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MINIMUM THICKNESS OF RCC SLAB IN ORDER TO PREVENT UNDESIRABLE FLOOR VIBRATION Submitted by Muntahith Mehadil Orvin Student No: 0804002 Submitted to the DEPARTMENT OF CIVIL ENGINEERING BANGLADESH UNIVERSITY OF ENGINEERING AND TECHNOLOGY In partial fulfillment of requirements for the degree of BACHELOR OF SCIENCE IN CIVIL ENGINEERING June, 2014

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Page 1: 0034_MINIMUM SLAB THICKNESS REQUIREMENT OF RCC SLAB IN ORDER TO PREVENT UNDESIRABLE FLOOR VIBRATION - Muntahith Mehadil Orvin.pdf

MINIMUM THICKNESS OF RCC SLAB IN ORDER TO

PREVENT UNDESIRABLE FLOOR VIBRATION

Submitted by

Muntahith Mehadil Orvin

Student No: 0804002

Submitted to the

DEPARTMENT OF CIVIL ENGINEERING

BANGLADESH UNIVERSITY OF ENGINEERING AND TECHNOLOGY

In partial fulfillment of requirements for the degree of

BACHELOR OF SCIENCE IN CIVIL ENGINEERING

June, 2014

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ii

DECLARATION

This is hereby declared that the studies contained in this thesis is the result of

research carried out by the author except for the contents where specific

references have been made to the research of others.

The whole thesis has been done under the supervision of Professor Dr. Khan

Mahmud Amanat (Department of Civil Engineering, BUET) and no part of this

thesis has been submitted to any University or educational establishment for a

Degree, Diploma or other qualification (except for publication).

Signature of author

(Muntahith Mehadil Orvin)

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iii

ACKNOWLEDGEMENT

Thanks to Almighty Allah for His graciousness, unlimited kindness and with the

blessings of Whom the good deeds are fulfilled.

The author wishes to express his deepest gratitude to Dr. Khan Mahmud Amanat,

Professor, Department of Civil Engineering, BUET, Dhaka, for his continuous

supervision all through the study. His systematic guidance, invaluable suggestions

and affectionate encouragement at every stage of this study have helped the author

greatly.

A very special debt of deep gratitude is offered to the author’s parents and his

younger sister for their continuous encouragement and cooperation during this

study.

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iv

ABSTRACT

Now-a-days, modern structures are becoming slender, irregular shaped and long

span structures which are susceptible to floor vibration phenomena. The purpose

of this study is to determine the minimum slab thickness of a reinforced concrete

slab to prevent undesirable vibration that will not cause discomfort to occupants.

Though American Concrete Institute (ACI) provided code for minimum slab

thickness requirement from static deflection criteria, it might not be sufficient for

dynamic serviceability like vibration. This study investigates this issue and its

findings may be helpful for preventing floor vibration of residential building

floors which include partition wall load. An investigation based on 3D finite

element modeling of a reinforced cement concrete floor subjected to gravity load

including partition wall load is carried out to study the natural floor vibration. The

ANSYS model verification was done previously and here this model is validated

by ETABS modeling and hand calculation. The variation of the floor vibration is

studied for several parameters such as different slab thickness, span length and

floor panel aspect ratio. Finally a graph correlating the slab thickness, span length

and floor panel aspect ratio is suggested that provides minimum slab thickness for

which the floor will not vibrate at less than 10 Hz. It is seen that with increase of

span length and floor panel aspect ratio, minimum slab requirement increases.

Variation with ACI limit can be observed from graphs that are provided aiding the

comparison with ACI serviceability limit of slab thickness requirement. For larger

span and aspect ratio, the ACI slab thickness requirement may not be sufficient for

vibration serviceability.

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v

TABLE OF CONTENTS

Page No.

DECLARATION ii

ACKNOWLEDGEMENT iii

ABSTRACT iv

Chapter 1 INTRODUCTION

1.1 GENERAL 2

1.2 BACKGROUND 2

1.3 OBJECTIVE AND SCOPE OF STUDY 5

1.4 BASIC ASSUMPTIONS 6

1.5 LAYOUT OF THE THESIS 7

Chapter 2 LITERATURE REVIEW

2.1 GENERAL 9

2.2 BASIC TERMS 9

2.2.1 What is Vibration 10

2.2.2 Fundamental Frequency 10

2.2.3 Amplitude 10

2.2.4 Cycle and Period 10

2.2.5 Damping 11

2.2.6 Critical Damping 11

2.2.7 Dynamic Loading 12

2.2.8 Resonance 13

2.2.9 Mass and Stiffness 13

2.2.10 Mode Shape 14

2.2.11 Modal Analysis 14

2.3 FACTORS AFEECTING FLOOR VIBRATION 15

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vi

2.3.1 Sources Of Building Vibration 16

2.3.2 Path Of Transmission Of Vibration 16

2.3.3 Receiver 16

2.4 CLASSIFICATION OF FLOOR VIBRATION 16

2.5 HUMAN RESPONSE CRITERIA TO BUILDING

VIBRATION 18

2.5.1 Modified Reiher and Meister Scale 18

2.5.2 Wiss and Parmelee 18

2.5.3 Canadian Standards Association Scale 19

2.5.4 Murray Criteria 21

2.5.5 ISO scale 22

2.5.6 Factors Influencing Vibration Perceptibility 23

2.6 OVERVIEW OF CURRENT CODE PROVISIONS

FOR VIBRATION 24

2.6.1 General Design Codes 24

2.6.2 Australian Standard 24

2.6.3 ISO Codes 24

2.6.4 BS Codes 25

2.6.5 Practice Guides 25

2.7 FLOOR VIBRATION PRINCIPLE 25

2.8 PAST RESEARCH 30

2.9 REMARKS 32

Chapter 3 DEVELOPMENT OF FINITE ELEMENT MODEL

3.1 GENERAL 34

3.2 SOFTWARE USED FOR FINITE 34

ELEMENT ANALYSIS

3.3 TYPES OF ANALYSIS OF STRUCTURES 35

3.4 CHARACTERIZATION OF STRUCTURAL 36

COMPONENT IN MODEL

3.4.1 BEAM4 (3-D Elastic Beam) for Beam, Column 36

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vii

3.4.2 SHELL43 For Slab 38

3.4.3 MASS21 (Structural Mass)

For Load Application 41

3.4.4 Support Condition 42

3.5 LOAD APPLICATION 43

3.5.1 Dead load 43

3.5.2 Live load 43

3.6 ANALYSIS METHOD 43

3.7 FINITE ELEMENT MODEL MESH 44

3.8 MODEL CHARACTERISTIC FOR ANALYSIS 45

AND TYPICAL RESULT

3.8.1 Different Parameters 45

3.8.2 Detail View of Model 48

3.8.3 Different Mode Shape 49

3.9 REMARKS 58

Chapter 4 VERIFICATION OF THE MODEL

4.1 VERIFICATION OF THE MODEL 60

4.1.1 Verification of the floor’s Total 60

Load with Hand Calculation

4.1.2 Verification of Model Floor Frequency by 62

ETABS 9.7

4.1.3 Verification Done by Rakib (2013) 64

4.2 REMARKS 65

Chapter 5 PARAMETRIC STUDY AND RESULTS

5.1GENERAL 67

5.2 SELECTED PARAMETERS 67

5.3VARIATION OF FLOOR VIBRATION WITH 69

CHANGE OF SPAN LENGTH

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viii

5.3.1 Analysis of floor vibration 69

5.3.2 Study of ‘Natural frequency v/s 85

Slab thickness’ curves

5.4 Comparison of ACI limit with 10 Hz limit 86

5.4.1 Study of ‘slab thickness v/s span 89

Curve for 10 Hz and ACI limit

5.5 MINIMUM SLAB THICKNESS

DETERMINATION 90

5.6 RESULT COMPARISON WITH 96

RAKIB (2013)

5.7 REMARKS 100

Chapter 6 CONCLUSION AND RECOMMENDATION

6.1 GENERAL 102

6.2 OUTCOME OF THE STUDY 102

6.3 PROPOSED GRAPH 103

6.4 LIMITATION AND RECOMMENDATION 103

FOR FURTHER STUDY

REFERENCES 104

APPENDIX 108

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ix

LIST OF NOTATIONS

Symbol Meaning

a/g Ratio of the floor acceleration

ap/g Estimated peak acceleration

a0/g Acceleration limit

A Maximum Amplitude

Ag Interior column gross area

D Damping Ratio

E Modulus of Elasticity

f Frequency

fstep Step Frequency

fn Natural Frequency of Floor System

Fy Steel yield strength

f’c Concrete compressive strength

g Gravity

h Height of Column

I Moment of Inertia

i Harmonic Multiple

k Stiffness

Ln Clear span length

m Mass

P Constant Force of Excitation

Pu Assumed Load on Column

R Mean Response Rating

R Reduction Factor

t Time Period

t Slab Thickness

W Weight of Floor

α Dynamic Coefficient

ø Phase Angle

to the acceleration of gravity

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x

β Modal Damping Ratio

β Floor panel aspect ratio

ρ Reinforcement ratio

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Chapter - 1

INTRODUCTION

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Introduction

2

1.1 GENERAL

Modern construction techniques make use of lightweight, high-strength materials

to create flexible, long-span floors. These floors sometimes result in annoying

levels of vibration under ordinary loading situations. Generally, these vibrations do

not present any threat to the structural integrity of the floor in extreme cases they

can render the floor unusable by the human occupants of the building if it creates

excessive discomfort to them.

Building structures designed without considering its dynamic properties such as

vibration, can create discomfort to occupants of the building. It is best practice to

solve the problem, which may create discomfort, before construction of the

structure. It will consume more money and time to retrofit a built structure to make

it serviceable to the users.

Modern buildings are now becoming more flexible, slender and irregular shape.

This fact increases the susceptibility to undesirable vibration. Most people are

sensitive to the frequency of vibration in the range of 4 Hz to 10 Hz (Wilson,

1998). If the structures have the frequency bellow 10 Hz, then it creates resonance

with human body. This resonance may cause discomfort to the people (Murray,

1997).

1.2 BACKGROUND

Today’s structures built to cater to the expectations of the community are

aesthetically pleasing and use high strength slender materials, irregular shapes and

larger span. These structures unfortunately exhibit vibration problems under

service loads causing discomfort to the occupants. At times, these vibrations have

also been cause of structural failure.

One such case of structural failure was the collapse of Hyatt Regency Hotel

walkway in Kansas City, US, which happened during a weekend ‘tea dance’ in

1981 (McGrath and Foote 1981) shown in Fig 1.1:

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Introduction

3

Fig 1.1: After Hyatt Regency Hotel Collapse in 1981

Another recent and well known case of vibration problem in a structure was the

Millennium Bridge in London, England shown in Fig 1.2. Two days after opening

in 2000 the bridge had to be closed to the public due to excessive sideway

movements that happened when large number of people crossed the bridge

(Dallard et al. 2001).

The Tacoma Narrows Bridge is a pair of twin suspension bridge that spans the

Tacoma Narrows strait of Puget Sound in Pierce County, Washington. The bridges

connects the city of Tacoma with the Kitsap Peninsula and carry State Route

16(known as Primary State Highway 14 until 1964) over the strait. The original

Tacoma Narrows Bridge opened on July 1, 1940. Its main span collapsed into the

Tacoma Narrows four months later on November 7, 1940, at 11:00 AM (Pacific

Time) due to a physical phenomenon known as resonance (Billah, 1990). The

bridge collapse had lasting effects on science and engineering. In many

undergraduate physics texts the event is presented as an example of elementary

forced resonance with the wind providing an external periodic frequency that

matched the natural structural frequency.

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Introduction

4

Fig 1.2: Millennium Bridge, London (2000)

Fig 1.3: Collapse of Tacoma Narrows Bridge (1940)

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Introduction

5

Similar concerns in vibration hazards have been also reported in human assembly

structures such as stadiums, grandstands and auditoriums; Some examples are the

Cardiff Millennium Stadium, Liverpool’s Anfield Stadium and Old Trafford

Stadium. The structure mentioned are all slender with natural frequency falls

within the frequency of human induced loads that causes discomfort.

Steel deck composite floors are also susceptible to vibration as they are slender.

These composite floors are designed using static methods which will not reveal

the true behavior under human induced dynamic loads. This vibration problems

has been identified and investigated by Bachmann et al. (1987), Allen and

Murray (1993), Williams and Waldron (1994), Da silva et al. (2007).

RCC building floor with less slab thickness with respect to span length and bay,

may create uncomfortable vibration. No codes provide specific guidelines for

RCC slab thickness to prevent undesirable floor vibration. The easiest way to

avoid the building floor vibration is to design the floor with physical

understanding. For this reason finite element method is developed to predict the

dynamic property of the floors. The objective of this study is to prevent

undesirable floor vibration by providing a minimum slab thickness. Rakib (2013)

investigated on this issue.

1.3 OBJECTIVES AND SCOPE OF THE STUDY

The aim of the current study is to enlarge knowledge regarding the vibration

response of a RCC building floor with change of different parameters. The

variation of floor vibration is analyzed with change of slab thickness, floor panel

aspect ratio and span length. From this study author tries to suggest a minimum

slab thickness which may prevent undesirable floor vibration and thus the

discomfort to occupant can be prevented. The principle features of this thesis are

summarized as follows:

To develop a 3D finite element model of floor having three span and bay.

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Introduction

6

Verify the built model with hand calculation and other structure analysis

software such as ETABS.

To perform modal analysis to determine mode shape and frequency.

Analysis the variation of natural floor frequency with variation of slab

thickness, span and floor panel aspect ratio.

Analysis of the slab thickness with variation of span length and floor panel

aspect ratio.

Determine a relationship among the slab thickness, span length and floor

panel aspect ratio from which minimum slab thickness to prevent

undesirable floor vibration with partition wall load can be found.

1.4 BASIC ASSUMPTIONS

The investigation is based on some assumptions, to avoid complexity in calculation.

These assumptions are as follows:

Material is linearly elastic and isotropic.

Live load is only considered to determine beam and column size, and is

not used in modal analysis.

Floor finished load is assumed to be 25×4.786×10-5

N/mm2

Partition wall load is assumed to be 50×4.786×10-5

N/mm2

1.5 LAYOUT OF THE THESIS: Six chapters are organized systematically to

describe the whole thesis. Chapter 1 is the current chapter, which introduces the

entire study which is performed in this thesis. Chapter 2 is the literature review

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Introduction

7

that provides the basic understanding about the relevant topic, submits the

provisions of different code, specifications as well as all the recent and past

publications relevant to the topic. In chapter 3, the description of the actual work

regarding the finite element modeling and parameter used related to the subject is

provided. In this chapter systematic descriptions are provided to have better

understanding about 3D floor modeling. Verification of the model is done in

Chapter: 4.Parametric study is done in Chapter 5. In this chapter, analysis and

output of the result based on various parameters are illustrated. Finally, the

summary of the organized outcomes of the thesis is provided and future research

works recommendations are proposed in Chapter 6. Flow Chart of the whole

thesis:

Conclusion And Recommendation

Parametric Study And Results

Verification of the Model

Development of Finite Element Model

Literature Review

Introduction

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CHAPTER – 2

LITERATURE REVIEW

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Literature Review

9

2.1 GENERAL

Modern construction techniques make use of lightweight, high-strength materials

to create flexible, long-span floors. The engineering community as whole growing

trend to use high-strength steel and concrete has reduced system mass without a

corresponding increase in elasticity, leading to an overall reduction in system

stiffness (AISC, 2001). Architects are continually pressing engineers for larger

column spacing. Modern floor systems can be more vibration-vulnerable due to

trends in design and construction leading to longer spans, lighter weight and

lower damping. While these vibrations do not present any threat to the structural

integrity of the floor, in extreme cases they can render the floor unusable by the

human occupants of the building (Murray, 1997)

In civil engineering dynamics, human-induced vibrations are becoming

increasingly vital serviceability and safety issues. Numerous researchers

examined these issues. The wide variety of scales and prediction techniques

available to engineers is an indication of the complex nature of floor vibrations.

