7 Thomas Murray-Vibraciones en Entrepisos

1

Vibraciones en pisos de

edificaciones con estructura

de al uso humano

Presented by

Thomas M. Murray, Ph.D., P.E.

Department of Civil and Environmental Engineering

Virginia Tech, Blacksburg, Virginia

[email protected]

26 October 2011

2 2 2 2

Floor Vibrations

A Critical Serviceability

Consideration

for Steel Framed Floors.

Humans are very sensitive to vertical

floor motion.

3 3 3 3

Topics

Basic Vibration Terminology

Floor Vibration Fundamentals

Walking Vibrations

Rhythmic Vibrations

Footbridges

Retrofitting

4

BASIC VIBRATION

TERMINOLOGY

5 5 5 5

Period And Frequency

Period t p

6 6 6 6

Natural Frequency

wL

tIsgE

2f

2/1

4n

7 7 7 7

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 = tficosPstepi 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 R

eaction (

kip

)

9 9 9 9

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 =

10 10 10 10

Effective Weight Floor Width

Flo

or

Le

ng

th

W

11

FLOOR VIBRATION

FUNDAMENTALS

12

The Power of Resonance

0 1 2

Flo

or

Resp

on

se

2 - 3% Damping

Natural frequency, fn

Forcing frequency, f

5 - 7% Damping

13 13 13 13

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.

14 14 14 14

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)

Walking

Speed

100 bpm

2nd Harmonic

3.33 Hz

System Frequency

5 Hz – 3rd Harmonic

Response from a Lightly

Damped Floor

15 15 15 15

A Tolerance Criterion has two parts:

• Prediction of the floor response to a

specified excitation.

• Human response/tolerance

Human Tolerance Criterion

16 16 16 16

FloorVibe v2.02 Software for Analyzing

Floors for Vibrations

Criteria Based on AISC/CISC Design

Guide 11

SEI

Structural Engineers, Inc.

537 Wisteria Drive

Radford, VA 24141

540-731-3330 Fax 540-639-0713

[email protected]

http://www.floorvibe.com

AISC/CISC Design Guide

17 17 17 17

_ _ _ _

_ _ _ _

_ _ _ _

_ ___ _

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

Pea

k A

ccel

era

tio

n (

% G

ravit

y)

Frequency (Hz)

Indoor Footbridges,

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

DG11 Uses

the Modified

ISO Scale for

Human

Tolerance

18

NATURAL FREQUENCY

OF

STEEL FRAMED

FLOOR SYSTEMS

19 19 19 19

Fundamental Natural Frequency

Uniformly Loaded – Simply

Supported Beam

(3.3)

(3.1)

(Hz.)

wL4

ItgEs2

f

2/1

n (Hz.)

/g18.0fn

ItE384 s/wL5 4

20 20 20 20

Member

Bay

System

Fundamental Frequencies

H/g18.0f zn

)/(g18.0f gbn

)/(g18.0f cgbn

21 21 21 21

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

22 22 22 22

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

25 25 25 25

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 FOR

WALKING EXCITATION

27 27 27 27

Walking Vibrations Criterion

g

a

W

)f35.0exp(P

g

a onop

Predicted Tolerance

28 28 28 28

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

g

a

W

)f35.0exp(P

g

a onop

Walking Vibrations Criterion

29 29 29 29

_ _ _ _

_ _ _ _

_ _ _ _

_ _ __ _

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

era

tio

n (

% G

ravit

y)

Frequency (Hz)

Indoor Footbridges,

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

ISO Baseline Curve for

RMS Acceleration

Modified

ISO Scale

30 30 30 30

Recommended Values of Parameters in Equation (4.1) and a /g Limits o

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 w ith few non-structural components (ceilings, ducts, partitions,

etc.) as can occur in open w ork areas and churches,

0.03 for floors w ith non-structural components and furnishings, but w ith only

small demountable partitions typical of many modular office areas,

0.05 for full height partitions betw een 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.

32 32 32 32

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

33 33 33 33

Equivalent Combined Mode

Panel Weight (W in Eqn. 2.3)

(4.4)

g

a

W

)f35.0exp(P

g

a onop

WWW g

gj

gj

gj

j

34 34 34 34

Beam and Girder Panel

Effective Weights

Beam Panel:

Girder Panel:

LjBj)S/wj(=Wj

LgBg)L avg,j/wg(=Wg

35 35 35 35

Beam Panel Width

Bj = Beam Panel

Width

36 36 36 36

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 j

4/1

j

37 37 37 37

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(D3es

fwE c5.1

c

38 38 38 38

Beam or Joist Panel

Effective Weights

For hot-rolled beams or joists

with extended bottom chords, Wj

can increased 50% if an adjacent

span is greater than 0.7 x the span

considered. That is,

Wj = 1.5(wj/S)BjLj

39 39 39 39

Effective Girder Panel Width

Bg = Girder

Panel

Width

40 40 40 40

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

41 41 41 41

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.

