Luận án tiến sĩ: Three dimensional strain measurement by a Bragg grating sensor subjected to axial and transverse load simultaneously

Abstract The objective of this thesis is to establish mathematical models for predicting strains in the axial and other two orthogonal transverse directions of a multi-parameter Bragg gr

THREE DIMENSIONAL STRAIN MEASUREMENT BY A BRAGG GRATING

SENSOR SUBJECTED TO AXIAL AND TRANSVERSE LOAD

SIMULTANEOUSLY

A DISSERTATION SUBMITTED TO THE DEPARTMENT OF MECHANICAL ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Tadamichi Mawatari November 2006

UMI Number: 3242591

Copyright 2007 by Mawatari, Tadamichi

All rights reserved

INFORMATION TO USERS

The quality of this reproduction is dependent upon the quality of the copy submitted Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction

In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted Also, if unauthorized copyright material had to be removed, a note will indicate the deletion

®

UMI

UMI Microform 3242591 Copyright 2007 by ProQuest Information and Learning Company

All rights reserved This microform edition is protected against unauthorized copying under Title 17, United States Code

ProQuest Information and Learning Company

300 North Zeeb Road P.O Box 1346 Ann Arbor, MI 48106-1346

© Copyright by Tadamichi Mawatari 2007

All Rights Reserved

il

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy

keV Mele

(Drew V Nelson) Principal Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in

scope and quality as a dissertation for the degree of Doctor of Philosophy

Abstract

The objective of this thesis is to establish mathematical models for predicting strains in the axial and other two orthogonal (transverse) directions of a multi-parameter Bragg grating sensor which is subjected to an axial and transverse loadings simultaneously Previous work by other researchers has established models for predicting axial strains, but not a combination of those three strain components

Test specimens consisted of polarization maintaining Bragg grating sensors created in optical fibers with bow-tie stress applying parts Each of the sensors had two Bragg wavelength peaks at around 1300 nm, and two others around 1550 nm, and those four peaks moved due to external stimuli such as applied loads and/or temperature changes The change in wavelength from a certain reference state, which is called a wavelength shift denoted by AA, was considered as the key to predict the strains due to the extemal stimuli experienced by the sensor The specimens underwent thermal, axial, transverse, and combined loading tests, and the relations between the wavelength shifts and external stimuli were investigated

In order to predict the strains experienced by the sensor from the data of wavelength

shifts obtained in the experiments, several mathematical models were investi gated Those

mathematical models were categorized into linear and non-linear models The linear model assumed a linear relation between the wavelength shifts and the loads applied, while the non-linear model tried to deal with more general conditions since a non-linear

IV

relation was observed in some of the data The computed strains were compared with ones obtained from finite element analyses, and the accuracies of the mathematical

models were studied

Some researchers hypothesize that the non-linearity between wavelength shifts and applied loads is caused by the rotation of optical axes of the sensor due to large transverse loads The related studies were reviewed, and some computations were conducted to understand a possible cause for the non-linear behavior

If a multi-parameter sensor is properly embedded in a material, it is expected that one might be able to measure the multi-axial strains in the material from data on wavelength shifts An example of an experiment that could check this possibility was formulated, and the related computational procedures were discussed Also, to improve the ability of the multi-parameter sensor to determine transverse strains, some different cross-sectional geometries of an optical fiber sensor were suggested

Acknowledgements

This thesis was only possible with the support of many individuals First of all, 1 would

like to thank my advisor, Professor Drew Nelson His academic advice, encouragement,

patience, and continuous financial support are greatly appreciated

Second, I would like to thank other reading committees, Professor Kosuke Ishii and Professor Sheri Sheppard, for helpful suggestions on my thesis

Third, I would like to thank Mr Stephen Kreger and Mr John Seim of Blue Road

Research (BRR), a fiber optic R&D firm in Fairview, Oregon for conducting experiments that provided data for this research

Furthermore, I would like to extend my thanks to my friends with whom | enjoyed my student life for many years Finally, I would like to thank my parents for supporting me during the entire of my life in Stanford

Vi

Table of Contents

2.1 IHTOUCHIOHN HT HT HH HH TH TH TT TH ng Hy 18 2.2 Multi-paraImet€T S€TSOF SH HH HH TH TT ng ca tấp 20

Vii

2.2.1 Basic concept of single-parameter SCNSOT :cccccseceesceeneeeeeees 22

2.2.2 Basic concept of mulfI-paramefer S€TSOT nhe 25

3.1 EXP€TIITTIS Ác HH“ HH HH HH HH HH nàn Hệ 29 3.1.1 SK specimen f©SfS «TH HH Họ HH nh HH ng 31 3.1.1.1 SK Thermal (€S( cu HH «HH HH HH ng gu 33 3.1.1.2 SK Axial loading f€SE - - HH ng yên 36 3.1.1.3 SK Transverse loading f€s - HH, 39 3.1.1.4 SK Combined loading fest Hee 54 3.1.2 C2 specimen đafa LH HH ng HH kp 69 3.1.2.1 C2 Axial loading test dala Hee, 70 3.1.2.2 C2 Transverse loading test data Hee, 70 3.1.2.3 C2 Combined loading test dafa cà ceecseeeeiei 72 3.1.3 C2 specImecn Ởafl cà kg HH 11111111316 76 3.1.3.1 C3 Axial loading test dafa - che 76 3.1.3.2 C3 Transverse loading test đafa nhe, 77 3.1.3.3 C23 Combined loading test data -.- cuoi 80 3.2 Finite element anaÏyS1S HH HH TH HH HH Hiện 34 3.2.1 Basics coordinate system for finite element analysis results 84 3.2.2 Axial ÏOading Hàng TH HT Hà Hàng tiện 86 3.2.3 Transverse ÌOading HH nh HT nh HH Hà vi, 86 3.2.4 Comparison with closed form solufion cu, 87 3.3 Determination of fast and SIOW aX€S SH HH ki, 88

4.1 APpTOACH cece cssseseseensessecsesssssssssssasceaseessssssassaaeseesessenenteneenensesgs 93 4.1.1 Determination of the second and third columns of the K matrix 94 4.1.2 Determination of the first column of the K matrix - 95 4.1.3 Reducing the K matrIX tO 3-DV-3 SIZ€ HH yên 96 4.2 Data ShiÍ[ LH HH TH HT nh HH TH Tà gu ch 98 4.2.1 Data shiÍt in transverse loading t€SfS ĩc HnHarrau 99 4.2.2 Data shift in combined loading test seesceseeeetecteeeneeeeees 101

Vili

4.3 Strain analysis and r©SuÏfS ng HH HT kg 103

43.1 SK data ccc 4 103

4.3.1.1 SK Axial loading f†€S «ch ki 106 4.3.1.2 SK Transverse loading test se 107 4.3.1.3 SK Combined loading tesfs cà coi 112 4.3.1.3.1 SK Combined loading test — Analysis I 113

4.3.1.3.2 SK Combined loading test — Analysis ÏI 119

4.3.1.3.3 SK Combined loading test — Analysis III 124

4.3.2 C2 afa HH HH HH HH TT cán nàng HH Hư 131 4.3.2.1 C2 Axial loading †€S( chớ 133 4.3.2.2 C2 Transverse loading f€sf cc.c not 134 4.3.2.3 C2 Combined loading f€St cớ 140 4.3.2.3.1 C2 Combined loading test — Analysis I 140

4.3.2.3,.2 C2 Combined loading test — Analysis II 146

4.3.2.3.3 C2 Combined loading test — Analysis III 151

=9 ốẽ ốố ố ốố ố.e 158

4.3.3.1 C3 A xial loading f€SI - HH HH Hee, 160 4.3.3.2 C23 Transverse loading ©Sf uc n neo 161 4.3.3.3 C3 Combined loading test 0.00 ccccecssessesseseeseesseseseeas 167 4.3.3.3.1 C3 Combined loading test — Analysis ! 167

4.3.3.3.2 C3 Combined loading test — Analysis IT 173

4.3.3.3.3 C3 Combined loading test — Analysis III 178

4.4 SUtImAFY - HH HH TT TH To TH TH HT HH nu 185 4.5 Other maihematical mod€Ì§ - ch HH HH HH TH HH ngà key 186 4.5.1 Model 1: Over-deterministiC sysfem các ccccceccercexes 188 4.5.2 Model 2: DeterministiC sYSf€m Ánh re, 194 4.5.3 Model 3: Singular value decomposition (SVP) 196

