Effect of warping‐pretwist coupling on the natural vibration of torsionally clamped‐pinned thin‐walled open‐profile bars

Exact natural frequencies are determined for the torsional vibration of pretwisted thin‐walled open‐profile bars having torsionally clamped and pinned end restraints. The equation of motion and boundary conditions are derived from a multifilament model of a pretwisted thin‐walled bar undergoing warp deformation and a moderately large twist about the elastic axis. Closed‐form torsional frequencies and mode shapes are derived from the linear terms of the resulting non‐linear equations. As the length‐to‐leg ratio (L/b) and leg‐to‐thickness ratio (b/t) of typical open‐profile bars are varied, the individual and collective effects of warping and pretwist on the non‐dimensional torsional frequencies are demarcated. In addition, the closed‐form solutions offered here reveal considerable destiffening of the torsional frequencies due to warping‐pretwist coupling present in thin‐walled bars having pretwisted, unsymmetrical, open‐profiles and torsionally clamped and pinned end restraints. This coupling effect, which have been ignored in thin‐walled bar vibration theories hitherto, is brought out by the Wagner effect and warping shear stresses acting on the profiles.