Seismic Vibration Control of Frame Structure Using Shape ...

Seismic Vibration Control of Frame Structure Using Shape Memory

Alloy

by

Md. Golam Rashed

MASTER OF SCIENCE IN CIVIL ENGINEERING (STRUCTURAL)

Department of Civil Engineering

BANGLADESH UNIVERSITY OF ENGINEERING AND TECHNOLOGY

February, 2013

Seismic Vibration Control of Frame Structure Using Shape Memory

Alloy

by

Md. Golam Rashed

A thesis submitted to the Department of Civil Engineering of Bangladesh University

of Engineering and Technology, Dhaka, in partial fulfilment of the requirements for

the degree of

MASTER OF SCIENCE IN CIVIL ENGINEERING (STRUCTURAL)

February, 2013

ii

The thesis titled “Seismic Vibration Control of Frame Structure Using Shape

Memory Alloy” submitted by Md. Golam Rashed, Roll No.: 0411042321, Session:

April 2011 has been accepted as satisfactory in partial fulfilment of the requirement

for the degree of M.Sc. Engineering (Civil and Structural) on 25th February, 2013.

BOARD OF EXAMINERS

______________________________________

Dr. Raquib Ahsan Chairman

Professor (Supervisor)


BUET, Dhaka.

______________________________________ Dr. Md. Mujibur Rahman Member

Professor and Head (Ex-officio)


BUET, Dhaka.

______________________________________

Dr. Tahsin Reza Hossain Member

Professor


BUET, Dhaka.

______________________________________ Dr. Sharmin Reza Chowdhury Member

Associate Professor (External)


AUST, Dhaka.

iii

DE C L A R A T I O N

It is hereby declared that this thesis or any part of it has not been submitted elsewhere

for the award of any degree or diploma.

_________________________________

Md. Golam Rashed

iv

DE D I C A T I O N

To my wife, for being patient and understanding.

v

AC K N O W L E D G E M E N T

At first I would like to express my whole hearted gratitude to Almighty Allah for each

and every achievement of my life.

I would like to express my great respect and gratitude to my thesis supervisor, Dr.

Raquib Ahsan, Professor, Department of Civil Engineering, BUET for providing me

continuous support and guideline to perform this research work and to prepare this

concerted dissertation. His contribution to me can only be acknowledged but never be

compensated. His consistent inspiration helped me to work diligently throughout the

completion of this research work and also contributed to my ability to approach and

solve a problem. It was not easy to complete this work successfully without his

invaluable suggestions and continuous help and encouragement. Despite many

difficulties and limitations he tried his best to support the author in every field related

to this study.

I would like to express my deepest gratitude to the Department of Civil Engineering,

BUET, The Head of the Department of Civil Engineering and all the members of

BPGS committee to give me such a great opportunity of doing my M.Sc. and this

contemporary research work on structural application of Shape Memory Alloy

(SMA).

I would like to render sincere gratitude to Dr. Toby Kim Parnell, USA and Dr. Furo

Jumbo, UK for providing useful knowledge on SMA simulation and advanced FEA. I

am grateful to Dr. Rafiqul A. Tarefder, UNM, USA for providing the experimental

test data and to Dr. Mehedi Ahmed Ansary, BUET, for providing required

computational facilities.

I would like to convey my gratefulness and thanks to my family, their undying love,

encouragement and support throughout my life and education. Without their

blessings, achieving this goal would have been impossible.

At last I would like to thank my respected supervisor Dr. Raquib Ahsan once again

for giving me such an opportunity, which has obviously enhanced my knowledge and

skills as a structural engineer to a great extent.

vi

AB S T R A C T

The use of Shape Memory Alloy (SMA) in mitigating the seismic vibration response

of civil infrastructure is gaining momentum. The name “Shape Memory” implies that

it remembers its original formed shape. SMA has two basic properties, Super-

Elasticity and Shape Memory Effect (SME). The “Super-Elastic” behaviour exhibited

by SMA materials, permits a full recovery of strains up to 8% from large cyclic

deformations, while developing a hysteretic loop. SME allows the material to recover

the initial shape or position which in turn can be used as re-centering mechanism. The

mechanism of shape recovery involves two crystallographic phases, Martensite and

Austenite, and the transformations between them. The Austenite phase provides more

stiffness than the Martensite phase. Phase transformation occurs between Martensite

& Austenite depending upon temperature & stress. These unique properties result in

high damping, combined with repeatable re-centering capabilities which can be used

to advantage in several civil infrastructure applications, especially in seismic vibration

control devices.

Super-Elastic response of SMA has historically been the primary mode of interest of

civil engineers as it occurs over a wide-range of temperatures; and also because SMA

reaches activation temperature and becomes Austenite at the ambient temperature of

civil engineering infrastructures. Thus the re-centering capability of SMA by

generation of an activation force is not utilized. The use of high temperature SMA has

enabled the re-centering mechanism to work. The SMA is heated by electrical current

flow and the use of constant current in this purpose will result in greater power

consumption which can be reduced significantly by passing pulsed current through

the SMA using Pulse Width Modulation (PWM) technique.

In this study, both the Super-Elastic and Shape Memory Effect has been taken into

account by using SMA with high activation temperature. A Thermo-Mechanical SMA

phenomenological constitutive model is used to simulate the SMA behaviour. The

dynamic response data of a frame structure has been obtained from FE analysis by

using the nonlinear FE software program MSC Marc. Then the frame is braced and

reanalyzed; first using standard steel wire and then later using SMA wire, the seismic

response of both the braced frames were measured. The SMA bracing is activated by

joule-heating due to electrical current flow. The SMA is first activated by constant

current, later using pulsed current. In this research work, From the FE solutions, the

effectiveness of SMA braces as a seismic vibration control device and guidelines to

optimum electrical input, considering appropriate stiffness and damping

characteristics; is established. From the simulation result, it is evident that the use of

pulsed current resulted in reduced energy consumption by the SMA, as well as

mitigating the seismic vibrations on the frame structure.

vii

CO N T E N T S

Page

No.

DECLARATION iii

DEDICATION iv

ACKNOWLEDGEMENT v

ABSTRACT vi

LIST OF FIGURES x

LIST OF TABLES xvi

LIST OF ABBREVIATIONS xvii

NOTATIONS xviii

CHAPTER 1 INTRODUCTION

1.1 General 19

1.2 Background and Present State of the Problem 20

1.3 Objectives of the Present Study 20

1.4 Scope and Methodology of the Study 21

1.5 Organization of the Thesis 21

CHAPTER 2 LITERATURE REVIEW

2.1 General 23

2.2 Basic Characteristics of SMA’s 25

2.2.1 Shape Memory Effect 27

2.2.2 Pseudo-Elasticity 27

2.2.3 Damping Properties 28

2.3 Constitutive Modeling of Shape Memory Alloys 29

viii

2.3.1 Phenomenological Modeling 29

2.3.2 Thermodynamics-Based Modeling 29

2.4 Structural Applications of SMA in Civil

Engineering

30

2.4.1 SMA in Building Structures 30

2.4.2 SMA in Bridge Structures 32

2.5 Limitations 33

2.6 Concluding Remarks 33

CHAPTER 3 CONSTITUTIVE MODELING

3.1 General 35

3.2 Overview of Constitutive Modeling of Shape

Memory Alloys

35

3.3 Saeedvafa Constitutive Model for Shape

Memory Alloy

37

3.4 Implementation 45


CHAPTER 4 VERIFICATION

4.1 General 49

4.2 Experimental Setup and Geometric Properties 49

4.3 Material Properties 51

4.4 Modeling Assumptions and Analysis Procedure 53

4.5 Verification 58

4.5.1 El Centro Case 59

4.5.2 Northridge Case 61


ix

CHAPTER 5 RESULTS AND DISCUSSIONS

5.1 General 64

5.2 Simulation Parameters and Procedures 64

5.3 Unbraced and Steel Braced Frame Parametric

Study

67

5.4 SMA braced Frame Parametric Study 73

5.4.1 Constant Current 74

5.4.2 Pulsed Current 87

CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS

6.1 Conclusions 104

6.2 Recommendations for Further Studies 105

REFERENCES 106

Appendix - A 112

x

L I S T O F F I G U R E S

Page No.

Figure 2.1 (a) SMA phases - Austenite 23

Figure 2.1 (b) SMA phases - twinned Martensite 23

Figure 2.1 (c) SMA phases - detwinned or deformed Martensite 23

Figure 2.2 Temperature-induced phase transformation of an SMA

without applied stress

25

Figure 2.3 Schematic of the shape memory effect of an SMA

showing the detwinning of the material with an applied

stress

26

Figure 2.4 Schematic of the shape memory effect of an SMA

showing the unloading and subsequent heating to

austenite under no load condition

26

Figure 2.5 Temperature-induced phase transformation in the

presence of applied load

26

Figure 2.6 (a) SME stress-strain diagrams of NiTi SMA 27

Figure 2.6 (b) PE stress-strain diagrams of NiTi SMA 27

Figure 2.6 (c) Ordinary plastic deformation stress-strain diagrams of

NiTi SMA

27

Figure 2.7 (a) Typical stress–strain curve of Superelastic SMA under

cyclic axial stresses

28

Figure 2.7 (b) Typical stress–strain curve of Martensite SMA under

cyclic axial stresses

28

Figure 2.8 (a) SMA braced frame 30

Figure 2.8 (b) Schematic of SMA connector for steel structures 30

Figure 2.8 (c) Steel beam-column connection using SMA tendons 30

Figure 2.9 (a) Schematic of the SMA isolation system for buildings 31

Figure 2.9 (b) Schematic of the SMA spring isolation device 31

Figure 2.9 (c) Schematic of a bell tower using SMA anchorage

retrofitting

31

Figure 2.10 (a) Schematic of the setup of SMA restrainer for a simple-

supported bridge

32

xi

Figure 2.10 (b) Schematic of the SMA damper for a stay-cable bridge 32

Figure 2.10 (c) Schematic of the SMA isolation device for elevated

highway bridges

32

Figure 3.1 Austenite to Martensite and Martensite to Austenite

Decomposition

37

Figure 3.2 Thermal History 39

Figure 3.3 G-function under 100% Martensite 42

Figure 3.4 Saeedvafa model implemented in MSC Marc 2012 –

Overview.

46


Austenite to Martensite.

