NASA/CR-2005-213929 

Investigation of the Thermomechanical 

Response of Shape Memory Alloy Hybrid 

Composite Beams 

Brian A. Davis 

North Carolina State University, Raleigh, North Carolina 

 

October 2005 
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NASA/CR-2005-213929 

Investigation of the Thermomechanical 

Response of Shape Memory Alloy Hybrid 

Composite Beams 

Brian A. Davis 

North Carolina State University, Raleigh, North Carolina 

National Aeronautics and 

Space Administration

Langley Research Center Prepared for Langley Research Center 

Hampton, Virginia 23681-2199 under Cooperative Agreement NCC-1-02043

 

October 2005 
Available from: 

NASA Center for AeroSpace Information (CASI) National Technical Information Service (NTIS) 

7121 Standard Drive 5285 Port Royal Road 

Hanover, MD 21076-1320 Springfield, VA 22161-2171 

(301) 621-0390 (703) 605-6000 

Acknowledgments 

I would like to especially thank North Carolina State University (NCSU) Mechanical and Aerospace 

Engineering Department, National Institute of Aerospace, and the Structural Acoustics Branch at NASA LaRC for 

funding my education. I am extremely grateful for the opportunity to be a part of the NIA educational program and 

to learn and conduct research in an environment like NASA LaRC. The amount that I have learned from all 

coursework at NCSU, and from being involved in research at NASA LaRC has been priceless. I would like to give 

special thanks to my committee members; Dr. Travis Turner, Dr. Stefan Seelecke, Dr. Richard Keltie, and Dr. 

Robert Nagel for their support and contributions to this work. I have benefited greatly from all courses and 

discussions I have had with you all, and feel privileged to be able to learn from you. I would especially like to thank 

my research mentor and advisor at NASA LaRC, Dr. Travis Turner, for the opportunity to work on a great research 

project, for all the valuable time spent teaching me, and for making this learning experience so enjoyable. The 

opportunity to work with and learn from you has allowed me to gain so much valuable experience, and I greatly 

appreciate all your guidance. I would also like to especially thank my advisor at NCSU for helpful insights and 

suggestions to this work and for all the support and encouragement through my NIA education and research. I 

would like to also thank my previous advisor, Dr. Richard F. Keltie, for all the guidance and help throughout my 

first year of my master’s program. 

I gratefully acknowledge the assistance of Gary Fleming (NASA LaRC) for calibration, and processing of 

the PMI data. Thanks are also due to Howard Jones and George Hilton (NASA LaRC) for their contribution to the 

static and dynamic test configurations. The assistance of Larry Becker and Brian Howerton (Lockheed-Martin) for 

the development of the LabVIEW-based thermal controller/data acquisition system is also greatly appreciated. 
Preface 

This research results presented herein constitute a master’s thesis prepared by Brian

Davis under the direction of Dr. Travis L. Turner of NASA Langley Research Center, 

Structural Acoustics Branch and Dr. Stefan Seelecke of North Carolina State University, 

Department of Mechanical and Aerospace Engineering. 

iii
TABLE OF CONTENTS

LIST OF TABLES VI

LIST OF FIGURES VII

LIST OF SYMBOLS XI

INTRODUCTION 1

1.1 SHAPE MEMORY ALLOYS 1

1.2 MOTIVATION/APPLICATIONS 3

1.3 OBJECTIVES 4

EXPERIMENTAL MEASUREMENT OF THE STATIC THERMOELASTIC 

RESPONSE OF THE SMAHC BEAM 7

2.1 BACKGROUND 7

2.2 EXPERIMENTAL SET-UP 8

2.3 PROJECTION MOIRE’ INTERFEROMETRY 9

2.4 STATIC THERMOELASTIC RESPONSE EXPERIMENTAL RESULTS 10

Laser Displacement Transducer Results 10

Projection Moire’ Interferometry Results 13

EXPERIMENTAL MEASUREMENTS OF THE DYNAMIC RESPONSE OF THE 

SMAHC BEAM 16

3.1 BACKGROUND 16

3.2 EXPERIMENTAL SET-UP 16

3.3 DYNAMIC RESPONSE DATA POST-PROCESSING OVERVIEW 17

3.4 SMAHC BEAM DYNAMIC RESPONSE EXPERIMENTAL RESULTS 19

THERMOELASTIC CONSTITUTIVE MODELING FORMULATION 21 

4.1 BACKGROUND 21

4.2 THERMOELASTIC CONSTITUTIVE MODEL/STIFFNESS PROPERTY FORMULATION 21 

4.3 STRESS-STRAIN RELATION FOR ORTHOTROPIC MATERIAL 25

NUMERICAL MODELING OF SMAHC BEAM SPECIMENS IN THE 

COMMERCIAL FINITE ELEMENT CODE ABAQUS 28

5.1 BACKGROUND 28

5.2 STATIC RESPONSE MODELING RESULTS IN ABAQUS 29

5.3 DYNAMIC RESPONSE MODELING RESULTS IN ABAQUS 31

CONCLUSIONS 34

REFERENCES 36

TABLES 38 

FIGURES 45

APPENDICES 82

iv
APPENDIX A 83

APPENDIX B 93

APPENDIX C 96

APPENDIX D 98

v
 LIST OF TABLES 

Table 3.1- Beam 2 Run 1 dynamic response RMS displacement values as

a function of temperature for positions 1, 2, and 3 along the

length of the beam.............................................................................................38 

Table 3.2- Beam 3 Run 2 dynamic response RMS displacement values as

a function of temperature for positions 1, 2, and 3 along the

length of the beam.............................................................................................39 

Table 3.3- Beam 2 dynamic response modal parameter estimates for runs 

1, 2, and 3. ..........................................................................................................40 

Table 3.4- Beam 3 dynamic response modal parameter estimates for runs 

1, 2, and 3. ..........................................................................................................41 

Table 4.1– Thermomechanical material property measurement of the 

Fiberite E-glass/934 epoxy material system in principal 

material direction..............................................................................................42 

Table 4.2– Thermomechanical material property measurements of the 

Nitinol Ribbon actuator material. ..................................................................43 

Table 4.3- Nitinol Ribbon Transformation Temperatures measured by 

DSC.....................................................................................................................44

vi
LIST OF FIGURES

Figure 1.1- SMA Lattice Structure at different phases....................................................45 

Figure 1.2– SMA body under tensile loading/unloading at low 

temperature, exhibiting the shape memory effect upon 

heating4

............................................................................................................45 

Figure 1.3– Schematic of martensitic and reverse transformations .

5 .............................46

Figure 1.4– Stress-Strain Curves for Shape Memory Alloys at various

intermediate temperatures6..........................................................................46 

Figure 1.5- Load-Deformation diagrams for shape memory alloy at

various temperatures (fully martensitic-quasiplastic) and 

(fully austenitic-pseudoelastic)4

....................................................................47 

Figure 1.6– Nitinol recovery stress vs. temperature and initial strain .

7 .........................47

Figure 1.7– Recovery stress vs. temperature, thermal cycle for nitinol 

ribbon samples at 4% pre-strain 8...............................................................48 

Figure 1.8– Variation of austenite start temperature with weight content 

(%) of Nickel in Nitinol Alloy10....................................................................48 

Figure 2.1- 22”x1”x0.080” Glass Epoxy/SMA Hybrid Beam (45/0/-

45/90)2s Lamination with SMA embedded in the 0o

 lamina 

only8 ................................................................................................................49 

Figure 2.2- Consolidated SMAHC structure after curing cycle, and

machined SMAHC beam specimens used for experiments8. ...................49 

Figure 2.3– Static Experimental Set-up: SMAHC Beam3 installed in the

beam fixture (front view)..............................................................................49 

Figure 2.4a– Mechanical and electrical grip of beam fixture (front view). ...................50

Figure 2.4b- Mechanical and electrical grip of beam fixture (back view).....................50

Figure 2.5- Static Thermoelastic Response Experimental Component 

Schematic........................................................................................................51 

Figure 2.6– Static Thermoelastic Response Experimental Configuration 

Overview.........................................................................................................51 

Figure 2.7- Simple PMI System Schematic35.....................................................................52 

Figure 2.8– SMAHC Beam2 Laser Displacement Experimental Results of 

Midspan for Runs 1-9. ..................................................................................52 

Figure 2.9– SMAHC Beam3 Laser Displacement Experimental Results

for Midspan Runs 1-10. ................................................................................53 

Figure 2.10– Experimental Configuration of initial imperfection 

measurements.................................................................................................53 

Figure 2.11- Laser Translation versus PMI initial imperfection 

measurements of SMAHC beams installed in beam fixture. ...................54 

Figure 2.12- Laser Translation Measurements of SMAHC Beam3 for

three different boundary condition configurations of the 

Garolite G-10 insulation blocks. ..................................................................54 

vii
Figure 2.13- Garolite G-10 Insulation Blocks Configurations for Laser

Displacement Transducer Experiments (Top view of beam

fixture boundaries)........................................................................................55 

Figure 2.14- Projection Moire’ Interferometry calibration plate . 34 ...............................55

Figure 2.15- PMI differential image data and line fit for the centerline 

out of plane deflection for Beam3 Run3 at T=160o

F.................................56 

Figure 2.16- PMI full field imaging of the SMAHC Beam2 Run1..................................56

Figure 2.17- PMI full field imaging of the SMAHC Beam 2 Run 4................................57

Figure 2.18- PMI full field imaging of the SMAHC Beam 2 Run 8................................57

Figure 2.19- PMI full field imaging of the SMAHC Beam 3 Run 1................................58

Figure 2.20- PMI full field imaging of the SMAHC Beam 3 Run 4................................58

Figure 2.21- PMI full field imaging of the SMAHC Beam 3 Run 8................................59

Figure 2.22- Beam2 PMI vs. Laser Static Deflection Measurements.............................59

Figure 2.23- Beam3 PMI vs. Laser Static Deflection Measurements.............................60

Figure 3.1– SMAHC beam dynamic response experimental

configuration. .................................................................................................60 

Figure 3.2– SMAHC beam specimen installed, with mounted beam

response transducers and fixture transducers (back view)......................61 

Figure 3.3– Acceleration PSD for the base excitation of the beam with a 

RMS value of .25g (96.5 in/s2

). .....................................................................61 

Figure 3.4- SMAHC beam dynamic response experimental component

schematic.........................................................................................................62 

Figure 3.5- Acceleration vs. Displacement PSD generated by attached mfile,which post-processes the acceleration time history data....................63 

Figure 3.6- SMAHC Beam 2 PSD’s for selected temperatures at position 

1........................................................................................................................63 

Figure 3.7- SMAHC Beam 2 PSD’s for selected temperatures at position 

2 (midspan).....................................................................................................64 

Figure 3.8- SMAHC Beam 2 PSD’s for selected temperatures at position 

3........................................................................................................................64 

Figure 3.9- SMAHC Beam 3 PSD’s for selected temperatures at position 

1........................................................................................................................65 

Figure 3.10- SMAHC Beam 3 PSD’s at selected temperatures for position 

2 (midspan).....................................................................................................65 

Figure 3.11- SMAHC Beam 3 PSD’s at selected temperatures for position 

3........................................................................................................................66 

Figure 3.12- Modal frequency variation of modes 1, and 3 as a function of 

temperature....................................................................................................66 

Figure 4.1- Representative volume element of SMAHC lamina . 18 .................................67

Figure 4.2– Applied stress vs. strain for .09x.006x10 inch sample with 0%

pre-strain at ambient temperature and 120o

C (248 oF)18. ........................67 

Figure 4.3– Applied stress vs. strain for .09x.006x10 inch sample with 4%

pre-strain at ambient temperature and 120o

C (248 oF)18. ........................68 

Figure 4.4– Recovery Stress vs. Temperature of the Nitinol ribbon (prestrained 4% and released) for thermal cycles 2, 4, and 5018. ...................68 

viii
Figure 4.5– Modulus for Nitinol ribbon prestrained 4% and released as a 

function of temperature18. ............................................................................69 

Figure 4.6- Differential Scanning Calorimetry (DSC) Results for the

Nitinol ribbon sample showing transformation 

temperatures. .................................................................................................69 

Figure 5.1- ABAQUS Finite Element Mesh of the SMAHC beam with

clamped boundary conditions......................................................................70 

Figure 5.2- Static Response SMAHC Beam 2 Run 6 ABAQUS vs

experimental results for original recovery force 

measurements, and 15 o

F shift in recovery force input data....................70 

Figure 5.3– Static Response SMAHC Beam 2 Run 7 ABAQUS vs. 

experimental results for original recovery force 

measurements, and 10 o

F shift in recovery force input data....................71 

Figure 5.4- Static Response SMAHC Beam 2 Run 8 ABAQUS vs. 

experimental results for original recovery force 

measurements, and 15 o

F shift in recovery force input data....................71 

Figure 5.5- Static Response SMAHC Beam 3 Run 3 ABAQUS vs. 

experimental results for original recovery force 

measurements, and 15 o

F shift in recovery force input data....................72 

Figure 5.6- Static Response SMAHC Beam 3 Run 6 ABAQUS vs. 

experimental results for original recovery force 

measurements, and 15 o

F shift in recovery force input data....................72 

Figure 5.7- Static Response SMAHC Beam 3 Run 8 ABAQUS vs. 

experimental results for original recovery force 

measurements, and 20 o

F shift in recovery force input data....................73 

Figure 5.8- Beam 2 Run 8, position 2 (midspan) PSD comparison at

ambient (∆f=.25 Hz). .....................................................................................73 

Figure 5.9- Beam 2 run 8, position 2 (Midspan) PSD comparison at 71.1 o

C (160 oF) (∆f=.25 Hz)..................................................................................74 

Figure 5.10- Beam 2 Run 8, position 2 (midspan) PSD comparison at 93.3 o

C (200oF) (∆f=.25 Hz)...................................................................................74 

Figure 5.11- Beam 2 Run 8, position 2 (midspan) PSD comparison at

121.1 o

C (250 oF).............................................................................................75 

Figure 5.12- Beam 3 Run 8, position 2 (midspan) PSD comparison at 21.7 o

C (71 oF) (∆f=.25 Hz)....................................................................................75 

Figure 5.13- Beam 3 Run 8, position 2 (midspan) PSD comparison at 71.1 o

C (160 oF) (∆f=.25 Hz)..................................................................................76 

Figure 5.14- Beam 3 Run 8, position 2 (midspan) PSD comparison at 93.3 o

C (200 oF) (∆f=.25 Hz)..................................................................................76 

Figure 5.15- Beam 3 Run 8, position 2 (midspan) PSD comparison at

121.1 oC (250 oF) (∆f=.25 Hz). ......................................................................77 

Figure 5.16- Beam 2 Run 8, position 1 PSD comparison for selected 

temperatures (∆f=.25 Hz). ............................................................................77 

Figure 5.17- Beam 3 Run 3, position 3 PSD’s comparison for selected 

temperatures (∆f=.25 Hz). ............................................................................78 

ix
Figure 5.18- Beam 3 Run 1, position 3 PSD’s comparison through the 

buckling range (T=71 oF)..............................................................................78 

Figure 5.19- Beam 3 Run 1, position 3 PSD’s comparison through the 

buckling range (T=100 o

F)............................................................................79 

Figure 5.20- Beam 3 Run 1, position 3 PSD’s comparison through the 

buckling range (T=130 o

F)............................................................................79 

Figure 5.21- Beam 3 Run 1, position 3 PSD’s comparison through the 

buckling range (T=140 oF)............................................................................80 

Figure 5.22- Beam 3 Run 1, position 3 PSD’s comparison through the 

buckling range (T=150 oF)............................................................................80 

Figure 5.23- Beam 3 Run 1, position 3 PSD’s comparison through the 

buckling range (T=160 oF)............................................................................81 

Figure 10.1- PMI full field imaging of the SMAHC Beam 2 Run 1................................84

Figure 10.2- PMI full field imaging of the SMAHC Beam 2 Run 2................................84

Figure 10.3- PMI full field imaging of the SMAHC Beam 2 Run 3................................85

Figure 10.4- PMI full field imaging of the SMAHC Beam 2 Run 4................................85

Figure 10.5- PMI full field imaging of the SMAHC Beam 2 Run 5................................86

Figure 10.6- PMI full field imaging of the SMAHC Beam 2 Run 6................................86

Figure 10.7- PMI full field imaging of the SMAHC Beam 2 Run 7................................87

Figure 10.8- PMI full field imaging of the SMAHC Beam 2 Run 8................................87

Figure 10.9- PMI full field imaging of the SMAHC Beam 2 Run 9................................88

Figure 10.10- PMI full field imaging of the SMAHC Beam 3 Run 1. ............................88

Figure 10.11- PMI full field imaging of the SMAHC Beam 3 Run 2. ............................89

Figure 10.12- PMI full field imaging of the SMAHC Beam 3 Run 3. ............................89

Figure 10.13- PMI full field imaging of the SMAHC Beam 3 Run 4. ............................90

Figure 10.14- PMI full field imaging of the SMAHC Beam 3 Run 5. ............................90

Figure 10.15- PMI full field imaging of the SMAHC Beam 3 Run 6. ............................91

Figure 10.16- PMI full field imaging of the SMAHC Beam 3 Run 7. ............................91

Figure 10.17- PMI full field imaging of the SMAHC Beam 3 Run 8. ............................92

x
LIST OF SYMBOLS 

Roman Symbols 

A Cross-Sectional area 

Af Austenite Finish Temperature 

As Austenite Start Temperature 

Au Gold 

Cd Cadmium 

d Distance between grid lines 

D Compliance matrix values 

E Elastic Modulus 

G Shear Modulus 

fn Natural frequency (cycles/sec) 

fk Frequency values 

i Row index for a pixel 

j Column index for a pixel

k Index number 

M+ Martensitic plus variant 

M- Martensitic minus variant 

Mf Martensite Finish Temperature 

Ms Martenstite Start Temperture 

N Total number of samples 

Ni Nickel 

n Index number of samples 

Pcr Critical Load 

Q Reduced stiffness matrix values 

t Time

T Temperature

To Initial temperature 

[T] Transformation matrix 

Ti Titanium

xi
W Width of SMAHC Lamina 

Xk Fourier components 

xn Time history sample value 

z(x,y) Out of plane deformation at a point specified by coordinates 

Greek Symbols 

α Coefficient of thermal expansion 

∆ Shearing deformation 

∆W Deformation in principle material direction 2. 

ε Strain

γ Engineering shear strain 

ν12 Poisson’s ratio 

π 3.14159265 

θ Orientation angle 

σ Stress 

σr Recovery Stress 

τ Shear stress 

υ Volume fraction 

ωn Natural angular frequency 

ζ Damping Ratio (

m n

c

2 ω

) 

Subscripts 

a Actuator material 

f Finish transformation temperature 

m Composite matrix material 

xii
r Recovery 

s Start transformation temperature 

Superscripts 

+ Positive orientation 

- Negative orientation 

-1 Inverse 

T Transpose

Abbreviations 

CCD Charge Coupled Device 

DSC Differential Scanning Calorimetry 

ECTEM Effective Coefficient of Thermal Expansion Model 

FFT Fast Fourier Transform 

FRF Frequency Response Function 

GPIB General Purpose Interface Bus 

LaRC Langley Research Center

LMA Lockheed Martin Astronautics 

NASA National Aeronautics and Space Administration 

PID Proportional, Integral, and Differential feedback

PMI Projection Moire’ Interferometry 

PSD Power Spectral Density 

RMS Root Mean Square 

SDOF Single Degree of Freedom 

SIM Stress Induced Martensite

xiii
SMAHC Shape Memory Alloy Hybrid Composites 

SME Shape Memory Effect 

VI Virtual Instrument 

Units of Measurement 

ft Foot 

in Inch 

m Meter 

K Degrees Kelvin 

V Volts 

W Watts 

s Second 

MPa MegaPascal (106

 N/m2) 

slug English unit of mass (=32.2 lbm pounds mass) 

μm Micrometer 

/ft Per foot 

o Degrees 

o

C Degree Celsius 

o

F Degree Farenheight 

Hz Hertz 

lbf Force pounds 

kg Kilograms 

N Newton 

xiv
Miscellaneous Symbols 

 Normalized quantity (ratio of rms and mean value) 

∂ Partial derivative 

d[ ] Derivative 

d/dt Time derivative of a quantity 

[ ] ∫ Integral 

∑ Summation of a quantity 

xv
SUMMARY

Previous work at NASA Langley Research Center (LaRC) involved fabrication and 

testing of composite beams with embedded, pre-strained shape memory alloy (SMA) ribbons

within the beam structures. That study also provided comparison of experimental results with 

numerical predictions from a research code making use of a new thermoelastic model for shape 

memory alloy hybrid composite (SMAHC) structures. The previous work showed qualitative 

validation of the numerical model. However, deficiencies in the experimental-numerical 

correlation were noted and hypotheses for the discrepancies were given for further 

investigation. The goal of this work is to refine the experimental measurement and numerical 

modeling approaches in order to better understand the discrepancies, improve the correlation 

between prediction and measurement, and provide rigorous quantitative validation of the 

numerical analysis/design tool. The experimental investigation is refined by a more thorough 

test procedure and incorporation of higher fidelity measurements such as infrared 

thermography and projection moiré interferometry. The numerical results are produced by a 

recently commercialized version of the constitutive model as implemented in ABAQUS and 

are refined by incorporation of additional measured parameters such as geometric imperfection. 

Thermal buckling, post-buckling, and random responses to thermal and inertial (base 

acceleration) loads are studied. The results demonstrate the effectiveness of SMAHC 

structures in controlling static and dynamic responses by adaptive stiffening. Excellent 

agreement is achieved between the predicted and measured results of the static and dynamic 

thermomechanical response, thereby providing quantitative validation of the numerical tool.

xvi
Chapter 1 

INTRODUCTION 

1.1 Shape Memory Alloys 

Shape Memory Alloys (SMA) have become increasingly researched over the past 30 

years, due to their fascinating functional properties. In the 1950’s the Shape Memory Effect 

(SME), the recovery of deformation to an original shape by heating to a critical temperature, 

was discovered in Gold (Au 47.5%) and Cadmium (Cd 52.5%) Alloy by Chang and Read1

. 

Then in 1963 the SME was discovered at the Naval Ordinance Laboratory in equiatomic 

composition Titanium-Nickel Alloy (Ni 52%-57%) by Beuhler and Wiley (Nitinol) 2. Since 

then other alloys have been discovered to possess the same attractive functionality. Although, 

the particular alloy, named Nitinol (Nickel, Titanium, Naval, Ordinance, Laboratory), is 

considered to be the SMA with the most potential for engineering applications because of its 

ductility at low temperature, high degree of shape-recovery capability (8% strain), large 

pseudoelastic hysteresis, corrosion and fatigue resistance, biomedical compatibility, and 

relatively high electrical resistance3

. 

SMA’s are defined as a class of materials that have the ability to recover a given 

deformation when heated above a certain temperature. There are two stable phases of the 

lattice structure for these metal alloys that may exist, termed martensite and austenite. 

