Solved BET107 MECHANICS OF STRUCTURES 1 Worksheet 1 - | Chegg.com - Free Printable

To determine the centroid \((\overline{x}, \overline{y})\) of the given composite shape, we will use the method of composite areas. The shape consists of:

1. A rectangular section.
2. A triangular section.
3. A circular hole.

Step 1: Define the individual components

#### Rectangular section:
- Dimensions: \(450 \, \text{mm} \times 100 \, \text{mm}\)
- Area: \(A_1 = 450 \times 100 = 45000 \, \text{mm}^2\)
- Centroid coordinates relative to the origin (bottom-left corner):
- \(x_1 = \frac{450}{2} = 225 \, \text{mm}\)
- \(y_1 = \frac{100}{2} = 50 \, \text{mm}\)

#### Triangular section:
- Base: \(350 \, \text{mm}\)
- Height: \(100 \, \text{mm}\)
- Area: \(A_2 = \frac{1}{2} \times 350 \times 100 = 17500 \, \text{mm}^2\)
- Centroid coordinates relative to the origin:
- \(x_2 = \frac{350}{3} \approx 116.67 \, \text{mm}\)
- \(y_2 = \frac{100}{3} \approx 33.33 \, \text{mm}\)

#### Circular hole:
- Diameter: \(90 \, \text{mm}\)
- Radius: \(r = \frac{90}{2} = 45 \, \text{mm}\)
- Area: \(A_3 = \pi r^2 = \pi (45)^2 \approx 6361.73 \, \text{mm}^2\)
- Centroid coordinates relative to the origin:
- \(x_3 = 150 + 150 + 45 = 345 \, \text{mm}\)
- \(y_3 = 100 - 45 = 55 \, \text{mm}\)

Step 2: Calculate the total area and weighted centroids

The total area \(A\) is the sum of the areas of the rectangle and triangle minus the area of the circular hole:
\[
A = A_1 + A_2 - A_3 = 45000 + 17500 - 6361.73 \approx 56138.27 \, \text{mm}^2
\]

#### Centroid in the \(x\)-direction:
The \(x\)-coordinate of the centroid \(\overline{x}\) is given by:
\[
\overline{x} = \frac{\sum (A_i x_i)}{A}
\]
Substitute the values:
\[
\overline{x} = \frac{A_1 x_1 + A_2 x_2 - A_3 x_3}{A}
\]
\[
\overline{x} = \frac{(45000 \times 225) + (17500 \times 116.67) - (6361.73 \times 345)}{56138.27}
\]
Calculate each term:
\[
45000 \times 225 = 10125000
\]
\[
17500 \times 116.67 \approx 2043725
\]
\[
6361.73 \times 345 \approx 2196666.85
\]
\[
\overline{x} = \frac{10125000 + 2043725 - 2196666.85}{56138.27} \approx \frac{9972058.15}{56138.27} \approx 177.47 \, \text{mm}
\]

#### Centroid in the \(y\)-direction:
The \(y\)-coordinate of the centroid \(\overline{y}\) is given by:
\[
\overline{y} = \frac{\sum (A_i y_i)}{A}
\]
Substitute the values:
\[
\overline{y} = \frac{A_1 y_1 + A_2 y_2 - A_3 y_3}{A}
\]
\[
\overline{y} = \frac{(45000 \times 50) + (17500 \times 33.33) - (6361.73 \times 55)}{56138.27}
\]
Calculate each term:
\[
45000 \times 50 = 2250000
\]
\[
17500 \times 33.33 \approx 583225
\]
\[
6361.73 \times 55 \approx 350095.15
\]
\[
\overline{y} = \frac{2250000 + 583225 - 350095.15}{56138.27} \approx \frac{2483129.85}{56138.27} \approx 44.23 \, \text{mm}
\]

Final Answer:

The coordinates of the centroid are:
\[
\boxed{(177.47, 44.23)}
\]