se

t

ieguBondPrimary cracksFinite element method

2009 Elsevier Ltd. All rights reserved.

1. Introduction

Tension stiffening is the contribution to the member stiffnessof the intact concrete between the primary cracks and it playsa significant role in the deformation of reinforced concrete atthe serviceability limit states, particularly in the case of lightlyreinforced members. Under typical in-service load levels, theconcrete between the primary cracks carries significant tensilestress and the actual member response is considerably stiffer thanthe response of the bare steel bar. To accurately simulate the in-service behavior of reinforced concrete, tension stiffening must beaccurately modeled.Consider the reinforced concrete prism in uniaxial tension

shown in Fig. 1(a). Prior to cracking, behavior is essentially linear-elastic. The first primary crack occurs at the weakest cross-sectionof the member at the cracking load Pcr and, at loads P > Pcr ,the global load-average strain response becomes non-linear, asshown in Fig. 1(b). Tension stiffening at this global level is oftenrepresented as the difference between the bare bar strain and theactual member strain (ts) at any particular load P . Recently,Fields and Bischoff [1] introduced the concept of a tensionstiffening factor t which is defined as the tension stiffening strainat load P(ts) divided by the maximum tension stiffening strain(ts.max) which occurs just prior to first cracking at P = Pcr .Fig. 1(c) shows the relationship between the tension stiffeningfactor and the average member strain.

Corresponding author. Tel.: +61 2 9385 6002; fax: +61 2 9313 8341.E-mail address: i.gilbert@unsw.edu.au (R.I. Gilbert).

A detailed explanation of the mechanism of tension stiffeningat different loading stages in a reinforced concretemember (beforeyielding of the steel reinforcement) is given by Gilbert andWu [2].The decay of tension stiffening with increasing load is mainlyattributed to the formation of new primary cracks during the crackformation phase and due to degradation of bond during the crackstabilization phase, as illustrated in Fig. 1(b) and (c). The formationof each newprimary crack causes a loss of concrete tensile stress inthe regions adjacent to the crack. After all the primary cracks havedeveloped, the crack stabilization phase begins. Cover-controlledcracks occur under increasing load at the steelconcrete interfacecausing a loss of bond and hence a loss of tension stiffening.The cover-controlled cracks are the internal cracks that radiatefrom the bar deformations and are contained within the concretecover [3].Fig. 1(d) shows a typical length of specimen between two

adjacent primary cracks during the crack stabilization stage. Ateach crack, the concrete stress is zero. The concrete tensilestress gradually increases as the distance from the nearest crackincreases, due to bond stress that develops at the steelconcreteinterface, reaching a maximum mid-way between the two cracks.The intact concrete between the two primary cracks remainselastic during the entire loading period and themaximumconcretestress is less than the tensile strength of concrete. Fig. 1(d) alsoshows the bond stress and slip distribution at the steelconcreteinterface between the two primary cracks. Bond stress is zero atthe section containing the primary crack, because the concrete andthe steel bar are not in contact. However, the slip at the primarycrack is at a maximum and decreases to zero mid-way betweenEngineering Structures

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Engineering

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Modeling short-term tension stiffening incontinuum-based finite element modelH.Q. Wu, R.I. Gilbert Centre for Infrastructure Engineering and Safety, School of Civil and Environmental Engin

a r t i c l e i n f o

Article history:Received 16 December 2008Received in revised form19 May 2009Accepted 19 May 2009Available online 13 June 2009

Keywords:Tension stiffeningReinforced concrete

a b s t r a c t

Tension stiffening is most oflaws of the tensile concrete.the steelconcrete interfaceis incorporated into a non-lserviceability limit states. Thto account for the local damaanalysis is undertaken to adjproposed model is shown toloaded tension members at t0141-0296/$ see front matter 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2009.05.01231 (2009) 23802391

ble at ScienceDirect

Structures

evier.com/locate/engstruct

reinforced concrete prisms using a

ering, The University of New South Wales, Sydney, Australia

en included in models of reinforced concrete by modifying the constitutiveIn reality, tension stiffening is caused by the bond stress that develops atbetween the primary cracks. In this paper, a modified CEBFIP bond modelnear finite element program to accurately model tension stiffening at thebondslip relationship at any point along the reinforcement bar ismodifiede of the surrounding concrete, as well as the level of steel stress. A non-localst the constitutive law of the bond interface element at each load step. Theaccurately predict the crack spacing, stresses and deformation in axiallyypical in-service load levels.

