Analytical solutions of refined plate theory for bending,buckling and vibration analyses of thick plates

Analytical solutions for bending, buckling, and vibration analyses of thick rectangular plates with various boundary conditions are presented using two variable refined plate theory. The theory accounts for parabolic variation of transverse shear stress through the thickness of the plate without using shear correction factor. In addition, it contains only two unknowns and has strong similarities with the classical plate theory in many aspects such as equations of motion, boundary conditions, and stress resultant expressions. Equations of motion are derived from Hamilton’s principle. Closed-form solutions of deflection, buckling load, and natural frequency are obtained for rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions. Comparison studies are presented to verify the validity of present solutions. It is found that the deflection, stress, buckling load, and natural frequency obtained by the present theory match well with those obtained by the first-order and third-order shear deformation theories.     

A new sinusoidal shear deformation theory for bending,buckling, and vibration of functionally graded plates

A new sinusoidal shear deformation theory is developed for bending, buckling, and vibration of functionally graded plates. The theory accounts for sinusoidal distribution of transverse shear stress, and satisfies the free transverse shear stress conditions on the top and bottom surfaces of the plate without using shear correction factor. Unlike the conventional sinusoidal shear deformation theory, the proposed sinusoidal shear deformation theory contains only four unknowns and has strong similarities with classical plate theory in many aspects such as equations of motion, boundary conditions, and stress resultant expressions. The material properties of plate are assumed to vary according to power law distribution of the volume fraction of the constituents. Equations of motion are derived from the Hamilton’s principle. The closed-form solutions of simply supported plates are obtained and the results are compared with those of first-order shear deformation theory and higher-order shear deformation theory. It can be concluded that the proposed theory is accurate and efficient in predicting the bending, buckling, and vibration responses of functionally graded plates.     

Size-Dependent Effects of Micro-Nano Mindlin Plates Based on the Couple Stress Theory北大核心CSCD

基于偶应力理论,建立了适用于微纳米结构的Mindlin板理论。考虑横向剪切变形和材料的尺度效应并引入长度尺寸参数,推导了各向同性微纳米Mindlin板的本构方程。根据板的平衡条件,进一步推导出用位移函数和转角函数表示的板的屈曲和振动控制方程。通过对位移和转角变量进行空间和时间域上的分离,得出了四边简支（SSSS）和对边简支、对边固支（SCSC）两种边界情况下微纳米板的屈曲和振动问题的解析解。然后利用MATLAB软件进行算例分析,获得了不同尺寸参数、长宽比、厚长比等情况下板的临界屈曲荷载和固有频率。研究结果与已有文献中的结果以及ABAQUS有限元仿真解进行对比,结果表明,不同参数下的三种方法得到的结果均十分接近。算例分析发现,尺度效应对屈曲载荷和固有频率都有显著影响。     

Free vibration of functionally graded rectangular plates using first-order shear deformation plate theory

The main objective of this research work is to present analytical solutions for free vibration analysis of moderately thick rectangular plates, which are composed of functionally graded materials (FGMs) and supported by either Winkler or Pasternak elastic foundations. The proposed rectangular plates have two opposite edges simply-supported, while all possible combinations of free, simply-supported and clamped boundary conditions are applied to the other two edges. In order to capture fundamental frequencies of the functionally graded (FG) rectangular plates resting on elastic foundation, the analysis procedure is based on the first-order shear deformation plate theory (FSDT) to derive and solve exactly the equations of motion. The mechanical properties of the FG plates are assumed to vary continuously through the thickness of the plate and obey a power law distribution of the volume fraction of the constituents, whereas Poisson’s ratio is set to be constant. First, a new formula for the shear correction factors, used in the Mindlin plate theory, is obtained for FG plates. Then the excellent accuracy of the present analytical solutions is confirmed by making some comparisons of the results with those available in literature. The effect of foundation stiffness parameters on the free vibration of the FG plates, constrained by different combinations of classical boundary conditions, is also presented for various values of aspect ratios, gradient indices, and thickness to length ratios.     

