5 February 2008 The Structural Engineer|37technical note: footfall vibrationFig 1. BBN/ASHRAE laboratory equipment check / Fig 2. Response factors a) 8m bay rc flat slab; b) 8m bay PT slabThe possibility of human footfall loadingleading to excessive vibration of structureshas long been recognised. Soldiers marchingacross a cast iron bridge in 1831 generated vibra-tions that caused the bridge to collapse; manybridges now display notices instructing soldiers tobreak step when crossing. Other collapses of floorsand stadium structures have been induced bycrowds dancing or jumping in unison.The introduction of lightweight long spancomposite construction and open plan offices inNorth America in the 1960s led to concerns notover safety, but that normal walking causeduncomfortable vibrations for occupants of thebuildings. Until this time, serviceability waschecked using only simple stiffness-based crite-ria, such as limiting imposed load deflections to aratio of the span.Market driversThere are numerous market forces causingclients to insist on floor vibration checks: Hospitals operating theatres require theutmost stability for delicate operations, whilenight wards are nearly as onerous. Laboratories equipment, such as opticaland electron microscopes to laser researchsystems, are very sensitive to vibrations.Such floors must comply with the BBN orAshrae standards (Fig 1). Airports airport owners require maximumresponse values for the passenger waiting areasas floor vibrations can upset nervous travellers. Retail many major retailers require amaximum liveliness for their display floors,such as where they are displaying glasses onglass shelves: if the floor is too lively then theglasses will rattle. Commercial on lively floors, computer userscomplain because their screens wobble,making it difficult to work.The vibration problemFor many years now, serviceability requirementshave been a part of structural design. Initially,these were just deflection limits to preventfinishes from cracking and building occupantsnoticing the floors sagging. These provedadequate for decades, until advances began to bemade into more efficient, lighter structures, suchas composite beam or post-tensioned slab floors.Unfortunately, the users of these buildings foundthat the floors could be rather lively.The first proposed remedy to this problem wasto restrict the natural frequency of the floorbeams, as it was felt that if this was kept abovewalking pace, then resonance should not occur.The fact that finding this frequency is a simplehand calculation encouraged this approach.Two problems emerged with this solutionhowever. The first was that floors are excited bythe harmonics of the pedestrians footstepfrequency; the second was that while short spanshad high natural frequencies, they had a lowmass in ratio to the weight of a person, incomparison to long span floors that may have alow frequency but also a large floor area, makingthem more difficult to excite. What the industryneeded was a solution for all materials and for allframing layouts.Industry solutionsIndustry experts recognised that what wasneeded was not a measure of the floor frequency,but of how much the floor responds to the foot-steps of a person walking over it: a footfallresponse calculation. Various trade organisa-tions, such as The Steel Construction InstituteFootfall vibration and finiteelement analysis2Vibration caused by human footfall has been recognised for some time and the adventof longer span and lighter structures has brought new issues to the fore. Peter Debneyof Oasys Software describes how modern design and analysis software can helpprovide engineers with the answers ab1SE3 Footfall vibs FEA(Arup):Layout 130/1/0816:30Page 3738|The Structural Engineer 5 February 2008technical note: footfall vibrationFig 3. Building vibration z-axis curves for acceleration (r.m.s) / Fig 4. Typical floor vibration modes a) Mode 1; b) Mode 2; c) Mode 3(SCI), the American Institute of SteelConstruction (AISC), and The Concrete Centre,have produced guides on how to find this floorresponse.Floor vibration problems are not restricted tojust steel floors; while flat slabs tended to avoidthese problems, thinner post-tensioned slabs donot have sufficient mass to resist the forcedvibrations (Fig 2).Measuring vibrationThere are a number of sources of vibration, orrather force excitation, in structures. Examplesinclude footfalls from walking individuals andgroups, rhythmic people inputs, such as dancingand aerobics, machinery, transportation (roadsand railways sometimes integrated with thestructure) and temporary inputs such as fromdemolition and construction.The three general characteristics of the waythe structure responds to the force inputs areImpulsive or Transient Response, ResonantResponse, and Radiation as an audible sound.Design criteria generally considers humancomfort (perceptible or tactile vibration or noise),interference with equipment or processes, andstructural damage.Response factorsThe response factor (R) is simply a multiplier onthe level of vibration at the average threshold ofhuman perception. Therefore a response factor of1 represents the magnitude of vibration that isjust perceptible by a typical human, a responsefactor of 2 is twice that, and a response factor of 8is eight times that (Fig 3). The response factorconsiders the amplitude (acceleration) andfrequency of the vibration.The difficulty with some floor vibration guide-lines was that in order to reduce the complexityand avoid finite element analysis, they offeredsolutions only for regular, rectangular floorlayouts; unfortunately, very few modern build-ings have such simple framing. Some softwaresuppliers have suggested that irregular framescannot experience resonant problems, but inpractice this is found to be untrue.The ideal solutionThe ideal methodology for