H.Q. Wu, R.I. Gilbert / Engineering

Notations

The following symbols are used in this paper:

Dc Damage parameter of concrete elements;Dc Spatial averaged damage parameter of concrete

elements;db Reinforcing bar diameter;dmax Maximum aggregate size;Ec Elastic modulus of concrete;Ecp Secantmodulus at peakuniaxial compressive stress;Gf Fracture energy of concrete;fcp Compressive strength of the in situ concrete;fct Uniaxial tensile strength of concrete;f c Concrete compressive strength;k Decay factor for post peak response of compressive

stressstrain curve;lt Element size;n Ratio used in defining the equivalent uniaxial

compressive stressstrain curve (Eq. (11));Pc Spatial averaged confinement stress of concrete

elements;pc Confinement pressure in the concrete elements

around the reinforcing bar;R Radius defining the region of neighborhood con-

crete element integration points;r Distance between the target bond element integra-

tion point and its neighborhood concrete elementintegration points;

s Slip;s1, s2, ds3 Values of slip defining CEBFIP bondslip model; Parameter defining the ascending part of CEB

bondslip curve;f Coefficient related to aggregate size;1, 2, 3 Parameters defining the tensile stressstrain rela-

tionship of concrete; Ratio between major and minor principal stresses; Scaling factor for compressive strength;t Tension stiffening factor;1, 2 Scaling factor for major and minor principal direc-

tions;cp Concrete compressive strain at peak stress corre-

sponding to fcp;tp Concrete tensile strain at peak stress corresponding

to fctcp Concrete compressive strain at peak stress corre-

sponding to f c ;1 Principal tensile strain; Strain ratio used in defining equivalent uniaxial

compressive stressstrain curve (Eq. (11));1, 2 Parameters in the proposed bond model defining

the effects of cracking and steel stress on bond;1c, 2c Concrete stresses at principal directions;s Steel stress at primary crack;sy Steel yield stress;b Bond stress;max Maximum bond stress or bond strength; (r) Non-local weight function; (r) Normalized weight function; Ratio between steel stress and yielding stress;the two cracks, as shown. Bond stress increases rapidly adjacent toeach crack and thendecreases to zeromid-waybetween the cracks.Structures 31 (2009) 23802391 2381

An increment of external load causes a decrease in the bond stress(Fig. 1(d)) and a widening of existing primary cracks associatedwith increasing slip.Themagnitude of the bond stress for a cracked tensionmember

depends on many factors, some of which vary with the appliedload, and accurate modeling of bond, and hence tension stiffening,is a complicated task. The local bond stress is not only dependenton the local slip, but also on the magnitude of stress and strainin the reinforcing steel, the duration of the applied load andthe level of drying shrinkage. In this paper, the time-dependenteffects of shrinkage are not considered, although restraint to dryingshrinkage greatly affects tension stiffening [4,2,5]. Early shrinkagecauses a reduction in the cracking load and, under sustained serviceloads; shrinkage initiates the formation of additional primarycracks with time and causes a time-dependent degradation of thesteelconcrete bond.Most of the tension stiffening models in both analytical and

continuum-based finite element methods incorporate tensionstiffening by either modifying the constitutive law of the tensileconcrete [613] or that of the tensile reinforcement [14,15].However, in reality, at any point in the vicinity of a crack inthe tensile concrete, the concrete stress and strain both reduce(elastically unload) at first cracking, with the level of reductionvarying with distance from the nearest crack to the point inquestion. Tension stiffening is in fact a result of elastic unloadingof the concrete after cracking rather than post-peak softening. Thetensile stress in the concrete between cracks is caused by bondstress at the steelconcrete interface and it is the deterioration inbond that causes a reduction in tension stiffening with increasingload. Tension stiffening is most rationally modeled using a wellcalibrated local bond stressslip relationshipIn order to better describe this mechanism in the finite element

