Footfall Analysis

Footfall Analysis calculates the vibration that human activity and rotating machines cause in a floor, using the vibration modes of a finite element model built from the floor plan. The calculation is a serviceability check, and its result is compared with the limit the chosen guide gives for the room type. The guide is chosen for each calculation and governs the whole method, from the modelling rules to the reported quantity.

Computed quantity

Guide	Reported quantity	Source of the limit	Activities
SCI P354	Response factor $R$ : the frequency-weighted RMS acceleration over the base value of 0.005 m/s²	Tables 5.2, 5.3, 8.1 (room multiplying factor), Tables 5.4 and 8.2 (VDV)	Walking, rhythmic activity, machines
CCIP-016	Response factor $R$ : the response over the guide's base curve	§3.3 (room multiplying factor)	Walking, machines
AISC DG11	Peak acceleration as a percentage of gravity (%g), and one-third octave band velocity in sensitive-equipment areas	Tables 4-1, 4-4 (peak acceleration), Table 6-2 (band velocity)	Walking, running, rhythmic activity, machines

Machine vibration is calculated by the same method under all three guides and assessed against the continuous vibration criterion. The method is described in Machine vibration.

The result is calculated at every node of the floor. The whole floor and each receiver area the user draws are separate assessment units. Each unit is compared with the limit of its own room type, and its worst point is reported.

Floor model

Elements

The slab is modelled with four-node shell elements. The mesh size is an input with a default of 0.7 m. Beams are frame elements in the mid-plane of the slab and pass through the mesh nodes. Structural walls are shell elements that share the slab's nodes. Partition walls are not elements of the model and enter only as mass, through their weight.

Columns and walls are extended below and above the slab. The length of the extension and the end condition depend on the guide.

Columns and walls extend to half the storey height (§6.1.1). At the far end of a column the three translations and the rotation about its own axis are held, and the rotations in bending are free. At the far edge of a wall the three translations are held. Where there is no storey above the slab or the column does not continue upwards, there is no upper extension.

Columns and walls extend over the full height of the storeys below and above (Annex A, item 9). All six degrees of freedom are held at the far end of the extension.

Columns and walls extend over the full height of the storeys below and above (§7.2). All six degrees of freedom are held at the far end of the extension.

A column with no storey below and nothing above becomes a pinned support at slab level. A wall in the same situation is modelled as a line of supports at slab level.

The in-plane translations of the slab are held at the nodes where columns and walls meet the slab. A floor modelled on its own has no lateral system, so an in-plane sway mode is not a floor mode. This restraint does not change the vertical modes of the floor.

Facade

The condition the guides give for the building's outer edge can be applied as an option, which is off by default. It applies only to the outer outline of the building, that is, to edges with no slab outside them.

The three translations are held at the nodes on the outer outline (§6.1.1).

The vertical translation is held at the nodes on the outer outline (Annex A, item 8).

The moment of inertia of the beams on the outer outline is multiplied by 2.5 (§7.2). Where the beam is composite, the factor applies to the composite moment of inertia.

Material

Vibration is a small and fast deformation, so concrete works with a higher modulus of elasticity than under static load. The guides define this modulus in different ways.

Concrete is solved with the dynamic modulus of elasticity from the material library (§4.1.3). For every concrete grade in the library this value is 38,000 MPa. A user-defined concrete must state its dynamic modulus.

Concrete is solved with the dynamic modulus of elasticity from the material library (Annex A). For every concrete grade in the library this value is 38,000 MPa. A user-defined concrete must state its dynamic modulus.

The static modulus of elasticity of concrete is multiplied by 1.35 (§3.1, §7.2). The shear modulus is increased in the same ratio.

Steel is used with the modulus of elasticity from the material library, without a factor.

Mass

The self-weight of the elements comes from the unit weight of the material. Loads on the slab are converted to mass with the share the calculation's mass source gives for each load case.

Superimposed dead load and imposed load spread over the whole slab are added to the slab's mass.
Area, line and point loads, beam and column loads and partition weights are attached as mass where they stand, through the shape functions of the element they stand on. No node is added for these loads and no load is moved to another element. An upward load gives no mass.
The whole weight of a piece of equipment is taken as mass spread over its footprint. The weight of the equipment is its own and permanent, so the mass source's share is not applied.

SCI P354 §4.1.2 recommends taking only a small part of the imposed load on a floor as mass. When more than 10 % of the imposed load in a floor is taken as mass, the calculation gives a warning. AISC DG11 takes the actual imposed load as mass (§7.2, Table 3-1), so under this guide no warning is given.

Partitions

Partitions always enter the model with their weight, as mass where they stand. A wall's weight is the product of the wall thickness, the unit weight of the wall material and the wall area with door and window openings taken off. On the upper bound of the range TS ISO 9194 Annex A gives for unplastered walls, brick walls are taken at 1,450 kg/m³ and aerated concrete walls at 1,100 kg/m³. You can include plaster by adding it to the wall thickness or by raising the unit weight in the Materials list. The library has no value for pumice, so the unit weight of pumice walls has to be entered.

As an option, the stiffness of the full-height partitions marked permanent in the model editor can also be counted. A wall whose top reaches the underside of the beam or slab above it is full-height. The option is off by default and is applied under all three guides with the rule of AISC DG11 §7.2.

A masonry wall standing on the slab is modelled with shell elements working in its own plane and connected to the slab's nodes. The top of the wall is free, and the shells carry no mass. Door and window openings are cut out of the model on their own edges, and the wall over and under an opening stays in the model. The modulus of elasticity comes from the wall's material.
A wall under the slab, and a partition that is not masonry, is modelled with vertical springs of 2.0 kip/in./ft (about 1,149 kN/m per metre of wall) along the line where it meets the slab. There are no springs over an opening that reaches that line.
Partitions that cross are joined along their line of intersection.

