Изгиб защемленной тонкой изотропной пластины методом Канторовича с использованием специальных полиномов

Dmitry P. Goloskokov; Alexander V. Matrosov; Igor V. Olemskoy

doi:10.21638/11701/spbu10.2023.401

Authors

Dmitry P. Goloskokov Emperor Alexander I St. Petersburg State Transport University, 9, Moskovsky pr., St. Petersburg, 190031, Russian Federation
Alexander V. Matrosov St. Petersburg State University, 199034, St. Petersburg, Russian Federation
Igor V. Olemskoy St. Petersburg State University, 199034, St. Petersburg, Russian Federation

DOI:

https://doi.org/10.21638/11701/spbu10.2023.401

Abstract

The problem of bending a thin isotropic rectangular plate clamped on all four sides under the action of a normal load uniformly distributed over its surface is considered. An analytical solution of the boundary value problem for the resolving differential equation with respect to the normal deflection of the plate is obtained by the method of L. V. Kantorovich using special-type polynomials satisfying homogeneous boundary conditions. A feature of these polynomials is the so-called “quasi-orthogonality” property of the first and second derivatives, which leads to the splitting of the system of ordinary differential equations of the L. V. Kantorovich method into separate ordinary differential equations that are easily solved analytically. However, this property of polynomials is only approximately fulfilled. Two solutions are compared: an analytical one under the assumption of “quasi-orthogonality” of the first and second derivatives of polynomials and a numerical-analytical one without this assumption. The stress-strain state in the neighborhoods of corner points has been studied. It is shown that the moments and shear forces tend to zero when approaching the corners of the plate, as well as a double change in the sign of the shear force on the edge of the plate in the neighborhoods of the corner points.

Keywords:

isotropic plate, bending of a thin isotropic plate, numerical-analytical methods, clamped plate, L. V. Kantorovich method, orthogonal polynomials, Jacobi polynomials

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References

Литература

Lurie S. A., Vasiliev V. V. The biharmonic problem in the theory of elasticity. London: Gordon and Breach Science Publ., 1995. 276 p.

Meleshko V. V. Selected topics in the history of the two-dimensional biharmonic problem // Appl. Mech. Rev. 2003. Vol. 56. N 1. P. 33–85.

Goloskokov D. P., Matrosov A. V. Bending of clamped orthotropic thin plates: polynomial solution // Mathematics and Mechanics of Solids. 2022. Vol. 27. N 11. P. 2498–2509.

Голоскоков Д. П., Голоскоков П. Г. Метод полиномов в задачах теории тонких плит. СПб.: Изд-во Санкт-Петербургского государственного университета водных коммуникаций, 2008. 251 c.

Голоскоков Д. П. Применение полиномов специального вида для расчета колебаний прямоугольной пластины // Журнал Университета водных коммуникаций. 2009. Т. 1. № 1. С. 185–188.

Голоскоков Д. П. Применение полиномов специального вида в задачах о колебаниях прямоугольных и секторальных пластин // Вестник Татарского государственного гуманитарного педагогического университета. 2010. Т. 3. № 21. С. 22–32.

Голоскоков Д. П. Колебания секторальных пластин // Вестник Инжекона. Сер. Техн. науки. 2008. Т. 8. № 27. С. 156–161.

Hassan A. H. A., Kurgan N., Can N. The correct derivation of the buckling equations of the shear-deformable FGM plates for the extended Kantorovich method // Meccanica. 2022. Vol. 57. N 2. P. 441–456.

Mamandi A. Bending deflection and stress analyses in a thin functionally graded material skew plate with different boundary conditions on the Winkler — Pasternak elastic foundation and under combined in-plane and uniform lateral loads using the extended Kantorovich method // Proceedings of the Institution of Mechanical Engineers. Pt C: Journal of Mechanical Engineering Science. 2022. Vol. 236. N 1. P. 330–353.

