The design of flexible footbridges for synchronous lateral loading
Neil Hawke
A dissertation submitted in partial fulfilment of the requirements for the Degree of Master of Science in Bridge Engineering
August 2008
1 Abstract
“The design of flexible footbridges for synchronous lateral loading” aims to give the designer practical guidance on the design of these types of structures for synchronous lateral loadings resulting from crowd-induced loads. A ‘flexible footbridge’ can be classed as a footbridge where the vertical natural frequency is at or below 5.0Hz, or the lateral natural frequency is at or below 1.5Hz.
There has been much written on the subject recently, and with the issues surrounding the Millennium Footbridge in London, attention has been focused on the dynamics of these structures.
Research by Pimentel identified shortfalls in code requirements, and suggests that these are re-assessed. (Pimentel, Pavic, & Waldron 2001)
This paper investigates the phenomena of sway of a structure under lateral walking loads. It provides the designer with guidance for designing these types of structures considering current knowledge and using finite element analysis of a typical flexible footbridge to assess the lateral loading effect.
Comparisons are made between the results of the finite element analysis and the code stipulated limiting vibration criteria, and guidance is given for assessing the synchronous lateral loading effect. Consideration is also given to the factors that affect the lateral movement of a footbridge, such as damping and mass. The study shows a positive correlation between the Scruton number approach and the FEA of a typical flexible footbridge, and suggests that the Scruton number approach is used in the first instance to assess the synchronous lateral loading effect.
4 Listoffigures
5 Listofequations
6 Acknowledgements
I wish to thank the following people for their help and assistance during the writing of this dissertation and for the support during the MSc programme itself. Rob Latham who actually motivated me to finish the dissertation. Phil Mansell, Paul Mullett and Liz Thompson for reviewing the work. Nick Fuchs for his tireless encouragement and technical assistance, and finally to Amanda Sheppard for her unending support, help and encouragement. I couldn’t have done it without you!
7 Definitionsandabbreviations
E
fcu
FEA
FEM
Youngs modulus (N/mm2)
Characteristic concrete strength (N/mm2)
Finite element analysis
Finite element method
fm Frequency of mode (Hz)
fn Natural frequency (Hz)
fw Walking frequency (Hz)
fy Characteristic steel strength (N/mm2)
Hz Frequency
M Mass (kg)
m Pedestrian mass (kg)
P Vertical applied pedestrian load (N)
PL Lateral portion of pedestrian load (N)
T Period of vibration (sec)
THA
Time history analysis
t Time (sec)
x Lateral translation (mm)
Sc Scruton number (non-dimensional)
Scp Pedestrian Scruton number (non-dimensional)
Y Longitudinal translation (mm)
Z Vertical translation (mm)
b Diameter of cylindrical object for Scruton number calculation (mm)
ζ Damping as a fraction of critical (%)
∆ Displacement (mm)
α Acceleration (m/s2)
µ Mode shape
ζ m Modal damping as a fraction of critical (%)
ά Amplitude of body movement of pedestrians to the amplitude of the pavement.
β Proportion of pedestrian synchronisation (%).
δ Logarithmic decrement of the decay of vibration
π Mathematical constant which represents the ratio of any circle's circumference to its diameter.
ρ Air density (kg/m3)
8 Introductionandaimsofresearch
Walking, running and jumping activities on flexible structures produce vibrations. These vibrations can disturb other users, and instability and excessive movements of the structures can occur, causing users to alter their gait to stabilize themselves.
Vibration of structures in this manner is not new. Tredgold (1882), wrote that girders over long spans should be “made deep to avoid the inconvenience of not being able move on the floor without shaking everything in the room”(Allen & Murray 1993). It is also common practice for soldiers to break step when passing over a bridge to avoid excessive vibrations and potentially dangerous movements.
Most studies into the vibration of footbridges have focused on the vertical component of the walking load and vertical vibrations. It was not until the Millennium Footbridge in London suffered problems on its opening day that the focus changed to the lateral component of the walking load. The effect caused by this lateral loading by a crowd has become known as ‘synchronous lateral loading’.
Synchronous lateral loading has come to widespread notice as a result of the Millennium Footbridge in London, where excessive lateral translations took place on the opening day. The bridge was crowded with visitors crossing the bridge. Videos taken on the day showed significant lateral movement, to such a degree that pedestrians were holding onto the hand rails to steady themselves, and could not walk a straight line across the bridge. It could be seen from some of this video footage that there was synchronous head movement throughout the crowd, thus implying that at least a portion of the crowd was ‘tuned’ to the movement of the structure and their walking frequency was locked into the bridge frequency.
Upon further investigation, it came to light that a number of other bridges had also experienced similar lateral movements under crowd induced loads. The designers of the London Millennium Footbridge, along with the then president of the Institution of Civil Engineers claimed that this loading effect was something not seen before. However, this loading and its possible effect had been identified in the literature some time ago, (Bachmann 1992). The problem with the Millennium Footbridge was
solved by installation of tuned mass dampers, however, as a result of this case interest in the synchronous lateral loading phenomenon and associated subjects has increased.
The aim of this research is to review the current knowledge on the issue of synchronous lateral loadings, and through a simple ‘process’ and finite element modelling of a flexible footbridge, give guidance on the assessment of a structures stability against this loading effect, without the need for testing and retrofit design.
This study collates the current thinking on this subject, and uses this along with FEA, to produce a practical design guide for the assessment of the synchronous lateral loading phenomenon.
