The Saint Venant Problem: A Kinematic Approach Based on Generalized Cross-Sectional Average Displacements

ABSTRACT:This paper presents an alternative formulation for the classic Saint Venant cylinder problem by embedding macroscopic kinematic parameters directly into the three-dimensional continuum representation from the outset. Unlike semi-inverse or tension-based traditional methods, the proposed approach decomposes the local displacement field into a rigid-body cross-sectional average kinematic field and a residual intra-sectional warping field featured by null average values and null average partial derivatives. By partitioning the out-of-plane warping into independent Dini-Neumann boundary sub-problems, the model highlights a critical epistemological dualism in structural mechanics. When cross-section rotations are defined via purely geometric integral means, the formulation yields unit shear deformability factors and restricts the torsional centre to the section centroid. Conversely, when macroscopic parameters are derived through energetic equivalence principles via the Virtual Work and Clapeyron theorems, the kinematic compatibility equations become coupled, shifting the torsional centre away from the centroid toward the shear centre. Crucially, this work demonstrates that while the choice between geometric and energetic metrics introduces an inherent arbitrariness in the macroscopic beam description, the underlying local stress and strain fields remain unique and invariant. Finally, by rigorously accounting for the energy contributions of the residual displacement fields, this formulation reconciles the apparent energy discrepancies found in classical beam theories, establishing a mathematically consistent bridge between 3D continuum mechanics and 1D structural models. HISTORY AND CREDITS:The study had its beginning in 2002, during the period of the PhD Course in Structural Engineering at the University of Pisa, as a consequence of the tutoring activity carried out within the Course in Structural Mechanics, run by Professor F. Cafarella. An initial handwritten version was issued in 2003, containing many of the principal aspects that feature the work in the present form. The study was developed in the following years, and underwent various modifications, due to the revisions of Professor S. Ligarò at first, and of Professor F. Cafarella subsequently. It assumed the present aspect approximately in 2009; but it has to be regarded as not completely fulfilled, and it is supposed to offer many hints for further studies. For this reason, any contribution or observation from the readers will be really appreciated. The author's academic collaboration and tutoring activities with Professor M. Sassu within the Structural Design Course at the University of Pisa continued seamlessly until approximately 2020, concluding upon the definitive closure of the course, which the Professor maintained until that year despite his relocation to the University of Cagliari in 2016.