SCI P354 counts partitions as damping (§2.4.2) and gives 4.5 % damping for a floor with partitions (§4.1.1). When the stiffness of partitions is counted and the damping ratio is 0.045 or more, the calculation gives a warning. Appendix A.5 recommends leaving partitions out of a finite element model unless tests show their effect, and the result notes say so.

CCIP-016 considers it appropriate to include full-height partitions in the analysis model (§3.3.2.4). Part-height and movable partitions are not counted. Relaxing the office limit by up to 1.5 applies only when partitions are not in the model (§3.3.2.1), and it is not applied in the room type table.

AISC DG11 counts partitions both in the model (§7.2) and in the damping (Table 4-2).

The stiffness of partitions can change the frequency and the response markedly. A wall stiffening one bay can move the vibration into the neighbouring bays, so the response does not fall in every case.

Composite slabs and beams

A concrete slab on a profiled deck is modelled as an orthotropic layer. The layer's thickness is the concrete depth above the deck. Across the ribs the modulus of elasticity of concrete is used. Along the ribs the modulus is defined so that the layer carries the bending stiffness of the section including the rib concrete and the deck (SCI P354 Equation 25):

E_x = \frac{12\,E_c\,I_{c,x}}{h_c^3}

Here $I_{c,x}$ is the moment of inertia per metre of the uncracked section with the deck transformed by the modular ratio, and $h_c$ is the concrete depth above the deck. A profiled deck without concrete is modelled in the same way with the deck's own moment of inertia.

The moment of inertia of a steel beam declared to act with the slab is calculated from the transformed section with the concrete over the effective width. The slab's own moment of inertia is already in the shell, so it is taken out of the beam section. The degree of shear connection does not reduce the vibration stiffness.

The effective width is the smaller of the width the beam carries and a quarter of the span (§6.1.2, Equation 27).

CCIP-016 gives no rule for the effective width. The rule of SCI P354 is therefore applied, and the effective width is taken as the smaller of the width the beam carries and a quarter of the span.

For an interior beam the effective width is the smaller of the width the beam carries and 0.4 times the span (§3.2). For an edge beam the smaller of half the spacing and 0.2 times the span is taken on the inner side, and the overhang is added in full. Where the ribs run parallel to the beam, half of the rib concrete is also included in the section.

The width a beam carries is half the distance to the neighbouring beam that is parallel within 20°, or the overhang to the slab edge. The span is the drawn length of the beam. A beam declared composite with no concrete beside it is modelled as bare steel, and the calculation gives a warning.

Column footprint and rigid zone

The cross-section of a reinforced concrete column is cut out of the slab mesh, and the nodes around it are tied rigidly to the column node. Steel sections do not cut the slab. This choice can be changed for each calculation. The rigid zone factor for the part of a beam inside a column is an input between 0 and 1 with a default of 0. At a column whose cross-section is cut out of the mesh, the beam ends at the column face whatever the factor.

Modes

Modes are calculated up to the frequency $\max(2 f_1,\ 40\ \text{Hz})$ , with at most 200 modes. For machine vibration, twice the highest machine frequency replaces 40 Hz where it is larger. The fundamental frequency $f_1$ is the lowest mode of the model. Mode shapes are mass-normalised, and the calculation uses the vertical ordinates at the slab nodes. Each guide works with a subset of these modes.

Floor classification and mode sets

A floor is low-frequency when its fundamental frequency is below the cut-off frequency, and high-frequency when it is above (Table 6.1). The cut-off is 10 Hz for general floors, 8 Hz for enclosed spaces and 24 Hz for rhythmic activity. On a low-frequency floor both the resonant and the transient response are calculated and the larger is taken. On a high-frequency floor only the transient response is calculated.

The resonant response takes the modes up to 2 Hz above the cut-off frequency (§6.3.2). The transient response takes the modes up to $\max(2 f_1,\ 40\ \text{Hz})$ (§6.3.3).

The resonant response is calculated when the fundamental frequency is below 4.2 times the highest pacing frequency (§4.2.2). The transient response is calculated when the fundamental frequency is above 4 Hz (§4.3.1). Where both are calculated, the larger is taken.

The resonant response takes the modes up to 15 Hz (§4.2.2), and the transient response the modes up to $2 f_1$ (§4.3.1).

A floor is low-frequency when its fundamental frequency is below 9 Hz (§1.5). On a low-frequency floor the resonant response is calculated. When the dominant frequency is between 9 and 20 Hz the transient response is also calculated (Table 7-1), and the larger of the two is taken. A floor with a fundamental frequency above 20 Hz is outside the scope of the guide and is not solved.

The resonant response is found from the frequency response function (FRF) between $f_1 - 1$ Hz and 9 Hz. The function is evaluated at every natural frequency in this range and at 25 evenly spaced points (§7.3). The transient response takes the modes up to 20 Hz. The dominant frequency is the natural frequency of the mode with the largest FRF value between $f_1 - 1$ Hz and 20 Hz.

Walking force

The force a walker applies to the floor is taken as the sum of harmonics at the pacing frequency and its multiples. The walker's weight is an input. When it is not given, the guide's value is used.