Furthermore, since each method inherently makes numerous assumptions about

the structure, not all methods are equally applicable to all situations.

Remedies for annoying floor vibrations are often cumbersome and expensive. It

is far better to design the floor properly the first time, rather than retrofit the

structure once a problem develops. The easiest way to avoid building a floor that

is susceptible to annoying vibrations is to design the floor with an adequate

understanding of the physical phenomena. But there is not enough design code for

designing slab thickness of reinforced concrete (RC) structures that will not cause

any disturbing vibration to the occupants as was provided for the slab to resist

deflection by ACI.

A finite element model is developed to predict the dynamic property and

fundamental natural frequency and determine the slab thickness that will not

cause resonance is studied here. Comparison with ACI limit is also done.

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Literature Review

10

2.2 BASIC TERMS

2.2.1 What is vibration: Oscillation of a system about its equilibrium position is

called vibration. An object vibrates when it moves back and forth, up and down,

or side to side, usually very rapidly. Vibration describes the physical energy from

a vibrating object, and also what we feel when that energy is transmitted to us.

Free vibration occurs when the system is excited and allowed to vibrate at a

natural frequency of that system. Forced vibration occurs when a system is

continually excited at a particular frequency and the system is forced to oscillate

at that frequency.

2.2.2 Fundamental frequency: Frequency describes the number of vibrating

movements in a given time period. Frequency is measured in cycles per second

or hertz (Hz). An object vibrating with a frequency of one hertz completes one

full vibrating cycle over one second. A cycle is the complete pattern of

movement of the vibrating object from start to finish. Natural frequency is the

frequency at which a body or structure will vibrate when displaced and then

quickly released. This state of vibration is referred to as free vibration. All

structures have a large number of natural frequencies; the lowest or "fundamental"

natural frequency is of most concern. Fundamental frequency is the function of

mass in the system as well as the stiffness of the system. Floors that oscillate in

the range of 4 to 8 Hz are of particular concern. (Sladki, 1999) From simple

harmonic motion, we can write,

(2.1)

Where,

T = time period = 1/frequency (f)

m = mass

k = stiffness.

Thus we find the equation, f =

(2.2)

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Literature Review

11

2.2.3 Amplitude: Amplitude is the intensity or magnitude of vibration. It is

measured as the maximum distance an object moves from a central point.

2.2.4 Cycle and Period: Cycle is the m o t i o n of the system from the time it

passes through a given point travelling in a given direction until it returns to that

same point and in the same direction. Time for one cycle is referred as period

usually measured in seconds. Time period is the inverse of frequency.

Fig 2.1: Amplitude v/s Frequency (Hz)

2.2.5 Damping: Loss of energy per cycle during the vibration of a system, usually

due to friction. Viscous damping, damping proportional to velocity, is generally

assumed. Damping is an important parameter in mitigation excessive vibration in

floor structures. A precise value for the damping of reinforced concrete structure is

mostly unknown. The use of partition wall in finished floor, increase damping of

the structure. Murray (2003) used damping of 3% for an office building.

2.2.6 Critical damping: The damping required to prevent oscillation of the

system. Damping is usually presented as a ratio of actual damping divided by

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Literature Review

12

critical damping. Log decrement damping is determined by taking the natural

logarithm of the ratio of successive peaks in the response curve. Modal damping

is determined from an analysis of the Fourier spectrum of the response or modal

analysis. Modal damping tends to be smaller than log decrement damping and

tends to more accurately match the actual damping present in a structure.

Fig 2.2: Amplitude and Period

2.2.7 Dynamic loading: Dynamic loadings can be classified as harmonic, periodic,

transient, and impulsive. Harmonic or sinusoidal loads are usually associated with

rotating machinery. Periodic loads are caused by rhythmic human activities such as

dancing and aerobics and by impact machinery. Transient loads occur from the

movement of people and include walking and running. Single jumps and heel-drop

impacts are examples of impulsive loads.

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Literature Review

13

If a structural system is subjected to a continuous harmonic driving force, the

resulting motion will have a constant frequency and constant maximum amplitude

and is referred as steady state motion. If a real structural system is subjected to a

single impulse, damping in the system will cause the motion to subside, this is one

type of transient motion.

Fig 2.3: Types of dynamic loading (Murray et al., 1997)

2.2.8 Resonance: If a frequency component of an exciting force is equal to a

natural frequency of the structure, resonance will occur. At resonance, the

amplitude of the motion tends to become large to very large, as shown in Figure

2.4. Resonance occurs it can lead to eventual failure of the system. Adding

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Literature Review

14

damping can significantly reduce the magnitude of the vibration. The magnitude

can be reduced if the natural frequency can be shifted away from the forcing

frequency by changing the stiffness or mass of the system.

2.2.9 Mass and Stiffness: mass is the term where force is divided by the

acceleration, mass is also equal to its weight divided by gravity. Stiffness of a body

is the measure of the resistance offered by an elastic body to deformation. Both

mass and stiffness is important in floor vibration criteria since natural frequency of

floor is significantly affected by these two parameters. Damping influences the

mass and stiffness of a system.

Fig 2.4: Resonance and Damping

2.2.10 Mode Shape: When a floor structure vibrates freely in a particular mode, it

moves up and down with a certain con-figuration or mode shape. Each natural

frequency has a mode shape associated with it. Figure 2.5 shows typical mode

shapes for a slab and a building.

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Literature Review

15

2.2.11 Modal Analysis: Modal analysis refers to a computational, analytical or

experimental method for determining the natural frequencies and mode shapes of a

structure, as well as the responses of individual modes to a given excitation.

Fig 2.5: Typical mode shapes for a slab and a building

2.3 FACTORS AFFECTING FLOOR VIBRATION: In any given situation

involving excessive or annoying vibration there are always three factors involves:

These are Source, Path, Receiver shown in Fig 2.6.

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Literature Review

16

2.3.1 Sources of Building Vibration: Building vibration can have several sources.

Earth quake, explosion can cause large vibration with structural damage and

sometimes total building failure. Slender and irregular shaped building can also

become wind sensitive and have large lateral movement when subjected to high

wind. Occupants of the building can feel this vibration when this vibration is at

low frequency (Setareh, 2010). Another source of building as well as floor

vibration is the building occupants’ movements such as walking, jogging, running,

dancing. These vibrations are generally small in amplitude but large enough to

cause problem such as annoying and discomfort to the occupants (Setareh, 2010).

Excessive vibration of building due to normal human activities such as walking

needs special attention to engineers and building owners as their occurrences have

recently become more common.

Accurate prediction, evaluation and assessment of vibration can greatly assist

engineers and architectures to design cost-effective structure without such

problems.

2.3.2 Path of Transmission of Vibration: The path is how the vibration is

transmitted to the receiver. In this case the path is the building structure from

which the vibration is transmitted.

2.3.3 Receiver: The receiver is he who receives the vibration. If the vibration

source is the building itself, then the receiver will be its occupants. On the other

hand, if the source is human, then the receiver will be the building.

2.4 CLASSIFICATION OF FLOOR VIBRATION

Murray (1975) classified the human perception of floor vibration in four

categories, such as

Vibration is not noticed by the occupants.

Vibration is noticed but do not disturb the occupants.

Vibration is noticed and disturbs the occupants.

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Literature Review

17

Vibration can compromise the security of the occupants.

Fig 2.6: Sources of Building Vibration

Vibration of floors with frequencies below about 10 Hz is much more likely to

cause discomfort of the occupants than in floor with frequencies above 10 Hz

(Murray et al., 1997; Smith et al., 2007). Based on this limit, floors can be

classified as low-frequency or high-frequency; In low-frequency floors, a walking

force harmonic frequency can match a natural frequency, resulting in resonant

response. In high-frequency floors, a walking force harmony frequency do not

match with natural frequency, so no resonant response will occur.

ISO classify human response to vibration into three categories:

Limit beyond which comfort is reduced.

Limit beyond which the working efficiency is impaired.

Limit beyond which the safety is endangered.

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Literature Review

18

2.5 HUMAN RESPONSE CRITERIA TO BUILDING VIBRATION

Floor vibration induced by human rhythmic activities like walking, running,

jumping consist on a very complex problem. This is due to the fact that the

dynamical excitation characteristics generated during these activities are directly

related to the individual body adversities and to the specific way in which each

human being executes a certain rhythmic task. In order to determine the dynamic

behavior of floor structural systems subjected to excitations from human

activities, various studies have tried to evaluate the magnitude of these rhythmic

loads. However the first pioneer on determining the human induced frequency was

Fisher (1895), a German mathematician.

2.5.1 Modified Reiher and Meister Scale: In 1931, Reiher and Meister

developed human response criteria. These criteria were developed by exposing a

group of standing people to a steady state vibration. The frequencies of these

vibrations ranged from 5 to 100 Hz with amplitudes ranging from 0.01 mm to 10

mm. As the people experienced these vibrations, the perceptibility level was then

noted in ranges from barely perceptible to intolerable. (Murray et al., 2003)

Following this development, Lenzen (1966) further applied the Reiher-Meister

scale. Lenzen used a single impact to excite the floor and then determined the

perceptibility. In 1974 and 1975, McCormick and Murray utilized this scale to

develop design criteria.

2.5.2 Wiss and Parmelee: In 1974, Wiss and Parmelee exposed 40 humans to a

vibratory force that was similar to that caused by a human footfall. The parameters

that were varied in the experiment included frequency, amplitude and damping.

The following human response criteria formula was developed:

R = 5.08 [ (fA)/D0.217

]0.265

(2.3)

Where,

R is the mean response rating, which is based on a numerical scale with the

following numerical designations:

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R=1, imperceptible vibration

R=2, barely perceptible

R=3, distinctly perceptible

R=4, strongly perceptible

R=5, severe vibration

f = frequency

A = maximum amplitude

D = damping ratio (Murray, 83)

Fig 2.7: Modified Reiher-Meister Scale (Murray, 63)

2.5.3 Canadian Standards Association Scale: Allen and Rainer developed these

criteria1976 by testing a series of long span floor systems with a heel drop load

test. The peak acceleration as a percent of gravity is a function of the frequency

and damping. In the figure 2.9 the scale is shown.

2.5.4 Murray Criteria: In 1981, Murray performed a similar study as Wiss and

Parmelee, with a heel drop loading and developed design criteria. Suggested

criteria is as follows:

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D > 35Af + 2.5 (2.4)

Comparison of Modified Reiher-Meister and Wiss-Parmelee scale is provided

below in fig 2.8

Fig 2.8: Comparison of Modified Reiher-Meister and Wiss-Parmelee scales

Here,

f = frequency

A = maximum amplitude

D = damping ratio. (Murray, 68)

2.5.5 International Organization for Standardization Scale: The International

Organization for Standardization’s standard ISO 2631-2: 1989 contains many

different human response criteria for numerous loading conditions.

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A baseline curve of peak acceleration versus frequency for transient loading was

developed. Depending on the use of the structure, the peak acceleration is

determined by an occupancy multiplier.

Fig 2.9: Canadian Standards Association scale developed by Allen and Rainer

Human response criteria for floor vibration for human activities (Murray, 1993)

provided below: The reaction of people who feel vibration depends very strongly

on what they are doing. People in offices or residences do not like ‘distinctly

perceptible’ vibration (peak acceleration of about 0.5 percent of the acceleration

of gravity, g), whereas people taking part in an activity will accept vibrations

approximately 10 times greater (5 percent g or more). People dining beside a

dance floor, lifting weight beside an aerobics gym, or standing in a shopping mall,

will accept something in ween (about 1.5 percent g).

Sensitivity within each occupancy also varies with duration of vibration and

remoteness of source. The above limits are for vibration frequencies between 4 Hz

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and 8 Hz. Outside this frequency range, people accept higher vibration

accelerations

ISO scale is shown in Fig 2.10:

Fig 2.10: ISO scale (Murray et al., 69)

2.5.6 Factors Influencing Vibration Perceptibility: Several factors influence the

level of perception. These include:

Position of the human body

Excitation source characteristics (amplitude, frequency, duration)

Exposure time

Floor system characteristics (mass, stiffness, damping)

Level of expectancy

Type of activity engaged in (walking, dancing, aerobics )

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Position of Human Body:

Considering the human body coordinate system defined in Fig 2.11, here, the x

axis defines the back to chest direction, y axis defines the right side to left side

direction and the z axis defines the foot to head direction. According to ISO9,10

,

the frequency range of maximum sensitivity to acceleration for humans is between

4 to 8 Hz for vibration along the z axis and 0 to 2 Hz for vibration along the x or y

axis. The z axis vibration is of most significant concern for office and work

places. For sleeping comfort in residence and hotels all three axis become

important. Human tolerance of vibration decreases in a characteristic way with

increasing exposure time. The more one expects vibration the less startling the

vibration becomes.

Fig 2.11: Direction of coordinate system for vibrations influencing human

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2.6 OVERVIEW OF CURRENT CODE PROVISIONS FOR VIBRATION

2.6.1 General Design Codes

The guidance provided in the Australian Standards AS3600 [2], AS4100 [3],

AS2327.1 [4], AS5100 [5], British Standards BS8110-1[6], BS5950 [7] and the

Structural Euro Codes EN 1992, EN 1993 and EN1994 [8] covering concrete,

steel, composite and bridge structures is generic and limited

to isolating the vibration source, increasing the damping and limiting frequencies

to control the effects of floor vibration induced by human activity.

2.6.2 Australian Standard

The Australian Standard that relates directly to vibration is AS2670 [9]. It

provides guidance on the evaluation of human exposure to whole-body vibration:

Part 1 gives general requirements, while Part 2 treats continuous and shock

induced vibration in buildings and presents base curves for acceleration limits.

2.6.3 ISO Codes

International Standardization Organization (ISO) Codes provide guidelines for

occupancy comfort and operating criteria for structures are subjected to vibration.

Currently there are three ISO publications: (i) general requirements for the

evaluation of human exposure to whole-body vibrations in ISO 2631-1[10], (ii)

evaluation of human exposure to vibrations in buildings (1-80Hz range) in ISO

2631-2 [11] and (iii) bases for the serviceability design of building structures and

walkways subjected to vibrations in ISO 10137 [12]. ISO 2631-1[10] suggests the

use of frequency-weighting functions to evaluate vibration for human

perception/discomfort in both the vertical and horizontal directions. It describes

the frequency weighting method and the method of determining the RMS

acceleration. ISO code also provides damping ratios for different types of floor

structures and walkways as discussed earlier.

