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Micromechanical Analysis and Design of an Integrated Thermal Protection System for Future Space Vehicles

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PAGE 9

= e = e = g = k = k = k

PAGE 14

{ } { } e k

PAGE 15

t y y

PAGE 16

S

PAGE 40

{ } [ ]{ } = = k g e q

PAGE 41

= = = q = = q q = = + =

PAGE 42

[ ] = = = { } { } = { }{ } g e e

PAGE 43

{ } { } = = k k k g e e k k k g e e { } { } = = k k k g e e k k k g e e

PAGE 44

g e e = e = e = e = e = = = = = = = k k k k

PAGE 45

{ } ( ) [ ]{ } = = = { } ( ) [ ] { } ( ) = = ( ) [ ] ( ) = = = =

PAGE 46

= D D D + D D D + t t = = t d d g + =

PAGE 47

= d d d = = d

PAGE 48

= = d d d + = d d d = = d d d d d g + + = + = = [ ] = k e k e [ ] [ ] { } { } ( ) k e s + =

PAGE 49

= e e e k g e = k = k q g g = q q q k k + = q q

PAGE 50

( ) + = g g g ( ) + = g g g ( ) = g g g = + = = = g g g g g = = = = g g g

PAGE 51

( ) = = g g g q = k

PAGE 52

= + + = + = + = =k q

PAGE 55

g g g g

PAGE 56

n = e = e = g = k = k = k

PAGE 57

[ ] [ ] = =

PAGE 60

=

PAGE 61

+ =

PAGE 66

g k

PAGE 67

s e s e s e s e = e s e s e = e

PAGE 68

G g G g = g

PAGE 69

s k s k s k s k = k

PAGE 70

s k s k s k s k = k

PAGE 71

G k G k = k

PAGE 74

[ ] { } D = a = k e

PAGE 75

D

PAGE 76

[ ] [ ]{ } = D = + + + = + + + = ( ) + + = + = =

PAGE 77

= D = a e s [ ] { } D = a s

PAGE 78

+ = + = k e

PAGE 79

+ + D = = + + D = = e [ ] = k e

PAGE 80

[ ] [ ] [ ] [ ] = sn

PAGE 84

D

PAGE 85

s s s s

PAGE 86

s s s s

PAGE 89

+ + = + + + = = =

PAGE 90

= + + + + + = = = = = = x =

PAGE 91

lll l = + + + + + x x x x x x l x = l l l l x + + + = = + + + + + lx l x l xl l x l x x = l l l l l l l l l l l l l l l l

PAGE 92

= l l l l l l l l l l l l l l l l l l l l l l l l xl l x + + =x xxxxn

PAGE 93

x

PAGE 94

x x x = n

PAGE 98

x

PAGE 99

x x

PAGE 100

x x

PAGE 102

= = = = = = = = p = = = = p p = = = p p = = = p p y = = = p p y y y

PAGE 103

= + + = + = + + = y y y y + + = y y = + + + + + + + p p p p p p p p p p p p

PAGE 104

+ + + =

PAGE 105

= = + + q [ ] = q q + + + = + + = q

PAGE 107

+ = k = q q + =

PAGE 108

= + + = s s = t s s

PAGE 109

= e

PAGE 110

= = = = = = = = = = = y y y = = = = = = p p p y y = = y

PAGE 111

= + + h h = + + h h h = + h h hp = = l a = = l b = = l gl = + + + + a g b h l h l h h l lh l lh h l =

PAGE 112

p = = = p = p p l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l a a a a a a

PAGE 113

= + + + = = = = = = = = = p = = + + p p

PAGE 114

= + + p p = + + + = -

PAGE 115

n

PAGE 116

n

PAGE 117

nn

PAGE 118

= =

PAGE 122

nn

PAGE 126

y y k y k y k + = = = [ ] k e s + = k e

PAGE 127

y y k y k y k + = = = [ ] k e s + = k e k

PAGE 128

+ +

PAGE 136

s s s s

PAGE 137

s s

PAGE 138

s s

PAGE 140

q q q q

PAGE 143

q r q + + = s s s q q

PAGE 144

q s

PAGE 145

nn

PAGE 146

=

PAGE 149

q q

PAGE 150

q q

PAGE 156

+ = q q q q = q q q q k = k = = = =

PAGE 157

= = e + + = = q q q k e = + = g = = k q k q q k k + = = = = k q q k e = { } { } = = k k k g e e q q q q q q q q q q k k k g e e { } { } + = = k k k g e e q q q q q q q q q q k k k g e e

PAGE 158

{ } { } = = k k k g e e q q q q u u q k k k g e e { } { } = = k k k g e e q q q q u u q k k k g e e

PAGE 159

= t q t t = + = q t + = e e q t e = k + = q q t = k k k

PAGE 160

= t q t q + = q q = + + + = q

PAGE 161

+ + + + = = = + = = + + = q q q q q q q q d + = = d + = d

PAGE 162

l l l l l l a a a a a a + + + + + = l l l l l l b b b b b b + + + + + = l l l l l l g g g g g g + + + + + = a a b = = a a g = = + + = h l = h l h l + = = h = l = + = = = + = + + + + = h

PAGE 163

+ + = h p h = + + = = =


Permanent Link: http://ufdc.ufl.edu/UFE0019802/00001

Material Information

Title: Micromechanical Analysis and Design of an Integrated Thermal Protection System for Future Space Vehicles
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0019802:00001

Permanent Link: http://ufdc.ufl.edu/UFE0019802/00001

Material Information

Title: Micromechanical Analysis and Design of an Integrated Thermal Protection System for Future Space Vehicles
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0019802:00001


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Full Text





MICROMECHANICAL ANALYSIS AND DESIGN OF AN INTEGRATED THERMAL
PROTECTION SYSTEM FOR FUTURE SPACE VEHICLES




















By

OSCAR MARTINEZ


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007




































2007 Oscar Martinez

































To my wife, Coleen, and son, Josh. I would not have been able to complete this without their
love and support.









ACKNOWLEDGMENTS

I would like to thank my wife, Coleen Martinez, for all of her support throughout my

research experience as well as through my graduate career. I would like to thank my son, Josh

Martinez, for reminding me why I completed this dissertation. I would like to thank my mother,

Martha Mendez, my father, Oscar Martinez, my stepfather, Carlos Mendez, and my stepmother,

Terry Martinez for their support. Finally, I would like to thank Dr. Bhavani Sankar, Dr. Raphael

Haftka, Dr. Peter Ifju, and Dr. David Bloomquist for participating and evaluating my research

work. Dr. Sankar and Dr. Haftka have served as my advisor's, mentor's, guide's, and professor's

and I could not have completed this dissertation without their guidance, support, and dedication.

I am also thankful to Dr. Max Blosser (NASA Langley) for his inputs and suggestions in my

research work, which kept us on track with the expectations of NASA. I am thankful to CUIP

and SEAGEP, for their financial support through my graduate career.









TABLE OF CONTENTS

page

ACKN OW LED GM EN TS ....................................................................................... 4

L IST O F T A B L E S .................................................................. ....................................... . 8

L IST O F F IG U R E S .................................................................. .................................... . 9

LIST OF ABBREV IA TION S............................................................... ........................... 13

LIST OF SYMBOLS ....................................................................... ........... 14

A B S T R A C T ...................................... ......................................... ............... 16

CHAPTER

1 INTRODUCTION ................................... ............................. ....... ......... 18

Therm al Protection System ..................................................... ................................. 21
Function of a Therm al Protection System ............................................... .............. 21
General Requirements of a Thermal Protection System .............................................22
Approach to Thermal Protection Systems....................... .................................23
History of Thermal Protection Systems .............. ............. ....... ...................... 25
Integral Therm al Protection System ............................................................ .............. 28
P u rp o se ......... ..... ............ ................... ........ ..................... ............ 3 1
O b je c tiv e s .............. ..... ............ ............................................................................ 3 1

2 MICROMECHANICAL ANALYSIS .................................. ............................ 33

S an d w ich S tru ctu re s ...........................................................................................................3 3
Literature Review on Corrugated Core Sandwich Panels.................................... .............33
A analytical A approach ........................................................................36
G eom etric Param eters ................................... ..................... ....40
Extensional and Bending Stiffness.................. ... ...............................41
Formulation of deformation transformation matrix for the facesheets ................ 42
Formulation of the web deformation transformation matrix ........................ 44
Stiffness matrix determination using the strain energy approach........................45
Transverse Shear Stiffness, A 55............................................................ .............. 45
Transverse Shear Stiffness, A 44...........................................................................46
Face Sheet and W eb Stress Determination................................. ........................ 48
M idplane micro shear strain in the web ........................................ .............. 50
Micro curvature in the y -direction for the webs............................ .............. 51
C o n clu sio n ...................................... .................................................... 52









3 FINITE ELEMENT VERIFICATION........................... ..............56

Extensional and B ending Stiffness ............................................... ............................ 56
Stress V verification .............................. ..... ......................................................59
Midplane Shear Strain and Curvature in the Webs ....................................................59
S tre ss V erific atio n .................................................................................................. 5 9
Transverse Shear Stiffness (A44) Verification .............. ................. ................................ 60
C o n clu sio n ......... ..... ...................................................... .................................6 1

4 THERMAL ANALYSIS OF AN ITPS UNIT CELL ......... ............. .................73

Intro du ctio n ................. ............. ................. ....................... 7 3
Therm al M icrom echanics Approach.......................................................... .............. 75
Thermal Force Resultants and M moments ................................................. .............. 75
T h e rm a l S tre ss ................................................................ ....................................7 7
Constrained case .................................. ............................ ... ........ 77
U nconstrained case ................................................................... 79
Finite E lem ent V verification .......................................................................................... 80
Thermal Force and Moment Resultants ................................................................ 80
Therm al Stress V verification ...................................................................... .............. 81
C o n strain ed case .................................................................................. 8 1
U ncon strained case ..................................................................................... 8 1
C o n c lu sio n .................................................................................................................... 8 2

5 BUCKLING ANALYSIS OF AN ORTHOTROPIC INFINITE STRIPS AND
A PPL IC A T IO N S T O IT P S ..................................... ................................... ................... ..... 87

B u ck lin g o f an IT P S ...................................................................................................... 8 7
M ethods of Critical Loads Calculation ..................................................................... ..... 88
Stability of an Infinite Strip under Compression or Shear .............................................. 89
R e su lts ................................... ....................................................................................... 9 2
U niaxial Com pression, Nx only.......................................................... ............. 93
U niaxial C om pression, N y only ........................................................................ ... ... 93
Biaxial Com pression, N = Ny.......................................................... .............. 94
Shearing Load, N y = 1 ................................................................. 94
C o n c lu sio n .................................................................................................................... 9 5

6 ITPS PANEL AS A TWO DIMENSIONAL PLATE .............................. 101

Intro du ctio n ....................... .. .............. .. .....01..........
A n a ly sis ............. ..... .. .... .............. .. ............................................................. 1 0 2
U uniform Pressure Loading..................................... ......................... .............. 102
Out-of-plane displacement........................... .............. 102
Top facesheet local deflection ........... .. ....... ................... .............. 104
Local stresses ........................ ............. ...................106





6









Un-symmetric ITPS Panel Configuration ............................................ ..............108
T em perature D distribution ................................................................................... 109
Out-of-plane displacement, First Order Shear Deformable Plate Theory............ 109
Out-of-plane displacement, Classical Laminate Plate Theory........................... 112
R results .................... .. .......... ..... ....... ...... ... ... ........... ........... ........... 114
ITPS Out-of-Plane Displacement, Pressure Load............................................. 114
ITPS Out-of-plane displacement, Temperature Distribution..............................117
ITPS Local Stress ..................................................................... ......... 121
Web Angle Sensitivity................. ........................... 122
C o n c lu sio n ........................................................................................... 12 4

7 OPTIMUM DESIGN OF THE INTEGRATED THERMAL PROTECTION ................... 141

In tro d u ctio n ...................................................................................................................... 1 4 1
The ITPS Optim ization Problem .......................................................... .............. 143
R e su lts ................... ................... ..................................................... .. 1 4 5
C onclu sion ............................................................................................ 147

8 CONCLUSIONS ........................................................................ ......... 151

Su m m ary ................... ................................. 15 1
C o n c lu sio n s ...................................................................................................................... 1 5 3
R ecom m endations .................................................................................................... 155

APPENDIX

A DEFORMATION TRANSFORMATION MATRICES................................ 156

B DETAILED DERIVATION OF A55 ................................................................159

C TRANSVERSE SHEARING STIFFNESS ............................................. ........... 161

D ED GE M OM EN TS ..................................................... 162

L IST O F R E FE R E N C E S .............................................................. ..................................164

BIOGRAPHICAL SKETCH ................................................................... ............ 167









LIST OF TABLES


Table page

3-1 Periodic displacement boundary conditions .........................................................62

3-2 Non-zero [A], [B], and [D] coefficients for an ITPS panel with [0 / 90]s layup ..............62

3-3 Non-zero [A], [B], and [D] coefficients for an ITPS panel with a [45 / -45]s layup.........63

3-4 Comparison of the non-zero [A], [B], and [D] coefficients between the refined
transformation matrix and the deformation transformation matrix ..............................63

3-5 Non-zero [A], [B], and [D] coefficients for an unsymmetric ITPS sandwich panel.........64

4-1 Non-zero Thermal Forces of the unit cell. ........ .......................................................... 83

5-1 Critical buckling load of plate II, N, = 1................................................... ........ 96

5-2 Critical buckling load of plate II, Ny = 1 ................................................... ........ 96

5-3 Critical buckling load of plate II, k= 1 ............... ................................... 96

5-4 Critical buckling load of plate II, N y 1 ........ ................................................. ......... 96

6-1 Thermal moments of Inconel under the 450 s reentry temperature distribution ........... 125

7-1 Ranges of the seven design variables for the ITPS optimization problem .................. 149

7-2 Optimum designs with a maximum deflection constraint only. ................................ 149

7-3 Optimum designs with a yield stress constraint only. ............................................. 149

7-4 Preliminary ITPS optimum designs with deflection and yield constraints ................150

7-5 Preliminary ITPS optimum designs with deflection, yield, and temperature
co n strains. ...................... ............................... ....................................... 15 0

7-6 Optimal design comparison between FE and analytical method. .............................. 150









LIST OF FIGURES


Figure page

1-1 Flight regim es for hypersonic vehicles ................. ..... ......... .................... 32

1-2 Corrugated-core sandwich panels for use as an ITPS ................................. .................32

2-1 Z-core and C-core sandw ich structure ............................................................................53

2-2 Sim plified unit cell dim ensions...................................................................... 53

2-3 Equivalent orthotropic thick plate for the unit cell corrugated core sandwich panel........53

2-4 Global and local coordinates of the unit cell, faces and webs. ........... .... ............... 54

2-5 Small element removed from a body, showing the stresses acting in the x-direction....... 54

2-6 Unit cell subjected to unit transverse shear and horizontal force................. ..............54

2-7 Half unit cell of the corrugated-core sandwich panel............. ...... .................. 55

2-8 Free body diagram of the top face sheet under the action of midplane shear strain .........55

2-9 Half unit cell under the action of end couples at the faces. ........... ........................ 55

3-1 Finite elem ent unit-cell m esh .. .... ......................................................... .............. 65

3-2 Boundary conditions imposed on the plate to prevent rigid body motion...................65

3-3 Deformations of the ITPS due to periodic boundary conditions............... .............. 66

3-4 Comparison of FEM and analytical micro strain for midplane shear strain and
curvature. ........... ......... .. ....................... ................... .......... 66

3-5 Stresses in the x- and y- direction of the ITPS for a unit cell strain of M =1 ...............67

3-6 Stresses in the x- and y- direction of the ITP S for a unit cell strain of = 1. ...............67

3-7 Stresses in the x- and y- direction of the ITPS for a unit cell strain of yM = 1 .............68

3-8 Stresses in the x- and y- direction of the ITPS for a unit cell strain of Kj = 1 ...............69

3-9 Stresses in the x- and y- direction of the ITPS for a unit cell strain of KrM = 1 ............... 70

3-10 Stresses in the x- and y- direction of the ITPS for a unit cell strain of KM =1 ................71









3-11 Truss core modeled as a cantilever beam with ten unit cells...................................71

3-12 Finite element and analytical result for the transverse shearing stiffness ...................72

4-1 Heating used for preliminary thermal load and stress analysis of an ITPS panel............. 83

4-2 Core temperature distribution and thermal force resultants and thermal moments. .........83

