1. sports
2. climbing

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
An in-situ case study of human induced vibrations on slender staircases
Christian Meinhardt1, Volkmar Zabel2, Hernando Gonzalez2
Gerb Vibration Control Systems, Germany, Roedernallee 174-176, 13407 Berlin, Germany
2
Institute of Structural Mechanics, Bauhaus-Universität Weimar, Marienstrasse 15, 99423 Weimar, Germany
email: [email protected], [email protected], [email protected]
1
ABSTRACT: Following the general trend in structural engineering, staircase structures tend to be more long-span, slender and
lightweight which leads to a higher susceptibility to human induced vibrations. Beside ensuring the comfort of the staircase
occupants, it also has to be ensured that the serviceability is maintained and that eventual damages, for example cracks at often
used glass panels that serve as guard rails. When predicting the resulting vibrations of staircases to assess the vibration
susceptibility, the application of load functions for footfall on plane surfaces does not lead to a good correlation with the values
that were experimentally obtained after the completion. To define an applicable load function for the use of stairways and to
define a critical frequency range for the vibration susceptibility, force plate experiments using 3D load cells have been
conducted and compared to existing experimental data. Furthermore the resulting load functions have been applied to a
numerical model of an existing structure and the results have been compared to experimental data from this structure. To
emphasize the relevance of the subject, experimental results from additional project examples and measures to reduce the
occurring vibrations will be introduced within this contribution.
KEY WORDS: Footfall Induced Vibrations, Load Function, Staircase Design
1
INTRODUCTION
Since slender and lightweight staircase designs become more
and more popular, the assessment of the vibration
susceptibility of these structures is a crucial part of the
structural analysis to avoid discomfort and limitations of the
serviceability. To determine the resulting staircase vibrations,
a realistic load function is required. In previous investigations
it was found, that footfall loading functions from persons
moving on even surfaces are not sufficiently describing the
process of ascending or descending stair cases (see [1] & [2]).
Also the critical frequency range for staircases differs from
the one for footbridges. Previous research into human loads
on staircases is limited, with few stair vibration research
studies. [1] & [2] are the most relevant to the vibration
analysis of slender staircases for which force plate
experiments have been conducted to derive vertical load
functions and to assess effects of group loading.
More recently, in [4] a thorough study on the footfall
characteristics of a large number of persons has been
investigated. Variations both in amplitude and step
frequencies were identified depending on whether a person
climbs the stairs stepping on each or on every second stair.
Furthermore a dependency of the load applied to the structure
on the geometry, in particular the inclination, of the stairs was
observed.
In most publications exclusively the vertical component of
the dynamic forces generated by persons ascending or
descending stairs were investigated. The horizontal
components in lateral and longitudinal direction were also
measured in the study described in [4] but not considered
further.
The limited information documented in literature, especially
with respect to horizontal components of the excitation forces,
motivated the study described in this contribution.
An experimental laboratory study was performed to identify
typical load histories with their components in vertical and
longitudinal direction. Furthermore, existing stairs were
investigated both by means of tests and numerical simulations.
Numerical simulations of the dynamic behavior of stairs are
often performed in the context with the design of measures to
reduce vibrations. Therefore, some considerations with
respect to practical applications of the simulation of human
induced vibrations of slender staircase structures are presented
here.
2
2.1
FORCE PLATE LABORATORY TESTS
Test Setup
It was intended to measure forces in vertical and longitudinal
direction that are generated by typical footfalls of different
persons ascending and descending stairs at different speed, or
with different step frequencies. For that purpose a staircase
was chosen on which a force plate was installed. The force
plate was built by placing two 3D force sensors of type Kistler
9251A between two steel plates applying a prestress
according to the technical specifications of the sensors. It was
placed at the middle step of a relatively stiff flight of stairs. To
ensure that the footfalls are not influenced by a change of
inclination of the stairs, wooden stairs with the height of the
force plate were placed at the two stairs above and below the
force plate, respectively, as shown in figure 1.