§ Non-linear model for strain analysis: Foliated Quasi Inverse (FQI) method 203

S.1 THh€OTY HH HH TH TH TT TT HH HT Hàn HH 203 5.1.1 Problem de€SCTIptIOH càng HT TH HH nh HH Hư Hy 203 5.1.1.1 Mathematical model for wavelength shift: AA=[AA,,AA,,

ix

5.1.1.2 Mathematical model for strain: e=[£¡,¿,£;, T] 204

5.1.1.3 Generalized K mafTiX óc tt tnsvseeeeree 205 5.1.1.4 Problem settings (Problem to be addressed) 205

5.1.1.5 Outline of this chapf€T cty 206 5.1.2 5ystem COndi[IOTNS - HH Hà 1H11 11 Hà Hàn ng Hệ 209 5.1.2.1 Relationships observed in experimenIs 20

5.1.2.2 Reasonable assumptions from mechanics of materials .209

5.1.2.3 Other assumptions cccccescccsseesesseseesesessecssesseseeeees 209 5.1.2.4 Available experimental đata cuc 210 5.1.2.5 Formulae to be determined -cc sec 210 5.1.3 Construction of matrix representation .cccccccssessessessssscessssees 211 5.1.3.1 Basic formulae for A and Bio ccccsesesssesserseaseeeseees 211 5.1.3.2 Basic formulae for Aw cecececscessssescsessessseesceseeseasseses 213 5.1.3.3 Basic formulae for E_ c.ccccccceseccec 214 5.1.3.4 Matrix forms of A and E -c HH sgk, 215 5.1.4 Best approximation of matrix represenfation 218

5.1.4.1 Theoretical properties of the best approximation 218

5.1.4.2 Experimental properties of the best approximation 219

5.1.4.3 Selection of the function for the best approximation 219

5.1.4.4 Selection of the basis - HH ưo 221 5.1.4.5 Determination of knof sequence coi 223 5.1.4.6 Derivation of composition mmatfiX - -5- - 224

5.1.4.7 Derivation of companion MAatriX ccecssesseeseeeeeeneens 226 5.1.4.8 Formulation into Mini-max problem 228

5.1.5 Foliation for an underdetermined system .- c5 230 5.1.5.1 Definition of ÍOliatiOH - cà xe cv, 230 5.1.5.2 Definition of wavelength and strain foliation 231

5.1.5.3 Foliation and simultaneous algebraic equations 232 3.1.6 Algorithm for the best apprOXimafOn ác con, 233

5.1.6.1 Determination of the maximum pOInt 233 5.1.6.2 Determination of the minmimum pOIt - 234 5.1.6.3 Algorithm for the best approximation - Summary - 240 5.1.7 ĐBragg grating senSOr th€OFY - ng HH ng nh 243 5.1.7.1 Mathematical representation for wavelength foliation 244 5.1.7.2 Mathematical representation for strain foliation 245 5.1.7.3 Derivation of the generalized K matrix and its inverse 246 5.1.7.4 Mathematical representations for mechanical loading 248 5.1.7.4.1 Mathematical representation for axial loading 248 5.1.7.4.2 Mathematical representation for transverse ÏOAdITBE Ăn HH HH H4 ky 250 5.1.7.5 Global coordinate system and 1{s SÍTuCfUTe 254 5.1.7.5.1 Introducing orthogonal global coordinate system

— 254 5.1.7.5.2 Definition of the coordinate transformation @ 256 5.1.7.5.3 BGS space and properties of data type 257 5.1.7.6 Local coordinate system and Its structfure 260

5.1.7.6.1 Local coordinate system in BGS A(Ð).{s ho

TH K1 TH HC TT TH Ti TH 018 8 1411011550 264 5.177.7 NoIse eliminatfiOn - HH niên 266 5.1.7.7.1 Erasing Noise TYype Í code 266 5.1.7.7.2 Erasing Noise Type Í| nen 267

5.1.7.8 Algorithm for {Ä,(*)}ˆ and {Š,(*)} 268

5.1.7.9 Basic structure of the theory for Bragg grating sensor 269 5.2 ResultS Of ạtaÏYS1S HH TH TH HT nọ TH gà Hà Ty 271 5.2.1 SK Ở4fA HT HH HH Hà HH TK TH Hàn Hàn Hà nàn 271 5.2.1.1 SK Axial loading test eececeecceceeeeeeesseseessseesaees 271

XI

5.2.1.2 SK Transverse loading teSt ccccccsccsscscseessesssesasenes 272

5.2.1.3 SK Combined loading tesi nhe 277 5.2.1.3.1 SK Combined loading test — Analysis I 277 5.2.1.3.2 SK Combined loading test — Analysis H 283 5.2.1.3.3 SK Combined loading test — Analysis II 288 5.2.2 C2 (afA Án LH HH HH TH TT Tà nọ TH TH ng 1e 294 5.2.2.1 C2 Axial loading f€SỂ - QnnH HH re 294 5.2.2.2 C2 Transverse loading †€sf - cv sec 295 5.2.2.3 C2 Combined loading t€f chen 300 5.2.2.3.1 C2 Combined loading test - Analysis Ï 300 5.2.2.3.2 C2 Combined loading test — Analysis IT 305 5.2.2.3.3 C2 Combined loading test - Analysis HỊ 311 5.2.3 C3 data icccsccscccssssesssscsesscsseceseescsseeeseeseeeseeceseesesssesseeesssassesseseees 317 5.2.3.1 C3 Axial loading f†€SỂ cung cư 317 5.2.3.2 C3 Transverse loading test co cv 318 5.2.3.3 C3 Combined loading test ác Hee 323 5.2.3.3.1 C3 Combined loading test — Analysis 1 323 5.2.3.3.2 C3 Combined loading test - Analysis I] 328 5.2.3.3.3 C3 Combined loading test - Analysis ITl 334

6.1 Computation Of ETTOFS cv HH HH TH HH TH HT KT Hàn ĐH HH 341 6.2 COimpaTISOT OÍ T€SUÏfS - HQ HH HH ng HH nh nụ 342 6.2.1 AXial loading f€SE à HH HH HH Ha TH HH HH vê 342 6.2.2 Transverse loading †€SiL - ong TH HH Hà Hy 343 6.2.3 Combined loading f€S[ nh HH TH H1 1 Hy HH nh nvệy 345 6.2.3.1 AnaÌySIS Ì HH HH HH TH nàng thay 345

6.2.3.3 Analysis lÏÍ - con TH HT TH HH Hà ru 348

7.1 Non-linear behavior of AÀ, vs CUTV€S HQ LH HH ngư 351

XH

9,1 EXP€TIITI€TS nh HH Tà TH ng HH HH TH TH TH 3n 380

9.1.1 Thermmal tess -.:: 2222222221 1211 111.rred 381

9.1.2 Axial loading f€SfS - HH HH TH Hà kh kh Hệ 381 9.1.3 Transverse loading †©SfS HH HH HH ray 381 9.1.4 Combined loading †€SES LH HH HH kh rệt 381

9.2.1 Linear mtOdeÌ cu HH HH HH TH TH HT TH Hệ 382 9.2.1.1 3-by-3 Linear mOdel xà x ng grkkirec 382 9.2.1.2 Other linear modeÌS ác nHt TH HH 1H Hy 383 9.2.2 Non-linear (FQ]) model .- - c LH HH HH KH vi 383 9.3 Rotation of Optical aX€S HH“ HH HH TT HT Tá gà Hà Hành ưu 384 9.4 FUtHf€ WOK HH HH TH ng TH nọ TH CC K2 60 ke 384

A.1 SK dafa LH“ HH TH HH HT TT Hàn TH Hà Hàng ky 386 A.2 C2 ỞAfA HH HH HH HH Tà TH HH ng TH TC Hư 424 A.3 CỔ dala LH nh TH Hà gu ng TH HC ng kg 427

B.O Definition of vector norm and condition number . - s5 430 B.1 Notes on Section S Í -o su nh HH HH HH kh TH TH Tp 434

XII

B3 Notes on Secbion S.Ì.3 ch HT TH ng ng ng kg 436 B.4 Notes on Sectlon 5 |.Ả càng HH kg Tư rưy 438 B.5 Notes on hat function eee eee 440 B.6 Derivation procedure of Eq (5.4.); LH HH Hà yêu 441 B.7 Derivation procedure of EQq (S.4.5)s TS HH4 11112 xe, 443 B.8 Notes on SecCflOn 5 Ì.S ác HT HH TH HH HH KHE Hy 444 B.9_ Notes on Singular Value DecompOSIfiOïi se, 446 B.10 Derivation procedure of Egs (5.6.9), and (S.6.Ĩ); keo 452

