Nonlinear vibration of laminated plates on an elastic foundation

This paper studies the effects of initial stresses on the nonlinear vibrations of laminated plates on elastic foundations. The nonlinear partial differential equations based on Mindlin plate theory are derived for the nonlinear vibration of laminated plates in a general state of nonuniform initial stress. Both rotary inertia and transverse stress are considered. Using the derived governing equations, the nonlinear vibration behavior of an initially stressed cross-ply plate on a Pasternak or Winkler elastic foundation is studied. The Galerkin's approximate method is applied to the governing partial differential equations to yield ordinary differential equations. The ordinary differential equations are solved by Runge–Kutta method to obtain the ratio of nonlinear frequency to linear frequency. The initial stress is taken to be a combination of a pure bending stress and an extensional stress in the plane of the plate. It is found that the foundation stiffness and initial stresses may result in a drastic change of the nonlinear vibration behavior. The frequency responses of nonlinear vibration are sensitive to the initial stress, vibration amplitude, modulus ratio and foundation stiffness. The effects of various parameters on the nonlinear vibration are discussed.     

Nonlinear vibration of initially stressed functionally graded plates

In this paper, nonlinear partial differential equations for the vibration motion of an initially stressed functionally graded plate (FGP) are derived. The formulations are based on the classical plate theory and derived for the nonlinear vibration motion of an FGP in a general state of arbitrary initial stresses. The material properties of the FGP are assumed to vary continuously from one surface to another, according to a simple power law distribution in terms of the constituent volume fractions. The motion of functionally graded ceramic/metal plate is obtained by employing the Galerkin method and then solved by Runge–Kutta method. The initial stress is taken to be a combination of pure bending stress plus an extensional stress in the plane of the ceramic/metal plate. It is found that the frequency responses of nonlinear vibration are sensitive of the state of initial stress, the amplitude of vibration and the volume fraction of FGP.     

弹性地基上正交异性板的振动分析

采用微分求积法对非线性弹性地基上的正交异性板的振动响应进行数值模拟。采用微分求积技术将偏微分方程转化为离散特征值问题。数值计算结果与相应文献的结果较为接近。数值计算结果表明,约束刚度、平板高宽比和地基刚度对正交异性板具有很大影响。     

Nonlinear free vibration of tapered Mindlin plates with edges elastically restrained against rotation using DQM

Large amplitude free vibration analyses of tapered Mindlin rectangular plates with elastically restrained against rotation edges are investigated using different differential quadrature method (DQM). The governing equations are based on the first-order shear deformation plate theory in conjunction with Green's strain and von Karman assumption. The spatial derivatives are discretized using DQM and the harmonic balance method is used to transform the resulting differential equations into frequency domain. A direct iterative method is used to solve the nonlinear eigenvalue system of equations. The convergence of the method is shown and their accuracy is demonstrated by comparing the results with those of the limiting cases, i.e. nonlinear free vibration analysis of plates with classical boundary conditions and also linear free vibration analysis of tapered plates. The effects of the elastic restraint coefficient at the edges and the geometrical parameters on the ratio of the nonlinear natural frequency to linear natural frequency of plates with linearly and bi-linearly varying thickness are studied.     

Nonlinear thermomechanical post-buckling of circular FGM plate with geometric imperfection

Nonlinear thermomechanical post-buckling of an imperfect functionally graded material (FGM) circular plate, subjected to both mechanical load and transversely non-uniform temperature rise, is presented. The material properties of FGM plates are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Based on von Kármán's plate theory, equilibrium equations governing a large axi-symmetric deformation of the FGM circular plate under thermomechanical loads are derived. In the analysis, the geometric imperfections of the plate are taken into account. By using a shooting method the nonlinear ordinary differential equations with immovably clamped boundary conditions are solved numerically. Responses for the nonlinear thermomechanical post-buckling responses of the FGM plate are obtained. Numerical examples are presented that relate to the performances of perfect and imperfect, homogenous and graded plates. Characteristic curves of the post-buckling deformation of the imperfect FGM circular plate varying with thermal loads, imperfection parameters and volume fraction index are plotted. And then effects of the load parameters, materials constitution, and the geometric imperfection of the plate on the deformation are discussed in detail.     

