Calculation of simply supported circular plates in the statement of the problem of contact interaction in a package of two plates

The possibility of applying the mechanicomathematical model of bending of a package of transversely isotropic plates to approximate calculations of thick plates is investigated. Within the frame work of this model, the problem of axisymmetric bending of a package of two identical plates with simply supported edges is considered. The conditions of rigid contact are given between the plates. Based on the analytical result obtained, the unknown distribution of stresses and displacements across the thickness of the plate is approximated by the distribution of corresponding parameters in the package of two plates. For an isotropic body, the results of numerical calculations are compared with those given by the 3D theory of elasticity. In the case of a transversely isotropic body, a comparison with the results found by the refined bending model of plates (taking into account the transverse compression and shear) and the Timoshenko model is carried out. The accuracy to which the boundary conditions in every model are satisfied is analyzed. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 5, pp. 603–616, September–October, 2007.     

Elasticity solutions for functionally graded rectangular plates with two opposite edges simply supported

England (2006) [13] proposed a novel method to study the bending of isotropic functionally graded plates subject to transverse biharmonic loads. His method is extended here to functionally graded plates with materials characterizing transverse isotropy. Using the complex variable method, the governing equations of three plate displacements appearing in the expansions of displacement field are formulated based on the three-dimensional theory of elasticity for a transverse load satisfying the biharmonic equation. The solution may be expressed in terms of four analytic functions of the complex variable, in which the unknown constants can be determined from the boundary conditions similar to that in the classical plate theory. The elasticity solutions of an FGM rectangular plate with opposite edges simply supported under 12 types of biharmonic polynomial loads are derived as appropriate sums of the general and particular solutions of the governing equations. A comparison of the present results for a uniform load with existing solutions is made and good agreement is observed. The influence of boundary conditions, material inhomogeneity, and thickness to length ratio on the plate deflection and stresses for the load x²yq are studied numerically.     

Levy-type solution for buckling analysis of thick functionally graded rectangular plates based on the higher-order shear deformation plate theory

In this article, an analytical approach for buckling analysis of thick functionally graded rectangular plates is presented. The equilibrium and stability equations are derived according to the higher-order shear deformation plate theory. Introducing an analytical method, the coupled governing stability equations of functionally graded plate are converted into two uncoupled partial differential equations in terms of transverse displacement and a new function, called boundary layer function. Using Levy-type solution these equations are solved for the functionally graded rectangular plate with two opposite edges simply supported under different types of loading conditions. The excellent accuracy of the present analytical solution is confirmed by making some comparisons of the present results with those available in the literature. Furthermore, the effects of power of functionally graded material, plate thickness, aspect ratio, loading types and boundary conditions on the critical buckling load of the functionally graded rectangular plate are studied and discussed in details. The critical buckling loads of thick functionally graded rectangular plates with various boundary conditions are reported for the first time and can be used as benchmark.     

Local radial basis function collocation method for bending analyses of quasicrystal plates

The local radial basis function collocation method (LRBFCM) is proposed for plate bending analysis in orthorhombic quasicrystals (QCs) under static and transient dynamic loads. Three common types of the plate bending problems are considered: (1) QC plates resting on Winkler foundation (2) QC plates with variable thickness and (3) QC plates under a transient dynamic load. According to the Reissner–Mindlin plate bending theory, there is allowed to simulate the behavior of the two excitations in QC plates, phonon and phason, by 2D strong formulations for the system of governing equations. The governing equations, which describe the phason displacements, are based on Agiasofitou and Lazar elastodynamic model. Numerical results demonstrate the effect of the elastic foundation, as well as plate thickness on the phonon and phason characteristics in this paper. For the transient dynamic analysis, the influence of the phason friction coefficients on the responses of QC plate to transient dynamic loads is also studied.     

二维横观各向同性厚板在空间变化热源及体力作用下的动力学响应

研究热源和体力作用下的横观各向同性厚板的二维问题,板的上表面无应力作用,但有规定的表面温度作用;板的下表面置于刚性基础之上,并处于绝热状态.采用Green和Naghdi提出的广义热弹性理论,通过Laplace和Fourier双重变换,在Laplace-Fourier变换域中,得到位移和温度场的控制方程.数值求解双重变换的逆变换,采用一个基于Fourier级数展开的方法,数值地求解Laplace变换的逆变换.对材料镁(Mg)进行数值计算,并用图形表示其结果.推演出各向同性材料铜(Cu)的数值结果,并用图形与横观各向同性材料镁进行比较.同时研究了体力的影响.     

