Moment of Inertia (Second Moment of Area) Calculator computes section stiffness for common shapes with metric/imperial toggling and multiple workflows (solid, hollow, and I-beam). Use it to size members, compare shapes, and feed values into bending stress and deflection checks.
About the quantity. The second moment of area (I) measures how a cross-section’s area is distributed about a reference axis. Larger I means the section resists bending more effectively around that axis.
About axes and orientation. For prismatic members, I is typically evaluated about centroidal principal axes (strong vs. weak). Rectangles and I-beams are usually checked about the strong axis parallel to the flange; circles are axis-independent in plane. Off-axis values are obtained using the parallel-axis theorem or principal-axis transformation where needed.
About units. Inputs are linear dimensions; I has dimensions of length to the fourth power (e.g., cm⁴ or in⁴). The tool shows the primary unit set for your system alongside the secondary unit for quick cross-checks.
About shapes and modeling assumptions. Solid rectangles and circles use closed-form expressions. Hollow rectangles subtract the inner void from the outer shell. I-beams are composed from flange and web parts using area additivity with parallel-axis terms around the common centroidal strong axis.
Calculation. The tool evaluates closed-form expressions for each selected shape about the centroidal strong axis and, where applicable, combines component contributions (flanges + web) using the parallel-axis theorem. Geometric input changes or workflow switches recompute results immediately without extra steps.
Formulas:
- Rectangle (about centroidal strong axis): I = b · h³ / 12 (solid rectangle)
- Circle (about centroid): I = (π · d⁴) / 64 (solid circle)
- Hollow rectangle (about centroidal strong axis): I = (b · h³ − bi · hi³) / 12 where bi = b − 2t, hi = h − 2t (outer minus inner window)
- I-beam (about centroidal strong axis):
I = 2 · [ (bf · tf³)/12 + bf · tf · (h/2 − tf/2)² ] + [ tw · (h − 2tf)³ ] / 12
(sum of two flanges via parallel-axis + web about its own centroid)
Examples:
- Rectangle. b = 200 mm, h = 400 mm ⇒
I ≈ 106,667 cm⁴ (≈ 2,562.68 in⁴) - Hollow rectangle. b = 300 mm, h = 500 mm, t = 20 mm ⇒
I ≈ 101,605 cm⁴ (≈ 2,441.08 in⁴) - I-beam. h = 450 mm, bf = 200 mm, tf = 20 mm, tw = 10 mm ⇒
I ≈ 42,750 cm⁴ (≈ 1,027.07 in⁴)
Corresponding tools. Use together with:
Bending Stress Calculator,
Beam Deflection Calculator,
Section Properties,
and
Steel Weight Calculator.
