Model for Analysis of Short Columns of Concrete Confined by Fiber-Reinforced Polymer

t columnsnfinedsecant axialr confinedt describesalidate the

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Model for Analysis of Short Columns of ConcreteConfined by Fiber-Reinforced Polymer

Severino Pereira Cavalcanti Marques1; Dilze Coda dos Santos Cavalcanti Marques2;Jefferson Lins da Silva3; and Marcio Andre Araujo Cavalcante4

Abstract: This paper presents a numerical model for evaluating the behavior of axially loaded rectangular and cylindrical shorof concrete confined by fiber-reinforced polymer~FRP! composites. The proposed formulation considers, for unconfined and cocompressed concrete, a uniaxial constitutive relation that utilizes the area strain as a parameter of measure of the materialstiffness. For unconfined concrete, the model adopts an explicit relationship between axial strain and lateral strain, while foconcrete, an implicit relation is considered. For this last case, the model employs a simple iterative-incremental approach thathe entire stress-strain response of the columns. The behavior of the FRP is considered linear elastic until the rupture. To vmodel, a number of columns were analyzed and the numerical results were compared with experimental values published by otThis comparison between experimental and numerical results indicates that the model provides satisfactory predictions of the sresponse of the columns.

DOI: 10.1061/~ASCE!1090-0268~2004!8:4~332!

CE Database subject headings: Concrete columns; Fiber reinforced polymers; Composite materials; Axial loads; ConfinConstitutive relations.

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Introduction

The effects of confinement on the behavior of concrete havestudied for many years. Many studies have demonstrated theral confinement in columns increases the compressive streductility, and energy absorption capacity of the concrete.practical applications, the confinement has been introdthrough conventional reinforcement of steel or by wrappingconcrete with steel jackets and FRP sheets or by filling fireinforced polymer~FRP! tubes with concrete. The use of FRPa material for confining columns seems very interesting duFRP’s mechanical and chemical properties, easiness of aption, and capacity of confinement.

Due to renewed interest in the behavior of confined concrenumber of theoretical and experimental works have beenlished in recent years, many of which are concerned with con

1Professor of Civil Engineering, Dept. de Engenharia Estrutural,versidade Federal de Alagoas, Rod. BR 104, 57072-970, Macei´-AL,Brasil. E-mail: [email protected]

2Professor of Civil Engineering, Dept. de Engenharia Estrutural,versidade Federal de Alagoas, Rod. BR 104, 57072-970, Macei´-AL,Brasil. E-mail: [email protected]

3Student of Civil Engineering, Universidade Federal de Alagoas,BR 104, 57072-970, Maceio´-AL, Brasil. E-mail: [email protected]

4Student of Civil Engineering, Universidade Federal de AlagRod. BR 104, 57072-970, Maceio´-AL, Brasil. E-mail: [email protected]

Note. Discussion open until January 1, 2005. Separate discumust be submitted for individual papers. To extend the closing daone month, a written request must be filed with the ASCE ManaEditor. The manuscript for this paper was submitted for review andsible publication on September 4, 2002; approved on August 21,This paper is part of theJournal of Composites for Construction, Vol. 8,No. 4, August 1, 2004. ©ASCE, ISSN 1090-0268/200

332 / JOURNAL OF COMPOSITES FOR CONSTRUCTION © ASCE / JULY/A

,

confined by transverse reinforcement, such as in the case oventional reinforced concrete columns. For these cases, sanalytical models are available in the literature~Mander et al1988; Cusson and Paultre 1995; Razvi and Saatcioglu 1!.Such models are frequently used to describe the behavior ocrete confined by other devices, including steel jackets andcomposites. However, previous studies have demonstratemodels developed for transverse reinforcement may not beservative for specimens confined by composite materials~Mirmi-ran and Shahawy 1997!. Recently, several models were propofor describing the behavior of FRP-confined concrete. Mosthese models are applicable to the cylindrical specimens~Samaanet al. 1998; Saafi et al. 1999; Spoelstra and Monti 1999; Tou1999; Fam and Rizkalla 2001!. Few models were formulated fthe case of specimens with rectangular cross sections~Rocheteand Labossie`re 2000; Wang and Restrepo 2001!, for which theconfinement effects are smaller than for circular cross sectio

