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

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]^eT =[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.