含裂纹功能梯度Euler-Bernoulli梁和Timoshenko梁的屈曲载荷计算与分析

含裂纹构件的屈曲载荷是结构是否安全的判定准则之一, 其计算与分析也是结构健康监测和安全评价中关注的重要问题。基于Euler-Bernoulli梁理论和Timoshenko梁理论, 建立了一种求解含裂纹功能梯度材料梁的屈曲载荷计算方法。首先裂纹导致的构件截面转角不连续性由转动弹簧模型进行模拟, 再根据功能梯度材料Euler-Bernoulli梁和Timoshenko梁的屈曲控制方程及其闭合解, 由传递矩阵法建立了求解含裂纹功能梯度材料梁在多种边界条件下屈曲载荷的循环递推公式和特征行列式, 使问题通过降阶的方法得到快速准确的解答。数值算例研究了剪切变形、 裂纹的不同数目及位置、 材料参数变化、 长细比和不同边界约束条件等对含裂纹功能梯度材料梁屈曲载荷的影响。结果表明该方法可以简单、 方便和准确地计算不同数目裂纹和任意边界条件下功能梯度材料梁的屈曲问题。      

Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory

Dynamic stability of microbeams made of functionally graded materials (FGMs) is investigated in this paper based on the modified couple stress theory and Timoshenko beam theory. This non-classical Timoshenko beam model contains a material length scale parameter and can interpret the size effect. The material properties of FGM microbeams are assumed to vary in the thickness direction and are estimated though Mori–Tanaka homogenization technique. The higher-order governing equations and boundary conditions are derived by using the Hamilton’s principle. The differential quadrature (DQ) method is employed to convert the governing differential equations into a linear system of Mathieu–Hill equations from which the boundary points on the unstable regions are determined by Bolotin’s method. Free vibration and static buckling are also discussed as subset problems. A parametric study is conducted to investigate the influences of the length scale parameter, gradient index and length-to-thickness ratio on the dynamic stability characteristics of FGM microbeams with hinged–hinged and clamped–clamped end supports. Results show that the size effect on the dynamic stability characteristics is significant only when the thickness of beam has a similar value to the material length scale parameter.     

一阶剪切理论下功能梯度梁与均匀梁静态解之间的相似关系

基于一阶剪切理论,研究了功能梯度材料Timoshenko 梁的静态弯曲解与对应的均匀材料梁的解的线性转换关系。通过比较功能梯度材料梁和均匀材料梁的无量纲控制方程,发现了它们弯曲解的线性相关性。在给定材料弹性模量沿横向非均匀变化规律后,可将功能梯度材料Timoshenko 梁在静载荷作用下的弯曲变形解用相同尺寸、相同载荷以及相同边界条件下的均匀材料Timoshenko 梁的弯曲变形解线性表示。这样,可将非均匀Timoshenko 梁弯曲问题的求解转化为对应的均匀材料Timoshenko 梁弯曲问题的求解和转换系数的计算,从而使得求解过程得以简化。     

Closed-form solutions for axially functionally graded Timoshenko beams having uniform cross-section and fixed–fixed boundary condition

In this paper, we study the free vibration of axially functionally graded (AFG) Timoshenko beams, with uniform cross-section and having fixed–fixed boundary condition. For certain polynomial variations of the material mass density, elastic modulus and shear modulus, along the length of the beam, there exists a fundamental closed form solution to the coupled second order governing differential equations with variable coefficients. It is found that there are an infinite number of non-homogeneous Timoshenko beams, with various material mass density, elastic modulus and shear modulus distributions having simple polynomial variations, which share the same fundamental frequency. The derived results can be used as benchmark solutions for testing approximate or numerical methods used for the vibration analysis of non-homogeneous Timoshenko beams. They can also be useful for designing fixed–fixed non-homogeneous Timoshenko beams which may be required to vibrate with a particular frequency.     

Vibration and buckling analysis of functionally graded sandwich beams by a new higher-order shear deformation theory

This paper proposes a new higher-order shear deformation theory for buckling and free vibration analysis of isotropic and functionally graded (FG) sandwich beams. The present theory accounts a new hyperbolic distribution of transverse shear stress and satisfies the traction free boundary conditions. Equations of motion are derived from Lagrange's equations. Analytical solutions are presented for the isotropic and FG sandwich beams with various boundary conditions. Numerical results for natural frequencies and critical buckling loads obtained using the present theory are compared with those obtained using the higher and first-order shear deformation beam theories. Effects of the boundary conditions, power-law index, span-to-depth ratio and skin-core-skin thickness ratios on the critical buckling loads and natural frequencies of the FG beams are discussed.     

