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CONTENTS
Volume 32, Number 1, May10 2009
 

Abstract
The present contribution addresses uncertainty quantification and uncertainty propagation in structural mechanics using stochastic analysis. Presently available procedures to describe uncertainties in load and resistance within a suitable mathematical framework are shortly addressed. Monte Carlo methods are proposed for studying the variability in the structural properties and for their propagation to the response. The general applicability and versatility of Monte Carlo Simulation is demonstrated in the context with computational models that have been developed for deterministic structural analysis. After discussing Direct Monte Carlo Simulation for the assessment of the response variability, some recently developed advanced Monte Carlo methods applied for reliability assessment are described, such as Importance Sampling for linear uncertain structures subjected to Gaussian loading, Line Sampling in linear dynamics and Subset simulation. The numerical example demonstrates the applicability of Line Sampling to general linear uncertain FE systems under Gaussian distributed excitation.

Key Words
uncertainty propagation; stochastic; Monte Carlo Simulation; dynamics; reliability.

Address
G.I. Schueller: Institute of Engineering Mechanics, University of Innsbruck, 6020 Innsbruck, Austria, EU

Abstract
An importance sampling method is presented for computing the first passage probability of elasto-plastic structures under stochastic excitations. The importance sampling distribution corresponds to shifting the mean of the excitation to an \'adapted\' stochastic process whose future is determined based on information only up to the present. A stochastic control approach is adopted for designing the adapted process. The optimal control law is determined by a control potential, which satisfies the Bellman\'s equation, a nonlinear partial differential equation on the response state-space. Numerical results for a single-degree-of freedom elasto-plastic structure shows that the proposed method leads to significant improvement in variance reduction over importance sampling using design points reported recently.

Key Words
adapted process; elasto-plastic; first passage problem; stochastic optimal control; reliability; stochastic dynamics.

Address
Siu-Kui Au: Department of Building and Construction, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong

Abstract
In this paper the problem of calculating the probability that the responses of a wind-excited structure exceed specified thresholds within a given time interval is considered. The failure domain of the problem can be expressed as a union of elementary failure domains whose boundaries are of quadratic form. The Domain Decomposition Method (DDM) is employed, after being appropriately extended, to solve this problem. The probability estimate of the overall failure domain is given by the sum of the probabilities of the elementary failure domains multiplied by a reduction factor accounting for the overlapping degree of the different elementary failure domains. The DDM is extended with the help of Line Sampling (LS), from its original presentation where the boundary of the elementary failure domains are of linear form, to the current case involving quadratic elementary failure domains. An example involving an along-wind excited steel building shows the accuracy and efficiency of the proposed methodology as compared with that obtained using standard Monte Carlo simulations (MCS).

Key Words
reliability analysis; wind excitation; domain decomposition method; line sampling.

Address
L.S. Katafygiotis and Jia Wang: Department of Civil Engineering, Hong Kong University of Science and Technology, Hong Kong, P.R. China

Abstract
A quasi ideal importance sampling simulation method combined in the conditional expectation is proposed for the structural reliability estimation. The quasi ideal importance sampling joint probability density function (p.d.f.) is so composed on the basis of the ideal importance sampling concept as to be proportional to the conditional failure probability multiplied by the p.d.f. of the sampling variables. The respective marginal p.d.f.s of the ideal importance sampling joint p.d.f. are determined numerically by the simulations and partly by the piecewise integrations. The quasi ideal importance sampling simulations combined in the conditional expectation are executed to estimate the failure probabilities of structures with multiple failure surfaces and it is shown that the proposed method gives accurate estimations efficiently.

Key Words
structural failure probability; simulation-based reliability method; conditional expectation; ideal importance sampling.

Address
Masaaki Yonezawa: Dept. of Mechanical Engineering, Kinki University, Higashi-Osaka 577-8502, Japan
Shoya Okuda: Dept. of Total System Engineering, Kinki University Technical College, Japan
Hiroaki Kobayashi: Dept. of Mechanical Engineering, Kinki University, Higashi-Osaka 577-8502, Japan

Abstract
In this paper a procedure for Monte Carlo simulation of univariate stationary stochastic processes with the aid of neural networks is presented. Neural networks operate model-free and, thus, circumvent the need of specifying a priori statistical properties of the process, as needed traditionally. This is particularly advantageous when only limited data are available. A neural network can capture the \"pattern\" of a short observed time series. Afterwards, it can directly generate stochastic process realizations which capture the properties of the underlying data. In the present study a simple feedforward network with focused time-memory is utilized. The proposed procedure is demonstrated by examples of Monte Carlo simulation, by synthesis of future values of an initially short single process record.

Key Words
Monte Carlo simulation; neural networks; stochastic processes.

