非线性弹性地基梁在随动载荷作用下的屈曲和振动

基于可伸长梁的几何非线性理论,建立了非线性弹性地基上梁在随动载荷作用下的屈曲问题和振动问题控制方程,分别采用打靶法分析了弹性地基梁的后屈曲行为以及后屈曲构形上的振动问题。给出了不同非线性弹性地基系数下,梁在随动载荷作用下的过屈曲平衡路径曲线以及过屈曲附近前三阶频率随载荷变化的曲线。研究表明:立方刚度系数K_2对梁的屈曲和振动影响较小,而线性刚度系数K_1对梁的过屈曲性态和固有频率都有影响。     

压电/压磁层合纳米梁屈曲及自由振动的研究

针对压电/压磁层合纳米梁屈曲、自由振动问题,基于非局部理论与正弦剪切型变形梁理论,建立了力学模型;利用哈密顿原理推导出层合梁运动方程与边界条件;通过数值解法求得层合梁临界屈曲载荷与自由振动频率。对数值结果分析可知:磁电弹夹层对压电/压磁层合纳米梁屈曲和自由振动的影响不能忽略;磁电弹夹层中压电或压磁材料的体积分数和夹层厚度为主要影响因素;分析得到的影响规律可为此类材料在工程中的应用提供理论参考。     

非局部应变梯度理论下纳米梁的力学特性研究

基于非局部应变梯度理论,建立了一种具有尺度效应的高阶剪切变形纳米梁的力学模型.其中,考虑了应变场和一阶应变梯度场下的非局部效应.采用哈密顿原理推导了纳米梁的控制方程和边界条件,并给出了简支边界条件下静弯曲、自由振动和线性屈曲问题的纳维级数解.数值结果表明,非局部效应对梁的刚度产生软化作用,应变梯度效应对纳米梁的刚度产生硬化作用,梁的刚度整体呈现软化还是硬化效应依赖于非局部参数与材料特征尺度的比值.梁的厚度与材料特征尺度越接近,非局部应变梯度理论与经典弹性理论所预测结果之间的差异越显著.     

INVESTIGATIONS ON THE MECHANICAL CHARACTERISTICS OF NANOBEAMS BASED ON THE NONLOCAL STRAIN GRADIENT THEORY1)

基于非局部应变梯度理论,建立了一种具有尺度效应的高阶剪切变形纳米梁的力学模型. 其中,考虑了应变场和一阶应变梯度场下的非局部效应. 采用哈密顿原理推导了纳米梁的控制方程和边界条件,并给出了简支边界条件下静弯曲、自由振动和线性屈曲问题的纳维级数解. 数值结果表明,非局部效应对梁的刚度产生软化作用,应变梯度效应对纳米梁的刚度产生硬化作用,梁的刚度整体呈现软化还是硬化效应依赖于非局部参数与材料特征尺度的比值. 梁的厚度与材料特征尺度越接近,非局部应变梯度理论与经典弹性理论所预测结果之间的差异越显著.     

BUCKING AND FREE VIBRATION OF PIEZOELECTRIC/PIEZOMAGNETIC LAMINATED NANO-BEAM 1)

针对压电/压磁层合纳米梁屈曲、自由振动问题，基于非局部理论与正弦剪切型变形梁理论，建立了力学模型；利用哈密顿原理推导出层合梁运动方程与边界条件；通过数值解法求得层合梁临界屈曲载荷与自由振动频率。对数值结果分析可知：磁电弹夹层对压电/压磁层合纳米梁屈曲和自由振动的影响不能忽略；磁电弹夹层中压电或压磁材料的体积分数和夹层厚度为主要影响因素；分析得到的影响规律可为此类材料在工程中的应用提供理论参考。     

基于微分求积单元法的层合梁热弹性分析

基于微分求积单元法,开展了非均匀温度场中层合梁的热弹性分析．首先基于Fourier导热定律,分析一般热边界条件下层合梁的二维稳态温度场;然后基于二维热弹性力学理论,分析层合梁的热应力和变形．为求解热传导和热应力问题,将层合梁沿各层界面划分为若干空间子域,采用微分求积法对每一子域的控制方程和边界条件进行离散并求解．数值算例验证了本方法的收敛性,与已有文献结果的对比验证了本方法的正确性．最后,算例分析了非均匀热边界条件对夹层梁温度分布的影响,以及端部支承条件和长厚比对梁内位移和应力分布的影响．     

