The seismic response characteristics of a lead-rubber-bearing(LRB) base-isolated structure under rare and very rare earthquakes were investigated. The acceleration, ductility coefficient, and shear strain of the LRB increase significantly under very rare earthquakes in comparison to rare earthquakes; in particular, the shear strain of the LRB may exceed the ultimate shear strain and cause damage to the base-isolated structure. The criterion selected for the optimum tuned inerter damper (TID) of the TID-LRB hybrid control system is the minimization of the mean value of the maximum shear strain of the LRB. For each inertance mass ratio of the TID, there exists an optimum tuning frequency ratio and damping ratio of the TID to minimize the shear strain of the LRB, and the effectiveness is increased with a higher inertance mass ratio. By equipping the TID with appropriate parameters, the safety of the LRB during rare and very rare earthquakes can be ensured. Finally, the pounding response of the base-isolated structure collision with the moat wall under very rare earthquakes was analyzed. It was observed that under very rare earthquakes, the ductility coefficients of the superstructure by equipping with the suitable TID were improved, and the shear strain of the LRB was reduced. In addition, equipping the TID can reduce the required width of the isolation joint to avoid collision between the isolation layer and the moat wall, and with an increase in the inertance mass ratio, the required width of the isolation joint is smaller.
