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.     

Elasticity solutions for functionally graded rectangular plates with two opposite edges simply supported

England (2006) [13] proposed a novel method to study the bending of isotropic functionally graded plates subject to transverse biharmonic loads. His method is extended here to functionally graded plates with materials characterizing transverse isotropy. Using the complex variable method, the governing equations of three plate displacements appearing in the expansions of displacement field are formulated based on the three-dimensional theory of elasticity for a transverse load satisfying the biharmonic equation. The solution may be expressed in terms of four analytic functions of the complex variable, in which the unknown constants can be determined from the boundary conditions similar to that in the classical plate theory. The elasticity solutions of an FGM rectangular plate with opposite edges simply supported under 12 types of biharmonic polynomial loads are derived as appropriate sums of the general and particular solutions of the governing equations. A comparison of the present results for a uniform load with existing solutions is made and good agreement is observed. The influence of boundary conditions, material inhomogeneity, and thickness to length ratio on the plate deflection and stresses for the load x²yq are studied numerically.     

Nonlinear bending analysis of FGM elliptical plates resting on two-parameter elastic foundations

Nonlinear bending analysis is first presented for functionally graded elliptical plates resting on two-parameter elastic foundations, and investigations on FGM elliptical plates with immovable simply supported edge are also new in literature. Material properties are assumed to be temperature-dependent and graded in the thickness direction. The governing equations of a functionally graded plate are based on Reddy’s high-order shear deformation plate theory that includes thermal effects. Ritz method is employed to determine the central deflection-load and bending moment-load curves, the validity can be confirmed by comparison with related researchers’ results, and it is worth noting that the forms of approximate solutions are well-chosen in consideration of both simplicity and accuracy. Influences played by different supported boundaries, thermal environmental conditions, foundation stiffness, ratio of major to minor axis and volume fraction index are discussed in detail.     

Bending,buckling and free vibration analysis of incompressible functionally graded plates using higher order shear and normal deformable plate theory

In the present study, higher order shear and normal deformable plate theory is developed for analysis of incompressible functionally graded rectangular thick plates. Also, The effect of incompressibility is studied on the static, dynamic and stability responses of thick plate. It is assumed that plate is incompressible and the incompressibility condition is considered in addition to the governing equations for determining the unknowns. Since the plate is thick, higher order shear and normal deformable theory is applied so that the Legendre polynomials are used for expansion of displacement field components in the thickness direction. Also, it is supposed that material properties vary through the thickness based on the power law function. Utilizing the variational approach, governing equations for static, stability and dynamic analysis of plate are derived. Resulted equations are solved analytically for simply supported plates. Finally, the effects of material properties and dimensions on the response of incompressible plates are investigated in details.     

An analytical solution for bending of transversely isotropic thick rectangular plates with variable thickness

In this study, the bending solution of simply supported transversely isotropic thick rectangular plates with thickness variations is provided using displacement potential functions. To achieve this purpose, governing partial differential equations in terms of displacements are obtained as the quadratic and fourth order. Then, the governing equations are solved using the separation of variables method satisfying exact boundary conditions. The advantage of the purposed method is that there is no limitation on the thickness of the plate or the way the plate thickness is being varied. No simplifying assumption in the analysis process leads to the applicability and reliability of the present method to plates with any arbitrarily chosen thickness. In order to confirm the accuracy of the proposed solution, the obtained results are compared with existing published analytical works for thin variable thickness and thick constant thickness plate. Also, due to the lack of analytical research on thick plates with variable thickness, the obtained results are verified using the finite element method which shows excellent agreement. The results show that the maximum displacement of the plates with variable thickness is moved from the center toward the thinner plate edge. In addition, results exhibit the profound effects of both thickness and aspect ratio on stress distribution along the thickness of the plate. Results also show that varying thickness has not a profound impact on bending and twisting moments in transversely isotropic plates. Five different materials consist of four transversely isotropic and one isotropic, as a special case, are considered in this paper, which it is shown that the material properties have a more considerable impact on higher thickness plate.     

Anisotropic transient thermoelasticity analysis in a two-dimensional decagonal quasicrystal using meshless local Petrov–Galerkin (MLPG) method

The meshless local Petrov–Galerkin (MLPG) method is employed for anisotropic transient thermoelasticity analysis of 2D decagonal quasicrystals (QCs) subjected to transient thermal and mechanical shock loadings. The wave type model and the elasto-hydrodynamic model are applied to derive the phonon and phason governing equations, respectively. The temperature affects only the phonon field. To find the temperature distributions on the assumed 2D domain, the anisotropic heat conduction problem is solved using the MLPG method. Also, the MLPG method is successfully employed to obtain the transient behaviors of both phonon and phason displacements by solving the governing equations in local integral equations (LIEs) forms. Making use a unit step function as the test functions in the local weak-form of governing equations, we derived the local integral equations (LIEs) considered on small subdomains identical with support domains of test functions around each node. The radial basis functions are used for approximation of the spatial variation of field variables. The Laplace-transform technique is utilized to discretize the time variations.     

