About Cross-Section Properties Calculator
The cross-section properties calculator computes the geometric section properties of eight standard structural shapes: rectangle, circle, I-beam, channel, angle, hollow rectangle, hollow circle, and T-section. It is used by structural engineers to obtain the inputs needed for bending, deflection, buckling, and torsion checks.
Pick a shape, enter its dimensions in millimetres, and the tool returns the area, centroid, second moments of area Ix and Iy, elastic and plastic section moduli, radii of gyration, and the torsion constant, alongside a scaled SVG diagram that updates as you type.
How It Works
- Select one of the eight shapes and enter its dimensions; composite shapes are split into simple rectangular or circular components.
- Compute the area and centroid, then the second moment of area I about each centroidal axis using the parallel axis theorem for built-up sections.
- Derive the elastic section modulus S = I / c from the distance to the extreme fibre and the radius of gyration r = sqrt(I / A).
- Compute the plastic section modulus and the torsion constant (closed-form for solid shapes, Bredt or thin-walled approximations for hollow and open sections), then draw the section to scale.
Worked Example
A 200 x 300 mm rectangle (width b = 200, height h = 300) has area A = b * h = 60,000 mm2, Ix = b * h^3 / 12 = 200 * 300^3 / 12 = 4.5e8 mm4, Sx = b * h^2 / 6 = 3.0e6 mm3, and rx = h / sqrt(12) = 86.6 mm, with the centroid at (100, 150) mm.
Formulas
- Area (rectangle)
A = b * h- Second moment of area (rectangle, x-axis)
Ix = b * h^3 / 12- Elastic section modulus
S = I / c- Radius of gyration
r = sqrt(I / A)- Parallel axis theorem
I = I_c + A * d^2
Standards & References
- Mechanics of materials (parallel axis theorem)
- Bredt thin-walled torsion theory
Frequently Asked Questions
Which cross-section shapes can the calculator handle?
It supports eight shapes: rectangle, circle, I-beam, channel (C), angle (L), hollow rectangle, hollow circle, and T-section, each with its own dimension inputs and exact property formulas.
What is the difference between elastic and plastic section modulus?
The elastic section modulus Sx = Ix / c relates to first yield at the extreme fibre, while the plastic section modulus Zx assumes the whole section has yielded and is larger; their ratio is the shape factor used in plastic design.
How is the torsion constant calculated?
For solid circular sections it equals the polar moment J = pi * d^4 / 32; for closed hollow sections the Bredt formula J = 4 * Am^2 * t / p is used; and for open and thin-walled sections a (1/3) * sum(b * t^3) approximation applies.
Why does the centroid matter for unsymmetric shapes?
For angles, channels, and T-sections the centroid is not at the geometric centre, so the section modulus uses the larger distance from the centroid to an extreme fibre, which governs the maximum bending stress.