1. sports
2. walking

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Porto, Portugal, 30 June - 2 July 2014
A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)
ISSN: 2311-9020; ISBN: 978-972-752-165-4
Prediction of Floor Vibration Induced by Walking Loads
using Response Spectrum Approach
Jun CHEN1,2, Ruotian XU2 and Mengshi ZHANG2
State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai, China.
2
Department of Building Engineering, Tongji University, Shanghai, Chian
email: [email protected], [email protected], [email protected]
1
ABSTRACT: A design-oriented acceleration response spectrum approach has been introduced to predict a floor’s response due
to a single person walking. More than two thousand measured footfall traces from 61 test subjects were used to generate tensecond peak root-mean-square acceleration response spectra, on which a piecewise mathematical representation is based. The
proposed piecewise response spectrum curve covers the frequency range of 1.0 to 10Hz and consists of five parts. The first part
is linear line from 1.0 to 1.5Hz, the second part is the first plateau ranging from 1.5 to 2.5Hz, the third part is another linear line
from 1.5 to 3.0Hz, the fourth part is second plateau ranging from 3.0 to 5.0Hz, and the final part is a descending curve going
from 5.0 to 10Hz. The first and second plateau correspond to, respectively, the first and second harmonics of the walking load.
The representative value of each plateau and the mathematical representation for the descending curve were determined
statistically for different confidence levels. Furthermore, the effects of factors, such as floor span, occupant stride length, mode
shapes and peak acceleration response, on the proposed spectrum have been investigated and a modification measure for each
factor is suggested. The proposed spectrum approach has been applied to four existing floors to predict their acceleration
responses. Comparison between predicted and field measured responses shows that the measured accelerations of the four floors
are generally close to or slightly higher than the predicted values for the 75% confidence level, but are all lower than the
predicted values for the 95% confidence level. The suggested spectrum-based approach can be used for predicting a floor’s
response subject to a single person walking.
KEY WORDS: Vibration serviceability; Human walking loads; Long span floor; Root-mean-square acceleration response
spectrum.
1
INTRODUCTION
In recent years, long span floors have been increasingly
popular in buildings like offices, shopping centers and stadia
due to the rapid development of construction techniques and
wide spread application of high-strength and light-weight
materials. As a result, the vibration serviceability problem has
become an important design issue. The problem is that the
floor experiences annoying large-amplitude vibration due to
the occupants’ daily activities, like walking and jumping.
Altering problematic floors to rectify this problem has proved
to be very difficult and costly. Thus, the vibration
serviceability assessment of long-span floors at the design
stage has attracted increasing attention from researchers and
engineers.
The response spectrum approach, which is very popular in
structural seismic design, is well known for its efficient and
ability to predict a structure’s maximum response to certain
extreme loads. Ungar et al. [1] presented an approach to
estimate the walking-induced vibrations of floor using
response curve, which describes variation of the peak
acceleration or peak RMS acceleration with the floors’
fundamental frequency. An idealized pulse model, which is
not very common, for the footfall was adopted in the paper to
calculate the response curve. The parameters of the pulse
model were determined using footfall records from different
literature [2-4] for only three walking rates of 1.25, 1.67 and
2.08Hz. Moreover, the response curve was applicable only for
floors with fundamental frequencies in the range 5-20 Hz
because the study focused mainly on special floors
accommodating vibration-sensitive equipment. Song et al. [5]
suggested a frequency-response curve to predict a floor’s peak
acceleration response based on the floor’s fundamental
frequency. They applied the walking force measurement
provided in Ellingwood et al. [6] and theoretical walking load
model suggested by Matsumoto et al. [7] to a 6-m simply
supported beam to determine three response curves for
walking frequencies 1.6, 1.9 and 2.2Hz. By fitting the three
calculated curves, two design response curves for damping
ration 3% and 5% were obtained for floors whose span was in
the range of 6 to 17 meter. Georgakis et al. [8] proposed a
response spectrum approach to calculate the peak acceleration
response of a footbridge subjected to crowd people walking.
A reference response spectrum was first developed by a series
of Monte Carlo simulation using a probabilistic walking load
model available in the literature. The prediction value from
the reference response spectrum was then modified to account
for factors as the real bridge span length, structural damping
ratio, pedestrian walking frequency and flow rate. The
response was further adjusted to represent specific return
period. Recently, Mashaly [9] used the response spectrum
approach to predict the vertical acceleration response of
footbridges subjected to walking load. They used the walking
load model provided by Murray et al. [10] for only three
walking frequencies as 1.1, 1.5 and 2.2Hz. The pedestrian
load was assumed stationary at the mid-span of the footbridge
and the pedestrian’s weight was assumed as fixed value of
1103
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
2
WALKING LOADS FOR DEVELOPING RESPONSE
SPECTRUM
Walking load experiments were conducted on seventy-three
test subjects using two fixed force plates and threedimensional motion capture technology (Vicon System). Each
subject completed seven test cases, which were: walking at
fixed paces (guided by a metronome) of 1.5, 1.75, 2.0 and
2.25Hz, and walking freely at self-chosen slow, normal and
fast speeds. For each test case, the subject walked six or seven
times. A summary of the body statistics of all the test subjects
is given in Table 1.
Table 1 Statistics of test subjects
Gender
Male
Female
Number
57
16
Weight
(39-88kg)


