Inelastic closed-form multilayer frame element with multiple axial-flexure-shear coupled discontinuities

The nonlinear interaction of axial force, bending moment and shear force plays an important role in the inelastic assessment of structures. Sophisticated frame finite element formulations have been developed to address this phenomenon. However, reliable inelastic responses are obtained by considering complex theories and approximation methods, which has led to a strong reliance on numerical techniques. Lately, closed-form methodologies have emerged to largely avoid this issue. Though, when this nonlinear difficult coupling is contemplated, some numerical dependencies persist. Therefore, an enhanced closed-form multilayer in-plane frame element is formulated in this work, in which the inelastic axial-flexure-shear interaction is modeled with multiple embedded axial, rotation, and transverse coupled discontinuities at arbitrary positions along the element length. For deeper understanding and for the sake of simplicity, small strains, uniform external loads and quasi-static problems are considered. The variational model is solved by pure mathematics, which naturally leads to an original boundary value problem. Consequently, flexibility and stiffness matrices are derived in closed form, which are symmetric, naturally condensed, positive definite in most cases and all their entries are non-zero values considering inelastic coupling using an innovative and simplified multilayer integration method for arbitrary cross-sections. This novel element drastically reduces drawbacks associated with traditional numerical simulations, thereby reducing computational cost and, when sufficient embedded discontinuities and layers are considered, providing high accuracy results with a single element, i.e., a mesh reduction method. In addition, the developed element is free from shear-locking, tension shift effects are naturally included, and no mathematical algorithms are necessary to guarantee internal equilibrium. Application examples validate the accuracy and numerical robustness of the formulated element.