# Efficient Seismic Fragility Analysis for Large-Scale Piping System Utilizing Bayesian Approach
## 3 Proposed Method  
This subsection proposes and describes a method that can calculate a more efficient and relatively accurate piping seismic fragility curve by introducing the Bayesian updating concept based on the existing piping seismic fragility methods (i.e., (1) IDA-based MLE method and (2) EPRI SOV method). In the Bayesian updating, the seismic fragility probability model parameters are estimated using the following formulation. \[f(\theta|\mathbf{y})=\kappa L(\theta|\mathbf{y})f(\theta) \tag{11}\]  
where \(\theta=\{A_{m};\beta\}\) represents the set of unknown parameters of the probability model of seismic fragility. \(\mathbf{y}\) denotes the data vector of \(n\) observations \(\mathbf{y}=\{y_{1};y_{2};\ldots,y_{n}\}\). \(f(\theta)\) is a prior probability model of seismic fragility. \(L(\theta|\mathbf{y})\) is the likelihood function that includes the information from the data vector \(\mathbf{y}\). \(\mathbf{f(\theta|y)}\) is the posterior probability distribution that reflects the updated state of information about the model parameter vector \(\theta\). \(\kappa\) is a normalizing factor as \(\kappa=\left(\int L(\theta|\mathbf{y})f(\theta)d\theta)\right)^{-1}\). The detailed procedures for Bayesian updates are detailed in other literature [25,32]. Figure 3 shows the main procedure of the proposed method. First, we establish a coupled numerical model of the building-piping system. In addition, an earthquake ground motion corresponding to the seismic intensity of the design basis earthquake is defined for the building foundation. A dynamic seismic response analysis is performed based on the building-piping coupled numerical model and the input ground motion of the design basis earthquake. Based on the calculated numerical results, the first step is to derive the prior piping seismic fragility curve using the EPRI SOV seismic fragility method. The next step is to select the additional seismic intensities required to improve the accuracy of the piping seismic fragility curve based on the acquired prior fragility curve and define earthquake ground motions associated with these. Based on the selected seismic intensities, the defined ground motions are scaled and the nonlinear seismic response analyses are iteratively performed based on these scaled ground motions according to the increase of seismic intensities. Based on these numerical results, conditional failure probabilities are calculated at selected seismic intensities using Equation (2). Finally, based on the conditional failure probability information at the selected seismic intensities obtained in the previous procedure, the prior piping seismic fragility curve information is re-evaluated using the Bayesian updating concept to derive the final seismic fragility curve (so-called a posterior seismic fragility curve). As a result, the proposed approach can improve the accuracy of the existing EPRI SOV seismic fragility method through additional nonlinear seismic response analysis results. Furthermore, the cost of dynamic nonlinear numerical analyses can be minimized by using the prior seismic fragility curve information, so that the efficiency can be improved, unlike the IDA-based MLE method. To be more specific, the IDA-based MLE method is based on the seismic fragility data obtained from the IDAs, and the key is to estimate the major variables of the cumulative probability distribution function (i.e., the cumulative log-normal distribution function) of the seismic fragility utilizing the statistical method of the MLE. As a result of such methodological features, the seismic intensity is usually divided evenly between small and large values, and the seismic fragility data is obtained in the corresponding all seismic intensities. On the other hand, the method proposed in this study efficiently derives the prior probability distribution for the seismic fragility using the EPRI SOV method at first. Next, to improve the accuracy of this prior probability distribution of the seismic fragility, additional seismic fragility data is secured.