List of Tables

Table 1.1 ~ Change In index of refraction Of sillca 8ÌaSS ào rớu 5 Table 3.1 — Strains in axial loading fest under P= 1 N Ăn re, 86 Table 3.2 — Strains in transverse loading test under Q = 1 N/MM .cccccccscsersereerseenss 87 Table 4.1 — Slope of AÀ, vs Q CUTV€ HH HH H0101112 110112111111 111 tru 103 Table 4.2 — Slope Of £ VS Q CUTV€ uc HH HH TH TH To TH TH TH TH TH ng Hiệp 103 Table 4.3 — Slope of AA vs PP CUTV€ HH TH TT HT HH TT KH như 103 Table 4.4 — Slope OÍ E vs PP CUTV€ LH HH Hư HH Hàng Hà ngà TH TH Hiệu 104 Table 4.5 — Condition numbers of 3-by-3 matTiC©S - HH nsh tri 105 Table 4.6 — Errors in strain analysis: SK axial loading test c.cccseceseseteessereerseeees 107 Table 4.7 — Errors in strain analysis: SK transverse loading tesi 112 Table 4.8 — Errors in strain analysis: SK combined loading †esf -.c cóc 130 Table 4.9 — Slope of A^, vs C CUTV€ LH HHH22 1111 TH HH1 HH ngư 131 Table 4.10 — Slope of € VS CQ CUTV HH H Hàng TH Hàng TH TH Hư tp 131 Table 4.11 — Slope of AA vs P CUTV€ - TS HH1 1111111 HH HT Hước 131 Table 4.12 ~ Condition numbers of 3-by-2 mafrIC€S TH HH xe, 132 Table 4.13 — Errors in strain analysis: C2 axial loading fest -c-ccccccsececeei 134 Table 4.14 — Errors in strain analysis: C2 transverse loading test cc cò 139 Table 4.15 — Errors in strain analysis: C2 combined loading test 157 Table 4.16 — Slope of ÁÀ, vs Q CUFV€ HH HH HH HH TH Hà Thư HH ni 158 Table 4.17 — Slope of AA vs PP CUTV€ HH HH T111 T11 HH Hàng HH tư 158

XV

Table 4.18 — Condition numbers of 3-by-3 matrices 0.0 eeseeesseeessesseeeeeseesseesessees 159 Table 4.19 — Errors In strain analysis: C3 axial loading f€SỂ nhe 161

Table 4.20 — Errors In strain analysis: C3 transverse loading †est -«- 166

Table 4.21 — Errors in strain analysis: C3 combined loading test c.cccc- 184 Table 5.1 — Errors in strain analysis: SK axial loading f€st -ccc sex 272 Table 5.2 — Errors ín strain analysis: SK transverse loading fest cceieeve 276 Table 5.3 — Errors in strain analysis: SK combined loading test 293

Table 5.4 — Errors in sfrain analysis: C2 axial loading f€St cty 294 Table 5.5 — Errors in strain analysis: C2 transverse loading (est ccccxcei 299 Table 5.6 — Errors in strain analysis: C2 combined loading test c.ccccocc 316 Table 5.7 — Errors in sirain analysis: C3 axial loading f€st - .-Sc<ccccs- 317 Table 5.8 — Errors in strain analysis: C2 transverse loading fest ccccecieeo 322 Table 5.9 — Errors in strain analysis: C3 combined loading †esf 339

Table 6.1 — Average errors in axial loading test for SK data co 342 Table 6.2 — Average errors in axial loading test for C2 dafa cccc ca 342 Table 6.3 ~ Average errors in axial loading test for C3 dafa ác nen 342 Table 6.4 — Average errors in transverse loading test for SK data 343

Table 6.5 — Average errors in transverse loading test for C2 đafa -.- co cesc 344 Table 6.6 — Average errors in transverse loading test for C3 đata - 344

Table 6.7 — Average errors in combined loading test analysis I for SK data 345

Table 6.8 — Average errors in combined loading test analysis Ï for C2 data 345

Table 6.9 — Average errors in combined loading test analysis I for C3 data 346

Table 6.10 ~ Average errors in combined loading test analysis II for SK data 347

Table 6.11 — Average errors in combined loading test analysis I] for C2 data 0 347

Table 6.12 — Average errors in combined loading test analysis II for C3 data 348

Table 6.13 — Average errors in combined loading test analysis III for SK data 349

Table 6.14 ~ Average errors in combined loading test analysis III for C2 data 349

Table 6.15 — Average errors in combined loading test analysis III for C3 data 350

XVI

List of Figures

Figure 1.1: Schematic of an optical fiDer sySI€m án HH HH HH nh, 2 Figure 1.2: Posstble paths Of lighi HH HH1 1 TH HH HH HT TH Hàng 2 Figure 1.3: Transmitting light by total internal refÏeCtiON ĩc cư, 3 Higure 1.4: (a) multi-mode and (b) single-mode optical fibers - cccccccssc<e 4 Figure 1.5: Flame hydrỌyS1S DFOC€SS - HH HH HH HH HH HT TH Hiệu 6 Figure 1.Ĩ: FIbeT ẢraWiT - TT TH HH TH TH TT TT TH KH TH TH Tu no TH 7 Figure 1.7: Intrinsic Fabry-Perot Ïn†€TÍ€rOIm€f€T - ĩc tà th SH 9n ng re 9 Figure 1.8: Transmission and reflection of light in Fabry-Perot Interferometer 9 Figure 1.9: Extrinsic Fabry-Perot Interferometer (EFFPI) LH ng 10 Figure 1.10: TL FE S€TSOT - 5 GB SH 1111141 111101 11 010cc ng cưa 11 Figure 1.11: System for absolute strain measurement using Fabry-Perot interferometer

Figure 1.12: Bragg gTaing SCTSOT TT H 901 H10 Hàn TH TH HH HH ưu 13 Figure 1.13 Transmission and reflection under Bragg condition 14 Figure 1.14: BIrefÍTing€nce cà HH H11 101110111 101 TH Hàn rà 15 Figure 1.15: Stress appÌying pATS ch n HH HH HH HH Hàn ky 16 Figure 1.16: Fabrication of bow-tie stress appÏying pAFT ác ni, 16 Figure 2.1: Arrangement of EFPI as an optical rosette strain gage 19 Figure 2.2: Schematic of resin-rich region that forms around an optical fiber embedded normal to the direction of reinforcing fibers in a composite laminate 20 Figure 2.3: Spliced grating pDAiT - HH HT TH TT TH HC ng 22

XVii

Figure 2.4: Coordinate system of a single-parameter sensor with a single Bragg grating 23 Figure 2.5: Coordinate system of a multi-parameter S€NSOF ccccsesseseeseressssesessrsees 26 Figure 3.1: Wavelength shifts in reflected spectra cccccccccsscsscsscsecsscsesscsecescaceceers 30 Figure 3.2: Schematic of optical set-up for SK (€SfS HH ng 32 Figure 3.3: Schematic of a fused 2 x 2 Coupler .ccccccsssssessscscacseessessscssesesseasassasanes 32 Figure 3.4: SK Thermal test — Raw Data ch HH5 1 01H HH ưệc 33 Figure 3.5: SK Thermal test — converted data 0 ccccccccssssssssssssesesesssesesescesscsssvsesesssnans 36 Figure 3.6: SK Axial lOading f€SE ch HH HH HH HH TH HH HT TT HH ng ryc 36 Higure 3.7: Schematic Of AÀ,.VS PP CUTV€ ST HH HH TH HH HT He rsep 38 Figure 3.8: SK axial loading †€S( cà HH 1011 H111 HH HH HT 39 Figure 3.9: Schematic of transverse loading f©sE - su LH HH ngay 39 Figure 3.10: Angle @ of transVerse ÏOad ác HT TH HH HH HT HH ng iyp 40 Figure 3.11: Test fixture of Wagreich ef đJ (1996) LH n HH HH HH Hee 40 Figure 3.12: Schematic of test set up for SK transverse loading tests 42 Figure 3.13: Top view of transverse loading (€SỂ nàn ey 43 Figure 3.14: Applied concentrated load Qu cccccssssssessssssesessesesssesscssesssssssssssesevavaescecees 44 Figure 3.15: Side view of loading plate along its centerline .c.ccccccsscesssessseseseesseees 45 Figure 3,16: Sample of transverse loading test data (0 = 72°) v.occcccccsuscscsceesesseserseseeees 46 Figure 3.17: Transverse loading (diametrical compression) .c.ccssssssssssssecseseeseneneees 48 Figure 3.18: SK transverse loading test (0 = 0° and 90°) cccccccesscsssecssssssessesseseevansees 49 Figure 3.19: SK transverse loading test (8 = 9° and 99°) ccccsssssssesessssssssssesssasscseeces 49 Figure 3.20: SK transverse loading test (9 = 18° and 108°), LH ngư 50 Figure 3.21: SK transverse loading test (Ø = 27° and 11177) -scs cuc set svz 50 Figure 3.22: SK transverse loading test (9 = 362 and 126°) Lee, 51 Figure 3.23: SK transverse loading test (9 = 45° and 135°) cty 51 Figure 3.24: SK transverse loading test (Ø = 54° and 144”) csc cung kayp 52 Figure 3.25: SK transverse loading test (8 = 63° and 153°) cccccscssscscsssssssssecesseeseees 52

Figure 3.26; SK transverse loading test (0 = 72° and 16/2°) is cty 53

Figure 3.27: SK transverse loading test (@ = 81° and 171°) ccccscsssscsseccsccesesesssvssscenees 53

XViil

Figure 3.28: SK transverse loading test (@ = 90° and 180°) cccscseccsssssesesssseseseeeees 54 Figure 3.29: À, vs dL curve in axial loading (€SỂ cv nành ngư 55 Figure 3.30: AA, vs P curve in axial loading †€S( cá Lành re dệt 55 Figure 3.31: SK combined loading test (9 = 0° and 9Q°, P = 0.427 N) coi 56

Figure 3.32: SK combined loading test (Ø9 = 9° and 999, P = 0.427 Nì 57

Figure 3.33: SK combined loading test (9 = 18° and 108°, P = 0.427N) cuc 57

Figure 3.34: SK combined loading test (8 = 27° and 117°, P= 0.427 N) 5

Figure 3.35: SK combined loading test (9 = 36° and 126°, P = 0.427 N) 58

Figure 3.36: SK combined loading test (0 = 45° and 135°, P = 0.427 N) 59

Figure 3.37: SK combined loading test (9 = 54° and 144, P = 0.427 N) ccuc 59

Eigure 3.38: SK combined loading test (9 = 63° and 153°, P = 0.427N) 60

Figure 3.39: SK combined loading test (9 = 72° and 1629, P = 0.427 N) coi 60 Figure 3.40: SK combined loading test (9 = 81° and 171°, P = 0.427 Nì) 61

Figure 3.41: SK combined loading test (ð = 90° and 1809, P = 0.427 N) 61

Figure 3.42: SK combined loading test (6 = 0° and 90°, P = 0.684 N) 62

Figure 3.43: SK combined loading test (9 = 9° and 99°, P = 0.684 N) co 62

Figure 3.44: SK combined loading test (9 = 18° and 108°, P = 0.684 N) 63

Figure 3.45: SK combined loading test (9 = 279 and 1179, P = 0.684 N) 63

Figure 3.46: SK combined loading test (9 = 36° and 126°, P = 0.684 N) 64

Rigure 3.47: SK combined loading test (8 = 45° and 135°, P = 0.684 N) 64

Higure 3.48: SK combined loading test (9 = 54° and 144°, P = 0.684 N) 65

Figure 3.49: SK combined loading test (9 = 63° and 153°, P = 0.684 N) 65

Figure 3.50: SK combined loading test (Ô = 72° and 162°, P = 0.684.N) 66

Figure 3.51: SK combined loading test (9 = 81° and 171°, P = 0.684N) 66

Figure 3.52: SK combined loading test (9 = 90° and 180°, P = 0.684.N) 67

Figure 3.53: Schematic of axial loading test set-up used for C2, C2 tests 68

Figure 3.54: Schematic of transverse loading fixture for C2, C3 data 69 Figure 3.55; C2 axial loading †©SE ch HH HH4 HH TH TH HT Hà Hàn Hư 70

xix

Eigure 3.56: C2 transverse loading test (9 = 75° and 135”) cceeheehee 70 Figure 3.57: C2 transverse loading test (9 = 90° and 1507) chen 71 Figure 3.58: C2 transverse loading test (9 = 105” and 165”) c eeee 71 Figure 3.59: C2 transverse loading test (9 = 120” and 180”) neo 72

Figure 3.60: C2 combined loading test (9 = 90° and 135°, P = 0.736 N) 72

Figure 3.61: C2 combined loading test (9 = 105° and 1500, P = 0.736 N) 73

Figure 3.62: C2 combined loading test (9 = 120° and 165”, P = 0.736 N) 73

Figure 3.63: C2 combined loading test (9 = 135° and 180”, P = 0.736 N) 74

Figure 3.64: C2 combined loading test (9 = 90° and 135°, P = 1.472 N) 74

Figure 3.65: C2 combined loading test (9 = 105° and 1500, P = 1.472N) eo 75 Figure 3.66: C2 combined loading test (9 = 120° and 165°, P= 1.472N) 75

Figure 3.67: C2 combined loading test (9 = 135° and 18OP, P = 1.472N) 76

Figure 3.68: C3 axial loading †©SÍ so Tnhh 77 Figure 3.69: C3 transverse loading test (8 = 60° and 120) Hoà 77 Figure 3.70: C3 transverse loading test (0 = 75” and 135°) cà, 78 Figure 3.71: C3 transverse loading test (9 = 90” and 150”) «HH, 78 Figure 3.72: C3 transverse loading test (Ø = 105” and 165”) -c chen 79 Figure 3.73: C3 transverse loading test (9 = 120” and 1807) che 79 Figure 3.74: C3 combined loading test (9 = 75° and 1200, P = 0.736 N) 80

Figure 3.75; C3 combined loading test (8 = 90° and 135°, P= 0.736 N) weer 80 Figure 3.76: C3 combined loading test (9 = 105° and 1500, P = 0.736 N) 81

Figure 3.77: C3 combined loading test (9 = 120” and 165”, P = 0.736 N) 81

Figure 3.78: C3 combined loading test (9 = 75° and 120”, P = 1.472 N) 82

Figure 3.79: C3 combined loading test (9 = 90° and 135°, P = 1.472 N} su 82

Figure 3.80: C3 combined loading test (9 = 105° and 150°, P= 1.472 N) 83

Figure 3.81: C3 combined loading test (8 = 120° and 1650, P= 1.472N) 83 Figure 3.82: Coordinate system for finite element artaÌyS1S cà ve, 84 Figure 3.83: Cross section Of SĨ SD€CHH€H HH Hàn HH nh ng 85

XX

Higure 3.84: Cross sectional geometry for finite element anaÌyS1S 85 Figure 3.85: Comparison of FEA data with closed form soÌutiOns 88 Figure 3.86: x, and x; axes on the cross-section of the test specimen 89 Figure 3.87: Fast and SÏow aXes OÝ tes( SD€CI€H LH HH Hào 91 Figure 4.1: Basic idea of data shIft In transverse loading tesf co ocese 99 Figure 4.2: Curve fit and determination of the value of AÀ, at Q = Ô re 100

Figure 4.4: Finding A” from axial loading test đata ác re 102 Figure 4.5: Basic idea of data shift in combined loading test (at P= P?) 102 Figure 4.6; SK axial loading Í€SÍ cu HH HH ng HH Ty Hy 106 Figure 4.7: SK transverse loading test (Ô = Ô) cuc HH ng ng kHy 108 Figure 4.8: SK transverse loading test (Ô = 9F) HH HH Ha HH HH 108 Figure 4.9: SK transverse loading test (Ø = 277) HH 1T ng Hành 109 Higure 4.10: SK transverse loading test (Ô = 45”) ác HH Han ng khay 109 higure 4.11: SK transverse loading test (Ö = '722”) c4 HH ng nàn rên 110 Figure 4.12: SK transverse loading test (9 = 106”) HH Hy 110 Higure 4.13: SK transverse loading test (Ô = 1355”) HH ri, 111 Figure 4.14: SK transverse loading test (Ô = 162”) HH HH 1H y1 Ty 111 Figure 4.15: SK combined loading test — Analysis I (9 = 0°, P= 0.427N) 114 Figure 4.16: SK combined loading test — Analysis I (@ = 459, P = 0.427N) 114 Figure 4.17: SK combined loading test — Analysis I (9 = 108°, P = 0.427N) 115 Figure 4.18: SK combined loading test — Analysis I (6 = 135°, P = 0.427N) 115 Figure 4.19: SK combined loading test — Analysis I (@ = 162°, P= 0.427N) 116 Figure 4.20: SK combined loading test - Analysis I (6 = 0°, P = O0,6S4N) 116 Figure 4.21: SK combined loading test — Analysis I (0 = 45°, P = 0.6§84N) 117 Figure 4.22; SK combined loading test —- Analysis I (@ = 108°, P = 0.684N) 117 Figure 4.23: SK combined loading test — Analysis I (0 = 135°, P = 0.6S4N) 118 Figure 4.24: SK combined loading test — Analysis I (@ = 162°, P = 0.684N) 118

