**2. Traffic engineering concepts**

The most common traffic characteristics are *flow* (sometimes called *flux* or *volume*), *density* (sometimes called *concentration*) and *mean speed*. Flow is inherently a *temporal* quantity (number of vehicles per unit of time), density a *spatial* one (number of vehicles per unit of length) and the *mean speed* can be either, depending on whether speed is averaged at a certain point over a time interval (*time mean speed, v*) or at an instant of time over a stretch of road (*space mean speed, vs*). Density is a key traffic variable for bridge-loading applications, as it is directly related to the number of vehicles present on a bridge at any one time. Speed is related to minimum inter-vehicle gaps: the lower the speed, the smaller the minimum safe distance between vehicles.

There are two main detector types for collecting traffic data: *point* detectors, which count the number of vehicles in a unit of time (the natural way to collect *flow* data) and *spatial* detectors, which count the number of vehicles in a unit of length (the natural way to collect *density* data). In practice, flow is far easier to measure than density, as it can be measured by means of common point detectors such as induction loops [14].

Clearly, the knowledge of vehicle positions is a prerequisite for any subsequent structural analysis. However, as traffic Data is available only at selected road cross-sections, the actual vehicle positions along a stretch of road can only be estimated from such point measurements, typically assuming constant speed. As will be shown later, this is a reasonable assumption in free-flowing traffic and therefore appropriate for short-span bridges. However, during congestion (relevant to long-span bridges), speeds may vary significantly, like in the common case of *stop-and-go waves*. In this case, the estimation of vehicle positions from point measurements may result in a significant loss of accuracy [15].

## **2.1. Traffic theory**

The Fundamental Equation of Traffic (FET) has been long used to relate *flow q, density k* and *space mean speed v*s [16, 17]:

$$q = k \cdot \nu\_s \tag{1}$$

Eq. (1) implicitly assumes that each vehicle maintains a constant speed, although individual speeds may be different. Given the large availability of point measurements, density is typically estimated from Eq. (1) from flow and speed data.1 Even when vehicles do not keep their speed, the FET might still be able to provide fairly accurate density estimates during congestion [18].

<sup>1</sup> The space mean speed vs can be reasonably approximated as the harmonic mean speed of individual vehicles collected at one point, although it is by definition a spatial quantity. The use of the time mean speed (arithmetic mean) in Eq. (1) is incorrect, although frequent, leading to a systematic underestimation of the actual density when traffic is not stationary, that is when there are large variations of speed, typical of most congestion events [18].

Single-vehicle data is usually aggregated over a time interval varying from 20 seconds to 5 minutes. Aggregated *macroscopic* variables, such as those implied in Eq. (1), are useful to obtain a global and concise description of the traffic stream. However, for bridge-loading applications, it is highly desirable to also have single-vehicle *microscopic* data, such as time stamps, so as to identify vehicle configurations and reconstruct vehicle positions.

**2. Traffic engineering concepts**

28 Structural Bridge Engineering

common point detectors such as induction loops [14].

ments may result in a significant loss of accuracy [15].

typically estimated from Eq. (1) from flow and speed data.1

that is when there are large variations of speed, typical of most congestion events [18].

vehicles.

**2.1. Traffic theory**

congestion [18].

1

*space mean speed v*s [16, 17]:

The most common traffic characteristics are *flow* (sometimes called *flux* or *volume*), *density* (sometimes called *concentration*) and *mean speed*. Flow is inherently a *temporal* quantity (number of vehicles per unit of time), density a *spatial* one (number of vehicles per unit of length) and the *mean speed* can be either, depending on whether speed is averaged at a certain point over a time interval (*time mean speed, v*) or at an instant of time over a stretch of road (*space mean speed, vs*). Density is a key traffic variable for bridge-loading applications, as it is directly related to the number of vehicles present on a bridge at any one time. Speed is related to minimum inter-vehicle gaps: the lower the speed, the smaller the minimum safe distance between

There are two main detector types for collecting traffic data: *point* detectors, which count the number of vehicles in a unit of time (the natural way to collect *flow* data) and *spatial* detectors, which count the number of vehicles in a unit of length (the natural way to collect *density* data). In practice, flow is far easier to measure than density, as it can be measured by means of

Clearly, the knowledge of vehicle positions is a prerequisite for any subsequent structural analysis. However, as traffic Data is available only at selected road cross-sections, the actual vehicle positions along a stretch of road can only be estimated from such point measurements, typically assuming constant speed. As will be shown later, this is a reasonable assumption in free-flowing traffic and therefore appropriate for short-span bridges. However, during congestion (relevant to long-span bridges), speeds may vary significantly, like in the common case of *stop-and-go waves*. In this case, the estimation of vehicle positions from point measure-

The Fundamental Equation of Traffic (FET) has been long used to relate *flow q, density k* and

Eq. (1) implicitly assumes that each vehicle maintains a constant speed, although individual speeds may be different. Given the large availability of point measurements, density is

their speed, the FET might still be able to provide fairly accurate density estimates during

 The space mean speed vs can be reasonably approximated as the harmonic mean speed of individual vehicles collected at one point, although it is by definition a spatial quantity. The use of the time mean speed (arithmetic mean) in Eq. (1) is incorrect, although frequent, leading to a systematic underestimation of the actual density when traffic is not stationary,

*<sup>s</sup> q kv* = × (1)

Even when vehicles do not keep

The motion of individual vehicles can be fully described by tracing their trajectory, plotted over space-time domains of the traffic stream, such as in the example of **Figure 1**. 2 Space-time domains can be generated with a rapid sequence of aerial photographs or a video [21, 22], or else with a dense installation of loop detectors or other point sensors [23]. In fact, both options are rarely practicable and currently limited to research applications.

**Figure 1.** Example of vehicle trajectories collected on the US-101 highway near Los Angeles (adapted with permission from [24]).

Let us consider the common case of a point detector. It would collect traffic data as per the straight line depicted at 200 m in **Figure 1**. Say the detector provides both macroscopic data aggregated every 60 s and microscopic single-vehicle data. Let us assume we are interested in inferring the density over a stretch of 100 m either sides of the point detector. In the first minute (08.13–08.14), the traffic stream flows quite steadily at about 45 km/h. Therefore, Eq. (1) applies and it is then possible to accurately infer density over the 200 m length, as well as individual vehicle positions from single-vehicle data. Afterwards, the traffic flow breaks down, developing *stop-and-go waves* (a stopped vehicle can be recognised when its trajectory is horizontal in the space-time domain). In the time interval 08.16–08.17, it may be possible to compute a reasonably accurate *average* density estimate over the 200 m length from Eq. (1), but this is likely to miss the critical *maximum* density occurring at some point within the 60 s aggregation

<sup>2</sup> Edie [19] generalised the FET for any space-time domain (like the one in **Figure 1**), making it adaptable to any kind of detector and traffic state. Edie's equations are exact and consistent by definition, but only within the regions where measurements can be effectively taken, for instance, density between two closely spaced loop detectors [20].

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 crossed the point detector.
