Solved problems for the moment of inertia.
Brief content of the first video-solved problem #1.
While for the expression for the radius of gyration for x, we have K^2 x= Ix/area, our Ix already obtained =0.0288 / area so 2x=3/25, we can get the kx, by taking the sqrt of 3 /25 which is = 0.3464m. This is a part of the video, which has a closed caption in English. This is a link to the first video via u-tube.
Solved problem#2-Second video
We are going to solve another solved problem, we call it problem #2. In this problem, It is required to estimate the following. moment of inertia Ix and to estimate the product of inertia of the given unequal angle.
In an application of the rectangle, we can constitute an equal angle, consisting of two rectangles, The first one with width=1.0 m and the height = 4 m This is a part of the video, which has a closed caption in English. If you wish to watch the second video on u-tube.
The First solved problem for a given rectangle.
We will solve our first problem, problem # 1. We have x and y axes, but we have a rectangular section. with width=0.40m. And the height is 0.60 m. This rectangle is apart from the y-axis by a distance of 0.20 m.
It is required to estimate the following for the given rectangle. IX, Kx, the radius of gyration at the x-axis, and b.
We want to estimate the moment of inertia at the y-axis and the radius of gyration for the y-axis.
We have our X and Y external axes. As we know that I x = the Ixg+ the area *(y^2).
We have, for the moment of inertia about the CG, for a rectangle section, we have b* h^3/12. Then we have this x-axis. We have to add the product of area multiplied by y^2, first our Ix g=b*h^3/12= (0.40*0.60^3/12, which will give us 0.0072 m4.
Our area is (0.40*0.60), which is 0.24 m2. For item A* y^2, we have A=b*h, and y̅ =h/2 All raised to the power of 2. So we have b*h, which is (0.40*0.60)*(h/2)^2,h/2=0.30 divided by 4, which will give us 0.0216 m4.
Our required moment of inertia about the x-axis will be the summation of these two items, the first item is Ixg=0.0072+ A*y̅^2, which is =0.0216 adding both will give us 0.0288 m4.
For k^2x=Ix/A=0.0288/0.24=3/25, then the Kx=sqrt(3/25)=sqrt(3)/5.
For part b of the first solved problem, Iy=Iyg+A*xbar^2. Iyg=h*b^3/12=0.60*?(0.40)^3/12=0.0032 m4.
Ax^2g=(0.40)(0.60)(0.20+0.50.40)^2=0.0384m4. The final Iy=(0.0032+0.0384)=0.0416m4.
k^2y=Iy/A=0.0416/(0.40*0.60)=13/75. ky=sqrt(13/75)=0.4163 m2.
The second solved problem for a given section.
We can consider the L shape as consisting of two rectangles. The first one of width=1.0 m and height = 4 m and the other rectangle, has a width of 2 m and a height is 1 meter, if we join them together, they will constitute an unequal angle with the same width, as we can see Unequal angle 3 by 4 and a width of 1.0m, can be evaluated as composed of two rectangles.
The first one A1 of width 1.0m and height=4.0 m, that’s why its area would become(1*4)=4.0 m^2 And its X1 will be=1/2 m. y1 from the x-axis will be. 4 / 2, which is 2 meters. For the second area, we have the A2=(3-1)*.this is the breadth, the height will be 1.0 m, so we have 2=(1*2)=2.0 m^2.
Our X2, measured to the external y-axis, will be (1m+1/2 of the difference between (3-1.0) Which will give us = 2 meters, our y2= 0.50 m.
The Ix for the unequal angle is the summation of Ix for the first rectangle+ the Ix for the second rectangle.
The moment of inertia Ix2 value of the solved problem.
The final Ix=22.00 m4 is part a) of the solved problem.
For part c) of the solved problem, for each rectangle we estimate the product of inertia about the external axes, and then add them together.
The final Ixy=6.00 m4 is part b) of the solved problem.
This is the PDF used for the illustration of this post.