Module 9 · Lesson 9.3
Second moment of area and section modulus
The flexure formula, and what makes one section better than another.
Why this matters
Two beams with identical amounts of steel can differ in capacity by a factor of three, purely because of where the material sits. This lesson explains why, and gives you the number that captures it.
By the end of this lesson you should be able to
- Use σ = My/I
- Calculate I for rectangles and I-sections
- Use Z to check stress quickly
What it calculates: The bending stress at any distance y from the neutral axis, and the peak stress at the extreme fibre.
- σ
- Bending stress (N/mm²)
- M
- Bending moment at the section (N·mm)
- y
- Distance from the neutral axis (mm)
- I
- Second moment of area about the bending axis (mm⁴)
- c
- Distance to the extreme fibre (mm)
- Z
- Elastic section modulus (mm³)
This assumes
- The material is elastic and follows Hooke's law
- Plane sections remain plane
- Bending is about a principal axis of the section
- Deflections are small
In plain terms: Stress grows with the moment and with distance from the neutral axis, and falls as I grows. Watch the units: a moment in kN·m must be multiplied by 10⁶ to become N·mm before dividing by mm³.
Second moment of area measures how the material is spread relative to the bending axis. Each bit of area counts in proportion to the square of its distance from the neutral axis, so material far out counts enormously more. For a rectangle, I = bd³/12 — notice the depth is cubed. Double the depth of a beam and you multiply I by eight.
That single fact explains why beams are deep rather than wide, and why an I-section puts most of its material in flanges top and bottom, joined by a thin web whose main job is to carry shear and hold the flanges apart.
Try it
Move material around the section
Keep the depth and watch what happens to I when material moves away from the neutral axis.
Shape
mm
mm
kN·m
- Area A
- 60,000 mm²
- Second moment I
- 8.000e+8 mm⁴
- Section modulus Z
- 4.000e+6 mm³
- Extreme fibre stress M/Z
- 15.0 N/mm²
- Weak axis I
- 1.125e+8 mm⁴
- I per unit area
- 13333 mm²
The I-section carries a similar moment with far less material, because the flanges sit where the bending stress is largest. Notice how much smaller the weak-axis value is.
Worked example
Bending stress in a rectangular beam
Given
- Rectangular section 200 mm wide × 450 mm deep
- Bending moment at the section = 90 kN·m
Find
The maximum bending stress, and where it occurs.
Assumptions
- Elastic behaviour
- Bending about the strong axis
- Plane sections remain plane
Practice
A rectangular timber beam is 100 mm wide and 300 mm deep. What is its second moment of area about the strong axis?
Practice
A solid circular section has a diameter of 200 mm. What is its second moment of area about a centroidal axis?
Practice
A rectangular section 120 mm wide and 360 mm deep carries a bending moment of 54 kN·m. What is the maximum bending stress?
Practice
A beam section has I = 4.5 × 10⁸ mm⁴ and an overall depth of 500 mm. What is its elastic section modulus?
Summary
- σ = My/I, and σmax = M/Z at the extreme fibre
- I = bd³/12 for a rectangle — depth cubed, so depth dominates
- I-sections put material in the flanges where the stress is greatest
- Convert kN·m to N·mm before dividing by a section modulus in mm³
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint