Vibration Serviceability Assessment of Slender Structures using VSATs
Software
S. Živanović, A. Pavic, J.M.W. Brownjohn
Vibration Engineering Section
Department of Civil and Structural Engineering, The University of Sheffield
Sir Frederick Mappin Building, Mappin Street, Sheffield, S1 3JD, UK
ABSTRACT
Slender structural forms have become common in contemporary design of civil engineering structures occupied
and dynamically excited by humans, such as footbridges, floors, staircases and grandstands. This slenderness
nowadays often leads to increased susceptibility to excessive structural vibrations induced by human activities
such as walking, jumping and running. As this increased slenderness is a trend which is likely to remain in civil
structural engineering, it is necessary to develop a reliable and efficient means for the vibration serviceability
assessment of the slender structures. Responding to this challenge, the Vibration Engineering Section at the
University of Sheffield (UK) has developed an in-house MATLAB-based and GUI driven suite of programmes
called VSATs (Vibration Serviceability Assessment Tools) for vibration serviceability assessment of slender
structures excited by humans. VSATs is a set of software tools that enables: 1) modelling of human-induced load
in probabilistic sense and according to some existing design guidelines, 2) reliable estimation of vibration
response of floor structure having an arbitrary level of complexity, 3) statistical treatment of human induced
dynamic loading and processing of calculated responses and 4) vibration serviceability assessment based on a
number of criteria defined in British and international standards. The main features of this software are presented
in this paper using an example of a long-span high-frequency floor structure.
1
INTRODUCTION
Problems with excessive vibration that has occurred on some high profile structures in recent years while they
were exposed to human-induced dynamic loading – the Millennium Bridge in London [1], Solférino passerelle in
Paris [2], the Millennium Stadium in Cardiff [3], Brooklyn Bridge in New York during the blackout [4] are only some
examples – highlighted the lack of adequate design guidance related to modelling human-induced loads (such as
walking, jumping, swaying, etc.) and estimation of the responses they generate. To deal with this problem, the
Vibration Engineering Section (VES) at the University of Sheffield in the UK has developed suite of programmes
called VSATs (Vibration Serviceability Assessment Tools) for vibration serviceability assessment of slender
structures excited by humans. This paper describes the main features of the software that has already been used
successfully by VES in various projects. The demonstration of the software will be explained on an example of a
long-span high frequency floor.
2
VSATs FEATURES
VSATs is a MATLAB-based GUI-driven software developed in VES with the aim to analyse and asses the
vibration serviceability limit state of large slender civil engineering structures (such as footbridges, long-span
floors, staircases, and stadia) that have been the subject of VES research for more than a decade. VSATs
incorporates the up-to-date models for vibration serviceability assessment recently developed either externally by
professional bodies/organisations that are developing design guidelines (such as the UK Concrete Society,
Concrete Centre, and the Steel Construction Institute), or internally within the group (see, for example, Živanović
et al. [5]). The software enables effective comparison of the results from different models, including experimental
response measurements, and could be used for vibration serviceability assessment of any slender civil
engineering structure with known geometry and modal properties estimated either analytically or experimentally.
In this paper a high-frequency floor will be used to demonstrate the main features of the software. Both exciters of
the floor and receivers of the vibration will be humans in this example (although the software can be used for
assessing the acceptability of vibration levels for sensitive equipment as well).
3
STRUCTURAL DESCRIPTION
The floor used for demonstrating capabilities of VSATs is the University of York, UK, bioscience research facility.
This building consists of three 3-storey blocks which have both office and laboratory space on each of the floors
[6]. It is the first floor that is analysed in this paper. The floor comprises three bays of different sizes: 6.6x7.3 m for
the laboratory, 6.6x9.2 m for the office and corridor, and a smaller central bay measuring 6.6x6.0 m (Figure 1a).
