Vestnik MGSU (Aug 2023)
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
Abstract
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.
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