thomas murray andres sanchez floor vibrations
TRANSCRIPT
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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
Developed by
Thomas M. Murray, Ph.D., P.E.
Department of Civil and Environmental Engineering
Virginia Tech
Blacksburg, [email protected]
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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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8888
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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12121212
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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13131313
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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14141414
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
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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19191919
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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21212121
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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24242424
Walking Vibrations Criterion
g
a
W
)f35.0exp(P
g
a onop
=
Predicted Tolerance
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25252525
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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30303030
Beam and Girder Panel
Effective Weights
Beam Panel:
Girder Panel:
LjBj)S/wj(=Wj
LgBg)L avg,j/wg(=Wg
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31313131
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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51515151
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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53535353
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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54545454
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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57575757
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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58585858
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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59595959
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!!