Behavior and Modeling of Fiber Reinforced Polymer-Confined Concrete

Behavior and Modeling of Fiber Reinforced
Polymer-Confined Concrete
J.G.Teng 1and m 2
Abstract:One important application of fiber reinforced polymer (FRP )composites in the retrofit of reinforced concrete structures is to provide confinement to columns for enhanced strength and ductility.As a result,many theoretical and experimental studies have been carried out on FRP-confined concrete.This paper provides a critical review of existing studies,with the emphasis being on the revelation of the fundamental behavior of FRP-confined concrete and the modeling of this behavior.Aspects covered in this paper include stress–strain behavior,dilation properties,ultimate condition,and stress–strain models.The paper concludes with a brief outline of issues which require further research.Although the paper is explicitly limited to concrete confined by FRP jackets in which the fibers are oriented only or predominantly in the hoop direction,many of the observations made in this paper are also applicable or relevant to concrete confined by FRP jackets with a significant axial stiffness,as found in concrete-filled FRP tubes as new columns.DOI:10.1061/(ASCE )0733-9445(2004)130:11(1713)
CE Database subject headings:Fiber reinforced polymers;Concrete structures;Retrofitting;Ductility;Confinement;Stress strain curves;Models .
Introduction
Fiber reinforced polymer (FRP )composites have found increas-ingly wide applications in civil engineering due to their high strength-to-weight ratio and high corrosion resistance.One impor-tant application of FRP composites is as wraps or jackets for the confinement of reinforced concrete (RC )columns for enhanced strength and ductility.In FRP-confined concrete,the FRP is prin-cipally loaded in hoop tension while the concrete is loaded in triaxial compression,so that both materials are used to their best advantages.Both the strength and the ultimate strain of concrete can be greatly enhanced as a result of FRP confinement,while the high tensile strength of FRP can be fully utilized.Instead of the brittle behavior exhibited by both materials,FRP-confined con-crete possesses greatly enhanced ductility.
In practical applications,reliable design of FRP jackets is only possible if the stress–strain behavior of FRP-confined concrete is well understood and accurately modeled.In early studies of FRP retrofit of RC columns,the stress–strain model of Mander et al.(1988)for steel-confined concrete was directly used in the analy-sis of FRP-confined concrete columns (Saadatmanesh et al.1994;Seible et al.1995).Subsequent studies however showed that this direct use is inappropriate.This is because in Mander et al.’s
(1988)model,a constant confining pressure is assumed,which is the case for steel-confined concrete when the steel is in plastic flow,but not the case for FRP-confined concrete.Consequently,many studies have been carried out on FRP-confined concrete in the past few years,resulting in a large number of stress–strain models of different levels of sophistication.
This paper provides a critical review of existing studies,with the emphasis being on the revelation of the fundamental behavior of FRP-confined concrete,and the modeling of this behavior.The paper is limited to concrete confined by FRP jackets in which the fibers are oriented only or predominantly in the hoop direction as such jackets are commonly used in column retrofit.Nevertheless,many of the observations made in this paper are also applicable to concrete confined by FRP jackets with a significant axial stiffness,as found in concrete-filled FRP tubes as new columns.
Basic Behavior of Fiber Reinforced Polymer-Confined Concrete
Confining Action of Fiber Reinforced Polymer Jacket When a concrete cylinder confined by an FRP jacket is subject to an axial compressive stress ␴c ,it expands laterally.This expan-sion is confined by the FRP jacket which is loaded in tension in the hoop direction.The confining pressure provided by the FRP jacket increases continuously with the lateral strain of concrete because of the linear elastic stress–strain behavior of FRP,in con-trast to steel-confined concrete in which the confining pressure remains constant when the steel is in plastic flow.Failure of FRP-confined concrete generally occurs when the hoop rupture strength of the FRP jacket is reached.The confining action in FRP-confined concrete can be schematically illustrated in Fig.1.The lateral (radial )confining pressure acting on the concrete core ␴r is given by
1
Professor,Dept.of Civil and Structural Engineering,The Hong Kong Polytechinic Univ.,Hong Kong,China.E-mail:cejgteng@.hk 2
Senior Research Fellow,Dept.of Civil and Structural Engineering,The Hong Kong Polytechnic Univ.,Hong Kong,China.E-mail:cellam@.hk
Note.Associate Editor:Yan Xiao.Discussion open until April 1,2005.Separate discussions must be submitted for individual papers.To extend the closing date by one month,a written request must be filed with the ASCE Managing Editor.The manuscript for this paper was submitted for review and possible publication on September 23,2003;approved on January 5,2004.This paper is part of the Journal of Structural Engi-neering ,V ol.130,No.11,November 1,2004.©ASCE,ISSN 0733-9445/2004/11-1713–1723/$18.00.
