Bending stress in beams
When a beam sags, its top fibres shorten and its bottom fibres lengthen. Somewhere between them is a layer that does not change length: the neutral axis. For a symmetric elastic section it passes through the centroid.
Simple bending theory assumes plane sections stay plane, the material is elastic and the beam bends about a principal axis. From these, strain and therefore stress grow linearly with distance from the neutral axis, reaching their peaks at the extreme fibres. The bending formula links the bending moment, the second moment of area I and the distance y to the fibre.
The ratio I / y_max is called the section modulus Z. It measures how efficiently a shape resists bending. This is why I-sections put most of their material in the flanges, far from the neutral axis: the same weight of steel gives a much larger Z than a solid square.
The formula applies to elastic behaviour. Reinforced concrete is cracked in tension and steel beams can go plastic, so design codes use modified approaches, but the idea of a lever arm between a compression zone and a tension zone carries through to every beam design.
Compare sections by Z, not by area: depth pays off as the square.
M / I = σ / y = E / Rσ_max = M / Z, with Z = I / y_maxRectangle b × d: I = b d³ / 12, Z = b d² / 6
Using d² instead of d³ in the second moment of area of a rectangle.
Problem. A timber beam 100 mm wide and 200 mm deep carries a bending moment of 4 kNm. Find the maximum bending stress.
- Z = b d² / 6 = 100 × 200² / 6 = 666 667 mm³.
- M = 4 × 10⁶ Nmm.
- σ_max = M / Z = 4 × 10⁶ / 666 667 = 6.0 N/mm².
Answer. 6.0 MPa, compression at top and tension at bottom for sagging
Quick quiz
1Where is bending stress zero?
At the neutral axis.
2Doubling the depth of a rectangular beam changes Z by what factor?
Four, since Z grows with d².
3Why are I-sections efficient in bending?
Material is concentrated far from the neutral axis, raising I and Z per unit weight.
References and further reading
Standard texts, listed in no order of merit. We do not copy text, figures, tables or solved problems from any book; our explanations and examples are original.
- J. M. Gere, B. J. Goodno, Mechanics of Materials, Cengage, latest edition.
- F. P. Beer, E. R. Johnston, J. T. DeWolf, Mechanics of Materials, McGraw Hill, latest edition.
- E. P. Popov, Engineering Mechanics of Solids, Prentice Hall, 2nd edition.
Free and official sources
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