scholarly journals DYNAMICS OF RECTANGULAR ORTHOTROPIC PLATES USING CHARACTERISTIC ORTHOGONAL POLYNOMIAL – GALERKIN’S METHODS

2016 ◽  
Vol 36 (1) ◽  
pp. 50-56
Author(s):  
NN Osadebe ◽  
CM Attama ◽  
OA Oguaghamba
Keyword(s):  
Orthogonal Polynomial ◽  
Rectangular Plate ◽  
Natural Frequencies ◽  
Rectangular Plates ◽  
Trial And Error ◽  
Shape Functions ◽  
Orthotropic Plates ◽  
Approximate Methods ◽  
Simply Supported ◽  
Polynomial Theory

The assumed deflection shapes used in the approximate methods such as in the Galerkin’s method were normally formulated by inspection and sometimes by trial and error, until recently, when a systematic method of constructing such a function in the form of Characteristic Orthogonal Polynomial (COPs) was developed by Bhat in 1985. In the vibrational analyses of orthotropic rectangular plates with different boundary conditions, the study used the characteristic orthogonal polynomial theory to obtain satisfactory approximate shape functions for these plates. These functions were applied to Galerkin indirect varational method to obtain new set of fundamental natural frequencies for these plates. The results were reasonable when compared with those in the previous work. All round simply supported thin rectangular plate (SSSS), rectangular clamped plated (CCCC) and rectangular plate with one edge clamped and all others edges simply supported (CSSS) gave 5.172, 9.429 and 6.202 natural frequencies in rad /sec respectively at 0.05%, 0.0% and 22.93% difference with the previous[3] results5.170rad/sec, 9.429rad/sec and 8.048rad/sec  for SSSS, CCCC and CSSS. For others like: rectangular plate with one edge simply supported and all other edges clamped (CCSC), rectangular plate simply supported at two opposite sides and clamped at the others (CSCS) and rectangular plate clamped at two adjacent sides and simply supported at the others (CCSS) with no available results, their natural frequencies obtained are 8.041rad/sec, 6.272rad/sec and 7.106rad/sec respectively. http://dx.doi.org/10.4314/njt.v36i1.8

BUCKLING AND VIBRATION OF ORTHOTROPIC PLATES WITH AN INTERNAL LINE HINGE

2002 ◽  
Vol 02 (04) ◽  
pp. 457-486 ◽  
Author(s):  
P. R. GUPTA ◽  
J. N. REDDY
Keyword(s):  
Solution Method ◽  
Natural Frequencies ◽  
The Other ◽  
Internal Line ◽  
Rectangular Plates ◽  
Vibration Frequencies ◽  
Decomposition Technique ◽  
Orthotropic Plates ◽  
Buckling Loads ◽  
Simply Supported

This paper presents the exact buckling loads and vibration frequencies of orthotropic rectangular plates with internal line hinge and having two opposite edges simply supported while the other two edges may have any combination of free, simply supported, and clamped conditions. An analytical method that uses the Lévy solution method and the domain decomposition technique is employed to determine the buckling loads and natural frequencies of rectangular plates with internal line hinge.

Vibration of Orthotropic Rectangular Plates With Variable Thickness

1989 ◽  
Vol 111 (1) ◽  
pp. 101-103 ◽  
Author(s):  
Wei-Cheun Liu ◽  
Stanley S. H. Chen
Keyword(s):  
Boundary Conditions ◽  
Computer Program ◽  
Variable Thickness ◽  
Energy Method ◽  
Natural Frequencies ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Rectangular Plates ◽  
Orthotropic Plates ◽  
Orthotropic Properties

The problem vibration of rectangular orthotropic plates with variable thickness and mixed boundary conditions are solved by a modified energy method. A general expression is written for the deflection of the plate without aiming at any particular combination of boundary conditions. Boundary conditions are satisfied approximately by adjusting a set of so-called fixity factors. A computer program has been developed to solve for natural frequencies of plates with variable thicknesses and having different orthotropic properties.

