Stress-Strain Model for Fiber-Reinforced Polymer Jacketed Concrete Columns

Stress-Strain Model for Fiber-Reinforced Polymer Jacketed Concrete Columns

Harajli, Mohamed H

The stress-strain behavior of fiber-reinforced polymer (FRP) confined concrete columns was experimentally and analytically investigated, with particular emphasis on rectangular column sections. A new design-oriented model of the stress-strain response of FRP confined columns was developed and an experimental study was carried out for deriving the model characteristic parameters. The test variables included the volumetric ratio of the FRP jackets, the aspect ratio of the column section, and the area of longitudinal and lateral steel reinforcement. It was found that jacketing rectangular column sections with FRP sheets increases their axial strength and ductility. In reinforced concrete columns, the FRP jackets prevent premature failure of the concrete cover and buckling of the steel bars, leading to substantially improved performance. The corresponding improvements become less significant as the aspect ratio of the column section increases. The rate of increase in concrete lateral strain with axial strain is influenced by the stiffness of the FRP jackets and aspect ratio of the column sections. Based on the results of this investigation, the main parameters that control the stress and strain characteristics of FRP-confined rectangular column sections were discussed, and a general design model of the stress-strain response of FRP-confined concrete was generated. The results predicted by the model showed very good agreement with the results of the current experimental program and other test data of FRP-confined circular and rectangular columns reported in the literature.

Keywords: columns; confined concrete; ductility; fiber-reinforced concrete; polymer; stress; strain.

(ProQuest-CSA LLC: … denotes formulae omitted.)

INTRODUCTION AND LITERATURE REVIEW

Many experimental and analytical investigations have been conducted in recent years to evaluate the axial load capacity and stress-strain response of concrete confined with fiber-reinforced polymer (FRP) laminates (ACI Committee 440 2002). These investigations have clearly demonstrated that confining concrete with FRP jackets leads to substantial improvement of the axial strength and energy absorption capacity of concrete columns under both static and cyclic loading.

Several confinement models were proposed in the literature to evaluate the axial strength and to describe the stress-strain response of FRP jacketed columns. A comprehensive review and assessment of existing models has been recently presented by Teng and Lam (2004). According to Teng and Lam, proposed stress-strain models of FRP-confined concrete can be classified mainly into two major categories: design-oriented and analysis-oriented models. In the designoriented models, the stress-strain curve is generated using a simple closed form solution based on evaluation and interpretation of experimental data. In the analysis-oriented models, the stress-strain curve is generated more rigorously using an iterative procedure by considering interaction between the concrete core and the confining FRP. Because of their relative complexity, analysis-oriented models are only suitable for incorporation in numerical computer analysis.

Irrespective of their classification, most of the proposed stress-strain relationships are based on the following confinement model proposed by Richart et al. (1928, 1929) from tests conducted on concrete specimens confined with hydrostatic pressure

… (1)

… (2)

where f’^sub cc^ and ε^sub cc^ are the confined concrete compressive strength and corresponding strain, respectively; f’^sub c^ and ε^sub o^ are the compressive strength and corresponding strain for unconfined concrete; k^sub 1^ is the confinement effectiveness coefficient and f’^sub l^ is the lateral hydrostatic pressure. Based on their test results, Richart et al. (1928, 1929) found values for k^sub 1^ = 4.1 and k^sub 2^ = 5.

Among the well-known expressions for evaluating the effect of confinement on the axial strength of concrete columns is the one proposed by Mander et al. (1988) for steel confined concrete. In this expression, the confined concrete compressive strength f’^sub cc^ and corresponding strain ε^sub cc^, calculated at the onset of yielding of the transverse steel, are expressed as a function of the effective constant lateral confining pressure f^sub l^ as follows

… (3)

… (4)

Different expressions were generated by Mander et al. (1988) for calculating f^sub l^ depending on the shape of the column section and configuration of longitudinal and lateral steel.

