简支梁大挠度弹性后屈曲载荷与变形的优化算法

针对简支梁结构大挠度后屈曲载荷与变形的计算问题,本文提出了一种直接求解其后屈曲载荷和变形的优化算法。在简支梁处于大挠度屈曲平衡状态下,将梁结构划分为有限子段,以待求后屈曲载荷为设计变量,根据起点的边界条件和每个子段满足的弯矩变形公式,累积计算出其他各个节点的坐标,以得到的终点坐标满足的边界条件构建目标函数模型。在此基础上,通过MATLAB编制优化程序分析了两个典型算例,并将理论结果与相关软件的计算结果进行对比,从而证明了本文算法的正确性。本文算法求解过程简单、快速,具有一定的实用性,为变截面结构大挠度弹性屈曲稳定性问题的研究提供了参考。     

微分求积方法对于桩基大变形分析的应用

本文基于大变形的理论,采用弧坐标首先建立了具有初始位移的桩基的非线性数学模型,一组强非线性的微分-积分方程,其中,地基的抗力采用了Winkeler模型;其次,引入变数变换将微分-积分方程转化为一组非线性微分方程,并用微分求积方法离散了方程组,得到一组离散化的非线性代数方程;最后用Newton-Raphson迭代方法对离散化方程进行了求解,得到了桩基变形前后的构形、弯矩和剪力.计算中选取了两种不同类型的初始位移,并考察了它们对桩基大变形力学行为的影响.     

薄壁细胞受微吸管与探针共同作用接触模型及数值模拟

建立了以典型的薄壁球型植物细胞为原型的细胞、微吸管及探针接触模型.模型的细胞壁采用封闭球形薄膜,其本构关系为体积不可压超弹性,膜球内充满有压流体以模拟细胞质.应用轴对称几何非线性方法得出了基本微分方程组,并应用龙格-库塔法进行了求解;同时,应用流固耦合有限元法进行了数值模拟以资比较.两种方法得出了较为一致的变形和应力分...     

弹性或弹塑性土体中桩基的大变形分析

采用弧坐标,首先建立了位于弹性地基或弹塑性地基上并具有初始位移的桩基大变形行为的非线性微分方程组,并采用Winkeler模型来模拟地基对桩基的抗力;其次,应用微分求积方法离散非线性微分方程组,得到一组离散化的非线性代数方程,并给出了利用Newn-Raphson方法求解非线性代数方程的步骤;作为应用给出了数值算例,得到了桩顶受组合载荷作用时,变形后桩基的构形、弯矩和剪力,考察了土的弹性和弹塑性性质、桩基初始位移、荷载等参数对桩基力学行为的影响.最后将模型进行简化,得到了小变形理论的解析解,并比较了由大变形理论与小变形理论所得结果的差别.     

索网天线的参变量变分及非线性有限元方法

针对大型周边桁架式索网天线由拉索拉压模量不同引起的本构非线性和结构大变形引起的几何非线性问题,给出了基于参变量变分原理的几何非线性有限元方法. 首先针对含预应力索单元拉压模量不同分段描述的本构关系,通过引入参变量,导出了基于参变量及其互补方程的统一描述形式,避免了传统算法需要根据当前变形对索单元张紧/松弛状态的预测,提高了算法收敛性. 然后利用拉格朗日应变描述索网天线结构大变形问题,结合几何非线性有限元法,建立了基于参变量的非线性平衡方程和线性互补方程;并给出了牛顿-拉斐逊迭代法与莱姆算法相结合的求解算法. 数值算例验证了本文提出的算法比传统算法具有更稳定的收敛性和更高的求解精度,特别适合于大型索网天线结构的高精度变形分析和预测.      

THE PARAMETRIC VARATIONAL PRINCIPLE AND NON-LINEAR FINITE ELEMENT METHOD FOR ANALYSIS OF ASTROMESH ANTENNA STRUCTURES

针对大型周边桁架式索网天线由拉索拉压模量不同引起的本构非线性和结构大变形引起的几何非线性问题,给出了基于参变量变分原理的几何非线性有限元方法. 首先针对含预应力索单元拉压模量不同分段描述的本构关系,通过引入参变量,导出了基于参变量及其互补方程的统一描述形式,避免了传统算法需要根据当前变形对索单元张紧/松弛状态的预测,提高了算法收敛性. 然后利用拉格朗日应变描述索网天线结构大变形问题,结合几何非线性有限元法,建立了基于参变量的非线性平衡方程和线性互补方程;并给出了牛顿-拉斐逊迭代法与莱姆算法相结合的求解算法. 数值算例验证了本文提出的算法比传统算法具有更稳定的收敛性和更高的求解精度,特别适合于大型索网天线结构的高精度变形分析和预测.     

一种几何大变形下的非线性气动弹性求解方法

非线性气动弹性的时域求解中, 涉及到非线性的流体动力学(CFD)和非线性的结构动力学(CSD)耦合问题. 基于Co-rotational理论, 推导了三维壳单元几何非线性下的切线刚阵和内力公式, 针对推进过程中的能量守恒, 引入预估-校正推进格式, 发展了一种近似能量守恒的非线性动态响应算法; 基于1/2时间步的交错耦合格式, 结合带有几何守恒律的双时间推进求解雷诺平均N-S方程的求解器, 发展了非线性气动弹性求解的高精度耦合格式. 通过结构几何大变形下的静力和动力分析验证了所发展的结构非线性求解器, 并通过AGARD445.6机翼的非线性气动弹性响应分析, 说明了所发展耦合求解方法的实用性.      

基于一阶剪切变形理论的变曲率曲梁的几何非线性方程

基于一阶剪切变形理论和轴线可伸长的精确几何非线性理论,推导了变曲率曲梁在热机载荷共同作用下的几何非线性控制方程。通过引入轴线伸长率,变形后的轴线弧长被当作基本未知量之一,基本未知量均被表示为变形前的轴线坐标的函数,使问题的求解区间仍为未变形时的曲梁轴线长度;给出了在给定曲梁轴线参数方程时,利用本文控制方程进行几何分析所需的初始曲率、变形前曲梁几何关系的数学表达式;介绍了几种常见的曲梁边界条件。所给数学模型可为轴线可伸长的变曲率曲梁的几何非线性分析和计算提供理论参考。     

一种O(n)算法复杂度的递推绝对节点坐标法研究

相比于传统的浮动坐标法,绝对节点坐标法(absolute nodal coordinate formulation,ANCF)在处理柔性体非线性大变形问题上具有显著优势,但是对于ANCF的求解目前主要依据拉格朗日方程等分析力学原理建立微分代数方程(differential algebraic equation,DAE)进行,其算法复杂度为O(n2)或O(n3)(n为系统自由度),且求解过程存在位置或速度的违约问题.据此,研究了一种O(n)算法复杂度的递推绝对节点坐标法(recursive absolute nodal coordinate formulation,RANCF).该方法采用ANCF描述大变形柔性体,借鉴铰接体递推算法(articulatedbody algorithm,ABA)思路建立多柔体系统逐单元的运动学和动力学递推关系,得到微分形式的系统动力学方程(ordinary differential equation,ODE).在ODE方程中,系统广义质量阵为三对角块矩阵,通过恰当的矩阵处理,可以得到逐单元求解该方程的递推算法.在此基础上,给出了RANCF算法的详细流程,并对流程中每个步骤进行了细致的算法效率分析,证明了RANCF是算法复杂度为O(n)的高效算法.RANCF方法保留了ANCF对大转动、大变形多柔体系统精确计算的优点,同时极大地提升了算法效率,特别在处理高自由度复杂多柔体系统中具有显著优势.并且该方法采用ODE求解,无DAE的违约问题,因此具有更高的算法精度.最后,在算例部分,通过MSC.ADAMS仿真软件、能量守恒测试、算法复杂度曲线对RANCF的正确性、计算精度和计算效率进行了验证.     

