4. Issues of Vibrations in Footbridges by Mahesh Tandon

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VI National Conference on Wind Engineering 2012, Dec. 14-15

ISSUES OF VIBRATIONS IN FOOTBRIDGES

Prof. Mahesh Tandon is an international expert in th field of Structural Engineering. He was appointed

Distinguished Visiting Professor at IIT Kanpur and IIT Roorkee by the Indian National Academy of

Engineering (INAE) and the All India Council for Technical Education (AICTE).

ABSTRACT

The phenomenon of vibrations in slender footbridges must be investigated to ensure safety and comfortof the pedestrians apart from its structural adequacy.

The Millennium Bridge in London which was designed adequately for wind as well as vertically applieddynamic loads of pedestrians witnessed the unique problem of lateral sway under pedestrian traffic.

The norm in present day designs of footbridges is to check its dynamic behaviour both for aerodynamicexcitation as well as that attributable to pedestrians.

The paper attempts to compile the approach to the possible solutions to the problem in a simplifiedmanner and also demonstrates the application of this approach to the design of 5 nos footbridges recentlyconstructed in Delhi.

Keywords :Footbridge, vibrations, steel, design, Delhi

Mahesh TandonManaging Director

Tandon Consultants Pvt. Ltd.New Delhi, India

E mail : [email protected]


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1 INTRODUCTION

Tall, long and slender structures are the order of the day. Additionally, footbridges are lightly loaded,which makes them even more susceptible to their vibrations due to smaller natural frequencies.

The dormant design issues relating to vibrations in footbridges are two-fold:

these created by aerodynamic excitation these created by footfalls of pedestrians

Usually, these vibrations impact only the serviceability of foot bridges when pedestrian comfort becomesof primary concern. Occasionally, the vibration of footbridges are seen to be excessive and could lead tostructural damage and in rare case to fatigue and failure.

The very nature of slender structures makes them susceptible to vibrations. However, it is resonancewhich is of primary concern, and can occur when the excitation frequency coincides with or is a multipleof one of the natural frequencies of the structure.

Issues relating to vibrations are complex because structures have different natural frequencies in the

various modes that can be excited by wind and footfalls as identified earlier. The modes of vibration thatcan be caused as a result can be:

flexural in horizontal direction (both lateral or longitudinal to the bridge) flexure in vertical direction torsion about the longitudinal axis

When confronted with the issue of resonance in footbridges, the first attempt should be to increase itsstiffness and mass or reduce its span so that its natural frequencies are outside the range that can beexcited by wind or footfalls. If such a remedy cannot be made to succeed on its own, then we must looktowards employing dampers which have been most effective in the past in controlling vibrations instructures such as very tall buildings.

The dynamic response of the structure can be evaluated only after its natural frequencies in vertical,lateral and torsional modes have been determined. A detailed computer analysis with appropriatesoftware that can permit identification of mode shapes clearly is therefore an essential requirement.

2 VIBRATIONS CAUSED BY AERODYNAMIC EXCITATION

2.1 Types of investigation

Most modern pedestrian bridges come in the category of wind sensitive structures. Though IRC:6 (Ref 1)has a chapter on Wind Load, such structures are excluded from its purview.

The British Standard BD 49/01 (Ref 2) is one code which deals with aerodynamic effects on suchstructures is some detail.

Aerodynamic excitation produce motions in isolated vertical bending or torsional modes, or, more rarely,in coupled vertical bending-torsional modes. The objection of the investigations is to ensure that thestructure under wind-excited vibrations has:

limited amplitude response for vortex excitation in bending and torsional modes of vibrations(clauses 2.1.1 and 3.1)


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limited amplitude response for dynamic turbulence loading, particularly if the calculatedfrequencies in bending and torsion are less than 1 Hz (clause 2.1.2)

divergent amplitude response which involves calculation of critical wind speed for galloping andstall flutter as well as for classical flutter (clause 2.1.3)

The code BD 49/01 defines a unique Aerodynamic Susceptibility Factor, Pb, which decides the categoryof the bridges as well as further investigations. This factor Pb is given in Table 1:

Table 1 Aerodynamic Susceptibility Factor (Ref 2)

The bridge categories from the point of view of susceptibility to wind excited vibrations are based onthe following criteria:

Pb < 0.04 : Insignificant effects in all forms of aerodynamic excitations

0.04 < Pb< 1.00 : Investigate limited amplitude response described in the foregoing

Pb > 1.00 : Potentially very susceptible to aerodynamic excitations and may require specialconsiderations including wind tunnel tests on scale models or computational fluiddynamics (CFD) procedures

2.2 Vortex excitation

The vortex excitation in bending and torsion first requires the evaluation of the critical wind speed (V cr) atwhich these phenomenon can occur, and, if found to be lower than the hourly mean wind speed at site,requires further investigation as per clause 3.1 of Ref 2.