4-3 Half-unit cell of the truss core sandwich panel with a temperature distribution ..............84

4-4 Free body diagram of the top face sheet and web. ...................................................... 84

4-5 Free body diagram of the webs subjected to a temperature distribution......................... 84

4-6 Web expansion for the constrained thermal problem....................................................85

4-7 Stress in the x- and y- directions of the ITPS for the constrained thermal problem.......... 85

4-8 Deformation of the unit cell due to the unconstrained boundary condition ........ ........ 86

4-9 Stress in the x- and y- directions of the ITPS for the unconstrained thermal problem......86

5-1 Local buckling of an ITPS panel ............ ........ ............. .. .............. 97

5-2 Infinite Strip under shear and compression loading. ................................................97

5-3 Critical buckling load flow chart for an infinite plate. ............ ..... .................97

5-4 Deformation of the quarter plate due to Nx=l ............. ........................... ..............98

5-5 Critical buckling load in the plate for Nx= ....... .......................................................... 98

5-6 Deformation of the quarter plate due to Ny= ........... ........................... ..............98

5-7 Critical buckling load in the plate for Ny = 1 ............ ............................... ..............99

5-8 Deformation of the quarter plate due to k=......... .............................................. 99

5-9 Critical buckling load in the plate for N, = Ny. ............................. ..... ......... 100

6-1 ITPS as a two dimensional orthotropic thick plate.................................... ............. 125

6-2 Half of the top face under the action of a uniform pressure loading ........................... 125

6-3 Local stress flow chart for an ITPS as a 2-D plate.............................................. 126

6-4 ITPS unit cell under the action of transverse shear force ............ ........................ 126

6-5 Local stress flow chart for an ITPS with transverse shear force effects consideration... 127









6-6 Free Body diagram of the top face under the action of a uniform pressure loading. ...... 127

6-7 Free body diagram of section BC of the top facesheet............................ .............. 128

6-8 Free body diagram of section AB of the top facesheet............................................ 128

6-9 ITPS orthotropic plate subjected to uniformly distributed thermal end moments......... 128

6-10 ITPS panel finite element model and mesh ........................................... ..............129

6-11 Finite element boundary conditions for the ITPS plate.............................................. 129

6-12 ITP S out-of-plane deform ation .......................................................................... .... 130

6-13 Out-of-plane displacement comparison between FEM and analytical solution of the
face s ................... ................... ................... ................................. .. 13 0

6-14 Finite element mesh of the 2-D plate and boundary conditions .................................. 131

6-15 Out-of-plane displacement of an isotropic plate subjected to uniform edge moments... 131

6-16 Out-of-plane displacement of an orthotropic plate subjected to uniform edge
moments at x = a/ 2......... ............................ ... ........... 132

6-17 Out-of-plane displacement of an orthotropic plate subjected to uniform edge
moments at y = b / 2......... ............................ ............... 132

6-18 Isotropic plate out-of-plane displacement contour for M = 1 and My = 1 .................... 133

6-19 Orthotropic plate out-of-plane displacement contour for M = 1 and My = 1................. 133

6-20 ITPS temperature distribution at 450 s reentry time...................................................... 133

6-21 ITPS out-of-plane thermal displacement contour. ...................................................... 134

6-22 ITPS out-of-plane displacement due to a temperature distribution for boundary
conditions. ................................................................ ......... ......... 134

6-23 ITPS center panel displacement for various L / h values .............................................. 135

6-24 ITPS out-of-plane displacement due to a temperature distribution with L / h = 18....... 135

6-25 Top facesheet stress in the x- andy- direction of the ITPS panel. ........... ............... 136

6-26 Bottom facesheet stress in the x- andy- direction of the ITPS panel............................ 136

6-27 Web stresses in the x- andy- direction of the ITPS panel. ............................................ 137

6-28 Out-of-plane displacement comparison for an unsymmetric ITPS.................. ......... 137









6-29 Top facesheet stress in the x- and y- direction of an unsymmetric ITPS ........................138

6-30 Bottom facesheet stress in the x- andy- direction of an unsymmetric ITPS................ 138

6-31 Web stresses in the x- and y- direction of and unsymmetric ITPS................. ............ 139

6-32 Behavior of the stiffness properties and deflection to a change in the web angle .......... 140

6-33 Exaggerated Deformed Mesh (Deformation Scale Factor = 2) .................................... 140









LIST OF ABBREVIATIONS

AFRSI advanced flexible reusable surface insulation

ARMOR advanced-adapted, robust, metallic, operable, reusable

CEV crew exploration vehicle

ELV expendable launch vehicle

FEA finite element analysis

FEM finite element method

FRCI fibrous refractory composite insulation

FSDT first order shear deformable plate theory

HRSI high-temperature reusable surface insulation

ISS International Space Station

ITPS integral thermal protection system

LEO low earth orbit

LRSI low-temperature reusable surface insulation

NASA National Aeronautics and Space Administration

RCC reinforced carbon-carbon

RLV reusable launch vehicle

SSTO single-stage-to-orbit

TPS thermal protection system

TPSS thermal protection support structure









LIST OF SYMBOLS

a panel length (x-direction)

a coefficient of thermal expansion (CTE)

A* inverse of the extensional stiffness matrix

A44 shearing stiffness (y-direction)

Ass shearing stiffness (x-direction)

[A] extensional stiffness matrix

[B] coupling stiffness matrix

b panel width (y-direction)

[D] bending stiffness matrix

D* inverse of the bending stiffness matrix

A temperature change

d height of the sandwich panel (centerline to centerline of the facesheets)

{D} e) deformation vector of the eth component (micro deformation)

{D y deformation vector of the unit cell (macro deformation)

e component index of the unit cell

so midplane strain

El equivalent flexural rigidity

F(m) nodal force in the FEM model

K curvature

k ratio between compressive force in the y-direction and x-direction.

I length of the cantilever beam

Mf thermal moment resultant









M, component moment resultant

No lowest compressive load before buckling.

N, component force resultant

N" thermal force resultant

2p unit cell length

Pz pressure load acting on the 2-D orthotropic panel

Qx, Qy shear force on the unit cell

Q transformed lamina stiffness matrix

R ratio between shear load and compressive load.

s web length

rx. shear stress in the web

tTF top face sheet thickness

tBF bottom face sheet thickness

tw web thickness

0 angle of web inclination

[TD ](e" deformation transformation matrix of the eth component of the ITPS

U unit cell strain energy

Vtp tip deflection of cantilever beam

w ITPS panel deflection

/x, Vy rotations of the plate's cross section

Length between the successive buckling waves in the plate

y local axis of the web









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

MICROMECHANICAL ANALYSIS AND DESIGN OF AN INTEGRATED THERMAL
PROTECTION SYSTEM FOR FUTURE SPACE VEHICLES

By

Oscar Martinez

May 2007

Chair: Bhavani Sankar
Major: Aerospace Engineering

Thermal protection systems (TPS) are the key features incorporated into a spacecraft's

design to protect it from severe aerodynamic heating during high-speed travel through planetary

atmospheres. The thermal protection system is the key technology that enables a spacecraft to be

lightweight, fully reusable, and easily maintainable. Add-on TPS concepts have been used since

the beginning of the space race. The Apollo space capsule used ablative TPS and the Space

Shuttle Orbiter TPS technology consisted of ceramic tiles and blankets. Many problems arose

from the add-on concept such as incompatibility, high maintenance costs, non-load bearing, and

not being robust and operable. To make the spacecraft's TPS more reliable, robust, and efficient,

we investigated Integral Thermal Protection System (ITPS) concept in which the load-bearing

structure and the TPS are combined into one single component.

The design of an ITPS was a challenging task, because the requirement of a load-bearing

structure and a TPS are often conflicting. Finite element (FE) analysis is often the preferred

method of choice for a structural analysis problem. However, as the structure becomes complex,

the computational time and effort for an FE analysis increases. New structural analytical tools

were developed, or available ones were modified, to perform a full structural analysis of the

ITPS. With analytical tools, the designer is capable of obtaining quick and accurate results and









has a good idea of the response of the structure without having to go to an FE analysis. A

MATLAB code was developed to analytically determine performance metrics of the ITPS such

as stresses, buckling, deflection, and other failure modes. The analytical models provide fast and

accurate results that were within 5% difference from the FEM results. The optimization

procedure usually performs 100 function evaluations for every design variable. Using the

analytical models in the optimization procedure was a time saver, because the optimization time

to reach an optimum design was reached in less than an hour, where as an FE optimization study

would take hours to reach an optimum design.

Corrugated-core structures were designed for ITPS applications with loads and boundary

conditions similar to that of a Space Shuttle-like vehicle. Temperature, buckling, deflection and

stress constraints were considered for the design and optimization process. An optimized design

was achieved with consideration of all the constraints. The ITPS design obtained from the

analytical solutions was lighter (4.38 lb / ft2) when compared to the ITPS design obtained from a

finite element analysis (4.85 lb / ft2). The ITPS boundary effects added local stresses and

compressive loads to the top facesheet that was not able to be captured by the 2-D plate

solutions. The inability to fully capture the boundary effects lead to a lighter ITPS when

compared to the FE solution. However, the ITPS can withstand substantially large mechanical

loads when compared to the previous designs. Truss-core structures were found to be unsuitable

as they could not withstand the large thermal gradients frequently encountered in ITPS

applications.









CHAPTER 1
INTRODUCTION

Throughout the past century there has been an exponential advancement in flight

technology. Since the dawn of the aerial age which began with the Wright brothers, mankind has

been pushing the envelope of flight which is to fly at faster speeds, longer distance, and higher

altitudes. In the 1950s, mankind entered the rocket propulsion era, which brought the space age

to life. With the advent of rocket propulsion, space vehicles can be accelerated to speeds in

excess of 11,000 m/s (24,606 mph) which is the minimum velocity needed to escape the earth's

gravitational pull.

In the next 20 years the National Aeronautics and Space Administration (NASA) plans to

send manned space missions to Mars. For vehicles traveling at hypersonic speeds through a

planetary atmosphere, the aerodynamic forces provide two things; unwelcome air resistance that

must be overcome by a powerful propulsion system or a welcome means of slowing down the

space vehicle. The latter is known as aerocapture. The aerocapture approach uses the planetary's

atmosphere as a welcome means of drag to alter the space vehicle's velocity and establish a

capture orbit. Propellant is not needed for the decelleration of the space vehicle, therefore a fuel-

free planetary entry method could reduce the overall mass and voyage cost of the space vehicle.

This reduction in mass allows for cheaper and smaller space vehicles for interplanetary voyages.

Aerocapture will have an impact on the space vehicles' heat shielding because of the excessive

aerdoynamic heating from the atmospheric friction due from air drag.

Excessive aerodynamic heating to the space vehicle's structure is a result of high velocity

flights through a planetary atmosphere. An aerospace vehicle traveling at high speeds through a

planetary atmosphere must push the atmospheric gas out of its way to continue with its intended

flight path. The faster the vehicle is traveling, the faster the atmospheric molecules travel along









the outer space vehicle structure. The acceleration of the molecules causes friction between the

vehicle's surface and atmospheric gas which can damage the vehicle's structure. The thermal

protection system (TPS) is the key feature that is designed into the space vehicle to protect it

from the extreme aerodynamic heating.

Various vehicles have pushed the flight envelope limit by flying at hypersonic speeds.

Space vehicles like the Space Shuttle have carried people into low earth orbit (LEO) and the

moon. Unmanned probes have penetrated alien planets in our solar system to gather scientific

information. Military and commercial planes are capable of reaching supersonic speeds for

sustained flight through the atmosphere. One of the key technologies required by all of these

high speed vehicles is a TPS to protect it from the extreme aerodynamic heating (Figure 1-1).

The major focus of the government and industries around the world are to provide cost

effective and reliable space transportation (Freeman, Talay, and Austin, 1997). The control of the

aerodynamic deceleration loads and the protection of the payload and crew from the induced

aerodynamic heating constitute the primary problems for a space vehicle designer (Stewart and

Greenshields, 1969). A number of new space vehicles are being developed to provide routine and

low cost access to space. Proposed vehicles include a reusable launch vehicle (RLV), hypersonic

air breathing vehicles (Hunt, Lockwood, Petley, and Pegg, 1997), military space planes (Blosser,

1996), crew exploration vehicle (CEV), and unmanned experiment return capsules (Christiansen

and Friesen, 1997).

Reducing the cost of launching a space vehicle is one of the critical needs of the space

industry. There is a major demand for space access from the government and industries. The

government launches satellites for weather, military, communications, scientific, and

reconnaissance purposes. The industries' thirst for space tourism is growing rapidly, and many









orbiting satellites and the International Space Station (ISS) require constant reconstruction and

servicing. If use of space is to become routine, future space vehicles must become fully reusable,

have greater operational flexibility, and have a lower operating cost than the current space

vehicles like the Space Shuttle (Bohon, Shideler, and Rummler, 1997). Reducing the cost of

delivering a pound of payload into space by an order of magnitude is one of NASA's objective at

achieving their low cost effective and reliable space access (Blosser, 2000). The space vehicle's

TPS is one of the most expensive and critical systems of the vehicle (Behrens and Muller, 2004).

The high cost of space transportation is viewed as one of the biggest obstacles to the growth of

space exploration, commercialization and research.

Expendable launch vehicles (ELV) such as Europe's Arianne, The United State's Delta,

and Russia's Soyuz provide the most economical means of delivering payload to space. The

ELV's estimated LEO payload cost per pound is averaged at $5,000 (Furtron Corporation, 2002).

However, a new launch vehicle is required for each every launch, making ELV's non-reusable

and not economical for space missions that require multiple launches. The Space Shuttle offers a

reusable space vehicle but at a very expensive cost. The estimated Space Shuttle cost of

delivering a pound of payload into LEO is about $4,000 and the price rises to about $23,000 for a

geosynchronous transfer orbit. Besides launch costs, the shuttle's external fuel tank has to be

replaced for every launch, the solid rocket boosters require extensive refurbishment, and the

maintenance time for the Space Shuttle is 40,000 man hours (Blosser, 2000). The recently

proposed Lockheed Martin VentureStarTM was aimed at proposing a low cost access to space by

reducing the payload cost per pound by an order of magnitude (Dorsey, Poteet, Wurster, and

Chen, 2004). The VenutreStarTM was going to be a single-stage-to-orbit (SSTO) RLV. The









proposed vehicle program and development was stopped due to technological and budget

constraints that could not be overcome.

Regardless of the problems with the VentureStarTM, NASA's RLV program is seeking a

replacement for the Space Shuttle. The future space vehicle technology will involve the research

and development of an all rocket, fully reusable, SSTO vehicle. Such vehicles will have a large

area to be covered with TPS, because it includes the fuel tanks required for launch. As a result of

the large TPS coverage area, a need for a lightweight TPS is necessary to keep the vehicle weight

and launch cost reasonable and affordable. The significant reduction of the payload

transportation costs is the rational in the development of a future space vehicle. The emphasis of

the vehicle would be lightweight, fully reusable, and easily maintainable. A key factor in the

reusability, life, and operational flexibility of such classes of future space vehicles is an efficient

and advanced TPS. Future space vehicles will require an efficient TPS that provide the vehicle

with the necessary protection from aerodynamic heating as well as lower the operational cost,

maintenance cost, and maintenance time.

Thermal Protection System

Function of a Thermal Protection System

The primary function of a TPS is to protect the space vehicle from extreme aerodynamic

heating and to maintain the underlying structure within acceptable temperature and mechanical

constraints (Zhu, 2004). The vehicle's structure acts as the thermal storage reservoir for the heat

that passes through the TPS. To protect the vehicle structure and payload the heat that passes

through the TPS must be minimized. The amount of heat that passes through the TPS is

dependant on many material properties such as thermal conductivity, mass of the TPS, specific

heat, and geometric parameters. Temperature limits are imposed on the space vehicle structure to

avoid material property degradation, excessive deformation, and critical thermal stresses. The









TPS is exposed to a variety of environmental conditions because it forms the external surface of

the space vehicle. Therefore, a set of requirements are needed for a TPS.

General Requirements of a Thermal Protection System

The TPS generally covers the entire exterior surface of the space vehicle and therefore it

defines the space vehicle's exterior shape. The TPS is subjected to a wide variety of

environments corresponding to all phases of flight (Dorsey et al., 2004). A set of general

requirements for a TPS are listed below.

