### Analysis of the Green SectionThe 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:**- \( I_x = 344 \times 10^6 \, \text{mm}^4 \)- \( I_y = 507 \times 10^6 \, \text{mm}^4 \)### TaskThe 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 ExplanationThe 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 \(

Given: A=13222 mm2Ix=344×106 mm4Iy=507×106 mm4x=119 mmy=100 mm By parallel axis theorem.…

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

6th Edition

ISBN:9781337630931

Author:KASSIMALI, Aslam.

Publisher:KASSIMALI, Aslam.

Chapter2: Loads On Structures

Section: Chapter Questions

Problem 1P

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Graphical Methods

The graphical method is a method of solving equations involving more than one variable using the Cartesian coordinate system. In civil engineering, the use of the graphical method is limited to the problems involving two variables. The graphical method has a great impact in solving the questions regarding engineering mechanics.

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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 \(

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