The walker's weight is $Q$ = 746 N (Equation 17). The force has four harmonics (Equation 12):

F(t) = Q\left[1 + \sum_{h=1}^{4} \alpha_h \sin\left(2\pi h f_p t + \phi_h\right)\right]

The dynamic load factors come from Table 3.1: $\alpha_1 = 0.436\,(f_p - 0.95)$ , $\alpha_2 = 0.006\,(2f_p + 12.3)$ , $\alpha_3 = 0.007\,(3f_p + 5.2)$ , $\alpha_4 = 0.007\,(4f_p + 2.0)$ . The phases are 0, $-\pi/2$ , $\pi$ and $\pi/2$ in turn. The table is defined for 1.8–2.2 Hz, and the calculation gives a warning for pacing frequencies outside this range.

The pacing frequency range is chosen by the type of use. It is 1.8–2.2 Hz for general use (Equation 14), 1.8 Hz for enclosed spaces (Equation 15) and 1.5–1.8 Hz near patients (§8.2.1). A range given by hand must lie within 1.5–2.5 Hz (§3.1.1).

When the path length $L_p$ of the walking area is given, the resonant response is multiplied by the factor that limits the response building up along the walker's path (Equation 37): $\rho = 1 - e^{-2\pi\zeta L_p f_p / v}$ . Here $v = 1.67 f_p^2 - 4.83 f_p + 4.50$ is the walking speed (Equation 16).

The walker's weight is 700 N. The force has four harmonics, and the dynamic load factors are the design values of Table 4.3. With the harmonic's frequency $f = h f_w$ , $\alpha_1 = \min\left[0.41\,(f - 0.95),\ 0.56\right]$ , $\alpha_2 = 0.069 + 0.0056 f$ , $\alpha_3 = 0.033 + 0.0064 f$ , $\alpha_4 = 0.013 + 0.0065 f$ .

The pacing frequency is swept from 1.0 Hz to the highest pacing frequency of the use (Table 4.2). This is 2.5 Hz for footbridges and corridors, 2.0 Hz for offices and residences and 1.8 Hz for laboratories and operating theatres. A range given by hand must lie within 1.0–2.8 Hz (Table 4.3).

When both the path length and the stride length are given, the resonant response of each harmonic is multiplied by the factor that limits the build-up (Equation 4.2): $\rho = 1 - e^{-2\pi\zeta N}$ , $N = 0.55\,h\,L/l$ .

The walker's weight is 168 lb (0.747 kN, Table 1-1). The pacing frequency is not swept. In the resonant response the load is represented by one harmonic at the dominant frequency, with the dynamic coefficient $\alpha = 0.09\,e^{-0.075 f_n}$ (Equation 7-2). In the transient response the pacing frequency is the dominant frequency divided by its harmonic number from Table 7-1.

The resonant response is multiplied by the build-up factor, which depends on the damping ratio (Equation 7-3). The factor is $50\beta + 0.25$ for $\beta < 0.01$ , $12.5\beta + 0.625$ for $\beta < 0.03$ , and 1.0 for higher damping.

Under SCI P354 and CCIP-016 the pacing frequency range is swept at evenly spaced points. The number of points is an input with a default of 9. The frequency $f_n / h$ ( $h = 1..4$ ) of every mode in the resonant set is added to the sweep when it falls inside the range. The frequency at which a harmonic coincides exactly with a mode is therefore also calculated. At each point the worst pacing frequency of the sweep is taken.

Resonant response

For each harmonic the modal contributions are added with their signs (Equations 29, 30, 36):

a_h = \sum_n \mu_{e,n}\,\mu_{r,n}\,\frac{\alpha_h Q}{M_n}\,D_n\,W(h f_p), \qquad D_n = \frac{\beta_n^2}{\sqrt{(1-\beta_n^2)^2 + (2\zeta\beta_n)^2}}, \qquad \beta_n = \frac{h f_p}{f_n}

For mass-normalised modes the product $\mu_e \mu_r / M_n$ is the product of the mode ordinates. The harmonics are combined by the square root of the sum of their squares, giving $a_{w,rms} = \sqrt{\sum_h a_h^2}\,/\sqrt{2}$ . The frequency weighting $W$ is read at each harmonic's own frequency.

Equation 36 adds the modal contributions within a harmonic with their signs and without their phases. The time-history solution the guide names as the most accurate method (Equation 31) carries the phases. The difference between the two, and Equation 36 staying on the conservative side under self excitation, are measured on the verification page.

For each harmonic the modal contributions are added with their phases, as complex numbers (Equations 4.4–4.6):

a_h = \sum_m \left(\frac{f_h}{f_m}\right)^2 F_h\,\mu_{e,m}\,\mu_{r,m}\,\rho\,\frac{A_m + i B_m}{A_m^2 + B_m^2}, \qquad A_m = 1 - \left(\frac{f_h}{f_m}\right)^2, \qquad B_m = 2\zeta\frac{f_h}{f_m}

The response of each harmonic is divided by the value of the guide's base curve at that harmonic's frequency (Equation 4.7), and the harmonics are combined by the square root of the sum of their squares (Equation 4.8).

The peak acceleration is calculated with the largest value of the frequency response function (Equation 7-1):

a_p = \mathrm{FRF}_{max}\,\alpha\,Q\,\rho, \qquad \mathrm{FRF}(f) = \sum_m \mu_{e,m}\,\mu_{r,m}\,\frac{(f/f_m)^2}{1 - (f/f_m)^2 + 2i\zeta f/f_m}

The modal contributions are added with their phases. The result is divided by the acceleration of gravity and given in %g.

Transient response

Each footstep excites the modes of the floor as an impulse, and the modes vibrate and decay until the next footstep. The transient response is the RMS value of this decaying vibration over one footstep. The integral for the RMS value is not sampled in time steps and is evaluated analytically.

