modelling floors with a constrained damping layer

MODELLING FLOORS WITH A CONSTRAINED

DAMPING LAYER

A thesis submitted to The University of Manchester for the degree of

Master of Philosophy

In Faculty of Engineering and Physical Sciences

2011

JOSEPH ANTAR

SCHOOL OF MECHANICAL, AEROSPACE & CIVIL

ENGINEERING

2

TABLE OF CONTENTS

ABSTRACT ...............................................................................................................................9

DECLARATION .....................................................................................................................10

COPYRIGHT STATEMENT ................................................................................................10

ACKNOWLEDGEMENTS ....................................................................................................11

INTRODUCTION ...................................................................................................................12

1.1 INTRODUCTION .............................................................................................................. 12

1.2 SCOPE ............................................................................................................................ 13

CHAPTER 2 LITERATURE REVIEW ...............................................................................14

2.1 INTRODUCTION ........................................................................................................ 14

2.2 LONG-SPAN CONCRETE FLOOR TYPES AND CHARACTERISICS .................. 16

2.3 HUMAN-INDUCED DYNAMIC LOADS ON FLOORS ........................................... 18

2.3.1 Walking ................................................................................................................ 19

2.3.2 Running / Jumping ............................................................................................... 23

2.3.3 Dancing / Aerobics ............................................................................................... 25

2.4 HUMAN RESPONSE TO THE STRUCTURE VIBRATIONS AND THE

CURRENT METHODS USED IN THE DESIGNED CODES ........................................... 26

2.4.1 Frequency weightings .......................................................................................... 26

2.4.2 RMS Acceleration................................................................................................. 27

2.4.2 Vibration dose value ............................................................................................ 28

2.5 DESIGN CRITERIA AGAINST FLOOR VIBRATIONS ........................................... 30

2.5.1 Acceleration limits ............................................................................................... 30

2.5.1.1 Acceleration limits for walking excitation ................................................ 31

2.5.1.2 Acceleration limits for rhythmic excitation ............................................... 34

2.5.2 Response factor method ....................................................................................... 36

2.5.3 Assessment of vibration design criteria ............................................................... 37

2.6 DETERMINATION OF NATURAL FREQUENCY .................................................. 38

2.6.1 General approaches ............................................................................................. 38

2.6.2 Sophisticated approach ........................................................................................ 41

2.7 EVALUATION OF DAMPING AND DAMPING COEFFICIENTS ......................... 42

2.8 REMEDIAL MEASURES AGAINST FLOOR VIBRATION .................................... 43

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2.9 DAMPING DEVICES FOR VIBRATION CONTROL............................................... 45

2.9.1 The use of passive damping devices ..................................................................... 45

2.9.1.1 Tuned Mass Dampers ................................................................................ 46

2.9.1.2 Passive control using advanced materials ................................................. 47

2.10 FINITE ELEMENT ANALYSIS ................................................................................ 49

2.10.1 Finite elements.................................................................................................. 51

2.10.2 Analysis techniques ........................................................................................... 52

2.10.2.1 Linear static analysis ............................................................................... 52

2.10.2.2 Non-linear static analyses ....................................................................... 52

2.10.2.3 Dynamic analysis .................................................................................... 53

2.11 SUMMARY ................................................................................................................. 55

CHAPTER 3 ENGINEERING PROPERTIES AND MODELLING OF RUBBER .......56

3.1 ENGINEERING PROPERTIES OF RUBBER ............................................................ 56

3.1.1 Preliminary remarks ............................................................................................ 56

3.1.2 Hyperelasticity and viscoelasticity of rubber ....................................................... 56

3.1.2.1 General theory of large elastic deformations1 ........................................... 57

3.1.2.2 Viscoelastic behavior ................................................................................ 58

3.1.3 Energy dissipation in rubber ................................................................................ 59

3.1.3.1 Friction ...................................................................................................... 59

3.1.3.2 Stress softening ......................................................................................... 60

3.1.3.3 Crystallization ........................................................................................... 60

3.1.3.4 Structural breakdown ................................................................................ 60

3.2.4 Rubber compounding ........................................................................................... 60

3.1.4.1 Sulfur curing .............................................................................................. 61

3.1.4.2 Filler systems............................................................................................. 61

3.1.5 Mechanical aspects of high damping rubber ....................................................... 62

3.1.6 Summary ............................................................................................................... 64

3.2 RUBBER MODELLING IN ABAQUS ....................................................................... 64

3.2.1 Hyperelasticity ..................................................................................................... 65

3.2.1.1 Hyperelastic materiel models .................................................................... 65

3.2.1.2 Modeling the Hyperelasticity in ABAQUS .............................................. 68

3.2.1.3 Modeling the Hyperelasticity of Rubber layer .......................................... 71

3.2.2 Viscoelasticity ...................................................................................................... 74

3.2.2.1 Viscoelastic Model .................................................................................... 75

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3.2.2.2 Modeling the Viscoelasticity in ABAQUS ............................................... 77

3.2.2.3 Modeling the Viscoelasticity of Rubber layer........................................... 77

CHAPTER 4 MODELLING THE CONSTRAINED DAMPING LAYER ......................79

4.1 INTRODUCTION ........................................................................................................ 79

4.2 CONSTRAINED LAYER MODELLING USING FINITE ELEMENT ..................... 80

4.3 FINITE ELEMENT MODELLING ............................................................................. 82

4.3.1 Finite element geometric models properties ........................................................ 83

4.3.2 Material properties ............................................................................................. 86

4.3.3 Types of Elements ............................................................................................... 86

4.3.3.1 Shell elements ........................................................................................... 87

4.3.3.2 3-D Solid continuum elements .................................................................. 88

4.3.3.3 Hybrid elements ........................................................................................ 89

4.4 DAMPING FOR FR MODELS .................................................................................... 90

4.4.1 Calculation of mass and stiffness proportional damping ................................... 91

4.5 MODELLING THE FLAT CONCRETE PLATE ........................................................ 93

4.6 RESULTS ..................................................................................................................... 99

4.6.1 Method of analysis .............................................................................................. 99

4.6.1.1 Eigenvalue analysis ................................................................................... 99

4.6.1.2 Simulation of forced vibration test .......................................................... 103

4.6.1.3 Case study ............................................................................................... 104

4.6.2 Effect of the slab thickness in relation to the rubber layer ............................... 106

4.6.3 Effect of the slab damping ................................................................................ 111

4.6.4 Effect of protection layer .................................................................................. 116

4.6.5 Discussion ......................................................................................................... 118

CHAPTER 5 CONCLUSION & FURTHER WORK .......................................................119

5.1 CONCLUSION .............................................................................................................. 119

5.2 FURTHER WORK RECOMMENDATIONS ......................................................................... 121

REFERENCES ......................................................................................................................122

APPENDIX A ........................................................................................................................129

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List of Tables

Table 2.1 pacing rate, pedestrian propagation and stride length for walking ................. 21

Table 2.2 Pacing rates for different events...................................................................... 21

Table 2.3 Fourier coefficients for walking (Bachmann, 1987) ...................................... 22

Table 2.4 Pacing rate, pedestrian propagation and stride length for running events

(Wheeler, J.E. 1982)........................................................................................................ 23

Table 2.5 VDVs at which various degrees of adverse comment may be expected (BS

6472) ............................................................................................................................... 29

Table 2.6 estimated loading during rhythmic events (Murray, Allen et al. 1997) .......... 35

Table 2.7 Response factor R for offices (Wyatt, 1989) .................................................. 37

Table 3.1 Ogden strain energy function with N=3 parameters ....................................... 73

Table 3.2 Polynomial strain energy function with N=2 parameters ............................... 73

Table 4.1 Main features of the composite FE models ..................................................... 82

Table 4.2 Geometric properties of the concrete plate, rubber and protection layers ...... 84

Table 4.3 Geometric properties of all the FE models cases ........................................... 85

Table 4.4 Material properties .......................................................................................... 86

Table 4.5 Damping levels used for the Concrete plate behaviour .................................. 90

Table 4.6 Mass proportional and stiffness proportional damping for the four concrete

plates ............................................................................................................................... 93

Table 4.8 Effect of meshing on the natural frequencies of the models ........................... 97

Table 4.9: Natural Frequency of all the FE models ...................................................... 101

Table 4.10 Natural Frequencies, displacement and damping of model 16………..…...104

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List of Figures

Figure 2.1 Floor spans achievable by in situ reinforced and prestressed concrete for

office loading of 5 (Stevenson, 1994). ................................................................ 17

Figure 2.2 Typical forcing patterns for walking (Galbraith & Barton, 1970)................. 19

Figure 2.3 Time history patterns for various modes of walking and jumping/running

excitation (wheeler, 1982) ............................................................................................... 20

Figure 2.4 Harmonic components of the walking force in (a) vertical, (b) lateral and (c)

longitudinal directions (Bachmann and Ammann, 1987) ............................................... 22

Figure 2.5 Idealized load time function for running and jumping (a) half sine model (b)

impact factor for depending on contact duration ratio (Bachmann.H and Ammann. W,

1987). .............................................................................................................................. 24

Figure 2.6 Weighted z-axis vibration RMS acceleration (BS 6841) .............................. 27

Figure 2.7 Modified Reiher-Meister Scales .................................................................... 37

Figure 2.8 Recommended peak acceleration limits design chart (Murray, Allen et al.

1997) ............................................................................................................................... 33

Figure 2.9 Canadian floor vibration perceptible scales .................................................. 34

Figure 2.10 floor mass–drop displacement responses (Ebrahimpour, 2005) .................. 47

Figure 2.11 Frequency response for the original floor Figure 2.12

Frequency response for the damped floor ....................................................................... 48

Figure 2.13 Finite elements used by ABAQUS .............................................................. 51

Figure 3.1 Maxwell and Voigt models for viscoelasticity .............................................. 58

Figure 3.2 Uniaxial tension-compression tests on HDR (Amin, 2002) .......................... 62

Figure 3.3 Relaxation tests on HDR ( Yoshida, 2004). .................................................. 62

Figure 3.4 Typical stress-strain curve for hyperelastic material ..................................... 64

Figure 3.5 Schematic of deformations in different test used to model hyperelasticity ... 68

Figure 3.6 Unixial tension-compression test on high damping rubber (Yoshida, 2004) 71

Figure 3.7 Biaxial test on high-damping rubber (Yoshida, 2004) .................................. 71

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Figure 3.8 Uniaxial model of ABAQUS ......................................................................... 72

Figure 3.9 Biaxial model of ABAQUS ........................................................................... 73

Figure 3.10 Creep and recovery for a viscoelastic material ............................................ 74

Figure 3.11 Relaxation shear test on high-damping rubber (Yoshida, 2004) ................ 77

Figure 3.12 Relaxation model of ABAQUS ................................................................... 77

Figure 4.1 The plate with constrained damping layer ..................................................... 79

Figure 4.2 The three FE models used to model the damping layer................................. 80

Figure 4.3 Cross section of the full FE model ................................................................ 83

Figure 4.4: Boundary conditions ..................................................................................... 86

Figure 4.5: 8 node continuum shell (SC8R).................................................................... 87

Figure 4.6 : 8-node continuum solid elements (C3D8) ................................................... 88

Figure 4.7 Shape of the fully interpolation first order element under the moment ......... 89

Figure 4.8 Shape of the fully interpolation first order element under the moment ......... 89

Figure 4.9 3-D solid elements models with different number of element through

thickness .......................................................................................................................... 96

Figure 4.10 First mode of shape of the model 16 ......................................................... 103

Figure 4.11 Second mode of shape of the model 16 ..................................................... 103

Figure 4.12 The natural frequencies of the concrete plate treated with the constrained

damping layer as a function of the damping layer thickness ........................................ 102

Figure 4.13 The half-power bandwidth method to calculate the damping ................... 103

Figure 4.14 Frequency responses for the concrete plate 150 mm with 1.6% damping

ratio ............................................................................................................................... 105

Figure 4.15 Frequency responses for the130 mm concrete plate with 1.6% damping

ratio damped with 2.5 mm rubber layer and 50 mm protection layer ........................... 105

Figure 4.16 Model damping of the FE models as a function of the thickness of the

damping layer with concrete base damping 1.6% ......................................................... 107

Figure 4.17 Model damping of the FE models as a function of the thickness of the


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Figure 4.18 : Model damping of the FE models as a function of the thickness of the


Figure 4.19 : Model damping of the FE models as a function of the thickness of the

damping layer with concrete base damping 12.0% ....................................................... 110

Figure 4.20 The percentage of reduction in the displacement response for a 130 mm

concrete plate and 50 mm protection layer ................................................................... 112







Figure 4.24 Percentage of reduction in the displacement response for a 150 mm concrete

plate and 5 mm rubber................................................................................................... 117

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ABSTRACT

People walking across modern large span floors with low damping can generate

vibrations that may prove uncomfortable for other users. This leads to a serviceability

problem that needs to be considered in design. The largest vibrations are produced when

the floor‟s fundamental frequency is an integer multiple of the walkers pacing frequency

thus producing a resonance phenomenon. Conventional engineering knowledge

addresses vibration problems by increasing stiffness; however, this study indicated that

these problems may be resolved much more efficiently by increasing the damping

perhaps through the use of the constrained damping layer technique. This forms the

background of this research.

This thesis describes the use of Finite Element Analysis in predicting the dynamic

behaviour of floors with a constrained damping layer. Three-dimensional finite element

models were developed for the constrained damping layer using the finite element

package, ABAQUS. The hyperelastic and viscoelastic behaviour were considered to

represent the material properties of the rubber layer. Due to the complexity of the

materials and the contact between the different layers that represent the full model, three

different types of simplified finite element models are used to model the system

proposed. The device dissipates energy through amplified strains in the viscoelastic

rubber. The models were analysed under dynamic loadings to understand the dynamic

behaviour. A parametric study was carried out on the constrained damping layer to

investigate the effect of different parameters such as concrete base, rubber layer and

protection layer on the overall dynamic performance of this system.

This thesis with the title Modelling Floors with a Constrained Damping Layer is

submitted to the University of Manchester by Joseph Antar for the degree of Master of

Philosophy in 2011.

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DECLARATION

No portion of the work referred to in the thesis has been submitted in support of

an application for another degree or qualification of this or any other university or other

institute of learning;

COPYRIGHT STATEMENT

i. The author of this thesis (including any appendices and/or schedules to this

thesis) owns certain copyright or related rights in it (the “Copyright”) and

s/he has given The University of Manchester certain rights to use such

Copyright, including for administrative purposes.

ii. Copies of this thesis, either in full or in extracts and whether in hard or

electronic copy, may be made only in accordance with the Copyright,

Designs and Patents Act 1988 (as amended) and regulations issued under it

or, where appropriate, in accordance with licensing agreements which the

University has from time to time. This page must form part of any such

copies made.

iii. The ownership of certain Copyright, patents, designs, trademarks and other

intellectual property (the “Intellectual Property”) and any reproductions of

copyright works in the thesis, for example graphs and tables

(“Reproductions”), which may be described in this thesis, may not be owned

by the author and may be owned by third parties. Such Intellectual Property

and Reproductions cannot and must not be made available for use without

the prior written permission of the owner(s) of the relevant Intellectual

Property and/or Reproductions.

iv. Further information on the conditions under which disclosure, publication

and exploitation of this thesis, the Copyright and any Intellectual Property

Rights and/or Reproductions described in it may take place is available from

the head of School of Mechanical, Aerospace and Civil engineering and for

its candidates.

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ACKNOWLEDGEMENTS

First, I am heartily thankful to my supervisor,Tianjian Ji, whose encouragement,

guidance and support from the initial to the final level enabled me to develop an

understanding of the subject.

Afterwards, I would like to thank my family for supporting me. Without their

help and support, I would probably not be able to make true my dreams. Therefore, I

would like to dedicate to my family this piece of work.

Chapter 1:Introduction

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Introduction

1.1 Introduction

Structural designers are faced with a variety of tasks in the design of modern-day

structure. Safety is clearly an area of paramount concern; buildings and bridges must be

able to carry all design loads with a reasonable factor of safety. In addition to safety, of

equal importance, serviceability is the structure designed to execute its function within

defined standards of performance? One such serviceability problem faced by structural

designers is that of motion control. Specifically, motion in the form of floor vibration, a

problem in horizontal structures exacerbated by lengthy column-free spans, is

considered to be one of the most common and persistent serviceability issues

encountered by today‟s structural designers.

The issue of floor vibration is relevant to a wide variety of structures. Such structures

prone to vibration-related problems include pedestrian bridges, manufacturing facilities,

shopping centers, health care and laboratory facilities, educational facilities, office

buildings, residential complexes, arenas and places of assembly such as convention

centers ( West and Fisher, 2003). Floor vibration due to pedestrian traffic in these

structures has been reported to be a nuisance to residents, causing an uncomfortable

work environment. In some instances occupants reported a fear for their safety in

vibration-prone structures even when engineers considered the structure to be sound.

Consequently, this problem is passed to the building owners in forms of lost rental

space and lowered commercial value of property (ASCE 7, 2002). In extreme cases,

excessive vibrations may adversely affect the productivity of sensitive manufacturing

equipment or drastically skew the accuracy of laboratory test results.

The reported floor vibration problematic in a variety of structural types has risen sharply

over the past 20 years. 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. Moreover, offices 30 or 40

years ago often contained floor-to-ceiling dividers and bulky filing cabinets, which

added damping and mass to the floor system.

It is commonplace these days to redesign such office layouts with cubicles replacing

full dividers and computers replacing full filing cabinets. As a result, the damping

Chapter 1:Introduction

13

capacity and mass of the floors system are effectively reduced, leading to reports of

vibration problems where vibrations were never before noticed (Hanagan, 2005)

1.2 Scope

This report aims to investigate the challenge of controlling motion in long span

horizontal structures. Generally, long span will be used in reference to structural

distances in excess of 6 metres. Horizontal structures considered include any structure

spanning a horizontal distance prone to vertical vibration problems, namely pedestrian

bridges and building floor systems.

This thesis is organized in five chapters of which the first is the introduction. The

literature review in the second chapter introduces a review over the present state of

knowledge in the area of floor vibration such as human induced loads, design criteria

against floor vibration, damping devices for vibration control…Chapters 3 reviews the

high damping rubber engineering properties and describes the rubber material modelling

to dissipate energy used in this thesis. Chapter 4 presents a detailed finite element

analysis conducted to predict the forced vibration response of constrained-layer damped

flat plate and investigate the effect of different parameters on the dynamic properties.

Finally, Chapter 5 summarizes the conclusion from this study and provides

recommendations for further work in the field of floor vibration control.

All the detailed results of the Finite Element analysis for all the models in terms of

natural frequencies, displacement and damping ratio are given in appendix A of the

thesis

Chapter 2: LITERATURE REVIEW

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CHAPTER 2 LITERATURE REVIEW

In civil engineering dynamics, human-induced vibrations are becoming

increasingly vital serviceability and safety issues. Numerous researchers examined these

issues and their findings of effect of dynamic loads on structures are reviewed in this

chapter. Investigations carried out on human induced loads; particularly on floor

structures, are subsequently reviewed. Then the current state of knowledge in the floor

design addressing the performance of human-induced dynamic loads and the human

response to these vibrations are also examined. The important design parameters, such

as activity/forcing frequencies, fundamental frequencies and damping and their

evaluations, are also observed. A review of predicting the fundamental frequency is also

included. The use of passive damping methods applied in mitigating the vibration

problems in floor structures is reviewed further. An introduction to finite element

methods is included in the last part of the literature review. Finally, this chapter

summarises the present state of knowledge, identifying the gaps in knowledge and the

contribution of this research to the current research in the area of floor vibration due to

human activity.

2.1 INTRODUCTION

Firstly, and most importantly, a structure has to be safe. Secondly, the safe

structure must be functional, i.e. it is able to perform satisfactorily in day-to-day service

throughout its life span. Serviceability is defined as “a state in which the function of a

building, its appearance, maintainability, durability, and comfort of its occupants are

preserved under normal usage” (Pavic, 2002). Although serviceability issues have

always been a design consideration, changes in codes and materials have added

importance to these matters. Serviceability is concerned with the day-to-day function of

the building. Violations of it mean that the owner or occupier has annoyance,

inconvenience or dissatisfaction with some aspect of the building‟s performance.

Traditional concrete designs for office building have been associated with either beam

and slab or flat slab floors, typically with 6 to 7.5 m spans. Occasionally, longer-span


15

floors have been designed using ribbed or waffle construction. In recent times, changes

in the requirements of end-users and in developers‟ specifications have led to more

open-plan offices and larger floors. This has increased spans from 6 to 9 m, even to 15

m and more which is known as long-span floor.

In recent years, there have been an increasing number of cases where serviceability

problems have occurred in floors with long spans. This problem is occurring due to the

increasing in length of the span and to the use of new types of concrete mixes such as

Fiber Reinforced Polymer and prestress techniques. Serviceability is a satisfactory

performance in service under common loadings, no cracking, no bouncing and

satisfactory appearance and function. The serviceability is concerned with deflection

and vibration where the deflection is depending on the dead load and live load carried

by the span, and vibration problems are noted after construction.

The problem of floor vibration induced by people walking is topical and, as modern

designs often produce relatively lively floors, this leads to a serviceability issue that

should be considered. The assessment of vibration is an increasingly important aspect of

structural design but in some areas serviceability requirements are poorly defined.

Serviceability related to human acceptance of vibration is an issue for many modern

structures, with problems ranging from the perception of vibration in quiet

environments to potentially intolerable vibrations generated by audiences jumping at

pop concerts (Ellis, 2000).

The problem of vibration serviceability of long-span concrete floors in buildings is

complex and interdisciplinary in nature.

Vibrations are a constitutive part of the environment and are unavoidable. In principle,

everything vibrates continuously. The problem with vibrations occurs when they

become excessive, causing annoyance, malfunction of sensitive equipment, damage or

structure failure. The human annoyance factor is, however, the most frequent vibration

serviceability problem.

Excessive floor vibrations due to human-induced loading have been characterized as

probably the most persistent floor serviceability problem encountered by designers

(Murray, 1991). Excessive floor vibration occurs when the floor system design has

inadequate stiffness, low damping and/or low mass..

In order to check the vibration serviceability of floors, two types of assessment exist :(1)

an analytical approach where the evaluation is performed by calculation at the floor

design stage, and (2) the evaluation by vibration measurement of already built full scale

floor structures (ISO 1992; Griffin,1996).


16

Problems with the vibration serviceability of floor structures have historically been

limited to lightweight types of floor construction, such as timber and composite steel-

concrete floors. However, there is a current trend for ever more slender long-span

concrete floors to be built, yielding benefits such as reduced building costs and

increased flexibility of use. This is particularly common in new office building

developments where the increased slenderness may be achieved by utilizing relatively

new technologies such as post tensioning and high-strength concrete. Unfortunately,

because these types of floor structures often have low natural frequencies that may be

excited by human pedestrian dynamic loads, it is possible that vibration serviceability

may quite easily become the critical design criterion (Pavic, 1999).

The current push towards stronger concrete materials and the use of prestressing is

resulting in increasing slenderness and liveliness of long-span concrete floors in

buildings. Although concrete floors currently have a good track record regarding their

vibration serviceability, this trend may lead to an increasing number of floors failing

their vibration serviceability (Pavic and Reynolds, 2002). This is particularly so because

there was lack of research in the vibration performance of this particular floor type

where the major focus of this thesis will be on the serviceability of long floor spans due

to walking loads.

2.2 LONG-SPAN CONCRETE FLOOR TYPES AND

CHARACTERISICS

From a construction point of view, suspended floors are usually made entirely of

concrete or of a combination of concrete and other construction materials, such as steel

or timber. Cast in situ concrete floors typically of elements made entirely of concrete

poured onto formwork on site together with the rest of the supporting building frame.

Such floors are also known as cast in-place or, simply, in situ concrete floors. On the

other hand, composite floors are a combination of a system of beams and building frame

which is made of concrete, steel or timber acting with a concrete slab or topping which

forms the floor surface.

The in situ concrete floors which are typically either reinforced or prestressed post-

tensioned, where the long-spans floors belong to the prestressed post tensioned type

because the reinforced concrete floors spans are limited to a certain range which cannot

exceed 8 m in the flat slab (Figure 2.1) and the long-spans under study in this thesis are

much larger and maybe exceed 20m.


17

The main advantage of utilizing post-tensioning in situ concrete slabs is the reduction in

thickness which leads to smaller column and foundation sizes. In practice, PT floors can

accommodate 50% greater spans than their classically reinforced counterparts of similar

thickness (Figure 2.1).

Figure 2.1 Floor spans achievable by in situ reinforced and prestressed concrete for office

loading of 5 (Stevenson, 1994).