XXI

Figure 4.25: SK combined loading test — Analysis II (9 = 0°, P= 0.427N) 119 Figure 4.26: SK combined loading test — Analysis H (9 = 45°, P = 0.427N) 120 Figure 4.27: SK combined loading test — Analysis ÍI (9 = 1089, P = 0.427N) 120 Figure 4.28: SK combined loading test — Analysis l (8 = 135°, P= 0.427N) 121 Figure 4.29: SK combined loading test — Analysis II (9 = 162°, P= 0.427N) 121 Figure 4.30: SK combined loading test — Analysis IT (0 = 0°, P = 0.6S4N) 122 Higure 4.31: SK combined loading test — Analysis II (0 = 459, P= 0.684N) 122 Figure 4.32: SK combined loading test - Analysis H (9 = 108, P = 0.684N) 123 Figure 4.33: SK combined loading test — Analysis IT (@ = 135°, P = 0.684N) 123 Figure 4.34: SK combined loading test - Analysis I] (@ = 162°, P = 0.684N) 124 Figure 4.35: SK combined loading test — Analysis ïII (6 = 0°, P = 0.427N) 125 Figure 4.36: SK combined loading test — Analysis III (@ = 45°, P = 0.427N) 125 Figure 4.37: SK combined loading test — Analysis III (6 = 108°, P = 0.427N) 126 Figure 4.38: SK combined loading test — Analysis III (@ = 135°, P = 0.427N) 126 Figure 4.39: SK combined loading test — Analysis III (@ = 162°, P = 0.427N) 127 Figure 4.40: SK combined loading test — Analysis HI (@ = 0°, P = 0.6§84N) 127 Figure 4.41: SK combined loading test — Analysis III (0 = 45°, P = 0.684N) 128 Figure 4.42: SK combined loading test — Analysis III (@ = 108°, P = 0.684N) 128 Figure 4.43: SK combined loading test - Analysis III (@ = 135°, P = 0.684N) 129 Figure 4.44: SK combined loading test ~- Analysis III (6 = 162°, P = 0.684N) 129 Figure 4.45: C2 axial loading f€SỂ - HH HT TH TT Hán HH gu cư 134 Figure 4.46: C2 transverse loading test (Ô = 75”) HH HH HH HH re, 135 Higure 4.47: C2 transverse loading test (Ø = 9Q) HH HH TT HH gu 136 Figure 4.48: C2 transverse loading test (Ø = 105”) HH HH HH Hy du 136 Figure 4.49: C2 transverse loading test (0 = 120Ó7) - LH“ ng kg ưưu 137 Figure 4.50: C2 transverse loading test (0 = 135°) oo cung HH HH kg Hài 137 Figure 4.51: C2 transverse loading test (9 = 150°) 00 cuc HH HH 41k ớt 138

XX

Figure 4.52: C2 transverse loading test (Ô = 165) HH 9 1H Hy HH Hay 138 Figure 4.53: C2 transverse loading test (9 = 180°) cán HH ky 139 Figure 4.54: C2 combined loading test — Analysis I (6 = 909, P = 0.736N) 141 Figure 4.55: C2 combined loading test — Analysis [ (9 = 105°, P =0.736N) 141 Figure 4.56: C2 combined loading test — Analysis I (9 = 135°, P= 0.736N) 142 Figure 4,57: C2 combined loading test - Analysis I (9 = 1659, P=0.736N) 142 Figure 4.58: C2 combined loading test — Analysis [ (9 = 180°, P=0.736N) 143 Figure 4.59: C2 combined loading test ~ Analysis I (6 = 900, P = 1.472N) 143 Figure 4.60: C2 combined loading test - Analysis [ (9 = 105°, P= 1.472N) 144 Figure 4.61: C2 combined loading test - Analysis I (8 = 1350, P = 1.472N) 144 Figure 4.62: C2 combined loading test — Analysis Í (9 = 1659, P = 1.472N) 145 Figure 4.63: C2 combined loading test —- Analysis Í (9 = 1809, P = 1.472N) 145 Figure 4.64: C2 combined loading test — Analysis II (9 = 90°, P = 0.736N) 146 Figure 4.65: C2 combined loading test — Analysis H (9 = 1059, P = 0.736N) 147 Figure 4.66: C2 combined loading test — Analysis IÍ (9 = 1359, P = 0.736N) 147 Figure 4.67: C2 combined loading test — Analysis Í (9 = 1659, P=0.7236N) 148 Figure 4.68: C2 combined loading test — Analysis IÍ (9 = 1809, P = 0.7236N) 148 Figure 4.69: C2 combined loading test — Analysis IÍ (9 = 9Q°, P = 1.472N) 149 Figure 4.70: C2 combined loading tesí — Analysis II (9 = 105°, P = 1.472N) 149 Figure 4.71: C2 combined loading test — Analysis IT (@ = 1359, P = 1.472N) 150 Figure 4.72: C2 combined loading test — Analysis II (9 = 165°, P = 1.472N) 150 Figure 4.73: C2 combined loading test - Analysis IÍ (9 = 18G, P = 1.472N) 151 Figure 4.74: C2 combined loading test — Analysis HI (9 = 909, P = 0.736N) 152 Figure 4.75: C2 combined loading test - Analysis ITI (@ = 105°, P = 0.736N) 152 Figure 4.76; C2 combined loading test — Analysis III (@ = 135°, P = 0.736N) 153 Figure 4.77: C2 combined loading test — Analysis HI (9 = 165°, P = 0.736N) 153 Figure 4.78: C2 combined loading test ~ Analysis II] (0 = 180°, P= 0.736N) 154

XXili

Figure 4.79: C2 combined loading test ~ Analysis ITI (@ = 90°, P= 1.472N) 154 Figure 4.80: C2 combined loading test - Analysis HI (9 = 105°, P= 1.472N) 155 Figure 4.81: C2 combined loading test — Analysis III (@ = 135°, P= 1.472N) 155 Figure 4.82: C2 combined loading test ~ Analysis III (@ = 165°, P= 1.472N) 60 156 Figure 4.83: C2 combined loading test — Analysis III (8 = 180°, P= 1.472N) 156 Figure 4.84; C3 axial loading f€SE - ác HH HH g1 HT TH HH HH tiếu 161 Figure 4.85: C3 transverse loading test (Ø = ÓO) cà HH HH ng ng Hy Hy 162 Figure 4.86: C3 transverse loading test (0 = 5”) HH2 0v Hàn Hy 163 Figure 4.87: C2 transverse loading test (Ø = 90) cv HH TH HH TH Hy 163 Figure 4.88: C3 transverse loading test (9 = 1057) HH HH ng Hy 164 Figure 4.89: C3 transverse loading test (Ô = 120”) LH n HH 1 11 411 ty 164 Figure 4.90: C3 transverse loading test (Ô = 135”) su cuc nh TH ngu 165 Figure 4.91: C3 transverse loading test (Ø = 15OP) LHHTH HH Hy Hy ru 165 Figure 4.92: C3 transverse loading test (Ô = 1807) HH H111 111 HH re, 166 Figure 4.93: C3 combined loading test — Analysis I (8 = 75°, P = 0.736N) 168 Figure 4.94: C3 combined loading test — Analysis I (9 = 90°, P = 0.736N) 168 Figure 4.95: C3 combined loading test— Analysis I (9 = 135°, P = 0.736N) 169 Figure 4.96: C3 combined loading test — Analysis I (9 = 150°, P = 0.7236N) 169 Figure 4.97: C3 combined loading test — Analysis I (@ = 165°, P = 0.736N) 170 Figure 4.98: C3 combined loading test - Analysis 1 (9 = 75°, P = 1.472N) 170 Figure 4.99: C3 combined loading test — Analysis I (@ = 909, P = 1.472N) 171 Figure 4.100: C3 combined loading test — Analysis I (9 = 1359, P = 1.472N) 171 Figure 4.101: C3 combined loading test — Analysis I (9 = 1509, P = 1.472N) 172 Figure 4.102: C3 combined loading test — Analysis I (9 = 1659, P = 1.472N) 172 Figure 4.103: C3 combined loading test — Analysis II (8 = 75°, P=0.736N) 173 Figure 4.104: C3 combined loading test - Analysis II (9 = 90, P = 0.736N) 174 Figure 4.105: C3 combined loading test — Analysis II (6 = 135°, P = 0.736N) 174