* High and Low velocity impact: When the vehicle is being fabricated, maintained,
assembled, transported, or just waiting to be launched, the TPS may be exposed to
handling damage from tools such as, an accidental dropping of tools. During launch and
landing the TPS must with-stand runway debris that is stirred up by the rocket exhaust or
winds. While in flight to space the TPS may experience bird strikes, hail, space vehicle
debris, and dust impacts that can cause catastrophic failure to the TPS as was evident from
the Space Shuttle Columbia disaster. When orbiting in space the TPS may encounter
hypervelocity impacts from micrometeorites or space debris. A TPS should withstand all
possible types of low and high velocity impact to prevent catastrophic TPS failure.

* Panel Deflection: The excessive aerodynamic heating causes extreme temperatures in the
TPS. The extreme temperatures result in thermal loads and moments which cause the panel
to deflect out of plane. During the portion of flight through the atmosphere the TPS will
encounter dynamic, acoustic, aerodynamic pressure, and shear loads that cause the panel to
deflect out of plane and vibrate. The deflection of the panel must be kept within acceptable
limits to prevent extreme local aerodynamic heating and maintain a smooth aerodynamic
profile of the vehicle. The vibration must be kept below its natural frequency to prevent
dynamic failure such as flutter.

* Chemical Deterioration: During maintenance, or fabrication, the TPS panel may be
exposed to unfriendly chemical agents that may alter the chemical composition or
deteriorate the material properties of the TPS panel. The TPS is exposed to rain, snow or
ice that can lead to possible rust or oxidation of the panels and degrade the structure. While
in space there are various chemical effects that must be considered for as well.

* Mechanical and Thermal Loads: During launch the TPS must withstand severe
acoustical and dynamic loading that is initiated from the rocket propulsion system. These
loads vary widely depending on its location on the vehicle. The TPS panels near the
exhaust of the vehicle will experience high thermal loads and acoustic loads. During flight
the TPS will be exposed to aerodynamic pressure, aerodynamic shear, and inertial loads.
The TPS must withstand all these loads to prevent failure and fracture.









There are also several desirable features of a TPS such as being lightweight. Low mass is

desirable on any item carried by a vehicle at any speed. As mass is added to the vehicle, more

energy is required to accelerate the vehicle. Thus the excess mass results in more fuel carried by

the vehicle or a decrease in payload (Blosser, 2000). The TPS weight is desirable for an SSTO

space vehicle because of its large surface area. Therefore, the TPS should be designed and

optimized for minimum mass to achieve the lost-cost space access goal. Finally, low cost,

robustness, and operability are other desirable features of a TPS. The consideration of all these

desirable features leads to a low life-cycle cost. The life-cycle cost of a TPS includes fabrication,

installation, and maintenance in acceptable operating conditions (Blosser, 2000). Cost is a

significant driver of any vehicle let along a space vehicle which can lead to launch costs in the

billions. Cost can be reduced by lowering initial fabrication cost, installation costs, required

maintenance time, and turnaround time between flights. A robust TPS is one that is not easily

damaged by its environment and can tolerate damage without requiring immediate repair and/or

replacement. An operable TPS is one that can be easily maintained, inspected, and replaced

should there be damage. Cost can be reduced by developing a TPS that requires minor

inspection, maintenance, repair between flights, and enable a widening operational envelope of

the space vehicle to maximize the time spent in space (Blosser et al., 2004). In summary, the

thermal protection system must have sufficient durability and reusability in order to withstand

repeated exposure to the adverse environments of space, launch, and reentry, as well as the abuse

due to normal ground maintenance and inspection.

Approach to Thermal Protection Systems

The approach to a TPS design depends on the magnitude and duration of the aerodynamic

heating. There are three approaches to a TPS; active, semi-passive, and passive (Blosser, 2000).

An active TPS contains external systems that provide coolant to the TPS during flight to









continually remove heat from the system and keep the temperature within acceptable limits.

Active TPS concepts are transpiration cooling, film cooling, or convective cooling. These

concepts use pumps, coolant, pipes, and coolant storage tanks. The added parts to the vehicle

increase the vehicle mass and only make it a viable option for small, highly heated areas of a

space vehicle. Semi-passive TPS have a working fluid that removes heat from the point of

application. They require no external system to provide the coolant. Semi-passive concepts are

ablation (Stewart and GreenShields, 1969) and heat pipes. Heat pipe concepts transfer the heat

through the pipes from a high heat transfer area to a low heat transfer area creating a smooth heat

transfer surface, therefore reducing any concentrated heat spikes on the space vehicle's TPS.

Ablative heat shields are the simplest and least inexpensive type of TPS. Ablation dissipates heat

from the plasma by allowing its outer layers to char, melt, and to some extent vaporize. Ablative

heat shields are heavier than conventional TPS, which would make it an uneconomic option for

space vehicles with a large surface area like the Space Shuttle or RLV. The simplest TPS

approach is a passive TPS. A passive TPS either radiates heat from the surface or absorbs it into

the structure. The advantages of using a passive TPS are that they are simple concepts and have

the highest reliability. There are three passive TPS concepts; heat sink, hot structure, and

insulated structure. A heat sink absorbs all the incident heat and stores it in the structure. The

amount of heat storage capability depends on the mass, specific heat capacity, and service

temperature of the material. A hot structure allows the structure's temperature to rise until the

heat being radiated from the surface is equal to the incident heating. This concept is limited by

the allowable temperature limit of the material. An insulated structure is both a heat sink and a

hot structure. The outer surface of an insulated structure keeps it near the radiation equilibrium

temperature. The radiation equilibrium temperature is the surface temperature resulting from the









given absorbed heat flux and surface emittance. Most of the incident heat is radiated out of the

structure and only a small fraction of incident heat is stored in the underlying structure.

History of Thermal Protection Systems

High speed flight and space travel has made significant advancements in the last half

century. Starting with the artificial satellite (Sputnik) in 1957 and then progressing rapidly to

manned space flight programs such as the Mercury, Gemini, and Apollo. All of these manned

spacecraft missions used ballistic blunt-body reentry vehicles and ablative heat shields to

dissipate the extreme aerodynamic heating. The cone-shaped capsules had heat shields attached

to its base. The materials used in the heat shield would carry excess heat away from the

spacecraft and its crew through vaporization. The Mercury and Gemini capsules of the early

1960's were protected by an all ablative heat shield made of silica-fiber resin, while the later

Apollo capsules had heat shields made of phenolic epoxy resin, a form of plastic. The low-

density ablation material (AVCOAT 5026-39/HC-GP) is bonded to its primary structure. The

ablation material chars at sufficiently high-heating rates. Apollo heat shields were nearly 7 cm

(2.7 in) thick and weighed 1,360 kg (3,000 lb).

During the 1960s, the Space Shuttle was created based on a need for a logistical spacecraft

to support orbital and space stations such as Mir and the ISS. The Space Shuttle's TPS design is

required to keep the structural temperatures less than 350F (449.81 K). The Space Shuttle uses a

passive TPS consisting of several materials selected for stability at high temperatures and weight

efficiency. These materials are as follows (Thornton, 1992):

* Reinforced Carbon-Carbon (RCC): RCC is used for the nose cap and wing leading edges
where temperatures are above 2300F (1533.15 K).

* High-temperature reusable surface insulation tiles (HRSI): HRSI are used in areas on the
upper forward fuselage, the entire underside of the fuselage where RCC is not used, and
some leading edges of the orbiter where temperatures are below 2300F (1533.15 K).
These tiles have black surface coatings necessary for entry emittance.









* Low-temperature reusable surface insulation tiles (LRSI): LRSI consists of 8 in2 (1.24
cm2) silica tiles and covers the top of the vehicle where temperatures are less than 1200F
(922 K).

* Fibrous refractory composite insulation tiles (FRCI): FRCI are high in strength and were
used to replace HRSI in some areas. The FRCI tiles have a density of 12 lb/ft3
(192.22 kg/m3) and provide improved strength, durability, resistance to coating cracking
and weight reduction.

* Advanced flexible reusable surface insulation blankets (AFRSI): AFRSI blankets replaced
the vast majority of LRSI tiles. The direct application of the blankets to the orbiter results
in weight reduction, improved durability, reduced fabrication and installation cost, and
reduced installation schedule time.

The greatest disadvantage of the tiles is that these materials make the space vehicle's

exterior brittle, susceptible to damage from small impact loads, high in maintenance, and

consequently its reusability may not be sufficient for advanced mission requirements. After each

flight the entire orbiter is waterproofed and each tile is inspected manually for any cracks and/or

failures. The tiles coefficient of thermal expansion (CTE) is less than the Space Shuttle's

structure CTE which is aluminum. As a result of that incompatibility, direct mounting of the TPS

to the structure was not possible. All these factors contributed to the Space Shuttle's launch cost

and maintenance.

The X-33 Venture Star was aimed at proposing a low cost access to space. One of the most

important goals of the X-33 was to eliminate the numerous problems relating to the Space

Shuttle's TPS. Metallic TPS was considered as the primary heat shield for the RLV. Metallic

TPS was considered as a much-needed alternative to the ceramic-based brittle tile and thermal-

blanket currently used on the Space Shuttle. Metallic TPS offers significant advantages such as;

(1) high temperature adhesive, sealants, or water proofing is not required, (2) metal ductility

promotes robustness, and good damage tolerance properties, and (3) weight savings when used

as part of an integrated aeroshell structural system (Harris, Shuart, and Gray, 2002).









Recently, Blosser (2004) designed an advanced-adapted, robust, metallic, operable,

reusable (ARMOR) TPS concept. The ARMOR TPS was designed to be attached to the

cryogenic tank structure of an RLV. The outer surface of the ARMOR TPS consists of a foil

gauge, Inconel 617 metallic honeycomb sandwich panel that is exposed to ascent, reentry

aerodynamic heating profiles, and acoustic and aerodynamic pressure. The thin titanium box

frame defines the edges of the panel's inner surface. The bottom panel is insulated from the top

by Saifill' insulation, which is made from alumina fibers. The ARMOR TPS is a combination of

a hot structure and an insulated structure. The ARMOR TPS is fully metallic which gives it good

impact resistance features. The ARMOR concept can also eliminate radiation in panel-to-panel

gaps, provide subsurface sealing and attachments, and decouple deformation and thermal

expansion between the inner and outer faces. The structural load is taken by the underlying foil-

gauge titanium TPS support structure (TPSS). However, the ARMOR TPS load bearing

capabilities are limited, and large in-plane loads cannot be accommodated under this design.

Selection of the optimum TPS for a particular space vehicle is a challenging task that

requires considerations in weight, operability, reusability, maintenance, durability, initial cost,

life-cycle cost, and compatibility. The common feature between the Apollo, Space Shuttle, and

X-33 VentureStarTM TPS concepts was that their TPS was an add-on to the vehicle's outer

structure. The add-on feature created incompatibility problems between the thermal structure and

the load-bearing structure of the vehicle as well as an increase in maintenance. The relatively

weak bonding of the Space Shuttle tiles to the vehicle can expose it to catastrophic failures like

tile loosening or tile detachment as was evident in the Columbia Space Shuttle disaster. The

Apollo heat shield was heavy and it was not practical for large surface areas space vehicles.

These various TPS concepts were also not load bearing members; they were purely thermal









structures. Fasteners, frames, and support brackets contributed to the overall weight of the TPS

and the TPS as an add-on feature added mass to the total vehicle weight.

Integral Thermal Protection System

There is a need for a new and efficient TPS concept. The new TPS concept will have to be

lightweight, robust, and operable. This concept can be accomplished by using recently developed

metallic foams and also innovative core materials, for example, corrugated cores and truss cores

(Figure 1-2). The Integral TPS/structure (ITPS) design can significantly reduce the overall

weight of the vehicle as the TPS/structure performs the load-bearing function. The ITPS is

expected to be multifunctional (offer insulation as well as load bearing capability in order to

reduce the mass of the vehicle). Advantages of an ITPS concept are as follows:

* ITPS panels can reduce the overall weight of the vehicle: The faces of an ITPS are made
from thin metallic plates which result in lightweight components and good impact
resistance.

* Greater flexibility for designer to create elegant curves: This is ideal for space vehicles
with conical or blunt bodies such as the CEV.

* Panels can be large in size thus eliminating the total number of panels to be used on a
space vehicle.

* Performs load bearing functions, making it multifunctional and robust.

* TPS will be integrated with the vehicles structure, which promotes low installation and
maintenance costs.

The sandwich structure will replace the structural skin and the insulation in the current

thermal structures. Since the sandwich construction is stiffer than single skin construction, the

number of frames and stringers will be significantly reduced. Furthermore, the insulation is

protected from foreign object impact and requires less or no maintenance such as water proofing.

An ITPS is a sandwich panel composed of two thin faces separated by a corrugated core

structure which can be of homogeneous materials such as metals or orthotropic materials such as









composite laminates. The sandwich panel is composed of several unit cells placed adjacent to

each other. The empty space in the corrugated core will be filled with a non load-bearing

insulation such as S fill'. The ITPS concept combines all three passive TPS concepts: heat sink,

hot structure and insulated structure. The top face sheet acts as the hot structure and radiates out

most of the heat. The insulation only allows a fraction of the incident heat to flow into the

underlying vehicle structure, and the whole ITPS panel acts as a heat sink. The corrugated-core

feature provides the load bearing characteristic of the multifunctional structure.

Combining the thermal and structural requirements of a structure is a challenging task for a

designer. Requirements of a load-bearing structure and a TPS are often conflicting. A TPS

requires the structure to have low conductivity, low density, and high service temperature.

Materials that meet these requirements are ceramics, thermal blankets and insulation materials.

Ceramics and insulation materials have poor strength, low fracture toughness, and poor impact

resistance. A load-bearing member requires the structures to have high strength in compression

or tension, high fracture toughness, and good impact resistance. Such materials that satisfy the

load bearing conditions are metals and alloys. A load bearing member possess high thermal

conductivity, high density, and low service temperatures. Integrating the thermal requirements

with the structural requirements is a great challenge to any designer. The coupling of the thermal

requirements with the load bearing requirements makes the vehicle design process complicated,

and it makes the choice of material for an ITPS challenging.

Integral Thermal Protection System Material Selection

The material selection for the ITPS depends largely on its multifunctional ability, location

on the space vehicle, and heating profile. Metals are the best material of choice for the top face









sheet of an ITPS rather than ceramic materials. The advantages of using metals for the top face

sheet over ceramic materials are listed:

* Metals have a higher ductility over ceramics and therefore tend to be more damage
resistant.

* Small gage thickness. Many metals can be made into foils of 0.001 in (0.0254 mm)
thickness.

* Many fabrication techniques are readily available such as welding, extrusion, brazing, and
machining.

* Resistance to oxidation.

The top face sheet panel of the ITPS is required to withstand extreme reentry temperatures,

have good impact resistance, high service temperatures, and high strength. Inconel 718 wrought

nickel-chromium alloy is a possible material that meets all the top face sheet standards. Inconel

718 wrought nickel-chromium alloy was used for the design of the ARMOR TPS (Poteet, Abu-

Khajeel, and Hsu, 2004). The service temperatures of Inconel 718 wrought nickel chromium is

1255 K (982 C). The bottom face sheet is expected to be a heat sink for the ITPS; therefore a

material that has a high heat capacity is needed for the bottom face sheet. The bottom face sheet

will also experience a major portion of the in-plane stresses because of the attachment

mechanisms of stringers and frames to the space vehicle. Therefore, a high Young's Modulus

material with a high heat capacity is suitable for the bottom face sheet. Possible material choices

for the bottom face sheet are titanium and aluminum alloys, beryllium, and carbon epoxy fiber

composite. The web acts as the heat conduction path from the top face sheet to the bottom face

sheet. It also acts as a load bearing member by supporting most of the transverse shearing loads.

To decrease the amount of heat that is conducted to the bottom face sheet, the web must either

have a small thickness or low heat conduction. Possible material candidates for the web are

titanium alloys because of their high stiffness and high service temperature.









Purpose

The purpose of this dissertation was to investigate the use of an ITPS panel on a future

space vehicle.

Objectives

The objectives of this study are listed as follows:

1. Identify the key failure mechanisms in the ITPS sandwich construction.

2. Develop new methods or modify available methods to perform thermo-mechanical analysis
of a full ITPS sandwich panel to estimate panel deflections and stresses.

3. Investigate the effects of various thermal and mechanical boundary conditions on the stresses
and deflections.

4. Investigate possible buckling and failure modes.

5. Design an optimization study of the advantages of the ITPS designs.

The analytical models will be compared with finite element results. The analytical models

will be refined so that the errors in prediction of critical metrics, (critical stresses) are within 5%.











