Exact solutions for free vibration of transversely isotropic piezoelectric circular plates

By employing the general solution for coupled three-dimensional equations of a transversely isotropic piezoelectric body, this paper investigates the free vibration of a circular plate made of piezoelectric material. Three-dimensional exact solutions are then obtained under two specified boundary conditions, which can be used for both axisymmetric and non-axisymmetric cases. Numerical results are finally presented. The project supported by the National Natural Science Foundation of China (No. 19872060)     

A new trigonometric shear deformation theory for isotropic,laminated composite and sandwich plates

A new trigonometric shear deformation theory for isotropic and composite laminated and sandwich plates, is developed. The new displacement field depends on a parameter “m”, whose value is determined so as to give results closest to the 3D elasticity bending solutions. The theory accounts for adequate distribution of the transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surface, thus a shear correction factor is not required. Plate governing equations and boundary conditions are derived by employing the principle of virtual work. The Navier-type exact solutions for static bending analysis are presented for sinusoidally and uniformly distributed loads. The accuracy of the present theory is ascertained by comparing it with various available results in the literature. The results show that the present model performs as good as the Reddy’s and Touratier’s shear deformation theories for analyzing the static behavior of isotropic and composite laminated and sandwich plates.     

A refined theory of elastic thick plates for extensional deformation

Based on elasticity theory, various two-dimensional (2D) equations and solutions for extensional deformation have been deduced systematically and directly from the three-dimensional (3D) theory of thick rectangular plates by using the Papkovich–Neuber solution and the Lur’e method without ad hoc assumptions. These equations and solutions can be used to construct a refined theory of thick plates for extensional deformation. It is shown that the displacements and stresses of the plate can be represented by the displacements and transverse normal strain of the midplane. In the case of homogeneous boundary conditions, the exact solutions for the plate are derived, and the exact equations consist of three governing differential equations: the biharmonic equation, the shear equation, and the transcendental equation. With the present theory a solution of these can satisfy all the fundamental equations of 3D elasticity. Moreover, the refined theory of thick plate for bending deformation constructed by Cheng is improved, and some physical or mathematical explanations and proof are provided to support our justification. It is important to note that the refined theory is consistent with the decomposition theorem by Gregory. In the case of nonhomogeneous boundary conditions, the approximate governing differential equations and solutions for the plate are accurate up to the second-order terms with respect to plate thickness. The correctness of the stress assumptions in the classic plane-stress problems is revised. In an example it is shown that the exact or accurate solutions may be obtained by applying the refined theory deduced herein.     

Asymptotic solutions for the Föppl – von Kármán equations governing deflections of thin axisymmetric annular plates

We are concerned with the deformation of thin, flat annular plates under a force applied orthogonally to the plane of the plate. This mechanical process can be described via a radial formulation of the Föppl – von Kármán equations, a set of nonlinear partial differential equations describing the deflections of thin flat plates. We are able to obtain analytical solutions for the radial Föppl – von Kármán equations with boundary conditions relevant for clamped, loosely clamped, and free inner and outer. This permits us to study the qualitative behavior of the out-of-plane deflections as well as the Airy stress function for a number of cases. Provided that an appropriate non-dimensionalization is taken, we find that the perturbation solutions are surprisingly valid for a wide variety of parameters, and compare favorably with numerical simulations in all cases (rather than just for small parameters). The results demonstrate that the ratio of the inner to outer radius of the annular plate will strongly influence the properties of the solutions, as will the specific boundary data considered. For instance, one may choose to fix the plate in place with a specific set of boundary conditions, in order to minimize the out-of-plane deflections. Other boundary conditions may result in undesirable behaviors.     

