...

State of the Art Report - IA

by user

on
Category: Documents
121

views

Report

Comments

Transcript

State of the Art Report - IA
3
2
BasicNotations
eυ
aao
aL
c
d
刀
s
u
Wc
l
e
n
g
t
ho
fc
r
a
.
c
ko
rc
r
a
c
kba
.
nd
一
.
c
kl
e
n
g
t
h
n
i
t
i
a
l(
p
r
e
e
x
i
s
t
i
n
g
)n
o
t
c
ho
rc
ra
ー i
r.
a
c
kl
e
n
g
t
h
一 elasticallyequivalent(eff .tive)c
も
w
o
・
d
i
m
e
n
s
i
o
n
a
l
s
p
e
c
i
m
e
n
t
h
i
c
k
n
e
s
s
o
f
一
.
c
ke
x
t
e
n
s
i
o
n
一 elasticallyequivalentcra
h
a
r
a
c
t
e
r
i
s
t
i
cd
i
m
e
n
s
i
o
n(
o
rs
i
z
e
)o
fs
t
r
u
c
t
u
r
e(
o
rs
p
e
c
i
m
e
n
)
ー c
,飢
一 tensilestrength
a
c
i
n
g
一 cracksp.
d
i
s
p
l
a
.
c
e
m
e
n
t
一
w
i
d
t
ho
fc
r
a
c
kba
.
nd
G1c
a
.
st
i
c
i
t
y
l
一 Young'smodulusofe
.
n
e抗 r
e
s
s
,
E'= E(
1-v2)f
o
rp
l
一 E forpla
e
l
e
a
s
er
a
t
e
一 energyr
.
t
ef
o
rMode1
一 criticalenergyreleasera
G
7
ー
ー
E
E'
G
,
G
姐
es
t
r
a
.
i
n
f
ra
.
c
t
u
r
ee
n
e
r
g
y
a
.
c
t
u
r
ee
n
e
r
gyo
b
t
a
i
n
e
dbyt
h
eRILEMMe
t
h
o
d
f
r
s
t
r
e
s
s
i
n
t
e
n
s
i
t
y
f
a
.
c
t
o
r
s
f
o
r
M
o
d
e
s
1
,
I
I
,
I
I
I
,
r
e
s
p
e
c
t
i
v
e
l
y
[
(
/
,K
I
I
J
<
I
I
/
,
f
r
a
c
t
u
r
e
t
o
u
g
h
n
e
s
s
c
r
i
t
i
c
a
l
s
t
r
e
s
s
i
n
t
e
n
s
i
t
y
f
a
c
t
o
r
f
o
r
M
o
d
e
1
K/
c
p
l
o
a
d
maximum(
u
l
t
i
m
乱t
el
o
a
d
)
Pu
一r
,
α/
d
e
l
a
t
i
v
ec
r
a
c
kl
e
n
g
t
h
α
一 brittlenessnumber
F
一
r
a
c
k
o
p
e
n
i
n
gd
i
s
p
l
ac
e
m
e
n
t
ー c
一
a
le
f
f
e
c
t
iv
ec
r
a
c
k
t
i
pop
e
n
i
n
gd
i
s
p
l
a
.
c
e
ment
dCTOD
一 critic.
r.
a
ck-moutho
p
e
n
i
n
gd
i
s
p
l
a
c
e
m
e
n
t
dCMOD
一c
も
l
o
a
d
p
o
i
n
td
i
s
p
l
a
c
e
m
e
n
hPD
正
s
t
r
a
i
n
r
e
s
s
σ
一 st
o
m
i
n
a
ls
t
r
e
s
sa
tu
l
t
i
m
a
t
el
o
a
d
σN
ー n
一
P
o
i
s
s
o
n
'
s
r
a
t
i
o
U
一
sa
一 extensionofcrackorcrackband
o
t
e
n
t
i
a
len
e
r
g
y
E
ー p
ι
F
r
a
c
t
u
r
eM
e
c
h
a
n
i
c
so
fC
o
n
c
r
e
t
e
:
C
o
n
c
e
p
t
s,
ModelsandD
e
t
e
r
m
i
n
a
t
i
o
n
o
fM
a
t
e
r
i
a
lP
r
o
p
e
r
t
i
e
s
R
e
p
o
r
tbyACIC
o
m
m
i
t
t
e
e4
4
6,
F
r
a
c
t
l
i
r
eM
e
c
h
a
n
i
c
s
.
ー
V
e
l
l
o
r
eS
.Gopalaratnam 1•3
(
S
e
c
r
e
t
町)
Zd
en
位 P
.Baiant1 •2
(C
凶 m 切)
堕豆盟単語
1
O
r
a
lBuyukozturk
1
L
u
i
g
iC
ed
o
l
i
n
3
DavidDa
.
r
w
i
n
e
lElices1 •3
M姐 u
S
h
u
J
i
nFang
.
l
t
e
rG
e
r
s
t
l
e
Wa
N
e
i
lM.Ha
.
w
k
i
n
s
H
i
d
e
y
u
k
iH
o
r
i
j
l
J
e
r
e
myI
s
e
n
b
e
r
g
V
i
c
t
o
rC
.L
i1
Feng-BωL
泊1
S
t
e
v
e
nL
.McC
a
.
be
3
S
h
e
n
g
TaurMau
3
J
a
c
k
yMa
.
z
a
.
r
s
S
i
d
n
e
yM
i
n
d
e
s
s
A
n
t
o
i
n
eE
.Na
.
a
m
an1
C.De
a
.
nNorman
Ph
i
1
l
i
pA
.P
f
e
i
f
f
e
r
Gi
U
e
sP
i
j
a
u
d
i
e
r
C
a
b
o
t
3
1,
a
.
3
V
i
c
t
o
rSaoum
1
aP.Sh
a
.
h
,
3
S
u
r
e
n
d
r.
R
ρ
b
e
r
t1
.S
i
e
r
a
.
k
o
w
s
k
i
1
WimalSu
a
.
r
i
s
S
t
u
a
r
tE.SWa.rtz1 •2
.Ts
ub
a
.
k
i
T
a
t
s
u
ya
C
.V
i
p
u
l
a
n
姐 d
姐 1
1
M
e
t
h
iWe
c
ha
.
r
a
t
a
n
a
∞gnizethe∞ntributionsofthefollowingnon-votingmembers:
Thecommitteew
i
s
h
e
st
or
e
Fa
.
r
hadA
n
s
a
r
i1
3
Ra
v
i
n
d
r
a
.G
e
t
t
u
1
AmeHi
1
l
e
r
b
o
r
g
B.L
.K
a
r
i
h
a
l
∞1
a
.
t
l
P
e
r
eC
.Pr
i
n
h
a
.
r
dt
l
H姐 sW.Re
1M
emberso
fS
u
b
c
o
m
m
i
t
t
e
e1(
c
h
a
i
r
e
db
yBa1組t
)w
h
i
c
hp
r
e
pa
.
r
e
dt
h
er
e
p
o
r
t
2
P
r
i
n
c
i
p
a
lAu
t
h
o
r
s
3
Con
t
r
i
b
u
t
i
n
gA
u
t
h
o
r
s
*
T
h
i
sr
e
p
o
r
tw剖a.
p
p
r
o
v
edbyav
o
t
eo
ft
h
ef
u
l
lACICommi
t
t
e
e4
4
6i
nDecember1
9
8
9
.I
t
d
o
e
sn
o
tr
e
f
l
.
e
c
もr
e
s
e
a
r
c
ha
f
t
e
rt
h
a
td.
a
t
e
.