assessing the suscepti-bility of a structure to footfall vibration would be: Applicable to as many structural forms aspossible, whether simple or complex (Fig 4) Straightforward to use, enabling the conse-quences of various design iterations to bereadily and quickly assessed Applicable to structures whose structuralproperties may be ascertained by: Hand calculation type procedures undertaken early in the design procedure or toverify more complex analysis Finite element analysis MeasurementMost of the existing published methods relyheavily on rules that classify different structuralforms and hence the details of the analysis to beused. If the structure does not readily fit into oneof these classifications, then approximationsmust be made. If the underlying methodologiesto the empirical rules are not fully understoodthen the assumptions made are likely to be inac-curate.GSA footfallFor many years, GSA has been one of the leadingPC based packages for structural analysis.Developed by Arup to meet the demanding anddiverse requirements of Arup, one of the worldsleading firms of international consulting engi-neers, GSAs capabilities are proven on thou-sands of complex and prestigious projectsworld-wide.GSA Footfall analyses structures to the Arupmethod (as adopted by The Concrete Centre),The Steel Construction Institute specification,and more. Not only that, but because it analyseswhich floor areas have high and low responses; itenables consultants to position sensitive equip-ment and services or to improve problem areas ina cost-effective way.BenefitsFull FE analysis is the only way to sensiblypredict the footfall response of any floor that isnot part of a regular rectangular frame. Evenwith a rectangular frame, the calculations arequicker using FE than hand calculations.Thusyou get quick and accurate predictions of floorresonant and transient response to footfall vibra-tions, including Response Factors, peak accelera-tions, and peak, RMS and RMQ velocities (Fig 5).GSA analysis enables you to locate regions ofhigh and low response to enable sensible loca-tions of sensitive equipment or activities and toassess localised modifications to floor structuresfor sensitive equipment or activities to minimisecost to the client.Also, because GSA is FE analy-sis not empirical examples, you can calculate thefootfall response of any structure, including34a4b 4cSE3 Footfall vibs FEA(Arup):Layout 130/1/0816:30Page 385 February 2008 The Structural Engineer|39technical note: footfall vibrationFig 5. Harmonic analysis results a) Displacement; b) Accelerationstairs, bridges, concrete flat slabs, and steelcomposite framesBecause you can define exactly where on amodel to examine, you can check particular areas,such as the effect of running down a corridor nextto an operating theatre.You can also examine theresonance of a structure to vibrating machineryand dance loads using harmonic analysisFeaturesModal analysis Choice of vibration analysis: Modal, Modal P-delta, Ritz, Ritz P-delta Choose number of vibration modes and startmode Include additional horizontal or verticalrestraints Specify mass or derive mass from loads andself weight Calculate modal damping Include stiffening effects of loadsFootfall analysis Quick or full excitation methods Check full model or set areas Damping by user input values, modal dampingor table Vary number of footfalls for resonantresponse Vary weight of walker Choice of excitation force methods: Arup /Concrete Centre, Steel Construction Institute,or Arup Stair Adjust minimum and maximum walkingfrequencies Detailed chart views of results5a5bSE3 Footfall vibs FEA(Arup):Layout 130/1/0816:30Page 3940|The Structural Engineer 5 February 2008technical note: footfall vibrationHarmonic analysis Displacement, velocity, and acceleration elasticresponse to steady state harmonic loadingStep-by-step guideSub-frame modelCreate a model of the floor in question (Fig 6). Asthe vibration strains are so small: connections are generally modelled as fullyfixed full height partitions and cladding can gener-ally modelled as vertical restraintsAs the footfall analysis is only concerned withvertical vibrations, eliminate vibrations thathave no vertical component. This may not bepossible for staircases and similar structures.Modal analysisRun a modal analysis to find the vibrations up tothe limit specified in the relevant design guideFig 6. Composite frame / Fig 7. Modal analysis a) Mode 1: b) Mode 2: c) Mode 3 /Fig 8. Footfall results / Fig 9. a) Hospital floor b) Helical stair c) Bridge(Fig 7). These are typically 10Hz, 15Hz or twicethe fundamental frequency. Options includecalculating the modal damping of the structureand converting applied loads to masses.Footfall analysisRun a footfall analysis (Fig 8). Options includemethod (including SCI and Arup), area of excita-tion, walking frequencies, and damping.SummaryGSA Footfall is for structural engineers who needto accurately predict the response of a structure tomomentary or vibrational loading. It is a finiteelement analysis program that provides the abilityto analyse any structure for footfall response,whether steel or concrete, bridge, floor or stair (Fig9). Unlike other programs or manual methods, itgives you the tools to assess your structure usingthe SCI, Concrete Centre and Arup methods.se Further information: Peter Debney; a free down-load 30 day trial version is available (tel: 0191238 7559; web: www.oasys-software.com/GSA).9b 9a8 7c Willford, M.R. and Young, P.: A Design Guide forFootfallInducedVibrationofStructures,TheConcrete Centre, 2007 Smith, A. L., Hicks, S. J., and Devine, P. J.: Designof Floors for Vibration: A New Approach, SCI, 2007 Murray, T. M., Allen, D. E., and Ungar, E. E.: FloorVibrationsDuetoHumanActivity,AISC,20031974, pp197207FURTHER READING7b 6 7a9cSE3 Footfall vibs FEA(Arup):Layout 130/1/0816:30Page 40