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.

Problems concerning the bending of isotropic thin elastic plates subject to various distributions of normal pressures

1958 ◽  
Vol 54 (2) ◽  
pp. 265-287 ◽  
Author(s):  
W. A. Bassali ◽  
H. P. F. Swinnerton-Dyer
Keyword(s):  
Circular Plate ◽  
Plate Theory ◽  
Normal System ◽  
Free Boundaries ◽  
Elastic Plates ◽  
Normal Loading ◽  
Complex Variable Methods ◽  
Concentrated Forces ◽  
Special Case ◽  
Thin Elastic Plates

ABSTRACTWithin the limitations of the small-deflexion plate theory, complex variable methods are used in this paper to obtain an exact solution for the problem of a thin circular plate supported at several interior or boundary points, and subjected to a certain normal loading spread over the area of an eccentric circle, the boundary of the plate being free. The load considered includes as a special case a linearly varying load over the circle and, as the radius of the loaded circle tends to zero, this load can be made to tend to a couple nucleus at its centre. As limiting cases the procedure adopted provides us with solutions appropriate to a circular plate, an infinitely large plate and a half-plane having free boundaries and acted upon by any normal system of concentrated forces and concentrated couples in equilibrium. Formulae for the moments, shears and deflexions relating to special examples are worked out in detail.

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.

General Solutions for the Statics of Anisotropic, Transversely Inhomogeneous Elastic Plates in Terms of Complex Functions

2006 ◽  
Vol 11 (6) ◽  
pp. 596-628 ◽  
Author(s):  
Kostas P. Soldatos
Keyword(s):  
General Solution ◽  
Elastic Plate ◽  
Plate Theory ◽  
High Order ◽  
Transverse Strain ◽  
Elastic Plates ◽  
General Solutions ◽  
Material Inhomogeneity ◽  
Complex Functions ◽  
Anisotropic Elastic

This paper develops the general solution of high-order partial differential equations (PDEs) that govern the static behavior of transversely inhomogeneous, anisotropic, elastic plates, in terms of complex functions. The basic development deals with the derivation of such a form of general solution for the PDEs associated with the most general, two-dimensional (“equivalent single-layered”), elastic plate theory available in the literature. The theory takes into consideration the effects of bending–stretching coupling due to possible un-symmetric forms of through-thickness material inhomogeneity. Most importantly, it also takes into consideration the effects of both transverse shear and transverse normal deformation in a manner that allows for a posteriori, multiple choices of transverse strain distributions. As a result of this basic and most general development, some interesting specializations yield, as particular cases, relevant general solutions of high-order PDEs associated with all of the conventional, elastic plate theories available in the literature.

Large-Amplitude Oscillations of Unsymmetrically Laminated Anisotropic Rectangular Plates Including Shear, Rotatory Inertia, and Transverse Normal Stress

1985 ◽  
Vol 52 (3) ◽  
pp. 536-542 ◽  
Author(s):  
K. S. Sivakumaran ◽  
C. Y. Chia
Keyword(s):  
Boundary Conditions ◽  
Normal Stress ◽  
Transverse Shear ◽  
Plate Theory ◽  
Orientation Angle ◽  
Nonlinear Frequency ◽  
Rectangular Plates ◽  
Elastic Plates ◽  
Rotatory Inertia ◽  
Transverse Normal Stress

This paper is concerned with nonlinear free vibrations of generally laminated anisotropic elastic plates. Based on Reissner’s variational principle a nonlinear plate theory is developed. The effects of transverse shear, rotatory inertia, transverse normal stress, and transverse normal contraction or extension are included in this theory. Using the Galerkin procedure and principle of harmonic balance, approximate solutions to governing equations of unsymmetrically laminated rectangular plates including transverse shear, rotatory inertia, and transverse normal stress are formulated for various boundary conditions. Numerical results for the ratio of nonlinear frequency to linear frequency of unsymmetric angle-ply and cross-ply laminates are presented graphically for various values of elastic properties, fiber orientation angle, number of layers, and aspect ratio and for different boundary conditions. Present results are also compared with available data.

On the Vibration of Layered Sandwich Elastic Plates

2008 ◽  
Author(s):  
Vincent O. S. Olunloyo ◽  
Charles A. Osheku
Keyword(s):  
Plate Theory ◽  
Impact Damage ◽  
Local Buckling ◽  
Recent Analysis ◽  
Sandwich Plates ◽  
Elastic Plates ◽  
Interface Pressure ◽  
Engineering Structures ◽  
Thin Plate Theory ◽  
Operational Methods

Sandwich elastic plates have found increasing applications in civil, aerospace, military and offshore industries to enhance superior resistance to fatigue crack propagation, impact damage, local buckling and are very effective for vibration damping and noise reduction. Such structural application has significantly led to reduction in vulnerability of warships to blasts, ballistics, bomb and fire attacks. In engineering structures, one of the effective ways of damping vibration and noise attenuation, is to exploit the occurrence of slip at the interface of structural laminates where such members are held together in a pressurised environment. Recent analysis and experimental investigation of vibration characteristics and damping properties of layered sandwich structures, are mostly limited to elastic beams. This paper is an attempt to extend such analytical investigations to layered sandwich plates. By employing contact mechanics and laminated thin plate theory, the generalised equation governing the vibration of two layered sandwich plates that are held together in pressurised environment is presented. In particular, by invoking operational methods for the case of linear interface pressure distribution, closed form analytical results for the system natural frequency and dynamic response under external excitation are reported for design analysis and applications.

Effect of Shear Deformation and Rotatory Inertia on Dynamic Buckling of Elastic Plates

1988 ◽  
Vol 110 (3) ◽  
pp. 282-286
Author(s):  
V. Birman
Keyword(s):  
Dynamic Response ◽  
Shear Deformation ◽  
Plate Theory ◽  
Deformation Mode ◽  
Dynamic Buckling ◽  
Rectangular Plates ◽  
Elastic Plates ◽  
Rotatory Inertia ◽  
Qualitative Effect

The influence of shear deformation and rotatory inertia on dynamic response of elastic rectangular plates subject to in-plane loads increasing with time is discussed using Mindlin’s plate theory. The qualitative effect of those factors on transverse displacements is estimated. It is shown that this effect becomes essential only if the plate is thick and the number of half-waves along the plate axes in the deformation mode is large.

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