4. Issues of Vibrations in Footbridges by Mahesh Tandon

VI National Conference on Wind Engineering2012, Dec. 14-15 ISSUES OF VIBRATIONS IN FOOTBRIDGES Mahesh Tandon Managing Director Tandon Consultants Pvt. Ltd. New Delhi, India E mail : [email protected] 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 comfort of the pedestrians apart from its structural adequacy. The Millennium Bridge in London which was designed adequately for wind as well as vertically applied dynamic 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 aerodynamic excitation as well as that attributable to pedestrians. The paper attempts to compile the approach to the possible solutions to the problem in a simplified manner and also demonstrates the application of this approach to the design of 5 nos footbridges recently constructed in Delhi. Keywords: Footbridge, vibrations, steel, design, Delhi 63 A detailed computer analysis with appropriate software that can permit identification of mode shapes clearly is therefore an essential requirement. such structures are excluded from its purview.1. However. Dec. The very nature of slender structures makes them susceptible to vibrations. Additionally. 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. 2 2.1 VIBRATIONS CAUSED BY AERODYNAMIC EXCITATION Types of investigation Most modern pedestrian bridges come in the category of wind sensitive structures. in coupled vertical bending-torsional modes.1 and 3. Though IRC:6 (Ref 1) has a chapter on Wind Load. Aerodynamic excitation produce motions in isolated vertical bending or torsional modes. more rarely. or. The dynamic response of the structure can be evaluated only after its natural frequencies in vertical. The modes of vibration that can 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. 14-15 1 INTRODUCTION Tall. and can occur when the excitation frequency coincides with or is a multiple of one of the natural frequencies of the structure. then we must look towards employing dampers which have been most effective in the past in controlling vibrations in structures such as very tall buildings. Occasionally. The objection of the investigations is to ensure that the structure under wind-excited vibrations has: – limited amplitude response for vortex excitation in bending and torsional modes of vibrations (clauses 2. the first attempt should be to increase its stiffness and mass or reduce its span so that its natural frequencies are outside the range that can be excited by wind or footfalls. If such a remedy cannot be made to succeed on its own. it is resonance which is of primary concern. long and slender structures are the order of the day. lateral and torsional modes have been determined. which makes them even more susceptible to their vibrations due to smaller natural frequencies. The British Standard BD 49/01 (Ref 2) is one code which deals with aerodynamic effects on such structures is some detail.1) 64 . footbridges are lightly loaded. these vibrations impact only the serviceability of foot bridges when pedestrian comfort becomes of primary concern. The dormant design issues relating to vibrations in footbridges are two-fold: – – these created by aerodynamic excitation these created by footfalls of pedestrians Usually.VI National Conference on Wind Engineering 2012. the vibration of footbridges are seen to be excessive and could lead to structural damage and in rare case to fatigue and failure. 00 : Insignificant effects in all forms of aerodynamic excitations : Investigate limited amplitude response described in the foregoing : Potentially very susceptible to aerodynamic excitations and may require special considerations including wind tunnel tests on scale models or computational fluid dynamics (CFD) procedures 2. which decides the category of the bridges as well as further investigations.00 Pb > 1.3) The code BD 49/01 defines a unique Aerodynamic Susceptibility Factor. 14-15 – limited amplitude response for dynamic turbulence loading.2) – divergent amplitude response which involves calculation of critical wind speed for galloping and stall flutter as well as for classical flutter (clause 2. and. if found to be lower than the hourly mean wind speed at site. 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 on the following criteria: Pb < 0.1 of Ref 2. Dec.1.04 < Pb< 1. requires further investigation as per clause 3.VI National Conference on Wind Engineering 2012. 