Geometric nonlinear formulation for curved beams with varying curvature

Based on exact Green strain of spatial curved beam, the nonlinear strain-displacement relation for plane curved beam with varying curvature is derived. Instead of using the previous straight beam elements, curved beam elements are used to approximate the curved beam with varying curvature. Based on virtual work principle, rigid-flexible coupling dynamic equations are obtained. Physical experiments were carried out to capture the large overall motion and the strain of curved beam to verify the present rigid-flexible coupling formulation for curved beam based on curved beam element. Numerical results obtained from simulations were compared with those results from the physical experiments. In order to illustrate the effectiveness of the curved beam element methodology, the simulation results of present curved beam elements are compared with those obtained by previous straight beam elements. The dynamic behavior of a slider-crank mechanism with an initially curved elastic connecting rod is investigated. The advantage of employing generalized-α method is pointed out and the special nonlinear dynamic characteristics of the curved beam are concluded.     

Geometrically exact curved beam element using internal force field defined in deformed configuration

This paper presents a study on the development of high-performance finite elements for geometrically nonlinear analysis of frame structures with curved members. Based on the geometrically exact beam theory, a highly efficient and accurate mixed finite element is developed. A new approach is proposed for constructing the independent internal force field by including major terms satisfying equilibrium conditions in the deformed configuration. An element-level equilibrium iteration procedure is employed for the condensation of element internal degrees of freedom during the nonlinear solution. Numerical results are presented to demonstrate the excellent performance of the element developed, and it is shown that even when each structural member is modelled with just one element, accurate solutions can still be achieved.     

Finite element analysis of 2D SMA beam bending

A thermomechanical model of a shape memory alloy beam bending under tip force loading is implemented in finite element codes.The constitutive model is a one dimensional model which is based on free energy and motivated by statistical thermodynamics.The particular focus of this paper is on the aspects of finite element modeling and simulation of the inhomogeneous beam bending problem.This paper extends previous work which is based on the small deformation Euler-Bernoulli beam theory and by treating an SMA beam as consisting of multi-layers in a twodimensional model.The flux terms are involved in the heat transfer equation.The simulations can represent both shape memory effect and super-elastic behavior.Different thermal boundary condition effect and load rate effect can also be captured.     

Geometric nonlinear analyses of functionally graded beams using a tailored Lagrangian formulation

This paper presents a geometric nonlinear analysis formulation for beams of functionally graded cross-sections by means of a Total Lagrangian formulation. The influence of material gradation on the numerical response is investigated in detail. Two examples are given that illustrate the main features of the formulation, in which the behavior of beams of graded cross-sections is compared with homogeneous material beams. A motivation for this work is the potential development of functionally graded risers for the offshore oil exploration industry.     

Geometric nonlinear formulation and discretization method for a rectangular plate undergoing large overall motions

In the present paper, the geometric nonlinear formulation is developed for dynamic stiffening of a rectangular plate undergoing large overall motions. The dynamic equations, which take into account the stiffening terms, are derived based on the virtual power principle. Finite element method is employed for discretization of the plate. The simulation results of a rotating rectangular plate obtained by using such geometric nonlinear formulation are compared with those obtained by the conventional linear method without consideration of the stiffening effects. The application limit of the conventional linear method is clarified according to the frequency error. Furthermore, the accuracy of the assumed mode method is investigated by comparison of the results obtained by using the present finite element method and those obtained by using the assumed mode method.     

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。     

Ultrasonic beam steering using Neumann boundary condition in multiplysics

The traditional one-dimensional ultrasonic beam steering has time delay and is thus a complicated problem. A numerical model of ultrasonic beam steering using Neumann boundary condition in multiplysics is presented in the present paper. This model is based on the discrete wave number method that has been proved theoretically to satisfy the continuous conditions. The propagating angle of novel model is a function of the distance instead of the time domain. The propagating wave fronts at desired angles are simulated with the single line sources for plane wave. The result indicates that any beam angle can be steered by discrete line elements resources without any time delay.     

Approximate and numerical analysis of nonlinear forced vibration of axially moving viscoelastic beams

Steady-state periodical response is investigated for an axially moving viscoelastic beam with hybrid supports via approximate analysis with numerical confirmation. It is assumed that the excitation is spatially uniform and temporally harmonic. The transverse motion of axially moving beams is governed by a nonlinear partial-differential equation and a nonlinear integro-partial-differential equation. The material time derivative is used in the viscoelastic constitutive relation. The method of multiple scales is applied to the governing equations to investigate primary resonances under general boundary conditions. It is demonstrated that the mode uninvolved in the resonance has no effect on the steady-state response. Numerical examples are presented to demonstrate the effects of the boundary constraint stiffness on the amplitude and the stability of the steady-state response. The results derived for two governing equations are qualitatively the same,but quantitatively different. The differential quadrature schemes are developed to verify those results via the method of multiple scales.     

