MATERIAL MODELLING OF REINFORCED CONCRETE...

MATERIAL MODELLING OF REINFORCED CONCRETE ATELEVATED TEMPERATURES

Master ThesisFebruary 2011

Fire Safety, Section for Building Design,Department of Civil Engineering, the Technical University of Denmark

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

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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 di!erent 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 sti!en-ing e!ect 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 sti!ening 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 e!ectsof 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 e!ecton 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 about500!C, 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 di!erent 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 sta! 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. José 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, Cristián Maluk, Nicolas Bal, Ste!en 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 e!ects on both the project at hand and on future choice ofoccupation.

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

Last but not least, gratitude is directed to the OTICON Foundation, Reinholdt W. Jorck’sFoundation, KAB’s 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 E!ects 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 Sti!ening Model as per the CEB-FIB Model Code . . . . . . . . 214.3.2 Tension Sti!ening Model by Cervenka et al. . . . . . . . . . . . . . . . . . 224.3.3 Tension Sti!ening 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 E!ect 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 Sti!ening 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 E!ective 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 sti!ening.

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].E"

p Slope of the descending branch inthe 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 energy

at ambient temperature.GfT Tensile fracture energy at elevated

temperatures.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 E!ective 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 sti!ness.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 coe"cient in

ABAQUS [8]."concrete Thermal expansion coe"cient of

concrete."steel Thermal expansion coe"cient 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.%in

c Inelastic strain in ABAQUS [8].%pl

c , %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 sti!ening as per the CEB-FIB Model Code [16].

%s,m Strain in the reinforcement withtension sti!ening as defined in theCEB-FIB Model Code [16].

%!T Instantanious stress-related strain%t Tensile strain.%ck

t Cracked strain in ABAQUS [8].%th Unrestrained thermal strain.%p

x 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.µ Parameter for visco-plastic regu-

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

( Poisson’s 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 e!ective 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 e!ective 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 tensionsti!ening 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 sti!en-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 di!erent elementside lengths, h [mm]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.6 The e!ect 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 di!erent di!erent re-inforced areas As [mm2] using the tension sti!ening 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 sti!ening 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 E!ect of changing the fraction of the ultimate tensile strength applied on theinteraction contribution of the tension sti!ening 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 di!erent 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 sti!ening and the compression model in ABAQUS. . . . . . . . . . . 38

4.20 The tension sti!ening 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 sti!ening 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 = 20!C and T = 300!C, (a),and T = 500!C and T = 700!C, (b). The ultimate stress is normalized by theultimate stress at ambient temperatures. . . . . . . . . . . . . . . . . . . . . . . 48

5.7 Illustration of the di!erence 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 = 20!C and T = 300!C, (a), andT = 500!C and T = 700!C, (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.9 Tensile stress-strain relationship as suggested by Terro [24] for concrete at tem-peratures of T = 20!C, T = 300!C, T = 500!C and T = 700!C. . . . . . . . . . 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 = 20!C, T =300!C and T = 500!C, (a), and T = 700!C, T = 900!C and T = 1100!C, (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 di!erent 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 coe"cient 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 sti!ening 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 = 20!C. . . . . . . . . . . . . . . . . . . 95

D.2 Compressive concrete model, (a), and the combined concrete and interactionstress contribution in tension, (b), for T = 100!C. . . . . . . . . . . . . . . . . . 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 Ståbi [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 di!erent 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 di!erent 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 e!ect 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, #p

1 . 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 e!ect 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 di!er 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 di!erent 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 isdi"cult 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-e!ective 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 bya!ecting the stress and material sti!ness 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 sti!ening 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 e!ective 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 e!ective 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 Rankine’s 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 for Kc = 1, 0, (b). Reproduced from ABAQUS Version6.7 Documentation [8] and Pankaj [9], respectively.

The di!erence 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 di!erent evolution of strength under tension and compression, Fenve’smodification of Lubliner’s yield function is used in ABAQUS [8]:

F =1

1" "

!q̄ " 3"p̄ + -

"%pl

##ˆ̄!max$ " .#"ˆ̄!max$

$" !̂c(%pl

c ) = 0 (2.1a)

where

" =(!b0/!c0)" 12(!b0/!c0)" 1

for 0 ! " ! 0.5 (2.1b)

- =!̄c(%pl

c )!̄t(%pl

t )(1" ")" (1 + ") (2.1c)

. =3(1"Kc)2Kc " 1

(2.1d)

In this, ˆ̄!max is the maximum principal e!ective 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 %pl

t and %plc . The tensile and compressive stresses corre-

sponding to these are computed from the input given by the tension sti!ening 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% = d%p +d%e,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 + q̄2 " p̄ tan, (2.2)

Here, , is the dilatation angle, !t0 is the uniaxial tensile stress at failure from the tension sti!en-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, µ

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 µ

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, µ, 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.

12

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 di!erent chemical composition of the two components, the response at micro-levelcauses di!erent 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 e!ectively 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 e!ects 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 E!ects 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 di!erent in itself, and the behaviour at elevated temperatures is thereforeneither easy to define nor to model. This di"culty arises from the fact that the di!erence inresponse of the components also a!ects 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 di!erent.

13

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 di!erent chemical com-pounds, which all react di!erently 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 di!erent 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 di!erentiated; 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 100-140!C, the water begins to evaporate, usually causing a buildupof pressure within the concrete.

• Once the temperature reaches 150!C 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 270!C and internal stresses arise. From above 300!C these stresses will result inmicro cracking and hence irreversible deformations are initiated.

• At about 400-600!C, 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 575-800!C strength loss due chemical changes of the aggregate is initiated. For quartz-based aggregates a mineral transformation at 575!C causes the aggregates to increase in

14


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

• Above 1150!C 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 e!ect 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 100!C di!ers 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 temperaturee!ects 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 400-600!C

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


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 e!ectivelymeans that the yield stress (or the 0.2% stress) will decrease with an increase in temperature.Moreover, within the first 200-300!C, a temperature increase also means that more new dislo-cations can be formed if stress is applied. Cold-working of steel utilizes this e!ect to increasethe ultimate strength.

Two di!erent 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. Thee!ect of cold-working is permanently lost if the reinforcement is exposed to temperatures beyond400!C 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 200-300!C, 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 250-300!C.

• Up until about 400!C 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 700!C 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 e!ective tensile strength of any of the steel reinforcement islost. However, concrete has low thermal conductivity and as a result the steel reinforcement ise!ectively 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 e!ect 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 e!ectiveness 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 500!C 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 e!ects 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 di!erence 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 wherefailure is mostlikely to occur

In the cold phase of a fireas the strength is further de-creased after exposure to ele-vated temperatures

In the hot phase of a fire,as the strength is signifi-cantly reduced when exposedto high temperature levelsand the strength is regainedupon cooling

Main mechanismscausing failure ofreinforced membersat hightemperatures

Spalling and cracking result-ing in exposure of the rein-forcement

Strength reduction as a resultof exposure to high temper-tures

18

Modelling of Uniaxial behaviourof Reinforced Concrete atAmbient Temperature

4.1 Introduction

As described in chapter 2, the uniaxial behaviour defines the evolution of the yield criterion ina FE-analysis. However, the uniaxial behaviour of plain and reinforced concrete is associatedwith a great degree of uncertainty at ambient temperatures.

At ambient conditions, it is generally recognized that in order to obtain mesh independent re-sults of models of reinforced concrete in FE-analysis, a fracture energy based material modelmust be adopted. In tension, such models are widely used and in most FE-codes, ABAQUSfor example, it is possible to define the tensile post-peak behaviour in three ways; through anelement size dependent stress-strain relation, through a stress-displacement formulation or bygiving the tensile fracture energy and letting ABAQUS define the behaviour. In compressionthe fracture energy based behaviour models are less used and the compressive fracture energyis, for example, not discussed in any current codes and it is generally examined by very few. Toapply a fracture energy based compressive model in a FE-analysis, an element size dependentstress-strain formulation must be used.

As the scope herein is to extend the current formulations for the ambient condition to ele-vated temperatures the existing models are examined. The evaluation of the models includesinvestigating their limits of application, which poses demands on the following:

• The minimum element size.

• The maximum element size.

• The minimum amount of reinforcement that can account for the interaction contributionfor the modelling of the tension sti!ening.

Of the above points, the limitation on the maximum element size is widely recognized whereasthe demand for a minimum element size and a minimum reinforcement ratio is novel research.

A discussion of how the strength level of the tension sti!ening contribution influences the overallbehaviour of a reinforced member is given and, to conclude the chapter, a brief summary of thechosen concrete model is provided. This is followed by three numerical benchmark examplesof a simple plate configuration subjected to uniaxial tension, uniaxial compression and pureshear.

19

Chapter 4: Modelling of Uniaxial behaviour of Reinforced Concrete at Ambient Temperature

4.2 Material Model of Reinforcement

As the material model used for the reinforcement a!ects the tensile behaviour of reinforecedconcrete, a brief discription of the reiforcement model considered herein is given prior to thediscussion on the tensile behaviour of reinforced concrete.

The reinforment model used herein is based on the CEB-FIB Model Code [16] as this aimsat combining research and technical finding and translating these to practical purposes. Asimple bi-linear elasto-plastic model is therefore employed with steel Grade 500, with elasticperformance until yield sets in at fy, after which perfectly plastic deformations occur. Further,it is assumed that the behaviour of the reinforcement is equal in tension and in compression andthat all reinforcement is in the form of either rods or grids of steel.

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

fy Es (

500 MPa 200 GPa 0.3

4.3 Reinforced Concrete in Tension

The behaviour of reinforced concrete in tension depends on the state of the concrete; cracked ornot cracked. Once cracking has commenced, the load-bearing ability of the structure is shiftedfrom the concrete to the reinforcement. However, this does not occur abruptly as all the concreteis not cracked simultaneously. Due to the presence of reinforcement the uncracked concrete con-tinues to carry additional forces and, as a result, there is thus no reduction of the load carryingcapacity in the composite. The contribution to the sti!ness of the structure by the uncrackedpart of the concrete is known as tension sti!ening and it decays as the stress increases due to thecontinued formation of cracks. Some tension sti!ening is always present until the reinforcementstarts to yield.

The load-displacement response of a reinforced concrete specimen in uniaxial tension is shown inFigure 4.1. The figure displays the response with and without tension sti!ening and illustrateshow the strength level of the tension sti!ening e!ect influences the load-displacement relation.

The tension sti!ening e!ect can be modelled in several ways, often using either a linear oran exponential model. In the following some of the procedures recommended in literature arediscussed.

20


Concrete has no

tensile strength after

cracking

Tensile stiffening, !"Pservice

Concrete does

not crack

Concrete is assumed

to be fully plastic

P

cr

Pure reinforcement

!

P

Figure 4.1: Illustrative load-displacement diagram explaining the concept of tension sti!eningof reinforced concrete members.

4.3.1 Tension Sti!ening Model as per the CEB-FIB Model Code

In this thesis, a concrete of grade C30 is considered for the computations. The material param-eters for such a concrete in tension, suggested in the CEB-FIB Model Code [16], are given inTable 4.2 for a maximum aggregate size of dmax = 32 mm.

Table 4.2: Tensile material parameters for concrete grade C30 with maximum aggregate sizedmax = 32 mm [16].

fct,m Ec Gf

2.9 MPa 33.5 GPa 0.095 N/mm

The tension sti!ening model in the CEB-FIB Model Code [16] defines the strain in the reinforce-ment, %s,m, as a piecewise linear function. Two states are defined; State I and State II-Naked,corresponding to the behaviour of the uncracked section and the behaviour when concrete hasno contribution in tension, and the strain is computed on the basis of these two. The strains ofthe two states are denoted %s1 and %s2, respectively.

0 0.5 1 1.5 2 2.5

x 10!3

0

100

200

300

400

500

600

!

" [

MP

a]

!s1

!s2

!s,m

!"##$%&'(&)*+,%"-&'(&)*+,

.*-/01-%/.0!!*-0-2

*!!*&.%0-&#"+*+,

!s2

!s1

!s,m

Figure 4.2: Behaviour of reinforced concrete member using the CEB-FIB Model Code [16].

21


Figure 4.2 shows the stress-modified steel strain diagram for a reinforced concrete member intension. Here, a plate of length and width b = 1000 mm and thickness t = 10 mm is considered,with a total area of the reinforcement of As = 300 mm2 and hence a reinforcement ratio of*s = 0.03. The material parameters from Table 4.1 and Table 4.2 are used for the reinforcementand the concrete, respectively. It is seen that the sti!ness of the reinforced member is graduallyshifted from the concrete to the reinforcement as cracking progresses.

In a FE-model, the tension sti!ening input is given in the concrete definition and this is donethrough a stress-strain diagram, as described in appendix B.1. However, as discussed in sec-tion 1.2, a stress-strain diagram cannot be defined as a material property for concrete as it isdependent upon the chosen element size. It is therefore necessary to look at possible ways ofincorporating the tension sti!ening based on fracture energy.

4.3.2 Tension Sti!ening Model by Cervenka et al.

Cervenka et al. [17] proposed that the tension sti!ening e!ect should not be considered as a partof the concrete constitutive law, but rather as a separate interaction contribution. This meansthat the total stress in a member becomes a sum of the three stress contributions from theconcrete, the reinforcement and the interaction. This makes it possible to define the concretestress contribution, as well as the tension sti!ening e!ect, by the tensile fracture energy and theelement side length, thus making the FE-model mesh insensitive.

The interaction contribution was defined as a trilinear function, where the first incline corre-sponds to the decreasing branch of the tension softening of the concrete. The descending partstarts when yielding of the reinforcement begins and in between these two, a constant part equalto a fraction of the tensile strength, ts, is given. It is convenient to define the strength of theinteraction contribution as a fraction of the tensile strength, i.e.. ts = "tsfct,m where "ts canvary between 0 and 1. Cervenka et al. [17] use a constant value of "ts = 0.4.

Concrete

SteelTension stiffening

cu

ts !ts fct,m=

"

#

"

#

"c "0 y

ts

"u"

(a) Stress-strain diagram for reinforced concrete member

Concrete

SteelTension stiffening

cu

ts !ts fct,m=

"

#

"

#

"c "0 y

ts

"u"

(b) Interaction contribution

Figure 4.3: Interaction contribution suggested by Cervenka et al. [17].

22


In Figure 4.3a a schematic stress-strain diagram for a reinforced concrete member is given, andin Figure 4.3b the interaction contribution is illustrated. Here, %c0 is the strain at which theconcrete reaches its maximum tensile strength, %cu is the ultimate concrete strain, i.e. when theconcrete contribution is zero, %u is the strain at which the yield stress for the reinforcement isreached in the member and %y is the strain at which the reinforcement starts to yield. The fourstrain components are defined by the following: The strain at peak stress is given by:

%c0 =fct,m

Ec(4.1a)

The ultimate concrete strain, %cu, can be determined if considering the tension sti!ening e!ectfrom the softening of the plain concrete as a function of the fracture energy, Gf , and elementside length, h.

ct,m

!cu

"

!!c0

Gf

fct,m

!cu h

"

f

#

(a) Stress-strain diagram

ct,m

!cu

"

!!c0

fct,m

!cu h

"

#

Gf

f

p

(b) Stress-plastic displacement dia-gram

Figure 4.4: Schematic plots of the stress-strain relation of pure concrete in tension, (a), andthe stress-plastic displacement diagram, (b), to illustrate the dependency of thefracture energy on the element size, h.

In Figure 4.4a a schematic stress-strain diagram for plain concrete in tension is given, assuminga linear post-peak softening. This can be transferred into a stress-plastic displacement diagram,as done in Figure 4.4b, simply by multiplying the ultimate concrete strain, %cu, with the lengthof the considered element, h. Beacause the fracture energy is defined as the area under thestress-plastic displacement diagram, which is seen in Figure 4.4b to be Gf = 1/2%cuhfct,m, anexpression for %cu can easily be obtained as:

%cu = 2Gf

hfct,m(4.1b)

Cervenka et al. [17] defines the ultimate strain as:

%u = %y ""tsfct,m

*s,effEs(4.1c)

The yield strain of the reinforcement is given by:

%y =fy

Es(4.1d)

Herein, it has become evident the that model suggested by Cervenka et al. [17] is prone to somelimitations on the element size and the minimum amount of reinforcement that must be presentfor the interaction contribution to be considered. In the following, expressions for the validityrange are formulated.

23


Minimum Element Size

A clear dependency of the ultimate concrete strain on the element length is seen in equation(4.1b). This means that the slope of the softening branch of the stress-strain curve decreaseswith decreasing h and the ultimate concrete strain is increased. It has been found herein, thatthis imposes a requirement on the minimum element side length as the ultimate concrete straincannot be more than the ultimate strain of the reinforced member, %u.

0 0.5 1 1.5 2 2.5

x 10!"

0

0.5

1

1.5

2

2.5

"

!

" [M

Pa]

h = 500

h = 250

h = 150

0 0.5 1 1.5 2 2.5

x 10!"

0

0.5

1

1.5

2

2.5

"

!

" [M

Pa]

h = 150

h = 50

(a) !cu < !u for h = 500 mm, h = 250 mm andh = 150 mm

0 0.5 1 1.5 2 2.5

x 10!"

0

0.5

1

1.5

2

2.5

"

!

" [M

Pa]

h = 500

h = 250

h = 150

0 0.5 1 1.5 2 2.5

x 10!"

0

0.5

1

1.5

2

2.5

"

!

" [M

Pa]

h = 150

h = 50

(b) !cu > !u for h = 50 mm

Figure 4.5: Combination of concrete and interaction stress contribution for di!erent elementside lengths, h [mm].

In Figure 4.5 examples of the combined concrete and interaction contributions are given fordi!erent values of h for a specimen with height and side length b = 1000 mm, thickness t = 100mm and reinforced in one direction with a total area of As = 300 mm2. Further, the materialproperties from Table 4.2 are used. Figure 4.5a shows, that by decreasing h, the ultimateconcrete strain is increased and Figure 4.5b illustrates the situation where %cu > %u. As it is notpossible for the ultimate concrete strain to exceed the ultimate strain, it is necessary to define alower limit for the element side length, h. Herein, this is done by equating the ultimate concretestrain (4.1b) with the ultimate strain (4.1c), yielding:

hmin =2Gf

fct,m

!%y "

"tsfct,m

*s,effEs

$ (4.2)

For the above example, equation (4.2) yields a minimum element side length of hmin = 116mm, which is violated by choosing an element side length of h = 50 mm as shown in Figure4.5b.

Maximum Element Size

It is also possible to choose an element size that is too large, a situation that will cause a snap-back on the plain concrete stress-strain diagram as illustrated in Figure 4.6a. Exactly the sameparameters are used for the example in Figure 4.6 as in the previous example, apart from theelement side length which is increased to h = 1000 mm.

24


! !"# !"$ !"% !"& ' '"#

()'!!$

!

!"*

'

'"*

#

#"*

+

!

" ),-./0

h)1)*!!

h)1)'!!!

! !"* ' '"* # #"*

()'!!+

!

!"*

'

'"*

#

#"*

+

!

" ),-./0

h)1)*!!

h)1)'!!!

(a) Concrete

! !"# !"$ !"% !"& ' '"#

()'!!$

!

!"*

'

'"*

#

#"*

+

!

" ),-./0

h)1)*!!

h)1)'!!!

! !"* ' '"* # #"*

()'!!+

!

!"*

'

'"*

#

#"*

+

!

" ),-./0

h)1)*!!

h)1)'!!!

(b) Concrete and interaction

Figure 4.6: The e!ect on the stress-strain curve of concrete, (a), and on the combined concreteand interaction contribution, (b), of snap-back of concrete for a model with toolarge element side length, here h = 1000 mm, compared to model without snap-back, h = 500 mm.

