martin-effect of bond, aggregate interlock and dowel action

Engineering Structures 23 (2001) 214–227www.elsevier.com/locate/engstruct

Effect of bond, aggregate interlock and dowel action on the shearstrength degradation of reinforced concrete

B. Martın-Perez a,*, S.J. Pantazopouloub

a Institute for Research in Construction, National Research Council, Ottawa, Ontario, Canada K1A 0R6b Department of Civil Engineering, Demokritus University of Thrace, Xanthi 67100, Greece

Received 21 June 1999; received in revised form 2 December 1999; accepted 29 December 1999

Abstract

The macroscopic shear strength contribution of concrete in reinforced concrete members is supported by dowel action, aggregateinterlocking across tension-shear cracks and the tensile stress field that becomes mobilised in concrete through reinforcement con-crete bond. The mechanics of this relationship are explored in this paper for the benefit of improved understanding of the degradationof shear strength in reinforced concrete as a function of imposed deformation demand. The mathematical formulation uses the non-linear smeared crack/smeared reinforcement approach to consider plane stress states in reinforced concrete elements. The criticalmodelling assumption in assessing the concrete contribution to shear resistance was the representation of force transfer from barto concrete, which requires establishing equilibrium both at crack locations as well as in a global sense. The significance of thismodelling approach on the overall shear strength was evaluated by the comparison of computed results with those obtained fromthe conventional model, wherein concrete participation is lumped artificially under the so-called ‘tension-stiffening’ property. Fromthis comparison the parametric dependence of tension stiffening on bar diameter, crack spacing and bond properties is illustrated.Through the mathematical formulation it was possible to identify and highlight the effect that compression softening of the concretestruts has on the contribution of the web reinforcement to shear resistance, which represents yet another source of globally observedstrength degradation. The proposed model was verified by comparison with experimental results and was subsequently used forparametric investigation of the associated design problem. 2000 Elsevier Science Ltd. All rights reserved.

Keywords:Aggregate interlock; Bond; Concrete contribution; Dowel action; Shear design; Shear strength; Strength degradation

1. Introduction

Recent trends towards developing a complete frame-work for earthquake design of reinforced concrete (r.c.)members that is driven by assessingdisplacement/deformation demands (rather than forcedemands) have forced the need to describe nominalstrength and to detail requirements in terms of defor-mation. For most limit states of r.c., linking strength todeformation is not an easy or straightforward task,because it requires a valid physical model in order tostate in mathematical terms the relationship between theimportant design parameters. For the case of shearstrength sources in r.c., most issues relating to physical

* Corresponding author. Tel.:+1-613-993-3788; fax:+1-613-954-5984.

E-mail address:[email protected] (B. Martı´n-Perez).

0141-0296/01/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved.PII: S0141-0296 (00)00004-3

interpretation are still fraught with considerable debate.For example, consensus is lacking as to the physical sig-nificance of the concrete contribution term, and to themathematical description of tension-based sources ofshear-strength and their relationship to strain intensityand cyclic displacement history. In general, the availableexpressions for shear strength are independent of themagnitude of imposed deformation, leading to overlyconservative estimates of capacity for low levels of dis-placement ductility demand, and becoming increasinglyunconservative as the displacement ductility demandincreases.

Except for a recent amendment in the Japanese Code[1,2], the effect of deformation demand on the primaryand secondary sources of resistance, which are supportedby the compression struts of the 45°-truss idealisation,has not been addressed in design codes. Conceptually,the degradation of shear strengthn and of its individualcomponentsnc andns (concrete and steel contributions,

215B. Martın-Perez, S.J. Pantazopoulou / Engineering Structures 23 (2001) 214–227

respectively), with increasing deformation in r.c. mem-bers undergoing cyclic loading, follows the qualitativepattern shown in Fig. 1(a) (taken from [3]). This modelhas been verified by experiments and has been used byresearchers to define thenc term [4–8]. In most of theabove studies, thens term was taken to be constant whenreducing the experimental data (this value being associa-ted to yielding of the transverse reinforcement), therebyattributing tonc whatever strength reduction is observedin the overall strength as a result of cyclic loads. It is,however, debatable whether the strength of the 45°-truss(the ns term) is indeed insensitive to deformationdemand. Thus, resolving this point through modellingof the related mechanisms is a pre-requisite in properlyquantifyingnc as well.

Furthermore, the parametric dependence of shear

Fig. 1. (a) ATC model for shear strength degradation (ATC 6-2,1983). (b)Truss analogy for the shear. (c) Free body equilibrium high-lighting the role of stirrups in the truss analogy.

strength and the pattern of its degradation with increas-ing deformation could be more transparent when themechanical problem is studied through a consistentmathematical model. There is a vast amount of literatureand diversity on the subject of modelling plane stressstates in r.c.; however, the basic framework of smearedmodels is now generally accepted as a successful model-ling tool for concrete members where reinforcement hasbeen detailed so as to provide adequate crack control. Inthese models, stresses and strains are usually evaluatedin principal directions using equilibrium and kinematicconsiderations. Principal stresses are calculated fromprincipal strains based on pertinent material constitutiverelations that usually represent results from standard uni-axial compression/tension tests of concrete. For this rea-son, softening in the basic constitutive relations isdirectly reflected in the macroscopic scale in the calcu-lated relationship between stress and strain resultants.However, whereas in general terms all models will pre-dict softening of shear resistance with increasing distor-tion, the exact magnitude will vary between modelsdepending on the sensitivity of the constitutive relationsand the associated assumptions of the critical parametersof the problem. With respect to the modelling pro-cedures, controversy is focused on a number of issues,ranging from the true relationship between angles ofprincipal stress and principal strain (the simplest optionis to use coincident principal axes for clarity of the for-mulation, but other options include coincident directionsfor principal incremental stress and strain, or setting theprincipal strain axes in concrete coincident with the prin-cipal directions of the boundary stresses); the source andnumerical definition of the residual tensile strength ofconcrete (to avoid mesh dependency of the mathematicalproblem, but to also reflect the size dependency of thephysical problem); and, the interpretation of the localstress state at the crack.

