---
title: "Principal MI of section with quarter circle"
subject: "Strength of Materials"
chapter: 2
key: "semester_3/strength_of_materials:principal-mi-quarter-circle-800-600"
source: https://www.bctengineer.com/q/semester_3/strength_of_materials/principal-mi-quarter-circle-800-600
site: BCT Engineering
---

# Principal MI of section with quarter circle

Strength of Materials · Chapter 2: Geometrical Properties of Sections

- Asked in: 2074 Asoj, 12 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/strength_of_materials/principal-mi-quarter-circle-800-600
- Whole chapter: https://www.bctengineer.com/md/semester_3/strength_of_materials/chapter-2

## Question

Find the principal moments of inertia and directions of principal axes for the section as shown in figure below. [Figure: section in mm: a triangle at the top left of height 600, a horizontal rectangle of 300 thickness, a quarter circle at the lower right; dimensions 800, 800, 800.]

## Short answer

I_max = 5.114×10^10 mm^4, I_min = 1.421×10^10 mm^4 about centroid. Principal axes: 42.40° (I_max) and -47.60° (I_min) from +x axis.

## Answer

**Dimensions and assumptions (read from the figure):**
- Dimensions in mm. Rectangle 800 × 300 (x 0 to 800, y 0 to 300); triangle above it with base 800 and height 600 (vertices (0,300), (800,300), (0,900)); quarter circle of radius 300 below the right end, centre (800, 0), occupying x 500 to 800, y −300 to 0.

#### 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 $mm^4$)
| Part | Sign | A | $\bar x$ | $\bar y$ | $I_{x,G}$ | $I_{y,G}$ | $I_{xy,G}$ |
|---|---|---|---|---|---|---|---|
| Rectangle 800×300 | + | $2.400\times10^{5}$ | 400.0 | 150.0 | $1.800\times10^{9}$ | $1.280\times10^{10}$ | $0$ |
| Triangle | + | $2.400\times10^{5}$ | 266.7 | 500.0 | $4.800\times10^{9}$ | $8.533\times10^{9}$ | $-3.200\times10^{9}$ |
| Quarter circle r = 300 | + | $70686$ | 672.7 | -127.3 | $4.445\times10^{8}$ | $4.445\times10^{8}$ | $-1.334\times10^{8}$ |

#### Centroid
$$
\begin{aligned}
A &= \sum A_i = 5.507\times10^{5}\ \text{mm}^2 \\
\bar X &= \frac{\sum A_i x_i}{A} = \frac{2.075\times10^{8}}{5.507\times10^{5}} = 376.9\ \text{mm} \\
\bar Y &= \frac{\sum A_i y_i}{A} = \frac{1.470\times10^{8}}{5.507\times10^{5}} = 266.9\ \text{mm}
\end{aligned}
$$

#### 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 $I_x = I_{x,G}+Ad_y^2$, $I_y = I_{y,G}+Ad_x^2$, $I_{xy} = I_{xy,G}+Ad_xd_y$:

| Part | $I_{x,G}+A d_y^2$ | $I_{y,G}+A d_x^2$ | $I_{xy,G}+A d_x d_y$ |
|---|---|---|---|
| Rectangle 800×300 (+) | $5.082\times10^{9}$ | $1.293\times10^{10}$ | $-6.486\times10^{8}$ |
| Triangle (+) | $1.784\times10^{10}$ | $1.145\times10^{10}$ | $-9.365\times10^{9}$ |
| Quarter circle r = 300 (+) | $1.143\times10^{10}$ | $6.629\times10^{9}$ | $-8.377\times10^{9}$ |
| **Total about centroid** | **$3.435\times10^{10}$** | **$3.101\times10^{10}$** | **$-1.839\times10^{10}$** |

#### Principal moments of inertia and principal axes
$$
\begin{aligned}
I_{max,min} &= \frac{I_x+I_y}{2}\pm\sqrt{\left(\frac{I_x-I_y}{2}\right)^2+I_{xy}^2} \\
&= 3.268\times10^{10}\pm1.847\times10^{10} \\
I_{max} &= 5.114\times10^{10}\ \text{mm}^4,\qquad I_{min} = 1.421\times10^{10}\ \text{mm}^4
\end{aligned}
$$

Orientation: $\tan 2\theta_p = \dfrac{-2I_{xy}}{I_x-I_y} = \dfrac{3.678\times10^{10}}{3.344\times10^{9}}$, so $2\theta_p = 84.80^\circ$ (and $-95.20^\circ$).

- The axis of $I_{max}$ makes 42.40° with the +x axis (anticlockwise positive); the axis of $I_{min}$ is at -47.60° (perpendicular), both passing through the centroid (376.9, 266.9) measured from the reference origin.

**Answer: $I_{max} = 5.114\times10^{10}\ \text{mm}^4$, $I_{min} = 1.421\times10^{10}\ \text{mm}^4$; principal axes through the centroid at 42.40° (for $I_{max}$) and -47.60° (for $I_{min}$) from the +x axis.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Why must the centroid (X, Y) of the whole section be found first?
   A. Because the principal axes always coincide with the reference axes through the origin
   B. Because the product of inertia about the origin is always zero
   C. Because the principal moments asked for are about axes through the centroid, and parallel-axis shifts need the centroid location
   D. Because the centroid is the point where Imax occurs
   Answer: C. I_x, I_y, I_xy are required about centroidal axes, so each part's distance d_x, d_y is measured from the overall centroid (376.9 mm, 266.9 mm). (Key point: Centroid first)

2. How is the product of inertia of one part transferred to the section's centroidal axes?
   A. I_xy = I_xy,G + A d_x^2
   B. I_xy = I_xy,G + A (d_x + d_y)
   C. I_xy = I_xy,G, because the product of inertia does not change with axis shift
   D. I_xy = I_xy,G + A d_x d_y, keeping the signs of d_x and d_y
   Answer: D. The parallel-axis theorem for product of inertia adds A times the product of the signed centroidal offsets. (Key point: Parallel-axis for Ixy)

3. Which expression gives the principal moments of inertia from I_x, I_y and I_xy?
   A. (I_x+I_y)/2 ± sqrt(((I_x-I_y)/2)^2 + I_xy^2)
   B. (I_x+I_y)/2 ± sqrt((I_x-I_y)^2 + I_xy)
   C. (I_x-I_y)/2 ± sqrt(((I_x+I_y)/2)^2 + I_xy^2)
   D. sqrt(I_x^2 + I_y^2) ± I_xy
   Answer: A. The principal values are the centre of Mohr's circle plus and minus its radius. (Key point: Principal moment formula)

4. With I_x = 3.435e10, I_y = 3.101e10 and I_xy = -1.839e10 mm^4, which statement matches the answer?
   A. I_max = 3.435e10 and I_min = 3.101e10 mm^4; the I_max axis is along the x axis
   B. I_max = 5.114e10 and I_min = 1.421e10 mm^4; the I_max axis is at 42.40° from +x and the I_min axis is perpendicular to it
   C. I_max = 1.421e10 mm^4 and I_min = 5.114e10 mm^4 with both axes at 45 degrees
   D. I_max and I_min are both equal to (I_x+I_y)/2, and the axes are not defined
   Answer: B. Imax and Imin come from the formula, and the principal axes are perpendicular, with the I_max axis at 42.40° (I_min axis at -47.60°). (Key point: Result and axis angle)

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