# A Fatigue Reliability Assessment for Rail Tension Clamps Based on Field Measurement Data
## 2 The Statistical Model for the Stress Range
Figure 3: The strain gauge used and the measurement locations (MLs): (**a**) The attachment of the rosette strain gauge to the tension clamp; (**b**) the MLs on the railway track. result of the goodness of fit test for all stress ranges (Figure 5). To confirm the most likely distribution type for the stress range, the likelihood functions were calculated for the five tested distribution types. The likelihood function \(L\) is defined as follows:  
\[L(X_{1},X_{2},\cdots,\ X_{n},\ ;\ \theta)=f_{X}(X_{1};\theta)\cdots f_{X}(X_{n}; \theta), \tag{2}\]  
where \(X_{i}\) and \(f_{X}\) indicate the \(i\)-th data and the probability density function (PDF) of a random variable \(X\), respectively, and \(\theta\) denotes the statistical parameters of an assumed distribution. The normal distribution was the most likely distribution type for the stress ranges at ML1, ML2, and ML3, while the lognormal distribution was the most likely for the stress range at ML4. Table 2 shows the distribution types for the stress ranges at the four MLs. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{1}{c}{**Measurement Location**} & \begin{tabular}{c} **Mean** \\ **(MPa)** \\ \end{tabular} & \begin{tabular}{c} **Standard Deviation** \\ **(MPa)** \\ \end{tabular} & \begin{tabular}{c} **COV 1** \\ \end{tabular} & \
\begin{tabular}{c} **Distribution** \\ **Type** \\ \end{tabular} &
\begin{tabular}{c} **Figure** \\ **Number** \\ \end{tabular} \\ \hline \multirow{2}{*}{ML1} & Inside track gauge & 90.01 & 11.83 & 0.13 & Normal & Figure 5a \\  & Outside track gauge & 76.98 & 10.46 & 0.14 & Normal & Figure 5b \\ \hline \multirow{2}{*}{ML2} & Inside track gauge & 83.85 & 16.38 & 0.20 & Normal & Figure 5c \\  & Outside track gauge & 75.02 & 10.87 & 0.14 & Normal & Figure 5d \\ \hline \multirow{2}{*}{ML3} & Inside track gauge & \multicolumn{4}{c}{\(\mathrm{N/A}\)} \\  & Outside track gauge & 88.38 & 12.31 & 0.14 & Normal & Figure 5e \\ \hline \multirow{2}{*}{ML4} & Inside track gauge & 95.56 & 14.55 & 0.15 & Lognormal & Figure 5f \\  & Outside track gauge & 96.70 & 12.48 & 0.13 & Lognormal & Figure 5g \\ \hline \hline \end{tabular}  
* COV, coefficient of variation.  
\end{table} 
Table 2: The statistical parameters of the principal stress ranges at the four MLs. Figure 5: The histograms and PDF plots of the principal stress ranges at the four MLs: (**a**) the inside track gauge at ML1; (**b**) the outside track gauge at ML1; (**c**) the inside track gauge at ML2; (**d**) the outside track gauge at ML2; (**e**) the outside track gauge at ML3; (**g**) the inside track gauge at ML4; (**f**) the outside track gauge at ML4. The mean and the coefficient of variation (COV) of the stress range were calculated by applying linear regression to the empirical cumulative distribution function on a probability paper. Figure 6 shows the cumulative probabilities obtained by the Weibull plotting positions, plotted on the normal and lognormal probability papers with centered symbols for the outside track gauges at ML3 and ML4, respectively, as representative cases. The straight line in each figure indicates the fitted normal and lognormal distributions for the stress ranges. The statistical parameters, such as the mean, standard deviation, and COV, of the stress ranges are summarized in Table 2 and the PDFs of the fitted distribution for the four MLs are exhibited in Figure 5. The stress ranges of the inside track gauges are larger than those of the outside track gauges for ML1, ML2, and ML4, as also confirmed by the mean values reported in Table 2. When comparing the stress ranges of the inner and outer rails, similar levels of the stress ranges were measured at ML1 and ML2, whereas that at ML4 was larger than at ML3.