65.4
51.6
9.85
6.87
Height
(1.55-1.85m)


1724.8
1616.9
146.71
30.81
In each test, thirty-nine reflective markers were attached to
the test subject. When the subject performed the test, the
spatial trajectories of all of the markers were monitored by a
Vicon Motion Capture System with ten cameras. Figure 1a
shows the experimental setup and Figure 1b shows a typical
1104
experimental record of walking load. The two dashed lines in
Figure 1b are the footfall force measured by each force plate
and the solid blue line is the resultant force curve for two
continuous steps. The footfall trace in Figure 1 was already
normalized by the test subject’s weight. Hereafter, we refer to
the resultant force curve as the single footfall trace. A detailed
description of the experimental setup, test procedure and data
processing can be found in Chen et al.[11,12]
a. Experimental setup
1.5
Vertical force/ Weigth
700N. Plots of response spectra, without a mathematical
representation, were provided in the study for design purpose
for three damping ratios 0.01, 0.02 and 0.03. The response
spectrum predictions were claimed to be more accurate than
the results obtained using the equation in AISC standards
(AISC Design Guideline #11). It is well known that real
earthquake records, rather than synthetic earthquake waves,
should be used for developing earthquake response spectra. In
all the above researches, however, numerical walking load
models, which are tenable only for certain walking frequency
range or several specific walking frequencies, have been
adopted.
Moreover,
the
suggested
mathematical
representations, if any, of the response spectra were generally
determined by curve fitting thus had not clear
physical/mechanical meaning. A design-oriented response
spectrum approach for calculating floor response under
occupant walking is still not available.
This paper presents an approach based on acceleration
response spectrum to predict the acceleration response of a
floor due to one person walking. Initially, walking load
experiments were conducted on sixty-one test subjects
resulting in more than two thousand footfall traces with
different step frequencies ranging from 1.3 to 3.0Hz. Each
recorded footfall trace was then applied to a generalized single
degree of freedoms system to calculate the 10 second peak
root-mean-square acceleration response spectrum, on which a
piecewise mathematical representation has been suggested.
The effects of the following key factors on the reference
spectrum have been investigated: floor span, occupant stride
length, higher modes of vibration, boundary conditions and
different vibration assessment indexes. Finally, the proposed
spectrum approach was applied to four existing floors. The
applicability and feasibility of the approach is shown by
comparing the spectrum predictions of four floors with their
field test results.
1
0.5
0
0
0.5
1
Time（ sec）
1.5
2
b. A typical walking load curve measured (male test subject,
walking frequency 2.0Hz)
Figure 1. Walking load experiment using force plates and 3D
motion capture technology and typical experimental records
Quality checks have been made on the original
experimental records and 2204 qualified single footfall trace
from 61 subjects were adopted in this study. The distribution
of the actual step frequencies of all the records is shown in
Figure 2. It roughly follows a normal distribution with a mean
value of 1.937 Hz and a variance of 0.296 Hz. From the
marker’s trajectories, the Vicon system can identify the start
and end time of each gait cycle of a test subject in the walking
test. Assume that the walking load curve in every gait cycle is
the same as the one measured by the force plate, the measured
single footfall trace can then be extended into a continuous
force curve. Figure 3 shows conceptually the procedure to
extend the single footfall curve into a continuous one using
the motion capture data. In-depth discussion on the footfall
curve extension can be found in Chen et al. [13] in which the
accuracy, applicability and parameter sensitivity of four
