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.

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.

A Composite Plate Theory With Rotatory Inertia, Transverse Shear, and Normal Stress Effects

1991 ◽  
Vol 113 (2) ◽  
pp. 127-132 ◽  
Author(s):  
G. Z. Voyiadjis ◽  
P. D. Panera
Keyword(s):  
Normal Stress ◽  
Transverse Shear ◽  
Plate Theory ◽  
Composite Plates ◽  
Offshore Structures ◽  
Refined Theory ◽  
Normal Strain ◽  
Rotatory Inertia ◽  
Stress Effects ◽  
Transverse Normal Strain

A refined theory for the flexural motions of composite plates is presented. The theory incorporates rotatory inertia in addition to the influence of transverse normal strain, transverse normal stress, and transverse shear. This is of primary interest for the analysis of offshore structures as well as piping analysis. The classical wave propagation problem is used to test the proposed theory. The results indicate the influence of the transverse normal strain on the wave speed for large values of h/λ. The shear coefficient obtained from the proposed theory has a constant magnitude as opposed to the undetermined coefficient form in previous flexural motion formulations.

scholarly journals An Exact Series Solution for the Vibration of Mindlin Rectangular Plates with Elastically Restrained Edges

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Xue Kai ◽  
Wang Jiufa ◽  
Li Qiuhong ◽  
Wang Weiyuan ◽  
Wang Ping
Keyword(s):  
Boundary Conditions ◽  
Plate Theory ◽  
Mindlin Plate ◽  
Rectangular Plates ◽  
Cross Sectional ◽  
Fourier Cosine ◽  
Cosine Series ◽  
The Matrix ◽  
Elastically Restrained Edges ◽  
Elastically Restrained

An analysis method is proposed for the vibration analysis of the Mindlin rectangular plates with general elastically restrained edges, in which the vibration displacements and the cross-sectional rotations of the mid-plane are expressed as the linear combination of a double Fourier cosine series and four one-dimensional Fourier series. The use of these supplementary functions is to solve the possible discontinuities with first derivatives at each edge. So this method can be applied to get the exact solution for vibration of plates with general elastic boundary conditions. The matrix eigenvalue equation which is equivalent to governing differential equations of the plate can be derived through using the boundary conditions and the governing equations based on Mindlin plate theory. The natural frequencies can be got through solving the matrix equation. Finally the numerical results are presented to validate the accuracy of the method.

Analytical Simulation of the Composite Ice Wall

1989 ◽  
Vol 111 (3) ◽  
pp. 174-180 ◽  
Author(s):  
P. Corder ◽  
T. Kozik
Keyword(s):  
Shear Stress ◽  
Closed Form ◽  
Normal Stress ◽  
Transverse Shear ◽  
Parametric Study ◽  
Beam Theory ◽  
Transverse Shear Stress ◽  
Transverse Normal Stress ◽  
Analytical Simulation ◽  
Sandwich Configuration

A system of linear, closed-form stress equations for a steel-concrete-steel sandwich configuration, i.e., the “Composite Ice Wall,” was derived incorporating a formulation of classical beam theory. The stress terms include the longitudinal normal stress, the transverse shear stress and the transverse normal stress. These equations were programmed using Pascal and a parametric study was conducted. Some of the results are included herein. The analytical model produces principal stress contours and centerline deflections very similar to those in the classical beam for comparable pressure loadings.

scholarly journals The Effect of Using Different Elastic Moduli on Vibration of Laminated CFRP Rectangular Plates

2019 ◽  
Vol 2 (1) ◽  
pp. 19-27
Author(s):  
Yoshihiro Narita ◽  
Michio Innami ◽  
Daisuke Narita
Keyword(s):  
Boundary Conditions ◽  
Natural Frequencies ◽  
Plate Theory ◽  
Linear Regression Analysis ◽  
Laminated Composite ◽  
Thin Layers ◽  
Test Accuracy ◽  
Rectangular Plates ◽  
Classical Boundary ◽  
Definition Of

