## 3 Strength Test
### Establishing the Fatigue Equation Based on the Weibull Distribution of Point Parameters
\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 & \(-5.5173\times 10^{6}\) & \(4.7240\times 10^{6}\) & 0.8152 \\  & Flexural tensile & \(-2.4064\times 10^{6}\) & \(1.9855\times 10^{6}\) & 0.6848 \\  & Indirect tensile & \(-3.4806\times 10^{6}\) & \(2.8836\times 10^{6}\) & 0.7094 \\ \hline \multirow{3}{*}{90} & Unconfined compressive & \(-7.7162\times 10^{6}\) & \(6.5691\times 10^{6}\) & 0.7258 \\  & Flexural tensile & \(-2.5573\times 10^{6}\) & \(2.1357\times 10^{6}\) & 0.7700 \\ \cline{1-1}  & Indirect tensile & \(-4.2797\times 10^{6}\) & \(3.5978\times 10^{6}\) & 0.7489 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Fitting parameters for the fatigue equation with a cement content of 5% under different stress states. \[ln\bigg{[}ln\bigg{(}\frac{1}{1-\rho}\bigg{)}\bigg{]}=mln(N_{i})-ln(t_{0}) \tag{4}\]  
where \(m\) and \(t_{0}\) are the shape and scale parameters, respectively, \(R\) is cyclic eigenvalues. Table 8 summarizes the data from the Weibull distribution of the fatigue life shown in Figures 6-8 according to Equation (4). Owing to space limitations, only one stress state and the cement content randomly selected under each stress ratio of the Weibull distribution of the fatigue life were listed. Results obtained from the Weibull distribution test summarized in Table 8 revealed that the fatigue life and equivalent fatigue life were subject to a two-parameter Weibull distribution, and the correlation coefficient was extremely good. By substituting the regression coefficient obtained by the Weibull distribution test into Equation (4), and then multiplying the result by \(R\) (cyclic eigenvalues), an equivalent fatigue life (\(N_{i}\)) of 95% for different stress states, different curing times, and different cement content was obtained. This relationship has been obtained for fatigue life and equivalent fatigue life at different stress states, different curing times and cement content. Then, Equation (5) was used, where a single logarithmic fatigue equation fitted the standardization equations for the fatigue characteristics for different cement content, different stress states, and different curing times with a guaranteed rate of 95%. \[lgN_{i}=aS+b \tag{5}\]  
where \(S\) is the stress ratio; \(a\), \(b\) are the regression coefficients; and \(N_{i}\) is the equivalent fatigue life (times). Figures 9-14 shows the P-S-N figure of standardization equations, where \(R^{2}\) is the correlation coefficient, \(lgN_{i}\) is the logarithm equivalent fatigue life, \(S\) is the stress ratio. Figure 11: Standardization equations for fatigue properties with a cement content of 5% at a curing time of 28 days and a guaranteed rate of 95% under different stress states. Figure 10: Standardization equations for the fatigue properties with a cement content of 4% at a curing time of 28 days and a guaranteed rate of 95% under different stress states. Figure 12: Standardization equations for the fatigue properties with a cement content of 3% at a curing time of 90 days and a guaranteed rate of 95% under different stress states. Figure 13: Standardization equations for fatigue properties with a cement content of 4% at a curing time of 90 days and a guaranteed rate of 95% under different stress states. Standardization fatigue equations were fitted with different cement content, different curing times, and different stress states. Table 9 summarize the fitting parameters and standardization equations. From the above Figures 9-14 and the Table 9. It can be observed that:  
1. The results obtained from the fatigue tests of cement-treated aggregate base materials under different stress states revealed a good linear relationship in the P-S-N coordinate system, and the fitting correlation coefficient was extremely good.