Nonlinear dynamic analysis on rigid-flexible coupling system of an elastic beam

Previous work examined the effect of the attached stiffness matrix terms on stability of an elastic beam undergoing prescribed large overall motion. The aim of the present work is to extend the nonlinear formulations to an elastic beam with free large overall motion. Based on initial stress method, the nonlinear coupling equations of elastic beams are obtained with free large overall motion and the attached stiffness matrix is derived by solving sub-static formulation. The angular velocity and the tip deformation of the elastic pendulum are calculated. The analytical results show that the simulation results of the present model are tabled and coincide with the one-order approximate model. It is shown that the simulation results accord with energy conservation principle.     

弹性地基双层梁理论下的混凝土路面力学分析

为建立混凝土路面结构受力分析计算模型,以Winkler弹性地基梁模型为基础,推导出了弹性地基双层梁理论的表达式;给定边界条件,利用MATLAB软件获得了无限长弹性地基梁在集中力作用下的挠度表达式。将混凝土路面结构简化为弹性地基上的双层梁,当车辆荷载作用于混凝土路面时,在集中载荷的作用下,建立了面层与基层的微分平衡方程。应用广义“初参数”法,得到了双层梁位移和应力的解析解。通过算例,对面层及基层的变形和应力进行了分析,结果表明:增大面层、基层的轴惯性矩和地基的弹性常数,可以有效地减少面层和基层的变形量,降低最大应力数值,但抗弯刚度对基层和面层的弯矩受力影响不大。最后将结果与ANSYS分析结果进行了比较,佐证了解的可靠性,研究结果可为混凝土路面结构设计提供依据。     

Transition wave in a supported heavy beam

We consider a heavy, uniform, elastic beam rested on periodically distributed supports as a simplified model of a bridge. The supports are subjected to a partial destruction propagating as a failure wave along the beam. Three related models are examined and compared: (a) a uniform elastic beam on a distributed elastic foundation, (b) an elastic beam in which the mass is concentrated at a discrete set of points corresponding to the discrete set of the elastic supports and (c) a uniform elastic beam on a set of discrete elastic supports. Stiffness of the support is assumed to drop when the stress reaches a critical value. In the formulation, it is also assumed that, at the moment of the support damage, the value of the ‘added mass’, which reflects the dynamic response of the support, is dropped too. Strong similarities in the behavior of the continuous and discrete-continuous models are detected. Three speed regimes, subsonic, intersonic and supersonic, where the failure wave is or is not accompanied by elastic waves excited by the moving jump in the support stiffness, are considered and related characteristic speeds are determined. With respect to these continuous and discrete-continuous models, the conditions are found for the failure wave to exist, to propagate uniformly or to accelerate. It is also found that such beam-related transition wave can propagate steadily only at the intersonic speeds. It is remarkable that the steady-state speed appears to decrease as the jump of the stiffness increases.     

热力耦合结构的弹性支撑分析与拓扑优化设计

弹性支撑是保证热力耦合载荷作用下结构有效承载的一种支撑设计方案, 可以在满足结构刚度设计要求的同时有效降低结构内的应力集中, 保证结构的热变形协调. 从理论分析与数值计算两个方面, 研究了热力耦合载荷作用下结构的弹性支撑优化设计. 首先以弹性支撑梁模型为例, 通过给出热力耦合载荷下应力计算的解析表达式, 从理论上说明弹性支撑对结构应力峰值的影响, 阐述了弹性支撑的承载与热变形协调的双重效果. 在此基础上, 提出了解决该类问题的弹性支撑通用优化数学模型和拓扑构型优化方法, 通过数值算例与优化结果展示了方法的有效性.     

考虑收缩徐变影响的新旧混凝土叠合梁单元刚度矩阵改进

混凝土自身的收缩徐变会在新旧混凝土叠合梁中使应力重分布.为了计算重分布应力,首先推导以挠度表达的叠合梁非线性微分方程,然后通过求解该微分方程,引入位移形函数、刚度形函数和等效节点载荷形函数,最后得出新混凝土梁、旧混凝土梁和Goodman弹性夹层三合一的叠合梁改进型单元刚度矩阵和等效节点载荷,从而为收缩徐变影响下的混凝土的内力计算提供了一种有效的新方法.文中还进行了实例验证分析,并从中得出了一些有益的结论.     

