Reliability Analysis of Bond Behaviour of CFRP–Concrete Interface under Wet–Dry Cycles

Effective bonding between adherents plays a key role in retrofitting concrete structures in civil engineering using fibre-reinforced polymers (FRPs). To ensure structural safety, it is critical to develop design codes, which account for uncertainties of materials, the environment, and load, to estimate bond behaviour under long-term exposure to harsh environments. Therefore, a reliability analysis was performed to study the bond behaviour of FRP–concrete interface under wet–dry cycles and sustained loading. Thirty double-lap, shear-bonded carbon FRP (CFRP)–concrete composite specimens were tested after wet–dry cycles and sustained loading exposure. The fracture energy Gf of the bond behavior between CFRP and concrete was directly obtained from the measured local bond-slip curves. Five widely used test methods were adopted to verify the possible distribution types of Gf. Based on the best fit distribution of Gf, a reliability index β was then calculated for the specimens. The effects of wet–dry exposure and sustained loading on β were analysed separately. The effects of the mean and standard deviation of the load on β were compared. It was found that the mean had a greater impact on reliability than the standard deviation, but neither changed the regulation of the exponential reduction of β with increasing wet–dry cycle time. Their impact was significant for a small number of wet–dry cycles but insignificant for more than 4000 wet–dry cycles.


Introduction
Externally bonded fibre-reinforced polymer (FRP) composites have been shown to be an effective strengthening method for existing engineering structures due to its excellent tensile strength, light weight, corrosion resistance, and easy tailoring [1,2]. The performance of this method depends on adequate bond strength between the FRP and the concrete, which ensures the stress transmission between and coordination of FRP and concrete [3]. Therefore, numerous studies have been conducted on the bond mechanism of the FRP-to-concrete interface, including experimental studies [4][5][6][7], theoretical analysis [8][9][10][11], and finite-element simulations [12,13]. The vast majority of these studies have used fracture energy, G f , as an indicator for evaluating the FRP-concrete interfacial bond strength. Several empirical models [14][15][16][17][18][19][20] have been proposed for predicting the G f based directly on the regression of test data or theoretical models based on fracture mechanics.
However, many uncertainties exist in the parameters of these models, such as concrete tensile strength, FRP stiffness, the width ratios of FRP, and concrete and the shear stiffness of the adhesive layer. Variation in mechanical properties and the geometry of FRP composites and adhesive layer can cause significant deviations from the intended design performance of an FRP rehabilitation, increase the probability of failure, and reduce the safety of the structure [2]. Kaiser and Karbhari [21] identified sources of uncertainty in wet layup composites such as wrinkled or broken fabric strands, the use of incorrect or tainted resin mixtures, poor site preparation, and the misplacement of fabric during

Experimental Programme
The details of the experimental programme can be found in our previously published article [3]. Here, only a brief description is provided to ensure the completeness of the paper. Thirty CFRPconcrete composite specimens were designed and tested as shown in Table 1. Each composite specimen consisted of two FRP strips and two concrete prisms, as shown in Figure 1a. The dimensions of the two concrete prisms were 75 × 75 × 220 mm and 75 × 75 × 150 mm, respectively. To simulate the actual mechanical situation of the CFRP strengthened concrete structures, sustained loads were applied through a specially designed and self-made frame, as shown in Figure 1b. After applying the sustained loads, some of the specimens were fully immersed in a sink filled with 5% NaCl solution for 12 h and then subjected to accelerative blow-drying for another 12 h per cycle to simulate tidal changes. Once the predetermined exposure cycles had been completed, the coupons were placed in the lab to air dry. A total of nine high precision foil strain gauges, including two strain gauges in unbonded zone and seven strain gauges in the bonded zone, were then attached to the FRP strip to record the tensile load and monitor the strain variation along the FRP strips during the loading process. The length and the width of the active gage are 3.0 mm and 2.2 mm, respectively. The center-to-center distance of the adjacent strain gauge in the bonded zone is 20 mm, as shown in Figure 1a. The CFRP-concrete composite specimens were tested according to ASTM D3039, as shown Figure 1c. All of the data were recorded by an electronic acquisition system. The average concrete strength (fcu), Young's modulus, and the ultimate strength of the CFRP are listed in Table 1.