The excessively large choice of h causes snap-back and this is a widely recognized problem. Itcan be avoided by controlling the softening modulus, H, as suggested by Pankaj [6].

H

ct,m

!!cu

"

p

f

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

If the softening is assumed to be linear, as illustrated in Figure 4.7, H can be expressed asa function of the tensile strength, fct,m, and the ultimate concrete strain, %cu. By using thedefinition of %cu from equation (4.1b), the following expression for the softening modulus can beobtained.

H = "hf2

ct,m

2Gf(4.3)

To avoid snap-back for softening plasticity, the softening modulus must be limited by the elasticmodulus of the concrete as it is ultimately the di!erence of the cracking strain of concrete, %c0,and the ultimate concrete strain, %cu, that needs to be limited to ensure that %cu " %c0 % 0. Interms of H, this means that the following inequality must be fulfilled.

" Ec ! H ! 0 (4.4)

Thus, by rewriting (4.3) and the left part of (4.4) an expression for the maximum element sidelength is found.

25


hmax =2EcGf

f2ct,m

(4.5)

For the example in Figure 4.6, equation (4.5) yields a maximum element side length of hmax =757.

Minimum Reinforcement Ratio

An equally critical situation arises if the ratio of reinforcement is so low that the ultimate concretestrain is higher than the ultimate strain, %cu > %u. If this is the case, the interaction contributioncannot be accounted for in the tension sti!ening definition. The previously described example isused to illustrate the phenomenon in Figure 4.8, where the element side length is held constantat h = 100 mm.

0 0.5 1 1.5 2 2.5

x 10!"

0

0.5

1

1.5

2

2.5

"

!

" [M

Pa]

A

s = 400

As#$#"%&

As = 250

Figure 4.8: Combined concrete and interaction stress contribution for di!erent di!erent rein-forced areas As [mm2] using the tension sti!ening model by Cervenka et al. [17].

A minimum e!ective reinforcement ratio is defined as a function of h by equating expressions(4.1b) and (4.1c):

*s,eff,min ="tsfct,m!

%sy "2Gf

hfct,m

$Es

(4.6)

For the current example (h = 100 mm), it would mean that a minimum e!ective reinforcementratio of *s,eff = 3.15 · 10#3, and hence As,min = 315 mm2, would be required.

As a result of the above discussion, it has been established herein, that the tension sti!eningmodel by Cervenka et al. [17] is only valid for a reinforced concrete model that fulfills the threedescribed requirements. To summarize, these are the previously recognised limit on the maxi-mum element size, (4.5), and the two limitations formulated herein; the limit on the minimumelement size, (4.2), and the limit on the minimum reinforcement ratio, (4.6).

4.3.3 Tension Sti!ening Model by Feenstra and de Borst

The tension sti!ening model by Feenstra and de Borst [18] is also based on the fundamentalassumption of the tension sti!ening being defined as an interaction contribution. The issue of

26


the limit on the minimum length of the element size has however been tackled by defining amodified fracture energy for reinforced concrete. Further, a specific value for the stress level ofthe interaction contribution is not given but it is left as a variable, "ts.

Fracture Energy for Reinforced Concrete

In reinforced concrete, the cracking process stabilizes after a number of cracks are developedand at this instant the crack spacing is determined by the amount of reinforcement. In thestudy by Feenstra and de Borst [18], it is taken into account that the average crack spacing, ls,is generally much larger than the element side length, h, by introducing a reinforced fractureenergy, Grc

f , released over the equivalent length, h.

Grcf = min

!Gf , Gf

h

ls

$(4.7)

The average crack spacing is defined as a function of the reinforcement ratio, position and thediameters of the rebars used, but is independent of the equivalent length, h. It is determinedby initially computing the equivalent reinforced diameter for a member that is reinforced in theorthogonal directions p and q:

+eq =+p*p + +q*q

*p + *q(4.8a)

The e!ective element size can then be evaluated by the following:

heff = min

&'(

')

2.5!

c ++eq

2

$

t

2

(4.8b)

In this, c is the cover layer of the reinforcement. Once the e!ective element size has beendetermined, the e!ective reinforcement ratio can be found.

*s,eff =As

Ac,eff=

As

bheff(4.8c)

Finally, the average crack spacing is determined by:

ls =23

!2s0 +

+s

*s,eff

$(4.8d)

Here, s0 is the minimum bond length which usually is taken as 25 mm. This value is thereforealso chosen herein.

In the following examples the validity range for the model by Feenstra and de Borst [18] is ex-plored for a plate that is reinforced in the p direction, which is the direction the load is appliedin. The same total steel area is used as in the examples of the model by Cervenka et al. [17],As = 300 mm2, and it is used that +p = 1.95 mm, yielding a cover layer thickness of c = 499.5mm. This yields an average crack spacing of ls = 250.5 mm.

If using the same example as done for the model by Cervenka et al. [17], and taking into con-sideration the layer of reinforcement, the combined concrete and interaction stress contributionis obtained, as shown in Figure 4.9.

27

Chapter 4: Modelling of Uniaxial behaviour of Reinforced Concrete at Ambient Temperature0 0.5 1 1.5 2 2.5

x 10!"

0

0.5

1

1.5

2

2.5

"

!

" [

MP

a]

h =500

h#$"%&

h =150

0 0.5 1 1.5 2 2.5

x 10!"

0

0.5

1

1.5

2

2.5

"

!

" [

MP

a]

h =500

h#$"%&

h =150

0 0.5 1 1.5 2 2.5

x 10!"

0

0.5

1

1.5

2

2.5

"

!

" [

MP

a]

Feenstra

Cervenka

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

From Figure 4.9 it is seen how the minimum limit on side length is increased. In fact it is foundthat by the introduction of the reinforced fracture energy, the requirement for the minimumelement side length is no longer needed.

However, with regards to the limit on element side length, the modification of the fracture en-ergy only tackles the problem of small element sizes, and does not take into consideration theerrors created by choosing an excessive value of h. Therefore, it is still necessary to implementthe herein formulated equation (4.5).

Further, it is still possible to have a reinforcement ratio that is too low. The reinforced fractureenergy complicates the definition of a minimum reinforcement ratio as a function of the elementside length as it makes the expression iterative. Therefore, it is much more convenient forpractical purposes simply to check that the following inequality is fulfilled for the chosen elementsize, h:

%cu ! %u (4.9)

! !"# !"$ !"% !"& ' '"#

()'!!$

!

!"*

'

'"*

#

#"*

+

!

" ),-./0

h)1)*!!

h)1)'!!!

! !"* ' '"* # #"*

()'!!+

!

!"*

'

'"*

#

#"*

+

!

" ),-./0

h)1)*!!

h)1)'!!!

(a) Concrete stress contribution for constantAs = 300 mm2 with h = 500 mm and 1000mm, respectively.

0 0.5 1 1.5 2 2.5

x 10!"

0

0.5

1

1.5

2

2.5

"

!

" [

MP

a]

A

s = 400

As = 200

As = 85

0 0.5 1 1.5 2 2.5

x 10!"

0

0.5

1

1.5

2

2.5

"

!

" [

MP

a]

A

s = 400

As = 250

As = 120

0 0.5 1 1.5 2 2.5 "

x 10!"

0

0.5

1

1.5

2

2.5

"

!

" [

MP

a]

de Borst

Cervenka

(b) Combined concrete and interaction stresscontribution for h = 100 mm and As ={120; 250; 400} mm2.

Figure 4.10: Examples of possible errors in the tension sti!ening 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


Figure 4.10 shows examples of the concrete stress contribution if an excessive element sidelength is chosen and the combined concrete and interaction contribution for di!erent levels ofreinforcement for h = 100 mm. It is seen, by comparing Figure 4.10a to Figure 4.6a, that theproblem for large values of the element side length is unchanged, and hence that the requirementfor the maximum element side length is still given by (4.5). Figure 4.10b illuatrates the problemof a reinforcement ratio that is too small. However, by the introduction of the reinforced fractureenergy, the limitation of the application is lowered. For the previously discussed example aminimum steel area of As = 125 mm2 is found by iteration. It is used with the reinforcedmember length b = 1000 mm, thickness t = 100 mm, and an element side length h = 100 mm.The material properties of Table 4.2, a minimum bond length of s0 = 25 mm are used and thecover layer is taken as a function of the diameter of the reinforcement c = (t" +p)/2.

Influence of Fraction on Strength of Interaction Contribution

The level of stress defined in the interaction contribution can be varied by changing the fractionof the tensile concrete strength, "ts. In Figure 4.11 the e!ect of increasing "ts from 0.0 to 1.0is shown for a specimen with length b = 100 mm, thickness t = 10 mm, As = 7 mm2, thus*s = 0.007 and a constant element side length of h = 100 mm.

0 0.5 1 1.5 2 2.5x 10!3

0

0.5

1

1.5

2

2.5

3

!

" [M

Pa]

0 0.5 1 1.5 2 2.5x 10!3

0

100

200

300

400

500

!

" [M

Pa]

#ts = 0.0#ts = 0.4

#ts = 0.7

ts = 1.0#

#ts = 0.0#ts = 0.4

#ts = 0.7

ts = 1.0#

(a) Combined concrete and interaction stress con-tribution

0 0.5 1 1.5 2 2.5x 10!3

0

0.5

1

1.5

2

2.5

3

!

" [M

Pa]

0 0.5 1 1.5 2 2.5x 10!3

0

100

200

300

400

500

!

" [M

Pa]

#ts = 0.0#ts = 0.4

#ts = 0.7

ts = 1.0#

#ts = 0.0#ts = 0.4

#ts = 0.7

ts = 1.0#

(b) Stress-strain diagram for combined concrete,interaction and steel

Figure 4.11: E!ect of changing the fraction of the ultimate tensile strength applied on theinteraction contribution of the tension sti!ening 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).

Figure 4.11a shows that "ts controls the strength level of the interaction contribution and how itis increased for increasing values of "ts. As expected, and seen in Figure 4.11b, the stress-strainrelation for the specimen is tri-linear for "ts = 1.0, which is the level suggested by Feenstraand de Borst [18]. However, this is considered to be an unreasonable model of the tensionsti!ening e!ect, as the concrete maintains at full strength after onset of cracking. Further, onthe stress-strain curve for the reinforced member with "ts = 0.4, which was the value suggestedby Cervenka et al. [17], it is seen that the shift from the concrete contribution to the interactioncontribution causes a slight decrease in strength. An intermediate value of "ts = 0.7 is taken inthe following for the level of strength on the interaction contribution.

29


4.4 Compressive Behaviour of Concrete

As there is no concept equivalent to tension sti!ening in compression, the compressive behaviourof reinforced concrete is simply a combination of the stress contributions of the plain concreteand the contribution of reinforcement. As a result, this section contains compressive models forplain concrete, which for the reinforced case simply must be added to the steel contribution.The compression model from the CEB-FIB Model Code [16] is described in section 4.4.1 and isfollowed by a overview of the concept of compressive fracture energy in section 4.4.2. Further,the compressive fracture energy based models suggested by Nakamura and Higai [19] and byFeenstra and de Borst [18] are discussed in sections 4.4.3 and 4.4.4, respectively. The threecompression models are compared in section 4.4.5.

According to usual sign convention, compressive stresses and strains are negative, but in thissection, they are considered to be positive. This is chosen to avoid confusion, as the compressiveproperties must be entered as positives in the FE-code utilized herein, as described in appendix2.

4.4.1 Compression Model in CEB-FIB Model Code

In the CEB-FIB Model Code [16], the material data for a range of di!erent concrete grades incompression is given. Herein, it is chosen to focus on a concrete of grade C30 and the relevantparameters for the compression model are given in Table 4.3.

Table 4.3: Parameters used for the compression model from the CEB-FIB Model Code [16] forconcrete grade C30.

fcm Eci %c1 %c,lim

38 MPa 33.5 GPa 2.2·10#3 3.3·10#3

The compression model in the CEB-FIB Model Code [16] is described by two functions, oneprior to and after the compressive strain, %c,lim, which has no significance for anything otherthan computational purposes.

Figure 4.12 shows that the CEB-FIB Model Code [16] models the compressive behaviour assomewhat linear until a certain yield point, after which hardening is evident until the peakstrength is reached. This peak is followed by a clear softening branch.

30


! !"!!# !"!$ !"!$# !"!% !"!%# !"!&!

$!

%!

&!

'!

! "

()*+,

! !"!!# !"!$ !"!$# !"!% !"!%# !"!&!

$!

%!

&!

'!

!

" ()*+,

G

c -+./01231f

c

Gc -+./01231G

f

! !"!!# !"!$ !"!$# !"!% !"!%# !"!&!

$!

%!

&!

'!

!

" ()*+,

G

c 452617238

Gc -+./01231f

c

Gc -+./01231G

f

! !"!!# !"!$ !"!$# !"!% !"!%# !"!&!

$!

%!

&!

'!

!

" ()*+,

9:;!<=;>+5+8?6+1@f

cA

0/1;25.B1@fcA

Figure 4.12: Compressive behaviour as defined by the CEB-FIB Model Code [16].

However, the compression model in the CEB-FIB Model Code [16] is based on a stress-strainrelation, and therefore the problems with localization of deformations, described in section 1.2,are bound to arise, resulting in a mesh sensitive FE model. As a result, it is necessary toinvestigate the possibility of using a compressive fracture energy based material model, whichis element-size dependent.

4.4.2 Compressive Fracture Energy

In tension the fracture energy, Gf , and the equivalent length has widely been used for FEmodels. In compression, on the other hand, the use of the compressive fracture energy, Gc, andthe equivalent length, h, is less established. In this section an overview of two distinctly di!erentdefinitions of the compressive fracture energy is given, proposed by Vonk [20] and Nakamuraand Higai [19], respectively. Common for the two, is that the compressive fracture energy isconsidered to be constant for a given reinforced member.

Compressive Fracture Energy by Vonk

It was suggested by Vonk [20] that the compressive fracture energy consists of two contributions;a local and a continuum. The local fracture energy is considered to be constant, whereas thecontinuum accounts for the size e!ects that were found to be significant in compression. Thecontinuum is considered to be linearly increasing with the specimen height as shown in Figure4.13. This approximation is arrived at by linear regression of test results with three di!erentspecimen heights.

If considering a specimen of height 100 mm, Figure 4.13 yields a compressive fracture energy ofGc = 15 N/mm and for a specimen height of 500 mm a value of Gc = 41 Nmm/mm2 is obtainedby extrapolating the regression line. However, it must be stressed that as the spread of the testresults evidently was increasing for increased specimen heights, and that no tests were carriedout for members taller than 200 mm, an extrapolation is very crude and a great uncertainty isassociated with compressive fracture energies for larger specimens.

However, it should be stressed that the defintion of the fracture energy requires the term to bea material constant. Therefore, it can be questioned whether the specimen size-dependency ofthe compressive fracture energy induces that the expression by Vonk [20] is really not a fractureenergy at all.

31


0 50 100 150 200 2500

5

10

15

20

25

30

Local Fracture Energy

Continuum Fracture Energy

Specimen height [mm]

Gc [N

mm

/mm

2]

Figure 4.13: Compressive post-peak fracture energies for di!erent specimen geometries. Re-produced from Vonk [20].

Compressive Fracture Energy as per Narakuma and Hiagai

In the study by Nakamura and Higai [19], experiments were carried out to investigate the na-ture of the compressive fracture energy for plain concrete, i.e. its dependency on the specimensize and shape, aggregate size and grading and compressive strength. It was found that thecompressive fracture energy is independent of the size and shape of the test specimens whenthe aggregate grading is the same, and hence that when the aggregate grading is constant, thecompressive fracture energy can be assumed as a material property. This is distinctly di!erentfrom conclusion of Vonk [20], where the specimen height is stated to have an e!ect on the com-pressive fracture energy.

It was further concluded by Nakamura and Higai [19] that the compressive fracture energyis a function of the compressive strength of the concrete and they provide the following rela-tion:

Gc = 8.8%

fcm (4.10)

Here, Gc is the compressive fracture energy and fcm the compressive strength (in MPa).

Further, a linear relation between the compressive and tensile fracture energies is suggestedas:

Gc = 250Gf (4.11)

For a given reinforced member with concrete grade C30, the compressive fracture energies ob-tained from Figure 4.13, expressions (4.10) and (4.11) are compared for member heights of 100mm and 500 mm in Table 4.4.

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

Member height Figure 4.13 Eq. (4.10) Eq. (4.11)

100 mm 15 54 24500 mm 41 54 24

32


From Table 4.4 it is seen that the two expressions from Nakamura and Higai [19] are notconsistent for this grade of concrete. As expression (4.11) is arrived at by comparing the obtainedcompressive fracture energies from expression (4.10) to the tensile fracture energies of the testspecimens, expression (4.10) is here taken for the computations of the compressive fractureenergy. It is further seen, that the compressive fracture energies obtained by Vonk [20], whichherein are argued not to be fracture energies at all, are much smaller than the ones observed byNakamura and Higai [19], using (4.10).

4.4.3 Compression Model of Narakuma and Higai

In addition to the methods for computing the compressive fracture energy, Nakamura and Higai[19] also suggested a compressive behaviour model for concrete.

!

"

! p

E

cmf

c E c

Gc / h

Figure 4.14: Illustrative stress-strain relationship for the compression model by Nakamura andHigai [19].

This model consists of a parabola prior to peak stress and a linear decrease after as indicatedschematically in Figure 4.14. The compressive fracture energy is considered to be the area underthe stress-plastic displacement curve, and the indicated area in Figure 4.14 is therefore equal toGc/h.

! !"!!# !"!$ !"!$# !"!% !"!%# !"!&!

$!

%!

&!

'!

!

" ()*+,

! !"!!# !"!$ !"!$# !"!% !"!%# !"!&!

$!

%!

&!

'!

!

" ()*+,

G

c -+./01231f

c

Gc -+./01231G

f

! !"!!# !"!$ !"!$# !"!% !"!%# !"!&!

$!

%!

&!

'!

!

" ()*+,

G

c 452617238

Gc -+./01231f

c

! !"!!# !"!$ !"!$# !"!% !"!%# !"!&!

$!

%!

&!

'!

!

" ()*+,

9:;!<=;>+5+8?6+1@f

cA

0/1;25.B1@fcA

(a) h = 100 mm

! !"!!# !"!$ !"!$# !"!% !"!%# !"!&!

$!

%!

&!

'!

!

" ()*+,

! !"!!# !"!$ !"!$# !"!% !"!%# !"!&!

$!

%!

&!

'!

!

" ()*+,

G

c -+./01231f

c

Gc -+./01231G

f

! !"!!# !"!$ !"!$# !"!% !"!%# !"!&!

$!

%!

&!

'!

!

" ()*+,

G

c 452617238

Gc -+./01231f

c

! !"!!# !"!$ !"!$# !"!% !"!%# !"!&!

$!

%!

&!

'!

!

" ()*+,

9:;!<=;>+5+8?6+1@f

cA

0/1;25.B1@fcA

(b) h = 500 mm

Figure 4.15: 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).

In Figure 4.15 the compressive material model suggested by Nakamura and Higai [19] is plottedfor concrete grade C30, using the parameters from the CEB-FIB Model Code [16] in Table 4.3for element lengths of h = 100 mm and h = 500 mm. From Figure 4.15 it is apparent that the

33


post peak behaviour in compression can be significantly di!erent depending on the model usedfor the compressive fracture energy. It is also seen, by comparing the two, that the behaviourdi!ers significantly with varying element side lengths.


Herein, it has been found that a limitation must be imposed on the maximum element size.Figure 4.15b illustrates the problem that occurs for large values of h. It is seen that when com-puting the compressive fracture energy based on the tensile fracture energy, Gf , using (4.11),a snap-back e!ect occurs in the stress-strain diagram. It must be understood that this is apossible problem that can occur no matter which expression is used for the compressive fractureenergy, if the element side length is chosen to be of excessive magnitude.

