PSI - Issue 78

Amirmahmoud Behzadi et al. / Procedia Structural Integrity 78 (2026) 513–520

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4. Results and Comparison An Empirical Cumulative Distribution Function (ECDF), denoted as F Fbf,L , is established to represent the cumulative distribution of the resultant braking forces derived from the simulations, based on WIM traffic data considered representative of the vertical traffic load on a specific road. The Probabilistic Braking Force Model (PBFM) assumes that p = F Fbf,L ( q ) defines the probability of non-occurrence for a braking force corresponding to the q(q-quantile) , meaning that 1− p gives the probability of occurrence. This non-occurrence probability p is then related to the return period T R and the nominal bridge life V N as indicated in Eq.1. For high return periods (500 – 1000 years), Scenario 2 yielded the highest braking forces. Compared to Eurocode predictions, the probabilistic model often produced higher forces for longer spans and lower nominal life spans, revealing that deterministic models may underestimate critical braking forces in such conditions. (1 1/ ) N V R p T = − (1) Dynamic amplification factors (DAFs), derived from SDOF bridge models, were also applied according to EC1-2 background document: 1.8 for one vehicle, 1.4 for two, and 1.2 for three or more vehicles. Overall, it can be observed from Figs. 4 to 6 (Scenario 2 as most conservative scenario) that the PBFM yields a braking force that increases with span length for a fixed VN. Similarly, for a constant V N , the braking force becomes larger as the return period ( T R ) increases. Conversely, when the return period T R is held constant, the braking force decreases with increasing V N , as the probability of non-occurrence p (refer to Eq. 1) declines. Although Figs. 4 to 6 represent Scenario 2, the relationship between braking force, return period, and nominal life is the same for other scenarios as well.

Fig 4. Scenario 2, V N = 5 years.