The installation of base isolation systems in nuclear power plants can improve their safety from seismic loads. However, nuclear power plants with base isolation systems experience greater displacement as they handle seismic loads. The increase in relative displacement is caused by the installed base isolation systems, which increase the seismic risk of the interface piping system. It was found that the failure mode of the interface piping system was low-cycle fatigue failure accompanied by ratcheting, and the fittings (elbows and tees) failed due to the concentration of nonlinear behavior. Therefore, in this study, the limit state was defined as leakage, and an in-plane cyclic loading test was conducted in order to quantitatively express the failure criteria for the SCH40 6-inch carbon steel pipe elbow due to low-cycle fatigue failure. The leakage line and low-cycle fatigue curves of the SCH40 6-inch carbon steel pipe elbow were presented based on the test results. In addition, the limit state was quantitatively expressed using the damage index, based on the combination of ductility and energy dissipation. The average values of the damage index for the 6-inch pipe elbow calculated using the force−displacement (P–D) and moment−relative deformation angle (M–R) relationships were found to be 10.91 and 11.27, respectively.