Refinement of a design-oriented stress-strain model for FRPconfined

Transcription

1 This is the Pre-Published Version. Refinement of a design-oriented stress-strain model for FRPnfined nrete J.G. Teng 1,*, T. Jiang 1, L. Lam 1 and Y.Z. Luo 2 1 Department of Civil and Strutural Engineering The Hong Kong Polytehni University, Hong Kong, China 2 Department of Civil Engineering Zhejiang University, Hangzhou, Zhejiang Provine, China *ejgteng@polyu.edu.hk Abstrat: This paper presents the results of a reent study nduted to refine the designoriented stress-strain model originally proposed by Lam and Teng for FRP-nfined nrete under axial mpression. More aurate expressions for the ultimate axial strain and the mpressive strength are proposed for use in this model. These new expressions are based on results from reent tests nduted by the authors group under well-defined nditions and on results from a parametri study using an aurate analysis-oriented stress-strain model for FRP-nfined nrete. They allow the effets of nfinement stiffness and the jaket strain apaity to be separately refleted and aunts for the effet of nfinement stiffness expliitly instead of having it refleted only through the nfinement ratio. The new expressions an be easily inrporated into Lam and Teng s model for more aurate preditions. Based on these new expressions, two modified versions of Lam and Teng s model are presented. The first version involves only the updating of the ultimate axial strain and mpressive strength equations. The send version aters for stress-strain urves with a desending branh, whih is not vered by the original model. Keywords: FRP; nrete; nfinement; stress-strain model; design 1 Introdution Various stress-strain models have been developed for fiber reinfored polymer (FRP)- nfined nrete under axial mpression. These models an be lassified into two main ategories (Teng and Lam 24): design-oriented models (e.g. Fardis and Khalili 1982; Karbhari and Gao 1997; Samaan et al. 1998; Miyauhi et al. 1999; Saafi et al. 1999; Toutanji 1999; Lillistone and Jolly 2; Xiao and Wu 2, 23; Lam and Teng 23; Berthet et al. 26; Harajli 26; Saenz and Pantelides 27; Wu et al. 27; Youssef et al. 27) and analysis-oriented models (e.g. Mirmiran and Shahawy 1997; Spoelstra and Monti 1999; Fam and Rizkalla 21; Chun and Park 22; Harries and Kharel 22; Marques et al. 24; Binii 25; Teng et al. 27a; Jiang and Teng 27). Design-oriented models are generally defined using simple losed-form expressions and are suitable for diret use in pratial design. By ntrast, analysis-oriented models generally predit stress-strain urves using an inremental-iterative proedure and are thus undesirable for diret use in design. Analysis-oriented models have a better preditive apability and are more versatile than designed-oriented models as the former aunt expliitly for the interation between the nfining material and the nrete re. Therefore, a rational approah for the development of a design-oriented model is to base it on extensive numerial results from an aurate 1

2 analysis-oriented model in addition to a more limited set of test results. A mprehensive review and assessment of existing analysis-oriented models has reently been nduted by Jiang and Teng (27) who also showed that Teng et al. s (27a) model, with a modifiation proposed by them, an provide aurate preditions of test results. Among the many different design-oriented stress-strain models, the model proposed by Lam and Teng (23) appears to be advantageous over other models due to its simple and familiar form as well as auray. This model, with some modifiation, has been adopted by the design guidane for the strengthening of nrete strutures using FRP issued by the Conrete Soiety (24) in the UK. More reently, this model has also been adopted by ACI- 44.2R (28) with only very slight modifiations. The model adopts a simple form that naturally redues to that for unnfined nrete when no FRP is provided. Its simple form also aters for easy improvements to the definition of the ultimate ndition (the ultimate axial strain and the mpressive strength) of FRP-nfined nrete, whih are the key to the aurate predition of stress-strain urves of FRP-nfined nrete by this model. Although Lam and Teng s (23) model was developed on the basis of a large database of tests on FRP-nfined irular nrete ylinders, a number of signifiant issues uld not be readily resolved using the test database available to them at that time. In partiular, there was nsiderable unertainty with the hoop tensile strain reahed by the FRP jaket at rupture failure, whih has an important bearing on the definition of the ultimate ndition. Arding to a subsequent study by the same authors (Lam and Teng 24), the distribution of FRP hoop strains is highly non-uniform around the irumferene of an FRP-nfined nrete ylinder and the hoop strains in the overlapping zone of the FRP jaket are muh lower than those measured elsewhere. The lower FRP hoop strains in the overlapping zone redue the average hoop strain but do not result in a lower nfining pressure in this zone beause the FRP jaket is thiker there (Lam and Teng 24). This observation suggests that hoop strain readings within the overlapping zone should be exluded when interpreting the behavior of FRP-nfined nrete, as these readings reflet neither the atual strain apaity of the nfining jaket nor the atual dilation properties of the nfined nrete. However, suh important proessing of the hoop strain readings was not possible with test data lleted by Lam and Teng (23) from the existing literature at that time, for whih the preise number and loations of strain gauges for measuring hoop strains were generally not reported. Apart from the unertainty assoiated with the FRP hoop strain, the different methods of axial strain measurements (e.g. strain gauges versus displaement transduers and the gauge lengths employed) adopted by different researhers may have also affeted the nsisteny of the test data of that database. To address these defiienies of that database and hene those of Lam and Teng s (23) stress-strain model, a large number of additional tests on FRP-nfined nrete ylinders have been nduted under standardized testing nditions by the authors group (Lam and Teng 24, Lam et al. 26, Teng et al. 27b, Jiang and Teng 27). In addition, further theoretial modeling work on FRP-nfined nrete has also been arried out by the authors group (Teng et al. 27a; Jiang and Teng 27). These reent test results and the new understandings from further researh have aumulated a solid basis for the refinement of Lam and Teng s (23) stress-strain model. This paper therefore presents refinements to the Lam and Teng (23) stress-strain model. In this paper, more aurate expressions for the ultimate axial strain and the mpressive strength of FRP-nfined nrete are first developed. Two modified versions of Lam and 2