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Figure 1. Test Setup with raised stairs and force plate.
In total 4-channels were employed for every test. The force
sensor had to be mounted under prestress because the
horizontal shear forces Fx and Fy are transmitted by static
friction from the base and cover plate to the faces of the force
sensor. The preload of 25 kN was applied to each force sensor
prior to testing using a hydraulic press prior to tightening the
screws fixing the sensors in their position. The sensor is
designed to measure the three orthogonal components of
dynamic or quasi-static forces, in a range from -2.5 kN to
2.5 kN in the two horizontal directions, Fx and Fy, and from 0
to 30 kN in vertical direction, Fz. As signal conditioners for
the load cells, Brüel & Kjær charge amplifiers of Type 2635
were employed.
The data acquisition system was based on a National
Instruments DAQ-CARD 6062E and a National Instruments
BNC-2120 connector block. The acquisition software was
developed by means of the programming platform NILabVIEW.
2.2
Performed Tests
During the tests, the horizontal and vertical components of the
force exerted by a person ascending and descending a
staircase with different footfall rates within a range between
1.0 and 6.0 Hz. To obtain information about the deviations of
the forces generated by different individuals, in total 9
different persons were employed, who passed the test stand
with different step frequencies. In total, 42 ascending traces
and 41 descending traces were acquired. In order to be able to
compare the data obtained from different tests, each footfall
trace was normalized with respect to the respective person’s
body weight. The sampling rate for the data recording of the
performed tests was chosen to be 1024 Hz.
2.3
Data Processing and Assessment
For the simulation of human induced vibrations it is common
to create signals based on a series of appropriate Fourier
coefficients. If measurements of the dynamic forces are
available from tests, as described in the previous section, one
can obtain the Fourier coefficients by means of Fourier
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transformation or a Fourier series decomposition. The
procedure applied in this study followed the recommendations
given in [1].
Before conducting the Fourier analysis, the recorded footfall
trace has to be converted into a continuous time domain signal
(see [1]). To develop a continuous force time history from the
single foot fall trace, each trace was overlapped by the exact
period between heel strikes. To estimate the period of the
footfall trace, the signal offset of the continuous trace must be
equal to 1 since the subjects' footfall traces were normalized.
This gives the coefficient A0 of a Fourier analysis. Therefore,
if a starting period has been assumed, through iteration of the
period, eventually a period will be found that produces a value
of A0 equal to 1. This period is then the overlap period
between traces. Three copies of the trace between the start and
end points were inserted into an array with an offset between
each trace of 0.5 seconds or 2 Hz. This was the initial default
footfall rate assumed in the program that was developed for
this study.
Once overlapped, the three traces were superimposed at
each point to produce a continuous time history (see figure 2).
The repeating period shown in figure 2 was the trace to which
the A0 analysis was applied. The calculation was repeated
with the overlap between the three signals increased or
decreased until the value of A0 was within a range from 0.995
to 1.005. After completing the calculations, the period of the
overlap became the period of the continuous trace, which in
turn became an accurate measure of the person’s footfall rate.
Figure 2. Generation of a continuous time history from a
single footfall trace [1]
3
RESULTING LOAD FUNCTIONS AND FOURIER
COEFFICIENTS
As described in section 2.3 Fourier coefficients that can be
used to reconstruct a force signal for simulation purposes were
derived by application of a Fourier transformation to the
measured time series. In this section the results of these
analyses are summarized distinguishing between data
acquired from tests with persons ascending or descending the
stairs, respectively. Furthermore, the force components in
vertical and longitudinal direction were considered separately.
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Results – Footfall Time Records / Ascending
3.1
P(dynamic)/P(static)
Figures 3 and 4 show typical normailized force signals that
were measured in vertical and longitudinal direction,
respectively, while a person was ascending the stairs at
different step frequencies. On can very well identify two
phases of the footstep: the phase of landing and the pushing
phase. The vertical force components shows two peaks while
from the longitudinal component the direction in which the
force is acting can be derived from the sign.
