## 3 Numerical Modeling of Full-Scale Bridge
### External Cross-Frame  
The external cross-frames used in the prototype bridge were K-type, as shown in Figure 1). It comprised of WT 7 \(\times\) 21.5 rolled sections for the upper and lower chords and L 5 \(\times\) 3.5 \(\times\) 0.375 angles for the diagonals. It was assumed that one of the two girders was fully fractured at its mid-span, and truck live loads were applied to the damaged side girder. Under these damage and loading conditions, transverse bending can occur to transfer the applied loads from the damaged girder to the intact girder, as shown in Figure 4a. As the external cross-frames were attached to the girders, it was presumed that they could resist the deformation of the damaged girder and contribute to the load transfer. Figure 4b shows the expected section forces of the external cross-frames caused by transverse bending. High bending moments can occur in the chord members, and a compression or tension force can develop in the diagonal members. Figure 3: FE bridge model. Potential failure modes of the WT section under flexure are material yielding, lateral torsional buckling, and local buckling. These failure modes could be simulated with fine meshes and material inelasticity. Note that the full-scale bridge modeling was constructed in this study, so it would not be efficient to apply fine meshes to detect the local failure modes of the frames. Furthermore, the local failure such as the buckling of the frames could introduce numerical ill-conditioning and difficulty in conversing to a solution. In this study, therefore, these potential failure modes were not directly simulated in the FE analysis. Instead, the nominal flexural strength of the WT section was calculated based on AISC 360-16 [25] and used as an upper limit of the frame elements (FRAME3D) for the chord members. The WT 21 \(\times\) 21.5 used in the prototype bridge comprised of a compact flange and stem. The yield strength and elastic modulus of the WT section were 248 MPa (36 ksi) and 200 GPa (2900 ksi), respectively. The longest unhraced length (\(L_{b}\)) was 1943 mm (76.5 in.) in the upper chord member, which was less than the limiting laterally unhraced length (\(L_{p}\)) specified in the AISC 360-16 sect. F9, as indicated in Equation (1). Therefore, the flexural strength of the chord was governed by the yield of the entire section and was calculated as 28.7 kN-m (253.8 kip-in.) based on the plastic moment (\(M_{p}\)) in Equation (2). Once the section force of a flexural member reaches its plastic moment, a plastic hinge forms. Therefore, perfect plastic behavior was assumed in the FE simulations for the chord members beyond the plastic moment:  
\[L_{p}=1.76r_{y}\sqrt{\frac{E}{F_{y}}} \tag{1}\]  
\[M_{p}=F_{y}Z_{x} \tag{2}\]  
where \(r_{y}\) is the radius of gyration about the \(y\) axis, \(E\) is the elastic modulus, \(F_{y}\) is the yield strength, and \(Z_{x}\) is the plastic section modulus about the x(strong) axis. The diagonal members of the external cross-frame were modeled using connector elements (CONN3D2). The single angles used for the diagonals were under compressive or tensile forces and, therefore, could fail by tensile yielding, flexural buckling, and(or) torsional buckling. Similar to the chord members, the failure of the L 5 \(\times\) 3.5 \(\times\) 0.375 angles was simulated with nominal strength based on the AISC specification. The yield strength and elastic modulus of the diagonals were identical to those of the chord members. Figure 4: Load transfer by transverse bending. Therefore, diagonal tensile failure occurred at 488 kN (109.8 kips). The compressive strength of an angle was calculated according to the provisions specified in AISC 360-16 sect. E5. The longer legs of the angles were welded to the chord members. The length of the angle \(L\), was 1524 mm (60 in.). To account for the bending caused by load eccentricity, the AISC specifications allow the use of a modified effective slenderness, as indicated in Equation (3), for unequal leg angles.