Allen-Murray dynamic pedestrian load model

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

Application

This formula forms the basis for the LRFD Guide Specifications for the Design of Pedestrian Bridges Vertical vibration comfort limits.

Derivaton

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:

Figure 1: Frequency spectrum for dynamic pedestrian loading
Figure 1: Frequency spectrum for dynamic pedestrian loading

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.

Figure 2: Curve fit to frequency spectrum
Figure 2: Curve fit to frequency spectrum

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).

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