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.

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.

scholarly journals Structural Imposed Load Analysis of Isotropic Rectangular Plate Carrying a Uniformly Distributed Load Using Refined Shear Plate Theory

2021 ◽  
Vol 6 (4) ◽  
Author(s):  
Festus C. Onyeka ◽  
Chidoebere D. Nwa-David ◽  
Emmanuel E. Arinze
Keyword(s):  
Shear Deformation ◽  
Rectangular Plate ◽  
Plate Theory ◽  
Plate Thickness ◽  
Variational Calculus ◽  
Present Theory ◽  
Numerical Comparison ◽  
Analysis And Design ◽  
Distributed Load ◽  
Uniformly Distributed Load

This presents the static flexural analysis of a three edge simply supported, one support free (SSFS) rectangular plate under uniformly distributed load using a refined shear deformation plate theory. The shear deformation profile used, is in the form of third order. The governing equations were determined by the method of energy variational calculus, to obtain the deflection and shear deformation along the direction of x and y axis. From the formulated expression, the formulars for determination of the critical lateral imposed load of the plate before deflection reaches the specified maximum specified limit  and its corresponding critical lateral imposed load before plate reaches an elastic yield stress  is established. The study showed that the critical lateral imposed load decreased as the plates span increases, the critical lateral imposed load increased as the plate thickness increases, as the specified thickness of the plate increased, the value of critical lateral imposed load increased and increase in the value of the allowable deflection value required for the analysis of the plate reduced the chances of failure of a structural member. This approach overcomes the challenges of the conventional practice in the structural analysis and design which involves checking of deflection and shear after design; the process which is proved unreliable and time consuming. It is concluded that the values of critical lateral load obtained by this theory achieve accepted transverse shear stress to the depth of the plate variation in predicting the flexural characteristics for an isotropic rectangular SSFS plate. Numerical comparison was conducted to verify and demonstrate the efficiency of the present theory, and they agreed with previous studies. This proved that the present theory is reliable for the analysis of a rectangular plate. Keywords— Allowable deflection, critical imposed load, energy method, plate theories, shear deformation, SSFS rectangular plate

Experimental studies on the dynamic response of thin rectangular plates subjected to moving mass

2020 ◽  
pp. 107754632093313 ◽  
Author(s):  
Sajjad Seifoori ◽  
Ahmad Mahdian Parrany ◽  
Sajjad Darvishinia
Keyword(s):  
Dynamic Response ◽  
Rectangular Plate ◽  
Plate Theory ◽  
Experimental Studies ◽  
Time History ◽  
Moving Mass ◽  
Rectangular Plates ◽  
Classical Plate Theory ◽  
Thin Rectangular Plate ◽  
Expansion Technique

This article presents experimental studies on the dynamic response of a thin rectangular plate with clamped boundary conditions subjected to a moving mass. The designed experimental setup is described in detail, and the obtained experimental results are compared with theoretical solutions. In this regard, the governing motion equation of the thin rectangular plate excited by a moving mass is formulated based on the classical plate theory, and the eigenfunction expansion technique is used to solve the equation. Parametric studies are carried out to investigate the effect of some parameters, including the moving object mass and velocity, as well as the plate’s aspect ratio and thickness, on the dynamic response of the plate based on the time history of the plate’s central point deflection.

EXACT SOLUTION FOR BENDING OF AN ELASTIC PLATE CONTAINING A CRACK AND SUBJECTED TO A CONCENTRATED MOMENT

2007 ◽  
Vol 04 (02) ◽  
pp. 265-281
Author(s):  
LALITHA CHATTOPADHYAY ◽  
S. SRIDHARA MURTHY ◽  
S. VISWANATH
Keyword(s):  
Stress Intensity Factor ◽  
Elastic Plate ◽  
Integral Transform ◽  
Plate Theory ◽  
Infinite Plate ◽  
Bending Moment ◽  
Single Line ◽  
Bending Stress ◽  
Single Crack ◽  
Classical Plate Theory

The problem of estimating the bending stress distribution in the vicinity of cracks located on a single line in an elastic plate subjected to concentrated moment is examined. Using classical plate theory and integral transform techniques, the general formulae for the bending moment and twisting moment in an elastic plate containing cracks located on a single line are derived. The solution is obtained in detail for the case in which there is a single crack in an infinite plate, and the bending stress intensity factor is determined in a closed form. Two examples are considered to illustrate the present approach.

Transverse bending of infinite and semi-infinite thin elastic plates. III

1958 ◽  
Vol 54 (2) ◽  
pp. 288-299 ◽  
Author(s):  
W. A. Bassali ◽  
M. Nassif ◽  
H. P. F. Swinnerton-Dyer
Keyword(s):  
Isotropic Plate ◽  
Plate Theory ◽  
Outer Boundary ◽  
Exact Expression ◽  
Transverse Displacement ◽  
Elastic Plates ◽  
Classical Plate Theory ◽  
Transverse Bending ◽  
Elastically Restrained ◽  
Limiting Case

ABSTRACTWithin the restrictions of the classical plate theory, complex variable methods are used in this paper to develop an exact expression for the transverse displacement of an infinitely large isotropic plate having a free outer boundary and elastically restrained at an inner circular boundary, the plate being subjected to a general type of loading distributed over the area of a circle. The limiting case of a half-plane clamped along the straight edge and acted upon normally by the same loading is also considered.