2.6.4 BS Codes: Currently there are two relevant British Standards. BS 6841 [13]

provides general requirements for the measurement and evaluation of human

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exposure to whole-body vibration and repeated shocks. BS 6472-1 [14] gives

guidance on the evaluation of human exposure to building vibrations (1- 80-Hz

range). It does not have the base curves but it recommends VDV as the only

method to evaluate vibration.

2.6.5 Practice Guides

The American Institute of Steel Construction (AISC) Design Guide 11 [15] and

the Commentary D of the National Building Code of Canada [16] are commonly

used in North America. They use the peak unweighted accelerations as the

acceptability criteria for vibration control in building floors for different

occupancy types. These limits are based on the recommendations made by Allen

and Murray [17] in a previous publication, and do not consider the influence of

vibration duration and frequency on the acceptable limits. AISC Design Guide 11

[15] provides a method to determine the fundamental frequency and peak

acceleration of concrete/steel framed floor structures which are then used to check

compliance. Walking and rhythmic activities are used in the analysis.

2.7 FLOOR VIBRATION PRINCIPLE

Although human annoyance criteria for vibration have been known for many

years, it has only recently become practical to apply such criteria to the design of

floor structures. The reason for this is that the problem is complex—the loading is

complex and the response complicated, involving a large number of modes of

vibration. Experience and research have shown, however, that the problem can be

simplified sufficiently to provide practical design criteria.

Where the dynamic forces are large, as they are for aerobics, resonant vibration is

generally too great to be controlled practically by increasing damping or mass. In

this case, the natural frequency of any vibration mode significantly affected by the

dynamic force (i.e. a low frequency mode) must be kept away from the forcing

frequency. This generally means that the fundamental natural frequency must be

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made greater than the forcing frequency of the highest harmonic force that can

cause large resonant vibration.

Most floor vibration problems involve repeated forces caused by machinery or by

human activities such as dancing, aerobics or walking, although walking is a little

more complicated than the others because the forces change location with each

step. In some cases, the applied force is sinusoidal or nearly so. In general, a

repeated force can be represented by a combination of sinusoidal forces whose

frequencies, f, are multiples or harmonics of the basic frequency of the force

repetition, e.g. step frequency, for human activities. The time-dependent repeated

force can be represented by the Fourier series:

F = P [1+∑αi cos(2πifstept + ø)] (2.5)

Where, P = person’s weight, αi = dynamic coefficient for ith

harmonic force, i =

harmonic multiple, fstep = step frequency of the activity, t = time, ø = phase angle

for the harmonic.

A time dependent harmonic force component which matches the fundamental

frequency of the floor, F = Pαicos(2πifstept). A resonance response function is of

the form:

a/g = (Rαi P/βW)×cos(2πifstept) (2.6)

where,

a/g = ratio of the floor acceleration to the acceleration of gravity

g = gravity

R = reduction factor

β = modal damping ratio

W = effective weight of the floor

For evaluation, the peak acceleration due to walking can be estimated from

Equation (2.6) by selecting the lowest harmonic, i, for which the forcing

frequency, can match a natural frequency of the floor structure. The peak

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acceleration is then compared with the appropriate limit in Figure 2.10. The

following simplified design criterion is obtained from equation (2.6):

ap/g = P exp(-0.35fn)/βW ≤ a0/g (2.7)

here,

ap/g = estimated peak acceleration (in units of g)

a0/g = acceleration limit from Fig 2.10

fn = natural frequency of floor system

P = Constant force representing the excitation

American Institute of Steel Constructions (AISC) Steel Design Guide, Series 11:

Floor Vibrations Due to Human Activity (Murray et al. 1997) states that the floor

system is satisfactory if the peak acceleration, due to walking excitation as a

fraction of the acceleration of gravity, g, determined from equation (2.7), does not

exceed the acceleration limit in Fig 2.10.

DG11 suggests the peak acceleration used as the threshold for human comfort in

offices or residences subjected to vibration frequencies between 4 Hz and 8 Hz is

0.005g, or 0.5% of gravity. The lower threshold within the frequency range of 4

to 8 Hz can be explained by studies showing humans are particularly sensitive to

vibrations with frequencies in the 5-8 Hz range. DG11 states that from experience

and records, if the natural frequency of a floor is greater than 9-10 Hz, significant

resonance with walking harmonics does not occur. Where the natural frequency of

the floor exceeds 9-10 Hz, resonance becomes less important for human induced

vibration. It indicates that people are most susceptible to vibrations in the 4 Hz to

8 Hz frequency range.

Design Guide 11 recommends designing floor structures to meet a minimum

natural frequency to prevent unacceptable vibrations based on peak acceleration

response of the structure.

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Bachman and Ammann (1987) recommend that concrete slab-steel framed floor

systems have a minimum first natural frequency of 9 Hz. Floors that have a

natural frequency at or near 4-8 Hz may exhibit an excessive response because the

input force component of the harmonic may coincide with the resonant frequency

of the floor. Since frequency is proportional to the square root of moment of

inertia, a substantial amount of material is required to satisfy the 9.0 Hz criterion.

Wyatt (1989), however, has recently proposed design criteria for walking

vibration for fundamental natural frequencies not less than 7 Hz. His

recommendations are more conservative than those used in North America.

Ohlsson (1988) has proposed criteria for light-weight floor systems. He

recommends that floors not to be designed with fundamental frequencies below 8

Hz.

The following figure 2.12 explains that human walking frequency mostly varies

from 1.6 Hz to 8.8 Hz (Setareh, 2010). So building floor modes with natural

frequencies in excess of 10 Hz is not usually excited by people walking. If the

natural frequency of floor is more than 10 Hz, resonance will not occur for human

excitation and discomfort to occupant is prevented.

Figure 2.12: Variation of the frequency weighting versus frequency (Setareh,

2010)

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If the natural frequency of floor system is greater than 10 Hz, the estimated peak

acceleration limit will not exceed the tolerable limits and thus the fundamental

natural frequency can be made greater than the forcing frequency of the highest

harmonic force due to walking excitation. Then the large resonant vibration will

not occur. Experience also shows that the higher modes of vibration need not to be

considered as they die out quickly and do not cause discomfort. ‘10 Hz limit’

criteria can be used in designing the floor system for vibration serviceability.

The natural frequency of a floor can be calculated by the following formula that

describes that with increase in stiffness, the natural frequency of the system

increases and frequency decreases with the increase in mass of the structure. The

formula follows:

Natural frequency,

(2.8)

There is not enough design code for designing slab thickness of reinforced

concrete structure that will not cause excessive vibration to the occupants. If span

are large and structure materials are light weight, vibration of RC slab must be

considered. It is better to design it as per the required stiffness so that the

fundamental natural frequency remains not less than 10 Hz rather than retrofitting

the structure later which will be costly.

ACI (American Concrete Institute) provided reinforced concrete (RC) slab

thickness requirement (Nilson et al., 2003) to resist deflection as follows:

Minimum Slab Thickness,

(2.9)

Where,

t = Slab thickness in mm

Ln = Clear span length in mm

β = Floor panel aspect ratio

Fy = Steel yield strength, psi

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The equation (2.9) is for resisting excessive deflection of a two way floor system

which is from static serviceability and should be minimum 90 mm.

In this study, determination of the minimum slab thickness is done by dynamic

analysis of the building using computer programming software that will satisfy the

vibration limit 10 Hz and comparing the thickness with ACI two way slab

thickness requirement criteria.

2.8 PAST RESEARCH ON FLOOR VIBRATION

Murray et al., (2011) studied on floor vibration characteristic of long span

composite slab system. They have done their experiment on a ‘laboratory floor

specimen’ and on a ‘full scale floor mockup’ and on ‘laboratory footbridge’. In

laboratory floor, they used 9.14m 9.14m single bay, single storied floor in

experimental setup. They used 222 mm composite slab of 24MPa normal weight

concrete. The floor was supported only in the perimeter with W530 66 girder and

W360 32.9 beam which framed in to W310 60 column. Then the natural floor

vibration is calculated, and found 4.98 Hz. The laboratory floor test indicates that

long span composite deck has very good resistance to floor vibration due to

walking. The floor had natural frequencies that are in the range of those measured

for composite slab and composite beam floor system and far above the 3 Hz limit

to avoid vandal jumping.

Petrovic and Pavic (2011), studied on effect of non structural partition wall on

vibration performance of floor structure. One of the most promising research

topics in vibration serviceability has been concerned with quantifying the effects

of different non-structural elements on dynamic behavior of floor systems.

Ignoring non-load bearing components is generally conservative and today there

is a need for dynamic floor analysis which will enable more realistic and accurate

assessment of the modal parameters. He has chosen a historical approach as the

most suitable one due to the fact there wasn’t enough systematic research in this

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field in the past. Therefore, serious research and extensive simulation and

verification studies are needed in this area.

Silva and Thambiratnam (2011), studied about the vibration characteristic of

concrete steel composite multi panel floor structure. Their result showed that the

potential for an adverse dynamic response from higher and multi modal excitation

influenced by human induced pattern loads. In their research, they have found

that, under pattern loading, the second and third modes of the structure can be

excited by higher harmonics of the activity frequency. These types of concrete

steel composite multi-panel floor structures often exhibit higher and multi modal

vibration under pattern loads, hence the simplified guidance for vibration

mitigation in the present codes. Under normal jumping activity, there was

additional peaks frequency.

Setareh (2010), studies the result of the model testing conducted on an office

building floor and analysis of the collected vibration measurements. It compares

the results with the structural response using computer analysis. It also studies a

sensitivity study to assess the importance of various structural parameters on floor

dynamic response. From the result it concluded that, for the structure used in this

study the raised flooring and non structural element act mainly as added mass and

did not contribute to floor damping. The addition of non structural element such as

partition wall increases the stiffness of the building.

Guilherme et al. (2009), studied about vibration analysis of long span joist floors

submitted to human rhythmic activities. In their work they investigated on the

dynamic behavior of composite floors when subjected to the rhythmic activities

corresponding to aerobics and dancing effects. The dynamic loads were obtained

through experimental tests conducted with individuals carrying out rhythmic and

non-rhythmic activities such as stimulated or non stimulated jumping. This study

also present that, dynamic loads can even generated considerable perturbation on

adjacent areas, where there is no human rhythmic activities of such kind.

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Williams and Waldron (2009), studies on structural behavior of concrete steel

composite floors, subjected to human induced loads. Some of these research

findings are used in developing the practice guide on steel structure vibration.

They also provide some simplified formula to be used in to limit the steel

structural vibration.

Murray et al. (2003), studied on floor vibration due to human activity. In their

work they provide basic principles and simple analytical tools to evaluate steel

framed floor systems and footbridges for vibration susceptibility to human

activity. Both human comfort and the need to control movement for sensitive

equipment are considered. Then developed a remedial measure for problematic

floor.

Rakib (2013), studied on the RCC floor vibration due to human excitation in his

paper ‘Minimum Slab Thickness of RC Slab To prevent Undesirable Floor

Vibration’. His investigation was based on 3D finite element modeling of a

reinforced concrete building of three story subjected to gravity load carried out to

study of the natural floor vibration. His purpose was to determine the minimum

slab thickness of a reinforced concrete building to prevent undesirable vibration

which will not cause discomfort to occupants. The variation of the floor vibration

was studied for different slab thickness, span length and aspect ratio. Empirical

equations were suggested which provide minimum slab thickness of a short span

RC building to prevent undesirable floor vibration, for both considering and

without considering partition wall load.

2.9 REMARKS

This part includes the basic of vibration terms as well as vibration principles. It

also discusses the human response and guideline available for floor vibration. 10

Hz limit criteria is included in this part as well. A lot of research works has been

done on human activity on composite structural vibration. There is not such work

on RC building, to estimate its vibration behavior. Computer modeling and

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analytical representation of building structural properties to predict the floor

response subjected to excitation due to human activity are important issues that

require further study. Vibration analysis of building structure can assist in more

accurate estimation of structural dynamic properties. Estimate of the stiffness,

mass and damping of building systems are needed to predict their dynamic

responses.

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Chapter - 3

DEVELOPMENT OF FINITE

ELEMENT MODEL

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3.1 GENERAL

The development of Finite Element method as an analysis tool was essentially

initiated with the advancement of digital computer, which replacing the analytical

methods, especially when it is required to do a specific task. Using Finite element

methods, it is possible to establish and solve the governing equation with complex

form in an easy and effective way. Finite element methods can be used to

accurately predict the dynamic properties of reinforced concrete structure.

This chapter describes the finite element modeling of a reinforced building with

different appropriate edge, corner, interior column and beam dimension.

Representation of various physical elements with the FEM (Finite Element

Modeling) elements, properties assigned to them, boundary condition, material

behavior, analysis type have also been discussed. The various material behavior

used and details of finite element meshing are also discussed in detail.

3.2 SOFTWARE USED FOR FINITE ELEMENT ANALYSIS

A number of finite element analysis tools or packages are readily available in the

civil engineering field. They vary in degree of complexity, usability and

versatility. Such packages are ABAQUS, DIANA, PATRAN, SDRC/I-DEAS,

ANSYS, EASE, COSMOC, NASTRAN, ALGOR, ANSR, MARC, ETABS,

STRAND, ADINA, SAP, FEMSKI, SAFE and STAAD etc. Some of these

programs are intended for special type of structures. Of these the packages

ANSYS 11.0 is used in this study for its relative ease of use, detailed

documentation, flexibility and vastness of its capabilities. ANSYS 11.0 is one of

the most powerful and versatile packages available for finite element structural

analysis. The verification of the model built in ANSYS 11.0 is done by building

the model in ETABS 9.7.

ANSYS Finite element analysis software enables engineer to perform the

following tasks:

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Build computer models or CAD models of structures, products,

components and systems.

Apply operating loads and other design performance conditions.

Study the physical responses, such as stress levels, temperature

distributions, or the impact of electromagnetic fields.

Optimize a design early in the development process to reduce

production costs.

Do prototype testing in environments where it otherwise would be

undesirable or impossible (for e.g., biomedical applications).

It is a finite element modeling package for numerically solving a wide

variety of structural and mechanical problems.

This problems include static and dynamic analysis of the structure

(both linear and nonlinear)

The ANSYS program has a comprehensive Graphical User Interface (GUI) that

gives user easy, interactive access to program functions, commands,

documentation and reference materials. Hence the ANSYS program is user

friendly.

3.3 TYPES OF ANALYSIS OF STRUCTURES

Structure can be analyzed for small deflection and elastic material properties

(Linear analysis), small deflection and plastic material properties (material non

linearity), large deflection and elastic material properties (geometric non

linearity), and for simultaneous large deflection and plastic material properties.

By plastic material properties, the structure is deformed beyond yield of the

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material, and the structural will not return to its original shape, when the applied

loads are removed. The amount of permanent deformation may be slight and

inconsequential, or substantial and disastrous.

By large deflection the shape of the structure has been changed enough that, the

relationship between applied load and deflection is no longer a simple straight

line relationship. This mean doubling the load will not double the deflection, the

material properties can however still be elastic. For the current study it is

assumed that the RC framed is linearly elastic, and materials are homogenous

and is always steel reinforced in reality. According to ACI recommendation, the

analysis result for RC frame is accurate enough for this assumption.

3.4 CHARACTERIZATION OF STRUCTURAL COMPONENT IN

MODEL

For modeling this building slab, beam, column separate elements are used. For

slab modeling SHELL43 and for beam, column BEAM4 (3-D elastic beam) is

used.

3.4.1 BEAM4 (3-D Elastic Beam) For Beam, Column

Element Description

For modeling beams and columns used element BEAM4, is an elastic, uniaxial,

3-dimensional element which can withstand tension, compressions, torsion and

bending. The element has two nodes with six degrees of freedom at each node;

translations in the nodal x, y, and z axes and rotations about the nodal x, y, z

axes. The geometry, node locations, and the coordinate system for this element

are shown in Fig. 3.1

The element is defined by two or three nodes, the cross-sectional area, two

area moments of inertia (Izz and Iyy), an angle of orientation (θ or ν) about the

element x-axis, and the material properties.

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Input Summary

Element Type : BEAM4

Nodes : I, J, K (K, orientation node is optional)

Degrees of Freedom : UX, UY, UZ, ROTX, ROTY, ROTZ

Real Constant : AREA, IZZ, IYY, HEIGHT, WIDTH, IXX

Material properties : EX, PRXY, DENS

Output Data

The solution output associated with the element is in two forms: Nodal

displacement include in the overall nodal solution and additional element

output.

Assumptions and Restrictions

The beam must not have a zero length or area. The moments of inertia,

however, may be zero if large deflections are not used.

Fig 3.1: BEAM4 Geometry

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The beam can have any cross-sectional shape for which the moments of

inertia can be computed. The stresses, however, will be determined as if the

distance between the neutral axis and the extreme fiber is one-half of the

corresponding thickness.

The element thicknesses are used only in the bending and thermal stress

calculations.

The applied thermal gradients are assumed to be linear across the thickness

in both directions and along the length of the element.

Eigen values calculated in a gyroscopic modal analysis can be very

sensitive to changes in the initial shift value, leading to potential error in

either the real or imaginary (or both) parts of the Eigen values.

If use the consistent tangent stiffness matrix take care to use realistic (that

is, "to scale") element real constants. This precaution is necessary because the

consistent stress-stiffening matrix is based on the calculated stresses in the

element. If use artificially large or small cross-sectional properties, the

calculated stresses will become inaccurate, and the stress-stiffening matrix will

suffer corresponding inaccuracies. (Certain components of the stress-stiffening

matrix could even overshoot to infinity.) Similar difficulties could arise if

unrealistic real constants are used in a linear pre-stresses or linear buckling

analysis.

3.4.2 SHELL43 for Slab

Element Description: SHELL43 is well suited to model linear, warped,

moderately-thick shell structures. The element has six degrees of freedom at each

node: translations in the nodal x, y, and z directions and rotations about the nodal

x, y, and z axes. The deformation shapes are linear in both in-plane directions. For

the out-of-plane motion, it uses a mixed interpolation of tension components. The

element has plasticity, creep, stress stiffening, large deflection and large strain.

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Input Summary

Element Type : SHELL43

Nodes : I, J, K, L

Degrees of Freedom : UX, UY, UZ, ROTX, ROTY, ROTZ

Real Constant : TK(I), TK(J), TK(K), TK(L)

Material properties : EX, PRXY, DENS

Output Data

The solution output associated with the element is in two forms: Nodal

displacements included in the overall nodal solution and Additional element

output.

Assumptions and Restrictions

Fig 3.2: SHELL43 Geometry

Zero area elements are not allowed. This occurs most often whenever the

elements are not numbered properly.

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Zero thickness elements or elements tapering down to a zero thickness at

any corner are not allowed.

Under bending loads, tapered elements produce inferior stress results and

refined meshes may be required.

Use of this element in triangular form produces results of inferior quality

compared to the quadrilateral form. However, under thermal loads, when the

element is doubly curved (warped), triangular SHELL43 elements produce

more accurate stress results than do quadrilateral shaped elements.

Quadrilateral SHELL43 elements may produce inaccurate stresses under

thermal loads for doubly curved or warped domains.

The applied transverse thermal gradient is assumed to vary linearly

through the thickness.

The out-of-plane (normal) stress for this element varies linearly through

the thickness.

The transverse shear stresses (SYZ and SXZ) are assumed to be constant

through the thickness.

Shear deflections are included.

Elastic rectangular elements without membrane loads give constant

curvature results, i.e., nodal stresses are the same as the centroid stresses.

For linearly varying results use SHELL63 (no shear deflection) or

SHELL93 (with mid side nodes).

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Triangular elements are not geometrically invariant and the element

produces a constant curvature solution.

3.4.3 MASS21 (Structural Mass) For Load Application

MASS21 is a point element having up to six degrees of freedom: translations in

the nodal x, y, and z directions and rotations about the nodal x, y, and z axes. A

different mass and rotary inertia may be assigned to each coordinate direction.

Fig 3.3: MASS21 Geometry

Input Summary

Element Type : MASS21

Nodes : I

Degrees of Freedom : UX, UY, UZ, ROTX, ROTY, ROTZ

Real Constant : MASSX, MASSY, MASSZ

Material properties : DENS

Output Data

Nodal displacements are included in the overall displacement solution. There is no

element solution output associated with the element unless element reaction forces

and/or energies are requested.

Assumptions and Restrictions

2-D elements are assumed to be in a global Cartesian Z = constant plane.

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The mass element has no effect on the static analysis solution unless

acceleration or rotation is present, or inertial relief is selected

The standard mass summary printout is based on the average of MASSX,

MASSY, and MASSZ.

In an inertial relief analysis, the full matrix is used. All terms are used

during the analysis.

3.4.4 Support Condition

Only one floor is considered in the study and the boundary condition is such that

at top of column joint the vertical translation is disallowed. At bottom of all

column ends are considered to act under fixed support condition with all degrees

of freedom of the support being restrained.

Fig 3.4: Support and Boundary condition

3.5 LOAD APPLICATION

The x-z plane is acting as the horizontal plane in global co-ordinate system.

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To find the dynamic behavior of multistoried RC framed building, basic

unfactored load case is considered as DL (self weight, partition wall, floor finish).

In calculation of column size determination, factored dead and live load is used.

3.5.1 Dead Load: Dead load is the vertical load due to the weight of permanent

structural and nonstructural components of a building. For the present study only

self weight of beams, columns and slabs and nonstructural load (Partition wall,

floor finish) are considered as dead load case of the structure.

All vertical loads except self weight of beams, columns and slab are applied as

mass on the structure. Total vertical load applied on the structure is 25×4.786×10-5

N/mm2 (25 psf) and 50×4.786×10

-5 N/mm

2 (50 psf) for floor finish and partition

wall load respectively. These two loads are applied as mass element of the

structure and applied at the nodes.

3.5.2 Live Load: Live load (LL) considered in the analysis is the load due to fixed

service equipment etc. Total live load applied on the structure is 100×4.786×10-5

N/mm2 (100 psf). This live load will only be used in determining the beam,

column size. In modal analysis live load is not used.

3.6 ANALYSIS METHOD

In this study author provide an extensive study on floor vibration of a RCC

building with partition wall, through ANSYS modeling software. In this study

author made reinforced floor model with ANSYS 11. The model was analyzed

with modal analysis to get the dynamic behavior such as frequency of that floor

for different span length and floor panel aspect ratio (beta). The model type and

degree of freedom depends on the complexity of the actual structure and the

results desired in the analysis modeling a building for a dynamic analysis is

currently more an art than a science. The overall objective is to produce a

mathematical model that will represent the dynamic characteristics of the

floor and produce realistic consistent with the input parameters. The floor

should be modeled as a three dimensional space frame with joints and nodes

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selected to realistically model the stiffness and inertia effects of the structure.

Each joints or nodes should have six degrees of freedom, three translational and

three rotational. At column top node, the vertical translation is disallowed. The

structural mass should be with a minimum of three translational inertia terms.

Only linear behavior is valid in Modal analysis. Nonlinear elements, if any, are

treated as linear. Both Young’s modulus (and stiffness in some form) and density

(or mass in some form) must be defined. Material properties can be linear,

isotropic or orthotropic, and constant or temperature-dependent. Nonlinear

properties, if any, are ignored.

The modal solution natural frequencies and mode shapes is needed to calculate in

this study. Type of modal analysis:

Subspace

Block Lancoz

Power dynamics

Ruduced

Unsymmetric

Damped

QR Damped

But in all of the method the subspace, Block Lanczos, or reduced method are used

to extract the modes. The number of modes extracted should be enough to

characterize the floor’s response in the frequency range of interest.

3.7 FINITE ELEMENT MODEL MESH

FEM is the outlines of the elements used to model the object of interest. To outline

of the mesh should give the appropriate view of the object being modeled. Mesh

size can vary in the analysis of a single structure. Different mesh arrangements

generally give slightly varying solutions. In fact in real life problems mesh is

constantly refined to get a consistent and representative solution.

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According to the literature of FEM the finer the mesh in an idealization, the smaller

are the elements and the better the solution. But this has practical constraints

because a very fine mesh requires tremendous computational effort which may

justified as difficult to deliver even by the mainstream desktop computers.

Another point is that, which element having higher order shape functions the

degree of gain in accuracy diminishes after a certain level of fineness in mesh

discretization. In fact there are a few elements which give exact solution even for

only one element discretization such as beam elements: Thus, it is not always

necessary to use a very fine mesh at the expanse of huge computer memory and

computational time.

It is to be noted here that no mesh is usually the ultimate one, giving exact solution.

A refinement of the mesh is within the scope of further studies and may be selected

on the basis of its approximation of the true result.

3.8 MODEL CHARACTERISTIC FOR ANALYSIS AND TYPICAL

RESULT

3.8.1 Different Parameters

In the present study the objective is to determine the minimum slab thickness for

natural vibration of floor for 10 Hz limit. The variation of frequency with variation

of slab thickness is studied for different floor panel aspect ratio. Slab thickness for

10 Hz limit for variation of span and variation of floor panel aspect ratio is also

discussed. Comparison of slab thickness with ACI limit is also studied for variation

of span and floor panel aspect ratio. For this study, a building frame of one floor

with three span and three bay (bay is the longer floor panel) has been analyzed

for 3m, 4m, 5m, 6m, 7m, 8m, 9m,10m span length. Floor panel aspect ratios are

considered as 1, 1.2, 1.4, 1.6, 1.8. The slab thickness is takes as 50 mm increment

starting from 50 mm. The column size is calculated from the load imposed on the

floor assuming that the floor is a typical floor of a 5 storied building. Beam size is

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taken as the function of slab thickness and span length meaning that with change in

slab thickness and span length, the beam size changes.

Slab Thickness: Slab thickness is taken as 50 mm increment starting from 50 mm

thickness.

Column: Columns are considered square for simplicity. There are three types of

column used in this study shown in Fig 3.4: Interior column, edge column and

Corner column.

Fig 3.5: Column and Beam (plan view)

Interior column size, Ag is determined from the following formula:

Pu = αø[0.85f’c+ρ(fy-0.85 f

’c)]Ag (3.1)

Inte

rio

r co

lum

n

Corner column

Edge C

olu

mn

Ed

ge Beam

Interior Beam

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Where,

α = 0.80

ø = 0.65

f’c = concrete compressive strength

ρ = Reinforcement ratio (taken as around 2%)

fy = steel yield strength

Ag = Interior column gross area

Pu = Assumed Load = (1.40*Dead load) + (1.7*Live load)

In the calculation, the assumed load is 30% increased due to the effect of lateral

load and 30% due to the self weight of column. Corner column size will be 75%

of interior column and edge column size will be 85% of interior column.

Minimum size of edge, corner and interior column must not be less than the beam

width.

Beam:

Two types of beam are used in the study (Interior beam and edge beam) as shown

in Fig: 3.4. Beam depth is taken as the function of span length and slab thickness.

Beam depth = [(span length/11) + (slab thickness/2)]. Beam width = (beam

depth/1.5). Minimum size of beam depth and beam width is taken as 300 mm.

Edge beams are taken as same size as interior beams.

Floor panel Aspect Ratio, Bay and Span:

In the study, Span is considered as the shorter panel length and bay as the longer

panel length. The ratio of longer panel to shorter panel is taken as the floor panel

aspect ratio. Span ranging from 3m to 10m and aspect ratio 1.0 to 1.8 are

considered for the study.

3.8.2 Detail View of Model: The plan, elevation and three dimensional view of

model are shown in following:

Tied column

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Fig 3.6: Plan View of the Model

Fig 3.7: Elevation of the Model

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Fig 3.8: Elevation view of the Model

3.8.3 Different Mode Shapes

This study is about floor vibration. Modal analysis is used to calculate dynamic

behavior of the floor (frequency). In the case of modal analysis different mode

shapes for probable vibration pattern are encountered. Every mode shape have a

particular frequency. We have to be careful in determining the proper mode

shape corrosponding to the natural frequency of the floor. Initial shapes

generally corrosponds to sway shapes. The minimum natural frequency from

mode shape must be defined carefully.

Different mode shapes are provided following from Fig 3.9 to Fig 3.22:

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Fig 3.9: 1st Mode Shape, frequency 5.43 Hz (3D)

Fig 3.10: 1st Mode Shape, frequency 5.43 Hz (elevation)

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Fig 3.11: 2nd

Mode Shape, frequency 5.44 Hz (3D)

Fig 3.12: 2nd

Mode Shape, frequency 5.44 Hz (elevation)

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Fig 3.13: 3rd

Mode Shape, frequency 6 Hz (3D)

Fig 3.14: 3rd

Mode Shape, frequency 6 Hz (elevation)

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Fig 3.15: 4th

Mode Shape, frequency 16.87 Hz (3D)

Fig 3.16: 4th

Mode Shape, frequency 16.87 Hz (elevation)

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Fig 3.17: 10th

Mode Shape, frequency 19.38 Hz (3D)

Fig 3.18: 10th

Mode Shape, frequency 19.38 Hz (elevation)

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Fig 3.19: 15th

Mode Shape, frequency 30 Hz (3D)

Fig 3.20: 15th

Mode Shape, frequency 30 Hz (elevation)

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Fig 3.21: 20th

Mode Shape, frequency 34.18 Hz (3D)

Fig 3.22: 20th

Mode Shape (elevation)

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3.9 REMARKS:

This chapter provides extensive information on how to build a 3-D Single floor

frame model in ANSYS and how to use & change different parameters of the

model. It also provides information regarding the validation of the ANSYS

model. This chapter also provides understanding about the mode shape. The

ANSYS built model is used to perform analysis of determining slab thickness of

floor that fulfills the 10 Hz criteria.

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Chapter - 4

VERIFICATION OF THE

MODEL

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4.1 VERIFICATION OF THE MODEL

Validation of the ANSYS model is necessary in order to check whether the result

found from ANSYS is accurate or not. Two checks are discussed here. Static load

calculation from ANSYS model is compared with hand calculation for the first

check. And then the same model is generated in software ETABS 9.7 and a

particular mode shape frequency is checked whether it matches with that

particular mode shape frequency of ANSYS model.

