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

In this paper, the dual boundary integral formulation is used to determine the stiffness and flexibility matrices for rods and beams by using the direct and indirect methods. 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- and double-layer) or displacement-moment (single- and triple-layer) formulations in the direct (indirect) method can yield the stiffness matrix except the degenerate scale and a special fundamental solution. 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. Both the rigid body mode and spurious mode can be extracted out from the right and left unitary vectors of the influence matrices by using SVD.