# Fatigue Performance of Tunnel Invert in Newly Designed Heavy Hauilway Tunnel
## 4 Fatigue Test of Tunnel Base Structures
### Load Condition  
The fatigue performance was influenced by maximum tensile stress, dynamic tensile stress, and ultimate flexural strength. The variable of stress level \(S\) was introduced to reduce the parameters of the \(S\)-\(N\) formula.  
The definition equations are presented in Equations (2)-(4):  
\[S_{\text{min}}=\sigma_{\text{min}}/f_{t}=F_{\text{min}}/F_{t} \tag{2}\]  
\[\Delta S=(\sigma_{\text{max}}-\sigma_{\text{min}})/f_{t}=(F_{\text{max}}-F_{ \text{min}})/F_{t}=\Delta F/F_{t} \tag{3}\]  
\[S_{\text{max}}=\sigma_{\text{max}}/f_{t}=F_{\text{max}}/F_{t} \tag{4}\]  
where \(S_{\text{min}}\) is the minimum stress/load level or static stress/load level, \(S_{\text{max}}\) is the maximum stress/load level or static stress/load level, \(\Delta S\) is the dynamic stress/load level, \(\sigma_{\text{min}}\) is the tensile stress caused by the surrounding rock pressure, \(\sigma_{\text{max}}\) is the tensile stress caused by the coupling load, \(F_{\text{min}}\) is the minimum value of the fatigue load, \(F_{\text{max}}\) is the maximum value of the fatigue load, \(\Delta F\) is the dynamic load of the fatigue load, and \(f_{t}\) is the ultimate flexural strength of the specimen.  
The fatigue load was simplified to a sinusoidal wave, as indicated in Figure 7. We applied 1.5 MPa lateral confining stress using the lateral hydraulic jack.  
The numerical simulation and the specimen strength test conditions are outlined in Table 5. The test was terminated when a fracture occurred in the specimen; the number of cycles was recorded as the fatigue life of the specimen. If no fracture occurred after 2 million cycles, the test was terminated. The mechanical properties of plain concrete specimens were discrete, so every test condition was conducted in triplicate.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Mechanical**} & \multicolumn{2}{c}{**Cube Specimens**} & \multicolumn{2}{c}{**Rectangular Specimens**} \\  & \multicolumn{2}{c}{**150 \(\times\) 150 \(\times\) 150 mm**} & \multicolumn{2}{c}{**100 \(\times\) 100 \(\times\) 300 mm**} & \\ \cline{2-5}  & **Static Elastic** & **Compression** & **Dynamic Elastic** & **Bending Strength** & **Bending Strength** \\  & **Modulus \(E\) (GPa)** & **Strength (MPa)** & **Modulus \(E\) (GPa)** & **(MPa)** & **Poisson Ratio \(\
u\)** \\ \hline Value & 21.7 & 42.3 & \(31.8\) & 2.5 & 0.23 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Mechanical parameters of specimens.  
Figure 7: Sketch map of simplified cyclic loading.  
Bilateral loads in all test conditions were 1.5 kN to reflect the horizontal confinement of the tunnel invert specimens. In dynamic load conditions (case 1-4), the static loads were the same and represented that the tunnel was under the same surrounding rock pressure and the dynamic loads were different to investigate the influence of the different axle load train. In static load conditions (case 5-8), the dynamic loads were the same and represented that the tunnel is under the action of the same axle load train and the static loads were different to study the influence of the different pressure of the surrounding rock.