Post on 02-Jan-2016
description
1
Vibraciones en pisos de edificaciones con estructura
de al uso humano
Presented byThomas M. Murray, Ph.D., P.E.
Department of Civil and Environmental EngineeringVirginia Tech, Blacksburg, Virginia
thmurray@vt.edu
26 October 2011
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Floor Vibrations
A Critical Serviceability Consideration
for Steel Framed Floors.
Humans are very sensitive to vertical floor motion.
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Topics
Basic Vibration Terminology Floor Vibration Fundamentals
Walking VibrationsRhythmic Vibrations
FootbridgesRetrofitting
7777
Damping
Loss of Mechanical Energy in a Vibrating System
Critical Damping
Smallest Amount of Viscous Damping Required to Prevent Oscillation of a Free Vibrating System
8
Harmonics
P3
1st Harmonic
2nd Harmonic
3rd Harmonic
Footstep = tficosP stepi 2
f1f step1
f2f step2
f3f step3
P1
P2
0 0.2 0.4 0.6 0.80
0.05
0.1
0.15
0.2
0.25
0.3
Time (sec.)
Gro
und
Rea
ctio
n (k
ip)
9999
Acceleration Ratio
Acceleration Of A System, ap
Acceleration Of Gravity, ag
Usually Expressed As %g.
0.5%g is the Human ToleranceLevel for Quite Environments.
Ratio =
12
The Power of Resonance
0 1 2
Flo
or
Res
po
nse
2 - 3% Damping
Natural frequency, fn
Forcing frequency, f
5 - 7% Damping
13131313
Phenomenon of Resonance
• Resonance can also occur when a multiple of the forcing function frequency equals a natural frequency of the floor.• Usually concerned with the first natural frequency.• Resonance can occur because of walking dancing, or exercising.
14141414
0 1 2 3 4 5 6 70
0.1
0.2
0.3
0.4
0.5
Frequency (Hz)
Measure
d A
uto
spectr
um
(P
eak,
%g)
WalkingSpeed100 bpm
2nd Harmonic3.33 Hz
System Frequency5 Hz – 3rd Harmonic
Response from a Lightly Damped Floor
15151515
A Tolerance Criterion has two parts:
• Prediction of the floor response to a specified excitation.
• Human response/tolerance
Human Tolerance Criterion
16161616
FloorVibe v2.02Software for AnalyzingFloors for Vibrations
Criteria Based on AISC/CISC Design Guide 11
SEI
Structural Engineers, Inc.537 Wisteria DriveRadford, VA 24141
540-731-3330 Fax 540-639-0713tmmurray@floorvibe.comhttp://www.floorvibe.com
AISC/CISC Design Guide
17171717
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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
ISO Baseline Curve forRMS Acceleration
Pea
k A
ccel
erat
ion
(%
Gra
vity
)
Frequency (Hz)
Indoor Footbridges,
. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .
DG11 Uses the Modified ISO Scale for Human Tolerance
19191919
Fundamental Natural FrequencyUniformly Loaded – Simply
Supported Beam
(3.3)
(3.1)
(Hz.)
wL4ItgEs
2f
2/1
n (Hz.)
/g18.0fn
ItE384 s/wL5 4
21212121
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
22222222
Section Properties - Beam/Girder
b (< 0.4 L)
• Fully Composite
• Effect Width
• n = Es/1.35Ec
23
Why is the full composite moment of inertia used in the frequency calculations even when the beam or girder is non-composite?
)/(g18.0f gbn
ItE384 s/wL5 4
A Frequently Asked Question
24
Why is the full composite moment of inertia used in the frequency calculations even when the beam or girder is non-composite?
Annoying vibrations have displacements of 1-3 mm. Thus, the interface shear is negligible, so its acts as fully composite.
A Frequently Asked Question
25252525
Minimum Frequency
To avoid resonance with the first harmonic of walking, the minimum frequency must be greater than 3 Hz. e.g.
fn > 3 Hz
28282828
ap = peak acceleration
ao = acceleration limit
g = acceleration of gravity
fn = fundamental frequency of a beam or joist panel, or a combined panel, as applicable
Po = a constant force equal to 65 lb for floors and 92 lb for footbridges
= 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
ga
W
)f35.0exp(Pga onop
Walking Vibrations Criterion
29292929
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_
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_ _ _
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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
Pea
k A
ccel
erat
ion
(%
Gra
vity
)
Frequency (Hz)
Indoor Footbridges,
. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .
ISO Baseline Curve forRMS Acceleration
Modified ISO Scale
_ _ _
_ _ _
_ _ _
_
30303030
Recommended Values of Parameters in Equation (4.1) and a /g Limitso
Occupancy Constant Force Damping Ratio Acceleration Limit ao/g x 100% Po
Offices, Residences, 65 lb (0.29 kN) 0.02 – 0.05
*0.5%Churches
Shopping Malls 0.02 1.5%
Footbridges - Indoor 0.01 1.5%
Footbridges - Outdoor 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
65 lb (0.29 kN)
92 lb (0.41 kN)
92 lb (0.41 kN)
31
Estimating Modal Damping, β
Structural System – 0.01 (1%)Ceiling and Ductwork – 0.01(1%)Electronic Office Fitout – 0.005 (0.5%)Paper Office Fitout – 0.01 (1%)Churches, Schools, Malls – 0% Dry Wall Partitions in Bay – 0.05 to 0.10
5% to 10%
Note: Damping is cumulative.
