# A Fatigue Reliability Assessment for Rail Tension Clamps Based on Field Measurement Data
## 3 Reliability Assessment of the Tension Clamp
The reliability indexes of 3.72, 3.09, and 2.33 corresponding to the probability of failures (\(P_{f}\)) of \(10^{-4}\), \(10^{-3}\), and \(10^{-2}\) are plotted with solid black lines. The years in service corresponding to the three reliability indexes are given in Table 4. The reliability indexes of the tension clamps decreased rapidly at the start of use and then decreased gradually. Since the reliability indexes of the tension clamps were reduced by half compared to the initial level within the first 10 years, it is necessary to inspect the tension clamps for fatigue failure. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Random Variable** & & **Mean** & **COV 1** & **Distribution Type** & **Reference** \\ \hline Miner’s critical damage & \(\Delta\) & 1.0 & 0.3 & Lognormal & [23] \\ \hline Error factor & \(e\) & 1.0 & 0.3 & Lognormal & [15] \\ \hline Detail category constant (MPa3) & \(A\) & 4.929 \(\times\) 1013 & 0.544 & Lognormal & [23] \\ \hline Equivalent stress range (MPa) & \(S_{re}^{m}\) & - & - & Normal/Lognormal & Present study \\ \hline \hline \end{tabular} \({}^{1}\) COV, coefficient of variation.  
\end{table} 
Table 3: The statistical parameters of the random variables. Figure 7a shows the reliability indexes for the inner and outer rails (ML1 and ML3) calculated for similar years in service. The reliability levels for the inside track gauges were always lower than those for the outside track gauges, which implies that the probability of the failure of the tension clamp on the inside rail in direct contact with the wheels of the train is high. Similarly, Figure 7b shows that the reliability index for the inside track gauge at ML4 was slightly smaller than that for the outside track gauge. In addition, the reliability indexes for the inner rail (ML4) were much smaller than those for the outer rail (ML3). These results indicate that the tension clamp on the inner rail has a higher probability of failure than that on the outer rail, which is consistent with the field observations of fatigue damage in the tension clamps. The variations in the reliability indexes against the accumulated number of stress cycles are presented in Figure 8. It can be clearly observed that the reliability index of the tension clamps decreased rapidly as the number of stress cycles increased. When the cumulative number of stress cycles reached 5 million, the reliability indexes of the tension clamps on the inside track gauges at ML1, ML2, and ML4 decreased to half of the initial value. After 10 million stress cycles, the reliability index of the tension clamps was reduced to less than half of the initial value. As the \(C_{s}\) values in Equation (4) were set as 11 for ML3 and 10 for ML4, the difference between the reliability indexes for ML3 and ML4 in Figure 8b is larger than that in Figure 7b. For equal years in service, ML3 had a larger cumulative number of stress cycles than ML4, so the rate of the decrease in the reliability index for ML3 appeared to be smaller. It is noteworthy that the cumulative number of stress cycles, as well as the years in service, should be considered when estimating the damage to tension clamps and making plans to replace them. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Measurement Location** & **Reliability Index** & **3.72** & **3.09** & **2.33** \\  & (\(P_{f}\) = 10\({}^{-4}\)) & (\(P_{f}\) = 10\({}^{-3}\)) & (\(P_{f}\) = 10\({}^{-2}\)) \\ \hline ML1 & Inside track gauge & 6.6 & 10.1 & 17.6 \\  & Outside track gauge & 9.8 & 15.5 & 27.5 \\ \hline ML2 & Inside track gauge & 5.9 & 9.6 & 17.6 \\  & Outside track gauge & 10.6 & 16.8 & 29.8 \\ \hline ML3 & Outside track gauge & 5.9 & 9.4 & 16.6 \\ \hline ML4 & Inside track gauge & 4.5 & 7.4 & 13.7 \\  & Outside track gauge & 5.0 & 7.9 & 14.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The number of years in service corresponding to the reliability index for the four MLs (unit: years). Figure 7: The variations in the reliability indexes with the number of years in service: (**a**) ML1 and ML2; (**b**) ML3 and ML4.