Structural Analysis Fundamentals / Bending of beams
Deriving the bending formula
Building σ = My/I from geometry, Hooke's law and equilibrium — not quoting it.
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Read the lesson register and outstanding review notesWhy this matters
Deriving the flexure formula shows how geometry, Hooke’s law and cross-section equilibrium fit together. Its assumptions determine where the result can be used.
By the end of this lesson you should be able to
- Explain why bending makes strain vary linearly through the depth
- Show that the neutral axis passes through the centroid
- Derive M/I = σ/y = E/R from equilibrium of the cross-section
- State every assumption and say what breaks if it fails
What you should already know
- Direct stress and strain, and Hooke's law (Module 7)
- Second moment of area (Module 9)
- Bending moment at a section (Module 3)
From first principles
Simple bending theory
We want to show: that bending stress varies linearly through the depth, and that M/I = σ/y = E/R.
Take a straight beam and bend it into a sag. The fibres near the bottom get longer; the fibres near the top get shorter. Between them there must be one layer whose length has not changed at all — the neutral surface. Now here is the key physical step. If a flat cross-section stays flat as the beam bends, and simply rotates, then how much a fibre stretches depends only on how far it sits from that neutral layer. Twice as far from the neutral axis means twice the stretch. Strain is therefore proportional to distance — and since stress follows strain through Hooke's law, so is stress. Everything else is bookkeeping: we add up the stresses over the cross-section and insist they carry no net axial force and exactly the applied moment.
Check yourself
In the derivation, which step tells us the neutral axis passes through the centroid?
Practice
A rectangular beam 150 mm wide and 400 mm deep carries a bending moment of 75 kN·m. Using the formula you have just derived, what is the maximum bending stress?
Practice
A rectangular beam 200 mm wide and 400 mm deep carries a bending moment of 120 kNm. Using σ = My/I, what is the maximum bending stress, in N/mm²?
Practice
For the same section, what is the elastic section modulus Z = I/ymax, in mm³?
Summary
- Geometry gives ε = y/R; Hooke's law gives σ = Ey/R
- Zero net axial force puts the neutral axis through the centroid
- Moment equilibrium over the section introduces I = ∫y² dA
- M/I = σ/y = E/R, from which σ = My/I and 1/R = M/EI