Stress strain model for concrete confined by FRP composites

Transcription

1 Composites: Part B 38 (27) Stress strain model for onrete onfined by FRP omposites Marwan N. Youssef, Maria Q. Feng, Ayman S. Mosallam * Department of Civil and Environmental Engineering, University of California, Irvine, CA , USA Reeived 2 Marh 26; aepted 6 July 26 Available online 27 Deember 26 Abstrat In this paper, a stress strain model for onrete onfined by fiber reinfored polymer (FRP) omposites is developed. The model is based on the results of a omprehensive experimental program inding large-sale irular, square and retangular short omns onfined by arbon/epoxy and E-glass/epoxy jakets providing a wide range of onfinement ratios. Ultimate stress, rupture strain, jaket parameters, and ross-setional geometry were found to be signifiant fators affeting the stress strain behavior of FRP-onfined onrete. Suh parameters were analyzed statistially based on the experimental data, and equations to theoretially predit these parameters are presented. Experimental results from this study were ompared to the proposed semi-empirial model as well as others from the literature. Ó 27 Elsevier Ltd. All rights reserved. Keywords: A. Laminates; A. Polymer-matrix omposites (PMCs); C. Analytial modeling; D. Mehanial testing; Conrete. Introdution The need for strengthening defiient existing reinfored onrete (RC) omns is triggered by many reasons. For years, engineers have been studying ways to retrofit or strengthen existing defiient RC omns to meet new ode requirements, espeially in earthquake prone areas. The use of fiber reinfored polymers (FRP) jakets as an external mean to strengthen existing RC omns has emerged in reent years with very promising results [,3,4,7,8]. Espeially for irular omns, suh tehnique has proved to be very effetive in enhaning their dutility and axial load apaity [2,2,25]. Reently many studies have been onduted to evaate the ultimate strength and strain enhanement of FRP-onfined onrete [4,5,2,22,26,28,3 33]. This field; however, remains in its infany stages and more studies are needed to explore its apabilities, limitations, and design appliability. Many researhers and design engineers are still using steel-onfined onrete models in prediting the behavior and * Corresponding author. address: mosallam@ui.edu (A.S. Mosallam). design of FRP strengthened omns [9]. This is aused by the fat that a general FRP-onfined onrete model is yet to be adopted by industry odes. As expeted, studies have shown that FRP-onfined onrete behaves differently from steel-onfined onrete [2]. As a result, several FRP-onfinement models have been developed to fill the gap and to better understand the behavior of the FRP jakets [9,,6,27,29]. The majority, if not all, of suh models have been developed from experimental data produed by testing small onrete speimens wrapped with FRP jakets. The most popular speimen size used is the onventional 52 mm 35 mm (6 in. 2 in.) standard onrete ylinders, and some researhers even used 76 mm 35 mm (3 in. 2 in.) [3]. Consequently models were developed based on limited test database, and therefore, its appliability is unlear, espeially with regard to a wide range of onfinement ratios. This paper presents a new general onfinement model developed by the authors for FRP-onfined onrete, based on testing large-sale axially loaded speimens with a wide range of onfinement ratios [36,37]. This model is appliable to irular as well as retangular onrete omns retrofitted by FRP jakets. The model predits the ultimate /$ - see front matter Ó 27 Elsevier Ltd. All rights reserved. doi:.6/j.ompositesb

