Analysis of Thin Beams and Cables Using the Absolute Nodal Co-ordinate Formulation

The purpose of this paper is to present formulations for beam elements based on the absolute nodal co-ordinate formulation that can be effectively and efficiently used in the case of thin structural applications. The numerically stiff behaviour resulting from shear terms in existing absolute nodal co-ordinate formulation beam elements that employ the continuum mechanics approach to formulate the elastic forces and the resulting locking phenomenon make these elements less attractive for slender stiff structures. In this investigation, additional shape functions are introduced for an existing spatial absolute nodal co-ordinate formulation beam element in order to obtain higher accuracy when the continuum mechanics approach is used to formulate the elastic forces. For thin structures where bending stiffness can be important in some applications, a lower order cable element is introduced and the performance of this cable element is evaluated by comparing it with existing formulations using several examples. Cables that experience low tension or catenary systems where bending stiffness has an effect on the wave propagation are examples in which the low order cable element can be used. The cable element, which does not have torsional stiffness, can be effectively used in many problems such as in the formulation of the sliding joints in applications such as the spatial pantograph/catenary systems. The numerical study presented in this paper shows that the use of existing implicit time integration methods enables the simulation of multibody systems with a moderate number of thin and stiff finite elements in reasonable CPU time.     

Bending analysis of functionally graded CNT reinforced doubly curved singly ruled truncated rhombic cone

The bending analysis of functionally graded carbon nanotube (CNT) reinforced doubly curved singly ruled truncated rhombic cone is investigated. In this study, a simple C⁰ isoparametric finite element formulation based on third order shear deformation theory is presented. To characterize the membrane-flexure behavior observed in a CNT reinforced truncated rhombic cone, a displacement field involving higher-order terms in in-plane fields is considered. The proposed kinematics field incorporates for transverse shear deformation and nonlinear variation of the in-plane displacement field through the thickness to predict the overall response of the CNT reinforced truncated rhombic cone in an accurate sense. The material properties of the CNT reinforced truncated rhombic cone are estimated according to the rule of mixture. The present model eliminates the need of shear correction factor and imposed zero-transverse shear strain at upper and lower surface of the truncated rhombic cone. The new feature in present model is simultaneous inclusion of twist curvature in strain field as well as curvature in displacement field that makes it suitable for moderately thick and deep truncated rhombic cone. The proposed new mathematical model is implemented in finite element code written in FORTRAN. The proposed model has been validated with analytical, experimental, and finite element results from the literature. This is first attempt to study bending response of CNT reinforced doubly curved singly ruled truncated rhombic cone. The effect of CNT distribution, boundary condition, loading pattern, and other geometric parameters are also examined.     

Nonlinear analysis of members curved in space with warping and Wagner effects

The centroidal axis of a member that is curved in space is generally a space curve. The curvature of the space curve is not necessarily in the direction of either of the principal axes of the cross-section, but can be resolved into components in the directions of both of these principal axes. Hence, a member curved in space is primarily subjected to combined compressive, biaxial bending and torsional actions under vertical (or gravity) loading. In addition, warping actions in particular may occur in curved members with an open thin-walled cross-section, and as the deformations increase, significant interactions of the compressive, biaxial bending and torsional actions occur and profoundly nonlinear deformations are developed in the nonlinear range of structural response. This makes the nonlinear behaviour of a member curved in space very complicated, making it difficult to obtain a consistent differential equation of equilibrium for the nonlinear analysis of members curved in space. In addition, because torsion is one of the primary actions in these members, when the torsional deformations become large, the Wagner effects including both Wagner moment and the conjugate Wagner strain terms are increasingly significant and need to be included in the nonlinear analysis. This paper takes advantage of the merits of so-called “geometrically exact beam theory” and the weak form formulation of the differential equations of equilibrium in beam theory, and it develops consistent differential equations of equilibrium for the nonlinear elastic analysis of members curved in space with warping and Wagner effects. The application of the nonlinear differential equations of equilibrium to various problems is illustrated.     

A coupled refined high-order global-local theory and finite element model for static electromechanical response of smart multilayered/sandwich beams

In the present study, a coupled refined high-order global-local theory is developed for predicting fully coupled behavior of smart multilayered/sandwich beams under electromechanical conditions. The proposed theory considers effects of transverse normal stress and transverse flexibility which is important for beams including soft cores or beams with drastic material properties changes through depth. Effects of induced transverse normal strains through the piezoelectric layers are also included in this study. In the presence of non-zero in-plane electric field component, all the kinematic and stress continuity conditions are satisfied at layer interfaces. In addition, for the first time, conditions of non-zero shear and normal tractions are satisfied even while the bottom or the top layer of the beam is piezoelectric. A combination of polynomial and exponential expressions with a layerwise term containing first order differentiation of electrical unknowns is used to introduce the in-plane displacement field. Also, the transverse displacement field is formulated utilizing a combination of continuous piecewise fourth-order polynomial with a layerwise representation of electrical unknowns. Finally, a quadratic electric potential is used across the thickness of each piezoelectric layer. It is worthy to note that in the proposed shear locking-free finite element formulation, the number of mechanical unknowns is independent of the number of layers. Excellent correlation has been found between the results obtained from the proposed formulation for thin and thick piezoelectric beams with those resulted from the three-dimensional theory of piezoelasticity. Moreover, the proposed finite element model is computationally economic.     

任意斜裂纹转子的耦合振动研究

对包含不同类型裂纹(横裂纹、横-斜裂纹以及任意斜裂纹)的转子的耦合振动进行研究,以揭示裂纹转子在不同方向上刚度参数的变化规律及其交叉耦合机理,特别是由此引发的振动特征. 对于包含不同类型裂纹的转子轴段,采用六自由度Timoshenko梁单元模型对其进行单元建模,并基于应变能理论推导计算柔度参数和刚度矩阵. 在此基础上, 采用纽马克-$\beta$数值算法求解裂纹转子的运动方程,获得裂纹转子在单故障或多故障激励(不平衡激励、扭转激励或不平衡激励加扭转激励)作用下的耦合振动响应,进而分析耦合振动谱特征. 与横裂纹和横-斜裂纹相比,任意斜裂纹使转子刚度矩阵的交叉耦合效应更显著,导致转子发生更强烈的弯-扭耦合甚至是纵-弯-扭耦合振动.无论是在不平衡激励还是扭转激励作用下, 弯曲振动与扭转振动幅度都更大. 而且,包含不同类型裂纹的转子的耦合振动特征频率,例如旋转基频与二倍频、扭转激励频率及其边带成分的幅值,对裂纹面方向角具有不同的敏感性. 所得的这些研究结果,可以为转子裂纹的特征参数辨识与诊断提供理论依据.      

张拉结构非线性分析两节点曲线单元有限元法

提出适合张拉结构几何非线性分析的两节点曲线单元有限元方法．假定索元的初始形状呈二次抛物线,根据单索的平衡条件、几何和物理关系建立了索元的位移函数;由拉格朗日应变的定义建立了可以考虑任意次高阶位移影响的索元轴向应变的精确表达式,并基于拉格明日描述方法和虚功原理得到了索元的非线性平衡方程与切线刚度矩阵．采用荷载增量法与Newton－Raphson法相结合的混合法进行了实例计算,结果表明：本文方法的精度明显优于两节点直线索单元,适合于大跨度索阿、索穹顶等张拉结构的几何非线性分析．     

A NEW SPATIAL THIN-WALLED BEAM ELEMENT INCLUDING TRANSVERSE AND TORSIONAL SHEAR DEFORMATION

基于Timoshenko梁及Benscoter薄壁杆件理论,建立了考虑剪切变形、弯扭耦合以及翘曲剪应力影响的空间任意开闭口薄壁截面梁单元. 通过引入单元内部结点,对弯曲转角和翘曲角采用三节点Lagrange独立插值的方法,考虑了剪切变形和翘曲剪应力的影响并避免了横向剪切锁死问题;借助载荷作用下薄壁梁的截面运动分析,在位移和应变方程中考虑了弯扭耦合的影响. 通过数值算例将该单元的计算结果与理论解以及商用有限元软件和其他文献中的数值解进行对比和验证,结果对比表明该薄壁梁单元具有良好的精度和收敛性.     

