Reinforced Concrete at Elevated Temperatures

MATERIAL MODELLING OF REINFORCED CONCRETE ATELEVATED TEMPERATURES

MasterThesisFebruary2011

FireSafety,SectionforBuildingDesign,DepartmentofCivilEngineering,theTechnicalUniversityofDenmark

Josephine Voigt Carstensen, s052204

Material Modelling of ReinforcedConcrete at Elevated Temperatures

M.Sc. in Civil Engineering - Master Thesis credited with 30 ECTS pointsProject Period: 2010.09.13-2011.02.11Language: English

Fire Safety at the Section for Building DesignDepartment of Civil EngineeringTechnical University of Denmark

In collaboration with:BRE Centre for Fire Safety EngineeringThe Univeristy of Edinburgh

Supervisor: External Supervisor:Dr. Grunde Jomaas Dr. Pankaj PankajAssistant Professor Senior LecturerDepartment of Civil Engineering School of EngineeringTechnical University of Denmark The University of Edinburgh

Handed in 2011.02.11 by:

Josephine Voigt Carstensen, s052204

i

Abstract

Previous disasters have elucidated that accurate and realistic modelling of concrete behaviourat elevated temperatures is fundamental for the safe design of, for example, nuclear and struc-tures exposed to fire. However, when the same model is evaluated with different mesh sizes, theexisting models for the behaviour of concrete at elevated temperatures are subject to problemswith convergence of results in the Finite Element (FE) analysis. These problems arise as a resultof the problem of localization of deformations associated with the post-peak response of concrete.

This current research focuses on the modelling of the uniaxial behaviours of reinforced concreteat elevated temperatures and in particular on the key issues associated with the post-peak be-haviour.

It is generally recognized that in order to obtain mesh independent results of models of rein-forced concrete in FE-analysis at ambient conditions, a fracture energy based material modelmust be adopted. In tension, such models are widely used and in most FE-codes, for exampleABAQUS, it is possible to define the tensile post-peak behaviour in three ways; either throughan element size dependent stress-strain relation, through a stress-displacement formulation orby giving the tensile fracture energy and letting ABAQUS define the behaviour. However, ifreinforced concrete is to be considered, the tensile definition must account for the tension stiffen-ing effect that gradually shifts the load-bearing capacity from the concrete to the reinforcementas the cracking progresses. This issue can be tackled by defining an element size dependentinteraction stress contribution that is combined with the concrete contribution for the definitionof the post-peak behaviour. In compression the fracture energy based behaviour models are lessused and the compressive fracture energy is, for example, not discussed in any current codesand it is generally examined by very few. To apply a fracture energy based compressive modelin a FE-analysis, an element size dependent stress-strain formulation must be used.

In this current research, the existing models for the ambient condition have been extended toelevated temperatures, largely by applying the material properties at a given elevated temper-ature to the current formulation. Therefore, the existing models have been evaluated prior tothe extension and it has been found necessary to express limits for their application. It is wellestablished that a limit on the maximum element size exists. However, herein it has been foundthat restrictions on the minimum element size and, if modelling the tension stiffening throughthe definition of an interaction stress contribution, on the minimum level of reinforcement ad-missible also apply.

As experimental data is currently not available on the evolution of the compressive and thetensile fracture energy with temperature, the fracture energies inherent in the existing elevatedtemperature models have been examined. It has been found that the tensile fracture energyinherent in the currently available model follows the decay function for material strength. The

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compressive fracture energy has been based on the models by four current compressive modelswhere two considers solely the instantaneous stress-related strain and two includes the effectsof the LITS. It has been established that the current compressive elevated temperature modelsdoes not agree on the post-peak behaviour and that the LITS does not seem to have an effecton the post-peak response.

The limits of application are extended to elevated temperatures by expressing a validity rangefor the element sizes and a minimum reinforcement ratio. It has been found that up to about500C, the maximum element size is typically governed by the tensile properties after whichthe compressive parameters are governing. Once the compressive model becomes governing, itonly provides meaningful results within a very limited range of mesh-sizes. This range shouldbe considered the new validity domain of the model.

This novel model for the uniaxial behaviours of reinforced concrete at elevated temperatures canreadily be applied for FE analysis, for example in ABAQUS, and, if the modelling is performedwithin the limits of application, it is possible to get mesh independent results of the analyseswith different mesh configurations.

Preface

This project is a M.Sc. thesis of 30 ECTS points created in the period September 13th 2010 toFebruary 11th 2011. A M.Sc. thesis is a compulsary project in order to fulfill the requirementsfor the M.Sc. programme in Civil Engineering at the Technical University of Denmark, (DTU).

The project has been carried out for the Fire Safety Group at the Section for Building Design,Department of Civil Engineering at the Technical University of Denmark in collaboration withat the BRE Centre for Fire Safety at the University of Edinburgh.

The internal supervisor of the project has been Dr. Grunde Jomaas (Assistant Professor, DTU)and the external supervisor has been Dr. Pankaj Pankaj (Senior Lecturer, Edinburgh).

The work presented in the thesis was conducted at the University of Edinburgh.

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Acknowledgements

First, a great amount of appreciation must be given to the BRE Center for Fire Safety En-gineering at the University of Edinburgh and especially to the students and staff in the JohnMuir Building for creating a welcoming and inspiring research environment. My visit there hasproved to be a highly educative experience, thanks both to the academic and the non-academicsupport received at the premises. A special expression of gratitude is given to Prof. Jos L.Torero for setting up the practical framework, without which this project would not have beenaccomplished.

A very special thanks is directed to Dr. Pankaj Pankaj for all his guidance and encouragement.I have immensely appreciated that he has always taken time to patiently explain the arisenproblems - no matter the magnitude. His ability to make even the most complex problemsunderstandable is something I profoundly admired. On this note appreciation is also dedicatedto Prof. Kristian D. Hertz (DTU) and Dr. Martin Gillie for their clarifications of puzzlingdefinitions.

Will Kingston is to be deeply thanked for the helpful discussions and useful hints throughoutthe project period. Especially his calm introduction to ABAQUS modelling at the project startand his patient answers to emerging ABAQUS related questions have been beyond compare.On this note Adam Ervine, Kate Andersson and Joanne Knox must also be recognized alongwith Rory Hadden, Cristin Maluk, Nicolas Bal, Steffen Kahrmann and Dr. Francesco Colella.

Further, a particularly gratefulness is given to Dr. Grunde Jomaas (DTU) for his friendly ap-proach and guidance. He must be recognized for creating the contact between the collaboratorsof the project and for being an tremendous source of inspiration. His mentoring and guidancethrough the project planning and execution, as well as through decision making about furtherprofessional career, have had great effects on both the project at hand and on future choice ofoccupation.

Lrke Mikkelsen (DTU) and Miki Kobayashi (DTU) are acknowledged for helping with retriev-ing literature and Mads Mnster Jensen (DTU) for his clarification of the mysteries of concretetechnology.

Last but not least, gratitude is directed to the OTICON Foundation, Reinholdt W. JorcksFoundation, KABs studielegat, the Department of Civil Engineering at the Technical Univer-sity of Denmark and BRE Center for Fire Safety Engineering at the University of Edinburghfor the received financial support.

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Contents

Abstract iii

Preface v

Acknowledgements vii

Nomenclature xiii

List of Figures xvii

List of Tables xxiii

1 Introduction 11.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Localization and Fracture Energy in Tension . . . . . . . . . . . . . . . . . . . . 21.3 Localization and Fracture Energy in Compression . . . . . . . . . . . . . . . . . . 31.4 Novelties and Milestones of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Finite Element modelling of Multiaxial Behaviour of Reinforced Concrete 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 FE-Modelling of Concrete Cracking . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Concrete Model in ABAQUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 Yield Surface Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Response of Reinforced Concrete to Fire Exposure 133.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Chemical and Physical Effects of Fire . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.1 Chemophysical Response of Concrete to Fire . . . . . . . . . . . . . . . . 143.2.2 Chemophysical Response of Reinforcing Steel to Fire . . . . . . . . . . . . 15

3.3 Typical Failures of Reinforced Members . . . . . . . . . . . . . . . . . . . . . . . 173.4 Choice of Analysis Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5 Overview of Concepts Involved in the Response of Reinforced Concrete to a Fire 18

4 Modelling of Uniaxial behaviour of Reinforced Concrete at Ambient Tem-perature 194.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Material Model of Reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 Reinforced Concrete in Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.3.1 Tension Stiffening Model as per the CEB-FIB Model Code . . . . . . . . 214.3.2 Tension Stiffening Model by Cervenka et al. . . . . . . . . . . . . . . . . . 224.3.3 Tension Stiffening Model by Feenstra and de Borst . . . . . . . . . . . . . 26

4.4 Compressive Behaviour of Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . 304.4.1 Compression Model in CEB-FIB Model Code . . . . . . . . . . . . . . . . 304.4.2 Compressive Fracture Energy . . . . . . . . . . . . . . . . . . . . . . . . . 31