3 Teng s model based on these new expressions are next presented. The first version involves only the updating of the ultimate ndition equations. The send version differs from the first version in that it is apable of prediting stress-strain urves with a desending branh when the level of nfinement is low. It should be noted that in this paper, the term stress-strain should be understood as axial stress-axial strain. The latter is used only when the axial stress-lateral strain responses of the nrete are also disussed. The following sign nvention is adopted: in the nrete, mpressive stresses and strains are positive, but in the FRP, tensile stresses and strains are positive. 2 Test database 2.1 General The test database used in the present study (i.e. the present test database) has previously been reported by Jiang and Teng (27) for the assessment of analysis-oriented stress-strain models. This database ntains results of 48 tests on nrete ylinders (152 mm 35 mm) nfined with varying amounts of arbon FRP (CFRP) and Glass FRP (GFRP), with the unnfined nrete strength (i.e. the mpressive strength of unnfined nrete) f ranging from 33.1 MPa to 47.6 MPa. All these tests were reently nduted under standardized test nditions at The Hong Kong Polytehni University by the authors group. As full details of these tests are available in Jiang and Teng (27), only the key features of the test database are summarized below: a) The FRP jakets were formed via the wet lay-up proess and had hoop fibers only. For eah bath of nrete, two or three 152 mm x 35 mm plain nrete ylinders were tested as ntrol speimens to determine the average values of the unnfined nrete strength f and the rresponding axial strain. b) The hoop strain h of the FRP jaket was found as the average reading of the five hoop strain gauges outside the 15 mm overlapping zone; these gauges had a 2 mm gauge length. ) The axial strain of nrete was found as the average reading of two linear variable displaement transduers (LVDTs) at 18 apart and vering the mid-height region of 12 mm. The lateral strain of nrete l was assumed to be equal to the jaket hoop strain in magnitude; they have different signs arding to the adopted sign nvention. d) The test database vers a wide range of FRP nfinement levels. The mpressive strength of the most heavily-nfined speimen inreased by about 32% due to nfinement while the most lightly-nfined speimen exhibited a stress-strain urve with a desending branh and a negligible strength inrease. For ease of disussion, three basi ratios are first defined: the nfinement ratio fl f, the nfinement stiffness ratio K and the strain ratio. The mathematial expressions of these three ratios are as follows: f f l 2E frp t h, rup fd (1a) K 3

4 2E frp t K (1b) f D hrup, (1) where f l is the nfining pressure provided by the FRP jaket when it fails by rupture due to hoop tensile stresses (i.e. the maximum nfining pressure possible with the jaket), is the elasti modulus of FRP in the hoop diretion, t is the thikness of the FRP jaket, is the hoop rupture strain of the FRP jaket, and D is the diameter of the nfined nrete ylinder. The nfinement ratio is a mmonly used parameter in the existing literature. The nfinement stiffness ratio represents the stiffness of the FRP jaket relative to that of the nrete re. The strain ratio is a measure of the strain apaity of the FRP jaket. The nfinement ratio is equal to the produt of the other two ratios. 2.2 Stress-strain urves The stress-strain urves as well as other key results of all these tests are given in Jiang and Teng (27). Only eight typial stress-strain urves are shown in Fig. 1 to illustrate the behavior of nfined nrete ylinders. Both the axial strain and the lateral strain l are normalized by the rresponding value of, while the axial stress is normalized by the rresponding value of f. All the asending type urves in Fig. 1 exhibit the well-known bi-linear shape, ending with the rupture of the nfining jaket at the ultimate point defined by the mpressive strength f and the ultimate axial strain u ; both the strength and the strain apaity of the nrete are signifiantly enhaned. By ntrast, for the desending type stress-strain urves, f is reahed before the rupture of the jaket with very limited strength enhanement. For ease of disussion, if the axial stress of FRP-nfined nrete at ultimate axial strain f u falls below the unnfined nrete strength f, the nrete is lassified as insuffiiently nfined in this paper. All other ases of FRP-nfined nrete are lassified as suffiiently nfined. The key information of these eight speimens, together with that of some other speimens referred to in the present paper, is given in Table 1. The names of the speimens are the same as reported in Jiang and Teng (27). 2.3 Ultimate ndition Using the analysis-oriented model for FRP-nfined nrete proposed by Spoelstra and Monti (1999), Lam and Teng (23) demonstrated that the stiffness of the FRP jaket affets both the ultimate axial strain and the mpressive strength of FRP-nfined nrete. The tests of the present database offer lear experimental evidene on the effet of jaket stiffness, as illustrated in Fig. 2. Fig. 2a shows the experimental stress-strain urves of speimens 31 and 45. The former had an unnfined nrete strength of 45.9 MPa, a GFRP jaket, and a nfinement ratio of.14, while the latter had an unnfined nrete strength of 44.2 MPa, a CFRP jaket and a nfinement ratio of.141. It an be seen that although the two speimens had very similar unnfined nrete strengths and nfinement ratios, their stress-strain urves are signifiantly different in the later stage due to the differene in the jaket nfinement stiffness. The stress-strain urve of speimen 31 with a smaller nfinement stiffness ratio terminates at a lower axial stress but a larger axial strain. E frp hrup, 4

