Free vibration analysis of moderately thick functionally graded plates by local Kriging meshless method

This paper mainly Presents free vibration analyses of metal and ceramic functionally graded plates with the local Kriging meshless method. The Kriging technique is employed to construct shape functions which possess Kronecker delta function property and thus make it easy to implement essential boundary conditions. The eigenvalue equations of free vibration problems are based on the first-order shear deformation theory and the local Petrov–Galerkin formulation. The cubic spline function is used as the weight function which vanishes on internal boundaries of local quadrature domains and hence simplifies the implementation. Convergence studies are conducted to examine the stability of the present method. Three types of functionally graded plates – square, skew and quadrilateral plates – are considered as numerical examples to demonstrate the versatility of the present method for free vibration analyses.     

Hierarchic finite elements for moderately thick to very thin plates

We consider the numerical solution of Reissner-Mindlin plates. The model is widely used for thin to moderately thick plates. Generally low order finite elements (with degree one or two) are used for the approximation. Unfortunately the solution degenerates very rapidly for small thickness (locking phenomenon). Non standard formulations of the problem are usually combined with low order finite elements to weaken or possibly eliminate the locking of the numerical solution. We introduce a family of hierarchic finite elements and we present a set of numerical results for the plate problem in its plain formulation. We show that reliable solutions are achieved for all thicknesses of practical interest by using high order finite elements. Moreover, the hierarchic structure allows a low cost assessment of the quality of the solution.     

Physical shape functions in finite element analysis of moderately thick plates

A new rectangular finite element for moderately thick plates is presented. The element is based on the concept of physical shape functions, i.e. functions which contain in themselves physical and geometrical properties of the element. The analytic formulae of stiffness, geometric stiffness and mass matrices are presented for isotropic material. The element is free from locking and zero energy modes not corresponding to rigid body motions. Several examples are presented for static, initial stability and free vibration problems.     

Large deformation analysis of moderately thick laminated plates on nonlinear elastic foundations by DQM

In this paper, the nonlinear behavior of symmetric and antisymmetric cross ply, thin to moderately thick, elastic rectangular laminated plates resting on nonlinear elastic foundations are studied using differential quadrature method (DQM). The first-order shear deformation theory (FSDT) in conjunction with the Green’s strain and von Karman hypothesis are assumed for modeling the nonlinear behavior. Elastic foundation is modeled as shear deformable with cubic nonlinearity. The differential quadrature (DQ) discretized form of the governing equations with the various types of boundary conditions are derived. The Newton–Raphson iterative scheme is employed to solve the resulting system of nonlinear algebraic equations. Comparisons are made and the convergence studies are performed to show the accuracy of the results even with a few number of grid points. The effects of thickness-to-length ratio, aspect ratio, number of plies, fiber orientation and staking sequence on the nonlinear behavior of cross ply laminated plates with different boundary conditions resting on elastic foundations are studied.     

Thermal buckling analysis of moderately thick functionally graded annular sector plates

In this article, thermal buckling analysis of moderately thick functionally graded annular sector plate is studied. The equilibrium and stability equations are derived using first order shear deformation plate theory. These equations are five highly coupled partial differential equations. By using an analytical method, the coupled stability equations are replaced by four decoupled equations. Solving the decoupled equations and satisfying the boundary conditions, the critical buckling temperature is found analytically. To this end, it is assumed that the annular sector plate is simply supported in radial edges and it has arbitrary boundary conditions along the circular edges. Thermal buckling of functionally graded annular sector plate for two types of thermal loading, uniform temperature rise and gradient through the thickness, are investigated. Finally, the effects of boundary conditions, power law index, plate thickness, annularity and sector angle on the critical buckling temperature of functionally graded annular sector plates are discussed in details.     

On the flexural response of moderately thick bimodular laminated plates on elastic foundations

An exact flexural analysis of rectangular simply supported bimodular laminated plates attached to Winkler-Pasternak elastic foundations and subjected to single sinusoidal heap loading is described. A computer implementation of the analysis is used to generate numerical results for the primary quantities of interest, e.g. deflections, etc., which are presented in dimensionless graphical format. The effects of thickness ratio and the magnitudes of the foundation constants on plate response are highlighted, particularly in respect of single and two-layer cross-ply plates made of aramid- and polyester-rubbers.     