As the snap-back is clearly not a realistic compressive behaviour, it is necessary to limit theelement side length. This will be di!erent from previously defined expression for the maximumelement size in tension, (4.5), as both the strengths and fracture energies di!er in tension andin compression. Herein, the expression in compression is arrived at by equating the ultimateconcrete strain, %cu, with the peak strain, %p, which gives:

hmax =23

GcEc

f2cm

(4.12)

By applying expression (4.12) to the material values from Table 4.3 and computing the compres-sive fracture energy by (4.11), a maximum element side length of hmax = 371 mm is obtained.The choice of h = 500 mm in Figure 4.15b is therefore clearly in violation of the validity range,which explains the snap-back behaviour.

4.4.4 Compression Model by Feenstra and de Borst

As compressive failure is initiated by a combination of shear and tensile stresses (described indetail in appendix A.1) Feenstra and de Borst [18] define the material behaviour by a damagemodel. The concrete behaviour is described by two internal parameters; &T in tension and &C incompression. These are related to the released energy per unit damaged area by an equivalentlength, h. Because the equivalent parameters are dependent on h, the material model is linkedto the element size and the ultimate parameters, &uC and &uT , are assumed to be constantelement-related material parameters as they can be calculated from the material properties andthe equivalent length, h; the latter related to the element area. In a monotonic loading cycle,it is not possible to distinguish the roles of damage and plasticity (or cracking strains and plas-tic strains), which only become apparent durring unloading. Here, the damage parameters, orequivalent cracking strains, are therefore regarded as plastic strains.

The material strength in compression is defined by two functions; pre- and post-peak. Theequivalent strain, &e, corresponding to peak is expressed as:

&e =4fcm

3Ec(4.13)

The ultimate compressive concrete strain is defined as the following and is seen to be dependentupon the compressive fracture energy.

&uC = 1.5Gc

hfcm(4.14)

34


In Figure 4.16 the compressive material behaviour by Feenstra and de Borst [18] is shown fora concrete grade C30, defined by the parameters in Table 4.3. Two di!erent values for thecompressive fracture energy are taken; the one obtained by Vonk [20] (see Figure 4.13) andthe chosen methods defined by Nakamura and Higai [19] computed by the compressive strengthusing (4.10). The compressive behaviour is shown for element side lengths of h = 100 mm andh = 500 mm.

! !"!!# !"!$ !"!$# !"!% !"!%# !"!&!

$!

%!

&!

'!

!

" ()*+,

! !"!!# !"!$ !"!$# !"!% !"!%# !"!&!

$!

%!

&!

'!

!

" ()*+,

G

c -+./01231f

c

Gc -+./01231G

f

! !"!!# !"!$ !"!$# !"!% !"!%# !"!&!

$!

%!

&!

'!

!

" ()*+,

G

c 452617238

Gc -+./01231f

c

! !"!!# !"!$ !"!$# !"!% !"!%# !"!&!

$!

%!

&!

'!

! "

()*+,

9:;!<=;>+5+8?6+1@f

cA

0/1;25.B1@fcA

(a) h = 100 mm

! !"!!# !"!$ !"!$# !"!% !"!%# !"!&!

$!

%!

&!

'!

!

" ()*+,

! !"!!# !"!$ !"!$# !"!% !"!%# !"!&!

$!

%!

&!

'!

!

" ()*+,

G

c -+./01231f

c

Gc -+./01231G

f

! !"!!# !"!$ !"!$# !"!% !"!%# !"!&!

$!

%!

&!

'!

!

" ()*+,

G

c 452617238

Gc -+./01231f

c

! !"!!# !"!$ !"!$# !"!% !"!%# !"!&!

$!

%!

&!

'!

!

" ()*+,

9:;!<=;>+5+8?6+1@f

cA

0/1;25.B1@fcA

(b) h = 500 mm

Figure 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 concretegrade C30 is considered and the material data of Table 4.3 are used.

It is clearly seen by Figure 4.16, that the chosen value for the compressive fracture energy hasgreat significance for the model, especially for low values of h. Further, it is illustrated that thevalue obtained by Vonk [20] is much lower than the one obtained using the approach suggested byNakamura and Higai [19]. In the following it is decided to take the value obtained by expression(4.10).


Herein, it has been established that this compression model also has problems with excessivevalues of the element side length, h. This is illustrated in Figure 4.17, where it is seen that theconcrete does not reach its full strength level for h = 2000 mm.

An expression for the maximum allowable element side length, can be arrived at by equatingthe equivalent strain at peak &e, obtained using (4.13), with the ultimate equivalent strain &uC ,obtained using (4.14).

hmax =98

GcEc

f2cm

(4.15)

In Figure 4.17, this imposes a maximum limit on the element side length of 1409 mm.

By comparing expression (4.15) to the obtained maximum element side length in the compressionmodel by Nakamura and Higai [19], obtained using (4.12), it is seen that the maximum limit isincreased by 69%, and thus, that the model by Feenstra and de Borst [18] allows considerablylarger element side lengths.

35


0 0.005 0.01 0.015 0.02 0.0250

10

20

30

40

!C

" [

MP

a]

h = 100 mm

h = 500 mm

h = 2000 mm

Figure 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.

4.4.5 Comparison of Compression Models

The agreement with the CEB-FIB Model Code [16] is evident for both the compressions modelpresented by Nakamura and Higai [19] and by Feenstra and de Borst [18]. The three are plottedfor the same material parameters in Figure 4.18.

0 0.005 0.01 0.015 0.02 0.025 0.030

10

20

30

40

!

" [

MP

a]

0 0.005 0.01 0.015 0.02 0.025 0.030

10

20

30

40

!

" [

MP

a]

G

c based on f

c

Gc based on G

f

0 0.005 0.01 0.015 0.02 0.025 0.030

10

20

30

40

!

" [

MP

a]

G

c from Vonk

Gc based on f

c

0 0.005 0.01 0.015 0.02 0.025 0.030

10

20

30

40

!

" [

MP

a]

CEB!FIBNarakuma (f

c)

de Borst (fc)

(a) h = 100 mm

0 0.005 0.01 0.015 0.02 0.025 0.030

10

20

30

40

!

" [

MP

a]

0 0.005 0.01 0.015 0.02 0.025 0.030

10

20

30

40

!

" [

MP

a]

G

c based on f

c

Gc based on G

f

0 0.005 0.01 0.015 0.02 0.025 0.030

10

20

30

40

!

" [

MP

a]

G

c from Vonk

Gc based on f

c

0 0.005 0.01 0.015 0.02 0.025 0.030

10

20

30

40

!

" [

MP

a]

CEB!FIBNarakuma (f

c)

de Borst (fc)

(b) h = 500 mm

Figure 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 = 500mm, (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).

It is easily seen by Figure 4.18b, that good agreement with the CEB-FIB Model Code [16] com-pression model is found for large values of element side length with both the model by Nakamuraand Higai [19] and the model by Feenstra and de Borst [18]. However, it is clearly evident thatboth fracture energy based models ignore the last segment of the softening branch where thestress slowly decreases. Further, Figure 4.18a illustrates the necessity for a compressive fractureenergy based material model when using small element side lengths.

As expected, the most significant di!erences in the softening branches are found in the com-pression models by Nakamura and Higai [19] and by Feenstra and de Borst [18]. Considering

36


the uncertainty associated with the value of the compressive fracture energy, the di!erence ofthe employed model is regarded as being quite insignificant for reinforced concrete. Herein,it is chosen to implement the compression model by Feenstra and de Borst [18], as its curvedsoftening resembles the shape of the compression model from the CEB-FIB Model Code [16] themost, and it allows for the use of greater element side lengths.

4.5 Chosen Uniaxial Concrete Models

Herein, the considered concrete material is of Grade C30 and the material properties of theCEB-FIB Model Code [16] are used. These are given in Table 4.2 and Table 4.3 for tension andcompression, respectively.

Because it has previously been found that the stress-strain relation cannot be taken as a materialproperty for concrete, it is here chosen to base the material model on the concept of fractureenergy. In both tension and compression the material models presented by Feenstra and de Borst[18] are used as these were found to have the largest span in choice of the element side length,h. It must still be ensured that the chocie of h falls within the acceptable range, where themaximum limit is expressed in tension by equation (4.5) and in compression by equation (4.15).Combining these two requirements yields:

hmax = min

&''''(

'''')

2EcG

rff

f2ct,m

for ls ! hmax

98

EcGc

f2cm

(4.16)

If the interaction contribution is to be defined as a part of the tension sti!ening, it must moreoverbe ensured that the reinforcement ratio is su"cient. This is most practically done by ensuringthat inequality (4.9) is fulfilled.

A value of "ts = 0.7 is taken for the strength level of the interaction contribution for the tensionsti!ening definition.

The compressive fracture energy is computed by expression (4.10) presented by Nakamura andHigai [19].

4.6 Numerical Test Examples

In this section the chosen uniaxial material models are applied to three benchmarck test exam-ples in ABAQUS. This is done to verify that the expected results of the FE-analysis correlateswith the obtained output. Therefore, a simple reinforced concrete plate is considered in uniaxialtension and in uniaxial compression. Further, a pure shear analysis is carried out.

The plate considered is square with side length b = 100 mm and thickness t = 10 mm and ismodelled by a single plane stress element. The plate is reinforced by a one dimensional rodthat has an area of As = 30 mm2 and hence *p = 0.03. The member consists of concrete gradeC30 and steel Grade 500 and the material properties of Table 4.1 (steel), Table 4.2 (concretein tension) and Table 4.3 (concrete in compression) are used. Further, the fraction of strength

37


of the interaction contribution is taken as "ts = 0.7 and the compressive fracture energy iscomputed by equation (4.10).

p

y

= 10 mmt

3

4

qx

1

2

100 mm

100 mm

= 30 mmA s2

Figure 4.19: FE-configuration of the reinforced member considered for uniaxial load tests ofthe tension sti!ening and the compression model in ABAQUS.

If applying the used parameters to equation (4.16) it is found that the maximum element size is756 mm and hence that h = 100 mm is admissible. Further, it is found that that the consideredlevel of reinforcemenet yields an ultimate strain of %u = 2.16 · 10#3 which is greather than theultimate concrete strain, %cu = 5.37 · 10#4. Therefore, the tension sti!ening can be modelled bythe interaction contribution as inequality (4.9) is fulfilled.

For the FE-analysis, the concrete plasticity damage model described in chapter 2 is applied,using the input parameters of Table 2.1.

4.6.1 Uniaxial Tension

The FE-configuration of the considered reinforced member subjected to uniaxial loads can beseen in Figure 4.19. The performed analysis is displacement controlled and the load is thereforeintroduced by positive displacements of node 2 and 3 in the y-direction when considering uniaxialtension.

0

t

! tck1 ! t

ck2!cu !u ! t!y

" ts f ct,m #!

" ts f ct,m

f ct,m

!c

$

Figure 4.20: The tension sti!ening 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.

When attempting to define the tension sti!ening by the combined concrete and interaction

38


contribution, it is found that an error occurs on the output load-displacement diagram. It isestablished that the error does not occur when there is a slope between the points in the tensionsti!ening input. Therefore, a small stress contribution, !, is subtracted from the last of the twopoints, forcing a slope. The modified combined concrete and interaction contribution used as thetension sti!ening input in the ABAQUS model, can schematically be seen in Figure 4.20.

0 0.05 0.1 0.15 0.2 0.250

2000

4000

6000

8000

! [mm]

P [

N]

Expected

ABAQUS output

0 0.05 0.1 0.15 0.2 0.250

2000

4000

6000

8000

! [mm]

P [

N]

Expected

ABAQUS output

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3x 10

4

! [mm]

P [

N]

Expected

ABAQUS output

(a) *CONCRETE TENSION STIFFENING without mod-ification

0 0.05 0.1 0.15 0.2 0.250

2000

4000

6000

8000

! [mm]

P [

N]

Expected

ABAQUS output

0 0.05 0.1 0.15 0.2 0.250

2000

4000

6000

8000

! [mm]

P [

N]

Expected

ABAQUS output

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3x 10

4

! [mm]

P [

N]

Expected

ABAQUS output

(b) *CONCRETE TENSION STIFFENING modified by! =0 .01 MPa

Figure 4.21: ABAQUS output of load-displacement diagram in the y-direction on node 3 forthe example plate subjected to uniaxial tension. The tension sti!ening is mod-elled as presented by Feenstra and de Borst [18], (a), and modified by ! = 0.01MPa to ensure a constant presence of slope, (b).

The output of the ABAQUS analysis for the uniaxial tension test can be seen in Figure 4.21 forboth the tension sti!ening defined with and without a modification of ! = 0.01 MPa. As theoutput coincides with the exptected theoretical results, it is seen that the modification does notinterfere with the expected theoretical results for this value of !.

4.6.2 Uniaxial Compression

The compressive model of Feenstra and de Borst [18] is used in a similar test example for uniaxialcompression. The compressive behaviour is defined by the *COMPRESSION HARDENING functionthat is described in appendix B. According to the ABAQUS Version 6.7 Documentation [8] boththe in-elastic strains and their corresponding stresses must be given as positives and thereforethe parameters from Table 4.3 are used to compute the stress-strain relation for the materialmodel. The inelastic strains for the point-wise input are then found. In this example, the in-elastic behaviour is defined by 15 points to check the agreement of the ABAQUS output withthe expected theoretical results.

The load is, as for uniaxial tension, introduced as displacements of node 2 and 3 in the y-direction(Figure 4.19), but this time as negatives.

39

Chapter 4: Modelling of Uniaxial behaviour of Reinforced Concrete at Ambient Temperature0 0.05 0.1 0.15 0.2 0.250

2000

4000

6000

8000

! [mm]

P [

N]

Expected

ABAQUS output

0 0.05 0.1 0.15 0.2 0.250

2000

4000

6000

8000

! [mm]

P [

N]

Expected

ABAQUS output

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3x 10

4

! [mm]

P [

N]

Expected

ABAQUS output

Figure 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].

In Figure 4.22 the considered output is the reaction force and the displacement in the y-directionof node 3. It is clearly seen that the output and the expected load-displacement diagram are ingood agreement.

4.6.3 Pure Shear

A similar plate as above is considered subjected to pure shear. The shear load is applied bydisplacing all four nodes as illustrated in Figure 4.23. Two conditions are evaluated; whenthe plate consists of plain concrete and when the plate is equally reinforced in the x and y-directions.

t100 mm

qx

100 mm

= 30 mmA s

32

1 4

p

y

2

= 10 mm

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

The outputs from the two analyses in form of the stresses and strains obtained in the integrationpoints and the forces and displacements in the nodes, are for some time increments enclosed inappendix C for the plate example with and without reinforcement. By comparison of the two,it is found that the output obtained with and without reinforcement are equal and thus thereinforcement parallel to the plate edges has no e!ect in pure shear.

40

Existing Models of the Behaviourof Reinforced Concrete atElevated Temperatures

5.1 Introduction

This chapter is concerned with the existing models of reinforced concrete at elevated tempera-tures and is therefore commenced with a brief description of the decay of material strength inthese models. This is followed by a discussion of some of the existing models for the behaviourof concrete in compression and in tension at elevated temperatures. Finally, the material modelfor the reinforcement at elevated temperatures is described.

5.2 Decay of Material Strength

As described in chapter 3, the material strengths of both the plain concrete and the reinforcementdecrease when the temperature is increased. One existing model for the change in materialproperties is suggested by Hertz [7], who defines a reduction parameter which is a S-shapedfunction of the temperature. It is developed so that the same function can be used for the decayof all material properties of a given material:

)(T ) = k +1" k

1 +T

T1+

!T

T2

$2

+!

T

T8

$8

+!

T

T64

$64 (5.1)

The parameter )(T ) is the ratio of the material property at ambient temperatures to thatat elevated temperatures and the constants k, T1, T2, T8 and T64 depends on the consideredmaterial.

5.2.1 Compressive Strength of Concrete

For computations of the decay of the material properties of concrete, using the method suggestedby Hertz [7], the parameters depend on the type of aggregate used. In the hot phase of a firethese are specified in Table 5.1 for siliceous, main group and light aggregate concretes.

41

Chapter 5: Existing Models of the Behaviour of Reinforced Concrete at Elevated Temperatures

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

k T1 T2 T8 T64

Siliceous concrete 0.00 15000 800 570 100000

Main group concrete 0.00 100000 1080 690 1000

Light aggregate concrete 0.00 100000 1100 800 940

The decay of compressive strength as a function of temperature is tabulated in Eurocode 2 [21].The code distinguishes between two di!erent types of aggregates; siliceous and calcerous. InTable 5.2 the reductions of compressive strength, fcT /fcm, recommended by Eurocode 2 [21],are given for temperatures ranging from 20!C to 1200!C. Further, the two parameters describingthe compressive stress-strain relationship, %c1T and %cu1T , are given.

Table 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).

(a)

T

20!C100!C200!C300!C400!C500!C600!C700!C800!C900!C1000!C1100!C1200!C

(b) Siliceous Aggregates

fcT /fcm %c1T %cu1T

1.00 0.0025 0.02001.00 0.0040 0.02250.95 0.0055 0.02500.85 0.0070 0.02750.75 0.0100 0.03000.60 0.0150 0.03250.45 0.0250 0.03500.30 0.0250 0.03750.15 0.0250 0.04000.08 0.0250 0.04250.04 0.0250 0.04500.01 0.0250 0.04750.00 - -

(c) Calcerous Aggregates

fcT /fcm %c1T %cu1T

1.00 0.0025 0.02001.00 0.0040 0.02250.97 0.0055 0.02500.91 0.0070 0.02750.85 0.0100 0.03000.74 0.0150 0.03250.60 0.0250 0.03500.43 0.0250 0.03750.27 0.0250 0.04000.15 0.0250 0.04250.06 0.0250 0.04500.02 0.0250 0.04750.00 - -

In Figure 5.1 the reduction factor suggested by Hertz [7], computed by (5.1), using the param-eters of Table 5.1, are compared with the strength ratio of Eurocode 2 [21], Table 5.2.

It is seen by Figure 5.1 that the approaches of Eurocode 2 [21] and of Hertz [7] generally correlate.However, Figure 5.1a shows that for siliceous aggregates exposed to temperatures above 500!C,the compressive strengths computed by Hertz [7] are lower than those recommended by Eurocode2 [21]. This is due to the fact that the strengths in Eurocode 2 [21] are based on transient straintests where the strain is held constant in the concrete and the temperature is varied. This typeof test is, according to Hertz [7], known to yield strengths of up to 25% greater magnitudes thanif data is collected by holding the temperature constant and varying the strain.

42


200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

1

T [oC]

fcT /

fcm

Eurocode 2

Hertz

200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

1

T [oC]

fcT /

fcm

Eurocode 2

Hertz

200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

1

T [oC]

fctT

/ f

ct,

m

Eurocode 2

Hertz, Sciliceous

Hertz, Main group

Hertz, Light aggregates

(a) Siliceous aggregates

200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

1

T [oC]

fcT /

fcm

Eurocode 2

Hertz

200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

1

T [oC]

fcT /

fcm

Eurocode 2

Hertz

200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

1

T [oC]

fctT

/ f

ct,

m

Eurocode 2

Hertz, Sciliceous

Hertz, Main group


(b) Calcerous aggregates

Figure 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 Table 5.1are used and the reduction presented in Eurocode 2 [21] is given in Table 5.2.

In Eurocode 2 [21] the residual properties after exposure to a temperature T is considered to beequal to the properties at high temperatures. The parameters used to compute the strength ofconcrete after temperature exposure, as suggested by Hertz [7], using equation (5.1), are givenin Table 5.3.