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␴r =
␴h t R
͑1͒
where ␴h =hoop stress in the FRP jacket;t =thickness of the FRP jacket;and R =radius of the confined concrete core,respectively.For FRP jackets with fibers only or predominantly in the hoop direction,due to the linearity of FRP,the hoop stress in the FRP jacket ␴h is related to the hoop strain ␧h by
␴h =E frp ␧h
͑2͒
where E frp =elastic modulus of FRP in the hoop direction.The lateral confining pressure reaches its maximum value f l at the rupture of FRP,with
f l =
␴h ,rup t R =E frp ␧h ,rup t
R
͑3͒
where ␴h ,rup and ␧h ,rup =hoop stress and strain of FRP at rupture,
respectively,which are generally not the same as the ultimate tensile strength and the ultimate tensile strain of the FRP respec-tively from tensile tests of flat coupons as discussed in some detail later in the paper.The ratio between the confining pressure f l when the jacket ruptures (i.e.,the maximum confining pressure )and the compressive strength of unconfined concrete f co Јis an important parameter and is referred to as the confinement ratio.
Axial Stress–Strain Behavior of Confined Concrete Fig.2shows typical stress–strain curves of carbon FRP (CFRP )-confined concrete and unconfined concrete,obtained by the present authors (Lam and Teng 2003a,2004;Lam et al.2004)from the compression tests of 152mm ϫ305mm concrete cylin-ders as well as those from compression tests conducted by Can-dappa et al.(2001)on 100mm ϫ200mm concrete cylinders with active confinement at three different lateral pressures (4,8,and 12MPa ).The stress–strain curve of unconfined concrete shown in Fig.2is for a concrete with a compressive strength of 41MPa at an axial strain of 0.00258(Lam et al.to be published ).The cyl-inders confined by one and two plies of CFRP had an unconfined concrete strength of 35.9MPa at an axial strain of 0.00203,while the cylinder confined by three plies of CFRP had an unconfined concrete strength of 34.3MPa at an axial strain of 0.00188(Lam and Teng 2003a,2004).The concrete cylinders tested with active confinement had an unconfined concrete strength f co Јof 42MPa at an axial strain of 0.0024.For the purpose of comparison,the axial stress ␴c is normalized by the compressive strength of the uncon-
fined concrete f co Јwhile the axial strain ␧c or lateral strain ␧r is normalized by the axial strain of the unconfined concrete at its peak stress ␧co .The CFRP had a nominal thickness of 0.165mm per ply and a material ultimate strain of 1.52%from flat coupon tensile tests (ASTM 1995).The elastic modulus of the CFRP was about 250GPa based on the nominal thickness (Lam and Teng 2004).
All three axial stress–axial strain curves of CFRP-confined concrete shown in Fig.2feature a monotonically ascending bilin-ear shape.By contrast,the axial stress–axial strain curves of ac-tively confined concrete feature a softening branch.This is be-cause in the case of FRP-confined concrete,as the axial stress increases,the confining pressure provided by the jacket also in-creases instead of remaining constant.If the amount of FRP pro-vided exceeds a certain threshold value,this confining pressure increases fast enough to ensure that the stress–strain curve is monotonically ascending.This bilinear phenomenon was also previously observed by Xiao et al.(1991)for concrete stub col-umns confined by steel tubes before the yielding of steel.His steel tubes were primarily used as transverse reinforcement and were not directly loaded in the axial direction.Monotonically ascend-ing stress–strain curves have been observed in the majority of existing tests on FRP-confined concrete.Naturally,if the amount of FRP is small,a descending branch is possible and has been observed in tests (Demers and Neale 1994;Xiao and Wu 2000;Aire et al.2001).
Dilation Properties
The dilation properties of unconfined concrete and actively con-fined concrete have been well established (Chen 1982;Panta-zopoulou 1995).Under axial compression,unconfined concrete has an initial Poisson’s ratio (the absolute value of the lateral-to-axial strain ratio at ␧c =0)between 0.15and 0.22and experiences a volumetric reduction or compaction up to 90%of the peak stress.Thereafter the concrete shows volumetric expansion or di-lation as a result of the rapidly increasing lateral-to-axial strain ratio.Unstable dilation after the initial compaction has also been observed in actively confined concrete in triaxial compression tests,although at a higher confining pressure,the dilation is less pronounced,as described by Pantazopoulou (1995).