无粘结预应力混凝土板火灾行为的非线性分析

火灾试验研究表明,火灾下无粘结预应力混凝土受弯构件的变形非常大,其中转动也很大,尤其是单面受火的板,是典型的材料非线性、几何非线性问题。如何对其进行精确、高效地求解是值得关注的问题。而基于S-R分解原理的更新拖带坐标有限元法有诸多优点:有利于跟踪变形物体中各点的变形;保证单元的质量守恒;在有限元增量法求解时,还可以避免对坐标的修正;而且将转动作为一个独立的自由度,提高了求解效率。在升温过程中,虽然预应力钢筋的应力处处相等,但其各点的温度不同,导致各点的温度应变、蠕变、塑性应变均不同。预应力筋必然产生滑移。本文采用该方法对火灾下无粘结预应力钢筋混凝土板进行编程分析。通过实际算例验证该算法的可靠性,该方法求解效率高,精度也比较好。     

功能梯度变曲率曲梁的几何非线性模型及其数值解

基于弹性曲梁平面问题的精确几何非线性理论,建立了功能梯度变曲率曲梁在机械和热载荷共同作用下的无量纲控制方程和边界条件,其中基本未知量均被表示为变形前的轴线坐标的函数。以椭圆弧曲梁为例,采用打靶法求解非线性常微分方程的两点边值问题,获得了两端固定功能梯度椭圆弧曲梁在横向非均匀升温下的热弯曲变形数值解,分析了材料梯度指数、温度参数、结构几何参数等对曲梁受力及变形的影响。     

Studies of nonlinear deformation and stability of noncircular cylindrical shells in transverse bending

The variational finite element method in displacements is used to solve the problem of geometrically nonlinear deformation and stability of cylindrical shells with a noncircular contour of the cross-section. Quadrangle finite elements of shells of natural curvature are used. In the approximations of element displacements, the displacements of elements as solids are explicitly separated. The variational Lagrange principle is used to obtain a nonlinear system of algebraic equations for the unknown nodal finite elements. The system is solved by the method of successive loadings and by the Newton-Kantorovich linearization method. The linear system is solved by the Crout method. The critical loads are determined in the process of solving the nonlinear problem by using the Sylvester stability criterion. An algorithm and a computer program are developed to study the problem numerically. The nonlinear deformation and stability of shells with oval and elliptic cross-sections are investigated in a broad range of variation of the elongation and ellipticity parameters. The shell critical loads and buckling modes are determined. The influence of the deformation nonlinearity, elongation, and ellipticity of the shell on the critical loads is examined.     

Analytical modeling of the flexible rim of space antenna reflectors

This paper presents a geometrically nonlinear analytical model of the flexible cylindrical rim of a deployable precision large space antenna reflectors made of shape-memory polymer composites. A nonlinear boundary-value problem for the rim in the deformed (folded) configuration is formulated and exact analytical solutions in elliptic functions and integrals describing the deformation modes of the rim are obtained. Exact analytical solutions based on the geometrically nonlinear model are obtained and can be used to determine preliminary geometric dimensions and optimal shape of the flexible rim along with the estimation of the accumulated energy.     

Refined geometrically nonlinear formulation of a thin-shell triangular finite element

A refined geometrically nonlinear formulation of a thin-shell finite element based on the Kirchhoff-Love hypotheses is considered. Strain relations, which adequately describe the deformation of the element with finite bending of its middle surface, are obtained by integrating the differential equation of a planar curve. For a triangular element with 15 degrees of freedom, a cost-effective algorithm is developed for calculating the coefficients of the first and second variations of the strain energy, which are used to formulate the conditions of equilibrium and stability of the discrete model of the shell. Accuracy and convergence of the finite-element solutions are studied using test problems of nonlinear deformation of elastic plates and shells. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 5, pp. 160–172, September–October, 2007.     

