**4. Short-span bridges**

Short-span bridges are governed by free-flowing traffic, plus an allowance for dynamic effects. Free-flowing traffic measurements can be used directly or as a basis to generate loading scenarios. The arrival of vehicles in free traffic is often idealised as a Poisson process, which can be described with a negative exponential distribution. If cars are neglected, as usually assumed in short-span bridge loading, other distributions may be more suitable to describe the arrival of truck traffic [76].

Firstly, the case of an individual lane is considered. Shorter bridge spans (< ≈30 m) are governed by a single heavy vehicle or, if very short, individual axles. Therefore, commonly available information about axle configuration and weight suffices to generate a realistic (static) load model, as there is no need to account for vehicle interaction.

For longer spans, in-lane multiple presence of heavy vehicles is a possible event. The headway between two following trucks is an important parameter to identify the number of trucks that may be simultaneously present on a bridge. Headways may be estimated from the WIM database, as discussed in Section 2.2, or from a calibrated headway model [76].

For the common case of multi-lane bridges, it is necessary to consider the multiple presence of Side-by-side heavy vehicles, whether across same-direction or opposing lanes. In the development of current European and North American codes, data about actual multiple presence of side-by-side vehicles was not collected but artificially reproduced. For instance, Reference [66] considers that one in 15 trucks has another truck side by side. This conservative assumption, used for the calibration of the current AASHTO load model [71], can be nowadays adapted to site-specific traffic conditions due to the availability of more accurate WIM data [77, 78]. Indeed, a 2-truck side-by-side meeting event is likely to strongly influence the bridge design for shorter spans (< ≈30 m) and sometimes it has also been the only considered event for longer spans [66, 79]. However, as spans get longer, the likelihood of events involving more than two trucks increases and such meeting events should be accounted for [75, 76, 80].

## **4.1. Dynamic amplification factor**

Fast-moving vehicles, typical of free-flowing traffic, interact dynamically with a bridge. The total load effect resulting from a loading event, LEt , is typically larger than would result from a static analysis, LEs. The *Dynamic Amplification Factor* (DAF) represents the ratio of these two load effects:

$$
\Delta AF = \frac{LE\_t}{LE\_s} \tag{16}
$$

Dynamic amplification varies significantly and depends on a number of factors, such as span length, vehicle class and speed, axle spacing and weight, suspension stiffness, or the road surface profile. As described in the next section, the main codes use DAFs in the range from 1.0 to 1.8.

Reference [81] presents a review about dynamic factors for highway bridges. In recent years, the estimation of dynamic allowance has shifted from a worst-case scenario point-of-view to a probabilistic approach. In fact, several studies have identified the potential for a large DAF reduction when the static traffic loading increases, as the maximum dynamic effect does not correspond to the maximum static effect. For instance, for a single-vehicle event, there is a significant probability that a vehicle will travel at a speed which excites the bridge, whereas for a two-vehicle meeting event it is much less likely that both vehicles contribute to dynamic amplification. Moreover, a heavier vehicle typically excites the bridge less than a lighter one.

To address this issue, the *Assessment Dynamic Ratio* (ADR) has been proposed as the ratio between the *total characteristic* load effect and the *static characteristic* load effect [2]. These two characteristic values may not necessarily arise from the same loading scenario. Numerical investigations on dynamic vehicle-bridge interaction have found that the ADR is in the order of 1.05 [2, 82]. This considerable difference from typical dynamic allowances has the impact that many longer bridges thought to be governed by free-flowing traffic could actually be governed by congestion conditions [4]. Therefore, the simulation of congested traffic, described in Section 5.1, may have a wider application to That originally thought.
