scholarly journals Analysis of inelastic buckling of rectangular plates with a free edge using polynomial deflection functions

2020 ◽  
Vol 11 (1) ◽  
pp. 15-21
Author(s):  
Uchechi G. Eziefula
Keyword(s):  
Boundary Conditions ◽  
Rectangular Plate ◽  
Polynomial Function ◽  
Free Edge ◽  
Rectangular Plates ◽  
Maclaurin Series ◽  
Inelastic Buckling ◽  
Aspect Ratios ◽  
Isotropic Plates ◽  
Deformation Plasticity

AbstractThe inelastic buckling behaviour of different rectangular thin isotropic plates having a free edge is studied. Various combinations of boundary conditions are subject to in-plane uniaxial compression and each rectangular plate is bounded by an unloaded free edge. The characteristic deflection function of each plate is formulated using a polynomial function in form of Taylor–Maclaurin series. A deformation plasticity approach is adopted and the buckling load equation is modified using a work principle technique. Buckling coefficients of the plates are calculated for various aspect ratios and moduli ratios. Findings obtained from the investigation are found to reasonably agree with data published in the literature.

Work principle in inelastic buckling analysis of axially compressed rectangular plates

2017 ◽  
Vol 14 (2) ◽  
pp. 95-100
Author(s):  
U.G. Eziefula ◽  
D.O. Onwuka ◽  
O.M. Ibearugbulem
Keyword(s):  
Rectangular Plate ◽  
Variational Principles ◽  
Error Function ◽  
Mathematical Formulation ◽  
Plasticity Theory ◽  
Maclaurin Series ◽  
Inelastic Buckling ◽  
Content Type ◽  
Aspect Ratios ◽  
Buckling Coefficient

Purpose The purpose of this paper is to analyze the inelastic buckling of a rectangular thin flat isotropic plate subjected to uniform uniaxial in-plane compression using a work principle, a deformation plasticity theory and Taylor–Maclaurin series formulation. Design/methodology/approach The non-loaded longitudinal edges of the rectangular plate are clamped, whereas the loaded edges are simply supported (CSCS). Total work error function is applied to Stowell’s plasticity theory in the derivation of the inelastic buckling equation. Mathematical formulation of the Taylor–Maclaurin series deflection function satisfied the boundary conditions of the CSCS rectangular plate. The critical inelastic load of the plate is then derived by applying variational principles. Findings Values of the plate buckling coefficient are calculated using various values of moduli ratio for aspect ratios ranging from 0.1 to 1.0, in intervals of 0.1. The accuracy of the proposed technique is validated by comparing the results obtained in the present study with solutions from a previous investigation. The percentage differences in the values of the buckling coefficient ranged from −0.122 to −4.685 per cent. Originality/value The results indicate that the work principle approach can be used as an alternative approximate method for analyzing inelastic buckling of rectangular thin flat isotropic plates under uniform in-plane compressive loads.

Development of Hybrid Semi-Analytical and Finite Element Analysis to Calculate Sound Radiation from Vibrating Structure

2013 ◽  
Vol 845 ◽  
pp. 71-75 ◽  
Author(s):  
Azma Putra ◽  
Nurain Shyafina ◽  
Noryani Muhammad ◽  
Hairul Bakri ◽  
Noor Fariza Saari
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Boundary Conditions ◽  
Rectangular Plate ◽  
Analytical Model ◽  
Complex Geometry ◽  
Sound Radiation ◽  
Vibration Velocity ◽  
Rectangular Plates ◽  
Element Analysis

Simple analytical model of plate dynamics usually applies for rectangular plate with simply supported edges. Analytical model of sound radiation from rectangular plate is also convenient, but not for other geometries and other boundary conditions. This paper presents a hybrid mathematical model which combines a semi-analytical model with the Finite Element Analysis (FEA) method to determine sound radiation from a vibrating structure. The latter is employed to calculate the vibration velocity of a structure with a rather complex geometry. The results are then used as the input in the semi-analytical model to calculate the radiated sound pressure through the Rayleigh integral. Results from the proposed model are presented here for the radiation efficiency of rectangular plates with different boundary conditions.

