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。     

一种考虑剪切变形的平行四边形厚/薄板通用单元

根据Timoshenko二广义位移梁理论，构造了深梁位移场的插值函数。利用斜坐标系与直角坐标系的变换关系、有限条带思想和深梁位移插值函数，构造了一种考虑剪切变形的平行四边形厚／薄板弯曲通用单元的位移(曲率、剪应变、转角、横向位移)插值函数，导出了刚度矩阵和非结点荷载等效力。并对简支阍支方板、Razzaque斜板、四边简支斜交板弯曲进行了数值计算。算例表明此单元有较好的精度，对于薄板不出现剪切闭锁，可适应于目前桥梁建设中大量采用的斜交板桥结构分析。     

一维区间B样条小波单元的构造研究

基于区间B样条小波及小波有限元理论,提出了一种区间B样条小波有限元方法。传统有限元多项式插值被一维区间B样条小波尺度函数取代,进而构造形状函数和单元。与小波Galer-kin方法不同,本文构造的区间B样条小波单元通过转换矩阵将无明确物理意义的小波插值系数转换到物理空间。转换矩阵在小波单元构造过程中起到关键作用,为了保证求解的稳定性,转换矩阵必须非奇异。构造了以区间B样条尺度函数为插值函数的一系列一维区间B样条小波单元。数值算例表明,本文构造的区间B样条小波单元与传统有限元方法相比,在求解变截面,变载荷等问题时具有收敛快和精度高等优势;有效地丰富了小波有限元法单元库。     

轴向功能梯度Timoshenko变截面梁的屈曲分析

运用插值矩阵法研究了不同边界条件下轴向功能梯度材料变截面Timoshenko梁的屈曲性能问题。基于Timoshenko梁基本理论,将轴向功能梯度变截面Timoshenko梁临界荷载的计算转化为一组变系数常微分方程特征值问题,然后运用插值矩阵法可一次性地计算出轴向功能梯度变截面梁在不同边界条件下的屈曲临界荷载。当区间划分点数n为80时,在不同的边界条件下均质材料等截面Timoshenko梁量纲为一的临界荷载的本文计算值与解析解有7位有效数字相同,轴向功能梯度Timoshenko锥形梁量纲为一的临界荷载的本文计算值与已有文献计算结果有3~5位有效数字相同,数值计算结果表明了本文方法的有效性和较高的计算精度。同时,本文方法可获取相应的挠度模态函数,而且对于材料梯度函数和截面几何轮廓的具体形式无任何限制条件。     

变截面梁动力响应的频域传递矩阵计算方法

为提高变截面梁振动分析的计算效率,提出了基于频域传递矩阵法的动力计算算法．首先,选择线速度、角速度、弯矩和剪力作为求解变量,通过Laplace变换将变截面梁的动力响应时域偏微分方程转换为频域常微分方程;然后,通过求解频域方程并结合协调和边界条件建立变截面梁的频域传递矩阵;通过构造傅里叶级数展开形式的时域响应函数,对变截面梁传递矩阵方法求解的频响函数进行Laplace逆变换,建立了变截面梁的固有特性计算和时域瞬态响应计算方法,最后,借助数值仿真软件,开发了变截面梁动力响应分析的计算程序．完成对算例的仿真计算和分析,并与有限元计算结果进行对比,数值结果验证了该方法的正确性和有效性．     

一个改进的平面梁单元

根据有限单元法基本原理 ,提出了一个变截面平面梁单元 ,推导了其单元钢度矩阵。这一改进的梁单元用于分析梁高呈线性变化及二次抛物线变化的矩形截面梁 ,将得到准确解。文中给出了一个变截面悬臂梁算例 ,计算表明 ,这一改进的梁单元使变截面梁的分析大大简化     

基于广义变分原理的梁板单元分析的数值流形方法

数值流形方法(NMM)是一种基于有限覆盖技术的新型数值方法.以该方法的覆盖位移模式为基础,利用广义变分原理中罚函数理论,详细推导了梁板流形单元的覆盖位移函数,刚度矩阵和应变矩阵,并建立了可应用于梁板单元分析的数值流形方法.最后通过算例分析表明,该方法在对梁板弯曲问题分析是有效的.     

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.     

变截面Timoshenko梁的单元刚度矩阵

变截面构件在工程中应用广泛,在对变截面梁进行数值计算时,需要建立变截面梁单元的刚度矩阵。该文采用势能驻值原理,考虑了轴力引起的几何非线性和剪切变形的影响,将梁截面刚度的变化率作为小量,得到了近似到二阶的单元刚度矩阵。在构造位移模式时,从梁的微分平衡方程出发,得到同样近似到二阶、分别以三次和五次多项式表示的剪切和弯曲位移模式。该文还证明了单元刚度矩阵的奇异性,给出了轴压刚度的表达式,定量论证了与某些精确解的误差,表明在一定范围内,该文的结果具有足够的精度。最后以一个计算实例说明该文的单元刚度矩阵具有较快的收敛性。     

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采用谱单元方法推导带阻尼梁的传递函数矩阵. 采用一维连续梁的解析解作为动力形状函 数. 与有限元法相比,谱单元方法的自由度和单元数目小且计算精度高. 传递函数表示为梁 的几何和物理参数的超越隐函数,不同于用模态参数表示的传统传递函数. 提出了采用遗传 算法的结构物理参数识别方法. 以变截面悬臂梁为数值算例,显示该方法的有效性.     