加权残数配点法解正交各向异性板的积分方程

本文推导了一般各向异性板弯曲的积分方程,运用加权残数配点法求解了正交各向异性板弯曲的积发方程,本文将部分配点取在边界上,另一部分配点取在域外,只用关于找度的基本积分方程,而不用关于转角的补充积分方程,简化了方程求解和计算程序,由于正交各向异性板没有争析形式的、实用的基本解,本文提出了两种新的近似基本解;加权双三角级数;广义各向同性板解析形式的基本解和加权双三角级数的叠加,算例表明,本文提出的解法和近似基本解适用于各类边界条件的正交各向异性板,具有简单、可靠、精度高等优点。     

BOUNDARY INTEGRAL EQUATIONS FOR BENDING PROBLEM OF REISSNER'S PLATES ONTWO-PARAMETER FOUNDATION

I.IntroductionThickplatesonelastict'oundationarewidelyusedinengineering,suchasthebottomplatesofoffShorestructures,surfaceplatesonrunwayofairportsandfoundationsofhigh-risebuildingsandthelike.Itisextremelydifficulttoobtainanalyticalsolutiontarathickplatewithcomplicatedshapeorcomplicatedboundaryconditiononelasticfoundation.Inrecentyears,theboundaryelementmethod(BEM)hasbeensuccessl'ullyusedtoanalyzethebendingproblemofplatesoneverykindofelasticfoundation(Ref.[l,2.3]).Butthereareonlyfewreferences…     

Symplectic system based analytical solution for bending of rectangular orthotropic plates on Winkler elastic foundation

This paper analyses the bending of rectangular orthotropic plates on a Winkler elastic foundation.Appropriate definition of symplectic inner product and symplectic space formed by generalized displacements establish dual variables and dual equations in the symplectic space.The operator matrix of the equation set is proven to be a Hamilton operator matrix.Separation of variables and eigenfunction expansion creates a basis for analyzing the bending of rectangular orthotropic plates on Winkler elastic foundation and obtaining solutions for plates having any boundary condition.There is discussion of symplectic eigenvalue problems of orthotropic plates under two typical boundary conditions,with opposite sides simply supported and opposite sides clamped.Transcendental equations of eigenvalues and symplectic eigenvectors in analytical form given.Analytical solutions using two examples are presented to show the use of the new methods described in this paper.To verify the accuracy and convergence,a fully simply supported plate that is fully and simply supported under uniformly distributed load is used to compare the classical Navier method,the Levy method and the new method.Results show that the new technique has good accuracy and better convergence speed than other methods,especially in relation to internal forces.A fully clamped rectangular plate on Winkler foundation is solved to validate application of the new methods,with solutions compared to those produced by the Galerkin method.     

Optimal control of the dynamic response of an anisotropic plate with various boundary conditions

The problem of minimising the dynamic response of an anisotropic rectangular plate with minimum possible expenditure of force is presented for various cases of boundary conditions. The plate has a principal direction of anisotropy rotated at an arbitrary angle relative to the coordinate axes. This orientation angle has been taken as an optimisation design parameter. The control problem is formulated as an optimisation problem by using a performance index, which comprises a weight sum of the control objective and penalty function of the control force. The explicit solutions for the closed-loop distributed control function is obtained by means of Liapunov-Bellman theory. To assess the present solution, numerical results are presented to illustrate the effect of anisotropy ratio, orientation angle, aspect ratio and boundary conditions on the control process.     

A new simple third-order shear deformation theory of plates

An improved simple third-order shear deformation theory for the analysis of shear flexible plates is presented in this paper. This new plate theory is composed of three parts: the simple third-order kinematics of displacements reduced from the higher-order displacement field derived previously by the author; a system of 10th-order differential equilibrium equations in terms of the three generalized displacements of bending plates; five boundary conditions at each edge of plate boundaries. Although the resulting displacement field is the same as that proposed by Murthy, the variational consistent governing equations and the associated proper boundary conditions are derived and identified in this work for the first time in the literature. The applications and accuracy of the present shear deformation theory of plates are demonstrated by analytically solving the differential governing equations of a twisting plate, a bending beam and two bending plates to which the 3-D elasticity solutions are available, and excellent agreements are achieved even for the torsion of a plate with square cross-section as well the local effects of stresses at plate boundaries can be characterized accurately. These analytical solutions clearly show that the simple third-order shear deformation theory developed in this work indeed gives better results than the first-order shear deformation theories and other simple higher-order shear deformation theories, since the present third-order shear flexible theory is based on a more rigorous kinematics of displacements and consists of not only a system of variational consistent differential equations, but also a group of consistent boundary conditions associated with the differential equations. The present simple third-order shear deformation theory can easily be applied to the static and dynamic finite element analysis of laminated plates just like the applications of other popular shear flexible plate theories, and improved results could be obtained from the present simple third-order shear deformable theories of plates.     

