# Impact of Road Traffic Tendency in Europe on Fatigue Assessment of Bridges
## 4 Case Studies
### Outcomes of the Analysis and Discussion  
Following the procedure described before, the appropriate equivalence factor \(k_{eq}-L\) curves have been determined for each case study, distinguishing the two subcases for fatigue resistance curves already identified in Section 4.1. The outcomes of the analysis are summarized in Figures 8-10 for the three case studies, referring to a total flow of \(N_{t}=10^{8}\) lorries. Figure 8: Equivalence factor \(k_{eq}-L\) curves for bending moment at midspan of simply supported beams (case study no. 1—total lorry flow: \(10^{8}\) lorries): (**a**) subcase 1—\(S-N\) curves as per Figure 7a,b; (**b**) subcase 2—\(S-N\) curves as per Figure 7c,d. More precisely, Figure 8 considers bending moment at midspan of simply supported beams (case study no. 1); Figure 9 refers to the hogging moment at intermediate support of two-span continuous beams (case study no. 2) and Figure 10 refers to the bending moment at midspan of three-span continuous beams (case study no. 3). In the figures, (a) refers to subcase 1 and (b) to subcase 2. To improve the readability of the diagrams, curves concerning fatigue load models of EN1991-2 are in blue; curves concerning Moerdijk traffic are in red; curve concerning Igualada traffic are in green. Figure 11 refers again to case study no. 1, but assuming the total lorry flow limited to \(5\times 10^{6}\) lorries, which is a common situation for roads with low flow rate of lorries. In EN1991-2, for example, annual flows of \(1.25\times 10^{5}\) and \(5\times 10^{4}\) lorries, respectively, are given for main roads and secondary roads with low flow rate of lorries. Figure 10: Equivalence factor \(k_{eq}-L\) for bending moment at midspan of three-span continuous beams (case study no. 3—total lorry flow: \(10^{8}\) lorries): (**a**) subcase 1—\(S-N\) curves as per Figure 7a,b; (**b**) subcase 2—\(S-N\) curves as per Figure 7c,d. Figure 9: Equivalence factor \(k_{eq}-L\) curves for hogging moment at intermediate support of two-span continuous beams (case study no. 2—total lorry flow: \(10^{8}\) lorries): (**a**) subcase 1—\(S-N\) curves as per Figure 7a,b; (**b**) subcase 2—\(S-N\) curves as per Figure 7c,d. With this low lorry flow, similar results have been obtained also for case studies no. 2 and no. 3, so it seems unnecessary further discussion about. Inspecting the curves, it can be remarked that:  
* if \(S-N\) curves are as per Figure 7c,d, i.e., details do not manifest constant amplitude fatigue limit (subcase 2), fatigue load model LM4 leads in all considered case studies to estimations of the fatigue life on the safe side;
* on the contrary, if \(S-N\) curves are as per subcase 1 (Figure 7c,d), i.e., details are characterized by fatigue limit, fatigue load models of EN1991-2 could result in being slightly on the unsafe side in comparison with Auxerre traffic, when \(L<15\) m and high flow rate is assumed, i.e., the total lorry flow is \(10^{8}\) lorries. In any case the error is generally smaller than 8%, except in case study no. 2 for \(L=1\) m (Figure 9a), when it attains about 15%. But, in the latter circumstance, besides observing that elements characterized by \(L\approx 1\) m are rather uncommon, it should be highlighted that the differences are emphasized by the particular shape of the influence line considered in case study no. 2 (Figure 6b): in fact, when \(L\approx 1\) m the typical inter-axle distances of tandem axles are close to the distance between the two absolute minima of that influence line, \(0.954L=0.954\) m. In the other cases, the discrepancies can be explained considering that, in the particular context of \(L<15\) m and high flow rate, the assessment is governed by the set of \"frequent\" lorries included in fatigue load model LM2. In fact, in such a condition, the unavoidable differences that exist between standardized lorries and actual lorries, in terms of axle loads; vehicle weight; inter-axle distances of tandem and tridem axles, clearly maximize their effects.