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.

Conversion between solid and beam element solutions of finite element method based on meta-modeling theory:development and application to a ramp tunnel structure

In this study, a new method for conversion of solid finite element solution to beam finite element solution is developed based on the meta-modeling theory which constructs a model consistent with continuum mechanics. The proposed method is rigorous and efficient compared to a typical conversion method which merely computes surface integration of solid element nodal stresses to obtain cross-sectional forces. The meta-modeling theory ensures the rigorousness of proposed method by defining a proper distance between beam element and solid element solutions in a function space of continuum mechanics. Results of numerical verification test that is conducted with a simple cantilever beam are used to find the proper distance function for this conversion. Time history analysis of the main tunnel structure of a real ramp tunnel is considered as a numerical example for the proposed conversion method. It is shown that cross-sectional forces are readily computed for solid element solution of the main tunnel structure when it is converted to a beam element solution using the proposed method. Further, envelopes of resultant forces which are of primary importance for the purpose of design, are developed for a given ground motion at the end.

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 finite-element 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.

COMPUTATION OF SUPER-CONVERGENT NODAL STRESSES OF TIMOSHENKO BEAM ELEMENTS BY EEP METHOD

The newly proposed element energy projection(EEP) method has been applied to the computation of super_convergent nodal stresses of Timoshenko beam elements.General formulas based on element projection theorem were derived and illustrative numerical examples using two typical elements were given.Both the analysis and examples show that EEP method also works very well for the problems with vector function solutions.The EEP method gives super_convergent nodal stresses,which are well comparable to the nodal displacements in terms of both convergence rate and error magnitude.And in addition,it can overcome the “shear locking” difficulty for stresses even when the displacements are badly affected.This research paves the way for application of the EEP method to general one_dimensional systems of ordinary differential equations.

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 threedimensional 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.

STOCHASTIC BOUNDARY ELEMENT METHOD FOR RELIABILITY ANALYSIS OF PLATE AND BEAMS COMPOSITE STRUCTURES

In this paper, the reliability of orthotropic plate and beams composite structures, which is under the actions of the stochastic loading and stochastic boundary conditions, have been analyzed by stochastic boundary element method. First, the boundary integral equation of orthotropic plate and beams composite structures is given in this paper, and then based on the stochastic boundary element method, the method for reliability analysis of stochastic structures is establishes and formulas for computation of reliability index of orthotropic plate and beams composite structures are obtained. The computed examples show the efficient of the method used in this paper.

A new 3-D element formulation on displacement of steel-concrete composite box beam

Slip of a composite box beam may reduce its stiffness, enlarge its deformation and affect its performance. In this work, the governing differential equations and boundary conditions of composite box beams were established. Analytic solutions of combined differential equations were also established. Partial degree of freedom was adopted to establish a new FEA element of three-dimensional beam, taking into account the slip effect. Slip and its first-order derivative were introduced into the nodes of composite box beams as generalized degree of freedom. Stiffness matrix and load array of beam elements were established. A three-dimensional nonlinear calculation program was worked out. The results show that the element is reliable and easy to divide and is suitable for special nonlinear analysis of large-span composite box beams.

Nonlinear finite element analysis of steel-concrete composite beams

Proposes a simplified finite element model for steel-concrete composite beams. The effects of slip can be taken into account by creating a special matrix of shear connector stiffness and using the iteration method. Meanwhile, the effect of material non-linearity of steel and concrete on rigidity and strength of composite beams is considered. With the age-adjusted effective modulus method, the analysis for the whole process of shrinkage and creep under long-term load can be performed. The ultimate load, deflection, stress and slip of continuous composite beams under short-term and long-term load are computed using the proposed finite element model. The numerical results are compared with the experimental results and existing values based on other numerical methods, and are found to be in good agreement.