3000T-


Y 2000-

3


E
- 1000-


Apollo


> lAblators

Mercury


Insulation

Space Shuttle


X-30


- Heat Sink

x-15 Hot Structure

YF-12


Exposure Time (h)


Figure 1-1. Flight regimes for hypersonic vehicles


Figure 1-2. Corrugated-core sandwich panels for use as an ITPS


I I









CHAPTER 2
MICROMECHANICAL ANALYSIS

Sandwich Structures

A sandwich panel is a three-layer element composed of two thin flat faces separated by a

thick, lighter, and flexible core. The thin flat faces are high in stiffness when compared to the

low average stiffness of the thick core. Sandwich constructions are frequently used because of

their high bending stiffness-to-weight ratio. The high bending stiffness is the result of the

distance of the facesheets from the neutral axis. The face sheets support the major portion of the

in-plane loads. Commonly used materials for the facesheets are composite laminates and metals,

while cores are made of metallic and non-metallic honeycombs, cellular foams, functionally

graded foams (Lee, 2006), balsa wood, trusses or corrugated core. The core helps stabilize the

facesheets and support the shear loads through the thickness. The corrugated core keeps the

facesheets apart and stabilizes them by resisting vertical deformations, transverse shear strains,

curvature in the longitudinal direction, and enables the structure to acts as a single thick plate.

Unlike soft honeycomb core, a corrugated core resists bending and twisting in addition to

vertical shear (Chang, Venstel, Krauthammer, and John, 2005). All these characteristics make

corrugated sandwich structures ideal for aviation, aerospace, civil engineering, and marine

applications, where weight and stiffness are important design drivers.

Literature Review on Corrugated Core Sandwich Panels

Honeycomb sandwich constructions are the earliest forms of sandwich cores. A number of

theoretical and experimental investigations on innovative sandwich panels have been published

in the literature. Libove and Hubka (1951) determined the elastic constants of corrugated core

sandwich plates through the force-distortion relationship. The authors homogenized the









sandwich panel and compared the behavior of the equivalent homogenized thick plate with that

of the corrugated sandwich panel.

Nordstrand (2004) determined the buckling coefficient for edge-loaded corrugated

orthotropic plates (including transverse shear) by including additional moments in the governing

moment equilibrium equation of the panel. The corrected analysis expression for buckling

coefficient was shown to reduce the classical formulation of an orthotropic plate without shear

deformation when the transverse shear stiffness became large. The critical buckling load

obtained from the corrected analysis was compared with predicted loads obtained from finite

element and experimental panel compression tests. The author found that the explicit equation

presented for the buckling load was accurate, (less than 0.5% deviation when compared with

finite element results). However the deviation was larger between theoretical and experimental

results, possibly due to non-linear material behavior.

Chang et al. (2004) did a comprehensive analysis of the linear elastic behavior of a

corrugated-core sandwich plate using the Mindlin-Reissner plate theory (Whitney, 1987). The

authors reduced the three-dimensional sandwich panel into an equivalent two-dimensional

structurally orthotropic thick plate continuum and used previously determined stiffness values

for their analysis. The authors investigated the plate behavior of a corrugated core sandwich

structure subjected to a uniform pressure load. Their findings indicated that lower ratios of core

thickness to web thickness, (or width to core thickness) made the plate stronger. The researchers

also found that rectangular corrugations provided better bending resistance but larger deflections

due to low shear stiffness. Local stress analysis of the corrugated core sandwich structure panel

was not done.









Lok, Cheng, and Heng (2000) derived analytical equations to predict the elastic stiffness

properties and behavior of truss-core sandwich panels through the force-distortion relationship.

The authors used the homogenous thick plate approach to represent the three-dimensional

structure into a two-dimensional thick plate. The findings included a closed-form solution to

determine maximum plate deflection of the continuum. Calculated results were in good

agreement with numerical 3-D finite-element results. The equivalent elastic constants revealed

that the shear stiffness in the y-direction was important, and that panels with vertical web angle

of corrugations possess weak shear stiffness. The authors concluded that for panels with

triangular corrugations, shear deformation can be ignored. The researchers did not do a local

stress analysis of the sandwich structure and only used stiffness equations for isotropic materials.

Fung, Tan, and Lok (1994) determined the elastic constants for Z-core and C-core

sandwich panels (Figure 2-1) by using the homogenous continuous approach and force-distortion

relationship. Excellent agreement was obtained between the analytical two-dimensional thick

plate model and finite element analysis.

Valdevit, Hutchinson, and Evans (2004) structurally optimized sandwich panels with

prismatic cores. The authors identified all failure mechanisms (face yielding, face buckling, core

yielding, and core buckling) of the prismatic core and analytical expressions for the critical loads

were derived. The authors' goal was to find geometric parameters that minimize weight per unit

width subject to a combination of moment and shear forces as a function of the load index. Their

findings indicated that the corrugated core panel performs best when loaded longitudinally

because in this orientation, the performance is limited by plate buckling, rather than beam

buckling.









Tian and Lu (2005) investigated the optimal design of compression corrugated panels. The

authors found that the minimum weight of a corrugated panel subjected to uniform axial

compressive load was calculated by using a sequential quadratic programming optimization

algorithm. The authors used analytical formulas to determine the constraints of the corrugated

panel. Finite element analysis was not used as a verification tool of their findings. The authors

concluded that from a weight standpoint, panels with hat-stiffeners are found to be most efficient

for a given boundary condition and that the square web configuration was the least efficient

sandwich panel.

Carlsson, Nordstrand, and Westerland (2001) reviewed previous analytical approaches to

the analysis of the elastic-stiffness of a corrugated-core sandwich panel into the first-order shear

deformation laminated plate theory. The authors found that the bending, twisting, in-plane

extensional, and shear stiffness were dominated by the extensional and shear stiffness of the face

sheets. Their predictions agreed with experimental measured data. The authors also found that

the block shear test constrained deformation of the face sheets which led to un-conservative

overestimation of the effective shear modulus.

Analytical Approach

Micromechanical analysis of a unit cell was performed to determine the structure's

extensional, bending, coupling shear stiffness, stresses, and unit cell behavior. Micro-scale

stresses are the local facesheet and web stresses of the ITPS. The micro-scale stresses within the

unit cell were computed using the micromechanical analysis. The relationship between the unit

cell macrostress and macrostrains provided the constitutive relations for the material. Thus,

constitutive characterization matrices [A], [B], [D] were found directly from micromechanics.

The stresses were also used to predict the failure of the corrugated core. A detailed formulation

and description of the extensional, coupling, bending, and shearing stiffness of the ITPS panel









were presented for a unit cell by representing the sandwich panel as an equivalent thick plate

which was homogeneous, continuous, and orthotropic. A strain energy approach and a

deformation transformation matrix were used in deriving the analytical equations of the

extensional, bending, coupling and shearing stiffness. Previous researchers adopted the force-

distortion relationship approach to determine the equivalent stiffness parameters (Fung, Tan, and

Lok, 1993; Fung, Tan, and Lok, 1994; Libove and Hubka, 1951; Lok, Cheng, and Heng, 1999).

However, the force-distortion relationship approach can become complicated and tedious if the

ITPS was composed of faces and webs with different materials and thickness. This problem can

be solved with the proposed strain energy approach and deformation transformation matrix. The

stiffness results can be used in the First Order Shear Deformable Plate Theory (FSDT) to

determine their response on an ITPS plate when subjected to mechanical and thermal loads. The

analytical models were compared with detailed finite element analysis for verification.

The proposed strain energy method approach used in the research for predicting stiffness

properties of a corrugated core was not adopted by the previous researchers that were mentioned.

The previous researchers adopted the force-distortion relationship which involved mechanics of

materials equations that do not provide the designer with in depth local stress results of either

component of the corrugated-core sandwich structure. The mechanics of materials equations

were only average stress equations of the facesheets and webs. hThe stiffness results obtained

from the strain energy were also able to predict results for isotropic and orthotropic materials

whereas the force-distortion method used by the mentioned researchers only predicted stiffness

results for isotropic materials. The only comparison between the force-distortion method and the

strain energy method is in the final prediction of the transverse shearing stiffness. The strain

energy method is capable of handling laminated composites and uses Castigliano's second









theorem rather than the unit load method but in the final step the force-distortion relationship

was adopted in the strain energy method. Improvements were made in the transverse shearing

results when compared to Lok et al.'s (1999) transverse shearing stiffness results and FEM

results. Furthermore, the strain energy method allows the designer of a corrugated core sandwich

structure to choose any kind of material for both the facesheet and web and still be able to obtain

an accurate stiffness matrix with the inclusion of the coupling stiffness matrix. The force-

distortion method becomes tedious and complex when the corrugated core structure is un-

symmetric.

One of the most significant developments in engineering over the last four decades was the

introduction of the finite element method (FEM). With FEM, virtually any complex structure can

be modeled with a high degree of accuracy. The capability to model complex structures with a

high degree of accuracy requires an increase in cost and time. The greater the accuracy, the

greater the computation time, and consequently the greater the cost will be. The finite element

method is commonly used to analyze sandwich structures. Shell elements are often preferred for

the faces and webs to construct a detailed three dimensional FEM model. However, the number

of elements and nodes needed to appropriately mesh the sandwich panel can be excessive; as a

result a 3-D FEM model is not economical for a quick preliminary analysis of an ITPS. Such

panels may also be represented as a thick plate that is continuous, orthotropic, and homogenous

for which analytical and 2-D FEM solutions (Tan, Fung, and Lok, 2003) are available.

The extensional stiffness matrix [A], coupling stiffness matrix [B], bending stiffness [D]

and the transverse shear stiffness terms A44 and Ass were calculated by analyzing the unit cell.

For bending analysis of the plate, a closed-form solution was obtained by using the FSDT

method. Thus, advanced knowledge of the orthotropic thick plate stiffness was essential for









successful implementation of the FSDT for plate analysis. Typically, plate analyses yield

information on deflections, force, and moment resultants at any point on the plate. The

micromechanical analysis procedures developed in this study were used to determine the stresses

in the face sheets and the webs. Then failure theories such as the Tsai-Hill criterion were used to

determine if the stresses were acceptable or not (Gibson, 1994).

In the derivation of the stiffness parameters the following assumptions were made:

* Assumption 1: The deformation of the panel was less than 5% when compared to the
panel thickness.

* Assumption 2: The panel dimensions in the y-direction were much three to six larger than
the unit cell width 2p depending on the number of unit cells.

* Assumption 3: The face sheets were thin with respect to the core thickness.

* Assumption 4: The core contributes to bending stiffness about the x-axis but not about the
y-axis.

* Assumption 5: The face and web laminates were symmetric with respect to their own mid-
planes.

* Assumption 6: The core was sufficiently stiff so that the elastic modulus in the z-direction
is assumed to be infinite for the equivalent plate. Local buckling of the facesheets does not
occur and the overall thickness of the panel was constant.

Previous researchers adopted these assumptions in the derivation of stiffness parameters of

sandwich panels with corrugated core (Libove and Hubka, 1951), C-core, (Fung et al., 1993),

and Z-core (Fung et al., 1994). The in-plane and out-of plane stiffness governing the elastic

response of a shear-deformable sandwich panel were defined in the context of laminated plate

theory incorporating FSDT described by Vinson (1999) and Whitney (1987). The appropriate

stiffness of the orthotropic plate may be obtained by comparing the behavior of a unit cell of the

corrugated core sandwich panel with that of an element of the idealized homogeneous

orthotropic plate (Figure 2-3). The in-plane extensional and bending response, and out-of-plane

(transverse) shear response of an orthotropic panel were governed by the constitutive relation









(Equation 2-1) where e and y were the normal and shear strains, K was the bending and twisting

curvatures, [A], [C], and [D] were the extensional, shear, and bending stiffness of the ITPS.

N [A] ~ ,
Q = [C] \
LJ!= [C] [D K 1 (2-1)

or {F= [K]{D}

Geometric Parameters

The corrugated sandwich panel for use as an ITPS was simplified to contain the least

number of design parameters. The simplified geometry provided a useful preliminary design that

can be improved upon with more design variables. The reduced design variables simplified the

micromechanical procedure. Consider a simplified geometry of the corrugated core unit cell in

Figure 2-2. The z-axis is in the thickness direction of the ITPS panel. The stiffer longitudinal

direction is parallel to the x-axis, and the y-axis is in the transverse direction. The unit cell

consists of two inclined webs and two thin face sheets. The unit cell is symmetric with respect to

the yz-plane. The upper face plate thickness (-TF) and material can be different from the lower

plate (tTB) as well as the web (tw). The unit cell can be identified by six geometric

parameters(p, d, tF, tBF, t, 0).

All of the geometric parameters had a direct or indirect effect on the thermal and

mechanical response of the ITPS. If the length of the unit cell (p) is increased then there would

be fewer unit cells in the panel and the stiffness of the panel decreased. However, the mass per

unit area of the unit cell decreased because of decreased number of unit cells which would make

the ITPS lightweight. The height between the top and bottom facesheets (d) dictates the thermal

response of the unit cell. If the height was large then the heat conduction path was increased

which decreased the maximum bottom face sheet temperature. The thickness of the webs and









facesheets dictated the thermal and structural response of the unit cell. A thick face or web

reduced the bottom face sheet temperature of the unit cell but increased the mass of the ITPS

panel. The web angle contributed to the thermal and structural response of the unit cell. A

rectangular web angle configuration would result in the stiffest unit cell in the x-direction, but in

contrast would result in large panel deflections due to shear effects from the web. Futhermore, a

rectangular corrugated core would also shorten the heat conduction path and increase the bottom

face temperature. A triangular web angle configuration would result in infinite shear stiffness,

minimum panel deflection, and an increase in the effective thermal conductivity of the web

would decrease the bottom face temperature. However, a triangular web configuration created

major buckling problems because of the long unsupported web lengths. Four other

dimensions (be, do, s, f) were obtained from geometric considerations (Equation 2-2). The

ratio f/p = 0, corresponds to a triangular corrugated core, and f/p = 0.5, corresponds to a

rectangular corrugated core.

1 1
d = d-- t-TF tBF
2 2 (2-2a)

1 d

2 tan 0 (2-2b)

bc =p-2f (2-2c)


s = dj +bf =
sin0 cos0 (2-2d)

Extensional and Bending Stiffness

An analytical method was developed to calculate the stiffness matrix of the corrugated core

sandwich panel. Consider a unit cell made up of four composite laminates (two facesheets and

two webs). Each laminate has its respective material properties, and ABD matrix. The ABD









matrix of each component was combined together in an appropriate manner to create the overall

stiffness of the sandwich panel. The formulas for determining the ABD matrix of a composite

laminate are given below (Gibson, 1994).


ABD (e) z2]dz (Q)k-Zr k kkl (Zi zk-1k (2-3)
k=1 2 3


In Equation 2-3, Nis the number of laminas in the composite laminae and Q," are the

components of the transformed lamina stiffness matrix. The range of e is e = 1-4, (1 top face

sheet, 2 = bottom face sheet, 3 = left web, 4 = right web). The overall stiffness of the unit cell

was determined by imposing unit mid-plane strains and curvature (macro deformation) to the

unit cell and then calculating the corresponding mid-plane strains and curvatures (micro

deformations) in each component. The unit cell components were the two face sheets and two

webs. A transformation matrix related the macro- and micro-deformations. The micro-

deformation was defined as the local midplane strain and curvature of the facesheets and the

webs under the action of a unit cell midplane strain and curvature.

{De) = [TD ](e)(D M (2-4)

In Equation 2-4, {D e) was the micro deformation in each component, {(D was the macro

deformation of the unit cell, and T"e) was the deformation transformation matrix that related

macro deformation to micro deformations.

Formulation of deformation transformation matrix for the facesheets

The deformation transformation matrix of the top face sheet was determined by first

considering the unit cell under the action of mid-plane macro strains, cx, o, and macro

curvature Kx, Ky, ry. Each strain and curvature was considered by itself and the resulting










midplane strains and curvatures in the face sheets also called micro strains and curvatures -

were derived (Equations 2-5 and 2-6).