analysis, a modified CEBFIP bondmodel is proposed in this paper.The modified bond model is incorporated into the continuum-based finite element program RECAP [16] to analyze a numberof uniaxially loaded tension members. The numerical resultsobtained using the proposed model are compared with both theexperimental results and numerical results obtained assuming theoriginal CEBFIP bondmodel [17]. The proposedmodel is shown toaccurately predict the behavior of the test specimens at all stagesof loading.

2. Concrete model

The biaxial concrete strength envelope proposed by Foster andMarti [18], and shown in Fig. 2, is used in the finite element model.In Fig. 2, 1c and 2c are the principal stresses, and fcp is the uniaxialcompressive strength of the in-situ concrete.For concrete in tension, the bilinear softening curve proposed

by Petersson [19] is used (Fig. 3(a)). Cracking of concrete isintroduced in the element as soon as the principal tensile stressexceeds the tensile strength of the concrete. In order to deal withthe issue ofmesh sensitivity inducedby localization of deformationin the vicinity of a crack in a concrete element, the crack bandtheory approach [20] is used to depict the softening branch of thetensile stressstrain relationship in the fracture zone immediatelyadjacent to the crack. The three parameters 1, 2, and 3, whichare defined in Fig. 3(a) and are required to define the tensilestressstrain relationship, are given as:

1 = 13 ; 2 =293 + 1; 3 = 185

EcGff 2ct lt

(1)where Ec is the elasticmodulus of concrete; lt is the element size; fctis the uniaxial tensile strength of concrete;Gf is the fracture energy

Fig. 1. Stresses and deformations in an axially loaded tension member.

Fig. 2. Biaxial concrete strength envelope of [18].

of concrete associated with tensile softening, which is here takenas specified in the CEBFIP model code [17]:

Gf = f(fcp10

)0.7(fcp in MPa and Gf in N/mm) (2)

and f is a coefficient related to themaximum aggregate size (dmaxin mm) and is taken as:

f = (1.25dmax + 10) 103. (3)In this study, the concrete compressive strength in one principal

direction is given by:

where is a scaling factor [21] which depends on the stress inthe orthogonal principal direction, as illustrated in Fig. 3(b). Theconcrete strain at peak stress is also modified by the same scalingfactor , so that:

cp = cp. (5)In the compressioncompression state, is greater than 1.0

and depends on the ratio of major and minor principal stresses = 1c/2c [21]:

2 = 1.01 /2.4 for 0 0.48 (6)

2 = 1.15(1+ 5.51)

1+ /5.5 for 0.48 < 1.0 (7)

where 2 is the scaling factor accounting for confinement in theminor principal direction. The confinement scaling factor for themajor principal direction 1 is given by multiplying 2 by :

1 = 2. (8)In the tensioncompression state, the major principal stress

is tensile and it reduces the compressive strength in the minorprincipal direction [2224]. The scaling factor developed byVecchio and Collins [21] is adopted here:

= 10.8+ 0.34 1

cp

1.0 (9)2382 H.Q. Wu, R.I. Gilbert / Engineeringf c= fcp (4)Structures 31 (2009) 23802391where 1 is the principal tensile strain.

