二维有限元单元角结点位移精度修正之初探

二维四边形有限元单元角结点位移相较于其他结点位移,有更高的收敛阶。对于足够光滑的问题,采用m次单元,其角结点位移收敛阶最高可达2 m阶。该文以二维Poisson方程为例,在有限元解的基础上,利用单元能量投影(EEP)法的超收敛解计算残余荷载向量,在不改变整体刚度矩阵的基础上,仅需进行代数方程组回代,即可得到具有更高精度的单元角结点位移。数值结果表明:当采用EEP简约格式解计算残余荷载向量时,单元角结点位移收敛阶最高可提高为2m+2阶。特别地,对于线性元,精度翻倍,效益十分显著。     

矩阵位移法分析桁架结构

矩阵位移法分析桁架结构,具有易于实现计算自动化的优点,得到广泛应用.它的主要解题思路是:首先将结构离散成为有限个独立的单元,进行单元分析,建立单元杆端力与单元杆端位移之间的关系式---单元刚度方程;然后利用结构的变形连续条件和平衡条件将各单元组成整体,建立结点力与结点位移之间的关系式---结构刚度方程,这一过程为整体分析;最后求得结构的位移和内力.矩阵位移法就是在一分一合,先拆后搭的过程中,把复杂结构计算问题转化为简单的单元分析和集合问题.     

基于单位分解法的实体壳广义单元模型

基于单位分解的广义有限元法的逼近空间由单位分解函数和局部覆盖函数构成,采用传统有限元形函数作为单位分解函数,其局部覆盖函数的定义不依赖于有限元网格.以十六结点六面体等参单元形函数作为单位分解函数,采用一阶多项式局部覆盖函数建立了十六结点六面体广义单元.在此基础上利用广义有限元法可以灵活构造各向异性逼近空间的特点,根据薄壳的变形特性,对壳体法向挠度和切向位移分别采用一阶和零阶多项式局部覆盖函数,构造了实体薄壳广义单元.计算结果表明：十六结点六面体广义单元和实体薄壳广义单元用于板壳结构分析时具有比相应的常规实体单元更高的收敛性和求解效率,且实体薄壳广义单元比十六结点六面体广义单元具有更高的求解效率.     

强震作用下立体桁架结构抗连续倒塌性能研究

地震灾害发生时,主震之后通常伴有多次余震。以往的震害事故表明,主震过程可能引起立体桁架结构薄弱部位发生初始失效,并导致结构在余震中发生连续倒塌破坏。针对以上问题,采用增量动力法确定立体桁架结构薄弱部分分布规律,然后采用预定义场法引入初始失效杆件,分析不同破坏模式下该类结构破坏加速度、倒塌极限位移和塑性杆件比例等响应的变化规律,最后提出了改善结构抗连续倒塌性能的有效措施。分析结果表明:在强震作用下立体桁架结构的薄弱部位为主桁架跨中3/8L范围内的上弦杆、下弦杆及跨中支撑桁架的上弦杆。引入初始失效后结构破坏加速度降低了13.3%～40.0%,倒塌极限位移降低了1.6%～30.9%。当初始失效杆件为受压杆件时,结构产生动力失稳破坏;当初始失效杆件为受拉杆件时,结构产生动力强度破坏。针对发生动力失稳破坏的立体桁架结构,增加侧向交叉支撑使破坏模式转变成动力强度破坏,破坏加速度提高69.4%～73.7%。     

基于微分求积法的Euler-Bernoulli梁的大变形力学行为研究

对工程中大量使用的细长柱的大变形力学行为进行了研究。基于杆件可伸长和忽略剪切变形的Euler-Bernoulli梁理论,建立了杆件大变形分析的几何非线性的控制方程,利用微分求积法直接从控制方程出发,把控制方程和边界条件转化为代数方程形式,采用改进的Newton迭代法即逆Broyden秩1方法进行了求解。计算表明微分求积法是用于结构大变形分析的一个有力的工具。     

斜放四角锥网架的拟夹层板分析法

本文把斜放四角锥网架连续化为构造上的夹层板来分析。在拟夹层板的计算模型中,既考虑了板的横向剪切变形,又考虑了板的弯曲变形与平面变形的偶合作用,从而较好地反映了这种网架的受力状态。文中给出了斜放四角锥网架拟夹层板的基本方程式,它可归纳为对于一个广义位移函数w的十阶偏微分方程式,并给出了拟板内力换算成网架杆件内力的一般计算公式。对于常用的周边简支网架,求得了基本方程式的解析解,并根据某些主要参数、编制了内力、位移计算用表。此外,指出了所谓假想弯矩法,是本文方法的一个退化了的特例。     

