## 4 Results and Discussions
### Probability Distributions of Pavement Performance  
Transfer functions are utilized to evaluate the allowed number of load repetitions which damage the pavement. Although many transfer functions have been developed by researchers, the basic forms of the transfer functions, including the fatigue function and rutting model, can be generally expressed by Equations (16) and (17), respectively. \[N_{f}=q_{1}\cdot\epsilon_{t}^{q_{2}}\cdot E_{1}^{q_{3}} \tag{16}\]  
\[N_{r}=q_{4}\cdot\epsilon_{y}^{q_{5}} \tag{17}\]  
where, \(N_{f}\) is the allowed number of load repetitions for fatigue failure; \(N_{r}\) is the allowed number of load repetitions for rutting; \(\epsilon_{t}\) is the tensile strain at the bottom of the asphalt layer; \(\epsilon_{v}\) is the vertical strain on the top of the subgrade; and \(q_{1}\), \(q_{2}\), \(q_{3}\), \(q_{4}\), and \(q_{5}\) are empirically determined constants. In Equation (16), \(E_{1}\) could be ignored because the magnitude is generally much larger in \(q_{2}\) than in \(q_{3}\). The transfer functions established by the Illinois Department of Transportation for fatigue failure [16] and the Asphal Institute for rutting [17] were adopted in this study, where \(q_{1}=5\times 10^{-6}\), \(q_{2}=-3\), \(q_{4}=1.365\times 10^{-9}\), \(q_{4}=-4.477\). The mean values and standard deviations of random variables listed in Table 4 were used for further discussion. According to transfer functions, the cumulative probability distributions of the allowed number of load repetitions for fatigue failure and rutting could be easily converted from the cumulative probability distributions of the longitudinal tensile and vertical compressive strains (Figure 5). The results are illustrated in Figure 11, in which the cumulative probability distributions of the allowed number of load repetitions for fatigue failure and rutting calculated through the combined RSM and FORM are similar to those using MCS. Compared with MCS, the probability calculation procedure based on the combined RSM and FORM is suggested for reasons of computational efficiency. For the cumulative probability distributions of the allowed number of load repetitions with standard deviations of the random variables equal to zero (i.e., the deterministic problem), deterministic results could be acquired. It is worth mentioning that these deterministic results could not completely reflect the random characteristics of pavement performance under realistic random moving loads because the damage to the pavement varies with  
Figure 10: Effect of the standard deviations of vehicle wandering for the probability of being less than 100 \(\mu\epsilon\) for vertical compressive strain at different calculation points. each random moving load and pavement temperature. Similarly, it can be found that large fatigue failure may appear with a high probability near the wheel tracks (\(A_{4}\)) and large rutting may occur with a high probability in the middle of the double rectangular load (\(B_{1}\)), according to Figure 6 and the transfer functions. The damage of fatigue failure and rutting under a single random load can be generally expressed by Equations (18) and (19), respectively. \[D_{f}=1/N_{f} \tag{18}\]  
\[D_{r}=1/N_{r} \tag{19}\]  
where, \(D_{f}\) is damage of fatigue failure under a single random load; \(N_{r}\) is damage of rutting under a single random load. Therefore, the cumulative probability distributions for the damage of fatigue failure and rutting under a single random load could be easily converted from the cumulative probability distributions of longitudinal tensile and vertical compressive strains by using Equations (18) and (19).