## 4 Optimum Design for a Single Earthquake Ground Motion  
The CSDBC-designed 5-story RC frame structure is optimized, implementing the proposed optimization method subjected to a ground motion (San Fernando, 1971, RSN68). Figure 5 illustrates _Cov_, _TR_ for each floor and the longitudinal reinforcement dimension of single columns for each floor from the CSDBC-designed structure tending to the final optimization design. As shown in Figure 5a,b, the proposed method eventually converges on a near-uniform distribution of _TR_ until the variable coefficient of _TR_ meets the constraint requirement with few iterative loops. It should be classified that the value of _TR_ sometimes is negative, which results from the non-monotonic and back-and-forth twisting behaviors of the IDA curves. Further elaboration of these behaviors can be found in the original investigation of IDA [45]. It is shown in Figure 5c that the longitudinal reinforcement dimension of single column on the fourth and fifth floors eventually converged on the minimum reinforcement based on the constraints in CSDBC [4]. It indicates the necessity of the consideration of the constrained provisions in the optimization procedure. During the optimization procedure, the reinforcements of the fourth and fifth floors are gradually transferred to the first three floors with most materials being moved to the second floor. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**No.** & **Earthquake** & **Year** & **Recording Name** & **Magnitude** & **PGA (g)** \\ \hline
1 & Northridge & 1994 & Canyon Country-WLC & 6.7 & 0.48 \\
2 & Imperial Valley & 1979 & EI Centro Array \#11 & 6.5 & 0.38 \\
3 & Kobe, Japan & 1995 & Shin-Osaka & 6.9 & 0.24 \\
4 & Kocaeli, Turkey & 1999 & Duzce & 7.5 & 0.36 \\
5 & Kocaeli, Turkey & 1999 & Arcelik & 7.5 & 0.22 \\
6 & Landers & 1992 & Yermo Fire Station & 7.3 & 0.24 \\
7 & Loma Prieta & 1989 & Gilroy Array \#3 & 6.9 & 0.56 \\
8 & Superstition Hills & 1987 & Poe Road (temp) & 6.5 & 0.45 \\
9 & Chi-Chi, Taiwan & 1999 & CHY101 & 7.6 & 0.44 \\
10 & San Fernando & 1971 & LA-Hollywood Stor & 6.6 & 0.22 \\
11 & Hector Mine & 1999 & Hector & 7.1 & 0.34 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Basic information of selected ground motions. Figure 4: Elastic response spectra of selected earthquake set and CSDBC design spectrum for soil type II (5% damping). The \(TR\) distribution along the height of the building under the earthquake intensity corresponding to the collapse state of the code-designed structure, the longitudinal reinforcement dimension of a single column for each story, and IDA curves for each story for the code-designed and optimized structure are compared in Figure 6. As shown in Figure 6a, this proposed optimization procedure is feasible, and can efficiently achieve a close-to-uniform distribution of \(TR\) along the height of the building. It is difficult to achieve an absolutely uniform distribution of \(TR\) because of the constraints setting in the iterative cycle. After optimization, the \(TR\) of each floor is greater than 20%. This implies that the optimized structure will not collapse under the collapse intensity of the code-designed building. In other words, the proposed optimization method can feasibly increase the structural collapse resistant capacity without a construction cost increase. For this ground motion, most reduced reinforcement in the fourth and fifth floor is moved to the second floor, and the reinforcement increase of the first and third floor is almost the same. It can be seen in Figure 6c that the first story of the code-designed structure collapses at earthquake intensity of 0.9 g, while the second story collapses for the optimized structure at 1.3 g. The collapse resistant capacity of the structure increases by 44.4% from 0.9 g to 1.3 g. At the same time, the maximum inter-story drift ratio response decreases for the first story at the inelastic range. It should be noted that the change of reinforcement of columns does not lead to a change of seismic demand in the elastic range.