Modeling of Bond of Sand-coated Deformed Glass Fibre-reinforced Polymer Rebars in Concrete

Transcription

1 Modeling of Bond of Sand-coated Deformed Glass Fibre-reinforced Polymer Rebars in Concrete Modeling of Bond of Sand-coated Deformed Glass Fibre-reinforced Polymer Rebars in Concrete Weichen Xue*, Yu Yang, Qiaowen Zheng, and Zhiqing Fang Department of Structural Engineering, Tongji University, Shanghai , China Received: 27 July 2014, Accepted: 12 March 2015 SUMMARY In this paper, 30 beam tests, 48 Losberg pullout tests and 6 standard pullout tests were conducted to investigate bond-slip response of sand-coated deformed glass fibre-reinforced polymer (GFRP) rebars in concrete. The specimen parameters varied were concrete strength, embedment length, bar diameter and test method. Two failure modes including pullout failure and splitting failure were observed. The experimental bond stress-slip (τ-s) curves of sand-coated deformed GFRP rebars in concrete in pullout failure included a micro-slippage branch, a slippage branch, a descending branch and a residual branch, while the τ-s curves in splitting failure included a micro-slippage branch and a slippage branch. Based on the tests, a new bond-slip constitutive model for sand-coated deformed GFRP rebars in concrete was presented. Compared with other available models, the model proposed in this paper can provide more reliable simulations of the test results. Keywords: Sand-coated deformed GFRP rebar; Bond strength; Model of bond-slip behaviour; Test 1. Introduction Compared with conventional steel bars, glass fibre reinforced polymer (GFRP) rebars perform outstanding characteristics, such as corrosion resistance, high tensile strength, lightweight, electromagnetic permeability, etc. To improve bond properties with concrete, which is essential to the cooperation between these two materials, GFRP bars are often produced with various surface shapes and treatments. According to ACI 440.1R-06 1, commercially available GFRP rebars mainly have three types of surface patterns, i.e., sand-coated, sand-coated deformed (wrapped and sand-coated) and ribbed. Among them, sand-coated deformed is the most common applied surface pattern of GFRP rebars in practical applications. An accurate bond stress-slip (τs) model is essential for GFRP rebars because it could completely characterize the bond behaviour of GFRP rebars and be used in non-linear full-range analysis of the reinforced concrete members 2. However, although numerous experimental studies have been conducted to investigate the bond behaviour of GFRP rebars in concrete, only several GFRP bond stress-slip models are available to date. A brief review of these available models is reported in the following. Based on extensive experimental researches on GFRP rebars characterized by different surfaces patterns, Malvar 3 proposed the first FRP bond stress-slip model, which depends on two empirical constants determined by curve-fitting the experimental τ-s curves. This model is represented by the following relationship: (1) where τ, s = average bond stress and the corresponding slip, respectively; τ u, s u = peak bond stress, corresponding slip; and F and G = empirical constants determined for GFRP rebars with different surface patterns. However, those empirical constants are only evaluated for FRP rebars with two types of rebar surface patterns, and the effect of rebar diameter is not considered 4. In 1995, Rossetti and Cosenza 5,6 applied the bond-slip law for steel rebars to FRP rebars, and proposed a new model known as the BPE model. It has an ascending curve, a plateau interval, a linear descending branch and a final horizontal branch. However, the second branch with constant maximum bond stress in the BPE Model is not in agreement with the experimental bond-slip curve 4. In 1996, Cosenza et al. 7 proposed an alternative analytical model obtained by modifying the BPE Model. Equation (2) describes the τ-s curve of the BPE modified model. * Corresponding author. Tel.: ; Fax: address: xuewc@tongji.edu.cn (Weichen Xue) Smithers Information Ltd., 2016 (2) Polymers & Polymer Composites, Vol. 24, No. 1,

