Condensation of independent variables in free vibration analysis of curved beams

In this paper, the condensation method which is based on the Rayleigh-Ritz method, is described for the free vibration analysis of axially loaded slightly curved beams subject to partial axial restraints. If the longitudinal inertia is neglected, some of the Rayleigh-Ritz minimization equations are independent of the frequency. These equations can be used to formulate a relationship between the weighting coefficients associated with the lateral and longitudinal displacements, which leads to "connection coefficient matrix". Once this matrix is formed, it is then substituted into the remaining Rayleigh-Ritz equations to obtain an eigenvalue equation with a reduced matrix size. This method has been applied to simply supported and partially clamped beams with three different shapes of imperfection. The results indicate that for small imperfections resembling the fundamental vibration mode, the sum of the square of the fundamental natural and a non-dimensional axial load ratio normalized with respect to the fundamental critical load is approximately proportional to the square of the central displacement.