基于遗传算法的离散型结构拓扑优化设计

采用遗传算法求解包括桁架结构和框架结构的离散型结构拓扑优化问题。在遗传算法的基础上,通过引入拓扑变量并修改被删除杆件的材料弹性模量,提出了一个受多工况荷载作用,能同时考虑应力、稳定及位移等约束的离散型结构拓扑优化问题统一数学模型。该模型不但能同时适用于桁架结构和框架结构等离散型结构拓扑优化问题,而且还能解决奇异最优解问题。结合上述统一数学模型和遗传算法,给出了求解离散型结构拓扑优化问题的优化方法。算例结果表明,采用该文提出的拓扑优化方法可有效、方便地对桁架结构、框架结构等离散型结构进行拓扑优化设计。     

一种新的集成非线性杆件单元刚度矩阵的方法

对于非线性杆件单元,本文提出一种新的简便有效的集成单元刚度矩阵的算法。该方法直接从结构力学中的位移法的概念出发,通过解析积分或数值积分求解积分算子,由积分算子线性组合,能快速求解考虑弯、剪、扭、轴压等各种非线形刚度的杆件单元的刚度矩阵。该方法具有广泛的普适性,能适用于所有多项式、插值多项式、解析式、离散点描述的变刚度、变截面直杆的单元刚度矩阵集成计算。文中通过求解线性直杆单元刚度矩阵验证了该方法的正确性。     

钢桁架结构稳定性与两层面强度优化设计研究

现行桁架结构优化设计方法难以同时满足构件和体系两层面强度需求。鉴于此,利用弹性模量缩减法研究建立了考虑稳定性影响的桁架结构两层面强度优化设计的改进均匀承载法。在桁架杆件的截面轴向强度中引入压杆稳定系数,据此建立考虑稳定性影响的桁架杆件单元承载比计算表达式。根据弹性模量缩减法迭代首步与末步的构件承载比分别确定桁架在构件和体系两个层面的强度系数,建立构件与体系两层面强度之间的显性关系式按照两层面强度系数的目标值调整各构件的截面强度,利用构件承载比均匀度建立桁架结构优化设计的迭代收敛判据。通过对比分析,验证了该文方法建立的结构设计方案,取得了较好经济效果,克服了现行桁架结构优化设计方法难以满足桁架两层面强度需求的缺陷。     

弹性支承连续梁与弹性地基梁的静力及动力分析

本文导出弹性支承连续梁各跨度的内力及形变递推计算公式,据以编制程序,用微型计算机计算。本程序对于解算变截面杆件的连续梁及绘制内力图特别方便;除用于静力计算外,还可推广应用于连续梁的动力计算。对于弹性地基梁来说,也可离散为若干跨弹性支承梁计算。对于一根具有不同弯曲刚度、基床系数的弹性地基梁,用本法计算,远较其他解析法,如微分方程法或Light-foot的弯矩分配法等直接便利得多。     

含机构位移起重机主副臂组合臂架结构几何非线性分析

工程中很多起重机械含有控制吊重位置的机构,这些附加机构约束了起重机臂架结构承载变形后的位置。机构的作用可以等效为施加到结构上的广义外力,这些外力与结构位移和机构自由度是高度耦合的。对臂架结构按长度方向划分了多个子结构,将子结构的内部位移凝聚到了左右端面上。基于共旋坐标法,建立了随子结构一起运动的随动坐标系,推导了子结构单元的广义节点内力。以履带式起重机的主副臂组合臂架结构为例,考虑到臂架需要通过变幅机构进行约束和控制,给出了机构位移附加在臂架上的广义节点外力及其导数。最后,通过起重机的副臂工况算例,给出了在不同载荷下臂架结构位移以及约束机构位移,验证了分析方法的正确性和合理性。     

三向类网架结构弯曲和振动分析的分解刚度法

把三向类网架结构简化为夹层板,采用考虑剪切变形的具有三个广义位移的平板弯曲理论进行分析。基于分解刚度思想提出了三向类网架(三角锥网架、三向网架)静力分析和固有振动分析的方法,给出了简便实用的计算公式。通过有限元法的验证,表明了分解刚度法作为一种简化的计算方法,其精度是比较高的,绝大多数的误差都小于5%。可以应用于工程结构设计,便于快速得出网架结构挠度和内力的计算结果及分布规律。此外,与其它的计算方法相比其计算公式大大简化了。     

A new implementation of the force method and the optimum design of large trusses

An implementation of the force method is proposed in which the forces and the displacements are simultaneously obtained by the solution of a sparse symmetric indefinite system. The matrix of coefficients is formed by just the concatenation of the element flexibility and equilibrium matrices. No computational procedure is required to generate the compatibility conditions (or the self-stress matrix) and no partitioning of the force vector is made into a basic set and a redundant set, unlike the conventional force method. A slightly modified sparse unsymmetric system can be written in which the stresses and the displacements are the unknowns. This is used as constraints in the formulation of the minimum weight design problem for large structures under static loading conditions. A sparse generalized reduced gradient package is used as the optimizer. A class of test problems involving large truss structures is solved. The results indicate that the present implementation of the force method is better than the displacement method for the optimum design of large structures.     

