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ABSTRACT: This paper investigates the vibration serviceability of a steel half turn staircase in a new office building. Based on
a finite element model, the vibration serviceability is assessed under moving load conditions simulating walking or running of a
single person ascending or descending the stairs. For reference purposes also simulations with stationary loading conditions are
performed. The predicted acceleration levels are evaluated using the response factor R, corresponding to the multiplier of the
base perception curve of BS6472 for vertical vibration. Response factors significantly higher than 32 are predicted which is the
criterion for single person excitation proposed by Bishop et al. to ensure negligible adverse comment for the case of light use in
an office environment. To validate the predicted results, measurements are carried out to identify the modal parameters of the
staircase and to evaluate the vibration levels under different conditions of usage.
KEY WORDS: Human induced vibrations; Vibration serviceability; Staircases.
1 INTRODUCTION
The design of slender and lightweight staircases leads to a
low stiffness to mass ratio resulting in structures prone to
human induced vibrations. Because of this trend, the natural
frequencies of structures tend to be in the range of human
induced loading frequencies. The phenomena of resonance
can cause high acceleration levels in these structures, even
under normal circumstances. This can cause comfort problems
which contributes to a feeling of insecurity or can lead to
damage in the worst case scenario.
An increased awareness to these problems during the design
stage can prevent these problems and small interventions can
make a difference to the vibration serviceability of structures
under human induced vibrations.
To assess the vibration serviceability, the acceleration
response needs to be calculated under different simulations
using a moving load, simulating walking or running of a
single person ascending or descending the stairs, and also
simulations with stationary loading are performed.
Kerr and Bishop [1,2] carried out numerous force plate
experiments to describe the vertical force in function of time
for one person walking on a staircase. According to these
measurements, the stationary load that consists of impact
footfalls can be described as a Fourier series with two
harmonics. The force-time history of a walking force can also
be described with a Fourier series as a stationary load, or as a
sequence of footsteps [3].
The vibration levels can be predicted using these force
models, simulating loading scenarios on a finite element
model. The load cases include a realistic scenario of walking
or running up- and downstairs of the staircase with one
person. Secondly, the vibration levels for a stationary load on
a vibration sensitive spot on the structure are investigated as a
reference.
The ISO Standard for serviceability of buildings and
walkways against vibrations [5] provides an evaluation of the
maximum vibration levels for staircases using the response
factor R. This rating number will be used to define a well-
founded judgment about the vibration serviceability of the
staircase in this outline of the paper.
2 DESCRIPTION OF THE STAIRCASE
Figure 1: Side view staircase
The atrium staircase of a new office building consists
essentially of steel tube profiles and will reach through three
floor levels, with a cantilever landing in between. This steel
half turn staircase consists of two flights of nine or ten steps
each, connected by a cantilever landing and bolted to the
concrete upper and lower floor. A side view of one module of
the staircase is shown in figure 1. The U-shaped steps and the
platforms between the floor levels contain a concrete infill.
Glass panels will be used between the handrails and the
stringers.
Vibration serviceability assessment of a staircase based on moving load simulations
and measurements
Charlotte Schauvliege1, Pieter Verbeke
2, Peter Van den Broeck
1,2, Guido De Roeck
2
1 KU Leuven @ KAHO, Department of Civil Eng., Technology Cluster Construction – Structural Mechanics
Gebroeders De Smetstraat 1, B-9000 Ghent, Belgium 2 KU Leuven, Department of Civil Eng., Structural Mechanics Division, Kasteelpark Arenberg 40, B-3001 Heverlee, Belgium
email: [email protected], [email protected], [email protected],
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014Porto, Portugal, 30 June - 2 July 2014
A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)ISSN: 2311-9020; ISBN: 978-972-752-165-4
1043
3 CALIBRATION OF THE FINITE ELEMENT MODEL
This section describes how the finite element is calibrated in
two steps, corresponding to the two construction stages in
which the modal parameters were experimentally identified.
First, the details of the operation modal analysis are
summarised. Secondly, the correspondence between the
measured and calculated modal parameters are investigated
for the initial and calibrated finite element model. This will be
done for both construction stages: (a) bare steel structure
without concrete infill on the steps and landing nor handrail,
(b) composite structure without handrail.