In this paper non-linear analysis of r.c. membraneelements using the basic ‘smeared’ stress/strain frame-work was conducted to investigate how modelling thedifferent sources of shear strength might affect the com-puted relationship between strength and deformationdemand. The parametric dependence of shear strengthdegradation on several design variables was investigatedby the authors [9], where the residual sources of concretestrength in tension were represented collectively by theconventional ‘tension-stiffening’ model that lumps allthese mechanisms in an artificially large post-crackingstiffness of cracked concrete [10]. It is the objective ofthe present work to model explicitly the residual strengthmechanisms of concrete, namely mobilisation of tensionstress fields in concrete by bond-slip at the steel–con-crete interface, dowel action of primary and secondaryreinforcing bars crossing the crack planes, as well asaggregate interlock at crack faces, while taking intoaccount the correct magnitude of post-cracking resist-

216 B. Martın-Perez, S.J. Pantazopoulou / Engineering Structures 23 (2001) 214–227

ance of concrete in order to study the effect of thesephenomena on the global shear-strength/deformationrelation.

2. Shear design of prismatic members forearthquake resistance

The principle underlying the current design practiceis still motivated by the original truss analogy that wasadvanced over a century ago by Ritter and Mo¨rsch[11,12]. In the original truss, compression struts wereassumed inclined at 45°, whereas horizontal and verticalreinforcement acted as tension ties (Fig. 1(b)). From theequilibrium of the free body shown in Fig. 1(c), it fol-lows that the nominal shear resistance associated withyielding of the transverse reinforcement isVn=nAsfy,wheren is the number of stirrups/ties intersecting a 45°-crack (the crack is assumed to run parallel to the com-pression strut),As is the total cross-sectional area of thelegs of a single transverse reinforcement layer inter-secting the crack plane, andfy is the yield stress of thatreinforcement. (The nominal shear force is converted tostress by dividing by the effective web areabwd.) Con-sidering thatn=d/s, wheres is the spacing of transversereinforcement in the longitudinal direction of the mem-ber, it follows that the nominal resistance of the truss,nn, is nn=rnfy where rn is the ratio of vertical shearreinforcement.

Although understanding and modelling the mechanicsof shear in r.c. has advanced greatly from the time whenthe above result was first established, the various codedesign equations that have been adopted over the yearsmay be viewed as variations of the same basic themeexpressed by the Ritter–Mo¨rsch equation. Because theestimatenn=rnfy has been shown by experiments to betoo conservative,nn is enhanced in most design codestoday by an additional empirical termnc, so-called ‘con-crete-contribution’, that is meant to account collectivelyfor the participation, in the actual function of the truss,of several secondary sources of resistance in r.c., suchas aggregate interlock, dowel action of reinforcementcrossing the crack planes, restraint in the form of com-pressive axial load, transfer of stress from reinforcementto concrete through bond, and the residual tensilestrength of concrete. The empiricism necessarilyinvolved in expressing the concrete contributionnc inmathematical terms has led to different expressions fornc in different codes. For example, the ACI Code [13],the simplified method in the CSA Standard [14], andthe CEB/FIB Model Code [15], respectively, propose thefollowing basic equations for the design shear strengthn:

ACI [13]:

n5f(ns1nc)5frnfy1f0.17Îf9cS11P

14AgD; f (1a)

50.85

CSA [14]:

n5fsns1fcnc5fsrnfy1fc0.20Îf9c; fs50.85,fc (1b)

50.6

CEB/FIB [15]:

nRd35fnw1nc50.9rnfy1l1nRd1; nRd15FtRdK(1.2 (1c)

140rl)10.15PAgG

Stress units are MPa. In Eqs. (1a) and (1b) the contri-bution of the axial loadP in the nc term is only con-sidered when compressive. The termtRd in Eq. (1c) isthe nominal shear resistance of the cross section at diag-onal tension cracking, taken as one-quarter of the designtensile strength of concrete, which is in turn related to(f9c)2/3 (whereas the nominal shear strength at crackingin Eqs. (1a) and (1b) is related to√f9c.) The other vari-ables in Eq. (1c) are as follows:K accounts for the effectof size of the cross section (K=1.62d$1 in m) andrl,0.02 is the percentage of adequately anchored longi-tudinal tension reinforcement in the cross section ofinterest. Parameterl1 regulates the magnitude of con-crete contribution depending on the proximity of thecross section to a plastic hinge zone and the magnitudeof the axial compression acting on it, i.e.,l1=1 every-where outside the critical regions of beams and columns,whereas within the critical regionsl1=0.9 for P$0.1Agf9c/gc (with gc=1.5) and l1=0.3 for smaller axialloads (these values apply to earthquake design only,otherwisel1 is always 1). In comparison, in earthquakedesign the ACI and CSA Codes neglect the concretecontribution entirely forP,0.05Agf9c and P,0.1Agf9c,respectively, but account for the value of Eq. (1a) andhalf the value of Eq. (1b), respectively, for higher valuesof axial load.

