Isaac Elishakoff

Antioptimization of structures with large uncertain-but-non-random parameters via interval analysis

Non-probabilistic, interval modeling of uncertain-but-non-random parameters for structures is developed in this paper for antioptimization analysis, consisting in determining the least favorable responses. The uncertain-but-non-random parameter is considered to be a deterministic variable belonging to a set modeled as an interval. The least favorable static displacement bound estimation for structures with uncertain-but-non-random parameters is transformed into solving interval linear equations. For small interval parameters (the width of interval being small), the uncertainties of interval parameters are treated as the perturbed quantities around the midpoint of interval parameters, by means of the interval matrix central notation and the natural interval extension. Interval perturbation method for estimating the static displacement bound of structures with interval parameters was presented in the recent study by Qiu et al. [1]. For large interval parameters, a subinterval perturbation method for estimating the static displacement bound of structures with interval parameters is put forward in the study. The numerical results show that a subinterval perturbation method yields tighter bounds than those yielded by the interval perturbation method.

Essay on uncertainties in elastic and viscoelastic structures : from A. M. Freudenthal's criticisms to modern convex modeling

Abstract Stochastic approach and non-stochastic, convex modeling of uncertainty are critically contrasted. First some drawbacks of the probabilistic methods are discussed, attributable to lack of sufficiently accurate data; then the effect of human error in constructing a probabilistic model for input quantities is elucidated. Extensive quotations from pertinent works of Freudenthal, who is rightfully considered as an architect of modern reliability theory, are utilized to explain some doubts he himself experienced about probabilistic methods. His hints are realized in modern convex modeling of uncertainty, which the writer is advocating and advancing. A set-theoretical, convex description of uncertainty is discussed in detail. Uncertainty is described as a set of constraints unlike the classical probabilistic approach. Moreover, instead of conventional optimization studies, where the minimum possible responses are sought, here an uncertainty modeling is developed as an “anti-optimization” problem of finding the least favorable response under the constraints within the set-theoretical description. The question of how the output quantities of such an anti-optimization process vary when the global knowledge on the uncertainties increases is considered in detail. Response variability of viscoelastic structures is evaluated. Combined probabilistic-convex modeling is proposed for situations where the input quantities should be modeled as stochastic ones, but some of their probabilistic characteristics are unknown but bounded.

Probabilistic Methods in the Theory of Structures

Keywords: probabilites ; structures ; vibration ; methodes de : calcul Reference Record created on 2005-11-18, modified on 2016-08-08

Structural design under bounded uncertainty-optimization with anti-optimization

Abstract In many cases precise probabilistic data are not available on uncertainty in loads, but the magnitude of the uncertainty can be bound. This paper proposes a design approach for structural optimization with uncertain but bounded loads. The problem of identifying critical loads is formulated mathematically as an optimization problem in itself (called anti-optimization), so that the design problem is formulated as a two-level optimization. For linear structural analysis it is shown that the antioptimization part is limited to consideration of the vertices of the load-uncertainty domain. An example of a ten-bar truss is used to demonstrate that we cannot replace the anti-optimization process by considering the largest possible loads.

Refined second-order reliability analysis

Abstract A refined second-order method is presented for structural reliability analysis. Exact and approximate reliability solutions are obtained for a circular shaft subject to random bending moments and a random torque. The comparison of the approximate results with exact ones shows that the first-order approximation is only applicable to the case where the failure surface is “far” from the origin, while the suggested second-order approximation yields quite accurate results even if the failure surface is “close” to the origin.

Combination of probabilistic and convex models of uncertainty when scarce knowledge is present on acoustic excitation parameters

Abstract This paper is devoted to a fundamental problem of accounting for parameter uncertainties in random vibrations of structures. In contrast to the overwhelming majority of random vibration studies where perfect knowledge is assumed for the parameters of the excitation, this crucial conjecture is dispensed with. The probabilistic characteristics of the excitation are assumed to be given as depending on some parameters which are not known in advance. We postulate that some imprecise knowledge is available; namely, that these parameters belong to a bounded, convex set. In the case where this convex set is represented by an ellipsoid, closed form solutions are given for the upper and lower bounds of the mean-square displacement of the structures. For the first time in the literature the system uncertainty in the random vibrations is dealt with as an ‘anti-optimization’ problem of finding the least favorable values of the mean-square response. The approach developed here opens a new avenue for tackling parameter uncertainty which is often encountered in various branches of engineering.

Bounds of eigenvalues for structures with an interval description of uncertain-but-non-random parameters

Abstract This paper deals with eigenvalue problems involving uncertain but non-random interval stiffness and/or mass matrices. If one views the deviation amplitude of the interval matrix as a perturbation around the nominal value of the interval matrix, one can solve the standard eigenvalue problem of the interval matrix by applying the interval extension to the matrix perturbation method. In this study, the interval perturbation approximating formula is presented for evaluating interval eigenvalues of the interval matrix. Inextensive computational effort is a characteristic of the proposed method. The illustrative numerical examples are provided.

Uncertain buckling: its past, present and future

An authoritative review on stochastic buckling of structures was written by Amazigo some quarter century ago. The present review summarizes some of the developments which took place since then. A brief overview of the effect of uncertainty in the initial geometric imperfections, elastic moduli, applied forces, and thickness variation is given. For the benefit of the thoughtful reader, the review is of critical nature.

Non-linear buckling of a column with initial imperfection via stochastic and non-stochastic convex models

Abstract Buckling of initial imperfection sensitive structure — column on a non-linear elastic foundation — is investigated. A criterion based on the concept of “modal buckling load” is proposed to determine which modes should be included in the analysis when the weighted residuals method is utilized to calculate the limit load — maximum load the structure can support — for a given initial deflection. For stochastic analysis, a random field model is suggested for the uncertain initial imperfection, and Monte Carlo simulations are performed to obtain the probability density of the buckling load and the reliability of the column. Finally, a non-stochastic convex model of uncertainty is employed to describe a situation when only limited information is available on uncertain initial deflection, and the minimum buckling load is obtained for this model. The results from both the stochastic and the non-stochastic approaches are derived and critically contrasted.

A new approximate solution technique for randomly excited non-linear oscillators—II

Abstract The method of weighted residuals is applied to the reduced Fokker-Planck equation associated with a non-linear oscillator, which is subjected to both additive and multiplicative Gaussian white noise excitations. A set of constraints are deduced for obtaining an approximate stationary probability density for the system response. One of the constraints coincides with the previously proposed criterion of dissipation energy balancing, and the others are useful for calculating the equivalent conservative force. It is shown that these constraints imply certain relationships among certain statistical moments; their imposition guarantees that such moments computed from the approximate probability density satisfy the corresponding exact equations derived from the original equation of motion. Moreover, the well-known procedure of stochastic linearization and its improved version of partial linearization are shown to be special cases of this scheme, and they are less accurate since the approximations are not chosen from the entire set of the solution pool of generalized stationary potential. Applications of the scheme are illustrated by examples, and its accuracy is substantiated by Monte Carlo simulation results.