()Submitted on 10 Apr 2014
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Multi-scale method for modeling thin sheet buckling under residual stress : In the context of cold strip rolling
Rebecca Nakhoul
To cite this version: Rebecca Nakhoul. Multi-scale method for modeling thin sheet buckling under residual stress : In the context of cold strip rolling. Other. Ecole Nationale Supérieure des Mines de Paris, 2014. English. NNT : 2014ENMP0005. pastel-00976843
MINES ParisTech CEMEF – UMR CNRS 7635
1, rue Claude Daunesse, CS 207 – 06904 Sophia Antipolis Cedex, France
École doctorale n° 364 : Sciences Fondamentales et Appliquées
présentée et soutenue publiquement par
Rebecca NAKHOUL
stress – In the context of cold strip rolling
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Doctorat ParisTech T H È S E
pour obtenir le grade de docteur délivré par
l’École nationale supérieure des mines de Paris
Spécialité “ Mécanique Numérique ”
T
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♣r♦♣♦s♥ s t ♠♦tt♥ st
t♦ t♥ r ♠ ♦r t ♣ ♥ ♥ ♦r
r♥ ♠ tr rs ♦ rsr
s ♦r ♦♥t ♦tt♥ s♦ r t♦t t ♥tt r♦r ♦♥trt♦♥ ♥
♦♦rt♦♥ ♦ ♦t Pr♦sss♦r P♦trrr ♥ Pr♦ss♦r ♠ r♦♥ ♦
♦ t♦ ♣rss ♠ rtt ♦r t♠
t♦ t♥ Pr♦ss♦r ♥ r ♥ ♥t t♥♥ ♦r ♥ ♠
t ♦♥♦r ♦ r♥ t r t♥ q Pr♦ss♦rs ♠♠ ♥ ♥P♣♣
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♦ t r t♥ s Pr♦ss♦r ♠ ♥ Pr♦ss♦r P♦trrr
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♦r t ♦♦ t♠s ♥ s t♦ ♠ r♥s ♥r r♦
♥ ♦♥ ♦ ♦♥ ss r♠ ♥ ♥ ♦ t s ♣sr
♠t♥ ♦ ♥ sr♥ ♦ t♠s ♥ ♠♠♦rs t ♦
♥ t♦♥ t♦ t♥ été ♦r ♥ tr ♥ ♥ srr♦♥♥ ♠ t
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♥tr♠t ♦
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a ♠♥s♦♥ss ♠♣t ♦ s♥s♦ t s ♥
t ♠ts ♠t♦
stt t♥s♦r
D ♣t ♦ t t
E ♦♥s ♠♦s
F r♦♥ ♦r
G ♥trt♦♥ ss ♣♦♥t
h str♣ t♥ss
it t♦♥♣s♦♥ trt♦♥
l str♣ t
L str♣♠♦♠♥ ♥t
Lbite t ♥t
Lt(.) t♥♥t ♦♣rt♦r s ♥
L0 t (.) ♥t t♥♥t ♦♣rt♦r s ♥
l ♥t ♦ s♥s♦ t
m n t ♥♠r ♦ s ♥ t xyrt♦♥s
♠ ♥♠r ♦ s ♦ t ♦rr srs s ♥ t ♠t
s ♠t♦
N σ.h
N0 T0.h
N0 N2 ♦rr ♦♥t ♦ N s ♥ t ♠ts ♠t♦
q ♥♠r ♥ t x−rt♦♥ s ♥ t ♠t
s ♠t♦
q1 q2 ♥♠r ♥ t x/y−rt♦♥ Qx Qy r ♦rs
R r♦s rs
S P♦r♦ strss t♥s♦r

T0 str♣ t♥s♦♥ strss
U ♠ r ♦♥t♥♥ t ♠♥tr rs ♦
♦r t ♠ts ♠t♦
U♠ ♦rr ♦♥t ♦ U
str♥ ♥r
m ♠♠r♥ str♥ ♥r
V t♦t ♥r
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rt♦♥s
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w0 w1 ♦rr ♦♥t ♦ w s ♥ t ♠ts ♠t♦
t r ♥ t rst ♥♦♣
α r♣ ♥ t♥ t str♣ ♥ t t♥ss r♦
αB ♣r♠tr s s t t♦ ♥ strss str
t♦♥
♦♠♥ t
γc ♦♠♣t ♣rt ♦ γ
γl ♥r ♣rt ♦ γ
γnl ♥♦♥♥r ♣rt ♦ γ
γ ♥♥ ♣rt ♦ γ
ε str♥ ♥r♠♥t ♦♠♣t tr ♦r t
♥ ♠♦
µf t ♣rtrt♦♥ ♦r s ♥
ε str♥ t♥s♦r
εms ♠sr str♥ t♥s♦r

♠
λcr rt ♦♥ ♣r♠tr
λres ♦♥ ♣r♠tr ♦r t rs strss s ♥
λten ♦♥ ♣r♠tr ♦r t str♣ t♥s♦♥ s ♥
λI λII ♥ str♥ ♦♠♣t ♥ t ♣r♥♣ r♠
♠
σres rs strss t♥s♦r
σcr ♦ rt strss
σMAN strss rstrt♦♥ tr ♥ ♦♠♣t
P♦t♥t ♥r

♠ ♦ ts st
♥ r ♦♥ t ♦♥ Pr♦ss
♦ r♦♥
r s♠♣ ♥ss ♦ t r♦ st s♥
tst♥ ♠s
♥♥ ♠sr♠♥t sst♠ ♦r t♥t ts
♥♥ ♠sr♠♥t sst♠ ♦r ♠♥st ts
♥♥ t♥ss ♦♥tr♦ ♥ tt♦rs
P♦str♦♥ t♥ss ♠♣r♦♠♥t
és♠é ♥ r♥çs
trtr r sts ♥ ♣ts ♥ t♦rs
sr♣t♦♥ ♦ t ♣r♦♠
♦♥ ♠t♦s ♦r ♥ ♥ ♣♦st♥ ♣r♦♠s
♥r ♠t♦s ♦r tr♠♥♥ t rt ♦
♦♠♥ ♥ qr♠ t♦
♦♠♥ ♥ ♥r t♦
P♦st♥ ♠t♦s
s♠♣t♦t ♠r t♦ s ♦♥ t Pr
trt♦♥ t♦r
ts t♦ s ♦♥ ♦rr srs ♣♥s♦♥
♥t t♦ s ♦♥ t ♠r♥ ♦r
♦ ♠t♦s ♣♣ t♦ r♦ str♣ t
♥♦♣ ♣♣r♦s ♦r ♦♥♥ ♣r♦♠s
♠♥t st s
♠♥t st sr
♥ ♥ss ♣r♦♠ t s ♥t ♠♥t ♠t♦
trt ♠t♦ ♦r ♣t ♥ ♥r rs
strsss
♥ s ♥t ♠♥ts ♦r♠t♦♥
♦♣ ♣♣r♦s ♦r ♦♥♥ ♣r♦♠s
♦♥♥ ♠♦ ♦♥
t♥ss tt♦rs ♦♥ ♥ s♥ ♠
rt♦♥ ♥ t♥ss
♥s♦♥ ♥ t♥ss
♦♥s♦♥
r♦♠ ♦♥ ár♠á♥ qt♦♥s t♦ ts s ♥ ♠♦
♦♥ ár♠á♥ qt♦♥s
♦♥ ár♠á♥ qt♦♥s ♦r ♥ ♣t ♥r rs strss
st♥ ♥ ♥♦♥♥q♥ss s♦t♦♥ ♦ t ♦♥ ár♠á♥ qt♦♥s
Ps♥rt ♦r♠t♦♥ ♦ t ♣r♦♠
r♦♠ ♦♥ ár♠á♥ qt♦♥s t♦ t ts ♣r♦♠
♥t ♠♥ts srtt♦♥

♦♥r ♦♥t♦♥s ♥ Pr♦♠
srtt♦♥ P♦st♥ Pr♦♠
♦♥r ♦♥t♦♥s P♦st♥ Pr♦♠
♠r rsts tt ♣♣rt♦♥
s ♠♣
♠r rsts ♥ttt t♦♥
♥ t♦♥ s♠♥t s♦t♦♥s
P♦st♥ t♦♥ t s ♠♥ts ♠♦
t♦♥ s♠♣ qs ♠♣
sss♦♥s
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♥♦♣ ♦♥♥ ♦
t♥ss ♥ rt♦♥
♦♣ ♦♥♥ ♦ ♠
sts ♠ s ♠ ♥ ♠

♦r♣

Mt sts r s ♥ ♠♥ r♥t ♥strs ♥ ♠♥ ♣♣
t♦♥s s r ♦♥strt♦♥ ♦r♥ r♦♦s ♦rt♦♥ t♦♠♦t r♦♥t
♥ ♠♦r ♥r tr♥s♣♦rt ♥str ♦♠st tr ♣♣♥s ♣♥ ♥ ♠♥
♦tr ♣♣t♦♥s ♥♥ ts st♦r ♥ts


