Thermal deflection of an elastic circular plate of variable thickness

The elastic behaviour of a solid circular plate of variable thickness under stationary temperature distribution is studied. The Young’s modulus and Poisson’s ratio are, however, supposed to be constants. Thermal deflections and moments per unit length and also the bending stresses have been determined for both clamped and simply-supported plates.     

几种典型板考虑初始荷载效应的基频近似解

基于两组板考虑初始荷载效应的动力控制微分方程:一般形式的动力控制微分方程和极坐标形式的动力控制微分方程,运用Galerkin（伽辽金)法求解得到了简支矩形板、固支矩形板、简支等边三角形板、固支椭圆形板、简支圆形板和固支圆形板6种典型板考虑初始荷载效应的自由振动基频（第一阶频率）近似解．通过与相关文献提出的有限元法计算结果对比,验证了公式的正确性．基频近似解表达式简单明了,物理意义明确,清楚地说明了初始荷载及相关因素对板自由振动基频的影响,直观地说明了板的初始荷载效应这一概念．计算分析表明:初始荷载的存在增加了板的弯曲刚度,提高了板的自振频率．这种初始荷载效应对频率的影响主要受初始荷载大小、跨厚比及边界条件等因素的影响．在计算分析和设计中应考虑并重视这种初始荷载效应对板计算分析带来的影响.     

Axisymmetric vibrations of circular plates of linearly varying thickness

Summary Free axisymmetric vibrations of circular plates of linearly varying thickness have been studied on the basis of classical theory of plates. The differential equation governing the motion of such plates have been solved by Frobenius method. The transverse displacement has been expressed as an infinite power series in the radial coordinate. Frequency parameters have been computed for the first three modes of vibration for clamped as well as simply supported circular plates corresponding to different values of taper constant.

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Zusammenfassung Auf Grund der klassischen Scheibentheorie werden die freien axialsymmetrischen Schwingungen der Kreisscheibe mit linear veränderlicher Dicke studiert. Die Differentialgleichung der Bewegung wird mit Hilfe der Methode von Frobenius gelöst. Die transversale Verschiebung wird als eine Reihe der radialen Koordinaten ausgedrückt. Die Schwingungsparameter werden für die ersten drei Eigenschwingungen sowohl für die eingespannte als auch für die aufgelegte Kreisscheibe für verschiedene Werte des Anzugs berechnet.

     

A semi-analytic approach for the nonlinear dynamic response of circular plates

This paper presents a new semi-analytic perturbation differential quadrature method for geometrically nonlinear vibration analysis of circular plates. The nonlinear governing equations are converted into a linear differential equation system by using Linstedt–Poincaré perturbation method. The solutions of nonlinear dynamic response and the nonlinear free vibration are then sought through the use of differential quadrature approximation in space domain and analytical series expansion in time domain. The present method is validated against analytical results using elliptic function in several examples for both clamped and simply supported circular plates, showing that it has excellent accuracy and convergence. Compared with numerical methods involving iterative time integration, the present method does not suffer from error accumulation and is able to give very accurate results over a long time interval.     

R-functions in problems of bending of three-layer plates

We study the application of the method of R-functions to the solution of problems of bending of elastic three-layer plates of arbitrary shape. We obtain new solution structures for plates that are freely supported over the whole boundary. We solve test problems and compare the experimental and theoretical results. All numerical results were obtained using the POLE programming system.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 77–79.     

3D thermo-mechanical bending solution of functionally graded graphene reinforced circular and annular plates

Within the framework of three-dimensional elasticity theory, this paper investigates the axisymmetric bending of novel functionally graded polymer nanocomposite circular and annular plates reinforced with graphene nanoplatelets (GPLs) whose weight fraction varies continuously and smoothly along the thickness direction. The generalized Mian and Spencer method is utilized to obtain the analytical solutions of nanocomposite circular and annular plates under a combined action of a uniformly distributed transverse load and a through-thickness steady temperature field. Three different distribution patterns of GPLs within the polymer matrix are considered. The present analytical solutions are validated through comparisons against those available in open literature for the reduced cases. A parametric study is conducted to examine the effects of GPL weight fraction, distribution pattern, plate thickness to radius ratio, and boundary conditions on the stress and deformation fields of the plate. The results show that GPL nanofillers with a low content can have a significant reinforcing effect on the bending behavior of the thermo-mechanically loaded plate.     

