isen

rsityeld

Revised 10 March 2015Accepted 11 March 2015Available online 17 April 2015

Keywords:ConnementColumns

lateral connement which may be provided by transverse steel reinforcement and/or a supplementalber-reinforced polymer (FRP) jacket. Despite extensive past research on conned concrete columns,

concrete connement is exploited are referred to as conned con-crete columns in this paper. Only circular columns are explicitlyconsidered in the paper, although the approach presented in thepaper is generally applicable to conned concrete columns of othercross-sectional shapes.

To understand the behavior of conned concrete columns,researchers have traditionally relied on experimental

hanisms betweenplored. For exam-e discretemns, and

assume that the connement provided to the concrete is invover the height. This assumption of uniformity over height lsignicant uncertainty when an FRP-jacketed RC column isconsideration: interaction between the non-uniformity of steel-hoop connement and the brittle FRP jacket may well haveundesirable consequences as the FRP jacket can be expected torupture when local high hoop strains reach its hoop rupture strain.

To overcome the inadequacy of uniaxial stressstrain modelsfor conned concrete in predicting the three-dimensional (3D)behavior of conned concrete columns, one may resort to a 3D

Corresponding author. Tel.: +852 27666012; fax: +852 23346389.E-mail address: cejgteng@polyu.edu.hk (J.G. Teng).

Engineering Structures 97 (2015) 1528

Contents lists availab

g

lsecially in structures designed to resist seismic loading. Differenttechniques, such as transverse steel bars and ber-reinforced poly-mer (FRP) jackets/tubes, have been used to achieve lateral conne-ment to concrete in columns. Columns in which the concept of

concrete columns, including the interaction mecthe conning material and the concrete, to be exple, these uniaxial stressstrain models ignore thof steel hoops in reinforced concrete (RC) coluhttp://dx.doi.org/10.1016/j.engstruct.2015.03.0300141-0296/ 2015 Elsevier Ltd. All rights reserved.natureinsteadariableeads tounderWhen concrete is under lateral connement (including bothactive connement by a constant conning pressure and passiveconnement by a conning pressure that depends on the dilationof concrete), it experiences signicant enhancement in bothstrength and ductility. As a result, the use of conned concrete incolumns has increased signicantly in the past few decades, espe-

stressstrain models for the conned concrete to facilitate theanalysis of such columns in design; many such models have beenproposed based on experimental observations (e.g. [1,3,4]). Whilethese uniaxial stressstrain models have played a useful role inthe approximate analysis of conned concrete columns for designpurposes, they do not allow the fundamental behavior of connedConcreteFRPFinite elementsPlastic-damage model

1. Introductionmost of the existing work has been either experimental or empirical, particularly when discrete steelhoops/spirals need to be considered. This paper instead is focused on the alternative approach ofthree-dimensional (3D) nite element (FE) analysis of circular FRP-conned RC columns, with the dis-crete nature of transverse steel reinforcement properly captured. The key to the success of such FE analy-sis lies in an accurate constitutive model for the concrete which is under 3D compressive stresses, andthis is achieved in the present study by building on an accurate plastic-damage model recently proposedby the authors group. In implementing this plastic-damage model, a local stressstrain model forconcrete under uniform connement, obtained by resolving a number of issues associated with 3D FEmodeling, is employed to generate data for the input parameters. The proposed FE approach is capableof providing accurate prediction for both FRP-conned RC columns and steel-conned RC columns asdemonstrated through comparisons with existing test data. FE results obtained for steel-conned circularRC columns are also examined in detail to gain an improved understanding of the connementmechanisms in these columns.

2015 Elsevier Ltd. All rights reserved.

investigations (e.g. [13]). A major aim of these experimental stud-ies has been to establish empirical or semi-empirical uniaxialArticle history:Received 8 March 2014

The strength and ductility of reinforced concrete (RC) columns can be substantially enhanced thoughThree-dimensional nite element analyscolumns with FRP and/or steel connem

J.G. Teng a,, Q.G. Xiao a, T. Yu b, L. Lam aaDepartment of Civil and Environmental Engineering, The Hong Kong Polytechnic Univeb School of Civil, Mining & Environmental Engineering, University of Wollongong, North

a r t i c l e i n f o a b s t r a c t

Engineerin

journal homepage: www.eof reinforced concretet

, Hong Kong, Chinas Avenue, Wollongong, NSW 2522, Australia

le at ScienceDirect

Structures

vier .com/ locate /engstruct

acteristics were introduced into these four components to elimi-nate the deciencies of the previous DP type plasticity models

g Strnite element (FE) model, with all details explicitly represented, inconjunction with appropriate constitutive models for the allmaterials involved. As constitutive models for steel bars and FRPjackets/tubes are well established, the key to the success of such3D models lies in the accuracy of the constitutive model forconcrete under general 3D compressive stresses.

For circular RC columns conned with transverse steel bars (i.e.steel-conned RC columns), 3D FE models have been presented byprevious researchers [5,6,810] based on different constitutivemodels for concrete. Abdel-Halim and Abu-Lebdeh [5] used a non-linear elastic model for concrete; their model succeeded in closelypredicting the axial stressaxial strain curves of core concrete butfailed to properly depict the dilation behavior of core concrete. Liuand Foster [6] employed an explicit microplane model for concrete[7]. The values of parameters for this constitutive model wereobtained from a back analysis of circular concrete cylinders undervarious conning pressures; using the calibrated parameters, themodel was shown to provide close predictions of the stressstrainbehavior of core concrete. Barros [8] used an associated plasticitymodel for concrete; a scaling technique was introduced for thehardening/softening rule, and as a result, the model was shownto provide accurate predictions for the peak stress and the slopeof the softening branch of axial stressstrain curves of coreconcrete. Because an associated ow rule was used by Barros [8],the model was unable to predict the dilation behavior of steel-conned concrete closely. Imran and Pantazopoulou [9] used anon-associated plasticity model for concrete; this model wasshown to provide reasonably accurate predictions for the dilationbehavior of steel-conned concrete. Grassl and Jirsek [10] useda plastic-damage model for concrete; the model was shown toprovide close predictions of both the axial stressstrain curveand the dilation behavior of steel-conned concrete. It should benoted that in predicting the axial stressstrain behavior ofsteel-conned concrete, the accuracy of the model in predictingthe dilation behavior is not so critical, as the conning pressuredoes not depend on the dilation of concrete once the steel hasyielded. For FRP-conned concrete and other passively-connedconcrete, this dilation behavior controls the variation of theconning pressure. Although some of the above-mentionedconstitutive models can predict the behavior of steel-connedconcrete closely, all of them have recently been shown to beincapable of properly depicting the dilation behavior of connedconcrete [16].

For circular FRP-conned RC columns where interactionsbetween the discrete steel hoops/spirals and the FRP are importantin the connement mechanism, 3D FE models have been presentedby Montoya et al. [11] who used the compression eld theory forconcrete; by Rougier and Luccioni [12] who used a plastic-damagemodel for concrete; and by Eid and Paultre [13], Karabinis et al.[14] and Doran et al. [15] who used a DruckerPrague (DP) typeplasticity model for concrete. The studies reviewed above haveachieved partial success in predicting the behavior of conned RCcolumns, but there is also considerable room for improvement.For instance, Montoya et al.s model [11] is inconvenient forimplementation in a FE analysis driven by strain or displacementincrements as a special algorithm is needed to determine the valueof Poissons ratio corresponding to the current conning pressure.In particular, the accuracy of these 3D models, as a predictive tool,is still limited. This is because they all suffer from the lack of anaccurate constitutive model [16] for FRP-conned concrete whichis predominantly under three-dimensional compressive stresses.For instance, the constitutive models used in [1215] failed toclosely predict the dilation behavior of FRP-conned concrete.