♦tt♦♥ ♦ ts st
T ♠♥tr♥ ♦ ts ♦♦s ♥s ♦ r♦♥ s ♠♦r st♣ ♣r♦♥
t s♠♥s ♣r♦ts r t♥ st♠♣ ♦r ♣r♥ ♥t ♥ t t
st ♠t ♦r♠♥ tr♥♥t qt rqr♠♥ts r ♠♣♦s ♦♥ ts s♠♥s
♣r♦ts ♦♠tr s ♦♥ ♦ t ♠♦st rt ♦r♠t ♦ t sts t
t♥ss ♠st ♦t♥ ♣rs ♥ t s tt ♦rs s ♣♦ss trt
s ♥ ♦r r ♠tr ♦st ♦r s t♦ tr♠♠ ♣r♦ss♥ ♦st ♥ ♠tr ♦ss
t♥ss ♠st r ♣rs ♥ ts s ♣rss ♥ µm ♦r♦r ♦r s ♦
♠♥♣t♦♥ t t st♦♠rs t t♥ss rt♦♥s r♦ss t t str♣ ♣r♦
♦r str♣ r♦♥ ♥ ♦♥ t ♥t ♠st ♣t t♦ ♠♥♠♠
r r♥t t♣s ♦ t♥ss ts ♥♦♥tr r♥ ♦ r♦♥ r♦♠ t t♦ rt s ♥ ♥tr s st
s♣ ♦r t♥ss s ♠♦st rt s ♥ ♣rtr r♦♠ t♥ss ♠ rst ♥
♣♦♦r rtr ♣r♦sst ♥ ♥♣t ts ♦ t ♥ ♣r♦t s s t s
t ♦ ts ♠♠♦r ♠♥ t♦rs ♦ t♥ss ts r s♦♥ ♥ r
s ♠♥ ♣r♦♠s ♦r ♦t ♣r♦rs ♥ st♦♠rs ♥rt ♠♥♣t♦♥
♦r ss♠♥ ts ♣t rtr ♣r♦ss♥ t rs st♠♣♥ ♠♦r♦r tr
s♣t ♠ r♠♥ ♥ tr ♣♥t♥ s ♥♦t ♣t t r ♦ ♥
♣♥s r ♦r ♥ r ♥
♦♠♠♦♥ ♥tr ♦ ts r♥t s♣s s st ♥ ♠♥ ♥stt
♥s s♣r♦s ♦t♦♣♥ s♣♠♥ts ♥ rt ♥♣♥ ♦♠♣rss
♦ s ♣♣ ♦♥ t♥ ♣r♦t r ♦r♥ s t r♦ st ♦r♠t♦♥ rsts
♥ tr♦♥♦s t♥ss rt♦♥ ♦ t st ♥ r♥t ♦♥t♦♥ s ♥
tr♥ ss s♥ rs strss s ♥ t ♣r ♥ ♣♦stt rs t ♣rt
♦ t st ♥ ♥ ♦♠♣rss♦♥ ♦♠♣rss♦♥ ♦ r t♥ st ♠♦st ♥ssr
♠♥s ♥


♥ ♥rt ts ts r t t♦ ♠♥t s rqrs r ♥s♣
t♦♥ rtr ♣r♦ss♥ s s ♥ t t ♦rrs♣♦♥♥ ♥st♠♥t ♥ ♦r
♦st t♥ss ts s♦ tr♦r ♣r♥t ♦r t st ♣t t♦ s ♦
tt tr ♦rrt♦♥ ♥♥ ♦♣rt♦♥ s s ♦♥ ♠♣♠♥t♥ ♣♦r
♥ ♦♠♣ ♦♥♥ ♠sr♠♥t sst♠s strss♠tr r♦s ♦♠♣ st♥s t ♥
♠r♦s tt♦rs r♦ ♥♥ st r♦s t ♥ t♦♠t ♦♥tr♦ t♥qs ♥
♦rt♠s ♥ ♦r ♦ s ♥ ♥ t ♥t ♣tr
ts q♣♠♥t s st s t ♠♥s ♦ t t♥ss ts ♥ t rs♣♦♥s
t♦ tt♦♥ r rt ♥♦♥ ♥ t♦ ♥ ♣rts ♥ ts ♠tt♦r
♦♣ ♣r♦♠ s t♦ ♦♣ r♦♥ ♠♦ ♣ ♦ ♣rt♥ t stt ♦ t ♦
r♦ str♣ s ♦♥ ♥♦r♠t♦♥ trt r♦♠ s ♠♦ ♦♥ ♦ st t
r♦♥ ♦♥t♦♥s ♥ ♣r♥t t♥ss ♣r♦♠s
♦ ♦♠♣t sr s♣t ♥ ♠♥ ♣r♦♣rts r ♦tr ♠♣♦rt♥t rqr
♠♥ts ttr s ♦ ♥trst t♦ s s ♥ rr ♦s r ♥ ♣t ♥ sr t♦ ♣
tr rs ♦r ♥r♦♥♠♥t rs♦♥s rr ♦s ♥ t♦ r r♦♥
♦s ♥ ♠♦r r♦ ♦r♠t♦♥ rst ♥ ♥♥ t♥ss ♦♥tr♦ ts
♥ r♠r ts t♥ ♦♥str♥ts ♠st ♦♠♥ t ♠♠♠
♣r♦tt ♥ ♦rr t♦ ♣r♦ ♥ ♣r
♠ ♦ ts st
Ar♦rtt ♥ ♦♥st♠ t♦ ♠♦r ♠t sts ♣r♦rs ♦♥ ♥t
♠♥t t♦ s r♦♥ ♠♦ ♥♠ ♠ ♥ t s ♦ st r♦♥
t ♠♦ ♦♣s r♦ tr♠♦st ♦r♠t♦♥ t str♣ ♣st ♦r♠t♦♥
s ♦s ♦♠♣t♥ t strsss ♥ str♥s ♥ t r♦ t r♦ ♦s ♥ t♦rqs r♦
♦r♠t♦♥ ♥ str♣ tr♥srs t♥ss ♣r♦ tr♠ s t s♦ tr♠♥s t
♦♠♣ ♣♦stt strss rst♥ r♦♠ tr♦♥♦s str♥
s ♣t s s ♣♦t ♦r t s♠t♦♥ ♦ r♦♥ ss r sts r♠♥ t
♦♥ t ♠ ♥r ♥trst♥ ♦r ♦♥ str♣ t♥s♦♥ ♦rs t t st♥r rs♦♥ ♦
♠ q♥ s ♥♦t q♣♣ t♦ ♣rt ts sts ♦ ♦♥ ♥
♥ s ss ♦♥ ♥s ♠♦r r♥s t♥ ♦♠♣t ♥ ♠sr ♣♦stt strss
♣r♦s s t ♥ t♦r♦ strss rstrt♦♥ ♦rs s ♥♦t ♣tr
♥ t s♥ ♦ ♥ ♦♠♣t♥ ♦rt♠ t s ♥ s♦♥


tt ♦rt♥t ts rr♦r ♠ ♦♥ t ♦t♦t s s ♥♦ s♥♥t ♠♣t ♦♥ t
♥t strss ♥ str♥ s ♦rts ♥ r♦ ♦r♠t♦♥ ♦r ♥ str♣ ♣r♦ t
♠t♦♥ s ♠t t♦ s rsts t s♦tr ♦rs ♣rt
t ♣♦stt strss s ♥ ♣t rst ♦r ♦rt♦r t♥ss ts r t
st t st♥r rs♦♥ s ♥♦ ♠♦r s♥t s ♠tt♦♥ s s ♦t
tr♥s♣♦rt♥ t ♥♦rrt strss ♥t♦ strtr ♠♥s t♦♦ tts
t ♥ rt♦♥ t♥ t ♦♥ ♦r ♥♦t t s ♣♦st♥
♣t t s♦ t s♣ ♦ t ts ♠♦ ♥t ♠♣t ♥ t
♦rrt strss ♦t tt ♦t t s♣s ♥ strsss ♥r str♣ t♥s♦♥ ♦rs
♦♥ t ♠ ♥ tr t♥s♦♥ ♥t♦♥ t t st♦♠rs r ♦ ♥trst
t♥ t ♣t ♦ ♠ t♦rs t♥ss ♣r♦♠s ♠
♣♠♥t♥ t♦ ♠♦s
t t strss tr♠♥t♦♥ ♥s ♠ str♦♥ ♦♣♥ t
♦♥sttt ♠♦ ♦s ♥ ♣♦ss st♣st ♦r♠t♦♥ ♥
♥trt♦♥s t♦ t♥ ♥t♦ ♦♥t ♥ t ♦tr ♥ ts ♠♦ s ♠t t♦
♥ ♦♥ ♥ ♥r t♥s♦♥ ♦rs t ♥ tt ♥ t ♦s ♥♦t t
s♣ ♦ t ts t ♦s ♥♦t rqr ♠ ♠♦r t♠ t♥ ♠ ts
♠ ♦♣s ♥ str s♠ t r♦♥ s♦tr t ♣♦r
s s♦tr ♦♥♥trt♥ ♦♥ ♥ ♥ ♣♦st♥ sts t♥s t♦ ♥
♥ ♥♦♥♥r s♦r s ♦♥ ♠t♦ s♠♣t♦t ♠r
t♦ ♦♣♥ ♣r♦r s ♥♦rt♥t ♦♠♣ ♥ t♦s ♥ t
rqrs ♦♥ ♦♠♣t♥ t♠s t t s ♦ q♥t ts ♥ rs♣ts
sr♥ t ♣♦st♥ stt ♦ t str♣ ♥r t♥s♦♥ s s tr t♥s♦♥
♥t♦♥ ♥ s♥ ♦♠♣tt♦♥
♠ ♦ ts st s t♦ ♦♣ ♥ ♠♦ ♦♠♥♥ t ♥ts ♦ ♦t
♠♦s t s♦ t♦ tt t t ♦♥ sr t t♥ss t ♠♦
♥ q♥t t ♠♣t ♥ ♥t ♣rt t ♣♦st stt ♦ t str♣
♥♥ r strsss t t s♦ s♠♣ ♥♦ t♦ ♦♠♣t ♠♠rs ♥
t r♦♥ ♠♦ ♠ ♥ str♦♥ ♦♣♥ t strss tr♠♥t♦♥ ♥ t
♦t s t♦ ♣r♦ t ♥str ♣rt♥rs t t♦ t♦♦s s ♦♥ t s♠ ♥
t♦r
♦♣ rs♦♥ ♥ s s ♣♦st♣r♦ss♦r ♦r t strss tr♠♥
♠
♦♣ rs♦♥ r♦♥♥ ♠♦ s♠r t♦ ♠ t
t ♥ t♦♥ ♣t t♦ q♥t t ♦♠tr ♦ t♥ss ts
trtr ♦ ts ♠♠♦r
♣tr ♣rs♥ts ♥ ♦r ♦ t r♦♥ ♣r♦ss ♦r ♥♠r rrs t
♦s ♦♥ t♥ss ts t r♦♥ ♠s ♥ s♦ r s tr trs r ♦♥♥t
t t♥ss ts t ♠♥ ♦r♥ ♦ t♥ss ts tr ♠sr♠♥t
♦♥tr♦ sst♠s
♣tr srs trtr ♦♥ t r♥t ♠t♦s ♥ t ♥ s♣
t♦s ♥ ♣♣ t♦ t♥ss ts ♥ r♦♥ ♣r♦sss
♣tr ♦♠♣ts t sr♣t♦♥ ♣♦t♥ ♦♥ ♦ t♦s t ♥ ♠♦ ♠
♣♠♥t ♥ ♠ ♥ q st ♦ t ts ♦ r♥t
r♦♥ ♠ tt♦rs ♦♥ t♥ss ts
♣tr srs t ♠ts ♥ ♠♦ t ♥tr ♦♣♠♥t
♦ ts P ♦r t♦rt ♣rs♥tt♦♥ s ♦♠♣t t s♠♣ ♣♣t♦♥s
t♦ ♠ ♠♣s t ♦♠♣rs♦♥ t♦ rsts ♥ t trtr
♣tr t ♠ts ♥ ♠♦ s s ♥ ♦♣ ♠♥♥r t♦ ♣rt
♦♥♥ t♥ss ts ♦r r r♦♥ ♦♣rt♦♥ tst♥ ♥ t♦♥ ts rs♣♦♥s
t♦ ♥♥ rt♦♥ ♥ ♦r r♦ ♥♥ ♦r
♣tr ♣♦rs t ♦♣♥ ♦ ♠ ♥ ♥ ♦♥r♦♥t♥ t
rsts t ♣r♦s ♦r
♥ ♥r ♦♥s♦♥s ♥ r♦♠♠♥t♦♥s r r♥



