*5.1.1. Identification of critical traffic states*

**5. Long-span bridges**

42 Structural Bridge Engineering

[36, 58].

Long-span bridges are governed by congested conditions, with no allowances for dynamic effects, due to the slow speed of the vehicles involved in critical loading events. Traditionally, long-span bridge loading has been based on the simulation of queues of vehicles [5–11, 27, 63, 64]. As outlined in Section 2.4, traffic microsimulation is a powerful tool to simulate more

An application of traffic microsimulation to long-span bridge loading is presented for a stretch of a two-lane same-direction 8000 m long highway, based on [60]. The microsimulation is carried out using the car-following IDM (Section 2.4.1) and the lane-changing model MOBIL (Section 2.4.2). In order to highlight the influence of several traffic features on bridge loading, a simplified vehicle stream made up of two classes of vehicle, cars and trucks, is used with the parameters shown in **Table 4**. Real sites are likely to have a more complex traffic stream, but

The car-following parameters are based on those calibrated and used in [36], who used only identical vehicles to successfully replicate obseved congestion patterns, as described in Section 2.4.1. Trucks are introduced here and assigned greater length and weight and slower desired speed [88, 89]. Other truck parameters are kept the same as the parameter set in [36], as consideration of different parameters is not strictly necessary to reproduce congested patterns

The desired speeds, *v*0, of both vehicle classes are uniformly randomly distributed. Although the desired speed governs the free traffic behaviour in the IDM, it is necessary to introduce speed randomness in order to correctly model lane-changing manoeuvres [59]. Trucks are assigned mean Gross Vehicle Weight (GVW) equal to the European minimum legal limit of 44 t [90, 91] and the GVW is considered normally distributed with a Coefficient of Variation (CoV: standard deviation divided by the mean) of 10%. Those two assumptions can be easily adapted

Cars and trucks are assigned the same MOBIL parameters, as calibrated in [15]. However, the assigned difference in the IDM desired speed, *v*0, has also the desirable effect of making trucks less prone to undertake a lane-changing manoeuvre. Notably, bridge loading is not found to

Two long-span bridges (200 and 1000 m long) are centred at 5000 m. The dynamic capacity Qout is 3070 veh/h for a flow with 20% trucks. A range of bottleneck strengths Δ*Q* is generated downstream of the bridge by locally increasing the safe time headway *T* from 1.6 s to the values of 1.9, 2.2, 2.8, 4.0 and 6.4 s, thereby inducing the following traffic states: SGW, OCT, HCT/OCT, HCT(1) and HCT(2). HCT(1) and HCT(2) differ for the average speed of traffic (approximately

8.7 and 5.0 km/h). The full-stop condition (FS) is also simulated, for which Δ*Q*= *Q*'out.

to a specific site, as truck GVWs and speed histograms can be computed from WIM.

be significantly sensitive to the lane-changing parameters [15].

realistic congested scenarios for long-span bridge loading.

**5.1. Application of microsimulation to bridge loading**

site-specific traffic data could be equally introduced [87].

The hourly maxima of total load for a flow of 3000 veh/h with 20% trucks on the two spans are plotted in **Figure 4**, along with the extrapolated characteristic values corresponding to return periods of 5, 75 and 1000 years.


**Table 4.** Model parameters.

In order to consider the case of the combination of congestions (Eq. 14), it is necessary to assign the congestion frequencies *f*<sup>j</sup> [38]. Firstly, the number of expected full-stop events, FSn is computed as follows:

$$FS\_n = FS\_r \frac{ADT \cdot L \cdot T}{10^6} \tag{17}$$

in which FSr is the full-stop rate (**Table 2**), ADT the Average Daily Traffic (veh/day), L the length of road observed (km) and T the duration of observations (days). Here it is assumed that FSr = 0.25 FS/MVkmT [45], ADT = 32000 veh/day, T = 250 days (equivalent to one year) and L = 5 km (i.e. incidents occurring up to 5 km downstream of the bridge will affect it), thus returning 10 expected full-stop events each year. Following the previous assumption of 250 congestion events per year, this corresponds to a frequency of 4% over the total number of congestion events.

**Figure 4.** Probability paper plots of maximum total load: (a) 200 m span; (b) 1000 m span.

The expected frequency of full-stop events is used as the basis for assigning congestion frequencies to the other bottleneck strengths generated (**Figure 5**). The distribution is taken to be exponential [38] and qualitatively agrees with available observations [37].

Several features can be noticed from the probability paper plots of **Figure 4**:


<sup>10</sup> Free traffic maxima are shown only for reference and have not been included in the statistical analysis.