XXIV

Figure 4.106: C3 combined loading test — Analysis II (@ = 150°, P=0.736N) 175 Figure 4,107: C3 combined loading test - Analysis II (6 = 165°, P = 0.736N) wc 175 Figure 4.108: C3 combined loading test — Analysis II (0 = 75°, P = 1.472N) 176 Figure 4.109: C3 combined loading test — Analysis II (8 = 90°, P = 1.472N) 176 Figure 4.110: C23 combined loading test — Analysis II (9 = 135, P = 1.472N) 177 Figure 4,111: C3 combined loading test — Analysis II (@ = 1509, P = 1.472N) 177 Figure 4.112: C3 combined loading test — Analysis II (@ = 165°, P = 1.472N) 178 Figure 4.113: C3 combined loading test — Analysis III (6 = 75°, P= 0.736N) 00 179 Figure 4,114: C3 combined loading test — Analysis III (9 = 90°, P= 0.736N) 179 Figure 4.115: C3 combined loading test — Analysis III (@ = 135°, P = 0.736N) 180 Figure 4.116: C3 combined loading test - Analysis III (@ = 150°, P= 0.736N) 180 Figure 4.117: C3 combined loading test — Analysis III (6 = 165°, P = 0.736N) 181 Figure 4.118: C3 combined loading test ~ Analysis III (6 = 75°, P = 1.472N) 181 Figure 4.119; C3 combined loading test — Analysis III (8 = 90°, P= 1.472N) 182 Figure 4.120: C3 combined loading test — Analysis III (Ô = 1359, P= 1.472N) 182 Figure 4.121: C3 combined loading test — Analysis II (9 = 1509, P= 1.472N) 183 Figure 4,122: C3 combined loading test — Analysis II (9 = 165, P = 1.472N) 183 Figure 5.0; Non-linear strain analysis by FQT method .ccccsesssscsseseessseseesrsescsseeseees 208 Figure 5.1: Foliation for our mOd€] ác xxx 3141111 21H nh HH nguy 232 Figure 5.2: Blocks on the (X;, Xa) — SPAC€ 0 HH HH HH HH HH ng ki 251 Figure 5.3: Bragg Grating Sensor space in the global coordinate system 258 Figure 5.4: Representation of combined loading ín local coordinate system 263 Figure 5.5: Bragg Grating Sensor space in the global coordinate system 265 Figure 5.6: SK axial loading t€S( ác HH 9101111111111 HH HT HT Tp 271 Figure 5.7: SK transverse loading test (Ð = O°) LH ng ng HH Hài 272 Figure 5.8: SK transverse loading test (Ô = 9°) HH HH Ho HH HH cườc 273 Figure 5.9: SK transverse loading test (0 = 27°) cccccsssssscssesseesesssssesesssssecnsasssssssenes 273 Figure 5.10: SK transverse loading (est (Ô = 45”) LH HH HH gu 274

XXV

Figure 5.11: SK transverse loading test (6 = 72”) cà HH HH TH 11111 1c 274

Figure 5.12: SK transverse loading test (0 = 108°) 0.0 ch HH 1 ra 275

Figure 5.13: SK transverse loading test (0 = 135°) cceccecsseseeeeessesecceenseuneesseeenenees 275 Figure 5.14: SK transverse loading test (Ø = 162”) ng HH nhờ 276 Figure 5.15: SK combined loading test ~ Analysis I (0 = 09, P = 0.427N) 278 Higure 5.16: SK combined loading test - Analysis I (8 = 45°, P = 0.427N) , 278 Figure 5.17: SK combined loading test — Analysis I (@ = 108°, P= 0.427N) 279 Figure 5.18: SK combined loading test - Analysis Ï (9 = 1350, P = 0.427N) 279 Figure 5.19: SK combined loading test - Analysis ï (8 = 162°, P= 0.427N) 280 Figure 5.20: SK combined loading test — Analysis Ï (0 = 0°, P = 0.6S4N) 280 Figure 5.21: SK combined loading test — Analysis I (9 = 45°, P = 0.684N) 281 Figure 5.22: SK combined loading test — Analysis I (6 = 108°, P = 0,6S4N) 281 Figure 5.23: SK combined loading test - Analysis I (9 = 1359, P = 0.684N) 282 Figure 5.24: SK combined loading test — Analysis I (@ = 162°, P = 0.6S4N) 282 Figure 5.25: SK combined loading test — Analysis II (Ô = 09, P= 0.427N!) 283 Figure 5.26: SK combined loading test — Analysis II (9 = 459, P = 0.427N) 284 Figure 5.27: SK combined loading test — Analysis II (0 = 108°, P = 0.427N) 284 Figure 5.28: SK combined loading test - Analysis II (9 = 135°, P = 0.427N) 285 Figure 5.29: SK combined loading test — Analysis IÏ (9 = 162°, P = 0.427N) 285 Figure 5.30: SK combined loading test — Analysis II (9 = 09, P = 0.684N) 286 Figure 5.31: SK combined loading test — Analysis lÏ (9 = 45°, P = 0.684N) 286 Figure 5.32: SK combined loading test — Analysis II (9 = 108°, P = 0.684N) 287 Figure 5.33: SK combined loading test — Analysis II (@ = 135°, P = 0.684N) 287 Figure 5.34: SK combined loading test — Analysis II (0 = 162°, P = 0.684N) 288 Figure 5.35: SK combined loading test — Analysis III (0 = 0°, P = 0.427N) 288 Figure 5.36: SK combined loading test — Analysis IT] (6 = 45°, P = 0.427N) ou ee 289 Figure 5.37: SK combined loading test — Analysis III (6 = 108°, P = 0.427N) 289

XXVI

Figure 5.38: SK combined loading test — Analysis III (@ = 135°, P = 0.427N) 290 Figure 5.39; SK combined loading test— Analysis HI (9 = 162°, P = 0.427N) 290 Figure 5.40: SK combined loading test — Analysis HH (9 = 0°, P = 0.684N) 291 Figure 5.41: SK combined loading test - Analysis HI (8 = 459, P = 0.684N) 291 Figure 5.42: SK combined loading test — Analysis III (9 = 1089, P = 0.684N) 292 Figure 5.43: SK combined loading test — Analysis III (6 = 135°, P = 0.684N) 292 Figure 5.44: SK combined loading test ~ Analysis III (0 = 162°, P = 0.684N) 293 Higure 5.45: C2 axial loading f€SL HH HH HH HH TH TH 294 Figure 5.46: C2 transverse loading test (Ø = 75”) cuc HH HH HH gi ng 0x ng 295 Figure 5.47: C2 transverse loading test (9 = 9Ó”) HH HH HH g 296 Figure 5.48: C2 transverse loading test (Ø = 105”) HH HH HH kh 296 Figure 5.49: C2 transverse loading test (Ô = 127) uc HH HH 121 1 re 297 Figure 5.50: C2 transverse loading test (Ø = 135) HH HH iu 297 Figure 5.51: C2 transverse loading test (Ô = 15OP) LH 2 1H HH tết 298 Figure 5.52: C2 transverse loading test (Ô = 165”) LH HH ng H2 1k vườn 298 Figure 5.53: C2 transverse loading test (0 = 180°) HH HH HH nà Ha 299 Figure 5.54: C2 combined loading test - Analysis I (6 = 909, P = 0.736N) 300 Figure 5.55: C2 combined loading test — Analysis I (9 = 105°,P=0.736N) 301 Figure 5.56: C2 combined loading test— Analysis I (9 = 1359, P= 0.736N) 301 Figure 5.57: C2 combined loading test — Analysis I (0 = 165°, P = 0/736N) 302 Figure 5.58: C2 combined loading test — Analysis Ï (9 = 1809, P= 0.736N) 302 Figure 5.59: C2 combined loading test — Analysis I (8 = 909, P = 1.472N) 303 Figure 5.60: C2 combined loading test — Analysis [ (9 = 105°, P = 1.472N) 303 Figure 5.61: C2 combined loading test — Analysis I (9 = 135°, P = 1.472N) 304 Figure 5.62: C2 combined loading test — Analysis I (9 = 1659, P = 1.472N) 304 Figure 5.63: C2 combined loading test — Analysis Ï (Ø9 = 1809, P = 1.472N) 305 Figure 5.64: C2 combined loading test — Analysis II (9 = 90, P = 0.736N) 306

