10.9:
Mohr's Circle for Moments of Inertia
Mohr's circle is a graphical method to determine an area's principal moments of inertia by plotting the moments and product of inertia on a rectangular coordinate system.
The average of the moments of inertia about the coordinate axes gives the circle's center from the origin and its radius is determined from the moments and the product of inertia.
The intersection points between the circle and the horizontal axis correspond to the principal moments of inertia. The product of inertia at these points is zero.
Consider a point on the circumference of the Mohr's circle. The line connecting this point to the center forms an angle with the axis of the positive moment of inertia.
This angle is twice the angle between the x-coordinate axis and the major principal axis. The minor principal axis for the area is perpendicular to the major principal axis.
The graphical representation of Mohr's circle remains the same when the moments of inertia and the product of inertia of the area with respect to a rotated axes are considered.
10.9:
Mohr's Circle for Moments of Inertia
Mohr's circle is a graphical method to determine an area's principal moments of inertia by plotting the moments and product of inertia on a rectangular coordinate system.
The center of Mohr's circle is obtained by averaging the moments of inertia about the x and y-axis. Its radius is determined from the moments and products of inertia.
The intersection points between this circle and the horizontal axis gives the value of the principal moments of inertia. The product of inertia is zero at these points.
Consider point B on the circumference of Mohr's circle. The line connecting this point to the center forms an angle with the axis of the positive moment of inertia. This angle is twice the angle between the x-coordinate axis and the major principal axis. The minor principal axis for the area is perpendicular to the major principal axis.
As all physical parameters remain unchanged regardless of orientation of the coordinate system, the same graphical representation can be used for calculation of the principal moments of area and the principal axes for rotated coordinates.
Suggested Reading
- R. C. , Hibbeler Engineering Mechanics Statics, Pearson. Pp. 555-556
- F.P. Beer, E.R. Johnston, D.F. Mazurek, P.J. Cornwell, B.P. Self, Vector Mechanics For Engineers Statics and Dynamics Engineering Mechanics Statics, Mc Graw-Hill Education. Pp. 523-524