46


Martensite to Austenite

46


Phase transformation parameters

47

Figure 3.8 Saeedvafa model implemented in MSC Marc 2012 -

Plasticity

47


Thermal expansion

47

Figure 4.1 A Schematic of the experimental setup 49

Figure 4.2 Picture of experimental setup 50

Figure 4.3 Stress-Strain curve for steel 51

Figure 4.4 (a) FE model of the frame - Front view 53

Figure 4.4 (b) FE model of the frame - Isometric view 53

Figure 4.4 (c) FE model of the frame - Side view 53

Figure 4.5 SMA brace consisting of three wires 53

Figure 4.6 Step one - Coupled Electrical-Thermal FE model of SMA

brace only

55

Figure 4.7 Step two - Structural FE model of Full frame 55

Figure 4.8 Acceleration at the top from experimental data 59

Figure 4.9 Acceleration at the top from simulation with no damping

in frame

60

xii

Figure 4.10 Acceleration at the top from simulation with 5%

damping in frame

60


damping in frame

61

Figure 4.12 Acceleration at the top from experimental data 61

Figure 4.13 Acceleration at the top from simulation with no damping

in frame

62


damping in frame

62


damping in frame

63

Figure 5.1 Acceleration time history of El Centro earthquake 65

Figure 5.2 Displacement time history of El Centro earthquake 65

Figure 5.3 Fourier Amplitude Spectrum of El Centro earthquake 66

Figure 5.4 Power Spectrum of El Centro earthquake 66

Figure 5.5 Predicted displacement at the top of the unbraced frame,

Earthquake scaled down by a factor of 10

67

Figure 5.6 Predicted displacement at the top of the steel braced

frame, Earthquake scaled down by a factor of 10

68



68



69



69



70



70

Figure 5.12 Predicted displacement at the top of the Steel braced


71

Figure 5.13 Maximum displacement of Unbraced and Steel braced

frame at different scaling factor

71

Figure 5.14 Plastic strain of steel brace at different scaling factor 72

Figure 5.15 Constant current input for 1A current 73

xiii

Figure 5.16 Pulsed current input for 1A current 74

Figure 5.17 Predicted maximum temperature history of the SMA

brace for 0A current

75

Figure 5.18 Predicted displacement at the top of the frame for 0A

current

75


brace for 1.4A current

76

Figure 5.20 Predicted displacement at the top of the frame for 1.4A

current

76



77


current

77



78


current

78



79


current

79



80


current

80



81


current

81



82


current

82



83


current

83



84

xiv


current

84



85


current

85



86


current

86

Figure 5.41 Optimum constant current 87



88


current

89



89


current

90



90


current

91



91


current

92



92


current

93



93


current

94



94


current

95

xv



95


current

96



96


current

97



97


current

98



98


current

99



99


current

100



100


current

101



101


current

102

Figure 5.70 Optimum pulsed current 102

Figure A1 SMA wires connected in parallel 112

Figure A2 Comparative study of equivalent and single wire

maximum temperature

114

xvi

L I S T O F TA B L E S

Page No.

Table 2.1 Typical properties of NiTi SMA with comparison to

Structural steel

24

Table 4.1 Geometric properties of frame 50

Table 4.2 Material properties for 304 steel 51

Table 4.3 Material properties for 90C Flexinol SMA wire 52

Table 4.4 Details of finite element models used in the study 57

Table 4.5 Mesh convergence study 58

Table 5.1 Modal frequencies of the unbraced and steel braced

frame

67

Table 5.2 Total electric energy consumption 103

xvii

L I S T O F AB B R E V I A T I O N

FE Finite Element

FFT Fast Fourier Transformation

NiTi Nickel-Titanium

PE Pseudo-Elasticity

PWM Pulsed Width Modulation

SMA Shape Memory Alloy

SME Shape Memory Effect

TRIP Transformation Induced Plasticity

xviii

NO T A T I O N S

𝐴𝑓 Austenite Finish temperature

𝐴𝑠 Austenite Start temperature

𝑀𝑓 Martensite Finish temperature

𝑀𝑠 Martensite Start temperature

𝜎𝑒𝑞 Von Mises equivalent stress

𝐶𝑚 Austenite to Martensite slope

𝐶𝑎 Martensite to Austenite slope

∆𝜺𝐸𝑙 Elastic strain

∆𝜺𝑇𝑕 Thermal strain

∆𝜺𝑃𝑕 Phase transformation strain

∆𝜺𝑃𝑙 Plastic strain

∆𝜺𝑇𝑟𝑖𝑝 Trip strain

∆𝜺𝑇𝑤𝑖𝑛 Twin strain

∆𝑓 Volume fraction Martensite

∆𝜆 Equivalent plastic strain increment

𝑕𝛼 Hardening coefficient

𝑲 Bulk modulus

𝜗 Poisson’s ratio of the aggregate

𝛼𝑎 Coefficient of thermal expansion for the Austenite

𝛼𝑚 Coefficient of thermal expansion for the Martensite

𝜇 Shear modulus

𝜀𝑒𝑞𝑇 Deviatoric part of transformation strain

𝜀𝑣𝑇 Volumetric part of the transformation strain

I Identity tensor

𝑇 Temperature

𝑔 G function

CH A P T E R 1

IN T R O D U C T I O N

1.1 General

Destruction of infrastructures due to strong ground motion warrants new and

innovative materials to be used in designing earthquake resistant structures. One of

the more promising techniques for anti-seismic resistance of structures involves the

use of shape memory alloys (SMA). This class of metal is known to exhibit several

exceptional properties. Shape-memory properties for nickel (Ni) titanium (Ti) alloy

were discovered in the 1960s, at the Naval Ordnance Laboratory (NOL); hence, the

acronym NiTi-NOL or Nitinol, which is commonly used when referred to Ni-Ti based

shape-memory alloys. SMA’s owe their unique properties to solid-solid phase

transformations that occur as a result of thermal or mechanical changes. SMA

materials undergo transformations between two stable phases, austenite and

martensite, and return to their original un-deformed position due to either a change of

temperature, the shape memory effect; or removal of stresses, the superelastic effect.

Since the material behavior is hysteretic and yet does not have any residual

displacement, a substantial amount of energy dissipation capacity and re-centering

ability is offered (Ozbulut, 2007).

In recent times, several researchers have explored the application possibility of SMA

in civil infrastructures. Fugazza (2003) proposed a uniaxial constitutive model for

Superelastic SMA’s that are candidates for installation into civil engineering

structures. Auricchio (1995) developed one and three dimensional thermo-mechanical

constitutive models for SMA materials. Both of these research studies included

numerical simulations to show the ability of their models and exemplify the dynamic

behavior of SMA’s. More recently, Penar (2005) studied NiTi shape memory material

with the goal of developing a re-centering beam-column connection for steel frames.

Also, McCormick (2006) investigated cyclical properties of large diameter shape

memory alloys for structural applications.

20

Despite the fact that recent studies have contributed significantly to comprehension of

the potential for using shape memory alloys in civil engineering applications, there

are still many questions to be answered before full-scale applications can be

implemented. To this end, this thesis adopts a thermo-mechanical model fully capable

of simulating the Super elasticity and shape memory effect. A framework has been

suggested on how to reduce the seismic vibration using high temperature SMA wire

as bracing system.

1.2 Background and Present State of Problem

Shape memory alloys (SMA) are widely used in different disciplines and it has

substantial potential for civil engineering applications (Alam et al., 2007, Song et al.,

2006, Janke et al., 2005). The name shape memory implies that it remembers its

original formed shape. The super-elastic behavior exhibited by SMA’s help material

to totally recover from large cyclic deformations, while developing a hysteretic loop

(Lagoudas, 2008). The mechanism of shape regaining works in two phases,

Martensite and Austenite. Austenite phase provides more stiffness than that of

Martensite and civil engineers can leverage the variation in stiffness depending upon

temperature and stress. These unique properties result in high damping, combined

with repeatable re-centering capabilities which can be used in civil infrastructures,

especially in seismic vibration control devices (Saadat et al., 2002).

Super-elasticity of SMA is primary interest of civil engineers till now. As the SMA

reaches activation temperature at the ambient temperature of civil engineering

infrastructures, shape memory effect does not take place. Thus the re-centering

capability of SMA is left unused (Ozbulut et al., 2011). Both the super-elasticity and

shape memory effect can be taken into account by using SMA of high activation

temperature (Churchill & Shaw, 2008). Proper numerical simulation of SMA can be

done by taking both the super-elasticity & shape memory effect into consideration

(Saeedvafa & Asaro, 1995).

1.3 Objectives of the Present Study

The objective of the present study includes,

21

To overview the state of the art of Shape Memory Alloy (SMA) usage in civil

engineering infrastructures.

To study the available constitutive theories and suitable numerical

implementation of Shape Memory Alloy (SMA).

To simulate the behavior of frame structure with Shape Memory Alloy (SMA)

considering earthquake oscillations.

The outcome from the study is,

To provide guidelines for use of Shape Memory Alloy (SMA) in civil

engineering infrastructures considering appropriate stiffness and damping

characteristics.

1.4 Scope and Methodology of the Study

Shake table vibration response data of a frame structure has been obtained from

literature where the response of the frame structure subjected to seismic vibration on

shake table was measured. The frame structure was braced using Shape Memory

Alloy (SMA) and later again using standard steel. The response of both the braced

frames was measured accordingly. The SMA bracing was activated by joule-heating

due to electrical current flow. In this study, a coupled Thermo-Mechanical SMA

phenomenological constitutive model will be used to simulate the SMA response. For

this purpose a program, MSC Marc based on nonlinear Finite Element Method (FEM)

will be used to analyze the braced frames. The FEM solutions will be compared with

the experimental data and thus the Thermo-Mechanical SMA phenomenological

model used in numerical simulation will be verified. Further guidelines to optimum

electrical input considering appropriate stiffness and damping characteristics will be

established, also the effectiveness of SMA braces as a seismic vibration control

device will be studied from numerical simulation data.

1.5 Organization of the Thesis

Apart from this chapter, the remainder of the thesis has been divided into five

chapters.

22

Chapter 2 presents literature review concerning properties of SMA materials, past

research on the field of SMA constitutive modeling, application of SMA in civil

engineering infrastructures.

Chapter 3 presents thermo-mechanical constitutive modeling of SMA using additive

decomposition. This model describes both the Super-elasticity and Shape Memory

Effect.

Chapter 4 presents the experimental test setup with results obtained from literature

and verification of the presented SMA phenomenological constitutive model.

Chapter 5 presents a semi-active vibration control framework and determination of

optimum current input required to reduce seismic vibration response of the frame

structure.

Finally, Chapter 6 presents the major conclusions of the study and also provides

recommendations for future study.

CH A P T E R 2

L I T E R A T U R E RE V I E W

2.1 General

The name “shape memory” implies that it remembers its original formed shape. The

super-elastic behavior exhibited by shape-memory alloys help material to totally

recover from large cyclic deformations, while developing a hysteretic loop. Due to its

hysteretic behavior and excellent re-centering capability, SMA can be used in a wide

variety of civil engineering applications. The other key features of SMA’s include

high strength, good fatigue and corrosion resistance, high damping capacity,

temperature-dependent Young’s modulus, ability to undergo large deformations, and

availability in many possible shapes and configurations.

(a) (b) (c)

Figure 2.1: SMA phases: (a) Austenite; (b) twinned Martensite; and (c) detwinned or

deformed Martensite (Key to Metals AG, 2008).

SMA’s have two main phases which have different crystal structures. One is called

Martensite that is stable at low temperatures and/or high stresses and the other

Austenite, which is stable at high temperatures and/or low stresses. Austenite, also

named as the parent phase, generally has a cubic crystal structure while Martensite

has a less-ordered crystal structure (Song et al., 2006). The un-deformed Martensite

phase is the same size and shape as the cubic austenite phase on a macroscopic scale,

so that no change in size or shape is visible in shape memory alloys until the

24

Martensite is deformed. Martensite also has two forms that are termed twinned and

detwinned (Figure 2.1). SMAs owe their peculiar characteristics to the solid-to-solid

transformations between these two phases. Austenite phase provides more stiffness

than that of Martensite and civil engineers can leverage the variation in stiffness

depending upon temperature & stress.

Table 2.1: Typical properties of NiTi SMA with comparison to Structural steel

(Santos, 2011).

