Shear Resistance of Reinforced Concrete Beams without Shear Reinforcement under Sustained Loading

Reza Sarkhosh

Reza Sarkhosh

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Reza Sarkhosh

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy

Faculty of Civil Engineering & Geosciences, Department of Structural Engineering,

Concrete Structures Delft, the Netherlands

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Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op donderdag 17 april 2014 om 15:00 uur door

Reza SARKHOSH

Master of Science in Civil/Earthquake Engineering,

Amirkabir University of Technology (Tehran Polytechnic)

Dit proefschrift is goedgekeurd door de promotor: Prof.dr.ir.Dr.-Ing. h.c. J. C. Walraven

Copromotor: Ir. J. A. den Uijl

Samenstelling promotiecommissie:

Rector Magnificus, voorzitter

Prof.dr.ir.Dr.-Ing. h.c. J.C. Walraven Technische Universiteit Delft, promotor Ir. J. A. den Uijl Technische Universiteit Delft, copromotor Prof.Dr.-Ing. H.W. Reinhardt University of Stuttgart

Prof.dr. A. Muttoni Ecole Politechnique Fédérale de Lausanne (EPFL) Prof.dr.ir. D.A. Hordijk Technische Universiteit Delft

Prof.dr.ir. J.G. Rots Technische Universiteit Delft

Dr.ir. A. de Boer Rijkswaterstaat

Prof.ir. A.Q.C. van der Horst Technische Universiteit Delft, reservelid

ISBN 978-94-6108-645-7

Copyright Ⓒ 2014 by R. Sarkhosh

All rights reserved. No part of this material protected by this copyright notice may be reproduced, stored in any retrieval system or transmitted in any form or by any means, electronic or mechanical, including photocopying or recording, without the prior written permission from the author.

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Shear failure of reinforced concrete (RC) beams without stirrups is an instantaneous brittle failure mode and is complicated by the behaviour of inclined shear crack, aggregate interlock and dowel action. The time-dependency of shear-critical beams is even more complicated by the effects of creep and shrinkage, development of concrete strength, crack opening displacements, creep of bond and stress redistribution in the RC member. In this research, a set of tests on plain and reinforced concrete beams is presented together with numerical analyses, in order to describe the behaviour of concrete members subjected to long-term sustained loads. Additionally, a model is proposed for predicting the shear capacity of RC beams without stirrups, including the effect of aggregate interlock and considering the time effects.

An experimental investigation that focused on the time-dependency of flexural cracks in concrete beams under sustained loads is carried out on small-size plain concrete beams with a notch. The load intensity (λ=Psus/Pu) is chosen between

0.72 and 0.89. The time to failure shows a logarithmic relation to the load intensity. The crack opening rate is simulated by an analytical model based on elasto-viscoplastic (EVP) behaviour for the prediction of the crack growth under long-term loading. In order to predict the corresponding time to fracture and with reference to the experimental data, an empirical expression is given for the crack opening limit. Based on this expression, the critical strain of the concrete at crack initiation is determined. Furthermore, a finite element model is employed in order to study its suitability for the simulation of time-dependent phenomena and to verify the results of the proposed EVP model.

The long-term behaviour of concrete under sustained loads is modelled by means of a bulk creep function (outside the fracture process zone). Under long-term

loading, the strain due to the creep effect in the high stress zone around the fictitious crack tip may be large enough to reach the critical strain, so that crack formation can occur below the static tensile strength. To that end, the criterion should be adjusted to account for the time effect.

Additionally, the model of Gastebled & May (2001) for the prediction of the shear resistance of concrete beams without stirrups is modified by introducing a bilinear crack pattern with special attention to the aggregate interlock mechanism. The proposed model is validated against 393 experiments and compared with the recommendations by ACI 318-08 (2008), BS 8110 (1997), Eurocode 2 (2005), Model Code 2010 (2013) Level II approximation as well as the analytical models presented by Bažant & Yu (2004), Gastebled & May (2001) and Xu et al. (2012). However, the prediction of shear resistance under sustained loading according to the model with bilinear crack pattern comprises a complicated model with time-dependent parameters. Therefore, in order to present the effect of each parameter on the shear resistance of the beam, a discussion is given on the shear resistance under sustained loading and the time-dependent parameters in this model.

