This load model is described in Design Criterion for Vibrations Due to Walking by D.E. Allen and T.M. Murray (Engineering Journal - American Institute of Steel Construction, 30, (4), pp. 117-129, 93)
\[\begin{align} F(t) & = 0.83 F_{0} e^{-0.35f_{n}} \sin 2 \pi f_{n} t \\ a_{peak} & = \frac{0.83 R F_{0} g e^{-0.35f_{n}}}{\zeta W} \end{align}\]
| Symbol | Description |
|---|---|
| \(F(t)\) | Cyclic load |
| \(a_{peak}\) | Peak vertical acceleration of floor due to cyclic load |
| \(F_{0}\) | Static load of single pedestrian (160 lbf or 0.7 kN) |
| \(f_{n}\) | Natural frequency of the floor/bridge |
| \(\zeta\) | Damping factor |
| \(R\) | Reduction factor (\(0.7\) for footbridges, \(0.5\) for two-way slabs) |
| \(W\) | Weight of floor structure / bridge |
This formula forms the basis for the LRFD Guide Specifications for the Design of Pedestrian Bridges Vertical vibration comfort limits.
Cyclic loadings can be represented by a truncated Fourier series:
\[ F(t) = F_{0}(1 + \sum_{n=1}^{N} \alpha_{n} \sin 2 \pi n f t)\label{a}\tag{Eq.1} \]
Allen empirically determined values for the above equation for a pedestrian walking:
| \(n\) | \(\alpha_{n}\) |
|---|---|
| \(1\) | \(0.50\) |
| \(2\) | \(0.20\) |
| \(3\) | \(0.10\) |
| \(4\) | \(0.05\) |
We can plot the range of possible frequencies that walking can excite using the above rules:
Allen then fit the expression \(0.83e^{-0.35f}\) to envelope this plot. Note that this expression is conservative, it very much overestimates the dynamic load factors in the frequencies between the harmonics.
The expression \(0.83e^{-0.35f}\) gives us the dyanmic load factor at any frequency, but we’re only interested in the harmonic at the natural frequency of our structure, namely where \(f = f_{n}\), as this harmonic will be the critical one, therefore we can simplify \(\ref{a}\) by removing all terms except for the term where \(\alpha_{n} = 0.83e^{-0.35f_{n}}\)
\[ F(t) = 0.83 F_{0} e^{-0.35f_{n}} \sin 2 \pi f_{n} t \]
We can then write an expression for the peak acceleration assuming a simple mass-spring-damper system (see derivation):
\[ a'_{peak} = \frac{0.83 F_{0} e^{-0.35f_{n}}}{2 \zeta m} \]
However, as this is not a simple mass-spring-damper system, Allen callibrated his model with a reduction factor \(R = 0.7\) for footbridges and an effective mass \(m = 0.5W / g\) (equal to the effective mass for a beam).
\[ a_{peak} = \frac{0.83 R F_{0} g e^{-0.35f_{n}}}{\zeta W} \]