3D Finite Element Numerical Simulation of Residual Stresses on Electron Beam Welded BT20 Plates

A three-dimensional finite-element model (FEM) used for calculating electron beam (EB) welding temperature and stresses fields of thin plates of BT20 titanium has been developed in which the nonlinear thermophysical and thermo-mechanical properties of the material has been considered. The welding temperature field, the distributions of residual stresses in as-welded (AW) and electron beam local post-weld heat treatment (EBLPWHT) conditions have been successfully simulated. The results show that: (1) In the weld center, the maximum magnitude of residual tensile stresses of BT20 thin plates of Ti alloy is equal to 60%- 70% of its yield strength σs. (2) The residual tensile stresses in weld center can be even decreased after EBLPWHT and the longitudinal tensile stresses are decreased about 50% compared to joints in AW conditions. (3) The numerical calculating results of residual stresses by using FEM are basically in agreement with the experimental results. Combined with numerical calculating results, the effects of electron beam welding and EBLPWHT on the distribution of welding residual stresses in thin plates of BT20 have been analyzed in detail.

THE ANALYSIS OF THIN WALLED COMPOSITE LAMINATED HELICOPTER ROTOR WITH HIERARCHICAL WARPING FUNCTIONS AND FINITE ELEMENT METHOD

In the present paper, a series of hierarchical warping functions is developed to analyze the static and dynamic problems of thin walled composite laminated helicopter rotors composed of several layers with single closed cell. This method is the development and extension of the traditional constrained warping theory of thin walled metallic beams, which had been proved very successful since 1940s. The warping distribution along the perimeter of each layer is expanded into a series of successively corrective warping functions with the traditional warping function caused by free torsion or free beading as the first term, and is assumed to be piecewise linear along the thickness direction of layers. The governing equations are derived based upon the variational principle of minimum potential energy for static analysis and Rayleigh Quotient for free vibration analysis. Then the hierarchical finite element method. is introduced to form a,. numerical algorithm. Both static and natural vibration problems of sample box beams axe analyzed with the present method to show the main mechanical behavior of the thin walled composite laminated helicopter rotor.

Vibration Analysis of Beams by Spline Finite Element

In this paper,the spline finite element method is developed to investigate free vibration problems of beams.The cubic B-spline functions are used to construct the displacement field.The assembly of elements and the introduction of boundary conditions follow the standard finite element procedure.The results under various boundary conditions are compared with those obtained by the exact method and the finite difference method.It shows that the results are in excellent agreement with the analytical results and much more accurate than the results obtained by the finite difference method,especially for higher order modes.

A New Non-uniform Beam Element and Its Application to Buckling Analysis for Framed Structures

The non-uniform beam components are commonly used in engineering,while the method to analyze such component is not too satisfactory yet. A new non-uniform beam element with high precision was developed based on the non-linear analysis and the static condensation. Based on the interpolation theory, the displacement fields of the three-node non-uniform Euler-Bernoulli beam element were constructed at first: the quintic Hermite interpolation polynomial was used for the lateral displacement field and the quadratic Lagrange interpolation polynomial for the axial displacement field. Then,based on the basic assumptions of non-uniform Euler-Bernoulli beam whose section properties were continuously varying along its centroidal axis, the linear and geometric stiffness matrices of the three-node non-uniform beam element were derived according to the nonlinear finite element theory. Finally,the degrees of freedom ( DOFs) of the middle node of the element were eliminated using the static condensation method, and a new two-node non-uniform beam element including axial-force effect was obtained. The results indicate that each bar needs to be meshed with only one element could get a fairly accurate solution when it is applied to the stability analyses.

