Configuration Design Sensitivity Analysis of Nonlinear Structural Systems with Elastic Material*

ABSTRACT

A continuum-based design sensitivity analysis (DSA) method is presented for configuration design of nonlinear structural systems using Mindlin plate and Tim-oshenko beam theories. Both displacement and critical load performance measures are considered. Configuration design variables are characterized by shape and orientation changes of structural components. The material derivative that is used to develop the continuum-based shape DSA method is extended to account for effects of configuration design variation. The piecewise linear design velocity field, i.e., C⁰-regular, is used to support configuration design changes for a broad class of built-up structures with beams and plates. To allow use of the C⁰-design velocity field, mathematical models of beam and plate bending must be second-order partial differential equations, so that only first-order derivatives appear in the integrand of the energy equation and, thus, in the integrand of the configuration design sensitivity expression. Since the Mindlin plate and Timoshenko beam theories use displacement and rotation to describe structural response, mathematical models of beam and plate bending are reduced to second-order partial differential equations. The isoparametric finite element formulations are used for numerical evaluation of continuum design sensitivity expressions.     

Geometric and Material Nonlinear Static and Dynamic Analysis of Space Truss Structures

A formulation of the nonlinear FEM truss element that takes into account both the geometric nonlinearity and material nonlinearity of the structure is derived. The associated iterative algorithm for the application is described. Based on two inelastic material models (isotropic hardening and dynamic hardening), three space truss structures are subjected to static and to dynamic loads in order to illustrate the application of nonlinear FEM. The nonlinear FEM described can accurately trace the complex structural behavior of space truss structures, including snap-through, and buckling. The FEM results match well with the theoretical results.     

Elastoplastic Analysis with Limited Plastic Deformations and Displacements*

ABSTRACT

This paper presents solution methods for elastoplastic and shakedown analysis of linearly elastic, perfectly plastic bodies for which the conventional classical formulations of these problems are completed by constraints on overall plastic deformation and elastoplastic displacement. The methods are described in terms of nonlinear mathematical programming and provide solutions when the plastic reserves of the body are not fully exhausted, and the plastic performance and the plastic deformations are controlled. Application of the method is illustrated by an example.     

Design Component Method for Sensitivity Analysis of Built-up Structures

ABSTRACT

A “design component method” that provides a unified and systematic organization of design sensitivity analysis for built-up structures is developed and implemented. Both conventional design variables, such as thickness and cross-sectional area, and shape design variables of components of built-up structures are considered. It is shown that design of components of built-up structures can be characterized and system design sensitivity expressions obtained by simply adding contributions from each component. The method leads to a systematic organization of computations for design sensitivity analysis that is similar to the way in which computations are organized within a finite element code.     

随机参数智能桁架结构动力特性分析

文中针对随机参数压电智能桁架结构研究了基于概率的结构动力特性分析方法。建立了压电主动檑和被动杆结构材料的质量密度、弹性模量同时具有随机性时结构的刚度矩阵和质量矩阵；从结构振动的瑞利商表达式出发，利用代数综合法推导出结构特征值随机变量数字特征的计算表达式。最后通过两个算例考察了智能桁架结构物理参数的随机性对其动力特性的影响，并获得了一些有意义的结论。     

Accuracy of Finite Element Based Shape Design Sensitivity Analysis*

ABSTRACT Accuracy of finite element based shape design sensitivity analysis, for use in shape design optimization, is analyzed and tested. A shape design sensitivity formulation that uses stress averaging to treat stress constraint func-tionals is presented. A smooth boundary parameterization is introduced to avoid the "Babuska paradox" and to improve accuracy of design sensitivity analysis. Accuracy of design sensitivity analysis with displacement and hybrid stress finite element formulations is compared. Good accuracy is obtained using higher order displacement finite elements, such as linear stress triangular and 8-noded isoparametric elements.     