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Running - 3.83 Hz
Walking - 2.29 Hz
Mixture - 3.70 Hz
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t [s]
Fig. 5.4 Typical Footfall Traces. Longitudinal Force. Ascending
Figure 3. Sample time history for vertical force component ascending.
The harmonic components for the traces given in fig. 5.4 are shown in table 5.1.
T able 5.1 T ypical har monic data for subject ascending a staircase. L ongitudinal For ce
T est
M ax. Normalized
f [H z] Force
12
2.29
1.33
74
3.7
1.17
86
3.83
1.46
1st
2nd
3rd
4th
H armonic H armonic H armonic H armonic
1.312
0.181
0.070
0.049
0.939
0.110
0.090
0.060
0.526
0.069
0.025
0.037
Figure 5. Fourier coefficients of vertical force component ascending.
Figure 5.5 shows all the harmonics components values as a function of the footfall trace,
normalized with respect to the subject weight. As it happened before, the harmonics are largely
spread depending on the footfall rate employed by the subject during the test, with a not clear
pattern and even a decrease of the mean value for the first harmonic component. The second
harmonic component is also scattered. For the higher harmonic data, the 3rd harmonic as a mean
value of approximately 0.06, and the 4th harmonic as a mean value of approximately 0.04,
Figure
Sample
time history
forwithlongitudinal
values
twice 4.
as big
as those previously
calculated
the 1D load cell.force
Still, as compoit was previous
- ascending.
noticed, the remaining harmonicsnent
are too
small to have any significant contribution to the
response.
The amplitudes of the force components in longitudinal
direction are in a range of up to approximately 10 per cent of
the vertical force components. This observation agrees with
the measured forces represented in [4].
28
Figure 6. Fourier coefficients of longitudinal force component
- ascending.
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
In Figure 5 the results of the Fourier analyses applied to the
measured force components in vertical direction are illustrated
by means of the coefficients that were identified for the first
three harmonic components in the case of ascending the stairs.
Since the data was generated from tests with several persons
the results show significant scatter. Nevertheless, one can
clearly see the decreasing trend with increasing step
frequency.
In contrast to the vertical component, the Fourier
coefficients calculated from the measured longitudinal force
components are not only significantly smaller but show also
an increasing trend with increasing step frequency as
indicated in Figure 6.
component tend to decrease while those of the longitudinal
component are rather increasing.
3.3
Discussion of the Results from the Load Tests
Generally it was observed that the Fourier coefficients of the
vertical force components are decreasing with increasing
footfall frequency while the longitudinal force components
show an opposite behavior. The scatter of the measured data is
still rather high due to the individual characteristics of the test
persons. Even though the number of samples is still rather low
to derive general conclusions the observations made in the
described tests suggest some trends. The confirmation and
quantification of these observations require further
investigations.
Results – Footfall Time Records / Descending
3.2
4
Similarly as for the case of ascending the stairs, the footfall
forces measured from a person descending the stairs also
show two phases. The vertical force components are acting in
both phases into the same direction, downwards, while the
directions of the longitudinal force components are again
changing during a single footfall. However, compared to the
process of ascending, the acting directions are now inverse.
The magnitudes of the forces are slightly higher than observed
for the process of ascending. Even if the magnitudes of the
longitudinal components are, as in the case of ascending the
stairs, relatively small, the values are also slightly higher in
relation to the vertical components, than those identified for
ascending.
NUMERICAL ANALYSIS AND EXPERIMENTAL
VERIFICATION
To verify the derived load functions a numerical analysis of
an existing staircase has been performed with the objective to
compare the results with data from an experimental analysis
of the staircase. Figure 9 shows the actual staircase structure
and the numerical model.
Figure 9. Investigated Staircase – left: in situ situation, right:
numerical model (SAP2000).
Figure 7. Typical time history for the vertical force
component - descending.
