Determination of design spectrum compatible evolutionary spectra via Monte Carlo peak factor estimation

The problem of generating ensembles of artificial non-stationary earthquake accelerograms compatible with a given (target) response/design spectrum is cast on a stochastic basis. The design spectrum of the European aseismic code provisions (EC8) for various soil conditions and damping ratios is used as a paradigm of a design/target spectrum. The generated accelerograms are construed as realizations of a non-stationary random process; they are char-acterized in the frequency domain by a parametrically defined evolutionary power spectrum (EPS). An appropriate least squared optimization problem is formulated for the determination of the parameters of the EPS. The solution of this problem involves the incorporation of a “peak factor” which is used to re-late the target spectrum to the EPS in a probabilistic context. To this end, a comprehensive Monte Carlo study is undertaken to estimate numerically the statistical properties of the peak factor from appropriately computed popula-tions, and to derive polynomial expressions for the median frequency-dependent peak factors (peak factor spectra). These expressions are used in conjunction with the herein adopted optimization problem to determine EPSs compatible with the EC8 design spectrum. The derived median peak factor spectra yield an excellent level of agreement between the EC8 spectrum and the ensemble average and median response spectra of simulated EPS-compatible ensembles of accelerograms

Derivation of Eurocode 8 spectrum-compatible time-histories from recorded seismic accelerograms via harmonic wavelets

A computationally efficient harmonic wavelet-based iterative procedure is proposed to modify suites of recorded accelerograms to be used in the aseismic design of critical structures regulated by the European code provisions (EC8). Special attention is focused on assessing the potential of appropriately defined orthogonal harmonic wavelet basis functions to derive design spectrum compatible time-histories which preserve the non-stationary characteristics of the original recorded signals. This is a quite desirable attribute in the practice of the aseismic design of yielding structures. In this regard, seven recorded accelerograms recommended for the design of base-isolated structures are modified via the proposed procedure and base-line adjusted to meet the pertinent EC8 compatibility criteria. The instantaneous energy (IE) and the mean instantaneous frequency (MIF) of the modified EC8 compatible time-histories extracted from appropriate wavelet-based signal time-frequency analyses are compared vis-à-vis the IE and MIF of the corresponding original accelerograms. Examining these numerical results, it is established that the herein proposed procedure is a useful tool for processing recorded accelerograms in cases where accounting for the time-varying energy content and frequency composition of strong ground motions associated with historic seismic events is deemed essential in aseismic design

Derivation of equivalent linear properties of Bouc-Wen hysteretic systems for seismic response spectrum analysis via statistical linearization

A newly proposed statistical linearization based formulation is used to derive effective linear properties (ELPs), namely damping ratio and natural frequency, for stochastically excited hysteretic oscillatorsinvolving the Bouc-Wen force-deformation phenomenological model. This is achieved by first using a frequency domain statistical linearization step to substitute a Bouc-Wen oscillator by a third order linear system. Next, this third order linear system is reduced to a second order linear oscillator characterized by a set of ELPs by enforcing equality of certain response statistics of the two linear systems. The proposed formulation is utilized in conjunction with quasi-stationary stochastic processes compatible with elastic response spectra commonly used to represent the input seismic action in earthquake resistant design of structures. Then, the derived ELPs are used to estimate the peak response of Bouc-Wen hysteretic oscillators without numerical integration of the nonlinear equation of motion; this is done in the context of linear response spectrum-based dynamic analysis. Numerical results pertaining to the elastic response spectrum of the current European aseismic code provisions (EC8) are presented to demonstrate the usefulness of the proposed approach. These results are supported by pertinent Monte Carlo simulations involving an ensemble of non-stationary EC8 spectrum compatible accelerograms. The proposed approach can hopefully be an effective tool in the preliminary aseismic design stages of yielding structures and structural members commonly represented by the Bouc-Wen hysteretic model within either a force-based or a displacement-based context

Statistical linearization based estimation of the peak response of nonlinear systems subject to the EC8 design spectrum

A stochastic approach is proposed to obtain reliable estimates of the peak response of nonlinear systems to excitations specified via a response∕design seismic spectrum. This is achieved without resorting to numerical integration of the governing nonlinear equations of motion. First, a numerical scheme is utilized to derive a power spectrum which is compatible in a stochastic sense to a given elastic design spectrum. This spectrum is then treated as the excitation spectrum in the context of the statistical linearization method to determine effective parameters, damping and stiffness, corresponding to an equivalent linear system (ELS). The obtained parameters are used in conjunction with the linear design spectrum, for various values of damping, to estimate the response of certain nonlinear systems. The case of single‐degree‐of‐freedom systems with cubic stiffness nonlinearity and hysteretic systems whose restoring force traces a bilinear law are considered in conjunction with the elastic design spectrum prescribed by the European aseismic code provisions (EC8). Monte Carlo simulations pertaining to an ensemble of non‐stationary EC 8 design spectrum compatible accelerograms are conducted to confirm that the average peak response of the nonlinear systems compare reasonably well with that of the ELS. This is true, even in cases where the response of the nonlinear oscillators deviates significantly from the linear one. In this manner, the proposed approach yields ELS which can reliably replace the original nonlinear systems in carrying out computationally efficient analyses in the initial stages of the aseismic design of structures under severe seismic excitations. Furthermore, the potential of this approach for developing inelastic design spectra from a given elastic design spectrum is established