65 . the bridge can be assumed to be stable in vortex excited vibrations. particularly if the calculated frequencies in bending and torsion are less than 1 Hz (clause 2. P b.1.2 Vortex excitation The vortex excitation in bending and torsion first requires the evaluation of the critical wind speed (V cr) at which these phenomenon can occur. If the fundamental frequency is greater than 5 Hz.04 0. In case the frequencies are not outside the critical range.3 Turbulence effects Because of the turbulent nature of wind the forces and moments in the structure can be magnified if any of the frequencies lie within a range.2 Hz. The typical pacing frequency for walking is around 2 steps per second. The bridge had to be closed down and retrofitted with damping devices before it could be put into service again. the excitation forces would come down. The range for slow to fast walking can be in the range of vertical forcing frequency of 1. 14-15 Clause 3.2 of BD 49/01. i. The bridge which had been designed for vertical excitation exhibited horizontal (i. magnification due to this effect may be ignored. 10) The first step in the evaluation of susceptibility of a footbridge to vibrations is to calculate its natural frequencies.. If the traffic is too dense. the third step would be to carryout a dynamic analysis with an appropriate structural model subjected to a pulsating th 66 .7 to 1. If the wind storm speed (Vwo) is found to be less than Vf and Vg. synderonisation of the bridge vibrations with the footfalls frequency can lead to resonance.. This involves the evaluation of critical wind speed for galloping and stall flutter (Vg) as well as classical flutter (Vf) in both vertical and torsional motion.VI National Conference on Wind Engineering 2012.4 Hz. The second step is to check whether these frequencies are in the critical range that can cause unacceptable vibrations. lateral) sway with amplitude and accelerations that were uncomfortable for the pedestrians.3 of BD 49/01.4 Divergent amplitude response Response of the structures to wind excitation also needs to be investigated for divergent amplitude response. Dec. 2. The age-old convention of requiring a marching army to break step while crossing a bridge to avoid resonance is well known. Until the opening of the Millennium Bridge in London on 10 June 2000 attention was paid to vertical forces and vibrations induced by pedestrians.2 to 2. which is fairly large.e. 8.e. Some types of bridge cross sections are exempt from this investigation as detailed in clause 2. The critical wind speed so calculated are then compared with a hypothetical wind storm speed (V wo) that could occur at site knowing the maximum wind gust speed (V d) and a specified coefficient of probability of occurrence (k1A). the structure can be considered to be stable for divergent amplitude response.1. as per clause 2. 9. Since the lateral component of the force is applied at half the footfall frequency it can be estimated to be in the range 0. However. It can therefore be concluded that both vertical and horizontal excitation due to pedestrians crossing the bridge must be investigated so as to avoid excessive vibrations.1.1 provides approximate formulations for determining amplitude ymax of vibrations and a dynamic sensitivity factor kD to assess the structural adequacy for withstanding the effects of ymax as well as for pedestrian comfort.5 persons per sqm of deck area is considered to be appropriate for the these considerations (for example Ref 7. in case the fundamental frequencies of the structure are greater than 1 Hz. 2. 3 3. A maximum of 1.1 VIBRATIONS CAUSED BY FOOTFALL OF PEDESTRIANS Steps for assessment Slender bridges with low mass and low damping when used by crowds may cause unacceptable vibrations. The density of pedestrian traffic crossing the bridge at a given time is important. A ‘lock-in effect. 6) which specify the maximum acceptable acceleration for human comfort for Footbridges. 8) on the subject which are possibly the fore-runners of modified criteria that may find place in future editions of codes. Hauksson (Ref 10) has made a comparative study of present codes and has tabulated the acceptable criteria as given in Table 2.25Hz < fi < 4. 3. the Indian Code (Ref 9) gives simplified method for evaluating frequencies and accelerations which can be applied in some cases. The revised British Standard BD 29/04 (Ref 4) mentions that the critical range should be considered as frequencies less than 5Hz for vertical vibrations and 1.25Hz < fi < 2. lateral vibrations are not affected by the second harmonic because of the very nature of load excitation. 