CLASSICAL AND HOMOGENIZED EXPRESSIONS FOR BUCKLING SOLUTIONS OF FUNCTIONALLY GRADED MATERIAL LEVINSON BEAMS

The relationship between the critical buckling loads of functionally graded material(FGM) Levinson beams(LBs) and those of the corresponding homogeneous Euler-Bernoulli beams(HEBBs) is investigated. Properties of the beam are assumed to vary continuously in the depth direction. The governing equations of the FGM beam are derived based on the Levinson beam theory, in which a quadratic variation of the transverse shear strain through the depth is included.By eliminating the axial displacement as well as the rotational angle in the governing equations,an ordinary differential equation in terms of the deflection of the FGM LBs is derived, the form of which is the same as that of HEBBs except for the definition of the load parameter. By solving the eigenvalue problem of ordinary differential equations under different boundary conditions clamped(C), simply-supported(S), roller(R) and free(F) edges combined, a uniform analytical formulation of buckling loads of FGM LBs with S-S, C-C, C-F, C-R and S-R edges is presented for those of HEBBs with the same boundary conditions. For the C-S beam the above-mentioned equation does not hold. Instead, a transcendental equation is derived to find the critical buckling load for the FGM LB which is similar to that for HEBB with the same ends. The significance of this work lies in that the solution of the critical buckling load of a FGM LB can be reduced to that of the HEBB and calculation of three constants whose values only depend upon the throughthe-depth gradient of the material properties and the geometry of the beam. So, a homogeneous and classical expression for the buckling solution of FGM LBs is accomplished.     

Free vibration of functionally graded beams based on both classical and first-order shear deformation beam theories

The free vibration of functionally graded material （FGM） beams is studied based on both the classical and the first-order shear deformation beam theories. The equations of motion for the FGM beams are derived by considering the shear deforma- tion and the axial, transversal, rotational, and axial-rotational coupling inertia forces on the assumption that the material properties vary arbitrarily in the thickness direction. By using the numerical shooting method to solve the eigenvalue problem of the coupled ordinary differential equations with different boundary conditions, the natural frequen- cies of the FGM Timoshenko beams are obtained numerically. In a special case of the classical beam theory, a proportional transformation between the natural frequencies of the FGM and the reference homogenous beams is obtained by using the mathematical similarity between the mathematical formulations. This formula provides a simple and useful approach to evaluate the natural frequencies of the FGM beams without dealing with the tension-bending coupling problem. Approximately, this analogous transition can also be extended to predict the frequencies of the FGM Timoshenko beams. The numerical results obtained by the shooting method and those obtained by the analogous transformation are presented to show the effects of the material gradient, the slenderness ratio, and the boundary conditions on the natural frequencies in detail.     

Free vibration of functionally graded material beams with surface-bonded piezoelectric layers in thermal environment

Free vibration of statically thermal postbuckled functionally graded material （FGM） beams with surface-bonded piezoelectric layers subject to both temperature rise and voltage is studied. By accurately considering the axial extension and based on the Euler-Bernoulli beam theory, geometrically nonlinear dynamic governing equations for FGM beams with surface-bonded piezoelectric layers subject to thermo-electro- mechanical loadings are formulated. It is assumed that the material properties of the middle FGM layer vary continuously as a power law function of the thickness coordinate, and the piezoelectric layers are isotropic and homogenous. By assuming that the amplitude of the beam vibration is small and its response is harmonic, the above mentioned non-linear partial differential equations are reduced to two sets of coupled ordinary differential equations. One is for the postbuckling, and the other is for the linear vibration of the beam superimposed upon the postbuckled configuration. Using a shooting method to solve the two sets of ordinary differential equations with fixed-fixed boundary conditions numerically, the response of postbuckling and free vibration in the vicinity of the postbuckled configuration of the beam with fixed-fixed ends and subject to transversely nonuniform heating and uniform electric field is obtained. Thermo-electric postbuckling equilibrium paths and characteristic curves of the first three natural frequencies versus the temperature, the electricity, and the material gradient parameters are plotted. It is found that the three lowest frequencies of the prebuckled beam decrease with the increase of the temperature, but those of a buckled beam increase monotonically with the temperature rise. The results also show that the tensional force produced in the piezoelectric layers by the voltage can efficiently increase the critical buckling temperature and the natural frequency.     

