vibrations of reinforced concrete floor...

1/14/2015

Vibration of Concrete Floor Systems 1

Vibrations of Reinforced Concrete Floor Systems

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© The Concrete Reinforcing Steel Institute

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About the Speaker

� Mike Mota» Member of ACI 318, 318B and 318R (2019)» Chair of ACI 314 “Simplified Design of Concrete

Structures”» Member of ASCE-7 (2016)» Board of Directors CIB/NYC-ACI Chapter» Fellow of ACI, ASCE and SEI

Vibration of RC Floor Systems

� First design guide solely dedicated to vibrations of RC floor systems

� Based on approximate methods

� Verified by FEA solutions

� Case study of actual voided slab structure

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Objectives

� Assist design professional when effects from vibration must be considered

� Provide simplified methods to determine key vibration characteristics of RC floor systems and compare against acceptance criteria

� Overview of Mitigation Strategies� Several detailed design examples

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Outline

� CRSI Vibration Design Guide» Acceptance Criteria» Vibration Characteristics of Reinforced

Concrete Floor Systems54

» Examples• Voided Flat Plate Vibration Analysis

– Walking

– Rhythmic

– Sensitive

» Case Study6

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Acceptance Criteria

� Human Comfort» Walking excitation

• Dependent on peak acceleration experienced by structure

» Rhythmic excitation• Combination of peak acceleration and relationship

between natural and excitation frequencies

� Sensitive Equipment» Vibrational velocity

� Not mutually exclusive

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Acceptance Criteria


• Recommended acceleration limits �� ⁄• Ref (Allen and Murray, 1993)

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Acceptance Criteria


• Peak acceleration less than or equal to recommended acceleration limit

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�� 65 �. ��

��

Acceptance Criteria

� Human Comfort» Rhythmic excitation

• k = 1.3 for dancing• k = 1.7 for concert or sporting event• k = 2.0 for aerobics or jumping

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�� 1 � �� ⁄

��

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Acceptance Criteria

� Human Comfort» Rhythmic excitation

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Activity HarmonicForcing

Frequency �

Weight of Participants �� (psf)

Dynamic Coefficient

!Dancing 1 1.5 – 3.0 12.5 0.50Concert or sporting event

1 1.5 – 3.031.0

0.25

2 3.0 – 6.0 0.05

Aerobics or jumping

1 2.0 – 2.754.2

1.502 4.0 – 5.5 0.603 6.0 – 8.25 0.10

Acceptance Criteria

� Sensitive Equipment» Maximum velocity " � Limiting "

• Maximum velocity " � Δ$%& ��⁄• Limiting "

– Manufacturers’ criteria

– Generic criteria

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Acceptance Criteria

� Sensitive Equipment» Footfall impulse parameters

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Walking Pace (steps/minute)

'((lb)

�) � */,)(Hz)

-.(lb Hz 2)

Fast (100) 315 5.0 25,000

Moderate (75) 280 2.5 5,500

Slow (50) 240 1.4 1,500

Acceptance Criteria

� Sensitive Equipment» Generic criteria

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Equipment or UseVibrational

Velocity (/12./3456

• Computer systems• Operating rooms• Surgery facilities• Bench microscopes at up to 100x

magnification

8,000

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Acceptance Criteria


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Velocity (/12./3456

• Bench microscopes at up to 400x magnification

• Optical and other precision balances• Coordinate measuring machines• Metrology laboratories• Optical comparators• Microelectronics manufacturing

equipment – Class A (inspection, probe test and other manufacturing support equipment

2,000

Acceptance Criteria


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Velocity (/12./3456

• Micro surgery• Eye surgery• Neuro surgery• Bench microscopes greater than 400x

magnification• Optical equipment at isolation tables• Microelectronics manufacturing

equipment – Class B (Aligners, steppers, and other critical equipment for photolithography with lines widths of 3 microns or more)

