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.