4
5
SYNOPSIS
Introduction
I
nt
h
ef
i
r
s
to
fi
t
55
e
r
i
e
5o
ff
o
u
rs
t
a
t
e
o
f
t
h
e
a
r
tr
e
p
o
r
t
su
n
d
e
rp
r
e
p
a
r
a
t
i
o
n,t
h
eC
o
m
m
i
t
t
e
ed
e
s
c
r
i
b
e
s
t
h
eb
a
s
i
cc
o
n
c
e
p
t
so
ff
r
a
c
t
u
r
em
e
c
h
a
n
i
c
so
fc
o
n
c
r
e
t
e
,
t
h
ee
x
i
s
t
i
n
gt
h
e
o
r
e
t
i
c
a
lm
o
d
e
l
s,
a
n
dt
h
em
e
t
h
o
d
s
仔e
r
s自v
er
e
a
s
o
n
sf
o
ri
n
t
r
o
d
u
c
i
n
gf
r
a
c
t
u
r
e
f
o
rd
e
t
e
r
m
i
n
i
n
gt
h
em
a
t
e
r
i
a
lf
r
a
c
t
u
r
ep
a
r
a
m
e
t
e
r
s
.C
h
a
p
t
e
r10
,i
n
c
l
u
d
i
n
gsomec
o
d
ep
r
o
v
i
s
i
o
n
s
:(
1
)a
m
e
c
h
a
n
i
c
si
n
t
oc
e
r
t
a
i
na
s
p
e
c
t
so
fd
e
s
i
g
no
fc
o
n
c
r
e
t
es
t
r
u
c
t
u
r
e
s
,i
.
e
.,e
l
i
m
i
n
a
t
e
t
h
e
o
r
e
t
i
c
a
le
n
e
r
g
ya
r
g
u
m
e
n
t
;(
2
)t
h
en
e
e
dt
oa
c
h
i
e
v
eo
b
j
e
c
t
i
v
i
t
yo
ff
i
n
i
t
ee
l
e
m
e
n
ts
o
l
u
t
i
o
n
s
刷 r
i
o
U
5meshs
e
n
s
i
t
i
v
i
t
y
;(
3
)t
h
ep
r
叩 悶i
v
e(
p
r
o
p
a
g
a
t
i
n
g
)n
a
t
u
r
eo
ff
a
i
l
u
r
e
,i
m
p
l
i
e
dw
h
e
n
e
v
e
rt
h
el
o
a
d
刑l
o
n
d
e
f
l
e
c
t
i
o
nd
i
a
g
r
a
ml
a
c
k
say
i
e
l
dp
l
a
t
e
a
u
;(
4
)t
h
en
e
e
dt
or
a
t
i
o
n
a
l
l
yp
r
e
d
i
c
td
u
c
t
i
l
i
t
ya
n
de
n
e
r
g
ya
b
s
o
,(
5
)t
h
ee
f
f
e
c
to
fs
t
r
u
c
t
u
r
es
i
z
eo
nt
h
en
o
m
i
n
a
ls
t
r
e
n
g
t
h(iιnominal
c
a
p
a
b
i
l
i
t
y
;a
n
dm
o
s
ti
m
p
o
r
t
a
n
t
l
y
Sa
tmaximumo
ru
l
t
i
m
a
t
el
o
a
d
)a
sw
e
l
la
so
nd
u
c
t
i
l
i
t
ya
n
de
n
e
r
g
ya
b
s
o
r
p
t
i
o
nc
a
p
a
b
i
l
i
t
y
.The
s
t問 5
仇c
ti
sd
u
et
os
t
o
r
e
de
n
e
r
g
yr
e
l
e
a
s
ei
n
t
ot
h
ef
r
a
c
t
u
r
ef
r
o
n
t,a
n
di
sn
o
tg
o
v
e
r
n
e
db
yWeibul
1
句p
e
s
l
z
ee
'
回 r
y
.E
xp
e
r
i
m
e
n
t
a
le
v
i
d
e
n
c
eo
nt
h
ee
x
i
s
t
e
n
c
eo
ft
h
es
i
z
ee
仔'
e
c
t,h
i
t
h
e
r
t
oi
g
n
o
r
e
di
nd
e
s
i
g
n
s
t
a
t
i
s
t
i
c
a
lt
h
p
r
a
c
t
i
c
ea
n
dc
o
d
ep
r
o
v
i
s
i
o
n
s
,i
5d
o
c
u
m
e
n
t
e
d
.
C
h
a
p
t
e
r2g
i
v
e
sab
r
i
e
fr
e
v
i
e
wo
ft
h
en
e
c
e
s
s
a
r
yb
a
s
i
cr
e
s
u
l
t
so
fl
i
n
e
a
re
l
a
s
t
i
cf
r
a
c
t
u
r
em
e
c
h
a
n
i
c
s
e
p
a
r
t
u
r
e
sf
r
o
mt
h
i
sc
1
a
s
5
i
c
a
lt
h
e
o
r
ya
r
ec
a
u
s
e
db
yt
h
ee
x
i
s
t
e
n
c
eo
fd
i
s
t
巾Ijt
e
d
(
l
E
F
M
)
.I
nc
o
n
c
r
e
t
e,d
c
r
a
c
k
i
n
g(
o
rd
a
m
a
g
e
)i
nap
r
o
g
r
e
s
s
i
v
e
l
ys
o
f
t
e
n
i
n
gf
r
a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
ew
h
i
c
hs
u
r
r
o
u
n
d
st
h
et
i
po
fac
o
n
t
i
n
u
o
u
sc
r
a
c
k.I
nC
h
a
p
t
e
r3n
o
n
l
i
n
e
a
rf
r
a
c
t
u
r
em
o
d
e
l
sc
h
a
r
a
c
t
e
r
i
z
i
n
gt
h
es
o
f
t
e
n
i
n
gs
t
r
e
s
s
d
i
s
p
l
a
c
e
m
e
n
t
s
t
凶 nr
e
l
a
t
i
o
n
s(
s
u
c
ha
st
h
o
s
eo
fH
i
l
l
e
r
b
o
r
g
'
sf
i
c
t
i
t
i
o
u
sc
r
a
c
km
o
d
e
l,
c
r
a
c
kb
a
n
dm
o
d
e
l,n
o
n
l
o
c
a
l
o
rs
t問 s
s
t
r
a
i
n
s
o
f
t
e
n
i
n
gm
o
d
e
l
s
,e
t
c
.
)a
r
ed
e
5
c
r
i
b
e
da
n
drandomp
a
r
t
i
c
l
es
i
m
u
l
a
t
i
o
no
fa
g
g
r
e
g
a
t
em
i
c
r
o
s
t
r
u
c
t
u
r
ei
sd
i
s
c
u
s
s
e
d
.Thep
r
i
n
c
i
p
l
e
so
fi
m
p
l
e
m
e
n
t
a
t
i
o
no
ft
h
e
s
em
o
d
e
l
si
nf
i
n
i
t
ee
l
e
m
e
n
tp
r
o
g
r
a
m
sa
r
ea
l
5
0
o
u
t
l
i
n
e
d
.C
h
a
p
t
e
r4p
r
e
s
e
n
t
ss
i
m
p
l
e
rn
o
n
l
i
n
e
a
rf
r
a
c
t
u
r
em
o
d
e
l
sw
h
i
c
hr
e
p
r
e
s
e
n
ta
d
a
p
t
a
t
i
o
n
so
fl
i
n
e
a
r
e
l
a
s
t
i
cf
r
a
c
t
u
r
em
e
c
h
a
n
i
c
s,s
u
c
ha
sJ
e
n
qa
n
dS
h
a
h
'
sm
o
d
e
Ja
n
dt
h
eRc
u
r
v
e
,a
l
o
n
gw
i
t
hd
e
t
e
r
m
i
n
a
t
i
o
n
f
T
e
c
tl
a
wp
r
o
p
o
s
e
db
yB
a
f
a
n
t
.T
h
i
sl
a
w
,d
e
s
c
r
i
b
i
n
g
o
fg
e
o
m
e
t
r
y
d
e
p
e
n
d
e
n
tR
c
u
r
v
e
sf
r
o
mt
h
es
i
z
ee
,i
sd
i
s
c
u
s
s
e
di
n
t
h
ea
p
p
r
o
x
i
m
a
t
ed
e
p
e
n
d
e
n
c
eo
ft
h
en
o
m
i
n
a
ls
t
r
e
s
sa
tmaximuml
o
a
do
ns
t
r
u
c
t
u
r
es
i
z
e
C
h
a
p
t
e
r5
,a
n
ds
t
r
u
c
t
u
r
a
lr
e
s
p
o
n
s
ei
5c
h
a
r
a
c
t
e
r
i
z
e
db
yt
h
eb
r
i
t
t
l
e
n
e
s
sn
u
m
b
e
r
.
C
h
a
p
t
e
r6p
r
e
s
e
n
t
si
nc
o
n
s
i
d
e
r
a
b
l
ed
e
t
a
i
lt
h
ec
u
r
r
e
n
tm
e
t
h
o
d
sf
o
re
x
p
e
r
i
m
e
n
t
a
la
n
da
n
a
l
y
t
i
c
a
ld
e
t
e
r
m
i
n
a
t
i
o
no
fm
a
t
e
r
i
a
lf
r
a
c
t
u
r
ep
a
r
a
m
e
t
e
r
s,
i
n
c
l
u
d
i
n
gt
h
eq
u
a
s
i
l
E
F
Mm
e
t
h
o
d
s,
RllEM(
w
o
r
k
o
f
f
r
a
c
t
u
問)
method,t
h
eJ
e
n
q
S
h
a
ha
n
dK
a
r
i
h
a
l
o
o
N
a
l
l
a
t
h
a
m
b
im
e
t
h
o
d
s
,a
n
dt
h
es
i
z
e
e
f
T
e
c
tm
e
t
h
o
d
.E
x
p
e
r
i
m
e
n
白n
i
n
g
t
a
ld
e
t
e
r
m
i
n
a
t
i
o
no
ft
h
ec
h
a
r
a
c
t
e
r
i
s
t
i
cl
e
n
g
t
hf
o
rn
o
n
l
o
c
a
lc
o
n
t
i
n
u
u
mm
o
d
e
l
sa
n
dt
h
es
t
r
a
i
n
5
0
f
n
dm
a
t
e
r
i
a
lp
a
r
a
m
e
t
e
r
sf
o
rmodes1a
n
d1
1
1,s
h
e
a
rf
r
a
c
t
u
r
e
sa
n
dm
i
x
e
d
p
r
o
p
e
r
t
i
e
si
st
h
e
ne
x
a
m
i
n
e
d,a
modef
r
a
c
t
u
r
ea
r
ea
l
s
od
i
s
c
u
s
s
e
d
.C
h
a
p
t
e
r7t
h
e
np
r
o
c
e
e
d
st
od
e
5
c
r
i
b
ev
a
r
i
o
u
si
n
f
l
u
e
n
c
i
n
gf
a
c
t
o
r
s
,s
u
c
h
a
st
h
el
o
a
d
i
n
gr
a
t
e,h
u
m
i
d
i
t
ya
n
dt
e
m
p
e
r
a
t
u
r
e
,a
sw
e
l
la
st
h
ee
f
T
e
c
to
fc
y
c
l
i
cl
o
a
d
i
n
g
.C
h
a
p
t
e
r8i
s
f
T
e
c
to
fr
e
i
n
f
o
r
c
i
n
gb
a
r
sa
n
dt
h
e
i
rb
o
n
ds
l
i
po
nf
r
a
c
t
u
r
ep
r
o
p
a
g
a
t
i
o
n,a
n
dt
of
r
a
c
t
u
r
e
d
e
v
o
t
e
dt
ot
h
ee
トr
e
i
n
f
o
r
c
e
dc
o
n
c
r
e
t
e
.C
h
a
p
t
e
r9d
e
a
l
sw
i
t
hmoret
h
e
o
r
e
t
i
c
a
lp
r
o
b
l
e
m
so
fm
o
d
e
l
i
n
gs
y
s
t
e
m
so
f
o
ff
i
b
e
i
n
t
e
r
a
c
t
i
n
gc
r
a
c
k
s
.A
t
t
e
n
t
i
o
ni
sf
o
c
u
s
e
do
ns
y
s
t
e
m
so
fp
a
r
a
l
l
e
lg
r
o
w
i
n
gc
r
a
c
k
s
.T
h
e
i
rs
t
a
b
i
l
i
t
yd
e
c
i
d
e
s
t
h
es
p
a
c
i
n
ga
n
dw
i
d
t
ho
ft
h
ec
r
a
c
k
sf
r
o
mt
h
em
e
c
h
a
n
i
c
sv
i
e
w
p
o
i
n
t
.
f
t
e
rad
e
c
a
d
eo
fr
a
p
i
dp
r
o
g
r
e
s
si
n問 s
e
a
r
c
h,t
h
et
i
m
ea
p
p
e
a
r
sr
i
p
ef
o
ri
n
t
r
o
d
u
c
i
n
g
I
ti
sc
o
n
c
l
u
d
e
dt
h
a
t,a
f
r
a
c
t
u
r
em
e
c
h
a
n
i
c
5i
n
t
od
e
s
i
g
np
r
a
c
t
i
c
e
.T
h
i
ss
h
o
u
l
dn
o
to
n
l
yb
r
i
n
ga
b
o
u
tmoreu
n
i
f
o
r
ms
a
f
e
t
ym
a
r
g
i
n
s
,
t
h
u
si
m
p
r
o
v
i
n
gs
a
f
e
t
ya
n
deconomyo
fd
e
s
i
g
n,
b
u
ta
l
5
0p
a
v
et
h
ewayf
o
rs
a
f
e
ra
n
dmoree
f
f
i
c
i
e
n
tu
s
eo
f
h
i
g
h
p
e
r
f
o
r
m
a
n
c
ec
o
n
c
r
e
t
e
sa
n
dp
e
r
m
i
td
e
s
i
g
ne
x
t
r
a
p
o
l
a
t
i
o
n
sb
e
y
o
n
dt
h
er
a
n
g
eo
fp
r
e
v
i
o
u
se
x
p
e
r
i
m
e
n
t
s
a
n
dd
e
s
i
g
n
.
且e
s
s,
c
o
n
c
r
e
t
e,
c
o
n
c
r
e
t
es
t
r
u
c
t
u
r
e
s,
c
r
a
c
ks
p
a
c
i
n
gandw
i
d
t
h,
c
r
a
c
k
i
n
g
,
d
a
.
r
n
a
g
e
KEYWORDS:B
r
i
t
t
l
e
mech
姐 i
ω,
d
e
s
i
g
nc
o
d
e
s,
d
u
c
t
i
l
i
t
y
,f
a
i
l
u
r
e
,f
i
b
e
r
r
e
i
n
C
o
r
c
e
dc
o
n
c
r
e
t
e
,nonlocalcontinuummodels,
r
e
i
n
f
o
r
c
e
dc
o
n
c
r
e
t
e,
s
i
z
ee
f
f
e
c
t,
s
t
r
a
i
ns
o
f
t
e
n
i
n
g,
s
t
r
u
c
t
u
r
a
ld
e
s
i
g
n,
t
e
s
t
i
n
gmethods,
u
l
t
i
m
a
t
el
o
a
d
s
.
Concretes
t
r
u
c
t
u
r
e
sa
r
efuUo
fc
r
a
c
k
s
.F
a
i
l
u
r
eo
fωucretes
t
r
u
c
t
u
r
e
st
y
p
i
c
a
l
l
yi
n
v
o
l
v
e
s
s
t
a
b
l
egrowtho
fl
a
r
g
ec
r
a
ε
k
i
n
g
z
o
n
e
渇a
ndt
h
ef
o
r
m
a
t
i
o
no
fl
a
r
g
ef
r
a
c
t
u
r
e
sb
e
f
o
r
et
h
emaximum
1
0
叫 I
Sr
e
叫 h
e
d
.Yetd
e
s
i
g
ni
sn
o
tbasedonf
r
a
c
t
u
r
emech
a
.
n
i
c
s,
eventhought
h
eb
a
s
i
cf
r
a
c
t
u
r
e
mech
a
.
n
i
c
st
h
e
o
r
yh
a
sbeena
.
v
a
i
l
a
b
l
es
i
n
c
et
h
emiddleo
ft
h
i
sc
e
n
t
u
r
y
.Sowhyhasn
o
tf
r
a
c
t
u
r
e
mecha
.
n
i
c
sbeeni
n
t
r
o
d
u
c
e
di
n
t
o∞n
c
r
e
t
ed
e
s
i
g
n
? Have∞n
c
r
e
t
ee
n
g
i
n
e
e
r
sbeeng
u
i
l
t
yo
f
l Theformsoff
r
a
c
t
u
r
emechanicswhichwerea
v
a
i
l
a
b
l
eu
n
t
i
lr
e
c
e
n
t
l
y
j
g
n
o
r
a
n
c
e
? Nota
ta
l.
werea
p
p
l
i
c
a
b
l
eo
n
l
yt
ohomogen
田 u
sb
r
i
t
t
l
em
a
t
e
r
i
a
l
ssuch制 g
l
a
s
s,
o
rt
ohomogeneous
b
r
i
t
t
l
e
-d
u
c
t
i
l
em
e
t
a
l
s
. Theq
u
e
s
t
i
o
no
fa
p
p
l
i
c
a
b
i
J
i
t
yo
ft
h
e
s
ec
l
a
s
s
i
c
a
lt
h
e
o
r
i
田 t
oc
o
n
c
r
e
t
e
wase
x
p
l
o
r
e
dl
o
n
gago-thei
d
伺 o
fu
s
i
n
gt
h
es
t
r
e
s
si
n
t
e
n
s
i
t
yf
a
.
c
t
o
rappeareda
l
r
e
a
d
yi
nt
h
e
帥 r
l
y1
9
5
0
'
s(
e
.
g
.,
B
r
e
s
l
e
r姐 dWollack,
1
9
5
2
)ands
e
r
i
o
u
si
n
v
e
s
t
i
g
a
t
i
o
n
ss
t
a
r
句 di
nthe1
9
6
0
'
s
(
e
.
g
.,Kapl
a
.
n
, 1961,
姐dothers). Butthe叩swerw出
,a
.
tt
h
a
.
ttime,
n
e
g
a
t
i
v
e(
e
.
g K
e
s
l
e
r
,
NausandLo
t
t,
1
9
7
1
)
.Asi
snowunderstood,
t
h
er
e
a
s
o
nw剖 t
h
a
ti
nc
o
n
c
r
e
t
es
t
r
u
c
t
u
r
e
sone
mustt
a
k
ei
n
t
oaccοunts
t
r
a
i
n
s
o
f
t
e
n
i
n
gduet
od
i
s
t
r
i
b
u
t
e
dc
r
a
c
k
i
n
g,
l
o
c
a
l
i
z
a
t
i
o
no
fc
r
a
c
k
i
n
g
i
n
t
ol
a
.
r
g
e
rf
r
a
c
t
u
r
e
sp
r
i
o
rt
of
a
i
l
u
r
e
, 組db
r
i
d
g
i
n
gs
t
r
e
s
s
e
sa
tt
h
ef
r
a
c
t
u
r
ef
r
o
n
t
. A formo
f
f
r
a
c
t
u
r
em民 h姐 iωthatc
組
bea
p
p
J
i
e
dt
osuchs
t
r
u
c
t
u
r
e
sh
a
sbeendevelopedo
n
J
yd
u
r
i
n
g
t
h
el
a
s
td
e
c
a
d
e
.
g
nh副
a
.
l
r
e
a
.
d
ys
田 nt
wor
e
v
o
l
u
t
i
o
n
s
.Thef
i
r
s
t,
whichmadet
h
et
e
c
h
n
o
l
o
g
y
Concreted田 i
o
f∞n
c
r
e
t
es
t
r
u
c
t
u
r
e
日p
o
s
s
i
b
l
e,
wast
h
edevelopmento
ft
h
ee
l
a
s
t
i
cno
・t
e
n
s
i
o
n姐 a
l
y
s
i
sd
u
r
i
n
g
1
9
0
0・1
9
3
0
.Thesecondr
e
v
o
l
u
t
i
o
n,
b
a
s
e
donat
h
e
o
r
yc
o
n
c
e
i
v
e
dc
h
i
e
s
yd
u
r
i
n
gthe1
9
3
0
'
s,
was
t
h
ei
n
t
r
o
d
u
c
t
i
o
no
fp
l
a
s
t
i
cl
i
m
i
t如 a
l
y
s
i
s,
whichoc
叩 r
r
e
dd
u
r
i
n
g1940-1970
.There釘 egood
r
e
a
s
o
n
st
ob
e
l
i
e
v
et
h
抗 t
h
ei
n
t
r
o
d
u
c
t
i
o
no
ff
r
a
c
t
u
r
emechanic8i
n
t
ot
h
ed
e
s
i
g
no
fc
o
n
c
r
e
t
e
s
t
r
u
c
t
u
r
倒
, bothreinforcedandunreiuforced,mightbethethirdmajorrevolution. The
t
h
e
o
r
y
,
f
o
r
m
u
l
a
t
e
dmostlyd
u
r
i
n
gt
h
eJ
a
s
tdozeny
e
a
r
s,
f
i
n
a
l
l
yappe
a
.
r
st
ober
i
p
e
.
Fr
】舵
a
.
c
t
旬
ur
,問
er
e
s
e
a
r
c
h
e
r
sha
.
vea
.
tt
h
e戸
pr
伺 e
叩n
tnodoubtt
凶
ha
tu
凶
も
hei
I
叫
北
1
trod
町
u
I
民
c
“
ω
も
ぬ
∞
o
i
nof合
f
r
悶a
拭c
t
旬u
r
,陀
'
eme
令ト
c
h
a
I
凶cs叫
m
も
ω
ot
h
ed
e
s
i
g
nc
r
i
t
e
r
i
af
o
ra
l
lb
r
i
川 ef
a
i
l
u
r
e
so
fr
e
i
n
f
o
r
c
e
dc
o
n
c
r
e
t
es
t
r
u
c
t
u
r
e
s(
s
u
c
h
a
sdiagon
a
.
lshω,punchingshear,t
o
r
s
i
o
no
rp
u
l
lout,o
rf
o
r∞n
c
r
e
t
edams),c
組
b
r
i
n
g
abouts
i
g
n
i
f
i
c
a
n
tb
e
n
e
f
i
t
s
.I
twi
1
lmakei
tp
o
s
s
i
b
l
et
oa
c
h
i
e
v
emoreuniforms
a
f
e
t
ymargins,
e
s
p
e
c
i
a
l
l
yf
o
rs
t
r
u
c
t
u
r
e
so
fd
i
f
f
e
r
e
n
ts
i
z
e
s
. This,
i
nturn,
wi
1
limproveeconomya
sw
e
l
la
s
s
t
r
u
c
t
u
r
a
lr
e
l
ia
.
b
i
l
i
t
y
.1
もw
i
l
lmakei
tp
o
s
s
i
b
l
et
oi
n
t
r
o
d
u
回 目wd
e
s
i
g
n
sandu
t
i
l
i
z
enew∞n
c
r
e
t
em
a
t
e
r
i
a
l
s
.F
r
a
c
t
u
r
em田 h
a
n
i
c
sw
i
l
lbep
a
r
t
i
c
u
l
a
r
l
yimportantf
o
rhighs
t
r
e
n
g
t
hc
o
n
c
r
e
t
e
s
t
r
u
c
t
u
r
e
s,
f
i
b
e
r
r
e
i
n
f
o
r
c
e
dc
o
n
c
r
e
t
es
t
r
u
c
t
u
r
e
s,
c
o
n
c
r
e
t
es
t
r
u
c
t
u
r
e
so
fu
n
u
s
u
a
l
l
yl
a
r
g
es
i
z
e
s,
a
.
ndf
o
rp
r
e
s
t
r
e
s
s
e
ds
t
r
u
c
t
u
r
e
s
. Thea
p
p
l
i
c
a
t
i
o
no
ff
r
a
c
t
u
r
emechanicsi
smostu
r
g
e
n
tf
o
r
s
t
r
u
c
t
u
r
回 s
uch槌 ∞n
c
r
e
t
edamsandn
u
c
l
e
a
rr
e
a
c
t
o
rv
e
s
s
e
l
so
rc
o
n
t
a
iQ.IIle
n
t
s,
f
o
rwhicht
h
e
s
a
f
e
t
ycon
白 血s
a
.
r
ep
a
r
t
i
c
u
l
a
r
l
yhigha
.
n
d出 econsequenαso
fap
o
t
e
n
t
i
a
ld
i
s
a
s
t
e
renormous.
S
u
r
v
e
y
so
fc
o
n
c
r
e
t
ef
r
a
c
t
u
r
emechanicshaver
e
c
e
n
t
l
ybeenpreparedbyv
a
r
i
o
u
scommitt
e
e
s(Wittmann,
1983,
叩dElfgren,1989).However,
duet
ot
h
er
a
p
i
d
l
yadvancingr
e
s
e
a
.
rch,
t
h
ec
o
n
t
e
n
t
so
ft
h
ep
r
e
s
e
n
ts
t
a
t
e
of
.
叶l
e
a
r
tr
e
p
o
r
t訂 eq
u
i
t
ed
i
f
f
e
r
e
n
t
.A u
n
i
f
i
e
d
,
s
y
s
t
e
m
a
t
i
c
p
r
e
s
e
n
t
a
'
も
i
o
n,
r
a
t
h
e
rthanac
o
m
p
i
l
a
t
i
o
no
fa
l
lt
h
ec
o
n
t
r
i
b
u
t
i
o
n
sbyva
.
r
i
o
u
sa
u
t
h
o
r
s,
i
sa
t
temptedi
nt
h
ep
r
e
s
e
n
ts
t
a
t
e
-o
f
a
r
trepo
比 Ther
e
p
o
r
ti
saimedp
r
i
m
a
r
i
l
ya
tr
e
s
e
a
.
r
c
h
e
r
s,
n
o
t
ne
偲 s
s
a
r
i
l
ys
p
e
c
i
a
l
i
s
t
si
nf
r
a
c
t
u
r
emech
組 i
c
s
.However,
i
ts
h
o
u
J
da
l
s
obeo
fi
n
t
e
r
e
s
tt
od
田 i
g
n
吋
“
7
6
e
n
g
i
n
田 r
sb
e
c
a
u
s
ei
td
e
s
c
r
i
b
e
sat
h
e
o
r
yt
h
a
.
ti
sl
i
k
e
l
yt
op
r
o
f
o
u
n
d
l
yi
n
f
l
u
e
n
c
et
h
ed
e
s
i
g
np
r
a
.
c
t
i
c
ei
nt
h
ene
a
.
rf
u
t
u
r
e
.S
u
b
s
e
q
u
e
n
tr
e
p
o
r
も
sd
e
a
l
i
n
gw
i
t
ha
p
p
l
i
c
も
.
ai
o
n
si
nd
e
s
i
g
n
,
f
i
n
i
t
ee
l
e
m
e
n
t
a
.
c
t
u
r
e
,
anddynamicf
r
a
.
c
t
u
r
ean
a
.
l
y
s
i
s,
a
r
ei
np
r
e
p
a
.
r
a
.
t
i
o
nbyACICommittee
a
n
a
l
y
s
i
so
ff
r
(0)
レ
4
4
6
.
Chapter1
. W H YFRACTUREMECHANICS?
ε
F
r
a
.
c
t
u
r
emech
組 i
c
s,
i
nab
r
o
a
ds
e
n
s
e
,i
sa
.f
a
.
i
l
u
r
et
h
e
o
r
ywhich(
1
)u
s
e
se
n
e
r
g
yc
r
i
t
e
,
ap
o
s
s
i
b
l
yi
nc
o
n
j
u
n
c
t
i
o
nw
i
t
hs
t
r
e
n
g
t
hc
r
i
t
e
r
i
a, 組d(
2
)w
h
i
c
ht
a
k
e
si
n
t
o配 ∞un
色f
a
i
l
u
r
e
r
i
p
r
o
p
a
g
a
t
i
o
nt
h
r
o
u
g
ht
h
es
t
r
u
c
t
u
r
e
.
.
.
E
t?LU_
j
_
t
1
.1F
i
v
eReasonsf
o
rF
r
a
c
t
u
r
eMechanicsApproach
S
i
n
田 ∞n
c
r
e
t
es
t
r
u
c
t
u
r
e
shaveb
田 nd
e
s
i
g
n
e
d姐 ds
u
c
c
e
s
s
f
u
l
l
yb
u
i
l
ta
c
c
o
r
d
i
n
gt
oc
o
d
e
s
w
h
i
c
ht
o
t
a
l
l
yi
g
n
o
r
ef
r
a
c
t
u
r
em
e
c
h
a
n
i
c
st
h
回 r
y
,i
tmight日eemunnec
閃 s
a
r
yt
och
姐 g
et
h
e
t
h
e
r
ea
r
ef
i
v
ec
o
m
p
e
l
l
i
n
gr
e
出 o
n
sf
o
rd
o
i
n
gs
o
.
c
u
r
r
e
n
tp
r
a
ι
t
i
c
e
.N
e
v
e
r
t
h
e
l
e
s
s,
U
A
B
~:I h
Reason1
:EnergyRequiredf
o
rCrackFormation
i
tmustber
e
c
o
g
n
i
民 dt
h
a
tw
h
i
l
ec
r
a
ε
ki
n
i
t
i
a
t
i
o
n
Fromt
h
es
t
r
i
c
t
l
yp
h
y
s
i
c
a
lv
i
e
w
p
o
i
n
t,
maydependons
t
r
e
s
s,
t
h
ea
c
t
u
a
lf
o
r
m
a
t
i
o
no
fc
r
a
c
k
sr
e
q
u
i
r
e
sac
e
r
t
a
i
ne
n
e
r
g
y t
h
ei
r
a
c
t
u
r
e
e
n
e
r
g
y-whichr
e
p
r
e
s
e
n
t
st
h
es
u
r
f
.
a
c
ee
n
e
r
g
yo
fas
o
l
i
d
. Hence
,energycriteriashouldbe
u
s
e
d
.T
h
i
s釘 gumentmights
u
缶c
et
oap
h
y
s
i
c
i
s
tb
u
tn
o
t乱 d
e
s
i
g
n
e
r
. Butt
h
e
r
ea
.
r
eo
t
h
e
r
r
e
a
s
o
n
s
.
←
園
・
(d)
・
嗣
.
自
・ .
・
.・
.・
・
3
固圃.
(
e
)
l
o
a
d
Reason2
:ObjectivityofC
a
l
c
u
l
a
t
i
o
n
s
o
r
ymustbeo
b
j
e
c
t
i
v
ei
nt
h
es
e
n
s
et
h
a
tt
h
er
e
s
u
l
to
fc
a
l
c
u
l
a
t
i
o
n
smade
Anyp
h
y
s
i
c
a
l山 e
w
i
t
hi
tmustn
o
tdependons
u
b
j
e
c
t
i
v
e
ω
p
e
c
t
ss
u
c
h舗 t
h
ec
h
o
i
c
eo
fc
o
o
r
d
i
n
a
t
e
s,
t
h
ec
h
o
i
c
e
o
fmesh,
e
t
c
.I
fathωryi
sf
o
u
n
dt
obeu
n
o
b
j
e
c
t
i
v
e
,
i
tmustber
e
j
e
c
七e
d
.Therei
snon
e
e
d
t
oe
v
e
n∞mparei
tもoe
x
p
e
r
i
m
e
n
t
s
.O
b
j
e
c
t
i
v
i
t
ycomesa
h
e
a
do
fe
x
p
e
r
i
m
e
n
t
a
lv
e
r
i
f
i
c
a
t
i
o
n
.
l
e
m
e
n
ta
n
a
l
y
s
i
so
fc
o
n
c
r
e
t
ec
r
a
c
k
i
n
gi
s出 ec
o
n
c
e
p
to
f
Ap
o
w
e
r
f
u
la
p
p
r
o
a
c
ht
of
i
n
iもee
sme
a
.
r
edc
r
a
c
k
i
n
g,
i
n
t
r
o
d
u
c
e
dbyRa
s
h
i
d(
1
9
6
8
)
.Aαordingt
ot
h
i
sa
p
p
r
o
a
c
h
,thes
t
r
e
s
si
na
a
t
e
r
i
a
l, ,
anda
f
t
e
rr
e
a
c
h
i
n
gt
h
i
s
f
i
n
i
t
ee
l
e
m
e
n
ti
sl
i
m
i
t
e
dbyt
h
et
e
n
s
i
l
es
t
r
e
n
g
t
ho
f出 em
s
t
r
e
n
g
t
hl
i
m
i
t,t
h
es
t
r
e
s
si
nt
h
ef
i
n
i
t
ee
l
e
m
e
n
tmustd
e
c
r
e
a
s
e
.Asi
n
i
t
i
a
l
l
yp
r
a
c
t
i
田 d
,出es
t
r
e
s
s
i
nav
e
r
t
i
c
a
ld
r
o
p
jb
u
t6
0
0
ni
七W剖 r
e
a
l
i
z
e
dt
h
a
t
wasassumedt
od
e
c
r
e
a
s
es
u
d
d
e
n
l
yt
oz
e
r
o,
b
e
t
t
e
randmorer
e
a
l
i
s
t
i
cr
e
s
u
l
t
sa
.
r
eu
s
u
a
l
l
yo
b
t
a
i
n
e
di
ft
h
es
t
r
e
s
si
sr
e
d
u
c
e
dg
r
a
d
u
a
l
l
y
,
i
.
e
.,
t
h
em
a
t
e
r
i
a
li
sassumedt
oe
x
h
i
b
i
ts
t
r
a
i
n
s
o
f
t
e
n
i
n
g(
S
c
a
n
l
o
n,
1
9
7
1
;L
i
nandS
c
o
r
d
e
l
i
s,
1
9
7
5
)
;
S田 F
i
g
.1
.1
a
.Thec
o
n
c
e
p
to
fs
t
r
a
i
n
s
o
f
t
e
n
i
n
g
,
t
h
o
u
g
h,
p
r
o
v
e
dt
obeamixedb
l
e
s
s
i
n
g
.A
f
t
e
r
a
.
i
n
s
o
f
t
e
n
i
n
gh吋 b
e
e
ni
m
p
l
e
m
e
n
t
e
di
nl
a
r
g
ef
i
n
i
t
ee
l
e
m
e
n
tp
r
o
g
r
a
m
.
sandw
i
d
e
l
ya
p
p
l
i
e
d,
s
t
r
i
twasd
i
s
c
o
v
e
r
e
dt
h
a
tt
h
e∞n
v
e
r
g
e
n
偲 p
r
o
p
e
r
t
i
e
sa
.
r
ei
n
c
o
r
r
e
c
tandt
h
ec
a
l
c
u
l
a
t
i
o
nr
e
s
u
l
t
s
.
n
a
l
y
s
t
'
sc
h
o
i
c
eo
f山 emesh,
i
.
e
.,
t
h
er
e
s
u
l
t
ss
i
g
n
i
f
i
白 川l
y
a
r
en
o
to
b
j
e
c
t
i
v
ew
i
t
hr
e
g
a
r
dt
ot
h
ea
c
h
a
n
g
ei
ft
h
emeshi
sr
e
f
i
n
e
d(B
話 姐t
,1976,1982jBa
.
i
a
n
tandC
e
d
o
l
i
n,
1
9
7
9,
1
9
8
0
,1
9
8
3
;
B
a
i
a
n
tandOh,1
9
8
3
a
;Darwin,1
9
8
5
jR
o
t
s,Nauta
,KustersandBlaauwendraad,1
9
8
5
)
.
.
r
ee
n
c
o
u
n
t
e
r
e
dwhenc
r
a
c
k
i
n
gi
smodeleda
sd
i
s
c
r
e
t
ei
n
t
e
r
e
l
e
m
e
n
tc
r
a
c
k
s,
S
i
r
n
i
l
a
rp
r
o
b
l
e
m
sa
・~・
load
A
:
f
d
e
f
L
(
f
)
,~
crackI
ength
…
energy
i
c
h
a
n
i
c
sor
nonl
o
c
a
lf
i
n
i
t
eelements
{
山
刊
N
o
.o
felements
c
r
i
t
e
r
i
o
nand
l
o
c
α
If
i
n
i
t
eelements
F
i
g
.1
.1S
p
u
r
i
o
u
sMeshS
e
n
s
i
t
i
v
i
t
y
9
8
(
α
)
b剖 吋 ont
h
e山 e
n
g
t
hc
o
n
c
e
p
t(
t
m
sa
p
p
r
o
a
.
c
hwasi
n
t
r
o
d
u
句 di
n
t
of
i
n
i
t
ee
l
e
m
e
n
ta
n
a
l
y
s
i
sby
1
9
6
2,
組dbyNgoandScordelis,
1
9
6
7
)
.
C
l
o
u
g
h,
e
n
s
i
も
ivityωnbei
l
l
u
s
t
r
a
t
e
d,
f
o
rexample
,
byt
h
er
e
c
t
a
n
Theproblemo
fs
p
u
r
i
o
u
sm包 hs
g
u
l
a
rp
a
n
e
li
nF
i
g
.1
.1
bandc
,whichi
ss
u
b
j
e
c
t
e
dt
oau
n
i
f
o
r
mv
e
r
t
i
c
a
ld
i
s
p
l
a
c
e
m
e
n
ta
tt
h
e
t
o
pb
o
u
n
d
a
r
y
.As
m
a
l
lr
e
g
i
o
nn
e
a
rt
h
ec
e
n
t
e
ro
ft
h
el
e
f
ts
i
d
ei
s掛 sumedt
ohaveas
l
i
g
h
t
l
y
s
m
a
l
l
e
rs
t
r
e
n
g
t
ht
h
回 出er
e
s
to
ft
h
ep
a
n
e
l,
andc
o
n
s
e
q
u
e
n
t
l
ya
.s
mearedc
r
a
.
c
kbands
ta
.
r
t
s
a
.
d
i
n
gw
i
t
htwof
i
n
i
t
e
g
r
o
w
i
n
gfroml
e
f
tt
or
i
g
h
t
.Thes
o
l
u
t
i
o
ni
so
b
t
a
i
n
e
dbyi
n
c
r
e
m
e
n
t
a
l
l
o
e
l
e
m
e
n
tm
e
s
h
e
so
fv
e
r
yd
i
f
f
e
r
e
n
tmeshs
i
z
e
sa
sshown.Bys
t
a
b
i
l
i
t
yc
h
e
c
k
si
ti
sf
o
u
n
dt
h
a
t
fs
i
n
g
l
ee
l
e
m
e
n
tw
i
d
t
ha
.
tt
h
ec
ra
.
c
k
i
n
gf
r
o
n
t
t
h
ec
r
a
c
k
i
n
gmusta
l
w
a
y
sl
o
c
a
l
i
z
ei
n
t
o乱 bando
(
F
i
g
.l
.
lb,
c
)
. Thet
y
p
i
c
a
ln
u
m
e
r
i
c
a
lr
e
s
u
l
t
sf
o
r出 向 回 w
e
l
l錨 v
a
r
i
o
u
so
t
h
e
rp
r
o
b
l
e
m
sa
r
e
a
.
t
e
di
nF
i
g
.1
.1
d,
e,
f
.l
nt
h
el
o
a
d
d
e
:
f
i
e
c
t
i
o
nd
i
乱g
ram(
F
i
g
.l
.
1d
),
i
ti
ss
伺 nt
h
a
.
tt
h
ep
e
a
k
i
l
l
u
s
t
r
h
ep
o
s
t
p
e
a
ks
o
f
t
e
n
i
n
gi
ss
t
r
o
n
g
l
yd
e
p
e
n
d
e
n
tont
h
emeshs
i
z
e
,
b
e
i
n
gr
o
u
g
h
l
y
l
o
a
da
sw
e
l
l剖 t
p
r
o
p
o
r
t
i
o
n
a
lt
oh-1/2 wherehi
st
h
ee
l
e
m
e
n
ts
i
z
e
.P
l
o
t
t
i
n
gt
h
el
o
a
d(
r
e
a
c
t
i
o
n
)v
e
r
s
u
st
h
e
l
e
n
g
t
ho
ft
h
ec
r
a
c
kb組 d,
l
a
r
g
ed
i
f
f
e
r
e
n
c
e
s制 a
g
a
i
nf
o
u
n
d(
F
i
g
.l
.
l
e
)
.
r
a
c
k
i
n
gdeαe
出回 w
i
t
ht
h
er
e
f
i
n
e
m
e
n
to
ft
h
ef
i
n
i
t
e
Thee
n
e
r
g
ywhichi
sd
i
s
s
i
p
a
t
e
ddueもoc
.l
f
)andc
o
n
v
e
r
g
e
st
o0a
sh→ O
.
e
l
e
m
e
n
tmesh(
F
i
g.l
Thef
o
r
e
g
o
i
n
gu
n
o
b
j
e
c
t
i
v
i
t
yi
sp
h
y
s
i
c
a
l
l
yu
n
a
c
c
e
p
t
a
b
l
e
.Theo
n
l
ywa
.
yt
oa
.
v
o
i
di
ti
ssome
formo
ff
r
a
c
t
u
r
emech
a
.
ni
c
s
.Bys
p
e
c
i
f
y
i
n
gもhee
n
e
r
g
yd
i
s
s
i
p
a
t
e
dbyc
r
a
c
k
i
n
gp
e
ru
n
i
tl
e
n
g
t
h
o
ft
h
ec
r
a
c
ko
rt
h
ec
r
a
c
kband,
t
h
eo
v
e
r
a
l
le
n
e
r
g
yd
i
s
s
i
p
a
t
i
o
ni
sf
o
r
c
e
dt
obei
n
d
e
p
e
n
d
e
n
t
o
ft
h
ee
l
e
m
e
n
ts
u
b
d
i
v
i
s
i
o
n(
t
h
eh
o
r
i
z
q
n
t
a
ld
a
s
h
e
dl
i
n
ei
nF
i
g
.l
.l
f
),
a
.
n
ds
oi
st
h
emaximum
1
0
0
0.
platecu
他
)
何
耐r
i
n
g
def.
l
F
i
g
.1
.
2LoadD
e
s
e
c
t
i
o
nDiagramo
fD
u
c
t
i
l
eandB
r
i
t
t
l
eS
t
r
u
c
t
u
r
e
s
Reason3
: LackofYieldPlateau
B剖 edonl
o
a
d
d
e
f
l
e
c
t
i
o
nd
i
a
g
r
a
.
ms,
onemayd
i
s
t
i
n
g
u
i
s
htwob
a
s
i
ct
y
p
e
so
fs
t
r
u
c
t
u
r
a
l
f
a
i
l
u
r
e
:p
l
a
s
t
i
candb
r
i
t
t
l
e
. Thet
y
p
i
c
a
lc
h
a
r
a
c
t
e
r
i
s
t
i
co
fp
l
a
s
t
i
cf
a
i
l
u
r
ei
st
h
a
tt
h
es
t
r
u
c
e
d
e
g
r
e
e
o
f
f
r
e
e
d
o
mmechanisms
u
c
ht
h
抗 f
a
i
l
u
r
ei
nv
a
r
i
o
u
sp
a
r
t
so
f
t
u
r
ed
e
v
e
l
o
p
sas
i
n
g
l
t
h
es
t
r
u
c
t
u
r
ep
r
o
c
田d
ss
i
m
u
l
t
a
n
e
o
u
s
l
y
,
i
np
r
o
p
o
r
t
i
o
nt
oas
i
n
g
l
ep
a
r
a
m
e
t
e
r
.Suchf
a
i
l
u
r
e
sa
r
e
.2
a
)
.
