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In this paper, a numerical–experimental study about natural frequencies of doubly unsymmetrical thin-walled and open cross-section beams is presented. The equations of motion are based on Vlasov's theory of thin-walled beams, which is modified to include the effects of shear flexibility, rotatory inertia in the stress resultants and variable cross-sectional properties. The formulation is also applicable to solid beams, constituting therefore a general theory of coupled flexure and torsion of straight beams. The differential equations are shown to be particularly suitable for analysis in the frequency domain using a state variables approach. A simplified theory, which excludes the analysis the warping constraint, is presented leading to a more simple theory that is used for comparison purposes in this paper. Finally, free vibration experimental tests about unsymmetrical beams are presented which allow to verify the theory presented in this paper and provide good quality data that can be used for checking the accuracy and reliability of different theories.

The first part of the paper develops a theory for the torsional analysis for open thin-walled beams of general cross-sections which accounts for shear deformation effects. Statically admissible stress fields are postulated in agreement with those resulting from the Vlasov thin-walled beam theory. The principle of stationary complementary energy is then adopted to formulate the governing field compatibility condition under the stress fields postulated. The naturally arising boundary terms are found to relate the warping deformations to the internal force fields. A torsion beam example is solved using the new theory in order to illustrate its applicability to practical problems. The second part of the paper implements the solution numerically in a force-based finite element context. Two finite elements are developed by assuming linear and hyperbolic bimoment fields. The FEA solutions are shown to provide lower bound representations of the stiffness when compared to those based on conventional beam theories founded on postulated kinematic assumptions. © 2006 Elsevier Ltd. All rights reserved.