The truss analogy is valid provided the diagonal strutshave adequate compressive strength to support the forcesthat develop in the longitudinal and transverse reinforce-ment. For this reason, all codes limit implicitly orexplicitly the design shearn by a lower bound estimateof the force required to crush the diagonal struts, i.e.,

ACI [13]:

ns#0.66Îf9c (2a)


CSA [14]:

fsns#0.8fcÎf9c; fs50.85,fc50.6 (2b)

CEB/FIB [15]:

n#nRd250.9S12l2f9c/gcD; gc51.5,l250.72

f9c

200(2c)

$0.5

In Eq. (2c) the resistancenRd2 is reduced if there is axialcompression on the cross section, in order to account forthe weakening influence it has on the residual compress-ive strength of the web. Coefficientl2 in Eq. (2c) reflectsthe reduction in concrete compressive strength due tothe weakening influence of cracks due to tension in theorthogonal principal direction. According to the AIJCode [1], further reduction from this reference value ofl2 is necessary to capture the strength loss in plastichinge regions of earthquake resistant members (i.e.,l29=t2l2, where the reduction factorr2 is a function ofthe lateral driftd in radians, i.e.,r2=1215d for d,5%,and r2=0.25 for d.5%).

3. Effect of damage on shear strength

All of the preceding definitions of shear strength (Eqs.(1a, 1b, 1c, 2a, 2b and 2c)), and in particular, the con-crete contribution componentnc, explicitly rely on mech-anisms of tensile resistance (marked by the root powersof f9c) that degrade and eventually break down withincreasing crack width. Yet, these expressions areentirely insensitive to parameters that would characterisethe width of cracks and the level of damage in the mem-bers, such as for example the intensity of deformationdemand and the number of imposed cycles in earthquakeloading. Similarly, the compressive strength of the diag-onal struts (appearing either implicitly or explicitly inEqs. (2a, 2b and 2c)) is affected by the presence of paral-lel cracks (i.e., softening due to orthogonal tensile strain[10]). Through nodal equilibrium, the force that couldbe developed in the longitudinal and transverse ties ofthe idealised truss is directly related to the force magni-tude of the diagonal compression strut. Therefore, propa-gation and widening of cracks not only affects thenc

term, but also the integrity and load carrying capacity ofthe overall truss mechanism as that is expressed throughthe ns term in Eqs. (2a) and (2b) and through thenRd2

term in Eq. (2c).In seismic design, where the aim is to control the

location and type of damage by controlling the relativemagnitudes of flexural and shear strengths in the critical

regions of frame and shear wall elements, this sensitivityof shear resistance to deformation may dramaticallyaffect the realisation of the design objectives. Note thatin adequately detailed, under-reinforced frame members,flexural response is very ductile, i.e., flexural resistancemay be sustained with minimal or no degradation up toexcessively large curvatures. In such a case the designshearVf=Mf/L, whereL is the length of the shear span,is insensitive to displacement demand, particularly if theshear span is slender (.3d, whered is the static depthof the beam’s cross section). In contrast, as the imposeddeformation increases, the shear strengthVn decreasesrapidly due to diagonal cracks extending normal to theprincipal tensile stress field and due to the subsequentwidening of these cracks, a circumstance that eventuallyleads to reversal of the inequalityVn.Vf, despite theinitial intentions of the design (Fig. 2). In a recent studyit has been suggested that shear failure is likely in shearwalls where the shear force required to cause flexuralhinging is greater than 60% of the nominal shear resist-ance [16].

4. Sources of shear strength

It was mentioned in the preceding section that shearresistance in r.c. comprises a primary contribution rep-resented by the truss analogy, and a secondary compo-

Fig. 2. Relationship between nominal shear capacity,Vn and sheardemand,Vf: (a) Calculation ofVf; (b) variation ofVn andVf with defor-mation demand.


nent that collectively represents all other contributingmechanisms, namely:

(a) Bond of reinforcement to concrete and the ten-sile stress field mobilised in the concrete masssurrounding the reinforcement through thisinteraction.

(b) Residual diagonal tensile strength of crackedconcrete.

(c) Dowel action of reinforcement intersecting theinclined cracks.

(d) Friction between crack faces and aggregateinterlock.

Despite the obvious significance bond has in determiningthe response of r.c. to shear stress, in the vast majorityof specimens used for the systematic study of shear inr.c. these have been tested with reinforcement and con-crete clamped at the ends, thereby ensuring compatibilityof deformations in a smeared sense [17,18]. This is animportant point of difference between experiments andthe actual circumstances that occur in plastic hinges ofstructural members, wherein global bond-slip may domi-nate the load–deformation response, the redistribution ofstress, and the rate of strength degradation.

For a uniform stress field (as in most experiments onr.c. panels, [17,18]), reinforcement stresses at all parallelcrack faces are constant throughout the specimen; thesereduce to an intermediate value between cracks due tobond transfer (Fig. 3). The ensuing difference between

Fig. 3. Variation of reinforcement stress and bond stress betweensuccessive cracks.

