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Procedia Structural Integrity 28 (2020) 287–294

1st Virtual European Conference on Fracture

1st Virtual European Conference on Fracture
Improvement of Staircases Vibration Serviceability to Human
Improvement of Staircases Vibration Serviceability to Human
Ergonomics: A Case Study
Ergonomics: A Case Study
Pedro Andradeaa, José Santosb,c,*, Lino Maiab,c
Pedro Andrade , José Santosb,c,*, Lino Maiab,c
a
University of Madeira, 9020-105 Funchal, Portugal
b a
University
University of Madeira, Faculty of Exact Sciences of Madeira,
and Engineering, 9020-105 Funchal,
Department Portugal and Geology, 9020-105 Funchal, Portugal
of Civil Engineering
b
University of Madeira, Faculty
c of Exact Sciences andFaculty
CONSTRUCT-LABEST, Engineering, Department
of Engineering of Civil
(FEUP), Engineering
University and Geology,
of Porto, Portugal9020-105 Funchal, Portugal
c
CONSTRUCT-LABEST, Faculty of Engineering (FEUP), University of Porto, Portugal

Abstract
Abstract
Contemporary, slender and lightweight monumental staircases are often highly susceptible to resonance phenomena, due to
Contemporary, slender andfrequencies,
typically low fundamental lightweightwhich
monumental staircasesamplify
can considerably are often
theirhighly susceptible
responses, to resonance
raising major phenomena,
serviceability dueand
problems to
typicallydiscomfort
causing low fundamental frequencies,
and unsafety which
concerns to can considerably
its users. amplify
This paper their responses,
presents a case study raising
of a major serviceability
low fundamental problemssteel
frequency and
causing
staircasediscomfort
with knownand highunsafety
levels ofconcerns
vibrationtosince
its users. This paper
the beginning presents
of its a case in
construction, study
whichof various
a low fundamental
improvementfrequency steel
solutions were
staircase
proposedwith known
in order high levels
to increase of vibration
its vibration since the beginning
serviceability. of its
In total, six construction,measures
improvement in whichwere
various improvement
proposed, solutions
being tested usingwere
the
proposed
Finite in order
Element to increase
(FE) softwareitsSAP2000.
vibration serviceability.
The initial FE In total, sixmodel
staircase improvement
was firstmeasures
calibrated were
withproposed, being tested
the vibrations using the
experimentally
Finite
measured Element
on the(FE)
real software
staircase.SAP2000.
Then, the The initial
original FEFE staircase
model model was
was modified first
with thecalibrated with themeasures
six improvement vibrationsand
experimentally
the resulting
measured
vibrations on thecompared
were real staircase. Then,initially
with those the original FE model
obtained and thewas modified
acceptable withsuggested
limits the six improvement
by the designmeasures
guide SCIand the resulting
P354, to verify
vibrations wereThe
their viability. compared with those
most efficient initiallyimprovements
numerical obtained and the wereacceptable
those thatlimits suggested
increased by the design
the staircase guide SCI
fundamental P354, tooff
frequency, verify
the
their viability.
range The most
of frequencies efficient
excitable numerical improvements
by pedestrians walking. were those that increased the staircase fundamental frequency, off the
range of frequencies excitable by pedestrians walking.
© 2020 The Authors. Published by ELSEVIER B.V.
© 2020 The Authors. Published by Elsevier B.V.
© 2020
This The Authors. Published by ELSEVIER B.V.
This isisan
an open
open access
access article
article under
under the BY-NC-ND
the CC CC BY-NC-ND licenselicense (https://creativecommons.org/licenses/by-nc-nd/4.0)
(https://creativecommons.org/licenses/by-nc-nd/4.0)
This is an
Peer-review open access
under article
responsibilityunder
of the CC BY-NC-ND
European license
Structural (https://creativecommons.org/licenses/by-nc-nd/4.0)
Integrity
Peer-review under responsibility of the European Structural Integrity Society Society (ESIS) ExCo
(ESIS) ExCo
Peer-review under responsibility of the European Structural Integrity Society (ESIS) ExCo
Keywords: Improvement Measures; Vibration Serveciability; Pedestrian’s Comfort; Resonance Phenomenon; Low Frequency Staircases.
Keywords: Improvement Measures; Vibration Serveciability; Pedestrian’s Comfort; Resonance Phenomenon; Low Frequency Staircases.

