变截面铁木辛柯梁振动特性快速计算方法

提出了一种快速计算变截面铁木辛柯梁横向振动特性的方法.基于铁木辛柯梁理论建立的变截面梁的横向振动方程,其梁的截面参数如有效剪切面积、密度、弯曲刚度、转动惯量等沿梁轴线连续或非连续变化；首先将变截面梁等效为多段均匀阶梯梁；然后基于相邻两段连接处的位移(位移、转角)和力(弯矩、剪力)连续条件,建立相邻两段模态函数间相互关系,并递推出首段段与末段模态函数相互关系,利用边界条件得到相应特征方程,使用Newton-Raphson方法计算其固有频率；最后针对梁常见边界条件,得到计算变截面铁木辛柯梁横向振动固有频率特征     

Spectral determination for a cantilever beam

The eigenvalue specification problem is discussed for a beam with one end clamped and the other end free, which is allowed to have some structural damping, and has a moment force at the free end acting as a control. The spectrum and eigenvectors of the uncontrolled system are first analyzed and are then used to construct the feedback element which solves the problem. It is shown how this feedback element can be numerically approximated, and a spillover result is proved for the approximation. In the case where there is no damping, this method yields a bounded feedback law which exponentially stabilizes the beam     

Identification and elimination of parasitic shear in a laminated composite beam finite element

A displacement-based finite element for the analysis of laminated composite beams is formulated using strain gradient notation. The definition of the beam’s longitudinal displacement possesses only the independent term (axial displacement) and a term which is linear in the thickness coordinate z. Thus, the finite element is first-order shear deformable. As strain gradient notation is physically interpretable, the contents of the coefficients of the polynomial expansions are identified a priori. Thus, the modeling capabilities as well as modeling deficiencies of the element are identified during the formulation procedure. A single parasitic shear term (spurious) is found to be present in the transverse shear strain expression of the element, which is responsible for locking. This parasitic shear term is also found to be the cause of a qualitative error existing in the representation of transverse shear strain along the length of a typical beam model. As the spurious term has been clearly identified, it can easily be removed to correct the element. The effectiveness of the procedure is shown through numerical analyses performed using the element containing the spurious term and then corrected for it. The beam model is validated by comparing numerical solutions with analytical solutions provided by the minimization of the total potential energy for a given laminated composite beam.     

基于欧拉 伯努利梁和铁木辛柯梁理论的功能梯度材料模量测定

为测定功能梯度材料的弹性模量和剪切模量,引入梁理论并将梁沿长度方向离散,建立单元平衡方程后可得到弹性模量和剪切模量分布;假设弹性模量为沿长度方向的线性函数或指数函数,用有限元软件仿真计算功能梯度材料梁单元节点处的挠度和转角,然后用插值法构造变形特征函数,并计算得出弹性模量和剪切模量,且计算值与理论值的误差较小.计算结果还表明,采用铁木辛柯梁理论不仅可以得到弹性模量,还可以计算剪切模量,且弹性模量计算结果比用欧拉-伯努利梁计算结果更接近真实值,但铁木辛柯梁理论中需测定转角,对测定过程的要求会更加严格。     

An exact high order beam element

This work gives the exact stiffness coefficients for an high order isotropic beam element. The terms are found directly from the solutions of the differential equations that describe the deformations of the cross-section according to the high order theory, which include cubic variation of the axial displacements over the cross-section of the beam. The model has six degrees of freedom at the two ends, one transverse displacement and two rotations, and the end forces are a shear force and two end moments. Also given are the equivalent end forces and moments for several cases of loading along the member. The components of the end moments are investigated, and are found for exact results. Comparison is made with the Bernoulli-Euler and Timoshenko beam models.     

A simple selectively integrated torsion—Flexure coupled tapered beam element with transverse shear deformation

A tapered beam element with torsional flexibility and transverse shear deformation is developed for use in swept plate studies. Recently established guidelines for selective reduced integration allow the use of independent interpolation functions for the transverse deflection $\text{w}$, rotation ψ and torsional rotation θ to obtain an element that does not lock in extremely thin situations encountered in thin beam/plate analysis. The element is tested for free vibration and for the static case of a severely swept cantilever plate of high aspect-ratio subjected to uniform distributed load. The results indicate very good performance of the element.     

Curved beam element stiffness matrix formulation

Stiffness matrix of order 12 × 12, for a curved beam element has been formulated involving all the forces together, using Castigliano's theorem. Effects of transverse shear forces and tangential thrust are also taken into account. In earlier works, stiffness submatrices for the two uncoupled systems of forces are formulated independently and then they are combined to give the overall 12 × 12 matrix. A program subroutine, NEWCBM for the stiffness matrix formulation of curved beams has been written in FORTRAN which can be added to the element library of general purpose computer programs like SAP-IV and its improved versions. Example problems have been worked to check the accuracy of this formulation.     