steel stresses at a crack from their smeared averagevalues is usually attributed to the so-called ‘tension-stiff-ening’ mechanism, which stands for the increase in stiff-ness of the tensile reinforcement effected by the contri-bution of concrete. Tension stiffening is often describedby the post-peak branch of the uniaxial tension stress–strain curve that models the behaviour in the principaltensile direction, i.e., it is usually considered as theresidual tensile strength of cracked concrete [10]. Nat-urally, the actual residual tensile strength in concrete isorders of magnitude smaller than what the tension-stiff-ening models assume, but what causes such a relativelyslow rate of decay in tests of r.c. specimens is the bondof reinforcement to concrete. Recently published ‘ten-sion-stiffening’ relationships have the form [10,19]:

fc15f9t

1+ÎBec1

(3)

where fc1 is the post-cracking principal tensile stress inconcrete,ec1 is the co-axial tensile strain, andB is anempirical constant (B ranges between 200 and 500[10,19], as experimental values differ from one test ser-ies to another). Thus, the commonly accepted models fortension stiffening are unnaturally insensitive to designparameters known to affect the concrete-reinforcing barinteraction (e.g., bar size, length of bar over which trans-fer takes place, degree of transverse confinement, bondstrength). Therefore, though tension stiffening is anindirect way of modelling the bond, it cannot realisti-cally account for the bond-slip action that occurs alongthe anchored reinforcement. (This is the most likely rea-son why tests from different investigators point to suchvastly different values for the post-cracking stiffness asexpressed through Eq. (3).) Furthermore, the post-crack-ing stiffness required to match experimental results usingthe tension-stiffening modelling option is unnaturallyhigh, the residual tensile strength also being attributedimplicitly to all the mechanisms, (a)–(d), mentioned pre-viously. In the following sections these mechanisms aremodelled explicitly in order to provide a vehicle for thestudy of their influence on the computed shear strength.

5. Analytical modelling of stress-states involvingshear

The mechanics of r.c. shear behaviour were estab-lished here from the equilibrium and compatibility con-siderations of a simple r.c. element, whose deformationis primarily characterised by in-plane shear distortiongxy

and where material stresses are approximated by a stateof plane stress (Fig. 4). With no loss of generality, itwas assumed that the element considered has an orthog-onal grid of reinforcement in directionsx andy, parallelto the element boundaries and uniformly distributed in


Fig. 4. Membrane element: (a) externally applied stresses; (b) average strains; (c) definition of termscqx, cqy, andd; (d) average stresses; (e) localstresses at a crack.

each principal reinforcing direction (under this conditionit is possible to assume that cracks are smeared over theelement). It was further assumed that the crack planescoincide with planes of maximum normal tensile strain,and that cracked concrete behaves as an orthotropicmaterial with the material axes being oriented along thedirection of principal stress. The direction of principalstrain was assumed to coincide with the direction ofprincipal stress. For specified normal boundary stressesnx, andny, the progressive evolution of the shear stressnxy that develops in the element in response to animposed history of shear distortiongxy was determined.

The states of average stress and average strainin con-

creteat every point over the element were described bythe second-order tensors:

e5Sex 0.5gxy

0.5gxy eyD s5Sfcx nxy

nxy fcyD (4)

In Eq. (4), fcx and fcy are the average concrete stressesin the x andy directions, respectively, andnxy the shearstress. The above obey all the characteristic propertiesof a second-order tensor (e.g., they possess invariants,eigenvalues (principal values) and eigenvectors(principal directions), and obey coordinatetransformations). The convention used here is tension


positive. With fsx and fsy representing the average axialstresses in the reinforcement, consideration of globalequilibrium with the normal boundary stresses yields(Fig. 4):

nx5fcx1rxfsx; ny5fcy1ryfsy (5)

where rx and ry are the reinforcement ratios in thexand y directions, respectively (neglecting the reductionin concrete area of concrete due to the presence of thereinforcing bars).

The kinematic relationship between deformations ofconcrete and steel (a compatibility requirement) is usu-ally defined by means of an interaction model for thetwo materials, where bond-slip is modelled explicitly. Inthis case, kinematics require compatibility betweenstrains in the reinforcement, slip, and average strains inconcrete. An additional equation is needed to state equi-librium between average bond stress and average andlocal stresses in the reinforcement. Tension stiffening isinconsistent with this modelling approach, and hence,the descending branch of the tensile stress–strainrelationship of concrete is taken to be several orders ofmagnitude steeper than what is used in conventionalmodels [10,19], because now this property only rep-resents progressive microcracking.

6. Local equilibrium at crack locations

At crack locations, tensile stresses in concrete arezero, and all the tension must be carried by the reinforc-ing bars. While tensile stresses in the reinforcementcrossing the cracks are higher than the average valuesobtained from the constitutive relations, the stresses inthe reinforcement between cracks are lower than theaverage values (Fig. 3). This non-uniform distribution ofstresses along the reinforcing bars is important becausethe load-carrying capacity of the cracked structuralmember may be governed by the reinforcement’s abilityto transmit stresses along the anchorage length, as wellas across the cracks. It was assumed here that bondstresses are distributed uniformly between cracks,whereas the average reinforcement stress value matchesthe actual stress in the bar at the quarter points withinthe crack spacing (Fig. 3). By establishing equilibriumalong the bar segment bound by two consecutive cracks,the stress in the reinforcement,fsxcr, at crack locations isobtained as a function of the average reinforcementstress value,fsx, and the intensity of bond stress,fbx,(similarly for the y direction):

f crsx5fsx1

cqxDbx

fbx; f crsy5fsy1

cqyDby

fby (6)

where cqx and cqy represent thex and y projections ofthe crack spacing (Fig. 4(c)), andDbx andDby the corre-

sponding bar diameters in the two principal reinforcingdirections, respectively. Average diagonal crack spacingalong the principal tensile direction is given by [19]:

cq51

sinqsx

+cosq

sy

; sx52Sc1l

10D10.25k1

Dbx

rx

(7)

whereq is the direction of average principal compressivestress measured from thex-axis,sx andsy are the averagecrack spacings that would result if the member was sub-jected to mere tension along thex and y axes, respect-ively, k1 is 0.4 for deformed bars and 0.8 for plain bars,c is the maximum distance from the reinforcement cross-ing the crack plane, andl is the maximum spacingbetween the reinforcing bars (,15Dbx) [15].