夹层圆板的大幅度振动

本文利用哈密顿原理导出了夹层圆板轴对称大幅度自由振动的基本方程,并给出了表板很薄情况下的简化形式.作为算例,利用修正迭代法求出了具有滑动固定边界条件夹层圆板对轴称大幅度自由振动的一种解析解,并由此导出了夹层圆板振幅和振频的解析关系式.     

Yield condition for circular cylindrical shells made of a fiber-reinforced composite

A yield condition is obtained for circular cylindrical shells made of a definite class of fiber-reinforced composite material whose components possess plastic properties. It is shown that, in the plane of generalized stresses — the axial bending moment and the circumferential force (when the axial force is absent) — the yield curve consists of two linear and four curvilinear sections. By approximating the curvilinear sections, we get a piecewise linear yield condition described by a hexagon in the plane indicated. The nonlinear equations and the corresponding piecewise linear equations of the yield condition for particular cases are given in the form of tables. In solving specific boundary-value problems, we consider a circular cylindrical shell simply supported at its ends and loaded with a uniform internal pressure, for which the load-carrying capacity is determined in relation to the mechanical properties of composite components and some characteristic geometrical parameters. The results of numerical calculations are represented in the form of graphs. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 5, pp. 655–666, September–October, 2006.     

Axisymmetric isospectral annular plates and membranes

A simpler ‘conditional’ form of the vibrating plateequation for which all the physical parameters may be spatiallyvarying is found when the relationship D(1 – v) = constantis satisfied between flexural rigidity and Poisson's ratio.A method of coordinate inversion is used to find a family ofinhomogeneous clamped annular plates (with axisymmetric spatiallyvarying density, thickness, Young's modulus, and Poisson's ratio)which have the same complete frequency spectrum as a standardhomogeneous clamped annular plate of the same inner and outerradii. In particular, these inhomogeneous plates have inversefourthpower radial dependence of density. General radial power-lawtransformations are also considered, leading to conclusionsfor the axisymmetric vibration spectra. Various results forcircular and annular membranes also emerge from the analysis.     

不用克希霍夫—拉夫假设的弹性圆板理论再探

本文在不用克希霍夫一拉夫假设的弹性板一般理论的基础上,建立了不用克希霍夫一拉夫假设的弹性圆板的一级近似理论,对圆板在四周固定和均布载荷的条件下,得到了具体的轴对称分析解,并和经典的圆薄板解进行了比较,证明本文新解更加接近实验结果,本文也具体地讨论了理论结果中厚度增大时的影响。     

弹性矩形薄板受迫振动的功的互等定理法（Ⅱ）：两邻边固定的矩形板

在本文,我们应用功的互等定理法给出了两邻边固定矩形板在分布和集中谐载作用下固定边弯矩幅值和自由边挠度幅值的分布.     

扇形板的富里哀—贝塞尔级数解

本文以加补充项的富里哀—贝塞尔双重级数的位移模式,对扇形弹性薄板在各种边界件条下的弯曲和振动问题,提出了一种应用范围比较广的、便于计算的、解析形式的解法.作为算例,文中给出了各种角度的径向边界简支或固定的扇形板在均布荷载或集中荷载作用下产生的挠度和弯矩的分布曲线,并与有关文献的数值结果进行了比较.本文推广了加补充项的富氏级数法的应用范围,并计算出各种角度的径向边界简支的扇形板的自振频率和节线的图表.     

Vibrations of an annular plate under the influence of a force rotating at a constant velocity

An exact solution is obtained for a dynamic problem concerning the bending of circular and annular plates under the influence of a force rotating at a constant angular velocity. It is shown that this problem is related to the problem of the bending of a plate under the influence of a harmonic load. Velocities leading to resonance are found and their first values are calculated for circular hinged and rigidly fastened plates.Translated from Dinamicheskie Sistemy, No. 4, pp. 62–65, 1985.     

