A simple finite element analysis of axisymmetric vibration of annular and circular plates

The axisymmetric vibration of annular and circular plates, isotropic or polar orthotropic, is analysed by an axisymmetric finite element. The features of the present study are that: (i) the formulation of the finite element is based on elasticity theory and has no assumptions as in the conventional plate theory-based analysis, yet is still simple and ready for use, and (ii) the boundary conditions are satisfied exactly and the significant effect of boundary conditions on the vibration frequency is demonstrated. Comparisons with alternative solutions show the accuracy of the present approach and the inadequacy of conventional methods in dealing with the vibration of annular and circular plates with simply supported boundary conditions.     

Natural frequencies of shear deformable rhombic plates with clamped and simply supported edges

The natural vibrations of thick and thin rhombic plates with clamped and simply supported edges are analyzed, using assemblages of nine-node Lagrangian isoparametric quadrilateral C⁰ continuous finite elements based on a higher-order shear deformable thick plate theory. Here, additional nodal displacement degrees of freedom are derived by retaining higher-order powers of the thickness coordinate in the in-plane displacement fields, which in turn allows for the proper representation of the transverse shear strains of thick plates. Essential rotary inertia terms are derived and included in the present analysis. Nondimensional frequencies are calculated for thick and thin rhombic plates having various combinations of clamped and simply supported edge conditions, and skew angles. The efficacy of using higher-order shear deformable plate finite elements for predicting the in-plane vibration modes of rhombic plates is found to increase as the span-to-thickness ratio decreases and the skew angle increases. The present work shows that higher-order shear deformable finite elements essentially eliminate the transverse shear over-correction of thick rhombic plate frequencies that is produced when finite elements based on the widely used first-order Reissner-Mindlin plate theory are utilized.     

Statistical energy analysis of non-resonant response of isotropic and orthotropic plates

The conventional SEA model considers only the resonant part of the structural response to an acoustic excitation. Therefore, this study investigates non-resonant responses of isotropic and orthotropic plates to acoustically induced vibrations in a reverberation chamber. A modified SEA model is introduced to predict the non-resonant plate response. The estimated non-resonant and resonant responses are then compared with those obtained experimentally, and good agreement is observed for isotropic and orthotropic plates. For an isotropic plate with a small dissipation loss factor, when the non-resonant part is ignored, the estimated response can lead to significant errors at frequencies near and above the critical frequency, while large errors may occur at frequencies below the critical frequency for an orthotropic plate with a high dissipation loss factor. The experimental study indicates that the non-resonant response component should be included in the estimated responses to enhance predictive accuracy.     

Plastic analysis of perforated plates for orthotropic yield criteria

The concept of an equivalent orthotropic material is used in evaluating the collapse loads for perforated plates. Uniformly loaded circular plates are investigated. Limit pressure solutions are obtained for both the simply supported and rigidly built-in edge conditions. Statically admissible radial and circumferential bending moment fields are found and the associated velocity fields are shown to be kinematically admissible. Numerical results are given in dimensionless form covering arbitrary plate geometries over the entire range of material orthotropy.These solutions are quite useful in the plastic analysis and design of perforated plates used as pressure vessel heads, tube sheets, reactor core support plates and the like. Perforated plates have considerably higher effective yield strengths when subjected to equi-biaxial loading than when subjected to loading of arbitrary biaxiality and orientation with respect to the penetration pattern. Thus, the equivalent orthotropic plastic material concept is ideally suited to the analysis of such plates. The resulting limit load pressures are substantially lower than the values obtained using isotropic Tresca yield criterion based on the yield strength for equi-biaxial loading. The results are substantially higher than the values obtained using the maximum isotropic Tresca yield condition falling entirely within the orthotropic yield surface.     