对带几何缺陷的FGM圆板的非线性热力学后屈曲现象的分析

带有几何缺陷FGM圆板的非线性热力学后屈曲现象是由力学荷载及横向不均匀温度升高导致的。基于组成材料的体积分数导致的能量分布法则,FGM板的材料特性假定在板厚度方向呈梯度变化。根据VonK偄rm偄n平板理论,推导出FGM板在温度荷载作用下的轴对称大变形的计算公式。在此分析中同时考虑了板的几何缺陷。通过运用目标法,解决了那些带有固定边界条件的非线性方程。数值算例中考虑了是否带有几何缺陷、均质或非均质的钢板。带缺陷FGM圆板的后屈曲变形特征曲线根据温度荷载、缺陷参数和体积分数的变化绘制而成。文中同时讨论了荷载参数、材料构成及板的几何缺陷对变形的影响。     

The effect of thermo-mechanical coupling for a simply supported orthotropic rectangular plate on non-linear dynamics

This paper is presented for the analysis of the large amplitude thermo-mechanically coupled vibration of the simply supported orthotropic rectangular thin plate. The governing partial differential equation of the large deflection orthotropic rectangular thin plate of thermo-mechanical coupling is derived and simplified to a set of three non-linear ordinary differential equations by the Galerkin method. The modeling result of numerical simulation indicated that the amplitude of the simply supported orthotropic rectangular thin plate decays under the various parameters of orthotropic materials. As these thermo-mechanical coupling factor, percentage of the Poisson ratio, percentage of the thermal conductance of the plate, percentage of the coefficient of thermal expansion, external in-plane force are decreased, the thermo mechanical damping of plate would be increased. The different thermal damping of the simply supported orthotropic plate can be controlled in the special range of the parameter.     

Non-linear free vibration of elliptic sector plates by a curved triangular p-element

A curved triangular p-element is developed and applied to geometrically non-linear free vibration of isotropic elliptic sector plates. The formulation takes into account shear deformation, rotary inertia, and geometric non-linearity. The element can describe the geometry of the elliptic sector plate exactly and is therefore suitable for this type of plate. The shifted Legendre orthogonal polynomials are used as enriching hierarchic shape functions. The element stiffness and mass matrices are integrated numerically using the Gauss–Legendre quadrature. The non-linear equations of free motion are obtained using the harmonic balance method and solved iteratively by the linearized updated mode method. Numerical results for the linear and non-linear frequencies of clamped elliptic sector plates are obtained. The efficiency of the proposed p-element is demonstrated through convergence studies and comparisons with published results. The effects of the ellipse eccentricity, thickness ratio, and sector angle on the backbone curves are examined. It is shown that these parameters influence the hardening behavior.     

Large deflection analysis of flexible plates by the meshless finite point method

The classical finite difference technique and methods based on series expansions can only be adopted for solving plates with simple geometry, loading and boundary conditions. In contrast, the finite element method has been widely used for general analysis of bending and flexible plates (coupled bending and in-plane effects). Lack of stress continuity and relatively expensive mesh generation and remeshing schemes have led to the emergence of meshless methods, such as the finite point method (FPM). FPM is a strong form solution which combines the moving least square interpolation technique on a domain of irregularly distributed points with a point collocation scheme to derive system governing equations. In this study, coupled nonlinear partial differential equations of fourth order are solved to analyse large deflection behaviour of plates subjected to lateral and in-plane loadings. Several plate problems are solved and compared with analytical solution and other available numerical results to assess the performance of the proposed approach.     

