Flexural-torsional vibration of a tapered C-section beam

Previous studies have shown that numerical models of tapered thin-walled C-section beams based on a stepped or piecewise prismatic beam approximation are inaccurate regardless of the number of elements assumed in the discretization. Andrade recently addressed this problem by extending Vlasov beam theory to a tapered geometry resulting in new terms that vanish for the uniform beam. (See One-Dimensional Models for the Spatial Behaviour of Tapered Thin-Walled Bars with Open Cross-Sections: Static, Dynamic and Buckling Analyses, PhD Thesis, University of Coimbra, Portugal, 2012, https://estudogeral.sib.uc.pt) In this paper, we model the coupled bending-twisting vibration of a cantilevered tapered thin-walled C-section using a Galerkin approximation of Andrade's beam equations resulting in an 8-degree-of-freedom beam element. Experimental natural frequencies and mode shapes for 3 prismatic and 2 tapered channel beams are compared to model predictions. In addition, comparisons are made to detailed shell finite element models and exact solutions for the uniform beams to confirm the validity of the approach. Comparisons to the incorrect stepped model are also presented.