er3. Bond interface model

The bond model must correctly predict the relationshipbetween local bond stress and slip, including the effects of concretecracking and load intensity (i.e. steel stress level). Results ofdifferent pull-out tests (e.g. [2732]) have indicated that the bondstress is influenced bymany parameters, such as concrete strength,bar diameter, bar spacing, transverse reinforcement, confiningstress, and so on. However, most of the pull-out tests have beenundertaken on short anchorage lengths from 1 db to 6 db (where dbis the reinforcing bar diameter) with variable boundary conditions,and cracking and steel stress states are usually ignored.Fig. 4 shows a typical pull-out test configuration. Since the

embedment length is normally shorter than the crack spacing, noprimary cracks can form in the test specimens. The bond stressslip

is based on the average bond stressslip relationship measured inpull-out tests by Eligehausen et al. [30]. The bond stress generallydecreases with increasing steel stress and Shima et al. [33] andKankam [34] suggested that the effect of steel strain should beintroduced into the bondslip relationship.In the following, the CEBFIP bond model is modified by

incorporating the effects of local damage, primary cracking andconcrete confinement pressure. At service load levels, when thesteel stressstrain relationship is linear-elastic (i.e. the steel stressis less than the yield stress), slip normally does not exceed s1 inFig. 5. The basic bondslip relationship (ignoring cracking and steelstress state) is therefore based on the ascending part of CEBFIPbond model, which can be expressed as:

b = max(ss1

)(14)n 1+ in which:

n = EcEc Ecp ; =

|c |cp

(11)

cp is the strain corresponding to the peak stress; and Ecp is the se-cantmodulus at the peak of the curve, Ecp = fcpcp . The parameter k inEq. (10) is the decay factor associated with the post peak responseand is given by Collins and Porasz [26] as:

k = 1.0 for |c | cp (12)k = 0.67+ fcp

62 1.0 for |c | > cp. (13)

Fig. 5. CEBFIP bond stressslip relationship.

stressslip relationships represent the average response. Thesetests take account of the global effects of the localized cover-controlled cracks (shown in Fig. 4(a)) on bond degradation and slip.The bondslip relationship specified in the CEBFIP code (Fig. 5)H.Q. Wu, R.I. Gilbert / Engineering

(a) Concrete in tension.

Fig. 3. Stressstrain rel

(a) Cover-controlled cracking at thesteel-concrete interface.

(b) Id

Fig. 4. Typical configu

The uniaxial compressive stressstrain relationship proposedby Thorenfeldt et al. [25] is used in this study. That is

c = fcp n nk (10)curves generated in the tests are based on the measured loadapplied to the bar and the slip at the free ends, so that the bondStructures 31 (2009) 23802391 2383

(b) Concrete in compression.

ationships for concrete.

alized averaged bond stress.

ation of pull-out tests.in which b is the bond stress, max is the maximum bond stresswhich depends on the size and pattern of deformations on the bar

2 = 1.0 when s < 250 MPa;2 = 2.0 0.004s when 250 MPa s 500 MPa; (15b)2 = 0.0 when s > 500 MPawhere s is the local steel stress, Dc is the non-local averageddamage parameter of concrete elements based on the concretemembrane model adopted. The product 12 (calculated usingEqs. (15a) and (15b)) has been calibrated within the finite elementformulation using the experimental data of Bischoff [4] and Gilbertand Nejadi [39].According to the basic concept of continuum damage theory,

the magnitude of damage of the concrete can be represented bya damage variable, Dc . In this study, only the cracking damage inthe major principal direction is considered as the parameter af-

Dc =V (r)Dc (r, ) dV (19a)

pc =V (r) pc (r, ) dV (19b)

where the normalized weight function (r) is:

(r) = (r)V (r) dV

. (20)

The weight function in this study is the dome shaped surface ofFig. 6(b) given by:[ (

2r)2]2384 H.Q. Wu, R.I. Gilbert / Engineering

(a) Concrete element integration points.

