STRUCTURAL VIBRATIONS DUE TO HUMAN ACTIVITY What’s Old? What’s New? What’s Hot? Presented by Thomas M. Murray, Ph.D., P.E. Emeritus Professor Virginia Tech, Blacksburg, Virginia [email protected]

61st Structural Engineering Conference University of Kansas March 3, 2016 1

What’s Old?

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What’s Old? The Speaker: KU Ph.D. 1970

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What’s New? 11

Steel Design Guide

Vibrations of Steel-Framed Structural Systems Due to Human Activity Thomas M. Murray, Ph.D., P.E. Virginia Tech Blacksburg, VA

David E. Allen, Ph.D. National Research Council Canada Ottawa, Ontario, Canada

Eric E. Ungar, Sc.D., P.E. Acentech Incorporated Cambridge, MA

D. Brad Davis, Ph.D., S.E. University of Kentucky Lexington, KY

AMERICAN INSTITUTE OF STEEL CONSTRUCTION

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SJI Technical Digest 5 - 2nd Ed. 2014 • Written by Prof. Tom Murray Emeritus Professor Virginia Tech • Prof. Brad Davis Assistant Professor University of Kentucky • • • • • • • 5

Follows AISC DG11 Walking Excitation Modified ISO Scale Resonant Build-Up Rhythmic Excitation Finite Element Analysis Retrofitting of Lively Floors 5

AISC Design Guide 11 - 2nd Ed. 2016 • New Co-Author Prof. Brad Davis Assistant Professor, University of Kentucky • • • • • • • •

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Steel Design Guide

Vibrations of Steel-Framed Introduction Structural Systems Evaluation Criteria for Human Comfort Due to Human Activity Natural Frequency of Steel Framed Floor Systems Design for Walking Excitation Design for Rhythmic Excitation Design for Sensitive Equipment and Sensitive Occupancies Finite Element Analysis Methods Evaluation of Vibration Problem Systems and Remedial Measures Thomas M. Murray, Ph.D., P.E.

Virginia Tech Blacksburg, VA

David E. Allen, Ph.D.

National Research Council Canada Ottawa, Ontario, Canada

Eric E. Ungar, Sc.D., P.E.

Acentech Incorporated Cambridge, MA

D. Brad Davis, Ph.D., S.E.

University of Kentucky Lexington, KY

AMERICAN INSTITUTE OF STEEL CONSTRUCTION

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AISC DG11

nd 2

Ed.

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Selected Topics from DG11 2nd Ed. • Low and High Frequency Floors (LFF and HFF) • Low Frequency Floor Assessment • Finite Element Analysis Overview • Assessment of Problem Floors

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Definition – Low Frequency Floor Low Frequency Floor (LFF) • A low frequency floor is one that can undergo resonant build-up due to walking. • A resonant build-up can occur if at least one natural mode has a frequency less than ~9 Hz. Meas. Acceleration (%g)

Resonant Build-up

1.5

Peak Accel. = 1.36%g

1 0.5 0 -0.5 -1 RMS Accel. = 0.531%g

-1.5 0

2 4 Time (sec.)

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Definition – High Frequency Floor High Frequency Floor (HFF) •

•

A high frequency floor is one that cannot undergo resonant build-up because the dominant frequency is greater than ~ 9 Hz. The response resembles a series of individual impulse responses to individual footsteps.

Impulse Responses

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Low Frequency Floor Assessment Walking Dynamic Loading • When humans walk, run, bounce, sway, or jump, inertial forces cause dynamic loads. Ground Reaction / Weight

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1.5

Heel Strike

Knee bends, weight shifts forward

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Push off

0.5

0 0

0.2

0.4

0.6

0.8

Time (sec.)

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Low Frequency Floor Assessment Walking Dynamic Loading • Series of footstep forces cannot be represented by simple equations. 0.4

Summation

Left Footstep

Ground Reaction (kip)

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0

Right Footstep 0.5

1

1.5

Need Simple Mathematical Representations

Fourier Series

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Time (sec.)

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Low Frequency Floor Assessment Walking Dynamic Loading • Specialized Fourier Series for Human Induced Forces Weight of Walker (lbf)

Harmonic Number

Phase Lag

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F ( t ) = 0 + ∑ Q ⋅ αh ⋅ sin(2 ⋅ π ⋅ h ⋅ fStep ⋅ t − φh ) h=1

“DC Offset” not needed.

Harmonic Amplitude

Dynamic Load Factor (DLF)

Step or Pacing Frequency

Need Q, DLFs, range of step frequencies, and phase lags.

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Low Frequency Floor Assessment Walking Step Frequencies • fStep between 1.6 Hz and 2.2 Hz (96 bpm and 132 bpm) • Average is about 1.9-2.0 Hz (114-120 bpm)

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Low Frequency Floor Assessment Resonant Responses • Practical examples - Pushing a child on a swing. - Jumping on a diving board.

• Same applies to floors: if hfStep matches a natural frequency  resonance, which causes the most severe response. Matches a Natural Frequency 4

F( t ) = ∑ Q ⋅ αh ⋅ sin(2 ⋅ π ⋅ h ⋅ fStep ⋅ t − φh )

Causes Resonance (Most Severe Response)

h=1

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Low Frequency Floor Assessment Resonant Response • Example: - Natural Frequency = 5.00 Hz. - Walking at 1.67 Hz (within normal walking range) - Responses at 1.67 Hz, 3.33 Hz, 5.00 Hz, 6.67 Hz 0.35 Peak Accel. = 1.36%g

Meas. Acceleration (%g)

Meas. Acceleration (%g)

1.5 1 0.5 0 -0.5 -1

Response to 3rd Harmonic (5.00 Hz)

0.3 0.25

Response to 2nd Harmonic (@ 3.33 Hz)

0.2

Response to 4th Harmonic (6.67 Hz)

0.15 0.1 0.05

Response to 1st Harmonic (@ 1.67 Hz)

RMS Accel. = 0.531%g

-1.5 0

2 4 Time (sec.)

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0 1

2

3

4

5

7

6

Frequency (Hz)

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Low Frequency Floor Assessment Why do some walkers cause more floor motion then other walkers? • Because their pace is a sub-harmonic of the floor dominant frequency. • That is a harmonic of their walking speed, • Generally, 2 or 3 times their walking speed matches the floor dominant frequency.

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Low Frequency Floor Assessment Specialize by defining P for walking. • Possible to write an equation that works regardless of which harmonic is applicable. 4

F ( t ) = 0 + ∑ Q ⋅ αh ⋅ sin(2 ⋅ π ⋅ h ⋅ fStep ⋅ t − φh ) h=1

α =0.83e −0.35 fn

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Low Frequency Floor Assessment Adjustments • Incomplete resonant build-up. • Walker and annoyed person are not at the same location at the same time. • Use a reduction factor, R = 0.5 for floors with two-way mode shapes • Use R = 0.7 for footbridges with a one-way mode.

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Low Frequency Floor Assessment Predicted Walking Acceleration aSteadyState

P = 2 ζM

DLF

Call this Po

Po ⋅ e −0.35f R ⋅ Q ⋅ 0.83e −0.35f R ⋅ P R ⋅ Q ⋅ 0.83e −0.35f g g= ap = = = β ⋅W β ⋅W 2 ζM 2 ⋅ β ⋅ 0.5W / g n

Po Po

n

n

= (0.5)(157 lb)(0.83) 65 lb for floors = (0.7)(157 lb)(0.83) 91.2 lb (use 92 lbf) for footbridges Po ⋅ e −0.35f ap = g β ⋅W n

Predicted Peak Acceleration Due to Walking in DG11

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Low Frequency Floor Assessment Predicted ≤ Tolerance a p P o exp(−0.35 f n) a o = ≤ g βW g where ap/g = predicted acceleration ratio ao/g = acceleration limit for the appropriate occupancy (Example: 0.005 or 0.5%g for quiet spaces.) Po = amplitude of the driving force, 65 lb or 92 lb W = effective weight supported by the beam or joist panel, girder panel, or combined panel, as applicable, lb fn = fundamental natural frequency of a beam or joist panel, a girder panel, or a combined panel, as applicable, Hz β = damping ratio 21

Low Frequency Floor Assessment Accuracy of the AISC DG11

Data from a study being conducted at the University of Kentucky by Dr. Brad Davis. 22

Low Frequency Floor Assessment Example. Determine the effect of Floor Length on the response of Bays A and B due to walking.

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Low Frequency Floor Assessment

Length

Length

ap g

W is a function of Floor Width and Length Bj = Cj (Ds / Dj)1/4Lj < 2/3 x Floor Width Bg = Cg(Dj /Dg)1/4 Lg < 2/3 × Floor Length

P o exp(−0.35 f n) a o ≤ βW g

Bays A Floor Width: 4x30’ = 120’ Floor Length 32.5’+16’+32.5’ = 81’ Bay B Floor Width: 4x30’ = 120’ Floor Length 32.5’+16’ = 48.5’

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Low Frequency Floor Assessment Bay A Response Bays A & B Bj is same for both Bays. Bg is different for each Bay because Floor Length is different. Bg = 59.9’ < 2/3 Floor Length

Girder Panel

Bays A: Floor Length = 81’ 2/3x81 = 54’ < 59.9’ ap/g = 0.46%g < 0.5%g OK

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Low Frequency Floor Assessment Bay B Response Bays A & B Bg = 59.9’ < 2/3 Floor Length

Girder Panel

Bays A: Floor Length = 81’ ap/g = 0.46%g < 0.5%g OK Bay B: Floor Length = 48.5’ 2/3x48.5 =32.3’ < 59.9’ ap/g =0.61%g > 0.5%g NG

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Finite Element Analysis

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Finite Element Analysis Overview of Evaluation Procedure • Develop 3D model. - Specialized for extremely low amplitude motion. • Predict natural modes. • Predict response to human activity. • Compare to tolerance limit.

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Finite Element Analysis Extent of Model • Horizontal Expanse – Avoid over-predicting area in motion. – Beware uniform framing over many bays.

• 15 Bays in Motion • Unrealistic • Unconservative.

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Finite Element Analysis Extent of Model • Horizontal – Motion should be limited to the bay being evaluated plus a few around it. – Option: Start with model of large area. Delete bays if necessary. Bay Being Evaluated

– Another Option: Model the bay being evaluated plus one bay each side. 30

Finite Element Analysis Natural Mode Prediction • Typical Eigenvalue Analysis • Number of Modes – All modes with single curvature within a bay. – Include modes up to about double the fundamental natural frequency.

• Many modes are often predicted. • Use Frequency Response Function (FRF) to determine dominate mode.

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Finite Element Analysis Frequency Response Function

1 lbf Sinusoidal Force, Frequency = f

Steady state response amplitude, %g

Accelerance FRF Mag. (%g/lbf)

1 lbf sinusoidal load

Dominant Frequency Natural Frequencies

0.1

0.05

0 0

Width Related to damping

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4 6 Frequency (Hz)

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Sinusoidal Response, Frequency = f 32

Finite Element Analysis Frequency Response Function • FRF Computed using “Steady State Analysis” in SAP2000 - Requires hysteretic damping. - Stiffness proportional coefficient = 2b - Mass proportional coefficient = 0

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Finite Element Analysis Walking Acceleration Prediction – FRF Method • FRF Magnitude – FRFMax = Highest magnitude under 9 Hz. – FRFMax in %g/lb • Harmonic Force Amplitude: Product of – Reference bodyweight, Q = 168 lb – Dynamic coefficients, ai. • ai = 0.4, 0.07, 0.06, 0.05 • a2 through a4 approximated by α =0.09e −0.075f n 34

Finite Element Analysis Walking Acceleration Prediction – FRF Method • Partial Build-Up Factor – Envelope Function

ρ = 1 − e −2 πf nβTBU = 1 −e

−2 πβHN

TBU = resonant build-up duration H = harmonic causing resonance N = number of footsteps

– For N = 6 (typical length of walking path in offices) ρ= 50β + 0.25 if β < 0.01

= ρ 12.5β + 0.625 if 0.01 ≤ β < 0.03

= ρ 1.0 if β ≥ 0.03

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Finite Element Analysis Walking Acceleration Prediction – FRF Method • In equation form,

= a p FRFMax αQ ρ • Bay is OK if

a p ≤ ao

Tolerance Limit from DG11 e.g., 0.5%g for offices.

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Problem Floor Evaluation

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Problem Floor Evaluation Simplified Testing Procedure • • • • •

Requiring minimal equipment, time, and cost. Handheld Analyzer Seismic Accelerometer Heel-Drop impact to obtain floor natural frequency. Walking at sub-harmonic frequency to obtain maximum response. • Filtering of response data: 1 Hz to 15-20 Hz • Comparison of Equivalent Sinusoidal Peak Acceleration (ESPA) to appropriate tolerance limit. 38

Problem Floor Evaluation Sample Equipment Set

Datastick VSA-1214, 1215 Metronome

Seismic Accelerometer and Cables 39

Problem Floor Evaluation Heel-Drop Tests • Raise onto balls of feet and drop forcefully. • Heel-drop forcing function contains up to 40 Hz and is ideal for determining floor frequency.

Time History

Frequency Spectrum 40

Problem Floor Evaluation Measured Walking Acceleration

• Problem floors are usually “low frequency floors”  an occupant causes resonance. • Want to cause resonance during tests. • Step frequency, fStep - In normal range: 1.6 Hz to 2.2 Hz. - Such that integer multiple of fStep = fn - Example: fn = 6 Hz. Select fStep = 2 Hz because third harmonic will match fn,  resonance.

• Metronome is used to monitor walking speed. • Multiple individual walkers • Multiple tests per walker.