This paper deals with effects of using different sets of material constants on the natural frequencies of laminated composite rectangular plates. The plate is symmetrically laminated by thin layers composed of recently developed carbon fiber reinforced plastic (CFRP) materials. Numerical experiments are conducted by using a semi-analytical solution based on the thin plate theory and the lamination theory. The displacements are assumed to accommodate any combination of classical boundary conditions. The material property is expressed by a set of four elastic constants, and some typical sets of values are cited from the recent literature. Furthermore, a new standard set of discretized constants is proposed to uncover the underlying characteristics of the existing constants. The convergence study is carried out first, and the lowest five natural frequencies are calculated for five sets of classical boundary conditions including totally free through totally clamped cases. Next, a new definition of frequency parameters is introduced to promote more physically meaningful comparison among the obtained results, and the effect of using slightly different constants is clarified for unified comparison and insights. It is also discussed to derive approximate frequency formulas by linear regression analysis and to test accuracy of the formulas.

scholarly journals A nonlocal sinusoidal plate model for micro/nanoscale plates

2014 ◽  
Vol 228 (14) ◽  
pp. 2652-2660 ◽  
Author(s):  
Huu-Tai Thai ◽  
Thuc P Vo ◽  
Trung-Kien Nguyen ◽  
Jaehong Lee
Keyword(s):  
Boundary Conditions ◽  
Shear Deformation ◽  
Transverse Shear ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Nonlocal Elasticity Theory ◽  
Small Scale ◽  
Plate Model ◽  
Proposed Model ◽  
Small Scale Effect

A nonlocal sinusoidal plate model for micro/nanoscale plates is developed based on Eringen’s nonlocal elasticity theory and sinusoidal shear deformation plate theory. The small-scale effect is considered in the former theory while the transverse shear deformation effect is included in the latter theory. The proposed model accounts for sinusoidal variations of transverse shear strains through the thickness of the plate, and satisfies the stress-free boundary conditions on the plate surfaces, thus a shear correction factor is not required. Equations of motion and boundary conditions are derived from Hamilton’s principle. Analytical solutions for bending, buckling, and vibration of simply supported plates are presented, and the obtained results are compared with the existing solutions. The effects of small scale and shear deformation on the responses of the micro/nanoscale plates are investigated.

Free and Forced Vibrations of Monolithic and Composite Rectangular Plates With Interior Constrained Points

2018 ◽  
Vol 141 (1) ◽  
Author(s):  
Arka P. Chattopadhyay ◽  
Romesh C. Batra
Keyword(s):  
Forced Vibration ◽  
Plate Theory ◽  
Translational Motion ◽  
Orientation Angle ◽  
Stationary Points ◽  
Width Ratio ◽  
Rectangular Plates ◽  
Fiber Reinforced ◽  
Mode Localization ◽  
Unidirectional Fiber

The restriction of deformations to a subregion of a system undergoing either free or forced vibration due to an irregularity or discontinuity in it is called mode localization. Here, we study mode localization in free and forced vibration of monolithic and unidirectional fiber-reinforced rectangular linearly elastic plates with edges either simply supported (SS) or clamped by using a third-order shear and normal deformable plate theory (TSNDT) with points on either one or two normals to the plate midsurface constrained from translating in all three directions. The plates studied are symmetric about their midsurfaces. The in-house developed software based on the finite element method (FEM) is first verified by comparing predictions from it with either the literature results or those computed by using the linear theory of elasticity and the commercial FE software abaqus. New results include: (i) the localization of both in-plane and out-of-plane modes of vibration, (ii) increase in the mode localization intensity with an increase in the length/width ratio of a rectangular plate, (iii) change in the mode localization characteristics with the fiber orientation angle in unidirectional fiber reinforced laminae, (iv) mode localization due to points on two normals constrained, and (iv) the exchange of energy during forced harmonic vibrations between two regions separated by the line of nearly stationary points that results in a beats-like phenomenon in a subregion of the plate. Constraining translational motion of internal points can be used to design a structure with vibrations limited to its small subregion and harvesting energy of vibrations of the subregion.

