剪切可变形梁热过屈曲解析解

该文导出了面内热载荷作用下,梁过屈曲问题的精确解。首先基于非线性一阶剪切变形梁理论,推导了控制轴向和横向变形的基本方程。然后,将3 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。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,得到了热过屈曲构形的闭合解,这个解是外加热载荷的函数。利用精确解,得到了临界屈曲载荷的一阶结果与经典结果的解析关系。为考察热载荷、横向剪切变形以及边界条件的影响,根据得到的精确解给出了两端固定、两端简支以及一端固定一端简支边界条件下的具体数值算例,讨论了梁在面内热载荷作用下的过屈曲行为,并与经典结果进行了比较。该文得到的精确解可以用于验证或改进各类近似理论和数值方法。     

非均匀热载荷作用下功能梯度梁的非线性静态响应

由于功能梯度材料结构沿厚度方向的非均匀材料特性,使得夹紧和简支条件的功能梯度梁有着相当不同的行为特征。该文给出了热载荷作用下,功能梯度梁非线性静态响应的精确解。基于非线性经典梁理论和物理中面的概念导出了功能梯度梁的非线性控制方程。将两个方程化简为一个四阶积分-微分方程。对于两端夹紧的功能梯度梁,其方程和相应的边界条件构成微分特征值问题;但对于两端简支的功能梯度梁,由于非齐次边界条件,将不会得到一个特征值问题。导致了夹紧与简支的功能梯度梁有着完全不同的行为特征。直接求解该积分-微分方程,得到了梁过屈曲和弯曲变形的闭合形式解。利用这个解可以分析梁的屈曲、过屈曲和非线性弯曲等非线性变形现象。最后,利用数值结果研究了材料梯度性质和热载荷对功能梯度梁非线性静态响应的影响。     

热环境中功能梯度材料圆板的自由振动

基于von Kaman经典板理论,建立了功能梯度材料圆板在升温场内的大挠度动力学控制方程.将控制方程的响应分解为热过屈曲静态解和振动解两部分,得到了功能梯度材料圆板在热过屈曲平衡构形附近小振幅线性自由振动的微分方程.采用打靶法同时求热过屈曲和振动问题的控制方程,得到了随温度载荷变化的热过屈曲平衡路径以及前三阶固有频率的数值解.分析和讨论了板的材料梯度参数、温度场分布参数、边界条件等因素对过屈曲变形和振动响应的影响.分析中考虑了功能梯度板的组份材料的物性参数对温度的依赖性.     

轴向随动分布载荷作用下简支梁杆的过屈曲

基于可伸长梁的大变形理论,建立了受沿轴线分布切向随动载荷作用简支梁的弹性过屈曲控制方程。这是一个包含七个未知函数的强非线性常微分方程两点边值问题,其中将变形后的轴线弧长也作为基本未知量之一。采用打靶法和解析延拓法数值求解所得非线性边值问题,获得了梁的过屈曲平衡路径和平衡构形。结果表明,非保守载荷作用梁的平衡路径与保守载荷作用梁的平衡路径有着明显不同。     

基于物理中面FGM梁的非线性力学行为

当坐标面置于功能梯度材料(FGM)梁的物理中面上时,其本构方程中,面内力与弯矩并不耦合,这使得问题的控制方程以及边界条件得以简化.该文利用物理中面概念,基于一阶非线性梁理论,导出了FGM梁的基本方程,分析研究了热载荷作用下FGM梁的过屈曲、弯曲以及在这些构形上的振动等问题.假设功能梯度材料性质只沿梁厚度方向,并按成分含...     

