## 4 Effect of External Cross-Frame on Bridge's Ultimate Behavior
### Section Forces of External Cross-Frame  
As described in Section 3.2, it was expected that the external cross-frame would contribute to the transfer of live loads from the fractured girder to the intact girder after one girder fractures. The analysis results shown in Figure 8 correspond to the expected behavior induced by this load transfer mechanism. The compressive force developed on the right-side diagonal, and the tensile force on the left diagonal. In the upper chord, a negative bending moment occurred where the right-side diagonal (dia_R) was connected, and a positive bending moment occurred where the left-side diagonal (dia_L) was connected, as shown in Figure 8a. Reverse bending moments occurred in the lower chord, as shown in Figure 8b. Figure 9a shows the maximum section force variations in Frame 2 along the damage level in Case 2. The bending moments normalized by the flexural strength \(M_{p}\) are plotted with blue and orange bars for the upper and lower chords, respectively. The compressive axial force normalized by the buckling strength \(P_{c}\), on the right-side diagonal (dia_R) is indicated by the gray bars. Figure 9b,c show the normalized section force variations of the chords and diagonals along the live loads after girder fracture. In Figure 9b, the solid blue and dashed orange lines indicate the bending moments of the upper and lower chords, respectively. The compressive force on the right diagonal (dia_R) is plotted with a solid line, and the tensile force on the left diagonal (dia_L) is plotted with a dashed line. As shown in Figure 9a, the section forces did not vary significantly until the bottom flange fractured; however, they increased as web fracture damage was applied. With the web fracture, the bending moment increased to 96% of the flexural strength in the upper chord. As the live loads increased, the normalized bending moment of the upper chord readily reached one and, therefore, it was presumed that a plastic hinge formed at that location. The compressive force on the right-side diagonal (dia_R), as shown in Figure 9c, increased and reached its buckling strength when the applied live load exceeded approximately five-times that of the HS-20 truck. At this stage, the cross-frame lost its resistance to laterally transferring live loads from the fractured girder to the intact girder. Therefore, the axial forces suddenly decreased, as shown in Figure 9b,c. This tendency conforms to the load-displacement behavior described in the previous section. Figure 10 shows the axial force variations of the diagonals for Cases 2-4 after the girder fracture. The dashed and solid gray lines indicate the axial forces of the left-side diagonals (dia_L) and right-side diagonals (dia_R) of Frame 2, respectively. The axial forces of Frame 1 are plotted as dashed orange lines for dia_L and solid orange lines for dia_R. Before applying live loads, the section force of dia_L was compression and that of dia_R was tension in Frame 1, which was the opposite of that in Frame 2. This can be attributed to the extensive downward girder deflection of the right-side girder. The fractured girder  
Figure 8: Bending moment of chords in Frame 2 (SM1). was discontinuous at the midspan due to the fracture of the girder, which caused extensive girder rotation. To resist such girder displacement, downward reactions occurred at the support of the fracture girder while leaving local upward deflection near Frame 1. However, this tendency disappeared as the live loads increased and reverted to the same pattern as that in Frame 2. As shown in Figure 10a,b, the loaded pattern of Case 2 was similar to that of Case 3. Initially, Frame 2 transferred only the live load from the fractured girder to the intact girder. However, as the live loads exceeded 1.35-times of the HS-20 truck, Frame 1 began to be involved in the load transfer. In Case 2, the bridge collapse was initiated by the buckling of the diagonal of Frame 2, as the applied live loads reached approximately five-times the HS-20 truck load.