随机桁架结构的非平稳随机动力响应分析

本文研究了随机桁架结构在非平稳随机激励下的动力响应问题。在利用随机因子法分析随机结构动力特性的基础上，给出了一种分析随机结构非平稳随机响应的新方法。从结构非平稳随机响应的表达式出发，同时考虑桁架结构的物理参数、几何尺寸的随机性，利用求解随机变量函数矩的方法和求解随机变量数字特征的代数综合法，导出了随机桁架结构在非平稳随机激励下位移响应均方值和应力响应均方值的均值、方差和变异系数的计算表达式。通过算例，分析了结构物理参数和几何尺寸的随机性对结构位移响应均方值和应力响应均方值随机变量随机性的影响。本文方法具有对随机结构进行一次动力分析便可求得动力响应的数字特征，且可以考察结构任一参数的随机性对结构非平稳随机响应分析结果的影响之优点。     

基于区间因子法的不确定性桁架结构动力响应分析

研究了不确定性桁架结构的动力响应分析问题.在桁架结构的物理参数、几何尺寸和荷载幅值均为区间变量时,从结构响应的Dulaamel积分式出发,利用区间因子法、区间运算和振型叠加法导出了结构动力响应区间变量的表达式.通过算例考察了结构参数和作用荷载的区间分散性对结构动力响应分散性的影响,经与蒙特卡洛数值模拟结果的比较,表明文中所提出的计算模型和分析方法的合理性和可行性.     

冗余桁架结构系统可靠性分析理论与算法

在阶段临界强度分枝-约界准则的基础上结合网络分析技术,提出了适用于大中型结构自动化可靠性分析的全局临界强度分枝-约界准则和相应的算法。该算法在一级和多级搜索纵深的条件下,可严格确保不遗漏结构系统的主要失效模式,数值计算结果表明,该算法具有良好的收敛效率。     

线性随机桁架结构的平稳随机响应分析

研究了随机桁架结构的平稳随机响应问题。同时考虑结构的物理参数、几何尺寸的随机性，从结构平稳随机响应在频域上的表达式出发，利用求解随机变量函数矩的方法，导出了随机桁架结构在平稳随机激励下位移响应均方值的均值、均方差和变异系数的计算表达式。通过算例考察了随机荷载激励下结构物理参数、几何尺寸的随机性对结构随机响应的影响。     

Sensitivity Analysis of Multiple Eigenvalues*

ABSTRACT

This paper is devoted to sensitivity analysis of eigenvalues of nonsym-metric operators that depend on parameters. Special attention is given to the case of multiple eigenvalues. Due to the nondifferentiability (in the common sense) of multiple roots, directional derivatives of eigenvalues and eigenvectors in parametric space are obtained. Sensitivity analysis is based on the perturbation method of eigenvalues and eigenvectors. The generalized eigenvalue problem and vibrational systems are also investigated. Strong and weak interaction of eigenvalues are distinguished and interactions in two- and three-dimensional space are treated geometrically. It is shown that the strong interaction of eigenvalues is a typical catastrophe. Simple examples that illustrate the main ideas are presented. The results obtained are important for qualitative and quantitative study of mechanical systems subjected to static and dynamic instability phenomena.     

桁架结构几何大变形分析的精确方法

以往对桁架结构的大变形非线性分析,都是应用最小势能原理建立关于节点位移的非线性联立平衡方程,求解的工作量大,尤其对多自由度的大型复杂桁架更为突出.为了克服这个困难,本文采用两步交替迭代线性逐步逼近法,使平衡状态与变形状态协调统一,建立并求出变形后的平衡方程及其解.第一步,由已知杆件内力建立计算节点位移的连续方程并求解;第二步,由已知节点位移建立计算杆件内力的平衡方程并求解.通过多次迭代求得平衡状态与变形状态协调统一的非线性大变形分析的精确解.若干例题计算证明,本法是有效、精确的.尤其是对几何大变形桁架结构的优化设计,可将结构分析的迭代过程与优化过程相结合,省去了多次结构重分析的迭代过程,只在一次结构分析的迭代过程中即可完成优化设计,大大节省了时间.本法对扁桁架尤其有用.     

