
Volumes of research has been produced on the topic of human-induced vibration
and in particular crowd-induced vibration. Research in this area has
intensified in the last 20 years as structures, in particular floor plates,
have become lighter and more slender. This has resulted in unacceptably large
occupant-induced vibrations being observed in service more frequently. There is
now a much greater understanding among engineers and other procurement
stakeholders about the need to carefully asses vibration serviceability at
design stage.

For the purposes of this analysis project, when we say human-induced vibration,
we're referring to the vibration induced by a single pedestrian as they walk.
Crowd-induced vibration will refer to the response induced by multiple people
simultaneously walking across a structure.

Human-induced vibration is a complex and fascinating dynamic problem because it
involves interaction between the structure and the occupying pedestrian, both
of which are dynamic systems. The problem gets even more interesting (and
complex) when we start to introduce interaction between pedestrians, and try to
capture the tendency for a pedestrian to be influenced by the behaviour of
those around them. We won't attempt to dive in to the detail too much here but
we do need some understanding of the problem before we start coding.

In addition to dividing the problem into human-induced and crowd-induced
vibration, we can further divide the problem into one that considers vertical
vibration and one that considers lateral vibration. Human locomotion response
and balance behaviour is very different in each case. We'll focus here on
vertical vibration and leave the juicy problem of lateral human-structure
interaction and synchronisation for another day!

## Footfall force modelling 101

In the simplest of terms, when a person walks across a structure, in our case a
linear footbridge, they will apply a vertical footfall force or ground reaction
force (GRF) with each footstep. We can crudely model this GRF using a sine
function, or more specifically a series of sine functions representing the
harmonic components of the GRF. This assumes that the GRF is perfectly periodic
which is itself a simplification but we can live with it. So, the GRF can be
represented as the static weight of the pedestrian, $G$, added to a Fourier series representing
the dynamic component of loading,

$$
F_v(t) = G + \sum_{i=1}^n G\:\text{DLF}_i\sin(2\pi i f_v t - \phi_i)
\tag{1}
$$

where, $\text{DLF}_i$ is the $i^{th}$ Fourier coefficient (also knows as a Dynamic Load Factor), $\phi_i$ is the phase of the $i^{th}$ harmonic and $f_v$ is the pacing frequency. We will only concern ourselves with the fundamental harmonic, i.e. that component of the GRF at the pacing frequency. We'll ignore the higher integer harmonics in this example, but these may be significant, depending on the dynamic characteristics of the structure in question.

In order to model the GRF we need to know the Fourier coefficient of the
fundamental harmonic. Again there has been a lot of research work on this over
the years but we'll use an early relationship proposed by Kerr [1] that relates
the $DLF$ to the pacing frequency, $f_v$,

$$
\text{DLF} = 0.41(f_v-0.95) \leq 0.56
\tag{2}
$$

This relationship is valid for pacing frequencies between $1\: Hz$ and $2.8 \:Hz$. Now, this raises the question of what should the pacing frequency be? For this we'll rely on a relationship presented by Bruno and Venuti [2] that was developed from the earlier work of Bertram & Ruinne [3],

$$
f_v = 0.35v_p^3 -1.59v_p^2+2.93v_p
\tag{3}
$$

where $v_p$ is the walking velocity. Of
course now we're presented with the task of working out what a sensible walking
velocity should be. Here we'll rely on work by Pachi and Ji [4] who observed
that walking velocities in a sample of pedestrians were normally distributed
with a mean value of $1.3\:m/s$ and a
standard deviation of $0.125\:m/s$.

An import feature in the modelling of crowd-induced vibration is the flow
behaviour of the pedestrian crowd and their evolving distribution across the
structure. As a pedestrian crowd becomes more dense, walking velocities are
reduced due to congestion. However in this analysis, we'll ignore all traffic
effects due to interaction between pedestrians. This is a reasonable assumption
for uncongested crowds where it's reasonable to expect that an individual's
desired walking velocity won't be constrained by neighbouring pedestrians.

We're now at a point where we can approximate the GRF imparted by a pedestrian
with a mass of $80\:kg$ walking with a
velocity of $1.3\:m/s$ in a straight line.
If we assume the pedestrian walks across a $60\:m$ long footbridge, it will take then $60\:m/1.3\:m/s = 76.9$ seconds to cross
the bridge. This gives us the duration of load application.

```python
L = 60 #(m) Bridge span
vp = 1.3 #(m/s) Pedestrian walking velocity
tMax = L/vp #(s) Crossing time
m = 80 #(kg) Pedestrian mass
G = 9.81*m #(N) Static weight of pedestrian

fv = 0.35*vp**3 - 1.59*vp**2 + 2.93*vp #(Hz) Pacing frequency
DLF = 0.41*(fv-0.95) #Dynamic load factor
print(f"- The DLF = {round(DLF,3)} and the pacing frequency is {round(fv,2)} Hz ({round(fv,2)} steps per second)")
print(f"- Duration of a single step is {round(1/fv,2)} seconds")
```

- `The DLF = 0.386 and the pacing frequency is 1.89 Hz (1.89 steps per second)`
- `Duration of a single step is 0.53 seconds`