This paper presents a numerical model for evaluating thhavior of axially loaded rectangular and cylindrical short coluof concrete confined by FRP composites. The proceduressider, for both unconfined and confined compressed concruniaxial constitutive relation that utilizes the area strain as arameter of measure of the material secant axial stiffness. Foconfined concrete, the model adopts an explicit relationshiptween axial strain and lateral strain, while for confined concan implicit relation is considered. For the last case, the memploys a simple iterative-incremental approach that provideentire stress-strain response of the columns. The behaviorFRP is considered linear elastic until the rupture. To validatemodel, a number of columns were analyzed and the numresults were compared with experimental values publisheother authors. This comparison between experimental anmerical results indicates that the model provides satisfactory

UGUST 2004

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The mechanical properties of concrete, such as strength andness, are influenced directly by microstructural aspects of theterial. For example, the system of voids in the microstructurconcrete, which defines its porosity, has a great influence onproperties. The porosity of concrete depends on factors relathe concrete’s material properties, such as water-cement ration the level of internal cracking. The internal cracks can beduced by rheological phenomena of the material, such as sage of the cement paste, and by external loading. In generamaterial stiffness is reduced when the degree of porosity omaterial is increased, whether this increase has been inducnatural or mechanical means~Pantazopoulou and Mills 1995!.When a concrete specimen is subject to an increasing unload, the existent cracks grow and new cracks appear in theterial, indicating a reduction of the axial stiffness.

In general, during the first stage of the loading, the reducof the stiffness is small and hence the axial stress versusstrain behavior is closely linear. In this initial stage, the specipresents a lateral strain that can be considered proportionalaxial strain. For subsequent stages of loading, the internal cing of the material grows at a faster rate, inducing nonlineahavior of the axial stress versus axial strain curve. In these sfor unconfined concrete, the Poisson’s ratio is always increaand the secant modulus of the material is always decreasingthe increase of the axial strain.

In the case of confined concrete, the presence of a 3Dpressive state of stress has a favorable effect on the strengstiffness of the material. Confinement is known to delay theof strength and stiffness of the concrete. This can be explainthe lateral kinematics restraint due to the confinement pressThat restraint tends to prevent volumetric dilation and to keeconcrete fragments together. Hence, confinement contributethe reduction of the rate of increase of the internal microcracof the concrete.

This paper assumes that the area strain of the compressecrete specimen is directly related to the change of porosity.area strain, however, is adopted as a control parameter of thestiffness of the material~Pantazopoulou and Mills 1995!. Thisproposition is considered valid for unconfined as well as conconcrete.

Proposed Axial Constitutive Model

Fig. 1 illustrates typical stress versus strain curves for uniax

Fig. 1. Typical stress versus strain curves for uniaxially loaunconfined concrete

loaded unconfined concrete. The curve to the right represents the

JOURNAL OF COMP

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axial stress (f c) versus axial strain («c) response, whereas tcurve to the left shows the axial stress versus lateral strain« l)relationship. The lateral strain is approximately proportionathe axial strain during the initial loading stage. This is valid uthe axial strain reaches the value«c

lim . For axial strain valuegreater than«c

lim , the existing level of cracking of the concreteseen as inducing nonlinear effects in the behavior of the matBased on the model proposed by Pantazopoulou and Mills~1995!,the relations between axial strain and lateral strain can bscribed by

« l5n«c for «c<«clim (1)

« l5n«c1122n

2a«c8S «c2«c

lim

«c82«climD 2

for «c.«clim (2)

wheren and«c85Poisson’s ratio and the axial strain at maximstressf c8 of the unconfined concrete, respectively. The parama is defined so thata«c8 represents the axial strain at the instanzero volumetric strain. As described by Pantazopoulou and~1995!, a approaches unity when the nominal strength of thecrete increases.