Fig. 6. Non-local averaging sc

surface (and is taken here to be 2.5fcp for deformed high-bond

bars and 1.0fcp for plain round bars); and s is the slip. Eq. (14)

is suitable for modeling the bondslip relationship if cracking isconsidered to be smeared throughout the tension zone, but it mustbemodifiedwhen the problem is considered at amoremicroscopiclevel, to reflect the effect on the local bondslip relationship ofthe proximity of discrete primary cracks to each particular bondinterface element.A number of investigators has proposed to incorporate concrete

cracking or damage into the bond constitutive law [3537],by reducing the bond stiffness as a result of damage to thesurrounding concrete elements. On the basis of experimentalresults, Maekawa et al. [38] proposed a bond model that is relatedto the local steel strain.As shown in Fig. 1, soon after first cracking, the drop in tension

stiffening is substantially attributed to the formation of primarycracks, where at each crack the bond stress drops to zero andthe slip is substantial. After the formation of the primary cracks(at the crack stabilization stage), the loss of tension stiffeningcan be best modeled by reducing the bond stress with increasinglocal steel stress (to model the development of cover controlledcracking). Two governing parameters 1 and 2 are introducedhere to take into account the local concrete damage at primarycracks (1) and the effect of steel stress on bond degradation (2).The parameter 1 varies from 1.0, when the concrete in the vicinityof the crack is undamaged and the average principal tensile strain isless than tp (see Fig. 3(a)), down to 0.0 when the average principaltensile strain exceeds 3tp (see Fig. 3(a)) as given by Eq. (15a).The parameter is a simple model of the phenomena previouslyidentified by others [3537]. The parameter 2 has been includedtomodel the observed change in the bondslip relationship at steelstresses exceeding 250 MPa and varies linearly from 1.0 at a steelstress of 250 MPa to 0.0 at a steel stress exceeding 500 MPa, asgiven by Eq. (15b).

1 = 1 Dc, and (15a)fecting the local bond deterioration. Based on the concrete tensilestressstrain law (Fig. 3(a)), the concrete remains intact at 1 tpStructures 31 (2009) 23802391

2r

(b) Weight function for non-local analysis.

heme for the bond elements.

(Dc = 0) and cracked at 1 > tp (0 < Dc 1). A simpleexpression to model the transition between undamaged (un-cracked) and extensively damaged has been developed for Dc asfollows:

Dc = 0 for 1 tp (16a)Dc = 1 1cEc1 for tp < 1 3tp (16b)Dc = 1 for 1 > 3tp. (16c)In Eq. (14), max and s1 are treated as constants, which

are related to the bond confinement condition. The test datain [32] shows that the bond strength envelope increases and itsradial deformation decreases with the application of confinementpressure. This test data is adopted here to alter the peak bondstrength max according to the average confinement stress pc ofconcrete around the steel bar elements. To model the test data, aparameter 3 is selected as a variable power of max as follows:

3 = 1+ pcfcp

(17)and, consequently, the modified bond stressslip law can bewritten as:

b = 12 3max(ss1

). (18)

In order to incorporate these parameters into the bondstressslip relationship in the finite element program, a non-localanalysis is undertaken at the end of each load step. As shown inFig. 6, at each integration point in the bond interface elements,the surrounding concrete damage value and confinement stresswithin a radius R is averaged by a weighting function (r), wherer is the distance between the bond element integration point andthe source concrete element integration points within the radiusR (as shown in Fig. 6(a)). Therefore, the weighted average damageparameter Dc and confinement stress pc can be written as: (r) = exp R

. (21)

and (17) to give the governing parameters 1 and 3. In this study,only the concrete within a distance of 1.5 db from the surface ofthe bar is assumed to influence the bondslip relationship, i.e. R =2 db in Fig. 6. This assumption has been found to give reasonableagreement with the available test data.

4. Finite element modeling of uniaxial tension tests

The 2-D finite element program RECAP [16,40,18], incorporat-ing the bondslip relationship of Eq. (18), was used to model aseries of short-term uniaxial tension tests conducted by Wu andGilbert [41]. Each test specimen consisted of a concrete prism ofsquare cross-section (100mmby100mm) and1100mm long, con-taining a single hot-rolled deformed reinforcing bar running longi-tudinally through the centroid of each cross-section, as shown inFig. 7. The steel bar was Grade 500 normal ductility steel, i.e. thenominal characteristic yield stress was 500 MPa. The tensile axialloadwas applied to the ends of the reinforcing bar protruding fromeach end of the concrete prism. The experimental and numericalresults from two prisms (STN12 and STN16) are presented here.The diameters of the deformed reinforcing bar in STN12 and STN16were 12mm and 16mm, respectively. Each prismwasmoist curedprior to testing, so that the magnitude of shrinkage in the concreteat the time of testingwas small (about25) and is ignored here.The finite elementmesh used in the analyses is also shown in Fig. 7.Taking advantage of symmetry, only one quarter of the test speci-men was analyzed.An element size of 5mmby 5mmwas adopted, so as to reliably

