A Symmetry transformation method for chiral boundary based on minimum substructure model of progressive collapse and least action principle

In the analysis of structural progressive collapse, the asymmetry of the chiral boundary¹ in the minimum substructure prevented the simplification of the model. This study proposed a method for converting chiral boundaries based on the principles of virtual work and least action. Firstly, common patterns in the area of direct collapse effect (ADCE) under failure conditions of corner, perimeter, and internal columns were extracted based on the finite element results of NIST buildings. A minimal substructure model was constructed with boundary conditions as the primary differentiating factor. Based on the principles of virtual work and least action, a theoretical calculation method for converting chiral boundaries into equivalent symmetric boundaries was derived. Energy and deformation difference coefficients were introduced to control the conversion error. Reliability validation of the simplification method was performed based on finite element method from both 2D and 3D perspectives. Comparative results from the 2D model showed that the error in the collapse response between substructures with equivalent and chiral boundaries was below 10 %. The spatial surface boundaries of the equivalent stiffness were characterized based on the Fourier series expansion. Constructed a three-dimensional spatial function of equivalent stiffness and a nonlinear interaction relationship of three stiffness. The comparison results of the 3D model further confirmed that the load-displacement response and energy dissipation differences between the equivalent and chiral constraints were less than 5 %, and exhibited similar substructure failure modes, thereby validating the reliability of the method in complex three-dimensional scenarios. Further parametric analysis revealed that an increase in horizontal constraint stiffness could alter the deformation modes and failure paths of the substructure, highlighting the coupling behavior between horizontal constraint stiffness and the catenary effect. The innovation of this study lies in the proposed chiral-to-equivalent symmetric transformation theory based on the energy method, which breaks through the traditional minimal substructure model’s dependence on symmetric boundaries and simplifies the calculation process for nonlinear boundaries. The study systematically quantified the impact mechanism of boundary constraints on the structural resistance to collapse.