横观各向同性板的弹性理论和弹性改进理论及一种新的厚板理论

本文从横观各向同性体弹性力学位移形式的基本方程出发,考虑板面承受横向荷载,建立了横观各向同性板弯曲的弹性理论.并由此建立了一个在板的每边能满足三个边界条件的弹性改进理论和一种新的厚板理论.文中求得了周边简支多边形板的弹性改进理论解,数值结果与三维弹性理论精确解的结果非常接近.新的厚板理论和以往的中厚板理论的系统比较表明,我们提出的厚板理论最靠近弹性理论的结果.     

Representation of Solutions of the Deflection Equilibrium Equations for Thick Transversely Isotropic Plates

A system of equilibrium equations for the deflection of thick transversely isotropic plates with constant thicknesses is derived by expanding the stress and displacement components as Fourier series in Legendre polynomials of the thickness coordinate. A method for constructing a general solution of this system is described. It is made up of three types of solutions: biharmonic, potential, and rotational.     

Analysis of Mindlin micro plates with a modified couple stress theory and a meshless method

A modified couple stress theory and a meshless method is used to study the bending of simply supported micro isotropic plates according to the first-order shear deformation plate theory, also known as the Mindlin plate theory. The modified couples tress theory involves only one length scale parameter and thus simplifies the theory, since experimentally it is easier to determine the single scale parameter. The equations governing bending of the first-order shear deformation theory are implemented using a meshless method based on collocation with radial basis functions. The numerical method is easy to implement, and it provides accurate results that are in excellent agreement with the analytical solutions.     

Bending,buckling and free vibration analysis of incompressible functionally graded plates using higher order shear and normal deformable plate theory

In the present study, higher order shear and normal deformable plate theory is developed for analysis of incompressible functionally graded rectangular thick plates. Also, The effect of incompressibility is studied on the static, dynamic and stability responses of thick plate. It is assumed that plate is incompressible and the incompressibility condition is considered in addition to the governing equations for determining the unknowns. Since the plate is thick, higher order shear and normal deformable theory is applied so that the Legendre polynomials are used for expansion of displacement field components in the thickness direction. Also, it is supposed that material properties vary through the thickness based on the power law function. Utilizing the variational approach, governing equations for static, stability and dynamic analysis of plate are derived. Resulted equations are solved analytically for simply supported plates. Finally, the effects of material properties and dimensions on the response of incompressible plates are investigated in details.     

Calculation of rigidly clamped circular plates in the statement of the problem on contact interaction in a two-layer package

The use of a mechanicomathematical model of bending of thick transversely isotropic plates is illustrated, where the plate is divided in an arbitrary number of equally thick conditional layers. This model allows one to approximately reduce the problem of determination of stresses and displacements in the thick plate to a corresponding contact problem for a bent pack age of layers. The axisymmetric bending of a rigidly clamped package consisting of two plates rigidly fastened together is considered. The results of numerical calculations are presented, which are compared with those obtained within the framework of a refined bending model of plates (with account of transverse compression and shear) and of the Timoshenko model, as well as in the statement of the three-dimensional theory of elasticity. The accuracy of satisfying the boundary conditions in each model is analyzed. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 1, pp. 93–108, January–February, 2007.     

横观各向同性层合矩形板弯曲、振动和稳定的三维精确分析

针对四边简支的横观各向同性矩形板的弯曲、振动和稳定给出了新的状态空间分析方法。从横观各向同性弹性力学的三维基本方程出发,通过引入位移函数和应力函数,构造了两类相互独立的状态空间方程,不仅使原方程得到解耦而且降低了阶数,十分有利于具体问题的求解。对于四边简支的矩形板,建立了层合板上下表面状态变量间的关系式。特别针对矩形板的自由振动(稳定)问题,发现存在两类彼此无关的形式,一类对应板的纯面内振动(稳定),而另一类则是一般意义上的板的弯曲振动(稳定)。给出了数值结果,并考察了相关参数的影响。     

