Predictions of transverse deflections and in-plane strains in impulsively loaded thin plates

This paper is an extension of earlier work [Nurick et al., Inelastic Behaviour of Plates and Shells. Springer Verlag, Berlin (1986)] in which experimental and theoretical studies of thin circular, square and rectangular plates subject to transverse impulsive loading were presented. Major interest in the experimental study was the range where the permanent displacement is large, up to 12 times the thickness of the plate. The theoretical model was based on a rigid-plastic membrane theory, using the concept of the instantaneous mode approximation technique. Predicted transverse displacements agreed well with experimental data. However, it was assumed that points on the plate surface move only in the direction normal to the initial plane of the plate: as a consequence of that, in-plane membrane strains are incorrectly predicted. In this paper the mode approximation is modified by assuming that points on the plate surface move in a direction which is instantaneously normal to the current plate surface. The modified theory continues to give good predictions of the transverse displacement, and gives quite good estimates of in-plane strains.     

Vibration of circular Mindlin plates with concentric elastic ring supports

This paper studies the vibration behaviour of circular Mindlin plates with multiple concentric elastic ring supports. Utilizing the domain decomposition technique, a circular plate is divided into several annular segments and one core circular segment at the locations of the elastic ring supports. The governing differential equations and the solutions of these equations are presented for the annular and circular segments based on the Mindlin-plate theory. A homogenous equation system that governs the vibration of circular Mindlin plates with elastic ring supports is derived by imposing the essential and natural boundary and segment interface conditions. The first-known exact vibration frequencies for circular Mindlin plates with multiple concentric elastic ring supports are obtained and the modal shapes of displacement fields and stress resultants for several selected cases are presented. The influence of the elastic ring support stiffness, locations, plate boundary conditions and plate thickness ratios on the vibration behaviour of circular plates is discussed.     

General sinusoidal stiffness matrices for buckling and vibration analyses of thin flat-walled structures

This paper is concerned with the derivation of stiffness matrices for the buckling or vibration analysis of any structure consisting of a series of long, thin, flat plates rigidly connected together at their longitudinal edges. Each plate is assumed to be subjected to a basic state of plane stress which is longitudinally invariant, and it is further assumed that the mode of buckling or vibration varies sinusoidally in the longitudinal direction. During buckling or vibration, the edges of any individual plate are subjected to additional systems of forces and moments which are sinusoidally distributed along the edges, and these give rise to sinusoidally varying edge displacements and rotations. Spatial phase differences between the forces and displacements are accounted for by defining them in terms of complex quantities. The sinusoidal edge forces and displacements are split into two uncoupled systems, corresponding to out-of-plane and in-plane displacements, and two stiffness matrices are defined. The out-of-plane stiffness matrix is shown to be in general complex, and Hermitian in form, but the inplane stiffness matrix is real and symmetrical. Explicit expressions are derived for the elements of the matrices, in which all the essential destabilizing effects of the basic stresses, as well as dynamic effects, are included. Finally, it is shown that buckling and vibration phenomena for any structure of this type are closely interrelated.     

Nonlinear vibration of plates by the hierarchical finite element and continuation methods

The hierarchical finite element (HFEM) and the harmonic balance methods are applied to analyse the geometrically nonlinear vibration of thin, isotropic plates. The von Kármán type of nonlinear strain–displacement relationships are used. Symbolic computation is employed in the derivation of the model. The equations of motion are solved by the Newton and continuation methods. Free and steady-state forced vibration are analysed. The excitations considered are harmonic plane waves at both normal and grazing incidence. The stability of the obtained solutions is investigated by studying the evolution of perturbations to the solutions. The convergence properties of the HFEM and the influence of the middle plane in-plane displacements are discussed and results compared with published experimental and numerical results.     

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.     