On the finite integral transform method for exact bending solutions of fully clamped orthotropic rectangular thin plates

Exact bending solutions of fully clamped orthotropic rectangular thin plates subjected to arbitrary loads are derived using the finite integral transform method. In the proposed mathematical method one does not need to predetermine the deformation function because only the basic governing equations of the classical plate theory for orthotropic plates are used in the procedure. Therefore, unlike conventional semi-inverse methods, it serves as a completely rational and accurate model in plate bending analysis. The applicability of the method is extensive, and it can handle plates with different loadings in a uniform procedure, which is simpler than previous methods. Numerical results are presented to demonstrate the validity and accuracy of the approach as compared with those previously reported in the bibliography.     

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.     

Uncovering the finite difference model equivalent to Hencky bar-net model for axisymmetric bending of circular and annular plates

As there are a few finite difference models in the literature for axisymmetric bending of plates, only one of these models is equivalent to the Hencky bar-net model (HBM) that comprises a finite number of rigid circular arcs and straight radial segments joined by frictionless hinges with elastic rotational springs. This paper is concerned with uncovering the one finite difference model (FDM) that is equivalent to the HBM. Based on the energy formulation, the governing equation for HBM is derived and it will be used to identify the FDM that has the same discrete set of equations. By using this equivalency between the HBM and the identified FDM, the expressions of edge spring stiffnesses of HBM are derived for various boundary conditions. As illustrative examples, the HBM is used to solve the bending problems of circular plates under uniformly and linearly increasing distributed loads. The analytical solutions of HBM avoids the singularity problem faced in FDM at the plate center. This paper also presents some benchmark bending solutions for annular plates with and without an internal ring support for different boundary restraints by using the HBM.     

弹性基上锥壳一般弯曲问题的精确解

本文应用Donnell的简化假定,从弹性基上锥壳位移型微分方程组出发,通过引入一个位移函数U(s,θ)(在极限情况下就退化成V.S.Vlasov对于圆柱壳所引的位移函数[5]),将基本微分方程组化成为一个八阶可解偏微分方程.这个方程的一般解用级数形式给出.对于在实际中有广泛应用价值的Winkler弹性基上锥壳的轴对称弯曲问题,本文给出了详细的数值结果,并求出了边缘荷载作用下的影响系数,这对计算弹性基上锥壳组合结构有着重要的意义.     

多变量样条元法分析弹性地基板的弯曲,振动与稳定问题

本文应用双三次乘积型二元B样条函数来构造弹性地基板的位移、弯矩和扭矩等多种场函数，由混合变分原理导出多变量样条无法方程．文中，对弹性地基板的弯曲、振动与稳定问题作了分析与计算．由于，本文方法设定了各自独立的场函数，因此，所算得的场未知量如位移、弯矩和扭矩值的精度均比较高。     

Nonlinear analysis of thin rectangular plates on Winkler–Pasternak elastic foundations by DSC–HDQ methods

This article introduces a coupled methodology for the numerical solution of geometrically nonlinear static and dynamic problem of thin rectangular plates resting on elastic foundation. Winkler–Pasternak two-parameter foundation model is considered. Dynamic analogues Von Karman equations are used. The governing nonlinear partial differential equations of the plate are discretized in space and time domains using the discrete singular convolution (DSC) and harmonic differential quadrature (HDQ) methods, respectively. Two different realizations of singular kernels such as the regularized Shannon’s kernel (RSK) and Lagrange delta (LD) kernel are selected as singular convolution to illustrate the present DSC algorithm. The analysis provides for both clamped and simply supported plates with immovable inplane boundary conditions at the edges. Various types of dynamic loading, namely a step function, a sinusoidal pulse, an N-wave pulse, and a triangular load are investigated and the results are presented graphically. The effects of Winkler and Pasternak foundation parameters, influence of mass of foundation on the response have been investigated. In addition, the influence of damping on the dynamic analysis has been studied. The accuracy of the proposed DSC–HDQ coupled methodology is demonstrated by the numerical examples.     

