## 4 The Calculation of Reliability Based on Fuzzy Mathematics
### The Calculation Method of Fuzzy Reliability  
Among the general set \(U_{1}=\{u_{1},u_{2},...,u_{m}\}\), the membership of every element is definite, which means that whether an element belongs to set \(U\)--or not--is certain. However, due to the natural characteristics of the thing itself, many sets do not have a definite boundary to provide a clear definition and evaluation standard. This uncertainty is called fuzziness and a subset that has a fuzzy boundary is called a fuzzy subset. [PERSON] (1965) proposed using a value in a closed interval [0, 1] to represent the subordinative degree of element \(u_{i}\) to fuzzy subset \(\widetilde{A}\), which is called the \"membership degree\", and expressed by \(\widetilde{A}(u_{i})\) and is usually denoted by \(u_{i,A}\). Obviously, the greater \(u_{i,A}\), the higher the subordinative degree. When \(u_{i,A}\) is equal to 1, element \(i\) belongs to subset \(A\). In contrast, when \(u_{i,A}\) is equal to 0, element \(i\) does not belong to subset \(A\) absolutely. The general set is a special case of fuzzy set.  
When using traditional reliability theory to judge the reliability of a structure, the state of the structure jumps directly from reliability to failure, as shown in Figure 1a. However, in practical engineering, the process of a structure's state from reliability to failure is gradual and there exists a fuzzy transition state, as shown in Figure 1b, so the deterministic failure criterion is neither scientific nor does it correspond to the actual engineering.  
To solve the problem that exists in traditional reliability theory, this paper adopted the membership function of fuzzy performance function \(\widetilde{Z}\) to express this gradual process, which considers the fuzziness  
Figure 1: The working state of the structure in the theory of reliability. (**a**) Structural working state in traditional reliability theory; and (**b**) structural working state in fuzzy reliability theory.  
existing in structural failure. A performance function that can consider the fuzziness of structural failure is shown in Equation (18):  
\[\widetilde{Z}=\widetilde{R}-\widetilde{S}. \tag{18}\]  
There are several methods to develop membership function, e.g., the fuzzy statistical method, the assignment method, using \"objective measurement\" for reference and dualistic contrast compositor, etc. Due to the necessity of a large body of fuzzy statistics and expert evaluation for the establishment of a reasonable membership function, the assignment method was adopted for the sake of simplicity and the half-ladder type distribution was assumed as the membership function. Based on this, the membership function of fuzzy performance functions is shown in Equation (19):  
\[\widetilde{A}_{\widetilde{Z}}(z)=\begin{cases}1&z\leq r_{1}\\ -\frac{z-r_{2}}{r_{2}-r_{1}}&r_{1}<z<r_{2}\\ 0&z\geq r_{2}\end{cases}. \tag{19}\]  
When the membership function of fuzzy performance function was established, the value of the upper and lower bounds \(r_{1},r_{2}\) of the fuzzy transition area needed to be determined, i.e., \"the tolerance\". There are many ways to determine \"the tolerance\", in which the amplification coefficient method in Reference [27] is commonly used in engineering and is also adopted in this paper. Based on conventional design experience, an amplifying coefficient \(\lambda\) (0.05--0.4 times of general allowable values) was introduced to determine \"the tolerance\" of fuzzy intervals in this method. For SRHPC frame structures, using coefficient \(\lambda\) to multiply the value of inter-story drift limit values corresponding to different performance levels are shown in Table 3, and \"the tolerance\" of the fuzzy membership function corresponding to the different performance level of SRHPC frame structures can be obtained.