State and prove parallel axis theorem

Parallel axis theorem

Statement: the moment of inertia of an area about any axis equals its moment of inertia about a parallel axis through its centroid plus the product of the area and the square of the distance between the two axes.

I = I_G + A d²

Proof: let G be the centroid of the area A, and let X′ be the centroidal axis. Take a parallel axis X at distance d from X′. For a small element dA at distance y′ from X′, its distance from X is y = y′ + d.

I_X = ∫ y² dA = ∫ (y′ + d)² dA
    = ∫ y′² dA + 2d ∫ y′ dA + d² ∫ dA

• ∫ y′² dA = I_G, the moment of inertia about the centroidal axis.
• ∫ y′ dA = A·ȳ′ = 0, because the distance of the centroid from the centroidal axis is zero.
• ∫ dA = A.

So I_X = I_G + A d². Similarly, I_Y = I_Gy + A d_x². Since A d² is never negative, the moment of inertia about the centroidal axis is the least of the moments of inertia about all parallel axes.

Note: the theorem also holds for the mass moment of inertia (I = I_G + M d²) and for the polar moment of inertia (J = J_G + A d²). It is used to find the moment of inertia of composite sections about their centroidal axes, and to shift the moment of inertia of a standard shape to another axis.