1. Introduction
1. Introduction
Nowadays, for aesthetic reasons, slender and lightweight monumental staircases are becoming major architectural
Nowadays,
features forbuildings,
of many aesthetic reasons, slender
hotels and otherand lightweight
public monumental
areas. Design staircases
requirements are becoming
for these major
are usually veryarchitectural
aggressive,
features of many buildings, hotels and other public areas. Design requirements for these are usually very aggressive,

* Corresponding author. Tel.: +351 291 705 197.

* Corresponding
E-mail address:author. Tel.: +351 291 705 197.
jmmns@fe.up.pt
E-mail address: jmmns@fe.up.pt
2452-3216 © 2020 The Authors. Published by ELSEVIER B.V.
2452-3216 © 2020
This is an open Thearticle
access Authors. Published
under by ELSEVIER
the CC BY-NC-ND B.V.(https://creativecommons.org/licenses/by-nc-nd/4.0)
license
This is an open
Peer-review access
under article under
responsibility CC BY-NC-ND
of the European license
Structural (https://creativecommons.org/licenses/by-nc-nd/4.0)
Integrity Society (ESIS) ExCo
Peer-review under responsibility of the European Structural Integrity Society (ESIS) ExCo
2452-3216 © 2020 The Authors. Published by Elsevier B.V.
This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0)
Peer-review under responsibility of the European Structural Integrity Society (ESIS) ExCo
10.1016/j.prostr.2020.10.034
288
2 Pedro et
P. Andrade Andrade et al. / Procedia
al. / Structural Integrity Structural
Procedia Integrity
00 (2019)28 (2020) 287–294
000–000

with long and unsupported spans being the standard, often resulting in flexible staircases that are highly susceptible
to human induced vibrations. In extreme cases, when the step frequency of pedestrians matches the staircase
fundamental frequency or is one of its submultiples, a resonance phenomenon can occur, which amplifies the
vibrations to a great extent and could cause discomfort and the feeling that the structure is not safe. In the traditional
design of steel staircases, loads caused by pedestrians were usually treated as static loads. However, when assessing
vibrations, taking into account only static loads, it will probably give rise to the conception of steel staircases subject
to an unsatisfactory dynamic behaviour. Although, there is a growing need to design staircases with human-structure
interaction in mind, in order to avoid excessive vibrations, scientific knowledge on this subject is still scarce. This
paper aims to study a low frequency steel staircase (i.e. susceptible to resonance effects), which had a well-known
level of liveness, since the beginning of its construction. In order to increase the vibration serviceability and,
consequently, reduce the discomfort felt by the occupants who walk it, several improvement measures are proposed.
All proposed solutions were numerically tested using the FE software SAP2000, but with the purpose of being
feasible employed in a real context. First, the vibrations on the steel staircase analysed in this study are experimentally
measured. A very realistic FE model of the steel staircase is created and then calibrated, so the vibrations numerically
calculated were close to those experimentally measured. After the FE model being calibrated, it is modified with the
various improvement measures and the vibrations are recalculated. In total, six different solutions are proposed that
could be applied to the actual staircase without changing the original structure, because it was intended to improve
the dynamic behaviour and not, by any means, demolish to rebuild again. In the end, the vibrations calculated
numerically through each improvement measure are compared with those initially obtained and with the serviceability
criteria proposed by the design guide SCI P354, in order to verify their effectiveness.

