PropIO — Free Online Arbitrary Cross-Section Properties Calculator

Polygon + holes engine • High-accuracy numerical meshing • Instant AISC-style section properties for any shape

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

• Custom Polygon Builder: Primary workflow — enter vertices directly or paste from a spreadsheet. Any simple closed polygon works (asymmetric tees, tapered legs, irregular plates).
• Holes: Any number of inner closed polygons subtracted from gross area. Edit hole vertices or place templated voids with X/Y offset.
• Shape Templates: Optional shortcuts that generate vertices from standard dimensions (HSS, W-shapes, angles). Edit points after applying.
• Quick Shapes: One-click AISC presets — W12×26, HSS tubes, C10×15.3, L4×4×½, NPS 6 pipe, rectangle.

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”)

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 to the I₁ (maximum inertia) axis, counter-clockwise positive, reported in [-90°, 90°]. Axis-aligned rectangles show φ ≈ 0°.
• After solving, the preview marks the centroid (green ×), shear center (orange ◇), I₁ axis (purple dashed), load vectors (P, Mx, My), and σ max/min markers when biaxial loads are set. Editing geometry clears overlays until you recalculate.

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 lie inside the outer polygon. Edge cutouts may share an outer edge — PropIO auto-insets those edges by 0.001 in at solve so the mesh can form.
• See the L-shape with rectangular cutouts example for a working edge-cutout pattern (flush geometry + auto-inset).
• 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. Biaxial flexure stress (after solve)

Enter axial force P (lb) and bending moments Mx, My (in-lb). Choose Global X / Y when loads are defined in PropIO coordinates (default), or Principal I₁ / I₂ when moments align with principal axes — both give the same stress for the same global Mx/My. Normal stress at each boundary vertex:

σ = P/A + [Mx(Iyy·y′ − Ixy·x′) + My(Ixy·y′ − Ixx·x′)] / (Ixx·Iyy − Ixy²)

x′, y′ measured from the centroid. σ_max and σ_min are taken over outer and hole vertices. Optional Fallow (psi) yields margin of safety MS = Fallow/|f| − 1 with |f| = max(|σ_max|, |σ_min|). Units: psi, lb, in-lb. Not an AISC code check — compare to your allowables.

8. 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).
• Elastic section moduli (S_x, S_y) and plastic moduli (Z_x, Z_y) are not reported — use biaxial stress at boundary vertices after solve.
• Very complex shapes with >500 points or extremely fine mesh may slow down browser calculation.