JAMP  Vol.4 No.8 , August 2016
Bending and Vibrations of a Thick Plate with Consideration of Bimoments
Abstract: The paper is dedicated to the development of the theory of orthotropic thick plates with consideration of internal forces, moments and bimoments. The equations of motion of a plate are described by two systems of six equations. New equations of motion of the plate and the boundary conditions relative to displacements, forces, moments, and bimoments are given. As an example, the problems of free and forced oscillations of a thick plate are considered under the effect of sinusoidal periodic load. The problem is solved by Finite Difference Method. Eigenfrequencies of the plate are determined, numeric maximum values of displacements, forces and moments of the plate are obtained depending on the frequency of external force. It is shown that when the value of the frequency of external effect approaches the eigenfrequency, there occurs an increase in displacement, force and moment values; that testifies a gradual transition of the motion of plate points into the resonant mode.
Cite this paper: K. Usarov, М. , М. Usarov, D. and T. Ayubov, G. (2016) Bending and Vibrations of a Thick Plate with Consideration of Bimoments. Journal of Applied Mathematics and Physics, 4, 1643-1651. doi: 10.4236/jamp.2016.48174.

[1]   Ambartsumyan, S.A. (1987) Theory of Anisotropic Plates. Nauka, Ch. Ed. Sci. Lit., Moscow, 360 p.

[2]   Galimov, K.Z. (1977) Theory of Shells with Account of Transverse Shear. Ed. Kazan University, Kazan, 212 p.

[3]   Galimov, Sh.K. (1976) Specified Theory of Calculation of Orthotropic Rectangular Plate under Lateral Load. Investigations in Theory of Plates and Shells, Sat. articles, Kazan, Vol. XII, 78-84.

[4]   Mushtari, Kh.M. (1990) Nonlinear Theory of Shells. Nauka, Moscow, 223 p.

[5]   Vlasov, B.F. (1952) On a Case of Bending of a Rectangular Thick Plate. Vestnik MGU. Mechanics, Mathematics, Astronomy and Chemistry, No. 2, 25-34.

[6]   Baida, E.N. (1983) Some Spatial Problems of Elasticity. Leningrad University, Leningrad, 232 p.

[7]   Karamooz Ravari, M.R. and Forouzan, M.R. (2011) Frequency Equations for the In-Plane Vibration of Orthotropic Circular Annular Plate. Archive of Applied Mechanics, 81, 1307-1322.

[8]   Soukup, J., Vales, F., Volek, J. and Skocilas, J. (2011) Transient Vibration of thin Viscoelastic Orthotropic Plates. Acta Mechanica Sinica, 27, 98-107.

[9]   Papkov, S.О. (2013) Steady-State Forced Vibrations of a Rectangular Orthotropic Plate. Journal of Mathematical Sciences, 192, 691-702.

[10]   Chang, H.-H. and Tarn, J.-Q. (2012) Three-Dimensional Elasticity Solutions for Rectangular Orthotropic Plates. Journal of Elasticity, 108, 49-66.

[11]   Zenkour, A.M., Allam, M.N.M., Shaker, M.O. and Radwan, A.F. (2011) On the Simple and Mixed First-Order Theories for Plates Resting on Elastic Foundations. Acta Mechanica, 220, 33-46.

[12]   Usarov, M.K. (2014) Calculation of Orthotropic Plates Based on the Theory of Bimoments. Uzbek Journal Problems of Mechanics, Tashkent, No. 3-4, 37-41.

[13]   Usarov, M.K. (2014) Bimoment Theory of Bending and Vibrations of Orthotropic Thick Plates. Vestnik NUU, No. 2/1, 127-132.

[14]   Usarov, M.K. (2015) Bending of Orthotropic Plates with Consideration of Bimoments. St. Petersburg, Civil Engineering Journal, 1, 80-90.

[15]   Usarov, M.K. (2015) On Solution of the Problem of Bending of Orthotropic Plates on the Basis of Bimoment Theory. Open Journal of Applied Sciences, 5, 212-219.