# Durability-Aimed Design Criteria of Cement-Stabilized Loess Subgrade for Railway
## 3 Results and Discussion  
### Fatigue Performance of Cement-Stabilized Loess  
The fatigue test results of cement-stabilized loess are shown in Table 4. According to previous studies, the Weibull distribution was used to analyze and process the above data [57]. The fatigue life \(N\) and the equivalent fatigue life \(\overline{N}\big{(}\overline{N}=N^{1-R}\big{)}\) of cement-stabilized loess obey a two-parameter Weibull distribution, and then the failure probability \(\rho\) is [58]:  
\[\rho=F\big{(}\overline{N}\big{)}=1-\exp\Big{[}-\big{(}\overline{N}/t_{0}\big{)} ^{m}\Big{]},N\geq 1;m,t_{0}>0 \tag{4}\]  
where  
\(m\) is shape parameter;  
\(t_{0}\) is scale parameter. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\
\begin{tabular}{c} **Cement** \\ **Dosage**, \\ **\(P_{\text{s}}\) (\%)** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Compaction** \\ **Level (K)** \\ \end{tabular} } & \multicolumn{4}{c}{**The Fatigue Life \(N\) of Cement-Stabilized Loess Specimens**} \\ \cline{4-7}  & & & **under the Following Stress Levels \(S\) (Time)** & \\ \cline{4-7}  & & & **0.80** & **0.75** & **0.70** & **0.65** & **0.60** \\ \hline \multirow{6}{*}{3} & \multirow{6}{*}{0.92} & 239 & 568 & 1326 & 4825 & 8143 \\  & & 344 & 782 & 2072 & 6843 & 11,629 \\  & & 656 & 996 & 2945 & 8848 & 14,401 \\  & & 897 & 1453 & 4177 & 10,268 & 19,955 \\  & & 1121 & 1882 & 5584 & 12,851 & 26,785 \\ \cline{2-7}  & & 416 & 1645 & 4060 & 9061 & 19,320 \\  & & 688 & 3098 & 7345 & 16,453 & 34,396 \\  & & 935 & 4552 & 8840 & 23,697 & 46,005 \\  & & 1210 & 6009 & 11,285 & 32,078 & 54,720 \\  & & 1720 & 7946 & 13,865 & 42,153 & 72,153 \\ \cline{2-7}  & & 578 & 3177 & 6835 & 13,645 & 51,421 \\  & & 896 & 4945 & 9275 & 22,787 & 76,553 \\  & & 1427 & 6362 & 12,951 & 30,597 & 85,648 \\  & & 2166 & 7761 & 15,784 & 35,780 & 100,990 \\  & & 3439 & 9581 & 17,896 & 49,668 & 121,764 \\ \hline \multirow{6}{*}{4} & \multirow{6}{*}{0.92} & 389 & 895 & 2846 & 8815 & 10,524 \\  & & 536 & 1539 & 3781 & 10,629 & 13,066 \\  & & 861 & 2262 & 4974 & 13,815 & 17,251 \\ \cline{1-1}  & & 1062 & 3122 & 5655 & 15,993 & 21,892 \\ \cline{1-1}  & & 1413 & 4528 & 7523 & 18,651 & 24,145 \\ \cline{1-1} \cline{2-7}  & & 575 & 3628 & 7225 & 15,729 & 40,548 \\ \cline{1-1}  & & 906 & 5043 & 9356 & 28,955 & 53,526 \\ \cline{1-1}  & & 1288 & 6215 & 13,023 & 35,726 & 67,262 \\ \cline{1-1}  & & 1662 & 8149 & 14,898 & 44,830 & 81,049 \\ \cline{1-1}  & & 2141 & 9352 & 17,653 & 56,004 & 89,553 \\ \cline{1-1} \cline{2-7}  & & 810 & 5682 & 11,617 & 36,299 & 78,219 \\ \cline{1-1}  & & 1621 & 7427 & 18,528 & 54,473 & 90,316 \\ \cline{1-1}  & & 2577 & 8932 & 26,743 & 60,338 & 99,886 \\ \cline{1-1}  & & 3705 & 10,055 & 34,622 & 72,455 & 114,190 \\ \cline{1-1}  & & 5106 & 11,947 & 43,850 & 81,637 & 130,925 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The fatigue test results of cement-stabilized loess. Assuming that there is a fatigue limit of cement-stabilized loess, the fatigue equation should satisfy the following two boundary conditions:  
when \(S=1\), \(N=1\);  
when \(S=0\), \(N\ \rightarrow\ \infty\);  
The above boundary conditions are properly relaxed to obtain a relatively reasonable fatigue equation, which is Equation (5). \[\lg N=m-n\lg S \tag{5}\]  
where  
\(m\) and \(n\) are undetermined regression coefficients of the fatigue equation. According to Equations (4) and (5), the data in Table 4 were analyzed to obtain the fatigue prediction equation of cement-stabilized loess with a 5% failure probability. The results are shown in Table 5.  
According to a 0.95 confidence level, the regression coefficient of the fatigue equation of cement-stabilized loess was calculated, which is shown in Equations (6) and (7). \[m_{0.95}=\overline{m}-1.645S \tag{6}\]  
\[n_{0.95}=\overline{n}-1.645S \tag{7}\]  
where  
\(\overline{m}\) and \(\overline{n}\) are the average of undetermined regression coefficients of the fatigue equation;  
\(S\) is the criteria deviation.