Mohr's Circle for Moments of Inertia: Problem Solving

Mohr's circle is a graphical method for determining an area's principal moments by plotting the moments and product of inertia on a rectangular coordinate system. This circle can also be used to calculate the orientation of the principal axes.

Consider a rectangular beam. The moments of inertia of the beam about the x and y axis are 2.5(10⁷) mm⁴ and 7.5(10⁷) mm⁴, respectively. The product of inertia is 1.5(10⁷) mm⁴. Determine the principal moments of inertia and the orientation of the major and minor principal axes.

The moments and products of inertia are plotted on a rectangular coordinate system.

The center of the circle from the origin, calculated by taking the average of the moments of inertia values about the x and y axis, is 5.0(10⁷) mm⁴. The radius of the circle can be estimated using  trigonometry and is given by the expression,

Substituting the value of the moments and products of inertia into the above expression yields the radius as 2.9(10⁷) mm⁴. A Mohr's circle with the obtained center and radius is drawn. The points of intersection between the circle and the moment of inertia axis give the principal moments of inertia. The sum of the circle's radius and the average moments of inertia gives the maximum moment of inertia. This value is calculated to be 7.9(10⁷) mm⁴. Similarly, subtracting the circle's radius from the average moments of inertia gives the minimum moment of inertia. This value is calculated to be 2.1(10⁷) mm⁴.

The line joining the center of Mohr's circle to a reference point on the circumference makes an angle with the horizontal axis that can be obtained using the trigonometric formula. The angle, obtained using trigonometry, is 31.1 degrees. So, the angle between the major principal axis and horizontal axes is 15.6 degrees. The principal axis corresponding to the maximum value of the moment of inertia is obtained by rotating the x-axis 15.6 degrees counterclockwise. Similarly, the principal axis corresponding to the minimum value of the moment of inertia can be obtained by rotating the y-axis through the same angle.