Address
Michael Beer: Dept. of Civil Engineering, National University of Singapore, Singapore
Pol D. Spanos: Ryon Endowed Chair in Engineering, Rice University, Houston TX, USA

Abstract
The present contribution addresses the parallelization of advanced simulation methods for structural reliability analysis, which have recently been developed for large-scale structures with a high number of uncertain parameters. In particular, the Line Sampling method and the Subset Simulation method are considered. The proposed parallel algorithms exploit the parallelism associated with the possibility to simultaneously perform independent FE analyses. For the Line Sampling method a parallelization scheme is proposed both for the actual sampling process, and for the statistical gradient estimation method used to identify the so-called important direction of the Line Sampling scheme. Two parallelization strategies are investigated for the Subset Simulation method: the first one consists in the embarrassingly parallel advancement of distinct Markov chains; in this case the speedup is bounded by the number of chains advanced simultaneously. The second parallel Subset Simulation algorithm utilizes the concept of speculative computing. Speedup measurements in context with the FE model of a multistory building (24,000 DOFs) show the reduction of the wall-clock time to a very viable amount (<10 minutes for Line Sampling and . 1 hour for Subset Simulation). The measurements, conducted on clusters of multi-core nodes, also indicate a strong sensitivity of the parallel performance to the load level of the nodes, in terms of the number of simultaneously used cores. This performance degradation is related to memory bottlenecks during the modal analysis required during each FE analysis.

Key Words
structural reliability; stochastic structural mechanics; parallel computing; Monte Carlo simulation.

Address
M.F. Pellissetti: Institute of Engineering Mechanics, University of Innsbruck, 6020 Innsbruck, Austria

Abstract
A novel approach is proposed to effectively estimate the quantile functions of normalized performance indices of reliability constraints in a reliability-based optimization (RBO) problem. These quantile functions are not only estimated as functions of exceedance probabilities but also as functions of the design variables of the target RBO problem. Once these quantile functions are obtained, all reliability constraints in the target RBO problem can be transformed into non-probabilistic ordinary ones, and the RBO problem can be solved as if it is an ordinary optimization problem. Two numerical examples are investigated to verify the proposed novel approach. The results show that the approach may be capable of finding approximate solutions that are close to the actual solution of the target RBO problem.

Key Words
reliability; reliability-based optimization; quantile function; stochastic simulation.

Address
Jianye Ching: Dept. of Civil Engineering, National Taiwan University, Taipei, Taiwan
Wei-Chi Hsu: Dept. of Construction Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan

Abstract
This study proposes a certain measure or investment strategy for decision making associated with seismic retrofitting. This strategy reduces the risk of a large-scale malfunction such as water supply loss under seismic risks. The authors developed a stochastic value index that will be used in the overall evaluation of social benefit, income gain, life cycle costs and failure compensation associated with existing lifeline systems damaged by an earthquake during the remaining service period. Optimal seismic disaster prevention investment of deteriorated lifeline systems is discussed. Finally, the present study provides a performance-based design method for seismic retrofitting strategies of existing lifelines which are carried out using the target probabilities of value loss and structural failure.

Key Words
stochastic value index; seismic risk management; seismic investment; existing lifeline.

Address
Takeshi Koike: Tokyo City University, Tokyo, Japan
Toshio Imai: JFE Engineering Corporation, Tokyo, Japan

Abstract
Initial imperfections, such as initial deflection or remaining stress, cause deterioration of buckling strength of structures. The Koiter imperfection sensitivity law has been extended to describe the mechanism of reduction for structures. The extension is twofold: (1) a number of imperfections are considered, and (2) the second order (minor) imperfections are implemented, in addition to the first order (major) imperfections considered in the Koiter law. Yet, in reality, the variation of external loads is dominant over that of imperfection. In this research, probabilistic evaluation of buckling loads against external loads subjected to probabilistic variation is conducted by extending the concept of imperfection sensitivity. A truss arch subjected to dead and live loads is considered as a numerical example. The mechanism of probabilistic variation of buckling strength of this arch is described by the proposed method, and its reliability is evaluated.

Key Words
buckling loads; imperfection sensitivity law; probabilistic analysis.

Address
Kiyohiro Ikeda: Dept. of Civil & Environmental Engineering, Tohoku University, Sendai 980-8579, Japan
Makoto Ohsaki: Dept. of Architecture & Architectural Engineering, Kyoto University, Kyoto 615-8540, Japan
Kentaro Sudo: Dept. of Civil & Environmental Engineering, Tohoku University, Sendai 980-8579, Japan
Toshiyuki Kitada: Dept. of Civil Engineering, Osaka City University, Sumiyoshi-ku, Osaka 558-8585, Japan

Abstract
A stochastic response spectrum method is proposed for simple evaluation of the structural response of an actively controlled aseismic structure. The response spectrum is constructed assuming a linear structure with an active mass damper (AMD) system, and an earthquake wave model given by the product of a non-stationary envelope function and a stationary Gaussian random process with Kanai-Tajimi power spectral density. The control design is executed using a linear quadratic Gaussian control strategy for an enlarged state space system, and the response amplification factor is given by the combination of the obtained statistical response values and extreme value theory. The response spectrum thus produced can be used for simple dynamical analyses. The response factors obtained by this method for a multi-degree-of-freedom structure are shown to be comparable with those determined by numerical simulations, demonstrating the validity and utility of the proposed technique as a simple design tool. This method is expected to be useful for engineers in the initial design stage for structures with active aseismic control.

Key Words
active control; stochastic response spectra; multi-DOF; peak factor.

Address
Takashi Mochio: Dept. of Intelligent Systems, Kinki University, Wakayama, Japan


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