If the fundamental frequency is greater than 5 Hz, the bridge can be assumed to be stable in vortexexcited vibrations.


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Clause 3.1 provides approximate formulations for determining amplitude ymax of vibrations and a dynamicsensitivity factor kD to assess the structural adequacy for withstanding the effects of ymax as well as forpedestrian comfort.

2.3 Turbulence effects

Because of the turbulent nature of wind the forces and moments in the structure can be magnified if anyof the frequencies lie within a range, which is fairly large, as per clause 2.1.2 of BD 49/01,

However, in case the fundamental frequencies of the structure are greater than 1 Hz, magnification due tothis effect may be ignored.

2.4 Divergent amplitude response

Response of the structures to wind excitation also needs to be investigated for divergent amplituderesponse. This involves the evaluation of critical wind speed for galloping and stall flutter (Vg) as well asclassical flutter (Vf) in both vertical and torsional motion. Some types of bridge cross sections are exemptfrom this investigation as detailed in clause 2.1.3 of BD 49/01.

The critical wind speed so calculated are then compared with a hypothetical wind storm speed (Vwo) that

could occur at site knowing the maximum wind gust speed (V d) and a specified coefficient of probability ofoccurrence (k1A).

If the wind storm speed (Vwo) is found to be less than V f and Vg, the structure can be considered to bestable for divergent amplitude response.

3 VIBRATIONS CAUSED BY FOOTFALL OF PEDESTRIANS

3.1 Steps for assessment

Slender bridges with low mass and low damping when used by crowds may cause unacceptablevibrations. A lock-in effect, i.e., synderonisation of the bridge vibrations with the footfalls frequency canlead to resonance. The age-old convention of requiring a marching army to break step while crossing abridge to avoid resonance is well known.

Until the opening of the Millennium Bridge in London on 10th

June 2000 attention was paid to verticalforces and vibrations induced by pedestrians. The bridge which had been designed for vertical excitationexhibited horizontal (i.e., lateral) sway with amplitude and accelerations that were uncomfortable for thepedestrians. The bridge had to be closed down and retrofitted with damping devices before it could be putinto service again.

The typical pacing frequency for walking is around 2 steps per second. The range for slow to fast walkingcan be in the range of vertical forcing frequency of 1.2 to 2.4 Hz. Since the lateral component of the forceis applied at half the footfall frequency it can be estimated to be in the range 0.7 to 1.2 Hz.

It can therefore be concluded that both vertical and horizontal excitation due to pedestrians crossing thebridge must be investigated so as to avoid excessive vibrations.

The density of pedestrian traffic crossing the bridge at a given time is important. If the traffic is too dense,the excitation forces would come down. A maximum of 1.5 persons per sqm of deck area is considered tobe appropriate for the these considerations (for example Ref 7, 8, 9, 10)

The first step in the evaluation of susceptibility of a footbridge to vibrations is to calculate its naturalfrequencies. The second step is to check whether these frequencies are in the critical range that cancause unacceptable vibrations. In case the frequencies are not outside the critical range, the third stepwould be to carryout a dynamic analysis with an appropriate structural model subjected to a pulsating


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load representing the stream of footfalls of the bridge. In case the acceleration determined from thedynamic analysis exceeds the prescribed limit, the fourth step would be to resort to dampers (such asTMD) to control the vibrations.

