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.