Optimal stiffener design of moderately thick plates under uniaxial and biaxial compression

In this paper, the optimal stiffener design of moderately thick plates under uniaxial and biaxial compression is investigated on the premise that the plate thickness and the required ultimate strength are given. As the theoretical basis of stiffener design, the ultimate strength formulations of weak stiffened thick panels under in-plane biaxial compression are first developed on the basis of large deflection orthotropic plate theory, in which the post-weld initial deflection is taken into account. The von Mises yield criterion is employed to determine the limit state of the panel, and the Nelder-Mead simplex algorithm is used to obtain the efficient solution of nonlinear differential equations. The optimization method presented is based on the stiffener design principles of the overall instability stress and of the working stress. In the optimization formulation, the numbers and geometric sizes of the stiffeners are defined as design variables; the weight ratio of stiffeners to plate is taken as a single objective function; requirements against overall buckling of the panel, local buckling of the plates between the stiffeners and local buckling of the stiffeners themselves are set as constraint functions. Results of both design examples and parameter studies show that, for moderately thick plates, the stiffener weight given by the proposed optimization method is much lower than the weight determined by the current stiffener design method on the premise of the same requirement of structural safety. Using the present optimization method to obtain the lightest and the most effective stiffener layout for moderately thick plates is proposed.     

Buckling of imperfect functionally graded plates under in-plane compressive loading

Buckling behavior of rectangular functionally graded plates with geometrical imperfections is studied in this paper. The equilibrium, stability, and compatibility equations of an imperfect functionally graded plate are derived using the classical plate theory. It is assumed that the nonhomogeneous mechanical properties of the plate, graded through thickness, are described by a power function of the thickness variable. The plate is assumed to be under in-plane compressive loading. Simultaneous solving of the stability and compatibility equations in conjunction with the equilibrium equations leads to the buckling relation of the plate. The critical buckling load of a sample plate is obtained and compared for different geometrical ratios. The results are reduced and compared with the results of perfect functionally graded and imperfect isotropic plates.     

Nonlinear vibration of thin plates with initial stress by spline finite strip method

The spline finite strip method and the incremental time–space finite element procedure are used to analyse large amplitude vibration of plates with initial stresses. Two improvements for the procedure are presented. The free vibration and the internal resonance of plates with initial stress as well as the forced vibration of plates with damping and initial stress are computed. The results compared favourably with those available in other publications.     

Vibration analysis of tall buildings with narrow rectangular plane configuration

In this paper, tall buildings of narrow rectangular plane configuration are treated as one-step or multi-step cantilever flexural-shear plates for the analysis of their free vibrations. The governing differential equations for the vibration of flexural-shear plates considering the effects of both flexural and shear deformation are established. The general solutions for one-step flexural-shear plates are derived and used to obtain the eigenvalue equation for multi-step cantilever flexural-shear plates. A new exact approach is presented which combines the transfer matrix method and a closed-form solution for a uniform flexural-shear plate. A numerical example demonstrates that the calculated natural frequencies and mode shapes of a tall building are in good agreement with the experimentally measured data. It is also shown that the effect of shear deformation on the fundamental natural frequency can be ignored, but its effect on the higher natural frequencies should be considered. Copyright © 1998 John Wiley & Sons, Ltd.     

Partial Hybrid Stress Element for Mindlin Laminated Plates Theory

Abstract: The partial hybrid stress method is applied to the Mindlin plate theory for orthotropic composite plates. The displacement formulation is adopted in the flexurel part and the hybrid formulation in the transverse shear part. Since the interface traction discontinuity is overcome, the present element performs much better than the displacement-based method. Although only the stress parameters in transverse shear part are required, the accuracy approaches that of the conventional hybrid method, The governing equations of plate are also derived from the modified Hellinger-Reissner variational principle.     