3.1 Operational modal analysis
In both construction stages an identical measurement setup
was used, using 12 tri-axial accelerometer sensors in a single
configuration as presented in figure 2. Since the mode shapes
of the initial finite element model indicated an important
movement of the landing of the staircase, it was equipped with
8 sensors. The monitoring of the supports was performed by
placing the sensor on the first/last step of the stairs.
Figure 2: Measurement setup
The output-only data was processed using the reference-based
covariance-driven stochastic subspace identification (SSI-cov)
[6]. The modal assurance criterion (MAC) is used to identify
matching modes [7] .
The measured mode and the calculated mode can be matched
if their MAC value is close to 1. An important note is that the
mass due to the weight of the sensors is taken into account in
the finite element model to approximate the measured
situation as good as possible.
3.2 Calibration method
To calibrate the finite element model, parameters are
calibrated by minimising a cost function that measures the
discrepancy between measured and computed data. For this
objective, a least squares cost function is used without
regularisation, as described by Van Nimmen et al [8]. Each
construction stage will be calibrated separately in order to
start with a calibrated basic model before moving on to a more
advanced building stage. The accuracy with which the modal
parameters can be determined are taken into account with
weight factors. During the calibration process, minimising the
discrepancy between the measured and calculated frequencies
will be considered as thousand times more important than the
discrepancy between mode shapes, because natural
frequencies can be determined more accurate than mode
shapes.
3.3 Calibration in construction stage 1: bare steel
structure
The initial finite element model representing the bare steel
structure was constructed using beam elements for the stair
steps and the main structure. The steel plate which is fixed to
the bottom of the beams of the landing was modeled by shell
elements. Each of the flight ends has two pinned supports
(with fixed translations and free rotations), representing the
bolted connection to the upper and lower concrete floor.
Figure 3: 3D representation of the staircase with boundary
conditions.
For the measurements in this construction stage, it has to be
noted that since the welded wire mesh was already fixed for
the reinforcement of the concrete, bolts were needed to lift the
sensors with the aim of correctly registering the vibrations of
the main structure of the staircase. The agreement between the
initial finite element model and the measurements is presented
in table 1. Only the finite element mode shapes that match
with the measurements are represented. The first measured
mode shape is a rigid body mode, which involves the
movement of the building, while the others pertain to the stair
construction itself.
Table 1: Comparison between the measured, initial and
updated modal parameters of the bare steel structure including
mass of the sensors.
Measured Initial Calibrated
f,s ξ,s f MAC Δf f MAC Δf
[Hz] [%] [Hz] [-] [%] [Hz] [-] [%]
4.53 4.75
6.86 1.96
7.61 0.53 7.88 0.969 3.5 7.61 0.968 0.0
10.44 0.41
11.36 1.44
12.33 0.45 12.50 0.870 1.4 12.33 0.871 0.0
15.78 0.44 16.42 0.900 4.1 15.78 0.897 0.0
16.23 0.63
For the calibration of this first model the support stiffnesses
are considered as updating variables. In the finite element
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
1044
model, each support is modelled with three translational
springs and three rotational springs. To reduce the complexity
of the calculations, characteristics of the springs are assumed
to be identical in the four supports.
From table 1 it is clear that due to the calibration process the
natural frequencies derived from the measurements and finite
element models correspond almost perfectly, while the effect
on the mode shapes is less noticeable.
3.4 Calibration in construction stage 2: composite
structure
In this construction stage the U-shaped steps and the landing
contain a concrete infill. The calibrated finite element model
from the previous construction stage is therefore extended
with shell elements to model the concrete at the landing and a
modified mass density to model the mass effect of the
concrete infill of the steps.
Table 2 shows the experimentally identified modal parameters
together with the results of the modified finite element model,
indicated as ‘initial’ for this construction stage. The addition
of the concrete to the cantilever landing and stair causes a
reduction of the natural frequencies due to the mass of the
concrete. Besides this fact, the concrete has an increasing
effect on the damping of the modes.
Table 2: Comparison between the measured, initial and
updated modal parameters of the composite structure.
