6. Modelling crowd-structure interaction

5.4. Simulation procedure

74 Bridge Engineering

tion given in Ref. [27]:

The simulation of a pedestrian flow requires the determination of four parameters: (i) the pedestrian density, d, (ii) the desired velocity of each pedestrian, vd, (iii) the phase shift among

Parameter Element Value Relaxation time tr 0.50 sec. Interaction strength pedestrians Ap 2000 N Interaction range pedestrians Bp 0.30 m Potential factor λ<sup>p</sup> 0.20 Contact strength pedestrians Cp 2000 N Sliding strength pedestrians Dp 4800 N Interaction strength boundaries Ab 5100 N Interaction range boundaries Bb 0.50 m Contact strength boundaries Cb 2000 N Sliding strength boundaries Db 4800 N Radius of pedestrian rp 0.20 m

First, the pedestrian density, d, is established according to the owner's requirements [12]. Second, the values of the desired velocity of each pedestrian can be obtained from the pedestrian step frequencies, f <sup>s</sup>, assuming that initially the pedestrian velocity, vp, is equal to the desired velocity, vd. For this purpose, the Gaussian distribution of the pedestrian step frequency, Nð Þ 1:87; 0:186 Hz, reported in Ref. [2], can be adopted as reference. After assigning a step frequency to each pedestrian, its desired velocity is determined from the empirical rela-

> <sup>3</sup> � <sup>1</sup>:<sup>59</sup> � <sup>v</sup><sup>p</sup>

Subsequently, the initial phase shift among pedestrians, ϕp, which allows estimating the number of pedestrian that arrive at the footbridge in phase, is determined considering that it follows a Poisson distribution [14]. Finally, the original distance among pedestrians is calculated considering the width of the footbridge, a predefined geometrical-shaped mesh of pedes-

<sup>a</sup><sup>p</sup> <sup>¼</sup> <sup>F</sup>pci

Finally, the evaluation of the remaining variables that govern the crowd model, v<sup>p</sup> and xp, are

<sup>2</sup> <sup>þ</sup> <sup>2</sup>:<sup>93</sup> � <sup>v</sup><sup>p</sup> 

(27)

<sup>m</sup> (28)

pedestrians, ϕp, and (iv) the distance among pedestrians, dp.

Table 5. Parameters of the crowd sub-model reported in Refs. [19, 20].

f <sup>s</sup> ¼ 0:35 � v<sup>p</sup> 

trians (triangular or rectangular) and the considered pedestrian density.

then performed using a multi-step method [14].

The acceleration vector, ap, that acts on each pedestrian may be determined as:

The crowd-structure interaction is usually modelled including additional behavioural conditions [20–22]. Concretely, two requirements have been included in this proposal: (i) a comfort and (ii) a lateral lock-in threshold [14, 20].

First, a comfort condition is usually included in the crowd-structure interaction model to take into account the modification of the behaviour of each pedestrian due to the change of his/her comfort level. For this purpose, a retardation factor has been applied to the pedestrian velocity. A minimum comfort threshold 0.20 m/s<sup>2</sup> is selected following the results provided by several researches [20, 28]. In this manner, if the lateral acceleration of each pedestrian, y€a, is above this value, the pedestrian velocity is reduced by a retardation factor, rv, which is a function of the acceleration experienced by the pedestrian. Following the intuitive assumption, reported in Ref. [20], that the pedestrians are likely to react more firmly as the lateral acceleration they feel is higher, a tri-linear function is considered, Eq. (29).

$$r\_v(\ddot{y}\_a) = \begin{array}{c c} 0.1 \text{ (1.05)} \cdot \ddot{y}\_a & \ddot{y}\_a \le 1.05 \text{ } m/s^2\\ r\_v(\ddot{y}\_a) = \begin{array}{c c} 0.9 - (0.3/0.65) \cdot \left(\ddot{y}\_a - 1.05\right) & \ddot{y}\_a \le 1.7 \text{ } m/s^2\\ 0.6 - (0.6/0.4) \cdot \left(\ddot{y}\_a - 1.7\right) & \ddot{y}\_a \le 2.1 \text{ } m/s^2\\ 0 & \ddot{y}\_a > 2.1 \text{ } m/s^2 \end{array} \tag{29}$$

On the other hand, a maximum lateral limit acceleration, <sup>y</sup>€lim <sup>¼</sup> <sup>2</sup>:10 m/s2 , have also been considered [29], so pedestrians stop walking, when the experienced acceleration becomes too high, to keep their balance, and they remain stopped until the footbridge reduces its accelerations. Both to stop walking and to remain stationary before starting to walk again, the same reaction time, trea ¼ 2:00 s, has been adopted. A linear variation has been considered to simulate the variation of the pedestrian velocity during the reaction time. Additionally, a practical lower limit of the pedestrian velocity has been established in order to avoid meaningless small values of this magnitude [20].

$$|v\_p| = \begin{cases} 0.1 \cdot |v\_d| & \text{if } \quad \ddot{y}\_a < \ddot{y}\_{\text{lim}} \cap |v\_p| \le 0.1 \cdot |v\_d|\\ 0 & \text{if } \qquad \ddot{y}\_a \ge \ddot{y}\_{\text{lim}} \end{cases} \tag{30}$$

Finally, as lateral lock-in threshold, the criterion suggested by the French standard [11] is usually adopted to simulate the synchronization phenomenon between the movement of the crowd and the footbridge. For this purpose, both the step frequency, f <sup>s</sup>, and phase shift, ϕp, of each pedestrian are modified to match the natural frequency of the structure, if the lateral acceleration experienced by each subject is above 0.15 m/s<sup>2</sup> and its step frequency is within �10% of the lateral natural frequency of the structure [30].