On scattering of an SH-wave by a corner comprised of two different elastic materials

The problem of the scattering of plane SH-waves by a corner comprised of two different elastic materials is considered. This problem is the analogy of the dielectric wedge problem in electromagnetics. A Fourier transform method is used to derive equations which can readily be solved numerically. The unknowns are the Fourier transforms of the displacements and tractions on the interfaces of the wedge. Fredholm integral equations of the second kind with simple continuous kernels are obtained after taking a physically reasonable representation of the unknown quantities. Numerical results for the diffraction coefficients are presented. The method can be generalized for incidence of Stonely, plane P-, or SV-waves.     

Analytical Full-field Solutions of a Piezoelectric Layered Half-plane Subjected to Generalized Loadings

The two-dimensional problem of a planar transversely isotropic piezoelectric layered half-plane subjected to generalized line forces and edge dislocations in the layer is analyzed by using the Fourier-transform method and the series expansion technique. The full-field solutions for displacements, stresses, electrical displacements and electric fields are expressed in explicit closed forms. The complete solutions consist only of the simplest solutions for an infinite piezoelectric medium with applied loadings. It is shown in this study that the physical meaning of this solution is the image method. The explicit solutions include Green's function for originally applied loadings in an infinite piezoelectric medium and the remaining terms are image singularities which are induced to satisfy free surface and interface continuity conditions. The mathematical method used in this study provides an automatic determination for the locations and magnitudes of all image singularities. The locations and magnitudes of image singularities are dependent on the piezoelectric material constants of the layered half-plane and the location of the applied loading. With the aid of the generalized Peach-Koehler formula, the image forces acting on dislocations are derived from the full-field solutions of the generalized stresses. Numerical results for the full-field distributions of stresses and electric fields in the piezoelectric layered half-plane and image forces for edge dislocations are presented based on the available analytical solutions.     

基于细化单胞模型的复合材料层合板强度预报方法

通用单胞模型常被应用于复合材料细观力学分析。但原始的通用单胞模型存在求解量大、计算效率低的问题。本文中对其改进, 建立了以子胞界面细观应力为未知量的细化单胞模型。该模型可以充分考虑纤维、基体和界面相等细观组分, 并实现单向板的宏-细观多尺度力学分析。通过将组分材料失效判据引入到模型中, 再与经典层合理论相结合, 提出了一种基于细化单胞模型的复合材料层合板强度预报方法, 并给出了基于试验数据的强度谱定量评测方法。通过与世界失效分析习题的失效理论和试验数据进行对比, 证明本文的预报方法具有很高的计算精度和广泛的普适性。     

Exact geometry four-node solid-shell element for stress analysis of functionally graded shell structures via advanced SaS formulation

Abstract

The exact geometry four-node solid-shell element formulation using the sampling surfaces (SaS) method is developed. The SaS formulation is based on choosing inside the shell N not equally spaced SaS parallel to the middle surface in order to introduce the displacements of these surfaces as basic shell unknowns. Such choice of unknowns with the use of Lagrange basis polynomials of degree N???1 in the through-thickness interpolations of displacements, strains, stresses and material properties leads to a very compact form of the SaS shell formulation. The SaS are located at Chebyshev polynomial nodes that make possible to minimize uniformly the error due to Lagrange interpolation. To implement efficient 3D analytical integration, the extended assumed natural strain method is employed. As a result, the proposed hybrid-mixed solid-shell element exhibits a superior performance in the case of coarse meshes. To circumvent shear and membrane locking, the assumed stress and strain approximations are utilized in the framework of the mixed Hu-Washizu variational formulation. It can be recommended for the 3D stress analysis of thick and thin doubly-curved functionally graded shells because the SaS formulation with only Chebyshev polynomial nodes allows the obtaining of numerical solutions, which asymptotically approach the 3D solutions of elasticity as the number of SaS tends to infinity.     

广义变分原理在高速铁路无缝道岔结构分析中的应用

在继承现有试验成果的基础上,将广义变分原理应用于铁路无缝道岔结构体系的分析,提出了一种新的铁路无缝道岔计算理论。建立了较为完善的计算模型,在假设钢轨纵向位移函数的基础上,计算了无缝道岔结构体系各部分的能量,通过广义变分法建立了结构体系的平衡方程,编制了计算程序,分析了固定辙叉无缝道岔钢轨温度力与位移。为无缝道岔计算理论和设计方法的研究开辟一条新途径。     

Analysis of nonlinear fully intrinsic equations of geometrically exact beams using generalized differential quadrature method

In this paper, the generalized differential quadrature (GDQ) method is presented for solving the nonlinear, fully intrinsic equations of geometrically exact rotating and nonrotating beams. The fully intrinsic equations of beams involve only moments, forces, velocity and angular velocity, and in these equations, the displacements and rotations will not appear explicitly. This paper presents the generalized differential quadrature method for solution of these equations. To show the accuracy, validity and applicability of the proposed generalized differential quadrature method for solving the fully intrinsic beam equations, different cases are considered. It is found that the GDQ method gives very accurate results with very few numbers of discrete points and also has very low computational cost as compared to some other conventional numerical methods and therefore this method is very efficient, accurate and fast for solving the fully intrinsic equations.