XXVH

Figure 5.65: C2 combined loading test - Analysis II (8 = 105°, P=0.736N) 306 Figure 5.66: C2 combined loading test - Analysis II (6 = 135°,P.=0.736N) 307 Figure 5.67: C2 combined loading test — Analysis II (@ = 165°, P= 0.736N) 307 Figure 5.68; C2 combined loading test — Analysis II (@ = 180°, P = 0.736N) 308 Figure 5.69: C2 combined loading test - Analysis II (0 = 909, P = 1.472N) 308 Figure 5.70: C2 combined loading test — Analysis II (@ = 105°, P = 1.472N) 309 Figure 5.71: C2 combined loading test — Analysis IT (9 = 135°, P = 1.472N) 309 Figure 5.72: C2 combined loading test - Analysis II (@ = 165°, P= 1.472N) xxx hd 310 Figure 5.73: C2 combined loading test — Analysis II (@ = 1809, P = 1.472N) 310 Figure 5.74: C2 combined loading test - Analysis III (9 = 909, P = 0.736N) 311 Figure 5.75: C2 combined loading test — Analysis III (@ = 105°, P = 0.736N) 311 Figure 5.76: C2 combined loading test - Analysis III (@ = 135°, P= 0.736N) 312 Figure 5.77: C2 combined loading test — Analysis III (9 = 165°, P = 0.736N) 312 Figure 5.78: C2 combined loading test ~ Analysis ITI (@ = 180°, P = 0.736N) 313 Figure 5.79: C2 combined loading test - Analysis IIT (9 = 90°, P= 1.472N) 313 Figure 5.80; C2 combined loading test — Analysis HI (@ = 105°, P= 1.472N) 314 Figure 5.81: C2 combined loading test — Analysis III (9 = 135°, P = 1.472N) 314 Figure 5.82: C2 combined loading test —- Analysis HI (8 = 165°, P= 1.472N) 315 Figure 5.83: C2 combined loading test — Analysis III (9 = 1800, P= 1.472N) 315 Figure 5.84: C3 axial loading f€SỂ nen HH HH TT TH HH rà 317 Higure 5.85: C3 transversc loading test (Ơ = ØO”) Q Q LH HH ng HH ngư krvcg 318 Figure 5.86; C3 transverse loading test (0 = '75) HH HT 1 Hàng ng TH àp 319 Figure 5.87: C2 transverse loading test (Ø = 9O”) HH HH ng HH tàu 319 Figure 5.88: C3 transverse loading test (Ơ = 105) LH HH HH Ha kh ru 320

Figure 5.89: C3 transverse loading test (Ừ = 120°) Q-Q TQ SH 2n ke 320

Figure 5.90: C3 transverse loading test (Ø = 125”) cuc HH Hưng diệu 321 Figure 5.91: C2 transverse loading test (Ø = 150”) Làn L HH ng HH du 321

XXVHI

Figure 5.92: C3 transverse loading test (9 = 1807) HH HH HH cư 322 Figure 5.93: C3 combined loading test - Analysis I (8 = 75°, P = 0./736N) 323 Figure 5.94: C3 combined loading test - Analysis l (9 = 90°, P= 0.736N) 324 Figure 5.95: C3 combined loading test — Analysis I (9 = 1359, P =0.736N) 324 Figure 5.96: C2 combined loading test — Analysis Ï (9 = 1500, P=0.736N) 325 Figure 5.97: C3 combined loading test — Analysis Ï (9 = 1659, P = 0./736N) 325 Figure 5.98: C3 combined loading test — Analysis Í (9 = 755, P = 1.472N) 326 Figure 5.99: C3 combined loading test - Analysis I (6 = 909, P = 1.472N) 326 Figure 5.100: C3 combined loading test ~ Analysis I (9 = 135°, P = 1.472N) 0 327 Figure 5.101: C3 combined loading test ~ Analysis I (@ = 1500, P = 1.472N) 327 Figure 5.102: C3 combined loading test — Analysis I (9 = 165°, P = 1.472N) 4 328 Figure 5.103: C3 combined loading test — Analysis II (6 = 759, P=0.736N) 329 Figure 5.104: C3 combined loading test — Analysis ÏI (9 = 909, P = 0.736N) 329 Figure 5.105: C3 combined loading test — Analysis II (9 = 1359, P = 0.726N) 330 Figure 5.106: C3 combined loading test — Analysis II (Ô = 150°, P = 0.736N) 330 Figure 5.107; C3 combined loading test — Analysis I] (@ = 165°, P = 0.736N) 331 Figure 5.108: C3 combined loading test — Analysis II (9 = 75°, P = 1.472N) 331 Figure 5.109: C3 combined loading test — Analysis II (9 = 909, P = 1.472N) 332 Figure 5.110: C3 combined loading test — Analysis II (9 = 135°, P = 1.472N) 332 Figure 5.111: C3 combined loading test— Analysis H (6 = 1500, P= 1.472N) 333 Figure 5.112: C3 combined loading test — Analysis II (9 = 165°, P = 1.472N) 333 Figure 5.113: C3 combined loading test — Analysis HI (8 = 75°, P = 0.736N) 334 Figure 5.114: C3 combined loading test — Analysis III (9 = 90°, P= 0.736N) 334 Figure 5.115: C3 combined loading test — Analysis III (6 = 135°, P = 0./736N) 335

Figure 5.116: C3 combined loading test — Analysis II] (9 = 150°, P = 0.736N) 335

Figure 5.117: C3 combined loading test — Analysis III (6 = 165°, P = 0.736N) 336 Figure 5.118: C3 combined loading test — Analysis III (@ = 75°, P = 1.472N) 336

XXIX

Figure 5.119: C3 combined loading test ~ Analysis III (@ = 90°, P= 1.472N) 337 Figure 5.120: C3 combined loading test - Analysis III (@ = 135°, P= 1.472N) 337 Figure 5.121: C3 combined loading test — Analysis III (@ = 150°, P= 1.472N) 338 Figure 5.122: C3 combined loading test —- Analysis III (@ = 165°, P = 1.472N) 338 Figure 7.1: SK transverse loading test data at ÖỀ HH 21103 1 re 352 Figure 7.2: SK transverse loading test data at 72° .ccccscsssssessscesesscsesessesessessseeesees 352 Higure 7.3: Rotation of optical axes in the study Of Carrara éf đỈ - 353 Figure 7.4: Coordinate system ÍOr anaÌVSIS HH H.9 1H Hà TH g0 HT bày 355 Figure 7.5: Stress state due to stress applying pDAFÉS ng Hee, 355 Figure 7.6: Stress state due to transVerse ÍOad óc HH 121 811211 11x 356 Figure 7.7: Estimation of 6, and Ơ, - - +5 s4 HH HH 0101111 Trào 359 Figure 7.8: AMA vs Q CUIVeS at 0 = O° LH HHHHHH010101110111111111 11111 pưệp 361 Figure 7.9: AMA vs Q curves at 0 = 9° eccsesessssssssssesssssesseseesesesseerscseseesessevssessesuneas 361 Figure 7.10: AAM/A vs Q CUIVeS at 0 = 18° cic ccueestetsessecsesesssesssessesterssesessessssessstsees 362 Figure 7.11: AA/A vs Q curves at 0 = 27° LH HT HH ng TH TH HH ky 362 Figure 7.12: AA/À vs Q curves at Ô = 36 HH HH HH HH ro 363 Figure 7.13: AA/A vs Q curves at Ô = 45” Q QQL LH HH HH HH HH HH HH cưy 363 Figure 7.14: AMA vs Q curves at 0 = 5⁄4” c0 110 0c ra 364 Figure 7.15: AMA vs Q curves at Ô = Õ3 LH HH HT 0 HH cư 364 Figure 7.16: AA/À vs Q CuTv€S af Ô = 772” HH 0111111 1111111111 1111111 tre 365 Figure 7.17: AMA vs Q curves at Ö = ÑÍ ch H112 11111101011 365 Figure 7.18: AM/A vs Q curves at Ö = OŨ”, LH HH TH HH HH ng, 366 Higure 8.1: Test specimen with an embedded multi-parameter sensor 368 Figure 8.2: Multi-parameter sensor embedded in the composite laminate specimen 369 Eigure 8.3: Computational pFOC€dUTC - sàn HH KH Hà Hàng The 370 Figure 8.4: Coordinate system for a unidirecfiOnal DỈY chờ 372 Eigure 8.5: Global coordinate sysfem HH HH HH TH nh HH nhờ 374 Figure 8.6: Schematic of cross-section Of Íarminaf€ cá cv LH ng nhe, 375 Figure 8.7: Optical fiber sensor with elliptical core and side holes 378