Pseudo-beam method for compressive buckling characteristics analysis of space inflatable load-carrying structures

This paper extends Le van＇s work to the case of nonlinear problem and the complicated configuration. The wrinkling stress distribution and the pressure effects are also included in our analysis. Pseudo-beam method is presented based on the inflatable beam theory to model the inflatable structures as a set of inflatable beam elements with a prestressed state. In this method, the discretized nonlinear equations are given based upon the virtual work principle with a 3-node Timoshenko＇s beam model. Finite element simulation is performed by using a 3-node BEAM189 element incorporating ANSYS nonlinear program. The pressure effect is equivalent included in our method by modifying beam element cross-section parameters related to pressure. A benchmark example, the bending case of an inflatable cantilever beam, is performed to verify the accuracy of our proposed method. The comparisons reveal that the numerical results obtained with our method are close to open published analytical and membrane finite element results. The method is then used to evaluate the whole buckling and the loadcarrying characteristics of an inflatable support frame subjected to a compression force. The wrinkling stress and region characteristics are also shown in the end. This method gives better convergence characteristics, and requires much less computation time. It is very effective to deal with the whole load-carrying ability analytical problems for large scale inflatable structures with complex configuration.

双参数层状地基中水平动荷载和扭转振动组合作用下受荷桩受力变形分析

为了更准确地分析桩顶在承受水平动荷载H(t)和扭转振动T(t)组合荷载作用下的受力变形特性,基于Pasternak模型,考虑多向荷载间的相互作用,建立层状地基中H(t)-T(t)受荷桩简化分析模型。利用有限杆单元法推导出桩身内力位移数值解,并与已有相关理论解、模型试验和有限元模拟结果进行对比验证。参数分析表明:(1)与Winkler模型计算结果相比,考虑土体剪切效应后,桩顶水平位移和桩身最大弯矩分别减小了11.8%和10.5%。(2)增大外荷载无量纲频率可减小桩身水平位移和弯矩,但同时会减小H(t)-T(t)耦合刚度。(3)在层状地基中,表层土对桩身内力位移的影响最大,且存在临界影响深度,表层硬土的临界影响厚度是表层软土临界影响厚度的3.5~6.5倍。(4)采用杆单元模型可减少单元划分数量和计算时间,有效提高计算效率。

FE Dynamic Analysis Using Moving Support Element on Multi-Span Beams Subjected to Support Motions

In the present study, finite element dynamic analysis or time history analysis of two-span beams subjected to asynchronous multi-support motions is carried out by using the moving support finite element. The elemental equation of the element is based on total displacements and is derived under the concept of the quasi-static displacement decomposition. The use of moving support element shows that the element is very simple and convenient to represent continuous beam moving, deforming and vibrating simultaneously due to support motions. The comparison between the numerical results and analytical solutions indicates that the FE result agrees with the analytical solution.

涉水高墩现浇箱梁支撑体系的承载能力分析和方案优化

混凝土现浇梁支架虽然属于临时结构,但仍具有承载功能,而且随着桥梁工程的发展,高大支架的受力也越来越复杂,传统的结构力学手算方法已无法全面反映和权衡特殊体型、特殊部位和特殊工况的受力情况。为保证工程的顺利开展,需采用数值方法进行整体承载力计算和方案设计。该文采用Midas/Civil有限元软件,从强度、刚度、稳定性和经济性(用钢量)等方面综合评价满堂支撑、梁式支撑体系的安全性能和通用性。结果表明:(1)满堂支撑虽然施工简单,用钢量小,下部支撑杆系较多,受力均匀,变形小,最大挠度仅为2.82 mm,但是强度和稳定性不及梁式支撑,对场地要求还高。当应用于高大桥梁时,需整体进行屈曲分析,谨防结构失稳破坏;(2)梁式支撑虽然不会发生强度和失稳性破坏,对场地和地基基本没有要求。但是其用钢量高、施工难度也大,并且因跨中缺乏竖向支撑,变形有所增加。因此,在应用于“预拱度”要求较高的复杂桥梁工程中时,需进行找形分析;(3)在力学指标均达标的前提下,可以适当调整贝雷梁横向净距或下弦杆加强等措施,以获取经济性。以该文案例工程为例,贝雷梁净距由80 cm调整到100cm,可以节省约10%的用钢量。

