Winkler地基上四边自由正交各向异性矩形薄板弯曲问题的辛叠加解

研究Winkler地基上正交各向异性矩形薄板弯曲方程所对应的Hamilton正则方程, 计算出其对边滑支条件下相应Hamilton算子的本征值和本征函数系, 证明该本征函数系的辛正交性以及在Cauchy主值意义下的完备性, 进而给出对边滑支边界条件下Hamilton正则方程的通解, 之后利用辛叠加方法求出Winkler地基上四边自由正交各向异性矩形薄板弯曲问题的解析解. 最后通过两个具体算例验证了所得解析解的正确性.     

Three-Dimensional Elasticity Solutions for Rectangular Orthotropic Plates

A state space approach for three-dimensional analysis of rectangular orthotropic elastic plates subjected to external loads on the top and bottom faces is developed. Through Hamiltonian variational formulation via Legendre’s transformation, the basic equations of elasticity are formulated into the state space framework in which the state equation exhibits Hamiltonian characteristics and the associated eigensystem possesses symplectic orthogonality. By means of separation of variables and eigenfunction expansion, three-dimensional elasticity solutions for orthotropic rectangular plates with two opposite edges simply supported and the other two arbitrary—which can be any combinations of simply-supported, clamped, and traction-free edges—are determined in a systematic way. The existing elasticity solution for the fully simply-supported plate is recovered. The through-thickness variations of the displacements and stresses are evaluated within the context.     

非均匀Winkler弹性地基上变厚度矩形板自由振动的DTM求解

针对非均匀Winkler弹性地基上变厚度矩形板的自由振动问题,通过一种有效的数值求解方法——微分变换法（DTM）,研究其无量纲固有频率特性。已知变厚度矩形板对边为简支边界条件,其他两边的边界条件为简支、固定或自由任意组合。采用DTM将非均匀Winkler弹性地基上变厚度矩形板无量纲化的自由振动控制微分方程及其边界条件变换为等价的代数方程,得到含有无量纲固有频率的特征方程。数值结果退化为均匀Winker弹性地基上矩形板以及变厚度矩形板的情形,并与已有文献采用的不同求解方法进行比较,结果表明,DTM具有非常高的精度和很强的适用性。最后,在不同边界条件下分析地基变化参数、厚度变化参数和长宽比对矩形板无量纲固有频率的影响,并给出了非均匀Winkler弹性地基上对边简支对边固定变厚度矩形板的前六阶振型。     

Analytical bending solutions of free orthotropic rectangular thin plates under arbitrary loading

This paper presents the analytical solutions for the bending of orthotropic rectangular thin plates by the double finite integral transform, which, as an effective tool in solving plate problems, should have received attention. As a representative and difficult problem in the theory of plates, free plates’ bending is successfully solved to demonstrate the accuracy of the method by comparing the present analytical solutions with those from the literature as well as those by the finite element method. With the proper integral transform kernels, the proposed solution procedure is applicable to the bending of orthotropic rectangular plates with all combinations of simply supported, clamped and free boundary conditions, which serves as an elegant approach to analytical solutions of plate bending problems.     

On new symplectic approach for exact bending solutions of moderately thick rectangular plates with two opposite edges simply supported

The symplectic geometry method is introduced for exact bending solutions of moderately thick rectangular plates with two opposite edges simply supported. The basic equations for the plates are first transferred into Hamilton canonical equations. The whole state variables are then separated. Using the method of eigenfunction expansion in the symplectic geometry, typical examples for plates with selected boundary conditions are solved and exact bending solutions obtained. Since only the basic elasticity equations of the plates are used, this method eliminates the need to pre-determine the deformation function and is hence more reasonable than conventional methods. Numerical results were presented to demonstrate the validity and accuracy of this approach as compared to those reported in other literatures.     