4. 8) the pedestrian excitation in the first harmonic leads to the following critical range for natural frequencies: – for vertical and longitudinal vibrations: 1. Also. Guidance. 14-15 load representing the stream of footfalls of the bridge.6Hz Incidentally. the critical range for vertical and longitudinal vibrations expands to: 1. 5Hz and 1. ie. There are also recent interesting research papers (Ref 7.5 Hz for lateral vibrations.5 Hz < fi < 1.3 Max accelerations acceptable for human comfort There are several codes of practice (Ref 3. 5. 3. 5.VI National Conference on Wind Engineering 2012. The recently published Indian code IRC:SP:56-2011 (Ref 9) on Steel Pedestrian Bridges has provisions that are identical to that of BD 29/04 for the critical range of vertical and lateral frequencies.2Hz In case the second harmonic of pedestrian excitation is taken into account. In both Refs 4 and 9 if the natural frequencies are not outside the critical range it is obligatory to determine the maximum accelerations caused by pedestrian footfalls. though is given only for simple cases in Refs 4 and 9. the fourth step would be to resort to dampers (such as TMD) to control the vibrations.2 Critical range of frequencies In most codes of practice (Ref 3.5 Hz respectively. Fig 1 Fig 2 Table 2: Acceleration Criteria for Pedestrian Comfort (Ref 10) 67 . 4. 6) and published literature (Ref 7.3Hz – for lateral vibrations: 0. In case the acceleration determined from the dynamic analysis exceeds the prescribed limit. Dec. Dec. 14-15 The curves of ISO 10137 (Ref 5) for vertical and lateral acceleration are reproduced here as Figs 1 and 2 respectively.VI National Conference on Wind Engineering 2012. Fig 1: Vertical vibration base curve for acceleration (Ref 5) Fig 2: Horizontal vibration base curve for acceleration (Ref 5) 68 . Dec.VI National Conference on Wind Engineering 2012. 14-15 Hauksson has also compared the provisions of various codal provisions graphically as shown in Figs 3 and 4 Fig 3: Comparison of acceptability of vertical vibration (Ref 10) Fig 4: Comparison of acceptability of horizontal vibration (Ref 10) 69 . With the codal provisions and experience in recent times it became imperative to check that the proposed pedestrian bridges would not only meet the structural design criteria for static loads but also the dynamic behviour 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.1 EXAMPLE Background Five pedestrian bridges of similar design were executed in the city of Delhi in the last couple of years. underground and overhead utilities and work in progress by other agencies. 14-15 Research studies made by HIVOSS (Ref 7) and SETRA (Ref 8) are of great interest as they not only indicate human comfort criteria in terms of accepted accelerations but also suggest the appropriate pulsating loads which could represent a stream of pedestrians crossing the foot bridge for the purpose of evaluation of accelerations. The latter consideration revealed that both from the aerodynamic excitation due to wind and pedestrian comfort points of view special attention would be required to the dynamic response of the bridges apart from the 70 . 9. The arch and the walkway in structural steel could be manufactured in a quality fabrications shop equipped with the requisite facilities and then shipped in transportable segments to site and erected by crane (Fig 10) fitted the bill perfectly. Median supports when provided become fairly massive in appearance and difficult to fit into the general aesthetics of a slender footbridge (Fig 11) which is an important consideration for a structure. being in prominent public view in the urban environment.VI National Conference on Wind Engineering 2012. Dec. (Figs 5. 10). 80m and 66m at the different road crossings and at the same time be capable of implementation without seriously disturbing the existing traffic. 9. The concept selected involved a steel arch bridge with a suspended walkway. The connections of steel segments were effected essentially by HSFG bolts to restrict to the minimum any site welding. Arch bridges by their very form are aesthetic to behold and can more easily span across wide roads. An accredited stainless steel bar system was selected for the suspenders which had the facility of length adjustment during construction so as to obtain the required deck profile and camber on completion. 