The effective impulse of each mode is calculated with Equation 18, and the acceleration history with Equation 33:

F_I = 60\,\frac{f_p^{1.43}}{f_n^{1.30}}\,\frac{Q}{700}, \qquad a(t) = \sum_n 2\pi f_n\sqrt{1-\zeta^2}\,\mu_{e,n}\,\mu_{r,n}\,\frac{F_I}{M_n}\,W(f_n)\,\sin\left(2\pi f_{d,n} t\right)e^{-2\pi\zeta f_n t}

Here $f_{d,n} = f_n\sqrt{1-\zeta^2}$ is the damped frequency. The weighted RMS acceleration is taken over the period $T = 1/f_p$ (Equations 12, 34). The transient response is calculated at every swept pacing frequency.

The transient response is calculated as a velocity (Equations 4.10–4.14):

v(t) = \sum_m \mu_{e,m}\,\mu_{r,m}\,\frac{I_{\text{eff},m}}{M_m}\,e^{-2\pi\zeta f_m t}\sin\left(2\pi f_m t\right), \qquad I_{\text{eff},m} = 54\,\frac{f_w^{1.43}}{f_m^{1.30}}\,\frac{Q}{700}

The RMS velocity is taken over the period $T = 1/f_w$ and is calculated only at the highest pacing frequency (§4.3.1). The response factor is the RMS velocity over the base velocity at the fundamental frequency (Equation 4.15). The base velocity is $5 \times 10^{-3}/(2\pi f_1)$ m/s for $f_1 < 8$ Hz and $1 \times 10^{-4}$ m/s at higher frequencies.

The effective impulse is calculated with Equation 1-6, and the acceleration history with Equations 7-4 and 7-5:

I_{\text{eff},n} = \frac{f_{step}^{1.43}}{f_n^{1.30}}\,\frac{Q}{17.8}, \qquad a(t) = \sum_n 2\pi f_n\,\mu_{e,n}\,\mu_{r,n}\,\frac{I_{\text{eff},n}}{M_n}\,e^{-2\pi\beta f_n t}\sin\left(2\pi f_n t\right)

The equivalent sinusoidal peak acceleration (ESPA) is $\sqrt{2}$ times the RMS acceleration over one footstep ( $1/f_{step}$ ) (Equation 7-6). The modes up to 20 Hz are taken. The pacing frequency is the dominant frequency divided by its harmonic number from Table 7-1. The harmonic number is 5 for a dominant frequency of 9–11 Hz, 6 for 11–13.2 Hz, 7 for 13.2–15.4 Hz, 8 for 15.4–17.6 Hz and 9 for 17.6–20 Hz.

Under SCI P354 and CCIP-016 the larger of the resonant and the transient response is taken at each point and each pacing frequency. Each point keeps its own worst pacing frequency. The reported resonant and transient responses are each the worst over all pacing frequencies.

Criteria

Frequency weighting and base value

The response factor $R$ is the frequency-weighted RMS acceleration over the base value. The base value is 0.005 m/s² on the vertical axis and 0.00357 m/s² for a person lying down (Equations 38, 39). The weighting curves are defined by Equations 21–23:

W_b = \begin{cases} 0.4 & f < 2 \\ f/5 & 2 \le f < 5 \\ 1 & 5 \le f \le 16 \\ 16/f & f > 16 \end{cases} \qquad W_g = \begin{cases} 0.5\sqrt{f} & f < 4 \\ 1 & 4 \le f \le 8 \\ 8/f & f > 8 \end{cases} \qquad W_d = \begin{cases} 1 & f < 2 \\ 2/f & f \ge 2 \end{cases}

$W_b$ is used for offices, residences and workshops (Table 5.1). $W_g$ is mandatory in hospital rooms (§8.2.2). $W_d$ is for a person lying down, and the vertical motion of the floor is read along the person's own x axis. When neither a room type nor a weighting curve is chosen, $W_b$ is used.

CCIP-016 assesses the floor against a single base curve, and no separate weighting curve is chosen. In the resonant response the acceleration of each harmonic is divided by the base curve's value at that harmonic's frequency (Equation 4.7):

a_{R=1}(f) = \begin{cases} 0.0141/\sqrt{f} & f < 4 \\ 0.0071 & 4 \le f \le 8 \\ 2.82\pi f \times 10^{-4} & f > 8 \end{cases} \quad \text{m/s}^2

The base value of the transient response is the velocity in Equation 4.15. The reported RMS acceleration is the response factor multiplied by 0.005 m/s².

The peak acceleration is assessed unweighted, as a percentage of gravity. The limit is an acceleration value in its own right (Tables 4-1, 4-4).

Limits by room type

The room type fills the limit and the weighting curve from the guide's table. Where the guide gives a range, the lower end is taken. Every value can be changed by hand.

Room type	Weighting curve	Limit $R$	VDV limit (m/s¹·⁷⁵)
Office	$W_b$	8	0.4
Shopping mall	$W_b$	4	–
Dealing floor	$W_b$	4	–
Residential (day)	$W_b$	2	0.2
Residential (night)	$W_b$	1.4	0.13
Workshop	$W_b$	8	0.8
Operating theatre	$W_g$	1	Not permitted
Hospital ward	$W_g$	2	0.2
General laboratory	$W_g$	4	0.4
Consulting room	$W_g$	8	0.4
Rhythmic activity area	$W_b$	120	–
VC-A / VC-B / VC-C / VC-D / VC-E	$W_g$	0.5 / 0.25 / 0.125 / 0.0625 / 0.0313	–

The sources are Tables 5.1–5.4, 5.6, 8.1, 8.2 and §5.3.4.