Free vibration analysis of moderately thick functionally graded plates on elastic foundation using the extended Kantorovich method

Free vibration analysis of moderately thick rectangular FG plates on elastic foundation with various combinations of simply supported and clamped boundary conditions are studied. Winkler model is considered to describe the reaction of elastic foundation on the plate. Governing equations of motion are obtained based on the Mindlin plate theory. A semi-analytical solution is presented for the governing equations using the extended Kantorovich method together with infinite power series solution. Results are compared and validated with available results in the literature. Effects of elastic foundation, boundary conditions, material, and geometrical parameters on natural frequencies of the FG plates are investigated.     

Exact solutions for variable-thickness inhomogeneous elastic plates under various boundary conditions

In this paper, an exact solution to the governing equations of the bending of a variable-thickness inhomogeneous rectangular plate is presented. The procedure is applicable to variable-thickness inhomogeneous rectangular plates with two opposite edges simply supported. The remaining ones subjected to a combination of clamped, simply supported, and free boundary conditions and between these two edges the plate may have varying thickness. The procedure is valuable in view of the fact that tables of deflections and stresses cannot be presented for variable-thickness inhomogeneous orthotropic plates as for uniform-thickness homogeneous isotropic plates even for commonly encountered loads because the results depend on the inhomogeneity coefficient and the orthotropic material properties instead of a single flexural rigidity. Numerical results, useful for the validation or otherwise of approximate solutions, are tabulated. The influences of the degree of the inhomogeneity, aspect ratio, thickness parameter and degree of non-uniformity on the deflections and stresses are investigated. This paper is partially supported by the Deanship of Scientific Research at King AbdulAziz University (Grant no. 172/427).     

Hamilton体系下矩形薄板受抛物线压力载荷的屈曲分析

针对四边简支矩形薄板在两对边相向的非线性分布压力下的面内应力分布以及屈曲问题,应用弹性力学的Hamilton体系和Galerkin法进行了研究.基于弹性力学的平面矩形域Hamilton体系,根据辛本征向量展开解法,得到了对应于零本征值和非零本征值的含待定常数的实数型面内应力分布通解.依据必须满足的应力边界条件,导出了矩形薄板在抛物线分布载荷下的面内应力分布.考虑到应力分布表达式的复杂性,用完全的解析方法得到屈曲载荷是不可能的.因此,运用基于虚功原理的Galerkin法,根据四边简支矩形薄板弯曲的位移边界条件,给出了不同长宽比矩形薄板受抛物线分布载荷的屈曲临界载荷.通过与已有文献中DQ法给出的数值计算结果比较,表明了本文求解方法的有效性和正确性.基于所给出的结果,可望为解决矩形薄板在非线性分布载荷下的面内应力分布以及屈曲问题提供一种新的研究方法.     

Nonlinear bending behavior of orthotropic Mindlin plate resting on orthotropic Pasternak foundation using GDQM

Rectangular plates resting on elastic foundations are operational activities of large transportation aircraft on runways, footings, foundation of spillway dam, civil building in cold regions, and bridge structures. Hence, in the present work, nonlinear bending analysis of embedded rectangular plates is investigated based on orthotropic Mindlin plate theory. The elastic medium is simulated by orthotropic Pasternak foundation. Adopting the nonlinear strain–displacement relation, the governing equations are derived based on energy method and Hamilton’s principle. The generalized differential quadrature method is performed for the case when all four ends are clamped supported. The influences of the plate thickness, shear-locking, elastic medium constants, and applied force on the nonlinear bending of the rectangular plate are studied. Results indicate that increasing the plate thickness decreases the deflection of the plate. It is also observed that increasing the applied force increases the deflection of the plate. Furthermore, considering elastic medium decreases deflection of the plate, and the effect of the Pasternak-type is higher than the Winkler-type on the maximum deflection of the plate. Also, it is found that the present results have good agreement with previous researches.     

