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

Szybiński Józef, Ruta Piotr

Studia Geotechnica et Mechanica

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2019

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Vol. 41, nr 1

1--12

EN

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).

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Local and Global Instability of the Slender Geometrically Nonlinear SystemWith Non-Prismatic Element Subjected to the Euler’s Load

Szmidla J., Skrzypczak T., Jurczyńska A.

Machine Dynamics Research

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2016

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Vol. 40, No. 4

79--88

EN

A theoretical considerations and numerical calculations concerning the issue of the stability of the geometrically nonlinear system with non-prismatic element are presented in this work. The analysed columns were subjected to the Euler’s load. On the basis of the minimum potential energy principle as well as the small parameter method, the differential equations of displacements were formulated and its solutions were obtained. The assumption that the approximation of the non-prismatic rod satisfies the condition of constant total volume resulting from the value of the coefficient of flexural stiffness distribution has been made. The results of the carried out numerical simulations refer to the local and global stability loss. It has been proved that taking into consideration in the geometrically nonlinear system appropriate shaped rod of variable cross-section causes an increase in the value of bifurcation load and in a consequence an „exit” from the area of the local instability (loss of rectilinear form of static equilibrium).

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Free Vibrations of the Slender Non-Prismatic Column Under the Specific Load Regarding the Joint Flexibility

Szmidla J., Jurczyńska A.

Machine Dynamics Research

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2016

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Vol. 40, No. 3

79--89

EN

The issue of the free vibrations of the non-prismatic rod subjected to the selected case of the specific load has been studied. In the carried out simulations, a flexibility of a constructional joints modelled by the translational and rotational springs at the point of mounting or on the free end of analysed system was taken into account. The shape of rod was approximated by linear function and by polynomial of degree 2, under the condition of constant volume of the column. After prior definition of total mechanical energy, a differential equations of motion as well as a boundary conditions were formulated on the basis of the Hamilton’s principle. The results of the numerical calculations refer to an influence of variable cross-section of the rod, joint flexibility and a geometry of a loading structure on the value of the frequency of free vibrations due to the external load (a characteristic curves) and on the critical load.