Unlike confinement by steel hoops where the confinement pressure becomes theoretically constant beyond yielding of the hoops, the linear stress-strain behavior of the FRP causes the confining pressure in FRP-confined concrete, associated with concrete dilation, to increase continuously with increasing lateral or axial strain. Provided there is a good bond between the concrete surface and FRP, the lateral strain in the FRP is often assumed to be equal to the lateral strain in concrete. Consequently, for FRP-confined circular column sections, the lateral confining pressure f^sub l^ is calculated as a function of the volumetric ratio ρ^sub f^ and lateral strain ε^sub l^ of the FRP using the requirements of lateral strain compatibility and force equilibrium between the concrete and confining FRP jacket as follows

… (5)

where

… (6)

in which n^sub f^ is the number of applications (layers); t^sub f^ is the design thickness of the FRP fabric; and D is the column diameter. Whereas it does not recognize increases in strength for FRP-confined rectangular column sections, ACI Committee 440 recommends evaluating the axial strength of FRP-jacketed circular columns using the expression of Mander et al. (1988) (Eq. (3)) in which the lateral confining pressure f^sub l^ is calculated using Eq. (5) corresponding to an effective lateral strain ε^sub l^ = ε^sub fe^ = 0.004 ≤ 0.75ε^sub fu^, where ε^sub fu^ is the ultimate tensile strain of the FRP material.

Numerous analytical and experimentally based confinement models were proposed to calculate the confinement effectiveness coefficient k^sub 1^ (refer to Eq. (1)) for FRP confined concrete. A summary of some of the proposed expressions, including the k1 equivalence of Eq. (3), is given in Table 1. Furthermore, whereas the confinement coefficient k^sub 2^ is constant for steel confined concrete (refer to Eq. (4)) in developing a stress-strain relationship for FRP-confined circular columns, Toutanji (1999) considered, based on the experimental results of Rey (1997), that it varies in proportion to the lateral strain in the FRP as follows

k^sub 2^ = 310.57ε^sub l^ + 1.9 (7)

Whereas the stress-strain behavior of FRP confined concrete in circular columns has been extensively studied Saadatmanesh et al. 1994; Samaan et al. 1998; Spoelstra and Monti 1999; Toutanji 1999; Fam and Rizkalla 2001), because of the many unknowns associated with the behavior of FRP-confined rectangular columns, only few analytical models have been proposed to evaluate their stress-strain response (Rochette and Labossiere 2000; Wang and Restrepo 2001; Lam and Teng 2003a,b).

Concerned in this study with the development of a designoriented stress-strain model for FRP-confined concrete, to the best of the authors’ knowledge, the latest design-oriented model to describe the stress-strain response of FRP-jacketed circular or rectangular columns is the one proposed by Lam and Teng (2003a,b). Because of its relevance to the current investigation and its simplicity in application, the model of Lam and Teng is presented in this study for comparative purposes. Shown schematically in Fig. 1, the model is composed of a parabolic first portion with its initial slope being the elastic modulus of unconfined concrete Ec, and a linear second portion with a reduced slope E^sub 2^ that intersects the stress axis at f^sub c^ = f’^sub c^, where f’^sub c^ is the axial strength of unconfined concrete. The model can be expressed in the following general form

… (8)

where ε^sub t^ is the axial strain at the intersection point between the first and second portions. The strain ε^sub t^ and the slope E^sub 2^ are calculated as

… (9)

… (10)

in which f’^sub cu^ and ε^sub cu^ are the axial stress and corresponding axial strain at ultimate. For the general case of rectangular columns, f’^sub cu^ and ε^sub cu^ are expressed taking into account the reduced efficiency of rectangular sections as follows

… (11)

… (12)

where

… (13)

in which k^sub s1^ and k^sub s2^ are shape factors; k^sub 1^ = 33 and k^sub 2^ = 12.0; ε^sub o^ = 0.002; and ε^sub h^,^sub rup^ is the hoop rupture strain of the FRP. According to Lam and Teng (2003a,b), due to the effect of nonuniform stress distribution and curvature in the FRP jacket, the rupture strain of the FRP confinement is lower than the ultimate tensile strain determined from direct coupon tests. Based on evaluation of experimental data, Lam and Teng suggested using a value of ε^sub h,rup^ for CFRP, GFRP, and AFRP equal, respectively, to 58.6, 62.4, and 85.1% of the ultimate tensile strain of the FRP material. For rectangular sections, the term D in Eq. (13) is the diameter of an equivalent circular column given as D = … , where b is the short side and h is the long side of the section. Finally, the shape factors are expressed as a function of the ratio of effectively confined concrete area A^sub e^ to the cross sectional area A^sub g^

… (14)

… (15)

… (16)

in which r is the radius of the corner, and ρ^sub s^ is the ratio of longitudinal steel reinforcement in the section. Note that for circular sections, the model remains exactly the same except that the shape factors k^sub s1^ = k^sub s2^ = 1.0. According to Lam and Teng (2003b), in using the previous model for rectangular sections, strength enhancement should not be expected if the value of k^sub s1^ = f^sub la^/f’^sub c^ is less than 0.07. Lam and Teng verified the accuracy of their model by comparing with their own test data of rectangular column sections. Also, in its application to circular columns, Teng and Lam (2004) concluded that their stress-strain model is more advantageous than the other available models in several aspects including accuracy and simplicity for direct use.