In a first step, a model updating has been done based on the
experimentally determined modal parameters of the staircase
(natural frequencies and corresponding mode shapes and
damping ratios). To obtain these parameters, an ambient
modal analysis has been performed at the staircase structure
and it was found that the lowest natural frequency with a
mode with vertical and longitudinal contribution was at 9.9
Hz (mode shape see Figure 10).
Figure 8. Typical time history for the longitudinal force
component - descending.
The trends of the Fourier coefficients are similar as
described for the case of ascending the stairs. With increasing
step frequency, the Fourier coefficients of the vertical
1030
Figure 10. Left: Relevant mode shape at 9.9 Hz Right:
stabilization diagram from the ambient modal analysis.
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
4.1
Assessment of Vibration Susceptibility
The experimental analysis showed, that the staircase displayed
relevant modes with a contribution in vertical and longitudinal
direction above the footfall frequency range so a resonancelike excitation of these modes caused by ascending or
descending persons on the staircase would occur at a 2nd, 3rd
or 4th sub-harmonic of the relevant natural frequency (similar
to the tests described in [1] & [2]). Also the structural
damping for these modes that has been determined was
comparatively high with a damping ratio of approx. D=3.0%.
So the resulting accelerations at the staircase were expected to
be relatively small.
4.2
content of the calculated accelerations consists of the footfall
components (3.3 Hz and higher harmonics) and the resonant
dynamic response of the staircase at the natural frequency.
Time History Analysis of Pedestrian Loading
A time domain analysis has been performed for which the
loading from a person descending the staircase either walking
(approx. 1.65 Hz) or running (3.3 Hz) has been simulated,
using the footfall traces for running and walking down the test
stand that where measured during the force plate experiments.
The footfall traces were transformed into a continuous time
series and the resulting Fourier coefficients were f1=0.57,
f2=0.24, f3= 0.14 and f4=0.12 for running down the stairways
resp. f1=0.273, f2=0.15 f3= 0.06 and f4=0.05 for walking down
the staircase (see Figure 11).
Figure 11. Measured footfall trace (left) and continuous time
history (right) used for the numerical simulation- top: for
running down the staircase (3.3 Hz), bottom: for walking
down the staircase (1.65 Hz)
Figure 12. Calculated dynamic response of the staircase at the
upper landing (time histories and corresponding mode shapes)
- top: for walking down the staircase (1.65 Hz), bottom: for
running down the staircase (3.3 Hz)
4.4
Measured Dynamic Response to one Person descending the staircase
To evaluate the numerical results, the occurring accelerations
at the stairs and landings have been recorded while a subject
was ascending and descending the staircase at a frequency of
1.65 Hz resp. 3.3 Hz. The footfall trace was directed with a
metronome. The weight of the subject was 83 kg (814 kN) in
both tests. The data was acquired using PCB piezoelectric
accelerometers of type 393A03 that recorded in lateral and
vertical directions.
Figure 13 shows the recorded
accelerations at the upper landing for ascending and
descending of one person with footfall rates analogue to those
of the numerical analysis.
In order to generate the time history as a moving load the
loads from one footfall trace was applied with specific arrival
times which corresponded to the step frequency and the
number of stairs to be walked.
4.3
Simulated Dynamic Response to one Person Descending the Staircase
Figure 12 shows the time histories and the corresponding
frequency spectra of the calculated accelerations at the upper
landing of the considered staircase for ascending with a
footfall frequency of 1.65 Hz and descending with 3.3 Hz. It
can be seen that ascending with a slower pace causes a
dominant dynamic response of the staircase in the relevant
vertical mode. The actual footfall frequency and high
harmonics contribute to the footfall excitation and cause a
distinct amplification in range of the natural frequency.