2 M.N. Youssef et al. / Composites: Part B 38 (27) Nomenlature A A j A s b D E E j f u f l f f jt f ju f f ross-setional area of onrete ross-setional area of FRP jaket total area of longitudinal steel in a omn width of retangular setion diameter of irular omn modus of elastiity of onrete tensile modus of FRP jaket in the hoop diretion ompressive strength of unonfined onrete (measured on the day of test) ultimate strength of FRP-onfined onrete lateral onfining stress onrete stress FRP jaket stress at transition from first to seond region tensile strength of FRP jaket lateral onfining stress at ultimate ondition of FRP jaket effetive lateral onfining stress at ultimate ondition of FRP jaket f t h k e r t j t l e e u e jt e ju e t q j q l axial stress at the boundary point of the first and seond region where the jaket is beginning to get fully ativated depth of retangular setion onfinement effetiveness oeffiient orner radius of retangular setion total thikness of FRP jaket thikness per layer of FRP jaket strain in onrete ultimate onfined onrete ompressive strain FRP jaket strain at transition from first to seond region =.2 ultimate tensile strain of FRP jaket axial strain at the boundary point of the first and seond region where the jaket is beginning to get fully ativated vometri ratio of FRP jaket area ratio of longitudinal reinforement strength, ultimate stain, and depits the entire stress strain diagram for FRP-onfined onrete speimens. 2. Experimental program 2.. Test speimen A total of 87 large unreinfored speimens were tested under pure axial load. In addition a total of 3 52 mm 35 mm (6 in. 2 in.) onrete ylinders were tested. Table shows the test program onduted in this study. Both arbon/epoxy and E-glass/epoxy jakets were used to onfine the omn speimens. Table 2 presents the omplete material properties used in the study. The FRP laminates were applied diretly to the pretreated surfaes of the speimens providing unidiretional lateral onfinement in the hoop diretion. An overlap of 52 mm (6-in.) was used to ensure the development of full omposite tensile strength. Ready-mix onrete with nominal 28-day strength of MPa (4 psi) to MPa (5 psi) was used. All speimens were instrumented with strain gages bonded to both the onrete surfaes and external laminates, at two perpendiular diretions. The strain gages were plaed at mid-height of the speimen. Two linear variable differential transduers (LVDT) per speimen were used, 8 degrees apart, to apture the vertial displaements over the middle half of the speimens. For the 52 mm 35 mm (6 in. 2 in.) ylinders, both ends of all ylinders were apped using sulfur apping to insure parallel surfaes and to distribute the load uniformly in order to redue load eentriity Testing mahine The 3, kn (7-million pound) ompression mahine used to rush the speimens is a 4-post up-ating hydrauli press. The bottom ylinder is designed to swivel in order to minimize eentriity. This 4.88 m (6 ft) high mahine, shown in Fig., is very unique in terms of its very large loading apaity; 73 m (68 in.) open daylight; 94 m (37 in.) diameter main ylinder, and an over all weight of 5, kg (, lbs) [34 36]. 3. Mehanis of onfinement Under low level of longitudinal strain, the onrete is known to behave elastially and the transverse strain is related proportionally to the longitudinal strain by the Poisson s ratio. As the load inreases, raks start to form leading to a large inrease in the transverse strain. Assuming deformation ompatibility, the lateral strain of the onfined speimens is equal to the strain in the FRP jaket. The tendeny of onrete to dilate after raking and the radial stiffness of the onfining jaket to restrain the onrete dilation, are onsidered to be two important fators affeting the onrete onfinement. By wrapping the onrete with an external ontinuous FRP jaket, the fibers in the hoop diretion resist the transverse expansion of the onrete providing a onfining pressure. At low levels of longitudinal stress; however, the transverse strains are so low that the FRP jaket indues little onfinement, if any. At higher longitudinal stress levels, the dramati inrease in transverse tensile strains ativates the FRP jaket and the onfining pressure beomes

3 66 M.N. Youssef et al. / Composites: Part B 38 (27) Table Test program and properties of unreinfored test speimens Speimen ross-setion mm mm (in. in.) Height mm (in.) No. of ontrol speimen No. of jaketed speimens No. of plies of arbon fiber No. of plies of E-glass Retangle ( 5) 762 (3) Cirular 46B 83 (6B 32) 83 (32) Square (5 5) 762 (3) Cylinder (6 2) 35 (2) more signifiant. The general onfining pressure indues a tri-axial state of stress in the onrete. It is well understood that onrete under tri-axial ompressive stress exhibits superior behavior, in both strength and dutility, as ompared to onrete in uniaxial ompression [24]. 3.. Lateral onfining stress, f The lateral onfining stress (f ) is produed in onfined onrete when the member is loaded suh that the onrete starts to dilate and expands laterally. The vae of suh stress depends on the ross-setion geometry of the onfined member and the amount and mehanial properties of onfining materials provided. For example, when onfining a irular member, the FRP jaket provides a uniform onfining stress around the parameter resulting in a great improvement in member s behavior under loading. On the other hand, onfining square or retangular members tends to produe onfining stress onentrated around the orners of suh members, as shown in Fig. 2. In fat, all of the square and retangular omns tested in this study failed by the rupture of the FRP jaket that started at one of the orners Effetive lateral onfining stress f To study and quantify the behavior of onrete members onfined by FRP jaket, the amount of onfining pressure (stress) provided by the jaket should be determined. Suh onfining pressure is a funtion of the omn s ross-setion, the stiffness of the FRP jaket, and the transverse expansion of the loaded onrete. To alate the lateral onfining stress, f, applied to the onrete by onfinement, a free-body diagram of a irular rosssetion onfined with an FRP jaket is onsidered as shown in Fig. 3. Based on stati analysis, equilibrium of fores, deformation ompatibility, and by onsidering ft (.3 m) setion along the omn height, the following expression an be written: f ¼ 2f jut j ðþ D Introduing q j as the onfinement ratio, whih is defined as the jaket vome divided by the onrete vome, i.e., q j ¼ 4t j D Substitute in Eq. (), we have f ¼ 2 q jf ju ð2þ ð3þ The effetive lateral onfining stress at ultimate ondition of the FRP jaket is defined as f ¼ k ef ð4þ

