= Second moment of area
{title2=$I$}
{wiki}
The second moment of area of a cross-section about an axis is $I=\int y^2\,dA$, where $y$ is the perpendicular distance from that axis. Its dimensions are length to the fourth power. For a circular section of radius $a$, polar integration gives $I=\int_0^a r^3dr\int_0^{2\pi}\sin^2\theta d\theta=\pi a^4/4$. A slender rod with <Young's modulus> $E$ has <filament bending modulus> $B=EI$: the axial <strain> caused by <curvature> $\kappa$ is $-y\kappa$, and integrating the <linear elasticity> energy density $E y^2\kappa^2/2$ over the section gives $B\kappa^2/2$ per unit length.
Back to article page