热过屈曲正交异性圆(环)板的自由振动响应

基于vonKármán薄板几何非线性理论,建立了以中面位移为基本未知量的加热极正交异性圆板轴对称大挠度动力学控制方程。然后,将控制方程的响应分解为热过屈曲静态解和振动解两部分,并在小振幅振动假设下得到了板在热过屈曲静平衡构形附近自由振动的线性微分方程。最后,采用打靶法获得了板在热过屈曲前后的固有频率与升温参数之间的特征关系曲线。结果表明,周边面内约束板的前两阶固有频率在热屈曲前随着温度升高而降低,而一旦板进入过屈曲平衡状态,前两阶频率都随着升温而单调增加。     

径向载荷作用下复合材料圆柱壳的非线性动力屈曲

采用半解析法求解径向阶跃载荷作用下复合材料圆柱壳的非线性动力屈曲。基于一阶剪切变形理论，由Hamilton原理推导出包含横向剪切变形以及几何初缺陷的圆柱壳的非线性动力方程，位移及载荷沿周向采用级数展开，由Galerkin方法得到微分方程组，通过有限差分法求解；根据响应情况，由B—R准则判定屈曲，确定屈曲临界载荷。     

热载荷作用下嵌入SMA丝复合材料梁的横向自由振动

基于形状记忆合金Brinson一维热力学本构方程,采用复合材料细观力学分析方法,建立了热载荷作用下嵌入SMA丝复合材料梁的一维热弹性本构关系。其次利用Euler-Bernoulli梁的轴线可伸长几何非线性理论和自由振动理论,建立了嵌入SMA丝复合材料梁在均匀升温场内自由振动的动力学控制方程,导出了热过屈曲构形附近嵌入SMA丝复合材料梁微幅横向自由振动的模型。最后通过打靶法求解了两端固定约束条件下嵌入形状记忆合金丝复合材料梁在加热过程中的振动响应,获得了梁的前四阶固有频率在不同SMA相对体积含量时随温度变化的特征关系曲线。数值结果表明,SMA丝相变过程中的回复应力和弹性模量变化对梁在过屈曲前后的各阶固有频率均有影响,是实现梁自振频率主动控制的一种有效方法。     

弹性地基上加热弹性圆板的热过屈曲及临界屈曲模态跃迁

基于von Kármán薄板理论建立了Winkler弹性基础上弹性圆板在均匀升温下的轴对称热过屈曲控制方程。这是一组以中面位移为基本未知量的非线性常微分方程,其中包含了温度载荷和弹性地基刚度两个参数。采用打靶法数值求解相应的非线性两点边值问题,获得了周边不可移简支圆板的热屈曲及热过屈曲响应。绘出了前三阶屈曲模态对应的临界温度载荷随地基参数连续变化的特性曲线,获得了反映临界热屈曲模态跃迁特性的地基参数值。给出了弹性圆板按一阶模态失稳后的热过屈曲平衡路径和平衡构形,分析了地基刚度参数对临界屈曲温度载荷以及过屈曲平衡构形的影响。     

考虑热对材料参数影响的FGM梁热后屈曲特性研究

功能梯度梁热后屈曲特性研究对于推进功能梯度材料在航天器热防护设计中的应用有着重要意义。基于经典梁的几何非线性理论和物理中面的概念,建立了热载荷作用下功能梯度梁的运动微分方程,通过化简得到一个仅关于挠度的四阶微分-积分方程,并与固支边界条件构成特征值问题,分析研究了功能梯度梁的热后屈曲及在此基础上的振动问题。首先证明了通过哈密顿原理推导的运动方程是轴线可伸长理论的近似形式。接着考虑热对材料物性参数的影响,并对梁的长细比、功能梯度指数和温度比作了详细分析,研究了这些参数对热后屈曲路径和后屈曲振动的影响规律。结果表明:只有在长细比较大时才可以不考虑温度对材料物性参数的影响,否则误差较大;长细比、功能梯度指数和温度比的增大会增大无量纲热屈曲载荷,同时使屈曲路径和频率-载荷曲线向热载荷增大的方向移动。     

