## 3 Modeling of Interactions between Non-Simultaneous Vehicles
In fact, as in reality the element typologies to be considered are often more than two, it can happen that a given element typology can interact with several element typologies, therefore participating in various combinations. Aiming to solve this crucial issue, a general procedure is proposed here to identify combinations of individual stress histories. As mentioned before, in the most general case the total stress history is formed by a random union of the individual stress histories caused by the elements of the non-interacting vehicle flow. Without loss of generality, it is useful to comment on the example of total stress history illustrated in Figure 12, where \(A\), \(B\), \(C_{i}\), \(C_{j}\) indicate the individual stress histories caused by the corresponding element typologies. Looking at the example in Figure 12, it results that the extremes of the individual stress histories are ordered as follows: \(\sigma_{A\mathrm{max}}\ >\ \sigma_{C\mathrm{/max}}\ >\ \sigma_{B\mathrm{max}}\ >\ \sigma_{C\mathrm{/max}};\sigma_{C\mathrm{/min}}\ <\ \sigma_{B\mathrm{min}}\ <\ \sigma_{A\mathrm{min}}\ <\ \sigma_{C\mathrm{/min}}\); consequently, according to conditions (46):  
* \(C_{j}\) can interact with \(A\);
* \(A\) can interact with \(C_{j}\) and \(B\);
* \(B\) can interact with \(A\);
* \(C_{i}\) cannot interact. Consider the potential combination of the occurrence no. \(n\) of \(A\) and the subsequent occurrence no. \(m\) of \(B\), assuming that rainflow method or reservoir method is adopted. Owing that \(A\) can also interact with \(C_{j}\) and that \(\sigma_{C\mathrm{/min}}\ <\ \sigma_{B\mathrm{min}}\), \(A\) combines with \(B\) only if the combination with \(C_{j}\) cannot take place. In fact, it can be remarked that:  
* Combination concerns, in turns, only two element typologies;
* the occurrence no. n of \(A\) combines with the occurrence no. \(m\) of \(B\) only if the latter takes place in the time interval \(\Delta t_{n}=t_{n+1}-t_{n}\) elapsing between the two consecutive occurrences \(n\) and \(n\ +\ 1\) of \(A\) and if any \(C_{j}\) satisfying the condition \(\sigma_{C\mathrm{/min}}\ <\ \sigma_{B\mathrm{min}}\) does not occur in \(\Delta t_{n}\);  
Figure 12: Example of combination of individual stress histories. * the combination is independent on the number of occurrences of any \(C_{i}\), different from \(A\) and \(B\), satisfying both conditions \(\sigma_{C_{i}\min}\ >\ \sigma_{B\min}\) and \(\sigma_{C_{i}\max}\ <\ \sigma_{A\max}\);
* a given occurrence of \(A\) can combine only once: In case it can participate to alternative combinations, the less favorable, i.e., the one characterized by the maximum stress range, will occur;
* the occurrence no. \(n\) of \(A\) will not combine at all, if all the stress histories \(C_{i}\) occurring during the time interval elapsing from occurrences no. \(n\ -\ 1\) and no. \(n\ +\ 1\) of \(A\), \(\Delta t\ ^{*}=\Delta t_{n}+\Delta t_{n-1}=t_{n+1}-t_{n-1}\), satisfy the conditions \(\sigma_{C_{i}\min}\ >\ \sigma_{A\min}\) and \(\sigma_{C_{i}\max}\ <\ \sigma_{A\max}\). Bearing in mind these observations and taking into account that combinations are governed by the extremes of the stress histories, an original procedure can be setup which, starting from a given non-interacting vehicle flow, allows to determine the number of occurrences of each combined stress history and the remaining number of occurrences of each individual stress history. For the sake of clarity, consider that the non-interacting vehicle flow is formed by \(h\) typologies of elements and indicate with \(Z_{i}\) the \(i\)-th element typology. Said \(\sigma_{Z_{i}\max}\) and \(\sigma_{Z_{i}\min}\) the extremes of the \(h\) individual stress histories \(Z_{i}\), it is possible to sort them in two different ways, taking the minima in increasing order and the maxima in decreasing order.