Vibrations of nonhomogeneous orthotropic rectangular plates with bilinear thickness variation resting on Winkler foundation

Free transverse vibrations of nonhomogeneous orthotropic rectangular plates with bilinear thickness variation resting on Winkler foundation are presented here using two dimensional boundary characteristic orthogonal polynomials in the Rayleigh-Ritz method on the basis of classical plate theory. Gram-Schmidt process has been used to generate orthogonal polynomials. The nonhomogeneity of the plate is assumed to arise due to linear variations in elastic properties and density of the plate material with the in-plane coordinates. The two dimensional thickness variation is taken as the Cartesian product of linear variations along the two concurrent edges of the plate. Effect of nonhomogeneity parameters, aspect ratio and thickness variation together with foundation parameter on the natural frequencies has been illustrated for the first three modes of vibration for four different combinations of clamped, simply supported and free edges correct to four decimal places. Three dimensional mode shapes for specified plate for all the four boundary conditions have been plotted. A comparison of results in special cases with published one has been presented.     

Variational analysis of gradient elastic flexural plates under static loading

Gradient elastic flexural Kirchhoff plates under static loading are considered. Their governing equation of equilibrium in terms of their lateral deflection is a sixth order partial differential equation instead of the fourth order one for the classical case. A variational formulation of the problem is established with the aid of the principle of virtual work and used to determine all possible boundary conditions, classical and non-classical ones. Two circular gradient elastic plates, clamped or simply supported at their boundaries, are analyzed analytically and the gradient effect on their static response is assessed in detail. A rectangular gradient elastic plate, simply supported at its boundaries, is also analyzed analytically and its rationally obtained boundary conditions are compared with the heuristically obtained ones in a previous publication of the authors. Finally, a plate with two opposite sides clamped experiencing cylindrical bending is also analyzed and its response compared against that for the cases of micropolar and couple-stress elasticity theories.     

热/机械载荷下功能梯度材料矩形厚板的弯曲行为

采用Reddy高阶剪切板理论，考虑材料物性参数随坐标和温度变化的特性，研究在均匀变化的温度场内功能梯度材料矩形板在面内与横向载荷共同作用下的横向弯曲问题，基于一维DQ法和Galerkin技术，给出了一对边固支，另对边任意约束时板弯曲问题的半解析解，以Si3N4/SUS304板为例考察了材料组份，温度场，面内载荷及边界约束条件等对功能梯度材料板弯曲行为的影响。     

环扇形薄板弯曲问题的环向辛对偶求解方法

根据平面弹性与薄板弯曲问题的相似性原理,极坐标系板弯曲的弯矩函数被引入作为原变量,并通过恰当的辛内积定义建立了环扇形薄板弯曲问题的一个辛几何空间. 然后应用类Hellinger-Reissner变分原理,导出了辛几何空间的对偶方程,从而将环扇形薄板弯曲问题导入到辛对偶求解体系. 于是,分离变量和本征展开的有效数学物理方法得以实施,给出环扇形薄板弯曲问题的一个分析求解方法. 具体讨论了两弧边简支和两弧边一边固支一边自由薄板的本征问题,分别导出它们对应的本征值超越方程和本征向量,并给出原问题本征展开形式的通解. 最后,给出了两个算例的分析解并与已有文献或数值方法的解进行了对比,结果表明该方法有很好的收敛性和精度.      

基于物理中面与高阶剪切变形理论双参数弹性地基上FGM梁的非线性弯曲分析

在本文中,基于物理中面概念与高阶剪切变形理论,用Ritz法给出双参数弹性地基上FGM梁非线性弯曲的近似解答,并且讨论不同温度场、地基参数、不同边界条件、以及体积分数变化对FGM梁力学行为的影响。值得进一步指出的是：在基准温度场中,Winkler地基FGM梁的挠度介于Pasternak型与无地基梁之间;在热传导场中固支FGM梁的挠度介于均匀热场与基准温度场之间,而简支FGM梁由于有初始热挠度的影响,并非总是如此。     

Thermal bending analysis of laminated orthotropic plates by the generalized differential quadrature method

The interlaminar stresses in a thin laminated rectangular orthotropic plate with four sides simply supported edges under bending was determined by using the generalized differential quadrature (GDQ) method involving the effects of thermal expansion strain and transverse load. The approximate stress and displacement solutions are obtained under the effects of thermal expansion force and uniform pressure load for eight-layer unidirectional laminates, symmetric cross-ply laminates. Numerical results on the dominant interlaminar stresses and displacement of bending analysis are compared to the Navier solution. The thermal induced forces have significant effect on the bending of plates.     

