scholarly journals A New Approach to the Analysis of Large Deflections of Plates

1955 ◽  
Vol 22 (4) ◽  
pp. 465-472
Author(s):  
H. M. Berger
Keyword(s):  
Boundary Conditions ◽  
Exact Solutions ◽  
Flat Plate ◽  
Strain Energy ◽  
Numerical Solutions ◽  
Middle Surface ◽  
Large Deflections ◽  
Rectangular Plates ◽  
New Approach ◽  
Various Boundary Conditions

Abstract Simplified nonlinear equations for a flat plate with large deflections are derived by assuming that the strain energy due to the second invariant of the middle-surface strains can be neglected. Computations using the solution of these simplified equations are carried out for the deflection of uniformly loaded circular and rectangular plates with various boundary conditions. Comparisons are made with available numerical solutions of the exact equations. The deflections found by this approach are then used to obtain the stresses for the circular plate and the resulting stresses are compared with existing solutions. In all the cases where comparisons could be made, the deflections and stresses agree with the exact solutions within the accuracy required for engineering purposes.

Double Fourier series and boundary value problems

1944 ◽  
Vol 40 (3) ◽  
pp. 222-228 ◽  
Author(s):  
A. E. Green
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Rectangular Plate ◽  
Strain Energy ◽  
Double Fourier Series ◽  
Energy Methods ◽  
Rectangular Plates ◽  
Simply Supported ◽  
Points Of View

1. Problems in elasticity which are concerned with isotropic rectangular plates have attracted the attention of many writers both from the theoretical and practical points of view. When the boundary conditions are of the simply supported type the solution of the problems is usually simple, although when double Fourier series are used the validity of such solutions is not very clearly shown in most cases. Satisfactory exact solutions for many classical problems in which the edges of the rectangular plate are clamped have only been obtained in recent years, but approximate strain energy methods often gave results which were useful for practical purposes.

Postbuckling Behavior of Rectangular Plates With Small Initial Curvature Loaded in Edge Compression

1959 ◽  
Vol 26 (3) ◽  
pp. 407-414
Author(s):  
Noboru Yamaki
Keyword(s):  
Boundary Conditions ◽  
Numerical Solutions ◽  
Thin Plates ◽  
Large Deflections ◽  
Effective Width ◽  
Rectangular Plates ◽  
Postbuckling Behavior ◽  
Initial Curvature ◽  
Special Cases ◽  
Fundamental Equations

Abstract The solution of Marguerre’s fundamental equations for large deflections of thin plates with slight initial curvature is presented for the case of a rectangular plate subjected to edge compression. The problem is solved under eight different boundary conditions, combining two kinds of loading conditions and four kinds of supporting conditions. Numerical solutions are obtained for square plates with and without initial deflection, and the connections of deflection, edge shortening, and effective width of the plate with applied loads are clarified. The solutions here obtained include as special cases those investigated by Levy and Coan.

Static Analysis of Moderately Thick Functionally Graded Plates With Various Boundary Conditions

2010 ◽  
Author(s):  
Nastaran Shahmansouri ◽  
Mohammad Mohammadi Aghdam ◽  
Kasra Bigdeli
Keyword(s):  
Boundary Conditions ◽  
Volume Fraction ◽  
Closed Form Solution ◽  
Shear Deformation Theory ◽  
Functionally Graded ◽  
Rectangular Plates ◽  
Functionally Graded Plates ◽  
Various Boundary Conditions ◽  
Fg Plates ◽  
Ode Systems

The present study investigates static analyses of moderately thick FG plates. Using the First Order Shear Deformation Theory (FSDT), functionally graded plates subjected to transversely distributed loading with various boundary conditions are studied. Effective mechanical properties which vary from one surface of the plate to the other assumed to be defined by a power law form of distribution. Different ceramic-metal sets of materials are studied. Solution of the governing equations, including five equilibrium and eight constitutive equations, is obtained by the Extended Kantorovich Method (EKM). The system of thirteen Partial Differential Equations (PDEs) in terms of displacements, rotations, force and moment resultants are considered as multiplications of separable function of independent variables x and y. Then by successful utilization of the EKM these equations are converted to a double set of ODE systems in terms of x and y. The obtained ODE systems are then solved iteratively until final convergence is achieved. Closed form solution is presented for these ODE sets. It is shown that the method is very stable and provides fast convergence and highly accurate predictions for both thin and moderately thick plates. Comparison of the normal stresses at various points of rectangular plates and deflection of mid-point of the plate are presented and compared with available data in the literature. The effects of the volume fraction exponent n on the behavior of the normalized deflection, moment resultants and stresses of FG plates are also studied. To validate data for analysis fully clamped FG plates, another analysis was carried out using finite element code ANSYS. Close agreement is observed between predictions of the EKM and ANSYS.

