Analysis of Clamped Rectangular Plates

1940 ◽  
Vol 7 (4) ◽  
pp. A139-A142
Author(s):  
Dana Young
Keyword(s):  
Middle Surface ◽  
Lateral Loading ◽  
Thin Plates ◽  
Superposition Method ◽  
Rectangular Plates ◽  
Lagrange’S Equation

Abstract This paper attempts to solve the problem of the bending action of rectangular plates clamped at all four edges and subjected to lateral loading. Analytical in nature, the author’s investigation is based upon the ordinary theory of bending of thin plates as treated in Lagrange’s equation of the middle surface. The superposition method is used and applied to a number of loadings not hitherto studied.

A General Analytical Solution for Free Vibration of Rectangular Plates Resting on Fixed Supports and With Attached Masses

1992 ◽  
Vol 114 (2) ◽  
pp. 239-245 ◽  
Author(s):  
R. K. Singal ◽  
D. J. Gorman
Keyword(s):  
Free Vibration ◽  
Analytical Procedure ◽  
Thin Plates ◽  
Mode Shapes ◽  
Experimental Program ◽  
Space Vehicles ◽  
Superposition Method ◽  
Rectangular Plates ◽  
Solar Panels ◽  
Good Agreement

A comprehensive analytical procedure based on the superposition method is described for establishing the free vibration frequencies and mode shapes of thin plates resting on rigid point supports and with attached masses. Effects of rotary inertia of the attached masses are incorporated into the analysis and are shown to be highly significant. Results of an extensive experimental program are reported and very good agreement is demonstrated between theory and experiment. The analytical procedure has application in numerous contemporary industrial problems, in particular, in the design of solar panels for space vehicles and in the field of electronic packaging.

scholarly journals The elastic stability of an annular plate

1924 ◽  
Vol 106 (737) ◽  
pp. 268-284 ◽  
Keyword(s):  
Annular Plate ◽  
Practical Interest ◽  
Middle Surface ◽  
Elastic Stability ◽  
Thin Plates ◽  
Thin Shells ◽  
Stability Equation ◽  
Inner Radius ◽  
Plane Plate ◽  
The Stability

1. Introduction and Summary. —This paper deals with the elastic stability of a circular annular plate under uniform shearing forces applied at its edges. Investigations of the stability of plane plates are altogether simpler than those necessary in the case of curved plates or shells. In the first place, as shown by Mr. R. V. Southwell, two of the three equations of stability relate to a mode of instability that is not of practical interest, and are entirely independent of the third equation which gives the ordinary mode of instability resulting in the familiar bending of the middle surface of the plate. Consequently with a plane plate there is only one equation of stability to be solved, as contrasted with the case of a shell where the three equations are dependent, and must all be solved. In the second place the theory of thin shells can be used with confidence in a plane plate problem, though a more laborious procedure is necessary to deal adequately with a shell. The only stability equation required for the annular plate is therefore deduced without trouble from the theory of thin shells, and its solution presents no difficulty in the case of uniform shearing forces. A numerical discussion is given of the stability of the plate under such forces, the “favourite type of distortion” and the stess that will produce it being obtained for plates with clamped edges in wich the ratio of the outer to the inner radius exceeds 3·2. To some extent to results have been checked by experiment, in which part of the work the viter is indebted to Prof. G. I. Taylor for his valuable help and advice. Distrtion of the type predicted by the theory took place in the two thin plates of rober different ratio of radii, which were used. The disposition of the loci of points which undergo maximum normal displace nt gives some idea of the appearance of the plate after distortion has taken pce. The points have been calculated for a plate in which the ratio of radii 4·18, and the loci are shown on a diagram, which may be compared with a potograph of a distorted plate in which this ratio is 4·3. The ratio of normal dplacements of points of the plate can be seen from contours drawn on the ne diagram. (See pp. 280, 281.)

New analytic buckling solutions of side-cracked rectangular thin plates by the symplectic superposition method

2021 ◽  
Vol 191 ◽  
pp. 106051
Author(s):  
Zhaoyang Hu ◽  
Xinran Zheng ◽  
Dongqi An ◽  
Chao Zhou ◽  
Yushi Yang ◽  
...  
Keyword(s):  
Thin Plates ◽  
Superposition Method

scholarly journals Analytic bending solutions of free rectangular thin plates resting on elastic foundations by a new symplectic superposition method

2013 ◽  
Vol 469 (2153) ◽  
pp. 20120681 ◽  
Author(s):  
Rui Li ◽  
Yang Zhong ◽  
Ming Li
Keyword(s):  
Finite Element Method ◽  
Symplectic Geometry ◽  
Analytic Solutions ◽  
Thin Plates ◽  
Superposition Method ◽  
The Finite Element Method ◽  
Simple Derivation ◽  
Elastic Foundations ◽  
Winkler Model ◽  
Geometry Method