Property NiTi SMA Structural

Steel Austenite Martensite

Density (g/cm3) 6.45 7.85

Recoverable elongation (%) up to 8 0.20

Young’s modulus (GPa) 30–83 21–41 200

Yield strength (MPa) 195–690 70–140 248 − 517

Ultimate tensile strength (MPa) 895–1900 448 − 827

Elongation at failure (%) 5–50 (typically 25) 20

Poisson’s ratio 0.33 0.27 − 0.30

Corrosion performance Excellent (similar to

stainless steel)

Poor

A comparison between structural steel and Nickel-Titanium (NiTi) SMA in its

Martensitic and Austenitic phases is presented in Table 2.1. It is seen that structural

steel is much stiffer than NiTi and that the Martensitic yield strength of NiTi is lower

than its Austenitic counterpart. However, the most important characteristic of NiTi is

its outstanding ability to recover from strains up to about 8%, without residual

deformations, while showing a mechanical hysteresis. This provides the material with

unique energy dissipation and re-centering capabilities (Santos, 2011).

Over the past two decades, SMA’s have been widely investigated for their possible

application in civil engineering structures (Rashed et al., 2012). This chapter tries to

cover some latest applications of SMA’s in civil engineering along with established

examples while also focusing on the limitations in application.

25

2.2 Basic Characteristics of SMA’s

SMA’s have two unique properties, The Shape Memory Effect (SME) which is the

phenomenon that the material returns back to their original shape upon heating and

the Pseudo-Elasticity (PE) which is the phenomenon that the material can undergo a

large amount of inelastic deformation and recover after unloading. These properties

are the result of reversible phase transformations between the Austenite phase and the

Martensite phase.

In the stress-free state, an SMA is characterized by four transition temperatures such

as Martensite start temperature Ms, Martensite finish temperature Mf, Austenite start

temperature As and Austenite finish temperature Af. At a temperature below Mf, the

SMA exhibits the SME and at a temperature above Af, the SMA exhibits the PE.

Phase transformations may be temperature-induced or stress-induced illustrated in

Figure 2.5. An SMA that is in the austenite state transforms to the Martensite state

upon cooling. A reverse transformation, from Martensite to austenite, takes place

when the material is heated (Figure 2.2).

Figure 2.2: Temperature-induced phase transformation of an SMA without applied

stress (Lagoudas, 2008).

26

Figure 2.3: Schematic of the shape memory effect of an SMA showing the detwinning

of the material with an applied stress (Lagoudas, 2008).

Figure 2.4: Schematic of the shape memory effect of an SMA showing the unloading

and subsequent heating to austenite under no load condition (Lagoudas, 2008).

Figure 2.5: Temperature-induced phase transformation in the presence of applied load

(Lagoudas, 2008).

27

(a) (b) (c)

Figure 2.6: Stress-strain diagrams of NiTi SMA, (a) SME; (b) PE; (c) Ordinary plastic

deformation (Qian et al., 2010).

2.2.1 Shape Memory Effect

Shape memory effect (SME) is a unique characteristic of SMA that exhibits thermo-

elastic Martensitic phase transformation. It is the ability of SMA material to recover

its original shape after being deformed through a thermal cycling. Through training,

the material has the ability to memorize a very specific physical configuration or

shape in either the Martensite or Austenite phase, which is called one-way shape

memory (Figure 2.6a). Also it is possible to train the material, such that it memorizes

two different configurations or shapes in Martensite and Austenite phases, which is

called two-way shape memory.

The key to the SME is the build-up of residual stress fields within the SMA, by

deforming the material plastically, and then these stress fields control the phase

transformation (Saadat et al., 2002).

2.2.2 Pseudo-Elasticity

Pseudo-Elasticity (PE), also known as super-elasticity, is described as the recovery of

large strain as a result of the stress-induced Martensitic phase transformations under

constant temperature. When T>Af, SMA is in its Austenite phase. If a sufficiently

high stress is applied to the material in the Austenite phase, the SMA transforms into

the detwinned Martensite shown in Figure 2.3. When the load is released, a reverse

transformation to the Austenite state takes place, which results in complete shape

recovery and a substantial hysteretic loop (Figure 2.4 & 2.6b). However, if the

temperature is below Af but above As, there will be only a partial shape recovery.

28

Also, if the temperature in the Austenite phase exceeds the maximum temperature at

which Martensite occurs, Md, the material is stabilized in the Austenite phase and the

Martensitic transformations cannot be induced by an applied load, thus the PE of

SMA is completely lost (Figure 2.6c).

2.2.3 Damping Properties

SMA used for damping can be both Martensitic as well as Austenitic. The damping

comes from either Martensite variations reorientation in the Martensitic material or

from stress-induced Martensite in Austenitic material. When an SMA specimen is

subjected to a cycle of deformation within its Superelastic strain range, it dissipates a

certain amount of energy without permanent deformation (Figure 2.7a). This results

from the phase transformation from Austenite to Martensite during loading and the

reverse transformation during unloading, ensuring a net release of energy. When an

SMA is loaded in the Martensite phase, it yields at a nearly constant stress after initial

elastic deformation and displays strain hardening at larger strains. When unloaded,

there remains some residual strain at zero stress. This Martensitic composition of

SMA’s generates a full hysteresis loop around the origin (Figure 2.7b). Thus,

Martensite SMA dissipates a much higher amount of energy compared with that of

Austenite SMA because of its larger hysteresis loop. But it has no re-centering

capability like the Austenitic SMA. In the Martensite phase under tension–

compression cycles, the maximum stress attained in compression has been found to be

approximately twice that in tension (Figure 2.7b). Although Superelastic SMA

dissipates less energy than Martensitic SMA, its advantage is that it can still dissipate

a considerable amount of energy under repeated load cycles with negligible residual

strain (Alam et al., 2007).

(a) (b)

Figure 2.7: Typical stress–strain curve of SMA under cyclic axial stresses: (a)

Superelastic SMA; (b) Martensite SMA; (Alam et al., 2007).

29

2.3 Constitutive Modeling of Shape Memory Alloys

The modeling of SMA behavior such as SME and PE has been an active area of

research. The modeling approaches can be categorized into phenomenological and

thermo-dynamical approaches.

2.3.1 Phenomenological Modeling

The phenomenological modeling is essentially a macroscopic approach which

attempts to capture the SMA response at the macroscopic level using phenomenology.

These models are based on setting up material constants of a model to match the

experimental data. A large number of phenomenological models had been proposed to

capture response of SMA, both mechanical and thermo-mechanical, due to their

relative simplicity and accuracy. In the field of civil engineering, SMA’s are mostly

used as bars and wires. For this reason it is convenient to use one-dimensional

phenomenological model. Phenomenological model of SMA have been implemented

in several finite element software packages such as Ansys, Abaqus, SeismoStruct,

MSC Marc; where the material models are included from Auricchio et al. (1997),

Auricchio & Taylor, 1996; Auricchio & Sacco, 1997 and Saeedvafa, 2001

respectively. Only the Saeedvafa model takes both SME and PE into consideration

while the rest only replicates PE (Choudhry & Yoon, 2004).

2.3.2 Thermodynamics-Based Modeling

The thermodynamics-based modeling is essentiality a microscopic approach which is

built on the laws of thermodynamics and energy considerations by pursuing closely

crystallographic phenomena within the material. Several Thermodynamics-based

models have been proposed, some of which are Patoor et al. (1994), Goo and

Lexcellent (1997), Huang and Brinson (1998). Thermodynamics-based models are

more complicated and computationally expensive than phenomenological models

because they present a highly sensible technique to derive accurate three-dimensional

constitutive law (Alam et al., 2007).

30

2.4 Structural Applications of SMA in Civil Engineering

The vibration suppression of civil structures to external dynamic loading can be

pursued by using active control, semi-active control, and passive control. The

applications of SMA in civil engineering structures are described below in two broad

categories, Building and Bridge structures.

2.4.1 SMA in Building Structures

Several studies have considered the use of SMA’s as diagonal braces in frame

structures (Saadat et al., 2001; Tarefder et al., 2006). The frame structures deform

under excitation, SMA braces dissipate energy through stress-induced Martensite

transformation (in the Superelastic SMA case) or Martensite reorientation (in the

Martensite SMA case) as shown in Figure 2.8a. Auricchio et al. (2006) investigated

the effectiveness of using large diameter NiTi bars as a bracing system for steel

structures and compared the SMA braces with buckling-restrained steel braces. The

outcome of numerical studies showed that SMA bracing systems can satisfactorily

limit the inter-story drifts in steel buildings and significantly reduce the residual drifts

(McCormick et al., 2007).

(a) (b) (c)

Figure 2.8: (a) SMA braced frame (Tarefder et al., 2006); (b) Schematic of SMA

connector for steel structures (Song et al., 2006); (c) Steel beam-column connection

using SMA tendons (Qian et al., 2010).

Several studies have been conducted on SMA beam-column connectors (Qian et al.,

2010). SMA connectors have been designed to provide damping and tolerate

relatively large deformations and found to be most effective in controlling structural

response under high levels of seismic intensity. Ocel et al. (2004) experimentally

31

evaluated the performance of partially restrained steel beam-column (Figure 2.8b &

2.8c) connections using Martensitic SMA’s. It was observed that the SMA

connections were able to recover 76% of the beam tip displacement.

(a) (b) (c)

Figure 2.9: (a) Schematic of the SMA isolation system for buildings (Qian et al.,

2010); (b) Schematic of the SMA spring isolation device (Song et al., 2006); (c)

Schematic of a bell tower using SMA anchorage retrofitting (Song et al., 2006).

Several studies have been made on SMA based isolation devices for seismic

protection of building structures by performing shake table tests (Song et al., 2006).

But majority of them are Superelastic SMA due to its zero residual strain after

unloading. Martensite SMA’s can be used to help dissipate more energy and further

improving the damping effect of the Superelastic SMA isolation devices. The re-

centering device by Dolce et al. (2001) is a good example of combining the

Superelastic and Martensitic SMA’s (Figure 2.9a).

SMA’s have been used to retrofit existing or damaged structures (Islam et al., 2012).

The San Giorgio Church Bell-Tower, which was damaged in 1996 by earthquake, was

retrofitted using SMA tie bars. As shown in Figure 2.9c, the SMA tie bars, which run

through the height of the tower and are anchored at its foundation, reinforce the

structure and increase its modal frequencies. That tower stood intact after a similar

earthquake in 2000 (Qian et al., 2010).

32

2.4.2 SMA in Bridge Structures

(a) (b) (c)

Figure 2.10: (a) Schematic of the setup of SMA restrainer for a simple-supported

bridge; (b) Schematic of the SMA damper for a stay-cable bridge; (c) Schematic of

the SMA isolation device for elevated highway bridges; (Song et al., 2006).

Several studies have been carried out to investigate the possibility of using SMA as

unseating prevention devices on multiple span bridges to overcome some of the

limitations of traditional devices such as steel cable restrainers, steel rods, and shock

transmission units (Qian et al., 2010). The schematic of SMA restrainer setup for

simple supported bridge is shown in Figure 2.10a.

Both Superelastic and Martensite SMA’s can be used as damper elements for bridges.

Li et al. (2004) theoretically studied the vibration mitigation of a combined cable-

SMA damper system which can be used on a stay-cable bridge (shown in Figure

2.10b). The dynamic responses of the SMA damped cable were simulated as it

vibrated at its first mode or at its first few modes respectively. They stated that the

proposed Superelastic SMA damper can suppress the cable’s vibration in both cases

(Song et al., 2006).