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 Frequency Equations and Modes of Free Vibrations of Rectangular Plates with Various Edge Conditions

1994 ◽  
Vol 208 (5) ◽  
pp. 307-319 ◽  
Author(s):  
C W Bert ◽  
M Malik
Keyword(s):  
Boundary Conditions ◽  
Exact Solutions ◽  
Free Vibrations ◽  
Edge Condition ◽  
Rectangular Plates ◽  
Simply Supported ◽  
Aspect Ratios ◽  
Edge Conditions ◽  
Classical Boundary ◽  
Free Edges

This paper considers linear free vibrations of thin isotropic rectangular plates with combinations of the classical boundary conditions of simply supported, clamped and free edges and the mathematically possible condition of guided edges. The total number of plate configurations with the classical boundary conditions are known to be twenty-one. The inclusion of the guided edge condition gives rise to an additional thirty-four plate configurations. Of these additional cases, twenty-one cases have exact solutions for which frequency equations in explicit or transcendental form may be obtained. The frequency equations of these cases are given and, for each case, results of the first nine mode frequencies are tabulated for a range of the plate aspect ratios.

scholarly journals LINEAR AND NONLINEAR FREE VIBRATION ANALYSIS OF RECTANGULAR PLATE

2021 ◽  
Vol 12 (1) ◽  
pp. 15-25
Author(s):  
Edward Adah ◽  
David Onwuka ◽  
Owus Ibearugbulem ◽  
Chinenye Okere
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Rectangular Plate ◽  
Large Deflection ◽  
Closed Form Solution ◽  
Middle Surface ◽  
Rectangular Plates ◽  
Nonlinear Free Vibration ◽  
Surface Displacements ◽  
Linear And Nonlinear

The major assumption of the analysis of plates with large deflection is that the middle surface displacements are not zeros. The determination of the middle surface displacements, u0 and v0 along x- and y- axes respectively is the major challenge encountered in large deflection analysis of plate. Getting a closed-form solution to the long standing von Karman large deflection equations derived in 1910 have proven difficult over the years. The present work is aimed at deriving a new general linear and nonlinear free vibration equation for the analysis of thin rectangular plates. An elastic analysis approach is used. The new nonlinear strain displacement equations were substituted into the total potential energy functional equation of free vibration. This equation is minimized to obtain a new general equation for analyzing linear and nonlinear resonating frequencies of rectangular plates. This approach eliminates the use of Airy’s stress functions and the difficulties of solving von Karman's large deflection equations. A case study of a plate simply supported all-round (SSSS) is used to demonstrate the applicability of this equation. Both trigonometric and polynomial displacement shape functions were used to obtained specific equations for the SSSS plate. The numerical results for the coefficient of linear and nonlinear resonating frequencies obtained for these boundary conditions were 19.739 and 19.748 for trigonometric and polynomial displacement functions respectively. These values indicated a maximum percentage difference of 0.051% with those in the literature. It is observed that the resonating frequency increases as the ratio of out–of–plane displacement to the thickness of plate (w/t) increases. The conclusion is that this new approach is simple and the derived equation is adequate for predicting the linear and nonlinear resonating frequencies of a thin rectangular plate for various boundary conditions.

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.

Large Deflections of Rectangular Plates

1971 ◽  
Vol 15 (02) ◽  
pp. 164-171
Author(s):  
Norman Jones ◽  
R. M. Walters
Keyword(s):  
Approximate Method ◽  
Rectangular Plate ◽  
Large Deflections ◽  
Rectangular Plates ◽  
Finite Deflections ◽  
Perfectly Plastic ◽  
Aspect Ratios ◽  
Plastic Analysis ◽  
Theoretical Procedure

An approximate rigid, perfectly plastic analysis which retains the influence of finite deflections is presented herein for a uniformly loaded, fully clamped rectangular plate. This theoretical procedure provides reasonable engineering estimates of the permanent deflections of rectangular plates according to the recent experiments of Hooke and Rawlings on plates with aspect ratios in the range 1/3 ≤ β ≤ 1. The approximate method also predicts values which agree fairly well with the tests of Young on long rectangular plates β = 1/3), and for large permanent deflections gives similar values to the analysis by Greenspon when β = 1.

