Failure Analysis of a Beam-Column Under Oblique-Eccentric Loading: Potential Failure Surfaces for Cervical Spine Trauma

For a cantilever beam-column with one end built-in and the free end subjected to an oblique-eccentric arbitrary concentrated force, general formulas to produce failure were derived. The original generalized uniform solution to the oblique-eccentric buckling problem was obtained. The Secant formula and Euler’s formula were proved to be specific cases in this general solution. The load ratio, F/aE, was derived as functions of the force acting direction, α, the slenderness ratio, L/r, as well as the eccentricity ratio, ec/r². Material and buckling failures aspects were combined in a uniform structural failure analysis. Safe regions for the load ratio, F/aE, were visualized in the three-dimensional (F/aE)-α-(L/r) space with the eccentricity ratios, ec/r², as a parameter. The column failure factor, kL, was shown to be a key index controlling both aspects of failure as well as the orientation of the second stiff est region. The angle α_E = tan⁻¹ (2L/πe) for kL = π/2 is the singular point for both strength and buckling failure, and α_II = tan⁻¹ (2L/3e) for kL = 0 is the upper bound of the second stiffest region. The feasible domain of the second stiffest region is bounded by α_E and α_II both of which are only functions of geometrical properties. The implications of these analyses for the experimental validation of cervical spine trauma are discussed.