For highway bridges, Comparative simulations of the SMA isolation system (shown

in Figure 2.10c) and a conventional isolation system were conducted with three

excitation levels. For small excitation level, the SMA isolation system firmly links the

pier and the deck, while the relative motion emerges in the case of the conventional

system. For a medium excitation level, the SMA bar undergoes a stress-induced

Martensitic transformation so that the soft stiffness allows a relative displacement

comparable to that of the conventional isolation system. At severe loading, the SMA

bar enters an elastic range of Martensite and the maximum displacement is one-fifth

as much as that of the conventional isolation system. The comparison shows that the

33

damage energy of the bridge with the SMA isolation system is smaller than with the

conventional system (Song et al., 2006).

2.5 Limitations

The price of the SMA’s is high in comparison to the conventional civil engineering

construction materials. However, a significant reduction in the price of SMA has

occurred over the last decade, recently developed Iron based SMA's are quite cheaper

than traditional NiTi, sometimes by ten folds (Alam et al., 2007). Fe-Mn-Si-X alloys

are an example of a potentially low cost SMA.

SMA’s can be heated by using electric current for actuation. But short activation

times in the range of seconds are not possible for large cross sections. A high capacity

power supply with a current of several hundred Ampere can reduce activation times

considerably. This may incur greater costs when setting up the actuator and keeping

up the high temperature state for long time (Janke et al., 2005).

Another difficulty regarding the application of SMA is the machining of large

diameter bars using conventional equipment, due to its hardness. The welding of

SMA’s is often difficult (Desroches & Smith, 2003).

2.6 Concluding Remarks

This chapter presents a review of the basic properties of SMA such as the SME and

the PE. The ability to change shape by application of heat can be used to move

objects. If recovery is resisted, the SMA generates force that is useful as an actuator

which can be activated by electrical Joule heating. Thus many applications are

possible. Numerous analytical and experimental studies point toward the feasibility

and superiority of SMA based devices over conventional methods for seismic

protection. The main characteristics of such devices are high energy dissipation and

re-centering capabilities.

This chapter also reviews the constitutive models developed to carry out numerical

simulation of SMA devices. These constitutive characteristics of SMA’s greatly

depend on variations in the alloy’s component and manufacturing method to begin

with. From discussion it is observed that phenomenological models are more adequate

34

for civil engineering applications because they are simple and easy to incorporate in

finite element programs and are not computationally demanding.

The high price of SMA is a burden for their use as construction materials. Hopefully,

with the advent of new manufacturing and processing techniques, the price of SMA is

reducing over the years.

CH A P T E R 3

CO N S T I T U T I V E MO D E L

3.1 General

The behavior of SMA’s shows a high level of complexity, since it depends on stress

and temperature, and is closely connected with the crystallographic phase of the

material and the thermodynamics underlying the transformation processes. For these

complications, any simulation taking advantage of the peculiar properties of the SMA

has to be based on a proper constitutive model. In this chapter, modeling of SMA by

various researchers has been discussed, followed by describing a thermo-mechanical

SMA phenomenological model. Finally, the implementation of the SMA model in the

non-linear FE program MSC Marc is discussed.

3.2 Overview of Constitutive Modeling of Shape Memory Alloys

The modeling of SMA behavior such as SME and PE has been an active area of

research for the last two decades. The modeling approaches can be categorized into

phenomenological and thermo-dynamical approaches.

Liang & Rogers (1990) reviewed the early phenomenological theories. They

presented a one-dimensional model and does not specifically account for the dual

effects of stress induced transformation and Martensite reorientation. The model

presented by Barrett & Sullivan (1995) have generalized one-dimensional constitutive

description of the shape memory phenomena, using a mixture of phenomenological

relations to describe the effects of multi-axial states of stress on the evolution of

Martensite volume fraction. But it does not lead to a full set of integrable constitutive

equations that may be used for numerical simulation.

The Brinson’s (1993) model was used by Brinson & Lammering (1993) in 3D version

to perform FEM simulation. The limitations of this model are, it describes the

transformation-induced plasticity as a form of non-linear elastic behavior, and it does

not distinguish between deformation caused by the transformation itself and

reorientation events.

36

Thermodynamics-based models to analyze the shape memory alloys are considered

next. However, thermodynamic models limit the possibilities of material behavior as

noted by Boyd & Lagoudas (1996). So, properly constructed phenomenological

theories are easily verified to have no physically objectionable characteristics and are

usually easier to understand and implement (Choudhry & Yoon, 2004).

Models based on more classical plasticity approaches have been presented by

Berveiller et al., 1991; Auricchio & Talyor, 1996; Lubliner & Auricchio, 1996 and

Trochu & Qian, 1997. The model of Lubliner & Auricchio (1996) is most complete in

that it formally accounts for the possibility of visco-plastic deformation in both

Martensite & Austenite phase. But the description of transformation induced plasticity

is limited as only dilatational transformation strains are accounted for and no

distinction is made between transformation due to stress based Austenite-to-

Martensite and those caused by reorientation of Martensite.

Based on the above discussion, while there has been considerable attention paid to the

constitutive modeling of shape memory alloys, no model yet exists that accounts for

the various phenomena of interest to properly simulate and at the same time is simple

enough form to be practicably implementable (Saeedvafa, 2001).

In this chapter, a computational model based on phenomenological approach is

recommended for the efficient and accurate simulation of shape memory alloy. The

framework of such a complete phenomenological model outlined by Saeedvafa (2001)

that provides a description of a wide range of the observed behavior, which are both

tractable from analytical as well as computational viewpoint is presented here. The

present constitutive model is designed to describe the deformation processes that

constitute essentially proportional loading. Transformation induced plasticity occurs

by the process of forming textured Martensite from the parent Austenite phase.

Texturing is in turn induced by the presence of deviatoric stress states. The presented

model describes the development of transformation induced inelastic strains by the

Austenite to Martensite transformation and by additional texturing of Martensite. It

also includes TRIP and TWIN strains with plasticity and thermal transformation

between Austenite and Martensite. The constitutive theory and its implementation are

presented in the following sections.

37

3.3 Saeedvafa Constitutive Model for Shape Memory Alloy

NiTi alloys with near equiatomic composition exhibit a reversible, thermo-elastic

transformation between a high-temperature, ordered cubic (B2) austenitic phase and a

low-temperature, monoclinic (B19) Martensitic phase. The density change and thus

the volumetric are small and on the order of 0.003. The transformation strains are,

thus mainly deviatoric, of the order of 0.07-0.085. However, these small dilatational

strains do not necessarily lead to a lack of pressure sensitivity in the phenomenology.

The behavior of Nitinol is different depending on whether the materials are subjected

to hydrostatic tension or compression (Marc User Documentation, 2012).

Figure 3.1: Austenite to Martensite and Martensite to Austenite Decomposition

(Miyazaki et al., 1981).

The curves indicate that upon cooling, the material transformation from Austenite to

Martensite begins once the temperature is reached. Upon further cooling, the volume

fraction on Martensite is a given function of temperature; the volume fraction

becomes 100% Martensite when the temperature is reached. Upon heating,

transformation from Martensite to Austenite begins only after 𝐴𝑠 temperature is

reached. This re-transformation is complete when the 𝐴𝑓 temperature is reached.

Finally, note that the four transformation temperatures are stress dependent. The

38

experimental data indicate the 𝑀𝑠 , 𝑀𝑓 , 𝐴𝑠 , and 𝐴𝑓 may be approximated from their

stress-free values, 𝑀𝑠0, 𝑀𝑓

0, 𝐴𝑠0, and 𝐴𝑓

0 by

𝑀𝑠 = 𝑀𝑠0 +

𝜎𝑒𝑞

𝐶𝑚

𝑀𝑓 = 𝑀𝑓0 +

𝜎𝑒𝑞

𝐶𝑚, and

𝐴𝑠 = 𝐴𝑠0 +

𝜎𝑒𝑞

𝐶𝑎

𝐴𝑓 = 𝐴𝑓0 +

𝜎𝑒𝑞

𝐶𝑎

Where 𝜎𝑒𝑞 is the Von Mises equivalent stress. At a sufficiently high temperature,

often called the 𝑀𝑑 temperature, transformation to Martensite does not occur at any

level of stress. The transformation characteristics such as the transformation

temperatures depend sensitively on alloy composition and heat treatment.

For the discussion of the thermo-mechanical response of NiTi, the data of (Miyazaki

et al., 1981) is shown in Figure 3.2. Following this thermal history, it is observed that,

when unstrained specimens with fully austenitic microstructures are cooled, the

transformation to Martensite begins at a temperature of 190K; the transformation is

complete at 128K. This established the so-called Martensite start (𝑀𝑠0) and Martensite

finish (𝑀𝑓0) temperatures at 190K and 128K, respectively. With the imposition of an

applied uniaxial tensile stress, the low temperature Martensite is favored and the 𝑀𝑠0

and 𝑀𝑓0 temperatures increase. Upon heating a specimen with fully martensitic

microstructure, the reverse transformation is observed to begin at a temperature of

188K and to be complete at 221K. These define the Austenite start (𝐴𝑠0) and Austenite

finish 𝐴𝑓0 temperatures, respectively. Uniaxial tension tests are carried out in

temperature ranges where T<𝑀𝑠 , 𝑀𝑠 < 𝑇 < 𝐴𝑓 , and 𝐴𝑓 < 𝑇 < 𝑇𝑐 where 𝑇𝑐 is defined

as the temperature above which the yield strength of the austenitic phase is lower than

the stress required to induce the Austenite-to-Martensite transformation.

In the temperature range where 𝑇 < 𝑀𝑓 , the microstructures are all Martensitic. The

stress versus strain curves display a smooth parabolic type of behavior which is

consistent with deformation caused by the movement of defects such as twin

39

boundaries and the boundaries between variants. Note that unloading occurs nearly

elastically and that the accumulated deformation, caused by the reorientation of the

existing Martensite and the transformation of any pre-existing Austenite, remains

after the specimen is completely unloaded. Note also that the accumulated

deformation is entirely due to oriented Martensite and this would be recoverable upon

heating to temperatures above the (𝐴𝑠 − 𝐴𝑓) range. This would, then, display the

shape memory effect.

Figure 3.2: Thermal History (Miyazaki et al., 1981).

Pseudo-elastic behavior is displayed in the temperature range 𝐴𝑓 < 𝑇 < 𝑇𝑐 . In this

range, the initial microstructures are essentially all austenitic, and stress induced

Martensite is formed, along with the associated deformation; upon unloading,

however, the Martensite is unstable and reverts to Austenite thereby undoing the

accumulated deformation.

40

As expected, the stress levels rise with increasing temperature. In this range, the

transformation induced deformation is nearly all reversible upon unloading. At

temperatures where 𝑇 > 𝑇𝑐 , plastic deformation appears to precede the formation of

stress induced Martensite. The unloading part of the stress versus strain behavior

displays nonlinearity and the unloading is now associated with permanent (plastic)

deformation. Permanent deformation, due to plastic deformation of the Austenite, is

Non-recoverable and, if such deformation is large, shape memory behavior is lost.

Below, the constitutive model based on Saeedvafa (2001) is presented. The

constitutive model described in this work is based on additive decomposition and the

kinematics of incremental small strains. The total incremental strain ∆𝜺 is expressed

as follows:

∆𝜺 = ∆𝜺𝐸𝑙 + ∆𝜺𝑇𝑕 + ∆𝜺𝑃𝑕 + ∆𝜺𝑃𝑙 (1)

In Eq. (1), the superscripts, “El”, “Th”, “Ph” and “Pl” mean Elastic, Thermal, Phase

transformation and Plastic, respectively.

