Some Thin-Plate Problems by the Sine Transform

1951 ◽  
Vol 18 (2) ◽  
pp. 152-156
Author(s):  
L. I. Deverall ◽  
C. J. Thorne
Keyword(s):  
Thin Plate ◽  
The Other ◽  
Rectangular Plates ◽  
Specific Load ◽  
General Solutions ◽  
Arbitrary Load ◽  
Sine Transform ◽  
Edge Conditions ◽  
Method Of Solution ◽  
Load Function

Abstract General expressions for the deflection of thin rectangular plates are obtained for cases in which two opposite edges have arbitrary but given deflections and moments. The sine transform is used as a part of the method of solution, since solutions can be found for an arbitrary load for each set of edge conditions at the other two edges. Even for the classical cases, the use of the sine transform makes the process of solving the problem much easier. The six general solutions given are those which arise from all possible combinations of physically important edge conditions at the other two edges. Solutions for a specific load function can be found by integration or by the use of a table of sine transforms. Tables useful in application of the method to specific problems are included.

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 Use of Higher Order Plate Theory in dynamic analysis of SSFS and CSFS thick rectangular plates in orthogonal polynomials

2021 ◽  
Vol 40 (2) ◽  
pp. 199-209
Author(s):  
I.C. Onyechere ◽  
U.C. Anya ◽  
O.M. Ibearugbulem ◽  
A.U. Igbojiaku ◽  
E.O. Ihemegbulem ◽  
...  
Keyword(s):  
Potential Energy ◽  
Governing Equation ◽  
Plate Theory ◽  
Total Potential Energy ◽  
The Other ◽  
Frequency Function ◽  
Rectangular Plates ◽  
Total Potential ◽  
Edge Conditions ◽  
Simple Supports

This study applied polynomial expressions as displacement and shear deformation functions in the free-vibration study of thick and moderately thick isotropic rectangular plates. Rectangular plates with two different edge conditions investigated in this work are: one with simple supports at three of its edges and with no support at the other edge denoted with the acronym (SSFS) and a rectangular plate with simple supports at opposite edges while the other opposite edges has a fixed support at one edge and no support at the other edge, this is denoted with the acronym (CSFS). The total potential energy of the plate was derived using the general theory of elasticity. The general governing equation of the plate was derived by minimizing the total potential energy equation of the plate. Edge conditions of the SSFS and CSFS plates were met and substituted into the general governing equation to obtain a linear expression which was solved to generate fundamental natural frequency function for the plates with various span-depth proportion (m/t) and planar dimensions proportion (n/m). The results obtained from this research were found to agree favourably with the results of similar problems in the literature upon comparison.

Exact Solutions for Free-Vibration Analysis of Rectangular Plates Using Bessel Functions

2005 ◽  
Vol 74 (6) ◽  
pp. 1247-1251 ◽  
Author(s):  
Jiu Hui Wu ◽  
A. Q. Liu ◽  
H. L. Chen
Keyword(s):  
Differential Equation ◽  
Free Vibration ◽  
Exact Solutions ◽  
Thin Plate ◽  
Vibration Analysis ◽  
Bessel Functions ◽  
Free Vibration Analysis ◽  
Rectangular Plates ◽  
Simply Supported ◽  
Edge Conditions

A novel Bessel function method is proposed to obtain the exact solutions for the free-vibration analysis of rectangular thin plates with three edge conditions: (i) fully simply supported; (ii) fully clamped, and (iii) two opposite edges simply supported and the other two edges clamped. Because Bessel functions satisfy the biharmonic differential equation of solid thin plate, the basic idea of the method is to superpose different Bessel functions to satisfy the edge conditions such that the governing differential equation and the boundary conditions of the thin plate are exactly satisfied. It is shown that the proposed method provides simple, direct, and highly accurate solutions for this family of problems. Examples are demonstrated by calculating the natural frequencies and the vibration modes for a square plate with all edges simply supported and clamped.

