve." Tech. Rep. No. SP-23, Structure Research Laboratory, Denver. Bažant, Z.P. and Oh, B.H. (1983), “Crack band theory for fracture of concrete”, Materials and Structures, 16 (1983) 155-177. Bažant, Z.P. and Planas, J. (1998), “Fracture and Size Effect in Concrete and Other Quasibrittle Materials”, CRC Press, Boca Raton. Bentz, E.C. (2005), “Explaining the riddle of tension stiffening models for shear panel experiments,” ASCE J. of Struct. Engng., 131(9), pp. 1422-1425. Bolander, J.E., and Saito, S. (1998) “Fracture Analysis Using Spring Networks with Random Geometry,” Engineering Fracture Mechanics, 61, pp. 569-591. Bolander, J.E., Hong, G.S., and Yoshitake, K., (2000). “Structural Concrete Analysis Using Rigid-Body-Spring Networks,” Computer-Aided Civil and Infrastructure Engineering, 15, pp. 120-133. Bolander, J.E. and Hong, G.S. (2002). “Rigid-Body-Spring Network Modeling of Prestressed Concrete Members”, ACI Structural Journal, 99(5), Sept-Oct, pp. 595-604. Cervenka, V. and Gerstle, K. (1971, 1972), “Inelastic analysis of reinforced concrete panels”: (1) Theory, (2) Experimental verification and application, Publications IABSE, Zürich, V.3100, 1971, pp.32-45, and V.32-II, 1972, pp.26-39. CEB-FIP Model Code 1990 (1993), Thomas Telford Services, Ltd., London, for Comité Euro-International du Béton, Bulletin d’Innformation No. 213-214, Lausanne, pp.437. Collins, M.P., (1978), “Towards a rational theory for RC members in shear,” ASCE J. of Struct. Engng., 104(4), pp. 649-666. DIN 1045-1 (1998), Tragwerke aus Beton, Stahlbeton und Spannbeton. Teil 1: Bemessung und Konstruktion. Eurocode 1 (2002), Actions on structures - Part 1-1: General actions -densities, self weight, imposed loads for buildings. CEN/TC 250/SC1. Eurocode 2 (2002) Design of concrete structures - Part 1: General rules and rules for buildings. CEN. 2nd draft. 2002. Foster, S.J., and Marti, P. (2002). “FE Modelling of RC Membranes Using the CMM Formulation”, Proceedings of the Fifth World Congress on Computational Mechanics

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(WCCM V), July 7-12, Vienna, Austria, Editors: Mang, H.A.; Rammerstorfer, F.G.; Eberhardsteiner, J., Publisher: Vienna University of Technology, Austria, ISBN 3-9501554-06, http://wccm.tuwien.ac.at Foster, S.J., and Marti, P. (2003). “Cracked Membrane Model: FE Implementation”, ASCE, Journal of Structural Engineering, V129, N9, September, pp. 1155 1163. Gilbert, R.I., and Warner, R.F. (1978), “Tension stiffening in reinforced concrete slabs,” ASCE J. of Struct. Engng., 104(ST12), pp. 1885-1900. Goto, Y. (1971), “Cracks formed in concrete around deformed tension bars,” ACI Journal Proceedings, 68(4), pp. 244-251. Harada, M., Onituka S., Adachi, M. and Matsuo, T. (2001), “Experimental study on deformation performance of cylindrical reinforced concrete structure”, Proc. of Japan Concrete Institute, 23(3). Harajli, M.H. (2004), “Comparison of bond strength of steel bars in normal and high strength concrete,” J. of Materials in Civil Engng., 16(4), pp. 365-374. Ingraffea, A. R. and Saouma, V. E. (1984), “Numerical modelling of discrete crack propagation in reinforced and plain concrete, in fracture mechanics of concrete, Structural Application and Numerical Calculation”, Sih and de Tomaso (Ed), Martinus Nijho_ Publ. JSCE (1999), “Recommendation for Structural Performance Verification of LNG Uuderground Storage Tanks”, Japan Society of Civil Engineers, Concrete Library, 98. JSCE (2002), “Standard Specification of Concrete Structures – Structural Performance Verificaiton ”, Japan Society of Civil Engineers. Kaufmann, W. and Marti, P. (1998), “Structural concrete: cracked membrane model", ASCE J. of Struct. Engng., ASCE, 124(12), pp.1467-1475. Kent, D.C., and Park, R. (1971), “Flexural members with confined concrete,” ASCE J. of Struct. Engng., 97(ST7), pp. 1969-1990. Kim, J.-K., Yi, S.-T., Yang, E.I. (2000), “Size effect on flexural compressive strength of concrete specimens,” ACI Structural Journal, 97(2), pp. 291-296. Kupfer, H., Hilsdorf, H.K., and Rusch, H. (1969), “Behavior of concrete under biaxial stresses,” ACI Journal Proceedings, 66(8), pp. 656-666. Marti, P., Alvarez, M., Kaufmann, W., and Sigrist, V. (1998). “Tension Chord Model for Structural Concrete”, Structural Engineering International, IABSE, 4/98, 287-298. Maekawa, K., Pimanmas, A. and Okamura, H. (2003), “Nonlinear Mechanics of Reinforced Concrete”, Spon Press, London. Maekawa, K., An, X., and Tsuchiya, S. (1999), “Application to fracture analysis of concrete structures”, recent development in mechanics of fracture for concrete structures, Concrete Journal, Japan Concrete Institute, 37(9), pp.54-60.


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Maekawa, K. and An, X. (2000), “Shear failure and ductility of RC columns after yielding of main reinforcement”, Engineering Fracture Mechanics, 65, pp. 335-368. Manacini, G. (2002), “Nonlinear analysis and safety format for practice”, Proceeding of the First fib Congress, Osaka, Japan, pp. 53-58. Morita, S. and Kaku, T. (1979), “Splitting bond failure of large deformed reinforcing bars”, ACI journal, Proceedings Vol.76, pp.93-110. Okamura, H. and Maekawa, K., “Nonlinear Analysis and Constitutive Models of Reinforced Concrete”, Gihodo-Shuppan Co. Tokyo, 1991. Ottosen N. S. (1977), “A failure criterion for concrete”, Journal of Engineering Mechanics, ASCE, 103, pp.527-535. Ozcebe, G., Ersoy, U., Tankut, T. (1999), “Minimum flexural reinforcement for T-beams made of higher strength concrete,” Canadian J. of Civil Engng., 26(5), pp. 525-534. Petrangeli, M. and Ožbolt, J. (1996), “Smeared crack approaches - Material modelling,” ASCE J. of Eng. Mech., 122(6), pp. 545-554. Rashid, Y. R. (1968), “Analysis of prestressed concrete pressure vessels”, Nuclear Engineering Design, Vol. 7(4), pp.334-344. Rashid, M.A., Mansur, M.A., Paramasivam, P. (2002), “Correlations between mechanical properties of high-strength concrete,” J. of Materials in Civil Engng., 14(3) pp. 230-238. Richart, F.E., Brandtzaeg, A., and Brown, R.L. (1928), “A Study of the Failure of Concrete Under Combined Compressive Stresses,” University of Illinois Engineering Experimental Station, Bulletin No.185, 104 pp. Rots, J. G. and Blaauwendraad, J. (1989), “Crack models for concrete: discrete or smeared? Fixed, multi-directional or rotating?”, HERON, 34(1). Scott, B.D., Park, R., and Priestley, M.J.N. (1982), “Stress-Strain Behavior of Concrete Confined by Overlapping Hoops at Low and High Strain Rates,” ACI Journal Proceedings, 79(1), pp. 13-27. Shima, H., Chou, L. and Okamura, H. (1987), “Micro and macro model for bond behavior in reinforced concrete”, Journal of The Faculty of Engineering, The University of Tokyo (B), 39(2), pp.133-194. Tsuchiya, S. and Maekawa, K. (2006), “Cross-sectional damage index for RC beam-column members subjected to multi-axial flexure”, Journal of Advanced Concrete Technology, 4(1) 179-192. Turneaure, F.E., Maurer, E.R. (1908), “Principles of Reinforced Concrete Construction,” 1st Edition, John Wiley and Sons, Inc, New York. van Mier, J. G. M., Schlangen, E. and Vervuurt, A. (1995), “Lattice type fracture models for concrete”, Chapter 10 in Continuum Models for Materials with Microstructure (Mühlhaus, H.B., ed.), John Wiley & Sons, Chichester, pp.341-377.

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Vecchio, F. and Collins, M. P. (1993), “Compression response of cracked concrete”, ASCE J. Struct. Engrg., 119(12), pp. 3590-3610. Vecchio, F. and Collins, M. P. (1986), “The modified compression field theory for reinforced concrete elements subjected to shear”, ACI Journal, 3(4), pp. 219-231.


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4

Analysis and design of frame structures using non-linear models

4.1

Introduction

In this chapter nonlinear models to be adopted in 1-D finite element procedures and/or fibre analysis procedures for the analysis of reinforced concrete structures are treated, with the aim of: "

performing (conventional and seismic) assessment on existing buildings,

"

assessing rehabilitation interventions on inadequate constructions,

"

identifying (potential) causes of damage or collapse in buildings.

These models can be used in order to determine the reserve strength and deformation capacity of existing healthy or damaged buildings. Once existing capacities are known along with anticipated strength and ductility demands, then a structural repair or upgrade that is consistent with the existing deficiencies of concrete structures and expected loading conditions can be devised. The determination of the structural behaviour of a reinforced concrete structure is based on a correct modelling of the constituent materials, that is, the concrete, the steel and the bond between each. Constitutive models for these materials presented in this section can be used to clearly identify the structural capacity that can be compared with both local and global demands. A simple example of one such case would be a reinforced concrete column damaged during strong ground motions. When performing accurate nonlinear analysis that leads the structure to its ultimate state, it is fundamental to rely on detailed models that can capture the strong nonlinear behaviour of concrete and steel reinforcing bars. As an example, reinforcing bar buckling models, presented in this chapter, can be used in conjunction with confinement models for concrete to quantify the available strength and ductility of a reinforced concrete element, also under cyclic action. In these cases, both the behaviour of buckled bars and the response of confined concrete must be modelled accurately, so that the remaining rotation capacity of a plastic hinge and strength of earthquake-damaged columns can be correctly evaluated. Retrofit of a corrosion-damaged concrete column would be another typical example where the models presented in this chapter can be used. The capacity reduction in concrete columns can be evaluated based on the amount of reinforcing bar loss due to corrosion. Once this reduction in the capacity is determined, the models presented in this section can be used in determining the feasibility of various repair techniques that can be used. This Chapter is organised in two sections: Section 4.3 Nonlinear Models of Frame Elements, and Section 4.4 Interpretation of Results. For material modelling, the reader could refer to Chapters 3 and 6, especially to the section where issues relevant to modelling of the materials concrete and steel are treated. Also, long-term effects of ageing and distress in both concrete and steel should be looked at as well as buckling of steel bars and details of lap splices. Section 4.3 Nonlinear Models of Frame Elements is devoted to the comparison among different approaches in modelling the response at the element level. Fibre elements are compared to strut-and-tie type of models. Modelling of shear and of bond-slip problems are given particular attention. ! fib Bulletin 45: Practitioners’ guide to finite element modelling of reinforced concrete structures

121


In Section 4.4 Interpretation of Results, important issues related to the localisation problem are treated, both for displacement- and force-based frame elements.

4.2

Notation

! $ x%

array containing the derivatives of the shape functions NU(x)

D bi $ x %

bond interface force between beam and bar i

E e ( x) F f ( x) G I K LIP Lp

Young’s modulus section deformations vector element flexibility matrix without rigid body modes section flexibility matrix shear modulus moment of inertia element stiffness matrix length pertaining to an integration point plastic hinge length

M M B $ x%

bending moment beam section bending moment

NB $ x%

vector of axial forces in the beam

Ni $ x %

vector of axial forces in bar i

NP $x%

force interpolation functions array

NU $ x %

shape functions

P py $ x %

element resisting forces transverse distributed load

V VB $ x %

shear beam section shear force

s ( x) U u(x) ub i $ x %

force distributions vector along the element nodal displacement vector section displacements vector bond slip between beam and bar i

u0

longitudinal displacement at the cross-section reference axis

v0

transverse displacement at the cross-section reference axis

wh

weight factor

yi

distance of bar i from element reference axis

:0

cross-section deformation due to bending

&

cross-section deformations due to shear

)0

axial strain at section centroid

f -

section curvature section rotation

122

4 Analysis! and design of frame structures using non-linear models


4.3

Nonlinear models of frame elements

4.3.1

Lumped versus distributed plasticity

Because the inelastic behaviour of reinforced concrete frames often concentrates in certain sections (typically the ends of girders and columns for seismic loads and midspan for distributed, static loads), one option for modelling the nonlinear behaviour of reinforced concrete frames is to used a lumped plasticity model, where nonlinear zero-length springs are placed at the critical points, connected by linear elastic elements. Depending on the formulation these models, they may consist of several springs that are connected either in series or in parallel. The earliest parallel component element was introduced by Clough and Johnston (1966) and allowed for a bilinear moment-rotation relationship: the element consists of two parallel elements, one elastic-perfectly plastic to represent yielding and the other perfectly elastic to represent strain-hardening. The stiffness matrix of the member is the sum of the stiffnesses of the components. Takizawa (1976) generalized this model to multi-linear monotonic behaviour allowing for the effect of cracking in reinforced concrete members. Filippou and Issa (1988) subdivide the element in different, parallel, sub-elements. Each subelement describes a single effect, such as inelastic behaviour due to bending, shear behaviour at the interface or bond-slip behaviour at the beam-column joint. The interaction between these effects is then achieved by the combination of sub-elements. This approach allows the hysteretic law of the individual sub-element to be simpler, while the member still exhibits a complex hysteretic behaviour through the interaction of the different sub-elements. The series model was formally introduced by Giberson (1967), although it had been reportedly used earlier. Its original form consists of a linear elastic element with one equivalent nonlinear rotational spring attached to each end. The inelastic deformations of the member are lumped into the end springs. This model is more versatile than the original Clough model, since it can describe more complex hysteretic behaviour by the selection of appropriate moment-rotation relationships for the end springs. This makes the model attractive for the phenomenological representation of the hysteretic behaviour of reinforced concrete members. Several other lumped plasticity constitutive models have been proposed. Such models include cyclic stiffness degradation in flexure and shear (Clough and Benuska, 1966, Takeda et al., 1970, Brancaleoni et al., 1983), pinching under reversal (Banon et al., 1981, Brancaleoni et al., 1983) and fixed end rotations at the beam-column joint interface due to bar pull-out (Otani, 1974, Filippou and Issa, 1988). Typically, axial-flexural coupling is neglected. Nonlinear rate constitutive representations have also been generalized from the basic endochronic theory formulation in Ozdemir (1981) to provide continuous hysteretic relationships for the nonlinear springs. An extensive discussion of the mathematical functions that are appropriate for such models is given by Iwan (1978). The dependence of flexural strength on the axial load under uniaxial and biaxial bending conditions has been explicitly included in the modelling of beams and structural walls. In most lumped plasticity models, the axial force-bending moment interaction is described by a yield surface for the stress resultants and an associated flow rule according to the tenets of classical plasticity theory (Prager and Hodge, 1951). The response is assumed to be linear for stress states that fall within the yield surface in which case the flexural and axial stiffness of the member are uncoupled and independent of the end loads. With the introduction of


123


multiple yield and loading surfaces and corresponding hardening rules, multilinear constitutive representations that include cracking and cyclic stiffness degradation are possible for the springs, as was originally suggested by Takayanagi and Schnobrich (1979). A lumped model is a simplification of the actual behaviour that involves the gradual spread of inelastic deformations into the member as a function of loading history. This modelling deficiency was recognized in several correlation studies, particularly those related to large resisting elements of flexural wall-frame structures (Charney and Bertero, 1982, Bertero et al., 1984). The basic advantage of the lumped model is in its simplicity in that it reduces storage requirements and computational cost and improves the numerical stability of the computations. Most lumped models, however, oversimplify certain important aspects of the hysteretic behaviour of reinforced concrete and are, therefore, limited in their applicability. One such limitation derives from restrictive a priori assumptions for the determination of the spring parameters. Parametric and theoretical studies of girders under monotonic loading by Anagnostopoulos (1981) demonstrate a strong dependence between model parameters and the imposed loading pattern and level of inelastic deformation. Neither factor is likely to remain constant during the dynamic response. The problem is further accentuated by the fluctuation of the axial force in the columns. Because of this history dependence, damage predictions at both the local and the global level may be grossly inaccurate. Such information can only be obtained with more refined models capable of describing the hysteretic behaviour of the section as a function of axial load. Another limitation of most lumped plasticity models, proposed to date, is their inability to describe adequately the deformation softening behaviour of reinforced concrete members. Such deformation softening can be observed as the reduction in lateral resistance of an axially loaded cantilever column under monotonically increasing lateral tip displacement. Again, more advanced models are needed in this case. The generalization of the rigid plastic theory concepts by Prager and Hodge (1951) to reinforced concrete column stress and strain resultant variables, such as bending moment and rotation and axial force and extension, limits the applicability of these models to well detailed members with large inelastic deformation capacity at the critical regions. For a reinforced concrete column section, the yield surface of the stress resultants is actually a function of a reference strain that couples the corresponding displacement components. This contradicts classical plasticity theory which does not account for deformation softening and assumes that the section deformability is unlimited. A refined resultant model has been proposed by El-Tawil and Deierlein (2001) that developed a bounding surface plasticity model implemented in the stress-resultant space. The model is generally applicable to steel, reinforced concrete and composite members. Two variations of the plasticity model are considered: a finite-surface and a degenerate-surface version. The former explicitly considers a fully elastic response region to exist within the inner surface and is, thus, applicable to steel members that typically have such behaviour. The degeneratesurface model shrinks the elastic region to a point and the section behaviour starts out as inelastic in any loading direction. This version is suitable for sections that have little or no elastic response region such as reinforced concrete and composite sections. Stiffness degradation is accounted for as a function of the plastic strain energy absorbed by the composite member. To overcome some of the limitations of classical plasticity theory in the description of the interaction between axial force and bending moments, Lai et al. (1984) proposed a fibre hinge model that consists of a linear elastic element extending over the entire length of the reinforced concrete member and has one inelastic element at each end. Each inelastic element is made up of one inelastic spring at each section corner, representing the longitudinal 124



reinforcing steel, and a central concrete spring that is effective in compression only. The five spring discretisation of the end sections is capable of simulating the axial force-biaxial bending moment interaction in reinforced concrete members in a more rational way than is possible by classical plasticity theory. In Lai's model, the force-deformation relationship for the effective steel springs follows that of Takeda et al. (1970) but the parameters that define the envelope are established from equilibrium considerations. A more accurate description of the inelastic behaviour of reinforced concrete members is possible with distributed nonlinearity models. These models are treated in more detail in the following section. 4.3.2

Distributed models

In distributed models, material nonlinearity can take place at any element section and the element behaviour is derived by a weighted integration of the section response. In practice, since the element integrals are evaluated numerically, only the behaviour of selected sections at the integration points is monitored. Either the element deformations or the element forces are the primary unknowns of the model and these are obtained by suitable interpolation functions from the global element displacements or forces, respectively. Discrete cracks in distributed models are represented as “smeared” over a finite length rather than treated explicitly. The constitutive behaviour of the cross-section is either formulated in accordance with classical plasticity theory, in terms of stress and strain resultants, or is explicitly derived by discretisation of the cross-section into fibres, as is the case in the spread plasticity fibre models. Frame models are usually based on either: i) Euler-Bernoulli beam theory in which plane sections remain plane and normal to the longitudinal axis of the beam; that is, there are no shear deformations (see Figure 4.1a); or ii) Timoshenko beam theory of plane sections remaining plane but not normal to the longitudinal axis with the difference between the normal and the plane section rotations being the shear deformation (see Figure 4.1b). Biaxial bending is a simple extension of the case of uniaxial bending, shown in Figure 4.1.

y

y

: 0 ( x) &

a' Deformed

b'

Deformed

dv0 dx

dv0 dx

v0 ( x)

a

Undeformed

v0 ( x)

Undeformed

b

(a) Euler-Bernoulli beam

b'

a

x

u0 ( x )

a'

x

b

u0 ( x )

(b) Timoshenko beam

Figure 4.1: Assumptions of Euler-Bernoulli and Timoshenko beam theories.


125


In the development of the theory that follows, the following notation is used with an element of length dx (Figure 4.2):

u0, v0 = longitudinal and transverse displacements, respectively, at the cross-section reference axis dv0 = total cross-section rotation dx

:0, & = cross-section deformations due to bending and shear, respectively. p

M

M+dM

dV dx dM V' dx p'

V+dV

V

dx

Figure 4.2: Beam segment of infinitesimal length.

Based on the forces acting on a segment of infinitesimal length (Figure 4.2), the following differential equations are derived: Euler-Bernoulli beam

d 2M Equilibrium: 'p dx 2

(4.1)

Section constitutive law: M ' EIf

(4.2)

Section compatibility: f ' Differential equation:

d 2v0 (curvature) dx2

(4.3)

d 2 e d 2 v0 b c EI 2 ` ' p dx 2 d dx a

Strain at any point of the cross-section: ) x '

(4.4)

d 2v du du0 ' 7 y 20 dx dx dx

(4.5)

Timoshenko beam

g dV hh dx ' 7 p Equilibrium: i hV 7 dM ' 0 hj dx

(4.6)

Section constitutive laws: V ' GAs&

126

M ' EIf

(4.7)



Section compatibility:

f'

d: 0 dx

(flexural deformation)

(4.8)

&'

dv0 7 :0 dx

(shear deformation)

(4.9)

g e d 2 v0 d: 0 b h GAs c 2 7 ` ' 7p dx dx a h d Differential equations: i d 2: 0 e dv0 b h 7 7 '0 GA EI : 0` h s cd dx dx 2 a j

(4.10)

In Eqs. 4.7 and 4.10, As is the “shear area”. To find the expression for As in the shear stress expression according to the Timoshenko beam theory, assume that the strain energy U) =

1 1 , xy& xy dA ' G G , xy2 dA G 2A 2 A

(4.11)

is the same as that found using the stresses and strains of the exact solution for an elastic section. For a rectangular cross-section it can be shown that As ' 1.2bh ' 1.2 A . Figure 4.3 compares the sectional shear stresses from Timoshenko beam theory for a rectangular, linearly elastic, section with that of the “exact” solution. SHEAR STRESSES IN RECTANGULAR SECTION

, xy

, xy

y x

h

b

, xy '

6V e h 2 2b c 7y ` bh3 d 4 a

"Exact" Theory

, xy ' const '

V GAs

Timoshenko Beam Theory

Figure 4.3: Shear stresses on a rectangular section.

Euler-Bernoulli beam elements

Two formulations are presented for frame elements, one based on the classical displacementbased approach, the other based on the force-based approach. Displacement-based formulation The displacement-based beam formulation uses the classical finite element approach to derive the element stiffness matrix and the element restoring force vector. The central step is the assumption of displacement fields along the element that are defined in terms of nodal displacements.


127


If U is the nodal displacement vector defined in Figure 4.4, the section displacements

u $ x % ' Hu0 $ x % v0 $ x %I are defined by: T

u $ x % ' NU $ x % U

(4.12)

where NU $ x % are the shape functions that define a linear axial displacement u0(x) and a cubic transverse displacement v0(x): Z x 0 X1 7 L X NU $ x % ' 2 3 X exb exb 1 7 3c ` 8 2 c ` X 0 dLa dLa Y

u1

0 xb e x c1 7 ` d La

x L

0

0

exb e xb 3c ` 7 2 c ` dLa d La

2

2

v1

v2

-1

W U U 2 e xb e xb U 7x c ` 8 x c ` U d La dLa V 0

3

u2

-2

U ' Hu1 v1 -1 u2

v2 - 2 I

T

Figure 4.4: Classical two-node 2D beam element.

Using the principle of minimum potential energy, or other equivalent variational principles, the following expressions are obtained for the element stiffness matrix K and for the element resisting forces P: K = G ! T $ x % k $ x % ! $ x % dx

(4.13)

P = G ! T $ x % s $ x % dx

(4.14)

L

L

where ! $ x % is an array containing the derivatives of the shape functions NU(x) (in particular the first row of B is the first derivative of the first row of NU, and the second row of B is the second derivative of the second row of NU) and k(x) is the cross-section stiffness matrix. The matrix k(x) depends on the cross-section model selected for the analyses and s(x) is the crosssection force vector, that is:

s $ x % = H N $ x % M $ x %I

T

(4.15)

The implementation of displacement-based frame elements in a general purpose finite element framework is straightforward, and this is the main reason why they have been widely used. On the other hand, the above frame element is exact only if the element cross-section is constant and the material behaviour is linear-elastic. In the case of reinforced concrete frames, where material nonlinearities become important, the element is approximate and this leads to the use of refined meshes to provide a good description of a frames response.

128



Force-based formulation The force-based beam formulation assumes force fields rather than displacement fields along the element. The element is developed here without rigid body modes (Figure 4.5), for reasons that will become clear later on in the chapter.

k1, -1

k 2, - 2

P = HM 1

M2

N, u

NI

T

Figure 4.5: Two-node 2D beam element without rigid body modes.

If P is the nodal force vector defined in Figure 4.5, the force distributions along the element are written as: s $ x% ' NP $ x% P

(4.16)

where P is the previously defined vector containing the cross-sectional forces and N P $ x % is the force interpolation functions array given by: 0 1W Z 0 X NP $ x% ' e x b e x b U Xc 7 1 ` c ` 0 U YXd L a d L a UV

(4.17)

Again, using the principle of minimum complementary potential energy, or other equivalent variational principles, the following expressions are obtained for the element flexibility matrix without rigid body modes F and for the corresponding nodal deformations U: F = G N TP $ x % f $ x % N P $ x % dx

(4.18)

U = G N TP $ x % e $ x % dx

(4.19)

L

L

where f(x) is the section flexibility matrix and e(x) is a vector containing the section deformations. That is:

e $ x % = H) 0 $ x % f $ x %I

T

(4.20)

The motivation and interest for force-based elements stem from fact that the equilibrium relationship (Eq. 4.16) is “exact” within the assumptions of the Euler-Bernoulli beam theory. In other words, the axial load and the bending moment remain constant and linear, respectively, irrespective of the beam cross-section variation and material behaviour. Element loads are included in the force-based formulation by modifying the element force distributions in Eq. 4.16.


129


One problem encountered with force-based elements is with their implementation in a general purpose finite element program. The element state determination, that is the computation of the element stiffness matrix K and resisting forced P, become complex. As for the stiffness matrix, this is “easily” computed by inverting the flexibility matrix:

K = F 71

(4.21)

and by adding the rigid body modes to K to obtain K. Computing the element resisting forces is a much more complex problem. The complexity stems from the fact that there is no way to directly relate section resisting forces and element resisting forces, as is the case with the displacement-based elements. Spacone et al. (1996) propose an iterative method to compute the “exact” element forces and deformations. The procedure is basically a Newton-Raphson iteration loop under imposed nodal displacements that adjusts the element forces and section deformations until there is compatibility between section deformations and imposed nodal deformations, as expressed by the weak statement of Eq. 4.19. The element is computationally more involved and expensive than the corresponding two-node displacement-based element but its precision leads to the use of a single element per structural member, thus leading to large savings in the global number of degrees of freedom to be solved for. Timoshenko beam elements

Similarly to the Euler-Bernoulli beam element, two formulations are presented for the Timoshenko beam element, one based on the classical displacement-based approach and the other based on the force-based approach. Displacement-based formulation The displacement-based formulation is similar to that of the Euler-Bernoulli beam but there are three independent fields: horizontal displacement, u0, vertical displacement, v0, and axis rotation, :0. A typical choice for these elements is to assume parabolic displacement fields for the three displacements. The resulting three-node element is shown Figure 4.6.

:1

v1

u1 1

:2

v2

u2 2

:3

v3

u3 3

Figure 4.6: Three-node 2D Timoshenko beam element.

In this case

U ' Hu1 v1 :1 u2

v2 : 2 u3 v3 : 3I

T

(4.22)

and

u $ x % ' Hu0 $ x % v0 $ x % : 0 $ x %I

T

(4.23)

Eq. 4.12 still holds true with the displacement interpolation vector NU(x) defined by

130



0 0 N2 $ x % 0 0 N3 $ x % 0 0 W Z N1 $ x % X U N1 $ x % 0 0 N2 $ x % 0 0 N3 $ x % 0 U NU $ x % ' X 0 XY 0 0 N1 $ x % 0 0 N2 $ x % 0 0 N 3 $ x % UV

(4.24)

The explicit expressions of the shape functions are given in natural coordinates in Figure 4.7.

N 2 $R % ' $1 7 R 2 %

1 N1 $R % ' R $R 7 1% 2 1

-1

Figure 4.7:

1 N 3 $R % ' R $R 8 1% 2

1

0

-1

1

R

Displacement interpolation functions for three-node 2D Timoshenko beam element in natural coordinates ( R ' 718 2 x L ).

The remainder of the element formulation is formally identical to that of the Euler-Bernoulli beam. The section force vector now becomes:

s $ x % = H N $ x % M $ x % V $ x %I

T

(4.25)

the corresponding section deformation vector is:

e $ x % = H) 0 $ x % f $ x % & $ x %I

T

(4.26)

and the vector B(x) changes accordingly. The resulting element is approximate even in the linear elastic case and several elements must be used to obtain a satisfactory approximation of the solution. Also, these elements may lock in the sense that too much energy goes into the shear deformation mode, leading to a stiff element. Force-based formulation The force-base formulation for a Timoshenko beam is identical to that of the Euler-Bernoulli beam with the exception that the expressions for the section forces s(x) and section deformations e(x) change according to the definitions given in the displacement-based formulations. The force interpolation functions become: Z 0 X ex b N P $ x % ' Xc 7 1 ` Xd L a X Y 1/ L

1W U exb U 0 c ` dLa U U 1/ L 0 V 0

(4.27)

The flexibility matrix F is once again “exact” within the assumptions of the Timoshenko beam theory, irrespective of the variation in beam section or material behaviour.


131


4.3.3

Section models: fibre elements vs. strut-and-tie

The definitions of the section stiffness matrix k(x) and flexibility matrix f(x) for both the Euler-Bernoulli and Timoshenko beams depend on the section models used in the formulations. Some of the models most commonly used in the published literature are discussed in this section. Rational modelling of reinforced concrete members is vital to perform structural health assessment on existing buildings, assess rehabilitation interventions on substandard constructions or in identification of potential causes of damage or collapse in buildings. Fibre analysis and STM procedures presented in this section can be used to evaluate the existing capacity of structures with or without damage. Once the behaviour and existing capacities of structural components are established, the deficient components can be upgraded in order to meet the capacity demands. Fibre (sectional) analysis is an appropriate tool that can be used to estimate behaviour of the B-regions of reinforced and prestressed concrete members. In D-regions, beam theory does not apply as the “plane sections remain plane” principle of the beam theory is not accurate or, necessarily, safe. In these regions strut-and-tie modelling (STM) can be used to estimate the capacity. It is important to appreciate that STM can be used to model Bernoulli beams in the way Ritter (1899) and Mörsch (1902) used concrete compression struts and tension ties (stirrups and longitudinal reinforcement) to explain the force flow and stress fields in a typical reinforced concrete beam. In this section fibre analysis procedures and strut and tie modelling techniques are treated in order to: "

perform (conventional and seismic) assessment on existing buildings,

"

assess rehabilitation interventions on inadequate constructions,

"

identify potential causes of damage or collapse in buildings.

Structural members can be divided into “B-regions” (i.e. Bernoulli regions or beam regions) and “D-regions” (i.e., disturbed regions or discontinuity regions.) In B-regions the beam theory applies and flexural behaviour of reinforced and prestressed concrete members can be established by using equilibrium, compatibility and constitutive relationships. Fibre (sectional) analysis is an appropriate tool that can be used to establish behaviour of the B-regions of reinforced and prestressed concrete members. In addition, through the use of the modified compression field theory (MCFT), it is possible to extend the use of fibre models to shear critical elements. In D-regions, strut-and-tie models, such as those proposed by Marti (1985a, 1985b) and Schlaich et al. (1987), can be used in order to estimate the capacity. Ritter and Mörsch planted the first seeds of what is currently known as STM by idealizing reinforced concrete beams with a series of diagonal concrete struts and reinforcing bars (ties). It is equally important to note that a significant amount of research has been conducted in this area since the original research of the aforementioned pioneers. Strut-and-tie modelling is a detailing and ultimate strength calculation procedure for discontinuity regions within structures. Strut-and-tie modelling represents a design method for complex structural details that has a basis in mechanics and is derived from the theory of plasticity but is simple enough to be readily applied in design. The method involves the idealization of a complex structural member into a simple collection of struts, ties, and nodes representing, in a general manner, the flow of stress paths within the member.

132



Fibre analysis and STM procedures presented in this section can be used to evaluate the existing capacity of structures with or without damage. Once the behaviour and existing capacities of structural components are established, the deficient components can be upgraded in order to meet the capacity demands. The methods presented in this section are of particular significance for this reason. Fibre analysis

Flexural behaviour of reinforced and prestressed concrete members can be estimated by using equilibrium, compatibility and constitutive relationships. Compatibility in fibre analysis is achieved through the “plane sections remain plane” hypothesis (Hooke, 1678, Bernoulli, 1705, and Navier, 1826). This geometric assumption forms the basis of the engineering beam theory that is used in sectional analysis of concrete members. Equilibrium is achieved by integrating stresses at any given section and equating them to the required sectional forces. Appropriate stress-strain relationships for concrete and reinforcing and/or prestressing steel need to be used in order to obtain the stress distribution through the depth of the section for a given strain profile. Concrete strain distribution can be defined by two variables (for example, two strains, a strain and curvature, etc.). If the strain profile is known, realistic stress-strain relationships for materials can be used to determine the stress distribution, which can subsequently be integrated to evaluate the resultant forces for concrete and reinforcing steel. In this way, the moment and axial load acting on the section can be determined. Conversely, if we know the strain profile we can evaluate the axial force and bending moment that caused these strains. The only detail that makes the evaluation of the response of flexural members a somewhat cumbersome task lies in the integration of the stress over the part of a concrete section in compression. Numerical integration can be performed to simplify this task with the section idealized as a series of rectangular (or trapezoidal) layers and with the stress in a given layer assumed as constant over the width of that layer. Figure 4.8 illustrates a typical example where the concrete section is subdivided into multiple layers to facilitate the numerical integration process or fibre analysis. The modified compression field theory (MCFT) of Vecchio and Collins (1982, 1986) has proven to be a simple, yet powerful, computational tool that can be used in FE analysis of reinforced concrete structures and in the sectional analysis for concrete members to predict the deformation and load carrying capacity. The latter will be discussed in this section.

yci

f’c , $ o ,

bi , hi , "ti

ysj

Asi , #$ pj , f ylj

f yt , Es

Figure 4.8: Fibre Analysis of a Reinforced Concrete Section.


133


The MCFT was based on reinforced concrete panel tests subjected to in-plane forces. Equilibrium, compatibility and stress-strain relationships based on test observations were used in the formulation of the models. Constitutive laws were used to link average stress-strain relationships both for the reinforcement and concrete. The MCFT is based on a rotating crack model that assumes that there is a gradual reorientation of principal stresses (and strains) in the concrete parallel to, and normal to, the direction of the cracks. Furthermore, the MCFT assumes enforced alignment of principal stress and strain directions and links the principal compressive stresses to principal compressive strains considering the effect of co-existing principal tensile strains. The use of MCFT within a layered section (fibre) analysis is discussed in Section 4.3.4. In this way the fibre analysis, discussed above, can be extended to include rational models to obtain shear force, axial force and bending moment interaction for reinforced and prestressed concrete sections and members. Strut-and-tie modelling

Strut-and-tie modelling (STM) has proven to be an efficient tool for analyzing structural members. The efficiency arises only in areas of discontinuity where plane sections do not remain plane, strains are not linear, and transverse strains may not be conservatively neglected. In these areas, typical layered sectional analysis does not provide a reasonable estimate to a section’s capacity due to violation of the basic assumptions made when analyzing the section. It should be noted, however, that any structural member may be analyzed to find its ultimate capacity using a STM. This is possible because the method results in a lower bound solution. For example, a typical Bernoulli beam may be analyzed using struts and ties as shown by Ritter (1899) and Mörsch (1902) in Figure 4.9. In his classic 1902 text, Mörsch explained the truss model shown in Figure 4.10 in more detail.

Figure 4.9: Ritter (1899) truss analogy.

(a) Truss Model

(b) Stirrup Forces

Figure 4.10: Morsch (1902): (a) truss model; (b) stirrup forces.

134



B-Regions D-Regions

h h-1.5h typ

h-1.5h typ

h

h-1.5h typ h1-1.5h1 typ

h2-1.5h2 typ

h

h

h1-1.5h1

h h2-1.5h2

h

Figure 4.11: Separation of Members into B- and D-regions.


135


According to the theory of plasticity, any statically admissible stress field that is in equilibrium with the applied loads and in which stress levels are on or within the material yield surface, constitutes a lower bound solution. However, the strain capacity of the materials is a fundamental requirement to fully satisfy that a lower bound solution occurs and, in that concrete is not an elasto-plastic material; some care is needed in determining an appropriate model. To begin a STM, the designer must first discretize the member into B- and D-regions. The B-regions are areas of the member where plane sections remain plane, strains are linear and transverse strains are negligible. D-regions are those in which sectional analysis does not apply because the aforementioned assumptions are not reasonable. These are areas under point loads, points of reaction, openings, re-entrant corners, frame joints, etc., as illustrated in Figure 4.11. The D-region is assumed to be 1 to 1.5 times the member height to either side of the disturbance, as per St. Venant’s principle. Once the D-regions are isolated they can be de-coupled from the rest of the member for analysis purposes. It is essential that the forces applied to the D-region are accurate. In some cases this step might prove to be most challenging to designers. A clear force path must be determined to find all external loads acting on the member. Once the external loads are determined then structural analysis must be completed to find external reactions and internal forces. If the member is not entirely comprised of single a D-region, then sectional analysis must be undertaken to determine internal forces from the adjacent B-regions, as shown in Figure 4.12. D region

B region

Figure 4.12: Forces on Boundaries of D-regions.

Once the forces are accurately placed on the de-coupled D-region, a STM or truss model may be applied. The truss model should accurately depict the flow of tensile and compressive forces within a statically determinate truss. The model may be chosen by experience and judgment of the designer, by following the elastic principal stresses produced by a finite element model (see Chapter 8), or by guidelines given by code bodies and/or the literature. The following considerations should be taken into account when choosing an appropriate STM for a D-region: "

136

Separate models may have to be conceived for different load cases. For example one model may be used when gravity loads control, and a separate model may be used when lateral loads control, with the region detailed to accommodate both models (Figure 4.13).



Tie

i. Gravity load conditions

Strut

ii. Seismic conditions

Figure 4.13: Alternate Models for Different Load Cases.

"

Serviceability of the member must be taken into account. The model chosen should account for serviceability by limiting cracking and provide sufficient stiffness to control deflections (Figure 4.14).

i. Service load model

ii. Ultimate load model

Figure 4.14: Model Variance on Service/Ultimate Conditions.

"

The physical geometry of the truss should be practical and free of congestion, with no overlapping struts or nodes.

"

When analyzing in-situ structural members, consideration shall be given to the existing condition of the concrete and physical placement of steel reinforcement. For example, if concrete is severely cracked in a location of a modelled strut, the strut capacity should be considerred negligible or zero. The centroid of modelled ties should coincide with the reinforcing steel placed in the member.

In general, optimizing the model used for final design and detailing of a D-region is a subjective and iterative process. Schlaich et al. (1987) proposed to follow the principles of minimum strain energy after cracking. That is, the model with the least and shortest amount of ties is the most appropriate based on the assumption that cracked concrete struts will deform little compared to steel-reinforced ties. However, as stated in FIP (1998) and mentioned above, this ultimate load model may not be valid when evaluating service conditions.


137


Once the nodal geometry is determined and, hence, the tie geometry, concentration is now given to determining the strut geometry. The struts at mid-height can be idealized using one of the stress fields shown in Figure 4.15.

bursting forces

Prismatic stress field

Fan stress field (no bursting forces)

Bottle stress field

(a) Types of concrete struts dc C

C/2

C/2

Tb /2

Tb /2 :

l b {bursting zone}

d c /4 d c /4

d c /4 d c /4

C

(b) Bursting forces in bottle shaped struts Figure 4.15: Concrete strut idealizations

The designer must exercise judgment in selecting strut geometry within the model. If a strut is located in the pure compression field of a member, or two nodes are modelled as close together, producing confined compression fields, a prismatic strut geometry should be considered. If the designer wishes to model a compression field at a beam support, in Mörsch’s fashion, then a fan of compression struts should be considered. If a strut is in a location that allows the compressive stresses to spread laterally then bottle shaped geometry needs to be considered.

138



In the case of bottle shaped struts, further model refinement is needed to account for the transverse tension associated the lateral spread of compression within the bottle shaped strut. For service loads, the spread of compression may be modelled at a slope of 2 longitudinal:1 transverse (tan : = 1/2) to the axis of the strut, whereas for the ultimate condition the dispersion angle narrows to tan : = 1/5 (Foster, 1998). When modelling the transverse tension, the designer should be cognizant of tie reinforcing anchoring the strut, the strut angle, and any reinforcing steel placed to confine the spread of compression; all of which influence the dispersion of compression and hence the model (Baumann, 1998). 4.3.4

Modelling of shear

Several older structures lack the shear reinforcement to guarantee that plastic hinge will form before shear failure of the member. Shear modelling in a frame analysis is dealt with at two levels, the element level and the section level. At the element level, the Timoshenko beam theory is typically used to model the shear behaviour. Displacement-based and force-based formulations for a Timoshenko beam are presented in Section 4.3.2. At the section level, different approaches can be followed. The simplest relies on stressresultant laws, where different nonlinear laws are given for the flexural, axial and shear response. One such model has been proposed by Martino et al. (2000), where a fibre model is used to model axial and bending responses, while a nonlinear law is used for the shear forceshear deformation response. While the shear response is decoupled from the other deformations at the section level, the implementation of this section model in a force-based element allows coupling between axial and bending responses at the element level. A fibre section model has been proposed for a Bernoulli beam by Petrangeli et al. (1999) that includes shear deformations. In this approach, each fibre has basically three deformations, axial strain, transverse strain (in the direction of the stirrups) and shear deformations, plus the corresponding stresses. Given the section deformations e(x), the axial strain and shear deformation of each fibre can be computed through compatibility. The third condition is that the stress in the direction of the vertical stirrups is zero; that is, the sum of the forces in the concrete and in the stirrups is zero. Given these three conditions (axial strain, shear deformation and vertical zero net stress), the fibre state determination consists of finding the corresponding axial stress, shear stress and vertical deformations and the fibre stiffnesses. The fibre stresses and fibre stiffness are added to give the section forces and the section stiffness. A concrete constitutive law based on the microplane theory is used to describe the concrete. The fibre section model with shear deformations proposed by Petrangeli et al. (1999) is implemented in a force-based element and has given results that correlate well with available experimental results. Finally, the MCFT can also be used to describe the shear deformation of a reinforced concrete member. The MCFT has proven to be successful in predicting the load deformation response of reinforced concrete beams, with different amounts of longitudinal and transverse reinforcement, using sectional analysis procedures (Vecchio and Collins, 1986). It has also been implemented into FE programs for similar analysis (for example, Vecchio, 1989, 1990, Foster and Gilbert, 1990, Foster, 1992, and many others). In the MCFT, constitutive relationships are used to link average stresses and strains for both reinforcement and concrete for analysis of reinforced concrete member behaviour. The stresses and strains used are average values realizing that the maximum and minimum values can be different to the average values. Conversely, local conditions may govern the failure but average conditions are likely to be more relevant in representing the member behaviour. Figure 4.16 illustrates the average stress and strain states employed in MCFT. ! fib Bulletin 45: Practitioners’ guide to finite element modelling of reinforced concrete structures

139


Figuure 4.16: Stresss and Strain Conditions: C M MCFT.

a assumed d to dependd only on thhe axial straains with The axiial stresses in the reinfforcement are any sheear stresses acting on the reinforcement neeglected. Thhe usual unniaxial streess-strain relation nship is used d for steel ass follows:

+ s ' Es) s ? f y

(4.28)

Strain hardening h b behaviour o the reinfforcement at of a high straain levels ccan be inco orporated using th he correspon nding stresss-strain relattionship fro om material tests. The aveerage stresss-strain relaationship fo or concrete is obtainedd from paneel tests and d can be found elsewhere e (V Vecchio an nd Collins, 1982). Prin ncipal stresss and principal strain axes are assumed d to be coin ncident as in rotating crack modeels. The beehaviour in panel tests showed that craacked conccrete subjeccted to hig gh tensile strains in the directiion normall to the compresssion is soffter and weeaker than in i standard cylinder teests. The cooncrete stresss in the principaal direction can be relatted to the prrincipal con ncrete strainn as follows: 2 e ) e ) 2 b b` c 2 c ` f c 2 ' f c 2 max c 2 7 c ` ` c )o d )o a ` d a

(4.29)

where fc2 is the priincipal com mpressive strress, "2 is th he principaal compressiive strain, and a "o is the straiin in concreete at peak compressive c e stress.

140



A number of models have been developed to determine the effect on concrete compressive strength, and stiffness, for elements under the influence of transverse tensile strains (fc2max in Eq. 4.29); for example, see Vecchio and Collins (1993), Belarbi and Hsu (1995), Kaufmann (1998) and Vecchio (2000), amongst others. One such relationship that has proved robust with time and can be used to compute fc2 is that of Vecchio and Collins (1986):

e b 1 `` f c 2 max ' f c# cc 0 . 8 0 . 34 ) ) 8 1 oa d

(4.30)

For concrete in tension, prior to cracking a linear elastic stress-strain relationship is used. The principal tensile stress fc1 is given as follows:

f c1 ' Ec)1

where )1 ? ) cr

(4.31)

where Ec is the tangent modulus of concrete, ) 1 is the principal tensile strain, fcr is the tensile strength of concrete, and ) cr is the cracking strain of concrete. After cracking, tension stiffening of concrete can be represented by:

f c1 '

f cr 1 8 500)1

(4.32)

The stress-strain relationships summarized above can be used in a sectional analysis to compute the load carrying and deformation capacity of reinforced concrete members. This will be discussed in the following section. A reinforced or prestressed concrete section can be discretized into a series of concrete strips with superimposed longitudinal steel components (Figure 4.8). Each concrete strip is defined by its width, b, depth, h, amount of transverse reinforcing steel, $t, and its position relative to the top of the beam, yc. Similarly, the longitudinal steel components are defined by their cross-sectional area, As, yield strength, fy, and position relative to the top of the beam, ys. Prestrain in reinforcement can also be considered in the analysis. Properties common to the entire cross-section are usually the concrete strength, f c# , concrete strain at peak stress, "o, yield strength of transverse steel, fy, and modulus of elasticity for reinforcing bars, Es. Sectional analysis is carried out assuming plane section remains plane with equilibrium and compatibility satisfied within each fibre. Hence, the estimates of the longitudinal strains and the shear flow distribution through the depth of the section are required to compute the stressstrain conditions in a reinforced concrete member subjected to a given set of sectional loads. First, an estimate of longitudinal strain distribution can be made. Strain in concrete for each strip is assumed to be constant. The stress conditions in steel layers can be directly computed from the assumed longitudinal strain distribution together with the uniaxial relationship of steel from material tests. Second, a shear stress distribution is assumed for the section such that the sum of shear stresses in each strip will be equal to the externally applied shear. Given the longitudinal strain and shear stress for a strip, equilibrium and compatibility must be satisfied to compute the compressive stress for the strip. The resultant of stresses for the strips must balance the applied sectional forces, axial force, N, moment, M, and shear force, V. For a section discretized with m concrete strips and n steel strips, equilibrium requires that


141


m

P i '1 m

P i '1

n

f li bi hi 8 P f slj Asj ' N j '1

n

f li bi hi y ci 8 P f slj Asj y sj ' M

m

Pv b h i '1

i i i

(4.33)

'V

where fli is the compressive stress in the longitudinal direction in the ith fibre, and vi is the shear stress in strip i. If these conditions are not satisfied, the longitudinal strains are adjusted until equilibrium is satisfied. To determine the correct shear stress distribution, a second section a small distance from the one considered can be analysed. Satisfying sectional equilibrium for each case, the assumed stress distribution can be checked using the free body diagram of the strip. The procedure summarized above is shown in a flow chart in Figure 4.17 with the stress and strain state of a section for a given set of loads computed using this procedure. Also, it is possible to analyse the overall response of a section by analysing different increments of longitudinal strains, shear stresses, and axial forces by using the procedure outlined above. Bayrak and Sheikh (2001) proposed a plastic hinge analysis technique that can be used in a sectional analysis platform in order to incorporate buckling of longitudinal bars in the analysis. This technique employs slightly different displacement compatibility requirements along with equilibrium considerations and constitutive relationships to evaluate the plastic hinge response of tied columns. The following is a step-by-step description of the procedure for the analysis of plastic hinges: 1) Standard sectional analysis procedure (Figure 4.17) is followed before the initiation of buckling of the longitudinal bars. 2) At the initiation of bar buckling, the tie forces generated as a result of core concrete bearing against the reinforcing bars are calculated. 3) By using proper boundary conditions and an assumed shape function for the forces acting on longitudinal bars, their outward deflection at the mid-height (e) between two sets of ties can be calculated. Dividing this deflection by the longitudinal bar diameter (d) yields an e/d ratio. 4) Using the e/d ratio calculated above and the ratio of the unsupported length of the longitudinal bars (l) to the bar diameter (l/d), the relevant stress-strain curve for reinforcing bars under compression can be selected and used as the constitutive relationship for compressed bars in a sectional analysis. Figure 4.18 illustrates experimentally determined stress-strain curves for reinforcing bars with initial imperfections tested under compression (Bayrak and Sheikh, 2001).

