Section properties
Solid rectangle (flat bar or plate on edge): A, I, Z, r
- where
- area (cm²)
- width, parallel to the bending axis (mm)
- depth, in the direction of load (mm)
- second moment of area (cm⁴)
- elastic section modulus (cm³)
- radius of gyration (cm)
Explain this equation
Works out the section properties of a solid rectangle, such as flat bar or plate on edge, for bending and buckling checks.
A is just width times depth. I measures stiffness in bending; for a rectangle it is bh³/12, so depth matters far more than width. Z is I divided by the distance to the outer fibre, h/2, giving bh²/6; bending stress is moment divided by Z. r, the radius of gyration, is √(I/A), which is h/√12 for a rectangle. The 100, 10⁴, 1000 and 10 convert mm units to cm units used in steel tables.
Standard results from integrating over a rectangle.
Sense check: 100 × 10 flat on edge has A = 10 cm², I ≈ 83.3 cm⁴ and Z ≈ 16.7 cm³. Laid flat (b = 100, h = 10) I drops to under 1 cm⁴.
Watch out: make sure h is the dimension in the direction of the load. Getting b and h the wrong way round gives a wildly wrong stiffness.
Properties of a solid rectangular section bending about the axis parallel to its width, for stiffness and stress checks on flats, plates and lugs.
Enter your values to see the result.
Good to know
Depth h is measured in the direction the load acts. Stiffness (I) goes up with the cube of depth, so a flat bar used on edge is many times stiffer than the same bar laid flat: 100 × 10 on edge has I = 83.3 cm⁴, laid flat only 0.83 cm⁴. Results are in cm units to match the Blue Book.
Common mistake
Swapping b and h. Always put the dimension in the direction of the load as h.
Limits
Elastic properties for a solid rectangle. Thin plates on edge can buckle sideways (lateral-torsional buckling) long before the stress from Z is reached.
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)
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.