Loss Factor for a Rectangular Plate of Parabolic Thickness Variation

1977 ◽  
Vol 99 (3) ◽  
pp. 799-801
Author(s):  
S. P. Nigam ◽  
G. K. Grover ◽  
S. Lal
Keyword(s):  
Rectangular Plate ◽  
Variable Thickness ◽  
Fundamental Mode ◽  
Loss Factor ◽  
Thickness Variation ◽  
Stress System ◽  
Rectangular Plates ◽  
Uniform Thickness ◽  
Aspect Ratios ◽  
Mode Loss

The importance of the internal damping and of the evaluation of the fundamental mode loss factor of structural members subjected to multiaxial stress system is well known. A good amount of work is available on the elastic vibrations of ectangular plates of uniform thickness but it appears that little work has been done on vibrations of rectangular plates of variable thickness, though such cases are of interest in the aeronautical field since they approximate to wing sections. In the present work, the fundamental mode loss factors for a simply supported rectangular plate with parabolic thickness variation in X direction have been evaluated for different combinations of the aspect ratios and the taper parameters. An approximate relationship has been obtained which correlates the loss factor for the plate of variable thickness with that of a plate of uniform thickness.

Three-Dimensional Vibration Analysis of Rectangular Plates With Mixed Boundary Conditions

2005 ◽  
Vol 72 (2) ◽  
pp. 227-236 ◽  
Author(s):  
D. Zhou ◽  
Y. K. Cheung ◽  
S. H. Lo ◽  
F. T. K. Au
Keyword(s):  
Boundary Conditions ◽  
Rectangular Plate ◽  
Mixed Boundary ◽  
Plate Thickness ◽  
Three Dimensional ◽  
Mixed Boundary Conditions ◽  
Boundary Function ◽  
Rectangular Plates ◽  
Characteristic Boundary ◽  
Fixed Boundaries

Three-dimensional vibration solutions are presented for rectangular plates with mixed boundary conditions, based on the small strain linear elasticity theory. The analysis is focused on two kinds of rectangular plates, the boundaries of which are partially fixed while the others are free. One of those studied is a rectangular plate with partially fixed boundaries symmetrically arranged around four corners and the other one is a rectangular plate with partially fixed boundaries around one corner only. A global analysis approach is developed. The Ritz method is applied to derive the governing eigenvalue equation by minimizing the energy functional of the plate. The admissible functions for all displacement components are taken as a product of a characteristic boundary function and the triplicate Chebyshev polynomial series defined in the plate domain. The characteristic boundary functions are composed of a product of four components of which each corresponds to one edge of the plate. The R-function method is applied to construct the characteristic boundary function components for the edges with mixed boundary conditions. The convergence and comparison studies demonstrate the accuracy and correctness of the present method. The influence of the length of the fixed boundaries and the plate thickness on frequency parameters of square plates has been studied in detail. Some valuable results are given in the form of tables and figures, which can serve as the benchmark for the further research.

Vibrations of Axially Moving Vertical Rectangular Plates in Contact with Fluid

2016 ◽  
Vol 16 (02) ◽  
pp. 1450092 ◽  
Author(s):  
Yan Qing Wang ◽  
Sen Wen Xue ◽  
Xiao Bo Huang ◽  
Wei Du
Keyword(s):  
Boundary Conditions ◽  
Plate Theory ◽  
Fluid Pressure ◽  
Added Mass ◽  
Vibration Characteristics ◽  
Rectangular Plates ◽  
Moving Plate ◽  
Bernoulli’S Equation ◽  
Aspect Ratios ◽  
Axially Moving

The vibration characteristics of an axially moving vertical plate immersed in fluid and subjected to a pretension are investigated, with a special consideration to natural frequencies, complex mode functions and critical speeds of the system. The classical thin plate theory is adopted for the formulation of the governing equation of motion of the vibrating plates. The effects of free surface waves, compressibility and viscidity of the fluid are neglected in the analysis. The velocity potential and Bernoulli’s equation are used to describe the fluid pressure acting on the moving plate. The effect of fluid on the vibrations of the plate may be regarded as equivalent to an added mass on the plate. The formulation of added mass is obtained from kinematic boundary conditions of the plate–fluid interfaces. The effects of some system parameters such as the moving speed, stiffness ratios, location and aspect ratios of the plate and the fluid-plate density ratios on the above-mentioned vibration characteristics of the plate–fluid system are investigated in detail. Various different boundary conditions are considered in the study.

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