32323232
Use very low live load (6-8 psf or 0.27-0.35 kPa) and low modal
damping (2% – 2.5%) for electronic office floor systems.
See Floor Vibration and the Electronic Office in Modern Steel
Construction August 1998
Important
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Equivalent Combined ModePanel Weight (W in Eqn. 2.3)
(4.4)
ga
W
)f35.0exp(Pga onop
WWW ggj
gj
gj
j
34343434
Beam and Girder PanelEffective Weights
Beam Panel:
Girder Panel:
LjBj)S/w j(=Wj
LgBg)L avg,j/wg(=Wg
36363636
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 j4/1
j
37373737
Section Properties - Slab
12” ´
_ _ _ _
de=dc-ddeck /2
A = (12 / n) de
n = Es/1.35 Ec
in4/ ft
f’c in ksi
)12/d)(n/12(D 3es
fwE c5.1
c
38383838
Beam or Joist PanelEffective Weights
For hot-rolled beams or joistswith extended bottom chords, Wj
can increased 50% if an adjacentspan is greater than 0.7 x the span considered. That is,
Wj = 1.5(wj/S)BjLj
40404040
Effective Girder Panel Width
Bg = Cg(Dj/Dg)1/4 Lg 2/3 × Floor Length
Cg = 1.6 For Girders Supporting Joists
Connected Only to a Girder Flange = 1.8 For Girders Supporting Beams Connected to a Girder Web
Dg = Ig/Lj,avg in4/ft
41414141
Constrained Bays
Girder Deflection Reduction Factor for Constrained Bays:
If Lg < Bj, substitute:
(4.5)
for g in Equation (4.4) and in Frequency Eq.
gj
gg
B
L5.0
B
L
j
g with
43434343
S
W24 × 55
W21 × 444 SPA @ 7´- 6´ =30´= L´ g
W21
× 4
4W
14 ×
22
W18
× 3
5
W14
× 2
2
L
= 4
5´j
W18 × 35
3.50”2.00”
d = 3.50 +e2.00
2= 4.50”
SectionW14
× 2
2
Floor Width = 30 ft Floor Length = 90 ft
Paper Office
44444444
Gravity Loads:LL : 11 psf (0.5 kPa) (For Vibration Analysis) Mech. & Ceiling : 4 psf (0.2 kPa)
Deck Properties:Concrete: wc = 110 pcf f’c = 4000 psi Floor Thickness = 3.50 in. + 2 in. ribs = 5.50 in.
Slab + Deck Weight = 47 psf
45454545
Beam Properties
W18 × 35 A = 10.30 in.2
Ix = 510 in.4
d = 17.70 in.
Girder Properties
W24 × 55 A = 16.20 in.2
d = 23.57 in.
Member Properties
Ix = 1350 in.4
46464646
Beam Mode Properties
Effective Concrete Slab Width = 7.5 ft < 0.4 Lj
= 0.4 x 45 = 18 ft.
n = modular ratio = Es/1.35Ec
= 29000 / (1.35 x 2307)
= 9.31Ij = transformed moment of inertia = 1799 in4
ksi23070.4110fwE 5.1c
5.1c
47474747
wj = 7.5 (11 + 47 + 4 + 35/7.5) = 500 plf
Equation (3.3)
Beam Mode Properties Cont.
.in885.017991029384
1728455005
EI384Lw5
6
4
j
4jj
j
jj
g18.0f
Hz76.3885.0
38618.0
48484848
Cj = 2.0
Bj = Cj (Ds/ Dj)1/4Lj
= 2.0 (9.79 / 240)1/4(45) = 40.4 ft > 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=Dj
ft/.in79.9 4=)12/50.4 3)(31.9/12(=)12/d( 3e)n/12(=Ds
49494949
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
50505050
wg = Lj (wj/S) + girder weight per unit length
= 45(500/7.5) + 55 = 3055 plf.
(3.3)
Girder Mode Properties Cont.
.in43.0=4436×10×29×384
1728×30×3055×5=
gIsE384
Lw5=Δ 6
44gg
g
.Hz37.5=433.0
38618.0=
Δ
g18.0=f
gg
.ft/.in6.98 4=45/4436=Lj/Ig=Dg
51515151
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
52525252
Combined Mode Properties
Lg = 30 ft < Bj = 20 ft Do Not Reduce
fn = Fundamental Floor Frequency
)+18.0= ΔΔ/(g gj
Hz08.3=
)433.0+885.0/(38618.0=
53535353
Combined Mode Properties Cont.
