elastic analysis of a circumferential crack in an isotropic curved beam using modified mapping ...

73 Pages·2013·1.12 MB·English

ELASTIC ANALYSIS OF A CIRCUMFERENTIAL CRACK IN AN ISOTROPIC
CURVED BEAM USING MODIFIED MAPPING-COLLOCATION METHOD
A THESIS SUBMITTED TO
THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES
OF
MIDDLE EAST TECHNICAL UNIVERSITY
AYDIN AMIREGHBALI
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR
THE DEGREE OF MASTER OF SCIENCE
IN
AEROSPACE ENGINEERING
FEBRUARY 2013

Approval of the thesis:
ELASTIC ANALYSIS OF A CIRCUMFERENTIAL CRACK IN AN ISOTROPIC
CURVED BEAM USING MODIFIED MAPPING-COLLOCATION METHOD
submitted by AYDIN AMIREGHBALI in partial ful(cid:28)llment of the requirements for the
degree of Master of Science in Aerospace Engineering Department, Middle East
Technical University by,
Prof. Dr. Canan (cid:214)zgen
Dean, Graduate School of Natural and Applied Sciences
Prof. Dr. Ozan Tekinalp
Head of Department, Aerospace Engineering
Assist. Prof. Dr. Demirkan ˙(cid:246)ker
Supervisor, Aerospace Engineering Department, METU
Examining Committee Members:
Prof. Dr. Altan Kayran
Aerospace Engineering Department, METU
Assist. Prof. Dr. Demirkan ˙(cid:246)ker
Aerospace Engineering Department, METU
Prof. Dr. Ozan Tekinalp
Aerospace Engineering Department, METU
Assoc. Prof. Dr. Serkan Da§
Mechanical Engineering Department, METU
Assist. Prof. Dr. Ercan G(cid:252)rses
Aerospace Engineering Department, METU
Date:
I hereby declare that all information in this document has been obtained and
presented in accordance with academic rules and ethical conduct. I also declare
that, as required by these rules and conduct, I have fully cited and referenced all
material and results that are not original to this work.
Name, Last Name: AYDIN AMIREGHBALI
Signature :
iv
ABSTRACT
ELASTIC ANALYSIS OF A CIRCUMFERENTIAL CRACK IN AN ISOTROPIC
CURVED BEAM USING MODIFIED MAPPING-COLLOCATION METHOD
Amireghbali, Aydin
M.Sc., Department of Aerospace Engineering
Supervisor : Assist. Prof. Dr. Demirkan ˙(cid:246)ker
February 2013, 55 pages
The modi(cid:28)ed mapping-collocation (MMC) method is applied to analyze a circumferential
crack in an isotropic curved beam. Based on the method a MATLAB code was developed to
obtain the stress (cid:28)eld. Incorporating the stress correlation technique, the opening and sliding
fracture mode stress intensity factors (SIF)s of the crack for the beam under pure bending
moment load case are calculated.
The MMC method aims to solve two-dimensional problems of linear elastic fracture mechan-
ics (LEFM) by combining the power of analytic tools of complex analysis (Muskhelishvili’s
formulations, conformal mapping, and extension arguments) with simplicity of applying the
boundary collocation method as a numerical solution approach.
Qualitatively, a good agreement between the computed stress contours and the fringe shapes
obtained from the photoelastic experiment on a plexiglass specimen is observed. Quantita-
tively, the results are compared with that of ANSYS (cid:28)nite element analysis software. The
e(cid:27)ect of crack size, crack position and beam thickness variation on SIF values and mode
mixity is investigated.
Keywords: linear elastic fracture mechanics (LEFM), modi(cid:28)ed mapping-collocation (MMC)
method, stress intensity factor (SIF), circumferential crack, curved beam, pure bending, com-
plex analysis, conformal mapping, Muskhelishvili formulation, boundary collocation method,
stress correlation technique, ANSYS Mechanical APDL
v
(cid:214)Z
MOD(cid:157)F(cid:157)YE HAR(cid:157)TALAMA-E(cid:147)D(cid:157)Z(cid:157)ML(cid:157)L(cid:157)K Y(cid:214)NTEM(cid:157) (cid:157)LE (cid:157)ZOTROP(cid:157)K E(cid:135)R(cid:157) K(cid:157)R(cid:157)(cid:147)
(cid:157)˙(cid:157)NDEK(cid:157) ˙EVRERSEL ˙ATLA(cid:135)IN ELAST(cid:157)K ANAL(cid:157)Z(cid:157)
Amireghbali, Aydin
Y(cid:252)ksek Lisans, Havac(cid:25)l(cid:25)k ve Uzay M(cid:252)hendisli§i B(cid:246)l(cid:252)m(cid:252)
Tez Y(cid:246)neticisi : Yrd. Do(cid:231). Dr. Demirkan ˙(cid:246)ker
(cid:147)ubat 2013 , 55 sayfa
(cid:157)zotropik e§ri kiri‡teki (cid:231)evresel (cid:231)atla§(cid:25)n elastik analizi i(cid:231)in modi(cid:28)ye haritalama-e‡dizimlilik
(MMC) y(cid:246)ntemi uygulanm(cid:25)‡t(cid:25)r. Y(cid:246)ntemi saf b(cid:252)kme y(cid:252)k(cid:252) alt(cid:25)nda olan bu nesne i(cid:231)in kullan-
mak(cid:252)zereMATLABkodugeli‡tirerek;gerilmealan(cid:25)eldeedilmi‡tir.Gerilmeba§(cid:25)nt(cid:25)tekni§ini
y(cid:246)nteme katarak (cid:231)atla§(cid:25)n a(cid:231)(cid:25)lma ve kayma k(cid:25)r(cid:25)lma modlar(cid:25) i(cid:231)in gerilme ‡iddeti fakt(cid:246)r(cid:252) (SIF)
de§erleri hesaplanm(cid:25)‡t(cid:25)r.
(cid:157)ki-boyutlu linear elastik k(cid:25)r(cid:25)lma mekani§i problemlerin (cid:231)(cid:246)z(cid:252)m(cid:252)n(cid:252) ama(cid:231)layan MMC y(cid:246)ntemi,
