Elastic Stability of Composite Column

Abstract

A vertical column possessing two different flexural rigidities is clamped at the bottom end and is free at the top. Both piecewise uniformly distributed axial loads (self-weights) and a concentrated load at the top are applied to the column. For the buckling load, the governing differential equation is a second-order nonlinear with variable coefficients. Even upon linearization, its solution involves linear combinations of the special functions. The equation for the eigenvalue problem is exact but tedious to solve numerically. Since the physical problem is fundamental in structural mechanics, this note develops some useful approximate formulas for the stability criterion. For the first mode, which is the most important one, elastic stability criterion is determined by a quadratic equation with variable coefficients via the Rayleigh-Ritz energy approach. The effects of ratios among various loads and between flexural rigidities on buckling loads are found. The comparison with the exact results available shows the formula giving satisfactory approximation.