A number of studies have been concerned with the dilation properties of FRP-confined concrete (Mirmiran and Shahawy 1997a,b;Samaan et al.1998;Xiao and Wu 2000;Harries and Kharel 2002).Mirmiran and his co-workers (Mirmiran and Sha-hawy 1997a,b;Samaan et al.1998)compared the volumetric responses of FRP-confined concrete with those of plain
concrete
Fig.2.Axial stress versus axial and lateral strains of fiber reinforced polymer-confined and actively confined
concrete
Fig. 1.Confining action of fiber reinforced polymer jacket to concrete
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and steel-confined concrete which behaves similarly to actively confined concrete after the yielding of the confining steel.They demonstrated that for steel-confined concrete,unstable dilation occurs when steel yields,but for FRP-confined concrete,the lin-early increasing hoop stress of the FRP jacket can eventually curtail the dilation if the amount of FRP is large enough.
In Fig.3,the lateral-to-axial strain ratio and the volumetric response of FRP-confined concrete under axial compression are compared with those of unconfined and actively confined con-crete using the test data of the specimens of Fig.2.In Fig.3(a ),the lateral-to-axial strain ratio,given in absolute values,is plotted against the normalized axial strain,while in Fig.3(b ),the normal-ized axial stress is plotted against the volumetric strain ␧V ,which is given by
␧V =␧c +2␧r
͑4͒
A positive volumetric strain indicates compaction while a nega-tive value corresponds to dilation.
Fig.3(a )shows that the lateral-to-axial strain ratio of FRP-confined concrete eventually stabilizes at values that depend on the amount of FRP provided.This is in contrast to unconfined and actively confined concrete for which this ratio continuously in-creases with the axial strain.Fig.3(b )shows that for unconfined and actively confined concrete,the change from compaction to dilation occurs at different stress levels depending on the confin-ing pressure,and thereafter the dilation tendency remains until failure.This unchanged dilation tendency is only observed for one of the three CFRP-confined concrete specimens under consider-ation which is confined by a one-ply CFRP jacket.For the speci-men confined by two plies of CFRP,dilation is taken over by compaction at a normalized axial stress of about 1.5.For the specimen with three plies of CFRP,no dilation is found during the
entire loading history.These distinct volumetric responses of FRP-confined concrete are due to the linear elastic behavior of FRP.The linearity of FRP leads to a continuously increasing con-fining pressure until FRP rupture,and limits the lateral strain of confined concrete to the hoop rupture strain of FRP.As a result,concrete confined by a large amount of FRP may not show dila-tion at all.
Ultimate Condition
As eventual failure of FRP-confined concrete is by the rupture of FRP jacket,the ultimate condition of the confined concrete,often characterized by its compressive strength and ultimate axial strain,is intimately related to the ultimate tensile strain or tensile strength of the confining FRP jacket in the hoop direction.In most existing theoretical models for FRP-confined concrete,it has been assumed that tensile rupture of FRP occurs when the hoop stress in the FRP reaches its tensile strength from material tests,either flat coupon tests (e.g.,ASTM 1995)or ring splitting tests (e.g.,ASTM 1992).However,extensive experimental results have shown that the material tensile strength of FRP cannot be reached in FRP-confined concrete as the hoop rupture strains of FRP mea-sured in FRP-confined cylinder tests have been found to be con-siderably smaller than those obtained from material tensile tests (e.g.,Shahawy et al.2000;Xiao and Wu 2000;Pessiki et al.2001).