Postbuckling and free vibrations of composite beams

An exact solution for the postbuckling configurations of composite beams is presented. The equations governing the axial and transverse deformations of a composite laminated beam accounting for the midplane stretching are derived. The inplane inertia and damping are neglected, and hence the two equations are reduced to a single nonlinear fourth-order partial–integral–differential equation governing the transverse deformations. We find out that the governing equation for the postbuckling of symmetric or asymmetric composite beams has the same form as that of beams made of an isotropic material. Composite beams with fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions are considered. A closed-form solution for the postbuckling deformation is obtained as a function of the applied axial load, which is beyond the critical buckling load. To study the vibrations that take place in the vicinity of a buckled equilibrium position, we exactly solved the linear vibration problem around the first buckled configuration. Solving the resulting eigen-value problem results in the natural frequencies and their associated mode shapes. Both the static response represented by the postbuckling analysis and the dynamic response represented by the free vibration analysis in the postbuckling domain strongly depend on the lay-up of the laminate. Variations of the beam’s midspan rise and the fundamental natural frequency of the postbuckling domain vibrations with the applied axial load are presented for a variety of lay-up laminates. The ratio of the axial stiffness to the bending stiffness was found to be a crucial parameter in the analysis. This control parameter, through the selection of the appropriate lay-up, can be manipulated to help design and optimize the static and dynamic behavior of composite beams.     

弹性地基上复合材料层合梁的自由振动

本文在文[1]建立的层合梁理论的基础上,得到了弹性地基上层合梁的动力控制方程,并将此控制方程简化,获得了不考虑剪切变形时的弹性地基上层合梁经典理论的动力控制方程.利用这二种不同的控制方程,文中获得了层合梁自由振动问题的相应的精确解.由这二种不同的结果,讨论了剪切变形对层合梁固有频率的影响.      

Analysis of large amplitude free vibrations of unsymmetrically laminated composite beams on a nonlinear elastic foundation

The large amplitude free vibration of an unsymmetrically laminated composite beam (LCB) on a nonlinear elastic foundation subjected to axial load has been studied. The equation of motion for the axial and transverse deformations of a geometrically nonlinear LCB is derived. Using the Ritz method, the governing equation is reduced to a time-dependent Duffing equation with quadratic and cubic nonlinearities. The homotopy analysis method (HAM) is used to obtain exact expressions for the dynamic response of the LCB. This study shows that the third-order approximation of the HAM leads to highly accurate solutions that are valid for a wide range of vibration amplitudes. The effects of different parameters on the ratio of nonlinear to linear natural frequency of beams and the post-buckling load-deflection relationship are studied.     

固定边界超临界轴向运动梁平面振动频率

通过数值方法研究超临界速度下,两端固定边界的轴向运动梁平面耦合非线性振动固有频率。发展有限差分法,确定在超临界范围轴向运动梁的径向与横向耦合平面内非平凡静平衡位形。基于非平凡静平衡位形,经坐标变换,建立超临界轴向运动梁连续陀螺系统的标准控制方程。运用高阶Galerkin截断,研究超临界运动状态下梁平面振动的固有频率;并研究Galerkin截断阶数对计算结果的影响。     

Transverse Shear and Normal Deformation Theory for Bending Analysis of Laminated and Sandwich Elastic Beams

The influence of transverse normal strain on bending analysis of cross-ply laminated and sandwich beams is presented. A higher-order shear deformation beam theory is developed. Euler-Bernoulli classical, Timoshenko first-order and simple higher-order theories have been also used in the analysis. The governing equations for a beam composed of orthotropic layers and subjected to any given mechanical load distribution are derived. Making use of Navier-like approach, exact solutions are obtained for cross-ply laminated and sandwich beams subjected to arbitrary loadings. Numerical results for beams with the simply-supported boundary conditions are presented. The effects due to transverse normal strain, transverse shear deformation and number of layers on the static response of the beams are investigated.     

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.     

Nonlinear free vibration analysis of generic coupled induced strain actuated piezo-laminated beams

Piezo-laminated thin beams have been analyzed with induced strain actuation using Kirchhoff’s hypothesis and von Kármán strain displacement relations. Extremizing the Lagrangian of the system derives the governing nonlinear partial differential equations for the beam. Eliminating the in-plane displacement, an integro-partial differential equation of motion is obtained in terms of the transverse displacement. A deflection function that satisfies the simply supported boundary conditions is assumed to get the system equation as a nonlinear second order ordinary differential equation in time, which is of Duffing’s type. The solution of the problem is obtained through exact integration. Results are presented for frequency and amplitude for surface bonded PZT-5A layer in composite beams with various stacking sequences.