Based on the assumption that the secant elasticity moduthe concrete under uniaxial compressive loading is reducedthe area strain«A is increased, the following constitutive relatis proposed:

f c5Ec

11b«Ac

«c (3)

whereEc5initial elasticity modulus. The parametersb andc areconstants of the material that depend only on the unconfinedcrete properties (Ec , f c8 , «c8 , n, and a!. Spoelstra and Mon~1999! proposed a similar constitutive model withc51 and bevaluated by the condition that («c8 , f c8) is a point of the axiastress versus axial strain curve of the unconfined concrete.ever, this last model does not guarantee that the point («c8 , f c8)corresponds to the peak of the curve. On the other handmodel proposed in this paper provides a more appropriatestress versus axial strain curve by assuming that when«c5«c8 ,the conditionsf c5 f c8 and d fc /d«c50 are satisfied in ordercalculate the constantsb andc. This is shown in Fig. 2 for unconfined concrete withf c8560 MPa. Based on these conditio

Fig. 2. Axial stress versus axial strain curves for unconfined con

the following expressions are achieved:

OSITES FOR CONSTRUCTION © ASCE / JULY/AUGUST 2004 / 333

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b5Ec«c82 f c8

f c8~«c8!2c (4)

c51

2

Ec«c8

~Ec«c82 f c8!

~«c82«clim!

@~12n!«c82n«clim#

(5)

In order to simplify, the valuea51 was adopted in the derivatiof Eqs.~4! and ~5!. The model assumes«c

lim50.001 andn50.2.The values of the lateral strain given by Eqs.~1! and ~2! are

only valid for unconfined concrete. In the case of confinedcrete, the lateral strain also depends on the lateral confinepressure, and then the problem is more complicated. For colwrapped with FRP or other materials, the lateral pressure offinement is a function of the lateral strain, which depends onthe axial strain and stiffness of the confining devices. In thisit is not possible to express the lateral strain as explicitly a ftion of the axial strain. For confined concrete, this study useiterative model that considers an implicit relationship betwaxial strain and lateral strain. Such a model is based on Spoand Monti~1999! and allows determination of the lateral pressof confinement for each value of the axial strain. The applicaof the model requires the use of a constitutive relation thascribes the response of the concrete subjected to both a vaaxial compressive stress and a fixed lateral pressure. Herstudy employs Popovics’s well-known equation~Popovics 1973!,which was suggested by Mander et al.~1988! for the case oconfined concrete, given by

f c5

f cc8 S «c

«cc8D r

r 211S «c

«cc8D r (6)

where

r 5Ec

Ec2Escc; Escc5

f cc8

«cc8(7)

and f cc8 and «cc8 5confined peak strength and the corresponstrain, respectively.

The peak stress of the confined concrete depends on theof the lateral confinement pressuref l . Hence, for the casepassive confinement, as for columns wrapped with FRP,f cc8 variesduring the loading stage. To predict the peak stress and theresponding strain, several models are proposed by differenthors ~Mander et al. 1988; Cusson and Paultre 1995; Kono e

Fig. 3. Lateral forces produced by confinement with circular jac

1998; Razvi and Saatcioglu 1999!. Most of them were developed

334 / JOURNAL OF COMPOSITES FOR CONSTRUCTION © ASCE / JULY/A

for cylinders of concrete confined by spirals and hoops. Ineral, such models consider the strengthf cc8 as a function of thlateral pressures and of the unconfined concrete strength.