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.     

Corotational nonlinear analyses of laminated shell structures using a 4-node quadrilateral flat shell element with drilling stiffness

A new 4-node quadrilateral flat shell element is developed for geometrically nonlinear analyses of thin and moderately thick laminated shell structures. The fiat shell element is constructed by combining a quadrilateral area co- ordinate method （QAC） based membrane element AGQ6- II, and a Timoshenko beam function （TBF） method based shear deformable plate bending element ARS-Q12. In order to model folded plates and connect with beam elements, the drilling stiffness is added to the element stiffness matrix based on the mixed variational principle. The transverse shear rigidity matrix, based on the first-order shear deformation theory （FSDT）, for the laminated composite plate is evaluated using the transverse equilibrium conditions, while the shear correction factors are not needed. The conventional TBF methods are also modified to efficiently calculate the element stiffness for laminate. The new shell element is extended to large deflection and post-buckling analyses of isotropic and laminated composite shells based on the element independent corotational formulation. Numerical re- sults show that the present shell element has an excellent numerical performance for the test examples, and is applicable to stiffened plates.     

双轴对称截面薄壁圆弧曲梁的弹性稳定平衡方程

基于薄壁构件分析的基本假定,采用双轴对称截面薄壁圆弧曲梁的精确翘曲位移表达式,导出了曲梁考虑几何非线性情况下的总势能,根据欧拉公式得到了曲梁的稳定平衡方程。推导中采用横截面线性和非线性总应变为零的假定,从而无需考虑横向应力的影响,对应变高阶项采用合理的简化处理,使理论推导过程简单明了。在理论推导的基础上分析了简支拱在均布径向荷载和两端等弯矩荷载作用下的平面内和平面外屈曲问题,并与其他研究者的结果进行了比较,追溯了各理论结果存在差别的根源,论证了本文理论推导过程的合理性。使用通用有限元软件ANSYS进行了模拟,与本文的分析结果一致,证明了所得公式的正确性。通过一些无碍结果的近似使所得公式形式简洁,便于在工程中应用。     

A new finite element of spatial thin-walled beams

Based on the theories of Timoshenko＇s beams and Vlasov＇s thin-walled members, a new spatial thin-walled beam element with an interior node is developed. By independently interpolating bending angles and warp, factors such as transverse shear deformation, torsional shear deformation and their Coupling, coupling of flexure and torsion, and second shear stress are considered. According to the generalized variational theory of Hellinger-Reissner, the element stiffness matrix is derived. Examples show that the developed model is accurate and can be applied in the finite element analysis of thinwalled structures.     

Nonlinear flexural–torsional dynamic analysis of beams of variable doubly symmetric cross section—application to wind turbine towers

In this paper, a boundary element solution is developed for the nonlinear flexural–torsional dynamic analysis of beams of arbitrary doubly symmetric variable cross section, undergoing moderate large displacements, and twisting rotations under general boundary conditions, taking into account the effect of rotary and warping inertia. The beam is subjected to the combined action of arbitrarily distributed or concentrated transverse loading in both directions and to twisting and/or axial loading. Four boundary-value problems are formulated with respect to the transverse displacements, to the axial displacement, and to the angle of twist and solved using the Analog Equation Method, a Boundary Element Method (BEM) based technique. Application of the boundary element technique yields a system of nonlinear coupled Differential–Algebraic Equations (DAE) of motion, which is solved iteratively using the Petzold–Gear Backward Differentiation Formula (BDF), a linear multistep method for differential equations coupled with algebraic equations. Numerical examples of great practical interest including wind turbine towers are worked out, while the influence of the nonlinear effects to the response of beams of variable cross section is investigated.     

