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.

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.

Characterization of the vibration, stability and static responses of graphene-reinforced sandwich plates under mechanical and thermal loadings using the refined shear deformation plate theory

2021 ◽  
pp. 109963622199386
Author(s):  
Hessameddin Yaghoobi ◽  
Farid Taheri
Keyword(s):  
Shear Deformation ◽  
Volume Fraction ◽  
Plate Theory ◽  
Micromechanical Model ◽  
Sheet Thickness ◽  
Analytical Investigation ◽  
Sandwich Plates ◽  
Simply Supported ◽  
Vibration Stability ◽  
Shear Deformation Plate Theory

An analytical investigation was carried out to assess the free vibration, buckling and deformation responses of simply-supported sandwich plates. The plates constructed with graphene-reinforced polymer composite (GRPC) face sheets and are subjected to mechanical and thermal loadings while being simply-supported or resting on different types of elastic foundation. The temperature-dependent material properties of the face sheets are estimated by employing the modified Halpin-Tsai micromechanical model. The governing differential equations of the system are established based on the refined shear deformation plate theory and solved analytically using the Navier method. The validation of the formulation is carried out through comparisons of the calculated natural frequencies, thermal buckling capacities and maximum deflections of the sandwich plates with those evaluated by the available solutions in the literature. Numerical case studies are considered to examine the influences of the core to face sheet thickness ratio, temperature variation, Winkler- and Pasternak-types foundation, as well as the volume fraction of graphene on the response of the plates. It will be explicitly demonstrated that the vibration, stability and deflection responses of the sandwich plates become significantly affected by the aforementioned parameters.

Enhanced First-Order Shear Deformation Theory for Laminated and Sandwich Plates

2005 ◽  
Vol 72 (6) ◽  
pp. 809-817 ◽  
Author(s):  
Jun-Sik Kim ◽  
Maenghyo Cho
Keyword(s):  
Shear Deformation ◽  
Deformation Theory ◽  
Plate Theory ◽  
Three Dimensional ◽  
Shear Deformation Theory ◽  
Higher Order ◽  
Laminated Plates ◽  
Sandwich Plates ◽  
First Order ◽  
Transverse Shear Strains

A new first-order shear deformation theory (FSDT) has been developed and verified for laminated plates and sandwich plates. Based on the definition of Reissener–Mindlin’s plate theory, the average transverse shear strains, which are constant through the thickness, are improved to vary through the thickness. It is assumed that the displacement and in-plane strain fields of FSDT can approximate, in an average sense, those of three-dimensional theory. Relationship between FSDT and three-dimensional theory has been systematically established in the averaged least-square sense. This relationship provides the closed-form recovering relations for three-dimensional variables expressed in terms of FSDT variables as well as the improved transverse shear strains. This paper makes two main contributions. First an enhanced first-order shear deformation theory (EFSDT) has been developed using an available higher-order plate theory. Second, it is shown that the displacement fields of any higher-order plate theories can be recovered by EFSDT variables. The present approach is applied to an efficient higher-order plate theory. Comparisons of deflection and stresses of the laminated plates and sandwich plates using present theory are made with the original FSDT and three-dimensional exact solutions.

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.

Three-Dimensional Plate Theory for Flexible Multibody Dynamics

2015 ◽  
Author(s):  
Shilei Han ◽  
Olivier A. Bauchau
Keyword(s):  
Plate Theory ◽  
Three Dimensional ◽  
Laminated Composite ◽  
Flexible Multibody Dynamics ◽  
Composite Plates ◽  
Complex Stress ◽  
Mindlin Plate ◽  
Stress States ◽  
Three Dimensional Elasticity ◽  
Material Line

In structural analysis, many components are approximated as plates. More often that not, classical plate theories, such as Kirchhoff or Reissner-Mindlin plate theories, form the basis of the analytical developments. The advantage of these approaches is that they leads to simple kinematic descriptions of the problem: the plate’s normal material line is assumed to remain straight and its displacement field is fully defined by three displacement and two rotation components. While such approach is capable of capturing the kinetic energy of the system accurately, it cannot represent the strain energy adequately. For instance, it is well known from three-dimensional elasticity theory that the normal material line will warp under load for laminated composite plates, leading to three-dimensional deformations that generate complex stress states. To overcome this problem, several high-order, refined plate theories have been proposed. While these approaches work well for some cases, they often lead to inefficient formulations because they introduce numerous additional variables. This paper presents a different approach to the problem: based on a finite element semi-discretization of the normal material line, plate equations are derived from three-dimensional elasticity using a rigorous dimensional reduction procedure.

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.

Three-Dimensional and Shell-Theory Analysis of Axially Symmetric Motions of Cylinders

1956 ◽  
Vol 23 (4) ◽  
pp. 563-568
Author(s):  
George Herrmann ◽  
I. Mirsky
Keyword(s):  
Shear Deformation ◽  
Shell Theory ◽  
Plate Theory ◽  
Wave Length ◽  
Three Dimensional ◽  
Approximate Theory ◽  
Axially Symmetric ◽  
Rotatory Inertia ◽  
Phase Velocities ◽  
Transition Wave

Abstract The frequency (or phase velocity) of axially symmetric free vibrations in an elastic, isotropic, circular cylinder of medium thickness is studied on the basis of the three-dimensional linear theory of elasticity and several different shell theories. To be in good agreement with the solution of the three-dimensional equations for short wave lengths, an approximate theory has to include the influence of rotatory inertia and transverse shear deformation, for example, in a manner similar to Mindlin’s plate theory. A shell theory of this (Timoshenko) type is deduced from the three-dimensional elasticity theory. From a comparison of phase velocities it appears that, to a good approximation, membrane and curvature effects on one hand, and on the other hand, flexural, rotatory-inertia, and shear-deformation effects are mutually exclusive in two ranges of wave lengths, separated by a “transition” wave length. Thus, in the full range of wave lengths, the associated lowest phase velocities may be determined on the basis of the membrane shell theory (for wave lengths larger than the transition wave length) and on the basis of Mindlin’s plate theory (for wave lengths smaller than the transition wave length).

Vibration Characteristics of Simply Supported Thick Skew Plates in Three-Dimensional Setting

1995 ◽  
Vol 62 (4) ◽  
pp. 880-886 ◽  
Author(s):  
K. M. Liew ◽  
K. C. Hung ◽  
M. K. Lim
Keyword(s):  
Elasticity Theory ◽  
Three Dimensional ◽  
Stress Singularity ◽  
Skew Angle ◽  
Simply Supported ◽  
Significant Difference ◽  
Simple Support ◽  
Elasticity Solutions ◽  
Support Conditions ◽  
Three Dimensional Elasticity

A procedure is presented for determining the three-dimensional elasticity solutions for free vibration analysis of simply supported thick skew plates. The exact expressions of strain and kinetic energies are derived from linear, small-strain, three-dimensional elasticity theory. To allow the treatment of soft and hard simple support conditions, sets of three-dimensional spatial displacement functions are expressed in terms of unit normals to the edges. By virtue of the three-dimensional elasticity theory, the present method does not require a special treatment for stress singularity at the obtuse corners. This method is also demonstrated to be free from shear locking phenomena. The significant difference in the vibration response of skew plates with soft and hard simple support conditions is highlighted. The influence of skew angle on the eigenvalues of thick skew plate is discussed in the context of the three-dimensional elasticity solutions.

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