5 Similarly, Fig. 2b shows the stress-strain urves of speimens 28 and 42. The former had an unnfined strength of 45.9 MPa and a nfinement ratio of.59 while the latter had an unnfined strength 44.2 MPa and a nfinement ratio of.55. It is interesting to note that speimen 28 with a smaller nfinement stiffness ratio exhibited a stress-strain urve of the desending type while speimen 42 exhibited a stress-strain urve of the asending type. These observations suggest that the effet of nfinement stiffness on the ultimate ndition of FRP-nfined nrete should be properly refleted in a design-oriented stress-strain model. The mparison shown in Fig. 2b also suggests that the nfinement stiffness ratio is important in determining whether the suffiient nfinement ndition is met. 3 Lam and Teng s stress-strain model for FRP-nfined nrete Lam and Teng s (23) design-oriented stress-strain model is based on the following assumptions: (i) the stress-strain urve nsists of a paraboli first portion and a linear send portion, as illustrated in Fig. 3; (ii) the slope of the parabola at zero axial strain (the initial slope) is the same as the elasti modulus of unnfined nrete; (iii) the nonlinear part of the first portion is affeted to some degree by the presene of an FRP jaket; (iv) the paraboli first portion meets the linear send portion smoothly (i.e. there is no hange in slope between the two portions where they meet); (v) the linear send portion terminates at a point where both the mpressive strength and the ultimate axial strain of nfined nrete are reahed; and (vi) the linear send portion interepts the axial stress axis at a stress value equal to the unnfined nrete strength. The justifiations for the above assumptions are given in Lam and Teng (23) and are thus not repeated herein. Based on these assumptions, Lam and Teng s stress-strain model for FRP-nfined nrete is desribed by the following expressions: E E E for t (2a) 4 f f E2 for t u (2b) where and are the axial stress and the axial strain respetively, E is the elasti modulus of unnfined nrete, and E 2 is the slope of the linear send portion. The paraboli first portion meets the linear send portion with a smooth transition at t whih is given by t 2 f E E 2 (3) The slope of the linear send portion E 2 is given by E 2 f f (4) u 5

6 This model allows the use of test values or values speified by design des for the elasti modulus of unnfined nrete E. Lam and Teng (23) proposed the following equation to predit the ultimate axial stra in u of nfined nrete: u K 1.45 (5) Lam and Teng s (23) mpressive strength equation takes the following form: f fl f 1 3.3, if l f f f.7 (6a) f fl 1, if f f.7 (6b) Eqs 6a and 6b are for suffiiently and insuffiiently nfined nrete respetively. A minimum value of.7 for f l f for suffiient nfinement was originally suggested by Spoelstra and Monti (1999) and was used in Lam and Teng s (23) model with justifiation using test data available to them. A mparison of Lam and Teng s model with the test data of the present database is shown in Fig. 4. In prediting the mpressive strength and the ultimate axial strain, a nstant value of.2 for was used. The experimental values of hrup, were used in making the preditions shown in Fig.4 and throughout the paper. The elasti modulus of unnfined nrete was taken to be E 473 f (in MPa) [ACI-318 (25)]. It an be seen from Fig. 4 that Eqs 5 and 6 overestimate the ultimate axial strain of nrete at high levels of nfinement and the mpressive strength of nrete at low levels of nfinement. In addition, the effet of nfinement stiffness is only expliitly and separately aunted for in the ultimate axial strain equation, but not in the mpressive strength equation. Refinement of these equations is therefore neessary to provide more aurate preditions. 4 Generalization of equations To take the effet of nfinement stiffness into proper aunt, the expressions for the ultimate axial strain and the mpressive strength of FRP-nfined nrete are generalized here. Eq. 5 an be ast into the following form: u C F( K ) f( ) (7) where F ( K ) and f ( ) are funtions of the nfinement stiffness ratio and strain ratio respetively, and C is a nstant. Sine the nfinement ratio is the produt of the nfinement stiffness ratio and the strain ratio, Eq. 6 an be ast into the following form whih is similar to Eq. 7: 6

