A mixed differential quadrature and finite element free vibration and buckling analysis of thick beams on two-parameter elastic foundations

As a first endeavor, a mixed differential quadrature (DQ) and finite element (FE) method for boundary value structural problems in the context of free vibration and buckling analysis of thick beams supported on two-parameter elastic foundations is presented. The formulations are based on the two-dimensional theory of elasticity. The problem domain along axial direction is discretized using finite elements. The resulting system of equations and the related boundary conditions are discretized in the thickness direction and in strong-form using DQM. The method benefits from low computational efforts of the DQ in conjunction with the effectiveness of the FE method in general geometry and systematic boundary treatment resulting in highly accurate and fast convergence behavior solution. The boundary conditions at the top and bottom surface of the beams are implemented accurately. The presented formulations provide an effective analysis tool for beams free of shear locking. Comparisons are made with results from elasticity solutions as well as higher-order beam theory.     

Three-dimensional layerwise-finite element free vibration analysis of thick laminated annular plates on elastic foundation

Using a three-dimensional layerwise-finite element method, the free vibration of thick laminated circular and annular plates supported on the elastic foundation is studied. The Pasternak-type formulation is employed to model the interaction between the plate and the elastic foundation. The discretized governing equations are derived using the Hamilton’s principle in conjunction with the layerwise theory in the thickness direction, the finite element (FE) in the radial direction and trigonometric function in the circumferential direction, respectively. The fast rate of convergence of the method is demonstrated and to verify its accuracy, comparison studies with the available solutions in the literature are performed. The effects of the geometrical parameters, the material properties and the elastic foundation parameters on the natural frequency parameters of the laminated thick circular and annular plates subjected to various boundary conditions are presented.     

Accurate vibration analysis of skew plates by the new version of the differential quadrature method

An accurate free vibration analysis of skew plates is presented by using the new version of the differential quadrature method (DQM). Eight combinations of simply supported (S), clamped (C) and free (F) boundary conditions are considered. Detailed solution procedures are given and key points for success by using the DQM are emphasized. A way to simplifying the programming in using the DQM is proposed. Convergence study is made for the simply supported skew plate with a large skew angle. Good convergence of frequencies is observed. The DQ results agree very well with the existing first known accurate upper bound solutions, obtained by using Ritz method taking into considerations of the bending stress singularities occurred at corners having obtuse angles. Since slight discrepancy between the DQ data and the known accurate solutions is observed for plates with large skew angles, the DQ results are also compared with data obtained by using finite element method with very fine meshes to verify their accuracy.     

Vibration analysis of functionally graded plate with a moving mass

A hybrid method is proposed to predict the dynamic behavior of functionally graded (FG) plate subjected to a moving mass. The governing equations of motion of FG plate are derived using the Kirchhoff plate theory and Lagrange equation. Improved Rayleigh–Ritz solution is used to treat the spatial partial derivatives. Penalty method is employed to deal with the constraints, and the energy terms due to boundary conditions are included in Lagrange, hence it is not necessary to particularly consider the constraints in the modeling process. And the combination of simple polynomials and trigonometric functions is selected as the admissible functions. The advantage of this improvement in Rayleigh–Ritz method is that it is not needed to find satisfied admissible functions for different boundary conditions while the convergence of the solution is improved. Meanwhile, the method can be used to handle the versatile boundary conditions. Differential quadrature method (DQM) as a step-by-step time integration scheme is employed for discretization of temporal derivatives. The validated results show that the presented method is very reliable and efficient, and its convergence and accuracy are also better compared to finite element method for solving the dynamic problems of FG plate with moving loads (force and mass). Moreover, the influences of material properties and boundary conditions on maximum dynamic deflections are investigated, as well as moving speeds and inertial effects of loads (mass and force). Although only four edge boundary conditions are addressed in the present work, the proposed procedure is applicable for any arbitrary edge boundary conditions.     