Lateral buckling of beams with shear deformations – A hyperelastic formulation

The equilibrium and buckling equations are derived for the lateral buckling of a prismatic straight beam. A consistent finite strain constitutive law is used, which is based on a hyperelastic model for an isotropic material. The kinematics of the cross-sectional deformations are based on a Timoshenko type beam displacement of the cross-sectional plane using Euler angles and two shear finite rotations coupled with warping taken normal to the displaced plane. Also derived are the second order approximations to the displacements, curvatures, twist and internal actions. The constitutive relationships for the internal actions reveal new coupling terms between the bending moments, torsion and bimoment, which are functions of the cross-sectional warping and shear deformations. New Wagner type nonlinear torsion terms are derived which are functions of the warping of the cross-sectional plane, and are coupled to the twisting and shear deformations of the cross-section. Solutions are determined for the lateral buckling of a prismatic monosymmetric beam under pure bending and the flexural–torsional buckling under axial compression. For the flexural–torsional buckling problem it is found that the Euler type column buckling formula is consistent with Haringx’s column buckling formula while the torsional buckling formula is different to conventional equations. The second variation of the total potential is also derived. The effects of shear deformations are explored by examining the non-dimensional lateral buckling equation for a simply supported beam.     

Modified Mindlin plate theory and shear locking-free finite element formulation

The basic equations of the Mindlin theory are specified as starting point for its modification in which total deflection and rotations are split into pure bending deflection and shear deflection with bending angles of rotation, and in-plane shear angles. The equilibrium equations of the former displacement field are split into one partial differential equation for flexural vibrations. In the latter case two differential equations for in-plane shear vibrations are obtained, which are similar to the well-known membrane equations. Rectangular shear locking-free finite element for flexural vibrations is developed. For in-plane shear vibrations ordinary membrane finite elements can be used. Application of the modified Mindlin theory is illustrated in a case of simply supported square plate. Problems are solved analytically and by FEM and the obtained results are compared with the relevant ones available in the literature.     

带旋转自由度C^0类任意四边形板（壳）单元

基于Ｒｅｉｓｓｎｅｒ－Ｍｉｎｄｉｌｉｎ板弯曲理论和Ｖｏｎ－Ｋａｒｍａｎ大挠度理论,采用单元域内和边界位移插值一致性的概念,将四节点等参弯曲单元与Ａｌｌｍａｎ膜变形二次插值模式相结合,对层合板壳的大挠度分析提供了一种实用的带旋转自由度的四节点Ｃ＾０类板单元。大量算例表明：该单元对板壳结构的线性强度、稳定性和后屈曲分析都表现出良好的收敛性和足够的工程精度。     

ANCF索梁单元应变耦合问题与模型解耦

索粱结构在土木工程、航空航天等领域有着广泛的应用.在各类索梁动力学建模方法中,由于绝对节点坐标方法(absolute nodal coordinate formulation,ANCF)能够描述柔性体的大变形和大转动问题,因此非常适合大变形索梁结构的动力学建模.对绝对节点坐标索梁单元的应变进行分析可知,弯曲变形会引起单元内部轴向应变的不均匀分布,即单元轴向应变与弯曲应变相互耦合.这种应变耦合效应使单元产生伪应变能,导致单元刚度增大,造成单元失真.分析不同弯曲角下的单元应变及应变能可知,弯曲变形越大,单元失真越严重.通过构造等效一维杆单元重新描述轴向应变,实现了轴向应变与弯曲应变解耦.在此基础上推导广义弹性力,得到了绝对节点坐标索梁单元的应变解耦模型.对解耦前后的两种梁模型进行静力学和动力学仿真,结果表明;解耦模型消除了单元伪应变,相比原模型表现出更好的收敛性和曲率连续性,在相同单元数目下具有更高的精度.同时由于解耦模型降低了单元刚度,因此相比原模型,速度曲线中不再有高频振动.     