However, as modern suspended floor structures made entirely of concrete become

increasingly slender, problems associated with occupant-induced vibrations are also

becoming a very important design issue. In particular, concerns are now being raised

about the vibration serviceability performance of post-tensioned (PT) concrete floors

since prestressing permits relatively light and flexible long span solutions (Pavic and

Reynolds, 2002).

Long-span PT floors constructed from internally prestressed in situ concrete may be up

to 30% thinner than the slabs containing normal unstressed reinforcement. In addition,

the spans of PT floors with band beams can be up to 70% greater than those made of

normal reinforced concrete, as shown in Figure 2.1. Although such slender slabs may be

designed to have sufficient strength, this reduced floor depth leads to a drastic reduction

in stiffness which could give rise to structural serviceability problems, such as

unacceptable levels of floor deflection, vibration or cracking. Concrete cracking and

excessive static deflection in a prestressed PT slab can be overcome to a large extent by

the careful choice of the amount and location of the prestress members. No amount of

prestressing, however, will significantly improve the floor dynamic behaviour since this

is governed largely by slab stiffness, mass and damping on which different levels of

prestressing do not have major influence. Therefore, although other serviceability

IN-S

ITU

Flat slab

Ribbed slab

Band beam & slab

Reinforced concrete

Prestressed concrete

6 8 10 12 14 16 18 20Span (m)


18

design requirements can generally be satisfied, vibration serviceability for PT floors

remains as a potential problem (Pavic and Reynolds, 2002).

In addition to increased slenderness and longer floor spans, building owners and

developers are increasingly specifying uninterrupted open-space environments with

little or no permanent partitioning. When compared with partitioned layouts, the

damping in unpartitioned floors is commonly considered to be lower. The introduction

of large open-plan offices `might harm‟ and is `bad practice‟ with regard to floor

vibration serviceability. However, the trend towards such practice continues, and this

may further impair the vibration performance of slender concrete floors (Bolton, 1994).

Unlike composite steel-concrete and timber floors, cast in situ concrete floors, which are

used widely in office construction, are heavier and have an excellent track record with

regard to their past vibration problems (Khan and Williams, 1995). Indeed, complaints

about their vibration behaviour are rare, in fact, almost non-existent. Therefore, the

focus of researchers in the past has been mainly on the lighter and more lively

composite steel-concrete and timber floors. However, an ongoing push to utilising

prestressed post-tensioned floors for longer more slender spans could result in similar

problems of excessive liveliness.

Designers of post-tensioned concrete floors world-wide are currently aware of the

potential problem if relatively long post-tensioned floor spans are required. However, it

has also been widely recognized that checking and assessing the vibration serviceability

of post-tensioned floors is, at the moment, far from being a routine design procedure.

2.3 HUMAN-INDUCED DYNAMIC LOADS ON FLOORS

Resonance vibration problems can occur in many types of structures such as

bridge structures or floor structures. The most problematic response is due to the

human-induced vibration reported in office building floors, shopping mall floors,

aerobic dance floors, gymnasiums and parking floors (Hanagan, 2003). Laboratory floor

structures which use sensitive laboratory equipment are among those, vibrations of

which caused by football are problematic (Ungar and White, 1979).

All movements by people result in a fluctuating reaction force on floors. For example,

the simplest walking causes a modest cyclic change in the height of the body mass

above the floor, and the product of the mass with the respective accelerations equates to

a cyclic force. Similarly, human actions including jumping, running and performing

aerobics on floors causes these cyclic forces, which create vibration problems.


19

Periodic or harmonic load functions have been used to describe the human actions and

the respective forces. The periodic or harmonic dynamic loads are repeated time and

nature of human induced dynamic loads depends on many factors, such as pacing rate,

floor type or surface condition, person‟s weight, type of footwear and person‟s gender.

Willford, (2001) showed that the dynamic loads induced by crowd jumping depends on

the factors describing how energetic their dancing is and the quality of their

coordination. Thus, it is clear that there are several factors, which contribute to vibration

of slender floors in response to human actions.

A lot of research had been done to obtain graphical and mathematical formulae to

understand human-induced loads. Galbraith et al. (1970), Wheeler (1982) and Bachman

et al. (1987) made research contributions to express the human-induced dynamic loads

in terms of different human actions.

Figure 2.2 Typical forcing patterns for walking (Galbraith & Barton, 1970).

2.3.1 Walking

The human motion of walking gives rise to considerable dynamic loading which

causes vibrations in slender floor structures (Ellis, 2001). The first known work dealing

the forces caused by walking excitation was from Harper (1962) as referenced in Pavic

and Reynolds (2002). Afterwards, Galbraith et al. (1970), Wheeler (1982) and Eriksson

(1994) published their work related to walking excitation forces acting on various

structures.


20

Galbraith et al.‟s (1970) involvement in developing an intruder detection system based

on earth micro-tremors, found that the loading fluctuation due to walking has its own

periods and speeds and gave typical forcing patterns describes in Figure 2.2.

From Figure 2.2, it can be observed that human actions such as walking and running

make observable differences in force magnitudes. The forces produced in walking are

smaller than the forces produced in running and each leg overlaps the periods of

walking. Thus, continuous ground contact in walking and flying time in running events

can be clearly seen.

Wheeler (1982) gave a reasonably good graphical representation for walking excitation

caused by six different modes of human actions as seen in Figure 2.3.

Figure 2.3 Time history patterns for various modes of walking and jumping/running excitation

(wheeler, 1982)

The terms „‟Pacing Rate‟‟, „‟Forward Speed‟‟ and the „‟Stride Lengths‟‟ have been used

to describe the human actions for walking and running events. The pacing rate in

walking or running are the number of footfalls per second (FF/s) which causes the

dynamic load. At most times, the pacing rate has expressed in Hz due to the nature of

the loading. The forward speed or pedestrian propagation is the actual frontal speed

of walking measured in m/s while stride length is the distance between ground

contacts on two successive foot falls.


21

The pacing rate of individual event is the most important phenomenon to understand,

since the walking induced dynamic loads applied to the floor structures are being in

phase with the pacing rate. Wheeler (1982) derived relationships for pacing rates,

forward speeds and stride lengths by averaging test results for various walking speeds.

The following Table 2.1 describes the pacing rates, forward speeds and stride lengths of

walking action measured by Wheeler (1982). Table 2.2 presents pacing rates for

different events such as walking, jogging or sprinting.

Event Pacing rate,

(Hz)

Pedestrian

propagation, (m/s)

Stride length

(m)

Slow walk ~1.7 1.1 0.60

Normal walk ~2.0 1.5 0.75

Fast walk ~2.3 2.2 1.00

Table 2.1 pacing rate, pedestrian propagation and stride length for walking

Event Pacing rate (Hz)

Normal walk on horizontal surfaces 1.5~2.5

Normal jogging 2.4~2.7

Sprinting About 5.0

Table 2.2 Pacing rates for different events

Load-Time Function

The load-time function describes the vertical or the horizontal load exerted on a

structure. This load-time function depends upon the pacing rate, floor type or surface

condition, person‟s weight, type of footwear and / or person‟s gender.

For mathematical representation of walking, a general Fourier series expression was

produced as outlined in equation 2.1:

(2.1)

Where is the dynamic load induced from walking,

is the weight of the person in (N) (static load of the person),

Fourier‟s coefficient of the harmonic


22

activity rate (Hz)

phase angle of the harmonic

Various research workers have arrived to this model, one of the best know Bachmann

and Ammann (1987), who proposed Fourier coefficients and phase angle as given in

Table 2.3 for walking at 2 Hz. They also gave the first Fourier coefficient (0.5) for

walking at 2.4 Hz with linear interpolation for other frequencies inside the 2.0–2.4 Hz

range.

Frequency coeffcient 1st harmonic 2nd harmonic 3rd harmonic

2.0 Hz 0.4 0.1 0.1

0

Table 2.3. Fourier coefficients for walking (Bachmann, 1987).

On the other hand, Bachmann & Ammann (1987) reported the first five harmonics for

vertical walking force and also harmonics for the lateral and longitudinal direction.

They reported that the 1st and 3rd harmonics of the lateral and the 1st and 2nd

harmonics of the longitudinal force are dominant (Fig. 2.4.).

Figure 2.4. Harmonic components of the walking force in (a) vertical, (b) lateral and (c)

longitudinal directions (Bachmann and Ammann ,1987).


23

It has been found that the longitudinal and lateral loadings due to walking are much

lower than that of the vertical loading and thus can be considered negligible. However,

Dallard et al. (2001) after investigating the lateral swaying of London Millennium

Bridge provided an evidence of the importance of understanding the horizontal loading

component. Taking into consideration the horizontal loading exerted on a structure is

important in instances where the lateral stiffness provided by the floor-columns

interaction is not preventing the lateral sway of the structure, which is not a common

occurrence in floor structures.

2.3.2 Running / Jumping

Running is another human action, which causes floor structure to vibrate. The

graphical representation of jumping is the one which is the most similar to running, for

details refer Figure 2.5. The load produced by running / jumping can be several times

higher than the load resulting from standing still. The normal frequency of pacing in

running / jumping varies from 1.8 to 3.4 Hz. The pacing rates, forward speeds and

strides lengths for running / jumping were proposed by Wheeler (1982) and are

described in the following Table 2.4.

Event Pacing rate,

(Hz)

Pedestrian

propagation, (m/s)

Stride length

(m)

Slow running (jogging) ~2.5 3.3 1.30

Fast running

(sprinting)

>3.2 5.5 1.75

Table 2.4 Pacing rate, pedestrian propagation and stride length for running events (Wheeler,

J.E. 1982)

Load-time function

The mathematical representation for the running / jumping events has been simplified as

discontinuous half sine wave described in Equation 2.2 (Smith, 2002) and Figure 2.5:


24

(2.2)

Where is the impact factor, is the peak dynamic load, G is the weight of the

person, is contact duration which can vary from 0 to and is the pace period or

step period derived from as .

The impact factor was obtained from Figure 2.5b and pace period is taken from

Table 2.4.

To reduce complexity in describing the running / jumping formulation the half sine

time-function described above was transformed into Equation 2.3.

(2.3)

Where G is the weight of the person, is the load component of the harmonic, n

is the number of harmonics, is the pacing rate and is the contact duration.

Figure 2.5 Idealized load time function for running and jumping (a) half sine model (b) impact

factor for depending on contact duration ratio (Bachmann.H and Ammann. W, 1987).


25

2.3.3 Dancing / Aerobics

Dancing and performing aerobics result in dynamic forces similar to those of

jumping (Ebrahimpour and Sack 1989). However, unlike jumping, the frequency of

these dynamic forces varies from 1.5 Hz to 3.4 Hz. Research by Wyatt (1985) showed

that the frequencies of dancing can vary from 1.2 Hz ~ 2.8 for individual jumping, 1.5

Hz ~ 2.5 Hz for small groups and 1.8 Hz ~ 2.5 Hz for large groups.

Load-Time Function

Early research done by Allen (1990) investigated the vibrational behaviour of a 20 m

span non-composite joist floor and recommended a loading function, using a

periodic function containing three sinusoidal harmonic components, to represent loads

induced by aerobics activity, as noted in Equation 2.4.

(2.4)

Where α are the dynamic load factors α1 = 1.5, α2 = 0.6, α3 = 0.1 for the third harmonics

respectively, is the maximum weight of participants over the loaded area, f is the

maximum jumping frequency, t is the time for various excitation frequencies induced by

the groups of people jumping dancing together, similar dynamic load factors were

presented by Willford (2001) through a parametric study, giving the confidence level

for use in the design of gymnasium floors and dance floors. The dynamic load factors

were 1.5 for excitation frequencies 1.5 ~ 3.0 Hz, 0.6 for excitation frequencies 3.0 ~ 6.0

Hz, 0.1 for excitation frequencies 6.0 ~ 9.0 Hz.

Theoretical research done by Ji et al. (1994) investigated floor response produced by

dancing and aerobics and provided possible resonance at higher harmonics. It suggested

an equation to calculate the number of Fourier terms or harmonics needed to be

considered in the analysis (refer to Equation 2.5). The number of Fourier terms needs to

be considered was determined as follows:

(2.5)

Where I is the number of Fourier terms, βi is the ratio of excitation frequency considered

and to the ith

natural frequency of the system.


26

2.4 HUMAN RESPONSE TO THE STRUCTURE VIBRATIONS

AND THE CURRENT METHODS USED IN DESIGN CODES

In recognition of the complex nature of vibrations to which humans are exposed, a

number of methods have been developed which allow the effects of complex vibration

to be assessed. These can be broadly divided into two classifications (Griffin, 1996):

• Rating methods are methods in which only the worst component of vibration is

assessed.

• Weighting methods are methods in which the complex vibration is weighted

according to differences in human response to vibrations at different

frequencies.

The frequency weighted complex vibration is then summed in some manner (e.g. RMS)

resulting in a single quantity that may be used for assessment. Weighting methods are

now widely considered to be more appropriate than rating methods (Griffin, 1996).

There are two parameters which are typically used in modern codes of practice for

assessing the amount of vibration and its effects on the human occupants of office-

floors (ISO 2631, 1997). These are root-mean-square (RMS) accelerations in ISO 2631,

and the more recently established so called `4th

power‟ methods, such as the root-mean-

quad and vibration dose value (VDV) methods in BS 6472.

2.4.1 Frequency weightings

Prior to assessment of the severity of the vibration response, it is necessary to

apply a frequency weighting to take account of the differing human perception of

vibration at different frequencies. A frequency weighing provides a model of the

response of the person to the vibration. People are sensitive to some frequencies of

vibration than others, and this frequency dependence is simulated using the frequency

weightings.

The human body is not equally sensitive to all frequencies of vibration. For example,

the body is more sensitive to whole body vibration at about 5 Hz than at 50 Hz;

therefore, the vibration at 50 Hz is weighted such that its relative contribution to the

total signal is reduced accordingly. In attempt to account for this variation in the

sensitivity of a human being with respect to the frequency of the vibration, the British

Standards has adopted a standard, BS 6841, which defines the manner in which the

vibration at various frequencies should be weighted in order to more closely


27

approximate human sensitivity. In BS 6841, such weighting curve is defined for vertical

vibration direction (Figure 2.6).

Figure 2.6 Weighted z-axis vibration RMS acceleration (BS 6841)

In principle, weightings do not amplify at any frequency. Therefore, the magnitude of

the frequency-weighted signal should not be more than the magnitude of the

unweighted signal. For the vertical vibration, the human is most sensitive in the 4-8 Hz

range, and the weighting function has a value of 1 in that region. As we move away

from this region, the human becomes less sensitive and so the weighting decreases by a

factor of:

in the range of 1-4 Hz

in the range of 8-80 Hz

2.4.2 RMS Acceleration

Vibrations in buildings are seldom simple sinusoids. Often, the vibration time

signatures are modulated, transient or random, and they contain a range of frequencies,

where a more or less narrow range of frequencies exists. After being weighted, the most

common method for mapping such vibrations into a single numerical („effective‟) value

to be compared with the vibration limit is to calculate the RMS of the weighted

acceleration time-history using the following Equation 2.6:

(2.6)


28

RMS acceleration is used as it is a measure of the total vibration causing distress to the

human body over the and measurement duration. Greater RMS accelerations

correspond to higher vibration magnitudes causing more annoyance. However, an

assessment of the human distress using the RMS relationship is appropriate for, as

Griffin defines them, “well behaved” vibrations which are steady-state long-lasting

periodic or stationary random. If the vibrations are short lived transients, then the RMS

acceleration no longer appears to be a reliable effective value (Griffin, 1996).

The measure RMS acceleration is affected by both the length of the record and by when

the recording is taken. Therefore, if this is to be used in assessing serviceability, the

choice of record timing and length needs to be made. This happens because the RMS

acceleration when calculated for the whole T length of the test it will combine the low

and high response of the structure, this is a major weakness for using RMS value to

check the serviceability when applied to walking vibrations. In order to use the RMS

value only part of the spectrum acceleration data should be used and limited only to the

time of action which corresponds to the interval (t2-t1) and not including the

measurement before and after. However, it may actually happen that the periods of high

pedestrian activity are sufficient to cause annoyance to human occupants of the floors

2.4.2 Vibration dose value

A method, which addresses this problem and is gaining acceptance internationally, is

the previously mentioned vibration dose value (VDV) method. This method is suitable

for assessing all types of vibratory motion (periodic, random and transient). The VDV is

a cumulative measure of the vibration transmitted to a human receiver during a certain

period.

When assessing intermittent vibration, use the vibration dose value (VDV). The VDV is

given by the fourth root of the integral with respect to time of the fourth power of the

acceleration after it has been weighted. This is the root-mean-quad approach. The use of

the fourth power method makes VDV more sensitive to peaks in the acceleration

waveform. VDV accumulates the vibration energy received over the daytime and night-

time periods. The vibration dose is fully described in BS 6472-1992. Acceptable values

of vibration dose are presented in Table 2.5.

Where vibration comprises repeated events, each of a similar value and duration, a

VDV may be calculated. The following formula requires the overall weighted

acceleration over the frequency range 1 to 80 Hz:


29

(2.7)

Where VDV is the vibration dose value in , is the frequency-weighted

acceleration ( ) and is the total duration of a measurement (seconds) during

which the vibration may occur. Thus from the human perspective, the relates to

both the magnitude of the vibrations and how many times they occur (Ellis, 2001).

From BS 6472 the VDVs required for evaluating the level of adverse comment are for a

16-hour day or 8- hour night. Often it will not be possible to take measurements over

such long periods, and assessments may need to be made to calculate the VDV for 24 h.

Therefore it is necessary to consider estimating daily VDV.In order to estimate the total

VDV throughout a day, it is reasonable to assume an equal chance of any event to

happen not just assuming the large one because it will be overestimated. Hence a

representative VDV estimated for 24h (Equation 2.8) would consist on the fourth root of

the sum of the fourth powers of the individual . Where is the individual dose

value at each separated interval of measurement for every event taken into consideration

to calculate the VDV required.

(2.8)

There is a low probability of adverse comment or disturbance to building occupants at

vibration values below the preferred values. Adverse comment or complaints may be

expected if vibration values approach the maximum values. Activities should be

designed to meet the preferred values where an area is not already exposed to vibration.

Where all feasible and reasonable measures have been applied, values up to the

maximum range may be used if they can be justified. For values beyond the maximum

value, the operator should feel severe vibration.

Place Low probability of

adverse comment

Adverse comment

possible

Adverse comment

probable

Critical areas 0.1 0.2 0.4

Residential buildings 0.2-0.4 0.4-0.8 0.8-1.6

Office 0.4 0.8 1.6

Workshops 0.8 1.6 3.2

Table 2.5 VDVs at which various degrees of adverse comment may be expected (BS 6472)

Evidence from research suggests that there are summative effects for vibrations at

different frequencies. Therefore, for the evaluation of vibration in relation to annoyance


30

and comfort, overall weighted RMS acceleration values of the vibration in each

orthogonal axis are preferred (BS 6472).

2.5 DESIGN CRITERIA AGAINST FLOOR VIBRATIONS

Design criteria against floors vibrations are based on the acceleration responses

from which human response scales were developed as primary design tools against

floor vibration problems. Attempts to achieve and quantify an accurate acceleration

response in floor motions have been tried for many years and still are being reviewed

in modern construction, which are being designed with a view of addressing future

complaints of annoying vibrations (Hewitt and Murray, 2004b).

Allen and Pernica (1998) quoted from a paper by Tredold published in 1828:

“Girders should always be made as deep as they can to avoid the inconvenience of

not being able to move on the floor without shaking everything in the room.”

(Allen and Pernica, 1998)

This statement describes that the early problems associated with floors vibration

problems were addressed for example by increasing the thickness of the slab / floor.

The Australian standards for design of concrete structures provide the following clause

against the effects of floor vibration.

Vibration of slabs, AS 3600 (2001), Clause 9.5 states:

“Vibration in slabs shall be considered and appropriate action taken, where

necessary, to ensure that vibrations induced by machinery or vehicular and

pedestrian traffic will not adversely affect the serviceability of the structure.”

In contrast, the design criteria of floors against the effects of vibration can be found in

subsidiary publications, AISC Steel Design Guide Series 11 (Murray, Allen et al. 1997)

and Design Guide on the Vibration of Floors (Wyatt 1989). In this context, different

scales were developed, such as acceleration limits and response factor method, to design

floor structures against human-induced vibrations. British Standards have give

acceleration limits by considering the human comfort under vibrations (BS 6472).

2.5.1 Acceleration limits

Acceleration limits provide a floor vibration assessment considering the

occupancy of the building. Bachmann et al. (1987), Allen et al. (1990a), (1998), Naeim


31

(1991), commentary A (1995) by national Research Council (NRC Canada), Murray et

al. (1997), presented design acceleration limits for floors and design charts for

buildings. These design limits and charts provide peak acceleration limits formulated

from frequencies and damping ratios of the floor structure, in the different human action

scenarios.

2.5.1.1 Acceleration limits for walking excitation

One of the first well-known and widely recognized criterion for acceleration

limits for walking was developed by Reiher and Meister in 1930‟s. Their research

involved a group of standing people subjected to steady-state vibration of frequencies 3

– 100 Hz and amplitudes of 0.01016 to 10.16 mm (Murray, 1990). The subjective

reactions by standing people, yielded a scale “slightly perceptible”, “distinctly

perceptible” , “strongly perceptible”, “disturbing”, and “very disturbing” to describe the

vibrations. However, several investigations by Lenszen in 1960‟s on joist-concrete floor

systems, gave a modified Reiher-Meister scale (refer to Figure 2.7). The original scale

was applicable only if it was scaled down by a factor of 10 for floor systems with less

than 5% critical damping (Naeim 1991).

Figure 2.7 Modified Reiher-Meister Scales

Although the modified Reiher-Meister scale is the basic and frequently used criterion, it

has been used with another additional method to pass on the judgment of perceptibility.

This is due to the lack of reflecting on the damping in this method. Murray in 1970‟s

showed that this scale can result in complaints from the occupants living in steel-beam

concrete floors with damping from 4% to 10% (Naeim 1991). Consequently, Murray et


32

al. (1989) have been involved in further development and after considering the

occupancy of the occupancy of the floor they presented the following inequality to be

used with modified Reiher-Meister scale to address the presence of damping (Murray,

1990).

(2.9)

Where D is the percentage of damping, is the initial amplitude of the heel impact

test in mm and is the first natural frequency.

The required damping percentages are describes in the Section 2.7. The assessment also

needs heel drop amplitude , which can be obtained from experimental analysis of the

floor. The natural frequency, can be calculated using finite element method for more

accuracy however, the simplified approaches describes in the Section 2.6 are adequate

for the above criterion.

In contrast, more recent publication by Murray et al. opposes the use of the modified

Reiher-Meister scale. The reason being that the developed criterion was calibrated

against 1960‟s and 1970‟s floor systems (Hewitt and Murray, 2004a). Thus, the need for

an improved criterion to be used for modern day slender floor systems has been

identified.

For cases of walking excitation, acceleration response criterion was published by Steel

Design Series 11 (Murray, Allen et al. 1997). In this context, an acceleration response

function due to walking excitation was presented by Murray, Allen et al. (1997) as

follows:

(2.10)

Where is the ratio of the floor acceleration response to the acceleration of gravity,

R is the reduction factor (0.7 for footbridges and 0.5 for floor structures), is the

dynamic coefficient of the harmonic force component, P is the persons weight, is

the modal damping ratio, W is the effective weight of the floor and is the step

frequency. For design purposes, the equation 2.10 was further simplified by

approximating the step relationship between the dynamic coefficient and frequency

f by as seen in Equation 2.11:


33

(2.11)

Where is the estimated acceleration response in a fraction of gravity, is the

natural frequency of the floor structure, is the constant force (0.29 KN for floors and

0.41 KN for footbridges), Is the modal damping ratio and W is the effective weight of

the floor. The peak acceleration due to walking excitation is then compared with the

appropriate limits describes in Figure 2.8.

Brand (1999) used Equation 2.11 in long-span joist floors, which in turn gave

affirmative results. By considering the above formulae presented by Murray et al.

(1997), Hanagan et al. (2001) made an effort to develop a simple design criterion for a

slab / deck profiled floor system. However, this simple design criterion was limited to a

single class of floors using grade 50 steel and thus needs to be expanded.

