Evolutionary Power Spectral Density Function Research Articles

Design sensitivity analysis of structural systems with damping devices subjected to fully non-stationary stochastic seismic excitations

In the framework of dynamic excitations to be considered during the design phase of structures, the most crucial one is the ground motion acceleration. To increase the structural performances against seismic actions, one of the most effective design criteria is to introduce damping devices.An efficient approach to define the optimal parameters of the damping system is based on the design sensitivity analysis, which provides a quantitative estimate of desirable design change, by relating the available design variables.In this paper a method to evaluate the sensitivities of stochastic response characteristics of structural systems with damping devices subjected to seismic excitations, modelled as fully non-stationary Gaussian stochastic processes, is proposed. The main steps are: i) to define the time–frequency varying response (TFR) function for non-classically damped systems; ii) to evaluate closed form solutions for the first-order derivatives of the TFR function as well as of the one-sided evolutionary power spectral density function of the structural response, with respect to damping parameters of devices; iii) to perform a design sensitivity analysis selecting as performance measure function the non-geometric spectral moments of nodal displacements.A numerical application demonstrates how the proposed approach is suitable to cope with practical problems of engineering interest.

Simulation of non-stationary wind velocity field on bridges based on Taylor series

The simulation of non-stationary wind velocity field based on the spectral representation method often requires significant computational efforts due to the summation of trigonometric functions usually involved in the simulation procedure. Some techniques which make use of FFT algorithm have been developed but most of these techniques deal with seismic ground motions. Limited effort has been devoted to the simulation of non-stationary wind velocity. Therefore, in this paper, a spectral-representation-based technique which takes advantage of FFT algorithm is proposed by combining Cholesky decomposition and Taylor series expansion. The approach consists of locating and expanding the time and frequency non separable part of the decomposed evolutionary power spectral density function by mean of Taylor series expansion to allow the application of the FFT algorithm. Samples of non-stationary wind velocity can be then generated through multiple executions of the FFT algorithm once the Taylor series expansion is successful. The present approach, which is primarily developed for the simulation of non-stationary wind velocity on long-span cable supported bridges, is very efficient since the summation of the trigonometric functions can be carried out through FFT algorithm which is well known for its higher efficiency. The approach was further improved by reformulating the simulation formulas where the order in which the summation operations are executed is imposed.

A stochastic harmonic function representation for non-stationary stochastic processes

The time-domain representation of non-stationary stochastic processes is of paramount importance, in particular for response analysis and reliability evaluation of nonlinear structures. In the present paper a stochastic harmonic function (SHF) representation originally developed for stationary processes is extended to evolutionary non-stationary processes. Utilizing the new scheme, the time-domain representation of non-stationary stochastic processes is expressed as the linear combination of a series of stochastic harmonic components. Different from the classical spectral representation (SR), not only the phase angles but also the frequencies and their associated amplitudes, are treated as random variables. The proposed method could also be regarded as an extension of the classical spectral representation method. However, it is rigorously proved that the new scheme well accommodates the target evolutionary power spectral density function. Compared to the classical spectral representation method, moreover, the new scheme needs much fewer terms to be retained. The first four moments and the distribution properties, e.g., the asymptotical Gaussianity, of the simulated stochastic process via SHF representation are studied. Numerical examples are addressed for illustrative purposes, showing the effectiveness of the proposed scheme.

Sensitivity of the stochastic response of structures coupled with vibrating barriers

The sensitivity of the stochastic response of linear behaving structures controlled by the novel Vibrating Barrier (ViBa) device is scrutinized. The Vibrating Barrier (ViBa) is a massive structure, hosted in the soil, calibrated for protecting structures by exploiting the structure–soil–structure interaction effect. Therefore the paper addresses the study of the sensitivity of soil–structure coupled systems in which the soil is modelled as a linear elastic medium with hysteretic damping. In order to accomplish efficient sensitivity analyses, a reduced model is determined by means of the Craig–Bampton procedure. Moreover, a lumped parameter model is used for converting the hysteretic damping soil model rigorously valid in the frequency domain to the approximately equivalent viscous damping model in order to perform conventional time-history analysis. The sensitivity is evaluated by determining a semi-analytical method based on the dynamic modification approach for the case of multi-variate stochastic input process. The ground motion is modelled as non-stationary zero-mean Gaussian random process defined by a given evolutionary Power Spectral Density function. The paper presents the sensitivity of the response statistics of a model of an industrial building, passively controlled by the ViBa, to relevant design parameters. Comparisons with pertinent Monte Carlo Simulation will show the effectiveness of the proposed approach.

