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.

Postbuckling Behavior of Rectangular Plates With Small Initial Curvature Loaded in Edge Compression—(continued)

1960 ◽  
Vol 27 (2) ◽  
pp. 335-342 ◽  
Author(s):  
Noboru Yamaki
Keyword(s):  
Stress State ◽  
Boundary Conditions ◽  
Ultimate Load ◽  
Numerical Solutions ◽  
Effective Width ◽  
Rectangular Plates ◽  
Postbuckling Behavior ◽  
Shear Theory ◽  
Title Problem ◽  
Square Plate

In the previous paper [1], the title problem is theoretically treated under eight different boundary conditions and numerical solutions are obtained for the deflection, edge shortening, and effective width of the square plate in edgewise compression. As a continuation of this work, the stress state in the buckled plate is investigated and numerical results for the square plate are given graphically. Further the formulas for the ultimate load of the square plate in each case are derived by using the maximum-shear theory for the beginning of yielding and comparison is made with the previous results and experiments.

Thin Rectangular Plates on Elastic Foundation

1952 ◽  
Vol 19 (3) ◽  
pp. 361-368
Author(s):  
H. J. Fletcher ◽  
C. J. Thorne
Keyword(s):  
Boundary Conditions ◽  
Rectangular Plate ◽  
Elastic Foundation ◽  
Numerical Solutions ◽  
The Other ◽  
Rectangular Plates ◽  
Transverse Loads ◽  
Sine Transform ◽  
Special Cases ◽  
Plates On Elastic Foundation

Abstract The deflection of a thin rectangular plate on an elastic foundation is given for the case in which two opposite edges have arbitrary but given deflections and moments. Six important cases of boundary conditions on the remaining two edges are treated. The solution is given for general transverse loads which are continuous in one direction and sectionally continuous in the other. By use of the sine transform the solution is obtained as a single trigonometric series. Numerical solutions are obtained for six special cases.

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.

Semi-Analytical Solutions for the Uniformly Compressed Simply Supported Plate with Large Deflections

2020 ◽  
Vol 20 (09) ◽  
pp. 2050108
Author(s):  
Mihai Nedelcu
Keyword(s):  
Analytical Solutions ◽  
Numerical Solutions ◽  
Thin Plates ◽  
Energy Methods ◽  
Large Deflections ◽  
Software Application ◽  
Simply Supported ◽  
Thin Walled Structures ◽  
Thin Walled ◽  
Fundamental Equations

The thin plane plates are largely used in practice as single elements or as components of the thin-walled structures, and their behavior under compression is characterized by large post-buckling load-carrying capacity. Various semi-analytical solutions of the uniformly compressed simply supported plate with large deflections were formulated almost a century ago, mainly solving the fundamental equations governing the deformation of thin plates, or using classic energy methods. Due to several shortcomings, none of these solutions were introduced in the design codes of thin-walled members. This paper presents new semi-analytical solutions based on classic energy methods. The main innovation is brought by the considered displacement field which is far more accurate than the ones used by the previous formulations. The initial geometric imperfections are considered, and the proposed solutions are validated against numerical solutions and experimental data. The validation is also supported by a publicly available software application.

Experiments on the Postbuckling Behavior of Square Plates Loaded in Edge Compression

1961 ◽  
Vol 28 (2) ◽  
pp. 238-244 ◽  
Author(s):  
Noboru Yamaki
Keyword(s):  
Boundary Conditions ◽  
Reasonable Agreement ◽  
Numerical Solutions ◽  
Experimental Results ◽  
Initial Deflection ◽  
Rectangular Plates ◽  
Maximum Deflection ◽  
Postbuckling Behavior ◽  
Different Boundary Conditions

In previous papers [1, 2], the postbuckling behavior of rectangular plates under edge compression has been studied theoretically under eight different boundary conditions, and numerical solutions are presented for square plates with and without initial deflection. To compare with these results, experiments were carried out by using aluminum square plates, and the relations of the maximum deflection, stresses, and strains at typical points in the plate with applied loads were determined under four different boundary conditions. It is found that the experimental results are in reasonable agreement with those theoretically predicted.

Buckling of the Circular Plate Beyond the Critical Thrust

1942 ◽  
Vol 9 (1) ◽  
pp. A7-A14
Author(s):  
K. O. Friedrichs ◽  
J. J. Stoker
Keyword(s):  
Circular Plate ◽  
Limit State ◽  
Radial Pressure ◽  
Critical Value ◽  
Radial Symmetry ◽  
Thin Plates ◽  
Large Deflections ◽  
Effective Width ◽  
Rectangular Plates ◽  
Mathematical Solution

Abstract In this paper a complete mathematical solution of the problem of the buckling with large deflections of a thin circular plate under uniform radial pressure is given, assuming radial symmetry. Buckling takes place as soon as the prescribed thrust pe at the edge reaches a certain critical value pE. Thin plates do not, however, fail if the thrust pe is increased beyond pE. It is of importance, then, to determine the stresses when the ratio Λ = pe/pE becomes greater than unity. This problem, which is a nonlinear one, is solved by two methods for finite values of Λ; also an asymptotic solution is given for the limit state when Λ tends to infinity. The most notable single result is that the membrane stresses for large values of Λ become tensions in the interior of the plate and change abruptly to compressions in a narrow “boundary layer” at the edge of the plate. Curves showing the behavior of the deflection and the stresses are given for a series of values of Λ. Such curves show clearly, in particular, that the limit state is approached quite closely for relatively small values of Λ, i.e., Λ > 5. In closing, the authors discuss the relation between their results and von Kármán’s theory of effective width for the buckling of rectangular plates.

On Nonuniqueness and Stability in Barber’s Model of Thermoelastic Contact

1996 ◽  
Vol 63 (3) ◽  
pp. 582-586 ◽  
Author(s):  
Z. S. Olesiak ◽  
Yu. A. Pyryev
Keyword(s):  
Boundary Conditions ◽  
Thermal Resistance ◽  
Numerical Solutions ◽  
Computational Simulation ◽  
Nonstationary Problem ◽  
Number Of Solutions ◽  
Thermoelastic Contact ◽  
Resistance Function ◽  
Special Cases ◽  
Unstable Solutions

In the recent our paper we have derived an algorithm which lets us find numerical solutions of special cases of a nonstationary problem of thermoelasticity with imperfect boundary conditions of Barber’s model. In this paper we show the results of computational simulation for a case of the thermal resistance function. We have obtained a number of solutions for different situations and have discussed the unique, nonunique, stable, and unstable solutions. We have found cases when the unique and nonunique solutions alternate. The results have been presented in the form of diagrams.

Deflection of Clamped Plates by Two-Step Membrane Analogue

1955 ◽  
Vol 59 (533) ◽  
pp. 358-360 ◽  
Author(s):  
V. Cadambe ◽  
R. K. Kaul
Keyword(s):  
Differential Equation ◽  
Numerical Methods ◽  
General Solution ◽  
Fourth Order ◽  
Biharmonic Equation ◽  
Numerical Solutions ◽  
Thin Plates ◽  
Geometrical Shape ◽  
Marked Contrast ◽  
Special Cases

The Classical Kirchhofif–Love Theory for the deflection of thin plates leads to fourth order Lagrange's differential equation,D△4w — q = 0 for which a general solution is not always possible. Exact solutions are known so far only for a few special cases and, therefore, numerical solutions have often been tried. The advantage of numerical solution is that it can be applied easily to any plate plan form which is in marked contrast to the analytical method where, for mathematical reasons, definite restrictions have to be imposed on the geometrical shape of the plate. Among the various numerical methods, relaxation is the easiest, but when applied to solving a biharmonic equation, the process becomes extremely difficult and laborious as convergence is very slow and the unit relaxation operator cumbersome to deal with.

FREE VIBRATION OF FUNCTIONALLY GRADED THIN RECTANGULAR PLATES RESTING ON WINKLER ELASTIC FOUNDATION WITH GENERAL BOUNDARY CONDITIONS USING RAYLEIGH–RITZ METHOD

2014 ◽  
Vol 06 (04) ◽  
pp. 1450043 ◽  
Author(s):  
S. CHAKRAVERTY ◽  
K. K. PRADHAN
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Elastic Foundation ◽  
Power Law ◽  
Ritz Method ◽  
Functionally Graded ◽  
Rectangular Plates ◽  
Aspect Ratios ◽  
Winkler Elastic Foundation ◽  
Special Cases

In this paper, free vibration of functionally graded (FG) rectangular plates subject to different sets of boundary conditions within the framework of classical plate theory is investigated. Rayleigh–Ritz method is used to obtain the generalized eigenvalue problem. Trial functions denoting the displacement components are expressed in simple algebraic polynomial forms which can handle any sets of boundary conditions. Material properties of the FG plate are assumed to vary continuously in the thickness direction of the constituents according to power-law form. The objective is to study the effects of constituent volume fractions, aspect ratios and power-law indices on the natural frequencies. New results for frequency parameters are incorporated after performing a test of convergence. Comparison with the results from the existing literature are provided for validation in special cases. Three-dimensional mode shapes are presented for FG square plates having various boundary conditions at the edges for different power-law indices. The present investigation also involves the rectangular FG plate to lay on a uniform Winkler elastic foundation. New results for the eigenfrequencies associated with foundation parameters are also reported here with the validation in special cases after checking a convergence pattern.

Three-dimensional static analysis of rectangular thick plates by using the meshless local Petrov—Galerkin method

2009 ◽  
Vol 223 (9) ◽  
pp. 1983-1996 ◽  
Author(s):  
R Vaghefi ◽  
G H Baradaran ◽  
H Koohkan
Keyword(s):  
Boundary Conditions ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
3D Analysis ◽  
Rectangular Plates ◽  
Weak Formulation ◽  
Heaviside Step Function ◽  
Thick Plates ◽  
Test Function ◽  
Aspect Ratios

In this article, a meshless local Petrov—Galerkin (MLPG) approach is developed for three-dimensional (3D) analysis of thick plates. Two different MLPG methods including MLPG1 and MLPG5 are employed to solve the elasto-static problems of thick plates. In MLPG1, a namely fourth-order spline function is considered as test function, while the Heaviside step function is employed as test function in MLPG5. Considering 3D equilibrium equations, the local symmetric weak forms are derived. The moving least-squares approximation is used to interpolate the solution variables and the penalty method is applied to impose the essential boundary conditions. In the present study, brick-shaped domains are chosen as local subdomains and support domains. The integrals appearing in the weak formulation are easily evaluated over brick-shaped subdomains and their boundaries. Considering the present approach, elasto-static deformations and stresses are analysed for thick rectangular plates with various boundary conditions and different aspect ratios. Excellent agreement is seen comparing the present results with the known analytical and numerical solutions in the literature.

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