Invariant 8-node hybrid-stress elements for thin and moderately thick plates

Eight-node hybrid-stress elements are developed for the analysis of plates ranging from arbitrarily thin to moderately thick. The displacement behaviour is characterized by a transverse displacement and independent cross-section rotations, which are interpolated using the 8-node Serendipity shape functions. All components of stress are included; alternative elements are developed which differe in the form of the inplane distribution of the stresses. Elements are sought for whic the stiffiness is invariant and of correct rank, and whic show on signs of deterioration in the thin-plate limit. A discussion of the prospects for developing a 4-node element with these characteristics is also presented. Example problems are used to compare the performance of the 8-node elements including convergence behaviour, intraelement stress distributions and optimal sampling locations, and range of applicability in terms of plate thickness ratio.     

A Serendipity cubic-displacement hybrid-stress element for thin and moderately thick plates

A hybrid-stress element is developed for the analysis of thin and moderately thick plates. The independent transverse displacement and rotations are interpolated by the 12-node cubic Serendipity shape functions. All components of stress are included and 36β stress assumption is used. The element stiffness possesses correct rank and numerical results indicate that the element does not lock in the thin-plate limit. Results obtained using the present element are compared with those obtained using a 12-node assumed-displacement based Mindlin plate element with reduced integration; the present hybrid-stress element is shown to yield superior accuracy for all cases considered. In addition, the accuracy of the present element is compared against that of analogous 4-node and 8-node hybrid-stress Mindlin plate elements.     

A modified model for the penetration into moderately thick plates by a rigid,sharp-nosed projectile

The deep penetration model based on cavity expansion analysis (Int. J. Impact Eng. 27 (2002) 619) is modified to take account of the increase of the contact area between the projectile nose and the target medium before the projectile nose fully penetrates into the target. The modified penetration model is applicable to those penetration problems where the length of the projectile nose is comparable to the penetration depth. The predicted results are in good agreement with experimental results from Diskshit and Sundararajan (Int. J. Impact Eng. 12(3) (1992) 373) and Diskshit et al. (Int. J. Impact Eng. 16(2) (1995) 293).     

Trefftz method applied to a moderately thick plate

This paper presents the Trefftz direct and indirect methods for analysing moderately thick plate bending problems based on an improved plate theory (IPT). By using a T‐complete set in the formulation, a non‐singular boundary integral equation is obtained. The results show that the present method is effective for both thin and thick plates. An alternative type of locking problem which is caused by the overflow of Trefftz functions has been observed and a so‐called variable‐reducing procedure for eliminating such a phenomenon is also discussed. Copyright © 1999 John Wiley & Sons, Ltd.     

Determining the temperature-dependence of the thermophysical properties of solids by the method of successive approximations

The nonlinear heat conduction problem is solved by the method of successive approximations and, in connection with it, the feasibility of determining the temperature characteristics of the thermophysical properties is also considered.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 22, No. 4, pp. 718–724, April, 1972.     

The method of value oriented successive approximations for the average reward Markov decision process

Summary We consider the Markov decision process with finite state and action spaces at the criterion of average reward per unit time. We study the method of value oriented successive approximations, extensively treated by Van Nunen for the total reward case. Under a strong aperiodicity assumption and various conditions which guarantee that the gain of the process is independent of the starting state we show that the method converges and produces nearly optimal policies.

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Zusammenfassung Wir betrachten Markoffsche Entscheidungsprozesse mit endlichem Zustands- und Aktionenräumen für das Kriterium des Durchschnittsertrags. Wir untersuchen die Methode der wertorientierten sukzessiven Approximation, die für das Kriterium des Gesamtertrags von Van Nunen ausführlich untersucht wurde. Unter einer starken Aperiodizitätsbedingung und verschiedenen Voraussetzungen, die eine Unabhängigkeit des optimalen Durchschnittsertrages vom Anfangszustand garantieren, beweisen wir die Konvergenz der Methode.