This uncertainty with FRP hoop rupture strains has led to dif-ficulties in predicting the ultimate condition of FRP-confined con-crete,particularly the ultimate axial strain.This is because the ratio between the FRP hoop rupture strain and the material tensile strain varies with the type of FRP (Lam and Teng 2003b ).De Lorenzis and Tepfers (2003)showed that of the models they re-viewed and assessed (Fardis and Khalili 1981;Saadatmanesh et al.1994;Miyauchi et al.1997;Kono et al.1998;Samaan et al.1998;Saafiet al.1999;Spoelstra and Monti 1999;Toutanji 1999;Xiao and Wu 2000),none was able to predict the ultimate axial strain with reasonable accuracy if the hoop strain in the FRP jacket at rupture is taken to be equal to the ultimate material tensile strain.Xiao and Wu (2000),based on their own test ob-servations,suggested that 50%of the flat coupon ultimate tensile strain be taken as the hoop rupture strain.Jin et al.(2003),how-ever,suggested using 0.96times the flat coupon ultimate tensile strain.Moran and Pantelides (2002)suggested using the hoop strain limits of 0.0085for CFRP and 0.0125for glass FRP (GFRP ).Lam and Teng (2003b )recently suggested that in devel-oping confinement models,the actual hoop rupture strain ␧h ,rup measured in the FRP jacket should be used to evaluate the stress in the FRP rather than simply using the FRP material tensile strength f frp .
Several causes have been suggested for the difference in the ultimate tensile strain between FRP tensile test specimens and FRP jackets confining concrete (Spoelstra and Monti 1999;Sha-hawy et al.2000;Xiao and Wu 2000;Pessiki et al.2001;Moran and Pantelides 2002;De Lorenzis and Tepfers 2003;Lam and Teng 2003b ).These suggestions are generally speculative without sound experimental m and Teng (2003a,2004)re-cently conducted the first carefully planned study involving com-parative experiments in an attempt to clarify the causes for the reduced strain capacity of FRP when used to confine concrete.The experimental program covered flat coupon tensile tests (ASTM 1995)and ring splitting tests (ASTM 1992)on CFRP and GFRP specimens,and compression tests on concrete cylinders wrapped with one to three plies of CFRP and GFRP.Based on
the
Fig.3.Dilation properties of fiber reinforced polymer-confined and actively confined concrete
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test observations,Lam and Teng (2004)concluded that the aver-age hoop rupture strains of FRP measured in FRP-confined con-crete cylinders are affected by at least three factors:(1)the cur-vature of the FRP jacket;(2)the deformation nonuniformity of cracked concrete;and (3)the existence of an overlapping zone in which the measured strains are much lower than strains measured elsewhere.The first factor results in a reduced strain capacity of FRP in confined concrete cylinders.The second and third factors result in a nonuniform strain distribution in the FRP jacket.While the effect of curvature is material dependent,which means that the curvature of the FRP jacket has a stronger detrimental effect on CFRP than on GFRP in the context of their study,the nonuni-formity of strain distribution is independent of the type of FRP.In addition,Harries and Carey (2003)recently investigated the effect of adhesive bonding on the hoop rupture strain ␧h ,rup in a GFRP jacket.Their test results showed that on the unbonded specimens which contained a 0.08mm thick plastic wrap between the FRP jacket and the concrete,the hoop rupture strains were nearly uniform around the circumference.However,the average hoop rupture strains of these specimens did not appear to be higher than those measured on the specimens with the FRP jacket bonded to the concrete.Thus,the effect of bonding on reducing the hoop rupture strain has not been established by these tests.
Stress–Strain Models Classification of Models
From the large number of studies on FRP-confined concrete,many stress–strain models have resulted.These models can be classified into two categories:(1)design-oriented models,and (2)analysis-oriented models.In the first category,stress–strain mod-els are presented in closed-form expressions,while in the second category,stress–strain curves of FRP-confined concrete are pre-dicted using an incremental iterative numerical procedure.It should be mentioned that a number of studies exist which have been concerned with the nonlinear finite element analysis of FRP-confined concrete (e.g.,Rochette and Labossiere 1996;Mirmiran et al.2000;Parent and Labossiere 2000).These finite element studies are not discussed in this paper.
Design-Oriented Models
Various simple stress–strain models in closed-form expressions have been proposed based directly on test stress–strain curves of FRP-confined circular concrete specimens.Most of the these models predict only the increasing type of stress–strain curve,but the models of Miyauchi et al.(1999),Jin et al.(2003),and Xiao and Wu (2000)predict both the increasing and the decreasing types of stress–strain curve.The model proposed by Shitindi et al.(1998)was based on the test results of rectangular specimens ͑100ϫ200ϫ400mm ͒confined by FRP sheets and spirals.Their model predicts only the descending type of stress–strain curves.All these models are suitable for direct application in design cal-culations by hand or spreadsheets and are thus referred to as design-oriented models,although they may also be used in non-linear computer analysis of structures with FRP confinement.In addition,Moran and Pantelides (2002)proposed a stress–strain model which may be classified as a design-oriented model as its many parameters are directly based on test results,but the model is far more complicated than other design-oriented models and
thus does not possess the necessary simplicity for direct design application.This model also predicts stress–strain curves of both the increasing and decreasing types.