4.1.1 Verification of the Floor’s Total Load with Hand Calculation

In the verification of the static analysis result given from ANSYS, one floor is

modeled with 3 spans and 3 bays. The total reaction of the whole building given

from the ANSYS is verified with hand calculation. The following data are used

in the ANSYS model:

No of span = 3

No of bay = 3

Span length = 3500 mm

Bay width = 3500 mm

No of floor = 1

Floor height = 3000 mm

Beam size = 250 mm 450 mm

Column size = 350 mm 350 mm

Slab thickness = 150 mm

Floor finish = 25×4.786E-5 N/mm2

Partition wall = 50×4.786E-5 N/mm2

Slab dead load = (slab area× slab thickness× unit weight× g)

= 9×35002×150×2.4×10

-9×9810

= 389.36 KN

Partition wall and floor finish load = (75×4.786×10-5

×9×35002) = 395.74 KN

Load from Beam = (beam area× beam length× no of beam× g× unit weight)

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= 250×450×24×3500×9810×2.4×10-9

= 222.49 KN

Column Load = (column area× floor height× no of column× g× unit weight)

= 350×350×3000×16×2.4×10-9

×9810

= 138.438 KN

Total load from hand calculation = 1146.028 KN and Total load from ANSYS

calculation = 1146.40 KN.

So, it almost meets the result from those two calculations with 0.03% error. The

ANSYS script has been enclosed in Appendix. The result provided by ANSYS is

shown in the following Fig 4.1:

Fig 4.1: Total load of Model from ANSYS

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4.1.2 Verification of Model Floor Frequency from ETABS 9.7

Dynamic result (mode frequency) given from ANSYS for a particular model is

verified with the result given by ETABS 9.7 for the same model.

For ANSYS 11 and ETABS 9.7, the following parameter of the model is

considered:

No of span = 3

No of bay = 3

Span length = 3500 mm

Bay width = 3500 mm

No of floor = 1

Floor height = 3000 mm

Beam size = 300 mm 400 mm

Column size = 400 mm 400 mm

Slab thickness = 150 mm

Floor finish = 25×4.786E-5 N/mm2

Partition wall = 50×4.786E-5 N/mm2

Result Obtained from ANSYS

From ANSYS, the natural frequency of the model floor (4th

mode shape

frequency) is 22.295 Hz,

Three dimensional and elevation views are shown in Fig 4.2 and 4.3.

Result Obtained from ETABS 9.7

From ETABS, the natural frequency of the model floor (4th

mode shape

frequency) is (1/0.0405) Hz = 23.25 Hz.

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Fig 4.2: ANSYS result, 4th

mode frequency 22.295 Hz (3D)

Fig 4.3: ANSYS result, 4th

mode frequency 22.295 Hz (3D)

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Fig 4.4: ETABS result, 4th

mode frequency 23.25 Hz (elevation)

4.1.3 Verification Done by Rakib (2013)

Dynamic result given from ANSYS was verified by Rakib (2013) with an

experimental result done by Murray and Sanchez (2011) in their research on

“Floor Vibration Characteristic Of Long Span Composite Slab System” has done

an experiment on a laboratory floor specimen. They measured the floor natural

vibration with a vibrato meter. They use a composite floor system with single

bay and span. They measured the vibration of the floor in the mid-section.

The measured frequency was 4.98 Hz through their experimental setup.

Rakib (2013) showed that the natural frequency of the model given by ANSYS

is at 2nd

mode. The frequency of the floor specimen is 4.72 Hz. This is close to

the experimental value; 4.98 Hz (Murray, 2011).

4.2 REMARKS

The validation is necessary to know that whether the dynamic result provided by

ANSYS is correct or not. For the model, the obtained value of natural frequency

from ANSYS is 22.295 Hz and from ETABS is 23.26 Hz. So, the 4th

mode

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frequency of the particular model calculated from ANSYS 11 is close to value

provided by ETABS 9.7. Total load from hand calculation = 1146.028 KN and

Total load from ANSYS calculation = 1146.40 KN. So, it almost meets the result

from those two calculations. Rakib (2013) also showed that the ANSYS model

result was close to experimental result of Murray and Sanchez (2011).

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Chapter - 5

PARAMETRIC STUDY AND

RESULTS

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5.1 GENERAL

In this chapter the analysis of determining minimum slab thickness for 10 Hz limit

(Murray, 1997) for a building floor, is modeled and parametric study with results

will be discussed. The parameters are taken based on practical values so that the

actual building behavior under vibration will be same as the 3-D modeling frame.

Current study is actually involved with large number of variables and parameters

but only the variation of floor vibration with respect to variation of slab thickness

and variation of span length with different aspect ratio is studied here. Our

focused is only confined with short span to medium span of building frame (3m to

10m). The material properties are assumed to be linearly elastic. Only natural

vibration of floor including 10 Hz criteria is studied. Comparison with ACI slab

requirement with 10 Hz criteria requirement is also discussed here.

5.2 SELECTED PARAMETERS

The study has performed for determining the minimum slab thickness which does

not create undesirable floor vibration. The limiting floor vibration frequency is 10

Hz given by Murray (1997). The parameter describe in table 5.1 used in the

analysis.

The analysis is done for determining the frequency for various slab thickness. This

process is done for various span length and floor panel aspect ratio. From the

study corresponding to slab thickness for 10 Hz (Murray, 1997) frequency is

plotted against span length. From these studies a relation of the minimum slab

thickness with variation of span and variation of floor panel aspect ratio is

discussed. Also a comparison is made with ACI slab requirement.

From equation number 2.2, the natural frequency of single degree of freedom is

dependent on stiffness and mass.

The stiffness of beam and column can be calculated from the following equation:

Stiffness, ∑

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Parametric Study And Results

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Where,

E = Modulus of elasticity, N/mm2

I = Moment of inertia, mm4

h = Height of the column or length of beam, mm

Table 5.1: Values and dimension of the parameters and structural

components

Parameter Values/ Dimension

Span length 3, 4, 5, 6, 7, 8, 9, 10 m

Aspect ratio 1, 1.2, 1.4, 1.6, 1.8

Bay width as per aspect ratio

Floor height 3 m

No. of story 1

No of span 3

No of bay 3

Slab thickness as per requirement

Floor finish load 25×4.786×10-5

N/mm2

Partition wall load 50×4.786×10-5

N/mm2

Live load 100×4.786×10-5

N/mm2

Beam width beam depth/1.5 ≥ mm

Beam depth (span/11)+(slab thickness/2)

≥ 300 mm

Interior Column as per requirement ≥ 300 mm

Corner Column 75% of interior column

Edge Column 85% of interior column

Gravitational acceleration 9810 mm/sec2

Line mesh size 5

Area mesh size 5

Beam, Column Element BEAM4

Slab Element SHEEL43

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Parametric Study And Results

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Concrete properties

Modulus of elasticity 20000 N/mm2

Poisson’s ratio 0.15

Density 2.4×10-9

Ton/ mm3

5.3 VARIATION OF FLOOR VIBRATION WITH CHANGE OF SPAN

LENGTH

The variation of floor vibration with respect to span length and aspect ratio is

studied here. The analysis is for with partition wall is done for normal residential

building where average number of population is expected to walk. It is tried to

find out the minimum slab thickness which produce vibration frequency more

than 10 Hz. If the structural vibration is larger than the vibration produce by

human walking, Murray (1997) told that will not cause much discomfort to the

occupants. Because humans walking frequency is in the ranges of 2-10 Hz

(Setareh, 2010). If the floor natural vibration is less than 10 Hz, it may create

resonance that will create much vibration and cause discomfort for the occupants.

5.3.1 Analysis of Floor Vibration:

The analysis for variation of floor vibration for different span length, slab

thickness and floor panel aspect ratio will be studied in this section of a building

structure including partition wall load. The modeled structure which has done in

the previous chapter has shown that the beam and column size is depended on the

self weight of the structure, span length and the superimposed load (live load).

With increasing load (for increase of slab thickness and span length) the beam and

column size of the structure also increase and thereby increase the moment of

inertia.

Hence increases the stiffness of the structure. From equation 5.1, the stiffness of

the structural element decrease with increasing length. Hence decreases the

natural frequency of the structure.

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The self weight and the partition wall load are imposed on the modeled structure

as mass element which is a point element. And it is imposed on a node of the

structure.

The graphs on variation of floor frequency with respect to slab thickness for

different span length and aspect ratio for present study are as follows along with

the graph provided by Rakib (2013).

Natural Frequency v/s Slab Thickness

for Aspect Ratio 1.0:

Fig 5.1: Frequency v/s Slab thicknesses for 3m span

0

5

10

15

20

25

30

35

50 100 150 200 250

Fre

quen

cy (

Hz)

Slab Thickness (mm)

Rakib (2013)

Present Study

span:bay=1:1.0

span = 3m

bay = 3m

10 Hz limit

ACI Serviceability limit

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Fig 5.2: Frequency v/s Slab thicknesses for 4m span

Fig 5.3: Frequency v/s Slab thicknesses for 5m span

0

5

10

15

20

25

50 100 150 200 250 300 350

Fre

qu

ency

(H

z)

Slab Thickness (mm)

Rakib (2013)

Present Study

span:bay=1:1.0

span = 4m

bay = 4m

0

5

10

15

20

25

50 100 150 200 250 300 350 400

Fre

quen

cy (

Hz)

Slab Thickness (mm)

Rakib (2013)

Present Study

span:bay=1:1.0

span = 5m

bay = 5m

10 Hz

limit

ACI Serviceability limit

10 Hz limit

ACI Serviceability limit

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Parametric Study And Results

72

Fig 5.4: Frequency v/s Slab thicknesses for 6m span

Fig 5.5: Frequency v/s Slab thicknesses for 7m span

0

5

10

15

20

25

50 100 150 200 250 300 350 400 450 500

Fre

qu

ency

(H

z)

Slab Thickness (mm)

Rakib (2013)

Present Study

span:bay=1:1.0

span = 6m

bay = 6m

0

5

10

15

20

25

50 150 250 350 450 550 650

Fre

quen

cy (

Hz)

Slab Thickness (mm)

Rakib (2013)

Present study

span:bay=1:1.0

span = 7m

bay = 7m

10 Hz limit

ACI Serviceability limit

10 Hz limit

ACI Serviceability limit

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Parametric Study And Results

73

Fig 5.6: Frequency v/s Slab thicknesses for 8m span

Fig 5.7: Frequency v/s Slab thicknesses for 9m span

0

5

10

15

20

25

50 150 250 350 450 550 650 750

Fre

qu

ency

(H

z)

Slab Thickness (mm)

Rakib (2013)

Present Study

span:bay=1:1.0

span = 8m

bay = 8m

0

5

10

15

20

25

50 150 250 350 450 550 650 750

Fre

quen

cy (

Hz)

Slab Thickness (mm)

Rakib (2013)

Present study

span:bay=1:1.0

span = 9m

bay = 9m

10 Hz limit

ACI Serviceability limit

10 Hz limit

ACI Serviceability limit

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Parametric Study And Results

74

Fig 5.8: Frequency v/s Slab thicknesses for 10m span

Natural Frequency v/s Slab Thickness for Aspect Ratio 1.2:

Fig 5.9: Frequency v/s Slab thicknesses for 3m span

0

5

10

15

20

25

50 150 250 350 450 550 650 750 850

Fre

qu

ency

(H

z)

Slab Thickness (mm)

Rakib (2013)

Present Study

span:bay=1:1.0

span = 10m

bay = 10m

0

5

10

15

20

25

50 100 150 200 250

Fre

quen

cy (

Hz)

Slab Thickness (mm)

Rakib (2013)

Present study

span:bay=1:1.2

span = 3m

bay = 3.6m

10 Hz limit

ACI Serviceability limit

10 Hz limit

ACI Serviceability limit

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Parametric Study And Results

75

Fig 5.10: Frequency v/s Slab thicknesses for 4m span

Fig 5.11: Frequency v/s Slab thicknesses for 5m span

0

5

10

15

20

25

50 100 150 200 250 300 350

Fre

qu

ency

(H

z)

Slab Thickness (mm)

Rakib (2013)

Present Study

span:bay=1:1.2

span = 4m

bay = 4.8m

0

5

10

15

20

25

50 100 150 200 250

Fre

quen

cy (

Hz)

Slab Thickness (mm)

Rakib (2013)

Present Study

span:bay=1:1.2

span = 5m

bay = 6m

10 Hz limit

ACI Serviceability limit

10 Hz limit

ACI Serviceability limit

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Parametric Study And Results

76

Figure 5.12: Frequency v/s Slab thicknesses for 6m span

Figure 5.13: Frequency v/s Slab thicknesses for 7m span

0

5

10

15

20

25

50 100 150 200 250 300 350 400 450 500

Fre

qu

ency

(H

z)

Slab Thickness (mm)

Rakib (2013)

Present Study

span:bay=1:1.2

span = 6m

bay = 7.2m

0

5

10

15

20

25

50 150 250 350 450 550 650

Fre

quen

cy (

Hz)

Slab Thickness (mm)

Rakib (2013)

Present

Study

span:bay=1:1.2

span = 7m

bay = 8.4m

10 Hz limit

ACI Serviceability limit

10 Hz limit

ACI Serviceability limit

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Parametric Study And Results

77

Fig 5.14: Frequency v/s Slab thicknesses for 8m span

Fig 5.15: Frequency v/s Slab thicknesses for 9m span

0

5

10

15

20

25

50 150 250 350 450 550 650 750

Fre

quen

cy (

Hz)

Slab Thickness (mm)

Rakib (2013)

Present study

span:bay=1:1.2

span = 8m

bay = 9.6m

0

5

10

15

20

25

50 150 250 350 450 550 650 750

Fre

quen

cy (

Hz)

Slab Thickness (mm)

Rakibul 2013

This Author

span:bay=1:1.2

span = 9m

bay = 10.8m

10 Hz limit

ACI Serviceability limit

10 Hz limit

ACI Serviceability limit

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Parametric Study And Results

78

Natural Frequency v/s Slab Thickness for Aspect Ratio 1.4:

Fig 5.16: Frequency v/s Slab thicknesses for 3m span

Fig 5.17: Frequency v/s Slab thicknesses for 4m span

0

5

10

15

20

25

50 100 150 200 250

Fre

qu

ency

(H

z)

Slab Thickness (mm)

Rakib (2013)

Present study

span:bay=1:1.4

span = 3m

bay = 4.2m

0

5

10

15

20

25

50 100 150 200 250 300 350

Fre

quen

cy (

Hz)

Slab Thickness (mm)

Rakib (2013)

Present study

span:bay=1:1.4

span = 4m

bay = 5.6m

10 Hz limit

ACI Serviceability limit

10 Hz limit

ACI Serviceability limit

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Parametric Study And Results

79

Fig 5.18: Frequency v/s Slab thicknesses for 5m span

Fig 5.19: Frequency v/s Slab thicknesses for 6m span

0

5

10

15

20

25

50 100 150 200 250 300 350 400

Fre

qu

ency

(H

z)

Slab Thickness (mm)

Rakib (2013)

Present study

span:bay=1:1.4

span = 5m

bay = 7m

0

5

10

15

20

25

50 150 250 350 450

Fre

quen

cy (

Hz)

Slab Thickness (mm)

Rakib (2013)

Present study

span:bay=1:1.4

span = 6m

bay = 8.4m

10 Hz limit

ACI Serviceability limit

10 Hz limit

ACI Serviceability limit

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Parametric Study And Results

80

Fig 5.20: Frequency v/s Slab thicknesses for 7m span

Natural Frequency v/s Slab Thickness for Aspect Ratio 1.6:

Fig 5.21: Frequency v/s Slab thicknesses for 3m span

0

5

10

15

20

25

50 150 250 350 450 550 650

Fre

quen

cy (

Hz)

Slab Thickness (mm)

Rakib (2013)

Present study

span:bay=1:1.4

span = 7m

bay = 9.8m

0

5

10

15

20

25

50 100 150 200 250

Fre

quen

cy (

Hz)

Slab Thickness (mm)

Rakib (2013)

Present study

span:bay=1:1.6

span = 3m

bay = 4.8m

10 Hz limit

ACI Serviceability limit

10 Hz limit

ACI Serviceability limit

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Parametric Study And Results

81

Fig 5.22: Frequency v/s Slab thicknesses for 4m span

Fig 5.23: Frequency v/s Slab thicknesses for 5m span

0

5

10

15

20

25

50 100 150 200 250 300 350

Fre

qu

ency

(H

z)

Slab Thickness (mm)

Rakib (2013)

Present study

span:bay=1:1.6

span = 4m

bay = 6.4m

0

5

10

15

20

25

50 100 150 200 250 300 350 400

Fre

quen

cy (

Hz)

Slab Thickness (mm)

Rakib (2013)

Present Study

span:bay=1:1.6

span = 5m

bay = 8m

10 Hz limit

ACI Serviceability limit

10 Hz limit

ACI Serviceability limit

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Parametric Study And Results

82

Fig 5.24: Frequency v/s Slab thicknesses for 6m span

Natural Frequency v/s Slab Thickness for Aspect Ratio 1.8:

Fig 5.25: Frequency v/s Slab thicknesses for 3m span

0

5

10

15

20

25

50 150 250 350 450

Fre

qu

ency

(H

z)

Slab Thickness (mm)

Rakib (2013)

Present study

span:bay=1:1.6

span = 6m

bay = 9.6m

0

5

10

15

20

25

50 100 150 200 250

Fre

quen

cy (

Hz)

Slab Thickness (mm)

Rakib (2013)

Present study

span:bay=1:1.8

span = 3m

bay = 5.4m

10 Hz limit

ACI Serviceability limit

10 Hz limit

ACI Serviceability limit

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Parametric Study And Results

83

Fig 5.26: Frequency v/s Slab thicknesses for 4m span

Fig 5.27: Frequency v/s Slab thicknesses for 5m span

0

5

10

15

20

25

50 100 150 200 250 300 350

Fre

qu

ency

(H

z)

Slab Thickness (mm)

Rakib (2013)

Present study

span:bay=1:1.8

span = 4m

bay = 7.2m

0

5

10

15

20

25

50 150 250 350 450

Fre

quen

cy (

Hz)

Slab Thickness (mm)

Rakib (2013)

Present study

span:bay=1:1.8

span = 5m

bay = 9m

10 Hz limit

ACI Serviceability limit

10 Hz limit

ACI Serviceability limit

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Parametric Study And Results

84

5.3.2 Study of ‘Natural Frequency v/s Slab Thickness’ Curves

From the above Fig 5.1 to Fig 5.27, the change of frequency with change of slab

thickness, span length and aspect ratio is shown. Here span is the short panel

length, and bay width is the large panel length. From the above figures the

following points are observed:

The natural frequency of floor is increasing with increase of slab thickness due

to the fact that with increasing slab thickness, the beam and column size is

also increased, due to self weight of the slab. Hence increase the moment of

inertia of structural elements and consequently increases the natural frequency

of the floor.

With increase of span length, the stiffness of the floor decreases, as a result the

floor frequency is reduced.

With increase of floor panel aspect ratio, the natural frequency of floor is

decreasing. Because increased aspect ratio increases the bay width of the floor.

Hence decreases the stiffness of the floor and thus the floor frequency

decreases.

Increase of slab thickness means the increase of weight of floor as the self

weight of slab, beam and column size is increasing and it indicates the

increase of mass. Increase of mass means decrease of natural frequency of the

structure. But increased slab thickness increases the beam and column size that

increases the moment of inertia and stiffness that increases the floor frequency

For a particular curve, due to these two contradictory conditions, the initial

part of the curves mentioned above in the figures are steeper when mass is

small and when the mass is higher the curve becomes less steep. The

frequency still increases due to the fact that the column and beam size is

increasing with increase of slab thickness which increases the stiffness. And

increased stiffness increases the moment of inertia.

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Parametric Study And Results

85

At low aspect ratio, the point of changing slope is achieved earlier (at small

slab thickness), but with increase of aspect ratio this point gradually shifted

toward the larger slab thickness.

This condition is also seen with increase of span length. This condition

prevails because initially the moment of inertia is high enough hence floor is

stiff enough to ignore the effect of reduced frequency due to increasing mass.

Also when slab thickness is small, column and beam size is small. Hence the

mass is also small and that’s why the reduction of frequency is lesser.

Gradually, reduced frequency due to increasing mass becomes prominent as a

result of increase of slab thickness, beam and column size. So, a less steep

curve is found at the latter part.

This effect is seen for both change of span and floor panel aspect ratio.

The typical shape of the curve provided by this author is almost same as

provided by Rakib (2013) but in comparison with Rakib (2013), it is seen that

natural frequency is in the higher range provided by this author than by Rakib

(2013).

In some curves provided by Rakib (2013), the frequency is very low resulting

unrealistic slab thickness requirement.

5.4 Comparison of ACI Serviceability limit

with 10 Hz limit

Comparison of 10 Hz limit with ACI serviceability limit is important issue. Slab

thickness fulfilling 10 Hz limit is plotted against various span length for various

aspect ratio for both Present study and Rakib (2013).

ACI Serviceability limit is also shown in the following figures:

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Parametric Study And Results

86

Fig 5.28: Slab Thickness v/s Span length for aspect ratio 1.0

Fig 5.29: Slab Thickness v/s Span length for aspect ratio 1.2

0

100

200

300

400

500

3 4 5 6 7 8 9 10 11 12

Sla

b T

hic

knes

s(m

m)

Span Length (m)

ACI Limit

10 Hz Limit

10 Hz limit

(Rakib,2013)

0

100

200

300

400

500

3 5 7 9 11 13

Sla

b T

hic

knes

s(m

m)

Span Length (m)

ACI Limit

10 Hz Limit

10 Hz limit

(Rakib,2013)

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Parametric Study And Results

87

Fig 5.30: Slab Thickness v/s Span length for aspect ratio 1.4

Fig 5.31: Slab Thickness v/s Span length for aspect ratio 1.6

0

100

200

300

400

500

3 4 5 6 7 8 9 10

Sla

b T

hic

knes

s(m

m)

Span Length (m)

ACI Limit

10 Hz Limit

10 Hz limit

(Rakib,2013)

0

100

200

300

400

500

3 4 5 6 7 8 9 10 11

Sla

b T

hic

knes

s(m

m)

Span Length (m)

ACI Limit

10 Hz Limit

10 Hz limit

(Rakib,2013)

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Parametric Study And Results

88

Fig 5.32: Slab Thickness v/s Span length for aspect ratio 1.8

5.4.1 Study of ‘Slab Thickness v/s Span’ curves for ACI limit and 10 Hz limit

The slab thickness required for satisfying the 10 Hz limit, is increasing with

the increase of span for a particular floor panel aspect ratio. The larger the

span, the higher the slab thickness required for satisfying 10 Hz criteria.

Slab thickness requirement as per ACI Serviceability limit is directly

proportional to span length for a specific floor panel aspect ratio (beta). For a

particular aspect ratio, the ACI slab requirement is increasing in proportion of

increasing span length.

Minimum slab thickness satisfying the deflection criteria provided by ACI is

90 mm.

For a particular aspect ratio, after a specific span length, the 10 Hz limit is

governing than ACI limit.

0

100

200

300

400

500

3 4 5 6 7 8 9 10

Sla

b T

hic

kn

ess(

mm

)

Span Length (m)

ACI Limit

10 Hz Limit

10 Hz limit

(Rakib,2013)

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Parametric Study And Results

89

When floor panel aspect ratio is small, slab thickness requirement is governed

by ACI limit for smaller span. With increase of span, 10 Hz limit governs. For

higher aspect ratio, generally 10 Hz limit is dominant.

In comparison with Rakib (2013), it is seen that the 10 Hz limit curves

provided by Rakib (2013) is higher than the curve provided by this author. It

means that minimum slab thickness requirement is larger for Rakib’s (2013)

analysis and if the span length is large, the minimum slab thickness is

sometimes found unrealistic. This problem is overcome with the curve

provided by present study for 10 Hz limit.

5.5 MINIMUM SLAB THICKNESS DETERMINATION

The minimum slab thickness, which is required to prevent vibration which causes

annoying to the occupants, will be determined here. It is known that human

walking vibration is in the range of 4-8 Hz (Murray, 1991). So the floor vibrating

in this range is perceived as uncomfortable. The slab thickness should be such that

vibration is greater than 10 Hz. The floor vibration should be greater than 10 Hz

as if resonance is not created.

Previously slab thickness required fulfilling the 10 Hz criteria for various span and

floor panel aspect ratio is determined from Fig 5.1 to Fig 5.27. And ACI limit is

compared with 10 Hz limit for various span and floor panel aspect ratio from Fig

5.28 to Fig 5.32.

For determining the desired slab thickness satisfying the vibration criteria of 10

Hz, slab thickness corresponding to 10 Hz limit is plotted against various floor

panel aspect ratios for various span lengths in a single graph.

From Fig 5.33, Minimum slab thickness required to fulfill the 10 Hz criteria can

be determined for span length of 3 to 10 meters and floor panel aspect ratio of 1.0

to 1.8. The thickness should be compared with ACI limit.

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Parametric Study And Results

90

Fig 5.33: Minimum Slab Thickness v/s Aspect Ratio for various

Span Lengths (Satisfying 10 Hz criteria)

The Fig 5.33 indicates that for larger span length and higher floor panel aspect

ratio, higher slab thickness is required. Initial portion of the curve is less steep

when the floor panel aspect ratio is small. But with increase of floor panel aspect

ratio, after a certain point the curve is steeper for a particular span length.

This is due to the fact that when aspect ratio is small, stiffness is high and mass is

small, so minimum slab thickness requirement to fulfill 10 Hz limit is small.

When aspect ratio is large, the mass is greater and stiffness is lesser contributing

to higher minimum slab requirement to satisfy 10 Hz limit. It is also seen that after

aspect ratio crosses 1.30 to 1.4, a certain rise in minimum slab thickness

requirement occurs.

0

100

200

300

400

500

600

700

800

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Sla

b T

hic

knes

s (m

m)

Floor Panel Aspect Ratio

3m

4m

5m

6m

7m 8m

9m

10

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Parametric Study And Results

91

Comparison of Minimum Slab Thickness

with ACI Serviceability limit for various Span Lengths and Aspect Ratio:

With a view to visualizing the deviation of minimum slab thickness requirement

of ACI Serviceability limit from 10 Hz limit, slab thickness is also plotted against

floor panel aspect ratio for a particular span. Thus the determined minimum slab

thickness from Fig 5.33 can be compared with ACI serviceability limit of slab

thickness requirement.

Minimum slab thickness v/s floor panel aspect ratio for span length 3m to 10m are

following:

Fig 5.34: Minimum Slab Thickness v/s Aspect ratio for

3m Span Length

25

45

65

85

105

125

1 1.2 1.4 1.6 1.8

Sla

b T

hic

knes

s

(mm

)

Floor Panel Aspect Ratio

ACI limit

10 Hz

limit

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Parametric Study And Results

92

Fig 5.35: Minimum Slab Thickness v/s Aspect ratio for

4m Span Length

Fig 5.36: Minimum Slab Thickness v/s Aspect ratio for

5m Span Length

50

100

150

200

250

1 1.2 1.4 1.6 1.8

Sla

b T

hic

kn

ess

(mm

)

Floor Panel Aspect Ratio

ACI limit

10 Hz limit

50

100

150

200

250

300

350

400

1 1.2 1.4 1.6 1.8

Sla

b T

hic

knes

s

(mm

)

Floor Panel Aspect Ratio

ACI limit

10 Hz limit

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Parametric Study And Results

93

Fig 5.37: Minimum Slab Thickness v/s Aspect ratio for

6m Span Length

Fig 5.38: Minimum Slab Thickness v/s Aspect ratio for

7m Span Length

100

200

300

400

500

600

1 1.2 1.4 1.6 1.8

Sla

b T

hic

kn

ess

(mm

)

Floor Panel Aspect Ratio

ACI limit

10 Hz limit

100

200

300

400

500

600

700

800

1 1.2 1.4 1.6 1.8

Sla

b T

hic

knes

s

(mm

)

Floor Panel Aspect Ratio

ACI limit

10 Hz limit

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Parametric Study And Results

94

Fig 5.39: Minimum Slab Thickness v/s Aspect ratio for

8m Span Length

Fig 5.40: Minimum Slab Thickness v/s Aspect ratio for

9m Span Length

150

250

350

450

550

650

750

850

1 1.2 1.4 1.6 1.8

Sla

b T

hic

knes

s

(mm

)

Floor Panel Aspect Ratio

ACI limit

10 Hz limit

150

200

250

300

350

400

450

500

550

600

1 1.2 1.4 1.6 1.8

Sla

b T

hic

knes

s

(mm

)

Floor Planel Aspect Ratio

ACI limit

10 Hz limit

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Parametric Study And Results

95

Fig 5.41: Minimum Slab Thickness v/s Aspect ratio for

10m Span Length

The figures show that for small span length, ACI limit criteria governs up to a

certain floor panel aspect ratio. For span larger than 5m, ACI slab requirement

may not be sufficient. From Fig 5.34 to Fig 5.41, minimum slab thickness

satisfying 10 Hz, found from Fig 5.33 can be compared with minimum slab

thickness requirement provided by ACI limit.

5.6 RESULT COMPARISON WITH RAKIBUL ISLAM (2013):

From the variation of the slab thickness with span length for aspect ratio, Rakib

(2013) formed a polynomial equation (equation 5.2):

Where,

t = Minimum slab thickness for around 10 Hz floor vibration, mm

x = Short span length in m

200

300

400

500

600

700

800

1 1.2 1.4 1.6 1.8

Sla

b T

hic

knes

s

(mm

)

Floor Panel Aspect Ratio

ACI limit

10 Hz limit

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Parametric Study And Results

96

β = aspect ratio.

Rakib (2013) formed polynomial equation to determine the tread line of the

analysis value. Those tread line for different aspect ratio is analyzed to give a

common equation.

Instead, in this present study, a graph Fig 5.33 is provided where minimum slab

thickness required to satisfy 10 Hz criteria is plotted against aspect ratio for

various spans. Comparison with ACI limit is also possible from Fig 5.34 to Fig

5.41.

