9+ Polar Moment Of Inertia Formulas [2024]

The polar moment of inertia, quantifies an object/section’s resistance to rotational motion about a specific axis.

This property plays an important role in structural and mechanical engineering, ranging from bolted steel connections to complex machinery analysis.

Every shape has a different polar moment of inertia. In this blog post, we’ll show the formulas for different sections, so you don’t have to calculate them yourself.

Let’s get started. 🚀🚀

What Is The Polar Moment Of Inertia?

The second polar moment of inertia, often denoted as I_p or J, is a measure of an object’s resistance to torsional, or twisting, deformation around an axis perpendicular to its longitudinal axis.

It is commonly used in engineering and mechanics to analyse and design structures, components and connections subjected to torsional loads, such as shafts, steel tubes and bolted connections.

Formula Of Polar Moment Of Inertia

The formula for the 2nd polar moment of inertia is the following:

$$I_p = \int \int_{A}^{} r^2 \,dA $$

With,

dA	Infinitesimal element
r	Radial distance to element dA

I know, I know, we all don’t like integrals. They are just too complicated, right?

So, the formula for the 2nd polar moment of inertia that we all understand better is:

$$I_p = I_y + I_z$$

Where

Iy	Moment of inertia strong axis
Iz	Moment of inertia weak axis

Now, the formula has become a lot easier. You can check out our blog post about moment of inertial formulas to calculate I_y and I_z quicker.

1. I- and H Section

Example calculation

$h$ = 300mm, $b$ = 150mm, $t_f$ = 10mm, $t_s$ = 7m

$I_p = \frac{w \cdot h^3}{12} – \frac{(w-t_w) \cdot (h-2\cdot t_f)^3}{12} + \frac{(h-2\cdot t_f) \cdot t_w^3}{12} + \frac{2\cdot t_f \cdot w^3}{12} $

$I_p = \frac{150mm \cdot (300mm)^3}{12} – \frac{(150mm-7mm) \cdot (300mm-2\cdot 10mm)^3}{12} \frac{(300mm-2\cdot 10mm) \cdot (7mm)^3}{12} + \frac{2\cdot 10mm \cdot (7mm)^3}{12} $

$I_p = 7.59 \cdot 10^7 mm^4$

2. Square Section

Example calculation

w = 120 mm

$J = \frac{w^4}{6} = \frac{(120mm)^4}{6} = 3.46 \cdot 10^7 mm^4$

3. Rectangular Section

Example calculation

w = 120 mm, h = 240mm

$J = \frac{wh (w^2 + h^2)}{12} = \frac{120mm \cdot 240mm \cdot ((120mm)^2 + (240mm)^2)}{12} = 1.73 \cdot 10^8 mm^4$

4. Circle/ Circular Section

Example calculation

D = 200 mm

$J = \frac{\pi \dot D^4}{32} = \frac{\pi \dot (200mm)^4}{32} = 1.57 \cdot 10^8 mm^4$

5. Circular Tube Section

Example calculation

D = 200 mm, d = 180mm

$J = \frac{\pi}{32} \cdot (D^4 – d^4) = \frac{\pi}{32} \cdot ((200mm)^4 – (180mm)^4) = 5.4 \cdot 10^7 mm^4$

6. Circular Section With Square Cutout

Example calculation

D = 200 mm, w = 100 mm

$J = \frac{\pi \cdot D^4}{32} – \frac{w^4}{6} = \frac{\pi \cdot (200mm)^4}{32} – \frac{(100mm)^4}{6} = 1.4 \cdot 10^8 mm^4$

7. Hexagon Section

Example calculation

D = 200 mm, w = 100 mm

$J = \frac{5 \cdot \sqrt{3}}{8} \cdot s^4 = \frac{5 \cdot \sqrt{3}}{8} \cdot (100mm)^4 = 1.08 \cdot 10^8 mm^4$

8. Hollow Rectangular Section

Example calculation

H = 240 mm, W = 120 mm, h = 200 mm, w = 90 mm

$J = \frac{WH (W^2 + H^2)}{12} – \frac{wh (w^2 + h^2)}{12} = \frac{120mm \cdot 240mm ((120mm)^2 + (240mm)^2)}{12} – \frac{90mm \cdot 200mm ((90mm)^2 + (200mm)^2)}{12} = 1.0 \cdot 10^8 mm^4$

9. Triangular Section

Example calculation

s = 100 mm

$I_p = \frac{\sqrt{3}}{48} \cdot s^4 = \frac{\sqrt{3}}{48} \cdot (100mm)^4 = 3.6 \cdot 10^6 mm^4$

10. Circular Section With Socket Head Cutout

Example calculation

D = 300 mm, s = 100 mm

$I_p = \frac{\pi D^4}{32} – \frac{5 \sqrt{3}}{8} \cdot s^4$

$I_p = \frac{\pi (300mm)^4}{32} – \frac{5 \sqrt{3}}{8} \cdot (100mm)^4$

$I_p = 6.9 \cdot 10^8 mm^4$

Conclusion

The polar moment of inertia is an important parameter in structural design. Therefore, it’s recommendable to know how to calculate it for different cross-sections. You do that by adding I_y and I_z together.

In the following blog posts, we show step-by-step, how to calculate these moments of inertias.

• Moment of inertia of an I-beam
• Moment of inertia calculation

If you are new to structural design, then check out our design tutorials about other structural elements. 👇👇

• Design of timber roof beams
• Buckling design of timber columns
• Buckling design of the collar beam

Do you miss any 2nd polar moment of inertia formula for any shape or cross-section that we forgot in this article? Let us know in the comments. ✍️✍️