The elastic strain is taken to be simply related to as set of elastic modulus, L, and the

Cauchy stress as:

∆𝜺𝐸𝑙 = 𝐿−1: 𝜎 (2)

Also, the thermal strain increment is related to a volume average thermal expansion

coefficient, α through

∆𝜺𝑇𝑕 = 𝛼∆𝑇𝑰 (3)

Where 𝑇 is the temperature and 𝑰 is the identity tensor.

In the work, it is assumed that the phase transformation strain is composed of trip

strain and twin strain as follows:

∆𝜺𝑃𝑕 = ∆𝜺𝑇𝑟𝑖𝑝 + ∆𝜺𝑇𝑤𝑖𝑛 (4)

In Eq. (4), TRIP strain is the deformation by the formation of oriented stress-induced

Martensite and TWIN strain is the deformation by the reorientation of randomly

oriented thermally induced Martensite.

41

For TRIP strain formulation, it is assumed that the extent to which Martensite forms

with preferred variants depends on the intensity of the deviatoric stress state. In

addition, under the assumption of isotropy, it is assumed that the deviatoric part of

∆𝜺𝑃𝑕 is co-axial with the deviatoric stress (Saeedvafa, 2001).

Accordingly, with 𝑓 being the volume fraction of Martensite, ∆𝜺𝑇𝑟𝑖𝑝 is expressed as

follows:

∆𝜺𝑇𝑟𝑖𝑝 = ∆𝑓𝑀𝑔 𝜎𝑒𝑞 𝜀𝑒𝑞𝑇 2

3

𝜎

𝜎𝑒𝑞+ ∆𝑓𝑀𝜀𝑣

𝑇𝐼 + ∆𝑓𝐴𝜀𝑃𝑕 (5)

In Eq. (5), ∆𝑓𝑀 ≥ 0 represents the increment at which Martensite is formed. Also, 𝜀𝑒𝑞𝑇

is deviatoric part of transformation in uniaxial tension and 𝜀𝑣𝑇 is the volumetric part of

the transformation strain. In addition, 𝜎𝑒𝑞 is the effective stress defined with Von-

Mises yield function as 𝜎𝑒𝑞 = 3

2𝜎 : 𝜎 and I is the identity tensor. The function

𝑔 𝜎𝑒𝑞 0 ≤ 𝑔 𝜎𝑒𝑞 ≤ 1 is a measure of the extent to which the Martensite

transformation strains are aligned with the deviatoric stress and is further discussed

later in this section. The first two terms in Eq. (5) describe the development of

transformation strains due to the formation of stress-induced Martensite. ∆𝑓𝐴(< 0) is

the increment of formation of Austenite (the decrease of Martensite volume fraction).

So, the last term in Eq. (5) represents the recovery of the accumulated phase

transformation strain.

Deformation due to the reorientation of Martensite occurs at fixed volume fraction, 𝑓.

With the assumption that the contribution to strain increment due to reorientation is

co-axial (Saeedvafa, 2001), twin strain is defined as:

∆𝜺𝑇𝑤𝑖𝑛 = 𝑓∆𝑔 𝜎𝑒𝑞 𝜀𝑒𝑞𝑇 2

3

𝜎

𝜎𝑒𝑞 ∆𝜎𝑒𝑞 {𝜎𝑒𝑞 − 𝜎𝑒𝑓𝑓

𝑔} (6)

In Eq. (6) the modified form of McCauley’s bracket is used. We also note that there is

no dilatational contribution to TWIN strain and the twinning strain increment is zero

when 𝜎𝑒𝑞 is less than 𝜎𝑒𝑓𝑓𝑔

or when the magnitude of the stress decreases (𝜎𝑒𝑞 < 0). It

is also often observed that there exists a threshold Equivalent stress level below which

de-twinning does not occur; this stress is referred as 𝜎𝑒𝑓𝑓𝑔

. The function 𝑔 in Eq. (5)

42

and (6) has the range of 0 ≤ 𝑔 𝜎𝑒𝑞 ≤ 1. Furthermore, the value of 𝑔 at the stress of

𝜎𝑒𝑓𝑓𝑔

is 𝑔 𝜎𝑒𝑓𝑓𝑔 = 0. Also, in practice the function 𝑔 tends to approach unity at a

finite equivalent stress level, called 𝜎0𝑔

. Thus 𝑔(𝜎0𝑔

) = 1, and the trial function is

chosen to be of the form:

𝑔 𝜎𝑒𝑞 = 1 − 𝑒𝑥𝑝 𝑎 𝜎𝑒𝑞 /𝜎0𝑔

𝑛 (7)

With a < 0. Figure 1 shows the plot of a = - 4.0 and n =2.0 under simple uniaxial

tension and f = 1.0. Then, Eq. (7) integrates to 𝜀𝑇𝑤𝑖𝑛 = 𝑔 𝜎𝑒𝑞 𝜀 𝑒𝑞𝑇 .

A better fit is achieved by replacing the term in the bracket with sum of polynomial of

stress and a general form of the function 𝑔 has been implemented in the present work:

𝑔 𝜎𝑒𝑞 = 1 − 𝑒𝑥𝑝 𝑔𝑎(𝜎𝑒𝑞

𝜎0𝑔 )𝑔𝑏 + 𝑔𝑐(

𝜎𝑒𝑞

𝜎0𝑔 )𝑔𝑑 + 𝑔𝑒(

𝜎𝑒𝑞

𝜎0𝑔 )𝑔𝑓 (8)

The coefficients 𝑔𝑎 ,𝑔𝑏 ,𝑔𝑐 ,𝑔𝑑 ,𝑔𝑒𝑎𝑛𝑑 𝑔𝑓 are input to the model. In general, the

variables 𝑔𝑎 < 0, 𝑔𝑏 = 3.0, 𝑔𝑐 ≥ 0, 𝑔𝑑 = 2.75, 𝑔𝑒 ≥ 0 and 𝑔𝑓 = 3.0 yield a good

match with the experimental results. It is often observed that there exists a threshold

equivalent stress level below which detwinning does not occur; this stress is referred

to as 𝜎𝑒𝑓𝑓𝑔

.

Figure 3.3: G-function under 100% Martensite.

The plastic strain increment of the two-phased aggregate as well as that of each phase

is assumed to be governed by J2 -flow theory, that is

∆𝜀𝑃𝑙 = ∆𝜆3

2

𝜎

𝜎𝑒𝑞 (9)

43

Where ∆𝜆 is the equivalent plastic strain increment. For the case to use von-Mises

yield function, by work equivalent theorem, ∆𝜆 = (23 ∆𝜀𝑃𝑙 :∆𝜀𝑃𝑙).

Self-consistent methods provide physically attractive averaging scheme for multi-

phase aggregate. The suggested model is a self-consistent method for a rate-dependent

material and is based on Eshelby’s (1957) solution for an inclusion in an elastic

media. For each phase, let the increment of equivalent plastic strain be related to the

applied stress via,

∆𝜆𝛼 = ∆𝜆0𝛼(

𝜎𝑒𝑞𝛼

𝜎𝑌𝛼 )𝑚 (10)

Where 𝜎𝑌𝛼 is the flow stress of the 𝛼 − 𝑡𝑕 phase and can be described by 𝜎𝑌

𝛼 =

𝜎𝑌𝛼 𝑇,𝜎𝑒𝑞 , 𝜆… . Also 𝛼 means 𝑀 (Martensite) and 𝐴 (Austenite). The current value

of 𝜎𝑌𝛼 for each phase can be calculated from

∆𝜎𝑌𝛼 = 𝑕𝛼∆𝜆𝛼 (11)

Where 𝑕𝛼 is the hardening coefficient. It is assumed that the increment of equivalent

plastic strain obeys the same rule as Eq. (10). Utilizing the definition of 𝛤 = 𝜎𝑒𝑞

3∆𝜆,

∆𝜆𝛼 and 𝜎𝛼can be derived from Eshelby’s (1957) solution as follows:

∆𝜆𝛼 =5𝛤

3𝛤+2𝛤𝛼 ∆𝜆 (12a)

𝜎𝛼 =5𝛤𝛼

3𝛤+2𝛤𝛼 𝜎 (12b)

Let 𝑥𝛼 = ∆𝜆𝛼/∆𝜆, = 𝛼 𝜎𝑌𝛼/𝜎𝑌 , then by rearranging the above two equations using

Eq. (10), the following equations are obtained as

𝑥𝛼 = 53 − 2

3 (𝑥𝛼) 1𝑚 𝛼 (13)

1 = 𝑥𝛼 𝑓𝛼𝛼 (14)

Eq. (13) and Eq. (14) can be expressed as the equivalent form as follows:

𝑥𝐴 = 53 − 2

3 (𝑥𝐴) 1𝑚 𝐴 (15a)

44

𝑥𝑀 = 53 − 2

3 (𝑥𝑀) 1𝑚 𝑀 (15b)

1 = 1 − 𝑓 𝑥𝐴 + 𝑓𝑥𝑀 (15c)

By substituting the relation, 𝑥𝛼 = ∆𝜆𝛼/∆𝜆 along with Eq. (10) in Eq. (15c), we get:

(𝜎𝑌) = 1 − 𝑓 𝜎𝑌𝐴 −𝑚 + 𝑓 𝜎𝑌

𝑀 −𝑚 1𝑚 (16)

If we assume that 𝜎𝑌 = 𝜎𝑌(𝑇,𝜎𝑒𝑞 ) and 𝑓 = 𝑓(𝑇,𝜎𝑒𝑞 ), the following variations can be

derived:

∆𝜎𝑌 = 𝜎𝑌 1

𝜎𝑌

𝜕𝜎𝑌

𝜕𝑇∆𝑇 +

1

𝜎𝑌

𝜕𝜎𝑌

𝜕𝜎𝑒𝑞∆𝜎𝑒𝑞 (17a)

= 𝜎𝑌(𝐻𝑇∆𝑇 + 𝐻𝑞∆𝜎𝑒𝑞 )

∆𝑓 = ∆𝑓𝑀+∆𝑓𝐴 =𝜕𝑓

𝜕𝑇∆𝑇 +

𝜕𝑓

𝜕𝜎𝑒𝑞∆𝜎𝑒𝑞 (17b)

= 𝐹𝑀 + 𝐹𝐴 ∆𝑇 − (𝐹𝑀

𝐶𝑀+𝐹𝐴

𝐶𝐴)∆𝜎𝑒𝑞

In Eq. (17a), 𝐻𝑇 and 𝐻𝑞 can be rewritten using the chain rule as:

𝐻𝑇 =1

𝜎𝑌

𝜕𝜎𝑌

𝜕𝑓

𝜕𝑓

𝜕𝑇 (18a)

𝐻𝑞 =1

𝜎𝑌

𝜕𝜎𝑌

𝜕𝑓

𝜕𝑓

𝜕𝜎𝑒𝑞 (18b)

In general, the elastic moduli of Austenite and Martensite are quite different from

each other. In fact, the Young’s modulus of Austenite is approximately 75 GPa, while

that of the Martensite is about 40 GPa. Thus, another scheme is required to compute

the elastic moduli in Eq. 2 from the elastic properties of each phase. Assuming that

each phase as well as the total aggregate is elastically isotropic, 𝑳 can be expressed as

𝐿𝐸 = 2𝜇𝑰𝑰 + 𝜆𝑰 (19)

Where 𝑰𝑰 is the fourth order identity tensor. 𝜇 is the shear modulus and 𝜆 = 𝐾 −2

