# A Fatigue Reliability Assessment for Rail Tension Clamps Based on Field Measurement Data
## 2 The Statistical Model for the Stress Range  
The statistical model for the stress range of the tension clamps was developed to estimate the fatigue reliability indexes using field measurement data. Figure 2(a) shows a rosette strain gauge attached to the top of the middle bend of a tension clamp and Figure 2(b) presents the four measurement locations (MLs) on the railway track. ML1 and ML3 were located on the outer rail of the railway track while ML2 and ML4 were positioned on the corresponding inner rail of the railway track. The field measurements were performed on both the inside and outside track gauges at the four MLs. [PERSON]'s law was employed to determine the principal stresses using the field measurements as follows:  
\[\sigma_{P,Q}=\frac{E}{2}\left[\frac{\varepsilon_{1}+\varepsilon_{3}}{1-\
u}\pm \frac{\sqrt{2}}{1+\
u}\sqrt{\left(\varepsilon_{1}-\varepsilon_{2}\right)^{2}+ \left(\varepsilon_{2}-\varepsilon_{3}\right)^{2}}\right], \tag{1}\]  
where \(\sigma_{P}\), \(\sigma_{Q}\), \(E\), and \(\
u\) are the maximum and minimum principal stresses, the elastic modulus, and Poisson's ratio, respectively, and \(\varepsilon_{1}\), \(\varepsilon_{2}\), and \(\varepsilon_{3}\) indicate the measured strains with the rosette strain gauge. As a representative case, the variations in the principal stresses for the outside track gauge at ML3 and ML4 during the train passage are illustrated in Figure 4; there seems to be a clear distinction between the main vibrations due to the tandem axle of each carriage and the resulting residual vibrations. The train was composed of six carriages and each carriage had two tandem axles. Thus, 12 stress peaks are shown in the figure. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{3}{*}{**Curve Radius (m)**} & \multirow{3}{*}{**Number of Fractures**} & \multicolumn{4}{c}{**Down Train**} & \multicolumn{4}{c}{**Up Train**} \\ \cline{3-10}  & & \multicolumn{2}{c}{**Outer Rail**} & \multicolumn{2}{c}{**Inner Rail**} & \multicolumn{2}{c}{**Outer Rail**} & \multicolumn{2}{c}{**Inner Rail**} \\ \cline{3-10}  & & **Inside** & **Outside** & **Inside** & **Outside** & **Inside** & **Outside** & **Inside** & **Outside** \\  & & **Track** & **Track** & **Track** & **Track** & **Track** & **Track** & **Track** & **Track** \\ \hline
1200 & 28 & 0 & 1 & 13 & 11 & 0 & 0 & 2 & 1 \\ \hline
1200 & 11 & 0 & 0 & 0 & 0 & 0 & 0 & 7 & 4 \\ \hline
800 & 335 & 1 & 0 & 16 & 4 & 3 & 17 & 164 & 130 \\ \hline
600 & 18 & 0 & 0 & 13 & 4 & 0 & 0 & 0 & 1 \\ \hline
600 & 5 & 0 & 0 & 5 & 0 & 0 & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The damage status of the tension clamps on an urban railway. The rainflow-counting algorithm [18] was adopted to calculate the stress range and the histograms of the stress ranges of the main vibrations are plotted for the four MLs in Figure 5. The histogram for the stress range of the inside track gauge at ML3 is not presented because there was a measurement error in the data. The stress ranges due to the main vibrations were calculated as 55 to 120 MPa, while the stress range due to the residual vibrations was 30 MPa or less. According to the findings from a previous study [19], a stress range of less than a quarter of the constant amplitude fatigue threshold (CAFT) can become truncated. Since the CAFT of the base metal is 165 MPa [20], the stress ranges that were less than 41 MPa could be excluded. Since most incidences of the fatigue damage to the tension clamps are observed on the inner rail, probabilistic models for the stress range were developed for the inner and outer rails and for the inside and outside track gauges individually for comparison purposes. The distribution type of the stress range was determined through the likelihood function and the Kolmogorov-Smirnov goodness of fit test [21] for five distribution types. Normal, lognormal, Gumbel, Frechet, and Weibull distributions were not rejected as a  
Figure 4: The variations in the principal stresses for the outside track gauge at ML3 and ML4.