## 1 Introduction  
In recent years, the application of cost-effectiveness criteria in the building sector has become more extensive; for example, [PERSON] and [PERSON] [1; 2] developed external insulation prefabricated panels based on cost-effectiveness criteria to reduce energy consumption in the process of building use. However, they only considered the costs and benefits of the structure from the aspect of energy; the seismic performance of the structure was not taken into account.  
A structural optimization design method based on cost-effectiveness criteria with consideration of seismic performance has been widely applied in the design of steel structures [3; 4; 5; 6; 7] and reinforced concrete structures [8; 9; 10]. In addition, the optimal design method of steel-concrete composite elements and structures has been studied by some researchers [11; 12; 13; 14; 15]. However, there are two problems with these optimization methods: (1) they only consider the initial cost of the structures as the optimization objective, leading to poor structural performance and poor ability to resist natural disasters; and (2) there are too many optimization variables and constraints in the optimization program, leading to a complicated optimization process.  
Reinforced concrete (RC) is widely applied in the building sector and, with the development of science and technology, many new concrete and structural forms have appeared (for example, fiber-reinforced concrete, high strength, high performance concrete, etc.). Steel is combined with high strength, high performance concrete to form steel reinforced high strength high performance concrete (SRHPC) composite structures [16; 17]. As a kind of steel-concrete composite structure, SRHPC composite structures can make full use of the excellent mechanical properties and durability of high-performance concrete and enhance the cooperative working ability of steel and concrete [18; 19; 20]. Compared with RC structures, SRHPC structures have high bearing capacity, stiffness, and ductility. When the bearing capacity is equal, the section size of the SRHPC structural member is smaller, thus the overall material cost can be reduced. Although numerous SRHPC structures have been erected worldwide over the past few decades, the research and application of the optimization design of SRHSC structures is relatively scarce, which often leads to poor structural seismic performance and the waste of materials. Thus, a more efficient design optimization framework for SRHPC buildings is required.  
Based on fuzzy mathematics, this paper puts forward a fuzzy theory and performance-based optimization method for the life-cycle cost of SRHPC frame structures to solve the problems discussed above. The specific research methods are as follows: (1) according to the seismic test results of the SRHPC frame columns of our research group and other researchers, the quantitative seismic performance indicators of SRHPC frame structures were determined. Based on this, the performance-based seismic design method was introduced into the structural optimization program; (2) Considering the fuzzy reliability of structures under each performance level, the fuzzy mathematics theory was introduced into the Monte Carlo method to establish the fuzzy reliability calculation method of the SRHPC frame structures; (3) Considering the fuzziness of the seismic response spectrum, a fuzzy seismic response spectrum was established, and used as the basis for the calculations of the seismic effect and reliability of the structure; (4) The reliability calculation was simplified with the explicit function of inter-story drift, and then structural fuzzy reliability calculation for different performance levels were achieved. Thereby, the fuzzy theory and performance-based life-cycle optimization model of SRHPC frame structures was established.