基于一种广义梁理论的弹性地基FGM简支梁自由振动解析解

基于一种扩展的n阶广义剪切变形梁理论(n-GBT),应用Hamilton原理,建立了以轴向位移、横向位移及转角为未知函数的Winkler-Pasternak弹性地基功能梯度材料(FGM)梁的自由振动方程,采用Navier法获得了弹性地基FGM简支梁自由振动的精确解。与多种梁理论预测结果进行比较,讨论并给出了GBT阶次n的理想取值;分析了梯度指标、跨厚比及地基刚度对FGM梁频率的影响。结果表明:本文方法有效且适用范围广,若采用高阶剪切梁理论模型,宜取n≥3的奇数;FGM梁的自振频率随材料梯度指标的增大而减小;随跨厚比的增加而增大,但当跨厚比大于20,跨厚比增加对频率的影响很小;随地基刚度的增加而增大,地基刚度足够大时,频率趋于收敛。     

THE FINITE ELEMENT ANALYSIS OF THE FLEXIBLE BEAMS AND PLATES

This paper studies large deflection problem of beam and plates by the finite elementmethod.The elongation of the middle surface caused by its rotation is considered in strain-displacement relations.The higher order terms will be reserved when strain energy iscalculated.The elastic stiffness matrix,linear and nonlinear initial stress stiffness metricesare derived by the principle of minimum potential energy.Examples show that precision willbe properly manifested although the total storage amount and the calculating time are notincreased.The iterative method with co-moving coordinate must be adopted to avoid parasiticrigid body motion.     

拼接式组合扁梁受弯性能分析

本文提出了一种拼接式组合扁梁,对其进行了受弯性能分析.组合扁梁属于内嵌式组合梁,其组成部分较为复杂,传统承载力计算公式偏保守.为便于此类梁的设计,本文在等效应力法的基础上使用等效矩形应力图法,对拼接式组合扁梁的正截面抗弯承载力公式进行推导.使用ABAQUS有限元软件对组合方法公式的准确性进行验证,并与等效应力法公式进行对比,结果表明组合公式得到的数值更为精准.在相同截面高度和用钢量下设计了一个深肋组合扁梁和一个拼接式组合扁梁,并进行了受弯性能对比;同时对不同钢强度和翼缘厚度影响下的拼接式组合扁梁抗弯承载性能进行分析.使用换算截面法对其进行了弹性刚度的计算,并和有限元进行了对比,二者吻合较好,研究结果为拼接式组合扁梁的设计和优化提供了一定参考依据.     

Analytical solutions for thermal vibration of nanobeams with elastic boundary conditions

A nonlocal Euler beam model with second-order gradient of stress taken into consideration is used to study the thermal vibration of nanobeams with elastic boundary.An analytical solution is proposed to investigate the free vibration of nonlocal Euler beams subjected to axial thermal stress.The effects of the nonlocal parameter,thermal stress and stiffness of boundary constraint on the vibration behaviors of nanobeams are revealed.The results show that natural frequencies including the thermal stress are lower than those without the thermal stress when temperature rises.The boundary-constrained springs have significant effects on the vibration of nanobeams.In addition,numerical simulations also indicate the importance of small-scale effect on the vibration of nanobeams.     

钢-混凝土组合梁收缩徐变效应的随机分析

钢-混凝土组合梁是由混凝土板和钢梁通过剪力键连接而成的组合结构。由于混凝土的收缩徐变,将引起结构内力和应力重分布。混凝土收缩徐变具有离散性大的特点,进而导致结构长期响应表现出随机性。本文综合考虑徐变模型、收缩模型、混凝土抗压强度、混凝土弹性模量、环境湿度、钢材弹性模量、荷载以及剪力键刚度的随机性对钢-混凝土组合梁结构响应的影响。利用拉丁超立方抽样技术和基于响应面方法的蒙特卡洛抽样,研究了钢-混凝土组合梁挠度和应力时变效应的概率问题。     

Tilting analysis of circular elastic layers interleaving with flexible reinforcements

Bonding with reinforcements can increase the stiffness of elastic layers in the normal direction. The flexibility effect of the reinforcement on the bonded elastic layers of a circular cross-section subjected to pure bending moment is analyzed through a theoretical approach. Based on two kinematics assumptions in the elastic layers, the closed-form solutions of the horizontal displacements in the elastic layers and the reinforcements are solved using the governing equations established by stress equilibrium in the elastic layers and the reinforcements. Through these solved displacements, the tilting stiffness of the bonded elastic layer, the shear stress on the bonding surfaces, and the internal forces of the reinforcements are derived in closed forms.     

A numerical model for static and free vibration analysis of elastic composite beams with end shear restraint

The end shear restraint, which is an un-classical type of end support, has a significant effect on the behavior of elastic composite beams. The principal aim of this paper is to present a numerical model for studying the effect of end shear restraint on static and free vibration behavior of elastic composite beams with various end conditions. The elastic composite beam, considered in this study, is composed of an upper concrete slab and a lower steel beam, connected at the interface by shear transmitting studs. This type of beam is widely used in constructions especially for highway bridges. The three types of end conditions considered in this study are simple, fixed and free supports. The numerical model is based on the combination of transfer matrix and analog beam methods. The field transfer matrices for the element of the elastic composite beam are derived. The present model is applied to the beam systems with and without end shear restraint and the static response and natural frequencies are calculated. the effect of shear stiffness between the upper slab and lower beam is also demonstrated.     

