Extension of Variational Principles for Non-conservative Greenhill’s Shafts

This paper lays out a procedure to quantify the buckling loads of Greenhill shafts using variational iteration method. Lack of approximate methods for Greenhill shafts is attributed to the non-self-adjointness property of its governing differential equation. For a Greenhill shaft, a series solution is assumed for the displacement in terms of polynomials, and a correction functional is applied, which requires a Lagrange multiplier function. An exact general Lagrange multiplier is difficult to identify for a fourth-order differential equation as it requires solution procedures for quartic polynomials. Hence, simplified approximate formulae for Lagrange multiplier are chosen from the literature, and the effectiveness is demonstrated for a pinned–pinned shaft, subjected to a conservative axial load, and a non-conservative axial torque.