commonly used extension methods are compared.
4
2
Gaussian fit
4.1
1.5
 =1.937Hz
 =0.296Hz
1
0.5
0
1.5
2
2.5
Step frequency(Hz)
3
Figure 2. Distribution of the step frequency of 2204 records.
Figure 3. Extended continuous walking load.
3
STANDARD
SPECTRUM
RESPONSE SPECTRUM
FOOTFALL TRACES
FROM
Consider the vertical dynamic response of a simply-supported
rectangular floor when a person walks along the floor’s
central line. Using the model decomposition theory, the
dynamic response of the floor can be decomposed as
summation as single degree of freedom system, whose
equation of motion is
(1)
Where a jref is the ratio of pedestrian’s weigh to the modal
(2)
Suppose S jk max   is the peak or peak root-mean-square
(RMS) acceleration response spectrum for GSDF, then the
maximum response in Eq. 2 will be
(3)
For convenience, S jk max   for GSDF is termed the
standard acceleration response spectrum. The next step is to
use the measured footfall traces to generate S jk max   and
find it a proper mathematical representation.
RESPONSE
The acceleration response spectrum for each of the 2204
extended walking loads is determined by the following three
steps. Firstly, for a measured walking load f (t) with the
associated pedestrian’s weight G, the generalized force Fj  t 
can be constructed. In the calculation, the stride length L is
selected as 0.75m, the walking distance (i.e. the floor’s span)
is taken as 42m and the first vibration mode is considered.
Secondly, apply Fj  t  to Eq. 2, the dynamic responses can be
calculated for a given frequency and damping ratio. The
running 10-second RMS curve of the time-history of
acceleration responses is then computed and the peak value of
the RMS curve, denoted as max_ a10sec
RMS ( ) , is taken as the
representative value for the current frequency and damping
ratio. Thirdly, change the frequency of GSDF (Eq. 2) from
0.05 to 10Hz with steps of 0.05 Hz, the 10-second peak RMS
acceleration response spectrum (hereafter the 10s-RMS
spectrum) can be constructed by connecting the representative
values max_ a10sec
RMS ( ) at each frequency. Five structural
damping ratios, 0.01 to 0.05 with an interval of 0.01, are
considered. Therefore, each recorded footfall trace will have
five response spectra. Figure 4 shows a typical 10s-RMS
spectrum for walking force record at a frequency of 2.3Hz
with a damping ratio of 0.01.
RECORDED
mass of the (j,k) th vibration mode of the target floor, Fj  t  is
the generalized force which is calculated by the measured
footfall curve and the mode shape.
Define the generalized single degree of freedom system
(GSDF) whose equation of motion is given in Eq. 2.
ACCELERATION
Root-mean-square Acceleration Response Spectrum
Peak 10s RMS acceleration（ m/s 2）
Probability density function
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
10
8
6
4
2
0
0
2
4
6
8
Structure frequency（ Hz）
10
Figure 4. A typical 10s-RMS spectrum for walking at 2.3Hz
(damping ratio 0.01)
4.2
Standard RMS Acceleration Response Spectrum
Since each test subject contributed several walking loads at
different walking frequencies in the experiment, there are
multiple response spectrum curves associated with the same
test subject. The envelope curve of all 10-s RMS spectra for
the same subject is taken as the representative spectrum for
this subject. Therefore, we have sixty-one representative
spectrum curves that are depicted in Figure 5 together with
two curves covering 75% and 95% of sixty-one representative
values at each frequency.
1105
Peak 10s RMS acceleration（ m/s 2）
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
I
15
II
III
probability P . The quantile (inverse cumulative distribution)
function for the Weibull distribution is
95%confidence level
75%confidence level
 rms    ln   P 
1/ k
Fitting the Eq. 5 to the curves shown in Figure 6 we can get
the proper distribution model parameters. Table 2 summarizes
the fitted  and 1/ k values for five damping ratios for the
two harmonic plateaus. The final expression for  rms for the
two harmonic plateaus is as Eq. 6.
10
5
0
1
3
5
Structure frequency（ Hz）