Several failure models were implemented in the modelposed in this work. Taking the numerical results into accoRazvi’s model~Razvi and Saaticioglu 1999! and Kono’s mode~Kono et al. 1998! were chosen for the analyses of cylindrcolumns and rectangular columns, respectively. For cylindcolumns, Razvi’s model can be given by

f cc8 5 f c81k1f l (8)

«cc8 5«c8S 115k1k3

f l

f c8D (9)

k156.7f l20.17; k35

40

f c8<1 (10)

In Eq. ~10!, f l and f c8 are expressed in megapascals. AccordinKono’s model, the peak stress and the corresponding straevaluated by

f cc8 5 f c8~110.0572f l ! (11)

«cc8 5«c8~110.280f l ! (12)

For most of the analyzed cases, the well-known failure mproposed by Mander et al.~1988! overestimated the effects of tconfinement, especially for the columns with rectangular csection.

Fig. 4. Arching action of rectangular columns with fiber-reinforpolymer jacket

Fig. 5. Lateral forces produced by confinement with rectangjacket

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When the experimental values ofEc and«c8 are unknown, thmodel adopts the following relationships proposed, respectby Carrasquillo et al.~1981! and Collins and Mitchell~1991!:

Ec53320Af c816900 (13)

«c85f c8

Ec

n

n21(14)

wheren50.81 f c8/17 andf c8 is expressed in megapascals.

Fig. 6. Iterative procedure of determination of axial strain veaxial stress

Table 1. Data for Applying Analytical Model in Cylindrical Short

ReferenceSpecimennumber Type of fiber

Toutanji ~1999! GE Glass

C1 CarbonC5 Carbon

Saafi et al.~1999! GE1 Glass

GE2 GlassC1 CarbonC2 Carbon

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Evaluation of Lateral Pressure of Confinement

Columns with Circular Cross Section

For the case of concentrically loaded cylindrical columns,lateral pressure of confinement can be evaluated by simplesiderations of equilibrium of force and compatibility of strbetween the concrete surface and the FRP composite. As thestress increases, the corresponding lateral strain increases aconfining jacket develops a tensile hoop stressf j balanced by thradial pressuref l , which reacts against the concrete lateral dtion ~Fig. 3!. By equilibrium considerations, the following eqution can be derived:

f l52t

Df j (15)

wheret5thickness of the jacket; andD5diameter of the concrecore. Considering the material of the jacket as linear elastictaking into account that the concrete radial strain is equal tjacket circumferential strain, the lateral pressure of confinecan be obtained by

f l52t

DEj« l (16)

whereEj represents the Young’s modulus of the FRP compo

Columns with Rectangular Cross Section

In the case of rectangular columns under concentric loadingconfinement action is more concentrated in regions close t

Fig. 7. Stress-strain curve of specimen GE tested by Toutanji~1999!

ns

a!D

~mm!t

~mm!Ej

~GPa!f r

~MPa!

1 76.0 0.24 72.6 1,5

1 76.0 0.22 230.5 3,41 76.0 0.33 372.8 2,95 152.4 0.80 32.0 4

5 152.4 1.60 34.0 55 152.4 0.11 367.0 3,35 152.4 0.23 390.0 3,5

Colum

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3

333

333

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corners. Fig. 4 shows the arching action, which takes place icross section, defining regions of unconfined concrete andgion of effectively confined concrete. It is assumed that themay be considered as a second degree parabola making anu with the edge of the rectangular section~Fig. 4!. Based onexperimental results for reinforced concrete columns conwith ties ~Sheikh and Uzumeri 1982!, the model assumesu545°.The area of concrete effectively confined by the jacket is givefollows:

Acc5axay24r c22pr c

221

3~wx

21wy2!tanu (17)

whereax anday5overall dimensions of the concrete core parato thex andy directions, respectively;wx andwy5widths of thestraight portion of the column’s sides in thex and y directionsrespectively; andr c5radius of the concrete column’s corners. Ttotal area of unconfined concrete is given by

Acu51

3~wx

21wy2!tanu (18)

The lateral pressures acting on the concrete core vary alonperimeter of the jacket, with their values increasing towardcorners. Based on the equilibrium of the lateral pressures athe tensile forces acting on the wall of the jacket~Fig. 5!, theaverage lateral pressures in thex and y directions, respectivelcan be found as

Fig. 8. Stress-strain curve of specimen C1 tested by Toutanji~1999!

