A Stability Criterion Model of Flexible Footbridges under Crowd-Induced Vertical Excitation

The excessive vibration caused by crowds walking across footbridges has attracted great public concerns in the past few decades. This paper presents, from considering the dynamic characteristics of the bipedal crowd model, a new stability limit criterion based on the bipedal excitation model. The stability limit can be used to estimate the upper boundary of crowd size. In addition, the dynamic stable performances of a structure, under a certain walking crowd size, can be predicted by the stability criterion. This proposed mechanism provides an alternative comprehension how crowd excite the excessive sway motion with a large-span structure.


Introduction
These structures with long-spans have been become prevalent all over the world and their dynamic vibrations are the crucial concern in its servicing period.A structure with the longer span is prone to trigger an excessive sway motion.The evidences from some vibrational accidents and investigations have showed how the bridge becomes flexible along with the extension of span (Newland, 2003;Fujino and Siringoringo, 2013).Some controlling techniques including the passive control (Soong and Spencer, 2002;Fan et al., 2010), active control (Nyawako and Reynolds, 2007) and semi-active control (Jalili, 2002) methods have been used to mitigate the effect of crowd-induced vibrations (Hudson and Reynolds, 2012).Moreover, some crowd-induced excited models (Matsumoto et al., 1978;Roberts, 2005;Piccardo and Tubino, 2008) are proposed to investigate the lateral resonance mechanism of flexible footbridges.Some further studies considering dynamic interaction between human and structure such as the kinetic crowd biomechanical model (Carroll et al., 2013)  In this paper, a stability limit criterion with the vertical vibration is studied based on the crowd excitation mechanism with the modeled bipedal pedestrians (Qin et al, 2013).Firstly, the dynamic equilibrium equation of a structure is established by considering the vertical ground reaction force (GRF) between footholds and pavement.In addition, an assumed uniform distribution of the walking crowds is used to calculate the crowd size by employed the Taylor Expansion.Finally, the stability limit is identified and the upper boundary of crowd size for a stable vibration can be estimated according to the stability limit.Some parameters about the stability limit criterion are analyzed and a numerical example is applied to assess the stable state of a footbridge under dynamic walking crowds.