The free body diagrams illustrated in Fig. 4(d, e) showthe average stresses of the element and the local stressesthat occur at a crack, respectively. The principal planewas defined here from average stresses, i.e., the averageshear stress component vanishes. However, at the crackplane, shear is transferred by means of aggregate inter-lock, nagg, and by dowel action of the reinforcing bars,fdx andfdy (fdx is the dowel shear stress along thex-axis).The two stress states illustrated in Fig. 4(d, e) are stati-cally equivalent. By summing the components of thestresses normal to the crack direction it follows that:

fc15sin2 q[rx(f crsx2fsx)1ryfdy]1cos2 q[ry(f cr

sy2fsy) (8a)

1rxfdx]

By substitution of Eq. (6) into Eq. (8a), the principaltensile stress takes the form given by Eq. (8b). Thus, inorder to ensure local equilibrium at the cracks, the prin-cipal average tensile stress in concrete given by Eq. (8b),

fc15sin2 q[rx

cqxDbx

fbx1ryfdy]1cos2 qFry

cqyDby

fby (8b)

1rxfdxGmust be less or equal at all times than the correspondingvalue that is specified through the constitutive relation-ship. This implies that the envelope of the constitutiverelationship for post-cracking tension of r.c. is controlledby the characteristics of the bond and dowel action. Asan extension to the above, note that for tension-stiffeningmodelling purposes, the residual tensile stress in anyprincipal direction may be calculated from Eq. (8b).Hence, any decay with an increasing value of strain mustbe linked to the associated degradation of the bond anddowel action.

7. Constitutive relations

The softening behaviour of concrete under cycliccompression, as well as the tensile/compressive behav-


iour of steel reinforcement under reversed cyclic loading,were modelled following the same procedure as thatused earlier by the authors [9], which relied on a ‘ten-sion-stiffening’ model (also see Fig. 5). Components ofthe constitutive response used in the present work, whichdepart radically from that classical ‘smeared model’, aredescribed below.

7.1. Constitutive model for bond

A simple average bond stress/strain relationship wasadopted to simulate the stress attenuation in steel withincreasing distance from a crack (Fig. 6). From the equi-librium of an infinitesimal segment of an embedded barof length dl, the bond stress was evaluated asfb=0.25Db(dfs/des)(des/dl) (Fig. 3), i.e., the bond demandon concrete is a function of the average stress and strain,fs, es, in the reinforcement. For the sake of simplicity,the rate of changein strain along the length of the bar,des/dl, was approximated here by the average strain inthe reinforcement normalised by the crack spacing in thedirection of the reinforcing bar (cqx or cqy respectively).

Fig. 5. Material models under cyclic load reversals: (a) concrete incompression and tension; (b) steel.

Hence, the resulting average bond stresses along thexand y directions are estimated from:

fbx5Dbs

4cqxEt

sesx; fby5Dby

4cqyEt

sesy (9)

whereEts is the tangent modulus of the average stress–

strain curve of the reinforcing steel (dfs/des), which wasobtained by differentiation of the associated mathemat-ical form [20]. The envelope of the bond-slip responsewas defined by the monotonic response curve shown inFig. 6(a), which was obtained according to Eq. (9). Theunloading path followed two linear segments: the firstwith a slope equal toEs, the elastic modulus of steel,i.e., fb=Es(es2xo) for uesu.uxou; the second segment withzero slope (fb=0 for uesu,uxou). Upon reloading, the pathfollowed was the same as under unloading conditionsuntil the envelope was reached again, i.e., there was noadditional energy dissipation in this phase of theresponse.

7.2. Concrete in tension

The relationship between average principal tensilestressfc1 and average principal tensile strainec1 in con-crete was assumed to be linear-elastic prior to cracking,with a slope equal toEc. After cracking (i.e.,ec1.f9t/Ec, f9t being the concrete tensile strength, taken hereas 0.33√f9c) the response was assumed to follow a linearstrain-softening branch (Fig. 6(b)). The softening modu-lus Et depends on the fracture energyGf, which isdefined as the energy dissipation due to localised crack-ing per unit length of fracture and corresponds to the areaunder the stress–strain curve [21]. The fracture energyis considered a material property of concrete and wasdetermined here by means of an empirical relationshipproposed by Bazant and Oh [22]:

Gf5(2.7213.10ft9)ft92da

EcN/mm (10)

whereda is the maximum aggregate size in the mix. Notethat in using the fracture energy as a cracking criterion,it is necessary to adjust the average stress–strain relationof concrete in tension so as to ensure the same energydissipationGf for any crack spacingcq (to avoid theproblem of dependency of the analysis results on thechoice of crack spacing). By equating the product of thearea under the uniaxial tensile stress–strain curve and thecrack spacingcq to the fracture energyGf [21], the ulti-mate straineu is calculated as (Fig. 6(b)):

eu52Gf

ft9cq(11)

Under cyclic loading, an open crack was assumed tohave closed when the principal strain across it becamecompressive, and, consequently, the concrete was able


Fig. 6. (a) Averaged bond-stress/axial strain relationship; (b) tension stress–strain curve in plain concrete; (c) in-plane shear stress due to aggregateinterlock; (d) unloading–reloading path; (e) dowel force versus transverse displacement relationship; (f) definition of dowel displacement.

to transmit compression loads in that direction again.The crack was assumed to reopen again when the princi-pal strain normal to it became tensile.