2. Experimental program

2.1. Staircase description

The steel staircase studied in this paper is located inside a building in Funchal, Madeira, Portugal and is known for
its vibration problems, causing discomfort and being the object of several adverse comments by its users. Hence, with
a clear need of an intervention to increase its serviceability.
The sample staircase is composed of four flight of steps, with identical geometry, which serve as a connection
between the three floors of the building, as represented in Fig. 1. Sample staircase: (a) complete drawing of project
(mm); (b) FE numerical model.a). The staircase is supported on each floor by a European wide flange beam HEB180,
which is connected to two hollow structural section (HSS) 120x60x4 mm stringers \that support the flight of steps, by
means of an 8 mm metal plate and an M 20x100 mm screw (area indicated with circles in Fig. 1a)). Due this solution,
rotational movement is possible, so the support could be assumed as pinned with the behaviour of the two upper flights
being independent of the two lower flights. The span between supports makes a total of 4.44 m. The stair steps have
a length of 1.15 m and a width of 0.32 m and are composed, as the intermediate landings, of a 3 mm thick metal plate
coated by a granite sheet stone of 30 mm thick.

2.2. Modal properties

A battery of experimental modal tests was conducted on the analysed staircase to determine its dynamic properties.
The natural frequencies and corresponding vibrations modes were obtained by applying multiple instant strikes along
the staircase and recording accelerations in free vibration near to the driving and other locations of interest, for
subsequent calculation using a subroutine developed in MATLAB. Table 1 shows the natural frequencies and shapes
of the vibrations modes experimentally measured. The half-power bandwidth method was applied to the free vibrations
of the staircase to estimate the damping coefficient. The damping was consistently estimated to be about 1.18 % of
critical, being in accordance with the authors Bishop et al. (1995), Davis et al. (2015; 2009) and González (2013),
who obtained in their measurements on steel staircases a value of approximately 1 %.
 Pedro Andrade et al. / Procedia Structural Integrity 28 (2020) 287–294 289
P. Andrade et al. / Structural Integrity Procedia 00 (2019) 000–000 3

2.3. Walking tests results

After measuring the modal properties, the vibrations to which the sample staircase was subjected were
experimentally measured, so later being able to verify the viability of the various improvement measures proposed.
Assuming that from the frequency of the 4th harmonic amplitude a resonant build-up cannot occur and that is possible
to walk staircases with step frequencies up to 4Hz (Kerr, 1998; Kerr and Bishop, 2001), according to Andrade et al.
(2017a) and Santos et al. (2019), the cut-off frequency for staircases should be considered equal to 16 Hz. Considering
that the fundamental frequency of the studied staircase is 13.9 Hz (see Table 1), this mean that descending and
ascending at 3.5 Hz (4th sub-multiple of the fundamental frequency) is a plausible scenario to originate resonant effects
and amplify its response. Therefore, various walking tests with this step frequency were performed to estimate the
staircase’s vibrations due ascents and descents, for a single pedestrian and a group of pedestrians. It was observed that
the maximum vibrations occurred for descents, reaching peak accelerations of approximately 2.0 m/s2 and 5.4 m/s2,
for a single pedestrian and a group of pedestrians, respectively.

 
a) b)

Fig. 1. Sample staircase: (a) complete drawing of project (mm); (b) FE numerical model.

Table 1 – Experimental and numerical vibration modes.

Modes Experimental Numerical

Nº Shape Frequency (Hz) Frequency (Hz)
1 Vertical 13.9 13.9
2 Vertical 14.5 14.9
3 Torsion 20.9 23.4
4 Torsion 21.9 26.5
5 Torsion 22.4 27.1
6 Torsion 23.2 29.3
290 Pedro Andrade et al. / Procedia Structural Integrity 28 (2020) 287–294
4 P. Andrade et al. / Structural Integrity Procedia 00 (2019) 000–000

