It has often been reported that, when building structures are subjected to near-fault earthquake ground motions, horizontal and vertical impulsive inputs may cause critical damage during the first few seconds. In practical design of building structures, however, the safety check, taking into account the effect of multi-component ground motions, is hardly conducted except the design of important structures such as high-rise buildings and nuclear power plants. Furthermore, it is not clear how the correlation of multi-component ground motions influences the actual safety of structures. In this paper, the detailed property of critical excitation is discussed in association with the relationship between the characteristics of ground motions and those of structures. The properties of various auto power spectral density (PSD) functions of the horizontal and vertical ground motions are investigated, and those of the critical cross PSD function of these two-directional ground motions are found by a devised algorithm in a feasible complex plane. A closed-form expression is derived from the critical relation of the auto PSD functions of the simultaneous inputs. This critical excitation method provides us with a new approach for earthquake-resistant design against the possible future earthquake which causes the critical damages to buildings. It has often been reported that, when building structures are subjected to near-fault earthquake ground motions, horizontal and vertical impulsive inputs may cause critical damage during the first few seconds. In practical design of building structures, however, the safety check, taking into account the effect of multi-component ground motions, is hardly assumed the design of important structures such as high-rise buildings and nuclear power plants. furthermore, it is not clear how the correlation of multi-component ground motions influences the actual safety of structures. In this paper, the detailed property of critical excitation is discussed in association with the relationship between the characteristics of ground motions and those of structures. The properties of various auto power spectral density (PSD) functions of the horizontal and vertical ground motions are investigated, and those of the critical cross PSD function of these two-directional ground motions are found by a devised algorithm in a feasible complex plane. A closed-form expression is derived from the critical relation of the auto PSD functions of the simultaneous inputs. This critical excitation method provides us with a new approach for earthquake-resistant design against the possible future earthquake which causes the critical damages to buildings.