Top face sheet:


{D 100d0{D





S 000010-
yo 0 0 1 0 0 d 7syo (2-5)
xK 0 0 0 1 0 0 KC
Ky 0 0 0 0 1 0 IC

KY 0 0 0 0 0 1 Ky

Bottom face sheet:



S(1) r i c (At
G 1 0 u d 0 0 co
mc 0 1 0 0 0
YO 2 YO
S1 d Yy (2-6)
xK 0 0 0 1 0 0 KC
Ky 0 0 0 0 1 0 Ky

Ky 0 0 0 0 0 1 Ky

There was a one-to-one relationship between mid-plane macro and micro strain as well as a

one to one relationship between macro and micro curvature as indicated by unity along the

diagonal of the transformation matrices. Using the assumptions that the in-plane displacements u

and v were linear functions of the z-coordinate and that the transverse normal strain Ez was

negligible (Gibson, 1994) the d/2 factor was used to relate the macro curvatures to the mid-plane

micro strains.









Formulation of the web deformation transformation matrix

Right Web

Formulation of the deformation transformation matrix for the webs was relatively

complicated because of the need for a coordinate transformation due to the inclination of the

webs. Consider a global xyz coordinate system and a local xyz coordinate system (Figure 2-4).

The origin of the web's local axis is at the top face sheet and web junction point. The

transformation from the global to local coordinate axes requires a rotation and translation. The

transformation from the global to local displacements only requires a rotation (Appendix A). In

Equation A-i, 0 was the angle of web inclination of the right web, the first matrix was the

rotation matrix and the second vector was a translation vector.

Consider the unit cell of the ITPS panel under the action of mid-plane macro

strains, exo, o, y o and macro curvature Kx, Ky, Kxy. From Assumption 4 given in the Analytical

Approach section of Chapter 2, it can be noted that (,4) = 0 and e 34) = 1 when the unit cell

was subjected to eM = 1 and e = 1. Derivation of the micro strains on the webs due to a macro

curvature was more complex to determine; therefore a detailed discussion was appropriate

(Appendix A). The micro strains and curvature in the right web due to a macro unit curvature

along the y-axis (Ky) was derived; all other curvatures were set equal to zero.

a2w a2w 2w
K= = O K ic =1= 2 = -2 = 0 (2-7)
X X2 Y y2 Y axay

Starting with Equation 2-7 and following the detailed derivation in Appendix A leads to

the transformation matrix of the left and right web (Equations A-6 and A-7).









Stiffness matrix determination using the strain energy approach

As the unit cell was deformed by the unit macro strains and curvatures, it stored energy

internally throughout its volume. The total strain energy in the unit cell was the sum of all strain

energies in the individual components (faces and webs). Since the deformations of the webs were

a function ofy integration of Equation 2-9 was done with respect toy. The integration limits

were from 0 to s, s being the length of the webs. By substituting Equation 2-4 into Equation 2-8

the strain energy of the web in terms of macro deformations from the unit cell was represented.

The strain energy in each laminate in terms of the global deformation was written as {D}M. The

stiffness matrix K of the idealized orthotropic panel was derived as the sum of individual

stiffness contributions from each component.

U 1 (2p)2 ({D}M[K]{D} ) U(e) (2-8)
2 e=l


U(e) =(2p) (TD K- e(T D (2-9)
0

K(e) 1 jL( T)' K(eT eldj7 (2-10)
2pK^ y (2-10)
2p0


K= ZK(e) = j(Te)j [K](e)(Te) (2-11)
e=l 2 e1

Transverse Shear Stiffness, Ass

For a corrugated core sandwich structure loaded in shear transverse to the corrugations (by

shear stress rxz or shear force Qx), it was recognized that the face sheets and core would undergo

bending deformation (Libove and Hubka, 1951). For the determination of A55 the shear stress in

the face sheets were neglected because of its small thickness and classical plate theory was used.

To determine the shearing stiffness due to Qx, the shear stress in the webs due to Qx was









determined. Figure 2-5 depicts a free body diagram of the corrugated core panel unit of length dx

in the x-direction where only the stress which act in the x-direction were shown and considered.

The stress values shown were average stresses over the faces of an element which was assumed

to be very small. A summation of the forces in the x-direction yielded Equation 2-12.

aF
(F + Ax- F)AyAz +2(zxAx( j))AyAz = 0 (2-12)


Following the procedure in Appendix B, the closed form equation of the shear stresses in

the webs (rx,) due to Qx was determined. The shear strain energy density (strain energy per unit

area of the sandwich panel) was calculated from either the web shear stress in Equation B-6, or

from the shear force Qx. Equating the two shear strain energy density terms resulted in Equation

2-13. In Equation 2-13, A5s was the only unknown term in the expression.


U, dy- (2-13)
P 0 GX 2A55

Transverse Shear Stiffness, A44

Formulation of the transverse shear stiffness (A44) of the panel was relatively complicated

because certain conditions needed to be fulfilled (Fung, Tan, and Lok, 1996). Figure 2-6A

depicts a sandwich panel of unit length in the x-direction subjected to unit transverse shear,

Qy 1. The horizontal force Y p /d provided equilibrium.

Point A in Figure 2-6A was assumed to be fixed to eliminate rigid body movements of the

unit cell. The relative displacements (6y and 6,) resulted from the transverse shearing and

horizontal force. Because the force was small, the displacements were proportional to Qy, thus an

average shear strain was represented as:


= --+ 5 (2-14)
d p









Due to antisymmetry only half of the unit cell was considered for analysis (Figure 2-7A).

The unit shear force resultant was divided into force P acting on the top face sheet and force R

acting on the lower face sheet. A shear force F was assumed to act on the top face sheet at point

A where there were no horizontal forces due to antisymmetry, and a force (1-F) was determined

through a summation of the forces in the z-direction. Under the action of all these forces in the

half unit cell the displacements are shown in Figure 2-7B.

From Figure 2-7 there were three unknown forces and five displacements that were solved

through the energy method. The total strain energy in the half unit cell was the sum of the strain

energies from each individual member (AB, BC, DE, BE, and EG). The strain energy due to

bending moments was considered while the strain energy due to shear and normal forces was

neglected (Appendix C).

Castigliano's second theorem (Equation 2-15) states that displacement is equal to the

first partial derivative of the strain energy in the body with respect to the force acting at the point

and in the direction of displacement (Hibbeler, 1999). Castigliano's second theorem was used to

find the unknown forces and displacements from Figure 2-7B.


,- a (2-15)


Since the overall thickness of the sandwich panel remained constant during distortion, the

boundary conditions were 3 = 3G and A = O. Since the half unit cell was under unit shear, then

P+R = 1. The two boundary conditions along with Castigiliano's second theorem lead to a

system of two linear equations with two unknowns.









"' =0 (2-16)
aF

(2-17)
8P 8R

A substitution of Equation C-l into Equations 2-16 and 2-17 resulted in a solution of the

unknown forces P, F, and R of the system of linear equations. Substitution of Equations C-l into

Equation 2-15 along with the values of the unknown forces yielded the solutions of the

displacements (Appendix C). The displacements of half the unit cell were 5, = 8 + ~G and

8, = 83 = G5 in the y- and z- directions. By a use of the force distortion relationship (Libove and

Batdorf,1948) the transverse shear stiffness A44 was determined (Equation 2-18).


A44 Q 1 (2-18)
Y Sy z 1 1 c
S + Y( +8)+ -,
d p d p

Face Sheet and Web Stress Determination

Equation 2-4 has shown that through use of the deformation transformation matrix, the

local strains and curvature of the faces and webs due to a unit cell deformation can be

determined. As a result, the stresses in each component were found by multiplication of the local

strains and curvatures in a particular component with the corresponding transformed lamina

stiffness matrix.
r (e) r (M)
S= [T] e){ (2-19)


[](7e) = [e) ( ) + Z{Ker)) (2-20)

The previously derived deformation transformation matrices for the webs were good for

stiffness prediction. However, they did not yield accurate stress results when compared with a









finite element analysis (FEA). For example; the assumption, C' = 0, constrained the webs from

expanding in the y direction due to Poisson effect. This led to stresses in the y direction that were

not present in the 3-D FE analysis. Therefore, corrections were applied to the deformation

transformation matrix to analytical predict accurate web stresses. The refined web stress

deformation transformation matrices (Appendix A) contained a poisson's ratio that took into

account the lateral contraction or elongation ( co) of the web due to a unit macro midplane strain

in the x-direction (Fo ) and a unit macro curvature in the x-direction (Ix). The micro midplane

strains ( cy) in the web due to either KM = 1 or K = 1 were removed because there was no

force in the y -direction that was causing a midplane strain. From Equation A-6 a relation

between macro midplane shear strain and macro curvature was derived as the product of the unit

cell deformation and a function.

(y = YI cosO (2-21)

K() = K (cos3 0 + 2 cos sin2 0) (2-22)

Equations 2-21 and 2-22 treated the web as an unresisting member to the unit cell when it

was deforming. For example, when the unit cell underwent a unit midplane shear strain or a unit

curvature, the faces were compliant with that deformation but the webs resisted that movement.

Equations 2-21 and 2-21 are true if the webs were at a right angle to the face sheet, however the

equations were proved incorrect in general and the assumption that the webs resisted

deformation was proved correct by conducting several FE analyses for various web angle

inclinations. An analytical procedure that took into account the web's resistance to deformation

was established to determine the micro midplane shear strain and micro curvature

(f(p,d, tF,tBF,tw,O) and g(p,d,tF,tBF, t,O)).









Midplane micro shear strain in the web

An analytical procedure was developed to relate macro midplane shear strain to micro

midplane shear strain. The analytical method took into account the resistance to shear that the

webs experienced when the unit cell was under midplane shear. The top and bottom face sheets

were investigated separately under the action of a shear force (Figure 2-8). The resistance to

shear by the webs was included as a shear force acting on the webs (F,).

The total top face, bottom face, and web shear strain under the action of the shear forces are as

follows:

l = (l + 2(P f)) (2-23)

72)= (yf + y3(P- f)) (2-24)


1 = (f y(p- f)) (2-25)

where

F F, F F + F, F
W71 = I 3 = 4 = B (2-26)
tTFGTF tTFGTF tBFGBF tBFGBF

The shear force in the webs was

F = Gt y, (2-27)

There were three unknown shear forces, FT, Fw, FB. The three unknown forces were

determined by solving the system of three linear equations with the three unknowns (Equation 2-

28).

y/l =1
(2) = 1 (2-28)
0 = G ty F









Equation 2-28 yielded the three shear forces that acted on the facesheets during a unit shear

strain. Substitution of the known shear forces from Equation 2-28 into the web shear strain

equation (Equation 2-29) yielded the macro to micro midplane shear strain relation of the web.

f(p,d, tTF,tBF, t,, 0)= Y( = (r,f Y(p- f)) (2-29)

Micro curvature in the y -direction for the webs

An analytical procedure was developed to relate macro y-direction curvature to micro y -

direction curvature in the webs. The analytical method took into account the resistance to the

curvature that the webs experienced when the unit cell experienced curvature in the y-direction.

Half the unit cell was investigated under the action of couples that acted on the faces (Figure 2-

9).

Consider the half unit cell under the action of an end couple that causes unit curvature in

the y-direction. The half unit cell end couple was represented as three end couples acting on the

faces and webs (Cr, Cw, and CB). The slopes of the faces and web due to an end couple were

obtained from beam theory (one-dimensional) formulas (Cook and Young, 1999). There were

three unknown couples (CT, Cw, and CB) in Figure 2-9. To solve the three couples, a system of

three linear equations was required. The three equations came from the boundary conditions. The

first two boundary conditions were that the slopes of the top and bottom face sheet must equal

the slopes of the faces when c- = 1. The last boundary condition was that the difference of slope

between the face and web junction point (A and B) must equal the slope of the web. After solving

the system of linear equations the curvature of the webs was determined by dividing the couple

acting on the web with the flexural stiffness of the web (equivalent El).









(C + C,)f C,(p- f)
+ =p (2-30)
(EI)TF (EI)TF

(C, -C,)(p- f) Cf
+ =p (2-31)
(EI)BF (EI)BF

(C -C)(p- f) (CB +C,)f Cs
(2-32)
(EI)BF (E),TF (EI),


g(p,d, tTF, tBF, t, ) = (3 (2-33)
Y (EI)

Conclusion

Finite element analysis was commonly used to analyze sandwich structures; however, a

full 3-D FEA was not economical for a preliminary analysis of a structure. Such panels can be

represented as an orthotropic thick plate for which analytical solutions can be derived. A method

to homogenize the corrugated sandwich panel into an orthotropic thick plate was presented. A

detailed formulation of the bending, extensional, coupling, and shear stiffness of the ITPS panel

was determined through an energy method. The analytical models were capable of handling

laminated composite materials for the face sheets and webs of the sandwich panel. Furthermore,

different materials can be used for the face sheets and web. For example, the hot side (outer) face

sheet can be composed of titanium, aluminum or other super alloys and the cool side (inner) face

sheet can be a polymer matrix composite. The webs can be made of other materials such as

titanium, aluminum, or composite. The resistance of the webs due to shear and curvature was

analytically determined. For a web configuration other than a rectangular configuration the webs

were resisting the unit cells strain or curvature.























Figure 2-1. Corrugated core. A) Z-core, B) C-core.


Figure 2-2. Simplified unit cell dimensions


Figure 2-3. Equivalent orthotropic thick plate for the unit cell corrugated core sandwich panel.























Figure 2-4. Global and local coordinates of the unit cell, faces and webs.







A F+
F + a F

A-

A


Z-F


Figure 2-5. Small element removed from a body, showing the stresses acting in the x-direction
only A) side view, B) isometric view.


Y=p/d


Qv=11

Y=p/d


Y=p/d
S 4-r
~jQy1


Figure 2-6. Unit cell subjected to unit transverse shear and horizontal force.


'=p/d









fF B


4- pld
p p/d
P
i. p/d


Figure 2-7. A) Half unit cell of the corrugated-core
unit cell.



Top Face Sheet


'


71 72



f p-f


6y,



r~r--'-3-1
L G
6zG B

sandwich panel. B) Deformations of the half




Bottom Face Sheet


Figure 2-8. A) Free body diagram of the top face sheet under the action of midplane shear strain.
B) Free body diagram of the bottom face sheet under the action of midplane shear
strain.


rCT


Figure 2-9. Half unit cell under the action of end couples at the faces.


Fw-FT


i 4I


I









CHAPTER 3
FINITE ELEMENT VERIFICATION

Extensional and Bending Stiffness

For verification of the effectiveness and prediction capability of the analytical models, an

ITPS sandwich panel with the following dimensions was analyzed by a FE analysis: p = 80 mm,

d = 80 mm, tTF = 1 mm, tBF = 1 mm, tw 1 mm, = 750, a = 0.65 m, b = 0.65 m. An AS/3501

graphite/epoxy composite (El = 138 GPa, E2 =9 GPa, 12 = 0.3, G12 = 6.9 GPa) with four

laminae in each component and a stacking sequence of [(0/90)2] was used as an example to

verify the analytical models. The facesheets and the webs were assumed to be made of

graphite/epoxy laminates. A representative volume element or unit cell (Figure 3-1) approach

was adopted to obtain the stiffness properties. An FE analysis was conducted on the unit cell

using the commercial ABAQUSTM finite element program. Eight node shell elements were used

to model the face sheets and webs of the unit cell. The shell elements have the capability to

include multiple layers of different material properties and thicknesses. Three integration points

were used through the thickness of the shell elements. The FEM model consisted of 18,240

nodes and 6,000 elements to guarantee convergence.

The ITPS plate stiffness was obtained by modeling the unit cell with shell elements and

forcing the unit cell to six linearly independent deformations (Marrey and Sankar, 1995). The six

linearly independent strains were: (1) c' = 1 and maintaining the rest of the macroscopic strains

and curvature zero; (2) ', = 1 and maintaining the remaining strains and curvature zero; and

similarly (3) y' = 1; (4) ," = 1; (5) ,M = 1; and (6) Kr = 1. Strains were imposed to the FEM

model by enforcing periodic displacement boundary conditions (Table 3-1). To prevent rigid

body motion and translation, the unit-cell (Figure 3-2) was subjected to minimum support









constraints. The top and bottom surfaces were assumed to be free of traction. The faces x = 0 and

x = a had identical nodes on each side as well as the other faces, y = 0 and y = b. The identical

nodes on the opposite faces were constrained to enforce the periodic boundary conditions. Figure

3-3 represents the deformations of the unit cell as a result of imposing the periodic boundary

conditions.

