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Design of PostDesign of Post--TensionedTensionedSlabs for VibrationsSlabs for VibrationsSlabs for VibrationsSlabs for Vibrations

Fl i A l iFlorian AalamiPresident, ADAPT Corporation

VIBRATION DESIGN OF CONCRETE FLOORS

Part 1 – Basic information and practical approachWhy are we concerned about vibration; yWhat are the causes of vibration;What are the allowable limits of vibration; How to evaluate the vibration acceptability of a

floorfloor Numerical Example

Why are we concerned about vibration ?

Vibrations may be considered as “annoying” by occupantsoccupantsThreshold of “perception”Threshold of “annoyance”

Vibrations may interfere with the functions of machines and instruments such as in laboratoriesmachines and instruments, such as in laboratories.Each lab/instrument has its own specific limits of

acceptable vibration


What are the causes of vibration ?What are the causes of vibration ?

Most common cause is “foot drop” (heel drop) occupants;

Dynamic impact from rolling object.

What are the allowable limits of vibration?

Perception of vibration depends on: Frequency (cycles per second, Hz); and Peak acceleration (expressed as percentage of

gravitational acceleration %g)gravitational acceleration %g)

Consensus is that humans are most sensitive to vibration for frequencies between 4 to 8 Hz. q

Higher accelerations can be tolerated at higheror lower frequencies.

PERCEPTION OF VIBRATION

What are the allowable limits of vibration? Frequency (Hz); and Peak acceleration

1 00

1.50

tion

(% g

)

0.50

1.00Office, residence, assembly hall

Operatingak A

ccel

erat

0.25

1 2 4 8 12 20

Operating roomsP

ea

Frequency (Hz)

ATC = Applied Technology Council

Frequency (Hz)

Threshold of Human Sensitivity to Vertical Vibration (ATC)

ATC Applied Technology Council


How to evaluate the vibration acceptability of a floor?

6 steps for complete evaluation

Step 1 - Determine natural frequencies (Hz)St 2 S l t iti f f ib tiStep 2 – Select exciting force of vibrationStep 3 - Select floor typeStep 4 – Calculate the weight of vibrating panelStep 5 – Calculate peak accelerationStep 5 Calculate peak accelerationStep 6 - Evaluate the floor

Numerical example

NATURAL FREQUENCIES

Step 1 - Determine natural frequencies (Hz)

Use Finite Element Method with plate elements, orempirical formulas

For specific areas, such as a lab or operating room,determine the “dominant” frequencies of the location of interest.

Observe vibration modes of the above floor system

EXITING FORCE OF VIBRATION

Step 2 - Select exciting force of vibration

Use the following graph to determine the excitingforce

1.0

Fact

or (D

LF)

0.6

0.8

ynam

ic L

oad

F

0 2

0.4

Dy

Frequency Hz

0 1.0 2.0 3.00

0.2

Dynamic Load Factor for First Harmonic of Walking Force

Frequency Hz

Exciting force = DLF * (Weight of Person)

TRANSMISSION PATH OF VIBRATION

Step 3 - Select floor type

Refer to the table to select damping factor (β); in most cases 0.03 applies

The recommended values vary from 2-3% for bare concrete floors to 5-8% for furnished rooms with partitions extending full height.partitions extending full height.

RECOMMENDED DAMPINGFACTORS FOR VARIOUS OCCUPANCIES

Occupancy Damping factorβ

B t fl 0 02Bare concrete floor 0.02Furnished, low partition 0.03Furnished, full height partition 0.05Shopping malls 0.02

PEAK ACCELERATION

Step 4 –Calculate weight of target panel (W)Include superimposed load that follows the vibration

(stones; tiles)( ; )

Step 5 – Calculate Peak Acceleration (ap/g)Use an empirical relationship, such as the one below for

footfallfootfall

(1)0.35fn

p 0a P e

( )

ap = peak acceleration;

g W

ap peak acceleration;g = gravitational acceleration [32.2 ft/sec2; 9.81 m/sec2 ];Po = constant force representing the walking force (from

Step 2 and weight of walking person);β = modal damping ratio, from previous tableW = effective weight of the panel and the superimposed

load; andf = first natural frequency (Hz)fn = first natural frequency (Hz).

