## 3 Modeling of Interactions between Non-Simultaneous Vehicles  
Recalling that distinct elements of non-interacting vehicle flow separately travel the bridge, the total stress history caused by the flow itself can be seen as a random assembly of the individual stress histories caused by each element. On the contrary, the total stress spectrum, i.e., the number of cycles and the associated stress variations, cannot be determined, in general, as a pure and simple sum of the individual stress spectra derived from the individual stress histories. Figure 9: Outline of an example of multilane traffic generation. When the maximum stress, \(\sigma_{\text{max}}\), and minimum stress, \(\sigma_{\text{min}}\), of the stress history are caused by distinct individual stress histories [11], the two stress histories interact generating a combined stress spectrum, whose effects, in terms of fatigue, are more relevant than the sum of those produced by the two individual stress histories. As already remarked in Section 1, several cycle counting methods identify this behavior, including, inter alia, rainflow method and reservoir method, currently adopted in fatigue assessment of bridges. In current practice, this topic is particularly relevant when the fatigue assessments are based on artificial samples of traffic, generated from a number of conventional vehicles' geometries, as in Eurocode EN1991-2 [21], where fatigue load models are given in terms of sets of standardized lorries and standardized train. The issue is investigated in the following. Let:  
* \(A\), \(B\), \(C\) individual element typologies of the non-interacting vehicle flow, be characterized by a number of annual occurrences \(N_{A}\), \(N_{B}\), \(N_{C}\), respectively;
* \(\sigma_{A}\ =\ \sigma_{A}(t)\), or simply \(A\), the individual stress history produced by \(A\);
* \(\sigma_{A\text{max}}\) and \(\sigma_{A\text{min}}\), respectively, the maximum and the minimum values of \(\sigma_{A}(t)\). Since two elements \(A\) and \(B\) of the non-interacting vehicle flow cannot travel the bridge at the same time, they cause individual stress histories at different times, so that phase shift effects are excluded. However, when one of the following conditions is satisfied,  
\[(\text{a})\ \left\{ \begin{array}{l}\sigma_{A\text{max}}\ >\ \sigma_{B\text{max}}\\ \sigma_{A\text{min}}\ >\ \sigma_{B\text{min}}\end{array} \right.\quad( \text{b})\ \left\{ \begin{array}{l}\sigma_{A\text{max}}\ <\ \sigma_{B\text{max}}\\ \sigma_{A\text{min}}\ <\ \sigma_{B\text{min}}\end{array} \right. \tag{46}\]  
the two stress histories interact, as illustrated in Figure 10, originating the combined stress history \(\sigma_{AB}\ =\ \sigma_{AB}(t)\), or \(AB\). In this case, the maximum stress range, \(\Delta\sigma_{AB}\ =\ \Delta\sigma_{AB}(t)\), pertains to \(AB\) and the overall stress spectrum comes from the analysis of \(AB\); if not, the maximum stress range still pertains to a single stress history and the overall stress spectrum is simply the sum of individual stress spectra (see Figure 11). Figure 11: Non-interacting individual stress histories. Figure 10: Combination of individual stress histories. If the non-interacting vehicle flow is composed only by element typologies \(A\) and \(B\) causing stress histories satisfying conditions (46), the expected number of annual occurrences of the combined stress history \(\sigma_{AB}\) trivially results, as the entry of \(A\) on the bridge is independent on the entry of \(B\),  
\[N_{AB}\ =\ \frac{N_{A(0)}N_{B(0)}}{N_{A(0)}\ +\ N_{B(0)}}; \tag{47}\]  
where \(N_{A(0)}\ =\ N_{A}\) and \(N_{B(0)}\ =\ N_{B}\) are the annual occurrences of \(A\) and \(B\), respectively. The remaining expected number of occurrences of \(A\) and \(B\) then are given by  
\[N_{A(1)}\ =\ \frac{N_{A(0)}^{2}}{N_{A(0)}\ +\ N_{B(0)}};\ N_{B(1)}\ =\ \frac{N_{B(0)}^{2}}{N_{A(0)}\ +\ N_{B(0)}}. \tag{48}\]  
Truly, actual assessment conditions can be not so simple.