The following figures (Fig 5.42 to Fig 5.47) are the comparison of minimum slab

thickness requirement between present study and Rakib (2013) for 10 Hz limit:

Fig 5.42: Slab thickness v/s aspect ratio

for 3m span

25

45

65

85

105

125

1 1.2 1.4 1.6 1.8

Sla

b T

hic

knes

s

(mm

)

Floor Panel Aspect Ratio

Rakib

(2013)

Present

study

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Parametric Study And Results

97

Fig 5.43: Slab thickness v/s aspect ratio for 4m span

Fig 5.44: Slab thickness v/s aspect ratio

for 5m span

50

100

150

200

250

1 1.2 1.4 1.6 1.8

Sla

b T

hic

kn

ess

(mm

)

Floor Panel Aspect Ratio

Rakib

(2013)

Present

study

50

100

150

200

250

300

350

400

1 1.2 1.4 1.6 1.8

Sla

b T

hic

knes

s

(mm

)

Floor Panel Aspect Ratio

Rakib (2013)

Present study

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Parametric Study And Results

98

Fig 5.45: Slab thickness v/s aspect ratio

for 6m span

Fig 5.46: Slab thickness v/s aspect ratio

for 7m span

100

200

300

400

500

600

1 1.2 1.4 1.6 1.8

Sla

b T

hic

kn

ess

(mm

)

Floor Panel Aspect Ratio

Rakib

(2013)

Present

study

100

200

300

400

500

600

700

800

1 1.2 1.4 1.6 1.8

Sla

b T

hic

knes

s

(mm

)

Floor Panel Aspect Ratio

Rakib

(2013)

Present

study

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Parametric Study And Results

99

Fig 5.47: Slab thickness v/s aspect ratio for 8m span

The minimum slab thickness requirement provided by Rakib (2013) is in the

higher side in comparison with this present study. The variation of shape of curves

for different spans is also visible in the graphical plots. The variation of minimum

slab thickness found from the curves of present study and Rakib (2013) may be

due to the following reasons:

This study only considers single floor for the vibration analysis. Only a single

floor vibration is analyzed here and understanding the proper mode shape and

corresponding frequency is easier. On the other hand Rakib (2013) considered 3

floors, building vibration characteristics may be merged with the floor vibration

phenomena. Also the column axial stiffness may be included in the analysis of

Rakib (2013).

Some parameters used in the analysis of vibration during this study are not

identical with Rakib (2013). For instance, the beam and column size increment

with increase of slab thickness.

150

250

350

450

550

650

750

850

1 1.2 1.4 1.6 1.8

Sla

b T

hic

knes

s

(mm

)

Floor Panel Aspect Ratio

Rakib

(2013)

Present

study

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Parametric Study And Results

100

5.7 REMARKS

From the above study, Fig 5.33 can be used in order to determine the minimum

slab thickness for normal residential building having partition wall that will not

vibrate at less than 10 Hz. The partition wall load used in the study is 50 psf. The

minimum slab thickness can be compared with ACI limit criteria with the help of

Fig 5.34 to Fig 5.41. The study is for span ranges from 3m to 10m and floor panel

aspect ratio from 1.0 to 1.8.

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Chapter – 6

CONCLUSIONS

AND

RECOMMENDATIONS

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Conclusion and Recommendation

102

6.1 GENERAL

In the present study an investigation has been done to determine the required

minimum slab thickness from dynamic serviceability. ACI provides a minimum

slab thickness (equation 2.9) equation in order to prevent excessive deflection

(static serviceability) of the slab. There is no such recommended RCC slab

thickness which can prevent undesirable floor vibration. This study provides a

guide line about RCC slab thickness with variation of span length and floor panel

aspect ratio that will prevent excessive uncomfortable vibration of the floor. This

minimum slab thickness for which floor frequency is greater than 10 Hz is

determined for normal residential building including partition wall load. The study

considers a floor of three spans and three bays with different floor panel aspect

ratio and span length. Finally minimum slab thickness determination and

comparison with ACI limit is done by establishing graphs.

6.2 OUTCOME OF THE STUDY

The general output and findings from the previous chapters are summarized

below:

The modal analysis of a floor is done and various modes of vibration

frequency were determined by this analysis. This is helpful for understanding

the dynamic behavior of floor under vibration.

Floor frequency is dependent to mass of floor as well as the stiffness of the

floor system. Increasing mass decreases the frequency and increasing

stiffness increases the floor frequency.

Floor frequency decreases with increase of span length and floor panel

aspect ratio. For span larger than 5m, minimum slab thickness requirement

provided by ACI may not be good enough for vibration serviceability.

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Conclusion and Recommendation

103

6.3 PROPOSED GRAPH

With increase of slab thickness, floor frequency increases. Based on the study,

finally a graph (Fig 5.33) is proposed to determine the minimum slab thickness for

variation of span and floor panel aspect ratio. Minimum slab thickness

requirement for ACI serviceability limit can also be compared from Fig 5.34 to

Fig 5.41.

6.4 LIMITATION AND RECOMMENDATION FOR FURTHER STUDY

The current study has some limitations. The results are not sufficient to apply for

all type of situations as so many other factors have not been considered.

Advancement of current study can be done combining some other variables. The

following fields related to this study can be considered for the further analysis:

The model was considered to be linearly elastic. To be more realistic with

the results a finite element analysis with nonlinearly material properties can

be performed.

The asymmetric floor frames can be studied under the variables considered

for symmetric frames.

Different number of span and bay other than three can be studied.

Study can be carried out for without partition wall load.

Effect of result due to floor height change can be another part of study.

Vibration effect due to other sources (machinery, traffic) can be studied.

This study can be continued for flat plate floors.

Only gravity load on floor is considered in the study.

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104

REFERENCES:

Allen D.E. and Murray T.M. (1993), “Design Criteria For Vibration Due To

Walking”, AISC Engineering J., 40(4), 117-129.

Bachmann H. and Ammann W. (1987), “Vibrations in Structures Induced by Man

and Machine”, Structural Engineering Document 3e, Chapter 1, 2, 3.

Billah K.Y. and Scanlan R.H. (1990), “Resonance Tacoma Narrows bridge failure

and undergraduate physics Textbooks”, Page 120.

British Standard Institution (BSI 2008), “Guide to evaluation of human exposure

to vibration in buildings – part I: Vibration sources other than blasting”, BS 6472-

1, London.

British Standards Institution (BSI) (1992), “Guide To Evaluation Of Human

Exposure To Vibration In Buildings, (1Hz To 80H)”, BS 6472, London.

Dallard P., Fitzpatrick A.J., Flint A., Bourva S.Le., Low A., Smith R.M.R.,

Willford M. (2001), “The London Millennium Footbridge”.

De Silva S.S. (2007), “Vibration Characteristics of Steel Deck Composite Floor

System under Human Excitation”, Chapter 1.

Guilherme, E., Faisca, R.G. (2009). “Characterization of Dynamic Loads due to

Human Activities” Page: 1-240.

International organization for Standardization (ISO), (2007), “Bases For Design

Of Structures- Serviceability Of Building And Walkways Against Vibratio”, ISO

10137, 2nd

Ed., Geneva.

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references

105

International Standard Organization - ISO 2631-2 (1989), “Evaluation of Human

Exposure to Whole-Body Vibration, Part 2: Human Exposure to Continuous and

Shock-Induced Vibrations in Buildings (1 to 80Hz), International Standard’’.

McGrath P. and Foote D. (1981), “What Happened at the Hyatt?” Newsweek,

Section: National Affairs, Page 26.

Murray T.M. and Sanchez T.A. (2011), “Floor Vibration Characteristics Of Long

Span Composite Slab System”, Journal of Structural Engineering, ASCE.

Murray T.M., Allen D.E. and Uger E.E. (2003), “Floor Vibration Due To Human

Activities.” Steel Design Guide Series, AISC, Chicago.

Murray T.M., Allen D.E. and Ungar E.E. (1997), Design Guide No. 11. “Floor

Vibrations Due to Human Activity”, American Institute of Steel Construction

(AISC), Chicago, IL, Chapter 2, 4.

Nilson A.H., Darwin D., Dolan C.W., “Design of Concrete Structures”,13th

edition, Chapter 13, Page 437.

Ohlsson S.V. (1988), “Springiness and Human-Induced Floor Vibrations- A

Design Guide”, D12:1988, Swedish Council for Building research, Sweden.

Petrovic S. and Pavic A. (2011), “Effects Of Non-Structural Partitions On

Vibration Performance Of Floor Structures”, Proceedings of the 8th International

Conference on Structural Dynamics, EURODYN 2011 Leuven, Belgium, 4-6 July.

Rakib M. (2013),“Minimum slab thickness of RC slab to prevent undesirable floor

vibration”, Chapter 3 and 4.

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references

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Setareh M. (2010), “Vibration Serviceability Of A Building Floor Structure I:

Dynamic Testing And Computer Modeling”, Journal Of Performance Of

Construction Facilities, ASCE.

Setareh M. (2010), “Vibration Serviceability Of A Building Floor Structure II:

Vibration Evaluation And Assessment”, Journal Of Performance Of Construction

Facilities, ASCE.

Setareh, M. (2011), “Vibration Studies Of A Cantilevered Structure Subjected To

Human Activities Using A Remote Monitoring System”, Journal of Performance of

Constructed Facilities, Vol.25, No.2, April1, ASCE, ISSN 0887-3828/2011/2-87–

97.

Silva S. and Thambiratnam D (2011), “Vibration Characteristic Of Concrete-Steel

Composite Floor Structures”, ACI Structural Journal, V. 108, No. 6, Novemnber-

December.

Smith A.L., Hicks S.J., and Devine P.J. (2007), “Design of Floors for Vibration: A

New Approach,” The Steel Construction Institute (SCI), Ascot, Berkshire.

Sladki M.J. (1999), “Prediction Of Floor Vibration Response Using The Finite

Element Method.” Master Degree Thesis, Virginia Polytechnic Institute and State

University.

Silva G. and Andrade S. (2009), “Vibration Analysis Of Long Span Joist Floors

Submitted To Human Rhythmic Activities”, UERJ, Brazil.

Wilson A. (1998), “Expected Vibration Performance Of Wood Floors As Affected

By MSR Vs. VSR Lumber E-Distribution,” Master Degree Thesis, Virginia

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references

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William and Waldron (2009), “Evaluation of Methods for Predicting Occupant-

Induced Vibrations in Concrete Floors”, Chapter 2.

Wyatt T.A. (1989), “Design Guide on the vibration of floors”, ISBM: 1 870004 34

5, The Steel Construction Institute, Berkshire, England.

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APPENDIX

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Appendix

109

ANSYS SCRIPT USED IN MODEL FLOOR ANALYSIS:

FINISH /CLEAR !3D FLOOR !parameters NSPAN = 3 !no of span NFLOOR = 1 !no of floor NBAY = 3 !no of bay SPANL= 4000 !span length (mm) BAY = SPANL*1.0 !bay width (mm) FHT= 3000 !floor height (mm) SLABT= 150 !slab thickness (mm) G=9810 !Gravity (N/mm2) !**defining load (N/mm2)** FF= 25*4.786E-5 !floor finish load (N-mm2) PW= 50*4.786E-5 !partition wall load (N-mm2) LL= 100*4.786E-5 !live load (N-mm2 DL= FF+PW !dead load (N-mm2) !**defining beam section (mm)** !beam depth BEAMD = ((SPANL/11) + (SLABT/2.0)) *IF, BEAMD, LT, 300, THEN BEAMD=300 *ENDIF *IF, BEAMD, GT, 1050, THEN BEAMD=1050 *ENDIF !beam width BEAMW=BEAMD/1.5

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110

*IF, BEAMW, LT, 300, THEN BEAMW=300 *ENDIF !**defining column section(mm)** BEAML=SPANL+BAY CAREA=SPANL*BAY !contributing area, mm2

SLABDL = (SLABT/25.4)*(150/12)*(CAREA*10.758E-6) !slab selfload in lb

DEADL = (DL*20894.275)*(CAREA*10.758E-6) !dealload in lb BEAMDL = ((BEAMW*BEAMD)/25.4**2)*(150/144)*(BEAML*0.00328)

!beam dead load in lb

TDL=(DEADL+BEAMDL+SLABDL)*5 !total dead load in lb LIVEL=(LL*20894.27)*(CAREA*10.76E-6)*5 !live load in lb PU=(1.4*TDL)+(1.7*LIVEL) !ultimate load in lb PUU=(1.3*1.3*PU)/1000 !30% increase due to !lateral load and 30% !due to column self wght !load in kip AG=PUU/2.35664 !gross area in in2

AGG=AG*(25.4**2) !area in mm2

!interior column width, length ICOLW = NINT (AGG**0.5) *IF, ICOLW, LT, BEAMW, THEN ICOLW = BEAMW

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111

*ENDIF ICOLL = ICOLW !corner column width, length CCOLW = NINT (ICOLW*0.75) *IF, CCOLW, LT, BEAMW, THEN CCOLW = BEAMW *ENDIF CCOLL = CCOLW !edge column width, length ECOLW = NINT (ICOLW*0.85) *IF, ECOLW, LT, BEAMW, THEN ECOLW = BEAMW *ENDIF ECOLL = ECOLW !minimum slab thickness as per ACI code for 2-way slab LL=(BAY-BEAMW) !clear span in long direction in mm LS=(SPANL-BEAMW) !clear span in short direction in mm !minimum slab thickness in mm IB=(1/12)*(BEAMW)*(BEAMD**3)!beam moment of inertia in mm4 IS=(1/12)*(LL+BEAMW)*(MINSLAB**3) !slab moment of inertia in mm4 BETA=LL/LS !ratio of long to short span FY=60 !steel yield in ksi ALPHA=(IB/IS) *IF, ALPHA, LT, 2, THEN, Q = 0.8 + ((FY*1000)/200000) R = 36 + (5*BETA*(ALPHA-0.2)) MINT=NINT(LL*(Q/R)) !minimum slab thickness in mm *IF, MINT, LT, 125, THEN MINT = 125 *ENDIF

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112

*ENDIF *IF, ALPHA, GT, 2, THEN, Q = 0.8 + ((FY*1000)/200000) R = 36 + (9*BETA) MINT=NINT(LL*(Q/R)) !minimum slab thickness in mm *IF, MINT, LT, 88, THEN MINT = 90 *ENDIF *ENDIF *IF, ALPHA, EQ, 2, THEN, Q = 0.8 + ((FY*1000)/200000) R = 36 + (9*BETA) MINT = NINT (LL*(Q/R)) *IF, MINT, LT, 90, THEN MINT=90 *ENDIF *ENDIF !**material properties** POIS=0.15 !poison ratio EC=20000 !young modulus of concrete DENSITY=2.4E-9 !concrete density (ton/mm3) !**meshing number** LDIV=5 !line division ADIV=5 !area division !**section calculation** ICOLA=ICOLW*ICOLL !interior col area ICOLIX=ICOLW*(ICOLL**3)/12 !interior col moment of inertia ICOLIY= ICOLL*(ICOLW**3)/12 !interior col moment of inertia ECOLA = ECOLW*ECOLL !exterior col area ECOLIX= ECOLW*(ECOLL**3)/12 !exterior col moment of inertia ECOLIY= ECOLL*(ECOLW**3)/12 !exterior col moment of inertia CCOLA = CCOLW*CCOLL !corner col area CCOLIX= CCOLW*(CCOLL**3)/12!corner col moment of inertia