PERCEPTIBILITY OF MOTION

Step 6 – Evaluate the floor Use natural frequency from Step 1; and Peak ground acceleration (ao/g) from Step 5; ando Chart below from ATC to determine acceptability

1.50

on (%

g)

0 50


Acc

eler

atio

0.25

0.50

1 2 4 8 12 20

Operating roomsP

eak

Threshold of Human Sensitivity to Vertical Vibration (ATC)

Frequency (Hz)

Vibration (ATC)

NUMERICAL EXAMPLE

Numerical Example

GiGiven Concrete floor system Slab thickness 8” (203 mm) Superimposed DL 20 psf (1 kN/m2) Concrete f’c 5000 psi (33 8 MPa) Concrete f’c 5000 psi (33.8 MPa) Modulus of Elasticity 1.2 Ec

Required Evaluate vibration compliance of the floor panel Evaluate vibration compliance of the floor panel

identified under foot drop

NUMERICAL EXAMPLE

TNO_123

Slab thickness8'' (200mm)

(a) Floor plan Column18'' x 24''(460mm x 610mm)

26'-3''(8.00m) ( )

30'-0'' (9.14m)

Identification of Panel for Vibration Evaluation

(b) Panel plan


Step 1 – Determine natural frequencies (Hz) Generate model of the entire floor Use modulus of elasticity 1.2Ec Include weight of objects attached to floor Obtain first few natural frequencies

Discretization of the Floor for a Reliable Frequency ValuesReliable Frequency Values


(a) First mode – frequency 5 79 Hz(a) First mode frequency 5.79 Hz

(b) Second mode – frequency 6 33 Hz(b) Second mode – frequency 6.33 Hz

(c) Third mode – frequency 6.44 Hz( ) q y

Demonstrate live


Step 2 - Select exciting force of vibration Weight of person 150 lb (667 N)Walking speed: 2 steps per secondWalking speed: 2 steps per second From the chart:

DLF = 0.53Po = 0.53 * 150 = 79.5 lb (354 N)

F) 0.8

1.0

d Fa

ctor

(DLF

0.6

Dyn

amic

Loa

d

0.2

0.4

Frequency Hz

0 1.0 2.0 3.00

Dynamic Load Factor for First Harmonic of Walking Force


Step 3 - Select floor type

From below select (β) = 0.03 From table

Office furnished; low partitions

Occupancy Damping factorβ

Bare concrete floor 0.02Furnished, low partition 0 03Furnished, low partition 0.03Furnished, full height partition 0.05Shopping malls 0.02


Step 4 –Calculate weight of target panel (W)

Dimensions of panel as shown below M fi l h d fl 20 f (1 kN/ 2) Mortar, stone, firmly attached to floor 20 psf (1 kN/m2) Concrete weight: 150 pcf; (25 kN/m3)

The total weight of the panel W is :

W = = 94.5 k (421 kN) 830 26.25 0.15 0.0212

TNO_123

Slab thickness8'' (200mm)

(a) Floor plan Column18'' x 24''(460mm x 610mm)

26'-3''(8.00m)

30'-0'' (9.14m)

(b) Panel plan


Step 5 – Calculate Peak Acceleration (ap/g) Use an empirical relationship, such as the one below for

footfall0 35fnP 0.35fn

p 0a P eg W

ap = peak acceleration;g = gravitational acceleration [32.2 ft/sec2; 9.81 m/sec2 ];Po = 0.53 * 150 = 79.7 lb (0.354 kN) (step 2)ββ = damping ratio 0.03 (step 3)W = 94.5 k ( 421 kN ) (step 4)fn = first natural frequency (Hz) = 5.79 Hz (step 1).

= = 0.00367 ; 0.37%

0.35 5.9779.5 e0.03 94.5 1000

pagg

Peak ground acceleration ( ap) is calculated to be 0.37% of gravitational acceleration (g)


Step 6 – Evaluate the floor Use natural frequency from Step 1; 5.79 Hz; and Peak ground acceleration from Step 5; 0.37% g; and Chart below from ATC to determine acceptability

1 00

1.50

tion

(% g

)

0.50


Operatingak A

ccel

erat

0.25Panelstatus

1 2 4 8 12 20

Operating roomsP

ea

Frequency (Hz)Threshold of Human Sensitivity to Vertical

Vibration (ATC)

The target panel is acceptable for office and residential i b f h i l i

Frequency (Hz)

occupancies, but not for hospital operating room


Eliminate the noise (disturbance) from non-targeted floor regions

TNO 123

Floor plan showing the target panel

_

First mode – frequency 5.79 Hz

Note that the first mode is primarily due to excitation of a non-targeted region of floorexcitation of a non targeted region of floor


Isolate the target area for vibration analysisThe response of the isolated region is greatly influenced by the boundary conditions specified

d th taround the cut If the cut does not extend adequately beyond the

target panel, the results may not be reflective of theprototype

Note: first frequency of the entirefloor is 5.79 Hz


(a) Selection of an extended region

(b) First mode of vibration – Frequency 6.07 Hz

Note: first frequency of the entirefloor is 5.79 Hz


How can we bring the vibration of a given floorto compliance?