The detailed description of this pre-stressed pre-cast hollow core floor construction, its modal testing, and FE
modelling (Figure 1b) in ANSYS FE code can be found elsewhere [6]. This paper concentrates on vibration
serviceability assessment of office areas (positioned in the longest bay and therefore most prone to vibration)
generated by people walking in them. Therefore, the main receivers of vibration in this case are people working in
these offices. The vibration serviceability assessment of areas containing the sensitive equipment is described by
Pavic et al. [6] and will not be repeated here.
3.6
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6.6
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6.6
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6.6
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6.6
(a)
walking path 1 (WP1)
6.6
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test points for modal testing
points for response measurements
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Z Y
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(b)
Figure 1: (a) Floor plan with test points and (b) the FEM model.
4
INPUT-PATH-OUTPUT-RECEIVER PHILOSOPY
VSATs software is based on ISO 10137 [7] philosophy. This means that it requires defining the vibration source
(i.e. human-induced force), and vibration path (i.e. modal properties of the structure) first, which then leads to the
vibration response calculation and assessment for the given vibration receiver. This response can than be
presented graphically, for example in form of R-factors. The acceptability of the vibration response can be
assessed depending if the receivers of vibration are humans or sensitive equipment.
After starting VSATs (Figure 2) and entering the program, the following menu options (with explanation of their
main purpose given in brackets) become available:
• File (contains options for: saving the screen in an appropriate image format, changing the working directory,
loading and saving projects and exiting from the software).
• Path=FEM+modes (allows for: importing the structural geometry and modal properties either from the FE
software or from the experimental model, as presented in Figure 3, modifying the modal properties if required
for, say, parametric studies, and defining a walking path along which walking takes place).
• Calculate Response (this option allows for calculation of the vibration response under the dynamic load, such
as a walking force, acting on the structure. This can be done by choosing between several walking force
models available).
• View Response (this allows for seeing the contour plots of the vibration response in form of R-factors over
the floor investigated, as well as comparison of vibration responses obtained by different methods).
• Assess Response (this allows for estimating the calculated responses according to, for example, BS6841
guideline).
• Options (allows for adjusting colours and font sizes).
Figure 2: The starting screen in VSATs.
(a)
(b)
Figure 3: The first mode shape as imported to VSATs from (a) experimental modal analysis and (b) FE software.
5
GEOMETRY AND MODAL PROPERTIES OF STRUCTURE
By measurement it was found that the lowest mode of vibration of the floor is at 12.7 Hz, meaning that the floor
belongs to the high frequency floor category according to Appendix G of Technical Report 43 published by the UK
Concrete Society [8]. This guideline will be referred to as ‘AppG’ in this paper. Most of seven measured vibration
modes had damping around 0.6% [6]. The FE model (Figure 2b) was successfully updated [6] based on these
measured modes and the updated FE modes are shown in Figure 4.
Mode 1 : f = 12.79 Hz
Mode 2 : f = 13.78 Hz
Mode 3 : f = 14.83 Hz
Mode 4 : f = 16.41 Hz
Mode 5 : f = 18.60 Hz
Mode 6 : f = 18.78 Hz
Mode 7 : f = 19.38 Hz
Figure 4: First seven modes of vibration from the updated FE model.
For vibration serviceability assessment in VSATs, either experimental or FE models could be used as long as
they contain all modes of interest and the walking surface is a flat (i.e. not inclined) plane. In this paper, the FE
model is used for vibration serviceability assessment since it offers possibility for taking into account all modes up
to 50 Hz, differently from the experimental model where the highest mode identified was at 19.7 Hz [6]. It was
assumed, for the purpose of this paper, that all modes have the damping ratio of 0.6%.
6
VIBRATION ASSESSMENT METHODS IN VSATs ACCORDING TO DIFFERENT MODELS
To calculate the vibration response in VSATs, several walking force models are available: 1) AppG model [8], 2)
force time histories measured on an instrumented treadmill [9], and 3) a probabilistic model based on a
combination of AppG and an internally developed design procedure [5]. Additionally, the experimentally measured
responses may be evaluated. Key features of these models and their outputs, as available in VSATs, will briefly
be explained in this section, starting with the measured responses.