Austenite phase represents the high temperature stable state, and possesses the symmetric, 

cubic crystal lattice structure (See Figure 1.1) with a high elastic modulus. Martensite 

represents the low temperature stable state, and possesses the less symmetric lattice structure of 

a monoclinic crystal with a lower elastic modulus. Additionally, the martensite phase may 

exist as twinned or de-twinned (M+

, M-) states (See Figure 1.1). The interactions of thermal 

and applied mechanical loads to this material system are the basis for driving the phase 

transitions. The phase transitions between these two states are what provide the unique and 

attractive functionalities of SMA. When cooled under no applied mechanical load the material 

transforms from the austenite phase to the self-accommodating martensitic twinned phase. 

There is no observable shape change associated with this transformation. The boundaries 

between the layers of twinned martensite are easily deformed under applied load; therefore 

deformation can easily be achieved in the martensitic state. Figure 1.2 shows the 

transformation behavior under heating/cooling and loading/unloading of an SMA element. 

Once a tensile load is applied in the martensitic state, the self-accommodated martensitetwinned configuration will transform, at a critical load, to M+

 or M- variants depending on the 

direction of the applied load. When the uniaxial load is removed the M+

 or M- phase will 

remain, leaving the given deformation to the SMA element. Upon heating of the martensitic 

phase the “reverse transformation” occurs in which the lattice structure returns to austenite and 

recovers any deformation. As previously stated this is termed the shape memory effect. The 

SME always refers to the free recovery of the deformation. It is important to mention that 

during the attempt to transform to austenite while constrained, high recovery forces may be 

generated. This makes SMA’s very attractive as actuators in mechanical systems. 

Associated with the martensitic and reverse transformations are four characteristic 

temperatures. Martensite start (Ms) is the temperature at which the SMA begins transforming 
from austenite to martensite; martensite finish (Mf) is the temperature at which the SMA

becomes fully martensitic and the transformation is complete; austenite start (As) represents the 

temperature at which the “reverse transformation” begins; and austenite finish (Af) represents 

the temperature at which the SMA is fully in austenitic phase. The difference in the phase 

transformation temperatures results in an associated hysteresis curve, which can be seen in

Figure 1.3. 

SMA’s can also exhibit transformation pseudoelastic behavior, which is associated with

the recovery of the strain upon unloading. If the material is above Af temperature and is loaded 

to a critical stress then the martensitic transformation can be induced, which is termed stress

induced martensite (SIM). SIM produces a large deformation in the material, which is fully 

recovered upon unloading when the material is above Af temperature (Superelasticity). 

Pseudoelastic behavior of SMA is attractive for use as an energy dissipation mechanism, and

can provide hysteretic damping to structures. The pseudoelastic behavior of SMA is

commonly exploited in medical and dental applications. 

 SIM can also occur when the material is at lower temperatures than Af temperature, but

only partial strain recovery occurs upon unloading at temperatures between As< T <Af. In

Figure 1.4 the stress-strain diagrams for various temperatures demonstrate the different effects

of SMA phase transitions. When the material is at a temperature below Mf and is in the 

twinned martensitic phase its yield stress to induce fully M+ de-twinned state, and thus a large

deformation, is quite low. Once the material is fully de-twinned and deformed to M+

, 

continued loading would produce elastic deformation until slip occurs. Once the material is at 

a higher temperature in the austenitic phase, the yield stress value is much higher than the

martensitic phase’s. Figure 1.5 shows the corresponding load-deformation curves for the 

material at various temperatures representing the quasi-plastic behavior (low temperature) and 

the pseudoelastic behavior (high temperature). 

As previously mentioned SMA’s are capable of producing large forces when

recovering strain. The SME or reverse transformations described above refer to a free recovery 

(under zero load). If the strain recovery occurred under load the SMA performs work and is

termed a restrained recovery. If the SMA is prevented from recovering strain once heated

above the transformation temperature that is termed constrained recovery. Recovery stresses

up to 700 MPa may be generated during constrained recovery. In this work the constrained 

recovery of SMA is used to induce large tensile forces in pre-strained SMA actuator ribbons. It 

is known that the recovery force generated during constrained recovery is a function of prestrain, temperature, and thermal cycle. Cross reported the recovery stress vs. temperature for

various pre-strain amounts, as seen in Figure 1.6 for Nitinol .1 in diameter rod7

. At the As

temperature the recovery stress increases with temperature and then levels off near Af

temperature. As pre-strain increases the recovery stress increases, but will never exceed the 

austenitic yield stress. Material characterization of Nitinol has been performed experimentally 

at NASA LaRC and Lockheed Martin Astronautics (LMA) to quantify the recovery stress vs.

temperature, among the other temperature dependent properties, of the Nitinol alloy in ribbon 

form used for this study8

. Figure 1.7 shows the recovery stress vs. temperature and thermal

cycles at 4% pre-strain. It can be seen that the recovery stress/actuation degrades with 

thermal cycle to near 50 cycles and then levels off. 

Factors such as Ni content, aging, thermo-mechanical treatment and addition of 

alloying elements, which affect the structure, are important for controlling the memory

behavior9

. The transformation temperatures for SMA (Ms, Mf, As, Af) can be altered by 

2
changing the composition of the alloy. It was reported by Beuhler and Wang that the As

temperature for Nitinol can be varied from –50 o

C (-122 oF) to 166 oC (330 oF) by changing 

the Ni content from 54.4% to 56.5% by weight as shown in Figure 1.810. When a nearly 

equiatomic Nitinol (Ni-Ti) alloy is fully annealed the SME consists of transformation directly 

from the austenite phase to the martensite phase. However, when the equiatomic Nitinol is

partially annealed, the austenitic phase transforms to a R-phase, which consist of a 

rhombohedral lattice structure, before transforming to the martensitic monoclinic structure. 

This R-phase structure is an intermediate martensitic transformation, which is reported to 

strengthen the austenitic lattice structure and improve the recoverability of the SME11,12. This 

additional transformation, being a thermoelastic, martensitic transformation itself, also

possesses a SME and pseudoelasticity but produces a much smaller strain recovery, and 

hysteresis. This is an attractive property for sensor applications. Due to the improved shape

memory effect and strengthening of the material, partial annealing is desirable for most

applications. The presence of the R-phase does not affect the performance of the alloy for most

applications, but will show up in material characterization measurements. 

Because there are so many aspects that may affect the transformation behavior of 

SMA’s it is important to have accurate material characterization measurements for experiment

and modeling agreement. The transformation temperatures are typically measured by

differential scanning calorimetry (DSC). This measurement technique results in plots of heat

flow vs. temperature, and shows endothermic peaks corresponding to the austenitic phase 

transformations, and exothermic peaks corresponding to the martensitic twinned transformation 

and R-phase transformation. Thus, from the DSC measurements one can observe the values of 

the transformation temperatures, and the state of the SMA. Then mechanical properties of the

SMA are measured by thermomechanical testing to quantify properties such as recovery stress, 

elastic modulus, and coefficients of thermal expansion vs. temperature to accurately model 

experimental behavior of SMA. 

1.2 Motivation/Applications 

The unique functionality of SMA’s described above make them a very attractive 

material for a variety of applications. Since their discovery, early applications involved 

coupling devices, fasteners, and electrical connectors13. The most well documented largescale application of SMA’s in 1971 was as a coupling to connect titanium tubing in the 

Grumman F-14 aircraft9

. More recent applications for SMA’s have been focused in the 

medical field, and smart material research. “Smart Structures” have been defined by C.A. 

Rogers as material systems with intelligence and life features integrated in the microstructure 

of the material system to reduce mass and energy and produce adaptive functionality14. 

Materials commonly classified as “smart materials” include piezoelectric materials, 

magnetostrictive materials, electrorheolgical fluids, magnetorheological fluids, and shape

memory alloys. All of these materials possess the ability to change their inherent properties as 

a result of a sensing mechanism. SMA’s have the ability to change stiffness, shape, natural

frequency, damping and other mechanical characteristics in response to a change in

temperature. The recent interest in smart structures has shown that SMA’s have great potential

for vibration, structural acoustic, and structural shape control15. Smart materials such as 

SMA’s can be used, as actuator/sensing devices in structures to accomplish desired adaptive 

properties. One such type of structure was discovered when Rogers and Robertshaw16

3
introduced the idea of embedding SMA actuators in a composite laminate. These structures

became known as Shape Memory Alloy Hybrid Composites (SMAHC). 

The effects of noise and vibration are present in almost every aspect of every day life. 

It has become common practice to find ways to reduce noise radiation and vibration of

structures. Common noise and vibration reduction techniques involve increasing the mass,

stiffness, or damping of a structure. This is typically done by passive treatments such as

constrained layer damping, acoustic absorption foam, tuned vibration absorbers, visco-elastic

vibration isolators, etc.17. The concept of employing smart structures that can adapt to the 

surrounding environment, to provide such previously mentioned functionality would be

extremely attractive in any field of applications where weight-penalty is of concern. Once such

application is in the field of high-performance aerospace vehicles where the excitation levels of

their structures are high due to engine noise and turbulent boundary layer fluctuating pressures.

Typical treatments for these effects are often limited to relatively low temperatures and high 

frequencies, and cannot adapt to a changing environment. The elevated thermal environment 

that aerospace vehicles are exposed to could activate SMAHC structures adaptive properties, 

rather than employ auxiliary hydraulic/electronic systems to facilitate actuation. Therefore 

smart structures such as SMAHC’s are of great interest. 

There has been much research done in recent years on SMAHC’s. At NASA LaRC a 

great deal of experimental work has been performed on SMAHC structures with embedded

Nitinol ribbon in which a new thermo-elastic model to predict the thermal buckling and postbuckling response of the structures was formulated18. Similar theoretical and experimental

work has been done by Baz, who used Nitinol SMA wires for controlling the natural 

frequencies of composite beams

19. The same has also been done for SMAHC plates20. A 

multitude of studies have been performed on the thermal buckling and post-buckling behavior

of composite beams and plates reinforced by SMA components21,22,23,24. Other work done in

the area has included active vibration control of cantilevered beams with the actuators external 

to the structure25. There have also been analytical formulations developed to demonstrate the

vibration/structural acoustic control of SMAHC panels 26,27

. 

There has been much work done to develop theoretical models that capture the thermomechanical behavior of SMA, as well as characterization of SMA material properties28,29,30,31

. 

There have also been many good comprehensive overviews of SMA characteristics and their

applications published6,9

. Although, as explained in the subsequent section, constitutive 

models that lend themselves easily to actual engineering property measurements are scarce. 

1.3 Objectives 

There has been previous investigation of SMAHC beam structures at NASA LaRC, 

which provided comparison of experimental results with a research code making use of a new

thermomechanical model for SMAHC structures. The previous work showed qualitative

validation of the model via the research code, but more definitive investigations are necessary. 

The goal of this work is to conduct experimental investigations on SMAHC beams to more 

accurately quantify the static buckling/post-buckling and dynamic response characteristics of 

the specimens. Numerical results will be generated in the commercial finite element code 

ABAQUS for comparison to experimental results, in contrast to the research code previously

used. Fabrication of SMAHC beam specimens was previously performed at NASA LaRC, in 

which pre-strained (4%) SMA (Nitinol) ribbons were embedded within layers of glass-epoxy 

matrix material. The glass-epoxy matrix material was chosen in order to allow electrical 

4
isolation of the SMA actuators, and flaw detection due to the material’s translucency. Once the 

SMA actuator ribbons are placed in the desired layers, the laminate is cured, and the SMA and 

glass-epoxy are integrated into one hybrid composite structure. Activation of the SMA 

actuators by resistive heating will serve as an adaptive stiffening mechanism to the structure by 

generating large recovery forces in the embedded pre-strained SMA ribbons. Accurate 

measurement techniques will be used to capture the adaptive stiffening effect of the SMA

actuators. Numerical modeling will be performed in order to show the correlation between the 

measured and predicted responses. 

Results from measurements of the static response of the SMAHC beam specimens, 

under thermal load by resistive heating, will be presented. This requires the application of an

electrical current through the embedded SMA ribbon. The As temperature (40 o

C (104 oF)) for 

the embedded SMA actuators is above ambient temperature thus once the temperature initially 

increases the SMAHC beam will be subjected to in plane compressive forces due to thermal 

expansion thus inducing buckling and post-buckling phenomena. The in plane compressive 

forces responsible for the post-buckled deflection will be overcome by the adaptive stiffening 

of the actuators once the temperature is high enough for activation of the SME. This highly 

nonlinear response behavior is quantified by a laser displacement transducer, Projection Moire’

Interferometry (PMI), and infrared thermal imaging measurements. These measurement

techniques also accurately determine initial geometric imperfections of the SMAHC beam 

specimens to help obtain accurate numerical models of the experimental response. 

Additionally, the steady-state dynamic response of the SMAHC beams, under random 

inertial loads is measured. Once the dynamic response time histories are captured for the beam 

specimens this data is post-processed into FFT, FRF, PSD, and coherence functions, which are

used for damping estimates, and displacement RMS values of the SMAHC’s vibration 

response. The results demonstrate the adaptive stiffening characteristics of the SMA actuators,

and provide accurate data to be used in the numerical simulation of the experiments. 

A new thermo-elastic model was recently developed, that makes use of engineering 

property measurement, by Turner32. This effective coefficient of thermal expansion model

(ECTEM) is relatively simplified, and easily integrated into commercial structural analysis 

codes such as ABAQUS. The other aforementioned constitutive models are based on the 

micro-mechanical behavior of the SMA and typically calculate the stress in SMA material 

based on three components; an elastic component, transformational component, and a thermal

component, that are all dependent on a martensitic volume fraction. The calculations of the

volume fractions differ from model to model but are usually functions of stress and 

temperature. In the ECTEM the stress in an SMA material is based purely on an elastic

component, and an effective thermal strain component. This effective thermal strain term

represents both the thermal and transformational components of other models. As a

consequence, the ECTEM model is limited to thermally activated transformations, but is

particularly attractive for SMAHC because it only requires the experimental measurement of 

fundamental engineering properties. Therefore it can easily be used in commercial structural 

analysis codes that have classical nonlinear thermo-elasticity capabilities. In the case of 

SMAHC structures, the micro-mechanical models are not efficient at modeling many spatially 

distributed SMA actuators. The ECTEM allows this to be done easily, by implementing the

composite properties from measurements of the material properties of each individual material

making up the composite. The formulation of the ECTEM model will be explained in detail in 

Chapter 5 in order to show how a model making use of thermo-elastic properties can be

5
implemented easily into a finite element formulation for efficient structural analysis. Also, the

reader is referred to the more detailed formulation, Turner32. 

6
Cha p t e r 2 

EXPERIMENTAL MEASUREMENT OF THE STATIC THERMOELASTIC 

RESPONSE OF THE SMAHC BEAM

In this chapter a background is given on the fabrication of the SMAHC beam

specimens, the buckling/postbuckling behavior, and geometric imperfections. Then the 

experimental set-up and instrumentation functionality are described. Then a brief overview of

the Projection Moire’ Interferrometry (PMI) measurement technique is given. Finally, the

experimental results of the laser displacement measurement and PMI measurements are 

presented and compared. 

2.1 Background 

The project consists of static/dynamic response experiments on two SMAHC beam

specimens, which were previously fabricated. The fabrication procedure will briefly be

described. For a more detailed explanation of the fabrication process the reader is referred to 

previous work at NASA LaRC32. The beam specimens were fabricated from glass-epoxy prepreg and Nitinol ribbon materials with a ply orientation sequence of (45/0/-45/90)2s. The glass 

epoxy pre preg has a thickness of 1.27e-4 m (.005 inches) and the Nitinol ribbon had cross 

section dimensions of .0023x1.5e-4 m (.09x.006 inches). Therefore, 5 pre-strained (4%) SMA 

ribbons were placed side by side in the middle of the 0o

 glass-epoxy layer to represent the 0.45-

inch actuator width called for by design. The schematic of the SMAHC beam specimen can be

seen in Figure 2.1 with dimensions of 0.5588x0.0254x0.0019 m (22x1x0.078 in). The SMA 

actuator ribbons had a length of 0.6604 m (26 in), therefore there is an excess of SMA actuator

material extending from the boundaries of the beam used as electrical leads for resistive 

heating. 

The SMAHC fabrication procedure consisted of pre-straining the SMA actuator

ribbons (4%), stacking of each glass-epoxy ply in the corresponding orientation while placing 

the actuators in the 0.45 inch width strips cut from the middle of 0o

 plys, curing the laminate

while restraining the actuators, and machining of the two beam specimens from the

consolidated SMAHC laminate structure. The curing cycle consists of placing the laminate

lay-up in vacuum drawn to an internal pressure of .0475 atm, and then subjected to an 

autoclave cure cycle in which a max temperature of 176.7 o

C (350 oF) is reached. Upon curing 

of the lay-up, the laminate constituents are integrated into one hybrid composite structure. The 

whole hybrid composite structure panel and the beam specimens after machining can be seen in 

Figure 2.2. 

Typically when a beam is subjected to compressive loads in the axial direction, the

beam loses stability. The Euler Load is the critical load at which the beam loses stability. 

Theoretically a lateral load is applied to cause deflection to a new position from which the 

beam will not return when the lateral load is removed. In reality this deflection occurs not as a 

result of an applied lateral load, but to geometric imperfections in the beam. The SMAHC 

beams were installed into a beam fixture with clamped boundary conditions. The first

experiments will involve applying only a thermal load to the SMAHC beam by electrically 

activating the embedded SMA actuators. The heating of the constituent materials with 

7
clamped boundary conditions will generate a compressive load within the SMAHC beam. 

Once this compressive load reaches the critical value (Euler buckling load) the beam buckles

undergoing a large deflection. In classical theory the stiffness of the beam goes to zero and 

deflection goes to infinity at the critical load (Pcr). Mathematically the solution path of the 

beam “bifurcates” to a new equilibrium position. 

As previously mentioned the laminate consists of geometric imperfections, which 

dictate the direction of the buckling phenomena. As temperature of the post-buckled beam

continues to increase, the embedded SMA actuators will be activated and generate in-plane 

tensile recovery forces that will overcome the compressive forces and render the post-buckled 

state of the beam back to its flat position. Therefore it is important to study these static 

responses of the SMAHC beams because the buckling/post-buckling states are part of the 

dynamic response as well. During the static response experiments accurate measurement

techniques using Projection Moire’ Interferometry, laser displacement transducer, and infrared

thermal imaging will be conducted to accurately capture the static response characteristics. 

2.2 Experimental Set-up 

In these static response experiments a thermal cycle, which consisted of increasing the

temperature of the SMA actuators from 23o

C (74 oF) to 121 oC (250 oF), was conducted 10 

times for 2 separate beam specimens for repeatability investigation. The SMAHC beam was 

installed into the beam fixture as shown in Figure 2.3 with clamped boundary conditions. Each 

SMAHC beam has an unsupported length of .4572m (18 in). The aluminum beam fixture 

consists of a mechanical grip, which applies fixed boundary conditions on each end, and 

electrical grips that apply the electrical current to the SMA actuators embedded within the

beam. Figure 2.4a shows the mechanical and electrical grips from the front of the beam

fixture. The positive power supply lead can be seen connected to the electrical grip block. 

This entire block is electrically activated but is isolated from the mechanical gripping block by 

a layer of Garolite G-10 insulation. The SMAHC beam is also thermally isolated from the 

beam fixture by layers of Garolite G-10 installed between the mechanical grip surfaces. The

Garolite G-10 is a glass fabric-epoxy insulator, which is a relatively hard/stiff material that 

should eliminate any compression under in-plane loads caused by the SMA recovery stress 

during testing. In similar tests a fibrous ceramic insulator was in service, which showed signs

of compression during testing, thus relieving some of the recovery stress in the activated SMA 

actuators. Type-T thermocouples were attached to the beam specimen on the center of the

beam 1 inch from the left mechanical grip, to both of the mechanical grips, and one is 

measuring the surrounding ambient air temperature. These thermocouples can be seen attached 

to the left mechanical grip and the beam specimen in Figure 2.2b, which shows the backside of 

the beam fixture. Thermocouple measurements are subject to large error due to contact

resistance, and “fin” effect, in which the attached thermocouple lead conducts heat away from 

the contact and the heat is lost into the surrounding air due to convection. Therefore, the

thermocouple measurements are used for monitoring purposes only. An infrared thermal

imaging camera (FLIR Thermacam SC2000) was used for control measurements of the beam 

specimen temperature. The thermal camera eliminates the measurement uncertainty of contact

resistance and “fin” effect, but does require calibration to the target emissivity. The average 

total hemispherical emissivity of the beam specimens was measured to be 0.8 and it was

assured that the beam radiates as a gray-body. 

8
A Hewlett Packard DC Power Supply, model #6652A, is connected to the two 

electrical grips of the beam fixture, and is controlled via a LabView virtual instrument (VI).

The LabView VI controller was previously configured and designed at NASA LaRC. The VI 

controller makes sensor measurements through the infrared thermal imaging camera, the typeT thermocouples, and controls the power supply current output through general-purpose 

interface bus (GPIB) commands. Optimal gains are set to use proportional-integral-derivative

(PID) control to supply current to raise the temperature of the SMAHC beam centerline to 

desired set points where data is recorded. The LabView VI system is capturing data from a 

laser displacement transducer (Keyence model LK-081). The laser displacement transducer

was attached to an optical positioning traverse, which was used to precisely align the laser to 

the mid-span point of the SMAHC beam specimen, and calibrate the initial laser

transducer/beam surface distance to zero. This was done before every run. Therefore as the VI 

controller runs through a thermal cycle, the beam temperature is sensed by the thermal camera, 

current is applied to raise the temperature from 23 o

C (74 oF) (ambient) to 121 oC (250oF)

capturing set points in increments of every 5.55o

C (10 oF), and the mid span displacement 

values are being recorded continuously approximately every 4.2 seconds during the thermal 

cycle except for beam 3 run 2 and beam 3 run 3, in which laser data was captured at set-point 

only. One entire cycle took approximately 480 seconds. As an added feature the VI controller 

was designed to automatically trigger the PMI imaging system to capture images at every set

point during a thermal cycle. Therefore, the laser displacement transducer is capturing the

mid-span displacement measurements, and the PMI system is capturing the full-field 

displacement of the SMAHC beam specimen, throughout the thermal cycle. Figure 2.5 is a

schematic showing an overview of the components and functionality of the experimental setup. Figure 2.6 is an image of the entire experimental set-up and all measurement components

can be seen. The PMI measurement of full-field displacement will be explained in more detail 

in the subsequent section.

The results of the full-field displacement measurement give great insight into the

buckling behavior of the beam, and initial imperfection measurements can be captured. Also 

the PMI displacement measurements can be compared to the laser displacement transducer

measurements of the SMAHC beam mid-span to reduce uncertainty. 

