Vibrations of Sectorial Plates Having Corner Stress Singularities

1993 ◽  
Vol 60 (1) ◽  
pp. 134-140 ◽  
Author(s):  
A. W. Leissa ◽  
O. G. McGee ◽  
C. S. Huang
Keyword(s):  
Outer Boundary ◽  
Full Range ◽  
Ritz Method ◽  
Trigonometric Polynomials ◽  
Mode Shapes ◽  
Vertex Angle ◽  
Stress Singularities ◽  
Simply Supported ◽  
Plate Vibrations ◽  
Continuous Boundary

A procedure is presented for determining the free vibration frequencies and mode shapes of sectorial plates having re-entrant corners (i.e., vertex angles exceeding 180 degrees). No correct results for such problems have been found in the vast literature of plate vibrations. The procedure is applicable to sectorial plates having arbitrary (but continuous) boundary conditions (e.g., clamped, simply supported, or free) along the two radial edges and the circular edge. It is based upon the Ritz method, but utilizes two sets of admissible functions simultaneously. One set consists of algebraic-trigonometric polynomials. The other is the set of corner functions derived by Williams (1952) to deal with the bending stress singularities which may arise at the corner when the vertex angle becomes large. The method is demonstrated for sectorial plates having all edges simply supported, which yields the strongest singularity in a re-entrant corner. Frequencies are compared with those obtained from an analytical solution involving Bessel functions. It is shown that the latter solution is invalid for re-entrant corners. Analytical solutions are also obtained for annular sectorial plates having very small ratios of inner to outer boundary radii. These solutions are found to be consistent with those using polynomials and corner functions. Accurate fundamental frequency data is presented for simply supported sectorial plates having three values of Poisson’s ratio (0, 0.3, 0.5) and the full range of vertex angles (0 < α ≤ 360 deg).

Accurate Vibration Analysis of Simply Supported Rhombic Plates by Considering Stress Singularities

1995 ◽  
Vol 117 (3A) ◽  
pp. 245-251 ◽  
Author(s):  
C. S. Huang ◽  
O. G. McGee ◽  
A. W. Leissa ◽  
J. W. Kim
Keyword(s):  
Ritz Method ◽  
Mode Shapes ◽  
Transverse Displacement ◽  
Algebraic Polynomials ◽  
Stress Singularities ◽  
Simply Supported ◽  
Free Transverse Vibration ◽  
Skewed Plates ◽  
Convergence Of Solution ◽  
Skew Angles

This is the first known work which explicitly considers the bending stress singularities that occur in the two opposite, obtuse corner angles of simply supported rhombic plates undergoing free, transverse vibration. The importance of these singularities increases as the rhombic plate becomes highly skewed (i.e., the obtuse angles increase). The analysis is carried out by the Ritz method using a hybrid set consisting of two types of displacement functions, e.g., (1) algebraic polynomials and (2) corner functions accounting for the singularities in the obtuse corners. It is shown that the corner functions accelerate the convergence of solution, and that these functions are required if accurate solutions are to be obtained for highly skewed plates. Accurate nondimensional frequencies and normalized contours of the vibratory transverse displacement are presented for simply supported rhombic plates with skew angles ranging to 75 deg. (i.e., obtuse angles of 165 deg.). Frequency and mode shapes of isosceles and right triangular plates with all edges simply supported are also available from the data presented.

Vibration Analysis of Anisotropic Simply Supported Plates by Using Variable Kinematic and Rayleigh-Ritz Method

2011 ◽  
Vol 133 (6) ◽  
Author(s):  
Erasmo Carrera ◽  
Fiorenzo Adolfo Fazzolari ◽  
Luciano Demasi
Keyword(s):  
Vibration Analysis ◽  
Ritz Method ◽  
Single Layer ◽  
Free Vibration Analysis ◽  
Mode Shapes ◽  
Virtual Displacement ◽  
Simply Supported ◽  
Order Expansion ◽  
Thickness Layer ◽  
Made In

This work deals with accurate free-vibration analysis of anisotropic, simply supported plates of square planform. Refined plate theories, which include layer-wise, equivalent single layer and zig-zag models, with increasing number of displacement variables are take into account. Linear up to fourth N-order expansion, in the thickness layer-plate direction have been implemented for the introduced displacement field. Rayleigh-Ritz method based on principle of virtual displacement is derived in the framework of Carrera’s unified formulation. Regular symmetric angle-ply and cross-ply laminates are addressed. Convergence studies are made in order to demonstrate that accurate results are obtained by using a set of trigonometric functions. The effects of the various parameters (material, number of layers, and fiber orientation) upon the frequencies and mode shapes are discussed. Numerical results are compared with available results in literature.