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

k T1 T2 T8 T64

Siliceous concrete 0.00 3500 600 480 680

Main group concrete 0.00 10000 780 490 100000

Light aggregate concrete 0.00 4000 650 830 930

As described in section 3.2.1, the strength of concrete is further reduced upon cooling afterexposure to a temperature elevation. The strengths computed by Eurocode 2 [21] are thereforebound to be higher than those modelled by Hertz [7], which also is evident in Figure 5.2.

43


200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

1

T [oC]

fcT /

fcm

Eurocode 2

Hertz

200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

1

T [oC]

fcT /

fcm

Eurocode 2

Hertz

200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

1

T [oC]

fctT

/ f

ct,

m

Eurocode 2

Hertz, Sciliceous

Hertz, Main group


(a) Siliceous aggregates

200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

1

T [oC]

fcT /

fcm

Eurocode 2

Hertz

200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

1

T [oC]

fcT /

fcm

Eurocode 2

Hertz

200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

1

T [oC]

fctT

/ f

ct,

m

Eurocode 2

Hertz, Sciliceous

Hertz, Main group


(b) Calcerous aggregates

Figure 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 computed by equation(5.1) with the parameters from Table 5.3 and the reduction from Eurocode 2 [21]is given in Table 5.2.

5.2.2 Tensile Strength of Concrete

The same considerations are largely valid for the reduction of the tensile strength of concreteafter exposure to elevated temperatures as for the compressive strength. However, as Eurocode2 [21] does not generally recommend that the tensile properties of concrete are taken intoconsideration, a simple bi-linear function for the reduction of tensile strength is given, as shownin Figure 5.3.

200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

1

T [oC]

fcT /

fcm

Eurocode 2

Hertz

200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

1

T [oC]

fcT /

fcm

Eurocode 2

Hertz

200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

1

T [oC]

fctT

/ f

ct,

m

Eurocode 2

Hertz, Sciliceous

Hertz, Main

Hertz, Light

(a) Hot phase of fire

200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

1

T [oC]

fcT /

fcm

Eurocode 2

Hertz

200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

1

T [oC]

fcT /

fcm

Eurocode 2

Hertz

200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

1

T [oC]

fctT

/ f

ct,

m

Eurocode 2

Hertz, Sciliceous

Hertz, Main

Hertz, Light

(b) Cold phase of fire

Figure 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 and Table5.3 are used.

It is seen by Figure 5.3 that because Eurocode 2 [21] does not consider the aggregate type used, itlargely yields conservative tensile strengths in comparrison to the strength levels obtained usingthe expression presented by Hertz [7] in both the hot and the cold phase of a fire. However, the

44


tensile strength of concrete by Eurocode 2 [21] is seen to be insensitive to temperatures below200!C.

5.2.3 Strength of Reinforcement

The parameters for equation (5.1) in Table 5.4 defines the reduction of strength in the hot phasefor hot-rolled and cold-worked reinforcement, respectively, as suggested by Hertz [14].

Table 5.4: Parameters describing decay functions for reinforcement in the hot phase of a fireas presented by Hertz [14].

k T1 T2 T8 T64

Hot-rolled bars, 0.2% stress 0.00 6000 620 565 1100


Cold-worked bars, 0.2% stress 0.00 100000 900 555 100000


The parameters for computations of the strength reductions are given in Table 5.4 for both0.2% stress and for 2.0% stress. Here, 0.2% corresponds to the yield stress of the steel and2.0% corresponds to the ultimate strength of steel. These are both defined as it was found byHertz [14] that the tabular values for strength reductions recommended by Eurocode 2 [21] wereassociated with the ultimate stress and not the yield stress as the code specifies.

200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

1

T [oC]

fsy

T /

fsy

Eurocode 2

Hertz, 0.2% stress

Hertz, 2.0% stress

200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

1

T [oC]

fsy

T /

fsy

Eurocode 2

Hertz, 0.2% stress

Hertz, 2.0% stress

0 0.05 0.1 0.15 0.20

100

200

300

400

500

T [oC]

fsy

T [

MP

a]

T = 20oC

T = 300oC

T = 500oC

T = 600oC

T = 700oC

T = 1100oC

(a) Hot-rolled

200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

1

T [oC]

fsy

T /

fsy

Eurocode 2

Hertz, 0.2% stress

Hertz, 2.0% stress

200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

1

T [oC]

fsy

T /

fsy

Eurocode 2

Hertz, 0.2% stress

Hertz, 2.0% stress

0 0.05 0.1 0.15 0.20

100

200

300

400

500

T [oC]

fsy

T [

MP

a]

T = 20oC

T = 300oC

T = 500oC

T = 600oC

T = 700oC

T = 1100oC

(b) Cold-worked

Figure 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.

In Figure 5.4 the decay of strength of hot-rolled and cold-worked reinforcement exposed to ele-vated temperatures are shown as functions of the temperature, computed by the tabular valuesof Eurocode 2 [21] and based on the method by Hertz [14]. The latter makes use of the pa-rameters in Table 5.4 and is plotted for both the yield stress (0.2% stress) and the ultimatestress (2.0% stress). The problem discovered by Hertz [14] is clearly illustrated, and thereforeit is chosen herein to compute the reduction of the strength of the reinforcement by equation(5.1), making use of the parameters specified for 0.2% stress to ensure that the strength of the

45


reinforcement is not overestimated.

The parameters used to compute the reduction of strength in the cold phase of a fire by themethod presented by Hertz [14], are given in Table 5.5.

Table 5.5: Parameters describing decay functions for reinforcement in the cold phase of fire aspresented by Hertz [14].

k T1 T2 T8 T64





The hot-rolled reinforcement regains it strength post-fire and the parameters of Table 5.5 there-fore yield a constant value of )(T ) =1 for all T , when applied to equation (5.1). As described insection 3.2.2, cold-worked reinforcement does not regain full strength after exposure to temper-atures above 400!C and the decay of the additional strength post-fire is illustrated in Figure 5.5.It is seen that the in method by Hertz [14] 50% of the yield strength (0.2% stress) is remainingat T = 700!C and above and 60% strength for the ultimate strength (2.0% stress) at 800!C andabove. The strength reduction of the cold-worked steel is not considered by Eurocode 2 [21],which therefore overestimates the post-fire strength of the reinforcement for high temperatures.After exposure to elevated temperatures, the lower strength of the reinforcement can thereforecause failure if the member is assumed to have regained its previous strength.

200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

1

T [oC]

fsy

T /

fsy

Eurocode 2

Hertz, 0.2% stress

Hertz, 2.0% stress

200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

1

T [oC]

fsy

T /

fsy

Eurocode 2

Hertz, 0.2% stress

Hertz, 2.0% stress

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

5.3 Uniaxial Compressive behaviour of Concrete at Ele-vated Temperatures

Youssef and Moftah [22] provide a review of the existing models for uniaxial compressive be-haviour of concrete and the choice of models to be reviewed herein is largely based on theirrecommendations. Common for most models is that the total strain of the mechanical stress-

46


strain relationship is considered to consist of a number of components. Li and Purkiss [23]considers three strain components and the total strain is therefore given by:

%tot = %!T + %th + LITS (5.2)

Here, %!T is the instantaneous stress-related strain from the applied load and %th is the unre-strained thermal strain arising from the expansion caused by the temperature elevation. TheLITS are the load induced thermal strains which is the e!ect on the thermal expansion causedby the presence of loads during first time heating.

5.3.1 Strain Components at Elevated Temperatures

Instantaneous Stress-Related Strain

The instantaneous stress-related strain, %!T , is a function of the applied stress and the temper-ature. It has its peak value at %0T and the initial modulus of elasticity, EciT , defines the shapeof the stress-strain curve. Expressions for these two parameters will therefore be given, beforethe stress-strain relationship is elaborated upon.

The strain value for the peak stress at a elevated temperatures, %0T , can account for di!erentlevels of applied compressive stress prior to heating if expressed by the formulation of Terro [24]:

%0T = (50'2L + 15'L + 1) · %01 + 20 · ('L " 5'2

L) · %02 + 5 · (10'2L " 'L) · %03 (5.3a)

Here, 'L is the initial compressive stress level and %01, %02 and %03 are expressed by:

%01 = 2.05 · 10#3 + 3.08 · 10#6 · T + 6.17 · 10#9 · T 2 + 6.58 · 10#12 · T 3 (5.3b)%02 = 2.03 · 10#3 + 1.27 · 10#6 · T + 2.17 · 10#9 · T 2 + 1.64 · 10#12 · T 3 (5.3c)%03 = 0.002 (5.3d)

The initial modulus of elasticity was proposed by Anderberg and Thelandersson [25] to be givenby:

EciT =2fcT

%0T(5.4)

Equation (5.4) enables to for the initial E-modulus to implicitly account for the type of aggregate,when implementing the reduced strength proposed by Hertz [7]. As a result of using expressions(5.3) and (5.4), it is possible to account for the e!ect of the initial compressive load and the ag-gregate, as well as the temperature, in the computation of the instantaneous stress-related strain.

There exists sereval models for the stress-strain relationship for the instantaneous stress-relatedstrain. In Youssef and Moftah [22] the models by Anderberg and Thelandersson [25], Lie andLin [26] and Schneider are described. However, as the model proposed by Scheneider does notconsider the post-peak behaviour, only the models by Anderberg and Thelandersson [25] andLie and Lin [26], will be evaluated herein.

The compressive stress-strain relationship for the instantaneous stress-related strain is by An-derberg and Thelandersson [25] modelled by a parabola for the ascending branch and assumes

47


linear softening:

!cT = EciT ·!

%!T "%2

!T

2%0T

$for %!T ! %1 (5.5a)

!cT = !1 " 880 · (%!T " %1) for %!T % %1 (5.5b)

In (5.5b) the parameter !1 must be entered in MPa and the parameters %1 and !1 are given bythe following, where the initial elastic modulus, EciT , also must be defined in MPa:

%1 = %0T ·!

1" 880EciT

$and !1 = EciT ·

!%1 "

%21

2%0T

$(5.5c)

The model for the stress-strain relationship for the instantaneous stress-related strain developedby Lie and Lin [26] assumes parabolic functions for both the ascending and descending branches:

!cT = fcT ·*1"

!%0T " %!T

%0T

$2+for %!T ! %0T (5.6a)

!cT = fcT ·*1"

!%!T " %0T

3%0T

$2+for %!T % %0T (5.6b)

The two considered models for the instantaneous stress-related strains are plotted in Figure5.6 for a siliceous concrete at ambient temperature and elevated temperatures of T = 300!C,T = 500!C and T = 700!C.

0 0.01 0.02 0.03 0.040

0.2

0.4

0.6

0.8

1

!"T

" c

T/f

cm

0 0.01 0.02 0.03 0.040

0.2

0.4

0.6

0.8

1

!"T

" c

T/f

cm

Lie and Lin

Anderberg and ThelanderssonT = 20oC

T = 300oC

T = 500oC

T = 700oC

(a) T = 20!C and T = 300!C

0 0.01 0.02 0.03 0.040

0.2

0.4

0.6

0.8

1

!"T

" c

T/f

cm

0 0.01 0.02 0.03 0.040

0.2

0.4

0.6

0.8

1

!"T

" c

T/f

cm

Lie and Lin

Anderberg and ThelanderssonT = 20oC

T = 300oC

T = 500oC

T = 700oC

(b) T = 500!Cand T = 700!C

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

From Figure 5.6, it is clear that the ultimate strains predicted by the models di!er significantly.However, due to the lack of expirimental investigations, it is, according to Youssef and Moftah[22], not possible to determine which model that provides the most accurate results.

Unrestrained Thermal Strain

The unrestrained thermal strain results from the thermal expansion caused by the elevatedtemperatures. This can therefore only contribute to the mechanical stress-strain relation if the

48


considered specimen is constrained against expansion.

A variety of models of the unrestrained thermal strain, %th, exists and Youssef and Moftah [22]concludes that the model in Eurocode 2 [21] correlates the best with the expirimental observa-tions. Herein, this model is therefore utilized.

Eurocode 2 [21] gives the following expression for concrete with siliceous aggregates:

%th = "1.8 · 10#4 + 9 · 10#6 · (T " 20!C) + 2.3 · 10#11 · (T " 20!C)3 ! 14 · 10#3 (5.7a)

For concrete with carbonate aggregates the expression is given by:

%th = "1.2 · 10#4 + 6 · 10#6 · (T " 20!C) + 1.4 · 10#11 · (T " 20!C)3 ! 12 · 10#3 (5.7b)

Load Induced Thermal Strains

Law and Gillie [27] explains how the LITS covers a number of di!erent strain componentsin heated concrete; the transitional thermal creep, the drying creep and the transient strain.The transitional thermal creep develops irrecoverably during the first time heating of sealedconcrete under load and is the largest component of the LITS. The drying creep is the shrinkageexperienced by the material due to the evaporation of water, whereas the transient strain refersto the sum of the transitional thermal strain and the drying creep, where the drying creepis most often omitted because it is very small comparred to transitional thermal strain. It istherefore chosen herein as well.

tot

C]o[100 200 300 400 500 600

T

Free thermal expansion

LITS

!

Net thermal expansion under pre!stress

Figure 5.7: Illustration of the di!erence between the total strain when heated with and withoutapplied stress. Reproduced from Law and Gillie [27].

Figure 5.7 shows how the LITS can have a significant e!ect on the total strain of a concretemember at elevated temperatures. The total strain is plotted as a function of the temperaturefor an unloaded specimen that thus experiences free thermal expansion and of a specimen thatis pre-loaded. The di!erence between the two curves illustrates the e!ect of the LITS.

As a result of the fact that the LITS only occurs during first time heating, and because it isirrecoverable upon cooling, Law and Gillie [27] argue that it is necessary to define the contri-bution to the strain by the LITS as a plastic deformation. However, this is not the case in thefew existing models for the mechanical stress-strain relationship where the LITS are included.According to Law et al. [28] there is no way to determine which of the existing methods thatprovides the most accurate results and therefore it is chosen herein to consider the two most

49


simple models; the model by Li and Purkiss [23] and the model presented in Eurocode 2 [21].

The model by Li and Purkiss [23] is a simplification of the Anderberg and Thelandersson modelincluding the transient strain e!ect. The initial load is taken as 30% of the compressive strengthand the model does not allow other levels to be considered. The compressive concrete behaviouris modelled by a simple bi-linear relationship and an empirical formula is developed for thestrain at peak stress, %"cu:

%"cu =2fcm

Ec+ 0.21 · 10#4 · (T " 20!C)" 0.9 · 10#8 · (T " 20!C)2 (5.8)

The tangent modulus in the descending branch at temperature T is expressed by:

E"p = "880 · ekp(T#20!C)2.15

(5.9)

where the parameter kp should be taken as 10#6.

Eurocode 2 [21] does not distinguish between any strain component, but it vaguely states thattransient e!ects are included to some extent. The stress-strain relation prior to peak stress isgiven by the following, after which linear softening is assumed:

!cT =3 · %cT fcT

%c1T ·*2 +

!%cT

%c1T

$3+ (5.10)

In expression (5.10), %c1T is the strain at peak stress. This and the ultimate strain, %cu1t, isgiven for a variety of temperatures in Table 5.2.

0 0.01 0.02 0.03 0.04

0

0.2

0.4

0.6

0.8

1

!"T

" c

T/f

cm

0 0.01 0.02 0.03 0.04

0

0.2

0.4

0.6

0.8

1

!"T

" c

T/f

cm

Li and Purkiss

Eurocode 2T = 20

oC

T = 300oC

T = 500oC

T = 700oC

(a) T = 20!C and T = 300!C

0 0.01 0.02 0.03 0.04

0

0.2

0.4

0.6

0.8

1

!"T

" c

T/f

cm

0 0.01 0.02 0.03 0.04

0

0.2

0.4

0.6

0.8

1

!"T

" c

T/f

cm

Li and Purkiss

Eurocode 2T = 20

oC

T = 300oC

T = 500oC

T = 700oC

(b) T = 500!C and T = 700!C

Figure 5.8: Compressive stress-strain relations as defined by Li and Purkiss [23] and Eurocode2 [21] for siliceous concrete at T = 20!C and T = 300!C, (a), and T = 500!C andT = 700!C, (b).

In Figure 5.8 the compressive model in Eurocode 2 [21] is compared to the model by Li andPurkiss [23] for a range of temperatures. It is seen that for T = 500!C and above, both thestrain at peak stress and the ultimate strain is significantly higher in the Eurocode 2 [21] modelthan in the model by Li and Purkiss [23]. However, common for both models is that the strain

50


at peak stress is increased as the temperature rises.

It is widely recognized that the LITS changes the E-modulus in the direction where the load ispresent, but it has been suggested by Law et al. [28] that the unloaded direction has a di!erentconstitutive relation. However, there is a severe lack of information on the response in the di-rections that the load is not applied in, as it has not been experimentally investigated.

For simplicity it is chosen herein to ignore the LITS in the formulation of a fracture energybased model at elevated temperatures. If initial load is present this assumption will result ina di!erent distribution of the stresses modelled within the considered member to the stressesactually occurring. However, this can prove to be a safe assumption in a structural context, if,for example, considering a line of columns where one column is more heavily loaded then theothers. The presence of the LITS will then result in a smaller expansion of the heavily loadedcolumn, which ultimately redistribes the loads to the other columns in the line. As the materialstrength decays, ignoring the LITS can in this case be considered a safer scenario.

5.4 Uniaxial Tensile behaviour of Concrete at Elevated Tem-peratures

The knowledge on the tensile performance of concrete at elevated temperatures is limited. Terro[24] suggests modelling the concrete behaviour as linear post-peak with an ultimate strain of%ctuT 0.004. This will yield the stress-strain relation illustrated in Figure 5.9 for a range oftemperatures.

0 1 2 3 4

x 10!"

0

0.2

0.4

0.6

0.8

1

!"T

"ctT

/fctm

T = 20oC

T = 300oC

T = 500oC

T = 700oC

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

As noted in section 3.2.2, the bond strength between the concrete and the reinforcement de-creases when the temperature is increased. Khoury [12] describes that the reduction of the bondstrength most often is modelled as a reduction of the tensile strength. However, according toYoussef and Moftah [22] few models exist and there is a large scatter in the available experi-mental results due to a lack of uniformity in the test procedures. Due to this uncertainty, andfor simplicity, the reduction of the bond strength is not considered herein.

51


5.5 Reinforcement Model at Elevated Temperatures

It has been shown by Hertz [14], and is it also implemented in the stress-strain relationshipdefined in Eurocode 2 [21], that the yield strain of steel is the same at all temperatures. Thismeans that the elastic modulus at elevated temperatures can be expressed as:

EsT =fyT

fy/Es= )(T ) · Es (5.11)

The reinforcement is still modelled as a bi-linear relationship at elevated temperatures as il-lustrated in Figure 5.10. The strength reduction is modelled using the procedure presented byHertz [14], and the model can therefore both consider hot-rolled and cold-worked steel bars. Tocompute the reduced yield strength, the parameters for hot-rolled or cold-worked steel stress0.2% of Table 5.4 are used.

0 0.002 0.004 0.006 0.008 0.01 0.012

0

100

200

300

400

500

!

"

T = 20oC

T = 200oC

T = 300oC

T = 400oC

T = 500oC

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

5.6 Overview of Relevant Assumptions for the Formulationof the Fracture Energy Based Material Models

In the subsequent chapter (chapter 6) the fracture energy based material models of reinforcedconcrete at elevated temperatures is formulated. The formulation is based on the existing ma-terial models at elevated temperatures and therefore it is subject to a number of assumptions.The assumptions include:

• The strength reductions of both concrete and reinforcement caused by temperature riseare computed using equation (5.1) as suggested by Hertz [7] and Hertz [14].

