Vertical vibration comfort limits — LRFD Guide Specifications for the Design of Pedestrian Bridges.

The LRFD Guide Specifications for the Design of Pedestrian Bridges includes the following criteria for pedestrian comfort against vibrations:

\[ f_{n} \geq 2.86 \ln \frac{180}{W} \]

Symbol Description
\(f_{n}\) Natural frequency of footbridge
\(W\) Weight of footbridge in kips

This equation is derived from the Allen-Murray Pedestrian Load Model, which tells us the peak vertical acceleration in the structure \(a_{peak}\) is given by the expression:

\[ a_{peak} = \frac{0.83 R F_{0} g}{\zeta W}e^{-0.35f_{n}} \]

We can then set the criteria that accelerations must be below some acceptable level \(a_{max}\)

\[\begin{align} a_{peak} & \leq a_{max} \\ a_{max} & \geq \frac{0.83 R F_{0} g}{\zeta W}e^{-0.35f_{n}} \\ e^{0.35 f_{n}} & \geq \frac{0.83 R F_{0}}{\zeta W \frac{a_{max}}{g}} \\ f_{n} & \geq 2.86 \ln \frac{0.83 R F_{0}}{\zeta W \frac{a_{max}}{g}} \end{align}\]

We can then substitute in the recommended values for footbridges

Symbol Description Value
\(R\) Reduction factor 0.7
\(F_{0}\) Static pedestrian load 0.157 kips
\(\zeta\) Damping factor 0.01
\(\frac{a_{max}}{g}\) Max allowable vertical acceleration 0.05

\[ f_{n} \geq 2.86 \ln \frac{180}{W} \]

Note that this formula assumes a 157 lb person walking across the bridge, so may not be suitable for situations where users may be running, jumping, climbing stairs, etc.