9.1
Introduction
The load exerted by a walking pedestrian has three translational components; vertical, longitudinal and lateral. The lateral component of the walking load applied at half the footfall frequency by a large crowd can cause the bridge to sway. If this sway is perceptible by others, a further effect can be seen. The crowd will start to walk in synchronisation with the swaying of the bridge, thus ensuring that the footfalls are applied at the resonant frequency of the bridge. This frequency ‘lock-in’ caused the excessive movement seen at the Millennium Footbridge, and the effect has been termed ‘synchronous lateral loading’.
Synchronous lateral loading was first identified in 1992 (Bachmann 1992). In his paper, Bachmann identifies that the lateral frequency of a test bridge was half the vertical forcing frequency, and a walking pedestrian could cause lateral sway. He also suggests that the pedestrians synchronise their step with the sway of the bridge thus exciting the bridge still further. Although Bachmann identified this effect in 1992, in general there is very little in the literature on this type of loading effect. It was not until the opening day of the Millennium Footbridge in London that it became of common interest to bridge engineers.
The lateral loading that causes initial bridge movement is applied at half the vertical frequency. For normal walking the lateral component of the force occurs at 1.0Hz, with the vertical natural frequency at 2.0Hz. However, the values vary and are dependant on the walking pace. It has been found that the lateral sway of a persons centre of gravity occurs at half the walking pace, and it is this sway that causes the lateral forces. (Newland 2003)
Resonance can also occur at the second and third harmonics, 2.0Hz and 3.0Hz, so simply providing a larger natural frequency may not necessarily solve the problem, unless the frequency is above the fourth harmonic of the load. (Allen & Murray 1993)
Figure 1 shows graphically the synchronous lateral loading effect (Fujino et al. 1993)
Figure 1 Synchronous lateral loading effect
1 First small lateral motion
2 Synchronisation of walking to movement
3 Resonant force on girder, increasing the movement
The degree of susceptibility of a flexible structure to the synchronous lateral loading effect is linked to the following factors (Anon 2002c).
1. The lateral frequency of the bridge.
2. The modal mass of the lateral frequency.
3. The damping coefficient of the frequency.
4. The number of pedestrians on the bridge.
It is suggested in the paper produced on the Millennium Footbridge in London (Dallard et al. 2001) that bridges with a lateral natural frequency of under 1.3Hz are most at risk to excessive movements and accelerations due to synchronous lateral loadings. It is noted in the text that crowd density and damping play a large role, as do the second and third harmonics.
9.3 Currentdesignmethods
The design methods currently employed for the design of flexible footbridges are as follows:
1. Frequency tuning
This method involves the avoidance of critical frequency ranges. As walking pace rate generally occurs between 1.6Hz to 2.4Hz, then the avoidance of these frequencies and the second harmonic frequencies of 3.2Hz to 4.8Hz is used to tackle vertical vibration serviceability.
Lateral frequencies for normal walking occur at 0.8Hz to 1.2Hz, therefore avoidance of these frequencies and the second harmonic frequencies of 1.6Hz to 2.4Hz is used to tackle the lateral vibration serviceability.
However, one of the problems with this method is that at the design stage it is not easy to predict natural frequencies with any degree of accuracy. (Pimentel, Pavic, & Waldron 2001). This method also ignores mass effects, and the energy required to produce vibrations.
2. Dynamic response analysis approach
In this method serviceability is evaluated for the load generating the largest response by exciting the structure at its fundamental natural frequency. The load used is a representative pedestrian load derived from codes of practice. The
usual measure of the response is acceleration, and the results are then compared to code specified limiting acceleration criteria. (Pimentel, Pavic, & Waldron 2001). This method has the advantage that energy and relative mass are considered in the analysis. The susceptibility to vibration depends on the energy required to accelerate the structure, which is related to the mass of the load and the mass of the structure itself, and also to the natural frequency. The natural frequency is related to mass, stiffness, and damping.
3. Retrospective design
Provision for mass dampers is made at the design stage, with their implementation based on post construction in-situ testing.
At present, there is no common design method for assessing the effects of synchronous lateral loading on flexible footbridges.
9.4 Coderequirements
Limits in codes are identified in terms of the natural frequencies of the structure and accelerations. The following limits are identified in BD37-01 Appendix B. (Anon 2002a).
The code states that for fundamental lateral frequencies of less than 1.5Hz, special consideration shall be given to the possibility of excitation by pedestrians of lateral movements of unacceptable magnitude.
The Eurocode BSEN 1990:2002 Basis of structural design (Anon 2002b), limits the acceleration as follows:
Table 1 BD37-01 Code requirements
Table 2 Eurocode acceleration limits
ModeofAmplitude
Vertical
Horizontal
0.7 m/s2
0.2 m/s2
Exceptional use (lateral) 0.4 m/s2
The design comfort criteria are expressed as acceptable accelerations.
The code also states that verification of the comfort criteria should be performed in the following circumstances:
1. If the fundamental natural frequency of the deck is less than 5.0Hz.
2. If the fundamental natural frequency for lateral and torsional modes is less than 2.5Hz.
Also mentioned is that the data used in calculating the vibration of the bridge is subject to significant uncertainties, so if the comfort criteria is not satisfied by a significant margin then it may be necessary to make provision for retrospective tuned mass dampers in the structure.
This last statement from the code alludes to retrospective testing and post installation of mass dampers being considered from the outset.
9.5 Pedestriansynchronisationwithbridgemovements
Under a lateral impulse, the amplitude of a bridge increases. As the amplitude increases the amount of pedestrians synchronised with the movement changes, which in turn changes the amplitude. One of the challenges in assessing the load contributing to the lateral movement is determining what percentage of the pedestrians on the bridge are synchronised to the movement.
