Section properties
Hollow rectangle (box section, square corners): A, I, Z, r
- where
- area (cm²)
- width (mm)
- depth, in the direction of load (mm)
- wall thickness (mm)
- second moment of area (cm⁴)
- elastic section modulus (cm³)
- radius of gyration (cm)
Explain this equation
Works out section properties of a hollow rectangle (box section) with square corners. Use it for fabricated boxes, or as a close estimate for RHS.
The box is a solid rectangle B × H with a smaller rectangle (B − 2t) × (H − 2t) taken out. Area and second moment are found for the outer shape minus the inner. Z is I divided by the distance to the outer fibre; with I in cm⁴ and H in mm, the outer fibre distance in cm is H/20. r is √(I/A).
Properties of hollow shapes come from subtracting the hole from the solid, because both share the same centroid.
Sense check: 100 × 50 × 5 gives about 14 cm², 174 cm⁴ and 34.7 cm³. Real RHS with rounded corners comes out slightly lower.
Watch out: square corners overstate the properties of rolled hollow sections a little. Use published tables for design.
Properties of a box section about its major axis, treating the corners as square. Use for fabricated boxes or a quick check on SHS/RHS.
Enter your values to see the result.
Good to know
The section is the outer rectangle minus the hole, so A and I are found by subtraction. Rolled SHS/RHS have rounded corners, so these square-corner figures run a few per cent higher than the Blue Book; use the published tables for design of standard hollow sections. Example: 100 × 50 × 5 gives I = 173.7 cm⁴ here against about 167 cm⁴ tabulated for the cold-formed section.
Common mistake
Bending about the wrong axis. A 100 × 50 RHS is about three times stiffer with the 100 mm side vertical.
Limits
Square corners; elastic properties only. Slender walls (large B/t) may buckle locally before the elastic stress is reached; check the section classification to BS EN 1993-1-1.
Sources
- SCI / Steel for Life Blue Book: Steel building design – design data (section property formulas and tables for UKB, UKC, hot-finished and cold-formed hollow sections)
- W. C. Young, R. G. Budynas, Roark's Formulas for Stress and Strain, 8th ed., Table 8.1 (simply supported beam: UDL M = wL²/8, δ = 5wL⁴/384EI; central point load M = WL/4, δ = WL³/48EI)
- BS EN 10219-2:2019 Cold formed welded steel structural hollow sections – Part 2: Tolerances, dimensions and sectional properties (mass from density 7.85 kg/dm³; corner radius rules and area formula)
Results are calculations from the published methods and figures shown, and depend on the data you enter. Follow the welding procedure specification (WPS), BS EN ISO 15614/9606, BS EN 1090, BS EN 1993 and manufacturers’ data. Structural welds need qualified welders and engineered designs.