## 3 Modeling of Interactions between Non-Simultaneous Vehicles
Finally, if the combined stress history \(Z_{i}Z_{j}\) is the result of the \(m\)-th combination of the stress history \(Z_{i}\) and of the \(n\)-th combination of the stress history \(Z_{j}\), the number of its occurrences is  
\[N_{Z_{i}Z_{j}}\ =\ \frac{N_{Z_{i}(m\ -\ 1)}N_{Z_{j}(n\ -\ 1)}}{N_{Z_{i}(m\ -\ 1)}\ +\ N_{Z_{j}(n\ -\ 1)}}, \tag{53}\]  
where \(N_{Z_{i}(m\ -\ 1)}\) and \(N_{Z_{i}(n\ -\ 1)}\) are the occurrences of the individual stress histories \(Z_{i}\) and \(Z_{j}\) remaining from the previous combinations, and not yet combined, while the remaining occurrences of the individual stress histories, available for subsequent combinations, if any, are given by  
Figure 10:0  
The expected residual number of individual stress histories \(Z_{i}\) still remaining after all combinations can be equivalently evaluated as  
Figure 10:1  
where the summation obviously includes all the occurrences of the \(p\) combined stress histories \(Z_{i}Z_{j}\), \(0\leq p<h\), in which \(Z_{i}\) enters. In conclusion, the total stress history is simply the union of \(N_{Z_{i}Z_{j}}\) combined stress histories and \(N_{Z_{i}(p)}\) individual stress histories, provided that all possible combinations are considered. Taking into account that these stress histories cannot interact, the overall stress spectrum is simply the summation of the stress spectra associated with \(N_{Z_{i}Z_{j}}\) repetitions of combined stress spectra and \(N_{Z_{i}(p)}\) individual stress spectra. To better illustrate the procedure, it is applied to a significant case study--Evaluation of Stress Spectrum in a Stiffener to Stiffener Joint of an Orthotropic Steel Deck. Case Study--Evaluation of Stress Spectrum in a Stiffener to Stiffener Joint of an Orthotropic Steel Deck  
The model previously proposed has been applied to a relevant case study: The evaluation, according to Eurocodes, of the stress spectrum in a stiffener to stiffener joint which is one the most fatigue sensitive details in orthotropic steel deck bridges (see, for example [42]). In short, the steel orthotropic deck is composed by a deck plate 14 mm thick and by trapezoidal longitudinal stiffener, 8 mm thick, and 300 mm height, spaced 600 mm (Figure 13), supported by transverse beams, 620 mm height, spaced 3500 mm. To evaluate the stress histories, the fatigue load model 4 (LM4) of EN1991-2 [21] has been adopted, considering medium distance traffic composition. LM4, which is a set of five standardized lorries with \"equivalent\" axle loads, is described in detail in Section 4.1. It is important to remind that axles of standardized lorries are characterized by different tires, so that three different axle typologies are distinguished, indicated with A, B, and C.  
On the orthotropic steel deck bridge a main road traffic with low flow rate has been hypothesized. According to EN1991-2 the traffic flow is \(1.25\,\times\,10^{5}\) lorries per year, so that, considering a 100 years design life, the total traffic volume results \(1.25\,\times\,10^{7}\) lorries. Generally, the most unfavorable situation for the stiffener is when the wheel is centered on its axis, but the stress history in the stiffener itself is also sensitive to variation of the lorry position. For that reason, in some cases, the varying position can be more demanding than that centered on the stiffener, as it will be shown below. According to EN1991-2, the transverse position of the lorry itself has been considered varying in a band of around 0.5 m, centered on the assumed theoretical position, as indicated in Figure 14, in which are also indicated as the percentage of vehicles running in each subinterval. Referring to Figure 14, (0) means that the wheel axis is centered on the stiffener, (1) means that the wheel axis is 100 mm distant from the axis of the stiffener, (2) that the wheel axis is 200 mm distant from the axis of the stiffener. On this basis, in the present case study, the lorry and its position are identified by means of the lorry number in LM4 followed by the position; for example, lorry 3 (1) indicates the lorry no.