Analytic bending solutions of free rectangular thin plates resting on elastic foundations, based on the Winkler model, are obtained by a new symplectic superposition method. The proposed method offers a rational elegant approach to solve the problem analytically, which was believed to be difficult to attain. By way of a rigorous but simple derivation, the governing differential equations for rectangular thin plates on elastic foundations are transferred into Hamilton canonical equations. The symplectic geometry method is then introduced to obtain analytic solutions of the plates with all edges slidingly supported, followed by the application of superposition, which yields the resultant solutions of the plates with all edges free on elastic foundations. The proposed method is capable of solving plates on elastic foundations with any other combinations of boundary conditions. Comprehensive numerical results validate the solutions by comparison with those obtained by the finite element method.

scholarly journals RBF-Based Meshless Method for Large Deflection of Elastic Thin Rectangular Plates with Boundary Conditions Involving Free Edges

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Mohammed M. Hussein Al-Tholaia ◽  
Husain Jubran Al-Gahtani
Keyword(s):  
Boundary Condition ◽  
Basis Function ◽  
Meshless Method ◽  
Iterative Procedure ◽  
Large Deflection ◽  
Free Edge ◽  
Thin Plates ◽  
Rectangular Plates ◽  
Numerical Examples ◽  
Complex Boundary

An RBF-based meshless method is presented for the analysis of thin plates undergoing large deflection. The method is based on collocation with the multiquadric radial basis function (MQ-RBF). In the proposed method, the resulting coupled nonlinear equations are solved using an incremental-iterative procedure. The accuracy and efficiency of the method are verified through several numerical examples. The inclusion of the free edge boundary condition proves that this method is accurate and efficient in handling such complex boundary value problems.

A New General Solution for the Bending and Vibration of Orthotropic Rectangular Thin Plates with Four Free Edges

2010 ◽  
Vol 168-170 ◽  
pp. 1158-1162 ◽  
Author(s):  
Hong Zhang ◽  
Hai Qun Que ◽  
Huan Ding
Keyword(s):  
General Solution ◽  
Vibration Analysis ◽  
Thin Plates ◽  
Physical Parameters ◽  
Winkler Foundation ◽  
Vertical Force ◽  
Rectangular Plates ◽  
Cosine Series ◽  
Vibration Problems ◽  
Free Edges

This paper firstly introduces a new general solution constructed by double trigonometric cosine series with supplementary terms for the bending and vibration analysis of orthotropic rectangular plates with four free edges on the Winkler foundation subjected to arbitrary vertical force. The general solution, which is fourth-order continuously differentiable with less undetermined coefficients, can be used to solve the bending and vibration problems of orthotropic rectangular plates on the Winkler foundation with various physical parameters requiring no classification and superposition. This makes the bending and vibration analysis of orthotropic rectangular plates with four free edges on the Winkler foundation more unified, simplified and regulated. This paper also gives a Series of analytical example to prove that the method is feasible.

scholarly journals EQUATION DECOMPOSITION METHOD FOR SOLVING OF PROBLEMS OF STATICS, VIBRATIONS AND STABILITY OF THIN-WALLED CONSTRUCTIONS

2020 ◽  
Vol 16 (2) ◽  
pp. 63-70
Author(s):  
Elena Koreneva ◽  
Valery Grosman
Keyword(s):  
Decomposition Method ◽  
Middle Surface ◽  
Free Vibrations ◽  
Rectangular Plates ◽  
Effective Equation ◽  
Transverse Loads ◽  
Approximate Analytical Solutions ◽  
Thin Walled ◽  
Definition Of ◽  
Elastically Supported

The work suggests the effective equation decomposition method (EDM) for solving of statics, vibrations and stability problems of thin-walled constructions. This method is based on the partition of the initial problem on the consideration of more simple auxiliary problems. The additional unknown functions are introduced for definition of the sought solutions. The paper shows the method’s advantages on the examples of the boundary value problems for rectangular areas. The problem of anisotropic plate resting on an elastic subgrade and subjected to an action of expanding forces acting in the middle surface and to transverse loads is under study. The plate’s edges are elastically supported. Also free vibrations of the rectangular plates of variable thickness with different boundary conditions were under consideration. The approximate analytical solutions with high exactness are obtained.

Thin Plates and Shallow Cylindrical Shells Subjected to Hot Spots

1964 ◽  
Vol 31 (1) ◽  
pp. 79-82 ◽  
Author(s):  
A. Jahanshahi
Keyword(s):  
Plane Stress ◽  
Hot Spots ◽  
Cylindrical Shells ◽  
Equation Of Motion ◽  
Thin Plates ◽  
Lagrange’S Equation ◽  
Singular Functions

Using Lagrange’s equation of motion for thin plates, the equations of plane stress, and the equations for shallow cylindrical shells, the deformations of thin plates and cylindrical shells subjected to bending and plane hot spots are studied. The interrelation between different singular functions associated with these problems has been indicated also.

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