An efficient direct analysis of fire-exposed spatial steel frames using flexibility-based beam-column elements

An efficient direct analysis method is developed for second-order inelastic analysis of fire-exposed spatial steel frames using flexibility-based beam-column elements. The method is implemented through a three-dimensional flexibility-based plastic-zone formulation that integrates force-based member equilibrium, fiber-level temperature-dependent constitutive updating, and sectional flexibility integration. Fire-induced free thermal strain is represented by generalized sectional thermal deformation, including axial thermal elongation and thermal curvature. The thermal curvature is incorporated into both the force-based compatibility relation and the recovered transverse displacement field for state-dependent force interpolation, enabling thermal bending to interact with initial member imperfections and member-level P-δ effects. During incremental fire analysis, the fiber state, sectional flexibility, and member resisting force are updated consistently. Numerical examples involving members, planar frames, and spatial frames show that the one-element-per-member model agrees well with refined ABAQUS beam models and captures nonlinear displacement amplification under uniform and non-uniform heating. The proposed method enables efficient analysis while retaining fiber-level inelasticity, thermal curvature, initial imperfections, and member-level second-order effects.