Flutter analysis of rotating beams with elastic restraints

The aeroelastic stability of rotating beams with elastic restraints is investigated. The coupled bending-torsional Euler-Bernoulli beam and Timoshenko beam models are adopted for the structural modeling. The Greenberg aerodynamic model is used to describe the unsteady aerodynamic forces. The additional centrifugal stiffness effect and elastic boundary conditions are considered in the form of potential energy. A modified Fourier series method is used to assume the displacement field function and solve the governing equation. The convergence and accuracy of the method are verified by comparison of numerical results. Then, the flutter analysis of the rotating beam structure is carried out, and the critical rotational velocity of the flutter is predicted. The results show that the elastic boundary reduces the critical flutter velocity of the rotating beam, and the elastic range of torsional spring is larger than the elastic range of linear spring.     

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.     

薄壁箱梁广义坐标法刚度矩阵的推导及应用

在符拉索夫广义坐标法初参数方程的基础上,推导出可用于均布扭转荷载作用下薄壁箱梁翘曲分析的刚度矩阵,该刚度矩阵具有较高的单元精度,可用于由较多薄壁箱梁组成的复杂结构的整体有限元分析。通过对广义坐标法刚度矩阵和乌曼斯基理论、修正乌曼斯基理论求得薄壁箱梁的位移和应力进行分析比较,为各方法在实际工程中的应用提供一定的参考。     

Dynamic stiffness method for free vibration of an axially moving beam with generalized boundary conditions

Axially moving beams are often discussed with several classic boundary conditions, such as simply-supported ends, fixed ends, and free ends. Here, axially moving beams with generalized boundary conditions are discussed for the first time. The beam is supported by torsional springs and vertical springs at both ends. By modifying the stiffness of the springs, generalized boundaries can replace those classical boundaries.Dynamic stiffness matrices are, respectively, established for axially moving Timoshenko beams and Euler-Bernoulli(EB) beams with generalized boundaries. In order to verify the applicability of the EB model, the natural frequencies of the axially moving Timoshenko beam and EB beam are compared. Furthermore, the effects of constrained spring stiffness on the vibration frequencies of the axially moving beam are studied. Interestingly, it can be found that the critical speed of the axially moving beam does not change with the vertical spring stiffness. In addition, both the moving speed and elastic boundaries make the Timoshenko beam theory more needed. The validity of the dynamic stiffness method is demonstrated by using numerical simulation.     

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.     

Monitoring structural damage of components using an effective modulus approach

This paper describes a novel nondestructive damage detection method that was developed to study the influence of a crack on the dynamic properties of a cantilever beam subjected to bending. Experimental measurements of transfer functions for the cracked cantilever beam revealed a change in the natural frequency with increasing crack length. A finite element model of a cracked element was created to compute the influence of severity and location of damage on the structural stiffness. The proposed model is based on the response of the cracked beam element under a static load. The change in beam deflection as a result of the crack is used to calculate the reduction in the global component stiffness. The reduction of the beam stiffness is then used to determine its dynamic response employing a modal analysis computational model. Euler–Bernoulli and Timoshenko beam theories are used to quantify the elastic stiffness matrix of a finite element. The transfer functions from both theories compare well with the experimental results. The experimental and computational natural frequencies decreased with increasing crack length. Furthermore the Euler–Bernoulli and Timoshenko beam theories resulted in approximately the same decrease in the natural frequency with increasing crack length as experimentally measured.     

一种梁柱非线性分析新方法

提出了一种等截面梁柱非线性分析的新方法，即采用按幂函数的变截面法模拟梁柱截面刚度，建立了梁柱非线性分析的简化数学计算模型，并研究了梁柱从弹性到塑性状态时梁柱截面刚度的变化规律，通过与Chen法的比较证明了截面刚度按幂函数模拟的变截面法的正确性。在此基础上进一步应用变截面法讨论了单边塑性情况下偏心受压梁柱随偏心轴力P值的变化规律。此方法为进一步研究非线性梁柱结构提供了一种新的思路。     

Elastic constants for granular materials modeled as first-order strain-gradient continua

Formulation of a stress–strain relationship is presented for a granular medium, which is modeled as a first-order strain-gradient continuum. The elastic constants used in the stress–strain relationship are derived as an explicit function of inter-particle stiffness, particle size, and packing density. It can be demonstrated that couple-stress continuum is a subclass of strain-gradient continua. The derived stress–strain relationship is simplified to obtain the expressions of elastic constants for a couple-stress continuum. The derived stress–strain relationship is compared with that of existing theories on strain- gradient models. The effects of inter-particle stiffness and particle size on material constants are discussed.