Measured Initial Calibrated
f,c ξ,c f MAC Δf f MAC Δf
[Hz] [%] [Hz] [-] [%] [Hz] [-] [%]
4.58 2.44
6.72 1.47 6.42 0.955 -4.5 6.71 0.951 -0.1
8.85 2.49
11.13 0.77 10.72 0.941 -3.7 11.20 0.931 0.6
14.28 0.99 13.63 0.949 -4.6 14.13 0.948 -1.1
20.99 0.98 20.83 0.877 -0.8 21.17 0.883 0.9
For this calibration step, the modulus of elasticity and the
mass density of the concrete are assumed to be updating
parameters. The initial and calibrated values of these
parameters are summarised in table 3. Again the mode shapes
that could be matched with measured mode shapes are taken
into account for this updating process. The natural frequencies
could be calibrated up to an accuracy of approximately 1%.
Table 3: Uncertain parameters during the calibration process
ρconcrete [kg/m³] Econcrete [N/m²]
Initial value 2.50 x 10³ 3.80 x 1010
Calibrated value 1.71 x 10³ 5.21 x 1010
4 VIBRATION SERVICEABILITY BASED ON
SIMULATIONS
To assess the vibration serviceability of the staircase,
numerous loading scenarios are tested by simulating the
acceleration response under a time varying walking or running
force along a predefined path on the structure. Additionally,
the response under stationary loading conditions is calculated
as a worst case reference. The acceleration response will be
compared to the vibration comfort levels defined by the
response factor R, corresponding to the multiplier of the base
perception curve of BS6472 for vertical vibration. If
necessary, vibration mitigation will be applied and evaluated.
4.1 Modal parameters of the finished staircase
The mode shapes and the natural frequencies of the finished
staircase are calculated starting from the calibrated finite
element model in construction stage 2 with the effect of the
handrail and the glass panels modeled as mass. The damping
ratios are taken from the measurements in construction stage
2. Table 4 summarises the calculated modal parameters used
for the numerical simulations. For the finished staircase no
measurement data are yet available.
Table 4: Modal parameters of the finished staircase
Mode shapes
3D view Front view Top view
Mode 1: f1 = 5.98 Hz ξ1 = 0.0147 = ξ,c,2
Mode 2: f2 = 9.92 Hz ξ2 = 0.0077 = ξ,c,4
Mode 3: f3 = 12.44 Hz ξ3 = 0.0099 = ξ,c,5
Mode 4: f4 = 18.66 Hz ξ4 = 0.0098 = ξ,c,6
4.2 Human induced loading on the staircase
The first step in the simulation process consists of defining
the load that will be used for simulations. Kerr and Bishop
[1,2] showed that the observed human induced forces on stair
steps are highly dependent on the pace, which will differ as
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
1045
the person is running up- or downstairs. Human induced
forces on stair steps consist of a bigger amplitude and
frequency-content than normal walking forces.
In case of a slow walking velocity, the person lands on the
sole of the foot after which the heel strikes the tread, followed
by a peak in the force diagram caused by the tip of the foot in
order to prepare for the next step. While in the case of walking
with a high velocity only the tip of the foot touches the tread
which will cause an impulse load.
Both load cases are represented during the simulations. For
the description of the walking or running force could be
referred to the definition of a single footstep [3] or a stationary
load [1,4] as defined in the literature.
First of all, the stationary load of a walking and running
force can be written as a Fourier series consisting of two
(running), or four or five (walking) harmonic components.
The properties of the force depend on the step frequency, the
weight of the person and the load type. Equation 1 describes
the Fourier series, while figure 4 and 5 show an example of
the stationary walking and running force as a function of time.
max
1
)2sin()(
h
h
hshe thfGGtF (1)
Figure 4: Stationary walking force (G = 700 N, fs = 1.99 Hz)
as a function of time, together with the components of the
Fourier series [4]
Figure 5: Stationary running force (G = 700 N, fs = 2.99 Hz)
as a function of time, together with the components of the
Fourier series [1,5].
Secondly, to simulate a moving and time varying force
along a realistic walking path on the finite element model,
defining a walking and running force of a single footstep is
required, since each force will be allocated to a footstep
location.
Li et al [3] derived the single step walking force from the
continuous walking force [4], and described this force by the
Fourier series of equation 2. An example of the single step
walking force is shown in figure 6.
e
h ene Ttt
T
nAGtF
0,sin)(
5
1
(2)
Figure 6: Single step walking force (G = 700 N, fs = 1.99 Hz),
according to Li et al [3].
The single step running force is derived from the stationary
running force by considering only the positive impulse force
downwards, as shown in figure 7.
Figure 7: Single step running force (G = 700 N, fs = 2.99 Hz).
