The Bending, Buckling, and Flexural Vibration of Simply Supported Polygonal Plates by Point-Matching

1961 ◽  
Vol 28 (2) ◽  
pp. 288-291 ◽  
Author(s):  
H. D. Conway
Keyword(s):  
Hydrostatic Pressure ◽  
Boundary Conditions ◽  
Flexural Vibration ◽  
Frequency Equation ◽  
Numerical Results ◽  
Special Technique ◽  
Point Matching ◽  
Buckling Loads ◽  
Simply Supported ◽  
Flexural Vibrations

The bending by uniform lateral loading, buckling by two-dimensional hydrostatic pressure, and the flexural vibrations of simply supported polygonal plates are investigated. The method of meeting the boundary conditions at discrete points, together with the Marcus membrane analog [1], is found to be very advantageous. Numerical examples include the calculation of the deflections and moments, and buckling loads of triangular square, and hexagonal plates. A special technique is then given, whereby the boundary conditions are exactly satisfied along one edge, and an example of the buckling of an isosceles, right-angled triangle plate is analyzed. Finally, the frequency equation for the flexural vibrations of simply supported polygonal plates is shown to be the same as that for buckling under hydrostatic pressure, and numerical results can be written by analogy. All numerical results agree well with the exact solutions, where the latter are known.

Triangular Plates Analyzed by Point Matching

1962 ◽  
Vol 29 (4) ◽  
pp. 755-756 ◽  
Author(s):  
H. D. Conway
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Numerical Results ◽  
Point Matching ◽  
Simply Supported ◽  
Special Adaptation ◽  
Matching Technique

This brief note analyzes uniformly loaded triangular plates with either clamped or simply supported edges using a special adaptation of the point-matching technique, the functions satisfying the differential equation, also being chosen to satisfy exactly the boundary conditions on one edge. Numerical results are tabulated for three geometries.

Analytical solution for the buckling of rectangular plates under uni-axial compression with variable thickness and elasticity modulus in the y-direction

2009 ◽  
Vol 224 (1) ◽  
pp. 33-41 ◽  
Author(s):  
M Saeidifar ◽  
S N Sadeghi ◽  
M R Saviz
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Power Series ◽  
Boundary Conditions ◽  
Variable Thickness ◽  
Elasticity Modulus ◽  
Plate Motion ◽  
Transverse Displacement ◽  
Buckling Loads ◽  
Simply Supported

The present study introduces a highly accurate numerical calculation of buckling loads for an elastic rectangular plate with variable thickness, elasticity modulus, and density in one direction. The plate has two opposite edges ( x = 0 and a) simply supported and other edges ( y = 0 and b) with various boundary conditions including simply supported, clamped, free, and beam (elastically supported). In-plane normal stresses on two opposite simply supported edges ( x = 0 and a) are not limited to any predefined mathematical equation. By assuming the transverse displacement to vary as sin( mπ x/ a), the governing partial differential equation of plate motion will reduce to an ordinary differential equation in terms of y with variable coefficients, for which an analytical solution is obtained in the form of power series (Frobenius method). Applying the boundary conditions on ( y = 0 and b) yields the problem of finding eigenvalues of a fourth-order characteristic determinant. By retaining sufficient terms in power series, accurate buckling loads for different boundary conditions will be calculated. Finally, the numerical examples have been presented and, in some cases, compared to the relevant numerical results.