Homogenized and classical expressions for static bending solutions for functionally graded material Levinson beams

The exact relationship between the bending solutions of functionally graded material (FGM) beams based on the Levinson beam theory and those of the corresponding homogenous beams based on the classical beam theory is presented for the material properties of the FGM beams changing continuously in the thickness direction. The deflection, the rotational angle, the bending moment, and the shear force of FGM Levinson beams (FGMLBs) are given analytically in terms of the deflection of the reference homogenous Euler-Bernoulli beams (HEBBs) with the same loading, geometry, and end supports. Consequently, the solution of the bending of non-homogenous Levinson beams can be simplified to the calculation of transition coefficients, which can be easily determined by variation of the gradient of material properties and the geometry of beams. This is because the classical beam theory solutions of homogenous beams can be easily determined or are available in the textbook of material strength under a variety of boundary conditions. As examples, for different end constraints, particular solutions are given for the FGMLBs under specified loadings to illustrate validity of this approach. These analytical solutions can be used as benchmarks to check numerical results in the investigation of static bending of FGM beams based on higher-order shear deformation theories.     

弹性地基上转动FGM梁自由振动的DTM分析

基于Euler-Bernoulli梁理论,利用广义Hamilton原理推导得到弹性地基上转动功能梯度材料(FGM)梁横向自由振动的运动控制微分方程并进行无量纲化,采用微分变换法(DTM)对无量纲控制微分方程及其边界条件进行变换,计算了弹性地基上转动FGM梁在夹紧-夹紧、夹紧-简支和夹紧-自由三种边界条件下横向自由振动的无量纲固有频率,再将控制微分方程退化到无转动和地基时的FGM梁,计算其不同梯度指数时第一阶无量纲固有频率值,并和已有文献的FEM和Lagrange乘子法计算结果进行比较,数值完全吻合。计算结果表明,三种边界条件下FGM梁的无量纲固有频率随无量纲转速和无量纲弹性地基模量的增大而增大;在一定无量纲转速和无量纲弹性地基模量下,FGM梁的无量纲固有频率随着FGM梯度指数的增大而减小;但在夹紧-简支和夹紧-自由边界条件下,一阶无量纲固有频率几乎不变。     

功能梯度梁与均匀梁静动态解间的相似转换

基于Euler-Bernoulli 梁理论, 研究了功能梯度材料梁的弯曲、屈曲和自由振动. 通过分析和比较功能梯度材料梁 和均匀梁的控制方程, 得到了功能梯度材料梁与均匀梁的解之间的相似转换关系. 在给定功 能梯度材料梁的材料性质在横向按幂函数分布的情况下, 导出了解之间的相似转换系数的解 析表达式. 该系数集中反映了功能梯度梁的材料非均匀性. 因此, 可将功能梯度材料梁的静 动态问题的求解转换为同样载荷和边界条件下均匀梁的静动态问题求解以及相似转换系数的 计算.     

多孔功能梯度材料Timoshenko梁的自由振动分析

基于Timoshenko梁理论研究多孔功能梯度材料梁(FGMs)的自由振动问题.首先,考虑多孔功能梯度材料梁的孔隙率模型,建立了两种类型的孔隙分布.其次,基于Timoshenko梁变形理论,给出位移场方程、几何方程和本构方程,利用Hamilton原理推导多孔功能梯度材料梁的自由振动控制微分方程,并进行无量纲化,然后应用微分变换法(DTM)对无量纲控制微分方程及其边界条件进行变换,得到含有固有频率的等价代数特征方程.最后,计算了固定-固定(C-C)、固定-简支(C-S)和简支-简支(S-S)三种不同边界下多孔功能梯度材料梁自由振动的无量纲固有频率.将其退化为均匀材料与已有文献数据结果对照,验证了正确性.讨论了孔隙率、细长比和梯度指数对多孔功能梯度材料梁无量纲固有频率的影响.     