Nonlinear bendings of symmetrically layered anisotropic rectangular plates

This paper has studied the nonlinear bending of symmetrically layered anisotropic rectangular plates under various supports. The uniformlys valid N-order asymptotic solutions of the deflection and stress function are derived by the singular perturbation methods offered in [1]. The analysis and calculations are given for simply and clamped supported, rectangular plates subjected to combined edge tensions and lateral loading in conjunction with the modified Galerkin procedure (a method of weighted residuals).     

Non-linear analysis of functionally graded plates under transverse and in-plane loads

Large deflection and postbuckling responses of functionally graded rectangular plates under transverse and in-plane loads are investigated by using a semi-analytical approach. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The plate is assumed to be clamped on two opposite edges and the remaining two edges may be simply supported or clamped or may have elastic rotational edge constraints. The formulations are based on the classical plate theory, accounting for the plate-foundation interaction effects by a two-parameter model (Pasternak-type), from which Winkler elastic foundation can be treated as a limiting case. A perturbation technique in conjunction with one-dimensional differential quadrature approximation and Galerkin procedure are employed in the present analysis. The numerical illustrations concern the large deflection and postbuckling behavior of functional graded plates with two pairs of constituent materials. Effects played by volume fraction, the character of boundary conditions, plate aspect ratio, foundation stiffness, initial compressive stress as well as initial transverse pressure are studied.     

A convergent method of solution for clamped rectangular plate

The conventional method of solution for clamped rectangular plate lies in releasing the constraints for rotation at the clamped sides and substituting them by equivalent distributed bending moments. This paper gives a general formulation of this method for arbitrary load with proof of the convergence, and discusses the rate of convergence for displacements and bending moments. The method and its convergence can be extended to the case of arbitrary boundary conditions and to plate on elastic foundation.The author expresses his sincere gratitude to Prof. Hu Hai-chang for guidance in this work.     

On new symplectic superposition method for exact bending solutions of rectangular cantilever thin plates

A novel superposition method based on the symplectic geometry approach is presented for exact bending analysis of rectangular cantilever thin plates. The basic equations for rectangular thin plate are first transferred into Hamilton canonical equations. By the symplectic geometry method, the analytic solutions to some problems for plates with slidingly supported edges are derived. Then the exact bending solutions of rectangular cantilever thin plates are obtained using the method of superposition. The symplectic superposition method developed in this paper is completely rational compared with the conventional analytical ones because the predetermination of deflection functions, which is indispensable in existing methods, is dispelled.     

On a method for determining the fixing conditions for a rectangular plate from the natural frequencies

We prove the duality of solutions for the problem of determining the boundary conditions on two opposite sides of a rectangular plate from the frequency spectrum of its bending vibrations. A method for determining the boundary conditions on two opposite sides of a rectangular plate from nine natural frequencies is obtained. The results of numerical experiments justifying the theoretical conclusions of the paper are presented. Rectangular plates are widely used in various technical fields. They serve as printed circuit boards and header plates, bridging plates, aircraft and ship skin, and parts of various mechanical structures [1–4]. If the plate fixing cannot be inspected visually, then one can use the natural bending vibration frequencies to find faults in the plate fixing. For circular and annular plates, methods for testing the plate fixing were found in [5–7], where it was shown that the type of fixing of a circular or annular plate can be determined uniquely from the natural bending vibration frequencies. The following question arises: Is it possible to determine the type of fixing of a rectangular plate on two opposite sides of the plate from the natural bending vibration frequencies if the other two sides are simply supported? Since the opposite sides of the plate are equivalent to each other, a plate with “rigid restraint—free edge” fixing will sound exactly the same as a plate with “free edge—rigid restraint” fixing. Hence we cannot say that the type of fixing of a rectangular plate on two opposite sides can be uniquely determined from its natural bending vibration frequencies. But it turns out that we can speak of duality in the solution of this problem. Here we observe an analogy with the problem of determining the rigidity coefficients of springs for elastic fixing of a string [8]: the rigidity coefficients of the springs are determined by the natural frequencies uniquely up to permutations of the springs.