Equilibrium equations for 3D critical buckling of helical springs

In most cases, the research on the buckling of a helical spring is based on the column, the spring is equivalent to the column, and the torsion around the axial line is ignored. A three-dimensional (3D) helical spring model is considered in this paper. The equilibrium equations are established by introducing two coordinate systems, the Frenet and the principal axis coordinate systems, to describe the spatial deformation of the center line and the torsion of the cross section of the spring, respectively. By using a small deformation assumption, the variables of the deflection can be expanded into Taylor’s series, and the terms of high orders are ignored. Hence, the equations can be simplified to the functions of the twist angle and the arc length, which can be solved by a numerical method. The reaction loads of the spring caused by the axial load subjected to the center point are also discussed, giving the boundary conditions for the solution to the equilibrium equations. The present work is useful to the research on the behavior of the post-buckling of the compressed helical spring.