Increase in stiffness increases the frequenciesIncrease in stiffness increases the frequenciesIncrease in weight reduces the frequenciesIncrease in frequency, increases the peak accelerationIncrease in weight reduces the peak accelerationI i t i t f b d ditiIncrease in restraint of boundary conditions,

reduction in span increases frequency

For the given example, increase slab thickness from 8” (203 mm) to 10” (250 mm)

Frequency fn = 7.72 HzW = 115.3 k (514 kN) 1.00

1.50

( ) Peak acceleration

(ap/g) = 0.154 %

ratio

n (%

g)

0.50

Office, residence, assembly hall

Operating rooms

eak

Acc

eler

0.25

1

2 4 8 12 20Pe

Frequency (Hz) Panelstatus


Part 2 – Past practice; alternative methods

Semi empirical and simple methods of Semi empirical and simple methods of vibration evaluation Numerical Example

Estimate the natural frequenciesusing empirical relationships

Make a good engineering judgment on theprobable shape of the first naturalfrequency

FORMULAS FOR NATURAL FREQUENCIES

Case BoundaryConditions Constant φ

FIRST NATURAL FREQUENCY CONSTANT Φ

Conditions φ

1 2φ 1.57 1 γ

a

b

2

3

2 4φ 1.57 1 2.5γ 5.14γ

2 4φ 1.57 5.14 2.92γ 2.44γ3

4

φ 1.57 5.14 2.92γ 2.44γ

2 4φ 1.57 1 2.33γ 2.44γ

5

6

2 4φ 1.57 2.44 2.72γ 2.44γ

2 41 57 5 14 3 13 5 146 2 4φ 1.57 5.14 3.13γ 5.14γ

rigidly supported, rotationally freerigidly supported, rotationally fixedg y y

a span length in x-directionb span l length in y-direction a/b

DETERMINATION OF NATURAL FREQUENCIES

The parameters for the Table are:

Where2

cf φa

3

2

Eh gc =q12 1-

f = first natural frequency [Hz];a = span length in X-direction;E = dynamic modulus of elasticity [1 25 static EE = dynamic modulus of elasticity [1.25 static E

in psi; MPa];h = slab thickness [in; mm];ν = Poisson’s ratio [0.2];[ ]g = gravitational acceleration [32.2 ft/sec2 ; 9810

mm/sec2]; andq = weight per unit surface area of the slab.

DETERMINATION OF NATURAL FREQUENCIES

Shape of first Natural Frequency Mode The first mode of vibration is affine to that of a

i l l i l t d l t Th h isingle panel simply supported plate. The shape isnot analogous to the deflected profile underselfweight

(a) Simple support (b) Fixed

(c) Continuous spans

First Mode Shapes and Deflection of Simple and Continuous Spans

(d) Deflection self weight

Simple and Continuous Spans

DETERMINATION OF NATURAL FREQUENCY

Shape of first Natural Frequency Use a simply supported boundary conditions along

the four sides of an interior panelthe four sides of an interior panel

Where columns are on a regular orthogonal grid,the first mode is likely to be in form of a one-wayslab deflecting in a cylindrical form.

Panels bounded by smaller spans, mayvibrate analogous to a rotationally fixed plate

(a) Simple support (b) Fixed(a) Simple support (b) Fixed

( ) C ti(c) Continuous spans

(d) D fl ti lf i ht

First Mode Shapes and Deflection of Simple and Continuous Spans

(d) Deflection self weight


Numerical Example

Evaluate the vibration compliance of the floor slabEvaluate the vibration compliance of the floor slab below, for foot drop having the same details in former example, using empirical formulas

View of the Floor System


Step 1Envisage the probable shape of the first mode of vibrationvibration.A column supported slab, such as the floor system under consideration is likely to vibrate in form of a cylinder (one-way system)

Probable Shape of First Mode


Step 2Select from the Table of frequency formulas, the case that can best simulate the envisaged shape of first mode of vibration

TNO_123

Find the First Natural FrequencyUsing the parameters of frequency table:

g 9.81 m/sec2 ;32.2 ft/sec2

First natural frequency, fn:

fn =

c =

2ca

3

dynE h gc =

Est = 29,000 MPa= 4287 ksi , using ACI-318

2 q12 1

Assume Edyn = 1.2 Est = 1.2* Est

CALCULATE NATURAL FREQUENCY

FIRST NATURAL FREQUENCY CONSTANT Φ

Case Boundary Constant φCase Conditions Constant φ

1a

2φ 1.57 1 γb

rigidly supported, rotationally free a span length in x-direction

a

b span l length in y-direction

a/b

3hc =

3

2

D Eh gm q12 1

2φ 1.57 1 γ

fn = ( c / a2 )Φ (frequency of first mode)

