Abstract
Multi-span beams and plates which are freely vibrating are analysed through suitable adaptive sets of global piecewise-smooth functions (or A-GPSFs). Such approximating functions were introduced in Messina (Int J Mech Sci 90:179–189, 2015) as an extension of an original work published in Messina (J Sound Vib 256(1):103–129, 2002), where GPSFs were in turn used in order to model physical quantities through the thickness of structural elements. Herein these functions are used on the middle plane of the structural elements to model free vibrations of thin multi-span beams and plates, thus showing the capability of the same functions to include previous classical formulations in several circumstances of engineering interest. Depending on the internal constraints in beams and plates, the A-GPSFs could require further supplementary conditions. As such functional sets are not immediately available, explicit subroutines are illustrated. The efficiency and capability of the proposed models result from the comparison between calculated eigen-parameters and those of other models presented in open literature.
Similar content being viewed by others
References
Messina, A.: Analyses of freely vibrating cross-ply laminated plates in conjunction with adaptive global piecewise-smooth functions (A-GPSFs). Int. J. Mech. Sci. 90, 179–189 (2015)
Messina, A.: Free vibrations of multilayered plates based on a mixed variational approach in conjunction with global piecewise-smooth functions. J. Sound Vib. 256(1), 103–129 (2002)
Love, A.E.H.: The small free vibrations and deformation of a thin elastic shell. Philos. Trans. R. Soc. Lond. A 179, 491–546 (1888)
Reissner, E.: The effects of transverse shear deformation on the bending of elastic plates. ASME J. Appl. Mech. 12, A69–A77 (1945)
Mindlin, R.D.: Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates. ASME J. Appl. Mech. 18, 31–38 (1951)
Timoshenko, S.P.: History of Strength of Materials. Dover publications, New York (1953)
Timoshenko, S.P.: On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philos. Mag. 41(s.6), 744–746 (1921)
Stephen, N.G.: The second spectrum of Timoshenko beam theory—further assessment. J. Sound Vib. 292, 372–89 (2006)
Levinson, M.: A new rectangular beam theory. J. Sound Vib. 74, 81–7 (1981)
Levinson, M.: An accurate, simple theory of the statics and dynamics of elastic plates. Mech. Res. Commun. 7(4), 343–350 (1980)
Bickford, W.B.: A consistent higher order beam theory. Dev. Theor. Appl. Mech. 11, 137–150 (1982)
Reddy, J.N.: A simple higher-order theory for laminated composite plates. ASME J. Appl. Mech. 51, 745–752 (1984)
Messina, A.: Free vibrations of multilayered doubly curved shells based on a mixed variational approach and global piecewise-smooth functions. Int. J. Solids Struct. 40, 3069–3088 (2003)
Amabili, M., Garziera, R.: A technique for the systematic choice of admissible functions in the Rayleigh–Ritz method. J. Sound Vib. 224(2), 519–539 (1999)
Eisenberger, M.: Exact vibration frequencies and modes of beams with internal releases. Int. J. of Struct. Stab. Dyn. 2(1), 63–75 (2002)
Wang, C.M., Xiang, Y., Wang, C.Y.: Buckling and vibration of plates with an internal line-hinge via the Ritz method. In: Proceedings of First Asian–Pacific Congress on Computational Mechanics, Sydney, pp. 1663–1672 (2001)
Xiang, Y., Reddy, J.N.: Natural vibration of rectangular plates with an internal line hinge using the first order shear deformation plate theory. J. Sound Vib. 263, 285–297 (2003)
Hu, Haichang: Variational Principles of Theory of Elasticity with Applications. Science Press and Gordon and Breach, Beijing (1984)
Whitney, M.J.: Structural Analysis of Laminated Anisotropic Plates. Technomic, Lancaster (1987)
MATHEMATICA Ver. 10.3.0.0, Wolfram Research Inc. 1988–2015
MATLAB 8.5.0.197613 (R2015a), The Mathworks Inc. 1984–2015
Sansone, G.: Moderna teoria delle funzioni di variabile reale III ed. Parte II (Sviluppi in serie di funzioni ortogonali). Nicola Zanichelli Editore, Bologna (1952). (In Italian)
Boyce, W.E., DiPrima, R.C.: Elementary Differential Equations and Boundary Value Problems. Wiley, New York (1996)
Gentile, A., Messina, A.: Detection of cracks by only measured mode shapes in damaged conditions. In: Proceedings of the 3rd International Conference on Identification in Engineering systems, Swansea, pp. 208–220 (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Messina, A. Modelling the vibrations of multi-span beams and plates through adaptive global piecewise-smooth functions (A-GPSFS). Acta Mech 229, 1613–1629 (2018). https://doi.org/10.1007/s00707-017-2066-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-017-2066-2