Orthogonal Expansion Method Of Engineering Stochastic Dynamic Loads And Its Application

The dynamic loads acting on engineering structures not only vary with time, butalso have apparent stochastic characteristics. In classical random vibration theory,stochastic dynamic loads are generally depicted by the power spectral densityfunction, which actually is the second-order statistical value of a stationary stochasticprocess. Therefore, the probabilistic information of the original stochastic process cannot be roundly reflected. Whereas, the classical random vibration analysis can onlygive numerical characteristic solutions of structural response, not obtain the precisesolution of structural reliability. This, consequently, leads to a bottleneck for thedevelopment of structural reliability theory. To solve this predicament, the orthogonalexpansion method of engineering stochastic dynamic loads and its application arethoroughly studied in this paper based on the rationale of Karhunen-Loevedecomposition.The Karhunen-Loeve (K-L) decomposition provides a feasible approach to studya stochastic process using a set of random variables. Its basic idea is to represent astochastic process as a linear combination of deterministic functions modulated byuncorrelated random coefficients. Practically, the K-L decomposition needs to solvethe Fredholm integral equation. However, the analytical solutions of the Fredholmintegral equation are generally unavailable except for few cases, thus an expansionmethod based on normalized orthogonal bases is first proposed to decomposestochastic process. It has been proved that this method is equivalent to theKarhunen-Loeve decomposition when expanding terms N→∞. Further, after thecomparative study of the Fourier orthogonal bases and Hartley bases, we choose theHartley orthogonal bases to expand the stochastic process.Utilizing the above method, the stochastic process for earthquake ground motionis carried out based on the Hartley orthogonal expansion bases. In order to capture themain probabilistic characteristics of seismic ground motion, we carry out theorthogonal expansion directly on the seismic displacement process. Further by using the principle of energy equivalence, the expanding expressions of seismicacceleration process is achieved with 10 random variables.This orthogonal expansion method is also applied to the research on thesimulation of random wind velocity fields. First the random wind velocity field isdecomposed into the product of a stochastic process and a random field, whichrepresent the temporal property and the spatial correlation of wind velocityfluctuations, respectively. The stochastic process for wind velocity fluctuations maybe represented as a finite sum of deterministic time functions with correspondinguncorrelated random coefficients by the orthogonal expansion. Similarly, the randomfield can be expressed as a combination form with 5 random variables by theKarhunen-Loeve decomposition. Finally, a numerical example is given todemonstrate the accuracy and effectiveness of this procedure using the numbertheoretical method.Since the recently developed probability density evolution method (PDEM) iscapable of capturing instantaneous probability density function and its evolution oflinear and/or nonlinear response of structures. So it is natural to combine the PDEMand the foregoing orthogonal expansion of seismic ground motion to study thenonlinear random earthquake response. Furthermore, the aseismatic reliability ofstructures is assessed using the idea of equivalent extreme-value, which can be usedaccurately to evaluate structural systems under compound failure criterion.Finally, further researches are briefly discussed.