Abstract Analytical seismic response of multi-storey buildings isolated by lead–rubber bearings (LRB) is investigated under near-fault motions. The superstructure is idealized as a linear shear type flexible building. The force–deformation behaviour of the LRB is modelled as bilinear with viscous damping. The governing equations of motion of the isolated structural system are derived and the response of the system to normal component of six recorded near-fault motions is evaluated by step-by-step numerical method. The variation of top floor absolute acceleration and bearing displacement of the isolated building is plotted under different system parameters such as superstructure flexibility, isolation period and bearing yield strength. The comparison of results indicated that for low bearing yield strength there is significant displacement in the bearing under near-fault motions. In addition, there also exists a particular value of the yield strength of the LRB for which the top floor absolute acceleration of the building attains the minimum value. Further, the optimum bearing yield strength is derived for different system parameters under near-fault motions. The criteria selected for optimality are the minimization of both the top floor acceleration and the bearing displacement. The optimum yield strength of the LRB is found to be in the range of 10%–15% of the total weight of the building under near-fault motions. In addition, the response of bridge seismically isolated by the LRB is also investigated and found that there exists a particular value of the bearing yield strength for which the pier base shear and deck acceleration attain the minimum value under near-fault motions.