During the investigation into the Millennium Footbridge in London, the designers, Arup, conducted tests at Imperial College London using a shaking table to asses the
probability of synchronisation of pedestrians at different lateral amplitudes. (Newland 2003)
The results of the tests carried out by Arup are presented in Figure 2.
These tests showed that the degree of synchronisation increases with increasing amplitude. Therefore, the assessment of the loading generated by a crowd on a bridge is not only a function of the crowd density and frequency, but also of the amplitude of movement of the bridge.
It is also interesting to note here that there was 30% synchronisation between the pedestrians on the bridge at zero bridge amplitude. This suggests that synchronisation is not just a function of amplitude, but also of the subconscious effect of walking in and amongst other people, and the instinct to fall into step with those around us.
Figure 2 Probability of synchronisation (Newland 2003)
Testing carried out by on a congested footbridge in Japan, (Fujino, Pacheco, Nakamura, & Warnitchai 1993), showed that 20% of the pedestrians were synchronised to the bridge amplitude of 10mm. The synchronisation was assessed by video camera and digitised. At the time, the bridge was congested with 2000 pedestrians, approximately 2.1people/m2. The crowded conditions on this bridge occurred during a boat race, and were a regular event. Many pedestrians did not complain about the vibrations because they had experienced the same thing at previous events on the bridge. This demonstrates the subjective nature of human perceptibility of vibrations.
9.6 Accelerations
Accelerations are the general parameter used to assess the serviceability of a flexible footbridge.
There is much written on the perception requirements of vertical vibrations, but very little on the magnitude of acceptable lateral accelerations.
BS6472:1992 presents perceptibility base curves for accelerations, depending on the frequency and orientation of the vibration with respect to the pedestrian. Curiously, in the 2008 version of the code the perceptibility curves have been removed.
For a frequency of 1.0Hz, perception of vertical movement occurs at an acceleration of 0.01m/s2
For the same frequency, perception of lateral movement occurs at 0.0035m/s2
This shows that lateral translation is 285% more perceptible than vertical translations, and clearly shows that specific acceleration limits must be used for lateral translations.
This phenomenon is reflected in the Eurocode comfort criteria limits, where the horizontal acceleration limit is 3.5 times less than the vertical limit.
It is worth noting here that the perceptibility of movement is very subjective. Cantieni, & Pietrzko carried out tests on a wooden footbridge and measured lateral accelerations of 1.3m/s2 when runners crossed the bridge. However, no remedial measures were taken in this instance, as there were no complaints from the bridge users. (Cantieni & Pietrzko 1993)
In 2003, Nakamura tested a long span footbridge in Japan that was reported to have significant lateral vibration when in use. (Nakamura 2003)
Using lateral accelerometers, the test showed the following results.
1. At an amplitude of 10mm (0.3m/s2), walking became uncomfortable.
2. At an amplitude of 25mm (0.75m/s2) some pedestrians had difficulty in walking, and some touched the handrail for support.
3. At an amplitude of 45mm (1.35 m/s2) pedestrians lost balance and had to stop walking to steady themselves.
4. At an amplitude of 75mm (2.1m/s2) as occurred on the Millennium Footbridge in London, pedestrians felt unsafe and were unable to walk.
A limit of serviceability has been suggested by Nakamura of 45mm amplitude with accelerations of 0.75 m/s2 .
The Eurocode limit for lateral acceleration of 0.2m/s2 seems to fit well with the first result, as the limit is set before the vibrations become uncomfortable for the pedestrians.
9.7 Influenceofcrowddensityandpacingfrequency
Crowd density is a significant factor in determining the applied lateral load and the frequency at which the load is applied. The synchronous lateral loading phenomenon only occurs when there is a crowd event on the bridge.
Natural walking frequency is normally about 2.0Hz. However in crowds this can vary between 1.4 and 2.2 Hz. (Anon 2000). An average value for design of 2.0Hz is normally taken.
Tests have show that a bridge becomes choked at 1.7 people/m2. Above this value, people are not constantly moving and generating the required lateral forces for the sway movement to occur. (Anon 2001)
Also of interest is that people not walking would reduce the sway as they act as dampers when they are stationary. However considering this as active damping in design is debatable, as quantifying a value for this could be problematic.
Testing on the Millennium Footbridge in London also found that restricting the flow of people on the bridge could make the flow of people more uniform instead of splitting it up, thus encouraging synchronisation. (Anon 2001)
9.8 Damping
Structural damping is the least discernable property in the dynamic design of structures.
Large amplitude oscillations activate available visco-elastic damping mechanisms leading to large energy dissipation. Such damping may come from loosely fitting handrails, expansion bearings and joints, composite material interface friction, cable connections and foundation movement.(Glanville, Kwok, & Denoon 1996)
Estimating a value for the damping of a bridge structure is difficult, but it has been suggested in the literature that a rule of thumb value for the damping of a cablestayed bridge can be estimated by the following formula. (Glanville, Kwok, & Denoon 1996)
Equation 1 Estimation of damping value (% critical) ζ = fn – 1
Glanville et al carried out in-situ testing on 27 footbridges in Sydney Australia to assess the damping values. The results are presented in Figure 3.
Figure 3 % of critical damping versus the natural frequency of footbridges (Glanville, Kwok, & Denoon 1996)
Category A bridges in Figure 3 are small amplitude bridges. Less than 0.2mm lateral amplitude.
Category B bridges are large amplitude bridges where amplitudes are larger than 0.2mm lateral amplitude.
It is not clear from the paper what loadings these amplitudes were generated from.
Figure 3 shows that actual measured damping values for the low natural frequencies are very low. In Allen and Murray’s paper it is suggested that a value of 0.01 (1% critical) is used as the damping coefficient for the natural frequency. (Allen & Murray 1993). This does not show good correlation with the test data of Table 3 and it is suggested that this value is not used.
