**2.3. Congestion**

time. Even when single-vehicle data is available, it is not possible to readily reconstruct the spatial distribution of vehicles, as vehicle speeds vary from those recorded when the vehicles

Traffic weight data is traditionally based on roadside truck surveys, or, more recently, WIM measurements. High-speed WIM stations are able to weigh axles and collect time stamps without stopping vehicles. Axle time headways can be then computed. *Double loop detectors* are used for supplementing information regarding speed. This enables the distance between axles to be computed and to reconstruct the vehicle configuration. The overall vehicle length

Data from paired WIM and loop stations has been widely used. Single-vehicle data is normally available for those stations, although sometimes only for heavy vehicles. Unfortunately, many WIM and loop detectors are not currently reliable at very low speeds [14]. As a consequence, data is largely collected during free-flowing traffic conditions, which also occur more frequently than congested conditions, whereas data about slow-moving vehicles is generally

The recorded traffic data may be directly used for subsequent structural analysis. Nevertheless, since the recorded dataset is rarely sufficiently large, it is preferable to use it as a basis to generate additional artificial traffic by means of common Monte Carlo simulations for further

Except for very short spans, the next step is to reconstruct the spatial distribution of vehicles from the recorded point measurements. This is equivalent to find the *headways* or gaps between

is an acceptable assumption only in free-flowing traffic. However, when load effects are calculated during congestion, large variations in speed may result in unrealistic spatial distributions or even vehicle overlapping. This aspect is particularly significant for long spans

 Here the headway is intended as the distance between same points of two consecutive vehicles, for instance, the front axle or the front bumper; the gap is the bumper-to-bumper distance between two vehicles. They can also be intended as time distances, and when so it will be specified. Note that different interpretations of these terms may be found.

 Other assumptions on the speed are also possible, and these will clearly affect the estimated headway. For instance, some studies assume a constant speed for all the vehicles in the traffic stream [26, 27], although the recorded speeds were different. This implies that vehicles are passed on the bridge with the same time headways as were recorded, but not with

the same space headways, which depend on the assumed speed. For a more detailed discussion, see [15].

When using individual recorded speeds, the headway *h* between the current vehicle

and the leading vehicle *i*-1 (that crossed the detector

1 1 ( ) *ii i i hv t t* =× - - - (2)

4

As seen in Section 2.1, this

crossed the point detector.

**2.2. Data collection**

30 Structural Bridge Engineering

can also be detected.

use in structural analysis – see for instance [25].

at a time stamp *t*i-1 and with speed *v*i-1) can be estimated as follows:

thereby assuming that the leading vehicle is keeping its speed *v*i-1 .

*i* crossing the detector at a time stamp *t*<sup>i</sup>

lacking.

vehicles.3

[15].

3

4

In simple terms, congestion forms whenever the *inflow Q*in (demand) is greater than the *dynamic capacity Q*out (supply). In reality, inflows greater than the dynamic capacity *Q*out are possible and the maximum flow that can be attained is named *static capacity Q*max, or simply *capacity*. However, in this case, the traffic flow is not stable and a significant perturbation (e.g. a braking vehicle) could break the flow down and generate a queue (*stop-and-go wave*), propagating backwards at a typical speed of 15 km/h. The flow coming out of such a queue is namely the dynamic capacity or *queue discharge rate*. The inflow *Q*in can be easily collected from point detectors, whereas several procedures are available to estimate the capacity *Q*max (see for instance [34]). The capacity depends on many factors, such as the road geometry; importantly for bridge-loading applications, it also depends on the truck percentage. The estimation of the dynamic capacity is not as straightforward, but research suggests it is 5–10% less than the static capacity *Q*max (see for instance [35]).

Both capacities can be further reduced by *bottlenecks* due to a variety of causes, as will be discussed in the next section. The *bottleneck strength*, Δ*Q*, can be defined as the difference between the dynamic capacity in normal conditions, *Q*out, and the reduced dynamic capacity when a bottleneck is in place, *Q'*out:

$$
\Delta \mathbf{Q} = \mathbf{Q}\_{out} - \mathbf{Q}'\_{out} \tag{3}
$$

Depending on the inflow *Q*in and the bottleneck strength Δ*Q* (and for a given traffic history), the traffic can take up any of the traffic states outlined in **Table 1** [36, 37]. Combinations of congested states may also occur.

In general, increasing inflow and/or bottleneck strength has the effect of moving down the table to a higher intensity of congestion. In addition, the greater the bottleneck strength, the lower the average speed and the lower the speed oscillations during congestion [38]. Congested states that occupy a significantly long stretch of road (so-called *extended* states), such as SGW, OCT and HCT, are of particular significance for long-span bridge-loading applications. For comparison with the common traffic loading assumption, the full-stop condition (FS) is also included.

Spatio-temporal speed plots are useful for visualising congestion patterns (**Figure 2**). The space mean speed is collected at four virtual detectors and aggregated over 60 s. **Figure 2(a)** shows a SGW state, where the waves are clearly visible as peaks. **Figure 2(b)** shows a combined HCT/ OCT state, where the upstream small oscillations typical of the OCT state fade away into a HCT state downstream, where there are essentially no oscillations.

**Figure 2.** Spatio-temporal speed plots for a simulated two-lane flow with 20% trucks and bottleneck strength: (a) 270 and (b) 1056 veh/h.

## *2.3.1. Causes and effects of congestion*

Congestion is due to insufficient road capacity (*recurrent congestion*, typically predictable and frequent) or other external causes (*non-recurrent congestion*, typically unpredictable and infrequent), such as inclement weather or incidents, of which incidents impact the traffic most [39, 40]. *Incident rates* are defined as number of incidents per million vehicle-km travelled [I/ MVkmT]; therefore, the expected number of incidents strongly depends on the flow. In the context of bridge loading, the cause of an incident is not relevant; instead, its effects on the traffic capacity and subsequent congestion are relevant. For instance, the Highway Capacity Manual [34] suggests that a lane closure of a two-lane motorway drops the overall capacity by 65%. However, it must be noted that a capacity reduction does not necessarily cause congestion, as this would depend on the inflow *Q*in.

**Table 2** presents data about incidents from selected literature. Since long-span bridges typically have two lanes in the same direction, incidents reported to cause the closure of two (or more) lanes are here considered to fully block the road; hence, the corresponding rate is named *full-stop rate FSr* (FS/MVkmT) [38]. Remarkably, while incident rates are spread over a wide range, full-stop rates cover a much smaller range.5


**Table 1.** Traffic states.

OCT state, where the upstream small oscillations typical of the OCT state fade away into a

**Figure 2.** Spatio-temporal speed plots for a simulated two-lane flow with 20% trucks and bottleneck strength: (a) 270

Congestion is due to insufficient road capacity (*recurrent congestion*, typically predictable and frequent) or other external causes (*non-recurrent congestion*, typically unpredictable and infrequent), such as inclement weather or incidents, of which incidents impact the traffic most [39, 40]. *Incident rates* are defined as number of incidents per million vehicle-km travelled [I/ MVkmT]; therefore, the expected number of incidents strongly depends on the flow. In the context of bridge loading, the cause of an incident is not relevant; instead, its effects on the traffic capacity and subsequent congestion are relevant. For instance, the Highway Capacity Manual [34] suggests that a lane closure of a two-lane motorway drops the overall capacity by 65%. However, it must be noted that a capacity reduction does not necessarily cause conges-

and (b) 1056 veh/h.

32 Structural Bridge Engineering

*2.3.1. Causes and effects of congestion*

tion, as this would depend on the inflow *Q*in.

HCT state downstream, where there are essentially no oscillations.


**Table 2.** Incident rates from selected literature and associated full-stop rates.

#### **2.4. Microsimulation**

The concept of modelling individual vehicles, namely *microsimulation*, is now well established for traffic studies [24, 46, 47]. Microsimulation takes account of the interaction between vehicles, as opposed to *macrosimulation*, which treats traffic as an aggregate flow. As traffic microsimulation is able to reproduce realistic spatial distributions of vehicles, it is a suitable tool to investigate load effects on bridges, without resorting to conservative assumptions about heavy-vehicle positions. Notably, widely available free-flowing traffic measurements can be used to generate initial and boundary traffic conditions. Congested data may be used to calibrate and validate the microsimulation parameters [48].

