<abstract xmlns="http://www.w3.org/1999/xhtml"><p>This article deals with the vibrations of a nonprismatic thin-walled beam with an open cross section and any geometrical parameters. The thin-walled beam model presented in this article was described using the membrane shell theory, whilst the equations were derived based on the Vlasov theory assumptions. The model is a generalisation of the model presented by <xref ref-type="bibr" rid="j_sgem-2019-0001_ref_022_w2aab3b7b2b1b6b1ab1b5c22Aa">Wilde (1968)</xref> in ‘The torsion of thin-walled bars with variable cross-section’, <italic>Archives of Mechanics</italic>, 4, 20, pp. 431–443. The Hamilton principle was used to derive equations describing the vibrations of the beam. The equations were derived relative to an arbitrary rectilinear reference axis, taking into account the curving of the beam axis and the axis formed by the shear centres of the beam cross sections. In most works known to the present authors, the equations describing the nonprismatic thin-walled beam vibration problem do not take into account the effects stemming from the curving (the inclination of the walls of the thin-walledcross section towards to the beam axis) of the analysed systems. The recurrence algorithm described in <xref ref-type="bibr" rid="j_sgem-2019-0001_ref_009_w2aab3b7b2b1b6b1ab1b5b9Aa">Lewanowicz’s work (1976)</xref> ‘Construction of a recurrence relation of the lowest order for coefficients of the Gegenbauer series’, <italic>Applicationes Mathematicae</italic>, XV(3), pp. 345–396, was used to solve the derived equations with variable coefficients. The obtained solutions of the equations have the form of series relative to Legendre polynomials. A numerical example dealing with the free vibrations of the beam was solved to verify the model and the effectiveness of the presented solution method. The results were compared with the results yielded by finite elements method (FEM).</p></abstract>

This article deals with the vibrations of a nonprismatic thin-walled beam with an open cross section and any geometrical parameters. The thin-walled beam model presented in this article was described using the membrane shell theory, whilst the equations were derived based on the Vlasov theory assumptions. The model is a generalisation of the model presented by Wilde (1968) in ‘The torsion of thin-walled bars with variable cross-section’, Archives of Mechanics, 4, 20, pp. 431–443. The Hamilton principle was used to derive equations describing the vibrations of the beam. The equations were derived relative to an arbitrary rectilinear reference axis, taking into account the curving of the beam axis and the axis formed by the shear centres of the beam cross sections. In most works known to the present authors, the equations describing the nonprismatic thin-walled beam vibration problem do not take into account the effects stemming from the curving (the inclination of the walls of the thin-walledcross section towards to the beam axis) of the analysed systems. The recurrence algorithm described in Lewanowicz’s work (1976) ‘Construction of a recurrence relation of the lowest order for coefficients of the Gegenbauer series’, Applicationes Mathematicae, XV(3), pp. 345–396, was used to solve the derived equations with variable coefficients. The obtained solutions of the equations have the form of series relative to Legendre polynomials. A numerical example dealing with the free vibrations of the beam was solved to verify the model and the effectiveness of the presented solution method. The results were compared with the results yielded by finite elements method (FEM).

Analysis of thin-walled beams with variable monosymmetric cross section by means of Legendre polynomials

Studia Geotechnica et Mechanica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

This article deals with the vibrations of a nonprismatic thin-walled beam with an open cross section and any geometrical parameters. The thin-walled beam model...

Analysis of thin-walled beams with variable monosymmetric cross section by means of Legendre polynomials

Article Category: Research Article

Published Online: Mar 26, 2019

Page range: 1 - 12

Received: Nov 20, 2018

Accepted: Jan 02, 2019

DOI: https://doi.org/10.2478/sgem-2019-0001

Keywords
thin-walled, free vibrations, nonprismatic, Legendre polynomials

© 2019 Józef Szybiński, Piotr Ruta, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Analysis of thin-walled beams with variable monosymmetric cross section by means of Legendre polynomials

Article Category: Research Article

Published Online: Mar 26, 2019

Page range: 1 - 12

Received: Nov 20, 2018

Accepted: Jan 02, 2019

DOI: https://doi.org/10.2478/sgem-2019-0001

Keywordsthin-walled, free vibrations, nonprismatic, Legendre polynomials

© 2019 Józef Szybiński, Piotr Ruta, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
thin-walled, free vibrations, nonprismatic, Legendre polynomials