# Finite Element Models of a Benchmark Footbridge
## 3 Finite Element Models
### Modelling Assumptions  
Both long-term effects on the concrete and the effect of the deck post-tensioning on the modal results were neglected, under the assumption of linear behavior of materials and perfect bond with tendons, since the variations in the deformed equilibrium configuration do not significantly modify the dynamic properties of the structure [35].  
The real variable cross-sections of the deck, made complex by the interior holes (see Figure 2), were simplified through the definition of five equivalent rectangular sections, whose tensor of moments of inertia were provided as input in Ansys. The dimensions \(b\) and \(h\) of the equivalent cross sections are chosen to match the moments of inertia of the real cross section (\(I_{x}=bh^{3}/12\), \(I_{y}=hb^{3}/12\)). The effective area (\(A_{\text{eff}}=bh\)) of the equivalent sections is larger than the real one (\(A_{\text{real}}\)), due to the presence of the holes. Table 3 shows the dimensions of the equivalent sections, referring to the pier names (from pier P3 to pier P10) shown in the elevation view (Figure 1).  
The elements of the arch, spandrels and piers are assigned tubular steel cross sections. Table 4 summarizes the dimensions of the assigned cross sections, where \(r_{1}\) and \(r_{2}\) are the internal and external radius, respectively.  
Due to the difference in the effective area of the deck cross-sections, a reduction in the density of the concrete was applied in order to preserve the total mass. Table 5 reports the implemented concrete properties. Note that the value of \(\rho_{\text{eff}}\) also accounts for the additional contribution due to the railings, the utilities and the wearing surface. Similarly, the density of structural steel was modified to account for the concrete infilling (Table 6).  
In both models, the deck and the pier nodes at every pier/spandrel location are connected using rigid body constraints between the centroid of the deck and the two upper nodes of every Y-shaped pier or spandrel (three nodes in total). Thus, all degrees of freedom (six DOFs) are constrained, in accordance with the bolted connection between deck and pipes.  
\begin{table}
\begin{tabular}{c c c c c} \hline  & **Arch** & **Pier** & **Y-Shaped Pier** & **Spandrel** \\ \hline \(r_{1}\) (m) & 0.143 & 0.122 & 0.1 & 0.074 \\ \(r_{2}\) (m) & 0.160 & 0.135 & 0.11 & 0.085 \\ \(A_{\text{s}}\) (m\({}^{2}\)) & 0.017 & 0.01 & 0.007 & 0.005 \\ \(A_{c}\) (m\({}^{2}\)) & 0.064 & 0.047 & 0.031 & 0.017 \\ \hline \end{tabular}
\end{table}
Table 4: Tubular steel cross sections of arch, spandrels and piers [36].  
\begin{table}
\begin{tabular}{c c c c c c} \hline  & **P6–P7** & **P5–P8** & **P4–P9** & **P3–P10** & **Legs** \\ \hline \(h\) (m) & 0.523 & 0.5349 & 0.5484 & 0.56 & 0.4826 \\ \(b\) (m) & 3.68 & 4.15 & 5.05 & 6.70 & 3.20 \\ \(A_{\text{eff}}\) (m\({}^{2}\)) & 1.93 & 2.22 & 2.77 & 3.75 & 1.54 \\ \(A_{\text{real}}\) (m\({}^{2}\)) & 1.46 & 1.74 & 2.24 & 3.21 & 1.28 \\ \(I_{x}\) (m\({}^{4}\)) & 0.04 & 0.05 & 0.07 & 0.10 & 0.03 \\ \(I_{y}\) (m\({}^{4}\)) & 2.17 & 3.20 & 5.90 & 14.02 & 1.31 \\ \hline \end{tabular}
\end{table}
Table 3: Equivalent rectangular cross-sections of the concrete deck [36].