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VIBRATIONS IN FLOOR SYSTEMS OF STEELSTRUCTURES DUE TO HUMAN USE
Presented by
Telmo Andres Sanchez, Ph.D.HDR Engineering, Inc.
Pittsburgh, PA
asanchez@hdrinc.com
Developed by
Thomas M. Murray, Ph.D., P.E.
Department of Civil and Environmental Engineering
Virginia Tech
Blacksburg, VAthmurray@vt.edu
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2222
Topics
Basic Vibration Terminology
Floor Vibration Fundamentals
Natural Frequency of Steel FramedFloor Systems
Design for Walking Excitation
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4444
Period And Frequency
Period tp
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5555
Natural Frequency
=
wLtIsgE
2f
2/1
4n
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6666
DampingLoss of Mechanical Energy in a
Vibrating System
Critical DampingSmallest Amount of Viscous Damping
Required to Prevent Oscillation of aFree Vibrating System
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Harmonics
P3
1st Harmonic
2nd Harmonic
3rd Harmonic
Footstep = tficosP stepi = 2
f1f step1
=
f2
f step2 =
f3f step3 =
P1
P2
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Acceleration Ratio
Acceleration Of A System, ap
Acceleration Of Gravity, ag
Usually Expressed As %g.
0.5%g is the Human Tolerance
Level for Quite Environments.
Ratio =
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9999
Effective WeightFloor Width
FloorLength
W
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FLOOR VIBRATION
FUNDAMENTALS
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The Power of Resonance
0 1 2
FloorResponse
2 - 3% Damping
Natural frequency, fn
Forcing frequency, f
5 - 7% Damping
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Phenomenon of Resonance
Resonance can also occur when a
multiple of the forcing functionfrequency equals a natural frequency of
the floor. Usually concerned with the first natural
frequency.
Resonance can occur because of walking
dancing, or exercising.
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0 1 2 3 4 5 6 70
0.1
0.2
0.3
0.4
0.5
Frequency (Hz)
Me
asuredAutospectr
um
(Peak,
%g)
Walking
Speed100 bpm
2nd Harmonic3.33 Hz
System Frequency
5 Hz 3rd Harmonic
Response from a Lightly
Damped Floor
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A Tolerance Criterion has two parts: Prediction of the floor response to a
specified excitation. Human response/tolerance
Human Tolerance Criterion
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FloorVibe v2.02Software for Analyzing
Floors for Vibrations
Criteria Based on AISC/CISC Design
Guide 11
SEI
Structural Engineers, Inc.
537 Wisteria DriveRadford, VA 24141
540-731-3330 Fax 540-639-0713
tmmurray@floorvibe.com
http://www.floorvibe.com
AISC/CISC Design Guide
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16161616
_ _ _ _
_ _ _ _
_ _ _ _
_ ___ _
1 3 4 5 8 10 25 40
25
10
5
2.5
1
0.5
0.25
0.1
0.05
Rhythmic Activities
Outdoor Footbridges
Shopping Malls,Dining and Dancing
Offices,
Residences
ISO Baseline Curve for
RMS Acceleration
PeakAcceleration(%G
ravity)
Frequency (Hz)
Indoor Footbridges,
. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .
DG11 Usesthe Modified
ISO Scale for
HumanTolerance
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NATURAL FREQUENCYOF
STEEL FRAMEDFLOOR SYSTEMS
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18181818
Fundamental Natural Frequency
Uniformly Loaded SimplySupported Beam
(3.3)
(3.1)
(Hz.)
=
wL4
ItgEs
2
f
2/1
n (Hz.)
/g18.0fn
ItE384 s/wL5 4
=
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Member
Bay
System
Fundamental Frequencies
H/g18.0f zn
)/(g18.0f gbn
)/(g18.0f cgbn
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20202020
Loads for Vibration Analysis
LDwItE384 s/wL5 4
D: Actual Load
L: 11 psf for Paper Office
6-8 psf for Electronic Office
6 psf for Residence
0 psf for Malls, Churches, Schools
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Section Properties - Beam/Girder
b (< 0.4 L)
Fully Composite
Effect Width
n = Es/1.35Ec
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Minimum Frequency
To avoid resonance with the firstharmonic of walking, the
minimum frequency must begreater than 3 Hz. e.g.
fn > 3 Hz
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DESIGN FOR
WALKING EXCITATION
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Walking Vibrations Criterion
g
a
W
)f35.0exp(P
g
a onop
=
Predicted Tolerance
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ap = peak acceleration
ao = acceleration limit
g = acceleration of gravity
fn = fundamental frequency of a beam or joist panel, or acombined panel, as applicable
Po = a constant force equal to 65 lb for floors and 92 lb forfootbridges
= modal damping ratio (0.01 to 0.05 or 1% to 5%)
W = effective weight supported by the beam or joist panel,
girder panel, or combined panel, as applicable
g
a
W
)f35.0exp(P
g
a onop
=
Walking Vibrations Criterion
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_ _ _ _
_ _ _ _
_ _ _ _
_ _ __ _
1 3 4 5 8 10 25 40
25
10
5
2.5
1
0.5
0.25
0.1
0.05
Rhythmic Activities
Outdoor Footbridges
Shopping Malls,
Dining and Dancing
Offices,
Residences
PeakAcceleration(%
Gravity)
Frequency (Hz)
Indoor Footbridges,
. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .
ISO Baseline Curve for
RMS Acceleration
Modified
ISO Scale
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Recommended Values of Parameters in Equation (4.1) and a /g Limitso
Occupancy Constant Force Damping Ratio Acceleration Limitao/g x 100%Po
Offices, Residences, 65 lb 0.02 0.05 * 0.5%
Churches
Shopping Malls 65 lb 0.02 1.5%
Footbridges - Indoor 92 lb 0.01 1.5%
Footbridges - Outdoor 92 lb 0.01 5.0%
Table 4.1
* 0.02 for floors with few non-structural components (ceilings, ducts, partitions,
etc.) as can occur in open work areas and churches,
0.03 for floors with non-structural components and furnishings, but with only
small demountable partitions typical of many modular office areas,
0.05 for full height partitions between floors.
Parameters
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Equivalent Combined ModePanel Weight (W in Eqn. 2.3)
(4.4)
g
a
W
)f35.0exp(P
g
a onop
=
WWW ggj
gj
gj
j
=
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Beam and Girder Panel
Effective Weights
Beam Panel:
Girder Panel:
LjBj)S/wj(=Wj
LgBg)L avg,j/wg(=Wg
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Beam Panel Width
Bj = Beam PanelWidth
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Effective Beam Panel Width
Floor Width
Cj = 2.0 For Beams In Most Areas= 1.0 For Beams at a Free Edge
(Balcony)
Dj = Ij/S (in4/ft)
3/2L)Dj/Ds(CjB j4/1j 2/3 (30) = 20 ft.
Wj = 1.5(wj/S)BjLj (50% Increase)
= 1.5 (500/7.5)(20.0 45) = 90,000 lbs = 90.0 kips
Beam Mode Properties Cont.
Bj
= 20 ft.
.ft/.in240 4=5.7/1799=S/Ij=Djft/
.in79.9 4
=)12/50.4 3
)(31.9/12(=)12
/d( 3
e)n/12(=D
s
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Girder Mode Properties
Eff. Slab Width = 0.4 Lg
= 0.4 x 30 x 12= 144 in. < Lj = 45 x 12 = 540 in.
b = 144
Ig = 4436 in4
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wg = Lj (wj/S) + girder weight per unit length
= 45(500/7.5) + 55 = 3055 plf.
(3.3)
Girder Mode Properties Cont.
.in43.0=44361029384
17283030555
=gIsE384
Lw5
= 6
44gg
g
.Hz37.5=433.0
386
18.0=
g
18.0=f gg
.ft/.in6.98 4=45/4436=Lj/Ig=Dg
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Cg = 1.8 (Beam Connected To Girder Web)
(4.3b)
= 1.8 (240 / 98.6)1/4 (30) = 67.4 ft > 2/3 (90) = 60
(4.2)
=(3055/45)(60 30) = 122,200 lb = 122 kips
Use
Girder Mode Properties Cont.
L)Dg/Dj(CgB g4/1
g=
LB)L/w(W ggjgg=
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Combined Mode Properties
Lg = 30 ft < Bj = 20 ft Do Not Reduce
fn = Fundamental Floor Frequency
)+18.0= /(g gj
Hz08.3=)433.0+885.0/(38618.0=
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Combined Mode Properties Cont.
W
W
g
gj
gj
gj
j
++
+=W
kips100=
)122(433.0+885.0
433.0+)90(
433.0+885.0
885.0=
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_ _ _ _
_ _ _ _
_ _ _ _
_ _ __ _
1 3 4 5 8 10 25 40
10
5
2.5
1
0.5
0.25
0.1
0.05
Rhythmic ActivitiesOutdoor Footbridges
Shopping Malls,Dining and Dancing
Offices,
Residences
PeakAcce
leration(%G
ravity)
Frequency (Hz)
Indoor Footbridges,
Extended by Allen
and Murray (1993). . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .
ISO Baseline Curve forRMS Acceleration
Original Design
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Original Design
W18x35 fb = 3.76 hz f n = 3.08 Hz
W24x55 fg = 5.37 hz ap/g=0.74%g
Improved Design
Increase Concrete Thickness 1 in.
W18X35 fb = 3.75 hz f n = 3.04 Hz
W24x55 fg = 5.28 hz ap/g=0.65%g
Original Design
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Original Design
W18x35 fb = 3.76 hz f n = 3.08 Hz
W24x55 fg = 5.37 hz ap/g=0.74%g
Improved Design
Increase Girder Size
W18X35 fb = 3.76 hz f n = 3.33 Hz
W24x84 fg = 7.17 hz ap/g=0.70%g
Original Design
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W18x35 fb = 3.76 hz f n = 3.08 Hz
W24x55 fg = 5.37 hz ap/g=0.74%g
Improved Designs
Increase Beam Size
W21x50 fb = 4.84 hz f n = 3.57 Hz
W24x55 fg = 5.29 hz ap/g=0.58%g
W24x55 fb = 5.22 hz f n = 3.71 Hz
W24x55 fg = 5.28 hz ap/g=0.50%g
Original Design
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Rule: In design, increase stiffnessof element with lower
frequency to improve
performance.
If beam frequency is less than the girderfrequency, increase the beam frequency to
the girder frequency first, then increase bothuntil a satisfactory design is obtained.
Final Thought
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Final Thought
Strength is essential but otherwiseunimportant.
Hardy Cross
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Thank You!!
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