In the force models mentioned, walking path 1 (WP1) along which the walking force is applied should be defined.
The walking path shown in Figure 1a is chosen since it is relevant for the assessment of the response in offices.
This walking path is defined in VSATs by graphically pointing out at its starting and ending point. However, if there
was a need a more complex walking path consisting of any number of straight line segments could be defined.
One example is shown in Figure 5. Walking path should be specified in VSATs after importing the structural
modal properties so that the interpolation of the vibration mode amplitudes along the walking path can be
performed. This interpolation is necessary for defining the modal force required for the response calculation.
Figure 5: An arbitrary piecewise-linear walking path can be specified in VSATs.
6.1
Measured responses
After the full-scale modal testing of the floor, the vibration responses (accelerations) of the floor at test points
(TPs) 3, 5, 10, 16, 18 and 19 (Figure 1a) were measured due to walking of two test subjects, one at a time [6].
They walked along WP1 (Figure 1a) at the following pacing rates: 1.90, 1.95, 2.00 (two times) and 2.12 Hz,
controlled by a metronome. The measured data were written in a MATLAB file in a format required for analysis in
VSATs. These data were then imported into VSATs and postprocessed. First, they were weighted according to a
chosen weighting curve, such as Wb in this example [10]. Then the vibration measures, such as MTVV, R-factor
and VDV were calculated. An example of the results obtained for TS1 walking at pacing rate of 2.00 Hz is
presented in Figure 6. Note that accelerations are much higher at TPs 5 and 3 that belong to the walking path
(first row of graphs in Figure 6), as would be expected.
Distributions of maxR factors (one value per weighted time history) and Running R factors (i.e. R factors
calculated at regular time intervals, such as 1 s, for all weighted time histories together) could be presented for
response time histories measured when walking at pacing frequencies in a chosen frequency range. Specifying a
wide frequency range in the way to incorporate all measured responses (such as 1-3 Hz in this structure), the
distributions shown in Figure 7 were obtained. It can be seen that maximum R factor measured is close to (very
low value of) 0.6 only.
Figure 6: Weighted response time histories and the corresponding vibration measures in a walking test.
Figure 7: Probability functions for maxR factor and Running R factor from all measurements.
6.2
AppG
Since the floor investigated belongs to the high-frequency floor category, AppG requires the input force to be
specified in the form of impulses acting at TPs along the WP1 (Figure 1a). After calculating responses to each
impulse at chosen points across the floor in each mode, the total responses could be obtained by summing up the
responses in the individual modes. VSATs offers possibility to review a time domain response at any chosen
response point, due to an impulse applied at a specified point. A VSATs screen showing the response at TP5 due
to the impulse at the same point is shown in Figure 8. VSATs allows for calculating the response across all points
on the floor, which offers opportunity for identification of the areas most prone to vibration. A plot of response
factors across the floor obtained for the case when the excitation and response points coincide is shown in
Figure 9. Maximum response factor is equal to 2.02 for walking at very fast (and improbable) frequency of 2.8 Hz,
suggested as an upper walking frequency limit in AppG.
Figure 9 also reveals quite clearly that the highest responses (red areas) are confined to the middles of 9.3 m and
7.3 m wide floor bays (Figure 1a) whereas the short 6.0 m wide bay does not move very much.
Figure 8: The response at TP3 due to an impulse at the same point.
Figure 9: Response factors across all points on the floor (pacing frequency 2.8 Hz, response points and excitation
points coincide).
6.3
Measured forces
A library of (currently available) 118 walking forces measured on a treadmill for test subjects walking at different
speeds (i.e. different walking frequencies) is available in VSATs to be utilised as a ‘model’ of walking-induced
excitation. These forces could be applied along the walking path of the floor investigated, one at a time. Then the
responses at specified points could be calculated and stored. The responses can be plotted for inspection. An
example presenting the force (generated by a test subject who walked at the average pacing rate of 2 Hz) and the
two responses of the structure at TPs 5 and 3 are presented in Figure 10 (from top to bottom, respectively). If
responses were calculated at all points across the floor, then contour plots of R factors of the kind presented in
Figure 9 would also be available.