     

Vibration of clamped moderately thick general cross-ply plates using a generalized Navier approach

A hitherto unavailable analytical solution to the free vibration problem of general cross-ply laminated rigidly clamped rectangular plates, incorporating first-order shear deformation, and rotatory and in-plane inertias into the formulation, is presented. A recently developed boundary continuous displacement-based generalized Navier solution technique is used to solve the five highly coupled linear second-order partial differential equations with constant coefficients, and the associated geometric boundary conditions. The assumed solution functions are in the form of double Fourier series, which satisfy the rigidly clamped boundary conditions a priori in a manner similar to the conventional Navier method. Convergence characteristics of the natural frequencies of both symmetric and antisymmetric cross-ply plates are numerically established. Other numerical results presented herein include (i) comparison with the corresponding available first-order shear deformation theory-based Galerkin and classical lamination theory-based boundary-discontinuous analytical solutions, and (ii) study of the effects of thickness and aspect ratio on the natural frequencies.     

Benchmark solution for free vibration of functionally graded moderately thick annular sector plates

In the present article, an exact analytical solution for free vibration analysis of a moderately thick functionally graded (FG) annular sector plate is presented. Based on the first-order shear deformation plate theory, five coupled partial differential equations of motion are obtained without any simplification. Doing some mathematical manipulations, these highly coupled equations are converted into a sixth-order and a fourth-order decoupled partial differential equation. The decoupled equation are solved analytically for an FG annular sector plate with simply supported radial edges. The accurate natural frequencies of the FG annular sector plates with nine different boundary conditions are presented for several aspect ratios, some thickness/length ratios, different sector angles, and various power law indices. The results show that variations of the thickness, aspect ratio, sector angle, and boundary condition of the FG annular sector plates can change the vibration wave number. Also for an FG annular sector plate with one free edge, in opposite to the other boundary conditions, the natural frequency decreases with increasing the aspect ratio for small aspect ratios. Moreover, the mode shape contour plots are depicted for an FG annular sector plate with various boundary conditions. The accurate natural frequencies of FG annular sector plates are presented for the first time and can serve as a benchmark solution.     

The derivation of equivalent constitutive equations of honeycomb structures by a two scale method

This paper evaluates the equivalent transverse shear and in-plane moduli of honeycomb cellular structures. The derivation is based upon a two scale method for the homogenization of periodic media. The equivalent two dimensional constitutive equations are evaluated analytically in terms of their geometry and material properties. The present results compare well with some of the existing analytical results obtained by conventional approaches and show the errors of some of the earlier results. The present method is a systemetic and rational technique for the homogenization of periodically inhomogeneous media. It allows us to derive the equivalent mechanical properties of honeybombs systemetically for the analysis and design of cellular structures of honeycomb. The structural efficiency of honeycombs will also be discussed.This work was supported by the GRC of Hong Kong Under Grant No. HKUST 019/91     

Solution of a boundary-value problem for a nonlinear differential equation by the method of successive approximations

A method is proposed for obtaining an approximate solution of a boundary value problem for the ordinary nonlinear heat-transfer equation in plane Poiseuille flow when the viscosity varies exponentially with the temperature.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 26. No. 3, pp. 539–543, March, 1974.     

中厚椭球壳自由振动动力刚度法分析

介绍精确动力刚度法分析中厚椭球壳自由振动具体实施方法,据环向波数不同将中厚椭球壳自由振动分解为一系列确定环向波数的一维振动;利用控制方程Hamilton形式建立动力刚度关系,用常微分方程求解器COLSYS求解控制方程获得单元动力刚度,用Wittrick-Williams算法求得该环向波数下椭球壳自振频率。数值算例给出中厚圆球壳及椭球壳不同边界条件的自振频率,验证动力刚度法高效、可靠、精确。     

Vibration analysis of nonhomogeneous moderately thick plates with point supports resting on Pasternak elastic foundation using element free Galerkin method

In this paper shear deformable plate theory in combination with Element-Free Galerkin Method (EFGM) is used for free vibration analysis of nonhomogeneous moderately thick plates with point supports resting on a two-parameter elastic foundation. It is shown that the vibration results obtained by this method are in a very good agreement with the available literatures in spite of using low numbers of nodes which can be considered as an inconvenience in some other methods to reach a satisfactory accuracy. Also, applicability of the method is demonstrated by solving numerical examples for different values of homogeneity variation parameter, aspect ratio, thickness to length ratio, foundation parameters, various types of boundary conditions and different numbers of point support. The numerical results present valuable information for engineers and designers in various structural applications and also prove useful to use as benchmarks for further references.