This paper evaluates the seismic structural and non-structural performance of selfcentering and conventional structural systems combined with supplemental viscous dampers. For this purpose, a parametric study on the seismic response of highly damped single-degree-of-freedom systems with self-centering flag-shaped or bilinear elastoplastic hysteresis is conducted. Statistical response results are used to evaluate and quantify the effects of supplemental viscous damping, strength ratio and period of vibration on seismic peak displacements, residual displacements and peak total accelerations. Among other findings, it is shown that decreasing the strength of nonlinear systems effectively decreases total accelerations, while added damping increases total accelerations and generally decreases residual displacements. Interestingly, this work shows that in some instances added damping may result in increased residual displacements of bilinear elastoplastic systems. Simple design cases demonstrate how these findings can be considered when designing highly damped structures to reduce structural and nonstructural damage.