A Closed-Form Buckling Formula for Open-Coiled and Properly Supported Circular-Bar Helical Springs

As a continuation of the author’s previous studies on the buckling analysis of helical springs, a closed-form formula having been obtained with the help of the artificial neural network (ANN) is proposed and discussed in detail for the first time for a cylindrical close/open-coiled helical spring with fixed ends and having a solid circular section. As far as the author knows there is no such a formula in the open-literature to consider the effects of all stress resultants (torsional and bending moments, axial and shearing forces), large helix pitch angles together with the axial and shear deformations on the buckled state. The present formula may be used in a wide range of the total number of active turns, the ratio of the free axial length to the mean helix diameter, and the spring index. It is yet again revealed that it is not appropriate to use the elementary theory to determine the critical buckling loads for open-coiled springs. The present formula may allow the deeper understanding of spring buckling mechanism and to be used directly and safely in the design processes of such closely/open-coiled springs.