We were discussing “The perpendicular axis theorem and its proof”, “The theorem of parallel axis about moment of inertia” and “Area moment of inertia” in our previous posts.

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Let us consider one rectangular section ABCD as displayed in following figure. Let us assume that centre of gravity of the given rectangular section is C.G and axis X-X passing through the center of gravity of the rectangular section as displayed in following figure.
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Now we will determine the moment of inertia of entire area of rectangular section about the X-X axis and it could be easily done by integrating the above equation between limit (-D/2) to (D/2).
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Similarly, we will determine the moment of inertia
of the rectangular section about the Y-Y axis and we will have

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Today we will see here the method to determine the area
moment of inertia for the rectangular section with the help of this post.

Let us consider one rectangular section ABCD as displayed in following figure. Let us assume that centre of gravity of the given rectangular section is C.G and axis X-X passing through the center of gravity of the rectangular section as displayed in following figure.

B = Width of the rectangular section ABCD

D = Depth of the rectangular section ABCD

I

_{XX}= Moment of inertia of the rectangular section about the X-X axis###
*Now
we will determine the value or expression for the moment of inertia of the
rectangular section about the X-X axis*

*Now we will determine the value or expression for the moment of inertia of the rectangular section about the X-X axis*

Let us consider one rectangular elementary strip
with thickness dY and at a distance Y from the X-X axis as displayed in
above figure.

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*Let
us determine first the area and moment of inertia of the rectangular elementary
strip about the X-X axis*

*Let us determine first the area and moment of inertia of the rectangular elementary strip about the X-X axis*

Area of rectangular elementary strip, dA = dY.B

Moment of inertia of the rectangular elementary
strip about the X-X axis = dA.Y

^{2}
Moment of inertia of the rectangular elementary
strip about the X-X axis = B Y

^{2}dYNow we will determine the moment of inertia of entire area of rectangular section about the X-X axis and it could be easily done by integrating the above equation between limit (-D/2) to (D/2).

Therefore, moment of inertia of the rectangular
section about the X-X axis after calculation, we will have

###
**I**_{XX}
=BD^{3}/12

Similarly, we will determine the moment of inertia
of the rectangular section about the Y-Y axis and we will have
_{XX}=BD

^{3}/12

###
**I**_{YY}
=DB^{3}/12

_{YY}=DB

^{3}/12

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**Reference:**

Strength of material, By R. K. Bansal

Image Courtesy: Google

We will see another important topic i.e. Determination
of the moment of inertia of the rectangular section about a line passing
through the base of rectangular section in the category of strength of
material.