Free vibration analysis of mindlin plates with parabolically varying thickness

The free transverse vibration characteristics of rectangular plates, including the effect of transverse shear deformation and rotary inertia, are presented. A method based on the variational procedure in conjunction with the finite difference technique is used to determine the natural frequencies and mode shapes. The convergence characteristics of the present method are studied by varying the thickness parameter and running a comparison with the classical thin plate solutions. The effects of the thickness parameter and taper thickness ratio on the vibration characteristics are studied. The predictions are presented for the simply supported and clamped plates and they are compared with existing solutions based on the thin plate theory.     

Three-dimensional vibration analysis of thick, circular and annular plates with nonlinear thickness variation

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, circular and annular plates with nonlinear thickness variation along the radial direction. Unlike conventional plate theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. Displacement components u_s, u_z, and u_θ in the radial, thickness, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the s and z directions. Potential (strain) and kinetic energies of the plates are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the plates. Numerical results are presented for completely free, annular and circular plates with uniform, linear, and quadratic variations in thickness. Comparisons are also made between results obtained from the present 3-D and previously published thin plate (2-D) data.     

Vibration and buckling analysis of axisymmetric polar orthotropic circular plates

The vibration and stability analysis of polar orthotropic circular plates using the finite element method is discussed. In order to formulate the eigenvalue problems associated with the vibration and stability analyses, the clement stiffness, mass, and stability coefficient matrices are presented. By assuming the static displacement function, which is an exact solution of the polar orthotropic circular plate equation, approximates the vibration and buckling modes, the mass and stability coefficient matrices are readily derived from the given displacement function. Results showing the effects of orthotropy on natural frequencies and buckling loads are compared with their isotropic counterpart.     

Free vibration analysis of orthotropic plates by the initial value method

A modified method of the classical initial value technique is presented for the free vibration analysis of rectangular orthotropic plates. The method employs finite difference operators in one direction and uses integration in the other direction. The solutions are superimposed to satisfy the boundary conditions. A detailed discussion of the method is given. A comparative study is presented of the initial value method and other numerical methods. Three examples that show the accuracy of the method are presented.     

Post-buckling behaviour of cylindrically orthotropic annular plates with uniform internal radial load

A finite element formulation for the study of the post-buckling behaviour of cylindrically orthotropic annular plates with uniform internal radial load is presented in this paper. The results for radial load ratios are presented in the form of empirical formulae in terms of the central deflection of the annulus to the thickness ratios of the plates for the first time in the literature for various values of the orthotropic parameters and radii ratios.     

Optimal design of rigid-plastic annular plates with piecewise constant thickness

Rigid-plastic stepped annular plates under uniform pressure load are considered. Both plate edges are supported. Four types of boundary conditions are studied. Tresca's yield condition is used. Such plate dimensions are sought for which the plate of constant volume has the maximal load carrying capacity.Notation a, d, R, h ₁, h₂ plate dimensions (Fig. 1) - Q ^* shear force - Q dimensionless shear force - M _r ^* radial bending moment - M ₁ dimensionless radial bending moment - M _t ^* circumferential bending moment - M ₂ dimensionless circumferential bending moment - M _k maximum value of bending moments in the rigid region - p ^* uniform pressure load - p dimensionless uniform pressure load - r radial coordinate - x dimensionless radial coordinate - , , dimensionless parameters for plate (5) - V plate volume - dimensionless plate volume - M ₀ ^* yield moment - ₀ yield stress - p ₀ load carrying capacity - p ₀ ^m maximum value of the load carrying capacity - p ₀ ^u load carrying capacity for uniform plate - ^m, ^m optimal parameters, which correspond to the maximum value ofp ₀ - s _i radius of circle between different plastic stages - () /x     

Post-buckling of cylindrically orthotropic linearly tapered circular plates by finite element method

The post-buckling behavior of tapered circular plates with cylindrically orthotropic material properties is studied in this paper through a finite element formulation. The results in the form of an empirical formula for the radial load ratios are presented for various values of the taper parameter and orthotropy parameter. Both simply supported and clamped boundary conditions are considered.     

Axisymmetric vibration of layered orthotropic spherical shells of variable thickness

Axisymmetric free vibration of thick orthotropic spherical shells with linearly varying thickness along the meridian is analysed. Both deep and shallow shells are considered for the analysis. The effect of thickness variation and lay-up are considered. The results are presented for clamped and hinged boundary conditions. A thick shell finite element is used for the analysis. It is observed that the thickness variation and lay-up have a pronounced effect on the natural frequencies and a considerable increase of the natural frequencies can be achieved by selecting a proper combination of lay-up and thickness variation.     

