George Deodatis

Simulation of Multi-Dimensional Gaussian Stochastic Fields by Spectral Representation

The subject of this paper is the simulation of multi-dimensional, homogeneous, Gaussian stochastic fields using the spectral representation method. Following this methodology, sample functions of the stochastic field can be generated using a cosine series formula. These sample functions accurately reflect the prescribed probabilistic characteristics of the stochastic field when the number of terms in the cosine series is large. The ensemble-averaged power spectral density or autocorrelation function approaches the corresponding target function as the sample size increases. In addition, the generated sample functions possess ergodic characteristics in the sense that the spatially-averaged mean value, autocorrelation function and power spectral density function are identical with the corresponding targets, when the averaging takes place over the multi-dimensional domain associated with the fundamental period of the cosine series. Another property of the simulated stochastic field is that it is asymptotically Gaussian as the number of terms in the cosine series approaches infinity. The most important feature of the method is that the cosine series formula can be numerically computed very efficiently using the Fast Fourier Transform technique. The main area of application of this method is the Monte Carlo solution of stochastic problems in structural engineering, engineering mechanics and physics. Specifically, the method has been applied to problems involving random loading (random vibration theory) and random material and geometric properties (response variability due to system stochasticity).

Non-stationary stochastic vector processes: seismic ground motion applications

A spectral-representation-based simulation algorithm is used in this paper to generate sample functions of a non-stationary, multi-variate stochastic process with evolutionary power, according to its prescribed non-stationary cross-spectral density matrix. If the components of the vector process correspond to different locations in space, then the process is also non-homogeneous in space (in addition to being non-stationary in time). The ensemble cross-correlation matrix of the generated sample functions is identical to the corresponding target. For the important application of earthquake ground motion simulation, an iterative scheme is introduced to generate seismic ground motion time histories at several locations on the ground surface that are compatible with prescribed response spectra, correlated according to a given coherence function, include the wave propagation effect, and have a specified duration of strong ground motion. Three examples involving simulation of earthquake ground motion are presented in order to demonstrate the capabilities of the proposed methodologies. In the first two examples, acceleration time histories at three points on the ground surface are generated according to a prescribed cross-spectral density matrix, while in the third example, the acceleration time histories are generated to be compatible with prescribed response spectra.

Stochastic process models for earthquake ground motion

Abstract In this paper we present a review of stochastic process models proposed for the simulation of seismic ground motion. The models reviewed include those based on filtered white noise processes, filtered Poisson processes, spectral representation of stochastic processes, and finally those based on stochastic wave theory. Mathematical expressions are provided for all models along with comments on their usefulness, advantages and disadvantages. Together with the review of auto-regressive moving-average models by F. Kozin in this PEM review series on earthquake engineering (June issue), this paper represents an overview of stochastic models of earthquake ground motion, which is hopefully of some use to researchers as well as practitioners.

Probabilistic Benefit-Cost Analysis for Earthquake Damage Mitigation: Evaluating Measures for Apartment Houses in Turkey

In the wake of the 1999 earthquake destruction in Turkey, the urgent need has arisen to evaluate the benefits of loss mitigation measures that could be undertaken to strengthen the existing housing stock. In this study, a benefit-cost analysis methodology is introduced for the comparative evaluation of several seismic retrofitting measures applied to a representative apartment building located in Istanbul. The analysis is performed probabilistically through the development of fragility curves of the structure in its different retrofitted configurations. By incorporating the probabilistic seismic hazard for the region, expected direct losses can be estimated for arbitrary time horizons. By establishing realistic cost estimates of the retrofitting schemes and costs of direct losses, one can then estimate the net present value of the various retrofitting measures. The analysis in this work implies that, even when considering only direct losses, all of the retrofitting measures considered are desirable for all but the very shortest time horizons. This conclusion is valid for a wide range of estimates regarding costs of mitigation, discount rates, number of fatalities, and cost of human life. The general methodology developed here for a single building can be extended to an entire region by incorporating additional structural types, soil types, retrofitting measures, more precise space- and time-dependent seismic hazard estimates, etc. It is hoped that this work can serve as a benchmark for more realistic and systematic benefit-cost analyses for earthquake damage mitigation.

Simulation of homogeneous nonGaussian stochastic vector fields

A spectral representation-based simulation methodology is proposed to generate sample functions of a multi-variate, multi-dimensional, nonGaussian stochastic vector field, according to a prescribed cross-spectral density matrix and prescribed (nonGaussian) marginal probability distribution functions. The proposed methodology starts by generating a Gaussian vector field that is then transformed into the desired nonGaussian one using a memoryless nonlinear transformation in conjunction with an iterative scheme. The generation of the Gaussian vector field is performed taking advantage of the Fast Fourier Transform technique for great computational efficiency. The special case of simulation of nonGaussian vector fields modeling material properties is examined, mainly from the point of view of certain simplifying assumptions that can be made for such random media. Finally, a numerical example involving a tri-variate, two-dimensional, nonGaussian stochastic vector field is presented in order to demonstrate the capabilities and the efficiency of the proposed methodology.