There is no guidance given on the damping of higher modes.
BD37/01 also suggests damping values in the form of a logarithmic decrement for the decay of vibration for various types of construction. Table 21 from BD37/01 is
reproduced below in Table 3. An extra column has been added, to show how these values translate to a percentage of critical damping.
The percentage of critical damping is obtained from the logarithmic decrement of decay of the vibration with the following formula:
Equation 2 Damping ratio from logarithmic decrement
Table 3 Logarithmic decrement and percentage critical damping from BD37/01
BridgeSuperstructure
The values presented in Table 3 for the percentage of critical damping are in general agreement with the data presented in Figure 3 only for natural frequencies below about 0.75Hz. However when frequencies are larger than this the values in Table 3 seem to be overly conservative when compared with the test data in Figure 3.
Also of significance to the damping is the mass of the pedestrians themselves. When the walking frequency becomes out-of-phase with the bridge, amplitude energy is extracted from the structure and the pedestrian becomes an active damper. (Glanville, Kwok, & Denoon 1996).
It has also been suggested in the literature that one stationary person could cancel out one walking person. The stationary person would provide sufficient damping for this effect. (Anon 2001).
For practical design it is advisable not to consider these effects, as there is little research or data available on possible damping values using these additional damping effects.
9.9 Scrutonnumberapproach
The pedestrian Scruton number has been shown to be a good gauge of the potential problems associated with synchronous lateral loading of flexible bridge structures (Newland 2003). The method does however rely on a number of experimental factors for which limited tests have been carried out. However, data so far presented in the literature shows a good correlation between the limiting Scruton number and problematic bridges.
It has been shown (Newland 2003), that the stability criterion for the limiting structural damping required to prevent instability occurring is;
Equation 3 Limiting damping to prevent instability
2ζ > α βm/ M
The Scruton number is a non-dimensional value that relates damping to the structural mass, and is generally used to examine the effect of vortex shedding on structures, and is defined as follows.
Equation 4 Scruton number
Sc = 4πζ M / ρb2
Large Scruton numbers are preferable.
In Newland’s paper, he points out that McRobie (Allan McRobie and Guido Morgenthal 2002), found that there is a correlation between the wind excitation of structures and the people excitation of bridges.
The following formula for the pedestrian Scruton number has been proposed by McRobie and is presented in Newland’s paper (Newland 2003).
Equation 5 Pedestrian Scruton number
Scp = 2ζ M /m
This formula is an attempt to represent the energy required to produce vibrations considering the mass of the load, the mass of the structure, and the damping.
Combining this with the formula for limiting damping from Equation 3, we can find that for lateral stability the following is required:
Equation 6 Limiting pedestrian Scruton number for stability
Scp > α β
α is the ratio of movement of a person’s centre-of-mass to movement of the pavement, which from test results has been found to be about 2/3 for lateral vibration in the frequency range 0.75 to 0.95 Hz.
β is the correlation factor for individual people synchronising with pavement movement, which is typically about 0.4 for lateral pavement amplitudes less than 10 mm. (Newland 2003)
Therefore, a limiting lower bound value for the pedestrian Scruton number is 0.266
10 Methodologyanddatacollectiontechniques
10.1 Methodology
The intention of this paper is to recommend the use of a simple process to assess the synchronous lateral loading effect. To do this a flexible footbridge has been modelled, and a load applied to represent the peak synchronous loading effect. The design data used to assess the loading, synchronisation and application frequency has been taken from the literature, after critical review and assessment.
In assessing performance requirements, reference has been made to current British Standards, Eurocodes, and data presented in the literature.
Dynamic and Time History Analyses have been carried out and the resulting accelerations and translations compared with results from the Scruton number approach. The Scruton number approach is used as a tool for assessing the synchronous lateral loading effect due to its simplicity at representing the energy to cause instability. Data presented in the literature shows good correlation when the Scruton number approach is used to assess some existing structures.
The process illustrated in Figure 5 has been followed to build the analysis model and interrogate the results. The following sections describe the various parameters and properties that have been used to build the model. It also details the design parameters that are used to test the sensitivity of the model to various changes in properties that have a significant effect on the dynamic properties of the bridge structure, such as damping. Manual methods such as the Scruton number have been calculated and compared with the analysis results in Section 11. Figure 4 shows the overall research process for this paper.
Start
Review Literature
Simple method to assess the synchronous lateral loading effect
Decide on SLL assessment method. From the review of the literature
Build bridge model in SAP 2000
Define application of the load. (Load time history)
Apply loading, and time history
Analyse bridge with various level of damping from 1.0% to 30%
Crowd loading
Review results for consistency
Compare simple approach from the literature to the results obtained from the time history analysis
Write conclusion and suggest analysis parameters and analysis process
Review dynamic properties such as natural frequencies
Damping from simple method to be compared to damping from the TH analysis to obtain the limiting acceleration values
Figure 5 Model analysis methodology for this paper
10.2 Modelanalysis
An analysis model has been constructed using SAP 2000 Version 11 for the assessment of the lateral response of the bridge.
SAP 2000 v11 is a powerful software package, with a graphical interface. It has sophisticated capabilities, and is able to handle the following analysis; Eigen and Ritz modal analysis, static and dynamic analysis, static nonlinear analysis for material and geometric effects, including pushover analysis; time history and nonlinear timehistory analysis by modal superposition or direct integration; and buckling analysis.