3𝜇,

Where 𝑲 is the bulk modulus. μ and K are defined as follows:

45

𝜇 =𝐸

2(1−𝜗 ) and 𝐾 =

𝐸

3(1−2𝜗 ) (20)

Then, the shear and bulk moduli of the total aggregate can be derived from those of

each phase following Eshelby’s (1957) solution as

𝜇

𝜇𝑎=

1−𝑆1 (1−𝜇𝑚𝜇𝑎

)

1−(𝑆1−𝑓)(1−𝜇𝑚𝜇𝑎

) (21a)

𝐾

𝐾𝑎=

1−𝑆2 (1−𝐾𝑚𝐾𝑎

)

1−(𝑆2−𝑓)(1−𝐾𝑚𝐾𝑎

) (21b)

Where

𝑆1 =2

15

4−5𝜗

1−𝜗 (22a)

𝑆2 =2

15

1+𝜗

1−𝜗 (22b)

And 𝜗 is the Poisson’s ratio of the aggregate,

𝜗 =3𝐾−2𝜇

6𝐾+2𝜇 (23)

Similarly, the coefficient of thermal expansion for the Austenite is about 𝛼𝑎 = 11 ×

10−6/°𝐶 and that of the Martensite is about 𝛼𝑚 = 6.6 × 10−6/°𝐶. The rule of

mixtures is used to calculate the coefficient of thermal expansion of the composite as:

𝛼 = 1 − 𝑓 𝛼𝑎 + 𝑓𝛼𝑚 (24)

3.4 Implementation

The above constitutive model has been implemented in MSC Marc (Marc User

Documentation, 2012). Figure 3.4 shows the Saeedvafa thermo-mechnical SMA

model implemented in the pre-processor of MSC Marc, Marc Mentat. The Young’s

Modulus & Poisson’s ratio for both the Austenite and Martensite phase are entered in

the overview window. Also there exits the options to input Austenite to Martensite

(Figure 3.5), Martensite to Austenite (Figure 3.6), Transformation Strains (Figure

3.7), Plasticity (Figure 3.8) and Thermal Expansion properties (Figure 3.9).

46

Figure 3.4: Saeedvafa model implemented in MSC Marc 2012 – Overview.

Figure 3.5: Saeedvafa model implemented in MSC Marc 2012 – Austenite to

Martensite.

Figure 3.6: Saeedvafa model implemented in MSC Marc 2012 – Martensite to

Austenite.

47

Figure 3.7: Saeedvafa model implemented in MSC Marc 2012 – Phase transformation

parameters.

Figure 3.8: Saeedvafa model implemented in MSC Marc 2012 - Plasticity.

Figure 3.9: Saeedvafa model implemented in MSC Marc 2012 – Thermal expansion.

48


The presented SMA model is based on phenomenological approach. It covers the

inelastic strains such as TRIP and TWIN including plasticity and temperature

dependency. The stress integration procedure is based on additive decomposition in a

co-rotational coordinate system using generalized stress and strain components. So,

the model can be used by both continuum and beam/shell elements. From the

discussion, it is seen that the material input data is relatively simple and can be

obtained from standard experimental tests. The model is implemented in MSC Marc

and is easy to use.

CH A P T E R 4

VE R I F I C A T I O N

4.1 General

In this chapter the various components of the simulation of the present study are

described. The various components are experimental setup & geometric properties,

material properties, the experimental procedure and assumptions in simulation. Shake

table vibration response data of a SMA braced frame structure has been obtained from

literature where the response of the frame structure subjected to seismic vibration on

shake table was measured (Ma et al., 2004). Then, using the Saeedvafa (2001) SMA

phenomenological model, the experiment is simulated. The experimental results were

then compared to the FEA predictions for the verification of the used SMA model.

4.2 Experimental Setup and Geometric Properties

The experimental setup consisted of a single storey frame structure with the option to

install various braces including SMA braces, a shake table with its control system, a

separate control system for controlling current applied to SMA braces, a

programmable power supply to actuate SMA braces, and various sensors (Figure 4.1).

A picture of the experimental setup is shown in Figure 4.2.

Figure 4.1: A Schematic of the experimental setup (Ma et al., 2004).

50

Figure 4.2: Picture of experimental setup (Ma et al., 2004).

The frame was 22.86-cm (9-inch) wide and 58.42-cm (23-inch) high. A 3.288 kg

mass was attached to the top floor of the frame. With a feedback control, the shake

table simulated various earthquakes. The accelerations of the ground and the top

levels of the frame were measured by two accelerometers. The dimension of the frame

members are summarized in Table 4.1. The SMA braces used three 0.384 mm (0.015-

inch)-diameter Nitinol wires that were in a diagonal configuration (Ma et al., 2004).

Table 4.1: Geometric properties of frame (Ma et al., 2004).

Member Length (mm) Width (mm) Thickness (mm)

Beam 228.60 101.6 3.175

Column 584.20 50.8 3.175

Brace 627.334 Three 0.384 mm Dia Nitinol wires

51

4.3 Material Properties

The beam and column member consisted of the most common grade type 304

stainless steel. 90C Flexinol SMA wire from Dynalloy, Inc. was used as the material

for the SMA brace. The term 90C means the temperature region where SMA activates

or fully converts to Austenite phase (Dynalloy, 2012). The material properties of 304

steel and 90C Flexinol SMA wire are presented in Table 4.2 and Table 4.3

respectively.

Table 4.2: Material properties for 304 steel (MatWeb, 2012).

304 Steel

Mass density (Kg/mm3) 8.00e-6

Young’s modulus (MPa) 193000

Poisson’s ratio 0.29

Yield strength (MPa) 215

Ultimate Tensile Strength (MPa) 505

Figure 4.3: Stress-Strain curve for steel.

52

Table 4.3: Material properties for 90C Flexinol SMA wire (Dynalloy, 2012; Churchill

& Shaw, 2008 and Harvey, 2010).

90C Flexinol

General Properties Austenite Martensite

Mass density (Kg/mm3) 6.45e-6

Young’s modulus (MPa) 83000 28000

Poisson’s ratio 0.33

Yield strength (MPa) 415 140

Melting temperature (°C) 1300

Thermal expansion co-efficient (mm/mm°C) 11e-6 6.6e-6

Thermal conductivity (W/mm °C) 0.018 0.0086

Specific heat (Joule/Kg °C) 837

Latent Heat (Joule/Kg) 24200

Convective cooling coefficient (W/mm2°C) 0.0006

Emissivity 0.5

Electrical Resistivity (Ohm-mm) 0.001 0.0008

Saeedvafa Model Properties Austenite Martensite

Start temperature (°C) 88 72

Finish temperature (°C) 98 62

Austenite to Martensite slope (MPa/°C) 7.9

Martensite to Austenite slope (MPa/°C) 8.2

Deviatoric transformation strain 0.085

Volumetric transformation strain 0

Twinning stress (MPa) 100

g-A -4.0

g-B 2.0

g-C 0.0

g-D 2.8

g-0 (MPa) 300

g-max 1.0

Stress at g-max 1.00e+20

53

4.4 Modeling Assumptions and Analysis Procedure

The SMA braced frame has been simulated using the Non-linear FE software MSC

Marc, the FE model of the frame is presented in Figure 4.4.

(a) (b) (c)

Figure 4.4: FE model of the frame - (a) Front view; (b) Isometric view; (c) Side view.

Figure 4.5: SMA brace consisting of three wires (Ma et al., 2004).

54

Because of the complex experimental procedure carried out by Ma et al. (2004) and

the presence of drawbacks, some assumptions have to be made in simulating the SMA

braced frame. The drawbacks & the corresponding assumptions are explained below-

1. The most significant drawback is using three SMA wires in the brace to get

larger cross sectional area. From Figure 4.5, the wires are touching each other

at various points. This may lead to no current at all in some wires because

electrical current will flow through the shortest distance and least resistance

path. So, there may be some wires which are not actuating at all, hence leaving

over-stressed situation in other wires. Also, it is a complex problem to model

the three wires independently because there is chance of electrical conduction

and short circuit among the wires. In order to simplify the model, the three

wires were replaced by a wire of equivalent cross-sectional area (0.3474 mm2)

and the input voltage adjusted to achieve an equivalent electrical resistance.

For this large Dia wire to achieve the same resistance (which results in same

amount of temperature), the input current has to be increased by three times.

The theoretical proof regarding this assumption is presented in Appendix A

and has also been verified numerically by MSC Marc.

2. No material property data was directly available in Ma et al., 2004. All

material properties have been obtained from other literature.

3. The beam-column connection is assumed to be 100% moment carrying.

4. The mass at the top is assumed to be a point mass at the middle of the beam.

5. The pre-stress load of the brace is unknown. An axial load of 35N is assumed.

6. The damping character of the steel frame is unknown. No external damping

has been applied in the simulation (e.g. numerical damping).

7. The resistance of the SMA wire increases with the increase of temperature.

But due to limitation in the constitutive model, temperature dependent

resistance is not considered here.

The simulation of the SMA braced frame has been carried out in two steps. First, only

the brace has been analyzed for coupled Electrical-Thermal analysis case (Figure 4.6).

The electrical problem has been solved first for the nodal voltages. Next, the thermal

problem has been solved to obtain the nodal temperatures. Then, the SMA braced

frame has been analyzed for Structural analysis case using the Saeedvafa SMA model

55

and importing the temperature of the brace from the first step (Figure 4.7). Here, the

structural problem is solved for the nodal displacements.

Figure 4.6: Step one - Coupled Electrical-Thermal FE model of SMA brace only.

Figure 4.7: Step two - Structural FE model of Full frame.

The coupled thermal-electrical analysis procedure can be used to analyze electric

heating problems. The coupling between the electrical problem and the thermal

SMA brace

Steel column Steel column

Steel beam

SMA brace

Copper wire

56

problem in a Joule heating analysis is due to the fact that the resistance in the electric

problem is dependent on temperatures, and the internal heat generation in the thermal

problem is a function of the electrical flow. However, temperature-dependent

resistivity (instantaneous resistivity at any given temperature) has not been used in

this thesis. A weak coupling between the electrical and thermal problems has been

assumed in the coupled thermo-electrical analysis, such that the distributions of the

voltages and the temperatures of the structure can be solved separately within a time

increment. A steady-state solution of the electrical problem (in terms of nodal

voltages) has been calculated first within each time step. The heat generation due to

electrical flow is included in the thermal analysis as an additional heat input. The

temperature distribution of the structure (obtained from the thermal analysis) is used

to evaluate the temperature-dependent resistivity, which in turn is used for the

electrical analysis in the next time increment. For output, voltage, current density, and

heat generation are available as integration point values. The Current density is the

electric current per unit area of cross section, while Ohmic current is the current going

through the total area (Marc User Documentation, 2012).

Temperature loss occurs to the environment which is a heat transfer problem. There

are three types of heat transfer - conduction (easy), convection (less easy) and

radiation (difficult). The first two are linear problems, but radiation is non-linear. A

temperature difference must exist for heat transfer to occur. Heat is always transferred

in the direction of decreasing temperature. Temperature is a scalar, but heat flux is a

vector quantity. Conduction takes place within the boundaries of a body by the

diffusion of its internal energy. Convection occurs in a fluid by mixing. Here we will

consider only free convection from the surface of a body to the surrounding fluid.

Radiation heat transfer occurs by electromagnetic radiation between the surfaces of a

body and the surrounding medium. It is a highly nonlinear function of the absolute

temperatures of the body and medium.