3. Numerical Analysis

3.1. Dynamic properties

In order to numerically calculate the accelerations, a FE model of the studied staircase was created using the
structural analysis software SAP2000. For the purposes of the analysis, only the two upper flights of steps were
modelled, since their behaviour is independent of the two lower flights, as explain in Subsection 2.1. A
comprehensively detailed FE model of the staircase was created using shell elements for all the structural elements
described in Subsection 2.1, i.e. HSS stringers, HEB180 beams, metal plates and granite sheet coating of the stair
steps and intermediate landings, and, using beam elements for the non-structural elements, i.e. the guardrails. Fig. 1.
Sample staircase: (a) complete drawing of project (mm); (b) FE numerical model.b) represents the FE staircase model
built.
The natural frequencies and corresponding modes shapes of the FE model created were predicted using the standard
eigenvalue analysis option. Table 1 shows the comparison between the first six vibrations modes numerically
computed and experimentally measured. From the 2nd vibration mode, the numerical natural frequencies begin to
differ from the experimental natural frequencies. However, the frequencies and shapes of the first two modes were
accurately predicted, and the difference for the higher modes is not expected to significantly change the numerical
results. Moreover, it was possible to estimate with close approximation the vibrations modes within the frequency-
band where a resonant build-up is plausible to occur, i.e. lower or equal to 16 Hz (Santos et al.(2017a; 2019)).

3.2. Numerical results

For the improvement measures could be reliably applied in practice, it was necessary that the initial FE staircase
model was calibrated, so the numerical accelerations obtained were close to those experimentally measured. Currently,
there are four main existing numerical methods to predict human induced vibrations on low frequency staircases: i)
footfall force time histories (GRFs), ii) Fourier series walking models, iii) steady-state analysis and iv) simplified
vibration evaluation. An extensive number of analysis were performed employing the four different numerical
methods, and it was verified that applying footfall forces force time histories to the FE model, realistically simulated
the pedestrian’s walking on the actual staircase, being the most accurate procedure. Hence, this was the method used
throughout this work to calculate the accelerations numerically. To date, the most comprehensive work conducted to
measure footfall time histories directly on stairs was developed by Kerr (1998; 2001). This researcher obtained more
than 500 footfall traces from 25 individuals ascending and descending the stair at different step frequencies on an
instrumented stair with a force plate. Footfall traces obtained by Kerr (1998; 2001) for a descent with a step frequency
of 3.5 Hz were used to calculate the accelerations numerically. Simulations for a single pedestrian and a group of
pedestrians descending the FE staircase model were performed applying footfall traces at increments of 1/3.5 Hz, to
obtain the maximum accelerations in resonance and to be comparable with the experimental results (see Subsection
2.3).
The accelerations were calculated performing time histories analysis in SAP2000, being obtained for a single
pedestrian and a group of pedestrians, peak accelerations of approximately 2.1 m/s2 and 6.6 m/s2, respectively. This
is in agreement with the walking tests results observed in Subsection 2.3 and, therefore, validating the initial FE model
and numerical method used.

4. Application of the improvement measures

The maximum peak accelerations measured for a single pedestrian and a group of pedestrians, as seen in Subsection
2.3, were approximately 2.0 m/s2 and 5.4 m/s2, respectively, which are significantly higher than the acceptable limits
proposed by design guides and researchers (SCI P354 (2009)/Bishop et al. (1995), AISC 11 (1997) Davis et al. (2015;
2009), and Zhou et al. (2011)). Considering the high level of vibration that the studied steel staircase is subjected,
various improvement measures have been proposed in order to reduce it. The different proposed measures were tested
by modifying connections and/or adding structural elements to the original FE model and then recalculating the
accelerations, to compare with the initially obtained and the acceptable limits, i.e. verifying their effectiveness. It
 Pedro Andrade et al. / Procedia Structural Integrity 28 (2020) 287–294 291
P. Andrade et al. / Structural Integrity Procedia 00 (2019) 000–000 5