Descending the staircase with a higher footfall frequency
causes in general higher peak accelerations. The frequency
Figure 13. Calculated dynamic response of the staircase at the
upper landing (time histories and corresponding mode shapes)
- top: for walking down the staircase (1.65 Hz), bottom: for
running down the staircase (3.3 Hz)
Similar to the numerical results, the occurring accelerations
when descending the staircases are approximately two times
higher than for ascending. Also the footfall frequencies and
the higher harmonics of the footfall rate can be identified from
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
the signal and a distinct dynamic response in the determined
relevant vertical natural frequency can be observed.
4.5
Consideration of Group Loading
Observations made in [2] at underground stations showed that
as people filed onto and up (down) stairways they tend to
walk at the same footfall rate. Therefore, a group simulation
loading was done with four subjects with identical footfall
with a phase shift between them. The footfall rate employed
was 3.88 Hz, in order to compare with the results from the
single person case, and given the authors in [3] pointed out
“that the group effects reduces with the footfall trace and
became negligible at around 2.5 footfalls per second.” The
simulation consisted of a subject starting to descend the
staircase at the chosen footfall rate, then when the subject was
between the third and fourth step or approximately 0.60 s
later, the second started descending as well, then at 1.30 s a
third subject and at eventually 1.85 s after the first one. The
time history trace for the group acting on the staircase can be
seen in Figure 14.
walking at the staircase than estimated for the numerical
analysis.
5
ADDITIONAL PROJECT EXAMPLES
To emphasize the relevance of assessing the vibration
susceptibility of staircases, results of dynamic tests from a
livelier staircase will be presented. The measured occurring
vibrations for different load scenarios will be compared with
estimations using the Fourier coefficients derived above. The
first example is a 21 m long staircase with two landings at
mid-span (see Fig. 16). The staircase has been reported to
cause discomfort especially when descending the structure. A
modal analysis of the staircase revealed, that the 1st vertical
bending mode of the staircase was at a natural frequency of
3.0 Hz (see Fig. 16, bottom). So a resonance like excitation
with a first and second harmonic of the footfall frequency for
ascending/descending people was very likely. In addition, the
structural damping of the structure was relatively low (D=0.8
– 1.1 %). To assess the vibration susceptibility of the
staircase, the previously described scenarios – walking down,
running down, walking up and running up were investigated.
For the loading by 1 person (weight: 90 kg) the occurring
accelerations at the landings have been recorded. For walking
a footfall frequency of 1.5 Hz (90 bpm) was attempted –
although descending with this footfall frequency is very
unlikely and not comfortable while ascending with such a
footfall rate is very likely. For running, a footfall rate of 180
bpm (3.0Hz) was attempted.
Fig. 14. Example for a continuous trace for group loading descending stairs at 3.3 Hz (running)
Figure 15. Top: calculated dynamic response of the staircase
for group loading (time history and corresponding frequency
spectrum) – bottom: measured dynamic response
The calculated accelerations for the upper landing are
shown in Fig.15, the peak accelerations are by factor 2 higher
than for a single person. This enhancement correlates with the
enhancement factors presented in [2] (Probability Distribution
0.7). The experimental results display a slightly higher peak
acceleration and a more distinct dynamic response in range of
the relevant vibration mode than the numerical results,
probably due to a higher synchronization of the people
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Figure 16. Top: Tested 21 m long lively staircase – bottom:
Relevant Vibration Mode at 3.0 Hz.
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
The recorded time histories and the corresponding
frequency spectra are shown in Figure 17. Since measures had
to be taken to reduce the occurring accelerations, a feasibility
study to assess how much reduction could be achieved by
implementing Tuned Mass Dampers to the staircase was
performed. For this analysis a realistic loading should be used
to ensure that certain comfort levels for the staircase
occupants could be maintained and to avoid any damages at
the glass-handrails. In order to do so, a numerical model has
been created based on the determined modal parameters. In
addition, footfall time histories with footfall frequencies of 1.5
Hz and 3.0 Hz were applied to the model. The resulting time
histories are shown in Fig. 17.
