https://file.aiquickdraw.com/mathbot/file/1725313174559o69puau0.jpg

To find the location of \( \bar{y} \) for a T-section, we need to calculate the centroid of the shape from a reference axis. The image provided includes the dimensions of a T-section with the following properties:

- **\( A \):** Cross-sectional area = 43.10 in\(^2\)
- **\( d \):** Depth of the section = 20.200 in
- **\( tw \):** Thickness of the web = 1.060 in
- **\( bf \):** Width of the flange = 12.000 in
- **\( tf \):** Thickness of the flange = 1.930 in

### Step-by-Step Calculation

To find \( \bar{y} \), the distance from the bottom of the T-section to the centroid of the section, we use the formula for the centroid of a composite area:

\[
\bar{y} = \frac{\sum A_i y_i}{\sum A_i}
\]

Where:
- \( A_i \) is the area of each component of the section.
- \( y_i \) is the distance from the reference axis (usually the bottom of the section) to the centroid of each component.

#### 1. Calculate the Areas of the Flange and Web:

- **Flange area (\( A_f \)):** 
  \[
  A_f = bf \times tf = 12.000 \, \text{in} \times 1.930 \, \text{in} = 23.16 \, \text{in}^2
  \]

- **Web area (\( A_w \)):** 
  \[
  A_w = tw \times (d - tf) = 1.060 \, \text{in} \times (20.200 \, \text{in} - 1.930 \, \text{in}) = 1.060 \, \text{in} \times 18.270 \, \text{in} = 19.3662 \, \text{in}^2
  \]

#### 2. Locate the Centroid of Each Component:

- **Centroid of the flange (\( y_f \)):**  
  The centroid is located at half the thickness of the flange from the top of the flange, which is \( \frac{tf}{2} \) above the bottom.
  \[
  y_f = d - \frac{tf}{2} = 20.200 \, \text{in} - \frac{1.930}{2} \, \text{in} = 20.200 \, \text{in} - 0.965 \, \text{in} = 19.235 \, \text{in}
  \]

- **Centroid of the web (\( y_w \)):**  
  The centroid is located at half the depth of the web from the bottom of the web, which is \( \frac{d - tf}{2} \).
  \[
  y_w = \frac{d - tf}{2} = \frac{20.200 \, \text{in} - 1.930 \, \text{in}}{2} = \frac{18.270 \, \text{in}}{2} = 9.135 \, \text{in}
  \]

#### 3. Compute \( \bar{y} \):

\[
\bar{y} = \frac{A_f y_f + A_w y_w}{A_f + A_w}
\]

Substitute the values:

\[
\bar{y} = \frac{(23.16 \, \text{in}^2 \times 19.235 \, \text{in}) + (19.3662 \, \text{in}^2 \times 9.135 \, \text{in})}{23.16 \, \text{in}^2 + 19.3662 \, \text{in}^2}
\]

Calculate the numerator and denominator:

\[
\bar{y} = \frac{(445.3146 \, \text{in}^3) + (176.9056 \, \text{in}^3)}{42.5262 \, \text{in}^2}
\]

\[
\bar{y} = \frac{622.2202 \, \text{in}^3}{42.5262 \, \text{in}^2}
\]

\[
\bar{y} \approx 14.63 \, \text{in}
\]

### Conclusion

The centroid \( \bar{y} \) of the T-section is approximately **14.63 inches** from the bottom of the section.

Would you like more details or have any questions?

### Related Questions

1. How is the moment of inertia calculated for this T-section?
2. What are the steps to determine the neutral axis of a composite beam section?
3. How does the location of \( \bar{y} \) affect the bending stress distribution in a T-section?
4. What are the differences between calculating centroids of T-sections and I-sections?
5. How does the thickness of the web influence the centroid location of the T-section?

### Tip

When calculating the centroid of composite shapes, breaking the shape into simpler components and systematically applying the centroid formula for each component makes the process more manageable.

Learn how to find the centroid (\( \bar{y} \)) of a T-section with a detailed step-by-step calculation. This problem covers key engineering concepts, including centroid calculation for composite shapes and area formulas. Suitable for advanced engineering studies.

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