16 J.G. Teng et al. / EngineerinYu et al. [17,18] recently assessed the accuracy of existing DPtype constitutive models in predicting the response of connedconcrete; their assessment showed that although one of the DPin modeling FRP-conned concrete. Yu et al.s model [18] has beensuccessfully implemented in ABAQUS [20] for the FE analysis oftwo types of FRP-conned concrete columns with uniform conne-ment over the column height and obtained accurate predictions:FRP-conned plain concrete cylinders and hybrid FRP-concrete-steel double-skin tubular columns (abbreviated as DSTCs in [21]).

Due to the uniformity of connement over height in FRP-con-ned plain concrete columns and FRP-concrete-steel double-skintubular columns, Yu et al.s FE analysis [18] employed a slice modelcontaining a single layer of solid elements. While such a slicemodelcan closely represent the mid-height region of these two types ofFRP-conned concrete columns, it is incapable of representing the3D behavior of FRP-conned RC columns and steel-conned RC col-umns (e.g. end constraints, transverse steel bars, and buckling oflongitudinal steel bars). This paper presents the rst accurate FEapproach for the 3D behavior of steel- and FRP-conned RC col-umns under concentric axial compression that employs the accu-rate plastic-damage model for conned concrete that wasrecently developed by Yu et al. [18]. This FE approach offers apowerful tool for understanding the complex behavior of concreteunder non-uniform connement as commonly found in real col-umns. It should be noted that the present study is concerned onlywith FRP-conned RC columns and steel-conned RC columnsunder monotonic loading as the constitutive model used here forconned concrete has only been veried under this type of loading.

The paper starts by providing a brief summary of the plastic-damage model proposed by Yu et al. [18], which is then followedby a calibration of the plastic-damage model for the three-dimen-sional behavior of FRP-conned circular plain concrete cylinderswith particular attention to the end restraint effects. The calibratedmodel is next veried using test data of steel-conned and FRP-conned RC columns. Finally, FE results obtained using the veriedFE approach are used to investigate the validity of the archingaction assumption, which has been commonly used in modelingthe behavior of concrete conned by transverse steel bars.

2. Constitutive models

2.1. Yu et al.s model for conned concrete and renement

Yu et al.s model [18] was based on a good understanding of thebehavior of conned concrete accumulated at The Hong KongPolytechnic University (e.g. [3,24]) over many years and formu-lated within the theoretical framework of the Concrete DamagedPlasticity Model (CDPM) provided in ABAQUS. A brief summaryof the model is provided below; further details of the model canbe found in [18]. In the model, a four-parameter Lubliner criterion[23] is adopted as the yield function. Three of the four parametersmodels (i.e. [19]) has the potential to provide accurate predictionsfor actively-conned concrete or steel-conned concrete, none ofthem can properly capture all the key features of FRP-connedconcrete. Based on Yu et al.s study [17,18], it can also be deducedthat other plastic-damage models available also suffer from similarproblems as they do not include the key features identied by Yuet al. [17,18] for the accurate modeling of concrete under a generalstate of connement. On the basis of this knowledge, Yu et al. [18]developed an improved plastic-damage model to provide accuratepredictions for FRP-conned concrete. This model includes fourcomponents, which are the yield surface, the hardening rule, theow rule, and the damage variable. Connement-dependent char-

uctures 97 (2015) 1528are related to the tensile strength, ft, uniaxial compressive strength,f 0co, and equal biaxial compressive strength, f

0b, of concrete. The

fourth parameter K is dened as the ratio of the square root of

shape of the yield surface in the deviatoric plane and controls the

general 3D compressive stresses), for a given ratio of f 0 =f 0 . In

g Strb co

the case of tri-axial compression, this yield surface can be repre-

sented in the I1 J2

q(I1 is the rst invariant of effective stresses,

and J2 is the second invariant of deviatoric effective stresses) planeby a linear line. This linear line corresponds to a strength equationin the form of f 0cc f 0co k1rl, where f 0cc is the peak stress of con-crete under a uniform lateral conning pressure, rl. Teng et al.[22] suggested a value of 3.5 for k1, and the corresponding valueof K can be shown to be 0.725 [18]. In addition, Yu et al. [17] foundthat most of the existing DP type plasticity models tend to predictalmost identical slopes for the second branch of stressstraincurves for concrete under different conning pressures if the hard-ening rule is related only to the uniaxial stressstrain curve ofunconned concrete, a phenomenon which is not consistent withexperimental observations. Therefore, Yu et al. [18] proposed aconnement-dependent hardening rule to eliminate this dis-crepancy. This connement-dependent hardening rule is obtainedfrom a set of stressstrain curves of concrete under uniform activeconnement generated by a so-called analysis-oriented stressstrain model [22] that accounts explicitly for the degree of activeor passive connement via an incremental analysis; the uniaxialstressstrain curve of unconned concrete is always included inthis set of stressstrain curves as a special case with its conningpressure being equal to zero. Similarly, the damage variable inYu et al.s model [18] is also set to be connement-dependentand is also obtained from the same set of stressstrain curves ofconcrete under uniform active connement. A DP hyperbolicfunction is used as the ow potential; which can be very close tothe DP linear function with proper parameter setting [18]. As aresult, the dilation angle in the ow rule becomes a major parame-ter. When a constant dilation angle is used, the volumetric plasticstrain increment and the deviatoric plastic strain increment are inproportion, which is also not consistent with experimental obser-vations. To eliminate this discrepancy, the dilation angle in Yuet al.s model [18] is taken as a function of the equivalent plasticstrain ~epl and the rate of the connement increment.

The preceding connement-dependent features require ananalysis-oriented stressstrain model for concrete under uniformactive connement that works well for both active and passiveconnement (e.g. [22]) to produce the necessary material parame-ters. In Yu et al.s model [18], Teng et al.s model [22] was adoptedto generate a connement-dependent hardening rule and a con-nement-dependent dilation angle for the CDPM.

In the present study, Teng et al.s model [22] was replaced byJiang and Tengs model [3] as the latter provides closer predictionsfor weakly conned concrete. In addition, the characteristics ofnon-uniform connement on the damage variable and the harden-ing rule are further accounted for by dening an effective conningpressure, rl,eff, as follows [18]:

rl;eff 2af0co r2af 0co r3

2af 0co r2 r3 af 0co 1

where r2 and r3 are the two principal lateral stresses respectively,and a is a constant to be determined based on test results. In thepredicted connement effect of concrete under tri-axial compres-sion (referring to axial compression in combination with equal lat-eral compressive stresses; to be differentiated from states ofthe second invariant of deviatoric stresses on the tensile meridianto that on the compressive meridian. This parameter describes the

J.G. Teng et al. / Engineerinpresent study, compressive stresses and strains are dened as nega-tive while tensile stresses and strains are dened as positive, unlessotherwise specied. It was suggested by Yu et al. [18] that the best-t value for a is 0.0039. Based on the denition of rl,eff, the rate ofthe connement increment, keq, is dened as follows:

keq 2rl;eff=e2 e3 2where e2 and e3 are the two principal lateral strains respectively. Inthe subsequent sections, keq is referred to as the equivalent conne-ment stiffness for brevity. While not explicitly given in the paper,Yu et al. [18] adopted the denition of equivalent plastic strain inABAQUS. In the present study, the following denition is adoptedfor the equivalent plastic strain increment so that the conne-ment-dependent characteristics incorporated into the model is lim-ited to the compressive zone:

d~epl d~eplmin 3

where ~eplmin is the principal plastic strain with the smallest algebraicvalue. This equivalent plastic strain increment is the same as thatdened in the CDPM model in ABAQUS for the compression zone.

