1 Tokyo Institute of Technology | Takeuchi Lab

Ben Sitler, PE

Arup Unified Method for Floors, Footbridges, Stairs and Other Structures

Footfall Induced Vibration

2 Tokyo Institute of Technology | Takeuchi Lab2

Outline

• Introduction to Footfall Induced Vibration• Computing the Structural Response• Project Examples• References

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Introduction to Footfall Induced Vibration

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When is footfall induced vibration an issue?Introduction to Footfall Induced Vibration

• Low Frequency (Resonance)• Low Mass (Acc=Force/Mass)• Low Damping• Large Dynamic Loads (Crowds)

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The design problemIntroduction to Footfall Induced Vibration :: Loading :: Modal Properties :: Response

Modal (structural) properties:

Loading:

consequences, acceptability, ?????

• Frequency

• Modal mass

• Mode shape

• Damping

• Amplitude

• Frequency

• Duration

Response

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What does a footfall time history look like?Introduction to Footfall Induced Vibration :: Loading :: Modal Properties :: Response

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What kinds of excitation frequencies are possible?Introduction to Footfall Induced Vibration :: Loading :: Modal Properties :: Response

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So how does this translate into a design load?Introduction to Footfall Induced Vibration :: Loading :: Modal Properties :: Response

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What is an appropriate amount of damping?Introduction to Footfall Induced Vibration :: Loading :: Modal Properties :: Response

0

5

10

15

20

25

0 0.5 1 1.5 2 2.5 3

Amplif

icatio

n Fact

or

(excitation frequency)/(natural frequency)

Steady State Resonant Amplification Factors for SDOF systems

0.02 damping0.05 damping0.10 damping0.20 damping

Damping :: Frequency :: Modal Mass :: Mode Shape

Be suspicious of >2%

Footbridges 0.5~1.5%Stairs 0.5%Floors 1~3%

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Modelling AssumptionsIntroduction to Footfall Induced Vibration :: Loading :: Modal Properties :: Response

Damping :: Frequency :: Modal Mass :: Mode Shape

…some tips (not exhaustive)…Loads Use best estimate, not code values LL: ~0.5kPa typically realistic SDL: upper/lower bound sensitivity studyModel Extent Typically model single floor only Floorplate should capture all modes of interestBoundary Conditions Fixed connections/supports unless true pin Façade vertical fixity may/may not be appropriateMember Modelling Orthotropic slab properties Composite beam (even if nominal studs) Ec,dynamic≈38GPa Subdivide 6+ elements per span

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What is a good design criteria?Introduction to Footfall Induced Vibration :: Loading :: Modelling :: Response

Situation R RMS Acc @ 5Hz

Low vibration 1 0.005 m/s2

Residential (night) 1.4 0.007 m/s2

Residential (day) 2-4 0.01~0.02 m/s2

Office (high grade) 4 0.02 m/s2

Office (normal) 8 0.04 m/s2

Footbridge (heavy) 24 0.12 m/s2

Footbridge (inside) 32 0.16 m/s2

Footbridge (outside) 64 0.32 m/s2

Stair (high use) 24 0.12 m/s2

Stair (light use) 32 0.16 m/s2

Stair (very light use) 64 0.32 m/s2

0.001

0.01

0.1

1 10 100Frequency (Hz)

rms a

cceler

ation (

m/s2 )

Approximate threshold of human perception to vertical vibration

T 2t tRMS RMS

Acc Peak,HarmonicRMS T

R=Acc Acc

AccAcc = =

2

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Computing the Structural Response

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Simplified vs Modal vs Time History MethodsComputing the Structural Response

vs

vs

Modal harmonic

response method

Explicit time history

method

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Force

Time

Analytical ModelComputing the Structural Response

Py

cKMy

Force

Time

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( ) ( ) ( ) ( ) i tmy t cy t ky t P t P e

Resonant ResponseComputing the Structural Response

Py

cKM

,F W DLF f harmonic

2 2

2 2 2

2 22 2 2 2

1 11 2

1 21 2 1 2

h hm m

h h h hm m m m

h h h hm m m m

disp f ff f

f f f ff f f ff f f ff f f f

kFRF k m ic i

i

SDOF Harmonic Steady State Response

excition reponseacc

FAcc FRFm

…Concrete Centre eq 4.4ƒh = harmonic forcing freq.ƒm = modal (structural) freq.