Effects of Boundary Conditions and Initial Out-of-Roundness on the Strength of Thin-Walled Cylinders Subject to External Hydrostatic Pressure

1956 ◽  
Vol 23 (3) ◽  
pp. 351-358
Author(s):  
G. D. Galletly ◽  
R. Bart
Keyword(s):  
Hydrostatic Pressure ◽  
Boundary Conditions ◽  
Semiempirical Methods ◽  
Test Results ◽  
External Hydrostatic Pressure ◽  
Simply Supported ◽  
Small Deflection ◽  
The Difference ◽  
Deflection Theory ◽  
Theoretical Results

Abstract Using classical small-deflection theory, an investigation was made of the effects of boundary conditions and initial out-of-roundness on the strength of cylinders subject to external hydrostatic pressure. The equations developed in this paper for initially out-of-round cylinders with clamped ends, and a slightly modified form of the equations previously derived by Bodner and Berks for simply supported ends, were applied to some actual test results obtained from nine steel cylinders which had been subjected to external hydrostatic pressure. Three semiempirical methods for determining the initial out-of-roundness of the cylinders also were investigated and these are described in the paper. The investigation indicates that if the initial out-of-roundness is determined in a manner similar to that suggested by Holt then the correlation between the experimental and theoretical results is quite good. The investigation also indicates that while the difference in collapse pressures for clamped-end and simply supported perfect cylinders may be quite considerable, this does not appear to be the case when initial out-of-roundnesses of a practical magnitude are considered.

Buckling of Short Cylinders Under Combined Loading

1978 ◽  
Vol 45 (3) ◽  
pp. 574-578 ◽  
Author(s):  
R. C. Tennyson ◽  
M. Booton ◽  
K. H. Chan
Keyword(s):  
Experimental Data ◽  
Hydrostatic Pressure ◽  
Boundary Conditions ◽  
Combined Loading ◽  
Circular Cylinders ◽  
Simply Supported ◽  
Interactive Behavior ◽  
Major Significance ◽  
Z Parameter ◽  
Simultaneous Loading

This report presents theoretical and experimental data on the buckling of short, homogeneous, isotropic circular cylinders subjected to simultaneous loading of axial compression and hydrostatic pressure. Of major significance is the drastic change from “concave” to “convex” interactive behavior as the Z parameter is decreased. This phenomenon is demonstrated for both clamped and simply supported boundary conditions.

A Set of Orthogonal Plate Functions for Flexural Vibration of Regular Polygonal Plates

1991 ◽  
Vol 113 (2) ◽  
pp. 182-186 ◽  
Author(s):  
K. M. Liew ◽  
K. Y. Lam
Keyword(s):  
Boundary Conditions ◽  
Vibration Analysis ◽  
Mixed Boundary ◽  
Flexural Vibration ◽  
Mixed Boundary Conditions ◽  
Mode Shapes ◽  
Two Dimensional ◽  
Computationally Efficient ◽  
Simply Supported ◽  
Free Boundary Conditions

A computationally efficient and very accurate numerical method is proposed for vibration analysis of regular polygonal plates with any combinations of clamped, simply-supported and free boundary conditions. The method involves the use of two-dimensional orthogonal polynomials generated by the Gram-Schmidt recurrence procedure. For the cases of simply supported and fully clamped hexagonal and octagonal plates, the results obtained agreed very well with those existing in the literature. The frequencies and mode shapes for several hexagonal and octagonal plates subjected to mixed boundary conditions are also presented.

On Vibrations of Pretwisted Rectangular Plates

1962 ◽  
Vol 29 (1) ◽  
pp. 30-32 ◽  
Author(s):  
R. P. Nordgren
Keyword(s):  
Shallow Shell ◽  
Shell Theory ◽  
Free Vibrations ◽  
Frequency Equation ◽  
Numerical Results ◽  
The Other ◽  
Torsional Vibrations ◽  
Rectangular Plates ◽  
Simply Supported

This paper contains an analysis of the free vibrations of uniformly pretwisted rectangular plates, utilizing the exact equations of classical shallow-shell theory. Specifically, solutions are given (a) for two opposite edges simply supported and the other two free, and (b) for all four edges simply supported. Numerical results obtained for case (b) are compared with previous results for the torsional vibrations of pretwisted beams. A simple frequency equation is obtained for case (b), permitting a detailed study of the effects of both pretwist and longitudinal inertia.

