Evaluation of the probability of the formation of ultimate deflections in a reinforced concrete beam with cracks on a stochastic base

The results of calculating a reinforced concrete beam on an elastic statistically inhomogeneous foundation after the formation of cracks in concrete are presented. The compliance coefficients of the base are considered as random stationary functions, and the load is assumed to be a random non-stationary function of the x coordinate. Strength parameters of concrete are taken as random Gaussian values. The distribution parameters are given for both the initial bending stiffness of a reinforced concrete beam and the stiffness of the beam after cracking, as a function of the random cube strength of concrete. The parameters of the distribution densities of the deflections of the beam before the formation of cracks in it, as well as after the formation of cracks, are determined. For an approximate solution of the differential equation for the bending of a reinforced concrete beam after the formation of cracks, the variational principle of stationarity of additional energy (the Castigliano functional) is used. This makes it possible to determine the probabilistic characteristics of the distribution of the equivalent constant stiffness of the beam, the probabilistic parameters of the distribution of beam deflections after cracking, as well as the total differential law of the distribution of deflections in a beam on an elastic foundation in an arbitrary section of the beam. The probability of occurrence of the limiting state in the form of exceeding the value of the deflections of the beam, the limiting standard value of the deflections, taking into account the possible formation of cracks in the beam, has been determined.

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