142



SPECIFY SEC TIO N PR O PER TIES

G IVEN ! l , v

SPECIFY C O N C RET E ST RIP PR O PERT IES

EST IM AT E ! d

SPEC IFY LO N G IT U D INAL ST EEL PRO PERT IES

EST IMAT E "’

SPECIFY SECT IO N LO AD S

D ET ERMINE ! t ,! dt , # lt , # m

EST IM AT E SH EAR ST RESS DIST R IBUT IO N

EST IM AT E LO NG IT U DIN AL ST RAIN D IST RIBUT IO N

D ET ERMINE f st , f t , f d , f dt , f l , f’d , "

C O M PU T E LO NG IT UD IN AL ST R ESSES DIST R IBUT IO N FO R EAC H REBAR LAYER

IS

" = "’ ?

CO MPU T E C O N C RET E ST RESSE/ST RAIN FO R EAC H C O N C RET E LAYER

C O MPU T E RESU LT ING SECTIO N LO ADS

NO

CH ECK SECT IO N AL EQ U ILIBRIU M ?

IS

fd = fd’ ?

NO

YES

C O M PU T E RESU LT ING SHEAR ST R ESS D IST R IBUT IO N

SH EAR ST R ESS D IST . SAME AS ASSU M ED ?

YES O U T PUT

YES

YES

REPEAT CALC U LAT IO N S FOR SECT IO N 2

NO

NO

Symbols: $ l : Longitudinal strain v : Shear stress $ d : Concrete principal compressive strain %’ : Assumed angle of inclination of principal compressive strain $ t : Transverse tensile strain $ dt : Principal tensile strain & lt : Normal shear strain & m : Maximum shear strain f st : Transverse tensile stress f t : Concrete transverse compressive stress f d : Concrete principal compressive strain f dt : Concrete principal tensile stress f l : Concrete longitudinal stress f’ d : Principal compressive stress computed using assumed shear stress % : Angle of inclination of principal compressive strain

Figure 4.17: Sectional Analysis for Shear, Flexure and Axial Loads.


143


Figure 4.18: Stress-strrain relationsh hips of reinforrcing bars.

4.3.5

M Modelling Bond Slip in Beams

Bond plays a centtral role in the response of reinfo orced concrrete membeers by allow wing the stress trransfer from m the steel bars b to the suurrounding concrete. Perfect P bondd is usually assumed a in the analysis off reinforcedd concrete structures and impliees full com mpatibility between concretee and reinfoorcement strrains. This assumption n is valid onnly in regionns where neegligible stress trransfer occuurs between the two com mponents (K Kauser and Mehlhorn, 1987) and can only take plaace at early loading stagges and at loow strain leevels. As thee load is inccreased, craacking as well as breaking of o bond unaavoidably occcurs and a certain am mount of bond-slip takees place, all of which w will, inn-turn, affecct the stresss distributions in both the t steel andd the concreete. Near the craccks, high boond-slips deevelop caussing relativee displacem ments betweeen concretee and the reinforccement steeel. Due to this bondd-slip, diffeerent strainns are obseerved in the steel reinforccing bars thhan in the surrounding s g concrete. At the cracks the loccalised straiin in the reinforccing steel caan be many times greatter that the average a straain and this is known as a “strain localisaation”.

144



Bond-slip affects the overall response of reinforced concrete structural members. In particular, two phenomena are worth discussing: 1. An increase in stiffness in the regions between two adjacent cracks. This effect, called tension stiffening, can be included by either considering the tensile resistance of the concrete or by increasing the tensile stiffness and strength of the steel reinforcing bars. Tension stiffening mostly affects the member response under serviceability conditions, since the effect of bond is completely lost when member failure occurs. 2. An increase of flexibility at the member ends, due to the pullout of the reinforcing bars at the interface either with beam-column joints or with the footings. Similar drops in stiffness may also be caused by insufficient lap splice lengths. These effects become particularly important and complex under seismic loading conditions, when bond gradually deteriorates due to large strains and damage caused by multiple load reversals. Rubiano-Benavides (1998) proposed the use of rotational springs at the element ends to account for the added flexibility due to bond-slip. This approach is suitable for lumped plasticity models but requires particular care in the selection of the rotational spring's mechanical properties. A similar approach is followed by Filippou et al. (1999). Another approach is followed by Monti and Spacone (2000), who modify the fibre section model to add the effects of bond slip, modelled according to Monti et al. (1997a, 1997b). The basic idea is fairly simple: the strain in the steel fibre is the sum of the actual strain in the reinforcing bar and the strain equivalent to the bar slip. This model accounts not only for the response of the reinforcing bar inside the beam but also for its anchorage outside the element, in either a structural joint or a footing. The steel fibre strain is given by the sum of the effects of the bar deformation and the anchorage slip. The response is still computed in terms of fibre stress and stiffness, which are needed for the fibre section state determination. The steel fibre strain is obtained from the section deformations using compatibility and is written as

) s8a ' ) s 8

1 ua ' ) s 8 ) a LIP

(4.34)

where )a is a strain-equivalent contribution of the anchorage pullout, condensed at the fibre level through the length LIP of the integration point. That is, the total steel fibre elongation is given by the sum of the reinforcing bar deformation and anchorage pullout. Compatibility is maintained between the concrete strain and the total steel fibre elongation ()c = )s+a), while concrete and steel strains are different ()c = )s). This procedure is illustrated in Figure 4.19 for the case of a slice with axial deformation only and no curvature. It is important to point out that the formulation proposed by Monti and Spacone (2000) provides a solution to the bond-slip problem within a single beam finite element. The element can capture the base rotation in RC columns due to bar slips and can also describe slip and bond failure at the bar splices. The fibre section model by Monti and Spacone (2000) is inserted in the force-based element by Spacone et al (1996).


145


Slice length LIP

Cross Section axial strain = ) curvature = 0

)5LIP = slice elongation Element with perfect bond

)5'5)c55'5)s

Element with partial bond

)5'5)c5'5)s 85)a

)a LIP

$/8)s) LIP

Figure 4.19: Slice response to axial deformation only (Monti and Spacone, 2000).

A different approach to modelling bond-slip was followed by Limkatanyu and Spacone (2002). In this model, the concrete element and the reinforcing bars are considered as different line elements connected at the nodes to give a frame model with bond slip. The procedure is demonstrated in Figure 4.20 with the free body diagram for a segment of length dx of a reinforced concrete frame element with n bars and with the bond interfaces shown. Only bond stresses tangential to the bars are considered in the formulation. Axial equilibrium in the beam component and in the bar i lead to:

dN B $ x % dx

n

8 P D bi $ x % ' 0 i '1

dN i $ x % 7 D bi $ x % ' 0, dx

(4.35)

i ' 1, n

where N B $ x % and N i $ x % are the axial forces in the beam and in bar i, respectively, and D b i $ x % is the bond interface force between the beam and bar i.

Vertical equilibrium of the infinitesimal segment dx yields:

dVB $ x % 7 py $ x % ' 0 dx

(4.36)

where VB $ x % is the beam section shear force and p y $ x % is the transverse distributed load. Finally, moment equilibrium gives:

dM B $ x % dx

146

7 VB $ x % 7

n

P y D $ x% i '1

i

bi

'0

(4.37)



py (x) R/C beam with bond-slip

beam reference axis

dx

= py (x)

concrete beam y1 yn

Db1(x) Mb (x) 8 dMb (x)

Mb (x)

beam reference axis

NB (x)

NB (x) 8 dNB (x) VB (x) 8 dVB (x)

VB (x)

Dbn(x)

+

bars bar 1 y1

Db1(x)

N1(x)

beam reference axis Dbn(x)

yn

bar n

N1(x) 8 dN1(x)

Nn (x) 8 dNn (x)

Nn (x)

Figure 4.20: Slice of RC frame element with bond slip (Limkatanyu and Spacone, 2002).

where M B $ x % is the beam section bending moment and yi is the distance of bar i from the element reference axis. The work by Limkatanyu and Spacone (2002) follows the EulerBernoulli beam theory and, thus, the shear deformations are neglected. The shear force VB $ x % is removed by combining the above equations to obtain:

d 2M B $ x% dx 2

n

dD bi $ x %

i '1

dx

7 p y $ x % 7 P yi

'0

(4.38)

Compatibility

As for the element compatibility, the reinforced concrete element is treated as a Bernoulli beam. On the other hand, the reinforcing bar slip are determined by the compatibility relationship between the beam and the bar displacements:

ubi $ x % ' ui $ x % 7 uB $ x % 8 yi

dvB $ x % dx

(4.39)

where ub i $ x % is the bond slip between the beam and bar i.


147


Starting from the above differential equations, Limkatanyu and Spacone (2002) proposed three formulations: a displacement-based, a force-based, and a two-field mixed formulation. While the details of the three formulations may be involved, it is important to point out that the displacement-based element with bond-slip, while computationally robust, is quite inaccurate and thus several elements per structural member must be used. On the other hand, force-based and mixed elements are much more precise and, though more complex to implement in a general purpose FE program, they can be used in a coarser mesh leading to computational cost savings. 4.3.6

Analysis of a section

All the tools needed for various steps described earlier are available at this point. Therefore, the response of a section located in the plastic hinge region of a concrete column can be predicted using the plastic hinge analysis procedure. At this point it should be appreciated that the equations presented herein are derived for cases where buckling over one tie spacing takes place. For cases where one or two tie sets rupture, the existence of ruptured tie sets can be ignored and spacing can be modified accordingly so that the equations presented herein can be used. In reality, according to the principal of minimum potential energy, the mechanism that would require minimum energy to be stored in the system is the governing mechanism of failure and, hence, that mechanism must be used in the analysis. Figure 4.21 illustrates the sectional response of four specimens (AS-3, AS-17, AS-18 and AS19) tested by Sheikh and Khoury (1993). The moment curvature predictions, obtained using the confined concrete stress-strain relationships suggested by Sheikh and Uzumeri (1982) (SU), Kent and Park (1971) (MKP) and Mander (1988) (MAN) are also shown in these figures. In the conventional sectional analyses performed to obtain the sectional responses, the aforementioned confined concrete stress-strain relationships are used for the core concrete. An unconfined concrete stress-strain relationship is used for the cover concrete and stressstrain relationships obtained from a tensile coupon test are used for longitudinal bars under tension and compression. Predictions obtained using the plastic hinge analysis procedure, with the Sheikh and Uzumeri model (SU+B) are also shown in Figure 4.21. The computer program, SecRes99, (Bayrak, 1999) was used to obtain the predictions shown in Figure 4.21. The use of the plastic hinge analysis procedure resulted in reasonably accurate predictions for the behaviour of sections located in the plastic hinge region of the test specimens. Conventional sectional analyses (fibre-analyses) could not provide a reasonable prediction for the ultimate curvatures. With the use of the plastic hinge analysis technique, the reduction in the load carrying capacity of longitudinal bars as a result of their buckling can be predicted. The ultimate curvature and, hence, the failure of a column can therefore be determined with reasonable accuracy.

4.4

Interpretation of results

4.4.1

Localisation problems

Reinforced concrete frame elements, similarly to concrete solid elements, can lead to numerical inconsistencies that derive from localization problems. When the cross-section response starts softening (as may be the case of reinforced concrete column sections), the element response becomes non unique and depends on the number of integration points per element and/or the number of elements per structural member. Localization issues, though similar in nature, tend to be different in displacement-based and in force-based frame

148



Figure 4.21: Experimental and predicted sectional responses .


149


elements. Localization in displacement-based solid finite elements has been studied by Bažant and Oh (1983), Bažant and Planas (1998) de Borst et al. (1994), among others and the theory extends to frame elements. Coleman and Spacone (2001), Scott and Fenves (2006) and Valipour and Foster (2007) discuss localization in force-based elements and propose various solution strategies. To illustrate the numerical problems encountered in force-based frame elements when plastic hinges form, we will consider the case of a steel cantilever beam under an imposed transverse tip displacement. A single force-based element is used for the entire member. As the applied tip displacement increases, a plastic hinge forms at the base where the maximum moment occurs. Figure 4.22 illustrates the response of the force-based element for the cantilever beam with an elastic-strain hardening section behaviour. Unloading is prescribed in the final steps to clarify the peak displacement and curvature demands. The base shear is plotted against the tip displacement on the right and the base curvature (i.e. the curvature of the first integration point) is shown on the left. The response is objective at both the element and the section levels for models with four or more integration points. Three integration points do not accurately integrate the element integrals leading to over-prediction of the stiffness in the strain-hardening region. Base Shear 3 IP 3 IP

4 IP 5, 6, 7, 8 IP

4, 5, 6, 7, 8 IP

O 3, 4, 5, 6, 7, 8 Integration Points (IP) Elastic-strain hardening moment-curvature Curvature

Displacement, O

Figure 4.22: Cantilever beam with elastic-strain hardening section response (Colemanand Spacone, 2001).

Figure 4.23 shows the response of the same force-based element to an imposed tip displacement with an elastic-perfectly plastic moment-curvature behaviour. Here, the prediction of the element force-displacement response remains objective while the peak curvature demand varies with the number of integration points. The loss of objective curvature prediction is due to the localization of the inelastic curvature at the base integration point. When this bottom section reaches the plastic moment, the column reaches its load carrying capacity. As the tip displacement increases, the curvature of the base integration point increases with constant (plastic) moment, while all the other integration points remain linear elastic and do not see any change in either curvature or moment. The length of the base integration point and, thus, the plastic hinge length, becomes a function of the number of integration points used. As the number of integration points increases, the plastic hinge length decreases and the curvature demand in the base integration point must increase to give the same prescribed tip displacement. From here the non-objective section prediction of Figure 4.23 is obtained.

150



Base Shear

3, 4, 5, 6, 7, 8 IP 8 IP

7 IP

6 IP 5 IP 4 IP 3 IP

O 3, 4, 5, 6, 7, 8 Integration Points (IP) Elastic-perfectly plastic moment-curvature Displacement, O

Curvature

Figure 4.23: Cantilever beam with elastic-perfectly plastic section response (Coleman and Spacone, 2001).

In the case of a softening moment-curvature response, the loss of objectivity is more pronounced. Softening section responses may take place in reinforced concrete columns or in reinforced concrete bridge piers that support a substantial dead load and are subjected to seismic forces. Figure 4.24 illustrates the response of a reinforced concrete column modelled with a single force-based element. Both the local base section moment-curvature response and the global base shear-displacement response lose objectivity. As the number of integration points increases from three to five, the length of the first integration point decreases and increasing curvatures are required to achieve the same prescribed tip displacement. The concrete fibre compressive strains in the hinge region quickly increase resulting in rapidly degrading material stiffness. For larger numbers of integration points, the post-peak response becomes brittle and, with increasing numbers of integration points, snap-back may even occur.

P (constant) O

3, 4, 5 Integration Points (IP)

Base Shear

3 IP R/C beam-column

3 IP

4 IP

4 IP

5 IP 5 IP

Curvature

Displacement, O

Figure 4.24: RC beam-column modelled with strain softening section response (Coleman and Spacone, 2001).


151


Figure 4.25 shows the schematic bending moment and curvature distributions for three, four and five integration points along the column height. This case refers to the elastic-perfectly plastic section response of Figure 4.23 and helps clarify the localization process in forcebased elements. The plastic curvature Cp occurs when the plastic moment Mp is first reached. Since equilibrium is strictly satisfied in the force-based element, the bending moment remains linear. When the plastic moment Mp is reached at the first (base) integration point, the applied force cannot increase and the tip displacement increases under constant applied load (and constant base moment). Because the bending moment cannot increase beyond Mp, adjacent integration points remain elastic and the inelastic curvature localizes at the first integration point. The tip displacement is computed as the weighted sum of the curvatures at the integration points. The first integration point has a finite length LIP=1 = w1L proportional to its integration weight w1. The larger the number of integration points, the shorter the length of the first integration point and the larger the curvature at the first integration point to obtain the same tip displacement, as shown in Figure 4.25.

Gauss-Lobatto integration point

Moment

Curvature

Mp C p A) 3 Integration Points

Mp

Cp

B) 4 Integration Points

Mp

Cp

C) 5 Integration Points

Figure 4.25: Moment and curvature profiles for an elastic-perfectly plastic cantilever modelled with a single force-based element (Coleman and Spacone, 2001).

In summary, it is the ability to capture a jump from elastic to inelastic behaviour that makes the force-based formulation both attractive and prone to unique numerical problems. One can observe that as the distance between the first (plastic) and second (elastic) integration point varies, the response also varies. This implies that the number and placement of the integration points not only influences the accuracy of the integration but also the post-peak response. For hardening materials, plasticity usually spreads beyond a single integration point and numerical problems are limited to a non-smooth response if too few integration points are used (Figure 4.22). For perfectly plastic and softening cross-section responses, the curvature tends to localize at a particular integration point and problems with objectivity arise (Figure 4.23 and Figure 4.24). For further reading on the state of research into the effects of localisation in fibre based elements and on methods for obtaining objective results, the reader is referred to Scott and Fenves (2006) and Valipour and Foster (2007). 4.4.2

Physical characteristics of localised failure in concrete

In order to introduce a method for re-establishing objectivity of the force-based element response, a brief discussion of the physical characteristics of concrete failure is necessary. Concrete is a heterogeneous material prone to localized failure. In tension, failure is characterized by numerous micro-cracks that bridge together to form a main crack. The 152



discontinuity induced by the crack opening basically prevents stresses normal to the crack. The failure is termed localized because material outside the fracture process zone remains practically undamaged. In compression, concrete failure is a different phenomenon to that of tension. Figure 4.26 illustrates a typical laboratory test of a concrete cylinder subjected to uniaxial compression. The test occurs under displacement-control in order to capture the post-peak response of the specimen. Region B in Figure 4.26 corresponds to a region of damaged concrete. Damage in this region, initially characterized by axial splitting, progresses until a sliding shear band forms and the cylinder stiffness rapidly degrades. The stress-strain curve in this damaged region enters the post-peak branch, where a drop in stress is associated with increasing strains. The concrete in the regions labelled A in Figure 4.26 is not severely damaged and has not reached the peak strength. To maintain equilibrium of axial force, the stress-strain response in regions A unloads elastically. +

O +

A

L OlL overall response

B

region B

)

region A

)

+

A

Figure 4.26: Uniaxial compression test under displacement control (Coleman and Spacone, 2001).

The concrete cylinder can be idealized as a series system, where one element (region B in Figure 4.26) represents the weak link. When region B reaches the peak strength and starts unloading, the concrete in regions A must unload in order to maintain equilibrium. From these observations it is concluded that concrete failure in compression, as with tension, occurs in a localized manner and requires special attention in a numerical model. 4.4.3

Regularisation techniques for force-based frame elements

Constant fracture energy criterion

The concept of constant fracture energy in tension is widely used to regularize mesh-sensitive smeared crack displacement-based elements in continuum FE analyses (Bažant and Oh, 1983, Bažant and Planas, 1998, among others). The concept is applied here to force-based beam elements that soften in compression. While the constant fracture energy concept is not as widely accepted for compression as it is for tension, experimental research (Lee and Willam, 1997, Jansen and Shah, 1997) and analytical investigations (Markeset and Hillerborg, 1995) show that the theory also holds true also for localization in compression. The main idea of the regularization process is to assume that the uniaxial stress-strain relationship for concrete is supplemented by an additional material parameter, the fracture energy in compression Gcf , defined as:


153


G cf ' G +dui

(4.40)

where + is the concrete stress and ui the inelastic displacement. The integral represents the area under the post-peak portion of the compressive stress-displacement curve. This relationship mimics the tensile fracture energy with a superscript c to indicate compression. To adapt the fracture energy concept to general use, Eq. 4.40 may be rewritten in terms of stress and strain:

G cf ' hG +d)i ' LIP G +d)i

(4.41)

where )i indicates inelastic strain and h is a length scale. For smeared crack elements, h represents the size of a single element in the crush band. For force-based frame elements, h becomes the length of the softening integration point LIP. Even though the proposed procedure is general, the regularization is applied here to the Kent and Park (1971) law used for the concrete fibres of the fibre section. The pre-peak behaviour is given by a parabola, followed by a linear post-peak softening branch until a stress of 0.2 f c# is reached at a prescribed strain labelled ) 20 . The residual strength is assumed to remain constant for strains larger than ) 20 . The fracture energy is defined here from the peak stress fc# until the end of the softening branch (Figure 4.27). This definition is similar to that of Jansen and Shah (1997), where the fracture energy is defined from the peak stress until 0.33 f c# . It follows that ) 20 must be calibrated to maintain a constant energy release. Assuming that G cf is known from experimental tests (Jansen and Shah, 1997), Eq. 4.41, together with the definition of G cf from Figure 4.27, leads to:

) 20

G cf

0.8 f c' ' 7 8 )0 E 0.6 f c' LIP

(4.42)

where E is Young’s modulus and ) 0 is the strain corresponding to peak stress as indicated in Figure 4.27. +

f c'

G cf h

E

0.2 f c' )o

)20

)

Figure 4.27: Kent and Park (1971) stress-strain law and compression fracture energy (Coleman and Spacone, 2001).

154



Theoretically, Eq. 4.42 implies that the constitutive law must be calibrated for each separate integration point in a structural model. In practice, plastic hinges generally form at the element ends where the extreme integration points lie. If all elements in a model are integrated with the same scheme and with a number of integration points, ) 20 only varies for elements of different length, L. Linking the constitutive law to the element length is a straightforward process. As the number of integration points increases, the extreme integration point must furnish larger inelastic strains to satisfy the constant fracture energy criteria. This is equivalent to assuming a constant stress-displacement relationship (Figure 4.28b), rather than a constant stress-strain law as shown in Figure 4.28a.

+ fc’

Given: L, f’ c, )p, G cf wIP varies

+ 4 IP 5 IP 6 IP

4, 5, 6 IP

)

)o (a)

O

(b)

Figure 4.28: Stress-strain curves with constant fracture energy: a) stress-strain curves for 4, 5, and 6 IP schemes; b) stress-displacement curve (Coleman and Spacone, 2001).

Curvature post-processing

Once the global force-displacement response is regularized, in some cases there is still a need to post-process the results to obtain an objective prediction of the curvature demand in the plastic hinge region. This is due to the fact that the plastic zone length is the length of the first integration point and does not necessarily correspond to the physical length of the plastic hinge. As with the inelastic strains in a smeared crack model, the constant fracture energy criterion is insufficient for regularization of the internal element deformations. Because different mesh sizes in a smeared crack model must produce the same crack opening displacements, the magnitude of inelastic strains must vary. While the inelastic strains in a smeared crack solid finite element do not have any physical meaning and can remain nonobjective, inelastic curvature demands may be of particular interest in frame elements. A simple procedure is presented below to obtain an objective prediction of the true curvature demand on the member. Figure 4.29 shows a deformed interior beam with plastic hinges forming at both ends. The plastic rotation - i and the relevant displacements O i are indicated at the near (N) and far (F) ends of the element. These are essentially fictitious quantities used to formulate the curvature scaling law. The bending moment diagram and the curvature profile corresponding to elasticperfectly plastic behaviour are also shown in Figure 4.29. Inelastic curvatures, indicated by Ci, are concentrated at the extreme integration points and spread over a length LIP.


155


Near Node (N)

Far Node (F)

O iF

- iN

- iF O iN

L Mp Moment diagram:

LIP

Mp

!i Curvature profile:

M EI

Ce ' LIP

Figure 4.29: Beam with plastic hinge at each end (Coleman and Spacone, 2001).

The total curvature in the plastic hinge region can be separated into elastic and inelastic curvature components as C ' Ce 8 Ci . Considering the geometry of Figure 4.29, the inelastic hinge rotation is - i ' Ci LIP . Neglecting the elastic curvature in the other integration points and using a small angle approximation for - i , the inelastic curvature can be approximated as:

CiMODEL m

Oi ZL L W LIP X 7 IP U 2 V Y2

(4.43)

MODEL

where Ci indicates the inelastic curvature that results from the analysis. Substituting the actual length of the plastic hinge Lp for LIP yields a similar approximation for the true curvature demand:

CiPREDICT m

Oi Z L Lp W Lp X 7 U Y2 2 V

(4.44)

PREDICT

indicates the inelastic curvature demand based on the assumed plastic hinge where Ci length Lp. Finally the model output can be scaled according to:

C ' Ce 8 (scalefactor ) CiMODEL

(4.45)

The scale factor is computed by solving for O i in Eq. 4.43 and substituting into Eq. 4.44 to obtain:

156



scalefactor '

wIP L2 (1 7 wIP ) Lp (L 7 Lp )

(4.46)

The double-curvature case shown in Figure 4.29 prevails in the analyses of buildings under lateral loads. On the other hand, some structural members, such as cantilevers, experience single curvature and the plastic hinge forms at one end only. Such is the case of bridge piers subjected to seismic loads in the transverse direction. The double-curvature derivation is readily recast in terms of single curvature by replacing the L 2 term in the denominator of Eqs. 4.43 and 4.44 with the full length L. Taking this single-curvature approach results in: scalefactor '

wIP L2 (2 7 wIP ) L p 2( L 7 L p )

(4.47)

From the previous discussion, it appears that if the length of the first integration point corresponds to the length of the plastic hinge, that is if LIP ' Lp , no post-processing of the curvature is needed because the curvature is objectively predicted by the element. One might then suggest to select the number of Gauss-Lobatto integration points in such a way that LIP [ L p . This may cause two problems: 1) the number of integration points may be too small for short elements (causing undesirable reduced integration) or too large for long elements (increasing the computational cost); and 2) in most cases, the length of the element would also have to be adjusted, thus introducing an additional element in each member and greatly increasing the computational cost of the analyses. Different integration schemes in which the user can define the length of the integration points would solve the issue but may compromise the accuracy of the numerical integration. The regularization approach described above is general and does not affect either the element formulation or the integration scheme. 4.4.4

Practical considerations

Some practical considerations are presented here to help analysts when using nonlinear elements for static and dynamic analyses of frames. When a fibre section model is used, one of the important question is how many fibres or layers are needed to discretize the section. Spacone (1994) studied the sensitivity of a simple circular section cantilever beam response to the number of section fibres (Figure 4.30). Their results, and those of similar parametric studies, indicate that using too many fibres overkills the problem and increases the computational cost of the analyses. While no precise indication on the number of fibres to use exists, it appears that for a rectangular section fifteen layers are a good compromise between accuracy and computational cost in a simple bending analysis. On the other hand, users should always test their section discretization by comparing the results of different, more and less refined, fibre meshes to assess the optimal fibre section discretization. Such initial tests should be performed on simple structures such as a cantilever beam with or without axial load.


157


200

1000 kips COMPRESSIVE AXIAL FORCE GAUSS-LOBATTO INTEGRATION

P,d

SHEAR P (kips)

100

55.78 ft

18 #24

4 integration points

0

-100

-200

-2

-1

0

1

2

TIP DISPLACEMENT d (ft)

Figure 4.30: Response of a cantilever beam with circular cross section for three different section mesh refinements.

A similar problem applies to the number of integration points to use in an analysis. Two integration schemes are typically used for line elements, Gauss and Gauss-Lobatto. The two schemes, applied to a natural [-1,1] domain, are given by 1

I '

G

m

71 1

I '

G

71

f $R % dR ' P wh f $Rh %

(4.48)

h '1

m71

f $R % dR ' w1 f $R1 ' 71% 8 P wh f $Rh % 8 wm f $Rm ' 1%

(4.49)

h'2

where Eq. 4.48 is the Gauss scheme, and Eq. 4.49 is the Gauss-Lobatto scheme (Stroud and Secrest, 1966). In Eq. 4.49, h denotes the monitored section and %& is the corresponding weight factor. The Gauss scheme with m integration points permits the exact integration of polynomials of degree up to (2m-1), while the Gauss-Lobatto scheme with m integration points integrates exactly polynomials of degree up to (2m-3). The Gauss scheme is preferred in displacement-based elements, while the Gauss-Lobatto scheme is preferred in force-based elements where monitoring the end sections is important (Spacone et al., 1996). Similar to the issue of the number of fibres, the selection of the number of integration points is important, because too few points may lead to inaccurate results, while too many integration points in computationally inefficient and can cause severe strain localization problems. Figure 4.31 and Figure 4.32 show a study on the sensitivity of the response of a circular cantilever beam to the number of integration points. Two to five Gauss-Lobatto and Gauss integration points are used in a single force-based element. From the figures, it appears that four or five Gauss-Lobatto or Gauss integration points are sufficient. Although these results have not been generalized, experience shows that more than five integration points are

158



200

1000 kips COMPRESSIVE AXIAL FORCE GAUSS-LOBATTO INTEGRATION

P,d

SHEAR P (kips)

100

55.78 ft

18 #24

0

nGL = 2 -100

nGL = 3 nGL = 4 nGL = 5

-200

-2

-1

0

1

2


Figure 4.31: Response of a cantilever beam with circular cross-section for different numbers of Gauss-Lobatto integration points.

200

P,d

SHEAR P (kips)

100

55.78 ft

18 #24

0

nG = 2 nG = 3 nG = 4

-100

nG = 5 1000 kips COMPRESSIVE AXIAL FORCE GAUSS INTEGRATION

-200

-2

-1

0

1

2


Figure 4.32: Response of a cantilever beam with circular cross-section for different numbers of Gauss integration points.


159


not needed for accuracy. Analysts are, however, strongly encouraged to run preliminary parametric studies on a simple structure (a cantilever beam, for example) to assess the number of integration points necessary to reach a converged solution. As discussed in the previous section on strain localization, the selection of the number of integration points is closely related to the length of the plastic hinge that may form in the structural member being considered. Without the inclusion non-localisation modelling (see Valipour and Foster, 2007), plastic hinges tend to localize at a single integration point in force-based elements, while plastic hinges tend to extend to entire elements in displacementbased elements. The details of the localization characteristics of force-based elements have been previously discussed and should guide analysts in selecting the number of integration points in an element that is likely to develop plastic hinges. Furthermore, it is important to note that the plastic hinge length Lp is a quantity that is not easily computed. The following expression suggested by Paulay and Priestley (1992) suggested that the plastic hinge length be taken as:

Lp ' 0.08 L 8 0.022 f y db (kN, mm)

(4.50)

where L is the distance from the plastic hinge to the nearest zero moment point, fy is the steel yield stress and db is the bar diameter. There exist other formulas in the published literature for estimating the plastic hinge length, mostly obtained from experimental results combined with analytical studies.

4.5

References

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Foster S.J., and Gilbert R.I. (1990), “Non-linear Finite Element Model for Reinforced Concrete Deep Beams and Panels”, UNICIV Report No.R-275, School of Civil Engineering, University of New South Wales, Kensington, December, 113 pp. Giberson, M. (1967). “The Response of Nonlinear Multi-Storey Structures Subjected to Earthquake Excitations”, Earthquake Engineering Research Laboratory, Pasadena. Hooke, R. (1678), “Lectures De Potentia Restitutiva, or of Spring Explaining the Power of Springing Bodies,” printed for John Martyn Printer to The Royal Society, at the Bell in St. Paul’s Church-Yard, 24 pp. Iwan, W. (1978), “Application of Nonlinear Analysis Techniques”, in, Iwan W. ed., Applied Mechanics in Earthquake Engineering, ASME, AMD, 8, New York, pp. 135-161. Jansen, D.C. and Shah, S.P. (1997), “Effect of length on compressive strain-softening of concrete.” J. Eng. Mech., ASCE, Vol. 123, No. 1., pp. 25-35. Kaufmann, W. (1998), “Strength and deformations of structural concrete subjected to in-plane shear and normal forces”, Report No. 234, Institute of Structural Engineering, ETH, Zurich, Switzerland. Kauser, M., and Mehlhorn, G. (1987), “Finite elements models for bond problems.” Journal of Structural Engineering, ASCE, Vol. 113, No. 10, pp. 2160-2173. Kent, D.C. and Park, R. (1971), “Flexural Members with Confined Concrete,” Journal of the Structural Division, ASCE, Vol. 97, ST7, July 1971, pp. 1969-1990. Lai, S., Will, G. and Otani, S. (1984), “Model for Inelastic Biaxial Bending of Concrete Members”, Journal of Structural Engineering, ASCE, 110(ST11), pp. 2563-2584. Lee, Y-H., and Willam, K. (1997), “Mechanical properties of concrete in uniaxial compression.” ACI Materials Journal, Vol. 94, No. 6, pp. 457-471. Limkatanyu, S. and Spacone, E. (2002), “R/C Frame Element with Bond Interfaces. Part 1: Displacement-Based, Force-Based and Mixed Formulations. Part 2: Element State Determination and Numerical Validation.” ASCE Journal of Structural Engineering, Vol. 128, No. 3, pp. 346-364. Mander, J.B., Priestley, M.J.N., and Park, R. (1988), “Observed Stress-Strain Behaviour of Confined Concrete,” Journal of Structural Division, ASCE, Vol. 114, No. 8, August 1988, pp. 1827-1849. Markeset, G., and Hillerborg, A. (1995), “Softening of concrete in compression – localization and size effects.” Cement and Concrete Research, Vol. 25, No. 44, pp. 702-708. Marti, P. (1985a), Basic Tools of Reinforced Concrete Beam Design, ACI Journal Proceedings, Vol. 7, No. 1, Jan.-Feb., pp. 46-56. Marti, P. (1985b), Truss Models in Detailing, Concrete International, American Concrete Institute, Vol. 7, No. 12, Dec. pp. 66-73.

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Martino, R., Spacone, E., and Kingsley, G. (2000), “Nonlinear Pushover Analysis of RC Structures.” ASCE Structures Congress, Advanced Technology in Structural Engineering, M. Elgaaly editor, Philadelphia, PA, May. 8 pp (CD-ROM). Monti, G., Filippou, F.C., and Spacone, E. (1997a), Finite element for anchored bars under cyclic load reversals. Journal of Structural Engineering, ASCE, Vol. 129, No. 5, May Monti, G., Filippou, F.C., Spacone, E. (1997b), Analysis of hysteretic behavior of anchored reinforcing bars. ACI Structural Journal, Vol. 94, No. 2, May-June Monti, G., and Spacone, E. (2000), “Reinforced Concrete Fiber Beam Element with BondSlip.” ASCE Journal of Structural Engineering, Vol. 126, No. 6, pp. 654-661. Mörsch, E. (1902), “Der Eisenbetonbau, seine Theorie und Anwendung (Reinforced Concrete, Theory and Application),” Stuggart, Germany. Navier, C-L. (1826), “Resume des lecons… de la resistance des corps solides,” Paris. Otani, S. (1974), “Inelastic Analysis of R/C Frame Structures”, Journal of the Structural Division, ASCE, 100(ST7). Ozdemir, H. (1981), “Nonlinear Transient Dynamic Analysis of Yielding Structures”, Ph. D. Thesis, Department of Civil Engineering, University of California, Berkeley. Paulay, T., and Priestley, M.J.N. (1992), “Seismic design of reinforced concrete and masonry buildings”, Wiley, New York Petrangeli, M., Pinto, P.E. and Ciampi, V. (1999), “Fiber element for cyclic bending and shear of RC structures. I: theory “ ASCE Journal of Engineering Mechanics, Vol. 125, No. 9, pp. 994-1001. Prager, W. and Hodge, P. (1951), Theory of Perfectly Plastic Solids, John Wiley and Sons, New York. Ritter, W. (1899), Die Bauweise Hennebique (Construction Techniques of Hennebique), Schweizerische Bauzeitung, Zurich, February. Rubiano-Benavides, N.R. (1998), “Predictions of the inelastic seismic response of concrete structures including shear deformations and anchorage slip.” Ph.D. dissertation, Department of Civil Engineering, University of Texas, Austin. Schlaich, J., Schäfer, K., and Jennewein, M. (1987), “Toward a Consistent Design of Structural Concrete,” Journal of the Prestressed Concrete Institute, Vol. 32, No. 3, May-June, pp. 74-150. Scott, M.H., and Fenves, G.L. (2006), “Plastic hinge integration methods for force-based beam-column elements”. Journal of Structural Engineering, Vol. 132, No. 2, pp. 244-252. Sheikh, S.A. and Khoury, S.S. (1993), “Confined Concrete Columns with Stubs,” ACI Structural Journal, V. 90, No. 4, July-August 1993, pp. 414-431.


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Sheikh, S.A. and Uzumeri, S.M. (1982), “Analytical Model for Concrete Confinement in Tied Columns,” Journal of the Structural Division, ASCE, Vol. 108, ST12, December 1982, pp. 2703-2722. Spacone, E. (1994), “Flexibility-Based Finite Element Models for the Nonlinear Static and Dynamic Analysis of Concrete Frame Structures” Ph.D. Dissertation, Department of Civil Engineering, University of California, Berkeley. Spacone, E., Filippou, F.C., and Taucer, F.F. (1996), “Fiber Beam-Column Model for Nonlinear Analysis of R/C Frames. I: Formulation. II: Applications.” Earthquake Engineering and Structural Dynamics, Vol. 25, No. 7, pp. 711-742. Takayanagi, T. and Schnobrich, W. (1979), “Non Linear Analysis of Coupled Wall Systems”, Earthquake Engineering and Structural Dynamics, Vol. 7, pp. 1-22. Takeda, T., Sozen, M.A., and Nielsen, N. (1970), “Reinforced Concrete Response to Simulated Earthquakes”, Journal of Structural Engineering, ASCE, 96(ST12), pp. 2557-2573. Takizawa, H. (1976), “Notes on Some Basic Problems in Inelastic Analysis of Planar RC Structures”, Trans. of Arch. Inst. of Japan, 240, Part I in February, pp. 51-62, Part II in March, pp. 65-77. Valipour H.R., and Foster, S.J. (2007), “A Novel Flexibility Based Beam-Column Element for Nonlinear Analysis of Reinforced Concrete Frames”, UNICIV Report No. R-447, School of Civil and Environmental Engineering, The University of New South Wales, September. Vecchio, F.J. (1989), “Nonlinear Finite Element Analysis of Reinforced Concrete Membrane Elements”, ACI Journal, Proceedings Vol. 86, No.1, Jan-Feb., pp. 26-35. Vecchio, F.J. (1990), “Reinforced Concrete Membrane Element Formulation”, Journal of Structural Engineering, Vol. 97, No. 1, pp.102-110. Vecchio, F.J. (2000), “Disturbed stress field model for reinforced concrete: formulation”, J. Struct. Eng., Vol. 126, No.9, 1070–1077. Vecchio, F., and Collins, M.P. (1982), “The Response of Reinforced Concrete to In-Plane Shear and Normal Stresses,” Publication No. 82-03, Department of Civil Engineering, University of Toronto, March, 332 pp. Vecchio, F.J., and Collins, M.P. (1986), “The Modified Compression Field Theory for Reinforced Concrete Elements Subjected to Shear,” ACI Journal, Vol. 83, No. 2, Mar.-Apr., pp. 219-231. Vecchio, F.J., and Collins, M.P. (1993), “Compression response of cracked reinforced concrete”. J. Struct. Eng., Vol. 119, No. 12, 3590–3610.

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5

Analysis and design of surface and solid structures using non-linear models

5.1

Introduction

As concrete can easily be moulded into any form desired by architects and/or structural designers, concrete structures and members, normally reinforced with steel bars or prestressing strands or wires, are usually the first choice of building materials when it comes to constructing 2D structures such as walls, slabs and shells. Simplified methods of analysis and design (e.g. strut and tie modelling as described in Chapter 8) for determining the ultimate loads of such structures have proved advantageous in many cases of structural design. However, as highlighted in Chapter 3, cases exist where more refined models are needed to adequately assess the behaviour of structures and structural members under service and ultimate loads and for ensuring cost effectiveness. The transition from 1D modelling to 2D or 3D modelling gives rise to additional complexity when dealing with a brittle material such as concrete. Examples of such complexities include: "

The material models used for concrete not only have to take account of the bi- and tri-axial behaviour of the materials, it is also necessary to formulate models in such a way that they provide, in combination with the underlying numerical model (e.g. FEM or BEM), stable and fast solution procedures.

"

More parameters are needed to determine the material behaviour. Strength limits and their statistical scatter, measured in laboratory tests, have to be adopted for calculating full scale structures.

"

Softening in cracked or crushed concrete can lead to solutions that depend on the kind of discretisation chosen, as described in Chapter 4. Proper regularisation techniques have to be applied.

"

When using displacement based finite elements in combination with smeared crack concepts, mesh dependencies such as directional bias of cracks, spurious kinematic modes and/or stress-locking may occur.

Depending on the task to be performed, the user has to choose the right method of computation in order to avoid over-sophistication on the one hand, yet including the decisive non-linear effects of a given problem on the other. This chapter deals only with the displacement based non-linear finite element method. Some approaches that combine the lower bound theorem of the theory of plasticity with linear optimisation algorithms and with other numerical solution techniques are described in Chapter 9.

5.2 Ec Eh

)c )cu )tu

Notation initial elastic modulus of concrete Complementary energy hardening modulus of steel concrete strain when concrete strength is reached ultimate compressive strain of concrete cracking strain of concrete


165


)t Gf ft fcm fc R ( H e C, C e Cs, Cs nx, ny mxx, myy, mxy mp, mpx, mpy e f e u e K d W q, p Rd Sd SdL SdNL aS

&R ^ wd

n

tensile strain fracture energy of concrete direct tensile strength of concrete mean value of concrete compressive strength concrete compressive strength vector of point loads vector of nodal force parameters deflection matrix matrix containing coefficients of a linearized yield criterion vector of yield moments shear forces per unit of length bending and twisting moment per unit of length plastic moment per unit of length vector of generalized nodal forces vector of nodal displacements stiffness matrix vector of design variables matrix of coefficients corresponding to d of a weight function distributed loads design value of resistance design value of action design value of cross-sectional action assuming linear elastic material design value of cross-sectional action assuming material nonlinearity area of reinforcement per unit of length partial safety factor for structural resistance load factor total deformation of compressive softening zone Poisson’s ratio of concrete

5.3

2D Structures with in-plane loading

5.3.1

Introduction

To demonstrate the practicalities of using non-linear FE analysis, we shall review the results of modelling of a two span deep beam tested by Cervenka and Gerstle (1971). The dimensions of the beam are shown in Figure 5.1a and the reinforcement arrangements for the beam are shown in Figure 5.1b with the reinforcement consisting of #3 (USA) deformed bars placed as shown (note that only one half of the specimen is shown with symmetry). The properties of the concrete and the reinforcing steel as used in the analysis are shown in Table 5.1.

166

5 Analysis and desing! of surface and solid structures using non-linear models


(a) Dimensions and loading.

(b) Reinforcement by no.3 bars.

Figure 5.1: Details of Cervenka and Gerstle (1971) panel W2. Table 5.1: Material properties for the Cervenka and Gerstle (1971) deep beam.

Concrete

Ec ' fcm "c wd ft Gf *

20 250 MPa 0.2 26.75 MPa 0.00264 0.5 mm 3.1 MPa 58 N/m

Reinforcement

Es fy Eh

190 000 MPa 353 MPa 700 MPa

Note: * Fracture energy (Gf ) calculated using the CEB-FIP Model Code 1990 (1993).

A plane stress analysis was performed using 4-node quadrilateral isoparametric finite elements, using 4 integration points per element and with the constitutive relationship for the concrete taken as the fracture-plastic model of Cervenka (2002). In this approach the tensile behaviour is described by the fracture energy-based smeared crack model and the compressive behaviour by a non-associated plastic flow rule. Softening of concrete behaviour in both tension and compression are considered. It is worthy of note that the tension stiffening effect is modelled indirectly through localization of strains and no specific tension stiffening parameter appears in the model. Perfect bond between concrete and reinforcement is assumed. The deep beam was modelled using two different FE meshes; a rectangular mesh (Figure 5.2a) and a triangular mesh (Figure 5.1b). The sensitivity of the results to the element size and shape was compared using three rectangular meshes with 5, 10 and 20 elements across the span and with one mesh of 10 triangular elements across the span. The results of the analyses are shown in the form of load versus displacement plots in Figures 5.3 and 5.4 and cracking patterns in Figures 5.5 and 5.6.


167


(a) Quadrilateral elements.

(b) Triangular elements.

Figure 5.2: Finite element meshes (Mesh 10). 140

Force P [kN]

120 100 80 Experiment

60

Mesh 5

40

Mesh 10

20

Mesh 20

0 0

1

2 3 Displacement [mm]

4

5

Figure 5.3: Load versus displacement: quadrilateral mesh results compared with the experimental data.

120

Force P [kN]

100 80 60

Experiment

40

Mesh 10 Quadrilateral

20

Mesh 10 Triangular

0

0

1

2

3

4

5

Displacement [mm] Figure 5.4- Effect of element type on deep beam response.

168



Figure 5.5: Comparison of experimental and FE crack patterns at failure (Mesh 10).

(a)

(b)

Figure 5.6: a) Cracks and compressive stresses (Mesh 20); b) plastic strains in bars (Mesh 10).

The behaviour of the deep beam includes the propagation of cracks from early stages of loading up to failure. At the maximum load, the horizontal bars at the bottom yield and deform plastically and the concrete at the top right corner fails in compression. The analyses show that the analytical models can reflect the real behaviour with accuracy for this failure mode of this case type. They, thus, provide for acceptable engineering solutions for this failure model for this case. To fully validate the model, other benchmark analyses need to be considered for other failure modes, as outlined in Chapter 7 of this report. In addition to strength results, the analyses can show the trends of various effects, which can be utilized in practical cases. For example, we observe the response during the crack propagation before maximum load is reached (that is, the ascending branch of loaddisplacement diagram). In this stage the response is dominantly affected by interaction of partially cracked concrete and reinforcement, the so called “tension stiffening” effect. This behaviour can be observed on the load displacement diagrams of Figures 5.2 and 5.3 and in the crack patterns, Figures 5.4 and 5.5. The study shows that the larger elements give lower stiffness in this stage and that smaller elements reflect better individual discrete cracks due to better modelling of strain localization. If we continue to increase the number of elements, so that the element size is several times smaller then the crack spacing, the crack pattern would be close to the experimentally observed discrete cracks. However, in practical cases extremely large numerical models are computationally expensive and the tension stiffening effect may be underestimated.


169


If we use triangular elements of a similar mesh size, the response is stiffer in the postcracking, serviceability, range (see Figure 5.4). The reason for the stiffer response of the triangular mesh is due to the effect of the shape function of the triangular element that produces a constant strain field, while the quadrilateral element has a semi-linear strain field. Next we examine the effect of the mesh size on the maximum load at failure. The load carrying capacity is exhausted by the attainment of the yield stress of reinforcement in the bottom portion of the beam and the surpassing of the compressive strength of the concrete in the top right corner. This is a typical bending mode of failure and larger elements give a higher load response. The higher response is also reached by triangular (more stiff) elements. In this particular case, the quadrilateral meshes 10 and 20 give near identical responses for the maximum load and indicates that mesh 10 is sufficiently fine, while mesh 5 and the triangular mesh 10 give slightly higher failure loads. The descending shape of the load displacement diagrams after the peak indicates a limited ductility, which is influenced by the softening model adopted for the concrete in compression. This limited ductility is not observed in experiments, where concrete behaviour in the panel corner may be influenced by the confining effect of the thickened central rib forming the central support. This difference can be reduced by assuming a perfectly plastic behaviour for the concrete, in this region, but such assumptions are not rationally founded since it applies only to a small region of a panel and must be used with care. This case study illustrates some of the general features valid for numerical simulation of reinforced concrete structures. In concluding, we note that: 1. Two distinct constitutive models for concrete (smeared cracks, fracture energy-based crack band) and for reinforcement (bi-linear) are capable of simulating reasonably well the behaviour of our reinforced concrete panel. Other models, however, can be adopted and it is for the user of such software to ensure that the models being used for their problem are capable of predicting the response with accuracy. Benchmarking is discussed in more detail in Chapter 7. 2. Finite element size and type can significantly influence the structural response. A mesh sensitivity study is required to show the objectivity of results. 3. A coarse mesh can underestimate the structural resistance in post-cracking stage of the analysis and overestimate the ultimate load capacity. 4. Low order elements overestimate the load response in the whole range. These conclusions are true for reinforced concrete structures with more or less uniformly distributed reinforcement, such as for the deep beam example illustrated. Structures with discrete reinforcement (for example, beams without web reinforcement) are less sensitive to the underestimation of tension stiffening but can be more dependent on the fracture model adopted for the concrete (see Cervenka, 1998).

5.4

Plate and shell structures

5.4.1

Layered elements

One possibility for modelling of 2D Structures is the application of volume elements. The reason why one should refrain from doing so is twofold; firstly, it is known from the nature of the 2D problem, that stresses and strains normal to the middle surface can be neglected. Using volume elements requires significant computational effort in calculating something that is known a priori. Secondly, it is necessary to choose the volume elements with lateral dimensions of the same magnitude – otherwise the stiffness matrix becomes ill conditioned

170



and resuults in an ennormous nuumber of volume elemeents as theirr characterisstic length has h to be of the order of the plate p or sheell thicknesss. o plane noormal stressees and assu uming a lineear variationn of the straains over By negllecting out of the elem ment heightt, one arrivees at a 5 DOF D shell ellement. Thee strain statte of the eleement is thus fully characterrized by 3 displacemen d nts and 2 in plane rotational DOF. Finite elem ments for a plates annd shells caan be conceiived as an assemblage of plane strress layers ((see Figure 5.7).