3.2 Critical range of frequencies

In most codes of practice (Ref 3, 4, 5, 6) and published literature (Ref 7, 8) the pedestrian excitation in thefirst harmonic leads to the following critical range for natural frequencies:

for vertical and longitudinal vibrations:1.25Hz < fi < 2.3Hz

for lateral vibrations:

0.5 Hz < fi < 1.2Hz

In case the second harmonic of pedestrian excitation is taken into account, the critical range for verticaland longitudinal vibrations expands to:

1.25Hz < fi < 4.6Hz

Incidentally, lateral vibrations are not affected by the second harmonic because of the very nature of loadexcitation.

The revised British Standard BD 29/04 (Ref 4) mentions that the critical range should be considered asfrequencies less than 5Hz for vertical vibrations and 1.5 Hz for lateral vibrations. The recently publishedIndian code IRC:SP:56-2011 (Ref 9) on Steel Pedestrian Bridges has provisions that are identical to thatof BD 29/04 for the critical range of vertical and lateral frequencies, ie, 5Hz and 1.5 Hz respectively. Inboth Refs 4 and 9 if the natural frequencies are not outside the critical range it is obligatory to determinethe maximum accelerations caused by pedestrian footfalls. Guidance, though is given only for simplecases in Refs 4 and 9.

3.3 Max accelerations acceptable for human comfort

There are several codes of practice (Ref 3, 4, 5, 6) which specify the maximum acceptable accelerationfor human comfort for Footbridges. Also, the Indian Code (Ref 9) gives simplified method for evaluatingfrequencies and accelerations which can be applied in some cases. There are also recent interestingresearch papers (Ref 7, 8) on the subject which are possibly the fore-runners of modified criteria that mayfind place in future editions of codes.

Hauksson (Ref 10) has made a comparative study of present codes and has tabulated the acceptablecriteria as given in Table 2.

Table 2: Acceleration Criteria for Pedestrian Comfort (Ref 10)

Fig 1 Fig 2


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The curves of ISO 10137 (Ref 5) for vertical and lateral acceleration are reproduced here as Figs 1 and 2respectively.

Fig 1: Vertical vibration base curve for acceleration (Ref 5)

Fig 2: Horizontal vibration base curve for acceleration (Ref 5)


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Hauksson has also compared the provisions of various codal provisions graphically as shown in Figs 3and 4

Fig 3: Comparison of acceptability of vertical vibration (Ref 10)

Fig 4: Comparison of acceptability of horizontal vibration (Ref 10)


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Research studies made by HIVOSS (Ref 7) and SETRA (Ref 8) are of great interest as they not onlyindicate human comfort criteria in terms of accepted accelerations but also suggest the appropriatepulsating loads which could represent a stream of pedestrians crossing the foot bridge for the purpose ofevaluation of accelerations.

4 EXAMPLE

4.1 Background

Five pedestrian bridges of similar design were executed in the city of Delhi in the last couple of years. Theconcept selected involved a steel arch bridge with a suspended walkway, (Figs 5, 6, 7, 8, 9, 10), Archbridges by their very form are aesthetic to behold and can more easily span across wide roads. Hithertopedestrian bridges crossing over wide and heavily trafficked roads were invariably provided with a centralsupport at the median of the bridge. Such a design is not entirely safe from the motorised traffic plying onboth sides of the median verge and they must be designed to cater to vehicle collision loads, Refs 6, 9.Median supports when provided become fairly massive in appearance and difficult to fit into the generalaesthetics of a slender footbridge (Fig 11) which is an important consideration for a structure, being inprominent public view in the urban environment.

Fig 5 Pedestrian Bridge: Elevation

The design concept (Fig 7) had to cater to arch spans of 90m, 80m and 66m at the different roadcrossings and at the same time be capable of implementation without seriously disturbing the existingtraffic, underground and overhead utilities and work in progress by other agencies. The arch and thewalkway in structural steel could be manufactured in a quality fabrications shop equipped with therequisite facilities and then shipped in transportable segments to site and erected by crane (Fig 10) fittedthe bill perfectly. The connections of steel segments were effected essentially by HSFG bolts to restrict tothe minimum any site welding. An accredited stainless steel bar system was selected for the suspenderswhich had the facility of length adjustment during construction so as to obtain the required deck profileand camber on completion.

With the codal provisions and experience in recent times it became imperative to check that the proposedpedestrian bridges would not only meet the structural design criteria for static loads but also the dynamicbehviour which is vital for the comfort and safety of the user.

Preliminary design stage investigations were done both for static as well as dynamic loading. The latterconsideration revealed that both from the aerodynamic excitation due to wind and pedestrian comfortpoints of view special attention would be required to the dynamic response of the bridges apart from the


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gravity loads. One of the important early decision was to convert the light steel plate decking of thewalkway to a heavier and stiffer concrete slab which would reduce the vibrations significantly. Theconcrete slab, Fig 6, cast on a prefabricated metal deck was made composite to the main longitudinalmembers of the truss below, which incidentally would also enhance its strength against accidents causedby vertical protrusions during passage of errant over-dimensioned vehicles.

Fig 6 Pedestrian Bridge: Section

The attempt of the engineering design was to create aesthetic cost effective and robust structures whichwould not only be durable but also require minimum inspection and maintenance during serviceconditions. The design therefore had to obviate the necessity of using dampers.

As the time available for the design and construction was short, the luxury of wind tunnel testing wasreplaced by a more conservative design approach.

For the extracts of calculations that follow the example of the 80m span arch bridge has been selected.The live load cases considered on the bridge for aerodynamic consideration were of three types:

Full live load 500 kg/sqm

Pedestrian density of 1.5 persons per sqm (pedestrian wt=70kg)

No live load

It was found in the investigation that in almost all the cases of aerodynamic excitations it is advantageousto have more mass and higher stiffness which led to higher frequencies.

For pedestrian excitation the live load density was assumed as 1.5 persons per sqm. The investigationsrelating to user comfort were carried out in accordance with HIVOSS, Ref 7, wherein this specifiedloading comes under Traffic class TC5 and described on exceptionally dense traffic.


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Fig 7 Sketch Showing Concept of Steel Arch Bridge with

Suspended Walkway

Fig 8 Photograph of one of the Completed Bridges


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Fig 9 Photograph of one of the Completed Bridges

Fig 10 Arch Bridge during Erection


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Fig 11 Pedestrian Bridge with Support at Median

4.2 Vibrations caused by wind excitation

4.2.1 As a first step we calculate the Aerodynamics susceptibility factor Pb. The data thatformed the basis of this calculation is given in Table 3.

Table 3 Basic Data

Notes: 1) The frequencies fB and fT were obtained from STAAD analysis.

2) The hourly wind speed has been taken from Ref 1.


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The type of cross section for aerodynamic excitation purposes can be taken as type IA as per Fig 1 of BD49/01.

The values of Pb thus determined were in the range of 0.39 to 0.77 with the three live load casesidentified earlier. As mentioned in para 2.1 above, we are in the range 0.44 < Pb < 1.00 and it becamenecessary to investigate limited amplitude response for various effects of aerodynamic excitation.

4.2.2 In the second step limited amplitude for vortex excitation was investigated. The critical windspeed for vortex excitation depends on the aspect ratio of the cross-section i.e., b/d4, where b= width ofsoffit (4.1m) and d4 depth of the section (0.85m). With b/d4 of 4.82 and the bending frequencies fBalready available, the critical wind speed was evaluated from the formula:

Vcr = 6.5 fB.d4,

Which gave values of 14.8 to 17.3 m/sec for the three live load cases, all of which are within the range of1.25x the reference wind speed, V r. Hence further investigations as per clause 3.1 for vertical bendingduring vortex excitation became necessary.

Incidentally, the torsional mode need not be investigated when the frequency fT is greater than 5 as wasin the present case.

Clause 3.1 of BD 49/01 indicates the displacement values ymax for both vertical and torsional vibrations.Applying these formula with logarithmic decrement due to structural damping s = 0.03, applicable tosteel, and the specified amplitude correction factor, we get ymax values ranging from 15 to 25mm forvertical vibrations.

The dynamic sensitivity factor, KD, given the code requires to be used with care and judgment particularlyin the case of pedestrian comfort.

4.2.3 In the third step we evaluate the limited amplitude response due to turbulence. Since thefundamental frequencies in both bending and in torsion are greater than 1Hz, the dynamic magnificationeffects can be ignored.

4.2.4 In the fourth step we first evaluate the critical wind speed for the cross section in vertical and

torsional motion for galloping and stall flutter (Vg) as well as classical flutter (Vf). As a matter of fact, thecross-section type IA, which is relevant to the present case is exempt form the calculation of Vg forvertical motion, while that for torsional motion can be calculated from

Vg= 3.3 fT b,

which gives values between 90.7 to 106.0 m/sec for the three live load cases.