Finally, it is noted that traffic microsimulation may be applied to free traffic as well. However, its computational requirements make this approach less attractive, compared to simpler models. Moreover, the IDM shows an excessive spreading of platoons when vehicles approach the desired speed *v*0 [15]. This has essentially no effect on the IDM capability of reproducing

**Figure 5.** Congestion frequencies.

= 0.25 FS/MVkmT [45], ADT = 32000 veh/day, T = 250 days (equivalent to one year) and L = 5 km (i.e. incidents occurring up to 5 km downstream of the bridge will affect it), thus returning 10 expected full-stop events each year. Following the previous assumption of 250 congestion events per year, this corresponds to a frequency of 4% over the total number of congestion

**Figure 4.** Probability paper plots of maximum total load: (a) 200 m span; (b) 1000 m span.

be exponential [38] and qualitatively agrees with available observations [37].

Several features can be noticed from the probability paper plots of **Figure 4**:

vehicles, although vehicles are spaced at the minimum jam distance *s*0.

10 Free traffic maxima are shown only for reference and have not been included in the statistical analysis.

**•** Free traffic maxima are much lower than the congested ones.10

The expected frequency of full-stop events is used as the basis for assigning congestion frequencies to the other bottleneck strengths generated (**Figure 5**). The distribution is taken to

**•** Full stop is not the most critical traffic state for the 200 m span (**Figure 4a**) . In fact, slowmoving HCT states give more combinations of vehicles (and subsequently more chances of finding an extreme loading scenario), whereas FS is the maximum of only one realisation of

**•** For the 1000 m span (**Figure 4b**), FS is the most critical condition even at small return periods.

events.

44 Structural Bridge Engineering

<sup>11</sup> It can be deduced from **Figure 6** that SGW and OCT occur in 57.3% of the working day. If the analysis is carried out neglecting SGW and OCT, the critical congested states HCT/OCT, HCT and FS occur on 106.7 working days per year. It can be shown that re-running the procedure above with adjusted congestion frequencies and target probabilities of nonexceedance returns nearly identical characteristic load values.

congested traffic states (as *v* << *v*0), but could potentially affect bridge loading in uncongested situations. To avoid this issue, a suitable modification of the IDM has been proposed [92].


**Table 5.** Parameters of the GEV distribution for HCT and FS conditions.

#### *5.1.2. Influence of some traffic features*

**Figure 6** shows the 5-year characteristic values for the inflows of 3000 (as described in Section 5.1.1), 2000 and 1250 veh/h, expressed as an Equivalent Uniformly Distributed Load (EUDL, total load divided by bridge length). EUDLs resulting from a flow of 1250 veh/h with 48% trucks are also plotted, in order to quantify the influence of truck percentage and car presence on loading (the 1250 veh/h flow with 48% trucks has the same truck flow of the 3000 veh/h flow with 20% trucks).12

In general, the loads resulting from different inflows with same truck percentage and same bottleneck strength are quite similar (within a range of ±11%), as long as the bottleneck is strong enough to trigger congestion for that inflow, that is *Q*in > *Q*'out. This suggests that the effect of the inflow on loading is not as strong as that of bottleneck strength and implies that critical loading events may occur also out of rush hours.13

Among the traffic features affecting bridge loading, the truck percentage is of particular significance. High truck percentages may occur at night time and early morning in correspondence to low overall flows. As such, it is rare to have congestion events, but if they occur, the resulting loading is likely to be quite heavy, as there are not many cars to keep heavy vehicles apart.

**Figure 6** also shows that at lower flows (1250 veh/h) FS governs for both spans and that the influence of truck percentage on the total load is expectedly large: +27.5% and +40.2% for FS, respectively, on the 200 and 1000 m span. On the other hand, the presence of cars reduces the loading by about 12% (200 m) and 29% (1000 m). Therefore, it is advised that traffic data collection should be based also on periods of low flow but with high truck percentage.

Finally, it is worth noting that the assignment of realistic congestion frequencies is certainly important for a correct bridge-loading analysis. As mentioned in Section 2.3.1, some sites are

<sup>12</sup> Note that the increased truck percentage reduces the dynamic capacity Qout, as mentioned in Section 2.3. Therefore, it is necessary to apply a greater modified safe time headway T' = 6.0 s to reproduce a HCT(2) state similar to that of a flow with 20% trucks [60].

