Mixed Formulation of Nonlinear Beam Finite Element

Recent studies show that beam finite elements that enforce equilibrium rather than compatibility along the element are better suited for the description of the nonlinear behavior of frame elements. This is particularly true for elements that exhibit loss of strength and stiffness under monotonic and cyclic loads. Existing formulations, however, fail to define a consistent way of implementing the force method in the context of imposed kinematic, rather than static, boundary conditions, as is the case for element models in a standard finite element program. This paper derives the general formulation of a beam finite element from a mixed approach which points the way to the consistent numerical implementation of the element state determination in the context of a standard finite element program. The element state determination centers on a new iterative solution algorithm that is based on residual deformations rather than residual forces at the section and element level. Equilibrium is enforced in a strict sense along the element, while the section constitutive relation is satisfied within a specified tolerance when the algorithm converges. The proposed algorithm is general and can be used with any section constitutive relation.