Principal Moment of Inertia

For analyzing the stresses in unsymmetrical bending of the beam section we required principal Moment of Inertia about bending axis. The product of inertia is used for calculating Principal Moment of Inertia.

1. Product of Inertia

The product of inertia of an area A relative to the indicated XY rectangular axes is

IXY = ∫ xy dA

The value of product of inertia depends on the position and orientation of selected axes

1. The value of the product of inertia may be positive negative or zero.

2. If the total area lies in the first and third quadrant it will be positive. If the total area lies in a second and fourth quadrant it will be negative.
3. If there has one axis of symmetry the product of inertia will be zero, because the left part of the axis is canceled the right part.
Parallel axis theorem for products of inertia:
The product of inertia with respect to any pair of the axis in its plane is equal to the product of inertia with respect to the parallel centroidal axes plus,the product of the area and the coordinate of the centroid with respect to the pair axis.

I xy = I xy + A.x.y

2. Moments of Inertia about inclined axis

In structural and mechanical design, it is sometimes necessary to calculate the moment of inertia with respect to a set of inclined x', y', axes when the values of θ, I x , I y , I xy are known.

Note :

To find the moment of inertia with respect to a set of inclined x', y' axis with an angle of θ as shown in fig. we will use transformation equations which relates the x, y, and x’, y’ coordinates. 

From Figure, these equation are

x' = x cosθ + y sinθ

y' = y cosθ - x sinθ

substitute these values in moment of Inertia equations, we get M.I. of the body Ix', Iy' and product of Inertia Ix'y' about inclined axis x' and y' and given below

Note : By adding the equations for Ix’ and Ix’ we can show that the polar moment of inertia about z axis passing through point o is independent of the orientation of x’ and y'
Ix’ + Iy’ = Ix + Iy = Iz (The Polar Moment of Inertia)

Thus the rotation of axes does not affect polar moments of inertia.

3. Principal Axes and Principal Moments of Inertia

Principal axes are the axes about which the product of Inertia is zero. The Moment of Inertia about the Principal axes are called Principal Moment of Inertia of the body/ section. 

For any symmetric axis (x or y axis) the product of Inertia is obviously zero, hence in the axis of symmetry  the both axes will represent  as Principal axes.

For unsymmetrical section we can find out principal axis by rotating the axes by an angle θ where the product of Inertia becomes zero. Ix'y' = 0   (equation no.3)

 =  0

                           

Alternatively the Angle which makes Ix’ and Iy’ either maximum or minmum can be found by setting the derivative of either Ix’ or Iy’ w.r.t. θ, and equal to zero.

The above equation have two solutions for θ, and they will differ by π/2

one value of θ will define the axis of maximum Moment of Inertia (major principal MI ) and the other defines the axis of minimum Moment of Inertia (minor principal MI).

These two rectangular axes are called the principal axes of inertia.

 

Substituting the angle θ in equations one and two, we get maximum Moment of Inertia and the minimum Moment of Inertia.

Note: the similarity between the transformation equations for principal moment of Inertia and products of inertia and the transformation equations of stress in the case of principal stresses.