The Contradicting Assumptions of Zero Transverse Normal Stress and Strain in the Thin Plate Theory: A Justification

1999 ◽  
Vol 68 (4) ◽  
pp. 660-662 ◽  
Author(s):  
K. Bhaskar and ◽  
T. K. Varadan
Keyword(s):  
Normal Stress ◽  
Thin Plate ◽  
Test Problem ◽  
Plate Theory ◽  
Stress And Strain ◽  
Elasticity Solution ◽  
Classical Plate Theory ◽  
Transverse Normal Stress ◽  
Thin Plate Theory

The need and validity of the contradicting assumptions of zero transverse normal stress and the corresponding strain in the classical plate theory are critically examined here. This is done by studying the relative magnitudes of these quantities with respect to other stresses and strains for a test problem amenable to an exact elasticity solution.

Flutter of Rectangular Plates in Three Dimensional Incompressible Flow With Various Boundary Conditions: Theory and Experiment

2012 ◽  
Author(s):  
Ivan Wang ◽  
Samuel C. Gibbs ◽  
Earl H. Dowell
Keyword(s):  
Boundary Conditions ◽  
Structural Model ◽  
Three Dimensional ◽  
Flexible Plate ◽  
Rectangular Plates ◽  
Elastic Plates ◽  
Test Bed ◽  
Trailing Edge ◽  
Various Boundary Conditions ◽  
Trailing Edge Flap

The aeroelastic stability of rectangular plates are well-documented in literature for certain sets of boundary conditions. Specifically, wing flutter, panel flutter, and divergence of a plate that is clamped on all sides are well-understood. However, the ongoing push for lighter structures and novel designs have led to a need to understand the aeroelastic behavior of elastic plates for other boundary conditions. One example is NASA’s continuous mold-line link project for reducing the noise generated by commercial transport aircraft during landing; in order to reduce the noise generated by vortex shedding from the trailing edge flap during landing, the project proposes to connect the gap between the trailing edge flap and the rest of the wing with a flexible plate. This paper summarizes the aeroelastic theory, numerical results, and experimental results of a study on the flutter and/or divergence mechanisms of a rectangular plate for different sets of structural boundary conditions. The theory combines a three-dimensional vortex lattice aerodynamic model with a plate structural model to create a high-fidelity frequency domain aeroelastic model. A modular experimental test bed is designed for this study in order to test the different boundary conditions. The test bed is also designed to test different plate thicknesses and sizes with only a small number of modifications. The well-understood boundary conditions are used as test cases to validate the analysis results, and then results of additional configurations that have not been extensively explored are presented. The results of this paper can be used to support the design efforts of projects involving plates or plate-membranes. In addition, the paper adds to the fundamental understanding of plate aeroelasticity and provides a wealth of experimental data for comparison and future validation.

The Effect of Transverse Shear Deformation on the Bending of Elastic Plates

1945 ◽  
Vol 12 (2) ◽  
pp. A69-A77 ◽  
Author(s):  
Eric Reissner
Keyword(s):  
General Solution ◽  
Rectangular Plate ◽  
Transverse Shear ◽  
Plate Theory ◽  
Infinite Plate ◽  
Present Theory ◽  
System Of Equations ◽  
Elastic Plates ◽  
Classical Plate Theory

Abstract A system of equations is developed for the theory of bending of thin elastic plates which takes into account the transverse shear deformability of the plate. This system of equations is of such nature that three boundary conditions can and must be prescribed along the edge of the plate. The general solution of the system of equations is obtained in terms of two plane harmonic functions and one function which is the general solution of the equation Δψ − (10/h2)ψ = 0. The general results of the paper are applied (a) to the problem of torsion of a rectangular plate, (b) to the problems of plain bending and pure twisting of an infinite plate with a circular hole. In these two problems important differences are noted between the results of the present theory and the results obtained by means of the classical plate theory. It is indicated that the present theory may be applied to other problems where the deviations from the results of classical plate theory are of interest. Among these other problems is the determination of the reactions along the edges of a simply supported rectangular plate, where the classical theory leads to concentrated reactions at the corners of the plate. These concentrated reactions will not occur in the solution of the foregoing problem by means of the theory given in the present paper.

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