Fig. 9. Stress-strain curve of specimen C5 tested by Toutanji~1999!

336 / JOURNAL OF COMPOSITES FOR CONSTRUCTION © ASCE / JULY/A

ef l ,x52

t

ayf j (19)

f l ,y52t

axf j (20)

where f j5tensile stress acting on the walls of the jacket.model assumes that the value of the stressf j is constant along thperimeter of the confining device and is obtained as a functiothe transverse strain« l .

To evaluate the peak stress and the corresponding strainEqs. ~11! and ~12!, f l is replaced by the average lateral strwhich is obtained by dividing the lateral resultant forces onjacket walls by the perimeter of the jacket cross section aslows:

f l5f l ,xay1 f l ,yax

ax1ay(21)

Numerical Procedures of Model

For determining the relation between the axial stress andstrain of the confined concrete, the model utilizes a simiterative-incremental approach shown in Fig. 6, which is base

Fig. 10. Stress-strain curve of specimen GE1 tested by Saafi~1999!

Fig. 11. Stress-strain curve of specimen GE2 tested by Saafi~1999!

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Spoelstra and Monti~1999!. During the analysis, a constantcrement of the axial strain,D«c , is adopted. Each incremenstep is concluded when the difference between the area straconsecutive iterations and the value of the area strain of thevious iteration is smaller than a tolerance value~suggested tequal 1026). The superscript~i! of the variables that appearFig. 6 represents the number of incremental steps. The analconcluded wheni . i max, where i max is the previously adoptetotal number of incremental steps, or when the rupture strathe FRP is attained in the hoop direction. Based on the asstion that all fibers of the jacket are oriented in the hoop direcand the load is applied only to the concrete core, the modelnot consider the effects of the axial strain, developed by thewith concrete, on the rupture strain of the FRP in the hoop dtion. For concrete columns confined by fiber-reinforced polytubes, Fam and Rizkalla~2001! showed that such effects are iportant when the axial load is applied to both the core andtube.

The value of the axial load, correspondent to the incremstepi, is obtained by the following equation:

P~ i !5 f cu~ i !Acu1 f c

~ i !Acc (22)

wheref cu( i ) and f c

( i ) are, respectively, the axial stress for the uncfined concrete and the axial stress for the confined concretecorresponding to the axial strain«c

( i ) . The areasAcc andAcu aregiven by Eqs.~17! and~18!, respectively. In the case of columwith a circular cross section, the first term on the right side of~22! does not exist.

Fig. 12. Stress-strain curve of specimen C1 tested by Saafi~1999!

Table 2. Data for Applying Analytical Model in Rectangular Sho

ReferenceSpecimennumber

Type offiber

f~MP

Rochette and Labossie`re ~2000! S25-C3 Carbon 4

S25-C4 Carbon 4S38-C3 Carbon 4

Mirmiran et al.~2000! 6 Plies Glass 4

10 Plies Glass 414 Plies Glass 4

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f

Verification of Proposed Model

Results of Cylindrical Columns

To verify the performance of the proposed model, severalcylindrical columns were analyzed. Table 1 shows a summathe properties of cylindrical columns tested by Toutanji~1999!and Saafi et al.~1999!. The parameterf r represents the ruptustress of the composite. Toutanji~1999! presents experimenresults of cylindrical specimens wrapped with two laps of unrectional FRP sheets, which were bonded to the concrete wtype of epoxy system. In these last specimens, three types owere used: two carbon~C1 and C5! and one glass~GE!. Saafiet al.~1999! tested several cylindrical columns with concrete cfined by FRP tubes. The FRP tubes were made of glass~GE1,GE2! and carbon-fiber~C1,C2! filament winding-reinforcepolymers with the fibers oriented in the circumferential direcof the cylinders. Figs. 7–13 show the experimental stress-curves together with the results obtained by the proposed mand by the analytical models due to other authors. In all theamples presented,Ec and «c8 were evaluated by Eqs.~13! and~14!, respectively.

