Dynamic Torsional Resistance in Complex-Geometry Structures A Multiscale Mathematical and Physical Framework for Non-Symmetric Architectural Forms

Contemporary architectural structures increasingly employ twisted, curved, folded, re-entrant, free-form, and non-symmetric geometries. These configurations generate strong coupling between translational motion,bending, torsional rotation, warping, diaphragm distortion, and nonlinear material response. Classical structural models based on independent bending and torsion are therefore insufficient for accuratelypredicting the dynamic resistance of complex structures.This paper presents an advanced mathematical and physical framework for analysing the dynamic torsional resistance of structures with complex geometries. The proposed methodology combines threedimensional continuum mechanics, coupled translational–torsional dynamics, warping torsion, nonlinear finite-element analysis, anisotropic constitutive modelling, and structural optimization. The fundamentalequations are derived from the balance of linear momentum, angular momentum, energy conservation,and constitutive relations.The results of the theoretical formulation show that torsional resistance is controlled not only by the torsional stiffness, but also by the spatial distribution of mass, the eccentricity between the centre of massand the centre of rigidity, the warping restraint, the damping matrix, the interaction between natural frequencies, and the anisotropy of the structural material. The framework is applicable to twisted towers,irregular buildings, free-form shells, spatial trusses, architected materials, and structural metamaterials.