Nonlinear dynamics of three-dimensional belt drives using the finite-element method

In this paper, new nonlinear dynamic formulations for belt drives based on the three-dimensional absolute nodal coordinate formulation are developed. Two large deformation three-dimensional finite elements are used to develop two different belt-drive models that have different numbers of degrees of freedom and different modes of deformation. Both three-dimensional finite elements are based on a nonlinear elasticity theory that accounts for geometric nonlinearities due to large deformation and rotations. The first element is a thin-plate element that is based on the Kirchhoff plate assumptions and captures both membrane and bending stiffness effects. The other three-dimensional element used in this investigation is a cable element obtained from a more general three-dimensional beam element by eliminating degrees of freedom which are not significant in some cable and belt applications. Both finite elements used in this investigation allow for systematic inclusion or exclusion of the bending stiffness, thereby enabling systematic examination of the effect of bending on the nonlinear dynamics of belt drives. The finite-element formulations developed in this paper are implemented in a general purpose three-dimensional flexible multibody algorithm that allows for developing more detailed models of mechanical systems that include belt drives subject to general loading conditions, nonlinear algebraic constraints, and arbitrary large displacements. The use of the formulations developed in this investigation is demonstrated using two-roller belt-drive system. The results obtained using the two finite-element formulations are compared and the convergence of the two finite-element solutions is examined.     

Nonlinear FE-based investigation of flexural damping of slacking wire cables

The flexural damping of wire cable due to the flexural hysteresis influences the dynamic behavior of slacking wire cables significantly. However, the details of the local model, accounting for the flexural hysteresis between the wire strands, are quite challenging to include in large-scale engineering applications. This paper addresses these difficulties by modeling the flexural damping of slacking wire cables using homogenized Rayleigh damping. By using the nonlinear finite element method and high-speed imaging technique, three aspects of the work were examined. First, the mechanical properties of the slacking cable were identified experimentally. Second, a sample cable was fixed at one end and allowed to vibrate freely at the other end. The shapes of the vibrating cable were captured by a high-speed digital camera and processed by photogrammetry. The cable demonstrated a high flexural damping at zero tension and its damping was measured to be as high as 37.7% of the critical damping. Third, the cable was modeled and analyzed using our newly developed nonlinear curved beam element with the Rayleigh damping. The finite element predictions of the cable motion agree well with the experimental measurement. These predictions demonstrate that the new element is capable of describing the dynamic response of the cable and that the Rayleigh damping is sufficient to model the flexural damping of slacking wire cables.     

张拉结构非线性分析的五节点等参单元

本文针对张拉结构的特点，提出了一种五节点等参数单元有限元模型，采用四次多项式作为位移插值函数及单元初始形状函数，并假定索是理想柔性的且满足虎克定律，基于修正的Ｌａｇｒａｎｇｉａｎ坐标描述法，建立了非线性有限元基本方程和切线刚度矩阵，利用Ｎｅｗｔｏｎ－Ｒａｐｈｓｏｎ法进行了实例计算。结果表明：本文方法精度极高，可供大跨度索网，索穹顶，拉线塔等张拉结构分析，设计时采用。     

A quadrilateral element based on refined global-local higher-order theory for coupling bending and extension thermo-elastic multilayered plates

In this paper a refined higher-order global-local theory is presented to analyze the laminated plates coupled bending and extension under thermo-mechanical loading. The in-plane displacement fields are composed of a third-order polynomial of global coordinate z in the thickness direction and 1,2–3 order power series of local coordinate ζ_k in the thickness direction of each layer, which is identical to the 1,2–3 global-local higher-order theory by Li and Liu [Li, X.Y., Liu, D., 1997. Generalized laminate theories based on double superposition hypothesis. Int. J. Numer. Methods Eng. 40, 1197–1212] Moreover, a second-order polynomial of global coordinate z in the thickness direction is chosen as transverse displacement field. The transverse shear stresses can satisfy continuity at interfaces, and the number of unknowns does not depend on the layer numbers of the laminate.Based on this theory, a quadrilateral laminated plate element satisfying the requirement of C¹ continuity is presented. By solving both bending and thermal expansion problems of laminates, it can be found that the present refined theory is very accurate and obviously superior to the existing 1,2–3 global-local higher-order theory. The most attractive feature of this theory is that the transverse shear stresses can be accurately predicted from direct use of constitutive equations without any post-processing method. It is also shown that the present quadrilateral element possesses higher accuracy.