0.20
 rms  0.20 0.76   ln 1  P   0.194



0.226 0.18
 sub
0.70
  ln 1  P  
 rms  0.11
10
Figure 5. The representative spectrum curves
for the sixty-one subjects.
Figure 5 shows that the first and second harmonics of the
walking load dominate the amplitude of the response
spectrum. As a result, the response spectrum can be broadly
divided into three bands, marked as I, II and III in Figure 5.
The first band I covers 1.5 to 2.5 Hz and it actually
corresponds to the first harmonic of walking load. The second
band II is from 3.0 Hz to 5.0Hz and it corresponds to the
second harmonic of walking load. The third band III is the
range above 5.0Hz where the amplitude of spectrum decreases
gradually with increasing structural frequency. Furthermore,
the spectrum amplitudes in the first and second band vary
significantly among different test subjects. Therefore a
statistical representative is needed for each band. Close
observation shows that all the spectrum curves broadly form a
plateau in the first band as well as in the second band, which
is named the first harmonic plateau and the second harmonic
plateau. For each harmonic plateau, a representative value
with a given confidence level will be selected.
Based on all the above observations, the proposed 10s-RMS
spectrum curve consists of the following five parts: part 1, an
ascending line from 1.0 to 1.5Hz; part 2, the first harmonic
plateau ranging from 1.5 to 2.5Hz); part 3, a descending line
ranging from 2.5 to 3.0Hz; part 4, the second harmonic
plateau ranging from 3.0 to 5.0Hz; and part 5, the descending
curve ranging from 5.0 to 10Hz.
4.3
4.3.1
P   f ( x)dx  1-exp((
 rms k
) )

(4)
Where k> 0 is the shape parameter and λ > 0 is the scale
parameter of the distribution.  rms is the value for given
(6)
f  3.0 Hz,5.0 Hz 
sub
level P , the representative peak RMS values  rms and  rms
of the first and second resonant platform can be directly
determined by Eq. 6.
4.3.2
Descending curve
An exponential function as Eq. 7, which is very popular in
seismic response spectrum, is adopted to represent the
descending curve ranging from 5 to 10Hz. That is

5

 f 
3
sub
 rms
 f    rms

f  5Hz,10 Hz 
(7)
where f is the natural frequency of the floor.
The parameter  in Eq. 7 is affected by the peak RMS
values at f  5.0Hz and f  10Hz , the damping ratio and the
confidence level. Extensive calculations have been conducted
on three confidence levels 95%, 75% and 50% and five
damping ratios. The results demonstrate that  varies in a
relatively small range for all the cases considered. Therefore,
the average value of  for all the computed cases is adopted;
which is   1.48 . The final expression for the descending
curve is therefore
3
 rms
 f , , P   
Representative value for each harmonic plateau
f  1.5Hz, 2.5Hz 
Therefore, for any given damping ratio  and confidence
1.48
5

 f 
Statistical Analysis of Spectrum Parameters
Statistical analysis was conducted in the first and second
harmonic plateaus to determine the probability desirubiton of
all the peak RMS values in the two bands. The resulting
cumulated probability function is shown in Figure 6 for the
first harmonic plateau for five damping rations, i.e. 1%, 2%,
3%, 4% and 5%. It is found that Weibull distributions fit the
calculated distributions quite well.
For Weibull distribution, its cumulative probability function
is shown in Eq. 4
1106
(5)
0.11 0.70   ln 1  P  
0.226 0.18
(8)
f  [5Hz,10 Hz ]
4.4
The Proposed
(1Hz~10Hz)
10s-RMS
Response
Spectrum
From the above analysis, the response spectrum can be
completed by connecting the functions in part I, II and III with
straight lines. The final piecewise mathematical representation
of the spectrum is given in Eq. 9. Figure 7 shows the proposed
spectrum for the 95% confidence level with 0.01 damping
ratio. Figure 8 compares the proposed 10s-RMS response
spectrum with the 61 representative response spectra.
Peak 10s RMS acceleration（ m/s 2）
Peak 10s RMS acceleration（ m/s 2）
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
15
 rms
10
 sub
rms
5
0
(5/f)1.48 sub
rms
1 1.5 2.5 3
5
Structure frequency（ Hz）
10
15
95%confidence level
75%confidence level
10
5
0
0
2
4
6
8
Structure frequency（ Hz）
Figure 8. Comparison of the proposed spectrum for different
confidence levels with representative spectra for the measured
footfall traces
Figure 7. The proposed acceleration spectrum for 95%
confidence level and damping ratio 1%
=0.02
real data
Gaussian fit
Weibull fit
0.8
Gaussian fit
=5.239m/s
 =3.335m/s 2
2
0.6
0.4
Weibull fit
=6.666m/s 2
0.2
0
Cumulative probability
Cumulative probability
=0.01
1
k=2.024m/s
0
5
2
10
real data
Gaussian fit
Weibull fit
1
0.8
= 3.364m/s 2
 =1.763m/s 2
0.4
Weibull fit
0.2
=4.060m/s 2
0
15
k=2.334m/s 2
0
2
4
Cumulative probability
Cumulative probability
=0.04
real data
Gaussian fit
Weibull fit
Gaussian fit
=2.475m/s 2
 =1.167m/s 2
0.6
0.4
Weibull fit
=2.922m/s 2
0.2
k=2.535m/s 2
0
1
2
3
8
(b) Damping ratio 2%
=0.03
0
6
10s RMS acceleration（ m/s2）
(a) Damping ratio 1%
0.8
Gaussian fit
0.6
10s RMS acceleration（ m/s2）
1
10
4
5
0.6
real data
Gaussian fit
Gaussian fit 2
=5.239m/s
Weibull fit
 =3.335m/s 2
Weibull fit
0.4
k=2.024m/s 2
1
0.8
Gaussian fit
=1.969m/s 2
 =0.861m/s 2
=6.692m/s 2
=2.310m/s
0
6
=6.692m/s 2
2
k=2.024m/s 2
k=2.267m/s 2
0
1
2
2
3
4
=5.239m/s 2
 =3.335m/s 2
Weibull fit
Weibull fit
0.2
Gaussian fit
5
10s RMS acceleration（ m/s2）
10s RMS acceleration（ m/s ）
(c) Damping ratio 3%
(b) Damping ratio 4%
Cumulative probability
=0.05
real data
Gaussian fit
Weibull fit
1
0.8
Gaussian fit
=1.639m/s 2
 =0.683m/s 2
0.6
0.4
Weibull fit
=1.889m/s 2
0.2
0
k=2.813m/s 2
0
1
2
3
4
10s RMS acceleration（ m/s2）
(e) Damping ratio 5%
Figure 6. Comparison of Gaussian and Weibull distributions with the probability distribution of peak RMS acceleration in the
frequency range 1.5 to 2.5Hz
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
f 1

