## 3 Performance-Based Optimization Model for the Life-Cycle Cost of SRHPC Frame Structures  
### Performance-Based Life-Cycle SRHPC Frame Optimization Mathematical Model  
The total cost of the structure over the life-cycle consists of three parts: (1) the initial cost of the structure; (2) the cost of structural inspection and repair; and (3) the expected loss of the structure. In general, the cost of the structural inspection and repair in the structural optimization design stage were not considered, but after the optimal design was established, we selected reasonable repair methods and maintenance time. As the cost of the structural inspection and repair is influenced by many factors such as the materials, methods, time interval of the inspection and the methods of maintenance and repair, this can make it difficult to determine cost in the design stage. Therefore, only the initial cost and the expected loss of the structure were considered in the optimization design stage.  
The aim of the performance-based seismic design was an economically feasible design method pursued through optimizing the life-cycle cost. Therefore, based on the seismic optimization model that [PERSON] and [PERSON] [25] put forward (discussed above), and taking into consideration the fuzziness existing in seismic analysis and the structural failure of SRHPC frame structures, a fuzzy- and performance-based life-cycle cost optimization model of the SRHPC frame structures was developed. The model is shown as follows:  
The design variables that needed to be solved are shown in Equation (1):  
\[X=\{x_{1},x_{2},\cdots,x_{n}\}^{T},X\in R. \tag{1}\]  
The optimization object function is shown in Equation (2):  
\[MinW(X)=\alpha_{1}C_{0}(X)+\alpha_{2}{\sum\limits_{i=1}^{n_{p}}C_{fi}\widetilde {P}_{fi}(X)}. \tag{2}\]  
Optimization constraints are shown in Equations (3)-(5):  
\[\widetilde{P}_{fi}(X)\leq\Big{[}P_{fi}\Big{]},i=1,...,n_{p} \tag{3}\]  
\[g_{j}(X)=0,j=1,2,...,p \tag{4}\]  
\[h_{k}(X)\leq 0,k=1,2,...,q. \tag{5}\]  
In the literature, the range of the ratio of the failure loss expectation to the initial cost of the structure under different damage levels is recommended [26]. In this paper, the mean values of the ratio ranges of each damage level were taken as the parameter to calculate the failure loss expectation and are shown in Table 4.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Damage Degree** & **Intact** & **Slight** & **Moderate** & **Extensive** & **Collapse** \\ \hline Ratio & 0.02 & 0.105 & 0.3 & 0.7 & 1.0 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The ratio of the failure loss expectation to the initial cost of the structure at different damage levels.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Earthquake Intensity** & **Minor Earthquake** & **Medium Earthquake** & **Major Earthquake** \\ \hline performance level & no damage & repairable & no collapse \\ economic acceptability for renovation of building safety & complete acceptance & acceptable & unacceptable \\ threshold for story drift angle & safe & safe & life safe \\ \hline \hline \end{tabular}
\end{table}
Table 3: The quantitative value for seismic performance objectives of SRHPC frame structures.  
In Equation (2), \(\alpha_{1}\) and \(\alpha_{2}\) are weighted coefficients used to reflect the importance of different structures. With the decrease of \(\alpha_{1}\), \(\alpha_{2}\) becomes larger, which means that the loss caused by structural failure is also larger and the structure is more important. By comparing the reliability of the SRHRC frame structure with the different weight coefficients shown in Section 5.2, it can be seen that the structure reliability index is stable when the weight coefficient \(\alpha_{1}\) is [0.4, 0.6]. Therefore, the weighting factors \(\alpha_{1}=0.4\), \(\alpha_{2}=0.6\) were recommended for the SRHPC frame structures.