Applications of the reciprocal theorem to solving the equations of the deflection surface of the rectangular plates with various edge conditions

1982 ◽  
Vol 3 (3) ◽  
pp. 353-364 ◽  
Author(s):  
Fu Bao-lian
Keyword(s):  
Reciprocal Theorem ◽  
Rectangular Plates ◽  
Edge Conditions ◽  
The Reciprocal Theorem

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.

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.

The Reciprocal Theorem

1992 ◽  
pp. 281-289
Author(s):  
J. R. Barber
Keyword(s):  
Reciprocal Theorem ◽  
The Reciprocal Theorem

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.

Boundary Conditions for Torsional Circular Shaft with Two-Dimensional Dodecagonal Quasicrystals

2012 ◽  
Vol 580 ◽  
pp. 411-414
Author(s):  
Bao Sheng Zhao ◽  
Di Wu
Keyword(s):  
Boundary Conditions ◽  
Necessary Conditions ◽  
Reciprocal Theorem ◽  
Interior Solution ◽  
Two Dimensional ◽  
Circular Shaft ◽  
Analysis Technique ◽  
State Boundary ◽  
The Reciprocal Theorem ◽  
Layer Solution

Through generalizing the method of a decay analysis technique determining the interior solution developed by Gregory and Wan, a set of necessary conditions on the end-data of torsional circular shaft in two-dimensional dodecagonal quasicrystals (2D dodecagonal QCs) for the existence of a rapidly decaying solution is established. By accurate solutions for auxiliary regular state, using the reciprocal theorem, these necessary conditions for the end-data to induce only a decaying elastostatic state (boundary layer solution) will be translated into appropriate boundary conditions for the torsional circular shaft in 2D dodecagonal QCs. The results of the present paper enable us to establish a set of boundary conditions.

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