Room type	Limit $R$
Office	8
Shopping mall	8
Premium office, busy corridor or seated public area	4
Residential (day)	4
Residential (night)	2.8
Hospital ward	2
Hospital ward (night)	1.4
Operating theatre	1
External footbridge	64
Indoor footbridge	32
Indoor footbridge, light, exposed or busy	24
VC-A / VC-B / VC-C / VC-D / VC-E	0.5 / 0.25 / 0.125 / 0.0625 / 0.03125

The sources are §3.3 and Table 3.2. The guide gives no numerical limit for residences and hospital wards. For residences twice the continuous vibration levels of BS 6472 are taken (§3.3.2.2, Table 3.1), and for hospital wards the residential levels of BS 6472 (§3.3.2.3).

Use	Limit (%g)
Offices, residences, churches, schools and quiet areas	0.5
Shopping mall	1.5
Dining	1.5
Weightlifting	1.5
Indoor footbridge	1.5
External footbridge	5
Rhythmic activity area	4

The sources are Tables 4-1, 4-4 and 5-1. The limits for sensitive-equipment areas are in Sensitive equipment.

Vibration dose value

The vibration dose value (VDV) is the accumulated effect of walks repeated over a day or a night. The estimated value and the number of walks allowed before the limit is exceeded are calculated with SCI P354 Equations 40 and 41:

\mathrm{VDV} = 0.68\,a_{w,rms}\,(n_a T_a)^{1/4}, \qquad n_a = \frac{1}{T_a}\left(\frac{\mathrm{VDV}_{lim}}{0.68\,a_{w,rms}}\right)^4

The duration $T_a$ of one walk is the path length of the walking area divided by the walking speed. The vibration dose value is therefore calculated only with a walking area that has a path length. A day is taken as 16 hours and a night as 8 hours, and these durations are used only for the number of walks allowed per hour. The vibration dose value is reported separately and takes no part in the pass or fail result. It is not calculated for rhythmic activity or in rooms where the guide does not permit it.

The VDV limit comes from the room type. The walking speed is taken from Equation 16 at the governing pacing frequency.

CCIP-016 gives no VDV table. The vibration dose value is calculated only when a VDV limit and the stride length are given. The walking speed is the stride length multiplied by the pacing frequency.

AISC DG11 has no vibration dose method. A VDV limit that is given is not used, and the result says so.

Pass and fail

The worst value of each assessment unit is compared with its own limit. For a unit without a limit only the value is reported. The calculation passes when every unit with a limit passes. The governing unit is the one with the largest ratio of its value to its limit.

Excitation and areas

Where the walking takes place is chosen for each calculation.

Walking at every point. Every node of the floor is both a walking point and a receiver point (SCI P354 §6.3.4).
From walking areas. Up to 20 walking areas can be drawn. Every node of the areas excites every node of the floor. The areas are one set of walkers, and at each receiver point the larger of the areas' results is taken, not their sum. Each area has its own path length.

A walking area takes the nodes inside it and on its outline. A receiver area takes all nodes of the elements it touches, and up to 50 receiver areas can be drawn. Areas read with the same weighting curve, axis and method are assessed together. When the number of excitation and receiver pairs exceeds the calculation limit, the calculation is not solved. In that case smaller areas or a larger mesh size should be used.

Running

Running is calculated only under AISC DG11 (§7.4.2). One runner stands at each point, and the runner's weight is 168 lb. The frequency response function is calculated between $f_1 - 1$ Hz and 17 Hz. The dominant frequency is the frequency within 1.6–16 Hz at which the function is largest. The harmonic number $h$ and the dynamic coefficient $\alpha_h$ come from Table 7-3 by the dominant frequency.

Dominant frequency (Hz)	$h$	$\alpha_h$
1.6–4	1	1.4
4–8	2	0.4
8–12	3	0.2
12–16	4	0.1

The peak acceleration is calculated with Equation 7-7:

a_p = \mathrm{FRF}_{max}\,\alpha_h\,Q\left[1 - e^{-2\pi\beta h N_{steps}}\right]\min\left(2,\ \sqrt{n}\right)

Here $\beta$ is the damping ratio, $N_{steps}$ the number of footsteps the runner takes to cross the span, and $n$ the number of people running in step. The number of footsteps is at most 10, and 10 is taken when it is not given.

Rhythmic activity

In rhythmic activity the crowd moves in phase over the activity area. The crowd is either on the whole floor or only in the activity areas drawn. The excitation is defined by the area integral of the mode ordinate ( $\int_A \phi\,dA$ ) over the elements whose centroid lies inside this area. In a calculation with activity areas, the floor result covers the nodes outside the activity area. The activity area itself is assessed against a separate limit.

The dynamic load factors and phases for low-impact aerobics, high-impact aerobics and normal jumping come from Table 3.3 with six harmonics. For jumping in a large group the factors of the first three harmonics are calculated by Equation 20 from the number of people in the group ( $2 \le p \le 64$ ), with the phases of normal jumping: $\alpha_1 = 1.61\,p^{-0.082}$ , $\alpha_2 = 0.94\,p^{-0.24}$ , $\alpha_3 = 0.44\,p^{-0.31}$ .

The distributed load is the crowd density multiplied by the person's weight. For aerobics a density of 0.25 persons/m² is taken when none is given (§3.1.3). For jumping the guide gives no density, so it must be given. The activity frequency is swept within 1.5–2.8 Hz for groups and 1.5–3.5 Hz for individuals.

The response is calculated in the same way as the resonant response to walking, with the modes up to 26 Hz. A floor with a fundamental frequency of 24 Hz or more is insensitive to rhythmic activity (Table 6.1). The activity area itself is assessed against $R = 120$ (§5.3.4, §8.1.3).

