Study of Stress-Strain State of Long Cracked Multi-Modulus Strip in Bending in Relation to Crack Formation in Tensile Zone of Concrete - Evgeniy M. Zveryaev (Structural Mechanics of Engineering Constructions and Buildings)
The problem of strength analysis of a multi-modulus strip, in contradiction to the existing standpoint of essential nonlinearity, may be formulated as a linear problem for a two-layer strip. First order differential equations of the theory of elasticity for the plane strip problem are transformed to dimensionless form and are replaced by integral equations with respect to the transverse coordinate, similar to how it is done in the Picard’s method of simple iterations. In this case, a small parameter appears as a multiplier in the integral equations before the integral sign, which ensures the convergence of solutions in accordance with the contraction mapping principle, also called the Banach fixed point theorem. The original system of equations of elasticity theory is splitted into integratable equations of bending, axial tension-compression and edge effect. The found solutions satisfy all boundary conditions of the elasticity theory problem. The formula determining the position of the neutral axis during bending is written. For a multi-modulus material, such as concrete, the neutral line shifts upward significantly in the compression region during bending, resulting in large displacements at the lower edge in tension and creating conditions for opening of vertical cracks. The occurrence of inclined cracks near supports is explained.