脱层梁-板的后屈曲分析

研究了含任意位置脱层的梁-板的后屈曲问题,采用小参数摄动法,导出了系统的一阶和二阶控制方程,边界条件,连续条件,平衡条件及相容条件,在算例中给出了梁-板的后屈曲响应曲线,且与有关文献进行了比较。     

一维六方准晶层合简支梁自由振动与屈曲的精确解

伪Stroh型公式能够将多场耦合材料的控制方程转化为线性特征系统来求解,从而获得多层结构简支边界条件的精确解.本文利用伪Stroh型公式,研究一维六方准晶层合简支梁的自由振动和屈曲问题,通过传递矩阵法,获得准晶层合梁自由振动固有频率与临界屈曲载荷的精确解.通过与已有梁的剪切变形理论结果比较,验证了本文伪Stroh型公式的正确性和有效性.通过数值算例,分析由两种不同准晶材料组成的三明治层合梁的叠层方式、高跨比、层厚比及层数对梁的固有频率、临界屈曲载荷及其模态的影响规律.结果表明,叠层顺序和梁的高跨比、层厚比对准晶层合梁的自由振动固有频率和临界屈曲载荷有很大影响,可通过调整梁的几何尺寸和叠层顺序得到准晶层合梁的最佳固有频率和临界屈曲载荷.本文给出的精确解可为工程上研究准晶梁的各种数值解法和实验方法提供理论参考.     

考虑前屈曲耦合变形时功能梯度简支梁的稳定性分析

在某些边界条件下,功能梯度材料（FGM）梁会由于拉弯耦合产生前屈曲耦合变形,而该变形对FGM梁的稳定性有影响。本文假设FGM梁的材料性质只沿厚度方向进行变化,基于经典非线性梁理论和物理中面概念,推导出FGM梁的平衡方程以及包含前屈曲耦合变形影响的屈曲控制方程,并用打靶法进行数值求解。讨论了前屈曲耦合变形、梯度指数以及材料性质的温度依赖等因素对FGM梁非线性变形和稳定性的影响。     

含脱层损伤复合材料层合梁的冲击动力屈曲

针对含初始缺陷和脱层损伤的复合材料层合梁的轴向冲击动力屈曲问题进行了分析。基于Hamilton原理导出了考虑初始缺陷、轴向和横向惯性、横向剪切变形以及转动惯性影响时含脱层损伤复合材料梁的非线性动力屈曲控制方程;基于B-R准则,采用有限差分方法求解了受轴向冲击载荷作用下含脱层损伤复合材料梁的动力屈曲问题;讨论了冲击速度、初始几何缺陷、铺层角度以及脱层长度等因素对复合材料层合梁动力屈曲的影响。     

Bending and buckling of thin strain gradient elastic beams

Bending of strain gradient elastic thin beams is studied adopting Bernoulli-Euler principle. Simple linear strain gradient elastic theory with surface energy is employed. The governing beam equations with its boundary conditions are derived through a variational method. It turns out that new terms are introduced, indicating the importance of the cross-section area in bending of thin beams. Those terms are missing from the existing strain gradient beam theories. Those terms increase highly the stiffness of the thin beam. The buckling problem of the thin beams is also discussed.     

Thermal buckling and post-buckling of pinned–fixed Euler–Bernoulli beams on an elastic foundation

In this article, both thermal buckling and post-buckling of pinned–fixed beams resting on an elastic foundation are investigated. Based on the accurate geometrically non-linear theory for Euler–Bernoulli beams, considering both linear and non-linear elastic foundation effects, governing equations for large static deformations of the beam subjected to uniform temperature rise are derived. Due to the large deformation of the beam, the constraint forces of elastic foundation in both longitudinal and transverse directions are taken into account. The boundary value problem for the non-linear ordinary differential equations is solved effectively by using the shooting method. Characteristic curves of critical buckling temperature versus elastic foundation stiffness parameter corresponding to the first, the second, and the third buckling mode shapes are plotted. From the numerical results it can be found that the buckling load-elastic foundation stiffness curves have no intersection when the value of linear foundation stiffness parameter is less than 3000, which is different from the behaviors of symmetrically supported (pinned–pinned and fixed–fixed) beams. As we expect that the non-linear foundation stiffness parameter has no sharp influence on the critical buckling temperature and it has a slight effect on the post-buckling temperature compared with the linear one.     

