# 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.