Crowd-Structure Interaction in Laterally Vibrating Footbridges: Comparison of Two Fully Coupled Approaches

Abstract

Two models that deal with the crowd-structure interaction have been developed. The first is a 1D continuous model and the other is a 2D discrete one. In this paper, a summary of the formulation of these two models is presented. Both approaches used to represent the pedestrian-structure coupling phenomenon are detailed and compared. We start by introducing the partial and ordinary differential equations that govern the dynamics of both the continuous and the discrete models. First, the equation of dynamics of the footbridge for the case of lateral vibrations is recalled. Then, the Kuramoto phase equation is implemented for describing the coupling between the pedestrians and the laterally moving deck of a footbridge. Results obtained from numerical simulations are presented and compared with available experimental data.

Appendices

Appendix

1.1 Coriolis Acceleration Norm in the 2D Discrete Model

We are interested here in the comparison between the amplitude of the Coriolis acceleration norm: and the amplitude of the relative acceleration one: $\ddot{u}^{rel}(t)$, in the 2D discrete model. For this purpose, let us consider a pedestrian of mass m walking with a constant tangential velocity $v_t$ on a circular trajectory of radius r. The pedestrian’s walking frequency is assumed to be constant: $\dot{\phi }^{dist}(t)=\omega$ and its value is chosen to be: $\omega =2\pi f=2\pi \,rd/s$. $\dot{\theta }=\frac{v_t}{r}$ represents the rotational velocity. The pedestrian applies a sinusoidal lateral force on the footbridge given by (see Fig. 1):

(10)

and the relative acceleration can be easily deduced:

(11)

The analytical expression of the velocity is deduced by time-integration of Eq. (11):

(12)

The Coriolis acceleration is then calculated:

(13)

The norm of and of can be easily deduced:

(14)

(15)

Upper bounds independent of t can be easily found for each previous norm:

(16)

In the numerical application, $m_i=75\,$kg, $v_t=1.34\,$m/s, $\omega =2\pi \,$rd/s, $F^{osc,T} =120\,$N and $F^{osc,N}=35\,$N. We study the case when $B_1\ge B_2$, corresponding to the maximum of greater than $10\,\%$ of the maximum of . This leads to: $r<(22\cdot v_t)/\omega$ and finally $r<4.8\,$m. The horizontal alignment of a footbridge is generally close to that of a straight line. The trajectory of a pedestrian shall admit a generally high radius of curvature and thus the Coriolis acceleration is assumed to be negligible.

1.2 Calculating $N_c$

The 1D Continuous Model (M1)

In the general case, where $\mu _{\omega }$ is different from the modal frequency of the system crowd-structure, it has been found in [3] that resolving a third order polynomial gives the value of $N_c$. By denoting $\varGamma =(N_cG\omega _0\varepsilon )/(2K)$, this polynomial is:

$$\begin{aligned} a_3\varGamma ^3+a_2\varGamma ^2+a_1\varGamma +a_0=0 \end{aligned}$$

(17)

where:

$$\begin{aligned} \begin{aligned} a_0&=16\pi ^2C\sigma _{\omega }^4 \\ a_1&=8\pi C\sigma _{\omega }^2n\omega _0^2-\frac{8\pi ^3\sigma _{\omega }^3nK}{\sqrt{2\pi }}+\frac{4\pi ^3nK}{\sqrt{2\pi }}(\mu _{\omega }-\omega _0)^2\\ a_2&=n^2\omega _0^4C-\frac{4\pi ^2\sigma _{\omega }n^2\omega _0^2K}{\sqrt{2\pi }}\,, \quad a_3=-\frac{n^3\pi K \omega _0^4}{2\sigma _{\omega }\sqrt{2\pi }} \end{aligned} \end{aligned}$$

(18)

The 2D Continuous Model (M2)

Similarly to the continuous model, for the general case ($\mu _{\omega }\ne \omega _T$), $N_c$ is the solution of the following polynomial [24]:

$$\begin{aligned} \gamma _6c_1N_c^3+\gamma _5N_c^3+\gamma _4c_1N_c^2+\gamma _3N_c^2+\gamma _2c_1N_c+\gamma _1N_c+\gamma _0=0 \end{aligned}$$

