 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
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| N.F. Morozov, P.E. Tovstik, and T.P. Tovstik, "Generalized Timoshenko-Reissner Model for a Multilayer Plate," Mech. Solids. 51 (5), 527-537 (2016) |
| Year |
2016 |
Volume |
51 |
Number |
5 |
Pages |
527-537 |
| DOI |
10.3103/S0025654416050034 |
| Title |
Generalized Timoshenko-Reissner Model for a Multilayer Plate |
| Author(s) |
N.F. Morozov (St.-Petersburg State University, Universitetskaya nab. 7-9, St. Petersburg, 199034 Russia; Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, Bol'shoy pr. 61, St. Petersburg, 199078 Russia, morozov@nm1016.spb.edu)
P.E. Tovstik (St.-Petersburg State University, Universitetskaya nab. 7-9, St. Petersburg, 199034 Russia; Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, Bol'shoy pr. 61, St. Petersburg, 199078 Russia, peter.tovstik@mail.ru)
T.P. Tovstik (Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, Bol'shoy pr. 61, St. Petersburg, 199078 Russia, tovstik_t@mail.ru) |
| Abstract |
A multilayer plate with isotropic (or transversally isotropic) layers strongly differing in rigidity is considered. This plate is reduced to an equivalent homogeneous transversally isotropic Timoshenko-Reissner plate whose deflections and free transverse vibration frequencies are close to those of the multilayer plate. By comparison with the exact solution of test three-dimensional problems of elasticity, the error of the proposed method is estimated both for the static problem and for free vibrations. This comparison can readily be carried out for the hinged edges of the plate, and explicit approximate formulas are obtained for the vibration frequencies. The scope of the proposed model turned out to be rather wide (the Young moduli of soft and rigid layers can differ by a factor of 1000). In the case of boundary conditions other than hinged support, a closed-form solution cannot be constructed in general. For rigidly fixed edges, the asymptotic method proposed by V. V. Bolotin is generalized to the case of a Timoshenko-Reissner plate. |
| Keywords |
multilayer plate, generalized Timoshenko-Reissner model, asymptotic integration, deflection, low-frequency transverse vibrations, Bolotin method |
| References |
| 1. | G. Kirchhoff,
Vorlesungen über Mathematische Physik: Mechanik
(Taubner, Leipzig, 1876). |
| 2. | A. E. H. Love,
A Treatise on the Mathematical Theory of Elasticity
(Cambridge Univ. Press, Cambridge, 1927; ONTI HKGiP SSSR, Moscow-Leningrad, 1935). |
| 3. | S. P. Timoshenko,
"On the Correction for Shear of the Differential Equation
for Transverse Vibrations of Prismatic Bars,"
Phil. Mag.
41 (6), 742-746 (1921). |
| 4. | E. Reissner,
"The Effect of Transverse Shear Deformation on the Bending of Elastic Plates,"
Trans. ASME. J. Appl. Mech.
12 (2), 69-77 (1945). |
| 5. | P. E. Tovstik and T. P. Tovstik,
"On the 2D Models of Plates and Shells Including Shear,"
ZAMM
87 (2), 160-171 (2007). |
| 6. | L. A. Agalovyan,
Asymptotic Theory of Anisotropic Plates and Shells
(Nauka, Moscow, 1997)
[in Russian]. |
| 7. | P. E. Tovstik,
"On Asymptotic Character of Approximate Models of Beams, Plates, and Shells,"
Vestnik SPbGU,
Ser. 1, No. 3, 49-51 (2007). |
| 8. | P. E. Tovstik and T. P. Tovstik,
"A Thin-Plate Bending Equation of Second-Order Accuracy,"
Dokl. Ross. Akad. Nauk
457 (6), 660-663 (2014)
[Dokl. Phys. (Engl. Transl.)
59 (8), 389-392 (2014)]. |
| 9. | E. I. Grigolyuk and G. M. Kulikov,
"Generalized Model of the Mechanics of Thin-Walled
Structures Made of Composite Materials,"
Mekh. Komp. Mater.,
No. 4, 698-704 (1988)
[Mech. Comp. Mater. (Engl. Transl.)
24 (4), 537-543 (1989)]. |
| 10. | V. A. Rodionova, B. F. Titaev, and K. F. Chernykh,
Applied Theory of Anisotropic Plates and Shells
(Izdat. SPbGU, St. Petersburg, 1996)
[in Russian]. |
| 11. | N. F. Morozov and P. E. Tovstik,
"Bending of a Two-Layer Beam with Non-Rigid Contact between the Layers,"
Prikl. Mat. Mekh.
75 (1), 112-121 (2011)
[J. Appl. Math. Mech. (Engl. Transl.)
75 (1), 77-84 (2011)]. |
| 12. | R. Kienzler and P. Shneider,
"Comparison of Various Linear Plate Theories in the Light of
a Consistent Second-Order Approximation,"
Shell Structures. Theory and Applications
3, 109-112 (2014). |
| 13. | V. V. Bolotin, B. P. Makarov, G. V. Mishenkov, and Yu. Yu. Shveiko,
"Asymptotic Method for Studying the Spectrum of Natural Frequencies
of Elastic Plates,"
in Strength Calculations, No. 6
(Mashgiz, Moscow, 1960),
pp. 231-253
[in Russian]. |
|
| Received |
09 June 2016 |
| Link to Fulltext |
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