Redistribution of bending moments in concrete slabs in the ...

Redistribution of bending moments in

concrete slabs in the SLS

Einar Óskarsson

August 2014TRITA-BKN. Master Thesis 434, 2014ISSN 1103-4297ISRN KTH/BKN/R�434�SE

c©Einar Óskarsson 2014Royal Institute of Technology (KTH)Department of Civil and Architectural EngineeringDivision of Structural Engineering and BridgesStockholm, Sweden, 2014

Summary

The �nite element method (FEM) is commonly used to design the reinforcement inconcrete slabs. In order to simplify the analysis and to be able to utilize the super-position principle for evaluating the e�ect of load combinations, a linear analysis isgenerally adopted although concrete slabs normally have a pronounced non-linearresponse. This type of simpli�cation in the modeling procedure will generally leadto unrealistic concentrations of cross-sectional moments and shear forces. Concretecracks already at service loads, which leads to redistribution of moments and forces.The moment- and force-peaks, obtained through linear �nite element analysis, canbe redistributed to achieve a distribution more similar to what is seen in reality.The topic of redistribution is however poorly documented and design codes, suchas the Eurocode for concrete structures, do not give descriptions of how to performthis in practice.

In 2012, guidelines for �nite element analysis for the design of reinforced concreteslabs were published in a joint e�ort between KTH Royal Institute of Technology,Chalmers University of Technology and ELU consulting engineers, which was �nan-cially supported by the Swedish Transport Administration. These guidelines aim toinclude the non-linear response of reinforced concrete into a linear analysis.

In this thesis, the guidelines mentioned above are followed to obtain reinforcementplans based on crack control, for a �ctitious case study bridge by means of a 3D�nite element model. New models were then constructed for non-linear analyses,where the reinforcement plans were implemented into the models by means of bothshell elements as well as a mixture of shell and solid elements. The results from thenon-linear analyses have been compared to the assumptions given in the guidelines.

The results from the non-linear analyses indicate that the recommendations given inthe aforementioned guidelines are indeed reasonable when considering crack widthcontrol. The shell models yield crack widths equal to approximately half the designvalue. The solid models, however, yielded cracks widths that were 15 - 20% lowerthan the design value. The results show that many factors attribute to the structuralbehavior during cracking, most noticeably the fracture energy, a parameter notfeatured in the Eurocode for concrete structures.

Some limitations of the models used in this thesis are mentioned as well as areas forfurther improvement.

Keywords: �nite element analysis, reinforced concrete, concrete slab, non-linearanalysis, crack control, fracture energy

i

Sammanfattning

Finita elementmetoden (FEM) används vanligtvis vid dimensionering av slakarmer-ade betongplattor. Som en förenkling och för att kunna använda superposition-sprincipen, för att beräkna dimensionerande snittkrafter, antas responsen ofta lin-järelastisk. Detta trotts att det är allmänt känt att betong är ett olinjärt heterogentmaterial. Problemet med en linjär materialmodell är att det existerar singularitetervilket i verkligheten försvinner när sprickor uppstår och snittkrafter omfördelas. Da-gens normer tillåtet därför att moment och tvärkrafter, som erhållits med dennaförenkling, omfördelas så att en mer realistisk spänningsbild erhålls. Dessvärreär ämnet mycket dåligt dokumenterat och normer, som t.ex. EN1992, ger ingenbeskrivning om hur denna omfördelning praktiskt kan genomföras.

Under 2012 publicerade KTH tillsammans med Chalmers och ELU Konsult en rap-port med rekommendationer för hur betongplattor bör dimensioneras. Rapporten�nansierades av Tra�kverket. Faktum är att, till skillnad från föregående rapporter,inkluderar rapporten tydliga riktlinjer för fördelningsbredder.

Syftet med föreliggande avhandling är att utvärdera om rekommendationer i ovannämnda rapport är rimliga. Detta kontrolleras genom att en 3D linjär �nita elementmodell upprättas och armeringen utformas enligt riktlinjer i rapporten. Därefterskapas en ny modell där olinjäritet för betongen och armeringen beaktas. Mod-ellen utförs både med skalelement och en blandning mellan solid- och skalelement.Resultatet från de både fallen har jämförts mot riktlinjerna.

Resultatet från de icke-linjära analyserna tyder på att ovannämnda riktlinjer förfördelningsbredder är rimliga när sprickvidder utvärderas. Den ickelinjära skalmod-ellen ger approximativt halva sprickvidden jämfört med den linjärelastiska modellen.Vad beträ�ar solidmodellen så blir mottsvarande värde 15 - 20 % lägre jämfört motden linjärelastiska modellen. Utöver detta visar resultaten att det är många faktorersom påverkar sprickvidden, varav den viktigaste parameterna är brottenergin, vilketär en parameter som inte beaktas i EN1992.

Söord: Finita elementmetoden, slakarmerad betong, betongplatta, icke-linjär analys,sprickkontroll, brottenergi

iii

Preface

This master thesis was initiated by the Division of Structural Engineering andBridges, Department of Civil and Architectural Engineering at the Royal Instituteof Technology, KTH. The thesis has been conducted under the supervision of Adj.Professor Costin Pacoste.

Many people have contributed to the work underlying this thesis and to them I amtruly grateful. My supervisor, Prof. Pacoste, for his engagement in the process. PhDstudent Christopher Svedholm for his endless support and patience with every aspectof the thesis work. A special thanks to Dr. Richard Malm and PhD student AbbasKamali for their insight into issues regarding the numerical modeling of concrete.

Stockholm, August 2014Einar Óskarsson

v

Contents

Summary i

Sammanfattning iii

Preface v

List of Abbreviations xi

1 Introduction 1

1.1 Previous studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Aims and scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 The case study bridge . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Theoretical background 3

2.1 Damaged Plasticity Model . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 Material properties and strength of concrete . . . . . . . . . . 4

2.1.2 Plasticity Theory . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.3 Fracture Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.4 Damage Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.5 Tension softening . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Crack propagation and crack control . . . . . . . . . . . . . . . . . . 19

2.2.1 Crack propagation . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.2 Crack control according to EC2 . . . . . . . . . . . . . . . . . 22

2.3 Moment and force redistribution from linear FE analysis . . . . . . . 24

2.3.1 Moment Peaks Over Columns . . . . . . . . . . . . . . . . . . 24

vii

2.3.2 Recommendations of redistribution widths . . . . . . . . . . . 25

2.4 Iteration procedure in non-linear analyses . . . . . . . . . . . . . . . . 29

3 Method 31

3.1 Modeling procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.1 General steps in modeling . . . . . . . . . . . . . . . . . . . . 31

3.2 Reinforcement dimensioning . . . . . . . . . . . . . . . . . . . . . . . 33

3.2.1 Redistribution widths for reinforcement moments . . . . . . . 33

3.3 The non-linear shell model . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3.1 Material parameters . . . . . . . . . . . . . . . . . . . . . . . 34

3.3.2 Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3.3 Elements and mesh . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4 The non-linear solid model . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4.1 Material parameters . . . . . . . . . . . . . . . . . . . . . . . 37

3.4.2 Elements and mesh . . . . . . . . . . . . . . . . . . . . . . . . 38

3.5 Numerical instability in Abaqus . . . . . . . . . . . . . . . . . . . . . 38

4 Results 41

4.1 Maximum downward de�ection . . . . . . . . . . . . . . . . . . . . . 41

4.2 Crack width growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3 Solid Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4 Reinforcement quantities . . . . . . . . . . . . . . . . . . . . . . . . . 49

5 Discussion and conclusions 51

5.1 Crack control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.1.1 Shell models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.1.2 Solid model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.1.3 Sensitivity of the results . . . . . . . . . . . . . . . . . . . . . 52

5.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.2.1 Recommendations by Pacoste et al. . . . . . . . . . . . . . . . 53

5.2.2 Fracture energy . . . . . . . . . . . . . . . . . . . . . . . . . . 54

viii

5.2.3 Limitations and possible improvements . . . . . . . . . . . . . 54

Bibliography 57

A Concrete Damaged Plasticity Input Data 61

A.1 Plasticity Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

A.2 Compressive Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 61

A.3 Tensile Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

A.3.1 Gf = 82.5 Nm/m2 . . . . . . . . . . . . . . . . . . . . . . . . 62

A.3.2 Gf = 105 Nm/m2 . . . . . . . . . . . . . . . . . . . . . . . . . 62

A.3.3 Gf = 143.7 Nm/m2 . . . . . . . . . . . . . . . . . . . . . . . . 63

B Reinforcement Design 65

B.1 Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

B.2 Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

B.3 Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

B.4 Reinforcement combinations . . . . . . . . . . . . . . . . . . . . . . . 72

ix

List of Abbreviations

CDM Concrete Damage ModelDOF Degree of FreedomFE Finite ElementFEA Finite Element AnalysisFEM Finite Element MethodFPZ Fracture Process ZoneSLS Serviceability Limit StateULS Ultimate Limit State

xi

Chapter 1

Introduction

The �nite element method (FEM) is commonly used to design the reinforcement inconcrete slabs. Despite the non-linear response usually exerted by concrete struc-tures, a linear analysis is generally adopted as to simplify the analysis and enable theuse of the super-position principle for evaluating the e�ect of load combinations. Inthese linear models, unrealistic concentrations of cross-sectional moments and shearforces will occur due to necessary simpli�cations in the model [1]. The Eurocodefor concrete structures (EC2) states that moments achieved through linear elasticanalysis, e.g. through linear �nite element analysis (FEA), may be redistributed,provided that the resulting distribution of moments remains in equilibrium with theapplied loads [2]. Detailed descriptions or recommendations of how to redistributethese moments are however, not found in the code.

1.1 Previous studies

Pacoste et al. [1] wrote a handbook with recommendations for FEA for the designof reinforced concrete slabs. The recommendations cover aspects of support condi-tion modelling, the choice of result sections and the choice of distribution widths.According to the authors, the topic of distribution widths was not well documentedand lacks extensive research. This is supported by the evident lack of coverage inEC2.

Blaauwendraad [3] writes about peak moments at column supports, the mesh sizedependency (see Chapter 2.3) as well as user- and program dependency. It is stressedthat the peak moment values are not of interest, rather the area under the bendingmoment diagram .

Studies in the Netherlands focused on the distribution of peak shear stresses inconcrete slabs over supports aiming at replacing old rules of thumb with methodssupported by research [4, 5]. Another study on shear distribution in concrete slabsutilized non-linear FEA to support a resulting shear force approximately 20% lowerthan that obtained through linear analysis [6].

1

CHAPTER 1. INTRODUCTION

1.2 Aims and scope

The aim of this masters thesis is to investigate the e�ect of distribution widths byexamining a case study bridge by means of linear as well as non-linear FE models.The recommendations by Pacoste, Plos and Johansson act as the basis for the designprocedure. The comparison of linear and non-linear models will hopefully give moreinsight into the e�ect of distribution widths as well as validate the recommendationsgiven by Pacoste et al [1].

1.3 The case study bridge

The case study bridge was originally dimensioned by Plos as part of the 10 millioneuro project, Sustainable Bridges - Assessment for Future Tra�c Demands andLonger Lives, aimed at assessing the readiness of railway bridges to meet expectedfuture tra�c loads [7]. The bridge is a two-span slab bridge supported by threecolumns, as shown in Figure 1.1. The bridge has a total length of 25 m and is 11m wide. It is supported by three roller bearings at each end as well as the threecolumns. The slab has a thickness of 0.6 m. The columns have a square cross sectionof 0.6 m sides and a length of 5.0 m. The two spans have a length of 12 m andthe bearings and columns are positioned with a 4.0 m spacing in the transversaldirection.

Figure 1.1: The two-span slab bridge supported by columns. All lengths are in me-ters. Reproduction from [7].