Some Archaic Gold Ornaments with representations of Sphinxes and Sirens

1911 ◽  
Vol 31 ◽  
pp. 263-265
Author(s):  
F. H. Marshall
Keyword(s):  
Seventh Century ◽  
The Other ◽  
Main Body ◽  
Rectangular Plates ◽  
Natural Size

1. In a recent description of an archaic Etruscan fibula here reproduced in natural size (Fig. 1), I regret that I failed to note certain interesting details with regard to the Sphinxes. The fibula is of pale gold, of a type peculiar to early Etruscan jewellery. It consists of two parts, each composed of four tubes ending in double female heads. In one case the outer tubes are furnished with long gold pins which fit into the hollow tubes corresponding to them in the other half of the fibula. There can be no doubt that these safety-pins were used for fastening a garment on the shoulder. The two halves were locked together by means of hooks and eyes soldered to rectangular plates hinged to the main body of the fibula. The tubes were also connected together by similar plates. The present fibula, which may be dated to the seventh century B.C., is said to have been found in the Roman Campagna. Upon the four rectangular plates already mentioned are seated sixteen Sphinxes in the round, four upon each plate. The eight Sphinxes on the outer plates are composed of the figure of a seated lion, with the head of a woman substituted for a wing.

Vibration of Rectangular Plates by the Ritz Method

1950 ◽  
Vol 17 (4) ◽  
pp. 448-453 ◽  
Author(s):  
Dana Young
Keyword(s):  
Ritz Method ◽  
Normal Modes ◽  
Approximate Solutions ◽  
The Other ◽  
Rectangular Plates ◽  
Elastic Plates ◽  
Modes Of Vibration ◽  
Ritz’S Method ◽  
Square Plate ◽  
Set Up

Abstract Ritz’s method is one of several possible procedures for obtaining approximate solutions for the frequencies and modes of vibration of thin elastic plates. The accuracy of the results and the practicability of the computations depend to a great extent upon the set of functions that is chosen to represent the plate deflection. In this investigation, use is made of the functions which define the normal modes of vibration of a uniform beam. Tables of values of these functions have been computed as well as values of different integrals of the functions and their derivatives. With the aid of these data, the necessary equations can be set up and solved with reasonable effort. Solutions are obtained for three specific plate problems, namely, (a) square plate clamped at all four edges, (b) square plate clamped along two adjacent edges and free along the other two edges, and (c) square plate clamped along one edge and free along the other three edges.

Combinations for the Free Vibration Behaviors of Anisotropic Rectangular Plates Under General Edge Conditions

1999 ◽  
Author(s):  
Yoshihiro Narita
Keyword(s):  
Free Vibration ◽  
Structural Design ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Technical Information ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Plate Models ◽  
Edge Conditions ◽  
Anisotropic Rectangular Plates

Abstract The free vibration behavior of rectangular plates provides important technical information in structural design, and the natural frequencies are primarily affected by the boundary conditions as well as aspect and thickness ratios. One of the three classical edge conditions, i.e., free, simple supported and clamped edges, may be used to model the constraint along an edge of the rectangle. Along the entire boundary with four edges, there exist a wide variety of combinations in the edge conditions, each yielding different natural frequencies and mode shapes. For counting the total number of possible combinations, the present paper introduces the Polya counting theory in combinatorial mathematics, and formulas are derived for counting the exact numbers. A modified Ritz method is then developed to calculate natural frequencies of anisotropic rectangular plates under any combination of the three edge conditions and is used to numerically verify the numbers. In numerical experiments, the number of combinations in the free vibration behaviors is determined for some plate models by using the derived formulas, and are corroborated by counting the numbers of different sets of the natural frequencies that are obtained from the Ritz method.