φ γ


Consider a region as shown inthe figure

b

a

a = 3*90*12 = 1080 in (42.5 m)

Φ = 1.57(1+2) = = 20.03

2901.57 126.25

3

2

D Eh gm q12 1

c =

E = 1 2 E = 1 2* 4287 = 5144 ksi (35 477 MPa)

Edyn = 1.2 Est = 1.2* 4287 = 5144 ksi (35,477 MPa)


h = 8 inch (203 mm)

q = weight/unit area = 150 * 8 2012 *144 144

35144 *1000 * 8 32.2 *12

= 0.833 lb/ in2 ( 5.74 *10-3 MPa )12 144 144

2

5144 1000 8 32.2 120.83312 1 0.2

c = = 325,653 (2.1*108 mm2/sec)

f

in2/sec

325653fn = = 5.59 Hz 2

325653 * 20.031080

0.35fna P ep 0a P e

g W

0 35 5 5979 5= = 0.004 ; 0.4 %

0.35 5.5979.5 e0.03 94.5 1000

pag

Note that the weight of one panel is used for W


Check the Results for A t bilitAcceptability

First natural frequency (5. 59 Hz) Peak response acceleration relative to

gravitational acceleration (0.4 %)

) 1.00

1.50

erat

ion

(% g

)

0.50

Office, residence, assembly hall

Operating roomsP

eak

Acc

ele

0.25Panelstatus

Acceptable for office and residential but not

1

2 4 8 12 20

roomsP

Frequency (Hz) Acceptable for office and residential, but not

for operating rooms


Part 3 – Impact of secondary factors Post-tensioning

Post-tensioned slabs are generally thinnerthan their conventionally reinforced counterparts; hence, they have lower frequencies

Precompression from post-tensioning inhibits,or reduces crack formation. Post-tensioned slabs are stiffer than RC slabs of the sameslabs are stiffer than RC slabs of the same thickness

Add picture of a PT slab???


Part 3 – Impact of secondary factors Cracking of concrete

Cracking of concrete, in particular in conventionally reinforced concrete, where hair cracks under service condition are common and numerous in number result in reduction of local stiffness.

Concentration of local cracks over the supportsand at midspan can result in a change in the natural mode shape of vibrationnatural mode shape of vibration


Part 3 – Impact of secondary factors Cracking of concrete

Neutral axis

Reinforcement

(a) Cracked beam elevation

(b) Applied moment

Cracking moment

(c) Effective moment of inertia Ie

le

(c) Effective moment of inertia Ie

Loss of stiffness due to cracking results in shorter frequenciesin shorter frequencies


Part 3 – Impact of secondary factors

Cracking of concrete

Extent of cracking under dead load. Max localreduction in stiffness 69%

EXTENT OF CRACKING SHOWN THROUGH REDUCTION IN STIFFNESS FOR MOMENTS ABOUT Y-Y AXIS


Part 3 – Impact of secondary factors

Cracking of concretegExtent of cracking under dead load. Max localreduction in stiffness 67%

EXTENT OF CRACKING SHOWN THROUGH REDUCTION IN STIFFNESS FOR MOMENTS ABOUT X-X AXIS


P t 3Part 3 – Impact of secondary factors Floor covers; tiles; stones Equipment that is fixed to the floor

The weight of the objects that are attached to afloor and follow its motion in harmony should be included in the analysis. These include tiles,

d i t fi d t fland equipment fixed to a floor. Objects attached to a floor will add weight, but

not necessarily stiffness. Addition of weight reduces the frequency

Depending on the method of fixity of an objectto a floor, analysis software allow users to define a fraction of the mass of each equipment to be active for motion in each of theequipment to be active for motion in each of thethree principal directions.

FREQUENCIES

Floor’s mass Modulus of elasticity

Factors affecting the natural frequencies:

Modulus of elasticity Damping Extent of cracking; post-tensioning

MassMass Expresses as (W/g),

“W” is the weight of the objects that are attached to the floor and faithfully follow its ydisplacement; the greater the weight the larger isthe period; the smaller is the frequency (Hz)

“g” is the gravitational acceleration taken as 32.2 ft/sec2 (9.81 m/sec2).( )

Modulus of Elasticity The elastic modulus for vibration analysis is larger

th th t ti l i ti l h hi hthan the static values, in particular when highstrength concrete is used.