横观各向同性饱和地基上刚性圆板的扭转振动

通过解析方法研究了横观各向同性饱和半空间上刚性圆板在简谐扭转荷载作用下的振动问题．运用Hankel变换求解了横观各向同性饱和土的动力控制方程,结合混合边界条件得出了刚性基础的扭转对偶积分方程,并将对偶积分方程转化为第二类Fredholm积分方程求解了基础的扭转振动问题,同时给出了动力柔度系数,基础的角位移幅值和基底接触剪应力的表达式．通过数值算例研究了地基的各向异性程度对基础扭转振动的影响．     

Static analysis of rectangular nano-plate using three-dimensional theory of elasticity

Analytical solution for bending of a simply supported rectangular graphene sheets based on three dimensional theory of elasticity, is studied employing non-local continuum mechanics. By applying the Fourier series solution to the both displacement and stress field along the in-plane rectangular coordinates direction, and to the governing equation and constitutive relations, the three-dimensional governing equations in term of displacement components can be obtained. Closed form solution for the bending behavior of nano-plate is obtained by exerting the surface tractions on the state equations. To validate the accuracy and convergence of the present approach, numerical results are presented and compared with the results available in the open literature. Effect of non-local parameter, aspect ratio, thickness to length ratio and half wave numbers in the bending behavior of plate are examined. Furthermore, these results may also serve as benchmark to further results into the two-dimensional plate theories.     

Free edge stress prediction in thick laminated cylindrical shell panel subjected to bending moment

A thick composite cylindrical shell panel with general layer stacking is studied to investigate the free edge and 3D stresses in the panel which is subjected to pure bending moment. To this aim, a Galerkin based layerwise formulation is presented to discretize the governing equation of the panel to ordinary differential equations. Employing a reduced displacement field for the cylindrical panel, the governing equations for thick panel are developed in terms of displacements and a set of coupled ordinary differential equations is obtained. The governing equations are solved analytically for free edge boundary conditions and applied pure bending moment. The accuracy of numerical results is examined and the distribution of interlaminar and in-plane stresses is studied. The free edge stresses are studied and the effect of radius to thickness ratio, width to thickness ratio and layer stacking on the distribution of stresses is investigated. The focus of numerical results is on the prediction of boundary layer and free edge stress distribution.     

横观各向同性饱和地基与中厚圆板的非轴对称动力相互作用

研究横观各向同性饱和土地基上中厚弹性圆板的非轴对称振动问题,即首先利用Fourier展开和Hankel变换技术,求解了简谐激励下横观各向同性饱和土地基的非轴对称Biot波动方程,然后按混合边值问题建立地基与弹性中厚圆板非轴对称动力相互作用的对偶积分方程,并将对偶积分方程化为易于数值计算的第二类Fredholm积分方程．文末给出了算例．数值结果表明,在一定频率范围内,地基表面的位移幅值随激振频率增加而增大,随距离的增大以振荡形式衰减变化．     

3D nonlinear variable strain-rate-dependent-order fractional thermoviscoelastic dynamic stress investigation and vibration of thick transversely graded rotating annular plates/discs

In the present article, the idea of using the variable-order fractional-derivative thermoviscoelastic constitutive laws in dynamic stress and vibration analysis of the engineering structures, the required implementation backgrounds, and the relevant numerical solution procedures are investigated for the first time. In this regard, dynamic 3D stress and displacement fields and radial/transverse vibrations of transversely graded viscoelastic spinning thick plates/discs exposed to sudden thermoelastic loads are investigated. Instead of using the approximate plate theories, the exact thermoviscoelasticity theory is employed in the development of the governing equations. Since the variable fractional order is dependent on the localized deformation rates, the resulting thermoviscoelastic integro-differential equations are nonlinear. These equations are solved by utilizing a combination of the second-order backward/central/forward finite difference discretization of the spatial and time domains, numerical evaluation and updating of the Caputo-type fractional derivatives, updating the growing number of terms of the governing equations, and Picard's iterations. Various edge conditions are considered. Finally, comprehensive sensitivity analyses and various 3D plots are presented and discussed regarding the effects of the variable fractional order of the constitutive law, time variations of the nonuniformly distributed transverse loads, and edge conditions on the distributions and damping of the resulting displacement and stress components.     