2 Weichen Xue, Yu Yang, Qiaowen Zheng, and Zhiqing Fang where t u, s u = maximum bond stress and the corresponding slip, respectively; t 3 = friction component of the bond resistance; α, p = parameters determined by the experimental curve. This model also gives a complete bond stress-slip constitutive curve for FRP rebars, but the effects of rebar diameter and fibre type are not taken into account in this model. To describe the bond stress-slip constitutive relation of FRP rebars at the serviceability state level, Cosenza et al. 7 proposed a refined model for only the ascending branch of the bond-slip law, known as the CMR Model. This model represents an alternative to the BPE Model and is defined by Equation (3): where t u = peak bond stress, s r0, β = parameters based on curve-fitting of the experimental data, respectively. Bond tests of FRP rebars with different fibre types and surface patterns were conducted to determine the parameters in the CMR Model, but some of the test results were scattered 4. Moreover, the effect of rebar diameter is still not considered in this model. In 2003 and 2008, Gao 8 and Xue 9 proposed bond stress-slip models for FRP rebars, respectively. However, the former is too complicated to apply, while the latter is designed especially for CFRP rebars. In 2014, Lee et al. 10 modified the BPE modified model and proposed a model for GFRP rebars with two surface patterns (helically wrapped and sand coated) under pullout failure. The model is described by Equation (4): where t m, s m = maximum bond stress and the corresponding slip, respectively; t r = friction component of the bond resistance; α = parameters determined by the surface pattern. It should be noted that none of these bond stress-slip models above was designed as special formula of bond stress-slip law for sand-coated deformed GFRP rebars, and the suitability and capability of these models still need to be validated by Table 1. Mechanical properties of GFRP bars and steel bars Sand-coated deformed GFRP bars a Nominal diameter (mm) Ultimate tensile strength (MPa) Elastic modulus (GPa) Deformed steel bars Nominal diameter (mm) Yield strength (MPa) 389 (3) b 400 (21) 404 (2) 368 (10) 369 (2) Ultimate tensile strength (MPa) 514 (10) 532 (11) 575 (3) 523 (23) 540 (11) Elastic modulus (GPa) 194 (4) 179 (7) 188 (3) 191 (7) 203 (5) Ultimate elongation % 22 (1) 19 (1) 33 (1) 40 (2) 31 (4) Note: a The mechanical properties of Aslan100 sand-coated deformed GFRP rebars are provided by Hughes Brothers Company, and the standard deviation data are not available b Standard deviation of the corresponding test data (3) (4) comparing with experimental results. Besides, the formulas of bond stresses as well as slips at the characteristic points in the τ-s curve are not given, and important parameters such as rebar diameter and embedment length are not considered in most models. Given that, this paper aims to investigate the bond behaviour of sand-coated deformed GFRP rebars in concrete through a series of tests, evaluate the influence of important parameters, including concrete strength, embedment length, bar diameter and test method, and then propose a constitutive bond stressslip model which covers the entire process of bond behaviour of sandcoated deformed GFRP rebars, with formulas of bond stresses and slips at characteristic points. This new model can provide a basis for the full range analysis of FRP-reinforced concrete structures. 2. Experimental Investigation 2.1 Material Properties Sand-coated deformed GFRP rebars used in the test were supplied by Hughes Brothers Company (Aslan 100 GFRP rebar, which is one of the most prevalent sand-coated deformed GFRP bar types). The sand-coated GFRP rebars have regularly spiral deformation, and the deformation spacing on the bars is around 15 mm, as can be seen in Figure 1. The measured mechanical properties of sand-coated deformed GFRP rebars and deformed steel bars are shown in Table 1. In the table, the data for the GFRP bars are characteristic values with a guarantee Figure 1. Sand-coated deformed GFRP bars. Beam specimen, Losberg pullout specimen, (c) Standard pullout specimen 46 Polymers & Polymer Composites, Vol. 24, No. 1, 2016

3 Modeling of Bond of Sand-coated Deformed Glass Fibre-reinforced Polymer Rebars in Concrete rate of 95%, and are provided by Aslan FRP Hughes Brothers Company, and the mechanical properties of deformed steel bars were tested according to the ASTM A The mechanical properties of concrete obtained immediately before conducting the bond experiments are listed in Table Specimens In the experiments, 30 beam specimens, 48 Losberg pullout specimens and 6 standard pullout specimens were constructed and tested. Test variables were bar diameter, embedment length, concrete strength, and test method. The details of specimens are presented in Table 3. Among the test variables, nominal diameters of GFRP bars were 6 mm, 9.5 mm, 12.7 mm, 16 mm and 19 mm. As contrast reinforcement material, deformed steel bars had nominal diameter of 6 mm, 10 mm, 12 mm, 16 mm and 20 mm. The