Finite element formulation of a thin-walled beam with improved response to warping restraint
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Linear elastic theory of torsion and flexure of thin-walled beams as developed by Vlasov and Timoshenko respectively are well known and commonly used in everyday engineering practice. However there are noticeable differences between calculations and experimental results. The difference is partly due to one of the basic assumption of classical theory, namely that the secondary shear strains due to warping are negligible.
In the present work a new three noded, with CO continuity, isoparametric beam finite element is developed based on a torsion theory by Benscoter. In the classical theory warping is assumed to be proportional to the rate of twist, whereas in Benscoter's theory it is assumed to be proportional to an independent quantity called the "warping function".
The exact form of this function can be evaluated from equilibrium equations. This assumption of Benscoter's allows the formulation of a Co element based on the assumed displacement method. The other advantage of Benscoter's theory is that it takes into account the effects of secondary shear strains. These effects are quite significant for closed sections. The element is validated for several cases of a cantilevered beam of rectangular cross section and in every case the results are in good agreement with the exact solution.
It is also shown that the element gives a very good representation of curved beams, for which there is torsional-flexural coupling. A number of cases of a curved I-beam under various loading and boundary conditions are analysed, and in every case the results agree closely with the analytical solution.
In order to represent the torsional response the element uses seven degrees of freedom per node. This seventh degree of freedom is the "warping function" mentioned earlier.
To make the element compatible with standard finite-element programs which have six degrees of freedom per node, static condensation is used.
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