Probabilistic Analysis of Random Structural Intensity for Structural Members under Stochastic Loadings

Abstract

The concept of structural intensity (SI) is extended to the random domain by introducing a physical quantity denominated random structural intensity (RSI). This quantity is formulated for mechanical systems whose dynamical responses are stochastic due to random excitations. In order to fully characterize the stochastic behavior of a system under random loadings, it is imperative to obtain the probability density function (PDF) of RSI. Based on the elastic theory and the definition of SI, RSI is expressed as functions of system responses. In general, the PDF of system responses is governed by Fokker–Planck–Kolmogorov (FPK) equation under the assumption that random dynamic loadings are idealized as white noise excitations. Therefore, the PDF of RSI is derived with the joint PDF of system responses. In the present study, four demonstrating cases of beams and plates under separately concentrated and uniform random loadings are studied to investigate the properties of RSI. Stationary and non-stationary PDFs of RSI at arbitrary section of beam and plate are obtained. Numerical results show that the PDF of RSI is transient at early stage of stationary loading and then converges to the exact stationary ones as time increases. With the obtained PDFs of RSI, energy transmission path over the beam and plate can be determined, which is guided from the locations with lower probabilities of RSI to the ones with higher probabilities of RSI. Furthermore, virtual energy flow sinks on the plate and beam can be found, which are identified by the locations with the maximum probabilities of RSI.