Why is second moment of area often referred as moment of inertia?

Question

In many references, mostly civil engineering, the second moment of area is referred as Moment of Inertia. Is that "really" correct? From my understanding, moment of inertia is analogous to mass in rotational motion, not the second moment of area.

Answer 1

I agree, the terminology is a total mess.

1. Statical Moment of Area or First Moment of Area $\int x {\rm d}A$
2. Second Area of Moment $\int x^2 {\rm d}A$
3. First Moment of Mass $\int x {\rm d}m$
4. Mass Moment of Inertia or Second Moment of Mass $\int x^2 {\rm d}m$

So what is correct? You can distinguish between area moments and mass moments, and then annotate if it is the first, second or polar version.

Answer 2

For a 2D object with density $\rho(\mathbf{r})$, the second moment of area is $$A=\iint\mathbf{r}^2d\mathbf{r}$$ and the moment of inertia is $$I=\iint\rho(\mathbf{r})\mathbf{r}^2d\mathbf{r}.$$ When $\rho(\mathbf{r})=\rho_0$ you get $$I=\rho_0A$$ and when $\rho_0=1$ you get equality. Otherwise, they are not necessarily the same.