2.3 Projection Moire’ Interferometry 

Projection Moire’ Interferometry is an optical surface deformation measurement 

technique used to measure the full-field deformation of a structure. PMI is an optically simple, 

non-contacting measurement technique used since the 1970’s for surface topology and shape

characterization33,34. A projection system with an installed Ronchi ruling is used to project a 

grid of equally spaced, parallel lines onto the structures surface. A CCD camera is positioned 

to view the structure at some angle θ inclined from the projector optical axis, which is 

perpendicular to the structure surface. Images are captured of the projected grid lines during 

the initial unloaded state and during loading of the structure. As previously mentioned for this 

experiment, images were captured every 5.55o

C (10oF) from 23-121oC (74-250oF). At each

set-point the PMI system is triggered, and it takes 10 images and averages them. Image

processing routines are used to remove camera perspective distortion and interfere the captured 

images with a numerically generated reference grid. This results in interferograms containing 

moire’ fringes. The fundamentals of PMI measurement and its applications have been studied

9
and quantified at NASA LaRC by G.A. Fleming for use in wind tunnels35. Figure 2.7 shows a

schematic of the PMI imaging system. 

Imagine that the grid lines are projected onto a flat surface located at coordinate zo in 

the schematic. The CCD camera captures this reference image. Then when the plate is

deformed or translates in the z direction the spatial locations of the gridlines change, thus 

producing a deformed image. This change can be represented mathematically by distorting the 

reference image grid by this phase function: 

( ) ( )

d

z x y i j π θ

φ

2 , tan , = (1) 

where φ(i, j) is the spatial phase at a pixel in the CCD camera image plane corresponding to

point (x,y) on the un-deformed object surface, z(x,y) is the out-of-plane deformation at that

point, and d is the distance between grid lines. It is known that taking the absolute difference

of the reference grid and the deformed grid produces a interferogram. The interferogram will 

contain moire’ fringes occurring when φ(i, j) = 2πn , thus: 

( ) tanθ , nd

z x y = (2)

Therefore, the moire’ fringes in each interferogram represent the contours of the deformation in

the z direction. The fringe contour interval is then defined as: 

tanθ

d ∆z = (3) 

This is the basic idea of how the PMI measurement theory is implemented. Further 

techniques of image processing such as phase shifting, phase unwrapping, choosing anchor

point, and grid-line nonlinearity issues are addressed in more detail in the following 

references35,36,37. 

2.4 Static Thermoelastic Response Experimental Results 

Laser Displacement Transducer Results 

The mid-span deflection measurements of the SMAHC beam2 and beam3, by the laser

displacement transducer, are presented in Figure 2.8, 2.9, respectively. The figures show the

mid-span deflection as a function of average temperature of the centerline along the length of 

the beam. Data from thermal cycles 1-9 on SMAHC beam specimen #2, and thermal cycles 1-

10 on SMAHC beam specimen #3 are presented. The adaptive stiffening mechanism is evident 

in all runs for both beams. Recall that the austenite start temperature for the Nitinol ribbon is

greater than the ambient temperature (As > To). Therefore, buckling of the clamped beam will

occur immediately when the temperature of the beam begins to change. As the temperature of

the beam is increased the midspan of the beam deflects until a critical temperature at which the 

SMA recovery forces overcome the in-plane compressive forces due to thermal expansion and 

the beam is rendered flat. The experimental data shows a relatively large variation between 

maximum buckling deflection. This is a highly nonlinear phenomenon and dependent upon 

many parameters such as initial deflection, initial stress, geometric imperfections, and 

10
boundary conditions. Each thermal cycle test and possible reasons for the behavior will be 

described in the next paragraph. 

In Figure 2.8 for beam 2, runs 1-9 are the first 9 thermal cycles on this beam specimen.

In Figure 2.9 for beam 3, runs 1-10, correspond to thermal cycles 5-14. Actuator degradation 

with thermal cycle reduces the maximum tensile recovery force in the beam at higher 

temperatures. Figure 4.4 shows the measured recovery stress as a function of temperature for

numerous thermal cycles of the pre-strained Nitinol ribbon used in this project. Notice, even at

high thermal cycles the temperature range that corresponds to where the post-buckled 

deflection is recovered (~32 o

C - 65.5 oC) shows no significant degradation with thermal cycle. 

Therefore thermal cycle degradation is not a factor in the observed variation of maximum 

deflection. In run 1 for beam 2, and runs 1, 2, and 10 for beam 3 a relatively large max 

deflection occurs. All these runs occurred as the first thermal cycle on the SMAHC beam after 

a relatively long period of storage time, and could be attributed to moisture absorption in the

beam. It is common for glass-epoxy composite materials to exhibit moisture absorption after 

relatively short periods of time. The absorption of moisture in the composite beam laminate 

could possibly cause swelling of the layers of the structure. The presence of moisture in the 

composite structure could enhance the thermal expansion effect at the initial temperature 

increase. As temperature continues to increase the maximum buckling deflection is increased 

and the point of deflection recovery is pro-longed to a higher temperature. It is likely that at 

some temperature the moisture is driven out of the structure, thus causing a rapid recovery of

the deflection. This could be the explanation for the rapid deflection recovery observed in the 

large deflection cases. Beam 3 Run 1 was the first thermal cycle on the beam in approximately 

one year. Beam 3 Run 2 occurred 9 days after the first run. Beam 3 Run 10 occurred 14 days

after the previous run. Beam 2 Run 1 was the first thermal cycle on the beam, which was

fabricated approximately one year before the experiment. So, it seems probable that the

buckling phenomenon and deflection behavior are very sensitive to hygothermal effects, and 

indicates that the numerical modeling to match the buckling phenomenon will require precise 

knowledge of all parameters that can affect the environmental state of the beam. 

Another contributing factor to the variation could also be a result of pre-tension on the

SMA actuator leads once installed. It is important to have no stress present in the installed

SMAHC beam initially, but it is also necessary to have the beam installed “taught” so the beam

is not initially buckled. The experimental procedure involved pulling the leads “taught” just

before clamping at the boundaries. It is a possibility that some pre-stress may be present in the 

SMA actuator ribbons initially before the run. Recall that the transformation temperatures are 

a function of stress, and typically vary linearly with applied stress. The buckling deflection is

very sensitive to the recovery force of the SMA, which would be effected by the shift in the As

transformation temperature. During the buckling and post-buckled state of the beam, there are 

two mechanisms competing. The compressive forces being generated due to thermal

expansion, and the recovery stress that is generated at As temperature of the embedded SMA 

actuators. A small change in either mechanism could result in a large change in the deflection 

behavior. 

It should also be mentioned that another possible contribution to the pre-stress variation

could perhaps be due to small deformation of the Aluminum beam fixture in which the

SMAHC beam is clamped within. During the experiment procedure thermal cycles were

performed in succession, once the infrared thermal imaging camera measured the beam’s

centerline cooling from elevated temperatures back to ambient (~74 o

F). In cases, where the 

11
thermal cycles were performed in quick succession, the Aluminum fixture may have been 

somewhat soaked with thermal energy, which it looses rather slowly relative to the SMAHC 

beam specimen. As a result, when the SMAHC beam was re-gripped during these cyclic runs

the fixture could be continuing to lose heat and contract. Thus, the beam would be under

compressive stress at the beginning of a thermal cycle, which was performed in quick 

succession from a previous one. 

Another observation from the laser displacement data is the variation of the initial 

imperfection of the mid-span. Beam 2 seems to exhibit a larger variation in initial deflection 

values between runs, than does beam 3. This could be attributed to the fact that beam 2 had 

never been subjected to a thermal cycle before, thus the internal stress state probably has a

large variation in the first couple of thermal cycles. After curing of the laminate, residual 

stresses are present in the composite matrix. Upon the first thermal cycle the largest recovery

forces are seen within the beam, causing a large chance of internal slipping, and settling of the 

composites components during early runs. Thus the initial deflection of the mid-span, which is

due to geometric imperfections of the composite, and also induced by the boundary condition 

imperfections of the fixture, could have larger variations if the internal state of the beam is

changing. Another interesting observation is that during all runs for both SMAHC specimens

the buckling phenomenon always occurred in the same direction towards the laser

displacement transducer. For Beam 2 run 9 the SMAHC specimen was flipped over in the

beam fixture, and still exhibited buckling towards the laser. This suggests that boundary

conditions of the beam fixture are contributing to the initial shape, and buckling direction of the 

beams. 

Following the static experiments in which the laser displacement data and PMI data 

was captured, another experiment was run to capture the geometric imperfections of the beam

and capture the full-field initial shape of the SMAHC beam installed in the fixture to confirm

the PMI results. The experiments uncovered some of the initial conditions contributing to the 

initial state of the beam. 

The experimental configuration consisted of a Klinger Scientific motorized linear 

traverse and controlled via the Labview VI. The translation stage was mounted parallel to the 

SMAHC beam fixture with installed beam as seen in Figure 2.10. The laser displacement

transducer was mounted to the translation stage, and traversed via a programmable stepper 

motor the length of the beam and recorded displacement measurements in .10 inch increments

along the 18 inch length of the beam. This data shows the initial imperfection of the beam, 

when mounted. The laser is calibrated to zero at the left end of the beam from a front view,

and all subsequent measurements during a run are relative to that first set point. Multiple cases 

were studied, and these experiments uncovered some interesting results. 

The initial imperfection data was captured on the beams in one direction and then 

flipped to the other side in the beam fixture. These results in Figure 2.11 show that the

SMAHC beams take on the same shape even once flipped. This figure shows the out of plane

deflection of the beam at ambient temperature as a function of the length of the beam. These

measurements represent the initial imperfection profile of the beam when installed in the

fixture. Notice that the blue colored line represents the out of planed deflection measurement 

for beam 2. Then beam 2 was flipped over and installed again in the fixture, which it would be

expected to see the opposite deflection profile. But the magenta line results for Beam 2 flipped 

produces the same deflection profile. This confirms that the boundary conditions are 

contributing to the initial shape of the beam. Approximately the same deflection profile results 

12
for beam 3 specimen as well. The laser data is also compared to one of the data sets for the

PMI absolute imperfection measurement. As will be explained in the next section the absolute

PMI measurement is the first image of the beam captured at ambient differenced from a 

reference image of a perfectly flat calibration plate. It is evident in Figure 2.11 that the laser 

displacement initial imperfection results are in excellent quantitative agreement with the black 

dashed line representing the PMI absolute measurement at ambient temperature. This is true 

for all runs of the PMI absolute measurements for both beams at ambient and confirms that the 

PMI absolute measurements are correct. 

An interesting observation is that the beam is skew in all cases from one grip to the 

other in Figure 2.11. At 18 inches represents the left boundary grip therefore it is expected that

deflection measurement should be zero relative to the other right boundary grip This was

noticed in the PMI results and first attributed to the possibility of the calibration plate not being 

perfectly aligned with the beam fixture. But, the laser displacement measurements confirm 

that there is an offset in the actual fixture from one grip to the other while a beam is installed.

A measurement of the distributed thickness of the Garolite G-10 insulation blocks that the 

beam is clamped within show that they are not symmetric. In fact the difference in thickness of

the inside edges of the two back G-10 insulation blocks that the beam is clamped against have

a difference of approximately .020 inches. This difference closely corresponds to the skewness

observed in the PMI, and laser displacement measurements of the initial imperfection. 

Subsequent measurements using the laser displacement transducer traverse with multiple G-10

block arrangements were performed to show that the effect the boundary conditions have on 

the initial shape of the beam. The results in Figure 2.12 show a totally different initial shape of 

the beam for different orientations. The corresponding configurations are best described by the 

schematic showing the G-10 block arrangement and each G-10 thickness at the edges in Figure 

2.13. 

Thus, based on the above observations the laser displacement transducer translation 

experiments prove that the boundary conditions are the primary factor to the initial shape of the 

beam, and confirm the PMI absolute measurements at ambient. The offset of the beam

between boundaries is a physical entity of the experimental results, but is just a rotation of the

beam. All numerical modeling in ABAQUS will be relative to a flat plane, so for comparison 

purposes all skewness present in the initial geometric imperfection experimental data has been 

subtracted before input into the ABAQUS input file. 

Projection Moire’ Interferometry Results 

The PMI measurements captured the full-field displacement of the beam and are 

represented as pixels. The full-field PMI images showed no variation in displacement in the 

vertical width direction, thus the centerline data points along the length of the beam are

representative of the full-field image of the beam. The PMI data presented for this experiment

shows the out of plane displacement of the centerline of the SMAHC beam for various SMA 

actuator temperatures. The x-axis represents the beam length. PMI data points were only 

captured for 0 - 17.83446 inches along the length of the beam due to the angle of the PMI

projector creating a shadow at the right boundary. PMI data is relatively noisy due to surface

roughness, and processing techniques. To adjust for this, experimental data was fit to a 6th

order polynomial of the form: , in which data

points from –1.006 to 0 and 18 to19.012 inches were manually zeroed to produce accurate line

fits representing the deformation profile of the beam. The zero padded data sections represent

2 3 4 5 6 Y = A + Bx + Cx + Dx + Ex + Fx + Gx

13
the x-locations in the clamped boundaries and it is a fair assumption that displacement values

are zero. Thus, the line fit was used to extrapolate data for the small beam length section from 

17.83446 - 18 inches. The positive sense deformation represents deflection away from the PMI 

imaging system. An example of the captured PMI data and corresponding line fit for a 

differential measurement at 48.8 o

C (120 oF) is presented in Figure 2.15 for Beam 2 Run 4. 

There are two types of PMI measurements made for these experiments. As explained 

the PMI data can only represent deformation from an initial reference image. Differential 

measurements use the initial ambient temperature image as this reference image. This 

reference image is considered the zero state. Subsequent images taken as temperature

increases are the deformations relative to the reference state. Therefore any initial imperfection 

of the SMAHC beam is not present in these measurements. Additionally, the skewness is not 

present in the differential measurements for the same reasons.

The other type of PMI measurement is the absolute measurement. In reality it is known 

that the SMAHC beam specimen in the initial ambient temperature is not perfectly flat, due to 

imperfections in the composite laminate lay-up, initial stress state within the beam, and 

imperfect boundary conditions. The absolute PMI measurements use a reference image of a 

flat plate in the plane of the beam fixture (See Figure 2.14). The first ambient temperature 

image captured from the SMAHC beam is then differenced with the flat plate image. This

results in a measurement representing the full-field initial imperfection of the SMAHC beam at 

ambient temperature. 

The Differential PMI measurements were post-processed by interfering the reference 

image at ambient with each elevated temperature image, which gives the difference. The fact

that the reference image was skewed, as well as all elevated temperature images it is 

differenced with, results in no offset present in the differential measurements. Therefore the 

confirmed initial imperfection, with skewness subtracted off, can be added to each differential 

measurement to give the absolute shape of the beam for all temperatures in order to show 

comparison with the numerical simulations. Each run’s absolute measurement, giving the 

initial geometric imperfection, was added to each differential measurement. Therefore, there 

now is a representative absolute measurement for every temperature in 5.55o

C (10oF)

increments between ambient and 121.11 o

C (250 oF). The PMI results are shown for selected 

runs 1, 4, and 8 for beam 2, and beam 3 respectively in Figures 2.16-2.21. The plots show the

full field out of plane deflection of the beam for a few individually selected temperatures. 

These presented results in the aforementioned figures show the same trends as all runs, which

are included in Appendix A. Notice that in all runs the initial shape of the beam takes on the 

same S-shaped geometric imperfection at ambient temperature. This will be the initial 

geometric imperfection data input into ABAQUS for accurate modeling of the experiments.

Also notice that the adaptive stiffening is also evident in all the PMI data and matches the laser 

displacement results for the midpoint of the beam.

The midspans of the centerline PMI deflection profiles were extracted for each

temperature and plotted with the laser displacement transducer midpoint deflection 

measurements in Figure 2.22, and Figure 2.23 for beam 2 and beam 3 respectively. It is

evident that there is good agreement between the two measurements. Note that there are fewer

data points making up the PMI data curve because there is only PMI midspan data at the 

temperature set points. Also, small differences in the data sets can be attributed to the fact that 

the PMI data system was triggered to take data once the temperature reached a certain set point. 

The PMI system is triggered after the set point temperature is reached and laser data is already 

14
written. Thus, 10 averages are collected as the thermal controller starts raising the temperature 

to achieve the next set-point. A similar scenario is likely in Beam 3 Run 1 because this is the 

only run where the control voltage was operated manually. Therefore there is a greater chance 

that the set point temperature drifted from the desired temperature by the time the PMI was 

manually triggered. This can explain the large difference in this particular data set. There are 

also small differences between data points at ambient temperature. This can be attributed to 

the fact that they are two separate measuring devices that are not referenced to the same initial 

position. The midspan data point for the PMI is referenced to the flat calibration plate image in 

the plane of the beam, whereas the laser displacement transducer’s initial data point is 

referenced to the beam’s flat position at 121.1 o

C (250 oF). 

Overall the PMI measurements prove to be extremely accurate, and in excellent

agreement with the laser displacement transducer measurements. This validates the accuracy 

of the PMI measurement technique for this experiment. The PMI measurements give

additional insight into the full-field deformation of the SMAHC beam throughout a thermal

cycle, and give the important initial imperfection data to be used for accuracy in numerical 

modeling. 

15
Cha p t e r 3 

EXPERIMENTAL MEASUREMENTS OF THE DYNAMIC RESPONSE OF THE 

SMAHC BEAM 

The dynamic response chapter is arranged as follows. First, a background is given on 

the dynamic response measurement of the SMAHC beams. Then the experimental set-up and 

instrumentation functionality are described. Then a brief overview of the post-processing of 

the captured data is presented. Finally, experimental results of the SMAHC beam specimen’s

responses are presented and compared for a repeatability investigation. 

3.1 Background 

All the SMAHC beam vibration results in this study result from random base

acceleration excitation. Resulting responses will be presented at various temperatures 

corresponding to the different stiffness states during the thermomechanical response. Using a

random inertial excitation requires a statistical analysis to be used to process the response data. 

Randomness in vibration analysis can be thought of as the result of a series of experiments all

performed in an identical fashion under identical circumstances, each of which produces a

different response42. Multiple time histories of the input and corresponding responses are 

sampled. From these multiple samples Fast Fourier Transforms can be done and averaged to

produce dynamic response characteristics. By demonstrating control over the dynamic 

response, i.e. adaptive stiffening, and RMS value reduction, implications of weight-efficient

vibration and acoustic radiation reduction methods are eluded too. Also analysis of the 

resulting frequency responses will result in beneficial modal parameter values to improve 

modeling attempts. The beam fixture was designed not to have any resonance in the excitation 

bandwidth used in the experiment. Transducers are placed on the fixture and the beam in order

to obtain the relative response of the SMAHC beam, and allow for comparison to response

spectra generated numerically by ABAQUS. During excitation a thermal load will be applied 

to the beam as in the previous static tests. The SMAHC beam will be brought to a desired 

temperature set point, in which dynamic data will be captured. One run will consist of 

capturing dynamic data at all the chosen set points through a thermal cycle from ambient to 

121.11 o

C (250 oF). 

3.2 Experimental Set-up 

The aluminum beam fixture with installed SMAHC beam specimen is mounted to a 

large electromechanical shaker (Model T1000) configured horizontally and attached to a

magnesium slip table, which oscillates over a bed of oil in a horizontal plane during operation,

as seen in Figure 3.1. The excitation of the beam during operation is measured by 3 Endevco 

Model 2250A-10 accelerometers mounted in 3 different positions along the length of the

SMAHC beam specimen. There are also 2 PCB model T352M92 accelerometers mounted to

each end the beam fixture itself. Two of the PCB accelerometers are used for control of the 

shaker table, and the other two are used to subtract off the base motion from the absolute 

motion measured by the accelerometers mounted to the beam. The dynamic response

16
experiments of each SMAHC beam specimen consisted of 3 thermal cycles where the beam

was heated from ambient to 121.11 o

C (250 oF) in increments of 5.55 oC (10 oF). At each set

point the dynamic acceleration data was captured. The experimental set up can be seen in 

Figure 3.1. The infrared thermal imaging camera is aligned at the end of the shaker table 

focused on the SMAHC beam, and used for control. A DC power supply voltage is applied to 

the beam via the Labview VI controller in the same manner as the static experiments described 

previously. The Labview VI controller is sensing measurements from thermocouples for 

monitoring, and the infrared thermal imaging camera for control. An addition to the Labview

controller was made to hold once a set point temperature is reached, and then can be released 

when specified by the user. This allowed the dynamic data to be captured for that temperature, 

and then the set point is released and the beam’s temperature is elevated to the next set point. 

In Figure 3.2, there are Endevco (2250A-10) accelerometers mounted on the SMAHC

beam in positions 1-3. Position 1 is 4.5 inches from the right mechanical grip from the back 

view, position 2 is at the midspan, and position 3 is 14.5 inches from the right mechanical grip.

There are also 2 PCB (model T352M92) accelerometers mounted on the back of each of the 

mechanical gripping blocks. One pair is used for control of the shaker table, and the other pair

is used to capture the fixture response. Time histories with a total length of 200 seconds were

captured with a sampling rate of 1024 Hz to allow 50-frame averages with a bandwidth of 0-

512 Hz and a frequency resolution of .25 Hz for every set point. The SMAHC beam was

excited by random base acceleration over a 10-400 Hz bandwidth with a RMS value of .25 g. 

The excitation’s power spectral density (PSD) is shown in Figure 3.3. All three response

transducers are capturing acceleration data, and one pair of the fixture transducers are being 

integrated to displacement data, while the other pair for control of the shaker is capturing 

acceleration data. All of the transducers are connected to a Hewlett Packard VXI mainframe

(model E8403A) with a 16 channel E1432A module data acquisition card. The dynamic data

is captured and processed using MTS IDEAS software. Upon capturing all of the time

histories for each temperature set point, the data is post-processed into relative response

quantities by subtracting off the base acceleration. Then response quantities are generated 

including frequency response functions (FRF) between the beam specimen’s response and the 

fixture’s input, power spectral densities (PSD) of the acceleration input and relative beam

responses, and coherence functions, which indicate data quality in general. A schematic of

the dynamic experimental set-up can be seen in Figure 3.4. 

Before the dynamic response experiment the beam specimen was weighed and installed

into the beam fixture. Beam 2’s mass without mounted transducers was 75.97e-3 kg (.1675 

lbf) and 79.15e-3 kg (.1745 lbf) with mounted transducers. Beam 3’s mass without mounted

transducers was 74.84e-3 kg (.1650 lbf), and 78.25e-3 kg (.1725 lbf) with transducers. Just

prior to activating the excitation to the base, the mechanical and electrical grips are torqued to 

13.56 N.

m (120 in.lbf). Once the random broadband excitation begins the thermal controller is 

activated and holds at the designated set point. The dynamic data acquisition system then 

acquires the acceleration data for all accelerometers. Following three runs for each beam the 

acceleration time histories are post-processed, which included averaging 50 times. 