Buckling of Rectangular Plates With General Variation in Thickness

1973 ◽  
Vol 40 (3) ◽  
pp. 745-751 ◽  
Author(s):  
D. S. Chehil ◽  
S. S. Dua
Keyword(s):  
Rectangular Plate ◽  
Variable Thickness ◽  
Perturbation Technique ◽  
Thickness Variation ◽  
Rectangular Plates ◽  
Bending Vibration ◽  
Simply Supported ◽  
Buckling Stress ◽  
General Variation ◽  
Plate Of Variable Thickness

A perturbation technique is employed to determine the critical buckling stress of a simply supported rectangular plate of variable thickness. The differential equation is derived for a general thickness variation. The problem of bending, vibration, buckling, and that of dynamic stability of a variable thickness plate can be deduced from this equation. The problem of buckling of a rectangular plate with simply supported edges and having general variation in thickness in one direction is considered in detail. The solution is presented in a form such as can be easily adopted for computing critical buckling stress, once the thickness variation is known. The numerical values obtained from the present analysis are in excellent agreement with the published results.

LINEAR BENDING ANALYSIS OF INHOMOGENEOUS VARIABLE-THICKNESS ORTHOTROPIC PLATES UNDER VARIOUS BOUNDARY CONDITIONS

2007 ◽  
Vol 04 (03) ◽  
pp. 417-438 ◽  
Author(s):  
A. M. ZENKOUR ◽  
M. N. M. ALLAM ◽  
D. S. MASHAT
Keyword(s):  
Boundary Conditions ◽  
Variable Thickness ◽  
Flexural Rigidity ◽  
Approximate Solutions ◽  
Rectangular Plates ◽  
Orthotropic Plates ◽  
Simply Supported ◽  
Various Boundary Conditions ◽  
Thickness Parameter ◽  
Space Concept

An exact solution to the bending of variable-thickness orthotropic plates is developed for a variety of boundary conditions. The procedure, based on a Lévy-type solution considered in conjunction with the state-space concept, is applicable to inhomogeneous variable-thickness rectangular plates with two opposite edges simply supported. The remaining ones are subjected to a combination of clamped, simply supported, and free boundary conditions, and between these two edges the plate may have varying thickness. The procedure is valuable in view of the fact that tables of deflections and stresses cannot be presented for inhomogeneous variable-thickness plates as for isotropic homogeneous plates even for commonly encountered loads because the results depend on the inhomogeneity coefficient and the orthotropic material properties instead of a single flexural rigidity. Benchmark numerical results, useful for the validation or otherwise of approximate solutions, are tabulated. The influences of the degree of inhomogeneity, aspect ratio, thickness parameter, and the degree of nonuniformity on the deflections and stresses are investigated.

Free Vibration of Plates of Various Shapes with Intermediate Point Supports by the Hp-Cloud Method and Lagrange Multiplier

2016 ◽  
Vol 16 (09) ◽  
pp. 1550055 ◽  
Author(s):  
Sajad Jamshidi ◽  
Mojtaba Azhari ◽  
Hossein Amoushahi
Keyword(s):  
Lagrange Multiplier ◽  
Natural Frequencies ◽  
Cell Structure ◽  
Ritz Method ◽  
Thin Plates ◽  
Plate Boundaries ◽  
Shape Functions ◽  
Intermediate Point ◽  
Simply Supported ◽  
Kronecker Delta

The Hp-Cloud meshless method was developed to study the dynamic analysis of arbitrarily shaped thin plates with intermediate point supports. By proposing a special pattern for the influence radius of nodes and a polynomial type of enrichment function, the Hp-Cloud shape functions with Kronecker delta property were constructed. They can satisfy the zero deflection conditions for the field nodes at the point supports. The results obtained from these shape functions agree well with the previous ones, showing good accuracy and convergence. For plates with sharp corners, it is not possible to construct the Hp-Cloud shape function with Kronecker delta property. To this end, the Lagrange multiplier method was used for enforcing the boundary conditions. The computations were carried out by the Ritz method, and the cell structure method is refined to improve the speed and accuracy of numerical integration on the subscription surface of clouds intersecting with the plate boundaries. Using the algorithm developed, the natural frequencies of plates of various shapes and support patterns were computed. By increasing the number of point supports on the plate edges, the natural frequencies computed of the plate tend to those of the simply supported plate. Appropriate pattern of point supports distribution was presented for modeling the simply supported plates of various shapes by comparing the corresponding natural frequencies.

