## 3 Results and Discussion
### Remaining Life Model Based on G-Value  
The remaining life of a concrete structure is defined as the useful life of that structure from the present time to the end of life, where the structure reaches an unacceptable condition that involves reconstruction or major rehabilitation [33]. To evaluate the remaining life of a concrete structure, different methodologies have been developed. Most of them are based on massive literature data and estimate the remaining life empirically; others are based on statistics, due to the wide range of uncertainty caused by the different parameters that may affect the deterioration of concrete structures. Destructive factors of concrete pavements can be divided into two categories: structural and environmental. Fatigue, due to traffic or consecutive changes in environmental conditions, is one of the main structurally destructive factors of concrete pavements. Various standards have used the Miner method to estimate the level of fatigue failure in concrete pavements [35; 66]. In addition, some of the most recent methods are based on live monitoring of the concrete structures [67]. Self-sensing concrete sensors embedded in concrete structures can monitor different parameters such as temperature, stress, strain, and damage during time continuously. These abilities will help to assess the remaining life based on proper models. As it is presented in Equation 1, the remaining life (RL) can be obtained by subtracting the partial number of load repetitions (n) from the total fatigue life (N). The number and intensity of loads applied to the structure can be counted by smart concrete sensors [68; 69; 70; 71]. Besides, total fatigue life can be estimated with different common methods or the model suggested from this study. \[\text{RL}=\text{N}-\text{n} \tag{1}\]  
In this study, to evaluate the remaining life concrete structure by a self-sensing concrete sensor embedded in it, the G-N model presented in Table 4 is used to estimate total life (N). On the other hand, the self-sensing concrete sensors can count the number of load repetitions on the structure (n). Therefore, by replacing the equation presented in Table 4 with Equation (1), the remaining life (in years or number of applied loads) is calculated based on sensor response in Equation (2). \[\text{RL}\ =10^{(2.617-0.205\times\text{Log(G)})}-\text{n} \tag{2}\]  
If the loading pattern on structure varies over time, the above equation is not helpful. To generalize the concept of the remaining life model to the real world where loads are varied over time, a solution could be to break down the time-history sensors' response into sub-segments (t\({}_{\text{i}}\)) to achieve a constant (G\({}_{\text{i}}\)) in each segment which represents constant damage rate. In this case, the remaining life is presented in Equation (3) as a percentage of \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Fatigue Model** & **SSE** & **Adjusted R-Square** & **RMSE** \\ \hline Log (G) = 12.77 \(-\) 4.881 \(\times\) log (N) & 1.576 & 0.9585 & 0.3138 \\ G = 160.5 \(-\) 64.25 \(\times\) log (N) & 2843 & 0.6839 & 13.33 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Calibrating different fatigue models. total life (N). In equation (3), \"i\" is the sub-segment number that ranges from 1 to c, where c is greater than zero and is a member of the set of natural numbers. \[\text{RL}=\left(1-\sum_{\text{i=1}}^{\text{c}}\frac{\text{n}_{\text{i}}}{10^{(2.6 17-0.205\times\text{Log}(\text{G}_{\text{i}}))}}\right)\times 100 \tag{3}\]  
The present model has been developed based on the results of the concrete beam fatigue test that presents the relationship between the slope of electrical changes of concrete sensor and fatigue life in the laboratory condition.