This analysis deals with the free torsional vibrations of bars of thin-walled open cross section for which the shear center and centroid coincide. Such sections include I-beams and Z-sections. The differential equation for torsional vibrations is derived and includes the effect of warping of the cross section. The effect of warping on the frequency of vibration and the shapes of the normal modes of vibration are determined for bars of single span with various end conditions. For a simply supported bar, a formula for the principal torsional frequencies and an expression for mode shape are derived. For other conditions of support, the frequency equations are derived and their solutions presented in graphical form. From these graphs the frequencies of vibration and the mode shapes may be obtained directly. The case of a cross section which does not warp is a limiting case of the general problem. For such shapes (for example, a cross-shaped or cruciform section) the formulas for torsional frequencies and modes of vibration are quite simple. These formulas also are valid for a circular shaft and may be used approximately for other solid sections.

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