Abstract
Mixed formulations of frame elements offer significant advantages over more traditional displacement formulations, particularly under large cyclic inelastic deformations including the effects of shear. This paper complements the recent proposal of a consistent variational basis for the mixed formulation of frame elements by supplying the explicit definition of the stress field over the crossโsection. The paper also addresses the numerical stability of the element state determination algorithm in the presence of an illโconditioned or even singular section stiffness matrix. The proposed algorithm is based on the eigendecomposition of the section stiffness matrix and uses the ShermanโMorrisonโWoodbury formula and the MooreโPenrose pseudoinverse to avoid the inversion of illโconditioned matrices in the element state determination. In the extreme case of uniform tension or uniform flexure the section flexibility matrix is split into an elastic and a plastic component before eigendecomposition. With the proposed method the inelastic response of the element under multiple perfectly plastic hinges can be successfully traced. Numerical examples demonstrate the capabilities of the approach. Copyright ยฉ 2008 John Wiley & Sons, Ltd.