Materially nonlinear free vibration analysis of reinforced concrete beam

The aim of this paper is to do materially nonlinear free vibration analysis of a beam by using the finite element method. In finite element modeling, two approaches are used. In the first approach, the material matrices of concrete and reinforcement are constructed separately, then superimposed to obtain the element stiffness matrix. In the second approach, the reinforcement is assumed to be uniformly distributed throughout the beam. So, the beam is modeled as a single composite element with increasing the modulus of elasticity of concrete by considering the reinforcement ratio. For these two approaches, elastic-perfectly plastic stress-strain relationship is used in compression. In tension, a stress-strain relationship considering tension softening is used. In order to obtain the stiffness and mass matrices of the beam at initial, yield, and failure, a computer program is coded in MATLAB. After these matrices are obtained, MATLAB is used to obtain the nonlinear frequency parameters and mode shapes. It is concluded that the approaches and the model considered in this study can effectively be used in the materially nonlinear free vibration analysis of reinforced concrete (RC) beams. It is also concluded that the effects of the change in the stiffness matrix from initial to yield on the nonlinear frequency parameters of the beam are always larger than those of the change from yield to failure.