Analysis of curved beams using a new differential transformation based curved beam element

Differential transformation method is used to obtain the shape functions for nodal variables of an arbitrarily non-uniform curved beam element including the effects of shear deformation considering axially functionally graded material. The proposed shape functions depend on the variations in cross-sectional area, moment of inertia, curvature and material properties along the axis of the curved beam element. The static and free vibration of axially functionally graded tapered curved beams including shear deformation and rotary inertia are studied through solving several examples. Numerical results are presented for circular, parabolic, catenary, elliptic and sinusoidal beams (both forms—prime and quadratic) with hinged-hinged, hinged-clamped and clamped-clamped and clamped-free end restraints. Three general taper types (depth taper, breadth taper and square taper) for rectangular cross section are studied. Out of plane vibration is studied and the lowest natural frequencies are calculated and compared with the published results. Out of plane buckling is investigated for circular beams due to radial load.     

Geometric nonlinear dynamic analysis of curved beams using curved beam element

Instead of using the previous straight beam element to approximate the curved beam,in this paper,a curvilinear coordinate is employed to describe the deformations,and a new curved beam element is proposed to model the curved beam.Based on exact nonlinear strain-displacement relation,virtual work principle is used to derive dynamic equations for a rotating curved beam,with the effects of axial extensibility,shear deformation and rotary inertia taken into account.The constant matrices are solved numerically utilizing the Gauss quadrature integration method.Newmark and Newton-Raphson iteration methods are adopted to solve the differential equations of the rigid-flexible coupling system.The present results are compared with those obtained by commercial programs to validate the present finite method.In order to further illustrate the convergence and efficiency characteristics of the present modeling and computation formulation,comparison of the results of the present formulation with those of the ADAMS software are made.Furthermore,the present results obtained from linear formulation are compared with those from nonlinear formulation,and the special dynamic characteristics of the curved beam are concluded by comparison with those of the straight beam.     

Large displacement analysis of shear deformable composite beams with interlayer slips

This paper presents a novel geometric non-linear finite element formulation for the analysis of shear deformable two-layer beams with interlayer slips. We adopt the co-rotational approach where the motion of the element is decomposed into two parts: a rigid body motion which defines a local coordinate system and a small deformational motion of the element relative to this local coordinate system. The main advantage of this approach is that the transformation matrices relating local and global quantities are independent to the choice of the geometrical linear local element. The effect of transverse shear deformation of the layers is taken into account by assuming that each layer behaves as a Timoshenko beam element. The layers are assumed to be continuously connected and partial interaction is considered by considering a continuous relationship between the interface shear flow and the corresponding slip. In order to avoid curvature and shear locking phenomena, the local linear element is formulated using “exact” displacement shape functions derived from the closed-form solution of the governing equations of a two-layer beam element. Finally, three numerical applications are presented in order to assess the performance of the proposed formulation.     

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.     

加权残值法在钢筋混凝土拱桥非线性有限元分析中的应用

本文用圆弧梁离散拱肋;用圆柱拖带坐标、三次位移插值函数及平截面假定来描述单元位形;用加权残值配点法来消除曲梁单元的剪力与膜力闭锁。按基于连续介质力学的Ｕ．Ｌ．列式建立单元增量平衡方程,以考虑几何非线性。假定钢筋为理想弹塑性材料。按三参数各向同性强化塑性模型,建立混凝土的弹塑性本构矩阵。将拱单元分段分块,根据钢筋及砼的本构特性,建立拱单元及梁段单元的弹塑性刚度矩阵,以考虑材料非线性。用编制的程序对两座模型拱桥进行计算,计算结果与模型测试结果接近     

钢筋混凝土双肢剪力墙静力弹塑性分析

建立了由墙肢单元模型、连粱单元模型和连接单元模型组成的钢筋混凝土双肢剪力墙的静力弹塑性分析计算模型。墙肢单元采用以有限元为基础的宏模型;按是否出现对角线剪切破坏,分别建立短连粱计算模型和长连粱计算模型;为计及连粱与墙肢连接界面的相对位移,建立用复合弹簧模拟的连接单元计算模型;给出了确定模型参数的方法。对有关文献的短连粱和长连粱双肢剪力墙试件进行了静力弹塑性分析,分析结果与试验结果符合较好。     

Contact dynamics of elasto-plastic thin beams simulated via absolute nodal coordinate formulation

