Seismic motions are likely to cause large displacements in nuclear power plants because the main mode of their piping systems is dominated by the low-frequency region. Additionally, large relative displacement may occur in the piping systems because their supports are installed in several places, and each support is subjected to different seismic motions. Therefore, to assess the seismic performance of a piping system, the relative displacement repeated by seismic motions must be considered. In this study, in-plane cyclic loading tests were conducted under various constant amplitudes using test specimens composed of SCH 40 3-inch pipes and a tee in the piping system of a nuclear power plant. Additionally, an attempt was made to quantitatively express the failure criteria using a damage index based on the dissipated energy that used the force–displacement and moment–deformation angle relationships. The failure mode was defined as the leakage caused by a through-wall crack, and the failure criteria were compared and analyzed using the damage index of Park and Ang and that of Banon. Additionally, the method of defining the yield point required to calculate the damage index was examined. It was confirmed that the failure criteria of the SCH 40 3-inch carbon steel pipe tee can be effectively expressed using the damage index.