In this study, experimental and analytical investigations were carried out for evaluating the stress-strain behavior of FRP-jacketed columns with particular emphasis on the response of rectangular column sections. Based on the results of this investigation, a new design-oriented stressstrain model of FRP-confined concrete is proposed. The model, which represents an improvement over an earlier approximate stress-strain model proposed by the first author for evaluating the axial load-moment interaction capacity of FRP-confined columns (Harajli 2005), takes into account most of the geometric and material properties that influence the stress-strain response. Verification of the model accuracy has been made by comparing with the experimental results obtained in this study and other experimental data reported in the literature.

RESEARCH SIGNIFICANCE

The results of this experimental and analytical investigation allow better understanding of the parameters that influence the stress-strain response of rectangular column sections when confined with FRP laminates. The stress-strain model developed in this study can be used for evaluating the axial strength and deformation capacity of FRP-jacketed circular or rectangular columns and can be employed for analyzing the load-deformation response of FRP-confined concrete under different types of load applications.

THEORETICAL STRESS-STRAIN RELATIONSHIP

A two-stage relationship of the stress-strain (f^sub cc^-ε^sub cc^) response of FRP confined concrete is proposed. In the first stage, because the lateral strains and the consequent lateral confinement pressure are small, the shape of the stress-strain response can be described using the ascending branch of the stress-strain equations developed earlier for unconfined or steel confined concrete (Sheikh and Uzumeri 1980, Scott et al. 1982, and Mander et al. 1988). In this study, the stressstrain response in the first stage is assumed, for simplicity, to follow a second-degree parabola similar to the one suggested by Sheikh and Uzumeri (1980) or Scott et al. (1982). The corresponding two-stage f^sub cc^-ε^sub cc^ relationship, taking into account confinement by internal steel ties or hoops, can be described in the following general form

… (17)

… (18)

in which f^sub co^ and εco are the stress and strain at the intersection point between the first stage (Eq. (17)) and the second stage (Eq. (18)); ε^sub cc^ = G(ε^sub l^) is the relationship between the axial strain and lateral strain in the FRP sheets; ε^sub cu^ and f^sub cu^ are the maximum limiting concrete strain and corresponding stress, respectively; ρ^sub st^ and f^sub yt^ are the volumetric ratio and yield strength of transverse confining steel, respectively.

In the second stage of the response, including the intersection point between the first and second stage (ε^sub cc^ = ε^sub co^; f^sub cc^ = f^sub co^), which is assumed to correspond to a lateral strain in the FRP sheets ε^sub l^ = ε^sub lo^ = 0.002 (Toutanji 1999), the confined concrete compressive strength f^sub cc^ and the relationship between the axial strain and lateral FRP strain, ε^sub cc^ = G(ε^sub l^), can be expressed explicitly as a function of the amount of reinforcement and material properties by making use of the concept of Eq. (1) and (2) as follows

… (19)

… (20)

where A^sub cc^ is the area of the concrete core confined with internal transverse hoops, measured to the centerline of the perimeter hoop, and ε^sub o^ is taken equal to 0.002. The terms f^sub lf^ and f^sub ls^ are the effective lateral confining pressure exerted by FRP and ordinary transverse steel on the concrete section, respectively

… (21)

… (22)

where

k^sub e^(k^sub ef^, k^sub es^) = A^sub e^/A^sub cc^ (23)

in which ρ^sub f^ is calculated using Eq. (6) where the equivalent value of D for rectangular column sections is taken in accordance with ACI Committee 440 (2002) as D = 2bh/(b + h). The terms k^sub e^ (k^sub ef^ or k^sub es^) and k^sub v^ (k^sub vf^ or k^sub vs^) account for the effectiveness of the lateral reinforcement in confining the concrete along the horizontal plane, and the concrete between transverse ties or FRP strips, respectively; and A^sub e^ is the effectively-confined concrete area. For circular columns, k^sub ef^ = k^sub es^ = 1.0. For columns confined with continuous FRP sheets k^sub vf^ = 1.0. Expressions for the coefficients k^sub e^ and k^sub v^ are given in Fig. 2 for rectangular columns based on the approach proposed by Sheikh and Uzumeri (1980), and more recently by Mander et al. (1988) and ACI Committee 440 (2002) (for evaluating the ductility of FRP confined sections). More details on the development of the expressions for k^sub v^ and k^sub e^ and also expressions for calculating k^sub v^ for circular column sections are described by Mander et al. (1988).

