# Efficient Seismic Fragility Analysis for Large-Scale Piping System Utilizing Bayesian Approach
## 4 Examples  
### Building-Piping Coupled Structural Model  
The numerical model of the building-piping coupled system used in this study was developed with OPENSEES (The Open System for Earthquake Engineering Simulation) code [33] and is constructed as shown in Figure 4. Specifically, a reinforced concrete moment resistance frame (called RC-SMRF: Special Moment Resisting Frame) was used for the building structural model. This structural model is the four-story RC-SMRF model modified from the original model [16], which was made in accordance with IBC (International Building Code) and ACI (American Concrete Institute) 318-14 [34] code by [PERSON] et al. [35]. The model was basically designed to have five transverse bays (7.4 m long), a single longitudinal bay (9.1 m long), and the same inter-story height (3.7 m height). The main feature of this building structural model was that the sum of the moment forces acting on the column was designed to be at least 20% greater than the sum of the moment forces of the beam, according to the \"strong column-weak beam design concept.\" The boundary condition of the building structural model is set to have a fixed condition at the base because the building is rigidly constrained to the solid foundation. The building structural model shown in Figure 4 is a two-dimensional structural model using only a single bay, and the detailed cross-sectional information and material properties of columns and beams are described in detail in other documents [16; 35]. The piping system structural model was targeted for a sprinkler piping system, for which dynamic cyclic testing was conducted as part of the \"Simulation of the Seismic Performance of Nonstructural Systems\" project at the University at Buffalo (UB), the State University of New York. Specifically, the utilized piping system model consists of a box-type ceiling, pedestal, vertical hangers, wire restraints, transverse sway braces, 1-inch pipe, 2-inch pipe, 4-inch pipe, etc., as shown in Figure 4. The main line and the vertical riser consist of 4-inch pipes, and the branches are composed of 1-inch and 2-inch pipes. Four-inch grooved fit joints were used for the connections for the main line and the vertical riser, and black iron threaded T-joints were used for the branch line (i.e., 1-inch and 2-inch pipes) connections. Detailed information on the layouts, material information, etc. of the piping system is provided in [PERSON]'s literature [6]. Figure 4: Building–piping coupled structural model. The numerical model of the piping system basically used a linear elastic frame element, and at the connections where nonlinear behavior was predicted during an earthquake, three different types of nonlinear rotating spring elements were used for each of the 1-inch, 2-inch, and 4-inch piping joints, respectively. Specifically, the hysteretic behavior of the 1-inch-pipe T-joints is characterized by a yield strength, initial stiffness, and hardening stiffness. The isotropic hardening parameter defines the increase in the compressive/tensile yield strength as the ratio of the yield strength after the plastic deformation. The hysteretic behavior of the T-joint connection components in the 2-inch and 4-inch pipes shows the behavior of the pinched moment-rotation responses. The numerical models of the connection components were verified with the results of the dynamic cyclic test for the T-joint component [6], as shown in Figure 5. Moreover, the accuracy of the overall piping system numerical model was verified through the leakage position record of the T-joint connections of the piping system, which was observed in the dynamic test regarding the entire piping system. Figure 6 compares the observed leakage location results (\"Test Leakage\") in the overall piping system during the dynamic cyclic test and the leakage location results (\"Analysis Leakage\") obtained from the numerical simulations of dynamic nonlinear seismic response analyses based on the numerical piping structural model.