Free vibration analysis of a thin rectangular plate with multiple circular and rectangular cut-outs

The present work is to investigate the variation in the natural frequency of a rectangular plate with the change in position of cut outs of different sizes along its diagonal and major axis line by considering different aspect ratios of plates. In this regard the free vibration analysis of an isotropic rectangular plate of various aspect ratios with different sizes of multiple circular and rectangular cut-outs is carried out by using an Independent coordinate coupling method (ICCM) under simply supported boundary condition. The ICCM utilizes independent coordinates separately for plate domain and hole domain, in which the reduced mass and stiffness matrices can be derived, by matching the deflection conditions for each hole imposed on the expressions. The resulting equation is useful for the calculation of the Eigen values. The position, size, and number of cutouts have been varied in all the possible ways to investigate their effects on the natural frequency of the rectangular plate.     

Vibrations of an initially stressed transversely isotropic circular thick plate

Equations of motion for a transversely isotropic circular thick plate in a general state of nonuniform initial stress where the effects of transverse shear and rotary inertia are included are derived. The natural frequencies of axisymmetric clamped plates and simply supported plates subjected to initial stresses are investigated. The initial stress is taken to be a combination of a pure bending stress plus an extensional stress in the plane of the plate. These equations are solved by the Galerkin method. The effects of various parameters on the natural frequencies are studied.     

Analysis of sandwich plates with unbalanced cross-ply faces

The analysis of rectangular sandwich plates constructed of an orthotropic core and unbalanced cross-ply face plates is presented. A double Fourier series approach is used for simply supported sandwich plates under lateral loads. Results are compared to the corresponding sandwich plate results with orthotropic faces. The results indicate that the effect of bending-membrane coupling depends mainly upon the relative thicknesses of the core and faces; other factors include the shear stiffnesses of the core, the degree of anisotropy of the individual plies, the total number and lay-up of plies in the faces and the aspect ratio of the plate.     

Post-buckling stability of orthotropic, linear elastic, rectangular plates under biaxial loads

The potential energy function in a neighborhood of the buckling point for a biaxially loaded, linear elastic, orthotropic, simply supported, rectangular plate is obtained from a perturbation analysis. The perfect system has two independent bifurcation parameters, the loads. The postbuckling behavior is stable, and a good estimate is found for the post-buckling deflection of the plate in terms of the loads. The orthotropic constitutive equation interacts with the biaxial loading to cause a bimodal buckling point for certain combinations of loads; this point is also stable. Away from the bimodal point, the equilibrium surface is described as a cusp cylinder. At the bimodal point, the equilibrium surface is a double cusp.     

Application of the singularity function method to semi-infinite orthotropic rectangular plates on an elastic foundation

The singularity function method is applied to solve the structural problems of semi-infinite orthotropic rectangular plates on a Winker-type elastic foundation. Previous application by Selek et al. (Int. J. Mech. Sci., 25 (1983) 397) of the singularity function to plates on an elastic foundation was limited only to isotropic cases. Foundation-free orthotropic plate problems have been previously solved Adewale et al. (Int. J. Mech. Sci., 31 (1989) 853). In this paper, the singularity function method technique is extended to orthotropic rectangular plate on Winkler-type foundation. The isotropic rectangular plate results of Selek et al. (Int. J. Mech. Sci., 25 (1983) 397) and the foundation-free orthotropic rectangular plate results provided by Adewale et al. (Int. J. Mech. Sci., 31 (1989) 853) are recovered as special cases.     

A study of the fundamental frequency characteristics of eccentrically and concentrically simply supported stiffened plates

The paper investigates the fundamental frequency characteristics of eccentrically and concentrically simply supported stiffened supported plates. As a first stage, a numerical procedure for the computation of the fundamental frequency is presented. The strain energy of the assembled plate/stiffener elements is derived in terms of generalized in- and out-of-plane displacement functions and Mathematical Programming is then used to determine the lowest natural frequency. The prediction of the described algorithm is verified with other numerical procedures like finite-element, finite-strip and finite-difference methods. Results are then presented showing the influence of the plate/stiffener geometric parameters on the fundamental frequency of the structure with various concentric and eccentric stiffening configurations.     

