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, alpha, 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)-alpha-(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 stiffest region. The angle alpha E = tan-1 (2L/pi e) for kL = pi/2 is the singular point for both strength and buckling failure, and alpha 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 alpha E and alpha 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.