## 1 Introduction  
Twin-steel box-girder bridges are a popular bridge type frequently used for highway interchanges because of their aesthetically pleasing appearance and high torsional resistance. However, two-girder bridges have been classified as fracture critical bridges according to the AASHTO LRFD bridge design specifications [1], which implies that the failure of one girder leads to bridge collapse. Based on this classification, twin-steel box-girder bridges are required to meet stricter provisions than non-fracture-critical bridges, such as the mandatory use of high-fracture-toughness materials, biennial hands-on inspections, and nondestructive inspections of welded connections [2]. These restrictions not only increase construction and maintenance costs but also discourage bridge owners from adopting this type of bridge despite its structural efficiency. Meanwhile, there have been several past incidents involving the significant failure of bridge components in steel plate-girder bridges classified as fracture critical, such as the I-79 Glenfield Bridges [3], US 422 [4], and I-794 Hoan bridge [5]. In these incidents, the applied loads were redistributed to the integral parts of the bridges without collapse, which suggests that inherent redundancy could exist in two-girder bridges. Various studies have been conducted to determine the redundancy sources in a two-girder bridge. [PERSON] et al. [6] proposed three different sources of redundancy for a bridge: internal, load-path, and structural redundancy. [PERSON] et al. [7] and [PERSON] and [PERSON] [8] suggested that even a two-girder composite bridge could be redundant owing to internal load redistribution. [PERSON] et al. [9] and [PERSON] and [PERSON] [10] found that I-shaped cross beams could contribute to an increase in the ultimate loading capacity of a steel plate girder bridge with one-girder fracture damage. [PERSON] et al. [11] and [PERSON] al. [12] showed that the concrete deck of a twin-steel trapezoidal box-girder bridge could provide load path-type redundancy. In addition to the concrete deck, [PERSON] et al. [13] found that the external cross-frames in a twin-steel box-girder bridge could contribute to load redistribution based on experimental tests. The external cross-frames in a twin-steel box-girder bridge have been neglected in redundancy evaluation because they are temporarily used during the bridge construction process. External cross-frames were installed between the steel girders, and the torsional resistance increased before the composite actions were obtained by casting a concrete deck [14; 15]. This is a common process in which they are removed after bridge construction. Furthermore, they comprise of angles and T-sections that may be vulnerable to buckling by compressive forces in the ultimate behavior of the bridges. Therefore, they are regarded as secondary members and neglected in the redundancy evaluation of twin-steel box-girder bridges. If external cross-frames could improve the redundancy of bridges, they could be utilized to enhance bridge safety without changing design practices and increasing costs significantly. In this study, the effects of K-type external cross-frames on the ultimate behavior of a twin-steel trapezoidal box-girder bridge were evaluated to investigate a possibility as the redundancy source of the bridge. For this purpose, finite element (FE) bridge models were constructed to simulate the possible failure modes of external cross-frame members, such as section yielding and buckling behaviors. In addition, various failure behaviors required for redundancy evaluation, such as large deflections, material inelasticity of concrete and steel, and stud connection failure, were incorporated into the models based on the modeling techniques proposed by [PERSON] and [PERSON] [16]. It was assumed that one of the two girders was fully fractured, and truck live loads were applied to the damaged girder side to generate the maximum bending moment at the fracture location.