WΔΔ
ΔW
ΔΔ
Δg
gj
gj
gj
j
++
+=W
kips100=
)122(433.0+885.0
433.0+)90(
433.0+885.0
885.0=
54545454
= 0.0074
= 0.03 from Table 4.1 (Modal Damping Ratio)
W = 0.03 × 100 = 3.0 kips
Evaluation
= 0.74% g > 0.50% g N.G.
3000
)08.335.0exp(65
W
)f35.0exp(Pga nop
55555555
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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
Pea
k A
ccel
erat
ion
(%
Gra
vity
)
Frequency (Hz)
Indoor Footbridges,
Extended by Allen and Murray (1993). . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .
ISO Baseline Curve forRMS Acceleration
56565656
Original DesignW18x35 fb = 3.76 hz fn = 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 fn = 3.04 Hz W24x55 fg = 5.28 hz ap/g=0.65%g
57575757
Original DesignW18x35 fb = 3.76 hz fn = 3.08 Hz W24x55 fg = 5.37 hz ap/g=0.74%g
Improved Design Increase Girder Size
W18X35 fb = 3.76 hz fn = 3.33 Hz W24x84 fg = 7.17 hz ap/g=0.70%g
58585858
W18x35 fb = 3.76 hz fn = 3.08 Hz W24x55 fg = 5.37 hz ap/g=0.74%g
Improved DesignsIncrease Beam Size
W21x50 fb = 4.84 hz fn = 3.57 Hz W24x55 fg = 5.29 hz ap/g=0.58%g
W24x55 fb = 5.22 hz fn = 3.71 Hz W24x55 fg = 5.28 hz ap/g=0.50%g
Original Design
59595959
Rule: In design, increase stiffnessof element with lower frequency to improve performance.
If beam frequency is less than the girder frequency, increase the beam frequency to the girder frequency first, then increase both until a satisfactory design is obtained.
64646464
Bay Floor
Width
Floor
Length
A 90 90
B 150 90
C 150 30 (45?)
D
Floor Width and Length Example
A
B
D
C
65656565
Bay Floor
Width
Floor
Length
A 90 90
B 150 90
C 150 30
D 30 90
Floor Width and Length Example
A
B
D
C
68
Bg = Cg(Dj/Dg)1/4 Lg 2/3 × Floor Length
Bays A & B Bg = 59.9’<2/3 Floor L Bays A:
Floor Length = 81’ e.g. (32.5’ + 16” + 32.5’)
Bg=2/3x81 = 54’ < 59.9’
ap/g=0.46%g < 0.5%
69
Bg = Cg(Dj/Dg)1/4 Lg 2/3 × Floor Length
Bays A & B Bg = 59.9’< 2/3Floor L
Bays A: Bg = 54’ ap/g=0.46%g < 0.5%
OKBay B: Floor Length = 48.5’ e.g. (32.5’ + 16’) 2/3x48.5 =32.3’ < 59.9’
ap/g=0.61%g > 0.5%g NG
73737373
b, g and c are beam, girder and columndeflections due to supported weight
Natural Frequency forRhythmic Excitation
Column deflections may be important foraerobic excitations.
)/(g18.0f cgbn
74747474
f
f2 n2
1f
fn2 2
w/w3.1
ga
stepstep
tpip
aa 5.1pa omax
5.1/1 (1.5 Power Rule)
Evaluation Using Acceleration
757575
g18.0nf
Note, for a given fn, Δ is constant.
Example. For fn = 5 Hz, g = 386 in/sec2
Δ = 0.5 in regardless of span length!!
Frequency versus Span
77777777
Be careful when designing foot-bridges and crossovers
• Very low damping
• Low frequency
• Lateral Vibrations
83838383
Methods To Stiffen Floors
Steel RodCover Plate
Cover Plates and Bottom Chord ReinforcingGenerally do not Work
87878787
Stiffening Of Girders SupportingCantilevered Beams and Joist Seats
CantileveredBeam orJoist Seat
Girder
Stiffener
88888888
Pendulum TMD
Large Mass ~ 2% Mass Ratio“Frictionless” Bearings
Coil Spring
Air Dashpot Damping
90909090
5th Floor - Response to Walk ing
-0.006
-0.004
-0.002
0.000
0.002
0.004
0.006
10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0
Tim e, seconds
Acc
eler
atio
n, g
's
Floor Acceleration w /o TMD
5th Floor - Response to Walk ing
-0.006
-0.004
-0.002
0.000
0.002
0.004
0.006
10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0
Tim e, seconds
Acc
eler
atio
n, g
's
Floor Acceleration w ith TMD
Without TMD
With TMD
Walking
91919191
Response to Walking
Results
5th Floor Response to Walking
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
0 1 2 3 4 5 6 7 8 9 10Frequency, Hz.
Ve
loci
ty, i
n/s
ec
0-p
k
Floor Velocity w/o TMD
Floor Velocity with TMD
5.25 Hz. , 0.01523 ips 0-pk
5.25 Hz. , 0.00756 ips 0-pk
50% Reduction