# Impact of Road Traffic Tendency in Europe on Fatigue Assessment of Bridges
## 5 Damage-Equivalence Factors
* 300 working days per year, corresponds to around \(N_{Aux}\approx 8\times 10^{5}\) lorries per year; \(N_{obs}\) is the annual lorry flow of the traffic considered and \(\overline{n}_{i}\) is the annual flow of lorries weighing \(Q_{i}\) in the traffic considered. In Equation (18), the equivalent lorry weight \(Q_{eq}\) of the actual traffic can be easily recognized: \[Q_{eq}=\Bigg{(}\frac{\sum_{i}\overline{n}_{i}Q_{i}^{m_{eq}}}{\sum_{i}\overline {n}_{i}}\Bigg{)}^{\frac{1}{m_{eq}}}.\] (19) Owing the fact that damage equivalence method refers to simplified linear \(S-N\) curves characterized by constant slope \(m_{eq}=5\), matching one of the inclined branches in Figures 12b and 13, in the current version of the Eurocodes \(\lambda_{2}\) is given by: \[\lambda_{2}=\frac{1}{Q_{eq,Aux}}\Bigg{(}\frac{\sum_{i}\overline{n}_{i}Q_{i}^{ 5}}{\sum_{i}\overline{n}_{i}}\Bigg{)}^{\frac{1}{5}}\Bigg{(}\frac{N_{obs}}{N_ {Aux}}\Bigg{)}^{\frac{1}{5}}.\] (20)  
Figure 12: Fatigue load model 3 (LM3) of EN1991-2 (dimensions are m [35], adapted from [1]). * \(\lambda_{3}\) considers the effects of the actual design working life, \(t_{d}\), in comparison with the reference design working life adopted for bridges, which is around 100 years: \[\lambda_{3}=\left(\frac{t_{d}}{100}\right)^{\frac{1}{n_{eq}}}.\] (21) Again, in the Eurocodes a fixed value of \(m_{eq}=5\) is assumed. * Finally, \(\lambda_{4}\) considers the effect of multilane traffic, also accounting for lorries simultaneously traveling the bridge. The evaluation of \(\lambda_{4}\) is a very complex issue, which is not solved in Eurocodes, where undoubtedly simultaneity is disregarded, and which will not be deepened here. Anyhow, for a more exhaustive discussion of the problem it is possible to refer to [35]. Since \(\lambda_{2}\) and \(\lambda_{3}\) express in terms of stress ranges the linear dependence of the damage on the lorry flow, they have the same structure, so that Equations (18) and (21) can be combined to obtain a single damage equivalent factor \(\lambda^{*}\):  
\[\lambda^{*}=\lambda_{2}\lambda_{3}=\frac{Q_{eq}}{Q_{eq,Max}}\left(\frac{t_{d} N_{obs}}{100N_{Max}}\right)^{\frac{1}{n_{eq}}}=\frac{Q_{eq}}{Q_{eq,Max}} \left(\frac{N_{t,obs}}{N_{t,Max}}\right)^{\frac{1}{n_{eq}}}, \tag{22}\]  
where \(N_{t,obs}\) and \(N_{t,Max}\) are the total lorry flows in the design working life of the bridge, derived from the actual traffic and from the Auxerre traffic, respectively. For the Auxerre traffic, the previous illustration indicates a global traffic volume \(N_{t,Max}\approx 8\times 10^{7}\), but in the following it has been assumed that \(N_{t,Max}=10^{8}\). From the context, it is evident that Equation (22) can be written in an equivalent way, directly referring to fatigue damage:  
\[\Delta\sigma_{eq}=\Delta\varphi_{fat}\Delta\overline{\sigma}\prod_{\lambda_{i }=1}^{n}\lambda_{i}=\Delta\varphi_{fat}\lambda\Delta\overline{\sigma}, \tag{14}\]0  
\[\Delta\sigma_{eq}=\Delta\varphi_{fat}\Delta\overline{\sigma}\prod_{\lambda_{i }=1}^{n}\lambda_{i}=\Delta\varphi_{fat}\lambda\Delta\overline{\sigma}, \tag{14}\]1 Damage-equivalence factor \(\lambda^{*}-L\) curves for various total lorry flows for bending moment at midspan of simply supported beams (case study no. 1—reference 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. where \(D_{obs}\) is the fatigue damage caused by the actual traffic and \(D_{Aux}\) is the reference fatigue damage, caused by a global flow of \(10^{8}\) lorries, fitting the Auxerre traffic composition. In any case, strictly speaking, for a given total flow \(m_{eq}\) is a function of \(L\). In fact, as discussed next, when bilinear or trilinear \(S-N\) curves are considered, \(\Delta\sigma_{C,\min}\) varies with the number of the stress cycles, which usually decreases as \(L\) increases.