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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.

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
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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### 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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