Figure 5.77: Layered sheell element

With a linear l strainn variation over o the element thickn ness, the strrain state "xxx(k), "yy(k), #xy(k) for layer k is i given by:

) xx (k ) ' ) xx 8 S xxx T z (k ) ) yy (k ) ' ) yy 8 S yyy T z (k )

(5.1)

& xy (k ) ' & xy 8 S xy T z (k ) T 2 F 5.8) are the strrains and thhe curvaturees in the where "xx, "yy, #xy, (xx, (yy annd (xy (see Figure middle plane and z(k) z designaates the disttance of lay yer k from the t referencce surface. With W the strains given g assum ming a planee stress statte, the distriibuted internnal forces oof each layeer can be calculatted via material modeels for the reinforcem ment and concrete. Seectional forrces and momentts are attainned by inteegrating thee layer conttributions thhrough the element th hickness. The elem ment stiffneess matrix can c be evaluuated in a similar way by b adding uup the layer stiffness matrices.

Fiigure 5.8: Nottations for thee strains and curvatures c of the t shell elemeent


171


To verify the approach described above, the orthogonally reinforced plate element M7, tested by Marti et al. (1987), is evaluated. The specimen had dimensions of 200 x 1700 x 1700 mm and was loaded with pure torsional moments in the reinforcement directions. Torsion was applied by couples of equal and opposing forces, P, applied at the corners B and D of the specimen (Figure 5.9a) with the specimen supported at the corners A and C. The reinforcement data is shown in Figure 5.9b (the reinforcement ratio is *x = *y = 0.25%) and the material properties are shown in Figure 5.9c. A FE analysis of plate ML7 was performed for a single, layered, shell element. The element was divided into concrete layers and reinforcement layers and the numerical computations were performed both with and without including tension stiffening in the formulation. Due to the low reinforcement ratios of the plate, tension stiffening is shown to have little influence on the results, as shown in the moment versus curvature relationships for the plate plotted in Figure 5.10.

Concrete: Ec = 35500 MPa n = 0.20 fc = 44.4 MPa ft = 4.0 MPa )c = 0.0025 )cu/)c = 1.5 )t/)ucr = 10 Reinforcement: fy = 479 MPa ES = 200 000 MPa Eh = 0.0 MPa

(a) (b)

(c) Loading and deformation of test specimen ML7 (Marti, 1987): Reinforcement; and c) material properties.

Moment - M [kNm/m]

Figure 5.9

a) applied load; b)

50 40 30 Experiment without TSE with TSE

20 10 0 0

5

10

15

20

25

30

35

Curvature 7 f [1/mm]*1E-6 Figure 5.10: Moment-curvature relationships for plate ML7.

172



5.5

Three dimensional solid structures

5.5.1

Introduction

There are many ways that 3D solid structures can be modelled and, as for any non-linear model, it is important that the model be calibrated to the problem being considered and verified using appropriate procedures. In this section, three of the more common methods for the development of models for the analysis of 3D structures are discussed: namely, models developed based on non-linear elasticity, models developed on the theories of plasticity and fracture and microplane models. Some examples of the implementation of such models are presented in the sections following. 5.5.2

Models based on non-linear elasticity

Although local stress/strain fields of a reinforced concrete plate or solid element subjected to uniform external forces are irregularly distributed, the load carrying mechanism can be reduced to a combination of space averaged one-dimensional stress flows developing in the cracked concrete and steel. The space-averaged constitutive models are the basis of 3D modelling of reinforced concrete solid elements. The compression-tension field modelling of 1D has been successfully applied to 2D modelling with the same concept extended to 3D problems. Shear stresses on a crack plane of a reinforced concrete element cause rotation in the direction of principal stresses from those at the initial cracked state. It has been shown that rotation of the principal stress may lead to formation of cracks in a new direction with almost full closure of the initial cracks (Vecchio and Collins 1982, Stevens et al. 1987). By taking the mechanical condition of the latest developed cracks, the concept of a rotating crack approach was implemented into FEM codes. In this approach, the local axes of cracked concrete in three directions (1,2,3) are considered to coincide with those of principal stress axes and information on previously developed cracks is not necessarily kept in the analysis. The compression field theory (CFT) (Mitchell and Collins, 1974) and subsequent modified compression field theory (MCFT) (Vecchio and Collins, 1986) provided a valuable contribution in the implementation of two-dimensional FE analysis of reinforced concrete. Vecchio and Selby (1991) extended the 2D rotating-crack-based constitutive models of the MCFT to the 3D case. In their approach, prior to cracking the reinforced concrete was taken to be a linear elastic, isotropic, material. After cracking, the concrete and reinforcement are considered to contribute separately to the structural element stiffness and its response to boundary tractions. Cracked concrete is modelled as a nonlinear orthotropic, pathindependent, material with the principal axes corresponding to the direction of the principal average strains. For concrete in the direction of the largest principal compressive strain ()3), the average stress-strain relationship is based on the MCFT (Figure 5.11a), in consideration of the effect of the coexisting tensile strain in the 1-direction ()1) and its cause of degradation of the concrete compressive stiffness and strength in the 3-direction. For concrete in tension and before cracking, in the direction of principal tensile strain ()1), a linear relationship is used. Whereas, after cracking, a decay function is adopted to represent tension stiffening effect (Figure 5.11b). The intermediate principal stress (in the 2-direction) is evaluated according to the two-dimensional MCFT, depending on its strain. If the strain is tensile, the tension stiffening model is used and if compressive the compression model of MCFT is used. Reinforcing bars are modelled using a tri-linear relationship with strain hardening.


173


2: intermediate principle direction

-fc3 f’c

fc3max

1: direction of normal to crack

Softening

3: direction of largest principle compressive strain

)0

-)3

(a)

fc1 fcr

Tension stiffening Local equilibrium

)c1

)cr (b)

Figure 5.11: Concrete material models adopted in the MCFT and applied to 3D. (Vecchio and Selby, 1991).

To validate their approach, Vecchio and Selby (1991) conducted series of numerical tests using the constitutive relationships described above. One of these tests is shown in Figure 5.12 for the analysis of hollow reinforced concrete box beam sections tested by Onsongo (1978). The test beams, measuring 410 mm by 508 mm in section and with a span of 2290 mm, were subjected to a constant moment and torque along their length. For the analysis, 750 mm lengths of the beams were modelled with a mesh of 1200 elements and 1760 nodes and resulting 5280 degrees of freedom. The torque was applied as nodal forces at the end of the bars. The longitudinal reinforcement was uniformly smeared over the respective areas. The comparison of analytical results in terms of capacity prediction and load-deformation is shown in Figure 5.12b and Figure 5.12c. It is seen that reasonably accurate predictions were obtained for the strength of both under-reinforced and overreinforced sections. An alternative approach to rotating crack approach of Vecchio and Selby (1991) is the multidirectional smeared crack model. This model allows for consideration of several cracks within a single element that need not necessarily be orthogonal to existing cracks. Here, information of all cracks assigned for each crack direction is stored in the analysis. The number of allowable cracks is usually limited, however, to reduce the complexity of computation (de Borst and Nauta, 1985, Barzegar 1989, Hofstetter and Mang, 1995, Fokuura and Maekawa, 1998). The occurrence of cracks in a 3D space can be approximated by quasi-two-dimensional stress fields. With this assumption Maekawa et al. (1997) proposed a strain decomposition and stress re-composition approach to obtain the 3D stress field (Figure 5.13) based on a 1D stress flow field. In each 2D subspace (after decomposition of strains) the partial stresses rooted in the crack projection are computed using path-dependent 2D constitutive laws (Okamura and Maekawa, 1991, Fokuura and Maekawa, 1998).

174



Figure 5.12: Comparison of three-dimensional analysis using a MCFT formulation with experimental results: a) experimental layout; b) and c) comparison of results (Vecchio and Selby, 1991).

composition of decompoded plane-base partial stresses

+// +.. +/.

in-plane constitutive modelling

(1,2)

(1,2) decomposed plane


3

+// +22 +/2

2 1

(1,3)


+.. +22 +.2

(2,3)

Figure 5.13: Breakdown and re-composition of load carrying mechanism of 3D cracked solids of concrete

The total stress is computed on the basis of the partial stress on the three subspaces. Hauke and Maekawa (1998) took into account the difference of concrete mechanics near and far from the reinforcing bars through consideration of two different zones in the reinforced concrete space (Figure 5.14a). One for the concrete near the steel bars and subjected to tension-stiffening behaviour, the so called reinforced concrete zone (due to influence of bond on the concrete stiffness), and a second for the concrete far from reinforcement the so called plain concrete zone which may have a localized crack in the concrete control volume (An et al., 2000, Hauke and Maekawa, 1998). The post cracking tension model in this approach is expressed as ! fib Bulletin 45: Practitioners’ guide to finite element modelling of reinforced concrete structures

175


+ t ' f t ( ) tu / ) t ) c

(5.2)

where c is a parameter representing the steepness of the descending branch and )tu is the cracking strain. In order to avoid mesh dependency of the results when dealing with localization of cracks in plain concrete zone, the softening parameter in Eq. (5.2) was defined according to fracture energy and the crack band width (Bažant and Oh, 1983). On the other hand, for the reinforced concrete zone, the softening parameter, c, is taken as constant and equal to 0.4, irrespective of element sizes. If such a crack is located in the plain concrete zone, the softening behaviour varies with the crack orientation. Also, for a control volume containing smeared reinforcing bars in one or several directions, the average softeningstiffening behaviour depends on the crack inclination. Hauke and Maekawa (1998) used a 2nd order interpolation function for considering the influence of crack inclination on the tension softening-stiffening parameter. The normalized interpolated fracture energy (area under the tension stiffening model as shown in Figure 5.14) in the desired direction, n, is computed as

G (n) ' * f

n12 G *f (1) 8 n22 G *f (2) 8 n32 G *f (3)

(5.3)

n12 8 n22 8 n32

from which the tension stiffening/softening parameter is computed. In Eq. 5.3, ni (i = 1, 2, 3) is the component of the crack normal unit vector and Gf*(i) is normalized released energy in direction i.

(a)

(b) 2

+t / ft

+t / ft

)t

1

(2)

3

Crack normal

n2 2 3

Mixed behaviour Gf*(n) {Gf*(1), Gf*(2), n } => c(n%5or c(-)

Softening Gf*(2) {Gf, lr,2, ft} => c(2)

)t

n

Directional angle

RE-bar

n1

3 2 1

Stiffening Gf*(1) {c(1) = 0.4} +t / ft

(1)

)t

imaginary orthogonal reinforcement system

1

Figure 5.14

a) Three dimensional zoning approach; b) interpolation scheme for tension stiffening model (Hauke and Maekawa 1998).

For the reinforcing steel, the average yield stress of the bars is taken as a function of an average uniaxial strain-stress relationship. In developing this relationship, the influence of crack inclination with respect to direction of reinforcing bar is considered (see Hauke and Maekawa, 1998, and Maekawa et al. 2003). As an example, a 1/5 scale model tested by Ichikawa et al. (1998) of a pier under constant axial load of 147 kN, applied at an eccentricity of 700 mm from the centre of cross section is considered (Tsuchia et al., 2001). The outline of the experiment is shown in Figure 5.15 with cyclic torsional loading applied to the specimen as shown in Figure 5.15d. In such a coupled action, cracks developed in three directions, i.e. two-way shear cracks due to reversed cyclic

176



torsion and shear as well as horizontal cracks due to flexural loading. The comparison of the horizontal load versus displacement relationships recorded from the experiment and calculated in the 3D FE analysis and, also, the torsion versus displacement relationships and longitudinal displacement are shown in Figure 5.16. A good correlation is observed for the numerical modelling when compared with the test results. 5.5.3

Fracture-plasticity modelling

Crack modelling based on the theory of plasticity permits a unified treatment of the compressive and tensile behaviour of concrete. This type of concrete modelling is usually developed from the rate constitutive equations (Hofstetter and Mang, 1995). Several constitutive models in 3D strain space exist on the fracture-plasticity formulation of plain concrete. For example, Pramono and Willam (1989) developed a fracture energy-based plasticity constitutive model that accommodates strain hardening and softening. They adopted a fracture energy release approach to describe the degradation of triaxial strength in tension and in compression in concrete with low levels of confinement. A unified treatment of plasticity and fracture can effectively make the return mapping algorithm possible (Wilkins, 1964) which guarantees the solution for all magnitudes of strain increments (Hofstetter and Mang, 1995, Cervenka and Cervenka, 1999). The main difficulty in the development of a unified model for both the compressive and tensile regions is the algorithm for the combination of the two models. On the basis of plasticity theory, fracture can be modelled by using a Rankine yield criterion (maximum tensile stress criterion) as:

Fi ' + i 7 f T ' 0

(5.4)

where +i (i =1, 2, 3) is the principal stress and fT is the tensile strength of the concrete. An appropriate softening law should be considered and can be taken as a function of the accumulated damage of the concrete (Feenstra, 1993) such that fT in Eq. 5.4 is given as a function of a damage variable. One such model is the crack band model of Bažant and Oh (1983), which can be used to describe mesh size-dependent softening laws. For hardening/softening plasticity of concrete there exists several models to describe concrete crushing. For example, the failure surface of Drucker and Prager (1952), the five-parameter ultimate strength surface of Willam and Warnke (1975), the four parameter model of Ottosen (1977) and three-parameter failure surface of Menetrey and Willam (1995). Cervenka et al. (1998) developed an algorithm for the combination of plasticity with fracture. The model is able to handle physical changes like crack closure and is not restricted to any particular shape of hardening or softening laws. The model can be used to simulate concrete cracking, crushing under high confinement and crack closure due to crushing in other material directions (Cervenka and Cervenka, 1999). The fracture is modelled by an orthotropic smeared crack model based on the Rankine tension criterion (Eq. 5.4). Further, it is assumed that strains, and stresses, are converted into the material directions that, in case of a rotating crack model, correspond to the principal directions and, in the case of fixed crack model, are given by the principal directions at the onset of cracking. The Menetrey and Willam (1995) three-parameter failure surface was used to model concrete crushing.


177


Figure 5.15: Experim mental verificaation of the multi-direction m n fixed crackk approach: aa) applied loa ads to the specimenn; b) geometryy of specimen, c) cross sectiional force; d)) loading methhod (Ichikawaa et al., 1998).

(a)

(b)

(c) Figure 5.16: Compariison of multi--directional force-displace f ement relationnships for a R RC pier: a) horizontal h loading versus horizzontal displaccement; b) force fo deviatioon torsion veersus displacement; c) displacem ment in the traansverse and longitudinal directions d (Tsuchia et al., 22001).

178



In the Cervenka et al. model, the strain is decomposed into elastic, plastic and fracture components, as introduced by de Borst (1986). The new stress state is then computed from the increment of each strain component based on the material used. The combined algorithm determines the separation of strains into plastic and fracturing components in an iterative scheme as the stress equivalence in each model must be preserved. Details of algorithm can be found in Cervenka et al. (1998). As an example of the verification of the model, an experiment test on pre-stress cable anchors is simulated (Cervenka and Cervenka, 2007). The pre-stressing force is transferred from the tendon to the concrete through special cylindrical anchors embedded into concrete. The anchor is surrounded by spiral reinforcement and loaded by compressive forces to simulate the action of the pre-stressing. A comparison of the load versus displacement obtained from the FE model and the capacity of the anchorage, as obtained from the experiment, is shown in Figure 5.17. A second example, presented in Figure 5.18, shows plastic strains at failure for a bridge over the Berounka River near Prague in the Czech Republic (Cervenka and Cervenka, 2007). The model includes the curved box girder, the piers and the foundation. The box girder and piers were modeled using higher order shell elements; the foundation by brick and tetrahedral volume solid elements and the pre-stressing cables by string elements. Various stages at service conditions were analysed as well as verification of the strength limit state. The example shows how nonlinear analysis can be applied to complex large structures for optimization of design and for verification at various limit conditions.


179


Figure 5.17: a) FE mesh m for pre-sstressing cablee anchor ana alysis; b) simuulated failure mode; and c) FE load versus displacement d compared with w peak loa ad recorded in the experriment (Cervvenka and Cervenka ka, 2007).

Figure 5.18: FE analyysis of the Berrounka River Bridge near Prague, P Czech Republic: vview from belo ow ground level shoowing plastic strains s at failuure (Cervenka a and Cervenkka, 2007).

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5.5.4

Microplane model

In dealing with plasticity of polycrystalline metals, Taylor (1938) proposed that the stressstrain relationship be specified independently on various planes in the material. The assumption was that the stresses on such a plane are the resolved components of the macroscopic stress tensor, called the static constraint, or that the strain components on such a plane are the resolved components of the macroscopic strain tensor, known as the kinematic constraint. This idea developed for plasticity (Batdorf and Budianski, 1949), is known as “slip theory”. Later, the model was extended by Bažant and his co workers to concrete and geomaterials and is referred to as the “microplane model”. This model is for analysis of quasibrittle materials that exhibit softening behaviour (Bažant and Gambarova, 1984, Bažant and Prat, 1988a, 1988b, Bažant and Ožbolt, 1990, Carol et al. 1992 and Bažant et al., 1996a, 1996b). Bažant and Prat (1988a) developed an advanced kinematic constraint version of microplane model which was extended by Ožbolt and Bažant (1992) to a more general cyclic form with rate sensitivity. The 3D version of the microplane model is constructed on the basis of uniaxial relationships between stress and strain components on planes of various orientations within a material (Figure 5.19). These planes may be imagined to represent the weak planes in the microstructure such as the inter particle contact planes, interfaces, planes of micro-cracks, etc. ! ! ! ! ! ! ! !

(b)

(a)

! ! !

(c)

!

Figure 5.19: Microplane model: a) integration sphere; b) microplane stress-strain components; and c) stress transfer through a number of contact planes (microplanes).

The strain components on the microplane are taken to be projections of the macroscopic strain tensor (kinematic constraint approach) from which the normal and shear strain component are computed. The normal microplane strain component is then decomposed into volumetric and deviatoric part to realistically model concrete dominated by compressive load and to control the initial elastic value of the Poisson’s ratio (Bažant and Prat 1988a). The simplicity of the model is due to the fact that constitutive properties are characterized entirely by only relationships between the stress and strain components of microplanes. Knowing the stressstrain law for each microplane component, the macroscopic stiffness and stress tensors are


181


calculated by the stress integration over all microplanes. Bažant and Prat produced the strain– stress microplane laws for the first loading and unloading part, while Ožbolt and Bažant (1992) introduced more general rules at the microplane level, including rate sensitivity (Figure 5.20).

(a)

(b)

Figure 5.20: Cyclic stress-strain relationship; a) volumetric microplane component; b) "Deviatoric microplane component (Ožbolt et al. 2001).

(b)

LB15

Displacement (inch)

LB13

LB16

In 3D analysis of reinforced concrete structures, the bond between reinforcement and concrete can be modelled by a layer of 3D elements around the reinforcing bars. Hoehler and Ožbolt (2001) adopted a microplane model for these bond elements such that the shear stress-strain relationship provides a realistic shear stress-slip response. Figure 5.21 shows the application of the microplane based FEM program in simulation of an experiment on a beam- column connection that was tested by Ma et al. (1976). The analysis was performed by a displacement controlled method as set out in Hoehler and Ožbolt (2001). The applied displacement history is shown in Figure 5.21b (note that in initial stage, the experiment was load-controlled). The test and numerically simulated results are compared in Figure 5.22 and a reasonable correlation is observed.

(a)

Figure 5.21: Details of by Ma et al. (1976) test: a) layout of experiment; b) time history of "applied displacement in the Hoehler and Ožbolt (2001) analysis.

182



Applied load (kips)

Experiment MASA

(c) Displacement (inches)

(a)

(b) (d)

Figure 5.22: 3D microplane FEM simulation and comparison with experimental data: a) section of FEM mesh; b) comparison of load-displacements; c) distribution of principal strains from analysis; and (d) axial strain in reinforcing bars from the analysis (Hoehler and Ožbolt, 2001).

A second example of the application of microplane models is that of Liu and Foster (2000) where 3D modelling was used to examine the effects of cover spalling in the design of high strength concrete columns under high axial loading. Figures 5.23a and 5.23b show the mesh used for the modelling of column CS-19 tested by Razvi and Saatcioglu (1996). One eighth of the column modelled due to symmetries about XY, YZ and XZ planes. The concrete was modelled using 20-node isoparametric solid elements with numerical integration on a 2 o 2 o 2 Gaussian quadrature. The in-situ mean concrete strength was taken as fc = 83 MPa, being 90% of the mean cylinder strength tested under standard conditions. The longitudinal and transverse reinforcement were modelled using 3-node truss elements with numerical integration on a 2 point Gauss quadrature. A perfect elasto-plastic stress-strain relationship was used for both the longitudinal and transverse reinforcement with a modulus of elasticity of 200 GPa. The yield strengths of the longitudinal and tie reinforcement was 450 MPa and 400 MPa, respectively. To include the effect of cover spalling in the analysis, a phenomenon commonly observed in laboratory tests of axially loaded columns, the cover was taken to be spalled from the crosssection when a threshold transverse tensile strain was reached at the interface between the cover concrete and the core. Such tensile strains occur due to the effects of the restraint of the Poisson’s effect provided to the core by the tie reinforcement (see Foster et al., 1998). In this analysis, the cover was taken as spalled if this transverse tensile strain between the cover and the core elements exceeded 750 B). The spalling was included in the analysis by setting the cover elements to a low stiffness once the threshold strain had been reached and, thus, the cover elements do not contribute to the axial capacity of the section once spalling of the element was deemed to have occurred. The failure of the column was by an axial localization of the concrete in compression around the XZ axis of symmetry, as shown in Figure 5.23c. In the Razvi and Saatcioglu experiment, the axial strain was measured over a gauge length of 300 mm (or 150 mm from the centreline with symmetry). The load versus axial strain results from the FE model is compared to the laboratory measurements in Figure 5.24, with and without cover spalling included in the model. A good correlation is observed between the FE results and the experimental data only when cover spalling is included. If the cover spalling phenomena, as observed in the laboratory tests, was not accounted for in the numerical analysis, the strength would be over predicted by some 20%. This example is used here to demonstrate the care needed when undertaking non-linear FE modelling that all influences significant to the failure condition can be captured in the analysis.


183


P = 3840 kN (descending)

Note: Cover elements not shown for clarity

(a)

(b)

(c)

Figure 5.23: Finite element mesh for Razvi and Saatcioglu (1996) column CS-19: (a) details and dimensions; (b) 3D view; (c) xz displacements at P = 3840 kN on the descending curve (dimensions in mm).

8000 7000

Load (kN)

6000 5000 4000 3000 Experimental FEM - without cover spalling FEM - cover spalling at 750 microstrain

2000 1000 0 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

Axial strain Figure 5.24: Finite element analysis of the Razvi and Saatcioglu (1996) column CS-19 with and without cover spalling.

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5.5.5

Examples of the application of 3D FE modelling

Example 1 – Optimisation of anchor system for prestressed concrete

As an example of the use of 3D non-linear modelling, the design of a pre-stressing anchoring device (shown in Figure 5.25a) is examined. Pre-stressing anchorage systems are subjected to three-dimensional stress fields of high intensity. The main issue is the increased compressive strength of concrete due to confinement and appropriate reinforcement detailing which facilitates anchoring. This example is taken from the development of Dywidag pre-stressing system where tendons are anchored in reinforced concrete wall element. The goal was to optimize the reinforcement of the anchoring region under a constraint of limited crack width.

(b) (a)

Figure 5.25: a) Wall with tendon duct; and b) details of anchorage.

The model of the wall represents a typical anchoring detail in bridge structures, where the prestressing tendon is anchored in a relatively thin wall. The reinforcement arrangement, schematically shown in Figure 5.25b, must provide sufficient resistance to a splitting action of the anchor as well as confinement effect to concrete and ensure safe tendon anchoring. The goal was to determine the horizontal force to be resisted by the reinforcement. The study included variations of the amount of tie and spiral reinforcement and was used as a basis to improve design of anchoring regions with optimal arrangements of reinforcement. To meet the design goals, a 3D solid analysis was employed. The model was reduced to using advantage of two symmetry axes as shown in Figure 5.26. It includes concrete, steel anchors and all reinforcing bars that are embedded in the concrete solid elements. A fracture-plastic constitutive model was used for the concrete and a Von-Mises plasticity model for steel of the anchoring body. The reinforcing bars were modelled using a bi-linear elastic-plastic law with hardening. The loading, representing the anchoring force of the prestressing tendon, was applied by prescribed forces at the top surface of the anchor. The force was applied in load steps up to the value corresponding to full prestressing and then progressively increased to failure.


185


Figure 5.26: FE mesh for prestressing anchor.

Examples of results of the analyses for the anchoring system are shown in Figure 5.27. The stress state of the anchoring region was evaluated in detail and the concrete, compressive stresses and plastic deformations were examined and crack widths were evaluated. The stresses in the reinforcement were also examined. The response of the anchor with spiral reinforcement (300mm in diameter) can be seen from the load-displacement diagram in Figure 5.28a and the stresses in the spiral reinforcement, Figure 5.28b, for different concrete quality (0.6 and 0.8 fc) and for different amounts of tie reinforcement. The response can be compared with the level of service load that is applied at the pre-stressing stage.

(a)

(b)

Figure 5.27: Results of FE analysis: a) cracks and compressive stress contours; b) stress in reinforcing bars.

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3.5

Anchoring force [MN]

3.0 Service load

2.5 2.0

SP300 C0.6

1.5

SP300 C0.8

1.0

SP300 C0.8 ties 1x

0.5

SP300 C0.8 ties 2x

0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Anchor displacement [mm]

(a)

Anchoring force [MN]

3.5 3 Service load

2.5 2 1.5

SP300 C0.8

1

SP300 C0.8 ties 1x

0.5

SP300 C0.8 ties 2x

0 0

100

200

300

400

Stress in spiral reinforcement [MPa]

(b)

Figure 5.28: Results from FE analyses: a) Anchoring force verses displacement diagram; b) stress in spiral reinforcement due to anchoring force.


187


Example 2 – Axisymmetric 3D solid analysis from design to practice

In this example, the performance of a hydro-electric powerhouse to stably support a generator is evaluated where the powerhouse is located inside the rock of a mountain (Figure 5.29). The hydraulic pressure applied to the inside casing pipes, the thermal gradient caused by the electricity generator and the deformation of the powerhouse restrained by the surrounding rocks, form the design loads and the boundary conditions. The limit states corresponding to the design requirements are specified in terms of cracking damage around the casing runner and transmission of higher frequency vibrations caused by the production of the hydroelectric power.

vibration protector generator machine base

23.0m

casing pile

plain concrete zone for piping devices hydro generator and arrangement of RC power house reinforcement

GHIJ

18.0m

ax symmetric finite element mesh

Figure 5.29: 3D design of reinforcement for RC powerhouse and casing runner

For design, the high pressure of running water inside of the casing is sustained by both the casing pipe, made of steel, and the surrounding reinforced concrete structure. If the hydrodynamic pressure was to be carried by the steel casing alone, its thickness becomes so large that welding quality could not be well assured. Thus, the reinforced concrete is expected to partially bear the pressure. At the same time, concrete cracking caused by the high pressure must be controlled so that the performance of powerhouse is maintained and able to support the weight of the plant and to act as a foil against underground water entering through the 188



rock joints. A conventional design, based on 3D linear analysis, shows that the quantities of hoop reinforcement needed is 3.3% around the casing pipes, as shown in Figure 5.29. The arrangement of the reinforcing bars in 3D is laborious and the high quantity makes high quality concreting works problematic as compaction is difficult. In reality, however, the thermal stress component declines significantly due to stiffness reduction by cracking and a linear analysis based design (strength based design) tends to bring about more reinforcement than is needed to meet serviceability and safety requirements. On the other hand, however, the steel stresses in the composite structures tend to be underestimated. Thus, a non-linear analysis for the performance assessment is required in view of both safety and cost benefits. Using a 3D non-linear FE analysis, structural engineers of an electric power firm investigated the behaviour of the reinforced concrete powerhouse coupled with the steel casing attached to the main body of the generator. The results of the analysis are shown in Figure 5.30 for the propagation of cracks and for the principal strain trajectories in the radial direction. The reinforcement ratio was 3.3%, corresponding to the strength design based on the linear FE analysis. The reinforcement ratio was then gradually reduced using a parametric study and finally a steel reinforcement ratio of 0.8% was adopted. As shown in Figure 5.30, the damage situation in the reinforced concrete powerhouse of 0.8% reinforcement ratio is almost the same as that with 3.3%. The stress level of steel casing was verified within the allowable stress level below the yield strength. However, if the casing reinforcement is reduced further, sharp crack localization is possible (as shown for P = 0.0) and increases the risk of yielding of

P= 3.3%

P= 0.8%

P =0.0%

Crack propagation analysis under hydro-pressure and thermal action (micro) 900 800 700 600 500 400 300 200 100 0

Profile of mean principal tensile strain (under operational load) Figure 5.30: Crack propagation and damage map


189


the steel casing. Localized damage may cause unexpected vibrations that would make the power generator unstable. An approximately 36% reduction of costs related to the casing runner was achieved by using non-linear 3D FE modelling. While this example shows some of the advantages of moving to higher methods of modelling, before adopting any non-linear analysis results great care is needed to ensure that the modelling accurately simulates the 3D crack propagation and size effects. Thus, tension softening of concrete around the plain concrete zone and tension stiffening of concrete close to the casing runner were taken into account and the accuracy of the 3D nonlinear analysis was verified by experiments (see Kato, 2000, and Kato, 2001).

5.6

References

An, X., Maekawa, K., Okamura, H. (1997), “Numerical Simulation of Size Effect in Shear Strength of RC Beams”, Journal of Material, Concrete structures and Pavements, 35(564), pp. 297-316. Batdorf, S.B., Budianski, B. (1949), “A Mathematical Theory of Plasticity Based on the Concept of Slip”, Technical Note. 1871, National Advisory Committee for Aeronautics, Washington DC. Barzegar, F. (1989), “Analysis of RC Membrane Element with Anistropic Reinforcement”, Journal of Structural Engineering, 115, pp/ 647-665. Bažant, Z.P., and Gambarova, P. (1984), “Crack Shear in Concrete: Crack Band Microplane Model”, Journal of Engineering Mechanics, ASCE, 110, pp. 2015-2035. Bažant, Z.P., and Prat, P.C. (1988a), “Microplane Model for Brittle-Plastic Material: I Theory”, Journal of Engineering Mechanics, ASCE, 114(10), pp. 1672-1688. Bažant, Z.P., and Prat, P.C. (1988b), “Microplane Model for Brittle-Plastic Material: II Verification”, Journal of Engineering Mechanics, ASCE, 114(10), pp. 1689-1702. Bažant, Z.P., and Oh, B.H. (1983), “Crack Band Theory for fracture of concrete, Material and Structures, RILEM, Paris, 16, pp. 155-177. Bažant, Z.P., and Ožbolt, J. (1990), “Nonlocal Microplane Model for Fracture, Damage and Size effect in Structures”, Journal of Engineering Mechanics, ASCE, 116(11), pp. 2485-2504. Bažant, Z.P., Xiang Y., and Prat, P.C. (1996a), “Microplane Model for Concrete- I. StressStrain Boundaries and Finite Strain”, Journal of Engineering Mechanics, ASCE, 122(3), pp. 245-262. Bažant, Z.P., Xiang, Y., Adley, M., Prat P.C., and Akers, S. (1996b), “Microplane Model for Concrete- II. Data Delocalization and Verification”, Journal of Engineering Mechanics, ASCE, 122(3), pp. 263-268. Carol, I., Prat, P., and Bažant Z.P. (1992), “New Explicit Microplane Model for Concrete: Theoretical Aspects and Numerical Implementation”, International Journal of Solids and Structures, 29(9), pp. 1173-1191.

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CEB-FIP Model Code 1990 (1993), Comité Euro-International du Béton, Thomas Telford, London. Cervenka, V. (1998), “Simulation of shear failure modes of R/C structures”, In: Computational Modelling of Concrete Structures (Euro-C 98), eds. R. de Borst, N. Bicanic, H. Mang, G. Meschke, A.A.Balkema, Rotterdam, The Netherlands, 1998, 833-838. Cervenka, V. (2002), “Computer simulation of failure of concrete structures for practice”, 1st fib Congress, Concrete Structures in 21 Century, Osaka, Japan, Keynote lecture in Session 13, 289-304 Cervenka, J., and Cervenka, V. (1999), “Three Dimensional Combined Fracture-Plastic Material Model for Concrete”, Proc. 5th US National Congress on Computational Mechanics, Boulder, Colorado. Cervenka, J., and Cervenka, V. (2007), Personal communications. Cervenka, J., Cervenka, V., and Eligehausen, R. (1998), “Fracture-Plastic Material Model for Concrete, Application to Analysis of Power Actuated Anchors, Proc. FRAMCOS, 3, pp. 1107-1116. Cervenka, V., and Gerstle, K. (1971), “Inelastic Analysis of Reinforced Concrete Panels: (1) Theory, (2) Experimental Verification and application”, Publications IABSE, Zürich, V.3100, 1971, pp.32-45, and V.32-II,1972, pp.26-39. De Borst, R. (1986), “Non-linear Analysis of Frictional Materials”, PhD Thesis, Delft University of Technology. De Borst, R., and Nauta, P. (1985), “Non-Orthogonal Cracks in a Smeared Finite Element Model”, Engineering Computations, 2, pp. 34-46. Drucker D.C., and Prager W. (1952), “Soil Mechanics and Plastic Analysis or Limit Design” Quarterly Journal of Applied Mathematics, 10(2), pp/ 157-165. Feenstra, P.H. (1993), “Computational Aspects of Biaxial Stress in Plain and Reinforced Concrete”, PhD Thesis, Delft. University of Technology, The Netherlands. Foster, S.J., Liu, J. and Sheikh, S.A. (1998), “Cover spalling in HSC columns loaded in concentric compression”, Journal of Struct. Engrg., ASCE, 124(12): 1431-1437. Fukuura, N., and Maekawa, K. (1998), “Multi-Directional Crack Model for In-Plane Reinforced Concrete Under Reversed Cyclic Actions-4 Way Fixed Crack Formulation and Verification”, in de Borst et al (eds.), Computational Modeling of Concrete Structures, Balkema, Rotterdam and Brookfield, pp. 143-152. Hauke B., and Maekawa, K. (1998), “Three-Dimensional Reinforced Concrete Model with Multi-Directional Cracking” in de Borst et al (eds.), Computational Modeling of Concrete Structures, Balkema, Rotterdam and Brookfield, pp. 93-102. Hoehler, M., and Ožbolt, J. (2001), ”Three-dimensional reversed-cyclic analysis of reinforced concrete members using the microplane model”, Otto Graf Journal, 12, pp. 93-113.


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Hofstetter, G., and Mang, H.A. (1995), “Computational Mechanics of Reinforced Concrete Structures”, Friedr. Vieweg & Sohn Verlagsesellschaft mbHg, Braunschweig/Wiesbaden. Ichikawa, H., Ogasawara, M., and Maekawa K. (1998), “Experimental Study of Seismic Performance of RC Piers Subjected to Eccentric Axial Force and Torsional Moment”, Proc. of JCI, 20(1), pp. 149-154. Kato, S., Iizuka, K., and Nanbu, S. (2000), “Nonlinear Behavior of Reinforced Concrete Structures Subjected to High Water Pressure”, Proc. of JCI, Vol.22, No.3, pp.67-72. Kato, S., Yamaya, A., Zhao, Y., and Ozawa, H. (2001), “Study on Structural Performance of Inner concrete using Ring-gage”, Proc. of JCI, Vol.23, No.1, pp.631-636. Liu, J., and Foster, S.J. (2000), “A Three Dimensional Finite Element Model for Confined Concrete Structures”, Computers and Structures, Vol. 77, No. 5, October, pp 441-451. Ma, S.Y.M., Bertero, V.V., and Popov E.P. (1976), “Experimental and Analytical Studies on the Hysteretic Behavior of Reinforced Concrete Rectangular and T-Beams, Earthquake Engineering Research Center (EERC), Report No. UBC/EERC 76-2, University of California, Berkeley. Maekawa, K., Irawan, P., and Okamura, H. (1997), “Path dependent three dimensional constitutive laws of reinforced concrete-formulation and experimental verification, Struct. Eng. Mech., 5(6), pp. 743-754. Maekawa, K., Pimanmas, A., and Okamura, H. (2003), ”Nonlinear Mechanics of Reinforced Concrete”, Spon Press, London. Marti, P., Leesti, P., and Khalifa, W.U. (1987), “Torsion Tests on Reinforced Slab Elements”, Journal of the Structural Division, ASCE. Menetrey, P., and Willam K.J., (1995), “Triaxial Failure Criterion for Concrete and its Generalization”, ACI Structural Journal, 92(3), pp. 311-318. Mitchell, D., and Collins, M.P. (1974), “Diagonal Compression Field Theory-A Rational Model for Structural Concrete”, ACI Journal Proceeding, 71(8), pp.396-408. Okamura, H., and Maekawa, K. (1991), “Nonlinear Analysis and Constitutive Models of Reinforced Concrete”, Gihodo-Shuppan Co., Tokyo. Ottosen, N.S. (1977), A failure criterion for concrete., Journal of Engrg. Mech. Div., ASCE, 103(EM4): 527-535. Onsongo, W. (1978), “The Diagonal Compression Field Theory for Reinforced Concrete Beams Subjected to Combined Torsion, Flexural and Axial Load”, Thesis presented to the University of Toronto, Toronto, Canada. Ožbolt, J., and Bažant, Z.P. (1992), “Microplane Model for Cyclic Triaxial Behavior of Concrete”, Journal of Engineering Mechanics, ASCE, 118(7), pp. 1365-1386. Ožbolt, J., Li, Y., and Kozar I. (2001), “Microplane Model for Concrete with Relaxed Kinematic Constraint”, International Journal of Solids and Structures, 38, pp. 2683-2711.

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Pramnono, E., and Willam, K.J. (1989), “Fracture Energy-Based Plasticity Formulation of Plain concrete”, Journal of the Engineering Mechanics, ASCE, 115, pp. 1183-1204. Razvi, S.R, and Saatcioglu, M. (1996), Tests of high strength concrete columns under concentric loading. Report OCEERC 96-03, Department of Civil Engineering, University of Ottawa, Canada. Stevens, N.J., Uzumeri, S.M., and Collins, M.P. (1987), “Analytical Modeling of Reinforced Concrete Subjected to Monotonic and Reversed Loadings”, Publication No. 87-01, Department of Civil Engineering, University of Toronto, Toronto. Taylor, G. I. (1938), “Plastic Strain in Metals”, J. Inst. of Metals, London, 62, pp. 307-324. Tsuchia, S., Tsuno, K., and Maekawa, K. (2001), “Nonlinear Three-Dimensional FE Solid Response Analysis of RC Columns Subjected to Combined Permanent Eccentric Axial Force and Reversed Cyclic Torsion and Bending/Shear, Journal of Material, Concrete structures and Pavements, 683(52), pp. 131-143. Vecchio, F.J., and Collins, M.P. (1982), “The Response of Reinforced Concrete to In-Plane Shear and Normal Stresses”, Publication No. 82-03, Department of Civil Engineering, University of Toronto, Toronto. Vecchio, F.J., and Collins, M.P. (1986), “The Modified Compression-Field Theory for Reinforced Concrete Elements Subjected to Shear”, ACI Journal, V. 83, No. 2, March-April, pp. 219-231. Vecchio, F.J., and Selby, R.G. (1991), “Toward Compression-Field Analysis of Reinforced Concrete Solids”, Journal of Structural Engineering, ASCE, 117(6), pp. 1740-1758. Willam, K.J., and Warnke, E.P. (1975), “Constitutive Model for the Triaxial Behavior of Concrete”, Proceeding of the International Association for Bridge and Structural Engineering, 19. Wilkins, M.L. (1964), “Calculation of Elastic-Plastic Flow”, Methods of Computational Physics, 3, Academic Press, New York.


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6

Advanced modelling and analysis concepts

6.1

Introduction

This chapter presents advanced details of material modelling in view of engineering application to practical design, construction and maintenance by means of computer software. First, the constitutive frameworks available for continuum solids will be reviewed as a basis for dealing with detailed materials modelling, followed by a discussion of specific issues relating to material models used for structural behavioural simulation. This chapter is also addressed to developers of design or analysis programs for RC structures, and to engineers who develop user options for large-scale nonlinear analysis codes. Chapter 3 served as an introduction to this chapter for practicing engineers and as a guide to use and/or select structural analysis software.

6.2

Constitutive frameworks

6.2.1

Non-linear elasticity

Arguably, non-linear elasticity is the most general and most popular of the approaches used to model reinforced or prestressed concrete structures, and most commercial FE packages have the ability for the user to use non-linear elastic models. Methods for analysis of reinforced concrete elements, based on concepts of linear or nonlinear elasticity, encompass a wide range of approaches, many of which have met with good success. In uncracked or confined states, concrete is generally modelled as an isotropic or orthotropic elastic material. In the cracked condition, concrete is considered to be an orthotropic material, and generally represented in a smeared crack context although a discrete crack representation is still sometimes used. Crack conditions can be represented in the manner of either fixed cracks or rotating cracks, with each approach having the potential to provide accurate modelling. Within the fixed crack concept, the crack direction and hence orientation of material orthogonality are defined by the formation of the first crack. In rotating crack models, the crack direction and material orthogonality are free to gradually realign depending on the prevailing stress and strain conditions. The computational solution algorithm required with both fixed and rotating crack models can be based on either an incremental-load tangent stiffness approach, or one based on a total-load secant stiffness formulation, often iterative in nature. Thus, the formulation of an elasticity-based constitutive model for concrete will depend on the combination of three main modelling decisions: smeared crack or discrete crack; fixed crack or rotating crack; tangent-stiffness or secant-stiffness. Presented below is the basic formulation for a smeared rotating crack model using a secant stiffness approach. An example of the formulation of a fixed crack model based on a tangent stiffness approach is provided by Maekawa et al. (1999). A hybrid formulation, combining elements of both fixed and rotating cracks, is described by Vecchio (1990, 2000, 2001). To simplify the discussion, the formulation presented will be limited to the case of twodimensional plane stress; all formulations are easily expandable to the general threedimensional case. Consider a reinforced concrete element, of unit dimensions, subjected to the plane stress condition +. The element contains ‘n’ components of reinforcement, where the i-th reinforcement component is oriented at angle :i to the reference x-axis (0 ? i ? n). The


195


amount of reinforcement in the i-th direction is denoted by the reinforcement ratio *i. The constitutive response of the element is described as +=D)

(6.1)

where + and ) are the total stress and total strain vectors for the element, and D is the composite secant stiffness of the material.

p

" ' +x

+ y , xy

qT ;

p

! ' )x

) y , xy

qT

(6.2)

At this point, it is being assumed that the strains are common for all components, meaning: perfect bond between concrete and reinforcement; no initial strains in the materials (e.g., no concrete shrinkage); and no subsequent volume changes due to thermal expansion, creep, or other types of elastic or plastic mechanisms (these are treated later). The composite material stiffness matrix D is comprised of contributions from the concrete and each reinforcement component; thus: n

D ' Dc 8 P D s i

(6.3)

i '1

For each reinforcement component, the material stiffness must first be formulated in the direction of the reinforcement, then transformed back to the reference axes (x, y). Thus, for reinforcement component i, Z * i E si X Ds i ' X 0 X 0 Y

0 0W U 0 0U 0 0UV

(6.4)

where Esi is the effective secant modulus of the reinforcement, evaluated as

Esi ' f si ) si .

(6.5)

The reinforcement stress f si is found from a suitable stress-strain model for reinforcing steel

(e.g. elastic-plastic with strain hardening), evaluated for the steel strain ) si . To transform back to the reference axes, the following transformation is used: Dsi ' TsT Dsi Tsi i

(6.6)

where

Z cos2 r X sin 2 r T 'X X7 2 cosr " sinr XY

sin 2 r cos2 r 2 cosr " sinr

cosr " sinr W U 7 cosr " sinr U (cos2 r 7 sin 2 r )UU V

(6.7)

and r = :i.

196

!

6 Advanced modelling and analysis concepts


Prior to cracking, in low confinement states, concrete can be sufficiently well represented as an isotropic material; thus Z1 n Ec X Dc ' n 1 1 7 n 2 XX Y0 0

W U 0 U (1 7 n ) / 2 UV 0

(6.8)

where Ec is the initial elastic modulus and n is the Poisson’s ratio. After cracking, concrete behaves as an orthotropic material, with the directions of orthotropy aligned with the crack direction. Thus, the concrete material stiffness matrix must be evaluated accordingly; that is, first defined with respect to the principal stress directions (defined by angle -, normal to the crack direction) and then transformed back to the reference axes. Thus,

Dc ' T T Dc T

(6.9)

where T is the transformation rotation matrix as before except that r = -. The concrete stiffness D is evaluated as Z Ec1 X D 'X 0 X 0 Y

0 Ec 2 0

0W U 0U Gc UV

(6.10)

where Ec1 and E c 2 are the effective secant moduli for the concrete in the principal tensile stress and principal compressive stress directions, respectively, and Gc is the effective secant shear modulus. Note that the absence of off-diagonal terms in D implies that post-cracking Poisson’s ratios are ignored; these can be included in a slightly more rigorous formulation described later. The effective secant moduli are evaluated as:

f Ecl ' c1 ;

)1

f Ec 2 ' c 2 ;

)2

Gc '

Ec1 T Ec2 Ec1 8 Ec2

(6.11)

where fc1 and fc2 are the post-cracking principal tensile stress and principal compressive stress in the concrete, respectively, evaluated based on the concrete principal tensile and compressive strains, )1 and )2, and using appropriate stress-strain relationships. Models for calculating fc1 can include the effects of tension softening, tension stiffening, and other influencing mechanisms (e.g. local stress conditions at crack locations). Models for calculating fc2 will typically consider compression softening and/or confinement effects. Note: The nature of the constitutive models for calculating fc1 and fc2 will depend on whether a fixed or rotating crack approach is being used, and some caution is needed in selecting these. The accuracy and range of application of this basic formulation can be considerably enhanced by introducing provisions for modelling elastic and plastic offset strains. With respect to the concrete, this will allow the consideration of such effects as dilation (post-cracking Poisson’s effects), thermal strains, shrinkage, and plastic residual strains (due to yielding and


197


unloading). For the reinforcement, it will permit the inclusion of prestressing strains, thermal strains, and plastic offsets (yielding). The general approach used is to first decompose, for each material, the total strains into elastic or plastic offset strains and net elastic strains. Thus, for concrete,

! ' !c 8 !co 8 !cp

(6.12)

where ) are the total element strains (as before), )c are the concrete net elastic strains, )co are the concrete elastic offset strains (e.g. due to shrinkage, thermal, or Poisson’s effects), and )cp are the concrete plastic offset strains (due to unloading, damage). Similarly, for the reinforcement

! ' !s 8 !so 8 !sp

(6.13)

where )si are the net elastic strains for the i-th component reinforcement, )soi are the reinforcement elastic offset (due to prestressing or thermal effects), and )spi are the plastic offsets (due to yielding, unloading). It is essential to note that it is the net elastic strains, )c and )si , that must be used within constitutive relations when determining concrete and reinforcement stresses, and when defining the effective secant moduli. The general constitutive relation for the element then becomes

" ' D T ! 7 "o

(6.14)

where the prestrain vector +o is defined as " o ' Dc ( !co 8 !cp ) 8

n

P Ds i (! soi 8 ! spi )

(6.15)

i '1

As an example, consider the formulation required to represent dilation effects in the concrete. Let n21 represent the strain in the 2-direction due to a stress in the 1-direction; similarly, n12 represents a strain in the 1-direction due to a stress in the 2-direction, where 1, 2 are the principal stress directions. Thus, define

p

o o o !co ' ) cx ) cy & cxy

q

T

(6.16)

where 1 2

1 2

(6.17a)

o ) cy ' ) co1 (1 7 cos 2- ) 8 ) co2 (1 8 cos 2- )

1 2

1 2

(6.17b)

o & cxy ' ) co1 T sin 2- 7 ) co2 T sin 2-

(6.17c)

o ) cx ' ) co1 (1 8 cos 2- ) 8 ) co2 (1 7 cos 2- )

and where, in this case, ) co1 ' n12 T ) c 2 , ) co2 ' n 21 T ) c1 and - is the inclination of the principal

198

!



stress direction. Stresses in the component materials are then evaluated as:

" c ' Dc (! 7 !co 7 !cp ) " si ' Dsi (! 7 ! so 7 ! sp )

(6.18)

Note that +si represents the contribution of the i-th reinforcement component to the total element stresses. The actual stress in the reinforcement must be evaluated as f si ' + si * i .

(6.19)

In this basic formulation, an iterative solution procedure is required. Some estimate of the material secant moduli is first made, and the element composite stiffness matrix is assembled. For a given element stress condition, the element total and net strains are then calculated. From the net elastic strains thus determined, the material secant moduli can be re-calculated. The process is repeated until the secant moduli have converged to stable values. This approach is applicable to post-peak strain/strain conditions with equal numerical stability as for pre-peak conditions. Additional details pertaining to this basic formulation can be found in Vecchio (1990). Equally viable formulations for a fixed crack approach, or for tangent-stiffness based models (either fixed or rotating crack based), can be found in the literature. More esoteric formulations based on multiple non-orthogonal cracks, microplane models, or hybrid crack models can also be found, and these generally provide equal capacity for accurate modelling of reinforced concrete based on elasticity principles. 6.2.2

Plasticity

The flow theory of plasticity is a natural generalization of the classical work of Tresca (1868), Saint-Venant (1870), Lévy (1870) and von Mises (1913). Its principal ingredients are the yield condition, the flow rule and the hardening law. The total strain is additively decomposed into the elastic strain and the plastic strain. It is assumed that the current state of an elementary material volume is fully described by the total strain, plastic strain, and by some additional hardening (or softening) variables that characterize the changes of microstructure and are collected in a vector k. The basic equations include the elastic-plastic split,

) ' )e 8 ) p

(6.20)

+ ' De) !

(6.21)

f ( + ,q ) ? 0

(6.22)

)! p ' ^! g( + , q )

(6.23)

the stress-strain law for the elastic part,

yield condition, flow rule,

and hardening law, which consists of two parts - the definition of the hardening variables, in general in the form of rate equations


199


f! ' ^!k( + ,f )

(6.24)

and the dependence of the parameters q appearing in Eqs. 6.22 and 6.23 on the hardening variables,

q ' h( f )

(6.25)

In the above, )={)11, )22, )33, &23, &13, &12}T is the column matrix of engineering strain components; )e and )p represent elastic and plastic strain, respectively; +={+11, +22, +33, +23, +13, +12}T is the column matrix of stress components; De is a square matrix of elastic moduli (the elastic material stiffness matrix); f is the yield function; g is a function specifying the direction of plastic flow; and ^ is the rate of the plastic multiplier. A dot over a symbol denotes differentiation with respect to time. However, recall that the theory is rateindependent. Time plays here only the role of a formal parameter that controls the loading process, and it does not need to have the meaning of real physical time. The rate equations, Eqs. 6.23 and 6.24 could also be presented in an incremental form, with rates replaced by infinitesimal increments. The yield function defines the elastic domain in the stress space, which is bounded by the yield surface. States for which f<0 are elastic, states corresponding to f=0 are plastic, and f>0 is not possible. Of course, as the variables q grow, the yield surface evolves - it can expand, shrink, translate and even change its shape. Plastic flow can take place only if the current state is plastic; this is expressed by the condition.

^! f $", q% ' 0

(6.26)

Indeed, if the material is in an elastic state (f<0), Eq. 6.26 implies ^! ' 0 and according to Eqs. 6.23 and 6.24, the plastic strain and the hardening variables remain constant. On the other hand, in a plastic state we have f=0 and Eq. 6.26 does not restrict the rate of the plastic multiplier. This rate should never be negative, because g in Eq. 6.23 specifies the oriented direction of evolution (e.g., the plastic strain rate in a uniaxial tensile test can be zero or positive). This is described by an additional restriction, ^! < 0 . Combining it with Eqs. 6.22 and 6.23 we obtain the loading/unloading conditions in the so-called Kuhn-Tucker form as, f ? 0, ^! < 0, ^! f ' 0

(6.27)

During plastic flow, the yield function must remain equal to zero, and so the rate of its change is also zero. This consideration leads to the consistency condition,

^! f! ' 0

(6.28)

Suppose that the current values of all variables as well as the rate of the total strain are given. The basic equations make it possible to compute the rates of all variables. If the current state is elastic (f < 0), the plastic multiplier, plastic strain and hardening variables remain constant, and the stress evolution is governed by the elastic law. If the current state is plastic (f = 0), the plastic flow can continue or the material can unload elastically. The former case is characterized by f! ' 0 and ^! @ 0 . In the latter case, we have ^! ' 0 and, as the subsequent stress state must be elastic, the yield function must decrease, which corresponds to f! 9 0 . The special case when f! ' 0 and ^! ' 0 is called neutral loading.

200

!



If plastic loading takes place, the rate of plastic multiplier can be calculated from the condition f! ' 0 . Applying the chain rule, we can write this condition as, T

T

e sf b e sf b f! ' c ` "! 8 cc `` q! d s" a d sq a

(6.29)

Combining the rate form of the elastic stress-strain law with the additive split and the flow rule we get

+! ' De ()! 7 ^! g )

(6.30)

The rate form of Eq. 6.25 combined with Eq. 6.24 yields q! '

sh sh f! ' ^! k sf sf

(6.31)

By substituting Eqs. 6.30 and 6.31 into Eq. 29 we have a linear equation for ^! , from which

^! '

f T+ De )!

(6.32)

f T" De g 7 f Tq Hk

where, we have denoted f+ ' sf / s+ , f q ' sf / sq , and H ' sh / sf . Substitution of this result back into Eq. 6.30 leads to the rate form of the elastoplastic stress-strain law as e

+! ' c De 7 c d

b ` )! f + De g 7 f Hk à De gf T+ De

T

(6.33)

T q

The matrix in parentheses is the elasto-plastic stiffness matrix. It is in general non-symmetric, except for the particular case when g ' f + . In this case, the flow rule is said to be associated, and it can be rewritten as

)! p ' ^!

sf (+ ,q ) s+

(6.34)

As the gradient sf ls+ is normal to the yield surface, Eq. 6.34 is sometimes called the normality rule. Yielding of metals is usually insensitive to the volumetric part of the stress tensor, and it can be conveniently described by the von Mises yield condition

f (+ ) t J 2 (+ ) 7, 0 ? 0

(6.35)

in which , 0 is the yield stress in shear and J2 is the second deviatoric invariant of the stress tensor. The J2-invariant can be computed from the principal stresses +1, +2 and +3 according to the following formula:

p

1 J 2 ' (+ 1 7 + 2 ) 2 8 (+ 2 7 + 3 ) 2 8 (+ 3 7 + 1 ) 2 6

q


(6.36)

201


It is customary to present yield conditions graphically in the principal stress space with axes +1, +2 and +3. The line on which all three principal stresses are equal is the hydrostatic axis, and planes perpendicular to this line are deviatoric planes. Intersections of the yield surface with deviatoric planes are deviatoric sections, and intersections with planes that contain the hydrostatic axis are called meridians. The yield surface corresponding to the Mises criterion is a cylinder rotationally symmetric about the hydrostatic axis and is generally not acceptable for concrete structures. All deviatoric sections are identical circles, and all meridians are identical straight lines parallel to the hydrostatic axis. Any normal to the Mises cylinder lies in a deviatoric plane. This means that the flow rule associated with the Mises condition gives purely deviatoric plastic flow, which is for metals in a good agreement with reality. Volume changes are in this case purely elastic. In Eq. 6.35, the shear yield stress , 0 is a constant material parameter, and so the yield surface does not evolve during plastic flow. This is the case of perfect plasticity (i.e., plasticity without hardening or softening), which does not use any hardening parameters. More advanced models take into account the changes in the microstructure induced by plastic flow and the resulting evolution of the yield surface. The simplest hardening model only expands or shrinks the yield surface radially. Such isotropic hardening can be described by a single scalar hardening variable, k. To characterize the total amount of plastic flow, the strainhardening model uses the accumulated plastic strain, defined by the rate equation as,

f! '

2 )! p 3

(6.37)

where | . | denotes the Euclidian norm of a tensor. The role of the scaling factor 2 3 is to make k coincident with the usual plastic strain under uniaxial loading, provided that the plastic flow is purely deviatoric. Alternatively, the work-hardening model defines k as the total plastic work, the rate of which is

f! ' + T )! p

(6.38)

By substituting from Eqs. 6.23 and 6.25 we can convert either of these definitions into the general form of Eq. 6.24. The dependence of the current shear yield stress on the hardening variable, , 0 ' h(f ) , can easily be identified from a uniaxial test. The yield function is then defined as

f (+ ,, 0 ) t J 2 (+ ) 7, 0

(6.39)

where ,0 is the only component of the vector q. The assumption of isotropic hardening is a rather crude simplification. If the specimen is subjected to tension, isotropic hardening has the same effect on the yield stress in compression as on the yield stress in tension. This does not reflect the well-known Bauschinger effect. An improvement is achieved by kinematic hardening, which translates the yield surface in the stress space without changing its size or shape. The current amount of translation is described by the back stress :. For example, the Mises yield function with kinematic hardening is written as

202

!



f (+ , : ) t J 2 (+ 7 : ) 7 , 0

(6.40)

A classical evolution law for the back stress was proposed by Prager (1955) and improved by Ziegler (1959). Realistic modelling of metals under cyclic loading requires even more complex models (Mroz 1967, Dafalias and Popov 1975, Krieg 1975). For pressure-sensitive materials such as concrete, rock, gravel or soil, it is important to take into account the effect of volumetric part of the stress tensor by the first stress invariant, I1 ' + 1 8 + 2 8 + 3

(6.41)

or by the mean (volumetric, hydrostatic) stress,

+ V ' I 1 / 3 ' (+ 1 8 + 2 8 + 3 ) / 3

(6.42)

By introducing a dependence on the volumetric stress into the Mises condition, we obtain the Drucker-Prager criterion as

f (+ ,, 0 ) t : I1 (+ ) 8 J 2 (+ ) 7, 0 ? 0

(6.43)

If the friction coefficient : is constant, the yield surface is a rotationally symmetric cone with straight meridians and circular deviatoric sections. Experimental measurements on concrete specimens indicate that the meridians are curved and the deviatoric sections have the form of a rounded triangle, whose shape changes from almost triangular for tensile and low compressive hydrostatic pressures to almost circular for high compressive hydrostatic pressures. As the deviatoric section is not circular, the yield function must depend on the third deviatoric invariant as, J 3 ' (+ 1 7 + V )(+ 2 7 + V )(+ 3 7 + V )

(6.44)

It is more convenient to transform J3 into the so-called Lode angle, which has a direct geometrical meaning. This angle, -, is defined by the relationship cos 3- '

3 3 J3 2 J 23 / 2

(6.45)

and its geometrical interpretation in the deviatoric plane is indicated in Figure 6.1a. A fairly general description of the strength envelope can be presented in the form: f ( I 1 , J 2 ,- ) ' c1 I 1 8 c 2 r (- ) J 2 8 c3 J 2 7 1 ' 0

(6.46)

where c1, c2 and c3 are material parameters and &()) is a suitable function of the Lode angle related to the shape of the deviatoric section. A failure criterion of the form given by Eq. 6.46 was first proposed by Ottosen (1977) as g Z1 W hh cos X 3 arccos( K cos 3- )U V r (- ) ' i Y 1 L Z hcos 7 arccos(7 K cos 3- )W UV hj XY 3 3

if cos 3- < 0 (6.47)

if cos 3- ? 0


203


where K is a shape factor affecting the “out-of-roundedness” of the deviatoric section; see Figure 6.1a. Another function &()) with the desired properties was constructed already by Willam and Warnke (1974) but they exploited it in a somewhat different manner. Their function, r (- ) '

4(1 7 e 2 ) cos 2 - 8 (2e 7 1)2 2(1 7 e 2 ) cos - 8 (2e 7 1) 4(1 7 e 2 ) cos 2 - 8 5e 2 7 4e

(6.48)

is derived from an ellipse in the deviatoric plane. Parameter e is the eccentricity, again related to the shape of the deviatoric section, as illustrated in Figure 6.1b. Menétrey and Willam (1995) presented a failure criterion of the form of with function &()) given by Eq. 6.48. Both definitions given by Eqs. 6.47 and 6.48 lead to a convex failure surface if the parameters K or e are chosen within a certain admissible range. The surface is smooth everywhere except for the intercept with the hydrostatic axis. If the parameters are determined from the same set of experimental data, there is usually only slight difference between the surfaces generated by the Ottosen criterion and the Menétrey-Willam criterion. As an example, Figure 6.2 shows the deviatoric sections of the Ottosen’s surface constructed with parameters c1 = 0.16 MPa-1, c2 = 0.5 MPa-1, c3 = 0.005 MPa-2 and K = 1. These values correspond to the recommendation of the CEB Model Code 1990 (1993) for concrete of grade C20. Finally, let us point out that associated flow rules usually mispredict plastic changes of volume in frictional materials such as concrete, and they have to be replaced by nonassociated ones, often written in the form

)! p ' ^!

sg s+

(6.49)

where g(),q) is a new function called the plastic potential. We might say that the flow rule is associated if the yield function is used at the same time as the plastic potential. A general plastic potential defines a set of equipotential surfaces such that the plastic strain always grows in the direction normal to the surface on which the current stress state is located. It is often possible to assume that the flow is associated in the projection onto the deviatoric plane and only its volumetric part is non-associated. A suitable plastic potential can derive from the yield function by modifying only the term that reflects the pressure sensitivity of the material. From the Drucker-Prager criterion (Eq. 6.43), we could derive a plastic potential as g ( I1 , J 2 ) ' :r I1 8 J 2

(6.50)

where :r is the dilatancy coefficient, in general different from the friction coefficient : that appears in the yield function. The most difficult component of a plasticity model for concrete is a realistic description of hardening and softening. This issue will be addressed later in the sections dedicated to particular models.

204

!



e=0.5 e=0.6 e=0.7

K=1.0 K=0.9 K=0.8

(b)

(a) Figure 6.1:

Influence of the shape parameter on the deviatoric section correspondingto zero mean pressure according to a) Ottosen, b) Menétrey and Willam.

80

60

250

0 MPa -10 MPa -20 MPa -30 MPa

200

-30 MPa -50 MPa -100 MPa -200 MPa

150

40 100

20

50 0

0

50

20

100

40 150

60

200 250

80

(b)

(a)

Figure 6.2: Deviatoric sections of Ottosen’s surface at a) low and b) high levels of mean pressure.

6.2.3

Continuum damage mechanics

The structure of a typical model based on damage mechanics will be explained using the simplest example - the isotropic damage model with a single scalar parameter. The stressstrain equations are written in the total form of

+ ' (1 7 N) De)

(6.51)

where N is the damage parameter. Initially, N is equal to zero, and the response of the material is linear elastic. As the material deforms, the initiation and propagation of microdefects such as voids or microcracks decreases the stiffness, which is reflected by the growth of the damage parameter. The model described by Eq. 6.51 is based on the simplified assumption


205


that the stiffness degradation is isotropic, i.e., stiffness moduli corresponding to different directions decrease proportionally, independently of the direction of loading. Similar to plasticity, we introduce a loading function f specifying the elastic domain and the states at which damage grows. In damage theory it is natural to work in the strain space. The loading function is therefore chosen to depend on the strain, ), and on an additional parameter k that describes the evolution of the elastic domain. States for which f(), k) < 0 are supposed to be below the current damage threshold. Damage can grow only if the state reaches the boundary of the elastic domain. This is described by the loading/unloading conditions that again have the Kuhn-Tucker form of,

f () , f ) ? 0,

f! < 0,

f! f () , f ) ' 0

(6.52)

It remains to link the damage threshold k to the damage parameter N. It would be possible to use a damage evolution law in a rate form. However, as both variables grow monotonically, it is more convenient to postulate an explicit relation between their total values as

N ' g (f )

(6.53)

The function g affects the shape of the stress-strain diagram and can be directly identified from a uniaxial test. An important advantage of the damage model is an easy evaluation of the stress corresponding to a given evolution of strain. The loading function usually has the form,

f () , f ) ' )# () ) 7 f

(6.54)

where, )~ is the equivalent strain. The formula defining the equivalent strain plays a similar role to the yield function in plasticity, because it directly affects the shape of the elastic domain. The simplest choice is to define the equivalent strain as,

)# ' )

or

)# ' ) T De) / E

(6.55) (6.56)

where, the normalization by E is introduced in order to obtain a strain-like quantity. Every definition of equivalent strain corresponds to a certain shape of the elastic domain in the strain space. For illustration, Figure 6.3a shows the elastic domains in the principal strain plane and in the principal stress plane for the case of plane stress and Poisson’s ratio n = 0.2. The domains are elliptical and symmetric with respect to the origin. Consequently, there would be no difference in the response to tensile and compressive loadings. For concrete and other materials with very different behaviours in tension and in compression, it is necessary to adjust the definition of equivalent strain. Microcracks grow mainly if the material is stretched, and it is natural to consider only normal strains that are positive and neglect those that are negative. This leads to the modified definitions

)# ' )

206

!

(6.57)



Figgure 6.3: Loaading surfacess for various definitions d of equivalent e straain.

or,

)# '

)

T

De ) / E

(6.58)

where McAuley M brackets < > denote thee “positive part of”. For F scalars, < x > = maax(0 , x), i.e., < x > = x for x positive annd < x > = 0 for x neg gative. For symmetric s ttensors, succh as the strain teensor ), the positive paart is a tensoor having th he same prinncipal axes as the origiinal one, with priincipal valuues replacedd by their poositive partss. Consequeently Eq. 6.57 can be rewritten r as

)~ '

3

P) i '1

2 i


(6.59)

207


where )i, i = 1, 2, 3 are the principal strains. The elastic domains corresponding to Eqs. 6.57 and 6.58 are shown in Figure 6.3b. If a model corresponding to the Rankine criterion of maximum principal stress is desired, one may use the definitions

)# '

1 max De) E i '1,2,3

(6.60)

i

or,

)# '

1 1 De) ' E E

3

P i '1

De)

2 i

(6.61)

where < De) >i, i = 1, 2, 3, are the positive parts of principal values of the effective stress tensor De). The former definition exactly corresponds to the Rankine criterion while the latter rounds off the corners in the octants with more than one positive principal stress; see Figure 6.3c. An important advantage of damage models is that the stress evaluation algorithm is usually explicit, without the need for an iterative solution. For a loading function in the form of Eq. 6.54, the variable f has the meaning of the largest value of equivalent strain that has ever occurred in the previous deformation history up to its current state. For a prescribed strain increment, the corresponding stress is computed simply by evaluating the current value of equivalent strain, updating the maximum previously reached equivalent strain and the damage parameter, and reducing the effective stress according to Eq 6.51. Depending on the definition of equivalent strain, we may have to extract the principal strains or principal stresses. This can be done relatively easily, since closed-form formulas for the eigenvalues of symmetric matrices of size 2x2 or 3x3 are available. Finally, let us emphasize that the purpose of the present brief introduction was to illustrate some of the basic aspects of the approach based on damage, using the simplest possible model. Damage mechanics literature contains a large number of models with various levels of complexity and sophistication, e.g., two-parameter scalar damage models in the compliance form (Ladevèze 1983, Mazars 1985, Borderie 1991) or in the stiffness form (Mazars 1985, Mazars 1986), models with two second-order damage tensors and one scalar parameter (Ramtani 1990, Papa and Taliercio 1996), a model characterizing damage by a fourth-order damage tensor (Chaboche 1979), or models characterizing damage directly by the compliance tensor or the stiffness tensor (Ortiz 1985, Simo and Wu 1987, Yazdani and Schreyer 1988) to name only a few. 6.2.4

Smeared crack models

Smeared crack models present the total strain as a sum of two parts - one corresponds to the deformation of the uncracked material, and the other is the contribution of cracking. The response of the uncracked material can be governed by a general nonlinear material law but usually is assumed to be linear elastic. The strain decomposition is written as

) ' )e 8 )c

(6.62)

and the elastic strain is related to stress by the linear law

208

!



+ ' De) e

(6.63)

The crack strain, )c, represents in a smeared manner the additional deformation due to the opening of cracks. A crack is initiated when the stress state reaches a certain failure surface. Models aiming at the description of fracture under shear and compression exploit more general criteria (Weihe 1995, Peekel Instruments 1994). The initiation criterion specifies also the orientation of the crack. According to the Rankine criterion the crack plane is normal to the direction of maximum principal stress. In a local coordinate system with axis n normal to c the crack and axes m and l lying in the crack plane, the only nonzero crack strains are ) nn, c c & nm and & nl. In other words, opening of the crack contributes to the normal strain in the direction normal to the crack plane, and sliding of the crack contributes to the shear strains in planes normal to the crack plane. The corresponding strain components in global coordinates are obtained by the transformation as g )11c u Z n12 h c h X 2 h) 22 h X n2 c 2 hh) 33 hh X n3 ' X i cv h& 23 h X 2n2 n3 h& 13c h X 2n1n3 h ch X jh& 12 wh XY 2n1n2

n1m1 n2 m2 n3 m3 n2 m3 8 m2 n3 n1m3 8 m1n3 n1m2 8 m1n2

W U n2l2 U c g ) nn u n3l3 U h c h U i& nm v n2l3 8 l2 n3 U h c h & nl n1l3 8 l1n3 U j w U n1l2 8 l1n2 UV n1l1

(6.64)

where ni, mi and li, i=1,2,3 are the components of unit vectors in the direction of axes n, m and l. In a compact notation, transformation rule (Eq. 6.64) can be written as

) c ' Tec

(6.65)

The components of ec are assumed to be directly related to the traction transmitted by the crack, s = {snn, snm, snl}T which is a projection of the stress on the crack plane. As s and ec form a work-conjugate pair, the same as + and )c, the stress transformation formula

s ' T T+

(6.66)

contains the transpose of the strain transformation matrix T from Eq. 6.65. The relationship linking s to ec is a smeared counterpart of a traction-separation law exploited by discrete (cohesive) crack models. Early smeared crack models assumed that the traction transmitted by the crack immediately drops down to zero. As explained in Chapter 3, such an approach leads to results that are not objective with respect to the mesh size. To ensure proper energy dissipation, and also to avoid unrealistic stress jumps, it is necessary to describe the loss of cohesion as a gradual process. Under pure Mode-I conditions, the normal traction can be considered as a decreasing function of the normal crack strain, c s nn ' f () nn )

(6.67)

where the function f is easily identified from a uniaxial experiment. The cracking law (Eq. 6.67) for normal components is sufficient for the rotating crack model, which assumes that the crack normal rotates and remains aligned with the current direction of


209


maximum principal strain. In contrast, the fixed crack model freezes the crack direction at the moment of crack initiation. The crack in general transmits shear that produces relative sliding of the crack faces, represented by shear components of crack strain. In simplistic versions of the fixed crack model, the shear traction is taken as proportional to the shear crack strain, with a proportionality factor (G, where G is the shear modulus of elasticity and ( < 1 is the socalled shear retention factor (Suidan and Schnobrich 1973). This is not very realistic because such a model crack can transmit large shear tractions even when it is widely open. If the shear retention factor is constant, a very small value has to be set, e.g. ( = 0.01, to limit the spurious stress transfer that may lead to the stress locking 2. A better remedy is to make ( variable and decrease it to zero as the crack opening grows (Cedolin and Poli 1977). It is also possible to formulate the relation between s and ec in the spirit of damage theory, with a scalar damage parameter depending on an equivalent crack strain that is computed from ec. Whatever choice is made, the relations can always be transformed into the incremental form with tangential stiffness matrix Dc as+ $ s! ' Dc e! c

(6.68)

The rate forms of Eqs. 6.62, 6.63 and 6.65, can be combined to yield

+! ' De)!e ' De ()! 7 )!c ) ' De ()! 7 Te!c )

(6.69)

By substituting this into the rate form of Eq. 6.66 and comparing with Eq. 6.68 we obtain a set of equations for e!c from e!c ' (T T DeT + Dˆ c ) -1T T De )!

(6.70)

It is possible to show that Dˆ e ' T T De T is a sub-matrix of the elastic stiffness matrix expressed in the local coordinates. If the elastic properties are isotropic, Dˆ e '

Z1 7 n E X 0 (1 8 n )(1 7 2n ) X XY 0

W U (1 7 2n ) / 2 0 U 0 (1 7 2n ) / 2UV 0

0

(6.71)

is the same in any coordinate system, and no matrix multiplication is needed. Moreover, if the cracking laws for the normal components and for the shear components are decoupled, then the crack stiffness matrix Dˆ c is diagonal, and the inversion of the diagonal matrix Dˆ e 8 Dˆ c is straightforward. Substituting Eq. 6.70 back into Eq. 6.69 we finally obtain the relation between the increments of stress and total strain in the form

+! ' D)!

(6.72)

D ' De 7 De T ( Dˆ e 8 Dˆ c ) 71 T T De

(6.73)

is the tangent material stiffness. An alternative expression is !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 2

In the present context, stress locking means spurious stresses build up around the band of cracking elements. This pollutes the numerical results and leads to an overestimated energy dissipation and residual strength of a cracked structure.

210

!



D ' (C c 8 TCˆ c T T ) 71

(6.74)

with C e ' De71 = elastic compliance matrix, and Cˆ c ' Dˆ c71 = tangent crack compliance matrix. Eq. 6.74 is formally simpler than Eq. 6.73 but it requires the inversion of a 6x6 matrix while in Eq. 6.73 we have to invert only a 3x3 matrix (sometimes even diagonal). So far we have tacitly assumed that the crack opening is monotonically increasing. However, it can also happen that the crack starts closing (unloading), and the tangent stiffness has to be replaced by the stiffness valid for unloading. Usually it is assumed that unloading takes place to the origin (Figure 6.4a), so that the crack strain completely disappears when the applied stress is removed. A model with some permanent strain (Figure 6.4b) is more realistic but in most practical simulations there is little difference between the two unloading models. When the applied normal stress changes sign to compression, the crack closes completely and becomes “locked”. The crack strain should not assume negative values because the crack faces cannot overlap. This is reflected by the vertical part of the diagram in Figure 6.4a. Upon complete crack closure, the material restores its original stiffness, and the overall response is purely elastic as long as the normal traction remains negative. If this traction changes sign again, the crack starts re-opening at constant secant stiffness, and when the crack opening reaches its previous maximum, the basic curve described by Eq 6.67 is followed.

!

!

"c

"c

(a)

(b)

Figure 6.4: Cracking law a) with unloading to the origin, b) with some permanent strain.

The approach outlined above can be generalized to the case of m cracks with different orientations. The basic equations remain formally the same but the column matrices s and ec now consist of blocks that correspond to individual cracks, the transformation matrix T is augmented accordingly, and the tangent crack stiffness matrix becomes block-diagonal:

s ' $s1 s2 .... sm %T ,

T ' pT1 T2 % Tm q ,

ec ' $ec1 ec 2 .... ecm %T Z Dˆ c1 X Xx ˆ Dc ' X X & Xx Y

x Dˆ c2

x

%

(6.75)

x W U

x U

& UU % Dˆ cm UV

(6.76)

However, for m cracks the number of possible combinations of loading and unloading is 2m, and it may be difficult to find the correct one. Another difficulty is a proper generalization of


211


the crack initiation criterion. If the standard criterion of maximum principal stress was kept, the normals of individual cracks could be arbitrarily close to each other. The number of generated cracks could be theoretically infinite. Then, it was proposed to impose additional restrictions such as the minimum threshold angle between a new crack and the already existing ones (Borst and Nauta 1985).+ As an alternative to the multiple fixed crack model, the rotating crack model takes into account the possible existence of cracks with various orientations by a continuous adjustment of the orientation of a single crack. The crack normal is assumed to always coincide with the direction of maximum principal strain. Here, the directions of principal stress and strain remain aligned and no shear tractions or shear crack strains appear in the formulation. This simplifies the cracking law because a relation between the normal components of type Eq. 6.67 is needed. On the other side, it is necessary to take into account that the transformation matrix T does not remain constant because the crack normal can rotate. For example, when differentiating Eq. 6.65 we have to write ! 8 Te! )!c ' Te c c

(6.77)

This complicates the derivation of the tangent stiffness matrix but the resulting formula is remarkably simple (Bažant 1983, Willam et al. 1987). In principal coordinates, we have: Z(C# + C# c ) -1 D'X e x Y

xW

(6.78)

U D# s V

This formula covers the general case, in which cracking can take place in all three principal planes. Matrix 7n 1

Z 1 1X ~ C e ' X7 n E XY7 n

7n

7n W 7 n UU 1 UV

(6.79)

~

is the block of the elastic compliance matrix that corresponds to normal components, and Cc is the diagonal crack compliance matrix with current compliances of three mutually orthogonal cracks on the diagonal. If some of the principal directions is not cracking, the corresponding crack compliance is set to zero. The block of the stiffness matrix (Eq. 6.78) that corresponds to shear is

Z s2 7 s3 X X 2$e2 7 e3 % ~ 0 Ds ' X X X X 0 XY

0 s3 7 s1 2$e3 7 e1 % 0

W U U U 0 U s1 7 s2 UU 2$e1 7 e2 % UV 0

(6.80)

where the symbols si and ei, i = 1, 2, 3, stand for principal stresses and principal strains. The tangent stiffness matrix D can be rotated into the global coordinates using the standard transformation rule.

212

!



The shear coefficients in Eq. 6.80 are properly defined only if the principal strains ei, i = 1, 2, 3, are mutually different. The case of two or three equal principal strains is a weak point of this particular form of the rotating crack model (Jirásek and Zimmermann 1997). An alternative formulation that avoids this deficiency is embodied within the Disturbed Stress Field Model, described by Vecchio (2000, 2001).+ 6.2.5

Microplane models

Unlike conventional tensorial models that relate the components of the stress tensor directly to the components of the strain tensor, microplane models work with stress and strain vectors on a set of planes of various orientations (so-called microplanes). The basic constitutive laws are defined on the level of the microplane and must be transformed to the level of the material point using certain relations between tensorial and vectorial components. The most natural choice is to construct the stress and strain vector on each microplane by projecting the corresponding tensors, i.e., by contracting the tensors with the vector normal to the plane. However, it is impossible to use this procedure for both the stress and the strain and still satisfy a general law relating the vectorial components on every microplane. The original slip theory for metals worked with stress vectors as projections of the stress tensor; this is now called the static constraint. Most versions of the microplane model for concrete and soils have been based on the kinematic constraint, which defines the strain vector e = (e1, e2, e3) on an arbitrary microplane with unit normal n = (n1, n2, n3,) as ei ' ) ij n j

(6.81)

where, )ij, i, j=1, 2, 3 are the components of the strain tensor. When dealing with tensorial components, we use the Einstein summation convention implying summation over twicerepeated subscripts in product-like expressions. For example, subscript j, on the right-hand side of Eq. 6.81 appears both in )ij and in nj, and so a sum over j running from 1 to 3 is implied. The microplane stress vector, s = (s1, s2, s3,), is defined as the work-conjugate variable of the microplane strain vector, e. The relationship between e and s is postulated as a microplane constitutive equation. A formula linking the microplane stress vector to the macroscopic stress tensor follows from the principle of virtual work, written here as

+ ij O) ij '

3 2L

G

y

s i Oei d y

(6.82)

where O) ij are components of an arbitrary (symmetric) virtual strain tensor, and

Oei ' O) ij n j

(6.83)

are components of the corresponding virtual microplane strain vector. Note that the summation convention implies summation over i and j on the left-hand side of Eq. 6.82, summation over i on the right-hand side of Eq. 6.82, and summation over j on the right-hand side of Eq. 6.83. Integration in Eq. 6.82 is performed over all microplanes, characterized by the components of their unit normal vectors, ni, i = 1, 2, 3. Because of symmetry, the integration domain y is taken as one half of the unit sphere, and the integral is normalized by the area of the unit hemisphere, 2L/3.


213


Substituting Eq. 6.83 into Eq. 6.82 and taking into account the independence of variations O), we obtain (after certain manipulations restoring symmetry) the following formula for the evaluation of macroscopic stress components:

+ ij '

3 4L

G

y

( si n j 8 s j ni ) dy

(6.84)

In summary, a kinematically constrained microplane model is described by the kinematic constraint Eq. 6.81, the stress evaluation formula Eq. 6.84, and a suitable microplane constitutive law that relates the microplane strain vector, e, to the microplane stress vector, s. If this law has an explicit form (Carol et al. 1992)+

s'~ s (e, n)

(6.85)

then the resulting macroscopic stress-strain law can be written as

+ ij '

3 4L

G p~s (e, n)n y

i

j

q

8 ni ~ s j (e, n) dy

(6.86)

Realistic models for concrete that take into account the complex interplay between the volumetric and deviatoric components of stress and strain (Bažant and Prat 1988, Ožbolt 1995, Bažant et al. 1996) usually lead to more general microplane constitutive laws of the type

s ' s# (e,n;+ )

(6.87)

that are affected by some components of the macroscopic stress, +, for example by its volumetric part. Instead of a direct evaluation of the explicit formula Eq. 6.86, the macroscopic stress is then computed as the solution of an implicit equation, and the stressevaluation algorithm involves some iteration.

6.3

Solution strategies

6.3.1

Introduction

As mentioned in Chapter 3, the most common solution algorithm used in FE analysis of concrete structures is one based on a Newton or modified-Newton technique. A few of the possible methodologies are discussed in this section. In Section 3.4, we started with a point on the equilibrium path (qo , & p) , as shown in Figure 3.18, where qo is the current displacement vector, & is a load magnification parameter and p is a reference vector of applied loads. Equilibrium of the discretized structure was then written as

^ p 7 f (q) ' 0

(6.88)

K( q )q 7 f (q) ' 0

(6.89)

or

214

!



where f is a vector of internal forces and is a function of the current displacement state. The solution of the equation system is then obtained from Kq – f = 0

(6.90)

and can be calculated directly for a linear system. For a non-linear system, however, an iterative approach is needed with successive solutions of the linear system until convergence is obtained. A number of algorithms have been introduced into finite element programs to trace equilibrium paths. The majority of these use either the Newton-Raphson (NR) or modified Newton (mNR) techniques as their basis. The NR and mNR algorithms work well when linear or bi-linear material relationships are being used, however, become more and more inefficient as higher degrees of non-linearity are introduced. A further disadvantage of the NR and mMR method is that without the addition of special techniques a falling load path cannot be handled. An early method of obtaining limit points was to use “displacement control” as given by Batoz and Dhatt (1979). This method has been used successfully on many occasions; however, it has a number of drawbacks, many of which are highlighted in Crisfield (1981). A further problem when trying to apply this method to finite elements which have a high degree of material nonlinearity (such as those modelling concrete) is that the initial solution, which forms the basis of further iterations, may be well away from the final equilibrium state and divergence and/or algorithm failure may occur. A major improvement to the standard NR – mNR techniques was introduced by Riks (1972) and Wempner (1971) and later modified by Crisfield (1981) and Ramm (1981) and involves control of the load/displacement path. The constant arc length method was later modified to include line searches and accelerations (Crisfield, 1983). 6.3.2

Newton-Raphson method

The most frequently used iteration scheme for the solution of non-linear equations is some form of the Newton-Raphson procedure. In the case where Eq. 6.88 cannot be solved exactly the residual forces at the ith iteration can be written as !

'()* + , -./ 0 1()* +

(6.91)

Eq. 6.91 may be differentiated with respect to q to obtain df (qi ) d (r (qi )) ' 7 ' 7 K (qi ) dq dq

(6.92)

If an approximate solution q ' qi is obtained, then the truncated Taylor expression may be written as r (qi 81 ) ' r (qi ) 8

dr (qi ) O qi ' 0 dq

(6.93)

where O qi ' _ qi 81 7 _ qi and, therefore,

r (qi ) ' K (qi ) O qi


(6.94)

215


Substituting Eq. 6.94 into Eq. 6.93 and rearranging yields

Oqi K (qi )71 (_^p 7 f (qi ))

(6.95)

Denoting K $qi % , the tangent stiffness matrix for the current displacement state, as K Ti , then 71

Oqi ' K Ti ( _^ p 7 f ( qi ))

(6.96)

The Newton-Raphson solution process is illustrated in Figure 3.18, noting that for every step the current stiffness matrix is formed and the linearized equations solved for O qi . 6.3.3

Modified Newton-Raphson method

From the above calculations it is seen that the formulation of a new tangent stiffness for each iterative cycle and the solution of a new system of equations must be undertaken. In computing terms, this can be time consuming and costly. To overcome this difficulty the approximation K Ti ' K To is often made. This modifies Eq. 6.96 to

O qi ' K To

71

( _^ p 7 f ( qi ))

(6.97)

and resolution of the same equation set is repeatedly used. The solution at each iteration is sped up; however, more iterations to convergence are required, as can be seen in Figure 3.19. The overall economy of the solution procedure is dependent on the problem size and nonlinear behaviour. An updated mNR approach may be adopted where the tangent stiffness matrix, K Ti , is updated if convergence is not obtained after a pre-determined number of iterations, n, and continues to be updated every n iterations following. 6.3.4

Incremental displacement method

The incremental displacement method was developed to overcome problems with load control solution schemes and their inability to deal with structural stability problems or problems where the loading decreases with increasing displacements. A discrete representation of a linear problem is given by Eq. 6.90 where the internal force vector is defined by Eq. 6.88. In a usual linear analysis the load vector, ^ p , is given and Eqs. 6. 88 and 6.90 can then be solved directly for the displacements, q. If, however, the jth component of q is chosen as unknown, the standard form of Eq. 6.90 needs to be modified to solve for & and (n-1) unknown displacements of q. Batoz and Dhatt (1979) developed an algorithm such that the structure’s stiffness matrix, K, is not modified, instead a solution of two vectors q a and q b is sought such that K q a ' r uh v K q b ' p hw

(6.98)

where r is the residual vector defined in Eq. 6.91. By using this method the structural stiffness matrix remains banded and symmetric.

216

!



If the stiffness matrix, K, of Eq. 6.90 is non-singular, then Eq. 6.98 maybe solved. The solution for & and q is given by

q ' q a 8 ^ qb

(6.99)

q j ' q aj 8 &q bj

(6.100a)

where for the jth component

^'

q j 7 q aj

(6.100b)

q bj

For a purely linear system, r = 0 and thus Eq. 6.99 leads to

q ' ^qb ^'

(6.101a)

qj

(6.101b)

q bj

The algorithm of Batoz and Dhatt can be applied to the Newton-Raphson solution procedure. Letting the quantities (qo , ô p) denote the displacement and load state at a point on the equilibrium path, as shown in Figure 6.5, then instead of increasing the load parameter, the jth component of q is incremented by _q j . The initial displacement vector, q, is modified so that

qi ' qo where qij ' qij 8 _q j . The residual force vector due to the modified initial displacements is computed and the displacement vectors due to the residual force, r, and unit load, p, are computed simultaneously from Eq. 6.98, that is,

p_q & q q' K (q ) a i

b i

i

71

pr (qi )& pq

(6.102)

where K $qi % is the non-linear tangent stiffness matrix corresponding to the current displacement state, q i . Eq. 6.100 maybe rewritten in the form

O qi ' _ qiq 8 O î qib

(6.103)

j It is desired that the jth element of O qi is equal to zero (that is, O qi ' 0) . This leads to

e

b

b

O î ' 7 c _ q aj q 8 _ q j ` j d

ai

(6.104)

The structure displacement vector can now be computed together with the loading parameter using ! fib Bulletin 45: Practitioners’ guide to finite element modelling of reinforced concrete structures

217


Figure 6.5: Schematic representation of incremental displacement method using mNR iteration.

_ qi 81 ' _ qi 8 O qi _î 81 ' _ î 8 O ^ i

(6. 105)

qi 81 ' qi 8 O qi If the algorithm is used with the modified Newton-Raphson method, then the structure stiffness matrix, K $qi % is updated only at selected iterations. A summary of the solution method is depicted in the schematic outlined in Figure 6.5.

218

!



6.3.5

The constant arc length method

The constant arc length method was developed to overcome some of the problems associated with relying on merely load or displacement control. The principle behind the arc length method is to increment the load-displacement path in such a manner as to derive the benefits of both load and displacement control. This was first introduced by Riks (1979) using a normal to the tangent stiffness and later modified by Crisfield (1981) to use a circular path; the latter method being slightly less likely to fail. Let the quantities $qo , ô p % denote the displacement and current load vector at a given point on the equilibrium path, as shown in Figure 6.6. The equation governing equilibrium is then given by Eq. 6.88 and the residual force vector, r, given by Eq. 6.91. Since the arc length method treats both & and q as variables, an extra equation is necessary to enable solution of Eqs. 6.88 and 6.91. This extra equation comes from the constraint relationship

_ qiT81 _ qi 81 ' _ qiT _ qi ' _ ' 2

(6.106)

where _' is the prescribed incremental length and _ qi is the incremental displacements after the (i-1)th iteration (see Figure 6.6).

Figure 6.6: Schematic representation of constant arc length method.


219


The solution procedure starts by incrementing the load parameter by _^ , and forming the current tangent stiffness pKa q . The initial change in displacement vector, O qo can be calculated by

H

Letting

HK

71 a .

I

Oqo ' _ ^1 K a71 . p

(6.107)

O q1 ' O qo ' _^1 # T

(6.108)

I

p ' # T gives

By now applying the constraint equation 6.106 the arc length can be set as #TT . #T

_' ' _^1

(6.109)

and remains constant for the remainder of the iterations within this load step. From Eq. 6.91 the residual force vector, r i , may now be calculated. The solution continues with the calculation of O q i from

H

O qi ' K a71 . p ri 8 O î . p q

I

(6.110)

Note that the load parameter & must be modified by O î in order not to violate the constraint equation (that is, _î 81 ' _î 8 Oî ). Letting the form of

HK

71 a . ri

I ' # , Eq. 6.110 may be rewritten in i

O qi ' # i 8 O î #T

(6.111)

Writing the updated displacement vector in the form of

_ qi 81 ' _ qi 8 Qi O qi

(6.112)

where Qi is an acceleration parameter obtained by line searches (if line searches are not used then Qi ' 1.0) and substituting Eqs. 6.112 and 6. 111 into Eq. 6.106 gives

p _q 8Qi p #i 8 O î #T qq T p _q 8Qi p#i 8 O î #T qq7 _'2 ' 0

(6.113)

Resolving Eq. 6.113 leads to the quadratic expression

a1 O î2 8 a2 O î 8 a3 ' 0

(6.114)

where

220

!



a1 ' Qi #TT #T

p

a2 ' 2 #TT _qi 8 Q1#TT #i T

T

a3 ' 2 #i _ qi 8 Qi # i # i

q

u h h h v h h h w

(6.115)

The two roots of Eq. 6.114 correspond to the two points of intersection between the equilibrium path and the surface described by Eq. 6.113. The appropriate root is the one that gives forward progression and may be obtained by ensuring an acute angle ) exists between _ qi and _ qi 81 where cos - ' 1 8

Qi Z T # _qi 8 Oî #TT _qi W 2 XY i UV _'

(6.116)

In most cases Eq. 6.116 will yield one positive value of cos ) (corresponding to an acute value for )) and one negative value and hence there is no problem finding the root that gives forward progression. In the event that both values for cos ) are positive, the appropriate root is the one that is closest to the linear solution – a3 a2 . Once the value for O^ i has been obtained, substitution into Eq. 6.111 yields Oqi and subsequently into Eq. 6.112 for O^ i 81 . As yet application of the arc length method to the modified Newton-Raphson technique has only been discussed. The constant arc length method can be applied equally well using the Newton-Raphson technique or any updated mNR method by recalculating the #T vector whenever the tangent stiffness matrix [Ka] is reformed. The arc length, _' , however, is not recalculated until the end of the current load/displacement increment. 6.3.6

Line searches

Irons and Elsawaf (1970) and Elsawaf (1979) first applied line search concepts to non-linear finite element problems. Since then many researchers have adopted the line search technique as a method of accelerating towards a solution. Crisfield (1983), as a method to avoid numerical difficulties caused by concrete cracking, introduced line search techniques into the arc length method. The line search concept seeks a scalar Qi such that the energy z at qi 81 is stationary in the direction of Qi , that is; sz sq sz ' 7 riT81 #i ' S j ' 0 ' sQi i 81 sqi i 81 sQi

(6.117)

In practice Eq. 6.117 cannot be met. Instead it is satisfactory to satisfy S j 9 r So

(6.118)

where


221


u h v T S j ' 7 # i . r i 81 hw S o ' 7 #Ti . r i

(6.119)

and * is the required solution tolerance. The work of Crisfield (1982, 1983), Wolfe (1975), Dixon (1972), Shanno (1978) and Foster (1992) indicate that a slack tolerance of * = 0.8 is optimum. For a complete mathematical formulation of the line search methods see Crisfield (1982, 1983). 6.3.7

Convergence criteria

In a finite element analysis of the non-linear behaviour of a structure, it is almost impossible to solve Eq. 6.88 exactly $ ie; r $qi % ' 0 % using the iterative solution techniques discussed. For this reason a measure of whether the solution has reached a predetermined accuracy is required. A simple method of monitoring convergence is to check either the out-of-balance force vector or the change in displacements after each iteration. A load step cycle may be considered complete whenever

r i 9 ) max r

j

( j ' 1,2,3... i )

(6.120)

T where r i ' O q i . O q i or r i ' r (q i )T . r (q i ) , as appropriate.

When modelling concrete structures, due to the changing of constitutive relationships from the uncracked to cracked states and the change in the material strain relationship as a function of the principal stress ratio, the displacement criteria to set the solution tolerance is often used. 6.3.8

Load-displacement incrementation

Ideally when using one of the Newton techniques to solve non-linear problems, the increase in load or displacement, as the case may be, from a converged state to a new state, should reflect the current degree of non-linearity. If the load step is too large then the problem will be slow to converge. If the load step is too small, then a larger number of steps is required to define the load-displacement state than needed. Crisfield (1983) remarked that, for modelling of reinforced concrete it may not be wise to relate the load increment size to the number of iterations required to achieve results at the previous load step. This is due to unpredictable nature of concrete and the large redistribution of forces when cracking occurs. Substantial change in the model occurs frequently and the number of iterations required to give convergence may increase dramatically. This, however, does not necessarily affect subsequent load steps and no apparent advantage is obtained by reducing the initial load increment (+&o) for the following load steps. In fact, substantial computer time may be taken to map a small aberration in a few elements that ultimately do not affect the results of the model. A more reliable method of automatic load incrementation is to monitor the performance of a key displacement component within the model and modify the load increments (for load control methods or the constant arc length method) within a set maximum and minimum in order to obtain optimum displacement increments.

222

!



6.4

Other issues

6.4.1

P peak response Post r off compressiion elemen nts

There are a many othher compliccating issuees related to specific orr special struuctures not covered in this report andd it is for the designner to pre-aassess theirr importancce before adopting a I the case of o columns,, for examp ple, if inform mation on a post-peak response r computeed results. In is desirred issues such s as spaalling of coover concreete and bucckling of thhe reinforceement in compresssion membbers needs consideratio c on. Figure 6.7 6 illustrates typical ffailure in th he plastic hinge reegion of a cyclically c looaded colum mn specimen n tested by Bayrak (1999). Since buckling b of the longitudinal l l bars is geenerally noot considereed in conveentional anaalytical pro ocedures, behavioour at large inelastic cuurvatures is generally over-predict o ed with resppect to stren ngth and ductilityy.

Figuure 6.7: Longiitudinal bar buuckling in speecimen RS-12H HT (Bayrak, 11999).

6.4.2

E Effects of ageing a and distress in concrete

Concrette is an ageeing materiaal, and its mechanical m properties change witth time. At an early age, concrete structure may be subjected to interrnal actionss due to innternal therm mal and m be affected by ex xternal condditions of lloads, restraint and moisturre gradientss, and it may surrounnding enviroonment. Thhe coupling of these actions a leadds to non-unniform defo ormation inside structural s m members, whhich resultss in the gen neration of internal strress and sometimes causes early e age cracking. Suuch initial defects d are likely to reeduce durabbility perforrmances, load carrrying capaccity, and duuctility of a concrete c strructure. The vollume changee of the matterial at an early age iss mainly cauused by tem mperature risse due to hydratioon reaction and autogeenous and drying d shrin nkage rootedd in the mooisture behaaviour in micro pores. p Here,, the internaal stress gennerated by the volumee change is strongly deependent on creeep behaviouurs. In addition, for a PC design n, considerration of crreep is esseential to evaluatee the loss of o prestress. Here, a coouple of evaaluation meethods for sshrinkage an nd creep are briefly reviewed. Several practical models m have been develloped to preedict creep and a shrinkaage deformaations. In y loaded such moodels, it hass been a customary proccedure to deefine the tottal strain off a uniaxially specimeen at time t after the casting of conncrete as folllows,


223


)$t % ' ) e $t % 8 ) c $t % 8 ) s $t % 8 ) T $t %

(6.121)

where, )(t) is the total strain, )e(t) the instantaneous strain, )c(t)the creep strain (which is divided into basic and drying creep), )s(t) is the shrinkage strain and )T(t) is the thermal dilatation. For stresses up to approximately 40% of the strength of concrete, creep can be assumed to be proportional to the stress (known as Davis-Granville relationship). Based on this assumption under constant stress +, the above equation can be transformed as, )$t % ' + T J $t , t #% 8 ) s $t % 8 ) T $t %

(6.122)

where, J(t, t’) is defined as a compliance function or creep function, which represents the strain at time t produced by a unit constant stress acting since time t’. It can be written as,

J $t , t #% '

1 1 8 C$t , t #% 8 C $t , t #% ' E $t #% E $t #%

(6.123)

where, E(t’)is the elastic modulus of concrete at time t # , C $t , t # % is the specific creep that represents the creep strain at time t produced by a unit constant stress acting since time t # and C(t, t # ) is a creep coefficient which represents the ratio of the creep strain to the elastic strain at time t. These creep functions have been formulated based on concrete properties, structural dimensions, and atmospheric conditions by many past researches and design codes. CEB-FIP Model Code 1990 In the CEB-FIP model code, the concrete creep function is ) c $t , t #% '

+ c $t #% + $t #% T C$t , t #% ' c pC 0 T ( c $t 7 t #%q Ec Ec

(6.124)

where, +c: constant stress applied at time t’, Ec: modulus of elasticity at the age of 28 days, C0 (t, t # ): notional creep coefficient and it depends on the ambient relative humidify, size of the member, and the mean compressive strength at the age of 28 days, and (c (t, t # ): coefficient describing the development of creep with time after loading and it depends on the ambient relative humidity and size of member. In the code, shrinkage is calculated by the following equation as, ) s $t , t s % ' ) so T ( s $t 7 t s %

(6.125)

where, )so is the notional shrinkage coefficient and it depends on the ambient relative humidity, the mean compressive strength at the age of 28 days, and the type of cement, (s is the coefficient to describe the development of shrinkage with time, which depends on the size of member, t is the age of concrete and ts is the age of concrete at the beginning of shrinkage or swelling.

224

!



National codes of practice All national codes of practice include simplified creep and shrinkage models in their contents that can be used in modelling. For example, the Japan Society of Civil Engineers (JSCE) Code 2002 has the following model for creep: )#c ' C T +#cp E ct

(6.126)

# the applied stress and Ect the modulus where, ) c# is the creep strain, C a creep coefficient, + cp

of elasticity at the loading age. In the code, the following equation to obtain specific creep is introduced as,

p

H

) #c $t , t #, t 0 % +#cp ' 1 7 exp 7 0.09$t 7 t #%

0.6

IqT )#

cr

) #cr ' ) #bc 8 ) #dc

(6.127)

# is the ultimate value of specific basic creep strain, which is a function of unit where, ) bc # is the ultimate value of specific weight of water and cement and water-to-cement ratio and ) dc drying creep strain, which is a function of ambient relative humidity, unit weight of water and cement, water-to-cement ratio, exposure surface area, and structural dimension.

The above models are generally used to predict a mean value of shrinkage and/or creep over the cross section of a certain member. For practical purposes, such simple treatments are easy to use, but their applicability should be clearly kept in mind. That is to say, they don’t give local properties within the cross section of a concrete member, such as the variation of internal stresses, moisture states, and generation of micro cracking. In addition, these models adopt the conventional separation between the shrinkage, drying creep, and basic creep in their formulations. Strictly speaking, however, these behaviours should not be treated separately but they are rather several aspects of one physical phenomenon.

Other models Recently, several models based on the microphysical phenomena in concrete, such as cement hydration, moisture transport/equilibrium, and microstructure of cement paste, have been proposed (Bažant and Prasannan, 1989, Bažant et al., 1997, Lokhorst and Breugel, 1997, Maekawa and Ishida, 2001). Since they try to simulate the actual rheological phenomena in the material and structure from the mesoscopic viewpoint, it is expected that they are a breakthrough giving a solution to classic research topics in the concrete engineering, i.e., creep and shrinkage. Figure 6.8 shows an example of such an approach for a creep test by Ross (1958) undergoing a complex loading and unloading history and modelled by Chong (2004) using the solidification formulation of Bažant and Prasannan (1989). 6.4.3

Effects of ageing and distress in reinforcing steel

As an effect of ageing/distress of reinforcement, corrosion may be main influential factor on structural performance. Corrosion of reinforcement is associated with formation of rusts. The volume of the oxide increases two to ten times at the steel/concrete interface (Broomfield 1997). This leads to cracking and spalling of cover concrete, and eventually it affects overall structural performances, such as reduction of load carrying capacity and ductility.


225


Figure 6.8: FE modelling of Ross (1958) creep test (Chong, 2004).

Figure 6.9 shows a schematic representation of the effect of corrosion on structural performance. When a reinforcing bar corrodes, it causes damage affecting structural behaviour, including: 1) damage of reinforcement itself (loss of bar section, reduction of strength, etc.); 2) damage of bond between reinforcement and surrounded concrete; and 3) damage of concrete around corroded reinforcement due to the expansion of the oxide (cracking and spalling of cover concrete). In order to evaluate the structural performance of a structure having corroded reinforcement, the above phenomena should be appropriately taken into account. Steel corrosion

Volumetric expansion accompanied with rust

Loss of bar section Reduction of strength Acceleration

Crack formation

Bond

Spalling Loss of concrete cross section

Load carrying capacity

Ductility

Structural performance Figure 6.9: Effect of corrosion on structural performance

226

!



6.4.4

Second ord der effects

Where second ordder effects are a importaant in assesssing the sttrength or sservice limiit states, these neeed to be inncluded in thhe modellinng. Many, iff not most, commerciaal packages used for modelling of concrrete structurres will havve some cap pability for modelling geometricaally nonp toogether with the non-linear mateerial modellling descriibed in the various linear problems chapterss of this repport. An exxample of a common sttructural eleement wherre designerss will be familiarr with the issue i of seccond order effects in their t treatm ment of slennder column ns where theeffecct of geometric non-linnearity is to magnify beending mom ments. A seccond, less common, c examplee is in creeep buckling of a beam m-column, su uch as thatt shown in Figure 6.10 0, where geometrric non-lineearity combbines with time effeccts and othher material non-lineaarities to engendeer failure. Other seecond orderr effects incclude the reesponse of concrete c eleements subjected to hig gh strain rates suuch as, for example, blaast and impaact. The treeatment of thhese speciaal cases, how wever, is not withhin the scoppe of this repport.

Steel ch hannel section eT Eccentric loadiing Strong wall

1250

Ten nsioning cable

Dial gaugee

Stirrups Ø10 at 15 50 Clearr cover 15 mm 1250

2N12 5000

Dial gaugee

A

Dial gaugee

2N12

1250

A

150 Section A-A

1250

Test co olumn

eB

Hydraaulic jack on loading arm I-sectio Load cell

Figuree 6.10: Modellling of concreete columns suubjected to creeep buckling (Chong, ( 2004,, Chong et al.,, 2008).

6.5

R Reference es

Batoz J., J and Dhattt G. (1979)), Incremenntal displaceement algorrithms for nnonlinear prroblems, Int. J. Num. N Meth. Engng., Vool 14, pp. 12262-1266. Bayrak,, O. (1999),, “Seismic Performanc P ce of Rectilinearly Connfined Highh Strength Concrete C Columnns”, Thesis submitted in conformitty with the requiremennts for the D Degree of Doctor D of Philosopphy in the University U o Toronto, pp.339. of Bažant Z. P. (19883), “Comm ment on orthotropic models forr concrete and geomaaterials”, E, 109, pp.8 849-865. Journal of Engineeering Mechaanics, ASCE


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Bažant Z. P. and Prat P. (1988), “Microplane model for brittle plastic materials. I: Theory, II: Verification”, Journal of Engineering Mechanics, ASCE, 114, pp.1672-1702. Bažant Z.P. and Prasannan S. (1989), “Solidification theory for concrete creep. I. Formulation, II. Verification and application”, Journal of Engineering Mechanics, 115(8), pp.1691-1725. Bažant Z. P., Xiang Y., and Prat P. C. (1996), “Microplane model for concrete. I: Stress-strain boundaries and finite strain”, Journal of Engineering Mechanics, ASCE, 122, pp.245-254. Bažant, Z. P., Hauggaard, A. B., Baweja, S., and Ulm, F.-J. (1997). “Microprestresssolidification theory for concrete creep. (I: Aging and drying effects, II: Algorithm and verification).” Journal of Engineering Mechanics, ASCE, Vol. 123, No. 11, pp. 1188-1201. Borderie C. La (1991), “Phénomènes unilatéraux dans un matériau endommageable: Modélisation et application à l’analyse de structures en béton”, PhD thesis, Université Paris VI., France. Borst R. and Nauta P. (1985), “Non-orthogonal cracks in a smeared finite element model”, Engineering Computations, 2, pp.35-46. Broomfield, J.P. (1997), “Corrosion of steel in concrete”, Understanding, investigation and repair, E&FN SPON. Carol I., Prat P., and Bažant Z. P. (1992), “New explicit microplane model for concrete: Theoretical aspects and numerical implementation”, International Journal of Solid and Structures, 29, pp.1173-1191. CEB-FIP Model Code 1990 (1993), Thomas Telford Services, Ltd., London, for Comité Euro-International du Béton, Bulletin d’Innformation No. 213-214, Lausanne, pp.437. Cedolin L. and Poli S. Dei (1977), “Finite element studies of shear-critical R/C beams”, Journal of Engineering Mechanics, ASCE, 103(3), pp.395-410. Chaboche J. L. (1979), “Le concept de contrainte e.ective, appliqué à l’élasticité et à la viscoplasticité en presence d’un endommagement anisotrope”, Grenoble, Editions du CNRS, Number 295 in Col. Euromech 115, pp.737-760. Chong, K.T. (2004). Numerical Modelling and Time Dependant Cracking and Deformation of Reinforced Concrete Structures, PhD Thesis, The University of New South Wales, School of Civil and Environmental Engineering, Australia, Chong, K.T., Foster, S.J., and Gilbert, R.I. (2008). Time-dependent modelling of RC structures using the cracked membrane model and solidification theory”, Computers & Structures, 86, pp.1305-1317. Crisfield, M.A. (1981), A fast incremental/iterative solution procedure that handles snapthrough, Comput. Struct., Vol 13, pp.55-63. Crisfield, M.A. (1982), Accelerated solution techniques and concrete cracking, Comp. Meth. Appli. Mech. Engng. Vol 33, pp.585-607.

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Crisfield, M.A. (1983), An arc length method including line searches and accelerations, Int. J. Num Meth. Engng. Vol 19, pp.1269-1289. Dafalias Y. F. and Popov E. P. (1975), “A model of nonlinearly hardening materials for complex loading”, Acta Mechanica, 21, pp.173-192. Dixon L.C.W. (1972), Nonlinear optimization, The English University Press Ltd, London. Elsawaf, A. (1979), The conjugate-Newton method for non-linear finite element problems. PhD. Thesis, University of Calgary, Canada. Foster S.J. (1992), “An application of the arc length method involving concrete cracking”, Int. J. Num. Methods Eng., Vol. 33, No. 2, pp. 269-285. Irons, B.M., and Elsawaf, A. (1970), “The conjugate-Newton algorithm for solving finite element equations”, in formulations and algorithms in finite element analysis, (Eds. K.J. Bathe, J.T. Oden and W. Wunderlich) MIT Cambridge, MA, pp. 656-672. Jirásek M. and Zimmermann T. (1997), “Rotating crack model with transition to scalar damage: I. Local formulation, II. Nonlocal formulation and adaptivity”, LSC Internal Report 97/01, Swiss Federal Institute of Technology, Lausanne, Switzerland. Krieg R. D. (1975), “A practical two-surface plasticity theory”, Journal of Applied Mechanics, ASME, pp.641-646. Ladevèze P. (1983), “Sur une théorie de l’endommagement anisotrope”, Rapport interne L.M.T. 34, E.N.S. de Cachan, Université Paris VI. Lévy M. (1870), “Mémoire sur des équations générales des mouvements intérieures des corps solids ductiles au delà des limites ou l’élasticité pourrait les ramener à leur premier état”, C. R. Acad. Arts Sci., 70, pp.1323. Lokhorst SJ and Breugel K. (1997), “Simulation of the effect of geometrical changes of the microstructure on the deformational behaviour of hardening concrete”, Cement and Concrete Research, 27(10), pp.1465-1479. Maekawa K and Ishida T. (2001), “Service-life evaluation of reinforced concrete under coupled forces and environmental actions”, Materials Science of Concrete, special volume, Ion and mass transport in cement-based materials, pp.219-238. Maekawa K, Chaube R. and Kishi T. (1999), “Modelling of Concrete Performance”, London: E & FN SPON. Mazars J. (1985), “A model of a unilateral elastic damageable material and its application to+ concrete” , In Fracture toughness and fracture energy of concrete, Elsevier, New York. Mazars J. (1986), “A description of micro and macroscale damage of concrete structures”,+ International Journal of Fracture, 25, pp.729-737. Menétrey Ph. and Willam K. J. (1995), “A triaxial failure criterion for concrete and its Generalization”, ACI Structural Journal, 92, pp.311-318.


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Mroz Z. (1967), “On the description of anisotropic work-hardening”, Journal of the Mechanics and Physics of Solids, 15, pp.163. Ortiz M. (1985), “A constitutive theory for the inelastic behavior of concrete”, Mechanics of Materials, 4(1), pp.67-93. Ottosen N. S. (1977), “A failure criterion for concrete”, Journal of Engineering Mechanics, ASCE, 103, pp.527-535. Ožbolt J. (1995), “Maßstabseffect und Duktilität von Beton- und Stahlbetonkonstruktionen”, IWB Mitteilungen 1995/2, Universität Stuttgart, Germany. Papa E. and Taliercio A. (1996), “Anisotropic damage model for the multiaxial static and fatigue behaviour of plain concrete”, Engineering Fracture Mechanics, 55(2), pp.163-179. Peekel Instruments, Rotterdam, The Netherlands. (1994). SBETA Program Documentation. Prager W. (1955), “Probleme der Plastizitätstheorie”, Birkhäuser, Basel, Stuttgart. Ramm E. (1981), “Strategies for tracing non-linear response near limit points”, in formulations and algorithms in finite element analysis, (Eds. K.J. Bathe, J.T. Oden and W. Wunderlich) MIT Cambridge, MA, pp. 656-672. Ramtani S. (1990), “Contribution à la modélisation du comportement multiaxial du béton endommagé avec description du caractère unilatéral”, PhD thesis, Université Paris VI. Riks E. (1972), “The application of Newtons method to the problem of elastic stability”, J. Appl. Mech., Vol. 39, pp.1060-1066. Ross, A.D. (1958), “Creep of Concrete under Variable Stress”, ACI Proceedings, 54(9), pp. 739-758. Saint-Venant B. (1870), “Mémoire sur l’établissement des équations différentielles mouvements intérieures opérés dans les corps solids ductiles au delà des limites ou l’élasticité pourrait les ramener à leur premier état”, C. R. Acad. Arts Sci., 70, pp.473. Shanno D.F. (1978) , “Conjugate gradient methods with inexact searches, Math O.R., Vol. 13, pp. 71-113. Simo J. C. and Wu J. W. (1987), “Strain and stress based continuum damage models: Part IFormulation”, International Journal of Solids and Structures, 23(7), pp.821-840. Suidan M. and Schnobrich W. C. (1973), “Finite element analysis of reinforced concrete”, Journal of the Structural Division, ASCE, 99(10), pp.2109-2122. Tresca. (1868), “Mémoire sur l’écoulement des corps solides”, Mémoires présentés par divers savants, 18, pp.733-799. Vecchio, F.J. (1990). “Reinforced Concrete Membrane Element Formulations”, ASCE Journal of Structural Engineering, Vol. 116, No. 3, pp. 730-750. Vecchio, F.J. (2000), “Disturbed Stress Field Model for Reinforced Concrete: Formulation”, ASCE J. Structural Engineering., Vol. 126, No. 9, pp. 1070-1077.

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Vecchio, F.J. (2001), “Disturbed Stress Field Model for Reinforced Concrete: Implementation”, ASCE J. Structural Engineering., Vol. 127, No. 1, pp. 12-20. Von Mises R. (1913), “Mechanik der festen Körper im plastisch-deformablem Zustand”, Göttinger Nachrichten (math.-phys. Klasse), 1, pp.582-592. Wolfe M.A. (1975), “Numerical methiods for unconstrained optimization – an introduction”, Van Nostrand-Reinhold, London. Yazdani S. and Schreyer H. L.ù (1988), “An anisotropic damage model with dilation for concrete”, Mechanics of Materials, 7(3), pp.231-244. Weihe S. (1995), “Modelle der fktiven Rißbildung zur Berechnung der Initiierung und Ausbreitung von Rissen”, PhD thesis, Universität Stuttgart. Wempner, G.A. (1971), “Discreet approximations related to nonlinear theories of solids”, Int. J. Solids Struct., Vol. 7, pp. 1581-1599 Willam K. J. and Warnke E. P. (1974), “Constitutive model for the triaxial behavior of concrete”, Concrete Structures Subjected to Triaxial Stresses, volume 19 of IABSE Report, pp.1-30, International Association of Bridge and Structural Engineers, Zurich. Willam K., Pramono E., and Sture S. (1987), “Fundamental issues of smeared crack models”, In S. P. Shah and S. E. Schwartz, editors, Fracture of Concrete and Rock, pp.192-207, Society for Experimental Mechanics, Bethel, Connecticut. Ziegler H. (1959), “A modification of Prager’s hardening rule”, Quarterly of Applied Mathematics, 17, pp.55-65.


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7

Benchmark tests and validation procedures

7.1

Introduction

Non-linear finite element analysis (NLFEA) programs for the design and analysis of structural concrete have not yet matured to the point where a user can simply develop a geometrically accurate model of the structure, specify basic material parameter values, and then expect that the program will accurately predict the response (capacity, deformation, distribution of internal stresses, etc.) of any structure to any prescribed loading. The goal of this chapter is to provide guidance for the efficient and safe use of NLFEA programs for the design and analysis of concrete structures. The development of this guidance is complicated by the variation in the types and capabilities of different programs and in the reasons for which numerical investigations are conducted. NLFEA programs range from concrete-specific to large commercial programs. These programs (or tools) employ a large variety of multiple material models, element types, and solution schemes. The level of required validation is program and task specific. If a concretespecific tool is used to examine the type of behaviour of a structure for which this tool was explicitly developed, then it is closer to being pre-calibrated. Some two-dimension frame and continuum analysis programs are close to being this mature. By contrast, large commercial programs are often not specifically designed to deliver reliable predictions of the behaviour of concrete structures but do provide the user with a selection of material models to choose from that need to be calibrated by user-defined values. In the use of these programs, the user must accept a greater responsibility for calibrating and validating the program in order to produce reliable estimates of the response of a modelled structure. NLFEA tools can be used for the design of a structure as well as for conducting a detailed investigation of a particular aspect of its behaviour such as the maximum stress in reinforcement, crack orientation and width, or energy absorbing capacity. For the purpose of general design and analysis, it is necessary that the practitioner evaluate the programs ability to predict the strength and behaviour of a well-selected group of experiments (benchmark tests) in order to determine a global safety (or strength reduction) factor to use in his or her design. The predicted capacity of a structure is then multiplied by this global safety factor to obtain a safe design capacity. Thus, a higher global safety factor is appropriate when the user has reliable material data for setting material characteristic values and when the program provides accurate predictions for the behaviour of physical experiments that capture the range in behaviours that are expected in the structure being modelled. The more comprehensive the validation activity, the greater confidence the designer can have in any calculated global safety factor. In the absence of less reliable material data, or when the NLFEA program does not provide a good estimate of the capacity of representative physical test specimens, or when an extensive validation procedure is not conducted, the results of the analysis should be penalized by the use of a lower global safety factor. Chapter 7 provides guidance for a designer who is trying to assess the appropriate global safety factor to apply to the results from a particular finite element prediction as well as for the user who is trying to investigate a particular aspect of the behaviour of a concrete structure. The most important segment of this chapter is in Section 7.2 which describes the steps in the calibration and validation of NLFEA programs. Section 7.3 discusses the factors to consider in the determination of a global safety factor. Section 7.4 addresses several important issues in the selection, use, and validation of NLFEA programs. The remainder of the document is devoted to the presenting case study examples. Section 7.5 presents an example of the design of a squat concrete shear wall with openings in which the global safety factor is first determined through a comparison of experimental test data with the predictions of a specific NLFEA program. In the second case study (Section 7.6), a comparison is made ! fib Bulletin 45: Practitioners’ guide to finite element modelling of reinforced concrete structures

233


between two different NLFEA programs for investigating the detailed behaviour of a deep beam. One of these programs is a large commercial program in which several behavioural models are available for the user to select between. The use of such a program requires additional care by the user as is illustrated in this example. This chapter concludes in Section 7.7 with a discussion of future trends in model validation.

7.2

Calibration and validation of NLFEA models

7.2.1

Overview of model calibration and validation process

The steps and flow of information in the calibration and validation of a NLFEA program is shown in Figure 7.1. The case of the design of a shear wall using a 2-Dimensional continuum finite element analysis program is used in this example in order to provide more specific details for the types of information that need be exchanged at each step.

Calibration

Constitutive Models for Uniaxial State of Stress for Concrete and Steel Reinforcement (User may need to choose from a number of models for compressive, tensile, and shear behavior)

Level 1

Mechanical Characteristics of Concrete and Reinforcement [Material Level Benchmark Tests]

Modelling Smeared Crack Model for Two-Dimensional State of Stress for Membrane Element

Calculation (Prediction)

Experimental Data from Panel Elements Under Biaxial State Validation & Calibration* of Stress (Collins 1982) [Element Level Benchmark Tests]

Used to Predict Capacity of Selected Membrane Elements Results from Comparison with Element Test Data may be used to Calibrate Additional Element Level Parameters Such as Crack Spacing Minor Adjustment to Element-Level Parameters*

Level 2

Application in 2-D FEM

Complete 2-Dimensional Continuum Model for Design and Analysis of Planar Walls

Calculation (Prediction)

Results from Comparisons Used to Assess Accuracy of 2-D Continuum Model for Predicting Strength and Stiffness Performance of Planar Walls; Global Safety Factor Selected Based on Comparisons

Validation Calibration*

Results from Lateral Load Tests on Reinforced Concrete Planar Walls [Structural-Level Benchmark Tests]

Level 3

*Any Calibration based on Element or Structural Level Experiments Must be Completed with Caution to Avoid Tuning a Model to Fit One Specific Class of Experiments; Importance of Validating Using a Comprehensive Series of Test Data that Captures the Range of Critical Behavior Characteristics cannot be Overemphasized

Figure 7.1: Overview of procedure for model calibration and validation (after Okamura and Maekawa, 1991).

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!

7 Benchmark tests and validation procedures


7.2.2

Level 1: model calibration with material properties

In the use of all NLFEA programs, it is necessary to specify the mechanical characteristics of materials. This is likely to include initial elastic stiffness as well as peak stress and strain values, concrete shrinkage strain, thermal expansion coefficients and other standard material properties. The more accurately the user can specify the required information, the more reliable will be the results of any analysis. In selecting material values, the user should use mean (average) material properties and recognize that the material properties in structural concrete may differ from those in material test specimens. For example, due to additional shrinkage in thin members and the restraint due to the presence of reinforcement it is common for structural concrete to crack under stresses that are considerably below what is obtained from material test data. It is also important to recognize that there can be a wide variation in some material properties and the reliability of analysis results is reduced if assumed values are used. To illustrate this latter point, Figure 7.2 plots the tensile cracking stress (fct) as a function of cylinder compressive strength (fcm). This wide variation illustrates the importance of using test data and not simply relying on typically assumed relationships, such as the one shown in Eq. 7.1, if the results of the analysis are expected to be sensitive to this value.

fct ' 0.33 fcm

(7.1)

where the values are in MPa.

Figure 7.2: Variation in tensile splitting strength as a function of cylinder compressive strength.


235


7.2.3 Level 2: validation and calibration with systematically arranged element–level benchmark tests

Since a NLFEA program does not provide the exact capacity of a structure, it is necessary to begin the use of any investigation with an evaluation of the program through a comparison of predicted and measured response of test structures that have been load-tested to failure. By examining the ratio of measured to predicted capacity of a well selected set of experiments, the user can determine a global safety factor to apply to the results of an analysis. This global safety factor, which will be labelled ', will be a fraction less than or equal to 1.0. The safe capacity predicted by the program for the design or analysis being conducted will then be the capacity predicted by the NLFEA program multiplied by this global safety factor, '. The global safety factor may be considered to be the end product of the model validation process. While Level 1 was purely a calibration activity in which material parameter values were set, Level 2 it not always a pure validation exercise. This is because the experimental test data that is most suitable for model evaluation is often the most suitable test data for setting structurallevel parameters that account for such effects as compression softening, tension stiffening, shear retention, loss in bond, crack spacing, etc. The tests that can serve this dual purpose are simple tests where there is little chance for misinterpretation of the test data. These tests may be referred to as fundamental or element-level tests. When evaluating a tool, or for that matter calibrating structural-level parameters, the user needs to predict the behaviour of a systematically arranged set of element-level benchmark tests that encompass the range in possible behaviours that may be experienced by the structure being designed or studied. Consider the following example to illustrate the evaluation of NLFEA programs at the element level. The construction of the Panel Element Tester at the University of Toronto (UofT) in 1979 allowed uniform tests on reinforced concrete shear panels to be performed for the first time. Based on the results of an IABSE colloquium in June of 1981, it was decided to hold an international competition (Collins, 1985) to test the quality of various numerical predictions for a collection of four panel elements (A-D). Panel D is shown in Figure 7.3.

Figure 7.3: Panel D after failure in the panel element tester.

236

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The tests were performed in August of 1981 and while the measured material properties were made available, the actual experimental results were kept confidential to ensure that the results of the contest would be true predictions. A total of 27 entries were received from 13 countries. The tests had been selected such that they would be difficult to predict; none of the elements failing in biaxial shear. It was found that the predictions varied substantially from the observed behaviour. Figure 7.4 presents the details of the panels. The best entry was able to predict all 4 panels to within 17% of the measured capacity. However, other predictions were in error by as much as a factor of 3 in the measured strength, as illustrated in Figure 7.5. What made the UofT competition so valuable is that each of the four selected panels evaluates an important aspect of the biaxial response of structural concrete. Panel A was selected so that the accuracy of the shear transfer relationship and the compression softening model would have little effect on the shear response of the test specimen. In Panel B, the softening of the compression model dominated the response. In Panel C, it was most important to accurately model both tension stiffening and shear transfer across cracks. In Panel D, it was important to accurately model all aspects of behaviour in order to make a good prediction of behaviour. When reviewing the accuracy of the 27 entries it was clear that using only 2 or 3 of the panel tests would have led to an inaccurate assessment of the capabilities of the numerical tools; thereby illustrating the importance of the selection of a systematically arranged set of validation experiments (benchmark tests). Since 1982 there has been significant advancements in the development of NLFEA programs for structural concrete. Figure 7.6 provides a comparison of the predicted versus measured shear behaviour of the four panels using a more recent NLFEA program and labelled as NLFEAP-1. Y

t

t

PANEL A t

PANEL B t sx

0 9 ( 9( u

+ x ' 70.7, + y ' 70.7, 0 9 ( 9( u

sy

X

f c' ' 20.5 MPa

$ 0 ' 0.00190

f c' ' 20.5 MPa

$ 0 ' 0.00190

f xy ' 442 MPa

* x ' 0.01785

f xy ' 442 MPa

* x ' 0.01785

f yy ' 442 MPa

* y ' 0.01785

f yy ' 442 MPa

* y ' 0.01785

Y

t

t

PANEL C t

PANEL D t sx

0 9 ( 9( u

+ x ' 7(, 7 3.9) 9 0 + y ' 7(, 7 3.9) 9 0 0 9 ( 9( u

sy

X

f c' ' 19.0 MPa

$ 0 ' 0.00215

f c' ' 21.7 MPa

$ 0 ' 0.00180

f xy ' 458 MPa

* x ' 0.01785

f xy ' 441 MPa

* x ' 0.01785

f yy ' 299 MPa

* y ' 0.00713

f yy ' 324 MPa

* y ' 0.00885

Figure 7.4: Description of panels A-D for the UofT competition.


237


12

Contest Results for Panel B

,

Number of Predictions


Contest Results for Panel A

,

10 8 6

*x5'5*y

4 2 0

0.7

0.8

0.9

1

1.1

1.2

1.3

8 7 6 5 4 3 2 1 0

, ,

+y *x5'5*y 0.3

1.4

0.5

0.7

Contest Results for Panel C

1.1

1.3

1.5

1.7

Contest Results for Panel D 6

,

Number of Predicitons


0.9

Predicted/Observed

Predicted/Observed

9 8 7 6 5 4 3 2 1 0

+x

,

*x5'52.55*y

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

,

5

,

4

+x

3

+y *x5'5 25*y

2 1 0 0.6 0.8

1

1.2 1.4 1.6 1.8

Predicted/Observed

2

2.2 2.4 2.6 2.8

Predicted/Observed

Figure 7.5: Distribution of measured to predicted capacities of four test specimens.

10 Observed-A Analysis-A Observed-B

Shear Stress(MPa)

8

Analysis-B Observed-C Analysis-C

6

Observed-D Analysis-D

4

2

Panel A

Panel B

Panel C

Panel D

Measured v test (MPa)

6.29

9.02

3.94

5.61

Predicted v nlfea (MPa)

6.597

8.244

4.136

7.2

Tatio v test v nlfea

0.95

1.09

0.95

0.78

0 0

5

10

15

20

25

Shear Strain (x0.001)

Figure 7.6: Comparison of measured and predicted behaviour of four test panels.

7.2.4

Level 3: validation and calibration at structural level

In order to make a thorough evaluation of a particular NLFEA program and thereby determine the global safety factor, it is necessary to make detailed comparisons with carefully selected test data of both element level tests (discussed in Section 7.2.4) and tests of complex structures. These complex – or structural-level tests – can account for the effects of complex geometries, loadings, and edge effects. In evaluating a model with structural-level tests, many types of comparisons can be made. The most common and often important comparisons are initial stiffness, capacity, deformation at peak load, and deformation at failure. The overall load-displacement response captures all of these measures. It is also possible to compare a wide range in other performance measures including the development, propagation, width and orientation of cracks, the strains in reinforcement, and the distribution of compressive strain in 238

!



concrete. These other comparisons are determined by the specific needs of a particular investigation.

7.3

Selection of global safety factor

In codes of practice, it is common to use a strength reduction factor, ', to determine the reliable design strength as a fraction of the code calculated nominal (or “predicted”) strength. The range of typical values for ' is between 0.6 and 1.0. This factor aims to account for variability in actual material strengths, imperfections in construction, the influence of load duration on structural behaviour, and inaccuracies in the relationships for calculating nominal strength when applied to structures that may be outside of the range of those experiments used to develop these relationships. For example, it is more common to use a lower safety factor in calculating the reliable shear strength as code provisions typically use coarse empirical relationships for calculating shear capacity. By contrast, a much less severe strength reduction factor (up to 0.95) is used for flexural capacity for which the model for behaviour is well established and verified. An example is now used to illustrate how global factors of safety are selected in a code-ofpractice; the equivalent approach can be used for determining ' for any particular NLFEA program. Figure 7.7 plots the ratio of the experimentally measured versus ACI318-02 code calculated shear strength of reinforced concrete beams with shear reinforcement, where the code calculated nominal shear capacity (Vn) is determined by

Vn ' Vc 8 Vs '

fc# 6

bwd 8

Av f y d

(7.2)

s

where bw is the width of the web, d is the member depth, Av is the area of shear reinforcement, and s is the spacing of the transverse reinforcement (units are in N, mm, and MPa).

3.0 2.5

Vtest / VACI

2.0 1.5 1.0 0.5 0.0 0

20

40

60

80

100

120

140

f'c (MPa)

Figure 7.7: Shear strength ratio of RC members with shear reinforcement.


239


As shown in Figure 7.7, Eq. 7.2 does not provide a very accurate estimate of the measured test strength nor does it always produce a conservative result. However, with the use of a suitable value of C, then this somewhat inaccurate equation can be safely used for design with the safe design strength being taken as CVn. In the development of codes of practice, the global safety factor is selected so that there is a small probability that the actual capacity is less than the design strength (C o nominal capacity). Based on an assumed normal distribution, there is a direct relationship between the mean (,), the standard deviation of the strength ratio (s), the strength reduction factor C and the percentage of the design strengths (or the Fractile level, F) that are expected to be less than C o nominal capacity. This relationship between these values is given by

eC 7u b F ' zc ` d s a

(7.3)

C ' z 71 ( F ) T s 8 u

(7.4)

where z is a Gaussian cumulative distribution function. To illustrate this relationship, consider the data described in Figure 7.7 for which the mean is , = 1.35 and the standard deviation is s = 0.37. The statistical values in Table 7.1 can be calculated using Eqs. 7.3 and 7.4. If no strength reduction factor is used (that is C = 1.0), then in 17.2% of the cases it would be expected that the actual strength is less than the design strength. When a strength reduction of C = 0.75 is applied, which is the current value for shear used in ACI318, then for only 5.24% of the cases is the actual strength expected to be less than the design strength (see Figure 7.8). Table 7.1: Example of relationships between global safety factor and fractile level.

Item

C F

1.00

0.95

0.1721 0.1398

Statistical Values 0.85 0.80 0.75

0.90 0.112

0.70

0.65

0.60

0.0883 0.0686 0.0524 0.0395 0.0293 0.0213

B = 1.35 s = 0.37 COV = 0.28

Fractile level F

C = 0.75 C = 1.00

B = 1.35

F= 0.0524 F= 0.1721

Figure 7.8: Relationship between level of safety and strength reduction factor.

240

!



7.4

Other issues in the use and validation of NLFEA programs

7.4.1

Problem definition and model selection

In many cases, the practitioner has limited choice in the NLFEA program that is available to apply to a particular problem due to financial, time, or technical constraints. In other cases, the practitioner has some time to evaluate and select the most suitable program for their short or long-term needs. There are many factors that influence which model is most appropriate to use in an analysis, ranging from technical to practical issues, as discussed below.

Capability: Are the analytical models (constitutive models and failure theories) incorporated in the NLFEA program capable of capturing the types of behaviour most relevant to the design challenges (or problems) to be studied? Some of the behaviours that are difficult and infrequently modelled well in structural concrete include the benefit of passive confinement, damage accumulation due to cyclic loading, dowel action across crack interfaces, fracture in plain concrete regions, interface shear transfer and compression softening. Transparency of Underlying Behavioural Models: Are the models, algorithms, assumptions, and solution routines employed in the NLFEA program clearly defined? Some programs, employ classical mechanics approaches, use a few well defined models for behaviour and employ reasonably strict convergence criteria. While this provides a transparent and wellfounded methodology, the inability to account for complexity of behaviour of cracked structural concrete coupled with frequent convergence difficulties limit the range of applicability of such programs. In addition, the user can be daunted by the impact that selected parameter values and convergence criteria have on the predictions of these tools. By contrast, less flexible NLFEA programs that are specifically designed to capture the complexity of structural concrete behaviour can provide reliable predictions of structural response without the need for extensive calibration or validation. Guidance for Selecting Parameter Values: What level of documentation and guidance does the NLFEA program provide for the determination of model input parameter values? It is frequently the case that there is insufficient material and structural test data information for defining all input parameters and, thus, guidance for selection of parameters values is often necessary. It is also useful if documentation is provided of the sensitivity of typical solutions to input parameter values. Benchmark Tests: What is the availability of benchmark tests for the type of problem for which the NLFEA program is to be utilized? Has the developer of the program provided a large and systematic set of experiments and analysis examples? If there is a good selection of benchmarks tests to use for the validation (and limited calibration) of the NLFEA program, then the user will be better able to more easily evaluate the global safety factor. Other Capabilities and Features of NLFEA Program: What is the ability of the NLFEA program to model other materials, physics, and complex geometry? What are the pre and post-processing capabilities of the program? Can the program automatically generate an appropriately dense finite element mesh for the structure under investigation? Many NLFEA programs are freely available while others cost up to tens of thousands of dollars per year.


241


7.4.2

Working within the domain of the program’s capability

A common problem in the application of NLFEA is that the program is used to investigate the type of behavior or type of structure that is outside the range of problems for which it was designed. For example, a program that uses a smeared-cracking model may not be able to predict the propagation of cracking and capacity of an unreinforced region. The range of application of a particular tool can be determined by a careful review of the documentation that should be delivered with the program. A less obvious source of error is that the practitioner has not properly assessed the accuracy of their program using a systematically selected set of elements and structural-level benchmark experiments. For example, research has shown that the common assumption that shear strength is proportional to the square root of the concrete strength does not hold when the concrete strength exceeds around 60 MPa. In this case, it would be unwise to scale up the results from tests on lower strength concrete by the square root of the compressive strength for the design of a structure for which high-strength concrete is to be used. Another example is the importance of scale. It is unsafe to evaluate the accuracy of a NLFEA program using small scale experimental test results when the structure to be designed is massive. This is illustrated in the following example. For many years the research community assumed that the shear stress at failure was not influenced by the depth of a member until the classic set of experiments were conducted on members up to 3 meters deep conducted by Shioya (1989) as shown in Figure 7.9. These tests demonstrated that the shear stress at failure of very large members could be less than half the shear stress at failure of smaller members (Figure 7.10). When selecting a set of experiments to use in any validation activity, it is important to use highly regarded and carefully conducted experiments the results of which are reliable measures of the actual performance of the test structure. Not all tests are equal and in many instances the researchers have failed to provide the information necessary to ensure that the results of the experiments are reliable. As an example, joint ASCE/ACI Committee 445 “Shear and Torsion” oversaw the establishment of a database of shear test results for reinforced concrete members that did not contain shear reinforcement. While more than 1000 experiments were identified, after criteria were applied to assess the reliability of the test data, only about 300 test results remained for use in the evaluation of code provisions. The main reason that beams were disqualified were that they were either too small, calculated to fail in flexure, calculated to have an anchorage failure, only specified material strengths were available, or neither of the supports was a roller.

242

!



Figure 7.9: Tests of Large Scale Bridge Girders by Shioya (1989).

d (mm) 0 3.2

1000 PCA Tests of Model Air Force Warehouse d Beams

2.8 2.4 V bwd fc#

2.0

(psi)

1.6

12”

6”

0

Air Force Warehouse Beams

" = As = 0.4% bwd fc# = 3500 psi (24.1 MPa) fy = 52 ksi (386 MPa)

a = 2.5 mm (0.1 in.)

0.8 8” d = 4 in.

0

20

0.20 0.15

V bwd fc # (MPa)

0.10 b = 39”

20” 39”

24”

0.25

12d

max. aggregate size a = 25 mm (1 in.) a=10 mm (0.39 in.)

3000 d

V calculated here

1.2

0.4

2000

d = 79”

b = 59” d= 118”

25.4 mm = 1”

40

60 80 100 d (inches)

120

0.05 0

Figure 7.10: Size effect in shear.


243


7.5

Case 1: Design of a shear wall with openings

7.5.1

Objective

The variation in strains in the squat wall with opening described in Figure 7.11 is very complex. Since neither engineering beam theory nor codes-of-practice provide detailed guidance for the design of this type of structure, the strut-and-tie method (STM, See Chapter 8) can be used for the design of this wall (Paulay, 1992). The designer may, therefore, wish to evaluate the capacity of the designed structure and to investigate its condition at service loading using a NLFEA program. In this example the design of this shear wall is completed using the 2-D continuum program (NLFEAP-1) that was used to predict the behaviour of the four Toronto Panels discussed in Section 7.2.3. As the performance of this program for predicting the response of element-level benchmark tests has already been completed for the grade of concrete and amount of reinforcement that is expected to be used in this wall, the next step is to validate the program using a full structural level benchmark test, or tests. Following this, all of the information obtained is used to establish the global safety factor. The reinforcement determined from the strut-and-tie design is shown in Figure 7.12. 7.5.2

Level 1 calibration

For this program, only Level 1 calibration is required, which is the setting of the mechanical characteristics of the concrete and reinforcement. This involved setting the following material parameter values: f-c = 31 MPa, "-c = 0.002, Es = 210000 MPa, fy = 414 MPa, "sh = 0.0025, "rupt = 0.1 and fu = 621 MPa 7.5.3

Level 2 and 3 validation

Level 2 element-level validation of the program was conducted in Section 7.2.3. Prior to using NLFEAP-1 for the design of the shear wall with multiple openings, a single structural level validation is conducted. The shear wall with an opening, as described in Figure 7.13 and which had been load tested to failure, was modelled in NLFEAP-1 as shown in Figure 7.14. A comparison of the measured and predicted load versus deformation response of this shear wall is shown in Figure 7.15 from which it can be determined that the shear strength ratio (measured strength / predicted strength) is 164.3/163.8 = 1.00. 7.5.4

Evaluation of global safetyf

Table 7.2 presents the ratio of the measured to predicted capacity of the four panel elements and the wall tested by Taylor (1998). The mean of the strength ratios B = 0.96 and the standard deviation of these ratios s = 0.114. Table 7.2: Measured and Predicted Capacities

244

Panel A

Panel B

Panel C

Panel D

Wall

Measured Capacity

6.29

9.02

3.94

5.61

164.3

Predicted Capacity

6.597

8.244

4.136

7.2

163.8

Ratio Measured/Predicted

0.95

1.09

0.95

0.78

1.00

!



15

30

15

3Q

15

3Q

15

30

15

30

60

30

2Q

Q

30

15

15

30

15

60

30

2Q

Q

30

6.00 2.52

8.83

7.10

4.25

Figure 7.11: Shear wall to be designed using NLFEAP-1.

D10-300

6-D16 0 0 7 R6 ties-300 0 0 1 2

2-D12 D12-300

0 0 1 sie t 6 R

4-D20 4-D12 0 0 1 -s ite 6 R

4-D10

4-D16

0 0 7

0 0 1 2

2-D12

8-D20 0 0 7 0 0 1 -s e ti 6 R

4-D16

8-D10

4-D10

0 0 1 2

2-D12 18-D12 14 -D10 4200

2100

R6 ties at 100

D 12 D10 D10

D12

700

700

D12 D16 D12

D 12

D16

Figure 7.12: Reinforcement in squat shear wall based on STM design (Paulay, 1992).


245


48 ” (1 21 9 mm)

H oop s

We b Bars

FOU RTH FLOOR

2 .5 ” (64 m m) 6 .5” (16 5 mm) 8@3 .75 ”=3 0” ( @95 =7 62 m m)

THIRD FL OOR 11 @7 .5 ”=8 2.5 ” (@ 19 1=20 96 mm ) A

A

9@ 7.5”=67 .5” (@19 1=17 15 mm)

SEC OND FL OOR

B

14 4” (36 58 mm)

B

FIR ST FLOOR

4@3 .75 ”=1 5” ( @95 =3 81 mm) C

C

27 .5 @2 ”=5 5” (@ 51 =1 39 7 mm)

5@ 5”=25 ” (@1 27 =6 35 m m)

GR OUN D LE VEL 27” (68 6 mm) 76 ” (1 93 0 mm)

RW3-O: Elevation view showing reinforcing

(a) 48 ” (1 21 9 mm) 3@ 2”=6” (@51 =1 52 m m) 0.7 5” 6” (1 9 mm ) (15 2 mm)

3@ 2”=6” (@51 =1 52 m m) 0.7 5” (1 9 mm )

6” (15 2 mm)

8-# 3

3@ 7.5”=22 .5” (@1 91=5 72 mm )

#2 @7 .5 ” (@1 95 mm)

lap spl ice = 6” (15 2 mm)

SECTION A-A 3@ 2”=6” (@51 =1 52 m m) 0.7 5” (1 9 mm )

3.3 8” (8 6 mm )

8-# 3

2.6 3” 6 7 mm) 3@ 5”=15 ” (@1 27 =3 81 mm)

#2 @5” (@12 7 mm)

2-# 3

3@ 2”=6” (@51 =1 52 m m) 0.7 5” 3@ 4.5”=13 .5” (1 9 mm ) (@ 114 =3 43 mm )

3/16 ”(5 mm ) h oo ps & cr oss tie s@2”( 51 mm )

SECTION B-B

3.3 8” (8 6 mm )

8-# 3

2.6 3” 6 7 mm) 3@ 5”=15 ” (@1 27 =3 81 mm)

#2 @5” (@12 7 mm)

0.7 5” (1 9 mm )

2-# 3

3@ 2”=6” (@51 =1 52 m m) 0.7 5” (1 9 mm ) 0.7 5” (1 9 mm )

12” (30 5 mm)

3/16 ”(5 mm ) h oo ps & cr oss tie s@2”( 51 mm )

SEC TION C -C RW3-O: C ross sections

2.5” (6 4 mm )

0.7 5” (1 9 mm )

0.7 5” (1 9 mm )

3@ 2”=6” (@51 =1 52 m m)

(b) Figure 7.13: Shear wall experiment (Taylor, 1998): (a) front element view and (b) reinforcement details.

246

!



(b) Vertical reinforcement ratio

(a) Horizontal reinforcement ratio

Figure 7.14: Finite element model of shear wall experiment.

180

Lateral load (KN)

150 120 90 60

NLFEAP-1

30

Experiment

0 0

10

20

30

40

50

60

70

Top displacement (mm) Figure 7.15: Predicted and measured load-deformation response of shear wall.


247


Using the method in Section 7.4, the global safety factor was evaluated based on it being acceptable in 5% of situations that the design strength (C o capacity) determined by the use of NLFEAP-1 were less than the actual strength. Adopting an assumed normal distribution of strength ratios, the global safety factor is evaluated by Eqs. 7.3 and 7.4 to be C = 0.76. Since only a few benchmark tests were used to evaluate the performance of NLFEAP-1 program, it may be desired to use a lower global safety factor than that calculated above. As additional comparisons are made, the updated value of the global safety factor will be more reliable. It is not surprising that NLFEAP-1 had such good success at predicting the capacity of the benchmark tests as this program was specifically written for the design of structural concrete with distributed reinforcement in two directions and thus the practitioner can anticipate that the developers of this program took considerable effort to model the complexities of the behaviour of structural concrete and to validate and calibrate this program based on large numbers of systematic element-level and structural level benchmark experiments. It is not surprising that NLFEAP-1 had such good success at predicting the capacity of the benchmark tests as this program was specifically written for the design of structural concrete and thus the practitioner can anticipate that the developers of this program took considerable effort to model the complexities of the behaviour of structural concrete and to validate and calibrate this program based on large numbers of systematic element-level and structural-level benchmark experiments. Design using a NLFEA program is of course an iterative procedure where the user proposes a selected pattern of reinforcement and then uses the program to calculate the capacity. In this case, the trial reinforcement pattern shown in Figure 7.12 was designed using the strut-and-tie method The finite element model used in this analysis is shown in Figure 7.16 and the predicted load deformation response of the squat wall with openings in shown in Figure 7.17. Based on the results of the analysis using NLFEAP-1, the yield strength of the wall is 108 kN while the ultimate capacity is approximately 136 kN. Using the global safety factor previously evaluated (C = 0.76), the design capacity of the wall is determined to be 0.76 o 108 = 82 kN which is just larger than the design load of 80 kN. With the availability of the completed analysis, the designer can also investigate the condition of the structure under serviceability loading or continue to refine the design to most efficiently use the reinforcement or to provide a desired level of overall ductility.

248

!



(a) Horizontal reinforcement ratio

(b) Vertical reinforcement ratio

Figure 7.16: Finite element model used to model squat wall with openings.

160

Q (KN)

120

80

40

NLFEAP-1 Factored design load(Qu)

0 0

2

4

6

8

10

12

14

Top displacement (mm) Figure 7.17: Load deformation response of squat wall by NLFEAP-1


249


7.6

Case study II: design of simply supported deep beam

7.6.1

Objective

The objective of this example is to illustrate the differences between the steps and capabilities of two different NLFEA programs in predicting the behaviour of a deep beam that supports a point load at mid-span. The first of these programs is NLFEAP-1 which was the concrete specific program used in Case Study I. The second of these is a large commercial finite element program and is denoted as NLFEAP-2. The commercial program provides several material models for structural concrete and uses a classical mechanics approach. 7.6.2

Calibration and validation of NLFEAP-1

For this program, only Level 1 calibration is required, which is the specification of the mechanical characteristics of the concrete and reinforcement. This involved setting the following material parameter values: f c# = 33.5 MPa, "-c = 0.002, Es = 210 000 MPa fy = 498 MPa, "sh = 0.01 ,"rupt = 0.1 and fu =549 MPa. 7.6.3

Calibration and validation of NLFEAP-2

The second program to be used for predicting the behaviour of a deep beam is a large commercial NLFEA Program. The process of designing and analyzing a structure using this type of program is substantially different from the use of the concrete specific tool as often these comprehensive programs were not designed to produce accurate predictions of the behaviour of concrete structures. Rather, they provide a choice of material models, solutions schemes, and convergence limits to be selected by the user and thus the user is primarily responsible for calibrating and validating these programs to ensure that they will produce reliable predictions. This requires that the user becomes familiar with the underpinnings of the program and evaluates the sensitivity of the program predictions to non-basic material parameter values. The analysis methods employed in NLFEAP-2 and the results of a sensitivity analysis are provided below. The calibration and validation of NLFEAP-2 follows these discussions.

Overview of concrete model used in NLFEAP-2 There are two different analysis methodologies that are used in NLFEAP-2. The implicit time integration scheme uses a traditional stiffness-based approach in which the tangent stiffness equation is solved using the Newton-Raphson method (refer Chapter 3). In the iterative approach, the number of time steps and convergence tolerances are used to control the solution process. In the explicit time integration scheme, both static and dynamic FEA are performed with the use of a stiffness matrix. At each time step, the process involves the solution of multiple vector equations in which no iterations are required or convergence criteria need be specified. A large number of very small time steps (more than 10,000) are often required in the use of an explicit analysis methodology. NLFEAP-2 supports two different concrete models, the smeared-crack model and the damage plasticity model. The smeared crack model is most applicable when concrete is subjected to monotonic loading and either tensile cracking or compressive crushing dominates the material response. The effect of cracking is taken into account by adjusting how the stresses and material stiffness values at integration points are calculated. Cracking is defined to occur when the tensile stress reaches a failure surface that NLFEAP-2 refers to as a “crack detection

250

!



surface”, shown in Figure 7.18. After cracking, anisotropic behaviour is assumed. Plastic straining in compression is controlled by a classical “compression” yield surface, as shown in 7.19. 5q

5+ u 5.

“Crack detection” surface

51

“Compression” surface

51

5.

5p

52

5+u

Figure 7.18: Tension failure surface.

“Crack detection” surface Uniaxial tension

5+2 5+1

Uniaxial compression “Compression” surface

Biaxial tension

Biaxial compression

Figure 7.19: Yield and failure surface for in-plane stress.

In order to achieve the best results from a NLFEA, it is often necessary to carefully select values for many parameters as well as to choose from a variety of material models and solution methodologies. The parameter values are used to control the behaviour and influence of many effects including: - crack spacing

- tension stiffening effect

- compression softening effect

- benefits of passive confinement

- interface shear transfer

- bond strength

- shear model

- shape of stress-strain response

- fracture energy and critical length

- dowel action

It can be very challenging to select values for the parameters that control the effects listed above unless specialized test data is available. Since the results of an analysis can be greatly influenced by the values selected for these parameters, it is necessary to conduct a sensitivity analysis to understand the effect of parameter values on the predicted response. In ! fib Bulletin 45: Practitioners’ guide to finite element modelling of reinforced concrete structures

251


NLFEAP-2, the concrete and reinforcement are modelled separately. Reinforcement is modelled as truss elements in which a one-dimensional strain model is used. The effects of tension stiffening and dowel effects are not accounted for in modelling the behaviour of the reinforcement but rather are accounted for in the concrete model through tension stiffening and shear retention parameters.

