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Über dieses Buch

From the preface: This volume is a collection of papers presented at the U.S. - Japan Joint Seminar on Stochastic Approaches in Earthquake Engineering held on May 6 and 7, 1987. The general theme of the two-day program was the application of probability and statistics to engineering problems related to strong ground motion. Within this general theme a great variety of subject matters were covered, including earthquake cataloging, ground motion modeling, system identification, failure mechanisms, response and reliability analyses, numerical techniques, and active control. The engineering systems considered included buildings, bridges and life-line networks.



Application of Probabilistic Approach to Aseismic Safety Analysis of Soil-Building Structure Systems

The fundamental equations of motion of soil-hysteretic building structural systems under earthquake-like random excitation are derived. Taking into account the non-stationary and non-white spectral characteristics of the excitation, the ordinary differential equations are derived for the covariance responses. Based on these responses, a new approach to estimation of the maximum ductility factor response is developed, and the probability of structural safety is examined. Numerical examples are presented and the accuracy of the proposed approach is demonstrated. The possible substitution of non-stationary spectral characteristics of the excitation for stationary ones is discussed through the examination of the response characteristics.
Koichiro Asano

Temporal and Magnitude Dependence in Earthquake Recurrence Models

The effects of temporal and magnitude dependence among seismic recurrences, which are ignored in the conventional Poisson earthquake model, are studied. The potential impact of non-Poissonian assumptions on practical hazard estimates are considered. A broad set of recurrence models with memory are analyzed using convenient second-moment time-magnitude statistics to parameterize a general class of semi-Markov models. The conventional time- and slip-predictable models are included and studied as special cases. Conditions are identified under which the Poisson model provides a sufficient engineering hazard estimate. i.e., either conservative or unconservative by a factor of no more than three. The Poisson approximation is found to be sufficient for all but what is expected to be a small subset of the cases encountered in practice.
C. Allin Cornell, Steven R. Winterstein

Response Statistics of Nonlinear Systems Subjected to Seismic Excitation

Reliability measures are evaluated for simple nonlinear structures subjected to nonstationary earthquake ground motions. Damage-related limit states are considered, and randomness in structural properties is taken into account. Response statistics are computed using a Latin hypercube sampling technique. The reliabilities found are relatively low in comparison with reliabilities of systems subjected only to gravity loads. The large uncertainty in the basic seismic hazard dominates the variability in structural response to earthquake ground motion and is the controlling factor in determining reliability.
Bruce Ellingwood, Jennifer O’Connor

Modal Interaction in Dynamical Systems with Closely-Spaced Natural Frequencies under Self-Excited and Random Forces

Certain dynamical systems have closely-spaced first and second natural frequencies. Multimodal response of such a system subject to nonlinear self-excited force is analytically studied modal selection and coexistence of two modes in self-excited oscillation are discussed in detail. Effect of random force on the modal interaction in self-excited system is also studied. A cable-stayed bridge tower which has almost coalescent first and second in-plane natural frequencies is used as an example structure. Wind tunnel experiments are conducted using three dimensional models of the bridge tower, and it is found that the results by analysis, can well explain the experimental observations regarding modal selection under self-excited and modal interaction under random and self-excited forces.
Yozo Fujino, Phoonsak Pheinsusom

Kalman Filtering of Versatile Restoring Systems

This paper presents an identification method on a hysteretic restoring system by appling the Extended Kalman filter incorporated with a weighted global iteration. By this method, a nonlinear versatile model applied to general hysteretic system may be identified in terms of the model’s parameters, at the stage of their convergency to optimal ones after weighted global iterations of the Extended Kalman filtering. The identification method is demonstrated for the estimation of response characteristics of a near surface ground subjected to a strong earthquake ground motion. Also, the justification of the method is investigated on numerically simulated data on responses of a known’ degrading system.
Masaru Hoshiya, Osamu Maruyama

Earthquake Failure Mechanism of Hysteretic Structures with Stress-Strain Based Modeling