Lastly, a series of experiments on shear-critical concrete beams without stirrups is conducted in order to investigate the sustained loading effects. The results of 28 short-term monotonic tests and 14 long-term sustained loading tests under high load intensity (load intensity between 0.83 and 0.98) are given. Time-dependent deflections, surface crack pattern, stress redistribution, crack length development and crack width development are studied and discussed based on the observation during the tests. In addition, the shear resistance of the concrete beams is tested at the end of the desired period of sustained loading, in order to investigate the influence of sustained loading on the final shear resistance.

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Bezwijken op afschuiving van gewapend betonnen balken zonder schuifwapening treedt plotseling op en wordt gekenmerkt door een bros karakter. Het is een gecompliceerd mechanisme, waarbij aspecten als het gedrag van een schuine afschuifscheur, scheurwrijving en deuvelwerking van de langswapening een rol spelen. Het tijdsafhankelijke gedrag is zelfs nog complexer door de effecten van kruip, krimp, de toename van de betonsterkte, de verplaatsingen van de scheurvlakken ten opzichte van elkaar, de aanhechtingskruip en de uit deze invloedsfactoren resulterende herverdeling van spanningen in het gewapende element. In dit onderzoeksprogramma zijn experimenten uitgevoerd op ongewapende en gewapende betonnen balken. Verder zijn numerieke analyses uitgevoerd, met als doel het bezwijkgedrag van betonelementen, onderworpen aan een langdurig aangrijpende belasting, te beschrijven. In aansluiting daarop is een model ontwikkeld om het afschuifdraagvermogen van gewapende betonbalken zonder schuifwapening te voorspellen, met speciale aandacht voor de effecten van scheurwrijving en tijdsafhankelijke vervorming.

Een experimenteel onderzoek naar het gedrag van buigscheuren in betonnen balken onder langdurige belasting is uitgevoerd op kleine betonbalken met een kerf. De verhouding tussen de langdurig aangebrachte belasting en de bezwijkbelasting bij snel toenemende belasting lag tussen 0.72 en 0.89. De tijd tot bezwijken bleek een logaritmische functie van deze verhouding te zijn. De snelheid van scheuropening kan worden beschreven op grond van een analytisch model, gebaseerd op het elasto-visco-plastische gedrag van het beton onder langdurige belasting. Om de corresponderende tijd tot breuk, in overeenstemming met de experimenten, te kunnen bepalen is een empirische uitdrukking voor de kritische scheuropening afgeleid. Gebaseerd op deze uitdrukking, kan de kritische rek in het beton bij scheurinitiatie worden bepaald.

Verder is de geschiktheid van een eindig elementen model onderzocht met betrekking tot het simuleren van de tijdsafhankelijke verschijnselen en het verifiëren van de resultaten van het voorgestelde EVP model.

Het langeduurgedrag van betonelementen onder langdurige belasting wordt gemodelleerd op grond van het kruipgedrag van het beton buiten het gebied waar de scheuruitbreiding plaatsvindt. Onder langdurige belasting kan de betonrek, als gevolg van het effect van kruip in de hoogbelaste zone rondom het scheuruitbreidingsgebied, groot genoeg worden om de kritische grenswaarde te bereiken, zodat scheurvorming op kan treden bij een spanning in het beton onder de korte duur betontreksterkte. Daarom moet het scheurcriterium worden aangepast om met het langeduur effect rekening te kunnen houden.

Vervolgens is het model van Gastebled en May (2001) voor het bepalen van het afschuifdraagvermogen van betonnen balken zonder schuifwapening uitgebreid door het introduceren van bilineaire scheuren, om hiermee het effect van scheurwrijving afdoende in rekening te kunnen brengen. Het voorgestelde model is gevalideerd aan 393 experimenten en vergeleken met de aanbevelingen volgens ACI 318-08 (2008), BS 8110 (1997), Eurocode 2 (2005), Model Code 2010 (2013) Level II benadering, en verder de analytische modellen van Bažant & Yu (2004), Gastebled & May (2001) en Xu et al. (2012). Niettemin is de bepaling van het afschuifdraagvermogen onder langdurige belasting met het model met bilineair scheurpatroon relatief gecompliceerd, gezien de tijdsafhankelijke effecten. Daarom is een parameterstudie uitgevoerd om de invloed van de tijdsafhankelijke parameters vast te stellen.

