Probability distribution estimation for harmonisable loads and responses of linear elastic structures

In probabilistic distribution-based structural dynamics representing joint probability density functions (JPDFs) of loads and those of structural responses at any collection of time instants is a challenging and important issue. In this study, based on the mean square (m.s.) stochastic calculus, a novel method for estimating the JPDFs of external loads and those of structural responses is proposed. In particular, an external load is modelled as a harmonisable process, and its two approximate representations are proposed by the discrete Fourier transform. The JPDFs of the Fourier coefficients are directly estimated from multiple load samples. Using the load JPDFs in the frequency domain can estimate the JPDFs of the load at any finite time instants. By applying the load representations to the dynamic governing equation of linear elastic structures, the JPDFs of the structural responses at any finite time instants can be estimated. The two load representations are proved to converge in m.s. to the exact load process under certain conditions. Correspondingly, the estimated JPDFs of the load and those of structural responses in the time domain converge to their corresponding exact ones. The proposed method is numerically verified using a linear elastic structure under a nonstationary earthquake ground motion.