## 4 Numerical Validation  
A simply supported beam [15] with a span of 40 m is used for the simulation. Its mass per meter is \(\overline{m}=2400\) kg/m. Vertical flexural stiffness is \(EI=9.96\times 10^{9}\) Nm\({}^{2}\). Damping ratio is \(\zeta=1.0\%\). Fundamental frequency of the beam is 2.0 Hz. Parametric values of pedestrians are listed in Table 1.  
To contrast the effect of crowd size and step frequency, the dynamic properties under different crowd sizes and step frequencies are analyzed. When the mass ratio \(M_{p}/M_{s}\) between crowd and structure increases from 0.1 to 0.5 under same step 1 Hz in Figure 4, the frequency approximately decreases from 1.985 Hz to 1.92 Hz and the damping ratio also shows an obvious change in Figure 4b. However, the change of step frequency under the same mass ratio \(M_{p}/M_{s}=0.5\) has a smaller effect on the model parameters. Figure 4c gives the detail of damping ratio under the \(M_{p}/M_{s}=0.1\) and \(f_{s}=1\) Hz, which shows that the structure damping ratio is periodically changed along with walking gait. The frequency in Figure 4a also shows this effect and model property is disturbed by periodic walking gaits. The crowd size has a more significant impact on model properties than the walking excitation frequency. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Values** & **References** \\ \hline \(m\) & 80 kg & \\ \(L_{0}\) & 1.0 m & \\ \(k_{leg}\) & 20 kN/m & [4] \\ \(\xi_{leg}\) & 8\% & \\ \(L_{s}\) & 0.7 m & [16] \\ \(A_{xx}\) & 0.005 m/s & [14] \\ \(A_{z}\) & 0.002 m & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the crowd. Figure 4: Effect of step frequency and crowd size on dynamic properties: (**a**) the larger mass ratio \(M_{p}/M_{s}\) results the lower natural frequencies of structure, which is altered following the step rate; (**b**) the larger mass \(M_{p}/M_{s}\) results the larger damping of structure; (**c**) damping of occupied structure is altered following the step rate 1 Hz under the mass ratio \(M_{p}/M_{s}=0.1\). The dynamic responses in mid-span are given in Figure 5. Under the same walking frequency \(f_{\rm s}\) = 1.0 Hz, the change of mass ratio between crowd and structure has a tiny effect on vibration amplitude. However, the variation of step frequency has an obvious impact on structural responses. Under the walking frequency \(f_{\rm s}\) = 2.0 Hz, which is the same as the structural natural frequency, the acceleration response is rapidly amplified, as shown in Figure 5b. This demonstrates that significantly varying the frequency of excitation from the structural natural frequency is a more efficient method for alleviating excessive vibration, than by limiting the crowd size. In order to explore the effect of crowd size, crowd mass was linearly increased from zero to half of the structure mass over 60 s. Figure 6 gives the frequency and damping ratio under the resonance excitation frequency \(f_{\rm s}\) = 2.0 Hz. Along with the increase of mass ratio between crowd and structure, crowd induced frequency approximately decreases from 2 Hz to 1.92 Hz and damping ratio approximately increases from 1% to 7%. It is noted that the both dynamic response displacement and acceleration amplitudes (Figure 7) increase along with the increase of mass ratio between crowd and structure under resonance excitation frequency. Thus it can be seen that the increase of crowd size under resonance excitation has the most prominent effect on structural dynamic properties.