7 f C F ( ) f ( ) K (8) f where F ( K ) and f ( ) are also funtions of the nfinement stiffness ratio and the strain ratio respetively, and C is a nstant. The above generalization allows the effets of nfinement stiffness and the jaket strain apaity to be separately refleted in both the ultimate axial strain and the mpressive strength equations and aunts for the effet of nfinement stiffness expliitly instead of having it refleted only through the nfinement ratio. 5 New equations for the ultimate ndition 5.1 Ultimate axial strain Based mainly on the interpretation of the test results in the present database, the following improved equation for the ultimate axial strain of FRP-nfined nrete is proposed: u (9) K where the strain at the unnfined nrete strength was taken to be.2 in determining the strain ratio, sine this value is mmonly aepted in existing design des for nrete strutures. It is therefore suggested that this value should be used when the proposed model is used in design. The first term on the right-hand side of Eq. 9 is taken to be 1.75 so that the equation predits a value of.35 for u when no FRP nfinement is provided; the effiient/exponents within the send term were determined by regression analysis of the present test results. It should be noted that u.35 is mmonly aepted for unnfined nrete; the nstant 1.75 may be adjusted to suit a different value for the ultimate axial strain of unnfined nrete in a speifi design de. In Fig. 5, the preditions of Eq. 9 are mpared with the present test database as well as the test database assembled by Lam and Teng (23) on whih the previous version of the ultimate ndition equations (Eqs 5 and 6) was based. Lam and Teng s (23) test database ntains 76 nonidential speimens from 14 independent soures. These speimens had diameters ranging from 1 mm to 2 mm and unnfined nrete strengths ranging from 26.2 to 55.2 MPa. Fig. 5 shows that Eq. 9 is very aurate for the present test database. The results of Lam and T eng s (23) database are niely sattered around the preditions (partiularly in the lower range of ultimate axial strains where there are more test data points), nfirming the general validity of the new ultimate axial strain equation; the wide satter of these test data are attributed to the defiienies of that database as disussed earlier in this paper. 5.2 Compressive strength The mpressive strength equation was refined on a mbined experimental and analytial basis. Fig. 1 shows that both the axial stress-lateral strain urves and the axial stress-axial strain urves exhibit a bi-linear shape, with the two portions smoothly nneted by a transition zone near the unnfined nrete strength. The shape of the send portion is very lose to a straight line. Fig. 1 also shows that the send portion of an axial stress-lateral 7

8 strain urve is loser to being a straight line than that of an axial stress-axial strain urve. A areful examination of the present tests data showed that the send portion of all the experimental axial stress-lateral strain urves interepts the axial stress axis at a stress value whih is very lose to the unnfined nrete strength when this portion is approximated by a best-fit straight line. These straight lines an be represented by the following equation: 1 K (1) l f whih means that the nstant C in Eq. 8 has been taken to be unity for the reason given a bove. K is the slope of the fitted straight line. It is obvious that the axial stress reahes f u when l = hrup,, and Eq. 1 thus bemes K (11) f u hr, up 1 K 1 f Comparing Eq. 11 with Eq. 8 leads to F ( ) = while the slope K = F ( K ) is yet to be determined. To define F ( K ) in Eq. 11, a parametri study was nduted using the refined version (Jiang and Teng 27) of the analysis-oriented stress-strain model of Teng et al. (27a) for FRP-nfined nrete. The parametri study vered nrete ylinders of 152 mm in diameter and nfined with either CFRP or GFRP, with f ranging from 2 to 5 MPa and a wide range of jaket thiknesses to represent different values of nfinement stiffness. The material properties used and parameters studied are given in Table 2. In the parametri study, 4 4 it was assumed f (Popovis 1973) instead of the more approximate val ue of.2. The value of has some effet on the auray of this analysis-oriented model, so Jiang and Teng (27) suggested the use of this expression for ases where the value of has to be assumed, for instane, in a parametri study. The parametri study nsisted of three steps: 1) produe a family of axial stress-lateral strain urves of a nrete ylinder nfined with a ertain type of FRP jaket for different nfinement stiffness ratios; 2) approximate the send portions of these axial stress-lateral strain urves using best-fit straight lin es that inter ept the vertial axis ( ) at unity ; and 3) identify F ( K ) by finding f the relationship between the slopes of these straight lines and the nfinement stiffness ratios. Fig. 6a demonstrates the first two steps for a nrete ylinder with f = 3 MPa nfined with a CFRP jaket for a range of jaket thiknesses. Eah stress-strain urve in Fig. 6a rresponds to a partiular value of the nfinement stiffness ratio. The portions of the stressstrain urves from l.5 onwards were fitted using straight lines (dashed lines in Fig. 6a). The slopes of the fitted lines are shown against the nfinement stiffness ratios in Fig. 6b, as a demonstration of the last step. A rresponding urve for the same nrete ylinder but with GFRP nfinement is also shown in Fig. 6b. It an be seen that these two urves are almost idential and an be losely represented using the following expression: 8

9 K F (12).9 ( K) 3.2 K.6 Eq. 12 also provides aurate preditions for other values of f studied. With F ( K ) defined, Eq. 11 bemes f u K.6 f (13) Fig. 7 shows that Eq. 13 mpares well with the present test results. In prediting fu for Fig. 7, experimental values of were used, as the aim was to verify the auray of Eq. 13 using experiments. A s the nonlinear relationship between F ( K ) and K in Eq. 13 is slightly innvenient for design use, the following simple linear equation is proposed as an approximation: fu 13.5 K.1 (14) f It should be noted that Eq. 14 predits the axial stress at the ultimate axial strain, but not the mpressive strength f of FRP-nfined nrete, although they are the same unless the stress-strain urve features a desending branh. Sine Lam and Teng s (23) model neglets the small amount of strength enhanement in insuffiiently nfined nrete and employs a horizontal line to approximate the desending branh of suh nrete, Eq. 14 needs to be modified for diret inrporation into Lam and Teng s model (23) for prediting f. It is proposed that f K, if K.1 (15a) f f 1 f, if.1 (15b) K The performane of Eq. 15 as assessed using the present test database and Lam and Teng s (23) test database is shown in Fig. 8. Eq. 15 is seen to provide aurate preditions of the results of the present test data. For the results of Lam and Teng s (23) database, Eq. 5 bemes more nservative but still mpares well with the test results. In prediting f for Fig. 8, =.2 was used for assessing the auray of Eq. 15 in design appliations. Eq. 15 def ines a minimum nfinement stiffness ratio K of.1 below whih the FRP is assumed to result in no enhanement in the mpressive strength of nrete. It should be noted that the above mparisons for the new ultimate ndition equations are based on small-sale speimens with diameters ranging from 1 mm to 2 mm. Speimens of suh sizes have been used in the vast majority of existing studies on the stress-strain behavior of FRP-nfined nrete. Only a small number of experimental studies have been 9