The nodal stresses of the boundary nodes were obtained from the finite element output

after the analyses. Nodal moments were obtained by multiplying the nodal forces with the

distance from the midplane. The nodal forces and moments of the boundary nodes were then

summed to obtain the force and moment resultants (Equation 3-1). By a substitution of the values

from Equation 3-1 into the plate constitutive relationship, the stiffness coefficients in the column

corresponding to the non-zero deformation were computed. The same procedure was repeated

for other deformation components to obtain and fully populate the unit cell stiffness coefficients.


[N,,M,]= [,z] (-(a,y,z) (3-1)


The finite element result indicated that Equation 2-11 provided an excellent prediction to

determine the extensional, coupling, and bending stiffness (Table 3-2). The finite element results

were in excellent agreement with the formulation of the derived stiffness parameters of the

corrugated core sandwich panel. All analytical results were within 2% difference when compared

to the finite element results from Equation 3-1.

For further verification, the stiffness matrix of an ITPS sandwich panel with a general

laminate stacking sequence other than a crossply laminate was determined and compared with

the finite element results. The ITPS sandwich panel had the same geometric dimensions and

material properties but each component was now composed of a different laminate stacking

sequence, [45/-45]s. The analytical procedure to determine stiffness was capable of providing









accurate stiffness results to within 3% difference when compared to the FEM results (Table 3-3).

The analytical stiffness prediction procedure does not limit the designer to just cross ply

laminates.

The ITPS sandwich panel can be composed of various materials that make it perform at its

optimum for mechanical and thermal applications. Supper alloys such as titanium and Inconel

are possible choices for the top face sheet due to their high service temperatures. Aluminum

alloys, beryllium, and composite materials are of preference for the bottom face sheet due to

their high specific heat properties. An ITPS with different materials for the top and bottom face

sheet resulted in an un-symmetric stiffness matrix with a non-zero coupling stiffness matrix, [B]

# 0. For verification of obtaining un-symmetric stiffness matrix properties from the analytical

model, the analytical stiffness matrix result of an ITPS sandwich unit cell with Inconel as the

material property for the top face sheet and graphite epoxy with a [(0/90)2] laminate stacking

sequence for the webs and bottom face sheet was compared with the finite element results from

the periodic boundary condition approach. The percentage difference between the finite element

results and the analytical results are less than 2% (Table 3-5). The analytical procedure for

predicting the stiffness matrix of an ITPS with different materials properties was capable of

predicting accurate stiffness results. The analytical procedure was robust and effective for all

possible material choices and laminate stacking sequences of the faces and webs.

The results indicated that the micromechanical procedure presented in Chapter 2 for

predicting the stiffness matrix of an ITPS sandwich panel provided excellent results for any type

of laminate stacking sequence. The analysis was robust because it was capable of providing

accurate stiffness results for an isotropic and orthotropic material of any laminate stacking

sequence.









Stress Verification

Midplane Shear Strain and Curvature in the Webs

For midplane shear strain verification, the same FEM unit cell element and mesh from

Figure 3-1 with the same material properties and cross ply layup was investigated. The web

angle inclination was changed from 550 to 900 and the unit cell was subjected separately to a

periodic unit midplane shear strain and a periodic y-direction curvature. The corresponding web

midplane shear strain and web curvature were extracted from the FEM output after analysis. The

results of micro midplane shear strain and micro curvature from FEM and Equations 2-29 and

2-33 were compared and plotted (Figure 3-4).

From Figure 3-4 the agreement between the FEM and analytical results of Equations 2-29

and 2-33 was less than 2%. The analytical equations that were derived previously accounted for

the resistance effect of the webs when the unit cell was subjected to midplane shear of bending in

the macro-scale sense. Accurate stress results were expected when compared with FEM.

Stress Verification

The refined web stress deformation transformation matrix was verified by an FE analysis.

A known strain was applied to the unit cell and the corresponding local stresses on the faces and

webs were obtained by multiplying the deformation vector with the refined web stress

deformation transformation matrix. The known strain was applied to the finite element model by

enforcing periodic displacement boundary conditions from Table 3-1. The stress results from

Equation 2-20 and the stresses from the FEM output after analysis were plotted

(Figures 3-5 to 3-10).

The analytical results were within 2% difference when compared to the finite element

stress output. Results from this dissertation verified the accuracy and validated the procedure that

predicts the local stresses of the ITPS sandwich panel. The refined web stress deformation









transformation matrix did an excellent job in predicting the correct strain in the webs which

resulted in stress results that were in good agreement with the FEM output. Furthermore, the

refined web stress deformation transformation matrix did not alter the stiffness matrix. The

refined web stress deformation transformation matrices in Appendix A were used to compute the

ITPS stiffness (Table 3-4). Analytical-1 results represent the stiffness values obtained from using

the facesheet deformation transformation matrices and the web deformation transformation

matrix (Appendix A). Analytical-2 results represent the stiffness values obtained from using the

facesheet deformation transformation matrices and the refined web stress deformation

transformation matrices (Appendix A). The refined web stress deformation transformation

matrices have the capability to accurately predict stresses in each ITPS component and

accurately predict the ITPS stiffness. The previously derived deformation transformation

matrices outputs excellent stiffness results but erroneous stress results.

Transverse Shear Stiffness (A44) Verification

The finite element verification of the A44 stiffness term consisted of a two part finite

element procedure. An assumption was made that the ITPS behaved like a cantilevered one-

dimensional beam. The equivalent cross sectional properties of the beam were: axial rigidity EA,

flexural rigidity EI, and shear rigidity A44. The beam consisted of 10 unit cells and was clamped

on the left end (Figure 3-11). Eight node solid elements were used to model the 1-D plate. First,

a known end couple was applied to the tip of the beam and the corresponding tip deflection was

determined from the finite element output after analyses. The tip deflection was also derived

analytical in terms of the moments and flexural rigidity as:

MI2
v 2E (3-2)
2El









The flexural rigidity of the beam (El) was determined using Equation 3-2. The couple was

then removed and a transverse force was applied at the tip of the cantilevered beam. The tip

deflections were obtained from the finite element output after analysis. The tip deflection can

also be determined by Equation 3-3.

F13 Fl
vt = + t (3-3)
3EI A4

The shear rigidity (A44) was determined by using finite element tip deflection in Equation

3-3 along with the flexural rigidity result from Equation 3-2. This finite element verification

procedure was done for various web angles. The finite element result along with the analytical

result from Equation 2-18 are illustrated in Figure 3-12. The finite element results were in good

agreement with the analytical formulation of A44. The percentage difference between the finite

element results and the analytical result did not exceed 7%. The finite element deformation of the

cantilever beam is shown in Figure 3-12B.

Conclusion

The stiffness results between the analytical model and the FE analysis were within 2%,

thus meeting the less than 5% requirement that was stated in the objective. The refined web

stress deformation transformation matrix made incremental percent improvements to the ITPS

stiffness when compared with FE results. Both the deformation transformation matrix for the

webs and the refined web stress deformation transformation matrix can be used in predicting

stiffness, but only the latter matrix can be used for stiffness and stress prediction. The refined

web stress deformation transformation proved to be a good method in determining stresses in the

webs and faces under the periodic displacement boundary conditions.










Table 3-1. Periodic displacement boundary conditions.
u(a,y) v(a,y) w(a,y) u(x,b) v(x,b) w (x,b) x,(a,y)- Oy(a,y)- ,(x, b)- Oy(x, b)-
,,( i,y) v(0,y) -. ,y) u(x,0) v(x,) wx,) (x, ) () (0, y) o0(x,0) oy(x, 0)

Exo a 0 0 0 0 0 0 0 0 0
=1
EyO 0 0 0 0 b 0 0 0 0 0
1
xyO 0 a/2 0 b/2 0 0 0 0 0 0
1
Kx= az 0 -a2/2 0 0 0 0 a 0 0
1
Ky 0 0 0 0 bz -b2/2 0 0 -b 0
1
Tcxy 0 az/2 -ay/2 bz/2 0 -bx/2 -a/2 0 0 b/2
1


Table 3-2. Non-zero [A], [B], and
layup.

Stiffness Analytical FE
An [N/m] 2.23E+08 2.20E+08

A12[N/m] 5.43E+06 5.43E+06

A22[N/m] 1.48E+08 1.48E+08


A66[N/m] 1.43E+07 1.41E+07

Dni[Nm] 2.76E+05 2.78E+05

D12[Nm] 8790 8690

D22[Nm] 2.37E+05 2.37E+05

D66[Nm] 2.23E+04 2.22E+04


[D] coefficients for an ITPS sandwich panel with [0 / 90]s


% diff.
S1.36%

0.00%

0.00%

7 1.42%

0.72%

1.15%

0.00%

1 0.57%









Table 3-3. Non-zero [A], [B], and [D] coefficients for an ITPS sandwich panel with a [45 / -45]s
layup.
Stiffness Analytical FE % diff.
Ani[N/m] 1.14E+08 1.13E+08 0.81%

A12[N/m] 6.29E+06 6.29E+06 0.00%

A22[N/m] 9.05E+07 9.04E+07 0.00%

A66[N/m] 7.24E+07 7.24E+07 0.71%

Dii[Nm] 1.57E+05 1.58E+05 0.35%

D12[Nm] 1.01E+05 1.01E+05 0.00%

D22[Nm] 1.45E+05 1.45E+05 0.00%

D66[Nm] 1.14E+04 1.14E+04 0.07%



Table 3-4. Comparison of the non-zero [A], [B], and [D] coefficients for an ITPS sandwich panel
between the refined transformation matrix and the deformation transformation matrix.
Stiffness Analytical-1 Analytical-2 FE % diff(FE-1) %diff (FE-2)
An [N/m] 2.24E+08 2.23E+08 2.20E+08 1.37% 1.33%

A12[N/m] 5.43E+06 5.43E+06 5.43E+06 0.00% 0.00%

A22[N/m] 1.48E+08 1.48E+08 1.48E+08 0.00% 0.00%

A66[N/m] 1.43E+07 1.40E+07 1.41E+07 0.98% 0.79%

Dji[N-m] 275920 275870 277640 0.62% 0.64%

D12[N-m] 8788.2 8691.5 8691 1.12% 0.01%

D22[N-m] 2.37E+05 2.37E+05 2.37E+05 0.08% 0.00%

D66[N-m] 2.23E+04 2.21E+04 2.21E+04 1.03% 0.08%









Table 3-5. Non-zero [A], [B], and [D] coefficients for an ITPS sandwich panel with Inconel for
the top face sheet and graphite epoxy for the webs and bottom face sheet.


Stiffness Analytical FE


Anl[N/m]


A12[N/m]


A22[N/m]


A66[N/m]


Dii[Nm]


D12[Nm]


D22[Nm]

D66[Nm]


Bil[Nm]


B12[Nm]


B22[Nm]


3.69E+08


6.54E+07


2.94E+08


8.58E+07


5.09E+05


1.05E+05


4.70E+05

1.37E+04


5.84E+06


2.40E+06


5.83E+06


3.66E+08


6.54E+07


2.94E+08


8.59E+07


5.11E+05


1.05E+05


4.69E+05

1.37E+05


5.84E+06


2.40E+06


5.85E+06


% diff.
0.89%


0.01%


0.14%


0.10%


0.31%


0.02%


0.14%

0.01%


0.02%


0.02%


0.29%


B66[Nm] 2.87E+06 2.92E+06 1.94%




















4;-.


Figure 3-1. Finite element unit-cell mesh


b


Figure 3-2. Boundary conditions imposed on the plate to prevent rigid body motion. An arrow
pointing at a black dot indicates that the displacement of that point is fixed in the
direction of the arrow.




















-Y =1


S=1 = 1 =1


Figure 3-3. Deformations of the ITPS due to periodic boundary conditions


0.5
E
E 0.4


T 0.3

CU
( 0.2
cU
- 0.1
0a


0 FEM
* Analytical





(S,


0.5


0.4

E
0.3


S0.2

0.1
0.1


0 FEM
* Analytical


0 1 1 0 1
40 60 80 100 50 60 70 80 90

A Web Angle Inclination degreee) Web Angle Inclination degreess) B

Figure 3-4. Comparison of FEM and analytical micro strain for A) midplane shear strain B)
curvature.


0" = 1 S, = 1













Face Stress x-direction (s =1)


1




S 0-
c 0
t-
0
IC-
H -0.5
ci,
(0
O
LL.
-1
1








S0.5


0
c,
-j
-g -0.5


Stress x (Pa)


x 1010


1

-5




0

0.5
(0
L(.
U-0


) 0.5


I 0


S-0.5


-1


TF-F
-- TF-A
SBF-F
X BF-A


-1 i
2.4 2.5 2.6 2.7 2.8 2.9
Stress y (Pa) x 109

Web Stress y-direction (P =1)


00-F
0 0-A
E 900-F
900-A





-2 -1 0 1 2 3


Stress y (Pa)


x 109


Figure 3-5. Stresses in the x- andy- direction of the top face, bottom face, and web for a unit cell

strain ofsc = 1. (Note: "A" are the analytical results and "F" are FEM results. The

"00" and "900"are the ply orientations of the web laminate. All values in the y-axis are

normalized with respect to the face thickness and web length.)


S
Face Stress x-direction (s yo=1)


Face Stress y-direction (s
Face Stress y-direction (s =1)


26 2t 28 29
Stress cx (Pa) x 109


Figure 3-6. Stresses in the x- andy- direction of the top face and bottom face for a unit cell strain

of cs = 1. (Note: "A" are the analytical results and "F" are FEM results. The "0" and

"900"are the ply orientations of the web laminate.)


Stress x (Pa) x 1010

Web Stress x-direction (xo =1)


SO0-F
00-A
E 900-F
S900-A






) 5 10 15


S05


0
-1-
ci

05



L -5


5 10
Stress o (Pa)


15
x 1010


Face Stress y-direction (s =1)















Face Stress xy-direction (xy =1)



STF-F
-- TF-A
0 BF-F
X BF-A


6 65 7
Stress Fy (Pa)


75 8
x 109


Web Stress xy-direction (
Web Stress xy-direction (y =1)
xyAJ


11 12 13 14 15 16
Stress F (Pa) x 109


Figure 3-7. Stresses in the x- andy- direction of the top face, bottom face, and web for a unit cell

strain ofyo = 1. (Note: "A" are the analytical results and "F" are FEM results. The

"00" and "900"are the ply orientations of the web laminate. All values in the y-axis are
normalized with respect to the face thickness and web length.)














Face Stress x-direction (KM=1)


MFace Stress y-direction (=
Face Stress y-direction (K 1)


05

1--



S-05
o



LL
I-
0) -05


Stress c (Pa) x 109

Web Sessx-direction 1)


0 TF-F
- TF-A
0 BF-F
- BF-A


-1 -05 0 05 1 15
Stress y (Pa) x 108

Web Stress y-direction (KM=)


-2 0 2 4 6
Stress x (Pa) x lo9


Figure 3-8. Stresses in the x- andy- direction of the top face, bottom face, and web for a unit cell

strain of Kc = 1. (Note: "A" are the analytical results and "F" are FEM results. The

"00" and "900"are the ply orientations of the web laminate. All values in the y-axis are

normalized with respect to the face thickness and web length.)


05C



0



- 05
c 0




CU
LO
L-
a) -05
o


05

t

r-

-05


-1


0
Stress y (Pa)














Face Stress x-direction (K M=1)
Y


0 TF-F
- TF-A
0 BF-F
BF-A


Face Stress y-direction (KM=1)
Y


r-
0.5

0)

0

Ic
r-

, -0.5

LL


1
-

- 0.5
N
()
()
- 0


, -0.5

LL
-1
--


-1.5








0.5


c0
I 0


-0.5


-1 -
-1
-2.5


-2 -1.5
Stress (Pa)


-1

x 105


Stress y (Pa)

Web Stress y-direction
Web Stress y-direction (K =1)


-4
Stress y (Pa)
y


Figure 3-9. Stresses in the x- andy- direction of the top face, bottom face, and web for a unit cell

strain of Ky = 1. (Note: "A" are the analytical results and "F" are FEM results. The

"0" and "90"are the ply orientations of the web laminate. All values in the y-axis are

normalized with respect to the face thickness and web length.)


1 -0.5 0 0.5 1 1.5
Stress x (Pa) x 108

Web Stress x-direction (KM=1)


0
x 106














Face Stress xy-direction (KM =1)
xy





0 TF-F
---TF-A
0O 0 BF-F
SBF-A




2 0 2 4


4 -2 0 2 4


Stress F (Pa)
xy


C 05



0) 0
0
c
-j

-05


x 108


Web Stress xy-direction (M 1)
xy





S00-F
S00A
E g ooF :].

