# 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

$F(t) = F_{0}(1 + \sum_{n=1}^{N} \alpha_{n} \sin 2 \pi n f t)\label{a}\tag{Eq.1}$

Allen empirically determined values for the above equation for a pedestrian walking:

• $$f$$ can be any value in the range $$1.6\textrm{ Hz}$$ to $$2.2\textrm{ Hz}$$
• the dynamic load factors $$\alpha_{n}$$ for the first four harmonics are as follows:
$$n$$ $$\alpha_{n}$$
$$1$$ $$0.50$$
$$2$$ $$0.20$$
$$3$$ $$0.10$$
$$4$$ $$0.05$$

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).

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