Buckling and vibration of the two-dimensional quasicrystal cylindrical shells under axial compression

In this paper, the buckling and the free vibration of the quasicrystal cylindrical shells under axial compression are investigated. Three quasi-periodicity cases of quasicrystal cylindrical shells are considered. The first-order shear displacement theory of the cylindrical shells is utilized to obtain the equations of motion and the boundary conditions. Numerical results for simply supported cylindrical shells at the two ends are calculated. The effects of the geometry, in-plane phonon and phason loads, and half-wave number of the quasicrystal cylindrical shells on both the buckling loads and the frequency are demonstrated.     

Elastic and inelastic local buckling of stiffened plates subjected to non-uniform compression using the Galerkin method

A solution for the elastic and inelastic local buckling of flat rectangular plates with centerline boundary conditions subjected to non-uniform in-plane compression and shear stress is presented. The loaded edges are simply supported, the longitudinal edges may have any boundary conditions and the centerline is simply supported with a variable rotational stiffness. The Galerkin method, an effective method for solving differential equations, is applied to establish an eigenvalue problem. In order to obtain plate buckling coefficients, combined trigonometric and polynomial functions that satisfy the boundary conditions are used. The method is programmed, and several numerical examples including elastic and inelastic local buckling, are presented to illustrate the scope and efficacy of the procedure. The variation of buckling coefficients with aspect ratio is presented for various stress gradient ratios. The solution is applicable to stiffened plates and the flange of the I-shaped beams that are subjected to biaxial bending or combined flexure and torsion and shear stresses, and is important to estimate the reduction in elastic buckling capacity due to stress gradient.     

Analysis of Mindlin micro plates with a modified couple stress theory and a meshless method

A modified couple stress theory and a meshless method is used to study the bending of simply supported micro isotropic plates according to the first-order shear deformation plate theory, also known as the Mindlin plate theory. The modified couples tress theory involves only one length scale parameter and thus simplifies the theory, since experimentally it is easier to determine the single scale parameter. The equations governing bending of the first-order shear deformation theory are implemented using a meshless method based on collocation with radial basis functions. The numerical method is easy to implement, and it provides accurate results that are in excellent agreement with the analytical solutions.     

Exact solutions for free flexural vibration of Lévy-type rectangular thick plates via third-order shear deformation plate theory

In this paper, exact closed-form solutions in explicit forms are presented for transverse vibration analysis of rectangular thick plates having two opposite edges hard simply supported (i.e., Lévy-type rectangular plates) based on the Reddy’s third-order shear deformation plate theory. Two other edges may be restrained by different combinations of free, soft simply supported, hard simply supported or clamped boundary conditions. Hamilton’s principle is used to derive the equations of motion and natural boundary conditions of the plate. Several comparison studies with analytical and numerical techniques reported in literature are carried out to demonstrate accuracy of the present new formulation. Comprehensive benchmark results for natural frequencies of rectangular plates with different combinations of boundary conditions are tabulated in dimensionless form for various values of aspect ratios and thickness to length ratios. A set of three-dimensional (3-D) vibration mode shapes along with their corresponding contour plots are plotted by using exact transverse displacements of Lévy-type rectangular Reddy plates. Due to the inherent features of the present exact closed-form solution, the present findings will be a useful benchmark for evaluating the accuracy of other analytical and numerical methods, which will be developed by researchers in the future.     

压电弯曲元的夹层梁解析模型

压电弯曲元是一类传感和作动器件,已得到广泛的应用．基于一阶剪切变形理论发展了压电弯曲元夹层梁解析模型,对梁截面采用统一转角并将耦合电势沿厚度的分布假设为二次函数,进一步修正了横向剪应变对电位移的影响．以弯曲元简支梁自由振动为例进行数值分析,解析模型解与二维精确解相比具有良好的精度,为分析弯曲元动力机电响应提供了良好的解析模型．     

Free vibration of a functionally graded annular sector plate integrated with piezoelectric layers