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s ♥♦t ♦t t♥ ♦r t st♦r♠ t♦ ♣sst s ♦t r♥♥ t♦
♥ ♥ t r♥
P♦ ♦♦
♦♥t♥ts
♦ r♦♥
r s♠♣ ♥ss ♦ t r♦ st s♥
tst♥ ♠s
♥♥ t♥ss ♦♥tr♦ ♥ tt♦rs
P♦str♦♥ t♥ss ♠♣r♦♠♥t
és♠é ♥ r♥çs

♣tr ♥ r ♦♥ t ♦♥ Pr♦ss
I♥ ts ♣tr q ♦r ♦ t r♦♥ ♣r♦ss s ♣rs♥t t♦ s♠♣ t s
t t♦ ♥♠r rrs ♦t ♦ ts r sr♣t♦♥ s ttr ♥rst♥♥
♦ t ♣r♦r ♥ t ♦r♥ ♦ t t♥ss ♣r♦♠s ♥ tr r♥t t♣s ♦r
♠♦r t sr♣t♦♥s t rrs r ♥t t♦ rr t♦ ♠♦r ♦♠♣t s♦rs s
s ♦rts ♦rts ♥r ♥ Ptt♥r ♠♥
r♦♠ strt t♦ ♥s t ♠t ♦r♠♥ ♣r♦ss s sq♥ ♦ ♦♥t♥♦s s♣r♦sss
tr♥s♦r♠♥ r ♠tr t♦ s♠♥s t♥ ♥s ♣r♦ts r t♦ s ♠♦st
♠♣♦rt♥t ♦♥s r
♦r t st r♥ tr r♥ ♥ r♥ r
♦r ♠♥♠ tr♦ss ♠t♦ ♥ ♠♥♠ ♦r s rst ♦♥rt ♥t♦ ♣r
♠♥♠ ♦ Al2O3 t r Pr♦ss ♥ t♥ tr♦ ♥t♦ ♠♦t♥ ♠♥♠
♦♥t♥♦s st♥
♦ r♦♥
t ♠♥ ♦s ♦♥ t st ♠♥ ♣r♦ss s ♥ ♠♣
r♦♠ st r♥ t♦ ♦ r♦♥ sr ♦
t ♠♥
I♥ t st r♥ r♦♥ ♦r ♦ ♥ ♠st♦♥ t r ♠trs ♦ st r
♦♥t♥♦s s♣♣ tr♦ t t♦♣ ♦ t r♥ r♦♠ ♦ ♦♥t♥♦s st
♦ s♣rt r s♦t♥ t r ♠trs ♥ t ♠ rt♦♥s tr♥s♦r♠ t♠
♥t♦ ♠♦t♥ r♦♥ ♠♦t♥ ♠t ♥ s r t♣♣ r♦♠ t ♦tt♦♠ ♥ st
ss t r♦♠ t t♦♣ ♥ st sr♣ t♦ ♠♦t♥ r♦♥ ♥t♦ ♥ ♦♥ r♥ t s
♥♦ tr♥s♦r♠ ♥t♦ ♠♦t♥ st s r ♠♦t♥ ♠t s ♦♥t♥♦s
st♥ ♣r♦ss ♦♥t♥♥ r♥t ♠♦s s♣s ♦ t ♠♦s tr♠♥ t
♦r♠s ♦ t s♠♥s ♣r♦ts tr ♦♦♥ strt♥♥ ♥ tt♥ ♦♦♠s

♣tr ♥ r ♦♥ t ♦♥ Pr♦ss
s s ♣ts sts rs ts ♥ strtr st♦♥s
♦rs st t♦ ♦r♠♥ ♥ ♥s♥ tr♠♥ ♥ ♠♦r ♦ t st r
trsts s♠♥s s♣s r r♥ t♦ ♥ s♣t♦♥s t♥ t♦ ♠♥
r♦♥ ♣r♦sss ♥t r♦♥ st♣s r ♦♥ ♦t ♦♥ t♥ ♦♦ ♦
♦♥ ♥ s r s ♣rs♥t st s t t♦ t r♦♥ ♣r♦sss
♥ ♣rtr t♦ ♦ r♦♥ ♦ t♥ str♣s ♥ t♦ t ♥str ts ♥♦♥tr
r♥ ts ♣r♦ss s s t♥ss ts r♦♠ ♥♦ ♦♥ ♦♥ t r♦♥ s ♦♥sr
♥ ♠t s rsst♥t t♦ s♣♥ ♥ ♦ t ♠st ♦t r♦ rst s
ss r rt ♥ r♥s r t ♦t♦♥ ♦ t ♣ t♠♣rtr s ♠♦♥t♦r
♥ t rt t♠♣rtr s r t ♣ s s t♦ r♠♦ ♥ t ♣ s
t♥ ♥t♦ ♦t r♦♥ ♠s r♥ ts st♣ ♣s r ♦r♠ ♦ tr rrs
tt♦♥ t♠♣rtr ♠♥ rs♦♥ ♦r ♦t r♦♥ s tt t t t♠♣rtrs
♠ts ♥ ♥ ♦♠ ♠♦r t s t t♥ss rt♦♥ ♠
sr
♦t r♦♥ ♦rs t ♣♦sst ♦ ♣r♦♥ r r s♣ ♥ ♥♦r rt♦♥
♥ s♥ ♦r♥ st♣ t♦t s♥ r ♠♦♥ts ♦ ♥tr♥ strss ♦r rs ♦t
r♦♥ ♥ r♠♦ s♦♠ ts tt ♦r ♥ st ♠ts t ♥ ♦s s ♣♦ts
s ♦r ♦s ♥ st t ♥ t ♠ s♦ r ♣ ♥♦♥♠t s s♦♠t♠s
t t ♥ t ♠t ♥s♦♥s
sr ♥s ♦ ♦t ♦r ♠t t♥s t♦ ♦rs ♥ r♦ s r♦s
r ♥ tr s t ♦♥st♥t ♣r♦♠ ♦ s ♦r♠t♦♥ ♦♥ t sr ♦ t ♦t
♠t t♦ ♦t♦♥
r♥ ♦t r♦♥ st♣s s s ♦r♠ ♦♥ t ♠t s ♠st r♠♦ ♦r
♦ r♦♥ ♦trs sr ts ♦r r ♣r♦sss r s ♥ t ♠♥♠
♥str t s♥ ♦ t st st ♣t s sr♣ ♠♥ ♥♦ rtr s
♥ ♦♣rt♦♥ s ♥ ♥ st ♦t r♦♥ tr t s♥ s r t ♠♦st rq♥t
♣rssr tr t rs ♠♣ts t s str♣ sr t rst♥ rt
♦♦♥ ♥ t t♠♣rtr r♥t ♣ r strsss rtr t ♦
♣ t ♦ ♥ t r♠♥ts r ♥ s t tr s s ♦♥ rst t
t t ♦ t rt♥ r♥s ♥ t♥ s♦♥ t♠ tr t r♦♥ ♠
♦r t ♥tr ♥t♦ t ♥s♥ ♠ s ♦ rr♦s ♥ t ♥s♥ ♠ ♥ ♦♥
t r♥♦t t s♥ s ♥ ♥ st ♦r ♦ r♦♥ s s s ♦♥

♦♦♠s rs 150 t♦ 400 mm2
s ♣r♣♣s 500 t♦ 1800 mm ♥ 50 t♦ 300 mm t ts rs 40 t♦ 150 mm2

r t ♠♥ ♣r♦ss tst st

♣tr ♥ r ♦♥ t ♦♥ Pr♦ss

♦ r♦♥
Mts r ♣st ♦r♠ t r♦♦♠ t♠♣rtr ♦ r♦♥ s ♣♣ tr
♦t r♦♥ t♦ ♦t♥ ♥ ♥ rt ♠♥s♦♥s ♦ t ♥ ♣r♦t s s
ttr sr ♥s ♥ ♦♦ ♠♥ ♣r♦♣rts
♦ r♦♥ ♣rs♥ts sr ♥ts
♥♥r s r ♣♦ss t♥ss < 0.5 ♠♠
♦♦♥
s tr s ♥♦ s♥♥t ♦t♦♥
♥ ♣r♦♣rts ♦ t sts ♥ ♦s ♦♥tr♦ ♥ sr t
str♥t ♦t♥ r♥ ♦ r♦♥ ♥ rt♥ tr♥t tt ♥
rst♦r ♥♥♥ ♥
r s strss ♥r t r♥ r♦♥ ♣r♦sss tr ♣♥ ♠
r ♦♥ ts s ♣♣r rs ♥ t ♥tr ♦ t str♣ rs ♥ t♦r♥ tr ♦s♦r
ts ♠ ♣rs♥t ♦♥ t srs ♦r ♥ t ♣r♦ts r ♥sr s
t ♦♠♣r♦♠s sr ♣♣r♥ ♥ rs t str♥t ♦r♠t ♥ ♦tr
♠♥tr♥ rtrsts r s♦s r♥t t♣s ♦ t ♦♥ r♥ ♦
r♦♥ ♦r ♠♣ rs r t♦ ♣♦♦r ♠tr tt t t r♦♥ t♠♣rtr
♦r t ♠♥ ♦s ♦ ts st s ss ♦ ♦♠tr ts t♥ss
ts t ♥ st♦♥ ♥ ♥ ♦ r♦♥ sts 0.1 mm ∼ 1 mm t♥ss
s♦ rt s♥stt t♦ rs strss ♥ ♥rt s♦♠ t♥ss t r
strts t ♣♣r♥ ♦ ♥♦♥♥♦r♠ rs strss ♥ t t♥ss ♦r ♥st♥
tr ♦ r♦♥ ♦s ♠ s♣♣ rt r♦♥ ♣r♦♣♦rt♦♥ ♦ st str♣s
s ♦r ♦t t♦ ♠♣r♦ rs sr ♣r♦♣rts ♥ ♠♦st ss str♣s ♥r♦

