The area of the green section is 208 in. It has the following moments of inertia with respect to the x- and y-axes: I, = 41,563 in* and I, = 64,881 in“.y'16 in13 intoDetermine the centroidal moments of inertia I,, I,, and J, in in“ and the corresponding radii of gyration ky, ky, and k, in inches.76715X in4IyX in4118129194844X in4kx'19.204X inKy'23.831X in=kc30.606X in=II

Given Data: The area of green section is A=208 in2. The moment of inertia with respect to Y-axis is…

Answered: The area of the green section is 208…

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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Question

The area of the green section is 208 in. It has the following moments of inertia with respect to the x- and y-axes: I, = 41,563 in* and I, = 64,881 in“.
y'
16 in
13 in
to
Determine the centroidal moments of inertia I,, I,, and J, in in“ and the corresponding radii of gyration ky, ky, and k, in inches.
76715
X in4
Iy
X in4
118129
194844
X in4
kx'
19.204
X in
Ky'
23.831
X in
=
kc
30.606
X in
=
II

Expert Solution

Step 1

Given Data:

The area of green section is A=208 in².

The moment of inertia with respect to Y-axis is I_X=41563 in⁴.

The moment of inertia with respect to X-axis is I_Y=64881 in⁴.

The distance of centroidal axis from X-axis is y=13 in.

The distance of centroidal axis from Y-axis is x=16 in.

Step 2

Calculate the centroidal moment of inertia.

The centroidal moment of inertia with respect to X-axis is I_Y'

$\begin{array}{rcl} I_{Y'} & = & I_{Y} - A \times x^{2} \\ = & (64881 {in}^{4}) - (208 {in}^{2}) \times {(16 in)}^{2} \\ = & 64881 - 53248 \\ = & 11633 {in}^{4} \end{array}$

The centroidal moment of inertia with respect to Y-axis is I_X'

$\begin{array}{rcl} I_{X'} & = & I_{X} - A \times y^{2} \\ = & (41563 {in}^{4}) - (208 {in}^{2}) \times {(13 in)}^{2} \\ = & 41563 - 35152 \\ = & 6411 {in}^{4} \end{array}$

The centroidal moment of inertia with respect to Z-axis is J_C

$\begin{array}{rcl} {\bar{J}}_{C} & = & I_{X'} + {\bar{I}}_{Y'} \\ = & (6411 {in}^{4}) + ({11633 in}^{4}) \\ = & 18044 {in}^{4} \end{array}$

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