<sup>13</sup> An exception is the HCT(2) state at 1250 veh/h for the 1000 m span. However, this is due to the slower congestion growth occurring at lower flows, which takes somewhat less than an hour to fill the bridge up, thus giving fewer realizations of critical events.

likely to be congested on a daily basis (e.g. urban corridors), whereas others can hardly experience any congestion (e.g. intercity motorways). Fortunately, the characteristic load is not largely sensitive to the congestion frequency [9, 15, 38]: it would generally suffice to consider a reasonable estimate. For instance, for the data reported above, halving the number of congestion events to 125 per year decreases the 5-year SEV to 6.44 (Eq. 13), thereby reducing the corresponding characteristic loading for the 200 m and the 1000 m respectively by 3.4% and 2.0%. If the expected congestion events are ten times less, which could make the difference between sites affected by recurrent or non-recurrent congestion, the 5-year SEV is 4.82; in this case, reductions become more significant (12.4% and 9.1%). Note that in the latter case, it is an *interpolation* of the simulated data that has been carried out.

**Figure 6.** 5-year EUDL for: (a) 200 m span; (b) 1000 m span.

#### **5.2. Main codes**

congested traffic states (as *v* << *v*0), but could potentially affect bridge loading in uncongested situations. To avoid this issue, a suitable modification of the IDM has been proposed [92].

**Figure 6** shows the 5-year characteristic values for the inflows of 3000 (as described in Section 5.1.1), 2000 and 1250 veh/h, expressed as an Equivalent Uniformly Distributed Load (EUDL, total load divided by bridge length). EUDLs resulting from a flow of 1250 veh/h with 48% trucks are also plotted, in order to quantify the influence of truck percentage and car presence on loading (the 1250 veh/h flow with 48% trucks has the same truck flow of the 3000 veh/h

In general, the loads resulting from different inflows with same truck percentage and same bottleneck strength are quite similar (within a range of ±11%), as long as the bottleneck is strong enough to trigger congestion for that inflow, that is *Q*in > *Q*'out. This suggests that the effect of the inflow on loading is not as strong as that of bottleneck strength and implies that critical

Among the traffic features affecting bridge loading, the truck percentage is of particular significance. High truck percentages may occur at night time and early morning in correspondence to low overall flows. As such, it is rare to have congestion events, but if they occur, the resulting loading is likely to be quite heavy, as there are not many cars to keep heavy

**Figure 6** also shows that at lower flows (1250 veh/h) FS governs for both spans and that the influence of truck percentage on the total load is expectedly large: +27.5% and +40.2% for FS, respectively, on the 200 and 1000 m span. On the other hand, the presence of cars reduces the loading by about 12% (200 m) and 29% (1000 m). Therefore, it is advised that traffic data collection should be based also on periods of low flow but with high truck percentage.

Finally, it is worth noting that the assignment of realistic congestion frequencies is certainly important for a correct bridge-loading analysis. As mentioned in Section 2.3.1, some sites are

12 Note that the increased truck percentage reduces the dynamic capacity Qout, as mentioned in Section 2.3. Therefore, it is necessary to apply a greater modified safe time headway T' = 6.0 s to reproduce a HCT(2) state similar to that of a flow

13 An exception is the HCT(2) state at 1250 veh/h for the 1000 m span. However, this is due to the slower congestion growth occurring at lower flows, which takes somewhat less than an hour to fill the bridge up, thus giving fewer realizations of

**Congested state HCT(1) HCT(2) FS HCT(1) HCT(2) FS** Location parameter, *μ* 5340 5908 5140 20334 23212 26519 Scale parameter, *σ* 328.7 366.4 694.5 811.1 954.0 1415.3 Shape parameter, *ξ* −0.028 −0.006 −0.132 −0.061 −0.179 −0.190

**Span 200 m 1000 m**

**Table 5.** Parameters of the GEV distribution for HCT and FS conditions.

loading events may occur also out of rush hours.13

*5.1.2. Influence of some traffic features*

flow with 20% trucks).12

46 Structural Bridge Engineering

vehicles apart.

with 20% trucks [60].

critical events.

The calibration studies of the Eurocode LM1 considered for jammed traffic (spans 75-200 m) a queue of heavy vehicles spaced at 5 m (axle to axle) in the slow lane [64]. National Annexes may extend the span limit of application of LM1. For instance, the UK National Annex extends its application up to 1500 m. Again, LM1 may be significantly reduced when using site-specific traffic data, with traffic microsimulation used to replicate congestion events [87].

The superseded British standard [84], which specified long-span bridge loading in a similar form to the HA loading for short spans (UDL + KEL), was greatly based on [9]: truck surveys and free-flowing traffic data were used to build up queues of heavy vehicles and cars, with gaps varying from 0.9 to 1.8 m; a simple modelling of lane choice of vehicles approaching a queue was also considered.

In North America, the American Society of Civil Engineers (ASCE) recommended a load model for the design of spans up to 1951 m [93]. A UDL is to be applied in conjunction with a point load, whose values depend on span length and truck percentage. The ASCE loading is mainly based on truck data from crossings of the Second Vancouver Narrows bridge [7]. Notably, 800 full-stop events per year were considered with vehicles spaced at 1.5 m.