Free Vibration and Tension Buckling of Circular Plates With Diametral Point Forces

1990 ◽  
Vol 57 (4) ◽  
pp. 995-999 ◽  
Author(s):  
E. F. Ayoub ◽  
A. W. Leissa
Keyword(s):  
Circular Plate ◽  
Tensile Force ◽  
Ritz Method ◽  
Compressive Force ◽  
Free Vibrations ◽  
Plane Elasticity ◽  
Mode Shapes ◽  
Simply Supported ◽  
Symmetry Classes ◽  
Tensile Forces

This paper presents the first known results for the free vibrations of a circular plate subjected to a pair of static, concentrated forces acting on the boundary at opposite ends of a diameter. The closed-form exact solution of the plane elasticity problem is used to provide the in-plane stress distribution for the vibration problem. A proper procedure using the Ritz method is developed for solving the latter problem for clamped, simply supported, or free boundary conditions. Numerical results are given for the vibration frequencies of a simply supported circular plate, which separate into four symmetry classes of mode shapes. Compressive buckling loads for each symmetry class are determined as a special case as the frequencies decrease to zero with increasing compressive force. Tracking the frequency versus loading data with increasing tensile forces shows that buckling due to tensile force can also occur, and the critical value of the force is found.

Buckling and Vibration of Orthotropic Nonhomogeneous Rectangular Plates With Bilinear Thickness Variation

2011 ◽  
Vol 78 (6) ◽  
Author(s):  
Yajuvindra Kumar ◽  
R. Lal
Keyword(s):  
Orthogonal Polynomials ◽  
Plate Theory ◽  
Ritz Method ◽  
Cartesian Product ◽  
Mode Shapes ◽  
Thickness Variation ◽  
Rectangular Plates ◽  
Two Dimensional ◽  
Classical Plate Theory ◽  
Simply Supported

An analysis and numerical results are presented for buckling and transverse vibration of orthotropic nonhomogeneous rectangular plates of variable thickness using two dimensional boundary characteristic orthogonal polynomials in the Rayleigh–Ritz method on the basis of classical plate theory when uniformly distributed in-plane loading is acting at two opposite edges clamped/simply supported. The Gram–Schmidt process has been used to generate orthogonal polynomials. The nonhomogeneity of the plate is assumed to arise due to linear variations in elastic properties and density of the plate material with the in-plane coordinates. The two dimensional thickness variation is taken as the Cartesian product of linear variations along the two concurrent edges of the plate. Effect of various plate parameters such as nonhomogeneity parameters, aspect ratio together with thickness variation, and in-plane load on the natural frequencies has been illustrated for the first three modes of vibration for four different combinations of clamped, simply supported, and free edges correct to four decimal places. Three dimensional mode shapes for a specified plate for all the four boundary conditions have been plotted. By allowing the frequency to approach zero, the critical buckling loads in compression for various values of plate parameters have been computed correct to six significant digits. A comparison of results with those available in the literature has been presented.

scholarly journals Transverse Vibrations of Clamped and Simply-Supported Circular Plates with Two Dimensional Thickness Variation

2012 ◽  
Vol 19 (3) ◽  
pp. 273-285 ◽  
Author(s):  
N. Bhardwaj ◽  
A.P. Gupta ◽  
K.K. Choong ◽  
C.M. Wang ◽  
Hiroshi Ohmori
Keyword(s):  
Ritz Method ◽  
Normal Modes ◽  
Mode Shapes ◽  
Thickness Variation ◽  
Two Dimensional ◽  
Circular Plates ◽  
Simply Supported ◽  
Modes Of Vibration ◽  
Dimensional Boundary ◽  
Special Case

Two dimensional boundary characteristic orthonormal polynomials are used in the Ritz method for the vibration analysis of clamped and simply-supported circular plates of varying thickness. The thickness variation in the radial direction is linear whereas in the circumferential direction the thickness varies according to coskθ, wherekis an integer. In order to verify the validity, convergence and accuracy of the results, comparison studies are made against existing results for the special case of linearly tapered thickness plates. Variations in frequencies for the first six normal modes of vibration and mode shapes for various taper parameters are presented.

ACCURATE FREQUENCIES AND MODE SHAPES FOR MODERATELY THICK, CANTILEVERED, SKEW PLATES

2007 ◽  
Vol 07 (03) ◽  
pp. 425-440 ◽  
Author(s):  
A. W. LEISSA ◽  
C. S. HUANG ◽  
M. J. CHANG
Keyword(s):  
Plate Theory ◽  
Ritz Method ◽  
The Other ◽  
Mode Shapes ◽  
Mindlin Plate ◽  
Transverse Displacement ◽  
Algebraic Polynomials ◽  
Stress Singularities ◽  
Dependent Variables ◽  
Mindlin Plate Theory

Accurate free vibration frequencies and mode shapes are presented for complete sets of moderately thick, cantilevered skew plates of triangular, trapezoidal and parallelogram shape. These accurate results are obtained by using the Ritz method applied to the Mindlin plate theory. Two sets of functions are employed simultaneously for each of the three dependent variables: transverse displacement (w) and bending rotations (ϕx and ϕy). One set is the widely used algebraic polynomials. The other is the set of corner functions which provide the proper stress singularities in the reentrant clamped-free corner, and accelerates the convergence of the solutions. The extensive frequencies presented are exact to the four digits shown. Corresponding mode shapes are also shown, by means of nodal patterns, most of which are novel in the published literature.

