Numerical Approach for Non-Stationary Covariance Analysis in Linear Random Vibrations Applied to a Multi-Storey Building Under Seismic Action

Real physical events, such as earthquakes, sea waves, and wind, are often random in nature and can be defined as a realization of a stochastic process. The simplest way to model them is to use stationary processes. However, in some cases, it’s necessary to consider their evolutionary nature over time to properly account for their non-stationary nature, as in the case of seismic records. In such circumstances, it’s common to assume the process as a non-stationary separable process modulated by a deterministic function that can represent the time variation of the physical event. Solving this problem is a complex task, and there are a few numerical approaches proposed with this aim. In this paper, the case of the dynamic structural response of a linear multi-degree-of-freedom system subject to non-stationary random Gaussian dynamic actions is analyzed. In the case of non-stationary inputs, the evaluation of second-order spectral moments requires the solution of a Lyapunov matrix differential equation. In this work, numerical schemes for its resolution are proposed. The numerical computational effort is minimized by taking into account the symmetry characteristic of the state space covariance matrix. As an application of the proposed method, a multi-storey building is analyzed to determine the reliability of ensuring that the maximum inter-storey drift does not exceed a specified acceptable limit.