Proceedings of the IMAC-XXVII
February 9-12, 2009 Orlando, Florida USA
©2009 Society for Experimental Mechanics Inc.
Design Spectra for Single Person Loading Scenario on Footbridges
Keith Wan1, Stana Živanović2, Aleksandar Pavic3
Department of Civil and Structural Engineering, The University of Sheffield
Sir Frederick Mappin Building, Mappin Street, Sheffield, S1 3JD, UK
NOMENCLATURE
amax
Maximum modal acceleration
L
Length of a footbridge
aspec
Spectral acceleration (determined directly from
design spectra)
ls
Step length
alim
Acceleration limit
m
Modal mass
DLF
Dynamic load factor
P
Probability
f0
Fundamental natural frequency
vp
Speed of a pedestrian
fn
Natural frequency
W
Weight of a pedestrian
fs
Step frequency
ζ
Modal damping ratio
km
Mass correction factor
ABSTRACT
Slender structural forms are common in modern footbridge design. This slenderness has increased the potential
for excessive vibrations in as-built structures. Popular design codes of practice, such as British Standard BS 5400,
no longer provide adequate guidance for designers assessing vibration serviceability of footbridges. The method
used in BS 5400 is based on an out-dated deterministic procedure and often produces results that are not
consistent with experimentally measured vibration responses. Its main drawback is that it does not account for the
intrinsic differences within a real pedestrian population. Therefore, a natural way forward is to include, now well
documented, inter- and intra-subject variability in the walking force modelling and response estimation. This paper
utilises a recently developed probabilistic procedure for estimation of vibration response to single person loading
scenario to develop a series of design spectra that could be used in vibration serviceability design as an
alternative to BS 5400. The design spectra are constructed for different probability of exceedance of various
vibration levels and are developed for the vertical direction. The application of design spectra was demonstrated
on examples of as-built footbridge.
1
INTRODUCTION
Modern footbridge structural design philosophy has led to a growing trend of slender structures, which are often
susceptible to vibration serviceability problems when under human-induced dynamic loading. As a result, there is
an increasing number of footbridges with excessive vibration problems caused by walking excitation. This has
presented a new challenge to the research community in recent years. Vibration problems on high-profile
structures such as the Millennium Footbridge in London [1] have attracted numerous studies in modelling lateral
vibration induced by crowd. However, current guidance for predicting human-induced vertical vibration under a
single pedestrian has not progressed in the UK much since the last research effort in the 1970s. There is little
guidance for civil engineers to use in design apart from the current British Standard BS 5400 [2]. Its approach to
vibration serviceability in vertical direction focuses on an average single-person who walks to match the
resonance of the bridge and has remained unaltered since
nce its introduction in 1978.
1978 Consequently, the BS 5400
method does not take into consideration now well-known
known phenomena such as interinter and intra-subject variability.
In addition to this, the BS 5400 guideline uses a binary pass-fail
pass
approach, which can hardly be considered
appropriate for vibration serviceability design when human are vibration receivers, as in case of footbridges.
This paper aims to develop design spectra as an alternative method to the current BS 5400 procedure for
vibration response prediction of footbridges under single pedestrian traffic condition.
condition Spectra are a convenient tool
for this purpose due to their visual
ual and intuitive application. These are important factors to consider since the
majority of users will be professional civil engineers that will (most likely) not have an in-depth
in
knowledge of
structural dynamics. The design spectra,, covering a range of footbridge structures (with different
diff
modal damping ζ
natural frequency fn and length L) will be generated in this paper using a probabilistic procedure that accounts for
inter- and intra-subject variability in human population [3].
2
2.1
DESIGN SPECTRA FOR VIBRATION SERVICEABILITY
ITY
Probabilistic
tic Procedure for Single Person Excitation
The design spectra were constructed by utilising a calculation procedure developed by Živanović et al [3]. This
novel procedure uses a probabilistic approach to account for inter- and intra-subject variability in human
population regarding the walking force induced. It does this by using a probabilistic model to describe walking
force in given pedestrian population. The model includes not only probability distributions of step frequency, step
length and force amplitude
mplitude in human population, but also takes into account that a pedestrian cannot repeat two
identical steps (intra-subject variability). Only the first harmonic of the walking excitation force is considered since
this harmonic is most often the one that generates the highest response of footbridges.
footbridges The model is then applied
to a given footbridge where the dominant mode of vibration is represented as a single-degree-of-freedom (SDOF)
system, to calculate the probability of having vibration response in a particular
p
acceleration range (Figure 1a). It
was assumed that this mode shape could be described as a half-sine
half
function. The procedure was proven to be
accurate based on observation of an as-built footbridge [4].