Interfacial Fracture Energy, G f
The interfacial fracture energy, G f , is one of the most important indicators used to characterise FRP-concrete interfacial bond behaviour. The above experimental results indicated that the failure mode of the control specimens (CC-CON) was debonding in concrete with much concrete becoming attached to the CFRP strip. As this outcome is caused by insufficient tensile strength f ct of concrete, so f ct was the key parameter of many common models for predicting G f of CFRP-concrete composite specimens. However, the strength of the adhesive decreased and f ct increased continuously with an increase in wet-dry cycle time. The failure position of the specimens after wet-dry cycles and sustained loading ageing transferred from the concrete to the adhesive-concrete interface, and with less and less concrete attached to the CFRP strip. Therefore, the models used to predict G f under normal circumstances could not be used to calculate the G f of composite specimens under long-term environmental exposure.
In this study, G f was directly obtained by measuring the area underneath of the local bond-slip curves. The average shear stress of one side between micro-units dx and the integral relative slip can be calculated by Equations (1) and (2), respectively.
where E f and t f are the elastic modulus and the thickness of FRP strip, respectively. ε i+1 and ε i are two adjacent strain readings at positions i + 1 and i, respectively. dx is the distance between the two adjacent gauge positions. Proceeding in this way for all of the gauge positions, the shear stress distribution along the bonded joint could be obtained and then integrated to get the shear stress of the overall FRP. The obtained local bond-slip curves of the specimens are shown in Figure 2.

Interfacial Fracture Energy, Gf
The interfacial fracture energy, Gf, is one of the most important indicators used to characterise FRP-concrete interfacial bond behaviour. The above experimental results indicated that the failure mode of the control specimens (CC-CON) was debonding in concrete with much concrete becoming attached to the CFRP strip. As this outcome is caused by insufficient tensile strength fct of concrete, so fct was the key parameter of many common models for predicting Gf of CFRP-concrete composite specimens. However, the strength of the adhesive decreased and fct increased continuously with an increase in wet-dry cycle time. The failure position of the specimens after wet-dry cycles and sustained loading ageing transferred from the concrete to the adhesive-concrete interface, and with less and less concrete attached to the CFRP strip. Therefore, the models used to predict Gf under normal circumstances could not be used to calculate the Gf of composite specimens under long-term environmental exposure.
In this study, Gf was directly obtained by measuring the area underneath of the local bond-slip curves. The average shear stress of one side between micro-units dx and the integral relative slip can be calculated by Equations (1) and (2), respectively. ) / where Ef and tf are the elastic modulus and the thickness of FRP strip, respectively. εi+1 and εi are two adjacent strain readings at positions i + 1 and i, respectively. dx is the distance between the two adjacent gauge positions. Proceeding in this way for all of the gauge positions, the shear stress distribution along the bonded joint could be obtained and then integrated to get the shear stress of the overall FRP. The obtained local bond-slip curves of the specimens are shown in Figure 2.   The mean and coefficients of variation (COVs) of G f after environmental exposure are listed in Table 1. As shown in the table, the mean of G f decreased as wet-dry cycle time increased. The mean of the G f of the CFRP-concrete composite specimens decreased by an average of 5.71%, 8.70%, and 13.09% after 90 d, 180 d, and 360 d, respectively, of wet-dry cycles exposure. Under 30% sustained loading, the reduction in G f increased to 9.47%, 13.44%, and 17.62%, respectively. Under 60% sustained loading, this reduction increased further to 16.50%, 17.69%, and 28.62%, respectively. In addition, the COVs of the G f of the specimens subjected to environmental exposure were significantly greater than those of the control specimens. The COV of CC-360-60% was 141% larger than that of CC-CON. Wet-dry cycles and sustained loading ageing were thus found not only to reduce the mean of G f of FRP-concrete interface, but also to increase the dispersion of G f .

