## 4 Two Relevant Applications
### Fatigue Load Models for Road Bridges in EN1991-2
Influence of the Reference Length \(L\) and of the Traffic Flow N on Simultaneity Effects on One Lane  
To study simultaneity effects on one lane, the probabilities \(\underline{P_{2}}\) and \(\underline{P_{3}}\) that two or three vehicles are traveling the lane at the same time have been derived applying Equation (11), depending on the reference length \(L\) and on the annual traffic flow \(N\). The curves \(\underline{P_{2}}\ -\ L\) and \(\underline{P_{3}}\ -\ L\) are parameterized with respect to the annual traffic flow \(N\) in Figure 26.  
Setting 5% the reference value of the probability above which effects of interaction can become relevant, investigation of curves in Figure 26 shows that:  
* The value of \(L\) above which effects of interactions between two vehicles become significant is about 60 m when the annual flow is \(N\ =\ 2\ \times\ 10^{6}\); increasing up to approximately 90 m for \(N\ =\ 10^{6}\); and up to 170 m for \(N\ =\ 5\ \times\ 10^{5}\); so that
* the probability \(\underline{P_{2}}\) that two vehicles are simultaneously on the same lane is important only when \(N\ \geq\ 5\ \times\ 10^{5}\);
* as anticipated in Section 2.1, the probability that three vehicles are traveling the lane at the same time can be generally disregarded: In fact, \(\underline{P_{3}}\) is just relevant when \(L\ >\ 140\) m and the traffic flow is high, \(N\ =\ 2\ \times\ 10^{6}\); on the contrary, \(\underline{P_{3}}\) is always negligible when \(N\ \leq\ 5\ \times\ 10^{5}\); while, when \(N\ \leq\ 10^{6}\), it attains values around 3% only when \(L\ \approx\ 200\) m;
* \(\underline{P_{2}}\ -\ L\) curves are slightly concave upward when \(N\ <\ 5\ \times\ 10^{5}\); are nearly linear when \(N\ =\ 5\ \times\ 10^{5}\); are concave downward when \(N\ >\ 5\ \times\ 10^{5}\); while the downward concavity of \(\underline{P_{3}}\ -\ L\) curves tends to increase as the traffic flow increases, so stressing once again that interactions involving more than two vehicles are relevant only for high values of \(L\) and \(N\). In addition, it must be considered that the real impact of these results in terms of stress history is usually less evident, since it depends not only on \(L\), but also on the shape of the influence line:  
Figure 25: Axle types (adapted from [21]). For instance, in a two span box girder continuous beam, the relevance of simultaneity effects decreases passing from hog bending moments at the intermediate supports, to sag bending moments in the span, despite the number of interacting vehicles can be the same. Influence of the Reference Length L and of the Traffic Flow N on Simultaneity Effects on Several Lanes  
Under the above-mentioned assumptions, the second example refers to the case of vehicles traveling simultaneously on two distinct lanes, discussed in Section 2.3. Of course, the less favorable situation occurs when the traffic flows on the two lanes are nearly identical: \(N_{1}\ =\ N_{2}\), so that the total flow is \(N^{\ast}\ =\ 2\ N_{1}\). Conditional at least that one vehicle is on the bridge, the probability \(\underline{P_{2,2}}\) that two lorries are traveling at the same time on the two lanes is given in Equation (37), from which the \(\underline{P_{2,2}}-L\) curves illustrated in Figure 27 are obtained, in the present case study. Examination of Figure 27 demonstrates that interactions due to simultaneity can be significant even if \(L\) is small, especially for high lorry flows, as they can be disregarded only for annual lorry flows \(N\ \leq\ 10^{5}\). These results can be further extended in view of future improvements of the so-called damage equivalence factor method or \(\lambda\)-coefficient method in Eurocodes [15; 43]. In fact, it can be also applied to refine the expression of the damage equivalence factor, denoted as \(\lambda_{4}\) in EN1993-1-9 [15], accounting for the multilane effects, in such a way to include also simultaneity.