In a slice model, the axial direction is always one of the princi-pal directions for both stresses and strains, and the other two prin-cipal directions (i.e. lateral directions) are always perpendicular tothe column axis. In addition, the principal stress and strain in theaxial direction are normally larger in magnitude than the principalstresses and strains in both lateral directions. In the specic casewhere the connement received by the concrete is uniform (e.g.FRP-conned circular concrete cylinders), e2 > 0 and e3 > 0 hold,and the keq given by Eq. (2) is always positive. For concrete undernon-uniform connement, however, e2 6 0 or e3 6 0 may happen,and this may lead to a negative or innite value of keq. For instance,when concrete is under uniaxial-strain compression, both e2 and e3are zero and rl,eff is positive, which results in an innite value ofkeq. To address this problem, the following assumptions wereadopted in implementing Yu et al.s model [18] in the presentstudy, without compromising the generality of the model for prac-tical applications:

(a) r2 and r3 in Eq. (1) are the two principal stresses with largeralgebraic values while e2 and e3 used in the ow rule are thetwo principal strains with larger algebraic values.

(b) r2 (or r3) is ignored in calculating rl,eff, if it becomes positive(i.e. r2 > 0 or r3 > 0), as the effect of tensile stresses on con-ned concrete is unclear.

(c) e2 + e3 > 0. If e2 + e3 is found to be negative or zero, it is takenas a very small positive value in determining the ow rule.

The purpose of these assumptions is to limit the revised part ofthe CDPM model to the compressive zone. In addition, two mate-rial parameters, the unconned concrete strength, f 0co, and thecorresponding axial strain, eco, are required to generate the inputparameters for Yu et al.s model [18].

It should also be noted that an analysis-oriented stressstrainmodel for concrete under uniform connement such as that ofJiang and Teng [3] was calibrated from experimental results ofFRP-conned circular concrete cylinders. The effect of endrestraints was generally not eliminated in the reported experimen-tal results. Therefore, such a stressstrain model has alreadyincluded the effect of end restraints implicitly. When this stressstrain model is used to generate material parameters for Yuet al.s model [18] in an FE slice model, the effect of end restraintsis also implicitly considered by the slice model. Due to this implicitconsideration, it is improper to use Jiang and Tengs model [3] togenerate material parameters for Yu et al.s model [18] when endrestraints have already been explicitly included in the FE model.

uctures 97 (2015) 1528 17Instead, a local analysis-oriented stressstrain model needs tobe found in which the effect of end restraints has been eliminated.This issue is addressed in detail in Section 3.

the normal practice for axial compression tests on FRP-conned

concrete cylinders, each with a diameter (D) of 150 mm and a

g Str2.2.1. FRP jacketsIn the FE models presented later, the FRP jacket is represented

using membrane elements and treated as a linear elastic brittlematerial. As the bers in the FRP jackets considered in the presentstudy were all oriented in the hoop direction, the Poissons ratio ofthe FRP jacket was set to zero. In addition, the no compressionoption available in ABAQUS was adopted to avoid the developmentof compressive stresses in the axial direction (i.e. the loading direc-tion). These modeling choices mean that the FRP jacket suppliesonly stiffness and strength in the hoop direction as a conningdevice in the FE model; the FRP jacket fails brittlely when its hooptensile rupture strain is reached. Perfect bonding between the FRPjacket and the concrete was assumed, and this was achieved bytying each of the 2-node axisymmetric membrane elementsrepresenting the FRP jacket to the adjacent 4-node solid elementfor concrete by means of the tie option in ABAQUS [20].

2.2.2. Steel reinforcementThe steel reinforcement in the FE model is represented using

either truss or membrane elements and is treated as an elastic-per-fectly plastic material; buckling of longitudinal steel bars underaxial compression is not considered. When modeled as an equiva-lent membrane, the membrane elements are subjected to biaxialstraining. To suppress the development of axial compressive stres-ses in the membrane elements, an orthotropic elastic model wasused to describe the elastic behavior of the equivalent membrane.The hoop direction was adopted as the principal material axis; themodulus of elasticity in the hoop direction was taken to be200 GPa which was assumed by the authors in their own analyticalworkwhile themodulus of elasticity in the axial direction and shearmodulus were assumed to have a very small value (e.g. 0.001 GPa);the Poissons ratio was set to zero. In addition, the von Mises yieldcriterionwith the associated ow rulewas used to describe the plas-tic behavior of the membrane elements. These modeling choicesmean that the transverse steel bars also supply only stiffness andstrength in the hoop direction as a conning device in the FEmodel.To achieve the assumed perfect bonding between the transversesteel bars and the concrete, the elements representing the trans-verse steel bars were embedded in the elements representing theconcrete by means of the embedded option available in ABAQUS[20]. The function of the embedded option is similar to that ofthe tie option: the former allows the embedding of lower-dimen-sional elements inside higher-dimensional elements (e.g. 3D ele-ments) while the latter is used between elements of the same order.

3. FRP-conned circular plain concrete columns

3.1. Local versus non-local stressstrain curves for concrete underuniform connement

A signicant issue that needs to be addressed for a 3D FE modelof a conned concrete column test specimen is the effect of endrestraints. Due to the restraints provided by the loading platens2.2. FRP and steel

The present study is focused on the performance of a concreteconstitutive model in the FE analysis of FRP-conned RC columns.FRP and steel are the other two materials which need to be consid-ered in the FE analysis. Therefore, constitutive models for thesetwo materials are also briey described here. In addition, a perfectbond is assumed between concrete and steel or FRP.

18 J.G. Teng et al. / Engineerinto the column ends, the column ends are not free to expand andmay be assumed to be fully prevented from lateral expansion. Asa result, the behavior of such a concrete column is non-uniformlength (L) of 300 mm, were considered. Yu et al.s model [18]described in the preceding section was adopted as the constitutivemodel for concrete. Based on the symmetry conditions of FRP-conned circular concrete cylinders, an axisymmetric model wasused and only half of the column height was included in the FEmodel. A 4-node axisymmetric solid element and a 2-node axisym-metric membrane element were employed to model the concreteand the FRP jacket respectively. Both the concrete and the FRPjacket had element sizes of about 6.25 mm, which was chosen onthe basis of a mesh convergence study. In all the FE models, axialdisplacements were uniformly imposed on the top surface of theconcrete cylinder until the maximum hoop strain in the FRP jacketreached its hoop tensile rupture value (i.e. 0.9%).

It was assumed for these numerical examples that the concretehas an unconned cylinder compressive strength, f 0co, of 40 MPaand an axial strain at peak stress, eco, of 0.0025; for the FRP jacket,its thickness, tfrp, modulus of elasticity, Efrp, and average tensilehoop rupture strain, eh,rup, are 0.34 mm, 240 GPa and 0.009,respectively.

