Moment of inertia

The second moment of area, usually called the moment of inertia I in structural work, measures how the area of a cross-section is spread about an axis. It is not about mass or motion here; it is purely geometry. Area far from the axis counts much more than area close to it, because each small piece contributes its area times the square of its distance. That is why an I-section, with most of its steel in the flanges, is so much stiffer in bending than a solid square bar of the same weight.

I appears in almost every beam and column formula. Bending stress is M y / I, beam deflection is inversely proportional to E I, and the buckling load of a column is proportional to E I. The section modulus Z = I / y_max combines the shape and the extreme fibre distance into one number for stress checks. The radius of gyration r = √(I/A) tells how slender a column is.

For a rectangle of width b and depth d about its centroidal axis, I = b d³ / 12, so doubling the depth gives eight times the stiffness while doubling the width only doubles it. For a circle, I = π D⁴ / 64. For built-up sections, the parallel axis theorem moves each part's own I to the common centroidal axis by adding its area times the square of the shift: I = I_c + A h². Add the parts, subtract any holes.

The polar moment of inertia J, the sum of the two perpendicular second moments, governs torsion of circular shafts. Products of inertia and principal axes become important for angles and unsymmetrical sections, which bend sideways as well as downward under vertical load. Mass moment of inertia, a different but related quantity, enters rotational dynamics of machine foundations.