Bending of a thin circular plate under hydrostatic pressure over a concentric ellipse

1959 ◽  
Vol 55 (1) ◽  
pp. 110-120 ◽  
Author(s):  
W. A. Bassali
Keyword(s):  
Exact Solution ◽  
Hydrostatic Pressure ◽  
Boundary Conditions ◽  
Circular Plate ◽  
Thin Plate ◽  
Plate Theory ◽  
Simply Supported ◽  
Thin Circular Plate ◽  
Thin Plate Theory

ABSTRACTAn exact solution in finite terms is derived within the limitations of the classical thin-plate theory, for the problem of a thin circular plate acted upon normally by hydrostatic pressure distributed over the area of a concentric ellipse, and subject to boundary conditions covering the usual rigidly clamped and simply supported boundaries.

The polygon-circle paradox and convergence in thin plate theory

1973 ◽  
Vol 73 (1) ◽  
pp. 279-282 ◽  
Author(s):  
N. W. Murray
Keyword(s):  
Boundary Conditions ◽  
Circular Plate ◽  
Thin Plate ◽  
Plate Theory ◽  
Series Solutions ◽  
Simply Supported ◽  
Convergence Of Series ◽  
Thin Plate Theory ◽  
Polygonal Plate ◽  
Image Position

AbstractThe solution for a simply supported many-sided polygonal plate does not agree with that for the corresponding circular plate. This paper describes the earlier work of Rao and Rajaiah on polygonal plates and then explains why best convergence of series solutions occurs when the boundary conditions are defined as

scholarly journals Thin circular plate uniformaly loaded over a concentric elliptic path and supported on columns

1986 ◽  
Vol 9 (1) ◽  
pp. 161-174
Author(s):  
W. A. Bassali
Keyword(s):  
Circular Plate ◽  
Thin Plate ◽  
Plate Theory ◽  
Thin Circular Plate ◽  
Thin Plate Theory

Within the limitations of the classical thin plate theory expressions are obtained for the small deflections of a thin isotropic circular plate uniformly loaded over a concentric ellipse and supported by four columns at the vertices of a rectangle whose sides are parallel to the axes of the ellipse. Formulae are given for the moments and shears at the centre of the plate and on the edge. Limiting cases are investigated.

NODAL INTEGRATION THIN PLATE FORMULATION USING LINEAR INTERPOLATION AND TRIANGULAR CELLS

2011 ◽  
Vol 08 (04) ◽  
pp. 813-824 ◽  
Author(s):  
X. Y. CUI ◽  
S. LIN ◽  
G. Y. LI
Keyword(s):  
Boundary Conditions ◽  
Thin Plate ◽  
Plate Theory ◽  
Weak Form ◽  
Linear Interpolation ◽  
Integration Domain ◽  
Nodal Integration ◽  
Thin Plate Theory ◽  
System Equations ◽  
Gradient Smoothing

This paper presents a thin plate formulation with nodal integration for bending analysis using three-node triangular cells and linear interpolation functions. The formulation was based on the classic thin plate theory, in which only deflection field was required and dealt with as the field variables. They were assumed to be piecewisely linear and expressed using a set of three-node triangular cells. Based on each node, the integration domain has been further derived, where the curvature in the domain was computed using a gradient smoothing technique (GST). As a result, the curvature in each integration domain is a constant whereby the deflection is compatible in the whole problem domain. The generalized smoothed Galerkin weak form is then used to create the discretized system equations where the system stiffness is obtained using simple summation operation. The essential rotational boundary conditions are imposed in the process of constructing the curvature field in conjunction with imposing the translational boundary conditions in the same way as undertaken in the standard FEM. A number of numerical examples were studied using the present formulation, including both static and free vibration analyses. The numerical results were compared with the reference ones together with those shown in the state-of-art literatures published. Very good accuracy has been achieved using the present method.

The Buckling of a Reinforced Circular Plate under Uniform Radial Thrust

1961 ◽  
Vol 12 (4) ◽  
pp. 337-342 ◽  
Author(s):  
I. T. Cook ◽  
H. W. Parsons
Keyword(s):  
Exact Solution ◽  
Circular Plate ◽  
Central Region ◽  
Clamped Plate ◽  
Simply Supported ◽  
Thin Circular Plate ◽  
Radial Thrust ◽  
Thickness Function

SummaryAn exact solution for the symmetrical buckling under uniform radial thrust is obtained for a thin circular plate having a particular type of thickness function for the cases in which the edge of the plate is either clamped or simply-supported. In both cases it is found that the critical thrust necessary to produce buckling can be increased from its value for the uniform circular plate of the same material and volume by concentrating material in the central region of the plate. For the clamped plate the increase is about 18 per cent and for the simply-supported plate about 29 per cent.

The Elastically Clamped Rectangular Plate With Arbitrary Lateral and Thermal Loading

1962 ◽  
Vol 29 (3) ◽  
pp. 578-580
Author(s):  
C. C. Chao ◽  
Max Anuliker
Keyword(s):  
Rectangular Plate ◽  
Thin Plate ◽  
Infinite Series ◽  
Series Solution ◽  
Thermal Loading ◽  
Plate Theory ◽  
Convergent Series ◽  
Clamped Plate ◽  
Simply Supported ◽  
Thin Plate Theory

Within the limits of classical thin-plate theory a variety of elementary problems have been solved for the rectangular plate3,4,5. In particular, the rectangular plate with edges simply supported or clamped has been dealt with at length and the solution to different loading cases given either in the form of a doubly infinite series or a single infinite series. In this paper a rapidly convergent series solution is outlined for the uniformly elastically clamped plate which is subjected to nonuniform lateral and thermal loading. The solution converges in the limit to those corresponding to the simply supported and rigidly clamped plate.