]. Further details of the procedure and its verification
can be found elsewhere [3, 4].
Figure 1: (a) Probability of certain acceleration level. (b) Cumulative probability that the acceleration is smaller than or equal to
the acceleration level considered.
Figure 1 shows the most relevant output (for this paper) from the probabilistic procedure: probability and
cumulative probability distributions of peak modal accelerations. Figure 1a shows the probability distribution of
peak modal accelerations for a particular bridge (m = 30000 kg, fn = 1.90 Hz, ζ = 0.30% and L = 65 m) produced
by a population of people crossing the bridge
idge one at a time.
time Figure 1b presents the cumulative probability
distribution of peak modal accelerations. Output of the kind presented in Figure 1b is used for constructing design
spectra in the following way. First a preferred cumulative probability of say 75% (0.75) is chosen for a particular
spectrum to be constructed, then for various bridges (i.e. spans and modal properties of the fundamental mode of
vibration) the corresponding peak modal acceleration is extracted. For example, in Figure 1b this value would
2
approximately be 0.28 m/s .
2.2
Design Spectra
An example set of design spectra, arranged into plots for specific span lengths, are presented in Figure 2. As can
be seen in Figure 2, all spectra plots are defined within the ranges 1.5 - 2.6 Hz for natural frequency fn and 0.1–
0.6% for modal damping ratio ζ, since these are common ranges of the two parameters for as-built footbridge
structures. In Figure 2 the top solid line denotes damping ratio of 0.1%, the solid line below corresponds to 0.2%,
and the further four lines below represent damping ratios of 0.3%, 0.4%, 0.5% and 0.6%, respectively. The same
order of lines applies to dash-dotted and dotted bands. To best outline the general dynamic behaviour of a
footbridge, the spectra are also defined for 50%, 75% and 95% (represented by dotted, dash-dot and solid lines
respectively in Figure 2) cumulative probability of occurrence. This illustrates the likelihood of exceedance of
given peak acceleration level for every 1 in 2, 1 in 4 and 1 in 20 crossings respectively. It should also be noted
that each spectra was produced using a “standard” modal mass of 10000kg to reduce the number of variables
involved. The response extracted from spectra could later be scaled to account for the actual modal mass.
All spectra were constructed for a pedestrian population in which the step frequency and step length can be
considered as normally distributed, with mean of 1.87Hz and 0.71m, respectively and the standard deviation of
0.187Hz and 0.071m, respectively. The distribution of DLFs was extracted from Kerr’s work [5].
3
APPLICATION OF DESIGN SPECTRA
This section describes the application process of the design spectra. The process is explained step by step using
an as-built footbridge in Montenegro (hereafter referred as Footbridge 1), where only the first harmonic was
relevant and the mode shape could be approximated as a half-sine function [4]. Footbridge 1 is a three-span steel
box girder footbridge shown in Figure 3a. Its total length is 104m, with 78m clear span between inclined columns.
The first mode of vibration dominates the response to normal walking excitation. This mode has natural frequency
of 2.04Hz [4]. The mode shape and modal properties (natural frequency fn, damping ratio ζ and modal mass m)
are shown in Figure 3b.
The first step in the application of the spectra is to select the spectra plot (from Section 2.2) which approximates
closest to the actual span length of Footbridge 1. Therefore, since the actual span length Lactual = 78m the most
relevant spectra plot is the 80m design spectra (Figure 2f). Using the 80m spectra plot and approximating the
damping ratio ζ (between 0.2% and 0.3% damping lines) for each probability band, the spectral acceleration
aspec,P can be deduced as shown in Figure 4. It can be seen from Figure 4 that aspec,95, aspec,75 and aspec,50 are
2
2
2
2.01m/s , 0.57m/s and 0.24m/s , respectively.
