# Efficient Seismic Fragility Analysis for Large-Scale Piping System Utilizing Bayesian Approach
## 2 Current Methods  
### IDA-Based MLE Method  
The seismic fragility of the structural system is a conditional probability of failure according to the seismic intensity (i.e., the intensity of earthquake ground motion), which is defined as the following equation:  
\[P_{f}(a)=P[D>C|a] \tag{1}\]  
where \(a\) denotes the seismic intensity; \(C\) denotes the capacity of the structural system, that is, a limit state; and \(D\) denotes the dynamic seismic response at the corresponding seismic intensity input. In piping systems, \(C\) can be represented by allowable stresses, critical displacements, etc., which define the collapse of a particular piping component. Based on the defined equation, in order for the probabilistic characteristics of the earthquake loading and structural model to be taken into account, \(D\) is calculated in multiple conditions of the loading and model according to the increase in the value of seismic intensity \(a\). Then, a conditional probability of failure is ultimately obtained by comparing such estimated values of \(D\) with \(C\). The conditional failure probability thus obtained is called a seismic fragility curve. Here, since the dynamic seismic response analyses must be performed according to the increase in seismic intensity when calculating \(D\), this is called an incremental dynamic analysis (IDA) method [29]. Under the IDA method framework, in order to obtain a seismic fragility curve, multiple nonlinear dynamic seismic analyses should be performed to take into account the nonlinear behavior characteristics of the structural model due to the increase in seismic input and the uncertainties of earthquake loading and structural model. Hence, the seismic fragility curve using the IDA method can be specifically expressed in the following equation:  
\[P_{f}(a)=\frac{\sum\limits_{i=1}^{N}1(D(a_{i})>C|a_{i})}{N} \tag{2}\]  
where \(N\) is the total number of seismic response analysis simulations at a specific seismic intensity. 1 (.) is an indicator function having a value of 0 or 1. Therefore, this equation represents, at a specific seismic intensity, the ratio of the total number of simulations conducted to the number of simulations in which a failure occurs. Due to this methodological nature, the seismic fragility curve is represented by a discrete curve, and numerous nonlinear dynamic seismic analyses must be performed to obtain a complete continuous curve. However, this approach has some issues, since the cost of numerical analyses that can be performed is limited, so it is usually assumed that the seismic fragility data obtained from Equation (2) follow the cumulative log-normal probability distribution of Equation (3). Then, based on the statistical method of the MLE technique, the median \(A_{m}\) and log-standard deviation \(\beta\) of the main parameters of the log-normal probability distribution are estimated [30].  
\[P_{f}(a)=\Phi\left[\frac{\ln(a)-\ln(A_{m})}{\beta}\right] \tag{3}\]Here, \(A_{m}\) represents the seismic intensity value for which the failure probability is 50% at the seismic fragility curve, which means the median seismic performance value of the corresponding structural system. \(\beta\) is the logarithmic standard deviation. The technical details on the process of estimating \(A_{m}\) and \(\beta\) are well described in the publication of [PERSON] et al. [31]. Based on the above descriptions and equations, the procedure for deriving the seismic fragility curve for the piping system, which is widely utilized in the current state, is briefly summarized in Figure 2. As we can see in Figure 2, the IDA-based MLE method, with respect to the seismic fragility estimation of piping systems, requires multiple nonlinear seismic response analyses; thus, this method still is limited in that it demands a substantial cost of numerical computations.