A New First-Order Shear Deformation Theory for Free Vibrations of Rectangular Plate

2015 ◽  
Vol 07 (01) ◽  
pp. 1550008 ◽  
Author(s):  
Wei Xiang ◽  
Yufeng Xing
Keyword(s):  
Shear Deformation ◽  
Rectangular Plate ◽  
Deformation Theory ◽  
Plate Theory ◽  
Shear Deformation Theory ◽  
Present Theory ◽  
Free Vibrations ◽  
Variable Theory ◽  
First Order ◽  
Geometrical Constraints

A new first-order shear deformation theory (FSDT) with pure bending deflection and shearing deflection as two independent variables is presented in this paper for free vibrations of rectangular plate. In this two-variable theory, the shearing deflection is regarded as the only fundamental variable by which the total deflection and bending deflection can be expressed explicitly. In contrast with the conventional three-variable first-order shear plate theory, present variationally consistent theory derived by using Hamiltonian variational principle can uniquely define the bending and the shearing deflections, and give two rotations by the differentiations of bending deflection. Due to more restrictive geometrical constraints on rotations and boundary conditions, the obtained natural frequencies are equal to or higher than those by conventional FSDT for the rectangular plate with at least one pair of opposite edges simply supported. This new theory is of considerable significance in theoretical sense for giving a simple two-variable FSDT which is variational consistent and involve rotary inertia and shear deformation. The relation and differences of present theory with conventional FSDT and other relative formulations are discussed in detail.

scholarly journals Transient analysis of laminated composite plates using NURBS-based isogeometric analysis

2014 ◽  
Vol 36 (4) ◽  
pp. 267-281
Author(s):  
Lieu B. Nguyen ◽  
Chien H. Thai ◽  
Ngon T. Dang ◽  
H. Nguyen-Xuan
Keyword(s):  
Numerical Solutions ◽  
Plate Theory ◽  
Shear Deformation Theory ◽  
Laminated Composite ◽  
Composite Plates ◽  
Present Theory ◽  
Response Analysis ◽  
Laminated Composite Plates ◽  
Analysis Model ◽  
Classical Plate Theory

We further study isogeometric approach for response analysis of laminated composite plates using the higher-order shear deformation theory. The present theory is derived from the classical plate theory (CPT) and the shear stress free surface conditions are naturally satisfied. Therefore, shear correction factors are not required. Galerkin weak form of response analysis model for laminated composite plates is used to obtain the discrete system of equations. It can be solved by isogeometric approach based on the non-uniform rational B-splines (NURBS) basic functions. Some numerical examples of the laminated composite plates under various dynamic loads, fiber orientations and lay-up numbers are provided. The accuracy and reliability of the proposed method is verified by comparing with analytical solutions, numerical solutions and results from Ansys software.

Flexural Stress Waves in an Infinite Elastic Plate Due to a Suddenly Applied Concentrated Transverse Load

1960 ◽  
Vol 27 (4) ◽  
pp. 681-689 ◽  
Author(s):  
Julius Miklowitz
Keyword(s):  
Shear Force ◽  
Plate Theory ◽  
Three Dimensional ◽  
Infinite Plate ◽  
Present Theory ◽  
Transverse Load ◽  
Phase Method ◽  
Stationary Phase Method ◽  
Flexural Stress ◽  
The Moment

The problem solved is that of an infinite plate subjected to a suddenly applied concentrated transverse shear load. The solution is derived from a plate theory that incorporates, in addition to bending, the effect of shear force and rotatory inertia on the deflection. These added effects give the present theory true wave character along with greater accuracy in the waves predicted. Numerical evaluation of the solution brings out the effects of dispersion and distortion on the moment and shear-force response of the plate. A criterion is developed for judging the accuracy of this response. It is based on a comparison, employing the stationary phase method, of the present approximate and exact (three-dimensional) theories.

Experimental Investigation of Nonlinear Vibration of a Thin Rectangular Plate

2019 ◽  
Vol 11 (06) ◽  
pp. 1950059 ◽  
Author(s):  
Sohayb Abdulkerim ◽  
Athanasios Dafnis ◽  
Hans-G Riemerdes
Keyword(s):  
Rectangular Plate ◽  
Plate Theory ◽  
Plate Thickness ◽  
Extraction Procedure ◽  
Time History ◽  
Laser Vibrometer ◽  
Classical Plate Theory ◽  
Thin Rectangular Plate ◽  
Sinusoidal Input ◽  
History Of

In this paper, the geometrical nonlinear vibrations of a rectangular plate have been investigated experimentally and numerically. The experiment was conducted on a thin rectangular plate. The plate was excited close to the first fundamental natural frequency. The time history of velocities of the central point has been measured by using a laser vibrometer. While the numerical investigation has been carried using the Finite Element Method (FEM), the numerical results are validated by analytical and experimental results. In order to develop and test the extraction procedure of the bifurcation plot of a dynamical system, a chaotic pendulum has been analyzed. Then, the same successful code has been used again for the experimental dynamics of the investigated plate. The plate has been forced with a sinusoidal input at a gradually stepped and increased amplitude. For every step, the phase portrait is determined, and then processed to extract the bifurcation map. The resulted map has shown successfully the linear range where the classical plate theory is adequate, and the boundary at which the transition to nonlinearity has occurred. The bifurcation has occurred when the lateral amplitude has reached 50% of the plate thickness.

Sign in / Sign up

Export Citation Format

Share Document