The second step in applying the design spectra is to calculate the mass correction factor km. The factor km is
required to transform the spectral acceleration (obtained directly from the spectra plot) to the maximum modal
acceleration amax,P appropriate for the modal mass of the bridge. This transformation is necessary because the
spectra plots were generated using a standard modal mass of 10000kg. The factor km is calculated as:
=
=
10000
= 0.1724
58000
where m0 is the standard modal mass (= 10000kg), m is the actual modal mass of the bridge. Finally the
maximum modal accelerations with 50%, 75% and 95% probability of occurrence (amax,50, amax,75 and amax,95
respectively) can be determined from:
,
=
Applying Equation (1), amax,95 can be calculated as below:
,
=
,
2
(1)
,
= 0.1724 × 2.01 = 0.35 m/s
2
By similar calculations amax,75 and amax,50 are 0.10 m/s and 0.04 m/s respectively. Therefore, we could expect that
2
2
the acceleration amplitude of 0.04 m/s is exceeded once in every two crossings, 0.10 m/s is exceeded once in
2
every four crossings while 0.35 m/s is exceeded only once in twenty crossings.
Figure 2: Design spectra for 95%, 75% and 50% chance of non-exceedance for span length of (a) 30 m, (b) 40 m, (c) 50 m, (d)
60 m, (e) 70 m, (f) 80 m, (g) 90 m and (h) 100 m.
Figure 3: Footbridge 1 – (a) Photograph and (b) modal properties of the fundamental mode of vibration.
3.0
2%
0.
2.01
2
Spectral Acceleration [m/s ]
2.5
2.0
0.
3%
1.5
1.0
95% exceedance
0.57
75% exceedance
0.5
50% exceedance
0.24
0.0
1.4
1.6
1.8
2.0 2.04
2.2
Modal Natural Frequency [Hz]
2.4
2.6
2.8
Figure 4: Design spectra for 80 m span lengths with spectral accelerations for Footbridge 1 (fn = 2.04 and ζ = 0.26%).
4
IMPLEMENTATION OF GRAPHIC USER INTERFACE
A GUI (Graphic User Interface) in MATLAB was constructed to drive the development of the design spectra. It
was implemented as a design aid to account for the different parameters for pedestrian population, bridge
properties and output probability. It gives the developer a visual tool to use when generating the spectra plots and
provides simple means to produce plots with different parameter values than those already used. As an example,
the GUI will be used to determine the exact values of amax,50, amax,75 and amax,95 for Footbridge 1. Figure 5 shows
the GUI with the parameters used to generate 50%, 75% and 95% probability spectrum corresponding to the
exact modal properties of Footbridge 1.
0.35
0.10
0.04
2.04
Figure 5: GUI displaying 50%, 75% and 95% probability spectra that correspond to exact modal properties of Footbridge 1, as
shown in “Bridge Properties” panel.
2
2
2
It can be seen from Figure 5 that the values of amax,50, amax,75 and amax,95 (0.04 m/s , 0.10 m/s and 0.35 m/s
respectively) that were calculated in Section 3 are the same as those found in Figure 4.
5
CRITICAL EVALUATION OF BRITISH STANDARD BS5400
It is interesting to compare the spectral response predictions for two footbridges to those estimated by current BS
5400 method. To do this Footbridge 1 and Footbridge 2 will be used, where Footbridge 1 is as described in
Section 3. Footbridge 2 is a three-span cable-stayed footbridge made of glass-reinforced plastic [4]. Its total span
is 113m and has a total mass of 20000 kg (Figure 6a). The main-span of the bridge is 63 m long and its mode
shape can be approximated to a half-sine wave. The properties of this span’s first mode of vibration are as follows:
fundamental natural frequency fn is 1.52 Hz, damping ratio ζ is 0.42 % and modal mass m is 2750 kg (Figure 6b).
The parameters corresponding to BS 5400 procedure are given in Table 1, which also shows the resulting peak
2
modal response in the last row of the table. The BS 5400 method estimates a peak modal response of 0.26 m/s
2
for Footbridge 1, and 3.12 m/s for Footbridge 2.
Figure 6: Footbridge 2 – (a) Photograph and (b) modal properties of the fundamental mode of vibration.
Table 1: Parameters for BS 5400 response simulation under a single pedestrian on two footbridges.
