Unified Energy Analysis and K-Factor for Columns with Elastic Rotational End Restraints

An energy-based variational reformulation is presented for the buckling analysis of columns with elastic rotational end restraints. The total potential energy functional includes bending strain energy, end spring energy, and load work. From the stationary condition of this functional, a unified stability characteristic equation valid for arbitrary end rotational stiffnesses is rederived and shown to degenerate consistently to the classical ideal limits of pinned, fixed, and propped cantilever boundary conditions. The contribution is deliberately framed as a systematic energy-based restatement of the classical restrained column problem, complemented by several practical analytical tools. These include a boundary rotational energy participation coefficient that quantifies the energetic role of end springs, an energy-coherent normalization convention that preserves the differential relations among deflection, rotation, and bending moment during mode comparison, and an approximate geometricmean relationship K [Formula: see text] for the effective length factor under asymmetric end restraints. A cross-check against finite element beam–spring models under the same ideal assumptions shows an average load error of 1.66%. Design contour maps are provided for practical stability design of ideal elastic columns within the stated assumptions.