精确频率约束下框架结构的尺寸优化

本文研究受到频率约束的框架结构优化设计问题,采用基于Bernoulli-Euler梁的动力刚度法求解框架结构的自振频率,采用W-W算法精确求解自振频率及频率对设计变量的灵敏度.分析比较了将框架结构的杆件按普通静刚度梁单元和动力刚度梁单元分别建立计算模型时得到的频率分析结果.以杆件面积为设计变量,以结构重量为优化目标,对基频约束下的框架结构进行了尺寸优化,比较了节点集中质量对优化结果的影响.通过2个数值算例,验证了动力刚度阵法在框架结构频率性能分析与优化设计中的适用性,并具有较高的计算精度.     

直接基于单元平衡的变截面梁反应分析方法

固定形状的单元位移插值函数不能合理地近似变截面梁内部的位移变化,从而影响了传统梁单元用于计算变截面梁的精度.采用直接基于单元平衡的思想给出了计算变截面梁反应的有限元方法,解决了单元位移插值函数局限性所带来的问题.导出了变截面梁单元的单元刚度矩阵、单元等效节点荷载和单元一致质量矩阵.在此基础上,利用编制的程序进行了算例验证与分析.算例验证了本文理论的正确性,表明本文方法具有很高的计算精度.     

轴向受载的Bernoulli—Euler薄壁梁固有振动和稳定性的理论研究

通过直接求解轴向受载的单对称均匀Bernoulli-Euler薄壁梁单元弯扭耦合振动的运动微分方程,推导了其动态传递矩阵,讨论了轴向载荷的变化对薄壁梁弯扭耦合振动固有频率的影响,并由此得到零频率振动（弹性屈出）发生时相应的轴向载荷,数值结果表明本文方法在其适应范围内是精确有效的。     

多分辨率Timoshenko梁单元

基于Timoshenko梁的假定,通过基本节点形函数的扩展,并在梁区域上进行伸缩和平移,本文构建了基函数系,由此生成相互嵌套、逐级包含的位移空间序列,最后,采用最小势能原理,得到梁的平衡方程,从而构造了多分辨率Timoshenko梁单元。该单元具有如下特性：1．可通过自由调节单元分辨率的大小来增减单元的网格节点数量,从而调整单元的计算精度,其精度与相同网格划分条件下任意多个传统梁单元计算的相一致;2．经重新划分网格后的单元整体刚度、质量矩阵及等效节点荷载向量可直接获得,不像传统梁单元那样需要重新生成;3．与传统梁单元一样可以很方便地处理各种边界条件。本文首次将多分辨率概念引入到传统的Timoshenko梁单元中,并构建了梁结构网格划分的数学依据—位移空间序列,同时,揭示了共同节点可以采用人工叠加方法形成整体刚阵的缘由—连续的节点扩展形函数。     

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.     

基于旋转软化的柔性梁动力学研究

建立了旋转柔性梁的非线性动力学模型,利用能量法及哈密顿原理导出了耦合的动力学方程,分析了转动惯性、Coriolis力、应力刚化、旋转软化、加速度、横向位移、弯曲刚度等作用效应;通过设置应力刚化及旋转软化等刚度矩阵和编制有限元程序,建立了梁单元有限元模型,对柔性梁在旋转软化状态下的振动模态进行了数值模拟与分析。计算表明:梁的旋转软化导致其沿旋转平面的弯振模态(摆振)频率随转速增大而相对下降,且对第一阶摆振频率的影响最显著,呈现非线性;梁的旋转软化对垂直于旋转平面的弯振频率几乎没有影响,此结果表明了旋转柔性梁动态特性的复杂性,因此在计算旋转柔性梁的振动特性时,必须同时设置平动、转动惯性质量矩阵,才能获得准确结果。此外,梁单元模型与实体单元模型计算结果误差小于等于5%,验证了本文梁单元模型求解方法的准确性。     

基于广义变分原理的梁板单元分析的数值流形方法

数值流形方法(NMM)是一种基于有限覆盖技术的新型数值方法.以该方法的覆盖位移模式为基础,利用广义变分原理中罚函数理论,详细推导了梁板流形单元的覆盖位移函数,刚度矩阵和应变矩阵,并建立了可应用于梁板单元分析的数值流形方法.最后通过算例分析表明,该方法在对梁板弯曲问题分析是有效的.     

????Timoshenko????????????

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.     

谱元法研究桁架结构的振动带隙特性

采用谱元法研究了桁架周期结构的带隙特性.从杆和梁的运动方程出发,推导出与频率相关的插值函数,得到了杆单元和梁单元的动力学刚度矩阵.在频域下将铰结构考虑为一个谱单元,并推导出铰单元的动力学刚度矩阵.将杆/梁单元和铰单元加以整合得到整体结构的动力学刚度阵,进而建立整体结构的运动方程.通过求解整体结构的动力学方程,获得结构的频域响应,进而研究结构的带隙特性.本文将谱元法求解得到的固有频率结果与有限元法进行了对比,分析了单胞数量和材料的变化对结构带隙特性的影响,拓展了谱元法的应用领域.     

