Complements to the Euler’s Linear Theory for Buckling of Long Columns: Modified Critical Stresses and Proposed Critical Deflections

The critical load of a long column was derived by Leonhard Euler in 1744. The corresponding deflection of a long column is arbitrary or indeterminate under small deformations. We demonstrate that this indeterminacy stems from an improper definition of critical stresses, which excludes bending stresses. By modifying the definition of critical stresses, we propose a formula to determine critical deflections. The previous critical stress is valid only if the slenderness ratio equals the critical slenderness ratio. The error of the critical deflection is determined by the perturbation method. The proposed method is illustrated using published experimental data of ZnO nanowires, a simple indeterminate two-bar macrostructure, a column with immovable ends, and a comparison with a nonlinear approach. The critical failure deflection is crucial in engineering, particularly for indeterminate structures, as it is required to establish deformation compatibility equations.