Principal MI of stepped section (cm)

Dimensions and assumptions (read from the figure):

• Dimensions in cm. Polygon (0,0), (120,0), (120,60), (75,60), (45,100), (0,100): height 100, base 120, top width 45, sloping edge 30 horizontal × 40 vertical, lower step 45 wide.

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

Centroid

$$\begin{aligned} A &= \sum A_i = 9600\ \text{cm}^2 \\ \bar X &= \frac{\sum A_i x_i}{A} = \frac{5.055\times10^{5}}{9600} = 52.66\ \text{cm} \\ \bar Y &= \frac{\sum A_i y_i}{A} = \frac{4.040\times10^{5}}{9600} = 42.08\ \text{cm} \end{aligned}$$

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

$$\begin{aligned} I_{max,min} &= \frac{I_x+I_y}{2}\pm\sqrt{\left(\frac{I_x-I_y}{2}\right)^2+I_{xy}^2} \\ &= 8.840\times10^{6}\pm3.438\times10^{6} \\ I_{max} &= 1.228\times10^{7}\ \text{cm}^4,\qquad I_{min} = 5.402\times10^{6}\ \text{cm}^4 \end{aligned}$$

Orientation: $\tan 2\theta_p = \dfrac{-2I_{xy}}{I_x-I_y} = \dfrac{5.346\times10^{6}}{-4.324\times10^{6}}$, so $2\theta_p = 128.97^\circ$ (and $-51.03^\circ$).

• The axis of $I_{max}$ makes 64.48° with the +x axis (anticlockwise positive); the axis of $I_{min}$ is at -25.52° (perpendicular), both passing through the centroid (52.66, 42.08) measured from the reference origin.

Answer: $I_{max} = 1.228\times10^{7}\ \text{cm}^4$, $I_{min} = 5.402\times10^{6}\ \text{cm}^4$; principal axes through the centroid at 64.48° (for $I_{max}$) and -25.52° (for $I_{min}$) from the +x axis.