3. New modelling framework of crowd-structure interaction

The most recent research on this topic proposes and further implements several crowdstructure interaction models to better characterize the dynamic response of footbridges under pedestrian action [14–17]. All these models, which share a common scheme, constitute a new modelling framework for this engineering problem. According to this new approach, the crowd-structure interaction is simulated by linking two individual sub-models (Figure 2): (i) a pedestrian-structure interaction sub-model and (ii) a crowd sub-model.

In the first sub-model, although there are several proposals [18, 22, 24, 25] to simulate the pedestrian action (single-degree-of-freedom (SDOF) system, multiple degrees of freedom (MDOF) system and inverted pendulum (IP) system), the most widely adopted alternative is to model the pedestrian either as a SDOF system in vertical direction [18] or as a IP system in lateral direction [20, 24], while the structure is characterized via its modal parameters [22, 26]. Recent Advances in the Serviceability Assessment of Footbridges Under Pedestrian-Induced Vibrations http://dx.doi.org/10.5772/intechopen.71888 67

Figure 2. Layout of the new modelling framework.

To estimate the considered natural frequency for each design scenario, the mass of pedestrians has to be taken into account (with a medium pedestrian weight about

, which avoids the occurrence of

70 kg) when its value is greater than 5% of the modal deck mass.

with the trigger acceleration amplitude, 0.10–0.15 m/s<sup>2</sup>

the lateral lock-in phenomenon.

66 Bridge Engineering

increase of the damping [12].

included in the next section.

vi. The dynamic response obtained for each considered design scenario must be compared

vii. The estimated dynamic acceleration is then compared with the specified comfort class. In case of non-compliance, the designer must adopt measures to improve the dynamic behaviour of the structure, such as for instance: (i) the modification of the mass of the deck, (ii) the modification of the natural frequencies of the structure and/or (iii) the

In spite of the fact that the Synpex design guidelines [12] were an important breakthrough, they still present several limitations, which originate that the numerical prediction of the dynamic response of footbridges, obtained using them, under- or over-estimates the values recorded experimentally. As main limitations, the following ones may be enumerated: (i) the change of the dynamic properties of the structure, due to the presence of pedestrians, is estimated in a simplified form, adding directly the pedestrian mass to the structural mass without considering any additional effect on the remaining modal parameters of the structure, (ii) the proposed methods do not fit well to the case where several vibration modes of the footbridge are affected by the pedestrian-induced excitations, (iii) the effect of the nonsynchronized pedestrians are not taken into account by these recommendations and (iv) the definition of the pedestrian load is performed under a deterministic approach which does not allow considering the inter- and intra-subject variability of the pedestrian action. In order to overcome these limitations, a new generation of models that configure a new modelling framework has been proposed. A brief description of this new modelling framework is

3. New modelling framework of crowd-structure interaction

(i) a pedestrian-structure interaction sub-model and (ii) a crowd sub-model.

The most recent research on this topic proposes and further implements several crowdstructure interaction models to better characterize the dynamic response of footbridges under pedestrian action [14–17]. All these models, which share a common scheme, constitute a new modelling framework for this engineering problem. According to this new approach, the crowd-structure interaction is simulated by linking two individual sub-models (Figure 2):

In the first sub-model, although there are several proposals [18, 22, 24, 25] to simulate the pedestrian action (single-degree-of-freedom (SDOF) system, multiple degrees of freedom (MDOF) system and inverted pendulum (IP) system), the most widely adopted alternative is to model the pedestrian either as a SDOF system in vertical direction [18] or as a IP system in lateral direction [20, 24], while the structure is characterized via its modal parameters [22, 26]. All the pedestrian-structure interaction models based on the use of a SDOF system share a common formulation to solve the pedestrian-structure interaction [22, 26] but, however, they differ in the values adopted to characterize the modal parameters of the SDOF systems. A wide summary of the pedestrian-structure interaction models proposed by different authors can be found in Ref. [18]. The main output obtained from this sub-model is usually the acceleration experienced by each pedestrian.