For the ultimate limit state the equivalent dynamic load is calculated as $F = q\left(1 + \sum_{h=1}^{3} \alpha_h D_{\delta,h}\right)$ (Equations 52–55). This load is required for a floor with a fundamental frequency below 8.4 Hz (§8.1.2).

CCIP-016 covers walking only. Rhythmic activity is not calculated under this guide.

For group dancing, lively concerts or sports events, aerobics and jumping exercises, the distributed load $w_p$ and the dynamic coefficients of the harmonics come from Table 7-4. When a crowd density is given, the distributed load is the density multiplied by the person's weight.

The frequency response function of a unit distributed load is calculated between 1 and 12 Hz. The smallest harmonic that covers the dominant frequency, where the function is largest, gives the activity frequency. The peak acceleration of each harmonic is found with Equation 7-9, and the harmonics are combined with Equation 7-10:

a_{p,i} = \mathrm{FRF}(i f_{step})\,\alpha_i\,w_p, \qquad a_p = \left[\sum_i a_{p,i}^{1.5}\right]^{1/1.5}

When the dominant frequency falls in no harmonic's range, the activity frequency is swept over the range of the first harmonic and the frequency giving the largest response is taken (Example 7.2). The activity area itself is assessed against 4 %g. This value is the lower end of the limit for rhythmic activity only in Table 5-1 (4–7 %g).

Sensitive equipment

Sensitive-equipment areas are assessed against the generic vibration criteria VC-A to VC-E.

The VC classes are response factor limits read with $W_g$ (Table 5.6). The area is assessed with the resonant and transient response to walking.

The VC classes are response factor limits against the base curve (Table 3.2). The area is assessed with the resonant and transient response to walking.

Areas given a VC class or the operating theatre are assessed by the one-third octave band velocity instead of the peak acceleration of walking (§7.5, Table 6-2).

Use	Limit (mips)
Operating rooms, laboratory robots, bench microscopes up to 100×	4,000
VC-A / VC-B / VC-C / VC-D / VC-E	2,000 / 1,000 / 500 / 250 / 125

The walking speed of a person passing the area is chosen from Table 6-1. The pacing frequency is 1.25 Hz for very slow walking, 1.60 Hz for slow, 1.85 Hz for moderate and 2.10 Hz for fast walking. The default is fast walking.

The resonant method (§7.5.2) applies when the dominant frequency is below $f_L$ , and the impulse method (§7.5.3) when it is above $f_U$ . $f_L$ and $f_U$ come from Table 6-1 by the walking speed. For very slow walking only the impulse method is used. In the zone between the two limits the band velocity is interpolated linearly between the results of the two methods by the dominant frequency. In the impulse method the acceleration history is sampled at 0.005 s intervals (Example 7.3). Each receiver point keeps its own worst walking point.

Machine vibration

Machine vibration is calculated by the same method under all three guides. A machine applies a vertical harmonic force at its running frequency to the floor node nearest to it, and the steady-state response of the floor is calculated.

Force

The force amplitude is given from one of three sources.

Manufacturer. The force amplitude the manufacturer gives is used directly.
ISO 21940-11. The force is calculated from the balance grade $G$ (mm/s) and the rotor mass as $F = m\,G\,\Omega/1000$ (N). There is no service factor in this route.
ACI 351.3R. The force is calculated from the eccentricity as $F = m_r\,e_r\,\omega^2 S_f$ (Equation 3-3). The service factor $S_f$ is at least 2.

For a variable-speed machine the lowest and highest speeds are given. The force is given at the highest speed and varies with the square of the speed (ACI 351.3R §3.2.2.1):

F(f) = F_{max}\left(\frac{f}{f_{max}}\right)^2

The speed range is swept at 41 evenly spaced points. Every natural frequency inside the range is added to the sweep. Each receiver point keeps its own worst speed.

Response

A unit force of the machine at node $i$ causes an acceleration at receiver point $j$ through the sum of the modes:

a_j = F\sum_m \phi_{i,m}\,\phi_{j,m}\,\frac{r_m^2}{1 - r_m^2 + 2i\zeta r_m}, \qquad r_m = \frac{f}{f_m}

The response takes every mode calculated up to $\max(2 f_1,\ 40\ \text{Hz},\ 2 f_{max})$ . Here $f_{max}$ is the highest running frequency of the machines. The RMS values of machines running at different speeds are combined as the square root of the sum of their squares. For machines swept over a speed range, the worst speed of each machine is taken as if they occurred at the same time.

Criterion

For people the criterion is continuous vibration, and it is the same under all three guides (BS 6472 Table 5, ISO 10137 Annex C, SCI P354 Table 5.2). The response factor is the weighted RMS acceleration over the base value of 0.005 m/s². In a calculation solved under AISC DG11 the machine result is also given as a response factor.

Room type	Weighting curve	Limit $R$
Operating theatre	$W_g$	1
Residential (day)	$W_b$	2
Residential (night)	$W_b$	1.4
Office	$W_b$	4
Workshop	$W_b$	8

For a room type with no row in the table the limit must be given by hand. The walking limits do not apply to machine vibration. In a sensitive-equipment area the RMS velocity in the one-third octave band of the machine's frequency is compared with the VC curve. The velocities of machines in the same band are combined as the square root of the sum of their squares.

Isolation

For a machine on isolators the isolators' natural frequency is calculated from the static deflection as $f_{iso} = \sqrt{g/\delta}\,/\,2\pi$ , or given directly. The isolators' damping ratio must be given. ACI 351.3R §4.5 gives 5–40 % for isolators with viscous dampers. The force reaching the floor is calculated by one of two methods.

