Extending Generalized Beam Theory to Allow Cross-Section Discontinuities: Application to Stiffened Thin-Walled Members

Thin-walled members with transversal/diagonal stiffeners (e.g., plate girders), from the point of view of the generalized beam theory (GBT), can be treated as beams with discontinuous cross-sections. Generally, it is hard to deal with these members with the classic GBT-based beam finite elements for the presence of stiffeners spoils the continuities of the mode amplitude functions, i.e., one of the essential assumptions in the GBT kinematics. This paper presents an extended GBT (XGBT)-based beam finite element through a combination of the classic GBT with the extended finite element method (XFEM). Specifically, the classic GBT kinematics are used to construct the standard approximation of the cross-section mid-line displacement fields of a beam finite element, and extra enrichment functions which describe the field discontinuities across the beam-stiffener joint edges are added to the standard approximation by using the partition of unity method (PUM). Furthermore, the stiffeners are modelled by shell finite elements, and they are integrated into the extended GBT-based beam model via a three-field mortar method, where the discrete Lagrange multiplier space on each beam-stiffener interface is spanned by the constant functions supported on coarsely partitioned segments over the interface. The implementation of the proposed XGBT formulation is performed via FORTRAN programming and several illustrative examples concerning the linear buckling and statically elastoplastic analyses of stiffened thin-walled beams are employed to accomplish the validation, where the XGBT results are verified by shell finite element results. Results show that the proposed XGBT approach exhibits much lower computational costs for producing equivalently accurate solutions and is much more structurally meaningful than the shell model.