Discussion of Fracture Energy, G f
Recently, the parameter uncertainties in the bond resistance of CFRP-concrete composite specimens, such as concrete compressive strength, concrete modulus, FRP strength, FRP modulus, and FRP thickness, have been explored and found to fit Normal, Lognormal, Weibull, or Gamma distribution [24,32,34,36]. However, few studies have been conducted on the type of statistical distribution that best fits G f . Therefore, the fit of these four typical distribution types to the G f data was investigated. In order to determine which distribution was the best fit for the data, five widely used test methods, the Shapiro-Wilk (S-W) test, the Kolmogorov-Smirnov (K-S) test, the Cramér-von Mises (C-M) test, the Anderson-Darling (A-D) test and the Chi-squared (χ 2 ) test, were adopted.
The G f of the specimens without environmental exposure, under wet-dry cycles exposure only, under wet-dry cycles exposure and 30% sustained loading, and under wet-dry cycles exposure and 60% sustained loading exposure, respectively, were considered to form one sample. The probability distributions of the four samples were analysed using the statistical software package SAS (Statistical Analysis System) and the goodness of fit (p), as listed in Table 2a-d. The size of p indicates the degree of coincidence for each probability distribution type. A larger p value indicates a greater probability that the sample will obey the given distribution. In this paper, α = 0.1 was used as the significance level, so when p > 0.1, the sample was considered to fit the distribution in question. As shown in Table 2a, the p values of the specimens without environmental exposure under the five test methods were all greater than 0.1 for the normal distribution. For the lognormal distribution, the p values were greater than 0.1 under all of the test methods except the χ 2 method. However, the p values for the lognormal distribution were larger than those for the normal distribution under all four test methods except the χ 2 method. The p values for the Weibull distribution were also lower than those for the lognormal distribution in most cases. The results for the gamma distribution could not be verified by most of the test methods due to the small sample size. So, in general, the lognormal distribution was considered to be the best fitting distribution type. As shown in Table 2b, the p values of the specimens under wet-dry exposure only were larger than 0.1 under every test method. Therefore, the G f of the specimens subjected to only wet-dry exposure followed the four tested distributions. However, the p values for the lognormal and gamma distributions were larger than those for the normal and Weibull distributions. So, the lognormal and gamma distributions were more appropriate to the G f of the specimens subjected only to wet-dry exposure, with similar goodness-of-fit values.
As shown in Table 2c, the G f of the specimens subjected only to wet-dry cycles and 30% sustained loads was considered to follow all four tested distributions, due to the large values of p. Thereinto, the lognormal distribution was again found to be the best fit for the G f of the specimens subjected to wet-dry cycles and 30% sustained loads because it had a larger p value than the other three distributions under all five test methods (except that in the K-S test, the p value for lognormal distribution was less than that for the gamma distribution). Table 2d shows that the G f of the specimens subjected to wet-dry cycles and 60% sustained loads followed the four distributions fairly closely, as the p values were far larger than 0.1 in all cases. Again, the p values for the lognormal and gamma distributions were larger than those for the normal and Weibull distributions. Although that the p values for the gamma distribution were larger than those for the lognormal distribution under the K-S and χ 2 methods, a lognormal distribution was considered to be the optimal probability distribution of the G f of the specimens subjected to wet-dry cycles and 60% sustained loads. This was because that the p value for the gamma distribution could not be calculated using the S-W test and the distribution type under other circumstances is preferably uniform to facilitate calculation. Another observation was that the p values of the specimens subjected to both wet-dry cycles and sustained loads were significantly larger than those of the specimens not exposed to wet-dry cycles. This may be due to the larger COV induced by wet-dry exposure.
In sum, for the specimens not exposed to wet-dry cycles and the specimens subjected to both wet-dry cycles and sustained loads, the optimal probability distribution of the G f of FRP-concrete bond strength was a lognormal distribution. Therefore, neither wet-dry exposure nor sustained loading changed the type of probability distribution of the G f of FRP-concrete bond strength. Figure 3 shows the probability distribution and cumulative distribution of typical specimens. exposed to wet-dry cycles. This may be due to the larger COV induced by wet-dry exposure.
In sum, for the specimens not exposed to wet-dry cycles and the specimens subjected to both wet-dry cycles and sustained loads, the optimal probability distribution of the Gf of FRP-concrete bond strength was a lognormal distribution. Therefore, neither wet-dry exposure nor sustained loading changed the type of probability distribution of the Gf of FRP-concrete bond strength. Figure  3 shows the probability distribution and cumulative distribution of typical specimens.

