Investigation of deflections and natural vibration frequencies of circular isotropic plates of variable thickness according to the law of square parabola with thickening to the support

Introduction. At the present time, round plates of variable thickness are used as load-bearing elements in buildings, which causes the necessity of their diagnostics and quality assessment. Such structures can be used as roofs of vertical cylindrical tanks, round silos and bunkers, hatches in the ceilings of buildings and structures. Professor V.I. Korobko revealed the relationship between the frequencies of their own transverse vibrations w and maximum deflections W0 from uniformly distributed load for isotropic plates of constant thickness at homogeneous support along the contour. The aim of the study is to establish the relationship between the maximum deflection and the frequency of their own transverse vibrations for plates of variable thickness according to the law of square parabola with thickening to the support. Based on the theoretical data obtained, it is possible to diagnose defects (change in the design scheme, destruction, reduction in the thickness of the plate as a result of corrosion, etc.) based on the results of comparison and analysis of theoretical and experimentally measured natural vibration frequencies and (or) maximum deflections in the center of the plate.

Materials and methods. The design structure is a steel round isotropic plate of variable thickness according to the law of square parabola with thickening to the support. The studies were carried out by the finite element method, hinged and rigid pinching.

Results. Maximum deflections and frequencies of natural vibrations of a circular isotropic plate with different ratio of the plate thickness on the support t1 to the thickness in the center t2 were determined. The relationship between the maximum deflections of uniformly distributed load W0 and the fundamental frequency of natural vibrations ω of the circular plate is considered. Based on the results of the study, graphs of dependence of maximum deflections and frequencies of natural transverse vibrations of the plate on the ratio t1/t2 are plotted.

Conclusions. As a result of numerical studies, the maximum deflections and the main vibration frequencies for circular isotropic plates of variable thickness according to the square parabola law with thickening to the support were determined. It was established that the K coefficient obeys within 5 % of the dependence of Professor V.I. Korobko only when the ratio of the thickness on the support to the thickness in the center t1/t2 = 55/50 < 1.1 for both support schemes. This is explained by the fact that the dependence (1) is derived for isotropic plates of constant thickness and the distribution of mass unevenly over the entire area of the plate leads to a significant error already at the stage of small difference between the thicknesses to the support and in the center. With the thickness ratio t1/t2 = 100/50 = 2, the discrepancy between the K coefficient and the analytical one is about 30 % for hinged support and 43.8 % for rigid support along the contour. This means a more significant influence of the uneven mass distribution for such homogeneous boundary conditions. It is also revealed that all values of the K coefficient for circular isotropic plates of variable thickness according to the law of the square parabola with thickening to the support give overestimated values of the K coefficient in comparison with theoretical values for hinged and rigid support.

1. Aleksandrov A.V., Potapov V.D. Fundamentals of the theory of elasticity and plasticity. Moscow, Vys-shaya shkola Publ., 1990; 398. (rus.).

2. Varvak P.M., Ryabov A.F. Handbook on the theory of elasticity. Kyiv, Budivelnik Publ., 1971; 419. (rus.).

3. Timoshenko S.P., Voinovsky-Krieger S. Plates and shells. Moscow, Nauka Publ., 1966; 636. (rus.).

4. Korobko V.I. On one “remarkable” regularity in the theory of elastic plates. Izvestiya vuzov. Construction and Architecture. 1989; 11:32-36. (rus.).

5. Korobko V.I., Boyarkina O.V. The relationship between the problems of transverse bending and free vibrations of triangular plates. Bulletin of SUSU. Series “Construction Engineering and Architecture”. 2007; 22(94):24-26. EDN KWYAWV. (rus.).

6. Korobko V.I. Application of the isoperimetric method to solving problems of the technical theory of plates. Khabarovsk, 1978; 66. (rus.).

7. Turkov A.V., Zhupikova K.A. Relationship between maximum deflections and natural frequencies of isotropic annular plates with hinged support along the external contour. Actual problems of construction, construction industry and architecture : collection of materials of the XX International Scientific and Technical Conference. 2019; 299-302. URL: http://dmitriy.chiginskiy.ru/publications/files/Materialy_XX_MNTK-2019_Tula.pdf (rus.).

8. Turkov A.V., Marfin K.V., Bazhenova A.V. The deflections and natural frequencies of the composite layered square isotropic plates with hinged supports along the contour at different stiffness values of shear ties. Building and Reconstruction. 2019; 4:64-69. DOI: 10.33979/2073-7416-2019-84-4-64-69 (rus.).

9. Turkov A., Аbashina N.S. Deflections and frequencies of natural oscillations of systems of composite two-layer isotropic plates of the round shape at the change of thickness of one of the layers. Istrazivanja i projektovanja za privredu. 2017; 15(3):389-394. DOI: 10.5937/jaes15-14669

10. Pisačić K., Horvat M., Botak Z. Finite difference solution of plate bending using Wolfram Mathematica. Tehnički glasnik. 2019; 13(3):241-247. DOI: 10.31803/tg-20190328111708

11. Sadigov I.R. Study of stability of multilayer round plates of variable thickness from nonlinear elastic material. International Research Journal. 2019; 7-1(85):33-37. DOI: 10.23670/IRJ.2019.85.7.006 EDN QPJLNM. (rus.).

12. Goldstein R.V., Popov A.L., Kozintsev V.M., Chelyubeev D.A. Non-axisymmetric loss of stability during axisymmetric heating of a round plate. PNRPU Mechanics Bulletin. 2016; 2:45-53. DOI: 10.15593/perm.mech/2016.2.04 EDN WBWPZT. (rus.).

13. Sudakova I.A. Natural vibrations of round plates made of orthotropic materials with different resistance. Topical issues and prospects for the development of mathematical and natural sciences : a collection of scientific papers based on the results of an international scientific and practical conference. Omsk, 2017; 4. (rus.).

14. Agalarov Dzh.G., Mamedova G.A. Oscillations of the plate pivotally fastened and elastic suspended on Vinklerov base. International Journal of Applied and Basic Research. 2018; 7:48-53. EDN XYQHKX. (rus.).

15. Gabbasov R.F., Uvarova V.N. Numerical Method of Calculation of Round Plates in a Geometrically Nonlinear Statement. Vestnik MGSU [Monthly Journal on Construction and Architecture]. 2017; 12(6):631-635. DOI: 10.22227/1997-0935.2017.6.631-635 EDN ZASZFL. (rus.).

16. Gabbasov R.F., Hoang Tuan Anh, Uvarova N.B., Lipatova O.N. Calculation of circular plates with constant stiffness for local loads. Industrial and Civil Engineering. 2015; 3:25-28. EDN TOBVXB. (rus.).

17. Akimov P.A., Mozgaleva M.L., Kaytukov T.B. Numerical solution of the problem of beam analysis with the use of b-spline finite element method. International Journal for Computational Civil and Structural Engineering. 2020; 16(3):12-22. DOI: 10.22337/2587-9618-2020-16-3-12-22

18. Kövesdi B. Finite element model-based design of stiffened welded plated structures subjected to combined loading. Periodica Polytechnica Civil Engineering. 2021. DOI: 10.3311/ppci.17229

19. Nadolski V., Rózsás Á., Sykora M. Calibrating partial factors — methodology, input data and case study of steel structures. Periodica Polytechnica Civil Engineering. 2019. DOI: 10.3311/PPci.12822

20. Semenov A.A., Gabitov A.I. Design and computer complex SCAD in the educational process. Moscow, ASV Publishing House, 2005; 152. (rus.).