The area of the green section is 13,222 mm2. It has the following moments of inertia with respect to the x- and y-axes: I, = 344 x 10° mm“ and I, = 507 x 10° mm 100 mm 119 mm Determine the centroidal moments of inertia I, I ,, and J, in mm“ and the corresponding radii of gyration k, ky, and k, in millimeters.

Structural Analysis
6th Edition
ISBN:9781337630931
Author:KASSIMALI, Aslam.
Publisher:KASSIMALI, Aslam.
Chapter2: Loads On Structures
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### Analysis of the Green Section

The diagram shows a green irregular section with its centroid marked as point \( C \). It is positioned with respect to the x- and y-axes. The axes \( x' \) and \( y' \) pass through the centroid \( C \).

**Key Measurements:**
- **Area of the green section**: \( 13,222 \, \text{mm}^2 \)
- **Coordinates of the centroid from the origin**: 
  - Horizontal distance: \( 119 \, \text{mm} \)
  - Vertical distance: \( 100 \, \text{mm} \)

**Moments of Inertia:**
- \( I_x = 344 \times 10^6 \, \text{mm}^4 \)
- \( I_y = 507 \times 10^6 \, \text{mm}^4 \)

### Task

The objective is to determine the centroidal moments of inertia and radii of gyration:
- **Centroidal moments of inertia** (\(\bar{I}_{x'}\), \(\bar{I}_{y'}\), and \(\bar{J}_C\)):
  - Provided: \( \bar{I}_{x'} = 4.7 \times 10^6 \, \text{mm}^4 \)
  - Needs calculation: \(\bar{I}_{y'}\), \(\bar{J}_C\)

- **Radii of gyration** (\(\bar{k}_{x'}\), \(\bar{k}_{y'}\), and \(\bar{k}_C\)):
  - Needs calculation: \(\bar{k}_{x'}\), \(\bar{k}_{y'}\), \(\bar{k}_C\)

### Diagram Explanation

The provided image includes:
- A green irregular shape with axes labeled \( x \)-\( y \) and centroidal axes \( x' \)-\( y' \).
- The centroid \( C \) is marked.
- Distances from the origin to the centroid in red.

**Error Indicator:**
- There is an incorrect check indicating a mistake in calculations involving the centroidal moment of inertia \( \bar{I}_{x'} \) which should be verified to be \( 4.7 \times 10^6 \, \text{mm}^4 \) instead of \(
Transcribed Image Text:### Analysis of the Green Section The diagram shows a green irregular section with its centroid marked as point \( C \). It is positioned with respect to the x- and y-axes. The axes \( x' \) and \( y' \) pass through the centroid \( C \). **Key Measurements:** - **Area of the green section**: \( 13,222 \, \text{mm}^2 \) - **Coordinates of the centroid from the origin**: - Horizontal distance: \( 119 \, \text{mm} \) - Vertical distance: \( 100 \, \text{mm} \) **Moments of Inertia:** - \( I_x = 344 \times 10^6 \, \text{mm}^4 \) - \( I_y = 507 \times 10^6 \, \text{mm}^4 \) ### Task The objective is to determine the centroidal moments of inertia and radii of gyration: - **Centroidal moments of inertia** (\(\bar{I}_{x'}\), \(\bar{I}_{y'}\), and \(\bar{J}_C\)): - Provided: \( \bar{I}_{x'} = 4.7 \times 10^6 \, \text{mm}^4 \) - Needs calculation: \(\bar{I}_{y'}\), \(\bar{J}_C\) - **Radii of gyration** (\(\bar{k}_{x'}\), \(\bar{k}_{y'}\), and \(\bar{k}_C\)): - Needs calculation: \(\bar{k}_{x'}\), \(\bar{k}_{y'}\), \(\bar{k}_C\) ### Diagram Explanation The provided image includes: - A green irregular shape with axes labeled \( x \)-\( y \) and centroidal axes \( x' \)-\( y' \). - The centroid \( C \) is marked. - Distances from the origin to the centroid in red. **Error Indicator:** - There is an incorrect check indicating a mistake in calculations involving the centroidal moment of inertia \( \bar{I}_{x'} \) which should be verified to be \( 4.7 \times 10^6 \, \text{mm}^4 \) instead of \(
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