Reliability Index of G f
To simultaneously consider accuracy and simplicity, the probability distribution of the G f of FRP-concrete interfacial was considered to fit a lognormal distribution. The limit state function or performance was represented by the following equation: where R is the random resistance and S is the load effect. The limit state equation for the degradation of G f can be expressed as follows: where G i f n (N) and G f c (N) are the resistance and load effect after N times wet-dry cycles respectively, and both obey a lognormal distribution. i was taken as 1, 2, and 3 for cases of unloading, 30% sustained loading, and 60% sustained loading, respectively. A simple transformation was undertaken as follows: where G i 1 (N) and G 2 (N) obey a normal distribution. Then Equation (3) can be converted to: In this study, the JC method was adopted to solve the reliability index (β). The direction cosine of the two coordinate vectors can be expressed as follows: where σ i 1 (N) is the standard deviation of G i 1 (N) and σ 2 (N) is the standard deviation of G 2 (N). The coordinates of the check point in the original coordinate system were as follows: where m i 1 (N) is the mean of G i 1 (N) and m 2 (N) is the mean of G 2 (N). Substituting the coordinates of the check point into Equation (4) gave: The β of the FRP-concrete interface was obtained iteratively using Equations (5)-(7) as follows: The corresponding interface degradation probability was calculated by P = Φ(−β). The mean m 2 (N) and standard deviation σ 2 (N) of the resistance were assumed to be constant and equal to the test value. m i 1 (N) and σ i 1 (N) were regressed from the fit of the experimental results as shown in Equations (11)     As shown in the figures, for all the specimens under a sustained load and those without sustained loading, the change in the β of G f with wet-dry cycle time were divided into two stages, comprising a fast turn-off and a slow turn-off, where the overall trend obeyed a logarithmic function. Taking the unloaded case as an example, the β of G f decreased rapidly from −0.2 to −1.35 after 1000 wet-dry cycles. In contrast, during the 1000 wet-dry cycles to 4000 wet-dry cycles, the β of G f decreased slowly from −1.35 to −1.55. Accordingly, the failure probability increased rapidly from 0.6 to 0.95 after 1000 wet-dry cycles and increased slowly from 0.95 to 0.97 from the 1000 wet-dry cycle times to the 4000 wet-dry cycles.
Under the dual action of wet-dry exposure and sustained loading, the β (or failure probability) decreased (or increased) markedly as sustained loading levels increased. The influence of the sustained loads was exaggerated as the number of wet-dry cycles increased. After 200 times of wet-dry cycles, β had decreased by 6% and 15% for specimens subjected to 30% and 60% sustained loads, respectively, relative to its values for unloaded specimens. After 2000 wet-dry cycles, the equivalent reductions reached up to 12% and 24%, respectively.

Effects of Load Distribution on Reliability of G f
In this section, to provide a theoretical basis for determining a reasonable design value, the effects of the mean value and standard deviation of the load on the β of FRP-concrete bond behaviour are analysed in detail.

Effects of Mean Values of Load
The mean values of the load were determined according to the design values and the experimental values of G f . The design values were calculated by several common models proposed in the literature [10,16,[18][19][20]. The range of calculated values of G f was 0.38 N/mm-1.10 N/mm. Given the experimental value of 1.54 N/mm, the mean values of the load in this study were set as 1.54 N/mm, 1.30 N/mm, 1.00 N/mm, 0.70 N/mm, and 0.40 N/mm, and the corresponding β and deterioration probability of G f were calculated by the formula in Section 5. Figures 6-8 show the calculated results, where the standard deviation of G f was set equal to the experimental value (0.2522 N/mm).
As displayed in Figures 6-8, for different mean values, the β of G f showed an exponential decrease as the number of wet-dry cycles increased, while the failure probability increased exponentially with increasing wet-dry cycle time. The rate of reduction of β (rate of increase of failure probability) was higher in the early period of wet-dry exposure but remained changeless beyond a certain wet-dry cycle exposure. For lower mean load values, the β is clearly greater than that of a higher mean load at the beginning of the wet-dry cycle. For example, the β of a specimen under a load with a mean of 0.40 N/mm is about 3.02 at the first wet-dry cycle, whereas the β of a specimen under a load with a mean of 1.00 N/mm is only 1.19 at the first wet-dry cycle. However, after 4000 wet-dry cycles, the difference in β of specimens under loads with different mean was very minor and their failure probability approaches 1.0. This is mainly due to the sharply degraded bond behavior under the continuous erosion of wet-dry cycles and sustained loading. For cases of 0, 30%, 60% sustained loads, the influence rules of mean values of the load were highly similar. exponentially with increasing wet-dry cycle time. The rate of reduction of β (rate of increase of failure probability) was higher in the early period of wet-dry exposure but remained changeless beyond a certain wet-dry cycle exposure. For lower mean load values, the β is clearly greater than that of a higher mean load at the beginning of the wet-dry cycle. For example, the β of a specimen under a load with a mean of 0.40 N/mm is about 3.02 at the first wet-dry cycle, whereas the β of a specimen under a load with a mean of 1.00 N/mm is only 1.19 at the first wet-dry cycle. However, after 4000 wet-dry cycles, the difference in β of specimens under loads with different mean was very minor and their failure probability approaches 1.0. This is mainly due to the sharply degraded bond behavior under the continuous erosion of wet-dry cycles and sustained loading. For cases of 0, 30%, 60% sustained loads, the influence rules of mean values of the load were highly similar.
(a) the change of β with wet-dry cycles (b) the change of P with wet-dry cycles

