Parametric Resonance in Concrete Beam-Columns

A dynamic instability, called parametric resonance, is exhibited by undamped elastic beam-columns when under the action of pulsating axial force. The scope of the existing theory of parametric resonance is restricted to physically linear beam-columns undergoing finite lateral displacements. In this Paper, the dynamic behaviour of physically nonlinear elastic cracked concrete beam-columns under pulsating axial force and constant lateral force is investigated. The constitutive equations derived earlier by Authors in the form of force-displacement relations are employed here to formulate equations of motion of the SDOF cantilever with mass lumped at its free end. The expected phenomenon of parametric resonance is exhibited in the form of regular subharmonic resonance at about the frequency ratio of two. Resonance peaks broaden with increase in pulsating force. Like damping, physical nonlinearity is also predicted to stabilize the dynamic response at resonance frequencies. In some particular statically unstable conditions, the loss of dynamic stability is shown to occur by divergence. Unexpectedly, similar phenomenon of parametric resonance is exhibited by these physically nonlinear beam-columns undergoing even small lateral displacements. The contribution made to the theory of parametric resonance and the potential relevance of the proposed theory to design of concrete beam-columns is discussed.