# Finite Element Models of a Benchmark Footbridge
## 5 Dynamic Response under Jumping Pedestrian Excitation  
The proposed application described in this section aims at presenting an example of simulation of dynamic response under human-induced excitation. In addition, it could be intended as a tentative validation of the dynamic behavior of the footbridge against experimental measurements of the acceleration response due to jumping pedestrians. Among the few experimental tests reported in the literature, which usually refer to pedestrians running along the south-east leg (Section 2.5), the ones described by [PERSON] [20] are considered since they are described in more detail, and time histories of the vertical acceleration are provided. In particular, in the test analyzed here, eight pedestrians jump at 3 Hz at about one-quarter of the way along the south-east leg for 30 s.  
A limited number of pedestrian load models are available in the literature (e.g., [37,38]). The pedestrian jumping load model proposed in ISO 10137 [39] has been adopted for the following application. The load exerted by one jumping pedestrian is modelled as a perfectly periodic half-sine function:  
\[F(t)=\left\{ \begin{array}{cc}\alpha G\sin(2\pi f)&0\leq t\leq t_{c}\\ 0&t_{c}<t\leq T\end{array} \right\}, \tag{1}\]  
where \(\alpha=2.1-0.15f\) is the dynamic load factor of the first harmonic of the load, \(G\) is the pedestrian weight, assumed equal to 700 N as usually is usually performed in the literature (e.g., [9]), \(f\) is the excitation frequency, \(T=1/f\) is the period and \(t_{c}\) is the contact time, assumed equal to \(T/2\). In order to account for the fact that it is very difficult for people to jump in phase with each other, even if they try to intentionally excite the bridge by jumping [40], a random phase shift is assigned among the eight pedestrian loads: specifically, pedestrians are assumed to start jumping with a time shift randomly assigned between 0 and 0.33 s (where 0.33 s is the step period). Moreover, the load amplitude is further reduced by 20% to take into account a lower correlation between pedestrians [39]. The eight forces are applied at eight nodes of the S model around the section at a quarter of the south-east leg span. A full transient analysis is performed in Ansys. Damping is modelled according to the Rayleigh formulation, with mass and stiffness coefficients \begin{table}
\begin{tabular}{l c c c c c} \hline
**Ref.** & \(f_{\text{exp}}\) **[Hz]** & \(f_{\text{B}}\) **[Hz]** & \(e_{\text{B}}\) **[\%]** & \(f_{\text{S}}\) **[Hz]** & \(e_{\text{S}}\) **[\%]** \\ \hline  & 3.11 & 3.0869 & 0. & 3.1197 & 0.31 \\ [PERSON] and [PERSON] [12,31] & 3.17 & 3.1568 & \(-0.42\) & 3.1451 & \(-0.95\) \\  & 3.72 & 3.708 & \(-0.32\) & 3.7129 & \(-0.19\) \\ \hline  & 3.08 & 3.0869 & 0.22 & 3.1197 & 1.29 \\ [PERSON] [20] & 3.08 & 3.1568 & 2.49 & 3.1451 & 2.11 \\  & 3.75 & 3.708 & \(-1.12\) & 3.7129 & \(-0.99\) \\ \hline  & 3.00 & 3.0869 & 2.90 & 3.1197 & 3.99 \\ [PERSON] et al. [22] & 3.00 & 3.1568 & 5.22 & 3.1451 & 4.84 \\  & 3.65 & 3.708 & 1.59 & 3.7129 & 1.72 \\ \hline \end{tabular}
\end{table} 
Table 9: Comparison between experimental and numerical frequencies. equal to 0.164 and 0.000153, respectively, estimated to obtain the experimental value of damping ratio of 0.57% [21] on the first numerical mode, and on the mode corresponding to a cumulated participation, mass in the vertical direction of around 60%. Time step is set equal to 0.02 s.  
Figure 14 plots the numerical time history of the vertical acceleration at a quarter of the south-east leg in comparison with the one measured by [PERSON] [20]. The numerical response is much more regular than the experimental one. Actually, the perfectly periodic force model is not able to capture the inter- and intra-subject variability [41; 42] that characterize pedestrian excitation. Moreover, the applied force does not account for human-structure interaction and for the related damping effect induced by human bodies [42].