Results of Rectangular Columns

Table 2 presents the properties of rectangular columns testRochette and Labossie`re ~2000! and Mirmiran et al.~2000!. Thecolumns of the first reference were wrapped with FRP shbonded to the concrete with an epoxy system, while the coluof the last reference were confined by square FRP tubes. A

Fig. 13. Stress-strain curve of specimen C2 tested by Saafi~1999!

umns

ax

~mm!ay

~mm!r c

~mm!t

~mm!Ej

~GPa!f r

~MPa!

152.0 152.0 25.0 0.90 82.700 1

152.0 152.0 25.0 1.20 82.700 1152.0 152.0 38.0 0.90 82.700 1152.5 152.5 6.35 1.45 37.233

152.5 152.5 6.35 2.21 40.336152.5 152.5 6.35 2.97 40.749

rt Col

c8a!

2.0

3.92.00.6

0.60.6

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338 / JOURNAL OF COMPOSITES FOR CONSTRUCTION © ASCE / JULY/A

Fig. 17. Stress-strain curve of specimen with 6 plies testedMirmiran et al.~2000!

Fig. 18. Stress-strain curve of specimen with 10 plies testeMirmiran et al.~2000!

Fig. 19. Stress-strain curve of specimen with 14 plies testeMirmiran et al.~2000!

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analyzed columns had rounded corners with a corner radiur c .Figs. 14–19 show the experimental stress-strain curves togwith analytical results obtained by the proposed model andmodel due to Wang and Restrepo~2001!.

Conclusions

A theoretical incremental-iterative model has been presenteevaluating the behavior of rectangular and cylindrical shortumns of concrete confined by fiber-reinforced polymer~FRP!composites. For confined concrete it was assumed that thestiffness is a decreasing function of the area strain of the colwhile in the case of unconfined concrete, an explicit parabrelation was adopted between the lateral strain and thestrain. For the rectangular column, the concrete of the crosstion was divided into parts of confined and unconfined concdefined by parabolic arcs making an initial angle of 45 degwith the side of the cross section. To evaluate the peak strescorresponding strain of the confined concrete, two distinct emcal models of failure have been selected. Several examplshort rectangular and cylindrical columns have been analThe concrete of the studied columns was confined with caFRP and glass FRP jackets. For these columns, the presentdescribed the axial stress versus axial strain response. In theof cylindrical columns, all the resulting stress versus strain cupresented an increasing behavior until a lateral strain is reacorresponding to rupture of the jacket. The stress-strain cwere compared to experimental values available in the literand analytical results due to models proposed by other auConsidering the columns presented in this study, it can becluded that the proposed model produced acceptable resuboth rectangular and cylindrical columns. Finally, it needs toemphasized that the literature still presents few experimentsults for rectangular columns confined by FRP. Hence, thefinement mechanism of these columns needs more detailedtigations.

Acknowledgments

The financial assistance provided by the Conselho NacionDesenvolvimento Cientı´fico e Tecnolo´gico–CNPq, Brasilia, Brazil, is gratefully acknowledged.