Vibration analysis of piezoelectric FGM sensors using an accurate method

In this study, free vibration analysis of moderately thick smart FG annular/circular plates with different boundary conditions is presented on the basis of the Mindlin plate theory. This structure comprised a host FG plate and two bonded piezoelectric layers. Piezoelectric layers are open circuit therefore this plate can be used as a sensor. According to power-law distribution of the volume fraction of the constituents, material properties vary continuously through the thickness of host plate while Poisson's ratio is set to be constant. Using Hamilton's principle and Maxwell electrostatic equation yields six complex coupled equations which are solved via an exact closed-form method. The accuracy of the frequencies is verified by the available literature, finite element method (FEM) and the Kirchhoff theory. The effects of plate parameters like boundary condition and gradient index are investigated and significance of coupling between in-plane and transverse displacements on the resonant frequency is proved.     

The load capacity of a squeeze film between curved porous rotating circular plates

The squeeze film between rotating circular plates is analysed. The curved upper plate approaches the flat non-porous plate normally. The Reynolds equations are uncoupled by using the Morgan—Cameron approximation and closed-form solutions are obtained. Expressions for the pressure and load capacity of the bearing are given. The effect of rotating fluid inertia is to reduce the load capacity of the bearing.     

Refined theory for vibrations and buckling of laminated isotropic annular plates

Hamilton's variational principle is used for the derivation of equations of transversally isotropic laminated annular plates motion. Nonlinear strain—displacements relations are considered. Linearized vibration and buckling equations are obtained for the annular plates uniformly compressed in the radial direction. The effects of transverse shear and rotational inertia are included. A closed form solution is given for the mode shapes in terms of Bessel, power and trigonometric functions. The eigenvalue equations are derived for natural frequencies and buckling loads of annular and circular plates elastically restrained against rotation along edges. Classical-type plate theory results are obtained then by letting the transverse shear stiffness go to infinity and rotational inertia go to zero. Numerical examples are presented by tables and figures for 2- and 3-layered plates with various geometrical and physical parameters. The transverse shear, rotational inertia and boundary conditions effects are discussed.     

Hygrothermal effects on the postbuckling of shear deformable laminated plates

The influence of hygrothermal effects on the postbuckling of shear deformable laminated plates subjected to a uniaxial compression is investigated using a micro-to-macro-mechanical analytical model. The material properties of the composite are affected by the variation of temperature and moisture, and are based on a micro-mechanical model of a laminate. The governing equations of a laminated plate are based on Reddy's higher-order shear deformation plate theory that includes hygrothermal effects. The initial geometric imperfection of the plate is taken into account. Two cases of the in-plane boundary conditions are considered. A perturbation technique is employed to determine buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of perfect and imperfect, antisymmetric angle-ply and symmetric cross-ply laminated plates under different sets of environmental conditions. The influences played by temperature rise, the degree of moisture concentration, the character of in-plane boundary conditions, transverse shear deformation, plate aspect ratio, total number of plies, fiber orientation, fiber volume fraction and initial geometric imperfections are studied.     

Buckling analysis of laminated plates using the extended Kantorovich method and a system of first-order differential equations

The extended Kantorovich method using multi-term displacement functions is applied to the buckling problem of laminated plates with various boundary conditions. The out-of-plane displacement of the buckled plate is written as a series of products of functions of parameter x and functions of parameter y. With known functions in parameter x or parameter y, a set of governing equations and a set of boundary conditions are obtained after applying the variational principle to the total potential energy of the system. The higher order differential equations are then transformed into a set of first-order differential equations and solved for the buckling load and mode. Since the governing equations are first-order differential equations, solutions can be obtained analytically with the out-of-plane displacement written in the form of an exponential function. The solutions from the proposed technique are verified with solutions from the literature and FEM solutions. The bucking loads correspond very well to other available solutions in most of the comparisons. The buckling modes also compare very well with the finite element solutions. The proposed solution technique transforms higher-order differential equations to first-order differential equations, and they are analytically solved for out-of-plane displacement in the form of an exponential function. Therefore, the proposed solution technique yields a solution which can be considered as an analytical solution.     