Tenslotte is een serie experimenten op betonnen balken zonder schuifwapening uitgevoerd om het tijdsafhankelijke effect op het afschuifdraagvermogen vast te stellen. Het afschuifdraagvermogen onder kortdurende belasting is vergeleken met het afschuifdraagvermogen onder langdurige belasting bij belastingniveaus tussen 0.83 en 0.98 van de korteduur sterkte. De tijdsafhankelijke doorbuigingen, scheurpatronen, spanningsherverdeling, scheurlengte- en scheurwijdte-ontwikkeling zijn geobserveerd en geanalyseerd. Aan het einde van de periode van langeduurbelasting is voor de niet op afschuiving bezweken balken de belasting tot breuk opgevoerd, om hiermee de invloed van de langeduurbelasting op het uiteindelijke afschuifdraagvermogen te bepalen.

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Background and Significance ... 1

Objective ... 4

Outline of the Research ... 4

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1.1. Shear-failure modes in concrete beams without shear reinforcement ... 7

1.1.1. Shear-flexure failure ... 8

1.1.2. Shear-tension failure ... 9

1.1.3. Shear-compression failure ... 9

1.2. Contributions to the shear bearing resistance ... 9

1.2.1. Uncracked compression area ... 9

1.2.2. Aggregate interlock ... 10

1.2.4. Direct strut action for loads near to support (as/d<2.5) ... 14

1.3. Other effects to be considered when describing the shear resistance .. 14

1.3.1. Size effect ... 14

1.3.2. Bond of reinforcing steel ... 15

1.3.3. Effect of an axial force ... 16

1.4. Influencing parameters on shear resistance of reinforced concrete beams without shear reinforcement ... 17

1.4.1. Concrete strength ... 17

1.4.2. Longitudinal reinforcement ratio ... 18

1.4.3. Shear span to depth ratio ... 19

1.5. Empirical relations ... 19

1.5.1. Eurocode 2 and CEB-FIP Model Code 1990 ... 20

1.5.2. Fib Model Code 2010 ... 20

1.5.3. ACI 318-08 ... 21

1.5.4. Canadian Standard (CSA) ... 21

1.5.5. British Standard ... 22

1.5.6. Rafla (1971) ... 23

1.6. Behavioural models for a reinforced concrete beam ... 23

1.6.1. Kani (1964) ... 23

1.6.2. Pruijssers (1986) ... 24

1.6.3. Reineck (1991) ... 26

1.6.4. Gastebled & May (2001) ... 26

1.6.5. Bažant and Yu (2008) ... 27

1.6.6. Xu, Zhang and Reinhardt (2012) ... 28

1.7. Summary of Chapter 1 ... 29

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2.1. Properties of concrete in time ... 31

2.1.1. Compressive and tensile strength ... 31

2.1.2. Modulus of elasticity ... 33

2.1.3. Fracture energy ... 33

2.1.4. Strength under sustained loading ... 33

2.2. Time-dependent effects ... 34

2.2.1. Shrinkage ... 34

Table of Contents | IX

2.2.3. Relaxation ... 39

2.2.4. Bond creep ... 40

2.2.5. Crack opening and sliding ... 41

2.3. Summary of Chapter 2 ... 42

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3.2.2. Variables and derivatives ... 45

