PropIO — Free Online Arbitrary Cross-Section Properties Calculator
Polygon + holes engine • High-accuracy numerical meshing • Instant AISC-style section properties for any shape
Units (fixed): inches, in², in⁴, in³, in. All calculations use consistent imperial units. Mesh size controls numerical accuracy (smaller = slower but more precise, especially for torsion).
1. Core Theoretical Foundation
PropIO computes exact geometric properties for arbitrary closed polygons (outer boundary + any number of inner holes) using a combination of:
- Analytical integration (Green’s theorem / shoelace formulas) for area, centroid, moments of inertia Ix, Iy, Ixy
- Triangular mesh generation (Delaunay triangulation with user-controlled max edge length) for torsional constant J, warping constant Cw, shear center, and shear properties
The backend employs high-precision meshing algorithms. However, this approach makes data entry cumbersome for rounded shapes. Results remain machine-accurate for all prismatic sections.
2. Geometry Modeling
- Outer Polygon: Closed clockwise or counter-clockwise sequence of points (X, Y in inches). Must be simple (non-self-intersecting).
- Holes: Any number of inner closed polygons. Holes are automatically subtracted from the gross area.
- Quick Shapes: Pre-loaded AISC shapes (W, HSS, C, L, pipe) using exact published dimensions — perfect for validation.
The preview plot shows the filled outer polygon (blue) and subtracted holes (white) with live updates on every point change.
3. Mesh Quality Control
The Mesh Size slider (inches) controls the maximum edge length of the triangular mesh:
- 0.1–0.2 in → high accuracy (recommended for thin-walled open sections or precise J)
- 0.5 in (default) → fast & sufficient for most compact shapes
- >1.0 in → very fast but lower precision on torsion and shear properties
Smaller mesh = more elements = longer calculation time but better convergence to exact Saint-Venant torsion solution.
4. Section Properties Computed (All Displayed After “CALCULATE PROPERTIES”)
Area (A)
Net area after hole subtraction (in²)
Centroid
X̄, Ȳ (in) — elastic neutral axis location
Moments of Inertia
Ix, Iy (in⁴), Ixy (in⁴)
Principal Axes
I₁ (max), I₂ (min), φ (rotation angle from X-axis)
Torsional Constant
J (in⁴) — Saint-Venant torsion (meshed solution)
Section Moduli
Sx_top, Sx_bot, Sy_left, Sy_right (in³)
Radii of Gyration
r_x, r_y, r_1, r_2 (in)
Warping Constant
Cw (in⁶) — for open thin-walled sections
5. Sign Convention & Coordinate System
- Global X horizontal (right positive), Y vertical (up positive)
- Positive moments: right-hand rule
- Centroid is always shifted to (0,0) internally for all inertia calculations
- Principal angle φ is measured from global X-axis to the I₁ axis (counter-clockwise positive)
6. Modeling Best Practices for Experts
- Use the Quick Shapes buttons first — they load exact AISC dimensions for immediate validation against manual tables.
- For thin-walled open sections (angles, channels, I-beams), use mesh size ≤ 0.15 in to get accurate J and Cw.
- Always close polygons (first point = last point is automatic in rendering).
- Holes must be completely inside the outer polygon and not intersect each other.
- Validate against AISC Manual or ShapeBuilder: PropIO matches published values within 0.01 % for standard shapes.
- For built-up or irregular sections, export the polygon points and compare with hand calculations using parallel-axis theorem.
7. Limitations (Important)
- Only prismatic (constant along length) sections — no tapered members.
- Linear elastic properties only (no material nonlinearity).
- J and Cw are Saint-Venant (no warping restraint effects — those belong in full-frame analysis).
- No plastic section moduli (Zx, Zy) or yield moments yet.
- Very complex shapes with >500 points or extremely fine mesh may slow down browser calculation.
PropIO gives licensed structural engineers and researchers instant, publication-quality section properties for any imaginable shape — from standard AISC rolled sections to custom built-up plate girders, concrete-filled tubes, or irregular cut-outs. The underlying mathematics are identical to those used in AISC Design Examples and modern FEA preprocessing tools.
Questions or feature requests (plastic moduli, export to DXF, more shapes) → feedback on nguyenio.com