model the local stress redistribution between adjacent primarycracks. The prisms are modeled using 1332 nodes, 1100 concrete

nodes by the zero-width bond interface elements (also shown inFig. 7). The finite element solution is dependent on the mesh size.Relatively small elements are necessary to accurately model thediscrete nature of cracking and bondslip at the steelconcreteinterface. However, for the specimen sizes considered here, afurther reduction in the element size resulted in relatively littleincrease in accuracy.During the experiments, as the applied load increased, primary

cracks formed at reasonably regular centers. At the crackstabilization stage, 5 primary cracks were observed along bothSTN12 and STN16, with average crack spacing of 195 mm and165 mm, respectively. In the finite element modeling, randomizedtensile strengths are generated in concrete elements with a 2%maximum difference being assumed. Therefore, first crackingwas initiated in the concrete element with the lowest assignedtensile strength and the tensile stress in surrounding elementsin the vicinity of the first crack dropped sharply. As the loadwas increased, additional cracking occurred subsequently inelements at distances far enough away from the existing cracksfor concrete stress levels not to be significantly affected. As thedistance from the nearest crack increases, the bond stress transfersthe tensile force from steel bar to the undamaged surroundingconcrete.The concrete and steel properties adopted in the study are

the properties measured and reported by Wu and Gilbert [41]:fct = 2.0 MPa (with 2% randomization), fcp = 21.6 MPa, Ec =22,400 MPa, fsy = 540 MPa, and Es = 200,000 MPa. The basicparameters for the CEBFIP bond stressslip relationship are inaccordancewith the code for good confinement conditions, i.e. =0.4, max = 11.6 MPa and s1 = 1.00 mm and typical valuesfor the parameters that define the tensile stress strain curve forH.Q. Wu, R.I. Gilbert / Engineering

Fig. 7. Dimensions and finite element mes

The weighting function is largest at the nearest integrationpoint and decreases as the distance r increases. The weightedaveraged parameters Dc and pc are then substituted into Eqs. (15)elements, 110 two node steel reinforcement truss elements and110 zerowidth bond interface elements. The concrete elements areStructures 31 (2009) 23802391 2385

h adopted for uniaxial tension specimens.

isoparametric quadrilateral plane stress elements with numericalintegration performed using 2 2 gauss quadrature. The steelelements are connected to the concrete elementswith overlappingconcrete (Fig. 3(a)) as recommended by Peterson [19] are assumed,i.e. 1 = 0.333, 2 = 50, 3 = 230.

5. Finite element analysis results

Fig. 8 compares the numerical and experimental load versusaverage strain graphs for STN12 and STN16. The average strainis calculated from the finite element model by dividing the axialdeformation between the two monitored nodes A and B, as shownin Fig. 7, by the distance between them. The proposed bond modelprovides an excellent correlation with the experimental results,especially at the cover-control crack stage. With increasing loads(i.e. with increasing steel stress during the crack stabilizationstage), the finite element model incorporating the proposed bondmodel predicts the gradual reduction of tension stiffening that wasobserved in the test specimens. By contrast, the CEB bond modeltends to overestimate tension stiffening under increasing loads.Fig. 9 shows the tension stiffening factor versus average strain

for STN12 and STN16. The tension stiffening factor predicted usingthe CEBFIP bond model tends to increase with increasing strainafter all the cracks have formed. Furthermore, this overestimationof stiffness is more significant in the specimen with the smaller

bond stress and slip along each member at different load levelsduring the crack stabilization phase. Slip is a vector quantity, withdirection either positive or negative in the direction of the bar. Alsoshown in Figs. 10 and 11, are themeasured variations of steel forcein each test specimen at the same two load levels.In the numerical solution, three primary cracks developed in

the half-length of STN12 at the locations indicated in Fig. 10(a)and four cracks in the half-length of STN16 as shown in Fig. 11(a).The numerical model provided excellent agreement with theexperiments in terms of both the number of cracks and thespacing between them. The principal tensile strains at the cracklocations are much greater than the strain corresponding to theonset of cracking (about 0.0001). In between the cracks; all theconcrete strains remain in the elastic range (