10 nduted on the axial mpressive behavior of large-sale irular lumns (Youssef 23; Carey and Harries 25; Mattys et al. 25; Roa et al. 26; Yeh and Chang 27). In these few studies, lumns with a diameter up to 61 mm were tested and the test results indiated that within this range, the stress-strain behavior does not vary signifiantly with the lumn diameter. The effet of lumn size is thus believed to be limited for irular lumns. 6 Modifiations to Lam and Teng s model-version I Eqs 9 and 15 an be diretly inrporated into Lam and Teng s (23) model. Fig. 9 shows a mparison between the preditions of the original model and those of the modified model (referred to as Version I to differentiate it from Version II presented in the next setion) for three sets of speimens of the present test database. The first two sets of speimens (speimens 22 and 23 in Fig. 9a and speimens 24 and 25 in Fig. 9b) had an unnfined nrete strength of 39.6 MPa and were nfined with 2 and 3 plies of GFRP, respetively. The nfinement ratio is.186 for the 2-ply GFRP jaket and.278 for the 3-ply GFRP jaket, while the nfinement stiffness ratio is.18 for the former and.27 for the latter. The last set of speimens (speimens 17, 18 and 19 in Fig 8) had an unnfined nrete strength of 38.9 MPa and were nfined with 2 plies of CFRP, with a nfinement ratio of.274 and a nfinement stiffness ratio of.55. The predited stress-strain urves were obtained using =.2. It is evident that the original model provides lose preditions for the speimens with two plies of CFRP (Fig. 9), but not for those with two and three plies of GFRP (Figs 9a and 9b). Note that the nfinement ratio of the 3-ply GFRP jaket is similar to that of the 2-ply CFRP jaket (.278 versus.274), but the nfinement stiffness ratio of the former (.27) is only half that of the latter (.55). The inauray of the original model in this ase is due to the omission of the nfinement stiffness as a parameter in addition to the nfining pressure in the mpressive strength equation (Eq. 6). It is lear that the use of Eqs 9 and 15 in Lam and Teng s (23) model improves its performane nsiderably for ases of low nfinement stiffness ratios. 7 Modifiations to Lam and Teng s model-version II Both the original model of Lam and Teng (23) and the modified version presented above simply approximate a post-peak desending branh due to a low level of nfinement using a horizontal line. As a result, when no FRP nfinement is present, the send branh of Lam and Teng s model redues diretly to the send branh of stress-strain models for unnfined nrete in many existing design des for nrete strutures suh as Eurode 2 (ENV ) although the ultimate strain value in eah de may be slightly different. This is learly an advantage. However, in appliations where dutility is the main nern, this simple approah may not be satisfatory. For example, in the seismi retrofit of nrete lumns, a small amount of FRP nfinement may be suffiient in enhaning the dutility of nrete although it is not suffiient to result in a signifiant enhanement in mpressive strength. In suh a situation, a more preise definition of the post-peak stress-strain response of FRP-nfined nrete is of interest to strutural engineers. For this reason, another modified version, referred to as Version II, of Lam and Teng s model is proposed in this setion. 1

11 In order to ater for both the asending and desending types of stress-strain urves of FRPnfined nrete, the stress-strain urve of unnfined nrete for design use is defined in a form similar to the well-known Hognestad s (1951) stress-strain urve. It is assumed that the stress-strain urve of unnfined nrete has a linear post-peak desending branh, whih terminates at an axial strain of.35 after a 15% drop from the mpressive strength of unnfined nrete. Note that in Hognestad s (1951) model, the ultimate axial strain of unnfined nrete was defined as.38, instead of.35. The latter is speified in design des suh as BS 811 (1997) and ENV (1991). In this alternative version, the paraboli first portion of Lam and Teng s model as given by Eq. 1a remains unhanged, while the linear send portion given by Eq. 1b is modified, leading to the following expressions: E E 2 E ( ) 2 t 4 f f E2 if K.1 f f u t u f ( ) if K.1 u ( ) where E 2, u and f are defined by Eqs 4, 9 and 15 respetively, and 14 but is subjeted to the following nditions: f u (16) is given by Eq. f f f (17) u.85, if K ; or.85, if K Eq. 17 limits the axial stress at the ultimate strain of FRP-nfined nrete f u to a minimum value of.85 f, beause the nrete is deemed to have failed at this stress level, regardless of whether the FRP jaket has failed by rupture or not. Moreover, it is proposed that if the value of f u predited by Eq. 14 is equal to or smaller than.85 f, the nrete should be taken as unnfined. In suh ases, the values of f u and u in Eq. 16 should be taken as.85 f and.35, respetively. It should be noted that Version II, shematially illustrated in Fig. 1, differs from Version I only for insuffiiently nfined nrete (i.e. with K.1). In Fig. 11, the performane of both modified versions for two pairs of insuffiiently nfined speimens is shown. These speimens were nfined with only 1 ply of GFRP. The first pair (speimens 2 and 21) had an unnfined nrete strength of 39.6 MPa, a nfinement ratio of.79 and a nfinement stiffness ratio of.9; the rresponding values for the send pair (speimens 28 and 29) were 45.9 MPa,.67 and.78. The predited stress-strain urves were obtained using.2. Note that for the speimens of Fig. 11a, the nfinement ratio is greater than the minimum value of.7 for suffiient nfinement defined by Lam and Teng (23), but for the speimens of Fig. 11b, the nfinement ratio is smaller than.7. In both ases, the nfinement stiffness ratios are smaller than the ritial value of.1 as defined in the present study. The original model predits a monotonially asending stress-strain urve for 11