Free vibration analyses of axially loaded laminated composite beams based on higher-order shear deformation theory

The dynamic stiffness matrix method is introduced to solve exactly the free vibration and buckling problems of axially loaded laminated composite beams with arbitrary lay-ups. The Poisson effect, axial force, extensional deformation, shear deformation and rotary inertia are included in the mathematical formulation. The exact dynamic stiffness matrix is derived from the analytical solutions of the governing differential equations of the composite beams based on third-order shear deformation beam theory. The application of the present method is illustrated by two numerical examples, in which the effects of axial force and boundary condition on the natural frequencies, mode shapes and buckling loads are examined. Comparison of the current results to the existing solutions in the literature demonstrates the accuracy and effectiveness of the present method.     

Thermal buckling analysis of functionally graded beam with longitudinal crack

The thermal buckling problem of functionally graded beam with longitudinal crack is presented in the paper. The whole beam is divided into four sub-beams and each one is modeled as a Timoshenko beam. The buckling governing equation of each sub-beam in thermal environment is established by using Hamilton Principle. Combining with the boundary conditions, the continuous conditions of the displacements and the forces, the buckling governing equations are solved by both the analytical and numerical methods. The buckling modes and critical buckling temperatures are obtained, and the effects of the functionally graded index, crack length, crack depth, and crack longitudinal location on the buckling characteristics of beams are discussed in numerical examples.     

Axial-flexural coupled vibration and buckling of composite beams using sinusoidal shear deformation theory

A finite element model based on sinusoidal shear deformation theory is developed to study vibration and buckling analysis of composite beams with arbitrary lay-ups. This theory satisfies the zero traction boundary conditions on the top and bottom surfaces of beam without using shear correction factors. Besides, it has strong similarity with Euler–Bernoulli beam theory in some aspects such as governing equations, boundary conditions, and stress resultant expressions. By using Hamilton’s principle, governing equations of motion are derived. A displacement-based one-dimensional finite element model is developed to solve the problem. Numerical results for cross-ply and angle-ply composite beams are obtained as special cases and are compared with other solutions available in the literature. A variety of parametric studies are conducted to demonstrate the effect of fiber orientation and modulus ratio on the natural frequencies, critical buckling loads, and load-frequency curves as well as corresponding mode shapes of composite beams.     

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.     

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针对连续Bernoulli-Euler和Timoshenko梁单元的动态刚度矩阵,分析了在使用连续梁单元 进行结构动态特性分析中的数值问题. 基于连续梁单元的运动方程,导出了连续 Bernoulli-Euler和Timoshenko梁单元的动态刚度矩阵. 分析了影响动态刚度矩阵中双曲函 数自变量的各个独立变量及其产生的影响,并给出了初估连续梁单元合理长度的方法. 使用 单一连续Bernoulli-Euler和Timoshenko梁单元的动态刚度矩阵分别进行了悬臂梁频响曲线 的数值求解. 研究表明,在合理选择连续梁单元的长度时,大多数工程结构的动态特性分析 中都不会产生数值问题.     

On the Buckling of an Infinite Beam of Finite Width Embedded in an Isotropic Elastic Solid∗

The present paper considers the problem of buckling of a beam of finite width that is embedded in bonded contact with an isotropic elastic solid. Analysis of the buckling problem is restricted to the class of slender beams of narrow width that exhibit flexure only in the longitudinal direction. The governing integral equations are solved in an approximate fashion. Numerical results presented indicate the manner in which the buckling load is influenced by the relative flexibility of the beam-elastic medium system.     

THERMAL POST-BUCKLING OF FUNCTIONALLY GRADED MATERIAL TIMOSHENKO BEAMS

Analysis of thermal post-buckling of FGM (Functionally Graded Material) Timoshenko beams subjected to transversely non-uniform temperature rise is presented. By accurately considering the axial extension and transverse shear deformation in the sense of theory of Timoshenko beam, geometrical nonlinear governing equations including seven basic unknown functions for functionally graded beams subjected to mechanical and thermal loads were formulated. In the analysis, it was assumed that the material properties of the beam vary continuously as a power function of the thickness coordinate. By using a shooting method, the obtained nonlinear boundary value problem was numerically solved and thermal buckling and post-buckling response of transversely non-uniformly heated FGM Timoshenko beams with fixed-fixed edges were obtained. Characteristic curves of the buckling deformation of the beam varying with thermal load and the power law index are plotted. The effects of material gradient property on the buckling deformation and critical temperature of beam were discussed in details. The results show that there exists the tension-bend coupling deformation in the uniformly heated beam because of the transversely non-uniform characteristic of materials.