Nonlinear Asymptotic Solution of the Reissner Plate Equations

1985 ◽  
Vol 52 (4) ◽  
pp. 907-912 ◽  
Author(s):  
L. A. Taber
Keyword(s):  
Plate Theory ◽  
Small Strain ◽  
Circular Plates ◽  
Exponential Form ◽  
Plate Edge ◽  
Reissner Plate ◽  
Plate Equations ◽  
Special Case ◽  
Clamped Edges ◽  
Layer Solution

Based on the Reissner plate equations for large displacements and rotations within the limits of small strain, asymptotic solutions are developed for circular plates under uniform pressure. With the boundary layer solution assumed in exponential form, the boundary conditions are applied directly at the plate edge without the need for matched asymptotic expansions. Results are presented for plates with clamped edges. When compared to the solution for the special case of von Ka´rma´n plate theory, stresses generally deviate by less than 10 percent for rotation angles up to about 30 deg.

Bending of fibre-reinforced composite circular plates of non-uniform thickness

1973 ◽  
Vol 8 (4) ◽  
pp. 253-259 ◽  
Author(s):  
H V Lakshminarayana ◽  
H Srinath
Keyword(s):  
Analytical Solutions ◽  
Epoxy Composite ◽  
Plate Theory ◽  
Composite Plates ◽  
Circular Plates ◽  
Uniform Thickness ◽  
Classical Plate Theory ◽  
Varying Thickness ◽  
Reinforced Composite ◽  
Plate Equations

By use of classical plate theory complete analytical solutions are obtained for the bending of thin fibre-reinforced composite circular plates of radially varying thickness exhibiting rectangular orthotropy. The plate equations are solved for three specific cases of axisymmetric loading and boundary conditions. The solutions are presented in non-dimensional form. Numerical results are presented graphically for glass-epoxy composite plates.

Natural Frequencies of Mindlin Circular Plates

1980 ◽  
Vol 47 (3) ◽  
pp. 652-655 ◽  
Author(s):  
T. Irie ◽  
G. Yamada ◽  
S. Aomura
Keyword(s):  
Natural Frequencies ◽  
Plate Theory ◽  
Mindlin Plate ◽  
Circular Plates ◽  
Simply Supported ◽  
Mindlin Plate Theory ◽  
Clamped Edges

The natural frequencies of vibration based upon the Mindlin plate theory are tabulated for uniform circular plates with free, simply supported, and clamped edges for the first several tens modes.

Influence of Large Amplitudes on Free Flexural Vibrations of Rectangular Elastic Plates

1956 ◽  
Vol 23 (4) ◽  
pp. 532-540
Author(s):  
Hu-Nan Chu ◽  
George Herrmann
Keyword(s):  
Isotropic Material ◽  
Previous Investigation ◽  
Plate Theory ◽  
Free Vibrations ◽  
Elastic Plates ◽  
Conservation Of Energy ◽  
Large Rotations ◽  
Plate Equations ◽  
Rectangular Elastic Plate ◽  
Special Case

Abstract In a recent paper (1) a set of plate equations was derived, which governs motions with small elongations and shears, but moderately large rotations, valid for an isotropic material obeying Hooke’s law. The resulting theory, which may be considered the dynamic analog of the von Karman plate theory, is applied presently to the study of free vibrations of a rectangular, elastic plate with hinged, immovable edges. The nonlinear equations are solved approximately by employing a perturbation procedure and also the principle of conservation of energy directly. The influence of large amplitudes on the period of free vibration and on the maximum normal stress is established. The free vibrations of a beam are studied as a special case and the resulting period compared with a previous investigation.

Boundary Layer Phenomena in Large Deflection, Small Strain Plate Theory

1993 ◽  
Vol 60 (1) ◽  
pp. 229-232
Author(s):  
L. J. Berg
Keyword(s):  
Boundary Layer ◽  
Boundary Conditions ◽  
Boundary Layers ◽  
Large Deflection ◽  
Large Deformations ◽  
Plate Theory ◽  
Small Strain ◽  
Thin Plates ◽  
Large Deflections ◽  
Layer Solution

Boundary layers exist at the edges of thin plates undergoing large deformations because the interior of the plate must assume a developable shape. The developable shape is sometimes incompatible with the force and moment resultants prescribed at the plate’s boundary, in particular when the edge of the plate is stress free. A boundary layer solution is presented which describes the shape of a boundary layer in a plate undergoing large deflections. The boundary layer is a slight perturbation of the interior shape which allows the appropriate boundary conditions to be satisfied. Since developable shells are applicable to a plane, the boundary layer is also appropriate for arbitrary developable shells.

