工程结构的随机特征问题研究及其在梁结构中的应用

采用子结构模态综合和摄动随机有限元相结合求解工程结构的随机特征问题。为求出随机特征对的方差，借助于模态截断概念推出诸特征值与特征向量对随机变量的偏导数。以染结构为典型算法，定量研究了子结构动模态的选取个数与随机特征对的计算精度间关系，以梁的长细比首次确定使用Timoshenko梁和Euler-Bernoulli梁两模型求解梁类工程结构随机特征问题的适用范围。     

局部损伤梁动力问题的近似计算方法

应用模态摄动法求解局部损伤梁动力特性。这一方法是在无损伤梁的模态子空间内实施,将含局部损伤梁的微分方程的求解转化为代数方程求解。通过算例,表明这一方法可有效的简化计算,同时计算结果具有较高的精度。     

任意边界条件下Timoshenko梁及其修正理论的自振特性分析

提出一种求解任意边界条件下经典Timoshenko梁以及修正Timoshenko梁自振频率和振型的新方法。利用改进的傅立叶级数消除传统傅立叶级数的边界不收敛问题,然后通过Rayleigh-Ritz法导出Timoshenko梁的拉格朗日泛函,根据Hamilton原理将原问题转化为求解矩阵广义特征值问题。通过与解析解对比,本文采用的方法具有较好的收敛性以及较高的计算精度;通过数值计算发现,经典Timoshenko梁的自振频率略高于修正的Timoshenko梁,随着振型阶数的提高,经典Timoshenko梁的计算结果逐渐偏离文献解和有限元结果,而修正的Timoshenko梁能够保持较好的一致性;对于不同边界条件下修正Timoshenko梁的计算结果均能与有限元的计算结果吻合得很好。最后运用MATLAB编程软件将程序设计为App,对于不同情形的梁只需要修改参数即可,可为实际工程提供高效便捷的计算方案和可靠理论依据。     

阶梯式Timoshenko梁自由振动的DCE解

本文基于微分容积法和区域叠加技术提出了微分容积单元法（Differential Cubature Element method，以下简称DCE方法），并用之求解阶梯式变截面Timoshenko梁的自由振动问题。根据梁的变截面情况将其划分为几个单元，在每个单元内应用微分容积法将梁的控制微分方程和边界约束方程离散成为一组关于该单元内配点位移的线性代数方程组，将这些方程组写在一起并在各单元之间应用连续性条件和平衡条件得到一组关于整个域内各点位移的齐次线性代数方程组，这是一广义特征值问题，由子空间迭代法求解该特征问题便可求得系统的自振动频率。数值算例表明，本方法能稳定收敛、并有较高的数值精度和计算效率。     

非线性MEMS微梁的重心有理插值迭代配点法分析

文章利用重心有理插值迭代配点法分析计算非线性MEMS微梁问题。通过处理MEMS微梁的几何通过假设初始函数,将微梁非线性控制方程转换为线性化微分方程,建立逼近非线性微分方程的线性化迭代格式。采用重心有理插值配点法求解线性化微分方程,提出了数值分析MEMS微梁非线性弯曲问题的重心插值迭代配点法。给出了非线性微分方程的直接线性化和Newton线性化计算公式,详细讨论了非线性积分项的计算方法和公式。利用重心有理插值微分矩阵,建立了矩阵-向量化的重心插值迭代配点法的计算公式。数值算例结果表明,重心插值迭代配点法求解微梁非线性弯曲问题,具有计算公式简单、程序实施方便和计算精度高的特点。     

卷积型伽辽金法求解任意边界梁的动力学问题

通过选取时间上采用级数形式,空间上采用梁振型函数的试函数,推导出求解任意边界条件 梁的卷积型伽辽金法. 计算了两端固定梁的动力问题,算例表明,方法计算精度 高,计算工作量少,是计算结构动力学问题的一种有效方法.     

拉普拉斯变换求解梁的弯曲变形

根据文献[1]介绍拉普拉斯变换求梁的挠度,本文将其推广到阶梯变截面梁、简单静不定问题和连续梁的求解。在熟悉积分变换的前提下,此法概念清楚,能使计算规范化,且工作量少。因而能为分析复杂问题提供方便。对于变截面梁以及复杂受载条件下的连续梁,应用拉氏变换求解,尤显其简化计算之优点。一、静定梁挠度曲线通过拉氏变换获得梁的挠度曲线,是解简单静不定问题及连续梁的基础。所以,首先介 ...     

时域自适应算法求解移动荷载下梁的多宗量反问题

通过一种时域自适应算法,建立了求解变速移动荷载下梁的多宗量反问题的数值模型,可同时识别移动荷载和梁的物性参数.正问题采用时域自适应算法和FEM建模,并可由此方便地推导敏度公式;在反问题求解中采用Levenberg-Marquardt法,计算表明该方法具有较好的抗不适定性.通过两个算例,对所提算法进行了数值验证,并探讨了噪声和测点的变化对反演结果的影响,结果令人满意.     

