## Probabilistic Models for Vehicle Interactions in Fatigue Assessment of Bridges  
**[PERSON](r)**  
Department of Civil and Industrial Engineering, University of Pisa; 56122 Pisa, Italy; [EMAIL_ADDRESS]; Tel.: +39-335-345611  
Received: 3 November 2019; Accepted: 5 December 2019; Published: 6 December 2019  
**Abstract:** Fatigue assessments of bridges depend on vehicle interactions, occurring when several vehicles travel simultaneously on the bridge or when two individual stress histories, caused by vehicles traveling in different times, generate a more damaging combined stress history. When interactions are significant, stress histories cannot be directly determined using conventional fatigue load models provided in Codes, unless suitable theoretical models for vehicle interactions are available. In the paper, original approaches are proposed to study different aspects of the problem. Concerning interactions due to simultaneity, the novelty is to consider the bridge a service system. Since the process of vehicle arrivals is a Markov process, vehicle interactions can be studied in the framework of the queuing theory. In this way, in the appropriate context, interacting vehicles are equivalent to queued requests (vehicles) in the service system. The method considers two subcases, to be tackled in the given sequence, so that the solution is noticeably simplified. The first subcase refers to vehicles traveling simultaneously in one lane; the second subcase to vehicle and vehicle convoys traveling simultaneously on two or more lanes. In the first subcase the problem is solved considering each lane as a single channel system with a waiting queue, where the number of vehicles in the queue and the waiting time, depending on the number of vehicles in the queue, are limited. A modified vehicle flow on each bridge lane is thus obtained, composed by vehicles and vehicle convoys separately traveling the lane, which is, if relevant, the input for the second subcase. In the second subcase the multilane bridge is modeled as a multichannel system without the waiting queue. When the number of requests exceeds the number of channels, \(r\), the surplus is lost and cannot reenter the system. The results regarding simultaneity are much more relevant than it appears at the first sight: Two relevant examples demonstrated that they can be fruitfully used to implement optimized Monte Carlo algorithms for artificial traffic generation, as well for adaptation of traffic measurements, when flows are modified. Finally, a \"non-interacting traffic flow\" is obtained, whose elements (vehicle, vehicle convoy, cluster of vehicles) travel individually on the bridge. The global stress history results thus a mere random assembly of the stress histories induced by each element of the non-interacting traffic flow. These stress histories can only combine, as simultaneity interactions are excluded for them. Combination of stress histories is a complex issue, especially when, as in the Eurocodes, fatigue load models are composed by a set of standardized lorries. In fact, questions concerning: Conditions for the combination; stress history which can combine; expected number of occurrences of combined stress histories and of the remaining individual ones; are still open. Really, they can be tackled resorting to sophisticated and time-consuming simulations based on Monte Carlo methods, but elementary solutions have not been proposed so far. The original method proposed here, whose practical application is illustrated referring to an important case study, allows to solve the problem providing simple recursive formulae. Finally, two relevant examples illustrate, with specific reference to the Eurocodes, some important implications of the study.  
**Keywords:** bridge loads; fatigue load models; fatigue; queueing; vehicle interactions; probabilistic models; Eurocodes +
Footnote †: journal: Applied