Figure 2.8 Recommended peak acceleration limits design chart (Murray, Allen et al. 1997)

Another approach was presented by Canadian Steelwork Association (CSA), which

provided a threshold for the peak acceleration due to walking (refer to Figure 2.9). This

is similar to the criterion provided by the AISC Steel Design Guide Series 11, except for

the acceleration response, which was measured by heel impact excitations done on the

floor system (Wyatt 1989). The “continuous vibration” line in the Figure 2.9 is used to

assess the response due to average peak acceleration due to walking, while “walking


34

vibration” curves asses the response by the peak acceleration due to a heel impact

excitation.

After variety of field tests done by Williams and Waldron (1994) in assessing the

applicability of the Concrete Society Method (CSA) of assessing the floor vibrations,

these gave unsatisfactory results. This was due to the heel drop used in the criterion, the

excitation force of which is unknown at most times.

Figure 2.9 Canadian floor vibration perceptible scales

2.5.1.2 Acceleration limits for rhythmic excitation

In the case of acceleration limits for rhythmic excitation on floors during

performance of aerobics, dancing and audience participation Ellingwood and Tallin

(1984), presented acceleration limit criterion using available literature. An equation

(refer to Equation 2.12) to calculate the maximum mid span acceleration of a floor

due to rhythmic activity of frequency f was developed

(2.12)


35

Where is the sinusoidal dynamic force, is the fundamental frequency, is the

static stiffness.

Due to lack of indication of the damping coefficient in the Equation 2.12, Allen (1990),

presented an incorporated acceleration response with damping ratio for rhythmic

excitation. This approach has been the basis for design against vibration due to rhythmic

excitation used in current design approaches such as Steel Design Series 11 for design

of the floor structure. Thus Murray, Allen et al. (1997) provided a criterion detailed in

equation 2.13 to incorporate this approach in the design of floor against rhythmic

excitation:

(2.13)

Where is the peak acceleration ratio in a fraction of gravity, dynamic

amplification factor for harmonic, is the effective weight per unit area of

participants distributed over the floor panel, is the effective distributed weight per

unit area of floor panel, including occupants, is the natural frequency of the floor,

is the forcing frequency (in terms of the step frequency) and is the damping ratio. The

dynamic coefficient , weight of participants, and excitation frequencies are

presented in the Table 2.6.

Activity

Forcing

Frequenc

y f, Hz

Weight of

Participants wp

Dynamic

coefficient

i

Dynamic load

alphai iwp

kPa psf kPa psf

Dancing : First Harmonic 1.5-3 0.6 12.5 0.5 0.3 6.2

Lively

concert or

sport event

:

First Harmonic 1.5-3 1.5 31 0.25 0.4 7.8

Second

Harmonic 3.0-5 1.5 31 0.05 0.075 1.6

Jumping

excercises :

First Harmonic 2-2.75 0.2 4.2 1.5 0.3 6.3

Second

Harmonic 4-5.5 0.2 4.2 0.6 0.12 2.5

Third Harmonic 6-8.25 0.2 4.2 0.1 0.02 0.42

Table 2.6 estimated loading during rhythmic events (Murray, Allen et al. 1997)


36

2.5.2 Response factor method

In an approach to design floors against the adverse effects of vibration, the use

of a response factor was recommended by Wyatt, (1989) and Murray et al. (1998). This

factor is then compared with a limit depending upon occupancy. The calculation of the

response factor depends upon whether the fundamental natural frequency of the floor

exceeds 7 Hz. In this context, two equations were developed to represent each case

when the fundamental natural frequency exceeds 7 Hz and when it does not.

If the fundamental natural frequency exceeds 7 Hz, in which case floors are of high

natural frequency the response factor R is given by (Wyatt, T.A. 1989):

(2.14)

Where is the floor mass in kg/m2, be is the minimum of either the floor beam spacing

b (m) or 40 times the average slab thickness, and is the floor beam span.

If the fundamental frequency is less than 7 Hz, referred to as floors of low natural

frequency the response factor is given by:

(2.15)

Where is the floor mass in kg/ m2, is the floor effective width, is the floor

beam effective span, Is the structural damping (critical damping ratio) and is the

Fourier component factor.

The critical damping ratio was considered to be 0.03 for normal, open plan and

furnished floors, 0.015 for unfurnished floor of composite deck construction and 0.045

for w floor with partitions.

The value of the Fourier component factor was found as a function of the floor

frequency f0 of which when:

(2.16)

The criteria for the response factor R for office floors are described in Table 2.7.


37

Type of office Response Factor, R

General Office 8

Special office 4

Busy Office 12

Table 2.7 Response factor R for offices (Wyatt, 1989)

For large public circulation areas such as pedestrian and shopping malls, lobbies and

assembly halls a response factor of four was proposed and this value should not increase

in case of residential floors.

2.5.3 Assessment of vibration design criteria

Many researchers using both experimental and analytical work assessed the floor

systems using the design criteria presented in Section 2.5.1 and 2.5.2. Number of case

studies has been done in this respect.

Osborne et al., (1990) analyzed a long-span lightweight composite slab of slab of 16 m

span and a thickness of 120 mm supported on 1.2 mm gauge steel deck. Using various

available methods they checked the acceptability of vibration characteristics both

experimentally and numerically. Their results provided clearly agreed with the AISC

Design Guide 11‟s acceleration limits. Later, Williams et al. (1994) tried to assess the

floor vibration dynamic characteristics, as reflected by fundamental frequency and

design methods using a set of full-scale vibration tests. As a part of the design approach,

they recommended the use of a high-power computational method, such as finite-

element analysis for more accurate results. Murray et al. (1998) evaluated the

differences between the procedures of the acceleration limits and the response factor

method in terms of a typical office floor and found them to be the same evaluation in

both cases.

Various researchers tried to understand the floor response due to human actions and as

described above, the acceleration response has been used as a design tool for floor

vibration in most design guidelines (Wyatt 1989, Murray, Allen et al. 1997). The

frequencies considered in these design tools, such as the first mode of natural

frequencies obtained from simplified equations, do not provide sufficient evidence of

high order mode shapes. In today‟s modern, long span floor constructions with lower

damping, the applicability of these design tools remains unsolved.


38

All these design guidelines provide responses of floor vibration due to human activities

where the activity has originated within the floor panel. However, there is no clue in the

design procedure as to the acceleration response due to pattern loading. None of the

design guidance has looked at the vibration measurements in either, the adjacent floor

panel or the behavior of the entire floor due to different type of loads. This is

particularly important, when a continuous floor is being used for different human-

activities with little or no permanent partition.

2.6 DETERMINATION OF NATURAL FREQUENCY

Every structure has its own natural frequencies. Particularly with floor

structures, there are various methods published in literature to determine the natural

frequencies. Some are simple methods and others are more sophisticated. To assess the

floor response to dynamic loads, an accurate calculation of the first natural frequency is

important to use in the design criteria against floor vibrations (refer to section 2.3.1 and

2.3.2 for design criteria). Research done by Wyatt (1989), Williams et al. (1994),

Bachmann and Pretlore (1995), Murray (1999) yielded the following methods to

estimate natural frequencies of floors:

1. Equivalent beam method.

2. Component frequency approach.

3. Concrete society method.

4. Self-weight deflection approach.

5. Finite element method of analysis.

These methods can be classified as general approaches and sophisticated approaches

and it must be noted that the natural frequencies of a floor depend upon numerous

factors including material property, structural type, slab thickness and boundary

conditions just to mention a few.

2.6.1 General approaches

The wquivalent beam method, component frequency approach, concrete society

method and self weight-deflection approach can be classified as general approaches.

Most of these scenarios in these approaches are used to predict the fundamental natural

frequency of a floor structure based on 1-way spanning approximation. The equivalent

beam method (EBM) given in Equation 2.17 and the concrete society method


39

approximate the behavior of floor to an equivalent simply supported beam to obtain the

first natural frequency (Williams and Waldrom, 1994):

(2.17)

Where E and I are the modulus of elasticity and second moment of area respectively, m

is the mass per unit length and l is the spam. Although EBM is a simple method,

Williams et al. (1994) provided an evidence of its non-applicability on concrete floors.

A similar approach to EBM is the one proposed by Murray, Allen et al. (1997) for a

beam or joist and girder panel to calculate fundamental natural frequency fn (Hz) as

stated here:

(2.18)

Where g is the acceleration of gravity, Es is the modulus of elasticity of steel, It is the

transformed moment of inertia, is the uniformly distribute load per unit length and L

is the span. This equation was further simplified using the mid span deflection equation

of a simply supported beam;

(2.19)

Where is the mid span deflection of a simply supported beam member and can be

derived from .

When a floor system has a significant interaction with the main beam deflection, a

change in fundamental mode shape and thus the natural frequency results. Therefore, a

modification factor CB was recommended to use with the above EBM in Equation 2.17

to calculate the natural frequencies. Wyatt (1989) presented the values for CB for a

single span, for both end pined to be 1.57, for one end pinned and the other fixed to be

2.45, for both end fixed to be 3.56 and for cantilever to be 0.56.

The Concrete Society method also uses the 1-way spanning approximation of equivalent

beam approach. In addition, it introduces modification factors to incorporate the

increased stiffness of 2-way spanning floors, yielding two independent natural

frequencies for the two perpendicular span directions.


40

In another publication by Wyatt (1989), the self weight deflection approach uses the

Equation 2.20 to determine the natural fundamental frequency f (Hz) of an un-damped

structural system:

(2.20)

Where K and m are the stiffness (KN/m) and the mass (tonnes) respectively.

Considering that in many plate and beam problems, weighted average of the defection

is taken at about ¾ of the maximum value of the self-weight deflection y0 (in mm),

and using the basic equation of motion, where , the fundamental

Equation 2.20 was rewritten for first natural frequency of a floor system as:

(2.21)

This has been used as a basic approach in many designs approaches for floor systems.

However, due to the fact that in joist floor systems resistance to floor vibration is not

only due to the slab itself but also due to the beams on girders supported by columns,

Murray (1999) used the following Equation 2.22 to take this into account:

(2.22)

where , and are the static deflections under weight, supported due to bending

and shear for the beam or joist, for girder and for column for axial strain respectively.

A similar approach to EBM was developed by Murray et al. (1997), to determine the

fundamental frequency of a floor consisting of a concrete slab or deck, supported on

steel beams or joists which were on steel girders or walls. In this context, the natural

frequencies of beam or joist and girder panel were calculated from the fundamental

natural frequency equation, and Dunkerley‟s relationship was used to estimate the

combined mode or system frequency.


41

Dunkerley‟s relationship

As floor systems usually comprises of three identifiable components floor slabs, floor

beams and main beams, in determining a natural frequency of a complete floor system it

is important to take account of the behavior these components individually. This was

done by considering each component separately using approximate methods. The

component frequencies are combined using the Dunkerley‟s method in Equation 1.23

for the total evaluation of the natural frequency for the floor system (Bachmann,

Pretlove et al. 1995), (Wyatt 1989) (Brand and Murray 1999):

(2.23)

Where , , are the component frequencies for each component of floor slab,

floor beams and main beams of the floor system.

2.6.2 Sophisticated approach

The general approaches described in section 2.6.1 yielded many erroneous

results. Consequently, the Institution of Structural Engineer‟s interim guidance report in

2001 concluded:

„Shortcut methods for determination of natural frequency based on the selected shape

under static loading or on rules of thumbs may not be adequate and can be very

misleading‟ ( Dougill, Blakeborough et al, 2001)

The most sophisticated and superior method to determine the natural frequency of a

structure is by finite element modeling (El-Dardiry and Ji, 2002). Commercially

available software such as ABAQUS, ANSYS and ALGOR can be used for this

purpose. The use of this approach not only provides greater accuracy but it also speeds

up calculations for more complex structures. However, it should be noted that the use of

FEM for obtaining the first natural frequency output, directly depends on the input of

structural properties.

The general methods described earlier use the conventional equation of dynamics or its

derivatives and have been idealized to a single degree of freedom system (SDOF) which

generates the first mode fundamental frequency. Williams et al. (1994) after

investigating the above methods with field exercises, adviced to use a computational

method such as finite element method. Pavic et al (2002) provided evidence that


42

currently popular in-situ cast concrete floors, modeled using SDOF systems based on

fundamental mode, was likely to produce erroneous results. With multi-degrees of

freedom structures in modern construction, the general approaches are not effective.

Hence, the ideal solution to determine the natural frequencies of a floor system is to use

the FEMs.

2.7 EVALUATION OF DAMPING AND DAMPING

COEFFICIENTS

Damping refers to the dissipation of vibrational energy. All physical systems

have some inherent damping, but the level of damping can be improved by increasing

energy dissipation. In this way, the response of a structure driven at a resonant

frequency can greatly decrease. Not only the components of the structural system but

also the non-structural components play a major role in damping, such as non-structural

elements, finishes, partitions, standing objects (Chen, 1999). Furthermore, the damping

can be either external or internal. The material or contact area within the structures such

as bearings and joints, are classified as internal damping materials while external

contacts such as non-structural elements are classified as external damping materials

(Bachmann 1995). The amount of damping in a structure is provided by a damping ratio

or damping factors.

Research done by Elnimeiri (1989), on composite floors recommended a damping

coefficient of 3.0% for open floors and 4.5%-6% damping for finished floors with

partitions. Murray et al., (1989) presented critical damping percentage requirements for

different floor design situations, where the critical damping for a typical office floor

system with hung ceiling and minimal mechanical duct work was estimated to be at

3.0%. Furthermore, Murray et al., (1989) states that if the required damping is between

3.5%~4.2% it is important to consider the configuration of the office and its intended

use. Obsorne et al., (1990) commented that measured damping of a floor was

considerably lower than the values generally assumed, verifying the difficulty of

estimating the damping. The damping of a floor also depends upon the usage or

occupancy. Thus, Maurenbrecher et al., (1997) after considering occupancy of

structures provided the following damping factors: damping 1% for footbridges, 2% for

shopping centers and 2%-5% for offices and residences. The partitions and other non-

structural components on a lightweight floor provide higher damping than those in


43

heavy floors. Thus, to observe the damping properties of lightweight floors with the

effect of non-structural components, Murray et al., (2004), in a recent publication

presented a damping criterion which provided damping ratio of 2%-2.5% for an

electronic office with limited number of cabinets and without full-height partitions. For

an open office space with cubicles with no full height partitions, they proposed a

damping factor of 2.5%~3.0% while for an office library with full height bookcases a

damping factor of 2%-4%.

In general, the damping coefficient appears to be a rage of 2.0%~6.0%. The human

presence on the floor area was neglected in the above mentioned research. Research

done by Browjohn (2001) proved that in the presence of humans on a floor system,

damping could increase up to 10%. Further, his full-scale experiments performed on a

function hall showed that the harmonic resonance was fully damped out by seated

humans. The review of the literature demonstrates that depending on the dynamic

properties of the empty structure has however the ability to increase damping in the

structure (Sachse, 2002).

Findings of the above studies reveal that damping in the floor structure is difficult to

determine and only approximate values can be provided for inherent damping in floor

systems.

2.8 REMEDIAL MEASURES AGAINST FLOOR VIBRATION

Several methods have been developed to rectify the vibration problems causing

human discomfort. It must be noted that such problems are often reported only after

construction and huge amount of money needs to be used for retrofitting. Thus, it is

critical to have a better understanding of the vibration response at the design phase prior

to construction, making the environment safer and diminishing future discomfort

problems. The following procedures and methodologies had been reported as remedial

measures.

1. Frequency tuning

2. Relocation of activities

3. Stiffening

4. Damping devices

5. Isolation


44

Generally, retrofitting can be done in two ways as active or passive (Hanagan and

Rottmann et al, 1996). Active methodology used an active control system while

servicing the structure for occupants. Normally, an active control system is present

physically, on a vulnerable floor system, which comprises of electromagnetic proof-

mass actuator, an amplifier, a velocity sensor and electromagnetic feedback controller

(Hanagan and Murray, 1997). This active control system reduces the floor vibrations by

adding forces and damping to counteract the resonant motion of the floor (Lichtenstein

2004). These forces are generated from the proof-mass actuator. The motion of the floor

is detected by the velocity sensor. The sensor takes the signals to the electromagnetic

feedback controller and then to the actuator to generate forces to counteract the resonant

motions.

1. Frequency tuning

Frequency tuning is adjusting or changing the natural frequency of a structure to avoid

the range of loading frequencies, which in turn helps to avoid resonance. The

fundamental structural frequency, natural frequency f1 ,can be either increased high

tuning or reduced low tuning to the relevant loading frequency f0. The success of tuning

is closely related to damping and frequency separation. The use of tuned mass dampers

(TMD) in frequency tuning is discussed in section 2.9.

2. Relocation of activities

Relocation of the source of vibration or sensitive occupancy may be an option to

remedy the problems of vibration. For example, aerobics exercises may be relocated

from the top floor of a building to floors below. Or complaints of a floor vibration can

be dealt with the positioning the source near to the column.

3. Stiffening

Increasing the stiffness of the structural elements can reduce the vibration caused by

walking or rhythmic activities. For example, introducing a new column from affected

floor to the foundations will increase the stiffness of the floor.

4. Damping devices

The added damping devices such as damper posts, tuned mass dampers or viscous

dampers, may be effective in reducing the resonance vibration. Hanagan, (1996)

reported the difficulty in improving the damping of a floor system with dampers, due to


45

the presence of multiple complex modes shapes and closely spaced natural frequencies

of the floor system.

5. Isolation

Isolation as a remedial measure against floor vibration means to isolate the excitation

paraphernalia from the structure. For example isolation of vibration machinery from the

floor by placing them it springs may be effective. This separates machine and the floor

structure reducing vibration transmitted from the machinery to the floor (Gordon, 2005)

2.9 DAMPING DEVICES FOR VIBRATION CONTROL

One of the common methods used to mitigate the excessive vibration is by

adding damping devices to the structural system. These damping devices can be

classified as passive, semi-active and active (Mackriell and Kwok, 1997).

Passive damping This approach consists of incorporating „passive” devices to the

structure. These passive devices may be visco-elastic

dampers, friction dampers, TMDs (Setareh and Ritchey,

2006). This the most common approach in vibration control

techniques.

Active damping This approach involves the use of actuators to produce a

counteract force to reduce the resonant vibration. The method

uses actuators, sensors and controllers, both analog and

digital to generate the counteract oscillation.

Semi-active damping This approach is combination of both active and passive

damping, also known as adaptive-passive damping. It uses a

self adjustable passive control scheme, where the response of

damping is adjusted to the oscillation of the structural system.

2.9.1 The use of passive damping devices

The most commonly used vibration control techniques are based on the use of

passive damping devices. These devices are capable of absorbing part of the energy


46

induced by the loads, while reducing the energy dissipation through the other structural

elements. As a result, the deflections and accelerations are controlled. The power

needed to counteract the vibrational effects is provided by the relative motion between

the two ends of the attachment, which is the damping device. This relative motion

determines the amplitude and the direction of the counteract force. Although these

systems provide supplemental damping to a structure, they are tuned to specific

structural responses or frequencies, making them unable to respond to structural

changes over time or any modelling and implementation error. An active control system

on the other hand, requires actuators, sensors, controllers and computer technology and

as result the installation is more complex and costly. For this reason, active damping

devices are not generally used in the floor systems subjected to human-induced loads.

Hence, passive systems are not guaranteed to be correctly tuned to the actual structural

response, potentially resulting in inefficient implementation of the passive control

system.

2.9.1.1 Tuned Mass Dampers

TMDs are other passive damping devices that have been tested on controlling

floor vibration (Hanagan, Rottmann et al. 1996). Webster et al. (Webster and Vaicaitis

2003) Setareh, Ritchey et al. (2006) presented cases studies of using TMDs to control

excessive floor vibrations. The TMDs consist of massive elements elastically connected

to the structure. This connection allows relative motion between the mass and the

structure, so that a large inertia force is produced. To gain such a large inertia force, the

natural frequency of the structure needs to be close to the fundamental frequency of the

structure. This mechanism of TMDs is most effective in controlling the first mode of

vibration. Setareh et al. (Setareh, Ritchey et al. 2006) presented an analytical and

experimental study of pendulum tuned mass damper to control excessive floor

vibrations due to human movements. Although it resulted in significant reductions in

the excessive vibrations caused by humans, due to the off-tuning caused by variations in

the floor live loads, the TMDs did not perform effectively. Consequently, TMDs were

found to be the most effective in addressing only the first mode of vibration. The floor

systems subjected to multi-modal vibrations did not produce such favorable results.

Thus, the current study did not use the TMDs as a retrofitting tool in floor system

subjected to multi-mode vibration. VE dampers were proposed instead as a more

suitable retrofitting tool in controlling vibration in composite floors.


47

2.9.1.2 Passive control using advanced materials

Ungar and Kerwin (1962) and Ungar (1963) used the concept of damping in

viscoelastic materials in terms of strain energy and developed a model that expressed

the overall damping in terms of the damping of individual layers. When viscoelastic

material is added to a system, its influence on the overall system damping depends on

how much strain energy is stored in the viscoelastic material under load. Therefore, the

amount of damping material can be minimized by placing it where it will store the most

strain energy.

Passive Visco-Elastic damper system is the most promising dissipation system that can

be used in floor structures although its technology is relatively new. The VE dampers

were first used in the USA in 1969, in the building of the World Trade Center twin

towers. Later in 1980, these dampers were used in Columbia Sea First and two union

Square buildings in Seattle and 1994 in Chein-Tan Railroad Station, Taipai . On most

occasions, these VE dampers were used to control the seismic response or to reduce

wind induced vibrations. Only a limited research has been conducted on using the VE

dampers in floor structures to control human induced vibrations.

Ebrahimpour and Sack (2005) retrofitted a laboratory-constructed floor to perform at

acceptable vibration levels, by laminates of carbon fibre reinforced polymer (CFRP) and

layers of Visco-Elastic material. In controlling the vibrations produced, the CFRP and

VE material act as a VE passive damping system. Using a mass drop test which is

similar to heel-drop test, the vibration response was observed. The damping ratio of the

floor increased from 2.4 % to 11.7 % was found and as result, 70 % deflection reduction

was achieved (Figure 2.10). This type of VE damping system in floor system is yet to

be studied comprehensively.


48

Figure 2.10 floor mass–drop displacement responses (Ebrahimpour, 2005)

Ellis and Ahmadi, treated a relatively lively 9 m by 6 m panel in the NE corner bay of a

floor of an eight-storey steel-framed building at BRE‟s Cardington laboratory with a

constrained damping layer, with the objective of increasing the damping and thus

changing the „feel‟ or acceptability of the floor. The selected area had been previously

used for a range of experiments related to response to human loading and it was chosen

because it had a clearly defined fundamental mode of vibration that was well separated

from other modes. The floor was also relatively lively, being adjudged on the

borderline of acceptability as regards its vibration response to walking. The objective of

this study was to show that current Finite Element Analysis (FEA) can be used in

predicting the dynamic behaviour of floors damped with a viscoelastic layer, and

establish design parameters. The FEA models consist on modeling a separate bay alone

with pinned boundary conditions to include the effect of the neighboring bays.

Figure 2.11 show the experimental result for a forced vibration testing for the original

floor, and Figure 2.12 presents the result of the tests after installing the damping layer.

With the damped floor (Figure 2.12) the situation was not quite so clear. Here the

frequency sweep showed two close peaks, rather than just one mode, with the higher

damping that had been anticipated.


49

Figure 2.11 Frequency response for the original floor Figure 2.42 Frequency response for the

damped floor

The experimental results of the treated floor showed the presence of two closely spaced

modes. FE analysis, however, predicted a resonant frequency well separated from other

modes (as with the original floor), and a first mode natural frequency substantially

higher than that observed. These discrepancies may be explained by the choice of

simplified boundary conditions. Modeling the entire 15 bays of the continuous floor (or

its equivalent) may reveal the presence of two closely spaced modes as observed

experimentally. Ideally future work should consider an isolated floor area before

examining the more complex problem of multi-bay systems, which are typically found

in buildings (Ellis and Ahamdi).

The agreement between the FEA predicted behaviour of a viscoelastic cantilever under

forced vibration and the analytical solutions indicates that a current commercial FEA

code can be used for designing structures damped with viscoelastic materials. There is

a need for the development of material models able to cater more realistically for the

in-elastic behaviour (strain softening, non-linearity) of some types of damping

materials (Ellis and Ahamdi) .

2.10 FINITE ELEMENT ANALYSIS

Finite element method (FEM) is known to have been developed during the early

1950s. It is a mathematical modelling technique used to determine the response of real

structures to external loads, sometimes also internal loads. Used in solving most of the

phenomena, the analysis has become one of the main computing tools for engineers and

scientists. Due to its cost and time-efficiency compared with physical experiments, it

plays an important role in engineering practice. Furthermore, it can be employed to


50

model and analyze simple structures as well as complex irregular structures, which are

more difficult to model using traditional analytical techniques. However, one must

understand that the accuracy of the results obtained from finite element analysis (FEA)

depends upon the quality of the input data. Thus, experimental calibration is needed to

guarantee acceptable results.

