Footfall Induced Vibration Arup Unified Method for Floors, - - PowerPoint PPT Presentation

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Footfall Induced Vibration Arup Unified Method for Floors, - - PowerPoint PPT Presentation

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Tokyo Institute of Technology | Takeuchi Lab Footfall Induced Vibration Arup Unified Method for Floors, Footbridges, Stairs and Other Structures Ben Sitler, PE 1 Tokyo Institute of Technology | Takeuchi Lab 2 Outline Introduction to

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Tokyo Institute of Technology | Takeuchi Lab

Ben Sitler, PE

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

Footfall Induced Vibration

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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 problem

Introduction 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

5 10 15 20 25 0.5 1 1.5 2 2.5 3

Amplification Factor (excitation frequency)/(natural frequency)

Steady State Resonant Amplification Factors for SDOF systems

0.02 damping 0.05 damping 0.10 damping 0.20 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 Assumptions

Introduction 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 study

Model Extent

  • Typically model single floor only
  • Floorplate should capture all modes of interest

Boundary Conditions

  • Fixed connections/supports unless true pin
  • Façade vertical fixity may/may not be appropriate

Member 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 10 100 Frequency (Hz) rms acceleration (m/s

2)

Approximate threshold of human perception to vertical vibration

T

2 t t

RMS RMS Acc Peak,Harmonic RMS T

R=Acc Acc Acc Acc = = 2

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Computing the Structural Response

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Simplified vs Modal vs Time History Methods

Computing the Structural Response

vs vs

Modal harmonic response method Explicit time history method

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Force Time

Analytical Model

Computing the Structural Response

P y c K M y

Force Time

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( ) ( ) ( ) ( )

i t

my t cy t ky t P t P e        

Resonant Response

Computing the Structural Response

P y c K M

 

, F W DLF f harmonic  

 

 

   

 

 

 

     

 

 

2 2 2 2 2 2 2 2 2 2 2

1 1 1 2 1 2 1 2 1 2

h h m m h h h h m m m m h h h h m m m m

disp f f f f f f f f f f f f f f f f f f f f

k FRF k m ic i i                     

SDOF Harmonic Steady State Response

excition reponse acc

F Acc FRF m        

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

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Transient Response

Computing the Structural Response

Force Time

SDOF Impulse – RMS Velocity Response

( ) ( ) ( ) ( )

i t

my t cy t ky t P t P e        

1.43 1.3

54 [Ns]

w eff n

f I f 

eff excition reponse Peak

I Vel m      ( ) sin

t Peak

Vel t Vel e t



 

 

( )

RMS m

Vel RMS Vel t 

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

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‘Bobbing’ Resonant Response

Computing the Structural Response

1 2 2 2

( ) 1

s a s s a a a disp a a a a a

k k m i c c k i c FRF K M iC k i c k m i c        

                   

s s s a p a p s s a p a p s a a a a a a a a p p p p p p p p

m y c c c c c y k k k k k y P m y c c y k k y P m y c c y k k y                                                                                            

2 2 2

1 (1,1) (1,2)

a disp a a a s a s a

m DMF FRF FRF FRF f f k D k D D f f                    

2

1 2

a a a a

f f D i f f          

2

1 2

s s s s

f f D i f f          

2 disp

Acc P DMF    ( ) ( ) ( ) ( )

i t

My t Cy t Ky t P t P e        

   

ƒ ,

a

P W GLF harmonic scenario    

MDOF Harmonic Response – crowd interacting with structure …IStructE Route 2 a = active crowd p = passive crowd s = structure

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Programming Implementation

Computing the Structural Response

foreach node // Get response of governing excitation frequency RNode = MAX(RNode(ƒ)) foreach excitation frequency // Get R Factor RNode(ƒ) = AccRMS,Node(ƒ) / AccRMS,0(ƒ) // Get SRSS acceleration AccRMS,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 // <participation factor(ρm)>,<static mass (W)>

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 acceleration AccRMS(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 Reading

Questions?

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 Vibration

Stadia & 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