# Impact of Road Traffic Tendency in Europe on Fatigue Assessment of Bridges
## 4 Case Studies
### Equivalent Fatigue Categories  
In the analyses, the verifications have been carried out both in terms of unlimited fatigue life as well as in terms of fatigue damage, according to the previously anticipated criteria. Regarding real traffics, the \"frequent\" value \(\Delta\sigma_{\max,f}\) has been determined, according to the already recalled Eurocode pre-normative studies, by using Equation (3). The fatigue damage, when relevant, has been calculated via the aforementioned Palmgren-Miner law [43; 44] (Equation (2)). For the sake of the comparison, an equivalent, but more expressive, format for fatigue verification can be introduced, in terms of detail category:  
\[\Delta\sigma_{C}\geq\Delta\sigma_{C,\min}, \tag{4}\]  
where \(\Delta\sigma_{C}\) is the actual detail category and \(\Delta\sigma_{C,\min}\) the minimum detail category satisfying fatigue verifications. For details which do not manifest constant amplitude fatigue limit, obviously, this results in:  
\[\Delta\sigma_{C,\min}=\Delta\sigma_{C,\alpha\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\[D=\sum_{\Delta\sigma_{i}\geq 0.737\Delta\sigma_{C,\alpha q}}\frac{n_{i}\Delta \sigma_{i}^{3}}{2\cdot 10^{6}\Delta\sigma_{C,\alpha q}^{3}}+\sum_{\Delta\sigma_{j}<0.737 \Delta\sigma_{C,\alpha q}}\frac{n_{j}\Delta\sigma_{j}^{5}}{5\cdot 10^{6}\big{(}0.737 \Delta\sigma_{C,\alpha q}\big{)}^{5}}=1; \tag{6}\]
* whereas, when the detail category \(\Delta\sigma_{C}\) corresponds to the fatigue strength at \(N^{*}\) cycles to failure, like in case of pre-stressing or reinforcing steel details: \(\bigcirc\) the abscissa of the knee of the \(S-N\) curve is located at \(N^{*}\); therefore, \(\Delta\sigma_{C,\alpha q}\) is the solution of the equation: \[D=\sum_{\Delta\sigma_{i}\geq\Delta\sigma_{C,\alpha q}}\frac{n_{i}\Delta\sigma _{i}^{m_{1}}}{N^{*}\Delta\sigma_{C,\alpha q}^{m_{1}}}+\sum_{\Delta\sigma_{j}< \Delta\sigma_{C,\alpha q}}\frac{n_{j}\Delta\sigma_{j}^{m_{2}}}{N^{*}\Delta \sigma_{C,\alpha q}^{m_{2}}}=1.\] (7) On the contrary, for details characterized by constant amplitude fatigue limit, this results in: \[\Delta\sigma_{C,\min}=\min\big{(}\Delta\sigma_{C,\alpha q};\Delta\sigma_{C,D} \big{)},\] (8) where \(\Delta\sigma_{C,D}\) is the detail category for which \(\Delta\sigma_{D}=\Delta\sigma_{\max,f}\); \(\Delta\sigma_{\max,f}\) is the already recalled maximum stress range relevant for unlimited fatigue life assessments and \(\Delta\sigma_{C,\alpha q}\) is the already defined detail category for which it results in \(D=1\). In the case of structural steel details, since \(\bullet\)\(m_{1}=3\); \(m_{2}=5\); \(\bullet\) the detail category \(\Delta\sigma_{C}\) is the fatigue strength at \(N=2\times 10^{6}\) cycles; \(\bullet\) the abscissas of the knees of the \(S-N\) curve are at \(N=5\times 10^{6}\) cycles and \(N=10^{8}\) cycles, respectively; It results in: \[\Delta\sigma_{C,D}\approx 1.357\Delta\sigma_{\max},\] (9) so that \(\Delta\sigma_{C,\alpha q}\) is the solution of the equation: \[D=\sum_{\Delta\sigma_{i}\geq 0.737\Delta\sigma_{C,\alpha q}}\frac{n_{i}\Delta \sigma_{i}^{3}}{2\cdot 10^{6}\Delta\sigma_{C,\alpha q}^{3}}+\sum_{0.405\Delta \sigma_{C,\alpha q}<\Delta\sigma_{j}<0.737\Delta\sigma_{C,\alpha q}}\frac{n_{ j}\Delta\sigma_{j}  
with obvious symbolism. In conclusion, an equivalence factor \(k_{eq}-L\) plot represents a very synthetic, but eloquent, representation of the fatigue resistance demand of a given traffic, in comparison with the fatigue resistance required by the Auxerre traffic. From this point of view, \(k_{eq}\) is also a tool to directly compare fatigue effects of different traffics or load models, since bigger fatigue effects correspond to higher values of \(k_{eq}\).