Moment of Inertia Calculator

The moment of inertia of a cross-section — properly called the second moment of area, symbol I — measures how the section's material is distributed relative to an axis through its centroid. It is the geometric quantity that decides how stiff and how strong a member is in bending:

• Deflection of a beam is inversely proportional to I: doubling Ix halves the sag (δ = 5qL⁴/384EI for a uniformly loaded simple span).
• Bending stress is inversely proportional to it too: σ = M·c/I, where c is the distance from the neutral axis to the extreme fiber.
• Buckling of a column depends on the weak-axis value: Pcr = π²E·Iy/L².

Because the contribution of each fiber grows with the square of its distance from the axis (I = ∫y²dA), area far from the centroid is enormously more effective than area near it. That is the entire reason the I-beam exists: it parks almost all of its steel in two flanges as far from the neutral axis as possible. The default section loaded above (an IPE 200-sized I-beam) puts 62% of its area in the flanges and gets Ix = 1 845.6 cm⁴ from just 27.2 cm² of steel — a solid square bar with the same area would manage only ~62 cm⁴, thirty times less.

Every shape here reports both axes. Ix (strong axis) governs bending of beams loaded vertically; Iy (weak axis) governs lateral-torsional behavior and column buckling. A section that looks generous in Ix can be dangerously soft in Iy — the I-beam above has Iy = 141.9 cm⁴, 13× smaller than its Ix.

For symmetric shapes like rectangles and tubes, the tabulated formula about the centroid is all you need. But the moment you compose a section from parts — a channel, an angle, a built-up girder — each part's own centroid no longer sits on the axis of the whole section. The parallel-axis theorem (Steiner's theorem) transfers each part's inertia to the common centroidal axis:

I = I₀ + A·d²

where I₀ is the part's inertia about its own centroid, A its area, and d the distance between the two parallel axes.

Take the default lipped-C section (h = 150, b = 60, c = 20, t = 2.66 mm) as a concrete example of what the calculator does internally, using thin-wall mid-line dimensions (h′ = 147.34 mm):

1. Web: I₀ = t·h′³/12 = 2.66 × 147.34³/12 = 70.9 × 10⁴ mm⁴, d = 0 → no transfer term.
2. Each flange: a thin strip of area A = b′·t = 57.34 × 2.66 = 152.5 mm² sitting at d ≈ h′/2 = 73.67 mm from the axis. Its own I₀ (≈ b′t³/12) is negligible — the A·d² term ≈ 152.5 mm² × 73.67² ≈ 82.7 × 10⁴ mm⁴ per flange is what matters.
3. Each lip: small area, still far from the axis → another meaningful A·d² contribution.

Summing all five parts gives Ix ≈ 277.9 × 10⁴ mm⁴ = 277.9 cm⁴ — exactly what the tool reports if you select the Lipped C shape. The lesson generalizes: in composed sections, the A·d² transfer terms dominate; the parts' own I₀ values are usually rounding noise. That is also why the lips on a cold-formed C profile are so cheap and so effective.

The calculator reports the complete first-order property set, because in real design you never use Ix alone:

• Elastic section modulus S = I/c (cm³ or in³) converts inertia into stress directly: σ = M/S. Sx uses the distance c from the neutral axis to the extreme fiber — for asymmetric sections (angle, channel) the calculator takes the larger c, which gives the conservative, governing stress side.
• Plastic section modulus Z is the fully-yielded equivalent, used for compact sections in LRFD/limit-state design (Mp = Z·fy). The ratio Z/S is the shape factor: 1.5 for a rectangle, ≈ 1.12–1.15 for I-beams, and ≈ 1.8–2.2 for angles. For the angle, the channel and the lipped-C the calculator computes the true plastic modulus by locating the equal-area (plastic neutral) axis of the exact rectangle decomposition — no shape-factor guessing.
• Radius of gyration r = √(I/A) (cm or in) is the column designer's number: slenderness λ = KL/r feeds every buckling curve in NBR 8800, AISC 360 and EC3. Note how the weak-axis ry of the default I-beam (2.28 cm) is nearly 4× smaller than rx — which is why columns buckle about their weak axis unless you brace it.
• Centroid (x̄, ȳ) locates the neutral axis. For doubly symmetric shapes it is trivially the middle; for the channel, lipped-C and angle the drawing above dimensions x̄ (and ȳ for the angle) explicitly — watch it shift as you change the flange width.
• Mass per meter (A × 7 850 kg/m³) turns any comparison into money: stiffness per kilogram is the real currency of steel design.

A moment of inertia number is rarely the end of the story — the practical question is "which profile do I actually buy?". That is what the closest standard profiles panel answers: your custom section is matched by Ix against CalcSteel's live 1,309-profile database — the same one behind the 3D editor: ABNT/Brazilian cold-formed (Ue, U, rounds), AISC (W, HSS, L, Pipe), European (IPE, HEA, HEB, HEM, UPN) and Indian (ISMB/ISMC) series — and the three nearest are linked straight to their profile pages, where you can run full design-code checks (NBR 8800, AISC 360, EC3) and export a PDF calculation report. Each match also shows its Iy delta and weight rank, and an optional max-depth constraint filters the catalog when your section has to fit a ceiling void or a masonry course.

The matching runs on the same section engine as those pages, so the numbers agree by construction. Sanity check you can reproduce right now: enter h = 200, b = 100, tw = 5.6, tf = 8.5 mm on the I/H shape. The engine returns Ix = 1 845.59 cm⁴ and the #1 match is IPE 200 at Δ 0.0% — because those are the IPE 200 dimensions.

One honest caveat baked into the tool: closed-form section formulas neglect root fillets (hot-rolled) and corner radii (cold-formed). Mill tables that include them run roughly 1–5% higher on Ix (tabulated IPE 200: 1 943 cm⁴ including its r = 12 mm fillets). For stiffness comparisons and preliminary sizing this is well inside engineering tolerance — and it is conservative.