♣tr ♥ r ♦♥ t ♦♥ Pr♦ss
rtr ♣st ♦r♠t♦♥ ♣r♦sss st ♠t ♦r♠♥ rqrs tt ♥
♥♥ s ♥tr♦ ♥ s ss ♦r r♦♥ st t ♠st ♦♦ t♠♣r
r♦♥ ♦r s♥ ♣ss r♦♥ s♠ rt♦♥ ♣ss ♠♣r♦s t♥ss ♥
♠♣rts s♣ sr t♦♣♦r♣ ♦♥ t str♣ sr
♦ r♦♥ ♠s
sr♣t♦♥ ♦ t ♠♥ st♥ ♦♠trs
T r r ♠♥ t♣s ♦ r♦♥ ♠s t♦ rt♥ t♥t ♦♥ ♥♥♦t ♥ t♦ ♦
♥ ♥ ♠♦r♥ ♦ r♦♥ ♣rt t♦ ♦r s ♥ strt♣
♠s ♥♥ t ♥♠r ♠s ♦♥sttt t ♣r♥♣ ♠♣s ♦ s♥st♥
♠s r t ♥♠r ♥ s ♦ tr r♦s
♦s r ss s ♦r r♦s ♥tr♠t r♦s ♦r ♣ r♦s
♣♥♥ ♦♥ tr r♦
♦r r♦s r t ♦♥s tt ♥tr rt ♥ ♦♥tt t t r♦ ♠tr ♥
♣♣ t ♦rs tt r t st♦♥ r ♠ ♦ ♠tr rr t♥ t
str♣ st r♦♥ st ♥ ♦r q♥ ♥ t♠♣r ♦t ♦r♥ t♦♦ sts
♣ ts ♦r ♦t r♦♥ r sts ♦r ♦ r♦♥ rr ♠
st rr ♠tr ♥ t ♥tr ♥ s t s s t♦ r♦♥ s
tr ♥ ts ♠♣t ♦♥ t♥ss sss ♥ st♦♥
♣ r♦s r ♥t♥ t♦ ♣r♦ r s♣♣♦rt t♦ t ♦r r♦s t♦ ♣r♥t ♥
♥ ♣♦♥ r♦♥ ♦ s rr ♠tr t♥ t ♦r r♦s ♣ t♦ t♦
♦r tr t♠s rtr
♦ ♠ strt ♥ r s t ♦st t♣ s ♦r ♦ rt♦♥
♦ st ♦s t s s ♠♥ ♦r t♠♣r r♦♥ ♦r s
trt ♥ r s r t ♠♦st rq♥t ♠ st♥ ♦r ♦ r♦♥ ♦ str♣ t
s ss♥t ♦r♠ t♦ ♦r r♦s ♣ t t♦ r r♦s r t♦♥ s
♦♠♣♠♥t ♣r♦ ♥ t♥ss tt♦rs t ♠♦st ♦♠♠♦♥ r t ♥♥ ♦ t
♠ r♦s ♦r ♦ ♥♥ ♦r ♣ ♦ ♥♥
s ♠ rr♥ ♥ s rs strt ♥ r ♥s ♣♣t♦♥ ♠♥
♥ ♥ ♣sss r♥ ♦ r♦♥ ♥ t str♣ ♦s tr♦ t ♣ t st t
♣r ♥tr♠t r♦s ♠♦ tr ♥ t♦ ♥ ♦r s♥ tr t t
str♣ s s♦♥ ♥ r t t♣r s♣ ♦ t ♥tr♠t r♦ ♥
♥♥♥ s t trt♠♥t ♥ t ♠t s t t♥ ♦♦ ♥r ♦♥tr♦ ♦♥t♦♥s t♦ r ♥tr♥ strsss ♥♦r s♦t♥ t ♠t ♥ rst♦r ts tt
♦r ♠♦r ts st♦♥

♣tr ♥ r ♦♥ t ♦♥ Pr♦ss
♦r ♠♦r s♣ ♦r t ♦r r♦ t♦ t ♥ ts ss rt♦♥ ♥r t ♥
ss ♥ss ♦♠♥ t t r♦ ♥♥ s t♦ ♦rrt ♥tr ts ts ♠
♦s ♦♣t♠ t sts
r ♦ tr ♦rts
r ♦r tr ♦rts


♣tr ♥ r ♦♥ t ♦♥ Pr♦ss
♦ t st s♠r ♠tr ♦r r♦s ♠♥t♥♥ rt t ♠st
s♣♣♦rt t ♥♦t ♦♥ ♥st rt ♥♥ t s♦ ♥ t ♦r♦♥t ♣♥
r♦ str ♠s ♦r♥ ♥ ♦r rs strt ♥ r r ♣t ♥t♦ s
♦r ts ♣r♣♦s ♦r tr s ♠t t♦ t rt♦ tt♥ t♦ str
♥r♥ t ♠st ♥♦t t♦ ♦tr ♠♥ ♦♥tt t t ♦
♦r♦♠ ts t st♥s t ♦r♥ ♥ sr rs ♥ s♥
s s t t ♠ strt ♥ r ♥ st♣ rtr t ♥♠r
♠s r ♦♣ ♥ t s r ♥ ♥ ♦r ♣t♥ ♦r
t♦ ♥♠♥t rs♦♥s rst ts ♠s ♥ t ♠♦r ♦♥♦♠ t♥ ♥♦♥
rrs ♦r ♠s ♦ ♦♠♣r rt♦♥ ♣t ♦♥ t s♠ ♦r r♦s
♠trs s ♠♠ ♦ ts str ♠ ♣r♦♠s ♦r ♦r♥ ♦rs ♥
♠ ♦sr r t♥ ♦tr ♠s ♦rts
r s♠♣ ♥ss ♦ t r♦ st s♥
W t t ♠ ♦ ♣♥♥ t r♥t ♥s ♦ st♥s ♥ tr ♦t♦♥ t s
♠♣♦rt♥t t♦ r ♦ r♦ ss t t r♦ ♦r ♥ ♦ r♦ ♦r ts st
r♦ ♦r♠t♦♥ ttr s rtt♥ s ♦♦s ♥ tr♠s ♦ R t r♦ rs h ♥ l
t str♣ t♥ss ♥ t Lbite t r♦str♣ ♦♥tt ♥t h t str♣ t♥ss
rt♦♥ µ t rt♦♥ ♦♥t ♥ σ0 t r strss ♦ t str♣
F = σ0.l.Lbite.(1 + κ.µ. Lbite
√ h.R
s qt♦♥ s ♥♦t s♥t t♦ rt ♣rt t r♦ ♦r t♦ ts ♦r s♠r
♦r♠s ♥ s ♦r ♦♥ ♥ ♠♦s t♥s t♦ t tt♥ t♦r κ t s ♦
♦♥ ts ♦t♦♥ t t r♥t ♥trst♥ ♣r♠trs s s t r♦ rs t s♦s
tt t r♦ ♦r s t st ♣r♦♣♦rt♦♥ t♦ t sqr r♦♦t ♦ t r♦ rs tr♦
Lbite ♦r ♥ rt♦♥ ♦♥t t ♥ t♥ss s♠r rs s ♦r r♦
♦r t♦ t s♠ rt♦♥ h
♦r r♦ ♦ ♠♥s tr tt t ♠ ♥ t ss str♦♥ ♥ tr♦r ♣r

♣tr ♥ r ♦♥ t ♦♥ Pr♦ss
r t ♦ t r♦ ♦r ♠t♦♥ ♥r t r♦♥ ♦ s♦♥ r♦ ♥♥ ♥ tt♥♥ t ♦♥tt rt♦s t
r r♦ tt♥♥ ♦♠♣♦♥♥t ♦ r♦ ♦r♠t♦♥ s r♦ ♣r♦♣♦rt♦♥ t♦ t r♦♥ ♦ ♦♥t♠t♦♥♥t
s♦♥ ♣♦♥t ♦ ♠♣♦rt♥ s r♦ ♦r♠t♦♥ s♠t sr ♥ r
♥r t ♦♥tt strss r♦s ♥r♦ ♦♠♣ st ♦r♠t♦♥
s ♦t♥ s♠♠r t♦ ♠♦r ♦♠♣♦♥♥ts
r♦ ♥♥ ♦♥sr♥ r♦ s qs ♥r ♠ ♦ t r♦♥
♦r F strt♦♥ ♦♥ ♣rt ♦ ts ♥t r ♥ st♠t ♦ t t♦♥
t ♥ ♣♦♥t x ♦♥ t ♠ ♥ ♥ r ♦♦♥♦
δx = F
− x(L− L′)4 + L(x− L′)4]
I = π4