3.3. Effect of end restraints

Fig. 1 shows that without the inclusion end restraints in the 3DFE model, the distributions of both axial displacements and stres-ses are uniform; however, when end restraints are included inthe 3D FE model, the distributions of both axial displacementsand axial stresses become highly non-uniform in both the axialand the radial directions. Fig. 2 shows that the 3D FE model with-circular concrete cylinders.As mentioned earlier, Yu et al.s model [18] relies on an analy-

sis-oriented stressstrain model for uniformly-conned concreteto generate the needed input data (e.g. [3]). When such a stressstrain model is used to derive the input parameters for a 3D FEmodel, it is implicitly assumed that this stressstrain model repre-sents the local behavior of a material point of concrete and doesnot reect the effect of end restraints. However, if this stressstrainmodel has been based on results of conventional tests on FRP-conned circular concrete cylinders with the presence of endrestraints, this assumption is violated (i.e. Jiang and Tengs model[3] is not a truly local stressstrain model). Therefore, when sucha stressstrain model is used to derive values for the input parame-ters, the end restraint effect is included twice: once through the 3Dmodeling and once through the use of a non-local stressstrainmodel. Obviously, to achieve an accurate 3D FE model for FRP-conned circular concrete cylinders, a truly local stressstrainmodel for material points in the 3D FE model is needed to generalvalues for the input parameters as explained later in this section.

3.2. 3D FE modeling of FRP-conned circular concrete cylinders

3D FE models were built to simulate the behavior of FRP-conned circular concrete cylinders using ABAQUS. Circularover the height. A slice model as that used by Yu et al. [18] canrepresent the average behavior of the mid-height region of sucha column, but cannot represent the deformation non-uniformityover height. Exactly because of this non-uniformity, in laboratorytests on circular concrete cylinders, it is commonly recommendedthat the axial compressive strain of concrete be based on the aver-age shortening in the mid-height region within a gauge length nomore than two thirds the height of the specimen [24]. This is also

uctures 97 (2015) 1528out end restraints reproduced the stressstrain curves predicted byJiang and Tengs model [3]. However, when end restraints wereadded, the FE analysis predicted a lower axial stress than Jiang

(a

g Strf'co=40MPa co=0.0025

Efrp=240GPa tfrp=0.34mm D=150mm L=300mm

D/2

w/o end restraints

J.G. Teng et al. / Engineerinand Tengs model [3] at the same hoop strain. In these gures, theaxial stress is the average value over the cylinder cross-section, thelateral strain is that obtained on the outer surface of the concretecylinder at mid-height (i.e. at the plane of symmetry), and the axialstrain is the axial displacement on the outer surface of the concretecylinder at a height of 60 mm divided by half of the gauge lengthwhich is equal to 60 mm in this case.

Figs. 1 and 2 show that end restraints have a negative effect onthe response of FRP-conned plain concrete cylinders in terms ofthe compressive strength and the strain capacity. This negativeeffect arises because end restraints result in non-uniform conne-ment, which was found to have an insignicant effect in increasingthe axial stress of FRP-conned concrete. By contrast, endrestraints prevent the FRP-conned concrete cylinder from lateralexpansion at the ends and lead to non-uniform straining of theFRP jacket over the height. When the FRP jacket reaches its tensilehoop rupture strain at the mid-height, its hoop strain is still muchsmaller away from the mid-height and can be taken to be zero atthe ends. As a result of this non-uniform straining of the jacket,the FRP-conned concrete cylinder reaches its ultimate axial stressearlier at a smaller value of axial strain. This explains why the FEmodel can reproduce the responses predicted by Jiang and Tengs(2007) model [3] when end restraints are not added, but doesnot do so when they are (see Fig. 2). In a 3D FE analysis, the effectof end restraints should be eliminated in the constitutive model for

(b

f'co=40MPa co=0.0025

Efrp=240GPa tfrp=0.34mm D=150mm L=300mm

D/2

w/o end restraints

Fig. 1. Distributions of axial displacements and axial stresses in axially-compressed FRP-cof axial displacements, (b) distribution of axial stresses.)

D/2

L/2

w/ end restraints

uctures 97 (2015) 1528 19conned concrete. To achieve this goal, a recalibration process isproposed in the next sub-section.

3.4. Recalibration of stressstrain model

To address the problem identied above, a local analysis-ori-ented stressstrain model for concrete under uniform connementis needed for determining the values of the input parameters. Thekey components in the stressstrain model of Teng et al. [22] orJiang and Teng [3] are the axial strain-lateral strain relationshipof concrete (i.e. the lateral strain equation) under various lateralconning pressures and the expressions dening the compressivestrength and the corresponding axial strain of actively-connedconcrete. In order to remove the effect of end restraints from thestressstrain model, these expressions were adjusted by a trial-and-error process. This process continued until the 3D FE modelincluding end restraints could reproduce the predictions of theoriginal stressstrain model (i.e. Jiang and Tengs [3] model inthe present study).

As a result of the recalibration, the lateral strain equationbecomes

ececo

1:05 10:75 eleco

0:7exp 7 el

eco

( ) 18 rl

f 0co

!4

)

D/2

L/2

w/ end restraints

onned circular concrete cylinders with and without end restraints. (a) Distribution

caused by end restraints. This underestimation is excluded fromthe local analysis-oriented stressstrain model, which leads to anactive-connement model capable of providing close predictionsfor actively-conned concrete. In addition, it is worth noting thatEq. (5) results in a K value equal to 0.699 instead of the originalvalue of 0.725 as suggested by Yu et al. [18].

Eqs. (4)(6), together with the stressstrain model of Popovics[27], form a new analysis-oriented stressstrain model which canbe considered as a local stressstrain model for concrete underuniform connement. Figs. 3 and 4 show the performance of the3D FE model with the input parameters produced by this new localstressstrain model. It is clear that by including the end restraintsin 3D modeling, the FE-model can very closely reproduce the pre-dictions of Jiang and Tengs model [3] (Fig. 3). In Fig. 4ad, the cir-cular plain concrete cylinders were wrapped with CFRP [28], GFRP[22], and Aramid FRP (AFRP) [29], respectively. Details of the speci-mens are provided in the gures. It is also evident that the 3D FEmodel including end restraints predicts the test results reasonablyclosely.

4. FRP-conned circular RC columns

4.1. Test columns

To further verify the capacity of the newly developed con-stitutive model, experimental results from [30] were chosen forcomparison with numerical results. In [30], a total of 21 large-scale

g Structures 97 (2015) 1528(a)

0

20

40

60

80

-0.015 -0.010 -0.005 0.000 0.005 0.010 0.015 0.020

Jiang and Teng (2007)FE w/o end restraintsFE w/ end restraints

55

60

65

7.5 8.0 8.5 9.0 9.5 10.0

h10-3

Strain

Axi

al st

ress

(MPa

)

f'co = 40 MPa co = 0.0025 Efrp = 240 GPa tfrp = 0.34 mm D = 150 mmL = 300 mm

-0.008

-0.006

-0.004

-0.002

0.0000 0.005 0.01 0.015 0.02

Jiang and Teng (2007)FE w/o end restraintsFE w/ end restraints

Lat

eral

stra

in

f'co = 40 MPa co = 0.0025 Efrp = 240 GPa tfrp = 0.34 mm D = 150 mmL = 300 mm

20 J.G. Teng et al. / Engineerinwhere ec and e1 are the axial strain and the lateral strain of concrete,and eco is the axial strain at the compressive strength of unconnedconcrete. In addition, the expression for the compressive strength ofactively-conned concrete becomes

f 0cc f 0co 4rl 5This change in the connement effectiveness coefcient also

leads to the following equation for the axial strain of concrete atecc:

ecceco

1 20 rlf 0co

!6

where ecc is the axial strain at the peak axial stress of concreteunder a lateral conning pressure, rl.