Modelling the Stationary Vibrations and Dissipative Heating of Thin-Walled Inelastic Elements with Piezoactive Layers

A coupled dynamic problem of thermoelectromechanics for thin-walled multilayer elements is formulated based on a geometrically nonlinear theory and the Kirchhoff–Love hypotheses. In the case of harmonic loading, an approximate formulation is given using the concept of complex moduli to characterize the cyclic properties of the material. The model problem on forced vibrations of sandwich beam, whose core layer is made of a passive physically nonlinear material, and face layers, of a viscoelastic piezoactive material, is considered as an example to demonstrate the possibility of damping the vibrations by applying harmonic voltage to the oppositely polarized layers of the beam. Substantiation is given for a linear control law with a complex coefficient for the electric potential, which provides damping of vibrations in the first symmetric mode at the linear and nonlinear stages of deformation. The stress–strain state and dissipative-heating temperature are studied     

基于近场动力学的薄板弯曲大变形及断裂分析

薄板结构仅在较小的荷载下就能产生较大的位移、旋转,甚至引发结构产生裂纹并扩展,进而发生结构整体断裂,因此,建立薄板结构在大变形过程中的裂纹扩展及断裂仿真模型,具有重要的工程实际意义.文章建立了用于薄板结构几何大变形和断裂分析的近场动力学(PD)和连续介质力学(CCM)耦合模型.首先基于冯·卡门假设,采用更新的拉格朗日法得到薄板在几何大变形增量步下的虚应变能密度增量公式,并利用虚功原理和均质化假设求出几何大变形微梁键的本构模型参数;接着分别建立几何大变形薄板PD模型与CCM模型的虚应变能密度增量,并建立了薄板几何大变形PD-CCM耦合模型;最后模拟了薄板结构在横向变形作用下的渐进断裂过程,得到与实验结果高度一致的仿真结果,验证了所提出的几何非线性PD-CCM耦合模型的精度.结果表明:本文所提出的薄板PD-CCM耦合模型具有简单高效,无需考虑材料参数限制和边界效应的特点,可以很好地用于预测薄板结构在几何大变形过程中的局部损伤和结构断裂,有利于薄板结构的断裂安全评价和理论发展.     

Dynamic Problems in the Theory of Sandwich Shells of Revolution with a Discrete Core under Nonstationary Loads

The equations of motion of sandwich shells of revolution with a discrete core are analyzed within the framework of the geometrically nonlinear theory of shells and plates. An efficient numerical algorithm is constructed to investigate nonstationary wave processes in elastic structures that are discrete-inhomogeneous across the thickness. Numerical examples are given     

Nonlinear resonance responses of geometrically imperfect shear deformable nanobeams including surface stress effects

In this work, a thorough investigation is presented into the nonlinear resonant dynamics of geometrically imperfect shear deformable nanobeams subjected to harmonic external excitation force in the transverse direction. To this end, the Gurtin–Murdoch surface elasticity theory together with Reddy’s third-order shear deformation beam theory is utilized to take into account the size-dependent behavior of nanobeams and the effects of transverse shear deformation and rotary inertia, respectively. The kinematic nonlinearity is considered using the von Kármán kinematic hypothesis. The geometric imperfection as a slight curvature is assumed as the mode shape associated with the first vibration mode. The weak form of geometrically nonlinear governing equations of motion is derived using the variational differential quadrature (VDQ) technique and Lagrange equations. Then, a multistep numerical scheme is employed to solve the obtained governing equations in order to study the nonlinear frequency–response and force–response curves of nanobeams. Comprehensive studies into the effects of initial imperfection and boundary condition as well as geometric parameters on the nonlinear dynamic characteristics of imperfect shear deformable nanobeams are carried out through numerical results. Finally, the importance of incorporating the surface stress effects via the Gurtin–Murdoch elasticity theory, is emphasized by comparing the nonlinear dynamic responses of the nanobeams with different thicknesses.     

A MESHLESS LOCAL PETROV-GALERKIN METHOD FOR GEOMETRICALLY NONLINEAR PROBLEMS

Nonlinear formulations of the meshless local Petrov-Galerkin （MLPG） method are presented for geometrically nonlinear problems. The method requires no mesh in computation and therefore avoids mesh distortion difficulties in the large deformation analysis. The essential boundary conditions in the present formulation axe imposed by a penalty method. An incremental and iterative solution procedure is used to solve geometrically nonlinear problems. Several examples are presented to demonstrate the effectiveness of the method in geometrically nonlinear problems analysis. Numerical results show that the MLPG method is an effective one and that the values of the unknown variable are quite accurate.