3.3 Dynamic Response Data Post-Processing Overview 

In general a transducer used to measure input and output during a vibration test usually 

contains noise. Additionally, performing a number of identical tests and averaging the results

increase confidence in a measurement of random vibration. In this experiment it was desirable

17
to obtain displacement response data, because it is more intuitive. Direct hardware integration 

was attempted during testing by PCB model 480B10 Signal Conditioners. In this case 4-

second time history samples were taken to construct a 200 second time history for each

temperature set point in MTS IDEAS. The integrated time signals were swamped with a low

frequency noise. The acceleration time signals were processed, and the FRF’s and PSD’s 

showed large low frequency peaks representing the noise. Therefore, upon integration of the

acceleration data in the frequency domain the low frequency noise dominates the resulting 

displacement time history. Thus, it was determined that all transducers would capture the 

acceleration data, and integration to displacement would be performed offline. A MATLAB

m-file was written to perform a Fast Fourier Transform (FFT) on the acceleration time history, 

integrate in the frequency domain to get displacement FFT values, “zero out” the low 

frequency noise below 10 Hz, and then perform an inverse FFT to convert to a displacement 

time history data file. Additionally this m-file computes the displacement PSD’s, and RMS

values from the generated displacement time history (See attached m-file in Appendix B). 

The Fast Fourier Transform is defined as 

( ) ∑ [ −

=

= ∆ − ∆

1

0

, exp 2

N

n

n X f t t x j πfn t] (4.1) 

assuming that there is a time history record x(t) that is sampled at N equally spaced points a 

with time increment ∆t, where ∆t has been selected to produce a sufficiently high cutoff 

frequency. Therefore, xn=x(n∆t) where n = 0, 1, 2,...., N-1. The selection of the

corresponding frequency values for the computation of X(f,t) is 

N t

k

T

k f k ∆ = = k = 0, 1, 2, ..., N-1 (4.2) 

At these particular frequencies, the transformed values give the Fourier components defined as 

∑

−

= ⎥

⎦

⎤ ⎢

⎣

⎡ = − ∆ =

1

0

2

exp

( ) N

n

n

k

k N

kn

x j

t

X f X π k = 0,1,2,...,N-1 (4.3) 

The Fourier components repeat after the index number k=N/2 since the Nyquist frequency 

occurs at this point. Thus, in the m-file the corresponding frequency vector generated for the

Fourier components is folded back on itself in order to retain the phase information to 

transform back to the time domain. 

Matlab generates the PSD’s using the specified time data, number of points per data

segment, sampling rate, window vector, and number of points to overlap. The output of the

PSD function in Matlab must then be normalized to the Nyquist frequency to obtain the proper

units of (in)2

/Hz. The RMS value is then calculated from the auto spectra (PSD) values. In

Figure 3.4, a particular displacement PSD at ambient temperature generated by the m-file is 

plotted with its corresponding acceleration PSD to confirm that the m-file is correct. It is 

observed that the low frequency noise is filtered out, and the frequency information is retained, 

providing excellent correlation with its corresponding acceleration PSD. 

18
3.4 SMAHC Beam Dynamic Response Experimental Results 

The experimental behavior of the SMAHC beam specimen observed during dynamic 

testing will briefly be described. As described in chapter 3, the beam exhibits a thermal postbuckling deflection over a particular range of temperatures. Within this temperature range the 

beam undergoes interesting dynamic responses because the stiffness of the beam is changing 

rapidly. Recall from the post-buckling analysis in chapter 2 that the stiffness theoretically 

drops to zero at the point where the beam loses stability and buckles. When the beam is in the 

post-buckled state the stiffness increases with deflection. When the SMA actuators are 

activated and return the beam flat, the stiffness again decreases but rapidly increases with 

continued heating from this state. For accurate numerical predictions in this temperature range 

a nonlinear dynamic analysis is needed, and will prove difficult to get good correlation for the 

simple fact that the beam’s dynamic characteristics are changing rapidly. During excitation of

the beam, its relative response was very noticeable at temperatures where the SMA actuators 

were not activated; where-as at high temperatures the relative response was small, and 

unnoticeable once the SMA ribbon is fully activated. Dynamic results will be presented for 

these temperature ranges. The SMAHC beam is returned to the flat configuration at

approximately 65.55 o

C (150 oF). 

In Table 3.1 and Table 3.2 the RMS displacement values are presented as a function of

temperature for all 3 positions on beam 2, and beam 3 respectively. The results show a 

decrease in RMS displacement of the beam with increasing temperature, which is evidence of

the adaptive stiffening of the beam. The modal parameter estimates for beam 2 and beam 3

were obtained and are presented in Table 3.3 and Table 3.4 for each run. The modal damping 

ratio estimates are particularly useful as input into the numerical modeling of the experiments

in ABAQUS. These modal damping ratio values are obtained in MTS IDEAS by performing a

single degree of freedom curve fit (SDOF method). In this method each peak is analyzed by 

assuming that it is the frequency response of a SDOF system. This assumes that near each 

resonant peak, the FRF is dominated by that single mode. The damping estimates for mode 1

and mode 3 show a decrease in damping from the non-activated state relative to the high 

temperature activated state of the SMA. Interestingly, at the higher temperatures near 121.1 o

C 

(250 o

F) the damping estimates begin to increase slightly. This may be attributed to the glass 

epoxy matrix material heating up and softening, and is not associated with the SMA actuator

material. 

In Figures 3.6-3.11 the displacement PSD’s for selected temperatures of 23.3 o

C (74 o

F), 71.1 oC (160 oF), 93.3 oC (200 oF), and 121.1 oC (250 oF) are presented for the various 

positions along the SMAHC beam specimens. The PSD’s for position 2 differ from the 

corresponding PSD’s for positions 1 and 3 in that the midspan of the beam does not contain the

anti-resonant peaks between mode 1 and mode 3 resonance peaks. Also position 2 contains a 

larger displacement RMS value, which makes sense at the midspan of the beam. The 

experimental results show excellent repeatability between runs 1, 2, and 3, as well as between 

beam 2 and beam 3. Therefore the presented data sets in Figures 3.6- 3.11 are representative

of all other captured dynamic data. The PSD’s for both beams at all positions demonstrate the

stiffening effect of the SMA actuators as the modal frequencies of the response are increased

with increasing temperatures above 60 o

C (140 oF). Figure 3.12 best shows the variation of the

modal frequencies with temperature. Notice that the fundamental frequency increases from 

30.1 Hz to 125.1 Hz from ambient temperature to 121.1 o

C (250 oF). Also, mode 3’s frequency

nearly is increased out of the bandwidth of excitation. 

19
Thus the dynamic response experiments demonstrate the adaptive stiffening mechanism 

of the embedded SMA actuator ribbons. The dynamic data gives a good overview of the full

dynamic behavior for multiple positions along the beam length. The fact that the RMS values 

decrease with temperature, and the beam is stiffened indicates the reduction in the dynamic

response. This has implications of improving sonic fatigue, and noise transmission reduction.

Finally, accurate modal parameter estimates are captured to include in the dynamic numerical

modeling of the SMAHC beam in ABAQUS. 

20
Chapter 4 

THERMOELASTIC CONSTITUTIVE MODELING FORMULATION 

4.1 Background 

The ECTEM developed by Turner32 makes use of experimentally measured

engineering properties. This makes the ECTEM easily implemented into a commercial code 

that is capable of nonlinear thermoelastic analysis of composite materials. This chapter will 

briefly summarize the development of this model, to give the understanding of how the

engineering property measurement becomes the determining parameter of the modeling. The 

model is based on a mechanics of materials approach and the reader is referred to numerous 

sources18,38,39,40

. 

4.2 Thermoelastic Constitutive Model/Stiffness Property Formulation 

Figure 4.1 shows a schematic of a representative volume element of the SMAHC 

laminate, within the plane of the structure. The principal material coordinates for the 

composite matrix are 1, and 2. Note that in general the figure can represent embedding 

between or within layers and no limitation is placed on SMA orientation relative to the 

composite matrix fiber directions. The following constitutive relation is the fundamental

equation of the ECTEM developed by Turner32 for the SMA element in the 1-direction. 

( ) ( ) ⎥

⎦

⎤ ⎢

⎣

⎡ = − ∫

T

To

σ a Ea T ε α a τ dτ 1 1 1 (4.1) 

where, Ea is the Young’s modulus of the SMA, ε1 is the strain in the 1-direction, α1a is the 

coefficient of thermal expansion (CTE). This equation represents the stress-strain relation 

specifically for the SMA material only (hence the subscript a for actuator). The key feature of

this constitutive relation is the term. This term embodies the effects of both 

the thermal strain and transformational strain of the SMA. In the constrained recovery 

application, this term is related to the recovery stress, and elastic modulus of the SMA i.e. 

σ

( ) ( ) ∫ −

T

To

Ea T α a τ dτ 1

r(T,εp), and Ea(T), still capturing the nonlinear effects when the temperature is above

austenitic start (T>As). 

( ) ( ) r

T

To

− Ea T α a τ dτ = σ ∫ 1 or ( ) ( )

E ( ) T

T

d

a

r p

T

To

a

σ ε

α τ τ ,

1 = − ∫ (4.2) 

When the temperature is below austenitic start (T<As) the thermoelastic relation remains the 

same, and the ECTE of the SMA is due to thermal expansion only, and can be measured 

21
experimentally. Therefore, the temperature dependent constitutive relation in the 1-direction 

for the SMA actuator specific to constrained recovery can be expressed as:

( ) ⎥

⎦

⎤ ⎢

⎣

⎡ = − ∫

T

To

σ a Ea T ε α a τ dτ 1 1 1 ( ) T < As (4.3) 

 ( )

( ) ⎥

⎦

⎤ ⎢

⎣

⎡ = +

E T

T

E T

a

r p

a a

σ ε

σ ε , ( ) 1 1 T ≥ As

Only the engineering properties of α1a(τ) for T < As and σr, Ea for T ≥ As need to be measured 

in experimental set-ups that imitate the application (pre-strain εp=4%, and boundary 

conditions=clamped) to implement the model. A similar constitutive relation evolves for 

principle material direction-2 (transverse). 

( ) ( ) ⎥

⎦

⎤ ⎢

⎣

⎡ = − ∫

T

To

σ a Ea T ε a α a τ dτ 2 2 2 (4.4) 

In this case the transverse CTE α2a(τ) is not related to the recovery stress, σr, and elastic 

modulus Ea(T), but is still nonlinear due to the changing of the martensite and austenite phases. 

Making the same assumptions, the same stress-strain relationship applies to the 

composite matrix material. Note, the 1,2-CTE’s of the composite matrix are temperature 

dependent. 

( ) ⎥

⎦

⎤ ⎢

⎣

⎡ = − ∫

T

To

σ m E m ε α m τ dτ 1 1 1 1 and ( ) (4.5, 4.6) ⎥

⎦

⎤ ⎢

⎣

⎡ = − ∫

T

To

σ m E m ε m α m τ dτ 2 2 2 2

Therefore, the elastic modulus, and CTE in both directions must be measured experimentally

as functions of temperature. 

Now, the classical mechanics of materials approach to stiffness38 can be implemented 

to define the material properties of the whole SMAHC lamina in terms of its constituents’ 

measured properties. Imagine that there is only stress present in the 1 direction. The 

assumption is made that the strains in the SMA fiber direction are the same, as the fibers in the 

composite matrix, thus making the plane lamina sections perpendicular to the 1-direction 

remain plane after stressing. The resultant force in the 1-direction can be defined as 

σ 1A1 = σ 1a Aa +σ 1m Am (4.7) 

22
where, A1, Aa, and Am are the cross-sectional areas of the SMAHC lamina element, SMA 

element, and the composite matrix element, respectively. Then the definition of the average 

stress for the SMAHC lamina element is 

σ 1 = σ 1aυ a +σ 1mυ m (4.8) 

where, υ a=Aa/A1 and υm=Am/A1 are the volume fractions of SMA and composite matrix, 

respectively. Then, the stress-strain relationship for each constituent is substituted resulting in: 

( ) ( ) ( ) ⎥

⎦

⎤ ⎢

⎣

⎡ ⎥ + −

⎦

⎤ ⎢

⎣

⎡ ⎥ = −

⎦

⎤ ⎢

⎣

⎡ − ∫ ∫ ∫

T

To

m m m

T

To

a a a

T

To

E ε α τ dτ E υ ε α τ dτ E υ ε α τ dτ 1 1 1 1 1 1 1 1 (4.9) 

From this expression the definition of the SMAHC lamina’s elastic modulus and CTE in the 1-

direction are formulated. 

E1 = Eaυ a + Emυ m and ( )

( ) ( )

∫

∫ ∫

+

+

=

T

To a a m m

T

To

T

To

a a a m m m

E E

E d E d

d

υ υ

υ α τ τ υ α τ τ

α τ τ

1

1 1 1

1 (4.10, 4.11) 

where, sgn( 

⎩

⎨

⎧

−

+ = 1

1

) α1a

s

s

T A

T A

≥

<

This formulation for the elastic modulus E1 is physically analogous to the stiffness 

calculation of two springs in parallel. The similar analogy can be made for the modulus 

definition E2 in the transverse direction of the SMAHC lamina. In this case, assume a uniform

stress in the 2-direction only. The strain differs in each constituent, therefore an expression 

representing the total elongation is written. 

m m aυ a ε ε υ ε 2 = 2 + 2 (4.12) 

Substituting for the strain, from equations (2.4, 2.6) results in: 

( ) [ ] ( ) ( ) ∫ ∫ + = + + +

T

To

a a m m

m

m

a

a

T

To

d

E E

d

E

α τ υ α τ υ τ

σ υ σ υ

α τ τ

σ

2 2

2

2 2

2

2

2 (4.13) 

Thus, the Young’s modulus and CTE for the SMAHC lamina in the 2-direction result: 

( ) m a a m

m a

E E

E E

E

υ + υ =

2

2

2 and ( ) [ ] ( ) ( ) (4.14) ∫ ∫ = +

T

To

T

To

α τ dτ α a τ υ a α m τ υ m dτ 2 2 2

23
In this case, the formulation for the Young’s Modulus in the 2-direction (E2) is analogous to 

two springs in series. The same mechanics of materials approach can be used to define the

SMAHC lamina’s Poisson’s ratio, ν12, and shear modulus of a lamina, G12. 

The Poisson ratio is similar to the derivation of the Young’s modulus E1. Assuming

that there is a uniform stress in the 1-direction only on the SMAHC lamina, there is a negative

deformation present in the 2-direction, ∆W, due to the strain in the 1-direction. Therefore this 

deformation in the 2-direction can be represented as 

2 Wε ∆W = − (4.15) 

where W is the width of the SMAHC lamina (Fig 4.1). The Poisson’s ratio is defined as 

1

2

12

ε

ε

ν = − , therefore the transverse deformation can be represented as 

12 1 Wν ε ∆W = (4.16) 

Additionally the same transverse deformation can be represented by the sum of the transverse 

deformations in the SMAHC lamina constituents, ∆mW and ∆aW. 

W 12 1 = ∆ mW + ∆aW ν ε (4.17) 

Similar in the analysis for the transverse Young’s modulus E2, the deformations and ∆ mW ∆aW

are approximately 

1 υ ν ε ∆ mW = W m m (4.18) 

1 υ ν ε ∆aW = W a a . 

Thus, upon substitution of (2.18) into (2.17) and division by Wε1, the expression for the 

SMAHC lamina Poisson’s ratio is 

ν 12 = υ mν m +υ aν a (4.19) 

Finally, the formulation of the in-plane shear modulus, G12, of the SMAHC lamina is 

formulated by the assumption that the shearing stresses on the SMA fiber and the matrix are 

the same. The basic assumption can be made that, 

m

m G12

τ

γ = and 

a

a G

τ

γ = (4.20) 

Let ∆ represent the total shearing deformation due to shear stress, then

∆ = γW (4.21) 

And similarly to the previous formulations, the shearing deformation can also be represented 

by the SMAHC lamina constituent’s deformations. 

m a mW m aW a ∆ = ∆ + ∆ = υ γ +υ γ (4.22) 

24
Therefore,

m m a a γ = υ γ +υ γ (4.23) 

Realization that 

G12

τ

γ = , and substitution of equation (4.20) into (4.23) yields the expression

for the in-plane shear modulus of the SMAHC lamina. 

m a a m

m a

G G

G G G

12

12

12

υ +υ = (4.24)

Now the engineering properties of the SMAHC lamina have all been formulated in 

terms of the individual material constituents (Glass-Epoxy Matrix, and NiTi-SMA), and 

include the thermo-elastic effects due to the ECTE. These properties of the constitutive 

materials can all be measured. Implementing the stress-strain relationships of an orthotropic 

material lamina under plane stress will complete the development of the constitutive relations 

of a SMAHC lamina. 

4.3 Stress-Strain Relation for Orthotropic Material 

In a lamina if there are two orthogonal planes of material property symmetry for a 

material, and symmetry exist relative to a third mutually orthogonal plane, the stress-strain 

relations in coordinates aligned with principal material directions are defined as an orthotropic

material. If an orthotropic material is subjected to a plane stress state (σ3=0, τ23=0, τ31=0), the

three-dimensional strain-stress relation below 

⎪

⎪

⎪

⎪

⎭

⎪

⎪

⎪

⎪

⎬

⎫

⎪

⎪

⎪

⎪

⎩

⎪

⎪

⎪

⎪

⎨

⎧

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

=

⎪

⎪

⎪

⎪

⎭

⎪

⎪

⎪

⎪

⎬

⎫

⎪

⎪

⎪

⎪

⎩

⎪

⎪

⎪

⎪

⎨

⎧

12

31

23

3

2

1

66

55

44

13 23 33

12 22 23

11 12 13

12

31

23

3

2

1

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0

0 0 0

0 0 0

τ

τ

τ

σ

σ

σ

γ

γ

γ

ε

ε

ε

D

D

D

D D D

D D D

D D D

 (4.25) 

where is the compliance matrix, is reduced to the following stress-strain relation upon 

inversion. 

[D]

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

=

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

12

2

1

66

12 22

11 12

12

2

1

0 0

0

0

γ

ε

ε

τ

σ

σ

Q

Q Q

Q Q

 (4.26) 

where is the reduced stiffness matrix of a composite lamina. This comes from Hooke’s

Law in its most basic form. This is then implemented for a SMAHC lamina, which includes 

the constitutive relations formulated previously in section 4.2. 

[Q]

25
( )

( ) ∫ ⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

−

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

=

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧ T

To Q

Q Q

Q Q

Q

Q Q

Q Q

0 0 0

0

0

0 0

0

0

2

1

66

12 22

11 12

12

2

1

66

12 22

11 12

12

2

1

α τ

α τ

γ

ε

ε

τ

σ

σ

 (4.27) 

The elements of the reduced stiffness matrix [Q] are related to the engineering properties 

through the following equations. 

12 21

1

11 1−ν ν = E Q , 

12 21

12 2

12 1 ν ν

ν

− = E Q , 

12 21

2

22 1−ν ν = E Q , Q66 = G12 (4.28) 

Recall, that these engineering properties of the SMAHC lamina have been defined in terms of

the lamina material constituents (glass-epoxy, SMA). 

E1 = Eaυ a + Emυ m ( ) m a a m

m a

E E

E E

E

υ + υ =

2

2

2 . 

ν 12 = υ mν m +υ aν a 

m a a m

m a

G G

G G G

12

12

12

υ +υ = . 

( )

( ) ( )

∫ ∫ ∫

+

+

=

T

To a a m m

T

To

T

To

a a a m m m

E E

E d E d

d

υ υ

υ α τ τ υ α τ τ

α τ τ

1

1 1 1

1 where sgn( . 

⎩

⎨

⎧

−

+ = 1

1

) α1a

s

s

T A

T A

≥

<

( ) [ ( ) ( ) ∫ ∫ = +

T

To

T

To

α τ dτ α a τ υ a α m τ υ m dτ 2 2 2 ] (4.29) 

The above SMAHC constitutive relation (4.27) is for a lamina. This lamina will now

be considered a general k-th orthotropic layer of a laminate, with an arbitrary orientation angle

θ, from the principal material direction-1, from Figure 4.1. Below are the transformation 

equations for expressing stress and strain in an x-y coordinate system in terms of stress and 

strain in a 1-2 principal material direction. 

[ ] ⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

=

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

−

12

2

1

1

τ

σ

σ

τ

σ

σ

T

xy

y

x

 and [ ]

⎪

⎪

⎭

⎪

⎪

⎬

⎫

⎪

⎪

⎩

⎪

⎪

⎨

⎧

=

⎪

⎪

⎭

⎪

⎪

⎬

⎫

⎪

⎪

⎩

⎪

⎪

⎨

⎧

−

2 2

12

2

1

1

12 γ

ε

ε

γ

ε

ε

y T

x

 (4.30) 

where . Using the above transformations 

(2.30), the thermoelastic constitutive equation for the k

[ ]

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

− −

= −

θ θ θ θ θ θ

θ θ θ θ

θ θ θ θ

2 2

2 2

2 2

sin cos sin cos cos sin

sin cos 2sin cos

cos sin 2sin cos

T

th layer of a SMAHC laminate in an

arbitrary orientation becomes 

26
[ ] [ ] ∫ ⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

−

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

=

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧ T

To

xy

y

x

k

k xy

y

x

k

xy

y

x

Q Q

α

α

α

γ

ε

ε

τ

σ

σ

 (4.31) 

where 

[ ] [ ] [ ] [ ] T Q k T Q k T − − = 1 (4.32) 

There are some attractive characteristics of this model, which will be mentioned. In 

equation 4.27 it can be seen that the effective CTE in the 1-direction has an effect on the stress 

in the transverse direction, and vice-versa for the SMA actuator. This indicates that above As

temperature the effects of the recovery stress in the 2-direction are included. Also, this model

not only includes the nonlinear effects of the CTE in the 1-direction, but also in the 2-direction 

for the SMA actuator. Thus, the behavior of the SMA actuator element in the composite is 

accurately represented in the ECTEM. 

Equation 4.31 represents the constitutive relation of the ECTEM in its final form for a 

SMAHC lamina layer. It is evident from the formulation that making use of the appropriate

temperature dependent engineering property measurements and defining a composite laminate

out of those material constituents is all that is needed for implementation of this model. Any 

commercial finite element code that is capable of analyzing composite laminates with 

temperature dependent material properties can then make use of the ECTEM. Work has been 

done in which an ECTEM research code was compared to ABAQUS, and MSC.Nastran and 

showed excellent correlation41. 

In this work, the commercial finite element code, ABAQUS, is used in this way to 

model experimental results of SMAHC beam specimens with dimensions of

0.5588x0.0254x0.0019 m (22x1x0.078 in). A finite element mesh of 36x4 elements is defined 

with the two center element widths representing the portion of the laminate containing SMA 

material, and the two outer edge element widths representing the portion of the laminate 

containing glass-epoxy matrix material only. Each element is assigned the appropriate material

stacking sequence and corresponding temperature dependent material properties. This is all

accomplished in a single input file, in which a number of dynamic/static response analyses can 

be performed to accurately represent the experimental conditions. The corresponding material

characterization, for the material constituents input into the modeling, was previously

performed at NASA LaRC18. These material property measurements used for the numerical

modeling in ABAQUS can be seen in Tables 4.1, 4.2, and 4.3. The Nitinol characterization 

stress-strain curves for 0% and 4% pre-strain, recovery force measurements as a function of 

thermal cycle at 4% pre-strain, and resulting elastic modulus as a function of temperature are

shown in Figures 4.2, 4.3, 4.4, and 4.5 respectively. These ABAQUS input files will be more 

thoroughly described in chapter 5. 

