## 1 Introduction  
Lateral torsional buckling (LTB) is a major failure mode induced by instability in flexural members that typically control the entire design in steel buildings and bridges. The moment capacity of the flexural members is enhanced by bracing elements to provide lateral and rotational supports to retain LTB in flexural elements. Thus, bracing members must possess sufficient strength and stiffness against internal forces affected by the LTB phenomenon. In general, bracing components in straight steel I-girder bridge systems are secondary structural members because they resist no direct applied forces, but secondary internal forces are induced by displacement changes in the top and bottom flanges of adjacent girders due to twists induced by the LTB. However, it is an essential structural component for the bridges to restrain against rotation with respect to their longitudinal axis at points of support and to prevent relative displacement (i.e., twist) of the top and bottom flanges. Therefore, current design specifications worldwide require the installation of bracing components such as torsional bracing, lateral bracing, or a combination of torsional and lateral bracing to prevent unexpected twists of the I-girder sections caused by the instability of the girders [1]. Furthermore, stiffness and strength requirements for the bracings in flexural members were stipulated to avoid the failure of flexural members and the bracing components due to the LTB. Since [PERSON] and [PERSON] [2] proposed an exact solution for the critical LTB strength of a doubly symmetric beam under a uniform moment with continuous torsional bracing, researchers have continued to refine their methods for computing the LTB strength of flexural members with torsional bracing. [PERSON] [3] demonstrated that cross-sectional distortion in beam members could be controlled by adopting an effective bracing stiffness (\(\beta_{T}\)). [PERSON] [4] expanded the effective bracing stiffness by considering the effect of stiffeners and in-plane gider flexibility in the braced flexural members. Later, [PERSON] [5] provided a comprehensive view of the beam bracing design. Furthermore, design factors affecting bracing requirements and design examples of the torsional bracing system were introduced. The design methodology developed by [PERSON] [5] was adopted in the third edition of the American Institute of Steel Construction Load and Resistance Factor Design Specification [6]. Based on research by [PERSON] [7] and [PERSON] [5], the ideal stiffness requirements obtained by eigenvalue analysis resulted in poor estimates of LTB behavior in beam bracing systems because the ideal stiffness requirements disregarded the effects of geometric nonlinearity due to initial imperfections. Therefore, [PERSON] and [PERSON] [8] investigated the effects of various types of imperfections for beam bracing systems based on geometric nonlinear analysis to determine the critical imperfection shapes for simulating the most critical LTB behavior in the beam bracing system. Thus, the stiffness requirements can be determined by numerous finite element analyses (FEAs). [PERSON] and [PERSON] [9] investigated the torsional brace strength requirements for steel I-girder systems. They conducted numerous parametric studies to evaluate the bracing moments stipulated in the current design specification [1,10]. [PERSON] and [PERSON] [11] investigated the elastic global LTB of straight I-girder bracing systems. Regarding internal forces created in the torsional bracings because of the LTB behavior, current design specifications provide conservative design values by assuming required rotational displacements of the bracing elements to ensure that the bracing elements possess sufficient strength and stiffness to avoid failure due to LTB. To estimate reasonable bracing internal forces, it is necessary to implement it through FEA analysis because the internal forces are generated by secondary effects due to LTB, which are challenging to solve using traditional mathematical models.