(b) Steel forcehalf sample (FEM). (c) Steel forcemiddle 600 mm of sample (experiment).

(d) Sliphalf sample (FEM). (e) Bond stresshalf sample (FEM).

Fig. 10. Local response of STN12 using the proposed bond model.

Table 1Measured and calculated maximum crack widths (mm).

STN12 STN16Load (kN) Steel stress at crack (MPa) Maximum crack width (mm) Load (kN) Steel stress at crack (MPa) Maximum crack width (mm)

Experiment FEM Experiment FEM

40.0 354 0.30 0.26 76.0 378 0.325 0.2550.0 442 0.375 0.35 90.0 448 0.375 0.34

strain as loading progressed. The steel force shown in the figuresis obtained directly from the measured strains by multiplying thelocal strain by the elastic modulus and the cross section area ofsteel bar. To minimize the impact of the strain gauges on thebond between the steel bar and the concrete, the gauges werecarefully attached to the longitudinal ridge that runs longitudinallyalong the bar and any contact between the gauge adhesive and thetransverse bar deformations was avoided.As the figure illustrates, at the cracks the steel force reaches its

more uniform along the bar (due to the degradation of bond), asthe tension stiffening effect decreases. The variation of steel forcemeasured during the tests is not smooth (as is predicted by thenumerical model) because the breakdown in bond is quite local atthe location of cover-controlled cracks and this is reflected by therelatively large differences in strains measured at adjacent straingauges. Nevertheless, the variation of steel forces predicted by thenumerical model is in reasonable agreement with the measuredvariation. The proposed bond model reduces the average bondstress with increasing steel stress, without capturing the very(a) Tensile strain contours on half sample (FEM) when P = 50 kN.H.Q. Wu, R.I. Gilbert / Engineeringmaximum and mid-way between cracks it is at a minimum. Thegraphs show that with increasing load the steel force tends to beStructures 31 (2009) 23802391 2387local effect of damage on the bond caused by cover-controlledcracking.

(b) Steel forcehalf sample (FEM). (c) Steel forcemiddle 600 mm of sample (experiment).

(d) Sliphalf sample (FEM). (e) Bond stresshalf sample (FEM).

Fig. 11. Local response of STN16 using the proposed bond model.

With regard to the slip at the steelconcrete interface and theaverage local bond stress (Fig. 10(d) and (e) for STN12 and Fig. 11(d)and (e) for STN16), the proposed bond model gives rise to reducedbond stress with increasing slip as the load is increased within theservice load range. Themaximum bond stress decreases by around1.77 MPa for STN12 as the load P is increased from 40 kN to 50 kNand by about 1.37 MPa for STN16 as P increases from 76 kN to90 kN. The bond stress distribution given by the proposed modelalso illustrates a rapid built-up of bond stress from zero at the crackto its maximum value near the cracks, dropping to zero mid-waybetween the cracks.The concrete tensile stress distributions along STN12 and STN16

calculated using the proposed bond model at different transversedistances z from the steel bar (at the bar surfacez = 0; atthe concrete surfacez = 50 mm; and mid-way betweenz =25 mm) at two different load levels are shown in Figs. 12 and 13,respectively. The tensile concrete stress at all locations reduces asthe applied load increases, i.e. the intact concrete is unloading asthe load increases during the crack stabilization stage. The crack

Table 2Average primary crack spacing (mm).

