圆薄板轴对称弯曲问题的基本方程讨论

首先通过级数展开和求极限运算的方法,确定了等厚度圆薄板轴对称大/小挠度问题的弯曲微分方程在圆心处的表达式.其次,根据轴对称问题的特点,推导出了实心圆薄板在圆心处应满足边界条件的数学表达式,使圆薄板轴对称大/小挠度问题的弯曲微分方程应满足的边界条件达到了应有的数量.本文工作进一步完善了圆薄板轴对称弯曲问题的微分方程形式和边界条件,从而使我们可以利用成熟的微分方程数值解法,对具有较复杂载荷的实心圆薄板轴对称弯曲微分方程进行数值求解.     

The membranal and flexural equations of thin elastic plates deduced exactly from the three-dimensional theory

Summary The classical field and boundary equations governing the in-plane stretching and the transversal bending of a thin plate are obtained by means of an exact mathematical procedure consisting in integrating over the thickness the equilibrium equations for a three-dimensional cylindrical body, made of a homogeneous, linearly elastic, transversely isotropic material, and subject to suitable internal constraints.

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Sommario Le classiche equazioni che descrivono il comportamento flessionale e membranale di una piastra, con le relative condizioni ai limiti, sono ottenute mediante un procedimento matematico esatto, consistente nell'integrazione sullo spessore delle equazioni di equilibrio per un corpo tridimensionale cilindrico, costituito da materiale linearmente elastico, omogeneo, trasversalmente isotropo e soggetto ad opportuni vincoli interni.

     

On the non-linear dynamic stability problem for thin elastic plates

In this paper the non-linear response of thin elastic plates under parametric excitation is investigated. A new analytical method is proposed. It gives the possibility to obtain all the characteristic features of the phenomenon considered, which are known from experiments—the existing of beats, their dependence on the excitation parameter, the influence of the initial conditions, the typical character of the vibrations in the different regions. Analog computer studies are carried out, and they show clearly the influence of different parameters on the output of the problem considered.     

薄板几何非线性中的精化元方法及膜闭锁问题

基于放松单元间协调条件的大变形变分原理和全局拉格朗日方法，推导了几何非线性精化三角形薄板单元。对几何刚度矩阵，通过引入特殊的单元位移函数，有效地消除了薄板弯曲问题中伴生的膜闭锁现象。数值结果表明该单元在几何非线性分析中既能消除膜闭锁又具有较高精度。     

Refined theory of orthotropic plates subject to tangential surface loads

The existence of two correction coefficients traditionally introduced to account for the effect of the distribution of tangential stresses over the thickness of a plate is discussed. The virtual-work principle is used to generalize the expressions for the coefficients to the case of arbitrary loading. These expressions and hypotheses for displacements help to derive equations for orthotropic rectangular plates subject to tangential surface loads. These equations account for the effect of the distribution of tangential stresses over the thickness of the plate. Numerical examples are given. The results obtained are compared with those produced by other theories __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 4, pp. 107–119, April 2008.     

Linearized equations of nonlinear elastic deformation of thin plates

A linearized system of equations governing elastic deformation of a thin plate with arbitrary boundary conditions at its faces in an arbitrary curvilinear coordinate system is proposed. This system of equations is the first approximation of a oneparameter sequence of equations of twodimensional problems obtained from the initial threedimensional problem by approximating unknown functions by truncated series in Legendre polynomials. The stability problem of an infinite plate compressed uniaxially is solved. The results obtained are compared with the existing solutions.     

General theory of thin plates on the basis of nonsymmetric theory of elasticity

The paper uses the asymptotically justified hypothesis method to construct three different general refined theories of micropolar thin elastic plates, depending on the values of physical dimensionless material parameters, involving: (i) independent displacement and rotation fields, (ii) constrained rotation, and (iii) low shear stiffness. All angular shear deformations are taken into account.     

On the modelling of dynamic problems for plates with a periodic structure

Summary The subject of analysis is the bending of elastic plates exhibiting a nonhomogeneous periodic structure and/or a periodically variable thickness in a certain direction parallel to the plate's midplane. The fundamental modelling problem is how to obtain an effective 2D-model of a plate under consideration, i.e., a 2D-model represented by PDEs with constant coefficients. This problem for periodic plates has been solved independently in [5] and [10], using asymptotic homogenization. However, homogenization neglects dynamic phenomena related to the plate's rotational inertia and cannot be applied to the analysis of higher-order vibration frequences. The main aim of this contribution is to formulate a new non-asymptotic effective 2D-model of a periodic plate which is free from the mentioned drawbacks and describes the dynamic behaviour of plates having the thickness of the order of the period length. The proposed model is applied to the analysis of some vibration problems.