2

Chapter 2

Theoretical background

In this section, the theory used in this thesis is described. First, an overview ofthe material modeling regarding the non-linear and inelastic behavior of concretewill be given. Secondly, the crack propagation process will be described along withthe methodology to limit cracking used in the EC2. Thereafter the redistributionmethods used in this thesis will be presented and �nally an overview of the iterationprocedure used in non-linear analysis.

Modeling Concrete

Concrete is a composite material. It consists of coarse aggregate and a continuousmatrix of mortar, which itself comprises a mixture of cement paste and smalleraggregate particles. Its physical behavior is very complex, being largely determinedby the structure of the composite material, such as the ratio of water to cement,the ratio of cement to aggregate, the shape and size of aggregate and the kind ofcement used [8]. Concrete may be modeled on di�erent levels of detail depending onwhether separate aggregates, pores or even cement particles are taken into account.For practical applications, a macroscopic level of observation is adopted. Thus, thematerial models are based on the assumption of homogeneity and isotropy untilcracking [9].

The �rst attempt to apply the FEM to a reinforced concrete structure was madeby Ngo and Scordelis in 1967 [8]. Since then, the modeling has advanced rapidlyalongside the rapid development and availability of computers with high compu-tational capacity. Commercial software programs o�er various concrete materialmodels which have di�erent �elds of application, e.g. static vs. dynamic analyses.The software used for modeling in this thesis is Abaqus and the applied materialmodel, Damaged Plasticity Model, is described in detail in the following subchapter.

3

CHAPTER 2. THEORETICAL BACKGROUND

2.1 Damaged Plasticity Model

The damaged plasticity model for concrete analysis provided by Abaqus aims tocapture the e�ects of irreversible damage associated with the failure mechanismsthat occur in concrete under fairly low con�ning pressures [10]. It is based on themodel proposed by Lubliner et al [11] and with modi�cations made by Lee andFenves [12].

The di�erent aspects of the model are presented in the following subchapters.

2.1.1 Material properties and strength of concrete

Concrete is a sort of brittle material. Its stress-strain behavior is a�ected by thedevelopment of micro- and macro-cracks in the material body. Particularly, concretecontains a large number of micro-cracks, especially at interfaces between coarse ag-gregates and mortar, even before the application of external load. These initialmicro-cracks are caused by segregation, shrinkage, or thermal expansion in thecement paste. Under applied loading, further micro-cracking may occur at theaggregate-cement paste interface, which is the weakest link in the composite sys-tem. The progression of these cracks, which are initially invisible, to become visiblecracks occurs with the application of external loads and contributes to the generallyobtained non-linear stress-strain behavior [8].

Uniaxial Compression

Experimental tests have shown that concrete is highly non-linear in uniaxial com-pression. A typical stress-strain diagram for a concrete sample loaded in uniaxialcompression is shown in Figure 2.1.

The stress-strain curve is linear elastic up to approximately 30% of the ultimatecompressive strength. After this, the stress increases gradually up to about 70-75% of the ultimate compressive strength. In this stage, strains orthogonal to theexternally applied load lead to additional bond cracks between the aggregate andthe cement paste in the direction of loading and therefore to a decrease of themacroscopic sti�ness. This leads to a non-linear stress-strain curve. Upon furtherloading the number of bond cracks increase and the matrix cracking starts. Afterreaching the peak value, the stress-strain curve descends. This is normally de�nedas softening. As the curve descends, crushing failure occurs at the ultimate strain[13].

Uniaxial Tension

Until tensile failure is reached, most material models assume concrete to behavein a linearly elastic manner, although some minor plastic deformations occur. As

4

2.1. DAMAGED PLASTICITY MODEL

Figure 2.1: Concrete behavior under uniaxial compression. Reproduction from [13].

mentioned above, the cracking of concrete is initiated by the formation of micro-cracks. These start to develop when the stress is close to the tensile strength of theconcrete. When the tensile strength is reached the micro-cracks start to localize to alimited area called the fracture process zone (FPZ), thereafter micro-cracking onlyoccurs within this zone. As the deformation increases the micro-cracks in the FPZincrease in number and start to merge with each other. This leads to lower stressesin the FPZ and the material exhibits a softening behavior, as illustrated in Figure2.2. The ultimate failure occurs when the micro-cracks eventually merge into a realcrack that splits the FPZ [14].

Figure 2.2: Concrete behavior under uniaxial tension. Reproduction from [13].

5


Multiaxial behavior

The behaviour of concrete at failure at low con�ning pressures di�ers from that athigher pressures. At low pressures, the failure is typically brittle in nature, which isthe case for tensile and compressive stresses at low hydrostatic pressures. If concreteis subjected to higher hydrostatic pressures on the other hand, the material candeform plastically on the failure surface like a ductile material before failure strainsare obtained [13].

Figure 2.3 illustrates a biaxial failure envelope for concrete and the concrete thatcorresponds to the stress state. It can be seen that tensile cracking occurs in the�rst, second and fourth quadrant. In the �rst quadrant the cracking occurs per-pendicularly to the principal tensile stress. In the second and the fourth quadrantthe crack is orientated perpendicularly to the tensile stress [14]. A biaxial compres-sion state is illustrated in the third quadrant. The uniaxial compressive strength,fc, increases approximately 16% under conditions of equal biaxial compression. Amaximum increase in uniaxial compressive strength of about 25% is obtained at astress ratio of σ1/σ2 = 0.5. It can also be observed that a state of simultaneouscompression and tension (second and fourth quadrant) reduces the tensile strength[13].

Figure 2.3: Yield criteria for biaxial stress state illustrated for plane stress state.Reproduction from [14].

6


2.1.2 Plasticity Theory

According to Chen and Han [8] the theory of plasticity has two tasks. The �rstis to set up relationships between stress and strain under a complex stress statethat can describe adequately the observed plastic deformation. The second is todevelop numerical techniques for implementing these stress-strain relationships inthe analysis of structures.

The classical theory of plasticity may be viewed as a translation of physical realityor as a model that approximates the mechanical behavior of solids under certaincircumstances. The previous view is often held with regard to ductile crystallinesolids, especially metals. With regard to concrete however, it is generally acknowl-edged that such prominent features of plasticity theory such as a well de�ned yieldcriterion and strictly elastic unloading are approximations at best. Nevertheless,many problems involving brittle materials have been quite successfully treated bymeans of plasticity theory [11].

Any plasticity model includes three aspects;

• An initial yield surface in stress space that de�nes the stress level at whichplastic deformations begins.

• A hardening rule that de�nes the change of the loading surface as well as thechange of the hardening properties of the material during the course of plastic�ow.

• A �ow rule that is related to a plastic potential function and gives an incre-mental plastic stress-strain relation.

These aspects will be described in the following sections.

Yield and failure functions

The yield criterion is described as a surface to account for biaxial and multiaxiale�ects. Normally, the yield surface is de�ned by the material strength at the pointwhere the material starts to exhibit non-linear behavior, while the failure surface isde�ned by the material's ultimate strength. In pure tension, the failure and yieldsurfaces coincide since concrete is assumed to be elastic up to the tensile strength[13].

Concrete can exhibit a signi�cant volume change when subjected to severe inelasticstates. In Figure 2.4a it can be seen that the increase in volume is more than twiceas large for the hydrostatic compressive state (σ1/σ2 = −1/ − 1) as for uniaxialcompression (σ1/σ2 = −1/0). The points marked on the stress-volumetric straindiagrams indicate the limit of elasticity, the point of in�ection in the volumetricstrain, the bendover point corresponding to the onset of instability or localizationof deformation and the ultimate load. The surfaces corresponding to these material

7


states are shown schematically in Figure 2.4b, which give an indication of the �ex-pansion� of the failure surface, i.e. the reserves of strength that concrete has fromthe moment its elastic limit is reached until it completely ruptures [15].

(a) (b)

Figure 2.4: (a) Volumetric strain in biaxial compression and (b) typical loadingcurves under biaxial stresses. Reproduction from [15].

The same result is not found in triaxial compression tests, at least not for su�cientlyhigh hydrostatic pressures, as illustrated in Figure 2.5a. This means that the yieldsurface is closed while the failure surface is open in the direction of hydrostaticpressure, as seen in Figure 2.5b [13].

(a) (b)

Figure 2.5: (a) Stress-displacement diagrams obtained from triaxial compression forthree di�erent levels of con�ning pressure σ2. (b) 3D failure surface andthe elastic limit. Reproduction from [13].

Being hydrostatic-pressure-dependent materials, concretes have a failure surfacewith curved meridians, indicating that the hydrostatic pressure produces e�ects

8


of increasing the shearing capacity of the material (Figure 2.6). The von Mises cri-terion (see Figure 2.7), commonly used for steel materials, is a two-parameter modelwith linear meridians and is therefor inadequate for describing the failure of concretein the high-compression range [8]. Some more common failure models are presentedin Figure 2.7. These failure models are sometimes modi�ed by using combinationsof di�erent models to accurately capture the materials behavior in various states ofstress [13].

Figure 2.6: Non-linear meridians of a failure surface. Reproduction from [8].

Hardening

As can be seen from a uniaxial stress-strain relation for concrete in compression(Figure 2.1), the concrete stresses continue to increase also after non-linear strainshave started to occur. The behavior is called strain hardening (or simply hardening)and it continues also after the maximum stress has been reached. In the descendingbranch of the curve, when the stress decreases, the hardening behavior is sometimescalled strain softening [9].

Within the framework of the theory of small strains, the strain tensor can be de-composed into an elastic part εe and an inelastic or plastic part εp, such that in therate form

ε̇ = ε̇e + ε̇p (2.1)

Plasticity theory permits description of the dependence of strain in the materialon its history through the introduction of an internal scalar variable, here notedas κ. As the internal variable usually describes irreversible material behavior, itsevolution is expressed by means of rate equations which are functions of the plasticstrain rate, ε̇p, i.e.

κ̇ = f(ε̇p) (2.2)

In both the work-hardening and the strain-hardening hypothesis, the internal pa-rameter, also called the hardening parameter, is integrated along the loading path

9


Figure 2.7: Some failure models and their respective meridians and deviatoric sec-tions. Reproduction from [8].

to give

κ =

∫κ̇dt (2.3)

The hardening rule de�nes the motion of the subsequent yield surface during loading.The yield condition generalizes the concept of yield stress to multiaxial stress statesand includes the history dependence through the scalar hardening variable [15].Thus, the yield surface may be generally expressed as a function of the currentstress state, the plastic strain and the hardening parameter [9]:

f(σ, εp, κ) = 0 (2.4)

Since the yield function is dependent on the loading history through κ it can onlyexpand or shrink in the stress space, not translate or rotate. Such hardening is calledisotropic hardening, irrespective of whether the work-hardening or strain-hardeningapproach is used. The direction of the plastic strain tensor ε̇p is determined fromthe derivative of the plastic potential function as illustrated in Figure 2.8 [15, 16].

Isotropic hardening is generally considered to be a suitable model for problems inwhich the plastic straining goes well beyond the initial yield state and where the

10


Bauschinger e�ect is not noticeable. The Bauschinger e�ect refers to a directionalanisotropy induced by plastic deformation in which an initial plastic deformation ofone sign reduces the resistance of the material with respect to a subsequent plasticdeformation of the opposite sign. Models using isotropic hardening are intended forproblems involving essentially monotonic loading, as distinct from cyclic loading.In a di�erent approach called kinematic hardening, the yield surface translates as arigid body in the stress space, therefor maintaining the size, shape and orientationof the initial yield surface. Kinematic hardening is more appropriate to use for casesof cyclic and reversed types of loading for materials with a pronounced Baushingere�ect [13].