XXX

Figure 8.8: Optical fiber sensor with non-CITCUÌaT B©OI€ÍTY se 379 Figure B.1.1: Parallelotope in several dimenSiOnS - ong 435 Figure B.4.1: AA VS X; CUTY ác HH ng HH HT HH HH0 Hàng 438

Figure B.4.3: Oy; VS Xs CUYVC LH HT HH HH HH HH cà HH 0 16 439 Higure B.5.1: Hat funCtiOn - - ch HH HT HH ng 440 Figure B.5.2: Linear combination of two hat ÍUnCHOTS nhe, 441

Figure B.8.2: A, Ag and Áz Ù Áu ào thu, 446 Figure B.12.1: Orthogonal prOJ©C[OT - HH HH ng TH HH TH tết 458

Figure B.14.1: HỆ)(x,) on the Xạ— AXÍS 2 tt 460

Figure B.14.2: HIỂ, (x;) on the Xạ— AXỈS ìcccccc t1 460 Figure B.14.3: Superposing H©)(x,) and H'),(2,) cssssssssssssessesssesssssnnnssssseceeseseees 460

XXXI

1.2 Optical fiber - How it functions

An optical fiber is a device used to transmit light from a source to a detector It in general consists of solid transparent cylinders, usually made of glass, called the cladding and the core Figure 1.1 shows a schematic of an optical fiber and how the light is transmitted in

its core.

CHAPTER I: BACKGROUND

Cladding Core

Figure 1.1: Schematic of an optical fiber system

The index of refraction n of a material is defined as the ratio of the speed of light in vacuum to the speed of light in the material

_ Speed of light in vacuum

Speed of light in the material

The index of refraction of the core (n,,,.) is different from that of the cladding (n,,,), So that possible behaviors of the incoming light at the boundary between the core and the cladding are as shown in Figure 1.2, depending upon the angle 0

Figure 1.2: Possible paths of light

When 9 is relatively small (as shown in (a) and (b)), the incoming light is split into two, and one part leaves the core through the cladding, and is eventually lost From the law of reflection and the law of refraction, the following relationships hold

CHAPTER I: BACKGROUND

Neore SiN = Ny, 7 $18, Law of Refraction (1.3)

When 0 is equal to a certain angle, 9, (in (c)), the part that was lost goes along the boundary This 9, can be derived by substituting 9 = 0, and @, = 90° in Eq (1.3)

Neore SING, = Neigg SiN9O” = 12,4

Figure 1.3: Transmitting light by total internal reflection

Optical fibers may be classified as multi-mode or single-mode Partial differential equations derived from the electromagnetic theory of light propagation in cylindrical waveguides show that multiple modes of light corresponding to certain discrete values of

9 will be transmitted (Cherin, 1983), as depicted in Figure 1.4 (a) The modes may be thought of as being analogous to the natural vibration modes of a beam As the core diameter is reduced, the number of propagating modes decreases When the diameter is

CHAPTER 1: BACKGROUND

small enough (e.g., 5 to 10 Hm in 125 um dia fiber), only the lowest order mode propagates along a single light path, as depicted in Figure 1.4 (b)

Cladding Cladding

Figure 1.4: (a) multi-mode and (b) single-mode optical fibers

In multi-mode fibers, light rays that enter a fiber at steeper angles travel along much longer paths than rays entering at a shallow angle The path-length difference of those rays leads to differences in the arrival time of each of the propagating waves Optical fibers are used mainly in telecommunications Typically a signal in the form of a stream

of pulses is transmitted (Cherin, 1983) In multi-mode fibers, pulse spreading and some signal distortion occur due to propagating waves having different path-lengths Pulse spreading does not occur in single-mode fibers because light is constrained to travel along a single path

1.3 Optical fiber — How it is made

CHAPTER 1: BACKGROUND

The manufacturing process of an optical fiber basically consists of two steps The first step is the fabrication of a specially constructed glass rod called a preform, and the second step is melting and drawing a preform to a thin optical fiber with the desired

diameter

1.3.1 Fabrication of preform

The main material used for an optical fiber is silica glass (SiO,) As described in section 1.2, choosing the correct indexes of refraction of the core and the cladding is very

important for transmitting light by total internal reflection The typical value of the index

of refraction of silica is 1.46, and it can be modified by a technique called doping, which means adding some special impurities to the material

Some examples of an impurity for doping, called dopant, are germanium (Ge),

phosphorus (P), fluorine (F), and boron (B) Some of impurities are in its glass form when added, and others are in elemental form If germanium in glass form (GeO,) or phosphorus in glass form (P,O,) is added to silica, the index of refraction is typically reduced If fluorine in elemental form (F) or boron in glass forrn (B,O;) is added, the index of refraction is typically increased Table 1.1 shows the change in the index of refraction of silica by each dopant (These data are quoted from the website of Cooper Union School, Engineering department given in the Bibliography.)

Table 1.1] — Change in index of refraction of silica glass

There are a variety of methods for fabricating a preform One common method developed

by Corning Glass is known as flame hydrolysis (Daly, 1984) In this method, vapors of

CHAPTER 1: BACKGROUND

purified chemical precursors, such as silicon tetrachloride, germanium tetrachloride,

boron trichloride and phosphorous oxychloride, are passed through a burner and

hydrolyzed at high temperature to produce mixtures of glass-forming oxides according

Figure 1.5: Flame hydrolysis process

The core material is deposited first, followed by the cladding material The rod is removed from the soot preform, and the preform is heated in a consolidation furnace to form a clear glass cylinder from which a fiber can be drawn

1.3.2 Fiber drawing

Although a preform can be fabricated with the same ratio in diameter of core to cladding

as the desired optical fiber, a preform is typically thousands of times larger in diameter than the desired fiber A preform is converted to an optical fiber with the desired size by drawing As shown in Figure 1.6, one end of the preform is heated, and it drops into a doughnut shaped furnace by gravitational pull The typical operational temperature is

CHAPTER 1: BACKGROUND

2100 °C The fiber is then fed onto a rotating bobbin The diameter of the fiber is

measured continuously by a laser scanner and that information is used to control the rotational speed of the bobbin which is pulling on the fiber as it cools Control of the draw speed is used to control fiber diameter To avoid undesirable corrosive attack of the glass by moisture, a primary coating of materials such as acrylate is applied to the fiber after drawing

Figure 1.6: Fiber drawing

CHAPTER 1: BACKGROUND

1.4 Optical fiber sensors

When light is guided through an optical fiber, its propagation can be altered by strain and/or temperature changes, causing changes in optical properties such as intensity, phase

or wavelength Those changes can be related to strain and/or temperature, thus forming the basis for optical fiber sensors Such sensors are categorized into the following two types depending on how the incoming light is transmitted in the fiber

(1) Extrinsic sensor

The light exits the fiber, is altered by strain and/or temperature, and then is transmitted to

a detector back through the same fiber or through a different fiber

(2) Intrinsic sensor

The light remains within the fiber and is affected by strains and/or temperature changes

Also, optical fiber sensors can be categorized into the following three types depending on the property of the applied light that is changed by the stains and/or temperature changes (1) Intensity-based fiber optic sensor: The intensity of the transmitted light is changed (2) Phase-based fiber optic sensor: The phase of the light is changed

(3) Wavelength-based fiber optic sensor: The wavelength of the transmitted or reflected light is changed

In this section, two of the most common fiber optic sensors are described: the Fabry- Perot Interferometer and the Bragg grating sensor

1.4.1 Fabry-Perot Interferometer (FPI)

1.4.1.1 Intrinsic Fabry-Perot Interferometer (IFPI)

As shown in Figure 1.7, the intrinsic Fabry-Perot Interferometer (IFPI) contains two parallel partially reflecting mirrors formed within an optical fiber