7.3. Shear interface

At crack locations, in-plane shear transfer wasaccomplished through aggregate interlock of the rough

crack surfaces and dowel action of the reinforcing barscrossing the crack planes. Based on experimental data itwas established that the shear stiffness of these mech-anisms decays rapidly after cracking. In this work, shearstress due to aggregate interlockingnagg was formulatedexplicitly as an inverse function of the principal tensilestrainec1, which is an indicator of the actual crack width.The proposed expression Eq. (12) was calibrated with


shear panel tests [17] to obtain the values given in theright-hand term (F represents the shear strength at crack-ing anda is a normalising constant, Fig. 6(c)):

nagg5F

1+ec1

a

50.1342×10−3Ec

1+ec1

0.0018

(12)

Eq. (12) defines the envelope of the aggregate interlockcyclic response adopted here. Unloading from the envel-ope was defined by two linear segments with slopesEn1andEn2 (Fig. 6(d)), whereEn1 is the initial tangent modu-lus of the proposed shear stress/principal tensile strainrelationship (En1=0.1342×1023 Ec), andEn2 is given as:

En250.1342×10−3Ec

En1emaxc1 −nmax

agg·En1 (13)

with emax1 the maximum principal tensile strain that con-

crete has undergone in a given direction andnmaxagg the

maximum in-plane shear stress developed in that direc-tion. Reloading towards the envelope followed the slopeof the straight line that passes through the points in thestress–strain curve corresponding to the maximum(emax

c1 ) and minimum (eminc1 ) principal tensile strains that

the r.c. element has undergone in the previous cycle, i.e.,

nagg5vminagg1Er(ec12emin

c1 ); Er5nmax

agg−vminagg

emaxc1 −emin

c1(14)

where nminagg is the minimum in-plane shear already

developed in a given direction.

7.4. Dowel action of reinforcement

The primary mechanism of resistance against slidingshear failure after the breakdown of aggregate interlockis the dowel action of the reinforcing bars. Although ithas been suggested that dowel strength across a shearplane is owing to a combination of direct shear, kinkingand flexure of the reinforcing bars, Millard and Johnson[23] have illustrated that flexure of the bars predomi-nates, since there is a significant amount of deformationin the underlying concrete cover. A physical model fordowel bars embedded in concrete after cracking is thatof a beam resting on an elastic foundation (the concreterepresenting the flexible foundation). The followingload–deformation response for dowel bars was adoptedin this study [23]:

Fd5FduF12expS−KiD

FduDG (15)

where Fd is the dowel force at a shear displacementacross the crack equal toD (Fig. 6(e)),Fdu is the ultimatedowel force, andKi is the initial dowel stiffness. ToevaluateFdu, a simple limit analysis model was usedassuming simultaneous formation of a plastic hinge in

the bar and a crushing zone in the concrete under thedowel:

Fdu51.30D2bÎf9cfy(1−A2) (N) (16)

with A the ratio of applied axial force to the yield axialforce of the bar with diameterDb. From experimentalevidence it is known that axial tension in the bars closeto yielding adversely affects the dowel resistance(causing a reduction in dowel stiffness and bendingcapacity [23]). This is why Eq. (16) results in zero dowelcapacity for rebar forces equal or exceeding yield (A.1).With regards to the initial stiffness, the followingexpressions were adopted from the literature [24,25]:

Ki50.166K0.75f D1.75

b E0.25s (N/mm); (17)

Kf5127bÎf9cS 1DbD2/3

(N/mm3); Es5200 GPa

whereKf is the foundation stiffness of concrete andb isa coefficient ranging from 0.6 for a clear bar spacingof 25.4 mm to 1.0 for larger bar spacings. The doweldisplacementD that is used in Eq. (15) is the componentof crack widthw in the direction normal to the reinforc-ing bar (w=cqec1, with the crack spacingcq as calculatedfrom Eq. (7)). Thus (Fig. 6(f)),

Dx5w sinq;Dy5w cosq (18)

Eq. (15) was used as the envelope of the dowel responsecurve. Unloading from the envelope followed a straightline with stiffnessKi (the initial stiffness) up to zeroingof the dowel force; at that point bars were assumed toact as dowels in the opposite direction following theenvelope (Eq. (15)) in that direction. Reloading followeda straight line with slope depending on the maximumand minimum dowel shear displacements attained in theprevious cycle (Fig. 6(e)), i.e.,

Fd5Fmind 5Er(D2Dmin); Er5

Fmaxd −Fmin

d

Dmax−Dmin (19)

with Dmin and Dmax the minimum and maximum trans-verse displacements that the dowel bar has undergone,and Fmin

d and Fmaxd the minimum and maximum dowel

shear forces developed previously in the bar.

8. Establishing the shear resistance of crackedconcrete

8.1. According to the proposed model

By establishing global equilibrium in terms of localstresses developed along the crack interface, bond,aggregate interlock and dowel action are introducedexplicitly (see free-body diagrams in Fig. 4(d, e)). Thus,the shear stressnxy of the r.c. element is evaluated from:


nxy5nagg1rxfdx1cotqryf crsy2cotqny (20)

Although angleq defines the principal plane in termsof average stresses (an assumption of the model), theintroduction of shear stresses at the crack implies thatthis direction changes when local conditions are con-sidered. By substituting Eq. (6) into Eq. (20), the shearresistancenxy becomes:

nxy5Fnagg1rxfdx1cotqSry

cqyDby

fby2nyDG1[cot qryfsy]5nc1np1ns; nc5nagg1rxfdx (21)

1cotqry

cqyDdy

fby

The truss mechanism contribution to shear resistance,ns=cotqryfsy, maintains the same form as given by thestandards, although expressed in terms of averagestresses. However, thenc term is now explicitly relatedto aggregate interlock, dowel action, and the bond mech-anism. Note the increase in thenxy term when compress-ive normal boundary stressesny (axial loads) are applied,marked by thenp component in Eq. (21) (np=2cotqny, ny,0 for compression); it is recalled that thisterm has been acknowledged explicitly by both the ACI[13] and CEB/FIP [15] Codes in Eqs. (1a) and (1c),through the terms 0.17√f9c(P/14Ag) and 0.15l1(P/Ag),respectively.