It should be noted that in the previously described theoretical stress-strain model, it is assumed that the stiffness of the FRP jackets is sufficiently large to produce a monotonically ascending stress-strain response until rupture of the FRP sheets. For insufficiently confined concrete, the stress-strain curve may experience a post-peak descending branch whereby the ultimate compressive strength is reached before the FRP ruptures, producing only slight or no increase in strength. Threshold values proposed by different investigators for the effective FRP confinement above which the stressstrain curve experiences ascending behavior were discussed by Lam and Teng (2003a).

For unconfined concrete or for the concrete cover of sections confined only internally with ordinary steel and for the purpose of validating the proposed model through comparisons with experimental data, the stress-strain relationship in the descending branch of the unconfined concrete stress-strain response is assumed to follow the equation proposed by Scott et al. (1982)

f^sub c^ = f’^sub c^ [1 – Z(ε^sub c^ – ε^sub o^)] ≥ 0.2f’^sub c^ for ε^sub c^ ≥ ε^sub o^ (24)

where

… (25)

Using the previously proposed model, the stress-strain response in the second stage of the response can be generated by incrementally increasing the lateral strain beyond ε^sub l^ = ε^sub lo^, and then calculating the compressive stress and corresponding strain from Eq. (19) and (20), respectively.

In this study, the characteristic parameters k^sub 1^ and k^sub 2^ of the proposed stress-strain model were determined using the test data of an experimental program designed specifically for the purpose of this investigation, as described in the next section.

EXPERIMENTAL PROGRAM

Test parameters and test specimens

Twenty-four small-scale column specimens of 300 mm height were tested. Section dimensions, specimens designation, and reinforcement details are provided in Table 2 and Fig. 3. The parameters investigated included the aspect ratio of the column sections (h/b of 1, 1.7, and 2.7, respectively), the area of FRP jackets, and the area of longitudinal and lateral steel reinforcement. The specimens were divided into three series depending on their aspect ratio. For each section aspect ratio, two groups of specimens were tested, one group corresponding to plain concrete and another group corresponding to reinforced concrete. In each group, four specimens were tested, one control specimen (without FRP) and three specimens with different areas of FRP jackets. While the specimens in the various test series have different aspect ratios, all column sections have approximately identical areas. Also, in conformity with ACI 440.2R-02 (2002) recommendations, round corners of 15 mm radius were provided in all specimens for FRP applications. Note that while the effect of corner radius on the stress-strain response is accounted for in the analytical modeling, the influence of this parameter on the test results was not evaluated in the current experimental program.

The FRP system consisted of monofilament carbon fiberreinforced polymer (CFRP) flexible sheets with impregnating epoxy resin. The design properties of the sheets as provided by the manufacturer are as follows: t^sub f^ = 0.13 mm per layer, E^sub f^ = 230,000 MPa, rupture strain ε^sub fu^ = 0.015, and ultimate strength f^sub fu^ = 3500 MPa. Preparation of the concrete surface, mixing of the epoxy resin, and application of the epoxy soaked FRP sheets were all carried out in accordance with the manufacturer specifications. Each FRP sheet was wrapped transversely around the circumference of the section with 100 mm overlap.

The longitudinal reinforcement in all the reinforced specimens consisted of four [straight phi]8 mm deformed Grade 60 steel (actual yield strength of 596 MPa), producing a longitudinal reinforcement ratio in the column sections of approximately 1.0%. The concrete cover over the longitudinal bars was maintained at 20 mm in all specimens. The transverse reinforcement in the same specimens consisted of plain 6 mm Grade 40 bars spaced at 100 mm, with the first tie located 50 mm from the top of the specimen, producing volumetric ratio ρ^sub st^ of the transverse steel (volume of ties per spacing to volume of concrete core measured to outside of tie) equal to approximately 1.1% for all specimens. The modulus of elasticity of both the longitudinal and the transverse steel is estimated at 2 × 10^sup 5^ MPa.

The plain concrete specimens and the reinforced specimens were cast together in two separate batches. The concrete mixture consisted of coarse aggregate having 10 mm maximum size, beach sand, and portland cement (Type I). The concrete compressive strength determined using three 150 x 300 mm cylinders for each batch was 18.3 MPa for the plain concrete specimens and 15.2 MPa for the reinforced specimens. Note that the selection of a relatively low concrete strength is necessitated by the limitation of the available test facilities.