Hassan A. H. A., Kurgan N. Bending analysis of thin FGM skew plate resting on Winkler elastic foundation using multi-term extended Kantorovich method // Engineering Science and Technology. 2020. Vol. 23. N 4. P. 788–800.

Singh A., Kumari P. Three-dimensional free vibration analysis of composite FGM rectangular plates with in-plane heterogeneity: An EKM solution // International Journal of Mechanical Sciences. 2020. Vol. 180. N 5. P. 1–6.

Fallah A., Kargarnovin M. H., Aghdam M. M. Free vibration analysis of symmetrically laminated fully clamped skew plates using extended Kantorovich method // Key Engineering Materials. 2011. Vol. 471–472. P. 739–744.

Kargarnovin M. H., Joodaky A. Bending analysis of thin skew plates using extended Kantorovich method // ASME 2010 10textsuperscriptth Biennial Conference on Engineering Systems Design and Analysis, ESDA2010. 2020. Vol. 2. P. 39–44.

Shufrin I., Rabinovitch O., Eisenberger M. A semi-analytical approach for the geometrically nonlinear analysis of trapezoidal plates // International Journal of Mechanical Sciences. 2010. Vol. 52. N 12. P. 1588–1596.

Farag A. M., Ashour A. S. Free vibration of orthotropic skew plates // Journal of Vibration and Acoustics, Transactions of the ASME. 2010. Vol. 122. N 3. P. 313–317.

Hassan A. H. A., Kurgan N. Buckling of thin skew isotropic plate resting on Pasternak elastic foundation using extended Kantorovich method // Heliyon. 2020. Vol. 6. N 6. Art. N 04236.

Rajabi J., Mohammadimehr M. Bending analysis of a micro sandwich skew plate using extended Kantorovich method based on Eshelby — Mori — Tanaka approach // Computers and Concrete. 2019. Vol. 23. N 5. P. 361–376.

Lopatin A. V., Morozov E. V. Buckling of a rectangular composite orthotropic plate with two parallel free edges and the other two edges clamped and subjected to uniaxial compressive distributed load // European Journal of Mechanics, A/Solids. 2020. Vol. 81. Art. N 103960.

Onyia M. E., Rowland-Lato E. O., Ike C. C. Galerkin — Kantorovich method for the elastic buckling analysis of thin rectangular SCSC plates // International Journal of Engineering Research and Technology. 2020. Vol. 13. N 4. P. 613–619.

Singh A., Kumari P., Hazarika R. Analytical solution for bending analysis of axially functionally graded angle-ply flat panels // Mathematical Problems in Engineering. 2018. Art. N 2597484.

Ruocco E., Mallardo V., Minutolo V., Di Giacinto D. Analytical solution for buckling of Mindlin plates subjected to arbitrary boundary conditions // Applied Mathematical Modelling. 2017. Vol. 50. P. 497–508.

Kumari P., Shakya A. K. Two-dimensional solution of piezoelectric plate subjected to arbitrary boundary conditions using extended Kantorovich method // Procedia Engineering. 2017. Vol. 173. P. 1523–1530.

Singhatanadgid P., Jommalai P. Buckling analysis of laminated plates using the extended Kantorovich method and a system of first-order differential equations // Journal of Mechanical Science and Technology. 2016. Vol. 30. N 5. P. 2121–2131.

Lopatin A. V., Morozov E. V. Approximate buckling analysis of the CCFF orthotropic plates subjected to in-plane bending // International Journal of Mechanical Sciences. 2014. Vol. 85. P. 38–44.

Singhatanadgid P., Singhanart T. The Kantorovich method applied to bending, buckling, vibration, and 3D stress analyses of plates: A literature review // Mechanics of Advanced Materials and Structures. 2019. Vol. 26. N 2. P. 170–188.pagebreak

Тимошенко С. П., Войновский-Кригер С. Пластинки и оболочки / пер. с англ. Г. С. Шапиро; под ред. И. К. Снитко. М.: Наука, 1966. 636 с. (Timoshenko S. P., Woinowsky-Krieger S. Theory of plates and shells. 2 ed. New York: McGraw-Hill Publ., 1987. 580 p.)

