Derivation of stiffness and flexibility for rods and beams by using dual integral equations

Abstract

In this paper, the dual boundary integral formulation is used to determine the stiffness and flexibility for rods and beams by using the direct and indirect methods. The stiffness and flexibility matrices derived by the dual boundary integral equations (DBIEs) are compared well with those derived by the direct stiffness and flexibility methods after considering the sign convention. Since any two boundary integral equations can be chosen for the beam problem, six options by choosing two from the four equations in dual formulation can be considered. It is found that only two options, either displacement-slope (single-layer and double-layer) or displacement-moment (single-layer and triple-layer) formulations in the direct (indirect) method can yield the stiffness matrix except the degenerate scale and a special fundamental solution. The rank deficiency is examined for the influence matrices. Not only rigid body mode in physics but also spurious mode in numerical implementation are found in the formulation by using SVD updating term and document, respectively.