kompleks analiz teorisinin analitik ara(cid:231)lar(cid:25)n(cid:25)n (Muskhelishvili form(cid:252)lasyonu, a(cid:231)(cid:25)-korur hari-
talama ve geni‡letme arg(cid:252)manlar(cid:25)) g(cid:252)c(cid:252)n(cid:252) s(cid:25)n(cid:25)r e‡dizimlilik metodunun n(cid:252)merik uygulanma
kolayl(cid:25)§(cid:25) ile birle‡tiriyor.
Hesaplanm(cid:25)‡olane‡y(cid:252)kseltigerilmee§rilerin,pleksiglasnumune(cid:252)zerindekifotoelastikdeneyin
frin(cid:231) ‡ekilleri ile nitel uyumu g(cid:246)zlemlenmi‡tir. ANSYS sonlu elemanlar yaz(cid:25)l(cid:25)m(cid:25) sonu(cid:231)lar(cid:25) ile
kar‡(cid:25)la‡t(cid:25)rarak nicel uyum ara‡t(cid:25)r(cid:25)lm(cid:25)‡t(cid:25)r. ˙atlak b(cid:252)y(cid:252)kl(cid:252)§(cid:252), konumu, ve kiri‡ kal(cid:25)nl(cid:25)§(cid:25)n(cid:25)n
SIF’ler ve onlar(cid:25)n oran(cid:25) (cid:252)zerindeki etkisi incelenmi‡tir.
Anahtar Kelimeler: linear elastik k(cid:25)r(cid:25)lma mekani§i (LEFM), modi(cid:28)ye haritalama-e‡dizimlilik
(MMC) y(cid:246)ntemi, gerilme ‡iddeti fakt(cid:246)r(cid:252) (SIF), (cid:231)evresel (cid:231)atlak, e§ri kiri‡, saf b(cid:252)kme y(cid:252)k(cid:252),
kompleksanaliz,a(cid:231)(cid:25)-korurharitalama,Muskhelishviliform(cid:252)lasyonu,s(cid:25)n(cid:25)re‡dizimlilikmetodu,
gerilme ba§(cid:25)nt(cid:25) tekni§i, ANSYS Mechanical APDL
vi
To my family, whom I met little during these years
vii
ACKNOWLEDGMENTS
I would like to express my sincere gratitude to my thesis supervisor, Professor Demirkan
Coker, whom this work would not have been possible without his superior guidance and also
encouragement.
I am deeply grateful to Professor Alan Zehnder for his brilliantly helpful non-pro(cid:28)t ebook [3].
My thanks also goes to the fellows who shared with me their o(cid:30)ce at the hangar building for
over a year; namely, friends of mine, Denizhan Yavas, for the photoelastic test result (Figure
3.1-(b)) and Miray Aydan Arca, Imren Uyar and Basar Kutukoglu, for their immense help on
enhancing my thesis defense presentation manner.
Words are inadequate in o(cid:27)ering my thanks to Mrs. Michelle Krummel, whose unique intro-
duction of the Texmaker editor [26] is highly recommended to everyone who starts learning
LATEX.
Finally; I wish to thank my parents, to whom I am indebted forever for all of their support.
viii
TABLE OF CONTENTS
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
(cid:214)Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
LIST OF SYMBOLS AND ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . xvi
CHAPTERS
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The problem perspective . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The MMC method, the big picture . . . . . . . . . . . . . . . . . . . . 3
1.3 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 FORMULATION AND SOLUTION . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Stress boundary condition equations . . . . . . . . . . . . . . . . . . . . 5
2.2 Force boundary condition equations . . . . . . . . . . . . . . . . . . . . 7
2.3 Moment boundary condition equations . . . . . . . . . . . . . . . . . . 8
2.4 The solution procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 RESULTS AND CONCLUSIONS. . . . . . . . . . . . . . . . . . . . . . . . . . 11
ix
3.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.1 Versus photoelastic caustics . . . . . . . . . . . . . . . . . . . 12
3.1.2 Versus FEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1.3 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1.4 Parametric study . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.4.1 The crack position e(cid:27)ect. . . . . . . . . . . . . . . 17
3.1.4.2 The beam thickness e(cid:27)ect . . . . . . . . . . . . . . 17
3.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
APPENDICES
A MUSKHELISHVILI’S COMPLEX REPRESENTATIONS . . . . . . . . . . . . 25
A.1 The stress function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
A.2 Stress (cid:28)eld equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
A.3 Stress boundary condition equations . . . . . . . . . . . . . . . . . . . . 28
A.4 Force boundary condition equations . . . . . . . . . . . . . . . . . . . . 30
A.5 Moment boundary condition equation . . . . . . . . . . . . . . . . . . . 31
B CONFORMAL MAPPING AND ITS AIM . . . . . . . . . . . . . . . . . . . . 33
B.1 The mapping function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
B.2 Kartzivadze’s continuation argument . . . . . . . . . . . . . . . . . . . 34
C THE LAURENT SERIES REPRESENTATION . . . . . . . . . . . . . . . . . 37
C.1 The Laurent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
C.2 The e(cid:27)ect of stress symmetry. . . . . . . . . . . . . . . . . . . . . . . . 37
D PURE BENDING OF A CURVED BEAM . . . . . . . . . . . . . . . . . . . . 43
x