基于改进Chebyshev级数的层合结构-振动分析新理论

提出了一种基于改进Chebyshev级数的层合结构高阶分层建模理论.该理论位移场由线性位移场和高阶位移场组成，线性位移场控制位移场的总体分布趋势，高阶位移场进行局部修正.高阶位移场由具有统一表达式的改进Chebyshev级数表示，通过改变高阶截断阶数可实现高阶位移场快速配置，能够满足不同建模精度需求.采用该高阶分层理论和广义谱方法推导了层合结构的自由振动特征方程，研究了一般边界条件下层合梁、板、壳的自由振动特性，并将计算结果与其他文献数据对比.结果表明:基于改进Chebyshev级数的层合结构高阶分层理论具有较高的建模精度和计算效率.     

Free-edge stress analysis of general composite laminates under extension, torsion and bending

In this study, based on the reduced form of elasticity displacement field for a long laminate, an analytical method is established to exactly obtain the interlaminar stresses near the free edges of generally laminated composite plates subjects to extension, torsion, and bending. The constant parameters being in the displacement field, which describe the global deformation of a laminate, are appropriately calculated by using the improved first-order shear deformation theory. Reddy’s layerwise theory is subsequently employed for analytical and numerical examinations of the boundary layer stresses within arbitrary laminated composite plates. Various numerical results are developed for the interlaminar normal and shear stresses along the interfaces and through the thickness of laminates near the free edges. Finally the effects of end conditions of laminates and geometric parameters on the boundary-layer stress are studied.     

An explicit exact analytical approach for free vibration of circular/annular functionally graded plates bonded to piezoelectric actuator/sensor layers based on Reddy’s plate theory

In the present study, a novel exact closed-form procedure based on the third order shear deformation plate theory is developed to analyze in-plane and out-of-plane frequency responses of circular/annular functionally graded material (FGM) plates embedded in piezoelectric layers for both close/open circuit electrical boundary conditions. Introducing a new analytical method, five governing partial deferential equations of motion beside Maxwell electrostatic equation are solved via an exact closed-form method. The high accuracy and reliability of the present approach is confirmed by comparing some of the present data with their counterparts reported in the literature. Finally, the effect of material properties, power law index and boundary conditions on the free vibration of the smart FGM plate are studied and discussed in detail.     

Haar wavelet discretization approach for frequency analysis of the functionally graded generally doubly-curved shells of revolution

This paper aims to investigate the free vibrational analysis of the generally doubly-curved shells of revolution made of functionally graded (FG) materials and constrained with different boundary conditions by means of an efficient, convenient and explicit method based on the Haar wavelet discretization approach. The FG materials of the shell consist of a combination of ceramic and metal, which four parameter power-law distribution functions have chosen for modeling of the smoothly and gradually variation of the material properties in the thickness direction. The theoretical model of the shell is formulated by employing of the first-order shear deformation theory. The rotation and displacement components of each point of the shell are expanded in the form of product of the Haar wavelet series in meridional direction as well as trigonometric series in the circumferential direction. By adding the boundary condition equations to the main system of equations, the constants appeared from the integrating of the Haar wavelet series are satisfied. In addition, with solving the characteristic equation, the vibrational results including the natural frequencies and the corresponding mode shapes are achieved. Then, the present results have been compared with those available in the literature. The results indicate that this method has high accuracy, high reliability and also a higher convergence rate in attaining the frequencies of the FG doubly-curved shells of revolution. Also, the effects of the main parameters such as power-law exponent, geometrical parameters, material distribution profiles and different types of boundary conditions, on the vibrational behavior of the FG doubly-curved shells of revolution, are investigated. Finally, taking into account the effects of geometrical parameters and material distribution profiles, for FG doubly-curved shells of revolution with different boundary conditions such as classic, elastic restraints and their combination, a variety of new frequency studies are provided which can be considered as proof results for further researches in this field.     

