Elishakoff, Isaac

In this paper, a periodic finite-span beam subjected to the stochastic acoustic pressure with bounded parameters
is investigated. Uncertainty parameters exist in this acoustic excitation due to the deviation or imperfection. First,
a finite-span beams subjected to the random acoustic pressure field are studied, the exact analytic forms of the
cross-spectral density of both the transverse displacement and the bending moment responses of the structure are
formulated. The combined probabilistic and convex modeling of acoustic excitation appears to be most suitable,
since there is an insufficient information available on the acoustic excitation parameters, to justify the totally
probabilitic analysis. Specifically, we postulate that the uncertainty parameters in the acoustic loading belong to
a bounded, convex set. In the special case when this convex set is an ellipsoid, closed form solutions are obtained
for the most and least favorable mean square responses of both the transverse displacement and bending moment
of the structure. Several finite-span beams are exemplified to gain insight into proposal methodology.

An approach to the optimum design of structures, in which uncertainties with a fuzzy nature
in the magnitude of the loads are considered, is proposed in this study. The optimization process
under fuzzy loads is transformed into a fuzzy optimization problem based on the notion of
Wemers' maximizing set by defining membership functions of the objective function and
constraints. In this paper, Werner's maximizing set is defined using the results obtained by
first conducting an optimization through anti-optimization modeling of the uncertain loads.
An example of a ten-bar truss is used to illustrate the present optimization process. The
results are compared with those yielded by other optimization methods.

The paper presents a novel approach to predict the response of earthquake-excited
structures. The earthquake excitation is expanded in terms of series of deterministic
functions. The coefficients of the series are represented as a point in N-dimensional
space. Each available accelerogram at a certain site is then represented as a point in
the above space, modeling the available fragmentary historical data. The minimum
volume ellipsoid, containing all points, is constructed. The ellipsoidal models of
uncertainty, pertinent to earthquake excitation, are developed. The maximum response
of a structure, subjected to the earthquake excitation, within ellipsoidal modeling of
the latter, is determined. This procedure of determining least favorable response was
termed in the literature (Elishakoff, 1991) as an antioptimization. It appears that
under inherent uncertainty of earthquake excitation, antioptimization analysis is a
viable alternative to stochastic approach.