# Experimental Investigation of the Fatigue Life of a Bridge Crane Girder Using S-N Method
### Assessment of Fatigue Life and Strength According to STN 270103
The parameters entering into the fatigue damage accumulation hypotheses are \(\sigma_{\text{c (m)}}=36\) MPa, \(N_{\text{c}}=2\cdot 10^{6}\) cycles and the Wohler curve given by the relation \(\sigma_{\text{a}}^{\text{m}}\cdot N=\text{const. }\) for the slope \(m=3,\ 5\). The total number of cycles to failure \(H_{\text{PM}}\) for the stress range corresponding to Modes 2 and 3 evaluated according to the Palmgren-Miner hypothesis, transformed from Figures 10 and 11, is given in \[R_{\text{fat, t}}(0)=\frac{5}{3}R_{\text{fat, t}}(-1)=\frac{5}{3}\cdot 63.6=106 \ \text{MPa}. \tag{9}\]1 The damage, according to the Palmgren-Miner hypothesis assuming \(\sigma_{\text{m}}=\text{const. }\), is determined by the relation  
\[R_{\text{fat, t}}(0)=\frac{5}{3}R_{\text{fat, t}}(-1)=\frac{5}{3}\cdot 63.6=106 \ \text{MPa}. \tag{9}\]2  
where \(n_{i}\) is the number of cycles with constant amplitude \(\sigma_{\text{ai}}\) in the spectrum, \(N_{i}\) is number of cycles to failure at constant amplitude \(\sigma_{\text{ai}}>\sigma_{\text{c}}\) (above the fatigue limit) and \(p\) is the number of levels of the distribution of the fundamental spectrum. The number of cycles to failure with constant amplitude \(\sigma_{\text{ai}}>\sigma_{\text{c}}\) is determined from the relation:  
\[R_{\text{fat, t}}(0)=\frac{5}{3}R_{\text{fat, t}}(-1)=\frac{5}{3}\cdot 63.6=106 \ \text{MPa}. \tag{9}\]3  
where \(\sigma_{\text{c}}\) is the fatigue limit of the material (MPa), \(\sigma_{\text{ai}}\) is spectrum amplitude (MPa), \(N_{\text{c}}\) is the number of cycles at fatigue limit and \(w\) is the exponent of the fatigue curve. The total number of cycles to failure according to the Palmgren-Miner hypothesis is given by:  
\[R_{\text{fat, t}}(0)=\frac{5}{3}R_{\text{fat, t}}(-1)=\frac{5}{3}\cdot 63.6=106 \ \text{MPa}. \tag{9}\]4  
It can be seen in Table 5 that the total number of cycles to failure according to the Palmgren-Miner hypothesis is the lowest on girder B at the measured locations B3 and B4. Compared to the corresponding cross-section on girder A (locations A1 and A2), it decreases by more than 50%. It was found that the number of cycles to failure at location B7 (i.e., 300 mm away from locations B3 and B4), was similar to that on girder A. This was due to the welded joint connecting the rail to the girder B top flange located precisely at the placement of the applied strain gauge transducers B3 and B4 (Figure 12). Figure 13 compares the calculated values of the normal stresses on girder A (Figure 13a) and on girder B (Figure 13b). From the course depicted in Figure 13a, it is evident that the distance of 300 mm between the measuring points for a total girder's length of 28,500 mm does not play a significant role. From the measurements performed on girders A and B, it can be stated that the stress values at locations A1, A2, A8 and B7 are very similar in character. At locations B3 and B4, the measured local values are sometimes higher by more than 50%. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Mode**} & \multicolumn{5}{c}{**Number of Cycles to Failure \(H_{\text{PM}}\) at Strain Gauge Locations**} \\ \cline{2-7}  & **A1** & **A2** & **B3** & **B4** & **B7** & **A8** \\ \hline
2 & 1,042,204 & 926,605 & **527,273** & **594,307** & 877,733 & 912,436 \\
3 & 905,820 & 886,978 & **417,921** & **482,169** & 857,737 & 888,170 \\ \hline \end{tabular}
\end{table} 
Table 5: Number of cycles to failure according to the Palmgren–Miner hypothesis.