## 2 Refined Seismic Design Methodology
The capacity of them is significantly influenced by the amount of longitudinal steel reinforcement. Accordingly, the longitudinal steel reinforcements of columns are selected as the optimization variables considered in developing the refined seismic design method in this study. The following procedure is presented to obtain a state of uniform \(TR\) distribution along the height of the building based on the IDA for a specific earthquake ground motion:1. Design a structure according to the current design code, such as CSDBC [4]. 2. Establish the numerical model of the designed building, and perform IDA to determine the collapse capacity under the selected earthquake ground motion. Then, the intensity of the earthquake ground motion relating to the state of structural collapse damage can be obtained and recorded as \(IM_{c}\). 3. Scale the intensity of selected ground motion to \(IM_{c}\), \(IM_{c-1}\), and \(IM_{1}\), and perform the dynamic analysis to derive the seismic demand of the structure. Calculate the \(TR\) of each floor using the Equations (1)-(3). 4. Based on the calculation results of \(TR\), calculate the coefficient of variation of \(TR\) based on the Equation (4). \[Cov=\frac{\sqrt{\frac{1}{n}\sum\limits_{i=1}^{n}\left(TR_{i,c}-TR_{\text{ average},c}\right)^{2}}}{TR_{\text{average},c}}\] (4) where \(Cov\) is the coefficient of variation of \(TR_{i,c}\); \(n\) is the number of the structure floor; and \(Tangent_{\text{average},c}\) is the average value of \(TR_{i,c}\). 5. If the value of \(Cov\) reduces to a pre-determined target value, that means the code-based structure has a relatively uniform \(TR\) distribution along the height of the building while the structure collapses, and no redesign is needed; otherwise, an iterative procedure is proceeded, where the longitudinal steel reinforcement of columns is sequentially modified using Equations (5) and (6) until the value of \(Cov\) reduces to small enough (e.g., less than 0.5). \[As_{i}^{j^{\prime}}=\left(\frac{TR_{i}^{j-1}}{TR_{\text{average}}^{j-1}} \right)^{\alpha}As_{i}^{j-1}\] (5) \[As_{i}^{j}=\frac{\sum\limits_{i=1}^{n}As_{i}}{\sum\limits_{i=1}^{n}As_{i}^{j ^{\prime}}}\] (6) where \(As_{i}^{j^{\prime}}\) and \(As_{i}^{j-1}\) are the dimension of the longitudinal steel reinforcement for one column on the \(i^{\text{th}}\) floor at the \(j^{\text{th}}\) and \((j-1)^{\text{th}}\) iteration, respectively; \(TR_{i,c}^{j}\) is the \(TR\) of \(i^{\text{th}}\) floor at \((j-1)^{\text{th}}\) iteration; \(TR_{\text{average},c}^{j}\) is the average of \(TR\) of each floor at \((j-1)^{\text{th}}\) iteration; and \(\alpha\) in Equation (5), which controls the convergence gradient, is a coefficient ranging from 0 to 1. Based on the previous studies [22; 23; 24; 25; 26; 27; 28; 29], \(\alpha\) is set as 0.02 for this study. In order to keep the same construction cost, the Equation (6) is utilized to adjust the sum of longitudinal steel reinforcement of candidate design to that of code-based structure. \(\sum\limits_{i=1}^{n}As_{i}^{j^{\prime}}\) is the sum of \(As_{i}^{j^{\prime}}\); \(\sum\limits_{i=1}^{n}As_{i}^{j}\) is the sum of longitudinal steel reinforcement for a code-based structure; \(As_{i}^{j}\) is the updated dimension after scaling the optimized results to the sum of longitudinal steel reinforcement for a code-based structure. It should be clarified that the diameters and numbers of the longitudinal steel bars in each side of the column section are assumed herein to be the same for simplification purposes and engineering practicability. Meanwhile, the dimension of the steel bars is assumed to be continuous during the iterations; then, the proper steel bars are selected with the bars available in commerce, trying to be close to the optimization results at the last step. A suite of constraints should be preliminarily checked for every candidate design to make sure that the candidate designs have a desirable performance under earthquake loads and satisfy the requirements set in the provisions of the conventional seismic design code.