# Fatigue Performance of Tunnel Invert in Newly Designed Heavy Hauilway Tunnel
## 4 Fatigue Test of Tunnel Base Structures
### Fatigue Performance of Tunnel Basal Structures  
Based on the fatigue lives of specimens in Table 7, a two-parameter fatigue life (\(S\)-\(N\)) equation of the tunnel bottom structure was obtained, as shown in Equation (6):  
\[\text{lg}N=-3.83S_{\text{max}}-13.57\Delta S+5.68\ (\text{Adjusted R-square}\ =\ 0.92) \tag{6}\]  
where \(\text{lg}N=2\) indicates that the specimen can bear 1 million cycles without fracture.  
Equation (6) is a two-parameter fatigue life equation with no corresponding \(S\)-\(N\) curve in the two-dimensional (2D) plane graph, so two independent \(S\)-\(N\) curves are drawn in Figure 11. Due to the lateral and bottom confinement, the fatigue lives of the specimens in this study were far longer than the results of related literature [(23; 24)].  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Case** & \(\Lambda\) & \(\alpha\) & \(\beta\) & \(\gamma\) & **Case** & \(\Lambda\) & \(\alpha\) & \(\beta\) & \(\gamma\) \\ \hline
2 & 0.7 & 1.5 & 1.01 & 0.18 & 6 & 0.7 & 2 & 1 & 0.17 \\
3 & 0.7 & 1.4 & 1.01 & 0.2 & 7 & 0.6 & 1.4 & 1.01 & 0.15 \\
4 & 0.8 & 1 & 1 & 0.25 & 8 & 0.5 & 1 & 1 & 0.14 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Fitting parameters.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Test Conditions**} & \multirow{2}{*}{\(S_{\text{max}}/\epsilon_{\text{max}}\)} & \multirow{2}{*}{\(\Delta S/\Delta\epsilon\)} & \multicolumn{4}{c}{**Fatigue Life**} \\ \cline{3-6}  & & & **Test 1** & **Test 2** & **Test 3** \\ \hline Case 1 & 0.65/56 \(\times\) 10\({}^{-6}\) & 0.056/16 \(\times\) 10\({}^{-6}\) & 1,661,391 & \textgreater{}2,070,000 & \textgreater{}2,030,000 \\ Case 2 & 0.70/68 \(\times\) 10\({}^{-6}\) & 0.082/28 \(\times\) 10\({}^{-6}\) & 937,982 & 863,045 & 424,988 \\ Case 3 & 0.75/74 \(\times\) 10\({}^{-6}\) & 0.118/34 \(\times\) 10\({}^{-6}\) & 142,565 & 171,412 & 182,940 \\ Case 4 & 0.80/85 \(\times\) 10\({}^{-6}\) & 0.143/45 \(\times\) 10\({}^{-6}\) & 62,348 & 73,915 & 107,242 \\ Case 5 & 0.60/52 \(\times\) 10\({}^{-6}\) & 0.082/28 \(\times\) 10\({}^{-6}\) & \textgreater{}2,000,000 & \textgreater{}2,000,000 & \textgreater{}2,000,000 \\ Case 6 & 0.65/56 \(\times\) 10\({}^{-6}\) & 0.082/28 \(\times\) 10\({}^{-6}\) & 1,873,947 & 1,632,586 & 1,836,432 \\ Case 7 & 0.70/68 \(\times\) 10\({}^{-6}\) & 0.082/28 \(\times\) 10\({}^{-6}\) & 937,982 & 863,045 & 424,988 \\ Case 8 & 0.80/84 \(\times\) 10\({}^{-6}\) & 0.082/28 \(\times\) 10\({}^{-6}\) & 339,625 & 298,581 & 380,517 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Fatigue lives of specimens.  
Figure 11: \(S\)–\(N\) curves (based on a single parameter): a 4-point bending fatigue test in dry condition, span: 450 mm, specimen size: 150 \(\times\) 150 \(\times\) 550 mm,\(f_{\text{cu}}\) = 57.5 MPa. b 3-point bending fatigue test in dry condition, span: 442 mm, \(R\) = 0.1, specimen size: 100 \(\times\) 100 \(\times\) 515 mm,\(f_{\text{cu}}\) = 51.7 MPa.  
When comparing the fatigue life of cases 1 and 6 and cases 4 and 8, the fatigue lifespan of specimens appeared to be more sensitive to the dynamic stress than static stress, because, under the same maximum stress level, the fatigue life decreased sharply with the increase in dynamic stress.  
In typical V-grade surrounding rock, under the impact of a 30 t axle load train, fatigue failure would not occur in the tunnel invert within 2 million cycles, according to results of the tests and numerical simulation. The fatigue life decreased with increasing dynamic (train load) and static (rock pressure) loads.  
The results also indicate that under the condition of 30 t axle loads, any break in the inverted arch in the newly designed heavy haul railway tunnel structure could be attributed to the excessive pressure of surrounding rock or the bedrock defects, rather than the dynamic train load.