Exact section properties for curved geometry

Draw a circular hollow section in most cross-section tools and you have not drawn a circle. You have drawn a 64-sided polygon that renders like a circle, and every property that follows — area, second moment of area, section modulus — carries the polygon’s error, not the circle’s.

For a fine enough facet count the error is small. It is also invisible, unbounded by anything you can see, and different from the closed-form value your reviewer will check it against. When your $I_x$ disagrees with the handbook in the fourth significant figure, you get to spend an afternoon finding out whether that is the faceting or a real mistake.

Two computation paths, chosen on purpose

GCS stores curved edges as bulge factors: each vertex optionally carries a value describing the circular arc leaving it, where $b = \tan(\theta/4)$ for included angle $\theta$. A bulge of zero is a straight segment; a magnitude of one is a semicircle. Centre and radius are never stored — they are derived on demand, because storing them duplicates information the endpoints already carry and creates a state that can drift out of agreement with itself.

From that representation, quantities split cleanly into two families.

Exact from the arc. Area, centroid, $I_{xx}$, $I_{yy}$, $I_{xy}$, elastic and plastic section moduli, extreme fibre distances, and perimeter are integrated analytically over the arc. A circle is a circle. Its $I$ is $\pi d^4 / 64$, to machine precision, and it matches the handbook because it is the handbook formula’s answer.

Faceted, because the method requires it. Fibre and ultimate-limit-state integration, neutral-axis sweeps, serviceability stress fields, and meshing work on a discretised section by their nature. There is no closed form to preserve — the discretisation is the method, and its refinement is a modelling decision with a known convergence behaviour.

The rule we hold ourselves to is that no analysis path may derive its own area or second moment from facet sums where the exact quantity is also displayed. One quantity, one computation path. Otherwise the properties panel and the capacity calculation drift apart by a fraction of a percent, and nothing in the product ever tells you which one to believe.

Where the approximation stays

Not every tool gets an exact tier, and that is deliberate rather than unfinished. Cold-formed steel sections are analysed as thin-walled strips — the quantities are inherently strip-discretised, and sub-dividing corners at a tolerance is standard practice in finite strip analysis. Adding an “exact” tier there would mean computing a number by a method the design standard does not use.

The distinction we care about is not exact-versus-approximate. It is whether the approximation is a choice you can see and reason about, or an artefact of how the geometry happened to be stored.

The practical version

If you draw an arc in GCS, the section properties you get back are the arc’s, not a polygon’s. If you export the report, the numbers agree with the panel. And if you check them by hand against a textbook, they will match — which is, after all, the only reason to compute them.