Non-Probabilistic Models of Uncertainty in the Nonlinear Buckling of Shells With General Imperfections: Theoretical Estimates of the Knockdown Factor

Abstract

A non-probabilistic, set-theoretical treatment of the buckling of shells with uncertain initial geometrical imperfections is presented. The minimum buckling load is determined as a function of the parameters which describe the (generally infinite) range of possible initial imperfection profiles of the shell. The central finding of this paper is a theoretical estimate of the knockdown factor as a function of the characteristics of the uncertainty in the initial imperfections. Two classes of set-theoretical models are employed. The first class represents the range of variation of the most significant N Fourier coefficients by an ellipsoidal set in N-dimensional Euclidean space. The minimum buckling load is then explicitly evaluated in terms of the shape of the ellipsoid. In the second class of models, the uncertainty in the initial imperfection profile is expressed by an envelope of functions. The bounding functions of this envelope can be viewed as a radial tolerance on the shape. It is demonstrated that a non-probabilistic model of uncertainty in the initial imperfections of shells is successful in determining the minimum attainable buckling load of an ensemble of shells and that such an approach is computationally feasible.