10.10:
Mohr's Circle for Moments of Inertia: Problem Solving
Consider the cross-sectional area of a beam for which the moments and product of inertia about the Cartesian coordinate axes are known. To determine the principal moments of inertia and the orientation of the principal axes, Mohr's circle can be used.
The moments and product of inertia are plotted on the x-axis and y-axis of a rectangular coordinate system, respectively.
The average of the rectangular moments of inertia gives the circle's center from the origin.
The circle's radius is determined from the moments and products of inertia.
Now, Mohr's circle is constructed with the obtained values of the circle's center and radius.
This circle intersects the moment of inertia axis at two points, corresponding to the minimum and maximum moments.
The values of these can be calculated using the circle's radius and the average moment of inertia.
The angle between a line joining the center to a reference point on the circumference and the horizontal axis is obtained using trigonometry.
Rotating the x-axis counterclockwise through half of this angle yields the major principal axis.
10.10:
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.
Suggested Reading
- R. C. , Hibbeler Engineering Mechanics Statics, Pearson. Pp. 557
- J.L., Meriam, L.G. Kraige, J.N. Bolton, Engineering Mechanics Statics, Wiley. Pp. 524-525