27
Cha p t e r 5 

NUMERICAL MODELING OF SMAHC BEAM SPECIMENS IN THE 

COMMERCIAL FINITE ELEMENT CODE ABAQUS

5.1 Background 

As previously mentioned, ABAQUS is a commercial finite element code capable of

analysis of laminated composite structures with temperature dependent material properties.

Recall that the beam specimen lamination involves the Nitinol ribbon material replacing the 

glass-epoxy matrix over approximately 0.45 inches of the beam width in the 0o

 layers. This 

can be conveniently modeled by using shell elements of two types; one having glass-epoxy 

layers only, and the other having alternating layers of 100% glass-epoxy and 100% Nitinol

material. Therefore the ECTE model can easily be implemented by formulating the effective

composite properties of each constituent material. This type of analysis has a variety of

element types and solution techniques available to provide sufficient freedom to accurately 

model a wide range of experimental configurations. In this project an initial commercial input 

file for ABAQUS is generated which initially defines the SMAHC structures corresponding 

element, laminate, and material property definitions. Therefore, only a single temperature

dependent material property data set for each constituent is defined, and used in the correct 

stacking sequence to define the laminate. Thus the laminated SMAHC materials, which are 

composed of orthotropic layers, are defined. The resulting input files in ABAQUS define the 

laminate by the *SHELL SECTION option, and the temperature dependent material properties 

are defined by the *MATERIAL, *ELASTIC, *EXPANSION options for each material

constituent (SMA, glass-epoxy matrix). The user is referred to the ABAQUS User’s and 

Reference Manuals and previous work done at NASA LaRC41. Once the element and material

definitions are defined, the ABAQUS input files specify thermal and dynamic loads, and 

solution techniques. The ABAQUS input files will be described briefly below, and a specific

input file representative of the presented modeling cases can be seen in the Appendix D.

The unsupported dimensions of the SMAHC beam are 0.4572x0.0254x0.0019 m 

(18x1x0.078 in). This section of the SMAHC beam is modeled by 36x4 type S4 quadrilateral

shell elements in ABAQUS. As seen in Figure 5.1, the two center element widths (black)

represent the hybrid composite stacking sequence (45/0/-45/90)2s with the SMA comprising the 

0o

 layers, and the two edge element widths (gold) represent the composite stacking sequence

(45/0/-45/90)2s consisting of all glass-epoxy material. Each glass-epoxy layer has a nominal 

thickness of 1.27e-4 m (.005 inches), and each Nitinol ribbon section has a thickness of 1.5e-4

m (.006 inches). The graphical representation of the beam shows the nodes at each end under

clamped conditions. Each defined node is represented by the red circle symbol. 

The temperature dependent material properties for each constituent can be seen in the

ABAQUS input file as seen in the Appendix D. These material properties are taken from the

material characterization measurements listed in Tables 4.1, and 4.2. It should be mentioned 

that ABAQUS uses secant CTE values as input for thermal expansion and not thermal strain

( ∫ = − =

T

T

r

T

o

d

E

α τ

σ

ε T>As). Recall, that the thermal strain is a function of temperature, and 

28
cycle number for the SMA material above the As temperature. Therefore the appropriate values

of recovery stress and elastic modulus measurements in Table 4.2 were used to calculate the

appropriate secant CTE for temperatures above As temperature. Another important detail worth 

noting is that these material characterization measurements were processed previously at

NASA LaRC for low thermal cycles 1-4 only. This will be discussed in the subsequent 

sections and impact the comparison of experimental results to the numerical simulations. The 

appropriate CTE values for the Nitinol ribbons below the As temperature were taken from 

published data as were the transverse direction CTE values for martensitic temperature 6.6e-6

/

o

C (3.67e-6 /oF) and austenitic temperature 11e-6 /oC (6.11e-6 /oF) . The transverse CTE 

values have been linearly interpolated between these two points to represent the values through 

the temperature range where the SME occurs. 

Once the geometry, finite element mesh, and material properties are defined, the 

physical conditions of each experiment can be simulated. Some of the experimental

measurements will serve as input into the numerical simulations. The ABAQUS input files 

consist of a sequence of several analysis steps progressing through the application of the

thermal load. At certain load levels a desired equilibrium state is achieved, where the random 

response analysis is simulated for those states, thus determining the dynamic response. In each

individual input file the actual measured thermal load from the thermal imaging camera for 

each thermal step to 121.1 o

C (250 oF) is applied to the SMAHC beam with clamped boundary

conditions. Throughout the application of the thermal load a dynamic inertial loading is 

defined that represents the .25g RMS base excitation used in the dynamic experiments. The

actual base excitation PSD measured from each experiment, see Figure 3.3 for an example, is 

input into the analysis by the *PSD-DEFINITION in ABAQUS. During the linear random 

response step analysis at the specified equilibrium states a modal approach is used

incorporating the first 10 modes, in order to account for residual effects. The modal parameter

estimates for mode 1 and 3 measured during the dynamic response experiments are used as

input for damping estimates through modes 1 through 10 using the *MODAL DAMPING

option. Additionally, the initial geometric imperfection measured by PMI measurements for

the full-field displacement of the beam is input into the analysis by the *IMPERFECTION 

option, to assist in the nonlinear static solution procedure of the buckling/post-buckling of the

beam. Recall the PMI measurements were for the centerline of the beam length only, but are

representative of the full-width of the beam. Therefore a MATLAB m-file was written which 

linearly interpolates the beam length centerline initial imperfection measurements to nearest 

nodes corresponding to the full field of the SMAHC beam (See m-file and corresponding data

set in Appendix C). 

In the subsequent sections the numerical simulation input files will contain the

previously discussed experimental measurements, which are beam specimen and run specific.

This should improve the accuracy of the numerical simulations. Results for the analysis of

nonlinear static, and linear random dynamic responses will be presented and compared to 

experimental results. 

5.2 Static Response Modeling Results in ABAQUS 

The plots of midspan deflection of the SMAHC beam vs. average centerline 

temperature are presented for the experimental PMI measurements and the numerical 

predictions in ABAQUS. Recall, from the static thermoelastic response experiments that the 

thermal buckling maximum midspan deflection varies greatly from run to run, and may be

29
attributed to moisture absorption, and varying pre-stress. Thus the specifically chosen 

numerical simulations will be given the initial imperfection data for runs that correspond to the 

middle of the varying deflection range where repeated experimental results occurred. 

As previously mentioned, during the numerical simulations the specific material 

property measurement data for each constituent (SMA, Glass-Epoxy) is input into the model.

Looking at figure 4.4 the recovery force data shows that As temperature begins when stress is 

starting to be generated at 28 o

C (82.4 oF). Isolength (constant strain) tests were performed on 

a nominal .254 m (10 inch) length of Nitinol ribbon material to obtain these recovery stress 

results as a function of pre-strain, and thermal cycle. Thermocouple’s were used to measure

the corresponding temperature. There is a large error associated with thermocouple

measurement at high temperatures due to ‘fin’ effect. There is also error due to ‘fin’ effect

even at low temperature if the thermocouple is attached to a small surface area, as the case with 

the SMA ribbon. This recovery stress data for thermal cycle 4 is used for numerical modeling,

but it is suspected that there is some error in the corresponding temperatures. This is validated 

by a discrepancy in the measured As temperature by differential scanning calorimetry (DSC)

measurements. 

Figure 4.6 shows the DSC results for small SMA ribbon samples (<10mg). The 

exothermic and endothermic peaks correspond to the transformation of the SMA material. 

During the first-heat there is an ill-behaved endothermic peak associated with the austenitic 

transformation recovery shear deformation induced from cutting the samples. This peak was 

not repeatable after the first heat, where as the other endothermic peak was. Therefore the 

measured As temperature is near 40 o

C (105 oF). These measurements are accurate and indicate 

a possible error with the recovery stress measurements. This will be shown in the results of the 

modeling. 

For each static thermoelastic response simulation the corresponding experimental

measurement for a given initial imperfection, and thermal load is plotted. The numerical 

simulations for the original material property measurements, and after shifting the recovery

force data 10-20 o

F to account for the possible measurement error are included. Material

property data is input into ABAQUS as elastic modulus data (E) and coefficient of thermal

expansion (CTE) data. Recall, from the model, that the Nitinol effective CTE values are 

obtained from the measured E and the measured recovery stress (σr) at temperatures above As. 

Therefore the σr data is shifted and the corresponding CTE values are obtained from equation 

4.2. These shifted CTE values are then used as input into ABAQUS to adjust for the

measurement error. 

In Figure 5.2 the predicted static response for Beam 2 Run 6 is shown. This numerical

prediction for clamped boundary conditions, and the original recovery stress (CTE)

measurements indicates that the SMA recovery force is overcoming the compressive forces 

due to expansion at a lower temperature than indicated by experiment, thus resulting in a lower

max deflection, and smaller post-buckled temperature range (see green dashed line). The 

recovery stress measurements were shifted 15 o

F and the corresponding new CTE values were

input instead. This results show great qualitative agreement between the numerical simulation 

and the experimental measurement (See blue dashed line). The same results were produced in 

multiple simulations for Beam 2 and Beam 3 (Figures 5.3 – 5.7). Some cases required varying

shifts between 10-20 o

F. These results indicate that the pre-stress initially present could be 

varying for run to run slightly affecting the recovery force behavior. These static experimental 

results are based on thermal cycles 5-14 for beam 3 and cycles 1-9 on beam 2. The recovery

30
force measurement data was only captured for thermal cycles 1-4. Therefore thermal cycle 4 

data was used in modeling. This could be attributed to the small differences between

experiment, and simulation. The recovery force data adjustment seems to give great qualitative 

agreement with experiment and indicates the sensitivity of the buckling behavior to the 

interaction between the SMA recovery stress generated and the compressive forces due to 

thermal expansion. 

The recovery stress temperature shift now activates the recovery forces at the correct 

corresponding temperature to more accurately capture the actuation, which returns the midspan 

deflection to zero. In all simulations the measurements of the initial imperfection, the thermal 

load, and material property measurements were input directly into the model to achieve

accuracy. These results verify the capability of the commercial finite element code ABAQUS, 

to accurately predict the static thermoelastic response of SMAHC structures by implementing 

the ECTE model. Additional material characterization measurements could be made for the

recovery stress of the constrained Nitinol actuator ribbons using infrared thermal imaging to 

produce even more accurate material property data to be used in the model. This could even 

further improve the accuracy of the numerical prediction with the experimental results. 

5.3 Dynamic Response Modeling Results in ABAQUS 

Experimentally measured and numerically predicted displacement PSD’s for 

measurement position 2, representing the midspan accelerometer, are shown for various 

temperature values in Figures 5.8-5.15. Results are presented only for beam 2 run 8 and beam

3 run 8, but are representative of all other dynamic results for that measurement position, and 

beam specimen. Dynamic response PSD’s are presented for both beam specimens at 21.7 o

C 

(71 o

F), 71.1 oC (160 oF), 93.3 oC (200 oF), and 121.1 oC (250 oF). These temperature values 

demonstrate the stiffening of the beam due to the SMA actuation and the ability of the 

numerical simulation to accurately capture the effect. For these numerical simulations the 

adjusted recovery stress material data was input for the material properties (15 o

F shift). 

Recall, from experimental results in chapter 2 it was discovered that the boundary

clamped G-10 insulation was unsymmetrical and the boundary conditions are contributing to 

the initial imperfection of the beam. Also the experimental initial imperfection measurements

show that the boundary conditions are dominating in the initial shape of the beam. The fact 

that the G-10 insulation thickness was not uniform for all samples indicates that the boundary

conditions are most likely not exhibiting perfectly clamped conditions, and would be difficult 

to accurately model. The unsymmetrical G-10 insulation clamps likely allow some torsional

rotation of the beam at the boundary. This was attempted to be accounted for in the model by 

applying torsional spring elements to the 5th degree of freedom (out of plane of the beam) of

the end nodes at the boundaries of the beam model. This is defined in ABAQUS by the

*SPRING function. Torsional springs with a rate of 60 (in·lbf/rad) were applied to match the

fundamental frequency. 

As indicated in previous work18, in the physical system the beam continues through the

mechanical grips and the Nitinol leads are terminated beyond that in the electrical grips.

Therefore the main restraint for the Nitinol is the mechanical termination at the electrical grips. 

Thus, the compliance of the entire support structure needs to be considered. Although, Garolite 

G-10 material was used as insulation between the electrical and mechanical grips, it is likely 

that there is still some compression occurring, partially relieving the recovery stress. This effect

was captured in the model by applying axial springs with a spring rate of 7000 (lbf/in) to each 

31
of the nodes at the boundary of the beam model in degree of freedom 1 (in-plane). This spring 

rate was obtained by matching the fundamental frequency of the simulation to the measured 

fundamental frequency at the elevated temperature of 250 o

F. 

The same explanations described in the previous static thermoelastic response modeling 

sections correspond to the over prediction of stiffness initially seen in dynamic results. The 

application of the boundary springs previously discussed produce the PSD comparison in

Figures 5.8-5.15, for beam 2 run 8 and beam 3 run 8 at position 2 (midspan). The displacement 

PSD predictions show excellent agreement with the experimental results. It can be seen that at

21.7 o

C (71 oF), 71.1 oC (160 oF), 93.3 oC (200 oF), and 121.1 oC (250 oF) temperatures, the 

ABAQUS predicted responses closely correspond with the experimental modal frequency 

values and RMS displacement values. In Figures 5.16 and 5.17, the same temperature values 

are compared to the experimental measured PSD’s for positions 1 and 3, along the length of the 

beam all in one plot. Again the ABAQUS predictions show excellent agreement with the 

experimental results in modal frequency values, and RMS displacement values. 

In Figures 5.18-5.23 displacement PSD’s of the numerical simulations and 

experimental dynamic responses are compared for temperatures corresponding to the buckled

and post-buckled range of the SMAHC beam specimen. These particular data sets correspond 

to beam 3 run 1 of the dynamic experiments. This is the only dynamic experimental run in 

which dynamic data was captured through this temperature range of buckling/post-buckling.

The static deflection behavior of the beam doesn’t correspond closely the numerical simulation 

of the static deflection for this particular run in that temperature range. Therefore the PSD 

comparison at these temperatures is expected to suffer as well. If dynamic data was captured 

for this temperature range for the thermal cycles presented in Figures 5.2-5.7 then the postbuckled temperature range PSD’s would be expected to compare reasonably well. Through 

this temperature range the beam goes through the buckling phenomenon in which the stiffness 

theoretically drops to zero. Once the beam is buckled the stiffness is increased, until the SMA 

recovery forces begin to return the buckling deflection back to a flat configuration. At this 

point the stiffness of the beam is decreasing. So during this temperature range the SMAHC 

beam exhibits complex dynamic behavior, and would be difficult to model accurately. The 

PSD comparison in Figures 5.18-5.23 demonstrates the difficulty in modeling this behavior

with the current analysis, and the complex dynamic behavior evident in the experimental

PSD’s can be seen. 

In Figure 5.18 the displacement PSD at ambient temperature 21.7 o

C (71 oF) is plotted 

for position 3 along the length of beam 3. This particular dynamic response demonstrates some

nonsymmetrical modes as seen by the additional spectral peaks in the experimental PSD. Also 

nonlinearity is demonstrated by peak shifting and peak broadening in the experimental PSD’s.

ABAQUS numerical model shows good agreement with the stiffness prediction even though a

linear dynamic response is predicted. Recall that a linear random analysis is being 

implemented in ABAQUS to predict the dynamic response. That explains the difference in the 

PSD’s comparison, but the modal frequency peaks still match indicating a good prediction of

the stiffness state of the SMAHC beam. In Figure 5.19 the displacement PSD comparison for a 

temperature of 37.7 o

C (100 oF) is shown for Beam 3 Run1 at position 3. At this temperature 

the beam has buckled. The experimental PSD’s show again the complex nonlinear behavior of 

the dynamic response, and shows a slight increase in the fundamental frequency from the

ambient state. The numerical simulation in ABAQUS does not agree with the experimental

results in that it is predicting a large decrease in stiffness by the decrease in the modal

32
frequency values. The same prediction error is shown in Figure 5.20 at a temperature of 43.3 o

C (130 oF). At this temperature, recovery stresses are being generated because it is above As

temperature, but they have not overcome the compressive forces due to thermal expansion, 

thus the beam is still in a buckled state. The experimental PSD is again showing complex 

dynamic behavior by the nonlinear peaks. In Figure 5.21 the PSD comparison at 60 o

C (140 o

F) shows a good agreement in modal frequency values. At this temperature state the beam has

been just been rendered flat by the SMA recovery forces. Since the beam has just been

returned flat it is still demonstrating some nonlinear dynamic behavior, but ABAQUS is able to 

make better predictions now using its linear analysis. Then in Figure 5.22 at 65.55 o

C (150 oC) 

the experimental PSD shows that the linear dynamic behavior has returned as the beam is flat

and an increase in stiffness is occurring. The ABAQUS numerical simulation now is in good

agreement with the behavior for the rest of the temperature range and accurately predicts the 

stiffness of the structure as seen in subsequent Figure 5.23 at 71.1 o

C (160 oF). 

So the dynamic response modeling results demonstrate the ability of ABAQUS to 

accurately model the dynamic behavior of the SMAHC beam specimens’ experimental results 

by using a linear random analysis. The adaptive stiffening effect and RMS values of the

vibration are accurately predicted by input of the measured modal parameters, the random

excitation from experiment, and the measured material properties of the constituents directly 

into the numerical model. Accurate simulations are produced in the temperature range below 

and above the buckling range, and demonstrate the stiffening effect of the embedded SMA

actuators. An attempt to model the dynamic behavior through the temperature range, where

buckling occurs, demonstrates that there is complex dynamic behavior occurring when the

SMAHC beam is in this state. If the static deflection behavior of the beam corresponds closely 

to the numerical simulations as in Figures 5.2-5.7, it expected that the numerical PSD 

prediction would correspond closely in this post-buckled temperature range except at the start

of the buckling and at the point of returning flat. These two locations are theoretically the only 

cases where linear analysis fails due to the nonlinear and complicated dynamics. This limits 

the accuracy of a linear random analysis to predict the behavior. It would be desirable perhaps 

for application advantages to use Nitinol actuator ribbons with a Ni and Ti composition which 

produces an As temperature below ambient temperature to avoid the buckling behavior of the

structure. 

33
Chapter 6 

CONCLUSIONS 

This study consisted of a thorough experimental investigation into the response 

behavior of SMAHC beams. The attractive functionality of SMA’s was demonstrated as an 

adaptive stiffening mechanism to composite beam specimens in all experimental cases. The 

ECTE constitutive model was implemented into the commercial finite element code ABAQUS, 

to accurately predict the response of the SMAHC structure. The experimental measurement

investigations uncovered important contributing factors to the behavior of the SMAHC beam,

such as initial geometric imperfections, contributions of the boundary conditions, distributed 

thermal load over the beam, and the real base excitation that was used to induce the dynamic

response. These measurements, and discoveries were able to be accounted for in the numerical

modeling to produce an accurate prediction of the SMAHC beam responses, and further

validate the use of the ECTE model to be used in the commercial finite element code 

ABAQUS. 

Numerous experimental tests were performed to determine the thermoelastic response

in a clamped boundary configuration with a thermal load applied via resistive heating. The out

of plane deflection of the midspan was measured as a function of temperature and uncovered a

possible dependency of maximum deflection on moisture absorption in the hybrid composite

structure. Excellent correlation was demonstrated between the PMI measurement technique 

and the laser displacement transducer measurements. The PMI measurements enabled the 

discovery of the initial geometric imperfections of the beam, and the extreme involvement of 

the beam fixture boundary conditions on this imperfection. This validates the accuracy of the 

PMI measurement technique for this application, and gives the valuable understanding of the

full-field displacement of the structure during testing. 

The SMAHC beam specimen’s dynamic response was investigated with a generated 

thermal load by resistive heating, and application of a 10-400 Hz random base excitation. The

dynamic measurements were post-processed into FRF’s, PSD’s, RMS values, and coherence

functions. The post-processing of the experimental results not only demonstrated the adaptive

stiffening mechanism of the SMAHC beams, but also enabled the modal parameter estimation 

of damping, and modal frequency of each mode. The displacement RMS values were shown 

to decrease with temperature. The modal damping estimates decreased once the Nitinol ribbon

was above As temperature, until higher temperatures near 121.1 o

C (250 oF) were reached,

where a slight increase in damping estimates occurred most likely due to the softening of the 

heated glass epoxy material. 

The precise experimental measurements were used as input into the commercial finite

element code ABAQUS to accurately predict the static and dynamic responses of the SMAHC

beam specimens. In the static thermoelastic response comparisons of numerical prediction and

experimental results, ABAQUS showed excellent qualitative agreement with experiment for 

multiple runs. It was discovered that there is most likely a temperature correlation error in the 

Nitinol recovery force measurements of 10 - 20 o

F. When this error is accounted for in the

CTE material property input values excellent agreement with the experimental results is

34
demonstrated. This indicates that it is critical that the appropriate material characterization 

measurements are accurately measured for the state that the embedded SMA material is in. 

The numerical prediction of the dynamic response of the SMAHC beams proved to be 

in excellent agreement with experimental measurements after the application of boundary

springs, and shifting of the CTE material property values. The displacement PSD’s

comparison showed agreement for all positions 1, 2, and 3 along the length of the beam for

both beam specimens. The results were achieved by a linear random analysis in ABAQUS 

using accurately measured modal parameters, material property data, thermal loads, and 

random base excitation as input into the model. An additional investigation into the dynamic 

behavior of the beam at temperatures corresponding to the buckling phenomena (80 o

F – 140 o

F) showed the limitations of the numerical model using a linear random analysis to capture the 

complex dynamic behavior which occurs through this temperature range. 

The motivation for the static and dynamic response investigation of SMAHC beams is

to demonstrate the decrease in the dynamic response amplitude when under random 

excitations. Random excitation of a structure can lead to sonic fatigue and undesirable noise

transmission characteristics of a structure. By demonstrating a reduction in the response, and 

the adaptive stiffening mechanism introduced by the SMA actuators, the implications of 

vibration and noise transmission reduction are evident. It is possible that finding an optimal

lay-up or embedding position of the SMA actuators to demonstrate most efficient control over

the structure could result in increased performance of the SMAHC structure. Also, the

fabrication of other SMAHC structures such as a panel specimen was previously done at

NASA LaRC, although experimental validation wasn’t completed. Numerical studies have 

shown that a weight-efficient SMAHC panel demonstrates superior dynamic response and 

sonic fatigue behavior over other conventionally noise-treated panels. Also, the SMAHC panel

numerically shows superior transmission loss (TL) characteristics in the low frequency,

stiffness-controlled region. The weight-efficient stiffening effect has significant implications 

for dynamic response and sonic fatigue reduction. For further research, the fabrication of 

another SMAHC panel could be done, in which an extensive experimental investigation could 

be performed measuring the vibration response, acoustic radiation, and transmission loss

characteristics using the more thorough measuring techniques used in this study. The acoustic

radiation and transmission loss experimental investigation could produce a multitude of 

additional discoveries. With the addition of increased commercial code capabilities the 

numerical simulation study could also be investigated to compare. 