4 M.N. Youssef et al. / Composites: Part B 38 (27) Table 2 Mehanial properties of the omposite systems System Thikness per layer mm (in.) Young s modus E j GPa (Msi) Tensile strength f ju MPa (ksi) Faire strain e ju (%) Carbon/epoxy.584 (.23) 3.8 (5.6) 246 (8.7).25 E-glass/epoxy.559 (.22) 8.5 (2.68) 425 (6.6) 2.6 E-glass/epoxy.43 (.45) 8.5 (2.68) 425 (6.6) 2.6 Fig.. Setup for testing large onrete omns. where k e is the onfinement effetiveness oeffiient. For irular omns, k e =.. Similarly, for retangular setions, it an be shown that f ¼ 2 q jf ju Unlike the irular setion, however, the onfinement effetiveness oeffiient for retangular setion is less than unity (i.e. k e is less than one). The vaes of k e are alated using the expression proposed by Restrepo and DeVino [23]. For retangular and square omns, k e ¼ h ðb 2rÞ2 þðh 2r Þ 2 3hb ð5þ i q l ð6þ q l where b and h are the width and depth of the ross-setion, respetively; r is the orner radius of the omn; q l is the longitudinal reinforement ratio of the setion = As, where bh A s is the area of the longitudinal steel. Fig. 2. Confinement of onrete omns with FRP omposite jakets. (a) Confined and unonfined portions of square setion. (b) Dilated square omn onfined with arbon/epoxy jaket Development of general FRP onfinement model In order to develop a theoretial stress strain model for onrete onfined by FRP jaket, the experimental results in the form of axial stress versus axial strain must be produed and analyzed arefully. As expeted, the stress strain diagram of the tested speimen indiates three different stages are enountered throughout the testing sheme. The three stages are: Stage : During this stage, the initial portion of the stress strain response of onfined onrete follows the path of unonfined onrete. Therefore, the slope

5 68 M.N. Youssef et al. / Composites: Part B 38 (27) results indiate that the urve exhibits a linear behavior up to the rupture of the jaket. This fat was also reported by other researhers [6,27,26]. This portion of the urve; however, was observed to be either asending or desending, depending of the ross-setional geometry of the omn and the amount of FRP onfinement provided. Fig. 3. Free body diagram of irular omn onfined by FRP jaket. of the urve in this stage is taken as, E, the onrete modus of elastiity. Other researhers used different formulas to alate E ; however, in this study, the experimental results orrelated very well with the vae for E reommended by ACI 38, where pffiffiffiffi E ¼ 47 f ðmpaþ ð7þ pffiffiffiffi E ¼ 57; ðpsiþ ð8þ f The proposed model was, therefore, developed based on this vae of E. Stage 2: After exeeding the unonfined strength of onrete, the stress strain urve starts to soften forming a transition zone within whih raks in the onrete starts to expand and the onrete start to dilate and bear against the jaket ativating it. Stage 3: The FRP jaket at this stage is fully ativated and the onfining stress provided by the jaket ontinues to inrease proportional to the applied load until faire. Within this stage, the experimental Based on these observations, the general stress strain urve shown in Fig. 4 will be used in the modeling proess for FRP-onfined onrete. The asending urve would represent situations where the slope of the urve, E 2, in the third zone is positive. This would be the ase of omns with moderate to high onfinement ratios whih also depends on the geometry of the ross-setion. Cirular omns, for instane, with moderate onfinement ratio usually have an asending stress strain urve. The desending urve represents low onfinement for irular ross-setions and low to moderate onfinement for retangular setions. The fat that the intent is to develop a general onfinement model for FRP-onfined onrete, both asending and desending ases are arefully addressed in this study. The essential points along the stress strain urve of interest are: Point A: The beginning point before the load is applied where the axial stress and axial strain are zero. Point B: Represents the end of stage 2 where the jaket is getting fully ativated under the effet of tensile stresses due to onrete dilation. From this point on, the stress provided by the jaket will ontinue to inrease until jaket rupture. This point will be represented by e t and f t as x and y oordinates, respetively. Point C: Point C represents the ultimate ondition where the jaket fails at an ultimate axial stress u, and an ultimate axial strain of e u. The term u Fig. 4. Proposed model for FRP-onfined onrete.

