Abstract: Theoretical researches on the stationary random vibration of a single-degree-of freedom structure with the superelastic shape memory alloy (SMA) damper subjected to earthquake excitations are carried out. The earthquake ground motion is assumed to be a stationary filtered color noise process, and the parameters of the random earthquake ground motion model (i.e. the stationary duration and the spectral intensity parameter) are determined according to the Code for seismic design of buildings (GB50011-2010). Random state equations of the SMA-damped structure under the stationary filtered color noise excitation are set up. Then, using an stochastic equivalent linearization method, the stationary random seismic responses of the SMA-damped structure are formulated, and the dynamic reliability of the structure is obtained on the basis of the first threshold crossing failure criterion. Numerical simulations prove that the proposed random vibration formulations are correct, and that the SMA damper can decrease the vibration of the structure effectively, and increase the dynamic reliability of the structure attributed to its capability of dissipating the earthquake energy.