MOMENT OF INERTIA OF RECTANGULAR SECTION ABOUT ITS BASE

We were discussing “The perpendicular axis theorem and its proof”, “The theorem of parallel axis about moment of inertia”, “Method to determine the area moment of inertia for the rectangular section” and “Area moment of inertia” in our previous posts.

Today we will see here the method to determine the area moment of inertia for the rectangular section about a line passing through the base of 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 one line is passing through the base of the rectangular section and let us consider this line as line CD and we will determine the area moment of inertia for the rectangular section about this line CD.

B = Width of the rectangular section ABCD

D = Depth of the rectangular section ABCD

I_CD = Moment of inertia of the rectangular section about the CD line

Now we will determine the value or expression for the moment of inertia of the rectangular section about the line CD

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

Let us determine first the area and moment of inertia of the rectangular elementary strip about the line CD

Area of rectangular elementary strip, dA = dY.B

Moment of inertia of the rectangular elementary strip about the line CD = dA.Y²

Moment of inertia of the rectangular elementary strip about the line CD = B Y² dY

Now we will determine the moment of inertia of entire area of rectangular section about the line CD and it could be easily done by integrating the above equation between limit 0 to D.

Therefore, moment of inertia of entire area of rectangular section about the line CD will be as displayed here in following figure

Therefore, moment of inertia of the rectangular section about the line CD

I_CD =BD³/3

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. Determination of the moment of inertia of a hollow rectangular section in the category of strength of material.