# Load Models Representative of Brazilian Actual Traffic in Girder-Type Short-Span Highway Bridges
## 6 Analysis of the Vehicle-Pavement-Structure Interaction
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Bridge**} & \multirow{2}{*}{**Effect**} & \multirow{2}{*}{
\begin{tabular}{c} **Value** \\ **(kN/N)** \\ \end{tabular} } & \multirow{2}{*}{**Scenario**} & \multicolumn{4}{c}{**Vehicle n\({}^{*}\) 1**} & \multicolumn{4}{c}{**Vehicle n\({}^{*}\) 2**} & \multicolumn{4}{c}{**Vehicle n\({}^{*}\) 3**} \\ \cline{4-14}  & & & & & & & & & & & & & & & & \\ \hline \multirow{2}{*}{S-10} & M+ & 587.4 & 3 & 1 & 2S3-L & 616.9 & 80 & 2 & 3C & 229.7 & 80 & - & - & - & - \\  & V & 514.5 & 3 & 1 & 2S3-S & 693.9 & 80 & 2 & 2S3-L & 411.6 & 80 & - & - & - & - \\ \hline \multirow{2}{*}{S-20} & M+ & 2100.2 & 1 & 1 & 2S3-L & 554.0 & 80 & 2 & 3S3-L & 632.6 & 60 & - & - & - & - \\  & V & 693.7 & 8 & 1 & 3T4 & 530.4 & 80 & 2 & 2S3-L & 625.2 & 80 & - & - & - & - \\ \hline \multirow{2}{*}{S-30} & M+ & 4927.7 & 8 & 1 & 3T4 & 727.5 & 100 & 2 & 3I3 & 541.1 & 80 & - & - & - & - \\  & V & 6981.6 & 5 & 1 & 3S3-L & 699.8 & 80 & 2 & 3S3-L & 511.2 & 80 & 3 & 3C & 144.3 & 60 \\  & V & 882.0 & 8 & 1 & 3T4 & 638.1 & 100 & 2 & 2S3-L & 575.9 & 60 & 3 & 2C & 83.4 & 100 \\ \hline \multirow{2}{*}{C-10} & M+ & 569.6 & 5 & 1 & 3S3-L & 798.0 & 80 & 2 & 2S3-L & 299.5 & 80 & 3 & 2 CB & 121.2 & 60 \\  & V & 531.1 & 5 & 1 & 3S3-L & 689.3 & 100 & 2 & 3S3-S & 471.2 & 80 & - & - & - & - \\ \hline \multirow{2}{*}{C-20} & M+ & 2051.7 & 6 & 1 & 2S3-S & 596.2 & 100 & 2 & 3C & 236.9 & 80 & 3 & 3T6 & 620.3 & 100 \\  & V & 724.8 & 7 & 1 & 3T4 & 719.1 & 100 & 2 & 3S3-L & 528.1 & 60 & - & - & - & - \\  & M\({}^{-}\) & 1932.5 & 6 & 1 & 3T6 & 798.5 & 80 & 2 & 3T6 & 783.5 & 60 & - & - & - & - \\ \hline \multirow{2}{*}{C-30} & M+ & 3922.2 & 5 & 1 & 3T6 & 751.7 & 80 & 2 & 3S3-L & 669.8 & 80 & - & - & - & - \\  & V & 856.1 & 8 & 1 & 3T4 & 726.3 & 80 & 2 & 3T6 & 741.5 & 80 & - & - & - & - \\  & M\({}^{-}\) & 3208.8 & 4 & 1 & 3S3-L & 464.8 & 80 & 1 & 3T6 & 686.9 & 80 & 2 & 2I3 & 426.7 & 80 \\ \hline \multirow{2}{*}{C-40} & M+ & 6062.4 & 7 & 1 & 3T4 & 808.4 & 80 & 2 & 2S3-S & 436.1 & 60 & - & - & - & - \\  & V & 934.2 & 7 & 1 & 3T4 & 704.5 & 60 & 2 & 3T6 & 705.7 & 80 & - & - & - & - \\  & M\({}^{-}\) & 4710.6 & 3 & 1 & 3T6 & 838.2 & 80 & 1 & 3T4 & 573.7 & 60 & - & - & - & - \\ \hline F-2.5 & M\({}^{-}\) & 595.9 & 8 & 1 & 3S3-L & 881.4 & 80 & - & - & - & - & - & - & - \\ \hline F-5.0 & M\({}^{-}\) & 1745.0 & 4 & 1 & 3S3-L & 806.9 & 80 & 2 & 2S3-S & 710.7 & 80 & - & - & - & - \\ \hline F-7.5 & M\({}^{-}\) & 2473.6 & 5 & 1 & 2S3-L & 483.4 & 80 & 2 & 2S3-L & 473.2 & 80 & - & - & - & - \\ \hline F-10 & M\({}^{-}\) & 3350.0 & 8 & 1 & 3S3-L & 697.0 & 80 & 2 & 2S3-S & 440.3 & 80 & 3 & 3I3 & 449.8 & 80 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Vehicle configurations that generated the extreme static effects on the bridges. coupled system equations are formulated, including the presence of the rough pavement. The motion equation associated to the structure \(j^{\text{th}}\) vibration mode is written as:  
\[m_{ej}\ddot{u}_{ej}+c_{ej}\dot{u}_{ej}+k_{ej}\dot{u}_{ej}=\sum_{i=1}^{3}\varphi_{ jk}F_{ei}, \tag{1}\]  
where \(m_{e}\), \(c_{e}\) and \(k_{e}\) correspond to the generalized mass, damping, and stiffness parameters and \(\varphi_{jk}\) is the \(j^{\text{th}}\) eigenvector component at node \(k\) of the structure where there is a tire contact point. At each vehicle-structure contact point, interaction forces of elastic (\(\dot{f}_{ei}\)) and damping (\(\dot{f}_{di}\)) components are generated as functions of the relative displacement and velocity between the axle mass \(m_{pi}\) and the structure contact point, considering the pavement roughness profile (\(u_{r}\)), as shown in Figure 6b.