A time history analysis of walking loads on the bridge has been carried out, applying the lateral component of the walking load to the structure. The bridge is a 57m span cable stayed footbridge. The pylon is concrete and 24m in height, whilst the edge and stiffening beams are steel I sections. The deck comprises a timber oak deck. The deck is assumed to provide zero lateral stiffness in the SAP model. Distance between stay cables is 6.5m, and the deck width is 5m. Stay cables have been modelled as tension members, and the sag in the cable has been accounted for using the Ernst formula. The scheme design for the bridge was carried out previously for a hypothetical project, based on current design trends. Figures 6, 7 and 8 show the structural arrangement of the bridge.
Figure 7 Pylon elevation
Figure 8 Deck Section
A visual of the structural model of the footbridge is shown in Figure 9.
The following assessment of the results has been carried out:
1.0 Assessment of modes and deflections using full dead and live UDL’s.
2.0 Assessments of modes and deflections using UDL dead load and adding walking loads including the lateral component. To assess these effects a time history analysis has been carried out.
3.0 Assessment of the parameters that can have a significant effect on the results of the lateral analysis such as damping.
The results of the analysis are presented in Section 11.
Oak Deck
10.3 Dynamiccharacteristicsofthemodel
The results presented in Section 11 show that the bridge structure is very flexible. A flexible structure has been produced in an attempt to correlate the pedestrian Scruton number of the structure with the results of the time history analysis. Various levels of damping have been assessed, as this parameter is critical to the dynamic characteristics of the bridge structure. The damping has been applied as proportional damping over the whole structure.
10.4 Derivationofloading
For vertical excitation, a pedestrian loading of 75kg has been taken. For a person walking at 2.0Hz, it has been shown that 36% of this load is the dynamic applied load. (Newland 2003)
For lateral excitation it has been shown that 4% of the static vertical load acts horizontally.(Newland 2003)
It has been taken that the bridge is congested with 1.7 people/m2 laterally and longitudinally, therefore the loading at each 0.75m interval in alternate directions is as follows.
Equation 7 Crowd load intensity
5m wide x 1.7 people @ 750N x 4% = 255N
It is assumed that 40% of the pedestrians are synchronized
Therefore the load to be applied is:
Equation 8 Synchronised crowd load intensity
255 x 0.4 = 102 N
This loading is applied at the natural frequency of the bridge (1.5Hz), over a time period of 10 seconds in both directions. The loading is applied at 0.75m centres along the entire length of the bridge.
10.5 Derivationofloadingtimehistory
Table 4, (Bachmann Ammann 1987) shows the loading application frequencies for a number of loading applications.
Table 4 Load application frequencies
It can be seen from Table 4 that the horizontal fundamental frequency for a pedestrian at a normal walk is 1.0Hz. Generally, this value is very close to the lateral fundamental natural frequency of a typical flexible footbridge.
From the analysis model the lateral natural frequency is 1.5Hz, The load has been applied to the structure at 0.75m intervals along the bridge as shown in Figure 10, at a frequency of 1.5Hz to match the fundamental lateral natural frequency of the bridge.
Although Table 4 shows that the horizontal fundamental frequency for normal walking is 1.0Hz, the load has been applied to the structure at the structures fundamental lateral frequency to encourage resonance. Therefore, the loading time history presented in Table 5 has been produced using an application frequency of 1.5Hz as follows:
1.5Hz = 1.5 cycles per second, which is a time period of 0.67 seconds. Therefore, the full 100% load will be applied every 0.67 seconds.
Figure 10 shows a graphical representation of the applied pulse load to replicate the peak loading encountered in a synchronous lateral loading event.
Table 5 Loading time history
Figure 10 Loading time history graphical representation
10.6 Applicationofloading
11 Application of lateral load
The lateral load has been applied in alternate directions as shown in Figure 11, at 0.75m intervals over the whole length of the bridge to simulate a crowd loading.
Figure
10.7 Analysisofthemodel
A modal analysis for the first 12 modes of vibration has been carried out. The results of this are presented in Section 11 and visuals of the mode shapes are presented in Section 15. A time history analysis has been calculated for a variety of damping values and the results of this are presented in Section 11.
10.8 Extractionofresults
Results of accelerations and translations have been extracted in numerical and visual form direct from SAP 2000.
Where graphs are required numerical data has been extracted from the SAP 2000 model into Microsoft Excel, and the graphs produced with the charting function.
Mode shape visuals have been generated directly by SAP 2000, and exported to the document as jpeg files.
Tabulated data has been reproduced from the results tables presented in SAP 2000.
The results of the study are presented in Section 11 of this work.
11.1 CalculationofthepedestrianScrutonnumber
The Scruton number for the analysis model is as follows for various levels of damping, using Equation 5.
Scp = 2ζ M /m
Damping, (ζ ) 1%, to 30% critical will be assessed.
Pedestrian mass, (m). It has been shown (Allen & Murray 1993) that a footbridge becomes congested, and the flow of pedestrians stops, when the number of pedestrians on the bridge exceeds 1.7people/m2
Bridge Mass, (M). The mass of the deck will be considered based on the deck geometry.
Table 6 shows the input values and the resulting pedestrian Scruton number.
Figure 12 shows the plot of damping against Scruton number.
Table 6 Pedestrian Scruton numbers for various damping ratios
11.2 Modalanalysisresults
Modal analysis has been carried out for the first 12 modes. Table 7 shows the frequency and time periods for each modes shape. Also shown is a description of the mode shape. Figure 13 shows a visual of the 1st modal response and Section 15 presents the visuals of the 12 modes analysed. Table 7
11.3 Accelerationsandtranslations
Through the application of the time history analysis accelerations and translations have been analysed for various damping values. Figures 14 and 15 show accelerations and translations plotted against damping. It can be seen from Figure 14 that damping has a significant effect on both accelerations and translations.