Rigid base. The force is multiplied by the transmissibility of a single-degree-of-freedom isolator on a rigid base (ACI 351.3R Figure 3.14). The machine's mass stays on the slab.
$T = \frac{\sqrt{1 + (2\zeta_{iso} r)^2}}{\sqrt{(1 - r^2)^2 + (2\zeta_{iso} r)^2}}, \qquad r = \frac{f}{f_{iso}}$
Mass-spring model. The machine is a separate mass connected to the floor through the isolators, and the flexibility of the floor at that point enters the calculation. The machine's weight is not added to the slab as mass as well.
$\frac{F_t}{F} = \frac{k^*}{k^* - m\omega^2 - k^* m\,\omega^2 H}, \qquad k^* = k + i\omega c, \qquad H = \sum_m \frac{\phi_{i,m}^2}{\omega_m^2 - \omega^2 + 2i\zeta\omega_m\omega}$
Here $k = m(2\pi f_{iso})^2$ is the isolators' stiffness, $c = 2\zeta_{iso}\sqrt{km}$ their damping, and $H$ the floor's displacement receptance at the machine's point.

When the rigid-base transmissibility exceeds 20 % at the machine's speeds, the calculation gives a warning (ACI 351.3R §3.4.4). When the isolators' frequency is above a third of the lowest forcing frequency, the result says so (SCI P354 B.3.2).

Resonance and support class

When a natural frequency of the floor is within 20 % of the machine's speed range, the calculation gives a warning. ACI 351.3R §3.4.3 gives 20–33 % for this margin, and the lower end is used. The support of each machine is classified by ISO 20816-3 §5.3. The support is rigid when the lowest natural frequency of the machine and support system is at least 1.25 times the highest running frequency, and flexible otherwise. The lowest natural frequency is the smaller of the floor's fundamental frequency and the isolators' frequency. The standard's zone values are for vibration measured on the machine's own bearings, so they are not compared with the vibration of the floor.

Stairs

A stair on the floor plan can also be the target of the analysis, assessed under SCI P354 or AISC DG11. CCIP-016 has no provision for stairs, so a stair is not assessed under it.

Stair model

As AISC DG11 §7.2 recommends, the stair is modelled apart from the structure that carries it, with its bearings as boundary conditions. Concrete flights and landings are four-node shell elements; the stringers, tread plates and landing beams of a steel stair are frame elements. Shell elements are at most 0.25 m, frame elements at most 0.5 m, and every member is at least two elements. The modulus of concrete follows the guide: the dynamic modulus under SCI P354 (§4.1.3) and 1.35 times the static modulus under AISC DG11 (§7.2).

On a concrete flight with a plane soffit the steps are added to the mass as the concrete above the shell: each step is its tread area times half its riser. On a folded soffit the steps are part of the shell, so no mass is added.

A steel tread is modelled in one of two ways. A plate connected to the stringers spans from stringer to stringer as a frame element whose section is the plate's thickness by the tread's going. A tread that does not connect the stringers (timber, glass, precast) enters as load and mass only, its weight shared equally by the stringers it sits on. The deck of a steel landing is mass only as well, each part of it carried to the nearest landing beam.

The loads on the stair become mass with their shares in the mass source. Loads on a flight's sloping area are converted to plan area. This mass adds no stiffness to the stair.

Bearings

The stair's bearings are found from the floor plan. Each is derived as fixed; on the stair card it can be made pinned or removed.

Flight ends. The foot and the head of a flight bear on the level or the landing they meet. A flight end that runs along a wall or a beam bears on that member.
Landing edges. Every piece of a concrete landing's edge that runs along a wall or a beam (within 5°) inside its body, at least 5 cm long, is a line bearing along that piece.
Steel landing beam ends. A steel landing carries its load through the connections at its beams' ends. A landing beam whose end lies inside a wall's or a beam's body, and which is not parallel to it (more than 5° off its axis), bears there at a point. A landing beam lying along a wall does not bear along its length.
Columns. A column whose top is at the landing and whose axis is under it is a frame element of its own section from its base, where it bears, up to the landing. Under a concrete landing the column's solid section is the landing's rigid zone, as for the columns of a floor: the landing nodes inside the section are tied rigidly to the column's top. On a steel landing the column meets the beam node above it.
Hangers. A hanger is a bar that carries axial force only, held at its top.

A landing that bears on no wall, beam or column and has no hanger is solved hanging from its flights. The calculation then gives a warning and asks for a wall or a beam under the landing's edge, a column under it, or a hanger.

Modes are computed up to $\max(2 f_1,\ 40\ \text{Hz})$ and normalised to the mass.

Stairs under SCI P354

Under SCI P354 every tread and every landing node is an assessment point, the walker and the person who feels the vibration standing at the same point. A tread's point on a stringer flight is the average of its seats; on a concrete or cantilevered flight it is the node nearest the tread's centroid.

The walking force uses the two harmonics of Table 3.2 ( $\alpha_1 = 1.1$ , $\alpha_2 = 0.22$ ), swept over pace frequencies of 1.2–4.5 Hz. The walker weighs 746 N unless entered. On a stair whose first mode is below 12 Hz the resonant response, with the modes up to 14 Hz, and the transient response are both computed and the larger is taken; on a stiffer stair only the transient response is computed. The transient response uses the effective impulse of Eq 18 over the same pace frequencies.

The response is weighted with the $W_b$ curve and divided by the base value of 0.005 m/s² to give the response factor $R$ , which is compared with the limit Table 5.3 gives for the stair's use: 32 for light use (offices) and 24 for heavy use (public buildings, stadia).

Stairs under AISC DG11

AISC DG11 §7.4.3 has one walker and one observer. The walker stands on the middle tread of the eight consecutive treads nearest a mode's largest ordinate. The observer stands where that mode is largest, on an intermediate landing where two or more flights meet, at midspan otherwise. The guide leaves the mode that sets this placement to engineering judgment, so every mode in the FRF band is tried in turn and the placement whose acceleration is nearest its limit is reported.

The FRF is the vertical acceleration at the observer for a unit vertical force at the walker's tread, computed from $f_1 - 1$ Hz to $\max(12\ \text{Hz},\ f_1)$ at the natural frequencies and 25 further frequencies. Its largest value is $FRF_{max}$ , at the dominant frequency $f_n$ . The peak acceleration is Eq 7-8:

\frac{a_p}{g} = 0.62\, e^{-\gamma f_n}\, FRF_{max}\, R\, Q\, \left(1 - e^{-100\beta}\right)

Descent	$\gamma$	$R$	Limit (Table 4-5)
Normal descent	0.29	0.7	1.7 %g
Rapid descent	0.19	0.5 if $f_n \le 8$ Hz, 0.7 otherwise	3.0 %g (vibration not perceptible) or 4.5 %g (perceptible)
Rapidly descending group	0.19	as the rapid descent	4.5 %g

$Q$ is the walker's weight, 168 lb (0.747 kN) unless entered, and $\beta$ the damping ratio. A rapidly descending group causes three times the acceleration of a rapidly descending individual. The normal descent is always computed; the rapid descent and the group when they are selected.

The dominant frequency is also checked against 5 Hz, and the first mode moving mostly horizontally against 2.5 Hz (§4.3, §7.4.3). Where the dominant frequency is below 3 Hz, AISC DG11 asks for the stair to be checked for group rhythmic loads as well (§7.4.4). That check is not part of this calculation, which reports the case as a note.

Points the guides leave open

Topic	Rule applied
End condition of column and wall extensions under CCIP-016 and AISC DG11	All six degrees of freedom are held at the far end.
Effective width of a composite beam under CCIP-016	The rule of SCI P354 is applied.
In-plane translation of the slab	Held at the nodes where columns and walls meet the slab.
Mode set of the SCI P354 transient response	A lower bound of 40 Hz is added to the $2 f_1$ limit.
SCI P354 load factors outside 1.8–2.2 Hz	The formulas of Table 3.1 are extended beyond the range, and the calculation gives a warning.
Limits the guide gives as a range	The lower end of the range is taken.
Residential and hospital ward limits under CCIP-016	Derived from the BS 6472 levels.
The zone between $f_L$ and $f_U$ in AISC DG11 §7.5	The results of the two methods are combined linearly.
Mode limit for machine vibration	Twice the highest machine frequency replaces 40 Hz where it is larger.
Sensitive equipment under machinery	The band velocity at the machine's frequency is compared with the VC curve.
Partition stiffness under SCI P354 and CCIP-016	Counted as an option, with the rule of AISC DG11 §7.2.
Top of a masonry wall on the slab	Left free.
Partitions that cross	Joined along their line of intersection.
Several machines swept over a speed range	Combined as the square root of the sum of squares, as if their worst speeds occurred at the same time.
The mode that places the walker and the observer under AISC DG11 §7.4.3	Every mode in the FRF band is tried in turn, and the placement whose acceleration is nearest its limit is taken.
Transient response of a stair under SCI P354	Eq 18 is swept over the stair's pace frequencies (1.2–4.5 Hz).
Resonant build-up on a stair under SCI P354	Full build-up is taken.
Bearings of a stair	Fixed; on the stair card each can be made pinned or removed.
A column under a landing	A frame element of its own section; under a concrete landing its solid section is the landing's rigid zone.

Out of scope

Different activities and machines are not assessed together. Each activity is assessed in a calculation of its own.
The floor is solved on its own. The storeys beyond the column and wall extensions are not in the model.
Horizontal vibration of the floor is not calculated. The $W_d$ curve reads the vertical motion of the floor along the axis of a person lying down.
A stair is solved apart from the structure that carries it; that structure's flexibility is not part of the stair's model.
A stair's horizontal vibration is not computed. Under AISC DG11 only its lateral natural frequency is compared with the 2.5 Hz lower bound.
The group rhythmic loads of AISC DG11 §7.4.4 are not computed on a stair.
Ascents are not assessed separately under AISC DG11, which states that descents are more severe.
The stiffness of guardrails and finishes is not included; given as loads, they add mass only.
Equipment manufacturers' own vibration criteria are not included. Sensitive equipment is assessed against the generic VC curves.
Only the vertical force of a machine is calculated. In the mass-spring model the machine is one mass and the isolators one spring, and the rocking motion of the machine is not included.
Machine start-up and run-down are not calculated. Only the running speeds are swept.
The joints between a wall and the slab, and between walls that cross, are taken as continuous at these small amplitudes. Their rotational continuity is not treated separately.
The orthotropy of masonry is not modelled, and one modulus of elasticity is used.

Computed quantity#

Floor model#

Elements#

Facade#

Material#

Mass#

Partitions#

Composite slabs and beams#

Column footprint and rigid zone#

Modes#

Floor classification and mode sets#

Walking force#

Resonant response#

Transient response#

Criteria#

Frequency weighting and base value#

Limits by room type#

Vibration dose value#

Pass and fail#

Excitation and areas#

Running#

Rhythmic activity#

Sensitive equipment#

Machine vibration#

Force#

Response#

Criterion#

Isolation#

Resonance and support class#

Stairs#

Stair model#

Bearings#

Stairs under SCI P354#

Stairs under AISC DG11#

Points the guides leave open#

Out of scope#

Sources