# Impact of Road Traffic Tendency in Europe on Fatigue Assessment of Bridges
## 5 Damage-Equivalence Factors
#### Damage-Equivalence Factors for Total Lorry Flow  
To assess the dependence of \(\lambda^{*}\) on the total traffic volume, fatigue effects caused by different volumes of Auxerre traffic have been compared in case study no. 1, i.e., considering the influence line of bending moment at the midspan of simply supported beams (Figure 6a). In the study, total flows \(N_{t}\) equal to \(5\times 10^{6}\); \(10^{7}\); \(2.5\times 10^{7}\); \(5\times 10^{7}\) and \(10^{8}\) lorries have been considered, thereby investigating the most relevant practical cases. As already said, \(N_{t,Aux}=10^{8}\) lorries has been assumed as reference global traffic flow. The need to have \"unbiased\" results, not influenced by the equivalent lorry weight, imposed to vary only the global flow, adopting in any case the Auxerre traffic composition. In fact, in this way Equation (22) reduces to:  
\[\lambda^{*}=\lambda_{2}\lambda_{3}=\left(\frac{N_{t,obs}}{N_{t,Aux}}\right)^{ \frac{1}{m_{eq}}}. \tag{24}\]  
being independent on \(Q_{eq}\). Recalling the definitions of \(\Delta\sigma_{C,\min}\) and the rationale of Equations (5) and (8), \(\lambda^{*}\) can be also interpreted as a correction factor for \(\Delta\sigma_{C,\min}\), accounting for modifications of the global flow:  
\[\lambda^{*}=\frac{\Delta\sigma_{C,\min}\big{(}N_{t,obs}\big{)}}{\Delta\sigma_ {C,\min}\big{(}N_{t,Aux}\big{)}}. \tag{25}\]  
where \(\Delta\sigma_{C,\min}(N_{t,Aux})\) is the minimum fatigue category for which fatigue verification is satisfied considering the \(N_{t,Aux}=10^{8}\) lorries reference global flow and \(\Delta\sigma_{C,\min}\big{(}N_{t,obs}\big{)}\) is the minimum fatigue category for which fatigue verification is satisfied considering the actual global flow. Evidently, as anticipated, when the \(S-N\) curve is linear with constant slope \(m\), \(m_{eq}=m\) and \(\lambda^{*}\) can be directly derived from Equation (24):  
\[\lambda^{*}=\left(\frac{N_{t,obs}}{N_{t,Aux}}\right)^{\frac{1}{m}}. \tag{26}\]  
In the general case, when the fatigue resistance \(S-N\) curve is bilinear or trilinear, as in subcases 1 and 2, the evaluation of \(\lambda^{*}\) is not so direct, since the exponent in Equation (24) is unknown. Anyway, the actual value of \(\lambda^{*}\) can be estimated introducing an appropriate equivalent slope \(m_{eq}\). Combining Equations (24) and (25), \(m_{eq}\) trivially results in:  
\[m_{eq}=\frac{\log N_{t,obs}-\log N_{t,Aux}}{\log(\Delta\sigma_{C,\min}(N_{t, Aux}))-\log\big{(}\Delta\sigma_{C,\min}\big{(}N_{t,obs}\big{)}\big{)}}=\frac{ \log N_{t,obs}-\log N_{t,Aux}}{\log\lambda^{*}}. \tag{27}\]  
It is important to remark that:  
* when bilinear \(S-N\) curves as per Figure 7d (subcase 2) are considered:\(\odot\)  it is \(m_{eq}\approx 3\), if fatigue damage is mostly caused by cycles with stress ranges \(\Delta\sigma_{i}\) satisfying the inequality \(\Delta\sigma_{i}\geq\Delta\sigma_{D}\); \(\odot\)  or \(m_{eq}\approx 5\), if the fatigue damage is mainly due to cycles for which \(\Delta\sigma_{i}<\Delta\sigma_{D}\);
* when bilinear or trilinear \(S-N\) curves as per Figure 7a,b (subcase 1) are considered, \(m_{eq}\) can result in being, even considerably, bigger than five if the number of cycles is so high that fatigue verification is governed by \(\Delta\sigma_{D}\) (unlimited fatigue life). The outcomes of the analysis are illustrated in Figures 13 and 14. The damage equivalence factor \(\lambda^{\star}-L\) curves, obtained applying Equation (25), are reported in Figure 13 and parameterized in terms of total flow, with, as said before, \(N_{t,Max}=10^{8}\) lorries being the reference flow. Curves in Figure 13a are produced referring to subcase 1, considering constant amplitude fatigue limit \(S-N\) curves as per Figure 7a,b. Curves in Figure 13b are produced referring to subcase 2, considering details without constant amplitude fatigue limit, characterized by \(S-N\) curves as per Figure 7c,d. The analogous inverse of the equivalent slope \(1/m_{eq}-L\) curves are illustrated in Figure 14a,b.