Figures 16 and 17 show for 10% damping the effect that the periodic load has on increasing the accelerations in both directions. Figure 16 shows the acceleration and its degradation over time for one loading cycle, whilst Figure 17 shows the accelerations for a 10-cycle time period.
Figure 14 Accelerations with damping 2% to 30% critical
Figure 15 Translations with damping 2% to 30% critical
Translation mm
Figure 16 Accelerations at 10% damping with one loading cycle
x10
Figure 17 Accelerations at 10% damping with ten loading cycles
From the assessment of the pedestrian Scruton number, it can be seen from Table 6 that a damping ratio of 13% critical would be required to provide lateral stability during a crowded event on the bridge.
The model analysis, Figure 13, shows that with 13% damping, a peak acceleration of just under 0.2m/s2 is generated, along with an amplitude of 2.75mm.
The time history analysis model also shows that with damping of 10%, the lateral accelerations and amplitudes are within acceptable levels as defined in Table 9. Table 8 shows the model results for the 10% and 13% damping cases.
Table 8 Acceleration and damping for 10% and 13% damping
Table 9 shows the limiting criteria from the literature and compares this with the time history analysis results and Scruton number approach.
Table 9 Limiting lateral acceleration and amplitude criteria comparison
In situ testing of a working footbridge(Cantieni & Pietrzko
The Scruton number approach shows excellent consistency with the time history analysis results, with the Scruton number approach being only 5% more conservative than the results obtained from the time history analysis. This would suggest that design could be carried out safely using the Scruton number approach, and if this method shows problems with accelerations and amplitudes, a more detailed analysis using a time history analysis could be carried out. In addition, the provision for any dampers could be made during the design stage instead of the need for retrospective design once the structure is built.
It is worth noting at this point that the London Millennium Footbridge was retro fitted with mass dampers to give damping of 20% (Dallard, Fitzpatrick, Flint, Le Bourva, Low, Ridsdill-Smith, & Willford 2001), well above the limiting Scruton value number determined for the Millennium Footbridge. However, it must be noted here that a proportional damping value and explicit damping provided at specific locations, as is the practical application of tuned mass dampers, cannot be compared directly in an FE model.
Table 10 shows the damping, Scruton number and acceleration from the time history analysis for the test bridge.
Table 10 Comparison of damping values from Scruton number and analysis
Bold text is the limiting Scruton number and the acceleration for that value of damping from the time history analysis. The limiting Scruton number is 0.266, and from the Eurocode the limiting acceleration is 0.2m/s2 .
12.2 Dynamicanalysisresults
The dynamic analysis results were much as would be expected for a flexible footbridge such as this, with mode 1, 3 and 7 being the dominant lateral modes and mode 2 and 4 being the dominant vertical modes. The lateral frequency 1.5Hz and the vertical frequency of 2.6Hz, are in keeping with other flexible footbridges presented in the literature.
It is worth noting here however, that the lateral frequency of 1.5Hz is at the limit set by BD37/01, whilst the vertical frequency of 2.6Hz is well below the BD37/01 limits. Although vertical accelerations and amplitudes have not been investigated in this paper, these results suggest that the bridge may also be susceptible to excessive vertical vibrations.
12.3 Timehistoryanalysis
The application of the lateral load at a time history of 1.5Hz was enough to show appreciable increases in accelerations and lateral translations. This is shown in Figures 15 and 16. The time history analysis shows that the acceleration increases by 50% when the load is applied as a time history over 10 cycles as opposed to a single pulse load. This suggests that a time history analysis is the most appropriate method for assessing this load effect.
It is interesting to note in Figure 17, that the largest accelerations occur in the first 2 seconds, after which they reduce to a steady value. It is not clear why this is the case, although it is suggested in the literature that the increase in acceleration is not infinite, and settles at some limiting value. A possible explanation for this could be the application of the load itself in the model. In the time history analysis model the lateral loads at each 0.75m are applied concurrently at time T=0.67 seconds as a pulse load. The load application could be improved by applying the loading as a sine curve, thus removing the instantaneous application of the full load as a pulse and replacing it with a gradual build up over the 0.67 second cycle.
Another factor that may affect this result is that the load would build up slowly over time as more pedestrians become synchronised, and amplitudes increase, eventually peaking and settling at some constant value. The literature also suggests that dynamic action in a flexible footbridge would need at least 20 seconds to build up. (Bachmann 1992)
Also of note is that the application of the lateral walking load is not a periodic event (Anon 2000) and thus a considerable assumption is made here. The effect of the non-periodic walking would be to engage the pedestrian as an active damper, as the walking frequency would be out of synchronisation with the bridge frequency.
12.4 Accelerationsandtranslationslimits
The applicability of the acceleration limits suggested in the Eurocode are debatable for the case of synchronous lateral loading. This is because the limit state considered, that of an extreme crowd-loading event, may have a return period of 50 years or so. In fact, in the literature there are examples of bridges that have performed in service for over 100 years without problems until an extreme crowdloading event caused excessive lateral vibrations. The Eurocode does not fix differing acceptance criteria based on the return period for the particular loading cases.
Testing carried out for a cable stayed footbridge in Japan (Nakamura 2003) showed that pedestrians have difficulty walking at accelerations of 0.75m/s2, and some lost balance and had to stop walking to steady themselves at 1.35m/s2 Nakamura suggested a serviceability limit of 45mm amplitude with accelerations of 0.75 m/s2. This may seem a sensible limit for the case of an extreme crowd-loading event that would cause synchronous lateral loading, with a return time of 50 years. Accelerations for general serviceability could then be fixed at a more stringent value of 0.2m/s2 as is now the case in the Eurocode.
