Hollow Rectangle (RHS) Section Properties Calculator
Enter the outer width, outer height and wall thickness of a rectangular hollow section and get its area, second moments, section moduli, radii 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
The rectangular hollow section is the box-beam of machine bases, crane girders and welded frames: a strong axis for bending, a closed wall for torsion, and flat faces to weld and bolt to. Its second moment is the outer rectangle minus the inner one, Ix = (b·h³ − bi·hi³)/12, which for thin walls is dominated by the two flanges placed far from the neutral axis.
Because the wall is uniform, the geometry is fully defined by b, h and t; the inner dimensions are b − 2t and h − 2t. The torsion constant follows Bredt’s closed-section formula on the wall mid-line.
How the hollow rectangle properties are calculated
- Inner sides
- bi = b − 2·t, hi = h − 2·t
- Area
- A = b·h − bi·hi
- Second moments of area
- Ix = (b·h³ − bi·hi³) / 12, Iy = (h·b³ − hi·bi³) / 12
- Elastic section moduli
- Sx = 2·Ix / h, Sy = 2·Iy / b
- Plastic section moduli
- Zx = (b·h² − bi·hi²) / 4, Zy = (h·b² − hi·bi²) / 4
- Torsion constant (Bredt)
- J = 4·Am²·t / pm, Am = (b − t)·(h − t), pm = 2·(b − t + h − 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 thickness on all four sides. Bredt’s torsion constant is a thin-walled approximation.
Other sections
Frequently asked questions
Should the long side be vertical or horizontal?
Vertical for bending stiffness: I scales with the cube of the depth, so a 100×60×4 box has I_x = 1.63·10⁶ mm⁴ on edge but only 0.70·10⁶ mm⁴ flat — the principal angle in the results switches from 0° to 90° when you swap b and h. Flat is preferred only when the load can come from any direction or when lateral-torsional stability governs.
How is the torsion constant of a box calculated?
With Bredt’s formula J = 4·A_m²·t / p_m, where A_m is the area enclosed by the wall mid-line and p_m its perimeter. It assumes the shear flow is uniform through the wall, which is accurate when the wall is thin compared with the sides; the calculator flags the value as an approximation for that reason.
Can I model a welded box from four plates of different thickness?
Not with this page, which assumes one wall thickness. A box with thicker flanges than webs is best handled as a sum of rectangles with the parallel-axis theorem — the I / H section page shows the same technique for an open profile.
References & further reading
- Second moment of area — Wikipedia — definition, parallel-axis theorem and the sign convention for the product of inertia.
- 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.
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