Beam Deflection in a Mezzanine Walkway
A warehouse mezzanine walkway spans 4.5 m between columns. The designer proposes a 200UB25.4 (I = 23.6×10⁶ mm⁴, E = 200 GPa) simply supported beam carrying a uniformly distributed dead load of 4 kN/m and a live load of 6 kN/m. The client complains that a trial installation 'bounces' noticeably when people walk across it. The designer points out the beam passes the strength check with a large margin: maximum bending moment is wL²/8 and the section capacity is well above demand. The walkway must satisfy a deflection limit of span/360 under serviceability loads and is subject to footfall excitation at roughly 2 Hz walking pace.
1. Calculate the maximum serviceability deflection and check it against span/360.
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Serviceability load w = 4 + 6 = 10 kN/m = 10 N/mm. L = 4500 mm. E = 200,000 MPa, I = 23.6×10⁶ mm⁴.
δ = 5wL⁴/(384EI) = 5×10×(4500)⁴/(384×200,000×23.6×10⁶).
L⁴ = 4.10×10¹⁴. Numerator = 5×10×4.10×10¹⁴ = 2.05×10¹⁶. Denominator = 384×200,000×23.6×10⁶ = 1.813×10¹⁵. δ = 11.3 mm.
Limit = L/360 = 4500/360 = 12.5 mm. So the beam passes the deflection limit - but only just, at 90% of the limit. The designer's strength argument is irrelevant to the complaint: the bounce is a dynamic serviceability issue, not a strength issue.
2. Explain why the walkway bounces despite passing the deflection check, and recommend a fix with justification.
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The span/360 limit is a static criterion for visual and finish damage; it does not control vibration. What people feel is the natural frequency of the beam-mass system. For a simply supported beam, f₁ = (π/2)×√(EI/(mL⁴)). Using the mass per metre from the 10 kN/m load (≈1020 kg/m): EI = 200,000×23.6×10⁶ = 4.72×10¹² N·mm²; mL⁴ = 1.02×4.10×10¹⁴ = 4.18×10¹⁴. f₁ = (π/2)×√(4.72×10¹²/4.18×10¹⁴) = (π/2)×√0.0113 ≈ 0.167 rad-based factor → f₁ ≈ 5.3 Hz.
A 5.3 Hz floor is in the range where walking harmonics (typically 1.6-2.4 Hz fundamental, with the second harmonic near 3-5 Hz) can excite resonance, and human perception is worst around 4-8 Hz. Passing span/360 says nothing about this. Fixes: increase I (deeper section, e.g. 310UB or adding a bottom flange plate) to raise f₁ above about 8-10 Hz, add mass to lower f₁ well below the excitation, or stiffen the supports. Increasing depth is the most efficient because f₁ scales with √I, so a section with roughly 4× the I doubles the frequency. Recommend the deeper section and re-check both static deflection and frequency.