should be emphasized that the improvement measures aim to be employed in practice for increasing the serviceability
of the actual staircase, not being intended to alter or demolish the existing structure.
To be consistent with Subsection 3.2, after employing the proposed measures, accelerations were recalculated using
the same footfall traces obtained by Kerr (1998; 2001) for a descent at 3.5Hz.
In total, six improvement measures were tested and are presented below:
 Improvement measure 1 – Weld or Screw an additional stringer, in the longitudinal direction, between the two
existing stringers.
In the first improvement measure, a stringer with a commercial steel hollow structural section (HSS) 250x100 mm
was connected, in the longitudinal direction, between the two existing stringers, modelled by a beam element. The
cross-section of the added stringer is approximately twice the height of the two existing stringers (120x60 mm),
however it was the necessary cross-section for the accelerations to be under the acceptable limits.
 Improvement measure 2 – Connect a steel cable between the HEB180 beam and the intermediate landing.
In second improvement measure, a steel cable was added to connect the HEB180 beam (located in the floor area)
and the intermediate landing area, with the aim of reducing the flight of steps deflection and, therefore, decreasing the
vibrations. The cable was modelled by a beam element with a circular cross-section. However, this measure proved
to be ineffective, not significantly decreasing the accelerations.
 Improvement measure 3 – Weld beams at the flight of steps midspans, in the transverse direction, supported by
an added column.
In the third proposed measure, the placement of two additional beams at the flight of steps midspans,
perpendicularly to the existing stringers, was tested. These beams are supported by an added column located between
the flights of steps. The beams employed in this measure consisted of a steel hollow structural section (HSS) 150x100
mm. This cross-section giving rise to the lowest accelerations. From this cross-section, accelerations cease to decrease
significantly. The employed column must consist of a European wide flange beam HEB180 or HEB160, not being
possible higher cross-sections, due to the reduced spacing between the flights of steps.
 Improvement measure 4 – Add an intermediate column in the landing area, simulated by a fixed support.
In this measure, it was decided to add a column on the intermediate landing in order to reduce the length of the
staircase span, thus decreasing the vibrations. The column was simulated by a fixed support and not by a beam element,
as opposed to improvement measure 3. The fourth proposed measure did not nearly affected the first vibration mode
frequency, which also caused the accelerations to not substantially reduce. Both this and the second improvement
measure did not generate the expected results.
 Improvement measure 5 – Duplicate the height of the sample staircase stringers.
In the fifth proposed measure, HSS steel stringers with a cross-section of 120x60 mm were added to the initial FE
model, placed under the existing stringers supporting the flight of steps. HSS stringers with a cross-section of 120x60
mm were used for two reasons: first, the aesthetically visual impact is less with the placement of stringers coinciding
with the dimensions of the existing HSS stringers (120x60 mm) and, second, the accelerations values obtained
considering this cross-section are considerably lower. In order to further reduce the accelerations, it is necessary to
use HSS stringers with much higher cross-section dimensions, which would be hardly feasible in practice.
This reinforcement measure implies that the sample staircase should have been initially designed with HSS
stringers about twice the height, to avoid excessive vibrations.
 Improvement measure 6 – Eliminate the connecting rod between the flight of steps and the HEB180 beam,
making the connection rigid.
As described in Subsection 3.1, the flight of steps is supported on a HEB180 beam, with the connection
between both elements allowing rotation to occur, its behaviour being assimilated to a pinned support. Consequently,
it is suggested as an improvement measure to transform the current pinned support, referring to the connection, into a
fixed support, i.e. welding or screwing the stringers directly into the HEB180 beam. Hence, increasing the rotational
292 Pedro Andrade et al. / Procedia Structural Integrity 28 (2020) 287–294
6 P. Andrade et al. / Structural Integrity Procedia 00 (2019) 000–000

stiffness between the flight of steps and the HEB180 beam, and, decreasing the vibrations. Through this measure, the
accelerations decreased to approximately half of the initially obtained, however continuing to be relatively higher than
the acceptable limits, mainly for a group of pedestrians.
In Figs. 2a) to 2f) are represented the FE models after applying the six improvement measures proposed.

 
a) b) 
  

c) d)

 
e) f) 
  
Fig. 2. FE numerical model with the: (a) first improvement measure; (b) second improvement measure; (c) third improvement measure; (d)
fourth improvement measure; (e) fifth improvement measure; (f) sixth improvement measure.
 Pedro Andrade et al. / Procedia Structural Integrity 28 (2020) 287–294 293
P. Andrade et al. / Structural Integrity Procedia 00 (2019) 000–000 7

5. Comparison of the accelerations after the improvement measures with an acceptance criteria

The accelerations obtained after employing the six improvement measures were compared with the acceptable
limits proposed by the SCI P354 (2009), since this is the only design guide that directly refers to an acceptance criteria
for staircases. The limits proposed in SCI P354 (2009) are given by a frequency-weighted base curve multiplied with
factors of 32 and 64, respectively, for a single pedestrian and a group of pedestrians. As the fundamental frequencies
of the FE models change when applying the various improvement measures, the acceptable limits are different for
each proposed solution. Table 2 presents the peak accelerations initially obtained and after each proposed measure
compared with the peak limits given by SCI P354 (2009).

