The method of compensating loads for solving of problems of cyclic symmetrical flexure of anisotropic plates, resting on an elastic subgrade

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The purpose of the study - receiving of exact analytical solutions of statics problems of anisotropic plates, resting on an elastic subgrade and subjected to an action of cyclic symmetrical loads. The method of compensating loads is used for solving of the formulated problems. The basic and the compensating solutions are determined. The new approach, connected with the use of Nielsen’s equation for receiving of the solutions, is applied. For the first time by means of the method of compensating loads the exact analytical solutions of the cycle symmetric flexure of anisotropic circular plates, resting on the elastic subgrade, are received. Various boundary conditions and the loads, distributed along circumferences and over ring surfaces, are considered. The problem of anisotropic infinite plate with the circular opening, resting on the elastic subgrade, is also examined. All the solutions are obtained in the closed form and expressed in terms of Bessel functions.

About the authors

Elena B. Koreneva

Moscow Higher Combined Arms Military Command School Holding the Order of Lenin, the Order of the October Revolution nd the Order of the Red Banner

Author for correspondence.
SPIN-code: 8804-7930
2 Golovacheva St, Moscow, 109380, Russian Federation

Professor of the Department of General Engineering Disciplines, Doctor of Technical Sciences


  1. Kovalenko A.D. Selected memoirs. Kiev: Naukova Dumka Publ.; 1976. (In Russ.)
  2. Korenev B.G. Some problems of the theory of elasticity and heat conductivity, solved in terms of Bessel functions. Moscow: Fizmatgiz Publ.; 1960. (In Russ.)
  3. Koreneva E.B. Analytical methods for calculation of plates with varying thickness and their practical application. Moscow: ASV Publ.; 2009. (In Russ.)
  4. Bank L.C., Yin J. Buckling of orthotropic plates with free and rotationally restrained unloaded edges. Thin-Walled Structures. 1996;24:83–96.
  5. Chen W.Q., Lüe C.F. 3D free vibration analysis of cross-ply laminated plates with one pair of opposite edges simply supported. Composite Structures. 2005;69:77–87.
  6. Civalek Ö. Fundamental frequency of isotropic and orthotropic rectangular plates with linearly varying thickness by discrete singular convolution method. Applied Mathematical Modelling. 2009;33:3825–3835. https://doi: 10.1016/j.apm.2008.12.019
  7. Karasev Al., Varianychko M., Bessmertnyi Ya., Krasovsky V., Karasev G. Numerical analysis on experimental research on buckling of closed shallow conical shells under external pressure. Journal of Theoretical and Applied Mechanics. 2020;58(1):117–126.
  8. Li C., Cheng H. Free vibration analysis of a rotating varying-thickness-twisted blade with arbitrary boundary conditions. Journal of Sound and Vibration. 2020;492:115791.
  9. Khakpour Komarsofla M., Jedari Salami S, Shakeri M., Khakpour Komarsofla A. Optimization of three-dimensional up to yield bending behaviour using the full layer-wise theory for FGM rectangular plate subjected to thermo-mechanical loads. Compos. Struct. 2021;257(1):113172.
  10. Vanskike W.P., Hale R.D. Comparative assessment of finite element modelling techniques for wind turbine rotors blades. AIAA Scitech 2020 Forum. Session: Wind Turbine Modeling. 2020.
  11. Sarafraz A., Sahmani S., Aghdam M.M. Nonlinear primary resonance analysis of nanoshells including vibrational mode interaction based on the surface elasticity theory. Math. Mech.-Engl. Ed. 2020;41:233–260.
  12. Javed S., Al Mukahal F.H.H., El Sayed S.B.A. Geometrical influence on the vibration of layered plates. Hindawi. Shock and Vibration. 2021;(3):1–17.
  13. Koreneva E.B., Grosman V.R. Equation decomposition method for solving of problems of statics, vibration and stability of thin-walled constructions. International Journal for Computational Civil and Structural Engineering. 2020;16(2):63–70.
  14. Koreneva E.B., Grosman V.R. The problems of computation of combined plates with piecewise variable thickness. Solutions in orthogonal polynomials. International Journal for Computational Civil and Structural Engineering. 2020;16(3):30–34.
  15. Burmistrov E.F. The simmetrical deformation of orthotropic shells of rotation. Saratov: Izd-vo Saratovskogo Universiteta Publ.; 1962. (In Russ.)
  16. Koreneva E.B. Method of compensating loads for solving of anisotropic medium problems. International Journal for Computational Civil and Structural Engineering. 2018;14(1):71–77. (In Russ.)
  17. Kamke E. The Handbook for ordinary differential equations. Moscow: Nauka Publ.; 1965. (In Russ.)
  18. Abramovitz M., Stigan I.A. Handbook of mathematical functions. 10th ed. National Bureau of Standards; 1972.
  19. Koreneva E.B., Grosman V.R. Analytical solution of the flexure of circular orthotropic plate of variable thickness, resting on an elastic subgrade. Vestnik MGSU. 2011;8:156–159. (In Russ.)
  20. Grosman V.R. Natural vibrations of circular orthotropic plates of variable thickness. The solutions in terms of Bessel functions. Stroitelnaya Mekhanika i Raschet Sooruzheniy. 2012;(3):52–54. (In Russ.)
  21. Koreneva E.B. The analytical method of oscillation problems of elastic anisotropic solids. Stroitelnaya Mekhanika i Raschet Sooruzheniy. 2018;(5):47–51. (In Russ.)
  22. Koreneva E.B. The analytical simulation of the certain problems of statics and oscillations of anisotropic solids. VII International Symposium APCSCE. 1–8 July 2018. Novosibirsk; 2018. p. 478. (In Russ.)
  23. Koreneva E.B., Grosman V.R. Forced vibrations of anisotropic elastic solids subjected to an action of complicated loads. International Journal for Computational Civil and Structural Engineering. 2019;15(3):77–83.



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