A note on seasonal Markov chains with gamma or gamma-like distributions

1979 ◽  
Vol 16 (1) ◽  
pp. 117-128 ◽  
Author(s):  
E. H. Lloyd ◽  
S. D. Saleem
Keyword(s):  
Markov Chain ◽  
Laplace Transform ◽  
Weighted Sums ◽  
Power Transformation ◽  
Exponential Form ◽  
Continuous State ◽  
Bessel Distribution ◽  
Storage Theory ◽  
Special Case ◽  
Type Of Transition

Weighted sums defined on a Markov chain (MC) are important in applications (e.g. to reservoir storage theory). The rather intractable theory of such sums simplifies to some extent when the transition p.d.f. of the chain {Xt} has a Laplace transform (LT) L(Xt+1; θ |Χ t=x) of the ‘exponential' form H(θ) exp{ – G(θ)x}. An algorithm is derived for the computation of the LT of Σat,Χ t for this class, and for a seasonal generalization of it.A special case of this desirable exponential type of transition LT for a continuous-state discrete-time MC is identified by comparison with the LT of the Bessel distribution. This is made the basis for a new derivation of a gamma-distributed MC proposed by Lampard (1968).A seasonal version of this process is developed, valid for any number of seasons.Reference is made to related chains with three-parameter gamma-like distributions (of the Kritskii–Menkel family) that may be generated from the above by a simple power transformation.

Asymmetric Thermal Buckling of Imperfect FGM Circular Plates with Rotationally Restrained Edge

2020 ◽  
Vol 20 (12) ◽  
pp. 2050127
Author(s):  
S. V. Levyakov
Keyword(s):  
Critical Temperature ◽  
Temperature Rise ◽  
Material Properties ◽  
Thermal Loading ◽  
Plate Theory ◽  
Nonlinear Eigenvalue Problem ◽  
Mode Shapes ◽  
Effect Of Temperature ◽  
Circular Plates ◽  
Governing Equations

The paper addresses the problem of asymmetric buckling of geometrically imperfect circular plates undergoing large axisymmetric deflections under thermal loading. The plate edge is assumed to be immovable in the radial direction and elastically restrained against bending rotation. The plate material is graded in the thickness direction and dependence of the material properties on temperature is taken into account. The governing equations are derived using the von Karman nonlinear plate theory and the concept of physically neutral surface. It is shown that, when subjected to increasing temperature, the plate initially bends into a figure of revolution and then buckles into asymmetric mode with local circumferential waves. To determine the critical temperature rise, a nonlinear eigenvalue problem is formulated by linearizing the governing equations about the axisymmetric state of equilibrium and solved using power-series expansions. The effect of temperature-dependent material properties, rotational spring stiffness and initial geometric imperfection on the critical temperature rise and buckling mode shapes is studied.

A closed form solution for bending/stretching analysis of functionally graded circular plates under asymmetric loading using the Green function

2010 ◽  
Vol 224 (6) ◽  
pp. 1153-1163 ◽  
Author(s):  
A R Saidi ◽  
A Naderi ◽  
E Jomehzadeh
Keyword(s):  
Green Function ◽  
Closed Form ◽  
Volume Fraction ◽  
Plate Theory ◽  
Closed Form Solution ◽  
Form Solution ◽  
Functionally Graded ◽  
Transverse Displacement ◽  
Circular Plates ◽  
Equilibrium Equations

In this article, a closed-form solution for bending/stretching analysis of functionally graded (FG) circular plates under asymmetric loads is presented. It is assumed that the material properties of the FG plate are described by a power function of the thickness variable. The equilibrium equations are derived according to the classical plate theory using the principle of total potential energy. Two new functions are introduced to decouple the governing equilibrium equations. The three highly coupled partial differential equations are then converted into an independent equation in terms of transverse displacement. A closed-form solution for deflection of FG circular plates under arbitrary lateral eccentric concentrated force is obtained by defining a new coordinate system. This solution can be used as a Green function to obtain the closed-form solution of the FG plate under arbitrary loadings. Also, the solution is employed to solve some different asymmetric problems. Finally, the stress and displacement components are obtained exactly for each problem and the effect of volume fraction is also studied.

Sign in / Sign up

Export Citation Format

Share Document