The structural analysis is a non-linear dynamic analysis. There are three sources of

nonlinearity - material, geometric, and nonlinear boundary conditions. Contact

problem is present in this case between the two diagonal SMA brace. However, the

contact has been assumed to be frictionless and there is no electrical or heat transfer

between the SMA wires. The steel beam-column has been simulated using Von Mises

57

yield criterion taking elastic-plastic material behavior. The SMA brace has been

simulated using Saeedvafa (2001) SMA model. The details of the FE model for both

the analysis cases are summarized in Table 4.4.

Table 4.4: Details of finite element models used in the study.

Member Element type Element

Formulation

No. of

Elements

Coupled Electrical-Thermal FE Analysis

SMA

brace

Element 36; simple linear straight link (heat

transfer element) with constant cross-sectional

area

Full

integration

40

Copper

wire

Element 36; simple linear straight link (heat

transfer element) with constant cross-sectional

area

Full

integration

8

Structural FE Analysis

SMA

brace

Element 9; simple linear straight truss with

constant cross section

Full

integration

40

Steel

column

Element 52; Straight Euler-Bernoulli beam in

space

Full

integration

40

Steel

beam

Element 52; Straight Euler-Bernoulli beam in

space

Full

integration

8

58

Table 4.5: Mesh convergence study.

A mesh convergence study has been carried out to determine optimum element

number for the simulation. The coarse mesh is consisted of Table 4.4 element data.

The fine mesh is created by doubling the number of elements. The study is

summarized in Table 4.5. The Fine mesh provides 0.118% of variation with respect to

the coarse mesh but consumes higher computational time and resource. So the coarse

mesh is selected for further analysis.

4.5 Verification

The presented constitutive model has been previously verified by several authors.

Choudhry & Yoon (2004) verified it for two cases. First, they simulated a one-

dimensional test of cubic specimen. Later, they simulated a stent used in medical

applications. For both cases, the Saeedvafa (2001) model provided good result.

Harvey (2010) accurately simulated coupled Electrical-Thermal-Structural behavior

of a SMA actuator. The simulation results were compared with physically obtained

experimental data of a small scale actuator. The results show that the finite element

simulations are in good agreement with measured test results.

For verification purpose of the SMA braced frame, two simulation cases are

performed. From the experimental data obtained from Ma et al., 2004, the SMA

braced frame is actuated by 1.8A current for a scaled down El Centro (1940)

earthquake and 2.2A for a scaled down Northridge (1994) earthquake. The shake table

acceleration time history for each earthquake case has been converted to displacement

time history using Digital signal Processing (DSP) techniques. For this purpose, a

strong ground motion processing software, SeismoSignal has been used

Mesh type Ground

motion

Maximum acceleration at the top

from simulation (mm/s2)

% Error

Coarse Mesh

Northridge

(1994)

43.119

0.118%

Fine Mesh 43.068

59

(SeismoSignal, 2012). First the input acceleration time history is corrected for base

line and then filtering has been applied to root out unwanted frequencies. Finally,

corrected displacement time history has been applied at the base as boundary

condition. Subsequently, the predicted absolute acceleration at the top of the frame

was compared with the corresponding experimental result.

4.5.1 El Centro Case

Figure 4.8 represent the acceleration at the top from experimental data and Figures 4.9

to 4.11 represent the acceleration at the top from simulation for 0%, 5% & 10%

damping in the frame respectively for the El Centro earthquake case. The initial

acceleration value matches but after some time they differ. The experimental data

shows high damping after around 20 seconds while the simulation data shows that the

frame is under free vibration. The reason the system has high damping is because of

the presence of clear plastic fixtures to hold the frame together (Figure 4.4), which is

absent in the FE model, and assumption for the simulation is that those joints are

perfectly rigid in moment transfer. Also, no external damping such as numerical

damping has been applied in the simulation.

Figure 4.8: Acceleration at the top from experimental data.

60

Figure 4.9: Acceleration at the top from simulation with no damping in frame.

Figure 4.10: Acceleration at the top from simulation with 5% damping in frame.

61


4.5.2 Northridge Case

Figure 4.12 represent the acceleration at the top from experimental data and Figures

4.13 to 4.15 represent the acceleration at the top from simulation for 0%, 5% & 10%

damping in the frame respectively for the Northridge earthquake case. This system

seems to be highly damped. Of the two earthquake cases, the El Centro earthqauke is

chosen for the parametric study to determine optimum current input in the next

chapter.

Figure 4.12: Acceleration at the top from experimental data.

62

Figure 4.13: Acceleration at the top from simulation with no damping in frame.


63



A procedure of the SMA braced frame analysis is detailed in this chapter. The

Analysis was done in two steps for simplicity. The SMA brace is heated by electric

current due to joule heating.

Due to the complexity of the experimental procedure, several assumptions had to be

made for proper simulation. Considering all the assumptions, from the comparative

study of the experimental data and simulation result, it is seen that the simulation

result is in acceptable aggrement with the experimental data of the acceleration at the

top of the frame. The saeedvafa (2001) SMA model works well in simulating the

SMA braced frame.

CH A P T E R 5

RE S U L T S A N D D I S C U S S I O N S

5.1 General

In this chapter, a framework has been presented about how semi-active vibration

control system works using electric current as the driving variable. The predicted

dynamic responses of three configurations of the frame (unbraced, steel braced and

SMA braced) were obtained using MSC Marc. The effectiveness of SMA as a

vibration control device is proved by the use of constant current input. However, the

use of constant current for the entire period of earthquake results in high electrical

power consumption. Then the SMA is activated by pulsed current and both constant

& pulsed current input is compared to determine the optimum solution for SMA

vibration control. It is evident that the use of pulsed current resulted in reduced

energy consumption by the SMA, as well as mitigating the seismic vibrations on the

frame structure. Guidelines to optimum electrical input, considering appropriate

stiffness and damping characteristics; are established for both the constant current and

pulsed current case by parametric study.

5.2 Simulation Parameters and Procedures

The same frame structure from chapter 4 has been considered for this chapter. All

geometric and material properties apply here from chapter 4. The ground motion

selected is the El Centro earthquake recorded at the Imperial Valley Irrigation District

substation in El Centro, California, during May 18, 1940, is shown in Figure 5.1. The

earthquake acceleration time history was scaled down and converted to displacement

time history using Digital signal Processing (DSP) techniques. For this purpose, a

strong ground motion processing software, SeismoSignal has been used

(SeismoSignal, 2012). First the input acceleration time history is corrected for base

line and then filtering has been applied to root out unwanted frequencies.

Acceleration time history is band-pass filtered between 0.10 and 25.00 Hz using

Butterworth filter. The corrected displacement time history for El Centro earthquake

is shown in Figure 5.2. Finally, corrected displacement time history has been applied

65

at the base of the frame as displacement boundary condition. The predicted vibration

response of the frame is obtained from dynamic FE structural analysis.

Figure 5.1: Acceleration time history of El Centro earthquake (CESMD, 2013).

Figure 5.2: Displacement time history of El Centro earthquake.

Figures 5.3 and 5.4 illustrate the Fourier Amplitude Spectrum and Power Spectrum of

El Centro earthquake respectively. Fast Fourier Transformation (FFT) and the Power

Spectrum are useful for measuring the frequency content of stationary or transient

signals. FFTs produce the average frequency content of a signal over the entire time

66

that the signal was acquired. From the spectrums, the dominant frequency of the

earthquake vibration is estimated.

Figure 5.3: Fourier Amplitude Spectrum of El Centro earthquake.

Figure 5.4: Power Spectrum of El Centro earthquake.

67

5.3 Unbraced and Steel Braced Frame Parametric Study

Four different scaled down El Centro earthquake acceleration time histories, by a

factor of 10, 20, 30 and 40; are processed and the displacement time histories are

applied at the base of the frame. The steel brace is pre-stressed with 5% of the yield

stress, 10.75 MPa. The pre-stressing is simulated by applying a 1 second static load-

case before the dynamic load-case. The predicted first three Eigen frequencies of

unbraced and steel braced frame obtained by applying modal load-case just after the

pre-stressing static load-case are summarized in Table 5.1.

Table 5.1: Predicted modal frequencies of the unbraced and steel braced frame

Modal frequency Unbraced frame (Hz) Steel braced frame (Hz)

1st 0.1284 0.4218

2nd

1.0972 0.6593

3rd

1.1252 0.9226

Figure 5.5: Predicted displacement at the top of the unbraced frame, Earthquake

scaled down by a factor of 10

68

Figure 5.6: Predicted displacement at the top of the steel braced frame, Earthquake




69





70





71



Figure 5.13: Maximum displacement of unbraced and steel braced frame at different

scaling factor.

72

Figure 5.14: Plastic strain of steel brace at different scaling factor.

Figures 5.5 to 5.12 show the predicted vibration response of the unbraced and steel

braced frame at different earthquake scaling factor. The unbraced frame is cross

braced with steel to reduce the earthquake vibration response, but the steel braced

frame experiences high displacement at the top than that of the unbraced frame for

high intensity earthquake vibration. With the lower intensity earthquake vibrations,

the steel braced frame succeeds in reducing the displacement at the top of the frame.

This phenomenon is summarized in Figure 5.13 which illustrates the maximum

displacement at the top of the frame for both the unbraced and the steel braced frame

at different scaling factor. The reason for the steel braced frame experiencing larger

displacement is, at the high intensity earthquake the steel brace experiences high

plastic deformation while for low intensity earthquake the steel brace frame

experiences low plastic deformation. Figure 5.14 illustrates the plastic strain for

different scaling factors and complements the observation. Due to plastic

deformation, the brace requires more displacement to develop enough tension force

on the brace. Also, due to elongated length of the brace because of the permanent

plastic deformation, the brace provides compressive force for small displacement,

resulting in amplifying the displacement at the top. Also, from Table 5.1 and from

Figures 5.3 & 5.4, it is evident that the dynamic amplification is also at play here

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 10 20 30 40 50

Scaling Factor

Pla

stic

stra

in

73

because of the closeness of natural frequency of the unbraced & steel braced frame

with the dominant frequency of the earthquake.

This parametric study indicates the weakness of steel brace system due to early

yielding in earthquake scenarios (for 304 steel). The predictions show that the steel

brace system works best when within the elastic limit throughout the entire

earthquake. As a result, the steel brace is pre-stressed by no more than 5% of the yield

stress of 215 MPa.

5.4 SMA braced Frame Parametric Study

The SMA brace is actuated by electric current due to joule heating. First the brace is

heated by constant current. Later, the brace is activated by pulsed current using

Pulsed Width Modulation (PWM) technique and the optimization gain using pulsed

current input is established. The El Centro earthquake acceleration time history is

scaled down by a factor of 30, processed and the displacement time history is applied

at the base of the SMA braced frame for both the constant and pulsed current input

case. The predicted displacement at the top of the frame is obtained from dynamic FE

structural analysis. The SMA brace is pre-stressed with 100 MPa. The pre-stressing is

simulated by applying a 1 second static load-case before the dynamic load-case. Point

to be noted here is that the SMA brace can be pre-stressed by far higher value than

the steel brace, and be still within the elastic limit. The constant current and pulsed

current input profile for 1A current is illustrated in Figures 5.15 & 5.16 respectively.

Figure 5.15: Constant current input for 1A current.

74

Figure 5.16: Pulsed current input for 1A current.