scholarly journals Vibration of Skew Sandwich Plates With Laminated Facings

1997 ◽  
Author(s):  
C. M. Wang ◽  
K. K. Ang ◽  
C. Wang
Keyword(s):  
Ritz Method ◽  
Composite Plates ◽  
Sandwich Plates ◽  
Rectangular Plates ◽  
Vibration Frequencies ◽  
Edge Conditions ◽  
System Vibration ◽  
Special Case ◽  
Length Limitation ◽  
Good Agreement

A Rayleigh-Ritz analysis is presented for the free vibration of skew sandwich plates composed of an orthotropic core and laminated facings. By proposing a set of Ritz functions consisting of the product of mathematically complete polynomial functions and the the boundary equations raised to appropriate powers, the Rayleigh-Ritz method can be automated to handle such composite plates with any combination of edge conditions. For convenience and better accurarcy, the Ritz formulation was derived in the skew coordinate system. Vibration frequencies of rectangular plates (a special case of skew plates) obtained via this method have been found to be in good agreement with previous researchers results. Owing to length limitation, only sample vibration frequencies for skew sandwich plates are presented.

The Plastic Buckling of Axially Compressed Square Tubes

1992 ◽  
Vol 59 (2) ◽  
pp. 276-282 ◽  
Author(s):  
S. Li ◽  
S. R. Reid
Keyword(s):  
Deformation Theory ◽  
Buckling Analysis ◽  
Rectangular Plates ◽  
Plastic Buckling ◽  
Buckling Mode ◽  
Important Conclusion ◽  
Simply Supported ◽  
Edge Conditions ◽  
Crushing Behavior ◽  
Incremental Theory Of Plasticity

A plastic buckling analysis for axially compressed square tubes is described in this paper. Deformation theory is used together with the realistic edge conditions for the panels of the tube introduced in our previous paper (Li and Reid, 1990), referred to hereafter as LR. The results obtained further our understanding of a number of problems related to the plastic buckling of axially compressed square tubes and simply supported rectangular plates, which have remained unsolved hitherto and seem rather puzzling. One of these is the discrepancy between experimental results and the results of plastic buckling analysis performed using the incremental theory of plasticity and the unexpected agreement between the results of calculations based on deformation theory for plates and experimental data obtained from tests conducted on tubes. The non-negligible difference between plates and tubes obtained in the present paper suggests that new experiments should be carried out to provide a more accurate assessment of the predictions of the two theories. Discussion of the results herein also advances our understanding of the compact crushing behavior of square tubes beyond that given in LR. An important conclusion reached is that strain hardening cannot be neglected for the plastic buckling analysis of square tubes even if the degree of hardening is small since doing so leads to an unrealistic buckling mode.

Large Elastic-Plastic Deformation Analysis of Rectangular Plates

2002 ◽  
Author(s):  
Reza Naghdabadi ◽  
Mohsen Shahi
Keyword(s):  
Lateral Displacement ◽  
Deformation Analysis ◽  
Plastic Behavior ◽  
Rectangular Plates ◽  
Computationally Efficient ◽  
Plastic Deformations ◽  
Elastic Plastic ◽  
Various Boundary Conditions ◽  
Method Of Solution ◽  
Suitable Combination

The purpose of this paper is to find a fast and simple solution for the large deformation of rectangular plates considering elastic-plastic behavior. This analysis contains material and geometric nonlinearities. For geometric nonlinearity the concept of load analogy is used. In this method the effect of nonlinear terms of lateral displacement is considered as suitable combination of additional fictitious lateral load, edge moment and in-plane forces acting on the plate. Variable Material Property (V.M.P.) method has been used for analysis of material nonlinearity. In this method, the basic relations maintain the form of stress-strain elastic formula, while material properties are modified to take into account the path-dependency involved in elastic-plastic deformations. Therefore, the solution of a von-Karman plate enduring large elastic-plastic deformations is reduced to that of an equivalent elastic plate undergoing small deformations. The method of solution employed in this study is computationally efficient and can easily be used for various boundary conditions and loadings.

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