## 4 CSA-BRB Ultimate Bearing Capacity Parasitic Bearing Restraint Ratio Threshold
### Restraint Ratio Thresholds for Bearing Elements  
Each group of cases in the critical state where the compressive strain of the core member exceeds 0.02, while the external restraining members' outermost edge fibers yield is chosen based on the results of static elastoplastic load carrying capacity calculations for the four sets of cases, as shown in Table 2, to determine its bending moment (\(M_{e,FE}\)) in the span under the action of the axial load \(M_{e,FE}\). The correction coefficients for the four groups of calculation examples can be found by comparing them to the Equation (5) calculation results. The maximum value is \(\alpha=1.8\) conservatively taken and recommended. Equation (6) gives an approximate formula for the restraint ratio threshold of the bearing CSA-BRB. The material over-strengthening factor \(\eta=1.37\) is considered for the core steel reinforced after yielding, and the axial compressive strain reaches 0.02. Take the first group of calculation examples in Table 1, the calculation example CSA-20-\(\zeta\) in Figure 6a as an example. For the calculation example CSA-20-2.47, the restraint ratio threshold \(\left[\zeta\right]_{\eta}=2.56\) is calculated by Equation (6), at which time its restraint ratio is less than its restraint ratio threshold. The external constraint stiffness is theoretically insufficient to cause instability damage, which is in line with the numerical analysis results, as shown in Figure 6a. As the restraint ratio increases, the restraint ratio threshold \(\left[\zeta\right]_{\eta}=2.46\) for CSA-20-2.68 is calculated by Equation (6), which is less than its restraint ratio. Therefore, this calculation case can meet the requirement of the core member reaching the entire section yielding and the axial compressive strain of the CSA-BRB reaching \(\varepsilon=0.02\), which is also consistent with the numerical analysis results, as shown in Figure 6a. The other three cases in Table 1 are similar to the first set of circumstances, and their numerical FEM results are also consistent with the results of Equation (6). Given this, the restraint ratio threshold of CSA-BRB can be determined more accurately by using Equation (6). During the preliminary design phase of the CSA-BRB, a conservative estimate of its restraint ratio threshold is required. Therefore, the restraint ratio threshold value can be obtained by plotting the relationship between the restraint ratio \(\zeta\) and the stability coefficient \(\varphi\) of the four groups of calculation examples in Figure 9.  
When the coefficient of stability of the CSA-BRB \(\varphi<1.0\), i.e., the CSA-BRB core, has not yet reached the initial yield, it has already failed, which belongs to the first type of failure mentioned above; as the restraint ratio \(\varphi\) of the CSA-BRB increases, the stability coefficient increases. When the CSA-BRB's stability coefficient is within the range of \(1.0\leq\varphi<1.37\), the CSA-BRB's kind of damage is the second type, and the restraint ratio is within the range of \(1.8\leq\zeta<2.8\) at the same time. When the restraint ratio of the CSA-BRB \(\zeta=2.8\), the axial compressive strain of the core member \(\varepsilon=0.02\), it happens that the overall instability of the CSA-BRB does not occur, so the restraint ratio threshold value of the CSA-BRB is approximately taken as \(\left[\zeta\right]_{\eta}=2.8\), so when the restraint ratio of the CSA-BRB \(\zeta\geq 2.8\), it can be considered that the CSA-BRB meets the requirements of the bearing type. It has a certain plastic deformation and strengthening capacity. The references [21; 22] give the restraint ratio threshold value of 1.8 and 2.0 for CS-BRB bearing members with solid web and wave web connections, respectively, where the restraint ratio threshold value for bearing members is defined as the core member reaching the entire section initial yielding and the CSA-BRB undergoing integral instability damage. At that point, the corresponding stability factor \(\varphi=1.0\).