# # Footfall Analysis

Footfall analysis (or in full, footfall induced vibration analysis) is
used to calculate the elastic vertical nodal responses (acceleration,
velocity, response factor etc.) of structures to human footfall loads
(excitations). The human footfall loads

where

After subtracting the static weight of the individual (since it does not vary with time and does not induce any dynamic response), the dynamic part of the footfall loads are the sum of a number of harmonic loads

There are two distinctive responses from the footfall excitation, the resonant (steady state) and transient. If the minimum natural frequency of a structure is higher than 4 times the highest walking frequency (see reference 24), the resonant response is normally not excited since the natural frequencies of the structure are so far from the walking (excitation) frequency, therefore the transient response is normally in control, otherwise, the resonant response is probably in control. Both resonant (steady state) and transient analyses are considered in GSA footfall analysis, so the maximum responses will always be captured.

## # Resonant response analysis

As footfall loads are composed of a number of harmonic loads
(components), harmonic analysis is used to get the responses for each of
the harmonic components of footfall loads and then to combined them to
get the total responses. From one of the harmonic components

where

Since the number of footfalls is limited and the full resonant response
from the equation above may not always be achieved, a reduction factor

Where

Applying this reduction factor to the dynamic magnification factors

where

For standard weighting factors see Table 3 of BS6841.

## # Transient response analysis

The transient response of structures to footfall forces is characterised by an initial peak velocity followed by a decaying vibration at the natural frequency of the structure until the next footfall. As the natural frequencies of the structure considered in this analysis is much higher the highest walking frequency, there is no tendency for the response to build up over time as it does in resonant response analysis. The maximum response will be at the beginning of each footfall. Each footfall is considered as an impulse to the structure, according to references 35 & 29, the design impulse can be calculated from

When walking on floor (Concrete Centre/Arup method)

When walking on floor (SCI P354 method)

When walking on stairs (Concrete Centre/Arup method)

When walking on stairs (SCI P354 method)

where

For this impulse, the peak velocity in each mode is given by

and the peak acceleration in each mode is given by

where

The variation of the velocity with time of each mode is given by

and the variation of the acceleration with time of each mode is given:

where

The final velocity and acceleration at the response node are the sum of the velocities and accelerations of all the modes that are considered

This gives the peak velocity and peak acceleration. The root mean square velocity and root mean square acceleration can be calculated from the period of the footfall

The response factor at time

where

The final transient response factor, based on the root mean square principle, is given by