Further, using the equations in clause 2.1.3.3 of BD 49/01, we can arrive at the critical wind speed forclassical flutter (Vf)which depends on the ratio of frequencies fB/fT. The critical wind speeds were found tobe in the range 341.4 and 310.9 m/sec for the three live load cases.

For the wind storm speed (Vwo) we first calculate the Maximum Wind Gust speed (V d) which was found tobe in the range 36.0 to 47.0 m/sec for the three live load cases. Taking the coefficient of probability of

occurrence K1A = 1.4 as specified for tropical cyclone conditions, Vwo can be evaluated as being in therange 55.4 to 72.4 m/sec for the three live load cases.

Since both Vg and Vf are much higher than Vwo evaluated for the site, the possibility of galloping and stallflutter as well as classical flutter can be ruled out.

4.3 Vibrations caused by pedestrian footfalls


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4.3.1 As a first step we calculate the natural frequencies of the bridge for the vertical (bending),longitudinal and lateral motions. The density of pedestrian traffic is assumed as 1.5 persons/sqm. Aseach type of motion mentioned above may be excited at different modes, they have been identifiedseparately in Table 4.

Table 4 Frequencies V/S Critical Range

4.3.2 In the second step we compare the natural frequencies with the critical range defined in para 3.2above.

From this criteria it was found that all the frequencies were outside the min/max range except the firstlongitudinal mode, refer Table 4, which must be investigated further.

4.3.3 In the third step we investigate whether the acceleration determined form dynamic analysis isacceptable from the point of view of pedestrian comfort. HIVOSS specifies maximum acceleration limitsfor different comfort classes as given Table 5.

Table 5 Comfort Classes


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HIVOSS also cautions against lock-in for lateral motions that could lead to a vanishing of the overalldamping response. The trigger lock-in phenomenon involving a sudden amplitude response could happenat as early as at an acceleration = 0.1 to 0.15 m/sec

2.

For the harmonic model, a uniformly distributed harmonic load p(t) is taken to represent the stream ofpedestrians as shown in Tables 6, 7 (units are N, m).

Table 6 Harmonic Load

Table 7 Parameters for Harmonic Loading (Ref 7)


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The max accelerations in three modes are evaluated through STAAD and depicted in Table 8.

Table 8 Max Acceleration V/S Comfort Class

It can be seen that in all the three motions the comfort class is CL1 which is acceptable without takingrecourse to external damping

5 REFERENCES


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[1]. IRC: 6 Standard Specifications and Code of Practice for Road Bridges, Section II: Load &Stresses. Indian Roads Congress, 2010.

[2]. Design Manual for Roads and Bridges. Design Rules for Aerodynamic Effects on Bridges: BD

49/01. Highways Agency, London. May 2001.

[3]. Eurocode, Basis of Structural Design pr Annex A2. EN 1990: 2002. European Committee forStandardization, Brussels, Belgium 2002.

[4]. Design Manual for Road and Bridges: Design Criteria for Footbridges: BD 29/04, HighwayAgency, London, February 2004.

[5]. ISO: Basis for design of structures. Serviceability of buildings and pedestrian walkways againstvibration, ISO/CD 10137, International Standardization organization, Geneva, Switzerland, 2005.

[6]. BRO 2004, Swedish Road Administration Standard, Stockholm, Sweden.

[7]. Human Induced Vibrations of Steel Structures (HIVOSS). Design of Footbridges (Guideline-EN

03). Publications office of the European Union, Luxemburg, Sept 2008.

[8]. Technical Guide for Footbridges: Assessment of Vibrational Behaviour of Footbridges underpedestrian loading. SETRA, France, Oct 2006.

[9]. IRC: SP: 56-2011, Guidelines for Steel Pedestrian Bridges, Indian Road Congress, New Delhi,May 2011.

[10]. Hauksson, F. Dynamic Behaviour of Footbridges subjected to Pedestrian-Induced Vibrations.Lund University, Sweden, November 2005.

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4. Issues of Vibrations in Footbridges by Mahesh Tandon