4.3 Simulation and results
An estimation of the expected vibration levels in real
conditions can be obtained by simulating the response due to
one person descending or ascending the staircase. Both
combinations of ascending or descending and walking or
running are discussed during this case study. The step
frequencies will assume a critical value, which means that one
of the harmonic components will be equal to a natural
frequency of the staircase. This resonance condition induces
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
1046
the highest acceleration levels. In the case of a stationary
walking force the range of step frequency varies from: 1.2 Hz
to 2.5 Hz and contains four harmonic components. This means
that the load frequency, which is a multiple of the step
frequency varies from 1.2 Hz to 10 Hz.
A stationary running force was described in the literature by
using two harmonic components with a range of step
frequencies between 1.2 Hz and 4.5 Hz corresponding to a
load frequency up to 9 Hz. The first two natural frequencies of
the staircase are within one of these critical ranges. Table 5
gives an overview of the step frequencies of interest.
Table 5: Critical step frequencies for simulations.
fj [Hz] fj/1 [Hz] fj/2 [Hz] fj/3 [Hz] fj/4 [Hz]
5.98 5.98 2.99 1.99 (1.50)
9.92 9.92 4.96 3.31 2.48
From table 5, three critical load scenarios are defined: walking
at fwalk =1.99 Hz or 2.48 Hz or running at frun = 2.99 Hz. The
walking path and the corresponding numbers of the steps that
were used during the simulations are graphically displayed in
figure 7. It takes into account a realistic scenario of a person
moving up- or downstairs. A closer look will be taken to two
extra points (27,28) at the corners of the platform, because of
the sensitivity of these cantilever positions.
Figure 7: Load path used for the simulations and reference
points 27 and 28 at the corners of the landing.
Figure 8 and 9 show the vertical and horizontal vibration
response at three different positions of the staircase while
running and walking downstairs at a critical step frequency.
The positions are situated in the middle of the upper flight,
in the middle and at one of the corner points of the cantilever
platform. From the time histories of the accelerations, derived
from the simulations at these step frequencies, the building up
of resonance is obvious. As a result, the calculated maximum
acceleration strongly depends on the damping ratio, which in
these simulations corresponds to construction stage 2, without
the handrails and the glass panels installed.
The running force causes overall a larger acceleration
response on the structure, which can be explained by the high
dynamic load factor of this type of force. The landing is also
more sensitive to vibrations than the stair flights, due to the
higher modal displacements at that area, causing higher
acceleration levels. The most critical points are situated at the
corners of the landing, which is not directly situated on the
walking path.
a)
b)
c)
Figure 8: Vertical acceleration at position (a) 6 – middle of the
upper flight, (b) 12 – middle of the landing and (c) 27 – corner
point landing, while running downstairs with frun = 2.99 Hz
and walking downstairs with fwalk = 1.99 Hz (left and right
column respectively).
a)
b)
c)
Figure 9: Horizontal acceleration along the length of the
landing at position (a) 6 – middle of the upper flight, (b) 12 –
middle of the landing and (c) 27 – corner point landing, while
running downstairs with frun = 2.99 Hz and walking
downstairs with fwalk = 1.99 Hz (left and right column
respectively).
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
1047
From figure 9 can be concluded that not only the vertical
accelerations, but also the horizontal accelerations reach a
significant high value.
4.4 Evaluation of the vibration serviceability
The vibration levels in the structure will be compared with
the user comfort criteria.
Vertical direction
In this case study the vertical acceleration levels are
evaluated using the response factor R, corresponding to the
multiplier of the base perception curve of BS6472 for vertical
vibration, as shown in figure 10. Since the sensitivity of
humans will depend on the frequency of the vibrations, not
every acceleration level will be rated in the same way.
Figure 10: Perceptible acceleration level in the vertical
direction, corresponding to R = 1.
Since acceleration levels in resonance conditions are
investigated, üR=1 will be evaluated corresponding to the
relevant natural frequency.
Figure 11 shows the response factor in the points 1-26 of the
load path (as defined in figure 7) and in points 27 and 28,
situated at the corners of the landing, for a single person
running downstairs with frun,downstairs=2.99 Hz. From these
results, it is clear that the highest vibration levels are
encountered at the landing with the maxima situated at the
corner points.
Also shown in figure 12 as a reference, is the response factor
due to a stationary running load at the most sensitive corner
point 27. From the difference in acceleration level caused by
this stationary load and the moving load of the simulations, it
can be concluded that resonance is not fully built up during
the passage of a person.
