# Time-Varying Reliability Evaluation of Concrete Based on Carbonation Depth
## 6 Time-Varying Failure Probability and Concrete Guaranteed Rate  
On the basis of the analysis of the distribution fitting, the exponential distribution describes the data distribution of carbonation depth. To develop a time-varying reliability for the assessment of structural safety, the eigenvalue of the exponential distribution function \(\lambda\) at different service ages was investigated. The eigenvalue could be determined from Figure 10 and was utilized to develop a function expressed by Equation (3):  
\[\lambda(t)=0.634\times\exp(-0.052\times t) \tag{3}\]  
To propose a time-varying reliability model regarding the carbonation depth, the concrete failure mode was defined as the carbonation depth reaching the concrete cover. The proposed model is presented in the following section.  
Figure 10: Fitting of the eigenvalue of the exponential distribution function.  
Figure 9: Time-varying model of the mean value of carbonation depth.  
The exponential distribution function is expressed as:  
\[f(x)=\left\{\begin{array}{cc}\lambda\exp(-\lambda x)&x>0\\ 0&x\leq 0\end{array}\right. \tag{4}\]  
where \(x\) is the random variable.  
By integrating Equation (4), the time-varying failure probability \(P_{f}(t)\) is revealed as:  
\[P_{f}(t)=\exp(-\lambda(t)\times c) \tag{5}\]  
where \(c\) is the thickness of the concrete cover (mm).  
However, only the data of carbonized concrete are considered in Equation (5). To have a more accurate evaluation of concrete safety, Equation (1) was considered to determine the failure probability. The formula of time-varying failure probability is:  
\[P^{\prime}_{f}(t)=\exp(-\lambda(t)\times c)\times CR(t) \tag{6}\]  
where \(P^{\prime}_{f}(t)\) is the time-varying failure probability considering both carbonized and uncarbonized concrete.  
On the basis of the building code calibration, concrete quality is determined by the guaranteed rate of above 95% [37]. Therefore, to observe the concrete failure time based on the time-varying failure probability, the guaranteed rate was evaluated in this paper. Concrete failure time can be observed when the qualified rate (\(p_{q}\)) is equal to the guaranteed rate (\(p_{g}\)). As shown in Figure 11, \(p_{q}\) is represented by the dimension of the shaded area.  
Figure 12 presents the time-varying failure probability of concrete with concrete cover assumed to be 25 mm. Tested by MC simulation, the errors of the calculated results were within 4%. It was found that \(P^{\prime}_{f}\) was far less than \(P_{f}\), because the carbonation rate was considered in \(P^{\prime}_{f}\). If concrete guaranteed rate was presumed to be 95%, the observed limit failure probability was 5%. Therefore, the concrete failure time was easily obtained as 44.3 years.  
Figure 11: Exponential distribution of \(\lambda=0.084\).  
Figure 13 shows the time-varying failure probabilities with different concrete covers. It indicates that the failure probability decreases with the increase of concrete cover. Moreover, the concrete cover has an effect on the changing rate of failure probability. The concrete failure time of other concrete covers can be predicted on the basis of the methodology applied in this paper.  
Figure 12: Time-varying failure probability, Monte-Carlo simulation, and concrete guaranteed rate.  
Figure 13: Time-varying failure probabilities of different concrete covers.