1,000

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Acceptance Criteria


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Velocity (/12./3456

• Electron microscopes at up to 30,000x magnification

• Microtomes• Magnetic resonance imagers• Microelectronics manufacturing

equipment – Class C (Aligners, steppers, and other critical equipment for photolithography with lines widths of 1 micron)

500

Acceptance Criteria


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Velocity (/12./3456

• Electron microscopes at greater than 30,000x magnification

• Mass spectrometers• Cell implant equipment• Microelectronics manufacturing

equipment – Class D (Aligners, steppers, and other critical equipment for photolithography with lines widths of ½ micron, including electron-beam systems)

250

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Acceptance Criteria


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Velocity (/12./3456

• Microelectronics manufacturing equipment – Class E (Aligners, steppers, and other critical equipment for photolithography with lines widths of ¼ micron, including electron-beam systems)

• Unisolated laser and optical research systems

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Vibration Characteristics

� Floor Stiffness» Direct effect on natural frequency of the

floor system» 789:

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� Floor Stiffness» Modulus of elasticity

• 78 � �8 ;.�33 �8=» Dynamic modulus of elasticity

• 1.278

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� Floor Stiffness» Effective moment of inertia

• ACI Eq. (9-8)

– 9: � ?@A?B

9C � 1 D ?@A?B

98E � 9C• Bischoff and Scanlon

– 9: � F@A; G@A

GBH ;I@AIJ

� 9C

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� Effective Floor Weight, �» Dead load of floor system» Superimposed dead load» Actual live load

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� Natural Frequency» Flat plate system

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� Natural Frequency» Voided slab system

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� Natural Frequency» Flat plate system / Voided slab system

• Assumptions– Thin, rectangular, isotropic plate

– Primary vertical deflection due to flexure

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• �� KHLMHNOℓQH

KQR@ST;NU ;VH

;/N

– ℓ; = longer of two center-to-center spans

– W = mass per unit area

– Based on Kirchoff Plate Theory

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• Rectangular plate, corner supports

• Dimensionless parameter X�N• Y*- long span

• Ref (Blevins – 2000)

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Y*/YZ [*Z [ZZ

1.0 7.12 15.8

1.5 8.92 21.5

2.0 9.29 27.5

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• Constant �;– Accounts for level of cracking in the concrete slab

– Estimated by 9: 9C⁄– Square panels: Average9: � 0.79d � 0.15e9:; � 9:N6– Rectangular panels:

9:|�g�:h � Fi|@j,lmFi|nj,o m Fi|@j,omFi|nj,lN

– Contribution of shrinkage restraint to cracking

�E � 4.5X �8=

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• Constant �N– Accounts for effect of rigidity at the joint between the

slab and column

– �N � q1.9foru; � 24in.2.1foru; x 24in.

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Examples

� Flat plate - Voided slab system » Walking» Rhythmic Excitation (Dancing)» Sensitive

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ConcreteCompressive strength �8= � 4,000psiDensity �8 � 150pcf

Reinforcing steel

Yield strength �| � 60,000psi

LoadsSuperimposed dead load = 20 psfLive load (design) = 100 psfLive load (vibration) = 12.5 psf

Examples

� Flat plate - Voided slab system » System properties

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Spans Typical bay: 40′−0"} 40′−0"

Member sizesVoided slab

Slab thickness ~� � 19.5"Void diameter � 14.125"Clear space between voids � 1.625"Number of void formers per sq ft = 0.58Stiffness reduction factor = 0.90

Columns 36" } 36"

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Design Data

� Section Properties

33Ref: Cobiax Engineering Manual

Examples

� Flat plate - Voided slab system » Concrete properties

• 78 � �8;.�33 �8= � 150 ;.� } 33 4,000 �3,834,254psi

• Dynamic78 � 1.278 � 4,601,104psi �662,559,007psf

• Reduced 78 due to voids � 0.90 } 4,601,104 �4,140,994psi � 596,303,078psf

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Examples

� Flat plate - Voided slab system » Concrete properties

• Modulusofrupture�E � 4.5X �8= � 4.5 }1.0 4,000 � 284.6psi (accounts for shrinkage restraint)(1)

