---
# System prepended metadata

title: Second moment of area
tags: [Materials Mechanics, Mechanical Mechanics]

---

# Second moment of area
## Common formulas
* Second moment of area : $I =\int r^2dA$
* parallel axis theorem : $I_s = I_x+AL^2$
* Section Modulus : $Z = \frac{I_x}{y_\max}$
* Radius of Gyration : $K = \sqrt{\frac{I}{A}}\ ,I = AK^2$
* Polar Second moment of area : $J = I_x+I_y$
* Polar Radius of Gyration : $K_J = \sqrt{\frac{J}{A}}$
## Second moment of area of different shapes
* **rectangle**
<div style="text-align: center;">
<img src="https://hackmd.io/_uploads/By3oOAU4ll.png" alt="image" width="250">
</div>

$$I_x = \frac{bh^3}{12}$$

$$I_y = \frac{hb^3}{12}$$

$$I_a = \frac{bh^3}{3}$$
* **triangle**
<div style="text-align: center;">
<img src="https://hackmd.io/_uploads/HJ7T_CINxx.png" alt="image" width="250">
</div>

$$I_x = \frac{bh^3}{36}$$

$$I_y = \frac{hb^3}{36}$$

$$I_a = \frac{bh^3}{12}$$

$$I_b = \frac{bh^3}{4}$$
* **circle**
<div style="text-align: center;">
<img src="https://hackmd.io/_uploads/S1WAd0INxe.png" alt="image" width="250">
</div>

$$I_x = I_y = \frac{\pi R^4}{4}$$

$$I_a = \frac54\pi R^4$$
* **half circle**
<div style="text-align: center;">
<img src="https://hackmd.io/_uploads/SklktAUElg.png" alt="image" width="250">
</div>

$$I_x = I_y = \frac{\pi R^4}{8}$$
* **Quarter circle**
<div style="text-align: center;">
<img src="https://hackmd.io/_uploads/HJs1FALVxl.png" alt="image" width="250">
</div>

$$I_x = I_y = \frac{\pi R^4}{16}$$