The basic methodology behind the finite element approach is to split complex problem

into simplified solvable ones (Clough and Penzien, 1993). Taking advantage of the high

computational power of modern computers and some advanced technique of matrix

mathematics, these large numbers of elements can be used to solve intractable problems.

It has become more simple and user-friendly to make use of FEM, with the use of finite

element (FE) software. There are many commercially available FE software such as

ANSYS, ABAQUS, MSC PATRAN, SAP. Every software analysis process involves

the following three major phases.

1. Pre-processing

The pre-processing involves defining an appropriate finite element mesh, assigning

suitable material properties and applying boundary conditions (restraint or constraints)

and loads. In general, pre-processing is used to build an input file.

2. Solution

The solution phase performs the execution of the input data field to from the output

results file. In this phase, the input file data are assembled into matrix format and solved

numerically. The assembly process depends upon the user‟s requirements i.e. static or

dynamic and model element types and properties, material properties, boundary

conditions and loads. The assembly process of a multi-degree of freedom system is

governed according to equation 2.24:

(2.24)

where:

(2.25)

Pre-processing Post-processingSolution


51

Where is the mass matrix, is the structural damping matrix, is the stiffness

matrix, is the accelaeration vector, is the velocity vector, is the

displacement vector and is the applied load vector.

The mass matrix , the structural damping matrix and stiffness matrix are

defined upon the element type and material properties, whilst acceleration vector ,

velocity vector and the displacement are developed based upon the

boundary conditions. The applied load vector is developed using the applied

external loads on the system. In the solution phase, the above equation is solved for

displacement and stretches to obtain internal/extrernal loads or stresses. This metod os

called displacement method or stiffness method.

3. Post-processing

The post-processing involves presenting the solved system to the end-user graphically

or numerically. In addition, it provides information on errors occurred in the solution.

Most advanced finite element software provides a log file, which gives information on

erroneous result and a quantitative measure of integrity.

In this research project, ABAQUS/ Standard version 6.8 is being used as the processor,

post processor and the solver.

2.10.1 Finite elements

The ABAQUS code contains a large number of finite elements, which can be

used to build complex structures. Most commonly used finite element are described in

Figure 2.13.

Figure 2.5 Finite elements used by ABAQUS


52

2.10.2 Analysis techniques

This section describes in detail the finite element solving techniques used in this

research approach.

2.10.2.1Linear static analysis

Most engineering design and practice rules are based on linear behaviour of

material and static load. In such approach it is expected that, if a given loading is

doubled, the resulting defections are doubled. Furthermore, it is assumed that all

deformations are recovered when the load is removed. However, it has been concluded

that linear analysis approximates the true behaviour of the structural system used for

basic design methods, and it is not adequate for research purposes (Hibbit Karlson

Sorensen Inc, 2001).

Linear static analysis is used in a case, where time independent loads are not applied to

the structural system. The Equation 2.24 can be modified to describe the linear static

analyse of structural system by omitting the mass matrix and the structural damping

matrix as depicted in Equation 2.26.

Thus,

(2.26)

2.10.2.2 Non-linear static analyses

Non-linear static analyses are of complex analyses which occur when the force-

displacement relationship of the system is non-linear. Thus, the force vector, and the

stiffness matrix are formulated on nodal displacements. This is mainly due to the

fact that real structures have a certain degree of non-linearity as a result of material non-

linearity, geometric non-linearity and boundary non-linearity (ABAQUS Analysis

User‟s Manual. 2008).

Material non-linearity

Structural materials like steel exhibit non-linearity in their behaviour and hence it is

desirable to add non linearity in modelling the material. Thus, the stress-strain


53

relationship of the material must be fed into the FE program. Usually, this has been

done by feeding an approximated stress-strain curve either bi-linearly or multi-linearly.

Geometric non-linearity

This type of non-linearity occurs when the system‟s internal forces are dependent upon

the final deformation. Thus, the original stiffness matrix is no longer valid and needs to

be adjusted accordingly. This can be illustrated by considering a wire hanging on its

own weight and loaded centrally. The system is said to be materially linear as the wire

relocates to its original shape when the load is removed, provided that the wire does not

exceed the elastic limit. However, it is geometrically non-linear since its ability carry to

the load depends upon the final deformation of „‟ V‟‟ shape.

Two methods have been suggested to rectify the problem. First one being an

approximate method, which assumes the size of the individual element representing the

system is constant, and reorientates the element stiffness matrix due to the elemental

deformation. Second method then recalculates the stiffness matrix with the calculated

displacements according to the preceding nodal coordinates.

Boundary Non-linearity

Boundary non-linearity occurs when the boundary conditions change during analysis,

Non-linear elastic springs, multi-point constraints are examples of sources of boundary

non-linearity (ABAQUS Analysis User‟s Manual, 2008).

2.10.2.3 Dynamic analysis

In dynamic analysis, the forced and the displacement experienced by the

structure are dependent upon the time history of the forcing function. Equation 1.24 has

been used to formulate dynamic analysis incorporating a time-dependent function.

However, when the structural system is materially or geometrically non-linear, time-

consuming step-by-step integration of the dynamic equation is required. ABAQUS

Standard offers a variety of dynamic analysis processes, which are briefly described

below.


54

Natural frequency analysis

The first step of a full dynamic analysis is the analysis of free vibration. This is to

observe the structure‟s natural frequency and mode shape. From an engineer‟s point of

view, it is important to understand the natural frequencies and the mode shapes as the

structure could resonate at such frequencies with externally applied dynamic loads,

causing excessive vibrations. Free vibration analysis depends on the structures mass and

stiffness and can be derived from Equation 2.24 by making force vector and

damping matrix equals to null vector and matrix respectively.

Thus it can be re-written:

(2.27)

Since free vibration is harmonic and therefore assuming:

(2.28)

Thus,

. (2.29)

By substituting for and in Equation 2.27, Equation 2.30 was derived. Herein,

represents the natural frequency of a structure. The number of natural frequencies

present in the structure is equal to the number of degree of freedom.

(2.30)

Direct-integration dynamic analysis

The direct-integration dynamic analysis provides response due to harmonic excitation.

This analysis assembles the mass, stiffness and damping matrices and solves the

equations of dynamic equilibrium detailed in Equation 1.24 at each point in time. The

ABAQUS uses the physical number of degrees of freedom of a model directly, to

calculate the steady-state response of a system. From this analysis, the structural

response due to human excitation can be extracted since all the human actions are

mathematically modelled using harmonic functions.


55

2.11 SUMMARY

The literature review presented in this chapter covered the effect of the dynamic

loads on floor structures particularly human-induced loads and the current state of

knowledge in designing floor systems against human induced vibration. Following this

literature review, the conclusions and arguments made are listed below:

Dynamic effects of the floor structures are an important design consideration

especially at or near resonance. Most design approaches use the first mode

frequency calculated using simplified approaches.

Human-induced loads, such as walking or performing aerobics or other dance-

type loads can create resonant vibration in floor systems.

All structures have some inherent damping, which depends on the construction

type, including cladding and partitions, and is being assumed in design.

Vibration effects can be mitigated by altering the structure‟s natural frequency,

or periods of vibration, by adding mass, or by increasing damping through

passive damping techniques using damping constrained layer or tuned mass

dampers.

FE element Analysis can be used to predict the dynamic behaviour of floors

damped with a viscoelastic layer, and establish design parameters.

Chapter 3: RUBBER ENGINEERING PROPERTIES AND MODELLING

56

CHAPTER 3 ENGINEERING PROPERTIES AND MODELLING of RUBBER

3.1 ENGINEERING PROPERTIES OF RUBBER

3.1.1 Preliminary remarks

The main property of elastomers like rubber is their capability to undertake large elastic

deformations, their capacity to stretch and return to their original form in a reversible

way. Elastomers, in their ordinary state, are not very useful engineering materials. Thus,

it becomes essentially to „synthesize‟ and introduce helpful properties into them.

Elastomers like natural rubber are amorphous, isotropic polymers to which various

ingredients are added and followed subsequently by heating and reactions. These

materials hold various advantageous properties such as damping and high tensile

strength, etc

The law of preservation of energy states that energy is neither created nor destroyed; it

is always transferred from one form to another. For this reason rubber has been used for

vibration isolation, due to its built-in ability to disperse huge amounts of energy due to

axial strains in addition to shear deformations. The magnitude of energy dissolute

depends on the components of rubber.

3.1.2 Hyperelasticity and viscoelasticity of rubber

Rubber consists of reasonably long network of polymeric shackles which

enclose a high degree of mobility and flexibility. The high deformability of rubber

comes from its mobility and flexibility (ability of chains to slide past one another).

When a stress is imposed on a rubbery material, the chains modify their configurations

immediately. The network structure of these shackles forces them to act monolithically.


57

Consequently, rubber can regularly be stretched up to ten times its original extent and

upon elimination of the force it returns to its original extent with modest undeviating

deformation. The molecular theories that form the basis of rubber elasticity are afar the

scope of this research.

Rubber is an exceedingly nonlinear material and firmly speaking no portion of the stress

tension curve follows the Hooke‟s law. Nevertheless, the stress-strain relation can be

supposed to be linear over small values of strains, but there is significant argument over

the limits of ‘small’ strains. Rubber vulcanizates (rubber compounds subjected to

vulcanization) enclose large amounts of reinforcing fillers and for this reason, it have

significant initial stiffness, which then softens before stiffening again, giving rise to a S-

shaped stress-strain curve which is typical of filled rubber as shown in Figure 3.2.

Material behaviour can be divided into two classes, the first class is the time dependent

behaviour (creep and viscoelasticity) which are discussed later in this chapter and the

second class is the Time independent behaviour (nonlinear elastic behaviour).

3.1.2.1 General theory of large elastic deformations1

A general theory of stress-strain relations for rubber like elastomers was

developed by Rivlin (1956), assuming that the material behavior is isotropic in elastic

behavior in the unstrained state and incompressible in bulk. The measures of strain, are

given by three strain invariants, given as follows

(3.1)

Where λ1,λ2,λ3 denote the principal stretch ratios, defined as the ratio of the stretched

length to the unstretched length of the edges of a cubical element. For incompressible

materials, the volume remains constant and hence . Hence, the strain energy

density ( ) is a function of 1 and 2 only. This results in the following Equation 3.2:

(3.2)

1 Further reference can be found in Engineering with rubber, How to design rubber

components, 2nd

edition, Alan Gent, chapter 3; pp 50-63.


58

Where C1 and C2 are constants. This particular form of strain energy function was

proposed by Mooney (1940) and is referred to as the Mooney-Rivlin equation. It is one

of the most commonly used strain energy relations for the finite element modeling of

rubber.

3.1.2.2 Viscoelastic behavior

Rubber shows time dependent behaviour and can be modelled as a viscoelastic

material with its properties depending on both time and temperature. Under conditions

of constant stress, rubber creeps (increase in deformation with time), and under

conditions of continuous strain there is relaxation (decay in stress with time), due to a

mixture of chemical and physical relaxation processes in rubber. The chemical process

is attributable to the modification of the crosslinks and alteration in the network of

shackles and occurs as a linear function of time. The physical process is caused by the

viscoelasticity of rubber and occurs more or less as a linear function of log time

An ideal linear elastic solid obeys Hooke‟s law; stress is proportional to strain. An ideal

viscous liquid obeys Newton‟s law: stress is proportional to rate of change of strain with

time. Viscoelasticity is a combination of an elastic and viscous behavior. A Hookean

solid can be expressed as a linear spring with the following relation:

(3.3)

Where, F is force, k is the spring rate and is the deformation.

Newton‟s law of viscosity can be written in the following form:

(3.4)

where is the damping constant. Viscoelastic behaviour has been expressed in the form

of two mechanical models, namely the Maxwell model and the Voigt (or Kelvin) model.

The Maxwell model consists of a spring and a dashpot in series, while the Voigt model

consists of a spring and a dashpot in parallel as shown in Figure 3.1.


59

Figure 3.1 Maxwell and Voigt models for viscoelasticity

In the Maxwell model, application of a load causes a sudden deflection in the elastic

spring which is followed by creep in the dashpot. In the case of deformation, the

reaction is first offered by the spring, followed by stress relaxation in the dashpot

according to the exponential law. In the case of the Voigt model, the spring and the

dashpot are in parallel and hence sudden application of load will not cause immediate

deflection in the spring due to the viscous behavior of the dashpot. Deformation builds

up gradually, with the spring taking a greater share of the load. The dashpot

displacement relaxes exponentially.

3.1.3 Energy dissipation in rubber

Under cyclic loading rubber dissipates energy due to hysteresis. Filled rubbers

undergo stress induced softening, due to progressive collapse of bonds that link one

polymer chain to another and due to separation between rubber molecules and the

reinforcing fillers. The most important factors causing the hysteresis effects in rubber

are discussed herein.

3.1.3.1 Friction

Rubber is composed of a network of chains and when it is loaded, the molecules

reorganize themselves due to the imposed load. This results in the sliding of the chains

relative to one another. This phenomenon is called internal friction (or) internal

viscosity and is a temperature dependent phenomenon. An enhancement in temperature

leads to a bigger mobility, resulting in a reduced viscosity and hence, reduced

hysteresis.


60

3.1.3.2 Stress softening

Stress softening refers to a reduced stiffness of rubber and change in damping

characteristics due to repeated loading. This is often referred to as the Mullins’ Effect

(Mullins, 1969). If an elastomer is subjected to a uniaxial strain, the stiffness remains

unchanged at strains higher than the previously applied strains; however, they have

lesser stiffness at strains lower than the previously applied strains. Stress softening

could be due to the rearrangement of the molecular network, micro structural damage

under stress and due to void formation.

3.1.3.3 Crystallization

Large extensions and retractions lead to the formation of crystallized regions in

the elastomer. Crystallization often results in increased strength. Natural rubber is one

such example of a material that has a low modulus at small strain with tensile strength

of thousands of pounds per square inch after crystallization at high strain.

3.1.3.4 Structural breakdown

In filled rubbers, the carbon black particles have a tendency to break down due

to shared interactions and this breakdown of the matrix/infill bond due to loading leads

to extensive hysteresis in rubber.

3.2.4 Rubber compounding

Compounding of rubber is a complex multidisciplinary science involving

materials physics, organic and polymer chemistry, inorganic chemistry and chemical

reaction kinetics. Compounded rubber has many unique characteristics not found in

other materials, such as high elasticity and dampening properties. An elastomer is a high

molecular weight liquid with low elasticity and strength. Vulcanization or curing is a

process of chemically linking the network of chains to form a tough elastic solid. This

results in an increase in stiffness and strength, while the hysteresis decrease. The


61

compounding of rubber begins with the choice of an elastomer, filler, cross linking

chemicals and various additives, which when added results in a „compound‟ having the

required characteristics (Freakley, 1978).

3.1.4.1 Sulfur curing

Sulfur is the most commonly used vulcanizing agent. It is carried out by heating

rubber mixed with sulfur under pressure. The rubber compounds subjected to this

treatment are called vulcanizates. For sulfur to effectively crosslink a rubber, an

elastomer must contain double bonds with allylic hydrogens. Commonly used

elastomers such as Butyl rubber, Nitrile rubber, Styrene Butadiene rubber satisfy this

requirement. Vulcanization can be done either using a soluble or an insoluble form of

sulfur (Freakley, 1978). Crosslinking with sulfur is usually ineffective and takes a long

time to cure. To increase the rate and efficiency of curing, accelerators such as

thiozoles, xanthates and thiurams are added. However, the accelerators should be

selected in such a way that they delay the onset of vulcanization so that the shaping

process is complete.

Mechanical aspects of rubber depend on the crosslink density. Modulus and hardness

increase proportional to the crosslink density. Crosslinking however, reduces the

hysteresis, because it reduces the sliding between the networks of chains. At high

crosslink levels, chain motions become restricted, and the network is incapable of

dissipating energy, resulting in brittle fracture at low elongation. The crosslink density

should be high enough such that it prevents failure by viscous flow and low enough to

prevent brittle fracture, providing the required dissipation at the same time.

3.1.4.2 Filler systems

Fillers or reinforcement aids, such as carbon black, clays and silicas are added to

rubber to improve material properties. Particle surface area is a very important

parameter for fillers. Particles with large surface area are useful since they have more

interaction with rubber and close particle-to-particle spacing. Carbon black and silica

are two of the most commonly used fillers.


62

Carbon black is chemically linked with rubber by shear mixing. The interactions

between rubber and carbon black vary in magnitude, with some chains chemically

bonded to the rubber while others have physical bonds of varying strength. To provide

the greatest strength, the carbon black must be broken down into fine aggregates and

dispersed thoroughly in the rubber, requiring mixing at high shear stresses. Carbon

black reduces the melt elasticity, increasing the processability, in addition to enhancing

the strength.

The addition of silica to a rubber compound improves tear strength, improves adhesion

of the compound to other components and reduction in heat build up. Silica, in

comparison to carbon black, does not provide the same level of reinforcement for the

same particle size. However, the addition of silica improves hysteresis and the tear

strength which is the force required to rip the rubber compound.

3.1.5 Mechanical aspects of high damping rubber

High damping rubber (HDR) is manufactured from the vulcanization of Natural

Rubber (NR) with the addition of carbon black, plasticizers, oils, resins and

consequently introduces specific characteristics such as hardening properties, energy

absorbing properties and maximum strain dependency of stress evolution (Yoshida et.

al, 2004).

Research on natural rubber for isolating buildings began in 1976 as a joint venture by

Earthquake Engineering Research Center (EERC), now called Pacific Earthquake

Engineering Research center (PEER) and the Malaysian Rubber Producers Research

Association (MRPRA). Furthermore, high damping rubber has been developed for

specific applications in base isolation from earthquakes (Kelly, 1997). The tensile and

shear strains are highly nonlinear for HDR. They show a high initial stiffness due to the

presence of high amounts of reinforcing filler, and the stiffness remains a constant

before increasing towards the end. This could be attributed to the finite extensibility of

the chains and also due to strain crystallization (Fuller et. al, 1996).

The high damping rubber, manufactured by Yokohama Rubber Co., used in this

research is based on experimental investigation by Yoshida et al (2004) and Amin et al.

(2002). The device undergoes cyclic deformations, and hence the results from tension


63

and compression tests are necessary. The experimental results used in modelling the

rubber are shown in Figure 3.2. The configuration of the device is such that there are

tensile and compressive strains induced simultaneously during any stage of the loading

cycle.

Figure 3.2 Uniaxial tension-compression tests on HDR (Amin, 2002)

HDR, like other elastomeric materials, exhibits a time-dependent behavior, referred to

as viscoelasticity. HDR creeps under the effect of constant stress and relaxes under the

effect of constant strain. Multi-step relaxation tests were conducted to determine the

time dependent behavior, the results of which are shown in Figure 3.3.

Figure 3.3 Relaxation tests on HDR ( Yoshida, 2004).


64

3.1.6 Summary

Natural rubber (NR) is a very versatile material and has been used in many

engineering applications such as dock fenders, bridge bearings etc. The properties of

NR can be modified by adding fillers, reinforcing materials, oils and resins referred to

as compounding.

Rubber dissipates energy due to various mechanisms such as friction, stress softening,

structural breakdown and crystallization. High damping rubbers have been developed

specifically for engineering applications and they involve modification of properties of

Natural Rubber based on the requirements

3.2 RUBBER MODELLING IN ABAQUS

Force equals stiffness times the deflection is probably the first equation that an

engineer encounters. This assumption is however, only valid for linearly elastic

materials. Rubber and other elastomers undergo large elastic deformations but they are

highly nonlinear in nature.

Hence, elastic modulus is almost never used in the modeling of rubber. The Poisson‟s

ratio for elastomers is between 0.499 and 0.5 which means that when a rubber block is

compressed, its volume remains virtually unchanged unless very high pressures are

applied, instead the block expands laterally and the volume remains unchanged. From

the equations of linear elasticity,

(3.5)

Where K is the bulk modulus, E is Young‟s Modulus (or) modulus of elasticity, G is the

shear modulus and ν is the Poisson‟s ratio.

In Equation 3.5, if the Poisson‟s ratio is assumed to be 0.5, corresponding to an

incompressible material, then the bulk modulus becomes infinity. This assumption also

dictates that E = 3G. This is however not true for most elastomers, making closed form

solutions impossible. Therefore, the equations of linear elasticity are no longer valid for

an elastomer.


65

3.2.1 Hyperelasticity

Elastomers, like rubber and foam, are classified as hyperelastic materials.

Hyperelastic materials have the ability to deform elastically up to large strains. The

stress-strain curve is highly nonlinear but elastic. Typical hyperelastic behavior of

rubber is shown in Figure 2.4.

Figure 3.4 Typical stress-strain curve for hyperelastic material

The constitutive behaviour of hyperelastic materials are usually derived from the strain

energy potentials which are discussed in detail in the following section. ABAQUS has

many built-in energy functions that can be used to model rubber hyperelasticity

accurately.

ABAQUS assumes the following assumptions for hyperelastic material

• Material behavior is isotropic.

• Material behavior is elastic.

• Material is incompressible by default (unless specified).

• Analysis includes nonlinear geometric effects.

3.2.1.1 Hyperelastic materiel models

To define the hyperelastic behavior of a material, a stored strain energy function,

is introduced. This function defines the strain energy stored in a material per unit


66

volume. The stress-strain relationship of a hyperelastic material can be obtained from

this function through the following relation:

(3.6)

where and are the stress and strain component, respectively.

Depending on the form of this function, there are many models or forms used to model

the hyperelasticity, such as Arrude-Boyce, Van der Waals, Mooney-Rivlin, Neo-

Hookean, Ogden, Polynomial, Reduced polynomial and Yeoh forms. Most of these

models are special cases from the polynomial form Raos (1992) and Hibbitt, Karlsson,

and Sorensen, Inc. (2002).

Raos (1992) investigated four models, Van der Waals, Mooney-Rivlin, Neo-Hookean,

and Ogden models in order to check the possibility of predicting the experimental

results by these models. The experimental results were obtained from uniaxial and

biaxial tension and compression tests on SBR rubber vulcanizate. These tests covered a

wide range of deformations (extension ratio, λ, from 0.5 to 4.0). According to the results

of this analysis, it was observed that Neo-Hookean and Mooney-Rivlin models can

predict experimental results in compression and moderate tension only (extension ratio,

λ less than 1.8) while Van der Waals and Ogden model can predict experimental results

for the whole range considered with satisfactory approximation.

In this chapter, the polynomial model is described as well as the Ogden model, which is

used in modeling the hyperelastic behavior of the viscoelastic material used in the

damping layer model.

1. Polynomial Model

The polynomial strain energy function per unit original volume, U (ε), is defined as

(3.7)

where and are temperature-dependent material parameters and and are the 1st

and 2nd

deviatoric strain invariants obtained from:


67

λ λ

λ

λ

λ

λ

(3.8)

And,

λ λ

(3.9)

where λ1, λ2 and λ3

are the principal stretches obtained from dividing the current length

over the initial one, λ are the deviatoric stretches, and J is the total volume ratio.

(3.10)

where is the original volume, J=1 for incompressible material,

is the elastic

volume ratio;

and

is the thermal volume ratio obtained from:

(3.11)

Where

is the linear thermal expansion strain.

It is worth noting that most elastomers are almost incompressible. This means that the

material volume almost doesn‟t change when it is stressed unless exposed to thermal

effects. This assumption is satisfactory for applications in which the material is not

highly confined. Accordingly, experimental data obtained from simple deformations

tests are used to define the material parameter, assuming material incompressibility.

2. Ogden Model

In this model, the strain energy function is defined as:

λ

λ λ

(3.12)

where , and

are temperature-dependent material parameters, is a material

parameter and λ are deviatoric principal stretches, which can be obtained from:


68

λ λ

(3.13)

where λ are the stretches in the principal directions.

The initial shear modulus, , can be obtained from

(3.14)

The bulk modulus, , is given by

(3.15)

3.2.1.2 Modeling the Hyperelasticity in ABAQUS

To obtain the hyperelastic material parameters that are necessary for defining the

material behavior, results of experimental tests are needed. ABAQUS can use the test

data from the following deformation modes:

Uniaxial tension and compression

Biaxial tension and compression

Planar tension and compression

Volumetric tension and compression

The schematic of these deformations are shown in Figure 3.5.


69

Figure 3.5 Schematic of deformations in different test used to model hyperelasticity

(ABAQUS/CAE, 2008)

Assuming material incompressibility and isothermal response, then,

λ λ λ (3.16)

Accordingly,

λ λ

λ

λ

λ

λ

(3.17)

Where λ and is the principal nominal strain.

The principal stretch, λ , is defined according to the deformation mode or in other

words the experimental test.

Uniaxial tests

For uniaxial tension or compression test,


70

λ λ

λ λ

λ

(3.18)

Where, λ is the stretch in the loading direction and

λ (3.19)

Biaxial tests

For biaxial tension or compression test,

λ λ λ

λ

λ

(3.20)

Where, λ is the stretch in the two perpendicular directions and

λ (3.21)

Planar tests

For planar tension or compression test,

λ λ

λ

λ

λ

(3.22)

Where, λ is the stretch in the loading directions and

λ (3.23)

The following steps summarize how the parameters of a hyperelastic material are

obtained and accordingly, how the stress-strain relationship is developed:

• Define number of terms to be used, N. In this model, N=3 was used. (N=1 is

sufficient for strains up to 100 %).

• According to the deformation mode or (experimental test), the principal stretch, λi,

is defined based on Equations from 3.18 to 3.22.


71

• The strain energy function is obtained according to the model used in the

analysis. The Ogden model was used for modeling the viscoelastic material in

the device. The material was assumed incompressible. Hence, the second term in

the strain energy function was set to be zero. Hence, the strain function for

Ogden model was obtained from:

λ λ

λ

(3.24)

• Based on the results of experimental test, ABAQUS determines the material

parameters in the last equation using a regression analysis, which is a statistical

technique applied to data to determine, for predictive purposes, the degree of

correlation of a dependent variable with one or more independent variables.

• The stress component can be obtained using Equation 3.6.

3.2.1.3 Modeling the Hyperelasticity of Rubber layer

In order to model the hyperelastic behavior of rubber in ABAQUS, results of

uniaxial and biaxial tension-compression tests are required. Yoshida et al. (2004)

presented experimental results of uniaxial tension and biaxial tension tests conducted on

high-damping rubber (HDR-A). In order to better model the hyperelastic behavior of the

rubber block, results of a compression test conducted on high-damping rubber specimen

is needed. Amin et al. (2002) presented the results of uniaxial compression test on high-

damping rubber. The different experimental results used in modeling the hyperelastic

behavior of rubber block are shown in Figures 3.6 and 3.7. Note that rubber compounds

in general can deform to large deformations in tension (strains up to 400 % to 500%),

however they cannot develop more than 50 % strains in compression.


72

Figure 3.6 Unixial tension-compression test on high damping rubber (Yoshida, 2004)

After modelling the results of uniaxial and biaxial tension-compression tests of rubber

from the test data sheet, I processed the data to find the equivalent parameters for

evaluating the Polynomial and Ogden (Table 3.1 and 3.2) models that will represent the

rubber material properties that will define the hyperelastic effect of rubber in function of

both Uniaxial and Biaxial models needed to be entered in ABAQUS as shown in

Figures 3.8 and Figure 3.9.

Figure 3.7 Biaxial test on high-damping rubber (Yoshida, 2004)


73

Figure 3.8 Uniaxial model of ABAQUS

1 5355576.453 2.878 0.0

2 28.893 9.696 0.0

3 14997.39 -3.0648 0.0

Table 3.1 Ogden strain energy function with N=3 parameters

D1 C10 C01 D2 C20 C11 C02

0.0 -61248 167508.98 0.0 43309.96 -12750.86 749.778

Table 3.2 Polynomial strain energy function with N=2 parameters


74

Figure 3.9 Biaxial model of ABAQUS

3.2.2 Viscoelasticity

Considering a specimen of rubber under loading, tension, compression or shear,

the specimen response differs according to the loading rate. If the load develops slowly,

larger strains are expected. Also, the strains increase gradually under the sustained loads

and this phenomenon is known as the material creep, which is a characteristic of

viscoelastic materials. To explain the viscoelastic material behavior, consider a rubber

specimen exposed to tensile stresses applied suddenly at time t=0. The material will

respond elastically with an elongation, OA, as shown in Figure 2.10. If the stress

remains constant for a time period, T, then the deformation increases with time, as

shown by curve segment AB. The rate of the increase in deformation depends on the

magnitude of the stress. If the load is removed suddenly at time t=T, the elastic

deformation will be instantaneously recovered and after that a gradual recovery will

take place, part BDE. At some point in time, the residual elongation will be zero.


75

Figure 3.10 Creep and recovery for a viscoelastic material

3.2.2.1 Viscoelastic Model

ABAQUS provides an isotropic rate-dependent viscoelastic material model. This

model can be used in large-strain problems and in conjunction with the hyperelastic

material behavior.

To define the model, consider a rubber specimen, which is exposed to a sudden small

shear strain, at time t=0 and then the strain is kept constant for a certain time.

Accordingly, the corresponding shear stress can be defined as

(3.25)

where is the time-dependent shear relaxation modulus, which characterizes the

response of the material.

The time-dependent shear relaxation modulus, , can be written in a non-

dimensional form:

(3.26)

Where is the time-dependent dimensionless shear modulus and is the

instantaneous shear modulus.

Thus,

(3.27)


76

Similarly, for large shear strains, the shear stress, , can be obtained by the

integration by parts as:

(3.28)

Or,

(3.29)

Where is the instantaneous shear stress at time .

Accordingly, to define the viscoelastic material behavior, it is necessary to define the

dimensionless shear relaxation modulus, . ABAQUS assumes that the modulus

can be defined by a Prony series expression as:

(3.30)

Where and are material constants and =1,2,.....N

Accordingly, for small shear strains

(3.31)

Where:

(3.32)

The corresponding creep strain can be calculated from:

(3.33)


77

3.2.2.2 Modeling the Viscoelasticity in ABAQUS

There are four ways to provide the viscoelastic material parameters in

ABAQUS. These methods are

Direct specification of Prony series parameters

Creep test data

Relaxation test data

Frequency-dependent data obtained from sinusoidal oscillation experiment

3.2.2.3 Modeling the Viscoelasticity of Rubber layer

For the device model, the viscoelastic behavior of rubber was modeled using the

experimental results of relaxation shear test conducted on a high-damping rubber

specimen (HDR-A) by Yoshida et al. (2004). The specimen was exposed to three-step

relaxation experiment. First, the specimen was stressed until 50% shear strains were

developed and then relaxed for 10 minutes. Then the specimen was stressed to develop

150 % shear strains. The specimen was relaxed again for 10 minutes before stressing it

again to 250 % shear strains then it was relaxed for the third time. ABAQUS uses only

one-step relaxation test to model the viscoelasticity of any material. Accordingly, the

first step (relaxation for the first 10 minutes) was used to model the viscoelasticity of

the rubber block. The experimental results of the shear relaxation test on the high-

damping rubber specimen for the first 10 minutes, which are used in modeling the

viscoelasticity, are shown in Figure 3.11. Where the Figure 3.12 represents the

relaxation model (blue line) in ABAQUS compared to the experimental data (red line).


78

Figure 3.11 Relaxation shear test on high-damping rubber (Yoshida, 2004)

Figure 3.12 Relaxation model of ABAQUS

Chapter 4: Modelling the constrained damping layer

79

Chapter 4 MODELLING THE CONSTRAINED DAMPING

LAYER

4.1 INTRODUCTION

Structural damping can be defined as the process by which a structure or

structural component dissipates mechanical energy or transfers energy into connected

structures or ambient media. These mechanisms have the effect of controlling the

amplitude of resonant vibrations and modifying wave attenuation and sound

transmission properties, increasing life through reduction in structural fatigue.

Passive damping treatments are widely used in engineering applications in order to

reduce vibration and noise radiation (Nashif et al. 1985; Sun and Lu 1995). Passive

layer damping, usually implemented as constrained damping, is the most common form

of damping treatment, where the damping layer deforms in shear, thus dissipating

energy in a more efficient way.

There are a number of ways in which an additional damping layer could be introduced

on to the structure. For this work a layer of high damping viscoelastic material was laid

on top of the concrete plate, and a constraining layer of concrete added over the

viscoelastic layer.

In this chapter, a detailed finite element analysis was conducted to predict the forced

vibration response of constrained-layer damped flat plate.


80

4.2 CONSTRAINED LAYER MODELLING USING FINITE

ELEMENT

In the past, researchers developed various FE approaches for structures with

constrained layer damping treatments. The easiest way of doing that was utilizing the

modelling capabilities of existing FE codes, such as NASTRAN, ANSYS, ABAQUS,

etc., to analyse constrained layer beams, plates and shells. Since these structural

viscoelastic damped models involve different physical layers with different material

properties, the aforementioned first approach, in an attempt to represent the real

physical distribution of the layers, was therefore often obtained by combining the

existing FE modelling capabilities available at that time into a "composite" FE model

involving various FE types.

Figure 4.1 The plate with constrained damping layer

It can find that there is many ways to model this type of problem due to the complexity

of the materials and the contact between the different layers that represent the full

model. The first model, model 1 (Figure 4.1), due to its simplicity and easiness of

implementation with commercial FE codes (e.g. ANSYS, ABAQUS), benefiting from

the usual standard discretization procedures available in these codes, was probably the

first approach to be utilized.

It uses standard linear (or higher-order) solid brick elements to model all the layers and,

for a three-layered configuration, yields four planes (layers) of nodal points.


81

Figure 4.2 The three FE models used to model the damping layer

Another more recent alternative to model 1 is the use of models 2 and 3; the latter two

models both have in common the use of solid brick elements to model the viscoelastic

core.

However, regarding the outer stiffer layers modeling, there are two distinct approaches.

In model 2 (Figure 4.2), the translational degrees of freedom of the plate are connected

to the brick ones by means of rigid links (Lu et al. (1979), Balmès and Germès (2002).

Using this model, the most complex one, it is possible to simulate bonding failure

between the viscoelastic layer and the adjacent plates by simply removing those links in

specific nodes of the FE method mesh. However, as in model 1, there are still four

planes of nodal points but the locking effects of the stiff thin external layers might be

more easily circumvented.

The model 3 (Figure 4.2), proposed by Johnson and Kienholz (1982) and further

employed by Kosmatka and Liguore (1993), Plouin and Balmès (1999) and Plouin and

Balmès (2000), is not as straightforward since while attempting to attach plate elements

to a solid element surface there are two problems that must be overcome: (i) it involves

offsetting the nodes from the plate mid-plane to the plate surface adjacent to the solid

element; (ii) it involves developing and incorporating a set of constraint equations used

to correct continuity discrepancies between the plate and solid elements. However, since

there are coincident nodes and translational degrees of freedom for the plate and the


82

adjacent face of the solid element, it allows the reduction of the total number of nodal

points requiring only two nodal planes. Table 4.1 compares between the three different

models in terms of application and the strengths and weakness of each in model in term

of the other models

Models Strengths Weaknesses

Composite models

Model 1

(s/s/s) Simplicity and easiness of

implementation

Standard and readily available

Fes

Standard discretization

procedures

3-D stress-strain state may be

captured

Brutal-force modelling

High computational cost

(many DoFs)

Uninteresting for large-

scale problems

Model 2

(p/r/s//r/p) Moderate computational cost

Less sensible to numerical

pathologies for thin external

layers

Ideal for sandwich structures

Possibility to simulate bonding

failure

Rigid links impose more

complexity

Non- standard

discretization and

assemblage procedures

Less-standard Fes

Model 3

(p+o/s/p+o) Less computational cost (allows

the reduction of DoFs)

Less sensible to numerical

pathologies for thin external

layers

Requires off-setting of

plate DoFs

Requires external

constraint equations

Non-standard

discretization and

assemblage procedures

Table 4.1 Main features of the composite FE models (Balmès, 2002 and Plouin, 2000)

4.3 FINITE ELEMENT MODELLING

In this study three different types of FE models (Figure 4.2) will be used in order

to try to find the best model that can be used in later studies to model correctly the

damping layer.

A typical cross section of the constrained damping layer used in this study for the three

FE models is shown in Figure 4.3. The structure is composed of a base core from

concrete, protection layer and one rubber block in between.

All the three models used in this FE study share a common representation of the

viscoelastic layer using 3D solid hybrid continuum elements (C3D8H). The base plate


83

and the protection layer of the surface treatments are both modelled by either shell plate

elements (SC8R) or 3D solid continuum elements (C3D8).

Figure 4.3 Cross section of the full FE model

The first model, model 1, uses solid continuum elements to model all the layers with

C3D8 for the concrete base and the protection layer, and C3D8H for the viscoelastic

layer. With this modelling approach, since all the models include solid continuum

elements, special care must be taken in order to avoid shear locking problems for the

3-D solid elements which means the FE elements will be overly stiff in bending

applications and modal analysis (section 4.3.3.2 for more details).

In the second model, model 2, the translational degrees of freedom of the shell plate

(SC8R) are connected to the solid continuum elements (C3D8H) ones by means of rigid

links. Using this model, the most complex, it is possible to simulate bonding failure

between viscoelastic layer and the adjacent plates by simply removing those links in

specific nodes of the FE mesh.

In the third model, model 3, the plate elements nodes are localized by offset of half of

the plate (SC8R) thickness to the plane in contact with the solid element, instead of

remaining in the standard mid-plane. This results in coincident nodes and translational

degrees of freedom for the plate and the adjacent face of the solid element.

4.3.1 Finite element geometric models properties

When dealing with composite layer structures like the constrained damping

layer, it is very important to determine the effect of each layer on the dynamic

behaviour of the structure. The effect of each of the three different layers will be

studied, in order to determine the dynamic effect of the thickness of the concrete plate


84

on vibration control when the constrained layer is applied on top of it, four different

thicknesses for the concrete plate will be used in this study ranging from 130 to 200 mm

(Table 4.2). After that the two layers that compose the constrained damping layer which

are the rubber and the protection layer will be examined to check the effect of each one

of them on the dynamic behaviour of the structures, for the rubber layer four different

cases were chosen range from 2.5 to 10 mm in thickness (Table 4.2) and finally for the

protection layer three different situations were selected 50, 40 and 30 mm thickness

(Table 4.2). Table 4.3 represents all the Geometric properties of the 48 FE models cases

that will be used in this study to determine the effect of the different parts in the

constrained damping layer. Pinned boundary conditions were imposed along the corners

of the models, by applying them to all the nodes through the corners (Figure 4.4).

Concrete plate

Length (mm)

Width (mm)

Thickness (mm)

Rubber layer

Thickness (mm)

Protection layer

Thickness (mm)

Model S130

7500 7500 130 Model R2.5

2.5 Model P50

50

Model S150

7500 7500 150 Model

R5 5

Model P40

40

Model S170

7500 7500 170 Model R7.5

7.5 Model P30

30

Model S200

7500 7500 200 Model

R10 10

Table 4.2 Geometric properties of the concrete plate, rubber and protection layers

Figure 4.4: FE models boundary conditions


85

FE Model Cases

Concrete

plate

(mm)

Protection

layer

(mm)

Rubber

layer

(mm)

FE Model

Concrete

plate

(mm)

Protection

layer

(mm)

Rubber

layer

(mm) Cases

1 130 50 10 25 170 50 10

2 130 50 7.5 26 170 50 7.5

3 130 50 5 27 170 50 5

4 130 50 2.5 28 170 50 2.5

5 130 40 10 29 170 40 10

6 130 40 7.5 30 170 40 7.5

7 130 40 5 31 170 40 5

8 130 40 2.5 32 170 40 2.5

9 130 30 10 33 170 30 10

10 130 30 7.5 34 170 30 7.5

11 130 30 5 35 170 30 5

12 130 30 2.5 36 200 30 2.5

13 150 50 10 37 200 50 10

14 150 50 7.5 38 200 50 7.5

15 150 50 5 39 200 50 5

16 150 50 2.5 40 200 50 2.5

17 150 40 10 41 200 40 10

18 150 40 7.5 42 200 40 7.5

19 150 40 5 43 200 40 5

20 150 40 2.5 44 200 40 2.5

21 150 30 10 45 200 30 10

22 150 30 7.5 46 200 30 7.5

23 150 30 5 47 200 30 5

24 150 30 2.5 48 200 30 2.5

Table 4.3 Geometric properties of all the FE models cases


86

4.3.2 Material properties

The FE model represents a rectangular plate made up of reinforced concrete, a

rubber layer on top of it and a protection layer of high strength concrete. For the

modelling of the concrete plate, the material properties of its elements used is the

normal reinforced concrete (Table 4.4). The rubber layer material used is the same

defined in the previous Section 3.2 (Rubber modelling in ABAQUS). The protection

layer on top of the rubber layer is modelled using high strength concrete. The material

properties used in the FE model analysis procedure are listed in (Table 4.4).

Material Young's modulus

(GPa)

Poisson's ratio

Density (kg/m3)

Rubber properties

Unixial tension and compression data

Relaxation test data

Concrete 33.5 0.2 2400 Nominal stress (MPa)

Nominal Strain

Stress (MPa)

Time (seconds)

High strength concrete

50 0.2 2350 -1.5 -0.5 3.1 0.1

Rubber - 0.499 1500 -1.1 -0.4 2.75 3

-0.8 -0.3 2.41 100

-0.55 -0.2 2.25 200

-0.32 -0.1 1.76 400

0 0

1.3 1

3 2

6 3

15 4

36 4.8

Table 4.4 Material properties (Yoshida, 2004)

4.3.3 Types of Elements

After the definition of the geometric and material properties of the FE models, it

is essential to consider the type of elements. Care should be taken in choosing the

element type to avoid the occurrence of singular stiffness matrices and produce reliable

results. (Roy, 1981; Cook, 1995)


87

4.3.3.1 Shell elements 2

Two types of shell elements are available in Abaqus: conventional shell

elements and continuum shell elements. Conventional shell elements discretize a

reference surface by defining the element‟s planar dimensions, its surface normal, and

its initial curvature. The nodes of a conventional shell element, however, do not define

the shell thickness; the thickness is defined through section properties. Continuum shell

elements, on the other hand, resemble three-dimensional solid elements in that they

discretize an entire three-dimensional body yet are formulated so that their kinematic

and constitutive behaviour is similar to conventional shell elements. Continuum shell

elements are more accurate in contact modelling than conventional shell elements, since

they employ two-sided contact taking into account changes in thickness no matter how

thick the elements are compared to other element dimensions

From that the shell elements that will be used in modelling the slab and the protection

layer are 3D Continuum thin Shell Elements (SC8R). SC8R is an 8-node (Figure 4.5),

quadrilateral, first-order interpolation, stress/displacement continuum shell element with

reduced integration .These elements are accurate in contact modelling since our model

will include a contact surface between the rubber block, protection layer and the

concrete plate.

Figure 4.5: 8 node continuum shell SC8R (ABAQUS, 2008)

2 Further reference can be found in ABAQUS/CAE Version 6.3 User‟s Manual (2008),

Hibbitt, Karlsson & Sorensen, Inc., 1080 Main St., Pawtucket, RI


88

4.3.3.2 3-D Solid continuum elements2

The used element for modelling the rubber block is 3-D solid continuum element

(C3D8). The solid (or continuum) elements in Abaqus can be used for linear analysis

and for complex nonlinear analyses involving contact, plasticity, and large

deformations. C3D8 is an 8-node (Figure 4.6), three-dimensional, fully integrated and

second order (quadratic) interpolation element.

Figure 4.6 8-node continuum solid elements C3D8 (ABAQUS, 2008)

Fully integrated linear brick elements of ABAQUS are overly stiff in bending

applications and modal analysis, this numerical problem is called shear locking. Fully

integrated first order elements such as solid elements may suffer from the locking

(Prathap, 2005). FEA codes could give false results when this type of element is used.

To correctly model the ideal shape change, an element should have the ability to assume

the curved shape. The edges of the fully integrated first order element are, however, not

able to bend to curves. The linear element will develop a shape shown in figure 4 under

a pure bending moment. The top surface experiences tensile stress, and the lower

surface experiences compressive stress. All dotted lines remain straight, but the angle A

can no longer stay at 90 degrees (Figure 4.7).


89

Fig 4.7 Shape of the fully interpolation first order element under the moment (ABAQUS, 2008)

To cause the angle A to change under pure moment, an incorrect artificial shear stress

has been introduced. This also means that the strain energy of the element is generating

shear deformation instead of bending deformation. The overall effect is that linear fully

integrated element becomes locked or overly stiff under the bending moment. Wrong

displacements, false stresses and natural frequencies may be reported because of the

locking.

The fully integrated second order element behaves differently since its edges are able to

bend to curves. Under a bending moment, the shape change of the element will correctly

assume that of the material block (Figure 4.8). The angle A continues to remain at 90

degrees after the bending. No artificial shear stress is introduced and the element can

correctly simulate the behaviours of the material block. There is no shear locking

associated with this type of element. Thus for that C3D8 type element has been selected

for modelling the 3-D solid elements in this thesis.

Fig 4.8 Shape of the fully interpolation second order element under the moment

(ABAQUS, 2008)

4.3.3.3 Hybrid elements

ABAQUS has a special family of elements to model hyperelastic behavior called

„hybrid‟ elements, which must be used to model fully the incompressible behavior in


90

hyperelastic materials. An incompressible material response cannot be modeled with

regular elements (except in the case of plane stress elements) because the pressure stress

in the element is indeterminate. If the material is incompressible, then it cannot undergo

any volume change. As a result, the stress cannot be calculated based on the

displacement in the nodes. Hybrid elements include an additional degree of freedom to

calculate the stress in the element directly. These „hybrid‟ elements are identified by the

letter „H‟ in their name, for example, the hybrid form of a 8-node brick element, C3D8,

is called C3D8H.

4.4 DAMPING FOR MODELS

In order to make the study effective in terms of understanding vibration control

using a constrained damping layer, the damping ratio is a major factor in determining

the dynamic behaviour of any structures under any type of load. From that in this study

we will apply damping ratio to the FE models based on the various investigations on

damping levels on floor system. Specific damping levels on floor systems are hardly

predictable before-hand as they tend to vary with cause. Thus, the current work looked

at four damping levels for the analysis narrowing the variation (refer Table 3.5).

Damping level Damping ratio (%)

Low damping 1.6

Mild damping 3.0

Medium damping 6.0

High damping 12

Table 4.5 Damping levels used for the Concrete plate behaviour

These four damping levels were identified after referring to investigations made in

various publications and their credibility are discussed.

Damping level of 1.6% was used for the damping level for a bare floor which can be

classified as low damping. In general, damping for bare composite floors has been

reported to be between 1.5% 1.8% (Bachmann and Ammann 1987). Obsorene et al.

(1990) used slightly higher damping values of 2% 3.0% for the bare floor (full

composite construction) for the super Holorib composite floor system. Damping

coefficient of 1.5% was used by Wyatt el al (1989) for a composite deck floor.


91

Another level of damping 3.0% had been identified as „mild damping‟ which has been

used to classify an office without permanent partitions or electronic/ paperless offices

Hewitt et. Al (2004) and Murray (2000).

Higher damping could arise in a floor with permanent (full height partitions), drywall

partitions where it could be a much as 5.0% 6.0% (Murray 2000). Elnimeiri (1989)

recommended a damping coefficient of 4.5% 6.0% for finished floors with partitions.

A floor with this situation can be classified as having a „‟ medium damping‟‟ condition.

Browjohn (2001) showed that the damping could increase to 10% depending on

occupant posture. This also would happen in an environment with an old office floor

with cabinets, bookcases and desks. On the other hand, Sachse (2002) proved that the

presence of stationary humans will increase the damping of the structure. This

phenomena is called human-structure interaction and previous investigations by Ji

(2003) has provided that this causes a significant increase in damping which could

increase the damping up to 12.0% and thus can be classified as floors with „‟high

damping‟‟. Although the damping ratio of 12% is unrealistically high for floors, it is

used to assess the effect of the rubber layer on floors with different damping ratio.

With this justification of four damping levels, the computer simulation incorporated

these four damping levels using the mass proportional damping, and the stiffness

proportional damping (the calculation is described in Section 3.3.1). The four models

gave different mass proportional damping and the stiffness proportional damping

are presented in Table 3.6. It was assumed that variation of damping ratios were

negligible for the first and the second natural frequencies when calculating the mass and

stiffness proportional damping coefficients.

4.4.1 Calculation of mass and stiffness proportional damping

The damping of structural system is more conveniently defined in terms of

modal damping ratios of levels (as described in section 4.3).

In solving the structural system the damping matrix cannot be expressed by the damping

ratios, instead an explicit damping matrix is used (Clough and Penzien 1993). Thus, a


92

method by Rayleigh which the damping is assumed to be proportional to the

combination of mass and stiffness matrix is used. This is described in the Equation 4.1.

(4.1)

Where is the system damping matrix, is the mass matrix, is the stiffness

matrix, is the mass proportional damping and is the stiffness proportional damping.