♥ L ♥ beam r rs♣t t ♥r ♠ ♥t ♥ ♠tr F
t r♦♥ ♦ ♣♣ ♦♥ ♣rt L′ ♦ ts ♥t E ♦♥s ♠♦s ♥ I t
r ♠♦♠♥t ♦ ♥rt
s s♦s tt r r♦s r ♠ ♠♦r rsst♥t t♦ ♥♥ ♥ s♠
r ♣rrr r ♥ssr ♦r♠ s♦ s♦s tt t ♣r♦♠ s
♠♦st rt ♦r ♠s r L ♥ ♦ ♦rs ♦r r ♠trs r ♦♥tt
strss
r t♦♥ ♦ s♠♣ s♣♣♦rt ♠ ♥r ♥♦r♠ ♦
r♦ tt♥♥ s t ♦ s♦rt♥♥ ♦ r♦ rs ♥r t ♦♠♣rss♦♥ ♥r♦♥
♥ t t t s ♠t t♦ t ♠♠t ♥t ♦ t ♦♥tt sr ♥ s t♦
rst ♣♣r♦♠t♦♥ ♣r♦♣♦rt♦♥ t♦ t ♦ ♦ strt♦♥ ♦r strss s r
♥ ♠♣♦rt♥t tr s tt t rr t r♦ rs t rr tt♥♥
s ♥ µ♠ ♥ t ♠♦r t t s tr♦r t♦ ♦t♥ ♣rs ♠♥s♦♥s ♥
♣rs s♣ r t s t tr♥st♦♥ t♥ t ♦♥tt r ♥ t
r♦ ♦ts t ♦♥tt s rt t♦ t r♦♣ t ♥ ss ♦ rt♦♥
s s♦ str♦♥ ♦♥♥t t t♥ss ts ♥ t s ♦ t♥ str♣s
♦ tt♥♥ ♥ ts t ♦♥ t ♦♥tt ♥t s ♦t♥ st♠t r
♣♣r♦♠t t♦s ♦r♠ s ♦♥ ♣♣r♦♠t♦♥s ♠ t
r♦ str♣ ♦♥tt ♠♥ t♦ rt st ♦♥tt t♦
R = R0
)

E ♥ ν r t r♦ st ♣r♠trs R0 t ♥t ♥ R t ♦r♠ r♦
rs F t r♦ ♦ ♣r ♥t t s F ♥ tr♥ ♣♥s ♦♥ R s s♦♥ ♦ ♥
trt s♠ s ♥ssr t♦ ♣♦t ts s♠♣ ♦r♠ r♣ s♠♣
♦♣♥ s ♥ ♥ s ♦r ♦♥ t♠ ♦r♠ s♦s tt ♦ rt♦♥s s
s ♦s r rt t♦rs ♦r r♦ tt♥♥ ♦r ♥st♥ ♦ rt♦♥
♦♥ r t♥ str♣ ♥r rtr rt♦♥ s s ♦rs ♥ t♠♣r r♦♥

♦♥trr t♦ ♥tt♦♥ r♠♣stättr
tst♥ ♠s
T t♦t rt♦♥ ♥ ♥♥♦t ♥ ♦♥ s♥ ♣ss ♥ t
♠♣ ♦ ♦t st str♣ r♦♥ strt♥ r♦♠ st s 250 ♠♠ t t ts ♦t
12 ♣sss t♦ r t♥ss t♥ 6 ♥ 1.5 ♠♠ r rt♦♥ ♣r ♣ss 27% t♦
35% ♦ r♦♥ ♦♥ t♦ 0.3 0.8♠♠ t♣ ts 5♠♦r ♣sss ♥ t♦ st♥s ♠
♥ssr t♦ r♦ t rtr ♦♥ t♦ 0.13 0.15 ♠♠ s ♥ ♦r ♥ ♠♥ Prt ♦
ts ♦t r♦♥ st ♥ ♠ ♦♥ rrs♥ ♠ s♥ st♥ ♥ t s
♣sss sr t♠s rrs♥ t r♦♥ rt♦♥ s ♠s t ♥st♠♥t ♦r t
t ♣r♦tt ♠♦rt ♦ ♣r♦ ♠♦♥s ♦ t♦♥s srs ♦ ♥♦♥rrs♥ st♥s
s ♣rr
rs♥ ♠ ♦♥♦♣ ♠tst♥ ♠ ♥♠ ♠
s tr t♣s ♦ ♦t str♣ r♦♥ ♥s ♥ ♦♥
t ♦ ♦t r♦♥ s s ♣r♦r♠ ♦♥ rrs♥ ♠
♥♦♥♦♣ ♠tst♥ ♠ ♦♦ st♥ t♥♠ ♠ s t st♥r ♦r
r ♣r♦t♦♥ st ♣♥ts
rrs♥ ♠ ♦r t r♦♥ st rst ♣sss ♦♥ t ss ♦♦
t♥♠ ♥s♥ ♠ ts t s ♠♦st ♦ t t♠ ♦r t ♦s st♥s
♥s♥ ♠ ♥ s♦♠t♠s ♦r st ♦r st♥s ♥s♥ ♠
♦r ♠♥♠ s s ♦r st t st♥r ♦ r♦♥ ♠ s t♥♠ ♠
st♥s ♦r st♥s ♦r r♦♥ st t st♥ss st ♦r r r♦♥ sts
♠♦r ♥r r t♥ str♣s r ♦ r♦ ♦♥ rrs♥ str ♠s s s ♥
♠r ♠s
t♥♠ ♥♦♥ t♥♠ rtr s ♠♣♦rt♥t s t ♦♥t♦♥s t ♣♣t♦♥ ♦

♣tr ♥ r ♦♥ t ♦♥ Pr♦ss
t str♣ ♣ t ♥ t ♥tr ♦ t ♠ ♥ rs♦♥ t t♦ rs r♦♥
♦s ♥ ♣stt t♥s strss s ♣♣ ♥ t ♦♠♣rss ♦♥tt strss
♥ rss t s♠ ♠♦♥t t♦ t ♣rt ♥ t♥ss ♦♥tr♦ ♥ rrs♥
♦ ♠s t♥s♦♥s r ♣♣ ♦rs rs ♥trst♥ t♥s♦♥s r ♣♣
t r♦ r♥t ♦t r♦♠ st♥ t♦ st♥ ♥ t♥♠ ♠s ♦rs ♦♥ ♥♦♥
♦♣ ♠tst♥ ♠ t♦t ♦rs s s ♦♥ t ♦t r♦♥ ♠ ♥♦ t♥s♦♥ ♥
♣♣ t s ♥♦t ♥ssr t ts st ♥
Pr♦♠ ♥t♦♥
I♥ ts st ♦r ♠s r s ♦r t s♠t♦♥s r r
t ♣♣ r♦♥t ♥ t♥s♦♥s r ss♥t t♦ ♥sr t t♥ss ♦ t r♦
str♣ t♦ ♣r♥t t str♣ r♦♠ str♦♥ ♥ t tr♥srs rt♦♥ ♥ t♦ r
t r♦♥ ♦r
♦♠♠♦♥ tt♦rs ♥ ts ♠ r ♥♥ ♦r ♥ ♣ r♦s s♣♦t ♦♦♥ sst♠
♥ t♥s♦♥ ♥♥ st♦♥ ♦r ♠♦r ♦♠♣t sr♣t♦♥
r ♠
As ♠♥t♦♥ rr t ♥t ♥ ♥ st♣ tr ♦ r♦♥ s t t
t♥ss ♠♣r♦♠♥t ♥ r ♥s♥ ♦r ♣r♦♥ ♦♥ ♠st ♥rst♥
t t s ♥s♣♥s t r t♥ss ts tr ♦r♥s ♥ ♦ t♦ ♣r♥t
tr ♣♣r♥
t♥ss ♣r♦♠s ♥ ♦ r♦♥
I♥ ts st♦♥ t ♣r♦♠ ♦ t♥ss ts ♥♦♥tr r♥ ♦ r♦♥ s
♣rs♥t s s t st ♥♦♥ t♥ss ♠sr♠♥t ♠t♦s ♥ r♥t
tt♦rs t♦ ♦♥tr♦ ♥ ♣r♥t tr ♣♣r♥ ♥ss s♣ ♦trs
t s♦rs ♦ ♥♦r♠t♦♥ s ♥ ts st♦♥ r ♦rts ♦rts ♥
♥r
P♥♦♠♥♦♦ ♦ t♥ss ts
T tr♠ t♥ss ♥ ♦ r♦♥ ♠♥s t t ♦ t str♣ t♦ t ♥ ♥♦
t♥s♦♥ s ♣♣
r♥ ♦ r♦♥ tr s ♥♦ s♥♥t t ♥rs ♦ t str♣ ♥♦r♠ t t
t♥ss rt♦♥ t♥ ♣ ♥ t t s t♦ ♣r♦♣♦rt♦♥ ♥rs ♥ t ♥t
♦ ♦t♥ t st str♣ t♥ss ♣♦ss ♦t t str♣ t♥ss ♣r♦ ♥ t r♦t
♣r♦ ♥r ♦ s♦ ♠t ♣rt trs t t♥ss rt♦♥ ♥ ts
♦rrs♣♦♥♥ ♥t ♥rss ♥♦♥♥♦r♠ r♦ss t t ♦ t str♣ ♥ tr♦r
t♥ss ts ♣♣r
♥♦tr ♠♣♦rt♥t ♥♦t♦♥ s t ♠♥st ♦r t♥t t s ♥tt♦♥ ♥
tr♦ str ♥ str stts tt t♥ss ♣r♦♠s ♠ tr
♥s ♥ t str♣ ♣♣rs t♦ t s strss ♠ tr♣♣ ♥s t
♠t ♥ ♥ ♥rt t♥ss t ♦♥ t str♣ s tr♠♠ ♦r ♥st♥
r t♥ss ts ♥ ♦ r♦♥ ♦ s♣ ts ♥ ♥ss ts st

