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Moments of Inertia of area: Parallel axis theorem

In many cases, the moment of inertia about an axis, particularly an axis passing through the centroid of  a common shape, is known (or relatively easier to calculate) and the moment of inertial of the area about a second axis parallel to the first axis is needed. The parallel axis theorem relates these two moments of inertia.

To derive the theorem, an area as shown in Fig. 10.9 is considered. The centroid of the area is denoted as C, the x' axis is an axis crossing the centroid (a centroidal axis), and the x axis is an arbitrary axis parallel to x'.

Fig. 10.9 Terms involved in deriving the parallel axis theorem.

If dA is a differential element of the area, its (perpendicular) distance to the x axis can be written as y=d_y+y' where d_y is the distance between the two parallel axes shown in Fig. 10.9. Therefore,

    \[\begin{split}I_x & = \int_A y^2 dA=\int_A (d_y+y')^2dA\\&=\int_A d_y^2dA + \int_A y'^2dA+\int_A 2d_y y'dA\end{split}\]

The term \int_A 2d_y y'dA equals zero because \int_A 2d_y y'dA= 2d_y \int_A y'dA= 2d_y\bar y A and \bar y=0 (measured from the x' axis) because x' passes through the centroid. Consequently,

    \[I_x = d_y^2\int_A dA + \int_A y'^2dA\]

With d_y^2\int_A dA=Ad_y^2 and \int_A y'^2dA=\bar I_{x'}, then

(10.7)   \[I_x = \bar I_{x'} + Ad_y^2 \]

which reads the moment of inertia about an axis x is equal to the moment of inertia about a parallel axis x' that crosses the centroid of A, plus the product of area A and the square distance between x and x'.

Equation 10.7 can be written for any two parallel axes with one crossing the centroid of the area. If a is an axis crossing C, and b a parallel axis to a as shown in Fig. 10.10a, we can write,

(10.8)   \[I_b = \bar I_{a} + Ad^2 \]

where d is the distance between the two parallel axes.

The parallel axis theorem also hold for the polar moment of inertia. If O is a point in the plane of an area and distant d from the centroid C of the area as shown in Fig. 10.10b, the polar moments of inertia about O and C are related as,

(10.9)   \[J_O = \bar J_{C} + Ad^2 \]

Fig. 10.10 The parallel axis theorem for (a) the rectangular and (b) polar moments of inertia.

EXAMPLE 10.3.1

Determine the moment of inertia of a rectangular area about the x and y axes shown. Use the moment of inertia about the centroidal axes parallel to its sides.

SOLUTION

Let x' and y' be two axes crossing the centroid of the area as shown. According to Example 10.1.1,

    \[I_{x'}=\frac{1}{12}bh^3 ,\quad I_{y'}=\frac{1}{12}hb^3\]

Therefore, by Eq. 10.8, we can write,

    \[I_x=\bar I_{x'} + Ad^2 = \frac{1}{12}bh^3 + (bh)(\frac{h}{2})^2=\frac{1}{3}bh^3\]

And,

    \[I_y=\bar I_{y'} + Ad^2 = \frac{1}{12}hb^3 + (bh)(\frac{b}{2})^2=\frac{1}{3}hb^3\]

EXAMPLE 10.3.2

If I_x=54\text{ m}^4 for the depicted triangular area, determine the moment of inertia about the axis crossing the centroid of the area and parallel to the x axis.

SOLUTION

Find the distance from the x axis to the centroid of the area, and use Eq. 10.8.

    \[I_x=\bar I_{x'} + Ad^2 \implies \bar I_{x'}= I_x - Ad^2\]

Therefore,

    \[\bar I_{x'}= 54 - (9)(2)^2=18\text {m}^4\]

Figure 10.11 demonstrates some basic areas and the formulations of their moments of inertia (see this document for some other areas).

Fig. 10.11 Basic areas and the formulations of their moments of inertia.