A simple method which can be applied in seismic codes to determine the critical angle of seismic incidence and the corresponding peak response of structures subjected to two horizontal components applied along any arbitrary directions and to the vertical component of earthquake ground motion, is proposed in this paper. The seismic components are given in terms of response spectra that may be equal or have different spectral shapes. The structures are discrete, linear systems with viscous damping. The method, which is based on the response spectrum method of analysis, requires the solution of standard cases of seismic analysis and therefore can be easily implemented in standard computer programs. For the general case of three arbitrary response spectra, the method requires the solution of five seismic loading cases, two for each horizontal component and one for the vertical component. If the horizontal response spectra have the same shape or if there is only one horizontal component, it is then required to solve just two seismic loading cases for the horizontal components and one for the vertical component. It can be shown that the formulas derived for the critical angles and the peak response are essentially identical to the ones obtained earlier by Smeby and Der Kiureghian using random vibration theory.
The application and the accuracy of the method is illustrated by means of numerical analysis of buildings, comparing the results with those obtained using other proposed methods. For the specific case of two horizontal spectra with identical shape and an arbitrary vertical spectra, the critical angle neither depends on the spectral ratio of the two horizontal components nor on the vertical spectrum. For the special case of equal horizontal spectra, the structural response does not vary with the angle of incidence and it is an upper bound for all possible responses.