Question

Locate the centroid y¯ for the beam's cross-sectional area.

Answer 1

The centroid
`\( \bar{y} \)` for the beam's cross-sectional area is the geometric center of the shape.

Step-by-step explanation:

The centroid
`(\( \bar{y} \))` of a beam's cross-sectional area is the point where the entire area can be considered to be concentrated. For simple and symmetric shapes, such as rectangles or circles, the centroid is at the center of the shape. This is the case when the shape has uniform density or material distribution.

In more complex cases, where the beam's cross-section is irregular or consists of multiple shapes, the centroid is found by calculating the weighted average of the individual centroids of each sub-area. Mathematically, the centroid
`(\( \bar{y} \))` is given by the sum of the products of each sub-area's centroid
`(\( y_i \))`and its corresponding area
`(\( A_i \)),` divided by the total area
`(\( A \)):`

`\[ \bar{y} = (\sum (A_i \cdot y_i))/(\sum A_i) \]`

Here,
`\( \bar{y} \)` represents the centroid,
`\( A_i \)` is the area of each sub-section, and
`\( y_i \)`is the distance from the reference axis to the centroid of each sub-area. This formula accounts for the distribution of material within the cross-section.

In conclusion, finding the centroid is crucial for structural analysis as it helps determine the point through which the resultant force of the entire cross-sectional area can be considered to act, simplifying calculations for various engineering applications.

Answer 2

The centroid
`\( \bar{y} \)` for the beam's cross-sectional area is calculated using the formula:

`\[ \bar{y} = (\sum A_i \cdot y_i)/(\sum A_i) \]`

Step-by-step explanation:

The centroid
`\( \bar{y} \)` represents the location of the center of mass of the beam's cross-sectional area. It is calculated as the weighted average of the individual centroids
`\( y_i \)` of each sub-area
`\( A_i \)`, divided by the total area
`\( \sum A_i \)`.

In the formula,
`\( A_i \)` represents the area of each sub-section, and
`\( y_i \)` is the distance from the reference axis to the centroid of that sub-section. The numerator calculates the sum of the moments of the sub-areas about the reference axis, and the denominator represents the total area of the cross-section.

This formula essentially considers each part of the cross-section as a weighted contribution to the overall centroid, where larger areas have a more significant impact on the final result. The centroid
`\( \bar{y} \)` is a crucial parameter in structural engineering, as it helps in analyzing and designing structures by providing insight into the distribution of mass within the cross-sectional area.