Flow rule

The shape of the yield surface at any given loading condition can be determined bythe hardening rule. The connection between the yield surface and the stress-strainrelationship is determined with a �ow rule.

Concrete can, as previously mentioned, exhibit a signi�cant volume change whensubjectd to severe inelastic states. This change in volume, usually referred to asdilation, caused by plastic distortion can be reproduced well by using an adequateplastic potential function G [11]. The evolution of the inelastic displacements in theFPZ is de�ned through the �ow rule. The �ow rule is de�ned as

ε̇p = κ̇∂G

∂σ(2.5)

where κ̇ ≥ 0 is a scalar hardening parameter which can vary throughout the strainingprocess. The gradient of the potential surface ∂G

∂σde�nes the direction of the plastic

strain increment vector ε̇p, and the hardening parameter κ̇ determines its length[13].

When the plastic potential function coincides with the yield surface, the plastic �owdevelops along the normal to the yield surface. This is called associated �ow rulebecause the plastic �ow is connected or associated with the yield criterion. Theother approach where two separate, non-coincided functions are used for the plastic�ow rule and the yield surface is called non-associated �ow rule. In this case theplastic �ow develops along the normal to the plastic �ow potential and not to theyield surface [17].

The plastic potential function used in the concrete damaged plasticity materialmodel is the Drucker-Prager hyperbolic function, illustrated in Figure 2.8 and givenby

G =√

(εft0 tanψ)2 + q̄2 − p̄ tanψ (2.6)

where

ε is the eccentricity, which de�nes the rate at which the plastic potentialfunction approaches the asymptote. Increasing value of ε provides morecurvature to the low potential.

ψ is the dilation angle, measured in the p-q plane at high con�ning pressure

11


Figure 2.8: The Drucker-Prager hyperbolic plastic potential function in the merid-ional plane. Reproduction from [13].

The dilation angle, ψ, is used as a material parameter in Abaqus. It measures theinclination of the plastic potential reaches for high con�ning pressures. Parametricstudies have proven the best value of ψ to be in the range of 25◦-40◦ to describeboth tension and compression in biaxial stress states. The �ow in the Drucker-Pragerfunction is non-associative in the meridional plane if the dilation angle di�ers fromthe material friction angle [16].

2.1.3 Fracture Mechanics

Fracture mechanics de�ne three di�erent types of fracture, namely tension (mode I),shear (mode II) and tear (mode III) as illustrated in Figure 2.9. Mode I is the mostcommon type of crack growth in concrete and can occur in its pure form. ModesII and III are however rarely obtained in their pure form. Combinations of thedi�erent modes often occur. The combination of modes I and II are most commonon concrete structures [13].

Figure 2.9: Di�erent types of failure. Reproduction from [13].

12


To determine whether a crack initiates and propagates, the fracture energy Gf ofthe material has to be considered. The fracture energy is a material property whichdescribes the energy consumed when a unit area of a crack is completely opened.The fracture energy is described in more detail in Section 2.1.5. In linear elasticfracture mechanics, mode I failure is considered to be reached when the maximumprinciple stress reaches the tensile strength of the material. This simplest form offracture mechanics is only valid for linear elastic materials with a sharp crack tip,which is not the case for concrete [14].

To be able to describe the tensile behavior of concrete, a non-linear fracture me-chanic approach has to be adopted. There are basically two di�erent concepts toachieve this; the discrete crack approach and the crack band approach; which willbe described in the following subsections.

Discrete Crack Model

Hillerborg et al [18] proposed the �rst non-linear theory of fracture mechanics. Itincludes the tension softening FPZ through a �ctitious crack ahead of the pre-existing face crack whose faces are acted upon by certain closing stresses such thatthere is no stress concentration at the tip of this extended crack, as illustrated inFigure 2.10. The stress increases from zero at the tip of the pre-existing traction-free macro-crack to the full tensile strength of the material, ft. The proposed modelassumes that the FPZ is of negligible thickness and thereof the alternative namediscrete crack model [15].

Two material parameters are essential to describe the material behavior in the dis-crete crack model; the stress-displacement relation σ(w) in the softening zone andthe fracture energy, which is de�ned as the area under the tension softening curve[15]. A more detailed description of the tension softening behavior and the fractureenergy is presented in Section 2.1.5.

Crack Band Model

As the micro-cracking and bridging in the FPZ is not continuous and as it does notnecessarily develop in a narrow discrete region in line with the continuous traction-free crack, it has been argued that the tension softening relation σ(w) can be equallywell approximated by a strain softening relation σ(ε), i.e. a decreasing stress withincreasing inelastic strain. However, this strain is now related to the inelastic defor-mation w and fracture energy Gf , so that the ultimate strain at complete rupture,εc, is related to wc. In other words, εc is now de�ned by a fracture criterion. Torelate the inelastic strain to w and Gf , it is necessary to introduce a gauge length,say h. It is then assumed that the micro-cracks in the FPZ are distributed over aband of width h, hence the name crack band model [15]. With this continuum ap-proach the local discontinuities are distributed, or smeared, over the element, whichgives rise to the name smeared crack approach. Contrary to the discrete crack con-cept, the smeared crack approach �ts the nature of the �nite element displacement

13


Figure 2.10: A traction free crack of length a0 terminating in a �ctitious crack withresidual stress transfer capacity σ(w) whose faces close smoothly nearits tip. Reproduction from [13].

method, as the continuity of the displacement �eld remains intact [19]. The crackband method was �rst introduced by Ba�zant [20] in 1976 and further developed byBa�zant and Oh [21] in 1983. The main conceptual di�erence between the smearedand discrete crack concepts are illustrated in Figure 2.11.

Figure 2.11: Tension softening relation based on stress-strain relations (above) andstress-crack opening displacement (below). The former is used in thesmeared crack approach, the latter in the discrete crack concept. Re-production from [19].

There are two approaches for determining the crack direction with the smearedcrack method. The �rst approach, called the �xed crack model is obtained if the

14


smeared cracks in the crack band are normal to the principal tensile stress at themoment of crack initiation. Their direction does not change during the subsequentgrowth of the crack band even when the direction of the principal stress changes.In the second approach the crack rotates as the direction of the principal stresschanges and the cracks are therefore always normal to the principal tensile stressdirection, hence the name rotated crack model. One main di�erence between thesetwo approaches is how the shear stresses and shear strains are treated at the smearedcrack bands. In the �xed crack model, a shear retention factor which reduces theshear modulus is needed to avoid convergence di�culties and to avoid physicallyunrealistic and distorted crack patterns. In concrete, the shear retention factorallows for the roughness of crack faces due to aggregate interlocking etc. The shearmodulus is normally reduced with increasing strain, which represents a reduction ofthe shear sti�ness due to crack opening. In the rotated crack model, no shear stressescan occur in the crack plane since the crack follows the direction of the principalstress. In this case an implicit shear modulus is calculated to provide co-axialitybetween the rotating principal stress and strain [13].

2.1.4 Damage Theory

The progressive evolution of micro-cracks and nucleation and growth of voids arerepresented in concrete damage models (CDMs) by a set of state variables whichalter the elastic and/or plastic behavior of concrete at the macroscopic level. Inpractical implementation, the damage models are very similar to the plasticity the-ory described in Section 2.1.2. In all CDMs, unloading leaves no residual damageirrespective of the degree of damage su�ered by concrete up to the instant of unload-ing. This means that there is no �plastic� deformation, which is rather unrealistic.For this reason, there are theories that couple the sti�ness reduction and plasticdeformation [15]. The di�erence is illustrated in Figure 2.12 where the e�ect ofloading and unloading is shown on stress-strain diagrams.

(a) (b)

Figure 2.12: Progressively fracturing models with (a) no �plastic� deformations and(b) permanent (�plastic�) deformations. Reproduced from [8].

15


In damage models, the total stress-strain relation has the following form

σ = Ds : ε (2.7)

where σ and ε are the stress and strain tensor, respectively. The secant sti�nesstensor Ds of the damaged material depends on a number of internal variables whichcan be tensorial, vectorial or scalar. The expression in Equation 2.7 di�ers fromclassical non-linear elasticity by a history dependence, which is introduced througha loading-unloading function F . This function vanishes upon loading and is negativeotherwise. It is the counterpart of the yield function in plasticity theory. For damagegrowth, F must remain zero for an in�nitesimal period, i.e. ∂F

∂t= 0. The theory is

completed by specifying the appropriate material dependent evolution equations forthe internal variables [22].

Isotropic Damage Models

For the case with isotropic damage evolution, the total stress-strain in Equation2.7 specializes so that the initial shear modulus and the initial bulk modulus aredegraded with separate scalar variables, d1 and d2, respectively. A simpli�cation ofthe isotropic model can be made by assuming that the degradation of the secantshear sti�ness, (1−d1)G, and the secant bulk moduli, (1−d2)K, degrade in the samemanner during damage growth, i.e. d ≡ d1 = d2. This means that the Poisson'sratio of the material remains unchanged during damage growth and leads to thefollowing expression

σ = (1− d)D0 : ε (2.8)

where the damage variable d grows from zero at an undamaged state to one atcomplete loss of integrity. The sti�ness tensor D0 represents the sti�ness of theundamaged material [22].

Damage-coupled plasticity theory

As mentioned above, the are no permanent strains present in a CDM. It is fullyrecovered upon unloading unlike the equivalent strain. This is known to be inac-curate in concrete, as permanent strains remain due to sliding and friction at themicro-cracks. Since failure mechanics of concrete in tension, as well as in compres-sion, for low levels of con�nement is associated with sti�ness degradation as wellas with inelastic deformations, models characterized by a coupling between damageand plasticity have been developed [13].

The simplest mode of coupling between damage and plasticity is a scalar damageelasto-plastic model based on the e�ective stress concept. The stress-strain equationfor the model is given as

σ = (1− d)D0 : (ε− εp) (2.9)

According to the e�ective stress concept, the plastic yield function is formulated interms of e�ective stress. The e�ective stress, σ̂, is calculated according to Equation

16


2.10 [13].σ̂ =

σ

1− d(2.10)

Implementation in the Concrete Damaged Plasticity model

The Concrete Damaged Plasticity model assumes that the elastic sti�ness degrada-tion is isotropic and characterized by a single scalar variable, d. The de�nition ofthe scalar degradation variable d must be consistent with the uniaxial monotonicresponses obtained in both tension and in compression, and it should also capturethe complexity of associated with the degradation mechanisms under cyclic loading.For the general multiaxial stress conditions, the model assumes that

(1− d) = (1− stdc)(1− scdt) (2.11)

where dc and dt are the scalar degradation variables in uniaxial compression and ten-sion, respectively, and sc and st are functions of the stress state that are introducedto represent sti�ness recovery e�ects associated with stress reversals [10].

The experimental observation in most quasi-brittle materials, including concrete, isthat the compressive sti�ness is recovered upon crack closure as the load changesfrom tension to compression. On the other hand, the tensile sti�ness is not recoveredas the load changes from compression to tension once crushing microcracks havedeveloped. This behavior is illustrated in Figure 2.13, where Γt = 0 corresponds tono recovery as the load changes from compression to tension and Γc = 1 correspondsto full recovery as the loading changes from tensile to compressive [10].

Figure 2.13: Uniaxial load cycle (tension-compression-tension) assuming default val-ues for the sti�ness recovery factors Γt = 0 and Γc = 1. Reproductionfrom [13].

17


2.1.5 Tension softening

Plain concrete is not a perfectly brittle material as it has some residual load-carryingcapacity after reaching its tensile strength. This experimental observation has ledto the replacement of purely brittle crack models by tension softening models, inwhich a descending branch was introduced to model the gradually diminishing tensilestrength of concrete upon further crack opening [22].