12 the speimens of Fig. 11a, but a stress-strain urve with a horizontal send portion for the speimens of Fig. 11b; the test urves are of the desending type in both ases. Version I of the modified Lam and Teng model predits a horizontal send portion for speimens of both Figs 1a and 1b, whih does not math the test urves well but provides a reasonably lose approximation. By ntrast, Version II of the modified Lam and Teng model is apable of prediting a desending branh and shows improved agreement with the test stress-strain urves. 8 Conlusions This paper has presented the results of a reent study aimed at the refinement of the designoriented stress-strain model for FRP-nfined nrete originally developed by Lam and Teng (23). Some weaknesses of this model have been illustrated through mparisons with reent test results obtained at The Hong Kong Polytehni University. New equations for prediting the ultimate axial strain and the mpressive strength of FRP-nfined nrete have been developed based on the interpretation of these test results and results from a parametri study using a reent analysis-oriented stress-strain model (Teng et al. 27a, Jiang and Teng 27). Two modified versions of Lam and Teng s (23) model have been proposed. Version I involves only the updating of the ultimate axial strain and mpressive strength equations. Version II is apable of prediting stress-strain urves with a desending branh, whih is not atered for by the original model of Lam and Teng (23). The following nlusions an be drawn from the present study: 1. The original model of Lam and Teng (23) overestimates the ultimate strain of nrete nfined with a large amount of FRP and the mpressive strength of nrete nfined with a small amount of FRP. 2. The new ultimate strain and mpressive strength equations aunt for the effets of nfinement stiffness and jaket strain apaity separately and provide lose preditions of test results. 3. Both modified versions of Lam and Teng s model provide muh loser preditions of test stress-strain urves than the original model. Version II performs better than Version I for test stress-strain urves with a desending branh, although Version I also provides a reasonably lose approximation of suh test stress-strain urves. 4. Both modified versions are suitable for diret use in pratial design and for inlusion in design des/speifiations. Version I is more mpatible with most urrent design des for nrete strutures as the send portion of the predited stress-strain urve redues naturally to a horizontal line for unnfined nrete. Aknowledgements The authors are grateful for the finanial support reeived from the Researh Grants Counil of the Hong Kong SAR (Projet No: PolyU 559/2E) and The Hong Kong Polytehni University (Projet Codes: RG88 and BBZH). 12

13 Referenes ACI-318 (25). Building Code Requirements for Strutural Conrete and Commentary, Amerian Conrete Institute, Farmington Hills, Mihigan, USA. ACI-44.2R (28). Guide for the Design and Constrution of Externally Bonded FRP Systems for Strengthening Conrete Strutures, Amerian Conrete Institute, Farmington Hills, Mihigan, USA. Berthet, J.F., Ferrier, E. and Hamelin, P. ( 26). Compressive behavior of nrete externally nfined by mposite jakets - Part B: modeling, Constrution and Building Materials, 2(5), Binii, B. (25). An analytial model for stress strain behavior of nfined nrete, Engineering Strutures, 27(7), BS 811 (1997). Strutural Use of Conrete, Part 1, Code of pratie for design and nstrution, British Standards Institution, London, UK. Carey, S.A. and Harries, K.A. (25). Axial behavior and modeling of nfined small-, medium-, and large-sale irular setions with arbon fiber-reinfored polymer jakets, ACI Strutural Journal, 12 (4), Conrete Soiety (24). Design Guidane for Strengthening Conrete Strutures with Fibre nd Composite Materials. Conrete Soiety Tehnial Report No. 55, 2 Edition, Crowthorne, Berkshire, UK. Chun, S.S. and Park, H.C. (22). Load arrying apaity and dutility of RC lumns nfined by arbon fiber reinfored polymer. Proeedings, 3rd International Conferene on Composites in Infrastruture (CD-Rom), San Franis. ENV (1991). Eurode 2: Design of Conrete Strutures Part 1: General Rules and Rules for Buildings, European Committee for Standardization, Brussels. Fam, A.Z. and Rizkalla, S.H. (21). Confinement model for axially loaded nrete nfined by irular fiber-reinfored polymer tubes, ACI Strutural Journal, 98(4), Fardis, M.N. and Khalili, H. (1982). FRP-enased nrete as a strutural material, Magazine of Conrete Researh, 34(122), Harries, K.A. and Kharel, G. (22). Behavior and modeling of nrete subjet to variable nfining pressure, ACI Materials Journal, 99(2), Hognestad, E. (1951). A Study of Combined Bending and Axial Load in Reinfored Conrete Members, Bulletin Series No. 399, Engineering Experiment Station, University of Illinois, Urbana, III. Kar bhari, V.M. and Gao, Y. (1997). Composite jaketed nrete under uniaxial mpression verifiation of simple design equations, Journal of Materials in Civil Engineering, ASCE, 9(4), Jiang, T., and Teng, J.G. (27). Analysis-oriented models for FRP-nfined nrete: a mparative assessment, Engineering Strutures, 29 (11), Lam, L., and Teng, J.G. (23). Design-oriented stress-strain model for FRP-nfined nrete, Constrution and Building Materials, 17(6-7), Lam, L., and Teng, J.G. (24). Ultimate ndition of fiber reinfored polymer-nfined nrete, Journal of Composites for Constrution, ASCE, 8(6), Lam, L., Teng, J.G., Cheng, C.H. and Xiao, Y. (26). FRP-nfined nrete under axial yli mpression, Cement and Conrete Composites, 28(1), Lillistone, D. and Jolly, C.K. (2). An innovative form of reinforement for nrete lumns using advaned mposites, The Strutural Engineer, 78(23/24),