0.2
0.06
0.04
[g]
[g]
0.1
0
0.02
-0.1
-0.2
0
5
10
15
Time[s]
20
25
0
30
0.2
0
5
10
15
20
Frequency [Hz]
25
30
0
5
10
15
20
Frequency [Hz]
25
30
0.06
0.1
[g]
[g]
0.04
0
0.02
-0.1
-0.2
0
10
20
30
Time[s]
40
0
50
Figure 17. Top: calculated dynamic response of the staircase
for one person descending (time history and corresponding
frequency spectrum) – bottom: measured dynamic response
-3
0.04
6
4
[g]
[g]
0.02
0
2
-0.02
-0.04
x 10
2
4
6
8
Time[s]
10
0
12
0
5
10
15
20
Frequency [Hz]
25
ascending at a second sub-harmonic footfall rate of
approximately 1.5 Hz).
The feasibility study showed that the occurring
accelerations could be reduced by 70-80 % with the
implementation of two Tuned Mass Damper Systems each
with an effective mass of 500 kg.
6
SUMMARY
To specify load functions of humans ascending and
descending staircases for a more precise prediction of the
occurring accelerations, force plate tests have been performed
to quantify the resulting loading in vertical and horizontal
direction. The test data was processed such that typical
footfall traces could be derived and Fourier coefficients for
ascending and descending were obtained for a certain range of
footfall frequencies. It was observed that the Fourier
coefficients of the vertical force components are decreasing
with increasing footfall frequency while the longitudinal force
components show an opposite behavior. The determined
maximum vertical forces compared to the static load as well
as the frequency bands for ascending and descending confirm
previous investigations [1].
To verify the established Fourier coefficients, a numerical
analysis of a sample staircase has been performed using the
determined values. The results were compared with data from
experimental tests and showed a good correlation with respect
to the peak acceleration and the frequency content.
To assess the effectiveness of passive control devices (Tuned
Mass Damper Systems) a similar analysis has been performed
for a lively staircase structure, which displayed high
acceleration levels. The use of a realistic load function
allowed an approximate prediction of the accelerations with
applied TMD systems and an optimization with regards to the
required effective mass and other TMD parameters.
30
0.04
4
0.02
3
[g]
[g]
-3
0
1
-0.04
0
20
22
Time[s]
24
26
REFERENCES
[1]
2
-0.02
18
x 10
[2]
0
5
10
15
20
Frequency [Hz]
25
30
Figure 18. Top: calculated dynamic response of the staircase
for one person descending (time history and corresponding
frequency spectrum) with activated TMD – bottom: measured
dynamic response
In addition, a simplified approach can be used- calculating
the deflection/acceleration of a SDOF system, which
represents the mode at 3.0 Hz and the corresponding
calculated modal mass of 12 tons, due to a dynamic loading
that results from a Fourier coefficient multiplied by a person’s
weight and the amplification that depends on the structural
damping. It could be seen that the resulting peak accelerations
correlate very well with the measured values when using the
previously introduced Fourier coefficients (0.6 - 0.7 for
descending at a footfall rate of approximately 3.0 Hz / 0.5 for
[3]
[4]
[5]
Kerr, S. C. and Bishop, N. W. M.: Human induced loading on flexible
staircases, Eng. Structures, 23, 37–45. 2001.
Kerr, S. C.: Human induced loading on staircases, Ph.D. thesis, Univ. of
London, London. 1998.
Bishop, N. W. M., Willford, M., and Pumphrey, R.: Human induced
loading of flexible staircases. Safety Sci., 18, 261–276. 1995.
Kasperski, M. and Czwilka, B.: A refined model for human induced
loads on stairs, in Topics on the Dynamics of Civil Structures,
Proceedings of the Int. Modal Analysis Conference – IMAC XXX, pp.
27-39, 2012.
Gonzalez, H.: Numerical simulation of human induced vibrations of
stairs, Master Thesis – Bauhaus Universität Weimar, Institute of
Structural Mechanics, 2013.
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