Under the frame of multibody dynamics, the contact dynamics of elasto-plastic spatial thin beams is numerically studied by using the spatial thin beam elements of absolute nodal coordinate formulation (ANCF). The inter-nal force of the elasto-plastic spatial thin beam element is derived under the assumption that the plastic strain of the beam element depends only on its longitudinal deformation. A new body-fixed local coordinate system is introduced into the spatial thin beam element of ANCF for efficient con-tact detection in the contact dynamics simulation. The linear isotropic hardening constitutive law is used to describe the elasto-plastic deformation of beam material, and the classical return mapping algorithm is adopted to evaluate the plastic strains. A multi-zone contact approach of thin beams previ-ously proposed by the authors is also introduced to detect the multiple contact zones of beams accurately, and the penalty method is used to compute the normal contact force of thin beams in contact. Four numerical examples are given to demonstrate the applicability and effectiveness of the pro-posed elasto-plastic spatial thin beam element of ANCF for flexible multibody system dynamics.     

两类基于局部标架梁单元的闭锁缓解方法

对于大转动、大变形柔性体的刚柔耦合动力学问题,基于李群SE(3)局部标架(local frame formulation, LFF)的建模方法能够规避刚体运动带来的几何非线性问题,离散数值模型中广义质量矩阵与切线刚度矩阵满足刚体变换的不变性,可明显地提高柔性多体系统动力学问题的计算效率. 有限元方法中,闭锁问题是导致单元收敛性能低下的主要原因, 例如梁单元的剪切以及泊松闭锁.多变量变分原理是缓解梁、板/壳单元闭锁的有效手段. 该方法不仅离散位移场,同时离散应力场或应变场, 可提高应力与应变的计算精度. 本文基于上述局部标架,研究几类梁单元的闭锁处理方法, 包括几何精确梁(geometrically exact beam formulation, GEBF)与绝对节点坐标(absolute nodal coordinate formulation, ANCF)梁单元. 其中, 采用Hu-Washizu三场变分原理缓解几何精确梁单元中的剪切闭锁,采用应变分解法缓解基于局部标架的ANCF全参数梁单元中的泊松闭锁. 数值算例表明,局部标架的梁单元在描述高转速或大变形柔性多体系统时,可消除刚体运动带来的几何非线性, 极大地减少系统质量矩阵和刚度矩阵的更新次数.缓解闭锁后的几类局部标架梁单元收敛性均得到了明显提升.      

带集中质量的旋转柔性曲梁动力学特性分析

本文对带集中质量的平面内旋转柔性曲梁动力学特性进行了研究.基于绝对节点坐标法推导出曲梁单元,其中该曲梁单元采用Green-Lagrangian应变,并根据曲梁变形前后的曲率变化和曲率的精确表达式计算了曲梁单元弹性力所作的虚功.通过虚功原理,利用$\delta$函数和中心刚体与悬臂曲梁之间的固支边界条件,建立了带集中质量的旋转柔性曲梁非线性动力学模型.基于该模型,本文仿真计算了悬臂曲梁的纯弯曲问题和带有刚柔耦合效应的旋转柔性曲梁动力学响应问题,以此分别讨论了所提出曲梁单元的收敛性和动力学模型的正确性.进一步应用D'Alembert原理,将旋转曲梁等效为带离心力的无旋转曲梁,通过线性摄动处理得到系统的特征方程,以此分别研究了旋转角速度、初始曲率和集中质量对曲梁动力学特性的影响.最后重点分析了旋转曲梁的频率转向和振型切换问题,并阐述了两者之间的相互关系.研究结果表明:随着旋转角速度的增大,曲梁的频率特性与直梁的频率特性相近,以及曲梁拉伸变形占主导的模态振型会提前.      

Natural frequencies of composite beams with a variable fiber volume fraction including rotary inertia and shear deformation

The present study is concerned with the vibration analysis of symmetric composite beams with a variable fiber volume fraction through thickness. First-order shear deformation and rotary inertia have been included in the analysis. The solution procedure is applicable to arbitrary boundary conditions. Continuous gradation of the fiber volume fraction is modeled in the form of an m-th power polynomial of the coordinate axis in the thickness direction of the beam. By varying the fiber volume fraction within the symmetric composite beam to create a functionally graded material （FGM）, certain vibration characteristics are affected. Results are presented to demonstrate the effects of shear deformation, fiber volume fraction, and boundary conditions on the natural frequencies and mode shapes of composite beams.     