各种边界条件平行四边形功能梯度板的动力特性解析解

本文基于斜坐标系,建立起平行四边形功能梯度板的基本微分方程及变分方程,用梁函数组合法对平行四边形及菱形功能梯度板进行动力特性分析,提出了适用于每边任取简支、固定、自由边界之一(包括36种边界)平行四边形功能梯度板固有频率与振型的解析解;在简化情况下,给出了各种边界条件平行四边形功能梯度板各阶固有频率解的统一表达式。     

The fundamental solution of Mindlin plates resting on an elastic foundation in the Laplace domain and its applications

The aim of this study is to investigate the method of fundamental solution (MFS) applied to a shear deformable plate (Reissner/Mindlin’s theories) resting on the elastic foundation under either a static or a dynamic load. The complete expressions for internal point kernels, i.e. fundamental solutions by the boundary element method, for the Mindlin plate theory are derived in the Laplace transform domain for the first time. On employing the MFS the boundary conditions are satisfied at collocation points by applying point forces at source points outside the domain. All variables in the time domain can be obtained by Durbin’s Laplace transform inversion method. Numerical examples are presented to demonstrate the accuracy of the MFS and comparisons are made with other numerical solutions. In addition, the sensitivity and convergence of the method are discussed for a static problem. The proposed MFS is shown to be simple to implement and gives satisfactory results for shear deformable plates under static and dynamic loads.     

SINGULAR SOLUTIONS OF ANISOTROPIC PLATE WITH AN ELLIPTICAL HOLE OR A CRACK

In the present paper, closed form singular solutions for an infinite anisotropic plate with an elliptic hole or crack are derived based on the Stroh-type formalism for the general anisotropic plate. With the solutions, the hoop stresses and hoop moments around the elliptic hole as well as the stress intensity factors at the crack tip under concentrated in-plane stresses and bending moments are obtained. The singular solutions can be used for approximate analysis of an anisotropic plate weakened by a hole or a crack under concentrated forces and moments.They can also be used as fundamental solutions of boundary integral equations in BEM analysis for anisotropic plates with holes or cracks under general force and boundary conditions.     

Exact solutions to one-dimensional transient response of incompressible fluid-saturated single-layer porous media

Based on the Biot theory of porous media,the exact solutions to onedimensional transient response of incompressible saturated single-layer porous media under four types of boundary conditions are developed.In the procedure,a relation between the solid displacement u and the relative displacement w is derived,and the well-posed initial conditions and boundary conditions are proposed.The derivation of the solution for one type of boundary condition is then illustrated in detail.The exact solutions for the other three types of boundary conditions are given directly.The propagation of the compressional wave is investigated through numerical examples.It is verified that only one type of compressional wave exists in the incompressible saturated porous media.     

A free rectangular plate on the two-parameter elastic foundation

This paper provides a rigorous solution of a free rectangular plate on the V.Z. Vlazov two-parameter elastic foundation by the method of superposition[1]. In this paper we derive basic solutions under the various boundary conditions. To superpose these basic solutions the most generally rigorous solution of a free rectangular plate on the two-parameter elastic foundation can be obtained. The solution strictly satisfies the differential equation of a plate on the two-parameter elastic model foundation, the boundary conditions of the free edges and the free corner conditions. Some numerical examples are presented The calculated results show that when the plane dimension of plate is given and the ratio between the laver depth and the plate thick is equal to 15, the two-parameter elastic model is near the Winkler’s. It shows that the Winkler model can be applied to the thinner layer.     

