# Section modulus formulas for different shapes

Last updated: November 8th, 2022

The Section modulus is usually used in the calculation of stresses in Cross-sections, however the Moment of inertia is often used instead, probably because designers are more familiar with Moment of inertia formulations.

While it’s very important to know how to derive and calculate the Section modulus, the further we get in our studies, the more we can use formulas.

In this post we’ll show, the most important and easiest formulas for Rectangular, I/H, Circular and hollow circular section but also formulas which involve more steps for L-, T- and U- Shapes.

All will be shown and explained by examples.

Now, before we get started, always remember that the unit of the Section modulus is the **third power of a length unit** [$length^3$].

If you would like to use $mm$ in your calculation, then the unit of the Section modulus is $mm^3$.

But now, let’s get started.

## 1. What is the section modulus?

The section modulus is a cross-sectional geometric property of structural elements such as beams, columns, slabs, etc. and it is used to calculate stresses in Cross-sections. In general, it can be said that the greater the dimensions of a cross-section under a given load, the greater the Section modulus and the smaller the bending stress.

## 2. Section modulus – Rectangular shape/section (formula)

#### Strong Axis

$W_y = \frac{1}{6} \cdot h^2 \cdot w$

#### Weak Axis

$W_z = \frac{1}{6} \cdot h \cdot w^2$

### Example calculation

h = 240mm, w = 120mm

##### Strong axis:

$W_y = \frac{1}{6} \cdot h^2 \cdot w = \frac{1}{6} \cdot (240mm)^2 \cdot 120mm = 1.15 \cdot 10^6 mm^3$

##### Weak axis:

$W_z = \frac{1}{6} \cdot h \cdot w^3 = \frac{1}{6} \cdot 240mm \cdot (120mm)^2= 5.76 \cdot 10^5 mm^3$

### Where is the Section modulus of a rectangular Cross-section used in real projects?

- Structural bending stress calculation of timber beams (here)
- Structural stress calculation of concrete beams

## 3. Section modulus – I/H shape/section (formula)

#### Strong Axis

$W_y = \frac{w \cdot h^2}{6} – \frac{(w-t_w) \cdot (h-2\cdot t_f)^3}{6h}$

#### Weak Axis

$W_z = \frac{w^2 \cdot 2t_f}{6} + \frac{t_w^3 \cdot (h-2t_f)}{6w}$

### Example calculation

$h$ = 300mm, $w$ = 150mm, $t_f$ = 10mm, $t_w$ = 7mm

##### Strong axis:

$W_y = \frac{w \cdot h^2}{6} – \frac{(w-t_w) \cdot (h-2\cdot t_f)^3}{6h} = \frac{150mm \cdot (300mm)^2}{6} – \frac{(150mm-7mm) \cdot (300mm-2\cdot 10mm)^3}{6h} = 5.06 \cdot 10^5 mm^3$

##### Weak axis:

$W_z = \frac{w^2 \cdot 2t_f}{6} + \frac{t_w^3 \cdot (h-2t_f)}{6w} = \frac{(150mm)^2 \cdot 2 \cdot 10mm}{6} + \frac{(7mm)^3 \cdot (300mm-2 \cdot 10mm)}{6\cdot 150mm} = 7.51 \cdot 10^4 mm^3$

### Where is the Moment of inertia of a I/H Cross-section used in real projects?

- Structural bending stress calculation of timber I-joists
- Structural bending stress calculation of steel I/H beams and columns

## 4. Section modulus – Circular shape/section (formula)

#### Strong Axis

$W_y = \frac{D^3 \cdot \pi}{32}$

#### Weak Axis

$W_z = \frac{D^3 \cdot \pi}{32}$

### Example calculation

D = 100mm

##### Strong axis:

$W_y = \frac{D^3 \cdot \pi}{32} = \frac{(100mm)^3 \cdot \pi}{32} = 9.82 \cdot 10^4 mm^3$

##### Weak axis:

$W_z = \frac{D^3 \cdot \pi}{32} = \frac{(100mm)^3 \cdot \pi}{32} = 9.82 \cdot 10^4 mm^3$