8.2. The significance of alternate assumptions forq

It was assumed here that the direction of principalaverage stresses and principal average strains coincide,allowing angle q to rotate. Alternative formulationsassume that the principal strain axes in concrete coincidewith the principal directions of the applied boundarystresses [26–28]. If this assumption is considered, themathematical form of Eq. (21) would remain the same,except nowq would be given by the initial crack patternresulting from the boundary stresses.

8.3. Shear resistance according to the tension-stiffening model

Alternatively, by representing the mechanism of shearresistance at the crack interface through the tension-stiff-ening model [9,10] and neglecting local conditions atcrack locations, the shear stressnxy is calculated in termsof average stresses from the equilibrium of the free bodyshown in Fig. 4(c):

nxy5cotq(fc12ny)1cotqryfsy5nc1np1ns; (22)

nc5cotqfc1

The shear contribution of the truss component,ns, aswell as thenp term, remain the same as in Eq. (21);

however, here the concrete shear contributionnc isdefined as the shear resisted by the tensile stresses car-ried by cracked concrete as given by the tension-stiffen-ing model.

9. Discussion of the various definitions fornc

According to Eqs. (21) and (22), the explicit math-ematical description of the concrete contribution term isdependent on the underlying assumptions of the formu-lation adopted and, thus, it is biased to reflect the pos-ition of the individual researcher in the ongoing debateabout the issue [26–28]. Still, the universally acknowl-edged role for thenc term is that it collectively representsall unaccounted for sources of residual shear resistancesupplied by the concrete in the form of bond (or tensionstiffening), aggregate interlock and dowel action. Notethat the truss componentns, in Eqs. (21) and (22) mustbe equilibrated by the stresses developed in the concretestruts and can therefore be expressed as:

ryfsy5(sin2 q21)fc12sin2 qfc21ny (23)

where fc1 (Eqs. (8a) and (8b)) andfc2 are the averageprincipal tensile and compressive stresses in concrete,respectively. According to Eq. (23), the truss contri-bution to shear resistance depends on the principal com-pressive stress,fc2, which, however, is limited by soften-ing due to parallel cracking [10,19] (fc2 max=f9c/(0.820.34ec1/eo),f9c).

10. Analytical results

The shear stressnxy required to resist a given angleof shear distortiongxy was calculated from Eqs. (4, 5,8a, 8b) and (21) using an incremental iterative algorithm[29]. At the first step, starting values for the algorithmwere obtained by solving the associated linear-elasticproblem (assuming that in the beginning concrete is iso-tropic and untracked):

HexeyJ5

1−n2c

EcH1+rxEs nc

nc 1+ryEsJ−1Hnx

nyJ (24)

For all subsequent steps, starting values forex and eywere the last converged values from the previous cycle.Specifically for the case of pure shear (nx=ny=0), theelastic problem becomes homogeneous and no uniquesolution can be obtained from Eq. (24). In this specialcase, starting values forex and ey were obtainedassuming cracked concrete (fc1=0), in which case theangle of inclination of principal tensile strain, along withthe needed values forex and ey, are obtained from thetensorial properties of the stress and strain tensors of r.c.:


tan4q51+

1nry

1+1

nrx

; n5Es

Ec

; ex5gxy

2(cotq2tanq)1ey; (25)

ey5gxy

2cotq−tanq

tan2 qry

rx

−1

(for rx=ry, ey=2gxy/2, based on L’Hospital’s rule).To illustrate the effect of the two alternative modelling

approaches, the monotonic envelope results obtainedfrom Eqs. (21) and (22) were compared for a r.c. panelwith f9c=45 MPa, eo=0.002, fyx=fyy=500 MPa,rx=ry=1.0%, and subjected to pure shear conditions (i.e.,nx=ny=0). Fig. 7 illustrates the calculated shear stressnxy

versus shear straingxy for both cases. Although theresponse given by Eq. (21) is much higher before crack-ing occurs (note that stresses here are given in terms oflocal values instead of average ones), the total responsesin both cases follow the same trend (cracked and failedat the same values ofgxy). However, when the differentcontributions to shear resistance are separated, the calcu-lated responses fornc are quite different in both cases.Consideration of local conditions at crack locations pro-vides a more realistic interpretation of the physicalbehaviour of r.c. Note that the contribution of aggregateinterlock to nc is definitely larger at low values ofgxy,whereas dowel action becomes significant after crackingtowards the latter stages of loading, both results beingentirely consistent with familiar experimental obser-vation. It is also apparent from Fig. 7 that the concretecontribution to shear strength decreases with increasingdeformation values. The difference between the steelcomponents in both cases is due to the contribution ofbond stresses in the first approach when calculating localstresses in the reinforcement crossing the crack.