Free vibration analysis of isotropic and orthotropic triangular plates

A highly accurate and computationally efficient numerical method is developed for the flexural vibration of isotropic and orthotropic triangular plates. A set of two-dimensional orthogonal plate functions is used as an admissible displacement function in the Rayleigh-Ritz method to obtain the natural frequencies and mode shapes for the plates. From a prescribed starting function satisfying the boundary conditions, the higher terms in the orthogonal plate functions are constructed using the Gram-Schmidt orthogonalization process. Natural frequencies and mode shapes are obtained for three triangular plates with different support conditions. The obtained numerical results are presented, and the isotropic case is verified with other numerical methods in the literature.     

Analytical solution of a two variable refined plate theory for bending analysis of orthotropic Levy-type plates

This paper presents analytical solutions of deflection and stress for orthotropic plates using a two variable refined plate theory. The theory accounts for parabolic variation of transverse shear stress through the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factor. Additional features of the theory are that it has strong similarity with classical plate theory in many aspects, and the number of involved variables is only two as against three in case of other shear deformation theories. The Levy-type solution procedure in conjunction with the state space concept is used to determine the closed-form solutions for orthotropic rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions. Comparison studies are performed to verify the validity of the present results. Finally, the effects of thickness ratio, modulus ratio and aspect ratio on the deflection and stress of orthotropic plates are investigated and discussed.     

Free vibration analysis of corroded steel plates

Vibration analysis of unstiffened/stiffened plates has long been studied due to its importance in the design and condition assessments of ship and offshore structures. Corrosion is inevitable in steel structures and has been so far considered in strength analysis of structures. We studied the free vibration of pitted corroded plates with simply supported boundary conditions. Finite element analysis, with ABAQUS, was used to determine the natural frequencies and mode shapes of corroded plates. Influential parameters including plate aspect ratio, degree of pit, one-sided/both-sided corroded plate, and different corrosion patterns were investigated. By increasing the degree of corrosion, reduction of natural frequency increases. Plate aspect ratio and plate dimensions have no influence on reduction of natural frequency. Different corrosion patterns on the surface of one-sided corroded plates have little influence on reduction of natural frequency. Ratio of pit depth over plate thickness has no influence on the reduction of natural frequency. The reduction of natural frequency in both-sided corroded plates is higher than one-sided corroded plates with the same amount of total corrosion loss. Mode shapes of vibration would change due to corrosion, except square mode shapes.     

A novel approach for in-plane/out-of-plane frequency analysis of functionally graded circular/annular plates

An exact closed-form frequency equation is presented for free vibration analysis of circular and annular moderately thick FG plates based on the Mindlin's first-order shear deformation plate theory. The edges of plate may be restrained by different combinations of free, soft simply supported, hard simply supported or clamped boundary conditions. The material properties change continuously through the thickness of the plate, which can vary according to a power-law distribution of the volume fraction of the constituents, whereas Poisson's ratio is set to be constant. The equilibrium equations which govern the dynamic stability of plate and its natural boundary conditions are derived by the Hamilton's principle. Several comparison studies with analytical and numerical techniques reported in literature and the finite element analysis are carried out to establish the high accuracy and superiority of the presented method. Also, these comparisons prove the numerical accuracy of solutions to calculate the in-plane and out-of-plane modes. The influences of the material property, graded index, thickness to outer radius ratios and boundary conditions on the in-plane and out-of-plane frequency parameters are also studied for different functionally graded circular and annular plates.     

Natural frequencies of stepped thickness rectangular plates

This paper presents the natural frequencies of stepped thickness square and rectangular plates together with the mode shapes of vibration. The transverse deflection of a stepped thickness plate is written in a series of the products of the deflection functions of beams parallel to the edges satisfying the boundary conditions, and the frequency equation of the plate is derived by the energy method. By use of the frequency equation, the natural frequencies (the eigenvalues of vibration) and the mode shapes are calculated numerically in good accuracy for square and rectangular plates with edges simply supported or elastically restrained against rotation, having square, circular or elliptical stepped thickness, from which the effects of the stepped thickness on the vibration are studied.     

