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

^{*}

## Abstract

**:**

_{f}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 G

_{f}. Based on the best fit distribution of G

_{f}, 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.

## 1. Introduction

_{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.

_{f}of the bond strength between FRP and concrete was directly obtained from measured local bond-slip curves. The possible distribution types of G

_{f}and the reliability index β of the specimens were then calculated. Subsequently, a detailed investigation was conducted of the effects of wet–dry cycles and the mean and standard deviation of sustained loads on β.

## 2. Experimental Programme

_{cu}), Young’s modulus, and the ultimate strength of the CFRP are listed in Table 1.

## 3. Interfacial Fracture Energy, G_{f}

_{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.

_{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.

_{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.

_{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}.

## 4. Discussion of Fracture Energy, G_{f}

_{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.

_{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.

_{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.

_{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).

_{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.

_{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.

## 5. Reliability Index of G_{f}

_{f}of FRP–concrete interfacial was considered to fit a lognormal distribution. The limit state function or performance was represented by the following equation:

_{f}can be expressed as follows:

^{2}values obtained from fitting Equations (11)–(16) were 0.8135, 0.8437, 0.7911, 0.8644, 0.8779, and 0.8035, respectively, which illustrated the regression equations were relatively accurate. The values of N in the equations were within the scope of the experimental research, ranging from 0 to 360 times. However, in the follow-up analysis of the β of G

_{f}, the equations were extended to predict the bond action beyond 360 times, but the precision of the prediction requires further examination by experimental data. Setting the load to the experimental values and substituting Equations (11)–(16) into Equation (10), β in cases of unloading, 30% loading, and 60% loading in our experimental conditions were predicted along with wet–dry cycle time. The results are shown in Figure 4 and Figure 5.

_{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.

## 6. Effects of Load Distribution on Reliability of G_{f}

#### 6.1. Effects of Mean Values of Load

_{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. Figure 6, Figure 7 and Figure 8 show the calculated results, where the standard deviation of G

_{f}was set equal to the experimental value (0.2522 N/mm).

_{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.

#### 6.2. Effects of Standard Deviation of the Load

## 7. Conclusions

_{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

## Acknowledgments

## Conflicts of Interest

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Specimens | Number of Pieces | Wet–Dry Cycle N (d) | Sustained Loading β (%) | Concrete Strength f_{cu} (MPa) | Young’s Modulus E_{f} (GPa) | Ultimate Strength f_{u} (MPa) | ${\mathit{G}}_{\mathit{f}}$ (N/mm) | |
---|---|---|---|---|---|---|---|---|

Mean | COV | |||||||

CC-CON | 3 | 0 | 0 | 24.8 | 236 | 3947 | 1.436 | 0.1635 |

CC-90-0% | 3 | 90 | 0 | 32.1 | 240 | 3765 | 1.354 | 0.2270 |

CC-180-0% | 3 | 180 | 0 | 37.3 | 240 | 3614 | 1.311 | 0.3083 |

CC-360-0% | 3 | 360 | 0 | 43.5 | 234 | 3579 | 1.248 | 0.3304 |

CC-90-30% | 3 | 90 | 30 | 32.1 | 240 | 3765 | 1.300 | 0.2782 |

CC-180-30% | 3 | 180 | 30 | 37.3 | 240 | 3614 | 1.243 | 0.3167 |

CC-360-30% | 3 | 360 | 30 | 43.5 | 234 | 3579 | 1.183 | 0.3310 |

CC-90-60% | 3 | 90 | 60 | 32.1 | 240 | 3765 | 1.228 | 0.2799 |

CC-180-60% | 3 | 180 | 60 | 37.3 | 240 | 3614 | 1.182 | 0.3165 |

CC-360-60% | 3 | 360 | 60 | 43.5 | 234 | 3579 | 1.025 | 0.3936 |

_{f}and f

_{u}are the Young’s Modulus and ultimate strength of CFRP.

Test Methods | Normal | Lognormal | Weibull | Gamma |
---|---|---|---|---|

(a) G_{f} for cases not exposed to environment | ||||

S–W test | 0.230 | 0.173 | - | - |

K–S test | >0.150 | >0.150 | - | - |

C–M test | >0.250 | >0.250 | >0.250 | - |

A–D test | >0.250 | >0.250 | 0.230 | - |

χ^{2} test | 0.123 | 0.092 | 0.079 | 0.073 |

(b) G_{f} for cases not subjected to sustained loading | ||||

S–W test | 0.358 | 0.383 | - | - |

K–S test | >0.150 | >0.150 | - | >0.500 |

C–M test | >0.250 | 0.378 | >0.250 | >0.250 |

A–D test | >0.250 | 0.445 | >0.250 | >0.500 |

χ^{2} test | 0.110 | 0.228 | 0.113 | 0.204 |

(c) G_{f} for 30% sustained loading cases | ||||

S–W test | 0.644 | 0.655 | - | - |

K–S test | >0.150 | >0.150 | - | >0.500 |

C–M test | >0.250 | >0.500 | >0.250 | >0.500 |

A–D test | >0.250 | >0.500 | >0.250 | >0.500 |

χ^{2} test | 0.423 | 0.623 | 0.454 | 0.617 |

(d) G_{f} for 60% sustained loading cases | ||||

S–W test | 0.557 | 0.639 | - | - |

K–S test | >0.150 | >0.150 | - | >0.500 |

C–M test | >0.250 | >0.500 | >0.250 | >0.500 |

A–D test | >0.250 | >0.500 | >0.250 | >0.500 |

χ^{2} test | 0.472 | 0.456 | 0.558 | 0.512 |

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**MDPI and ACS Style**

Liang, H.; Li, S.; Lu, Y.; Yang, T. Reliability Analysis of Bond Behaviour of CFRP–Concrete Interface under Wet–Dry Cycles. *Materials* **2018**, *11*, 741.
https://doi.org/10.3390/ma11050741

**AMA Style**

Liang H, Li S, Lu Y, Yang T. Reliability Analysis of Bond Behaviour of CFRP–Concrete Interface under Wet–Dry Cycles. *Materials*. 2018; 11(5):741.
https://doi.org/10.3390/ma11050741

**Chicago/Turabian Style**

Liang, Hongjun, Shan Li, Yiyan Lu, and Ting Yang. 2018. "Reliability Analysis of Bond Behaviour of CFRP–Concrete Interface under Wet–Dry Cycles" *Materials* 11, no. 5: 741.
https://doi.org/10.3390/ma11050741