Table 2 – Comparison of the accelerations obtained after employing the several improvement measures with the SCI P354 (2009).

Peak acc. Peak Lim.

Office Building Staircase Acc. Verif.
[m/s2] [m/s2]
Initial 2.07 0.39 KO!
Imp. Meas. 1 0.37 0.63 OK!
Single Pedestrian

Imp. Meas. 2 1.82 0.39 KO!

Imp. Meas. 3 0.48 0.61 OK!
Imp. Meas. 4 2.32 0.42 KO!
Imp. Meas. 5 0.35 0.60 OK!
Imp. Meas. 6 1.19 0.44 KO!
Initial 6.62 0.79 KO!
Imp. Meas. 1 1.27 1.27 OK!
Group of Pedestrians

Imp. Meas. 2 5.82 0.79 KO!

Imp. Meas. 3 1.60 1.21 KO!
Imp. Meas. 4 7.03 0.83 KO!
Imp. Meas. 5 1.28 1.20 KO!
Imp. Meas. 6 3.85 0.88 KO!

As can be observed, measures 1, 3 and 5 gave rise to peak accelerations lower than the peak limits for a single
pedestrian. Of these, only measure 1 meets the proposed limit for a group of pedestrians. However, given the
significantly high accelerations initially obtained for a group of pedestrians, measures 3 and 5 resulted in peak
accelerations substantially lower, being only slightly higher than the acceptable limits. Improvement measures 2 and
4, as can be seen from Table 2, do not present peak accelerations relatively different from those initially calculated,
being the most ineffective solutions. From Table 2, it is also possible to verify that the peak accelerations obtained
with measure 6 are approximately half of the originally generated, but continuing to be higher than the peak limits,
mainly for group simulations.

6. Summary and conclusions

The low frequency steel staircase studied in this paper presents a well-known level of liveness, as demonstrated in
Subsection 2.3, thus various improvement measures being proposed to reduce its vibrations and pedestrians comfort.
With the aim of realistically apply the different measures on the actual staircase, the initial FE model was calibrated,
so the numerical accelerations obtained through footfall time histories were close to the experimental accelerations.
In total, six improvements measures were tested. Measures 2 and 4 were the less effective solutions to be employed
in practice, while measure 6 decreased to half the accelerations initially obtained, although still being relatively higher
than the peak limits from SCI P354 (2009). Measures 1, 3 and 5 were the most accurate, since reduced the accelerations
and increased the fundamental frequencies, placing it off the range of submultiples excitable by pedestrian’s step
frequencies, where a resonant-build is possible. These measures also being the more technically feasible on site. The
294 Pedro Andrade et al. / Procedia Structural Integrity 28 (2020) 287–294
8 P. Andrade et al. / Structural Integrity Procedia 00 (2019) 000–000

improvement measure 5 is of particular relevance, since it demonstrates that the staircase stringers should have been
initially designed with a HSS cross-section twice the height, in order to verify the serviceability limit state.
All proposed measures were developed with the possibility to be implemented in practice, without changing the
existing staircase, however, inevitably with additional costs which are not predict in the original design. Hence, future
designs of low frequency staircases should be perform considering the dynamic behaviour and employing different
numerical methods to estimate human induced vibrations, consequently, avoiding pedestrian’s unsafety and later
modifications to the structure after construction. This being already observed and a reported concern in previous works
developed by the authors (Andrade et al. (2017b) and Santos et al. (2019)).

Acknowledgements

This work was financially supported by: Base Funding - UIDB/04708/2020 of the CONSTRUCT - Instituto de
I&D em Estruturas e Construções - funded by national funds through the FCT/MCTES (PIDDAC).

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