embedment lengths were designed as multiples of the Table 2. Mechanical properties of concrete Batch a Tensile strength f t (MPa) 2.5 (0.12) b 2.5 (0.35) 2.6 (0.33) 4.6 (0.45) 4.4 (0.41) Cubic compressive strength f cu (MPa) 30.2 (1.3) 30.0 (2.3) 31.2 (2.6) 63.3 (4.2) 64.1 (3.9) Elastic modulus E c (MPa) 2980 (25) 2950 (77) 2710 (65) 3880 (83) 4050 (79) Note: a The tensile strength f t was obtained from split test with concrete cube of 150 mm 150 mm 150 mm b Standard deviation of the corresponding test data Table 3. Main data of specimens Code a Batch d b (mm) l db (mm) Failure Mode b t u (MPa) s u (mm) Beam specimens B-G P P P B-G S S P B-G S S S B-G S S S B-G S S S B-S P P P B-S P P P B-S P S S B-S P P S B-S P S P Losberg pullout specimens LP-G P P P LP-G S S S LP-G S S S LP-G S S S LP-S P P P LP-S S P S LP-S S S S LP-S P P P LP-G P P P LP-G P P P LP-G P P P LP-G P P P LP-S P P P LP-S P P P LP-S P P P LP-S P P P Standard pullout specimens SP-G P P P SP-S P P P Note: a TM-T-BD-K-CSG: TM = test method(b: beam test, LP: Losberg pullout test, SP: Standard pullout test); T = bars type (G:GFRP bars, S: steel bars); BD = bar diameter; K = embedment length / bar diameter; CSG = concrete strength grade b. a b c are the failure mode of three nominal identical specimens for each specimen type, in which S and P represent splitting failure and pullout failure, respectively Polymers & Polymer Composites, Vol. 24, No. 1,

4 Weichen Xue, Yu Yang, Qiaowen Zheng, and Zhiqing Fang bar diameter to facilitate comparisons between bars with different diameters. In order to compare with standard test method in ACI 440.3R-04 12, which adopts 5 d b (d b is the bar diameter) as the embedment length, and to keep multiple relationships in test parameters, 2.5 d b, 5 d b, 10 d b and 20 d b were adopted in the test. With regard to concrete strength, 72 specimens in the test adopted C30 concrete. In order to investigate the influence of different concrete strength on the bond behaviour, 12 specimens adopted C50 concrete. The concrete cover depth of the pullout specimens is 4.5 d b (d b is the bar diameter) according to the RILEM specifications (1978) 13. In order to compare with the results of pullout tests, the beam specimens correspondingly adopt a relatively large value of concrete cover depth (42 mm). It should be emphasized that the tests mainly focus on the bond behaviour between sand-coated GFRP bars and concrete without restraint from stirrups. Therefore, no stirrup was placed in either the pullout or the beam specimens Experimental Procedure The investigation of bond strength of rebars in concrete is usually carried out using the pullout specimen because it is economically feasible, and the influencing parameters on the bond behaviour can be considered conveniently. The Losberg (1963) 14 and the standard (1978) 13 are two options for pullout tests. However, in the pullout test, splitting of the concrete is avoided by the thickness of the concrete cover, which leads to an upper bond value. In beam tests, the concrete surrounding the rebar is under tension, which is more close to the actual situation, and can realistically simulate the stress conditions of reinforced concrete elements subjected to bending 15,16. In order to investigate the effect of the test method on the bond strength of sand-coated deformed GFRP rebars, 30 beam tests, 48 Losberg pullout tests and 6 standard pullout tests were conducted. The dimensions and test setup of specimens are shown in Figure 2 and Figure 3, respectively. In the beam test, the specimens consisted of two rectangular blocks of reinforced concrete joined at the top by a steel ball joint and at the bottom by the reinforcement (GFRP or steel bar). The dimensions of the beam specimen were 1250 mm 150 mm 240 mm, as seen in Figure 2. In Losberg pullout test, the barconcrete contact areas both at the free end and at the loaded end along the debonded length of the Losberg pullout specimens were broken by soft Figure 2. Dimensions of Test Specimens (mm). Beam test, Pullout test (c) 48 Polymers & Polymer Composites, Vol. 24, No. 1, 2016