Now we can define the time vector and plot the GRF. Note that we're only
interested in the positive value of the GRF as the footfall force never acts to
pull the walking surface upwards! This means that a single sine wave actually
models 2 footsteps - this is why the frequency is divided by 2 in the code
below.

```python
delT = 0.005 #(s) Time-step
time = np.arange(0, tMax+delT, delT) #Time vector
Fv = G + abs(G*DLF*np.sin(2*math.pi*(fv/2)*time)) #Static + Dynamic GRF

fig = plt.figure()
axes = fig.add_axes([0.1,0.1,2,1])
axes.plot(time,Fv, '-', label='GRF')

#Housekeeping
axes.set_xlabel('time (s)')
axes.set_ylabel('Force (N)')
axes.set_title('Vertical ground reaction force')
axes.legend(loc='lower right')
axes.set_xlim([0,5])
plt.grid()
plt.show()
```

<Image
	src="/images/courses/simulating-crowd-induced-vibrations-using-the-duhamel-integral/an-introduction-to-dynamic-crowd-loading/img1.jpg"
	alt="Simplified ground reaction force signal. | EngineeringSkills.com"
	width="1000"
	height="377"
	caption="Fig 1. Simplified ground reaction force signal."
/>

Each of the half-sine peaks represents the force generated by one foot hitting
the surface. Remember, a more rigorous model would need to consider the
influence of varying walking speed. Even when we make the assumption of
constant walking speed, the GRF is not a perfectly periodic function so there
is some approximation involved in representing it as a Fourier series.

Furthermore, we're only representing the fundamental harmonic component here -
an actual recorded GRF resembles a square wave more than a half-sine wave.
Nevertheless, this will still give us a reasonably realistic first
approximation of human-induced vibration response when we apply it to a
structure.

It's also worth pointing out that in our modelling we're going to assume that
there is no human-structure interaction. This is generally a bigger problem for
lateral oscillations, due to the fact that humans tend to exhibit higher
sensitivity to lateral support motion versus vertical motion. This is likely a
result of our bi-pedal nature and inherent instability in the lateral
direction. So, just to be clear, we're assuming the vertical bridge motion does
not influence the pedestrians behaviour (and their GRF) in any way.

## The bridge/beam structure

Now we can turn our attention to the bridge structure. We'll keep things simple
here and assume that our bridge can be modelled as a simply supported beam.
This means that the modal properties are easily obtained. In particular we're
interested in the mode shapes, $\phi_n(x)$
which can be obtained from,

$$
\phi_n(x) = \sin \frac{n\pi x}{L}
\tag{4}
$$

where $n$ is the mode number and $x$ is the longitudinal coordinate along the bridge/beam.
For a deeper dive into modal analysis, refer to our [Multi-Degree of Freedom Dynamics course](/course/multi-degree-of-freedom-dynamics-modal-analysis-and-seismic-response-simulation-in-python). We're going to assume that the fundamental vertical mode of the bridge has a frequency of $2.5\:Hz$. This is sufficiently close to the average pacing frequency that if this were a real bridge we would be concerned that unacceptable vibrations may be observed in service.

We'll assume that the bridge has a damping ratio of $2.5\:\%$ and that the mass per unit
length, $M$ is $2000\:kg/m$. We're going to make another
simplifying assumption here; we'll assume that the mass of pedestrians on the
bridge has no influence on the modal mass of the dynamic system being
simulated. Our justification here is that the mass of pedestrians is
insignificant compared to the mass of the bridge. If this were not the case we
could consider modifying the modal mass to reflect the evolving distribution of
pedestrians on the bridge. I published a paper [5] on this some time ago that
you can read if you want to explore this further. Thanks to this
simplification, the modal mass, $m_n$ remains
constant and can be calculated from the expression,

$$
\begin{align}
m_n &= M\int_0^L  \phi_n(x)^2 \mathrm{d}x \\
&= M\int_0^L  \sin^2\frac{\pi x}{L} \mathrm{d}x \\
&=  \frac{ML}{2}
\end{align}
\tag{5}
$$

So, for the first mode, the modal mass is simply half the actual mass of the structure.

<Callout type='info' heading='References'>

- [1] S.C. Kerr, Human Induced Loading on Staircases, Ph.D. Thesis, University College London, Mechanical Engineering Department, London, UK, 1998.

- [2] F. Bruno, L.and Venuti. Crowd-structure interaction in footbridges: Modelling, application to a real case-study and sensitivity analyses. Journal of Sound and Vibration, 323:475–493, 2009.

- [3] J. E. A. Bertram and A. Ruina. Multiple walking speed-frequency relations are predicted by constrained optimization. Journal of Theoretical Biology, 209:445–453, 2001.

- [4] A. Pachi and T. Ji. Frequency and velocity of people walking. The Structural Engineer, 83:36–40, 2005.

- [5] S.P. Carroll, J.S. Owen, M.F.M. Hussein. Modelling crowd-bridge dynamic interaction with a discretely defined crowd. The Journal of Sound and Vibration, 331: 2685-2709, 2012

</Callout>


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