Brief data for post-7-inertia

7-Moment of inertia for right-angle-Ix-Case-1.

Moment of inertia for right-angle-Ix-Case-1.

Moment of inertia for right-angle-Ix-Case-1 video.

We have an objective, which is how to determine the moment of inertia for right-angle-Ix about the X-axis, by using a horizontal strip parallel to the external X-axis, we have a right-angle triangle in two shapes. In the first shape, we have the triangle AB, the adjacent side at the base, and the opposite side at the left-hand side of the triangle, this is the hypotenuse side.

We are going to make the X-axis coincide with the adjacent side and the Y-axis will coincide with the opposite side. We are going to use a horizontal strip parallel to the x-axis at the adjacent base and with the same result, we can obtain it by using a strip vertical strip, which is perpendicular to the external x-axis. This is a part of the video, which has a closed caption in English.

Case no.1 is the one for which the x-axis coincides with the base and the y-axis coincides with the opposite side, while case no.2 is the one for which the x-axis coincides with the base and the y-axis is at the intersection point between the adjacent side and the hypotenuse side.

Case no.1, the moment of inertia for right-angle-Ix is the second case in the reference attached table from FE Exam reference manual 3-5. This is the list of the first moment of area and inertia for the common plane shapes.

list of inertia for plain shapes

The next slide image represents the difference between Case -1 and Case-2 of the right-angle Triangle. for case -1 the opposite of the triangle is at the left side and for case-2, the opposite of the triangle is at the right.

The difference between case-1 and case-2.

How to get inertia for right-angle-Ix by using a horizontal strip?

Using the following steps, we will estimate the inertia for right-angle-Ix using a horizontal strip:
1-for line CB, we will write the equation as y=-h*x/b+h, we can check the validity of this equation by substituting the value of x coordinates of both C& b and get the corresponding value.

2- Our strip is a horizontal strip with width=dy and base=x. 3-dA, which is the area of the horizontal strip will be=x*dy, to get the dy, we will differentiate the y equation, which we have already estimated.
3- Our moment of inertia due to the strip is dIx=dA*y^2, remember that dA=x*dy.
4- we can write dIx=b*(h-y)/h*(y^2*dy), after substitution by the value of dy.  

5-Ix=∫dIx=∫b(h-y)/h(y^2dy), from y=0 to y=h.6- after integration and substitution we get Ix=bh^3/12.
The value of inertia for a right-angle-Ix=base*height^3/12.

                                                                         

Moment of inertia for a right-angle-Ix the right-angle triangle-case 1.

How to get the inertia for right-angle-Ix by using a vertical strip?

Using the following steps, we will estimate the Ix the moment of inertia for the right-angle triangle by using a vertical strip: 1- we are going to move this strip horizontally, so we have to substitute the value of y=-h/b*x+h, as we will see later.
2- Our strip is a vertical strip with width=dx and height=y.
3- The strip with an area dA, will be=y*dx.                                 

4-The moment of inertia for the small element -dIx from our study of a rectangular section can be estimated as dx*y^3.
5- The moment of inertia for the small element – dIx value is shown as per the next slide picture.

Moment of inertia for a right-angle-Ix , by using vertical strip.

6-Integrate from x=0 to x=b. 

7- We will get the same value of Ix as estimated by using a horizontal strip, which is=bh^3/12.8- k^2x=Ix/A=bh^3/(12(0.50b*h))=h^2/6.

Moment of inertia  for the right-angle- Ix at the CG.

How to get inertia for right-angle-Ix but at the CG?

Ixg at the CG can thus be obtained from the theorem of parallel axes, Ixg=Ix-A*ybar^2=(b*h^3/12)-(0.50*b*h)*(h/3)^2=Ixg=b*h^3/36. The radius of gyration at the Cg, Kxg=sqrt(IxG/Area)=sqrt(bh^3/36)/(0.50bh))=h/(3sqrt(2)).

The polar moment of inertia for the rectangular section.

This is the PDF used for the illustration of this post.
This is a link for the next post,  Moment of inertia – for the right-angle triangle-case-1-Iy.

  This is a useful external link for the moment of inertia- The Moment Of Inertia Of a Rectangle.

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