Moment of Inertia for Standard Cross-Sections
Calculate second moment of area (moment of inertia), section modulus, and radius of gyration for rectangle, circle, I-beam, T-section, and more.
The moment of inertia (or second moment of area) of a cross-section measures how resistant that shape is to bending about a given axis. A deeper beam is exponentially stiffer than a wider one because the formula cubes the height dimension. This is why I-beams are shaped the way they are — material is placed as far as possible from the neutral axis, where it does the most work resisting bending.
Symbol: I (mm⁴ or cm⁴ or in⁴) Key formula for bending stress: σ = M × y / I = M / Z
Where:
- M = bending moment (N·mm or lb·in)
- y = distance from neutral axis to extreme fiber
- Z = I / y_max = Section Modulus (mm³) — what engineers actually use
Formulas by Section Type:
| Section | I_x (strong axis) | Area |
|---|---|---|
| Solid Rectangle | b×h³/12 | b×h |
| Hollow Rectangle | (b×h³ − (b−2t)×(h−2t)³)/12 | b×h − (b−2t)×(h−2t) |
| Solid Circle (d=diameter) | π×d⁴/64 | π×d²/4 |
| Hollow Circle (D=outer, d=inner) | π×(D⁴−d⁴)/64 | π×(D²−d²)/4 |
| I-Beam (bf=flange width, H=total height, tf=flange thick, tw=web thick) | bf×H³/12 − (bf−tw)×hw³/12 | bf×2tf + hw×tw |
Where hw = H − 2×tf for I-beams.
Section Modulus Z = I / y_max For a rectangle: y_max = h/2, so Z = b×h²/6 For a circle: y_max = d/2, so Z = π×d³/32
Radius of Gyration r = √(I/A) Used in column buckling. The Euler critical load formula is: P_cr = π²×E×I / (K×L)² Where K×L is the effective column length. A higher r means a column can be longer before buckling.
Real Example — W8×31 Steel I-Beam (AISC): Actual properties: Ix = 110 in⁴, Sx = 27.5 in³, rx = 3.47 in If a beam carries M = 50 kip·ft = 600 kip·in: Bending stress σ = M/S = 600/27.5 = 21.8 ksi (well under A36 yield of 36 ksi)
Why Section Shape Matters: For the same cross-sectional area (same amount of steel), an I-beam has roughly 3–5× the moment of inertia of a solid rectangle. This is why wide-flange beams dominate structural steel construction — they are far more efficient than square bars.
Units Tip: Use consistent units throughout. If dimensions are in mm, I will be in mm⁴, and stress will be in MPa when moment is in N·mm.