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4.4.3 Compression Model of Narakuma and Higai . . . . . . . . . . . . . . . . . 334.4.4 Compression Model by Feenstra and de Borst . . . . . . . . . . . . . . . . 344.4.5 Comparison of Compression Models . . . . . . . . . . . . . . . . . . . . . 36

4.5 Chosen Uniaxial Concrete Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.6 Numerical Test Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.6.1 Uniaxial Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.6.2 Uniaxial Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.6.3 Pure Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5 Existing Models of the Behaviour of Reinforced Concrete at Elevated Tem-peratures 415.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.2 Decay of Material Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.2.1 Compressive Strength of Concrete . . . . . . . . . . . . . . . . . . . . . . 415.2.2 Tensile Strength of Concrete . . . . . . . . . . . . . . . . . . . . . . . . . 445.2.3 Strength of Reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.3 Uniaxial Compressive behaviour of Concrete at Elevated Temperatures . . . . . . 465.3.1 Strain Components at Elevated Temperatures . . . . . . . . . . . . . . . . 47

5.4 Uniaxial Tensile behaviour of Concrete at Elevated Temperatures . . . . . . . . . 515.5 Reinforcement Model at Elevated Temperatures . . . . . . . . . . . . . . . . . . . 525.6 Overview of Relevant Assumptions for the Formulation of the Fracture Energy

Based Material Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6 Fracture Energy Based Uniaxial Material Models at Elevated Temperatures 556.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.2 Fracture Energy Based Compressive behaviour Model for Concrete at Elevated

Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.2.1 Compressive Fracture Energy at Elevated Temperatures . . . . . . . . . . 566.2.2 Application of the Elevated Temperature Model by Anderberg and The-

landersson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.2.3 Application of the Elevated Temperature Model by Lie and Lin . . . . . . 586.2.4 Compressive Fracture Energies at Elevated Temperatures for Models In-

cluding the Effect of the LITS . . . . . . . . . . . . . . . . . . . . . . . . . 596.2.5 Comparison of Compressive Fracture Energies at Elevated Temperatures . 60

6.3 Formulation of Fracture Energy Based Tensile Model for Concrete at ElevatedTemperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626.3.1 Tensile Fracture Energy at Elevated Temperatures . . . . . . . . . . . . . 626.3.2 Fracture Energy Based Tensile Model of Plain Concrete . . . . . . . . . . 636.3.3 Fracture Energy Based Tensile Model for Reinforced Concrete . . . . . . . 63

6.4 Limits of Fracture Energy Based Models at Elevated Temperatures . . . . . . . . 656.4.1 Limitations on the Element Size . . . . . . . . . . . . . . . . . . . . . . . 656.4.2 Minimum Reinforcement Ratio . . . . . . . . . . . . . . . . . . . . . . . . 66

7 Numerical Example of a Reinforced Concrete Slab at Elevated Temperatures 697.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697.2 Parameters for the Uniaxial Material Models . . . . . . . . . . . . . . . . . . . . 707.3 Material Properties for the Thermal Analysis . . . . . . . . . . . . . . . . . . . . 717.4 FE-Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

7.4.1 Element size h = 129 mm . . . . . . . . . . . . . . . . . . . . . . . . . . . 737.4.2 Element size h = 73 mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

8 Conclusion 798.1 Remarks in Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798.2 Suggestions for Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Bibliography 84

A Detailed Description of Cracking and the Post-Peak Response of Concrete 85A.1 Crack Propagation and Softening . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

B ABAQUS Functions for Definition of Uniaxial Behaviour, Embedment ofReinforcement and Load Steps 87B.1 Tension Stiffening and Compression Hardening Models . . . . . . . . . . . . . . . 87B.2 Embedment of Reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88B.3 Load Step Definition for Static Analysis . . . . . . . . . . . . . . . . . . . . . . . 88

C ABAQUS Output from Pure Shear Example of Simple Plates with andwithout Reinforcement 91C.1 Simple Shear Example without Reinforcement . . . . . . . . . . . . . . . . . . . . 91C.2 Simple Shear Example with Reinforcement . . . . . . . . . . . . . . . . . . . . . 93

D Uniaxial Material Models for the Numerical Example of a Reinforced Slab 95

Nomenclature

Ac,eff Effective cross-sectional area ofthe concrete.

As Total area of the reinforcement.As,min Mimimum area of the reinforce-

ment if the interaction contribu-tion must be considered as a partof the tension stiffening.

b Length of reinforced concretespecimen.

c Cover layer of the reinforcement.dmax Maximum aggregate size.Ec E-modulus of concrete at ambient

temperature.EciT Initial E-modulus at elevated

temperatures.EciT,EC Initial E-modulus at elevated

temperatures in Eurocode 2 [21].Ep Slope of the descending branch in

the elevated temperature modelby Li and Purkiss [23].

Es E-modulus of reinforcement atambient temperature.

EsT E-modulus of reinforcement at el-evated temperatures.

fcm Compressive strength of concreteat ambient temperature.

fcT Compressive strength of concreteat elevated temperatures.

fct,m Tensile strength of concrete atambient temperature.

fctT Tensile strength of concrete at el-evated temperatures.

fy Yield strength of reinforcement atambient temperature.

fyT Yield strength of reinforcement atelevated temperatures.

F Yield function in ABAQUS [8].

G() Flow potential function inABAQUS [8].

Gc Compressive fracture energy atambient temperature.

(Gc/h)AT Compressive fracture energy di-vided by the corresponding ele-ment size inherent in the elevatedtemperature model by Anderbergand Thelandersson [25].

(Gc/h)EC Compressive fracture energy di-vided by the corresponding ele-ment size inherent in the elevatedtemperature model in Eurocode 2[21].

(Gc/h)model Compressive fracture energydivided by the corresponding ele-ment size inherent in a given ele-vated temperature model.

(Gc/h)LL Compressive fracture energy di-vided by the corresponding ele-ment size inherent in the elevatedtemperature model by Lie and Lin[26].

(Gc/h)LP Compressive fracture energy di-vided by the corresponding el-ement size inherent in the ele-vated temperature model by Liand Purkiss [23].

GcT Compressive fracture energy at el-evated temperatures.

GcT,AT Compressive fracture energy at el-evated temperatures as inherentin the model by Anderberg andThelandersson [25].

GcT,EC Compressive fracture energy at el-evated temperatures as inherentin the model of Eurocode 2 [21].

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Nomenclature

GcT,model Compressive fracture energy in-herent i a given elevated temper-ature model.

GcT,LL Compressive fracture energy at el-evated temperatures as inherentin the model by Lie and Lin [26].

GcT,LP Compressive fracture energy at el-evated temperatures as inherentin the model by Li and Purkiss[23].

Gf Tensile fracture energy at ambienttemperature.

Grcf Reinforced tensile fracture energyat ambient temperature.

GfT Tensile fracture energy at elevatedtemperatures.

h Element size.hAT Element size corresponding to the

compressive fracture energy in-herent in the elevated temper-ature model by Anderberg andThelandersson [25].

hEC Element size corresponding to thecompressive fracture energy in-herent in the elevated tempera-ture model of Eurocode 2 [21].

heff Effective element size.hmax Maximum element size at ambient

temperature.hmaxT Maximum element size at elevated

temperatures.hmin Minimum element size at ambient

temperature.hminT Minimum element size at elevated

temperature.hmodel Element size corresponding to the

compressive fracture energy in-herent i a given elevated temper-ature model.

hLL Element size corresponding to thecompressive fracture energy in-herent in the elevated tempera-ture model by Lie and Lin [26].

hLP Element size corresponding to thecompressive fracture energy in-herent in the elevated tempera-ture model by Li and Purkiss [23].

H Softening modulus.k, T1, T2, T8, T64 Constants describing the

decay function.

kp Parameter describing the stress-strain relationship suggested by Liand Purkiss [23].

Kc Parameter determining the shapeof the yield surface in ABAQUS[8].

Kt Tangential stiffness.LITS Load induced thermal strains.P Load.Pcr Load at which macrocracking of

concrete is initiated.PE11 Output from ABAQUS of the

plastic strains in the x-direction.ls Average crack spacing.p Pressure invariant in ABAQUS [8]s0 Minimum bond length.S11 Output from ABAQUS of stresses

in the x-direction.t Thickness of reinforced concrete

specimen.tFE Time in an FE-analysis.ts Strength of the interaction con-

tribution as defined by Cervenkaet al. [17].

T Temperature.Ta Ambient temperature.w Displacement.wpeak Displacement at peak stress. Thermal expansion coefficient in

ABAQUS [8].concrete Thermal expansion coefficient of

concrete.steel Thermal expansion coefficient of

steel.ts Strength level of the interaction

contribution as defined by Feen-stra and de Borst [18] (fraction ofthe tensile strength).

Displacement.p Plastic displacement. Stress adjustment necessary

in order to evaluate the com-bined concrete and interac-tion contribution through the*TENSION STIFFENING functionin ABAQUS.

$ Flow potential eccentricity. Strain.0T Strain at peak compressive stress

for concrete at elevated tempera-tures.

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Nomenclature

01, 02, 03 Parameters used in the compu-tation of strain at peak compres-sive stress as defined by Terro [24].

1,1 Parameters describing the instan-taneous stress-related strain sug-gested by Anderberg and The-landersson [25].

c Compressive strain.inc Inelastic strain in ABAQUS [8].plc ,

plt Hardening variables in ABAQUS

[8].c0 Strain at peak tensile stress of

concrete at ambient temperature.c1, c,lim Constants used to define the

compressive behavior as sug-gested by the CEB-FIB ModelCode [16].

c1t Strain at peak compressive stressas defined by Eurocode 2 [21].

ctuT Ultimate tensile strain of concreteat elevated temperatures.

cu Ultimate strain of concrete at am-bient temperature.

cu Ultimate strain of concrete in theelevated temperature model sug-gested by Li and Purkiss [23].

cu1t Ultimate compressive strain asdefined by Eurocode 2 [21].

cuT Ultimate compressive strain ofconcrete at elevated tempera-tures.

cuT,AT Ultimate compressive strain fromthe elevated temperature modelby Anderberg and Thelandersson[25].

cuT,model Ultimate compressive strainfrom a given elevated temperaturemodel.

cuT,LL Ultimate compressive strain fromthe elevated temperature modelby Lie and Lin [26].

cuT,LP Ultimate compressive strain fromthe elevated temperature modelby Li and Purkiss [23].

cT Compressive strain at elevatedtemperatures.

e Elastic strain.p Peak strain the in compressive

material model by Nakamura andHigai [19]

p Plastic strain.p0 Plastic strain corresponding to

peak compressive stress.

s1, s2 Strain states used to compute thetension stiffening as per the CEB-FIB Model Code [16].

s,m Strain in the reinforcement withtension stiffening as defined in theCEB-FIB Model Code [16].

T Instantanious stress-related straint Tensile strain.ckt Cracked strain in ABAQUS [8].th Unrestrained thermal strain.px Plastic strain in the x-direction.u Strain in the interaction contri-

bution at which the yield stressof the reinforcement is reached atambient temperature.

y Strain at yield stress of reinforce-ment at ambient temperature.

C ,T Internal parameters describingthe behavior at ambient tempera-ture as suggested by Feenstra andde Borst [18].

e Equivalent strain correspondingto peak compressive stress as sug-gested by Feenstra and de Borst[18].

eT Equivalent strain correspondingto peak compressive stress at el-evated temperatures.

uC Ultimate compressive concretestrain at ambient temperatureas suggested by Feenstra andde Borst [18].

uCT Ultimate compressive concretestrain at elevated temperatures.

L Initial compressive stress level. Parameter for visco-plastic regu-

larization of the concrete consti-tutive equations in ABAQUS [8].

Poissons ratio.(T ) Decay function for material prop-

erties defined by Hertz [7].p Reinforcement ratio in the direc-

tion of the load.q Reinforcement ratio in the direc-

tion orthogonal to the loading.s Reinforcement ratio.s,eff,min Minimum effective reinforce-

ment ratio for the interactioncontribution defined by Cervenkaet al. [17] to be considered atambient temperature.

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Nomenclature

s,eff,minT Minimum effective reinforce-ment ratio for the interaction con-tribution to be considered at ele-vated temperatures.

Stress.1,2,3 Primary stress axis.1, 2 Primary stress axis for plane

stress.b0/c0 Ratio of the equibiaxial compres-

sive yield stress and the initialuniaxial compressive yield stressin ABAQUS [8].

c0 Initial compressive yield stressused in the *COMPRESSIVEHARDENING option in ABAQUS[8].

cT Compressive stress.cT Compressive stress at elevated

temperatures.

cu Ultimate compressive stress inABAQUS [8].

max Maximum principle stress inABAQUS [8].

peak Peack compressive stress.x Stress in the x-direction.t Tensile stress.t0 Uniaxial tensile peak stress used

for the definition of the tensionstiffening in ABAQUS [8].

eq Equivalent reinforcement diame-ter.

p Diameter of the reinforcement inthe direction of the load.

q Diameter of the reinforcement inthe direction orthogonal to theloading.

Dilation angle.

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List of Figures

1.1 Examples of concrete subjected to elevated temperatures. . . . . . . . . . . . . . 11.2 Uniaxial tension test of pure concrete element with strain gauges at A, B and C,

(a), and the corresponding load-displacement diagrams, (b). Reproduced fromvan Mier [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Stress-displacement curve in localized region, (a), to illustrate construction ofthe load-plastic displacement diagram and of the fracture energy as the areaunder the curve, (b). Reproduced from Pankaj [6]. . . . . . . . . . . . . . . . . . 2

1.4 Illustrations of a typical temperature variation caused by a fire, (a), of the tem-peratures in the hot and cold phases of a fire, (b), and the strength ratio as afunction of the temperature, (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Descriptions of hot and cold phases of a fire, as defined by Hertz [7]. . . . . . . 4

2.1 Stress-strain relation for material undergoing hardening post-peak, (a), and ini-tial and subsequent yield surfaces in deviatoric plane, (b). Reproduced fromPankaj [9]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Discrete, (a), and smeared crack, (b), approaches. Reproduced from Pankaj [6]. 82.3 Drucker-Prager yield criteria in the deviatoric plane for Kc = 2/3 and Kc = 1.0,

(a), and in three dimensions for Kc = 1, 0, (b). Reproduced from ABAQUSVersion 6.7 Documentation [8] and Pankaj [9], respectively. . . . . . . . . . . . . 10

2.4 Yield surface in plane stress. Reproduced from ABAQUS Version 6.7 Documen-tation [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 Illustration of the plastic potential in relation to a yield surface. Reproducedfrom Pankaj [9]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.1 Schematic overview of deterioration of plain concrete as the temperature is in-creased. Based on Fletcher et al. [10] and Hertz [7]. . . . . . . . . . . . . . . . . 14

3.2 Dehydration of calcium hydroxide to calcium oxide and evaporable water caus-ing shrinking, (a), and rehydration upon cooling of calcium oxide to calciumhydroxide resulting in increased cracking, (b). Based on Hertz [7]. . . . . . . . . 15

3.3 Schematic overview of deterioration of reinforcement as the temperature is in-creased. Based on Fletcher et al. [10]. . . . . . . . . . . . . . . . . . . . . . . . . 16

4.1 Illustrative load-displacement diagram explaining the concept of tension stiffen-ing of reinforced concrete members. . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2 Behaviour of reinforced concrete member using the CEB-FIB Model Code [16]. 214.3 Interaction contribution suggested by Cervenka et al. [17]. . . . . . . . . . . . . 224.4 Schematic plots of the stress-strain relation of pure concrete in tension, (a), and

the stress-plastic displacement diagram, (b), to illustrate the dependency of thefracture energy on the element size, h. . . . . . . . . . . . . . . . . . . . . . . . 23

xvii

List of Figures

4.5 Combination of concrete and interaction stress contribution for different elementside lengths, h [mm]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.6 The effect on the stress-strain curve of concrete, (a), and on the combined con-crete and interaction contribution, (b), of snap-back of concrete for a model withtoo large element side length, here h = 1000 mm, compared to model withoutsnap-back, h = 500 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.7 Stress-plastic strain diagram for concrete assuming linear softening to illustratethe softening modulus, H. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.8 Combined concrete and interaction stress contribution for different different re-inforced areas As [mm2] using the tension stiffening model by Cervenka et al.[17]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.9 Combined concrete and interaction contribution for h = 50 mm as defined byCervenka et al. [17] and by Feenstra and de Borst [18]. . . . . . . . . . . . . . . 28

4.10 Examples of possible errors in the tension stiffening model by Feenstra andde Borst [18], arising from the selection of a too large element side length, whichcauses snap-back, (a), and a too low ratio of reinforcement, (b). . . . . . . . . . 28

4.11 Effect of changing the fraction of the ultimate tensile strength applied on theinteraction contribution of the tension stiffening model by Feenstra and de Borst[18] on the combined concrete and interaction stress contribution, (a), and thetotal stress-strain relation for the specimen, (b). . . . . . . . . . . . . . . . . . . 29

4.12 Compressive behaviour as defined by the CEB-FIB Model Code [16]. . . . . . . 314.13 Compressive post-peak fracture energies for different specimen geometries. Re-

produced from Vonk [20]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.14 Illustrative stress-strain relationship for the compression model by Nakamura

and Higai [19]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.15 Compressive stress-strain relations for h = 100 mm, (a), and h = 500 mm,

(b), for a concrete defined by the variables in Table 4.3 model as suggested byNakamura and Higai [19]. The compressive fracture energies are computed basedon the compressive strength fcm, (4.10), and based on the tensile fracture energyGf = 0.095 N/mm, (4.11). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.16 Compressive behaviour model suggested by Feenstra and de Borst [18] plottedusing compressive fracture energies as defined by Vonk [20], Figure 4.13, andNakamura and Higai [19] based on the compressive strength fc, (4.10). A con-crete grade C30 is considered and the material data of Table 4.3 are used. . . . 35

4.17 Stress-equivalent strain diagram for compression model by Feenstra and de Borst[18] for concrete grade C30 with fracture energy by expression (4.10), elementside lengths h = 100 mm, h = 500 mm and h = 2000 mm. . . . . . . . . . . . . 36

4.18 Stress-strain diagrams for compressive behaviour as defined by the CEB-FIBModel Code [16], Nakamura and Higai [19] and Feenstra and de Borst [18] forconcrete grade C30 with element side length h = 100 mm, (a), and h = 500 mm,(b). The material data is taken from Table 4.3 and the compressive fractureenergy is computed based on the compressive strength as defined in (4.10). . . . 36

4.19 FE-configuration of the reinforced member considered for uniaxial load tests ofthe tension stiffening and the compression model in ABAQUS. . . . . . . . . . . 38

4.20 The tension stiffening is defined in ABAQUS as the combination of the concreteand interaction contributions and must be forced to constantly have a slope, bysubtracting from the stress at the input, defining u. . . . . . . . . . . . . . . 38

xviii

List of Figures

4.21 ABAQUS output of load-displacement diagram in the y-direction on node 3for the example plate subjected to uniaxial tension. The tension stiffening ismodelled as presented by Feenstra and de Borst [18], (a), and modified by =0.01 MPa to ensure a constant presence of slope, (b). . . . . . . . . . . . . . . . 39

4.22 ABAQUS output of load-displacement diagram in the y-direction on node 3 forthe plate example subjected to uniaxial compression. The compressive propertiesare modelled as presented by Feenstra and de Borst [18]. . . . . . . . . . . . . . 40

4.23 FE-configuration for numerical test element subjected to pure shear. . . . . . . 40

5.1 Comparison of the decay function for compressive strength presented by Hertz[7] with the compressive decay function from Eurocode 2 [21] for a concrete withsiliceous, (a), and calcerous aggregates, (b). For computation of the decay ofstrength as suggested by Hertz [7], equation (5.1) and the parameters of Table5.1 are used and the reduction presented in Eurocode 2 [21] is given in Table 5.2. 43

5.2 Residual compressive strength of concrete after exposure to temperature level T ,as presented by Eurocode 2 [21] and Hertz [7], for siliceous, (a), and calcerous,(b), aggregates. The strength reduction presented by Hertz [7] is computedby equation (5.1) with the parameters from Table 5.3 and the reduction fromEurocode 2 [21] is given in Table 5.2. . . . . . . . . . . . . . . . . . . . . . . . . 44

5.3 Comparison of decay of tensile strength of concrete in the hot, (a), and the cold,(b), phase of a fire from Eurocode 2 [21] and the method presented by Hertz [7]with siliceous, main group and light weight aggregates. For the computations ofthe strength by Hertz [7], equation (5.1) and the parameters of Table 5.1 andTable 5.3 are used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.4 Decay functions from the Eurocode [21] and Hertz [14] for hot-rolled, (a), andcold-worked, (b), reinforcement bars when exposed to high temperatures. . . . . 45

5.5 Residual strength of cold-worked reinforcement steel after exposure to elevatedtemperature level, T , as presented by Eurocode 2 [21] and Hertz [14]. . . . . . . 46

5.6 Instantaneous stress-related strain as presented by Anderberg and Thelandersson[25] and by Lie and Lin [26] for temperatures of T = 20C and T = 300C, (a),and T = 500C and T = 700C, (b). The ultimate stress is normalized by theultimate stress at ambient temperatures. . . . . . . . . . . . . . . . . . . . . . . 48

5.7 Illustration of the difference between the total strain when heated with andwithout applied stress. Reproduced from Law and Gillie [27]. . . . . . . . . . . 49

5.8 Compressive stress-strain relations as defined by Li and Purkiss [23] and Eu-rocode 2 [21] for siliceous concrete at T = 20C and T = 300C, (a), andT = 500C and T = 700C, (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.9 Tensile stress-strain relationship as suggested by Terro [24] for concrete at tem-peratures of T = 20C, T = 300C, T = 500C and T = 700C. . . . . . . . . . 51

5.10 Example of reinforcement models at ambient and elevated temperatures for hot-rolled reinforcement with the material characteristics of Table 4.1. . . . . . . . . 52

6.1 The compressive fracture energy is inherent in the existing elevated temperaturemodels for the compressive behaviour of concrete. . . . . . . . . . . . . . . . . . 56

6.2 Compressive material model by Anderberg and Thelandersson [25] and fractureenergy based formulation with an element size of h = 65 mm for concrete gradeC30 at ambient temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.3 Compressive material model by Lie and Lin [26] and fracture energy based for-mulation with an element size of h = 300 mm for a concrete grade C30 atambient temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

xix

List of Figures

6.4 Comparison of the evolutions with temperature of the compressive fracture ener-gies obtained when applying the methods of Anderberg and Thelandersson [25],Lie and Lin [26], Li and Purkiss [23] and Eurocode 2 [21] to equation (6.6), forthe previously described example. . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.5 Illustration of how the tensile fracture energy changes due to the decrease of thetensile strength, fctT , at an elevated temperature, T , compared to the strengthat the ambient temperature, fct,m. . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.6 Comparison of fracture energy based tensile formulation of the tensile stress-strain relationship of plane concrete to the model suggested by Terro [24]. Anelement of size h = 16.5 mm is considered at temperatures of T = 20C, T =300C and T = 500C, (a), and T = 700C, T = 900C and T = 1100C, (b). . 63

6.7 Combined concrete and interaction stress contributions for a concrete grade C30with steel Grade 500 for a reinforced member with element size h = 100 mm. . 64

6.8 Evolution of the maximum element size, hmaxT , with temperature as defined byequation (6.15) for an example with a reinforced concrete member of grade C30. 66

6.9 Illustration of how the modelling of the combined concrete and interaction stresscontributions at different temperatures yields unrealistic results if the reinforce-ment ratio is too small. The temperature of the steel is assumed to be equal tothat of the concrete. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.10 Evolution of minimum reinforcement ratio for the example of a reinforced mem-ber from Figure 6.9 as a function of the temperature. . . . . . . . . . . . . . . . 67

7.1 Illustration of the dimensions of the considered reinforced slab. . . . . . . . . . . 697.2 Illustration of the reinforced concrete slab considered in this example. . . . . . . 707.3 Temperature profile within the considered slab. . . . . . . . . . . . . . . . . . . 707.4 Overview of the time in the FE-analysis of the considered reinforced slab. . . . . 707.5 Thermal expansion coefficient for concrete, concrete, as a function of the tem-

perature for the considered example of a reinforced concrete slab. . . . . . . . . 727.6 Limits on the maximum and minimum element size, equation (6.15), as functions

of the temperature for the considered example of a reinforced slab. . . . . . . . 727.7 Verification of the requirement to the minimum level of reinforcement (equation

(6.16)) that can be considered for validity of the interaction stress contributionof the tension stiffening for the considered example of a reinforcement slab withelement sizes of h = 73 mm, (a), and h = 129 mm, (b). . . . . . . . . . . . . . . 73

7.8 Material models for compression, (a), and tension, (b), for the reinforced slabwith an element size of h = 129 mm. . . . . . . . . . . . . . . . . . . . . . . . . 74

7.9 Position of the considered element for the post-processing of the contour plotsfrom ABAQUS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

7.10 Output from ABAQUS analysis of stress in the x-direction (S11) at varioustimes, where tFE = 1.00 coresponds to the onset of the temperature load. . . . . 75

7.11 Output from ABAQUS analysis of plastic strain in the x-direction (PE11) atvarious times, where tFE = 1.00 coresponds to the onset of the temperature load 76

7.12 Position of element 2 and an indication of the location of the integration pointswithin it. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

7.13 Evolution of the stress and the plastic strain the x-direction in the integrationpoints of element 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

7.14 Stress in the x-direction through the thickness of the slab at the left fixed endat times tFE = 1.00, (a), and tFE = 2.00, (b), for element configurations ofh = 129 mm and h = 73 mm, respectively. . . . . . . . . . . . . . . . . . . . . . 77

xx

List of Figures

A.1 Idealization of stresses around a single aggregate particle. Reproduced fromMindess et al. [11]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

A.2 Characteristic nominal stress-deformation relation of a loaded specimen in com-pression under displacement controlled test. Reproduced from Mindess et al.[11]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

B.1 Definition of cracking and inelastic strain. Reproduced from the ABAQUS Ver-sion 6.7 Documentation [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

D.1 Compressive concrete model, (a), and the combined concrete and interactionstress contribution in tension, (b), for T = 20C. . . . . . . . . . . . . . . . . . . 95

D.2 Compressive concrete model, (a), and the combined concrete and interactionstress contribution in tension, (b), for T = 100C. . . . . . . . . . . . . . . . . . 96








xxi

List of Figures

xxii

List of Tables

2.1 Input parameters used for *CONCRETE DAMAGED PLASTICITY in ABAQUS. . . . 12

3.1 Overview of the response of of the concrete and the reinforcement in reinforcedmembers upon exposure to a fire. . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.1 Material properties for reinforcing steel Grade 500 using the simplified materialmodel from the CEB-FIB Model Code [16]. . . . . . . . . . . . . . . . . . . . . . 20

4.2 Tensile material parameters for concrete grade C30 with maximum aggregatesize dmax = 32 mm [16]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.3 Parameters used for the compression model from the CEB-FIB Model Code [16]for concrete grade C30. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.4 Compressive fracture energies in N/mm for a reinforced members of height 100mm and 500 mm, fcm = 38 MPa and Gf = 0.095 N/mm, obtained using themethods presented by Vonk [20] (Figure 4.13) and Nakamura and Higai [19](4.10 and 4.11). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.1 Parameters describing decay functions for concrete in the hot phase of a fire aspresented by Hertz [14]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.2 Parameters describing the compressive behaviour of concrete at temperature T ,(a), as defined by Eurocode 2 [21] for siliceous, (b), and calcerous aggregates, (c). 42

5.3 Parameters describing decay functions for concrete in the cold phase of a fire aspresented by Hertz [7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.4 Parameters describing decay functions for reinforcement in the hot phase of afire as presented by Hertz [14]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.5 Parameters describing decay functions for reinforcement in the cold phase of fireas presented by Hertz [14]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6.1 Element sizes obtained corresponding to the compressive fracture eneregies in-herent in the elevated temperature models by Anderberg and Thelandersson [25],hAT , Lie and Lin [26], hLL, Li and Purkiss [23], hLP , and Eurocode 2 [21], hEC ,for the considered example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.1 Parameters at ambient temperature used for the thermal analysis of concrete asrecommended by Teknisk Stbi [29]. . . . . . . . . . . . . . . . . . . . . . . . . . 71

C.1 Output from ABAQUS for a simple shear example without reinforcement attime increments 7, 19, 22 and 410. . . . . . . . . . . . . . . . . . . . . . . . . . . 91

C.2 Output from ABAQUS for a simple shear example with reinforcement at timeincrements 7, 19, 22 and 410. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

xxiii

List of Tables

xxiv

Introduction

1.1 General

Concrete is a widely used construction material and has therefore been subjected to abundantresearch. Much of this is aimed at developing accurate formulations for computer models, whichare becoming an increasingly utilized tool in the design phase of structures. A commonly usednumerical modelling method is the Finite Element (FE) analysis where the considered memberor structure is divided into smaller elements in which the response to a given load is evaluated.The FE-model is evaluated for the tri-axial stress state by a defined yield criterion, where theuniaxial tensile and compressive stress-strain relations determines the evolution of the criterion.It is generally recognized that concrete is subject to localization of stresses due to the formationof cracks which means that continued deformation upon crack initiation localizes in the formedcrack. This means that the uniaxial material models must be defined based on the size of theelements in order to obtain convergence of the model response for different meshes.

(a) Diablo Canyon Nuclear Power Plant in SanLuis Obispo County, California [1].

(b) The fire in the Mont Blanc Tunnel in France/I-taly 1999 [2].

Figure 1.1: Examples of concrete subjected to elevated temperatures.

Concrete subjected to high temperatures are, for example, present in nuclear facilities, such asthe Diablo Canyon Nuclear Power Plant, illustrated in Figure 1.1a, or when fire occurs like thein Mont Blanc Tunnel shown in Figure 1.1b. Hence, concrete failure at elevated temperaturesis potentially strongly related to public safety. In light of previous disasters, for example theChernobyl nuclear disaster (1986) [3] and more recently the Mont Blanc Tunnel fire (1999) [4]that resulted in more than 50 and 41 fatalities, respectively, the understanding and accuratemodelling of the behaviour at elevated temperatures have gained importance. However, in spiteof the potentially large risks at stake, the knowledge base for concrete behaviour at elevatedtemperatures is very limited and the existing models are subject to convergence problems in theFE-analysis, when the same model is evaluated with different mesh sizes. Therefore, it is relevantto take a material model formulation that at ambient temperatures is generally considered toyield converging results and expand this to elevated temperatures.

1

Chapter 1: Introduction

1.2 Localization and Fracture Energy in Tension

Studies at ambient temperatures, among others by van Mier [5] and Pankaj [6], have shownthat as a result of the complex and highly heterogenous nature of concrete, it can be establishedthat the stresses and deformations occurring in concrete localizes in the formation of cracks.A detailed describtion of crack initiation and progression at microlevel is given in appendix A.Continued loading localizes the stresses in the formed cracks, which grow until failure occurs.It is argued by van Mier [5] and Pankaj [6] that since the descending branch of a stress-straincurve for concrete in tension is due to localized cracking (or strain localization), its slope cannotbe a local material property such as the E-modulus. In fact the slope of the softening branch isa function of the specimen size. This can be illustrated by the means of a simple uniaxial tensiletest of a plain concrete member with three strain gauges; at A, B and C, as seen in Figure 1.2a.The gauge at A measures the average strain in the region with a localized crack, whereas Bmeasures the strains in the uncracked part of the specimen. The gauge at C measures the strainover the entire specimen. In this case, gauge A will indicate strain softening, gauge B unloadingand gauge C an average, as shown in Figure 1.2b.

!

ACB

P

A

B

C

(a) Uniaxial tension test!

ACB

P

A

B

C

(b) Load-displacement curve for gauge A, B and C.

Figure 1.2: Uniaxial tension test of pure concrete element with strain gauges at A, B and C,(a), and the corresponding load-displacement diagrams, (b). Reproduced from vanMier [5].

The size of the elements in a FE-model will have an effect on the material definition because theload-displacement diagram for concrete depends on where the gauge is placed and the size of it.As a result, the stress-strain relation cannot be taken as a material property for concrete.

fctm

!u!p1

"1

fctm

!u!p1

"

"

!

"

!

1

Gf

(a) Stress-displacement diagram in localizedregion

fctm

!u!p1

"1

fctm

!u!p1

"

"

!

"

!

1

Gf

(b) Stress-plastic displacement diagram in lo-calized region

Figure 1.3: Stress-displacement curve in localized region, (a), to illustrate construction of theload-plastic displacement diagram and of the fracture energy as the area under thecurve, (b). Reproduced from Pankaj [6].

2


It has, however, been accepted that the stress-displacement curve can be taken as a materialinvariant, see Figure 1.3a. Here, the stress-displacement relation is plotted for a concrete speci-men with the tensile strength, fctm, and the ultimate displacement, u. The considered stress,1, corresponds to the plastic displacement, p1 . If plotting the plastic displacement, p, as afunction of the stress, , then the fracture energy, Gf , is defined as the area under the curve, asindicated in Figure 1.3b. The fracture energy is the specific energy required for fracture growthin an infinitely large specimen and, hence, the energy required to form a new fully separatedcrack surface. In FE-modelling of concrete in tension, Gf is taken as the material property whichin turn implies that the decending branch of the stress-strain curve is a function of element size.It has been shown by van Mier [5] that modelling based on Gf leads to mesh independentresults.

1.3 Localization and Fracture Energy in Compression

The considerations described above are also valid for concrete in compression, as compressivecrushing, like tensile cracking, is occuring in a localized region. However, the compressivefracture energy based models are rarely implemented, in part because very few have investigatedor discussed the compressive fracture energy, Gc. It is, for example, not included in any of thecurrent codes.

1.4 Novelties and Milestones of the Thesis

Currently, reinforced concrete models are not fracture energy based at elevated temperatures.In fact, even at ambient temperature, the existing compressive fracture energy models are rarelyimplemented. As the underlaying assumption for structural modelling is that the modelled ma-terial behaviour predicts the actual behaviour, it is evident that if this is not the case, the outputof an analysis will have little or no value. Therefore, the novelty of the current work lies in in-vestigating the existing fracture energy based models, especially in compression, and expandingthese to elevated temperatures. While doing so, it is possibile to examine the evolution of boththe compressive and the tensile fracture energy with an increase in temperature.

Further, the limits of application imposed by the fracture energy based models at ambient tem-perature are reviewed. As these are not currently defined, formulations of the limits are madeherein. This makes it possible to investigate how these limits evolve as functions of the temper-ature, which is crucial to keep in mind, to ensure that they are not violated when the elevatedtemperature model is applied.

Prior to extending a material model formulation to elevated temperatures, it is essential tohave knowledge about both the modelling of the behaviour of reinforced concrete at ambienttemperature and the physiochemical reactions caused by the temperature variation. Herein,normal strength concrete is considered and for brevity, the elevated temperature caused by afire will be simplified into a hot and a cold phase. A typical fire course consists of a heatingphase to a certain temperature peak, followed by a cooling phase until the ambient conditionsare reached again as schematically illustrated in time-temperature plot in Figure 1.4a. Figure1.4b shows how the hot phase refers to the reinforced concrete behaviour during exposure tothe maximum temperature of the fire and the cold phase refers to the residual behaviour afterexposure. The effect that the temperature elevation has on the strength of a considered material,for example in the hot phase, is illustrated in Figure 1.4c, where three possible decay curvesare given; one where the strength at elevated temperatures remains as at ambient, one where

3


it decays rapidly and one intermediate. The rate of the decay depends on the physiochemicalresponse of the considered material to the temperature elevation.