含旋转运动效应裂纹梁横向振动特性的研究

针对开口裂纹作用下旋转运动欧拉-伯努力梁的振动特性进行了研究。文中使用裂纹梁连续等效刚度模型模拟裂纹效应,将含裂纹旋转运动梁视为弯曲刚度沿梁长度方向连续变化的梁,并应用传递矩阵法推导了求解其振动特性的特征方程。考虑不同裂纹深度和位置、不同旋转速度,分析了梁的一阶和二阶固有频率的变化情况。研究结果表明:旋转运动效应和裂纹效应并非独立影响梁的固有频率,两者间具有耦合作用效应;转速提升使由裂纹导致的频率衰减幅度变小,同时裂纹加深使得由速度升高带来的阶频提升更加显著;相比于二阶频率,耦合作用效应对于一阶频率更加显著。     

An efficient procedure to find shape functions and stiffness matrices of nonprismatic Euler–Bernoulli and Timoshenko beam elements

Nonprismatic beam modeling is an important issue in structural engineering, not only for versatile applicability the tapered beams do have in engineering structures, but also for their unique potential to simulate different kinds of material or geometrical variations such as crack appearing or spreading of plasticity along the beam. In this paper, a new procedure is proposed to find the exact shape functions and stiffness matrices of nonprismatic beam elements for the Euler–Bernoulli and Timoshenko formulations. The variations dealt with here include both tapering and abrupt jumps in section parameters along the beam element. The proposed procedure has found a simple structure, due to two special approaches: The separation of rigid body motions, which do not store strain energy, from other strain states, which store strain energy, and finding strain interpolating functions rather than the shape functions which suffer complex representation. Strain interpolating functions involve low-order polynomials and can suitably track the variations along the beam element. The proposed procedure is implemented to model nonprismatic Euler–Bernoulli and Timoshenko beam elements, and is verified by different numerical examples.     

Out-of-plane responses of a circular curved Timoshenko beam due to a moving load

For the cases of using the finite curved beam elements and taking the effects of both the shear deformation and rotary inertias into consideration, the literature regarding either free or forced vibration analysis of the curved beams is rare. Thus, this paper tries to determine the dynamic responses of a circular curved Timoshenko beam due to a moving load using the curved beam elements. By taking account of the effect of shear deformation and that of rotary inertias due to bending and torsional vibrations, the stiffness matrix and the mass matrix of the curved beam element were obtained from the force–displacement relations and the kinetic energy equations, respectively. Since all the element property matrices for the curved beam element are derived based on the local polar coordinate system (rather than the local Cartesian one), their coefficients are invariant for any curved beam element with constant radius of curvature and subtended angle and one does not need to transform the property matrices of each curved beam element from the local coordinate system to the global one to achieve the overall property matrices for the entire curved beam structure before they are assembled. The availability of the presented approach has been verified by both the existing analytical solutions for the entire continuum curved beam and the numerical solutions for the entire discretized curved beam composed of the conventional straight beam elements based on either the consistent-mass model or the lumped-mass model. In addition to the typical circular curved beams, a hybrid curved beam composed of one curved-beam segment and two identical straight-beam segments subjected to a moving load was also studied. Influence on the dynamic responses of the curved beams of the slenderness ratio, moving-load speed, shear deformation and rotary inertias was investigated.     

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.     

Transverse vibration of a multiple-Timoshenko beam system with intermediate elastic connections due to a moving load

Based on Timoshenko beam theory, the dynamic response of an elastically connected multiple-beam system is investigated. The identical prismatic beams are assumed to be parallel and connected by a finite number of springs. Assuming n parallel Timoshenko beams, the motion of the system is described by a coupled set of 2n partial differential equations. The method involves a change of variables and modal analysis to decouple and to solve the governing differential equations, respectively. A case study is solved in detail to demonstrate the methodology and several plots of the midpoint deflections of beams are given and investigated for different values of moving load velocity and the stiffness of elastic connections. From the numerical results it is observed that the maximum deflection of the multiple Timoshenko beam system is always smaller than one of a single beam.     

Dynamic analysis of beams on viscoelastic foundation

This study is intended to analyze dynamic behavior of beams on Pasternak-type viscoelastic foundation subjected to time-dependent loads. The Timoshenko beam theory is adopted in the derivation of the governing equation. Ordinary differential equations in scalar form obtained in the Laplace domain are solved numerically using the complementary functions method to calculate exactly the dynamic stiffness matrix of the problem. The solutions obtained are transformed to the real space using the Durbin's numerical inverse Laplace transform method. The dynamic response of beams on viscoelastic foundation is analyzed through various examples.     

