Section Modulus Calculator

Use this tool to calculate the section modulus, second area moment, and neutral axis of many structural profiles, given their dimensions.

Section modulus S83.3333 cm³8.33333e-5 m³ — I ÷ c, where c is the distance to the extreme fibre

Second moment of area I4,166,666.6667 cm⁴bh³/12, S = bh²/6

Cross-sectional area5,000 mm²

Radius of gyration28.8675 mm√(I/A) — the figure that governs buckling

Bending stress60 MPaM ÷ S

Factor of safety in bending4.1667

Moment capacity at yield20.8333 kN·m

Euler critical buckling load913.8523 kN913.852 kN with K = 1 and an effective length of 3 m

Critical buckling stress182.7705 MPa

Slenderness ratio103.92intermediate — neither pure yielding nor pure Euler buckling applies, and an empirical column formula is needed

Slenderness where buckling takes over88.9π√(E/σy) — below this the column yields first, above it buckles first

Governing failure modebucklingEuler load 913.85 kN

At twice the length228.4631 kNa quarter of the load — buckling goes as the inverse SQUARE of effective length, which is why bracing a column is so effective

With both ends fixed instead3.6554 MNfour times the pinned-end capacity, for the same material

The formula

S = I ÷ c; Euler P_cr = π²EI ÷ (KL)²

Depth matters more than area

A beam's resistance to bending comes from its second moment of area, which for a rectangle goes as the cube of depth. Turning a 50 × 100 mm joist on edge rather than flat multiplies its stiffness by four and its strength by two, with no change of material. This is why beams are deep and why an I-section puts its material as far from the neutral axis as possible.

Section modulus is I divided by the distance to the outermost fibre, and it is the figure to use for strength: bending stress is simply M/S. Second moment of area is the figure for stiffness and deflection. Two sections can share a section modulus and deflect quite differently.

Buckling is a stability failure, not a strength one, and it depends on the inverse square of effective length. Halving an effective length quadruples the critical load, which is why a single mid-height brace transforms a slender column. End conditions matter as much: fixing both ends rather than pinning them also quadruples capacity.

The Euler formula only governs slender columns. A stocky one simply squashes, and the crossover is at a slenderness of about π√(E/σy) — roughly 89 for mild steel. Between about 50 and 120 neither pure model applies and an empirical column curve is needed, so treat the Euler figure as an upper bound rather than a prediction there.