## 2 Refined Seismic Design Methodology  
In the Code for Seismic Design of Buildings in China (CSDBC) [4], the structures are designed to be undamaged under earthquake intensity corresponding to a 63.2% probability of exceedance, repairable under the earthquake intensity with a 10% probability of exceedance, and to not collapse subjected to earthquake intensity with a 2-3% probability of exceedance, all referring to a 50-year return period. These seismic fortification objectives are called three levels of seismic fortification. The code-based seismic fortification objectives are expected to be satisfied in this study  
Collapse evaluation is the basis of this study. IDA is generally applied for the seismic collapse evaluation of structures. The IDA can efficiently describe the whole process of a structure from elastic state to inelastic range until collapse damage occurs. IDA involves subjecting the structure to a set of ground motions and incrementally scaling the records to multiple levels of intensity, subsequently developing a group of curves of seismic demand parameters versus earthquake intensity levels [42]. Inter-story drift ratio and the 5% damped spectral acceleration, \(S_{a}\)(T\({}_{1}\), 5%), corresponding to the structural fundamental period, are applied as the structural seismic demand parameter and earthquake intensity parameter, respectively. As suggested in FEMA P695 [43], the point on the IDA curve, which has a tangent slope that reduces to 20% of the initial elastic slope of the curve, is defined as the structural collapse capacity point. Therefore, the tangent slope of adjacent points on the IDA curve is a crucial parameter which can directly represent collapse state. Accordingly, as shown in Figure 1, _Tangent_ratio_ (_TR_) is formulated as the ratio of the tangent slope of the adjacent points on the IDA curve of the story and the initial elastic slope of the curve, and it can be calculated by Equations (1)-(3). \[TR_{i}=\frac{Tangent_{i,c}}{Tangent_{i,\text{initial}}} \tag{1}\]  
\[Tangent_{i,c}=\frac{IM_{i,c}-IM_{i,c-1}}{EDP_{i,c}-EDP_{i,c-1}} \tag{2}\]  
\[Tangent_{i,\text{initial}}=\frac{IM_{i,1}}{EDP_{i,1}} \tag{3}\]  
where \(IM_{i,c}\) and \(IM_{i,c-1}\) are the earthquake intensities of the structural collapse point and the point on the IDA curve before the collapse point for \(i^{\text{th}}\) story, respectively; \(TR_{i}\) is the \(TR\) of the \(i^{\text{th}}\) story; \(Tangent_{i,c}\) is the tangent slope of the collapse point and the step before the collapse damage state on the IDA curve for the \(i^{\text{th}}\) story; \(Tangent_{i,\text{initial}}\) is defined as the initial slope of the IDA curve of the \(i^{\text{th}}\) story; \(EDP_{i,c}\) and \(EDP_{i,c-1}\) are the seismic demands of the \(i^{\text{th}}\) story under the ground motion intensity \(IM_{i,c}\) and \(IM_{i,c-1}\), respectively; and the \(IM_{i,1}\) and \(EDP_{i,1}\) are the seismic intensity and the seismic demand of the \(i^{\text{th}}\) story at the first step of IDA, separately. Referring to the evaluation of structural collapse in FEMA P695 [43], the structure collapses when the \(TR\) reduces to less than 20%. In this study, in order to visually represent whether on each floor of the structure there occurs collapse damage, \(TR\) is selected as the story damage index for refining the seismic design method. The refined design is to achieve a state tending to be \(TR\) uniformly distributed along the height of the building when the building collapses. The structural configuration is pre-decided according to the structural space and occupancy requirements. It is impractical to change the concrete and the sizes of the components due to the structural construction practice limitation. Furthermore, the collapse evaluation of the structure under earthquake ground motions based on IDA is relatively time-consuming, so the number of optimization variables influences the efficiency of the optimization procedure. The columns are crucial components for resisting lateral loads.