Early stress–strain models for FRP-confined concrete (Fardis and Khalili 1981,1982;Ahmad et al.1991;Saadatmanesh et al.1994)were based on models for steel-confined concrete by Rich-art et al.(1928,1929),Ahmad and Shah (1982a,1982b ),and Mander et al.(1988),respectively.These models do not feature the characteristic bilinear shape of the stress-strain curves of FRP-confined concrete.This might be due to the fact that the charac-teristic bilinear stress–strain behavior was not clearly displayed in the early tests,which were performed on concrete cylinders con-fined by filament-wound FRPs (Fardis and Khalili 1981;Ahmad et al.1991).Bilinear stress–strain curves were observed in subse-quent studies on concrete cylinders confined by wrapped FRP sheets (e.g.,Harmon and Slattery 1992;Demers and Neale 1994;Howie and Karbhari 1994,Nanni and Braford 1995).Nanni and Braford (1995)suggested that the stress–strain curves of FRP-confined concrete be approximated using two straight lines,with the transition point being at a stress equal to the compressive strength of unconfined concrete and an axial strain of 0.003.This suggestion was applied by Nanni and Norries (1995)to the analy-sis of FRP-confined concrete under combined flexure and com-pression,in which the ultimate point of the stress–strain curve was determined using the equations of Fardis and Khalili (1982).Such strictly bilinear models were also proposed by Karbhari and Gao (1997)and Xiao and Wu (2000)for FRP-confined concrete.Trilinear models have also been proposed recently by Wu et al.(2003)and Chaallal et al.(2003),with the latter being for FRP-confined concrete in rectangular sections only.In other existing models,at least the first portion of the stress–strain curve is curved.Two expressions have been used most frequently in mod-eling FRP-confined concrete:the general expression proposed by Sargin (1971)and the four-parameter stress–strain curve proposed by Richard and Abbot (1975).Design-oriented models using these two expressions are first summarized below,which is then fol-lowed by the brief description of a recent model proposed by the authors.
It should be mentioned that apart from models predicting the stress–strain curve of FRP-confined concrete,a number of studies have been concerned with only the ultimate condition of FRP-confined concrete (Kono et al.1998;Lin and Chen 2001;Vintzi-leou 2001;Ilki and Kumbasar 2002,2003;Lam and Teng 2002;Pan et al.2002;Shehata et al.2002;De Lorenzis and Tepfers 2003).
Models Based on Sargin’s General Expression
The general expression proposed by Sargin (1971)has the follow-ing form:
␴c f co
Ј=
A ␧c
␧co +͑D −1͒
ͩ␧c ␧co
ͪ21+͑A −2͒␧c ␧co +D
ͩ␧c ␧co
ͪ
2
͑5͒
where A and D =constants controlling the initial slope and the descending path of the stress–strain curve,respectively.Ahmad and Shah (1982a,b )used Eq.(5)to model the stress–strain curve of concrete confined by steel spirals.This equation has also been used in Eurocode 2(CEN 1991)to represent the stress–strain curve of unconfined concrete for structural analysis.However,the stress-strain curve for design use in the same document takes the following form which is a special case of Eq.(5)with A =2and
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D =0,which is usually referred to as Hognestad’s (1951)parabola
␴c f co
Ј=ͫ2␧c ␧co −
ͩ␧c ␧co ͪ2
ͬ
͑6͒
where ␧co is assumed to be 0.002in Eurocode 2(CEN 1991).Ahmad et al.(1991)modified the model of Ahmad and Shah (1982a,b )for application to FRP-confined concrete,with f co Јand ␧co being replaced by the stress and strain at the peak stress of FRP-confined concrete.As mentioned above,this model does not feature a bilinear shape.Toutanji (1999)and Saafiet al.(1999)characterized the stress–strain curve of FRP-confined concrete using two segments with a smooth transition between them at a lateral strain of ␧r =0.002.These two models are identical in form but were calibrated with test results of FRP-wrapped concrete cylinders and concrete-filled FRP tubes respectively.The first seg-ments in their models are described by modified versions of Eq.(5)with ␴c /f co Јand ␧c /␧co replaced by ␴c and ␧c ,and with the constants redefined.The second segments in their models are given by separate equations for axial stresses and strains both of which are functions of the lateral strain ␧r .Jin et al.(2003)pro-posed a stress–strain model for concrete confined by wrapped FRP in which the first segment of the stress–strain curve is de-scribed by the same equation as given in Toutanji (1999)[i.e.,a modified version of Eq.