# Load Models Representative of Brazilian Actual Traffic in Girder-Type Short-Span Highway Bridges
## 4 Structural Models
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Element**} & \multirow{2}{*}{**Dimension**} & \multicolumn{3}{c}{**Supply Supported**} & \multicolumn{3}{c}{**Cantilever**} & \multicolumn{3}{c}{**Continuous**} \\  & & \multicolumn{3}{c}{**Span Lengths (m)**} & \multicolumn{3}{c}{**Span Lengths (m)**} & \multicolumn{3}{c}{**Span Lengths (m)**} \\ \cline{3-16}  & & **10** & **20** & **30** & **40** & **2.5** & **5.0** & **7.5** & **10** & **10** & **20** & **30** & **40** \\ \hline \multirow{2}{*}{Gider} & \(h\) & 1.00 & 2.00 & 3.00 & 3.50 & 0.90 & 1.80 & 2.50 & 3.00 & 0.90 & 1.80 & 2.50 & 3.00 \\  & \(b_{w}\) & 0.35 & 0.40 & 0.45 & 0.50 & 0.35 & 0.40 & 0.45 & 0.50 & 0.35 & 0.40 & 0.45 & 0.50 \\ \hline \multirow{2}{*}{Cross beam} & \(h_{T}\) & 0.80 & 1.60 & 2.40 & 2.80 & 0.70 & 1.40 & 2.00 & 2.40 & 0.70 & 1.40 & 2.00 & 2.40 \\  & \(b_{T}\) & 0.30 & 0.35 & 0.40 & 0.45 & 0.30 & 0.35 & 0.40 & 0.45 & 0.30 & 0.35 & 0.40 & 0.45 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Dimensions of the structural elements. Figure 3: (**a**) Finite element model for the 30-m span simply supported bridge (S-30). (**b**) Equivalent grid model. (**c**) Stick model for dynamic analyses. Figure 2: (**a**) RC \(\Pi\) cross section adopted (units: cm) and (**b**) structural schemes. stick model shown in Figure 3c, which represents the bridge in the analyses of the vehicle-pavement-structure dynamic interaction (Section 6). For each bridge structure, a Finite Element Model (FEM) was developed with four node quadrilateral shell elements whose formulation is based on Mindlin/Reissner theory, which accounts for shear behavior, to represent girders, slab, and crossbeams (Figure 3a). In the grid model the girders and cross beams are modeled with spatial frame elements. The geometric properties of the girders are calculated for a T cross-section, considering the portion of the slab that contributes to the stiffness. The effective width of the flange is defined by geometric conditions and the rules given by the Brazilian standard for design of concrete structures [34]. Girders and cross beams have a constant cross-section. Diagonal and vertical elements were modeled with a polygonal section and dimensions defined by the adjustment to the shell model. Diagonal elements arranged in \"X\" in the grid plane were created to simulate the deck slab; vertical and diagonal elements forming a \"X\" on the planes of the end cross beams were created to simulate the eccentricity of the girders' supports in relation to their center of mass. The adopted modeling procedures and the technique by which a 3D-grid model as the one described above is developed in adjustment to a 3D numerical model composed of shell elements were validated by comparison to experimental results in [PERSON] et al. [35]. Given the transverse configuration of the bridge (IT cross-section) and the presence of cross beams, an analytical solution for the transverse load distribution from the Courbon-Engesser method could be considered, as recently applied in [36], since its fundamental hypotheses are expected to be valid. In fact, comparisons between analytical and numerical (grid model) load factor distribution showed a maximum difference of only 5.2% for load eccentricities on the unfavorable side for the considered girder. However, the analytical solution for the longitudinal structural scheme considering a simple beam does not provide a faithful representation of the bridge stiffness. This way, the described numerical models were applied for static and dynamic analyses. Table 3 presents the comparison between results from the detailed shell model--Figure 3a--and the equivalent grid model--Figure 3b--in terms of dead loads of the structures, midspan displacements due to the passage of a class 3C truck (Figure 4) along the center line of the bridge, and the right girder (G1), as well as the natural frequencies in vertical bending and longitudinal torsion modes of simply supported bridges (S).