**3. Statistics concepts**

(6)

(7)

in performing a lane change must be greater

must be greater than the sum of the acceleration

*<sup>n</sup>* ( ) *safe at b* % ³ - (8)

8

This means that the acceleration advantage −

criterion for a *fast-to-slow* lane change is as follows:

In this case, the acceleration advantage −

and therefore applicable to short-span bridges.

target follower an−ãn in the fast-to-slow lane change.

of view, they are not often used for several practical reasons.

to the *safe braking* value *b*safe:

36 Structural Bridge Engineering

**2.5. Summary**

8

than the sum of the *acceleration threshold* Δ*a*th, which prevents overtaking with a marginal advantage, the *bias acceleration* Δ*a*bias, which acts as an incentive to keep in the slow lane, and the imposed disadvantage to the new follower in the fast lane − , weighted through a *politeness factor p*, to account for the driver aggressiveness. On the other hand, the incentive

threshold Δ*a*th, minus the bias acceleration Δ*a*bias (which acts as an incentive to move back to the slow lane), plus the disadvantage imposed to both new follower *n* in the slow lane and to

Finally, the safety criterion limits the imposed deceleration to the follower *n* in the target lane

Traffic data for bridge-loading applications is typically collected at high-speed WIM stations. Free-flowing traffic measurements are unbiased, reliable and now commonly available. Generally speaking, if traffic information is available at a point detector, such as a WIM station, it is possible to accurately reconstruct vehicle positions from single-vehicle data only when traffic characteristics do not change significantly. This is the usual case in free-flowing traffic

On the contrary, there is a shortage of data during congestion, mainly Due to current technological limitations. In addition, the analysis of traffic data can pose some issues: in fact, a vehicle's speed is likely to fluctuate, e.g. as a result of *stop-and-go waves*, making the estimation of vehicle positions from point measurements problematic. The use of spatial detectors (such as cameras) over a stretch of road allows the collection of vehicle positions during congestion, without resorting to estimation. Although cameras are the best solution from a theoretical point

Eq. (7) is based on [61] and preferred to the formulation in [59], which does not include the disadvantage to the new

the current follower *o* in the fast lane, weighted through the politeness factor *p*.

In bridge loading, extreme events with small frequencies of occurrence are of interest, rather than frequent scenarios. However, it is not generally practicable to simulate the long periods required to identify such rare events, even when the recorded database is expanded through Monte Carlo techniques. Therefore, the data is typically fitted with a statistical distribution and then extrapolated to determine characteristic loading values.

While pioneering studies focussed on wors-case scenarios [5, 6], a probabilistic approach to bridge loading is now common [7–11, 27, 33, 38, 60, 62–66]. The probability *F* that a load level *z* is not exceeded (*probability of non-exceedance*) is commonly expressed in terms of *return period, T*(*z*) [67]. The two variables are linked through the relation:

$$T\left(z\right) = \frac{1}{1 - F(z)}\tag{9}$$

Importantly, the return period is different from the bridge lifetime and, instead, should be seen as a measure of safety. For instance, the *characteristic* and *frequent* values of the Load Model 1 in Eurocode 1 are based on return periods of 1000 years and 1 week, respectively [68]. The superseded British HA loading was based on a return period of 2400 years [69]. The AASHTO load model is based on 75 years [70, 71]. For assessment, it is accepted that lower return periods should be used: in Europe, a value of 75 years may be considered, whereas AASHTO [72] suggests a return period of 5 years. As will be shown in Section 5.1, such large differences in the return period do not imply equally large differences in the extrapolated characteristic values.

## **3.1. Extreme value statistics**

*Extreme value theory* is a branch of statistics Appropriate for the probabilistic modelling of a range of civil engineering problems. including bridge loading. A popular approach is the *Block Maxima*: the maximum events in each block (e.g. maximum load effect per day) are selected as representative, then fitted with a statistical distribution, and finally extrapolated to determine characteristic values. While other methods are possible, the database sample has generally more influence on the results than the extrapolation method, thus highlighting the importance of long-run simulations [73].

Firstly, an empirical frequency to each sample data point, *z*<sup>i</sup> (*plotting position*), is assigned:

$$\hat{F}\left(z\_{i}\right) = \frac{i}{N+1} \tag{10}$$

in which *i* = 1, 2, …, *N* is the index of the sample ordered decreasingly.9 The *Generalised Extreme Value* (GEV) distribution is then fitted to the simulated maxima. Its *Cumulative Distribution Function* (CDF), which expresses the probability of non-exceedance *F*(*z*), is as follows [67]:

$$F(z) = \exp\left\{-\left[1 + \xi\left(\frac{z-\mu}{\sigma}\right)\right]^{-\frac{1}{\tilde{\epsilon}}}\right\} \tag{11}$$

in which *μ* is the location, *σ* the scale and *ξ* the shape parameter. Eq. (11) is defined for any value *z* for which 1 + > 0. When *ξ* = 0, the GEV distribution reduces to the Gumbel (or *Type I*) distribution:

$$F(z) = \exp\left\{-\exp\left[-\left(\frac{z-\mu}{\sigma}\right)\right]\right\} \tag{12}$$

When *ξ >* 0 and *ξ* < 0, the GEV distribution is named, respectively, Fréchet (or *Type II*) and Weibull (or *Type III*). The latter is more commonly found in bridge-loading applications. The GEV parameters can be inferred through *maximum likelihood estimation* (details can be found in [67]).

