Hong Life CalculatorsUnderstanding the section properties calculator — concept and principle
Before you can work out how much a beam bends or how much stress it carries, you need the geometric properties of its cross-section. This calculator gives the area, the second moment of area I, the elastic section modulus Z, the radius of gyration and, for round shapes, the polar moment J, for a rectangle, a solid circle, a tube or a symmetric I-beam.
How it is calculated
- Rectangle b × h: A = bh, I = bh³/12, Z = bh²/6.
- Solid circle, diameter d (enter d in the b field): A = πd²/4, I = πd⁴/64, Z = πd³/32, J = πd⁴/32.
- Tube, outer D and inner d: A = π(D² − d²)/4, I = π(D⁴ − d⁴)/64, Z = π(D⁴ − d⁴)/(32D), J = π(D⁴ − d⁴)/32.
- Symmetric I-beam: A = 2b·t_f + (h − 2t_f)·t_w and I = [bh³ − (b − t_w)(h − 2t_f)³]/12, with Z = I ÷ (h/2).
- Radius of gyration: r = √(I/A). All lengths are in mm, so I is in mm⁴ and Z in mm³.
The principle
When a beam bends, fibres far from the neutral axis stretch or compress the most, and their resistance grows with the square of that distance. Summing area × distance² over the section gives I, which controls stiffness. Bending stress is σ = M·y/I, largest at the extreme fibre y = c, so Z = I/c turns a moment directly into maximum stress: σ = M/Z. The I-beam formula is simply a full b × h rectangle minus the two empty rectangles beside the web. Because height enters as a cube, putting material far from the centre is the most efficient way to gain stiffness.
Worked example
- Inputs: rectangle, b = 100 mm, h = 200 mm (flange and web values are ignored for this shape).
- Area: 100 × 200 = 20,000 mm².
- I = 100 × 200³ ÷ 12 = 800,000,000 ÷ 12 = 66,666,666.67 mm⁴.
- Z = 100 × 200² ÷ 6 = 666,666.6667 mm³.
- r = √(66,666,666.67 ÷ 20,000) = √3,333.33 = 57.73503 mm. J is shown as "—" because it applies only to round shapes.
| b / D (mm) | Area (mm²) |
|---|---|
| 25 | 5,000 |
| 50 | 10,000 |
| 75 | 15,000 |
| 100 | 20,000 |
| 125 | 25,000 |
| 150 | 30,000 |
| 200 | 40,000 |
How to read the result
Compare sections by I for stiffness and by Z for strength. Turning the same 100 × 200 board flat (b = 200, h = 100) cuts I by a factor of four. Radius of gyration is used in buckling: a small r means a slender, buckling-prone member.
Common mistakes
- Mixing up b and h; h is always the depth in the direction of bending.
- Using J of a rectangle or an I-beam; their torsion constant is not the polar moment.
- Forgetting unit conversions: 1 cm⁴ = 10,000 mm⁴ and 1 cm³ = 1,000 mm³.
Frequently asked questions
What is the difference between I and Z?
I measures resistance to bending deformation; Z (I divided by the distance to the outermost fibre) relates bending moment to maximum stress.
Why is the I-beam efficient?
Most of its material sits in the flanges, far from the neutral axis, so it gets a large I for little weight.
What is the plastic section modulus?
It is a different property used for full yielding of the section; for a rectangle it is bh²/4, 1.5 times the elastic Z.
Sources: W. C. Young and R. G. Budynas, Roark's Formulas for Stress and Strain, McGraw-Hill · J. M. Gere and B. J. Goodno, Mechanics of Materials, Cengage