变截面Timoshenko悬臂梁自由振动分析

为考虑剪切变形和转动惯量的影响，基于模态摄动法基本原理，提出了一种求解变截面Timoshenko悬臂梁自由振动问题的近似解法。这一方法是利用等截面Euler梁的特征值和模态，将变截面Timoshenko梁特征方程的偏微分方程组转化为代数方程组进行求解，从而得到变截面Timoshenko梁的特征值和模态。该方法适用于求解任意复杂截面型式梁的动力特性，无论梁的截面变化是否连续。随后对截面阶跃变化和线性变化2类变截面梁进行算例分析，数值分析结果表明，这一方法简单、实用，具有良好的精度。      

平面弹性悬臂梁剪切挠度比较分析

采用在经典梁单元基础上引入剪切影响的梁单元和由精确积分得的Timoshenko梁单元和单位荷载法计算了平面悬臂梁在集中力、力偶和均布荷载作用下梁的挠度,对比分析了不同解法、泊松比和剪切系数对剪切挠度的影响。     

充气梁的有限元分析

采用包括Timoshenko动力学、有限转动和小应变的虚功原理定义充气梁弯曲和屈曲的离散非线性计算公式。随后推导出预应力参考构型的线性计算公式,从而得出新的充气梁有限元分析方法。刚度矩阵包括剪切系数和内压力。采用三节点梁单元建模分析悬臂梁的弯曲和屈曲、带箍环面的变形以及承受放射状压力的环面屈曲。采用这种梁元计算的结果与采用三维膜单元有限元计算结果很相近。结合充气梁的冲撞力或者蠕变压力概念,讨论了数值计算结果的有效性。     

FRP加固梁模态分析的摄动解法

采用基于Ritz函数展开的直接模态摄动法,求解纤维复合材料包覆钢筋混凝土（FRP-RC）梁的模态特性.在不考虑剪切变形和转动惯量的情况下,它将FRP-RC梁的无阻尼自由振动的变系数微分方程的求解转化成一组非线性代数方程组的求解,使得求解过程得以简化.在此基础上,计算了不同加固工况下FRP-RC梁在各种约束条件下的自振频率,并与其他方法进行了对比.数值结果表明,利用直接模态摄动法不仅可以简化计算过程,而且计算结果具有较高的精度和较强的适用性.     

预应力梁横向振动分析的模态摄动方法

以预应力简支梁为例,分析了预应力在梁的横向振动过程中的变化,建立了预应力梁横向弯曲振动的微分方程。采用模态摄动法,进一步推导出预应力梁模态特性的近似分析方法,把复杂的变系数微分方程的求解转化为线性代数方程组的求解,从而有效地简化了计算过程。最后通过算例,讨论了预应力对梁的横向振动特性的影响。计算结果表明：当施加预应力的位置有较大的偏心距时,预应力对梁的自振特性有较大的影响。     

索-梁组合结构的动力学建模理论及其内共振分析

利用索 -梁组合结构的连接条件及边界条件 ,建立了索 -梁组合结构的约化运动学控制方程。利用Galer kin模态截断 ,得到了该系统的多模态离散动力学方程。利用多尺度摄动方法 ,分析了索 -梁组合结构可能出现的内共振模式 ,并对该系统的内共振行为进行了数值模拟     

基于精细梁模型的向量式有限元分析

精细梁不同于Euler梁和Timoshenko梁,该模型在考虑剪切变形的同时还考虑了横向弯曲时截面转动产生的附加轴向位移及横向剪切变形影响截面抗弯刚度后产生的附加横向位移。推导了适用于向量式有限元分析的精细梁单元应变和内力表达式,采用FORTRAN自编了向量式有限元程序。对悬臂梁、两端固支梁和门式框架进行了算例分析,对比了采用不同梁单元模型下结构的竖向位移。结果表明:当高跨比较小时,3种梁单元的竖向位移相差不大;当高跨比较大时,精细梁单元的竖向位移较Euler梁和Timoshenko梁明显增大,表明剪切变形及刚度折减引起的附加轴向位移、附加横向位移不能忽略。精细梁单元模型对高跨比较大的梁进行分析可望得到更精确的结果。     

铁摩辛柯梁弯曲问题的精细积分法

Timoshenko梁弯曲问题的哈密顿对偶方程,是关于梁截面上的广义力和广义位移的一阶常微分方程组,可与现代控制理论的一些问题相比拟。由于系统矩阵具有辛矩阵的特性,数值计算具有良好的稳定性,可将Timoshenko梁弯曲问题的两端边值问题转化成初值问题,用精细积分法求得高精度的数值解。算例计算结果表明,本方法具有较高的精度和适用性,并可方便地用于变截面梁的计算。     