Based on the first order shear deformation theory, free vibration behavior of functionally graded (FG) annular sector plates integrated with piezoelectric layers is investigated. The distribution of electric potential along the thickness direction of piezoelectric layers which is assumed to be a combination of linear and sinusoidal functions, satisfies both open and closed circuit electrical boundary conditions. Through a reformulation of governing equations and harmonic motion assumption, a novel decoupling method is suggested to transform the six second order coupled partial differential equations of motion into two eighth order and fourth order equations. A Fourier series method is then employed to present analytical solutions for free vibration of smart FG annular sector plates with simply supported radial edges and arbitrarily supported circular edges. The results, which can be used as a benchmark and suitable for design purposes, are verified with those reported in the literature. Finally, by presenting extensive ranges of frequencies, the effects of geometric parameters, power law index, FG and piezoelectric materials, electrical and mechanical boundary conditions as well as the piezoelectric layer thickness on vibration response of smart annular sector plates are discussed in detail.     

Post-bucking of angle-ply laminated plates under thermal loading

Notionsa. b, h Plate dimensionsL', [-. [1- mid-plane displacement componentsu- v- Ic dboensionless mid-plane displacement componentsVy., ac'~ slOPeS in xo and gi plane, ropectivelyJll, N number of terms in Cheby-shev series in x and y directions, respectivelyCCCC all edges clampedSSSS all edges simply supportedCCCS three edges (x = fi and y = 1) clamped and one (y = --1) simply supportedCCSS two edges (x = 11) clamped and two (y = fi) simply supportedCSSS one edge (x = --1) clamped …     

Free Vibration Analysis of Laminated Composite Plates Resting on Elastic Foundations by Using a Refined Hyperbolic Shear Deformation Theory

The free vibration of laminated composite plates on elastic foundations is examined by using a refined hyperbolic shear deformation theory. This theory is based on the assumption that the transverse displacements consist of bending and shear components where the bending components do not contribute to shear forces, and likewise, the shear components do not contribute to bending moments. The most interesting feature of this theory is that it allows for parabolic distributions of transverse shear stresses across the plate thickness and satisfies the conditions of zero shear stresses at the top and bottom surfaces of the plate without using shear correction factors. The number of independent unknowns in the present theory is four, as against five in other shear deformation theories. In the analysis, the foundation is modeled as a two-parameter Pasternak-type foundation, or as a Winkler-type one if the second foundation parameter is zero. The equation of motion for simply supported thick laminated rectangular plates resting on an elastic foundation is obtained through the use of Hamilton’s principle. The numerical results found in the present analysis for free the vibration of cross-ply laminated plates on elastic foundations are presented and compared with those available in the literature. The theory proposed is not only accurate, but also efficient in predicting the natural frequencies of laminated composite plates.     

Levy-type solution for buckling analysis of thick functionally graded rectangular plates based on the higher-order shear deformation plate theory

In this article, an analytical approach for buckling analysis of thick functionally graded rectangular plates is presented. The equilibrium and stability equations are derived according to the higher-order shear deformation plate theory. Introducing an analytical method, the coupled governing stability equations of functionally graded plate are converted into two uncoupled partial differential equations in terms of transverse displacement and a new function, called boundary layer function. Using Levy-type solution these equations are solved for the functionally graded rectangular plate with two opposite edges simply supported under different types of loading conditions. The excellent accuracy of the present analytical solution is confirmed by making some comparisons of the present results with those available in the literature. Furthermore, the effects of power of functionally graded material, plate thickness, aspect ratio, loading types and boundary conditions on the critical buckling load of the functionally graded rectangular plate are studied and discussed in details. The critical buckling loads of thick functionally graded rectangular plates with various boundary conditions are reported for the first time and can be used as benchmark.     

A new refined simple TSDT-based effective meshfree method for analysis of through-thickness FG plates

In this paper a novel numerical method based on the Moving Kriging (MK) interpolation meshfree method, integrated with a simple higher-order shear deformation plate theory for analysis of static bending, free vibration and buckling of functionally graded (FG) plates is presented. In the proposed technique, the shape functions are built by the Kriging technique which possesses the property of Kronecker delta function which makes it easy to enforce essential boundary conditions. The present formulation is based on a refined simple third-order shear deformation theory (R-STSDT), which is based on four variables and it still accounts for parabolic distribution of the transverse shearing strains and stresses through the thickness of the plate present in the original simple third-order shear deformation theory (STSDT). In this theory, instead of assuming a specific distribution for the displacement field, the theory of elasticity is used for obtaining the kinematics of the plate deformation. We first propose the formulation, and then several numerical examples are provided to show the merits of the proposed approach.     

