Polar Properties of Solid and Hollow Circular Sections

For a solid circular section of diameter $d$:

$$J_p=\frac{\pi d^4}{32},\qquad W_p=\frac{J_p}{d/2}=\frac{\pi d^3}{16}.$$

For a hollow circular section with outer diameter $D$ and inner diameter $d$:

$$J_p=\frac{\pi(D^4-d^4)}{32},\qquad W_p=\frac{J_p}{D/2}=\frac{\pi(D^4-d^4)}{16D}.$$

• $J_p$ — polar second moment of area;
• $W_p$ — polar section modulus;
• $D$, $d$ — section diameters.

The calculator below is for a solid circular section.