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Problem Floor Evaluation Processing • Individual peaks not representative. • Tolerance limits are sinusoidal accelerations. • So compute “Equivalent Sinusoidal Peak Acceleration” (ESPA) - Compute rolling two second Root Mean Square (RMS)

acceleration. - For example: at t =1.8 sec., compute RMS from 0.8 sec. to 2.8 sec. - Maximum rolling RMS x 2 = ESPA.

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Problem Floor Evaluation

3’8” 32’4”

57’4”

21’4”

Case Study

22’11”

32’4”

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Problem Floor Evaluation Measured Natural Frequencies • Heel-drop Tests at Stations 1-7. • Stations 1 and 5 of primary interest.

8.69 Hz 6.88 Hz

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Problem Floor Evaluation Measured Walking Accelerations. Walking at 2.17 Hz (4)(2.17)= 8.69 Hz ESPA = 1.79%g Retrofit Required Station 1

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What’s Hot (Cool)? • New Approaches to Evaluate Floors Supporting Sensitive Equipment • Analysis Techniques for Evaluating Slender Monumental Stairs

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Sensitive Equipment

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Sensitive Equipment Tolerance Limits • Manufacturer’s requirements are generally in terms of velocity, but sometimes acceleration. • Generic requirements are available. • Requirements are usually very strict. • Short span, very stiff floor systems are required.

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Sensitive Equipment Sensitive Equipment Tolerance Limits • Peak velocity or acceleration specific limit. • Narrowband spectral velocity or acceleration specific limit. • One-third octave spectral velocity or acceleration generic limit. 3

Velocity (mips, rms)

10

2

10

1

10

0

10 Time (sec)

Acceleration Waveform

Narrowband Spectral Acceleration

4

5

6.3 8 10 12.5 Frequency (Hz)

16

20

One-third Octave Spectral Velocity 49

Sensitive Equipment Example Waveform Acceleration Limit • GE Open MRI Pre-installation Manual

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Sensitive Equipment Example Waveform Acceleration Limit

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Sensitive Equipment Example HFF Waveform Acceleration Limit Limit = 0.01 m/s2 aPeakToPeak = 0.0998%g (predicted) = 0.00979 m/s2

High Frequency Floor Response 52

Sensitive Equipment Walking Speeds in AISC DG11 2nd Ed. Walking Speed Very Slow Slow Moderate Fast

fstep (Hz) 1.25 1.6 1.85 2.1

Walking (bpm) 75 96 110 125

• Very Slow: One or two persons in a lab or exam room. • Slow: Three to five persons in a lab or exam room. • Moderate: More than five persons in a lab or exam room. • Fast: Large lab or exam room or near a corridor. 53

Sensitive Equipment Peak Velocity and Acceleration Predictions 1.43 19 × 109 f step vp = W f n1.3

v p = max

1.3x109 −γf n if f n ≤ f 4 max e βWf n 9

19x10 W

1.43 f step f n1.3

1.43 a p 310 f step = g W f n0.3

ap g

= max

22 −γf n if f n ≤ f 4 max e βW 1.43 310 f step W f n0.3

• Top equations for very slow walking. • Bottom equations for other walking speeds with - First expressions for low frequency floors - Second expressions for high frequency floors 54

Sensitive Equipment Generic Limits • Specific limits are often not available during the design phase. Therefore Generic Limits can be used. • Generally expressed as onethird octave spectral velocity magnitudes. • Most common are Velocity Critical, VC, curves. From Ungar et al. (2004) 55

Monumental Stairs

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Slender Monumental Stairs Architectural Features • In high-end buildings. • Aggressive requirements. – Long spans. – Slender stringers. – Low mass.

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Slender Monumental Stairs Susceptible to Vibrations • Natural frequencies usually below 9-10 Hz. • Stair descent step frequency up to 4 Hz. - 2nd harmonic can match fn up to 8 Hz. - 3rd harmonic can match fn up to 12 Hz.

• Harmonic force amplitudes are high. • Resonance  High Accelerations

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Slender Monumental Stairs Group Amplifications • Groups can cause much higher accelerations than an individual. • Same velocity, fixed stride length  Same fstep. • Three times higher accelerations.

Prediction Methods • Finite Element Analysis • “Manual” Calculations

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Slender Monumental Stairs “Manual” Calculations • Stairs are often linear elements. • Idealize as a beam on a slope

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Slender Monumental Stairs Acceptance Criteria • Frequency:

fn

1/2 π  gE s I t  = 4  2  wLs 

Ls = Stringer Length

• Predicted Acceleration: 2 a p RαQ cos φ(1− exp(−100β)) ≤ a o βW g g Descending Acceleration Tolerance Limits Step Frequency Acceleration Remarks Hz Limit (%g)