Derive elastic section modulus S = I/c and radius of gyration r = √(I/A) from a section's properties, with an optional required-S check from moment and allowable stress.
Enter moment of inertia I, extreme-fibre distance c, and area A for S = I/c and r = √(I/A).
Optional: enter a moment and allowable stress to size the required section modulus.
About this tool
The Section Modulus & Radius of Gyration Calculator is a free, in-browser tool that turns a cross-section's properties into the two numbers used to size members: the elastic section modulus S = I/c and the radius of gyration r = √(I/A). Enter the area moment of inertia I, the extreme-fibre distance c, and the area A. Everything is computed locally in your browser, so your inputs stay on your device.
The section modulus S = I/c relates bending moment to stress through σ = M/S, so a beam with a larger S carries more moment for the same stress. The radius of gyration r = √(I/A) measures how far the area is effectively spread from the axis and feeds directly into the column slenderness ratio KL/r. In the optional sizing mode, enter a bending moment M and an allowable stress σ_allow and the tool computes the required section modulus S_req = M/σ_allow, then reports whether the section you entered (S) passes or fails against it. Choose SI (mm⁴, mm, mm², N·m, MPa) or imperial (in⁴, in, in², lbf·in, psi).
As a worked example, a section with I = 10,000,000 mm⁴, c = 100 mm, and A = 5,000 mm² has S = I/c = 100,000 mm³ and r = √(I/A) = √2,000 ≈ 44.7 mm. If it must carry M = 8,000 N·m at an allowable stress of 100 MPa, the required modulus is S_req = 8,000 × 1000 / 100 = 80,000 mm³ — so the 100,000 mm³ section passes with margin. Section modulus is the elastic value; plastic design uses the plastic modulus Z instead.
Frequently asked questions
What is the section modulus?
The elastic section modulus is S = I/c, where I is the area moment of inertia and c is the distance to the extreme fibre. It links bending moment to peak stress through σ = M/S, so a larger S means a section resists bending at a lower stress. Units are mm³ or in³.
What is the radius of gyration used for?
The radius of gyration r = √(I/A) describes how far a section's area is spread from an axis. It is the key term in a column's slenderness ratio KL/r: a larger r means a stockier, more buckling-resistant column for the same length and end conditions.
How does the required-S check work?
Enter a bending moment M and an allowable stress σ_allow and the tool computes the minimum section modulus needed, S_req = M/σ_allow. It then compares your section's actual S = I/c against S_req and reports PASS if S ≥ S_req or FAIL otherwise. Leave M and σ_allow blank to skip this check.
Is this elastic or plastic section modulus?
This is the elastic section modulus, S = I/c, valid while stresses stay below yield. Plastic (limit-state) design of steel uses the plastic section modulus Z, which is larger; if you are designing to a plastic capacity, use Z rather than the elastic S computed here.