Shakedown Analysis of Nonsandwich Structures*

This paper presents a simple method of evaluating the alternating plasticity limit of structures with nonsandwich cross sections under combined stresses. Mathematical programming and matrix formulations are used. A procedure that enables the actual shakedown limit of the nonsandwich structure to be found is then given. Illustrative numerical examples are presented.     

结构可靠性优化中的灵敏度分析

文中构建了基于可靠性的工程结构优化设计数学模型，并利用二阶矩理论将结构位移和单元应力可靠性约束进行了等价化处理，在此基础上导出了位移和应力的可靠性指标函数对设计变量的灵敏度计算表达式。这一表达式简洁规范并具有普遍性，对于类似以二阶矩形式表出的可靠性指标函数的灵敏度求解均成立。算例表明本文给出的基于可靠性的结构敏度计算公式是有效和正确的。     

Sensitivity Analysis of Problems of Elasticity with Unilateral Constraints*

ABSTRACT A unified approach is developed for sensitivity analyses of unilateral problems for discrete and continuous elastic structures. Sensitivity analyses are crucial in problems of redesign or optimization of structures. The topic covers, in this paper, the problem of determining the variation of structural response, subject to variations of ihe design, material, and loading. This type of problem is inherently nonlinear and nondifFerentiable for structures with unilateral constraints and, in general, only directional sensitivities can be obtained. Employing modern methods of functional analysis and the principle of minimum potential energy, a general result is presented for determination of sensitivities, and the implications for discrete or discretized structures are discussed in some detail. Four example problems are used to illustrate different aspects of the sensitivity analysis and to show how to handle variations of unilateral constraints.     

模糊桁架结构有限元分析的随机因子法

研究模糊桁架结构在模糊力作用下的有限元分析方法。考虑桁架结构材料物理参数、几何尺寸和外荷载同时为模糊变量，利用信息熵将模糊变量转变为随机变量。基于随机因子法，利用代数综合法推导出结构位移和应力响应的均值、方差的计算表达式。通过算例，分析了结构物理参数、几何尺寸和外荷载的模糊性对结构位移和应力响应的影响，并验证了本文模型和方法的合理性与可行性。本方法的优点是能够反映结构某一参数的模糊型对结构响应的影响。     

任意光滑梯度变化的功能梯度材料纯弯曲梁的弹塑性分析

假设功能梯度材料为理想弹塑性材料,其屈服强度和弹性模量均沿梁的高度方向按任意光滑函数连续变化,在小变形及平截面假定下,导出了功能梯度材料纯弯曲梁弹性极限弯矩及塑性极限弯矩的解析表达式,建立了弹塑性应力状态下截面弯矩和截面的弹、塑性应力分布之间的解析关系．研究表明,功能梯度材料梁存在多种可能的屈服模式,其最先屈服的点不一定位于截面应力最大处,而可能位于截面的其他任意位置;屈服强度及弹性模量的梯度变化对梁的弹塑性力学性能有很大影响．研究结果可为功能梯度材料纯弯曲梁的弹塑性问题研究提供一定的参考．     

Optimal Material Layout in Three-Dimensional Elastic Structures Subjected to Multiple Loads*

A method is presented for the optimal layout of material in three-dimensional elastic structures subjected to multiple loads. The problem is solved using a moment formulation to characterize optimal anisotropy of the material at each point of the domain. The method has important practical applications, e.g., in design of cross sections of prismatic bars and in optimal design of laminates.     

广义增量法在三维扁平桁架结构的弹塑性屈曲分析中的应用

本文利用广义逆矩阵,提出了解决非线性屈曲问题的广义增量法。并用该法分析了非线性荷载-位移关系曲线(其中包括极值点)。最后,给出一数值算例—三维扁平桁架结构的弹塑性屈曲分析,证明了本文方法的正确性和实用性。