The stiffness of a reinforced concrete beam with random parameters the cracks formation

Abstract

The work of a reinforced concrete beam lying on an elastic foundation is considered, the deformation characteristics of which are described by a model with two Vlasov-Pasternak bed coefficients. The bed coefficients are considered as stationary random normally distributed functions of the x coordinate. Strength characteristics of concrete: cube strength and concrete tensile strength are assumed to be random variables with Gaussian probability density distributions. The concentrated forces approximating the columns of the building are applied at regular intervals and are assumed to be random normally distributed values. After the formation of cracks in the sections under concentrated forces in the beam, areas with a reduced bending stiffness are formed, and therefore the stiffness of the beam becomes variable along the length of the beam. The solution of the differential equation of bending of a beam with variable stiffness lying on an elastic foundation is proposed to be sought by a variational method minimizing the functional of the additional energy principle. The deflections of a beam with variable stiffness will be approximated by a function that satisfies the boundary conditions and the equilibrium condition. From the equality of the deflections of a beam with constant equivalent bending stiffness and deflections of a beam with variable bending stiffness due to cracking, the constant equivalent stiffness and its statistical parameters are determined. Knowing the constant equivalent bending stiffness of a concrete beam and using the known solutions of the bending equation of a beam lying on an elastic foundation with random properties, it becomes possible to find the probabilistic parameters of bending moments and shear forces and to determine the probability of the onset of limit states of the beam along the normal and inclined sections.