In conclusion, this study uses fundamental engineering property measurement to 

accurately model SMAHC beam specimens. The experimental investigation of the static and 

dynamic response of the SMAHC structures was performed using accurate measuring 

techniques, which uncovered interesting and complex behavior of the SMAHC beam

specimens. The SMA actuators demonstrated an adaptive stiffening effect to the structure, 

which was successfully, modeled in the commercial finite element code ABAQUS. 

35
Cha p t e r 7 

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22. Thompson S P and Loughlan J 2000 “The control of the post-buckling response in thin composite plates using smart

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35th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conf., Hilton Head, SC, April 18-

21, 1994, pp. 136-146. 

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31. Boyd, J. G. and Lagoudas, D. C.: “A Thermodynamical Constitutive Model for Shape memory Materials –Part I: The 

Monolithic Shape Memory Alloy,” Int. J. of Plasticity, 12(7), 1996, pp. 805-842. 

32. Turner, T.L. “A New Thermoelastic Model for Analysis of Shape Memory Alloy Hybrid Composites,” Journal of 

Intelligent Material Systems and Structures, Vol.11, May 2000. 

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Deformation Measurements of Active Twist Rotor Blades”, American Helicopter Society 58th Annual Forum, 

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34. G.A. Fleming, S. M. Bartram, M.R. Waszak, and L.N. Jenkins, “Projection Moire’ Interferometry Measurements of

Micro Air Vehicle Wings”, Proc. SPIE Vol. 4448, pp. 90 –101, (2001). 

35. G.A. Fleming, H.L. Soto, B. W. South, and S. M. Bartram, “Advances in Projection Moire’ Interferometry 

Development for Large Wind Tunnel Applications”, SAE Paper No. 1999-01-5598, SAE World Aviation Congress, 

San Francisco, CA, October 19-21, 1999. 

36. D. Phillips, “Image Processing in C, Part 14: Warping and Morphing”, C/C++ Users Journal, pp. 55-68, October,

1995. 

37. L. Pirrodda, “Shadow and projection moire’ techniques for absolute or relative mapping of surface shapes”, Optical 

Engineering 21(4), pp. 640-649, July/August (1982). 

38. Jones, R. M.: Mechanics of Composite Materials, 2nd Edition, Hemisphere Publishing Corporation, New York, NY,

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39. Shames, I. H.; and Dym, C.L: Energy and Finite Element Methods in Structural Mechanics, Hemisphere Publishing 

Corporation, New York, NY, 1985.

40. Whitney, J. M.: Structural Analysis of Laminated Anisotropic Plates, Technomic Publishing Company, Inc., 

Lancaster, Pennsylvania, 1987. 

41. T. L. Turner, and H. Patel, “Analysis of SMA hybrid composite structures using commercial codes,” Smart 

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37
Cha p t e r 8 

TABLES 

Table 3.1- Beam 2 Run 1 dynamic response RMS displacement values as a function of 

temperature for positions 1, 2, and 3 along the length of the beam. 

 Pos1 Pos2 Pos3 

Temp

o

F 

RMS 

Displ., in

RMS 

Disp, in 

RMS 

Displ., in

73 .838e-2 1.30e-2 .645e-2

150 .656e-2 .985e-2 .586e-2

160 .523e-2 .757e-2 .477e-2

170 .441e-2 .627e-2 .404e-2

180 .473e-2 .607e-2 .435e-2

190 .343e-2 .482e-2 .317e-2

200 .320e-2 .701e-2 .297e-2

210 .297e-2 .415e-2 .275e-2

220 .267e-2 .373e-2 .247e-2

230 .260e-2 .364e-2 .241e-2

240 .254e-2 .355e-2 .235e-2

250 .254e-2 .356e-2 .235e-2

38
Table 3.2- Beam 3 Run 2 dynamic response RMS displacement values as a function of 

temperature for positions 1, 2, and 3 along the length of the beam. 

 Pos1 Pos2 Pos3 

Temp

o

F 

RMS 

Displ., in

RMS 

Disp, in 

RMS 

Displ., in

73 .941e-2 1.69e-2 .812e-2

150 .652e-2 1.03e-2 .669e-2

160 .376e-2 .573e-2 .336e-2

170 .366e-2 .551e-2 .389e-2

180 .394e-2 .587e-2 .427e-2

190 .343e-2 .508e-2 .381e-2

200 .331e-2 .489e-2 .347e-2

210 .294e-2 .434e-2 .278e-2

220 .263e-2 .388e-2 .259e-2

230 .251e-2 .370e-2 .223e-2

240 .243e-2 .358e-2 .228e-2

250 .239e-2 .352e-2 .218e-2

39
Table 3.3- Beam 2 dynamic response modal parameter estimates for runs 1, 2, and 3. 

 Run1 Run2 Run3 

Temp, oF f1, Hz ζ1, % f3, Hz ζ3, % f1, Hz ζ1, % f3, Hz ζ3, % f1, Hz ζ1, % f3, Hz ζ3, %

73 30.1 1.06 160.2 .66 30.9 .81 163.3 .68 30.5 .87 162.1 .73

150 59.6 .31 223.2 .66 67.1 .17 240.0 .39 66.1 .22 237.8 .43

160 80.0 .17 271.9 .26 81.7 .17 276.1 .25 81.9 .20 276.7 .29

170 91.3 .14 302.0 .27 92.3 .15 304.5 .26 92.9 .14 306.4 .29

180 99.5 .15 323.7 .19 100.8 .13 327.3 .18 101.5 .13 329.1 .18

190 106.9 .13 343.9 .15 107.5 .15 345.5 .16 108.1 .13 347.1 .14

200 112.9 .16 360.2 .17 113.1 .16 360.7 .16 112.6 .17 359.6 .14

210 117.2 .18 371.9 .18 116.9 .16 371.2 .16 116.1 .16 369 .15

220 120.3 .20 380.9 .27 119.5 .17 378.5 .20 118.4 .17 375.4 .17

230 122.5 .21 387.4 .24 121.1 .18 383.1 .27 120.0 .18 379.9 .25

240 124.1 .21 391.6 .19 122.0 .18 385.8 .23 121.1 .19 383.1 .27

250 125.1 .21 394.2 .18 122.8 .21 388 .22 121.2 .19 384.7 .25

40
Table 3.4- Beam 3 dynamic response modal parameter estimates for runs 1, 2, and 3. 

 Run1 Run2 Run3 

Temp, oF f1, Hz ζ1, % f3, Hz ζ3, % f1, Hz ζ1, % f3, Hz ζ3, % f1, Hz ζ1, % f3, Hz ζ3, %

73 30.3 .72 162.6 .56 29.9 .81 161.9 .59 31.5 .76 163.9 .66

150 60.5 .24 222.4 .68 62.8 .24 225.2 .97 67.3 .21 237.9 .84

160 79.8 .29 269.4 .21 80.7 .29 272.3 .40 82.7 .33 276.5 .42

170 91.3 .12 298.9 .34 92.5 .20 301.9 .35 93.5 .21 304.1 .42

180 100.5 .10 324.5 .42 101.3 .15 324.6 .58 101.9 .14 326.7 .75

190 107.1 .09 342.5 .18 108.2 .14 346.4 1.13 108.3 .14 346.2 1.18

200 112.1 .11 356.3 .14 113.7 .16 362.4 .38 113.0 .15 360 .52

210 116.5 .15 368.5 .12 117.8 .16 372.2 .29 116.8 .18 369.8 .34

220 119.1 .18 375.6 .12 120.3 .20 378.7 .44 119.4 .23 375.9 .35

230 121.1 .21 381.2 .12 122.2 .21 383.8 .86 121 .23 380 .74

240 122.5 .24 384.6 .17 123.3 .22 388 1.10 121.8 .22 383.1 1.12

250 123.9 .25 388.8 .28 123.8 .20 389.9 1.31 122.3 .21 384.9 .78

41
Table 4.1– Thermomechanical material property measurement of the Fiberite Eglass/934 epoxy material system in principal material direction. 

42
Table 4.2– Thermomechanical material property measurements of the Nitinol Ribbon 

actuator material. 

43
Table 4.3- Nitinol Ribbon Transformation Temperatures measured by DSC. 

 Temperature 

Transformation Marker oC oF 

Austenite Start 45 113 

Austenite Finish 60 140 

Martensite Start 17 62.6 

Martensite Finish 0 32 

44
Cha p t e r 9 

FIGURES 

 Martensite (twinned) Martensite (M+) (detwinned) Austenite 

Figure 1.1- SMA Lattice Structure at different phases. 

Figure 1.2– SMA body under tensile loading/unloading at low temperature, exhibiting 

the shape memory effect upon heating4

. 

45
Figure 1.3– Schematic of martensitic and reverse transformations5

. 

Figure 1.4– Stress-Strain Curves for Shape Memory Alloys at various intermediate 

temperatures

6

. 

46
Figure 1.5- Load-Deformation diagrams for shape memory alloy at various 

temperatures (fully martensitic-quasiplastic) and (fully austenitic-pseudoelastic)

4

. 

Figure 1.6– Nitinol recovery stress vs. temperature and initial strain7. 

47
Figure 1.7– Recovery stress vs. temperature, thermal cycle for nitinol ribbon samples at 

4% pre-strain 8

. 

Figure 1.8– Variation of austenite start temperature with weight content (%) of Nickel 

in Nitinol Alloy10. 

48
Figure 2.1- 22”x1”x0.080” Glass Epoxy/SMA Hybrid Beam (45/0/-45/90)2s Lamination 

with SMA embedded in the 0o

 lamina only8 . 

SMA Actuator

Glass Epoxy 

Laminate 

Figure 2.2- Consolidated SMAHC structure after curing cycle, and machined SMAHC 

beam specimens used for experiments8

. 

Laser 

SMAHC Beam

Beam Fixture

Figure 2.3– Static Experimental Set-up: SMAHC Beam3 installed in the beam fixture 

(front view) 

49
Mechanical Grip Electrical Grip 

Figure 2.4a– Mechanical and electrical grip of beam fixture (front view). 

TC

G-10 Insulation 

Figure 2.4b- Mechanical and electrical grip of beam fixture (back view). 

50
GX-270 

PC 

Labview 

VI 

Controller

NI 2345 Signal

Conditioning Box 

HP DC Power 

Supply

PMI CCD 

Camera 

Laser Disp. 

Transducer

Flir 

Thermal 

Camera PMI 

Projector 

Laser 

Power

Supply

Thermal 

Cam 

Parallel 

Interface

-

PMI Trigger Ch 5 

Ch1 

Ch2 

RS 232 Data Cable 

RS 422 Data Cable 

Beam Fixture 

Ch3 

+ SMAHC Beam

TC TC 

TC 

TC 

Ch4 

FrameGrabber Card

Com-port 

I/O Card

G-PIB 

PMI Data 

Acquisition 

PC

Figure 2.5- Static Thermoelastic Response Experimental Component Schematic.

Beam

Infrared Camera

PMI Projector

Laser

CCD Camera

Figure 2.6– Static Thermoelastic Response Experimental Configuration Overview. 

51
Figure 2.7- Simple PMI System Schematic35. 

Average Beam Temperature (

o

F)

Mid

s

p

a

n

Dis

placement (in)

100 150 200 250

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14 Run 1, Heating

Run 2, Heating

Run 3, Heating

Run 4, Heating

Run 5, Heating

Run 6, Heating

Run 7, Heating

Run 8, Heating

Run 9, Heating

Figure 2.8– SMAHC Beam2 Laser Displacement Experimental Results of Midspan for 

Runs 1-9. 

52
Average Beam Temperature (

o

F)

Midspa

n

Dis

pla

c

e

m

e

nt (in)

100 150 200 250

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14 Run 1, Heating

Run 2, Heating

Run 3, Heating

Run 4, Heating

Run 5, Heating

Run 6, Heating

Run 7, Heating

Run 8, Heating

Run 9, Heating

Run 10, Heating

Figure 2.9– SMAHC Beam3 Laser Displacement Experimental Results for Midspan 

Runs 1-10. 

SMAHC Beam

Motorized Linear Traverse 

Laser 

Figure 2.10– Experimental Configuration of initial imperfection measurements. 

53
Figure 2.11- Laser Translation versus PMI initial imperfection measurements of 

SMAHC beams installed in beam fixture. 

Figure 2.12- Laser Translation Measurements of SMAHC Beam3 for three different 

boundary condition configurations of the Garolite G-10 insulation blocks. 

54
.247 

.242 

.247 G-10 

G-10 

G-10 

G-10 

SMAHC BEAM

.233 

.236 

.247 .247 

.244 

G-10 BC Configuration 1 

.247 

.242 

G-10 .247 G-10 

SMAHC BEAM

.236 G- .233 10 G-10

.247 .247 

.244 

G-10 BC Configuration 2 

.244 

G-10 .247 G-10 .247 

SMAHC BEAM

G-10 .242 G-10 .233 

.247 .247 

.236 

G-10 BC Configuration 3 

Figure 2.13- Garolite G-10 Insulation Blocks Configurations for Laser Displacement 

Transducer Experiments (Top view of beam fixture boundaries) 

 

Figure 2.14- Projection Moire’ Interferometry calibration plate34. 

55
-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

-4 -2 0 2 4 6 8 10 12 14 16 18 20 22

Beam Length (in)

Out of Plane Deformation (in)

Data @120F

Fit @120F

Figure 2.15- PMI differential image data and line fit for the centerline out of plane 

deflection for Beam3 Run3 at T=160o

F. 

Figure 2.16- PMI full field imaging of the SMAHC Beam2 Run1 

56
Figure 2.17- PMI full field imaging of the SMAHC Beam 2 Run 4 

Figure 2.18- PMI full field imaging of the SMAHC Beam 2 Run 8 

57
Figure 2.19- PMI full field imaging of the SMAHC Beam 3 Run 1 

Figure 2.20- PMI full field imaging of the SMAHC Beam 3 Run 4 

58
Figure 2.21- PMI full field imaging of the SMAHC Beam 3 Run 8 

Figure 2.22- Beam2 PMI vs. Laser Static Deflection Measurements 

59
Figure 2.23- Beam3 PMI vs. Laser Static Deflection Measurements 

Electromechanical 

Shaker 

Mounted Beam

Fixture 

Thermal Camera 

Figure 3.1– SMAHC beam dynamic response experimental configuration. 

60
Pos 3

Pos 2 Fixture 

accels Pos 1 

Figure 3.2– SMAHC beam specimen installed, with mounted beam response 

transducers and fixture transducers (back view). 

Frequency (Hz)

PS

D (in/s2

)

2/Hz

0 100 200 300 400 0

10

20

30

40

50

Figure 3.3– Acceleration PSD for the base excitation of the beam with a RMS value of

.25g (96.5 in/s2

). 

61
Acccel 

GX-270 

PC 

Labview 

VI 

Controller

NI 2345 Signal

Conditioning Box 

HP DC Power 

Supply

Flir 

Thermal 

Camera 

Thermal 

Cam 

Power 

Supply

G-PIB 

I/O Card

Com-port 

FrameGrabber Card

Ch1 

Ch2 

Ch3 

Ch4 

SMAHC Beam

Beam Fixture 

RS 422 Data Cable 

RS 232 Data Cable 

+ -

Kistler 

5010B

SC

Ch7 Ch6 Ch5 Ch4 Ch3 Ch2 Ch1 

VXI/IDEAS 

Data 

Acquisition

Ch7 Ch6 Ch5 Ch4 Ch3 Ch2 Ch1 

PCB SC 

Integrate 

Displacemt

PCB SC 

Integrate 

Displacemt

TC 

Figure 3.4- SMAHC beam dynamic response experimental component schematic. 

62
frequency (Hz)

Displace

m

e

nt P

S

D (m2/H

z)/A

c

c

eleratio

n

P

S

D ( (m/s2

)

2/H

z)

0 100 200 300 400 10-14

10-9

10-4

101

106

accel_psd

disp_psd

Figure 3.5- Acceleration vs. Displacement PSD generated by attached m-file,which postprocesses the acceleration time history data. 

Figure 3.6- SMAHC Beam 2 PSD’s for selected temperatures at position 1. 

63
Figure 3.7- SMAHC Beam 2 PSD’s for selected temperatures at position 2 (midspan). 

Figure 3.8- SMAHC Beam 2 PSD’s for selected temperatures at position 3. 

64
Figure 3.9- SMAHC Beam 3 PSD’s for selected temperatures at position 1. 

Figure 3.10- SMAHC Beam 3 PSD’s at selected temperatures for position 2 (midspan). 

65
Figure 3.11- SMAHC Beam 3 PSD’s at selected temperatures for position 3. 

Figure 3.12- Modal frequency variation of modes 1, and 3 as a function of temperature. 

66
Figure 4.1- Representative volume element of SMAHC lamina18. 

L

W 

Figure 4.2– Applied stress vs. strain for .09x.006x10 inch sample with 0% pre-strain at 

ambient temperature and 120o

C (248 oF)18. 

67
Figure 4.3– Applied stress vs. strain for .09x.006x10 inch sample with 4% pre-strain at 

ambient temperature and 120o

C (248 oF)18. 

Figure 4.4– Recovery Stress vs. Temperature of the Nitinol ribbon (pre-strained 4%

and released) for thermal cycles 2, 4, and 5018. 

68
Figure 4.5– Modulus for Nitinol ribbon prestrained 4% and released as a function of 

temperature

18. 

SME endotherm

MT exotherm

Figure 4.6- Differential Scanning Calorimetry (DSC) Results for the Nitinol ribbon 

sample showing transformation temperatures. 

69
Figure 5.1- ABAQUS Finite Element Mesh of the SMAHC beam with clamped 

boundary conditions. 

Figure 5.2- Static Response SMAHC Beam 2 Run 6 ABAQUS vs experimental results 

for original recovery force measurements, and 15 o

F shift in recovery force input data. 

70
Figure 5.3– Static Response SMAHC Beam 2 Run 7 ABAQUS vs. experimental results 

for original recovery force measurements, and 10 o

F shift in recovery force input data. 

Figure 5.4- Static Response SMAHC Beam 2 Run 8 ABAQUS vs. experimental results 

for original recovery force measurements, and 15 o

F shift in recovery force input data. 

71
Figure 5.5- Static Response SMAHC Beam 3 Run 3 ABAQUS vs. experimental results 

for original recovery force measurements, and 15 o

F shift in recovery force input data. 

Figure 5.6- Static Response SMAHC Beam 3 Run 6 ABAQUS vs. experimental results 

for original recovery force measurements, and 15 o

F shift in recovery force input data. 

72
Figure 5.7- Static Response SMAHC Beam 3 Run 8 ABAQUS vs. experimental results 

for original recovery force measurements, and 20 o

F shift in recovery force input data 

Figure 5.8- Beam 2 Run 8, position 2 (midspan) PSD comparison at ambient (∆f=.25 

Hz). 

73
Figure 5.9- Beam 2 run 8, position 2 (Midspan) PSD comparison at 71.1 o

C (160 oF) 

(∆f=.25 Hz).

Figure 5.10- Beam 2 Run 8, position 2 (midspan) PSD comparison at 93.3 oC (200oF) 

(∆f=.25 Hz).

74
Figure 5.11- Beam 2 Run 8, position 2 (midspan) PSD comparison at 121.1 o

C (250 oF). 

Figure 5.12- Beam 3 Run 8, position 2 (midspan) PSD comparison at 21.7 oC (71 oF) 

(∆f=.25 Hz).

75
Figure 5.13- Beam 3 Run 8, position 2 (midspan) PSD comparison at 71.1 o

C (160 oF) 

(∆f=.25 Hz).

 Figure 5.14- Beam 3 Run 8, position 2 (midspan) PSD comparison at 93.3 oC (200 oF) 

(∆f=.25 Hz).

76
Figure 5.15- Beam 3 Run 8, position 2 (midspan) PSD comparison at 121.1 o

C (250 oF) 

(∆f=.25 Hz).

Figure 5.16- Beam 2 Run 8, position 1 PSD comparison for selected temperatures 

(∆f=.25 Hz).

77
Figure 5.17- Beam 3 Run 3, position 3 PSD’s comparison for selected temperatures 

(∆f=.25 Hz).

Figure 5.18- Beam 3 Run 1, position 3 PSD’s comparison through the buckling range 

(T=71 o

F). 

78
Figure 5.19- Beam 3 Run 1, position 3 PSD’s comparison through the buckling range 

(T=100 o

F). 

Figure 5.20- Beam 3 Run 1, position 3 PSD’s comparison through the buckling range 

(T=130 o

F). 

79
Figure 5.21- Beam 3 Run 1, position 3 PSD’s comparison through the buckling range 

(T=140 o

F). 

Figure 5.22- Beam 3 Run 1, position 3 PSD’s comparison through the buckling range 

(T=150 o

F). 

80
Figure 5.23- Beam 3 Run 1, position 3 PSD’s comparison through the buckling range 

(T=160 o

F). 

81
Chapter 10 

APPENDICES 

82
APPENDIX A 

Projection Moire’ Interferrometry Thermoelastic Static Response Results 

83
Figure 10.1- PMI full field imaging of the SMAHC Beam 2 Run 1. 

Figure 10.2- PMI full field imaging of the SMAHC Beam 2 Run 2. 

84
Figure 10.3- PMI full field imaging of the SMAHC Beam 2 Run 3. 

Figure 10.4- PMI full field imaging of the SMAHC Beam 2 Run 4. 

85
Figure 10.5- PMI full field imaging of the SMAHC Beam 2 Run 5. 

Figure 10.6- PMI full field imaging of the SMAHC Beam 2 Run 6. 

86
Figure 10.7- PMI full field imaging of the SMAHC Beam 2 Run 7. 

Figure 10.8- PMI full field imaging of the SMAHC Beam 2 Run 8. 

87
Figure 10.9- PMI full field imaging of the SMAHC Beam 2 Run 9. 

Figure 10.10- PMI full field imaging of the SMAHC Beam 3 Run 1. 

88
Figure 10.11- PMI full field imaging of the SMAHC Beam 3 Run 2. 

Figure 10.12- PMI full field imaging of the SMAHC Beam 3 Run 3. 

89
Figure 10.13- PMI full field imaging of the SMAHC Beam 3 Run 4. 

Figure 10.14- PMI full field imaging of the SMAHC Beam 3 Run 5. 

90
Figure 10.15- PMI full field imaging of the SMAHC Beam 3 Run 6. 

Figure 10.16- PMI full field imaging of the SMAHC Beam 3 Run 7. 

91
Figure 10.17- PMI full field imaging of the SMAHC Beam 3 Run 8. 

92
APPENDIX B 

Dynamic Acceleration Data Post Processing MATLAB code 

93
%This code converts an acceleration time history data file into 

%displacement time history data file by performing an FFT and then 

%integrating to get displacement and transforming back to time history. 

