## 5 Summary and Conclusions  
External cross-frames are utilized to increase the torsional resistance during the construction of a trapezoidal box-girder bridge. They are temporary members that are removed after a composite action in the bridge is achieved. Recent studies demonstrated that they can contribute to load redistribution as redundant sources [9; 13]. In this study, the effectiveness of K-type external cross-frames as a redundant source was investigated. For this purpose, FE bridge models were constructed and the effects of the cross-frames were evaluated. The FE model was validated based on the results of a full-scale bridge fracture test conducted by [PERSON] et al. [11]. The load-displacement response of the FE model matched well with the test results, and the estimated ultimate capacity showed about a one percent difference compared to the test results.  
Four different FE models (Cases 1 to 4) were constructed, depending on the presence of the external cross-frames and failure modes of the frame members considered in the analysis. In Case 1, external cross-frames were not incorporated into the bridge model. Possible failure modes, such as yielding and buckling of the frame members were considered in Case 2. To investigate the effects of the failure sequence of the frame members, the elastic behavior of the diagonals was assumed for Case 3 and the chords for Case 4. The ultimate capacities of the bridge models were evaluated by applying live loads after one of the two girders fractured. The axle configuration of an HS-20 truck load was utilized for live loads. The axle loads were modeled using point loads and placed above the damaged girder to maximize the live load effect.  
The analysis results showed that the ultimate capacity of Case 1 was 3.85-times of the HS-20 truck. The load-displacement response of Case 2 showed a stiffer behavior than that of Case 1. The ultimate capacity of Case 2 was approximately 30% higher than that of Case 1. Case 3 exhibited a load-displacement response similar to that of Case 2, and the ultimate capacity was nine percent higher than that of Case 2. In Case 4, the ultimate capacity was similar to that of Case 1, but the load-displacement response exhibited an unstable behavior compared with the other cases. This was caused by the buckling of the external cross-frames.  
From these results, it is concluded that external cross-frames can increase the stiffness and ultimate capacity of a bridge with one girder fracture. The external cross-frames contributed to the transverse redistribution of the applied live loads on the fractured girder. Therefore, external cross-frames can be considered a redundancy source. The failure  
Figure 11: Equivalent plastic strain and Von Mises equivalent stress.  
sequence of the frame members can affect the ultimate capacity and load-displacement response. It is not recommended to use a heavy section for chords, such that frame failure is initiated by the buckling of diagonals. In this study, only K-type external cross-frames were considered. Various factors, such as the frame type, location, connection, and number of frames, might affect the ultimate capacity of a bridge. Therefore, further research is required to utilize external cross-frames as reliable sources to increase bridge redundancy.  
J.K. performed FE simulations and wrote the draft for the article; B.W.K. gave advice on the simulations and the data analysis; S.K. supervised this research program and finalized the article. All authors have read and agreed to the published version of the manuscript.  
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2021R1I1A3049510).  
Not applicable.  
The authors declare no conflict of interest. The funders did not contribute to the study design; collection, analyses, or interpretation of data; writing of the manuscript; or decision to publish the results.