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Monday, 27 February 2017

MOMENT OF INERTIA OF A CIRCULAR SECTION


Today we will see here the method to determine the moment of inertia of a circular section with the help of this post.

Let us consider one circular section, where we can see that R is the radius and O is the centre of the circular section as displayed in following figure.
 
Let us consider one small elementary circular strip of thickens dr and radius r as displayed in following figure.

R = Radius of the circular section
D = Diameter of the circular section
O = Center of the circular section

r = Radius of the small elementary circular strip
dr = Thickness of the small elementary circular strip

XX and YY are the axis which is passing through the center O of the circular section.

ZZ is the axis which is passing through the center O of the circular section and perpendicular to the plane of paper.

IXX = Moment of inertia of circular section about XX axis
IYY = Moment of inertia of circular section about YY axis
IXX = IYY (due to concept of symmetry)
IZZ = Moment of inertia of circular section about ZZ axis

Now we will determine the value or expression for the moment of inertia of circular section about XX axis and also about YY axis

First of all we will have to find out the moment of inertia of circular section about ZZ axis and after that we will use the principle of perpendicular axis i.e. the perpendicular axis theorem and its proof in order to secure the moment of inertia of circular section about XX axis and also about YY axis.

Let us determine the moment of inertia of small elementary circular strip about an axis ZZ which is passing through the center O of the circular section and perpendicular to the plane of paper.

Area of small elementary circular strip = 2П * r * dr
Moment of inertia of small elementary circular strip about axis ZZ = 2П * r * dr * r2
Moment of inertia of small elementary circular strip about axis ZZ = 2П r3 dr

Moment of inertia of the entire circular section about the axis ZZ will be determined by integrating the above equation between limit 0 to R and it as displayed here in following figure.
Therefore, moment of inertia of circular section about ZZ axis, IZZ = ПR4/2
Or we can also say that IZZ = ПD4/32

Let us use and recall the principle of perpendicular axis i.e. the perpendicular axis theorem and its proof in order to secure the moment of inertia of circular section about XX axis and also about YY axis.

IZZ = IXX + IYY

As we have already discussed above that IXX = IYY   

Therefore, moment of inertia of circular section about XX axis and moment of inertia of circular section about YY axis could be easily concluded as mentioned here.

IXX = IYY = IZZ /2
IXX = IYY = ПD4/64

IXX = ПD4/64
IYY = ПD4/64

Do you have any suggestions? Please write in comment box

Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google
We will see another important topic i.e. in the category of strength of material.

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