5. Modelling crowd dynamics

Fp ¼ m � g

pedestrians and nf is the total number of contributing harmonics.

purpose of the case under study.

70 Bridge Engineering

Xnf i¼1

α<sup>i</sup> � sin π � i � f <sup>s</sup> � t � φ<sup>i</sup> � ϕ<sup>p</sup> � �

where m ¼ ma þ ms is the total pedestrian mass, g is the acceleration of gravity, α<sup>i</sup> is the Fourier coefficients of the ith harmonic of the lateral force, f <sup>s</sup> [Hz] is the pedestrian step frequency, φ<sup>i</sup> is the phase shifts of the ith harmonic of the lateral pedestrian force, ϕ<sup>p</sup> is the phase shift among

According to this formulation, the deterministic or stochastic character of the pedestrian-structure interaction sub-model can be considered depending on the way in which the parameters of the model are defined. If a fixed value is assigned to the parameters, the sub-model will be deterministic; however, if the parameters are defined as random variables, the sub-model will be stochastic. Finally, Table 4 shows the values reported in Ref. [14] for the characterization of the pedestrian-structure interaction model. Additionally, a wide summary of the parameters proposed by other researchers can be found in Ref. [18]. These values (Table 4) allow defining the pedestrian-structure interaction model in either a deterministic form (considering the average values) or a stochastic form (considering the probabilistic distribution), depending on the

Pedestrian modal parameters

Walking pedestrian force

Lateral

Lateral

Table 4. Parameters of the pedestrian-structure interaction sub-model reported in Refs. [14, 15], where N μ; σ � � is a

Definition Parameter Value Pedestrian total mass m Nð Þ 75; 15 kg Pedestrian sprung mass ma Nð Þ 73:216; 2:736 % Pedestrian damping ratio ζ<sup>p</sup> Nð Þ 49:116; 5:405 % Pedestrian natural frequency fp Nð Þ 1:201; 0:178 Hz

Definition Parameter Value

First phase shift φ<sup>1</sup> 0� Second phase shift φ<sup>2</sup> 0� Third phase shift φ<sup>3</sup> 0�

Gaussian distribution with mean value, μ, and the standard deviation, σ.

First harmonic α<sup>1</sup> Nð Þ 0:086; 0:017 Second harmonic α<sup>2</sup> Nð Þ 0:094; 0:009 Third harmonic α<sup>3</sup> Nð Þ 0:040; 0:019 (14)

The pedestrian moving inside a crowd may be modelled using either a macroscopic [10] or a microscopic model [15]. The second option is currently the most utilized and it has been successfully implemented by several authors [15, 19–22]. According to this approach, the movement of each pedestrian is governed by the dynamic balance among particles [14]. This model assumes that the different motivations and influences experimented by the pedestrians are described by different social forces [19]. The model is based on Newton dynamics and is able to represent the following rules in relation with the natural pedestrian movement (see Ref. [19] for a more detailed description): (i) the fastest route is usually chosen by pedestrians, (ii) the individual speed of each pedestrian follows a probabilistic distribution function and (iii) the distance among pedestrians in a crowd depends on the pedestrian density, the spatial configuration of the crowd and the pedestrian speed. As an example, the different social forces acting between two pedestrians in a crowd are illustrated in Figure 4.

In this manner, the multi-agent model that simulates the behaviour of the crowd consists of the sum of three partial forces: (i) the driving force, Fdri, (ii) the repulsive force generated by the interaction among pedestrians, Fped, and (iii) the repulsive force generated by the interaction with the boundaries, Fbou. A detailed description of these three forces is carried out in the next sub-sections. The sum of these three forces generates the overall pedestrian-crowd interaction

Figure 4. Biomechanical pedestrian-structure interaction model [14].

force, Fpci, that describes the movement and direction of each pedestrian in the crowd. This resultant force is defined as follows:

$$\mathbf{F}\_{pci} = \mathbf{F}\_{dri} + \mathbf{F}\_{pel} + \mathbf{F}\_{lou} \tag{15}$$

divided in other two components: (i) the body force, Fphy\_nor

Fphy

Fphy\_nor

Fphy\_ tan

normalized tangential vector (which is perpendicular to np); Δv<sup>t</sup>

between two pedestrians; and Hð Þ • a function which may be defined as [22]:

ped <sup>¼</sup> <sup>F</sup>phy\_nor

ped ¼ Dp � H 2 � rp � dp

ped the normal component of the physical interaction force; <sup>F</sup>phy\_ tan

component of the physical interaction force; Cp the body force strength due to the contact between pedestrians; Dp the sliding force strength due to the contact between pedestrians; t<sup>p</sup> a

the relative pedestrian velocity in tangential direction; Δv<sup>p</sup> the vector of differential velocities