8. Refs 6. Such a design is not entirely safe from the motorised traffic plying on both sides of the median verge and they must be designed to cater to vehicle collision loads. 6. Hitherto pedestrian bridges crossing over wide and heavily trafficked roads were invariably provided with a central support at the median of the bridge. 7. Fig 5 Pedestrian Bridge: Elevation The design concept (Fig 7) had to cater to arch spans of 90m. 4 4. 5 persons per sqm. Fig 6. For pedestrian excitation the live load density was assumed as 1. 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. The investigations relating to user comfort were carried out in accordance with HIVOSS. 71 . which incidentally would also enhance its strength against accidents caused by vertical protrusions during passage of errant over-dimensioned vehicles. The design therefore had to obviate the necessity of using dampers.VI National Conference on Wind Engineering 2012. Ref 7. cast on a prefabricated metal deck was made composite to the main longitudinal members of the truss below.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 advantageous to have more mass and higher stiffness which led to higher frequencies. One of the important early decision was to convert the light steel plate decking of the walkway to a heavier and stiffer concrete slab which would reduce the vibrations significantly. The concrete slab. 14-15 gravity loads. the luxury of wind tunnel testing was replaced by a more conservative design approach. wherein this specified loading comes under Traffic class TC5 and described on “exceptionally dense traffic”. Dec. Fig 6 Pedestrian Bridge: Section The attempt of the engineering design was to create aesthetic cost effective and robust structures which would not only be durable but also require minimum inspection and maintenance during service conditions. For the extracts of calculations that follow the example of the 80m span arch bridge has been selected. As the time available for the design and construction was short. VI National Conference on Wind Engineering 2012. Dec. 14-15 Fig 7 Sketch Showing Concept of Steel Arch Bridge with Suspended Walkway Fig 8 Photograph of one of the Completed Bridges 72 . Dec. 14-15 Fig 9 Photograph of one of the Completed Bridges Fig 10 Arch Bridge during Erection 73 .VI National Conference on Wind Engineering 2012. 2 Vibrations caused by wind excitation 4. 2) The hourly wind speed has been taken from Ref 1.1 As a first step we calculate the Aerodynamics susceptibility factor Pb. Dec. 74 .2.VI National Conference on Wind Engineering 2012. The data that formed 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. 14-15 Fig 11 Pedestrian Bridge with Support at Median 4. The critical wind speed for vortex excitation depends on the aspect ratio of the cross-section i.2. As mentioned in para 2.1 above.3 In the third step we evaluate the limited amplitude response due to turbulence.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).25x the reference wind speed.03. while that for torsional motion can be calculated from Vg= 3.1 of BD 49/01 indicates the displacement values ymax for both vertical and torsional vibrations. The critical wind speeds were found to be in the range 341. the dynamic magnification effects can be ignored. Hence further investigations as per clause 3.4 and 310. Further.39 to 0.0 m/sec for the three live load cases.3.1. where b= width of soffit (4.3 fT b.4 m/sec for the three live load cases. the torsional mode need not be investigated when the frequency f T is greater than 5 as was in the present case.3 of BD 49/01. Vr.4 to 72. given the code requires to be used with care and judgment particularly in the case of pedestrian comfort.0 to 47.9 m/sec for the three live load cases. 4.00 and it became necessary to investigate limited amplitude response for various effects of aerodynamic excitation.7 to 106. 4. KD.5 fB. As a matter of fact.3 Vibrations caused by pedestrian footfalls 75 . Which gave values of 14. applicable to steel. 14-15 The type of cross section for aerodynamic excitation purposes can be taken as type IA as per Fig 1 of BD 49/01.85m). 