(19)

where

$$\begin{aligned} \begin{aligned} \gamma _0&=-16\sigma _{\omega }^4n\bar{m}\,, \quad \gamma _1=\frac{8\pi n\sigma _{\omega }\bar{N}\bar{\omega }\bar{\varepsilon }M}{\sqrt{2\pi }C}-16\sigma _{\omega }^4n\bar{m}\\ \gamma _3&=-\frac{4n^2\pi \sigma _{\omega }\bar{N}^2\bar{\varepsilon }^2M}{\sqrt{2\pi }CK}-\frac{n^2\bar{N}^2\bar{\varepsilon }^2}{K}+\frac{8n^2\pi \sigma _{\omega }\bar{N}\bar{\omega }\bar{\varepsilon }\bar{m}}{\sqrt{2\pi }C}\\ \gamma _4&=\frac{4n^2\pi \bar{m}\sigma _{\omega }\bar{N}\bar{\varepsilon }(2\sigma _{\omega }^2-\bar{\omega }^2)}{\sqrt{2\pi K} C}\,,\quad \gamma _5=\frac{4n^3\pi \bar{m}\sigma _{\omega }\bar{N}^2\bar{\varepsilon }^2}{\sqrt{2\pi }CK}\\ \gamma _6&=\frac{n^3\pi \bar{N}^3\bar{\varepsilon }^3\sqrt{K}}{2\sqrt{2\pi }C\sigma _{\omega }K^2}\,,\quad c_1=\sqrt{M+n\bar{m}N_c} \end{aligned} \end{aligned}$$

Notation

$A =$ the displacement amplitude of the footbridge at the steady-state;

$A(t) =$ the amplitude of the displacement of the footbridge;

$C =$ damping of the first lateral mode of the footbridge;

$\bar{F}^{osc,N} =$ the mean of the maximum amplitudes of the normal component of the oscillatory forces generated by the pedestrians;

$F^y(t) =$ the lateral modal force generated by the pedestrians;

$G =$ the mean amplitude of the lateral force induced by a pedestrian;

$K =$ stiffness of the first lateral mode of the footbridge;

$M =$ modal generalized mass of the first lateral mode of the footbridge;

$M_p(t) =$ the modal mass of the pedestrians;

$N =$ the total number of pedestrians on the footbridge;

$N_c =$ the critical number;

$U_y(t) =$ lateral modal displacement of the footbridge;

$\ell (x) =$ the footbridge’s width at position;

$m_p(t,x) =$ the local mass of pedestrians;

$m_{1p} =$ the average mass of a pedestrian;

$q_{OC_i}^x(t) =$ the position of pedestrian i on the footbridge and $\theta _i$ is the angle between and ;

$\ddot{u}^{cor,y}(t) =$ the Coriolis acceleration;

$\ddot{u}^{ent,y}(t) =$ the acceleration of the non-inertial frame of reference with respect to that generates a d’Alembert force [18];

$\ddot{u}^{rel,y}_i(t) = \ddot{u}^{osc,y}_i(t) =$ the acceleration with respect to due to the pedestrian’s oscillations about his trajectory resulting from the natural way of walking projected on the y-axis;

$\ddot{u}^{tr,y}_i(t) =$ the part of $\ddot{u}^{ent,y}(t)$ independent from the bridge motion;

$\varPsi _s(t) =$ the instantaneous phase of the lateral modal displacement of the footbridge;

$\varepsilon =$ quantifies a pedestrian’s sensitivity to the bridge’s acceleration;

$\bar{\varepsilon }_i =$ quantifies the influence of the bridge’s displacement on each pedestrian;

$\eta (x,t) =$ the pedestrians’ density at position x and time t;

$\phi ^{acc}(x,t) =$ the total phase of the lateral force induced by the pedestrians at position x and time t when they are sensitive to the bridge’s acceleration;

$\phi ^{dis}_i(t) =$ the phase of the lateral force induced by the ith pedestrian at time t when pedestrians are considered to be sensitive to the bridge’s displacement;

$\psi _1 =$ the first lateral modal shape of the footbridge;

$\omega (x) =$ the lateral angular frequency for free walking;

$\omega _i =$ the natural frequency of the pedestrian;

$\omega _{sync} =$ the frequency of synchronization;

$\omega _T =$ the modal frequency of the system crowd-structure