Stability and vibration analyses of carbon nanotube-reinforced composite beams with elastic boundary conditions: Chebyshev collocation method

This article aims to investigate stability and vibration behavior of carbon nanotube-reinforced composite beams supported by classical and nonclassical boundary conditions. To include significant effects of shear deformation and rotary inertia, Timoshenko beam theory is used to formulate the coupled equations of motion governing buckling and vibration analyses of the beams. An effective mathematical technique, namely Chebyshev collocation method, is employed to solve the coupled equations of motion for determining critical buckling loads and natural frequencies of the beams with different boundary conditions. The accuracy and reliability of the proposed mathematical models are verified numerically by comparing with the existing results in the literature for the cases of classical boundary conditions. New results of critical buckling loads and natural frequencies of the beams with nonclassical boundary conditions including translational and rotational springs are presented and discussed in detail associated with many important parametric studies.     

The modified couple stress functionally graded Timoshenko beam formulation

In this paper, a size-dependent formulation is presented for Timoshenko beams made of a functionally graded material (FGM). The formulation is developed on the basis of the modified couple stress theory. The modified couple stress theory is a non-classic continuum theory capable to capture the small-scale size effects in the mechanical behavior of structures. The beam properties are assumed to vary through the thickness of the beam. The governing differential equations of motion are derived for the proposed modified couple-stress FG Timoshenko beam. The generally valid closed-form analytic expressions are obtained for the static response parameters. As case studies, the static and free vibration of the new model are respectively investigated for FG cantilever and FG simply supported beams in which properties are varying according to a power law. The results indicate that modeling beams on the basis of the couple stress theory causes more stiffness than modeling based on the classical continuum theory, such that for beams with small thickness, a significant difference between the results of these two theories is observed.     

多孔功能梯度梁的热-力耦合屈曲行为

采用经典欧拉梁理论和高阶三角剪切变形理论,研究了多孔功能梯度梁的热-力耦合屈曲行为。分析中考虑了材料物性与温度的相关性,采用含孔隙率的Voight混合模型描述了多孔功能梯度的材料属性。利用迭代算法求解结构在均匀、线性和非线性温升(考虑热传导效应)下的热-力耦合屈曲临界温度,讨论了材料非均匀参数、孔隙率和长细比等参数对屈曲临界温度的影响。ABAQUS数值模拟结果和文献对比结果验证了理论的可靠性,同时表明高阶剪切变形理论较经典欧拉梁理论精确。结果表明,功能梯度材料梁的热屈曲分析必须考虑物性与温度的相关性,否则可能高估热屈曲临界温度10%~30%;随着孔隙率增大,材料的等效弹性模量减少,即结构刚度有所弱化,但屈曲临界温度反而大大增高。     

湿-热-机-弹耦合FGM梁的稳定性及振动特性

研究了初始轴向机械载荷作用下Winkler-Pasternak弹性地基上功能梯度材料（FGM）梁在湿-热环境中的稳定性及振动特性。假设温度和湿度沿梁厚度方向稳态分布,材料的物性依赖于温度且按Voigt混合幂律模型连续分布。首先,基于一种扩展的n阶广义梁理论,应用Hamilton原理,统一建立了以轴向位移、弯曲变形项挠度及剪切变形项挠度为基本未知函数FGM梁的屈曲及自由振动方程,采用Navier解法获得了FGM简支梁静动态响应的精确解。其次,通过算例验证并给出了该广义梁理论阶次n的理想取值,丰富梁理论的同时,可供验证或改进其他各种剪切变形梁理论。最后,着重探讨了3种湿-热分布下湿度与温度增加、初始轴向机械载荷、跨厚比、地基刚度、梯度指标等诸多参数对FGM梁稳定性和振动特性的影响。     

Mechanical buckling of functionally graded material cylindrical shells surrounded by Pasternak elastic foundation

In this study, the mechanical buckling of functionally graded material cylindrical shell that is embedded in an outer elastic medium and subjected to combined axial and radial compressive loads is investigated. The material properties are assumed to vary smoothly through the shell thickness according to a power law distribution of the volume fraction of constituent materials. Theoretical formulations are presented based on a higher-order shear deformation shell theory (HSDT) considering the transverse shear strains. Using the nonlinear strain–displacement relations of FGMs cylindrical shells, the governing equations are derived. The elastic foundation is modelled by two parameters Pasternak model, which is obtained by adding a shear layer to the Winkler model. The boundary condition is considered to be simply-supported. The novelty of the present work is to achieve the closed-form solutions for the critical mechanical buckling loads of the FGM cylindrical shells surrounded by elastic medium. The effects of shell geometry, the volume fraction exponent, and the foundation parameters on the critical buckling load are investigated. The numerical results reveal that the elastic foundation has significant effect on the critical buckling load.     