6 will be defined as the ultimate strength of FRPonfined onrete, and e u as the ultimate onfined onrete ompressive strain. In the analysis, the stress strain urve is divided into two regions, 6 e 6 e t, and e t 6 e 6 e u. In the first region, the experimental data, for both irular and retangular omns, trae that of unonfined onrete path. In the seond region; however, the urve for irular setions with moderate onfinement level was linear with a positive (asending) slope. The seond portion of the urve, in most ases of retangular and square omns, maintained a negative (desending) slope for most of the onfinement ratios used. Even irular omns with low onfinement ratio exhibit a desending behavior. In order to develop a unified model, both ases, E 2 >, and E 2 < are onsidered. In order to desribe the analytial proposed urve, several parameters were defined and established. The most important parameters are the ultimate strength of FRPonfined onrete, f u, and the ultimate onfined onrete ompressive strain, e u. Other parameters needed are f t and e t, whih are the axial stress and axial strain at the transition boundary point at the end of stage 2 and the beginning of stage 3, respetively. Using a similar approah used by Hoshikuma et al. [] to develop a onfinement model for onrete onfined by transverse steel, it is proposed that the onrete stress in region ( 6 e 6 e t ) be modeled by the following polynomial funtion: f ¼ C e n þ C 2e þ C 3 where C, C 2, C 3, and n are onstants to be determined from the following boundary onditions. First region where 6 e 6 e t, and E 2 >, () f =@e =, (2) df /de = e =, (3) df /de = E e = e t, (4) f = f e = e t. Now, substituting the four boundary onditions in Eq. (9) one obtains " f ¼ E e n E # n 2 e ðþ E e t n ¼ ðe E 2 Þe t ðþ E e t f t First region where 6 e 6 e t, and E 2 <, When E 2 <, the third boundary ondition above is hanged to reflet the fat that, from experimental observation, a tangent to the urve at (e t,f t ) is a horizontal straight line. df =de ¼ e t ð2þ Substituting the four boundary onditions, with the revised third ondition (Eq. (2)) into Eq. (9), one obtains M.N. Youssef et al. / Composites: Part B 38 (27) " f ¼ E e # n ð3þ n ð9þ e e t n ¼ E e t ð4þ E e t f t Seond region where e t 6 e 6 e u. Appliable to both asending and desending ases, the stress strain diagram is modeled as a straight line between two points desribed as f ¼ ae þ b ð5þ where a and b are onstants determined from the following boundary onditions: () f = f e = e t, (2) f ¼ f e ¼ e u. Substituting the two boundary onditions in Eq. (5), one obtains f ¼ f t þ E 2 ðe e t Þ ð6þ 3.4. Evaation of model parameters In the proposed model, the following ontrol parameters are of interest: (i) the ultimate strength of FRP-onfined onrete, u ; (ii) the ultimate onrete ompressive strain, e u ; (iii) the axial stress at the boundary point between the first and seond region, f t ; (iv) the axial strain that orresponding to f t, e t ; (v) and slope of the stress strain urve within the seond region, E 2. The effet of onfinement provided by the external jaket is determined using regression analysis of the experimental data produed from this study, as well as data from other researhers [5,,3]. Although the literature ontains many experimental data on the subjet of FRPonfined onrete, the majority of the data is obtained from standard 52 mm 35 mm (6 in. 2 in.) ylinders tests. In this modeling task, large and small speimens are onsidered to over a larger range of sizes and onfinement ratios. The main parameters that are likely to infene the onfinement effet produed by an external FRP jaket are the following: (i) vometri ratio of FRP jaket, q j ; (ii) tensile strength of FRP jaket, f ju ; (iii) tensile modus of elastiity of FRP jaket in the hoop diretion, E j ; (iv) ultimate tensile strain of FRP jaket, e ju ; (v) ompressive strength of unonfined onrete, ; (vi) ross-setion of the onrete speimen and its dimensions Ultimate strength of FRP-onfined onrete, f u This parameter is one of the two most important parameters of a stress strain model for onfined onrete. The ultimate strength is the stress reahed by the onfined onrete just before faire. For retrofitted irular omns, the ultimate stress, for moderate to high onfinement ratios, is always higher than the unonfined onrete