All specimens were capped using a 5 mm thick sulfur layer. The average axial strain was measured using two linear variable differential transformers (LVDTs) attached on either side over a 200 mm gauge length in the middle portion of the specimens (refer to Fig. 3). The average longitudinal strain was also measured using one LVDT connected between the actuator head and base of the specimens. Average lateral concrete strains were measured using two LVDTs attached on either side parallel to the long dimension at the midheight of the specimens. The gauge lengths of the LVDTs were 75, 115, 150 mm for the specimens of aspect ratio 1.0, 1.7, and 2.7, respectively.

Discussion of experimental results

All specimens mobilized monotonically increasing stressstrain response until fracture of the FRP sheets, except Specimens C3FP1, C3FP2, C3FP3, and CS3FP1. Because of their high aspect ratio, these specimens experienced a postpeak descending branch before tensile breaking of the FRP sheets. Fracturing of the FRP sheets, which took place mostly at the junction between the corners and the flat sides of the specimens (refer to Fig. 4), resulted in a sudden and almost total loss of axial strength. Typical load or stress versus axial strain and average lateral strain responses are given in Fig. 5. It should be indicated that because of the large curvature of the FRP sheets at the corners, it is likely that the actual FRP strains at the location where the sheets fractured (at the corners) are lower than the ultimate material tensile strain. This observation is even true for circular column sections and represent the basis upon which the design-oriented model proposed by Lam and Teng (2003a,b) is developed (Eq. (8) to (16)).

Some of the important and direct observations that can be drawn from the test results are: a) increasing the area of FRP reinforcement increased the axial stress and axial strain that can be mobilized at failure of the column sections; b) improvements in axial strength and strain were most significant for square columns and tended to decrease as the aspect ratio of the column section increased (refer to Table 2). For instance, considering the plain concrete specimens confined with three FRP wraps, the stress attained a 230, 190, and 143% increase for the column sections with aspect ratios of 1.0, 1.7, and 2.7, respectively; and c) For the steel reinforced specimens, external confinement by FRP prevented spalling of the concrete cover and premature buckling of the longitudinal steel bars that would otherwise occur, leading to superior improvements of the axial load and axial strain capacities when compared with the control unconfined specimens in the same test series. For the steel reinforced columns confined with three FRP wraps, the axial strength attained a sizable 330, 252, and 190% increase for the specimens with aspect ratio of 1.0, 1.7, and 2.7, respectively.

One of the most important observations in the current experimental study, which will be analyzed in more detail, is that the rate of increase of the measured average lateral strain with the axial strain tended to decrease as the aspect ratio of the column section and also as the area or stiffness of the FRP jacket increased. This observation, which is similar to the observation reported earlier by Chaallal et al. (2003) and which has been disregarded in the development of earlier stress-strain models, has a substantial implication on the derivation of the characteristic parameters k^sub 1^ and k^sub 2^ of the proposed stress-strain model as illustrated in the following.

PROPOSED EXPRESSIONS FOR k^sub 1^ AND k^sub 2^

Using the experimentally measured axial stress and lateral strains, values of k^sub 1^ for the various FRP-confined specimens were estimated in the second stage of the response (beyond ε^sub l^ = ε^sub lo^ = 0.002) from Eq. (19) as a function of the proposed confinement parameters f^sub lf^ and f^sub ls^(k^sub 1^ = [f^sub cc^ – f’^sub c^]/[f^sub lf^ + f^sub ls^A^sub cc^/A^sub g^]) and plotted as a function of f^sub l^/f’^sub c^= ([f^sub lf^ + f^sub ls^A^sub cc^/A^sub g^]/f’^sub c^) , as shown in Fig. 6. Shown also in Fig. 6, for the purpose of comparison, are the predictions of the various expressions summarized in Table 2. For the equation proposed by Samaan et al. (1998), a value of f’^sub c^ = 18.3 MPa, to correspond to the plain concrete specimens in the current investigation, is assumed.