优化设计中梁截面参数间函数关系的形状乘子法

提出了处理优化设计中梁截面参数间函数关系的形状乘子法。这种方法可以适用于各种复杂的梁截面，并且灵活地运用在含有梁截面的优化设计中。利用这种方法可以使含梁结构的优化更广泛地运用到工程实际问题中。     

变截面多稳态梁结构减振吸能分析与优化设计

基于多稳态梁结构具有吸能且可重复使用的特点,本文研究包含变截面多稳态梁的单胞结构及其周期性排布的减振吸能效应及其优化设计方法。对多稳态结构进行考虑几何非线性的位移加载/卸载有限元仿真,根据其载荷-位移曲线分析多稳态结构的减振吸能原理,并研究串联与并联周期性排布形式对结构整体吸能特性的影响规律。研究基于多参数调控的变截面梁结构形状表征方法,根据多稳态结构储能特点建立变截面多稳态单胞结构的结构优化模型,通过求解优化问题获得总质量不变条件下最优变截面梁结构形状。进一步地通过对优化结果的有限元分析验证优化的有效性,并对结构进行瞬态冲击荷载下动响应分析,证明多稳态结构的冲击保护作用。     

A novel beam finite element with singularities for the dynamic analysis of discontinuous frames

In this work, a model of the stepped Timoshenko beam in presence of deflection and rotation discontinuities along the span is presented. The proposed model relies on the adoption of Heaviside’s and Dirac’s delta distributions to model abrupt and concentrated, both flexural and shear, stiffness discontinuities of the beam that lead to exact closed-form solutions of the elastic response in presence of static loads. Based on the latter solutions, a novel beam element for the analysis of frame structures with an arbitrary distribution of singularities is here proposed. In particular, the presented closed-form solutions are exploited to formulate the displacement shape functions of the beam element and the relevant explicit form of the stiffness matrix. The proposed beam element is adopted for a finite element discretization of discontinuous framed structures. In particular, by means of the introduction of a mass matrix consistent with the adopted shape functions, the presented model allows also the dynamic analysis of framed structures in presence of deflection and rotation discontinuities and abrupt variations of the cross-section. The presented formulation can also be easily employed to conduct a dynamic analysis of damaged frame structures in which the distributed and concentrated damage distributions are modelled by means of equivalent discontinuities. As an example, a simple portal frame, under different damage scenarios, has been analysed and the results in terms of frequency and vibration modes have been compared with exact results to show the accuracy of the presented discontinuous beam element.     

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。     

A new efficient method to evaluate exact stiffness and mass matrices of non-uniform beams resting on an elastic foundation

In this paper, a new efficient method to evaluate the exact stiffness and mass matrices of a non-uniform Bernoulli–Euler beam resting on an elastic Winkler foundation is presented. The non-uniformity may result from variable cross-section and/or from inhomogeneous linearly elastic material. It is assumed that there is no abrupt variation in the cross-section of the beam so that the Euler–Bernoulli theory is valid. The method is based on the integration of the exact shape functions which are derived from the solution of the axial deformation problem of a non-uniform bar and the bending problem of a non-uniform beam which are both formulated in terms of the two displacement components. The governing differential equations are uncoupled with variable coefficients and are solved within the framework of the analog equation concept. According to this, the two differential equations with variable coefficients are replaced by two linear ones pertaining to the axial and transverse deformation of a substitute beam with unit axial and bending stiffness, respectively, under ideal load distributions. The key point of the method is the evaluation of the two ideal loads which in this work is achieved by approximating them by two polynomials. More specifically, the axial ideal load is approximated by a linear polynomial while the transverse one by a cubic polynomial. The numerical implementation of the method is simple, and the results are compared favorably to those obtained by exact solutions available in literature.     

对称变截面平面曲杆的单元刚度矩阵

本以一种新的思路和做法对变截面曲杆进行单元分析,直接把变截面曲杆作为单元,采用弹性中心法导出了单元刚度矩阵通用公式,确定了截面按余弦规律变化时,轴线分别为圆弧形、抛物线形、半椭圆形平面曲杆的单元刚度矩阵精确式,供结构设计参考使用。     

Inelastic axial-flexure–shear coupling in a mixed formulation beam finite element

In this paper a beam element that accounts for inelastic axial-flexure–shear coupling is presented. The mathematical model is derived from a three-field variational form. The finite element approximation for the beam uses shape functions for section forces that satisfy equilibrium and discontinuous section deformations along the beam. No approximation for the beam displacement field is necessary in the formulation. The coupling of the section forces is achieved through the numerical integration of an inelastic multi-axial material model over the cross-section. The proposed element is free from shear-locking. Examples confirm the accuracy and numerical robustness of the proposed element and showcase the interaction between axial force, shear, and bending moment.