For the aseismic design of structures, the probabilistic approach would be absolutely necessary, because earthquake excitations are considered to be essentially random. In this case the most realistic idealization of the analytical model will be three-dimensional representations of the input as well as the structure. In this sense this paper describes the stochastic response of the spacial structure subjected to three-dimensional earthquake excitations with assumptions of the stationary random process and the linear system, focused on how to express the cross spectral density matrix of inputs in the calculation of the cross spectral density matrix of outputs through the frequency response function matrix of a system.

This paper mainly consists of the following contents. Section 2 briefly describes the analytical process to estimate the response of three-dimensional structure due to three-dimensional earthquake motions by means of the probabilistic theory. In section 3 general expressions of earthquake excitations required in this method are discussed and proposed. This approach is applied to a simple three-dimensional structure as an example and results are examined in section 4.

*Chief of Dynamics Section, Structures Division, B.R.I. Dr.Eng