## A theory of torsional waves and vibrations in beams of thin-walled open section |

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analysis arbitrary constants bar velocity Boley and Chao boundary conditions centroidal Channel Beam circular cylindrical bar contour cosh coupled equations cross section parameter cross-section D'Alembert'e principle defined by equation effects of shear Equations of Motion evaluated exact theory flange flexural forced vibrations Gere given gives group velocities I-BEAMS impulsively applied initial conditions initial value problem integrand elements Jt P+a-L Jt l.lla Laplace transform lineal paths loads longitudinal inertia Love's correction mn mn Modal Functions mode of wave mode shape modes of vibration natural boundary conditions natural frequencies neutral axis non-dimensionalised obtained phase and group phase velocity present theory radius of gyration rotation s-plane shear and longitudinal shear center shear deformation shear force shear stresses short wave lengths simply supported beam sinh solution step torque strain energy Timoshenko beam Timoshenko torsion theory torsional waves unit length variable Venant's warping constant zero