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

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 Footfall Induced Vibration • Computing the Structural Response • Project Examples • References 2

3 Tokyo Institute of Technology | Takeuchi Lab Introduction to Footfall Induced Vibration 3

Tokyo Institute of Technology | Takeuchi Lab 4 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) 4

Response • Frequency • Duration • Frequency • Amplitude • Damping • Mode shape • Modal mass acceptability, ????? Tokyo Institute of Technology | Takeuchi Lab consequences, Loading : Modal (structural) properties : Introduction to Footfall Induced Vibration :: Loading :: Modal Properties :: Response The design problem 5 5

6 Tokyo Institute of Technology | Takeuchi Lab What does a footfall time history look like? Introduction to Footfall Induced Vibration :: Loading :: Modal Properties :: Response 6

7 Tokyo Institute of Technology | Takeuchi Lab What kinds of excitation frequencies are possible? Introduction to Footfall Induced Vibration :: Loading :: Modal Properties :: Response 7

8 Tokyo Institute of Technology | Takeuchi Lab So how does this translate into a design load? Introduction to Footfall Induced Vibration :: Loading :: Modal Properties :: Response 8

Floors 1~3% Damping :: Frequency :: Modal Mass :: Mode Shape Stairs 0.5% Tokyo Institute of Technology | Takeuchi Lab Be suspicious of >2% Footbridges 0.5~1.5% Introduction to Footfall Induced Vibration :: Loading :: Modal Properties :: Response What is an appropriate amount of damping? 9 Steady State Resonant Amplification Factors for SDOF systems 25 20 Amplification Factor 15 0.02 damping 0.05 damping 0.10 damping 10 0.20 damping 5 0 0 0.5 1 1.5 2 2.5 3 (excitation frequency)/(natural frequency) 9

Model Extent Boundary Conditions 10 Modelling Assumptions Introduction to Footfall Induced Vibration :: Loading :: Modal Properties :: Response Damping :: Frequency :: Modal Mass :: Mode Shape …some tips (not exhaustive)… Loads Member Modelling Tokyo Institute of Technology | Takeuchi Lab  Use best estimate, not code values  LL: ~0.5kPa typically realistic  SDL: upper/lower bound sensitivity study  Typically model single floor only  Floorplate should capture all modes of interest  Fixed connections/supports unless true pin  Façade vertical fixity may/may not be appropriate  Orthotropic slab properties  Composite beam (even if nominal studs)  E c,dynamic ≈38GPa  Subdivide 6+ elements per span 10

2 t Stair (high use) 24 Tokyo Institute of Technology | Takeuchi Lab Stair (light use) 32 0.16 m/s 2 Stair (very light use) 64 0.32 m/s 2 Approximate threshold of human perception to vertical vibration 2 t 64 RMS RMS Acc Peak,Harmonic RMS T R=Acc Acc Acc Acc = = 0.32 m/s 2 0.12 m/s 2 Footbridge (outside) Situation 1.4 0.007 m/s 2 Residential (day) 2-4 0.01~0.02 m/s 2 1 Office (high grade) 4 0.16 m/s 2 Low vibration RMS Acc @ 5Hz R Introduction to Footfall Induced Vibration :: Loading :: Modelling :: Response 0.005 m/s 2 What is a good design criteria? 11 0.02 m/s 2 Office (normal) 8 0.04 m/s 2 Footbridge (heavy) 24 0.12 m/s 2 Footbridge (inside) 32 Residential (night) 0.1 2 ) rms acceleration (m/s 0.01 0.001 1 10 100 Frequency (Hz)  T 11

12 Tokyo Institute of Technology | Takeuchi Lab Computing the Structural Response 12

Tokyo Institute of Technology | Takeuchi Lab 13 Simplified vs Modal vs Time History Methods Computing the Structural Response vs vs Modal harmonic response method Explicit time history method 13

Tokyo Institute of Technology | Takeuchi Lab 14 Analytical Model Computing the Structural Response P y y M K c Force Force Time Time 14