Meleshko V. V.,Gomilko A. M., Gourjii A. A. Normal reactions in a clamped elastic rectangular plate // Journal of Engineering Mathematics. 2001. Vol. 40. P. 377–398.

Алцыбеев Г. О., Голоскоков Д. П., Матросов А. В. Метод суперпозиции в задаче изгиба защемленной по контуру тонкой изотропной пластинки // Вестник Санкт-Петербургского университета. Прикладная математика. Информатика. Процессы управления. 2022. Т. 18. Вып. 3. С. 347–364. https://doi.org/10.21638/11701/spbu10.2022.305

References

Lurie S. A., Vasiliev V. V. The biharmonic problem in the theory of elasticity. London, Gordon and Breach Science Publ., 1995, 276 p.

Meleshko V. V. Selected topics in the history of the two-dimensional biharmonic problem. Appl. Mech. Rev., 2003, vol. 56, no. 1, pp. 33–85.

Goloskokov D. P., Matrosov A. V. Bending of clamped orthotropic thin plates: polynomial solution. Mathematics and Mechanics of Solids, 2022, vol. 27, no. 11, pp. 2498–2509.

Goloskokov D. P., Goloskokov P. G. Metod polinomov v zadachakh teorii tonkikh plit [ The method of polynomials in the problems of the theory of thin plates]. St. Petersburg, St. Petersburg State University for Water Communication Press, 2008, 251 p. (In Russian)

Goloskokov D. P. Primenenie polinomov spetcial’nogo vida dlia rascheta kolebanii priamougol’noi plastiny [The use of special-type polynomials for calculating the vibrations of a rectangular plate]. Journal of the University of Water Communications, 2009, vol. 1, no. 1, pp. 185–188. (In Russian)

Goloskokov D. P. Primenenie polinomov spetcial’nogo vida v zadachakh o kolebaniiakh priamougol’ny’kh i sektoral’nykh plastin [Application of special-type polynomials in the problems of oscillations of rectangular and sectoral plates]. Bulletin of the Tatar State Humanitarian Pedagogical University, 2010, vol. 3, no. 21, pp. 22–32. (In Russian)

Goloskokov D. P. Kolebaniia sektoral’nykh plastin [Vibrations of sectoral plates]. Bulletin of the Engineering. Series Technical Sciences, 2008, vol. 8, no. 27, pp. 156–161. (In Russian)

Hassan A.H. A., Kurgan N., Can N. The correct derivation of the buckling equations of the shear-deformable FGM plates for the extended Kantorovich method. Meccanica, 2022, vol. 57, no. 2, pp. 441–456.

Mamandi A. Bending deflection and stress analyses in a thin functionally graded material skew plate with different boundary conditions on the Winkler — Pasternak elastic foundation and under combined in-plane and uniform lateral loads using the extended Kantorovich method. Proceedings of the Institution of Mechanical Engineers, Pt C: Journal of Mechanical Engineering Science, 2022, vol. 236, no. 1, pp. 330–353.

Hassan A. H. A., Kurgan N. Bending analysis of thin FGM skew plate resting on Winkler elastic foundation using multi-term extended Kantorovich method. Engineering Science and Technology, 2020, vol. 23, no. 4, pp. 788–800.

Singh A., Kumari P. Three-dimensional free vibration analysis of composite FGM rectangular plates with in-plane heterogeneity: An EKM solution. International Journal of Mechanical Sciences, 2020, vol. 180, no. 5, pp. 1–6.

Fallah A., Kargarnovin M. H., Aghdam M. M. Free vibration analysis of symmetrically laminated fully clamped skew plates using extended Kantorovich method. Key Engineering Materials, 2011, vol. 471–472, pp. 739–744.