Dynamic excitation mechanism
A pedestrian, as the basic unit of a pedestrian flow, is modeled with the bipedal biomechanics model of mass-spring-dampers (Qin et al., 2013) as shown in Fig. 1.The structure is simulated by a simply supported Euler-Bernoulli beam with a uniform section.LB is the span length.EI means the flexural stiffness of the beam and is the mass percent unit length along in the longitudinal direction.The left end of the beam is defined as the origin of a planar coordinate system x-0-y.The q th ( = 1, ⋯ , ) pedestrian from the crowd size is simulated by the bipedal model with the lump mass ( ) .( ) and ( ) are the leading and trialing leg stiffness coefficients, respectively.( ) and ( ) are the leading and trialing leg damping coefficients, respectively.( ) and ( ) are the leading and trialing footholds positions in longitudinal direction, respectively. ( )and ( ) are the longitudinal and vertical positions of pedestrian center of mass (CoM), respectively.
( ) means the intersection angle between the beam and leading leg; means the intersection angle between the beam and trialing leg.
( ) is the ground reaction force between the leading foothold and pavement along the leg axial.
( ) is the ground reaction force between the trialing foothold and pavement along the leg axial.Where, subscripts 'l' and 't' denote leading and trialing legs, respectively.There is an assumption that there is no slip between feet and ground.
The dynamic equilibrium equation of the beam can be obtained as where δ(∎) is the Dirac function; is damping coefficient the beam.( , ) is the vertical displacement of the beam at the position x and time point t and its expression is where ( )is the modal function satisfying boundary condition and ( )is the generalized co-ordinate of the i th mode for the beam; n is total modal number.
Defining ( ) = sin( ⁄ ) and substituting the formula of ( ) into the Eq.(1) yields Multiplying ( ) both sides of the Eq. ( 3) and integrating along the whole span of the beam, then divided by 2 ⁄ , one can obtains where = ( ⁄ ) ⁄ and are the i th circular frequency and damping ratio of structure, respectively; = 2 is structural damping coefficient, = 2 ⁄ is modal mass.
The ground reaction forces from the leading and trialing footholds are obtained by where ( ) is the relax length of leg; ( ) and ( ) are the lengths of leading and trialing legs, respectively; ( ) and ( ) are axial velocities of leading and trialing legs, respectively.The leading and trialing leg where, and are the first modal damping ratio and circular frequency of footbridge.The series of the Eq. ( 19) is simplified by employing the Taylor expansion so that where ( ) means the infinitesimal of higher order based on the assumption of ≫ 1.
Substituting Eq. ( 20) into the Eq. ( 19), the dynamic equation becomes where and ̅ are respectively the frequency and damping ratio of structure including the walking crowd contribution.
where, is structural fundamental frequency., and are mass, damping parameters and external force, respectively.Their expressions are listed as The decrease processing is due to the positive damping from crowd-structure system.In the second stable boundary case of the = , the damping of crowd structure system is zero and corresponding stable response only from the crowd excitation.In the third case of > , the response behaves an enlarged effect.This is due to that the oversize crowd induces a negative effect.The walking velocity is changed from 1.0m/s to 1.5m/s.It is noted that the damping of structure under crowd is decreased along with the increase of crowd size.In addition, the damping is also decreased along with the increase of walking velocity.These show that the larger crowd size or faster speed can deteriorate the damping dissipating performance.The crowd size with stability limit is increased along with the lower walking speed.
The effect of crowd size on the frequency of the structure is plotted in the Fig. 6.Except for the body mass mp, other parameters of pedestrians are defined with the Tab. 1.The larger crowd size make the structure become more flexible because of its smaller frequency.In addition, the larger body mass also leads to the lower frequency.The crowd size with the Eq.(25) corresponding with the stability limit shows the upper boundary of structural dynamic stability is related with the structural damping, leg damping and walking gaits.
The larger structural damping would support a larger crowd size.This is due to the better dissipating performance.However, the increase of leg damping leads to the decrease of the structure carrying capacity.The larger leg damping may be caused by the faster walking velocity (Hong et al., 2013), which tend to excite a more drastic response.However, these results are deduced according to the bipedal excitation mechanism.Some experiments or investigations about this excitation theory are needed in the future.

Conclusion
In this paper, a new excitation mechanism is proposed to motivate the vertical vibrational motion caused by crowds walking across slender footbridges, based on a bipedal walking model including the crowd dynamic characteristics.The footbridge is studied as a crowd excited dynamic system and a stability boundary limit identified, depending on the ratio between the structural and crowd damping, on the ratio between step and leg lengths.In addition, the frequency and damping of structure can also be identified and they are deteriorated by walking crowd.The excitation mechanism opens a window how walking crowd influence the vibrational performances of these large-span structures.
have been recommended to investigate the lateral vibration mechanism of footbridges.However, the investigations about the vertical structural vibration under crowd-induced excitations are rare.Zhou et al. (2006 and 2016) and Yang et al. (2013) studied the vertical dynamic characteristics of structure under a modeled human oscillator.Zivanovic (2015) reviewed the experimental and numerical developments of lightweight structures under human actions.Qin et al. (2013 and 2014) studied dynamic performances of footbridge under a walking biomechanical bipedal pedestrian model based on a constant walking energy level.However, the vertical vibrational stability of structure under dynamic crowds is none.
( ) is the vertical displacement of CoM due to pedestrian-self vibration. ( )is the vertical displacement of CoM due to structural vibration.

Fig. 5 .
Fig. 5.The effect of crowd size on the damping of structure

Fig. 6 .
Fig. 6.The effect of crowd size on the frequency of structure