Vibration characteristics of composite/sandwich laminates with piezoelectric layers using a refined hybrid plate model

The free vibration characteristics of laminated composite and sandwich plates with embedded and/or surface-bonded piezoelectric layers is studied, where a hybrid plate theory is proposed for modelling the structural system. It involves a problem of coupled electro-mechanical field. The variation of mechanical/structural displacements across the thickness are modelled by an efficient plate theory, which ensures inter-laminar shear stress continuity as well as stress free condition at the plate top and bottom surfaces. At the same time, this plate model has the distinct advantage of involving unknowns at the reference plane (plate mid-plane) only. For the electrical potential, the through thickness variation is modelled by the layer-wise theory, where the unknowns are taken at all the interfaces including top and bottom surfaces of the plate. However, the degrees of freedom for the electric potential are finally eliminated from the element matrices through condensation. In order to have generality in analysis, the finite element technique is used to approximate the in-plane variation of displacement parameters at the reference plane and electric potential at the different interfaces. For this purpose, an attempt has been made to develop a simple and efficient rectangular element, which satisfies C¹ continuity of transverse displacement at the inter-element boundaries. To validate the proposed model, some numerical examples are solved and the results obtained are compared to the published results. As the number of such published results is not sufficient, new examples covering various features are solved to study different aspects of the proposed model.     

Buckling of antisymmetric cross- and angle-ply laminated plates

A system of three well-known equations of equilibrium governing the buckling response of arbitrarily laminated composite plates is reduced to a single eighth order partial differential equation in terms of a displacement function. This equation is then solved in closed form to predict the buckling response of antisymmetric cross- and angle-ply plates for different boundary conditions. The effect of various plate parameters including the effect of coupling between inplane extension and out of plane bending upon the buckling response of composite plates is discussed. The results are presented in nondimensional graphical form.     

带硬涂层圆盘构件的固有振动特性的精确解法

研究了在广义弹性简支边界条件下的具有硬涂层的圆盘构件的自由振动的量纲一固有频率的精确解.首先利用多铁性多层圆盘的解析分析的多层板弹性理论,导出带硬涂层的圆盘结构的状态方程,其中以位移、电势、磁势、应力、电位移和磁感应强度为状态变量.利用有限Hankel变换和传播矩阵法,得到考虑压电和压磁效应的带硬涂层的圆盘的量纲一固有频率的精确解.根据算例结果,比较了压电、压磁两类硬涂层材料在单面涂层、双面涂层和不同涂层厚度的结构配置下的固有频率变化规律.     

Geometric non-linear analysis of thin flat plates under end shortening, using different versions of the finite strip method

Two finite strip methods are developed for predicting the geometrically non-linear response of rectangular thin plates with simply supported ends when subjected to uniform end shortening in their plane. Although the formulations of both finite strip methods are based on the concept of the principle of minimum potential energy, the first finite strip method utilizes a semi-energy finite strip, whereas in the second finite strip method (which is designated by the name full-energy finite strip method) all the displacements are postulated by the appropriate shape functions. It is noted that in the semi-energy finite strip approach, the out-of-plane displacement of the finite strip is the only displacement which is postulated by a deflected form while the von Kármán's compatibility equation is solved to obtain the corresponding in-plane displacement forms. The developed finite strip methods are then applied to analyze the post-local-buckling behavior of some representative thin flat plates for which the results are also obtained through the application of finite element method, employing general purpose MSC/NASTRAN package. Through the comparison of results and the appropriate discussion, the knowledge of the level of capability of different versions of finite strip method is significantly promoted.     

Determination of material properties of orthotropic plates with general boundary conditions using Inverse method and Fourier series

Determination of material properties of orthotropic plates with general elastic boundary supports using the Inverse method is presented in this paper. The material properties were identified through updating four parameters in the governing equation of a symmetrically laminated thin plate. The sum of the squared difference of the natural frequencies obtained from modal testing was minimized using the Forward method. The displacement function was expressed as a 2-D Fourier cosine series supplemented with several terms in the form of 1-D series. A classical solution was derived by letting the series exactly satisfy the governing differential equation and all the boundary conditions at every field and boundary point. In the Inverse method, the series expansions for all the relevant derivatives of the eigen frequencies were obtained through term-by-term differentiations of the displacements. Many simulations were done but one numerical example is presented to demonstrate the accuracy and convergence of the solutions.     