The only difference between the original lateral strain equationproposed by Teng et al. [22] (and employed by Jiang and Teng [3])and Eq. (4) is that a coefcient of 0.85 in the former has been chan-ged to 1.05 in the latter. Similarly, Eq. (5) is slightly different fromthe original equation proposed by Teng et al. [22] and employed byJiang and Teng [3], and it is closer to the equation of Richart et al.[1] with a connement effectiveness coefcient of 4.1. The changeof Eq. (6) from the equations used in [22] and [3] is a result of thechange in the connement effectiveness coefcient, following theapproach of Richart et al. [25] who used a factor of ve times theconnement effectiveness coefcient in the equation for the axialstrain at peak stress. As the effect of end restraints is included inJiang and Tengs model [3], an active-connement model whichslightly underestimates the peak axial stress [26] and thecorresponding axial strain is used to remedy the underestimation

circular RC columns (303 mm 1200 mm) were tested and for tenof them, stressstrain curves for the conned concrete were

(b)

-0.010

Axial strain

Fig. 2. Comparison between Jiang and Tengs analysis-oriented stressstrain modeland FE simulation with the input parameters derived from the same stressstrainmodel. (a) Axial stressaxial strain curves, (b) axial strainlateral strain curves.

0

70

80Jiang and Teng(a)

(b)

0.000 0.005 0.010 0.015 0.020Axial strain

-0.010

-0.009

-0.008

-0.007

-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0.0000.000 0.005 0.010 0.015 0.020

Jiang and Teng

FE model

Axial strain

Lat

eral

stra

in

f'co=40MPaco=0.0025

Efrp=240GPaD=150mmL=300mm10

20

30

40

50

60FE model tfrp=0.34mm

tfrp=0.17mm

Axi

al st

ress

(MPa

)

f'co=40MPaco=0.0025

Efrp=240GPaD=150mmL=300mmFig. 3. Recalibration of the analysis-oriented stressstrain model. (a) Axial stressaxial strain curves, (b) axial strainlateral strain curves.

60)

tfrp = 0.51mm 60

g Str(a)

0

10

20

30

40

50

60

70

80

90

-0.02 -0.01 0 0.01 0.02 0.03

Test

FE model

f'co = 38.9 MPa co = 0.0025

Efrp = 250 GPa D = 150 mmL = 300 mm

Axi

al st

ress

(MPa

)

Strain

tfrp = 0.33mm

20

30

40

50

60

70

Test

Axi

al st

ress

(MPa

)

f'co = 41.1 MPa co = 0.00256

Efrp = 250 GPa D = 150 mm

tfrp = 0.165mm

10

20L 300

30 T tEfrff prr 250 GPa 0 0025 40

50

60 t 0 33

70

80

90

20 TestD = 150 mm

0 00256 30

40

50

60

70

J.G. Teng et al. / Engineerinreported. These large-scaled RC columns were designed to examinethe connement provided by the transverse steel reinforcement,the FRP jacket, or both in combination. Moreover, two types oftransverse steel reinforcement, hoops and spirals, were used toexamine their corresponding connement effects. Each end ofthese specimens was locally strengthened using two curved steelplates covering a length of 300 mm to ensure that the failure ofthese specimens would occur at their mid-height. These steel jack-ets were not directly modeled in FE analysis; instead, their effectwas indirectly considered by the imposition of end restraints inFE analysis. Most of these specimens (18 out of 21) were testedwith load control, but three specimens (C2MP0C, C2MP2C andC2MP4C) were tested with displacement control. The diametersof all the longitudinal bars were 16 mm; other details of the speci-mens used for comparison can be found in Table. 1.

To obtain the stressstrain curves just for the conned concrete,the axial load carried by the longitudinal steel bars was removed bythe authors [30] from the total load carried by the whole column.For the 10 specimens whose stressstrain curves were reported,the axial load carried by the longitudinal steel bars was calculatedbased on the steel stressstrain curves obtained from theircorresponding tension tests. As the results of these tension testswere not reported by these authors, the following two steps wereused in the comparison process. First, a specimen labeled A5NP2Cwas simulated using a 3D FE model, in which the concrete wasmodeled using 3D solid elements, both longitudinal and transversesteel bars were model using truss elements, and the FRP jacket wasmodeled using membrane elements. As the stressstrain curves ofthe longitudinal bars of this specimen were unknown, an elastic-

(b)

0

10

-0.015 -0.01 -0.005 0 0.005 0.01 0.015

FE model

Strain

L = 300 mm 10

Fig. 4. FE predictions versus test results for FRP-conned circular plain concrete cylindconned concrete specimens from [28] with tfrp = 0.165 mm, (c) GFRP-conned concrete(c)

0

10

20

30

40

50

Test

FE model

Strain

Axi

al st

ress

(MPa

f'co = 39.6 MPa co = 0.00263

Efrp = 80.1 GPa D = 150 mmL = 300 mm

tfrp = 0.34 mm

40

60

80

100

TestAxi

al st

ress

(MPa

)t=0.169mm

t=0.338mm

f'co = 39.2 MPa co = 0.0028

-0.03 -0.02 -0.01 0 0.01 0.02 0.03

10

20frff pr

30

40

50

40

60 0 169

80

10070

80

70

80

uctures 97 (2015) 1528 21perfectly-plastic model was utilized in the FE analysis to considertheir contribution. The effect of the end restraints for such alarge-scale specimen was also examined during this step. Then,based on the conclusions obtained from this step, axisymmetricFE models with a height equal to half the steel hoop/spiral spacingwere adopted for the specimens to reduce the size of the FEmodels,and the predicted stressstrain results for conned concrete werecompared with experimental results except for specimen C4NP0C.The results of this specimen were not used for comparison due tothe following two reasons. First, the axial stress of specimenC4NP0C without FRP connement is even higher than that of speci-men C4NP4C which was conned with a 4-layer FRP jacket. Thisabnormally high axial stress casts doubt on the reliability of the testresults of this column. Secondly, as this specimen was tested withload control, the descending branch of its stressstrain curve wasnot well captured and may not be suitable for comparison withthe numerical results. The details of the FE model used for compar-ison are given in the following sub-sections.

For concrete, the material constants f 0co and eco are given inTable. 1. The yield stress of the longitudinal bars, fylh, the yieldstress of the transverse steel bars, fyh, and the elastic modulus ofthe FRP jacket, Efrp, are also summarized in Table. 1.

4.2. FE models

4.2.1. Three-dimensional FE analysis of specimen A5NP2CIn the FE model, due to geometric and loading symmetry, only

1/24 of this specimen (1/12 of the top half) was included. The

(d)

0

20FE model

Strain

Efrp = 115 GPa D = 150 mmL = 300 mm

-0.040 -0.020 0.000 0.020 0.040 0.060

20Efrff prr 115 GPa

ers. (a) CFRP-conned concrete specimens from [28] with tfrp = 0.33 mm, (b) CFRP-specimens from [22], (d) AFRP-conned concrete specimens from [29].

then an FE axisymmetric model can be used to reduce the com-putational effort. The ten specimens tested by Eid et al. [30] wereused to verify the capability of the FE model by comparing theexperimental stressstrain curves with those obtained from FEanalysis. Among these ten specimens, seven of them were rein-forced with steel spirals. The actual geometry of steel spirals inFE analysis leads to complicated models, which may weaken thereliability of the FE result. Therefore, previous researchers (e.g.[11,13]) have opted to use steel hoops to represent connementfrom steel spirals so that an axisymmetric FE model can be usedto reduce the computational effort; the steel spirals were replacedby equivalent steel hoops with the same steel spacing in such amodel. Mander et al. [4] suggested an equation to calculate the

Table 1Geometry and material properties of FRP-conned RC columns.