12.5 Crowdloadingandsynchronisationassumptions
In this paper, an attempt has been made to model the synchronous lateral loading effect with a time history analysis. In doing this analysis, the following parameters have been assumed:
1. The application of the load and the frequency of application are a representation of the estimated peak loads and frequencies that may result from a synchronous lateral loading event.
2. The synchronisation effect itself is not practical to model. The bridge amplitude and acceleration are all functions of load, which is in turn linked to the bridge amplitude, frequency, and synchronisation of the pedestrians, which is again related to amplitude.
3. The load application and frequency has been simplified to obtain an anticipated peak value as follows:
a. 40% of the crowd eventually becomes synchronised with the lateral amplitude of the bridge.
b. The bridge is fully crowded with 1.7 people/m2. This is the assumed vertical mass.
c. The walking frequency is 2.0Hz. The lateral frequency is half of this value. For analysis, the lateral component of the load is applied at the lateral natural frequency of the bridge.
d. The lateral load in each direction is applied along the length of the bridge as a pulse load at T=0.67 intervals for a period of 10 seconds.
e. The lateral component of the walking load is 4% of the vertical static value of 75kg.
f. The same lateral loading has been applied along the full length of the bridge. As pedestrian synchronisation is related to bridge amplitude, the synchronised load along the length of the bridge will change, as the amplitude of the bridge will reduce near the end supports.
By using these considered assumptions, it is possible to estimate bridge response to a peak crowd load.
It is also clear that these parameters can easily change. The walking frequency for example, can reduce as the congestion on the bridge increases, thus making bridges with lower natural frequencies much more susceptible to this type of effect.
The flow of pedestrians on and off the bridge can also have an effect on the synchronisation of the crowd. The literature suggest (Anon 2001) that the splitting of a crowd on a bridge can lead to further synchronisation as the flow of pedestrians change.
It has also been reported in tests carried out by Arup (Newland 2003) that pedestrians will synchronise their walking frequencies with each other without the effect of lateral translations. See Figure 2 by Arup, which indicates that during testing 30% synchronisation was recorded with zero amplitude. This suggests that the synchronisation effect is not wholly due to the perceptibility of movement, but to the subconscious effects of walking with and around other people.
13 Conclusions
The pedestrian Scruton number gives a good indication of potential problems with synchronous lateral loadings at the design stage. It gives good correlation with the results from the time history analysis. It is therefore suggested that the design of any flexible footbridge should be approached considering the pedestrian Scruton number in the first instance. Should this show problems with the flexibility of the structure then a time history analysis should be carried out to calculate accelerations and translations under the crowd induced loads. Methods can then be employed to provide the correct amount of damping to attain reasonable pedestrian comfort levels such as including mass dampers, or making changes to the geometry.
The ‘synchronous lateral load’ effect should be separated from the general comfort criteria proposed by most codes of practice for the design of footbridges. Design should address the probability of this type of event occurring, and assess the need to design for this loading condition by a risk assessment.
At present, there is no other real practical method for assessing the susceptibility of bridge structures to synchronous lateral loading other than the pedestrian Scruton number, or an advanced THA making the design parameter assumptions as presented in Section 12.5. This paper recommends the practical method summarized in Figure 18 for the design of a flexible footbridge subject to synchronous lateral loadings.
Presented in Figure 19 is practical guidance on loading levels, acceleration limits, structural damping, and technical guidance on design parameters to use in conjunction with Figure 18.
There are many areas of this subject still under research and Section 14 presents some key issues that require further investigation for a full understanding of this loading phenomenon.
The recommendations in this paper for the design of these structures is a pragmatic starting point considering research carried out to date, however there is still a lot of
work to be done to develop a code of practice for the assessment of this loading effect during the design stage. The focus of further research should be the development of a code of practice for design.
Figure
No
Carry out Time History Analysis
Configure damping to structural model to satisfy acceleration limits Decide on method to provide the required damping to produce acceptable accelerations
Analyse and design bridge
Start
Fix geometry based on BD37-01 requirements
Calculate pedestrian Scruton Number Is the Scruton Limit satisfied?
Analyse and design bridge Yes
Time history analysis of the lateral portion of the walking load
Carry out analysis for various damping ratios to achieve acceleration limits
This can be achieved by changing the bridge geometry or including tuned mass dampers
A time history analysis can be carried out to check accelerations
It should be noted here that proportional damping as expressed in terms of critical damping ratios and explicit damping in the case of TMD’s are not interchangeable values. Care should be taken if TDM’s are to be used to provide the damping. A separate study should be carried out to assess the explicit damping provided by the TMD’s.
Figure 19 Design parameters for use with Figure 18
14 Areasforfurtherresearch
14.1 Humanperceptibilitylimits
Human perceptibility of movement is a subjective issue. Some pedestrians are concerned and frightened by a small movement or acceleration and others are not. In addition, the activity of the pedestrian has a significant effect on the perception of movement, as does the pedestrians previous experience of the bridge vibration. As described in the literature review, torsional and lateral translations are more perceptible than vertical movements. Further research can be carried out here, as flexible footbridges, by their nature, will move. There are many suspension bridges in parts of Nepal for instance, that move significantly when crossed. In these cases, it is clear to the users that the bridge will move a significant amount.
A relaxation in the acceleration limits may be justified if signage is installed informing pedestrians that the bridge will move when crossed. However, this may not suit all users or the Client’s requirements. Certainly, consideration should be given to reviewing the acceleration limits for events that may have a 50-100 year return period, such as opening day of the bridge, or a significant sporting event that produces a crowd on the bridge of 1.7 people/m2 .