Free vibration characteristics of sectioned unidirectional/bidirectional functionally graded material cantilever beams based on finite element analysis

Advancements in manufacturing technology, including the rapid development of additive manufacturing (AM), allow the fabrication of complex functionally graded material (FGM) sectioned beams. Portions of these beams may be made from different materials with possibly different gradients of material properties. The present work proposes models to investigate the free vibration of FGM sectioned beams based on onedimensional (1D) finite element analysis. For this purpose, a sample beam is divided into discrete elements, and the total energy stored in each element during vibration is computed by considering either the Timoshenko or Euler-Bernoulli beam theory. Then, Hamilton's principle is used to derive the equations of motion for the beam. The effects of material properties and dimensions of FGM sections on the beam's natural frequencies and their corresponding mode shapes are then investigated based on a dynamic Timoshenko model (TM). The presented model is validated by comparison with three-dimensional (3D) finite element simulations of the first three mode shapes of the beam.     

Modal analysis of cracked continuous Timoshenko beam made of functionally graded material

Abstract

The article addresses development of the Transfer Matrix Method (TMM) for free vibration of cracked continuous Timoshenko beam made of Functionally Graded Material (FGM). The governing equations of free vibration are established for the beam based on the power law of material grading, actual position of neutral plane and double spring model of crack. There is conducted frequency equation of the beam with intermediate rigid supports using the TMM after the transverse displacements at rigid supports have been disregarded. Therefore, the frequency equation is simplified and becomes more useful to compute natural frequencies of continuous FGM Timoshenko beam with a number of cracks. The obtained numerical results show the essential effect of cracks, material properties and also number of spans on natural frequencies of the beam.     

基于一种广义梁理论的弹性地基FGM简支梁自由振动解析解

基于一种扩展的n阶广义剪切变形梁理论(n-GBT),应用Hamilton原理,建立了以轴向位移、横向位移及转角为未知函数的Winkler-Pasternak弹性地基功能梯度材料(FGM)梁的自由振动方程,采用Navier法获得了弹性地基FGM简支梁自由振动的精确解。与多种梁理论预测结果进行比较,讨论并给出了GBT阶次n的理想取值;分析了梯度指标、跨厚比及地基刚度对FGM梁频率的影响。结果表明:本文方法有效且适用范围广,若采用高阶剪切梁理论模型,宜取n≥3的奇数;FGM梁的自振频率随材料梯度指标的增大而减小;随跨厚比的增加而增大,但当跨厚比大于20,跨厚比增加对频率的影响很小;随地基刚度的增加而增大,地基刚度足够大时,频率趋于收敛。     

Winkler-Pasternak弹性地基梁自由振动的二维弹性分析Two-dimensional elastic analysis for free vibration of beams set on winkler-pasternak elastic foundations

基于二维线弹性理论,应用Hamilton原理,获得Winkler-Pasternak弹性地基梁自由振动的控制微分方程,应用微分求积法(DQM)数值研究了梁自由振动的无量纲频率特性。计算结果与已有的结果(Bernoulli-Euler梁和Timoshenko梁)比较表明,本文的分析方法对弹性地基长梁和短梁自由振动的研究都有效。最后考虑了几何参数对梁频率的影响,以及不同边界条件下地基系数对频率的影响和收敛性。     

