The main aim of the present work is to develop a formulation and some strategies for the instability analysis of slender columns under an axial harmonic force this phenomenon is known as parametric ressonance. An excitation is said to be parametric if it appears as timedependent - often periodic - coefficients in the equations governing the motion of the system,and not as an inhomogeneous term.The column is described by Navier classical formulation. The present work consider the column with one or three degrees of freedom with or without nonlinearities. The equations governing the motion are obtained by the Ritz method.The linear equation (Mathieu equation) and the Duffing equation with small damping are solved in an approximate way using multiple scales techniques, revealing the possibility of destabilizing the static equilibrium position in certain regions of the control space. A similar conclusion is obtained by employing numerical methods for the solution of linear and nonlinear equation systems with or without initial geometrical imperfections.This enables one to obtain time response, phase space, projections Poincaré sections and bifurcation diagrams. These numerical results show that the column with nonlinearities and loaded by a periodic longitudinal force can present various solutions with the same period as the forcing and subharmonic e superharmonic oscillations, as well as chaotic motions.