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
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)
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.
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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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
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
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
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
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
x
β Modal Damping Ratio
β Floor panel aspect ratio
ρ Reinforcement ratio
Chapter - 1
INTRODUCTION
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:
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.
Introduction
4
Fig 1.2: Millennium Bridge, London (2000)
Fig 1.3: Collapse of Tacoma Narrows Bridge (1940)
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.
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
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
CHAPTER – 2
LITERATURE REVIEW
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.
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)
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
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.
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
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.
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.
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.
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.
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:
Literature Review
19
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:
Literature Review
20
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.
Literature Review
21
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
Literature Review
22
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 )
Literature Review
23
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
Literature Review
24
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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31
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.
Chapter - 3
DEVELOPMENT OF FINITE
ELEMENT MODEL
Development of Finite Element Model
35
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:
Development of Finite Element Model
36
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
Development of Finite Element Model
37
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.
Development of Finite Element Model
38
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
Development of Finite Element Model
39
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.
Development of Finite Element Model
40
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.
Development of Finite Element Model
41
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).
Development of Finite Element Model
42
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.
Development of Finite Element Model
43
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.
Development of Finite Element Model
44
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
Development of Finite Element Model
45
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.
Development of Finite Element Model
46
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
Development of Finite Element Model
47
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
Development of Finite Element Model
48
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
Development of Finite Element Model
49
Fig 3.6: Plan View of the Model
Fig 3.7: Elevation of the Model
Development of Finite Element Model
50
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:
Development of Finite Element Model
51
Fig 3.9: 1st Mode Shape, frequency 5.43 Hz (3D)
Fig 3.10: 1st Mode Shape, frequency 5.43 Hz (elevation)
Development of Finite Element Model
52
Fig 3.11: 2nd
Mode Shape, frequency 5.44 Hz (3D)
Fig 3.12: 2nd
Mode Shape, frequency 5.44 Hz (elevation)
Development of Finite Element Model
53
Fig 3.13: 3rd
Mode Shape, frequency 6 Hz (3D)
Fig 3.14: 3rd
Mode Shape, frequency 6 Hz (elevation)
Development of Finite Element Model
54
Fig 3.15: 4th
Mode Shape, frequency 16.87 Hz (3D)
Fig 3.16: 4th
Mode Shape, frequency 16.87 Hz (elevation)
Development of Finite Element Model
55
Fig 3.17: 10th
Mode Shape, frequency 19.38 Hz (3D)
Fig 3.18: 10th
Mode Shape, frequency 19.38 Hz (elevation)
Development of Finite Element Model
56
Fig 3.19: 15th
Mode Shape, frequency 30 Hz (3D)
Fig 3.20: 15th
Mode Shape, frequency 30 Hz (elevation)
Development of Finite Element Model
57
Fig 3.21: 20th
Mode Shape, frequency 34.18 Hz (3D)
Fig 3.22: 20th
Mode Shape (elevation)
Development of Finite Element Model
58
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.
Chapter - 4
VERIFICATION OF THE
MODEL
Verification of the Model
60
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)
Verification of the Model
61
= 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
Verification of the Model
62
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.
Verification of the Model
63
Fig 4.2: ANSYS result, 4th
mode frequency 22.295 Hz (3D)
Fig 4.3: ANSYS result, 4th
mode frequency 22.295 Hz (3D)
Verification of the Model
64
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
Verification of the Model
65
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).
Chapter - 5
PARAMETRIC STUDY AND
RESULTS
Parametric Study And Results
67
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, ∑
Parametric Study And Results
68
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
Parametric Study And Results
69
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.
Parametric Study And Results
70
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
Parametric Study And Results
71
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
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
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
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
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
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
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
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
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
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
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
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
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
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.
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:
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)
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)
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)
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.
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
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
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
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
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
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
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
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
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
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
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.
Chapter – 6
CONCLUSIONS
AND
RECOMMENDATIONS
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.
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.
104
REFERENCES:
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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.
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
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Rakib M. (2013),“Minimum slab thickness of RC slab to prevent undesirable floor
vibration”, Chapter 3 and 4.
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Setareh M. (2010), “Vibration Serviceability Of A Building Floor Structure I:
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Setareh M. (2010), “Vibration Serviceability Of A Building Floor Structure II:
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5, The Steel Construction Institute, Berkshire, England.
APPENDIX
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
Appendix
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
Appendix
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
Appendix
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
Appendix
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
Appendix
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
Appendix
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
Appendix
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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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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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