g

j

gg

B

L5.0

B

L

j

gwith

42 42 42

Example

43 43 43 43

S

W24 × 55

W21 × 44 4 SPA @ 7´- 6´ =30´= L ´ g

W2

1 ×

44

W

14

× 2

2

W1

8 ×

35

W1

4 ×

22

L =

45

´ j

W18 × 35

3.50”

2.00”

d = 3.50 + e

2.00

2 = 4.50”

Section W1

4 ×

22

Floor Width = 30 ft

Floor Length = 90 ft

Paper Office

44 44 44 44

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

45 45 45 45

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

46 46 46 46

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)

= 9.31

Ij = transformed moment of inertia = 1799 in4

ksi23070.4110fwE5.1

c5.1

c

47 47 47 47

wj = 7.5 (11 + 47 + 4 + 35/7.5) = 500 plf

Equation (3.3)

Beam Mode Properties Cont.

.in885.017991029384

1728455005

EI384

Lw56

4

j

4jj

j

j

j

g18.0f

Hz76.3885.0

38618.0

48 48 48 48

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

49 49 49 49

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

50 50 50 50

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

51 51 51 51

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 g

4/1

g

LB)L/w(W ggjgg

52 52 52 52

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=

53 53 53 53

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=

54 54 54 54

= 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(P

g

a nop

55 55 55 55

_ _ _ _

_ _ _ _

_ _ _ _

_ _ __ _

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

era

tio

n (

% G

ravit

y)

Frequency (Hz)

Indoor Footbridges,

Extended by Allen

and Murray (1993) . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

ISO Baseline Curve for

RMS Acceleration

56 56 56 56

Original Design

W18x35 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

57 57 57 57

Original Design

W18x35 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

58 58 58 58

W18x35 fb = 3.76 hz fn = 3.08 Hz

W24x55 fg = 5.37 hz ap/g=0.74%g

Improved Designs

Increase 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

59 59 59 59

Rule: In design, increase stiffness

of 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.

60 60 60

DG11 Floor Width and Length

61 61 61 61

Bay Floor

Width

Floor

Length

A

B

C

D

Floor Width and

Length Example

A

B

D

C

62 62 62 62

Bay Floor

Width

Floor

Length

A 90 90

B

C

D

Floor Width and

Length Example

A

B

D

C

63 63 63 63

Bay Floor

Width

Floor

Length

A 90 90

B 150 90

C

D

Floor Width and

Length Example

A

B

D

C

64 64 64 64

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

65 65 65 65

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%

OK

Bay 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 FOR

RHYTHMIC EXCITATION

71 71 71 71

Aerobics

72 72 72 72

Balcony Video

73 73 73 73

b, g and c are beam, girder and column

deflections due to supported weight

Natural Frequency for

Rhythmic Excitation

Column deflections may be important for

aerobic excitations.

)/(g18.0f cgbn

74 74 74 74

f

f2 n2

1f

fn2

2

w/w3.1

g

a

stepstep

tpip

aa 5.1pa omax

5.1/1 (1.5 Power Rule)

Evaluation Using Acceleration

75 75 75

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

77 77 77 77

Be careful when designing foot-

bridges and crossovers

• Very low damping

• Low frequency

• Lateral Vibrations

78 78 78 78

Troubled Bridge

Over Water

79

VIDEO

80

VIDEO

81

EVALUATION AND

REMEDIAL MEASURES

82 82 82 82

Methods To Stiffen Floors

Damping Post

Added

Posts

Damping

Element

83 83 83 83

Methods To Stiffen Floors

Steel Rod Cover Plate

Cover Plates and Bottom Chord Reinforcing

Generally do not Work

84 84 84 84

Queen Post Hanger Stiffening

HVAC

Added Queen Post Hanger

85 85 85 85

Queen Post Hanger Stiffening

86 86 86 86

Queen Post Hanger Stiffening

87 87 87 87

Stiffening Of Girders Supporting

Cantilevered Beams and Joist Seats

Cantilevered

Beam or

Joist Seat

Girder

Stiffener

88 88 88 88

Pendulum TMD

Large Mass ~ 2% Mass Ratio “Frictionless” Bearings

Coil Spring

Air Dashpot Damping

89 89 89 89

Pendulum TMD

90 90 90 90

5th Floor - Response to Walking

-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

Time, seconds

Accele

rati

on

, g

's

Floor Acceleration w /o TMD

5th Floor - Response to Walking

-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

Time, seconds

Accele

rati

on

, g

's

Floor Acceleration w ith TMD

Without TMD

With TMD

Walking

91 91 91 91

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 10

Frequency, Hz.

Ve

locity, in

/se

c 0

-pk

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

93 93 93

Final Thought

Strength is essential but otherwise

unimportant.

Hardy Cross

Thank You!!