Dynamic analysis of beam-cable coupled systems using Chebyshev spectral element method

The dynamic characteristics of a beam–cable coupled system are investigated using an improved Chebyshev spectral element method in order to observe the effects of adding cables on the beam. The system is modeled as a double Timoshenko beam system interconnected by discrete springs. Utilizing Chebyshev series expansion and meshing the system according to the locations of its connections,numerical results of the natural frequencies and mode shapes are obtained using only a few elements, and the results are validated by comparing them with the results of a finiteelement method. Then the effects of the cable parameters and layout of connections on the natural frequencies and mode shapes of a fixed-pinned beam are studied. The results show that the modes of a beam–cable coupled system can be classified into two types, beam mode and cable mode, according to the dominant deformation. To avoid undesirable vibrations of the cable, its parameters should be controlled in a reasonable range, or the layout of the connections should be optimized.     

Dynamic stability analysis of shear-flexible composite beams

Dynamic stability behavior of the shear-flexible composite beams subjected to the nonconservative force is intensively investigated based on the finite element model using the Hermitian beam elements. For this, a formal engineering approach of the mechanics of the laminated composite beam is presented based on kinematic assumptions consistent with the Timoshenko beam theory, and the shear stiffness of the thin-walled composite beam is explicitly derived from the energy equivalence. An extended Hamilton’s principle is employed to evaluate the mass-, elastic stiffness-, geometric stiffness-, damping-, and load correction stiffness matrices. Evaluation procedures for the critical values of divergence and flutter loads of the nonconservative system with and without damping effects are then briefly introduced. In order to verify the validity and the accuracy of this study, the divergence and flutter loads are presented and compared with the results from other references, and the influence of various parameters on the divergence and flutter behavior of the laminated composite beams is newly addressed: (1) variation of the divergence and flutter loads with or without the effects of shear deformation and rotary inertia with respect to the nonconservativeness parameter and the fiber angle change, (2) influence of the internal and external damping on flutter loads whether to consider the shear deformation or not.     

压电层合梁静动力学分析的高精度梁单元

给出了一个对复合材料压电层合梁进行数值分析的高精度压电层合梁单元。基于Shi三阶剪切变形板理论的位移场和Layer-wise理论的电势场,用力-电耦合的变分原理及Hamilton原理推导了压电层合梁单元列式。采用拟协调元方法推导了一个可显式给出单元刚度矩阵的两节点压电层合梁单元,并应用于压电层合梁的力-电耦合弯曲和自由振动分析。计算结果表明,该梁单元给出的梁挠度和固有频率与解析解吻合良好,并优于其它梁单元的计算结果,说明了本文所给压电层合梁单元的可靠性和准确性。研究结果可为力-电耦合作用下压电层合梁的力学分析提供一个简单、精确且高效的压电层合梁单元。     

层状分数阶黏弹性饱和地基与梁共同作用的时效研究

饱和地基与梁共同作用问题的研究在力学领域及工程界都具有重要意义. 采用分数阶Merchant模型研究饱和地基的流变固结, 该模型比常用整数阶黏弹性模型更能精确反映地基的时变特征. 基于层状正交各向异性黏弹性饱和地基的固结解答, 采用有限元法与边界元法耦合的方法, 研究梁与分数阶黏弹性饱和地基的共同作用问题. 依据Timoshenko梁理论将梁离散为若干单元, 进而得到梁的总刚度矩阵方程; 将黏弹性地基固结问题的精细积分解答作为边界积分的核函数, 采用边界元法建立地基柔度矩阵方程; 结合梁与地基接触面的位移协调条件以及力的平衡条件, 通过有限元法与边界元法的耦合, 最终求得层状分数阶黏弹性饱和地基与Timoshenko梁共同作用的解答. 将本文地基退化为Kelvin地基进行计算, 并与已有文献中的算例进行对比, 二者具有很好的一致性. 在此基础上, 探讨分数阶次和地基成层性对梁与黏弹性饱和地基共同作用的影响. 结果表明: 分数阶次高的黏弹性饱和地基的固结速率明显更快; 对于层状地基, 加固表层土体能有效控制地基整体沉降, 并减小差异沉降. 实际工程中, 应充分考虑饱和地基流变及土体分层性的影响, 以准确分析梁与地基的共同作用过程.      

Nonlinear behaviour of different flexible size-dependent beams models based on the modified couple stress theory. Part 1: Governing equations and static analysis of flexible beams

In the first part of the paper we employ the Sheremetev-Pelekh-Reddy-Levinson hypotheses, which yield a non-linear mathematical model of a beam taking into account geometric and physical non-linearity as well as transverse shear based on the modified couple stress theory. The general model includes both Bernoulli-Euler and Timoshenko models with/without geometric/physical non-linearity, and the size-dependent beam behaviour.In addition, we present results of the development of the relaxation method for solution to numerous static problems. The influence of the size-dependent coefficient on the load-deflection and stress-strain states of the Bernoulli-Euler, Timoshenko, and Sheremetev-Pelekh-Reddy-Levinson mathematical models has been also studied.