Vibration of Stiffened Rectangular Plates

1963 ◽  
Vol 67 (629) ◽  
pp. 305-307 ◽  
Author(s):  
S. Mahalingam
Keyword(s):  
Boundary Conditions ◽  
Present Note ◽  
Ritz Method ◽  
Arbitrary Parameter ◽  
Rectangular Plates ◽  
Various Boundary Conditions ◽  
Beam Function ◽  
Rayleigh Method ◽  
The Given ◽  
Free Edges

The free flexural vibrations of rectangular plates with various boundary conditions have been considered by Warburton. The natural frequencies were calculated by the Rayleigh method, the mode assumed being the product of the characteristic beam functions for the given boundary conditions. Comparison with experimental results shows that the method gives reasonably good approximations. The present note describes a method of obtaining the approximately equivalent characteristic beam functions to enable Warburton's method to be extended to plates having one or more stiffeners parallel to an edge. As a numerical example expressions for the frequencies are derived for a plate, simply supported along two opposite edges, and having a central stiffener parallel to the other two free edges. The results are compared with those given in a recent note by Kirk, who solved the same problem by the Rayleigh-Ritz method, using a mode with one arbitrary parameter. In the case of the fundamental frequency of the unstiffened plate, the characteristic beam function in a direction perpendicular to the free edges is simply a constant, and the solution is less accurate than that given by the Rayleigh-Ritz method. However, numerical analysis of a square plate shows that above a certain stiffener depth the characteristic beam function method is more accurate than the Rayleigh-Ritz method. The two methods are also compared for the 2/2 mode.

Analytical Solutions of Refined Plate Theories of Cross-Ply Composite Laminates

1991 ◽  
Vol 113 (4) ◽  
pp. 570-578 ◽  
Author(s):  
A. A. Khdeir ◽  
J. N. Reddy
Keyword(s):  
Boundary Conditions ◽  
Exact Solutions ◽  
Composite Laminates ◽  
Laminated Composite ◽  
Composite Plates ◽  
Third Order ◽  
State Space Approach ◽  
Various Boundary Conditions ◽  
Shear Deformation Theories ◽  
Space Approach

Exact solutions of rectangular laminated composite plates with different boundary conditions are studied. The Le´vy-type solutions of the classical, first-order and third-order shear deformation theories are developed using the state-space approach. The finite-element solutions for the three theories are also computed and compared with the exact solutions for various boundary conditions.

LINEAR BENDING ANALYSIS OF INHOMOGENEOUS VARIABLE-THICKNESS ORTHOTROPIC PLATES UNDER VARIOUS BOUNDARY CONDITIONS

2007 ◽  
Vol 04 (03) ◽  
pp. 417-438 ◽  
Author(s):  
A. M. ZENKOUR ◽  
M. N. M. ALLAM ◽  
D. S. MASHAT
Keyword(s):  
Boundary Conditions ◽  
Variable Thickness ◽  
Flexural Rigidity ◽  
Approximate Solutions ◽  
Rectangular Plates ◽  
Orthotropic Plates ◽  
Simply Supported ◽  
Various Boundary Conditions ◽  
Thickness Parameter ◽  
Space Concept

An exact solution to the bending of variable-thickness orthotropic plates is developed for a variety of boundary conditions. The procedure, based on a Lévy-type solution considered in conjunction with the state-space concept, is applicable to inhomogeneous variable-thickness rectangular plates with two opposite edges simply supported. The remaining ones are subjected to a combination of clamped, simply supported, and free boundary conditions, and between these two edges the plate may have varying thickness. The procedure is valuable in view of the fact that tables of deflections and stresses cannot be presented for inhomogeneous variable-thickness plates as for isotropic homogeneous plates even for commonly encountered loads because the results depend on the inhomogeneity coefficient and the orthotropic material properties instead of a single flexural rigidity. Benchmark numerical results, useful for the validation or otherwise of approximate solutions, are tabulated. The influences of the degree of inhomogeneity, aspect ratio, thickness parameter, and the degree of nonuniformity on the deflections and stresses are investigated.

Post-Buckling of Rectangular Plates with Various Boundary Conditions

1970 ◽  
Vol 21 (2) ◽  
pp. 163-181 ◽  
Author(s):  
K. R. Rushton
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Relaxation Method ◽  
Rectangular Plates ◽  
Dynamic Relaxation ◽  
Various Boundary Conditions ◽  
Post Buckling ◽  
Dynamic Relaxation Method ◽  
Theoretical Results

SummaryThe Dynamic Relaxation method is used to analyse the post-buckling of flat rectangular plates. Extensive results are obtained for two types of problem in which the transverse edges are unloaded; in one case the transverse edges remain straight, in the other they are free to wave. Comparisons are made with alternative experimental and theoretical results. The potentiality of this approach to post-buckling problems is demonstrated by considering a variable thickness plate with mixed boundary conditions.

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