5.4.1 Constant Current

The constant current is kept on for the entire earthquake duration of 53.74 seconds.

The SMA brace temperature increases because of joule heating. Due to conduction at

both ends of the SMA brace, the minimum temperature is always 23°C which is the

room temperature. The maximum temperature depends on the applied current. The

SMA will actuate over the length of the brace having varying temperature depending

on the phase transformation temperature. Figures 5.17 & 5.18 illustrate the predicted

maximum temperature and displacement at the top of the frame respectively for 0A

current. In this case, the SMA brace is in fully Martensite phase and there is no phase

transformation. Then constant current of 1.4A is applied and subsequently increased

by 0.1A till 2.4A. The simulated maximum temperature history and displacement at

the top of the frame is illustrated in Figures 5.19 to 5.40. The maximum displacement

of the frame is plotted against each constant current input case in Figure 5.41 and the

optimum constant current which results in minimum displacement at the top is

determined. The optimum constant current is 2.0A from Figure 5.41. For this

earthquake vibration, the maximum displacement for unbraced frame is 12.0121 mm,

for steel braced frame is 9.0833 mm and for SMA braced frame is 6.1644 mm with

2.0A constant current as the driving force. The SMA brace experiences no plastic

deformation while the steel brace suffers maximum plastic strain of 0.0038. The

result proves the effectiveness of SMA brace over conventional steel brace in

mitigating the seismic vibration response.

75

Figure 5.17: Predicted maximum temperature history of the SMA brace for 0A

current.

Figure 5.18: Predicted displacement at the top of the frame for 0A current.

76

Figure 5.19: Predicted maximum temperature history of the SMA brace for 1.4A

current.

Figure 5.20: Predicted displacement at the top of the frame for 1.4A current.

77


current.


78


current.


79


current.


80


current.


81


current.


82


current.


83


current.


84


current.


85


current.


86


current.


87

Figure 5.41: Optimum constant current.

5.4.2 Pulsed Current

Ma & Song (2003) demonstrated that the use of discrete pulse signal to drive SMA

actuators has the advantage of reducing power consumption without sacrificing

performance. In this study, the pulsed current is kept on for a cycle of 1.5 second

period with a 0.15 second duty time, for the entire earthquake duration of 53.74

seconds. The reason for using the pulse driving mode in this study is to achieve a

larger damping while maintaining a wide range for frequency adjustment. The

periodically applied pulse current ensures shape recovery if the SMA braces

experience a large strain. In a sense, pulsed actuation combines the advantages of

high stiffness and high damping (Ma et al., 2004). Due to conduction at both ends of

the SMA brace, the minimum temperature is always 23°C which is the room

temperature. The maximum temperature depends on the amplitude of the applied

pulsed current. The SMA brace temperature increases because of joule heating for the

0.15 second duty time and decreases due to heat loss to the environment for the rest

1.35 second off duty time. The on and off duty cycle occurs throughout the entire

earthquake duration. This ultimately increases the temperature of the SMA brace and

creates a fairly uniform temperature band of 30°C. The SMA actuates over the entire

length of the brace having varying temperature depending on the phase

transformation temperature.

88

Pulsed current with 4.1A amplitude is applied and subsequently increased by 0.1A till

5.4A. The simulated maximum temperature history and displacement at the top of the

frame is illustrated in Figure 5.42 to Figure 5.69. The maximum displacement of the

frame is plotted against each pulsed current input case in Figure 5.70 and the

optimum pulsed current which results in minimum displacement at the top is

determined. The optimum pulsed current is 5.2A from Figure 5.70. For this

earthquake vibration, the maximum displacement for optimum constant current

actuated SMA braced frame is 6.1644 mm and for optimum pulsed current actuated

SMA braced frame is 8.3641 mm.


current.

89



current.

90



current.

91



current.

92



current.

93



current.

94



current.

95



current.

96



current.

97



current.

98



current.

99



current.

100



current.

101



current.

102


Figure 5.70: Optimum pulsed current.

103

Table 5.2: Total electric energy consumption

Optimum current

case

Amplitude, I

(A)

Voltage, V

(volts)

Effective

duration, t (s)

Total electric

energy, E (J)

Constant current 2.0 17.3823 53.74 1868.2496

Pulsed current 5.2 39.9792 5.40 1122.6159

The constant current is on duty for the whole 53.74 seconds while the pulsed current

is 35.3266 cycles with 36 on duty peaks resulting in 5.40 second effective on duty

duration. The total electric energy is calculated by Eq. 5.1.

𝐸 = 𝐼𝑉𝑡 (5.1)

The total electric energy consumption by both the optimum constant and optimum

pulsed current is summarized in Table 5.2. Pulsed current actuated SMA brace has

39.91% less electric energy consumption than that of constant current actuated SMA

brace. From the result it is evident that the use of pulsed current not only resulted in

better seismic response control, but also reduced electric energy consumption.

CH A P T E R 6

CONCLUSIONS AND RECOMMENDATIONS

6.1 Conclusions

The present research work describes the various aspects of Shape Memory Alloy

(SMA) properties, usage of SMA in civil engineering structures and the constitutive

modeling approaches. An effective thermo-mechanical model is presented which can

be used in the numerical simulation of element having any dimensionality. Two new

approaches in vibration reduction of a frame structure using active SMA braces are

presented. The first approach uses a constant electrical current to actuate the SMA

braces to ensure partial phase transformation and the co-existence of both Martensite

and Austenite phases. The second approach uses specially designed pulsed current to

actuate the SMA braces. Both of these currents increase damping by using SMAs in

Martensite phase, which has better damping than its austenite phase. Both approaches

use electrical current to ensure the shape recovery after the SMA braces experience

large strain. Simulation results show that the two approaches using SMA braces are

effective in reducing structural vibrations induced by earthquake excitations, as

compared with the case of steel brace.

Under the scope of the present study, following conclusions can be made:

1. Unbraced frame can only be effectively stiffened using steel brace against

earthquake vibration when the intensity is low, ensuring that the steel brace

stays in elastic limit.

2. The displacement of the two dimensional (2D) frame under earthquake

loading can be reduced significantly using SMA bracing which steel brace

fails to achieve under severe earthquake scenarios.

3. For El Centro earthquake, the optimum value of the constant current is 2.0A

for minimum displacement of the system. The optimum current is believed to

result in a partial phase transformation, a co-existence of both Martensite and

austenite phases.

4. For El Centro earthquake, the optimum value of the pulsed current for

minimization of the displacement of the building frame is found to be 5.2A,

105

which also results in a partial phase transformation, a co-existence of both

Martensite and austenite phases.

5. While both constant & pulsed currents are somewhat effective in reducing

structural vibrations induced by earthquake excitations, the periodic pulsed

current used to stiffen the SMA brace is more effective than that of the

constant current in terms of cost optimization. In this research work, electric

energy savings of 39.91% is achieved using pulsed current over constant

current.

6.2 Recommendations for Further Studies

It is recommended that the study can be extended further in the following fields

focusing on both the shape memory effect and the super-elasticity properties:

1. Application of SMA in various types of seismic vibration control devices

(Dampers, Re-centering devices).

2. Application of pulsed current actuated SMA braces in active vibration control

with the help of feedback sensors.

3. Application of SMA as active reinforcing bar in regular concrete and

prestressed concrete structures.

4. Application of SMA in active confining of concrete member such as column,

bridge pier.

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APPENDIX-A

Proof of Equivalent Cross Sectional Area SMA Wire

Figure A1: SMA wires connected in parallel.

Figure A1 shows three wires connected in parallel, having cross sectional area of

0.1158 mm2 each and resistivity of 0.0008 Ohm-mm will be replaced by 1 wire

having cross sectional area of 0.3474 mm2.

Here,

𝑅 = 𝑅𝑒𝑠𝑖𝑠𝑡𝑒𝑛𝑐𝑒 𝑜𝑓 𝑒𝑎𝑐𝑕 𝑠𝑖𝑛𝑔𝑙𝑒 𝑤𝑖𝑟𝑒

𝑅𝑇 = 𝑇𝑜𝑡𝑎𝑙 𝑟𝑒𝑠𝑖𝑠𝑡𝑒𝑛𝑐𝑒 𝑜𝑓 𝑡𝑕𝑒 𝑐𝑖𝑟𝑐𝑢𝑖𝑡 𝑜𝑟 𝑅𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝑙𝑎𝑟𝑔𝑒 𝐷𝑖𝑎 𝑤𝑖𝑟𝑒

𝑉 = 𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑜𝑓 𝑡𝑕𝑒 𝑐𝑖𝑟𝑐𝑢𝑖𝑡

𝐼1 = 𝐼2 = 𝐼3 = 𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑝𝑎𝑠𝑠𝑖𝑛𝑔 𝑡𝑕𝑟𝑜𝑢𝑔𝑕 𝑒𝑎𝑐𝑕 𝑠𝑖𝑛𝑔𝑙𝑒 𝑤𝑖𝑟𝑒

𝜌 = 𝑅𝑒𝑠𝑖𝑠𝑡𝑖𝑣𝑖𝑡𝑦

Resistivity of Large Dia wire:

The linear resistance of each wire is 𝑅

𝐿=

𝜌

𝐴=

0.0008

0.1158= 0.00691 𝑂𝑕𝑚/𝑚𝑚

Total linear resistance of the parallel circuit,

113

𝑅𝑇

𝐿=

1

1𝑅1

𝐿 +

1𝑅2

𝐿 +

1𝑅3

𝐿

=1

1𝑅

𝐿 +

1𝑅

𝐿 +

1𝑅

𝐿

=1

3 ×1

𝑅𝐿

=1

3 ×1

0.00691

= 0.0023 𝑂𝑕𝑚/𝑚𝑚

The parallel circuit will be replaced by a Large Dia wire. For the large Dia wire,

𝑅𝑇

𝐿=

𝜌

𝐴

𝑜𝑟, 0.0023 =𝜌

𝐴

𝑜𝑟, 𝜌 = 0.0023 × 𝐴 = 0.0023 × 0.3474 = 0.0008 Ohm − mm

So, Resistivity will remain same for the Large Dia wire.

Input current for large Dia wire:

We know for parallel circuit,

𝐼1 = 𝐼2 = 𝐼3 =𝑉

𝑅

𝑉1 = 𝑉2 = 𝑉3 = 𝑉

And, 𝐼𝑇 = 𝐼1 + 𝐼2 + 𝐼3

Now, 𝐼𝑇 =𝑉

𝑅𝑇

𝑜𝑟, 𝐼1 + 𝐼2 + 𝐼3 =𝑉

𝑅𝑇

𝑜𝑟, 3 ×𝑉

𝑅=

𝑉

𝑅𝑇

𝑜𝑟, 𝑅 = 3 × 𝑅𝑇

114

𝐴𝑔𝑎𝑖𝑛,𝑉 = 𝐼 × 𝑅

𝑜𝑟, 𝑉 = 𝐼 × 3 × 𝑅𝑇

𝑜𝑟,𝑉 = 3 × 𝐼 × 𝑅𝑇

So, to achieve same amount of resistance, the input current for the large Dia wire has

to be increased by 3 times or by the number of wires connected in parallel

configuration.

Figure A2: Comparative study of equivalent and single wire maximum temperature.

Coupled Electrical-Thermal FE analysis is conducted for both the single wire and

equivalent cross section wire and the maximum temperature against constant current

input cases for both the wire is illustrated in Figure A2. The resultant graph ensures

the correctness of the input current assumptions for the equivalent cross section.

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