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 $f(t)$ are taken as periodic loads. Using to Fourier Series, the period footfall loads can be expressed as:

$$f(t) = G\left\lbrack 1 + \sum_{h = 1}^{H}{r_{h}\sin\left( \frac{2\pi h}{T}t \right)} \right\rbrack$$

where $G$ is the body weight of the individual, and $r_{h}$ are the Fourier coefficients (or dynamic load factor), the actual values of dynamic load factors can be found from reference 24, 25 and 26 in the bibliography, $T$ is the period of the footfall (inverse of walking frequency) and $H$ the number of Fourier (harmonic) terms to be considered, 4 is used for walking on floor using CCIP-016 method, 3 is used for walking on floor using SCI method and 2 is used for walking on stairs.

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

$$f(t) = G\sum_{h = 1}^{H}{r_{h}\sin\left( \frac{2\pi h}{T}t \right)}$$

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 $(h)$ of the footfall loads in equation above and the given walking frequency $\left( \frac{1}{T} \right)$, the following dynamic equation of motion can be obtained

$$\mathbf{M}{\ddot{\mathbf{u}}}_{h} + \mathbf{C}{\dot{\mathbf{u}}}_{h} + \mathbf{K}\mathbf{u}_{h} = \delta_{k}Gr_{h}\sin\left( \frac{2\pi h}{T}t \right)$$

where $\delta_{k}$ is a unit vector used to define the location of the harmonic load. All the components in this vector are zero except the term that corresponds to the vertical direction of the node subjected footfall load.

Since the number of footfalls is limited and the full resonant response from the equation above may not always be achieved, a reduction factor $\rho_{h,m}$ for the dynamic magnification factors $\mu$ is needed to account for this non-full resonant response. The reduction factor can be calculated from

$$\rho_{h,m} = 1 - e^{- 2\pi\zeta_{m}N}$$

Where $\zeta_{m}$ is damping ratio of mode m and $N = 0.55hW$ with $h$ the harmonic load number and $W$ the number of footfalls.

Applying this reduction factor to the dynamic magnification factors $(\mu)$ in Harmonic Analysis, this equation can be solved using the method described in Harmonic Analysis Theory section. Repeating this analysis, the responses from the other harmonic loads of the footfall can also be obtained. The interested results from this analysis are the total vertical acceleration and the response factor from all harmonic loads of the footfall. The total vertical acceleration is taken as the square root of the sum of squares of the accelerations from each of the harmonic analyses. The response factor for each of the harmonic loads is the ratio of the nodal acceleration to the base curve of the Root Mean Square acceleration given in reference 25 as shown below. This total response factor is then taken as the square root of the sum of squares of the response factors from each of the harmonic loads. According to this, the total acceleration and response factor can be calculated from

$$\begin{aligned}a_{i} &= \sqrt{\sum_{h = 1}^{H}{{\ddot{u}}_{i,h}}^{2}} \\ f_{i} &= \sqrt{\sum_{h = 1}^{H}{f_{i,h}}^{2}} = \sqrt{\sum_{h = 1}^{H}\left( \frac{{{\ddot{u}}_{i,h}}^{2}w_{i}}{0.005\sqrt{2}} \right)}\end{aligned}$$

where

$a_{i}$ is the maximum acceleration at node $i$

${\ddot{u}}_{i,h}$ is the maximum acceleration at node i by the excitation of harmonic load$h$

$H$ is the number of harmonic components of the footfall loads considered in the analysis

$f_{i}$ is the response factor at node $i$

$f_{i,h}$ is the response factor at node $i$ by the excitation of harmonic load$h$

$w_{i}$ is the frequency weighting factor and it is a function of frequency

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 (CCIP-016/Arup method)

$$I_{des,m} = 54\frac{f^{1.43}}{{f_{m}}^{1.3}}$$

When walking on floor (SCI P354 method)

$$I_{des,m} = 60\frac{f^{1.43}}{{f_{m}}^{1.3}}\frac{Q}{700}$$

When walking on stairs (CCIP-016/Arup method)

$$I_{des,m} = \frac{150}{f_{m}}$$

where $f_{m}$ is the frequency of the mode under consisderation.

When walking on stairs (SCI P354 method)

$$I_{des,m} = 0$$

where

$I_{des,m}$ is the design impulse for mode $m$in NS

$f$is the walking frequency in Hz

$f_{m}$is the natural frequency of the structure in mode $m$in Hz

$Q$is the weight of the walker in N

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

$${\widehat{v}}_{m} = u_{e,m}u_{r,m}\frac{I_{des,m}}{{\widehat{m}}_{m}}$$

and the peak acceleration in each mode is given by

$${\widehat{a}}_{m} = 2\pi f_{m}{\widehat{v}}_{m} = 2\pi f_{m}u_{e,m}u_{r,m}\frac{I_{des,m}}{{\widehat{m}}_{m}}$$

where

${\widehat{v}}_{m}$ is the peak velocity in mode $m$ by the footfall impulse

${\widehat{a}}_{m}$ is the peak acceleration in mode $m$ by the footfall impulse

$u_{e,m},u_{r,m}$ are the vertical displacements at the excitation and response nodes respectively in mode $m$

${\widehat{m}}_{m}$ is the modal mass in mode $m$

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

$$v_{m}(t) = {\widehat{v}}_{m}e^{- 2\pi\xi f_{m}t}\sin\left( 2\pi f_{m}t \right)$$

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

$$a_{m}(t) = {\widehat{a}}_{m}e^{- 2\pi\xi f_{m}t}\sin\left( 2\pi f_{m}t \right)$$

where $\xi$ is the damping ratio associated with mode $m$

The final velocity and acceleration at the response node are the sum of the velocities and accelerations of all the modes $\left(M\right)$ that are considered

$$\begin{aligned}v(t) &= \sum_{m = 1}^{M}{v_{m}(t)} \\ a(t) &= \sum_{m = 1}^{M}{a_{m}(t)}\end{aligned}$$

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

$$\begin{aligned}v_{RMS} = \sqrt{\frac{1}{T}\int_{0}^{T}{v^{2}(t)dt}} \\ a_{RMS} = \sqrt{\frac{1}{T}\int_{0}^{T}{a^{2}(t)dt}}\end{aligned}$$

The response factor at time $t$ ($t$ is from $0$ to $T$ and $T$ is the period of the footfall loads) can be calculated from

$$f_{R}(t) = \frac{1}{0.005}\sum_{m = 1}^{M}{a_{m}(t)w_{m}}$$

where

$w_{m}$ is the frequency weighting factor corresponding to the frequency of mode $m$

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

$$f_{R} = \sqrt{\frac{1}{T}\int_{0}^{T}{{f_{R}}^{2}(t)dt}}$$