Certified Saint-Venant torsional stiffness of solid, open and hollow polygonal sections: An exact-arithmetic primal–dual method based on closed-form analytical formulae for engineering section analysis

Abstract

The Saint-Venant torsional stiffness underpins the torsional design of prismatic structural members, yet for most of the cross-sections met in practice – cold-formed, welded and built-up profiles – no closed-form solution exists, and design falls back on family-specific handbook formulas (Bredt for tubes, the thin-strip sum for open shapes) whose accuracy is not quantified. This paper proposes a primal–dual procedure that encloses the torsion constant Jt (the stiffness being GJt) within a two-sided certified interval: the classical Ritz lower bound is complemented by a constructive upper bound built on an equilibrated polynomial stress field, and the trial spaces are chosen so that every energy integral reduces to a rational moment over a rational triangle. For any polygon with rational vertex coordinates – that is, for any cross-section drawn on a CAD plane – both endpoints and the width of the interval are therefore exact rational numbers , free of floating-point round-off; on rational triangulations the same construction covers non-convex open profiles (C-, L-, T-sections) and multiply connected hollow sections. What the engineer gains is a guaranteed and exactly reproducible reference value: the interval brackets Jt where no closed form exists, its width measures the residual uncertainty, and it serves both to verify finite-element and boundary-element torsion solvers and to bound the error of the design formulas. The method encloses the unit square within a rational interval of width ∼10−4 around Jt=0.140577; it shows that the thin-strip formula overestimates the compact rectangle by +137% at aspect ratio b/t=1 and the reference channel by +0.6% to +1.3%, whereas Bredt’s formula underestimates rectangular tubes by −1.5% to −9.2% as the wall thickness grows — errors of opposite sign that the certified interval pins down exactly. As a by-product, an exact criterion singles out the equilateral triangle as the only triangle admitting an elementary Prandtl solution.