 rms ( , P)
f  [1.0 Hz,1.5Hz ]

0.5

 rms ( , P)
f  [1.5Hz, 2.5Hz ]

 f  2.5
sub
sub
( rms ( , P)   rms
( , P))   rms
( , P) f  [2.5Hz,3.0 Hz ]
 ( f ,  , P)  
0.5

sub
 rms
( , P)
f  [3.0 Hz,5.0 Hz ]


5
sub

( )1.48  rms
( , P)
f  [5.0 Hz,10 Hz ]

f
4.5

Factors affecting the spectrum
This section investigates the influence of several key factors,
including higher vibration modes, boundary conditions, floor
span and walking step length, on the proposed spectrum.
Accordingly, a modification procedure for each factor is
suggested.
4.5.1
Mode shape
The model shape is used to determine the generalized force in
Eq. 2. Only the first vibration mode is adopted in above
calculations. It is therefore necessary to learn the effect of
higher vibration mode and the effect of boundary conditions
on the vibration mode shape. Extensive calculation shows the
proposed response spectrum function in Eq. 9 is tenable for
higher vibration modes and for different boundary conditions
and will give conservative predictions [14]. For real
applications it is convenient to use the same response
spectrum for different vibration modes. For computational
convenience, it is also suggested that the same function as Eq.
9 should be used for different boundary conditions.
4.5.2
Floor Span and stride length
It is found that the effect of floor and stride length can be
addressed together by the flowing equation
  1 e
(
0.075
L)
dl
(10)
where, dl is real stride length and L is the floor span in unit
meters.
4.5.3
a peak  2.0a10sec
RMS
(11)
APPLICATION PROCEDURE FOR THE PROPOSED
RESPONSE SPECTRUM
The key steps for calculating the total response of a floor
using the proposed response spectrum are summarized as
follows:
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
Step 1: calculate the natural frequency f jk and modal
mass M jk of the target floor by the finite element
method (FEM), or use the modal properties from field
measurement, where subscribe jk means the (j,k)th mode
of vibration.
Step 2: calculate the generalized 10s RMS acceleration
response value  ( f ,  , P) for each mode of vibration by
the proposed spectrum function in Eq. 9 for the given
floor frequency f jk , damping ratio  and confidence
level P.
Step 3: calculate the 10s RMS value for the jkth mode
using Eq. 12
a jk ,rms  wjk jk a jkref   f ,  , P 
(12)
Where wjk is the maximum mode shape value along the
walking route, it is unity if the pedestrian walks along the
central line of the floor;  jk is the value of the (j,k)th mode
shape at a specific floor point for the vibration analysis. The
parameter is introduced to account for the situation that other
location rather than the central point is considered;
a jkref  G M jk is the reference acceleration, G is the person’s
weight, M jk is the modal mass.

