## 2 Proposed Enhancements to Prevent Local Bulging Failure  
### Local Bulging Failure Mechanism of the Buckling-Restrained Braces  
BRBs with a restrainer filled with mortar typically have an unbonding material layer around the steel core designed to prevent bonding between the mortar filler and the steel core. The unbonding material layer is highly flexible, forming a space in which the steel core can deform freely before being restrained by the hardened mortar filler. When the compressive axial force acting on the steel core exceeds the primary mode buckling load of the steel core, higher-mode buckling occurs in the aforementioned space, as shown in Figure 1a, after which a local outward force acts at the contact point between the steel core and the mortar filler. As the axial compressive force increases, the outward force causes the mortar filler to break, and the restrainer undergoes local out-of-plane deformation; that is, bulging failure occurs. [PERSON] et al. (2016) derived expressions for the outward force demand induced by the steel core and the outward-force-resisting capacity of the restrainer, as described by Equations (1)-(3) [9].  
The outward force demand caused by the higher-mode buckling of the steel core can be determined using the following equation based on the moment equilibrium between the transverse and longitudinal forces formed between the two contact points, as shown in Figure 1b:  
\[P_{d,w}=\frac{4N_{cu}\big{(}2s_{rw}+v_{p}t_{c}\epsilon_{t}\big{)}}{L_{p,w}} \tag{1}\]  
where \(P_{d,w}\) is the outward force due to the higher-mode buckling of the steel core, \(N_{cu}\) is the compressive axial force acting on the brace, \(t_{c}\) is the thickness of the steel core, \(s_{rw}\) is the gap between the core and the filler, and \(v_{p}\) is the Poisson's ratio. In addition, \(L_{p,w}\) is the wavelength of higher-mode buckling, which can be calculated by considering the deformation shape, as shown in Figure 1b, using [PERSON]'s buckling equation as follows:  
\[L_{p,w}=\sqrt{\frac{4\pi^{2}(EI)_{eff}}{P_{y}}} \tag{2}\]  
where \(P_{y}\) is the yield axial force of the steel core and \((EI)_{eff}\) is the effective flexural stiffness after yielding, for which 5.5% of the elastic flexural stiffness, as recommended by [PERSON] et al. (2016), is applied [9]. The outward-force-resisting capacity of the restrainer can be calculated based on yield line theory, as follows:  
\[P_{c,w1}=\frac{2-B_{c}/B_{r}}{1-B_{c}/B_{r}}t_{r}^{2}F_{yr} \tag{3}\]  
Figure 1: Higher-order buckling of the steel core in BRB: (**a**) Higher-order buckling of steel core about weak axis; (**b**) Illustration of higher-order buckling of steel core about weak axis in detail.  
where \(P_{c,w1}\) is the outward-force-resisting capacity of the restrainer; \(B_{c}\), \(B_{r}\), and \(t_{r}\) are the width of the core and the width and thickness of the restrainer, respectively; and \(F_{yr}\) is the yield strength of the restrainer.  
The restrainer thickness needs to be increased to enhance the outward-force-resisting capacity based on Equation (3). However, a large amount of steel material is allocated to the restrainer, which does not directly contribute to the axial force resistance of the brace and is uneconomical. In this study, instead of increasing the overall thickness of the restrainer, enhancements that can resist bulging failure more efficiently are proposed.