Using confined shape memory alloy (SMA) bar or plate, this study proposes an innovative self-centering damper. The damper is essentially properly machined SMA core, i.e., bar or plate, that encased in buckling-restrained device. To prove the design concept, cyclic loading tests were carried out. According to the test results, the damper exhibited desired flag-shape hysteretic behaviors upon both tension and compression actions, although asymmetric behavior is noted. Based on the experimental data, the hysteretic parameters that interested by seismic applications, such as the strength, stiffness, equivalent damping ratio and recentering capacity, are quantified. Processed in the Matlab/Simulink environment, a preliminary evaluation of the seismic control effect for this damper was conducted. The proposed damper was placed at the first story of a multi-story frame and then the original and controlled structures were subjected to earthquake excitations. The numerical outcome indicated the damper is effective in controlling seismic deformation demands. Besides, a companion SMA damper which represents a popular type in previous studies is also introduced in the analysis to further reveal the seismic control characteristics of the newly proposed damper. In current case, it was found that although the current SMA damper shows asymmetric tension-compression behavior, it successfully contributes comparable seismic control effect as those having symmetrical cyclic behavior. Additionally, the proposed damper even shows better global performance in controlling acceleration demands. Thus, this paper reduces the concern of using SMA dampers with asymmetric cyclic behavior to a certain degree.