考虑几何非线性的杆单元应变公式

考虑几何非线性的杆单元应变公式范志良（同济大学桥梁系，上海２０００９２）有限单元法在结构几何非线性分析中应用已十分广泛。在目前的本科与研究生教学中已将一般杆、梁及板单元的几何非线性列式予以介绍￣［１］，但其中对有些问题阐述不够清晰。譬如对一般的平面杆...     

刚接与铰接混合连接杆系结构的几何非线性分析

本文提出用子结构原理解决具有刚接与铰接混合连接空间杆系结构的几何非线性分析,实现其非线性稳定性分析的载荷-位移全过程跟踪。该法无须单独推导刚接、铰接以及一端刚接一端铰接单元的弹性刚度矩阵和几何刚度矩阵,而可以直接由空间梁单元退化得到,而且可以将平面问题与空间问题、刚接与铰接混合连接体系进行统一处理,算例表明,本文方法对于杆系结构的统一和整体分析是有效的。     

空间杆系与叠层板组合结构的非线性稳定分析

本文用非线性有限元法研究一种空间杆系与复合材料叠层板组合结构的非线性稳定问题，将杆系离散成空间梁柱单元，将复合材料叠层板离散成复合材料叠层板单元，并考虑了几何非线性和板的成层正交各向异性性质。为了获得较高的计算效率并顺利通过稳定临界点，求解非线性平衡方程采用了变增量步长和柱面弧长法迭代策略。在考证了程序的正确性之后，具体分析了几个工程模型，得出了于工程设计有益的结论。     

空间杆系与叠层板组合结构的非线性稳定分析

本文用非线性有限元法研究一种空间杆系与复合材料叠层板组合结构的非线性稳定问题。将杆系离散成空间梁柱单元，将复合材料叠层板离散成复合材料叠层板单元，并考虑了几何非线性和板的成层正交各向异性性质。为了获得较高的计算效率并顺利通过稳定临界点，求解非线性平衡方程采用了变增量步长和柱面弧长法迭代策略。在考证了程序的正确性之后，具体分析了几个工程模型，得出了于工程设计有益的结论。     

张拉结构非线性分析两节点曲线单元有限元法

提出适合张拉结构几何非线性分析的两节点曲线单元有限元方法．假定索元的初始形状呈二次抛物线,根据单索的平衡条件、几何和物理关系建立了索元的位移函数;由拉格朗日应变的定义建立了可以考虑任意次高阶位移影响的索元轴向应变的精确表达式,并基于拉格明日描述方法和虚功原理得到了索元的非线性平衡方程与切线刚度矩阵．采用荷载增量法与Newton－Raphson法相结合的混合法进行了实例计算,结果表明：本文方法的精度明显优于两节点直线索单元,适合于大跨度索阿、索穹顶等张拉结构的几何非线性分析．     

含可动机构的杆系结构非线性力法分析

推导了基于矩阵奇异值分解的小变形条件下杆系结构的力法分析（LFM）过程，并采用杆件内力修正的方法，提出了考虑大位移的几何非线性力法分析（NFM）方法思想和步骤。该方法有效地解决了结构中同时存在刚体位移、弹性位移的结构分析问题，适合计算机编程，结构的强度计算、机构分析和形态分析，讨论了力法与位移法的优缺点，给出的算例表明所提出的方法是正确的、有效的。     

正装结构非线性应力应变累加规律研究

基于有限位移理论的正装结构非线性分析理论对解决超大跨度、柔性结构的非线性计算有重大意义.文中探讨了正装结构非线性的分析特点,研究了其应变场与应力场的Kirchhoff应力张量与Lagrange应变张量的适用性,提出了正装结构非线性分析中应力场与应交场的累加规律,导出了拖动坐标法的虚功增量方程,以此对杆系结构非线性分析常用的CR法和UL列式进行了精度比较分析.文中研究成果可为正装结构的非线性分析提供理论指导.     