Dynamic stability of a porous rectangular plate

The study is devoted to a axial compressed porous-cellular rectangular plate. Mechanical properties of the plate vary across is its thickness which is defined by the non-linear function with dimensionless variable and coefficient of porosity. The material model used in the current paper is as described by Magnucki, Stasiewicz papers. The middle plane of the plate is the symmetry plane. First of all, a displacement field of any cross section of the plane was defined. The geometric and physical (according to Hook's law) relationships are linear. Afterwards, the components of strain and stress states in the plate were found. The Hamilton's principle to the problem of dynamic stability is used. This principle was allowed to formulate a system of five differential equations of dynamic stability of the plate satisfying boundary conditions. This basic system of differential equations was approximately solved with the use of Galerkin's method. The forms of unknown functions were assumed and the system of equations was reduced to a single ordinary differential equation of motion. The critical load determined used numerically processed was solved. Results of solution shown in the Figures for a family of isotropic porous-cellular plates. The effect of porosity on the critical loads is presented. In the particular case of a rectangular plate made of an isotropic homogeneous material, the elasticity coefficients do not depend on the coordinate (thickness direction), giving a classical plate. The results obtained for porous plates are compared to a homogeneous isotropic rectangular plate. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Development of a higher order finite element on a Winkler foundation

Analyzing thick plates as a construction component has been of interest to structural engineering research for several decades. In particular, thick plates resting on elastic foundations are more specific. Mindlin's plate theory for thick plate analysis and the Winkler theory for elastic foundation analyses have wide applications. The current research considers analysis of isotropic plates on a Winkler foundation according to Mindlin's plate theory. The analysis uses a higher order plate element to avoid shear locking phenomena in the plate. The main features of this element are representation of real displacement functions of the plate perfect and shear locking do not occur at the plates modeled with this element. Derivation of the equations for finite element formulation for thick plate theory uses fourth-order displacement shape functions. A computer program using the finite element method, coded in C++, analyzes the plates resting on an elastic foundation. The analysis involves a 17-noded finite element. The study's graphs and tables assist engineers' designs of thick plates resting on elastic foundations. The study concludes with the computer-coded program, which allows effective use for the shear locking-free analysis of thick Mindlin plates resting on elastic foundations.     

On the natural frequency of transversely isotropic magneto-electro-elastic plates in contact with fluid

In this paper, the vibration characteristics of transversely isotropic magneto-electro-elastic (MEE) rectangular plates in contact with fluid are investigated. The mathematical formulation on the determination of added virtual mass for water-contacting MEE rectangular plates with uniform thickness is performed. Based on the recently proposed differential equation governing the dynamical responses of the MEE rectangular plates, a fluid-structure interaction model is established and analyzed. First of all, the added virtual mass incremental (AVMI) factor of the system is calculated by using the proposed approach and the added virtual mass can then be obtained, furthermore, the natural frequencies of the MEE rectangular plates in contact with fluid with different boundary conditions are also investigated. It is noted that the natural frequencies based on the proposed method are very useful for those engineers or researchers who are engaged in the vibration analysis and design of the MEE plate in contact with fluid.     

Buckling and free vibration analysis of thick rectangular plates resting on elastic foundation using mixed finite element and differential quadrature method

In this article, a combination of the finite element (FE) and differential quadrature (DQ) methods is used to solve the eigenvalue (buckling and free vibration) equations of rectangular thick plates resting on elastic foundations. The elastic foundation is described by the Pasternak (two-parameter) model. The three dimensional, linear and small strain theory of elasticity and energy principle are employed to derive the governing equations. The in-plane domain is discretized using two dimensional finite elements. The spatial derivatives of equations in the thickness direction are discretized in strong-form using DQM. Buckling and free vibration of rectangular thick plates of various thicknesses to width and aspect ratios with Pasternak elastic foundation are investigated using the proposed FE-DQ method. The results obtained by the mixed method have been verified by the few analytical solutions in the literature. It is concluded that the mixed FE-DQ method has good convergancy behavior; and acceptable accuracy can be obtained by the method with a reasonable degrees of freedom.