<sup>H</sup>ð Þ� ¼ • • if • <sup>&</sup>gt; <sup>0</sup>

The interaction with the boundaries gives rise to forces, Fbou. These forces are equivalent to the ones resulting from the interaction with other pedestrian, so they can be formulated in a

<sup>F</sup>bou <sup>¼</sup> <sup>F</sup>nor

rp � db Bb 

bou ¼ Db � H rp � db

0 if • ≤ 0

bou <sup>þ</sup> <sup>F</sup>tan

� <sup>v</sup>p; <sup>t</sup><sup>b</sup>

bou the component of the boundary interaction force in normal direction; Ftan

component of the boundary interaction force in tangential direction; Ab the pedestrianboundary interaction strength; Bb the pedestrian-boundary repulsive interaction range; db the pedestrian-boundary distance; Cb the body force strength due to the contact with the boundary; Db the sliding force strength due to the contact with the boundary; n<sup>b</sup> the normalized normal vector between the pedestrian and boundary; t<sup>b</sup> the normalized tangential vector

All the parameters for the considered crowd sub-model, based on the social force model, may be obtained from the results provided by different authors [19, 20] as summarized in Table 5.

þ Cb � H rp � db � <sup>n</sup><sup>b</sup> (25)

ped ¼ Cp � H 2 � rp � dp

quickly at small distances [19]. It is defined as [22]:

being Fphy\_nor

5.3. Interactions with boundaries

Fnor

bou ¼ Ab � exp

Ftan

(which is perpendicular to nb) and hi denotes the scalar product [22].

similar fashion as [22]:

being Fnor

The first component simulates the counteracting body action that each pedestrian performs to avoid physical damage in case he/she gets in physical contact with other individuals. The second component considers the pedestrians' tendency to avoid overtaking other subjects

ped <sup>þ</sup> <sup>F</sup>phy\_ tan

� <sup>Δ</sup>vt

Recent Advances in the Serviceability Assessment of Footbridges Under Pedestrian-Induced Vibrations

ped , and (ii) the sliding force, <sup>F</sup>phy\_ tan

http://dx.doi.org/10.5772/intechopen.71888

ped (20)

<sup>p</sup> � t<sup>p</sup> (22)

ped the tangential

(23)

bou the

the component of

� <sup>n</sup><sup>p</sup> (21)

<sup>p</sup> ¼ Δv<sup>p</sup> � t<sup>p</sup>

bou (24)

� <sup>t</sup><sup>b</sup> (26)

ped .

73

#### 5.1. Driving force

Each pedestrian has a certain motivation to reach his/her desired destination [19], dd, with his/ her desired velocity, vd, which is represented by the driving force, Fdri, as:

$$\mathbf{F}\_{dri} = m \cdot \left(\frac{v\_d \cdot \mathbf{e}\_d}{t\_r} - \frac{\mathbf{v}\_p}{t\_r}\right) \tag{16}$$

where e<sup>d</sup> is the desired direction vector, v<sup>p</sup> is the pedestrian step velocity and tr is the relaxation time (the time it takes a pedestrian to adapt its motion to its preferences).

#### 5.2. Interactions among pedestrians

The interaction among pedestrians originates a repulsive force [19], Fped, with two components, a socio-psychological force, Fsoc ped, and a physical interaction force, <sup>F</sup>phy ped , as:

$$\mathbf{F}\_{ped} = \mathbf{F}\_{ped}^{\text{sc}} + \mathbf{F}\_{ped}^{\text{yly}} \tag{17}$$

The socio-psychological force reflects the fact that the pedestrians try to maintain a certain distance to other pedestrians in the crowd. This socio-psychological force depends on the distance between pedestrians, reaching its maximum value at the initial distance between two pedestrians, dp, and tending to zero as such distance increases. The socio-psychological force is defined as:

$$\mathbf{F}\_{pcd}^{\rm soc} = A\_p \cdot \exp\left(\frac{2 \cdot r\_p - d\_p}{B\_p}\right) \cdot \mathbf{n}\_p \cdot \mathbf{s}\_p \tag{18}$$

where Ap is the interaction strength between two pedestrians; Bp is the repulsive interaction range between pedestrians; rp is the so-called pedestrian radius; n<sup>p</sup> is the normalized vector pointing between pedestrians and sp is a form factor to consider the anisotropic pedestrian behaviour [19], whose value may be obtained from:

$$s\_p = \lambda\_p + \left(1 - \lambda\_p\right) \cdot \frac{1 + \cos\left(\varphi\_p\right)}{2} \tag{19}$$

being, λp, a coefficient that takes into account the influence of the pedestrians placed in front of the subject on his/her movement, and, φp, the angle formed between two pedestrians.