Response and eigenvalue analysis of stochastic finite element systems with multiple correlated material and geometric properties

Abstract The variability of the random response displacements and eigenvalues of structures with multiple uncertain material and geometric properties are studied in this paper using variability response functions. The material and geometric properties are assumed to be described by cross-correlated stochastic fields. Specifically, two types of problems are considered: the response displacement variability of plane stress/plane strain structures with stochastic elastic modulus, Poissons ratio, and thickness, and the eigenvalue variability of beam and plate structures with stochastic elastic modulus and mass density. The variance of the displacement/eigenvalue is expressed as the sum of integrals that involve the auto-spectral density functions characterizing the structural properties, the cross-spectral density functions between the structural properties, and the deterministic variability response functions. This formulation yields separate terms for the contributions to the response displacement/eigenvalue variability from the auto-correlation of each of the material/geometric properties, and from the cross-correlation between these properties. The variability response functions are used to compute engineering-wise very important spectral-distribution-free realizable upper bounds of the displacement/eigenvalue variability. Using this formulation, it is also possible to compute the displacement/eigenvalue variability for prescribed auto- and cross-spectral density functions.

Variability response functions for stochastic plate bending problems

In this paper, the weighted integral method and the concept of variability response function are successfully extended to plate bending problems where the elastic modulus of the structure is considered to be a two-dimensional, homogeneous stochastic field, overcoming earlier computational problems associated with the large number of terms in the expression for the variability response function. The concept of the variability response function is used to compute spectral-distribution-free upper bounds of the response variability. Such bounds are of paramount importance for the majority of real-life problems where only first and second moments of the stochastic material properties can be estimated with reasonable accuracy. Under the assumption of a prespecified power spectral density function of the stochastic field describing the elastic modulus, it is also possible to compute the response variability (in terms of second moments of response quantities) and the reliability (in terms of the safety index) of the stochastic plate. The use of a variability response function based on the local averaging method reduces the computational effort associated with the weighted integral method, with only a small loss of accuracy in most cases. Numerical examples are provided to demonstrate all of the above capabilities. One of the conclusions is that the coefficient of variation of certain response quantities can become larger than the coefficient of variation of the elastic modulus (the input quantity).

Modeling, synthetics and engineering applications of strong earthquake wave motion

State of the art in modeling, synthetics, statistical estimation, and engineering applications of strong ground motion is reported in this paper. In particular, models for earthquake wave motion are presented, in which uncertainties both in the earth medium and the seismic source are taken into consideration. These models can be used to synthesize realistic strong earthquake ground motion, specifically near-field ground motion which is quite often not well recorded in real earthquakes. Statistical estimation techniques are also presented so that the characteristics of spatially-correlated earthquake motion can be captured and consequently used in investigating the seismic response of such large scale structures as pipelines and long-span bridges. Finally, applications of synthesized strong ground motion in a variety of engineering fields are provided. Numerical examples are shown for illustration.

San Francisco Monastery, Quito, Equador: characterisation of building materials, damage assessment and conservation considerations

Abstract Founded in 1535, the Monastery of San Francisco in Quito is one of the oldest monastic complexes in South America. Due to the large scale of the monument, this work is limited to the principal church and the first cloister, which are the oldest ones and most frequently visited. Samples were taken from the adobe, brick and stone structures and mortar joints and analysed by X-ray diffraction, optical microscopy, mercury intrusion porosimetry, calcimetry and scanning electron microscopy. The building materials were characterised, their decay mechanisms were studied and conservation strategies were proposed. Adobe samples exhibit the most severe weathering, while bricks and mortars suffer from water percolation and past conservation treatments, correspondingly. The andesitic facade is covered by a dark, red to black patina, consisting mainly of gypsum and apatite. Cleaning with dilute acid or laser cleaning would be effective techniques for the stone facade. In the case of adobe bricks though, consolidation treatment is more difficult to be executed, since they are usually painted.

Reliability of aircraft structures under non-periodic inspection: a Bayesian approach

This paper proposes a Bayesian analysis methodology to determine appropriate non-periodic inspection intervals of fatigue-sensitive aircraft structures, so that their reliability remains above a prespecified minimum level throughout their service life. The methodology is based on assumptions about the probability distribution function of the time to crack initiation, the law of crack propagation, the probability of crack detection and the failure rate before and after crack initiation. The Bayesian approach proposed in this paper is unique and novel in that it allows one to utilize judiciously the results of earlier inspections for the purpose of determining the time of the next inspection and estimating the values of several parameters involved in the problem that can be treated as uncertain. Numerical simulations verify the above-mentioned capabilities of the Bayesian method.