In the case of a shaft or axle in torsion, the polar moment of inertia is required to calculate the torsional shear stress occurring on the shaft or to determine the angle of twist in a shaft.
Read the full article to know the steps for finding the polar moment of inertia, direct formulae for the standard shapes, and solved numericals.
What is the Polar moment of inertia?
Polar moment of inertia is the ability of a shape of a crosssection of an object to resist a torsional deformation (caused by the torque acting along the axis perpendicular to the crosssection). It is also known as a moment of inertia about the axis perpendicular to the plane of a crosssection of an object.
The high value of the polar moment of inertia indicates the crosssection has more ability to resist torsional deformation.
It is denoted by the symbol `\mathbf{I_{z}}` or J and has an SI unit of m^{4}.
As per the perpendicular axis theorem, the polar moment of inertia at any point can be calculated by adding the moment of inertia about two mutually perpendicular axes that lie into the plane of the crosssection and are concurrent at the same point.
The above figure shows the plane lamina with two perpendicular axes (x & y) lying into the plane and the zaxis is perpendicular to the plane and all three axes (x, y, z) are concurrent at ‘O’.
Thus by using the perpendicular axis theorem, the equation for the polar moment of inertia about the zaxis (Jₒ) is given by,
J_{o} = 𝙸_{z} = 𝙸_{x} + 𝙸_{y}
Where
𝙸_{x} = Moment of inertia about the xaxis
𝙸_{y} = Moment of inertia about the yaxis
Therefore by finding the moment of inertia about the x and yaxis and adding them together we can find the polar moment of inertia.
To know how the polar moment of inertia is different from the moment of inertia, read our this article.
How to calculate the polar moment of inertia?
The polar moment of the standard shape can be found using a direct formula listed the below table. But in the case of irregular shapes, it is difficult to get a standard formula.
In such cases use the following steps to obtain a polar moment of inertia.
Step 1] Divide the complex shape into the simple geometric shapes that has a standard formula for the moment of inertia:
Step 2] Find the position of a centroid of the shape, and mark x and y axis:
In case, the polar moment of inertia has to be found at the centroid, then it is necessary to find the position of a centroid first. Use the below formula for calculating the position of the centroid.
`\bar{X} = \frac{\sum A_{i}.x_{i}}{A_{\text{total}}}`
`\bar{Y} = \frac{\sum A_{i}.y_{i}}{A_{\text{total}}`
Step 3] Find the moment on inertia about x and y axis (Ix and Iy) passing through the centroid:
Use the parallel axis theorem for calculating the area moment of inertia about the centroidal axis X_{c} (I_{x} or I_{xc}) and Y_{c} (I_{y} or I_{yc}).
We have an article for finding the moment of inertia using the parallel axis theorem.
Step 4] Find the polar moment of inertia at the centroid using the below formula:
I_{z} = I_{x} + I_{y}
Check the numerical to clearly understand the above steps.
Polar moment of inertia formulae for different shapes:
Following are the equations to find the polar moment of inertia of the standard shapes:
Read below to know how to derive above formulae,
1] For solid circular shaft:
The moment of inertia of the circular profile about the centroidal xaxis is given by,
𝙸_{x} = `\frac{\pi }{64}` d^{4}
The moment of inertia of the circular profile about the centroidal yaxis is given by,
𝙸_{y} = `\frac{\pi }{64}` d^{4}
Thus the polar moment of inertia about ‘O’ is given by,
J_{o} = 𝙸_{x} + 𝙸_{y}
J_{o} = `\frac{\pi }{64}` d^{4} + `\frac{\pi }{64}` d^{4}
J_{o} = `\mathbf{\frac{\pi }{32}}` d^{4}
This is the equation for finding the polar moment of inertia for the circular shaft.
2] For hollow circular shaft:
The above figure shows the crosssection profile of a hollow circular shaft with an outer diameter (do) and inner diameter (di).