This Rayleigh‟s damping leads to the following relationship (refer Equation 4.2)

between damping ratio and frequency of modes.

(4.2)

The solution for the mass proportional damping coefficient, and stiffness proportional

damping, were obtained by a pair of Equation 4.2 simultaneously for the and

mode.

Thus,

(4.3)

Assuming, the variation of damping with frequency is minor; the proportional factors

were given by, (i.e.

). Thus, Equation 4.3 can be rearranged to give

Equation 4.4 as formulae for calculation for mass and stiffness proportional damping.

(4.4)

Given the damping level and two natural frequencies and , mass and stiffness

proportional damping can be calculated. The natural frequencies can be found by using

natural frequency analysis while damping needed to be assumed considering structural

use and its intended purpose. Table 4.2 and Table 4.6 presents the geometric properties

and the Mass proportional and Stiffness proportional damping for the four different

concrete plates that will be used later in this study.


93

Concrete

plate

Damping 1.6% Damping 3.0% Damping 6.0% Damping 12%

a B a b a b a b

Model

S130 0.07947 0.002819 0.149014 0.005285 0.298029 0.010569 0.596057 0.021139

Model

S150 0.09146 0.002452 0.171488 0.004597 0.342977 0.009193 0.685954 0.018386

Model

S170 0.10339 0.00217 0.193871 0.00407 0.387741 0.008139 0.775482 0.016278

Model

S200 0.12121 0.001854 0.227275 0.003476 0.454551 0.006952 0.909102 0.013904

(a- mass proportional damping, b - stiffness proportional damping)

Table 4.6 Mass proportional and stiffness proportional damping for the four concrete plates

4.5 MODELLING THE FLAT CONCRETE PLATE

After looking to the three different models, geometric properties and material

properties, types of element that will be used to model the constrained damping layer,

the first issue to look at is the modelling of the base concrete plate. In the suggested FE

models for the constrained damping layer the concrete plate will be modelled using 3-D

solid element in the first model and using the shell element in the other two models. In

this part the FE models of the concrete plate are constructed to investigate the

appropriate model that can be used for modelling of such type of concrete plate. Also

these models will be used to study the effect of type of element used, mesh sizes and the

number of element through thickness on the dynamic behaviour of the concrete plate.

All the models are analysed using the ABAQUS finite element software.

The FE models of the concrete plate can be explained as follows

A. 3-D solid element models

In these models the 3-D solid element (C3D8) will be used to model the concrete

plate, these models also will be used to study the effect of the number of element

through thickness, the use of shell elements and the mesh size effect on the dynamic

behaviour of the plate.


94

B. Shell element model

In this model the concrete slab is modelled using Continuum thin Shell Elements

(SC8R). The effect of using 3-D solid elements to model the concrete plate instead of

the shell elements on the dynamic behaviour of the plate is identified by comparing

the two types of models.

The relationships between the above models to be analysed in detail are summarized in

Table 4.7. The used meshes in the models are the coarse mesh, the intermediate meshes

and the fine mesh.


95

Table 4.7 The relationship between the FE models used in the analysis

Model

Model number A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3

Model description

Dimensions of the

FE models

Mesh Number [1] [2] [3] [1] [2] [3] [1] [2] [3] [1] [2] [3]

Mesh Type Coarse MeshIntermediate

MeshFine Mesh Coarse Mesh

Intermediate


Intermediate


Intermediate

MeshFine Mesh

Number of

Elements66 1212 2020 66 1212 2020 66 1212 2020 66 1212 2020

Size of element 1.251.25 0.6250.625 0.3750.375 1.251.25 0.6250.625 0.3750.375 1.251.25 0.6250.625 0.3750.375 1.25 0.6250.625 0.3750.375

Element Through

Tickness1 1 1 2 2 2 3 3 3

Total number of

Element36 144 400 72 288 800 108 432 1200 36 144 400

Shell model [D]

the shell element to represent the

concrete plate

In this models we use 3-D solid element

to represent the concrete plate with one

element through thickness

3-D solid element Model [A] 3-D solid element Model [B]


to represent the concrete plate with two

element through thickness

3-D solid element Model [C]


to represent the concrete plate with

three element through thickness

Table 5 The relationship between the models used in the analysis

Width = 7.5m, Length = 7.5 and thickness =0.15 m


[96]

Also the effect of the number of element through thickness is studied in the 3-D solid

elements models where three different number of element through thickness (Figure

4.9) is used, the number of element used range from one to three elements.

One element through thickness Two element through thickness

Three elements through

Figure 4.9 3-D solid elements models with different number of element through thickness

The computed natural frequencies of the first three modes for the four different

models with different mesh density are summarized in Table 4.8.

The Table 4.8 shows the convergence of the natural frequencies of the plate with the

refinement of mesh, the type of elements used and the number of element through

thickness. The differences of the natural frequency of any mesh to that of the finest

mesh range from 0.28% up to 12.02 % for the first three mode shapes.


[97]

Table 4.8 Effect of meshing on the natural frequencies of the models presented in Table 4.7

The frequencies vary so much with different mesh sizes and this can be seen from the

results that the refinement of the mesh plays the major role in the difference of

calculated frequencies. Also, the fine mesh provides more accurate frequency values,

which is useful for comparison between the frequency measurements and

predictions.

From the results of the dynamic study of the plate it is clear that both the elements

type are working effectively with fine mesh and only 0.6% difference between the

two models when using one solid element through thickness compared to the shell

element model. Therefore, both the 3-D solid element and the shell elements with

the fine mesh can be used to represent the concrete plate in the following

investigations for the dynamic behaviour of the flat plate, i.e. natural frequencies and

mode shape of the plate.

Freq % Freq % Freq % Freq %

1 4.252 0.60% 4.238 0.28% 4.234 0.10% 4.226 0.00%

2 8.876 0.56% 8.854 0.31% 8.844 0.20% 8.8263 0.00%

3 10.767 0.47% 10.761 0.41% 10.746 0.28% 10.716 0.00%


1 4.4162 4.50% 4.396 4.02% 4.3909 3.90% 4.381 3.66%

2 9.1567 3.74% 9.129 3.42% 9.117 3.29% 9.094 3.03%

3 11.153 4.07 11.148 4.03% 11.129 3.85% 11.096 3.54%


1 4.734 12.02% 4.703 11.28% 4.69 10.97% 4.681 10.76%

2 9.781 10.81% 9.736 10.30% 9.72 10.12% 9.67 9.55%

3 11.705 9.22% 11.699 9.17% 11.684 9.03% 11.661 8.81%

3-D solid element models Shell model

Mode

shape No

Fine Mesh

Mesh 2020

A3 B3 C3 D3

Mode

shape No

Intermediate Mesh

Mesh 1212

A2 B2 C2 D2



Mode

shape No

Coarse Mesh

Mesh 66

A1 B1 C1 D1


[98]

Verification of natural frequency with analytical solution

The Finite Element method is used in this thesis to determine the dynamic behaviour

of the concrete plate. The results obtained with this method for the fundamental

natural frequencies in Section 4.5 were compared with a traditional mathematical

methodology to check the matching between the two results. The natural frequency

for a point supported rectangular plate from Blevins 2 text:

(4.5)

Where

length of the plate & the width of the plate

1.0 9.19

1.5 11.51

2 11.81

2.5 11.95

The plate stiffness factor is given in Equation 4.6:

(4.6)

Where

Mass/area Elastic modulus

Poisson‟s ratio Plate thickness

The frequency calculated using the analytical method is 4.29 Hz. When comparing

the results with the finite element method which gives 4.226 Hz using the shell

element with fine mesh, it gives approximate results with a small difference of 1.5 %.

2 R. Blevins, Formulas for Natural Frequency and Mode Shape, Krieger, Malabar,

Florida, 1979. See Table 11-8.


[99]

4.6 RESULTS

The aim of the finite element analysis was to establish the main factors

governing the damping behaviour of the constrained damping layer. This information

is required for the determination of the most practical method for modifying the

dynamic behaviour of a structure. During this stage, the finite element method can

characterize the structure response and analyse the effects of the treatment design

parameters on the structural dynamic performance. ABAQUS Finite element code

was used to predict the response of 48 different cases of three different FE models of

the constrained damping layer shown in Figure 4.1 under forced harmonic vibrations.

An isotropic linear viscoelastic material model was used to represent the damping

layer.

A parametric study was carried out on the constrained damping layer to investigate

the effect of different parameters considered in this study. The parameters considered

are:

• Effect of Host structure

• Effect of Protection layer

• Effect of the damping rubber layer

4.6.1 Method of analysis

4.6.1.1 Eigenvalue analysis

Conventional eigenvalue analysis of the model in Figure 4.2 is concluded by

solving the following matrix equation

(4.7)

Where is the stiffness matrix, is the mass matrix, is the natural circular

frequency and is the mode of shape. The influence on the natural frequencies and

the mode of shapes of the plate due to the introduction of the constrained layer in the

three models presented in section 4.1 was investigated. Table 4.9 and figure 4.12

compares the natural frequencies variations with that of the concrete before applying

the damping layer.


[100]

From the table 4.8 we can clearly see the matching of the eigenvalue analysis

between all the 3 FE models presented in Section 4.1 and used throughout this study.

Figure 4.10: First mode of shape of the model 16

Figure 4.11: Second mode of shape of the model 16

Figure 4.10 and Figure 4.11 show respectively the first and second mode of shape of

the model 16 (Table 4.3) with 150 mm as a concrete, 2.5 mm of rubber layer and 50

mm of protection layer.


[101]

Table 4.9: Natural Frequency of all the FE models

Model 1 3,669 4,323 5,336 5,05 4,821 Model 1 4,226 5,292 5,751 5,612 5,425

Model 2 3,669 4,221 5,211 4,913 4,726 Model 2 4,226 5,22 5,725 5,659 5,45

Model 3 3,669 4,155 5,23 4,9 4,751 Model 3 4,226 5,153 5,721 5,59 5,352

Model 1 3,669 4,215 5,125 4,971 4,775 Model 1 4,226 5,067 5,475 5,301 5,136

Model 2 3,669 4,103 5,021 4,824 4,687 Model 2 4,226 4,955 5,386 5,151 5,032

Model 3 3,669 4,056 4,956 4,852 4,7 Model 3 4,226 4,992 5,412 5,172 5,015

Model 1 3,669 4,042 4,95 4,761 4,59 Model 1 4,226 4,665 5,198 4,97 4,84

Model 2 3,669 3,98 4,876 4,632 4,456 Model 2 4,226 4,564 5,09 4,832 4,72

Model 3 3,669 3,986 4,853 4,626 4,418 Model 3 4,226 4,592 5,126 4,966 4,756

Model 1 4,782 6,302 5,625 6,202 6,25 Model 1 5,613 7,124 6,995 6,491 6,842

Model 2 4,782 6,269 5,595 6,05 6,059 Model 2 5,613 7,01 6,859 6,345 6,722

Model 3 4,782 6,122 5,525 6,096 6,122 Model 3 5,613 6,978 6,902 6,253 6,679

Model 1 4,782 6,026 5,321 5,887 5,723 Model 1 5,613 6,851 6,712 6,211 6,592

Model 2 4,782 5,995 5,312 5,703 5,601 Model 2 5,613 6,702 6,612 6,125 6,412

Model 3 4,782 5,997 5,295 5,685 5,542 Model 3 5,613 6,652 6,592 6,098 6,445

Model 1 4,782 5,79 5,122 5,61 5,492 Model 1 5,613 6,576 6,372 5,892 6,161

Model 2 4,782 5,721 5,036 5,512 5,354 Model 2 5,613 6,456 6,213 5,741 6,068

Model 3 4,782 5,695 5,045 5,562 5,413 Model 3 5,613 6,447 6,158 5,695 5,987

With 10

(mm)

rubber

50 (mm) protection layer



Natural frequency (Hz)

Concrete

plate 130

(mm)

Without

constrained

layer

With 2,5

(mm) rubber

layer

With 5

(mm) rubber

layer

With 7,5

(mm)

rubber


Concrete

plate 170

(mm)

Without

constrained

layer

With 2,5

(mm) rubber

layer

With 5

(mm) rubber

layer

With 7,5

(mm)

rubber

With 10

(mm)

rubber

With 7,5

(mm)

rubber layer








With 10

(mm) rubber

layer

Concrete

plate 150

(mm)

Without

constrained

layer

With 2,5

(mm) rubber

layer

With 5

(mm) rubber

layer

With 7,5

(mm)

rubber layer

With 10

(mm) rubber

layer





Concrete

plate 200

(mm)

Without

constrained

layer

With 2,5

(mm) rubber

layer

With 5

(mm) rubber

layer


102

Figure 4.12 The natural frequencies of the concrete plate treated with the constrained damping layer as a function of the damping layer thickness

Figure 9 The natural frequency of the concrete plate treated with the constrained damping layer as a function of the thickness of the damping layer.

3,5

4

4,5

5

5,5

6

2,5 5,0 7,5 10,0

Firs

t n

atu

ral f

req

uen

cy (H

z)

Thickness of damping layer (mm)

Concrete base core thickness 130 mm

4

4,5

5

5,5

6

6,5

2,5 5,0 7,5 10,0

Firs

t n

atu

ral f

req

ue

ncy

(Hz)



4,5

5

5,5

6

6,5

7

2,5 5,0 7,5 10,0

Firs

t n

atu

ral f

req

ue

ncy

(Hz)



5,5

6

6,5

7

7,5

8

2,5 5,0 7,5 10,0

Firs

t n

atu

ral f

req

ue

ncy

(Hz)



Protection layer 50 mm Protection layer 40 mm Protection layer 30 mm


[103]

4.6.1.2 Simulation of forced vibration test

The forced vibration can be simulated using the finite element method by

solving the following matrix equation in the time domain

(4.8)

Where P is the load vector, the damping matrix is determined using a combination of

the host structure, the protection layer where the damping is determined using a linear

combination of the mass matrix and stiffness matrix and the rubber layer where the

damping is represented through the viscoelastic and hyperelastic effects.

A 1000N harmonic force was applied to the centre of the plate and its displacement

response calculated over the frequency range covering the first mode of the floor. The

frequency response curves were generated, and used to calculate the modal damping of

all the FE models in its first mode using the half-power bandwidth method (Figure

4.13).

The solution in the time domain for the equation 4.8 is then converted into the

frequency domain.

Figure 4.13 The half-power bandwidth method to calculate the damping


[104]

4.6.1.3 Case study

In this part I present the dynamic response of one of the models under the excitation of a

1000 N harmonic load. The case represents the model number 16 in Table 4.3 with the

properties listed below:

The thickness of the base slab: 150 mm

The damping ratio: 1.6%

The thickness of the rubber layer: 2.5 mm

The thickness of the protection layer: 50 mm

Mesh size: 20 × 20 fine mesh

Model: Model 1 with 3-D solid elements

The Frequency Response Function (FRF) is plotted in Figures 4.14 and 4.15 shows the

FRF for displacement for both the slab and the slab-rubber-protection layer systems

respectively. The fundamental natural frequencies, the peak responses and the damping

values of the two systems can be identified from figures 4.14 and 4.15 as follows:

Damping

Slab 4.226 1.98 1.6%

Slab + rubber + protection layer 5.292 0.53 7.52%

Table 4.10 Natural Frequencies, displacement and damping of model 16

It can be noticed from Table 4.10 that the constrained damping layer reduced the total

displacement by about 73% from 1.98 mm to 0.53 mm under a 1000 N harmonic force.

As for the damping ratio it can be seen also the increase from 1.6% as initial system

damping to 7.52% damping for the whole system due to the application of the

viscoelastic rubber layer.


[105]

Figure 4.14 Frequency responses for the concrete plate 150 mm with 1.6% damping ratio

Figure 4.15 Frequency responses for the150 mm concrete plate with 1.6% damping ratio

damped with 2.5 mm rubber layer and 50 mm protection layer


[106]

4.6.2 Effect of the slab thickness in relation to the rubber layer

In this section the effect of the initial host structure thickness is considered on

the overall performance of the constrained damping layer. From the FE model results

the first thing we can conclude that the host structure thickness plays the major role in

determining the rubber layer thickness needed to achieve the peak damping ratio.

It can be noticed for the concrete plate 130 and 150 mm thicknesses, that the damping

ratio remains constant even with increasing the rubber layer thickness over 5 mm, and

for both cases the peak damping ratio is achieved at a thickness of 2.5 mm (Figure 4.16

to 4.17). On the other hand for the remaining two other concrete plates the 170 and 200

mm thickness it can be observed that they both needed a thicker layer of rubber to

achieve the peak damping compared to the much thinner plates (Figure 4.16 to 4.19).

From that we can conclude given the same boundary condition and for two different

plates the thicker plate needed more damping material to reach its peak damping

compared to the thinner plate.

As for the initial damping of the host concrete structure it can be seen that it did not

affect the thickness needed of the rubber layer to achieve the maximum damping ratio,

it can be seen when comparing for example the concrete base core thickness of 130mm

with initial damping ratio1.6% (Figure 4.16) and 3.0% (Figure 4.17) that for both the

peak overall damping ratio still achieved with 2.5 mm thickness of rubber layer.

Also related to the concrete plate we can conclude form the results that when the

thickness of the concrete plate increases the effect of the constrained layer in terms of

increasing the damping ratio is less this can be noticed when comparing the peak

damping ratio achieved (Figure 4.16 to 4.17).


[107]

Figure 4.16 Model damping of the FE models as a function of the thickness of the damping layer with concrete base damping 1.6%

Figure 10 modal damping as a function of the thickness of the damping layer with concrete base damping 1.6%

5

5,5

6

6,5

7

7,5

8

8,5

9

2,5 5,0 7,5 10,0

% o

f cr

itic

al d

am

pin

g



4

4,5

5

5,5

6

6,5

7

7,5

8

2,5 5,0 7,5 10,0

% o

f cri

tica

l dam

pin

g



4,5

5

5,5

6

6,5

7

7,5

8

2,5 5,0 7,5 10,0

% o

f cri

tica

l dam

pin

g


Concrete base core thickness 170mm

3,5

4

4,5

5

5,5

6

6,5

7

7,5

2,5 5,0 7,5 10,0%

of c

riti

cal d

amp

ing




Concrete base damping 1.6%


[108]

Figure 4.17 Model damping of the FE models as a function of the thickness of the damping layer with concrete base damping 3.0%

Concrete base damping 3%

6

6,5

7

7,5

8

8,5

9

9,5

10

2,5 5,0 7,5 10,0

% o

f cr

itic

al d

amp

ing



5,5

6

6,5

7

7,5

8

8,5

9

2,5 5,0 7,5 10,0

% o

f cri

tica

l dam

pin

g



5

5,5

6

6,5

7

7,5

8

8,5

9

2,5 5,0 7,5 10,0

% o

f cri

tica

l da

mp

ing



4,5

5

5,5

6

6,5

7

7,5

8

8,5

2,5 5,0 7,5 10,0

% o

f cri

tica

l da

mp

ing




Figure 11 modal damping as a function of the thickness of the damping layer with concrete base damping 3%


[109]

Figure 4.18 : Model damping of the FE models as a function of the thickness of the damping layer with concrete base damping 6.0%

Concrete base damping 6%

7

7,5

8

8,5

9

9,5

10

10,5

11

2,5 5,0 7,5 10,0

% o

f cri

tica

l dam

pin

g



7

7,5

8

8,5

9

9,5

10

2,5 5,0 7,5 10,0

% o

f cr

itic

al d

amp

ing



6,5

7

7,5

8

8,5

9

9,5

10

2,5 5,0 7,5 10,0

% o

f cri

tica

l dam

pin

g



6

6,5

7

7,5

8

8,5

9

9,5

10

2,5 5,0 7,5 10,0

% o

f cr

itic

al d

am

pin

g



Figure 12 modal damping as a function of the thickness of the damping layer with concrete base damping 6% Protection layer 50 mm Protection layer 40 mm Protection layer 30 mm


[110]

Figure 4.19 : Model damping of the FE models as a function of the thickness of the damping layer with concrete base damping 12.0%

Concrete base damping 12 %

12

12,2

12,4

12,6

12,8

13

13,2

13,4

13,6

13,8

14

2,5 5,0 7,5 10,0

% o

f cri

tica

l dam

pin

g



12

12,2

12,4

12,6

12,8

13

13,2

13,4

13,6

13,8

14

2,5 5,0 7,5 10,0

% o

f cr

itic

al d

am

pin

g



12

12,2

12,4

12,6

12,8

13

13,2

13,4

13,6

13,8

14

2,5 5,0 7,5 10,0

% o

f cr

itic

al d

amp

ing



12

12,2

12,4

12,6

12,8

13

13,2

13,4

13,6

13,8

14

2,5 5,0 7,5 10,0

% o

f cri

tica

l dam

pin

g



Figure 13 modal damping as a function of the thickness of the damping layer with concrete base damping 12% Protection layer 50 mm Protection layer 40 mm Protection layer 30 mm


[111]

4.6.3 Effect of the slab damping

The second major factor affecting the overall performance of the constrained damping

layer is the initial host structure damping. Indeed, the initial thickness of the host

structure affects the rubber layer thickness which is needed to achieve the peak damping

ratio (Section 4.6.2). However, the initial host structure damping influences the

reduction percentage of the displacement achieved after applying this technique.

The percentage of reduction can be expressed with the following relation:

(4.5)

The displacement reduction registers the maximum average as can be seen from Figure

4.20 to Figure 4.23 when the damping ratio of the host structure is 1.6% and when the

damping increases the percentage of reduction decreases.

It was observed that the structural damping present in the system significantly

influenced the overall reduction in displacement response. Consequently structural

systems with higher structural damping, fitted with constrained damping layer when

compared with those lower structural damping gave smaller reductions in displacement.

Thus, the higher structural damping in the structure, the lower the effects of the

constrained damping layer.

It also can be seen that the maximum reduction in vibration control is associated with

the smallest thickness of the concrete plate when compared with same damping ratio but

with thicker plate (Figure 4.20 and Figure 4.23).