Footbridge
1
2
Pedestrian weight, W (N)
700
700
Step frequency, fs (Hz)
2.04
1.52
DLF
0.257
0.257
1.84
1.37
0.26
3.12
Walking speed, vp (m/s)
2
Peak modal response (m/s )
BS 5400 predicts acceleration levels that are exceeded by only 7.0% and 0.8% of subjects crossing Footbridge 1
and Footbridge 2, respectively (as indicted in Figure 7). Regarding Footbridge 1 a 7.0% probability of exceedance
(as shown in Figure 7a) might be considered as adequate estimation for single pedestrian traffic since this
estimate covers 93% of events, which is close to the preferred 95 percentile often used in engineering
applications. Yet, the peak modal response for Footbridge 2 has a 0.8% probability of exceedance (as shown in
Figure 7b) which is overly conservative for vibration serviceability. The inconsistency of probability estimation
(between Footbridge 1 and Footbridge 2) may mainly be attributed to the following factors: a constant DLF of
0.257 regardless of walking frequency, overestimated step length and neglecting the importance of probability for
a pedestrian to walk at a resonant pacing rate. Namely, studies have shown that the mean DLF, for the first
harmonic of human walking, is a function of walking step frequency fs [5]. Furthermore, the mean step length ls
has been found to be closer to 0.71m [3] than to BS value of 0.9m. Finally, the probability that a person will walk
at pacing frequency of 2.04Hz (to cause the resonance of Footbridge 1) is quite different (and considerably
greater) than the probability that they will walk at 1.52Hz (to cause the resonance of Footbridge 2).
Continuing the BS 5400 procedure the limiting acceleration levels alim, for Footbridge 1 and Footbridge 2 are
2
2
0.71m/s and 0.62m/s respectively. It is interesting that alim for Footbridge 1 is almost never achieved. Yet,
observations on the as-built bridge showed that the structure was perceived, by about 30% of pedestrians, as
lively although vibration levels were all the time lower than the limiting acceleration [3]. Conversely, alim for
Footbridge 2 is exceeded in 19% of crossings (according to
Figure 7b) and hence would be perceived as lively by some people if the BS limit is correct. However, this bridge
is considered as lively by most people crossing it. These examples illustrate how unreliable performance of the
well-known empirical formula 0.5
, used to calculate the limiting acceleration in BS 5400, is.
Figure 7: (a) Footbridge 1: Cumulative probability distribution of peak acceleration levels. (b) Footbridge 2: Cumulative
probability distribution of peak acceleration levels.
6
CONCLUSION
The BS 5400 procedure is an inadequate method for vibration serviceability due to the inconsistency of the
methods for calculating the peak modal response and limiting acceleration. In addition, the use of a single limiting
peak acceleration value (as specified by BS 5400) does not give the engineer adequate discretion in designing for
vibration serviceability and also ignores that the British Standard limiting acceleration alim has been shown to be
inconsistent with real life pedestrian opinion.
In this paper, design spectra are presented as a feasible alternative to popular current design guidelines in the
British Standard BS 5400 for vibration serviceability of footbridges in vertical direction. The spectra are well suited
to use during design as they can be collected as a set and used as a reference manual, which are common forms
of literature in civil engineering consultancies. The probabilistic approach offered by the design spectra gives civil
engineers a more realistic estimation and also an indication of the behaviour of the structure during single person
traffic.
REFERENCES
[1] Dallard, P., Fitzpatrick, A.J., Flint, A., Le Bourva, S., Low, A., Ridsdill-Smith, R.M. & Wilford, M. The London
Millennium Footbridge, Structural Engineer, 79(22), p.17-33, 2001.
[2] BSI. Steel, concrete and composite bridges. Specification for loads, BS 5400: Part 2, British Standard
Institution 1978; London.
[3] Živanović, S., Pavic, A. & Reynolds, P. Probability Based Estimation of Footbridge Vibration due to Walking,
25th International Modal Analysis Conference (IMAC XXV), Orlando, Florida, USA, 19-22 February, 2007.
[4] Živanović, S. Probability-Based Estimation of Vibration for Pedestrian Structures due to Walking, PhD Thesis,
Department of Civil and Structural Engineering, University of Sheffield, February, 2006.
[5] Kerr, S.C. Human Induced Loading on Staircases, PhD Thesis, Mechanical Engineering Department,
University College London, UK, 1998.