In situations of physical contact among pedestrians (dp ≤ 2 � rp) and high pedestrian density (≥0.80 P=Person/m<sup>2</sup> ), the physical interaction force, Fphy ped , must be considered. This force may be divided in other two components: (i) the body force, Fphy\_nor ped , and (ii) the sliding force, <sup>F</sup>phy\_ tan ped . The first component simulates the counteracting body action that each pedestrian performs to avoid physical damage in case he/she gets in physical contact with other individuals. The second component considers the pedestrians' tendency to avoid overtaking other subjects quickly at small distances [19]. It is defined as [22]:

$$\mathbf{F}\_{ped}^{phys} = \mathbf{F}\_{ped}^{phys\text{\textquotedblleft nor}} + \mathbf{F}\_{ped}^{phys\text{\textquotedblright}} \tag{20}$$

$$\mathbf{F}\_{\text{ped}}^{\text{phys\\_nor}} = \mathbf{C}\_p \cdot H(\mathbf{2} \cdot r\_p - d\_p) \cdot \mathbf{n}\_p \tag{21}$$

$$\mathbf{F}\_{\text{ped}}^{\text{phys\\_tan}} = D\_p \cdot H(\mathbf{2} \cdot r\_p - d\_p) \cdot \Delta v\_p^t \cdot \mathbf{t}\_p \tag{22}$$

being Fphy\_nor ped the normal component of the physical interaction force; <sup>F</sup>phy\_ tan ped the tangential component of the physical interaction force; Cp the body force strength due to the contact between pedestrians; Dp the sliding force strength due to the contact between pedestrians; t<sup>p</sup> a normalized tangential vector (which is perpendicular to np); Δv<sup>t</sup> <sup>p</sup> ¼ Δv<sup>p</sup> � t<sup>p</sup> the component of the relative pedestrian velocity in tangential direction; Δv<sup>p</sup> the vector of differential velocities between two pedestrians; and Hð Þ • a function which may be defined as [22]:

$$H(\bullet) \cdot = \begin{cases} \bullet & \text{if} \quad \bullet > 0 \\ 0 & \text{if} \quad \bullet \le 0 \end{cases} \tag{23}$$

#### 5.3. Interactions with boundaries

force, Fpci, that describes the movement and direction of each pedestrian in the crowd. This

Each pedestrian has a certain motivation to reach his/her desired destination [19], dd, with his/

where e<sup>d</sup> is the desired direction vector, v<sup>p</sup> is the pedestrian step velocity and tr is the relaxation

The interaction among pedestrians originates a repulsive force [19], Fped, with two components,

The socio-psychological force reflects the fact that the pedestrians try to maintain a certain distance to other pedestrians in the crowd. This socio-psychological force depends on the distance between pedestrians, reaching its maximum value at the initial distance between two pedestrians, dp, and tending to zero as such distance increases. The socio-psychological force is

where Ap is the interaction strength between two pedestrians; Bp is the repulsive interaction range between pedestrians; rp is the so-called pedestrian radius; n<sup>p</sup> is the normalized vector pointing between pedestrians and sp is a form factor to consider the anisotropic pedestrian

�

the subject on his/her movement, and, φp, the angle formed between two pedestrians.

being, λp, a coefficient that takes into account the influence of the pedestrians placed in front of

In situations of physical contact among pedestrians (dp ≤ 2 � rp) and high pedestrian density

<sup>F</sup>ped <sup>¼</sup> <sup>F</sup>soc

ped, and a physical interaction force, <sup>F</sup>phy

ped <sup>þ</sup> <sup>F</sup>phy

2 � rp � dp Bp 

1 þ cos φ<sup>p</sup>

tr

� vp tr

<sup>F</sup>dri <sup>¼</sup> <sup>m</sup> � vd � <sup>e</sup><sup>d</sup>

her desired velocity, vd, which is represented by the driving force, Fdri, as:

time (the time it takes a pedestrian to adapt its motion to its preferences).

