More general expression for the torsional warping of a thin-walled open-section beam

In this study, an analytical formulation for the torsional warping function of a thin-walled open-section beam is built based on the combination of the Vlasov assumption and the Kirchhoff assumption of plate/shell theory, essentially due to Goodier and Gjelsvik. Vlasov, Timoshenko and many authors (follow Vlasov) only consider the contour warping function as the real warping function. Goodier and Gjelsvik (follow Goodier) not only consider the contour warping but also the thickness warping. For some cross-sectional constants related to the warping function such as the torsional constant, the adoption of the warping function based on Vlasovʹs theory or Timoshenkoʹs theory may cause them incorrectness. On the other hand, the torsional constant based on the Goodierʹs theory is consistent with the one based on the membrane analogy. Thus, Goodierʹs theory is a good approximation for the torsional warping of a thin-walled open-section beam. To the authors’ limited knowledge, the analytical expression for the complete torsional warping of a thin-walled open-section beam has not been found in the literature. In this study, a more general expression for the torsional warping of a thin-walled open-section beam is presented. The position formulas to determine the twist center are also given.