This paper presents the earthquake response and failure mechanism of inelastic reinforced concrete structures with varying axial forces. Material nonlinearity of an RC element is evaluated by using a so-called “fiber model” based on stress-strain relation. Accuracy of constitutive laws is examined from comparison of analytical and experimental moment-curvature relations. Failure mechanism is then examined with a damage index that includes effects of stiffness degradation and accumulated hysteretic energy absorption. A step-by-step integration of the equation of motion is used to calculate earthquake response of a structure. The method gives a very accurate estimation of inelastic earthquake response and deterioration process of RC structures.
Hirokazu Iemura, Yoshikazu Yamada

The Stochastic Response of Strongly Yielding Systems

A new look at an old problem
The bilinear hysteretic model is used to examine the nature of the stochastic response of strongly yielding (nearly elasto-plastic) systems. Both second-order and third-order equivalent linearization schemes are examined. It is shown that although these approximate techniques generally give acceptable results for second-order response statistics they fail to yield an accurate description of the nature of the response in the frequency domain. The influence of the excitation Power Spectrum on the response is examined. The desirability of decomposing the response into an elastic and inelastic component is demonstrated.
Wilfred D. Iwan, Leonidas G. Paparizos

Applications of Statistics and Probability to Seismic Disaster Mitigation Researches

There are many sources of uncertainties in the problems of earthquake engineering. These sources may be classified into three groups. They are (a) uncertainties In earthquake occurrences, (b).uncertainties In earthquake ground motions, and (c) uncertainties in structural response and seismic damage.
Tsuneo Katayama

Stochastic Seismic Response Sensitivity of Soil-Structure Interaction System

The seismic response sensitivity of soil-structure interaction system is statistically investigated in relation to the development of a reasonable seismic design methodology of a structural system. In the above analysis, the earthquake ground motion model is presented by the convolution of Green’s function of a random medium and random rupture process function which is expressed by the dynamic behavior of flexible membrane, and the inertia interaction of a rectangular foundation on a semi-infinite random medium is estimated in terms of the dynamic ground compliance. The seismic response characteristics of soil-structure interaction system are discussed through the variables describing the ground motion model and the dynamic ground compliance.
M. Kawano, T. Kobori

Active Stochastic Control of Seismic Structures

In a continuing effort to determine the feasibility of applying optimal control to building structures, a comprehensive experimental study was carried out using a standardized structural model under base excitation produced by the Earthquake Simulator at SUNY/Buffalo. Based upon computer simulations and experimental results, this paper presents a comparison of effectiveness of several optimal control algorithms using instantaneous optimal criteria, including time delay compensation. All investigations were done under similar conditions to permit a systematic evaluation of efficiency. Comparisons were also made between analytical and experimental results and between instantaneous control algorithms and classical closedloop linear feedback. Conclusions drawn regarding relative merits of the control algorithms considered under varying conditions are presented herein.
R. C. Lin, T. T. Soong, A. M. Reinhorn

Evolutionary Kanai-Tajimi Type Earthquake Models

A versatile mathematical framework based on the concept of random pulse train is proposed for the modeling of hypothetical ground acceleration in a future earthquake for engineering design purposes. This framework is potentially capable of incorporating various physical features arising from propagation, reflection and refraction of seismic waves in the ground. Three specific simplified models are then investigated: an evolutionary Kanai-Tajimi model, a one-dimensional elastic model, and a one-dimensional Maxwell model. Artificial seismograms are generated from these models to simulate the 1985 Mexico earthquake, and the results are compared with an actual record. It is shown that all the random pulse train models have an evolutionary spectral representation which permits variation of both mean-square intensity and frequency contents, and that the random vibration analyses of linear and nonlinear structures under such excitations can be simply formulated.
Y. K. Lin, Y. Yong