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Transient ResponseComputing the Structural Response

Force

Time

SDOF Impulse – RMS Velocity Response

( ) ( ) ( ) ( ) i tmy t cy t ky t P t P e 1.431.354 [Ns]weffn

fI feff excition reponse

PeakIVel m

( ) sintPeakVel t Vel e t

( )RMS mVel RMS Vel t

ƒw = walking forcing freq.ƒm = modal (structural) freq.…Concrete Centre eq 4.10~4.13

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‘Bobbing’ Resonant ResponseComputing the Structural Response

122 2

( )1 s a s s a a adisp

a a a a a

k k m i c c k i cFRF K M iC k i c k m i c

0 00 0 0 00 0 0 0 0s s s a p a p s s a p a p s

a a a a a a a ap p p p p p p p

m y c c c c c y k k k k k y Pm y c c y k k y P

m y c c y k k y

22 21(1,1) (1,2) adisp

aa a s a sa

mDMF FRF FRF FRF ffk D k D Df f

2

1 2a aa a

f fD if f

21 2s s

s s

f fD if f

2dispAcc P DMF

( ) ( ) ( ) ( ) i tMy t Cy t Ky t P t P e

ƒ ,aP W GLF harmonic scenario

MDOF Harmonic Response – crowd interacting with structure

…IStructE Route 2

a = active crowdp = passive crowds = structure

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Programming ImplementationComputing the Structural Response

foreach node// Get response of governing excitation frequencyRNode = MAX(RNode(ƒ))foreach excitation frequency// Get R FactorRNode(ƒ) = AccRMS,Node(ƒ) / AccRMS,0(ƒ)// Get SRSS accelerationAccRMS,Node(ƒ) = √(∑Mode∑HarmonicAccRMS(n,m,h,ƒ*h)2)foreach mode// Get modal properties// modal frequency (ƒ), damping (ξ) & mass (m)// displacement at excitation(φne) & response(φnr) nodes// ,ƒm,ξm,mm,φne,φnr,Wm,ρm = ...foreach harmonic// Get harmonic properties// dynamic amplification factor(DLF), harmonic freq (ƒ)DLFƒ,h,ƒm,h = ...// Get RMS accelerationAccRMS(n,m,h,ƒ*h)=RMS(F(ƒm,h,ξm,mm,φne,φnr,DLFƒ,h,Wm,ρm))

2000 nodes x 20 modes x 4 harmonics x 1.5Hz frequency range = 2.4E7 calculations

…not practical back in ’90s, hence the simplified methods in AISC DG11, AS 5100-2, etc

6000 nodes x 40 modes x 2 harmonics x 6Hz frequency range (running) = 2.8E8 calculations

…Excel VBA is single threaded so no parallel processing…Suggest building with parallel libraries in C#, VB, Python, etc

Footbridge GSA model

Library floor GSA model

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Some Examples

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How can we improve response?Examples

• Increase Damping• Increase Frequency

• Increase Stiffness• Decrease Mass

• Increase Mass• Isolate

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Further ReadingQuestions?

General Structures (Vertical Resonant or Transient of Pedestrians)

• Concrete Centre CCIP-016 A design guide for Footfall Induced Vibration of Structures• SCI P354 Design of Floors for VibrationStadia & Concert Hall (Vertical Resonance of Bobbing Crowd)

• IStructE Dynamic Performance Requirements for Permanent Grandstands Subject to Crowd Action

• C. Jones, A. Pavic, P. Reynold, R. Harrison Verification of Equivalent Mass-Spring-Damper Models for Crowd-Structure Vibration Response Prediction

High Use Footbridges (Lateral Synchronous Lock in and Vertical Crowd)

• P. Dallard The London Millenium Footbridge• BSI PD 6688-2:2011 Background to National Annex to BS EN 1991-2: Traffic loads on

bridges

• Setra Footbridges: Assessment of Vibrational Behaviour of Footbridges under Pedestrian Loading

contact: sitler.b.aa@m.titech.ac.jp