定积分法求解弯曲变形问题

本文采用定积分方法求解梁的弯曲变形问题。该方法不需要采用边界条件来确定积分常数，有效地简化了问题的求解过程；该方法以梁的转角微元为逻辑起点，清晰地刻画了梁弯曲变形的累加过程，便于深刻理解载荷作用下梁的变形历程。     

丝束轴向变角度复合材料梁的弹性分析

丝束变角度复合材料具有变刚度的特点,因此其结构分析具有相当难度．本文采用状态空间法和微分求积法联合的半解析数值方法对丝束轴向变角度复合材料梁的弯曲问题进行研究．假设纤维方向角沿梁的轴向按照任意连续函数变化,选取位移和位移的一阶导数作为状态变量,建立了丝束轴向变角度复合材料梁弹性分析的状态空间方程,将状态变量对轴向坐标的导数采用微分求积法进行求解,进而可得问题的半解析数值解．通过与现有文献及ABAQUS计算结果的比较,验证了本文方法的正确性,并对微分求积法求解本问题的收敛性进行了分析．通过数值算例研究了纤维方向角沿梁轴向的变化对丝束轴向变角度复合材料梁的位移及应力分布的影响,研究结果可为该种结构的设计提供一定的参考．     

无约束修正Timoshenko梁的冲击问题

介绍了修正后的Timoshenko梁运动方程,并比较了修正Timoshenko梁与经 典Timoshenko梁的运动方程. 推导了考虑剪切变形引起的转动惯量的修正Timoshenko 梁的正交条件,推导了集中质量对无约束修正Timoshenko梁的正碰撞对梁所引起的瞬态冲 击响应公式,并用算例进行了分析,且与集中质量对经典的无约束Timoshenko梁的正碰撞 对梁所引起的冲击响应进行了比较,另外还用算例分析了梁的刚度的变化和冲击质量比对其 冲击响应产生的影响.     

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将理性有限元法引入到Timoshenko梁问题中,提出了一种理性Timoshenko梁单元,克服了 剪切锁死现象. 在推导控制方程时,与传统有限元方法采用Lagrange插值不同, 理性有限元法用Timoshenko梁弯曲问题的基本解逼近单元内部场. 运用该梁单元分析 Timoshenko梁时,无需缩减积分,就能避免剪切锁死,并且极大地提高了计算精度,说明 理性有限元法具有广泛的应用前景.     

Transient wave analysis of a cantilever Timoshenko beam subjected to impact loading by Laplace transform and normal mode methods

This study applies two analytical approaches, Laplace transform and normal mode methods, to investigate the dynamic transient response of a cantilever Timoshenko beam subjected to impact forces. Explicit solutions for the normal mode method and the Laplace transform method are presented. The Durbin method is used to perform the Laplace inverse transformation, and numerical results based on these two approaches are compared. The comparison indicates that the normal mode method is more efficient than the Laplace transform method in the transient response analysis of a cantilever Timoshenko beam, whereas the Laplace transform method is more appropriate than the normal mode method when analyzing the complicated multi-span Timoshenko beam. Furthermore, a three-dimensional finite element cantilever beam model is implemented. The results are compared with the transient responses for displacement, normal stress, shear stress, and the resonant frequencies of a Timoshenko beam and Bernoulli–Euler beam theories. The transient displacement response for a cantilever beam can be appropriately evaluated using the Timoshenko beam theory if the slender ratio is greater than 10 or using the Bernoulli–Euler beam theory if the slender ratio is greater than 100. Moreover, the resonant frequency of a cantilever beam can be accurately determined by the Timoshenko beam theory if the slender ratio is greater than 100 or by the Bernoulli–Euler beam theory if the slender ratio is greater than 400.     

确定性载荷作用下Timoshenko薄壁梁的弯扭耦合动力响应

建立了一种普遍的解析理论用于求解确定性载荷作用下Timoshenko薄壁梁的弯扭耦合动力响应。首先通过直接求解单对称均匀Timoshenko薄壁梁单元弯扭耦合振动的运动偏微分方程，给出了计算其自由振动的精确方法，并导出了Timoshenko弯扭耦合薄壁梁自由振动主模态的正交条件。然后利用简正模态法研究了确定性载荷作用下单对称Timoshenko薄壁梁的弯扭耦合动力响应，该弯扭耦合梁所受到的荷载可以是集中载荷或沿着梁长度分布的分布载荷。最后假定确定性载荷是谐波变化的，得到了各种激励下封闭形式的解，并对动力弯曲位移和扭转位移的数值结果进行了讨论。     

A microstructure-dependent Timoshenko beam model based on a modified couple stress theory