3.2.3. Experimental procedure ... 46

3.2.4. Curing condition ... 46

3.2.5. Compressive strength of the concrete ... 47

3.2.6. Measurement ... 47

3.2.7. Loading procedure ... 48

3.3. Experimental results ... 48

3.3.1. Test setup and equipment ... 48

3.3.2. Handling of experimental data ... 48

3.3.3. Measured displacement in short-term tests ... 49

3.3.4. Fracture energy ... 50

3.3.5. Measured displacements in long-term tests ... 52

3.4. Analytical model for crack growth ... 56

3.4.1. Elasto-visco-plastic model ... 56

3.4.2. Modelling of the present tests ... 58

3.5. Application of the EVP model for crack opening rate ... 63

3.5.1. Step-by-step method to employ the proposed model ... 63

3.5.2. Prediction of failure ... 64

3.5.3. Cracking strain limit ... 67

3.6. Comparison with other test results ... 69

3.7. Summary of Chapter 3 ... 70

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4.1.1. Elastic-ideal-plastic model of Dugdale ... 72

4.1.2. Fictitious crack model of Hillerborg et al. ... 73

4.1.3. Crack band model ... 75

4.1.4. Two parameter fracture model ... 75

4.1.5. Effective crack model ... 77

4.1.6. Size effect model of Bažant ... 77

4.1.7. Conclusions ... 77

4.2. Assumptions of fictitious crack model ... 77

4.3. Nonlinear analysis by means of linear elastic fracture model ... 78

4.3.1. Time dependency of the FPZ ... 79

4.3.2. Finite element modelling ... 79

4.3.3. 2D beam in plane-stress ... 82

4.4. Modelling of monotonic short-term 3-point bending beam with flexural notch ... 83

4.4.1. Load-deflection curve of finite element analysis ... 84

4.4.2. Comparison of FE results and experiments ... 85

4.4.3. Effect of tensile strength on FE results ... 86

4.4.4. Effect of modulus of elasticity ... 88

4.4.5. Effect of softening function ... 89

4.4.6. Verification of the FE model ... 91

4.5. Modelling of long-term sustained loading (3-point bending) of a beam with a flexural notch ... 94

4.5.1. Development of strain in front of the notch tip ... 96

4.5.2. Verification with the experiments ... 99

4.5.3. Comparison of the results with the EVP model ... 99

4.6. Summary of Chapter 4 ... 101

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5.1. Shear failure model with linear inclined shear crack ... 105

5.1.1. Depth of the compression zone under sustained loading ... 105

5.1.2. Shear resistance according to Gastebeld & May and Xu et al. 109 5.1.3. Shear crack under sustained loading ... 113

5.1.4. Effect of bond creep ... 114

5.1.5. Parametric study for a shear-critical beam under sustained loading ... 116

Table of Contents | XI

5.2. Shear failure model with bilinear inclined shear crack ... 118

5.2.1. Proposed model ... 118

5.2.2. Verification of the analytical results with experiments ... 123

5.2.3. Extension of the bilinear crack model to the case of sustained loading ... 127 5.2.4. Parametric Study ... 128 5.3. Summary of Chapter 5 ... 129

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6.1. Test arrangement and setup ... 132

6.1.1. Details of the reinforced concrete beams ... 132

6.1.2. Variables ... 133

6.1.3. Material properties ... 134

6.1.4. Standard compressive tests on concrete ... 135

6.1.5. Test setup ... 135

6.1.6. Deflection measurement ... 137

6.1.7. Crack width measurement ... 138

6.2. Test Programme ... 138

6.2.1. Short-term monotonic loading ... 141

6.2.2. Long-term sustained loading ... 141

6.3. Results of short-term monotonic loading ... 141

6.3.1. Type of failure ... 141

6.3.2. Shear resistance ... 145

6.3.3. Midspan deflection ... 146

6.3.4. Diagonal deformation ... 147

6.3.5. Crack pattern ... 150

6.3.6. Summary of short-term tests ... 150

6.4. Results of long-term sustained loading ... 156

6.4.1. Load intensity ... 157

6.4.2. Time-dependent deflections ... 158

6.4.3. Crack pattern ... 164

6.4.4. Crack length development in time ... 165

6.4.5. Crack width development in time ... 172

6.4.6. Stress redistribution in time ... 176

6.4.8. Summary of long-term sustained loading tests ... 188

6.5. Comparison of the experiments with the analytical model ... 189

6.6. Summary of Chapter 6 ... 194

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7.1. Summary ... 195

7.2. Conclusions ... 198

7.3. Recommendations for future research ... 200

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Latin upper case letters

A1/2/3 Areas under load-deflection curve for measurement of fracture energy

Ac Concrete cross-sectional area

As Reinforcement cross-sectional area

C0(t,t0) Compliance function for basic creep

CRd, c Design factor to the shear resistance according to EC2

Cd(t,t0,tC) Additional compliance function for drying creep

Cf Aggregate effectivity factor

CMOD Crack mouth opening displacement COD Crack opening displacement CTOD Crack tip opening displacement CTODc Critical crack tip opening displacement

