Computing the reliability of anisotropic shells from the probability of infrequent excursions of a random vector field beyond the limiting surface

Abstract

The stress and strain fields that develop in the structural elements of composites while in service are usually random in nature. In specific cases, this may be governed by the stochastic nature of external effects or initial geometric imperfections (the latter is more significant for thin-wall plates and shells subject to compressive loads), by the random character of the elastic or relaxation characteristics of the material, etc. The problem encountered in this case, when computing the reliability, reduces to calculation of the moments of the number of random-field excursions beyond the limiting region of permissible states [I]. In analyzing highly reliable systems, it is sufficient to restrict ourselves merely to the first-order moment (mathematical expectancy) of the number of excursions. Current excursion theory makes it possible to estimate, with an accuracy sifficient for practical purposes, the reliability of a system, the evolution of state of which can be represented in the form of a random vector process or random scalar field in a certain phase space [2-4]. At the same time, a satisfactory description of the behavior of a system can be obtained in a number of cases only when random fields that are statistically related one to the other are considered. To evaluate the reliability of these systems, it is necessary to solve the problem of the excursion of a random vector field beyond a multidimensional limiting surface. In our study we briefly outline the approximate method of computing the probability of infrequent excursions of a random vector field beyond the limiting surface confining a multidimensional parallelepiped. This method is then used to evaluate the reliability of anisotropic shells with a random field of initial shape imperfections under certain forms of dynamic loading. i. Computing the Probability of Infrequent Excursions of the Random Vector Field. Let the behavior of a certain mechanical system be described by a random vector field o(r) {of(r), o2(r) ..... oh(r)}; r={xl,x2}~S, which is defined by a two-dimensional Euclidian region S twice differentiated with respect to all arguments. The region of permissible states of the system is the k-dimension parallelepiped ~k: