结构矩阵分析中的“平衡-几何”互伴定理

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

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{FE}e和单元变形向量{}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。

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（=[G]e）;ii） [H]e and [G]e are not adjoint matrices（≠[G]e）.3） An adjoint theorem between equilibrium matrix [H]e and geometric matrix [G]e is established.If the element internal force vector e and the element deformation vector [？]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.