Fsoc

behaviour [19], whose value may be obtained from:

ped ¼ Ap � exp

sp ¼ λ<sup>p</sup> þ 1 � λ<sup>p</sup>

), the physical interaction force, Fphy

Fpci ¼ Fdri þ Fped þ Fbou (15)

ped , as:

ped (17)

� n<sup>p</sup> � sp (18)

<sup>2</sup> (19)

ped , must be considered. This force may be

(16)

resultant force is defined as follows:

5.2. Interactions among pedestrians

a socio-psychological force, Fsoc

defined as:

(≥0.80 P=Person/m<sup>2</sup>

5.1. Driving force

72 Bridge Engineering

The interaction with the boundaries gives rise to forces, Fbou. These forces are equivalent to the ones resulting from the interaction with other pedestrian, so they can be formulated in a similar fashion as [22]:

$$\mathbf{F}\_{bou} = \mathbf{F}\_{bou}^{nor} + \mathbf{F}\_{bou}^{tan} \tag{24}$$

$$\mathbf{F}\_{bbu}^{nr} = \left\{ A\_b \cdot \exp\left(\frac{r\_p - d\_b}{B\_b}\right) + \mathbf{C}\_b \cdot H\left(r\_p - d\_b\right) \right\} \cdot \mathbf{n}\_b \tag{25}$$

$$\mathbf{F}\_{bou}^{\tan} = D\_b \cdot H(r\_p - d\_b) \cdot \left< \mathbf{v}\_p, \mathbf{t}\_b \right> \cdot \mathbf{t}\_b \tag{26}$$

being Fnor bou the component of the boundary interaction force in normal direction; Ftan bou the component of the boundary interaction force in tangential direction; Ab the pedestrianboundary interaction strength; Bb the pedestrian-boundary repulsive interaction range; db the pedestrian-boundary distance; Cb the body force strength due to the contact with the boundary; Db the sliding force strength due to the contact with the boundary; n<sup>b</sup> the normalized normal vector between the pedestrian and boundary; t<sup>b</sup> the normalized tangential vector (which is perpendicular to nb) and hi denotes the scalar product [22].

All the parameters for the considered crowd sub-model, based on the social force model, may be obtained from the results provided by different authors [19, 20] as summarized in Table 5.


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

#### 5.4. Simulation procedure

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

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 relation given in Ref. [27]:

$$f\_s = 0.35 \cdot \left| \mathbf{v}\_p \right|^3 - 1.59 \cdot \left| \mathbf{v}\_p \right|^2 + 2.93 \cdot \left| \mathbf{v}\_p \right| \tag{27}$$

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

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

$$\mathbf{a}\_p = \frac{\mathbf{F}\_{pci}}{m} \tag{28}$$

6. Modelling crowd-structure interaction

is higher, a tri-linear function is considered, Eq. (29).

vp � � �

rv y€<sup>a</sup> � � <sup>¼</sup>

and (ii) a lateral lock-in threshold [14, 20].

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

Recent Advances in the Serviceability Assessment of Footbridges Under Pedestrian-Induced Vibrations

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

> � ð Þ� 0:1=1:05 y€<sup>a</sup> :<sup>9</sup> � ð Þ� <sup>0</sup>:3=0:<sup>65</sup> <sup>y</sup>€<sup>a</sup> � <sup>1</sup>:<sup>05</sup> � � :<sup>6</sup> � ð Þ� <sup>0</sup>:6=0:<sup>4</sup> <sup>y</sup>€<sup>a</sup> � <sup>1</sup>:<sup>7</sup> � � 0

ered [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

On the other hand, a maximum lateral limit acceleration, <sup>y</sup>€lim <sup>¼</sup> <sup>2</sup>:10 m/s2

been established in order to avoid meaningless small values of this magnitude [20].

if if

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

7. Application example: Lateral lock-in phenomenon on a real footbridge

In order to illustrate the potential of this new modelling framework, the analysis of the lateral lock-in phenomenon on a real footbridge, the Pedro e Inês footbridge (Coimbra Portugal) has been

y€<sup>a</sup> < y€lim ∩ vp � � �

y€<sup>a</sup> ≥ y€lim

� <sup>¼</sup> <sup>0</sup>:<sup>1</sup> � j j vd 0

8 < :

�10% of the lateral natural frequency of the structure [30].

if

y€<sup>a</sup> ≤ 1:05 m=s<sup>2</sup> y€<sup>a</sup> ≤ 1:7 m=s<sup>2</sup> y€<sup>a</sup> ≤ 2:1 m=s<sup>2</sup> y€<sup>a</sup> > 2:1 m=s<sup>2</sup>

http://dx.doi.org/10.5772/intechopen.71888

� ≤ 0:1 � j j vd

(29)

75

(30)

, have also been consid-

Finally, the evaluation of the remaining variables that govern the crowd model, v<sup>p</sup> and xp, are then performed using a multi-step method [14].