3D free vibration analysis of laminated cylindrical shell integrated piezoelectric layers using the differential quadrature method

This paper addresses the free vibration problem of multilayered shells with embedded piezoelectric layers. Based on the three-dimensional theory of elasticity, an approach combining the state space method and the differential quadrature method (DQM) is used. The shell has arbitrary end boundary conditions. For the simply supported boundary conditions closed-form solution is given by making the use of Fourier series expansion. Applying the differential quadrature method to the state space formulations along the axial direction, new state equations about state variables at discrete points are obtained for the other cases such as clamped or free end conditions. Natural frequencies of the hybrid laminated shell are presented by solving the eigenfrequency equation which can be obtained by using edges boundary condition in this state equation. Accuracy and convergence of the present approach is verified by comparing the natural frequencies with the results obtained in the literatures. Finally, the effect of edges conditions, mid-radius to thickness ratio, length to mid-radius ratio and the piezoelectric thickness on vibration behaviour of shell are investigated.     

The Robin problem for bending of elastic plates

The solution of the Robin problem in a finite domain for the system of equations modeling the bending of elastic plates with transverse shear deformation is approximated by means of a generalized Fourier series method closely connected to the structure of the boundary integral equation treatment of the problem. The theory is exemplified by numerical computation that shows a high degree of accuracy and efficiency.     

Analytical solution for buckling of Mindlin plates subjected to arbitrary boundary conditions

The study investigates the buckling behavior of isotropic plates subjected to axial, biaxial and pure shear loads. The effect of transverse shear deformation is taken into account by adopting the Mindlin first order shear theory. By applying the extended Kantorovich method, an exact solution is presented without any approximation on the boundary conditions. The procedure is proposed for thin, moderately thick and thick isotropic plates. The obtained results are in good agreement with those available in literature and they demonstrate the accuracy of the proposed procedure.     

Analysis of laminated composite plates using higher-order shear deformation plate theory and node-based smoothed discrete shear gap method

This paper presents a novel finite element formulation for static, free vibration and buckling analyses of laminated composite plates. The idea relies on a combination of node-based smoothing discrete shear gap method with the higher-order shear deformation plate theory (HSDT) to give a so-called NS-DSG3 element. The higher-order shear deformation plate theory (HSDT) is introduced in the present method to remove the shear correction factors and improve the accuracy of transverse shear stresses. The formulation uses only linear approximations and its implementation into finite element programs is quite simple and efficient. The numerical examples demonstrated that the present element is free of shear locking and shows high reliability and accuracy compared to other published solutions in the literature.     

On the stress singularities and boundary layer in moderately thick functionally graded sectorial plates

In this paper, the stress analysis of moderately thick functionally graded sector plate is developed for studying the singularities in vicinity of the vertex and effects of boundary layer. Based on the first-order shear deformation plate theory, the governing partial differential equations are obtained. Using an analytical method, the stretching and bending equilibrium equations are decoupled. Also, introducing a function, called boundary layer function, the three bending equations are converted into two decoupled equations. These equations are solved analytically and the effects of boundary layer function on stress components are shown. Also, the singularities of shear force, moment resultants and boundary layer function are discussed for both salient (α?180)$(\alpha ?180)$ and re-entrant (α>180)$(\alpha >180)$ sectorial plates. In order to verify the accuracy of the results, the governing equations are also solved using differential quadrature method (DQM). By comparing the results of exact method with DQM, a good agreement can be seen.     

Analytical solution of piezolaminated rectangular plates with arbitrary clamped/simply-supported boundary conditions under thermo-electro-mechanical loadings

This paper extends an analytical method for static analysis of general cross-ply piezolaminated rectangular plates with any combination of clamped/simply-supported boundary conditions under uncoupled thermo-electro-mechanical loadings. This method is based on the novel superposition method and the first-order shear deformation theory (FSDT). The FSDT enables this expanded method to consider the effect of shear deformation of the plate. The process of applying electrical and thermal resultants causes some advantages due to its simplicity and less computational process. In this analysis displacement components are written in terms of unknown force and moment resultants. Using Fourier series for displacement components, mechanical, thermal, and/or electrical stress resultants, the complex governing differential equations of the plate are reduced to a set of linear algebraic equations with non-trivial solution. The obtained equations may be solved analytically to determine the unknown stress resultants. Several examples are proposed, and their obtained numerical results are compared with those available in the literature to verify the convergence, high accuracy, and the capability of the present method to analyze the static behavior of piezolaminated plates. It is found that there is high agreement between the present results with those obtained by other investigators.     

考虑高阶横向剪切正交各向异性板非线性弯曲的微分求积分析

采用微分求积方法（DQ方法）讨论了计及高阶横向剪切的正交各向异性弹性板的非线性弯曲问题．导出了非线性控制方程的DQ形式,利用推广的DQWB技巧处理了高阶矩的边界条件．进一步推广并运用新的分析技术简化了非线性方程的计算．为说明该方法的可靠性和有效性,将考虑剪切变形及不计剪切变形的薄板的数值结果与三维弹性解析解及其它数值解进行了比较,同时研究了数值结果的收敛性,并考察了不同的节点分布对收敛速度的影响A·D2还考察了几何、材料参数及横向剪切效应对正交各向异性板非线性弯曲的影响．分析结果表明横向剪切效应对正交各向异性中厚板的影响是显著的．     

Geometrically nonlinear static and dynamic analysis of functionally graded skew plates

The present paper deals with nonlinear static and dynamic behavior of functionally graded skew plates. The equations of motion are derived using higher order shear deformation theory in conjunction with von-Karman’s nonlinear kinematics. The physical domain is mapped into computational domain using linear mapping and chain rule of differentiation. The spatial and temporal discretization is based on fast converging finite double Chebyshev series and Houbolt’s method. Quadratic extrapolation technique is employed to linearize the governing nonlinear equations. The spatial and temporal convergence and validation studies have been carried out to establish the efficacy of the present solution methodology. In case of dynamic analysis, the results are obtained for uniform step, sine, half sine, triangular and exponential type of loadings. The effect of volume fraction index, skew angle and boundary conditions on nonlinear displacement and moment response are presented.     

The Dual Singular Function Method for 2D Boundary Integral Equations

The accuracy of standard boundary element methods for elliptic boundary value problems deteriorates if the boundary of the domain contains corners or if the boundary conditions change along the boundary. Here we first investigate the convergence behaviour of standard spline Galerkin approximation on quasi-uniform meshes for boundary integral equations on polygonal domains. It turns out, that the order of convergence depends on some constant describing the singular behaviour of solutions near corner points of the boundary. In order to recover the full order of convergence for the Galerkin approximation we propose the dual singular function method which is often used for improving the accuracy of finite element methods. The theoretical convergence results are confirmed and illustrated by a numerical example.     

Nonlinear flexural vibrations of composite shallow open shells

This paper addresses geometrically nonlinear flexural vibrations of open doubly curved shallow shells composed of three thick isotropic layers. The layers are perfectly bonded, and thickness and linear elastic properties of the outer layers are symmetrically arranged with respect to the middle surface. The outer layers and the central layer may exhibit extremely different elastic moduli with a common Poisson's ratio ν. The considered shell structures of polygonal planform are hard hinged supported with the edges fully restraint against displacements in any direction. The kinematic field equations are formulated by layerwise application of a first order shear deformation theory. A modification of Berger's theory is employed to model the nonlinear characteristics of the structural response. The continuity of the transverse shear stress across the interfaces is specified according to Hooke's law, and subsequently the equations of motion of this higher order problem can be derived in analogy to a homogeneous single-layer shear deformable shallow shell. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Análise de vigas laminadas utilizando o natural neighbour radial point interpolation method

In this work, a meshless method, “natural neighbour radial point interpolation method” (NNRPIM), is applied to the one‐dimensional analysis of laminated beams, considering the theory of Timoshenko.The NNRPIM combines the mathematical concept of natural neighbours with the radial point interpolation. Voronoï diagrams allows to impose the nodal connectivity and the construction of a background mesh for integration purposes, via influence cells. The construction of the NNRPIM interpolation functions is shown, and, for this, it is used the multiquadratic radial basis function. The generated interpolation functions possess infinite continuity and the delta Kronecker property, which facilitates the enforcement of boundary conditions, since these can be directly imposed, as in the finite element method (FEM).In order to obtain the displacements and the deformation fields, it is considered the Timoshenko theory for beams under transverse efforts. Several numerical examples of isotropic beams and laminated beams are presented in order to demonstrate the convergence and accuracy of the proposed application. The results obtained are compared with analytical solutions available in the literature.