Buckling and post-buckling analysis of beams with geometric imperfections: force-based formulation

This study investigated the nonlinear stability of slightly curved beams with geometric imperfections using Euler–Bernoulli beam theory. Existing formulations adopt varying kinematic assumptions that yield inconsistent nonlinear governing equations for the same problem. The present study resolved this ambiguity by deriving the governing equations from Newton’s force equilibrium on differential elements, incorporating the Green–Lagrange strain tensor. This derivation strategy, applied here to slightly curved beams within a kinematic framework in which the slope of the geometric imperfection is explicitly retained in the axial displacement field, revealed nonlinear coupling terms that are absent in displacement-only formulations. For static stability, two analytical solution methodologies were developed: a classical approach employing homogeneous solutions and a non-classical approach employing assumed displacement forms. Dynamic stability analysis via the Galerkin method confirmed that two-term truncation provided sufficient accuracy, with bifurcation diagrams consistent with static solutions in the steady state. The critical buckling loads yield Pc = ² Pc = π 2 for hinged–hinged and Pc = 4 ² Pc = 4 π 2 for fixed–fixed supports, in exact agreement with established results. The analysis established that geometric imperfections transformed the classical pitchfork bifurcation into a transcritical bifurcation, generating three distinct equilibrium branches in the supercritical regime. Closed-form expressions further revealed that the maximum critical load (2 λ 2 ) and the zero-load imperfection threshold ( a = 4 /η ) are governed solely by the slenderness coefficient, independently of the boundary conditions across all cases examined. These results provide a theoretically rigorous characterization of how initial curvature alters the bifurcation topology of compressed beams.