Moment of Inertia Calculator

What is Moment of Inertia?

Moment of inertia (second moment of area, I) quantifies a cross-section's resistance to bending. A higher I value means the section is stiffer and deflects less under load. It depends on the shape and how material is distributed relative to the bending axis.

Key section properties: Area (A) for axial capacity, Moment of Inertia (I) for bending stiffness, Section Modulus (S = I/c) for bending stress capacity, Plastic Section Modulus (Z) for ultimate strength, and Radius of Gyration (r = √I/A) for buckling analysis.

I-beams are efficient because material is concentrated in flanges far from the neutral axis, maximizing I with minimum material. A rectangle's I = bh³/12 — doubling height increases I by 8× (cubic relationship), while doubling width only doubles I.

Formula: Rectangle: I = bh³/12, S = bh²/6 Circle: I = πd⁴/64, S = πd³/32 Hollow Rectangle: I = (BH³ - bh³)/12 Radius of Gyration: r = √(I/A)

Example Calculation

A 200×300mm rectangular beam: I = 200 × 300³/12 = 450,000,000 mm⁴ = 4.5 × 10⁸ mm⁴. S = 200 × 300²/6 = 3,000,000 mm³. If allowable bending stress = 10 MPa, max moment = S × σ = 3.0 × 10⁶ × 10 = 30 kN·m.

When to Use This Calculator

• Structural engineers calculating section properties for preliminary beam and column sizing before selecting standard steel shapes
• Mechanical engineers determining moment of inertia for custom-fabricated sections, built-up beams, or composite cross-sections
• Civil engineering students verifying hand calculations of section properties for structural analysis coursework
• Design engineers checking whether a proposed section modification (adding plates, changing flange thickness) provides sufficient bending capacity

Common Mistakes to Avoid

• Calculating Ix when Iy is needed (or vice versa) — for vertical bending, Ix (about the horizontal axis) is the relevant property; lateral buckling checks use Iy
• Using the parallel axis theorem incorrectly — when combining shapes, the transfer distance is from each component's centroid to the composite centroid, not to an arbitrary reference
• Confusing elastic section modulus (S = I/c) with plastic section modulus (Z) — S is for elastic design (working stress), Z is for plastic/ultimate strength design; Z/S ratio (shape factor) is typically 1.1-1.7
• Not converting units consistently — mixing mm and m in the same calculation produces errors by factors of 10⁶ or 10¹²; keep all dimensions in the same unit system

How to Interpret Results

• Ix controls stiffness and deflection for strong-axis (vertical) bending — when comparing candidate sections of similar area, the one with the higher Ix deflects less under the same load
• Sx is the elastic section modulus: multiply it by the allowable bending stress to get the moment capacity (M = Sx × σ) — note this is the elastic value, not the plastic modulus Z used in ultimate-strength design
• rx and ry feed the slenderness ratio KL/r for column buckling checks — the smaller radius (usually ry) governs unless the weak axis is braced
• Centroid Y locates the neutral axis from the bottom fiber — for asymmetric T-sections the tool reports the governing (smaller) Sx based on the farther fiber, so the opposite face sees a lower bending stress
• All inputs and outputs are in millimeters (mm², mm⁴, mm³) — with moments in N·mm and stresses in MPa the units stay consistent; mixing in meters introduces 10⁶-10¹² scale errors
• I-beam and channel results use idealized sharp-cornered geometry — values for rolled sections differ slightly (typically a few percent) from AISC or mill tables that include fillets and root radii

Related Standards & References

• AISC Steel Construction Manual — section property tables and design provisions for standard steel shapes
• Roark's Formulas for Stress and Strain — authoritative reference for section property and stress formulas
• Timoshenko, Mechanics of Materials — classical derivations of the second moment of area, section modulus, and radius of gyration

Frequently Asked Questions

Why is depth more important than width for beams?

Moment of inertia scales with height³ but only linearly with width (I = bh³/12). Doubling the height increases I (and stiffness) by 8×, while doubling the width only doubles I. This is why beams are always tall and narrow, and why I-beams concentrate material in the flanges.

What is the difference between Ix and Iy?

Ix is the moment of inertia about the X-axis (horizontal), resisting bending in the vertical plane. Iy is about the Y-axis (vertical), resisting lateral bending. For beams loaded vertically, Ix is the primary design value. Iy matters for lateral-torsional buckling checks.