It can be observed in Fig. 6 that the magnitude of k^sub 1^ decreases consistently from a relatively high value in the early stage of the response during which the effective lateral confining pressure is low, to a value close to 2.0 as the confining pressure increases. Note that the magnitudes of k^sub 1^ at low values of f^sub l^/f’^sub c^ were slightly lower for the reinforced specimens in comparison with the plain concrete specimens. Part of this difference may be attributed to the fact that the values of f^sub cc^ for the plain specimens are extracted directly from the experimental data, while the values of f^sub cc^ for the reinforced specimens had to be estimated indirectly by taking into account the force carried by the longitudinal steel. It can also be seen in Fig. 6 that, while the experimental data falls well within the range of the predictions of the various expressions proposed in the technical literature, it agrees best with the value of k^sub 1^ = 2.0 derived earlier by Lam and Teng (2002) based on statistical analysis of experimental data, particularly at high values of f^sub l^/f’^sub c^.

The use of regression analysis of the data presented in Fig. 6 produced a best-fit expression for k^sub 1^ given by k^sub 1^ = 1.13(f^sub l^/f’^sub c^)^sup -0.69^ with a coefficient of correlation of 0.88 for the plain specimens, and k^sub 1^= 1.3(f^sub l^/f’^sub c^)^sup -0.41^ with a coefficient of correlation of 0.63 for the reinforced specimens. Based on regression analysis of all the data combined, the following equation is proposed for calculating the confinement effectiveness coefficient k1 (refer to Fig. 6)

… (26)

where 2 ≤ k^sub 1^ ≤ 7.

Figure 7 shows a variation of measured concrete lateral strain with axial strain and Fig. 8 shows a variation of k^sub 2^ calculated from the experimental results using Eq. (20) (k^sub 2^ = [ε^sub cc^/ε^sub co^ – 1]/[f^sub cc^/f’^sub c^ – 1]) with measured lateral strain beyond ε^sub lo^ = 0.002 for the specimens with different section aspect ratios.

The results presented in Fig. 7 show that, irrespective of the aspect ratio of the column sections or area of the FRP jackets, the rate of increase of lateral strain with axial strain (slope of the ε^sub l^-ε^sub cc^ relationship) was small in the early stage of the response but experienced a large increase and, consequently, a change of behavior beyond a lateral strain of approximately 0.002. Hence, the selection of a lateral strain ε^sub l^ = ε^sub lo^ = 0.002 to correspond to the intersection point between the two stages of the stress-strain response as suggested in the study of Toutanji (1999) and as adopted in this study appears to be reasonably validated. Beyond a lateral strain of 0.002, all specimens, including the control unconfined specimens, mobilized an approximately linear (ε^sub l^-ε^sub cc^) relationship. Because the FRP confinement curtails the dilation rate of concrete (Mirmiran and Shahawy 1997), however, the rate of increase in lateral strain with axial strain tended to decrease as the area of FRP jackets increased (mostly evident for the specimens with h/b = 1 and 1.7). Another interesting observation in Fig. 7 is that while the rate of increase in lateral strain with axial strain for the unconfined control specimens was not influenced by the shape of the column section, it dropped noticeably for the FRP confined specimens as the aspect ratio of the section increased.

Similar to the (ε^sub l^-ε^sub cc^) behavior shown in Fig. 7, the results in Fig. 8 clearly demonstrate that the magnitude of k^sub 2^ increases almost linearly with the increase in lateral strain. Also, it can be seen in Fig. 8 that the rate of increase in k^sub 2^ with lateral strain tends to decrease with increase in the volumetric ratio of the FRP jackets but increases significantly with increase in the aspect ratio of the column section. Based on these observations, accurate evaluation of the stress-strain response or ultimate load capacity of FRP confined columns should take into account that the confinement coefficient k^sub 2^ is not constant or only a function of the lateral strain as suggested in Eq. (7) but also a function of the volumetric ratio and modulus of elasticity of the FRP jackets, and most importantly, the aspect ratio h/b of the column section. At present and until more data becomes available to develop a more accurate relationship between the axial strain and lateral strain as a function of the stiffness of the FRP jacket and aspect ratio of the column section, the trend of the experimental results presented in Fig. 8 is only consistent enough to allow the generation of the following approximate, mathematically simple, and yet reasonably accurate expression for estimating k2 as a function of the control parameters

… (27)

In which E^sub f^ is expressed in MPa. Note that Eq. (27) is applicable within the range of values of the experimental parameters used in this investigation, that is, h/b ≤ 2.7 and is assumed to be applicable for circular column sections for values of h/b = 1.0. It is interesting to point out that for circular sections (h/b = 1), and for a value of ρ^sub f^ E^sub f^ = 840 MPa, Eq. (27) coincides with the experimentally-based equation (Eq. (7)) used by Toutanji (1999) to derive a stress-strain model for circular columns.