基于分层法的石墨烯增强功能梯度Timoshenko梁动力特性有限元分析

基于分层法对石墨烯增强功能梯度Timoshenko梁的动力特性问题进行有限元分析。在有限元建模过程中,首先将Timoshenko梁沿厚度方向分成若干层,然后利用改进的Halpin-Tsai细观力学模型计算各层的弹性模量,相应的泊松比和密度则根据混合率法则进行计算,最后对各层均采用4节点四边形板单元离散。通过数值算例分析分层数和单元尺寸比例的合理性,探究石墨烯片的分布形式、质量含量、几何形状和尺寸等因素对Timoshenko梁动力特性的影响。结果表明,添加少量的石墨烯能显著提高Timoshenko梁自由振动的频率,尤其在梁的上下部位分布较多正方形石墨烯片时效果最佳。随着石墨烯片长厚比的增加,Timoshenko梁的自由振动基频不断增大。当石墨烯片长厚比超过1 000之后,梁的基频变化不明显。

轴向移动内压力梁动力学的动网格法

面向轴向移动内压力梁结构动力学特性研究,采用Lagrangian方法建立了轴向移动内压力梁的运动控制方程,基于有限元动网格法对轴向移动梁运动控制方程进行离散。将内压力作为边界条件,采用Newmark-β时间积分方法计算轴向移动梁动力学响应,并开展动力学特性研究。结果表明,建立的有限元动网格模型能够有效地实现轴向移动内压力梁动力学响应计算。轴向移动内压力梁在收缩运动过程中,自由端振动频率将逐渐增加。悬臂长度对梁自由端振动频率具有显著影响。同时,结构振动频率随内压力的增加而增加。本文中建立的有限元动网格模型能够为轴向移动内压力梁动力学研究提供新方法。

模块化防屈曲钢板墙抗震性能

考虑经济性和便利性,提出了模块化防屈曲钢板墙及其常用模数,然后基于已有的防屈曲钢板墙的简化计算模型,推导了在不同墙数量情况下,模块化防屈曲钢板墙充分发挥抗震性能时,周边梁的承载力和刚度需求,并提出了周边梁的设计方法.进而,对模块化防屈曲钢板墙和整体墙进行了对比验证试验及其有限元模拟分析.研究结果表明:模块化防屈曲钢板墙能够达到与整体钢板墙相同的抗震性能,且周边梁未发生明显转动和破坏,说明所提出的周边梁设计方法是合理的;在层间位移角为1/50时,防屈曲钢板墙上下周边梁均未进入塑性,且承载力和初始刚度等的有限元值与试验值误差小于6%,从而进一步验证了其抗震性能.

钢-混凝土组合简支桥面连续结构横向应力分析

针对钢-混组合简支梁桥的桥面连续结构开裂等病害,围绕桥面连接板的横桥向应力问题,采用线弹性理论和板的偏微分方程进行分析,得出了桥面连接板挠度和应力的分布函数,建立非线性有限元模型,并进行实桥荷载试验.通过比较理论解、有限元解和实测试验结果,证实了理论解和有限元的有效性.根据得到的分布函数,发现横桥向和纵桥向上的最大拉应力出现在钢梁端部位置的连接板的上缘.此外,还分析了连接板区域尺寸变化对横桥向应力峰值的影响,包括纵梁端部距支座的长度、纵梁的间距以及连接板区域整体尺寸变化.结果表明:较小的纵梁间距和较长的纵梁端部与支撑之间的距离会导致连接板中的横向拉应力峰值增加,并提高横向拉应力在总应力中的占比,从而导致桥面连接板早于设计荷载开裂.因此对于纵梁间距较小、梁端长度较长的钢-混组合简支梁桥桥面连续结构,仅计算其纵桥向受力性能会导致计算结果偏危险,建议按照本文方法考虑横桥向应力对桥面连接板的影响.