This crack opening procedure, i.e. the formation of micro-cracks to macro-cracks,can be de�ned in terms of fracture energy, Gf , or by means of a stress-strain orstress-displacement relationship, see Figure 2.14. The fracture energy is a materialparameter that describes the amount of energy [Nm/m2] that is needed to open aunit area of a crack, to obtain a stress free crack. The area under the unloadingpart of the σ − w graph in Figure 2.14 corresponds to the fracture energy [13].Recommendations of the numerical values of fracture energy based on the concrete'squality and found aggregate size can be found tabulated in [23].

Figure 2.14: Crack opening with fracture energy. Reproduction from [13].

For models where there is no reinforcement in signi�cant regions of the model, theapproach based on a stress-strain relationship will introduce unreasonable meshsensitivity into the results. For these cases it is better to de�ne the fracture energyor the stress and crack opening displacement curve [10].

The most simple way to introduce the crack opening law is to use a linear ap-proximation. When de�ning the fracture energy in FE programs, a linear crackopening law is usually used. The linear softening behavior (Figure 2.15a) can inmost cases provide a solution that gives accurate results, even though the materialresponse tends to be slightly too sti� [13]. The crack opening that corresponds to astress free crack with a linear tension softening model is calculated as shown in the

18

2.2. CRACK PROPAGATION AND CRACK CONTROL

following equation

wc = 2Gf

ft(2.12)

Other, more detailed expressions can be used to describe the softening response.One commonly used is the bilinear expression derived by Hillerborg [24], illustratedin Figure 2.15b. Another is the exponential function experimentally derived byCornelissen, Hordijk and Reinhardt [25]. According to [15] this exponential function,illustrated in Figure 2.15c, is by far the best and most accurate model.

(a) Linear (b) Bilinear (c) Exponential

Figure 2.15: Di�erent functions to describe the tension softening of concrete. (a)reproduced from [14], (b) and (c) reproduced from [13].

2.2 Crack propagation and crack control

Cracking in reinforced concrete usually has a limited in�uence on its load-bearingcapacity as crack propagation is accounted for during design. Cracks can howeverhave a signi�cant e�ect on a concrete structure's waterproo�ng, sound isolation anddurability with regard to both the concrete's degradation as well as corrosion ofthe reinforcing steel [26]. In this sub-section, the crack propagation process will becovered followed by a description of the crack control measures used in the EC2.

2.2.1 Crack propagation

The crack propagation in reinforced concrete members in direct tension is well doc-umented, both theoretically as well as experimentally. Normally, this is performedwith a single, centric reinforcement bar.

Assume such a long reinforced concrete member where the rebars extrude as to allowfor the application of a tensile load (Figure 2.17). The member is loaded by a tensileload N which is not su�ciently large to produce tensile cracking. The steel- andconcrete stresses and strains at the member's ends can be calculated as

σs =N

As⇒ εs =

σsEs

and σc = 0 ⇒ εc = 0

19


where σ and ε are the stress and strain, respectively.

This means that the reinforcement needs to stretch with respect to the concrete,i.e. slip. The bond between the steel and concrete will however prevent slipping.Consequently, the load is transferred from the reinforcement to the concrete throughthe steel bar's combs and the concrete is forced to take part in the load carrying. Theconcrete- and steel strains are equal at a certain distance from the member's endsand no forces are transmitted from the reinforcement to the surrounding concrete.This stretch, where the force transmission takes place through bound stress (τb), iscalled the transmission length, lt.

A local bond failure occurs at the member's ends due to the loading towards thefree end as shown in Figure 2.16. This means that the bond stress is zero over ashort length denoted as ∆r.

Figure 2.16: A local bond failure at the reinforced concrete member's ends. Repro-duction from [26].

An increased tensile load causes the transmission length to elongate as well as in-creasing the concrete and reinforcement stresses in the center part of the member.This is illustrated in Figure 2.18. The reinforced member is at the verge of crackingwhen the concrete stresses reach the concrete's tensile strength. The �rst crack canemerge anywhere in the member's middle span as denoted in the bottom of Figure2.18. The cracking load Ncr is calculated according to the following equation

Ncr = fct · [Ac + (α− 1)As] (2.13)

whereα =

EsEcm

is the ratio between the elasticity modulus of the reinforcement and the concrete'ssecant modulus of elasticity, fct is the concrete's uniaxial tensile strength and Asand Ac are the cross sectional area of the reinforcement and concrete, respectively.

20


Figure 2.17: Distribution of concrete-, steel- and bond stresses in a reinforced con-crete member loaded by a tensile load less than the cracking load. Re-production from [26].

A cracking process is initiated when the cracking load is reached. New cracks emergewith a varying spacing although the load is not increased signi�cantly. A local bondfailure occurs at each new crack with a consequent redistribution of stresses.

The crack spacing, sr is limited by this redistribution of stresses as the concretestress will not reach the tensile strength until after a length equal lt,max + ∆r froma free edge as illustrated in Figure 2.18. Similarly, once a new crack emerges, theconcrete stress will have to build up towards the tensile strength over the samedistance. If two cracks emerge at a distance less than 2 · (lt,max + ∆r), the concretestress will not reach the tensile strength between the two cracks. Hence, a newcrack will not emerge between them. The upper and lower limits for crack spacingis presented in Equation 2.14 and illustrated in Figure 2.19.

lt,max + ∆r ≤ sr ≤ 2 · (lt,max + ∆r) (2.14)

It is evident that the crack spacing is dependent on the transmission length whichin return is a�ected by such factors as the reinforcement's surface and distributionin the concrete structure [26].

21


Figure 2.18: Distribution of concrete stresses in a reinforced concrete member loadedby an increasing tensile load up to the cracking load. Reproduction from[26].

2.2.2 Crack control according to EC2

The EC2 [2] proposes that a limiting calculated crack width, wmax, should be es-tablished, given the function and nature of the structure and the costs of limitingcracking. The proposed values of wmax lie between 0.2 and 0.4 mm. Formulas fordirect calculations are presented in EC2 as well as tables for which to achieve adesign, i.e. rebar diameter and spacing, that meets the requirements set by a givenwmax.

A method to calculate crack widths is given in EC2 as a function of the maximumcrack spacing, sr,max, which is given by

sr,max = k3c+ k1k2k4φ/ρp,eff (2.15)

where c is the cover to the reinforcement, φ is the rebar diameter and k1 − k4 arecoe�cients that account for bond properties and strain distribution.

22


Figure 2.19: Crack spacing's upper and lower limits. Reproduction from [26].

Where the spacing of the reinforcement exceeds the value of 5(c+ φ/2) (see Figure2.20), an upper bound to the crack width may be found assuming a maximum crackspacing of:

sr,max = 1.3(h− x) (2.16)

Figure 2.20: Crack width, w, at concrete surface relative to distance from bar. Re-production from [2].

23


Figure 2.20 shows how crack widths vary between two parallel rebars. `A' refers tothe neutral axis, `B' to the surface of concrete in tension, `C' refers to the crackspacing predicted by Equation 2.16, `D' to the crack spacing predicted by Equation2.15 and `E' is the actual crack width.

2.3 Moment and force redistribution from linear FE

analysis

As mentioned in Chapter 1.2, the EC2 allows the redistribution of peaks in cross-sectional moments obtained through linear elastic analysis. However, the code doesnot state a clear methodology as how to obtain these redistributed values.

2.3.1 Moment Peaks Over Columns

Blaauwendraad [3] examined a plate supported by four columns and acted upon by auniformly distributed load (Figure 2.21). A FEA was conducted with three di�erent

Figure 2.21: Plate examined by Blaauwendraad. Reproduction from [3].

element sizes; 0.25, 0.5 and 1.0 m. The resulting distribution of the moment, mxx,over a section above the two right columns is shown in Figure 2.22. It can be seenthat the peak moment values are sensitive to the mesh size. The peak moment valueof the �nest mesh is 56% higher than of the coarsest mesh (designated as 100%).The exact value of the integral of the moment mxx over the section of the plate isknown; it is determined from the free body equilibrium of the plate part right ofthe section under consideration and the load acting on it. This exact value can thenbe compared to the integrals obtained from the FEA, represented in Figure 2.22 asthe area under the moment curve. This comparison shows that the �nest mesh fallsless than 1% short from the exact value and the coarsest mesh only 2.4%. Althoughthere is a great di�erence in peak moment values between mesh sizes, the integraldi�ers less than 2% [3].

24

2.3. MOMENT AND FORCE REDISTRIBUTION FROM LINEAR FE ANALYSIS

Figure 2.22: Moment distribution for di�erent mesh �neness, from coarsest (bottom)to �nest (top). Reproduction from [3].

2.3.2 Recommendations of redistribution widths

Blaauwendraad [3] recommends that the integral over a section equal to 5 times thecolumn width should be used to �smear� out the moment peak. This is illustratedin Figure 2.23.

Figure 2.23: Smearing out of moment peak. Reproduction from [3].

25


By introducing a distribution width, w, a general equation can be obtained for thesmearing out of moments:

m′xx =1

w

∫ w

0

mxxdy (2.17)

where m′xx is the smeared moment used to dimension the reinforcement over thedistribution width, w. The point of the peak moment is centered at w/2. Note thatthe moments are redistributed along a line perpendicular to the force vector [27].

Recommendations by Pacoste et al.

Pacoste et al. [1] gave recommended values for redistribution widths. The recom-mended values were given for the ultimate limit state (ULS) as well as the SLS. Thedistribution widths are dependent of such features as the depth of the neutral axisat the ULS after redistribution, the e�ective depth of the section, the geometry ofthe slab and the concrete strength class.

The choice of an appropriate redistribution width for SLS is by far more intricatethan for ULS and there are very few recommendations available in literature. Thisis mainly due to the fact that the it is di�cult to determine the degree to whichmoment redistribution will take place in the SLS. Therefor, a conservative choice ofdistribution widths is recommended as given by Equation 2.18.

min

(3h,

LC10

)≤ w ≤ min

(5h,

LC5

)(2.18)

In Equation 2.18, h is the height of the section and LC is the characteristic spanwidth, determined di�erently for one-way, two-way and predominantly one-wayspanning slabs (Figures 2.24 and 2.25). A formal distinction between a one-wayand two-way spanning slab is given by EC2, but generally it refers to whether a slabtransfers loads in one direction or two. As many typical slab bridges fall betweenthese two groups, such as the case study bridge does, Pacoste et al. distinguishedthe third category, i.e. the predominantly one-way spanning slab [1]. Such a slab isillustrated in Figure 2.25 along with the notations used in the following recommen-dations.

26

2.3. MOMENT AND FORCE REDISTRIBUTION FROM LINEAR FE ANALYSIS

(a) (b)

Figure 2.24: (a) One-way and (b) two-way spanning slabs. Reproduction from [1].

Figure 2.25: A predominantly one-way spanning slab with notations used in the rec-ommendations given by [1]. Reproduction from [1].

The characteristic span width di�ers for the longitudinal, main load carrying direc-tion of the structure (x -axis) and the transversal direction (y-axis). The distributionwidth wy for the reinforcement moment mx can be determined with the character-istic span width computed according to

LC = L1(x) (2.19)

andLC =

L1(x) + L2(x)

2(2.20)

for supports in line S1 and S2, respectively. In addition to the restrictions presentedin Equation 2.18, the distribution width should also respect the condition

wy ≤ By/2 (2.21)

27


The distribution width wx for the reinforcement momentmy can be determined withthe characteristic span width computed according to

LC =L1(x) + L1(y)

2(2.22)

LC =L1(x) + Lm(y)

2(2.23)

LC =Lm(x) + L1(y)

2(2.24)

LC =Lm(x) + Lm(y)

2(2.25)

for supports S11, S12, S21 and S22, respectively, where

Lm(x) =L1(x) + L2(x)

2and Lm(y) =

L1(y) + L2(y)

2.