7 62 M.N. Youssef et al. / Composites: Part B 38 (27) stress, and the stress strain urve is almost always asending. The inrease in the ultimate stress is due to the uniform onfinement around a irular ross-setion provided by the external jaket. For square and retangular speimens, on the other hand, the stress strain urve is mostly desending exept in ases where the onfinement ratio is very high. Even then, the urve is only slightly asending. All of these statements were observed in the experimental phase of this study, and were onsidered in the model development. A formula relating the ultimate strength of FRP-onfined onrete to parameters that would impat its vae is usually referred to as a Strength Model. Suh formula usually represents a linear relationship between the onfinement effetiveness fator, f u as follows: f u f ¼ þ k f l f, and the onfinement ratio, f l f, ð7þ where k is the effetiveness oeffiient desribed earlier. Using regression analysis, the onfinement effetiveness fator, f u, and the onfinement ratio, f, were related for irular and retangular speimens using linear relationship as shown in Figs. 5 and 6. The following strength model is proposed, depending on the geometry of the ross-setion: 8 For irular setions u : þ 2:25 f 5 4 >< ¼ ð8þ For retangular setions :5 þ :225 f 3 >: 5 f The orrelation oeffiient for the irular setions is 93%, using 63 data points, and that for the retangular setions is 94% using 38 data points Ultimate onrete ompressive strain, e u The ultimate onrete ompressive strain is onsidered also to be a very important parameter of the stress strain urve of onfined onrete. In order to alate the available ultimate rotation apaity at a plasti hinge in a reinfored onrete flexural member, it is neessary to be able to predit the ultimate onrete ompressive strain e u. Similar to strength models, the formula for determining the ultimate onrete ompressive strain is some time referred to as a Strain Model. The effetive onfining stress, f, a funtion of speimen s ross-setion and jaket harateristis, as well as the mehanial properties of the jaket, will be used in the regression analysis to determine e u. Figs. 7 and 8 show the onfinement effetiveness fator f fju 2 E j versus the ultimate ompressive strain, e u. f The relationship between e u and fju 2 E j may be approximated by a linear funtion. The following relationships are obtained from regression analysis: 8 For irular setions :3368 þ :259 f fju 2 >< E j e u ¼ ð9þ For retangular setions :4325 þ :2625 f >: fju 2 f The orrelation oeffiient for the irular setions is 94%, using 63 data points, and that for the retangular setions is 96% using 38 data points Axial stress, f t Another parameter needed for prediting the entire stress strain diagram is the axial stress at the boundary point between the first and seond regions, f t. Although u and e u are onsidered to be the most important parameters of onfined onrete, the ability to predit the entire f Fig. 5. Relation between onfinement effetiveness and ultimate stress for irular omns.

8 M.N. Youssef et al. / Composites: Part B 38 (27) Fig. 6. Relation between onfinement effetiveness and ultimate stress for retangular omns. Fig. 7. Relationship for the ultimate strain for irular omns. stress strain urve is also important to be able to plot a load displaement urve or a moment-interation diagram for onfined member. The loation of f t along the stress strain urve is a funtion of the onrete strength, f, whih has a signifiant impat as the onrete starts to dilate and therefore bears against the jaket. Other important terms that would impat the vae of f t is the stress in the jaket at that point and the amount of jaket used, and the geometry of the ross-setion. The stress in the jaket, f jt, is equal to e jt E j. The portion of the jaket used and the ross-setion of the omn are represented by the vometri ratio of FRP jaket, q j. No other apparent fators were shown to affet the loation of f t. Regression analysis is then performed between ft and q je jt E j. Figs. 9 and show the results for suh regression analysis for irular and retangular omns, respetively. q je j e jt f 5 4 may be approx- The relationship between ft and imated by a linear funtion. The following relation are obtained from regression analyses: 8 For irular setions f t ¼ >< >: 5 4 : þ 3: q je j e jt For retangular setions 5 4 : þ :35 q je j e jt f ð2þ

9 622 M.N. Youssef et al. / Composites: Part B 38 (27) Fig. 8. Relationship for the ultimate strain for retangular omns. Fig. 9. Relationship for the axial stress f t for irular omns. The orrelation oeffiient for the irular setions is 87%, using 5 data points, and is 9% using 38 data points for the retangular setions Axial strain, e t The last parameter to be determined by regression analysis of the experimental data is the axial strain e t orresponding to the axial stress f t. This axial strain ats along the stress strain urve loated at the boundary between region and region 2. After deiding the regression terms for f t, i.e. q je jt E j, the use of the same term was attempted in the regression analysis of e t. It was very obvious that ft and e t are in good relation with the hoop tension of the FRP jaket. Figs. and 2 show the onfinement effetiveness fator versus the axial strain, e t 6 7 fju 2. q j E j e jt f E j q 6 je j e jt 7 fju 2 The relationship between e t and E j may be approximated by a linear funtion. The following relations are obtained from regression analyses: 8 For irular setions 6 7 fju 2 :2748 þ :69 q je j e >< jt e t ¼ For retangular setions >: :2 þ :775 q je j e jt f 6 7 fju 2 f E j ð2þ The orrelation oeffiient for the irular setions is 85%, using 5 data points, and is 88% using 38 data points for the retangular setions. Tables 3 and 4 show the equations for the model parameters, with orrelation oeffiients and number of observations for irular and retangular setions.