A stochastic approach for deriving effective linear properties of bilinear hysteretic systems subject to design spectrum compatible strong ground motions

A novel statistical linearization based approach is proposed to derive effective linear properties (ELPs), namely damping ratio and natural frequency, for bilinear hysteretic oscillators subject to seismic excitations specified by an elastic response/design spectrum. First, an efficient numerical scheme is adopted to derive a power spectrum, satisfying a certain statistical criterion, which is compatible with the considered seismic spectrum. Next, the thus derived power spectrum is used in conjunction with a frequency domain higher-order statistical linearization formulation to substitute a bilinear hysteretic oscillator by a third order linear system. This is done by minimizing an appropriate error function in the least square sense. Then, this third-order linear system is reduced to a second order linear oscillator characterized by a set of ELPs by enforcing equality of certain response statistics of the two linear systems. The ELPs are utilized to estimate the peak response of the considered hysteretic oscillator in the context of linear response spectrum-based dynamic analysis. In this manner, the need for numerical integration of the nonlinear equation of motion is circumvented. Numerical results pertaining to the European EC8 elastic response spectrum are presented to demonstrate the applicability and usefulness of the proposed approach. These results are supported by Monte Carlo analyses involving an ensemble of 250 non-stationary artificial EC8 spectrum compatible accelerograms. The proposed approach can hopefully be an effective tool in the preliminary aseismic design stages of yielding structures following either a force-based or a displacement-based methodology

Algorithmic options for joint time-frequency analysis in structural dynamics applications

The purpose of this paper is to present recent research efforts by the authors supporting the superiority of joint time-frequency analysis over the traditional Fourier transform in the study of non-stationary signals commonly encountered in the fields of earthquake engineering, and structural dynamics. In this respect, three distinct signal processing techniques appropriate for the representation of signals in the time-frequency plane are considered. Namely, the harmonic wavelet transform, the adaptive chirplet decomposition, and the empirical mode decomposition, are utilized to analyze certain seismic accelerograms, and structural response records. Numerical examples associated with the inelastic dynamic response of a seismically-excited 3-story benchmark steel-frame building are included to show how the mean-instantaneous-frequency, as derived by the aforementioned techniques, can be used as an indicator of global structural damage

A stochastic approach to synthesizing response spectrum compatible seismic accelerograms

Regulatory agencies require the use of artificial accelerograms satisfying specific criteria of compatibility with a given design spectrum, as input for certain types of analyses for the aseismic design of critical facilities. Most of the numerical methods for simulating seismic motions compatible with a specified design (target) spectrum proposed by various researchers require that a number of real recorded seismic accelerograms of appropriate frequency content is available. To by-pass this requirement, a previously established in the literature probabilistic approach to yield simulated earthquake records whose response spectrum achieves on average a certain level of agreement with a target spectrum is employed in the present paper. At the core of the above method lies the adoption of an appropriate parametric power spectrum model capable of accounting for various site-specific soil conditions. In this regard, the potential of two different, commonly, used spectral forms is evaluated for this purpose in context with the design spectrum defined by the European Code provisions. Next, an iterative wavelet-based matching procedure is applied to the thus acquired records to enhance, individually, the agreement of the corresponding response spectra with the targeted one. Special attention is paid to ensure that the velocity and the displacement time histories associated with the finally obtained artificial accelerograms are physically sound by means of appropriate baseline correction techniques

Derivation of response spectrum compatible non-stationary stochastic processes relying on Monte Carlo-based peak factor estimation

In this paper a novel approach is proposed to address the problem of deriving non-stationary stochastic processes which are compatible in the mean sense with a given (target) response (uniform hazard) spectrum (UHS) as commonly desired in the aseismic structural design regulated by contemporary codes of practice. The appealing feature of the approach is that it is non-iterative and “one-step”. This is accomplished by solving a standard over-determined minimization problem in conjunction with appropriate median peak factors. These factors are determined by a plethora of reported new Monte Carlo studies which on their own possess considerable stochastic dynamics merit. In the proposed approach, generation and treatment of samples of the processes individually on a deterministic basis is not required as is the case with the various “two-step” approaches found in the literature addressing the herein considered task. The applicability and usefulness of the approach is demonstrated by furnishing extensive numerical data associated with the elastic design UHS of the current European (EC8) and the Chinese (GB 50011) aseismic code provisions. Purposely, simple and thus attractive from a practical viewpoint, uniformly modulated processes assuming either the Kanai-Tajimi (K-T) or the Clough-Penzien (C-P) spectral form are employed. The Monte Carlo studies yield damping and duration dependent median peak factor spectra, given in a polynomial form, associated with the first passage problem for UHS compatible K-T and C-P uniformly modulated stochastic processes. Hopefully, the herein derived stochastic processes and median peak factor spectra can be used to facilitate the aseismic design of structures regulated by contemporary code provisions in a Monte Carlo simulation-based or stochastic dynamics-based context of analysis