### Where is the Section modulus of a circular Cross-section used in real projects?

- Structural steel wind bracing tension rods
- Structural concrete column

## 5. Section modulus – Hollow circular tube Section (formula)

#### Strong Axis

$W_y = \frac{(D^4 – d^4) \cdot \pi}{32D}$

#### Weak Axis

$W_z = \frac{(D^4 – d^4) \cdot \pi}{32D}$

### Example calculation

D = 100mm, d = 90mm

##### Strong axis:

$W_y = \frac{(D^4 – d^4) \cdot \pi}{32D} = \frac{((100mm)^4 – (90mm)^4) \cdot \pi}{32 \cdot 100mm} = 3.376 \cdot 10^4 mm^3$

##### Weak axis:

$W_z = \frac{(D^4 – d^4) \cdot \pi}{32D} = \frac{((100mm)^4 – (90mm)^4) \cdot \pi}{32 \cdot 100mm} = 3.376 \cdot 10^4 mm^3$

### Where is the Section modulus of a circular Cross-section used in real projects?

- Structural steel wind bracing tension rods
- Steel columns

## 6. Section modulus – Hollow rectangular tube Section (formula)

#### Strong Axis

$W_y = \frac{W \cdot H^2}{6} – \frac{w \cdot h^3}{6H}$

#### Weak Axis

$W_z = \frac{W^2 \cdot H}{6} – \frac{w^3 \cdot h}{6W}$

### Example calculation

W = 120mm, H = 240mm, w = 100mm, h = 220mm

##### Strong axis:

$W_y = \frac{W \cdot H^2}{6} – \frac{w \cdot h^3}{6H} = \frac{120mm \cdot (240mm)^2}{6} – \frac{100mm \cdot (220mm)^3}{6\cdot 240mm} = 4.126 \cdot 10^5 mm^3$

##### Weak axis:

$W_z = \frac{W^2 \cdot H}{6} – \frac{w^3 \cdot h}{6W} = \frac{(120mm)^2 \cdot 240mm}{6} – \frac{(100mm)^3 \cdot 220mm}{6 \cdot 120mm} = 2.704 \cdot 10^5 mm^3$

### Where is the Section modulus of a hollow rectangular Cross-section used in real projects?

- Structural Columns

## 7. Section modulus – U profile/C channel (formula)

#### Strong Axis

$W_y = \frac{w \cdot h^2}{6} – \frac{(w – t_w) \cdot (h – 2t_f)^3}{6H}$

### Example calculation

w = 100mm, h = 80mm, $t_f$ = 5mm, $t_w$ = 5mm

##### Strong axis:

$W_y = \frac{w \cdot h^2}{6} – \frac{(w – t_w) \cdot (h – 2t_f)^3}{6H} = \frac{100mm \cdot (80mm)^2}{6} – \frac{(100mm – 5mm) \cdot (80mm – 2 \cdot 5mm)^3}{6 \cdot 80mm} = 3.878 \cdot 10^4 mm^3$

### Where is the Moment of inertia of a U Cross-section used in real projects?

- Wind bracing of steel or timber structures

## 8. Section modulus – T section/profile (formula)

#### Weak Axis

$W_z = \frac{t_f \cdot w^2}{6} + \frac{h \cdot t_w^3}{6\cdot w}$

### Example calculation

w = 100mm, h = 100mm, $t_f$ = 5mm, $t_w$ = 5mm

##### Weak axis:

$W_z = \frac{t_f \cdot w^2}{6} + \frac{h \cdot t_w^3}{6 \cdot w}$

$W_z = \frac{5mm \cdot (100mm)^2}{6} + \frac{100mm \cdot (5mm)^3}{6 \cdot 100mm}$

$W_z = 8.354 \cdot 10^3 mm^3$

If you are new to structural design, then check out our design tutorials where you can learn how to use the moment of inertia and section modulus to design structural elements such as

Do you miss any Section modulus formulas for any shape or Cross-section that we forgot in this article? Let us know in the comments✍️

## Section Modulus FAQ

**How do you calculate the section modulus?**

The Section modulus S is calculated by dividing the Moment of Inertia I by the distance z from the Cross-section centre to the edge.

**What is the unit of the section modulus?**

The unit of the section modulus is mm^3 [milimeter^3].

**Is section modulus first moment of area?**

Yes, the section modulus can also be called first moment of area.