## 3 Strength Test
### Establishing the Fatigue Equation Based on the Weibull Distribution of Point Parameters  
Notably, from the fatigue test results under different stress states in Figures 6-8, the fatigue life exhibited discreteness and variability under different stress states by fitting the conventional S-N fatigue equation. However, even at the same stress ratio and the same cement content, the fatigue life of the specimen can be considerably discrete, which was related to the molding method of the specimen; curing and test conditions; and internal structural cracks of the material. In this study, to decrease the effect of the stress state, specimen shape, and size on the analysis and evaluation of fatigue characteristics and objectively investigate the laws of fatigue life and achieve standardization, statistical analysis and reliability principles were introduced to analyze these data. Current studies have suggested that fatigue life generally obeys a lognormal distribution; however, in recent years, some studies proposed very sophisticated modeling approaches by two-parameter or three-parameter Weibull distribution from statistical viewpoint [18; 23; 25; 26] to simulate the probability distribution of fatigue life. According to the Weibull distribution theory, the safe life or the minimum safe life in the range of an extremely high guarantee rate (99-100%) is still in agreement with the actual situation. For long-lived zones with a fatigue life of greater than \(10^{6}\) cycles, some test results were also approximately in accordance with the Weibull distribution, given a safe life in the long-life zone. The Weibull distribution exhibited a fatigue life with a 100% survival rate, which was better from the degree of fitting with empirical probability and more suitable for brittle materials. Currently, the fatigue equation with a guarantee rate of 95% is generally selected as the design basis for a road base layer in China. Therefore, the Weibull distribution theory is utilized to process the data obtained from the fatigue tests of semi-rigid base materials. From current research [3; 7; 19; 20; 21; 24], the fatigue life \(N_{f}\) and equivalent fatigue life \(N_{i}\) of cement-treated aggregate base materials obey the two-parameter Weibull distribution, and then the guarantee rate \(\rho\) should satisfy the following formula:  
\[N_{i}=N^{1-R} \tag{2}\]  
\[\rho\ =F\ (N_{i})\ =1-exp\ (\frac{N_{i}\ ^{m}}{t_{0}})\ N_{i}\geq 1,\ m,t_{0}>0 \tag{3}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Curing Times (days)** & **Stress States** & \(\mathbf{a}\) & \(\mathbf{b}\) & \(\mathbf{R^{2}}\) \\ \hline \multirow{3}{*}{28} & Unconfined compressive & \(-4.4579\times 10^{6}\) & \(3.6650\times 10^{6}\) & 0.5965 \\  & Flexural tensile & \(-1.7026\times 10^{6}\) & \(1.4041\times 10^{6}\) & 0.7053 \\  & Indirect tensile & \(-2.9069\times 10^{6}\) & \(2.3819\times 10^{6}\) & 0.5853 \\ \hline \multirow{3}{*}{90} & Unconfined compressive & \(-5.1242\times 10^{6}\) & \(4.2909\times 10^{6}\) & 0.6611 \\  & Flexural tensile & \(-2.2351\times 10^{6}\) & \(1.8415\times 10^{6}\) & 0.6934 \\ \cline{1-1}  & Indirect tensile & \(-3.5573\times 10^{6}\) & \(2.9092\times 10^{6}\) & 0.5976 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Fitting parameters for the fatigue equation with a cement content of 4% under different stress states.