T

phase

hot phase

cooling

T

t

cold

/

No decay of strength with T

strength with TRapid decay of

Tt

heating

fcT

fcm

(a) Schematic temperature vari-ation in a typical fire course.

T

phase

hot phase

cooling

T

t

cold

/



Tt

heating

fcT

fcm

(b) Temperatures in the hot andthe cold phases of a fire.

T

phase

hot phase

cooling

T

t

cold

/



Tt

heating

fcT

fcm

(c) Schematic stength ratios as func-tions of the temperature.

Figure 1.4: Illustrations of a typical temperature variation caused by a fire, (a), of the tem-peratures in the hot and cold phases of a fire, (b), and the strength ratio as afunction of the temperature, (c).

As the hot phase and cold phase will be referd to in the following, they are scematically illustratedin Figure 1.5.

Hot Phase Cold Phase

Properties of materials when astructure or a member is exposedto elevated temperatures

Residual material properties of astructure or a member after ex-posure to elevated temperatures

Figure 1.5: Descriptions of hot and cold phases of a fire, as defined by Hertz [7].

As a result of the above, this thesis comprises the following:

A discussion on the damaged plasticity formulation in ABAQUS used for the multiaxialanalysis of concrete (Chapter 2).

A literary study of the physiochemical response of reinforced concrete exposed to fire(Chapter 3).

A study of the existing uniaxial fracture energy based behaviour models for the ambientcondition including formulations of the on the limits of application (Chapter 4). Further, achoice of the material model formulations to expand to elevated temperatures is made andnumerical benchmark test are conducted to ensure correlation of the ABAQUS analysiswith the expected response.

A literary study of the existing models for concrete behaviour at elevated temperatures(Chapter 5). This includes the decay of strengths and a discussion on the formulations ofthe stress-strain relationship.

Formulations of fracture energy based uniaxial material models for reinforced concrete atelevated temperatures (Chapter 6). This includes an investigation of the modifications ofthe compressive and tensile fracture energies caused by temperature elevation. Further,the evolution of the limits of application is studied.

Numerical examples where the fracture energy based elevated temperature models areimplemented (Chapter 7).

4


Concluesion and recommendation for future work (Chapter 8).

5


6

Finite Element modelling ofMultiaxial Behaviour ofReinforced Concrete

2.1 Introduction

In FE-analysis the triaxial states of stress are evaluated and a yield criterion is used to de-termine whether the deformation occuring in an element should be considered to be elastic orplastic. It is generally accepted that concrete is a pressure sensitive material, which causes forconical yield criterion in three dimensions. However, a variety of criterions exists, some morecomplicated than others. Typically, a criterion that is very specific depends on several param-eters and as each parameter to be defined is associated with a degree of uncertianty, this islikely to accumulate. Herein, the FE-code ABAQUS Version 6.7 [8] is used for all finite elementcomputations and therefore this chapter commences with a description of how concrete crackingcan be considered in FE-computations, followed by a description of the model ABAQUS utilizes.

It is possible for the yield surface to change in size and shape as the plastic deformations evolve.This is a necessity in order to account for hardening or softening behaviour in a model asillustrated in Figure 2.1, where the uniaxial stress-strain relation is given in Figure 2.1a and theyield surface of the initial yield point as well as a subsequent indicated yield point, is given inin the deviatoric plane in Figure 2.1b.

Concrete has distinct strength assymetry, meaning that the uniaxial tensile and compressivebehaviours differ and, even at the ambient condition, there is still a great level of uncertaintyassociated with material modelling of the uniaxial behaviours. The uniaxial tensile and com-pressive behaviours of reinforced concrete will therefore be discussed in chapter 4.

A brief discription of how to define the uniaxial input parameters in ABAQUS is provided inappendix B, along with explaniations of some of the ABAQUS functions used for the FE-models.

The derivation of the FE-equations will not be given and a detailed description of concreteplasticity is also omitted as both are out of the current scope.

7

Chapter 2: Finite Element modelling of Multiaxial Behaviour of Reinforced Concrete

!3!2

!1

!

"

Initial yield surface

Subsequent yield surface

(a) Stress-strain relation

!3!2

!1

!

"

Initial yield surface

Subsequent yield surface

(b) Deviatoric plane

Figure 2.1: Stress-strain relation for material undergoing hardening post-peak, (a), and initialand subsequent yield surfaces in deviatoric plane, (b). Reproduced from Pankaj[9].

2.2 FE-Modelling of Concrete Cracking

Generally, there exist two distinctly different ways of modelling cracking in FE analysis; thediscrete and the smeared approach. The discrete approach models cracking as seperation ofelements, whereas the smeared approach models the solid cracked continuum, as described byPankaj [6].

In the discrete crack approach, Figure 2.2a, the nodes are separated during propagation of acrack and each crack is therefore considered separately. The smeared crack model, illustratedin Figure 2.2b, is a damage or plasticity model where the damage zone coincides with thedimensions of the elements. The cracking of the concrete is therefore modelled by adjusting thematerial properties in the regions of cracking or strain localisation. This can be adopted as thecracking is assumed to consist of a set of densely populated or smeared cracks and is simulatedby altering the constitutive relation in the damaged region.

(a) Discrete crack model (b) Smeared crack model

Figure 2.2: Discrete, (a), and smeared crack, (b), approaches. Reproduced from Pankaj [6].

It is not possible to determine which type of crack modelling method that is best suited withoutconsidering the context it is to be employed in. For example, the discrete crack approach isdifficult to use on large scale arbitrary structures as it requires a very fine mesh because theseparation takes place around the elements. This can be circumvented by redefining the originalmesh, but either way, the discrete crack approach imposes a large CPU-demand. This meansthat the model will demand a lot of computer power due to the large number of computations re-

8


quired, and this may not be cost-effective when considering the level of accuracy that the modelpredicts. For large arbitrary structures it is therefore often better suited to use the smearedcrack approach where it is possible to obtain mesh insensitive results, granted that the localmaterial softening law is made mesh dependent based on the fracture energy.

The concrete damage plasticity model in ABAQUS [8] is a smeared crack model in the sense thatit does not track individual macro cracks. Constitutive calculations are performed independentlyat each integration point of the FE-model and the presence of cracks enters the calculations byaffecting the stress and material stiffness associated with the integration point.

2.3 Concrete Model in ABAQUS

In ABAQUS [8] it is assumed that the main two failure mechanisms are tensile cracking andcompressive crushing. When using the *CONCRETE DAMAGED PLASTICITY option the yield crite-rion is defined and it is required to define the suboptions *CONCRETE TENSION STIFFENING and*CONCRETE COMPRESSION HARDENING and through these, the evolution of the yield surface withcontinued plastic loading.

In uniaxial tension the stress-strain relation is assumed to be linear until the failure stress, t0,which corresponds to the onset of macrocracking, is reached. This is most often followed bysoftening which induces strain localization. In uniaxial compression it is also assumed that theresponse is linear until the initial yield stress, c0, after which a plastic regime follows, typicallycharacterized by strain hardening until the ultimate stress, cu, and thereafter softening. Thedefinition of the tension stiffening and compressive behaviour in ABAQUS is described in ap-pendix B.

The damage model in ABAQUS [8] is based on the assumption that the uniaxial stress-strainrelations can be converted into stress-equivalent plastic strain curves and this is automaticallydone from the user-provided inelastic strain data. The effective tensile and compressive cohesionstresses are then computed to determine the current state of the yield surface that is used toanalyze multiaxial load cases.

2.3.1 Yield Surface Definition

A yield surface is a surface in the stress space enclosing the volume of the elastic region. Thismeans that the state of stress inside the surface is elastic, while stress states on the surface havereached the yield point and have become plastic. Further deformation causes the stress stateto remain on the surface, as the states that lie outside are non-permissible in rate-independentplasticity.

Several formulations of yield surface criterions exist and the Drucker-Prager yield criterion [9]is used for concrete in ABAQUS [8], because it makes it possible to determine failure both bynormal and shear stress. It is a pressure dependent criterion based on the two stress invariantsof the effective stress tensor; the hydrostatic pressure, p, and the Mises equivalent stress, q.

It is possible for the user to somewhat determine the shape of the yield surface, by the inputparameter Kc in the *CONCRETE DAMAGED PLASTICITY function. Kc is the ratio of the secondstress invariant on the tensile median to that on the compressive median at initial yield for any

9


given value of the pressure invariant, p, such that the maximum principal stress is negative,max < 0. It must be fulfilled that 0.5 < Kc 1.0 and the factor is per default 2/3, making theyield criterion approach Rankines formulation [9].

2

!3

!1

= 2/3Kc

= 1.0Kc

!1

!3

!2

!

(a) Yield surface in deviatoric plane

2

!3

!1

= 2/3Kc

= 1.0Kc

!1

!3

!2

!

(b) Yield surface in three dimen-sions for Kc = 1.0

Figure 2.3: Drucker-Prager yield criteria in the deviatoric plane for Kc = 2/3 and Kc = 1.0,(a), and in three dimensions forKc = 1, 0, (b). Reproduced from ABAQUS Version6.7 Documentation [8] and Pankaj [9], respectively.

