Section property formulas
- Rectangle: I = bh³ / 12,Z = bh² / 6
- Circle: I = πd⁴ / 64,Z = πd³ / 32
- Hollow circle: I = π(D⁴ − d⁴) / 64
- Hollow rectangle: I = (BH³ − bh³) / 12
- I-beam: outer rectangle minus the two side cutouts, about the strong axis
The radius of gyration follows fromr = √(I / A) and the section modulus from Z = I / y. These feed directly into the beam calculator: I sets the deflection, Z sets the bending stress, and r governs buckling checks.
Frequently asked questions
What is the second moment of area?+
The second moment of area (moment of inertia) I measures how far a cross-section's area is spread from its neutral axis. Bending stiffness is proportional to EI, so a beam with a larger I deflects less under the same load. For a rectangle I = bh³/12 — note the height is cubed, which is why depth does so much more work than width.
What is the section modulus?+
The elastic section modulus Z = I/y, where y is the distance from the neutral axis to the extreme fibre. Maximum bending stress is simply σ = M/Z, so a larger section modulus means lower stress for the same moment.
What is the radius of gyration?+
The radius of gyration r = √(I/A) is the distance at which the whole cross-sectional area could be concentrated to give the same second moment of area. It governs buckling: compression members are checked with the slenderness ratio L/r.
Which axis is the strong axis?+
The strong axis is the one with the larger second moment of area — for an open I-section this is the horizontal (x) axis, about which the flanges do most of the work. Bending about the weak axis uses Iy.