跨高比对梁力学性能影响规律研究

梁为结构的基本构件,它的力学特性直接影响结构整体的力学性能.随着梁跨高比的变化,梁的变形方式也将发生改变.考虑剪切变形的Timoshenko梁计算所得频率普遍小于Euler-Bernoulli梁计算所得频率,这种差距随着跨高比的减小逐渐增大,且对高阶频率影响比低阶大.理论解与Timoshenko梁理论计算第一频率差值在跨高比为5时出现第一个较大差异值,表明深梁划分范围在特征参量上得到证明.     

Euler梁和修正Timoshenko梁的地震反应对比分析

基于修正Timoshenko梁的动力方程,推导出其自振频率、动位移、弯距及剪力计算式。并与Euler梁模型进行对比,分析了影响两种梁模型的修正系数μi,此系数随阶数、高跨比及剪切影响系数的增大而减小并且逐渐远离1的位置,致使两种梁模型的计算结果偏差增大。同时,对比分析了矩形截面梁的波数及梁高对两种梁模型的地震反应影响,分析结果表明:梁高较大或者波数较多时,两种梁模型的计算结果差别较大。同时注意到Euler梁的频率与波数始终呈非线性增长;而当梁高达到一定值时,修正Timoshenko梁的频率将会与波数呈线性增长。     

Bending and buckling of inflatable beams: Some new theoretical results

The non-linear and linearized equations are derived for the in-plane stretching and bending of thin-walled cylindrical beams made of a membrane and inflated by an internal pressure. The Timoshenko beam model combined with the finite rotation kinematics enables one to correctly account for the shear effect and all the non-linear terms in the governing equations. The linearization is carried out around a pre-stressed reference configuration which has to be defined as opposed to the so-called natural state. Two examples are then investigated: the bending and the buckling of a cantilever beam. Their analytical solutions show that the inflation has the effect of increasing the material properties in the beam solution. This solution is compared with the three-dimensional finite element analysis, as well as the so-called wrinkling pressure for the bent beam and the crushing force for the buckled beam. New theoretical and numerical results on the buckling of inflatable beams are displayed.     

浅析支座区钢筋和砼变形对悬臂梁挠度的影响

通过对钢筋混凝土悬臂梁支座区钢筋和混凝土的变形分析,建立了由支座变形产生的悬臂梁附加挠度计算公式,可直接运用于悬臂梁的附加挠度计算。     

A new approach for free vibration analysis of nonuniform tall building structures with axial force effects

Dynamic analysis of beam‐like structures is significantly important in modeling actual cases such as tall buildings and several other related applications as well. This article studies free vibration analysis of tall buildings with nonuniform cross‐section structures. A novel and simple approach is presented to solve natural frequencies of free vibration of cantilevered tall structures with variable flexural rigidity and mass densities. These systems could be replaced by a cantilever Timoshenko beam with varying cross‐sections. The governing partial differential equation for vibration of a nonuniform Timoshenko beam under variable axial loads is transformed with varying coefficients to its weak form of integral equations. Natural frequencies can be determined by requiring the resulting integral equation, which has a nontrivial solution. The presented method in this study has fast convergence. Including high accuracy for the obtained numerical results as well. Numerical examples including framed tube as well as tube‐in‐tube structures are carried out in the study and compared with available results in the literature, and also with the results obtained from finite element analysis in order to show the accuracy of the proposed method in the study. Obtained results indicate that the presented method in this study is powerful enough for the free vibration analysis of tall buildings.     

Out-of-plane behaviour of partially composite or sandwich beams by exact and Finite Element Methods

