# Impact of Road Traffic Tendency in Europe on Fatigue Assessment of Bridges
## 4 Case Studies
### Assessment of Fatigue Effects  
For each case study and for each value of the span length \(L\) considered in the study, the bending moment histories produced by Auxerre, Moerdijk and Igualada traffics as well as by the fatigue load models LM2 and LM4 have been determined. As anticipated, the possibility that two vehicles were simultaneously travelling at the same lane, has been properly taken into account, when relevant. Regarding LM4, coherently with the features of real traffic measurements considered here, long-distance compositions have been adopted. In the analyses, fatigue effects have been assessed considering two distinct subcases, depending on the shape of the \(S-N\) curve:  
* Subcase 1, which deals with details and materials endowed with constant amplitude fatigue limit; like most of the structural steel details (EN1993-1-9 [23]), characterized by fatigue resistance bilinear or trilinear \(S-N\) characteristic curves;
* Subcase 2, which deals with details and materials characterized by bilinear \(S-N\) characteristic curves, lacking constant amplitude fatigue limit. \(S-N\) curves typically adopted in subcase 1 are illustrated in Figure 7a,b. They are characterized in the bi-logarithmic plane by two or three linear branches:  
* when the assessment refers to unlimited fatigue life, a bilinear \(S-N\) curve is adopted for steel details. The slope of the inclined branch is commonly assumed \(m=3\)[23], but only the horizontal branch, whose ordinate is the constant amplitude fatigue limit \(\Delta\sigma_{D}\), corresponding to \(N=5\times 10^{6}\) cycles to failure, is significant for the verification (Figure 7a);
* when, instead, the assessment requires calculation of fatigue damage, the trilinear \(S-N\) curve is adopted, characterized by two inclined linear branches; for steel details, the slopes of the branches are \(m_{1}=3\), for \(\Delta\sigma\geq\Delta\sigma_{D}\), and \(m_{2}=5\), for \(\Delta\sigma_{D}>\Delta\sigma\geq\Delta\sigma_{L}\), where \(\Delta\sigma_{L}\), corresponding to \(N=10^{8}\) cycles to failure, is the so-called cut-off limit, given by the ordinate of the horizontal branch (Figure 7b). Figure 7: Fatigue strength \(S-N\) curves – Typical \(S-N\) curves for most structural details (subcase 1): (**a**) bilinear; (**b**) trilinear—General bilinear \(S-N\) curves (subcase 2): (**c**) all-purpose; (**d**) for some steel details. Bilinear fatigue resistance \(S-N\) curves typically adopted in subcase 2 are characterized by two inclined branches. According to Eurocodes [23; 26], the slopes of the two linear branches of these curves are \(m_{1}\) and \(m_{2}\) and the abscissa of the knee is \(N^{*}\) cycles (see Figure 7c). In the present study, it has been assumed that \(m_{1}=3\) and \(m_{2}=5\) (see Figure 7d) to match the assumptions adopted in the calibration of fatigue load models of Eurocodes (see, for example, [33]), but it is also the case of some reinforcing steel details [25; 26]. Evidently, provided that the slopes of the inclined branches and the abscissa of the knees, i.e., the number of cycles to failure corresponding to the knees, are preset, the \(S\)-\(N\) curve can be univocally determined by means of one parameter in both the aforementioned subcases. This parameter is the so-called fatigue category \(\Delta\sigma_{C}\) of the detail. For structural steel details, \(\Delta\sigma_{C}\) represents the fatigue resistance, expressed in MPa, at \(N=2\cdot 10^{6}\) cycles, characterized by 95% probability of exceedance; for reinforcing and pre-stressing steel details, \(\Delta\sigma_{C}\) represents the fatigue resistance at \(N^{*}\) cycles, characterized by 95% probability of exceedance, or, in other words, the ordinate of the knee of the \(S-N\) curve. The aforementioned representations of the fatigue resistance are strictly valid when the fatigue resistance does not depend on the mean stress of the cycles; but they can be easily generalized, if parameterized in terms of mean stress.