5 Modeling of Bond of Sand-coated Deformed Glass Fibre-reinforced Polymer Rebars in Concrete Figure 3. Test setup. Pullout failure Splitting failure In the table, a b c are the results of three nominal identical specimens for each specimen type. It was observed that the embedment length greatly affected the failure mode: the specimens whose embedment length was less than five times the bar diameter failed by GFRP rebar pulling out; for the specimens whose embedment length was more than five times the bar diameter, splitting failure occurred; and those specimens with the embedment length of five times the bar diameter failed in either pullout or splitting failure. plastic sleeves, and the bonded length was between two debonded lengths (see Figure 2) 17. As shown in the Figure 3, one spherical joint was applied to ensure that the bars were vertical to the surface of bottom plates because small irregularities at the top surface of the cube might introduce accidental bending on the bars during loading or movements caused by local crushing. The Losberg pullout tests conducted by other scholars 16,18,19 adopted both the cube and the cuboid specimens, and the side lengths of the Losberg pullout specimens generally ranged from 150 mm to 200 mm. If the embedment length was more than five times the bar diameter or the effect of the concrete casting position on the bond behaviour was considered, the side length of the concrete block could be increased correspondingly. So in the test, the dimensions (length width height) of the Losberg pullout specimens were 10d b 10d b 10d b (l db 5d b, l db is the embedment length) and 200 mm 160 mm 160 mm (l db > 5 d b, l db is the embedment length), respectively. The standard pullout tests were performed in accordance with the RILEM specifications 13. The dimensions of the standard pullout specimen were 10 d b 10 d b 10 d b and the embedment length was 5 d b. The standard pullout specimens are shown in Figure 2(c). In order to eliminate the influence of the area at the loaded end, a plastic sleeve was placed around the bars to isolate the bars from the concrete. Similarly to the Losberg pullout test, one steel ball joint was applied to avoid the irregularities at the bottom surface. The loading rate was calculated as v f = 0.03d b 2 (kn/min), in which d b is bar diameter according to the test standard 20 in China, and v f was no greater than 20kN/min, as specified by ACI440.3R-12. The slips of the bars relative to concrete at both free end and loaded end were measured with linear variable differential transformers (LVDTs). 3. Test Results 3.1 Failure Mode The test results are presented in Table 3, in which S and P represent splitting failure and pullout failure, respectively. 3.2 Average Bond Stress-slip (τ-s) Curve The average bond stress, is defined as the maximum pullout load per unit surface area of the bars, and computed as: (5) where τ = average bond stress; d b = effective bar diameter; l db = embedment length; F = P for pullout tests, F = 3P for beam tests, P is the tensile load. Bond strength (τ u ) is obtained by Equation (5) when tensile load reaches its maximum value (F max ) during the tests, while slips are measured at the free end of the rebars. Typical experimental τ-s diagrams for sand-coated deformed GFRP rebars and deformed steel bars are given in Figure 4. Note the last letter in the specimen codes stands for reinforcement material, i.e., f for GFRP and s for steel. As may be seen in Figure 4, the τ-s curve for deformed steel bars in pullout failure mode includes three branches: an ascending branch, a descending branch and a residual branch, which are divided by two characteristic points, i.e., the peak point and the residual point. As for the sand-coated deformed GFRP rebars, there are four Polymers & Polymer Composites, Vol. 24, No. 1,

6 Weichen Xue, Yu Yang, Qiaowen Zheng, and Zhiqing Fang Figure 4. Typical bond-slip envelopes for sand-coated deformed GFRP bars and steel bars In Losberg pullout tests, the ratio of bond strength of specimens with C30 concrete to that of the corresponding specimens with C50 concrete is 57%. Higher concrete strength can lead to better chemical adhesion and wedging effect, and postpone cracking of concrete surrounding the bars, so higher bond strength is obtained 22. branches, including a micro-slippage branch, a slippage branch, a descending branch and a residual branch, which are divided by the three characteristic points of the elastic point, the peak point and the residual point. In the τ-s curve of sand-coated deformed GFRP rebars, the micro-slippage branch is characterized by an initial steep increase in the bond stress in the slip range of 1-2 mm, followed by the slippage branch in which the bond stress gradually increases in the slip range of 2-5 mm. The oscillating behaviour of bond of sand-coated deformed GFRP rebars can also be observed in the tests, and the residual branch is similar to a sine curve. This phenomenon is supposed to be related to the surface patterns of the sand-coated deformed GFRP bars. The period of oscillations in the τ-s curve