An overview of the maxima, situated at the corner points of
the landing, for a single person walking or running upstairs or
downstairs at the critical step frequencies is shown in figure
12.
It is clear that the running force causes higher accelerations
than the walking force because of its larger force amplitudes.
Also there is no significant difference in the vibration
response between ascending and descending the staircase.
Figure 11: Response factor R along the walking path (1-26)
and the corner points of the landing (27-28) for
frun,downstairs=2.99 Hz, plus the response factor at the corner
points for fstationary =2.99 Hz at the critical corner point on
position 27 and criteria Rmax, light use and Rmax, heavy use according
to Bishop et al [2].
Figure 12: Maximum global response factor R in the vertical
direction and criteria Rmax, light use and Rmax, heavy use according to
Bishop et al [2].
An assessment of the vibration serviceability can be carried
out by comparing the calculated response factors with the
comfort criteria proposed by Bishop et al [2]. A distinction is
made between frequently used staircases (e.g. public
buildings, stadia), for which a maximum response factor
Rmax, heavy use of 24 is proposed, and less intensively used
staircases (e.g. offices), for which a response factor Rmax, light
use of 32 is proposed. These comfort criteria are proposed to
ensure negligible adverse comment and are also shown in
figures 11 and 12. It can be concluded that the vibration levels
of the staircase, as predicted by the simulations, do not meet
these comfort criteria both for walking and running
conditions. More specifically, the landing is very sensitive to
the human induced vibrations which may be important since it
can be used by people pausing to look around or allowing
others to pass.
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
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Horizontal direction
A similar procedure could be applied for the evaluation of
the horizontal vibration levels, but to the author’s knowledge
no criteria specific for stairs are available yet in literature.
Since the sensitivity of humans is higher for horizontal
vibrations, a more negative evaluation is expected. A future
measurement campaign will validate the simulations. The
progress of the human induced vibration levels during several
load cases will be registered and analyzed to get more
information about the accelerations of the structure in real
situations.
5 CONCLUSIONS
The vibration serviceability of a staircase was investigated
based on simulations of a single person walking or running
downstairs or upstairs.
A calibrated finite element model of the staircase in
construction stage was developed in two steps corresponding
to the two construction stages in which the modal parameters
were experimentally identified. For the modeling of the
finished staircase, as used in the simulations, the calibrated
finite element model in construction stage 2 was modified by
adding mass to account for the effect of the handrail and the
glass panels, which were not yet installed. The damping ratios
were assumed to correspond to the experimentally identified
values in this construction stage.
Critical load scenarios were selected by identifying critical
step frequencies giving raise to resonance when the loading
frequency, as a multiple of the step frequency, corresponds to
a natural frequency of the staircase. With two natural
frequencies below 10 Hz for the investigated staircase, two
critical loading scenarios corresponded to walking and one
critical loading scenario corresponded to running. Due to the
high impulsive loading of the running, this last loading
condition caused the highest vibration levels both in the
vertical and the horizontal direction, especially at the
cantilever landing.
Based on the simulations, the vibration serviceability
assessment shows that the response factors in the vertical
direction are higher than the tentative comfort criteria for
single person excitation as proposed by Bishop et al. in 1995
[2]. For the horizontal vibration levels, no comfort criteria are
available.
In the near future, a measurement campaign will be carried
out on the finished staircase. The first objective of these
measurements is to identify the modal parameters. Especially
the effect of the glass panels on the damping ratios is of great
interest since damping is governing the vibration levels at
resonance conditions. The second objective is to measure the
vertical and horizontal acceleration levels under various
loading conditions. Based on these measurements, a validation
of the simulations as presented in this paper will be carried
out. Furthermore, these measurements will determine whether
vibration mitigation measures are necessary.
ACKNOWLEDGMENTS
The results of this paper were partly obtained within the
framework of the research project, OPTRIS ‘Optimization of
structures prone to vibrations’, financed by the Flemish
government (IWT, agency for Innovation by Science and
Technology). The authors would like to thank the engineering
offices and parties concerned for their cooperation and
provided information on the investigated staircase.
REFERENCES
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[3] Q. Li, J. Fan, J. Nie, Q. Li and Y. Chen. Crowd-induced random
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[4] M. R. Willford and P. Young. A Design guide for footfall induced
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