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.     

Analysis of regular and chaotic dynamics of the Euler-Bernoulli beams using finite difference and finite element methods

Chaotic vibrations of flexible non-linear Euler-Bernoulli beams subjected to harmonic load and with various boundary conditions(symmetric and non-symmetric)are studied in this work.Reliability of the obtained results is verified by the finite difference method(FDM)and the finite element method(FEM)with the Bubnov-Galerkin approximation for various boundary conditions and various dynamic regimes(regular and non-regular).The influence of boundary conditions on the Euler-Bernoulli beams dynamics is studied mainly,dynamic behavior vs.control parameters { ωp,q0 } is reported,and scenarios of the system transition into chaos are illustrated.     

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.     

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.     

Detection of crack location using cracked beam element method for structural analysis

The local flexibility introduced by cracks changes the dynamic behavior of the structure and, by examining this change, crack position and magnitude can be identified. In order to model the structure for FEM analysis, a special finite element for a cracked Timoshenko beam is developed. Shape functions for rotational and translational displacements are used to obtain the consistent mass matrix for the cracked beam element. Effect of the crack on the stiffness matrix and consistent mass matrix is investigated. Proposed is a procedure for identifying cracks in structures using modal test data.     

Free vibration of horizontally curved thin-walled beams with rectangular hollow sections considering two compatible displacement fields

A finite element is presented for vibration analyses of horizontally curved thin-walled rectangular hollow beams. Eight cross-section deformation modes are employed to describe the mid-surface contour displacement field with the modal superposition method. Focused on the in-plane moment equilibrium condition and the displacement continuity condition, two compatible displacement fields are constructed to calculate the strain energy and the kinetic energy of the beam, respectively. With the application of Hamilton’s principle the dynamic governing equations are formulated, and then approximated for the finite element implementation. Finally, numerical examples are illustrated to verify the validity of the present theory.     

钢管混凝土拱稳定分析的三维退化层合曲梁单元

为计算钢管混凝土拱的屈曲荷载,本文在文[1]三维退化梁单元的基础上,采用等效数值积分法,构造,出120－20结点三维退化层合曲梁单元,并考虑几何非线性影响,给出用于层合梁或拱线弹性稳定性分析的有限元列式,最后,以绍兴轻纺大桥为工程背景,计算出轻纺大枯钢管混凝土拱面内及面外屈曲的稳定系数。     

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.     

ANALYSIS OF COMPOSITE LAMINATE BEAMS USING COUPLING CROSS-SECTION FINITE ELEMENT METHOD

Beams and plates manufactured from laminates of composite materials have distinct advantages in a significant number of applications. However, the anisotropy arising from these materials adds a significant degree of complexity, and thus time, to the stress and deformation analyses of such components, even using numerical approaches such as finite elements. The analysis of composite laminate beams subjected to uniform extension, bending, and/or twisting loads was performed by a novel implementation of the usual finite element method. Due to the symmetric features of the deformations, only a thin slice of the beam to be analysed needs to be modelled. Conventional three-dimensional solid finite elements were used for the structural discretization. The accurate deformation relationships were formulated and implemented through the coupling of nodal translational degrees of freedom in the numerical analysis. A sample solution for a rectangular composite laminate beam is presented to show the validity and accuracy of the proposed method.     

Dynamics of three-dimensional beams undergoing large overall motion

In the previous linear formulation of flexible multibody system, the neglect of stiffening terms may cause significant error in case of high rotating speed. In this paper, a geometric nonlinear formulation of three-dimensional beams is proposed based on virtual power principle. Frequency results of a rotating spatial beam using the present nonlinear model are compared with those using the linear model without stiffening. An influence ratio, which is related to non-dimensional axial base acceleration and lateral angular velocity, is put forward to clarify the limit of the linear formulation. It is shown that the relative frequency error is closely related to the influence ratio. Finally, simulation of a flexible spatial manipulator is carried out to verify the effectiveness of the criterion.     

一种考虑初始缺陷影响的非线性梁单元

在目前广泛应用的梁单元中,尚缺乏全面考虑以下四种因素影响的粱单元：(1)轴力的影响;(2)剪切交形的影响;(3)初始弯曲的影响;(4)弯曲变形对轴向应变的影响,即弓形效应。事实上,以上四种非线性因素都会对钢框架结构的稳定和极限承载力有影响,需同时考虑。本文将致力于推导同时考虑以上四种因素影响的平面梁单元的平衡微分方程,最后得到精确的粱单元刚度矩阵,并研究以上四种因素对钢框架构件及钢框架结构的影响。