Sensitivity Analysis A sensitivity analysis was conducted on the parameter values that control tension stiffening and shear retention on the predicted response by NLFEAP-2 of a reinforced concrete panel subjected to in-plane shear loading. As presented in previous chapters, the tension stiffening effect accounts for the contribution of cracked concrete in tension for carrying tension in the concrete between cracks. The tension stiffening relationship used in NLFEAP-2 is given in Figure 7.20 in which the average tensile stress is shown to decrease linearly from the tensile strength of concrete at "cr to zero at "ts. It is possible to have this tension degrade in a several ways, but a linear relationship was selected for this sensitivity analysis. The shear retention parameter, which is expressed as a ratio from S.R. = 0 to S.R. = 1.0, is used to define shear stiffness after cracking as a fraction of the uncracked shear stiffness. It decreases from the selected S.R. value to zero at a tensile strain at $max. See Figure 7.21. Note that the user must select values for .cr, "cr, S.R., and "max .

Stress, + Failure point

5+cr

“Tension stiffening” curve

5)

5+cr

=

) ts

Strain, )

5E

Figure 7.20: Tension Stiffening relationship used in NLFEAP-2.

S.R. 1.0

0.5

)max ) Figure 7.21: Shear Retention relationship used in NLFEAP-2.

252

!



As shown in Figure 7.22, the results of an NLFEAP-2 prediction can be quite sensitive to the selected tension stiffening strain limit. In this evaluation, the shear retention factor was set to 0.5 and the corresponding maximum shear retention strain was set to "max = 0.006. The sensitivity of the shear response predicted by NLFEAP-2 as a function of the values used in the shear retention parameters is shown in Figure 7.23. In this comparison, the value of the tension stiffening parameter was held constant at 0.0012. Four different shear retention values are chosen. In three of these four cases, the shear retention parameter is set so that the shear modulus just after cracking is set to 50% of the uncracked modulus (S.R. = 0.5) and the tensile strain at which the modulus reduces to zero ($max) is varied from 0.003 to 0.012. The results in Figure 7.23 illustrate that as the maximum strain is increased, the stiffness of the post cracking response and the progression in the convergence also increases. 6

Shear stress(MPa)

5

4

T-S=0.0012, S-R=0.5, 0.006 T-S=0.0018, S-R=0.5, 0.006

3

T-S=0.0036, S-R=0.5, 0.006 T-S=0.0048, S-R=0.5, 0.006

2

1

0 0

1

2

3

4

5

6

7

8

Shear strain(x10^3)

Figure 7.22: Sensitivity of the NLFEAP-2 prediction to the tension stiffening model.

6

Shear stress(MPa)

5

T-S=0.0012, No S-R

4

T-S=0.0012, S-R=0.5, 0.012 T-S=0.0012, S-R=0.5, 0.006

3

T-S=0.0012, S-R=0.5, 0.003

2

1

0 0

1

2

3

4

5

6

7

8

Shear strain(x10^3)

Figure 7.23: Sensitivity of NLFEAP-2 prediction to the shear retention parameter.


253


Now that the models used in NLFEAP-2 have been reviewed and a sensitivity analysis has been performed, the next step is the calibration of NLFEAP-2 for the deep beam example. The basic material parameters for the deep are first selected, as illustrated for the use of NLFEAP-1 in Section 7.6.2. Since NLFEAP-2 was not specifically designed for modelling the behaviour of structural concrete, it is necessary to calibrate the program in Level 2 and as the test data is used for model calibration, it cannot then be used for model validation. One element-level calibration example is given below for NLFEAP-2 where the implicit solution scheme and smeared cracking model is calibrated so to fit the measured shear stress versus shear strain response of panel PV20 of Vecchio and Collins (1982), shown in Figure 7.24, by adjusting the tension stiffening and shear retention values. The values for these parameters that create a good fit with the test data are a strain at which the tension stiffening effect drops to zero of "ts = 0.0036, a shear retention factor S.R. = 0.1, and a strain at which the shear modulus goes to zero of )max = 0.011. The measured and computed (by NLFEAP-2) shear stress versus shear strain response for PV20 are compared in Figure 7.25. v

Loading: Monotonic pure shear Specimen Dimension = 890 mm x 890 mm x 70 mm f’c =19.6 MPa,

)’c = 0.00180

v Maximum Aggregate Size = 6 mm

v

Yonung's Modulus of the steel, Es = 200,000 MPa for all reinforcement *x = 1.79 %, (f6.35 mm @ 50 mm on each face), Clear Cover = 6 mm, fxy = 460 MPa *y = 0.89 %, (f4.47 mm @ 50 mm on each face), Clear Cover = 12.35 mm, fyy = 297 MPa Observations:

v

Plan View

vcr

= 2.21 MPa

vu

= 4.26 MPa

Shear failure of concrete prior to yielding of longitudinal steel Cross Section

Figure 7.24: Description of Panel Element PV20.

5

Shear stress(MPa)

4

3 Experiment

2

T-S=0.0036, S-R=0.1, 0.011

1

0 0

1

2

3

4

5

6

7

8

Shear strain(x10^3) Figure 7.25: Calibration of NLFEAP-2 using results from PV20.

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Since the required number and diversity of experiments for model calibration increases as the number of parametric values to set increase, it is desired to fix as many of the parameter values as possible using available material test data and from other experience. In some cases, it may be desired to calibrate one computational model, such as a comprehensive commercial program, with the predictions of more specialized program. One of the noticeable challenges to the use of commercial packages is that the softening behaviour and stiffness degradation in concrete materials can lead to convergence difficulties when implicit solution routines are utilized. One way of overcoming this difficulty is by using damaged plasticity models in which a viscoplastic regulation of the constitutive relationships is employed, but this topic is not further explored in this report. 7.6.4

Analysis of deep beam

The goal of this analysis is to compare the behaviour of the deep beam described in Figure 7.26 by the two NLFEA programs. This side-by-side comparison is a useful exercise for the practitioner in choosing between analysis programs. In this evaluation, a comparison is made of the ability of the programs to predict the capacity of the beam (P), the deformation at failure (+), the diagonal compressive strain just above and inside of the support ("d), the strain in the longitudinal reinforcement just inside of the support ("ls) and the strain in the stirrups mid-way between the point of loading and the support ("esv). P 2-10M bars 10M Stirrups

300

4 sp. @ 40

200

1000

300

PL 300x200x25

12-#6 bars

Cross Section

120

P/2

PL 200x200x25

P/2

2800 mm

120

Elevation View Loading: A single mid-span load Specimen Dimension = 3040 mm x 1000 mm x 200 mm

)’c = 0.0024 Maximum Aggregate Size = 10 mm f’c =33.5 MPa,

*l = 2.01 %, (12 - # 6 Bars + 2 - 10M bars), Clear Cover = 30 mm *v = 0.50 %, (2 - 10M Bars @ 200 mm), Clear Cover = 15 mm Observations: vcr = 6.79 MPa (Vcr = 1100 kN)

Bar As Size (mm ) 10M 100 #6 284 2

fy

Es

(MPa)

(MPa)

529 498

197,300 216,800

vu = 11.94 MPa (V u = 1935 kN)

Crushing failure of the entire end block over the west support followed by a complete loss in re-bar anchorage and spalling of large pieces of concrete

Figure 7.26 :Description of deep beam (Lee, 1982).


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Modelling of the behaviour is made difficult by the complex state of stress and distribution of cracking that is expected to occur in the structure. Examining the behaviour from a strut-andtie perspective, the spreading of the diagonal compressive strut between the point of loading and support is expected to produce cracks along this axis and well as to result in a concentration of compressive stresses near the points of loading and reaction. In addition, there will be a heavy demand placed on main tension reinforcement at the inside of the supports. The behaviour of this region may also be viewed from a sectional perspective in which this is a deep beam subjected to a combination of shear and flexure in which a combination of flexure, web-shear, and flexure shear cracking is to be expected. Based on the considerations described above, and in order to contrast the use and capabilities of two very different types of NLFEA programs, NLFEAP-1 and NLFEAP-2 are both used to predict the structural response of the deep beam described in Figure 7.26. The specific comparisons that should and can be made depend on the objective for the use of the NLFEA, the characteristics of the modelling activity, and the availability of data from the experiment. In this example, five items will be compared: (i)

capacity (point load “P” at midspan)

(ii)

the deformation at “peak” load (98% of ultimate)

(iii)

the diagonal compressive strain just above and inside the face of the support. This is interesting as it is expected to be the large compressive strain in the test structure and an indicator of local crushing that could lead to the failure of the deep beam.

(iv)

the strain in the longitudinal reinforcement just inside of the face of the support. This is an important measure as the level of straining in this location would be dramatically different if one is looking at the behaviour from a strut-and-tie perspective or from a sectional behaviour perspective.

(v)

the strain in the stirrups near mid-height and between the support and the point of load application.

These five comparisons are presented below.

(i) & (ii) Capacity (P) and deformation at midspan The relationship between the mid-span loading (P) and the mid-span displacement (/) is presented in Figure 7.27. There are two different measured responses shown. From the test data, it appears that the member was cracked prior to the recording of the data. That is, the uncracked response was not recorded. Based on the cracking load reported in the test report, a second “measured” response is shown in which the data points are shifted to the left by an amount corresponding to the additional displacement with is estimated to have occurred in the pre-cracked member. The predicted versus measured/calculated capacity and deformation at peak load is summarized in Table 7.3.

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2500

Point load (kN)

2000

1500

1000 Experiment(Original) Experiment(accounting for precrack)

500

NLFEAP-1 NLFEAP-2

0 0

2

4

6

8

10

12

14

Midspan displacement (mm) Figure 7.27: Load-deformation response of deep beam. Table 7.3: Comparison of Results for Deep Beam Procedure TEST Result (Original) TEST Result (Adjusted) NLFEAP-1 NLFEAP-2 ACI318-99 EuroCode 2

Capacity P (kN) 1935 1935 1950 2120 695 1062

Deformation at Midspan at Failure (mm) 12 10.5 8.4 8.2 N.A. N.A.

It is useful to note that the prediction of capacity by the three NLFEA methods ranged from only 1950 to 2120 kN (8%). Program NLFEAP-1provided the best estimate of the capacity of the deep beam to within 1% of the measured strength. NLFEAP-2 overestimated the strength by 10%. By contrast, the strength of the member calculated by code relationship were only 35% and 55% of the measured strength. Therefore, it can be concluded that the NLFEA programs provided a more accurate, albeit less conservative, estimate of the capacity. The stiffness of the load-deformation response predicted by the analysis tools were quite similar - differing by only a few percent. However, they both overestimated the stiffness.

(iii) Diagonal compressive strain Figure 7.28 compares the predicted versus measured diagonal compressive strain just above and inside the face of the support. All analysis programs provided a similar and accurate prediction of the compressive straining up until the cracking of the test specimen in the region of the measured diagonal strain. NLFEAP-2 did not effectively predict the influence of cracking on the concentration of straining at this location and in this direction. NLFEAP-1was better able to predict the concentration of straining along this direction with the total strain predicted by NLFEAP-1 providing a reasonable estimate of the measured behaviour.


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2500

Point Load(KN)

2000

1500

Experiment NLFEAP-1(Concrete net strain) NLFEAP-1(Total strain)

1000

NLFEAP-2

P

500

P/2

0 0

P/2

0.5

1

1.5

2

Compressive strain (millistrain) Figure 7.28: Compressive strain above left side support.

(iv) Strain in reinforcement at inside face of support It is interesting to examine the strain in the reinforcement at the inside face of the support as this would be poorly predicted by engineering beam theory. A comparison between the predicted and measured straining at this location is presented in Figure 7.29. Program NLFEAP-1 provides two values for the strain in the reinforcement, one which represents an average value and assumes perfect bond between the reinforcement and the concrete in which the concrete is supporting an average tensile stress determined by the tension stiffening relationship using the MCFT. The other is the strain at a crack location, which is calculated by assuming that there is no tensile stress in the concrete at the location that the strain is being calculated. Since it is not known if the strain gage on the reinforcement reported in this figure is at a crack location, it is to be expected that the measured strain is somewhere between that predicted at a crack and on average. Both analyses provided a reasonable estimate of the straining in the reinforcement at this location.

(v) Strain at location of shear reinforcement in left side of deep beam The predicted and measured straining at the location of a stirrup in the deep beam is compared in Figure 7.30. The strain in the shear reinforcement was not directly measured in the test, but the surface strain over a 200 mm gauge on the surface of the concrete, immediately above a stirrup, at the location shown in Figure 7.30 was measured at each loading stage. NLFEAP-1was found to provide a reasonable estimate of the strain at this location while NLFEAP-2 underestimated the strain.

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2500

Point load (KN)

2000 Experiment(HS2)

1500

NLFEAP-1(average) NLFEAP-1(at crack) NLFEAP-2

P

1000

Longitudinal Strain

500 P/2

P/2

0 0.0

0.5

1.0

1.5

2.0

Tensile Strain (millistrain) Figure 7.29: Strain in reinforcement at inside face of left support

2500

Point load (KN)

2000

1500 P

Experiment(S4)

1000

NLFEAP-1(average) NLFEAP-1(at crack) NLFEAP-2

500 P/2

P/2

0 0

5

10

15

20

Tensile Strain (millistrain) Figure 7.30: Strain at stirrup location on left side of deep beam.


259


The observations from this validation activity are as follows: (i)

NLFEAP-1 was able to accurately predict the strength of the deep beam and provide a reasonably accurate estimate of the overall deformation at failure even when the default values were used for parameters. The program was also able to predict well the compressive straining and reinforcement stresses at the locations examined in the example.

(ii)

NLFEAP-2 provided a reasonable estimate of overall load-deformation response up until peak loading. Prior to using this program to predict the response of structures with similar types of behaviour, it would be desirable to compare the prediction of the model used in this comparison (with the same parameter values) with other tests. This is particularly important in programs where predictions are sensitive to defined parameter values and selected convergence criteria, as was found to be the case for NLFEAP-2.

Since a large number of mechanisms of resistance control the behaviour/strength of structural concrete, it is important to validate NLFEA programs for the type of behaviour anticipated in the structure being designed or analyzed. This is best achieved by the development of a database of personally conducted and well-documented model calibration and validation efforts on a suite of NFLEA programs. Over time, the user will develop a measure of the strengths and limitations of each NLFEA program and develop appropriate global safety factors for each program.

7.7

Summary and future trends in model validation

7.7.1

Summary

The first step in the use of a computational program for the design and analysis of concrete structures is a thorough evaluation of the capabilities and limitations of the selected program. This requires a review of the underlying principles and behavioural models upon which the program is based. To this end, the analyst should examine the user options, including what constitutive relationships and failure theories are available, and make an assessment of the intended applications of the program and identify what parameter values are to be set by the user and the guidance provided in manuals for their selection. If the program is deemed to be potentially suitable for the analysis of the structure under examination, then the next step is to evaluate the accuracy and reliability of the program by using it to predict the response of structures that are expected to exhibit similar types of behaviour to that being designed and for which well documented experimental test data is available. This will provide the user with a quantitative assessment of the program’s ability to predict overall strength, stiffness, ductility and more specific information such cracking characteristics and reinforcement strains. While there is a spectrum of computational programs that provide a wide variety of features and capabilities, it is useful to characterize them as being of two types. One type of program is specifically developed for the analysis of certain forms of structural concrete in which the authors of the program have utilized their expertise to set default parameter values. To use these tools, the user should follow the guidance provided in the program manuals and need only input basic material property and geometric data. At the other end of the spectrum is a general-purpose analysis tool in which the user can greatly influence the predicted response by selected parameter values. These programs require a more thorough evaluation, which should involve a sensitivity study to evaluate the effect of selected values on predicted 260

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response. In this chapter, the steps used in the setting parameter values and in evaluating the accuracy of a computational program were referred to as calibration and validation as briefly summarized below. At the material level, the mechanical characteristics of materials are specified based on the results of standard compressive, tensile, and fracture tests. At the element level, calibration and validation can be done for specific behaviour such as tension stiffening, compression softening, crack spacing, as well as stiffness and strength degradation with cyclic loading. For this, a systematically arranged set of benchmark tests should be utilized that covers the range in behaviours that are expected to be exhibited by the structure under investigation. From these comparisons and for the purpose of making capacity predictions, the accuracy of the computational tool can be assessed as the ratio of the experimental measured strength to the predicted strength. This data can then be used to evaluate the mean test/predicted strength ratio and the coefficient of variation from which a global safety factor can be calculated to obtain the level of safely desired from the design. This is equivalent to what is done in the development of strength relationships in codes of practice. This calibration and validation procedure was illustrated in two separate examples; for the design of a shear wall and the analysis of a deep beam in which a different type of computational program was used in each example. NLFEAP-1 was a concrete-specific tool in which the user may rely upon the element level calibration predefined by the developers whereas NLFEAP-2 utilized a general type of material model that required an evaluation of the sensitivity of the results to non-material parameter values and calibration at the element level. The shear wall example was used to illustrate the use of computation tools for strength evaluation while the deep beam example was also used to illustrate how computational program can be used to also predict deformations and local strains.

7.8

Future trends in model validation

Model validation is the process of evaluating the suitability and accuracy of a particular computational program for a specific design and analysis task. This is done through a comparison of predicted and experimentally measured behaviour. Major challenges in model validation are a lack of suitable and well-documented test data, uncertainty as to what quantities to compare other than overall strength and deformation, and how to weigh the significance of a program’s effectiveness for each of these comparisons. Many efforts are underway that will lead to improved and more reliable validation procedures, some of which are described below. In 2001, the American Society of Mechanical Engineering (ASME) established a committee on Verification and Validation in Computational Solid Mechanics (VNVCSM). The charter of this group is “to develop standards for assessing the correctness and credibility of modelling and simulation in computational solid mechanics”. The membership includes individuals from US National Laboratories, professional practice, and from academia, including civil engineers. This group is focused on establishing a formal architecture for the process of model validation and for the development and use of benchmark tests. This group uses the interrelationship between three entities to describe the issues and methods for VNVCSM, as described in Figure 7.31. The entities consist of the real world problem, the conceptual model, and the computer model. In the field of structural engineering, this is the structure, a particular theory such as, for example, the modified compression field theory, and a specific non-linear finite element program such as NLFEAP-1 which was used in an example presented in this chapter. The relationships between the entities define questions: is the conceptual model


261


appropriate to use for the structure under analysis, is the conceptual model properly implemented in the computer model (verification) and can the computer model be expected to provide accurate predictions of the behaviour of the real work problem (validation)? At the core of the process is the data required for making these assessments.

Conceptual Model Validation

Reality of Interest

Abstraction and Modeling Validation or Prediction

Conceptual Model

Simulation Software Implementation

Verification

Computational Model

Figure 7.31: Verification and validation in CSM Figure 1 (after ASME, 2006).

The data for validation consists of experimental test results, such as experiments on beams, columns, walls, and joints. There are several shortcomings to test data including its unavailability, lack of basic material data and geometric details, breadth of tests, and density of measurements. In addition, only a brief summary of experimental test results are made available in technical journals. While more detailed information is usually presented in a longer report, typically a student thesis, this document is often more difficult to obtain, and often not all of the collected information provided. Another shortcoming in that basic material test data is not always collected and all aspects of experiments and experimental test set-ups may not be well described. Examples of missing basic information include compressive stress-strain relationships for concrete, fracture strengths, cover between longitudinal reinforcement and side walls, support conditions, and location of measurement devices. Another concern with laboratory test data is that what researchers test in laboratories does not represent the size and complexity of structures that are built in the field; very similar tests are often repeated in many institutions at the expense of conducting the types of experiments that are more greatly needed. This is often due to a lack of awareness of what has previously been conducted and the tendency to address similar issues to what has been studied by others. A final shortcoming is that few measurements are made of the detailed internal deformation and straining within structures such that for the majority of tests little more than overall loaddeformation response is available and this is insufficient for the types of validations that are necessary for assessing the accuracy of comprehensive models. Fortunately, progress is being made in many ways to improve experimental research practices. This includes the development of national and international databases of test results that are managed by design code and other committees. This provides the opportunity for 262

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serious scrutiny of previous test data, the evaluation of design code requirements, and the identification of areas of greatest research need. These databases are also available for the validation of computer-based numerical models, but are typically only useful for comparison predicted and measured strengths. There are however some database efforts that are more sophisticated in which all collected data from an experiment is archived in a searchable digital repository. One of the these databases is being developed as part of the US Network for Earthquake Engineering Simulation in which there is the requirement that the information collected from all funded projects be ingested into this digital archive with appropriate metadata in accordance to an predefined model such that all content is searchable. The development of similar archives and data-models are under development by other nations and there are international efforts to coordinate these activities. One of the most critically important advances has been in measurement systems. Coordinate measurement machines, digital photoelastic methods, photogrammetric methods, laser scanners, other optical methods, and image analysis methods are enabling researchers to collect an unprecedented density and accuracy of test data. One of the challenges has been to develop effective data processing and visualization methods that can deal with the difficulties of data from multiple sources that provide different densities, accuracies and, sometimes, conflicting information. Efforts to develop the necessary post-processing tools are also underway. As these databases become populated, the information will be available for the use of more comprehensive validation procedures. A further and very relevant development is that design code committees are now discussing how to facilitate the use of computational programs in structural design and analysis. As discussed in Section 7.3, the basis for design code provisions is experimental test data in which design code relationships are selected so that there is a specific low probability that the design strength will be less than the factored load. Through the establishment of consensus databases of benchmark tests, code committees can directly facilitate the validation and use of computer-based numerical models. As has been described in this chapter, validation of computational tools is essential for their safe and effective use in practice. While examples for assessing global safety factors for different types of computational tools were provided, efforts have only recently been started to develop more formal validation procedures, test databases, and means of supporting the use of computations tools in practice.

7.9

References

ACI Committee 318-02. (2002), “Building Code Requirements for Structural Concrete”, American Concrete Institute, Detroit, 443 pp. Collins, M.P., Vecchio, F.J., and Mehlhorn, G. (1985), “An International Competition to Predict the Response of Reinforced Concrete Panels”, Canadian Journal of Civil Engineering, 12, No.3, pp. 624-644. Lee, D.K., (1982) "An Experimental Investigation in the Effects of Detailing on the Shear Behavior of Deep Beams ", Master Thesis, Toronto, Ontario Okamura, H., and Maekawa, K. (1991), “Nonlinear Analysis and Constitutive Models of Reinforced Concrete”, University of Toyko, ISBN 7655-1506-0, pp 182.


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Paulay, T., and Priestley, M.J.N. (1992), Seismic design of reinforced concrete and masonry buildings, New York: Wiley. Shioya, T. (1989), “Shear Properties of large reinforced concrete member”, Special Report of Institute of Technology, Shimizu Corporation, No. 25, 198 pp. Taylor, C.P., Cote, P.A., and Wallace, J.W., (1998), “Design of Slender Reinforced Concrete Walls with Openings”, ACI Structural Journal, Vol. 95 No.4, pp. 420-433. Vecchio, F.J. (2000), “Disturbed Stress Field Model for Reinforced Concrete: Formulation”, ASCE Journal of Structural Engineering, 126, No. 9, pp. 1070-1077. Vecchio, F.J., and Collins, M.P. (1986), “The Modified Compression Field Theory for Reinforced Concrete Elements Subjected to Shear”, Journal of the American Concrete Institute, 83, No.2, pp. 219-231. Vecchio, F.J., & Collins, M.P. (1982), “The Response of Reinforced Concrete to in-plane Shear and Normal Stresses”, Publication No. 82-03, Department of Civil Engineering, University of Toronto, Canada, March, 332 pp. ASME (2006), American Society of Mechanical Engineers PTC 60 Committee, “Guide for Verification and Validation in Computational Solid Mechanics”, American Society of Mechanical Engineers, New York.

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8

Strut-and-tie modelling

8.1

Introduction

Since being promoted by Marti (1985) and Schlaich et al. (1987), strut-and-tie modelling has been increasingly accepted as a powerful tool for the analysis and design of reinforced and prestressed concrete structures. It has been used simply as a tool to identify and describe the flow of forces in a cracked concrete continuum as well as a quantitative tool for performing a complete design, that is, to evaluate force distributions and to proportion and detail structural concrete members. It has served as a basis in full-member design procedures, which is the focus of this chapter, and is referred herein as the strut-and-tie method (STM). Although the STM can be applied to all parts of a structure, it is usually applied to structural concrete regions near statical or geometrical discontinuities, commonly referred to as D(disturbed or discontinuity) regions. This is because the STM provides a more rational and consistent design procedure than empirical approaches historically applied to those regions. It was originally developed to replace these empirical rules, and at the same time was introduced as a method that is simpler than other procedures for D-regions, such as those based on finite element analysis. The other reason for using the STM in D-regions is that the other portions of a structure, commonly referred to as B- (Bernoulli or beam) regions, have been traditionally designed using sectional design procedures which are based on well-established principles, for example, the Bernoulli beam theory for flexural design and the truss analogy method for shear design. Provisions for designing using the STM have been incorporated into major international codes for structural concrete. These include, among others, the Canadian Building Code (1984), CEB-FIP Model Code 1990 (1993), and AASHTO LRFD Bridge Design Specifications (1994). Most recently, STM design provisions have been included as an alternative design procedure in the 2002 edition of the ACI318 Building Code Requirements for Structural Concrete (2002). Guidelines for using the STM are also available in the FIP report entitled “Practical Design of Structural Concrete” (1999). The design concept in the STM is simple but the implementation can be time consuming. As described later in Section 8.4, the design process is iterative and involves graphical representations of strut-and-tie models. A few special purpose computer applications implementing the STM have been developed to address the time consuming aspect of the calculations and therefore enable engineers to focus on proportioning and detailing the concrete members. However, most of the programs were developed in universities and have been primarily used for research purposes; only a few of them are currently available for practitioners. It is anticipated that additional computer-based STM tools will be available for the practicing community as the method is becoming well established. Another critical drawback of the STM is that the method focuses on design for only strength; serviceability requirements, such as deflections and crack widths, are not explicitly accounted for in the procedure. Extension of the STM that considers serviceability and predict the behaviour for the entire loading range would be an added value to the STM, and therefore the computer-based STM, as the same model used in proportioning and detailing may also be used in response evaluation. This chapter begins with an overview of the STM (Section 8.3), which includes brief discussions on strut-and-tie models, steps in STM design and complications in using the STM. Additional considerations in the use of the STM are then discussed in Section 8.5. Section 8.6 presents an overview of available computer-based STM tools and Section 8.7 describes some aspects in strut-and-tie modelling using the tools. The latter section is also intended to provide ! fib Bulletin 45: Practitioners’ guide to finite element modelling of reinforced concrete structures

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some insight on various procedures adopted in the available computer-based STM to obtain valid solutions. The discussion is limited to planar structures with uniform thickness in the out-of-plane dimension. A design example using a computer-based STM is provided to complete the discussion (Section 8.8).

8.2

Notation

Ac

effective cross-sectional area of at one end of a strut

As

area of non-prestressed reinforcement

A ps

area of prestressed reinforcement

b1 , b2 , ...

effective flange width of a structural member

C , C1 , C 2 , ...

compression force in a strut-and-tie model

Eco

initial tangent modulus of elasticity of concrete

Es

modulus of elasticity of steel

f c , f c1, fc 2 , ... normal stress in the concrete fc#

specified compressive strength of concrete

fcu

uniaxial compressive stress limit of a strut

ft#

specified tensile strength of concrete

fy

specified yield strength of non-prestressed reinforcement

f py

specified yield strength of prestressed reinforcement

F , F1 , F2 , ...

forces in strut-and-tie model components (general notation)

Fcu

capacity of a strut

Ftu

capacity of a tie

h, h1 , h2 , ...

depth of a structural member

L

T , T1, T2 , ...

length of the struts or ties thickness (out-of-plane dimension) of D-region tension force in a strut-and-tie model

vc

shear stress in concrete

w, w1 , w2 , ...

effective width

_f p

remaining stress in non-prestressed reinforcement to resist tension

n

effectiveness factor angle between a strut and a tie

t

-s +1 , + 2

principal (maximum and minimum) stresses (general notation); + 1 is always algebraically greater than + 2

+ x, c , + y ,c

normal stresses carried by concrete and defined in the x-y coordinate system

, xy, c , , yx, c

shear stresses carried by concrete and defined in the x-y coordinate system

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8 Strut-and-tie modelling


8.3

Overview of the STM

8.3.1

Strut-and-tie models

The STM is based on the lower-bound theory of limit analysis (for example, Nielsen 1971, Chen and Han 1988, Muttoni et al. 1997). In the STM, an internal truss is envisioned to carry the applied loading through the D-region being considered to its supports or boundaries. This truss is termed a strut-and-tie model and is a statically admissible stress field in lower bound solutions. Examples of strut-and-tie models for a few typical D-regions are shown in Figure 8.1.

!

(b) (a)

(c)

(d)

(f) (e) Strut

Figure 8.1:

Tie

Node

Examples of strut-and-tie models for common D-regions: a) beam-column joint; b) coupling beam; c) non-slender wall; d) dapped-beam end; e) pile cap under combined axial load and moment; and f) prestressed concrete beam end.

A strut-and-tie model consists of struts, ties, and nodes, where struts are the compression members, ties are the tension members and nodes are the meeting points of struts and ties. In this chapter, struts are represented using broken lines and ties by solid lines. 8.3.2

Components of strut-and-tie models

A strut represents a concrete compressive stress field where the principal stresses are predominantly along the centreline of the strut. The shape of strut stress fields in planar Dregions may be prismatic, bottle-shaped, or fan-shaped, as shown in Figure 8.2 (Schlaich et al. 1987).


267


! fc1
fc

w

w

w

Figure 8.2:

fc

fc

fc

(a) (b) (c) Idealized stress fields in struts: a) prismatic; b) bottle-shaped; and c) Fan-shaped (after Schlaich et al., 1987).

In general, the stress limit of a strut is not the same as the uniaxial concrete compressive strength obtained from cylinder tests, f c# . It is defined as f cu ' n f c# , where: fcu = stress limit of a strut, commonly referred to effective strength, and n = effectiveness factor less than or equal to 1.0 (refer to Section 8.5.3). The critical extent of a strut stress field perpendicular to the strut centreline in the plane of the structure under consideration, usually at the ends of the strut, is termed the effective width, w. The extent of the stress field in the out-of-plane dimension is the same as the thickness of the structure, t. If uniformly distributed stress field across the effective area is assumed (see Section 8.3.3), the capacity of struts is simply Fcu = Acfcu, where Ac is the effective cross-sectional area of the strut and is given as Ac = wt. Ties typically represent one or multiple layers of reinforcing steel or prestressing steel or, on occasions, concrete tensile stress fields. The stress in the ties is limited by the yield strength of ordinary reinforcing steel, fy, or prestressing steel, fpy. The effective prestressing force and its effects may be introduced as a set of statically equivalent external forces acting on the D-region under consideration (Schlaich et al. 1987). When this approach is applied to bonded prestressing, the capacity of a tie consisting of non-prestressed and prestressed reinforcement is given as Ftu = fy As + _fp Aps, where As = area of reinforcing steel, Aps = area of bonded prestressing steel, and _fp = remaining prestressing steel stress available to resist tension. Nodes (nodal zones or nodal regions) represent regions in which forces are transferred between struts and ties. Depending on the types of forces being connected, there are four basic types of nodes; namely CCC, CCT, CTT, and TTT, as illustrated in Figure 8.3, where C and T denote compression and tension forces, respectively. In 2D problems, the stresses in the nodal zones are biaxial and are limited to the yield criteria for plane stress problems. The stress distribution in the nodal regions depends on the idealized shapes that, in turn, depend on the effective width and direction of the strut or tie stress fields framing into the nodes. The simplest idealized shape of a nodal zone is that formed by the intersection of actual dimensions of struts and ties whose centrelines coincide at that node. An example of this nodal zone construction is shown in Figure 8.4. Variants of constructing the nodal zone shapes have been developed to simplify the state of biaxial stresses in the nodal regions. These include hydrostatic approach (Marti, 1985, Nielsen, 1999) or modified hydrostatic approach (Schlaich and Anagnostou, 1990). Further discussion about these last two approaches is presented in Section 8.7.6.

268

!



C

C C

T

C

C

T C

T

(a)

T

T

(b)

T

(c)

(d)

Figure 8.3: Basic types of nodes: a) CCC; b) CCT; c) CTT; and d) TTT.

fc2

! fc3 C3

t ru St

A

C1

fc3

3

fc4

fc1

fc1

B

2

fc1

fc4 fc1 PStrut 1 1

fc2 fc3 fc5

0

+

3 PStrut 3

2

fc2

fc4

C ut Str

C2

PStrut 2

fc5

+

PNode

fc5

Strut 1

,

fc2

, fc3

(b)

(a)

Figure 8.4:

8.3.3

(c)

Example of a node with three struts meeting: a) force acting on the node; b) simple nodal region shape; and c) Mohr’s circle describing biaxial stresses in the nodal zone.

Admissible strut-and-tie models

As a statically admissible stress field in the lower bound method, a strut-and-tie model has to be in equilibrium externally with the applied loading and reactions (the boundary forces) and internally at each node. Reinforcing or prestressing steel is selected to serve as the ties, the width of each strut is selected, and the shape of each node is constructed, such that the yield criteria for all components are not violated. Lower-bound solutions of limit analysis permit various stress distributions across the effective width of stress fields. The simplest and most commonly selected distribution is one that is uniformly distributed. An example of this distribution is illustrated in Figure 8.5a. With this selection, a stress discontinuity exists along the boundary of a stress field. This boundary is commonly referred to as a line of stress discontinuity. As can be observed from Figure 8.5b, the stress in Strut 3 of Figure 8.5a changes abruptly from a stress of fc3 to zero and vice versa. Nevertheless, equilibrium at any point along the lines of discontinuity is still always satisfied (Figure 8.5c).


269

fc1

w1

Stress-Free Region

fc2

A

fc1

(1)

Boundary of the Structure

(4) Strut 2

(2) t3 ru St

Line of Stress Discontinuity

!

w2

fc3

fc2

(5)

Stress-Free Region

w3

0

Lines of Stress Discontinuity

fc3

fc3

Stress-Free Region

Stress-Free Region

(3)

Stress Region of Strut 3

Line of Stress Discontinuity


0

Section A-A

w3

A

(a)

fc3

fc3

(b) fc4

fc5

vc vc

vc vc

(1)

fc2 fc4

fc5

fc1

fc3

fc2

fc2

fc2

fc3 fc1

(2)

(3)

(4)

(5)

(c)

Figure 8.5:

8.4

Examples of stress discontinuities in a typical D-region: a) free-body diagram showing the stress fields underneath the applied loading; b) stress distribution across the effective width of Strut 3 (along Section A-A); and c) equilibrium state of stress in the selected infinitesimal regions near the stress discontinuity lines.

STM design steps

Design using the STM for a D-region may be summarized into the five steps illustrated in Figure 8.6. These are:

270

"

Define the D-region to be considered and then evaluate the boundary and body forces. Boundary forces include the concentrated and distributed forces acting on the D-region boundaries. Boundary forces can also come from sectional forces (moment, shear, and axial load) at the interface of D- and B-regions. Body forces include those that result from D-region self-weight or the reaction forces of any members framing into the D-region.

"

Sketch a strut-and-tie model and solve for the truss member forces.

"

Select the reinforcing steel that is necessary to provide the required tie capacity and ensure that this reinforcement is properly anchored.

"

Evaluate the dimensions of the struts and nodes, i.e., select the effective width of the struts and construct the shapes of the nodal regions, such that the capacity of these components is sufficient to carry the design forces.

"

Ensure sufficient ductility capacity in the D-region by providing distributed reinforcement.

!



STEP 2 383 kN

kN

454 kN

L

356 kN

kN

356 kN

596 kN 356 kN

309 kN-m

44 2

407 kN

71 kN

N

B

74 7k

D

h

kN

48 9

2 32

kN

121 kN

50 3

!STEP 1

71 kN

810 kN

h STEP 5

m m

50

m m

75 m

8 #16

4 #19 12 5

mm

m

2 #13 bars two-legged closed stirrups

8 #13

75

37.5 mm

STEP 3 & 4

4 #16 two-legged, 4 #16 two-legged closed stirrups stirrups @ 100 mm @ 50 mm

Framing bars 2 #19 2 #13 U bars 4 #19 bars welded to angle 2 #19 U bars 3 #25 bars

3 #25

Figure 8.6: Steps in STM design process (after Tjhin and Kuchma 2002).

8.4.1

Complications in STM design

Several complications can arise in executing this five step design process. The selection and sketching of the initial strut-and-tie model is sometimes a challenge, particularly when dealing with complex D-regions. The selected strut-and-tie model often needs to be adjusted and refined graphically in order to satisfy stress limit criteria in the strut-and-tie model components, to fit the truss dimensions within the boundaries, to capture secondary loadresisting elements, or to optimize the design. Furthermore, truss dimensions, i.e., the widths of struts, the shapes of nodal regions and the area and configuration of steel ties, often need to be iteratively adjusted for similar reasons. Further, complications in the design using the STM may arise from the need to solve for truss member forces in statically indeterminate strut-andtie models. In many designs, multiple load cases and load combinations have to be considered. The latter may increase the time required to complete a design using the STM by a considerable factor because different strut-and-tie models have to be prepared to handle each different loading situation and load cases usually cannot be superimposed directly to form load combinations as a result of strain compatibility requirements. All these problems can make hand-based solutions prohibitively time consuming. Computer-based strut-and-tie design programs, as described in Section 8.6, can overcome many of these challenges.

8.5

Some considerations in using the STM

8.5.1

Rules in defining D-regions

Saint-Venant’s principle is usually used to determine the extent of the D-region under consideration. Based on this principle, the extent of a D-region is taken within a distance equal to the largest of the depth or width of the member from the discontinuity or the size of the disturbance. If two D-regions determined in this manner touch or overlap, they are considered one D-region. Once the D-region is determined, the internal forces at the interface between the B- and D-regions become boundary forces for that D-region.


271


Examples of division of B-regions and D-regions are shown in Figure 8.7. In the figure, the unshaded area with notation B indicates a B-Region and the shaded area with notation D indicates a D-Region. The notations h1 , h2 , ... and b1, b2 ,... are used to denote the depths and effective flange widths of structural members, respectively.

!

h1

D

D B

B

h2

h2 h2

D

h2

h3 h4

h2

B

B D

h1

D

h1

B

D

B

D

h3 h3

h1

B

B

D h4 h4

B h4

D D h5

h3

B h3

h1

h5

D

D

h5

D

(a) b1

!

b1 > h1

b2

h1 b1

1

D D

B h1

D

D

h2

B

B

Section 2-2

D

b1

D

1

h4

Section 1-1

b1

h2

D D

2

D

B b2

h3

h3 h2

b2 > h4

h4

b2

B b2

2

D b2

D

B h3

D D

(b)

Figure 8.7:

8.5.2

Examples of division of B- and D-regions: a) building structure b) bridge structure (after Tjhin and Kuchma, 2002, and Kuchma and Tjhin, 2002).

Two- and three-dimensional D-regions

D-region problems may also be classified as either 2D or 3D problems. The associated strutand-tie models are also 2D or 3D. Examples of 2D D-regions with their strut-and-tie models have been given in Figure 8.1. Examples of 3-D D-regions are pile caps with more than one row of piles in each direction and slabs nearing the vicinity of columns. 272

!



Some 3D D-regions may be conveniently separated into several 2-D regions. In the case of the simply supported T-section under a vertical load as in Figure 8.8, for example, the structure can be decomposed into two 2D regions, one associated with the web and the other associated with the flange. The strut-and-tie model for each region is also shown in the figure. In this case, equilibrium at the interface between the web and flange has to be maintained, that is, the top struts of the web become boundary forces for the strut-and-tie model in the flange region and vice versa.

(a)

(a)

(b)

(b)

(c)

Figure 8.8:

8.5.3

Example of separating a 3-D D-region into multiple 2-D regions: a) 3D view of the beam; b) strut-and-tie model in the web region; and c) strut-and-tie model in the flange region (after MacGregor, 1997).

Capacity of struts

Limit analysis was developed based on a rigid perfectly plastic material. This is a reasonable assumption for reinforcing steel in tension but it is a significant simplification of the typical uniaxial stress-strain relationships of concrete. Under increasingly uniaxial compressive stress, concrete softens and then weakens after reaching its post peak stress. In addition, its deformation capacity is usually limited. To account for this behaviour the STM, a parameter called the effectiveness factor, n , is introduced in the determination of the capacity of struts.


273


The effectiveness factor, n , is also used to account for strength reductions caused by the shape of strut stress fields and disturbances due to cracks and tensile strains along the path of a strut. It is also used to consider strength enhancements as a result of the use of distributed reinforcement and confinement surrounding struts (see Figure 8.9). Strength degradation in struts may also arise in the case of reversed cyclic loading. This degradation may occur, for example, when two strut-and-tie models associated with different loading arrangements share a common region (Figure 8.10) or when forces change direction from compression to tension or visa-versa.

(f)

! (d)

(e)

Figure 8.9:

(c)

(a)

(b)

Types of struts in a typical D-region: a) prismatic strut in an uncracked field; b) prismatic strut in a cracked field where struts are parallel to cracks; c) prismatic strut in a cracked field where struts are not parallel to cracks;d) bottle-shaped strut with bursting control reinforcement; e) bottle-shaped without bursting control reinforcement; and f) confined strut (after Tjhin and Kuchma, 2002).

!

Figure 8.10: Strength degradation of struts in the shaded region due to reversed cyclic loading.

Many of the factors influencing the compressive strength of concrete struts have been identified. However, the variety of factors considered in quantifying the strength of struts has resulted in different strength values specified in codes and proposed by researchers (e.g., Yun and Ramirez, 1996, ASCE-ACI Committee 445 on Shear and Torsion, 1998, and Foster and Malik, 2002).

274

!



8.5.4

Uniqueness of strut-and-tie models

As a strut-and-tie model is a statically admissible stress field, it follows that it is required only to satisfy equilibrium and the yield criteria. The third requirement in solid mechanics framework, that is, strain compatibility, does not have to be satisfied and, thus, the method is a lower bound plastic method. As a result of these relaxed requirements, there is no unique strut-and-tie model for a given problem. More than one admissible strut-and-tie model may be developed for each load case as long as the selected truss is in equilibrium with the boundary forces and the stresses in the struts, ties, and nodes of the truss are within acceptable limits. In theory, the lower-bound method should dictate that the capacity obtained from all statically admissible stress fields is lower than or equal to the collapse load. This conservative prediction of STM designs has been verified experimentally (e.g., Maxwell and Breen, 2000, Aguilar et al., 2002, Chen et al., 2002, Mitchell et al., 2002). In practice, however, the capacity may be over-calculated because of limited ductility in structural concrete. This limits the number of viable solutions for each design and suggests that feasible strut-and-tie models should be those that best match strain conditions and require small load redistribution before failure. Because it approximates the principal stress flow in a structure and is in equilibrium with the boundary forces, a strutand-tie model may be thought as an idealization of the actual flow of forces in the structure. Based on this consideration, feasible strut-and-tie models can be identified by examining the linear-elastic load path in the structure. In Figure 8.11, the importance of selecting appropriate strut-and-tie models in STM is illustrated. The figure shows nonlinear finite element prediction of load-deformation responses associated with three STM designs of a short cantilever subjected to a point load at its tip (Ali and White, 2001). The strut-and-tie model selected for each design is different but the design load capacity was the same for all. Figure 8.11 demonstrates, however, that a high variability in the predicted response and load capacity may exist between the designs. Design 1 gives the worst load and deformation capacity because the selected strut-and-tie model is not compatible with strain conditions; excessive cracking occurs in the top region near the support after the diagonal tie yields and the structure fails without much load redistribution. Design 2 shows an improved behaviour in which load redistribution from tie to the diagonal strut can occur before the structure fails due to diagonal splitting. Design 3 gives the largest capacity of the three designs as the additional loading is taken by both the diagonal tie and diagonal strut after yielding of the horizontal tie. Figure 8.12 illustrates a further example in which one solution is preferable to another. Due to the point load at the tip of the cantilever portion, the upper part of the beam is likely to develop horizontal tensile stresses along the beam. Therefore, the model with the upper horizontal tie (Figure 8.12a) is preferable to that shown in Figure 8.12b. The latter only effectively resists the tension in the upper region near the middle support. Further discussion on stress redistribution and ductility demand is presented in Section 8.5.9. Guidance to select appropriate strut-and-tie models is presented in Section 8.7.1.


275


! Load

Design 3

Severe Cracking Design 2 Severe Cracking (extending to the whole section at failure)

Design 1

Tip Deflection

Figure 8.11: Predicted load-deformation responses of a short cantilever with different strut-and-tie models having the same computed capacity (after Ali and White, 2001).

!

(a)

(b)

Figure 8.12: Two statically admissible strut-and-tie models for a cantilevered deep beam under vertical loading: a) workable truss; b) less favourable truss due to excessive ductility demands.

8.5.5

Strain incompatibility of struts and ties

In constructing a strut-and-tie model, the angle between struts and ties acting on a node, -s, should be reasonably large to minimize the level of strain incompatibilities that are caused by strut shortening and tie lengthening in almost the same direction (Figure 8.13a). Similar situation occurs when a tie crosses a strut at an angle (Figure 8.13b). As the angle decreases, the transverse tensile strain in the vicinity of the struts increases and this results in a reduction of the effective strength of the struts, as observed by Vecchio and Collins (1986).

276

!



! -s

-s

(a)

(b)

Figure 8.13: Examples of strain incompatibility between a strut and a tie: a) near a nodal zone region; and b) case where the tie crossing the strut.

The importance of the selecting the angle between struts and ties acting on a node has been recognized and discussed by many researchers (e.g., Rogowsky and MacGregor, 1986, Muttoni et al., 1997). In general, there are two approaches to handle this problem. Rogowsky and MacGregor (1983), for example, recommend that the angle be between 25 and 65 degrees for slender beam design. Vecchio and Collins (1986) include the angle parameter in the formulation of effective strength of struts. 8.5.6

Tension stiffening in ties

It has been long recognized that concrete surrounding ties in the region between cracks contributes to carrying tension forces. This is known as the tension stiffening effect. This effect reduces as the level of tensile strain in the ties increases. Tension stiffening is normally small when a tie reaches its yield strength and tension stiffening does not contribute to equilibrium at the crack. Because strut-and-tie models represent the strength limit state, tension stiffening is neglected in calculating the tie capacity. However, it should be considered if used for predicting load-deformation behaviour. 8.5.7

Influence of tie anchorages

In the STM, the steel ties have to be properly and effectively anchored beyond the nodal regions so that tensile forces can be fully developed and forces in the nodes can be fully transferred. There are still uncertainties in the current STM about the anchorage requirements and the need to distribute the tie reinforcement throughout the nodal region. Figure 8.14 illustrates how the distribution and anchorage of tie reinforcement clearly influence the ability to transfer the horizontal component of a diagonal strut to the tie at the end of a simply supported member. In addition to end anchorage of bars, other known factors that influence tie anchorage requirements include bar size and roughness, lateral and vertical spacing between bars, the angle of the incoming strut, width of bearing plate in nodal zone, amount of confinement, length of bar and use of fibres.


277


(a)

(b)

(c)

(d)

Figure 8.14: Examples of end anchorage of ties in a simply-supported deep beam: a) straight bar; b) standard 90-degree hook; c) T-headed bar; and d) bearing plate (after Tjhin and Kuchma, 2002).

8.5.8

Size, geometry, and strength of nodal zones

As discussed above, nodal zones are regions where the forces in a strut-and-tie model are redirected. Consequently, these regions are subject to multidirectional states of stress. The anchorage detail discussed in the preceding section is another example that illustrates the complexity of load transfer in nodal regions. Many factors may influence the strength of nodal zones. These include the shape of the nodal zone, the type of truss members (struts or ties), number of intersecting truss members, distribution of tie reinforcement, confinement and use of fibres, level of transverse straining, volume and condition of surrounding concrete and anchorage conditions of ties. Some work in quantifying the effects of these factors has been initiated (e.g., Lee, 1982, Jirsa et al., 1991, Polla, 1992), but it is still limited. As a result, different strength values are specified in codes and proposed by researchers, as summarized, for example, by Yun and Ramirez (1996) and ASCE-ACI Committee 445 on Shear and Torsion (1998). 8.5.9

Load redistribution and ductility requirements

Under lower load levels (before cracking), the distribution of stress throughout a D-region is well predicted by linear elastic methods. As the loading increases, the concrete cracks, softens, and may weaken, and the reinforcing steel may yield and strain harden, resulting in continuous adjustments of stress distribution to maintain equilibrium within the continuum. This phenomenon is termed load redistribution. The ability of a structure to perform load redistribution after yielding of reinforcing steel is known as ductility. As previously discussed, inappropriate selection of strut-and-tie model may require excessive ductility demand and thus reduce the load capacity. Failure of tie anchorages and inappropriate estimation of effective strength may also lead to premature failure as excessive load redistribution occurs to create new equilibrium systems. Therefore, it is necessary that all structures be detailed to possess sufficient ductility capacity. In this regard, the capacity of the strut-and-tie models should be governed by the capacity of ties. Also, the struts and nodes must have sufficient strength to include the over-strength of the tie steel due to strain hardening.

278

!