♦ ♦rrt s♥ r♥t ♣♣r♦s
♦ s♣ ts ♦ st r♦ss ♦ ♥ t♦rs♦♥ ♠♥ rs♦♥ ♦r ts
t♣s ♦ ts s ♦rrs♣♦♥♥ s♠♠tr t♥ t ♦r r♦s ♦ st rrs
t♦ ss♠♠tr σxx(z) rs strss ♣r♦ ♣♦ss t♦ r♥ t♥ t
r♦s s♣s ♦r ♠trs r♦ss ♦ trs ss♠♠tr σxx(z) strss ♣r♦ rs
t♦rs♦♥ s♦s ♦♠♣ r♥ts ♦ strss s ♦♣ ts ♥ t ♠t♠t
s♥s ♦ sr ♦♠tr r rrs ♥ ♥ ♦rrt ♣ss♥ t str♣ tr♦
s♠♣ ♣st ♥♥ ♣r♦ss
♥ss ts ♥♦♥♦♣ s ♦♥s s ♦rs
♥ ♥tr s r t ♠♦st ♦♠♣ t♣s ♦♥t♦♥ r♥t ♦♠♥ t
♥♦♥♥♦r♠ rs strss strt♦♥ ♥rts s r♦ss t str♣ ♥t
r♥ ♥ ♥t s ♥ rs strss ♥ s ♣♣r ♦♥ t ♠♦♥t ♦
♦ ♦♠♣rss rs strss s t ♥ ♠t ♦ t str♣ s
♥ ♥tr s r t ♠♦st ♦♠♠♦♥ ♦♥s r♥ ♦ r♦♥
r r♦♠ ♥tr r♦♥ r♦ t♦ s ♥ ♥rs ♥ r♦♥ ♦r ss ♠♦r ♦r r♦ t♦♥ t s♣ t♥ t r♦s s s♠r ♥r t s ♥ s rs t♦ r rt♦♥ ♦♥ t ♥ t♦ ♦♠♣rss rs strss ♦♥ t ss ♥ ♥ s ♦♥ t str♣ s tr ♣♥ ♠
♥ s r t rst ♦ ♠s♠t ♦ str♣ r♦♥ ♥ ♦r r♦s r♦♥
t♦ r♦ tt♥♥ ♥r t ♥rr♦ s ♦r t♦ ss r♦ ♥♥ s
♥r r ♦ rt♦♥ s ts ♥♦t ♥♦r♠ t str♣ s t♥♥r ♦♥ t s
s ♠♥s tt rs strsss r ♦♠♣rss ♦♥ t s ♥ r t♥ t
♥ ♠t t s s r
♣♣♥
♣tr ♥ r ♦♥ t ♦♥ Pr♦ss
♥ ♥tr s t ♦♥rs ♦ s ♥ ♥♦♥♥♦r♠ rt♦♥ ♥rts
♦♠♣rss strsss ♥ t ♥tr ♦t tt ♥ss ♥ s♦ ♦r ♥ t t ♦r
♦♥ ♥ t str♣ s st s ♦r ♦t ♦♥ t♦ ♥ss ♦rr♥ ♥ t
r♦♥ rt♦♥ r♦ss t ♥t ♦ t str♣ s strt ♥ r ts
♥ss t♣s ♥♥♦t ♠♥t s♠♣ ♥ rqr ♣♦str♦♥ ♥ ♣r♦sss
s st♦♥
♠♥ ♦r ♥ ♥rs♥ ♣r♦tt t♥♥r sts ♥ ♦♣t♠ qt t
t s♠ t♠ rqsts ♥ssr ttr ♥rst♥♥ ♥ ♦♥tr♦ ♦♥ ♣♣r♥ ♦
t♥ss ts ♥ t rst ♣ ♦ ts ♠ r♥t ♠t♦s ♥ ♦♥tr♦ tt♦rs
r ♣rs♥t ♥t ♥ st♦♥
t♥ss sr♠♥ts
As ♠♥t♦♥ rr t ♦♥♥t♦♥ ♣rss♦♥ ♦ t♥ss s t ♠♠♠ ♦
t♦♥ r♦♠ ♦r♦♥t t sr r sr tr♦ t
H ♥ ♥t l t ♦r ♣t ♦ t t ♥♦t D s ♥tr♦
♥ ♥ t ♥ s t♦ ♥ rr rtrt♦♥ ♦r ♣r♦
♠s t ♦ t s s ♥ ♥trst♥ ♣r♠tr ♣rtr ♥ r♦♥ r
♣r♦rs ♠ t♦ ♠♥♠ s ♣♦ss t t ♦ ts t♣s ♦ ts ♥ t♦♥
t♦ ts ♦♥♥t♦♥ ♣rss♦♥ t♦ t♥ss ♣r♠trs r ♥tr♦ ♦r rtr
♥ t♥ss ♦ sts ♣rs♥t♥ ♦♥t♥ s t st♣♥ss ♥ t t♥ss ♥
r ♣rs♥tt♦♥ ♦ st str♣ s♦♥ s
r ♠t r♣rs♥t t♦♥ ♦ t st ♠t♦ r t str♣ s ♦♥sr s srs ♦ ♦♥ t♥ ts ♦ r♥ ♥t
st♣♥ss ♥ S s ♥ ♦r♥ t♦ q ♦r st ♣rs♥t♥ s
♦ t H ♥ ♥t l
l

♣tr ♥ r ♦♥ t ♦♥ Pr♦ss
srs ♦ ♦♥t♥ ts ♥ ♠ t st strss s r s ♥ r
s♥ t ♥t ♦ ♦♥ ♦ ts str♣s s rr♥ Lref s t s♦rtst ♦♥ t
♥t s ♥ s ♦♦s
Lref
.105
r L s t r♥ t♥ t ♦♥r♥ t ♥ t rr♥ t s♥
t 105 s ♥tr♦ ♦r ts t♥ss ♥t ♥t ♥t♦♥ ♥ t s ♠♣♦ss
t♦ s r str♣ t♦ ssss ts qt t s ♠♣♦rt♥t t♦ t♦ ♦♥t♥♦s ♠sr
ts t♥ss t ♦♥♥ ♥ t s ♦ t♥t t t t ♣♣t♦♥ ♦ t♥s♦♥ t
♠ ♥♦t ♣♣r ♥ t ttr ♠t♦ s ♥ t♦ ♠sr t♥ss t♦t s♦♠
st♠♥ts
♦♦s ♣rsss L ♥ tr♠ ♦ Fe (N) t ♦r ♥ ♠sr rqr
t♦ ♦♥t t ts ♥t L r♦♠ t rr♥ ♥t Lref
E L
Lref
= Fe
A
r A s t r♦ss st♦♥ r ♥ E t ♦♥s ♦s
♦♠♥♥ qs ♥ t t♥ss ♥ I ♦r t♦ ♥t ts i ♥ j s
t s
Iij = Lij
♥♥ ♠sr♠♥t sst♠ ♦r t♥t ts
t♥ss ♦s r♥ r♦♥ s♦♠ ts ♠ ♥♦t s♦ s ♦ t ♣♣
t♥s♦♥ ♥ t str♣ t ♣♣r♥ st t♦ ssss t t♥ss ♦ t
str♣ ♣♦str♦♥ tr rs♥ t t♥s♦♥ s t♦ t♦ ♠sr t rs
strss strt♦♥ t♥ss r♦s sst♠ s ♠♥ ♦r♠ r♦ s s t♦r
r♦ ♥ t ♠ q♣♣ t srs ♦ s♥s♦rs ♣ t♦ 60 s♥s♦rs t rs♦t♦♥ ♦
52 mm ♥ ♠sr♠♥ts ♦♥ ♦♥ r♦ s♦ ♣ t st
t♦♥ t t str♣ t r ♦r s ♠sr ♥r t ♣♣ t♥s♦♥ t♥
t strss ♥ s♥s♦r i s tr♠♥ s
σi xx =

r α s t r♣ ♥ t♥ t str♣ ♥ t r♦ hi yi F i c ♥ σi
xx r rs♣
t t t♥ss t t t ♦r ♦ ♦♥tt ♥ t strss tr♠♥
♦♥ t ♦♥ i
1I − unit = 10
♣tr ♥ r ♦♥ t ♦♥ Pr♦ss
r ♥ t t ♠t r♣rs♥tt♦♥ ♦ t t♥ss r♦s ♠sr♠♥t sst♠ ♥ t rt ♥ ♠♣ ♦ ♠sr♠♥t
♥ t strss strt♦♥ s tr♠♥ ♦♥ ♥ ♣rt t ♣♦str♦♥ t♥ss
t♥ t t♥s♦♥ T0 t ♣r♦ ♦ σxx − T0 strt ♥ r s♦s
♦♠♣rss♦♥ ♥ t ♥tr ♦r ♥st♥ t ♣t t s ♥tr s
trss ♠sr♥ sst♠s ♣rs♥t s♦♠ ♥rt♥ts t♦ sr ♥s ♦r
♥st♥ ♥ t s ♦ tr♠ ♦r♥t strss strss strt♦♥ s ♠sr ♥
♥ t♦ ♣rt t t♣ ♦ t t♥ss t ♥ t str♣ ♦♦s ♦♥ t s♠
strt♦♥ s ♥♦ ♦♥r ♥ t ♣t t s ♥♦t ♥ssr t s♠ s
♣rt rr ♥♦tr ♠tt♦♥ s t♦ ♥s♥t ♥♠r ♦ s♥s♦rs ♦♥ t
s r ♠♦st ♦ t ♣r♦♠s ♦rs r♥ ♦ r♦♥
♣ ♥s♦r tr rst t♥ss sst♠ s ♦♥ s♣ s♥s♦rs s ♦♣
Prs♦♥ ♥ Prs♦♥ t s s ♦♥ t ♦ ♥♦♥♥♦r♠ ♦r♠t♦♥ ♥ ♦
r♦♥ ♦♥trts ♥ r♥ ♥ s♣ ♥ ♥t r♦♠ ♦♥ t♦ ♥♦tr
s♥ t t♥♥t s♣ t♥ss s tr♠♥ ♥ tr♠s ♦ V
Vref
Vref
Lref
♦♥ ♥ t
t♥ss ♥ ♠sr s♥ q
♦t tt ts ♠sr♠♥ts sst♠ ♥ s str t♦ s♥ t♥
t r♦s ♥ t r♦ ♣ ♥ t♦♥ t s ♥♦t t ♥ t s ♦ ♠♥st
ts t ♠sr♠♥ts r str t ts tr♥st s ♥♦s t♥
t ♦rrs♣♦♥♥ t
♥♥ ♠sr♠♥t sst♠ ♦r ♠♥st ts
♣♥ s♣♥ sst♠ strt ♥ r s s ♦♥ sr ♥
tr♥t♦♥ ♥ s s♣r ♥ ♦♥ ♠♥s♦♥ t♦ ♠ ♥ ♦ t s♣