Vibration of a Fluid Conveying Pipe Carrying a Discrete Mass

1974 ◽  
Vol 96 (4) ◽  
pp. 268-272 ◽  
Author(s):  
T. T. Wu ◽  
P. P. Raju
Keyword(s):  
Dynamic Response ◽  
Boundary Conditions ◽  
Flow Velocity ◽  
General Expression ◽  
Normal Modes ◽  
Critical Value ◽  
Numerical Results ◽  
Simply Supported ◽  
Various Boundary Conditions ◽  
Discrete Mass

This paper presents a method to predict the dynamic response of a fluid conveying pipe carrying a discrete mass when the flow velocity is less than its critical value. A general expression for the normal modes of a vibrating pipe with various boundary conditions is newly derived herein. Also presented for a particular case are the numerical results of eigenfunctions and eigenvalues which can be used to calculate the dynamic response of a simply-supported pipe with an attached discrete mass at its mid-span.

Bending of a thin circular plate under hydrostatic pressure over a concentric ellipse

1959 ◽  
Vol 55 (1) ◽  
pp. 110-120 ◽  
Author(s):  
W. A. Bassali
Keyword(s):  
Exact Solution ◽  
Hydrostatic Pressure ◽  
Boundary Conditions ◽  
Circular Plate ◽  
Thin Plate ◽  
Plate Theory ◽  
Simply Supported ◽  
Thin Circular Plate ◽  
Thin Plate Theory

ABSTRACTAn exact solution in finite terms is derived within the limitations of the classical thin-plate theory, for the problem of a thin circular plate acted upon normally by hydrostatic pressure distributed over the area of a concentric ellipse, and subject to boundary conditions covering the usual rigidly clamped and simply supported boundaries.

Buckling Under Locally Hydrostatic Pressure

1946 ◽  
Vol 13 (3) ◽  
pp. A198-A200
Author(s):  
G. H. Handelman
Keyword(s):  
Differential Equation ◽  
Hydrostatic Pressure ◽  
Boundary Conditions ◽  
Pressure Distribution ◽  
Buckling Loads ◽  
The Third ◽  
Basic Assumptions ◽  
Basic Differential Equation ◽  
Discrete Values ◽  
Definition Of

Abstract The instability of a simple pin-supported beam when subject to a locally hydrostatic pressure distribution, the intensity of which is a function of distance along the beam only, is discussed in this paper. In particular, the pressure is assumed to follow a polynomial law given by Equation [1]. The problem, basic assumptions, and precise definition of locally hydrostatic pressure are stated in the first section of the paper. The second section contains a discussion of the equations of equilibrium from which the basic differential equation is derived. The solutions of this differential equation satisfy the boundary conditions only for certain discrete values of the parameters involved, and these values in turn define the buckling loads. In the third section the buckling loads are tabulated for several cases in comparison with the buckling loads for the same beam subject only to an end thrust. The appendix contains a mathematical discussion of the solution of the basic differential equation and a derivation of the formula for the buckling loads.

Buckling of Polar Orthotropic Annular Plates Under Uniform Inplane Compressive Forces

1981 ◽  
Vol 48 (3) ◽  
pp. 643-653 ◽  
Author(s):  
G. K. Ramaiah
Keyword(s):  
Free Boundary ◽  
Boundary Conditions ◽  
Ritz Method ◽  
Buckling Loads ◽  
Simply Supported ◽  
Free Boundary Conditions ◽  
Annular Plates ◽  
Design Engineers ◽  
Direct Use ◽  
Polar Orthotropic

The problem of buckling of polar orthotropic annular plates under various types of inplane compressive forces along the radial edges has been analyzed in detail by the Rayleigh-Ritz method for eight different combinations of clamped, simply supported, and free boundary conditions. Accurate estimates of critical buckling loads have been obtained for various values of hole ratios and for various values of rigidity ratios. The numerical results are presented in the form of data sheets for direct use by the design engineers.

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