Levy-type solution for free vibration analysis of orthotropic plates based on two variable refined plate theory

Closed-form solutions for free vibration analysis of orthotropic plates are obtained in this paper based on two variable refined plate theory. The theory, which has strong similarity with classical plate theory in many aspects, accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. Equations of motion are derived from the Hamilton’s principle. The closed-form solutions of rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions are obtained by applying the state space approach to the Levy-type solution. Comparison studies are performed to verify the validity of the present results. The effects of boundary condition, and variations of modulus ratio, aspect ratio, and thickness ratio on the natural frequency of orthotropic plates are investigated and discussed in detail.     

横观各向同性层合矩形板弯曲、振动和稳定的三维精确分析

针对四边简支的横观各向同性矩形板的弯曲、振动和稳定给出了新的状态空间分析方法。从横观各向同性弹性力学的三维基本方程出发,通过引入位移函数和应力函数,构造了两类相互独立的状态空间方程,不仅使原方程得到解耦而且降低了阶数,十分有利于具体问题的求解。对于四边简支的矩形板,建立了层合板上下表面状态变量间的关系式。特别针对矩形板的自由振动(稳定)问题,发现存在两类彼此无关的形式,一类对应板的纯面内振动(稳定),而另一类则是一般意义上的板的弯曲振动(稳定)。给出了数值结果,并考察了相关参数的影响。     

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.     

层合板一个新的高阶理论解析解

本文以复合材料的Reddy高阶理论为基础,引进一个位移函数Φ,将原来求解的微分方程组转化为一个高阶微分方程,得到了四边简支情况下的Navier型解,和一对边简支另一对边任意情况下的Levy型解.文中列举了算例进行比较,其数值结果和文献上已有结果相吻合,表明本文采用的解法是可靠的.Reddy高阶理论未知数不多,但精度比一阶剪切变形理论要好,计算时无需用剪切修正系数,计算较为简单.     

On the effects of coupling between in-plane and out-of-plane vibrating modes of smart functionally graded circular/annular plates

In recent years many articles concerned with the mechanics of functionally graded plates have been published. The variation in material properties through the thickness of the plate introduces a coupling between in-plane and transverse displacements, the coupling is important in the vibration of functionally graded plates (FGPs), but none have produced an exact closed-form solution for the in-plane as well as transverse vibrations of smart circular/annular FGPs. Therefore, this paper develops an exact closed-form solution for the free vibration of piezoelectric coupled thick circular/annular FGPs subjected to different boundary conditions on the basis of the Mindlin’s first-order shear deformation theory. Through the comparison of present results with those available, the accuracy of the present method was verified. The effects of coupling between in-plane and transverse displacements on the frequency parameters are proved to be significant. It is concluded that the developed model can describe vibrational behavior of smart FGM plates more realistic. Due to the inherent features of the present solution, all findings will be a useful benchmark for evaluating other analytical and numerical methods developed by researchers in the future.     

Calculation of simply supported circular plates in the statement of the problem of contact interaction in a package of two plates

The possibility of applying the mechanicomathematical model of bending of a package of transversely isotropic plates to approximate calculations of thick plates is investigated. Within the frame work of this model, the problem of axisymmetric bending of a package of two identical plates with simply supported edges is considered. The conditions of rigid contact are given between the plates. Based on the analytical result obtained, the unknown distribution of stresses and displacements across the thickness of the plate is approximated by the distribution of corresponding parameters in the package of two plates. For an isotropic body, the results of numerical calculations are compared with those given by the 3D theory of elasticity. In the case of a transversely isotropic body, a comparison with the results found by the refined bending model of plates (taking into account the transverse compression and shear) and the Timoshenko model is carried out. The accuracy to which the boundary conditions in every model are satisfied is analyzed. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 5, pp. 603–616, September–October, 2007.