The moment of inertia of the hollow circular shaft about the centroidal xaxis is given by,
𝙸_{x} = `\frac{\pi }{64}. (d_{o}^{4} – d_{i}^{4})`
The moment of inertia of the hollow circular shaft about the centroidal yaxis is given by,
𝙸_{y} = `\frac{\pi }{64}.(d_{o}^{4} – d_{i}^{4})`
Now the polar moment of inertia for the hollow circular shaft about point ‘O’ is given by,
Jₒ = 𝙸_{x} + 𝙸_{y}
Putting the values of 𝙸_{x} and 𝙸_{y},
Jₒ = `[\frac{\pi }{64}. (d_{o}^{4} – d_{i}^{4})] + [\frac{\pi }{64}. (d_{o}^{4} – d_{i}^{4})]`
J_{o} = `\mathbf{\frac{\pi }{32}}.\mathbf{(d_{o}^{4} – d_{i}^{4})}`
This is the equation for finding a polar moment of inertia of hollow shafts, pipes, or tubes.
3] For rectangular plate:
The above figure shows the profile of the thin rectangular plate with a width of ‘b’ and a height of ‘h’.
The moment of inertia of a rectangular shape about the centroidal xaxis is given by,
𝙸_{x} = `\frac{bh^{3}}{12}`
The moment of inertia of a rectangular shape about the centroidal yaxis is given by,
𝙸_{y} = `\frac{hb^{3}}{12}`
Now the polar moment of inertia for the rectangular shape is given by,
Jₒ = 𝙸_{x} + 𝙸_{y}
J_{o} = `\frac{bh^{3}}{12}+ \frac{hb^{3}}{12}`
J_{o} = `\mathbf{\frac{1}{12}}` [bh³ + hb³]
This is the equation of the polar moment of inertia for a rectangular shape.
4] For square:
The square is a rectangle with all sides the same. Hence by putting b = d = a in the above equation of rectangle, we can get the formula for the polar moment of inertia of a square about the centroid.
J_{o} = `\frac{1}{12}.[a.a^{3} + a.a^{3}]`
J_{o} = `\mathbf{\frac{a^{4}}{6}}`
Solved example:
1] The hollow circular pipe has an outer diameter of 40 mm and an inner diameter of 35 mm. Find the polar moment of inertia for the pipe.
Given :
dₒ = 40 mm
d𝐢= 35 mm
Solution:
Using the formula of the polar moment of inertia for a hollow circular crosssection,
J_{o} = `\frac{\pi }{32} \times [d_{o}^{4} – d_{i}^{4}]`
J_{o} = `\frac{\pi }{32}` [40^{4} – 35^{4}]
J_{o} = 104003.89 mm⁴
2] Find polar moment of inertia about the axis passing through the centroid of the below Tsection.
Solution:
Step 1 and 2: Divide the T shape into simple shape and Find the position of centroid:
From the given diagram,
A_{1} = 2 × 1 = 2, x_{1} = 1, y_{1} = 0.5
A_{2} = 1 × 2 = 2, x_{2} = 1, y_{2} = 2
The position of the centroid is given by,
`\bar{Y} = \frac{A_{1}.y_{1}+A_{2}.y_{2}}{A_{1}+A_{2}}`
`\bar{Y} = \frac{2\times0.5 + 2\times2}{2+2}`
`\mathbf{\bar{Y}}` = 1.25
`\bar{X} = \frac{A_{1}.x_{1}+A_{2}.x_{2}}{A_{1}+A_{2}}`
`\bar{X} = \frac{2\times1 + 2\times1}{2+2}`
`\mathbf{\bar{X}}` = 1
Step 3: Moment of inertia about Xc and Yc:
From the above figure,
d_{1c} = `\bar{Y} – y_{1}` = 0.75
d_{2c} = `y_{2} – \bar{Y}` = 0.75
The Moment of inertia about the centroidal axis, X_{C} axis is given by,
`I_{xc} = I_{1xc} + I_{2xc}`
Applying the parallel axis theorem,
`I_{xc} = [I_{1x} + A_{1}.d_{1c}^{2}] + [I_{2x} + A_{2}.d_{2c}^{2}]`
`I_{xc} = [(\frac{b.h^{3}}{12})_{1} + A_{1}.d_{1c}^{2}] + [(\frac{b.h^{3}}{12})_{2} + A_{2}.d_{2c}^{2}]`
`I_{xc} = [\frac{2\times1^{3}}{12} + 2\times0.75^{2}] + [\frac{1\times2^{3}}{12} + 2\times0.75^{2}]`
`\mathbf{I_{xc}}` = 3.083
The Moment of inertia about the centroidal axis, Y_{C} axis is given by,
`I_{yc} = I_{1yc} + I_{2yc}`
As axis Y_{C}, y_{1}, y_{2} are same,
`I_{yc} = I_{1y} + I_{2y}`
`I_{yc} = [\frac{h.b^{3}}{12}]_{1} + [\frac{h.b^{3}}{12}]_{2}`
`I_{yc} = [\frac{1\times2^{3}}{12}] + [\frac{2\times1^{3}}{12}]`
`\mathbf{I_{yc}}` = 0.833
Step 4: Find the polar moment of inertia:
J_{o} = `I_{xc} + I_{yc}`
J_{o} = 3.083 + 0.833
J_{o} = 3.916
FAQs:

Where is a polar moment of inertia mostly used?
The polar moment of inertia is used for finding shear stresses and twisting deformation in objects subjected to the twisting load.

What mean by a high polar moment of inertia?
A higher polar moment of inertia indicates the mass is concentrated much away from the polar axis.

How angle of twist related to the polar moment of inertia?
The angle of twist is inversely proportional to the polar moment of inertia.