Specimen no. D (mm) c (mm) f0co (MPa) eco FRP comp

t (mm)

C4NP4C 303 25 31.7 0.002 1.524C2NP2C 303 25 31.7 0.002 0.762C4NP2C 303 25 31.7 0.002 0.762C2N1P2N 253 0 36 0.002 0.762A3NP2C 303 25 31.7 0.002 0.762C2MP4C 303 25 50.8 0.0024 1.524C2MP2C 303 25 50.8 0.0024 0.762B4NP2C 303 25 31.7 0.002 0.762A5NP2C 303 25 29.4 0.002 0.762

;

-0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008

Strain

Fig. 6. FE predictions and test results for specimen A5NP2C.

22 J.G. Teng et al. / Engineering Strelement sizes were determined based on a mesh converge study.The positions of the steel bars, the FRP jacket and the concreteare indicated in Fig. 5.

The periodic symmetry condition was imposed at the twoboundary planes perpendicular to the circumferential direction.This boundary condition was achieved by preventing the displace-ments in the circumferential direction of these two planes. In addi-tion, the mid-height horizontal plane, being a plane of symmetry,was restrained against vertical displacements. Furthermore, thetop plane was imposed with appropriate displacements to loadthe column in compression. Similar to the case of FRP-conned cir-cular concrete cylinders, the effect of end restraints on the overallaxial forcestrain behavior of specimen A5NP2C was investigated.

The axial forcestrain curves obtained from FE analysis with orwithout end restraints are given in Fig. 6, and these numericalresults are compared with the corresponding experimental resultsin the same gure. In this gure, the axial strain is for a mid-heightgauge length of 300 mm and the hoop strain is for the mid-heightof the column, which are the same as the experimental conditions.The curves obtained from FE analysis using different end restraintconditions are almost identical. The agreement between these twocases suggests that for such a relatively long column, the effect ofend restraints on their axial forcestrain behavior is negligible.Fig. 6 further indicates that although the FE model slightly under-estimates the axial force, the FE axial forcestrain curves correlateclosely with the experimental results. In addition, as end restraintshave negligible effects on the predicted behavior, periodic condi-tions of symmetry in the axial direction can also be applied andonly a column segment covering half the steel hoop spacing needsto be modeled in FE analysis.

C2MP0C 303 25 50.8 0.0024 0.0

Note: c = concrete cover; D = diameter of specimens; fyh = yield strength of steel bars4.2.2. Axisymmetric FE analysis of FRP-conned RC columnsThe presence of the longitudinal steel bars is not expected to

have any signicant effect on the behavior of the conned con-crete. If these longitudinal steel bars are ignored in the FE model,

Posion of steel bars

Mesh for the

concrete

Fig. 5. FE mesh of specimen A5NP2C and positions of steel bars.osite Transverse steel

Efrp (MPa) eh,rup Type fyh (MPa) S (mm) (mm)

78,000 0.0119 S 456 100 11.378,000 0.0059 S 456 65 11.378,000 0.0062 S 456 100 11.378,000 0.0084 S 456 65 11.378,000 0.0090 H 602 70 9.578,000 0.0107 S 456 65 11.378,000 0.0086 S 456 65 11.378,000 0.0104 H 456 100 11.378,000 0.0044 H 456 150 11.3 S 456 65 11.3

= diameter of steel bars; S = spirals; H = hoops.

0

500

1000

1500

2000

2500

3000

3500

4000

Test

FE w/ end restraints

FE w/o end restraints

Axi

al lo

ad (k

N)

uctures 97 (2015) 1528cross-sectional area, Aeq, for the equivalent steel hoops. The equa-tion is as follows:

Aeq KshAs 7where As is the cross-sectional area of the original steel spirals, andKsh is the conversion factor given by:

Ksh 11 s02ds8

where s0 is the clear spacing between two spiral bars and ds is thediameter of the spiral circle from bar center to bar center. The def-initions of the two parameters are also illustrated in Fig. 7.

Eq. (8) indicates that when the center-to-center diameter andthe spacing between steel bars are the same, steel spirals providemore effective connement than steel hoops. Fig. 8 shows theexperimental stressstrain curves of the conned concrete for bothspecimens C4NP2C and B4NP2C. The only difference betweenthese two specimens is the type of transverse steel reinforcement

used. These experimental results show the same phenomenon assuggested by Eq. (8). That is, the concrete conned by steel spiralshas a larger axial stress at a given axial or lateral strain than theconcrete conned by steel hoops.

Taking into account the symmetry features of the FE model asdiscussed above, an axisymmetric model of a column segment cov-ering half the steel hoop spacing was built with ABAQUS. The con-crete was modeled using 4-node solid elements; the transversesteel bars and the FRP jacket were both modeled using 2-nodeaxisymmetric membrane elements. A typical example of the FEmesh for the concrete is shown in Fig. 9 for specimen B4NP2C.

What should also be noted is that there is no axisymmetrictruss element (a ring element that carries only an axial force) inABAQUS, and thus the transverse steel bars were modeled usingthe axisymmetric membrane element. Each steel hoop was repre-sented by a steel membrane element with the same cross-sectionalarea: its height is equal to the radius of the steel bar Rts due to the

stressstrain curves for concrete conned only with the FRP jacket(curves labeled as FE w/o steel in Fig. 10) are also shown in thesegures to illustrate the contribution of transverse steel bars to con-nement. It is clear that the transverse steel bars have a signicanteffect on both the strength and the ductility of FRP-conned RC col-umns. The above comparisons verify the capability of the FEapproach in predicting the behavior of FRP-conned RC columnsalthough the improved concrete model itself was only recalibratedusing test results of FRP-conned circular plain concrete cylinders.

In Fig. 10ch, the stressstrain responses have an approxi-mately trilinear shape; this feature is different from that of thesmooth experimental stressstrain curves. In addition, the experi-mental axial stress in the transition region between the rstbranch and the second branch of the axial stressaxial strain curveis slightly underestimated by the FE model. It should be noted thatthe contribution of transverse steel reinforcement to connementin these specimens is generally substantial. This differencebetween experiments and FE analysis can be attributed to theexpression adopted for the compressive strength of actively-con-ned concrete (i.e. Eq. (4)) in the analysis-oriented stressstrainmodel [3] used to determine the material properties for the con-stitutive model for concrete. This linear equation was proposedfor simplicity at the expense of underestimating the compressivestrength of actively-conned concrete at low connement levels[3]; as a result, the axial stress of steel-conned concrete duringthe early stage of loading, when the level of connement is low,is also underestimated. The use of a more complex expression forthe compressive strength of actively-conned concrete is expectedto increase the accuracy of prediction of the FE model, but this willalso increase the complexity of the constitutive model for concrete.

Fig. 10e shows the stressstrain response of specimen C2MP2C.

J.G. Teng et al. / Engineering Strsymmetry condition at the mid-height and its thickness, tts,eq, canbe deduced from the following equation:

tts;eq AtsRts 9

where Ats is half of the cross-sectional area of the steel transversebar, Aeq.

For these axisymmetric FE models, the bottom horizontalboundary was restrained in the vertical direction to reect thesymmetry condition there and axial displacements were imposedon the top boundary to exert axial loading. For FRP-conned RCcolumns, axial displacements were uniformly imposed on the topsurface until the maximum hoop strain within the FRP jacketreached its hoop tensile rupture strain.