An additional condition which the distribution width must respect is given by

wx ≤ Lm(y) (2.26)

Mesh Dependency and Critical Cross Sections

In the FEA of the plate by Blaauwendraad [3], a clear mesh dependency was ob-served in the peak moment values although the area under the curves di�ered onlyslightly (Figure 2.22). Another observation made by Blaauwendraad is that thelarge di�erence in moment value rapidly disappears in the neighborhood of the col-umn. The moment in the �eld between two columns is within 1% for all three mesh�nenesses. The same result was reached by Plos [7], stating that the e�ect of thesingularity due to the modeling of the column in a single point has a negligible a�ecton the cross-sectional moments and shear forces only two element lengths away.

Figure 2.26: Critical sections for the determination of cross sectional moment (left)and shear force (right) at the connection between a slab and a column.Reproduction from [7].

Critical cross sections for the reinforcement moments and shear forces must be deter-mined based on possible failure modes in the reinforced concrete slab (Figure 2.26).

28

2.4. ITERATION PROCEDURE IN NON-LINEAR ANALYSES

Plos [7] recommended that the mesh density should be chosen such that there are atleast two element lengths between the support point and the critical cross section.Furthermore it was recommended that the moments and shear forces in the criticalcross sections should be used for design.

2.4 Iteration procedure in non-linear analyses

In an analysis with non-linear material properties, the governing equations can notbe solved directly as the solution requires an iterative process. Thus, the load isdivided into increments in order to increase the load gradually up to the desiredload level. Assume a structure with non-linear material properties acted upon byan external force P has a displacement response shown in Figure 2.27.

Figure 2.27: A non-linear load and de�ection curve. Reproduction from [28].

In order for the structure to be in equilibrium, the external forces must be balancedby the internal forces, I, i.e.

P − I = 0 (2.27)

To determine the non-linear response of a structure subjected to a small load in-crement ∆P , the tangential sti�ness K0 determined at the previous load increment,u0, is used to extrapolate a displacement correction ca for the structure. By usingthe displacement correction ca the structure's con�guration is updated to ua, asillustrated in Figure 2.28.

Subsequently, the internal forces Ia in the structure are calculated at the displace-ment ua and compared to the total applied load as

Ra = P − Ia (2.28)

where Ra is the iteration's force residual.

If the force residual is zero in every degree of freedom (DOF) in the model, point ain Figure 2.28 would lie on the load-de�ection curve and the structure would be inequilibrium. However, this is never the case in a non-linear analysis, i.e. equilibriumis never achieved in a non-linear analysis [28].

29


Figure 2.28: Iteration of an increment in a non-linear analysis. Reproduction from[28].

Instead, the force residual is compared to a predetermined tolerance value. If theforce residual is less than this value, the iteration is accepted and and the structureis assumed to be in equilibrium. However, before the solution is accepted, a checkof the last displacement correction ca is performed. Typically, ca has to be smallerthan a given tolerance, which is a predetermined fraction of the total incrementaldisplacement ∆ua = ua − u0. In Abaqus, the default tolerance values for the forceresidual and the displacement correction are 0.5% and 1.0%, respectively. Bothchecks must be satis�ed before a solution is determined to have reached convergence[10, 28].

30

Chapter 3

Method

3.1 Modeling procedure

The software used for the FEA, Abaqus, o�ers a graphical user interface to createmodels, simulate loading and review results. This section aims to describe thegeneral modeling procedure in Abaqus as well as some di�culties in the process.

3.1.1 General steps in modeling

Geometry

The two main structural elements of the bridge, the slab and the columns, are createdand dimensioned (see Chapter 1.3). The slab consists of 4-node shell elements andthe columns of 3-node beam elements. The two di�erent element types are coupledtogether through a user de�ned constraint, where the displacement and rotation ofthe respective elements are coupled. The slab has to be partitioned at points ofintersection, e.g. at the position of the roller supports as well as where the di�erentloads are to be applied as illustrated in Figure 3.1.

Figure 3.1: A screen-shot from Abaqus: the bridge modeled by shell and beam ele-ments for the slab and columns, respectively.

31

CHAPTER 3. METHOD

Materials and sections

The next step is to create a material model. For the linear model, linearly elasticmaterial models were de�ned for concrete and steel as presented in Table 3.1. Thesections are dimensioned according to [7] (see Chapter 1.3). The slab has a thicknessof 0.6 m and the columns have a square section with a side length of 0.6 m. Thematerial model is then assigned to the sections.

Table 3.1: Linearly elastic material model for steel and concrete.

Property SymbolMaterial

Concrete Steel

Young'smodulus

E 34 GPa 200 GPa

Poisson's ratio ν 0.2 0.3

Boundary conditions and loads

The roller supports located at the middle of the slabs width are restrained frommovement in the transversal direction, while the other roller supports are restrainedin the vertical direction only. The columns are �xed at their base, i.e. fully restrainedfrom both translation and rotation.

The bridge is assumed to be loaded by its self weight and with tra�c loads only.The load case used in the study is critical for the support moment at one of thecolumns closest to the edge. The loads were chosen according to the Swedish bridgecode, Bro 2004. The load case consists of three 3.0 m lanes and two vehicles, asshown in Figure 3.2. The lanes; numbered i = 1, 2, 3; are loaded by a uniformlydistributed load qi and the vehicles have three axles with an axle load of Pi. Theload from each axle is applied to the slab at two single nodes, i.e. each wheel acts ata single point on the slab. Although the application of point loads can result in localdisturbances, it will not a�ect the results of interest, namely the moment �eld overthe columns. If the shear stresses close to the wheel loads were of interest, the pointloads would indeed skew the results which would call for a di�erent load applicationmethod. The self-weight of the concrete including reinforcement is assumed to beρself , applied to the slab as a uniformly distributed load. Two safety factors areused, γself and γtraffic, for the self weight and the applied tra�c loads, respectively.The values of the tra�c loads, self weight and safety factors are tabulated in Table3.2.

32

3.2. REINFORCEMENT DIMENSIONING

Table 3.2: Numerical values of the loads and safety factors used in the model.

Load Value Safety factor Value

q1 4 kN/m2 γself 1.0q2 3 kN/m2 γtraffic 1.5q3 2 kN/m2

P1 250 kNP2 170 kNρself 2500 kN/m3

Figure 3.2: The tra�c load positions on the slab bridge. Reproduction from [7].

3.2 Reinforcement dimensioning

With the results from the linear FEA of the bridge (Figure 3.3), a reinforcementplan could be dimensioned based on the crack control criteria in EC2 where themaximum crack width was chosen as wmax = 0.30 mm. The dimensioning was donefor bending moments in both longitudinal and transversal directions for the top andbottom of the slab.

3.2.1 Redistribution widths for reinforcement moments

The concentrations of moments were redistributed over a width according to therecommendations given by Pacoste et al. [1]. In addition to the recommendedredistribution widths, two other cases were also investigated. The widths used inthe di�erent cases are presented in Table 3.3. Case 3 is the most extreme case, i.e.the upper limit given by Equation 2.26 for wx and the full distance between columnsfor wy.

33

CHAPTER 3. METHOD

Figure 3.3: The design moment mx obtained from the linear model before redistri-bution.

Table 3.3: Distribution widths used in the analyses.

Case wy [m] wx [m]

1 1.2 0.82 2.4 1.23 4.0 2.0

3.3 The non-linear shell model

Based on the reinforcement design, new models were constructed for non-linearanalyses. The concrete slab was modeled with shell elements that include embeddedlayers of steel reinforcement, according to the aforementioned reinforcement design.

3.3.1 Material parameters

In addition to the elastic parameters, described earlier, the damaged plasticity modelincludes �ve new parameters (Table 3.4) as well as a numerical description of thecompressive and tensile behavior (see Appendix A).

Table 3.4: Plasticity Parameters used in the CDP model.

DilationAngle

Eccentricity fb0/fc0 KViscosityParameter

35◦ 0.1 m 1.16 0.667 1 · 10−10

34

3.3. THE NON-LINEAR SHELL MODEL

Fracture energy

The concrete's fracture energy Gf was chosen according to the recommendationsby the CEB 1990 Model Code [23] for two di�erent maximum aggregate sizes, 16mm and 32 mm. They correspond to fracture energy values of 82.5 Nm/m2 and105 Nm/m2, respectively. In the CEB 2010 Model Code [29], the fracture energy isgiven as a function of the characteristic compressive strength (fcm) alone as stated inEquation 3.1. This yielded a value ofGf = 143.7 Nm/m2. A comparison between therecommendations between the 1990 and 2010 CEB model codes is made graphicallyin Figure 3.4. The fracture energy is not considered in the EC2.

Gf = 73 · f 0.18cm (3.1)

Figure 3.4: A comparison of the fracture energy as given by the CEB 1990 and 2010model codes.

3.3.2 Loading

Non-linear models are loaded incrementally, as covered in Chapter 2.4. The modelwas loaded gradually, �rst by the self weight, secondly by the uniformly distributedlane loads and �nally by the concentrated axle loads. The self weight was appliedlinearly from the �rst step increment (t = 0) to a time step increment of t = 1, afterwhich the self weight was held constant. The lane loads were applied linearly fromt = 1 to t = 2.5 and held constant thereafter. Finally the axles loads were appliedlinearly from t = 2.5 to t = 5.

The maximum step increment was set at ∆tmax = 0.005 but the lower limit at∆tmin = 1 · 10−15.

35

CHAPTER 3. METHOD

3.3.3 Elements and mesh

The slab was modeled with 4-node shell elements with embedded layers of reinforce-ment, as previously mentioned. An advantage of this method is that it requires noadditional DOFs nor extra elements. A drawback is however that the reinforcementis assumed to be completely bonded to the concrete, i.e. no slip can occur. Assign-ing reinforcement in this manner is a tedious task, as the slab has to be partitionedat intersections of all four reinforcement assignments, i.e. top and bottom rein-forcement in both x- and y-directions. This resulted in a total of over 100 di�erentcombinations of reinforcement in the three cases. This would not be approved as apractical design for a real construction but will work �ne for this theoretical study.The partitioned slab used in Case 1 is illustrated in Figure 3.5 and the di�erentcombinations are tabulated in Table B.1 in Appendix B.4. The reinforcement plansand the di�erent combinations used in all three cases can be found in Appendix B.

Figure 3.5: The slab partitioned to include embedded reinforcement in the top andbottom in both x- and y- directions. Each color represents a certaincombination of reinforcement as tabulated in Table B.1.

The top one meter of the columns were modeled with solid elements to overcomeconvergence problems, due to the singularity caused by modeling the connectionbetween the slab and columns in single points These solid elements in the top ofthe columns were given elastic material parameters, as they are mainly subjected tocompressive forces and their non-linear behavior is not of interest in these analyses.

As the cracks are of interest in the non-linear models, the mesh size in the slab abovethe columns must be �ne to extract su�ciently many result points. The crackingin the middle of the slab, i.e. above the column supports, is of most interest so themaximum mesh size there is chosen as l = 0.1 m, while else where the mesh sizedoes not exceed l = 0.3 m. Some irregularities appeared in the mesh as a result ofthe partition points due to the reinforcement groups and the tra�c load.

36

3.4. THE NON-LINEAR SOLID MODEL

Reduced and full integration

Abaqus o�ers two di�erent types of 4 node, �rst-order, �nite strain shell elements;S4 and S4R. The former uses a full integration scheme while the latter uses re-duced integration. Fully integrated �rst-order shell elements can tend to show overlysti� behavior in a well documented phenomena known as shear locking where non-existent (parasitic) shear forces develop in pure bending problems. This problemcan be overcome by applying the reduced integration scheme where only one Gausspoint is used for the calculations. The reduced integrated elements can however ex-perience a di�erent type of di�culty known as hour-glass modes. Since the elementshave only one integration point, it is possible for them to distort in such a way thatthe strains calculated at the integration point are all zero, which in turn leads touncontrolled distortion of the mesh [10, 30].