It is worth noting that the ultimate shear strength ofthe panel represented in Fig. 7 is limited under both

Fig. 7. Calculated shear stressnxy versus shear straingxy accordingto Eq. (21) (bond model) and Eq. (22) (tension-stiffening model).

modelling assumptions by crushing of the concrete strutsand hence by the load carrying capacity of the overalltruss mechanism (Eq. (23)). This was further exploredby modelling the cyclic shear problem on the basis ofEq. (23) as illustrated in Fig. 8, in which the change inthe angle of the compressive strutsq is plotted againstductility mg for various reinforcement ratiosry (rx waskept constant at 1.00%). The ductility indexmg wasdefined here as the maximum shear strain attained up tothe point considered in the displacement history normal-ised by the shear strain corresponding to the first yieldingof the reinforcement in either direction. It is observedfrom the graph that the angleq decreases with increasingmg when the r.c. element is heavily reinforced in onedirection compared to the other. As the crack planesbecome flatter with increasing ductility demand, the trusscontribution to shear resistance, as given byns, will belimited by softening of the diagonal compression struts.

11. Sensitivity analysis

To investigate the response of the model to differentinput parameters, the model was used to correlate experi-mental results obtained from a series of r.c. panel tests[17]. All specimens had standardised dimensions 890mm square by 70 mm thick. Experimental variableswere: the uniaxial compressive strengthf9c, the straineoat peak uniaxial stress, the amounts and yield stresses ofthe longitudinal and transverse reinforcementrx, ry, fyx,fyy, and the type of load condition examined in the tests,namely, pure shear (PS), combined shear and biaxialcompression (SBC) and combined shear and biaxial ten-sion (SBT). From this series of tests (30 in total), panelPV22, tested in pure shear, was chosen as a referencecase to study the sensitivity of the model. This panel hadproperties: f9c=19.5 MPa, eo=0.002, fyx=458 MPa,fyy=420 MPa,rx=1.785%,ry=1.524%,da=10 mm, andhad failed by compression crushing of concrete. Corre-

Fig. 8. Influence ofry on q with increasing ductilitymg (rx was keptconstant at 1%).


Fig. 9. Correlation of experimental and analytical values for PV22.

lation of measured response with the analytical modelis illustrated in Fig. 9, along with the sensitivity of thecalculated response to alternative values off9c. Themodel also appeared to correlate well with all panels ofthe experimental study, regardless of the failure mech-anism involved (yield of reinforcement or crushing ofconcrete). Other design variables considered in the para-metric sensitivity investigation were:eo which rangedfrom 0.0018 to 0.003;fyy which ranged from 250 to 450MPa; ry which was varied from 0.25 to 2%; the aggre-gate interlock variablea, which was varied from 0.0005to 0.005; and, the dowel action parameter that character-ises the stiffness of the concrete cover,Kf, which wasvaried from 100 to 500 N/mm3. The effect that thesevariables have on the shear strength and deformationcapacity of the r.c. element are summarised in Fig. 10.The reference case, panel PV22, witha=0.0018 andKf=170 N/mm3 is represented by the common point ofthe intersecting lines. Its peak strength was estimated asnmax

xy =1.54√f9c and the corresponding shear distortion was7.3×1023, whereas shear distortiong0..5, correspondingto a 50% loss in shear strength, was 8.1×1023. (In Fig.

Fig. 10. Parametric sensitivity of the model.

10 they-axis represents the estimated shear strength nor-malised with respect to√f9c, whereas thex-axis is theshear distortiong0..5 at 50% strength degradation.) Thearrows indicate the directions of change innmax

xy andg0..5

as a particular design variable is increased. With theexception off9c, an increase in any of the variables stud-ied causes an increase in the shear capacity of the r.c.element. The shear straing0..5 increases with an increasein either f9c or eo. The reverse effect is seen with anincrease in the amount of reinforcement (addition ofreinforcement renders failure by crushing of the concretestruts, the more likely mode of failure). The variablesaffecting the shear transfer at crack locations (a andKf)do not have a direct influence on the deformationcapacity of the element, but they cause an increase ininitial stiffness and strength.

12. Conclusions

The dependence of shear resistance of r.c. elementson deformation demand was investigated in this paperby means of a non-linear smeared-crack/smeared-reinforcement plain-stress model. Cracked concrete wastreated as an orthotropic material, with the material axesbeing oriented along the directions of principal stresses.In a global sense the model is formulated in terms ofaverage stresses and average strains, but local equilib-rium at crack locations was also enforced explicitly. Thiswas achieved by modelling the mechanics of load trans-fer through reinforcement concrete bond between cracks,and through aggregate interlock and dowel action at thecrack faces. Results of the formulation were comparedwith those of conventional models that use a tension-stiffening idealisation to implicitly account for the abovementioned sources of secondary resistance in concrete.Thus, it was possible to establish the parametric depen-dence of the tension stiffening property (i.e., the post-cracking stiffness of r.c.), which was fraught with con-siderable empiricism, to reinforcing bar diameter, bondstrength, dowel strength, and aggregate interlock. Com-parisons between both approaches revealed that, evenwhen tension stiffening is properly calibrated so that theestimated shear strengths from the two models are inagreement, definitions of the individual components ofshear resistance, namely the concrete and steel contri-butions, were dependent upon the modelling assump-tions adopted. It was observed from the analytical resultsthat shear strength degradation occurs because of thereduction of thenc term with increasing imposed defor-mation and because of the limitation of thens term withsoftening of the compression struts. The primary sourceof these degradations is the susceptibility of concretetensile and compressive strengths to increasing crackwidths, represented by principal tensile strains in thiswork. The proposed model was verified by comparisonwith experiments and through a sensitivity analysis.


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martin-effect of bond, aggregate interlock and dowel action

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