Hollow Square (SHS) Section Properties Calculator
Enter the outer side and wall thickness of a square hollow section and get its area, second moments, section moduli, radius of gyration and torsion constant, drawn to scale.
Section dimensions
Enter the dimensions in the selected unit — the drawing and the results update as you type. The drawing uses centroidal coordinates: the bold gridlines are the x and y axes through the centroid C.
Section
- Section with its dimensions
- Centroid C, centroidal axes x and y
- Principal axes 1 and 2, angle θp
- Shear centre S
Square hollow sections combine equal stiffness about both axes with a closed profile, which is what makes them the default choice for columns, machine frames and anything loaded in torsion. A closed section resists torsion by a shear flow around its wall, and its torsion constant — from Bredt’s formula — is orders of magnitude higher than that of an open section of the same weight.
The calculator treats the section as an outer square minus an inner square of side a − 2t, with sharp corners. Real cold-formed SHS have rounded corners (outer radius about 2t), which reduces area and I by a few percent; hot-finished sections come closer to the sharp-cornered idealisation.
How the hollow square properties are calculated
- Inner side
- ai = a − 2·t
- Area
- A = a² − ai²
- Second moment of area
- Ix = Iy = (a⁴ − ai⁴) / 12
- Elastic section modulus
- S = (a⁴ − ai⁴) / (6·a)
- Plastic section modulus
- Z = (a³ − ai³) / 4
- Torsion constant (Bredt)
- J = 4·Am²·t / pm = (a − t)³·t, mid-line area Am = (a − t)², mid-line perimeter pm = 4·(a − t)
Principal moments and axes follow from Ix, Iy and Ixy with I1,2 = (Ix + Iy)/2 ± √[((Ix − Iy)/2)² + Ixy²] and tan 2θp = −2·Ixy/(Ix − Iy); the radii of gyration are r = √(I/A) and the elastic section moduli S = I/c for each extreme fibre. Definitions of every property are in the glossary on the section properties overview.
Assumptions. Sharp corners, uniform wall. Bredt’s torsion constant is a thin-walled result; it is within a few percent for a/t above about 10.
Other sections
Frequently asked questions
Why is a hollow square so much better in torsion than an I-beam?
Because it is closed. In a closed section the shear stress flows around the whole wall with a lever arm equal to the section size, giving J = 4·A_m²·t/p_m. An open section can only develop shear stress across the thickness of each plate, giving J = Σb·t³/3 — for the same material a closed 100×100×5 SHS has a J about 500 times larger than the same four plates left open.
How does the plastic modulus of a box compare to its elastic modulus?
The shape factor Z/S of a thin-walled box is close to 1.12–1.20, much lower than the 1.5 of a solid rectangle, because most of the material already sits at the extreme fibres where it yields first. The calculator gives both values so you can read the ratio directly.
Does the calculator account for the corner radii of cold-formed sections?
No — it uses sharp corners, which is also what most textbook tables assume. For a cold-formed SHS with an outer corner radius of 2t the area and I are typically 2–4% lower than the sharp-cornered value; use the manufacturer’s table when that matters.
References & further reading
- Torsion constant — Wikipedia — why J differs from the polar moment for non-circular sections, with the thin-walled formulas.
- Section modulus — Wikipedia — elastic and plastic section modulus, with a table of formulas by shape.
- List of second moments of area — Wikipedia — closed-form Ix and Iy for the common shapes, used to check this calculator.
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