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113

CCOLIY= CCOLL*(CCOLW**3)/12 !corner col moment of inertia BEAMA = BEAMW*BEAMD !beam area BEAMIX= BEAMW*(BEAMD**3)/12 !beam moment of inertia BEAMIY= BEAMD*(BEAMW**3)/12 !beam moment of inertia !**dl slab contribution** MASSPF= DL*(NSPAN*SPANL)*(NBAY*BAY)/G !mass per floor FLOORA= (NBAY*BAY)*(NSPAN*SPANL) !floor area MASSPA= MASSPF/FLOORA !mass per area X1=SPANL/ADIV Z1=BAY/ADIV CONTA= X1*Z1 !contributing area MASSIN= MASSPA*CONTA !mass per internal node MASSEN=MASSIN/2 !mass per external node MASSCN=MASSIN/4 !mass per corner node /PREP7 ET, 1, BEAM4 ET, 2, SHELL43 !**declaring material property** MP, EX, 1, EC !for beam, column MP, PRXY, 1, POIS MP, DENS, 1, DENSITY MP, EX, 2, EC !for slab MP, PRXY, 2, POIS MP, DENS, 2, DENSITY !**declaring real constant** R, 1, ICOLA, ICOLIX, ICOLIY, ICOLL, ICOLW R, 2, ECOLA, ECOLIX, ECOLIY, ECOLL, ECOLW R, 3, CCOLA, CCOLIX, CCOLIY, CCOLL, CCOLW R, 4, BEAMA, BEAMIX, BEAMIY, BEAMW, BEAMD R, 5, SLABT !***ASSIGNING KEY POINT**** K, 1, 0, 0, 0

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114

K, 2, 0, FHT, 0 K, 3, SPANL, 0, 0 K, 4, SPANL, FHT, 0 K, 5, 0, 0, BAY K, 6, 0, FHT, BAY K, 7, SPANL, 0, BAY K, 8, SPANL, FHT, BAY L, 1, 2 L, 2, 4 L, 3, 4 L, 5, 6 L, 7, 8 L, 2, 6 L, 4, 8 L, 6, 8 LSEL, S, LOC, X, 0, 0 LSEL, R, LOC, Z, 0, 0 LGEN, NSPAN+1, ALL, , , SPANL !col generation along span LGEN, NBAY+1, ALL, , , , , BAY !col generation along bay LSEL, S, LOC, Y, 0, 0 LGEN, NSPAN, ALL, , , SPANL LGEN, NBAY, ALL, , , , , BAY LSEL, ALL LSEL, S, LOC, Y, FHT, FHT LGEN, NSPAN, ALL, , , SPANL LGEN, NBAY, ALL, , , , , BAY LSEL, ALL LGEN, NFLOOR, ALL, , , , FHT ALLSEL, ALL NUMMRG, KP NUMMRG, NODE !**area generation** A, KP (0, FHT, 0), KP (SPANL, FHT, 0), KP (SPANL, FHT, BAY), KP (0, FHT, BAY) ASEL, ALL AGEN, NFLOOR, ALL, , , , FHT

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115

AGEN, NSPAN, ALL, , , SPANL AGEN, NBAY, ALL, , , , BAY ALLSEL, ALL NUMMRG, KP !**area meshing** TYPE, 2 MAT, 2 REAL, 5 ASEL, ALL LESIZE, ALL, , , ADIV, , 1 AMESH, ALL !**edge beam meshing* !FACE Z = 0 LSEL, NONE *DO, EBEAM, 1, NFLOOR, 1 YEBEAM = EBEAM*FHT LSEL, A, LOC, Y, YEBEAM, YEBEAM LSEL, R, LOC, Z, 0, 0 *ENDDO LESIZE, ALL, , , LDIV, , 1 TYPE, 1 MAT, 1 REAL, 4 LMESH, ALL !FACE Z = BAY LSEL, NONE *DO, EBEAM, 1, NFLOOR, 1 YEBEAM = EBEAM*FHT LSEL, A, LOC, Y, YEBEAM, YEBEAM LSEL, R, LOC, Z, NBAY*BAY, NBAY*BAY *ENDDO LESIZE, ALL, , , LDIV, , 1 LMESH, ALL !FACE X=0 LSEL, NONE

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116

*DO, EBEAM, 1, NFLOOR, 1 YEBEAM = EBEAM*FHT LSEL, A, LOC, Y, YEBEAM, YEBEAM LSEL, R, LOC, X, 0, 0 *ENDDO LESIZE, ALL, , , LDIV, , 1 LMESH, ALL !FACE X=SPANL LSEL, NONE *DO, EBEAM, 1, NFLOOR, 1 YEBEAM = EBEAM*FHT LSEL, A, LOC, Y, YEBEAM, YEBEAM LSEL, R, LOC, X, NSPAN*SPANL, NSPAN*SPANL *ENDDO LESIZE, ALL, , , LDIV, , 1 LMESH, ALL !**interior beam meshing** LSEL, NONE *DO, IBEAM, 1, NFLOOR, 1 YIBEAM = IBEAM*FHT LSEL, A, LOC, Y, YIBEAM, YIBEAM LSEL, U, LOC, X, 0, 0 LSEL, U, LOC, Z, 0, 0 LSEL, U, LOC, X, NSPAN*SPANL, NSPAN*SPANL LSEL, U, LOC, Z, NBAY*BAY, NBAY*BAY *ENDDO LESIZE, ALL, , , LDIV, , 1 LMESH, ALL ALLSEL, ALL !**column meshing** !CORNER Column: !corner col1 LSEL, NONE LSEL, S, LOC, X, 0, 0 LSEL, R, LOC, Z, 0, 0 LESIZE, ALL, , , LDIV, , 1 TYPE, 1 MAT, 1

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117

REAL, 3 LMESH, ALL ALLSEL, ALL LSEL, NONE !corner col2 LSEL, S, LOC, X, NSPAN*SPANL, NSPAN*SPANL LSEL, R, LOC, Z, 0, 0 LESIZE, ALL, , , LDIV, , 1 LMESH, ALL ALLSEL, ALL LSEL, NONE !corner col3 LSEL, S, LOC, X, NSPAN*SPANL, NSPAN*SPANL LSEL, R, LOC, Z, NBAY*BAY, NBAY*BAY LESIZE, ALL, , , LDIV, , 1 LMESH, ALL ALLSEL, ALL LSEL, NONE !corner col4 LSEL, S, LOC, X, 0, 0 LSEL, R, LOC, Z, NBAY*BAY, NBAY*BAY LESIZE, ALL, , , LDIV, , 1 LMESH, ALL ALLSEL, ALL !EDGE COLUMN: !FACE X=SPANL LSEL, NONE *DO, ECOL, 1, NBAY, 1 XECOL = ECOL*BAY LSEL, A, LOC, Z, XECOL, XECOL LSEL, R, LOC, X, NSPAN*SPANL, NSPAN*SPANL LSEL, R, LOC, Z, BAY, BAY*(NBAY-1) *ENDDO TYPE, 1 MAT, 1 REAL, 2 LESIZE, ALL, , , LDIV, , 1 LMESH, ALL !FACE X=0 LSEL, NONE *DO, ECOL, 1, NBAY, 1

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118

XECOL = ECOL*BAY LSEL, A, LOC, Z, XECOL, XECOL LSEL, R, LOC, X, 0, 0 LSEL, R, LOC, Z, BAY, BAY*(NBAY-1) *ENDDO LESIZE, ALL, , , LDIV, , 1 LMESH, ALL !Z=0 FACE LSEL, NONE *DO, EC, 1, NSPAN, 1 ZEC = EC*SPANL LSEL, A, LOC, X, ZEC, ZEC LSEL, R, LOC, Z, 0, 0 LSEL, R, LOC, X, SPANL, SPANL*(NSPAN-1) *ENDDO LESIZE, ALL, , , LDIV, , 1 LMESH, ALL !Z=BAY FACE LSEL ,NONE *DO, EC, 1, NSPAN, 1 ZEC = EC*SPANL LSEL, A, LOC, X, ZEC, ZEC LSEL, R, LOC, Z, BAY*NBAY, BAY*NBAY LSEL, R, LOC, X, SPANL, SPANL*(NSPAN-1) *ENDDO LESIZE, ALL, , , LDIV, , 1 LMESH, ALL !**interior column meshing** LSEL, NONE *DO, ICOL1, 1, NSPAN, 1 XCOL = ICOL1*SPANL LSEL, A, LOC, X, XCOL, XCOL *DO, IC, 1, NFLOOR, 1 YC = IC*FHT LSEL, U, LOC, Y, YC, YC LSEL, U, LOC, Y, 0, 0 LSEL, U, LOC, Z, 0, 0 LSEL, U, LOC, X, 0, 0 LSEL, U, LOC, Z, NBAY*BAY, NBAY*BAY LSEL, U, LOC, X, NSPAN*SPANL, NSPAN*SPANL *ENDDO

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*ENDDO LESIZE, ALL, , , LDIV, , 1 TYPE, 1 MAT, 1 REAL, 1 LMESH, ALL ALLSEL, ALL NUMMRG, NODE !**mass element generation** ET, 3, MASS21 MP, DENS, 3, DENSITY R, 6, MASSIN, MASSIN, MASSIN !mass per interior node R, 7, MASSEN, MASSEN, MASSEN !mass per edge node R, 8, MASSCN, MASSCN, MASSCN !mass per corner node ALLSEL, ALL ! INTERNAL MASS ELEMENT TYPE, 3 MAT, 3 REAL, 6 *DO, IN, 1, NFLOOR, 1 YY = IN*FHT NNSPAN = (ADIV*NSPAN) + 1 !total no of node along span *DO, N1, 2, NNSPAN-1, 1 XX = (N1-1)*X1 NNBAY = (ADIV*NBAY) + 1 !total no of node along span *DO, N2, 2, NNBAY-1, 1 ZZ = (N2-1)*Z1 E, NODE (XX, YY, ZZ) *ENDDO *ENDDO *ENDDO ALLSEL, ALL ! EDGE MASS ELEMENT

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TYPE, 3 MAT, 3 REAL, 7 *DO, EN, 1, NFLOOR, 1 YY = EN*FHT NNSPAN = (ADIV*NSPAN) + 1 NNBAY = (ADIV*NBAY) + 1 XX=0 *DO, N3, 2, NNBAY-1, 1 ZZ = (N3-1)*Z1 E, NODE (XX, YY, ZZ) *ENDDO XX = NSPAN*SPANL *DO, N3, 2, NNBAY-1, 1 ZZ = (N3-1)*Z1 E, NODE(XX, YY, ZZ) *ENDDO ZZ=0 *DO, N3, 2, NNSPAN-1, 1 XX = (N3-1)*X1 E, NODE (XX, YY, ZZ) *ENDDO ZZ = NBAY*BAY *DO, N3, 2, NNSPAN-1, 1 XX = (N3-1)*X1 E, NODE (XX, YY, ZZ) *ENDDO *ENDDO ! CORNER MASS ELEMENT ALLSEL, ALL TYPE, 3 MAT, 3 REAL, 8

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*DO, EN, 1, NFLOOR, 1 YY = EN*FHT E, NODE (0, YY, 0) E, NODE (NSPAN*SPANL, YY, 0) E, NODE (NSPAN*SPANL, YY, NBAY*BAY) E, NODE (0, YY, NBAY*BAY) *ENDDO ALLSEL, ALL /SOLU NSEL, S, LOC, Y, 0, 0 D, ALL, ALL, 0 NSEL, NONE !corner col1 NSEL, S, LOC, X, 0, 0 NSEL, R, LOC, Z, 0, 0 *DO, CC1, 1, NFLOOR, 1 YY = CC1*FHT NN = NODE (0, YY, 0) D, NN, UY, 0 *ENDDO NSEL, NONE !corner col2 NSEL, S, LOC, X, NSPAN*SPANL, NSPAN*SPANL NSEL, R, LOC, Z, 0, 0 *DO, CC2, 1, NFLOOR, 1 YY = CC2*FHT NN = NODE (0, YY, 0) D, NN, UY, 0 *ENDDO NSEL, NONE !corner col3 NSEL, S, LOC, X, NSPAN*SPANL, NSPAN*SPANL NSEL, R, LOC, Z, NBAY*BAY, NBAY*BAY *DO, CC3, 1, NFLOOR, 1 YY = CC3*FHT NN = NODE (0, YY, 0) D, NN, UY, 0 *ENDDO

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NSEL, NONE !corner col4 NSEL, S, LOC, X, 0, 0 NSEL, R, LOC, Z, NBAY*BAY, NBAY*BAY *DO, CC4, 1, NFLOOR, 1 YY = CC4*FHT NN = NODE (0, YY, 0) D, NN, UY, 0 *ENDDO NSEL, NONE !edge col *DO, ECOL, 1, NSPAN, 1 !(Z=0 FACE) ZECOL = ECOL*SPANL NSEL, S, LOC, X, ZECOL, ZECOL NSEL, R, LOC, Z, 0, 0 *DO, ECC1, 1, NFLOOR, 1 YY = ECC1*FHT NN = NODE (0, YY, 0) D, NN, UY, 0 *ENDDO *ENDDO NSEL, NONE !edge col *DO, ECOL, 1, NSPAN, 1 !(z=bay face) ZECOL = ECOL*SPANL NSEL, S, LOC, X, ZECOL, ZECOL NSEL, R, LOC, Z, NBAY*BAY, NBAY*BAY *DO, ECC2, 1, NFLOOR, 1 YY = ECC2*FHT NN = NODE (0, YY, 0) D, NN, UY, 0 *ENDDO *ENDDO NSEL, NONE !EDGE COL *DO, ECOL, 1, NBAY, 1 !(X=0 FACE) XECOL = ECOL*BAY NSEL, S, LOC, Z, XECOL, XECOL NSEL, R, LOC, X, 0, 0 *DO, ECC3, 1, NFLOOR, 1 YY = ECC3*FHT NN = NODE (0, YY, 0) D, NN, UY, 0

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*ENDDO *ENDDO NSEL, NONE !edge col *DO, ECOL, 1, NBAY, 1 !(x=spanl face) XECOL = ECOL*BAY NSEL, S, LOC, Z, XECOL, XECOL NSEL, R, LOC, X, NSPAN*SPANL, NSPAN*SPANL *DO, ECC4, 1, NFLOOR, 1 YY = ECC4*FHT NN = NODE (0, YY, 0) D, NN, UY, 0 *ENDDO *ENDDO NSEL, NONE !interior column support *DO, ICOL, 1, NSPAN, 1 XC = (ICOL)*SPANL *DO, IC, 1, NBAY, 1 ZC = IC*BAY NSEL, S, LOC, X, XC, XC NSEL, R, LOC, Z, ZC, ZC NSEL, U, LOC, X, 0, 0 NSEL, U, LOC, Z, 0, 0 NSEL, U, LOC, Z, NBAY*BAY, NBAY*BAY NSEL, U, LOC, X, NSPAN*SPANL, NSPAN*SPANL NSEL, U, LOC, Y, 0, 0 D, NODE (0, FHT, 0), UY, 0 *ENDDO *ENDDO !**modal analysis** ANTYPE, 2 MODOPT, LANB, 4 MXPAND, , , , YES, 1.0E-5 ALLSEL, ALL SOLVE FINISH /POST1 /VIEW, 1, 1, 1, 1

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/ANG, 1 /REP, FAST ESEL, U, ENAME, , MASS21 *DO, ISET, 1, 4 SET, 1, ISET PLDISP, 1 *ASK, YN, PRESS ENTER *IF, YN, EQ, 1.0, THEN *GO, :END *ENDIF *ENDDO LCWRITE, 1 FINISH