Vibrations of an Incompressible Linearly Elastic Plate Using Discontinuous Finite Element Basis Functions for Pressure

2019 ◽  
Vol 141 (5) ◽  
Author(s):  
Lisha Yuan ◽  
Romesh C. Batra
Keyword(s):  
Finite Element ◽  
Hydrostatic Pressure ◽  
Boundary Conditions ◽  
Plate Theory ◽  
Mass Matrix ◽  
Basis Functions ◽  
Mode Shapes ◽  
Normal Strain ◽  
Simply Supported ◽  
Transverse Normal Strain

Abstract We numerically analyze, with the finite element method, free vibrations of incompressible rectangular plates under different boundary conditions with a third-order shear and normal deformable theory (TSNDT) derived by Batra. The displacements are taken as unknowns at the nodes of a 9-node quadrilateral element and the hydrostatic pressure at four interior nodes. The plate theory satisfies the incompressibility condition, and the basis functions satisfy the Babuska-Brezzi condition. Because of the singular mass matrix, Moler's QZ algorithm (also known as the generalized Schur decomposition) is used to solve the resulting eigenvalue problem. Computed results for simply supported, clamped, and clamped-free rectangular isotropic plates agree well with the corresponding analytical frequencies of simply supported plates and with those found using the commercial software, abaqus, for other edge conditions. In-plane modes of vibrations are clearly discerned from mode shapes of square plates of aspect ratio 1/8 for all three boundary conditions. The magnitude of the transverse normal strain at a point is found to equal the sum of the two axial strains implying that higher-order plate theories that assume null transverse normal strain will very likely not provide good solutions for plates made of rubberlike materials that are generally taken to be incompressible. We have also compared the presently computed through-the-thickness distributions of stresses and the hydrostatic pressure with those found using abaqus.

scholarly journals Semi-analytical static analysis of nonlocal strain gradient laminated composite nanoplates in hygrothermal environment

2021 ◽  
Vol 43 (5) ◽  
Author(s):  
Giovanni Tocci Monaco ◽  
Nicholas Fantuzzi ◽  
Francesco Fabbrocino ◽  
Raimondo Luciano
Keyword(s):  
Thin Plate ◽  
Strain Gradient ◽  
Plate Theory ◽  
Laminated Composite ◽  
Equilibrium Equations ◽  
Thin Plate Theory ◽  
Bending Behavior ◽  
Hygrothermal Environment ◽  
Load Types ◽  
Novel Applications

AbstractIn this work, the bending behavior of nanoplates subjected to both sinusoidal and uniform loads in hygrothermal environment is investigated. The present plate theory is based on the classical laminated thin plate theory with strain gradient effect to take into account the nonlocality present in the nanostructures. The equilibrium equations have been carried out by using the principle of virtual works and a system of partial differential equations of the sixth order has been carried out, in contrast to the classical thin plate theory system of the fourth order. The solution has been obtained using a trigonometric expansion (e.g., Navier method) which is applicable to simply supported boundary conditions and limited lamination schemes. The solution is exact for sinusoidal loads; nevertheless, convergence has to be proved for other load types such as the uniform one. Both the effect of the hygrothermal loads and lamination schemes (cross-ply and angle-ply nanoplates) on the bending behavior of thin nanoplates are studied. Results are reported in dimensionless form and validity of the present methodology has been proven, when possible, by comparing the results to the ones from the literature (available only for cross-ply laminates). Novel applications are shown both for cross- and angle-ply laminated which can be considered for further developments in the same topic.

The Bending, Buckling, and Flexural Vibration of Simply Supported Polygonal Plates by Point-Matching

1961 ◽  
Vol 28 (2) ◽  
pp. 288-291 ◽  
Author(s):  
H. D. Conway
Keyword(s):  
Hydrostatic Pressure ◽  
Boundary Conditions ◽  
Flexural Vibration ◽  
Frequency Equation ◽  
Numerical Results ◽  
Special Technique ◽  
Point Matching ◽  
Buckling Loads ◽  
Simply Supported ◽  
Flexural Vibrations

The bending by uniform lateral loading, buckling by two-dimensional hydrostatic pressure, and the flexural vibrations of simply supported polygonal plates are investigated. The method of meeting the boundary conditions at discrete points, together with the Marcus membrane analog [1], is found to be very advantageous. Numerical examples include the calculation of the deflections and moments, and buckling loads of triangular square, and hexagonal plates. A special technique is then given, whereby the boundary conditions are exactly satisfied along one edge, and an example of the buckling of an isosceles, right-angled triangle plate is analyzed. Finally, the frequency equation for the flexural vibrations of simply supported polygonal plates is shown to be the same as that for buckling under hydrostatic pressure, and numerical results can be written by analogy. All numerical results agree well with the exact solutions, where the latter are known.

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