Last Updated on October 5, 2026 by Maged kamel
Mohr’s Circle of inertia-third case.
Content of the Post.
In this Post, we will discuss Mohr’s Circle of Inertia—the third case. In this case, the Moment of Inertia about the x-axis is bigger than the Moment of Inertia about the y-axis, and the Product of Inertia Ixy is negative.
We expect tan 2θp to be positive from the equation tan 2θp, since Ixy is negative. Two examples of shapes that satisfy the requirements of Ix, Iy, and Ixy are the unequal angles with b>a. We consider the Ix, Iy, and Ixy about the Cg. In this case, we raise b to the third power to increase Ix.
The second example is the right-angle Triangle case 1, where Ix > Iy and Ixy is negative at the Cg. We have Ix at Cg = b*h^3/36, while Iy at Cg = h*b^3/12. The Product of Inertia Ixy will be found to be equal to -b^2*h^2/72.
What is the orientation of the principal axes for Mohr’s Circle of Inertia, third case?
We have the general Expression for the Moment of Inertia for any oriented axis, which we call Ix’. In Mohr’s Circle for the third case of Inertia, the Value can be a maximum when the angle 2θ has a positive cosine and a positive sine; this angle is called 2θ.
For a positive 2θ, the sine and cosine are positive. This angle is measured from the X-axis to axis U in the anticlockwise direction. If we check Ix’s Expression when Ix is bigger than Iy, the first term (Ix+Iy)/2 will be positive. The second term, (Ix-Iy)/2 cos 2θ, will also be positive.
For the third term (-Ixy*sin(2θ), it will be multiplied by a negative, and the result will be positive. The following slide image shows how to derive the Expression.
For Mohr’s Circle of the inertia-third case, the Value of Ix’ can be a minimum Value when the angle 2θp2 has a negative cosine Value, and the sine Value is also negative. This angle is measured from the X-axis to axis V, in the clockwise direction.
According to the next slide, both negative values are obtained if the angle 2θp2 lies in the third quadrant. Both the U and V axes are shown in the next slide.
When is Iy’s Value maximum or minimum?
For Mohr’s Circle of the inertia-third case, the Value of Iy’ can be a maximum Value when the angle 2θp2 has a negative cosine Value, and the sine Value is also negative. This angle is measured from the Y-axis to the U-axis. The following slide image shows how to derive the Expression.
For Mohr’s Circle of the inertia-third case, the Value for Iy’ can be minimal when the angle 2θp1 has a cosine and also a sine of a positive Value. This angle is measured from the Y-axis to the V-axis. The following slide image shows how to derive the Expression.
You can view or download the PDF data for this Post and the next. Please review the first five slides for the current Post content.
In the next Post, we will continue with part 2 and show how to draw Mohr’s Circle of Inertia for case 3.
This is a link to a useful external resource—a calculator for cross-section, Mass, Axial and Polar Area Moment of Inertia, and Section Modulus.