STN12 STN16Experiment FEM Experiment FEM

195 181 165 145

250 MPa to 400 MPa. For STN12, the maximum tensile stress mid-way between two primary cracks at the level of the steel bar dropsfrom1.9MPa at an applied load of P = 40 kN (when the steel stressat a crack s = 354 MPa) to approximately 1.3 MPa at P = 50 kN(when s = 442 MPa). For STN16, the corresponding tensile stressdrops from1.35MPa to 0.61MPa as the applied load increases from76 kN (s = 378 MPa) to 90 kN (s = 448 MPa).Tables 1 and 2 summarize the calculated and experimental

maximum cracks width and average crack spacing. The maximumcrack width was measured at the concrete surface. The crackwidth predicted by the numerical model at the steel level canbe obtained simply by adding up the two maximum slips either2388 H.Q. Wu, R.I. Gilbert / Engineering

(a) Tensile strain contours on half sample (FEM) when P = 90 kNstabilization stage usually occurs at typical service load levels, withsteel stresses at the primary cracks typically in the range fromStructures 31 (2009) 23802391

.side of the primary crack. The slight increase in crack width withdistance z from the steel bar can be obtained by integrating the

(b) At P = 50 kN.

Fig. 12. Concrete tensile stress distribution calculated using proposed bond model (STN12).

reduction in elastic tensile strain in the concrete (from that atthe level of the bar) over the half crack spacing on either sideof the crack. The proposed bond model provides a reasonableprediction of maximum crack width and average crack spacing;slightly underestimating both the observedmaximum crack widthand the average crack spacing in both test specimens.The CEBFIP bond model, therefore, fails to correctly predict

tension stiffening at short-term service loads in both global andlocal scale, because it fails to include the effects of concrete damageand increasing steel stress. The bond stressslip relationship inpractice must be correlated with either concrete damage or steelstress evolution if an accurate model of tension stiffening isrequired over the full service load range.

6. Summary and conclusions

The loss of tension stiffening in reinforced concrete membersunder monotonically increasing load is caused by the formationof primary cracks and the gradually decreasing bond stress underincreasing loads after all the primary cracks have formed. Thebond stressslip relationship at the interface between the concreteand the steel reinforcement is therefore of paramount importancewhen modeling tension stiffening in reinforced concrete. A bond

cracking and bond degradation under increasing steel stress hasbeen proposed and incorporated into an existing finite elementmodel of reinforced concrete. Two uniaxially loaded tensionmembers have been analyzed numerically using both the proposedbondmodel and the CEBFIP bondmodel and the numerical resultshave been compared with the experimental observations.The proposed bond model gives good correlation with the

test results at service loads (when steel stresses remain in theelastic range), while the CEBFIP bond model gives a significantlystiffer response and tends to overestimate tension stiffening. Theexperiments indicate that the tension stiffening factort decreaseswith increasing load after the primary cracks have developed andthis is predicted well using the proposed bond model. In contrast,the CEBFIP bond model does not predict the reduction of twith increasing load after the formation of the primary cracks.The proposed bond model provides reasonable predictions of thevariation of steel stresses, bond stresses and slip between theconcrete and the steel. In addition, the model provides a goodestimation of thewidth of the primary cracks, aswell as the spacingbetween them.This paper has not considered the variation of tension stiffening

with time, which is an important consideration in engineeringpractice. Future research will concentrate on the effects of creepH.Q. Wu, R.I. Gilbert / Engineering

(a) At P = 40 kN.stressslip relationship based on the CEBFIP bond model suitablymodified to include the effects of concrete damage due to primaryStructures 31 (2009) 23802391 2389and shrinkage on tension stiffening and the inclusion of theseeffects in the proposed bond model.

(b) At P = 90 kN.

Fig. 13. Concrete tensile stress distribution calculated using proposed bond model (STN16).

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Modeling short-term tension stiffening in reinforced concrete prisms using a continuum-based finite element modelIntroductionConcrete modelBond interface modelFinite element modeling of uniaxial tension testsFinite element analysis resultsSummary and conclusionsReferences