Thermo-electrical buckling of piezoelectric functionally graded material Timoshenko beams

In this article, buckling analysis of functionally graded material (FGM) beams with or without surface-bonded piezoelectric layers subjected to both thermal loading and constant voltage is studied. Thermal and mechanical properties of FGM layer is assumed to follow the power law distribution in thickness direction, except Poisson’s ratio which is considered constant. The Timoshenko beam theory and nonlinear strain-displacement relations are used to obtain the governing equations of piezoelectric FGM beam. Beam is assumed under three types of thermal loading and various types of boundary conditions. For each case of boundary conditions, existence of bifurcation-type buckling is examined and for each case of thermal loading and boundary conditions, closed-form solutions are obtained which are easily usable for engineers and designers. The effects of the applied actuator voltage, beam geometry, boundary conditions, and power law index of FGM beam on critical buckling temperature difference are examined.     

Free vibration of axially loaded rectangular composite beams using refined shear deformation theory

Free vibration of axially loaded rectangular composite beams with arbitrary lay-ups using refined shear deformation theory is presented. It accounts for the parabolical variation of shear strains through the depth of beam. Three governing equations of motion are derived from the Hamilton’s principle. The resulting coupling is referred to as triply axial-flexural coupled vibration. A displacement-based one-dimensional finite element model is developed to solve the problem. Numerical results are obtained for rectangular composite beams to investigate effects of fiber orientation and modulus ratio on the natural frequencies, critical buckling loads and load–frequency curves as well as corresponding mode shapes.     

A further discussion of nonlinear mechanical behavior for FGM beams under in-plane thermal loading

Due to the variation in material properties through the thickness, bifurcation buckling cannot generally occur for plates or beams made of functionally graded materials (FGM) with simply supported edges. Further investigation in this paper indicates that FGM beams subjected to an in-plane thermal loading do exhibit some unique and interesting characteristics in both static and dynamic behaviors, particularly when effects of transverse shear deformation and the temperature-dependent material properties are simultaneously taken into account. In the analysis, based on the nonlinear first-order shear deformation beam theory (FBT) and the physical neutral surface concept, governing equations for both the static behavior and the dynamic response of FGM beams subjected to uniform in-plane thermal loading are derived. Then, a shooting method is employed to numerically solve the resulting equations. The material properties of the beams 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, and to be temperature-dependent. The effects of material constants, transverse shear deformation, temperature-dependent material properties, in-plane loading and boundary conditions on the nonlinear behavior of FGM beams are discussed in detail.     

Dynamic buckling of FGM truncated conical shells subjected to non-uniform normal impact load

Dynamic buckling of functionally graded materials truncated conical shells subjected to normal impact loads is discussed in this paper. In the analysis, the material properties of functionally graded materials shells 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. Geometrically nonlinear large deformation and the initial imperfections are taken into account. Galerkin procedure and Runge–Kutta integration scheme are used to solve nonlinear governing equations numerically. From the characteristics of dynamic response obtain critical loads of the shell according to B-R criterion. From the research results it can be found that gradient properties of the materials have significant effects on the critical buckling loads of FGM shells.     

Parametric instability of functionally graded beams with an open edge crack under axial pulsating excitation

This paper studies the parametric instability of functionally graded beams with an open edge crack subjected to an axial pulsating excitation which is a combination of a static compressive force and a harmonic excitation force. It is assumed that the materials properties follow an exponential variation through the thickness direction. Theoretical formulations are based on Timoshenko beam theory and linear rotational spring model. The governing equations of motion are derived by using Hamilton’s principle and transformed into a set of Mathieu equations through Galerkin’s procedure. The natural frequencies with different end supports are obtained. The boundary points on the unstable regions are determined by using Bolotin’s method. Numerical results are presented to highlight the influences of crack location, crack depth, material property gradient, beam slenderness ratio, compressive load, and boundary conditions on both the free vibration and parametric instability behaviors of the cracked functionally graded beams.     