Notation

The following symbols are used in this paper:Acc 5 area of concrete effectively confined by jacket;Acu 5 total area of unconfined concrete;

ax ,ay 5 dimensions of concrete core;D 5 diameter of concrete core;

Ec 5 initial elasticity modulus of concrete;Ej 5 Young’s modulus of FRP composite;

Escc 5 concrete secant modulus corresponding to peakstrength;

f c 5 axial stress in concrete;f c8 5 strength of unconfined concrete;f c

i 5 axial stress for confined concrete of incrementalstepi;

f cc8 5 strength of confined concrete;f cu

i 5 axial stress for unconfined concrete of

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t

ls

-

f j 5 stress acting on jacket;f l 5 lateral pressure;

f l ,x , f l ,y 5 average lateral pressures inx andy directions,respectively;

f r 5 rupture stress of composite;i 5 number of incremental step;

i max 5 total number of incremental steps;k1 ,k3 5 constants of Razvi’s model;

n 5 parameter of unconfined concrete;P( i ) 5 axial load of incremental stepi;

r 5 modular ratio of Popovic’s equation;r c 5 radius of concrete column corners;t 5 thickness of jacket;

wx ,wy 5 width for straight portion of sides of column;a«c8 5 axial strain corresponding to zero volumetric

strain;b 5 constitutive constant of concrete;

D«c 5 axial strain increment;«A 5 area strain of column cross section;«c 5 axial strain of column;

«clim 5 limit axial strain beyond which microcracking

starts to occur;«c8 5 axial strain corresponding to unconfined

concrete compressive strengthf c8 ;«cc8 5 axial strain corresponding to confined concrete

compressive strengthf cc8 ;« l 5 lateral strain of column;u 5 arching angle that defines effectively confined

concrete;n 5 Poisson’s ratio of concrete; andc 5 constitutive constant of concrete.

ReferencesCarrasquillo, R. L., Nilson, A. H., and Slate, F. O.~1981!. ‘‘Properties o

high-strength concrete subjected to short term loads.’’ACI J., 78~3!,171–178.

Collins, M. P., and Mitchel, D.~1991!. Prestressed concrete structur,Prentice-Hall, Englewood Cliffs, N.J.

Cusson, D., and Paultre, P.~1995!. ‘‘Stress-strain model for confinehigh–strength concrete.’’J. Struct. Eng.,121~3!, 468–477.

Fam, A. Z., and Rizkalla, S. H.~2001!. ‘‘Confinement model for axiallloaded concrete confined by circular fiber-reinforced polymer tubACI Struct. J.,98~4!, 451–461.

Kono, S., Inazumi, M., and Kaku, T.~1998!. ‘‘Evaluation of confiningeffects of CFRP sheets on reinforced concrete members.’’Proc., 2ndInt. Conf. on Composites in Infrastructure, Chalmers Univ. of Technology, Division of Building Technology, Go¨teborg, Sweden, 343355.

Mander, J. B., Priestley, M. J. N., and Park, R.~1988!. ‘‘Theoreticalstress-strain model for confined concrete.’’J. Struct. Eng.,114~8!,1804–1826.

Mirmiran, A., and Shahawy, M.~1997!. ‘‘Behavior of concrete columnconfined by fiber composites.’’J. Struct. Eng.,123~5!, 583–590.

Mirmiran, A., Zagers, K., and Yuan, W.~2000!. ‘‘Nonlinear finite elemenmodeling of concrete confined by fiber composites.’’Finite ElemAnal. Design,35, 79–96.

Pantazopoulou, S. J., and Mills, R. H.~1995!. ‘‘Microstructural aspects othe mechanical response of plain concrete.’’ACI Mater. J., 92~6!,605–616.

Popovics, S.~1973!. ‘‘Numerical approach to the complete stress-strelation for concrete.’’Cem. Concr. Res.,3~5!, 583–599.

Razvi, S., and Saatcioglu, M.~1999!. ‘‘Confinement model for highstrength concrete.’’J. Struct. Eng.,125~3!, 281–289.

Rochette, P., and Labossie`re, P. ~2000!. ‘‘Axial testing of rectangulacolumn models confined with composites.’’J. Compos. Constr.,4~3!,

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l.’’

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UGUST 2004