In Fig. 10, the FE stressstrain curves of the conned concreteobtainedwith the recalibrated plastic-damagemodel are comparedwith the test results [30]. Both axial stressaxial strain curves andaxial stresslateral strain curves are considered in the comparison.For the numerical results, the axial stress was obtained by dividingthe load carried by the concrete by the total cross-sectional area;the axial strain was calculated as the average value over the wholeheight of the FE model; and the hoop strain was obtained from theouter edge of the column where the lateral displacement is the lar-gest (i.e. at the middle point between the two steel hoops/spirals).Fig. 10ah show that the predicted stressstrain responses are gen-erally in reasonably close agreementwith the test results. These g-ures indicate the FE approach developed in this study can provideclose predictions for FRP-conned RC columns. Moreover, the FE

Cover concrete

Ineffectively confined part of concrete core

s s

ds-s/2

dsFig. 7. Spacing between steel hoops and arch action.0

10

20

30

40

50

60

-0.015 -0.01 -0.005 0 0.005 0.01 0.015

B4NP2C, steel hoops

C4NP2C, steel spirals

Strain

Axi

al st

ress

(MPa

)

Fig. 8. Effect of the type of transverse steel bars on connement.

D/2=151.5mm

S/2=50mm

Steel bar

Axis of symmetry FRP jacket

Fig. 9. Axisymmetric FE model for column B4NP2C.

uctures 97 (2015) 1528 23The axial stressstrain curve features a curvature change at anaxial strain of around 0.3%. This curvature change is also due tothe analysis-oriented stressstrain model [3] used to determine

(b)

(c)

0

5

10

15

20

25

30

35

40

45

50

-0.01 -0.005 0 0.005 0.01 0.015

Test

FE model

FE model w/o steel

Axi

al st

ress

(MPa

)

Strain

0

20

40

60

80

-0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02

Test

FE model

FE model w/o steel

Strain

Axi

al st

ress

(MPa

)

(a)

0

20

40

60

80

-0.02 -0.01 0 0.01 0.02 0.03

Test

FE model

FE model w/o steel

Strain

Axi

al st

ress

(MPa

)

(d)

0

20

40

60

80

100

-0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025

Test

FE model

FE model w/o steel

Strain

Axi

al st

ress

(MPa

)

(e)

0

20

40

60

80

-0.01 -0.005 0 0.005 0.01 0.015

Test

FE model

FE model w/o steel

Axi

al st

ress

(MPa

)

Strain

(f)

0

10

20

30

40

50

60

-0.01 -0.005 0 0.005 0.01 0.015

Test

FE model

FE model w/o steel

Strain

Axi

al st

ress

(MPa

)

(g)

0

10

20

30

40

50

60

-0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02

Test

FE model

FE model w/o steel

Strain

Axi

al st

ress

(MPa

)

(h)

0

10

20

30

40

50

60

-0.015 -0.01 -0.005 0 0.005 0.01 0.015

Test

FE model

FE model w/o steel

Strain

Axi

al st

ress

(MPa

)

Fig. 10. Comparison of stressstrain behavior for FRP-conned RC column. (a) Specimen C4NP4C, (b) specimen C4NP2C, (c) specimen C2N1P2N, (d) specimen C2MP4C, (d)specimen C2MP2C, (f) specimen C2NP2C, (g) specimen A3NP2C, (h) specimen B4NP2C.

24 J.G. Teng et al. / Engineering Structures 97 (2015) 1528

to the discrete locations of steel hoops, the connement provided

Fig. 11 shows that the FE model led to smaller axial stressesthan both the test results and the empirical analytical results. Inthe Yu et al.s model [18], K and a are the two parameters that con-trol the connement effectiveness for concrete under non-uniformconnement. As mentioned earlier, the value of K is derived fromthe strength ratio between equal biaxial compression and tri-axialcompression. Therefore, when Eq. (5) is already given, changing the

value of K leads to changes in the value of the f0b

f 0coratio, but this ratio

has been assigned a xed value of 1.16 [32] in the yield function ofthe CDPMmodel as adopted in the present study. According to testresults, this ratio varies from 1.16 to 1.2. Changing the ratio withinthis small range has little effect on the FE predictions. Therefore, itis better to treat K as a constant rather than a variable. If K is keptconstant, only a may be recalibrated to provide closer predictionsof the test results. The stressstrain curves obtained with a recali-brated value for a are shown in the same gure (labeled as FE

g Structures 97 (2015) 1528 25by steel hoops to the concrete in an RC column is highly non-uni-form over the height; this differs from the connement to concreteprovided by an FRP jacket. To model this non-uniform conne-ment, two different approaches have previously been explored[2,4,33]. Besides the FE approach, the effective connementmethod (e.g. [4]) has also been used to estimate the amount of lat-eral connement acting on the concrete. The reduced connementeffect between the upper steel hoop and the lower steel hoop isconsidered through the arching action assumption: a parabolawith an initial slope of 45 covering a clear vertical spacing, s0,between steel hoops or spirals (see Fig. 7) is dened to separatethe effectively conned part from the ineffectively conned partof the concrete core [2]. The smallest cross-sectional area of theeffectively conned part, Ae, is thus located at the mid-height ofthe gap between two adjacent steel hoops [4]. In the studies ofMander et al. [4] and Saadatmanesh et al. [31], it was assumed thatthe ineffectively conned concrete annulus has a radial thicknessof a quarter of the clear vertical spacing between the two adjacentsteel hoops (i.e. s

0/4) and receives no connement, while the effec-

tively conned part within the area Ae is uniformly conned (alsoshown Fig. 7). The area Ae is given by

Ae p4 ds s0

2

210

In Fig. 11, the axial stressstrain curves of specimen C2MP0Cobtained from the above empirical model, the present FE modeland the test results [30] are compared. It should be noted that,the material parameters of the plastic-damage model for concrete.Such analysis-oriented stressstrain models typically predict axialstressstrain curves with a curvature change when the normalizedconnement stiffness of the FRP jacket is small [3]. When thesemodels are used to determine the material parameters for the plas-tic-damage model, the predicted axial stressstrain curve has thesame feature. Indeed, this curvature change can also been notedin the axial stressaxial strain curve for FRP-conned concreteshown in the same gure (i.e. the curve labeled as FE model w/o steel), although here it is not so obvious.

Fig. 10h shows the stressstrain response of specimen B4NP2C.For the early stage of the stressstrain response, the axial stressobtained from the FE model without considering the effect oftransverse steel bars is larger than that obtained from the FE modelconsidering this effect, which implies a negative effect of the steelconnement and is counterintuitive. Indeed, end restraints werealso found to have such a negative effect as discussed in the pre-vious section. In both cases, the end restraints or the transversesteel bars not only provide additional connement to the column,but also lead to non-uniform distributions of stresses and deforma-tion in the column. As a result, the connement provided by theFRP jacket becomes non-uniform, which weakens the overall bene-t of FRP connement for the column. When the additional con-nement is sufciently small, the detrimental effect ofconnement non-uniformity exceeds the benecial effect of addi-tional connement from end restraints/transfer steel bars, andthe net overall effect is negative.

5. Connement mechanism in RC columns

In specimen C2MP0C, the cover concrete does not receive anyconnement and the concrete core (the concrete contained bythe circular cylindrical surface on which the centerlines of steelhoops lie) receives connement only from the steel hoops. Due

J.G. Teng et al. / Engineerinfor a steel-conned RC column without FRP connement, onlythe stressstrain behavior of the concrete core is of interest.Therefore, only the concrete core was considered in the FE analysis.model-II). In this recalibrated FE model, the value of a was givena value of 0.1 to achieve closer predictions of the test results. Ingeneral, Yu et al.s model may be used in one of two ways in theFE analysis of steel-conned RC columns: (1) if the model is usedto make predictions, a should be taken as a constant and thedefault value of 0.039 suggested by Yu et al. [18] should be used;(2) if the model is used to explain experimental results, a can berecalibrated to closely reproduce the experimental results.