The difference of the yield surfaces in the deviatoric plane, i.e. where 1 + 2 + 3 = constant,for Kc = 2/3 and Kc = 1.0 is shown in Figure 2.3a. For comparison, the Rankine criterion isusually triangular whereas the Drucker-Prager criterion is circular in the deviatoric plane. Here,Kc is set to unity, which corresponds to using the traditional Drucker-Prager yield criterion,where the yield surface is cone shaped in the three-dimensional space as illustrated in Figure 2.3b.

Yield Function in ABAQUS

In order to account for the different evolution of strength under tension and compression, Fenvesmodification of Lubliners yield function is used in ABAQUS [8]:

F =1

1 (q 3p+ (pl)max max) c(plc ) = 0 (2.1a)

where

=(b0/c0) 12(b0/c0) 1 for 0 0.5 (2.1b)

=c(plc )t(plt )

(1 ) (1 + ) (2.1c)

=3(1Kc)2Kc 1 (2.1d)

In this, max is the maximum principal effective stress and b0/c0 is the user specified ratio ofthe equibiaxial compressive yield stress and the initial uniaxial compressive yield stress, whichper default is set to 1.16.

It is seen from the expressions above, (2.1a-2.1d), that the evolution of the yield surface iscontrolled by the hardening variables plt and plc . The tensile and compressive stresses corre-sponding to these are computed from the input given by the tension stiffening and compression

10


hardening definitions.

The yield surface in plane stress is illustrated in Figure 2.4, where the enclosed area of the figurerepresents the elastic states of stress. If a given member is loaded in tension in both the 1 andthe 2 directions, the stress state is in the first quadrant of the coordinate system. Likewise, ifit is loaded in compression in both directions, the stress state is in the third quadrant. For loadcases where a combination of tensile and compressive forces are applied (e.g. shear), the stressstates will be either in the second or the fourth quadrant.

Biaxial compression

^2

!^1

c0!b0

!(b0! ),

!t 0

TENSION

TENSION

COMPRESSION

TENSION

COMPRESSION

COMPRESSION

TENSION

COMPRESSION

1 ! "

1!c0

q p"3 !^

2 =( ! + # )

1 ! "

1q p"3 !

c0=)( !

1 ! "

1!^

1# )+p"3q( ! !c0=

Uniaxial tension

Biaxial tension

!

Uniaxial compression

Figure 2.4: Yield surface in plane stress. Reproduced from ABAQUS Version 6.7 Documen-tation [8].

Flow Potential Function in ABAQUS

Infinitely small strain increments can be divided into an elastic and plastic part, d = dp+de,and experimental results suggest that the plastic strain increment is normal to the yield surface[9]. Sometimes plastic strain increments are assumed to be normal to a surface other than theyield surface and this surface is referred to as the plastic potential and is illustrated in Figure 2.5.

The flow potential function, G(), used in ABAQUS [8] is the Drucker-Prager hyperbolic func-tion given by:

G() =($t0 tan)2 + q2 p tan (2.2)

Here, is the dilatation angle, t0 is the uniaxial tensile stress at failure from the tension stiffen-ing definition and $ is the eccentricity that defines the rate at which the function approaches the

11


asymptote. Both and $ are given as input parameters in the *CONCRETE DAMAGED PLASTICITYfunction.

d !p

= 0)("GPlastic potential

= 0)(" ,YFYield surface

2"

3" 1"

Figure 2.5: Illustration of the plastic potential in relation to a yield surface. Reproduced fromPankaj [9].

Input Parameters

The flow potential, yield surface and viscosity parameters for the concrete damaged plasticitymodel are, as described, defined through the *CONCRETE DAMAGED PLASTICITY input.

*CONCRETE DAMAGED PLASTICITY, $, b0/c0, Kc,

Herein, the parameters in Table 2.1 are used for all FE-computations.

Table 2.1: Input parameters used for *CONCRETE DAMAGED PLASTICITY in ABAQUS.

$ b0/c0 Kc

31.0 0.1 1.16 1.0 0.0

The dilation angle, , controls the amount of plastic volumetric strain developed during plasticshearing and is assumed constant during plastic yielding. Typically, for normal strength con-crete a dilation angle of = 31 is used, and this is therefore also chosen herein.

The flow potential eccentricity is per default $ = 0.1, meaning that the material has almost thesame dilatation angle over a wide range of configuring pressure stress values.

The ratio b0/c0 is set to the default value of 1.16 and it is chosen that Kc = 1.0 so that theyield surface has a perfect cone shape in the three dimensional space, as previously described.

The viscosity parameter, , is used for the visco-plastic regularization of the concrete constitutiveequations. The default value is 0.0 which means that a rate-independent analysis is carriedout.

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Response of Reinforced Concreteto Fire Exposure

3.1 Introduction

In relation to temperature, a fire typically means an increase to high levels followed by a decayuntil the amient conditions are reached again. The rate at which the elevation and decay is oc-curring can vary considerably and depends on a number of factors, such as the type and amountof fuel and the availability of oxygen. Herein, it is mainly the exposure to the temperatureelevation that is of concern and a detailed definition of the fire phases will therefore not be given.

The changes that reinforced concrete members undergo during fires are occuring at a micro-leveland are associated with the separate responses of the concrete and the steel reinforcement. As aresult of the different chemical composition of the two components, the response at micro-levelcauses different thermal properties at macro-level. Therefore internal stresses are generated,resulting in formation of cracks and potentially failure of the bond between the concrete andthe reinforcement. This effectively means that the material properties of the concrete and thesteel are reduced by the physiochemicak processes induced by temperature elevation, as Fletcheret al. [10] describes.

In this chapter the chemical and physical responses to a fire of reinforced concrete members toa fire are described. The effects of a fire on the concrete and on the reinforcement are explainedseparately and the choice of analysis for the fracture energy based material model at elevatedtemperatures (described in chapter 6) is discussed.

3.2 Chemical and Physical Effects of Fire

Concrete is, as described in appendix A, a heterogeneous material consisting of cement paste,aggregate and, for reinforced concrete, steel. The response to thermal exposure of each ofthese components is different in itself, and the behaviour at elevated temperatures is thereforeneither easy to define nor to model. This difficulty arises from the fact that the difference inresponse of the components also affects the overall response. Fletcher et al. [10] explaines how,for example, the thermal response of the aggregate may be straight forward, but in context itcan be substantially different.

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Chapter 3: Response of Reinforced Concrete to Fire Exposure

3.2.1 Chemophysical Response of Concrete to Fire

According to Mindess et al. [11] the cement paste consists of a range of different chemical com-pounds, which all react differently to exposure to high temperatures. A detailed description ofthe chemical composition of the cement paste is beyond the current scope, and only the mostimportant compounds in the context of elevated temperatures will be considered; namely hy-drated calcium silicate (commonly refered to as C-S-H) which typically makes up 50-60% of thepaste and calcium hydroxide (Ca(OH)2 but herein refered to as CH in accordance with commonterminology), which usually accounts for 20-25% of the solid paste volume. Finally, the watercontents of the concrete is of importance for the deterioration process at elevated temperatures.The water contents in a given concrete member depends on the w/c (water/cement) ratio usedin the mixture of the concrete and the conditions of the curing process. The water is held inthe pores of the concrete and generally two different kinds of pores are defined; capillary poresand gel pores. The two types are distinguished by their sizes, the size of the capillary poresvary from 10-104 nm, whereas the gel pores are less than or equal to 10 nm in size. The waterheld in the two kind of pores is also differentiated; the water in the capillary pores is consideredto be evaporable water, whereas the water held in the gel pores is regarded as a part of the C-S-H.

Due to of the complexity of the concrete composition, an exact temperature for a given chemo-physical change in any concrete cannot be given. However, the general response of concrete toelevated temperatures has described by various researchers, for example by Fletcher et al. [10]and by Hertz [7].

and to decompose

Calcium hydroxid (CH)cement paste turn into a glass phase

Feldspar melts and the minerals of the

400 600

575

to vaporiseWater starts

100 140

800

C][T

Ta

1150

150

Chemically bound water inhydrated calcium silicate (C!S!H)

initiates vaporisationincrease in volume and

Aggregates starts to

begins to dehydrate

Figure 3.1: Schematic overview of deterioration of plain concrete as the temperature is in-creased. Based on Fletcher et al. [10] and Hertz [7].

In Figure 3.1 an overview of the most important chemical processes occurring in concrete dueto temperature rise is given and in the following these processes are elaborated upon:

When concrete reaches 100-140C, the water begins to evaporate, usually causing a buildupof pressure within the concrete.

Once the temperature reaches 150C the chemically bound water is released from the C-S-H and hence the cement matrix begins to dehydrate and shrink. This process has a localpeak at 270C and internal stresses arise. From above 300C these stresses will result inmicro cracking and hence irreversible deformations are initiated.

