Timber joist passes the stress check but sags too much
A timber floor joist in a school verandah spans 4.0 m between simple supports and carries a uniformly distributed load of 6 kN/m, which already includes its own weight. The section chosen is 150 mm wide × 250 mm deep. For the species and grade available, the permissible bending stress is 8.5 N/mm², the permissible average shear stress is 1.0 N/mm², the modulus of elasticity is 5600 N/mm², and the deflection under service load must not exceed span/360. A plastered ceiling is fixed below the joist, so noticeable sag will crack the plaster. The contractor has already bought the timber and argues that because the bending stress is 'well within' the permissible value the joist is safe. The owner will not accept a mid-span support because it would obstruct the verandah. Assume the joist is simply supported, rectangular in section, and behaves elastically.
1. Calculate the maximum bending moment, the maximum shear force and the resulting bending and shear stresses, and state whether the section is adequate in strength.
Read the worked answer
Bending. M = wL²/8 = 6 × 4²/8 = 12 kN·m = 12 × 10⁶ N·mm. Section modulus Z = bd²/6 = 150 × 250²/6 = 1.5625 × 10⁶ mm³. Bending stress σ = M/Z = 12 × 10⁶ / 1.5625 × 10⁶ = 7.68 N/mm² < 8.5 N/mm² → OK, but 90% of the permissible value is used up; the margin is thin, not comfortable.
Shear. V = wL/2 = 6 × 4/2 = 12 kN. For a rectangle the maximum shear stress is 1.5 times the average: τ_max = 1.5 V/(b·d) = 1.5 × 12 000 / (150 × 250) = 0.48 N/mm² < 1.0 N/mm² → OK, and comfortably so; shear is not critical for a long, lightly loaded timber beam.
Interim conclusion. The contractor is right about strength and wrong to stop there: bending and shear are only two of the three checks, and the third is usually the one that governs timber floors. The 'well within' claim is also an exaggeration - at 90% utilisation any hidden defect in the timber would push it over.
2. Check the deflection against the serviceability limit and state what governs the design.
Read the worked answer
Section properties. I = bd³/12 = 150 × 250³/12 = 195.3 × 10⁶ mm⁴.
Deflection. δ = 5wL⁴/(384EI), with w = 6 kN/m = 6 N/mm and L = 4000 mm: 5 × 6 × (4000)⁴ = 5 × 6 × 2.56 × 10¹⁴ = 7.68 × 10¹⁵ 384 × 5600 × 195.3 × 10⁶ = 4.20 × 10¹⁴ δ = 7.68 × 10¹⁵ / 4.20 × 10¹⁴ = 18.3 mm.
Limit. L/360 = 4000/360 = 11.1 mm.
Verdict. 18.3 mm exceeds 11.1 mm by 65%. The joist is unsafe in service even though it is safe in stress. The plastered ceiling will crack, the floor will feel springy, and long-term creep and moisture in a humid climate will increase the sag well beyond the calculated elastic value.
What governs. Serviceability, not strength. This is the standard trap: a beam can satisfy σ < σ_permissible and τ < τ_permissible and still be unfit for purpose, because deflection grows with the cube of the span and the inverse cube of the depth. Always check all three criteria.
3. The owner refuses a mid-span support. Recommend the cheapest workable change and verify it numerically.
Read the worked answer
How much more stiffness is needed. Required factor = 18.3 / 11.1 = 1.65. Since I = bd³/12, depth is the efficient lever: increasing b by 65% would use far more timber than increasing d. For d: d_new/d_old = (1.65)^(1/3) = 1.18, so d_needed ≈ 250 × 1.18 = 295 mm.
Adopt 150 × 300 mm. I = 150 × 300³/12 = 337.5 × 10⁶ mm⁴. δ = 7.68 × 10¹⁵ / (384 × 5600 × 337.5 × 10⁶) = 7.68 × 10¹⁵ / 7.26 × 10¹⁴ = 10.6 mm < 11.1 mm → OK. Check strength with the extra self-weight: Z = 1.5 × 10⁶ mm³ × ... more precisely 150 × 300²/6 = 2.25 × 10⁶ mm³, so σ falls to about 5.8 N/mm² even with a slightly heavier joist. Shear stress also falls. Both are satisfied.
Practical notes for the answer sheet.
- Deepening by 50 mm adds about 20% of timber volume, which is cheaper than a steel mid-support and loses no verandah space.
- Alternately, specify a higher-E species or a seasoned, graded section, but the depth change is the reliable fix.
- Whatever is chosen, the timber must be seasoned and protected from rain, and the ceiling fixings should be detailed to tolerate the residual deflection.