• The strain at peak compressive stress is evaluated by using equation (5.3) as recommendedby Terro [24].

• The initial E-modulus at elevated temperatures of concrete is modelled by equation (5.4)as per Anderberg and Thelandersson [25].

• The unrestrained thermal expansion is modelled as defined by Eurocode 2 [21] in equation(5.7).

• The e!ect on the mechanical stress-strain relation caused by the LITS is ignored.

52


• The strain at yield stress of the reinforcement is assumed to be constant for all tempera-tures.

• The reduction of the bond strength between the concrete and the reinforcement is notincorporated in the tensile concrete model formulation.

53


54

Fracture Energy Based UniaxialMaterial Models at ElevatedTemperatures

6.1 Introduction

In this chapter the fracture energy based material models at elevated temperatures are formu-lated and examined. The examination includes an investigation of the evolution of the tensileand the compressive fracture energies at elevated temperatures as well as of the limits of appli-cation that were developed in chapter 4.

The formulations made in this chapter are based on models for the concrete behaviour at elevatedtemperatures and therefore the models are only valid for concrete exposed to high temperatures.Although out of the scope of the current investigation, it should be possible to extend the formu-lations to the residual stress-strain relationship, if existing models for such are applied.

6.2 Fracture Energy Based Compressive behaviour Modelfor Concrete at Elevated Temperatures

A compressive fracture energy based model for concrete behaviour at elevated temperaturesis developed based on the model for ambient temperature by Feenstra and de Borst [18]. Ascurrently no experimental data is available on the evolution of the fracture energy with temper-ature, the existing models for concrete behaviour at high temperatures are used to determinethe compressive fracture energy.

As concluded in chapter 5, the fracture energy based compressive behaviour model at elevatedtemperatures will consist solely of the instantaneous stress-related strain as the e!ects of theLITS are ignored. However, it is still interesting to investigate the e!ect of the LITS on thecompressive fracture energy at elevated temperatures. Therefore, expressions for the compressivefracture energy are developed for all the four previously described models (chapter 5). Herein,the two exiting models that solely consists of the instantaneous stress-related strain are:

• The model by Anderberg and Thelandersson [25] expressed by equation (5.5).

• The model suggested by Lie and Lin [26] given by equation (5.6).

55

Chapter 6: Fracture Energy Based Uniaxial Material Models at Elevated Temperatures

The existing models that includes the e!ects of the LITS are:

• The formulation by Li and Purkiss [23] expressed by equations (5.8-5.9).

• The Eurocode 2 [21] formulation given in equation (5.10).

6.2.1 Compressive Fracture Energy at Elevated Temperatures

The fracture energy based model for ambient temperature by Feenstra and de Borst [18] canreadily be extended to elevated temperatures as the equivalent strain at peak stress is definedby the E-modulus and the strength by equation (4.13). Extending this to elevated temperaturesyields:

&eT =43

fcT

EciT(6.1)

If not considering the e!ects of the LITS on the compressive behaviour at elevated tempera-tures, this is found to correlate well with the strain at peak stress from the existing elevatedtemperature models for all temperatures.

The ultimate strain in the model by Feenstra and de Borst [18] is defined by equation (4.14).Herein, this equation is extended to elevated temperatures, so that the ultimate strain at a giventemperature, &uCT , is a function of the compressive fracture energy at the same temperature,GcT :

&uCT = 1.5GcT

hfcm(6.2)

As a result, it is necessary to use the compressive fracture energy at elevated temperatures.In lack of experimental evidence on this subject, the compressive fracture energy at elevatedtemperatures is estimated by computing the compressive fracture energies that are intrinsic inthe existing elevated temperature models.

0

!

"

"

"

)(model

"0

cT

cm

cuT,modelT

cT

Gc / h

p

f

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

Figure 6.1 illustrates the stress-strain relation for a given considered existing elevated tempera-ture model where linear softening is assumed. If wanting to describe the model in terms of thecompressive fracture energy, it has been established in section 4.4, that the indicated grey area isequal to the compressive fracture energy divided by a corresponding element size, (GcT /h)model.To determine the size of the area, the plastic strain at peak stress, %p0, must be found. As theline that connects the points (%p0, 0) and (%0T , fcT ) has the slope of the initial E-modulus, itcan be expressed as:

!cT = EciT %cT " EciT %p0 (6.3a)

56


By insertion of the point (%0T , fcT ) into equation (6.3a), the plastic strain, %p0, can be estab-lished:

%p0 = %0T "fcT

EciT(6.3b)

The ultimate strain, %cuT,model, is determined from the given considered elevated temperaturemodel. When this is found, the compressive fracture energy divided by the correspondingelement size can be arrived at. For a model that considers linear softening, the enclosed areacan be expressed as:

(Gc/h)model =12fcT (%cuT,model " %p0) =

12fcT

!%cuT,model " %0T +

fcT

EciT

$(6.4)

For the considered elevated temperature model, the corresponding element size can be deter-mined by considering the ambient condition. Herein, the compressive fracture energy is com-puted by equation (4.10), and therefore the following expression is used:

hmodel =Gc

(GcT /h)model=

8.8&

fcm

(GcT /h)model(6.5)

When the corresponding element size is found, the fracture energy at elevated temperaturesis:

GcT,model = (GcT /h)modelhmodel (6.6)

6.2.2 Application of the Elevated Temperature Model by Anderbergand Thelandersson

As described in section 5.3, the material model by Anderberg and Thelandersson [25] for thecompressive behaviour of concrete at elevated temperatures is defined by equation (5.5). Theultimate strain in the model by Anderberg and Thelandersson [25], %cuT,AT , must be found for%1 ! %!T and, hence, by equation (5.5b). An expression can therefore be arrived at by letting!cT = 0 and substituting %!T = %cuT,AT :

%cuT,AT =!1

880+ %1 (6.7a)

In this, !1 and %1 are still expressed by (5.5c), where the considered temperature is implicitlyaccounted for.

As the model by Anderberg and Thelandersson [25] assumes linear softening, the compressivefracture energy divided by the corresponding element size can be found by combining equations(6.7a) and (6.4):

(GcT /h)AT =fcT

2

!%1 " %0T +

!1

880+

fcT

EciT

$(6.7b)

To determine the corresponding element size, a concrete grade C30 is considered with the ma-terial parameters at ambient temperature taken from Table 4.3. The strain at maximum stress,%0T , is calculated by equation (5.3), where no initial load is considered, i.e. 'L = 0, and the ini-tial E-modulus is found by equation (5.4). As a result, the following element size correspondingto the compressive fracture energy inherent in the model by Anderberg and Thelandersson [25](denoted with the subscript AT ) is found to be

hAT = 65 mm.

57


Figure 6.2 illustrates the fracture energy based formulation for the compressive behaviour ofconcrete at ambient temperature for the h = 65 mm element. A comparison with the originalmodel by Anderberg and Thelandersson [25] shows that the areas corresponding to (GcT /h) areequal, as they should be.

0 2 4 6 8

x 10!"

0

0.2

0.4

0.6

0.8

1

!

fcT/f

cm

Lie and Lin

Fracture Energy Based

0 0.01 0.02 #$#" 0.040

0.2

0.4

0.6

0.8

1

fcT/f

cm

Anderberg and Thelandersson


!

Figure 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.

6.2.3 Application of the Elevated Temperature Model by Lie andLin

In section 5.3 it is described that the compressive behaviour model of concrete at elevatedtemperatures as suggested by Lie and Lin [26] is expressed by equation (5.6). As the descendingbranch is not linear in this model, the indicated area in Figure 6.1 cannot be defined by equation(6.4). Instead, it is found as:

(GcT /h)LL =12(%0T " %p0) +

, "cuT,LL

"0T

fcT

*1"

!%!T " %0T

3%0T

$2+d%!T (6.8a)

The ultimate strain will occur in the descending branch, and hence equation (5.6b) will be usedto derive an expression for the ultimate strain based on the model by Lie and Lin [26], %cuT,LL.Substituting !cT = 0 and %!T = %cuT,LL and solving for %cuT,LL yields:

%cuT,LL = 4%0T (6.8b)

When carrying out the integration in equation (6.8a), and inserting the ultimate strain fromequation (6.8b) along with the plastic strain from equation (6.3b), the compressive fractureenergy divided by the corresponding element size of the Lie and Lin [26] model can be writtenas:

(GcT /h)LL =12

f2cT

EciT+ 2fcT %0T (6.8c)

For the previously considered example, it is found by equation (6.5), using (6.8c), that theelement size corresponding to the compressive fracture energy inherent in the model suggestedby Lie and Lin [26] is:

hLL = 300 mm

58


In Figure 6.3, the material model by Feenstra and de Borst [18] is compared to the model byLie and Lin [26] for an element size of h = 300 mm at ambient temperature. The comparison ismade using the same material properties as for the model based on Anderberg and Thelandersson[25] in Figure 6.2. The areas corresponding to the compressive fracture energy divided by theelement size are indicated for both formulations and it is, as expected, seen that they areequivalent.

0 2 4 6 8

x 10!"

0

0.2

0.4

0.6

0.8

1

!

fcT/f

cm

Lie and Lin


0 0.01 0.02 #$#" 0.040

0.2

0.4

0.6

0.8

1

fcT/f

cm



!

Figure 6.3: Compressive material model by Lie and Lin [26] and fracture energy based formu-lation with an element size of h = 300 mm for a concrete grade C30 at ambienttemperature.

6.2.4 Compressive Fracture Energies at Elevated Temperatures forModels Including the E!ect of the LITS

As the models by Li and Purkiss [23] and Eurocode 2 [21] include the e!ects of the LITS, it isnot possible to directly modify them into a stress-strain relation in the form of the model byFeenstra and de Borst [18]. However, it is still possible to define the plastic strain at peak stress,%p0, and thus to find an expression for the compressive fracture energy.

As described in section 5.3.1, the model by Li and Purkiss [23] is of triangular shape. As aresult, the plastic strain will equal zero, %p0 = 0, when the initial E-modulus reaches the peakstress. The compressive fracture energies will therefore simply be the areas under the triangularshaped curves illustrated in Figure 5.8.

The strain at peak stress, %"cu, is defined by equation (5.8) and the slope of the descendingbranch, E"

p , is given in equation (5.9). As a result, the descending branch can be described bythe line:

!cT = (%cT " %"cu)E"p + fcT (6.9a)

The ultimate strain of the elevated temperature model by Li and Purkiss [23] can thus be foundby substituting %cT = %cuT,LL and !cT = 0 into equation (6.9a), yielding:

%cuT,LP = %"cu "fcT

E"p

(6.9b)

Combining equations (6.9b) and (6.4) for a situation where %p0 = 0 yields the following expres-sion for the compressive fracture energy divided by the corresponding element size for the model

59


by Li and Purkiss [23]:

(GcT /h)LP =12

· fcT

!%"cu "

fcT

E"p

$(6.9c)

Considering the same example as previously, it is found by equations (6.5) and (6.9c) that theelement size corresponding to the compressive fracture energy inherent in the model by Li andPurkiss [23] is:

hLP = 63 mm

The model suggested in Eurocode 2 [21] is, as described in section 5.3.1, defined by the tabulatedvalues for the strain at peak stress, %c1t, and the ultimate strain, %cu1t, given in Table 5.2. Theascending branch is described by equation (5.10) and linear softening is assumed between thestrain peak stress, %c1t, and the ultimate strain, %cu1t. As the initial E-modulus is not includedin the formulation, it is necessary to develop an expression for it. This is done by computingthe tangent to equation (5.10) at the origin of the stress-strain relation:

EciT,EC =32

fcT

%c1t(6.10a)

By substituting %0T = %c1t and EciT = EciT,EC in equation (6.4), the compressive fracture energydivided by the corresponding element size, for the model in Eurocode 2 [21] becomes:

(GcT /h)EC =12

!%cu1t " %c1t +

fcT

EciT,EC

$(6.10b)

For the previously considered example, equation (6.5) and equation (6.10b) yields an elementsize, corresponding to the inherent fracture energy of

hEC = 150 mm

6.2.5 Comparison of Compressive Fracture Energies at Elevated Tem-peratures

From the equations obtained for the compressive fracture energy divided by the correspondingelement size for each of the four considered existing models, (GcT /h)model, and the found corre-sponding element sizes, hmodel, the compressive fracture energies at elevated temperatures arefound by equation (6.6). The element sizes obtained for the considered example are given inTable 6.1.

Table 6.1: Element sizes obtained corresponding to the compressive fracture eneregies inherentin the elevated temperature models by Anderberg and Thelandersson [25], hAT , Lieand Lin [26], hLL, Li and Purkiss [23], hLP , and Eurocode 2 [21], hEC , for theconsidered example.

hmodel

hAT 65 mmhLL 300 mmhLP 63 mmhEC 150 mm

From Table 6.1 it is seen that the magnitudes of the elements sizes that corresponds to theinherent fracture energies of the four discussed models di!er significantly. This highligts how

60


the current knowledge about the post-peak compressive behaviour is insu"cient and supportsthe necessecity of an experimental study of the matter.

In Figure 6.4 the obtained compressive fracture energies for the considered example of a con-crete grade C30, computed from the four models, are plotted as functions of the temperature.The strength reduction caused by temperature elevation is computed using equation (5.1), assuggested by Hertz [7], with the reduction parameters of Table 5.1 for a siliceous concrete.

Figure 6.4 shows that the compressive fracture energy obtained using the method suggestedby Li and Purkiss [23], yields results that are very similar to those obtained by the model ofAnderberg and Thelandersson [25]. However, this was expected as the model by Li and Purkiss[23] is based on the model by Anderberg and Thelandersson [25].

Further, it is seen that in the models by Lie and Lin [26] and Eurocode 2 [21] the compressivefracture energy increases until a certain temperature, after which a steep descent is visible. Forthe model by Lie and Lin [26], it is not until the temperature reaches 600!C that the compressivefracture energy becomes lower than the one at ambient temperature, whereas for the Eurocode 2[21] model, this is seen to occur at about 500!C. The model by Anderberg and Thelandersson [25]does not incorporate the increase at all, as it is evident that the compressive fracture energy isreduced for increasing temperatures. This again illustrates how there is thus a clear discrepancybetween the assumptions of the softening behaviour in the existing models. As there currently isno experimental evidence suggesting that one existing model is better than another, it is highlyencouraged that some experiments are made to investigate this further.

200 400 600 800 10000

0.5

1

1.5

T [oC]

GcT/G

c

200 400 600 800 10000

500

1000

1500

T [oC]

hm

axT [

mm

]

Lie and Lin


Li and Purkiss

Eurocode 2

Figure 6.4: Comparison of the evolutions with temperature of the compressive fracture energiesobtained when applying the methods of Anderberg and Thelandersson [25], Lieand Lin [26], Li and Purkiss [23] and Eurocode 2 [21] to equation (6.6), for thepreviously described example.

Furthermore, it is interesting that the presence of the LITS does not seem to have an e!ecton the compressive fracture energy, as the models including the e!ects of the LITS, the Li andPurkiss [23] and the Eurocode 2 [21] models, cannot be said to deviate significantly from themodels that consider solely the instantaneous stress-related strain. Therefore, this is also anissue it would be interesting to see researched experimentally.

61


6.3 Formulation of Fracture Energy Based Tensile Modelfor Concrete at Elevated Temperatures

The fracture energy based model of the tensile behaviour of concrete at elevated temperatures isbased on the material model by Terro [24] for plain concrete. This model has a linear softeningbranch and assumes an ultimate tensile strain of %ctuT = 0.004 for all temperatures.

6.3.1 Tensile Fracture Energy at Elevated Temperatures

As described in section 1.2, the tensile fracture energy for plain concrete members is the areabeneath the stress-plastic displacement curve. As the plastic displacement of an element isrelated to the plastic strain by #p = %ph, the area beneath the stress-plastic strain curve is equalto GfT /h, as illustrated in Figure 6.5.

= 0.004

G

p

ctuT

fT / h

fct,mfctT

Figure 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 strength atthe ambient temperature, fct,m.

Figure 6.5 clearly indicates that for an element of constant size h, the tensile fracture energydecays as the tensile strength of the concrete decreases. By the simple geometric relation forthe area of a triangle, the tensile fracture energy at elevated temperatures can be expressedas:

GfT =12fctT · %ctuh (6.11)

The tensile strength of concrete at an elevated temperature is given by the tensile strength atambient temperatures multiplied by the decay function, fctT = )(T )fct,m. This means thatequation (6.11) can be written as:

GfT =12)(T )fct,m%ctuh (6.12)

Equation (6.12) can now be expressed in terms of the tensile fracture energy at ambient tem-perature, Gf , as this is given by Gf = 1/2fct,m%ctuh. Rewriting yields:

GfT = )(T )12fct,m%ctuh = )(T )Gf (6.13)

From equation (6.13) it is clearly seen that the evolution of the tensile fracture energy is describedby the same S-shaped function that describes the decay of the strength.

62


6.3.2 Fracture Energy Based Tensile Model of Plain Concrete

For plain concrete at ambient temperature it is found, using equation (4.1b) for the ultimatestrain of concrete, that for an element size of h = 16.5 mm, the material model by Terro [24]yields the same result as the fracture energy based model by Feenstra and de Borst [18].

0 1 2 3 4

x 10!"

0

0.2

0.4

0.6

0.8

1

!

"ctT

/fct,

m

0 1 2 3 4

x 10!"

0

0.05

0.1

0.15

0.2

!

"ctT

/fct,

m


Terro

T = 500oC

T = 300oC

T = 20oC

T = 900oC

T = 1100oC

T = 700oC


Terro

(a) T = 20!C, T = 300!C and T = 500!C

0 1 2 3 4

x 10!"

0

0.2

0.4

0.6

0.8

1

!

"ctT

/fct,

m

0 1 2 3 4

x 10!"

0

0.05

0.1

0.15

0.2

!

"ctT

/fct,

m


Terro

T = 500oC

T = 300oC

T = 20oC

T = 900oC

T = 1100oC

T = 700oC


Terro

(b) T = 700!C, T = 900!C and T = 1100!C

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

In Figure 6.6 an element of plain concrete of grade C30 with size h = 16.5 mm is considered fora range of temperatures. The material parameters at ambient temperature are given by Table4.2 and the decay of strength with temperature elevation is computed using equation (5.1), assuggested by Hertz [7]. Further, the initial E-modulus is found by equation (5.4), as Terro [24]recommends, where the strain at compressive peak stress is found by equation (5.3), with noinitial load.

It is seen that the fracture energy based formulation correlates well with the model by Terro[24] for all temperatures.

6.3.3 Fracture Energy Based Tensile Model for Reinforced Concrete

As a result of the fact that the reinforced fracture energy complicates the expressions for thevalidity range of the fracture energy based models, it is herein chosen for simplicity not to ex-tend the formulation to elevated temperatures. This e!ectively means that the validity rangeof the fracture energy based elevated temperature model can be described by a lower and anupper limitation on the element size and a mimimum reinforcement ratio requirement for theconsideration of the interaction contribution of the tension sti!ening.

