Principal MI and orientation of L-section

Dimensions and assumptions (read from the figure):

• Dimensions in mm; origin O at the bottom-left corner. Horizontal leg 60 × 10 (x 0 to 60, y 0 to 10); vertical leg 12 × 50 above it (x 0 to 12, y 10 to 60). Both legs are 60 mm long overall.

Properties of the parts (x, y measured from the reference origin; $I_G$ about the part's own centroid; area in $mm^2$, I in $mm^4$)

Centroid

$$\begin{aligned} A &= \sum A_i = 1200\ \text{mm}^2 \\ \bar X &= \frac{\sum A_i x_i}{A} = \frac{21600}{1200} = 18.00\ \text{mm} \\ \bar Y &= \frac{\sum A_i y_i}{A} = \frac{24000}{1200} = 20.00\ \text{mm} \end{aligned}$$

Moments and product of inertia about the centroidal axes

With $d_x = x_i-\bar X$, $d_y = y_i-\bar Y$ and the parallel-axis theorems $I_x = I_{x,G}+Ad_y^2$, $I_y = I_{y,G}+Ad_x^2$, $I_{xy} = I_{xy,G}+Ad_xd_y$:

Principal moments of inertia and principal axes

$$\begin{aligned} I_{max,min} &= \frac{I_x+I_y}{2}\pm\sqrt{\left(\frac{I_x-I_y}{2}\right)^2+I_{xy}^2} \\ &= 3.800\times10^{5}\pm2.169\times10^{5} \\ I_{max} &= 5.969\times10^{5}\ \text{mm}^4,\qquad I_{min} = 1.631\times10^{5}\ \text{mm}^4 \end{aligned}$$

Orientation: $\tan 2\theta_p = \dfrac{-2I_{xy}}{I_x-I_y} = \dfrac{4.320\times10^{5}}{40000}$, so $2\theta_p = 84.71^\circ$ (and $-95.29^\circ$).

• The axis of $I_{max}$ makes 42.35° with the +x axis (anticlockwise positive); the axis of $I_{min}$ is at -47.65° (perpendicular), both passing through the centroid (18.00, 20.00) measured from the reference origin.

Values about the given origin

About the axes through the origin O: $I_x = 8.800\times10^{5}$, $I_y = 7.488\times10^{5}$, $I_{xy} = 2.160\times10^{5}\ \text{mm}^4$ (see parts table with the parallel-axis theorem). Principal values about O: $I_{max} = 1.040\times10^{6}$, $I_{min} = 5.887\times10^{5}$ at -36.55° to the x axis.

Answer: $I_{max} = 5.969\times10^{5}\ \text{mm}^4$, $I_{min} = 1.631\times10^{5}\ \text{mm}^4$; principal axes through the centroid at 42.35° (for $I_{max}$) and -47.65° (for $I_{min}$) from the +x axis.