Purpose: To investigate the stress-strain state of a thin homogeneous isotropic plate in the form of a sector by numerical-analytical method. To consider the possibility of using computer algebra systems (CAS) to calculate sectoral plates operating under bending due to a transverse load. To demonstrate the effectiveness of applying one of these CASs through the example of the Maple system for calculations using the Ritz method — performing analytical transformations when calculating the integral that determines the total potential energy functional, forming and solving the main resolving system of linear algebraic equations with respect to unknown numerical coefficients in the formula approximating the deflection of the plate, visualization of the obtained solution. Methods: A direct method is used to solve the variational problem of minimizing the functional of the total potential energy of deformation of a thin homogeneous isotropic plate in the form of a sector — the Ritz method. The solution is constructed in the form of a series in terms of basis functions. As basis functions, polynomial functions are chosen that exactly satisfy all boundary conditions. Results: An approximate numerical-analytical solution has been obtained for the problem of bending a sectoral plate in the form of a quarter of a circle, clamped along the contour and loaded with a uniformly distributed load. The effectiveness of using the Maple analytical computing system for solving the problem of bending a sectoral plate by the variational Ritz method is demonstrated. It is shown that the resulting solution quickly converges both for deflection and for bending moments and stresses. Practical significance: The algorithm proposed in the article for solving the problems of bending sectoral plates can be recommended for practical use in determining the stress-strain state of such plates.
Sectoral plate, plate bending, plate total potential energy functional, Ritz method, computer algebra systems
1. Noor A. K. Computerized symbolic manipulation in structural mechanics-progress and potential / A. K. Noor, C. M. Andersen // Computers and Structures. - 1979. - Iss. 10. - Pp. 95-118.
2. Brumberg V. A. Specialized celestial mechanics systems for symbolic manipulation / V. A. Brumberg, S. V. Tarasevich, N. N. Vasiliev // Celestial Mechanics. - 1989. - Iss. 45. - Pp. 149-162.
3. Pavlovic M. N. Computers and structures: non-numerical applications / M. N. Pavlovic, E. J. Sapountzakis // Computers & Structures. - 1986. - Vol. 24. - Iss. 3. - Pp. 455-474.
4. Eisenberger M. Application of Symbolic Algebra to the Analysis of Plates on Variable Elastic Foundation / M. Eisenberger // Symbolic Computation - 1990. - Iss. 9. - Pp. 207-213.
5. Ye Z. Application of Maple V to the nonlinear vibration analysis of circular plate with variable thickness / Z. Ye // Computers and Structures. - 1999. - Iss. 71. - Pp. 481-488.
6. Pavlovic M. N. Symbolic computation in structural engineering / M. N. Pavlovic // Computers and Structures. - 2003. - Iss. 81. - Pp. 2121-2136.
7. Kirsanov M. N. Maple i Maplet. Resheniya zadach mehaniki / M. N. Kirsanov. - SPb.: Lan', 2012. - 512 s.
8. Goloskokov D. P. Postroenie bazisa dlya odnomernyh kraevyh zadach v sistemah simvol'nyh vychisleniy / D. P. Goloskokov // Prostranstvo, vremya i fundamental'nye vzaimodeystviya. - 2017. - Vyp. 1. - C. 77-87.
9. Kirsanov M. N. Matematika i programmirovanie v Maple / M. N. Kirsanov. - M.: IPR MEDIA, 2020. - 164 s.
10. Timoshenko S. P. Plastinki i obolochki / S. P. Timoshenko, S. Voynovskiy-Kriger. - M.: Nauka, 1966.
11. Bubnov I. G. Trudy po teorii plastin / I. G. Bubnov. - M.: Gosudarstvennoe izdatel'stvo tehniko-teoreticheskoy literatury, 1955.
12. Goloskokov D. P. Izgib rebristoy plastiny pri slozhnom nagruzhenii / D. P. Goloskokov, A. V. Matrosov // Vestnik SPbGU. Prikladnaya matematika. Informatika. Processy upravleniya. - 2021. - T. 17. - Vyp. 2. - S. 120-130.
13. Goloskokov D. P. Metod nachal'nyh funkciy v raschete izgiba zaschemlennoy po konturu tonkoy ortotropnoy plastinki / D. P. Goloskokov, A. V. Matrosov // Vestnik SPbGU. Prikladnaya matematika. Informatika. Processy upravleniya. - 2021.- T. 17. - Vyp. 4. - C. 330-344.
14. Goloskokov D. P. Bending of clamped orthotropic thin plates: polynomial solution / D. P. Goloskokov, A. V. Matrosov // Mathematics and Mechanics of Solids. - 2022. - Vol. 27(11). - Pp. 2498-2509. - DOI:https://doi.org/10.1177/10812865221075280.
15. Alcybeev G. O. Metod superpozicii v zadache izgiba zaschemlennoy po konturu tonkoy izotropnoy plastinki / G. O. Alcybeev, D. P. Goloskokov, A. V. Matrosov // Vestnik SPbGU. Prikladnaya matematika. Informatika. Processy upravleniya. - 2022. - T.18. - Vyp. 3. - C. 347-364.
