Bending stress and deflection

Why it matters

The moment diagram tells you how hard a beam is being bent. Two questions follow. Strength: is the bending stress below what the material can safely take? Serviceability: does the beam sag so much that floors bounce, plaster cracks or doors jam? A beam can pass the first check and still fail the second, and for long spans deflection often governs. This lesson connects the moment from the last lesson and the section property I from lesson 3 to both checks.

The flexure formula

When a beam sags, its top fibres shorten and its bottom fibres lengthen. Somewhere between them is the neutral axis, which passes through the centroid of the section and has zero bending stress. Stress grows linearly with distance from that axis:

σ = M · y / I

The largest stress is at the fibre farthest from the neutral axis, at distance c:

σ_max = M · c / I = M / S,     where S = I / c is the section modulus

For a rectangle, S = b·h²/6. Steel section tables list S directly because it is what designers use.

Worked example: a timber beam

A rectangular timber beam, 50 mm wide and 150 mm deep, spans 3.0 m between simple supports and carries a 5.0 kN point load at mid-span. Take E = 10,000 MPa (a typical figure for softwood along the grain). Ignore the beam's self-weight.

Bending moment.

M = P·L / 4 = 5,000 N × 3,000 mm / 4 = 3.75 × 10⁶ N·mm   (3.75 kN·m)

Section properties.

I = b·h³/12 = 50 × 150³ / 12 = 14.06 × 10⁶ mm⁴
c = h / 2 = 75 mm

Bending stress.

σ_max = M·c / I = 3.75 × 10⁶ × 75 / 14.06 × 10⁶ = 20.0 MPa

The top fibre is at 20 MPa compression and the bottom at 20 MPa tension.

Deflection

Integrating the curvature M/(E·I) twice gives the deflected shape. You do not need to repeat that derivation for common cases; engineers use standard results:

For the timber beam:

δ = 5,000 × 3,000³ / (48 × 10,000 × 14.06 × 10⁶) = 20.0 mm

Deflection limits are usually written as a fraction of the span. As an illustration only, L/360 is a limit often quoted for floors under live load; here L/360 = 8.3 mm, so this beam would sag well over twice that much. The fix is more depth: because I grows with h³, a 225 mm deep beam of the same width has I = 47.5 × 10⁶ mm⁴ and deflects only 5.9 mm, while its stress drops to 8.9 MPa. (A 200 mm deep beam deflects 8.4 mm, still just over the limit.)

The real limits, load combinations and allowable stresses come from the applicable building code and material standard (for timber, values also depend on species, grade, moisture and load duration), so treat these numbers as practice, not design values.

Why deflection depends on L³ or L⁴

Doubling the span of a uniformly loaded beam multiplies its deflection by 2⁴ = 16. That is why long spans are expensive: you need much deeper sections, and why deflection, not strength, often controls the design of long beams.

Strength check versus stiffness check

• The stress check depends on M, c and I (or S), and is compared with an allowable or design strength of the material.
• The deflection check depends on E and I, and is compared with a limit like L/240 or L/360 from the code.
• Changing to a stronger material of the same E (a higher grade of steel, say) helps strength but does nothing for deflection.

Common mistakes

• Mixing kN·m with mm. Convert M to N·mm (× 10⁶) when I is in mm⁴, so σ comes out in MPa.
• Using the wrong formula for the support and load case. P·L³/48EI is only for a central point load on a simple span.
• Forgetting self-weight. Real beams carry their own weight as a uniform load; this example left it out for clarity.
• Ignoring other checks. Real design also checks shear, bearing at supports, lateral-torsional buckling of slender beams, and vibration. These are covered in design courses and codes.

Recap

• σ = M·y/I, maximum at the extreme fibre: σ_max = M·c/I = M/S.
• Deflection uses standard formulas in E·I; it grows with L³ or L⁴.
• Check both strength and deflection; deeper sections help both.
• Any real beam is sized to the governing code and checked by a licensed engineer.

Try it

On the Do tab, finish the deflection line of a small beam checker and use it on this lesson's timber beam.