…Concrete Centre eq 4.4 Resonant Response SDOF Harmonic Steady State Response Computing the Structural Response Tokyo Institute of Technology | Takeuchi Lab 15 P y P e         i t my t ( ) cy t ( ) ky t ( ) P t ( ) M 1 1 k   FRF     disp     2 2 k m ic K  f   f 1 i 2 h h c f f m m       2 2   f   f 2 1 f   f h h 2 h h f f f f m m   i m m       2     2   2 2 2 2       f f f f 1 2 1 2 h h h h f f f f m m m m     F W DLF f harmonic ,       F excition reponse   Acc FRF acc m ƒ h = harmonic forcing freq. ƒ m = modal (structural) freq. 15

…Concrete Centre eq 4.10~4.13 16 Transient Response Computing the Structural Response Tokyo Institute of Technology | Takeuchi Lab SDOF Impulse – RMS Velocity Response P e         i t my t ( ) cy t ( ) ky t ( ) P t ( ) Force f 1.43  I 54 w [Ns] Time eff 1.3 f n     I  eff excition reponse Vel Peak m      t Vel t ( ) Vel e sin t Peak     Vel RMS Vel t ( ) RMS m ƒ w = walking forcing freq. ƒ m = modal (structural) freq. 16

s = structure Tokyo Institute of Technology | Takeuchi Lab 17 ‘Bobbing’ Resonant Response Computing the Structural Response p = passive crowd a = active crowd …IStructE Route 2 MDOF Harmonic Response – crowd interacting with structure P e         i t My t ( ) Cy t ( ) Ky t ( ) P t ( )                      m 0 0  y c c c c c y  k k k k k y  P  s s s a p a p s s a p a p s                    0 m 0  y c c 0 y  k k 0 y P               a a a a a a a a                   0 0 m y c 0 c y k 0 k y 0               p p p p p p p p  1      2       k k m i ( c c ) k i c 1    FRF s a s s a a a  disp   2        2   K M iC k i c k m i c   a a a a a   2 m 1     DMF FRF (1,1) FRF (1,2) a disp 2 FRF 2     f f   k D k a D D     a a s a s f f     a 2 2     f f f f         D 1 i 2 D 1 i 2     a a s s f f f f     a a s s         P ƒ W GLF harmonic scenario , a    2 Acc P DMF disp 17

2000 nodes x 20 modes x 4 harmonics x 1.5Hz Tokyo Institute of Technology | Takeuchi Lab 18 Programming Implementation Computing the Structural Response …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 frequency range = 2.4E7 calculations …Excel VBA is single threaded so no parallel processing …Suggest building with parallel libraries in C#, VB, Python, etc foreach node // Get response of governing excitation frequency R Node = MAX(R Node(ƒ) ) foreach excitation frequency Library floor GSA model // Get R Factor R Node(ƒ) = Acc RMS,Node(ƒ) / Acc RMS,0(ƒ) // Get SRSS acceleration Acc RMS,Node(ƒ) = √(∑ Mode ∑ Harmonic Acc RMS(n,m,h,ƒ*h)2 ) foreach mode // Get modal properties // modal frequency (ƒ), damping (ξ) & mass (m) // displacement at excitation(φ ne ) & response(φ nr ) nodes Footbridge GSA model // < participation factor(ρ m )>,<static mass (W)> ƒ m ,ξ m ,m m ,φ ne ,φ nr, W m ,ρ m = ... foreach harmonic // Get harmonic properties // dynamic amplification factor(DLF), harmonic freq (ƒ) DLF ƒ,h ,ƒ m,h = ... // Get RMS acceleration Acc RMS(n,m,h,ƒ*h) =RMS(F(ƒ m,h ,ξ m ,m m ,φ ne ,φ nr ,DLF ƒ,h ,W m ,ρ m )) 18

19 Tokyo Institute of Technology | Takeuchi Lab Some Examples 19

Tokyo Institute of Technology | Takeuchi Lab 20 How can we improve response? Examples • Increase Damping • Increase Frequency • Increase Stiffness • Decrease Mass • Increase Mass • Isolate 20