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

Abstract

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 αE = tan−1 (2L/πe) for kL = π/2 is the singular point for both strength and buckling failure, and αII = 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 αE and αII both of which are only functions of geometrical properties. The implications of these analyses for the experimental validation of cervical spine trauma are discussed.