♣tr ♥ r ♦♥ t ♦♥ Pr♦ss
trs t t♥ss ♠sr s tr♠♥ s♥ t t♦♥ ♦ t ♥
r ♠t r♣rs♥tt♦♥ ♦ t s♣♥ t♥ss ♠sr♠♥t sst♠
r ♥ ♠♣ ♦ t ♠r ♣♦t♦ ♦ t t ♥ t t♦♣♦r♣ sr♥ t s♣ ♦ t st
s ♣r♦r ♥ r♣t ♣r♦ ♥ srs ♦ t s♣♥s ♦sr

♦ré t♦ s rst ♦♣ ♦s t ♥ ♦s t ♥
s s t♥ ♣♣ t♦ t♥ss ♠sr♠♥ts r♥ str♣ r♦♥
♥ ♥t♦ t ♥ Pr s sst♠ ♣rs♥t ♥ r
♦♥ssts ♥ ♣r♦t♥ rt♥ ♦♥ t r♦ str♣ t s rr♥ ♣♥ rt♥
♥ t ♦r♠ rt♥ s ♣tr ♠ s s♣r♣♦s♥ t rr♥ ♠
♥ t ♠ ♦ t t str♣ rst s ♦ré ♠ r s
trt ♥ ♦♥rt ♥t♦ t♦♣♦r♣ ♠ r♣rs♥t♥ t str♣ ♦♥t♦r ♥
t♥ss
♣tr ♥ r ♦♥ t ♦♥ Pr♦ss
r ♠t r♣rs♥tt♦♥ ♦ t ♦ré ♣r♦t♦♥ t♥ss ♠sr♠♥t ss t♠
r ♣r♦t rr♥ r ♣tr ♦r♠ r ♦ré ♣♦t♦ s♣r♣♦s♥ t ttr t♦ rs ♥ strt♥ t stt ♦ t t st
♣♥ ♦ré s s ♥ ♦tr sr s ♠sr♠♥t ♠t♦ s♦ sr♦s
ts ♦r s♣ r♦♥ ♣r♦sss ts ts t♣s ♦ ♠sr♠♥t
sst♠s r ♥♦r♠ ♥st ♦♥ ♥s♣t♦♥ ♥s ♦r ♥ t s ♦ ♦ s♣ r♦♥
♣r♦ss
r ♣ ♠sr♠♥t sst♠ ♥t st
r ♥ ♠♣ ♦ ♠♥st t♥ss t ♠♣ ♠sr ♣ r♦r♠tt st
♦♣ r♦r♠tt ♣ s ♥ ♦♥ttss ♠sr♠♥t ♦r str♣
t♥ss ♦♥♥ s r t s s srs ♦ ♦♥♥srs ♥ t♦ ♠t
t t ♦ t r♦ st st♥t ♦ ♥t♠trs 5 ♠ ∼ 10 ♠ r♦♠
♦tr str♣♦♥♥sr s st♥ s♦ ♦s♥ r ♥ t s ♦ st
♣rs♥t♥ str♦♥ ♦♠tr ts s♦ tt t ♦s ♥♦t r♥ ♣ ♥st t
str♣ ♥ t r♦ str♣ s ♣ss♥ ♣ r♦rs t rt♦♥ ♦ t tr

♣tr ♥ r ♦♥ t ♦♥ Pr♦ss
s rt♦♥s r tr♥s♦r♠ s♥ s♣ s♦tr ♥t♦ ♦♠tr
sr♣t♦♥ ♦s ♥ t stt ♦ t str♣ ♠♠t r♥ r♦♥
r s♣t ts ♣ts ts s ♦♥sr ♣♥s ♦♠♣r t♦ t
ttr ♦♥s ♥ s♦s rt♦♥ ts s stt rr t st♥ t♥
t str♣ ♥ t ♦♥♥srs s♦ ♦s♥ r ♦r str♣ ♣♥♥ ♦♥
t t♣ ♥ t ♠♥t ♦ t ♠♥st t
♥♥ t♥ss ♦♥tr♦ ♥ tt♦rs
As ♣♥ ♥ st♦♥ ♥♥ r♦ ♣ s t♦ ♥♦♥♥♦r♠ rt♦♥
r♥t ♥t ♥rs r♦♠ str♣ r t♦ ♥♦tr s t♥ss ts t ♣
♦ st t r♦ ♣ sr ♠t♦s ♦♥sst♥ ♥ st♥ t ♦r r♦ s♣ r
♥ ts ♠♣r♦♥ t s♣ ♥ t♥ss ♦ t str♣ t
♦ ♥♥ ♦ ♥♥ s t ♠♦st ♣♦r ♠t♦ t♦ ♥ t s♣ ♦ t
♦r r♦ ♣♦st ♥♥ rsts ♥ ♥t r♦ r♦♥ t ♦rs ss rt♦♥
♥ t ♥tr ♥ tr♦r s s t♦ ♣r♥t t ♣♣r♥ ♦ ♥tr s
♥t ♥♥ ♦r s ♣♦st r♦♥ ss rt♦♥ ♦♥ t s ♥ tr♦r
s s t♦ ♦rrt s
♦t tt t ♦r r♦ r♦♥ ♥
st s♦ t s ♦ ♥
♥ ♣ r♦s ♦r ♥tr♠t
st ♦r ♥♦t r♦s ♥ t
tr ts t♦ tt♦rs ♥
♥ ♥ ♥♥ ♥ ♦♠
♦♠♣ t♣ ♦ ♥ss s s
qrtr s
♥♥ ♥s♦♥ ♥♥ t ♣♣ str♣ t♥s♦♥ s ♦♥sr s♦ ♥ ♠♣♦r
t♥t ♥ r t tt♦r ♦r ♦♥♥ t♥ss ♦ t♥ str♣s 0.5 ♠♠ ♦rts
stts ♥ ♦rts tt ♦ ♦ t t♥s♦♥ ♣♥s ♦♥ t ♦s♥ rt♦♥
♦♥t ♥ t s♣ ♦ t ♦♥tt ♦♥ t t♥s♦♥ ♥rss r♦ s♣rt♥
♦r rss ♥ rsts ♥ ♠♦r r♦♥ r♦ s t str♣ ♣rs♥ts ♥tr
s t ♥tr s t♦♦ ♦♥ t s♦t♦♥ s t♦ rs t ♦r r♦ r♦♥
t ♣♣t♦♥ ♦ s♠r t♥s♦♥ strss ♦t tt ♦r ♥ ♦♥t ♦ r

r♥tr♥ str♣ ♣r♦♠s
t ♥tr♠t r♦ss ♦r♥ s♥ ♥tr♠t r♦s ♥ q♣♣
t st r♦s sst♠ ♦r ♥st♥ ♥ t s ♠ strt ♣r♦s ♥
r s r♦s ♥ ♠♦ ♥ ♦rt ♥ t rt♦♥ st♥ tr
s ♦♥ t s ♦ t str♣ ♦r ♦r r♦ ♥ ♥ r ♥t t ♣♣r♥ ♦
ts
♥♥ r♥t r♦♥ r♥t s ♥♦tr ♠♣♦rt♥t tt♦r r
t♦♥ rs rt♦♥ t♥ t r♦ str♣ ♥ t ♦r r♦s s ♦♥sq♥
t rs t r♦ s♣rt♥ ♦r ♥ tr♦r r♦s ♣rs♥t ss tt♥♥ ts
♥ r♦ r ♠♣r♦s t sr ♦ t r♦ str♣ rt s ♠♣r♦
♥ t r♦ t ♦r ♥st♥ r♦ tt♥♥ s r ♥ t r♦ ♠tr ♥rss
♥ ss ♦ ♦rr♦♥ s ♥ r s t♦ ♦r♥rr♦ rt♥
t♥♥rtr rs t♦ ♥sr ♥♦r♠ rt♦♥ ♥ ♣r♥t t ♣♣r♥ ♦
s
r♥t tr♥♦♦♥ r♦s ♦ r♦♥ ♠s s r ♠♦♥ts ♦ ♥r
♥ t s tr♥s♦r♠ tr ♦♥ ♥t♦ t tr♣♣ ♥ t ♦r r♦s s rst ♦r
r♦s ♣rs♥t tr♠ r♦♥ ss ♦♥tr♦ ♥ t♥ss ♣r♦♠s ♦
♦♦♥ s♣r♥ tr ♦♥ t ♣r♦♠t ♦♥ r♥t ♣ttr♥s ♣rssr ♦
♥ t♠♣rtrs s t st s♦t♦♥ ♦r ts rrrt t rs t r♦ r♦♥
♥ ♠♥♠s ♦♥tr♦ ♣r♦♠s
P♦str♦♥ t♥ss ♠♣r♦♠♥t
As ♦r t♥ss ♠♣r♦♠♥t ♠♥ ♣r♦rs r ♠♣r ♦r s♥ ♣ss
r♦♥ ♥ ♥ ♣r♦sss
♠♣r r♦♥ ♣r♦ss s t ♣r♦ss ♦ t ♦ r♦♥ st st rt♦♥
0.3% ∼ 3% t s s t♦ ♠♣r♦ t♥ss t♦ ♠♥♠ strt♥ ♥ str♥♥ t♦
t ♦♥sst♥ ♥ t♦ r♥ss t♦ t st st ♥ ♥ t♦ ♦t♥
s♣ sr ttr ♦ ♠s r ♣rt ♠t ♦r ts ♣r♦ss
♦ tt♥♥ ♠♥s t r♦ ♠tr s s♠r t♥ ts ♥t ♥ ts s t s ♦ ♥rr♦♥

