Moment of Inertia Calculator

The moment of inertia (I) for a cross-section, also called the second moment of area, measures a section's resistance to bending about a given axis. A larger I means more resistance to bending. It is used in beam deflection and bending stress calculations: bending stress = M x y / I, deflection = f(M, E, I, L). Units are in^4 or mm^4.

For a solid rectangle about its centroidal axis: I = (b x h^3) / 12, where b is the width (in the direction of bending) and h is the height (depth in the plane of bending). For bending about the weak axis: I = (h x b^3) / 12. A 6x12 rectangular beam bent about its strong axis has I = (6 x 12^3) / 12 = 864 in^4.

For a solid circle: I = pi x d^4 / 64, where d is the diameter. For a hollow tube: I = pi x (d_o^4 - d_i^4) / 64, where d_o is the outer diameter and d_i is the inner diameter. A 4-inch solid rod has I = pi x 4^4 / 64 = 12.57 in^4.

The parallel axis theorem allows calculating the moment of inertia of a shape about an axis parallel to its centroidal axis: I = I_centroid + A x d^2, where A is the area and d is the distance between the axes. This theorem is used when combining multiple shapes (like a T-beam or compound section) to find the total moment of inertia about a common axis.

Beam deflection is inversely proportional to the product of Young's modulus (E) and moment of inertia (I). For a simply supported beam with uniform load: maximum deflection = (5 x w x L^4) / (384 x E x I). Doubling I halves the deflection. Deeper beams with the same material (larger I) deflect less, which is why I-beams are efficient in bending.