Dynamic buckling of thin isotropic plates subjected to in-plane impact

The dynamic stability behaviour of imperfect simply supported plates subjected to in-plane pulse loading is investigated. For the calculation of dynamic buckling loads a stress failure criterion is applied. The large-deflection plate equations are solved by a Galerkin method by using Navier's double Fourier series. In this paper the dynamic load factor (DLF) is redefined and plots that are useful for the design of plate structures are presented. Parametric studies are performed in which the influences of the pulse duration, shock function, imperfection, geometric dimensions and limit stress of the material are discussed. Comparison between the dynamic buckling loads, which are obtained by the commonly used criterion of Budiansky and Hutchinson [Proceedings of the 11th International Congress of Applied Mechanics (1964) 636], and the dynamic elastic limit loads, which are computed by the stress failure criterion, shows that the latter criterion is more useful for the design of lightweight structures.     

Bending analysis of thin annular sector plates using extended Kantorovich method

Highly accurate approximate closed-form solution is presented for bending of thin sector plates with clamped edges subjected to uniform and non-uniform loading using the extended Kantorovich method (EKM). Successive application of EKM converts the governing equation which is a forth-order partial differential equation (PDE) to two separate ordinary differential equations (ODE) in terms of r and θ. The obtained ODE systems are then solved iteratively with very fast convergence. In every iteration, exact closed-form solutions are obtained for both ODE systems. It is shown that the method provides sufficiently accurate results not only for deflections but also for stress components. Comparison of the deflection and stresses at various points of the plate shows very good agreement with results of other analytical and numerical analyses.     

An analytical study on post-buckling behaviour of imperfect sandwich panels subjected to uniform thermal stresses

The governing equations for determining thermal buckling of imperfect sandwich plates are developed using the large deflection theory and considering first shear deformation principles. The equations are then solved via an analytical method comprising infinite summation of trigonometric series and defining in-plane and out-of-plane displacement. After verification of the proposed method and convergence studies, the effects of parameters like plate aspect ratio, material properties and layer setting angles on buckling stress and the post-buckling path are studied. Numerical results show that the number of trigonometric terms does not affect the buckling thermal stress or the post-buckling path of sandwich plates having face layer setting angles less than 15°. Results also show that the buckling stress and post-buckling path for complementary layer setting angles are similar and that the rate of variation of bifurcation points for different face layer setting angles of sandwich panels strongly depend on material properties, especially on the face and core stiffness. Furthermore, for aspect ratios greater than five, buckling thermal stress remains about constant.     

Periodicity in the response of nonlinear plate, under moving mass

The dynamics of nonlinear thin plates under influence of relatively heavy moving masses is considered. By expansion of the solution as a series of mode functions, the governing equations of motion are reduced to an ordinary differential equation for time development of vibration amplitude, which is Duffing's oscillator with time varying coefficients. Through the application of Banach's fixed-point theorem, the periodic solutions are predicted. The method presented in this paper is general so that the response of plate to moving force systems can also be considered.     

Elastic stability of orthotropic plates

A semi-numerical method is presented for the analysis of orthotropic rectangular plates subjected to uniform, linearly-varying and partial inplane loads. The solution procedure is based on the classical method of separation of variables. The basic functions are chosen as the eigenfunctions for straight prismatic beams in free vibration. The two-dimensional governing partial differential equation is first reduced to an ordinary differential equation. The classical unidirectional finite difference method is then employed to solve the resulting ordinary differential equation. Results are presented for plates with different edge and loading conditions.     

Random response of antisymmetric angle-ply laminated plates

The random response of an antisymmetric angle-ply composite plate subjected to random lateral load on its surface is obtained. The first-order transverse shear deformation theory, involving five coupled partial differential equations, is used. To evaluate the effect of shear deformation, anisotropy and other problem parameters on the response, the numerical results obtained with this theory are compared with the results obtained by the classical plate theory. Significant differences are observed in the two sets of results obtained by the two theories especially for thick plates, thus demonstrating the importance of shear deformations and need for utilization of plate theories which properly account for such deformations.