approximately corresponds to the deformation spacing on the bar. The similar behaviour was also reported by Okelo et al. 21. However, deformed steel bars do not exhibit this oscillating behaviour in the residual branch, and the average bond stress gradually decreases with the slip increasing. For sand-coated deformed GFRP rebars, the average bond stress at the residual point appeared to be about 50 to 60 percent of the bond strength, while the corresponding ratio for deformed steel bars was about 30%. This could be explained by the reason that the ribs on the deformed steel bars are rigid, which leads to the cracking of the concrete surrounding the bars, and results in the decreased bond. Figure 4 shows the τ-s curve both for GFRP bars and steel bars in splitting failure mode. Unlike the ascending branch of deformed steel bars, the experimental τ-s curve of sand-coated deformed GFRP rebars in concrete can be divided into two branches: one micro-slippage branch, the other slippage branch. 3.3 Bond Strength The influences of parameters on the bond strength of sand-coated deformed GFRP rebars are summarized in the following. With higher concrete strength, higher bond strength (τ u ) of sand-coated deformed GFRP rebars was obtained. When the embedment length or the bar diameter increased, the bond strength (τ u ) of sand-coated deformed GFRP rebar to concrete decreased. Of four different embedment lengths, the specimens with l db = 2.5d b developed the highest bond strength, and the values of bond strength of sand-coated deformed GFRP specimens with the embedment lengths of 5d b, 10d b, 20d b were about 89, 66, 25 percent of the corresponding values of the specimens with l db = 2.5d b, respectively. Similar phenomenon was also observed for steel bars and was supposed to be a result of the nonlinear distribution of bond stress on the bar 19,23. Compared to the embedment length, diameter variation only had a minor influence on the bond strength of sandcoated deformed GFRP rebars. The bond strength of GFRP specimens with the bar diameters of 9.5 mm, 12.7 mm, 16 mm, 19 mm were 88%, 89%, 77%, 61% that of the GFRP specimens with d b = 6 mm. Similar results were obtained in the experiments conducted by Nanni et al. 24 and Tighiouart et al. 25. Poisson s effect, and shear lag may be responsible for this phenomenon 19. In order to investigate the effect of the test method on bond strength of sand-coated deformed GFRP rebars in concrete, comparisons of the bond strength obtained by three different test methods were made (see Table 4). In Table 4, 9 specimens are selected (3 for each method, marked as 1#, 2#, 3#), and all of them had the same nominal diameter of 16 mm and the same embedment length of 80 mm;, S i *2 and S wi (i = 1,2,3) are the mean value, standard deviation and weighted mean value, respectively. The symbolic statistical method is used to analyze the effect of the test method. 50 Polymers & Polymer Composites, Vol. 24, No. 1, 2016

7 Modeling of Bond of Sand-coated Deformed Glass Fibre-reinforced Polymer Rebars in Concrete Table 4. Bond strength obtained by different test methods *2 Code Test method Bond strength (MPa) S i S iw 1 # 2 # 3 # (MPa) (MPa 2 ) (MPa) I30-1-f Beam test II30-2-f Losberg pullout test III30-1-f Standard pullout test The statistical values are calculated by t method 26 as: for Losberg and standard pullout tests, and, for beam and standard pullout tests. In Equation (6) and Equation (7), n i (i =1, 2, 3) is the number of specimens for each test method. In Schaum s Outline of Theory and Problems of Probability and Statistics (Second Edition, 2000) 27 when the statistical parameter a = 0.10, the critical parameter t 0.95 is equal to As a result, t 1 <t 0.95 (4) and t 2 <t 0.95 (4). It can be concluded that there is small difference between bond strength of sand-coated deformed GFRP rebars in concrete obtained in beam tests, Losberg pullout tests and standard pullout tests. 