结构矩阵分析中的&ldquo;平衡几何&rdquo;互伴定理

龙驭球

结构矩阵分析中的“平衡几何”互伴定理

龙驭球

清华大学土木工程系, 北京 100084

详细信息

中图分类号: TU311.4
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出版历程
- 收稿日期: 2012-05-08
- 修回日期: 2012-05-08
- 刊出日期: 2012-05-24

AN ADJOINT THEOREM BETWEEN EQUILIBRIUM MATRIX AND GEOMETRIC MATRIX IN STRUCTURAL ANALYSIS

LONG Yu-qiu

Department of Civil Engineering, Tsinghua University, Beijing 100084, China

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Corresponding author:
LONG Yu-qiu: 龙驭球

摘要

摘要: 在结构矩阵分析中,“外力-内力”之间的平衡分析及其平衡矩阵[H],“位移-变形”之间的几何分析及其几何矩阵[G],是两大主题和两个主要矩阵。该文提出并论证平衡矩阵[H]与几何矩阵[G]之间的互伴定理。分四点论述:1) 建立杆件单元e 的平衡矩阵[H]^e和几何矩阵[G]^e,指出[H]^e和[G]^e的表示形式不是唯一的,有多种方案可供选择(该文给出方案I 和方案II 两种不同形式);2) 指出[H]^e和[G]^e可形成多种组合,其中有的是互伴组合(即[H]^e与[G]^e互为转置矩阵),有的不是互伴组合;3) 建立“平衡-几何”互伴定理:如果所选取的单元内力向量{F_E}^e和单元变形向量{Λ}^e 互为共轭向量,则其平衡矩阵[H]^e 和几何矩阵[G]^e 必为互伴矩阵;4) 应用虚功原理可导出“平衡-几何”互伴定理。虽然两者的表述形式不同,但两者是互通的。
- 结构矩阵分析 /
- 互伴定理 /
- 平衡矩阵 /
- 几何矩阵 /
- 虚功原理
Abstract: In structural matrix analysis, the equilibrium matrix [H] and the geometric matrix [G] are two basic matrices. In this paper, an adjoint theorem between the equilibrium matrix [H] and the geometric matrix [G] is presented and proved. The discussion is divided into four parts: 1) The equilibrium matrix [H]^e and the geometric matrix [G]^e for the element e are established. There exist several different expressions for [H]^e and for [G]^e. In this paper two different expressions (version I and version II) are given for examples. 2) The relationship between [H]^e and [G]^e can be classified into two different cases: i) [H]^e and [G]^e are adjoint matrices ( [H]^{e^T} =[G]^e); ii) [H]^e and [G]^e are not adjoint matrices ( [H]^eT ≠[G]^e). 3) An adjoint theorem between equilibrium matrix [H]^e and geometric matrix [G]^e is established. If the element internal force vector [F_E]^e and the element deformation vector [Λ]^e are conjugate vectors, then the equilibrium matrix [H]^e and the geometric matrix [G]^e are adjoint matrices. 4) The adjoint theorem between [H]^e and [G]^e is proved by the principle of virtual work.
- structural matrix analysis /
- adjoint theorem /
- equilibrium matrix /
- geometric matrix /
- principle of virtual work

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参考文献(11)

[1]	龙驭球, 包世华, 匡文起, 等. 结构力学I——基本教程[M]. 第2 版. 北京: 高等教育出版社, 2006, 第9 章. Long Yuqiu, Bao Shihua, Kuang Wenqi, et al. Structural mechanics I-Fundamental course [M]. 2nd ed. Beijing: Higher Education Press, 2006, Chapter 9. (in Chinese)
[2]	龙驭球, 包世华, 匡文起, 等. 结构力学II——专题教程[M]. 第2 版. 北京: 高等教育出版社, 2006, 第13 章. Long Yuqiu, Bao Shihua, Kuang Wenqi, et al. Structural mechanics II-Advanced course [M]. 2nd ed. Beijing: Higher Education Press, 2006, Chapter 13. (in Chinese)
[3]	袁驷. 程序结构力学[M]. 第2 版. 北京: 高等教育出版社, 2008, 第4 章-第6 章. Yuan Si. Programming structural mechanics [M]. 2nd ed. Beijing: Higher Education Press, 2008, Chapter 4-6. (in Chinese)
[4]	Zienkiewicz O C, Taylor R L. The Finite Element Method [M]. Vol. I. Oxford: Butterworth Heinemann, 2000.
[5]	龙驭球, 龙志飞, 岑松. 新型有限元论[M]. 北京: 清华大学出版社, 2003. Long Yuqiu, Long Zhifei, Cen Song. High performance finite element method [M]. Beijing: Tsinghua University Press, 2003. (in Chinese)
[6]	Long Yuqiu, Cen Song, Long Zhifei. Advanced finite element method in structural engineering [M]. Springer, Beijing: Tsinghua University Press, 2009.
[7]	龙驭球, 刘光栋, 何放龙, 等. 能量原理新论[M]. 北京: 中国建筑工业出版社, 2007, 第8 章. Long Yuqiu, Liu Guangdong, He Fanglong, et al. New monograph of variational principles [M]. Beijing: China Architecture & Building Press, 2007, Chapter 8. (in Chinese)
[8]	Wang C K. Intermediate structural analysis [M]. New York: Mc-Graw-Hill, 1985.
[9]	Wang C K. Structural analysis on microcomputers [M]. Oxford: Macmillan, 1986.
[10]	Wang C K, Salmon C G. Introductory structural analysis[M]. London: Prentice-Hall, 1984.
[11]	龙驭球, 包世华, 匡文起, 等. 结构力学II——专题教程[M]. 第3 版. 北京: 高等教育出版社, 2012, 第14 章. Long Yuqiu, Bao Shihua, Kuang Wenqi, et al. Structural mechanics II-Advanced course [M]. 3rd ed. Beijing: Higher Education Press, 2012, Chapter 14. (in Chinese)