10 M.N. Youssef et al. / Composites: Part B 38 (27) Fig.. Relationship for the axial stress f t for retangular omns. Fig.. Relationship for the axial stress e t for irular omns Slope of seond branh, E 2 In this analysis, the portion of the stress strain urve in region 2 was modeled as a straight line between the points (e t,f t ) and ðu ; e uþ as follows: f ¼ f þ E 2ðe e t Þ ð22þ This is appliable to both asending and desending ases, where E 2 is positive or negative, respetively. With all of the onfinement parameters known, the omplete stress strain behavior of the FRP-onfined onrete members an now be desribed. 4. General onfinement model The parameters of the proposed onfinement model that is appliable to irular and retangular omns, are desribed below in their general form by the following equations: u f ¼ a þ b f ð23þ " e u ¼ k þ :26 f # fju 2 ð24þ " # 5 4 E j f t ¼ : þ g q je j e jt ð25þ " e t ¼ l þ w q 6 je j e 7 # jt f 2 ju ð26þ E j where a, b,, g, l, and w are shape fators that takes into effet the atual shape of the omn, and are given in Table 5.

11 624 M.N. Youssef et al. / Composites: Part B 38 (27) Fig. 2. Relationship for the axial stress e t for retangular omns. Table 3 Equations and orrelation oeffiients for irular setions Model parameter f u f Table 4 Equations and orrelation oeffiients for retangular setions Model parameter u 5. Performane analysis of proposed model 5.. Model parameters Proposed equation R 2 :5 þ :225 f fju e u :4325 þ :2625 f f t f e t Proposed equation R 2 : þ 2:25 f fju e u :3368 þ :259 f f t e t : þ 3: q jejejt Ej fju :2748 þ :69 q jejejt f f : þ :35 q 5 jejejt 4 f 9 38 :2 þ :775 q 6 jejejt 7 fju 2 f The four parameters of the proposed model ðf u ; e u ; f t ; e t Þ were analyzed, experimental versus theoretial f % % No. of observations No. of observations Table 5 Proposed general model shape fators a b k g l w Cirular Retangular Ej vaes, to determine their orrelation. The omparison was performed using the normalized effetive lateral onfining stress at ultimate onditions, f =f. The normalized fator represents the amount of onfinement provided to the omns. A good orrelation between the experimental and theoretial vaes was observed. The majority of the points fell between.8 and.2 along the y-axis indiating good preditions The proposed model versus experimental results Fig. 3 shows the experimental stress strain urves for the irular omns onfined with arbon/epoxy jaket ompared with the theoretial stress strain urve from the proposed model. As shown, exellent orrelation is ahieved in prediting the atual performane of the large-sale irular omns with FRP jakets Comparison between proposed versus existing models The performane of the proposed model versus other models available in the literature is ompared to the experimental data. In order to show the effetiveness of the proposed model, the stress strain relations predited by previous models were omputed for seleted test speimens and ompared with the experimental results. The omparison is performed based on the entire stress strain urve as well as for the atual vaes of the ultimate strength of FRP-onfined onrete, u, and the ultimate onrete ompressive strain, e u Cirular speimens A total of five existing onrete onfinement models were studied and ompared to the proposed model. Those models are in [9,,7,9,27].

12 M.N. Youssef et al. / Composites: Part B 38 (27) Fig. 3. Proposed model versus experimental data for irular omns. Fig. 4. Comparison of onfinement models to stress strain urve of C6LC2. Fig. 4 ompares the theoretial stress strain urves generated from the five published models with the experimental urve, for seleted speimen, produed from the study. As shown in this figure, Mander s model [8,9] seems to always overestimate the onfined onrete stress, u, where as its ultimate onfined onrete strain, e u, predition varies depending upon the onfinement ratio used. It should be noted that the theoretial urves of both the Hoppel [9] and Samaan [27] models were omitted from the presented stress strain urves. The reason behind this omission is that these two models, in ertain ases, predited urves that were not lose to the experimental vaes. These models, however, are shown on the bar harts of the speimens for omparison. Model omparisons are presented in Figs. 4 and 5. Fig. 4 indiates that the proposed model best predits the experimental results as ompared to other models Retangular speimens Due to the limited available onfinement models that are appliable to square and retangular omns, only two existing onrete onfinement models were evaated and ompared to the proposed model. These models are (i) Hosotani s model [], and (ii) Mander s model [8,9]. Figs. 6 and 7 ompare the entire theoretial stress strain urves from the two published models and ompare these two models with the experimental urves, for seleted speimens, produed from the study. Similar to irular omns, Mander s model appears to always overestimate the onfined onrete stress, u. In addition, the estimation

13 626 M.N. Youssef et al. / Composites: Part B 38 (27) Fig. 5. Comparison of onfinement models to C6LC2 results. Fig. 6. Comparison of onfinement models to stress strain urve of S4LC2. Fig. 7. Comparison of onfinement models to S4LC2 results.