A microstructure-dependent Timoshenko beam model is developed using a variational formulation. It is based on a modified couple stress theory and Hamilton's principle. The new model contains a material length scale parameter and can capture the size effect, unlike the classical Timoshenko beam theory. Moreover, both bending and axial deformations are considered, and the Poisson effect is incorporated in the current model, which differ from existing Timoshenko beam models. The newly developed non-classical beam model recovers the classical Timoshenko beam model when the material length scale parameter and Poisson's ratio are both set to be zero. In addition, the current Timoshenko beam model reduces to a microstructure-dependent Bernoulli-Euler beam model when the normality assumption is reinstated, which also incorporates the Poisson effect and can be further reduced to the classical Bernoulli-Euler beam model. To illustrate the new Timoshenko beam model, the static bending and free vibration problems of a simply supported beam are solved by directly applying the formulas derived. The numerical results for the static bending problem reveal that both the deflection and rotation of the simply supported beam predicted by the new model are smaller than those predicted by the classical Timoshenko beam model. Also, the differences in both the deflection and rotation predicted by the two models are very large when the beam thickness is small, but they are diminishing with the increase of the beam thickness. Similar trends are observed for the free vibration problem, where it is shown that the natural frequency predicted by the new model is higher than that by the classical model, with the difference between them being significantly large only for very thin beams. These predicted trends of the size effect in beam bending at the micron scale agree with those observed experimentally. Finally, the Poisson effect on the beam deflection, rotation and natural frequency is found to be significant, which is especially true when the classical Timoshenko beam model is used. This indicates that the assumption of Poisson's effect being negligible, which is commonly used in existing beam theories, is inadequate and should be individually verified or simply abandoned in order to obtain more accurate and reliable results.     

A micro scale Timoshenko beam model based on strain gradient elasticity theory

A micro scale Timoshenko beam model is developed based on strain gradient elasticity theory. Governing equations, initial conditions and boundary conditions are derived simultaneously by using Hamilton's principle. The new model incorporated with Poisson effect contains three material length scale parameters and can consequently capture the size effect. This model can degenerate into the modified couple stress Timoshenko beam model or even the classical Timoshenko beam model if two or all material length scale parameters are taken to be zero respectively. In addition, the newly developed model recovers the micro scale Bernoulli–Euler beam model when shear deformation is ignored. To illustrate the new model, the static bending and free vibration problems of a simply supported micro scale Timoshenko beam are solved respectively. Numerical results reveal that the differences in the deflection, rotation and natural frequency predicted by the present model and the other two reduced Timoshenko models are large as the beam thickness is comparable to the material length scale parameter. These differences, however, are decreasing or even diminishing with the increase of the beam thickness. In addition, Poisson effect on the beam deflection, rotation and natural frequency possesses an interesting “extreme point” phenomenon, which is quite different from that predicted by the classical Timoshenko beam model.     

基于等效黏性弹簧的黏弹性Timoshenko裂纹梁弯曲解析解

考虑裂纹的缝隙和黏性效应,将梁中横向裂纹等效为黏弹性扭转弹簧,利用广义Delta函数,给出了Laplace变换域内裂纹梁的等效抗弯刚度,得到了具有任意开闭裂纹数目且满足标准线性固体黏弹性本构的Timoshenko梁在时间域内的弯曲变形显式解析通解．在此基础上,通过两个数值算例,分析了时间、梁跨高比和裂纹深度等参数对黏弹性Timoshenko开裂纹梁弯曲变形的影响．结果表明：裂纹黏性对Timoshenko裂纹梁的弯曲具有显著的影响．相比于裂纹的弹性扭转弹簧模型,考虑裂纹黏性效应的黏弹性Timoshenko裂纹梁在裂纹处挠度尖点和转角跳跃现象十分明显．另外,由于横向剪切引起的附加变形,Timoshenko裂纹梁的稳态挠度与Euler-Bernoulli梁挠度的差值为常数,其大小与裂纹模型、梁跨高比或裂纹深度无关,这些结果对梁裂纹无损检测具有指导意义．     

饱和多孔弹性Timoshenko梁的大挠度分析

基于微观不可压饱和多孔介质理论和弹性梁的大挠度变形假设,考虑梁剪切变形效应,在梁轴线不可伸长和孔隙流体仅沿轴向扩散的限定下,建立了饱和多孔弹性Timoshenko梁大挠度弯曲变形的非线性数学模型.在此基础上,利用Galerkin截断法,研究了两端可渗透简支饱和多孔Timoshenko梁在突加均布横向载荷作用下的拟静态弯曲,给出了饱和多孔 Timoshenko梁弯曲变形时固相挠度、弯矩和孔隙流体压力等效力偶等随时间的响应.比较了饱和多孔Timoshenko梁非线性大挠度和线性小挠度理论以及饱和多孔 Euler-Bernoulli梁非线性大挠度理论的结果,揭示了他们间的差异,指出当无量纲载荷参数q＞l0时,应采用饱和多孔Timoshenko梁或Euler-Bernoulli梁的大挠度数学模型进行分析,特别的,当梁长细比λ＜30时,应采用饱和多孔Timoshenko梁大挠度数学模型进行分析.     

从一个怪例看细长梁理论的基本假定

分别采用欧拉和铁木辛柯梁理论分析了均匀分布力偶作用下的两端固支等截面匀质细长 梁, 并通过ABAQUS有限元分析了一个实例, 验证了铁木辛柯梁理论分析的结果. 对比证明在 这种载荷及边界条件下即使细长梁, 也必须考虑剪切效应的影响.