# A distortional semi-discretized thin-walled beam element

Michael Joachim Andreassen, Jeppe Jönsson

Research output: Contribution to journalJournal articleResearchpeer-review

## Abstract

Due to the increased consumption of thin-walled structural elements there has been increasing focus and need for more detailed calculations as well as development of new approaches. In this paper a thin-walled beam element including distortion of the cross section is formulated. The formulation is based on a generalized beam theory (GBT), in which the classic Vlasov beam theory for analysis of open and closed thin-walled cross sections is generalized by including distortional displacements. The beam element formulation utilizes a semi-discretization approach in which the cross section is discretized into wall elements and the analytical solutions of the related GBT beam equations are used as displacement functions in the axial direction. Thus the beam element contains the semi-analytical solutions. In three related papers the authors have recently presented the semi-discretization approach and the analytical solution of the beam equations of GBT. In this approach a full set of deformation modes corresponding to the homogeneous GBT equations are found. The deformation modes of which some are complex decouple the state space equations corresponding to the reduced order differential equations of GBT and allow the determination of the analytical solutions. Solutions of the non-homogeneous GBT differential equations and the distortional buckling equations have also been found and analyzed. Thus, these related papers are not dealing with an element but dealing with analytical solutions to the coupled differential equations.

To handle arbitrary boundary conditions as well as the possibility of adding concentrated loads as nodal loads the formulation of a beam element is needed. This paper presents the formulation of such a generalized one-dimensional semi-discretized thin-walled beam element including distortional contributions. It should be noticed that we are only dealing with a basic generalized beam theory and not an extended finite element formulation of an approximate beam element, which allows the addition of special (transverse extension and shear lag) modes. Illustrative examples showing the validity and the accuracy of the developed distortional semi-discretized thin-walled beam element are given and it is shown how the novel approach provides accurate results making it a good alternative to the traditional and time consuming FE calculations.
Original language English Thin-Walled Structures 62 142-157 0263-8231 https://doi.org/10.1016/j.tws.2012.07.011 Published - 2013

## Keywords

• Thin-walled beams
• Beam theory
• Distortion
• Warping
• Distortional beam theory
• Generalized beam theory
• Semi-discretization
• Beam element
• one-dimensional

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