14 Marques, S.P.C., Marques, D.C.S.C., da Silva J.L. and Cavalante, M.A.A. (24). Model for analysis of short lumns of nrete nfined by fiber-reinfored polymer, Journal of Composites for Constrution, ASCE, 8(4), Mattys, S., Toutanji, H., Audenaert, K. and Taerwe, L. (25). Axial behavior of large-sale lumns nfined with fiber-reinfored polymer mposites, ACI Strutural Journal, 12(2), Mirmiran, A. and Shahawy, M. (1997). Dilation harateristis of nfined nrete, Mehanis of Cohesive-Fritional Materials, 2 (3), Miyauhi, K., Inoue, S., Kuroda, T. and Kobayashi, (1999). Strengthening effets of nrete lumns with arbon fiber sheet, Transations of the Japan Conrete Institute, 21, Popovis, S. (1973). Numerial approah to the mplete stress-strain relation for nrete, Cement and Conrete Researh, 3(5), Roa, S., Galati, N. and Nanni, A. (26). Large-size reinfored nrete lumns strengthened with arbon FRP: experimental evaluation, Proeedings, 3 rd International Conferene on FRP Composites in Civil Engineering, Deember , Miami, Florida, USA. Saafi, M., Toutanji, H.A. and Li, Z. (1999). Behavior of nrete lumns nfined with fiber reinfored polymer tubes, ACI Materials Journal, 96(4), Samaan, M., Mirmiran, A., and Shahawy, M. (1998). Model of nrete nfined by fiber mposite, Journal of Strutural Engineering, ASCE, 124(9), Saenz, N. and Pantelides, C.P. (27). Strain-based nfinement model for FRP-nfined nrete, Journal of Strutural Engineering, ASCE, 133 (6), Spoelstra, M.R., and Monti, G. (1999). FRP-nfined nrete model, Journal of Composites for Constrution, ASCE, 3(3), Teng, J.G., and Lam, L. (24). Behavior and modeling of fiber reinfored polymer- nfined nrete, Journal of Strutural Engineering, ASCE, 13(11), Teng, J.G., Huang, Y.L., Lam, L., and Ye, L.P. (27a). Theoretial model for fiber reinfored polymer-nfined nrete, Journal of Composites for Constrution, ASCE, 11(2), Teng, J.G., Yu, T., Wong, Y.L., and Dong, S.L. (27b). Hybrid FRP-nrete-steel tubular lumns: nept and behavior, Constrution and Building Materials, 21(4), Toutanji, H.A. (1999). Stress-strain harateristis of nrete lumns externally nfined with advaned fiber mposite sheets, ACI Materials Journal, 96(3), Wu, G., Wu, Z.S. and Lu, Z.T. (27). Design-oriented stress-strain model for nrete prisms nfined with FRP mposites, Constrution and Building Materials, 21(5), Xiao, Y. and Wu, H. (2). Compressive behavior of nrete nfined by arbon fiber mposite jakets. Journal of Materials in Civil Engineering, ASCE, 12(2), Xiao, Y. and Wu, H. (23). Compressive behavior of nrete nfined by various types of FRP mposites jakets. Journal of Reinfored Plastis and Composites, 22(13), Yeh, F.Y. and Chang, K.C. (27). "Confinement effiieny and size effet of FRP nfined irular nrete lumns", Strutural Engineering and Mehanis, 26(2), Youssef, M.N. (23). Stress-strain Model for Conrete Confined by FRP Composites, Ph.D. Dissertation, University of California, Irvine. Youssef, M.N., Feng, M.Q. and Mosallam, A.S. (27). Stress-strain model for nrete nfined by FRP mposites, Composites Part B Engineering, 38(5-6),

15 Table 1 Properties of test speimens used in this paper Soure Lam et al. (26) Teng et al. (27b) Jiang and Teng (27) Speimen D H f Fiber t f ( f u ) u (mm) (mm) (MPa) (%) type (mm) (GPa) (%) (MPa) (%) Carbon Carbon Carbon Glass (38.8) Glass (37.2) Glass Glass Glass Glass Glass (4.5) Glass (4.5) Glass Carbon Carbon Carbon Carbon Carbon Carbon E frp h,rup 15

16 Table 2 Parameters used in the parametri study Conrete f (MPa) 2 to 5 at an interval of 5 CFRP E (GPa) 23 GFRP frp h,rup.75 t (mm) to 1 at an interval of.1 E (GPa) 8 frp h,rup.15 t (mm) to 1.5 at an interval of.1 16

