A dynamic instability, called parametric resonance, is exhibited by undampedelastic 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.
An axial follower force acting on the free end of a beam-column is known to remain tangential to its elastica at that point. Elastic beam-columns exhibit infinitely high buckling resistance to static compressive follower load. Loss of their dynamic stability is known to occur at critical follower loads, by flutter characterized by vanishing lateral displacement and infinitely high natural frequency. Classical theory deals with physically linear nonconservative beam-columns. Physical nonlinearity exhibited by concrete beam-columns under service loads is caused by the closing and reopening of the extant transverse cracks. In this Paper, analytical expressions for the lateral displacement and lateral stiffness of such concrete beam-columns are derived. Using these expressions, the stability of physically nonlinear elastic flanged concrete beam-columns under the action of a follower compressive axial force and a lateral force is investigated. The significance of the analytical approach and the theoretical predictions is discussed.