A so called hour-glass control is applied to elements with reduced integration schemeswhere a small amount of energy is introduced to the element to avoid zero-energymodes. This is an unattractive solution for this study case, as it is hard to controlthe exact amount of �ctitious energy introduced into the model. Furthermore, �niteelement slabs do not tend to experience shear locking when the thickness-to-spanratio is greater than approximately 1/40 [31]. As the case study bridge's ratio is0.6/12 = 1/20, shell elements with the full integration scheme were chosen.

3.4 The non-linear solid model

To further investigate the behavior of the concrete over the columns and in anattempt to validate the results obtained from the non-linear shell models, a newmodel was constructed using both shell and solid elements to model the slab. Inthe area over column 1, the shell elements were removed and replaced by solid andtruss elements to model the concrete and reinforcement, respectively, as shown inFigure 3.6. An original model replaced the shell elements over the entire width ofthe bridge over the columns. The immediate results of that model indicated that thecracking occurred over column 1, as anticipated. It was concluded that by reducingthe number of solid elements used, one could obtain reliable results with signi�cantlyless CPU time.

3.4.1 Material parameters

The material properties of both the concrete and the steel reinforcement are identicalto those used in the shell model and presented in Chapters 3.1 and 3.3.

37

CHAPTER 3. METHOD

Figure 3.6: Solid model: A close up of the area above column 1 where the shellelements were replaced by solid and truss elements to model the concreteand reinforcement, respectively.

3.4.2 Elements and mesh

The model utilizes 4-node shell and 8-node solid elements with full integration tomodel the slab. The top of the columns are modeled with 8-node solid elementswhile the rest of the columns are modeled with 3-node beam elements. The steelreinforcement in the solid part of the slab is modeled with 2-node truss elements.

The shell elements are connected to the solid through a Shell-to-Solid coupling,where the edges of the shell elements are coupled to the side surfaces of the re-spective solid elements. The truss elements, used to model the steel reinforcement,are embedded in the solid elements through an Embedded region constraint. Thisleads to a full bond between the reinforcement and the concrete and is thereforecomparable to the non-linear shell model described in Chapter 3.3.

Obtaining the crack width w

To obtain a crack width w from the shell models or the solid model, the plasticstrains have to be examined. For the shell models, the crack width is obtainedby integrating the plastic strain over the maximum crack spacing, sr,max, in thedirection normal to the crack. This is due to the fact that the cracking in the shellelements is �smeared" over each element, as described in Chapter 2. The crackwidth is also obtained by integrating the plastic strains in the solid model, but afully developed crack will be isolated to a single element row, so the integration canbe performed over the width of the element alone.

3.5 Numerical instability in Abaqus

As Abaqus is a multi-physics software, it has not been developed specially withconcrete behavior in mind. Malm [13] stated that obtaining a converging solution

38

3.5. NUMERICAL INSTABILITY IN ABAQUS

in Abaqus can be di�cult for those who are unexperienced with the software. Hesuggests some actions to avoid convergence di�culties such as the introduction ofarti�cial damping to the system, which can implemented through the automatic

stabilization function in the Static General solver. Another measure is to imple-ment a discontinuous analysis, where relatively many iterations are allowed beforethe program begins to check the convergence rate. This commando can be used whenconsiderable non-linearity is expected in the response, including the possibility ofunstable regimes as the concrete cracks.

Visco-plastic regularization

Material models exhibiting softening behavior and sti�ness degradation often leadto severe convergence di�culties in implicit analysis programs, such as the StaticGeneral solver in Abaqus. A common technique to overcome some of these con-vergence di�culties is the use of a visco-plastic regularization of the constitutiveequations, which causes the consistent tangent sti�ness of the softening material tobecome positive for su�ciently small time increments. The concrete damaged plas-ticity model can be regularized using visco-plasticity by permitting stresses to beoutside of the yield surface. This is done through the viscosity parameter, µ. Thebasic idea is that the solution of the visco-plastic system relaxes as t/µ→∞, wheret represents time. By default, Abaqus does not implement visco-plastic regulariza-tion, i.e. µ = 0. Thus, introducing a positive value of µ, which is small compared tothe characteristic time increment, will help overcome convergence problems withoutcompromising the accuracy of the results [10, 13].

39

Chapter 4

Results

By examining the distribution of forces and strains throughout the model with thegraphical user interface in Abaqus, key areas could be identi�ed for data extraction.As suspected, the largest plastic strains developed over the column supports and inthe left side span in the top and bottom layers, respectively. The largest downwardde�ection occurred in the left side span, where two of the three axle loads arelocated. The extracted data was processed numerically with Matlab for calculationsand graphical presentation.

4.1 Maximum downward de�ection

The maximum downward de�ection is plotted over the step increment, i.e. the loadapplication, for the three di�erent values of fracture energy Gf in Figures 4.2-4.4 forshell models 1-3, respectively. The vertical dashed lines indicate the points where theloading changes from self weight to lane loads and then to axle loads, as mentionedin Chapter 3.3.2.

Figure 4.1: Vertical de�ection of the model after loading. The deformed shape hasbeen exaggerated multiple times for clarity.

41

CHAPTER 4. RESULTS

Figure 4.2: Downward de�ection of the left side span during the load application fordi�erent values of Gf .


42

4.2. CRACK WIDTH GROWTH


4.2 Crack width growth

Figures 4.5 and 4.6 show the plastic strain in the x-direction in the bottom andtop layer, respectively. The crack width over the supporting column in lane 1 wascalculated by means of numerical integration, as described in Chapter 3.4.2 withvalues of sr,max as listed in Table 4.1. The crack spacing is calculated accordingto the guidelines given in the EC2. The progression of the crack width growthduring the load application is illustrated in Figures 4.7-4.9 for shell model cases 1-3,respectively. The results for cases 1 - 3 are compared graphically in Figure 4.10 andnumerically in Table 4.2. The results for the crack width at midspan are presentedsimilarly in Figure 4.11 and Table 4.3.

Table 4.1: Values of maximum crack spacing, sr,max [mm], used in the crack widthcalculations.

Mid span Over Column 1

Case 1 274 225

Case 2 277 218

Case 3 277 214

43

CHAPTER 4. RESULTS

Figure 4.5: The bottom layer plastic strains in the x-direction plotted on the model'sdeformed shape for Case 1, Gf = 143.7 Nm/m2.

Figure 4.6: The top layer plastic strains in the x-direction plotted on the model'sdeformed shape for Case 1, Gf = 143.7 Nm/m2.

44


Figure 4.7: Evolution of the crack width over the column in lane 1 during the dura-tion the load application for di�erent values of Gf .


45

CHAPTER 4. RESULTS


Figure 4.10: Comparison of the crack width over column 1 between cases and witha varying fracture energy.

46


Table 4.2: Crack widths [mm] over column 1 obtained by the shell models for di�erentfracture energy values Gf .

Gf [Nm/m2]

82.5 105 143.7

Case 1 0.1425 0.1389 0.1229

Case 2 0.1583 0.1648 0.1433

Case 3 0.1653 0.1558 0.1339

Figure 4.11: Comparison of the crack width in the left side span between cases andwith a varying fracture energy.

Table 4.3: Crack widths [mm] in the left side span obtained by the shell models fordi�erent fracture energy values Gf .

Gf [Nm/m2]

82.5 105 143.7

Case 1 0.0354 0.0333 0.0342

Case 2 0.0505 0.0419 0.0381

Case 3 0.1732 0.1207 0.0364

47

CHAPTER 4. RESULTS

4.3 Solid Case

The plastic strains development in the solid elements is shown in Figure 4.12. Thecenter to center distance between the two closest cracks is 200 mm.

(a) t = 1 (b) t = 2.5 (c) t = 5

Figure 4.12: Progression of the plastic strains (cracking) in the solid elements abovecolumn 1.

The maximum crack width, obtained by numerical integration of the solid model, ispresented in Table 4.4 and compared to the crack width obtained in the shell modelsgraphically in Figure 4.13.

Table 4.4: Crack widths [mm] obtained by numerical integration of the plastic strainsfrom the solid model with a varying fracture energy.

Gf [Nm/m2]82.5 105 143.7

0.2458 0.2531 0.2331

48

4.4. REINFORCEMENT QUANTITIES

Figure 4.13: Comparison of the crack width over column 1 between the solid andshell models.

4.4 Reinforcement quantities

The amount of steel reinforcement [kg/m3 concrete] was examined between cases.The average amount of steel for the entire slab is listed in Table 4.5. The averageamount in a 4 m wide strip over the columns is listed in Table 4.6.

Table 4.5: Average amount of steel reinforcement per cubic meter concrete [kg/m3]for Cases 1 - 3.

Case 1 Case 2 Case 3

55.8 57.8 58.0

Table 4.6: Average amount of steel reinforcement per cubic meter concrete [kg/m3]in a 4 m strip over the columns for Cases 1 - 3.

Case 1 Case 2 Case 3

74.7 76.5 78.0

49

Chapter 5

Discussion and conclusions

5.1 Crack control

5.1.1 Shell models

As the slab was dimensioned for crack control according to the EC2 for a load casedesigned to produce moment peaks over column 1, the result of most interest is thecracking in exactly that area.

Figures 4.7 - 4.9 show how the crack width evolves as the loads are applied to thestructure. The crack width growth is very limited during self weight loading andeven after the lane loads are applied, the crack width is only approximately 10%of the designed value of wmax = 0.3 mm. The e�ect of the fracture energy, Gf ,becomes evident as the axle loads are applied. As can be seen from Figures 4.10 and4.11, the general trend is that the crack width decreases with an increased fractureenergy level. Another trend that can be seen from the same �gure is that the crackwidth increases between cases, i.e. with an increased distribution width.

When compared to the design value, it is clear that the shell models yield resultswell below the design value or at least 42% lower. This was perhaps expected assome factors, such as material partial coe�cients and a factor to account for creep,are not implemented in the non-linear FEM.

5.1.2 Solid model

The solid model yielded considerably higher crack widths than the shell models, ascan be seen in Figure 4.13. The crack widths are still 15 - 22% lower than the designvalue wmax.

One possible explanation of these higher crack widths is that the solid elementsbehave too sti�y and thus attract forces. This would lead to an increased reactionforce at column 1 and concurrently lowered reaction forces at columns 2 and 3.

51

CHAPTER 5. DISCUSSION AND CONCLUSIONS

This is however not the case. The reaction force at column 1 decreased compared tothe shell models. This was not completely unexpected, as the cracking of the solidelements tends to lead to a softer behavior, thus not attracting the same amount offorces as it would otherwise. It is however evident, that the constraining equationsbetween the solid and shell elements need careful attention and they could indeedin�uence the results.

Crack spacing

The crack spacing, sr, can be extracted directly from the solid model by viewing thecenter to center distance between two parallel cracks. The crack spacing obtainedfrom the solid model �ts the calculated values from the EC2. However, the crackspacing will be a multiple of the solid elements' mesh size, in this case 0.1, 0.2, 0.3m, . . .. A considerably �ner mesh is needed to obtain a more accurate value of thecrack spacing which in return costs a signi�cantly higher amount of computationaltime. As the solid model used in this thesis had a computational time of 2 days,further re�nement of the mesh size was omitted due to time pressure.

5.1.3 Sensitivity of the results

Maximum crack spacing

As the crack width in the shell models is obtained by integrating the plastic strainsover the maximum crack spacing sr,max, it is obvious that the crack width will varydepending on the maximum crack spacing used. In this thesis, an algorithm wasused to compute the crack width by locating the point of maximum plastic strainand integrating over a length from −sr,max/2 to sr,max/2 with respect to that point.This is shown graphically in Figure 5.1 where three di�erent maximum crack spacingvalues are used; 0.2, 0.3 and 0.3 m; yielding crack widths of 0.129, 0.180 and 0.233mm, respectively.

(a) sr,max = 0.2 m;w = 0.129 mm

(b) sr,max = 0.3 m;w = 0.180 mm

(c) sr,max = 0.4 m;w = 0.233 mm

Figure 5.1: Plastic strain over column 1 (blue line) and the length over which theplastic strain is integrated to obtain the crack width (red line). Thevarying maximum crack spacing sr,max yields di�erent crack widths w.

52

5.2. CONCLUSIONS

It is evident from the above that the maximum crack spacing has a signi�cant impacton the crack width. One can argue that such an important factor could be de�nedby more detailed means than a simple equation such as that de�ned in EC2.

The maximum crack spacing will not a�ect the crack width in the solid models asthe plastic strains in the fully developed cracks are only present in a single row ofelements and the strains in the adjacent elements will be zero.

Stabilized crack pattern

The calculations of the maximum crack spacing sr,max assumes that a state of stabi-lized cracking has been reached, as covered in Chapter 2.2. If the cracking has notreached this stabilized state, the emergence of new cracks will alter the stress andstrain distribution in the model and can thus a�ect the results.

Mesh size

The mesh size might a�ect the results of the shell model at single nodes to someextent, but as pointed out previously in Chapter 2.3.1, the results over a section willnot be a�ected to a signi�cant degree by the mesh size.

The mesh size in the solid model has however to be checked as the plastic strainsdevelop in a single row of elements, hence, the crack width is estimated by integratingthe strain over the element length. As previously mentioned, further re�nement ofthe mesh size was omitted due to time pressure.

5.2 Conclusions

5.2.1 Recommendations by Pacoste et al.

Given the results obtained from the non-linear shell and solid models, one can con-clude that the recommendations by Pacoste et al. [1] are reasonable. By examiningthe shell models alone, one could perhaps conclude that the recommendations wereconservative, but the results from the solid model give rise to tread carefully. Al-though the crack width over column 1 did not increase dramatically between cases2 and 3, there was a signi�cant increase in the crack width in the mid span for thelower two values of Gf (Figure 4.11). Altering the cross-sectional design in a singlesection can a�ect the global structural response. With this in mind, it is importantto stress that the behavior of the structure as a whole has to be considered whenchoosing appropriate redistribution widths.

One of the motivations for redistributing the peaks that occur in moment diagramsis economical. But increasing the redistribution width actually raised the averageamount of reinforcement used in the slab, as seen in Chapter 4.4. This applies

53

CHAPTER 5. DISCUSSION AND CONCLUSIONS

for both the average reinforcement in the entire slab as well as the 4 m strip overthe column supports. Although this is the case for this particular load case andsubsequent moment distribution, the redistribution might indeed lead to a reducedoverall reinforcement amount for di�erent load cases.

5.2.2 Fracture energy

The fracture energy Gf has a signi�cant e�ect on the structural response in thenon-linear models. The CEB model codes [23, 29] o�er guidelines for choosingappropriate values of Gf if experimental results are unavailable. These guidelineso�er a single value, e.g. Gf = 105 Nm/m2, but the shape of the softening curve(Figure 2.15) is not speci�ed.

The EC2 does not however cover the topic of fracture energy. As the fracture energydescribes the structure's ability to withstand crack opening in the tensile post peakregions, omitting it can be considered as a conservative design approach. In alinear analysis, the fracture energy can of course not be implemented directly. Evenso, the complete absence of coverage of this important phenomenon in the EC2 isquestionable, to say the least.

5.2.3 Limitations and possible improvements

The models constructed and used for this thesis are not without their imperfections.As mentioned above, the solid model would require considerable mesh re�nementto be able to give a more accurate result of crack spacing as well as to investigatethe e�ect on the computed crack width. This will in turn require considerablecomputational time.

The material model

The e�ect of the tension softening curve's shape could be of interest. As mentionedabove, a given fracture energy value can be represented by many di�erent shapes.In this thesis, a bilinear curve was adapted as it can be easily de�ned by three pointson the stress-displacement curve. A more accurate shape would be the exponentialcurve, as described by Malm [13] and illustrated in Figure 2.15.

Certain long term e�ects such as creep and thermal loadings are not accounted forin the FEMs. In fact, the Concrete Damaged Plasticity model in Abaqus can notbe combined with an appropriate material model to describe creep [10].

Reinforcement plans

The reinforcement plans obtained in this thesis would not be considered feasible asactual reinforcement plans for a real structure. The intention was to give insight into

54

5.2. CONCLUSIONS

the behavior of the concrete slabs when designed according to the recommendations,i.e. comparing theory with theory. Further simpli�cations, such as rounding o�widths, are necessary in engineering practice to obtain reinforcement plans that canactually be constructed. A model based on such a feasible reinforcement would bean interesting comparison to the mere theoretical models.

User dependency

A �nal point worth of stressing is the user dependency in �nite element modeling.Blaauwendraad [3] pointed out that given the same information and software todisposal, 10 civil engineers came up with 8 di�erent reinforcement plans for a simpleslab structure. The choice of di�erent parameters in a FE model is highly dependenton the designer's previous experience of both the design of reinforced concrete aswell as with the software at hand.

Further research

As stated by Pacoste et al. [1], the topic of moment redistribution is not welldocumented. The models created and analyzed in this thesis were intended toapproximate the structural behavior of concrete slabs during out-of-plane loading,using the best tools available at the time. Full scale testing of concrete slabs isneeded to verify the accuracy of these models and consequently the software's abilityto simulate the structural response and behavior of reinforced concrete.

55

Bibliography

[1] C. Pacoste et al., �Recommendations for �nite element analysis for the designof concrete slabs,� tech. rep., KTH, Department of Civil and ArchitecturalEngineering, 2012.

[2] �Eurocode 2: Design of concrete structures - Part 1-1: General rules and rulesfor buildings. En 1992-1-1..� En 1992-1-1.

[3] J. Blaauwendraad, Plates and FEM. Surprises and Pitfalls. Springer, 2006.

[4] E. O. L. Lantsoght, C. van der Veen, J. C. Walraven, �Shear Capacity of Slabsand Slab Strips Loaded Close to the Support,� ACI Special Publication, vol. 287,2012.

[5] E. O. L. Lantsoght, C. van der Veen, J. C. Walraven, �Peak Shear Stress Distri-bution in Finite Element MModel of Concrete Slabs,� Research and Applicationsin Structural Engineering, Mechanics and Computation, 2013.

[6] P. S. Hakimi, �Distribution of Shear Force in Concrete Slabs,� Master's thesis,Chalmer University of Technology, 2012.

[7] M. Plos, �Redistribution of Moment and Forces from Linear FE Analysis,�in Non-Linear Analysis and Remaining Fatigue Life of Reinforced ConcreteBridges, 2007.

[8] W. F. Chen, D. J. Han, Plasticity for Structural Engineers. Springer-Verlag,1988.

[9] M. Plos, �Finite Element Analysis of Reinforced Concrete Structures,� tech.rep., Chalmers University of Technology, 1996.

[10] Abaqus Theory Manual (6.12). Chapter 4.5.2.

[11] J. Lubliner, J. Oliver, S. Oller, E. Onate, �A Plastic-Damage Model for Con-crete,� International Journal of Solids and Structures, vol. 25, no. 3, pp. 299�326, 1989.

[12] J. Lee, G. L. Fenves, �Plastic-Damage Model for Cyclic Loading of ConcreteStructures,� Journal of Engineering Mechanics, vol. 124, no. 8, pp. 892�900,1998.

[13] R. Malm, �Shear Cracks in Concrete Structures Subjected to In-Plane Stresses.�Royal Institute of Technology. Licentiate Thesis, 2006.

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[14] J. Björnström, T. Ekström, M. Hassanzadeh, Spruckna Betongdammar - Över-sikt och beräkningsmetoder. Elforsk, 2006.

[15] B. Karihaloo, �Failure of Concrete,� in Comprehensive Structural Integrity Vol.2.10, 2003.

[16] R. Malm, Predicting Shear Type Crack Initiation and Growth in Concrete WithNon-Linear Finite Element Method. PhD thesis, KTH, Royal Institute of Tech-nology, 2009.

[17] J.C. Gálvez, J. �ervenka, D. A. Cendón, V. Saouma, �A Discrete Crack Ap-proach to Normal/Shear Cracking of Concrete,� Cement and Concrete Re-search, vol. 32, pp. 1567�1585, 2002.

[18] A. Hillerborg, M. Modéer, P. E. Petersson, �Analysis of Crack Formation andCrack Growth in Concrete by Means of Fracture Mechanics and Finite Ele-ments,� Cement and Concrete Research, vol. 6, pp. 773�782, 1976.

[19] J. G. Rots et al., �Smeared crack approach and fracture localization in concrete,�Heron, vol. 30, no. 1, 1985.

[20] Z. Ba�zant, �Instability, ductility and size e�ect in strain-softening concrete,�Journal of Engineering Mechanics Division, vol. 102, pp. 331�344, 1976.

[21] Z. Ba�zant, B. H. Oh, �Crack band theory for fracture of concrete,� Materialsand Structures, vol. 16, no. 93, pp. 155�177, 1983.

[22] R. de Borst, �Fracture in Quasi-Brittle Materials: a Review of ContinuumDamage-Based Approaches,� Engineering Fracture Mechanics, vol. 69, pp. 95�112, 2002.

[23] Comité Euro-International du Béton, �CEB-FIP Model Code 1990,� 1993.

[24] A. Hillerborg, �The Theoretical Basis of a Method to Determine the FractureEnergy Gf of Concrete,� Materials and Structures, no. 108, pp. 291�296, 1985.

[25] H. A. W. Cornelissen, D. A. Horduk, H. W. Reinhardt, �Experimental Deter-mination of Crack Softening Characteristics of Normalweight and LightweightConcrete,� Heron, vol. 31, no. 2, 1986. Delft, The Netherlands.

[26] M. Al-Emrani, B. Engström, M. Johansson, P. Johansson, Bärande konstruk-tioner. Chalmers University of Technology, 2007.

[27] C. Johansson, P. Söderström, FEMDIM - Manual, 2012.

[28] Richard Malm, �Non-linear analyses of concrete beams with Abaqus,� RoyalInstitute of Technology, KTH, 2014. Compendium for Concrete Structures,advanced course.

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[30] O.C. Zienkiewicz, R.L. Taylor, The Finite Element Method, vol. 1. ButterworthHeinemann, Fifth ed., 2000.

[31] Costin Pacoste, �ELU Consult.� Personal meeting, June 3rd 2014.

59

BIBLIOGRAPHY

60

Appendix A

Concrete Damaged Plasticity Input

Data

A.1 Plasticity Parameters

Table A.1: Plasticity Parameters used in the CDP model.

DilationAngle

Eccentricity fb0/fc0 KViscosityParameter

35◦ 0.1 m 1.16 0.667 1 · 10−10

A.2 Compressive Behavior

The indata used to de�ne the non-linear compressive behavior of the Concrete Dam-aged Plasticity model is presented both numerically and graphically below.

