How Cross-Section Properties Work
Every structural part has a cross-section. The shape of that cross-section tells you how well the part resists bending, twisting, and buckling. Four numbers describe the behavior.
Moment of Inertia (I)
I tells you how stiff the cross-section is in bending. Higher I means less deflection. I depends on the shape and how far material is from the center. Material far from the center counts more than material close to the center.
Section Modulus (S)
Section modulus equals I divided by c, where c is the distance from the neutral axis to the outer fiber. S is used to calculate bending stress. Stress equals bending moment divided by S. Higher S means lower stress for the same load.
Pro tip: For the same weight, an I-beam has much higher Ixx than a solid rectangle. That is why structural steel is shaped like an I. The flanges carry most of the bending load.
Radius of Gyration (r)
Radius of gyration equals the square root of I divided by A. It is used in column buckling. A bigger r means the column is harder to buckle. Tall thin columns buckle because they have a small r.
Strong Axis vs Weak Axis
Most shapes have a strong axis (x-x) and a weak axis (y-y). An I-beam is much stiffer in the direction that matches its depth. That is why you see I-beams oriented with the tall dimension up. Bending it about the weak axis uses only the flanges, which is much less efficient.