14 M.N. Youssef et al. / Composites: Part B 38 (27) of the ultimate onfined onrete ultimate strain, e u, using this model varies depending on the onfinement ratio of the speimen under onsideration. 6. Consions A general unified FRP-onfined onrete model is proposed. The proposed model was verified using both the experimental results generated from this study as well as other previously published data. The results of this study indiated that this model an effetively predit the behavior of both irular as well as retangular omns. The majority of the experimental data used in developing this semi-empirial model were generated from large-sale speimens under axial load. Based on the results of the analytial proedures developed in this study, the following onsions are made:. The stress strain urve for onrete onfined by FRP omposites behaves bilinearly. The first portion of the stress strain urve traes that of unonfined onrete until the jaket start to get ativated. At this point, the urve beame either asend or desend, depending on the geometry of the ross-setion and the onfinement ratio provided. 2. Mander s model [8,9], was found, in most ases, to overestimate the onrete ultimate onfined strength, regardless to the onfinement ratio. On the other hand, results indiated that the variation of the predited ultimate strain using Mander s model depends largely on the onfinement ratio. 3. In most of the ases, Samaan s model [27] overestimated the ultimate onrete strain. 4. It was observed that although the Lam and Teng model [7] predited the asending portion of the stress strain urve of onfined onrete for ertain onfinement ratios, it did not do well in traing the initial portion of the stress strain urve. Also, the onrete modus of elastiity reommended by the Lam and Teng model does not orrelate well with the experimental results of this study and others. 5. Some of the onfinement models evaated in this study were effetive within ertain range of onfinement ratios, and not within others. The proposed model was suessful in overing a wide range of onfinement ratios whih was possible to generate by testing large-sale omns as opposed to the majority of the published tests that were onduted on standard 52 mm 35 mm (6 in. 2 in.) standard ylinders. As a result, a great orrelation is ahieved for all tested speimens. 6. The proposed unified onfinement model generated from this study proved to be a very effetive predition tool as ompared to experimental data, whereas other models either overestimated or under estimated both the ultimate stress and ultimate strain vaes. The proposed model was found also very effetive in prediting the entire stress strain diagram of the tested speimens. Aknowledgements The FRP materials used in the study was provided by Edge Strutural Composites In. The authors would like to aknowledge the ontributions of Mr. J. Kieh, Mr. J. Vargas, Mr. D. Mikhael, and Mr. T. Tietz to the experimental verifiation program. Referenes [] ACI 44.2R-2. Guide for the design and onstrution of externally bonded FRP systems for strengthening onrete strutures. Amerian Conrete Institute, Farmington Hills, Mihigan, 23. [2] De Lorenzis L. A omparative study of models on onfinement of onrete ylinders with FRP omposites. Division of Building Tehnology, Chalmers University of Tehnology, Work No. 46, Publiation: :4. [3] Demers M, Hebert D, Labossiere P, Neale KW. The strengthening of strutural onrete with aramid woven fiber/epoxy resin omposite. In: Proeedings of advaned omposite materials in bridges and strutures II, Montreal, August 995. p [4] Fam A, Flisak B, Rizkalla S. Experimental and analytial modeling of onrete-filled fiber-reinfored polymer tubes subjeted to ombined bending and axial loads. ACI Strut J 23;(4): [5] Harries KA, Kestner J, Pessiki S, Sause R, Riles J. Axial behavior of reinfored onrete omns retrofit with FRP jakets. Seond international onferene on omposites in infrastruture ICCI 98, Tuson, Arizona, USA, 5 7 January 998. [6] Harmon T, Slattery K. Advaned omposite onfinement of onrete. Advaned omposite materials in bridges and strutures. CSCE; 992, p [7] Haroun MA, Feng MQ, Elsanadedy HM, Mosallam AS. Composite jakets for the seismi retrofit and repair of bridge omns. In: Proeedings of the seventh US national onferene on earthquake engineering, Boston, USA, 22. [8] Haroun MA, Feng MQ, Youssef MN, Mosallam AS. Seismi retrofit of reinfored onrete omns using FRP omposite laminates. In: Proeedings of seond onferene on seismi repair and rehabilitation of strutures (SRRS2), Fullerton, CA, USA, Marh 2 22, 2. [9] Hoppel CR, Bogetti TA, Gillespie Jr JW, Howie I, Karbhari VM. Analysis of a onrete ylinder with a omposite hoop wrap. Proeedings of the ASCE materials engineering onferene. New York, NY: ASCE; 994, p [] Hoshikuma J, Kawashima K, Nagaya K, Taylor AW. Stress strain model for onfined reinfored onrete in bridge piers. J Strut Eng 997. [] Hosotani M, Kawashima K, Hoshikuma Jun-ihi. A stress strain model for onrete ylinders onfined by arbon fiber sheets. Report no. TIT/EERG 98-2, Tokyo Institute of Tehnology, Tokyo, Japan, pp [in Japanese]. [2] Hosotani M, Kawashima K, Hoshikuma Jun-ihi. A stress strain model for onrete ylinders onfined by arbon fiber sheets. J Conr Eng JSCE 998;39(592):37 52 [in Japanese]. [3] Karabinis AI, Rousakis TC. Carbon FRP onfined onrete elements under axial load. Pro FRP Compos Civil Eng 2;:39 6. [4] Kono S, Inazumi M, Kaku T. Evaation of onfining effets of CFRP sheets on reinfored onrete members. In: Proeedings of the seond international onferene on omposites in infrastruture ICCI 98, Tuson, Arizona, 5 7 January 998. p [5] Labossiere P, Neale K, Demers M, Piher F. Repair of reinfored onrete omns with advaned omposite materials onfinement. Advaned omposite materials in bridges and strutures. CSCE; 992. [6] Lam L, Teng JG. Strength models for fiber-reinfored plastionfined onrete. J Strut Eng ASCE 22(May): [7] Lam L, Teng JG. A new stress strain model for FRP-onfined onrete. Pro FRP Compos Civil Eng 2;:

15 628 M.N. Youssef et al. / Composites: Part B 38 (27) [8] Mander JB, Priestley MN, Park R. Theoretial stress strain model for onfined onrete. J Strut Div ASCE 988;4(8): [9] Mander JB, Priestley MN, Park R. Observed stress strain behavior of onfined onrete. J Strut Div ASCE 988;4(8): [2] Mirmiran A, Shahawy M. Model of onrete onfined by fiber omposites. J Strut Div ASCE 988;23(5). [2] Piher F, Rohette P. Confinement of onrete ylinders with CFRP. In: Proeedings of first international onferene on omposite infrastrutures, Tuson, Arizona, 996. p [22] Priestley MJN, Fyfe E, Seible F. Comn retrofit using fiberglass/ epoxy jakets. In: Proeedings of the first annual seismi researh workshop, California Department of Transportation, Division of Strutures, Deember 3 4, 99. p [23] Restrepo JI, DeVino B, Enhanement of the axial load arrying apaity of reinfored onrete omns by means of fiberglass epoxy jakets. In: Proeedings of advaned omposite materials in bridges and strutures II, Montreal, August 995. p [24] Rihart FE, Brantzaeg A, Brown RL. A study of the faire of onrete under ombined ompressive stresses. Bulletin No. 85, Engineering Experiment Station, University of Illinois, Urbana, IL, p. [25] Rousakis T. Experimental investigation of onrete ylinders onfined by arbon FRP sheets, under monotoni and yli axial ompressive load. Researh report, Chalmers University of Tehnology, Gteborg, Sweden. [26] Saafi M, Toutanji HA, Li Z. Behavior of onrete omns onfined with fiber reinfored polymer tubes. ACI Mater J 999;96(4):5 9. [27] Samaan M, Mirmiran A, Shahawy Mohsen. Model of onrete onfined by fiber omposites. J Strut Eng ASCE 998;24(9): [28] Seible F, Hegemier GA, Priestley MJN, Innamorato D, Ho F. Carbon fiber jaket retrofit test of retangular flexural omn with lap splied reinforement. Advaned Composites Tehnology Transfer Consortium Report No. ACTT-95/2, UCSD, Marh p. [29] Spoelstra MR, Monti G. FRP-onfined onrete model. J Compos Construt ASCE 999;3(3):43 5. [3] Toutanji H. Stress strain harateristis of onrete omns externally onfined with advaned fiber omposite sheets. ACI Mater J 999;96(3): [3] Xiao Y. Compression tests of onrete ylinders onfined by arbon fiber omposite jaket. USC Strutural Engineering Researh Report No. USC-SERP 98/6, September 998. [32] Xiao Y, Wu H. Compressive behavior of onrete ylinders onfined by arbon fiber omposite jakets. Am So Civil Eng, ASCE J Mater Civil Eng 2;2(2): [33] Xiao Y, Wu H. Compressive behavior of onrete onfined by various types of FRP omposite jakets. J Reinf Plast Compos 23;22(3): [34] Youssef MN, Haroun MA, Feng MQ, Mosallam AS. Experimental study on RC bridge omns retrofitted using fiber omposite materials. In: Proeedings of the 45th international SAMPE symposium, May 2 25, 2. [35] Youssef MN, Mosallam AS, Feng MQ. Experimental investigation on FRP-onfined onrete omns. In: Proeedings of the ninth international onferene on omposites engineering, ICCE/9, San Diego, USA, July 6, 22. [36] Youssef MN. Stress strain model for onrete onfined by FRP omposites. Ph.D. Dissertation, University of California, Irvine, Marh 23. [37] Youssef MN, Mosallam AS, Feng MQ. Experimental investigation on large-sale FRP-onfined axial members. In: Proeedings of the international onferene on ivil engineering infrastruture systems (CEIS 26), Amerian University of Beirut (AUB), Beirut-Lebanon, June 2 4, 26.