Free Vibrations of Thick, Complete Conical Shells of Revolution From a Three-Dimensional Theory

2005 ◽  
Vol 72 (5) ◽  
pp. 797-800 ◽  
Author(s):  
Jae-Hoon Kang ◽  
Arthur W. Leissa
Keyword(s):  
Ritz Method ◽  
Three Dimensional ◽  
Axial Direction ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Vibration Frequencies ◽  
Shells Of Revolution ◽  
Dimensional Theory ◽  
Conical Shells ◽  
Cylindrical Coordinate

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, complete (not truncated) conical shells of revolution in which the bottom edges are normal to the midsurface of the shells based upon the circular cylindrical coordinate system using the Ritz method. Comparisons are made between the frequencies and the corresponding mode shapes of the conical shells from the authors' former analysis with bottom edges parallel to the axial direction and the present analysis with the edges normal to shell midsurfaces.

Natural Frequencies and Mode Shapes of Elliptic Plates With Boundary Characteristic Orthogonal Polynomials As Assumed Shape Functions

1991 ◽  
Author(s):  
C. Rajalingham ◽  
R. B. Bhat ◽  
G. D. Xistris
Keyword(s):  
Coordinate System ◽  
Orthogonal Polynomials ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Mode Shapes ◽  
Polar Coordinate System ◽  
Free Boundaries ◽  
Natural Modes ◽  
Polar Coordinate ◽  
Boundary Characteristic

Abstract The natural frequencies and natural modes of vibration of uniform elliptic plates with clamped, simply supported and free boundaries are investigated using Rayleigh-Ritz method. A modified polar coordinate system is used to investigate the problem. Energy expressions in Cartesian coordinate system are transformed into the modified polar coordinate system. Boundary characteristic orthogonal polynomials in the radial direction, and trigonometric functions in the angular direction are used to express the deflection of the plate. These deflection shapes are classified into four basic categories, depending on its symmetrical or antisymmetrical property about the major and minor axes of the ellipse. The first six natural modes in each of the above categories are presented in the form of contour plots.

Natural Frequencies of Parallelogrammic Plates Using Characteristic Orthogonal Polynomials in Rayleigh-Ritz Method

1992 ◽  
Author(s):  
L. T. Lee ◽  
W. F. Pon
Keyword(s):  
Boundary Conditions ◽  
Coordinate System ◽  
Orthogonal Polynomials ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Energy Functional ◽  
Comprehensive Treatment ◽  
Energy Functionals ◽  
Simply Supported ◽  
Cartesian Coordinate

Abstract Natural frequencies of parallelogrammic plates are obtained by employing a set of beam characteristic orthogonal polynomials in the Rayleigh-Ritz method. The orthogonal polynomials are generalted by using a Gram-Schmidt process, after the first member is constructed so as to satisfy all the boundary conditions of the corresponding beam problems accompanying the plate problems. The strain energy functional and kinetic energy functionals are transformed from Cartesian coordinate system to a skew coordinate system. The natural frequencies obtained by using the orthogonal polynomial functions are compared with those obtained by other methods with all four edges clamped boundary conditions and greet agreements are found between them. The natural frequencies for parallelogrammic plates with other boundary conditions, such as four edges simply supported, clamped-free and simply supported-free, are also obtained. This method is considered as a better and accurate comprehensive treatment for this type of problems.

An analytical method for full-range mechanical behavior of continuous slab-deck in multi-span simply supported concrete bridges

2021 ◽  
pp. 136943322110427
Author(s):  
Xiang Zhang ◽  
Quan-Sheng Yan ◽  
Bu-Yu Jia ◽  
Zheng Yang ◽  
Ying-Hao Zhao ◽  
...  
Keyword(s):  
Nonlinear Analysis ◽  
Full Range ◽  
Scale Model ◽  
Axial Stiffness ◽  
Spring Model ◽  
Analysis Model ◽  
Rotational Spring ◽  
Simply Supported ◽  
Continuous Slab ◽  
Rotational Spring Model

Connecting the ends of girders with a continuous slab-deck to make a multiple-span simply supported girder bridge provides many benefits, but there is no suitable nonlinear analysis model which considers continuous slab-deck cracking under tension and bending. In this article, the rotational spring model is further refined to replace the restraining effects at both ends of the girder by the simplified mechanical model associated with axial stiffness, bending stiffness, and shear stiffness. Then, it is introduced into the analysis of continuous slab-deck. The more accurate rotations and displacements of both ends of continuous slab-deck are obtained to investigate the more precise moment and tension of the continuous slab-deck. Furthermore, this article presents an improved nonlinear analysis model of continuous slab-deck based on a detailed boundary rotational spring model. The displacements of important positions and the strain of key components in continuous slab-deck after cracking are investigated by numerical analysis and full-scale model test to verify the accuracy of the proposed nonlinear analysis model. The result shows that the nonlinear analysis model presented in this article could successfully evaluate the depth of cracks and the stress of rebars in continuous slab-deck, and it is instructional in predicting the cracking state of the continuous slab-deck and the reinforcement design.

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