## 6 Conclusions  
This study investigated the application of DNN algorithms to estimate internal forces, with two design variables, namely bracing moments (\(M_{br}\)) and bracing force (\(F_{br}\)), respectively. Based on this study, DNN-based prediction models were proposed for estimating the internal forces acting along the bracing elements in steel I-girder bracing systems. The conclusions are summarized below. * This study constructed structural response data based on three-dimensional geometrically nonlinear analysis with imperfections. The structural response data include the bracing moments (\(M_{br}\)) and bracing forces (\(F_{br}\)) acting on the bracing elements at the states in which LTB behavior occurs. This study introduces critical imperfections at the target bracing element to ensure that the maximum bracing internal forces occur in the bracing elements of the steel I-girder bracing systems. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Model** &
\begin{tabular}{c} **Hidden** \\ **Layers** \\ \end{tabular} & **Neurons** & **Epochs** & \(\mathbf{R^{2}}\) & **RMSE** & **MSE** & **MAE** \\ \hline H4N50E200 & 4 & 50 & 200 & 0.988 & 65.3 & 4267.5 & 45.3 \\ H4N100E100 & 4 & 100 & 100 & 0.995 & 35.8 & 1284.7 & 24.2 \\ H4N100E200 & 4 & 100 & 200 & 0.993 & 41.0 & 1683.5 & 26.0 \\ H4N200E100 & 4 & 200 & 100 & 0.995 & 35.8 & 1284.0 & 23.1 \\ H4N200E200 & 4 & 200 & 200 & 0.962 & 122.0 & 14891.1 & 68.8 \\
**H5N50E100** & **5** & **50** & **100** & **0.997** & **25.4** & **643.8** & **15.2** \\ H5N50E200 & 5 & 50 & 200 & 0.993 & 41.7 & 1736.5 & 29.3 \\ H5N100E100 & 5 & 100 & 100 & 0.995 & 35.4 & 1256.4 & 22.7 \\ H5N100E200 & 5 & 100 & 200 & 0.995 & 35.8 & 1284.4 & 21.4 \\ H5N200E100 & 5 & 200 & 100 & 0.992 & 44.0 & 1931.8 & 24.7 \\ H5N200E200 & 5 & 200 & 200 & 0.995 & 35.3 & 1249.3 & 20.2 \\ \hline \hline \end{tabular} Note that the best prediction model for \(F_{br}\) is H5N50E100.  
\end{table} 
Table 10: _Cont._  
Figure 13: Comparisons of predicted bracing forces (\(F_{br}\)) of training and validation data sets: (**a**) training data and (**b**) test data. * The bracing moments obtained from FEA were compared to the design equation stipulated in the AISC provisions [1]. The FEA results indicate that the design provisions provide considerably conservative design values for torsional bracing steel I-girder systems, regardless of the types of loading conditions. The built-up sections provide more conservative results compared to the W-sections. It may be concluded that the design bracing moments computed by AISC [1] provide conservative design values when it has wide flange sections, compared to the girder sections that have relatively smaller flange widths. * This study proposed DNN models to estimate the bracing moments and forces acting along the bracing elements. Based on the parametric study of the hyperparameters, such as the hidden layers, neurons, and epochs, several DNN models, which exhibit the best prediction results, were proposed. The accuracy of the presented DNN models was evaluated based on four different statistical performance indices (i.e., RMSE, MSE, MAE, and R\({}^{2}\)). For estimating bracing moments, the DNN models require five hidden layers with 200 neurons and 200 epochs to ensure the higher accuracy of the models. For bracing force estimations, the DNN models require five hidden layers with 50 neurons and 100 epochs to show the best prediction results compared to other models. * Although the bracing forces have minor effects on the bracing internal forces, it was concluded that using a prediction model for bracing forces may lead to a more accurate prediction of the internal forces acting on the bracing elements. Furthermore, the proposed models can be used for preliminary design phases to determine reasonable bridge layouts. It is recommended that the simultaneous use of the DNN models for bracing moments and forces may provide a more accurate estimation of the bracing internal forces. * The proposed DNN models were constructed for twin steel I-girder bracing systems.