In the second sub-model, the crowd is usually simulated via a behavioural model [19] that provides a description of the individual pedestrian position, xp, pedestrian velocity, vp, and step pedestrian frequency, f <sup>s</sup>. Additionally, in order to take into account the synchronization among pedestrians, an additional parameter must be included. A common manner to simulate this phenomenon is to add a different phase shift, ϕp, in the definition of the ground reaction load generated by each pedestrian [14].

The linking between the two sub-models is usually achieved in the different proposals by taking into account the modification of the pedestrian behaviour in terms of the vibration level that he/she experiences [15, 17, 20–24]. Two additional conditions are commonly included for this purpose: (i) a retardation factor, which reduces the pedestrian velocity in terms of the accelerations experienced by each pedestrian; and (ii) a lateral lock-in threshold, which allows simulating the synchronization among the pedestrians and the structure by the modification of both their step frequencies and the phases [20–23]. This new approach has only been implemented, to the best of the authors' knowledge, in vertical and lateral direction, since there are few reported cases of pedestrian-induced vibration problems in longitudinal direction. In order to illustrate briefly this new modelling framework, one of the most recent crowdstructure interaction models, which has been proposed by the authors, is described in the next sections [23]. Subsequently, the potential of the approach to accurately assess the vibration serviceability limit state of footbridges under pedestrian action is illustrated via its implementation for the analysis of a case study. For clarity, the model is described here only for the lateral direction, although it may be easily generalized to the vertical direction [14].

### 4. Modelling pedestrian-structure interaction

The pedestrian-structure interaction model may follow from the application of dynamic equilibrium equations between a SDOF-system (Figure 3a) and the footbridge (Figure 3b). The pedestrian mass is divided into sprung, ma, and unsprung, ms, components [kg].

As result of this dynamic equilibrium, the following coupled equation system may be obtained [22]:

$$M\_i \ddot{y}\_i + \mathbb{C}\_i \dot{y}\_i + K\_i y\_i = \phi\_{\text{num\\_i}}(\mathbf{x}\_p) \cdot F\_{\text{int}} \tag{4}$$

Thus, Fint follows from the above Eq. (6) to yield:

Miy€<sup>i</sup> þ Ciy\_

velocities, y\_

w x € <sup>p</sup>; <sup>t</sup> � � <sup>¼</sup> <sup>X</sup>nm

the amplitude yi

where ϕ<sup>0</sup>

i¼1 y€i

num\_ið Þx and ϕ<sup>00</sup>

equations to yield in matrix form:

Fint ¼ Fp � msy€<sup>s</sup> � cp y\_

<sup>i</sup> þ Kiyi ¼ ϕnum\_i xp

structure, the following expressions may be obtained:

w x \_ <sup>p</sup>; <sup>t</sup> � � <sup>¼</sup> <sup>X</sup>nm

ð Þ� t ϕnum\_i xp

i¼1 y\_ i

� � þXnm

i¼1

vibration mode and nm, is the number of considered vibration modes.

expressed in terms of a Fourier series decomposition [8, 9, 12] as:

and substituting this equation into the equilibrium equation of the structure:

w xp; <sup>t</sup> � � <sup>¼</sup> <sup>X</sup>nm

ð Þ� t ϕnum\_i xp

2 � y\_ i

i¼1 yi

<sup>s</sup> � y\_ a � � � kp ys � ya

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

� � � Fp � msy€<sup>s</sup> � cp <sup>y</sup>\_

� � þXnm

ð Þ� t vp,x � ϕ<sup>0</sup>

It is assumed that the lateral displacement of the footbridge may be decomposed in terms of

time variation of the pedestrian velocity is neglected due to its low contribution: Subsequently, the above relations Eqs. (10)–(12) may be substituted in the overall dynamic equilibrium equation of the footbridge, obtaining the following pedestrian-structure interaction model

In this manner, the pedestrian-structure interaction model may be represented by a system with ð Þ nm þ 1 equations (being nm the number of the considered vibration modes and 1 the SDOF system that simulates the pedestrian behaviour). In case of a group of k pedestrians (Figure 3b), the number of equations of system increases to ð Þ nm þ k , maintaining the same scheme. A more detailed description of this pedestrian-structure interaction model may be found in Ref. [22].

The lateral ground reaction force, Fp, generated by each pedestrian, may be defined under either a deterministic [8] or a stochastic approach [15]. The second approach allows taking into account the inter- and intra-subject variability of the pedestrian action [15]. Although there are more complex ways [15] to define the lateral ground reaction force, however, it is usually

<sup>s</sup> <sup>¼</sup> w x \_ <sup>p</sup>; <sup>t</sup> � � and accelerations, <sup>y</sup>€<sup>s</sup> <sup>¼</sup> w x € <sup>p</sup>; <sup>t</sup> � �, between the SDOF system and the

ð Þ� t ϕnum\_i xp

i¼1 yi

num\_i xp

num\_ið Þx the first and second spatial derivatives of the ith numerical

Mð Þ� t y€ð Þþ t Cð Þ� t y\_ð Þþ t Kð Þ� t yðÞ¼ t Fð Þt (13)

ð Þt and the modal coordinates of the nm vibration modes, ϕnum\_ið Þx , and the

Applying, at the contact point, the equations of compatibility of displacements, ys <sup>¼</sup> w xp; <sup>t</sup> � �,

<sup>s</sup> � y\_ a � � � kp ys � ya � � � � (9)

ð Þ� t vpx � ϕ<sup>0</sup>

� � þXnm

i¼1 yi ð Þ� t vp, <sup>x</sup>

� � (8)

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

� � (10)

� � (11)

<sup>2</sup> � <sup>ϕ</sup><sup>00</sup>

num\_i xp � �<sup>Þ</sup>

(12)

69

num\_i xp

$$m\_a \ddot{y}\_a + c\_p (\dot{y}\_a - \dot{y}\_s) + k\_p (y\_a - y\_s) = 0 \tag{5}$$

$$m\_s \ddot{y}\_s + c\_p \left(\dot{y}\_s - \dot{y}\_a\right) + k\_p \left(y\_s - y\_a\right) = F\_p - F\_{\text{int}} \tag{6}$$

where yi is the amplitude of the vibration mode ith of the footbridge [m]; ya is the displacement of the pedestrian sprung mass [m]; ys is the displacement of the pedestrian unsprung mass [m]; kp is the pedestrian stiffness [N/m]; cp is the pedestrian damping [sN/m]; Fp is the ground reaction force [N]; Fint is the pedestrian-structure interaction force [N]; Mi is the mass associated with the ith vibration mode [kg]; Ci is the damping associated with the ith vibration mode [sN/m]; Ki is the stiffness associated with the ith vibration mode [N/m]; ϕnum\_i is the modal coordinates of the ith vibration mode; xp ¼ vpx � t is the pedestrian's longitudinal position on the footbridge [m], being t the time [sec.] and vpx the longitudinal component of the pedestrian velocity vector [m/s]; dp is the distance among pedestrians [m] and w xð Þ ; t is the lateral displacement of the footbridge at position x [m].

The numerical vibration modes, ϕnum\_ið Þx , may be obtained by a numerical modal analysis of the structure based on the finite element method:

$$\phi\_{mm\\_i}(\mathbf{x}) = \sum\_j \phi\_i^j \cdot \mathbf{N}\_j(\mathbf{x}) \tag{7}$$

where Njð Þ<sup>x</sup> is the beam shape functions and <sup>ϕ</sup><sup>j</sup> <sup>i</sup> is the nodal values of the vibration modes.

Figure 3. Biomechanical pedestrian-structure interaction model in lateral direction [14]. (a) SDOF-system and (b) Footbridge.

Thus, Fint follows from the above Eq. (6) to yield:

4. Modelling pedestrian-structure interaction

[22]:

68 Bridge Engineering

bridge.