S900-F I
S900-A






4 -35 -3 -25 -2 -1 5
StressF (Pa) x 10


Figure 3-10. Stresses in the x- andy- direction of the top face, bottom face, and web for a unit

cell strain of ic = 1. (Note: "A" are the analytical results and "F" are FEM results.

The "0" and "90"are the ply orientations of the web laminate. All values in they-

axis are normalized with respect to the face thickness and web length.)









^ <1


Figure 3-11. Truss core modeled as a cantilever beam with ten unit cells.


1



- 05
N
/)
(/)
c 0
0

I-
8 -05
LL-

-1















I



-C-~--I


100------------
1020 40 60 80
Web Angle Inclination degreee)


Figure 3-12. A) Finite element and analytical result for the transverse shearing stiffness B)
Deformation of the beam due to the couple and transverse force


o Analytical
* FEM


E
S300
. 300









CHAPTER 4
THERMAL ANALYSIS OF AN ITPS UNIT CELL

Introduction

Thermal analysis of an ITPS involved the modeling of complex heat transfer mechanisms

in a severe thermal environment. The parameters of the severe thermal environment were: (1)

pressure variation and (2) temperatures variation (Blosser, 2004). The ITPS was a

multifunctional structure that possessed load bearing capabilities as well as provided insulation

for the space vehicle. The heating rates on an ARMOR TPS during ascent for the windward

surface of the Space Shuttle were investigated by Dorsey et al. (2004).

The heat flux caused the ITPS temperature to rise dramatically and as a result it caused

panel thermal loads and stresses, panel thermal deflection, and panel thermal buckling. Knowing

the behavior of the ITPS to a change in temperature is a critical design need because panel

deflection, panel buckling (local or global), panel temperature, and panel yielding are critical

functions of an ITPS that influence the sizing design. Typically, simplified, one-dimensional

models (Poteet, Abu-Khajeel, and Hsu, 2004) are used to predict the thermal performance of a

thermal protection system when subjected to realistic temperature distributions. Therefore, a one-

dimensional finite element heat transfer analysis was done by Bapanapalli, Martinez, Sankar,

Haftka, and Blosser (2006).

The heat transfer that was considered for a preliminary analysis of the ITPS thermal loads

and moments was at x=205.7 in (5.22 m) from the tip of the nose (Figure 4-1). The heat transfer

analysis determined the maximum bottom face sheet temperature of the ITPS and the core

temperature distribution at any particular reentry time after analysis. The core temperature

distribution was plotted for three re-entry times (450 s, 1575 s, and 1905 s) (Figure 4-2). The








1905 s reentry time corresponds to the time when the maximum bottom facesheet temperature

was reached for that particular ITPS.

Each temperature distribution resulted in thermal force resultants and thermal moments

that caused the sandwich structure to deform. In the case of laminate composites the thermal

force resultants and moments are computed as the product of the lamina stiffness matrix, CTE,

temperature change, and height of the laminate (Equation 4-1).


[N',M ]= [Q]{af}ATdz (4-1)


In Equation 4-1, [Q] is the transformed lamina stiffness matrix, {a} is the column matrix

of the coefficient of thermal expansion (CTE) and AT is the temperature change from the

reference temperature. However, these equations do not apply in predicting thermal forces and

moments to the present ITPS structure because there are no layers in the ITPS unit cell.

Therefore, a micromechanics (homogenization) approach was used to predict the ITPS thermal

forces and moments from a given reentry temperature distribution. Instead of using Equation 4-1

to predict the thermal forces and moments, consider the thermo-elastic laminate constitutive

relation (Equation 4-2).


M} [K]{}- {N (4-2)

In Equation 4-2, NTand MA are the ITPS' thermal force resultant and moment due to a

temperature change. The thermal force resultants and moments are the forces and moments that

act on the unit which causes the thermal expansions.









Thermal Micromechanics Approach

Thermal Force Resultants and Moments

An analytical method was developed to predict the thermal force resultants and moments

of an orthotropic ITPS sandwich panel composed of four composite laminates (two face sheets

and two webs) as an example. Each laminate has its respective material properties and ABD

matrix. The ITPS unit cell was subjected to a core temperature distribution (AT(y)) where

was the local axis of the inclined web whose origin was at the top facesheet and web junction

point. The change in temperature distribution equation was determined by fitting a fourth order

polynomial to the temperature distribution result (Figure 4-2). A reference temperature at which

the laminate was assumed to be stress free was assumed and the temperatures in the faces were

considered to be constant because they were thin when compared to the ITPS thickness. Due to

symmetry only half the unit cell was analyzed. The thermal forces and moments of the ITPS

were predicted through a hold and relax method. The half-unit cell was constrained to prevent

displacement and strain in the x- and y- directions. The top face sheet had roller supports in the z-

direction which allowed the webs to expand (Figure 4-3A). The resulting forces and moments

needed to constrain the ITPS from thermally expanding were equal to the ITPS thermal forces

and moments resultants. The thermal problem was broken down into two problems. The first

problem was the constrained thermal problem in which force resultants that were equal and

opposite to the components (face or web) thermal forces were applied to the unit cell to prevent

expansion in the x- and y- direction (Equation 4-3). The equal and opposite forces prevent any

expansion in the half-unit cell when it was exposed to a fourth order temperature distribution.

The expansion prevention leads to zero strains. This situation was represented as Equation 4-3.










[N, ) ABD= f- Z (Qe)k(zk zk 1)AT(y)dy (4-3)
L k=-I
2

The second problem was an unconstrained half-unit cell with no temperature distribution

and the forces developed in the constrained problem are "relaxed" and reversed. The "relaxed"

force resultants are equal and opposite of the force resultants obtained from Equation (4-3). The

constraints were represented by the reaction forces (Figure 4-3C). The constraints were unknown

reaction forces that were determined via Castigliano's second theorem (Hibbeler, 1999). The

strain energy due to a bending and normal force was considered. The strain energy of each

component was determined and then summed to obtain the overall strain energy of the half-unit

cell. There were seven unknown reactions forces to be determined. To determine the seven

unknown reaction forces, seven boundary conditions were imposed to the unit cell which were

that the displacement and rotations due to each reaction force was zero. The seven boundary

conditions along with Castigliano's second theorem (Hibbeler, 1999) lead to a system of seven

linear equations. Solving the system of linear equations led to the solution of the seven unknown

reactions. By summing the reaction forces in problems one and two the desired thermal force

resultant and moment for an ITPS sandwich panel were obtained (Equations 4-4 to 4-7).

Ny = R +R6+(N1) + N2)) (4-4)
d d

M, =R4 + R + (N N ))+ (R6 -R3) (4-5)
2 Y 2

Nx = (N1 2p + N 22p + N 3)2s) (4-6)


Mx = (N) N )2p + 2(z, N ()() (4-7)
21=1









In Equation 4-7, Nis the number of discretization points in the web length. The force

resultant and moments that were needed to constrain the unit cell during a change in temperature

was equal to the negative of the thermal force resultants and moments of the ITPS

(Equation 4-8).

[NT,MT] = [-N,-M] (4-8)

Thermal Stress

The change in temperature due to reentry aerodynamic heating caused the ITPS panel to

produce thermal forces and moments to the unit cell, which led to thermal stresses in the faces

and webs. The thermal stresses could lead to thermal yielding, thermal buckling, or thermal out-

of-plane displacement of the ITPS panel. An analytical procedure was derived to obtain the

thermal stresses in each component due to the reentry temperature variation. According to

classical laminate plate theory, the equation needed to determine thermal stresses was:

a = [Q]( aAT) (4-9)

To determine thermal stresses in either the faces or the webs, the micro thermal deformation of

each component must be known. The micro deformation of each component was determined

which relates macro to micro deformation (Equation 2-4).

Constrained case

There were two cases used to predict the thermal stresses of an ITPS from a reentry

temperature distribution. Each case had two different boundary conditions. The first case that

was investigated was the constrained case where strains in the x- and y- directions were zero,

however the webs were free to expand in the web length direction and constrained in the x-

direction. Equation 4-10 predicts stresses for a constrained thermal expansion problem.

[]= -[Q]({a}AT) (4-10)









Equation 4-10 was only valid for the top and bottom face sheets because the faces were

fully constrained and the strains were zero. The webs however were not fully constrained and

were allowed to expand in the y direction only. Therefore an analytical solution for the web

expansion under a fourth order polynomial temperature distribution was derived. From Figure 4-

3A, the constrained thermal problem for the half-unit cell was broken down into two individual

problems. Problem 1 is Figure 4-3B and Problem 2 is Figure 4-3C. The web strain in

the y direction from Problems 1 and 2 was determined and then summed to obtain the total web

strain for the constrained thermal problem that took into account the web expansion.

The web strain from Problem 2 was obtained by determining the midplane strain and

curvature in the webs due to the reactions and relaxed forces (Figure 4-4). The equation that

characterizes the force and moment at any location on the web were obtained by summing the

forces and moments in the y -direction (Appendix B). Then by inverting the webs ABD matrix

and multiplying it by the force vector the midplane strain and curvature in the web was obtained

(Equation 4-11).


eCo(y) = + A22 N(y)
( 2 (4-11)
(y)= D2 (y)


The web strain from Problem 1 was determined by first modeling the free body diagram of

the web from Figure 4-5. The webs were constrained by a force from Equation 4-3, which was

the average force needed to constrain the web in the web length direction. The average

displacement of the web was zero but the local displacements and local strains were not zero

because of the fourth order temperature distribution polynomial which caused local thermal

strain. Use of the constitutive relation in the y -direction and substitution of that expression into









the differential equation of equilibrium and then double integrating, resulted in the local v

displacement in the y -direction due to a fourth order temperature distribution (Equation 4-12).


v(y N) AT(y)ddy + Dy + E (4-12)


The two unknown constants (D and E) were solved by using two boundary conditions. The two

boundary conditions were that the web displacement at both ends of the web ( y = [0, s]) were

equal to zero. The web strain due to a fourth order temperature distribution was solved by taking

the first partial derivative of the web displacement from Equation 4-12.

av N a[ ]a+
CYo) AT(y)dydy + Dy +E (4-13)
ay CL JA22 ay

Summing the strain obtained from Equations 4-11 and 4-13 yielded the web strain in the

web length direction for the constrained problem with consideration of web expansion.

Unconstrained case

In this section the stresses in the faces and webs due to the thermal forces were determined

from Equation 4-8. Multiplying the thermal force vector (Equation 4-8) with the ITPS stiffness

(Equation 2-9) yielded the thermal strain and curvature for the unit cell.


{ [KJ{N (4-14)


Multiplying the ITPS' unit cell's thermal strain and curvature with the deformation

transformation matrix for the faces and the refined web stress deformation transformation matrix

for the webs yielded the micro deformation of the faces and webs (Equation 2-4). The micro

deformations were the local strains and curvature that the faces or the webs experienced due to a

reentry temperature distribution. The face and web stresses were determined by multiplying the

micro deformation with its respective transformed lamina stiffness matrix (Equation 4-15).










[]e = [Q [I )TD([K1 NT] (4-15)


A thermal micromechanics approach was developed to determine the thermal forces and

moment resultants of an ITPS sandwich panel when subjected to any reentry temperature

distribution profile. The resulting thermal stresses were determined by solving the ITPS for two

cases, constrained and unconstrained. The web expansion was taken into account for the

constrained cases.

Finite Element Verification

Thermal Force and Moment Resultants

For verification of the effectiveness of the analytical models to predict thermal force and

moment resultants of an ITPS, a corrugated core sandwich panel unit cell with the following

dimensions was analyzed: p = 50 mm, d= 100 mm, tTF 1 mm, tBF =1 mm, tw 1 mm, = 75.

An AS/3501 graphite/epoxy composite (EI = 138 GPa, E2 9 GPa, 112 =0.3, G12 6.9 GPa),

with four laminae in each component and a stacking sequence of [(0/90)2] was used as an

example to verify the analytical models. The same FEM model and unit cell described in Chapter

three was used for the thermal FE analysis. The strains and curvatures in Table 3-1 were set to

zero (the unit cell was not allowed to expand in the x- and y- directions but it was allowed to

expand in the z direction). The FEM ITPS model was exposed to the 450 s reentry time

temperature distribution. The resulting force resultant and moment needed to constrain the unit

cell was equal to the negative of the thermal forces. The force resultants and moments from the

FE analysis were determined from Equation 3-1 and Table 4-1.

The finite element result indicated that using the novel micromechanics approach to

determine the ITPS panel thermal forces was adequate for a thermal stress problem. The finite

element results were in excellent agreement with the formulation of the derived thermal forces









equation of an ITPS. The analytical results were within a less than 5% difference when compared

to the FE results.

Thermal Stress Verification

Constrained case

The same FEM unit cell representative volume element and mesh from Figure 3-1 with the

same material properties and angle ply layup was investigated for stress verification. The FEM

unit cell was exposed to the 450 s reentry temperature distribution. The reference temperature of

the FEM model was room temperature (294 K). The ITPS unit cell strains in the x- and

y-direction were zero but the web was allowed to expand in the y -direction. Figures 4-6 and 4-7

illustrate the finite element face and web strain and stress results with the analytical face and web

strain and stress results.

The results in Figure 4-6 indicated that the analytical equation for strain in the web length

direction that accounts for the free expansion of the webs (Equations 4-11 and 4-13) provided

accurate strain results when compared to the FEM results. The less than 1% prediction in thermal

strain in the webs yielded accurate stress results when compared to the finite element results. The

percentage difference between the analytical results and the finite element results did not exceed

4%.

Unconstrained case

The same FEM unit cell representative volume element and mesh from Figure 4-1 with the

same material properties and angle ply layup was investigated for the unconstrained problem.

For this example, periodic thermal strains from Equation 4-14 were applied to the finite element

model. The analytical and finite element ITPS stress results in the x- and y- directions were

plotted. The results from Equation 4-14 were substituted in Table 3-1 to obtain the appropriate

periodic displacement boundary conditions for the unconstrained thermal problem. Figure 4-8









illustrates the FEM deformation of the ITPS unit cell after appropriately applying the periodic

thermal boundary conditions and Figure 4-9 compares the unconstrained finite element and

analytical stresses in the x- and y- direction.

Conclusion

The stress results between the analytical solution and the finite element method were

within 5% difference of each other. The results indicated that the new refined web stress

deformation transformation provided less than 2% difference in strain results compared to FEM.

The hold and relax method was an efficient and fast way to determine thermal forces and

moments. All results were less than 5% different when compared with the finite element results.












Table 4-1. Non-zero Thermal Forces of the unit cell.

Stiffness Nx[N/m] Ny[N/m] Mx[Nm/m] My[Nm/m]


Analytical 578.65 317.48 15.35 11.41


FE 563.88 316.77 15.08 11.45


% diff. 2.62% 0.22% 1.79% 0.35%


9M0
7ffl~


450 1575 2175

Time of Re-entry (sec)


Figure 4-1. Heating used for preliminary thermal load and stress analysis of an ITPS panel.





Temperature Distribution


1

05
rF-
o -
S-05

-1
z


---450
91575
x 1905
200 0 600( 80


Temperature (K)


Figure 4-2. A) Core temperature distribution at three reentry times. B) Resulting thermal force
resultants and thermal moments.










R2


Figure 4-3. A) Half-unit cell of the truss core sandwich panel with a temperature distribution. B)
Constrained thermal problem C) Unconstrained "relaxed" expansion.


(3)
R2 R4


z*


M(y)


SN(y)


Figure 4-4. Free body diagram of the top face sheet and web.


AT(y)




y


4-


Figure 4-5. Free body diagram of the webs with an average constraining force and fourth order
temperature distribution.













x10-6
4


35 D
r


3


25


C 2


2 15


* FEM
SAnalytical


'

W
Wg
w
w


1 **


05 0 03 4
0 01 02 03 04 05 06 07 08 09 1
Normalized Web Length (ybar/s)

Figure 4-6. Web expansion for the constrained thermal problem


Face Stress, x-direction


Face Stress, y-direction


S1
cN
v
N 0.5
C,,
C,,
- 0
0
O




u-
L_


Stress Cx (Pa)

Web Stress, x-direction
,


OU.F
i0u.F
* i0.


