# A Fatigue Reliability Assessment for Rail Tension Clamps Based on Field Measurement Data
## 3 Reliability Assessment of the Tension Clamp  
The limit state function for the reliability analysis was defined as the critical cumulative damage index minus the cumulative fatigue damage obtained by the damage rule as follows [15]:  
\[g=\Delta-e\frac{N\cdot S_{re}^{m}}{A}, \tag{3}\]  
where \(g\), \(\Delta\), and \(n\) are the limit state function, Miner's critical damage cumulation index, and the number of cycles in the stress range, respectively. In Equation (3), \(A\) stands for the detail category constant (MPa3) of the S-N curve in the AASHTO Bridge Design Specifications [20]. Since the tension clamps were made from a plain base metal without any notches, the value for \(A\) is determined by the Category A. \(S_{re}^{m}\) indicates an equivalent stress range for \(m=3\), and \(e\) denotes an error factor, which includes the uncertainties in estimating the stress value. The damage cumulation index, fatigue index, equivalent stress range, and error factor were considered as random variables in the reliability analysis. The number of cycles in the stress range corresponding to the years in service was calculated by applying the following relationship [16]:  
\[N(Y)=365\cdot ADTT\cdot C_{s}\cdot Y, \tag{4}\]  
where \(N(Y)\) and \(Y\) present the accumulated number of the stress cycles and the number of years in service, respectively. \(ADTT\) refers to the average daily train traffic on a rail. The  
Figure 6: The stress ranges of the outside track gauges plotted on probability papers: (**a**) ML3 on the normal probability paper; (**b**) ML4 on the lognormal probability paper. actual _ADDT_ was equal to 160 for the urban railway track where the field measurements were performed. \(C_{s}\) denotes the number of stress cycles per train passage. As a result of the rainflow-counting, \(C_{s}\) became 10 for ML1, ML2, and ML4 and 11 for ML3. The advanced first-order second-moment reliability method [17] is usually adopted to calculate the reliability index, which is obtained by solving the following minimization problem:  
\[\underset{\mathbf{X}}{\text{Min}}\ \beta^{2}=\|\mathbf{X}\|_{2}^{2}\ \text{ subject to}\ g(\mathbf{X})=0, \tag{5}\]  
where \(\beta\) is the reliability index, \(\mathbf{X}\) denotes the vector of the random variables (\(\mathbf{X}=(\Delta,\ e,\ A,\ S_{re}^{m})^{T}\)), and \(\|\mathbf{X}\|_{2}^{2}\) indicates the 2-norm of the vector \(\mathbf{X}\). The Hasofer-Lind-Rackwitz-Fiessler algorithm with the gradient projection method [22] was adopted for the solution scheme to solve the minimization problem. Since the limit state function in Equation (3) was nonlinear with respect to the random variables, the most probable failure point could be calculated as follows. \[X_{k+1}^{\prime\prime}=\frac{1}{\big{|}\
abla g\big{(}X_{k}^{\prime\prime} \big{)}\big{|}^{2}}\Big{[}\
abla g\Big{(}X_{k}^{\prime\prime}\big{)}\cdot X_{ k}^{\prime\prime}-g\Big{(}X_{k}^{\prime\prime}\Big{)}\Big{]}\
abla g\Big{(}X_{k}^{ \prime\prime}\Big{)}, \tag{6}\]  
where \(X^{\prime}\) indicates the vector of the random variables in the reduced coordinate system, which is obtained by the Rackwitz-Fiessler algorithm. The asterisk after derivative indicates the most probable failure point, i.e., \(X_{k}^{\prime\prime}\) represents the minimum distance point on the limit state function at the \(k\)-th iteration, and \(\
abla g\Big{(}X_{k}^{\prime\prime}\Big{)}\) is the gradient vector of the limit state function at \(X_{k}^{\prime\prime}\). Note that \(k\) denotes the iteration count. The statistical parameters of the random variables quoted from previous studies [15; 23] are summarized in Table 3.  
Figure 7 shows the reliability indexes of the tension clamps according to the number of years in service at the four MLs. In the figure, the blue lines indicate the reliability indexes of the outer rails (ML1 and ML3) and the red lines represent those of the inner rails (ML2 and ML4). The solid and dotted lines show the reliability indexes for the inside and outside track gauges of the rail, respectively.