The pedestrian-structure interaction model may follow from the application of dynamic equilibrium equations between a SDOF-system (Figure 3a) and the footbridge (Figure 3b). The

As result of this dynamic equilibrium, the following coupled equation system may be obtained

where yi is the amplitude of the vibration mode ith of the footbridge [m]; ya is the displacement of the pedestrian sprung mass [m]; ys is the displacement of the pedestrian unsprung mass [m]; kp is the pedestrian stiffness [N/m]; cp is the pedestrian damping [sN/m]; Fp is the ground reaction force [N]; Fint is the pedestrian-structure interaction force [N]; Mi is the mass associated with the ith vibration mode [kg]; Ci is the damping associated with the ith vibration mode [sN/m]; Ki is the stiffness associated with the ith vibration mode [N/m]; ϕnum\_i is the modal coordinates of the ith vibration mode; xp ¼ vpx � t is the pedestrian's longitudinal position on the footbridge [m], being t the time [sec.] and vpx the longitudinal component of the pedestrian velocity vector [m/s]; dp is the distance among pedestrians [m] and w xð Þ ; t is the lateral dis-

The numerical vibration modes, ϕnum\_ið Þx , may be obtained by a numerical modal analysis of

j ϕj

Figure 3. Biomechanical pedestrian-structure interaction model in lateral direction [14]. (a) SDOF-system and (b) Foot-

<sup>ϕ</sup>num\_ið Þ¼ <sup>x</sup> <sup>X</sup>

<sup>i</sup> þ Kiyi ¼ ϕnum\_i xp

� � � <sup>F</sup>int (4)

� � <sup>¼</sup> <sup>0</sup> (5)

<sup>i</sup> � Njð Þx (7)

<sup>i</sup> is the nodal values of the vibration modes.

� � <sup>¼</sup> Fp � <sup>F</sup>int (6)

pedestrian mass is divided into sprung, ma, and unsprung, ms, components [kg].

<sup>a</sup> � y\_ s � � <sup>þ</sup> kp ya � ys

<sup>s</sup> � y\_ a � � <sup>þ</sup> kp ys � ya

Miy€<sup>i</sup> þ Ciy\_

may€<sup>a</sup> þ cp y\_

msy€<sup>s</sup> þ cp y\_

placement of the footbridge at position x [m].

the structure based on the finite element method:

where Njð Þ<sup>x</sup> is the beam shape functions and <sup>ϕ</sup><sup>j</sup>

$$F\_{\rm int} = F\_p - m\_s \ddot{y}\_s - c\_p \left(\dot{y}\_s - \dot{y}\_a\right) - k\_p \left(y\_s - y\_a\right) \tag{8}$$

and substituting this equation into the equilibrium equation of the structure:

$$M\_i \ddot{y}\_i + \mathcal{C}\_i \dot{y}\_i + K\_i y\_i = \phi\_{\text{num\\_i}}(\mathbf{x}\_p) \cdot \left(\mathcal{F}\_p - m\_s \ddot{y}\_s - \mathcal{c}\_p(\dot{y}\_s - \dot{y}\_a) - k\_p (y\_s - y\_a)\right) \tag{9}$$

Applying, at the contact point, the equations of compatibility of displacements, ys <sup>¼</sup> w xp; <sup>t</sup> � �, velocities, y\_ <sup>s</sup> <sup>¼</sup> w x \_ <sup>p</sup>; <sup>t</sup> � � and accelerations, <sup>y</sup>€<sup>s</sup> <sup>¼</sup> w x € <sup>p</sup>; <sup>t</sup> � �, between the SDOF system and the structure, the following expressions may be obtained:

$$w(\mathbf{x}\_p, t) = \sum\_{i=1}^{n\_w} y\_i(t) \cdot \phi\_{num\\_i}(\mathbf{x}\_p) \tag{10}$$