基于有限单元柔度法的材料与几何双重非线性空间梁柱单元

从有限单元柔度法的基本思想出发,基于完全拉格朗日格式(TL格式),建立了能考虑材料与几何双重非线性的一般化空间梁柱单元,适用于满足Euler-Bernoulli梁柱二阶分析理论假定的空间杆系结构非线性分析。另外,在梁柱单元截面分析中引入纤维模型,使其适用于解决钢筋混凝土这类复合材料结构的非线性分析问题。对钢筋混凝土双向偏心受压柱试验结果的模拟分析表明,本文所提方法是正确、可靠的,能有效地分析钢筋混凝土框架柱的材料与几何双重非线性问题。     

非线性材料杆系结构的内力求解

随着科技的发展,已研发出了许多非线性材料,这些非线性材料组成的结构已在工程实际中得到了广泛应用．本文研究了静不定非线性材料结构杆件内力的求解这一关键问题．利用非线性材料结构余能及静力平衡条件,构造了新泛函数,对新泛函数进行一阶求导,即可方便求得静不定非线性材料结构杆件的内力．同时,指出了非线性材料杆系结构的内力求解时存在漏解的原因：结构材料的应力与应变本构关系为非线性幂函数,把结构余能对杆系的内力求一阶偏导,令所得到的位移方程等于零,该方程为一元二次方程,应有两个实数解,而有关文献却遗漏了一个实数解．     

拉索穹顶结构非线性分析的混合有限元增量法

拉索穹顶结构是由受压桅杆和拉索组成的新型柔性大跨度空间组合结构,几何上表现为极强的非线性特性,计算困难,本文应用有限元法,结合拉索穹顶结构特征,假定拉索和桅杆的受力满足虎克宣定律,建立了可以直接考虑拉索垂度影响的两节点索单元模型,并与两节点直杆单元相结合,基于修正的拉格朗日描述方法和虚功原理建立了拉索穹顶结构非线性分析的混合有限元增量方程。采用荷载增量法与Ｎｅｗｔｏｎ－Ｒａｐｈｓｏｎ法相结合的求解     

含裂纹梁刚塑性动态断裂的非线性线弹簧模型分析

用非线性线弹簧模型分析了带裂纹梁的刚塑性动态断裂问题.在塑性势理论基础上,建立了全塑性状态下的弹簧本构关系,并用此关系导出带裂纹梁刚塑性动态断裂分析的基本方程,计算了在冲击载荷作用下,裂纹梁的动态断裂响应.     

The nonlinear numerical analysis method for frames

This paper gives the direct formulas of stiffness matrixes of two kinds of Kirchhoff nonlinear elements under total-Lagrange coordinate. For the first one, it includes not only the quadric terms of increments of strain and displacement but also the influence of rotations. For the second one, it is simplified and its nonlinear is considered by taking into account the influence of axial force on the equilibrium equation in the linear beam theory. The nonlinear equation obtained from both of the above-said elements is solved by mixed Newton-Raphson method, and by comparing the results obtained from two kinds of nonlinear beam some important conclusions that we can know how to use them right are given in our paper.     

斜拉桥合理成桥状态确定的一阶分析法

本文建立了斜拉桥索力优化的非线性规划模型，其中以斜拉桥主梁和索塔的弯曲应变能为目标函数，以各斜拉索的索力为设计变量，结构的应力及索力为约束条件，采用一阶分析法进行求解，用以确定成桥合理状态的索力。在计算中，考虑大跨度斜拉桥各种几何非线性因素的影响，并列出了优化模型的具体表达式及优化过程中的关键求解策略。应用该法和空间非线性有限元分析程序分析了某千米级斜拉桥方案的合理成桥状态，计算结果表明：该方法简单、有效。     

基于非线性梁理论的有限质点法

有限质点法是以向量式力学为基础,用有限数量的质点来模拟结构的变形行为,质点的运动由牛顿运动定律来计算。在有限质点法中,质点通过构件相连,构件约束着质点的运动,并且其内力由质点的运动变量来描述。基于向量式力学的基本思想和非线性梁理论,提出了一种新的有限质点法,该方法在共旋单元坐标系中描述梁的非线性变形。以空间梁系结构为例,推导了计算构件内力的非线性公式,并考虑了弯扭耦合变形。通过两个连续欧拉角的变换公式得到共旋坐标系的旋转矩阵。与传统的有限质点法相比,本文提出的方法避免了刚体虚转动分析。通过四个结构的数值求解,验证了本文方法在计算结构大变形响应时具有较高的精度。     

Generation of higher electron cyclotron frequency harmonics in plasma with a beam

We examine nonlinear excitation of the higher electron-cyclotron frequency harmonics for waves propagating perpendicular to an external uniform magnetic field in a Maxwell plasma for the case of low-density electron beam passage through the plasma. It is shown that the nonlinear excitation mechanism leads to the possibility of generating cyclotron harmonics for plasma parameters for which generation does not occur from the linear theory viewpoint. The nonlinear cyclotron harmonic generation increments are calculated for nonlinear scattering by the beam and plasma electrons of the high frequency longitudinal waves excited in the plasma by the beam.Translated from Zhurnal Prikladnoi Mekhaniki Tekhnicheskoi Fiziki, Vol. 10, No. 6, pp. 40–51, November–December, 1969.The author wishes to thank V. N. Tsytovich for posing the problem and for many discussions of the questions touched upon in the article.     

A Theory of Anharmonic Lattice Statics for Analysis of Defective Crystals

This paper develops a theory of anharmonic lattice statics for the analysis of defective complex lattices. This theory differs from the classical treatments of defects in lattice statics in that it does not rely on harmonic and homogenous force constants. Instead, it starts with an interatomic potential, possibly with infinite range as appropriate for situations with electrostatics, and calculates the equilibrium states of defects. In particular, the present theory accounts for the differences in the force constants near defects and in the bulk. The present formulation reduces the analysis of defective crystals to the solution of a system of nonlinear difference equations with appropriate boundary conditions. A harmonic problem is obtained by linearizing the nonlinear equations, and a method for obtaining analytical solutions is described in situations where one can exploit symmetry. It is then extended to the anharmonic problem using modified Newton–Raphson iteration. The method is demonstrated for model problems motivated by domain walls in ferroelectric materials.      

Nonlinear electro- and magneto-elastostatics: Material and spatial settings

The material and spatial settings of the nonlinear coupling problem of electro- and magneto-elastostatics are discussed in this paper. The governing equations and variational formulations of the problem derived in these two settings using basic equations of electricity, magnetism and elasticity allow the consideration of material defects by the material force method.     

The Nonlinearly Elastic Equilibrium of a Compound Rectangular Plate with Friction Between the Layers

We consider a spatial boundary-value problem in the physically nonlinear theory of elasticity for a multilayered rectangular plate with friction between the layers. The nonlinear relationships between stresses and small strains are assumed to have Kauderer's form. The nonlinear boundary-value solution is constructed as series in powers of a physical dimensionless parameter. The physically nonlinear problem posed is reduced to a sequence of linear inhomogeneous problems. The effect of physical nonlinearity on the elastic equilibrium of a two-layer thick plate with friction between the layers is studied.     

Solution of strongly nonlinear problems in the mechanics of a solid deformable body

A new algorithm for constructing a Lagrangian formulation of the incremental theory on the basis of finite-element digitization of the equilibrium equations of strongly nonlinear mechanical systems subject to static loads is proposed. Example solutions of problems of nonlinear deformation of a solid that confirm the validity of the applied relations between increments in stresses and increments in deformations is presented. Technical University of Construction and Architecture, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 3, pp. 22–26, March, 1999.