To find \( d\bar{x}_i \) for the T-section shape, we first need to understand what \( d\bar{x}_i \) represents. Typically, \( d\bar{x}_i \) is used to denote the distance from a reference axis to the centroid of each individual component area within a composite section, in the horizontal direction (along the x-axis). However, in the context of structural shapes like the T-section shown, we usually discuss the vertical axis distance \( y_i \) from a reference line to the centroid of the section or its components.

Since the T-section is symmetric about the vertical axis, \( d\bar{x}_i = 0 \) for both the flange and the web. This is because both the flange and web's centroids align horizontally with the centerline of the entire T-section.

### Confirming \( d\bar{x}_i \) for Symmetry

Given the provided T-section:
- **Width of the flange (\( bf \))** = 12.000 in.
- **Thickness of the web (\( tw \))** = 1.060 in.

#### Flange Centroid in the X-Direction (\( \bar{x}_f \))
The centroid of the flange (\( \bar{x}_f \)) is at half the width of the flange since it is centered along the T-section:

\[
\bar{x}_f = \frac{bf}{2} = \frac{12.000 \, \text{in}}{2} = 6.000 \, \text{in}
\]

#### Web Centroid in the X-Direction (\( \bar{x}_w \))
The centroid of the web (\( \bar{x}_w \)) is at half the thickness of the web since it is centered along the T-section:

\[
\bar{x}_w = \frac{tw}{2} = \frac{1.060 \, \text{in}}{2} = 0.530 \, \text{in}
\]

Given that the T-section is symmetric and both centroids lie on the central axis (Y-axis) of the T-section, both are considered horizontally centered, aligning them along the same axis.

Thus, the **horizontal distance** \( d\bar{x}_i = 0 \) for both the flange and the web in a symmetric T-section.

### Summary

For the T-section:
- **\( d\bar{x}_f = 0 \, \text{in} \)**
- **\( d\bar{x}_w = 0 \, \text{in} \)**

These values indicate that there is no horizontal offset for the centroids of the flange and the web from the central vertical axis of the T-section.

Would you like to explore more about this shape or need additional explanations?

### Related Questions

1. How do we calculate the moment of inertia for this T-section about the y-axis?
2. What are the typical applications of T-sections in structural engineering?
3. How would the centroid location change if the T-section were not symmetric?
4. What is the significance of the centroid in structural analysis?
5. How does the width of the flange affect the overall properties of a T-section?

### Tip

When analyzing symmetric shapes, remember that centroids in the horizontal direction will align with the axis of symmetry, simplifying many calculations related to balance and distribution.

Explore how to determine the horizontal distance (dx_i) for T-section shapes in structural engineering. Learn about centroids, symmetry, and their impact on structural analysis. Detailed explanation for engineers and students interested in structural mechanics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation