Footfall Analysis — Verification

Footfall Analysis is tested against the worked examples of SCI P354, CCIP-016 and AISC DG11. A worked example is the guide's own numerical example repeated from start to finish; it shows that the method is coded correctly. The response chain is also compared with ETABS's time-history solution of the same floor, and machine vibration is tested against the guides' formulas and independent closed-form solutions. For stairs the calculation is compared with SAP2000's solution of the same stairs.

Results

Example	Reference	Quantities compared	Largest deviation
SCI P354 D.1, composite office floor	Worked example	3	0.03 %
SCI P354 D.2, lightweight residential floor	Worked example	3	−0.15 %
SCI P354 D.3, rhythmic activity	Worked example	5	−0.02 %
CCIP-016 Example 1, footbridge	Worked example	53	0.82 %
CCIP-016 Example 2, post-tensioned floor	Worked example	34	0.83 %
CCIP-016 Example 3, ribbed laboratory floor	Worked example	4	equal to the printed precision
AISC DG11 Example 7.1, steel composite floor	Worked example	9	1.3 %
AISC DG11 Example 7.2, live concert	Worked example	3	0.16 %
AISC DG11 Example 7.3, sensitive equipment	Worked example	6	0.50 %
AISC DG11 Examples 4.1 and 7.1, composite section	Worked example	6	2.4 %
Office floor, ETABS time-history solution	Another program's solution	7	12.4 %
Machine vibration	Closed form	13	10⁻⁶
Partitions	Guide value and model rules	5	10⁻⁹
AISC DG11 Example 4.6, stair	Worked example	3	equal to the printed precision
SCI P354 resonant response on a stair	Closed form	1	10⁻¹⁵
Three stairs, SAP2000 comparison	Another program's solution	11	1.26 %

The deviation is the calculation's value divided by the guide's value, minus 1. The guides print their tables to two or three significant figures, so the deviations given for those tables are the part of the difference beyond the rounding of the printed last digit.

SCI P354

SCI P354 D.1, composite office floor

The guide's example D.1 computes, with Eq 50, the resonant response to walking of a composite office floor with a natural frequency of 9.30 Hz and 4.68 % damping. The modal mass is 10,226.8 kg and the walker weighs 746 N.

Quantity	Guide's value	Calculation	Deviation
Weighted RMS acceleration $a_{w,rms}$ (m/s²)	47.39 × 10⁻³	47.404 × 10⁻³	+0.03 %
Response factor $R$	9.48	9.481	+0.01 %
Allowed number of events $n_a$	2,405	2,405.1	+0.01 %

The guide rounds the frequency weighting $W = 8/9.30 = 0.86022$ to 0.86, which is where the 0.03 % comes from.

SCI P354 D.2, lightweight residential floor

The guide's example D.2 computes, with Eq 51, the transient response of a lightweight residential floor with a natural frequency of 13.6 Hz to a single footfall. The modal mass is 1,181.26 kg and the pace is 2.2 Hz.

Quantity	Guide's value	Calculation	Deviation
Weighted RMS acceleration $a_{w,rms}$ (m/s²)	199.95 × 10⁻³	199.648 × 10⁻³	−0.15 %
Response factor $R$	39.99	39.93	−0.15 %
Allowed number of events $n_a$	3,239	3,239.2	+0.01 %

The guide rounds the effective impulse $60 \times 2.2^{1.43} = 185.275$ to 185 (+0.15 %) and the frequency weighting $8/13.6 = 0.58824$ to 0.59 (−0.30 %). The two roundings together give the −0.15 %.

SCI P354 D.3, rhythmic activity

The guide's example D.3 computes, with Eq 52–55, the ultimate limit state load of a floor with a natural frequency of 8.1 Hz and 1.6 % damping under normal jumping.

Quantity	Guide's value	Calculation	Deviation
Governing activity frequency $f_p$ (Hz)	2.7 ( $h = 3$ )	2.7 ( $h = 3$ )	0
Dynamic magnification $D_{\delta,1}$ / $D_{\delta,2}$ / $D_{\delta,3}$	1.12 / 1.80 / 31.25	1.1249 / 1.7987 / 31.25	equal to the printed precision
Equivalent load $F$ (kN/m²)	20.94	20.937	−0.02 %

The guide's load coefficients (Tables 3.1 and 3.3, Eq 20), effective impulse (Eq 18), frequency weighting curves (Eq 21–23) and walking velocity (Eq 16) are also compared with their printed values one to one.

CCIP-016

CCIP-016 Example 1, footbridge

The guide's Example 1 gives the resonant response of a two-span footbridge at 1.55 Hz and 2.2 Hz pacing rates, mode by mode and harmonic by harmonic (Tables 5.2 and 5.3). Every cell of the two tables is compared.

Quantity	Guide's value	Calculation	Deviation
Table 5.2, total $R$ (1.55 Hz)	5.65	5.644	−0.10 %
Table 5.3, total $R$ (2.2 Hz)	6.38	6.347	−0.51 %
Tables 5.2 and 5.3, cells	50 cells		at most 0.82 %
Hand check, $R$ at 2.32 Hz (p. 44)	8.5	8.46	−0.4 %

The ordinate of the second mode at mid-span is computed from the shape of the fixed-pinned span mode the guide describes in §5.1.1. It comes out at 0.957, and its square (0.917) agrees with the ratio of Table 5.2's mode-2 cells to a unit ordinate (0.916–0.921).

CCIP-016 Example 2, post-tensioned floor

The guide's Example 2 gives the resonant response of a post-tensioned floor bare (2.2 Hz, Table 5.6) and finished (2.35 Hz, Table 5.5). The excitation and response points are different.

Quantity	Guide's value	Calculation	Deviation
Table 5.6, total $R$ (bare floor)	5.64	5.648	+0.14 %
Table 5.6, cells	33 cells		at most 0.83 %
Table 5.5, total $R$ (finished floor)	5.30	5.266	−0.64 %

Table 5.5's cells are consistent with a frequency of about 7.05 Hz rather than with the inputs given in the text (7.0 Hz, 24,100 kg, 3 %), so the finished floor is compared on its total $R$ . The governing third harmonic is little affected, and the total deviates by 0.64 %.

CCIP-016 Example 3, ribbed laboratory floor

The guide's Example 3 computes the high-frequency (transient) response of a ribbed laboratory floor both with hand-calculated modes and with finite element modes (Tables 5.8 and 5.9).

Quantity	Guide's value	Calculation
RMS velocity $v_{rms}$ , hand modes (mm/s)	0.10	0.104
Peak velocity, hand modes (mm/s)	0.40	0.406
RMS velocity $v_{rms}$ , finite element modes (mm/s)	0.07	0.0715
Response factor $R$	1.04	1.041

The guide's printed values come from the ten of Table 5.8's eleven modes below $2 f_1 = 22.2$ Hz. The calculation likewise takes the modes below $2 f_1$ for the transient response, as §4.3.1 says.

AISC DG11

AISC DG11 Example 7.1, steel composite floor

The guide's Example 7.1 computes the resonant and transient response to walking of a composite office floor on steel beams with the 38 modes of Table 7-2. The result is a peak acceleration as a percentage of gravity (%g).

Quantity	Guide's value	Calculation	Deviation
Resonant step, 3.49 Hz (%g)	0.374	0.3753	+0.35 %
Resonant step, 4.89 Hz (%g)	0.351	0.3556	+1.3 %
Resonant step, 7.05 Hz (%g)	0.173	0.1729	−0.04 %
Resonant step, 8.85 Hz (%g)	0.242	0.2423	+0.13 %
Dominant frequencies (Hz)	12.6 and 8.85	12.6 and 8.85	0
Effective impulse of mode 22, $I_{eff}$ (lb·s)	1.01	1.012	equal to the printed precision
Peak acceleration of mode 22 (%g)	0.206	0.2059	−0.05 %
Equivalent sinusoidal peak acceleration, ESPA (%g)	0.314	0.3153	+0.40 %
Total peak acceleration of the 38 modes (%g)	0.865	0.857	−0.94 %

The resonant steps are computed with the guide's printed frequency response function (FRF) values. The 4.89 Hz step comes out at 0.3556 %g from the printed inputs, and the 1.3 % is not explained by their rounding; the other three steps are within 0.35 %. The FRF built from the mode ordinates of Table 7-2 is about 11 % below the curve of Figure 7-10 at both peaks.

With the guide's 0.005 s sampling step of Eq 7-6, the ESPA is 0.3140 %g. The −0.94 % in the total peak acceleration comes from the mode ordinates being printed to three significant figures.

AISC DG11 Example 7.2, live concert

The guide's Example 7.2 computes the response of an 18.7 psf crowd at a 3 Hz live concert harmonic by harmonic and combines the harmonics with the 1.5-power rule of Eq 7-10.

Quantity	Guide's value	Calculation	Deviation
First harmonic $a_{p,1}$ (%g)	1.11	1.113	+0.24 %
Second harmonic $a_{p,2}$ (%g)	1.88	1.879	−0.03 %
Combined peak acceleration (%g)	2.41	2.414	+0.16 %

AISC DG11 Example 7.3, sensitive equipment

The guide's Example 7.3 computes the one-third octave band velocity at a sensitive instrument on a floor walked on at a fast pace, and compares it with the generic criteria of Table 6-2.

Quantity	Guide's value	Calculation	Deviation
Dominant frequency (Hz) and step frequency (Hz)	10.9 and 2.18	10.9 and 2.18	0
Effective impulse of mode 19, $I_{eff}$ (lb·s)	1.32	1.320	+0.03 %
Peak acceleration of mode 19 (%g)	0.0843	0.08425	−0.06 %
Total peak acceleration (%g)	0.156	0.1568	+0.48 %
Calibrated peak acceleration (%g)	0.234	0.2351	+0.48 %
One-third octave band velocity $V_{1/3}$ (mips)	2,850	2,864	+0.50 %

The intermediate values agree with the guide's printed three significant figures. The 0.50 % in the band velocity lies within the rounding of the printed mode table.

AISC DG11 Examples 4.1 and 7.1, composite section

The composite section values of steel beams acting with the slab are compared with those of the guide's Examples 4.1 and 7.1. The slab of Example 7.1 is 5¼ in. of lightweight concrete on a 2 in. deck.

Quantity	Guide's value	Calculation	Deviation
Slab moment of inertia across the ribs (in⁴/ft)	34.3	34.33	+0.08 %
Slab moment of inertia along the ribs (in⁴/ft)	102	101.6	−0.43 %
W16x26 interior beam, composite moment of inertia (in⁴)	1,100	1,126	+2.4 %
W16x26 edge beam, composite moment of inertia (in⁴)	1,010	1,027	+1.7 %
W24x55 girder, 24 ft span (in⁴)	3,970	4,061	+2.3 %
W24x55 girder, 16 ft span (in⁴)	3,670	3,727	+1.5 %

Example 7.1 shows no working for its section values and refers to Example 4.1; the same calculation gives Example 4.1's beam within 0.5 %. Moments of inertia 1.5–2.4 % above Example 7.1's make the floor slightly stiffer and its natural frequency slightly higher.

VC criteria

The vibration criteria for sensitive equipment (VC curves) are taken from each guide's own table.

Guide	Criterion	VC-A	VC-B	VC-C	VC-D	VC-E
SCI P354 Table 5.6	$R$ on the $W_g$ base curve	0.5	0.25	0.125	0.0625	0.0313
CCIP-016 Table 3.2	$R$ on its base curve	0.5	0.25	0.125	0.0625	0.03125
AISC DG11 Table 6-2	One-third octave band velocity (mips)	2,000	1,000	500	250	125

In a calculation under AISC DG11, an area with a VC class or an operating theatre (4,000 mips) is read as a velocity, and the pace of the person walking past it is taken from Table 6-1.

Office floor, ETABS time-history solution

For this comparison an office floor with flat-plate and 12 m beam bays, walls and openings is built in ETABS with its own objects. A 1 kN vertical sine force, standing for one walking harmonic, acts at the floor's governing point for $R$ , and ETABS's modal time-history solution is taken at 35 frequencies from 4 to 18 Hz.

Quantity	Measured
Steady-state sum with ETABS's modes / ETABS time history, at the peaks (8.69 / 16.29 / 17.52 Hz)	at most 0.11 %
Same ratio, whole sweep	at most 2.8 % (below the first modes)
8.7 Hz peak with MindFEM's modes / ETABS	amplitude +3.6 %, frequency −2.1 %
16.3 Hz and 17.5 Hz peaks with MindFEM's modes / ETABS	+12.4 % and −11.5 %
Static deflection under 1 kN at the same point / ETABS	+5.8 %
SCI P354 $R$ (walking at 2.2 Hz), with MindFEM's modes / with ETABS's modes	7.44 / 7.30 (+1.95 %)
SCI P354 Eq 36 / ETABS time history, at the peaks	1.148 / 1.230 / 1.141

The first two rows show that MindFEM's steady-state calculation, given ETABS's modes, reproduces ETABS's time-history solution. The other differences between MindFEM and ETABS come from the two programs' modelling — the shell formulation, the mesh and the way mass is distributed. The two close modes at 16.3 and 17.5 Hz have the same shape in both programs (MAC 0.97 and 0.92), but different amplitudes at this free-edge point.

Within one harmonic, SCI P354 Eq 36 adds the modal contributions with their signs and without their phases. The time-history solution, which the guide names the most accurate method in §6.4.1 (Eq 31), carries each mode's phase against the force. At resonance the resonant mode lags the force by a quarter period while modes far from it move in phase or in antiphase, so their peaks do not occur at the same instant, and Eq 36 gives 14–23 % more at the peaks. Under self excitation, where the walker and the receiver are the same point, every modal contribution is positive, so Eq 36 is never below the time-history solution and the result is on the safe side. Under full excitation, with the walker and the receiver apart, that order does not hold. The calculation applies Eq 36 as the guide recommends.

Machine vibration

The force a rotating machine applies to the floor and the floor's steady-state response are tested against the guides' formulas and independent closed-form solutions.

What was tested	Reference	Result
Force from the balance grade, $F = m\,G\,\Omega/1000$	ISO 21940-11 Eq 6	The same by two routes from the permissible residual unbalance
Force from eccentricity, $F = m\,e\,\omega^2 S_f$ , $S_f \geq 2$	ACI 351.3R Eq 3-3	Equal; $S_f < 2$ refused
Steady-state acceleration of one mode	Closed form	Equal
Two machines in one one-third octave band, combined by SRSS	Closed form	Equal
One-third octave band limits	AISC DG11 Table 7-5	Equal
Natural frequencies within 20 % of the running frequency	ACI 351.3R §3.4.3	Equal
Speed range sweep, force with the square of the speed	ACI 351.3R §3.2.2.1	Peak of a mode inside the range equal to the closed form to 10⁻⁹
Rigid-base transmissibility crossing 1 at $r = \sqrt{2}$ for every damping	ACI 351.3R Fig. 3.14	Equal
Undamped transmissibility 1/8 at $r = 3$ (about 90 % isolation)	SCI P354 B.3.2	Equal
Isolator frequency from static deflection, 3.15 Hz for 25 mm	$f = \sqrt{g/\delta}/2\pi$	Equal
Mass-spring model on a very stiff floor	Rigid-base transmissibility	10⁻⁶
Mass-spring model on a one-mode floor	Direct solution of the two-degree-of-freedom system	10⁻¹⁰
Support class	ISO 20816-3 §5.3	Equal

The guides do not include a worked example that solves a floor together with a machine on it, so this section rests on the formulas and closed-form solutions. Two successive solves of the same model can differ in $R$ by up to 0.12 %; the finite element mesh is not built bit for bit the same each time, and the difference shows for a machine running right next to a mode.

Partitions

When the stiffness of partitions is counted, the spring value of AISC DG11 §7.2 and the rules of the model are tested.

What was tested	Basis	Result
The sum of the springs along a wall, 2.0 kip/in./ft (1,149 kN/m per metre) × the length meeting the slab	AISC DG11 §7.2	Equal, 10⁻⁹
No spring over an opening that reaches the edge meeting the slab	Model rule	Equal
The shells of a masonry wall massless, its top free, the fundamental frequency rising	Model rule	Holds
A slab node at each edge of a door opening, and wall over the opening	Model rule	Holds
Two crossing walls sharing one node at every height along their intersection	Model rule	Holds

A comparison with a floor measured with its partitions is not part of this page.

Stairs

AISC DG11 Example 4.6, stair

AISC DG11 Example 4.6 computes, by the hand method (Eq 4-8), the peak acceleration of three descents on a steel stair spanning 36.9 ft, weighing 15.5 kips, with a fundamental frequency of 7.02 Hz and 3 % damping. The calculation solves the same stair with the finite element method's equation (Eq 7-8) and the first mode of a simply supported beam.

Descent	Guide value	Calculated	Deviation
Normal descent	1.44 %g	1.440 %g	0.02 %
Rapid descent	2.08 %g	2.076 %g	−0.20 %
Rapidly descending group	6.23 %g	6.228 %g	−0.04 %

On a single-mode stair Eq 7-8 equals Eq 4-8, so the deviations come from the guide printing three significant figures.

SCI P354 resonant response on a stair

On a single-mode stair with a 6 Hz fundamental frequency, 5 t modal mass and 2 % damping, the second harmonic of Table 3.2 is resonant at a 3 Hz pace. The response factor computed by hand from the two harmonics' $W_b$ -weighted responses is $R = 116.14$ ; the calculation gives the same value within 10⁻¹⁵ and the governing pace frequency of 3 Hz.

Three stairs, SAP2000 comparison

The modes and response of a steel U stair, a concrete U stair and a spiral stair drawn on the floor plan are compared with SAP2000. The fundamental frequencies agree within 1.26 %, the SCI P354 response factor within 0.87 % and the AISC DG11 peak accelerations within 0.18 %. The models, the comparison tables and the SAP2000 model files are on the Stair vibration, comparison with SAP2000 page.

Model files

The ETABS model that produced the office floor comparison can be downloaded. It opens in ETABS 20 and later with File > Import > ETABS .e2k File.

Office floor, ETABS model (.e2k)

What this calculation covers

Topic	Status
Vibration of a measured floor	The comparisons on this page are against the guides' examples and another program's solution. There is no comparison with a measured floor.
SCI P354's stair method	Applied with the coefficients of Table 3.2 and full resonant build-up, which is on the safe side.
AISC DG11's stair method	The method of §7.4.3 is for slender stairs.
SCI P354's finite element examples (D.1.7, D.2.6) and the measured floors of Appendix A	Their model inputs (sections, masses, supports) are not in the guide, so they are not rebuilt.
SCI P354 load coefficients outside 1.8–2.2 Hz	Table 3.1 gives values for this range; outside it the coefficients are extended with the table's formula.
$W_b$ or $W_g$ weighting in offices and dwellings	SCI P354 Table 5.1 and the guide's base curves differ. The weighting curve is the room type's preset and can be changed by hand.
AISC DG11 §7.5, between $f_L$ and $f_U$	The guide names neither the resonant nor the impulse method there. The calculation blends the two linearly in frequency across that range.
Sensitive equipment under machinery	The velocity in the one-third octave band of the machine's frequency is compared with the VC curve. The guides calibrate the VC curves for walking.
Several machines swept over speed ranges	Each machine's worst speed is combined by SRSS as if they coincided, which is on the safe side.
Mass-spring model	Vertical only. The machine is one mass at one node, the isolators one spring, and rocking is not included.
Machine start-up and run-down	Only the running speeds are swept. On start-up a machine passes briefly through its isolators' resonance; ISO 10137 §5.2.1 asks that this be considered and gives no numerical criterion.
ISO 20816-3 zone values	Only the support class is given. The standard's zone values are for vibration measured on the machine's own bearings and are not compared with the floor's vibration.

Results#

SCI P354#

SCI P354 D.1, composite office floor#

SCI P354 D.2, lightweight residential floor#

SCI P354 D.3, rhythmic activity#

CCIP-016#

CCIP-016 Example 1, footbridge#

CCIP-016 Example 2, post-tensioned floor#

CCIP-016 Example 3, ribbed laboratory floor#

AISC DG11#

AISC DG11 Example 7.1, steel composite floor#

AISC DG11 Example 7.2, live concert#

AISC DG11 Example 7.3, sensitive equipment#

AISC DG11 Examples 4.1 and 7.1, composite section#

VC criteria#

Office floor, ETABS time-history solution#

Machine vibration#

Partitions#

Stairs#

AISC DG11 Example 4.6, stair#

SCI P354 resonant response on a stair#

Three stairs, SAP2000 comparison#

Model files#

What this calculation covers#

Sources