%Note that the code also deletes the low frequency noise present in our 

%specific data file. 

%Written by Brian Davis 

%_______________________________________________________________________ 

clear; 

%loads data file, generates acceleration and time vectors, and defines 

%important constants 

dataname=input('Enter the name of the data file: ','s'); 

data=[dataname,'.dat']; 

fid=fopen(data,'r'); 

%filname=fscanf(fid,'%s',1); 

%variables=fscanf(fid,'%s',2); 

[n, gh]=fscanf(fid,'%g %g',[2,inf]); 

%fmin=input('0 to ? frequency value is the noise present in the data?'); 

fmin=10; 

dt=n(1,2); 

fnyq=1/(2*dt); 

t=n(1,:)'; 

a=n(2,:)'; 

N=length(a); 

ahat=fft(a); 

%generates frequency vector 

kmin=fix(fmin*N*dt); 

knyq=fix(fnyq*N*dt); 

k1=0:knyq; 

k2=(knyq):-1:0; 

k=[k1,k2]'; 

f=(1/(N*dt)).*k; 

%generates displacement coefficients by integrating, and "zeros-out" the 

%low frequency values. 

dhat=(1./-(2*pi.*f).^2).*ahat; 

for j=1:(kmin+1) 

 dhat(j,:)=0; 

end 

for j=(N-(kmin+1)):N 

 dhat(j,:)=0; 

end 

%inverse fft generates the displacement values 

x=ifft(dhat); 

%------------------------------------------------------------------------- 

%Compute PSD and RMS values from the generated displacement values 

fs=1/dt; 

df=1/(4096*dt); 

[xspec,freq]=psd(x,4096,fs,hanning(4096),0); 

94
xpsd=xspec/fnyq; 

xrms=sqrt(sum(xpsd)*df); 

%------------------------------------------------------------------------- 

%injects displacment and time vector into a matrix which is exported to a 

%new data file. note that the 'w' writes to data file and the 't' keeps in 

% text format so the carriage returns aren't deleted 

data_file=[dataname '_disp.dat']; 

data2=[t x]'; 

fid=fopen(data_file,'wt'); 

%fprintf(fid,'variables = "t (sec)" "V (volts)" \n'); 

fprintf(fid,'%g %g\n',data2); 

fclose(fid); 

%Write the displacement fft coefficents to data file form 

data_file=[dataname '_fft.dat']; 

data2=[f dhat]'; 

fid=fopen(data_file,'wt'); 

%fprintf(fid,'variables = "t (sec)" "V (volts)" \n'); 

fprintf(fid,'%g %g\n',data2); 

fclose(fid); 

%Write the displacement psd values to data file form 

data_file=[dataname '_psd.dat']; 

data2=[freq xpsd]'; 

fid=fopen(data_file,'wt'); 

%fprintf(fid,'variables = "t (sec)" "V (volts)" \n'); 

fprintf(fid,'%g %g\n',data2); 

fclose(fid); 

%Write the displacement rms values to data file form 

data_file=[dataname '_rms.dat']; 

data2=[xrms]'; 

fid=fopen(data_file,'wt'); 

%fprintf(fid,'variables = "t (sec)" "V (volts)" \n'); 

fprintf(fid,'%g %g\n',data2); 

fclose(fid); 

%------------------------------------------------------------------------- 

95
APPENDIX C 

Initial Imperfection Data Linearly Interpolated to Node Set, MATLAB code. 

96
%Brian Davis 

%M-File that interprets experimental PMI data (deflection values in the z 

%direction) to a defined node set. 

%-------------------------------------------------------------------------

- 

%loads experimental data file 

%-------------------------------------------------------------------------

- 

dataname=input('Enter the name of the datafile:','s'); 

data=[dataname,'.dat']; 

fid=fopen(data,'r'); 

[n,gh]=fscanf(fid,'%g %g %g %g',[2,inf]); 

x_data=n(1,:)'; 

w_data=n(2,:)'; 

%----------------------------------------------------------------------- 

%generate node set 

%----------------------------------------------------------------------- 

nodes=(1:185)'; 

for i=1:185 

 if i <= 37 

 x(i)=(i-1)*.5; 

 y(i)=0; 

 elseif i > 37 & i <= 74 

 x(i)=(i-38)*.5; 

 y(i)=.275; 

 elseif i > 74 & i <= 111 

 x(i)=(i-75)*.5; 

 y(i)=.5; 

 elseif i > 111 & i <= 148 

 x(i)=(i-112)*.5; 

 y(i)=.725; 

 else 

 x(i)=(i-149)*.5; 

 y(i)=1; 

 end 

end 

mdl=[nodes,x',y']; 

%----------------------------------------------------------------------- 

% interprets the data file values to a matrix corresponding to the defined 

% node sets initial deflection values (u,v,w). 

%----------------------------------------------------------------------- 

w=interp1(x_data,w_data,mdl(:,2)); 

uvw=[nodes,mdl(:,2:3).*0,w]; 

%------------------------------------------------------------------------ 

% exports to a new data file representing the initial imperfection of the 

% defined node set. 

%------------------------------------------------------------------------ 

data_file=[dataname '_nodeset.dat']; 

data2=[uvw]'; 

fid=fopen(data_file,'wt'); 

%fprintf(fid,'variables = "node" "x coordinate" "y coordinate" "interp z 

coordinate(imperfection)\n'); 

fprintf(fid,'%g, %f, %f, %f\n',data2); 

fclose(fid); 

97
APPENDIX D 

ABAQUS Static and Dynamic Analysis Input File

98
*HEADING 

Thermal post-buckling/random response of a SMAHC beam with initial 

imperfections 

** 

*NODE, NSET=ALL_NODES 

 1, 0., 0., 0. 

 2, 0.5, 0., 0. 

 3, 1., 0., 0. 

 4, 1.5, 0., 0. 

 5, 2., 0., 0. 

 6, 2.5, 0., 0. 

 7, 3., 0., 0. 

 8, 3.5, 0., 0. 

 9, 4., 0., 0. 

 10, 4.5, 0., 0. 

 11, 5., 0., 0. 

 12, 5.5, 0., 0. 

 13, 6., 0., 0. 

 14, 6.5, 0., 0. 

 15, 7., 0., 0. 

 16, 7.5, 0., 0. 

 17, 8., 0., 0. 

 18, 8.5, 0., 0. 

 19, 9., 0., 0. 

 20, 9.5, 0., 0. 

 21, 10., 0., 0. 

 22, 10.5, 0., 0. 

 23, 11., 0., 0. 

 24, 11.5, 0., 0. 

 25, 12., 0., 0. 

 26, 12.5, 0., 0. 

 27, 13., 0., 0. 

 28, 13.5, 0., 0. 

 29, 14., 0., 0. 

 30, 14.5, 0., 0. 

 31, 15., 0., 0. 

 32, 15.5, 0., 0. 

 33, 16., 0., 0. 

 34, 16.5, 0., 0. 

 35, 17., 0., 0. 

 36, 17.5, 0., 0. 

 37, 18., 0., 0. 

 38, 0., 0.275, 0. 

 39, 0.5, 0.275, 0. 

 40, 1., 0.275, 0. 

 41, 1.5, 0.275, 0. 

 42, 2., 0.275, 0. 

 43, 2.5, 0.275, 0. 

 44, 3., 0.275, 0. 

 45, 3.5, 0.275, 0. 

 46, 4., 0.275, 0. 

 47, 4.5, 0.275, 0. 

 48, 5., 0.275, 0. 

 49, 5.5, 0.275, 0. 

 50, 6., 0.275, 0. 

 51, 6.5, 0.275, 0. 

 52, 7., 0.275, 0. 

99
 53, 7.5, 0.275, 0. 

 54, 8., 0.275, 0. 

 55, 8.5, 0.275, 0. 

 56, 9., 0.275, 0. 

 57, 9.5, 0.275, 0. 

 58, 10., 0.275, 0. 

 59, 10.5, 0.275, 0. 

 60, 11., 0.275, 0. 

 61, 11.5, 0.275, 0. 

 62, 12., 0.275, 0. 

 63, 12.5, 0.275, 0. 

 64, 13., 0.275, 0. 

 65, 13.5, 0.275, 0. 

 66, 14., 0.275, 0. 

 67, 14.5, 0.275, 0. 

 68, 15., 0.275, 0. 

 69, 15.5, 0.275, 0. 

 70, 16., 0.275, 0. 

 71, 16.5, 0.275, 0. 

 72, 17., 0.275, 0. 

 73, 17.5, 0.275, 0. 

 74, 18., 0.275, 0. 

 75, 0., 0.5, 0. 

 76, 0.5, 0.5, 0. 

 77, 1., 0.5, 0. 

 78, 1.5, 0.5, 0. 

 79, 2., 0.5, 0. 

 80, 2.5, 0.5, 0. 

 81, 3., 0.5, 0. 

 82, 3.5, 0.5, 0. 

 83, 4., 0.5, 0. 

 84, 4.5, 0.5, 0. 

 85, 5., 0.5, 0. 

 86, 5.5, 0.5, 0. 

 87, 6., 0.5, 0. 

 88, 6.5, 0.5, 0. 

 89, 7., 0.5, 0. 

 90, 7.5, 0.5, 0. 

 91, 8., 0.5, 0. 

 92, 8.5, 0.5, 0. 

 93, 9., 0.5, 0. 

 94, 9.5, 0.5, 0. 

 95, 10., 0.5, 0. 

 96, 10.5, 0.5, 0. 

 97, 11., 0.5, 0. 

 98, 11.5, 0.5, 0. 

 99, 12., 0.5, 0. 

 100, 12.5, 0.5, 0. 

 101, 13., 0.5, 0. 

 102, 13.5, 0.5, 0. 

 103, 14., 0.5, 0. 

 104, 14.5, 0.5, 0. 

 105, 15., 0.5, 0. 

 106, 15.5, 0.5, 0. 

 107, 16., 0.5, 0. 

 108, 16.5, 0.5, 0. 

 109, 17., 0.5, 0. 

100
 110, 17.5, 0.5, 0. 

 111, 18., 0.5, 0. 

 112, 0., 0.725, 0. 

 113, 0.5, 0.725, 0. 

 114, 1., 0.725, 0. 

 115, 1.5, 0.725, 0. 

 116, 2., 0.725, 0. 

 117, 2.5, 0.725, 0. 

 118, 3., 0.725, 0. 

 119, 3.5, 0.725, 0. 

 120, 4., 0.725, 0. 

 121, 4.5, 0.725, 0. 

 122, 5., 0.725, 0. 

 123, 5.5, 0.725, 0. 

 124, 6., 0.725, 0. 

 125, 6.5, 0.725, 0. 

 126, 7., 0.725, 0. 

 127, 7.5, 0.725, 0. 

 128, 8., 0.725, 0. 

 129, 8.5, 0.725, 0. 

 130, 9., 0.725, 0. 

 131, 9.5, 0.725, 0. 

 132, 10., 0.725, 0. 

 133, 10.5, 0.725, 0. 

 134, 11., 0.725, 0. 

 135, 11.5, 0.725, 0. 

 136, 12., 0.725, 0. 

 137, 12.5, 0.725, 0. 

 138, 13., 0.725, 0. 

 139, 13.5, 0.725, 0. 

 140, 14., 0.725, 0. 

 141, 14.5, 0.725, 0. 

 142, 15., 0.725, 0. 

 143, 15.5, 0.725, 0. 

 144, 16., 0.725, 0. 

 145, 16.5, 0.725, 0. 

 146, 17., 0.725, 0. 

 147, 17.5, 0.725, 0. 

 148, 18., 0.725, 0. 

 149, 0., 1., 0. 

 150, 0.5, 1., 0. 

 151, 1., 1., 0. 

 152, 1.5, 1., 0. 

 153, 2., 1., 0. 

 154, 2.5, 1., 0. 

 155, 3., 1., 0. 

 156, 3.5, 1., 0. 

 157, 4., 1., 0. 

 158, 4.5, 1., 0. 

 159, 5., 1., 0. 

 160, 5.5, 1., 0. 

 161, 6., 1., 0. 

 162, 6.5, 1., 0. 

 163, 7., 1., 0. 

 164, 7.5, 1., 0. 

 165, 8., 1., 0. 

 166, 8.5, 1., 0. 

101
 167, 9., 1., 0. 

 168, 9.5, 1., 0. 

 169, 10., 1., 0. 

 170, 10.5, 1., 0. 

 171, 11., 1., 0. 

 172, 11.5, 1., 0. 

 173, 12., 1., 0. 

 174, 12.5, 1., 0. 

 175, 13., 1., 0. 

 176, 13.5, 1., 0. 

 177, 14., 1., 0. 

 178, 14.5, 1., 0. 

 179, 15., 1., 0. 

 180, 15.5, 1., 0. 

 181, 16., 1., 0. 

 182, 16.5, 1., 0. 

 183, 17., 1., 0. 

 184, 17.5, 1., 0. 

 185, 18., 1., 0. 

** 

*NSET, NSET=CENTER 

93, 

*NSET, NSET=ACCELS 

84, 93, 104 

** 

** Read geometric imperfection data from external file 

** 

*IMPERFECTION, INPUT=beam2_run7_pmi_imp_nodeset.dat 

** 

** --------------------------------------------------- 

** --- Include temp. dependent material properties --- 

** --------------------------------------------------- 

** 

** GLEP 

** Date: 13-Nov-03 Time: 16:50:42 

** 

*MATERIAL, NAME=GLEP 

*ELASTIC, TYPE=LAMINA 

** E1, E2, nu12, G12, G13, G23, T 

7.15e+6, 2.90e+6, 0.29, 1.40e+6, 1.40e+6, 1.40e+6, 60. 

7.15e+6, 2.90e+6, 0.29, 1.40e+6, 1.40e+6, 1.40e+6, 70. 

7.15e+6, 2.90e+6, 0.29, 1.40e+6, 1.40e+6, 1.40e+6, 80. 

7.13e+6, 2.82e+6, 0.29, 1.34e+6, 1.34e+6, 1.34e+6, 100. 

7.11e+6, 2.75e+6, 0.29, 1.29e+6, 1.29e+6, 1.29e+6, 120. 

7.08e+6, 2.68e+6, 0.29, 1.24e+6, 1.24e+6, 1.24e+6, 140. 

7.07e+6, 2.64e+6, 0.29, 1.22e+6, 1.22e+6, 1.22e+6, 150. 

7.07e+6, 2.58e+6, 0.29, 1.20e+6, 1.20e+6, 1.20e+6, 160. 

7.06e+6, 2.47e+6, 0.29, 1.15e+6, 1.15e+6, 1.15e+6, 180. 

7.05e+6, 2.35e+6, 0.29, 1.10e+6, 1.10e+6, 1.10e+6, 200. 

7.05e+6, 2.22e+6, 0.29, 0.98e+6, 0.98e+6, 0.98e+6, 220. 

7.04e+6, 2.09e+6, 0.29, 0.87e+6, 0.87e+6, 0.87e+6, 240. 

7.04e+6, 2.03e+6, 0.29, 0.81e+6, 0.81e+6, 0.81e+6, 250. 

7.05e+6, 1.95e+6, 0.29, 0.75e+6, 0.75e+6, 0.75e+6, 260. 

7.06e+6, 1.80e+6, 0.29, 0.62e+6, 0.62e+6, 0.62e+6, 280. 

7.08e+6, 1.65e+6, 0.29, 0.50e+6, 0.50e+6, 0.50e+6, 300. 

*DENSITY 

0.00019, 

102
*EXPANSION, TYPE=ORTHO, ZERO=71. 

** a1, a2, a3, T 

 0.2919E-05, 0.6095E-05, , 60. 

 0.2968E-05, 0.6334E-05, , 70. 

 0.3138E-05, 0.7169E-05, , 80. 

 0.3455E-05, 0.9097E-05, , 100. 

 0.3664E-05, 0.1060E-04, , 120. 

 0.3750E-05, 0.1149E-04, , 140. 

 0.3762E-05, 0.1177E-04, , 150. 

 0.3758E-05, 0.1196E-04, , 160. 

 0.3728E-05, 0.1215E-04, , 180. 

 0.3691E-05, 0.1219E-04, , 200. 

 0.3666E-05, 0.1220E-04, , 220. 

 0.3665E-05, 0.1222E-04, , 240. 

 0.3673E-05, 0.1225E-04, , 250. 

 0.3687E-05, 0.1230E-04, , 260. 

 0.3723E-05, 0.1244E-04, , 280. 

 0.3763E-05, 0.1261E-04, , 300. 

** 

** NITI 

** Date: 13-Nov-03 Time: 16:50:42 

** 

*MATERIAL, NAME=NITI 

*ELASTIC, TYPE=LAMINA 

** E1, E2, nu12, G12, G13, G23, T 

3.94000e+6, 3.94000e+6, 0.3, 1.51535e+6, 1.51535e+6, 1.51535e+6, 60. 

3.94000e+6, 3.94000e+6, 0.3, 1.51535e+6, 1.51535e+6, 1.51535e+6, 70. 

3.59667e+6, 3.59667e+6, 0.3, 1.38334e+6, 1.38334e+6, 1.38334e+6, 80. 

3.25333e+6, 3.25333e+6, 0.3, 1.25128e+6, 1.25128e+6, 1.25128e+6, 90. 

2.91000e+6, 2.91000e+6, 0.3, 1.11923e+6, 1.11923e+6, 1.11923e+6, 100. 

3.72800e+6, 3.72800e+6, 0.3, 1.43385e+6, 1.43385e+6, 1.43385e+6, 110. 

4.54600e+6, 4.54600e+6, 0.3, 1.74846e+6, 1.74846e+6, 1.74846e+6, 120. 

5.36400e+6, 5.36400e+6, 0.3, 2.06308e+6, 2.06308e+6, 2.06308e+6, 130. 

6.18200e+6, 6.18200e+6, 0.3, 2.37769e+6, 2.37769e+6, 2.37769e+6, 140. 

7.00000e+6, 7.00000e+6, 0.3, 2.69231e+6, 2.69231e+6, 2.69231e+6, 150. 

7.95600e+6, 7.95600e+6, 0.3, 3.06000e+6, 3.06000e+6, 3.06000e+6, 160. 

8.91200e+6, 8.91200e+6, 0.3, 3.42769e+6, 3.42769e+6, 3.42769e+6, 170. 

9.31200e+6, 9.31200e+6, 0.3, 3.58154e+6, 3.58154e+6, 3.58154e+6, 180. 

9.15600e+6, 9.15600e+6, 0.3, 3.52154e+6, 3.52154e+6, 3.52154e+6, 190. 

9.00000e+6, 9.00000e+6, 0.3, 3.46154e+6, 3.46154e+6, 3.46154e+6, 200. 

9.27000e+6, 9.27000e+6, 0.3, 3.56538e+6, 3.56538e+6, 3.56538e+6, 210. 

9.54000e+6, 9.54000e+6, 0.3, 3.66923e+6, 3.66923e+6, 3.66923e+6, 220. 

9.81000e+6, 9.81000e+6, 0.3, 3.77308e+6, 3.77308e+6, 3.77308e+6, 230. 

10.0800e+6, 10.0800e+6, 0.3, 3.87692e+6, 3.87692e+6, 3.87692e+6, 240. 

10.3500e+6, 10.3500e+6, 0.3, 3.98077e+6, 3.98077e+6, 3.98077e+6, 250. 

10.2740e+6, 10.2740e+6, 0.3, 3.95154e+6, 3.95154e+6, 3.95154e+6, 260. 

10.1980e+6, 10.1980e+6, 0.3, 3.92231e+6, 3.92231e+6, 3.92231e+6, 270. 

10.1220e+6, 10.1220e+6, 0.3, 3.89308e+6, 3.89308e+6, 3.89308e+6, 280. 

10.0460e+6, 10.0460e+6, 0.3, 3.86385e+6, 3.86385e+6, 3.86385e+6, 290. 

9.97000e+6, 9.97000e+6, 0.3, 3.83462e+6, 3.83462e+6, 3.83462e+6, 300. 

9.97000e+6, 9.97000e+6, 0.3, 3.83462e+6, 3.83462e+6, 3.83462e+6, 320. 

*DENSITY 

0.0005349, 

*EXPANSION, TYPE=ORTHO, ZERO=71. 

** a1 a2 a3 T 

 0.3670E-05, 0.3670E-05, , 60. 

 0.3670E-05, 0.3670E-05, , 70. 

103
-0.1430E-04, 0.3670E-05, , 80. 

-0.2016E-04, 0.3670E-05, , 90. 

-0.2692E-04, 0.3670E-05, , 100. 

-0.3082E-04, 0.3670E-05, , 110. 

-0.5461E-04, 0.3708E-05, , 120. 

-0.7336E-04, 0.3794E-05, , 130. 

-0.7746E-04, 0.3908E-05, , 140. 

-0.7345E-04, 0.4037E-05, , 150. 

-0.6568E-04, 0.4177E-05, , 160. 

-0.5786E-04, 0.4324E-05, , 170. 

-0.5466E-04, 0.4476E-05, , 180. 

-0.5452E-04, 0.4617E-05, , 190. 

-0.5356E-04, 0.4735E-05, , 200. 

-0.5033E-04, 0.4836E-05, , 210. 

-0.4667E-04, 0.4923E-05, , 220. 

-0.4348E-04, 0.4999E-05, , 230. 

-0.4051E-04, 0.5066E-05, , 240. 

-0.3765E-04, 0.5126E-05, , 250. 

-0.3634E-04, 0.5179E-05, , 260. 

-0.3496E-04, 0.5226E-05, , 270. 

-0.3375E-04, 0.5269E-05, , 280. 

-0.3257E-04, 0.5308E-05, , 290. 

-0.3159E-04, 0.5343E-05, , 300. 

-0.3159E-04, 0.5343E-05, , 320. 

** 

** --------------------------------------------------- 

** --- SHELL elements (with directional properties) -- 

** --------------------------------------------------- 

** 

*SHELL SECTION, COMPOSITE, ELSET=SMAHCELEM 

** total thickness = 0.0825 

0.004875, 3, GLEP, 45. 

0.006, 3, NITI, 0. 

0.004875, 3, GLEP, -45. 

0.004875, 3, GLEP, 90. 

0.004875, 3, GLEP, 45. 

0.006, 3, NITI, 0. 

0.004875, 3, GLEP, -45. 

0.004875, 3, GLEP, 90. 

0.004875, 3, GLEP, 90. 

0.004875, 3, GLEP, -45. 

0.006, 3, NITI, 0. 

0.004875, 3, GLEP, 45. 

0.004875, 3, GLEP, 90. 

0.004875, 3, GLEP, -45. 

0.006, 3, NITI, 0. 

0.004875, 3, GLEP, 45. 