Consistent curved beam element

Curved beam finite elements with shear deformation have required the use of reduced integration to provide improved results for thin beams and arches due to the presence of a spurious shear strain mode. It has been found that the spurious shear strain mode results from an inconsistency in the displacement fields used in the formulation of these elements. A new curved beam element has been formulated. By providing a cubic polynomial for approximation of displacements, and a quadratic polynomial for approximation of rotations a consistent formulation is ensured thereby eliminating the spurious mode. A rotational degree of freedom which varies quadratically through the thickness of the element is included. This allows for a parabolic variation of the shear strain and hence eliminates the need for use of the shear correction factor k as required by the Timoshenko beam theory. This rotational degree of freedom also provides a cubic variation of displacements through the depth of the element. Thus, the normal to the centroidal axis is neither straight nor normal after shearing and bending allowing for warping of the cross section. Material nonlinearities are also incorporated, along with the modified Newton-Raphson method for nonlinear analysis. Comparisons are made with the available elasticity solutions and those predicted by the quadratic isoparametric beam element. The results indicate that the consistent beam element provides excellent predictions of the displacements, stresses and plastic zones for both thin and thick beams and arches.     

On the vectorial nature of the rotational degrees of freedom in spatial beam elements

In this note, it is shown that the rotational degrees of freedom at a node of a spatial finite beam element are the components of a vector regardless of their magnitudes, as long as the structure to be analysed is composed of slender beams, and what may prevent them from being considered as the components of a vector is the presence of significant transverse shear strains, which usually appear in short beams, rather than the magnitudes of the rotations.     

Midpoint difference method for analysing beam structures

The midpoint difference method is organized for analyzing beam structures by approximating the solution of the differential equation of the elastic deflection curve. The method requires the generation and solution of a system of simultaneous, algebraic equations which yield the deflection, slope, bending moment and shear at specified points along the span of the beam. The method is rather simple and well suited for programming on the digital computer.

All of the basic equations necessary for developing an analysis of a beam structure are established and the application of the method is given in the paper. Some general comments are made about the development of a computer program and about the solution of the system of simultaneous equations peculiar to this method of analysis.     

Timoshenko beam finite element with four dof and convergence of O(h)

In situations where transverse shear deformations and rotary inertia in beams are important, elements based on the Timoshenko beam theory are useful. Among the two-noded, four DOF elements derived from the minimum total potential energy principle, the HTK. element proposed by Hughes et al. using linear displacement functions for both w and θ and the T1CC4 element proposed by Tessler et al. using quadratic displacement function for w and linear displacement function for θ are well known in the literature. The convergence of the HTK element in the thin beam situation has been too poor due to shear locking but by using selective integration this element can be shown to be equivalent to the T1CC4 element which has a rate of convergence of O(h²). In this paper a five DOF element with w and θ at the end nodes and θ at the middle node and based on the cubic displacement function for w and the quadratic displacement function for θ is first developed. Statically condensing the middle rotational DOF, the well-known (4 × 4) stiffness matrix using the φ-factor defined as φ = 12EI/kGAL² and hitherto obtained only through a flexibility approach or closed-form solution of the governing equations of the Timoshenko beam theory is derived. This element based on cubic displacement function for w has rate of convergence of O(h⁴), is completely free of shear locking and performs equally well in thin as well as thick beam situations.     

Stiffness matrix of parabolic beam element

The stiffness coefficient matrix is derived for a parabolic beam element of constant section. Effects of flexural, axial and shear deformation are taken into account. A numerical example is presented to illustrate the application of the parabolic beam element.     

Nonlinear finite element computation of the equilibrium,stability and motion of the extensional beam and ring

Discrete Gauss integration of the element total potential energy is applied to the formation of a cubic-cubic C¹ extensible completely nonlinear curved beam finite element. The versatility, accuracy, effectiveness, and robustness of the element, and the Newton-Raphson technique used to solve the nonlinear algebraic stiffness equation set up with it is numerically demonstrated by computations of the nonlinear equilibrium stability and motion of beams and rings.     

Lumped parameter model for timoshenko beam based on mechanical impedance

A lumped parameter model for Timoshenko beam clamped at both ends based on mechanical impedance is devised at low frequencies. Computations of the transient shear force and bending moment due to a low frequency ground acceleration show the usefulness of this model for shock analysis. The effect of correction due to shear deformation and rotary inertia on Timoshenko beam is discussed.     

