Damping of reinforced concrete structures under free vibrations

Abstract

This article considers the free vibrations of a reinforced concrete structure with nonlinear flexural stiffness due to the formation and closure of normal cracks and with energy dissipation represented by viscous damping. The calculation model is a cantilever bar loaded with a constant longitudinal force N and an instantly removed transverse load S. The problem is divided into three stages. In the first stage, a nonlinear static problem of bar deflection with a length-varying stiffness dependent on the bending moment level is numerically solved; the coefficient ψs is used to describe the behavior of tensile concrete. In the second stage, the critical damping and logarithmic decrement of vibrations are determined within the framework of a linearly rigid dynamic model with external viscous dampers using the state-space method. In the third stage, a nonlinear dynamic problem is solved, in which the stiffness matrix depends on the current headwall deflection, and the damping matrix is generated taking into account the presence of cracks. The calculations were performed using the MATLAB software package. It was found that for small and moderate transverse loads, the damping coefficient is virtually independent of crack formation. However, as the load approaches its limit, a sharp, power-law-like increase in the dissipative properties of the structure is observed. The resulting relationship allows for a more accurate assessment of damping in reinforced concrete elements when calculating dynamic loads.