# Efficient Seismic Fragility Analysis for Large-Scale Piping System Utilizing Bayesian Approach
## 4 Examples
### Seismic Response Analysis and Seismic Fragility Assessment  
In this study, a time history analysis was employed as a seismic response analysis of the piping system to consider the nonlinear behaviors of the piping joint components. In the numerical model of the building-piping coupled system, a 5% damping ratio was applied to the building structural model, and a 2% damping ratio was applied to the piping system structural model. Here is the technical background for determining these damping ratios: For RC structures, the USNRC [36] recommended that the 5-7% damping ratio be used for the response prediction under strong earthquakes. In addition, most design codes and standards do not distinguish between variations in the damping ratio due to structural materials, and a 5% damping ratio is implicitly used in the seismic forces and design response spectra defined by the criteria [37]. Such damping ratio values have also been demonstrated in the measurement and various testing of damping ratios concerning the structural system subjected to the seismic load [38]. On the other hand, for piping systems, it is widely recommended to use a 2-5% damping ratio during an earthquake loading [36]. Besides, the damping ratio values for piping systems have been found to be between 2% and 5% in several tests [39; 40]. Under this background, in this study, the damping ratio of the RC building structure was set at 5%, and the damping ratio of the piping system was determined to be 2%. In particular, the validity verification of the damping of the piping system was demonstrated through the piping system dynamic test results [6]. As such, due to the difference in the damping ratio between the RC structural frame and the piping, the frame-piping integrated system has a non-classical damping matrix, unlike the general structural system. In this case, the damping matrix for the entire system is constructed by assembling the damping matrix for the two subsystems: the first is for the RC structural frame and the second one is for the piping system. In these matrices, the part related to the common degree of freedom at the interface between the two subsystems includes contributions to both subsystems. The damping matrix of each numerical model of subsystems was derived by a method of the Rayleigh damping matrix generation. Based on this non-classical damping matrix, the results of the seismic response were derived at all points of the target piping system through the direct time history analysis using the Newmark method. The seismic fragility analysis of the piping system was performed based on the T-joint element responses. This numerical analysis was conducted through OPENSES code [33], which has been proven to be accurate for the earthquake time history analysis. To perform the piping seismic fragility analyses, a total of 21 recorded earthquake ground motions, which were proposed in the ATC-63 project [41], were utilized in this study. In order to reflect the uncertainty characteristics of the earthquake ground motion, the selected ground motion inputs included normal, reverse, and strike-slip fault phenomena, which are various fault mechanisms of earthquakes. In addition, to account for various earthquake occurrence conditions, earthquakes in various countries such as the United States, Taiwan, Japan, Iran, Turkey, etc. were selected, and less than two ground motions were only chosen regarding each earthquake. The selected ground motions were recorded from earthquakes having a moment magnitude of larger than 6.5 in all cases, and the peak ground acceleration (PGA) of the ground motions was distributed between 0.210 and 0.822 g. Detailed information on the selected ground motions such as the time series shape, response spectrum shapes, energy distribution, etc., are presented in the literature of [PERSON] [42]. The limit state of the piping system was defined for the T-joint where the nonlinear behavior was concentrated during an earthquake and, thus, leakage occurred.