DD Diagonal deformation

E0 Modulus of elasticity of elastic element in EVP model

Ec Tangent (asymptotic) modulus of elasticity of concrete at 28 days

Ec(t) Tangent modulus of elasticity of concrete at time t

Ecm Secant modulus of elasticity of concrete at 28 days

Ecm(t0) Secant modulus of elasticity of concrete at time t0 of loading

Es Modulus of elasticity of steel

Fa Horizontal component of aggregate interlocking force

FS Tensile force in steel

FC Tensile force in concrete due to softening

GIf , Gf Fracture energy of concrete (Mode I)

GIIf Pure Mode II fracture energy of concrete

Gf0 Energy required to propagate a tensile crack of unit area

Gs Shear modulus of steel

H(t) Spatial average of pore relative humidity

J(t, t0) The compliance function that represents the total stress-dependent

strain by unit stress

Jve Viscoelastic creep compliance function

Jvp Viscoplastic creep compliance function s

Ic

K Critical stress intensity factor

KIIc Mode II fracture toughness according to Reinhardt & Xu (1998)

L Span length

Lꞌ Beam length

M Bending moment

M0 Cross-sectional moment resistance

Mcr Critical bending moment

Mu Flexural moment at ultimate state

N Axial force

Psus Sustained load

Pu Peak point in load-deflection curve (Ultimate load)

Pu,mean Mean value of the ultimate load

Q(t,t0) Approximate binomial integral for the aging viscoelastic term of

creep

Qf(t0) Approximate binomial integral for the aging viscoelastic term of

creep

R Radius of aggregates idealized as spheres

RH Relative humidity of the ambient environment in % Ssc(t) Creep slip of reinforcing bar in concrete

Notation | XV

Ue Potential Elastic Energy

V Shear force

Va Shear resistance by aggregate interlocking (vertical component force)

Vcc Shear resistance of compression zone

Vd Contribution of the dowel action of the longitudinal reinforcement to

shear resistance

Vdu Ultimate shear force component carried by dowel action

Vsus Sustained shear load

Vu Shear resistance at ultimate state

Vu,calc Calculated shear resistance

Vu,calc 5% Lower confidence limit (5% fractile) or characteristic value of the

calculated shear resistance

Vu,calc 95% Upper confidence limit (95% fractile) of the calculated shear

resistance

Vu,exp Experimental shear resistance

Vu,mean Mean value of the experimental shear resistance of the beams in the

same series

W Water content in kg/m3 of concrete

W(i, j) Widening of the crack at node i when unity load is acting at node j W′(i) Widening of the crack at node i when applied load equals unity load Wext External work done

Latin lower case letters

ac Effective crack length

acc Length of the horizontal part of shear crack with a bilinear pattern

ag Aggregate content in kg/m

3

of concrete ai Distance between two node pairs

as Shear span

ax , ay Aggregate contact areas in x and y directions

b Beam width

bw Web width (effective beam width)

c Cement content in kg/m3 of concrete cf Critical effective crack extension

d Effective depth of beam

d0 Empirical parameter depending on the type of concrete according to

Bažant and Yu (2008) db Bar diameter

dg Largest nominal maximum aggregate size

ds Effective depth of concrete tooth

dv Effective shear depth, taken as the greater of 0.9d or 0.72h

fc Compressive strength of concrete

fck Characteristic strength of concrete at 28 days

fck, c Characteristic strength of confined concrete

fcm Mean value of compressive cylinder strength of concrete at 28 days

fcm, 52 days Mean value of compressive strength at 52 days

fcm(t) Mean concrete compressive strength at age of t days

fct Axial tensile strength of concrete

fctk Characteristic axial tensile strength of concrete

fctk, sus Characteristic tensile strength of concrete under sustained loading

fctm Mean value of axial tensile strength of concrete

fctm(t) Mean value of axial tensile strength of concrete at age of t days

fnet Net section flexural strength

fr Rupture modulus according to ACI 318

fyk Characteristic yield strength of reinforcement

f(w) Assumed unique softening stress-displacement function for infinitely slow loading