Step 4: use the method of square root of the sum of the
squares (SRSS) to predict the floor response for all
modes.
arms   a 2jk ,rms
(13)

Step 5: use Eq. 11 to estimate the peak acceleration
response if necessary.
6
ASSESSMENT OF THE RESPONSE SPECTRUM
METHOD
Peak response
Sometimes peak response rather than 10s-RMS acceleration is
adopted as an index to evaluate the vibration performance of a
floor. To this end, peak response spectra have been calculated
and compared with the 10s RMS acceleration response
spectrum. Statistical analysis shows that the peak response
spectrum can be approximately obtained by the following
equation.
5

(9)
To validate the proposed response spectrum approach for
predicating floor’s response, it has been applied to four real
floors, which are denoted as Floor A[15], B[16], C[17] and
D[18]. These floors are selected because they have fieldmeasured acceleration responses due to a single person
walking. Details of the four floors are summarized in Table 3
including source, natural frequencies, modal mass and field
measured acceleration response. The predicted acceleration
responses are then compared with the field measurements.
Figure 9 compares the measured RMS values with the
calculated values for the 95% and 75% confidence levels. For
Floors B, C and D, the length modification was considered
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Table 2

and
1/ k for the first and second resonant platforms
First resonant platform
Damping
ratio(ξ)
0.01
0.02
0.03
0.04
0.05
Second resonant platform

1/ k

1/ k
6.666
4.060
2.922
2.310
1.889
0.494
0.428
0.394
0.375
0.355
2.824
1.759
1.314
1.064
0.908
0.508
0.452
0.420
0.398
0.381
Table 3 Structural parameters of the four selected floors
Floor
Number
A
B
C
Prestressed
concrete
floor[15]
Concrete
floor[16]
Concrete
floor[17]
Composite
floor[18]
Peak 10s RMS acceleration（ % g）
D
Floor type
Floor
span(m)
Natural
frequency(Hz)
Damping ratio
Modal mass(103Kg)
Measured
acceleration(%g)
30
2.22,2.82,3.23,4.13
0.015,0.015,0.015,0.015
891, 992, 898, 1024
0.052(10s RMS)
42
4.67,5.11,6.10,7.51
0.02,0.02,0.02,0.02
1140,1210,1190,1350
0.013(10s RMS)
12.7
6.20
0.03
20.6
0.70(Peak)
9.2
7.57
0.0025
20
3.57(Peak)
Measured response
75%confidence level
95%confidence level
0.026
0.076
0.052 0.053
Floor A
0.019
0.865*
*
0.013
Floor B
0.700
*
0.623
Floor C
Floor number
3.57*
4.07*
2.46*
Floor D
Figure 9. Measured accelerations and calculated acceleration responses for the selected floors
using Eq.10, and for Floors C and D the peak responses were
determined by Eq.11. Results in Figure 9 demonstrate that for
all the four floor the measured acceleration responses are
close to, or slightly larger, than the predicted values with 75%
confidence level, but all are lower than the predicted values
for the 95% confidence level. Therefore, the proposed
response spectrum gives conservative result and can be used
at design stage to predict the peak 10s-RMS acceleration
responses due to a single person walking.
7
CONCLUSIONS
This study proposes a 10 second peak root-mean-square
acceleration spectrum for predicting a floor’s response subject
to a single person walking with given structural damping ratio
and response confidence level. Experimentally measured
footfall traces have been used in developing the response
spectrum. The suggested spectrum has three main parts The
first harmonic plateau, the second harmonic plateau and a
descending curve. Each part has a clear physical meaning and
corresponds to, respectively, the contribution of the first,
second and higher harmonics of the walking load. The
representative values for each plateau and the curve for the
descending part are statically determined using the
experimental data for various confidence levels, damping
ratios and structural frequencies. A piecewise mathematical
expression of the proposed spectrum is given along with
modification measures to consider the effect of the floor’s
span, boundary conditions, higher vibration modes, the
persons stride length and different vibration assessment
1109
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
indexes. The proposed spectrum has been used to calculate the
responses of four existing floors whose actual acceleration
responses to single person walking have been measured.
Comparison between the calculated responses and the field
measured responses proves the applicability of the proposed
spectrum-based approach.
ACKNOWLEDGMENTS
Joint financial support for this paper from the National
Science Foundation of China (NSFC, No. 51178338) and
Shanghai Natural Science Foundation (11ZR1439800) is
gratefully acknowledged.
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