Structural Analysis
6th Edition
ISBN: 9781337630931
Author: KASSIMALI, Aslam.
Publisher: Cengage,
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![**Problem: Calculation of Centroidal Moments of Inertia and Radii of Gyration**
The area of the green section is \(208 \, \text{in}^2\). It has the following moments of inertia with respect to the x- and y-axes:
\[
I_x = 41{,}563 \, \text{in}^4 \quad \text{and} \quad I_y = 64{,}881 \, \text{in}^4
\]
**Diagram Explanation:**
- The diagram illustrates a green irregular shape positioned in a coordinate plane.
- The origin of the x and y axes is marked with \(O\).
- A centroid \(C\) of the shape is indicated by a bold cross.
- The \(y'\) and \(x'\) axes are parallel to the y and x axes, passing through the centroid \(C\).
- The distance between the origin \(O\) and the centroid \(C\) is given as:
- \(16 \, \text{in}\) horizontally from the origin to \(y'\).
- \(13 \, \text{in}\) vertically from the origin to \(x'\).
**Objective:**
Determine the centroidal moments of inertia \(\bar{I}_{x'}\), \(\bar{I}_{y'}\), and \(\bar{J}_C\) (in \(\text{in}^4\)), and the corresponding radii of gyration \(\bar{k}_{x'}\), \(\bar{k}_{y'}\), and \(\bar{k}_C\) (in inches).
\[
\begin{align*}
\bar{I}_{x'} = & \, \, \, \, \, \, \, \, \text{in}^4 \\
\bar{I}_{y'} = & \, \, \, \, \, \, \, \, \text{in}^4 \\
\bar{J}_C = & \, \, \, \, \, \, \, \, \text{in}^4 \\
\bar{k}_{x'} = & \, \, \, \, \, \, \, \, \text{in} \\
\bar{k}_{y'} = & \, \, \, \,](https://content.bartleby.com/qna-images/question/bf4a0e0b-2125-467b-b1bc-8bc81e995df8/8bf5b5af-bc6d-4583-93eb-220624b59f98/kvybtx_processed.png)
Transcribed Image Text:**Problem: Calculation of Centroidal Moments of Inertia and Radii of Gyration**
The area of the green section is \(208 \, \text{in}^2\). It has the following moments of inertia with respect to the x- and y-axes:
\[
I_x = 41{,}563 \, \text{in}^4 \quad \text{and} \quad I_y = 64{,}881 \, \text{in}^4
\]
**Diagram Explanation:**
- The diagram illustrates a green irregular shape positioned in a coordinate plane.
- The origin of the x and y axes is marked with \(O\).
- A centroid \(C\) of the shape is indicated by a bold cross.
- The \(y'\) and \(x'\) axes are parallel to the y and x axes, passing through the centroid \(C\).
- The distance between the origin \(O\) and the centroid \(C\) is given as:
- \(16 \, \text{in}\) horizontally from the origin to \(y'\).
- \(13 \, \text{in}\) vertically from the origin to \(x'\).
**Objective:**
Determine the centroidal moments of inertia \(\bar{I}_{x'}\), \(\bar{I}_{y'}\), and \(\bar{J}_C\) (in \(\text{in}^4\)), and the corresponding radii of gyration \(\bar{k}_{x'}\), \(\bar{k}_{y'}\), and \(\bar{k}_C\) (in inches).
\[
\begin{align*}
\bar{I}_{x'} = & \, \, \, \, \, \, \, \, \text{in}^4 \\
\bar{I}_{y'} = & \, \, \, \, \, \, \, \, \text{in}^4 \\
\bar{J}_C = & \, \, \, \, \, \, \, \, \text{in}^4 \\
\bar{k}_{x'} = & \, \, \, \, \, \, \, \, \text{in} \\
\bar{k}_{y'} = & \, \, \, \,
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