Yield Stress [MPa] Inelastic Strain [-]

17.200000 028.901424 0.0004941237.761298 0.0009941242.398582 0.0014941242.377912 0.0019941237.201944 0.0024941226.299915 0.00299412

61

APPENDIX A. CONCRETE DAMAGED PLASTICITY INPUT DATA

A.3 Tensile Behavior

The concrete is assumed to behave linearly elastic up to the tensile strength, ft. Theindata used to de�ne the post-peak tensile behavior of the CDP model is presentedboth numerically and graphically below. The area under the graph represents thefracture energy Gf .

A.3.1 Gf = 82.5 Nm/m2


3.200 01.067 2.063·10−5

0.032 9.281·10−5

A.3.2 Gf = 105 Nm/m2


3.200 01.067 2.625·10−5

0.032 1.1813·10−4

62

A.3. TENSILE BEHAVIOR

A.3.3 Gf = 143.7 Nm/m2


3.200 01.067 3.5925·10−5

0.032 1.6166·10−4

63

Appendix B

Reinforcement Design

The reinforcement plans obtained for the three di�erent distribution width casesare presented below. The �nal reinforcement layout for each case was obtained bymerging all four layouts (X- and Y-directions in both the upper and lower layer) toobtain a division of the slab's geometry for modeling purposes in Abaqus. The �g-ures of the merged layout for each case are somewhat uninformative but are thoughtto underline the fact that the reinforcement plans are theoretical and such a com-plicated rebar plan would never be adopted in engineering practice. Each patternand color represent a di�erent combination of reinforcement. These combinationsare presented in section B.4 and are intended for those who might further developthe models.

B.1 Case 1

Figure B.1: Reinforcement plan for the top layer in the x-direction.

65

APPENDIX B. REINFORCEMENT DESIGN

Figure B.2: Reinforcement plan for the bottom layer in the x-direction.

Figure B.3: Reinforcement plan for the top layer in the y-direction.

Figure B.4: Reinforcement plan for the bottom layer in the y-direction.

66

B.1. CASE 1

Figure B.5: The combined reinforcement plan for Case 1.

67


B.2 Case 2




68

B.2. CASE 2



69


B.3 Case 3




70

B.3. CASE 3



71


B.4 Reinforcement combinations

Table B.1: The di�erent combinations of reinforcement used in the model.

Reinforcementcombination

x-direction y-direction

Top Bottom Top Bottom

1. φ12 s240 φ12 s240 φ12 s240 φ12 s240

2. φ12 s240 φ16 s200 φ12 s240 φ12 s240

3. φ12 s240 φ20 s160 φ12 s240 φ12 s240

4. φ12 s240 φ20 s107 φ12 s240 φ12 s240

5. φ12 s240 φ12 s240 φ12 s240 φ12 s240

6. φ16 s200 φ12 s240 φ12 s240 φ12 s240

7. φ20 s160 φ12 s240 φ12 s240 φ12 s240

8. φ20 s107 φ12 s240 φ12 s240 φ12 s240

9. φ20 s80 φ12 s240 φ12 s240 φ12 s240

10. φ20 s80/320 φ12 s240 φ12 s240 φ12 s240

11. φ16 s200 φ16 s200 φ12 s240 φ12 s240

12. φ16 s200 φ20 s160 φ12 s240 φ12 s240

13. φ20 s160 φ16 s200 φ12 s240 φ12 s240

14. φ12 s240 φ12 s240 φ16 s260 φ12 s240

15. φ12 s240 φ12 s240 φ16 s130 φ12 s240

16. φ12 s240 φ16 s200 φ16 s260 φ12 s240

17. φ12 s240 φ20 s160 φ12 s260 φ12 s240

18. φ12 s240 φ20 s107 φ16 s260 φ12 s240

19. φ12 s240 φ16 s200 φ16 s130 φ12 s240

20. φ12 s240 φ20 s160 φ16 s130 φ12 s240

21. φ12 s240 φ20 s107 φ12 s130 φ12 s240

22. φ20 s160 φ12 s240 φ12 s260 φ12 s240

23. φ20 s107 φ12 s240 φ12 s260 φ12 s240

24. φ20 s80 φ12 s240 φ16 s260 φ12 s240

25. φ20 s80/320 φ12 s240 φ16 s260 φ12 s240

26. φ20 s80/160 φ12 s240 φ16 s260 φ12 s240

27. φ16 s200 φ12 s240 φ16 s130 φ12 s240

28. φ20 s160 φ12 s240 φ16 s130 φ12 s240

29. φ16 s107 φ12 s240 φ16 s130 φ12 s240

Continued on next page

72

B.4. REINFORCEMENT COMBINATIONS

Table B.1 � continued from previous page




30. φ20 s80 φ12 s240 φ16 s130 φ12 s240

31. φ20 s80/320 φ12 s240 φ16 s130 φ12 s240

32. φ20 s80/160 φ12 s240 φ16 s130 φ12 s240

33. φ20 s107 φ12 s240 φ16 s87 φ12 s240

34. φ20 s80 φ12 s240 φ16 s87 φ12 s240

35. φ20 s80/320 φ12 s240 φ16 s87 φ12 s240

36. φ20 s80/160 φ12 s240 φ16 s87 φ12 s240

37. φ20 s80 φ12 s240 φ16 s65 φ12 s240

38. φ20 s80/320 φ12 s240 φ16 s65 φ12 s240

39. φ20 s80/160 φ12 s240 φ16 s65 φ12 s240

40. φ12 s240 φ16 s200 φ12 s240 φ16 s260

41. φ12 s240 φ20 s160 φ12 s240 φ16 s260

42. φ12 s240 φ20 s107 φ12 s240 φ16 s260

43. φ12 s240 φ20 s80 φ12 s240 φ16 s260

44. φ12 s240 φ16 s200 φ12 s240 φ16 s130

45. φ12 s240 φ20 s160 φ12 s240 φ16 s130

46. φ12 s240 φ20 s107 φ12 s240 φ16 s130

47. φ12 s240 φ20 s80 φ12 s240 φ16 s130

48. φ16 s200 φ16 s200 φ12 s240 φ16 s260

49. φ16 s200 φ16 s200 φ12 s240 φ16 s130

50. φ16 s200 φ16 s200 φ16 s260 φ12 s240

51. φ16 s200 φ16 s200 φ16 s130 φ12 s240

52. φ20 s160 φ16 s200 φ16 s260 φ12 s240

53. φ20 s160 φ16 s200 φ16 s130 φ12 s240

54. φ20 s160 φ20 s160 φ16 s130 φ12 s240

55. φ16 s200 φ20 s160 φ16 s130 φ12 s240

56. φ16 s200 φ16 s200 φ16 s260 φ16 s260

57. φ16 s200 φ16 s200 φ16 s130 φ16 s260

58. φ16 s200 φ16 s200 φ16 s260 φ16 s130

59. φ16 s200 φ16 s200 φ16 s130 φ16 s130

60. φ16 s200 φ20 s160 φ16 s130 φ16 s130


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61. φ20 s160 φ20 s160 φ16 s130 φ16 s260

62. φ20 s160 φ16 s200 φ16 s130 φ16 s130

63. φ20 s160 φ20 s160 φ16 s130 φ16 s130

64. φ20 s160 φ16 s200 φ16 s130 φ16 s260

65. φ20 s160 φ16 s200 φ16 s260 φ16 s130

66. φ20 s160 φ16 s200 φ12 s240 φ16 s130

67. φ16 s107 φ12 s240 φ16 s260 φ16 s260

68. φ12 s107 φ12 s240 φ16 s130 φ16 s260

69. φ20 s107 φ12 s240 φ16 s260 φ16 s130

70. φ20 s107 φ12 s240 φ16 s130 φ16 s130

71. φ20 s80 φ12 s240 φ16 s260 φ16 s260

72. φ20 s80 φ12 s240 φ16 s260 φ16 s130

73. φ20 s80 φ12 s240 φ16 s130 φ16 s260

74. φ20 s80 φ12 s240 φ16 s130 φ16 s130

75. φ20 s80/320 φ12 s240 φ16 s130 φ16 s260

76. φ20 s80/320 φ12 s240 φ16 s87 φ16 s260

77. φ16 s107 φ12 s240 φ12 s240 φ16 s260

78. φ20 s80 φ12 s240 φ12 s240 φ16 s260

79. φ20 s80 φ12 s240 φ12 s240 φ16 s130

80. φ12 s240 φ20 s107 φ16 s260 φ16 s260

81. φ12 s240 φ20 s107 φ16 s260 φ16 s130

82. φ12 s240 φ20 s107 φ16 s130 φ16 s130

83. φ20 s160 φ16 s200 φ16 s260 φ16 s260

84. φ12 s240 φ16 s200 φ16 s260 φ16 s130

85. φ12 s240 φ20 s160 φ16 s260 φ16 s130

86. φ12 s240 φ16 s200 φ16 s130 φ16 s260

87. φ12 s240 φ16 s200 φ16 s130 φ16 s130

88. φ12 s240 φ20 s160 φ16 s130 φ16 s260

89. φ12 s240 φ20 s160 φ16 s130 φ16 s130

90. φ12 s240 φ16 s200 φ16 s260 φ16 s260

91. φ12 s240 φ20 s160 φ16 s260 φ16 s260


74

B.4. REINFORCEMENT COMBINATIONS





92. φ12 s240 φ20 s107 φ16 s260 φ16 s130

93. φ12 s240 φ20 s107 φ16 s130 φ16 s130

94. φ12 s240 φ20 s107 φ16 s130 φ16 s260

95. φ20 s320 φ20 s160 φ12 s240 φ16 s260

96. φ20 s320 φ20 s160 φ12 s240 φ16 s130

97. φ20 s320 φ20 s160 φ16 s260 φ12 s240

98. φ20 s320 φ20 s160 φ16 s130 φ16 s260

99. φ20 s320 φ20 s107 φ16 s260 φ16 s260

100. φ20 s320 φ20 s160 φ16 s260 φ16 s260

101. φ20 s320 φ20 s107 φ16 s130 φ16 s130

102. φ20 s160 φ20 s160 φ12 s240 φ12 s240

103. φ20 s160 φ20 s160 φ16 s260 φ12 s240

104. φ20 s80 φ12 s240 φ16 s87 φ16 s260

105. φ20 s80/320 φ12 s240 φ16 s130 φ16 s130

106. φ20 s80/320 φ12 s240 φ16 s260 φ16 s130

107. φ16 s80/160 φ12 s240 φ16 s87 φ16 s130

108. φ20 s107 φ12 s240 φ12 s240 φ16 s130

109. φ12 s240 φ12 s240 φ16 s260 φ12 s240

110. φ12 s240 φ12 s240 φ16 s130 φ12 s240

111. φ12 s240 φ20 s80 φ16 s260 φ16 s260

112. φ12 s240 φ20 s80 φ16 s260 φ16 s130

113. φ12 s240 φ20 s80 φ16 s130 φ16 s130

114. φ20 s160 φ16 s200 φ16 s87 φ16 s130

115. φ20 s107 φ16 s200 φ16 s260 φ16 s260

116. φ20 s107 φ16 s200 φ16 s260 φ16 s130

117. φ16 s107 φ16 s200 φ16 s130 φ16 s130

118. φ20 s107 φ16 s200 φ16 s87 φ16 s130

119. φ20 s80/160 φ12 s240 φ16 s130 φ16 s260

120. φ20 s80/160 φ12 s240 φ16 s130 φ16 s130

121. φ20 s80/160 φ12 s240 φ16 s260 φ16 s130

122. φ20 s80/320 φ12 s240 φ12 s240 φ16 s260


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123. φ20 s80/320 φ12 s240 φ12 s240 φ16 s130

124. φ20 s80/160 φ12 s240 φ12 s240 φ16 s260

125. φ20 s80/160 φ12 s240 φ12 s240 φ16 s130

126. φ20 s80/160 φ12 s240 φ12 s240 φ12 s240

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