Replacing the values of k^sub 1^ and k^sub 2^ from Eq. (26) and (27) and the values of f^sub lf^ and f^sub ls^ from Eq. (21) and (22), respectively, into Eq. (19) and (20) leads to the following general expressions for generating the stress-strain relationship of FRP-confined concrete in the second stage of the response, including the intersection point between the first and second stage (ε^sub l^ = ε^sub lo^ = 0.002; ε^sub cc^ = ε^sub co^; f^sub cc^ = f^sub co^)

… (28)

… (29)

Because of the limitation imposed on the value of k^sub 1^ in Eq. (26), the value of (f^sub cc^/f’^sub c^ – 1) calculated using Eq. (28) shall not be taken less than 2.0[f^sub lf^ + f^sub ls^]/f’^sub c^ or more than 7.0[f^sub lf^ + f^sub ls^]/f’^sub c^. The stress-strain curve in the second stage can be generated by incrementally increasing ε^sub l^ and calculating f^sub cc^ from Eq. (28), and then calculating the corresponding ε^sub cc^ from Eq. (29). Because it is well established that the actual rupture strain in FRP is lower than the ultimate tensile strain of the FRP materials (Lam and Teng 2003a), it is recommended to estimate the ultimate axial stress f^sub cu^ and corresponding ultimate axial strain ε^sub cu^ using Eq. (28) and (29) by substituting a value for the lateral strain ε^sub l^ = Fε^sub fu^, where F is a strain reduction factor equal to 0.6 for CFRP and GFRP and 0.85 for AFRP in accordance with the values derived by Lam and Teng (2003a).

COMPARISON OF PROPOSED MODEL WITH EXPERIMENTAL RESULTS

The predictions of the proposed stress-strain model were compared with the experimental data of the current investigation as well as the predictions of the design-oriented model of Lam and Teng (2003a,b) presented in Eq. (8) corresponding to the plain concrete specimens. Results are shown in Fig. 9 and 10. Despite some discrepancy, the analytical results predicted using the proposed model are generally in good agreement with the experimental stress-strain response of the specimens in the current investigation. Because the FRP-confined plain concrete specimens with aspect ratio h/b = 2.7 experienced a post-peak descending branch before fracturing of the FRP sheets, the agreement between the analytical predictions and the test data was not as good in comparison with the remaining specimens.

The analytical model was also compared with test results of circular column specimens and rectangular specimens reported in the technical literature as shown in Fig. 11. A summary of test parameters for the specimens is provided in Table 3. It can be seen from the comparisons that the proposed model was able to reproduce other test data with reasonable accuracy. Note that while the model of Lam and Teng (Eq. (8)) is easier to apply and allows more direct use for design applications, it was generally less accurate in reproducing the experimental results when compared with the design-oriented model developed in this study.

CONCLUSIONS

The stress-strain response of FRP-confined rectangular concrete column sections was experimentally and analytically investigated. A theoretical stress-strain model is developed and an experimental study was carried out to derive the model characteristic parameters. Based on this investigation, the following conclusions and observations can be drawn:

1. Confining rectangular columns with FRP jackets leads to substantial improvement in the axial strength and ductility of compression failure of the columns. For square column sections without longitudinal reinforcement (plain specimens), the increases in axial strength were 154, 213, and 230% for the specimens confined with one, two, or three CFRP wraps, respectively;

2. The improvement of column axial strength and ductility due to FRP confinement becomes less significant as the aspect ratio (h/b) of the column section increases. For the plain column sections with aspect ratio of 2.7, the increases in axial strength were 133, 133, and 143% for the specimens confined with one, two, or three CFRP wraps, respectively;

3. For reinforced concrete columns, external confinement by FRP jackets prevents premature compression failure of the concrete cover and buckling of the longitudinal steel bars that normally occur in steel confined concrete, leading to substantial improvement in axial strength. For the square steel reinforced columns, the increase in axial strength in comparison with the control unconfined specimen in the same series attained a sizable 188, 255, and 310% increase for the specimens confined with one, two, or three CFRP wraps, respectively;

4. For a given aspect ratio of rectangular column section, the rate of increase in lateral strain with axial strain decreases as the stiffness ρ^sub f^ E^sub f^ of the FRP jackets increases. Also, for a given ρ^sub f^ E^sub f^ of the FRP jackets, the rate of increase of lateral strain with axial strain decreases with increase in the aspect ratio of the section; and

5. Irrespective of the h/b of the column section or ρ^sub f^ E^sub f^ of the FRP jackets, the stress-strain response of FRP confined columns experiences a considerable increase in lateral strain, and, consequently, a distinct change in behavior, beyond a confined lateral strain of approximately 0.002.