♣tr ♥ r ♦♥ t ♦♥ Pr♦ss
♦r ♥ ♣r♦ss s s t♦ ♠♣r♦ t♥ss ♦ t str♣ tr ♦t r♦♥
str♣ s ♣ss tr♦ ♥♠r ♦ ♦r r♦s ♦r♥ ♥t♦ t♦ rs ♥ ♥ ♦♥
♦♠♠♦♥ ♥tr ♣♦♥t strt ♥ r r♦s ♥ t str♣ st
t♦ ♦r♠ t ♣st ♥ t♦ ♥♦r♠ t♥ss ♦r t str♣ ♦s
r ♦sr t♦tr t t ♥tr t♥ t t♥ s s rr♥♠♥t ♥srs ss
♥♥ t♦r t t ♥ ♣r♥ts tr♦r ♥tr♦♥ t♦♥ rs strss
t♦ t str♣ rst str♣ s strt♥ ♥ r ♦ ts
♥s♦♥ ♥ ♣r♦ss trts sts t♦ ♠♣r♦ t♥ss tr ♦ r♦♥
♦♠♥t♦♥ t♥s♦♥♥♥ ♣s t st ♦♥ ts ♣♦♥t ♣ ♠♣r
t♦♥s r ♦rrt ♥ ts ♣r♦ss r t str♣ s ♣ss r♦♥ srs ♦ r♦s t♦
♥tt ♥♥ ♥ strt♥ ♥ t str♣ s tt♥ ♦t t♦ ♦rrt t ♥
r rst str♣ s t ♥ s r ♦ rs strsss
r ♦r ♥ ♣r♦ss tr ♦rts
r ♥s♦♥ ♥ ♣r♦ss tr ♦rts
és♠é ♥ r♥çs
D♥s ♣tr ♥ ♥tr♦t♦♥ é♥ér ♣r♦éé ♠♥ st ♣rés♥té
♣♦r s♠♣r st trs ♣ ♠rs t tt rè sr♣t♦♥ st
♥ ♠r ♦♠♣ré♥s♦♥ s ér♥ts t♣s éts ♣♥été t rs ♦r♥s
r♥t ♥ ♦♣ért♦♥ ♠♥ à r♦
♥♦t♦♥ ♠ s♦s ♦♥tr♥ts réss été ♥tr♦t ♥ té♦rs♥t s
éts ♥ t♣s ét ♠♥st t ét t♥t s s♣♦sts ♠sr
♣♥été s ♣s ♦♥♥s r♦ ♣♥été ♠♦ré s♣♥ ♦♥t été ♣rés♥tés ♥
♣s s t♦♥♥rs ♣♥été s♣♦♥s ♥ ♥ ♣♦r ♦rrr t étr ♣♣rt♦♥
s éts ♦♥t été és ♥♥ qqs ♣r♦éés ♣♦sttrt♠♥t tsés ♣♦r


P
r rtst ♥ss s ♥ ♥ ♣ ♠♦st rt♥ t♦ s
s s t♦ tr st ♦♥ ♠♦r t♠
♦♠s s♦♥
♦♥ ♠t♦s ♦r ♥ ♥ ♣♦st♥ ♣r♦♠s
♥r ♠t♦s ♦r tr♠♥♥ t rt ♦
P♦st♥ ♠t♦s
♥♦♣ ♣♣r♦s ♦r ♦♥♥ ♣r♦♠s
♦♣ ♣♣r♦s ♦r ♦♥♥ ♣r♦♠s
♦♥s♦♥

♣tr trtr r sts ♥ ♣ts ♥ t♦rs
As sr ♣r♦s t ♦r♠ ♦ t ♣ t♥ t r♦s ts t s♣ ♦
t r♦ str♣ ♠♥ rs♦♥s r t ♦r r♦s st ♦r♠t♦♥s s s t
r♦s r♦♥ tt♥♥ ♥ ♥♥ s ♦r♠t♦♥s rt ♥♦♥♥♦r♠ rt♦♥
♥ t t ♦ t str♣ ♥ ♥rt tr♦♥♦s rs strss strt♦♥ t
♦♠♣rss rs strss ♦s ♣st t rt t♦rt t str♣ t s
♥ ♦♠s ♥ t t s ♠♥st t ♣ t♥ t rt ♥
t ♦♠♣rss strss s ♥ t ♣♣r♥ ♦ ♦♠tr s♣♠♥ts
rs strss s ♦♥ ♠♦r ♠♦ t♦ rs♣t t qr♠ stt tr ♥ ♥
♥ t t ♣♣♥s ♥ t t t st ♥ ♠♣r♦ ♣♦str♦♥
♦♣rt♦♥s s t♠ ♥ ♥r ♦♥s♠♥ tr t♥ ♦rrt♥ ts t s
ttr t♦ ♦ t♠ t♦ ♦♥tr♦ t♥ss r♥ r♦♥ ♣tr ♥ ♣r♠trs
r ♦r tt ♦rrs♣♦♥♥ t♦ t rst ♦ t ♣s ♣♥♦♠♥ ♥ t
t♥ss tt♦rs ♥st ♦♥ t r♦♥ ♠s r ♦♠♣t rqrs ♠♦s ♦
♣rt♦♥ ♥ ♦♥ ♦♠♣tt♦♥
tr ts s♠♠r ♦♥ ♥ tt t r♦♥ ♣r♦ss s ♦♠♣ ♥ r s♣t
t♥ t rs strss s♦ ♥ r♦♥ ♠♦ s♦ ♦♣
♠♥ s♣ts
♥ ♦t ♦ t t ♥ ts t ♦♥ t t
tr♠ ♦r♠t♦♥ ♥ tr♠ strss ♦♣♠♥t
♦t tt s♥ ts st s t t♦ t ♣♣r♥ ♦ t♥ss ts r♥ ♦
r♦♥ tr♠ ts r ♥t ♣tr st♦♥ s ♦r ♦♣♥
t r♦♥ ♦ t t st ♦r♠t♦♥ t st♥ ♦r r♦♦r♠t♦♥ ♠♦
s ♥ ♦♣♥ t ♠ ♦t s t♦ ♥ st ♥
♠♦ ♣ ♦ sr♥ t stt ♦ t t str♣ ♥ qt t♦ ♦♣
t t r♦♥ ♠♦ ♠
trtr ♣r♦♣♦ss sr ♣♣r♦s ♦ r ♠♦st ♦ ts ♠♦s r s ♦♥ ♥
♥♦♣ ♣♣r♦ ♥ t t ♣r♦♠ ♥ t♦ ♣rts
♦♠♣tt♦♥ ♦ t r♦♥ ♣r♦ss t♦t ♥ ♥ t rs strsss
t♦♥ ♦ t r♦ st ♥ ♥r t t ♦ ts rs strsss
♥ ♥t♦ ♦♥t tt ♥ s rt ♦♥sq♥ ♦ t ♦rs ♥r t

♣tr trtr r sts ♥ ♣ts ♥ t♦rs
♦ t ♦♠♣t ♦ ts ♣r♦ss ♦♥ ♣t♦♥s trt♥ ♠♦ ♦ ♥
♥trt ♥ r♦♥ r t ♠♦s ♦ ♦♥ ♦♥ ♥
♥ ts ♣tr sr ♦ t ♠♦♥ ♠t♦s s ♣r
s♥t t t ♠ ♦ ♥♥ ♣♦r ♣♣r♦s t♦ rr ♦t ♥ ♥ ♣♦st♥
♦♠♣tt♦♥ t ♣t t♦ ♦♣ t t st♥ r♦♥ ♠♦ ♠
rst q sr♣t♦♥ ♦ t trt ♣r♦♠ s ♣rs♥t ♥t r♥t ♠♦♥
♠t♦s ♦r ♥r ♥ ♥ ♣♦st♥ ♣r♦♠s r ♣♦r ♥ t st
♣rt s ♦t t♦ sr♥ t ♣♣r♦s ♣♣ t♦ r♦ str♣ t♥ss
sr♣t♦♥ ♦ t ♣r♦♠
T ♠♦♥ ♦ ♥ ♥ ♣♦st♥ s r t ♣r♦♠ ♠ s
t♦ ♥ ♠t♦s t♦ sr t stt ♦ t ♣r♦♠ tt ♥ ♥ s
t ♣t ♦ t sst♠ t♦ rtr♥ t♦ ♥ ♥ st str ♦r ♥st♥
♥ t s ♦ str♣s ♥r ♦♠♣rss strss t s♦t♦♥ st♦♣s ♥ st ♦♥ t
rs strss s t rt ♥♦t σc ♦ sr t ♣r♦♠ t s ♥ssr
t♦ st t ♦r ♦ t sst♠ t ♥ ♦♥ ts rt ♣♦♥t
♥ t♦rs ♦r strtrs ♥ t t st ♦ r ♦r ♠ ♥ r
t ♦♥ ár♠á♥ ♦r♠t qt♦♥s sr♥ ♥ ♦ t♥ st ♣ts ♥r
♦♠♣rss strss ♥ ts qt♦♥s r s ♥♠ ♦♥ ár♠á♥ ♥ sts
r s ♦♥ r♥t ♦r♠ ♦ ts qt♦♥s ♥ ♣rs♥t r♥t ♠t♦s ♥
s♦t♦♥s ♦r ♠♥ r♥t ♣♣t♦♥s ♠♥t♦♥ ♥ ♣rtr t s♠♥t
♣♣r♦s ♣rs♥t sr sr t ♥ s s t t ♣
♣t♦♥ t♦ str♣ r♦♥
tr ♦♥ ♥ t♦r ♦r ♥ ♦ st strtrs t♥ ♥t♦ ♦♥t t ♥t
♠♣rt♦♥s s ♣rs♥t ♦tr ♥ ♦tr
♦tr s ♥ s st t rt♦♥ ♥ stt t♦rs rst ♣♣r ♥ t
♠t♠t sts ♦ P♦♥ré P♦♥r ♥ ♠t ♠t r
s♠♣t♦t ♣♥s♦♥s r s ♥ ♠♦st r ♥ ♥ ♣♦st♥ t♦rs
♥ r♥t sts t ♦♥s ♣rs♥t P♦trrr ♥ P♦trrr ♥
♥s ♥ ♥s r ♦♣ s♥ ♠♦r♥ ♠t♠t ♣♣r♦s
♦ t rt♦♥ t♦r ♥ ♣rs♥t sr ♣♣t♦♥s ♥ t ♦♠♥ ♦ ♥ ♥
♣♦st ♥ ♣r♦♠s

♣tr trtr r sts ♥ ♣ts ♥ t♦rs
♦♥ ♠t♦s ♦r ♥ ♥ ♣♦st♥
♣r♦♠s
T trtr sr ♣rs♥t ♥t ♥tr♦s ♠t♦s ♣ ♦
sr♥ t ♥ ♣r♦♠ tr♠♥♥ t rt ♦ ♦ stt ♥ t ♠♦
♦ t t s s t ♣♦st♥ ♣r♦♠ t ♠♥t ♦ t t
♥r ♠t♦s ♦r tr♠♥♥ t rt ♦