Dependence of Axisymmetric Bending of a Circular Plate on Boundary Conditions and Pressure on Its Surface
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Dependence of Axisymmetric Bending of a Circular Plate on Boundary Conditions and Pressure on Its Surface M. A. Il’gamov1* and V. E. Moiseeva1** (Submitted by A. M. Elizarov) 1
Institute of Mechanics and Engineering, Federal Research Center Kazan Scientific Center, Russian Academy of Sciences, Kazan, Tatarstan, 420111 Russia Received March 4, 2020; revised March 16, 2020; accepted March 22, 2020
Abstract—We study how the fixing conditions along edge of a circular plate under pressure effects its axisymmetric bending. In our calculations we take into account refined transverse distributed load, which depends on a gas pressure and on an area difference between convex and concave surfaces. Furthermore we take into consideration a compression force of a middle surface, which is created by compression of the plate over the thickness. The deflections are defined in linear and nonlinear settings. DOI: 10.1134/S1995080220070197 Keywords and phrases: circular plate, fixing conditions, average pressure, linear bending, nonlinear bending.
1. INTRODUCTION The classical theory of thin plates bending [1–12] states that the stress-strain state does not depend on an average pressure pm = (p1 + p2 )/2, where p1 and p2 are overpressures applied to bottom and top surfaces respectively, pe = p2 − p1 . A more accurate expression describing the transverse distributed load applied on circular plate is [13] 1 d d 2 2 r , (1) q = pe + pm h∇ w, ∇ = r dr dr where h is a thickness of the plate. Expression (1) is derived assuming no action on the plate edge area of pressures p1 and p2 at r = c . Here c is the radius of the circular plate. In the thin plate model the forces in the middle surface do not depend on the average pressure pm , but appear as a result of a nonlinear bending. If we take into account the deformations in the plane of the middle surface induced by compression along the thickness under the action of the average pressure pm , then in the absence of displacement on the supports, compressing forces occur. For a rectangular plate without deformation along y axis (εy = 0) the longitudinal force acting along axis x is induced Nx = σh = −pm h [14]. The linear bending of a rectangular plate with finite ratio of sides was studied in [15]. In this work an equation for a transverse distributed load is derived based on the theory of elasticity. The variability effect of the curvature of the middle surface and different boundary conditions on the value of the transverse distributed force, deflection, and critical forces was also studied. The effect of nonlinear terms under the same conditions was researched in [16]. * **
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DEPENDENCE OF AXISYMMETRIC BENDING
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2. PROBLEM DEFINITION We consider the static axisymmetric elastic bending of a circular plate with a radius c and a thickness h, on the lower and upper surfaces of which the gases act with pressures p0 + p1 and p0 + p2 , where p0 is atmospheric pressure, p1 , p2 are overpressures. The pressures p1 and p2 can be both positive a
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