8.6

Computer-based STM

At the time of writing this report, most of the work on computer-based STM in Europe has been done at the Swiss Federal Institute of Technology (ETH) and the University of Stuttgart. In North America, a series of computer-based STM tools have been developed at Cornell University, Purdue University and the University of Illinois. A computer-based STM tool typically consists of one or more main modules listed below. The modules are linked together and share the same database. Graphical User Interface (GUI) module This module serves as the interface between users and computer. It is an essential module in a computer-based STM tool because the design process in the STM is graphically extensive. In this module, users are allowed to interactively define the boundary of D-regions, loading and restraint conditions and strut-and-tie models. It also allows user to interactively modify the strut-and-tie model and other design parameters. The GUI module also serves as postprocessor to enable users to visually check the model and display the analysis and design results. It also acts as an interface with other output units or other computer applications, e.g., CAD software (Tjhin and Kuchma, 2002, 2007a). Module for identifying strut-and-tie models This module is developed to provide guidance in selecting the appropriate strut-and-tie models. Thus, it can be considered as the pre-processor of a computer-based STM tool. All guidance available is based on the use of elastic solutions for identifying strut-and-tie models, a technique suggested by Schlaich et al. (1987). The elastic solutions are given in the form of principal stress plots, optimal topographies, or automatic generation of simple strutand-tie models. Principal stress trajectories for this purpose have been obtained from linear elastic finite element analysis (e.g., Rückert, 1991, Mish et al., 1995, Alshegeir and Ramirez, 1992) and nonlinear finite element analysis of plain concrete (e.g., Yun and Ramirez, 1996). The optimal topographies are obtained from topology optimization of continuum structures (e.g., Liang et al., 2002). The automatic generation of simple strut-and-tie models is obtained from statistical results of principal stress trajectories (Harisis and Fardis, 1991) or topology optimization of trusses (e.g., Kumar, 1978, Biondini et al., 1999, Ali and White, 2001). Module for obtaining truss member forces and for dimensioning of model components This module consists of a truss analysis solver coupled with the routine for dimensioning struts, selecting tie reinforcement and constructing nodal regions. Truss analysis solvers that have been developed include elastic truss analysis (e.g., Alshegeir and Ramirez, 1992), nonlinear truss analysis (e.g., Yun, 2000), and combination of lowerbound approaches and plastic truss analysis (Anderheggen and Schlaich, 1990). Module for predicting load capacity and load deformation responses This module performs nonlinear analyses of the STM designed structure to obtain load deformation responses and confirm that the design loads are achievable.


279


8.7

Modelling aspects using computer-based STM

8.7.1

Identifying strut-and-tie models

As discussed in Section 8.5.4, there is no unique strut-and-tie model for a design but it has to be selected appropriately and used consistently throughout the design process. In general, a strut-and-tie model for a loading system can be identified by the following techniques: Following common and similar models Strut-and-tie models of various D-regions have been developed by researchers as part of their research studies. These strut-and-tie models have usually been verified and are available in the literature. Designers can benefit from the available strut-and-tie models by selecting models that match the loading conditions of their design and use them with appropriate adjustments. In some cases, an appropriate strut-and-tie model may also be taken from one that has been derived for a different structure but having similar conditions. Figure 8.15 shows an example in which the strut-and-tie model for the cantilevered deep beam (Figure 8.15a) is used for the design of a single corbel (Figure 8.15b).

(a)

(b)

Figure 8.15: Example of selecting a strut-and-tie model used in different structure: a) Deep beam shown in Figure 8.12; and b) Single corbel utilizing the same strut-and-tie model.

Developing load paths This approach requires some intuition and experience on how the forces flow within the Dregion. The concept that forces tend to take the shortest path to reach the support is utilized. A curvature in the load path introduces unbalanced equilibrium system and this must be balanced by introducing forces in the vicinity of the curvature. The load path and the balancing forces are then replaced by struts, if the forces are compression, and or ties, if the forces are tension. Identifying a strut-and-tie model in this way is shown in the example shown in Figure 8.16 of a deep beam under two point loads C1 and C2. The loads are positioned such that the left and

280

!



right reaction forces are C1 and C2, respectively. As shown in Figure 8.16a, two load paths, each has two curvatures, are developed: one is associated with the compression force C1 and the other with compression force C2. The curvature near the support system is balanced with a tension force T and the curvature near mid-height of the beam is balanced with force C3. Figure 8.16b shows the strut-and-tie model after replacement of the load paths and balancing forces by struts and ties.

C1

C 1> C 2

C3

C2

C2

C1

C3

Load Path

Load Path

T

T

C1

C2 (a)

C1

C2 (b)

Figure 8.16: Example of identifying a strut-and-tie model through load path: a) selected load paths; and b) strut-and-tie model (after Schlaich et al. 1987).

Fitting crack patterns This technique is based on the observation that principal compressive stresses are parallel to cracks because the principal tensile stresses that cause the cracks are always orthogonal to the principal compressive stresses (Figure 8.17).

Figure 8.17: Principal stresses of an infinitesimal stress field region passing through a crack.

In using this technique, pictures of crack pattern of experimentally tested or similar D-region under consideration must be available and the locations of ties must be known. The crack pattern pictures are used as the backdrop for drawing strut-and-tie models. The struts are arranged to be parallel between cracked regions. Figure 8.18 shows an example of a strut-andtie model constructed in this way. In the absence of this information, figures of crack pattern


281


in the D-region D unnder a loading system may m still bee obtained from f visuallizing the deformed d configuuration and then t identiffying the pootential cracck locationss. The latterr, however, requires some faamiliarity with w the behaaviour of loaaded structu ures.

Figure 8.18: Examplee of identifyingg a strut-and-ttie model by fitting f crack paattern Schlaicch and Schäfeer, 1991).

Following elasttic solutionss Originallly proposedd by Schlaiich et al. (19987), this ap pproach is based b on thhe premise that t both serviceaability and ultimate u lim mit states of a D-region will w be satissfied if its sttrut-and-tie model is arrangedd accordingg to elastic stress s distribbution. An advantage a inn this approoach is that only o one model is required to t satisfy booth limit staates. The dev viation of thhis arrangem ment should d be kept within 15 1 degrees of o the elasticc solution, as a suggested d by Schlaichh et al. (19887). To use this technique, the elaastic stress distribution n needs to be presenteed in the fo ormat of stress trrajectories or plots off principal stresses. Liike the crack pattern pictures, th he stress trajectorries or plotts of princippal stresses are used as a a backdroop for draw wing an app propriate strut-annd-tie modell. The strutts are arrannged paralleel with the principal p coompressive stresses and the potential tiee locations are identifieed by the prrincipal tenssile stressess. In compputer-basedd STM toolls, isotropicc linear finite elementt analysis oof plain con ncrete is usually used to obbtain the pllots of prinncipal stressses. Exampples of strutt-and-tie models m of various D-regions are a providedd in Figuress 8.19 to 8.2 22. It is woorth nothingg that plots of o principall stresses fro om nonlinear finite eleement analy ysis, also based on o plain conncrete, are also a used foor laying ou ut strut-and--tie models (Yun and Ramirez, R 1996). This is useeful when the t actual capacity c off a D-regionn using struut-and-tie model m is t accuracy of such ann approach is arguable because requiredd to be betteer assessed. However, the the nonlinear stresss distributioon is affecteed by the reesponse of the t reinforcing steel ass soon as the conccrete crackss.

282

!



(a)

(b)

Figure 8.19:

Strut-and-tie models of a deep beam supported by a column and a squat wall: a) load on top; and b) load on bottom.

!

Model 1

C L

Model 2

Figure 8.20: Strut-and-tie model of a multicolumn bridge pier.


283


!

Model 1

C L

Model 2

Figure 8.21: Strut-and-tie model of a prestressed hammerhead bridge pier.

Figure 8.22 - Strut-and-tie model of a three storey coupled shear wall subjected to lateral loading.

284

!



Recently, topology optimization of continuum structure has been introduced to the STM to identify the appropriate strut-and-tie models. As an example, Liang et al. (2002) proposed this type of optimization based on assumption that the concrete continuum is linear elastic. The optimal solution is obtained by minimizing the weight of the structure while maintaining the elastic strain energy within a prescribed value. A performance index as a function of the strain energy and structural weight was devised and is used to determine when the optimization process is terminated. A finite element procedure for plain concrete (plane stress problem) is used to calculate the elastic strain energy in the structure at each step in the optimization process. To achieve the optimization objective, a small number of finite elements that have the lowest strain energy density are removed from the continuum at each step in the optimization process. An example of this work is presented in Figure 8.23 with the associated performance index given in Figure 8.24.

!

2500 mm

2500 mm

2500 mm

500 mm

3750 mm

2250 mm

500 mm

3000 mm (a)

(b)

(c)

(d)

Figure 8.23: Example of identifying a strut-and-tie model of a bridge pier under vertical loading using topology optimization scheme: a) geometry and loading; b) topology at the 20th iteration; c) topology at the 40th iteration; and d) optimal topology at the 49th iteration (after Liang et al., 2002).


285


! Performance Index

1.5

1

0.5

0 0

20

40

60

80 Iteration

Figure 8.24: Performance index of the bridge pier example of Figure 8.23!(after Liang et al., 2002).

Automatic generation of strut-and-tie models The use of principal stress plots for guiding the construction of strut-and-tie models has also been extended for automatic generation of strut-and-tie models. Harisis and Fardis (1991) used statistical analysis of principal stress data obtained from linear finite element analysis to identify locations of struts and ties. The strut-and-tie models generated using this approach consists of triangles. Another example of automatic strut-and-tie model generation is the work of Rückert (1991). Based on the observation that regions bounded by the mesh of principal stress trajectories represent finite elements subjected to only normal stresses on each side, a strut-and-tie model can be formed using these finite elements. One example of a strut-and-tie model generated using stress trajectories is shown in Figure 8.25. Kumar (1978), Biondini et al. (1999) and Ali and White (2001) used topology optimization of trusses for automatic generation of strut-and-tie models. In this optimization technique, the so-called ground truss (or basic truss) is developed for the D-region under consideration. This ground truss consists of user-defined locations of nodes and truss members which interconnect the nodes. The truss members in the ground truss represent all possible struts or ties. The optimal solution in this work is said to give the most appropriate strut-and-tie model. In the work by Kumar (1978), the optimal solution is obtained from minimizing the elastic strain energy of the ground truss. Biondini et al. (1999) use similar approach, except that optimum criterion is improved by introducing weighted functions to account for different deformability characteristics in the struts and ties and to account for deviation in the orientation of struts and ties from the corresponding principal stresses. In the work by Ali and

286

!



! “Strip” Stress Trajectories

(a)

(b)

Figure 8.25: Automatic generation of the strut-and-tie model of a deep beam under uniformly distributed load: a) stress trajectories; b) strut-and-tie model (after Rückert 1991).

!

Figure 8.26:

(a)

(b)

(c)

(d)

Example of automatic generation of a strut-and-tie model of a two-span deep beam: a) ground truss; b) topology at the 3rd iteration; c) topology at the 6th iteration; and d) topology at the 30th iteration (after Ali and White, 2001).


287


White (2001), optimal truss solution can be generated following the elastic, minimum reinforcement volume, or composite criterion. Using an energy approach, they introduce the elastic strain compatibility error (SCER) concept to measure how a selected strut-and-tie model deviates from the elastic solution; the lower the SCER value, the closer it is to the elastic stress distribution. An example of obtaining the strut-and-tie model of a two-span deep beam is illustrated in Figure 8.26. 8.7.2

Refining strut-and-tie models

An appropriate strut-and-tie model should at least capture the key load-resisting elements in the problem under consideration. In many cases, strut-and-tie models need to be refined to capture more elements contributing to carrying the loading system. A representative example of refining a strut-and-tie model to higher degrees is shown in Figure 8.27. Another example of a refined a strut-and-tie model is shown in Figure 8.28 for the double corbel example shown in Figure 8.18. In the latter example, the vertical ties are utilized in the model to provide a better load capacity estimate. Figure 8.29 shows similar examples in which strut-and-tie models used in design are adapted to give a higher load capacity. 8.7.3

Other considerations

Besides the various techniques of choosing strut-and-tie models described above, Schlaich et al. (1987) suggest selecting a strut-and-tie model in which the total length of ties is a minimum. This recommendation is based on the principle of minimizing strain energy and the observation that ties are more deformable than struts. For practical reasons, ties should be laid out to form an orthogonal reinforcing net with the bars parallel to the boundaries of D-regions. Stress peaks much higher than the average values are normally observed in the corner regions of a structure. These peaks can be several times higher than the tensile strength of concrete, causing cracks to be easily initiated. Since corner cracks may have definite influence on the load capacity and serviceability behaviour, Almási (1992) recommends that the corner regions be reinforced to enhance the behaviour at both ultimate and serviceability states. The appropriate positions of this reinforcement can be identified by including the corresponding ties in the selected strut-and-tie model.

!

(a)

(b)

(c)

Figure 8.27: Various degrees or refinement of strut-and-tie models for a prestressed anchorage zone: a) simple model; b) refined model; and c) further refined model (after FIP Commission 3, 1999).

288

!



Figure 8.28: Refined strut-and-tie model of Figure 8.18 (after Schlaich and Schäfer, 1991).

A

A

Section A-A

(a)

A

A

Section A-A

(b)

Figure 8.29: Different selection of strut-and-tie models for a deep beam example under top loading and another deep beam under bottom loading: a) models used for design; and b) models for estimating capacity (after Leonhardt and Walther, 1966).


289


8.7.4

Static indeterminacy of strut-and-tie models

Like steel trusses, depending on how the reaction and member forces are obtained, strut-andtie models can be classified as either statically determinate or statically indeterminate. Statically indeterminate cases are further subdivided into internally indeterminate and externally indeterminate. A strut-and-tie model is said to be statically determinate when all of the reaction forces and member forces can be obtained solely from statics. If the total number of strut-and-ties exceeds the number required by statics, the strut-and-tie model becomes internally statically indeterminate. The extra number of struts and ties is commonly referred to the degree of internal redundancy. As an example, the strut-and-tie model in Figure 8.30 is internal redundant to one degree. Similarly, when a strut-and-tie model has more than three nonconcurrent supports in the restraint system, it becomes externally statically indeterminate; the extra number of support components is commonly referred to as the degree of external redundancy. In all cases, as a result of equilibrium requirement in the STM, sufficient restraints are to be provided in all directions and the struts and ties are arranged properly. !

Figure 8.30: Example of a statically indeterminate strut-and-tie model redundant to one degree.

Besides statics, additional requirements are usually needed to solve for the reaction and member forces in a statically indeterminate truss. These requirements are related to the relative elastic stiffness between the truss members. Member forces are then analyzed considering the strain compatibility between truss members and boundary conditions. However, difficulties arise in determining appropriate relative stiffness values for the struts and ties. These uncertainties have lead to the development of several approaches for obtaining the distribution of forces, some of which are described below. 8.7.5

Procedures to solve statically indeterminate strut-and-tie models

Linear elastic analysis To obtain the relative elastic stiffness for each strut-and-tie component, Yun and Ramirez (1996) and Yun (2000) proposed to use a truss-bar elastic stiffness of Eco Ac/L for a strut and EsAs/L for a tie, where Ac = the effective cross-sectional area of the strut, As = the required area of the tie, Eco = initial tangent modulus of elasticity of concrete, Es = modulus of elasticity of steel and L = length of the struts or ties. The strut effective cross-sectional area and required tie area are determined iteratively to satisfy the corresponding stress limits.

290

!



Plastic truss method The plastic truss method is a classic way to handle a statically indeterminate case in lower bound methods of limit analysis. In this method, several ties in the truss, usually those that are most heavily loaded, are assumed to have yielded and the redundancies removed so that the tie forces become known and the truss becomes statically determinate.

A s fy

As fy

To apply this method to the strut-and-tie model of Figure 8.30, the vertical tie is chosen as the yielded tie. If the reinforcement area is taken as As, the yield strength of the tie is As fy. The statically determinate truss system of the beam is shown in Figure 8.31. In analyzing the structure, the member is removed and replaced with a statically equivalent set of forces.

Figure 8.31: Plastic truss model of the strut-and-tie model shown in Figure 8.30.

Another way of applying this method is to decompose the statically indeterminate truss under consideration into several statically determinate trusses. Figure 8.32 illustrates the truss decomposition process for the strut-and-tie model of Figure 8.30.

!

Truss (1) + (2)

Truss (1)

Truss (2)

Figure 8.32 - Decomposition of the strut-and-tie model of Figure 8.30 into two statically determinate trusses.


291


Procedures based on lower bound solutions These procedures include rigid-plastic optimal design (RPOD) and shakedown optimal design (Anderheggen and Schlaich, 1990). In the RPOD, member forces of a statically indeterminate strut-and-tie model are determined in such a way that minimum tie resistances, corresponding to the minimum weight of steel ties, are obtained. The process is done using linear programming. Determining the statically redundant forces by minimizing steel weight has been traditionally done in limit analysis. For beam problems, it can be shown that this leads to minimum strain energy (Nielsen, 1999). The second procedure is based on the static shakedown theorem. This procedure is similar to RPOD except that elastic truss solutions have to be considered. It is particularly useful for multiple load cases as the computational time is less than for RPOD. The solution obtained lies between the elastic solution and the rigid-plastic solution. 8.7.6

Dimensioning nodal regions

There are several rules in defining the idealized geometry of a nodal zone. The dimensions and shape of a nodal zone affect its in-plane stress distribution and are dependent on the number, direction, and the effective widths of struts and ties framing into the node. The effective width of a strut is usually selected in order that the effective strength is not exceeded. The rules in determining the effective width of a tie are less clear and usually follow those for constructing nodal zones. Some methods for constructing the nodes are:

Simple method In this method, the shape of a node is formed by the intersection of actual dimensions of struts and ties whose centrelines are concurrent and meet at the node. An example of this node is shown in Figure 8.4. In this example, the node is triangular as the node is formed by the intersection of three members. The biaxial stress distribution in the node is uniform and can be presented in the form of Szmodits’ Mohr’s circle, as illustrated in Figure 8.4c. For nodes formed by more than three intersecting members (Figure 8.33), the biaxial stress distribution is no longer uniform. In these cases, the adequacy of the node strength may be simply checked by comparing the in-plane stresses acting on all sides of the nodes with the corresponding stress limits (Schlaich and Schäfer, 1991). This approach, however, is only applicable for nodes with typical configuration and must be used discriminately for complex node configurations. The finite element method becomes a choice to check the adequacy of complex node region shapes constructed using the simple method. The generic model for the finite element analysis is shown in Figure 8.34. The forces of the struts and ties framing into node become the external forces acting on the node. Tie forces are treated as compressive forces acting from behind the node. Alshegeir and Ramirez (1992) proposed to verify the bearing capacity of the nodal zones by comparing the principal stresses obtained from the finite element analysis with the Mohr-Coulomb failure criterion. In the later version of their program, nonlinear finite element analysis was used with failure criteria determined from experimental test data of a 2D plain concrete (Yun and Ramirez, 1996).

292

!



!

fc2

C1

C2

C3

fc1

fc3

C4

fc4 (a)

(b)

Figure 8.33: Simple nodal region bounded by four struts (after Tjhin and Kuchma 2002).

ru St

St ru

t

Tie

t Figure 8.34: Checking of nodal regions using finite element analysis (after from Alshegeir and Ramirez 1992)

The state of stress of node zones having triangular shapes (i.e., those formed when three struts or ties intersecting) is constant, as shown in Figure 8.4. For nodal zones consisting of more than three sides, similar stress conditions may be obtained by breaking them down into several stress triangles. The triangles are separated by lines of stress discontinuity (refer to Section 8.3.3) and are arranged such that the state of stress in all triangles is constant and equilibrium along the lines of stress discontinuity is satisfied. The lines of stress discontinuity must be introduced at the vertices of the nodal zone polygon to achieve these conditions. This technique was implemented by Tjhin and Kuchma (2007b). Figure 8.35 shows how this approach is applied to the nodal zone of Figure 8.33a. Any boundary forces acting on nodal zones divided into constant stress triangles are treated as if the force components acted on the sides of the nodal zones. An example of how this is done is illustrated in Figure 8.36. With this assumption, a body force that acts on a node will not have direct influence on the stress distribution in the nodal zone itself but it will affect the stress distribution in all other nodal zones and in all of the struts and ties. The assumption that body forces have no direct influence on the stress distribution of nodal zones carrying them can be justified by the fact that body forces mostly come from conservative lumping of selfweight or are reaction forces of other members framing into and forming integral parts within the D-region being considered.


293


fc6

fc7

fc2

fc5

fc8

fc1

fc1

fc9

fc3 fc4

fc10

Lines of Stress Discontinuity

fc4

Figure 8.35: Nodal zone of Figure 8.33 divided into constant stress triangles.

!

ft2

ft2 F2

ft1

F

fc3

F3

F4

ft1 F

F1 F2 F4

F3

fc3

F1 fc4 (a)

fc4 (b)

(c)

Figure 8.36: Example of how a body force acting on a node consisting of constant stress triangles is treated: a) nodal zone; b) decomposition of the body force according to the head-to-tail rule; and c) statically equivalent forces acting on the sides of the node.

An equilibrium approach of limit analysis is used for estimating the contribution of concrete and reinforcing steel in carrying the stresses in the constant stress triangles of CCT, CTT, and TTT nodal zones. The reinforcing steel is assumed to extend inside and beyond the nodal zones. This steel is also assumed to be evenly distributed in the nodal zones, i.e., smeared steel is assumed. See the CCT nodal zone example provided in Figure 8.37. The Mohr-Coulomb-Rankine yield condition, as shown in Figure 8.38, is used to check the strength adequacy of the constant stress triangles of CCC nodes (Schlaich and Anagnostou, 1990). A linearized version of the Mohr-Coulomb yield condition (Hajdin, 1990) is used for checking the adequacy of constant stress triangles of CCT, CTT, and TTT nodes. (Figure 8.39). For reinforcing steel, an elasto-plastic yield condition is used. The stress distribution between the concrete and reinforcing steel follows the approach by Müller (1975).

294

!



Figure 8.37: Example of how the steeel distributionn assumed in nodal zone co onsisting of coonstant stress triangles.

+1

!

Sliding Failure

fcu

Se eparation Failure

(-1, 0) -11/15

0

,

ft , fc

+2 fcu

,

ft , fc

-11/15 (0 0, -1)

Figure 8.38: Concretee yield criteriaa for checkingg the adequaccy of CCC nod dal zones dividded into constant stress triangless.

,xy,c

! (-3/4, -1/4 , 1/2)

(-1, 0, 1/6)

fcu

+y,c fcu

(-1/4 4, -3/4, 1/2) 0

(0,, -1, 1/6)

+x,c fcu c

(-1, -1, 0)

Figure 8.39: Concrette yield criteriia for checkingg the adequaccy of CCT, CT TT, and TTT nodal zones divvided into constantt stress trianglles.


295


Hydrostatic node construction This is the classic method for dimensioning a node. In this method, the shape of a node is arranged such that the stresses on all sides of the node, from the truss member forces as well as from the boundary forces meeting at the node, are equal. The biaxial state of stress inside the node is hydrostatic in two dimensions, that is, the in-plane stresses are isotropic, homogeneous, and equal to those on the sides. Arranging the node in this shape can be done by sizing the boundaries of the node so that they are proportional and perpendicular to the forces acting on them. In defining the effective width of a tie, the tie force is treated as a compressive force acting from behind the node (Figure 8.40). This type of node is termed a hydrostatic node. Figure 8.41a shows how the CCC node of Figure 8.4a looks if it is arranged using the hydrostatic approach. As shown in Figure 8.41b, the Mohr’s circle for the nodal zone collapses to a point because the stresses are hydrostatic. However, the forces framing into the node may not be concurrent (Figure 8.42).

!

fc

fc

T

fc

fc

fc

fc

fc fc

fc

fc fc fc

T

fc

fc

fc fc

fc

fc

T

T

T

fc

T

(d)

(c)

(b)

(a)

Figure 8.40: Hydrostatic construction of nodes shown in Figure 8.3: a) CCC; b) CCT;c) CTT; and d) TTT (after Tjhin and Kuchma 2002).

! w3

ru St

A Strut 1

w1

,

fc

t3

+

fc fc

C

fc

0

+

ut Str

fc

,

B

2

fc

w2 (a)

(b)

Figure 8.41: a) Construction of nodal zone shape of Figure 8.4 using the hydrostatic procedure; b) Mohr’s circle representing the stresses in the nodal zone.

296

!



Figure 8.42: Examplee of the node in i Figure 8.333a constructed d using the hyydrostatic procedure (after Tjhin and Kuchma,, 2002).

Modifieed hydrostatic node con nstruction This appproach waas introduceed by Schlaaich and Anagnostou A (1990) and basically y utilizes hydrostaatic node ruule whilst maintaining m g a set of co oncurrent forces. fo A noode with more m than three members m inttersecting iss handled by b breaking g down the node into several hyd drostatic nodes of o triangularr shapes coonnected byy short prissmatic struts. Figure 8.43 shows how the CCC noode of Figurre 8.33a is arranged a usiing this app proach. Trapezooidal transittion stress zones z betweeen a node and the inttersecting trruss membeers were also forrmulated using this appproach, alloowing differrent stress intensities i oof truss mem mbers to act on thhe node. Thhe length off these transsition zones is determinned using thhe separation failure criteria of the Mohr-Coulomb--Rankine yiield conditio on (Figure 8.38). 8

Figure 8.43: 8 Examplee of modified hydrostatic h no ode (after Tjhiin and Kuchm ma, 2002).


297


8.8

Design example using computer-based tools

8.8.1

Problem statement

The propped cantilever deep beam with an opening shown in Figure 8.44 is to be designed using the STM. The beam supports a factored column reaction force, P, of 5000 kN and a factored uniformly distributed load of 120 kN/m. The distributed load includes the self weight assumed, for simplicity, to act on the top of the beam. The compressive strength of concrete, f c# , and the yield strength of the steel reinforcement, f y , are taken as 35 MPa and 420 MPa, respectively.

!

4000 mm

P = 5000 kN

w = 120 kN/m

1000 mm

t = 600 mm

1500 mm

4000 mm

1000 mm

Shear Wall 1500 mm

2750 mm

500 mm 11000 mm

Figure 8.44: Propped cantilever deep beam with an opening.

8.8.2

Solution

The deep beam is statically indeterminate to the first degree, as schematically shown in Figure 8.45. For simplicity and to fit into the strut-and-tie model to be selected, the distributed load is lumped at locations indicated in the figure. A 2D linear finite element analysis was conducted to estimate the reaction forces. The principle stress vectors for the FE solution are plotted in Figure 8.46. A comparison with a beam model, with and without considering the shear deformation along the beam length, is given in Table 8.1 and shows that the vertical reaction at end A, VA, is not very sensitive to the analysis conducted. In the subsequent discussion, the value of VA equal to 3000 kN will be used and treated as an additional external load acting on the structure, thus removing the external redundancy. Force redistribution was assumed to be adequate to handle any deviation in the distribution of reaction forces.

298

!



!

480 kN

5000 kN 330 kN

330 kN

156 kN

A

VA

MA

B

2000 mm

2000 mm

1375 mm

1375 mm

4250 mm

VB

Figure 8.45: Magnitude and location of the lumped uniformly distribution loading.

!

Figure 8.46: Plot of principal stresses from linear finite element analysis.

Table 8.1: Reaction forces from four different analyses

Analysis

Reaction VA (kN)

VB (kN)

MA (kN-m)

Linear finite (quadrilateral) element

3000

3310

9430

Beam (flexure deformation only, uniform stiffness)

2890

3420

10530

Beam (flexure deformation only, considering opening)

2930

3380

10000

Beam (flexure and shear deformations, uniform stiffness)

2970

3340

9620


299


The selected strut-and-tie model is shown in Figure 8.47a. The model is statically determinate and was laid out based on the plot of elastic principal stresses shown in Figure 8.46. The force transfer from the column loading to the left support is provided by two diagonal struts AF and BG, connected by a vertical tie AG and horizontally equilibrated by strut FG at the top and ties FG and GH at the bottom. The main force transfer in the right side of the column force is also provided by diagonal struts, except that a smaller truss system extending from node C to node I was used in the vicinity of the opening, replacing the single diagonal strut CI. The truss system around the opening is shown in a greater scale in Figure 8.47b. For ease of construction, the ties of this truss system were laid out such that the resulting reinforcement is parallel to the boundaries of the opening. A stress limit equal to 0.6 f c# was used for dimensioning struts and nodes. A strength reduction factor of 0.75 was applied to all struts, ties, and nodes to account for variations in construction quality and workmanship. This corresponds to an effectiveness factor of 0.75 o 0.6 f c# ' 0.45 f c# . Figure 8.48 shows the distribution of the forces in the strut-and-tie model. Table 8.2 lists the required effective width and the corresponding capacity of the struts. The reinforcement for the ties is summarized in Table 8.3. To increase ductility capacity in the beam, additional distributed reinforcement 0.14% of the cross-sectional area is provided in the horizontal direction. A standard 90-degree hook is used to anchor tie FG while straight lengths are used for anchoring all other ties. The final reinforcing details are shown in Figure 8.49. Table 8.2: Strut forces, selected effective widths and strut capacities.

Strut ID IJ BC AB AF BG BH DK UI TU QT NQ CN SI PS MP LM CL QR RU OP LO FF'

300

Force (kN) -1327 -2180 -1714 -3455 -2902 -2665 -4192 -2947 -1396 -1974 -2712 -2209 -2378 -2928 -2111 -1492 -2986 -288 -2084 -330 -2000 -3000

Effective Width (mm) 400 270 212 427 358 329 517 364 172 244 335 273 294 361 261 184 369 200 257 200 247 370

!

Capacity (kN) 3240 2184 1714 3455 2902 2665 4192 2947 1396 1974 2712 2209 2378 2928 2111 1492 2986 1620 2084 1620 2000 3000



250 mm

5000 kN

480 kN

330 kN 78 kN

B

A

C

L

O

P K

G

R

S U

T

H

kN

Q

3917 kN

E

92 41

3500 mm

D

M

N

F 250 mm

330 kN

I

J

1327 kN

78 kN 2000 mm

1375 mm

2000 mm

1375 mm

4250 mm

3000 kN (a)

1275 mm

850 mm

850 mm 78 kN

M

N

850 mm

L P

1300 mm

O

1300 mm

1150 mm

C

955 mm

S

R U

T

I

78 kN 950 mm

850 mm

1150 mm

850 mm

Q

850 mm

1275 mm

Figure 8.47: a) Selected strut-and-tie model and boundary forces; b) detail ofstrut-and-tie model around the opening.


301


Figure 8.48: Force distribution in the struts and ties.

Table 8.3: Tie forces, required and provided reinforcing steel, and tie capacities.

302

Tie ID

Force (kN)

Required Steel Area (mm²)

FG GH HI CD DE GA HC TR OM ID NO RS QL UP

1714 3154 2180 1327 3917 2520 2480 1474 1414 2966 1084 1185 730 798

5442 10014 6921 4212 12434 8000 7873 4678 4490 9416 3442 3762 2319 2535

Provided Steel Configuration

Area (mm²)

3 Layers of 4 #29 4 Layers of 4 #29 3 Layers of 4 #29 2 Layers of 4 #32 4 Layers of 4 #32 2-Legged Stirrups #13 @100 mm 2-Legged Stirrups #16 @100 mm 2-Legged Stirrups #16 @100 mm 2-Legged Stirrups #16 @100 mm 2-Legged Stirrups #16 @100 mm 1 Layer of 5 #32 1 Layer of 5 #32 1 Layer of 3 #32 1 Layer of 3 #32

7740 10320 7740 6552 13104 8772 7960 5572 5572 9552 4095 4095 2457 2457

!

Capacity (kN) 2438 3251 2438 2064 4128 2763 2507 1755 1755 3009 1290 1290 774 774



! Two-Legged Stirrups #16 @100 mm

500 mm

1750 mm

2000 mm

5 #32

3 Layers of 4 #29

1375 mm

4250 mm

Two-Legged Stirrups #16 @ 100 mm

3 #32

5 #32

4 Layers of 4 #29

4 Layers of 4 #32

3 #32


4000 mm

3 Layers of 4 #29

2 Layers of 4 #32


2 #32 (Framing Bars)

4 #29 (Framing Bars) 1125 mm

10750 mm Note: Column reinforcement is not shown Horizontal Web Reinforcement #13 @ 300 mm Each Face (Typical)

Figure 8.49: Reinforcing details for deep beam example.

8.9

References

ACI Committee 318 (2002), Building Code Requirements for Structural Concrete (ACI 31802) and Commentary (ACI 318R-02), American Concrete Institute, Farmington Hills, Michigan, 443 pp. Aguilar, G., Matamoros, A.B., Parra-Montesinos, G.J., Ramirez, J.A., and Wight, J.K. (2002), “Experimental Evaluation of Design Procedures for Shear Strength of Deep Reinforced Concrete Beams”, ACI Structural Journal, Vol. 99, No. 4, July-August, pp. 539-548. Ali, M.A., and White, R. N. (2001), “Automatic Generation of Truss Model for Optimal Design of Reinforced Concrete Structures”, ACI Structural Journal, Vol. 98, No. 4, JulyAugust, pp. 431-442. Almási, J. (1992), “Cracks as Important Constituents of Strut and tie Models”, Periodica Polytechnica Civil Engineering, Budapest University of Technology and Economics, Vol. 36, No. 3, pp. 251-270. Alshegeir, A., and Ramirez, J. A. (1992), “Computer Graphics in Detailing Strut-Tie Models”, Journal of Computing in Civil Engineering, Vol. 6, No. 2, April, pp. 220-232. American Association of State Highway and Transportation Officials (1994), AASHTO LRFD Bridge Specification, 1st ed., Washington, DC, 1091 pp.


303


Anderheggen, E., and Schlaich, M. (1990), “Computer Aided Design of Reinforced Concrete Structures Using the Truss Model Approach”, Proceedings of the Second International Conference on Computer Aided Analysis and Design of Concrete Structures, N. Bicanic and H. Mang, eds., Zell am See, Austria, April 4-6, pp. 539-550. ASCE-ACI Committee 445 on Shear and Torsion (1998), “Recent Approaches to Shear Design of Structural Concrete”, Journal of Structural Engineering, ASCE, Vol. 124, No. 12, December, pp. 1375-1417. Biondini, F., Bontempi, F., and Malerba, P.G. (1999), “Optimal Strut-and-Tie Models in Reinforced Concrete Structures”, Computer Assisted Mechanics and Engineering Sciences, Institute of Fundamental Technological Research, Polish Academy of Sciences, Vol. 6, No. 34, pp. 279-293. Comité Euro-International du Béton (1993), CEB-FIP Model Code 1990, Thomas Telford Services, Ltd., London, 437 pp. Chen, B. S., Hagenberger, M.J., and Breen, J.E. (2002), “Evaluation of Strut-and-Tie Modeling Applied to Dapped Beam with Opening”, ACI Structural Journal, Vol. 99, No. 4, July-August, pp. 445-450. Chen, W.F., and Han, D.J. (1988), Plasticity for Structural Engineers, Springer-Verlag, Inc., New York, 606 pp. CSA Technical Committee on Reinforced Concrete Design (1994), Design of Concrete Structures, A23.3-94, Canadian Standards Association, Rexdale, Ontario, December, 199 pp. FIP Commission 3 (1999), Practical Design of Structural Concrete, fédération internationale du béton (fib), Lausanne, September, 114 pp. Foster, S.J., and Malik, A.R. (2002), “Evaluation of Efficiency Factor Models used in Strutand-Tie Modeling of Nonflexural Members”, Journal of Structural Engineering, ASCE, Vol. 128, No. 5, May, pp. 569-577. Lee, D.D.K. (1982), “An Experimental Investigation in the Effects of Detailing on the Shear Behaviour of Deep Beams”, M.A.Sc. Thesis, Department of Civil Engineering, University of Toronto, 138 pp. Hajdin, R. (1990), “Computerunterstützte Berechnung von Stahlbetonscheiben mit Spannungsfeldern”, Report No. 175, Institut für Baustatik und Konstruktion, Eidgenössische Technische Hochschule, Zürich, August, 115 pp. Harisis, A., and Fardis, M.N. (1991), “Computer-Aided Automatic Construction of Strut-andTie Models”, Structural Concrete, IABSE Colloquium, Stuttgart 1991, International Association for Bridge and Structural Engineering, Zürich, March, pp. 533-538. Kuchma, D.A., and Tjhin, T.N. (2002), “Design of Discontinuity Regions in Structural Concrete Using a Computer-Based Strut-and-Tie Methodology”, Paper No. 02-3000, Transportation Research Record 1814, Transportation Research Board of the National Academies, pp. 72-82. Kumar, P. (1978), “Optimal Force Transmission in Reinforced Concrete Deep Beams”, Computers and Structures, Vol. 8, No. 2, pp. 223-229.

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Jirsa, J.O., Bergmeister, K., Anderson, R., Breen, J.E., Barton, D., and Bouadi, H. (1991), “Experimental Studies of Nodes in Strut-and-Tie Models”, Structural Concrete, IABSE Colloquium, Stuttgart 1991, International Association for Bridge and Structural Engineering, Zürich, March, pp. 525-532. Leonhardt, T., and Walther, R. (1966), “Wandartiger Träger”, Deutscher Ausschuss für Stahlbeton, Bulletin No. 179, Wilhem Ernst & Sohn, Berlin, 159 pp. Liang, Q.Q., Uy, B., and Steven, G.P. (2002), “Performance-Based Optimization for Strut-Tie Modeling of Structural Concrete”, Journal of Structural Engineering, ASCE, Vol. 128, No. 6, pp. 815-823. MacGregor, J. G. (1997), Reinforced Concrete: Mechanics and Design, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 3rd ed., 939 pp. Marti, P. (1985), “Basic Tools of Reinforced Concrete Beam Design”, ACI Journal, Proceedings, Vol. 82, No. 1, January-February, pp. 45-56. Maxwell, B.S., and Breen, J.E., “Experimental Evaluation of Strut-and-Tie Model Applied to Deep Beam with Opening”, ACI Structural Journal, Vol. 97, No. 1, January-February, 2000, pp. 142-148. Mish, K., Nobari, F., and Liu, D. (1995), “An Interactive Graphical strut-and-Tie Application”, Proceedings of the Second Congress on Computing in Civil Engineering, Mohsen, J. P., ed., American Society of Civil Engineers, New York, pp. 788-795. Mitchell, D., Cook, W.D., Uribe, C.M., and Alcocer, S.M. (2002), “Experimental Verifcation of Strut-and-Tie Models”, Examples for the Design of Structural Concrete with Strut-and-Tie Models, SP-208, K.-H. Reineck, ed., American Concrete Institute, Farmington Hills, Michigan, pp. 41-62. Müller, P. (1978), “Plastische Berechnung von Stahlbetonscheiben und –balken”, Report No. 83, Institut für Baustatik und Konstruktion, Eidgenössische Technische Hochschule, Zürich, July, 160 pp. Muttoni, A., Schwartz, J., and Thürlimann, B. (1997), Design of Concrete Structures with Stress Fields, Birkhauser, Boston, Massachusetts, 143 pp. Nielsen, M. P. (1971), “On the Strength of Reinforced Concrete Discs”, Acta Polytechinca Scandinavica, Civil Engineering and Building Construction Series No. 70, Copenhagen, 261 pp. Nielsen, M. P. (1999), Limit Analysis and Concrete Plasticity, CRC Press LLC, 2nd ed., 908 pp. Polla, M. (1992), “A Study of Nodal Regions in Strut-and-Tie Models”, M.A.Sc. Thesis, Department of Civil Engineering, University of Toronto, 130 pp. Rogowsky, D.M., and MacGregor, J. G. (1983), “Shear Strength of Deep Reinforced Concrete Continuous Beams”, Structural Engineering Report No. 110, Department of Civil Engineering, University of Alberta, Edmonton, November, 178 pp.


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Rogowsky, D.M., and MacGregor, J. G. (1986), “Design of Reinforced Concrete Deep Beams”, Concrete International, August, pp. 49-58. Rückert, K.J. (1991), “Design and Analysis with Strut-and-Tie Models – Computer-Aided Methods”, Structural Concrete, IABSE Colloquium, Stuttgart 1991, International Association for Bridge and Structural Engineering, Zürich, March, pp. 379-384. Schlaich, J., Schäfer, K., and Jennewein, M. (1987), “Toward a Consistent Design of Structural Concrete”, Journal of the Prestressed Concrete Institute, Vol. 32, No. 3, May-June, pp. 74-150. Schlaich, J., and Schäfer, K. (1991), “Design and Detailing of Structural Concrete Using Strut-and-Tie Models”, The Structural Engineer, Vol. 69, No. 6, March, pp. 113-125. Schlaich, M., and Anagnostou, G. (1990), “Stress Fields for Nodes of Strut-and-Tie Model”, Journal of Structural Engineering, ASCE, Vol. 116, No. 1, January, pp. 13-23. Tjhin, T.N., and Kuchma, D.A. (2002), “Computer-Based Tools for Design by the Strut-andTie Method: Advances and Challenges”, ACI Structural Journal, Vol. 99, No. 5, SeptemberOctober, pp. 586-594. Tjhin, T. N., and Kuchma, D. A. (2007a), “Integrated Analysis and Design Tool for the Strutand-Tie Method”, submitted to Engineering Structures”, Journal of Engineering Structures [accepted for publication]. Tjhin, T. N., and Kuchma, D. A. (2007b), "Limit State Assessment of Nodal Zone Capacity in Strut-and-Tie Models”, Vol. 4, No. 4, August, pp. accepted for publication to the journal of Computers and Concrete Vecchio, F.J., and Collins, M.P. (1986), “The Modified Compression Field Theory for Reinforced Concrete Elements Subjected to Shear”, ACI Journal, Vol. 83, No. 2, MarchApril, pp. 219-231. Yun, Y.M., and Ramirez, J.A. (1996), “Strength of Struts and Nodes in Strut-Tie Model”, Journal of Structural Engineering, ASCE, Vol. 122, No. 1, January, pp. 20-29. Yun, Y.M. (2000), “Computer Graphics for Nonlinear Strut-Tie Model Approach”, Journal of Computing in Civil Engineering, ASCE, Vol. 14, No. 2, April, pp. 127-133.

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9

Special purpose design methods for surface structures

9.1

Introduction

Up to and including Chapter 7 this Guideline has discussed nonlinear analysis procedures on the basis of realistic constitutive material models for reinforced concrete and finite element techniques. It appears that the approach is of interest for both the design of structures and for purposes of final performance checking. Chapter 8 about strut-and-tie modelling, primarily focussing on design, differs fundamentally from the previous chapters. Apart of these methods some other design-oriented approaches are advocated for special types of surface structures like plates loaded-in-plane (shear-walls, etc.) and plates loaded out-of-plane (slabs). Three such approaches are discussed in this chapter. They all apply a computational elementbased technique to construct an equilibrium stress-state, of which two (developed in Denmark and Switzerland) offer an automated design on basis of optimizing procedures in combination with ideal plasticity and one (the Netherlands) an interactive design tool which links up with the modified compression-field theory. The Danish method is discussed for both plates loaded in their plane and slabs, the Swiss one for slabs only, whereas the Dutch one is restricted to plates loaded in their plane.

9.2

Notation

In this chapter a notation is applied which is as much as possible the same for the three methods. As a consequence the notation of the original documents will differ from the one chosen here.

d f f

e

fel fi B

set of design variables load vector; global vector of all f element nodal forces elastic global vector f

e

c se

global element force vector corresponding with ( i kinematic matrix follows from vector of yield limits

C

matrix that relates ( to C s ; global matrix of all C e

Ce

matrix that relates f e to c se

Cd

matrix that relates d to C s ; matrix depending on the linearization of the yield criteria array of yield moments for all nodes; global vector of all c se

Cs Ec fc fi

complementary energy compressive strength panel edge forces (i = 1, 2, 3, 4)

ft x

tensile strength in x-direction

y

tensile strength in y-direction

ft

F H K

matrix of all f i ; flexibility matrix; fictitious plastic strain distributions

matrix that relates ( to Rc 8 ^ R global stiffness matrix


307


Ke l mxx m yy

element stiffness matrix span of slab bending moment in x-direction bending moment in y-direction

m xy

twisting moment

mp

plastic yield moment

m8px

positive yield moment in x-direction

m7px

negative yield moment in x-direction

m8py

positive yield moment in y-direction

m7py

negative yield moment in y-direction

M Mp

moment in beam positive plastic moment in beam

M 7p

negative plastic moment in beam

pL N Nc Nt R Rc ui

limit distributed load on slab normal force in bar limit compressive normal force in bar

ue

element nodal displacements elastic global vector u

8

uel vx vy

V W

(

(i ^ r +xx

limit tensile normal force in bar set of variable concentrated loads set of fixed concentrated loads panel edge displacements (i = 1, 2, 3, 4)

shear force in x-direction shear force in y-direction set of dual variables related to vector Re matrix to compute weight of structure from design variables d ; matrix depending on the linearization of the yield criteria. vector of unknown stress and/or moment parameters; vector of all free amplitudes (i free amplitude of strain distribution; generalized panel stress (i = 1, 2, 3, 4, 5) scale factor set of variables corresponding to the yield moments C s

+ yy

stress in x-direction for in-plane loads stress in y-direction for in-plane loads

+ xy

shear stress for in-plane loads.

308

9! Special purpose design methods for surface structures


9.3

Design of slabs and shear walls: perfect plastic approach

Plates loaded in their plane (shear-walls) and loaded perpendicular to their plane (slabs) can be considered as perfectly plastic solids and designed using limit analysis. Two main approaches exist, based on either an assumed stress state or an assumed displacement mode, resulting in a lower bound solution or an upper bound solution, respectively. Solutions have been published for many plate types but in general cases hand calculations are time consuming. In this section, a Danish approach is described in which use is made of a computer analysis. The aim of the method is to provide a safe lower bound solution and information on the deformations at failure. The approach can be used to determine not only the ultimate load for a given geometry and the reinforcement but also to optimize the reinforcement for a given loading. First, the highlights of the method will be explained for plates subjected to bending and then for plates acting as membranes. For more details of the methodology the reader is referred to Damkilde and Krenk (1997), Krabbenhoft et al. (2002), Poulsen et al. (2002). 9.3.1

Slabs subjected to bending loads

For slabs subjected to bending loads, similar to finite elements, the plate is divided into elements triangular in plan. Within each element, a moment field is assumed which is linear for both the bending moments mxx and myy and for the twisting moment mxy. Consequently, the shear forces vx and vy are constant over the area of an element. Nine moment parameters are required to define this stress state, three in each corner. An equilibrating bending moment field is constructed for which the bending moment normal to the edge is continuous at the inter-element boundaries. The same is done for the twisting moments, which, strictly speaking, is not necessary. Figure 9.1 shows one element and the moments on the boundary, which are made continuous. As a result, a vector $ with unknown moment parameters is achieved for the whole plate.

Figure 9.1: Continuous moments on the boundary of triangular elements.

It should be noted that a triangle, with a linear moment field and constant shear forces, can be loaded only by constant distributed loads along the edges and these loads are replaced by equivalent corner point loads. One could say a homogeneously distributed load on the slab is replaced by a statically equivalent set of point loads at the joints. This is fully acceptable if the


309


mesh is sufficiently fine. The set of point loads is named Rc + &R, where Rc is a fixed load and &R a variable load, scaled by the factor &. For equilibrium, the bending moments are related to the nodal loads by

H( ' Rc 8 ^R

(9.1)

where the matrix H is assembled from element matrices in a similar way as a global stiffness matrix is assembled in the standard finite element method from element stiffness matrices. The matrix H is rectangular, because the number of moment parameters in $ is three times the number of nodes, which defines the length of the vectors Rc and R. The moments must obey the yield criterion for reinforced concrete slabs. For this purpose the yield criterion as shown in Figure 9.2 is applied, which was published by Nielsen (1964) and Wolfensberger (1964). The yield criteria is then represented by 2 7 (m8px 7 mxx )(m8py 7 myy ) 8 mxy ?0 2 7 (m7px 7 mxx )(m7py 7 myy ) 8 mxy ?0

(9.2)

where m 8px and m 7px are positive and negative yield moments respectively in the x-direction, and m 8py and m 7py are positive and negative yield moments in the y-direction. The yield criterion can be linearized using section planes. The minimum linearization contains eight planes, and a more accurate one is a replacement by sixteen planes. After such a linearization the yield condition can be written in the form:

C( ? Cs

(9.3)

where the values in C depend on the linearization planes and the global array Cs contains the yield moments for all nodes, based on the material strengths.

Figure 9.2: Yield criterion for reinforced concrete slab.

310



9.3.2

Ultimate load determination

A slab of given dimensions and reinforcement is considered for which we want to determine the maximum value of the load parameter &. In language of mathematical programming, ) is the objective function and Eqs. 5.13 and 5.14 are conditions to be met.. The objective function is to maximize:

g( u ZYOT 1WV i v j^ w

(9.4)

7 R W g ( u ' g Rc u i v i v O UV j ^ w ? j C s w

(9.5)

with the conditions:

ZH XC Y

From this mathematical programming problem, we receive a solution for $ and &. The procedure to solve this linear programming problem is not discussed here. Rather, the reader is referred to the reference source literature “Limit Analysis and Optimal Design of Plates with Equilibrium Elements” by Krenk et al. (1994). Here it suffices to mention that the number of unknown 0-values can be reduced to the number of static indeterminacies of the structure. The computing time on modern PC’s is in the order of tens of seconds. It is reported that a linearization of the yield condition with eight planes yields almost the same results as with sixteen planes; the differences are in the order of 5%.

Example 1 The first example is the simply supported square slab under uniform load shown in Figure 9.3 and the slab is to be isotropically reinforced equally for both the upper and lower layers. The plastic yield moment is mp. For this case the exact solution for the upper-bound limit load on basis of the yield line method is: p L ' 24.0

mp l2

l

l Figure 9.3: Simply supported square slab.


311


When an element mesh is chosen that contains the crossing diagonals, the same solution is obtained independently of the number of elements. Now it is a lower-bound solution. If no negative moment capacity in the corner region is available, the exact upper-bound solution is: p L ' 22.0

mp l2

A linearization of the yield function with eight planes yields a coefficient 21.9 for a negative moment capacity that is one-eight of the positive capacity, which is about five percent smaller. For a zero negative moment capacity the coefficient will even be smaller (expectation 21.3). It is reasonable that the value is less than 22.0, because the element-based method is based on the lower-bound theorem and the yield-line method on the upper-bound theorem.

Example 2 In this example, a square plate is considered that is clamped at all edges (see Figure 9.4). The exact upper bound solution for a homogeneous reinforcement arrangement in two directions, top and lower layers, is given by Fox (1974) as p L ' 42.85

mp l2

The results of the optimal design yields a coefficient of 40.8 for eight yield planes and 41.7 for 16 yield planes.

l

l Figure 9.4: Square slab with clamped edge.