Based on the results of the experimental investigation, analytical expressions for the characteristic parameters that influence the stress and strain behavior were derived and a general analytical model for generating the stress-strain response and evaluating the ultimate axial strength and deformation capacity of FRP-jacketed columns was developed. The model takes into account almost all the design variables that control the axial stress and strain characteristics of FRP-confined columns. Results predicted by the model were generally in good agreement with the experimental results of the current investigation and other test data of FRP-confined circular and rectangular column sections reported in the literature.

ACKNOWLEDGMENTS

This research was supported by the Lebanese National Council for Scientific Research (LNCSR). The authors are grateful for that support and to the Faculty of Engineering and Architecture at the American University of Beirut for providing the test facilities.

NOTATION

A^sub cc^ = area of concrete core

A^sub e^ = area of effectively confined concrete section

A^sub g^ = area of gross section

A^sub s^ = area of column longitudinal reinforcement

b = section short dimension

D = diameter or equivalent diameter of column section

E^sub c^ = modulus of elasticity of unconfined concrete

E^sub f^ = modulus of elasticity of transverse FRP

f^sub c^ = stress in unconfined concrete

f’^sub c^ = compressive strength of unconfined concrete

f^sub cc^ = stress in confined concrete

f’^sub cc^ = compression strength of confined concrete

f^sub co^ = stress at intersection point between first and second stage of stress-strain curve

f^sub cu^ = stress corresponding to a limiting strain ε^sub cu^

f^sub fu^ = rupture stress of FRP sheets

f^sub l^ = effective lateral confining pressure

f’^sub l^ hydrostatic confining pressure

f^sub lf^ = effective lateral confining pressure provided by FRP

f^sub ls^ = effective lateral confining pressure provided by steel hoops

f^sub yt^ = yield stress of transverse hoops

h = section long dimension

k^sub 1^, k^sub 2^, k^sub e^, k^sub v^ = confinement effectiveness coefficients

n^sub f^ = number of transverse FRP layers

r = corner radius

s’ = clear spacing between transverse hoops

t^sub f^ = thickness of one FRP layer

w = clear distance between adjacent longitudinal bars

w^sub xi^, w^sub yi^ = i-th clear distance between adjacent longitudinal bars along horizontal x- and y-dimensions, respectively

x, y = concrete core dimensions to center line of peripheral hoop

ε^sub c^ = axial strain in unconfined concrete

ε^sub cc^ = axial strain in confined concrete

ε^sub co^ = concrete strain at intersection point between first and second stage of stress-strain curve

ε^sub cu^ = limiting concrete strain

ε^sub fe^ = effective lateral strain in FRP

ε^sub fu^ = ultimate tensile strain of FRP material

ε^sub h,rup^ = hoop rupture strain of FRP sheets

ε^sub l^ = lateral concrete strain

ε^sub lo^ = lateral concrete strain at intersection point between first and second stage of stress-strain curve

ε^sub o^ = strain at maximum stress for unconfined concrete

ε^sub yt^ = yield strain of transverse hoops

ρcc = steel ratio relative to concrete core section

ρf = volumetric ratio of FRP reinforcement

ρs = ratio of column longitudinal reinforcement

ρst = volumetric ratio of hoop reinforcement

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ACI member Mohamed H. Harajli is a Professor of Civil Engineering at the American University of Beirut, Beirut, Lebanon. He is a member of ACI Committees 408, Bond and Development of Reinforcement, and 440, Fiber Reinforced Polymer Reinforcement. His research interests include the design and behavior of reinforced, prestressed, and fiber-reinforced concrete members, and strengthening and repair of concrete structures.

Elie Hantouche is a Consultant Engineer, Samir Khairalla and Partners, Lebanon. He received his MS from the American University of Beirut.

ACI member Khaled Soudki is the Professor and Canada Research Chair in Innovative Structural Rehabilitation at the University of Waterloo, Waterloo, Ontario, Canada. He is a member of ACI Committees, 215, Fatigue of Concrete; 222, Corrosion of Metals in Concrete; 440, Fiber Reinforced Polymer Reinforcement; and 546, Repair of Concrete. He is also a member of Joint ACI-ASCE Committee 550, Precast Concrete Structures. His research interests include prestressed concrete, durability of concrete, rehabilitation, and strengthening of concrete structures using fiber-reinforced polymer composites.

Copyright American Concrete Institute Sep/Oct 2006

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