The tensile formulation is now expanded to include the interaction contribution described insection 4.2 as per Cervenka et al. [17]. The interaction contribution is still considered to be atri-linear function (Figure 4.3b) and is at elevated temperatures defined by the four strains; thestrain at peak stress, %0tT , the ultimate concrete strain, %cuT , the ultimate strain, %uT , and theyield strain of the reinforcement, %yT . These four are simply computed using the formulationsat ambient temperatures, with the material properties of the concrete and the reinforcement ata given elevated temperature. This means that they can be expressed as:

63


• The strain at peak tensile stress:

%0tT =fctT

EciT(6.14a)

• The ultimate concrete strain (extension of equation (4.1b)):

%cuT =2GfT

hfctT(6.14b)

• The ultimate strain (extension of equation (4.1c)):

%uT = %yT ""tsfctT

*s,effEsT(6.14c)

• The yield strain of the reinforcement:

%yT =fyT

EsT(6.14d)

It is apparent from equation (6.14), that the interaction contribution depends on both thetemperature of the concrete as well as the temperature of the reinforcement. As explained insection 3.3, the temperature of the concrete is not necessarily uniform due to the low thermalconductivity of concrete. Ultimately, this means than a thermal analysis must be conductedto determine the temperature of the reinforcement, before the interaction contribution can bedefined.

0 0.5 1 1.5 2 2.5

x 10!"

0

0.2

0.4

0.6

0.8

1

!

"ctT

/fct,

m

T = 20oC

#$%$"&&oC

T = 500oC

T = 700oC

0 0.5 1 1.5 2 2.5

x 10!"

0

0.2

0.4

0.6

0.8

1

!

"ctT

/fct,

m

T = 20oC

#$%$"&&oC

T = 500oC

T = 700oC

Figure 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.

Figure 6.7 illustrates the combined concrete and interaction contribution to the stress when thesteel and the concrete both are at temperature T . A reinforced member of concrete of gradeC30 is considered, with the material properties at ambient temperatures given in Table 4.2,with reinforcement consisting of steel Grade 500, Table 4.1. The considered member has thedimensions of b = 100 mm and t = 10 mm, and is assumed to be reinforced in one directionwith an reinforcement area of As = 7 mm2. This yields a reinforcement ratio of *s = 0.007.The strength level of the interaction contribution is taken as "ts = 0.7 and the element sizeis chosen as h = 100 mm. The strength reductions at elevated temperatures are computedas suggested by Hertz [7], utilizing the parameters from Table 4.1 and Table 4.2, respectively,for a siliceous concrete and hot-rolled reinforcement with 0.2% stress. The initial E-modulus

64


is computed using equation (5.4), in which the strain at maximum compressive stress is foundfrom equation (5.3), assuming no initial load, i.e. 'L = 0.

It should be noted that the loss of bond strength between the concrete the reinforcement isbeyond the current objective and therefore is not accounted for herein. If it had been, thestrength level of the interaction contribution, "ts, would most likely have been a!ected byassuming a lower value for higher temperatures.

6.4 Limits of Fracture Energy Based Models at ElevatedTemperatures

The ambient temperature validity range of the material models by Cervenka et al. [17] andFeenstra and de Borst [18] at ambient temperatures was developed in chapter 4. At elevatedtemperatures, the fracture energy based model is based on the models by Cervenka et al. [17]and Feenstra and de Borst [18] and, as a result, it is very relevant to examine the evolution ofthe limits caused by an increase in the temperature.

6.4.1 Limitations on the Element Size

As it is chosen herein not to considered the reinforced fracture energy at elevated temperatures,the limitation on the minimum element size is found by extending equation (4.2). The limitaitionof the minimum and the maximum element size imposed by the uniaxial tensile model is foundsimply by extension of equation (4.16) and the limit imposed by the uniaxial compressive modelis redefined by equating expressions (6.1) and (6.2) and solving for h. Combining the limitationsyields the following:

2GfT

fctT

!%yT "

"tsfctT

*s,effEsT

$ ! h ! min

&'''(

''')

2EciT GfT

f2ctT

98

EciT GcT

)(T )f2cm

(6.15)

In Figure 6.8, the limits on the maximum element size imposed by both the tensile (Figure 6.8a)and the compressive (Figure 6.8a) models are illustrated for the previously described example.The compressive limits are illustrated for the formulations of the compressive fracture energy atelevated temperature based on the four existing models considered.

65


200 400 600 800 10000

200

400

600

800

1000

T [oC]

hm

axT [

mm

]

hmaxT

ls

200 400 600 800 10000

500

1000

1500

T [oC]

hm

axT [

mm

]

Lie and Lin

Anderberg

and Thelandersson

Li and Purkiss

Eurocode 2

(a) hmaxT = 2EciT GfT

f2ctT

, where ls ! hmax

200 400 600 800 10000

0.5

1

1.5

T [oC]

GcT/G

c


Lie and Lin

Li and Purkiss

Eurocode 2

200 400 600 800 10000

500

1000

1500

T [oC]

hm

axT [

mm

]

Lie and Lin

Anderberg and

Thelandersson

Li and Purkiss

Eurocode 2

(b) hmaxT =9

8

EciT GcT

"(T )f2cm

Figure 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.

From Figure 6.8 it is seen that, for the considered reinforced concrete example, the maximumelement size is governed by the tensile model until about 500!C as the lowest value of hmaxT isfound in Figure 6.8a, provided that the compressive fracture energy is computed based on theLie and Lin [26] or the Eurocode 2 [21] formulation. It is evident that for this investigated sam-ple it will be valid to use an element of size h = 300 mm, which seems reasonable for practicalmodelling purposes of structures.

However, it is evident for all the found compressive fracture energies that the models are onlypractically applicable until a certain temperature, after which a very fine element configurationis required. It is seen that the models by Anderberg and Thelandersson [25] and Li and Purkiss[23] only can be applied for temperatures up to about 450!C using reasonable element sizes.Only very small elements can be applied in the range 450-600!C, after which it is clearly seenthat the model will no longer be able to produce meaningful results. When computing thecompressive fracture energies inherent in the Eurocode 2 [21] model or the model suggested byLie and Lin [26], it is seen that the validity range is increased. However very small element sizesare still required for temperatures above 600!C.

6.4.2 Minimum Reinforcement Ratio

As at ambient temperatures, there are restrictions on the minimum level of reinforcement thatcan be considered for the interaction formulation. Problems arise when the ultimate strain islarger than the ultimate concrete strain. The minimum reinforcement ratio at elevated temper-atures can be arrived at by extending equation (4.6):

*s,eff,minT ="tsfctT!

%syT " 2GfT

hfctT

$EsT

(6.16)

Figure 6.9 illustrates the combined concrete and interaction stress contribution from the previ-ously desrcibed example of a reinforced concrete member at elevated temperatures where thetemperature of the concrete and the steel are assumed to be equal. The member in the figure is

66


reinforced in the direction of the applied load, the p-direction, and two reinforcement ratios areconsidered, *p = 0.007 and *p = 0.0025.

0 0.5 1 1.5 2 2.5

x 10!"

0

0.2

0.4

0.6

0.8

1

!

"/f

ct,

m

#p

= 0.0025

#p

= 0.0025

#p

= 0.007

#p

= 0.007

0 0.5 1 1.5 2 2.5

x 10!"

0

0.2

0.4

0.6

0.8

1

!

"/f

ct,

m

#p

= 0.0025

#p

= 0.0025

#p

= 0.007

#p

= 0.007

T#$#"%%oC

T = 20oC

T = 500oC

T = 700oC

T = 20oC :

T#$#"%%oC :

T = 500oC :

T = 700oC :

0 0.5 1 1.5 2 2.5

x 10!"

0

0.2

0.4

0.6

0.8

1

!

"/f

ct,

m

#p

= 0.0025

#p

= 0.0025

#p

= 0.007

#p

= 0.007

0 0.5 1 1.5 2 2.5

x 10!"

0

0.2

0.4

0.6

0.8

1

!

"/f

ct,

m

#p

= 0.0025

#p

= 0.0025

#p

= 0.007

#p

= 0.007

T#$#"%%oC

T = 20oC

T = 500oC

T = 700oC

T = 20oC :

T#$#"%%oC :

T = 500oC :

T = 700oC :

Figure 6.9: Illustration of how the modelling of the combined concrete and interaction stresscontributions at di!erent 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.

From Figure 6.9 it is clear that the reinforcement ratio of *p = 0.0025 is too low for all tem-peratures. Further, the ultimate strain, %uT , is reduced as the temperature is increased andhence the problem evolves as the temperature increases. It is therefore of interest to plot therestriction of the minimum reinforcement ratio as a function of the temperature. This is donein Figure 6.10.

0 0.5 1 1.5 2 2.5x 10!3

0

0.2

0.4

0.6

0.8

1

!

"/f ct,

m

#s,eff = 0.0025#s,eff = 0.007#s,eff = 0.0025#s,eff = 0.007

0 0.5 1 1.5 2 2.5x 10!3

0

0.2

0.4

0.6

0.8

1

! "

/f ct,m

#s,eff = 0.0025#s,eff = 0.007#s,eff = 0.0025#s,eff = 0.007

200 400 600 800 1000 12004

6

8

10

12

14

x 10!3

T [oC]

# s,eff

#s,eff,minT#p = 0.0025#p = 0.007

Figure 6.10: Evolution of minimum reinforcement ratio for the example of a reinforced memberfrom Figure 6.9 as a function of the temperature.

A plot like the one presented in Figure 6.10 e"ciently checks whether the requirement of equationof (6.16) is fulfilled as it can readily be detected if the used reinforcement ratio, *s,eff , is lowerthan the minimum reinforcement ratio, *s,eff,minT , or not and hence whether the interaction

67


contribution can be considered or not. From Figure 6.10 it is seen that the considered reinforce-ment ratio of *p = 0.0025 is too low for all temperatures, whereas the ratio of *p = 0.007 canreadily be applied. It is interesting to note the distinct increase in the minimum reinforcementratio that occurs for large values of T . This tendency was also seen for other values of *s, notincluded herein. This illustrates the necessity to check if the requirement of equation (6.16) isfulfilled for all considered temperatures as a case where a reinforcement ratio would be su"cientat most temperatures can easily be imagined.

68

Numerical Example of aReinforced Concrete Slab atElevated Temperatures

7.1 Introduction

In this chapter, the fracture energy based material formulations at elevated temperatures areapplied to an example of a reinforced slab. The slab (Figure 7.1) spans over L = 4 m, has a withof W = 1 m and a depth of D = 150 mm. It is reinforced in the top of the cross-section withlongitudinal bars with the diameter +p = 8 mm and in the bottom with a diameter of +p = 6mm. Both at top and bottom, a spacing of s = 150 mm is used, yielding a reinforcement ratioof *p = 0.0071. The cover layer of all the reinforcement is c = 25 mm.

= 8 mm!

= 6 mm

= 1 mW

= 4 mL

= 150 mms

= 150 mmD

c = 25 mm

c = 25 mm

L = 4 m

= 10.7 kN/mq

50 mm

p

!p

x

y

Figure 7.1: Illustration of the dimensions of the considered reinforced slab.

The considered slab is fixed in both ends and loaded with a uniformly distributed load ofmagnitude q = 10.7 kH/m as illustrated in Figure 7.2.

69

Chapter 7: Numerical Example of a Reinforced Concrete Slab at Elevated Temperatures

= 8 mm!

= 6 mm

= 1 mW

= 4 mL

= 150 mms

= 150 mmD

c = 25 mm

c = 25 mm

L = 4 m

= 10.7 kN/mq

50 mm

p

!p

Figure 7.2: Illustration of the reinforced concrete slab considered in this example.

Further, the slab is exposed to a elevated temperature from its bottom surface which yields thetemperature distribution in Figure 7.3 through the depth of the slab. The source of the temper-ature elevation is assumed to be purely convective and radiation is hence not considered.

0 200 400 600 8000

50

100

150

T [oC]

D [mm]

715oC

388oC

196oC

137oC

101oC

72oC

50oC

Figure 7.3: Temperature profile within the considered slab.

The analysis is essentially divided into two parts; one where the mechanical load is appliedand one where the thermal load is applied. Both are applied linearly and over a time step oftFE = 1.00, which means that the total length of the analysis is tFE = 2.00. An overview of theanalysis time is given in Figure 7.4.

FE

1.000.00 2.00

Linear application of themechanical load

Linear application of thetemperature profile

Constant mechanical load

t

Figure 7.4: Overview of the time in the FE-analysis of the considered reinforced slab.

7.2 Parameters for the Uniaxial Material Models

The slab is considered to consist of concrete grade C30 and therefore the material propertiesgiven in Table 4.2 and Table 4.3 are used for the ambient condition. The reinforcement consistsof steel Grade 500 and at hence the material properties of Table 4.1 are describing the behaviour

70


at ambient temperature.

The compressive concrete model is computed by the extension of the formulation suggested byFeenstra and de Borst [18], where the equivalent plastic strain at peak, &eT , is given by equation(6.1) and the ultimate strain, &uCT , by equation (6.2). The tensile model is also based on thefound extension of the model by Feenstra and de Borst [18], given by equations (6.14a-6.14d)and the reinforcement is assumed to be bi-linear in both tension and compression.

At elevated temperatures, the decay of the tensile and compressive strength of concrete arecomputed as suggested by Hertz [7], using the decay parameters of Table 5.1 for a siliceousconcrete in equation (5.1). The initial E-modulus for concrete at elevated temperatures is foundas suggested by Anderberg and Thelandersson [25] by equation (5.4), where the strain at peakcompressive stress is found by the formulation of Terro [24] using equation (5.3).

The compressive fracture energy is computed as inherent in the Eurocode 2 [21] model for com-pressive concrete behaviour at elevated temperatures using equations (6.6) and (6.10b). Thetensile fracture energy is found by equation (6.13).

The reduction of the yield stress and the E-modulus of the reinforcement are found by equation(5.1), as suggested by Hertz [14], using the parameters for 0.2% stress of Table 5.4 for hot-rolledreinforcement.

7.3 Material Properties for the Thermal Analysis

A thermal analysis must be conducted to account for the stresses that arises within the slabas a result of the restriction of thermal expansion imposed by the fixed ends. As a results, itis necessary to define the thermal properties of the materials. As per common practice, theconductivity and the density of the concrete is taken as defined in Table 7.1.

Table 7.1: Parameters at ambient temperature used for the thermal analysis of concrete asrecommended by Teknisk Ståbi [29].

Density Conductivity Specific Heat

2400 kg/m3 1.7 W/(m!C) 900 J/(kg!C)

The thermal expansion is in ABAQUS computed by the function *EXPANSION which uses theuser-defined expansion coe"cient, ". According to Youssef and Moftah [22] the thermal expan-sion of concrete is typically found by the following linear function using the thermal expansioncoe"cient, "concrete:

%th = "concrete · (T " 20!C) (7.1a)

However, as it here is desired to evaluate the thermal expansion, %th, by equation (5.7a), asdescribed in section 5.3.1, an expression for the thermal expansion coe"cient is arrived at byrewriting equation (7.1a):

"concrete =%th

(T " 20!C)(7.1b)

71


For the concrete considered herein, the thermal expansion coe"cient as a function of the tem-perature is presented in Figure 7.5.

200 400 600 800 1000 1200

0

2

4

6

8

x 10!"

T [oC]

!"!

[oC!#]

concrete

Figure 7.5: Thermal expansion coe"cient for concrete, "concrete, as a function of the temper-ature for the considered example of a reinforced concrete slab.

The expansion of the steel is typically assumed to be linear with an expansion coe"cient of"steel = 1.248 · 10#5 at T = 100!C [29].

7.4 FE-Analysis

In the depth of the slab it is necessary to make a division into six elements in order to applythe temperature profile with reasonable accuracy. However, the magnitude of the element depthhas no e!ect on the definition of the material models, as this is not the direction where thelocalization of deformation is taking place. Nevertheless, it is necessary to investigate the validityrange for the element sizes of the material models in order to make an appropriate elementdivision along the span on the slab. By application of equation (6.15) as a functions of thetemperature, the validity range can be illustrated as the grey area indicated in Figure 7.6.

200 400 600 800 10000

500

1000

1500

T [oC]

h

[mm

]

hmaxT

, Compressive Parameters

hmaxT

, Tensile Parameters

hminT

Admissible values of h

Figure 7.6: Limits on the maximum and minimum element size, equation (6.15), as functionsof the temperature for the considered example of a reinforced slab.

72


From Figure 7.6 it is seen that it is not possible to perform analysis for slab temperatures above800!C. As the maximum temperature slab regarded in this example is 715!C, it is possible toobtain meaningful results. However, it is evident from Figure 7.6, that only very small elementscan be evaluated for temperatures exceeding 600!C. For the temperature considered herein,715!C, it is found that the element size must be chosen within the range;

72.5 mm ! h ! 129.6 mm

Further, it is necessary to verify that the interaction stress contribution can be evaluated withthe level of reinforcement that is present in the slab. This is done in Figure 7.7 by plottingequation (6.16) as functions of the temperature for the two extreme element sizes of h = 73 mmand h = 129 mm.

200 400 600 800 10000

0.005

0.01

0.015

0.02

0.025

T [oC]

!s,

eff

!

s,eff,minT

!p = 0.0071

200 400 600 800 10000

0.005

0.01

0.015

0.02

0.025

T [oC]

!s,

eff

!

s,eff,minT

!p = 0.0071

(a) h = 73 mm

200 400 600 800 10000

0.005

0.01

0.015

0.02

0.025

T [oC]

!s,

eff

!

s,eff,minT

!p = 0.0071

200 400 600 800 10000

0.005

0.01

0.015

0.02

0.025

T [oC]

!s,

eff

!

s,eff,minT

!p = 0.0071

(b) h = 129 mm

Figure 7.7: 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 sti!ening for the considered example of a reinforcement slab withelement sizes of h = 73 mm, (a), and h = 129 mm, (b).

From Figure 7.7 it is clearly seen that it is valid to consider the interaction stress contributionas a part of the tension sti!ening as the e!ective reinforcement ratio in the slabs is higher thanthe minimum requirement for both the extreme element sizes.

7.4.1 Element size h = 129 mm

The least CPU-heavy FE-analysis is made where the length of the slab is divided into 31 ele-ments, yielding an element size of h = 129 mm.

The uniaxial material models are defined for temperatures up to 715!C and are given in fullin appendix D. As examples, the compressive stress-strain relation and the tensile combinedconcrete and interaction stress contributions used for the FE-analysis are illustrated in Figure7.8 for T = 20!C, T = 300!C, T = 500!C and T = 715!C.

73


0 0.5 1 1.5 2 2.5

x 10!"

0

0.5

1

1.5

2

2.5

"

!cT

"ctT

[M

Pa]

0 0.005 0.01 0.015 0.020

10

20

"#

40

!cT

"cT [

MP

a]

T = 20oC

T$%$"##oC

T = 500oC

T = 715oC

T = 20oC

T$%$"##oC

T = 500oC

T = 715oC

(a) Compression

0 0.5 1 1.5 2 2.5

x 10!"

0

0.5

1

1.5

2

2.5

"

!cT

"ctT

[M

Pa]

0 0.005 0.01 0.015 0.020

10

20

"#

40

!cT

"cT [

MP

a]

T = 20oC

T$%$"##oC

T = 500oC

T = 715oC

T = 20oC

T$%$"##oC

T = 500oC

T = 715oC

(b) Tension

Figure 7.8: Material models for compression, (a), and tension, (b), for the reinforced slab withan element size of h = 129 mm.

The analysis is performed as a plane strain analysis where the slab is modelled using beamelements and the *REBAR function is used to define both layers of the reinforcement within theconcrete elements.

As the largest stresses in the slab will occur at the fixed ends, the elements in this region are thefocus of the post-processing. For the analysis of the contour plots obtained from the FE-analysisin ABAQUS, the three rows of elements neer the right hand sided fixed end, shown in Figure7.9, are considered.

Figure 7.9: Position of the considered element for the post-processing of the contour plots fromABAQUS.

In Figure 7.10 the obtained stresses in the x-direction for the elements in Figure 7.9 are shownat the analysis times where the temperature load is applied (tFE = 1.00), an intermediate time(tFE = 1.50) and when the temperature load is fully applied (tFE = 2.00).