-3 -2
Stress Cx (Pa)


STF-F
TF-A
SBF-F
- BF-A


x105 Stress y (Pa) x 105

Web Stress, y-direction


-1 0

x 105


& 1

0-
-0.5


O -


I -os


1-
z-4


-2 0
Stress G (Pa)
y


2 4

x 105


Figure 4-7. Stress in the x- and y- directions of the top face,
constrained thermal problem.


bottom face and web for the


(c
- 1-


S0.5
C,,

0


- -0.5

CO
ILL
-1
-4
-4




Si 1


) 0.5


S 0
-a 0


S-0.5


E
S-1-4
z -





























Figure 4-8. Deformation of the unit cell due to the unconstrained boundary condition


Face Stress, x-direction


Stress x (Pa) x105


Web Stress, x-directic


0 2 4
Stress ax (Pa)


Face Stress, y-direction


Stress cy (Pa) 105


,n





-F

-A

-A -Q

E
0
o
6 z

x 105


-1 -0.5 0 0.5 1
Stress a (Pa) 104


Figure 4-9. Stress in the x- and y- directions of the top face, bottom face and web for the
unconstrained thermal problem.


LI_
..n I'
~,. \.LI i
1
LL: i.

-~









CHAPTER 5
BUCKLING ANALYSIS OF AN ORTHOTROPIC INFINITE STRIPS AND APPLICATIONS
TO ITPS

Buckling of an ITPS

During launch and reentry, the TPS is subjected to various mechanical and thermal loads.

The thermal and mechanical loads initiate compressive forces to each component (webs or face

sheets) on the ITPS. The faces of a sandwich structure are designed to withstand in-plane loads,

while the core is designed to withstand transverse shearing loads. One of the advantages of an

ITPS was that the panels can be large in size thus reducing the number of panels needed to cover

a certain area of a space vehicle and reducing the overall mass of an ITPS. Large sized panel

leads to unit cells that are large in length. The increase in unit cell length results in long

unsupported or partially supported sections of the ITPS' thin facesheets or corrugated core which

were subjected to in plane and transverse shearing loads.

The ITPS used in this study was composed of thin plates which made them susceptible to

buckling when exposed to compressive forces. The thin plates reduced the overall weight but

decreased the buckling resistance of the plates. The ITPS can undergo two types of buckling,

local and global buckling. Local buckling is limited to a part of the ITPS such as the faces or the

webs. Global buckling is when the ITPS plate buckles as a whole. Global buckling was not of a

concern because the ITPS plate was thick thus increasing the global buckling resistance.

Local buckling of an ITPS plate was an undesirable failure mode because a local dimple

due to a buckled face or corrugated core led to a change in the aerodynamic heating profile and

instability and collapse of the entire ITPS. The change in the local aerodynamic heating profile

resulted in local excess aerodynamic heating, which elevated the local temperature of the

facesheet past its temperature limit and caused catastrophic failure. Buckling of the ITPS was

one of the major design drives because of the thin plates. A local buckling analysis of an ITPS









plate began by an assumption that the plate was composed of three thin plates that were

susceptible to buckling (Figure 5-1).

Each thin plate had a different a to b ratio that was dependent on the unit cell length, angle of

corrugations, and unit cell thickness. The value of a / b varied from 10 to 50 depending on the

unit cell's geometric parameters. Due to the high a /b ratio, each plate was considered to be an

infinite strip. The infinite direction was the x-direction and the finite direction was the

y-direction. Solutions have been presented for plates with various boundary conditions under

compressive loading in the x direction. Those solutions converged to a constant value of the

critical buckling load for long plates (a >> b). Solutions have also been investigated for plates

with various boundary conditions under shear loading. The shear stability solutions were

approximate because of implementation of the Galerkin method (Reddy, 1997) and cannot be

conveniently applied to long plates. An infinite plate solution was implemented for the

determination of the critical buckling load of an ITPS facesheet or web.

Methods of Critical Loads Calculation

There are different methods to calculate the critical buckling load of a structure. The

methods used for determining critical buckling values of compressive forces applied to bars are

applicable for determining critical buckling values of plates. There are three methods to

determine the critical buckling values of plates (Timoshenko and Gere, 1963).

* Method 1: The critical buckling values of the forces acting on a plate at the middle surface
are obtained by assuming that from the beginning the plate has some initial curvature or
some loading in the longitudinal direction. Those values of forces at which the plate
deflections grow indefinitely are the critical buckling values.

* Method 2: For this case it was assumed that the plate buckles slightly under the action of a
compressive force applied to the plate's midplane and then the magnitudes that the forces
must have to keep the plate stable in a slightly buckled shape are the critical buckling
loads. The differential equation of the deflection shape was obtained from Equation 5-1,
assuming that there are no body forces and lateral load on the structure.









S4W 04W 04W 02W 02W 02W
Da-+ 2(D + 2D66) +D-- = N -+N --+2N- (5-1)
Sa24 12 a2 Y x y x2 x y


The simplest case was when all three compressive forces were constants and where a ratio

between the forces was assumed, Ny = kNx and Nxy = RN,. After solving Equation 5-1 for w for

a given plate boundary condition, it was evident that the critical buckling load for the plate was

possible only for a certain definite value of N. The smallest value of Nx determines the desired

critical buckling value.

S Method 3: The energy method can also be used to determine the critical buckling values
of plates. The energy method was useful where rigorous solutions of Equation 5-1 was
unknown or where a plate was reinforced by stiffeners and it was required to find only an
approximate value to the critical buckling load (Timoshenko and Gere, 1963). This method
assumed that the plate exhibited a small lateral bending due to the compressive forces
consistent with its boundary conditions. Only the energy of bending and the corresponding
work done by the compressive forces was considered. If the work done by the compressive
forces was smaller than the strain energy of bending for every possible buckling mode,
then the plate was stable and buckling did not occur. If the work done by the compressive
forces was greater than the strain energy of bending for every possible buckling mode, then
the plate was unstable and buckling occurred.

Stability of an Infinite Strip under Compression or Shear

The stability of a infinite strip in the x- direction and subjected to either a uniform shear

loading or compression loading in the x- and y- directions (Figure 5-2), was considered for an

analytical stability analysis. These exact solutions provided the limiting cases for long plates

under shear and in plane compressive loading.

An infinite plate with the edges at y = +b/2 and either clamped or simply supported was

considered for the stability analysis. The boundary condition of the facesheet and web junction

point varied according to the angle of corrugations. It was not known specifically what the true

boundary condition at the facesheet and web junction was. It was assumed that the true boundary

condition was in between the simply supported and clamped boundary condition. The ITPS panel









is composed of three infinite strips. Each infinite strip had a different a/b ratio. Correction

factors can be used accordingly to accurately represent the true boundary condition between the

facesheet and web junction point. The infinite strip was subjected to a compressive uniform

shear and biaxial loading. The faces of the ITPS were subjected to in-plane loads while the web

was subjected to transverse shearing loads. All three loads were needed for the analysis. The

governing differential equation for the applied uniform biaxial compression in the x- and y-

direction is shown in the form of one compressive load and two ratios (N,= -No, N= -kNo, and

N, = -RNo) (Equation 5-2).

a4w a4w a4w a 4w a4w
Di + 4DI W + 2(D2 + 2D66 )2 + 4D26,, + D2
9x 9x 9y 9x 9y 9x"y ~ /y
(5-2)
N, 2RN, kN,
a2 xy o 2

The boundary conditions along the edges y = b/2 are

(1) Simply supported edges

w=0 (5-3)
M D aw a~2w a~2w
My = -D12 D- 2 2D2 = 0 (5-4)
ax 2y 2x y

(2) Clamped edges

w=0 (5-5)

= 0 (5-6)
ay

Using the approach suggested by Whitney (1987), the solution in the form of Equation 5-7

was considered for the deflection shape.

w = f(y)e2 lb (5-7)









Substituting Equation 5-7 into the governing differential equation of equilibrium Equation 5-2,

the following differential equation was obtained in the form of the unknown functionf(y)

(Equation 5-8). Substituting Equation 5-9 into Equation 5-8 the solution of the governing

differential equation was represented by Equation 5-10. In Equation 5-10, Ai, A2, A3, and A4 were

the real or complex roots of the following characteristic equation:


D f(y) + 4D16 + 2(D12 + 2D66) 22


4D26 + =O -kN 2RN
^ b )y- 8 y b8 b )a b 8y

f(y)= Aec lbe2 )ylb (5-9)

w = e 2xb (Ae2ly/bb +Be24y/b + Ce24/b +De 2"'y/b) (5-10)

D114 + 4D16 3A + 2(D12 + 2D66 )22 + 4D26 3 + D22A

No ( kNo 2Z2 2RNo 0 (5-11)


The solution from Equation 5-10 was used in conjunction with the boundary conditions

which resulted in four linear homogeneous equations with the unknowns A, B, C, and D. Since a

non-trivial solution was desired, the determinant of the coefficients must be zero, which was a

sufficient condition to determine the buckling load. For the simply supported edges and clamped

edges the four conditions took the following matrix forms:

e e e e A
e e e e B e
det = (5-12)
M1e Me Me '1 M4e
-Me M M 12e Z, M3e-13 M z14 D 0









e e '2 e e 1
e e e e
det e e 1 = (5-13)
Ale IA 2e 2 Az3e 4e C 0
e 2 Ae '2 'e /' 4e D 0

In Equation 5-12, M = D122 + D2222 + 2D26 4 To determine the actual critical buckling

load for any combination of applied compressive loads or shear loads, the roots of the

characteristic equation must be determined in conjunction with Equations 5-12 or 5-13 for a

given value of E, where E characterizes the length between the successive buckling waves in the

plate. The solution from Equation 5-10 was found to depend on and the critical buckling load

corresponded to the value of which yielded the lowest compressive load (No). The procedure

was repeated for a clamped boundary condition. The flow chart (Figure 5-3) illustrates the

process for determining the critical buckling load for a given value of (.

Results

A FE analysis was used to verify the analytical procedure for determining critical buckling

loads of infinitely longs plates subjected to compressive and shear forces. An ITPS unit cell with

the following dimensions: p = 50 mm, d= 100 mm, tTF 1 mm, tBF 1 mm, tw= 1 mm, 0 = 75,

a =1 m, b 1 m was considered for the analysis. An AS/3501 graphite/epoxy composite

(El = 138 GPa, E2= 9 GPa, v12 =0.3, G12 = 6.9 GPa, with four laminae in each component and a

stacking sequence of [(0/90)2] was used as an example to verify the buckling analytical models.

Plate II from Figure 5-1 with the a/b11 ratio of 9.88 was modeled using the commercial

ABAQUST finite element program. Eight node shell elements were used to model the long plate.

The shell elements have the capability to include multiple layers of different material properties

and thicknesses. Three integration points were used through the thickness of the shell elements.

The FEA model consisted of 6,321 nodes and 2,002 elements. Due to symmetry of the plate only









a quarter of the plate was modeled with the appropriate symmetric boundary conditions. The

buckling load was obtained from ABAQ UST by applying either compressive loads or shearing

loads equal to one and obtaining the eigenvalues from the FEA output after analysis. The

eigenvalues were equal to the critical buckling loads if the compressive or shearing loads were

equal to one. Several critical buckling loads were investigated such as uniaxial compression in

the x and y direction, biaxial compression, and shear loading with simply supported and clamped

boundary conditions.

Uniaxial Compression, Nx only

For this case, a compressive load ofNx=l was applied to the finite element model and the

corresponding eigenvalue was obtained from the FEM output after analysis (Figure 5-4).

The finite element results (Table 5-1) indicated that the analytical procedure for

predicting the stability of an orthotropic infinite plate resulted in an accurate prediction of the

critical buckling loads for a simply supported and clamped boundary condition. The percentage

differences between the analytical and FEM results was less than 2%. The critical buckling load

for any value of is shown in Figure 5-5. As expected, the critical buckling load for a clamped

boundary condition was greater than the buckling load for a simply supported boundary

condition. The minimum data point of the two curves in Figure 5-5 was the desired critical

buckling value.

Uniaxial Compression, Ny only

A compressive load ofNy=l was applied to the finite element model and the corresponding

eigenvalue was obtained from the FEM output after analysis. The finite element deformation

contours for a simply supported and clamped boundary condition are shown below in Figure 5-6.

The minimum of the two curves from Figure 5-7 was the critical buckling load for the

infinite plate. The percentage difference between the analytical and finite element critical









buckling values was less that 2% (Table 5-2). The minimum of the curves from Figure 5-7 was at

S= 0 because from Figure 5-6, the length between the successive buckling waves in the x

direction was zero. To create a buckling mode in they direction only one buckling wave was

needed in the y direction which resulted in only one wave in the x direction and a E of zero since

there was no length between wavelengths.

Biaxial Compression, Nx = Ny

Two compressive forces were applied to the finite element model and the corresponding

eigenvalue was obtained from the FEM output after analysis. The finite element results and

analytical results are illustrated in Figures 5-8 and 5-9. The compressive load in the x- direction

was equal to the compressive load in they direction (k = 1). The percentage difference between

the analytical and finite element results was less than 1% (Table 5-3). The minimum of the

simply supported curve (Figure 5-9) was at = 0 because from Figure 5-8 the length between the

next successive wavelengths in the x- direction was zero because there was only one buckling

wave.

Shearing Load, Ny = 1

For verification of the shear buckling value, a long plate was subjected to a unit shear load.

The critical buckling load obtained from Equation 5-12 was compared with another established

analytical equation (Equation 5-14).


Nc = K EK (5-14)


The value of K from Equation 5-14 was dependent on the aspect ratio of the plate. The value of

K for a large aspect ratio was 5.34 for a simply supported boundary condition and 8.96 for a

clamped boundary condition. The comparison of the critical shear buckling loads obtained from

Equations 5-12 and 5-14 are compared (Table 5-4). An FE analysis was unsuccessful for this









case due to the lack of knowledge on boundary conditions for an infinite plate under shear

loading. The analytical results for an infinite plate with an a /b = 9.88 with steel properties and a

thickness of 1 mm (0.039 in) was considered for comparison with the buckling results obtained

from the infinite plate solution.

Conclusion

The comparison of the critical buckling value of the analytical model was less than a 2%

difference when compared to the critical buckling load from the FEA. The analytical model was

capable of obtaining accurate critical buckling values for any combined loading. The analytical

procedure resulted in the lower and upper bound of the true critical buckling value or the ITPS.

Correction factors can be used to obtain the true critical buckling value. Buckling is a major

design driver of the ITPS because of the thin faces and webs.









Table 5-1. Critical buckling load of plate II, N, = 1.
Critical Analytical [N/m] FEM [N/m] % diff


Simply Supported 11620 11591 0.25%


Clamped 23500 23907 1.70%



Table 5-2. Critical buckling load of plate II, Ny = 1.
Critical Analytical [N/m] FEM [N/m] % diff


Simply Supported 2030.3 2057.9 1.34%


Clamped 8124 8141 0.21%



Table 5-3. Critical buckling load of plate II, k = 1.

Critical Analytical [N/m] FEM [N/m] % diff


Simply Supported 2036.9 2036.8 0.00%


Clamped 7788 7823 0.45%



Table 5-4. Critical buckling load of plate II, Nxy 1.
Equation (5-13)
Critical Analytical [N/m] [N/m] % diff


Simply Supported 97600 98262.6 0.67%


Clamped 164000 164813 0.49%

















Figure 5-1. Local buckling of an ITPS panel.


N, /N////y


Nx


(_ Y/b/2

///////////////N ,


Figure 5-2. Infinite Strip under shear and compression loading.


Figure 5-3. Critical buckling load flow chart for an infinite plate.

















A 4*


Figure 5-4. Deformation of the quarter plate due to Nx=l. A) Simply supported boundary
condition B) Clamped boundary condition.


50


o
Z 20
15


Simply Supported
Clamped
0
0 0.5 1 1.5 2 2.5 3 3.5


Figure 5-5. Critical buckling load as a function of the length between the successive buckling
waves in the plate for Nx=l.


A


B


Figure 5-6. Deformation of the quarter plate due to Ny=l. A) Simply supported boundary
condition B) Clamped boundary condition.


Ct,


./L


001-












- Simply Supported


Clamped


0.1 0.2


Figure 5-7. Critical buckling load as a function of the length between the successive buckling
waves in the plate for Ny = 1.


Figure 5-8. Deformation of the quarter plate due to k=l. A) Simply supported boundary
condition B) Clamped boundary condition.


2.3


E 2.2
7Z.
o 2.1
z


. 8.2
0
Z 8.1

8
0


ft /













T6
z
0
z2


--Simply Supported
0
0 0.5 1 1



Clamped

E


z


Figure 5-9. Critical buckling load as a function of the length between the successive buckling
waves in the plate for Nx = Ny.