At about 400-600C, the CH (chemically denoted Ca(OH)2) in the cement begins to dehy-drate, generating calcium oxide (CaO) and more vapor (H2O). This dehydration processcauses the strength to decrease significantly.

At 575-800C strength loss due chemical changes of the aggregate is initiated. For quartz-based aggregates a mineral transformation at 575C causes the aggregates to increase in

14


volume and for limestone aggregates decomposition is commenced at 800C; most oftencausing the concrete to be crumbled to gravel.

Above 1150C feldspar melts and the remaining minerals of the cement paste turn into aglass phase yielding high brittleness and almost no strength.

It must be noted that not only the composition of the concrete has an effect on the response, butalso environmental factors influence the chemical processes occurring at elevated temperatures.The above process overview is for unsealed concrete, whereas the behaviour of externally moistsealed concretes at temperatures above 100C differs significantly, as Khoury [12] explains. Thisis caused by the fact that the chemophysical response in unsealed conditions is dominated bythe loss of various kinds of water, whereas the process is dominated by hydrothermal chemicalreactions in sealed concrete.

Further, if the concrete is loaded in compression during heating, the loading compacts theconcrete and inhibits the development of cracks. Khoury [12] describes how this can decreasethe reduction of both the elastic modulus and the compressive strength due to temperatureeffects significantly. Hertz [7] further explains that this is due to the fact that the compressivestresses in the concrete must be unloaded before any tensile stresses can be established, andhence before microcracking can be initiated. The strain contributions is called the load inducedthermal strains, LITS, and they are only occurring during first the heating cycle.

C!S!H

CaO

CaO

2OCaO + H

C!S!H

(a) Hot phase of a fire, minimum 400-600C

C!S!H

CaO

CaO

2OCaO + H

C!S!H

(b) Cooling phase of a fire

Figure 3.2: Dehydration of calcium hydroxide to calcium oxide and evaporable water causingshrinking, (a), and rehydration upon cooling of calcium oxide to calcium hydroxideresulting in increased cracking, (b). Based on Hertz [7].

It is necessary to emphasize that an important chemical reaction occurs after the exposureto elevated temperatures, i.e. in the cooling phase of a fire. Figure 3.2 shows, and Hertz [7]describes, how the calcium oxide expands during the cooling phase, as it absorbs water from theambient air. This process can reduce the compressive strength by another 20% after exposureto elevated temperatures and the importance of considering the concrete strength in the coldphase of a fire (for design purposes) must therefore be stressed.

3.2.2 Chemophysical Response of Reinforcing Steel to Fire

Nielsen [13] explains how iron is a crystalline solid, in which plastic deformations are causedby mechanical distortions of the crystal lattice. Essentially, steel is iron with small quantities

15

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of carbon added to it and the carbon atoms creates small irregularities in the lattice. The ir-regularities inhibits the movements of the lattice by acting as anchors for the dislocations, andthereby increasing the strength of the material but also making it more brittle.

As the temperature is related to the movement of the atoms, a temperature increase reducesthe external energy necessary to move the dislocations. Hertz [14] describes how this effectivelymeans that the yield stress (or the 0.2% stress) will decrease with an increase in temperature.Moreover, within the first 200-300C, a temperature increase also means that more new dislo-cations can be formed if stress is applied. Cold-working of steel utilizes this effect to increasethe ultimate strength.

Two different types of reinforcement bars generally exists; hot-rolled and cold-worked. The dif-ference between the two is, as Nielsen [13] explains, that cold-worked reinforcement has beentwisted, stretched or a combination of the two to obtain a more chaotic system of dislocationsand many sources for formations of new dislocations. Cold-working will therefore increase theyield strength of the steel at ambient temperatures and the material becomes less ductile. Theeffect of cold-working is permanently lost if the reinforcement is exposed to temperatures beyond400C as this is above the temperature of recrystallization for steel.

It should be noted that also pre-stressed reinforcement exists, however this is out of the currentscope and will therefore not be discussed.

400

C][T

300250 700

to about 20% of design valueLoad!bearing capacity reduced

low carbon contentsBlue brittleness if

expansion than concreteSignificantly larger thermal

Ta

Figure 3.3: Schematic overview of deterioration of reinforcement as the temperature is in-creased. Based on Fletcher et al. [10].

The performance of steel during a fire is generally better understood than that of concrete andhas, for example, been described by Fletcher et al. [10] and Hertz [14]. An overview of the mostsignificant processes occurring in reinforcement as a result of increase in temperature is given inFigure 3.3 and a brief summary is given below:

At temperatures of 200-300C, steels with low carbon contents show blue brittleness. Thesteel takes a blue color and the strength of the material is increased, but the materialalso loses its ductility and becomes very brittle. To avoid this, it is therefore generallyrecommended that the reinforcement is protected from temperatures higher than 250-300C.

Up until about 400C the thermal expansions of steel and concrete are fairly similar, butat higher temperatures the expansion of steel is significantly larger than the expansion ofconcrete. This causes increased interface stresses and hence a great risk of bond failure.

At temperatures in the order of 700C the load-bearing capacity of steel reinforcementwill be reduced to about 20% of its design value.

16


3.3 Typical Failures of Reinforced Members

It is evident by the discussion of response to temperature changes that the concrete remainsstrong at high temperatures where the steel is weak and that the properties of the steel areregained upon cooling whereas the concrete strength is further reduced. Hertz [7] suggests todefine a hot and a cold phase of a fire for design purposes, which both must be investigated forconcrete members to determine the which of the states that is most likely to induce failure.

Fletcher et al. [10] states that structural failure in the hot phase of a fire often only occursdue to bond failure or when the effective tensile strength of any of the steel reinforcement islost. However, concrete has low thermal conductivity and as a result the steel reinforcement iseffectively protected from exposure to the highest levels of temperature. It is therefore crucialthat the concrete keeps its integrity which can be lost by two mechanisms; either by extensivecracking of the outer layer of the concrete or by spalling of the concrete surface. Khoury [12]explains that spalling is a phenomenon involving ejections of chunks of concrete from the surfaceof the material and is generated by the thermal stresses and the increased pore water-pressurein a concrete member. It may occur under a variety of conditions where strong temperaturegradients are present. The presence of reinforcement enhances the risk of spalling, as it has alarge effect on the transport of water within a member because the water is forced around thebars, increasing the pore pressure in some regions of the concrete. However, as normal strengthconcrete is considered herein and it is assumed that microsilica is not used to densify it, it issafe to ignore spalling, provided that the moisture content is low [15].

It is further noted by Fletcher et al. [10] that compressive failures often are associated withtemperature-related loss of compressive strength of the concrete in the compressive zone. Thistype of failure is therefore most likely to arise in the cold phase of a fire.

3.4 Choice of Analysis Type

For concrete structures it is necessary to analyse the response of the entire exposed member,as the structural effectiveness of a member is not lost until it reaches the critical temperaturewhere the material strength is deteriorated excessively. This due is to the fact that concrete haslow thermal conductivity and therefore strong temperature gradients are generally generatedwithin fire exposed concrete.

Khoury [12] argues that it is necessary to perform a thermal analysis that computes the tem-perature distribution within the considered member for all types of analysis involving exposureto elevated temperatures. In a simplified limit state analysis the 500C isotherm, obtained bythe thermal analysis, is used to reduce the cross-section. Hereafter the load-bearing capacity iscarried out with the mechanical properties at ambient temperatures. A more accurate methodis the thermomechnical finite element analysis, where the thermal analysis is carried out forthe entire duration of the fire and then the a mechanical analysis is performed. However, asthe hydral problem of the deterioration process is simplified out of this analysis type, an exactprediction for all types of structures cannot be obtained. Therefore, a comprehensive thermo-hydromechanical finite element analysis has been developed, that includes a thermal, a hydraland a mechanical analysis in a fully integrated and interactive model.

The use of the thermomechanical finite element analysis does, according to Khoury [12], predictthe response to heating and loading with reasonable accuracy for the type of concrete members

17


considered herein. The implication of a fire on a given member will therefore be modelled by thedeterioration of the material properties caused by the temperature variation. This means thatthe physiochemical changes in the concrete will be simplified into deterioration of macroscopicmechanical properties. As a results, some effects cannot be considered by the model. Forexample, the determination of explosive spalling is governed by the pore pressure and becausethis is not computed, spalling will not be detected. H owever, as specified in section 3.3, it isassumed that spalling safely can be neglected.

3.5 Overview of Concepts Involved in the Response of Re-inforced Concrete to a Fire

In Table 3.1 an overview of some of the physical concepts involved when reinforced concreteis exposed to a fire are provided. The properties and the response of the concrete and of thereinforcement steel is considered seperately to emphasise the significant difference between thetwo, which contributes to the deterioation upon exposure.

Table 3.1: Overview of the response of of the concrete and the reinforcement in reinforcedmembers upon exposure to a fire.

Concrete Reinforcement

Conductivity Low High

Temperaturegradient within thematerial

Must be considered due thelow conductivity

Can be ignored because of therelatively high conductivity

Load type typicallycausing failure Compression Tension and flexure

Phase of fire wherefa

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Reinforced Concrete at Elevated Temperatures