Recommended values are 25% higher than the static modulushigher than the static modulus.

TRANSMISSION PATH OF VIBRATION

Extent of Cracking

Cracking reduces the stiffness of a floor and consequently lowers its natural frequency.

For conventional RC allow for cracking

For RC flat slab construction; with span to d th ti 30 l ll 30% d ti idepth ratio 30 or larger, allow 30% reduction in stiffness due to cracking

For post-tensioned floors designed using ACI (IBC), no reduction in stiffness is due

For post-tensioned floors based on the European code EC2 and most other majorEuropean code EC2 and most other major non-US codes, reduction in stiffness may be necessary


People are most sensitive to vibration when

What is considered unacceptable?

pengaged in sedentary activity while seated or lying.

Much more is tolerated by people who are standing, walking, or active in other ways

The response acceleration is sometimes compared with the minimum acceptable value fn using the empirical formula developed for steel.p p

nKf 2.86lnW

K = a constant, given in [Table next slide];β = modal damping ratio [Table earlier];;

β = modal damping ratio [Table earlier];;W = weight of area of floor panel affected by

the point load (heel drop); andfn = minimum acceptable frequency.n


The response acceleration is compared with the minimum acceptable value fn .

nKf 2.86lnW

CONSTANT K FOR MINIMUM ACCEPTABLE FREQUENCY

Occupancies Kki kNOccupancies kips kN

Offices, residences, assembly halls 13 58

4 5 20Shopping malls 4.5 20

Using the values of the previous example:

K = 13 kips (58 kN) (empirical constant)K = 13 kips (58 kN) (empirical constant)W = 94.5 kips (421 kN) (weight of panel)β = 0.03 (damping)

f = = 4 36 Hz < 5 97 OK 132.86ln

0 03 94 fn = = 4.36 Hz < 5.97 OK0.03 94.5

RHYTHMIC MOTIONS

rhythmic motions; dance halls; gyms;

In each instance, the evaluation starts by estimating the following: Maximum number of individuals that are likely

to be in step harmony Number of foot drops per second, assume 3,

in absence of information Estimate of total weight of persons in step Estimate of total weight of persons in step

harmony (P)

The remainder of the analysis and evaluation follows exactly the same procedure outlined in foregoing examples

PERCEPTION OF VIBRATION

What are the allowable limits of vibration? Frequency (Hz); and Peak acceleration

g)

15.0

erat

ion

(% g

10.0

5 0

Rhythmic activitiesoutdoor footbridges

Pea

k A

ccel

e

0.0

5.0

1

2 4 8 12 20

Frequency (Hz)

Threshold of Human Sensitivity to Vertical Vibrationfrom Rhythmic Activities (ATC)

ATC = Applied Technology Council

from Rhythmic Activities (ATC)

ATC Applied Technology Council


References

ADAPT Technical Note TN290, 2010, “Vibration Design of Concrete Floors for Serviceability ” www adaptsoft com 20Concrete Floors for Serviceability, www.adaptsoft.com, 20 pp., 2010

ADAPT Technical Note TN388, 2010,” Vibration Evaluation of a Floor System for Footfall ” www adaptsoft com 9 ppof a Floor System for Footfall, www.adaptsoft.com, 9 pp., 2010

AISC/CISC, (1997) ,”Steel Design Guide Series 11, Floor Vibrations Due to Human Activity,” American Institute ofVibrations Due to Human Activity, American Institute of Steel Construction, Chicago, IL, 1997.

ATC, (1999) “ATC Design Guide 1,” Minimizing Floor Vibration,” Applied Technology Council, Redwood City, CA, , pp gy , y, ,1999, 49 pp.

Bares, R., (1971), “Tables for the Analysis of Plates, Slabs and Diaphragms Based on the Elastic Theory,” Bauverlag GmbH, Wiesbaden und Berlin, 1971, pp. 626

Mast, F. R., (2001),”Vibration of Precast Prestressed Concrete Floors,” PCI Journal, November-December 2001, 2001 76 862001, pp. 76-86.


References

Source: Allen D E and Murray T M (1993) “DesignSource: Allen, D. E., and Murray, T. M., (1993) Design Criterion for Vibrations Due to Walking,” Engineering Journal, Fourth Quarter, American Institute of Steel Construction, 1993, pp. 117-129.

TR43, (2005),” Post-tensioned concrete floors: Design Handbook,” Second edition, The Concrete Society , Surrey GU17 9AB, UK.

Szilard, R., (1974), “Theory and Analysis of Plates-Classical and Numerical Methods,” Prentice-Hall, Inc., New Jersey, 1974, 724 pp.

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