74


(Avg: 75%)PE, PE11

+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00

(Avg: 75%)PE, PE11

!"#$%&'!&(

!"#)%&'!&(

!"#*%&'!&(

!"#(%"'!&(

!"#"%"'!&(

!$#%"('!&+

!)#%",'!&+

!*#%%&'!&+

!(#%%('!&+

!"#%%,'!&+

-)#)"%'!&*

-%#),.'!&+

-+#),*'!&+

(Avg: 75%)PE, PE11

!+#+(&'!&(

!(#$.&'!&(

!(#*(&'!&(

!(#&.&'!&(

!%#,(&'!&(

!%#".&'!&(

!"#)(&'!&(

!"#%.&'!&(

!.#(&,'!&+

!(#.&)'!&+

-,#$%+'!&*

-*#"$"'!&+

-$#,$"'!&+

(Avg: 75%)S, S11

!"#).*'-&,

!"#+.$'-&,

!"#"$%'-&,

!.#$+$'-&*

!*#$."'-&*

!(#&"('-&*

!+#*(+'-&(

-%#$%('-&*

-*#.$"'-&*

-.#.*$'-&*

-"#".('-&,

-"#+)$'-&,

-"#)),'-&,

(Avg: 75%)S, S11

!,#$(%'-&)

!,#(+,'-&)

!*#),&'-&)

!*#")*'-&)

!+#*.$'-&)

!+#&&('-&)

!(#+")'-&)

!%#.("'-&)

!%#%+,'-&)

!"#,,&'-&)

!"#&)+'-&)

!+#..('-&,

-$#)+)'-&*

(Avg: 75%)S, S11

!*#+)%'-&)

!*#&+('-&)

!+#,"('-&)

!+#".('-&)

!(#)*('-&)

!(#(%+'-&)

!%#.$+'-&)

!%#+,+'-&)

!%#&(*'-&)

!"#,&*'-&)

!"#")*'-&)

!)#+**'-&,

!(#"*.'-&,

(a) tFE = 1.00

(Avg: 75%)PE, PE11

+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00

(Avg: 75%)PE, PE11

!"#$%&'!&(

!"#)%&'!&(

!"#*%&'!&(

!"#(%"'!&(

!"#"%"'!&(

!$#%"('!&+

!)#%",'!&+

!*#%%&'!&+

!(#%%('!&+

!"#%%,'!&+

-)#)"%'!&*

-%#),.'!&+

-+#),*'!&+

(Avg: 75%)PE, PE11

!+#+(&'!&(

!(#$.&'!&(

!(#*(&'!&(

!(#&.&'!&(

!%#,(&'!&(

!%#".&'!&(

!"#)(&'!&(

!"#%.&'!&(

!.#(&,'!&+

!(#.&)'!&+

-,#$%+'!&*

-*#"$"'!&+

-$#,$"'!&+

(Avg: 75%)S, S11

!"#).*'-&,

!"#+.$'-&,

!"#"$%'-&,

!.#$+$'-&*

!*#$."'-&*

!(#&"('-&*

!+#*(+'-&(

-%#$%('-&*

-*#.$"'-&*

-.#.*$'-&*

-"#".('-&,

-"#+)$'-&,

-"#)),'-&,

(Avg: 75%)S, S11

!,#$(%'-&)

!,#(+,'-&)

!*#),&'-&)

!*#")*'-&)

!+#*.$'-&)

!+#&&('-&)

!(#+")'-&)

!%#.("'-&)

!%#%+,'-&)

!"#,,&'-&)

!"#&)+'-&)

!+#..('-&,

-$#)+)'-&*

(Avg: 75%)S, S11

!*#+)%'-&)

!*#&+('-&)

!+#,"('-&)

!+#".('-&)

!(#)*('-&)

!(#(%+'-&)

!%#.$+'-&)

!%#+,+'-&)

!%#&(*'-&)

!"#,&*'-&)

!"#")*'-&)

!)#+**'-&,

!(#"*.'-&,

(b) tFE = 1.50

(Avg: 75%)PE, PE11

+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00

(Avg: 75%)PE, PE11

!"#$%&'!&(

!"#)%&'!&(

!"#*%&'!&(

!"#(%"'!&(

!"#"%"'!&(

!$#%"('!&+

!)#%",'!&+

!*#%%&'!&+

!(#%%('!&+

!"#%%,'!&+

-)#)"%'!&*

-%#),.'!&+

-+#),*'!&+

(Avg: 75%)PE, PE11

!+#+(&'!&(

!(#$.&'!&(

!(#*(&'!&(

!(#&.&'!&(

!%#,(&'!&(

!%#".&'!&(

!"#)(&'!&(

!"#%.&'!&(

!.#(&,'!&+

!(#.&)'!&+

-,#$%+'!&*

-*#"$"'!&+

-$#,$"'!&+

(Avg: 75%)S, S11

!"#).*'-&,

!"#+.$'-&,

!"#"$%'-&,

!.#$+$'-&*

!*#$."'-&*

!(#&"('-&*

!+#*(+'-&(

-%#$%('-&*

-*#.$"'-&*

-.#.*$'-&*

-"#".('-&,

-"#+)$'-&,

-"#)),'-&,

(Avg: 75%)S, S11

!,#$(%'-&)

!,#(+,'-&)

!*#),&'-&)

!*#")*'-&)

!+#*.$'-&)

!+#&&('-&)

!(#+")'-&)

!%#.("'-&)

!%#%+,'-&)

!"#,,&'-&)

!"#&)+'-&)

!+#..('-&,

-$#)+)'-&*

(Avg: 75%)S, S11

!*#+)%'-&)

!*#&+('-&)

!+#,"('-&)

!+#".('-&)

!(#)*('-&)

!(#(%+'-&)

!%#.$+'-&)

!%#+,+'-&)

!%#&(*'-&)

!"#,&*'-&)

!"#")*'-&)

!)#+**'-&,

!(#"*.'-&,

(c) tFE = 2.00

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

From the stress contours in Figure 7.10 it is evident that the stress in x-direction (S11) doesnot continuously increase with the application of the elevated temperatures. It is seen that itdecreases after a certain peak, because the deformations occurring are in the plastic regime andthus that irreversible macro-cracking is occuring.

Figure 7.11 shows the obtained plastic strains in the x-direction for the elements in Figure7.9 are shown at the analysis times where the temperature load is applied (tFE = 1.00), anintermediate time (tFE = 1.50) and when the temperature load is fully applied (tFE = 2.00).When considering contours of the plastic strains in the x-direction (PE11), the evolution of theirriversible deformations are seen to be progressing as the temperature profile is applied.

75


(Avg: 75%)PE, PE11

+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00

(Avg: 75%)PE, PE11

!"#$%&'!&(

!"#)%&'!&(

!"#*%&'!&(

!"#(%"'!&(

!"#"%"'!&(

!$#%"('!&+

!)#%",'!&+

!*#%%&'!&+

!(#%%('!&+

!"#%%,'!&+

-)#)"%'!&*

-%#),.'!&+

-+#),*'!&+

(Avg: 75%)PE, PE11

!+#+(&'!&(

!(#$.&'!&(

!(#*(&'!&(

!(#&.&'!&(

!%#,(&'!&(

!%#".&'!&(

!"#)(&'!&(

!"#%.&'!&(

!.#(&,'!&+

!(#.&)'!&+

-,#$%+'!&*

-*#"$"'!&+

-$#,$"'!&+

(Avg: 75%)S, S11

!"#).*'-&,

!"#+.$'-&,

!"#"$%'-&,

!.#$+$'-&*

!*#$."'-&*

!(#&"('-&*

!+#*(+'-&(

-%#$%('-&*

-*#.$"'-&*

-.#.*$'-&*

-"#".('-&,

-"#+)$'-&,

-"#)),'-&,

(Avg: 75%)S, S11

!,#$(%'-&)

!,#(+,'-&)

!*#),&'-&)

!*#")*'-&)

!+#*.$'-&)

!+#&&('-&)

!(#+")'-&)

!%#.("'-&)

!%#%+,'-&)

!"#,,&'-&)

!"#&)+'-&)

!+#..('-&,

-$#)+)'-&*

(Avg: 75%)S, S11

!*#+)%'-&)

!*#&+('-&)

!+#,"('-&)

!+#".('-&)

!(#)*('-&)

!(#(%+'-&)

!%#.$+'-&)

!%#+,+'-&)

!%#&(*'-&)

!"#,&*'-&)

!"#")*'-&)

!)#+**'-&,

!(#"*.'-&,

(a) tFE = 1.00

(Avg: 75%)PE, PE11

+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00

(Avg: 75%)PE, PE11

!"#$%&'!&(

!"#)%&'!&(

!"#*%&'!&(

!"#(%"'!&(

!"#"%"'!&(

!$#%"('!&+

!)#%",'!&+

!*#%%&'!&+

!(#%%('!&+

!"#%%,'!&+

-)#)"%'!&*

-%#),.'!&+

-+#),*'!&+

(Avg: 75%)PE, PE11

!+#+(&'!&(

!(#$.&'!&(

!(#*(&'!&(

!(#&.&'!&(

!%#,(&'!&(

!%#".&'!&(

!"#)(&'!&(

!"#%.&'!&(

!.#(&,'!&+

!(#.&)'!&+

-,#$%+'!&*

-*#"$"'!&+

-$#,$"'!&+

(Avg: 75%)S, S11

!"#).*'-&,

!"#+.$'-&,

!"#"$%'-&,

!.#$+$'-&*

!*#$."'-&*

!(#&"('-&*

!+#*(+'-&(

-%#$%('-&*

-*#.$"'-&*

-.#.*$'-&*

-"#".('-&,

-"#+)$'-&,

-"#)),'-&,

(Avg: 75%)S, S11

!,#$(%'-&)

!,#(+,'-&)

!*#),&'-&)

!*#")*'-&)

!+#*.$'-&)

!+#&&('-&)

!(#+")'-&)

!%#.("'-&)

!%#%+,'-&)

!"#,,&'-&)

!"#&)+'-&)

!+#..('-&,

-$#)+)'-&*

(Avg: 75%)S, S11

!*#+)%'-&)

!*#&+('-&)

!+#,"('-&)

!+#".('-&)

!(#)*('-&)

!(#(%+'-&)

!%#.$+'-&)

!%#+,+'-&)

!%#&(*'-&)

!"#,&*'-&)

!"#")*'-&)

!)#+**'-&,

!(#"*.'-&,

(b) tFE = 1.50

(Avg: 75%)PE, PE11

+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00+0.000e+00

(Avg: 75%)PE, PE11

!"#$%&'!&(

!"#)%&'!&(

!"#*%&'!&(

!"#(%"'!&(

!"#"%"'!&(

!$#%"('!&+

!)#%",'!&+

!*#%%&'!&+

!(#%%('!&+

!"#%%,'!&+

-)#)"%'!&*

-%#),.'!&+

-+#),*'!&+

(Avg: 75%)PE, PE11

!+#+(&'!&(

!(#$.&'!&(

!(#*(&'!&(

!(#&.&'!&(

!%#,(&'!&(

!%#".&'!&(

!"#)(&'!&(

!"#%.&'!&(

!.#(&,'!&+

!(#.&)'!&+

-,#$%+'!&*

-*#"$"'!&+

-$#,$"'!&+

(Avg: 75%)S, S11

!"#).*'-&,

!"#+.$'-&,

!"#"$%'-&,

!.#$+$'-&*

!*#$."'-&*

!(#&"('-&*

!+#*(+'-&(

-%#$%('-&*

-*#.$"'-&*

-.#.*$'-&*

-"#".('-&,

-"#+)$'-&,

-"#)),'-&,

(Avg: 75%)S, S11

!,#$(%'-&)

!,#(+,'-&)

!*#),&'-&)

!*#")*'-&)

!+#*.$'-&)

!+#&&('-&)

!(#+")'-&)

!%#.("'-&)

!%#%+,'-&)

!"#,,&'-&)

!"#&)+'-&)

!+#..('-&,

-$#)+)'-&*

(Avg: 75%)S, S11

!*#+)%'-&)

!*#&+('-&)

!+#,"('-&)

!+#".('-&)

!(#)*('-&)

!(#(%+'-&)

!%#.$+'-&)

!%#+,+'-&)

!%#&(*'-&)

!"#,&*'-&)

!"#")*'-&)

!)#+**'-&,

!(#"*.'-&,

(c) tFE = 2.00

Figure 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

The evolution of the plastic deformations can also be illustrated by considering the output of asingle element. As an example, element 2 is considered, the position of which is illustrated inFigure 7.12 along with the locations of its integration points.

P3 P4

P2P1

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

The obtained stresses in the x-direction, !x, and plastic strains, %px, in element 2 from the

FE-analysis are given in Figure 7.13 as functions of the analysis time, tFE .

It is seen that during the application of the temperature load at tFE , the deformations becomesincreasingly plastic. Furthermore, it is, as expected, seen that the position of the element andthe integration points within it, has an e!ect on the plastic deformations as the peak compressivestress is observed to be reached at a later time for integration points deeper into the slab.

76


0 0.5 1 1.5 2

!"

!#

!$

!%

0

x 107

tFE

!x

P1

P2

&#

&"

0 0.5 1 1.5 2

!'

!"

!#

!$

!%

0

x 10!#

tFE

"p x

P1

P2

&#

&"

(a) Stress in the x-direction

0 0.5 1 1.5 2

!"

!#

!$

!%

0

x 107

tFE

!x

P1

P2

&#

&"

0 0.5 1 1.5 2

!'

!"

!#

!$

!%

0

x 10!#

tFE

"p x

P1

P2

&#

&"

(b) Plastic strain in the x-direction

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

7.4.2 Element size h = 73 mm

The underlying assumption that states that the fracture energy based material model is meshindependent is briefly investigated by analyzing the slab using a finer mesh. The span is thereforedivided into 55 elements yielding an element size of h = 73 mm. The material models aredeveloped for this element configuration can be seen in full in appendix D.

!! !" # " !

$%"#&

#

'#

"##

"'#

!x

D ())*

h%+%"!,%))

h%+%-.%))

!& !' !/ !. !! !" #

$%"#-

#

'#

"##

"'#

!x

D ())*

h%+%"!,%))

h%+%-.%))

(a) tFE = 1.00

!! !" # " !

$%"#&

#

'#

"##

"'#

!x

D ())*

h%+%"!,%))

h%+%-.%))

!& !' !/ !. !! !" #

$%"#-

#

'#

"##

"'#

!x

D ())*

h%+%"!,%))

h%+%-.%))

(b) tFE = 2.00

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

Figure 7.14 shows the output from the two FE-analyses using h = 129 mm and h = 71 mm,respectively. The considered output is the stresses in x-direction through the depth of the slabat the left fixed end. Generally, except the dosagreement at the top of the slab at analysis timetFE = 2.00, the outputs of the two analyses are seen to be alike and better than the one obtainedwas not to be expected.

77


78

Conclusion

8.1 Remarks in Conclusion

Herein, the scope has been to extend the existing fracture energy based models for tensionand compression from ambient conditions to elevated temperatures. The existing models wereevaluated prior to the extension and it was found that it was necessary to define limits ofapplication in order to ensure meaningful results of a FE-analysis. A restriction on the admissibleelement size has therefore been developed by the formulation of a maximum element size, givenby equation (4.16). Further, if modelling the tension sti!ening through the definition of aninteraction stress contribution, it has been established that a su"cient level of reinforcementmust be ensured. This is most e"ciently verified by making sure that inequality (4.9) is fulfilled.The two requirements have been computed for a concrete grade C30 to examine the practicalimplications and it has been found that:

• For a concrete grade C30 a maximum element size of 756 mm is admissible which seemsreasonable for practical application.

• For a considered example of a reinforced plate with a reinforcement ratio of *p = 0.03(typically a reinforcement level of 2-3% is used in practice), the requirement on the mini-mum level of reinforcement is found to be fulfilled when the element size is 100 mm.

The extension of the fracture energy based models to elevated temperatures is largely done byconsidering the ambient models with the material properties of a given elevated temperature.The compressive concrete model is given by equations (6.1-6.2) and the tensile model for theconcrete and the interaction stress contribution of the tension sti!ening e!ect is expressed byequation (6.14). As experimental data on this is not currently available on the evolution ofthe tensile and compressive fracture energies at elevated temperatures, the fracture energiesinherent in the existing elevated temperature models have been computed. The compressivefracture energy has been found based on the models by Anderberg and Thelandersson [25], Lieand Lin [26], Li and Purkiss [23] and Eurocode 2 [21]. Of these models, the two first considerssolely the instantaneous stress-related strain whereas the two latter includes the e!ects of theLITS. The most important findings upon comparison of the four investigated compressive modelsare:

• A significant spread in the post-peak behaviours of the existing models has been found.

• It has been seen that the LITS cannot be said to have an e!ect on the compressive fractureenergy.

• Further, it is seen that the compressive fracture energy inherent in the models by Eurocode2 [21] and Lie and Lin [26] operates with an increase until a certain temperature in therange of 500-600!C after which the fracture energy will be lower than at ambient condition.

79

Chapter 8: Conclusion

• The evolutions of the compressive fracture energy computed using the models by Ander-berg and Thelandersson [25] and the Li and Purkiss [23] are seen not to incorporate anyincrease, but to decrease continuously following a S-shaped curve. The two have beenseen to be very closely related, however this is not accredited any special significanceas the model by Li and Purkiss [23] is based on the Anderberg and Thelandersson [25]formulation.

When investigating the current models for tensile behaviour of concrete at elevated temperaturesit is found that the knowledge base is extremely limited. One material formulation has beenidentified; the one suggested by Terro [24] and when computing the inherent fracture energy itis found that:

• The tensile fracture energy follows the decay function for the material strength and thuscan be described by equation (6.13).

The validity range for the element sizes of the fracture energy based elevated temperature modelhas been formulated and is given by expression (6.15). To illustrate the impact of a temperaturevariation on the validity range, the evolution of the admissible element sizes with temperaturehave been investigated for an example of concrete grade C30.

• It is found that up to about 500!C the maximum element size is governed by the tensileproperties after which the compressive parameters are governing;

• as a result, the maximum element size above 500!C depends greatly upon the chosenmethod for the computation of the compressive fracture energy. When applying the modelsby Anderberg and Thelandersson [25] and by Li and Purkiss [23], a very fine mesh isrequired for analysis of temperatures in the range 500-700!C, and analysis beyond 800!Cis not possible. However, the same material can be evaluated for temperatures up to1100!C, when the compressive fracture energy is computed based on the models suggestedby Eurocode 2 [21] and by Lie and Lin [26], but a fine mesh requirement is still demandedfor temperatures above 600!C.

At elevated temperatures, a requirement on the minium reinforcement ratio that has to bepresent if the tension sti!ening must include the interaction contribution has been defined byequation (6.16). For an example of a reinforced slab with *p = 0.0071, this is found to be fulfilledfor all temperatures.

It is found that the model developed herein can readily be applied for a FE-analysis of reinforcedconcrete at elevated temperatures, for example in ABAQUS, and, if the modelling is performedwithin the limits of application, it is possible to get mesh independent results of the analysiswith di!erent mesh configurations.

8.2 Suggestions for Future Research

In order to ensure proper use of the developed fracture energy based models at elevated tem-peratures, experimental studies of the evolution of the fracture energies with temperature arerecommended as future research:

• An experimental study of the compressive fracture energy at ambient and at elevatedtemperatures, including the e!ect of the LITS; and

• An experimental study of the tensile fracture energy at elevated temperatures.

80


Further, it should be noted, that the fracture energy based model developed herein is subjectto some simplifications. An expansion of the model at elevated temperatures to include someof the phenomena that is not considered herein is therefore also suggested as scope of futureinvestigations:

• The e!ect of the LITS on the compressive behaviour; both on the behaviour prior andpost-peak, and considering both the direction where the load is present as well as theresponse of the unloaded directions.

