Distortional theory of thin-walled beams

Distortional eigenmodes and homogeneous solutions for semi-discretized thin-walled beams

“…GBT was generally not so known in the international research community until Davies, see [11], presented first-order GBT analysis. A distortional theory which generalizes Vlasov beam theory by including the modified definition of the warping function and one distortional mode was presented by Jönsson [12]. In that work the analytical solution of coupled torsional and distortional equations were found by reduction of order and solution of the related eigenvalue problem as in the present work.…”

“…Note that d α is related to the coordinate vector of the shear center, see [12]. The coupling terms in the axial stiffness between translations and rotation are found as follows (in the subspace):…”

“…This method is equivalent to the one used for the solution of the coupled homogeneous problem of one-mode distortion and torsion analyzed in [12]. After we have changed the order of the equations, we can recognize that the pure torsional St. Venant displacement modes with a constant or a linear variation of the angle of twist are eigensolutions.…”

“…It makes it possible to analyze thin-walled members with cross-section distortion and local plate behavior in a one-dimensional formulation through the linear combination of pre-established modes of deformation. However in this paper we find the analytical homogeneous solution to the differential GBT equations (obtained by semi-discretisation), using methods similar to Hanf [10] and Jönsson [12], this also (through the magnitude of the eigenvalues) gives a much better knowledge of the length scales of the modes. Alternatively the GBT equations may just as well have been solved using the approximate engineering methods (in which the shear coupling terms are neglected) producing orthogonal axial and transverse normal stress modes.…”

Isogeometric Methods in Dynamic Analysis of Curved Beams Including Warping and Distortional Effects

“…Schardt [5][6] developed an advanced formulation known as Generalized Beam Theory (GBT) which is a generalization of the classical Vlasov beam theory in order to incorporate flexural and torsional distortional effects. Further developments of GBT avoid some of its cumbersome procedures through eigenvalue cross sectional analysis [7][8][9][10][11][12]. Ferradi and Cespedes [2] presented the formulation of a 3D beam element solving an eigenvalue problem for the distortional behavior of the cross section (in-plane problem) and computing warping functions separately by using an iterative equilibrium scheme.…”