Probabilistic analysis of a reinforced concrete beam on a stochastic elastic foundation with consideration of cracking

Abstract

The article considers a methodology for the probabilistic analysis of a finite-length reinforced concrete beam freely resting on a stochastic elastic foundation with two bed coefficients under the action of a randomly distributed vibration load. The bed coefficients and the external load are treated as random Gaussian stationary functions of the beam coordinate with prescribed correlation functions. To solve the differential equation of the beam’s forced vibrations, the generalized Bubnov-Galerkin method is used in combination with the statistical Monte Carlo method. Cracking is taken into account using a two-stage scheme. At the first stage, the stress-strain state of the uncracked beam is determined, and the sections in which the cracking condition is satisfied are identified. At the second stage, a piecewise-variable stiffness function is formed for each realization, after which the deflections and bending moments are recalculated. As a result, the mathematical means, standard deviations, cracking probabilities, and probabilities of exceeding the allowable deflection along the beam length are determined. The numerical analysis showed that, within the considered example, accounting for cracking has only a minor effect on the probabilistic characteristics of deflections; however, it substantially changes the mathematical means and standard deviations of bending moments due to the redistribution of internal forces.