In this study, we consider the problem of nonlinearly tapered annular plate with a free edge. The supported edge may be simply supported, clamped or elastically restrained against rotation. Exact expressions of deflection, moment-resultants, and stresses are presented for nonuniform thickness. We compare the results of the Kirchhoff plate theory and the Mindlin plate theory. It is shown that the Kirchhoff plate theory and the Mindlin plate theory provide approximately the same results for the positive values of the thickness factor, but the difference between the deflections diverges as the thickness increases at the inner edge. We also propose that the Kirchhoff plate theory may be used in the region of −0.4 ≤ α < 1 and the Mindlin plate theory must be used for α < −0.4.Keywords Tapered annular plate · Shear deformation · Bending relationships
IntroductionCircular plates of nonuniform thickness are used in many branches of technology such as clutches, diaphragms of steam turbines, pistons of engines, and essential parts of numerous engineering systems. Through suitable tapering of the plate thickness, the bending stiffness, buckling and vibration capacities of the plate may be significantly increased over the uniform thickness counterparts [5]. There are so many studies related to a constant thickness circular plates problem. Convay [2,3] solved linearly tapered plates analytically by using the thin plate theory. Timoshenko and Krieger [14] studied circular plates of nonuniform thickness by using the Kirchhoff plate assumptions. Liu et al.[5] analyzed axisymmetric bending for linearly tapered, annular Mindlin plates. Wang [16] obtained a relationship between Mindlin and Kirchhoff bending solutions for tapered circular and annular plates. Chung and Wang [1] studied optimal design of stepped circular plates with allowance for the effect of transverse shear deformation by using the Newton method. There are also some comparative studies based on the classical thin plate theory and shear deformation theories for plates of uniform thickness [6,11,16]. However, the influence of nonlinear thickness variation has not received sufficient attention in the context of the effect of transverse shear deformation, and it still remains open.In a nonuniform thickness plate, some parts of plate are thicker than the other parts; therefore the effect of transverse shear deformation is important in a tapered plate. The shear deformation cannot be ignored in a thicker part of the plate. The Kirchoff plate theory underpredicts the plate deflection because of that it ignores the transverse shear and transverse normal effects. Many shear deformable theories have been proposed for the analysis of thick plates [4,8,10,13]. The most widely used and fundamentally simpler displacement-based theory for thick plates was developed by Mindlin [7].
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