g Gravity acceleration h Total height of the beam

h1 Notch depth

Notation | XVII

hτ Effective shear depth

k Size factor in shear resistance according to EC2

kdg Crack spacing parameter that allows for influence of aggregate size

according to MC2010

ks Cross-section shape-correction factor

kv ,kv,I , kv,II Parameters accounting for shear resistance of cracked concrete

according to MC2010 and CSA-04 lch Characteristic length

lc,max Maximum crack length at failure

ls,unb unbonded length of reinforcement

m, n Empirical parameters, for normal concrete m=0.5 and n=0.1 mb Mass of the beam

s Coefficient that depends on the strength class of cement sc Maximum crack spacing

scr Crack spacing

sz Crack spacing parameter dependent on crack control characteristics of

longitudinal reinforcement according to CSA-04

sze Equivalent value of sz that allows for influence of aggregate size

t Time being considered tꞌ Dummy integration variable

t0 Age of concrete at the time of loading in days

tc Age of concrete when drying began or end of moist curing in days

tcr Duration of sustained loading until failure

u Perimeter of the member in contact with the atmosphere vRd, c Nominal design shear resistance

vcalc Calculated nominal shear resistance

vexp Experimental nominal shear resistance

vu Nominal shear resistance

vu,m Mean nominal shear resistance

w, wi Crack width

0

w& Constant value of loading rate

wc,R Critical crack width for friction capacity according to Reineck (1991)

equal to 0.9 mm

cc

w Dimension free critical crack width x Depth of compression zone

z Internal lever arm between the resulting compressive force FC in

concrete and tensile force FS in steel

zꞌ Distance from the resulting compressive force FC to the top of the

compression zone Greek upper case letters ∆ Midspan deflection

∆ua , ∆va Opening of crack at mid-height in x and y directions

∆us , ∆vs Opening of crack at the level of reinforcement in x and y directions

∆u Midspan deflection corresponding to peak load Pu

∆t Time increment

∆x Distance between concrete teeth

∆0 Midspan deformation when the force has fallen to zero

∆εel Incremental elastic strain

∆εve Incremental viscoelastic strain

∆εvp Incremental viscoplastic strain

∆σ Stress increment ∆w Widening of the crack

Σs Reduced cross section of reinforcement

Greek lower case letters

α Constant which depends on the type of cement and the type of curing

α0 Constant

Notation | XIX

αc Coefficients regarding the crack spacing

αsus Reduction factor for the tensile strength under sustained loading

αu Size effect factor according to Rafla

β Constant which depends on the type of cement and the type of curing β1/2 Constants related to the cement type and the curing conditions

βbc(t,t0) Coefficient to describe the development of basic creep with time

βcc(t) Time development function according to EC2 and MC2010

βdc(t,t0) Coefficient to describe the development of drying creep with time

βH Coefficient depending on the relative humidity and the notional size

βc,sus(t, t0) Reduction factor for compressive strength which depends on the time

under sustained loads

βH Coefficient depending on the relative humidity and the notional size

γ Effectiveness parameter

γc Safety factor, =1.5 for persistent and transient and =1.2 for accidental

loading situations. Recommended 1.5 for design of shear resistance δ Shear displacement or sliding

ε1/2/3/4/5 Empirical material constitutive parameters given by formulae based

on concrete strength and composition, according to B3 Model εc Strain of concrete at the top fibre