Although there are differences between the numerical predic-tions (including both FE model and FE model-II) and theempirical analytical predictions, the test results fall between theFE and the analytical results, and are in good agreement with thesetwo types of predictions. Similar to the FE models, the empiricalanalytical model provides close predictions of stressstrain curves.However, the empirical analytical model does not account explic-itly for variations in the axial stress and the connement pressureover the cross-section and the height of the column. To investigatethe connement mechanism of transverse steel bars, the actualstress distributions obtained from FE analysis are illustrated inFigs. 1214. Fig. 12 shows the distribution of axial stress withinthe concrete core when the peak axial stress is reached. This gureindicates that within a section close to the level of a steel hoop, theconcrete stress achieves its largest enhancement near the outeredge, and the effect of this enhancement decreases in the heightdirection away from the steel bar. This variation of axial stress issimilar to the assumption of the arching action.

The FE predictions for the axial stress, however, differ from thearching action assumption. With the arching action assumption,the cross-section at the height of the steel hoop centerline isassumed to be uniformly stressed and is considered to be

0

20

40

60

80

-0.015 -0.010 -0.005 0.000 0.005 0.010 0.015 0.020

TestFE modelEmpirical modelFE model-II

Strain

Axi

al st

ress

(MPa

)Fig. 11. Comparison of stressstrain behavior for steel-conned RC columns(specimen C2MP0C).

effectively conned, while the FE analysis predicts a large stressnear the steel hoop, a much smaller stress at a small distance awayfrom the steel hoop, and nally a moderate stress near the axis ofaxisymmetry. The FE distribution is obviously more reasonable asthe FE analysis takes into account the balance of axial stresses inthe height direction, which is neglected by the arch action assump-tion. In addition, the FE results indicate that a non-uniform zone ofstress exists near the outer edge of the concrete core but within acertain radius, there is a zone of uniformly distributed axial stress;this distribution is similar in nature to the result from elasticanalysis conducted by Eid and Dancygier [33]. Zones of uniformlydistributed stress can also be observed in the distribution of con-ning pressure (Figs. 13 and 14). Fig. 13 shows that the largestvalue of conning pressure (radial stress) occurs adjacent to thesteel bar on the outer edge of the concrete core and the smallestvalue occurs at the mid-height plane on the outer edge of the con-crete core. Unlike the conning pressure, the largest stress in the

hoop direction appears at a plane between the steel hoop andthe mid-height plane. The different distributions of radial and hoopstresses result in a complicated axial stress distribution near theouter edge of the concrete core. Figs. 13 and 14 indicate that, azone of uniformly distributed radial stresses exists within a certainradius. This zone results in uniformly distributed axial stresses. Theexistence of a uniformly-stressed concrete zone is an importantreason why the empirical analytical approach based on the archingaction assumption shows good accuracy in predicting the responseof steel-conned concrete columns although it overlooks the bal-ance of axial stresses. As long as the radius of the effectively-con-ned concrete zone predicted by the empirical analytical approachis not too far from the actual radius of uniformly conned concrete,the results predicted by the empirical approach can be quiteaccurate.

To further illustrate the stress distributions, Figs. 1517 showrespectively the predicted distributions of axial, radial and hoop

Section-III

Section-II

Section-I

Fig. 12. Distribution of axial stress in a steel-conned concrete column.

n fo

26 J.G. Teng et al. / Engineering Structures 97 (2015) 1528Fig. 13. Distribution of stress in the radial directioFig. 14. Distribution of hoop stress inr transverse steel bar conned concrete columns.a steel-conned concrete column.

g Str65

70

75

80

85

90

Section-I

Section-II

Section-III

ial s

tres

s (M

Pa)

J.G. Teng et al. / Engineerinstresses in the radial direction at three selected heights. The threeselected heights are indicated in Fig. 12. It is again obvious that alarge zone of uniform stresses exists near the column center. Thewidth of this uniform stress zone is about 2/3 of the column radius,which is similar to the result of Eid and Dancygiers [33] elasticanalysis. Within this zone, all the three stress components showinsignicant variations. However, near the outer edge, the stressesvary much more rapidly due to the inuence of arching action. Thecapability of predicting stress variations over the whole column isone of the advantages of this 3D FE model over a slice model asemployed by Yu et al. [18].

50

55

60

0 20 40 60 80 100 120 140Distance from the center (mm)

Ax

Fig. 15. Radial distributions of axial stress in a steel-conned concrete column.

-5

0

5

10

15

20

25

30

35

40

0 50 100 150

Section-I

Section-II

Section-III

Rad

ial s

tres

s (M

Pa)

Distance from the center (mm)

Fig. 16. Radial distributions of radial stress in a steel-conned concrete column.

-5

0

5

10

15

20

25

30

0 50 100 150

Section-I

Section-II

Section-III

Distance from the center (mm)

Hoo

p st

ress

(MPa

)

Fig. 17. Radial distributions of hoop stress in a steel-conned concrete column.6. Conclusions

A three-dimensional FE approach for modeling the behavior ofFRP-conned circular plain concrete cylinders and RC columnsbased on Yu et al.s plastic-damage model [18] has been presentedin this paper. In implementing Yu et al.s plastic-damage model[18] for concrete, a stressstrain model for concrete under uniformconnement is needed to determine values for some of the inputparameters. The FE approach is capable of modeling non-uniformdeformation in the axial direction due to factors such as endrestraints and discrete transverse steel reinforcement. Numericalresults obtained with the FE approach have revealed that endrestraints lead to a smaller axial strain at a given lateral strain.This effect necessitates the use of a local stressstrain model ingenerating input parameters for the concrete constitutive modelto achieve precise predictions for FRP-conned short circular con-crete cylinders, but this issue becomes less important for longerspecimens and RC columns. With the use of a local stressstrainmodel, the proposed FE approach has been shown to provide accu-rate predictions for both FRP-conned RC columns and steel-con-ned RC columns although the concrete model was onlycalibrated using results of FRP-conned circular concrete cylinders.

The veried FE approach was used to examine the rationalebehind the method of effective connement widely used in theanalytical modeling of steel-conned concrete columns. The stressdistributions in the concrete core conned by transverse steelhoops obtained using the FE approach shed considerable light onthe connement mechanism and are in general in support of thearching action assumption adopted by the widely used empiricalanalytical model. The detailed FE results presented for steel-con-ned concrete columns have also demonstrated that the presentFE approach offers a useful tool for the exploration of complex con-nement mechanisms in various conned concrete columns in thedevelopment of simple stressstrain models for design purposes.For example, the 3D nite element approach can be applied to studyconnement mechanisms in FRP-conne rectangular RC columns.

Acknowledgements

The authors acknowledge the nancial support received fromthe Research Grants Council of the Hong Kong SpecialAdministrative Region (Project No: PolyU 5285/10E) and theNational Basic Research Program of China (973 Program) (ProjectNo. 2012CB026201).

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28 J.G. Teng et al. / Engineering Structures 97 (2015) 1528

Three-dimensional finite element analysis of reinforced concrete columns with FRP and/or steel confinement1 Introduction2 Constitutive models2.1 Yu et al.s model for confined concrete and refinement2.2 FRP and steel2.2.1 FRP jackets2.2.2 Steel reinforcement

3 FRP-confined circular plain concrete columns3.1 Local versus non-local stressstrain curves for concrete under uniform confinement3.2 3D FE modeling of FRP-confined circular concrete cylinders3.3 Effect of end restraints3.4 Recalibration of stressstrain model

4 FRP-confined circular RC columns4.1 Test columns4.2 FE models4.2.1 Three-dimensional FE analysis of specimen A5NP2C4.2.2 Axisymmetric FE analysis of FRP-confined RC columns

5 Confinement mechanism in RC columns6 ConclusionsAcknowledgementsReferences