A Work-Conjugate Matrix Formulation for the Saint-Venant Problem: Compliance Symmetry, Stiffness Coupling, and Reduction to Technical Beam Theory

In a previous study, the classical Saint-Venant problem was solved through a direct kinematic approach based on cross-sectional geometric average displacements and rotations. While this geometric representation preserves an uncoupled constitutive form, it formally yields unit shear factors and forces the center of twist to coincide with the section centroid.Moving forward from those analytical foundations, this paper establishes a comprehensive matrix framework to trace the transition from geometric averages to the classical work-conjugate energetic formulation. By introducing explicit kinematic transformation and static transport operators, the evolution of the cross-sectional compliance matrix is mapped across three distinct stages: starting from a diagonal symmetric form referred to the shear center, passing through a transient asymmetric energetic configuration, and finally recovering a pseudo-symmetric centroidal structure. It is shown that the residual asymmetry in the centroidal formulation is governed exclusively by Poisson’s ratio effects separating the static shear center from the energetic center of twist.