Second moment of area, explained

What it measures

I = ∫y² dA adds up every bit of area weighted by the square of its distance from the neutral axis. Area far from the axis counts far more than area near it, so moving material outward is the cheapest way to make a beam stiffer.

It is often called the area moment of inertia, and in Hebrew מומנט התמד or מומנט אינרציה של שטח. It is not the mass moment of inertia used in dynamics, which has units of kg·m². This one has units of length to the fourth power, mm⁴ or m⁴.

How it is used

Bending stress is σ = My/I, so a larger I means lower stress for the same moment. Deflection formulas all have EI in the denominator, so a larger I means a stiffer beam. Euler buckling uses the smallest I of the section.

For composite sections, find the centroid, then add each part’s own I plus its area times the squared distance from the centroid. That is the parallel axis theorem, and it is how I-beams, T-sections and box sections are calculated.

A worked number

A 20 × 100 mm plank standing on edge has I = 20 × 100³/12 ≈ 1.67 × 10⁶ mm⁴. Laid flat it has I = 100 × 20³/12 ≈ 6.7 × 10⁴ mm⁴. Same material, same area, 25 times the stiffness, which is (100/20)². That ratio is why joists stand on edge.

Common mistakes

• Cubing the wrong dimension. h is the depth in the direction of bending.
• Adding the parts’ own I values and forgetting the A·d² terms.
• Confusing the area moment (mm⁴) with the mass moment of inertia (kg·m²).