Winkler-Pasternak弹性地基梁自由振动的二维弹性分析

基于二维线弹性理论,应用Hamilton原理,获得Winkler-Pasternak弹性地基梁自由振动的控制微分方程,应用微分求积法(DQM)数值研究了梁自由振动的无量纲频率特性。计算结果与已有的结果(Bernoulli-Euler梁和Timoshenko梁)比较表明,本文的分析方法对弹性地基长梁和短梁自由振动的研究都有效。最后考虑了几何参数对梁频率的影响,以及不同边界条件下地基系数对频率的影响和收敛性。     

Nonlinear dynamic response of an edge-cracked functionally graded Timoshenko beam under parametric excitation

This paper investigates the nonlinear flexural dynamic behavior of a clamped Timoshenko beam made of functionally graded materials (FGMs) with an open edge crack under an axial parametric excitation which is a combination of a static compressive force and a harmonic excitation force. Theoretical formulations are based on Timoshenko shear deformable beam theory, von Karman type geometric nonlinearity, and rotational spring model. Hamilton’s principle is used to derive the nonlinear partial differential equations which are transformed into nonlinear ordinary differential equation by using the Least Squares method and Galerkin technique. The nonlinear natural frequencies, steady state response, and excitation frequency-amplitude response curves are obtained by employing the Runge–Kutta method and multiple scale method, respectively. A parametric study is conducted to study the effects of material property distribution, crack depth, crack location, excitation frequency, and slenderness ratio on the nonlinear dynamic characteristics of parametrically excited, cracked FGM Timoshenko beams.     

An analytical study on the nonlinear vibration of functionally graded beams

Nonlinear vibration of beams made of functionally graded materials (FGMs) is studied in this paper based on Euler-Bernoulli beam theory and von Kármán geometric nonlinearity. It is assumed that material properties follow either exponential or power law distributions through thickness direction. Galerkin procedure is used to obtain a second order nonlinear ordinary equation with quadratic and cubic nonlinear terms. The direct numerical integration method and Runge-Kutta method are employed to find the nonlinear vibration response of FGM beams with different end supports. The effects of material property distribution and end supports on the nonlinear dynamic behavior of FGM beams are discussed. It is found that unlike homogeneous beams, FGM beams show different vibration behavior at positive and negative amplitudes due to the presence of quadratic nonlinear term arising from bending-stretching coupling effect.     

Bending and free vibrational analysis of bi-directional functionally graded beams with circular cross-section

The bending and free vibrational behaviors of functionally graded (FG) cylindrical beams with radially and axially varying material inhomogeneities are investigated. Based on a high-order cylindrical beam model, where the shear deformation and rotary inertia are both considered, the two coupled governing differential motion equations for the deflection and rotation are established. The analytical bending solutions for various boundary conditions are derived. In the vibrational analysis of FG cylindrical beams, the two governing equations are firstly changed to a single equation by means of an auxiliary function, and then the vibration mode is expanded into shifted Chebyshev polynomials. Numerical examples are given to investigate the effects of the material gradient indices on the deflections, the stress distributions, and the eigenfrequencies of the cylindrical beams, respectively. By comparing the obtained numerical results with those obtained by the three-dimensional (3D) elasticity theory and the Timoshenko beam theory, the effectiveness of the present approach is verified.     

功能梯度材料微梁的热弹性阻尼研究

基于Euler-Bernoulli梁理论和单向耦合的热传导理论,研究了功能梯度材料(functionally graded material,FGM)微梁的热弹性阻尼(thermoelastic damping,TED).假设矩形截面微梁的材料性质沿厚度方向按幂函数连续变化,忽略了温度梯度在轴向的变化,建立了单向耦合的变系数一维热传导方程.热力耦合的横向自由振动微分方程由经典梁理论获得.采用分层均匀化方法将变系数的热传导方程简化为一系列在各分层内定义的常系数微分方程,利用上下表面的绝热边界条件和界面处的连续性条件获得了微梁温度场的分层解析解.将温度场代入微梁的运动方程,获得了包含热弹性阻尼的复频率,进而求得了代表热弹性阻尼的逆品质因子.在给定金属-陶瓷功能梯度材料后,通过数值计算结果定量分析了材料梯度指数、频率阶数、几何尺寸以及边界条件对TED的影响.结果表明:(1)若梁长固定不变,梁厚度小于某个数值时,改变陶瓷材料体积分数可以使得TED取得最小值;(2)固有频率阶数对TED的最大值没有影响,但是频率阶数越高对应的临界厚度越小;(3)不同的边界条件对应的TED的最大值相同,但是随着支座约束刚度增大对应的临界厚度减小;(4)TED的最大值和对应的临界厚度随着金属组分的增大而增大.     

质量-弹簧装置连接的双梁结构固有频率计算

双梁结构被用作一种新型的减振器来控制梁式结构的振动,在土木、机械和航空航天等工程中受到广泛应用。本文研究了两个平行的轴向功能梯度梁相互连接的双梁结构固有频率的计算问题,在这种双梁结构中,梁的端部受到平移和旋转两种弹性约束,同时,双梁结构通过质量-弹簧装置相互连接。基于Euler-Bernoulli梁的基本理论,将非经典边界条件下双梁结构自由振动固有频率的计算转化为一组常微分方程特征值问题,运用插值矩阵法可一次性计算出双梁结构的所有固有频率。数值算例表明,本文双梁结构量纲为一的固有频率的计算值与已有文献计算结果吻合良好。研究了弹簧刚度、质量系数和梯度参数对双梁系统的影响。数值计算结果表明,随着梯度系数?和悬挂物块的质量系数?的增大,第1阶固有频率?1逐渐减小。     

基于修正偶应力和高阶剪切变形理论的变截面微梁的自由振动

基于修正偶应力和高阶剪切理论建立了仅含有一个尺度参数的Reddy变截面微梁的自由振动模型,研究了变截面微梁自由振动问题的尺度效应和横向剪切变形对自振频率计算的影响。基于哈密顿原理推导了动力学方程与边界条件,并采用微分求积法求解了各种边界条件下的自振频率。算例结果表明,基于偶应力理论预测的变截面微梁的自振频率均大于经典梁理论的预测结果,即捕捉到了尺度效应。另外,梁的几何尺寸与尺度参数越接近,尺度效应就越明显,而梁的长细比越小,横向剪切变形对自振频率的影响就越明显。     

经典梁热过屈曲问题的解析解

基于非线性经典梁理论,建立了控制轴向和横向变形的基本方程,将两个非线性方程化简为一个关于横向挠度的四阶非线性积分-微分方程。对于本文所考虑的三类边界条件,该方程与相应的边界条件构成了微分特征值问题;直接求解该问题,得到热过屈曲构形的解析解,该解是外加热载荷的函数。为考察热载荷以及边界条件的影响,根据得到的解析解给出了一些数值算例,讨论了梁过屈曲行为的性质。本文得到的解析解可用于验证或改进各类近似理论和数值方法。     

Analytical test functions for free vibration analysis of rotating non-homogeneous Timoshenko beams

In this paper, the governing equations for free vibration of a non-homogeneous rotating Timoshenko beam, having uniform cross-section, is studied using an inverse problem approach, for both cantilever and pinned-free boundary conditions. The bending displacement and the rotation due to bending are assumed to be simple polynomials which satisfy all four boundary conditions. It is found that for certain polynomial variations of the material mass density, elastic modulus and shear modulus, along the length of the beam, the assumed polynomials serve as simple closed form solutions to the coupled second order governing differential equations with variable coefficients. It is found that there are an infinite number of analytical polynomial functions possible for material mass density, shear modulus and elastic modulus distributions, which share the same frequency and mode shape for a particular mode. The derived results are intended to serve as benchmark solutions for testing approximate or numerical methods used for the vibration analysis of rotating non-homogeneous Timoshenko beams.