Choosing a Beam Section for a Mezzanine Walkway
A workshop mezzanine walkway spans 4.0 m between simple supports and must carry a uniformly distributed live load of 3.0 kN/m plus its own self-weight. Two candidate sections are available in stock: a 150×75 mm solid hardwood timber beam (E = 11 GPa, allowable bending stress 8 MPa, density 750 kg/m³) and a 200 mm deep × 100 mm wide glued-laminated section (E = 13 GPa, allowable bending stress 12 MPa, density 550 kg/m³). Deflection under service load must not exceed span/300. The fabricator prefers timber because it can be cut on site, but the client has noticed visible sag in an existing similar walkway and wants the stiffer option. A third option, a 150 mm deep steel channel, is available but would require crane access and hot-work permits.
1. Determine whether the solid timber beam satisfies both the bending stress and deflection criteria, showing your working.
Read the worked answer
Loads. Self-weight of timber: 0.150 × 0.075 × 750 × 9.81 = 82.8 N/m ≈ 0.083 kN/m. Total w = 3.0 + 0.083 = 3.083 kN/m.
Bending. M = wL²/8 = 3.083 × 4.0²/8 = 6.17 kN·m = 6.17 × 10⁶ N·mm. Section modulus Z = bd²/6 = 75 × 150²/6 = 281 250 mm³. σ = M/Z = 6.17 × 10⁶ / 281 250 = 21.9 MPa.
This exceeds the 8 MPa allowable by a factor of about 2.7, so the section fails in bending regardless of deflection.
Deflection (for completeness). I = bd³/12 = 75 × 150³/12 = 21.09 × 10⁶ mm⁴. δ = 5wL⁴/(384EI) = 5 × 3.083 × 4000⁴ / (384 × 11 000 × 21.09 × 10⁶) = 3.94 × 10¹⁵ / 8.91 × 10¹³ = 44.2 mm. Limit = 4000/300 = 13.3 mm. Deflection also fails, by more than three times.
Conclusion. The solid timber beam is inadequate on both criteria; the observed sag in the existing walkway is consistent with an under-designed section rather than a workmanship fault.
2. Evaluate the glulam option against the same criteria and recommend a course of action, justifying your reasoning.
Read the worked answer
Loads. Self-weight: 0.200 × 0.100 × 550 × 9.81 = 108 N/m ≈ 0.108 kN/m. w = 3.108 kN/m.
Bending. Z = 100 × 200²/6 = 666 667 mm³. M = 3.108 × 16/8 = 6.22 kN·m. σ = 6.22 × 10⁶ / 666 667 = 9.3 MPa, which is below the 12 MPa allowable (utilisation 0.78). Satisfies.
Deflection. I = 100 × 200³/12 = 66.67 × 10⁶ mm⁴. δ = 5 × 3.108 × 4000⁴ / (384 × 13 000 × 66.67 × 10⁶) = 3.97 × 10¹⁵ / 3.33 × 10¹⁴ = 11.9 mm. Limit 13.3 mm, so it satisfies with about 10% margin.
Judgement. The glulam section works on both criteria, but the deflection margin is thin and service deflection is the client's actual complaint, so a designer should not treat a 10% margin as generous - long-term creep of timber under sustained load, not captured in this short-term check, will push deflection past the limit. The honest recommendation is either to increase the glulam depth (say 250 mm, which roughly halves deflection) or accept the steel channel if crane access and hot-work permits can be scheduled without disrupting production. Choosing the glulam purely because it is the cheapest compliant option ignores the mechanism of the original failure and the client's stated concern.