English
Język Polski
Stability and vibrations of micro-nonhomogeneous plate band resting on elastic subsoil
DOI:
https://doi.org/10.24425/ace.2021.136480Abstract
The paper presents a procedure of calculation of natural frequencies and critical buckling forces of a micro-nonhomogeneous plate band resting on nonhomogeneous elastic subsoil and having any given boundary conditions. The band consists of N parts - cells - called elements, having a constant width l = L/N. Each band element consists of three parts - subelements with variable widths. The two of these subelements are matrix, the third - inclusion placed symmetrically relative to the matrix. Each band element is built of two isotropic materials. The matrix and inclusion bands have the stiffness and mass per area unit as well as they rest on the subsoil. The model has been derived with use of the classical displacement method. The stiffness matrix of any band element and then the band stiffness matrix have been built. An appropriate computer program has been written to calculate natural frequencies and critical buckling forces. A number of tests have been performed to check the working of the program and several calculative examples has been presented in the paper.References
[1] Z. Kączkowski, Płyty. Obliczenia statyczne. Arkady, Warszawa 2000.
[2] G. Jemielita, Solutions to the problems of mechanics of non-homogeneous beams and plates. In: K. Wilmański (ed.), Mathematical methods in continuum mechanics, Technical University of Łódź, 383-402, 2011.
[3] W. Szcześniak, Wpływ dwuparametrowego podłoża sprężystego na drgania własne płyty o średniej grubości, Rozprawy Inżynierskie, 37(1), 87-115, 1989.
[4] G. Jemielita, Z. Kozyra, Statics of beam with arbitrary stiffness resting on a variable, unidirectional, two-parameter foundation, Theoretical Foundations of Civil Engineering, Polish-Ukrainian-Lithuanian Transactions, Warsaw, 143-150, 2010.
[5] M. Chalecki, G. Jemielita, Free vibrations of micro-non-homogeneous plate band, Scientific Review Engineering and Environmental Sciences, 23, 317-331, 2014 (in Polish).
[6] M. Chalecki, G. Jemielita, Stability of micro-nonhomogeneous plate band, Archives of Civil Engineering, LXI, 91-106, 2015.
[7] R. Solecki, J. Szymkiewicz, Układy prętowe i powierzchniowe. Obliczenia dynamiczne, Arkady, Warszawa 1964.
[8] D.H.Y. Yen, S.C. Tang, On the vibration of an elastic plate on an elastic foundation, Journal of Sound and Vibrations 14(1), 81-89, 1971.
[9] S.F. Bassily, S.M. Dickinson, Buckling and vibration of in-plane loaded plates treated by a unified Ritz approach, Journal of Sound and Vibration, 59(1), 1-14, 1978.
[10] M.S. Cheung, A simplified finite element solution for the plates on elastic foundation, Computers and Structures, 8, 139-145, 1978.
[11] J. Jędrysiak, Free vibrations of micro-heterogeneous thin plate band. In: G. Jemielita (ed.), Modelling of Engineering Structures and Constructions, Warsaw University of Life Sciences (SGGW), 133-139, 2014 (in Polish).
[12] E. Wierzbicki, D. Kula, M. Mazewska, On Fourier realization of tolerance modelling of heat flow problems in simple periodic composites. In: G. Jemielita (ed.), Modelling of Engineering Structures and Constructions, Warsaw University of Life Sciences (SGGW), 253-66, 2014 (in Polish).
[13] B. Michalak, Stability of elastic slightly wrinkled plates, Acta Mechanica, 130, 111-119, 1998.
[14] E. Baron, On a certain model of uniperiodic medium thickness plates subjected to initial stresses, Journal of Theoretical and Applied Mechanics, 43, 93-110, 2005.
[15] J. Rychlewska, Cz. Woźniak, M. Woźniak, Modelling of functionally graded laminates revisited, Electronic Journal of Polish Agricultural Universities, 9(2), #06, 2006.
[16] J. Jędrysiak, On stability of thin periodic plates, European Journal of Mechanics - A/Solids,19, 487-502, 2000.
[17] J. Jędrysiak, Dynamic of thin periodic plates resting on a periodically non-homogeneous Winkler foundation, Archive of Applied Mechanics, 69, 1999.
[18] B. Michalak, Vibrations of plates with initial geometrical imperfections interacting with periodic elastic foundation, Archive of Applied Mechanics, 70, 2000.
[19] J. Jędrysiak, A. Paś, Vibrations of medium thickness homogeneous plates resting on a periodic Winkler subsoil. In: Selected Topics in the Mechanics of Inhomogeneous Media, Cz. Woźniak (ed.), R. Świtka, M. Kuczma, Zielona Góra, 133-146, 2006.
[20] J. Jędrysiak, Application of the tolerance averaging method to analysis of dynamical stability of thin periodic plates, Journal of Theoretical and Applied Mechanics, 42(2), PSTAM (PTMTS), 357-379, 2004.
[21] B. Michalak, Stability of elastic slightly wrinkled plates interacting with an elastic subsoil, Engineering Transactions, 47, 269-283, 1999.
[22] Z. Naniewicz, On the homogenized elasticity with microlocal parameters, Bulletin of Polish Academy of Sciences, Technical Sciences, 35, 399-409, 1987.
[23] T. Lewiński, On the interrelation between microlocal parameter and homogenization approaches to the periodic elastic solids method of modelling solids, Bulletin of Polish Academy of Sciences, Technical Sciences, 35, 383-391, 1987.
[24] T. Lewiński, Homogenizing stiffnesses of plates with periodic structure, International Journal of Solids and Structures, 29(3), 1992.
[25] T. Lewiński, Effective stiffnesses of transversely non-homogeneous plates with unidirectional periodic structure, International Journal of Solids and Structures, 32 (22), 1995.
Downloads
Published
How to Cite
Issue
Section