Ultimate capacity of plates containing holes under linearly varying edge displacements

An approximate method is suggested for evaluating the strength of plates containing circular holes when subjected to linearly varying edge displacements. The peak load is evaluated as the one causing the first yield at the edges. The proposed method of solution is validated by a large number of tests on scaled-down models carried out on a specially fabricated rig. Methods of obtaining design charts for practical plates with varying sizes of diameters and locations of openings are proposed.     

Asymmetric vibration and dynamic stability of bimodulus thick annular plates

The free vibration and dynamic stability problems of asymmetric bimodulus thick annular plates are studied. The annular element with Lagrangian polynomials and trigonometric functions as shape function is developed. The element is based on the Mindlin plate theory so that the effect of transverse shear deformation is included. The dynamic stability of an annular plate subjected to a combination of a pure dynamic bending and a uniform dynamic extensional stress in the plane of the plate is investigated. The non-axisymmetric modes are shown to have significant effects in the annular bimodulus plates.     

Prebuckling optimal design of orthotropic variable thickness plates for inplane loading

This paper is concerned with the optimal design of orthotropic rectangular plates for shear and compressive buckling loads. Expressions for the total potential as series summations are derived for double cosine thickness varying plates and hybrid double sine thickness varying plates. An efficient Fortran coded program for analysis of simply supported plates that utilizes the Rayleigh-Ritz method until convergence has been developed. It incorporates an optimization module that reactivates analysis as required. Four types of orthotropic plates were optimized in a one parameter optimization search. Results indicate a 33% increase in capacity for shear loading and an 89% increase in capacity for compressive loading.Notation A matrix of generalized eigenvalue problem - a vector of buckled shape amplitudes - a plate dimension inx-direction - a _mn amplitude of buckled shape of plate - B matrix of generalized eigenvalue problem - B ₁ –B ₁₆ coefficients of cube of plate thickness - B ₁ –B ₉ coefficients of square of plate thickness - b plate dimension iny-direction - CCEF parameter of the FORTRAN program - COEF parameter of the FORTRAN program - COEFFICIENTS module of the FORTRAN program - D ₁,D _x,D _y,D _xy flexural rigidities for orthotropic plate - flexural rigidities for orthotropic plate per cubic thickness - DI1-DI16 modules of the FORTRAN program - DI1A-DI16A modules of the FORTRAN program - DIO1-DIO9 modules of the FORTRAN program - DIO1A-DIO9A modules of the FORTRAN program - E Young's modulus - EIGEN module of the FORTRAN program - F total potential energy of buckled plate - F _c total potential for double-cosine shape representation - F _s total potential for hybrid shape representation - FK1-FK16 modules of the FORTRAN program - FK1A-FK16A modules of the FORTRAN program - FKS1-FKS16 modules of the FORTRAN program - FKS1A-FKS16A modules of the FORTRAN program - t _avr average thickness of nonuniform plate - t(x, y) thickness varying withx andy - t ₁,t ₂,t ₃ shape coefficients - I ₁ –I ₁₆ types of integrals - J ₁ –J ₁₆ types of integrals - K ₁ –K ₅ numerical factors - L ₁ –L ₉ types of integrals - L(t) differential operator - M ₁ –M ₉ types of integrals - m number of waves inx-direction - MAIN module of the FORTRAN program - N _xy shearing force per unit distance in middle surface of plate - NWRK parameter of the FORTRAN program - n number of waves iny-direction - N _x normal force inx-direction in middle - OPTIMIZER surface of plate - OPTIMIZER module of the FORTRAN program - p integer - POSTMAN module of the FORTRAN program - q integer - S area of plate - T work of external forces - U strain energy of bending - V volume of plate - w lateral displacement of plate - {W} vector of derivatives (–w, _xx ; –w _yy ;w _xy ) - x coordinate direction - y coordinate direction - z coordinate direction - constant of proportionality - combination of integers - numerical factor - variation - numerical factor - strain energy density - Poisson's ratio On leave from the Technion-Israel Institute of Technology, Technion City, Haifa 32000, Israel     

Modeling vibration frequencies of annular plates by regression based neural network

In this paper, a regression based artificial neural network model for multi input single output (MISO) systems has been developed. The time devoted in training this model is considerably less in comparison with the traditional ANN model and the number of neurons in the hidden layer can be fixed by choosing a regression polynomial of desired degree. The proposed model has been used and simulated for an example problem of transverse vibrations of plates viz. vibration of circular and elliptic annular plates. There exist nine different boundary conditions for the present example problem which are all simulated using the model. The training and testing with unseen patterns show the efficacy and reliability of the proposed technique for the MISO systems. Comparison of the proposed model with the traditional ANN model shows the former to be better and much efficient.     