Opening of a thin axisymmetric shell in a flow of ideal fluid

This article examines the opening, in an ideal fluid, of an axisymmetric impermeable shell modeling the canopy of a circular parachute. The form of the shell is determined in the class of shapes representing the combination of a truncated cone and an ellipsoid. The forces of interaction of the shell and the fluid are calculated on the basis of the method of discrete vortices. A Lagrange equation of the second kind is used to determine the time-varying coordinates of the generatrix. Numerical results are presented.Translated from Dinamicheskie Sistemy, No. 5, pp. 37–41, 1986.     

均布载荷作用下夹层圆板的非线性振动

给出了均布载荷作用下夹层圆板的大幅度振动方程.按假设的时间模态导出了该问题的非线性耦合的代数和微分特征方程组,并利用修正迭代法求出了该方程组的近似解析解,得到了周边固定夹层圆板振动的幅频-载荷特征关系.讨论了载荷对非线性振动性态的影响.     

Stress state of freely supported multilayered elliptical plates of anisotropic materials

The problem of a stressed state in elliptic plates has been considered in general for a rigid contour fixation. It is much more difficult to obtain a solution for the freely supported plates, even for isotropic materials. In this paper we suggest an approach for defining the stressed state of thin elliptic plates with layered structure under the condition of a freely supported contour. The solution is obtained in a rectangular cartesian coordinate system. The displacements, which are the fundamental unknowns, are given in the form of polynomials with unknown coefficients defined by a system of algebraic equations. The resolving equations and three out of the four boundary conditions are satisfied precisely. One boundary condition, is satisfied by means of collocation method of separate points of the contour. Estimation of the accuracy of the suggested approach is carried out by comparing the obtained results with the known ones. The problem of deformation of a twolayered plate has been discussed, in which the principal direction of elasticity does not coincide with the coordinate directions.S. P. Timoshenko Institute of Mechanics, National Academy of Science of the Ukraine, Kiev. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 4, pp. 496–504, July–August, 1997. Original article submitted March 19     

Korn’s Inequalities for Junctions of Elastic Bodies with Thin Plates

We prove the anisotropic Korn inequalities for an elastic junction of a massive body with thin plates clamped along parts of the lateral surfaces. The distribution of the weight factors in the norms under consideration depends essentially on the disposition of the plates, the way they are clamped to the body, and their relative rigidity (as compared with the body).Original Russian Text Copyright © 2005 Nazarov S. A.The author was supported by the Russian Foundation for Basic Research (Grant 03-01-00835).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 876–889, July–August, 2005.     

Numerical Prediction of the Strain State of Thermoset Matrix/Mineral Filler Composite Plates

The numerical prediction of the fields of inelastic strains (the linear invariant of the tensor of inelastic strains) in thermoset polyester/marble filler composite plates is discussed. A uniformly distributed load is applied to the plates, which lie on a steel base. The strain fields are predicted by means of the boundary element method by using creep test data for the composites and the polyester matrix itself. Identical creep tests were performed for two ages of the materials (1 month and 13 years), which allowed evaluating the aging effect. The study is carried out in two stages. At the first stage, the application of the generalized Maxwell-Gurevich equation to the thermoset matrix/mineral filler composite is demonstrated. The model parameters determined from the experimental creep data is used for the second stage, where the state of inelastic strains in the plates is predicted by applying the boundary element method. The influence of composite formulation (filler content) and physical aging of the polyester matrix on the state of inelastic strains in the plates is shown.__________Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 41, No. 2, pp. 145–156, March–April, 2005.     

Temperature stresses in plates made of reinforced laminated material

The heat conduction equations and the basic equations of the quasi-static axisymmetric problem of thermoelasticity are obtained for thin plates with heat transfer and cylindrical anisotropy. The nonsteady temperature stresses are determined for an infinite plate with a circular opening and an annular plate, both made of reinforced laminated material.Physicomechanical Institute, Academy of Sciences of the Ukrainian SSR, L'vov. Translated from Mekhanika Polimerov, No. 4, pp. 721–726, July–August, 1970.     

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.