Example 3 The element-based method is of course not intended for elementary cases as were discussed in the first two examples. These examples were done to get confidence in the approach. In this example a more complex slab is analysed to illustrate the use of the element-based method. It is a structure, which originally was designed by a traditional yield-line calculation. The geometry, supported edges and yield-line pattern are shown in Figure 9.5. Where the edges are supported, they are simply supported. The load is uniform and the reinforcement arrangement is isotropic. The limit load from a hand calculation is called 100% here. 312



Damkilde and Krenk (1997) later referred to a fully automatic yield-line calculation, in which a limit load of 89.6% of the hand calculation resulted. The element-based method gives a result of 83.5% if eight planes are used to linearize the yield function and 88.3% for a linearization with 16 planes. Finally, a calculation with the non-linear yield criterion without any approximation gives 88.8% of what was originally calculated by hand. This is very close to the 89.6% of the fully automated upper-bound calculation.

(b)

(a)

Figure 9.5: Slab of irregular shape: a) yield line pattern; and b) FE mesh.

9.3.3

Failure mode determination

In the preceding section the optimization problem was stated as follows: maximize:

with the conditions:

g( u ZYOT 1WV i v j^ w ZH X YC

7 R W g ( u ' g Rc u i v i v O UV j ^ w ? j C s w

(9.6)

(9.7)

This is called the primal linear programming problem. In the linear programming theory there corresponds a dual problem to each primal problem. The variables of the dual problem correspond to each of the equations of the primal problem much like the principal of virtual work in mechanics. Thus, there is a set of variables V corresponding to the load vector Rc and another set of variables r corresponding to the yield moments Cs. The dual problem is obtained from the primal by transposing the matrix and interchanging the roles of the righthand side and the vector in the objective function. In the present case this leads to the dual problem: minimize

ZY RcT

g 7V u CsT WV i v jr w


(9.8)

313


Z HT conditions: X T Y7 R

C T W g 7V u ' gO u U i v i v; OT V j r w ' j 1 w

r
(9.9)

Only the dual variables % associated with the inequalities of the primal problem are restricted to positive values. In the present problem, V can be interpreted as the displacements of the nodes, while % can be interpreted as rotation discontinuities associated with the corresponding linear part of the yield surface. It is established within the theory of linear programming that the solutions of the primal and dual problems produce the same value of the objective function and that the solution of the one may be constructed directly from the other. This fact is important in the present context, because it establishes equivalence between the static model and a corresponding kinematic model. Thus, the notion of upper and lower bound solutions in classical plasticity theory is replaced by a common approximate solution that may give preference to the representation of the static or kinematic fields. For the structural engineer it means that he does not just receive the limit load but also the deformation mode that is associated with the failure load. The displacement field is shown in Figure 9.6 for the structure of Example 3. It is seen that the assumed yield-line pattern of the original hand calculation is not confirmed. The deformation mode associated with the limit load is quite different.

(a)

(b)

Figure 9.6: Irregularly shaped slab: a) original yield-line pattern; and b) displacement lines.

9.3.4

Material optimization

In case of material optimization the load Rc is kept constant and no & occurs. We add a number of design variables d and make the yield criteria linear in the design variables:

C $ + Cd d * Cs

(9.10)

where Cd depends on the linearization of the yield criteria. The quantities C and Cs have the same meaning as in the previous section. Now the equilibrium equations and yield conditions combine to the linear programming problem:

g( u T minimize: ZYO W WV i v jd w 314

(9.11)



ZH conditiions: X YC

O W g ( u g Rc u i v' i v; Cd UV j d w j C s w

r <0

(9.12)

minimized (volume, ( where W dependss on the chhoice made regarding the quantiity to be m blem gives the design parameterss and the weight, etc.). The solution off this optimiization prob T method d can easilyy be extendded to seveeral load correspoonding bennding momeent field. The cases buut this will not n be discuussed here.

Examplle 4 In this example, e ann optimal reeinforcemennt arrangem ment is sought for the sllab in Figurre 9.7. It is rectanngular and a/b = 2/3. One O side a is i clamped and one sidde b is simpply supporteed, while the rem maining two sides are free. f At thee corner off the two free sides, thhe slab supp port is a column. A uniform m load is appplied. The cost of eacch yield mooment is assumed prop portional nts. If all with itss magnitudee and the saame proporrtionality faactor is usedd for all yieeld momen four plaastic momeents (layers in the x- and a y-directions; uppeer layer andd bottom laayer) are taken eqqual and hoomogeneous over the full f plate, only o one deesign param meter occurss. In this case thee total costt is called 100%. If the t four plastic momeents can bee different but still homogeenous over the t full plate, four design parameters occur. Now N the cosst decreases to 89%. A thirdd optimization run is done d in whhich three regions r are distinguishhed in the plate, p as i Figure 9.7 and each with its own material group. g In eaach region ffour differen nt plastic shown in momentts can deveelop, so 12 design paraameters now w occur. Now the costt reduces fu urther to 64%. The total cosst of four layers reinfoorcement peer unit areaa is for the three regio ons 4.07, a four plasttic momentts are substaantial (highest layer 1.73 andd 3.02, resppectively. Inn region I all cost 1.226), in regioon II mainlyy the bottom m layers are important (highest ( layyer cost 0.77 7) and in region III I only a hiigh amountt of reinforccement is reequired in the t upper laayer in the direction d of side b (layer cosst 2.98).

Figure 9.7: 9 Element mesh m and diffe ferent materiall groups.

Remarkk In somee codes of practice, ruules are givven about th he distance that is perrmitted betw ween the elastic and a plastic solution in order to guuarantee sufficient defformation caapacity. This fits in directlyy in the LP-sscheme just as another restriction.


315


9.3.5

Plates subjected to in-plane loads

Shear-walls and deep beams are plates with in-plane loads. For this type of structure a similar procedure can be followed. Here we restrict ourselves to a short indication of the possibilities. Now triangular elements are combined with bar and beam elements. The triangles represent the distributed reinforcement and the bar elements the concentrated reinforcement. The stresses in the triangles are .xx, .yy and .xy. The stress resultant in the bar is a normal force N and in the beam a bending moment M. Again, equilibrium is fully satisfied by choosing a linear stress state in the triangles, a parabolic distribution for the normal force in the bars and a cubic moment distribution in the beams. The linear stress distribution in the triangle (parameters .) results in four generalized forces on each side, as shown in Figure 9.8. Through these generalized forces, transfer takes place to adjacent plate elements, bars (see Figure 9.9) and beams (see Figure 9.10). The unknown stress parameters . in the triangles, N in the bars and M in the beams are assembled again in the vector $.

Figure 9.8: Triangular element with generalized forces on the edges.

Figure 9.9: Stress parameters and generalized forces on the edges.

Figure 9.10: Stress parameters and generalized forces for a beam element.

316



In the case of in-plane loads the yield criterion is chosen proposed by Nielsen (1984) and given by the equations

7( f t x 7 + xx )( f t y 7 + yy ) 8 + xy2 ? 0 7( f c 8 + xx )( f c 8 + yy ) 8 + xy2 ? 0

(9.13)

where fc is the compressive strength and f t x and f t y are tensile strengths (limit tensile forces in reinforcement per unit length divided by the thickness of the plate). The yield criterion is drawn in Figure 9.11.

Figure 9.11: Yield criterion for a plate loaded in plane stress.

Here we skip the discussion how to linearize the yield criterion. It suffices to say that it is done differently for load optimization and for material optimization. The linear yield criteria for a bar and a beam , respectively, are given by

7 Nc ? N ? Nt 7 M p7 ? M ? M p8

(9.14)

where Nc is the limit compression force, Nt the limit tension force, M p8 the positive plastic moment and M p7 the negative plastic moment. In a beam with both normal and bending forces, a combination of N and M is necessary. In the case of load optimization where the strengths are known, a linearization as shown in Figure 9.12a is used. Whereas for the case of material optimization, the linearization shown in Figure 9.12b is used in order for the restriction to be linear in the material parameters.


317


!

(a)

(b)

Figure 9.12: Linearization of the yield surface for:(a) load optimization, and (b) material optimization

9.4

Design of slabs using the reinforcement field approach

In this Swiss modelling methodology, the aim is to provide a safe lower bound solution and providing a practical reinforcement arrangement with a minimal reinforcement weight. Only the two basic concepts and their application to slabs subject to bending loads are discussed here and the reader is referred to Anderheggen et al. (1995) and Steffen (1996) for more details of the method. Furthermore the discussion is restricted to the subject of material optimisation. For ultimate load determination, an incremental approach is used on basis of non-linear constitutive materials law, which is outside the scope of this report. A related method for plates loaded in plane is not discussed here. For this application is referred to Despot (1995). 9.4.1

Linear yield conditions for element nodal forces

The approach is based on the standard linear finite element method. Each finite element is regarded as an independent dimensioning unit that has to withstand the forces transmitted to it from its neighbouring elements (Figure 9.13).

!

Figure 9.13: Nodal forces for a four node plate bending element

318



This means that unlike the usual method of formulating dimensioning constraints on the stress components, the dimensioning conditions are imposed directly upon the (generalized) element e e e e nodal forces f . With f = K u these forces, which satisfy equilibrium with the external e (nodal) loads exactly, are easily obtained from the nodal displacements u and the element e stiffness matrix K . Starting from a linearized yield criterion, the yield condition for a finite e e e e element can now be written as C f ? cs , with cs corresponding to a vector of yield limits e e e based on the material strength. Assembling C , cs and f of all finite elements into the global matrix C and the global vectors Cs and f,the yield condition for the slab now reads: C f * Cs

(9.15)

In Eq. 9.15, f corresponds to the stress state in the structure. In the case of a linear-elastic analysis f is equal to fel, which is easily calculated from the elastic nodal displacements uel. 9.4.2

Material optimisation through stress redistribution

According to the lower bound theory of plasticity, the superposition of any self-equilibrating stress state on a stress state, which satisfies equilibrium with the external loads does not alter the load-carrying capacity of the structure. This opens the way to optimisation through stress redistribution; as long as equilibrium and yield conditions are satisfied, the stress state and thus the global nodal force vector fel obtained by linear-elastic analysis can be modified at will to reduce the total amount of steel. This is done with the introduction of fictitious plastic strain distributions within the finite elements. For each considered strain distribution of free amplitude $i, a global element nodal force vector fi is determined by linear-elastic analysis. Assembling fi and $i of all strain distributions into the matrix F and the vector $ the condition of Eq. 9.15, including stress redistribution, now reads: C fel + CF 0 * Cs

(9.16)

Adding a number of design variables d, corresponding to the steel content of considered reinforcement domains, the total reinforcement weight of the structure is written as Wd. With Cs = Cd d we now make the yield criteria linear in the design variables and get the following optimisation problem:

g( u T minimize: ZYO W WV i v jd w conditions: pCF

g( u 7Cd q i v ? C fel jd w

(9.17a)

(9.17b)

This method is easily applied to multiple load combinations if instead of Cfel the corresponding envelope values of all considered load combinations are used. Minimum values for the design parameters can also be specified, if required for strength limit or serviceability conditions. It should be noted here that the concepts presented above are very general and can be applied to all structure types like slabs, walls or shells. For the procedure to derive linear yield conditions for different types of finite elements and the strategy of generating useful fictitious plastic strain distributions is referred to Anderheggen et al. (1995).


319


9.4.3

Slab subjected to bending loads

The reinforcement layout of a slab consists of a series of domains of reinforcement, called reinforcement fields, with known locations and directions of reinforcing bars but unknown steel content d. The possibly overlapping reinforcement fields can be independently defined for the top and bottom steel. The yield condition Eq. 9.15 is derived from the yield criteria of Figure 9.2 through linearization. Assuming a constant inner lever arm per finite element and reinforcement direction, Cd d in Eq. 9.17b then corresponds to the plastic yield moments in each of the elements. 9.4.4

Dimensioning procedure

Following the goal of the method, which is providing a method for practicing engineers, the dimensioning procedure is implemented as a postprocessor to a standard linear-elastic finite element analysis. The three steps of the procedure are the following:

Step 1: Standard linear-elastic analysis The structure is first analysed for all independent load cases by means of linear-elastic analysis. Then, based on the condition Eq. 9.15, the required elastic reinforcement content in every finite element is calculated in both directions on the top and bottom layers.

Step 2: Elastic reinforcement layout Next, the engineer defines the conceptual reinforcement layout by graphically entering the location and direction of a series of reinforcement fields for the top and bottom steel, in certain zones possibly specifying the minimum reinforcement content according to the code. The choice of reinforcement fields should mainly be based on his experience and knowledge of a practical reinforcement layout and not exclusively on the results of Step 1as, at this point, no fictitious plastic strain distributions are to be considered, Eq. 9.17 can be written as minimise:

Wd

conditions: - Cd d # - C fel

(9.18a) (9.18b)

The solution of this optimisation problem is the elastic reinforcement layout that, due to undesirable elastic stress peaks, in general, results in an excessively large reinforcement content.

Step 3: Optimised reinforcement layout In order to optimise the elastic solution the program now automatically generates fictitious plastic strain distributions F and solves the optimisation problem of Eq. 9.17, minimizing the total steel weight. The optimisation algorithm works in steps, meaning that the full F is not introduced at once. In fact, starting from the elastic solution new plastic strain distributions are added in increments. After each carefully selected increment, an optimal solution is obtained. New optimisation increments are added until the reinforcement appears to be satisfactory to the engineer or until relevant savings are no longer possible. The methodologies described above have been incorporated into commercial software and is just one example of how optimisation techniques can be implemented into the design of slab

320



and wall structures. As for any non-linear approach to design, however, the methods should only be applied to appropriately verified problem types.

9.5

Design of shear-walls: the stringer-panel approach

The stringer panel method, developed in the Netherlands, is intended for the design of plane structures like shear-walls, beam-column joints, beams with openings and dented beam. The method can be used for the serviceability limit state (crack-widths) and the ultimate limit state (ultimate strength). Moreover it can handle multiple load conditions. Basically, just two elements are used, a stringer element (straight bar) and a panel element (rectangle or quadrilateral) as shown in Figure 9.14. This corresponds to the observation that main reinforcement often occurs in bundles at the edges of structures and around holes (represented by stringers) and distributed net reinforcement is applied between the bundles over large parts of the structure (represented by panels). The idea to apply stringers and panels has been used earlier in limit design for plane stress states, see Nielsen (1999).

reinforcement net

Panel

panel normal force panel shear force

reinforcement bar Stringer

stringer force

Figure 9.14: Stringer and panel element

9.5.1

Linear-elastic version

In a simple orthogonal version of the method, the material behaviour of stringers and panels is kept as linear-elastic; constant shear forces occur in the rectangular panels and normal forces occur in the stringers. Full equilibrium is assured in the panels, the stringers and in the interface between panels and stringers. The latter implies that constant shear panels must go together with linearly varying normal forces in the stringers. The method is generalised to quadrilateral panels, which gives more freedom for irregular structure shapes. Figure 9.15 shows this general panel configuration.


321


f3 u3 y f4

u2

u4 x

: Q

R

f2

u1 f1

Figure 9.15: Quadrilateral constant shear panel.

The stress state in the total structure is fully equilibrating and the principal of minimum complementary energy (Ec) is applied. This principal is modified in order to work in the framework of the stiffness method by adding the inter-element equilibrium equations, multiplied by a Lagrange multiplier, to the energy functional. This extended functional must be minimised with respect to both the stress parameters and the Lagrange multipliers. The multipliers can be interpreted as “average” displacements u in the interface between the panels and stringers and displacements in the joints where adjacent stringers meet. For the total system, the modified energy functional is written as: Ec '

1 T ( F ( 7 ( T Bu 7 f T u 2

(9.19)

where $ is a vector of stress parameters, F is a flexibility matrix, B a kinematics matrix which is dependent on the topology of the element mesh and f is the load vector. Load can be applied in the direction of each degree of freedom, so f and u are associated. The functional must be stationary with respect to the stress parameters $ and displacements u: sE c ' F( 7 Bu ' 0 s(

(9.20)

sE c ' BT ( 7 f ' 0 su

(9.21)

From these equations, we solve the relationship between the stress parameters and the displacements and the relationship between the stress parameters and the load:

( ' DB u

(9.22)

BT ( ' f

(9.23)

where the rigidity matrix D is the inverse of the flexibility matrix F. Substitution of the first equation into the second yields the set of equations and the stiffness matrix K of the global system:

322

Ku ' f (global set of equations)

(9.24)

K ' BT DB (global stiffness matrix)

(9.25)



The global stiffness matrix can be assembled form the individual element stiffness matrices. After solving the equations Ku = f, the stress parameters can be calculated from the equation $ = DB u. For details reference is made to Blaauwendraad et al. (1996) and Hoogenboom (1998). 9.5.2

Non-linear version

In the non-linear version of the model, the panel concrete material can both crack and crush and the panel reinforcement can yield. In this version, the normal forces must be included in the panels. Again, equilibrium is maintained within the panels, the stringers and on the interface of the elements. This is achieved by choosing an appropriate stress state in the panels as shown in Figure 9.16 for a square element. The stress field varies linearly in the xand y-directions giving 2x Z 1 0 0 Z+ xx W X a U X X 0 X+ yy U ' X0 1 0 X+ xy U X V X0 0 1 7 2 y Y XY a

WZ ( W 0 UX 1 U (2 2 y UX U U X (3 U b UX U 2x (4 7 UU XX UU b V Y (5 V

(9.26)

In Eq. 5.38, +xx and +yy are normal stresses and +xy is the shear stress in the panel material. Variables (1 to (5 are generalised stresses and a, b are panel dimensions. Five independent stress modes occur that are in equilibrium. A generalisation to quadrilaterals has been made.

1

2

3

4

5

Figure 9.16: Stress modes of a square panel with non-linear behaviour.

The non-linear stringer behaviour in tension can account for tension stiffening after cracking and yielding of the reinforcement at ultimate, see Eurocode (1991). The non-linear panel behaviour is according the modified compression field theory of Vecchio and Collins (1986). In Figure 9.17a the characteristic force-strain curve for a stringer in tension is drawn. In Figure 9.17b an impression is given of the relationship between the shear stress and the shear strain in the panels at given values of the normal strains. 9.5.3

A three-step design procedure

The dimensioning procedure for shear-walls and/or beam details consists of three steps, one of which is elastic and two are non-linear, Hoogenboom (1998), Blaauwendraad et al. (2002). Hereafter the subsequent three steps are explained.


323


force

step 1

step 2 step 3

shear force

step 1

step 2 and step 3

yielding cracked reinforcement only uncracked

shear strain

strain

(a)

(b)

Figure 9.17: Non-linear characteristics for stringers and panels: a) characteristic force-strain curve for a stringer in tension; b) relationship between the shear stress and the shear strain at each steps.

Step 1: Elastic analysis In this step the simple linear model is used. Thus, normal forces occur in the stringers and shear forces occur in the panels, which is fine for the elastic phase. For this first step it does not matter so much what sizes of cross-sections are assigned to the stringers. Application of rules of thumbs is sufficient. The panels have the wall thickness and a linear-elastic analysis for all load combinations must be performed. The force distribution achieved in this way is a first indication for the determination of the reinforcement in the tensile stringers and the mesh reinforcement in the panels.

Step 2: Non-linear Now one can prepare for a non-linear analysis in which account is taken of cracking in the tensile stringers and cracking and yielding in the panels. In this second step, the non-linear models are used and, thus, the stress state in the panels is extended to shear and normal stresses. In the non-linear step it does matter that one enters the correct cross-sections of the stringers. In compressed stringers the cross-section determines the compressive force, which has to be compared with the ultimate design strength. In tensile stringers, the assigned concrete area determines the contribution of the concrete to the tension stiffening of the stringer. For this analysis, the results from the elastic step are used as a first estimate. All input quantities being determined now and entered into the program, the non-linear calculation is performed. The loading is increased incrementally until the ultimate design load is reached (load factor is 1) using a Newton-Raphson procedure to solving the non-linear equations. Because cracking in the stringers and yielding in the panels occur, a non-linear load-displacement diagram will result. In this step no yielding of the reinforcement in tensile stringers is considered in order to receive at a robust design tool. In case a tensile stringer would reach its tensile yield strength, the cracked branch in the force-strain diagram of the stringer is artificially extended. From the analysis results it will become clear whether or not the ultimate tensile strength of a stringer has been surpassed for any of the load combinations and, if so, the reinforcement has to be increased. The crack-widths at service loads can also be inspected and the reinforcement adapted, if needed. Due to redistribution of stresses and the enriched capacity of the panels in this second step, it also may occur that the reinforcement in a tensile stringer can be reduced.

324



Step 3: Final simulation All decisions made on basis of the non-linear analysis are entered in the model and then the final step is started. The load is incremented until failure occurs. In this final simulation, no restrictions on the non-linear responses are made; the reinforcement in panels and stringers can yield and concrete can crush. If the ultimate load factor is greater than 1.0 and the crack widths are within serviceability bounds, no further iterations are needed. If done in this way, a robust and fast program can be made. The elastic analysis (step 1) is done more or less instantaneously and the non-linear analysis (step 2) and final simulation (step 3) only require a few minutes on moderns PCs. In fact, the time involved with the initial modelling of the structure and the professional decisions to be made by the engineer are determining for the duration of the design process. Finally, particular attention must be paid to detailing and anchoring of the steel reinforcement in both the stringer and panel elements. 9.5.4

Example

The stringer-panel method is demonstrated with the design of a structure, which has been previously used in a Swiss study, see Despot (1995). The structure is shown in Figure 9.18. The stringer-panel model is shown in Figure 9.19. In principle only stringers are chosen along edges of the structure, along edges of holes, where line loads or point loads are applied and where supports occur.

0.85

3.00

1.00

8.00

2.15

150 kN

1000 kN 150 kN/m 2.15

3.00 100 kN/m

5.00 100 kN/m

2.35

2.00

0.50 100 kN/m 0.50

1.00 1.50

6.30

0.90

4.80

15.00 m

Figure 9.18: Deep beam example (dimensions in m).


325


113 kN

240 kN

1356 kN

332 kN

281 kN

324 kN

356 kN

190 kN

190 kN

110 kN

310 kN 185 kN

238 kN

221 kN

398 kN

236 kN

Figure 9.19: Stringer-Panel model for the deep beam of Figure 9.18.

The first elastic step in the analysis results in the stress distribution shown in Figure 9.20, which shows linearly varying normal forces in the stringers and constant shear stresses in the panels. On basis of this elastic stress distribution reinforcement is chosen. The reinforcement in the stringers is concentrated and in the panels a two-way mat reinforcement is adopted. This reinforcement scheme is given in Figure 9.21. In general, codes of practice will prescribe minimum reinforcement percentages for wall-type structures that is sufficient for large parts of the structure. Only local additional panel reinforcement may be needed. Next step 2 is started, in which cracking of the stringers is permitted and yielding and crushing in the panels is allowed for. It is important to have selected proper cross-sectional areas for the stringers at the conclusion of step 1, at least for the tensioned stringers, because the tension stiffness depends on the cross sectional area. Figure 9.22 shows the resulting load-displacement diagram for the structure. The load factor is stepwise increased until a load factor of 1 is reached. The resulting stress distribution is shown in Figure 9.23. Substantial redistribution does occur if compared with the stress distribution for the elastic step in Figure 9.20. This means that the reinforcement has to be adapted. More reinforcement is required at the top of the hole, while in the lower edge of the structure the reinforcement can be reduced. Figure 9.24 shows crack-widths for the serviceability state. Clearly too large crack-widths do occur if one wants to limit them to 0.3 mm. This is another reason to adapt the reinforcement. Figure 9.25 shows a revised (adapted) reinforcing scheme. On basis of this reinforcement, the final simulation of step 3 is made in which the load factor is increased until failure. Figure 9.26 shows the load-displacement curve for the resulting analysis and. Figure 9.27 shows the crack-widths at the serviceability load. Finally, in Figure 9.28 the deformation mechanism at failure load is plotted. As the crack-widths are within the specified permissible range and the ultimate load factor is larger than 1, the design is completed. In this example, more than half of the reinforcement weight is due to the distributed reinforcement in the panels. Other examples also suggest that the distributed minimum reinforcement is, largely, the determining factor of the amount of reinforcement. Therefore, the total amounts of reinforcement in the elastic layout and after redistribution do not differ substantially. However, the position where the concentrated reinforcement is required changes substantially.

326



552 kN 88

353

659

354

36

142

779 kN 313 kN 582

235

749 kN 342

26

647

189

359

649 kN

Figure 9.20: Elastic stress distribution.

2Ø16

Ø12-150

3Ø20+2Ø16

4Ø20+2Ø16

2Ø16 A

Ø12-180

2Ø20

2Ø16

A

A

A

A

Ø12-300 at both faces

4Ø20+Ø16

Figure 9.21: Reinforcement scheme on basis of elastic stresses.

1.1 1.0

load factor

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

displacement [mm] 2

4

6

8

10

12

14

16

18

20

Figure 9.22: Load-displacement diagram in non-linear design step 2.

566 kN 3.9

522 kN 2.5 MPa 1.6

347 kN 3.1

1233 kN

2.9

2.0 540 kN

Figure 9.23: Non-linear stress distribution in design step 2.


327


0.04 mm

1.10 0.34 0.90 0.44

0.25 mm

Figure 9.24: Crack-widths in SLS in design step 2.

Ø12-150

6Ø20+4Ø16

2Ø20+3Ø16

4Ø16

A

A

A

A

A

Ø12-180

2Ø16

2Ø20

Ø12-300 at both faces

6Ø16

A: Ø7-150 in both directions on both faces

Figure 9.25: Reinforcement arrangement on the basis of the non-linear stresses of step 2.

1.1 1.0

load factor

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

displacement [mm] 2

4

6

8

10

12

14

16

18

20

Figure 9.26: Final load-displacement curve in design step 3.

328



0.13

0.04 mm 0.19

0.28 0.21

0.28 mm

Figure 9.27: Crack-widths at serviceability state in design step 3.

Figure 9.28:Deformation mechanism at failure load in design step 3.

9.6

References

Anderheggen, E., Despot, Z., Steffen, P., Tabatabai, S., and Tabatabai, M.R. (1995), Finite Elements and Plasticity Theory: Integration in Optimum Reinforcement Design”, 6th Int’l Conference on Computing in Civil Engrg and Building Engrg, Berlin, pp. 653-660. Blaauwendraad J, and Hoogenboom P.C.J. (1996), "Stringer Panel Model for Structural Concrete Design", ACI Structural Journal, Vol. 93 No. 3, pp. 295-305 Blaauwendraad, J. and Hoogenboom, P.C.J. (2002), “Design instrument SPANCAD for Shear Walls and D-regions”, Proceedings of the first fib Congress 2002, Concrete Structures in the 21st Century, Osaka, Japan, October, 2003, Vol 2, p. 411 - 416, in CD ROM. Damkilde, L., and Krenk, S. (1997), “LimitS – A System for Limit State Analysis and Optimal Material Layout”, Computers and Structures, Vol. 64, Issues 1-4. Despot, Z. (1995), “Methode der finiten Elemente und Plastizitätstheorie zur Bemessung von Stahlbetonscheiben“ (Finite Element Method and Plasticity Theory for Dimensioning of Reinforced Concrete Disks), Institut für Baustatik und Konstruktion, ETH Zürich, IBK Bericht Nr. 215, Birkhäuser Verlag, Basel, (In German). Eurocode (1991), Common Unified Rules for Concrete Structures, ENV1992-1-1: pp. 171174. Hoogenboom, P.C.J. (1998), “Discrete Elements and Non-linearity in Design of Structural Concrete Walls”, doctoral thesis, Delft University of Technology.


329


Krabbenhoft, K., and Damkilde, L. (2002), “Lower Bound Limit Analysis of Slabs with Nonlinear Yield Criteria”, Computers and Structures, 80 (220), pp. 2043-2057. Krenk, S., Damkilde, L., and Hoyer, O. (1994), “Limit Analysis and Optimal Design of Plates with Equilibrium Elements”, J. Engrg. Mech., Vol. 1200, Issue 6, pp 1237-1254. Nielsen, M.P. (1964) “Limit Analysis of Reinforced Concrete Slabs”, Copenhagen, Acta Polytech. Scand., Civil Eng. Build. Constr., Serial No. 26, 167 pp. Nielsen, M.P. (1999), Limit Analysis and Concrete Plasticity, second edition, ISBN 0-84939126-1, Prentice-Hall, London, (first edition 1984). Poulsen, P.N., and Damkilde, L. (2000), “Limit State Analysis of Reinforces Concrete Plates Subjected to In-plane Forces”, International Journal of Solids and Structures, 37, pp. 60116029. Steffen, P. (1996), “Elastoplastic Dimensioning of Reinforced Concrete Slabs by Finite Dimensioning Elements and Linear Programming”, PhD-thesis ETH Zurich, Switzerland (in German; original title: Elastoplastische Dimensionierung von Stahlbetonplatten mittels Finiter Bemessungselementen und Linearer Optimierung). Vecchio, F.J., and Collins, M.P. (1986), “The Modified Compression-Field Theory for Reinforced Concrete Elements Subjected to Shear”, ACI Journal, V.83, No. 2, pp. 219-231. Wolfensberger, R. (1964), “Traglast und optimale Bemessung von Platten“, Zurich, PhD dissertation.

330



10 Concluding remarks 10.1 Introduction This Practitioners’ Guide to Computer Based Modelling of Structural Concrete is intended to summarize the basic knowledge required for use of nonlinear analysis methods as applied to practical design, construction and maintenance of concrete structures. The contents of this guideline will be reviewed, once again, using several examples of performance based design schemes to illustrate how each chapter is applied. Note that in this guide, short-term static loads under normal ambient conditions are primarily considered and the mechanical properties of structural concrete and steel are assumed unchanged during the service life of the structure. Note also that nonlinear analysis and mechanical models of structural concrete can also be applied to the performance assessment of existing structures in service, although the input analysis data and boundary conditions would differ from those for newly constructed or future planned structures. This chapter summarizes the manner in which nonlinear modelling can be practically applied; design limit states and safety factors are also briefly discussed in relation to nonlinear structural analysis.

10.2 Structural performance based design in practice Figure 10.1 shows a flowchart of a limit state design or performance assessment scheme in which nonlinear modelling and analysis may function. In general, there exists a set of design requirements that must be satisfied for the various design loads acting on the structure (including natural ambient conditions such as temperature, humidity, radiation, rainfall and water supply). Structural shapes, sizes and dimensioning must be selected such that they satisfy requirements relating to safety, serviceability, durability and fatigue. Elegance (i.e., aesthetics) and environmental issues must also taken into account. Basically, this decision making process proceeds on a “trial-and-error” basis since there may exist an infinite solutions meeting the specified requirements and design conditions. If we begin with overlysafe details in the initial stage of the design cycle or, on the contrary, if very unsafe ones are chosen, then it may take many iterations to reach the solution that enables us to achieve a technically, economically, and environmentally efficient design. If a particular requirement of safety is known to be the governing one, we can apply strut-and-tie and stress-flow based nonlinear modelling to decide the arrangement of the reinforcement and associated details in a reasonable manner (Chapters 2, 8 and 9). Mechanics-aided design procedures present a means of realizing the load-carrying mechanism which was assumed in advance. The inversely decided design details regarding reinforcement and thickness of members can be shown to provide sufficient capacity that is not below the assumed static design loads, provided that some ductility is accomplished. Then, if other limit states are inconsequential, subsequent performance assessment by nonlinear analyses with respect to safety limit states is not necessary. There is a possibility of over-design in capacity if an appropriate stress flow is not assumed, and, in some cases, dense reinforcement may cause heavy construction of concreting works and elevate costs. At this point, the capacity can be verified by running nonlinear analyses and re-design is possible to provide a refined assessment of strength and improved detailing and dimensioning. As the successful strut-tie modelling under static loads is carried out under a single specified load, we have to check other performances required if those are thought to be possibly substantial. For example,


331


Nonlinear Modelling $Performance Assessment load+required performance 1

load+required performance 2

…

load+required performance N

Chapter 2,5,8

design process (creative jobs) stress-flow imagination, experiences, free-hand drawing, etc….

shapes, shapes,dimensions, dimensions,detailing, detailing,B.C., B.C., materials, arrangements, etc.. decided materials, arrangements, etc.. decided performance check needed?

No

END

Yes load+required performance 1

load+required performance 2

…

load+required performance N

…

selection of appropriate structural & material modeling for case-1 selection of appropriate structural & material modeling for case-N … Chapter 3,4,5,6

…

behavioral simulation output behavioral simulation output structural responses structural responses and material states for case-1 and material states for case-N

…

No

performance case-1 satisfied?

Chapter 3,4,5,6 Chapter 7

performance case-2 satisfied?

Yes

Yes

performance case-N satisfied?

No

Yes

END

Figure 10.1: Flowchart of structural design and performance assessment.

fatigue and durability requirements under service loads have to be examined after deciding design details by the strut-tie model. The performance assessment shown in the lower part of Figure 10.1 is conducted for each set of structural requirements and factored designed loads. Here, the decision making process of shape, dimension, detailing and material arrangement has nothing to do with the following behavioural simulation of the structure. Designers must select the most appropriate structural (Chapter 4 and Chapter 5) and material modelling (Chapter 3 and Chapter 6) for computing responses based on which the performance requirement is to be verified. After computing the design responses of the structure and constituent concrete and steel, judgment on acceptability of the computationally simulated behaviours has to be made. Here, the following issues have to be decided in practice: i) What sort of indicators are mechanically appropriate for limit states or design criterion in order to quantitatively express safety and functionality requirements?

332

!

10 Concluding remarks


ii) What is the appropriate limit value corresponding to the level of required performance? iii) How reliable is the computed indicator and the limit value for performance verification? In other words, what is the reasonable safety factor of nonlinear analysis? For items (i) and (ii), we can use the same set of indicators and limit values as that used in current design schemes based on linear and nonlinear analyses, that is, sectional forces and capacities, stress and strength, etc. In this case, nonlinear analysis can be used for computing the sectional forces as well as section capacities that cannot be evaluated only by on-going codified equations and formula in practice (Chapter 1). Here, the capacity computation by 2D and 3D nonlinear analyses (Chapters 4 and 5) can contribute to decisions. As multidimensional nonlinear analyses convey more detailed information on internal damage and material states, as well as displacements and section capacities, that that obtained by more conventional methods of analysis, more mechanics friendly and consistent indicators may be used in making decisions when adopting a non-linear modelling approach. This is discussed further in Section 10.3. For item (iii), one must quantitatively evaluate the reliability of computed responses obtained from nonlinear analysis in terms of each performance indicator. More specifically, the safety factor, which is applied to the indicator derived from the utilized nonlinear modelling, has to be specified by conducting a systematic experimental verification. Evaluation of the overall (global) reliability of the computer simulation is crucial and high-quality database of experiments (Chapter 7) is of great importance. In public works, these highly technological issues are the tasks of in-house engineers who are responsible for public financing and project/risk management, or code writers who produce statements on the use of nonlinear analyses.

10.3 Benefits of non-linear modelling and analyses Rational design may realize both reasonable cost and performances of concrete structures. When some technology is advanced and can be properly used, the total cost may be generally reduced if the targeted performance is specified. For conventional beams, slabs and columns that occur in the majority of buildings and bridges, room of improving cost performance is minimal since these widely used members are more or less reasonably efficiently designed in practice. However, there exist opportunities of rational design of structures where there is a shortage of past experiences and there exists a strong nonlinearity with the surrounding media. Figure 10.2 shows a cost benefit analysis of one of the worlds largest underground LNG storage tanks (JSCE, 1999). It consists of thin reinforced concrete side walls to bear horizontal static/dynamic loads of soil and underground water pressures and a large circular reinforced concrete tank base subjected to high static shear forces induced by underground water pressure and its dead weight. After the Hyogo-ken Nanbu earthquake (Kobe, Japan) in 1995, the design seismic loads were re-evaluated and increased. If conventional design procedures are applied without revision, the increased loading would lead to about a 25% increase in the cost of construction. This situation led to an engineering motivation to drastically change the design scheme from a capacity method by simplified elastic analysis to a performance based one with full 3D nonlinear coupled soil-RC analysis. This makes for a substantial cost reduction while safety and durability performances remain unchanged. A more than 20% cost reduction was realized


333


S 4/0+(.8-'

K125L L2H

120

X 140-30-8+YQ+ 0-, F41)*+Y)11(-1/

K120L L2H

A -/(7.+/-(/, (3+*4)8 Z F71)8-8+)90-1+$[[# \41+C%] +3*)//

110

K100L

80

conventional design

100

90

V99-30+49 M -8W3-8+ /0(99.-//

L2L L2L

X 140-30-8+YQ+ 0-, F41)*+ Y)11(-1/ M -8W3-8+ /0(99.-//

O )9-0Q+9)3041+ Z F71)8-8+941 O P-)1+1-(.9"

O P-)1+3)F)3(0Q \41, W*)+4.+/(N->-99-30

By K. Miyamoto, TEPCO

M )0(4.)*(N-8+ O P-)1+3)F)3(0Q+ R M S T/4(*+.4.*(.-)1(0Q U+?A +).)*Q/(/

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Figure 10.2: Cost performance analysis of nonlinear analysis based design for underground LNG storage tanks (JSCE 1999).

even though the required safety performance against earthquake was very much raised, as shown in Figure 10.2. For designing the sidewalls, nonlinearity of 3D in-plane shear and transverse flexure (Chapter 5) of cracked RC shell elements was explicitly taken into account and interaction with surrounding soil foundation was also analyzed using 3D FEM. This nonlinear simulation resulted in reduced internal forces of the main body and a relatively slight increase in the reinforcement even though a significantly high earthquake design load was imposed. A key factor of the base slab design was the new 3D size-effect modelling of punching shear failure because of its thickness (8-10m) with the disc span of approximately 100m. Thus, it was crucial to rationally estimate the transverse shear capacity for deciding on the quantity and arrangement of shear reinforcement (Chapter 1). As the total construction cost of an underground tank is massive, just a small cost reduction, relative to the overall cost of the project, more that offsets the increased expense of the works undertaken in the design office for a full 3D nonlinear analysis of the structure. In-house engineers in charge of financing and energy development conducted experiments for verifying their modelling of both the element and the structure. Finally, the global safety factor of overall structural analysis was discussed and authorized by an independent academic sector in terms of seismic performance limit state enabling re-use after the near and far field great earthquakes of 1000-years return period (JSCE, 1999). The expense of the large scale experiments was easily covered within the actualized cost saving obtained from results of the nonlinear analyses.

334

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10.4 Code provisions As stated in Chapter 3, practitioners are required to assess the accuracy of their nonlinear analyses and modelling as a design tool using a proper safety factor. Reliability or accuracy of nonlinear analysis has to be examined as a set of material and structural models used because of their nonlinearity. In other words, accuracy of individual modelling does not always stand for the accuracy of entire computational system. It should also be emphasized that functioning of computer programs or nonlinear analysis methods should be checked for each outcome. For example, the appropriate safety factor for computed capacity is not necessarily the same as that for ductility (deformability). Thus, the overall evaluation of program applicability is one of the main concerns for practice and experimental verification of systematically arranged parameters is essential. This guideline can not present quantitative assessment of a set of nonlinear models for particular design items of limit states but an outline is provided. Some design codes or recommendations include clauses on nonlinear structural and material models. At this point in time, conservative modelling is advocated due to shortage of experience. This guideline is also presented so as to facilitate in further codification of non-linear modelling.

10.5 Specification of design loads In some cases, the design loads are specified in consideration of the characteristics of the structural analysis and the ease of the design process and do not, necessarily, represent the realistic loading conditions. Thus, when practitioners apply load-path dependent nonlinear analysis for performance assessment, careful checking of specified loads must be made as code specified design loads are generally defined on the basis of load-path independent structural analysis, such as for linear elasticity. Figure 10.3 shows the bending moment diagram computed by the nonlinear smeared crack analysis of a prestressed concrete viaduct. For the examination of the ultimate limit state, the heavy loads are placed solely on the centre span of the bridge deck. If these specified loads are just put on the centre span monotonically, the moment at the section of the span centre is computed to be small, because the right and the left wing spans have greatly higher stiffness. But, this loading history is unrealistic. As the design load of bridges represents the traffic, the heavy live loads cannot reach the centre span without passing through the wing elements. If we apply the specified ultimate limit state load after operating the serviceable loads over the span, the computed flexural moment becomes larger due to the reduced stiffness of the wing spans with bending cracks caused by the service pre-loads. It should be noted that the linear elastic analysis results in reasonably better moment profiles because the sectional stiffness profile over the whole span is assumed uniform and is closer to eventual reality than the case of a nonlinear analysis with an unrealistic load history, no matter how unrealistic is the linear elastic analysis.


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specified patterns of design loads

Moment diagram: The specified Load is monotonically applied just on the middle span.

Moment diagram: The specified Load is finally applied after passing through the whole span of the bridge.

Figure 10.3: Loading pattern and path-dependency on nonlinear analysis based design.

10.6 Maintenance As introduced in Chapter 1, nonlinear analyses of collapsed real structures have greatly contributed to investigation of the causes of failure. In reality, lessons from bitter experiences have been driving forces for further improvement of design methods. Currently, nonlinear analysis is expected to play a role of maintenance issues for real structures in use. Owing to recent advances in non-destructive testing and sensing technology, much information related to mechanical damage of constituent materials becomes available (e.g., crack location, crack depth, crack width, corrosion level of steel inside concrete, yield hysteresis of steel, permanent displacement, elastic wave velocity, etc.). Here, engineers are expected to answer questions such as what potential capacity remains and how long the damaged structure may continue to function for planning of maintenance? This guideline does not methodically handle the performance assessment of damaged real structures under natural or artificial environments, although some knowledge included in this guideline may assist the practitioner in evaluating this issue. For example, damaged reinforced concrete beams subjected to flexure-shear are shown in Figure 10.4. Some part of main reinforcement was computationally decayed by volumetric expansion of corroded substances. This, induced, corrosion created pre-cracking as well as influencing the formation of subsequent diagonal shear cracks. This is introduced here merely to demonstrate the future potential of nonlinear analysis in the field of maintenance engineering. Guidelines on structural and service-life designs are thought to be an important development issue of the future. The issue of analysis for maintenance assessment is a target of some importance that involves the coupling of structural and materials engineering. Equally, performance assessment is needed in providing more rational tools for the planning of maintenance, as well as design of newly constructed structures. At present, nonlinear structural analysis and modelling technology are under development for meeting these challenges.

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No corrosion damage: after shear failure

corrosion induced at the span center: after shear failure

corrosion induced in shear span: just before failure Initiation of shear crack from pre-horizontal cracks

corrosion induced in shear span: after shear failure

Experiment Failure Crack Pattern

Analysis Failure Crack Pattern

Light corrosion beam Figure 10.4: Partially corroded RC beams in shear and cracking pattern (Maekawa et al. 2003).

10.7 References JSCE (1999), “Recommendation for Structural Performance Verification of LNG Uuderground Storage Tanks”, Japan Society of Civil Engineers, Concrete Library, 98. JSCE (2002), “Standard Specification of Concrete Structures – Structural Performance Verificaiton ”, Japan Society of Civil Engineers. Maekawa, K., Ishida, T. and Kishi, T. (2003), “Multi-scale modeling of concrete performance – Integrated material and structural mechanics -”, Journal of Advanced Concrete Technology, JCI, 2(1).


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fib Bulletins published since 1998 N° 1

Title Structural Concrete – Textbook on Behaviour, Design and Performance; Vol. 1: Introduction - Design Process – Materials Manual - textbook (244 pages, ISBN 978-2-88394-041-3, July 1999)

2

Structural Concrete – Textbook on Behaviour, Design and Performance Vol. 2: Basis of Design Manual - textbook (324 pages, ISBN 978-2-88394-042-0, July 1999)

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Structural Concrete – Textbook on Behaviour, Design and Performance Vol. 3: Durability - Design for Fire Resistance - Member Design - Maintenance, Assessment and Repair - Practical aspects Manual - textbook (292 pages, ISBN 978-2-88394-043-7, December 1999)

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Lightweight aggregate concrete: Extracts from codes and standards State-of-the-art report (46 pages, ISBN 978-2-88394-044-4, August 1999)

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Protective systems against hazards: Nature and extent of the problem Technical report (64 pages, ISBN 978-2-88394-045-1, October 1999)

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Special design considerations for precast prestressed hollow core floors Guide to good practice (180 pages, ISBN 978-2-88394-046-8, January 2000)

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Corrugated plastic ducts for internal bonded post-tensioning Technical report (50 pages, ISBN 978-2-88394-047-5, January 2000)

Lightweight aggregate concrete: Part 1 (guide) – Recommended extensions to Model Code 90; Part 2 (technical report) – Identification of research needs; Part 3 (state-of-art report) – Application of lightweight aggregate concrete (118 pages, ISBN 978-2-88394-048-2, May 2000) 9 Guidance for good bridge design: Part 1 – Introduction, Part 2 – Design and construction aspects. Guide to good practice (190 pages, ISBN 978-2-88394-049-9, July 2000) 10 Bond of reinforcement in concrete

8

State-of-art report (434 pages, ISBN 978-2-88394-050-5, August 2000)

11 Factory applied corrosion protection of prestressing steel State-of-art report (20 pages, ISBN 978-2-88394-051-2, January 2001)

12 Punching of structural concrete slabs Technical report (314 pages, ISBN 978-2-88394-052-9, August 2001)

13 Nuclear containments State-of-art report (130 pages, 1 CD, ISBN 978-2-88394-053-6, September 2001) 14 Externally bonded FRP reinforcement for RC structures Technical report (138 pages, ISBN 978-2-88394-054-3, October 2001)

15 Durability of post-tensioning tendons Technical report (284 pages, ISBN 978-2-88394-055-0, November 2001)

16 Design Examples for the 1996 FIP recommendations Practical design of structural concrete Technical report (198 pages, ISBN 978-2-88394-056-7, January 2002)

17 Management, maintenance and strengthening of concrete structures Technical report (180 pages, ISBN 978-2-88394-057-4, April 2002)

18 Recycling of offshore concrete structures State-of-art report (33 pages, ISBN 978-2-88394-058-1, April 2002)

19 Precast concrete in mixed construction State-of-art report (68 pages, ISBN 978-2-88394-059-8, April 2002)

20 Grouting of tendons in prestressed concrete Guide to good practice (52 pages, ISBN 978-2-88394-060-4, July 2002)

21 Environmental issues in prefabrication State-of-art report (56 pages, ISBN 978-2-88394-061-1, March 2003)

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22 Monitoring and safety evaluation of existing concrete structures State-of-art report (304 pages, ISBN 978-2-88394-062-8, May 2003)

23 Environmental effects of concrete State-of-art report (68 pages, ISBN 978-2-88394-063-5, June 2003) 24 Seismic assessment and retrofit of reinforced concrete buildings State-of-art report (312 pages, ISBN 978-2-88394-064-2, August 2003)

25 Displacement-based seismic design of reinforced concrete buildings State-of-art report (196 pages, ISBN 978-2-88394-065-9, August 2003)

26 Influence of material and processing on stress corrosion cracking of prestressing steel - case studies. Technical report (44 pages, ISBN 978-2-88394-066-6, October 2003) 27 Seismic design of precast concrete building structures State-of-art report (262 pages, ISBN 978-2-88394-067-3, January 2004)

28 Environmental design State-of-art report (86 pages, ISBN 978-2-88394-068-0, February 2004) 29 Precast concrete bridges State-of-art report (83 pages, ISBN 978-2-88394-069-7, November 2004) 30 Acceptance of stay cable systems using prestressing steels Recommendation (80 pages, ISBN 978-2-88394-070-3, January 2005)

31 Post-tensioning in buildings Technical report (116 pages, ISBN 978-2-88394-071-0, February 2005) 32 Guidelines for the design of footbridges Guide to good practice (160 pages, ISBN 978-2-88394-072-7, November 2005)

33 Durability of post-tensioning tendons Recommendation (74 pages, ISBN 978-2-88394-073-4, December 2005)

34 Model Code for Service Life Design Model Code (116 pages, ISBN 978-2-88394-074-1, February 2006)

35 Retrofitting of concrete structures by externally bonded FRPs. Technical Report (224 pages, ISBN 978-2-88394-075-8, April 2006)

36 2006 fib Awards for Outstanding Concrete Structures Bulletin (40 pages, ISBN 978-2-88394-076-5, May 2006)

37 Precast concrete railway track systems State-of-art report (38 pages, ISBN 978-2-88394-077-2, September 2006)

38 Fire design of concrete structures – materials, structures and modelling State-of-art report (106 pages, ISBN 978-2-88394-078-9, April 2007)

39 Seismic bridge design and retrofit – structural solutions State-of-art report (300 pages, ISBN 978-2-88394-079-6, May 2007)

40 FRP reinforcement in RC structures Technical report (160 pages, ISBN 978-2-88394-080-2, September 2007)

41 Treatment of imperfections in precast structural elements State-of-art report (74 pages, ISBN 978-2-88394-081-9, November 2007)

42 Constitutive modelling of high strength / high performance concrete State-of-art report (130 pages, ISBN 978-2-88394-082-6, January 2008)

43 Structural connections for precast concrete buildings Guide to good practice (370 pages, ISBN 978-2-88394-083-3, February 2008)

44 Concrete structure management: Guide to ownership and good practice Guide to good practice (208 pages, ISBN 978-2-88394-084-0, February 2008)

45 Practitioners’ guide to finite element modelling of reinforced concrete structures State-of-art report (344 pages, ISBN 978-2-88394-085-7, June 2008) Abstracts for fib Bulletins, lists of available CEB Bulletins and FIP Reports, and an order form are given on the fib website at www.fib-international.org/publications.

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