Last Updated on September 22, 2024 by Maged kamel

Mohr’s Circle of Inertia part-2.

Point of minimum value of inertia.

We will continue our discussion about Mohr’s circle of inertia part -2.

The point of the minimum value of inertia is point Z’, where the Ixy value equals zero. What about the direction of the  V axis?

The direction is obtained from Mohr’s circle of inertia by an enclosed angle of (180-2θp) in the anti-clockwise or (2θp )from the Y-axis in the clockwise direction.

While in the Normal view, it has an angle of (90-θp) from the x-axis in the anti-clockwise direction.

The expression of tan 2θp.

In Mohr’s Circle of Inertia part-2, the value 2θp can be estimated from the third equation by setting Ix’y’ to zero. This will give the 2θp value equal to (-Ixy/(Ix-Iy/2)). The negative sign will indicate that the angle is in the clockwise direction.

The expression of Imax using Mohr’s circle.

The value of Imax or Iu can be estimated as equal to the distance between Ixy and the center of the circle, which equals (Ix+Iy)/2 plus the radius value. Checking the tan 2θp expression, we can draw an angle and get the radius value for Ixy, Ix, and Iy.

Please refer to the next slide image for deriving the value of the radius R.

The expression of Imin using Mohr’s circle.

From the next slide in Mohr’s Circle of Inertia part-2, the value of I minimum or Imin or Iv can be estimated as equal to shift which is (Ix+Iy)/2 minus the radius value.

 Checking the tan 2θp expression, we can draw an angle, get the radius value in terms of Ixy, Ix, and Iy, and deduct the value of (Ix+Iy)/2.

The expression of pole point.

The pole point in Mohr’s circle of inertia is the point from which you can get direction, and the circle’s center is the pole point. When we join to point x, it will give the X-axis. 

While joining with point y, you can get the Y direction. Similarly, a line from the center of the circle to point Z will give the principal maximum direction or the U direction.

The pole point in Mohr’s circle for the normal view.

From the last slide, we can get the pole of the standard view as point Z’ joining to the mirror point of x, which will give the U direction in the normal view, and joining to the mirror of point y will give the V direction.

Thanks a lot, and peace be upon you all.

A video covers the topic of the previous post and this Post.

The next post is Easy Approach to Mohr’s Circle of inertia-First Case.

This link to a useful external resource: a calculator for Cross-Section, Mass, Axial and polar area moment of Inertia, and Section Modulus.