Stochastic Estimates of Nonlinear Dynamic Systems

Stochastic estimates of nonlinear dynamic systems including hysteretic structural systems are formulated in the form of the Ito stochastic differential equations. Based on the theory of continuous Markov vector process, differential forms of conditional probability density functions given observations during a finite time interval are presented by making use of the Kolmogorov differential operators and innovations process. Differential forms of conditional expectations of an arbitrary differentiable function of state variables are also presented for filtering, smoothing and prediction problems arising in a general class of nonlinear dynamic systems. From these Ito-Dynkin formulas, truncated conditional moment equations are derived by introducing an approximate conditional probability density functions expressed in the form of a finite mixed type series expansion in terms of different sets of orthogonal polynomials. Special cases of the general results obtained in the present study are found to coincide with the known results for nonlinear filtering as well as linear filtering and smoothing. The solution techniques to obtain conditional probability density functions as well as optimal stochastic estimators associated with filtering and smoothing are described for a general class of nonlinear dynamic systems.
Ryoichiro Minai, Yoshiyuki Suzuki

Importance Analyses for Upgrading Seismic Reliability of Large Scale Lifeline Networks

We have developed an algorithm which has only polynomial complexity to enumerate paths in the network. Using this developed algorithm we propose three measures for specifying critical components in lifeline networks. For each measure the sequence that strengthens each component is defined for the upgrading of the seismic reliability of the total system. Lifeline components are assumed to fail because of ground liquefaction. The theory for fatigue failures is used to analyze liquefaction phenomena. The intensity of stress and its rate are estimated using a response spectrum description of the excitation at the base rock level.The seismic environment of a lifeline is defined by the fault rupture model through the relation of the seismic moment to the rupture area. Calculation of the seismic reliability of an example network and evaluation of a component’s importance show that this new technique can be used with large networks.
T. Sato

Digital Simulation of Seismic Ground Motion

The method of spectral representation for uni-variate, one-dimensional, stationary stochastic processes and multi-dimensional, uni-variate (as well as multi-dimensional, multi-variate) homogeneous stochastic fields has been reviewed in detail, particularly from the viewpoint of digitally generating their sample functions. This method of representation has then been extended to the cases of uni-variate, one-dimensional, nonstationary stochastic processes and multi-dimensional, uni-variate nonhomogeneous stochastic fields, again emphasizing sample function generation. Also, a fundamental theory of evolutionary stochastic waves is developed and a technique for digitally generating samples of such waves is introduced as a further extension of the spectral representation method. This is done primarily for the purpose of developing an analytical model of seismic waves that can account for their stochastic characteristics in the time and space domain. From this model, the corresponding sample seismic waves can be digitally generated. The efficacy of this new technique is demonstrated with the aid of a numerical example in which a sample of a spatially two-dimensional stochastic wave consistent with the Lotung, Taiwan dense array data is digitally generated.
M. Shinozuka, G. Deodatis, T. Harada

Seismic Design of Secondary Systems

The ever increasing demand of providing accurate yet practical solution to the problem of seismic analysis of secondary system in industrial units has led to the development of some very novel analytical techniques. The paper describes the evolution of the methods which have been used to analyze the secondary systems in the past two decades. The developments starting with the direct generation of floor response spectra up to the recent introduction of the cross floor response spectra as the seismic inputs for the analysis of multiply supported secondary systems are discussed. lately, significant research efforts have also been directed to incorporate the effect of the dynamic interaction between the primary and secondary systems in their analyses. The latest developments in this area, utilizing the component mode synthesis approaches, are also described.
M. P. Singh

Evolutionary Power Spectrum Estimation of September 19, 1985 Mexico Earthquake Accelerograms

Some preliminary results of an approach to estimating seismic evolutionary power spectra are discussed. It is assumed that the available accelerograms belong to a broad-band stochastic process. The power spectrum is estimated by determining the statistical moments of the energy of a lightly damped linear system (filter) excited by the stochastic seismic process. Some preliminary numerical data from the September 19, 1985 Mexico earthquake are presented.
P. Spanos, J. Roesset, M. Donley

Application of stochastic differential equations to seismic reliability analysis of hysteretic structures

