## 3 Finite Element Analysis (FEA)
### Outlines of FEA  
In this study, geometrically and materially nonlinear analysis with imperfection (GM-NIA) was conducted using the ABAQUS program [23] with the Riks method. The bridges are modeled with a three-dimensional shell element (S4R) for the I-girders and a three-dimensional truss element (T3D2) for z-type bracing elements, as shown in Figure 1. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Type** & **Section** & \(\mathbf{b_{f}}\) & \(\mathbf{d}\) & \(\mathbf{t_{f}}\) & \(\mathbf{t_{w}}\) & \(\mathbf{L}\) & **No. of** \\  & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **Bracings** \\ \hline  & W920 \(\times\) 201 & 305 & 902 & 20.6 & 15.9 & & \\  & W920 \(\times\) 420 & 422 & 943 & 39.7 & 22.2 & & \\ W-section & W1000 \(\times\) 314 & 298 & 1000 & 36.5 & 19 & & \\  & W1000 \(\times\) 371 & 400 & 1000 & 36.5 & 19 & & \\  & W1100 \(\times\) 390 & 400 & 1100 & 36.5 & 20.6 & & \\  & W1100 \(\times\) 343 & 400 & 1090 & 31.8 & 17.5 & \multirow{2}{*}{25\(d\)} & \multirow{2}{*}{\(0,1,3,5,7\),} \\  & BT-1 & 667 & 2000 & 62 & 32 & & \\  & BT-2 & 500 & 2000 & 46 & 32 & & \\ Built-up & BT-3 & 400 & 2000 & 36 & 32 & & \\ section & BT-4 & 933 & 2800 & 86 & 46 & & \\  & BT-5 & 700 & 2800 & 64 & 46 & & \\  & BT-6 & 560 & 2800 & 52 & 46 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dimensions of I-girder bracing systems in this study (Unit: SI). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Elements** & **_E (MPa)_ & **_v_** \\ \hline Flange & & \\ Web & 205,000 & 0.3 \\ Torsional bracings & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material properties used in FEA models. \begin{table}
\begin{tabular}{c c c} \hline \hline \(\beta_{b}\) & \(\mathbf{\alpha}\) & \(\mathbf{\beta}\) \\ \hline \hline \(\alpha\times\beta\) & \(0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,\) & \(\beta=\frac{1}{n}\frac{\big{(}M_{2,b_{b}}^{2}-M_{2,b_{s}}^{2}\big{)}L}{E_{I}}\) \\  & where, \(M_{n,l_{b}}=\min\big{(}M_{y},M_{cr,l_{b}}\big{)}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Torsional stiffness of bracings (\(\beta_{b}\) ) used in the FEA model. The girder models are supported with a fork-end condition for each end, which means that it is simply supported for flexure, fixed for torsion, and free for warping at each end, as illustrated in Figure 2. For loading conditions, we apply various loading types to estimate internal forces because of the LTB behavior in steel I-girder bracing systems. Hence, various loading conditions with pure bending and moment gradients were considered in the FEA models. For pure bending conditions, equal end moments are applied at each end of the girder to ensure that the same moment is applied in the entire girder section. For the moment gradient condition, two different types of loading conditions are applied in the models, which comprise concentrated loads at the bracing points of the mid-span and bracing points near the quarter span. Therefore, the target segments between brace points, where the maximum bending moments are created, are near the mid-span for pure bending conditions and near the mid-span or the quarter span for moment gradients, as shown in Figure 3. The target segments with the maximum bending moment are critical segments for realistic bridge design. Therefore, in this study, the bracing forces generated due to the LTB are obtained at brace points near the target segments. Figure 1: FEA model used in this study: (**a**) FEA model; (**b**) cross-section of I-girder; (**c**) plane view of the bracing system; (**d**) top view of bracing system. Based on an earlier report [8], the initial imperfection significantly influences the LTB behavior and internal forces produced in the torsional bracing. Furthermore, it was concluded that the critical imperfection acting on the brace locations is essential for estimating accurate bracing forces and realistic lateral torsional behavior in the steel-I girder bracing systems.