Effects of Standard Deviation of the Load
Due to the lack of specific formula and regulations governing the standard deviation of the load, the standard deviations of the load were set as 0.1000 N/mm, 0.2522 N/mm, 0.4000 N/mm, and 0.5000 N/mm according to the experimental results. The β values and failure probability were calculated as explained in Section 5, and the results are shown in Figures 9-11. The curves were obtained using the experimentally obtained average value of the load, 1.54 N/mm.
As shown in Figures 9-11, in every loading case, the standard deviations of the load did not change the β or the exponential growth of failure probability with increasing wet-dry cycle times. However, in the first stage with rapid changes, a larger standard deviation corresponded to a lower β and a higher failure probability, and the differences induced by different standard deviations of the load was smaller than those induced by different mean values of the load. After 2000 wet-dry cycles, the differences induced by different standard deviations of the load were negligible.
In sum, neither the mean values nor the standard deviations of the load changed the regulation of β, which showed exponential growth with increasing wet-dry cycle times, but both had a significant influence on the β values. This influence was significant in the stage of a small number of

Effects of Standard Deviation of the Load
Due to the lack of specific formula and regulations governing the standard deviation of the load, the standard deviations of the load were set as 0.1000 N/mm, 0.2522 N/mm, 0.4000 N/mm, and 0.5000 N/mm according to the experimental results. The β values and failure probability were calculated as explained in Section 5, and the results are shown in Figures 9-11. The curves were obtained using the experimentally obtained average value of the load, 1.54 N/mm.
As shown in Figures 9-11, in every loading case, the standard deviations of the load did not change the β or the exponential growth of failure probability with increasing wet-dry cycle times. However, in the first stage with rapid changes, a larger standard deviation corresponded to a lower β and a higher failure probability, and the differences induced by different standard deviations of the load was smaller than those induced by different mean values of the load. After 2000 wet-dry cycles, the differences induced by different standard deviations of the load were negligible.
In sum, neither the mean values nor the standard deviations of the load changed the regulation of β, which showed exponential growth with increasing wet-dry cycle times, but both had a