4. Bond-Slip Constitutive Model 4.1 Model Based on the analysis of the τ-s curves obtained in the tests and the existing models, a whole model of the bond-slip (τ-s) behaviour represented by Equation (8) is proposed for the sand-coated deformed GFRP rebars in pullout failure. This model presents a linearly ascending branch, i.e. a micro-slippage branch, from (0, 0) to (s e, t e ); a slippage branch which is indeed the ascending branch of the BPE Model; a linearly descending branch from (s u, t u ) to (s r, t r ); and a residual branch in the shape of the sinusoid for s>s r. Seen from Figure 5, there are three characteristic points in the τ-s curve, including the elastic point A e, the peak point A u and the residual point A r. The bond stresses at the three points are t e, t u and t r, respectively; the corresponding slips are s e, s u and s r respectively. (6) (7) where Dt = amplitude of the bond stress in the residual branch, and it is proposed to be equal to 3 MPa; R d = space between the spiral deformation on the outer surfaces of the sand-coated deformed GFRP rebars; α = parameter obtained from curve-fitting of the experimental data based on the Curve Expert software. Similarly, the τ-s curve for the sandcoated deformed GFRP rebars in splitting failure mode only includes a micro-slippage branch and a slippage branch, which can be represented by the first two branches in Figure 5 and expressed as the first two formulations in Equation (8). 4.2 Slips at the Characteristic Points of τ-s Curve Based on the experimental results, the formulas of the slips at the three characteristic points (A e, A u and A r ) of τ-s curve of sand-coated deformed GFRP rebars were obtained, with the effects of the bar diameter and the embedment length considered. Here it should be emphasized that the relationship between the slip and the bar diameter or the embedment length varied with failure modes and the test methods. In the following, the formulas of the slips at the three characteristic points both for the pullout specimens and the beam specimens with two failure modes are given, respectively Pullout Tests (Losberg Pullout Tests and Standard Pullout Tests) By means of curve fitting of the experimental data, the slip at the peak point which is defined as s u can be expressed as the function of the bar diameter and the embedment length in Equation (9a) for pullout failure and in Equation (9b) for splitting failure (see Figure 6). (8) Polymers & Polymer Composites, Vol. 24, No. 1,

8 Weichen Xue, Yu Yang, Qiaowen Zheng, and Zhiqing Fang Figure 5. τ-s curve for sand-coated deformed GFRP bars Figure 6. S u as function of l db & d b of GFRP bars in pullout tests Pullout failure: Splitting failure: (9a) (9b) In order to obtain the values of s e and s u, the effects of two factors are considered. Figure 7 illustrate the relationship between (s e ) and l db, while the relationship of s r and l db is presented in Figure 8. In order to account for the effect of the bar diameter, plots of (s e ) and (s r ) versus d b are shown in Figure 9 and Figure 9, respectively. Therefore, s e and s r can be expressed as: (10) (11) For pullout specimens with GFRP bars whose bar diameter is no more than 20 mm, if l db < 5d b, pullout failure occurs and the slips at three characteristic points of τ-s curve can be calculated in Equation (12); if l db > 5d b, splitting failure occurs and the slips can be calculated in Equation (13); otherwise (i.e. l db = 5d b ), either failure might occur and both Equation (12) and Equation (13) should be considered. (12) (13) Beam Tests For beam specimens with GFRP bars whose bar diameter is no more than 20 mm, the slips at the characteristic points of τ-s curve in pullout failure mode and splitting failure mode can be calculated in Equation (14) and Equation (15), respectively. When l db = 5d b, both failure might occur, and it is proposed that both of the two equations be considered. When l db > 5d b, splitting failure occurs and Equation (15) should be applied. (14) (15) 4.3 Average Bond Stresses at the Characteristic Points of τ-s Curve As mentioned above, the average bond stresses at the three characteristic points of τ-s curve are the elastic bond stress t e, the maximum bond strength t u and the residual bond stress t r. The test results show that the effects of Figure 7. l db versus s e. pullout failure splitting failure Figure 8. l db versus s r. d b versus s e / s u d b versus s r 52 Polymers & Polymer Composites, Vol. 24, No. 1, 2016

9 Modeling of Bond of Sand-coated Deformed Glass Fibre-reinforced Polymer Rebars in Concrete bar diameter, embedment length and concrete strength must be considered in the formula of bond strength of sand-coated deformed GFRP rebars, so the bond strength can be expressed as t r = f(d b )f(l db )f t, in which f(d b ) and f(l db ) are functions of bar diameter and embedment length, respectively, and f t is tensile strength of concrete. Figure 9. d b versus s e and s r. τ u versus f t, τ u versus d b (c) τ u versus l db Plots of t u versus f t, t u versus d b and t l u versus db developed for sand-coated deformed GFRP rebars are shown in Figure 10, Figure 10 and Figure 10(c), respectively. By using linear regression, the best-fit lines passing through all data