• The loss of bond strength between the concrete and the reinforcement caused by a tem-perature elevation. This could perhaps be modelled through a decay of the stress level ofthe interaction contribution.

• Expanding the model to the cold phase of a fire.

• Spalling of the concrete surface. This would demand for the model to be extended to acomprehensive thermo-hydro-mechanical finite element analysis in order to include calcu-lations on the pore water pressure.

81


82

References

[1] http://www.alphabetics.info/international/?p=3031 (Accessed 2011.01.07).

[2] fhm.fhsd.k12.mo.us (Accessed 2011.01.07).

[3] Ho!man, M. and Fleming, M. Chernobyl: The True Scale of the Accident. Press re-lease, International Atomic Energy Agency, World Health Organization and United NationsDevelopment Programme (2005).

[4] http://www.atmb.net/atmb/en/tunnel/25/the-mont-blanc-tunnel/the-history-of-the-tunnel/the-fire-of 1999.html (Accessed 2011.01.07).

[5] van Mier, J. G. M. Strain-Softening of Concrete Under Multiaxial Loading Conditions.Ph.D. thesis, University of Technology Eindhoven (1984).

[6] Pankaj, P. Finite Element Analysis in Strain Softening and Localisation Problems. Ph.D.thesis, University of Wales (1990).

[7] Hertz, K. D. Concrete Strength for Fire Safety Design. Magazine of Concrete Reseach,57(8), (2005), 445–453.

[8] SIMULIA. ABAQUS Version 6.7 Documentation. Dassault Systemes (2010). URL http://abaqus.civil.uwa.edu.au:2080/v6.7/.

[9] Pankaj, P. Real Structural Behavior and Its Analysis - Material Nonlinearity. Lecture Note(Accessed 2010.11.22). URL www.see.ed.ac.uk/~pankaj/Real_Structural_Behavior/.

[10] Fletcher, I. A., Welch, S., Torero, J. L., Carvel, R. O., and Usmani, A. Behaviourof Concrete Structures in Fire. Thermal Science, 11(2), (2007), 37–52.

[11] Mindess, S., Young, J., and Darwin, D. Concrete. Prentice Hall, Pearson Education,Inc. (2003).

[12] Khoury, G. A. E!ect of Fire on Concrete and Concrete Structures. Progress in StructuralEngineering and Materials, 2(4), (2000), 429–447.

[13] Nielsen, A. Bygningsmaterialer - Metallære. Polyteknisk Forlag (1998).

[14] Hertz, K. D. Reinforcement Data for Fire Safety Design. Magazine of Concrete Reseach,56(8), (2004), 453–459.

[15] Hertz, K. D. Limits of Spalling of Fire-Exposed Concrete. Fire Safety Journal, 38(2),(2003), 103–116.

[16] CEB. CEB-FIB Model Code. Thomas Telford (1993).

[17] Cervenka, V., Pukl, R., and Eligehausen, R. Computer Simulation of Anchoring Tech-nique in Reinforced Concrete Beams. In Bi"ani", N. and Mang, H., editors, ComputerAided Analysis and Deign of Concrete Structures, volume 1. Pineridge Press (1990), 1–21.

83

http://abaqus.civil.uwa.edu.au:2080/v6.7/

http://abaqus.civil.uwa.edu.au:2080/v6.7/

Appendix 8: Post-Peak Response of Concrete

[18] Feenstra, P. and de Borst, R. Constitutive Model for Reinforced Concrete. Journal ofEngineering Mechanics, ASCE, 121, (1995), 587–595.

[19] Nakamura, H. and Higai, T. Compressive Fracture Energy and Fracture Zone Lengthof Concrete. In Shing, P. B. and Tanabe, T.-A., editors, Modeling of Inelastic Behaviorof RC Structures under Seismic Loads. ASCE (2001), 471–487.

[20] Vonk, R. Softening of Concrete Loaded in Compression. Ph.D. thesis, Eindhoven Univer-sity of Technology (1992).

[21] CEN. Eurocode 2: Design of Concrete structures - Part 1-2: General rules - Structuralfire design, BS EN 1992-1-2:2004. Brithish Standard (2004).

[22] Youssef, M. and Moftah, M. General Stess-Strain Relationship for Concrete at ElevatedTemperatures. Engineering Structures, 29(10), (2007), 2618–2634.

[23] Li, L.-Y. and Purkiss, J. Stress-Strain Constitutive Equations of Concrete Material atElevated Temperatures. Fire Safety Journal, 40(7), (2005), 669–686.

[24] Terro, M. J. Numerical Modeling of the Behavior of Concrete Structures in Fire. ACIStructural Journal, 95(2), (1998), 183–193.

[25] Anderberg, Y. and Thelandersson, S. Stress and Deformation Characteristics of Con-crete at High Temperatures - 2. Expirimental Investigation and Material Behaviour Model.Lund Inistitute of Technology (1976).

[26] Lie, T. T. and Lin, T. D. Fire Performance of Reinforced Concrete Columns. Fire Safety:Science and Engineering, ASTM STP, 882, (1985), 176–205.

[27] Law, A. and Gillie, M. Load Induced Thermal Strain: Implications for Structual Be-haviour. SiF, Singapore (2008).

[28] Law, A., Gillie, M., and Pankaj, P. Incorporation of Load Induced Thermal Strain inFinite Element Models. Application of Structural Fire Engineering, Prauge, Czech Republic(2009).

[29] Teknisk Ståbi. Nyt Teknisk Forlag, 19 edition (2007).

84

Appendix A:

Detailed Description of Crackingand the Post-Peak Response ofConcrete

This appendix contains a detailed overview of the crack propagation causing the softening of thepost-peak response and the failure of the concrete. The softening arises due to the initiation ofcracks and their propagation and the post-peak response is the reduced load carrying capacitywith increasing deformation.

A.1 Crack Propagation and Softening

As described by Mindess et al. [11], concrete is a composite material with a microstructureconsisting of aggregate and cement paste separated by a interfacial transition zone. The het-erogeneous nature causes the behaviour to di!er significantly in tension and in compression asthe localized stresses in the material become very di!erent from the nominal stresses. In mostcases, the interfacial transition zone acts as a weak link when stresses are applied. The strengthof this zone is lower than the strength of the cement paste and bond failure can arise as a con-sequence of di!erences in elastic moduli between hardened paste and aggregate, but also due todi!erent coe"cients of thermal expansion and di!erent responses to change in moisture content;ultimately causing microcracking.

Bond failure

due to tension

and shear

Bond failure

due to tension

Compressive force

Intact bond

Potential shear plane

Tensile strain

Cement matrix

Aggregate

particle

Figure A.1: Idealization of stresses around a single aggregate particle. Reproduced from Min-dess et al. [11].

85

Appendix A: Post-Peak Response of Concrete

Crack propagation is more prone to occur under tensile loads than compressive as there arelow frictional forces, and hence the tensile strength is lower than the compressive. Vonk [20]describes how one of the prevailing beliefs is that failure in compression is really a secondarytensile failure induced by the application indirect tension. This is illustrated in Figure A.1,where compressive failure on the microscopic level is illustrated. Once microscopic cracking hascommenced, internal crack growth continues with force application to the element or structure.

On a macroscopic level the internal crack growth will result in reduced load carrying capacityoften termed as strain softening. This is evident in displacement-controlled test specimens,where it is clearly detected as the post-peak decrease of mechanical resistance.

peak

Kt

Kt < 0

peakw w

Pre Peak Post Peak

!

Microcracking

!

Formation of macrocracks

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

In Figure A.2 a characteristic nominal stress-deformation relation of a concrete test specimenundergoing a displacement controlled test is shown. The tangential sti!ness, Kt, continuouslydecreases in the pre-peak regime as the load is increased until it reaches zero at the peak load.The continuous post-peak decrease of the mechanical resistance caused by the continued increaseof the deformations is called softening and it is characterized by the descending branch of thestress-deformation curve, which has a negative tangential sti!ness. The softening at macroscopiclevel is caused by the crack growth as the material is gradually weakened when the internal bondsare broken at microscopic level. Initially the cracking starts as distributed microcracking. Thisis a stable process, which means that the cracking only grows when the load is increased. Aroundpeak load the formation of macrocracks begins. These cracks are unstable, which means thatthe load has to decrease to avoid unstable growth.

86

Appendix B:

ABAQUS Functions forDefinition of Uniaxial Behaviour,Embedment of Reinforcementand Load Steps

This appendix contains a brief description of the *TENSION STIFFENING and the *COMPRESSIONHARDENING functions that has to be defined in order to utilize the ABAQUS concrete damagedplasticity model. Further, the functions used to embed the reinforcement in the concrete surfacesand the to define the load steps are described.

B.1 Tension Sti!ening and Compression Hardening Mod-els

When creating a FE-model in ABAQUS, the softening of concrete is defined through *CONCRETETENSION STIFFENING for tension and in compression by the use of *CONCRETE COMPRESSIONHARDENING. The behaviour in both tension and compression are defined by equivalent damagedstrain parameters; the cracking strain, %ck

t , in tension and the inelastic strain, %inc , in compression.

These are defined as the total strain at the considered point minus the corresponding elasticstrain for the undamaged material, and are thus described by the following relations [8]:

%ckt = %t " !t/Ec (B.1)

%inc = %c " !c/Ec (B.2)

The cracking strain and the in-elastic strain for the ABAQUS definitions of tension sti!eningand compression hardening are illustrated in Figure B.1.

87

Appendix B: ABAQUS Functions

c

Ec

Ec

! tck

"t "c

!cin

Ec

"cu

"c0

Ec

! t

"t0

!

(a) Cracking strain for definition of tension sti!en-ing

c

Ec

Ec

! tck

"t "c

!cin

Ec

"cu

"c0

Ec

! t

"t0

!

(b) Inelastic strain for definition of compressionhardening

Figure B.1: Definition of cracking and inelastic strain. Reproduced from the ABAQUS Version6.7 Documentation [8].

For tension sti!ening, ABAQUS has the possibility of a fracture energy definition via specifyingTYPE=GFI, where the fracture energy is defined as a constant and hence the material tensiledefinition does not need to be altered for changes in the element side length. However, as the areaunderneath the stress-plastic strain curve is deemed to be triangular, the interaction contributionof the tension sti!ening e!ect defined by Feenstra and de Borst [18] (section 4.3.3) cannot beincluded in the model because the tension sti!ening input must be the combined concrete andinteraction contribution (tri-linear). It is therefore necessary to utilize the TYPE=STRAIN option,where the stress-strain diagram for the post-peak relation is point-wise defined.

B.2 Embedment of Reinforcement

As previously described, the reinforcement is modelled as rods embedded in the concrete sur-faces. This means that the end nodes of the steel rods are considered to be slave nodes to theconcrete master nodes, and thus, that the steel nodes follow the deformations of the concretenodes.

The function *TIE is used to define the slave and the master surfaces.

*TIE-Slave Surface-, -Master Surface-

B.3 Load Step Definition for Static Analysis

A load step in ABAQUS is created by *STEP and *END STEP, and in between these, the type ofanalysis is defined and the relevant load is applied.Herein, only static analyses are performed, and all loads are therefore defined using *STATIC.

Newton’s method is used in ABAQUS to solve nonlinear static problems [8]. This means thateach defined step is divided into increments in which a force is applied. When a force is applied,iterations are performed within it until the residual forces equals the applied force with a defined

88

Appendix B: ABAQUS Functions

margin of error. The sti!ness correlation of the member is updated for each iteration.

The function *STATIC makes it possible for the user to control the chosen incrementation some-what.

*STATIC-Initial Increment-, -Period-, -Min. Increment-, -Max. Increment-

The parameters to be user-defined controls the following:

-Initial Increment- the initial increment

-Period- the period of the load step

-Min. Increment- the minimum allowed increment

-Max. Increment- the maximum allowed increment

It is possible to select either an automatic or a direct increment division when defining a step.The automatic choice is generally quicker and less CPU demanding and as a lot of experienceis required to implement the direct increment division, the automatic incrementation division ischosen herein.

The specification of the minimum and maximum time increments allowed are only used if auto-matic time incrementation is used. If nothing is specified for the maximum time increment, noupper limit will be imposed.

89

Appendix B: ABAQUS Output from Pure Shear Example of Simple Plates

90

Appendix C:

ABAQUS Output from PureShear Example of Simple Plateswith and without Reinforcement

This appendix contains the output from the ABAQUS run of the simple shear example fromsection 4.6.3.

By examination of the outputs from the two runs, it can be seen that they are exactly alikeand thus that the reinforcement has no e!ect for the case which the plate is subjected to pureshear.

C.1 Simple Shear Example without Reinforcement

Table C.1: Output from ABAQUS for a simple shear example without reinforcement at timeincrements 7, 19, 22 and 410.

Node Integration Point Direction # % ! P

[mm] [-] [MPa] [N]

Increment no 7

11 -0.00375 -3.44·10#23 -0.00277 -1251.021 1 22 -0.00375 0 -0.00277 -1251.02

12 0.00015 2.5048

11 0.00375 -3.44·10#23 -0.00277 -1253.792 3 22 -0.00375 0 -0.00277 -1253.79

12 0.00015 2.5048

11 0.00375 3.44·10#23 -0.00277 1251.023 4 22 0.00375 2.12·10#22 -0.00277 1251.02

12 0.00015 2.5048

11 -0.00375 3.44·10#23 -0.00277 -1253.794 2 22 0.00375 2.12·10#22 -0.00277 1253.79

12 0.00015 2.5048

91

Appendix C: ABAQUS Output from Pure Shear Example of Simple Plates

Increment no 19

11 -0.52104 -5.39·10#20 -9.82075 -375.8961 1 22 -0.52104 -9.11·10#21 -9.82075 -375.896

12 0.02084 10.5725

11 0.52104 5.39·10#20 -9.82075 10196.62 3 22 -0.52104 0 -9.82075 -10196.6

12 0.02084 10.5725

11 0.52104 -5.39·10#20 -9.82075 375.8963 4 22 0.52104 6.69·10#20 -9.82075 375.896

12 0.02084 10.5725

11 -0.52104 -5.39·10#20 -9.82075 -10196.64 2 22 0.52104 3.52·10#20 -9.82075 10196.6

12 0.02084 10.5725

Increment no 22

11 -1.02605 -9.04·10#19 -3.54567 222.1111 1 22 -1.02605 1.97·10#19 -3.54567 222.111

12 0.041042 3.10145

11 1.02605 -9.04·10#19 -3.54567 3323.562 3 22 -1.02605 -1.84·10#19 -3.54567 -3323.56

12 0.041042 3.10145

11 1.02605 9.04·10#19 -3.54567 -222.1113 4 22 1.02605 -2.21·10#20 -3.54567 222.111

12 0.041042 3.10145

11 -1.02605 9.04·10#19 -3.54567 -3323.564 2 22 1.02605 1.60·10#19 -3.54567 3323.56

12 0.041042 3.10145

Increment no 410

11 -2 -6.70·10#18 -0.122829 41.351 1 22 -2 3.50·10#18 -0.122829 41.35

12 0.08 0.040115

11 2 -6.70·10#18 -0.122829 81.472 3 22 -2 -3.45·10#18 -0.122829 -81.47

12 0.08 0.040115

11 2 6.70·10#18 -0.122829 -41.353 4 22 2 -3.49·10#18 -0.122829 -41.35

12 0.08 0.040115

11 -2 6.70·10#18 -0.122829 -81.474 2 22 2 3.46·10#18 -0.122829 81.47

12 0.08 0.040115

92


C.2 Simple Shear Example with Reinforcement

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

Node Integration Point Direction # % ! P

[mm] [-] [MPa] [N]

Increment no 7

11 -0.00375 -3.44·10#23 -0.00277 -1251.021 1 22 -0.00375 0 -0.00277 -1251.02

12 0.00015 2.5048

11 0.00375 -3.44·10#23 -0.00277 -1253.792 3 22 -0.00375 0 -0.00277 -1253.79

12 0.00015 2.5048

11 0.00375 3.44·10#23 -0.00277 1251.023 4 22 0.00375 2.12·10#22 -0.00277 1251.02

12 0.00015 2.5048

11 -0.00375 3.44·10#23 -0.00277 -1253.794 2 22 0.00375 2.12·10#22 -0.00277 1253.79

12 0.00015 2.5048

Increment no 19

11 -0.52104 -5.39·10#20 -9.82075 -375.8961 1 22 -0.52104 -9.11·10#21 -9.82075 -375.896

12 0.02084 10.5725

11 0.52104 5.39·10#20 -9.82075 10196.62 3 22 -0.52104 0 -9.82075 -10196.6

12 0.02084 10.5725

11 0.52104 -5.39·10#20 -9.82075 375.8963 4 22 0.52104 6.69·10#20 -9.82075 375.896

12 0.02084 10.5725

11 -0.52104 -5.39·10#20 -9.82075 -10196.64 2 22 0.52104 3.52·10#20 -9.82075 10196.6

12 0.02084 10.5725

93


Increment no 22

11 -1.02605 -9.04·10#19 -3.54567 222.1111 1 22 -1.02605 1.97·10#19 -3.54567 222.111

12 0.041042 3.10145

11 1.02605 -9.04·10#19 -3.54567 3323.562 3 22 -1.02605 -1.84·10#19 -3.54567 -3323.56

12 0.041042 3.10145

11 1.02605 9.04·10#19 -3.54567 -222.1113 4 22 1.02605 -2.21·10#20 -3.54567 222.111

12 0.041042 3.10145

11 -1.02605 9.04·10#19 -3.54567 -3323.564 2 22 1.02605 1.60·10#19 -3.54567 3323.56

12 0.041042 3.10145

Increment no 410

11 -2 -6.70·10#18 -0.122829 41.351 1 22 -2 3.50·10#18 -0.122829 41.35

12 0.08 0.040115

11 2 -6.70·10#18 -0.122829 81.472 3 22 -2 -3.45·10#18 -0.122829 -81.47

12 0.08 0.040115

11 2 6.70·10#18 -0.122829 -41.353 4 22 2 -3.49·10#18 -0.122829 -41.35

12 0.08 0.040115

11 -2 6.70·10#18 -0.122829 -81.474 2 22 2 3.46·10#18 -0.122829 81.47

12 0.08 0.040115

94

Appendix D:

Uniaxial Material Models for theNumerical Example of aReinforced Slab

This appendix contains the unixial material models of the concrete used for the numerical anal-ysis of the reinforced concrete slab described in chapter 7. For each of the given temperatures,inputs are made in ABAQUS; the compressive behaviour illustrated in the plot is defined throughthe *COMPRESSION HARDENING function and the combined concrete and interaction contributionis given by the *TENSION STIFFENING function.

The uniaxial material models are illustrated for element sizes of both h = 129 mm and h = 73mm.

! !"# $ $"# % %"#

&'$!!(

!

!"#

$

$"#

%

%"#

(

!cT

"ctT

)*+,-

! !"# $ $"# % %"#

&'$!!(

!

!"#

$

$"#

%

%"#

(

!cT

"ctT

)*+,-

h'.'$%/'00

h'.'1('00

h'.'$%/'00

h'.'1('00

! !"!$ !"!% !"!(!

$!

%!

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Figure D.1: Compressive concrete model, (a), and the combined concrete and interaction stresscontribution in tension, (b), for T = 20!C.

95

Appendix D: Material Models for the Reinforced Slab Example! !"# $ $"# % %"#

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Appendix D: Material Models for the Reinforced Slab Example

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97

Appendix D: Material Models for the Reinforced Slab Example

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(b) Tension


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MATERIAL MODELLING OF REINFORCED CONCRETE...

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Transcript of MATERIAL MODELLING OF REINFORCED CONCRETE...