Mechanics of Materials · §5.2
Grid
How a bending moment becomes stress that grows straight-line from the neutral axis — the one fact that sizes every beam.
What bending does
compression · tension
The concave top is squeezed, the convex bottom pulled. Between them lies the neutral surface — its length never changes.
Plane sections stay plane · the premise
ε(y) = y / ρ → σ = Eε
Each cross-section stays flat and simply rotates, so both strain and stress are proportional to the distance y from the axis: σ is linear in y.
The flexure formula · σ = My / I
Zero at the neutral axis, growing straight-line to a maximum at each face — compression on top, tension below.
Where the neutral axis sits
∫ y dA = 0 → through the centroid
Pure bending carries no net axial force, so the stresses sum to zero — which holds only when y is measured from the centroid.
Second moment of area · why depth wins
I = bh³ / 12
Each fibre is weighted by y², so far material counts most. Double the depth and I grows 2³ = 8× — the cube on h is the payoff.
Section modulus · one number to size a beam
σmax = M / S, S = I / c
Collect the geometry into S = I/c. Design becomes one line: require S ≥ M / σallow.
Worked example · size the stress
σmax = 64 MPa
A 50 × 150 mm beam under M = 12 kN·m: S = bh²/6 = 1.875×10⁻⁴ m³, so σmax = M/S = 64 MPa.
The payoff · put material where the stress is
Fibres far from the axis carry nearly all the load. An I-beam banks its area in the flanges — raising I, and S, for free.
§5.2 — conclusion
σ = My / I
One straight line — zero at the neutral axis, largest at the extreme fibre. Deepen the section or move area outward to raise S and drop σmax.