• Modularratio� � 7� 1.278⁄ �29,000,000 4,140,994 � 7.0⁄

• Poisson’s ratio � � 0.2 (conservatively takes into account dynamic effects)

35(1) ACI Structural Journal 105(4):498-506 (2008)

Examples

� Flat plate - Voided slab system » Effective and gross moments of inertia

• 9:|�g�:h � 0.79m +0.15(I e1 + I e2)(1)

• 9:|�g�:h � 143,751in.�• 9C|�g�:h � 266,936in.�• Crack coefficient

– �; � ;� ,��;N��,� � � 0.54

36(1) Fanella, D.A. 2011, Reinforced Concrete Structures – Analysis and Design, The McGraw-Hill Companies, NY

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Examples

� Flat plate - Voided slab system » Properties of voided slab

• Volumeofonesphericalvoid ��O

��N

� �O };,�N�

;�.;N�N

� 0.854ft

• Concretedisplacement �volumeofonesphericalvoid }numberofvoidspersqft � 0.854 } 0.58 �0.50ft /ftN

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Examples


• Volumeofconcrete � ;�.�;N D 0.50 � 1.13ft /ftN

• Slabweight � 1.13 } 150 � 169.5psf• Deadloadreduction � ;�.�

;N } 150 D 169.5 �74.3psf

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Examples


• Assuming an average void area in the slab of 70%:

– Averagedeadloadofslabwithvoids � ;�.�;N } 150 D

0.7 } 74.3 � 191.7psf• Total average weight of a panel �

� 191.7 } 40N � 306,720lbs

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Examples


• Mass � ��,�N�m N�m;N.� }��H N.N � 11,140lbsecN/ft

• W � ¡�¢¢£��

;;,;��H � 6.96slugs/ftN

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Examples

� Flat plate - Voided slab system » Natural frequency (1st Harmonic)

• �; � KHLQHNOℓQH

KQR@ST;NU ;VH

;/N(1)

• �; � N.;e�.;N6NO��H

�.��e��, � ,��6eQ¤.¥QH 6T;Ne�.��6 ;�.NH

;/N� 6.2Hz

– �N � 2.1 since the column is greater than 24 in.

– X;N � 7.12 from Table 4.1 for ℓ;/ℓN � 1.0

41(1) Blevins, R.D. 2001 “Formulas for Natural Frequency and Mode Shapes”Krieger Publishing Company, Malabar, FL

Examples

� SAFE Analysis of voided-slab (3-span)» ¨~0.17ªue�; � 5.9Hz ~ 5% error)

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Examples

� Voided slab system » Natural frequency (2nd Harmonic)

• �N � KHLHHNOℓQH

KQR@ST;NU ;VH

;/N

• �N � ;�.��.;N ∗ 6.2 � 13.8Hz

– �N � 2.1 since the column is greater than 24 in.

– XNN � 15.8 from Table 4.1 for ℓ;/ℓN � 1.0

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Examples

� Voided slab system – Walking Excitation» Check acceptance criterion

•g¬C � ��: ®.T¥¯�

°± � g²C

– For office occupancy, � � 0.03 for offices that have some nonstructural components but no full-height partitions

•g¬C � ��: ®.T¥}³.H

�.� } ��,�N� ~0.001• �� 0.10%� µ 0.50%� for office occupancies

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Examples

� Check walking acceptance criteria » Ok for office/residential application

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Examples

� Voided slab system – Rhythmic Excitation» Check acceptance criterion

• Since �� x �� E:¶=·, implies that unacceptable vibrations are not expected due to rhythmic excitations

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Examples

� Voided slab system – Rhythmic Excitation» Highest harmonic causing resonance

• �� E:¶=· x � 1 � Kg² C⁄

¸M¹¬¹º

– �� ⁄ � 0.02– �� weightofparticipants � 12.5psf– �» � 191.7 � 12.5 � 204.2psf– k=1.3fordancing