Kargarnovin M. H., Joodaky A. Bending analysis of thin skew plates using extended Kantorovich method. ASME 2010 10$^th$ Biennial Conference on Engineering Systems Design and Analysis, ESDA2010, 2020, vol. 2, pp. 39–44.

Shufrin I., Rabinovitch O., Eisenberger M. A semi-analytical approach for the geometrically nonlinear analysis of trapezoidal plates. International Journal of Mechanical Sciences, 2010, vol. 52, no. 12, pp. 1588–1596.

Farag A. M., Ashour A. S. Free vibration of orthotropic skew plates. Journal of Vibration and Acoustics, Transactions of the ASME, 2010, vol. 122, no. 3, pp. 313–317.

Hassan A. H. A., Kurgan N. Buckling of thin skew isotropic plate resting on Pasternak elastic foundation using extended Kantorovich method. Heliyon, 2020, vol. 6, no. 6, art. no. 04236.

Rajabi J., Mohammadimehr M. Bending analysis of a micro sandwich skew plate using extended Kantorovich method based on Eshelby — Mori — Tanaka approach. Computers and Concrete, 2019, vol. 23, no. 5, pp. 361–376.

Lopatin A. V., Morozov E. V. Buckling of a rectangular composite orthotropic plate with two parallel free edges and the other two edges clamped and subjected to uniaxial compressive distributed load. European Journal of Mechanics, A/Solids, 2020, vol. 81, art. no. 103960.

Onyia M. E., Rowland-Lato E. O., Ike C. C. Galerkin — Kantorovich method for the elastic buckling analysis of thin rectangular SCSC plates. International Journal of Engineering Research and Technology, 2020, vol. 13, no. 4, pp. 613–619.

Singh A., Kumari P., Hazarika R. Analytical solution for bending analysis of axially functionally graded angle-aly flat panels. Mathematical Problems in Engineering, 2018, art. no. 2597484.

Ruocco E., Mallardo V., Minutolo V., Di Giacinto D. Analytical solution for buckling of Mindlin plates subjected to arbitrary boundary conditions. Applied Mathematical Modelling, 2017, vol. 50, pp. 497–508.

Kumari P., Shakya A. K. Two-dimensional solution of piezoelectric plate subjected to arbitrary boundary conditions using extended Kantorovich method. Procedia Engineering, 2017, vol. 173, pp. 1523–1530.

Singhatanadgid P., Jommalai P. Buckling analysis of laminated plates using the extended Kantorovich method and a system of first-order differential equations. Journal of Mechanical Science and Technology, 2016, vol. 30, no. 5, pp. 2121–2131.

Lopatin A. V., Morozov E. V. Approximate buckling analysis of the CCFF orthotropic plates subjected to in-plane bending. International Journal of Mechanical Sciences, 2014, vol. 85, pp. 38–44.

Singhatanadgid P., Singhanart T. The Kantorovich method applied to bending, buckling, vibration, and 3D stress analyses of plates: A literature review. Mechanics of Advanced Materials and Structures, 2019, vol. 26, no. 2, pp. 170–188.

Timoshenko S. P., Woinowsky-Krieger S. Theory of plates and shells. 2 ed. New York, McGraw-Hill Publ., 1987, 580 p. (Russ. ed.: Timoshenko S. P., Woinowsky-Krieger S. Plastinki i obolochki. Moscow, Nauka Publ., 1966, 636 p.)

Meleshko V. V., Gomilko A. M., Gourjii A. A. Normal reactions in a clamped elastic rectangular plate. Journal of Engineering Mathematics, 2001, vol. 40, pp. 377–398.

Alcybeev G. O., Goloskokov D. P., Matrosov A. V. Metod superpozitcii v zadache izgiba zashchemlennoi’ po konturu tonkoi’ izotropnoi’ plastinki [The superposition method in the problem of bending of a thin isotropic plate clamped along the contour]. textit Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 2022, vol. 18, iss. 3, pp. 347–364. https://doi.org/10.21638/11701/spbu10.2022.305 (In Russian)