Simulation of Spatially Correlated Nonstationary Response Spectrum–Compatible Ground Motion Time Histories

AbstractA new methodology is presented for the generation of spatially correlated nonstationary ground motion time histories that are compatible with a prescribed response spectrum using the spectral representation method. The method introduces two important improvements over the current state of the art in that it preserves the coherence among the ground motion histories and enables the incorporation of both time and frequency modulation by upgrading the evolutionary power spectral density function with random pulse-like perturbations. An example is provided for the simulation of design spectrum–compatible, uniformly modulated nonstationary acceleration time histories at three locations on the ground, and the results are compared directly with an existing state-of-the-art methodology.

Simulation of Nonstationary Stochastic Processes by Spectral Representation

This paper presents a rigorous derivation of a previously known formula for simulation of one-dimensional, univariate, nonstationary stochastic processes integrating Priestly's evolutionary spectral representation theory. Applying this formula, sample functions can be generated with great computational efficiency. The simulated stochastic process is asymptotically Gaussian as the number of terms tends to infinity. This paper shows that (1) these sample functions accurately reflect the prescribed probabilistic characteristics of the stochastic process when the number of terms in the cosine series is large, i.e., the ensemble averaged evolutionary power spectral density function (PSDF) or autocorrelation function approaches the corresponding target function as the sample size increases, and (2) the simulation formula, under certain conditions, can be reduced to that for nonstationary white noise process or Shinozuka's spectral representation of stationary process. In addition to derivation of simulation formula, three methods are developed in this paper to estimate the evolutionary PSDF of a given time-history data by means of the short-time Fourier transform (STFT), the wavelet transform (WT), and the Hilbert-Huang transform (HHT). A comparison of the PSDF of the well-known El Centro earthquake record estimated by these methods shows that the STFT and the WT give similar results, whereas the HHT gives more concentrated energy at certain frequencies. Effectiveness of the proposed simulation formula for nonstationary sample functions is demonstrated by simulating time histories from the estimated evolutionary PSDFs. Mean acceleration spectrum obtained by averaging the spectra of generated time histories are then presented and compared with the target spectrum to demonstrate the usefulness of this method.

Simulation of non-stationary ground motion processes (I)

This paper presents a spectral representation method for simulation of non-stationary ground motion processes on the basis of Priestley’s evolutionary spectral theory. Following this method, sample processes can be generated using a cosine series formula. It is shown that, these sample processes accurately reflect the prescribed characteristics of the evolutionary power spectral density function when the number of the terms in the cosine series is large enough; and the ensemble expected value and the ensemble autocorrelation function approach the corresponding target functions, respectively, as the sample size increases; and these sample processes are asymptotically normal as the number of the terms in the series tends to infinity. Finally, a few special cases of the formula are discussed, one of which is non-stationary white noise process, and other one is reduced to the formula for simulation of stationary stochastic processes.

Non-geometric spectral moments for frequency varying filtered input processes

The spectral moments have been introduced in the literature for the case of structural systems subjected to stationary Gaussian input processes and were defined as the geometric moments of the one-sided power spectral density function of the response. It has been proved that the spectral moments play an important role for the safety prediction of structural systems. Extension of the definition of spectral moments for non-stationary Gaussian input processes has been provided by means of the geometric interpretation applied to the one-sided evolutionary power spectral density function of the response. In the case of white input process, such a geometric approach leads to divergences and needs adjustments in order to be exploited for the safety assessment in the non-stationary case. The exact formulation for the spectral moments has been proved to be based rather on a non-geometric time domain approach aiming at the evaluation of the co-variances of the pre-envelope response process. The geometric and the non-geometric approaches lead to the same spectral moments for the stationary case only. In this paper, a procedure for the evaluation of the non-geometric spectral moments is presented with particular attention to the case of non-stationary filtered Gaussian white noise which is both amplitude and frequency-modulated. The presented procedure is based on the formulation of the pre-envelope co-variance differential equations, whose solution is obtained by means of an approximated expression of the time-dependent transition matrix of the system constituted by the structure and the time varying frequency filter.

Fully Nonstationary Analytical Earthquake Ground-Motion Model

A versatile, nonstationary stochastic ground-motion model accounting for the time variation of both intensity and frequency content typical of real earthquake ground motions is formulated and validated. An extension of the Thomson's spectrum estimation method is used to adaptively estimate the evolutionary power spectral density (PSD) function of the target ground acceleration record. The parameters of this continuous-time, analytical, stochastic earthquake model are determined by least-square fitting the analytical evolutionary PSD function of the model to the target evolutionary PSD function estimated. As application examples, the proposed model is applied to two actual earthquake records. In each case, model validation is obtained by comparing the second-order statistics of several traditional ground-motion parameters and the probabilistic linear-elastic response spectra simulated using the earthquake model with their deterministic counterparts characterizing the target record.