17 4.5 Normalized Axial Stress Speimens 2,21 K =.9, =8.7 Speimens 24,25 K =.27, =9.1 Speimens 38,39 K =.23, =4.4 Speimens 34,35 K =.11, = Normalized Axial Strain (a) Axial stress-axial strain urves Normalized Axial Stress Speimens 2,21 K =.9, =8.7 Speimens 38,39 K =.23, =4.4 Speimens 34,35 K =.11, =4.9 Speimens 24,25 K =.27, = Normalized Lateral Strain l (b) Axial stress-lateral strain urves Fig. 1 Typial stress-strain behavior of FRP-nfined nrete ylinders 17

18 Normalized Axial Stress /f Speimen 31 Speimen Normalized Axial Strain / (a) Comparison between speimen 31 and speimen 45 Normalized Axial Stress /f Speimen 28 Speimen Normalized Axial Strain / (b) Comparison between speimen 28 and speimen 42 Fig. 2 Effet of nfinement stiffness on stress-strain behavior of FRP-nfined nrete 18

19 f Axial Stress f Unnfined nrete (ENV 1992) FRP-nfined nete (Lam and Teng) t.35 u Axial Strain Fig. 3 Illustration of Lam and Teng s model 19

20 25 Normalized Ultimate Axial Strain u / - Predited Lam and Teng (24), CFRP Lam and Teng (24), GFRP Lam et al. (26), CFRP 5 Teng et al. (27b), GFRP Jiang and Teng (27), GFRP Jiang and Teng (27), CFRP Normalized Ultimate Axial Strain u / - Test (a) Ultimate axial strain Normalized Compressive Strength f /f - Predited Lam and Teng (24), CFRP Lam and Teng (24), GFRP Lam et al. (26), CFRP Teng et al. (27b), GFRP Jiang and Teng (27), GFRP Jiang and Teng (27), CFRP Normalized Compressive Strength f /f - Test (b) Compressive strength Fig.4 Performane of Lam and Teng s model against the present test results 2

21 Normalized Ultimate Axial Strain u / - Predited Lam and Teng (24), CFRP Lam and Teng (24), GFRP Lam et al. (26), CFRP Teng et al. (27b), GFRP Jiang and Teng (27), GFRP Jiang and Teng (27), CFRP Lam and Teng (23) Normalized Ultimate Axial Strain u / - Test Fig. 5 Performane of Eq. 9 21

22 Normalized Axial Stress / f Jiang and Teng (27) Fitted Line Normalized Lateral Strain l / (a) First two steps.2 CFRP-nfined GFRP-nfined Proposed Equation -.2 Slope K Confinement Stiffness Ratio K (b) Last step Fig. 6 Demonstration of the parametri study 22

23 Normalized Axial Stress at Ultimate Axial Strain f u /f - Predited Lam and Teng (24), CFRP Lam and Teng (24), GFRP Lam et al. (26), CFRP 1 Teng et al. (27b), GFRP Jiang and Teng (27), GFRP Jiang and Teng (27), CFRP Normalized Axial Stress at Ultimate Axial Strain f u /f - Test Fig. 7 Performane of Eq. 13 Normalized Compressive Strength f /f - Predited Lam and Teng (24), CFRP Lam and Teng (24), GFRP Lam et al. (26), CFRP Teng et al. (27b), GFRP Jiang and Teng (27), GFRP Jiang and Teng (27), CFRP Lam and Teng (23) Normalized Compressive Strength f /f - Test Fig. 8 Performane of Eq

24 7 6 Axial Stress (Pa) f = 39.6MPa E frp = 81MPa t =.34mm R = 76mm h,rup =.25 Lam and Teng (23) 1 Modified Lam and Teng Model - Vesion I Test (2 Speimens) Axial Strain (a) Speimens 22 and Axial Stress (MPa) Lam and Teng (23) 1 Modified Lam and Teng Model - Vesion I Test (2 Speimens) Axial Strain f = 39.6MPa E frp = 81MPa t =.51mm R = 76mm h,rup =.181 (b) Speimens 24 and 25 24

25 9 8 Axial Stress (MPa) f = 38.9MPa 4 E frp = 247MPa t =.33mm 3 R = 76mm 2 h,rup =.994 Lam and Teng (23) 1 Modified Lam and Teng Model - Vesion I Test (3 Speimens) Axial Strain () Speimens 17,18 and 19 Fig. 9 Performane of Version I of the modified Lam and Teng model 25

26 f K >.1 Axial Stress f f u.85f Unnfined K <.1 K =.1.35 u u u Axial Strain Fig. 1 Shemati of Version II of the modified Lam and Teng model 26

27 Axial Stress (MPa) Lam and Teng (23) Modified Lam and Teng Model - Version I 5 Modified Lam and Teng Model - Version II Test (2 Speimens) Axial Strain (a) Speimens 2 and 21 f = 39.6MPa E frp = 81MPa t =.17mm R = 76mm h,rup = Axial Stress (MPa) 35 3 f = 45.9MPa 25 E frp = 81MPa t =.17mm 2 R = 76mm 15 h,rup = Lam and Teng (23) Modified Lam and Teng Model - Version I 5 Modified Lam and Teng Model - Version II Test (2 Speimens) Axial Strain (b) Speimens 28 and 29 Fig. 11 Predition of dereasing type of stress-strain urves 27