Generation of time and frequency dependent random processes compatible with recorded seismic accelerograms

Strong earthquakes, caused by a sudden release of stress along faults in the earth's crust, are among the most damaging and deadly natural phenomena. The energy released by an earthquake travels in the form of waves, known as seismic waves. When the seismic waves reach the ground surface, the produced shaking induces dynamic effects on structural and geotechnical systems that can severely compromise their safety level and stability. The induced dynamic action at a given site depends on several factors such as: the strength, and duration of shaking and the mechanical properties of the soil layers crossed by the seismic waves. The knowledge of ground motion attributable to earthquakes is essential for the design of earthquake-resistant structures, and the evaluation of the seismic vulnerability of existing ones. Among all possible sources of uncertainty stemming from the structural and soil material properties, the selection of the earthquake-induced ground motions has the highest effect on the variability observed in the response history analysis of structures and geotechnical systems. The characteristics of the design ground motion, representing the level of shaking for which satisfactory performance is expected, are influenced by the characteristics of seismic source, the rupture process, the source-site travel path, the local site conditions, and the importance of the structure or facility for which the ground motion is to be used. When the local, geologic and tectonic conditions of the site of interest is similar to those of sites where actual strong motions have previously been detected, the recorded time histories can be used directly as input motions in the dynamic analyses. Otherwise, the use of artificial accelerograms, having characteristics consistent with those of actual earthquakes, could represent a valid alternative. However, the generation of artificial accelerograms might not be easy: many motions that appear reasonable in the time domain may not be so when examined in the frequency domain, and vice versa. Furthermore, many reasonable-looking time histories of acceleration produce, after integration, unreasonable time histories of velocity and/or displacement. The main aim of this Ph.D. thesis is to propose two novel procedures for the generation of artificial accelerograms having the same time and frequency contents of recorded time histories. The purpose of the Chapter 1 is to illustrate the basic concepts of seismic engineering that may be useful in reading this thesis. Specifically, this Chapter describes the recording instruments used to detect strong ground motion and the signals processing techniques by which measured motions are corrected. Finally, a brief overview of the main intensity measures that can be used to characterize the amplitude, frequency content and duration of strong ground motions, is presented. Chapter 2 highlights the limitations of the classical Fourier analysis in describing non-stationary signals whose statistical parameters vary with time. Therefore, an introduction to joint time-frequency signal representation through the short time Fourier transform and the wavelet transform, is presented. More details will be given about a particular kind of harmonic wavelet, called “circular”, and the related theory. In Chapter 3, after a brief introduction on stochastic variables and processes, the discrete circular wavelet transform is proposed to randomly generate an arbitrary number of records with the same non-stationary characteristics of the target accelerogram. The influence of a novel correlation structure for the definition of the wavelet random phases and a different subdivision of the earthquake record in frequency bands are highlighted and discussed. Through the proposed stochastic generation method, an effective trade-off is identified between localisation in the frequency domain and in the time domain of the generated signals. In Chapter 4, a novel method for generating samples of a fully non-stationary zero-mean Gaussian process, having a target acceleration time-history as one of its own samples, is described. The evolutionary power spectral density (EPSD) function of the proposed fully non-stationary model is evaluated as the sum of uniformly modulated processes. These are defined in each time interval, as the product of deterministic modulating functions per stationary zero-mean Gaussian sub-processes, whose unimodal power spectral density (PSD) functions are filtered by high pass and low pass Butterworth filters. In each time interval the parameters of the modulating functions are estimated by least-square fitting the expected energy of the proposed model to the energy of the target accelerogram, while the parameters of the PSD functions of stationary sub-processes are estimated once both occurrences of peaks and zero-level up-crossings of the target accelerogram, in the various intervals, are counted. Chapter 4 concludes with the application of an iterative procedure aimed to obtain the compatibility between the mean spectrum of the generated samples and a target one. Depending on the aim to be achieved, it is possible to obtain the spectrum-compatibility in terms of response spectrum or Fourier spectrum, using two different corrective PSD function terms. In Chapter 5, a new approach which takes into account the inherent random nature of the ground motion acceleration as well as epistemic uncertainties affecting the definition of its power spectrum, is presented. Specifically, seismic excitation is modelled as a zero-mean stationary Gaussian random process fully characterized by an imprecise PSD function, i.e. with interval parameters. The ranges of such interval parameters are determined by analysing a large set of accelerograms recorded on rigid soil deposits. To discard outliers, the Chauvenet’s Criterion is applied iteratively. The proposed imprecise PSD function may be viewed as representative of the actual accelerograms recorded on rigid soil deposits. Due to imprecision of the excitation, the fractile of order p of the structural response turn out to have an interval nature. The bounds of the fractile order p are here used to define the range of structural performance. In this Ph.D. thesis, several numerical applications will be done in order to test the effectiveness of the proposed procedures.