$$\dot{\boldsymbol{w}}\left(\mathbf{x}\_{p},t\right) = \sum\_{i=1}^{n\_{m}} \dot{\boldsymbol{y}}\_{i}(t) \cdot \boldsymbol{\phi}\_{mm\underline{\boldsymbol{u}}\_{i}}\left(\mathbf{x}\_{p}\right) + \sum\_{i=1}^{n\_{m}} \boldsymbol{y}\_{i}(t) \cdot \boldsymbol{v}\_{p\mathbf{x}} \cdot \boldsymbol{\phi}\_{mm\underline{\boldsymbol{u}}\_{i}}'\left(\mathbf{x}\_{p}\right) \tag{11}$$

$$\ddot{w}\left(\mathbf{x}\_{\mathcal{P}},t\right) = \sum\_{i=1}^{n\_{\text{m}}} \ddot{y}\_{i}(t) \cdot \phi\_{\text{num\\_i}}\left(\mathbf{x}\_{\mathcal{P}}\right) + \sum\_{i=1}^{n\_{\text{m}}} 2 \cdot \dot{y}\_{i}(t) \cdot \boldsymbol{\upsilon}\_{\mathcal{P},\text{x}} \cdot \phi'\_{\text{num\\_i}}\left(\mathbf{x}\_{\mathcal{P}}\right) + \sum\_{i=1}^{n\_{\text{m}}} y\_{i}(t) \cdot \boldsymbol{\upsilon}\_{\mathcal{P},\text{x}} \,^{2} \cdot \phi''\_{\text{num\\_i}}\left(\mathbf{x}\_{\mathcal{P}}\right) \tag{12}$$

where ϕ<sup>0</sup> num\_ið Þx and ϕ<sup>00</sup> num\_ið Þx the first and second spatial derivatives of the ith numerical vibration mode and nm, is the number of considered vibration modes.

It is assumed that the lateral displacement of the footbridge may be decomposed in terms of the amplitude yi ð Þt and the modal coordinates of the nm vibration modes, ϕnum\_ið Þx , and the time variation of the pedestrian velocity is neglected due to its low contribution: Subsequently, the above relations Eqs. (10)–(12) may be substituted in the overall dynamic equilibrium equation of the footbridge, obtaining the following pedestrian-structure interaction model equations to yield in matrix form:

$$\mathbf{M}(t) \cdot \ddot{\mathbf{y}}(t) + \mathbf{C}(t) \cdot \dot{\mathbf{y}}(t) + \mathbf{K}(t) \cdot \mathbf{y}(t) = \mathbf{F}(t) \tag{13}$$

In this manner, the pedestrian-structure interaction model may be represented by a system with ð Þ nm þ 1 equations (being nm the number of the considered vibration modes and 1 the SDOF system that simulates the pedestrian behaviour). In case of a group of k pedestrians (Figure 3b), the number of equations of system increases to ð Þ nm þ k , maintaining the same scheme. A more detailed description of this pedestrian-structure interaction model may be found in Ref. [22].

The lateral ground reaction force, Fp, generated by each pedestrian, may be defined under either a deterministic [8] or a stochastic approach [15]. The second approach allows taking into account the inter- and intra-subject variability of the pedestrian action [15]. Although there are more complex ways [15] to define the lateral ground reaction force, however, it is usually expressed in terms of a Fourier series decomposition [8, 9, 12] as:

$$F\_p = m \cdot g \sum\_{i=1}^{n\gamma} \alpha\_i \cdot \sin\left(\pi \cdot i \cdot f\_s \cdot t - \varphi\_i - \phi\_p\right) \tag{14}$$

5. Modelling crowd dynamics

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

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

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

71

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

acting between two pedestrians in a crowd are illustrated in Figure 4.

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

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 pedestrians and nf is the total number of contributing harmonics.

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 purpose of the case under study.


Table 4. Parameters of the pedestrian-structure interaction sub-model reported in Refs. [14, 15], where N μ; σ � � is a Gaussian distribution with mean value, μ, and the standard deviation, σ.