An analytical method of stochastic seismic response and reliability analysis of hysteretic structures based on the theory of Markov vector process is presented, especially from the methodological aspect. To formulate the above analysis in the form of stochastic differential equations, the differential formulations of general constitutive laws for a class of hysteretic characteristics are derived. The differential forms of the seismic safety measures such as the maximum ductility ratio, cumulative plastic deformation, low-cycle fatigue damage are also derived. The state equation governing the whole nonlinear dynamical system which is composed of the shaping filter generating seismic excitations, hysteretic structural system and safety measures is determined as the Itô stochastic differential equations. By introducing an appropriate non-Gaussian joint probability density function, the statistics and joint probability density function of the state variables can be evaluated numerically under nonstationary state. The merit of the proposed method is in systematically unifying the conventional response and reliability analyses into an analysis which requires knowledge of only first order (single time) statistics or probability distributions.
Yoshiyuki Suzuki, Ryoichiro Minai

Site Effects on the Non-Stationarity of Earthquake Excitations and Structural Responses

A non-stationary spectral density is defined by a local time variance of component wave which passed through a narrow band filter. Using the non-stationary spectral ratio of input and output relations of structural system, time variant propagation characteristics of incident wave in multi-layered stratum are examined. The theoretical analysis of an amplification effects of incident wave in multi-layered strata are discussed with respect to a multi-layered reflection of body wave and a constructive interference of surface wave. The evaluation results by these theoretical analyses are compared with that of the spectral analysis of actually recorded accelerograms. The dynamic characteristics of large scale foundation subjected to earthquake excitation is-also examined.
Yoshihiro Takeuchi

Simulation of Fault Rupture Process by the Stochastic Finite Element Method

It has been showed that the finite element method which utilizes the joint element for representing the fault plane is a promising tool to analyze the fault rupture process and to predict the near field ground motion. In such analyses, however, the spatial distribution of stress and strength along the fault must be known and has been assumed to be deterministic. Then, as a next step of the analysis, the spatial distribution of stress and strength on the fault are considered to be random variables and the effects of spatial variation are discussed by making use of the Monte Carlo simulation and the first-order approximation method.
Kenzo Toki, Sumio Sawada, Yoshihiro Okashige

Statistical Analysis of Earthquake Catalogs for Seismic Hazard

In the use of historical earthquake catalogs for seismic hazard, the main objective is to estimate the rate of earthquakes of various sizes in a region around the site of interest. This is a complicated problem, in part because of deficiencies of the catalog such as missing data, nonuniformity of the scale in which earthquake sizes are measured and errors in reported epicentral locations and magnitudes, in part because earthquake activity is non-Poissonian and varies in space, when not also in time. Standard methods of analysis are simple but often inadequate: they ignore errors in the data, use regressions among size measures as if they were functional relationships, ignore the incomplete part of the catalog, and assume homogeneous Poisson occurrences inside given “seismogenic provinces”. The scope of the present paper is to challenge these assumptions and approximations. In particular, it is proposed that seismicity be represented as the Superposition of a nonhomogeneous Poisson process of main or parent events and a process of secondary and smaller events clustered around the parent earthquakes. Various models of catalog incompleteness are also proposed. Methods aie developed for the identification of secondary events and for the simultaneous estimation of spatially varying recurrence rates of main events and of catalog incompleteness, using any desired portion of the historical record. As a special case, the model includes the assumption of homogeneous Poisson sources. More in general, seismogenic provinces are considered as geographical regions inside which the variation of seismicity possesses some regularity. Other problems addressed in this study are the estimation of earthquake size on a common scale and the automatic identification of quasi-homogeneous earthquake sources. The proposed methods are exemplified through application to Northeastern Italy.
D. Veneziano, J. Van Dyck

On Fast Integration for Time Variant Structural Reliability

In evaluating structural reliability under stochastic loadings, the system parameters such as stiffness, damping, strength, excitation frequency content and duration, etc., are usually assumed given. In reality, however, these quantities are seldom perfectly known. Their uncertainties may play a major role as far as the overall structural reliability is concerned. This paper reviews currently available methods to include this parameter uncertainties and a new method is also proposed. The accuracy and analytical and numerical efforts required are compared. Through numerical examples on systems under dynamic excitation, collapse of ductile or brittle redundant systems, the advantages of the proposed method are demonstrated.
Y. K. Wen, H.-C. Chen


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