The area of the green section is 208 in“. It has the following moments of inertia with respect to the x- and y-axes: I̟ = 41,563 in* and I, = 64,881 in*. 16 in 13 in Determine the centroidal moments of inertia I„, Īy, and J, in in* and the corresponding radii of gyration k, ky, and k, in inches. 4 in in %3D in4 in %3D ky %3D in in
The area of the green section is 208 in“. It has the following moments of inertia with respect to the x- and y-axes: I̟ = 41,563 in* and I, = 64,881 in*. 16 in 13 in Determine the centroidal moments of inertia I„, Īy, and J, in in* and the corresponding radii of gyration k, ky, and k, in inches. 4 in in %3D in4 in %3D ky %3D in in
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
Related questions
Question
![**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'} = & \, \, \, \,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbf4a0e0b-2125-467b-b1bc-8bc81e995df8%2F8bf5b5af-bc6d-4583-93eb-220624b59f98%2Fkvybtx_processed.png&w=3840&q=75)
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'} = & \, \, \, \,
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, civil-engineering and related others by exploring similar questions and additional content below.Recommended textbooks for you
Structural Analysis (10th Edition)
Civil Engineering
ISBN:
9780134610672
Author:
Russell C. Hibbeler
Publisher:
PEARSON
Principles of Foundation Engineering (MindTap Cou…
Civil Engineering
ISBN:
9781337705028
Author:
Braja M. Das, Nagaratnam Sivakugan
Publisher:
Cengage Learning
Structural Analysis (10th Edition)
Civil Engineering
ISBN:
9780134610672
Author:
Russell C. Hibbeler
Publisher:
PEARSON
Principles of Foundation Engineering (MindTap Cou…
Civil Engineering
ISBN:
9781337705028
Author:
Braja M. Das, Nagaratnam Sivakugan
Publisher:
Cengage Learning
Fundamentals of Structural Analysis
Civil Engineering
ISBN:
9780073398006
Author:
Kenneth M. Leet Emeritus, Chia-Ming Uang, Joel Lanning
Publisher:
McGraw-Hill Education
Traffic and Highway Engineering
Civil Engineering
ISBN:
9781305156241
Author:
Garber, Nicholas J.
Publisher:
Cengage Learning