This study develops new 3d frame element formulations for steel, reinforced concrete and composite structures within the framework of the three field Hu-Washizu variational principle.

The study proposes an explicit definition of the stress field over the element cross section to complement the recent proposal of a consistent variational basis for the mixed formulation of 3d frame elements. The consistent linearization of the governing equations of global equilibrium, element compatibility, and section constitution results in a robust algorithm for the element state determination.

Two features of the proposed element state determination algorithm require special attention: (a) the potential for numerical instability in the presence of an ill-conditioned or singular section stiffness matrix, and (b) the lack of objectivity for softening response. The study solves the potential numerical instability problem by eigendecomposition of the section stiffness matrix and the use of the Sherman-Morrison-Woodbury formula and the Moore-Penrose pseudoinverse to formulate a numerically robust inversion of the section stiffness matrix. 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. The success of the proposed solution is confirmed with several numerical examples of the inelastic response of perfectly plastic structural elements.

To address the lack of objectivity in the presence of softening response, this study proposes a new numerical integration of the inelastic element response for the case that inelastic deformations arise only at the element ends, a case common under extreme lateral loading conditions of building and bridge structures. The proposed scheme uses a variable inelastic zone length under strain hardening conditions, and a fixed damage zone length under softening conditions. The proposed element thus has only one monitoring point in each inelastic zone and is of comparable computational efficiency with concentrated hinge models of structural members. With its ability to monitor the spread of inelastic deformations into the structural member the new element gives accurate estimates of local response, as demonstrated with several validation examples.

By enhancing the proposed Hu-Washizu variational formulation with fields for the interface bond-slip among several components of a structural member this study formulates a new 3d frame element for the inelastic response of structural members consisting of concrete and steel, reinforcing or prestressing steel and concrete, and FRP and concrete components. The enhanced mixed formulation uses additional degrees of freedom for the relative slip at the element nodes that are formulated in the local reference system, thus facilitating the transformation of the system variables to the global reference system under large displacements on the basis of the corotational formulation.

Three alternative field formulations are investigated for the inclusion of the bond-slip effect within the framework of the enhanced mixed formulation: a mixed displacement, a mixed force, and a mixed formulation. Numerical studies of the cyclic response of steel-concrete composite members, and reinforced concrete columns with pull-out show that the mixed force formulation is superior in the representation of the local response allowing for a jump in the relative slip at the element-joint interface. The mixed force formulation is then validated further by correlation studies with experimental results of the cyclic response of 2d steel-concrete composite frame specimens, and the cyclic response of 3d reinforced concrete cantilever columns with pull-out from the foundation.