Buckling of a slender elastic column under static compressive load is a fundamental and classical problem in structural stability. It has been examined over more than 250 years, starting with Euler and up to present day. For combinations of the ideal boundary conditions – simple, clamped, sliding and free — exact analytical solutions for critical loads are available. In real structures however, end fixations of columns are not ideal but rather restrained by elastic end supports, two at each end. For elastic supports, critical loads have been determined numerically for several cases, yet a comprehensive stability analysis covering all possible combinations of elastic end conditions is still missing. Mathematically, the buckling loads are obtained as roots of transcendental equations that include four constants of the end restraint springs.
The present research aims at a complete (once and for all) mapping of the buckling loads for entire possible combinations of elastic boundary conditions. Specifically, compact design charts, in a user-friendly format, are presented along with several elegant formulae for buckling loads within definite limits of validity. In addition, we extend the findings for elastic buckling into the plastic range, where pre-buckling stresses are higher than the yield stress. Analysis is within the framework of Euler buckling theory of homogeneous columns with constant bending stiffness.
We start with an earlier reformulation of the governing transcendental equation where the four spring constants are regrouped into just three parameters: two equivalent spring parameters and one nondimensional measure of supports asymmetry. The equation is then factorized into two distinct branches that are examined separately. New closed form expressions are derived for several cases and numerical charts are supported by accurate asymptotic expansions. Comparison with available solutions confirms the efficiency of suggested approach to the buckling problems.
A detailed sensitivity analysis reveals the nonlinear effect of elastic end supports and ends asymmetry on the buckling load. Among the cases examined in detail are rigid body instability, weak columns (applicable to MEMS), symmetric torsional springs and multiple roots. Also discussed are the buckling energy partition and influence of supports age deterioration.
