Buckling of an Elliptical Plate With Simply Supported Edge Under Uniform Compression

1976 ◽  
Vol 43 (3) ◽  
pp. 455-458 ◽  
Author(s):  
Kenzo Sato
Keyword(s):  
Plate Theory ◽  
Mathieu Functions ◽  
Elliptical Plate ◽  
Buckling Mode ◽  
Uniform Compression ◽  
Simply Supported ◽  
Aspect Ratios ◽  
Thin Plate Theory ◽  
The Stability ◽  
Circular Functions

On the basis of the ordinary thin plate theory, the stability of a simply supported elliptical plate subjected to uniform compression in its middle plane is considered by the use of circular functions, hyperbolic functions, Mathieu functions, and modified Mathieu functions which are solutions of the equilibrium equation of the buckled plate. The first five eigenvalues for the buckling mode symmetrical about both axes are calculated numerically for a variety of aspect ratios of the ellipse. The limiting cases of a circular plate and of an infinitely long strip are also discussed.

scholarly journals Stability plate with longitudinal constructive discontinuity

2011 ◽  
Vol 9 (1) ◽  
pp. 23-33
Author(s):  
Snezana Mitic ◽  
Ratko Pavlovic
Keyword(s):  
Exact Solutions ◽  
Analytical Approach ◽  
Plate Theory ◽  
Elastic Stability ◽  
The Other ◽  
Uniform Thickness ◽  
Simply Supported ◽  
Thin Plate Theory ◽  
The Stability ◽  
Different Thickness

The influence of longitudinal constructive discontinuity on the stability of the plate in the domain of elastic stability is solved based on the classical thin plate theory. The constructive discontinuities divide the plate into fields of different thickness. The plate has two opposite edges simply supported while the other two edges can take any combination of free, simply supported and clamped conditions. The Levy method is used for the solution of the problem of stability, with the aim of developing an analytical approach when researching the stability of plates with longitudinal constructive discontinuities and also with the aim of obtaining exact solutions for plates with non-uniform thickness. The exact solutions for stability presented herein are very valuable as they may serve as benchmark results for researches in this area.

scholarly journals A Method for Determining the Thicknesses of the Reserved Protective Layers in Complex Goafs Using a Joint-Modeling Method of GTS and Flac3D

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Xin Zhao ◽  
Dianshu Liu ◽  
Shenglin Li ◽  
Meng Wang ◽  
Shuaikang Tian ◽  
...  
Keyword(s):  
Thin Plate ◽  
Plate Theory ◽  
Three Dimensional ◽  
Joint Modeling ◽  
Modeling Method ◽  
Underground Cavity ◽  
Protective Layers ◽  
Safe Production ◽  
Thin Plate Theory ◽  
The Stability

In this study, a C-ALS underground cavity scanner was used to detect the shapes of mining goafs. In addition, GTS software was adopted to establish a three-dimensional geological model based on the status of the stopes, geological data, and mechanical parameters of each rock mass and to analyze the roof areas of the goafs. In regard to the morphology of the study area, based on a thin plate theory and the obtained field sampling data, a formula was established for the thicknesses of the reserved protective layers in the goafs. In addition, a formula for the thicknesses of the protective layers in the curved gobs was obtained. The thickness formula of the protective layers was then successfully verified. The detection results showed that the roof shapes of the goafs in the Yuanjiacun Iron Mine were mainly arc-shaped, and the spans of the goafs were generally less than 20 m. The stability of the arc-shaped roofs was found to be greater than that of the plate-shaped roofs. Therefore, by reducing the thicknesses of the protective layers in mining goafs, the ore recovery rates can be increased on the basis of safe production conditions. The formula of the thickness of the security layers obtained through the thin plate theory was revised based on the statistical results of the roof shapes of the goafs and then combined using GTS and FLAC3D. The modeling method successfully verified the stability of the mined-out areas. It was found that the verification results were good, and the revised formula was able to improve the recovery rate of the ore under the conditions of meeting safe production standards. Also, it was found that the revised formula could be used in the present situation. At the same time, it was also determined that the complexity of the rock masses obstructed the full identification of the joints and fissures in the present orebodies. Therefore, it is necessary to incorporate C-ALS underground cavity scanners to regularly observe the shapes of the goafs in order to ensure that stability and safety standards are maintained.

