Probabilistic representation of the critical factor coefficient in the reliability problems of structures

Abstract

This paper deals with the actual problems of the practical calculation of the steel storage capacities. The research defines generalized coefficient of the critical factor. It is the ratio of the generalized values of the internal forces and reliability, which are represented by random processes. The methods of the  рrobability theory and of the mathematical statistics were used for the solution. The value of the critical factor was expressed through its statistic characteristics, and also through the differential and integral distribution functions. The linearization of the non-linear function of the random values in the area of the expected values was made to determine the standard deviation and the coefficient of variation. At the same time correlation at the non-linearity was considered when calculating the dispersion. The density distribution of the critical factor coefficient was determined when using the normal law of the distribution for the random value of the generalized reliability. The probabilistic process of the generalized force was schematized by two laws of distribution. The normal law was used to describe the pressure of the bulk material on the body walls of the storage capacities. Double exponential distribution of Gumbel is used to describe maximums of the snow and wind loads. At the same time classical calculation was made. Thus, the finite analytical decision in two variants was obtained. Accordingly to the given algorithm, engineering calculations is very complicated. It requires using special mathematical packages to calculate integral expressions. To avoid this, a simulation procedure was used. This allowed to solve the problem of finding the probability distribution function in the range of the argument values, when the ordinates are close to one. It was proposed to express the probabilistic features of coefficient of the critical factor by the features of another random value. On the basis of polygon and distribution function of this value, the approximating expressions for a given range of probability variations are chosen. The obtained values for the critical factor allow us to solve the problem of probabilistic calculation analytically, what does not require using of complicated calculating procedures.