(5)]while the second segment,which is linear,may be ascending or descending.Miyauchi et al.(1997,1999)used the Hognestad parabola to describe the first portion of the stress–strain curve of concrete confined by wrapped CFRP and a straight line to describe the second portion.Lillistone and Jolly (2000)represented the first portion of the stress–strain curve of concrete-filled FRP tubes using the Hognestad parabola plus an additional term related linearly to the axial strain which accounts for the contribution of the FRP tube.The second portion in this model is linear,with the axial stress being that given by the same additional term plus the unconfined concrete strength.It should be mentioned that Lillistone and Jolly’s (2000)model predicts a con-stant ultimate axial strain for any amount of confinement.Li et al.(2003),however,used Eq.(6)for the response of FRP-confined concrete until the confined concrete strength f cc Јis reached.Obvi-ously,this model does not lead to a stress-strain curve of the bilinear shape.The use of Eqs.(5)and (6)for modeling FRP-confined concrete is illustrated in Fig.4.
Models Based on Richard and Abbot’s Four-Parameter Curve
The four-parameter curve of Richard and Abbot (1975),which was proposed to describe the elastic–plastic behavior of structural systems,is given by
␴=
͑E 1−E 2͒␧
ͭ1+
ͯ
͑E 1−E 2͒␧f o
ͯn ͮ
1/n
+E 2␧͑7͒
where ␴and ␧=stress and the strain;f 0=reference stress;E 1=initial modulus;E 2=plastic modulus;and n =shape parameter controlling the transition from the first portion to the second por-tion of the stress–strain curve.The definitions of f 0,E 1,E 2,and n are shown in Fig.5.This four-parameter curve was first used by Samaan et al.(1998)for FRP-confined concrete.An advantage of this four-parameter curve is that a bilinear stress–strain curve can be simulated using a single expression.Consequently,a number of models for FRP-confined concrete have been proposed using this expression (Arduini et al.1999;Yu 2001;Cheng et al.2002;Moran and Pantelides 2002;Xiao and Wu 2003),as well as a model by Toutanji and Saafi(2002)for concrete confined by FRP-reinforced polyvinyl chloride tubes.Some of these models predict both axial stress–axial strain and axial stress–lateral strain relationships (Samaan et al.1998;Yu 2001;Moran and Pantelides 2002;Xiao and Wu 2003),but others predict only the axial stress–axial strain relationship.A model proposed by Campione and Miragia (2003)is also modified from Eq.(8),in which nor-malized stresses and strains are used instead and no reference stress is defined.While the model of Samaan et al.(1998)was calibrated with test results of concrete-filled FRP tubes,the mod-els of Arduini et al.(1999),Xiao and Wu (2003),and Campione and Miragia (2003)were calibrated with test results of FRP-wrapped concrete cylinders.The models of Yu (2001)and Moran and Pantelides (2002)were calibrated with test results of both types of FRP-confined concrete.
Lam and Teng’s Model
The present authors recently developed a design-oriented model (Lam and Teng 2003b,c )based on a careful interpretation of a large test database assembled by them from the open literature.An earlier version of this model can be found in Lam and Teng (2001).The new version (Lam and Teng 2003b,c )includes sev-eral improvements over the previous version,including a more accurate and rational definition of the ultimate condition based on the actual hoop rupture strain and the explicit account taken of the effect of jacket strain capacity on the ultimate axial strain.It can be seen in Fig.6that this model consists of a parabolic first portion with its initial slope being the elastic modulus of uncon-fined concrete E c ,and a linear second portion which has a slope E 2and intercepts the stress axis at f 0=f co Ј.The parabolic first portion meets the linear second portion with a smooth transition at ␧t .This model allows the use of test values or values suggested by design codes for the elastic modulus of unconfined
concrete
e of Sargin’s general expression for stress–strain curves of fiber reinforced polymer-confined
concrete
Fig.5.Richard and Abbot’s four parameter stress–strain curve
D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y G u a n g d o n g U n i v e r s i t y o f T e c h n o l o g y o n 03/13/14. C o p y r i g h t A S C
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