On canonical bending relationships for plates

In recent years, a series of papers have appeared on algebraic relationships between the solutions (e.g., deflections, buckling loads and frequencies) of a given higher-order plate theory and the classical plate theory. The bending relationships, for example, can be used to generate the transverse deflection of a plate according to the particular higher-order theory from the known deflection of the same problem according to the classical plate theory. In the present study relationships between the bending solutions of several higher-order plate theories and the classical plate theory are obtained in a canonical form (i.e., one set of relationships contain several theories and they can be specialized to a specific theory by assigning values to the constants appearing in the relationships). Numerical examples of bending solutions for rectangular plates with various boundary conditions are presented to show how the relations can be used to determine the deflections and bending moments for various theories. The relationships are validated by comparing the numerical results obtained using the relationships for the Mindlin plate theory against those computed using the ABAQUS finite element program.     

Differential quadrature method for bending of orthotropic plates with finite deformation and transverse shear effects

Based on the Reddy ‘s theory of plates with the effect of higher-order shear deformations, the governing equations for bending of orthotropic plates with finite deformations were established. The differential quadrature ( DQ ) method of nonlinear analysis to the problem was presented. New DQ approach, presented by Wang and Bert ( DQWB), is extended to handle the multiple boundary conditions of plates. The techniques were also further extended to simplify nonlinear computations. The numerical convergence and comparison of solutions were studied. The results show that the DQ method presented is very reliable and valid. Moreover, the influences of geometric and material parameters as well as the transverse shear deformations on nonlinear bending were investigated. Numerical results show the influence of the shear deformation on the static bending of orthotropic moderately thick plate is significant.     

Deflection of sandwich plates in terms of corresponding Kirchhoff plate solutions

Summary This study presents exact relationships between the deflections of isotropic sandwich plates and their corresponding Kirchhoff plates. The governing equilibrium equations for the sandwich plates are derived on the basis of the Reissner-Mindlin shear deformation plate theory. The considered plates are either (i) simply supported, of general polygonal shape and under any transverse loading condition or (ii) simply supported and clamped circular plates under axisymmetric loading. As the relationships are exact under the assumptions used in the plate theories, one may obtain exact deflection solutions of sandwich plates if the Kirchhoff plate solutions are exact. The relationships should also be useful for the development of approximate formulas for plates with other shapes, boundary and loading conditions, and may serve to check numerical deflection values computed from sandwich plate analysis software.     

A differential quadrature procedure for three-dimensional buckling analysis of rectangular plates

This paper presents an application of the differential quadrature (DQ) method for three-dimensional buckling analysis of rectangular plates. The governing equations of the plate model are first presented in terms of displacement, stress displacement relationship, and boundary conditions with three-dimensional flexibility. These equations are then normalised and discretised using the DQ procedure. Example problems pertaining to the buckling of rectangular plates with generic boundary conditions are selected to illustrate the efficiency and simplicity of implementing the DQ procedure. The convergence characteristics of the method are first conducted based on numerical studies. The DQ solutions are then compared, where possible, with exact or approximate solutions. It is found that the differential quadrature method yields accurate results for the plate problems under the current investigation. In addition to the above, some parametric studies are carried out by varying the plates aspect ratio, boundary conditions and thickness to width ratio under axial and biaxial loading.     

Global structural behaviour of thin and moderately thick monoclinic spherical shells using a mixed shear deformation model

Summary The static and dynamic responses of anisotropic spherical shells under a uniformly distributed transverse load are investigated. Analytical solutions using the mixed variational formulation are presented for spherical shells subjected to various boundary conditions. Numerical results of a refined mixed first-order shear deformation theory for natural frequencies, critical buckling, center deflections and stresses are compared with those obtained using the classical shell theory. A variety of simply-supported and clamped boundary conditions are considered and comparisons with the existing literature are made. The sample numerical results presented herein for global structural behaviour of monoclinic spherical shells should serve as references for future comparisons.