The Resonant Response of a Rectangular Plate With an Elastic Edge Restraint

1972 ◽  
Vol 94 (2) ◽  
pp. 517-525 ◽  
Author(s):  
D. M. Egle
Keyword(s):  
Plate Theory ◽  
Time History ◽  
Resonant Response ◽  
Concentrated Load ◽  
Simply Supported ◽  
Single Term ◽  
Thin Plate Theory ◽  
Wide Range ◽  
Mode Solution ◽  
Elastically Restrained

An analysis of a plate, simply supported on three edges and elastically restrained on the fourth, excited by a concentrated load with a harmonic time history, is used to study the peak resonant response of the plate for several configurations and a wide range of edge restraint. Classical thin plate theory is employed with a complex elastic modulus to account for energy dissipation. An approximate method, based on a single term normal mode solution, is developed for calculating the limits of the peak resonant response for arbitrary edge restraint.

Effect of Nonhomogeneity on Vibration of Orthotropic Rectangular Plates of Varying Thickness Resting on Pasternak Foundation

2009 ◽  
Vol 131 (1) ◽  
Author(s):  
Roshan Lal ◽  
Dhanpati
Keyword(s):  
Differential Equation ◽  
Plate Theory ◽  
The Other ◽  
Thickness Variation ◽  
Order Partial Differential Equation ◽  
Rectangular Plates ◽  
Simply Supported ◽  
Varying Thickness ◽  
Aspect Ratios ◽  
Displacement Mode

Free transverse vibrations of nonhomogeneous orthotropic rectangular plates of varying thickness with two opposite simply supported edges (y=0 and y=b) and resting on two-parameter foundation (Pasternak-type) have been studied on the basis of classical plate theory. The other two edges (x=0 and x=a) may be any combination of clamped and simply supported edge conditions. The nonhomogeneity of the plate material is assumed to arise due to the exponential variations in Young’s moduli and density along one direction. By expressing the displacement mode as a sine function of the variable between simply supported edges, the fourth order partial differential equation governing the motion of such plates of exponentially varying thickness in another direction gets reduced to an ordinary differential equation with variable coefficients. The resulting equation is then solved numerically by using the Chebyshev collocation technique for two different combinations of clamped and simply supported conditions at the other two edges. The lowest three frequencies have been computed to study the behavior of foundation parameters together with other plate parameters such as nonhomogeneity, density, and thickness variation on the frequencies of the plate with different aspect ratios. Normalized displacements are presented for a specified plate. A comparison of results with those obtained by other methods shows the computational efficiency of the present approach.

Buckling of Circular Plates Based on Reddy Plate Theory

1997 ◽  
Author(s):  
C. M. Wang ◽  
K. H. Lee ◽  
J. N. Reddy
Keyword(s):  
Shear Deformation ◽  
Plate Theory ◽  
Three Dimensional ◽  
Edge Condition ◽  
Circular Plates ◽  
Simply Supported ◽  
Thin Plate Theory ◽  
Elastically Restrained ◽  
Three Dimensional Elasticity ◽  
Limiting Values

Treated herein is the elastic buckling of circular plates based on the Reddy plate theory. This plate theroy extends the Kirchhoff (or the classical thin) plate theory to allow for the effect of transverse shear deformation. Unlike the Mindlin’s shear deformation plate theory, there is no need for a shear correction factor in the Reddy plate theory. In this paper, exact buckling solutions are derived for circular plates whose edges are simply supported and elastically restrained against rotation as well. This general edge condition includes the classical simply supported and clamped edges at the limiting, values of the elastic rotational restraint constant. The buckling solutions are expressed in terms of the well-known Kichhoff buckling solutions. A comparison of buckling loads between the Mindlin, Reddy and three-dimensional elasticity plates is also given.