14.2 Finiteelementmodelupdating
Brownjohn carried out testing of a footbridge (Brownjohn 1997) and found that the natural frequencies predicted by the analysis were far too low when compared with the natural frequencies recorded on the bridge during testing. Also, a variation of temperature alone can cause a 7% change in the natural frequency of a bridge structure, (Ventura et al. 2002). This leads to the view that the only reliable method to assess the vibration performance of a footbridge is to test it after construction and retrofit dampers if required. Analysis by the finite element method would seem, from the literature to provide higher fundamental natural frequencies, and therefore a more conservative design if the recommendations in this paper are followed. Further research on existing structures to compare actual values with model predicted values
would be of value to give a better understanding of the design assumptions and results produced during the analysis process.
14.3
Studiesonthedampingofstructures
The damping of structures is also open to further research. It has been suggested (Anon 2001) that for every one person standing on the bridge deck the loads exerted by one pedestrian walker are nullified. This could allow a constant to be developed to reduce the synchronised crowd load further, based on the mass of the pedestrians themselves.
The actual damping of the bridge structure at the design stage is also difficult to predict, and most guidance on this design parameter comes from in-situ testing of completed bridges. Further real time testing of footbridges should be carried out and a study made of the damping, along with descriptions of construction details and nonstructural items such as handrails and kerbs to provide more useful data to engineers when considering the damping values to use during the design stage.
Currently damping values presented in BD37/01 seem to be very low, compared to values obtained from testing. Further investigations should be carried out to understand the difference in the code requirements to that of the test results, and modifications should be made if necessary to the code values.
14.4
Crowdinducedloadingandsynchronisation
The literature suggests (Anon 2001) that a bridge becomes congested at 1.7 pedestrians/m2. In the design method suggested in this paper, this load intensity is applied along the full length of the bridge, assuming 40% of the pedestrians synchronise their walking with that of the structure. This gives a high applied load. The return period of this abnormal loading should be considered and the acceleration limit re-assessed accordingly. There is evidence in the literature of bridges performing well for 100 years then being subjected to an extreme crowd event and vibrating. (Anon 2000). Therefore, the acceleration limits for this extreme load case
should be less onerous, and not included as a general serviceability comfort criteria as is now the case.
15 Bridgemodeshapevisuals
Mode 1 – 1.5183 Hz
Mode 2 – 2.6618 Hz
Mode 3 – 3.0030 Hz
Mode 4 – 3.4686 Hz
Mode 5 – 3.7099 Hz
Mode 6 – 4.2987 Hz
Mode 7 - 4.6195 Hz
Mode 8 – 6.1650 Hz
Mode 9 – 6.1672 Hz
Mode 10 – 6.4803 Hz
Mode 11 – 7.7829 Hz
Mode 12 – 8.0135 Hz
16 Glossary
Criticaldamping
Damping (proportional)
Ernstformula
ExplicitDamping
Finiteelementanalysis
This is the minimum amount of damping to prevent oscillation of a system.
To decrease the amplitude of an oscillation or wave. In this study, the damping is presented as a percentage of critical damping. Most floors and bridges have damping in the range of 1% to 5% critical.
A formula used to calculate the reduction in the modulus of elasticity of a cable to account for the sag in the cable.
Damping provided by a TMD at a particular point in the structure.
A computer simulation technique used in engineering analysis to simulate a structural element. It uses a numerical method called the finite element method.
Finiteelementmethod
Flexiblefootbridge
Flexiblestructure
The finite element method (FEM) is used for finding approximate solutions of partial differential equations as well as of integral equations.
A footbridge with a vertical natural frequency below 5.0Hz, and a lateral natural frequency below 1.5Hz.
A structure that when subjected to a dynamic load, undergoes significant translations or
Frequency
Frequencylockin
Harmonicload
Harmonicmotion
Hertz(Hz)
Humanperceptibility limits
Modalanalysis
Modeshape
Naturalfrequency (fundamentalfrequency)
PedestrianScruton number
vibrations.
A measure of the number of occurrences of a repeating event per unit time.
Phenomenon that occurs when the walking frequency synchronises with the structural frequency.
A load that causes harmonic motion.
This is a system which, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x according to Hooke’s law.
The SI measurement of frequency in cycles per second.
Limits of perception for humans subjected to accelerations in structures.
The study of the dynamic properties of structures under vibration excitation.
The oscillation or rotation of a structure subjected to vibration.
This is the lowest frequency in a harmonic series.
A non dimensional number relating modal damping to mass.
Periodofvibration
Resonance
Serviceability
Synchronouslateral loading
Timehistoryanalysis
Timeperiod
Tunedmassdamper
The time taken for one oscillatory cycle measured in seconds.
If a dynamic system is excited close to one of its natural frequencies then the eventual amplitude will be significantly higher than that due to the same excitation at a much higher or lower frequency.
Ability of the structure to perform in service, for vibration and deflections.
The phenomenon that occurs when the lateral walking frequency locks in with structures lateral natural frequency.
Application of a varying load over time.
See Period of vibration.
A tuned mass damper is a device mounted in structures to prevent discomfort, damage, or structural failure caused by vibration. A tuned damper balances the vibration of a system with comparatively lightweight component so that the worst-case vibrations are less intense.
Ultimatelimitstate Limit state of strength.
VortexShedding
An unsteady flow that takes place in special flow velocities. In this flow, vortices are created at the back of the body and detach periodically from either side of the body. Eventually, if the frequency of vortex shedding matches the
Young’smodulus
resonance frequency of the structure, the structure will begin to resonate.
Describes tensile elasticity, or the tendency of an object to deform along an axis when opposing forces are applied along that axis; it is defined as the ratio of tensile stress to tensile strain. It is often referred to simply as the elastic modulus, E.
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