εc2 Compressive strain of concrete at reaching the maximum strength

εcr Creep strain

εct Tensile strain of concrete at stress equal to fctm

εct Ultimate tensile strain of concrete

εcu2 Ultimate compressive strain of concrete

εc,max (t) Critical tensile strain of concrete at time t

εc(y) Strain of concrete at a distance y from the top fibre of the section t

εel Elastic strain at time t

εel Elastic strain due to instantaneous loading

εs Strain in reinforcing steel

εsh Shrinkage strain

εvp Viscoplastic strain at time t

η, ξ Natural coordinates of a quadrilateral element η0 Viscosity of viscoplastic element

ηi Viscosity of the i

th

Voigt element

θ0 Angle of crack at peak load under short-term monotonic loading

θ, θc Crack opening angle or angle of rotation

θ0 Crack opening angle due to instantaneous loading

θcreep Crack opening angle due to creep effect

θc,max,long-term Crack opening angle limit under long-term loading

κ Constant value for calculation of the crack opening rate according to Bažant & Li (ranges from 0.01 to 0.05 for concrete)

κ0 Empirical parameter depending on the type of concrete

κ1 Positive constant value representative for the shape of softening curve

λ Load intensity factor (ratio of the sustained load to the ultimate load) λc Critical load intensity, where failure occurs beyond that

µ Mean value of the data set ν Poisson’s ratio

ρs Reinforcement ratio

ς Shear safety margin

σ0 Stress corresponding to wi in the static σ-w relation

σ2 (= σ3) Effective lateral compressive stress at the ULS due to confinement

σR Stress relaxation

σc(y) Stress in concrete at a distance y from the top fibre of the section

σprincipal,1 First principal stress

σsus Axial stress due to sustained loading

σts Mean stress in the concrete regarding tension stiffening

,1...4 xx

σr _{Element’s nodal stresses in x direction} ,1...4

yy

σr _{Element’s nodal stresses in y direction} ,1...4

xy τ

r

Notation | XXI

σ(t) Stress at time t

σ(w) Resistant stress along the crack face, depending on the crack width w τ, τi Retardation/relaxation time

τsh Shrinkage half-time in days

ϕ Angle of the inclined shear crack with respect to the horizontal axis ϕ1 Dimensionless component for calculation of tensile strength in time

ϕ2 Dimensionless component for calculation of E modulus in time

ϕc Safety factor for concrete strength according to CSA, equals to 0.65

φ(t, t0) Creep coefficient, defining creep between times t and t0, related to

elastic deformation at 28 days φ0, bc Basic creep coefficient

φ0, dc Drying creep coefficient

φb(t) Bond creep coefficient

φs Creep coefficient of the crack slide δ

φw Creep coefficient of the crack width w

ωi , χi Parameters depending on the relaxation time of the ith Voigt element

ψ1…6 Coefficients to determine the time-dependent crack displacements

Acronyms

ACI American Concrete Institute BS British Standard

CBM Crack band model

CSA Canadian Standards Association

EC Eurocode

ECM Effective crack model EVP Elasto-visco-plastic EIP Elastic-ideal-plastic FE Finite element

FCM Fictitious crack model FPZ Fracture process zone

MC10 fib Model Code 2010 MC90 CEB-FIP Model Code 1990 TPFM Two parameter fracture model

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Background and Significance

Numerous investigations on reinforced concrete structures have been conducted since a long time, leading to several new theories, which generally are in good agreement with test results. However, the behaviour of reinforced concrete members is complicated by several parameters such as the structure material properties and the way of loading. This is probably the major reason why the discussion regarding the bearing capacity of concrete structures continues in a so varied and widespread way. The importance of the discussions is even increased by the demand for the extension of the lifetime of the existing structures.

The majority of the existing reinforced concrete structures in the Netherlands road network were built before 1975. The shear and bending resistance of the structures such as bridges, viaducts and tunnels in the major highway network or major waterways network, as well as their remaining lifetime, have been recently assessed by the Dutch Ministry of Infrastructure and the Environment (Rijkswaterstaat). When these structures are assessed for shear, they are often found not to satisfy the criteria for structural safety for two reasons:

• The traffic loads and volumes have increased over the past decades, resulting in more severe load models prescribed by the recently implemented Eurocodes.

• The shear provisions in codes are basically meant to be applied for new structures. For the assessment of existing structures they are often conservative, because they ignore a part of the residual capacity (Walraven, 2010).