[112]

Figure 4.20 The percentage of reduction in the displacement response for a 130 mm concrete plate and 50 mm protection layer Figure 14 Percentage of reduction in the displacement response for a 130 mm concrete plate and 50 mm protection layer

70%

53%

29%

18%

0%

20%

40%

60%

80%

100%

ξ = 1.6 % ξ = 3.0 % ξ = 6.0 % ξ = 12.0 %

Pe

rce

nta

ge o

f re

du

ctio

n

Damping (%)

Rubber layer 5 mm

73%

56%

33%

20%

0%

20%

40%

60%

80%

100%

ξ = 1.6 % ξ = 3.0 % ξ = 6.0 % ξ = 12.0 %

Pe

rce

nta

ge o

f re

du

ctio

n

Damping (%)

Rubber layer 7.5 mm

74%

57%

34%

20%

0%

20%

40%

60%

80%

100%

ξ = 1.6 % ξ = 3.0 % ξ = 6.0 % ξ = 12.0 %P

erc

en

tage

of r

ed

uct

ion

Damping (%)

Rubber layer 10mm

Model 1 Model 2 Model 3

78%

61%

36%

21%

0%

20%

40%

60%

80%

100%

ξ = 1.6 % ξ = 3.0 % ξ = 6.0 % ξ = 12.0 %

Pe

rce

nta

ge o

f re

du

ctio

n

Damping (%)

Rubber layer 2.5 mm


[113]

Figure 4.21 The percentage of reduction in the displacement response for a 150 mm concrete plate and 50 mm protection layer

Figure 15 Percentage of reduction in the displacement response for a 150 mm concrete plate and 50 mm protection layer

69%

52%

27%

15%

0%

20%

40%

60%

80%

100%

ξ = 1.6 % ξ = 3.0 % ξ = 6.0 % ξ = 12.0 %

Pe

rce

nta

ge o

f re

du

ctio

n

Damping (%)

Rubber layer 5 mm

71%

54%

32%

18%

0%

20%

40%

60%

80%

100%

ξ = 1.6 % ξ = 3.0 % ξ = 6.0 % ξ = 12.0 %

Pe

rce

nta

ge o

f re

du

ctio

nDamping (%)

Rubber layer 10mm

73%

56%

34%

19%

0%

20%

40%

60%

80%

100%

ξ = 1.6 % ξ = 3.0 % ξ = 6.0 % ξ = 12.0 %

Pe

rce

nta

ge o

f re

du

ctio

n

Damping (%)

Rubber layer 2.5 mm

71%

53%

32%

17%

0%

20%

40%

60%

80%

100%

ξ = 1.6 % ξ = 3.0 % ξ = 6.0 % ξ = 12.0 %

Pe

rce

nta

ge o

f re

du

ctio

n

Damping (%)

Rubber layer 7.5 mm



[114]

Figure 4.22 The percentage of reduction in the displacement response for a 170 mm concrete plate and 50 mm protection layer

75%

57%

32%

20%

0%

20%

40%

60%

80%

100%

ξ = 1.6 % ξ = 3.0 % ξ = 6.0 % ξ = 12.0 %

Pe

rce

nta

ge o

f re

du

ctio

n

Damping (%)

Rubber layer 5 mm

71%

55%

30%

19%

0%

20%

40%

60%

80%

100%

ξ = 1.6 % ξ = 3.0 % ξ = 6.0 % ξ = 12.0 %

Pe

rce

nta

ge o

f re

du

ctio

n

Damping (%)

Rubber layer 7.5 mm

73%

54%

31%

18%

0%

20%

40%

60%

80%

100%

ξ = 1.6 % ξ = 3.0 % ξ = 6.0 % ξ = 12.0 %

Pe

rce

nta

ge o

f re

du

ctio

nDamping (%)

Rubber layer 10mm


68%

54%

27%

16%

0%

20%

40%

60%

80%

100%

ξ = 1.6 % ξ = 3.0 % ξ = 6.0 % ξ = 12.0 %

Pe

rce

nta

ge o

f re

du

ctio

n

Damping (%)

Rubber layer 2.5 mm

Figure 16 Percentage of reduction in the displacement response for a 170 mm concrete plate and 50 mm protection layer


[115]

Figure 4.23 The percentage of reduction in the displacement response for a 200 mm concrete plate and 50 mm protection layer Figure 17 Percentage of reduction in the displacement response for a 200 mm concrete plate and 50 mm protection layer

57%

39%

19%11%

0%

20%

40%

60%

80%

100%

ξ = 1.6 % ξ = 3.0 % ξ = 6.0 % ξ = 12.0 %

Pe

rce

nta

ge o

f re

du

ctio

n

Damping (%)

Rubber layer 2.5 mm

65%

48%

24%15%

0%

20%

40%

60%

80%

100%

ξ = 1.6 % ξ = 3.0 % ξ = 6.0 % ξ = 12.0 %

Pe

rce

nta

ge o

f re

du

ctio

n

Damping (%)

Rubber layer 5 mm

64%56%

31%

19%

0%

20%

40%

60%

80%

100%

ξ = 1.6 % ξ = 3.0 % ξ = 6.0 % ξ = 12.0 %

Pe

rce

nta

ge o

f re

du

ctio

n

Damping (%)

Rubber layer 7.5 mm

60%52%

27%

17%

0%

20%

40%

60%

80%

100%

ξ = 1.6 % ξ = 3.0 % ξ = 6.0 % ξ = 12.0 %

Pe

rce

nta

ge o

f re

du

ctio

nDamping (%)

Rubber layer 10mm



[116]

4.6.4 Effect of protection layer

As for the protection layer it can be seen based on the FE models results

presented in sections 4.6.2 and 4.6.3 that plays a minor role in the process of the

constrained damping layer when compared to the base structure and the rubber layer

from the results of the three different protection thicknesses.

It can be seen from Figure 4.24 that the effect of the protection layer is decreasing from

about 5~7% more in displacement reduction by 10 mm of thickness to about 1~3%

when the damping of the core concrete plate damping ratio is increasing from 1.6 %to

12%.

For the protection layer it can be seen that when its thickness increases the damping

ratio also increases slightly, and its effect decreases when the damping ratio of the host

structure increases.


[117]

Figure 4.24 Percentage of reduction in the displacement response for a 150 mm concrete plate and 5 mm rubberFigure 18 Percentage of reduction in the displacement response for a 150mm concrete plate and 5 mm rubber layer

61%

48%

26%

16%

0%

20%

40%

60%

80%

100%

ξ = 1.6 % ξ = 3.0 % ξ = 6.0 % ξ = 12.0 %

Pe

rce

nta

ge o

f re

du

ctio

n

Damping (%)

Protection layer 40 mm

68%

52%

27%

17%

0%

20%

40%

60%

80%

100%

ξ = 1.6 % ξ = 3.0 % ξ = 6.0 % ξ = 12.0 %

Pe

rce

nta

ge o

f re

du

ctio

n

Damping (%)


56%

45%

23%

13%

0%

20%

40%

60%

80%

100%

ξ = 1.6 % ξ = 3.0 % ξ = 6.0 % ξ = 12.0 %

Pe

rce

nta

ge o

f re

du

ctio

n

Damping (%)




[118]

4.6.5 Discussion

This chapter has assessed the different parameters that affect the constrained damping

layer ranging from the initial structure, rubber layer to the protection layer. The major

conclusions that can be drawn as summarized as follows:

The effect of rubber layer thickness has a peak effect in increasing the damping

after which it remains constant even if the thickness of the rubber layer

increased.

To achieve the peak performance of the constrained damping layer it is only

related to the dynamic properties of the initial structure.

The initial structure plays the major role in determining the overall performance

of the constrained damping layer.

The higher structural damping in the structure, the lower the effects of the


With the same initial damping ratio but different concrete plate thickness the

thinner the concrete plate the better the performance in vibration reduction of the


As for the initial damping ratio it did not affect the thickness needed of the

rubber layer to achieve the maximum damping ratio, for example the same

concrete thickness with different damping ratios need the same thickness of

rubber layer to achieve the peak performance.

The protection layer plays the minor role when influencing the overall

performance of the constrained damping layer technique when compared with

the rubber layer and initial structure.

The higher the natural frequency of the initial structure the thicker the rubber

layer needed to achieve the peak ratio in vibration control.

All the three types of model presented in this study can be used to model

correctly the constrained damping layer system.

CHAPTER 5 CONCLUSION & FURTHER WORK

[119]


5.1 Conclusion

This thesis has assessed FE-based analytical strategies to model the damping

behaviour of viscoelastic materials, which might be used (surface mounted, constrained

or embedded) as damping treatments in structures to control noise and vibration levels.

The first part of this study was related to the rubber material properties to get a better

understanding about their effect on the dynamic behaviour when introduced into a

structure. After that we presented a detailed study on modelling material models able to

cater more realistically for the in-elastic behaviour (strain softening, non-linearity) of

rubber damping materials based on non-linear viscoelasticity, currently available in

ABAQUS. At the end of this study we have presented three different ways that can be

used to model the constrained damping layer, each FE model present different results

than the other, and in order to check the effect of each part of the constrained damping

layer on the overall response of the structure.

The conclusions obtained can be summarized as follows:

In modelling concrete plate either 3-D solid elements or shell elements can be

used to investigate the dynamic behaviour of the represented plate but the

boundary conditions should be taken into consideration. The mesh size

significantly affects the prediction of the natural frequencies of the plate using

the 3-D solid element with a difference of 12.02 % and 0.6% for both

respectively coarse mesh and fine mesh when compared to shell model. Also, for

the effect of the number of element through thickness it can be seen that it will

not increase the accuracy of the predicted natural frequencies when compared to

the increase in CPU time after multiplying the number of elements by three

where the predicted values differ only by 1%. .

The constrained damping layer can be effectively used to mitigate problems in

floor systems, which can achieve up to 80% reduction in deflection response.

Based on the results the three different FE models presented in this study, it was

observed that the models that use the 3-D solid element to represent all the


[120]

structure element use a lot of CPU time to assess the transient response when

compared to the two other simplified models that significantly save in CPU time

in compare to the 3-D element model.

The three simplified models can be used to investigate the dynamic behaviour of

the constrained damping layer. The predicted frequencies, reduction ratio in

displacement and damping ratio between the simplified models and the 3-D

solid element model reasonably match each other with a difference range up to

5% (Table 4.9).

From this study we can conclude that the constrained damping performance

depends on many factors not only on the rubber layer but it is also related to the

base structure itself in terms of the dynamic properties of the structure itself .

The main target of this study was to present a good understanding and represent an

approach to model the constrained damping layer, so these modelling techniques can be

used to model the constrained damping layer in a full structure.

At the end we can conclude that the performance of the constrained damping layer in

controlling the vibration in any structure is related directly to the dynamic properties of

the structure itself in terms of the damping material needed to be applied and the

reduction ratio that can be achieved. In this situation we cannot tell which model is the

best to represent the real dynamic behaviour of the rubber layer as this can only be

determined after an experimental study.


[121]

5.2 Further work recommendations

The main aim of this project is to generate fundamental research knowledge on

the vibration characteristic and efficiency of the method used for vibration mitigate such

as the constrained damping layer subjected to human-induced loads in order to evaluate

their compliance against serviceability and comfort requirements in the current design

standards for long-span floors.

The work done in this study is the first step into studying the controlling vibration

induced by human walking load on long-span floors. This step includes the

understanding of the damping layer properties and the modelling techniques after

applying it on a basic structure which is a flat plate. Ideally future work should consider

examining the more complex problem of multi-bay systems, which are typically found

in buildings which has not been studied before due to it modelling complexity.

The future work should be to apply this technique of the constrained damping layer

studied in the Chapter 4 on a full structure with multi bays to really predict the dynamic

behaviour of the structure. Also to use of Finite Element Analysis (FEA) in predicting

the dynamic behaviour of floors damped with a viscoelastic layer, and establish design

parameters that can be used to determine the material properties needed to be used.

From there we will also be able to compare this technique of increasing the damping of

the structure by introducing the constrained damping layer with the other vibration

control methods such as TMD..., to see which one is more efficient in terms of cost and

construction time for different type of human excitations. To recommend suitable

occupancies of the floor slabs and their operating conditions that would not cause

discomfort to the occupants of adjacent floor panels.

Also additional recommendations are to develop comprehensive finite element (FE)

models to carry out dynamics computer simulations and to study the influence of

parameters, such as structural damping, activity type, load intensity, load frequency and

location of activity in terms of influence on the efficiency of the vibration control

methods studied.

References

[122]

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Appendix

[129]

APPENDIX A

The natural frequencies, displacement and damping ratio for all the FE models

Appendix

130

Model

Natural

Frequency

(Hz)

RatioDisplacement

(mm/1000N)Ratio Damping Model

Natural

Frequency

(Hz)

RatioDisplacement

(mm/1000N)Ratio Damping

Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Concrete plat thickness t= 13 cm Concrete plat thickness t= 15 cm

Flat plate 3,669 100% 2,91 100% 1,60% Flat plate 4,226 100% 1,98 100% 1,60%

Model 1 4,323 118% 0,647 22,23% 8,75% Model 1 5,292 125% 0,539 27,21% 7,52%

8,64% Model 2 5,22 99% 0,551 27,85% 7,47%

Model 3 4,155 113% 0,673 23,12% 8,58% Model 3 5,153 122% 0,559 28,23% 7,42%

Model 2 4,221 115% 0,666 22,87%



Model 1 6,302 132% 0,459 32,58% 6,10% Model 1 7,124 127% 0,382 43,44% 4,65%

Model 2 6,269 131% 0,468 33,21% 6,01% Model 2 7,01 125% 0,389 44,15% 4,57%

Model 3 6,122 128% 0,465 32,97% 6,05% Model 3 6,978 124% 0,390 44,35% 4,55%

Table 7 The thickness of rubber layer 2.5 mm and protection layer 5 cm

Concrete plate damping 1.6%

Table 1 : The thickness of rubber layer 2,5 mm and protection layer 5 cm

Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement




Model 1 5,336 145% 0,858 29,50% 6,55% Model 1 5,751 136% 0,620 31,33% 6,73%

Model 2 5,211 142% 0,869 29,86% 6,52% Model 2 5,725 135% 0,631 31,86% 6,68%

Model 3 5,23 143% 0,889 30,54% 6,48% Model 3 5,721 135% 0,642 32,41% 6,65%



Model 1 5,625 118% 0,350 24,82% 7,45% Model 1 6,995 125% 0,311 35,32% 5,61%

Model 2 5,595 117% 0,354 25,12% 7,42% Model 2 6,859 122% 0,320 36,42% 5,54%

Model 3 5,525 116% 0,354 25,08% 7,43% Model 3 6,902 123% 0,318 36,12% 5,57%

Table 8 The thickness of rubber layer 5 mm and protection layer 5 cm


Table 2 : The thickness of rubber layer 5 mm and protection layer 5 cm

Appendix

131

Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement




Model 1 5,05 138% 0,793 27,25% 7,09% Model 1 5,612 133% 0,581 29,35% 7,05%

Model 2 4,913 134% 0,827 28,42% 7,02% Model 2 5,659 134% 0,591 29,87% 7,00%

Model 3 4,9 134% 0,811 27,86% 6,96% Model 3 5,59 132% 0,599 30,25% 7,03%



Model 1 6,202 130% 0,398 28,22% 6,65% Model 1 6,491 116% 0,231 26,21% 6,87%

Model 2 6,05 127% 0,415 29,45% 6,57% Model 2 6,345 113% 0,235 26,75% 6,83%

Model 3 6,096 127% 0,407 28,87% 6,60% Model 3 6,253 111% 0,240 27,23% 6,80%




Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement



Model 3

Concrete plat thickness t= 13 cm

Model 1


0,769

0,764129%

129%4,751

4,726 26,25%

26,41%

7,10%

7,03%

7,01%

1,60%

4,821 131% 0,731 25,12%

Flat plate 3,669 100% 2,91 100%


Flat plate 4,226 100% 1,98 100% 1,60%

6,75%

Model 3 6,122 128% 0,389 27,62% 6,72%

Model 2 60,59 1267% 0,384 27,25%

1,60%

Model 1 6,25 131% 0,375 26,57% 6,82%

Flat plate 4,782 100% 1,41 100%

Model 2

6,97%


Flat plate 5,613 100% 0,88 100% 1,60%

Model 3 5,352 127% 0,602 30,42%

7,06%

Model 2 5,45 100% 0,591 29,87% 7,01%

Model 1 5,425 128% 0,581 29,32%

6,04%Model 3 6,679 119% 0,268 30,42%

6,12%

Model 2 6,722 120% 0,263 29,87% 6,08%

Model 1 6,842 122% 0,259 29,41%



Appendix

132

Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement




Model 1 4,215 115% 0,886 30,45% 7,92% Model 1 5,067 120% 0,683 34,52% 7,09%

7,85% Model 2 4,955 117% 0,690 34,84% 7,05%

Model 3 4,056 111% 0,915 31,45% 7,84% Model 3 4,992 118% 0,702 35,46% 7,02%

Model 2 4,103 112% 0,912 31,33%



Model 1 6,026 126% 0,566 40,11% 5,43% Model 1 6,8512 122% 0,441 50,12% 4,11%

Model 2 5,995 125% 0,576 40,87% 5,37% Model 2 6,702 119% 0,444 50,45% 4,08%

Model 3 5,997 125% 0,574 40,72% 5,39% Model 3 6,652 119% 0,443 50,33% 4,09%




Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement




Model 1 5,125 140% 1,122 38,54% 5,75% Model 1 5,475 130% 0,781 39,42% 5,65%

5,73% Model 2 5,386 127% 0,797 40,23% 5,60%

Model 3 4,956 135% 1,131 38,86% 5,73% Model 3 5,412 128% 0,803 40,56% 5,58%

Model 2 5,021 137% 1,131 38,86%



Model 1 5,321 111% 0,456 32,33% 6,84% Model 1 6,712 120% 0,371 42,12% 5,12%

Model 2 5,312 111% 0,463 32,86% 6,79% Model 2 6,612 118% 0,374 42,51% 5,15%

Model 3 5,295 111% 0,467 33,12% 6,77% Model 3 6,592 117% 0,373 42,44% 5,14%




Appendix

133

Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement




Model 1 4,971 135% 1,002 34,44% 6,19% Model 1 5,301 125% 0,721 36,41% 5,90%

Model 2 4,824 131% 1,015 34,87% 6,14% Model 2 5,151 97% 0,745 37,64% 5,82%

Model 3 4,852 132% 1,011 34,75% 6,15% Model 3 5,172 122% 0,750 37,88% 5,80%



Model 1 5,887 123% 0,527 37,35% 6,12% Model 1 6,211 111% 0,296 33,65% 6,35%

Model 2 5,703 119% 0,534 37,84% 6,08% Model 2 6,125 109% 0,298 33,84% 6,33%

Model 3 5,685 119% 0,530 37,62% 6,10% Model 3 6,098 109% 0,300 34,12% 6,30%




Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement




Model 1 4,775 130% 0,999 34,32% 6,21% Model 1 5,136 122% 0,735 37,12% 5,95%

Model 2 4,687 128% 1,014 34,85% 6,15% Model 2 5,032 98% 0,742 37,45% 5,91%

Model 3 4,69 128% 1,025 35,23% 6,16% Model 3 5,015 119% 0,751 37,95% 5,93%



Model 1 5,723 120% 0,505 35,85% 6,42% Model 1 6,592 117% 0,318 36,15% 5,31%

Model 2 5,601 117% 0,509 36,12% 6,37% Model 2 6,412 114% 0,322 36,62% 5,27%

Model 3 5,542 116% 0,509 36,08% 6,38% Model 3 6,445 115% 0,326 37,09% 5,24%




Appendix

134

Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement




Model 1

4,94% Model 1 6,5769 117% 0,494 56,10% 3,88%

7,05% Model 2 4,564 108% 0,8074 40,78% 6,48%

Model 3

Model 2 3,98 108% 1,107 38,05%

6,49%

4,04 110% 1,069 36,75% 7,15% Model 1

Flat plate


5,613 100%

4,665 110% 0,7964 40,22% 6,53%

1,60%

Model 1 5,79 121% 0,673 47,75%

Model 3

Model 2 5,721 120% 0,680 48,21% 4,89% Model 2


4,782 100% 1,41

3,83%

4,592 109% 0,8088 40,85%

6,456 115%

Model 3 5,695 119%


3,79%


0,88 100%

0,683 48,45% 4,91% Model 3 6,447 115% 0,503 57,21%

0,500 56,85%

100% 1,60% Flat plate

3,986 109% 1,103 37,89% 7,06%


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement




Model 1 4,95 135% 1,346 46,25% 5,15% Model 1 5,1984 123% 0,900 45,43% 4,82%

5,01% Model 2 5,09 120% 0,910 45,95% 4,77%

Model 3 4,853 132% 1,374 47,21% 5,07% Model 3 5,126 121% 0,891 45,01% 4,78%

Model 2 4,876 133% 1,400 48,12%

Model 3 6,158 110% 0,407 46,22% 4,90%



Model 1 5,122 107% 0,551 39,11% 6,33% Model 1 6,372 114% 0,403 45,81% 4,95%

Model 2 5,036 105% 0,555 39,35% 6,30% Model 2 6,213 111% 0,410 46,54% 4,88%

Model 3 5,045 105% 0,557 39,52% 6,28%




Appendix

135

Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement




Model 1 4,761 130% 1,241 42,65% 5,38% Model 1 4,97 118% 0,866 43,72% 5,32%

45,22% 5,25%

0,878 44,34% 5,27%

Model 3 4,626 126% 1,284 44,12% 5,27% Model 3 4,966 118% 0,892 45,05% 5,24%

0,351 39,85% 5,95%

Model 2 4,632 43,35% 5,33% Model 2 4,832 114%

Model 3 5,562 116% 0,645 45,75% 5,20% Model 3 5,695 101% 0,357

Flat plate 5,613 100% 0,88


Flat plate 4,782 100% 1,41 100% 1,60%

126% 1,261

100% 1,60%

Model 1 5,61 117% 0,638

Model 2 5,512 115% 0,650 46,12% 5,18% Model 2 5,741 102% 0,354 40,21% 5,89%

Model 1 5,892 105%


40,56% 5,87%



Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement




Model 1 4,59 125% 1,226 42,12% 5,42% Model 1 4,84 115% 0,861 43,51% 5,35%

1,257 43,21% 5,35% Model 2 4,72 112% 0,874 44,12% 5,28%

Model 3 4,418 120% 1,248 42,89% 5,38% Model 3 7,756 184% 0,881 44,52% 5,25%

Model 2 4,456 121%

Model 3 5,413 113% 0,610 43,25% 5,59% Model 3 5,987 107% 0,377 42,84% 4,96%



Model 1 5,492 115%

Model 2 5,354 112% 0,608 43,10% 5,61% Model 2 6,068 108% 0,377 42,85% 4,96%

Model 1 6,161 110% 0,372 42,23% 5,02%0,600 42,53% 5,67%




Appendix

136

Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement



Model 2 6,269 131% 0,361 47,21% 7,46% Model 2 7,01 125% 0,295 61,21% 5,85%

Model 3 6,122 128% 0,365 47,65% 7,43% Model 3 6,978 124% 0,296 61,44% 5,81%



Model 1 6,302 132% 0,356 46,58% 7,52% Model 1 7,124 127% 0,292 60,52% 5,92%

9,48% Model 2 5,22 124% 0,497 44,21% 8,56%

Model 3 4,155 113% 0,697 40,75% 9,45% Model 3 5,153 124% 0,493 43,85% 8,58%

Model 2 4,221 115% 0,689 40,31%



Model 1 4,323 118% 0,674 39,42% 9,55% Model 1 5,292 125% 0,487 43,25% 8,62%



Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement



Model 2 5,595 117% 0,343 44,87% 8,70% Model 2 6,859 122% 0,259 53,75% 6,81%

Model 3 5,525 116% 0,346 45,25% 8,67% Model 3 6,902 123% 0,257 53,42% 6,83%



Model 1 5,625 118% 0,338 44,12% 8,75% Model 1 6,995 125% 0,255 52,87% 6,87%

Model 2 5,211 142% 0,813 47,52% 7,57% Model 2 5,725 135% 0,555 49,35% 7,42%

Model 3 5,23 143% 0,814 47,62% 7,56% Model 3 5,721 135% 0,557 49,54% 7,40%



Model 1 5,336 145% 0,801 46,85% 7,62% Model 1 5,751 136% 0,545 48,42% 7,49%



Appendix

137

Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement



Model 2 6,05 127% 0,354 46,23% 7,88% Model 2 6,345 113% 0,216 44,75% 8,08%

Model 3 6,096 127% 0,355 46,44% 7,86% Model 3 6,253 111% 0,218 45,21% 8,04%



Model 1 6,202 130% 0,348 45,52% 7,95% Model 1 6,491 116% 0,213 44,13% 8,12%

Model 2 4,913 134% 0,773 45,21% 8,15% Model 2 5,659 134% 0,523 46,52% 7,78%

Model 3 4,9 134% 0,781 45,65% 8,12% Model 3 5,59 132% 0,526 46,77% 7,76%



Model 1 5,05 138% 0,753 44,05% 8,22% Model 1 5,612 133% 0,517 45,94% 7,85%



Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


7,91%

Model 2 5,45 100% 0,527 46,85% 7,85%

Model 1 5,425 128% 0,518 46,02%

7,28%Model 3 6,679 119% 0,235 48,75%

7,38%

Model 2 6,722 120% 0,233 48,43% 7,31%

Model 1 6,842 122% 0,229 47,55%

Flat plate 4,782 100% 0,765 100%

Model 2

7,79%


Flat plate 5,613 100% 0,482 100% 3,00%

Model 3 5,352 127% 0,531 47,21%

43,95%

Flat plate 3,669 100% 1,71 100%


Flat plate 4,226 100% 1,125 100% 3,00%

7,90%

Model 3 6,1 128% 0,356 46,51% 7,92%

Model 2 6,05 127% 0,357 46,72%

3,00%

Model 1 6,25 131% 0,349 45,67% 7,97%



Model 3


Model 1


0,779

0,761129%

129%4,751

4,726 44,52%

45,56%

8,25%

8,18%

8,18%

3,00%

4,821 131% 0,752


Appendix

138

Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement



Model 2 5,995 125% 0,389 50,85% 6,79% Model 2 6,702 119% 0,306 63,45% 5,26%

Model 3 5,997 125% 0,387 50,65% 6,71% Model 3 6,652 119% 0,302 62,75% 5,29%



Model 1 6,026 126% 0,383 50,02% 6,85% Model 1 6,851 122% 0,300 62,23% 5,33%

8,85% Model 2 4,955 117% 0,542 48,21% 8,16%

Model 3 4,056 111% 0,801 46,85% 8,83% Model 3 4,992 118% 0,544 48,33% 8,15%

Model 2 4,103 112% 0,795 46,52%



Model 1 4,215 115% 0,791 46,25% 8,89% Model 1 5,067 120% 0,535 47,56% 8,21%



Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement



Model 2 5,312 111% 0,366 47,85% 8,20% Model 2 6,612 118% 0,263 54,55% 6,33%

Model 3 5,295 111% 0,368 48,12% 8,18% Model 3 6,592 117% 0,264 54,78% 6,29%



Model 1 5,321 111% 0,363 47,42% 8,23% Model 1 6,712 120% 0,259 53,82% 6,39%

6,96% Model 2 5,386 127% 0,601 53,45% 6,62%

Model 3 4,956 135% 0,914 53,43% 6,88% Model 3 5,415 128% 0,604 53,68% 6,59%

Model 2 5,021 137% 0,904 52,86%



Model 1 5,124 140% 0,891 52,12% 7,02% Model 1 5,475 130% 0,586 52,13% 6,72%



Appendix

139

Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement



Model 2 5,703 119% 0,379 49,54% 7,45% Model 2 6,125 109% 0,228 47,21% 7,45%

Model 3 5,685 119% 0,377 49,33% 7,46% Model 3 6,098 109% 0,230 47,63% 7,42%



Model 1 5,887 123% 0,371 48,52% 7,52% Model 1 6,211 111% 0,224 46,42% 7,52%

Model 2 4,824 131% 0,847 49,56% 7,41% Model 2 5,151 122% 0,561 49,85% 6,99%

Model 3 4,852 132% 0,837 48,96% 7,46% Model 3 5,172 122% 0,561 49,84% 6,99%



Model 1 4,971 135% 0,825 48,25% 7,52% Model 1 5,301 125% 0,553 49,12% 7,05%



Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement



Model 2 5,601 117% 0,378 49,42% 7,48% Model 2 6,412 114% 0,239 49,52% 6,53%

Model 3 5,542 116% 0,380 49,66% 7,46% Model 3 6,445 115% 0,240 49,85% 6,49%



Model 1 5,723 120% 0,373 48,75% 7,55% Model 1 6,592 117% 0,236 48,86% 6,59%

Model 2 4,687 128% 0,842 49,25% 7,47% Model 2 5,032 119% 0,567 50,41% 7,03%

Model 3 4,69 128% 0,832 48,64% 7,50% Model 3 5,015 119% 0,561 49,85% 7,05%



Model 1 4,775 130% 0,826 48,30% 7,54% Model 1 5,136 122% 0,554 49,20% 7,11%



Appendix

140

Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


0,482 100%

0,409 53,42% 6,28% Model 3 6,447 115% 0,312 64,74%

6,456 115% 0,311 64,54%


3,986 109% 0,899 52,56% 8,18% Model 3

Model 2 5,721 120% 0,411 53,75% 6,26% Model 2


4,782 100% 0,765

4,90%

Model 3 5,695 119%


4,88%

4,592 109% 0,582 51,69% 7,52%

4,04 110% 0,882 51,57% 8,26% Model 1

Flat plate


5,613 100%

4,665 110% 0,575 51,12% 7,55%

3,00%

Model 1 5,79 121% 0,404 52,85% 6,33% Model 1 6,5769 117% 0,308 63,85% 4,95%

8,22% Model 2 4,564 108% 0,582 51,75% 7,51%

Model 3

Model 2 3,98 108% 0,890 52,05%

Model 1





Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement



Model 2 5,036 105% 0,389 50,84% 7,72% Model 2 6,213 111% 0,268 55,65% 6,04%

Model 3 5,045 105% 0,393 51,43% 7,65% Model 3 6,158 110% 0,269 55,87% 6,02%



Model 1 5,122 107% 0,383 50,13% 7,79% Model 1 6,372 114% 0,265 54,92% 6,12%

6,42% Model 2 5,09 120% 0,626 55,66% 5,93%

Model 3 4,853 132% 0,991 57,95% 6,44% Model 3 5,126 121% 0,629 55,89% 5,91%

Model 2 4,876 133% 0,994 58,12%



Model 1 4,95 135% 0,979 57,23% 6,47% Model 1 5,198 123% 0,622 55,25% 5,96%



Appendix

141

Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model 2 5,512 115% 0,399 52,14% 6,66% Model 2 5,741 102% 0,237 49,21% 6,98%

Model 1 5,892 105%


49,44% 6,96%

114%

Model 3 5,562 116% 0,397 51,85% 6,70% Model 3 5,695 101% 0,238

Flat plate 5,613 100% 0,482


Flat plate 4,782 100% 0,765 100% 3,00%

126% 0,938

100% 3,00%

Model 1 5,61 117% 0,392 51,21% 6,75%

0,610 54,22% 6,38%

Model 3 4,626 126% 0,935 54,65% 6,85% Model 3 4,966 118% 0,620 55,11% 6,32%

0,234 48,65% 7,05%

Model 2 4,632 54,85% 6,84% Model 2 4,832



Model 1 4,761 130% 0,924 54,05% 6,91% Model 1 4,97 118% 0,602 53,55% 6,44%



Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement



0,399 52,12% 6,75% Model 2 6,068 108% 0,245 50,87% 7,07%

Model 1 6,161 110% 0,242 50,21% 6,04%0,393 51,42% 6,81%

Model 3 5,413 113% 0,397 51,86% 6,78% Model 3 5,987 107% 0,247 51,23% 6,90%



Model 1 5,492 115%

Model 2 5,354 112%

0,935 54,68% 6,85% Model 2 4,72 112% 0,611 54,32% 6,42%

Model 3 4,418 120% 0,938 54,85% 6,83% Model 3 4,756 113% 0,615 54,66% 6,39%

Model 2 4,456 121%



Model 1 4,59 125% 0,923 53,98% 6,93% Model 1 5,84 138% 0,604 53,65% 6,49%



Appendix

142

Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement



Model 2 6,269 131% 0,295 74,52% 8,55% Model 2 7,01 125% 0,199 81,54% 7,29%

Model 3 6,122 128% 0,293 73,98% 8,59% Model 3 6,978 124% 0,200 81,78% 7,27%



Model 1 6,302 132% 0,291 73,48% 8,62% Model 1 7,124 127% 0,198 81,25% 7,32%

10,64% Model 2 5,22 124% 0,385 66,75% 9,77%

Model 3 4,155 113% 0,579 65,42% 10,61% Model 3 5,153 122% 0,387 67,11% 9,74%

Model 2 4,221 115% 0,576 65,12%



Model 1 4,323 118% 0,569 64,25% 10,70% Model 1 5,292 125% 0,382 66,15% 9,81%



Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement



Model 2 5,595 117% 0,275 69,45% 9,86% Model 2 6,859 122% 0,185 75,68% 8,07%

Model 3 5,525 116% 0,276 69,76% 9,83% Model 3 6,902 123% 0,186 76,12% 8,02%



Model 1 4,625 97% 0,272 68,75% 9,92% Model 1 6,995 125% 0,184 75,37% 8,12%

Model 2 5,211 142% 0,639 72,23% 8,74% Model 2 5,725 135% 0,423 73,32% 8,65%

Model 3 5,23 143% 0,648 73,24% 8,68% Model 3 5,721 135% 0,428 74,12% 8,59%



Model 1 5,336 145% 0,633 71,52% 8,81% Model 1 5,751 136% 0,420 72,87% 8,71%



Appendix

143

Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement



Model 2 6,05 127% 0,281 70,86% 8,98% Model 2 6,345 113% 0,1705316 69,89% 9,44%

Model 3 6,096 127% 0,280 70,65% 9,01% Model 3 6,253 111% 0,171654 70,35% 9,39%



Model 1 6,202 130% 0,278 70,22% 9,04% Model 1 6,491 116% 0,1686528 69,12% 9,50%

Model 2 4,913 134% 0,605 68,32% 9,28% Model 2 5,659 134% 0,399 69,12% 8,92%

Model 3 4,9 134% 0,602 68,05% 9,31% Model 3 5,59 132% 0,403 69,76% 8,86%



Model 1 5,05 138% 0,598 67,52% 9,35% Model 1 5,612 133% 0,395 68,42% 8,97%



Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


9,05%

Model 2 5,45 129% 0,397 68,87% 9,03%

Model 1 5,425 128% 0,395 68,52%

8,57%Model 3 6,679 119% 0,178 72,87%

8,61%

Model 2 6,722 120% 0,178 73,12% 8,53%

Model 1 6,842 122% 0,177 72,48%

Flat plate 4,782 100% 0,396 100%

Model 2

8,97%


Flat plate 5,613 100% 0,244 100% 6,00%

Model 3 5,352 127% 0,400 69,33%

67,55%

Flat plate 3,669 100% 0,885 100%


Flat plate 4,226 100% 0,577 100% 6,00%

9,02%

Model 3 6,1 128% 0,285 71,85% 8,98%

Model 2 6,05 127% 0,283 71,42%

6,00%

Model 1 6,25 131% 0,279 70,52% 9,10%



Model 3


Model 1


0,613

0,606129%

129%4,751

4,726 68,45%

69,21%

9,40%

9,32%

9,29%

6,00%

4,821 131% 0,598


Appendix

144

Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement



Model 2 5,995 125% 0,301 75,97% 7,90% Model 2 6,702 119% 0,203 83,12% 6,68%

Model 3 5,997 125% 0,302 76,34% 7,86% Model 3 6,652 119% 0,204 83,45% 6,66%



Model 1 6,026 126% 0,298 75,21% 7,95% Model 1 6,851 122% 0,202 82,67% 6,73%

10,06% Model 2 4,955 117% 0,394 68,21% 9,16%

Model 3 4,056 111% 0,585 66,12% 10,70% Model 3 4,992 118% 0,394 68,33% 9,13%

Model 2 4,103 112% 0,586 66,21%



Model 1 4,125 112% 0,582 65,74% 10,11% Model 1 5,067 120% 0,390 67,53% 9,22%



Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement



Model 2 5,312 111% 0,280 70,75% 9,35% Model 2 6,612 118% 0,189 77,56% 7,74%

Model 3 5,295 111% 0,279 70,54% 9,37% Model 3 6,592 117% 0,189 77,41% 7,75%



Model 1 5,321 111% 0,278 70,15% 9,39% Model 1 6,712 120% 0,188 76,85% 7,79%

8,21% Model 2 5,386 117% 0,430 74,52% 7,70%

Model 3 4,956 135% 0,648 73,21% 8,18% Model 3 5,412 128% 0,433 75,12% 7,63%

Model 2 5,021 137% 0,645 72,87%



Model 1 5,124 140% 0,640 72,28% 8,25% Model 1 5,475 130% 0,425 73,65% 7,77%



Appendix

145

Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement



Model 2 5,703 119% 0,298 75,13% 8,60% Model 2 6,125 109% 0,175 71,54% 7,07%

Model 3 5,685 119% 0,299 75,45% 8,57% Model 3 6,098 109% 0,176 71,98% 6,90%



Model 1 5,887 123% 0,295 74,45% 8,65% Model 1 6,211 111% 0,173 70,87% 9,05%

Model 2 4,824 131% 0,609 68,87% 8,91% Model 2 5,151 122% 0,406 70,41% 7,70%

Model 3 4,852 132% 0,614 69,41% 8,87% Model 3 5,172 122% 0,405 70,21% 7,58%



Model 1 4,971 135% 0,605 68,41% 8,95% Model 1 5,301 125% 0,403 69,83% 8,21%



Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement



Model 2 5,601 117% 0,291 73,45% 8,65% Model 2 6,412 114% 0,181 74,25% 7,07%

Model 3 5,542 116% 0,293 73,87% 8,60% Model 3 6,445 115% 0,183 75,12% 6,90%



Model 1 5,723 120% 0,288 72,61% 8,71% Model 1 6,592 117% 0,180 73,67% 7,95%

Model 2 4,687 128% 0,612 69,12% 8,92% Model 2 5,032 119% 0,406 70,35% 8,18%

Model 3 4,69 128% 0,615 69,45% 8,89% Model 3 5,015 119% 0,409 70,86% 8,12%



Model 1 4,775 130% 0,606 68,47% 8,96% Model 1 5,136 122% 0,403 69,87% 8,25%



Appendix

146

Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


0,244 100%

0,3044052 76,87% 7,55% Model 3 6,447 115% 0,207 84,88%

6,456 115% 0,206 84,54%


3,986 109% 0,596 67,32% 9,67% Model 3

Model 2 5,721 120% 0,3053952 77,12% 7,52% Model 2


4,782 100% 0,396

6,34%

Model 3 5,695 119%


6,30%

4,592 109% 0,402 69,75% 8,62%

4,04 110% 0,591 66,81% 9,72% Model 1

Flat plate


5,613 100%

4,665 110% 0,397 68,77% 8,73%

6,00%

Model 1 5,79 121% 0,302346 76,35% 7,59% Model 1 6,5769 117% 0,205 83,86% 6,38%

9,64% Model 2 4,564 108% 0,399 69,21% 8,69%

Model 3

Model 2 3,98 108% 0,599 67,65%

Model 1





Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement



Model 2 5,036 105% 0,361 91,15% 8,85% Model 2 6,213 111% 0,191 78,35% 7,51%

Model 3 5,045 105% 0,361 91,19% 8,86% Model 3 6,158 110% 0,192 78,52% 7,49%



Model 1 5,122 107% 0,282 71,27% 8,92% Model 1 6,372 114% 0,189 77,65% 7,57%

7,59% Model 2 5,09 120% 0,435 75,32% 6,94%

Model 3 4,853 132% 0,632 73,86% 7,52% Model 3 5,126 121% 0,434 75,21% 6,95%

Model 2 4,876 133% 0,629 73,54%



Model 1 4,95 135% 0,623 72,86% 7,65% Model 1 5,198 123% 0,432 74,85% 6,98%



Appendix

147

Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model 2 5,512 115% 0,294 74,21% 7,87% Model 2 5,741 102% 0,178 72,87% 8,46%

Model 1 5,892 105%


73,25% 8,41%

114%

Model 3 5,562 116% 0,296 74,66% 7,82% Model 3 5,695 101% 0,179

Flat plate 5,613 100% 0,244


Flat plate 4,782 100% 0,396 100% 6,00%

126% 0,618

100% 6,00%

Model 1 5,61 117% 0,291 73,56% 7,95%

0,413 71,54% 7,62%

Model 3 4,626 126% 0,621 70,12% 8,18% Model 3 4,966 118% 0,415 71,95% 7,58%

0,176 72,21% 8,51%

Model 2 4,631 69,85% 8,21% Model 2 4,832



Model 1 4,761 130% 0,613 69,29% 8,26% Model 1 4,971 118% 0,411 71,21% 7,65%



Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement



0,293 74,11% 7,86% Model 2 6,068 108% 0,184 75,32% 7,41%

Model 1 6,161 110% 0,181 74,22% 7,49%0,292 73,65% 7,91%

Model 3 5,413 113% 0,295 74,52% 7,82% Model 3 5,987 107% 0,185 75,66% 7,37%



Model 1 5,492 115%

Model 2 5,354 112%

0,621 70,21% 8,22% Model 2 4,72 112% 0,413 71,65% 7,62%

Model 3 4,418 120% 0,618 69,87% 8,24% Model 3 4,756 113% 0,415 71,84% 7,59%

Model 2 4,456 121%



Model 1 4,59 125% 0,613 69,31% 8,29% Model 1 4,84 115% 0,411 71,25% 7,67%



Appendix

148

Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement



Model 2 6,269 131% 0,178 84,97% 12,69% Model 2 7,01 125% 0,117 90,12% 12,58%

Model 3 6,122 128% 0,179 85,32% 12,62% Model 3 6,978 124% 0,118 90,45% 12,53%



Model 1 6,302 132% 0,177 84,15% 12,75% Model 1 7,124 127% 0,116 89,57% 12,65%

13,85% Model 2 5,22 124% 0,250 80,52% 13,67%

Model 3 4,155 113% 0,369 79,41% 13,80% Model 3 5,153 122% 0,252 81,23% 13,58%

Model 2 4,221 115% 0,366 78,78%



Model 1 4,323 118% 0,364 78,25% 13,90% Model 1 5,292 125% 0,250 80,64% 13,65%



Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement



Model 2 5,595 117% 0,172 82,12% 13,47% Model 2 6,859 122% 0,112 86,12% 12,84%

Model 3 5,525 116% 0,172 82,13% 13,47% Model 3 6,902 123% 0,112 86,09% 12,86%



Model 1 5,625 118% 0,171 81,47% 13,51% Model 1 6,995 125% 0,111 85,62% 12,90%

Model 2 5,211 142% 0,383 82,45% 12,69% Model 2 5,725 135% 0,261 84,21% 12,91%

Model 3 5,23 143% 0,384 82,67% 12,67% Model 3 5,721 135% 0,265 85,41% 12,82%



Model 1 5,336 145% 0,380 81,75% 12,75% Model 1 5,751 136% 0,259 83,54% 12,98%



Appendix

149

Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement



Model 2 6,05 127% 0,174 82,95% 12,88% Model 2 6,345 113% 0,107 82,45% 13,32%

Model 3 6,096 127% 0,175 83,55% 12,81% Model 3 6,253 111% 0,108 83,15% 13,26%



Model 1 6,202 130% 0,173 82,45% 12,95% Model 1 6,491 116% 0,106 81,25% 13,41%

Model 2 4,913 134% 0,375 80,55% 13,13% Model 2 5,659 134% 0,259 83,56% 13,34%

Model 3 4,751 129% 0,378 81,24% 13,07% Model 3 5,59 132% 0,259 83,42% 13,36%



Model 1 5,05 138% 0,371 79,85% 13,25% Model 1 5,612 133% 0,255 82,12% 13,45%



Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model 2 5,45 129% 0,258571 83,41% 13,32%

Model 1 5,425 128% 0,254665 82,15%

12,96%Model 3 6,679 119% 0,123 94,86%

13,11%

Model 2 6,722 120% 0,123266 94,82% 12,96%

Model 1 6,842 122% 0,109135 83,95%

Model 3 6,1 128% 0,173 82,52% 13,14%

Model 2 6,05 127% 0,175 83,12%

12,00%

Model 1 6,25 131% 0,172 81,97% 13,21%

Flat plate 4,782 100% 0,21 100%

4,821 131% 0,371 79,87%

Flat plate 3,669 100% 0,465 100%


Flat plate 4,226 100% 0,31 100% 12,00%

13,09%

Model 2

13,25%


Flat plate 5,613 100% 0,13 100% 12,00%

Model 3 5,352 127% 0,260555 84,05%

13,47%



Model 3


Model 1


0,378

0,376129%

129%4,751

4,726 80,85%

81,25%

13,28%

13,19%

13,15%

12,00%


Appendix

150

Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement



Model 2 5,995 125% 0,180 85,62% 12,58% Model 2 6,702 119% 0,118 90,74% 12,52%

Model 3 5,997 125% 0,180 85,85% 12,57% Model 3 6,652 119% 0,118 90,85% 12,50%



Model 1 6,026 126% 0,177 84,47% 12,66% Model 1 6,851 122% 0,117 90,12% 12,57%

13,68% Model 2 4,955 117% 0,253 81,76% 13,48%

Model 3 4,056 111% 0,371 79,87% 13,65% Model 3 4,992 118% 0,255 82,32% 13,42%

Model 2 4,103 112% 0,369 79,41%



Model 1 4,215 115% 0,365 78,53% 13,75% Model 1 5,067 120% 0,251 81,12% 13,52%



Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement



Model 2 5,312 111% 0,175161 83,41% 13,29% Model 2 6,612 118% 0,113 87,12% 12,66%

Model 3 5,295 111% 0,174216 82,96% 13,33% Model 3 6,592 117% 0,114 87,31% 12,64%



Model 1 5,321 111% 0,171675 81,75% 13,39% Model 1 6,712 120% 0,112 85,95% 12,79%

12,52% Model 2 5,386 127% 0,263 84,84% 12,78%

Model 3 4,956 135% 0,387 83,25% 12,48% Model 3 5,412 128% 0,264 85,24% 12,71%

Model 2 5,021 137% 0,387 83,14%



Model 1 5,124 140% 0,382 82,13% 12,58% Model 1 5,475 130% 0,260 83,95% 12,84%



Appendix

151

Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement



Model 2 5,703 119% 0,175 83,12% 12,74% Model 2 6,125 109% 0,107 81,94% 13,25%

Model 3 5,685 119% 0,175 83,25% 12,72% Model 3 6,098 109% 0,107 82,13% 13,21%



Model 1 5,887 123% 0,174 82,65% 12,85% Model 1 6,211 111% 0,106 81,56% 13,32%

Model 2 4,824 131% 0,378 81,25% 13,01% Model 2 5,151 122% 0,258 83,25% 13,22%

Model 3 4,852 132% 0,382 82,11% 12,96% Model 3 5,172 122% 0,259 83,45% 13,18%



Model 1 4,971 135% 0,374 80,35% 13,08% Model 1 5,301 125% 0,256 82,67% 13,31%



Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement



Model 2 5,601 117% 0,174 82,65% 13,05% Model 2 6,412 114% 0,110 84,96% 12,98%

Model 3 5,542 116% 0,175 83,45% 12,98% Model 3 6,445 115% 0,111 85,26% 12,91%



Model 1 5,723 120% 0,173 82,27% 13,09% Model 1 6,592 117% 0,109 84,05% 13,03%

Model 2 4,687 128% 0,380 81,65% 13,05% Model 2 5,032 119% 0,259 83,45% 13,22%

Model 3 4,69 128% 0,380 81,75% 13,04% Model 3 5,015 119% 0,259 83,41% 13,23%



Model 1 4,775 130% 0,374 80,52% 13,12% Model 1 5,136 122% 0,256 82,71% 13,34%



Appendix

152

Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


0,13 100%

0,180 85,76% 12,47% Model 3 6,447 115% 0,119 91,32%

6,456 115% 0,119 91,21%


3,986 109% 0,373 80,13% 13,51% Model 3

Model 2 5,721 120% 0,179 85,42% 12,50% Model 2


4,782 100% 0,21

12,44%

Model 3 5,695 119%


12,42%

4,592 109% 0,253 81,65% 13,37%

4,04 110% 0,368 79,13% 13,55% Model 1

Flat plate


5,613 100%

4,665 110% 0,252 81,35% 13,41%

12,00%

Model 1 5,79 121% 0,178 84,81% 12,58% Model 1 6,5769 117% 0,117 90,35% 12,50%

13,48% Model 2 4,564 108% 0,256 82,45% 13,34%

Model 3

Model 2 3,98 108% 0,374 80,42%

Model 1





Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement



Model 2 5,036 105% 0,174 82,86% 13,25% Model 2 6,213 111% 0,113 87,12% 12,65%

Model 3 5,045 105% 0,175 83,15% 13,21% Model 3 6,158 110% 0,114 87,45% 12,62%



Model 1 5,122 107% 0,172 82,05% 13,30% Model 1 6,372 114% 0,112 86,21% 12,71%

12,29% Model 2 5,09 120% 0,265 85,42% 12,68%

Model 3 4,853 132% 0,392 84,23% 12,87% Model 3 5,126 121% 0,265 85,33% 12,69%

Model 2 4,876 133% 0,391 84,12%



Model 1 4,95 135% 0,384 82,56% 12,42% Model 1 5,198 123% 0,262 84,37% 12,74%



Appendix

153

Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model 2 5,512 115% 0,177 84,21% 12,69% Model 2 5,741 102% 0,108 83,15% 13,16%

Model 1 5,892 105%


82,85% 13,19%

114%

Model 3 5,562 116% 0,177 84,31% 12,67% Model 3 5,695 101% 0,108

Flat plate 5,613 100% 0,13


Flat plate 4,782 100% 0,21 100% 12,00%

126% 0,378

100% 12,00%

Model 1 5,61 117% 0,175 83,53% 12,76%

0,260 83,94% 13,14%

Model 3 4,626 126% 0,378 81,27% 12,81% Model 3 4,966 118% 0,261 84,14% 13,09%

0,107 82,05% 13,25%

Model 2 4,632 81,24% 12,79% Model 2 4,832



Model 1 4,761 130% 0,375 80,65% 12,91% Model 1 4,97 118% 0,258 83,07% 13,19%



Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement


Model

Natural

Frequency

(Hz)

RatioDisplacement


Natural

Frequency

(Hz)

RatioDisplacement



0,175 83,12% 12,91% Model 2 6,068 108% 0,111 85,45% 12,91%

Model 1 6,161 110% 0,109 84,21% 12,95%0,174 82,98% 13,00%

Model 3 5,413 113% 0,175 83,45% 12,85% Model 3 5,987 107% 0,111 85,67% 12,88%



Model 1 5,492 115%

Model 2 5,354 112%

0,378 81,24% 12,88% Model 2 4,72 112% 0,261 84,15% 13,16%

Model 3 4,418 120% 0,377 81,13% 12,89% Model 3 4,756 113% 0,262 84,55% 13,12%

Model 2 4,456 121%



Model 1 4,59 125% 0,376 80,79% 12,94% Model 1 4,84 115% 0,258 83,21% 13,21%

