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

2222

Floor Vibrations

A Critical Serviceability Consideration

for Steel Framed Floors.

Humans are very sensitive to vertical floor motion.

3333

Topics

Basic Vibration Terminology Floor Vibration Fundamentals

Walking VibrationsRhythmic Vibrations

FootbridgesRetrofitting

4

BASIC VIBRATIONTERMINOLOGY

5555

Period And Frequency

Period tp

6666

Natural Frequency

wL

tIsgE

2f

2/1

4n

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 =

10101010

Effective Weight

11

FLOOR VIBRATION FUNDAMENTALS

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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_ _

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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

18

NATURAL FREQUENCYOF

STEEL FRAMEDFLOOR SYSTEMS

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

20202020

Member

Bay

System

Fundamental Frequencies

H/g18.0f zn

)/(g18.0f gbn

)/(g18.0f cgbn

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

26

DESIGN FORWALKING EXCITATION

27272727

Walking Vibrations Criterion

ga

W

)f35.0exp(Pga onop

Predicted Tolerance

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

_ _ _ _ _ _ _ _ _ _ _

_ _ _

_ _ _

_

_ _ _ _

_ _ _ _ _ _ _ _ _ _ _

_ _ _

_ _ _

_

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

_ _ _ _ _ _ _ _ _ _ _

_ _ _

_ _ _

_

_ _ __ _

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

33333333

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

35353535

Beam Panel Width

Bj = Beam Panel Width

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

39393939

Effective Girder Panel Width

Bg = Girder Panel Width

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

424242

Example

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.

606060

DG11 Floor Width and Length

61616161

Bay Floor

Width

Floor

Length

A

B

C

D

Floor Width and Length Example

A

B

D

C

62626262

Bay Floor

Width

Floor

Length

A 90 90

B

C

D

Floor Width and Length Example

A

B

D

C

63636363

Bay Floor

Width

Floor

Length

A 90 90

B 150 90

C

D

Floor Width and Length Example

A

B

D

C

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

66

Bg = Cg(Dj/Dg)1/4 Lg 2/3 × Floor Length

67

Bg = Cg(Dj/Dg)1/4 Lg 2/3 × Floor Length

Bays A & B Bg = 59.9’<2/3Floor

Length

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

70

DESIGN FORRHYTHMIC EXCITATION

71717171

Aerobics

72727272

Balcony Video

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

76

FOOTBRIDGES

77777777

Be careful when designing foot-bridges and crossovers

• Very low damping

• Low frequency

• Lateral Vibrations

78787878

Troubled BridgeOver Water

Troubled BridgeOver Water

79

VIDEO

80

VIDEO

81

EVALUATION ANDREMEDIAL MEASURES

82828282

Methods To Stiffen Floors

Damping Post

AddedPosts

DampingElement

83838383

Methods To Stiffen Floors

Steel RodCover Plate

Cover Plates and Bottom Chord ReinforcingGenerally do not Work

84848484

Queen Post Hanger Stiffening

HVAC

Added Queen Post Hanger

85858585

Queen Post Hanger Stiffening

86868686

Queen Post Hanger Stiffening

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

89898989

Pendulum TMD

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

92

939393

Final Thought

Strength is essential but otherwise

unimportant.

Hardy Cross

Thank You!!