On minimum compliance problems of thin elastic plates of varying thickness

The paper deals with two minimum compliance problems of variable thickness plates subject to an in-plane loading or to a transverse loading. The first of this problem (called also the variable thickness sheet problem) is reduced to the locking material problem in its stress-based setting, thus interrelating the stress-based formulation by Allaire (2002) with the kinematic formulation of Golay and Seppecher (Eur J Mech A Solids 20:631–644, 2001). The second problem concerning the Kirchhoff plates of varying thickness is reduced to a non-convex problem in which the integrand of the minimized functional is the square root of the norm of the density energy expressed in terms of the bending moments. This proves that the problem cannot be interpreted as a problem of equilibrium of a locking material. Both formulations discussed need the numerical treatment in which stresses (bending moments) are the main unknowns.     

Flexural vibration of shear deformable circular and annular plates on ring supports

This paper presents an analysis of free flexural vibration of circular and annular Mindlin plates lying on multiple internal concentric ring supports. The Rayleigh-Ritz method with an admissible displacement function expressed in terms of a set of simple polynomials is used to obtain the governing eigenvalue equation. The natural frequencies of circular and annular Mindlin plates with one or two ring supports having inner and outer peripheries with different combinations of free, simply supported and clamped conditions are determined. A comprehensive set of vibration results is presented and, where possible, these have been verified with existing solutions published in the open literature. No previous theoretical results for thick annular plates supported with internal rings are known to exist. Thus the study should be useful and valuable to engineers and researchers.     

Deflection and stress analysis of orthotropic plates in flexure

A finite difference (FD) method is developed for computing deflections and stresses in rectangular orthotropic plates subjected to transverse flexural loadings. The plate material is elastic with symmetry axes parallel to the plate edges, and plate deflections are assumed to be small. The FD equations are solved iteratively using successive over relaxation techniques. The numerical procedure converges rapidly and is simple enough to be implemented in a minicomputer program for the design analysis of composite plates. As an example the program is applied to the analysis of simply supported and clamped square plates under uniform pressure or point loads, and the effect of material anisotropy on plate behaviour is discussed.     

On the vibration of annular, circular and sectorial plates with cut-outs or on partial supports

Natural frequency parameters are given for the free vibration of annular, circular and sectorial plates which may be supported along only parts of their boundaries or may have sectorial cut-outs. The results are obtained by using a Ritz solution previously presented for the study of the vibration of cracked circular or annular plates and which makes use of a minimum number of sectorial elements joined together by means of artificial springs, the stiffnesses of which are permitted to become very high, thus satisfying the required continuity conditions.     

Optimization of inelastic annular plates with cracks

Stepped plates made of a fibre-reinforced composite material are studied. The plates are subjected to the initial impulsive loading, which imparts the plates given kinetic energy. Optimal designs corresponding to the minimum of the maximal residual deflections are established making use of the variational methods of the theory of optimal control. Maximal deflections are evaluated with the method of mode form motions.     

Nonlinear axisymmetric static analysis of orthotropic thin circular plates with elastically restrained edge

This study deals with the nonlinear axisymmetric static analysis of elastic orthotropic thin circular plates with elastically restrained edges for rotation as well as in-plane displacement. Von-Karman equations have been employed and spatial discretisation has been done by using the method of orthogonal point collocation. Numerical results have been presented for orthotropic plates with hinged edge, simply supported edge, movable clamped edge and immovable clamped edge. Results are given for the whole range of elastic edge restraints from the movable simply supported edge to the immovable clamped edge. The effect of a prescribed inplane force or an inplane displacement at the edge, on the static response under a uniformly distributed transverse load has been investigated. Several new results are presented. The present results are in good agreement with the available results.     

Numerical analysis of nonsymmetric postbuckling behaviour of elastic annular plates

A modified shooting method is used combined with the Kantorovich orthogonalization method. An appropriate formulation of equations and an algorithm are given. Numerical examples illustrate the efficiency of the proposed method.     

Fundamental frequencies of annular plates with internal cracks

The Rayleigh method with a simple sub-sectioning technique has been used to obtain fundamental frequencies of annular plates with internal concentric cracks. Single-term deflection functions which satisfy the respective boundary conditions for the bending of an annular plate are used as admissible functions. The predicted fundamental frequencies are in good agreement with the reported analytical results for annular plates without any internal cracks. Numerical results for fundamental frequencies are presented for annular plates with internal cracks simply supported and clamped on both the outside and inside edges of the plates.