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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 the structural matrix analysis, the balance analysis of “external force - internal force ” and its geometric analysis [G] and the geometric matrix [G] of the balance matrix [H], “displacement- deformation And two main matrices. This paper presents and demonstrates the co-association theorem between the equilibrium matrix [H] and the geometric matrix [G]. The following four points are discussed: 1) The equilibrium matrix [H] e and the geometric matrix [G] e of the bar element e are established. It is pointed out that the representation of [H] e and [G] e is not unique. (Which gives two different forms of Scheme I and Scheme II); 2) it is pointed out that [H] e and [G] e can form various combinations, some of which are interdependent combinations [ie, e is a transposed matrix), and some are not intercompound combinations. 3) Establish the ”equilibrium-geometry“ co-operation theorem: if the selected element force vector {FE} e and element deformation vector { The balance matrix [H] e and the geometrical matrix [G] e will be the mutual matrices; 4) The ”equilibrium-geometry" companion theorem can be deduced from the principle of virtual work. Although the two are in different forms, the two are interoperable.