** 

*ELEMENT, TYPE=S4, ELSET=SMAHCELEM 

** 

37, 38, 39, 76, 75 

38, 39, 40, 77, 76 

39, 40, 41, 78, 77 

40, 41, 42, 79, 78 

41, 42, 43, 80, 79 

42, 43, 44, 81, 80 

43, 44, 45, 82, 81 

104
44, 45, 46, 83, 82 

45, 46, 47, 84, 83 

46, 47, 48, 85, 84 

47, 48, 49, 86, 85 

48, 49, 50, 87, 86 

49, 50, 51, 88, 87 

50, 51, 52, 89, 88 

51, 52, 53, 90, 89 

52, 53, 54, 91, 90 

53, 54, 55, 92, 91 

54, 55, 56, 93, 92 

55, 56, 57, 94, 93 

56, 57, 58, 95, 94 

57, 58, 59, 96, 95 

58, 59, 60, 97, 96 

59, 60, 61, 98, 97 

60, 61, 62, 99, 98 

61, 62, 63, 100, 99 

62, 63, 64, 101, 100 

63, 64, 65, 102, 101 

64, 65, 66, 103, 102 

65, 66, 67, 104, 103 

66, 67, 68, 105, 104 

67, 68, 69, 106, 105 

68, 69, 70, 107, 106 

69, 70, 71, 108, 107 

70, 71, 72, 109, 108 

71, 72, 73, 110, 109 

72, 73, 74, 111, 110 

73, 75, 76, 113, 112 

74, 76, 77, 114, 113 

75, 77, 78, 115, 114 

76, 78, 79, 116, 115 

77, 79, 80, 117, 116 

78, 80, 81, 118, 117 

79, 81, 82, 119, 118 

80, 82, 83, 120, 119 

81, 83, 84, 121, 120 

82, 84, 85, 122, 121 

83, 85, 86, 123, 122 

84, 86, 87, 124, 123 

85, 87, 88, 125, 124 

86, 88, 89, 126, 125 

87, 89, 90, 127, 126 

88, 90, 91, 128, 127 

89, 91, 92, 129, 128 

90, 92, 93, 130, 129 

91, 93, 94, 131, 130 

92, 94, 95, 132, 131 

93, 95, 96, 133, 132 

94, 96, 97, 134, 133 

95, 97, 98, 135, 134 

96, 98, 99, 136, 135 

97, 99, 100, 137, 136 

98, 100, 101, 138, 137 

99, 101, 102, 139, 138 

100, 102, 103, 140, 139 

105
101, 103, 104, 141, 140 

102, 104, 105, 142, 141 

103, 105, 106, 143, 142 

104, 106, 107, 144, 143 

105, 107, 108, 145, 144 

106, 108, 109, 146, 145 

107, 109, 110, 147, 146 

108, 110, 111, 148, 147 

** 

*SHELL SECTION, COMPOSITE, ELSET=GLEPELEM 

** total thickness = 0.078 

0.004875, 3, GLEP, 45. 

0.004875, 3, GLEP, 0. 

0.004875, 3, GLEP, -45. 

0.004875, 3, GLEP, 90. 

0.004875, 3, GLEP, 45. 

0.004875, 3, GLEP, 0. 

0.004875, 3, GLEP, -45. 

0.004875, 3, GLEP, 90. 

0.004875, 3, GLEP, 90. 

0.004875, 3, GLEP, -45. 

0.004875, 3, GLEP, 0. 

0.004875, 3, GLEP, 45. 

0.004875, 3, GLEP, 90. 

0.004875, 3, GLEP, -45. 

0.004875, 3, GLEP, 0. 

0.004875, 3, GLEP, 45. 

** 

*ELEMENT, TYPE=S4, ELSET=GLEPELEM 

** 

1, 1, 2, 39, 38 

2, 2, 3, 40, 39 

3, 3, 4, 41, 40 

4, 4, 5, 42, 41 

5, 5, 6, 43, 42 

6, 6, 7, 44, 43 

7, 7, 8, 45, 44 

8, 8, 9, 46, 45 

9, 9, 10, 47, 46 

10, 10, 11, 48, 47 

11, 11, 12, 49, 48 

12, 12, 13, 50, 49 

13, 13, 14, 51, 50 

14, 14, 15, 52, 51 

15, 15, 16, 53, 52 

16, 16, 17, 54, 53 

17, 17, 18, 55, 54 

18, 18, 19, 56, 55 

19, 19, 20, 57, 56 

20, 20, 21, 58, 57 

21, 21, 22, 59, 58 

22, 22, 23, 60, 59 

23, 23, 24, 61, 60 

24, 24, 25, 62, 61 

25, 25, 26, 63, 62 

26, 26, 27, 64, 63 

27, 27, 28, 65, 64 

106
28, 28, 29, 66, 65 

29, 29, 30, 67, 66 

30, 30, 31, 68, 67 

31, 31, 32, 69, 68 

32, 32, 33, 70, 69 

33, 33, 34, 71, 70 

34, 34, 35, 72, 71 

35, 35, 36, 73, 72 

36, 36, 37, 74, 73 

109, 112, 113, 150, 149 

110, 113, 114, 151, 150 

111, 114, 115, 152, 151 

112, 115, 116, 153, 152 

113, 116, 117, 154, 153 

114, 117, 118, 155, 154 

115, 118, 119, 156, 155 

116, 119, 120, 157, 156 

117, 120, 121, 158, 157 

118, 121, 122, 159, 158 

119, 122, 123, 160, 159 

120, 123, 124, 161, 160 

121, 124, 125, 162, 161 

122, 125, 126, 163, 162 

123, 126, 127, 164, 163 

124, 127, 128, 165, 164 

125, 128, 129, 166, 165 

126, 129, 130, 167, 166 

127, 130, 131, 168, 167 

128, 131, 132, 169, 168 

129, 132, 133, 170, 169 

130, 133, 134, 171, 170 

131, 134, 135, 172, 171 

132, 135, 136, 173, 172 

133, 136, 137, 174, 173 

134, 137, 138, 175, 174 

135, 138, 139, 176, 175 

136, 139, 140, 177, 176 

137, 140, 141, 178, 177 

138, 141, 142, 179, 178 

139, 142, 143, 180, 179 

140, 143, 144, 181, 180 

141, 144, 145, 182, 181 

142, 145, 146, 183, 182 

143, 146, 147, 184, 183 

144, 147, 148, 185, 184 

** 

*ELSET, ELSET=ALL_ELEMENTS 

SMAHCELEM, 

GLEPELEM, 

** 

*INITIAL CONDITIONS, TYPE=TEMPERATURE, 

INPUT=beam2_run7_thermal_t71_nodeset.dat 

** 

** 

*BOUNDARY 

CFCF, 1,6, 0. 

** 

107
** 

*NSET, NSET=CFCF 

1, 37, 38, 74, 75, 111, 112, 148, 

 149, 185 

** 

*PSD-DEFINITION, NAME=S10TO400, G=1.0, INPUT=input_accel_psd.abq, 

TYPE=BASE 

** 

** --------------------------------------------------- 

** ------ STEP data for Eigensolution at 71 F 

** --------------------------------------------------- 

*STEP, PERTURBATION 

Eigensolution for SMAHC beam at 71 F 

** 

*FREQUENCY, EIGENSOLVER=LANCZOS, NORMALIZATION=MASS 

10, , 1500. 

** 

** Eigensolution output 

** 

***OUTPUT, FIELD 

***NODE OUTPUT 

**U, 

*END STEP 

** 

** --------------------------------------------------- 

** ------ STEP data for Linear Random Response at 71 F 

** --------------------------------------------------- 

*STEP, PERTURBATION 

Linear random response of SMAHC beam at 71 F 

** 

** 0-400 Hz with 400 points between eigenfrequencies, 1.0 bias and linear 

frequency 

** 

*RANDOM RESPONSE 

0., 400., 400, 1.0, 1 

** 

** This is only needed in the event that a subset of the modes 

** from the *FREQUENCY card are used 

** 

*SELECT EIGENMODES, GENERATE 

1, 10 

** 

*MODAL DAMPING, MODAL=DIRECT 

1,2, .0081 

3,10, .0068 

** 

*CORRELATION, PSD=S10TO400, COMPLEX=NO, TYPE=CORRELATED 

2, 1. 

** 

*DLOAD 

** 

*BASE MOTION, DOF=3, LOAD CASE=2, TYPE=ACCELERATION 

** 

** Random response PSD at three accel nodes 

** 

*OUTPUT, FIELD 

*NODE OUTPUT, NSET=ACCELS 

108
U, 

** 

*END STEP 

** --------------------------------------------------- 

** ------ STEP data for post-buckling 70-80 F 

** --------------------------------------------------- 

*STEP, NLGEOM, INC=33.3333 

Post-buckling solution 70-80 degrees F 

** 

*STATIC, DIRECT 

0.03,1.0 

** 

** input thermal load 

** 

*TEMPERATURE, INPUT=beam2_run7_thermal_T80_nodeset.dat 

** 

** Load deflection data at center node (propagates to subsequent static 

steps) 

** 

*OUTPUT, HISTORY 

*NODE OUTPUT, NSET=CENTER 

U3, 

** 

*END STEP 

** --------------------------------------------------- 

** ------ STEP data for post-buckling 80-90 F 

** --------------------------------------------------- 

*STEP, NLGEOM, INC=33.3333 

Post-buckling solution 80-90 degrees F 

** 

*STATIC, DIRECT 

0.03,1.0 

** 

** Uniform thermal load 

** 

*TEMPERATURE, INPUT=beam2_run7_thermal_T90_nodeset.dat 

** 

** Load deflection data at center node (propagates to subsequent static 

steps) 

** 

*OUTPUT, HISTORY 

*NODE OUTPUT, NSET=CENTER 

U3, 

** 

*END STEP 

** --------------------------------------------------- 

** ------ STEP data for post-buckling 90-100 F 

** --------------------------------------------------- 

*STEP, NLGEOM, INC=33.3333 

Post-buckling solution 90-100 degrees F 

** 

*STATIC, DIRECT 

0.03,1.0 

** 

** Uniform thermal load 

** 

*TEMPERATURE, INPUT=beam2_run7_thermal_T100_nodeset.dat 

109
** 

** Load deflection data at center node (propagates to subsequent static 

steps) 

** 

*OUTPUT, HISTORY 

*NODE OUTPUT, NSET=CENTER 

U3, 

** 

*END STEP 

** --------------------------------------------------- 

** ------ STEP data for post-buckling 100-110 F 

** --------------------------------------------------- 

*STEP, NLGEOM, INC=33.3333 

Post-buckling solution 100-110 degrees F 

** 

*STATIC, DIRECT 

0.03,1.0 

** 

** Uniform thermal load 

** 

*TEMPERATURE, INPUT=beam2_run7_thermal_T110_nodeset.dat 

** 

** Load deflection data at center node (propagates to subsequent static 

steps) 

** 

*OUTPUT, HISTORY 

*NODE OUTPUT, NSET=CENTER 

U3, 

** 

*END STEP 

** --------------------------------------------------- 

** ------ STEP data for post-buckling 110-120 F 

** --------------------------------------------------- 

*STEP, NLGEOM, INC=33.3333 

Post-buckling solution 110-120 degrees F 

** 

*STATIC, DIRECT 

0.03,1.0 

** 

** Uniform thermal load 

** 

*TEMPERATURE, INPUT=beam2_run7_thermal_T120_nodeset.dat 

** 

** Load deflection data at center node (propagates to subsequent static 

steps) 

** 

*OUTPUT, HISTORY 

*NODE OUTPUT, NSET=CENTER 

U3, 

** 

*END STEP 

** --------------------------------------------------- 

** ------ STEP data for post-buckling 120-130 F 

** --------------------------------------------------- 

*STEP, NLGEOM, INC=33.3333 

Post-buckling solution 120-130 degrees F 

** 

110
*STATIC, DIRECT 

0.03,1.0 

** 

** Uniform thermal load 

** 

*TEMPERATURE, INPUT=beam2_run7_thermal_T130_nodeset.dat 

** 

** Load deflection data at center node (propagates to subsequent static 

steps) 

** 

*OUTPUT, HISTORY 

*NODE OUTPUT, NSET=CENTER 

U3, 

** 

*END STEP 

** --------------------------------------------------- 

** ------ STEP data for post-buckling 130-140 F 

** --------------------------------------------------- 

*STEP, NLGEOM, INC=33.3333 

Post-buckling solution 130-140 degrees F 

** 

*STATIC, DIRECT 

0.03,1.0 

** 

** Uniform thermal load 

** 

*TEMPERATURE, INPUT=beam2_run7_thermal_T140_nodeset.dat 

** 

** Load deflection data at center node (propagates to subsequent static 

steps) 

** 

*OUTPUT, HISTORY 

*NODE OUTPUT, NSET=CENTER 

U3, 

** 

*END STEP 

** --------------------------------------------------- 

** ------ STEP data for post-buckling 140-150 F 

** --------------------------------------------------- 

*STEP, NLGEOM, INC=33.3333 

Post-buckling solution 140-150 degrees F 

** 

*STATIC, DIRECT 

0.03,1.0 

** 

** Uniform thermal load 

** 

*TEMPERATURE, INPUT=beam2_run7_thermal_T150_nodeset.dat 

** 

** Load deflection data at center node (propagates to subsequent static 

steps) 

** 

*OUTPUT, HISTORY 

*NODE OUTPUT, NSET=CENTER 

U3, 

** 

*END STEP 

111
** --------------------------------------------------- 

** ------ STEP data for post-buckling 150-160 F 

** --------------------------------------------------- 

*STEP, NLGEOM, INC=33.3333 

Post-buckling solution 150-160 degrees F 

** 

*STATIC, DIRECT 

0.03,1.0 

** 

** Uniform thermal load 

** 

*TEMPERATURE, INPUT=beam2_run7_thermal_T160_nodeset.dat 

** 

** Load deflection data at center node (propagates to subsequent static 

steps) 

** 

*OUTPUT, HISTORY 

*NODE OUTPUT, NSET=CENTER 

U3, 

** 

*END STEP 

** --------------------------------------------------- 

** ------ STEP data for Eigensolution at 160 F 

** --------------------------------------------------- 

*STEP, PERTURBATION 

Eigensolution for SMAHC beam at 160 F 

** 

*FREQUENCY, EIGENSOLVER=LANCZOS, NORMALIZATION=MASS 

10, , 1500. 

** 

** Eigensolution output 

** 

***OUTPUT, FIELD 

***NODE OUTPUT 

**U, 

*END STEP 

** 

** --------------------------------------------------- 

** ------ STEP data for Linear Random Response at 160 F 

** --------------------------------------------------- 

*STEP, PERTURBATION 

Linear random response of SMAHC beam at 160 F 

** 

** 0-400 Hz with 400 points between eigenfrequencies, 1.0 bias and linear 

frequency 

** 

*RANDOM RESPONSE 

0., 400., 400, 1.0, 1 

** 

** This is only needed in the event that a subset of the modes 

** from the *FREQUENCY card are used 

** 

*SELECT EIGENMODES, GENERATE 

1, 10 

** 

*MODAL DAMPING, MODAL=DIRECT 

1,2, .0017 

112
3,10, .0025 

** 

*CORRELATION, PSD=S10TO400, COMPLEX=NO, TYPE=CORRELATED 

2, 1. 

** 

*DLOAD 

** 

*BASE MOTION, DOF=3, LOAD CASE=2, TYPE=ACCELERATION 

** 

** Random response PSD at three accel nodes 

** 

*OUTPUT, FIELD 

*NODE OUTPUT, NSET=ACCELS 

U, 

** 

*END STEP 

** --------------------------------------------------- 

** ------ STEP data for post-buckling 160-180 F 

** --------------------------------------------------- 

*STEP, NLGEOM 

Post-buckling solution 160-180 degrees F 

** 

*STATIC, DIRECT 

0.5,1.0 

** 

** Uniform thermal load 

** 

*TEMPERATURE, INPUT=beam2_run7_thermal_T180_nodeset.dat 

** 

** Load deflection data at center node (carries forward from above) 

** 

*END STEP 

** --------------------------------------------------- 

** ------ STEP data for post-buckling 180-200 F 

** --------------------------------------------------- 

*STEP, NLGEOM 

Post-buckling solution 180-200 degrees F 

** 

*STATIC, DIRECT 

0.5,1.0 

** 

** Uniform thermal load 

** 

*TEMPERATURE, INPUT=beam2_run7_thermal_T200_nodeset.dat 

** 

** Load deflection data at center node (carries forward from above) 

** 

*END STEP 

** --------------------------------------------------- 

** ------ STEP data for Eigensolution at 200 F 

** --------------------------------------------------- 

*STEP, PERTURBATION 

Eigensolution for SMAHC beam at 200 F 

** 

*FREQUENCY, EIGENSOLVER=LANCZOS, NORMALIZATION=MASS 

10, , 1500. 

** 

113
** Eigensolution output 

** 

***OUTPUT, FIELD 

***NODE OUTPUT 

**U, 

*END STEP 

** 

** --------------------------------------------------- 

** ------ STEP data for Linear Random Response at 200 F 

** --------------------------------------------------- 

*STEP, PERTURBATION 

Linear random response of SMAHC beam at 200 F 

** 

** 0-400 Hz with 400 points between eigenfrequencies, 1.0 bias and linear 

frequency 

** 

*RANDOM RESPONSE 

0., 400., 400, 1.0, 1 

** 

** This is only needed in the event that a subset of the modes 

** from the *FREQUENCY card are used 

** 

*SELECT EIGENMODES, GENERATE 

1, 10 

** 

*MODAL DAMPING, MODAL=DIRECT 

1,2, .0016 

3,10, .0016 

** 

*CORRELATION, PSD=S10TO400, COMPLEX=NO, TYPE=CORRELATED 

2, 1. 

** 

*DLOAD 

** 

*BASE MOTION, DOF=3, LOAD CASE=2, TYPE=ACCELERATION 

** 

** Random response PSD at 3 accel nodes 

** 

*OUTPUT, FIELD 

*NODE OUTPUT, NSET=ACCELS 

U, 

** 

*END STEP 

** --------------------------------------------------- 

** ------ STEP data for post-buckling 200-220 F 

** --------------------------------------------------- 

*STEP, NLGEOM 

Post-buckling solution 200-220 degrees F 

** 

*STATIC, DIRECT 

0.5,1.0 

** 

** Uniform thermal load 

** 

*TEMPERATURE, INPUT=beam2_run7_thermal_T220_nodeset.dat 

** 

** Load deflection data at center node (carries forward from above) 

114
** 

*END STEP 

** --------------------------------------------------- 

** ------ STEP data for post-buckling 220-250 F 

** --------------------------------------------------- 

*STEP, NLGEOM 

Post-buckling solution 220-250 degrees F 

** 

*STATIC, DIRECT 

0.5,1.0 

** 

** Uniform thermal load 

** 

*TEMPERATURE, INPUT=beam2_run7_thermal_T250_nodeset.dat 

** 

** Load deflection data at center node (carries forward from above) 

** 

*END STEP 

** --------------------------------------------------- 

** ------ STEP data for Eigensolution at 250 F 

** --------------------------------------------------- 

*STEP, PERTURBATION 

Eigensolution for SMAHC beam at 250 F 

** 

*FREQUENCY, EIGENSOLVER=LANCZOS, NORMALIZATION=MASS 

10, , 1500. 

** 

** Eigensolution output 

** 

***OUTPUT, FIELD 

***NODE OUTPUT 

**U3, 

*END STEP 

** 

** --------------------------------------------------- 

** ------ STEP data for Linear Random Response at 250 F 

** --------------------------------------------------- 

*STEP, PERTURBATION 

Linear random response of SMAHC beam at 250 F 

** 

** 0-400 Hz with 400 points between eigenfrequencies, 1.0 bias and linear 

frequency 

** 

*RANDOM RESPONSE 

0., 400., 400, 1.0, 1 

** 

** This is only needed in the event that a subset of the modes 

** from the *FREQUENCY card are used 

** 

*SELECT EIGENMODES, GENERATE 

1, 10 

** 

*MODAL DAMPING, MODAL=DIRECT 

1,2,.0021 

3,10, .0022 

** 

*CORRELATION, PSD=S10TO400, COMPLEX=NO, TYPE=CORRELATED 

115
2, 1. 

** 

*DLOAD 

** 

*BASE MOTION, DOF=3, LOAD CASE=2, TYPE=ACCELERATION 

** 

** Random response PSD at 3 accel nodes 

** 

*OUTPUT, FIELD 

*NODE OUTPUT, NSET=ACCELS 

U, 

** 

*END STEP

116
REPORT DOCUMENTATION PAGE Form Approved

OMB No. 0704-0188

2. REPORT TYPE 

Contractor Report

 4. TITLE AND SUBTITLE

Investigation of the Thermomechanical Response of Shape Memory Alloy

Hybrid Composite Beams

5a. CONTRACT NUMBER

 6. AUTHOR(S)

Brian A. Davis

 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Langley Research Center North Carolina State University

Hampton, VA 23681-2199 Raleigh, NC 27695

 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

 8. PERFORMING ORGANIZATION

 REPORT NUMBER

10. SPONSOR/MONITOR'S ACRONYM(S)

NASA

13. SUPPLEMENTARY NOTES

NASA Langley Technical Monitor: Travis L. Turner

An electronic version can be found at http://ntrs.nasa.gov

12. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified - Unlimited

Subject Category 39

Availability: NASA CASI (301) 621-0390

19a. NAME OF RESPONSIBLE PERSON

STI Help Desk (email: help@sti.nasa.gov)

14. ABSTRACT

Previous work at NASA Langley Research Center (LaRC) involved fabrication and testing of composite beams with embedded, pre-strained

shape memory alloy (SMA) ribbons. That study also provided comparison of experimental results with numerical predictions from a

research code making use of a new thermoelastic model for shape memory alloy hybrid composite (SMAHC) structures. The previous work

showed qualitative validation of the numerical model. However, deficiencies in the experimental-numerical correlation were noted and

hypotheses for the discrepancies were given for further investigation. The goal of this work is to refine the experimental measurement and

numerical modeling approaches in order to better understand the discrepancies, improve the correlation between prediction and

measurement, and provide rigorous quantitative validation of the numerical model. Thermal buckling, post-buckling, and random responses

to thermal and inertial (base acceleration) loads are studied. Excellent agreement is achieved between the predicted and measured results,

thereby quantitatively validating the numerical tool.

15. SUBJECT TERMS

Shape memory alloys, Nitinol, embedded actuators, hybrid composites, adaptive stiffening, thermal post-buckling control,

vibration control, ABAQUS, finite element analysis, infrared thermography, projection moire interferometry, geometric

imperfections

18. NUMBER

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135

19b. TELEPHONE NUMBER (Include area code)

(301) 621-0390

a. REPORT

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b. ABSTRACT

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17. LIMITATION OF 

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Prescribed by ANSI Std. Z39.18

Standard Form 298 (Rev. 8-98)

3. DATES COVERED (From - To)

Aug 2003-Aug 2004

5b. GRANT NUMBER

NCC-1-02043

5c. PROGRAM ELEMENT NUMBER

5d. PROJECT NUMBER

5e. TASK NUMBER

5f. WORK UNIT NUMBER

23-781-10-13

11. SPONSOR/MONITOR'S REPORT

 NUMBER(S)

NASA/CR-2005-213929

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PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS.

1. REPORT DATE (DD-MM-YYYY)

01- 10 - 2005