# Load Models Representative of Brazilian Actual Traffic in Girder-Type Short-Span Highway Bridges
## 8 Load Models Representative of the Actual Traffic  
Once the target values of the effects are defined, the physical and geometric configurations of load models that adequately represent the real traffic were sought by optimization via exhaustion, so that, with the loading of these fictitious models statically applied to the structures considered, they reproduce approximately the target values at the critical cross-sections of each one, automatically including the dynamic amplification. Defining geometric features such as number of axles, wheelbases, transverse distances between tires, number and width of the design lanes, and considering the values of concentrated and uniformly distributed loads as optimization parameters, a scan is performed in all possible combinations inside the domain of each variable, discretizing the total weight of the design vehicle (concentrated forces) in multiples of 50 kN and the distributed in \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Original** & \(\mathbf{d_{12}}\) & \(\mathbf{d_{23}}\) & \(\mathbf{P_{1}}\) & \(\mathbf{P_{2}}\) & \(\mathbf{P_{3}}\) \\
**Vehicle** & **(m)** & **(m)** & **(kN)** & **(kN)** & **(kN)** \\ \hline
2S3-S & 4.63 & 1.25 & 217.85 & 213.61 & 213.61 \\
2S3-L & 8.11 & 1.25 & 188.33 & 181.54 & 181.54 \\ \hline
3S3-L & 7.96 & 1.25 & 298.72 & 213.64 & 213.64 \\ \hline
3T4 & 7.30 & 4.74 & 268.75 & 206.82 & 217.90 \\ \hline
3T6 & 11.06 & 8.84 & 238.17 & 367.36 & 190.59 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: GVWs and wheelbases of standardized 3C equivalent vehicles for each truck class. multiples of 0.5 kN/m\({}^{2}\). Geometric information about the bridges' superstructure and the target values (Table 7) are also provided. The load models sought consisted of a vehicle composed of some concentrated forces, as well as distributed loads. The variable loads do not act in a favorable fashion; therefore, forces distributed by area only act in the regions of the deck that generate adverse effects to the analyzed element (in this case, the right girder). If there is more than one design lane, the effective width of the deck (region where the influencing surfaces only generate unfavorable effects) is divided so that the first lane is located at the most severe transversal position for the structural element considered. The remaining area of the deck is considered for the remaining width--vehicles do not load it. For the deck type shown in Figure 2a, a weight factor (\(W_{ij}\)) is defined according to the structural system and the internal force [24]:  
\[W_{ij}=W_{j}^{struct.}W_{ij}^{effect} \tag{6}\]  
Sub-indexes \(i\) and \(j\) refer, respectively, to internal forces and structures. Values adopted for \(W_{j}^{struct}\) were 2.0 for simply supported and continuous systems and 1.5 for cantilevers. For \(W_{ij}^{effect}\), 2.0 for moments and 1.5 for shear forces were adopted. The objective function is the weighted error: the sum of all relative differences between each effect generated by the candidate load model, and the corresponding target value. According to Table 7, to compose the objective function, there are 24 effects in 12 structures. Steps for seek are defined as follows [11]:  
* calculate each internal force \(i\) generated in each structure \(j\) by the load model \(k\) (\(E_{ijk}\));
* calculate the relative difference (\(\epsilon_{ijk}\)) between \(E_{ijk}\) and the corresponding target value of the effect (\(T_{ij}\)): \[\epsilon_{ijk}=\left|\frac{E_{ijk}-T_{ij}}{T_{ij}}\right|\] (7)
* calculate the weighted error for load model \(k\): \[\overline{\epsilon}_{k}=\frac{\sum_{i}\sum_{j}W_{ij}\epsilon_{ijk}}{\sum_{i} \sum_{j}W_{ij}}\] (8)  
For each set of geometric features, the load model with the smallest weighted error (m\({}^{\text{th}}\), 1 \(\leq\) m \(\leq\) NLM) is chosen as the most appropriate (NLM: number of load models). Several hypotheses were tested; each one considered a different set of geometric characteristics and search restrictions in the relative and absolute values of the loads per design range.