Lukianchenko O. Numerical implementation of the finite element method in the probabilistic formulation of problems of reliability and safety of thin shells with shape imperfections

A new numerical approach is developed. The approach is based on the finite element method and the main provisions of V.V. Bolotin on how to determine in the probabilistic formulation the design reliability of the stability of thin shells with shape imperfections. A new modified scheme of the finite element method is constructed to solve the problem of nonlinear deformation and stability of thin shells with arbitrary shape imperfections in the general curvilinear coordinate system. The stiffness matrix of a curvilinear finite element is constructed by means of a vector approximation of the displacement function represented by the Maclaurin series. Effective algorithms of computer modeling of shape imperfections of thin shells with the use of the NASTRAN program. The program of forming new coordinates of nodes of the shell model with the maximum amplitude of imperfections, developed by the author and adapted to NASTRAN program are applied as well. An algorithm for constructing curves (surfaces) of design reliability of thin shells under different types of loads according to given functions of the shape imperfections probability distribution is proposed. This algorithm uses curves (surfaces) of critical values of loads obtained by solving geometrically nonlinear statics problem by Newton-Raphson method. A new numerical approach to determining the operational reliability of the stability of thin shells with real shape imperfections is proposed using spline curves of the NASTRAN software with the possibility of their visualization at a given scale is presented and applied. An effective probabilistic approach to risk assessment of thin shell failures with real shape imperfections due to loss of general stability has been developed. The risk is estimated on the basis of probability theory theorems, fuzzy logic approaches, decision-making methods in conditions of uncertainty as exceeding the actual probability of a theoretical accident, the distribution law of which corresponds to the one-parameter Rayleigh distribution. Areas of acceptable values of accident risk are determined by the value of information entropy. The cumbersome procedure for determining the mathematical expectation of the reliability of the tank using statistical processing of data on the state of structures by the Monte Carlo method has been replaced by an effective method for determining the operational reliability of the design reliability curve. New solutions of practical problems of influence of geometry, boundary conditions, corrosion of metal, rings of rigidity and defects of welds on the general stability, reliability and safe operation of thin shells are received. A new algorithm for computer finite element modeling of thin shells with weld defects has been developed, taking into account their development for multiclass recognition and prediction of their technical condition. The following neural network classifiers are determined at the locations of sensors: deformations, stresses and natural frequencies of oscillations.