Influence of boundary constraints on the appearance of asymmetrical equilibrium states in circular plates under normal pressure

Unsymmetrical buckling of nonuniform circular plates with elastically restrained edge and subjected to normal pressure is studied in this paper. The unsymmetric part of the solution is sought in terms of multiples of the harmonics of the angular coordinate. A numerical method is employed to obtain the lowest load value, which leads to the appearance of waves in the circumferential direction. The effect of material heterogeneity and boundary on the buckling load is examined. It is shown that if the outer edge of a plate is elastically restrained against radial deflection, the buckling load for unsymmetrical buckling is larger than for a plate with a movable edge. The elasticity modulus decrease away from the center of a plate leads to sufficient lowering of the buckling pressure if the outer edge can move freely in the radial direction.

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On non-axisymmetric buckling modes of inhomogeneous circular plates

Vestnik of Saint Petersburg University Mathematics Mechanics Astronomy ◽

10.21638/spbu01.2021.201 ◽

2021 ◽

Vol 8 (2) ◽

pp. 204-211

Author(s):

Svetlana M. Bauer ◽

◽

Eva B. Voronkova ◽

Keyword(s):

Elasticity Modulus ◽

Normal Pressure ◽

Mode Number ◽

Circular Plates ◽

Plate Edge ◽

Buckling Modes ◽

Axisymmetric Buckling ◽

Flexibility Coefficient ◽

Elastically Restrained Edge ◽

Elastically Restrained

Unsymmetrical buckling of nonuniform circular plates with elastically restrained edge and subjected to normal pressure is studied in this paper. The asymmetric part of the solution is sought in terms of multiples of the harmonics of the angular coordinate. A numerical method is employed to obtain the lowest load value at which waves in the circumferential direction can appear. The effect of material heterogeneity and boundary on the buckling load is examined. For a plate with elastically restrained edge, the buckling pressure and mode number increase with a rise of spring stiffness. Increasing of the elasticity modulus to the plate edge leads to increasing of the buckling pressure, but the mode number does not change. If the translational flexibility coefficient is small, decreasing of the elasticity modulus to the shell (plate) edge leads to sufficient lowering of the buckling pressure.

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Frequencies of Circular Plates Weakened Along an Internal Concentric Circle and Elastically Restrained Edge Against Translation

Journal of Applied Mechanics ◽

10.1115/1.4006938 ◽

2012 ◽

Vol 80 (1) ◽

Cited By ~ 1

Author(s):

Lokavarapu Bhaskara Rao ◽

Chellapilla Kameswara Rao

Keyword(s):

Natural Frequencies ◽

Radial Crack ◽

Concentric Circle ◽

Outer Edge ◽

Circular Plates ◽

Plate Vibrations ◽

Rotational Restraint ◽

Method Of Solution ◽

Elastically Restrained Edge ◽

Elastically Restrained

The present study deals with the derivation of an exact solution for the problem of obtaining the natural frequencies of the vibration of circular plates weakened along an internal concentric circle due to the presence of a radial crack and elastically restrained along the outer edge of the plate against translation. The frequencies of the circular plates are computed for varying values of the elastic translational restraint, the radius of the radial crack, and the extent of the weakening duly simulated by considering the radial crack as a radial elastic rotational restraint on the plate. The results for the first six modes of the plate vibrations are computed. The effects of the elastic edge restraint, the radius of the weakened circle, and the extent of the weakening represented by an elastic rotational restraint on the vibration behavior of thin circular plates are studied in detail. The internal weakening due to a crack resulted in decreasing the fundamental frequency of the plate. The exact method of solution and the results presented in this paper are expected to be of specific use in analyzing the effect of a radial crack on the fundamental natural frequency of the circular plate in the presence of a translational restraint existing along the outer edge of the plate. These exact solutions can be used to check the numerical or approximate results.

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Splines in vibration analysis of non-homogeneous circular plates of quadratic thickness

Journal of Applied Analysis ◽

10.1515/jaa-2021-2078 ◽

2022 ◽

Vol 0 (0) ◽

Author(s):

Robin Singh ◽

Neeraj Dhiman ◽

Mohammad Tamsir

Keyword(s):

Radial Direction ◽

Outer Edge ◽

Thickness Variation ◽

Circular Plates ◽

Plate Material ◽

Quintic Spline ◽

Simply Supported ◽

Edge Conditions ◽

Modes Of Vibration ◽

First Time

Abstract Mathematical model to account for non-homogeneity of plate material is designed, keeping in mind all the physical aspects, and analyzed by applying quintic spline technique for the first time. This method has been applied earlier for other geometry of plates which shows its utility. Accuracy and versatility of the technique are established by comparing with the well-known existing results. Effect of quadratic thickness variation, an exponential variation of non-homogeneity in the radial direction, and variation in density; for the three different outer edge conditions namely clamped, simply supported and free have been computed using MATLAB for the first three modes of vibration. For all the three edge conditions, normalized transverse displacements for a specific plate have been presented which shows the shiftness of nodal radii with the effect of taperness.

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