Figure 10: The walking force as measured on a treadmill and the structural response under it at two TPs.
6.4
Probabilistic force model
Finally, the probabilistic force model offers possibility to generate walking forces by using Monte Carlo
simulations. This model allows for generating different forcing amplitudes, step lengths, and walking frequencies
for different pedestrians. In this case 100 pedestrians walking along WP1 were generated, and the results were
stored in form of maximum R-factor for each TP, due to each pedestrian passing. For each pedestrian modelled,
the contribution of low frequency and high frequency modes (and their spectra) to the total response of the
structure can be presented, as shown in Figure 11 for a pedestrian generated to walk at pacing frequency of
2.01 Hz. Contribution of low frequency modes, high frequency modes and the their sum are presented in the time
domain in the left column of the Figure 11 while the right column presents the corresponding spectra. In the
structure under study, the contribution of the low frequency modes is clearly zero, since there are no modes
below 10 Hz [8]. Contour plot of R factors of the kind presented in Figure 9 could also be found for this
probabilistic model if the responses were calculated at all points across the floor.
(b)
Figure 11: The response of the structure at TP5 under a randomly generated pedestrian.
7
PRESENTATION OF RESPONSE RESULTS
The responses calculated according to different models could be compared and plotted in the form of probability
functions related to the R factor and Running R factor (calculated at the chosen TP when the excitation pacing
frequency is in a specified range). Probability function and the corresponding cumulative probability for R factor
according to experimentally measured response at TP3 (when walking at, say, step frequency in the range 1.82.0 Hz) and the ones estimated based on AppG and on the forces measured on the treadmill are presented in
Figure 12 in the upper row. It could be seen that the maximum R factor according to AppG goes up to 1.1, the one
calculated for measured forces is up to 2.2, while the experimental one is around 0.6. This is because the AppG
accounts for a pedestrian producing the force which has 25% chance of being exceeded, covering a larger
population of test subjects than that used in experimental measurements. Based on Figure 12, it seems that
forces measured on treadmill are generated by some very efficient dynamic exciters, having in mind that the
vibration response under some of these forces exceeds the one based on AppG (supposed to cover 75% of
population). In the lower row the probability and the cumulative probability functions for Running R factors for TP3
are presented, based on measured responses and those calculated under measured forces. This form of the
response cannot be calculated in AppG because the response is calculated for one step only (which does not last
long enough to allow for calculation of Running R factors). Finally, the response distribution for probabilistic model
could also be presented. It is omitted from the plots only to improve the clarity of the graph in Figure 12.
Finally, it should be said that the real-life high frequency floor structure used as an example in this paper was
unclad and completely bare during modal testing. Therefore, it can be assumed that the measured and calculated
responses, although very low, would be even lower in a completely finished building. This is due to increased
damping and stiffness of a floor which can be expected in a fully clad and partitioned building of this kind.
Figure 12: Comparison of probability functions for R factors from different models.
8
CONCLUSIONS
This paper outlines main features of recently developed (and still developing) software for vibration serviceability
assessment of large slender civil engineering structures dynamically excited by human activities. It was shown
that the software could be used for both the calculation of the structural vibration response generated by humans
and for effective presentation and comparison of results.
ACKNOWLEDGEMENTS
The authors acknowledge the contribution of Dr Zoran Miskovic and Mr Chris Middleton in programming the
function in the software that imports geometry and modal properties of the structure from a model that is either
estimated experimentally or modelled in FE-based software. Also the authors would like to thank to the UK
Engineering and Physical Sciences Research Council (grant reference GR/S14924/01: Investigation of the AsBuilt Vibration Performance of System Built Floors and GR/T03000/01: Stochastic Approach to Human-Structure
Dynamic Interaction) for their support.
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