Effect of transverse shear and rotatory inertia on the forced axisymmetric response of linearly tapered circular plates

The forced axisymmetric response of linearly tapered circular plates, based on the shear theory is analyzed by the eigenfunction method. Clamped and simply supported plates subjected to constant and half-sine pulse loads, uniformly distributed over a symmetric portion of the plate, are solved as example problems. Numerical results computed for transverse deflection and radial stress of the plate are compared with the corresponding results of classical theory. Results obtained for a plate of constant thickness, as a particular case, are compared with closed form solutions and a very good agreement is found.     

Buckling analysis of FGM plates under uniform,linear and non-linear in-plane loading

The paper investigates the buckling responses of functionally graded material (FGM) plate subjected to uniform, linear, and non-linear in-plane loads. New nonlinear in-plane load models are proposed based on trigonometric and exponential function. Non-dimensional critical buckling loads are evaluated using non-polynomial based higher order shear deformation theory. Navier’s method, which assures minimum numerical error, is employed to get an accurate explicit solution. The equilibrium conditions are determined utilizing the principle of virtual displacements and material property are graded in the thickness direction using simple Voigt model or exponential law. The present formulation is accurate and efficient in analyzing the behavior of thin, thick and moderately thick FGM plate for buckling analysis. It is found that with the help of displacement-buckling load curve, critical buckling load can be derived and maximum displacement due to the instability of inplane load can be obtained. Also, the randomness in the values of transverse displacement due to inplane load increases as the extent of uniformity of the load on the plate is disturbed. Furthermore, the parametric varying studies are performed to analyse the effect of span-to-thickness ratio, volume fraction exponent, aspect ratio, the shape parameter for non-uniform inplane load, and non-dimensional load parameter on the non-dimensional deflections, stresses, and critical buckling load for FGM plates.

     

Large deformations of thin curved plane beam of constant initial curvature

A theory for the large deformations of a thin curved plane beam of constant initial curvature is presented, based on the hypothesis that only the longitudinal component of the strain tensor exists in the beam. In this case, five of the six compatibility equations are identically satisfied, while the remaining one requires the axial strain to vary parabolically over the cross-section of the curved beam. The non-linear strain-displacement relations are then solved for two in-plane displacement components in terms of two arbitrary functions of the longitudinal co-ordinate, angle . It is shown that, as a consequence of our hypothesis, the displacements vary linearly over the cross-sections of the beam. The obtained functional form of the displacement components leads to equilibrium equations of a beam on a deformed configuration expressed in terms of deformation functions. For the special case of a linearly elastic material and circular cantilever beam subjected to conservative and non-conservative point loads these equations have been numerically integrated. A number of numerical examples, including the bending of a C-shaped spring, are presented.     

Anti-plane piezoelastic study on plane layered media

Singularity problems in a multilayered piezoelectric body composed of arbitrary number of dissimilar plane layers perfectly bonded are analyzed in the framework of linear piezoelectricity. Each layer of the composite is assumed to posses the symmetry of a hexagonal crystal in the 6 mm class and subject to electromechanical loadings, which produce in-plane electric fields and out-of-plane displacement. Based on the complex variable theory and the method of alternating technique, the solution of electric and displacement fields is expressed in terms of explicit series form. The boundary shear stress and electric field for a practical case are discussed graphically. Some certain piezoelectric constant combination to make the composite perform the largest response of electric field is found and can be used to build a relatively effective sensor in a piezoelectric composite material system.     

Exact solutions for the free in-plane vibrations of rectangular plates

All classical boundary conditions including two distinct types of simple support boundary conditions are formulated by using the Rayleigh quotient variational principle for rectangular plates undergoing in-plane free vibrations. The direct separation of variables is employed to obtain the exact solutions for all possible cases. It is shown that the exact solutions of natural frequencies and mode shapes can be obtained when at least two opposite plate edges have either type of the simply-supported conditions, and some of the exact solutions were not available before. The present results agree well with FEM results, which show that the present solutions are correct and the direct separation of variables is practical. The exact solutions can be taken as the benchmarks for the validation of approximate methods.