平弯圆弧连续梁的矩阵位移法

本从力法出发建立了平弯圆弧梁单元的刚度矩阵,通过矩阵位移法计算平弯圆弧连续梁,方法简便、易于编程,便于工程设计人员使用。     

高效的三维曲梁单元

三维井眼中延伸数千米的三维细长圆截面钢钻柱应力分析问题是一个复杂的力学问题,通常使用有限元数值分析方法对其进行受力分析。而在进行有限元分析时,现有的圆弧曲粱单元和空间直粱单元在几何上都不能很好地模拟三维曲线形状的钻柱。为了确保计算精度．其单元划分势必不能过大,结果是计算时间长,收敛性差。为了解决这一问题,显然必须构建一种新的较有效的曲梁单元。基于自然坐标系,依据圆截面空间曲粱单元节点有6个自由度——3个线位移和3个角位移,利用包含全部刚体位移模式和常应变的形函数,忽略剪切变形,假设变形后的梁轴线的弯曲曲率改变为线性变化,建立起了保证收敛性的具有12个自由度的有初始曲率和挠率的圆截面空间曲梁的有限元模型。为了证明给出的有限元模型的高效性,分析了几个静态问题,并与现有文献中的解析解或数值结果进行了比较。基于所给出的结果,可望该有限元模型可以作为分析三维空间曲粱结构的有效工具。     

建立基于力的变截面梁单元质量矩阵的新方法

基于位移的有限梁单元中三次Hermite插值函数不能有效地描述变截面梁单元内部位移变化,只能通过加密网格增加单元数解决,会造成计算量增大。基于力的有限梁单元由于使用的力插值函数不受截面形状变化的影响,在处理变截面梁时有很大优势,可以得到精确的位移插值函数,利用较少的单元可以达到很高的精度,解决了基于位移的有限梁单元在处理变截面梁时的不足。本文得到了考虑剪切变形的位移插值函数和考虑转动惯量的一致质量矩阵。利用算例验证了本文理论的正确性和高效性。     

Large amplitude free flexural vibration of rings using finite element approach

Here, the large amplitude free flexural vibrations of isotropic/laminated orthotropic rings are investigated, using a shear flexible curved beam element based on field consistency principle. A laminated refined beam theory is introduced for developing the element, which satisfies the interface transverse shear stress and displacement continuity, and has a vanishing shear stress on the inner and outer surfaces of the beam. The formulation includes in-plane and rotary inertia effects, and the non-linearity due to the finite deformation of the ring. The governing equations obtained using Lagrange's equations of motion are solved through the direct integration technique. Amplitude-frequency relationships evaluated from the dynamic response history are examined. Detailed numerical results are presented considering various parameters such as radius-to-thickness ratio, circumferential wave number and ovality for isotropic and laminated orthotropic rings. The nature and degree of the participation of various modes in non-linear asymmetric vibration of oval ring brought out through the present study are useful for accurate modelling of the closed non-circular structures.     

Modified shooting approach to the non-linear periodic forced response of isotropic/composite curved beams

Large amplitude periodic forced vibration of curved beams under periodic excitation is investigated using a three-noded beam element. The element is based on the higher-order shear deformation theory satisfying interlayer continuity of displacements and transverse shear stress, and top-bottom conditions on the latter. The periodic responses are obtained using shooting technique coupled with Newmark time marching and arc length continuation algorithm developed. The second order governing differential equations of motion are solved without transforming to the first order differential equations thereby resulting in a computationally more efficient algorithm. The effects of excitation amplitude, support conditions and beam curvature on the frequency versus response amplitude relation are highlighted. The typical frequency response curves for isotropic and cross-ply laminated curved beams are presented. Phenomenon of strong modal interactions is observed.     

基于能量平衡原理的有限单元耦合方法

结构非线性数值计算分析应真实反映局部损伤破坏细节,以作为损伤演化全过程分析的依据。对同类构件,有限单元耦合方法可以解决破坏细节与整体模拟的空间尺度差异问题。基于能量平衡原理,建立了梁与实体单元、梁与壳单元以及壳与实体单元的耦合方程,适用于结构的损伤数值计算。对某RC框架结构原位推覆试验的损伤数值分析表明,有限单元耦合模型能正确反映整体结构的承载力和变形性能,并且能准确反映局部损伤破坏细节。     

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。