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20 in 8 in y ↓ r = 5in k 6 in C -8 in- * 6 in kist X C is the centroid of the shaded area. Calculate y (distance to the centroid) Note: the semi-circle is not shaded

20 in 8 in y ↓ r = 5in k 6 in C -8 in- * 6 in kist X C is the centroid of the shaded area. Calculate y (distance to the centroid) Note: the semi-circle is not shaded

Structural Analysis
Structural Analysis
6th Edition
ISBN:9781337630931
Author:KASSIMALI, Aslam.
Publisher:KASSIMALI, Aslam.
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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### Centroid Calculation of a Composite Shape **Problem:** Find the centroid (C) of the shaded area in the given composite shape. The shaded area consists of a rectangular and triangular segment, with a semi-circle subtracted from the rectangle. ### Diagram Explanation: - The overall height of the shape is 20 inches, split into two segments: 8 inches above the semi-circle and 12 inches below it. - The width of the rectangle is 20 inches, divided into three parts: 8 inches in the center and two 6-inch segments on either side. - The semi-circle’s radius is 5 inches and it is positioned on the left side of the shape with its diameter lying horizontally. - The triangular segment on the right has a base of 8 inches and tapers to a point at the top. ### Instructions: - The shaded area needs to find the centroid, marked as point C. - The y-coordinate of the centroid (denoted as \(\bar{y}\)) must be calculated. ### Given: - The overall height of the structure: 20 inches - Rectangle height above the semi-circle: 8 inches - Rectangle height below the semi-circle: 12 inches - Width of the rectangular part: 20 inches - Semi-circle radius: 5 inches ### Objective: Calculate the y-coordinate of the centroid (\(\bar{y}\)) of the shaded area. **Note:** The semi-circle is not shaded and thus subtracted from the overall shape when calculating the centroid. To find \(\bar{y}\), break the area down into simpler shapes (rectangle, triangle, and semi-circle) and use the centroid formula for composite areas.
Transcribed Image Text:### Centroid Calculation of a Composite Shape **Problem:** Find the centroid (C) of the shaded area in the given composite shape. The shaded area consists of a rectangular and triangular segment, with a semi-circle subtracted from the rectangle. ### Diagram Explanation: - The overall height of the shape is 20 inches, split into two segments: 8 inches above the semi-circle and 12 inches below it. - The width of the rectangle is 20 inches, divided into three parts: 8 inches in the center and two 6-inch segments on either side. - The semi-circle’s radius is 5 inches and it is positioned on the left side of the shape with its diameter lying horizontally. - The triangular segment on the right has a base of 8 inches and tapers to a point at the top. ### Instructions: - The shaded area needs to find the centroid, marked as point C. - The y-coordinate of the centroid (denoted as \(\bar{y}\)) must be calculated. ### Given: - The overall height of the structure: 20 inches - Rectangle height above the semi-circle: 8 inches - Rectangle height below the semi-circle: 12 inches - Width of the rectangular part: 20 inches - Semi-circle radius: 5 inches ### Objective: Calculate the y-coordinate of the centroid (\(\bar{y}\)) of the shaded area. **Note:** The semi-circle is not shaded and thus subtracted from the overall shape when calculating the centroid. To find \(\bar{y}\), break the area down into simpler shapes (rectangle, triangle, and semi-circle) and use the centroid formula for composite areas.
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