Higher-order theory for bending and vibration of beams with circular cross section

This paper presents an efficient and simple higher-order theory for analyzing free vibration of cylindrical beams with circular cross section where the rotary inertia and shear deformation are taken into account simultaneously. Unlike the Timoshenko theory of beams, the present method does not require a shear correction factor. Similar to the Levinson theory for rectangular beams, this new model is a higher-order theory for beams with circular cross section. For transverse flexure of such cylindrical beams, based on the traction-free condition at the circumferential surface of the cylinder, two coupled governing equations for the deflection and rotation angle are first derived and then combined to yield a single governing equation. In the case of no warping of the cross section, our results are exact. A comparison is made of the natural frequencies with those using the Timoshenko and Euler–Bernoulli theories of beams and the finite element method. Our results are useful for precisely understanding the mechanical behavior and engineering design of circular cylindrical beams.     

Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory

Investigated herein is the free vibration characteristics of microbeams made of functionally graded materials (FGMs) based on the strain gradient Timoshenko beam theory. The material properties of the functionally graded beams are assumed to be graded in the thickness direction according to the Mori–Tanaka scheme. Using Hamilton’s principle, the equations of motion together with corresponding boundary conditions are obtained for the free vibration analysis of FGM microbeams including size effect. A detailed parametric study is performed to indicate the influences of beam thickness, dimensionless length scale parameter, and slenderness ratio on the natural frequencies of FGM microbeams. Moreover, a comparison between the various beam models on the basis of the classical theory (CT), modified couple stress theory (MCST), and strain gradient theory (SGT) is presented for different values of material property gradient index. It is observed that the value of gradient index play an important role in the vibrational response of the microbeams of lower slenderness ratios. It is further observed that by increasing the length-to-thickness ratio of the microbeam, the value of dimensionless natural frequency tends to decrease for all amounts of the gradient index.     

用Timoshenko梁修正理论研究功能梯度材料梁的动力响应

采用Timoshenko梁修正理论研究了功能梯度材料梁的动力响应问题,利用静力方程确定了功能梯度材料梁的中性轴位置,在此基础上应用Timoshenko梁修正理论建立了功能梯度材料梁的振动方程,求得其自振频率表达式及其在简谐荷载作用下强迫振动的解析解。讨论分析了中性面位置、梯度指数等因素对功能梯度材料梁的动力响应的影响,并用有限元法验证了Timoshenko梁修正理论。通过实例计算,得到了中性轴位置对功能梯度材料梁动力响应有较大影响的结论。     

基于n阶GBT热-机载荷作用下FGM梁的振动特性和稳定性分析

研究了热-机载荷耦合作用下弹性地基FGM梁的振动特性与稳定性。考虑到材料的物性依赖于温度变化且组分沿梁厚按幂律分布。首先,基于一种扩展的n阶广义剪切变形梁理论(n-th GBT),应用Hamilton原理,统一建立了系统自由振动及屈曲问题力学模型的控制方程,采用一种改进型广义微分求积法(MGDQ)获得FGM梁静动态响应的数值解。其次,通过算例验证GBT的有效性并给出阶次n的理想取值,在丰富梁理论的同时,也可验证或改进其他各种剪切变形梁理论。最后,讨论并分析了升温、边界条件、初始轴向机械载荷、梯度指标、地基刚度、跨厚比等诸多参数对FGM梁振动特性和稳定性的影响。     

Non-linear active control of FG beams in thermal environments subjected to blast loads with integrated FGP sensor/actuator layers

Non-linear active control of dynamic response of functionally graded (FG) beams with rectangular cross-section in thermal environments exposed to blast loadings is presented. Two FG piezoelectric layers are bonded to the beam surfaces to act as sensor and actuator. Non-linear equations of motion of the smart beam are derived based on the first-order shear deformation theory and the von Karman geometrical non-linearity. Constant velocity feedback algorithm is used to control the dynamic response of the FG beam actively through closed loop control. The generalized differential quadrature method together with the Newmark-beta scheme is utilized to solve the non-linear partial differential equations in spatial and time domains. The resulted non-linear algebraic equations are then solved using the modified Newton–Raphson method. A detailed analysis of the influence of the geometric non-linearity, material parameters and temperature field on the active vibration control of FG beams subjected to various impulsive loads is carried out.