Analysis of Natural Vibrations of Round Flexing Membrane-Type Piezoelectric Transducers with an Arbitrary Dimension Ratio

The problem of natural vibrations of a round exing membrane-type piezoelectric transducer (piezoelectric plate) with an arbitrary dimension ratio is solved in the exact formulation using the finite element method for two variants of fastening the membrane (simply supported and rigidly fixed plate). The natural frequency spectra are analyzed in the resonance and antiresonance regimes. The displacement distributions over the plate surfaces are also analyzed and the dependences of the dynamic electromechanical coupling coefficient are investigated as functions of the geometric dimensions of the piezoelectric transducer, the plate material, and the piezoelectric ceramic composition. The optimum geometric dimensions of the transducer which ensure the maximum dynamic electromechanical coupling coefficient are determined and the value of the coefficient is estimated.     

两对边简支两对边夹支矩形薄板裂纹诊断研究初探

提出一种用无因次的频率系数、裂纹长度系数及裂纹位置系数来表示含裂纹矩形薄板的裂纹损伤与自然频率关系的方法。采用计算机仿真技术建立了两对边得支两对边夹支矩形板裂纹损伤的标准谱，并研制出相应的普于人工神经网络的裂纹诊断系统。该系统采用子网络集成方式，各子网独立训练，训练过程应用改进的遗传算法，有效地克服了传统的遗传算法随机选取初始种群和遗传素质低的缺点，该系统对相似结构具有通用性。     

A DXDR large deflection analysis of uniformly loaded square, circular and elliptical orthotropic plates using non-uniform rectangular finite-differences

A finite-difference analysis of the large deflection response of uniformly loaded square, circular and elliptical clamped and simply-supported orthotropic plates is presented. Several types of non-uniform (graded) mesh are investigated and a mesh suited to the curved boundary of the orthotropic circular and elliptical plate is identified. The DXDR method-a variant of the DR (dynamic relaxation) method-is used to solve the finite-difference forms of the governing orthotropic plate equations. The DXDR method and irregular rectilinear mesh are combined along with the Cartesian coordinates to treat all types of boundaries and to analyze the large deformation of non-isotropic circular/elliptical plates. The results obtained from plate analyses demonstrate the potential of the non-uniform meshes employed and it is shown that they are in good agreement with other results for square, circular and elliptical isotropic and orthotropic clamped and simply-supported plates in both fixed and movable cases subjected to transverse pressure loading.     

Vibration of bilayered cylindrical shells with layers of different materials

In this analysis, a comparative study for natural frequencies of two-layered cylindrical shells was presented with one layer composed of functionally graded material and the other layer of isotropic material. Love’s thin shell theory was exploited for the strain-displacement and curvature-displacement relationships. For governing frequency equations, the Rayleigh-Ritz method was utilized to minimize the Lagrangian functional in the form of an eigenvalue problem. Frequency spectra were computed for long, short, thick, and thin cylindrical shells by varying the nondimensional geometrical parameters, length-to-radius and thickness-to-radius ratios for a simply supported end condition. Influence of different configurations of cylindrical shells on the shell frequencies was studied. For validity, the results obtained were compared with some results of isotropic and single-layered functionally graded cylindrical shells from the literature.     

Elastic large deflection analysis of isotropic rectangular Mindlin plates

Governing equations for the large deflection analysis of isotropic rectangular Mindlin plates are introduced and their solution using the DR algorithm is briefly outlined. Two computer programs, based respectively on interlacing and non-interlacing finite-differences, have been developed for the numerical solution of these equations. The programs have been verified by analysing a variety of thin and moderately thick plate problems for which alternative solutions are available. Sample results comparisons are presented in order to quantify the accuracy of the DR results. The non-interlacing finite-difference DR program is then used to compute new results for uniformly loaded square moderately thick plates with simply supported, clamped and combined simply supported and clamped edges.