## 3 Calculation Method of the Restraint Ratio of CSA-BRB  
The steel tube-constrained kernel split BRB restraint ratio is defined regarding the definition and design theory of the single-cored BRB restraint ratio. In order to ensure that the external restraining members of a steel tube restrained core separated BRB will not be unstable before the complete section of the core member yields, the restraint ratio \(\zeta\) should be greater than a particular value, the restraint ratio threshold \([\zeta]\). A design that satisfies Equation (1) means that the overall instability of the CSA-BRB does not precede the yielding of the core. \[\zeta=\frac{P_{cr,e}}{P_{y,e}}\geq[\zeta] \tag{1}\]  
where \(P_{cr,e}\) is the elastic buckling load corresponding to the restrained external member, excluding the contribution of the core, and \(P_{y,c}\) is the initial yield load of the core. To derive the elastic buckling load of CSA-BRB, it is simplified as a quasi-frame model shown in Figure 2, in which the middle column of the frame is facilitated by the external restraining members, including two steel tube chords and connecting two chords. The two cores are columns on both sides of the frame, connected by internal core ribs. In the structure, the lateral deflection deformation of the middle column and the columns on both sides of the frame is shown in Figure 2.  
Figure 1: Composition of the proposed CSA-BRB. sides is the same. The axial displacement between the three is independent of each other. Elastic buckling loads for kernel-separated BRBs are derived from a quasi-frame model. Since the core's moment of inertia around the \(x\)-axis is substantially smaller than the external restraining members', the material's elastic modulus rapidly decreases to zero after the core yields; the lateral stiffness of the core around the \(x\)-axis is minor compared to that of the external restraining members and can be ignored. Therefore, the elastic buckling load of the steel tube-restrained separated BRB external restraining members [20] is:  
\[ \begin{split} P_{cr,e}&=\frac{(4{+}3.192\beta)}{(1{ +}3.192\beta)}\frac{\pi^{2}E_{e}I_{e}}{l^{2}};\\ \beta&=\frac{2(2E_{c1}I_{c1}{+}E_{e}I_{e})}{E_{c1}A_{ c1}h_{e}^{2}}\end{split}  \tag{2}\]  
where the \(E_{e}I_{e}\) is the lateral stiffness of the combined section around the \(y\)-axis of the external restraining members, calculated without taking into account the contribution of the filled lightweight material; \(E_{c1}I_{c1}\) is the lateral stiffness of the core around the \(x\)-axis of its section, \(E_{c1}A_{c1}\) is the compressive stiffness of the core, \(h_{e}\) is the axial distance between the two cores and \(l\) is the CSA-BRB length. The larger the \(\beta\) value, the stronger the combined effect of the two cores. Among them, the physical meaning \(\beta\) is the ratio of the sum of the bending stiffnesses of the two cores (\(2E_{c1}I_{c1}+E_{e}I_{e}\)) and the external restraining members to the bending stiffness calculated by the two cores according to the axis-shift theorem (\(E_{c1}A_{c1}h_{e}^{2}/2\)). The larger the value of \(\beta\), the stronger the combined effect of the two cores. According to the overall instability damage mechanism of the CSA-BRB, when the outermost fibers of the span section of the external restraining members reach their yield stress, the lateral stiffness of the span section of the external restraining members begins to decrease. With the continuous development of the yield zone of the external restraining members, the restraint stiffness of the restraint system is continuously reduced until the overall failure of the CSA-BRB occurs. Therefore, based on the edge-yielding criterion of the combined section of the external restraining members, the formula of its restraint ratio threshold is derived. According to the flexural bearing capacity \(M_{u}\) of the cross-section of  
Figure 2: The simplified analytical and calculation model of a CSA-BRB.