Critical Compressive Stress of a Beam-Column to a Second Order

Buckling of a beam-column under axial loads at the ends is considered as a three-dimensional stability problem. Displacement and stress fields are written in terms of Neuber-Papkovich potentials. These are then approximated to forms containing one more higher-order term than those corresponding to the Euler-Bernoulli equations. These expressions yield a modified formula for the bending moment. The energy method applied to the equations yield a modified form of Euler critical load and Shanley tangent modulus formula. The effect of the second-order modification is to decrease the critical load. For mild steel this decrease exceeds 5 percent for the slenderness ratio less than 25.