## 4 Discussion of the Corresponding Remedy Measures
### Procedure  
Nonlinear static analysis is used to consider the nonlinear effect, and Figure 15 shows the design procedure of the dual structural system. First, pushover of the initial frame structure is carried out. The inter-story shear and drift when the maximum inter-story drift reaches 1:50, which is the drift limit to prevent structural collapse according to the Chinese seismic design code, could be obtained. Thus, the inter-story stiffness could be easily calculated. Then, a target maximum inter-story drift of the dual structural system needs to be set manually. The target inter-story stiffness could be obtained by dividing the inter-story shear force by the target maximum inter-story drift. Then, the needed lateral stiffness could be obtained by the target inter-story stiffness minus existing inter-story stiffness. The dimensions of lateral force-resisting members could be designed by the needed lateral stiffness, which means a new dual structure is designed. The only issue to note here is that the used inter-story shear force when we calculated the target inter-story stiffness is from the previous frame structure, not from the present dual structure, so some error is inevitable. Thus, repeated iterative design of the new dual structure is needed until all the inter-story drifts of the designed structure are close to the set target inter-story drift.  
The inter-story drifts of improved structure is expected to be close to the target inter-story drift under the seismic intensity when the maximum inter-story drift of the initial RC frame reaches 1:50.  
Figure 15: Design procedure of dual structural system.  
Therefore, an important issue is how to determine this corresponding seismic intensity. Response method in the code for seismic design of buildings in China is used in this paper (Figure 16). In the general case, the natural period of multi-story RC frames is in the stage of \(T_{\text{g}}\)-\(5T_{\text{g}}\) in the seismic effect coefficient curve. The seismic effect coefficient could be calculated based on Equation (4).  
\[\alpha=\left(\frac{T_{g}}{T}\right)^{\gamma}\eta_{2\alpha}\ \max \tag{4}\]  
\[\gamma=0.9+\frac{0.05-\zeta}{0.3+6\zeta} \tag{5}\]  
\[\eta_{2}=1+\frac{0.05-\zeta}{0.08+1.6\zeta} \tag{6}\]  
where \(\alpha\) represents the seismic effect coefficient, \(\alpha_{\max}\) is the maximum seismic effect coefficient, \(T_{\text{g}}\) denotes the characteristic period of site, \(T\) is the structural natural period, \(\zeta\) represents the structural damping ratio and could be calculated as 0.05 for reinforced concrete structures.  
The structural base shear force \(F_{EK}=\alpha G_{eq}\), where \(G_{eq}\) denotes the representative value of gravity load. The base shear force when the maximum inter-story drift of the initial frame reaches 1:50 could be got in the first pushover analysis. The structural natural period changed as lateral force-resisting members were added, and the seismic effect coefficient and base shear force changed accordingly. The relationship of base shear in the two structures could be calculated based on Equation (7), thus the expected base shear under the according seismic intensity is obtained. Only the inter-story drift of the improved dual structure needs to be checked with the target value under this base shear in the design procedure.  
\[\frac{F_{EKI}}{F_{EK0}}=\frac{\alpha_{i}G_{eq}}{\alpha_{0}G_{eq}}=\frac{\left( \frac{T_{g}}{T_{i}}\right)^{\gamma}\eta_{2\alpha}\ \max\ \ G_{eq}}{\left(\frac{T_{g}}{T_{i}}\right)^{\gamma}\eta_{2\alpha}\ \max\ \ G_{eq}}=\left(\frac{T_{0}}{T_{i}}\right)^{\gamma}=\left(\frac{T_{0}}{T_{i}} \right)^{0.9} \tag{7}\]  
where \(F_{EK0}\) denotes the base shear when the maximum inter-story drift of the initial frame reaches 1:50, \(F_{EKI}\) denotes the base shear when the \(i\)th pushover analysis is terminated, \(T_{0}\) is the natural period of the initial frame structure, and \(T_{i}\) is the natural period of the dual structure designed after the \(i\)th pushover analysis.