Novel Methods for Calculating Natural Frequencies and Buckling Loads of Columns With Intermediate Multiple Elastic Springs

Numerical methods for calculating both the natural frequencies and buckling loads of columns with intermediate multiple elastic springs are developed. In formulating the governing equations of the column, each elastic spring is modeled as a discrete Winkler foundation of the finite longitudinal length. By using this model, the differential equations governing both the free vibration and buckled shapes of the column are derived, which are solved numerically. The Runge–Kutta method is used to integrate the differential equations, and the determinant search method combined with Regula–Falsi method is used to determine the eingenvalues, namely, the natural frequencies and buckling loads. In the numerical examples, fixed–fixed, fixed-hinged, hinged-fixed and hinged–hinged end constraints are considered. The numerical results including the frequency parameters, mode shapes of free vibrations and buckling loads are presented in non-dimensional forms.