The out-of-plane vibrations of composite beams with interlayer slip or three-layer sandwich beams are theoretically and numerically investigated in this paper for general boundary conditions. The governing dynamics equations are derived by applying the Hamilton's principle. A Finite Element Resolution is presented for general boundary conditions, and compared to the exact solution based on the resolution of a tenth-order differential equation. The Finite Element Method may exhibit slip locking phenomenon for very stiff connection, a phenomenon widely investigated in the past for the in-plane behaviour of partially composite beams or sandwich beams. This slip locking, analogous to the shear locking for Timoshenko beams, can be faced with some relevant interpolation shape functions of the same order for each kinematics variables, namely the deflections and the torsion angle. The numerical results are presented for layered wood beams and laminated glass beams, with particular emphasis on the rate of convergence of the natural frequencies with respect to the number of Finite Elements. It is theoretically and numerically shown that the elastic spectra of the symmetrical composite beam are composed of two independent spectrums. One spectrum is independent of the connection parameter and can be studied using the solution of the non-composite action, whereas the second spectrum can be obtained from the resolution of a third-order polynomial equation using the Cardano's method. We show the phenomenon of cut-on frequency for this out-of-plane problem, a phenomenon already noticed for the in-plane Timoshenko beam vibrations. The exact method associated to a 10 degrees-of-freedom shape function can be formally associated with the dynamics stiffness method. The numerical and the exact approaches lead to the same dimensionless spectra, up to four digits.     

Timoshenko beam effect in flexural resonant column tests

Current methods for deriving Young’s modulus (E) from flexural resonant column tests do not account for the Timoshenko beam effect. Hence, the extent of the Timoshenko beam effect is investigated in this study using a series of numerical experiments performed by finite element program LS-DYNA. Prior to conducting the numerical experiments, LS-DYNA is verified for a cantilever beam to ensure that it is capable of accounting for the Timoshenko beam effect. The Young’s modulus determined with the existing interpretation method for flexural resonant column tests, which does not consider the Timoshenko beam effect, is underestimated on average by about 11% for a typical resonant column specimen. A correction factor is proposed here to account for the Timoshenko beam effect using the existing interpretation method for flexural resonant column tests.     

弹性地基上铁摩辛柯梁的压弯问题

从哈密顿力学出发,由勒让德变换引入对偶变量,导出了弹性地基上铁摩辛柯梁压弯问题的哈密顿对偶求解体系,将梁的控制微分方程转化为哈密顿对偶方程,给出了问题的矩阵指数函数解,可用本征向量展开法求问题的解析解,也可用精细积分法求问题的高精度数值解,由于导出的系统矩阵具有辛矩阵的特性,数值计算具有良好的稳定性.     

Timoshenko versus Euler beam theory: Pitfalls of a deterministic approach

The selection criteria for Euler-Bernoulli or Timoshenko beam theories are generally given by means of some deterministic rule involving beam dimensions. The Euler-Bernoulli beam theory is used to model the behavior of flexure-dominated (or “long”) beams. The Timoshenko theory applies for shear-dominated (or “short”) beams. In the mid-length range, both theories should be equivalent, and some agreement between them would be expected. Indeed, it is shown in the paper that, for some mid-length beams, the deterministic displacement responses for the two theories agrees very well. However, the article points out that the behavior of the two beam models is radically different in terms of uncertainty propagation. In the paper, some beam parameters are modeled as parameterized stochastic processes. The two formulations are implemented and solved via a Monte Carlo-Galerkin scheme. It is shown that, for uncertain elasticity modulus, propagation of uncertainty to the displacement response is much larger for Timoshenko beams than for Euler-Bernoulli beams. On the other hand, propagation of the uncertainty for random beam height is much larger for Euler beam displacements. Hence, any reliability or risk analysis becomes completely dependent on the beam theory employed. The authors believe this is not widely acknowledged by the structural safety or stochastic mechanics communities.     

Behavior of steel-concrete composite cantilever box beams under negative moment

Four steel-concrete composite cantilever beam specimens were tested to investigate their mechanical behavior under negative moment induced by concentrated loads at the ends of the beams. The failure modes, serviceability and ultimate bearing capacities of the composite beams with full shear connection were studied. The crack initiation and propagation were investigated with consideration of two types of shear connectors. Three kinds of longitudinal reinforcement ratios were also examined. The experimental results indicate that an increase in the reinforcement ratio is beneficial to the bearing capacity of the composite beams to some extent and that the shear stud connector is superior to the steel block connector with regards to the serviceability of the beams. Two numerical models, which were based on a concrete material model and an elasto-plastic material model, were employed to simulate the behavior of steel-concrete composite beams. The numerical calculation results show that the combination of the two models can be used to predict the longitudinal cracking load and ultimate bearing capacity of composite cantilever beams. Based on the experimental and numerical results, it was found that the ultimate bearing capacity of a steel-concrete composite beam under negative moment can be significantly affected by longitudinal cracks in the concrete slabs. An equation to predict the longitudinal cracking load of a composite cantilever beam under negative moment by concentrated load was proposed and found to have good accuracy.