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.

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
IXX = 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.

#### 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.Y2
Moment of inertia of the rectangular elementary strip about the X-X axis = B Y2 dY

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

### IXX =BD3/12

Similarly, we will determine the moment of inertia of the rectangular section about the Y-Y axis and we will have

### IYY =DB3/12

Do you have any suggestions? Please write in comment box

### Reference:

Strength of material, By R. K. Bansal