Determining the location of additional supports of a pivotally supported plate under harmonic loading

DOI: 10.34759/trd-2023-128-03

Аuthors

Borshevetskiy S. A.

PJSC Yakovlev , 68, Leningradskiy prospect, Moscow, 125315, Russia

e-mail: wrdeww@bk.ru

Abstract

The article proposes a method for determining the location of such additional supports for two models of plate movement: Kirchhoff and Timoshenko. A rectangular thin pivotally supported plate of known dimensions of constant thickness, which has additional supports in area, is considered. Additional supports installed with the same pitch along the coordinate axes, forming equal segments. The harmonic concentrated force acts on a random place of the plate. It is necessary to determine the location of additional supports based on the stiffness condition: the maximum deflection does not exceed the set value. To determine the location of a set of supports, the segment size satisfying the stiffness condition is first determined. The solution soughting using the influence function, as a reaction of the system to a single impact. Since the harmonic load is representable by Euler, the problem reducing to a stationary one. Additional supports replaced by compensating loads. The influence function decomposing into double Fourier series satisfying the hinge support at the edges. Unknown reactions in the supports are determined from a system of linear algebraic equations according to Kramer’s rule. Then everything substituting into the condition of structural rigidity and this equation solving. At the end of the article, a numerical example and verification calculation giving, which show the fulfillment of the structural rigidity condition. The main advantage of the proposed methodology is the analytical form of the solution. This allows you to substitute any characteristics of the material, the geometry of the plate, as well as the magnitude of the desired load. In general, the technique is also applicable for shells using a local coordinate system that allows the shell to expand into a plate.

Keywords:

Kirchhoff plate, Timoshenko plate, structural rigidity, harmonic load, pivotally supported plate, influence function

References

1. Lizin V.T., Pyatkin V.A. Proektirovanie tonkostennykh konstruktsii (Design of thin-walled structures), Moscow, Mashinostroenie, 1994, 384 p.
2. Pechnikov V.P., Zakharov R.V., Tarasova A.V. Inzhenernyi zhurnal: nauka i innovatsii, 2017, no. 11 (71). DOI: 10.18698/2308-6033-2017-11-1703
3. Firsanov V.V., Fam V.T. Trudy MAI, 2019, no. 105. URL: https://trudymai.ru/eng/published.php?ID=104174
4. Firsanov V.V., Vo A.Kh. Trudy MAI, 2018, 102. URL: https://trudymai.ru/eng/published.php?ID=98866
5. Serdyuk A.O., Serdyuk D.O., Fedotenkov G.V. Vestnik Samarskogo gosudarstvennogo tekhnicheskogo universiteta. Seriya: Fiziko-matematicheskie nauki, 2021, vol. 25, no. 1, pp. 111-126. DOI: 10.14498/vsgtu1793
6. Lokteva N.A., Serdyuk D.O., Skopintsev P.D., Fedotenkov G.V. Trudy MAI, 2021, no. 120. URL: https://trudymai.ru/eng/published.php?ID=161423. DOI: 10.34759/trd-2021-120-09
7. Serdyuk A.O., Serdyuk D.O., Fedotenkov G.V., Hein T.Z. Green’s Function for an Unbounded Anisotropic Kirchhoff-Love Plate, Journal of the Balkan Tribological Association, 2021, vol. 27, no. 5, pp. 747-761.
8. Lokteva Natalia A., Serdyuk Dmitry O., Skopintsev Pavel D. Non-Stationary Influence Function for an Unbounded Anisotropic Kirchhoff-Love Shell, Journal of Engineering and Applied Sciences, 2020, vol. 18, no. 4, pp. 737-744. DOI:10.5937/jaes0-8205
9. Levitskii D.Yu., Fedotenkov G.V. Trudy MAI, 2022, no. 125. URL: https://trudymai.ru/eng/published.php?ID=168157. DOI: 10.34759/trd-2022-125-05
10. Ventsel’ E.S., Dzhan-Temirov K.E., Trofimov A.M., Negol’sha E.V. Metod kompensiruyushchikh nagruzok v zadachakh teorii tonkikh plastinok i obolochek (Method of compensating loads in problems of the theory of thin plates and shells), Khar’kov, B. i., 1992, 92 p.
11. Lokteva N.A., Serdyuk D.O., Skopintsev P.D. Materialy XII mezhdunarodnoi nauchno-prakticheskoi konferentsii, posvyashchennoi 160-letiyu Belorusskoi zheleznoi doroge «Problemy bezopasnosti na transporte», Gomel’, BelGUT, 2022, vol. 2, pp. 205–207.
12. Koreneva E.B. Metod kompensiruyushchikh nagruzok dlya resheniya zadachi o nesimmetrichnom izgibe beskonechnoi ledyanoi plity s kruglym otverstiem, International Journal for Computational Civil and Structural Engineering, 2017, vol. 13, no. 2, pp. 50-55. URL: https://doi.org/10.22337/2587-9618-2017-13-2-50-55
13. Koreneva E.B. Metod kompensiruyushchikh nagruzok dlya resheniya zadachi ob anizotropnykh sredakh, International Journal for Computational Civil and Structural Engineering, 2018, vol. 14, no. 1, pp. 71-77. URL: https://doi.org/10.22337/2587-9618-2018-14-1-71-77
14. Borshevetskii S.A., Lokteva N.A. Materialy XXVII Mezhdunarodnogo simpoziuma im. A.G. Gorshkova «Dinamicheskie i tekhnologicheskie problemy mekhaniki konstruktsii i sploshnykh sred». Moscow, OOO TRP, 2021, vol. 2, pp 19-20.
15. Borshevetskii S.A., Lokteva N.A. XI Mezhdunarodnaya nauchno-prakticheskaya konferentsiya «Problemy bezopasnosti na transporte»: sbornik trudov. Gomel’, BelGUT, 2021, vol. 1, pp. 256–257.
16. Gorshkov A.G., Medvedskii A.L., Rabinskii L.N., Tarlakovskii D.V. Volny v sploshnykh sredakh (Waves in continuous media), Moscow, FIZMATLIT, 2004, 472 p.
17. Ryabukhin A.K., Leier D.V., Lyubarskii N.N. Dinamika i ustoichivost’ sooruzhenii (Dynamics and stability of structures), Krasnodar, KubGAU, 2020, 171 p.
18. Chernina V.S. Statika tonkostennykh obolochek vrashcheniya (Statics of thin-walled shells of revolution), Moscow, Nauka, 1968, 456 p.
19. Korn G., Korn T. Spravochnik po matematike dlya nauchnykh rabotnikov i inzhenerov (Mathematical handbook for scientists and engineers), Moscow, Nauka, 1974, 832 p.
20. Basov K.A. ANSYS: spravochnik pol’zovatelya (ANSYS: user guide), Moscow, DMK Press, 2005, 640 p.

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