points are obtained. Finally, t u can be expressed as: Figure 10. Relationship between bond strength and test parameter. Pullout failure, standard pullout specimens Pullout failure, Losberg pullout specimens, (c) Splitting failure, Losberg pullout specimens (16) Using the method mentioned above, the formulas of the elastic bond stress t e and the residual bond stress t r can be obtained as well. So for GFRP rebars with the bar diameter less than 20 mm and the embedment length less than 20 times the bar diameter, the bond stresses at the three characteristic points of τ-s curve can be computed as: (c) (17) 5. Models versus Experimental Results Among the available models proposed by other scholars, the BPE Modified Polymers & Polymer Composites, Vol. 24, No. 1,

10 Weichen Xue, Yu Yang, Qiaowen Zheng, and Zhiqing Fang Model appears the most reliable for reproduction of the whole τ-s curve of GFRP rebars in concrete 7. Thus, a comparison is made of the experimental curve and the curve obtained by model proposed in this paper and the BPE Modified Model. Figure 11 shows the comparison of the two models and the results obtained in this test. Figure 11 and Figure 11 are for the pullout failure while Figure 11(c) is for the splitting failure. Figure 12 shows a comparison of the two models and the results obtained by Okelo et al. 21. As seen from these figures, it can be concluded that compared with the BPE Modified Model, the model proposed in this paper yields better correlation of the experimental τ-s curve of sand-coated deformed GFRP rebars in concrete. 6. Conclusions In this paper, pullout tests and beam tests of bond behaviour in sandcoated deformed GFRP rebars were performed systematically with multiple parameters, and a new constitutive bond stress-slip model was proposed. Compared with available bond stressslip models for GFRP rebars, the new model covers the entire process of bond behaviour of sand-coated deformed GFRP rebars, and provides formulas of bond stresses and slips at the characteristic points, which take into account the influences of embedment lengths, bar diameters. This new model could provide a basis for the full-range analysis of FRP-reinforced concrete structures. Figure 11. Comparison of models and test results obtained in this test (c) Figure 12. Comparison of models and test results obtained by Okelo et al. Based on the work above, the following conclusions can be drawn: 1. The bond stress-slip curve for sand-coated deformed GFRP rebars in pullout failure includes a microslippage branch, a slippage branch, a descending branch and a residual branch, with three characteristic points, i.e., the elastic point, the peak point and the residual point, while the τ-s curve in splitting 54 Polymers & Polymer Composites, Vol. 24, No. 1, 2016

11 Modeling of Bond of Sand-coated Deformed Glass Fibre-reinforced Polymer Rebars in Concrete failure comprises a micro-slippage branch and a slippage branch; 2. For sand-coated deformed GFRP rebars, the oscillating behaviour of bond can be observed in the tests when pullout failure occurred, and the residual branch is sinusoidal in shape. The period of the oscillations corresponds approximately to the deformations spacing of the sandcoated deformed GFRP rebar; 3. With increasing concrete strength, higher bond strength of sandcoated deformed GFRP rebars can be obtained. The bond strength of sand-coated deformed GFRP rebars in concrete decreases when the embedment length or the bar diameter increase. Based on the results, there are small difference between the bond strength of GFRP rebars obtained in the beam tests, Losberg pullout tests and standard pullout tests; 4. A new model of bond-slip relationship for sand-coated deformed GFRP rebars in concrete in both pullout failure and splitting failure was proposed and formulas for the bond stresses as well as the corresponding slips at the characteristic points were given, accounting for the effects of the concrete strength, test method, the bar diameter and the embedment length. The results of this research have been adopted by the Shanghai technical code for concrete structures reinforced with FRP bars (to be published in 2015). Acknowledgments The authors gratefully acknowledge the financial support of the Project of Shanghai Science and Technology Commission (No. 14DZ ) and the National Basic Research Program of China (Project No.2012CB026201). REFERENCES 1. ACI Committee 440, Guide for the Design and Construction of Concrete Reinforced with FRP Bars, Farmington Hills, Michigan, USA (2006). 2. 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