A corotational procedure that handles large rotations of spatial beam structures

A practical motion process of the three dimensional beam element is presented to remove the restriction of small rotations between two successive increments for large displacement and large rotation analysis of space frames using incremental-iterative methods. In order to improve convergence properties of the equilibrium iteration, an n-cycle iteration scheme is introduced.

The nonlinear formulation is based on the corotational formulation. The transformation of the element coordinate system is assumed to be accomplished by a translation and two successive rigid body rotations: a transverse rotation followed by an axial rotation. The element formulation is derived based on the small deflection beam theory with the inclusion of the effect of axial force in the element coordinate system. The membrane strain along the deformed beam axis obtained from the elongation of the arc length of the beam element is assumed to be constant. The element internal nodal forces are calculated using the total deformational nodal rotations. Two methods, referred to as direct method and incremental method, are proposed in this paper to calculate the total deformational rotations.

An incremental-iterative method based on the Newton-Raphson method combined with arc length control is adopted. Numerical studies are presented to demonstrate the accuracy and efficiency of the present method.     

Natural vibration analysis of rail track as a system of elastically coupled beam structures on Winkler foundation

An analytical model for analyzing the vertical free vibration of a rail track is presented. The track structure is represented as a system of elastically coupled beam structures resting on a Winkler foundation. The rail and the tie beams are described by any combination of the two existing beam theories, the Bernoulli-Euler type, and the Timoshenko type, while the rail is assumed to be periodically supported at discrete points on cross-track tie beams. A generalized track element, which consists of a rail span (beam segment), two adjacent ties, and the coupling spring stiffnesses, is established to discretize the track system into identical units. A concept of an equivalent frequency-dependent spring coefficient for the rail support system is introduced to formulate the dynamic stiffness matrix of the track element. Solutions are provided for the natural frequencies of the track and the associated mode shapes of the rail and the ties under transversely (cross-track) symmetric vibration. The free vibration results are used to obtain the dynamic receptance response of a typical field track and to compare them with an existing model and field experimental data.     

A thin-walled box beam finite element for curved bridge analysis

Practical design of single and multispan curved bridges requires an analysis procedure which is easy and economical to use, and provides a physical insight into structural response under general loading conditions. In the work presented, the thin-walled beam theory has been directly combined with the finite element technique to provide a new thin-walled box beam element. The beam element includes three extra degrees-of-freedom over the normal six degrees-of-freedom beam formulation, to take into account the warping and distortional effects as well as shear. The beam may be curved in space and variable cross-sections may be included. The performance of the box beam element has been compared favourably against results obtained from full 3D shell element analysis, differential equation solutions and experimental results.     

Deflection analysis of expanded open-web steel beams

The finite element method is very appropriate for calculating stresses at isolated areas of expanded open-web steel beams. For deflection analysis, however, the entire beam, or one-half the beam if the system is symmetrical, must be included in the idealization and therefore it is not practical to obtain deflections by a direct application of the finite element method. In this paper, the authors present a method of deflection analysis which treats the castellated beam as an assemblage of typical segments and utilizes the finite element method to form the stiffness matrix for the typical segment. The beam deflections at the panel points are then computed by the conventional stiffness method. Two different idealizations were tried for the finite element analysis of a typical segment. These idealizations resulted in a 13 × 13 and a 7 × 7 stiffness matrix. Deflection values calculated by using the 7 × 7 stiffness matrix showed close agreement with those obtained experimentally.     

The application of the thin-walled box beam element to multibox bridge analysis

An application of the recently developed thin-walled box beam element to the analysis of multibox bridges which arises in practical design, is presented. The thin-walled box beam element, which also takes account of warping distortional effects, when combined with traditional beam elements into a grillage model may adequately represent the three-dimensional behaviour of multibox superstructure. Equivalent sectional properties for the transverse grillage members across individual boxes are computed from a frame analysis. A numerical iterative procedure is introduced to take account of the interaction due to distortion.Comparisons with other numerical methods and model experiments demonstrate the accuracy and economy of the proposed method.     

几种不同的方法在GPS 大数据 探查中的应用分析

GPS 定位系统对车辆的运行调控以及拥堵性分析具有重要意义。但定时采样的GPS 数据难免存在坏点的情 况,而坏点的存在对分析结果容易产生较大错误,从而影响交通管理决策。本文通过高斯混合模型、K-均值聚类分析以及 SOM 自组织神经网络三种方法完成对原始数据时间段划分、字段提取以及坏值清理的操作。这三种方法主要用于对数据进 行聚类分析,根据分析结果识别孤立点从而进行清理。结果显示,高斯聚类与K-均值聚类算法的坏点识别精度小于SOM 自 组织神经网络,但前两种算法的运行效率较后者高。