4. Taking the coefficient of probability of occurrence K1A = 1. Dec. For the wind storm speed (Vwo) we first calculate the Maximum Wind Gust speed (V d) which was found to be in the range 36. Applying these formula with logarithmic decrement due to structural damping δ s = 0.77 with the three live load cases identified earlier.2 In the second step limited amplitude for vortex excitation was investigated.44 < P b < 1.8 to 17.d4.82 and the bending frequencies fB already available. the cross-section type IA. all of which are within the range of 1. Since both Vg and Vf are much higher than Vwo evaluated for the site. we get ymax values ranging from 15 to 25mm for vertical vibrations. we can arrive at the critical wind speed for classical flutter (Vf) which depends on the ratio of frequencies f B/fT. the possibility of galloping and stall flutter as well as classical flutter can be ruled out. the critical wind speed was evaluated from the formula: Vcr = 6. b/d4.VI National Conference on Wind Engineering 2012.1 for vertical bending during vortex excitation became necessary. 4. which is relevant to the present case is exempt form the calculation of Vg for vertical motion.e.0 m/sec for the three live load cases.2. With b/d4 of 4. The dynamic sensitivity factor. using the equations in clause 2. Incidentally.1m) and d4 depth of the section (0. Since the fundamental frequencies in both bending and in torsion are greater than 1Hz. The values of Pb thus determined were in the range of 0. which gives values between 90. V wo can be evaluated as being in the range 55.3 m/sec for the three live load cases.. we are in the range 0.2.4 as specified for tropical cyclone conditions. and the specified amplitude correction factor. Clause 3. 14-15 4.3. 4.5 persons/sqm. As each type of motion mentioned above may be excited at different modes.3. longitudinal and lateral motions. Table 4 Frequencies V/S Critical Range 4.2 In the second step we compare the natural frequencies with the critical range defined in para 3. From this criteria it was found that all the frequencies were outside the min/max range except the first longitudinal mode.1 As a first step we calculate the natural frequencies of the bridge for the vertical (bending). Table 5 Comfort Classes 76 .3. they have been identified separately in Table 4.VI National Conference on Wind Engineering 2012. refer Table 4. HIVOSS specifies maximum acceleration limits for different comfort classes as given Table 5. The density of pedestrian traffic is assumed as 1. Dec.2 above. which must be investigated further.3 In the third step we investigate whether the acceleration determined form dynamic analysis is acceptable from the point of view of pedestrian comfort. m). For the harmonic model. 7 (units are N. Dec. 14-15 HIVOSS also cautions against lock-in for lateral motions that could lead to a vanishing of the overall damping response.15 m/sec . a uniformly distributed harmonic load p(t) is taken to represent the stream of pedestrians as shown in Tables 6.VI National Conference on Wind Engineering 2012. The trigger lock-in phenomenon involving a sudden amplitude response could happen 2 at as early as at an acceleration = 0.1 to 0. Table 6 Harmonic Load Table 7 Parameters for Harmonic Loading (Ref 7) 77 . VI National Conference on Wind Engineering 2012. 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 taking recourse to external damping 5 REFERENCES 78 . Dec. 14-15 The max accelerations in three modes are evaluated through STAAD and depicted in Table 8. Design Rules for Aerodynamic Effects on Bridges: BD 49/01. Geneva. Dec. [3]. International Standardization organization. European Committee for Standardization. Basis of Structural Design – pr Annex A2. ISO: Basis for design of structures.VI National Conference on Wind Engineering 2012. ISO/CD 10137. [2]. France. SETRA. [9]. New Delhi. November 2005. [8]. Section II: Load & Stresses. 79 . Sweden. Luxemburg. [7]. Hauksson. IRC: 6 Standard Specifications and Code of Practice for Road Bridges. Sweden. Dynamic Behaviour of Footbridges subjected to Pedestrian-Induced Vibrations. Belgium 2002. Switzerland. Design Manual for Roads and Bridges. May 2001. 14-15 [1]. 2010. [4]. Lund University. Eurocode. EN 1990: 2002. Design Manual for Road and Bridges: Design Criteria for Footbridges: BD 29/04. May 2011. February 2004. Highways Agency. Technical Guide for Footbridges: Assessment of Vibrational Behaviour of Footbridges under pedestrian loading. BRO 2004. IRC: SP: 56-2011. Publications office of the European Union. Design of Footbridges (Guideline-EN 03). Highway Agency. Human Induced Vibrations of Steel Structures (HIVOSS). Brussels. F. Swedish Road Administration Standard. [5]. Guidelines for Steel Pedestrian Bridges. [10]. London. 2005. Indian Road Congress. Sept 2008. Serviceability of buildings and pedestrian walkways against vibration. Indian Roads Congress. London. [6]. Oct 2006. Stockholm.