Gumbel probability paper plots are useful to illustrate the extrapolation procedure [74]. On this plot, data from a Gumbel distribution appears as a straight line. The ordinate, or *Standard Extremal Variate* (SEV), is given by:

$$SEV\left(z\right) = -\log\left[-\log\left(F(z)\right)\right] \tag{13}$$

**Table 3** reports the target probabilities of exceedance and SEVs for typical return periods, under the common assumption that maxima are collected every day of a year with 250 working days.

#### **3.2. The law of total probability**

The GEV distribution applies under the assumption that individual events are independent and identically distributed. However, this assumption is not necessarily met in bridge-loading

<sup>9</sup> Alternative plotting position formulae are possible, but differences become small as the sample size increases.

applications: load effects can be the result of a number of quite different loading events, involving different numbers of trucks or different traffic states.

A generalisation of the *law of total probability* can be used to combine maximum load events resulting from different event types. The probability, *P*, that the combined maximum load effect, *z*, in a given reference period (e.g. a day or an hour) is not exceeded, i.e. the combined CDF, is:

$$P(z) = \sum\_{j=1}^{n\_r} F\_j(z) \cdot f\_j \tag{14}$$

in which *nt* is the number of event types, *F*<sup>j</sup> is the CDF of the maximum load effects for the *j*event type (Eq. 11) and *f*<sup>j</sup> is the probability of occurrence of the *j*-event type. Clearly, ∑ = 1 = 1. Equating *P*(*z*) to the target probability (e.g. as in **Table 3**) gives the characteristic combined load.


**Table 3.** Typical target probabilities of exceedance and SEV.

( ) <sup>1</sup>

*Value* (GEV) distribution is then fitted to the simulated maxima. Its *Cumulative Distribution Function* (CDF), which expresses the probability of non-exceedance *F*(*z*), is as follows [67]:

x

ï ï é ù æ ö - = -+ í ý ê ú ç ÷ ï ï ë û è ø î þ

in which *μ* is the location, *σ* the scale and *ξ* the shape parameter. Eq. (11) is defined for any

When *ξ >* 0 and *ξ* < 0, the GEV distribution is named, respectively, Fréchet (or *Type II*) and Weibull (or *Type III*). The latter is more commonly found in bridge-loading applications. The GEV parameters can be inferred through *maximum likelihood estimation* (details can be found

Gumbel probability paper plots are useful to illustrate the extrapolation procedure [74]. On this plot, data from a Gumbel distribution appears as a straight line. The ordinate, or *Standard*

**Table 3** reports the target probabilities of exceedance and SEVs for typical return periods, under the common assumption that maxima are collected every day of a year with 250 working

The GEV distribution applies under the assumption that individual events are independent and identically distributed. However, this assumption is not necessarily met in bridge-loading

Alternative plotting position formulae are possible, but differences become small as the sample size increases.

1

x

> 0. When *ξ* = 0, the GEV distribution reduces to the Gumbel (or

ì ü ï ï é ù æ ö - = -- í ý ê ú ç ÷ ï ï î þ ë û è ø (12)

*SEV z log log F z* ( ) = - é- ù ( ) ( ) ë û (13)

m

m

s

s

ì ü -

*<sup>N</sup>* <sup>=</sup> <sup>+</sup> (10)

The *Generalised Extreme*

(11)

<sup>ˆ</sup> *<sup>i</sup> <sup>i</sup> F z*

exp 1 *<sup>z</sup> F z*

( ) exp exp *<sup>z</sup> F z*

in which *i* = 1, 2, …, *N* is the index of the sample ordered decreasingly.9

( )

value *z* for which 1 +

*Extremal Variate* (SEV), is given by:

**3.2. The law of total probability**

*Type I*) distribution:

38 Structural Bridge Engineering

in [67]).

days.

9

Eq. (14) may be applied to the combination of maximum load effects resulting from different *j* traffic states, each of which occurs with the assigned probability *f*<sup>j</sup> [38]. It implies that, within the reference period, only *one* maximum loading event, *z*, can occur due to any of the *j* traffic states.

On the other hand, when considering load effects deriving from different *j*-truck meeting events, relevant to short-span bridge loading, Eq. (14) cannot be readily applied [75]. In fact, within any reference period, there will be *n*<sup>t</sup> maximum loading events, each of which due to a *j-*truck meeting event. In this case, and only when using the GEV distribution (Eq. 11), it can be demonstrated that the probability, *P*, that the combined maximum load effect, *z*, is not exceeded in the reference period is given by [75]:

$$P\left(z\right) = \prod\_{j=1}^{n\_{\gamma}} F\_j(z) \tag{15}$$

in which *nt* is the number of event types (typically 4) and *F*<sup>j</sup> is the CDF of the GEV distribution for the maximum load effects of the *j*-event type.
