Section Modulus & Moment of Inertia Calculator
Area, I, section modulus and weight per metre for the common structural shapes.
How it works
Second moment of area — I — is how a section resists bending, and it is almost entirely about how far the material sits from the neutral axis. For a rectangle it is bh³ ÷ 12: the cube on the depth is why a joist on edge is four times stiffer than the same joist laid flat, and eight times stiffer again if you double its depth.
Section modulus S = I ÷ c is the strength number rather than the stiffness number. Bending stress is simply M ÷ S, so if you are asking whether a beam breaks, S is what you want; if you are asking how far it sags, I is.
This is the argument for tubes and I-beams. Material near the neutral axis contributes almost nothing, so removing it costs little stiffness and a great deal of weight. A 100 × 50 × 3 rectangular tube has roughly 45% of the area of the solid bar it came from and around 80% of its I — the whole reason structural steel is shaped the way it is rather than sold as billet.
The weak-axis number is the one that gets forgotten. An I-beam is superb in the plane it was drawn in and startlingly poor at ninety degrees to it, which is why long unrestrained beams buckle sideways under loads they could easily carry if something held them straight.
Common questions
Strong axis or weak?
Strong is bending about the horizontal axis with the section standing as drawn — the deep direction. Weak is the same section pushed sideways. Nearly every efficient section is dramatically worse in the weak direction, and the ratio here says by how much.
Why is my I-beam's I lower than the table says?
Rolled sections have fillets between web and flange that add material, and tabulated values include them. This uses the idealised three-rectangle shape, typically 2–5% conservative — the right direction to be wrong in.
How do I use this with the deflection calculator?
Take I from here into the beam calculator, and use half the depth as the distance to the extreme fibre for any symmetric section. For a channel the centroid is not at mid-depth, so use the larger of the two distances — which is what this reports.
Does the material matter?
Not for I, S or area — those are pure geometry. It matters for deflection through E and for weight through density, both of which the beam calculator and the mass figure here handle.