Fundamental Frequency of Simply Supported Rectangular Plates With Linearly Varying Thickness

1965 ◽  
Vol 32 (1) ◽  
pp. 163-168 ◽  
Author(s):  
F. C. Appl ◽  
N. R. Byers
Keyword(s):  
Rectangular Plate ◽  
Fundamental Frequency ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Rectangular Plates ◽  
Simply Supported ◽  
Varying Thickness

Upper and lower bounds for the fundamental eigenvalue (frequency) of a simply supported rectangular plate with linearly varying thickness are given for several taper ratios and plan geometries. These bounds were determined using a previously published method which yields convergent bounds. In the present study, all results have been obtained to within 0.5 percent maximum possible error.

scholarly journals Analysis on free vibration and critical buckling load of a FGM porous rectangular plate

2021 ◽  
Vol 39 (2) ◽  
pp. 317-325
Author(s):  
Zhaochun Teng ◽  
Pengfei Xi
Keyword(s):  
Differential Equation ◽  
Elastic Modulus ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Rectangular Plate ◽  
Natural Frequencies ◽  
Gradient Index ◽  
Rectangular Plates ◽  
Buckling Loads ◽  
Governing Differential Equation

The properties of functionally gradient materials (FGM) are closely related to porosity, which has effect on FGM's elastic modulus, Poisson's ratio, density, etc. Based on the classical theory of thin plates and Hamilton principle, the mathematical model of free vibration and buckling of FGM porous rectangular plates with compression on four sides is established. Then the dimensionless form of the governing differential equation is also obtained. The dimensionless governing differential equation and its boundary conditions are transformed by differential transformation method (DTM). After iterative convergence, the dimensionless natural frequencies and critical buckling loads of the FGM porous rectangular plate are obtained. The problem is reduced to the free vibration of FGM rectangular plate with zero porosity and compared with its exact solution. It is found that DTM gives high accuracy result. The validity of the method is verified in solving the free vibration and buckling problems of the porous FGM rectangular plates with compression on four sides. The results show that the elastic modulus of FGM porous rectangular plate decreases with the increase of gradient index and porosity. Furthermore, the effects of gradient index and porosity on dimensionless natural frequencies and critical buckling loads are further analyzed under different boundary conditions with constant aspect ratio, and the effects of aspect ratio and load on dimensionless natural frequencies under different boundary conditions.

Stability of Rectangular Plates Under Shear and Bending Forces

1936 ◽  
Vol 3 (4) ◽  
pp. A131-A135 ◽  
Author(s):  
Stewart Way
Keyword(s):  
Critical Load ◽  
Rectangular Plate ◽  
Bending Moment ◽  
The Other ◽  
Rectangular Plates ◽  
Bending Stress ◽  
Simply Supported ◽  
Tension And Compression ◽  
The Way

Abstract The author first discusses the problem of a plane, simply supported rectangular plate loaded by shearing forces in the plane of the plate on all four edges. There are two stiffeners attached one third and two thirds of the way along the plate. The critical load is calculated for various stiffener rigidities. Also, the rigidity necessary to keep the stiffeners straight when the plate buckles is found. This stiffener rigidity is found to be slightly larger than that necessary for a plate with one stiffener and the same panel dimensions as the plate with two stiffeners. The second problem discussed by the author is that of a plane, simply supported rectangular plate loaded by uniformly distributed edge shearing forces in the plane of the plate and linearly distributed tension and compression in the plane of the plate at the ends. The end forces vary from tension hσo, at one corner to—hσo, at the other corner, so that their resultant is a bending moment. The presence of the edge shearing forces is found to diminish the critical bending stress in this case. Calculations are made for various magnitudes of bending and shearing forces for plates of various proportions.

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