Variationally consistent state determination algorithms for nonlinear mixed beam finite elements Original Research Article

This paper presents and evaluates several variationally consistent state determination algorithms for geometric and material nonlinear beam finite elements based on the Hellinger–Reissner (HR) assumed stress variational principle. The specifically targeted element type is derived from the two-field (displacement and generalized stress) form of this principle. Momentum balance and strain compatibility are expressed in a weak-form, and the constitutive equations are satisfied directly at each integration point (i.e., section) along the element length. A specific formulation of one element of this type is presented in a companion paper. Interelement compatibility is not enforced in the dual (stress resultant) variables for this element, and thus various state determination algorithms can be formulated at the local (i.e., element and section) levels. Four state determination algorithms, referred to as the L–L, L–N, N–L and N–N procedures, are considered. The first symbol in these names indicates the element level and the second denotes the section level. Furthermore, the symbol N indicates that local nonlinear iteration is performed prior to returning to the higher level, whereas the symbol L indicates that only the linearized equations are satisfied at the corresponding local level. The L–L algorithm corresponds to conventional Newton iteration on both the element and section levels, whereas the N–N algorithm entails the iterative satisfaction of the nonlinear element compatibility and section constitutive equations for each global iteration. The performance of these algorithms is evaluated and compared. It is found that the algorithms in which the local nonlinear strain–displacement compatibility conditions are solved iteratively at the element level for every global iteration lead to fewer global iterations at the expense of iterations at the element level. In particular, the N–N algorithm, is overall the most efficient of the four algorithms for the problems tested. Also, the N–N algorithm has substantially smaller element storage requirements. The N–N algorithm is expected to be particularly advantageous for cases in which a large number of elements remain elastic and inelasticity is concentrated in a few elements.