Effects of Standard Deviation of the Load
Due to the lack of specific formula and regulations governing the standard deviation of the load, the standard deviations of the load were set as 0.1000 N/mm, 0.2522 N/mm, 0.4000 N/mm, and 0.5000 N/mm according to the experimental results. The β values and failure probability were calculated as explained in Section 5, and the results are shown in Figures 9-11. The curves were obtained using the experimentally obtained average value of the load, 1.54 N/mm.
As shown in Figures 9-11, in every loading case, the standard deviations of the load did not change the β or the exponential growth of failure probability with increasing wet-dry cycle times. However, in the first stage with rapid changes, a larger standard deviation corresponded to a lower β and a higher failure probability, and the differences induced by different standard deviations of the load was smaller than those induced by different mean values of the load. After 2000 wet-dry cycles, the differences induced by different standard deviations of the load were negligible.
In sum, neither the mean values nor the standard deviations of the load changed the regulation of β, which showed exponential growth with increasing wet-dry cycle times, but both had a significant influence on the β values. This influence was significant in the stage of a small number of wet-dry cycles but insignificant for more than 4000 times of wet-dry cycles.
the standard deviations of the load were set as 0.1000 N/mm, 0.2522 N/mm, 0.4000 N/mm, and 0.5000 N/mm according to the experimental results. The β values and failure probability were calculated as explained in Section 5, and the results are shown in Figures 9-11. The curves were obtained using the experimentally obtained average value of the load, 1.54 N/mm. As shown in Figures 9-11, in every loading case, the standard deviations of the load did not change the β or the exponential growth of failure probability with increasing wet-dry cycle times. However, in the first stage with rapid changes, a larger standard deviation corresponded to a lower β and a higher failure probability, and the differences induced by different standard deviations of the load was smaller than those induced by different mean values of the load. After 2000 wet-dry cycles, the differences induced by different standard deviations of the load were negligible.
In sum, neither the mean values nor the standard deviations of the load changed the regulation of β, which showed exponential growth with increasing wet-dry cycle times, but both had a significant influence on the β values. This influence was significant in the stage of a small number of wet-dry cycles but insignificant for more than 4000 times of wet-dry cycles.
(a) Change of β with wet-dry cycles (b) Change of P with wet-dry cycles (a) Change of β with wet-dry cycles (b) Change of P with wet-dry cycles Figure 11. The effects of standard coefficients of the load on the 60% sustained loading specimens.

Conclusions
Thirty improved double-lap shear bond CFRP-concrete composite specimens were tested after experiencing a maximum of 360 days of wet-dry cycles and sustained loading to a maximum of 60% of the ultimate load. The fracture energy Gf obtained from the experimental results was used as an index to evaluate the bond strength of the FRP-concrete interface. The study analysed the probability distribution type of Gf and the reliability index β of FRP-concrete interface subjected to the dual action of wet-dry cycles in salty water and sustained loading. The results showed that the mean values of Gf and β decreased with increasing wet-dry cycle times. Meanwhile, wet-dry cycles and sustained (a) Change of β with wet-dry cycles (b) Change of P with wet-dry cycles Figure 11. The effects of standard coefficients of the load on the 60% sustained loading specimens.

Conclusions
Thirty improved double-lap shear bond CFRP-concrete composite specimens were tested after experiencing a maximum of 360 days of wet-dry cycles and sustained loading to a maximum of 60% of the ultimate load. The fracture energy Gf obtained from the experimental results was used as an index to evaluate the bond strength of the FRP-concrete interface. The study analysed the probability distribution type of Gf and the reliability index β of FRP-concrete interface subjected to the dual action of wet-dry cycles in salty water and sustained loading. The results showed that the mean values of Figure 11. The effects of standard coefficients of the load on the 60% sustained loading specimens.

Conclusions
Thirty improved double-lap shear bond CFRP-concrete composite specimens were tested after experiencing a maximum of 360 days of wet-dry cycles and sustained loading to a maximum of 60% of the ultimate load. The fracture energy G f obtained from the experimental results was used as an index to evaluate the bond strength of the FRP-concrete interface. The study analysed the probability distribution type of G f and the reliability index β of FRP-concrete interface subjected to the dual action of wet-dry cycles in salty water and sustained loading. The results showed that the mean values of G f and β decreased with increasing wet-dry cycle times. Meanwhile, wet-dry cycles and sustained loading exposure increased the dispersion (COV) of G f . Five widely used test methods were adopted to verify the possible distribution types of G f . The best-fitting distribution type was found to be a Lognormal distribution, and neither wet-dry exposure nor sustained loading exposure changed the probability distribution type of G f . Based on the best-fit distribution of G f , the β of the specimens was then calculated. The changes in the β of G f with wet-dry exposure were divided into two stages, comprising a fast turn-off and a slow turn-off where the overall trend obeyed a logarithmic function. In cases of dual action of wet-dry cycles and sustained loading, β (or failure probability) decreased (or increased) markedly with higher sustained loading level. Neither the mean values nor the standard deviations of the load changed the regulation of the exponential reduction of β with wet-dry cycles, but both had a significant influence on the β values. This influence was significant for a small number of wet-dry cycles but insignificant for more than 4000 wet-dry cycles.
Author Contributions: S.L. and Y.L. conceived and designed the experiments; H.L. and T.Y. performed the experiments; all the authors analyzed the data and wrote the paper.