# Reliability Study on FRP Composites Exposed to Wet-Dry Cycles
## 3 Results
### Reliability Analysis of FRP Strength  
The limit state function or performance equation for FRP strength can be represented by the following equation:  
\[Z=g(R,S)=R-S=0 \tag{1}\]  
in which, \(R\) is the random resistance and \(S\) is the load effect. Meanwhile, the limit state equation for the degradation of FRP strength can be expressed as:  
\[Z=g(T_{n},T_{c})=T_{n}(N)-T_{c}(N)=0 \tag{2}\]  
in which, \(T_{n}(N)\) and \(T_{c}(N)\) are the resistance and load effect after wet-dry cycles and sustained loading exposure, respectively, and they all obey the Normal distribution according to Section 3.3. Therefore, the probability of degradation of the FRP strength after \(n\) time wet-dry cycles is:  
\[P(N)=P(g(T_{n},T_{c})\leq 0)=P(T_{n}(N)\leq T_{c}(N)) \tag{3}\]  
When the mean and standard deviation of \(T_{n}(N)\) are assumed to be \(m_{n}(N)\) and \(\sigma_{n}(N)\), and the mean and standard deviation of \(T_{c}(N)\) are \(m_{c}(N)\) and \(\sigma_{c}(N)\), respectively, \(T_{n}(N)\) and \(T_{c}(N)\) can be converted into a standard normal distribution as:  
\[\hat{T}_{n}(N)=\frac{T_{n}(N)-m_{n}(N)}{\sigma_{n}(N)} \tag{4}\]  
\[\hat{T}_{c}(N)=\frac{T_{c}(N)-m_{c}(N)}{\sigma_{c}(N)} \tag{5}\]  
Thus, the reliability index (\(\beta\)) is obtained according to second-order moment reliability method (SORM):  
\[\beta=\frac{m_{n}(N)-m_{c}(N)}{\sqrt{\sigma_{n}{}^{2}(N)+\sigma_{c}{}^{2}(N)}} \tag{6}\]  
The corresponding failure probability was calculated by \(P=\Phi(-\beta)\), and the mean \(m_{c}(N)\) and standard deviation \(\sigma_{c}(N)\) of the resistance were assumed to be constant and equal to the test value. According to the Arrhenius method, a prediction equation has already been proposed to estimate the long-term deterioration of FRP materials, which has been proven in accuracy [7; 11; 31; 32]. The equation for predicting the tensile strength of FRP materials is in the form:  
\[f(N)=\frac{f_{0}}{100}[A\ln(N)+B] \tag{7}\]  
where \(f(N)\) and \(f_{0}\) are the tensile strengths at \(N\) times cycles and without environment exposure, respectively. \(A\) is a constant denoting degradation rate; and \(B\) is a material constant, which reflects the early effects of post-cure progression. Based on test results, the \(m_{n}(N)\) and \(\sigma_{n}(N)\) can be regressed from the fit of the experimental results as shown in Equations (8)-(11):  
\[m_{nC}(N)=\frac{m_{nC}(1)}{100}[-1.520\ln(N)+100] \tag{8}\]  
\[m_{nG}(N)=\frac{m_{nG}(1)}{100}[-1.798\ln(N)+100] \tag{9}\]  
\[\sigma_{nC}(N)=\frac{\sigma_{nC}(1)}{100}[0.057\ln(N)+100] \tag{10}\]  
\[Z=g(T_{n},T_{c})=T_{n}(N)-T_{c}(N)=0 \tag{2}\]0where \(m_{nc}(N)\) and \(m_{nc}(N)\) are the mean of CFRP and GFRP materials, respectively. \(\sigma_{nc}(N)\) and \(\sigma_{nc}(N)\) are the standard deviation of CFRP and GFRP materials, respectively. The \(R^{2}\) values obtained from fitting Equations (8)-(11) were 0.9083, 0.9131 0.9151, and 0.8092, 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 1 to 360 times. However, in the follow-up analysis of \(\beta\), 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 ultimate tensile strength without wet-dry cycle exposure and substituting Equations (8)-(11) into Equation (6), the reliability index (\(\beta\)) and the corresponding failure probability \(P\) with the increase of wet-dry cycle times can be obtained and are plotted in \[Z=g(T_{n},T_{c})=T_{n}(N)-T_{c}(N)=0 \tag{2}\]1  
As shown in the figure, the curves of CFRP and GFRP can be divided into two stages. In the first stage, \(\beta\) rapidly reduced with the increase of wet-dry cycles, and then \(\beta\) decreased very slowly with the increase of wet-dry cycles when \(\beta\) reduced to a small value. The boundary value of two stages for both CFRP and GFRP is 1000 times.