– k =2.0aerobicsorjumping

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Acceptance Criteria

� Voided slab system – Rhythmic Excitation» Rhythmic excitation

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Activity HarmonicForcing

Frequency �e¾¿6

Weight of Participants �� (psf)

Dynamic Coefficient

!Dancing 1 1.5 – 3.0 12.5 0.50Concert or sporting event

1 1.5 – 3.031.0

0.25

2 3.0 – 6.0 0.05

Aerobics or jumping

1 2.0 – 2.754.2

1.502 4.0 – 5.5 0.603 6.0 – 8.25 0.10

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Acceptance Criteria

� Voided slab system – Rhythmic Excitation» Criteria for dancing (1st harmonic)

» Criteria for aerobics (1st harmonic)

» Criteria for aerobics (2nd harmonic)

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�; � 3 1 � ;. �.�N

�.�e;N.�6N��.N = 5.2 < 6.2 Hz (ok)

�; � 2.75 1 � N.��.�N

;.�e�.N6;��.� = 5.6 < 6.2 Hz (ok)

�N � 5.5 1 � N.��.�N

�.�e�.N6;��.� = 8.3 < 13.8 Hz (ok)

Examples

� Voided slab system – Design for Sensitive Equipment» Check acceptance criterion

• " � À¬ÁÂ�� Limiting" � 1,000Äin./sec

• Δ� � Å�ÆuÇÈÉ�É��ÆÉÉÊªËªÇÌÍÎÇÉÎ�ÈÇÆÉ�Í• %& � ÏÐd��N (basedon185lb personatvariouswalkingpaces)

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Walking Pace

(steps/minute)'((lb)

�) � */,)

(Hz)

-.(lb

Hz2)

Fast (100) 315 5.0 25,000

Moderate (75) 280 2.5 5,500

Slow (50) 240 1.4 1,500

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Examples


• Maximum deflection of floor system subjected to a unit concentrated load

• Rectangular plate of uniform thickness with simply supported edges with point load at geometric center

– Δ� ��ℓH

H ;VH

KQR@SiTOT

∑;

dTtanh�d D ¸n

Ò�¢ÓH ¸nÔdÕ;, ,�⋯

– �d � dON

ℓQℓH , ℓ; � ℓN

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Examples


• Series in this equation converges quickly– �d � ÌÏ/2– Series is equal to 0.72

• Δ� � �} ��};N H ;�.NH�.��}�,;��,��};�.�T}OT } 0.72 � 1.86 } 10�in./lb

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Examples


• Slow walking pace

– " �;.��};�³};,��

�.N� 449Äin./sec

• Moderate walking pace

– " �;.��};�³}�,��

�.N� 1,649Äin./sec

• Fast walking pace

– " �;.��};�³}N�,��

�.N� 7,494Äin./sec

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Case Study

� University of Wisconsin» Experimental measurement of vibration

properties of flat plate voided slab while building was under construction

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Case study

� Edge floor panel

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Case study

� Impulsive loading and response

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0.0278 g

-0.0205 g

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Acc

ele

rati

on

(g

)

Time (sec)

Impact

Lift Off

Signal NoiseVertical Offset

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0 0.5 1 1.5 2 2.5 3

Acce

lera

tion

(g)

Time (sec)

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Case study

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� FFT of response

Case study

� Dominant modes from experiment» f1 ~ 12 Hz» f2 ~ 22 Hz» f3 ~ 35 Hz

� Conclusions» Experimental frequencies are higher than

approximate analysis:• Shorter (stiffer) spans • Small damping since building was under construction

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References

� CRSI Design Guide for Vibrations of Reinforced Concrete Floor Systems

� ATC Design Guide 1 “Minimizing Floor Vibration”

� Blevins “Formulas for Natural Frequency and Mode Shape”

� AISC DG 11 ‘Floor Vibrations Due to Human Activity”

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Questions?

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