流固冲击下加筋板的非线性动态屈曲

分析了流固冲击下加筋板的非线性弹性动态屈曲．考虑板与筋的膜力,忽略面内位移,运用Hamilton变分原理,得出非线性控制方程,采用双级数形式的挠度假设,由Galerkin方法得到离散方程组,根据Budiansky-Roth(B-R)曲线,判断加筋板的动态屈曲．     

Analysis of the dynamic response of 3D-braided rectangular plates on an elastic foundation

A new micromacromechanical model of unit cells is suggested. In this model, a 3D-braided composite plate can be a cell system, and the geometry of each cell strongly depends on its position in the cross section of the plate. Based on the Reddy higher-order shear deformation theory of plates and the general von Karman-type equations, analytical solutions for the dynamic response of simply supported 3D-braided rectangular plates resting on a two-parameter (Pasternak-type) elastic foundation and subjected to transverse dynamic loads are found by using the perturbation method. The numerical illustrations concern the effects of geometric parameters, fiber volume fraction, braiding angle, and foundation stiffness. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 6, pp. 861–880, November–December, 2008.     

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.     

分析各向异性板的各向同性化样条积分方程法

本文分别按Reissner理论和Kirchhoff理论导出各向导性板的各向同性化控制方程,并论证了它们间在正交各向异性简支矩形板中的相通性.在用样条积分方程法求解中采用的只是些简单的各向同性板基本解,在稀疏剖分下也能有良好的计算精度.对双参数弹性地基上的板也只需在板上虚载的取值上附加某些项而不致增加多大的工作量.     

功能梯度板的柱面弯曲弹性力学解

在推广后的England-Spencer功能梯度板理论基础上,研究了功能梯度板在不同荷载作用下的柱面弯曲问题.采用该理论中的位移展开公式,并且材料参数沿板厚方向可以任意连续变化,并将材料由各向同性推广到正交各向异性.假设板在y方向无限长,最终建立了一个从弹性力学理论出发的正交各向异性功能梯度板在横向分布荷载作用下柱面弯曲问题的板理论.通过算例分析,讨论了边界条件、材料梯度及板厚跨比等因素对功能梯度板静力响应的影响.     

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．     

Chaotic and bifurcation dynamics of a simply-supported thermo-elastic circular plate with variable thickness in large deflection

This paper presents the conditions that can possibly lead to chaotic motion and bifurcation behavior for a simply-supported large deflection thermo-elastic circular plate with variable thickness by utilizing the criteria of fractal dimensions, maximum Lyapunov exponents and bifurcation diagrams. The governing partial differential equation of the simply supported thermo-elastic circular plate with variable thickness is first derived by means of Galerkin method. Several different features including Fourier spectra, phase plot, Poincar’e map and bifurcation diagrams are numerically computed. These features are used to characterize the dynamic behavior of the plate subjected to various excitations of lateral loads and thermal loads. Numerical examples are presented to verify the conditions that lead to chaotic motion and the effectiveness of the proposed modeling approach. Numerical modeling results indicate that large deflection motion of a thermo-elastic circular plate with variable thickness possesses chaotic motions and bifurcation motion under different lateral loads and thermal loads. The simulation results also indicate that the periodic motion of a circular plate can be obtained for the convex or the concave circular plate. The dynamic motion of the circular plate is periodic for the cases including (1) the lateral loading frequency is within a specific range, (2) thermal and lateral loadings are operated in a specific range and (3) the thickness parameter is less than a specific critical value for the convex circular plate or greater than a specific critical value for the concave circular plate. The modeling results show that the proposed method can be employed to predict the non-linear dynamics of any large deflection circular plate with variable thickness.     

3D elasticity analytical solution for bending of FG micro/nanoplates resting on elastic foundation using modified couple stress theory

This paper addresses a 3D elasticity analytical solution for static deformation of a simply-supported rectangular micro/nanoplate made of both homogeneous and functionally graded (FG) material within the framework of modified couple stress theory. The plate is assumed to be resting on a Winkler–Pasternak elastic foundation, and its modulus of elasticity is assumed to vary exponentially along thickness. By expanding displacement components in double Fourier series along in-plane coordinates and imposing relevant boundary conditions, the boundary value problem (BVP) of plate system, including its governing partial differential equations (PDEs) of equilibrium are reduced to BVP consisting only ordinary ones (ODEs). Parametric studies are conducted among displacement and stress components developed in the plate and FG material gradient index, length scale parameter, and foundation stiffnesses. From the numerical results, it is concluded that the out-of-plane shear stresses are not necessarily zero at the top and bottom surfaces of plate. The results of this investigation may serve as a benchmark to verify further bending analyses of either homogeneous or FG micro/nanoplates on elastic foundation.