<sup>5</sup> This may be due to the fact that many small incidents can be unnoticed, while it is unlikely for a large incident causing lane closures to go unrecorded.

Many microsimulation models have been developed in the last few decades. The choice of a suitable microsimulation model mainly relies on the traffic features of interest, e.g. during freeflowing traffic or congestion. For bridge-loading applications, the microsimulation model should be able to reproduce the range of traffic states likely to occur on a bridge. Once calibrated, microsimulation enables the modelling of a large number of congestion events (and the subsequent identification of extreme loading events), which would be extremely difficult to record in the real world.

Microsimulation models divide into *car-following* (single-lane) and *lane-changing* (multi-lane) models. While in short-span load models lane-changing manoeuvres can be neglected, they play an important role in the context of long-span bridge loading. In fact, when overtaking is allowed, the car-truck mix for congestion is likely to be different to that for free traffic, since car drivers do not feel comfortable following trucks As traffic slows down and therefore tend to overtake them [49]. This typically results in longer truck-only platoons in congested traffic than in free-flowing traffic.

#### *2.4.1. The Intelligent Driver Model*

The *Intelligent Driver Model* (IDM) is a car-following model which has been shown to successfully replicate observed multi-lane congestion patterns on several motorways With few parameters using a single-lane traffic stream made up of identical vehicles with deterministic parameters [36, 50]. The IDM has also been proven able to replicate observed single-vehicle trajectory data [51–55]. The motion of each vehicle is simulated through an acceleration function:

$$\frac{\text{d}\mathbf{v}(\mathbf{t})}{\text{d}\mathbf{t}} = \mathbf{a} \left[ \mathbf{l} - \left( \frac{\mathbf{v}(\mathbf{t})}{\mathbf{v}\_0} \right)^4 - \left( \frac{\mathbf{s}^\*(\mathbf{t})}{\mathbf{s}(\mathbf{t})} \right)^2 \right] \tag{4}$$

in which *a* is termed the *maximum acceleration, v*0 the *desired speed, v*(*t*) the current speed, *s*(*t*) the current gap to the front vehicle and *s*\*(*t*) the *desired minimum gap*, given by:

$$\mathbf{s}^\*(\mathbf{t}) = \mathbf{s}\_o + \max\left\{ \mathbf{T}\mathbf{v}(\mathbf{t}) + \frac{\mathbf{v}(\mathbf{t})\Delta\mathbf{v}(\mathbf{t})}{2\sqrt{\mathbf{a}\mathbf{b}}}; \mathbf{0} \right\} \tag{5}$$

in which *s0* is termed the *minimum jam* (bumper-to-bumper) *distance, T* the *safe time headway*, Δ*v*(*t*) the speed difference between the current vehicle and the vehicle in front and *b* the *comfortable deceleration*. 6 There are five parameters in this model (*v*0, *s*0, *T, a, b*) to capture driver behaviour, which are relatively easy to measure. For simulation purposes, the length of the

<sup>6</sup> The desired minimum gap s\* is limited to be greater than or equal to the minimum jam distance s0, as noted in Eq. (5); otherwise, an inconsistent driver behaviour in multi-lane scenarios may be generated when the front vehicle is faster [18, 56].

vehicles must also be known. The congested states in **Table 1** can be effectively generated by applying an *inhomogeneity*, for instance, by increasing the safe time headway, *T*, downstream to – say – *T*'. For a comprehensive discussion, the reader is referred to [18] or [36].

Here, it is worth mentioning that for bridge-loading applications, the parameters *T* and *s*<sup>0</sup> greatly regulate the distance a vehicle keeps when following its leader, such that the smaller those parameters, the closer the vehicles and the Greater the loading. As speed tends to zero, the influence of the safe time headway *T* decreases and the spacing tends to the minimum jam distance *s*0. Therefore, *s*0 is a crucial parameter for bridge-loading applications [57].

The desired speed, *v*0, regulates the behaviour in free traffic, Whereas the traffic stability7 is mainly determined by *a, b* and *T*. Finally, it must be noted that the adoption of variable parameters among vehicles is not strictly necessary for reproducing the congested patterns [36, 58]. Further details about the application of the IDM to bridge-loading analysis can be found in [38].