scholarly journals Composite plate analysis made in an unsymmetric configuartion

2021 ◽  
Vol 2130 (1) ◽  
pp. 012014
Author(s):  
K Falkowicz
Keyword(s):  
Linear Analysis ◽  
Second Step ◽  
Torsional Buckling ◽  
Buckling Mode ◽  
Uniform Compression ◽  
Simply Supported ◽  
Plate Analysis ◽  
Flexural Torsional Buckling ◽  
Methods Numerical ◽  
Made In

Abstract The work presents a thin-walled plate element with the central rectangular cut-out which can be use as an elastic or load-bearing element. Plates were made of carbon epoxy laminate and subjected to uniform compression. Plates were simply supported on shorter edges, and loaded axial load. The study included analysis of the critical and weakly post-critical behavior using experimental and numerical methods. Numerical analysis was performed with using linear analysis of eigenvalue problem to determination critical loads. The second step connected nonlinear analysis of structure with initiated geometrically imperfection corresponding to the flexural-torsional buckling mode of the plate. To the numerical calculations the commercial ABAQUS program was used.

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.

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 Stability of Simply Supported Rectangular Surfaces in Uniform Subsonic Flow

1973 ◽  
Vol 40 (1) ◽  
pp. 68-72 ◽  
Author(s):  
C. H. Ellen
Keyword(s):  
Aspect Ratio ◽  
Three Dimensional ◽  
Stability Result ◽  
Fluid Flows ◽  
Inviscid Fluid ◽  
Low Frequencies ◽  
Critical Flow Velocity ◽  
Simply Supported ◽  
Aspect Ratios ◽  
The Stability

A study is made of the stability of a simply supported flat plate set in an infinite rigid baffle when an inviscid fluid flows uniformly at subsonic speed past one side of the surface. The generalized pressures are derived for low frequencies with two and three-dimensional flows. The three-dimensional generalized pressures are expanded asymptotically for high and low aspect ratios, and analytic forms derived for the critical flow velocity at instability. The asymptotic expansions enable the effect of aspect ratio on stability to be determined. It is shown that the incompressible limit, for two-dimensional flows, is singular but the stability criterion is associated with first-mode divergence and is identical with the three-dimensional high aspect ratio stability result, although there are certain detailed differences in the nature of the instability.

scholarly journals Nonlinear Parametric Vibration and Chaotic Behaviors of an Axially Accelerating Moving Membrane

2019 ◽  
Vol 2019 ◽  
pp. 1-11 ◽  
Author(s):  
Mingyue Shao ◽  
Jimei Wu ◽  
Yan Wang ◽  
Qiumin Wu
Keyword(s):  
Differential Equations ◽  
Nonlinear Vibration ◽  
Plate Theory ◽  
Velocity Variation ◽  
Mean Velocity ◽  
Thin Plate Theory ◽  
Time Histories ◽  
Axially Moving ◽  
The Stability ◽  
The Galerkin Method

Nonlinear vibration characteristics of a moving membrane with variable velocity have been examined. The velocity is presumed as harmonic change that takes place over uniform average speed, and the nonlinear vibration equation of the axially moving membrane is inferred according to the D’Alembert principle and the von Kármán nonlinear thin plate theory. The Galerkin method is employed for discretizing the vibration partial differential equations. However, the solutions concerning to differential equations are determined through the 4th order Runge–Kutta technique. The results of mean velocity, velocity variation amplitude, and aspect ratio on nonlinear vibration of moving membranes are emphasized. The phase-plane diagrams, time histories, bifurcation graphs, and Poincaré maps are obtained; besides that, the stability regions and chaotic regions of membranes are also obtained. This paper gives a theoretical foundation for enhancing the dynamic behavior and stability of moving membranes.

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