m
a
n
i
f
e
s
t
e
dbyt
h
ee
x
i
s
t
e
n
c
eo
fal
o
n
gy
i
e
l
dp
l
a
t
e
a
uont
h
el
o
a
d
d
e
f
l
e
c
t
i
o
ndiagram(
F
i
g
.1
I
ft
h
el
o
a
d
d
e
f
l
e
c
t
i
o
ndiagramd
o
e
sn
o
tha
.
v
es
u
c
hap
l
a
t
e
a
u,
t
h
ef
a
i
l
u
r
ei
sn
o
tp
l
a
s
t
i
cbu
t
b
r
i
t
t
l
e(
o
rb
r
i
t
t
l
e
d
u
c
t
i
l
e
)(
F
i
g
. 1
.2
b
)
.I
ft
h
e
r
ea
r
enos
i
g
n
i
f
i
c
a
n
tgωmetrice
仇c
t
ss
u
c
h制
t
h
eP-se
f
f
e
c
ti
nb
u
c
k
l
i
n
g,
t
h
ea
b
s
e
n
c
eo
fap
la
.
t
e
a
ui
m
p
l
i
e
st
h
ee
x
i
s
t
e
n
c
eo
fs
o
f
t
en
i
n
gi
nt
h
e
a
.
1due七of
r
a
c
t
u
r
e,
c
r
a
c
k
i
n
go
ro
t
h
e
rda
.
m
a
g
e
ji
ti
m
p
l
i
e
st
h
a
t山 ef
a
.
i
l
u
r
ep
r
o
c
e
s
sc
a
.
n
n
o
t
m
a
t
e
r
i
d
e
v
e
l
o
pas
i
n
g
l
ed
e
g
r
e
e
-o
f
・
f
r
e
e
d
o
mmechanismb
u
tc
o
n
s
i
s
t
so
fp
r
o
p
a
g
a
t
i
o
no
ft
h
ef
a
i
l
u
r
ez
o
n
e
t
r
u
c
t
u
r
e
.Soもhef
a
i
l
u
r
ei
sn
o
n
s
i
m
u
l
t
a
n
e
o
u
sa
.
ndp
r
o
p
a
g
乱t
i
n
g
.
t
b
r
o
u
g
h
o
u
t七hes
Toi
l
l
u
s
t
r
a
t
et
h
i
sb
e
h
a
v
i
o
r,
c
o
n
s
i
d
e
rt
h
ep
u
n
c
h
i
n
gs
h
e
a
rf
a
.
i
l
u
r
eo
fas
l
a
b(
F
i
g
.1
.3
)
.The
t
y
p
i
c
a
l(
a
p
p
r
o
x
i
m
a
t
e
)d
i
s
t
r
i
b
u
t
i
o
n
so
ft
e
n
s
i
l
es
t
r
e
s
s
σ
a
l
o
n
gt
h
ef
a
.
il
u
r
es
u
r
f
a
.
c
ea
r
edrawni
n
ft
h
em
a
t
e
r
i
a
li
sp
l
a
s
t
i
c,
t
h
ec
r
o
s
ss
e
c
t
i
o
ng
ra
.
d
u
a
l
l
yp
l
a
s
t
i
c
i
z
e
su
n
t
i
la
1
1i
t
sp
o
i
n
t
s
t
h
ef
i
g
u
r
e
.I
a
r
ea
tt
h
ey
i
e
l
dl
i
m
i
t.However
,
i
ft
h
em
a
t
e
r
i
a
le
x
h
i
b
i
t
ss
o
f
t
e
n
i
n
g,
t
h
e
nt
h
es
も
r
e
s
speakmoves
a
c
r
o
s
st
h
ef
a
i
l
u
r
ez
o
n
e,
l
e
a
v
i
n
gar
e
d
u
c
e
ds
t
r
e
s
s(
s
o
f
t
e
n
i
n
g
)i
ni
t
swa
.
k
e
.Thes
t
r
e
s
sr
e
d
u
c
t
i
o
n
,
i
nw
h
i
c
h日 s
e出 ep
I
舗 t
i
cl
i
m
i
ta
.
n
a
l
y
s
i
si
sn
o
t5
0f
a
ro
f
f
.
i
sm
i
l
do
n
l
yi
ft
h
es
t
r
u
c
t
u
r
ei
ss
m
a
l
l
I
ft
h
es
t
r
u
c
t
u
r
ei
sl
a
r
g
e,
h
o
w
e
v
e
r,
t
h
es
t
r
e
s
sp
r
o
f
i
l
ed
e
v
e
l
o
p
sa
.s
t
e
e
ps
t
r
e
s
sd
r
o
pb
e
h
i
n
dt
h
e
p
e
a
k
s
t
r
e
s
sp
o
i
n
t,
andt
h
e
r
e
f
o
r
et
h
el
i
m
i
ta
.
n
a
l
y
s
i
ss
o
l
u
t
i
o
n
sg
r
o
s
s
l
yo
v
e
r
e
s
t
i
m
a
.
t
et
h
ef
a
.
i
l
u
r
e
(b)
slze:
plasticmoteri
al
(d)
平
large
)L/
μt
eigm
~--
一、、/〆~~
ε
"
)
i
L
I/
、
.
.
.
.E
↑
!
↑
↑
↑
F
i
g
.1
.3P
r
o
g
r
e
s
s
i
v
eN
a
t
u
r
eo
fF
a
i
l
u
r
eI
l
l
u
5
t
ra
.
t
e
df
o
rPunchingS
h
e
a
ro
f
aS
l
a
b
1
1
10
1
0
0
0
.
Reason4
:EnergyAbsorptionC
a
p
a
b
i
l
i
t
yandD
u
c
t
i
l
i
t
y
Thea
r
e
aundert
h
ee
n
t
i
r
el
o
a
dd
e
f
l
e
c
t
i
o
nd
i
a
g
r
a
mr
e
p
r
e
s
四 t
st
h
ee
n
e
r
g
ywhicht
h
es
t
r
u
c
t
u
r
ew
i
l
la
b
s
o
r
bd
u
r
i
n
gf
a
i
l
u
r
eandmustt
h
e
r
e
f
o
r
ebeB
u
p
p
l
i
e
dbyt
h
el
o
a
d
s
.C
o
n
s
i
d
e
r
a
t
i
o
n
o
f出 i
s四 位gyi
simpor
同且te
s
p
e
c
i
a
l
l
yf
o
rdynamicl
o
a
d
i
n
g
,anddeterminestheductilityof
t
h
es
t
r
u
c
t
u
r
e
.P
l
a
s
t
i
cl
i
m
i
ta
n
a
l
y
s
i
sc
孤
g
i
v
enoi
n
f
o
r
m
a
t
i
o
nont
h
ep
o
s
t
p
e
a
kd
e
c
l
i
n
eo
ft
h
e
sp
r
o
α別 . Someformo
ff
r
a
c
t
u
r
em
e
c
h
a
n
i
c
si
sne
四 四 位y
.
l
o
a
dandt
h
ee
n
e
r
g
yd
i
s
s
i
p
a
t
e
di
n山 i
円1
0stlab
tests
Reason5
:S
i
z
eE
f
f
e
c
t
Thes
i
z
ee
f
f
e
c
ti
s,
f
o
rd
e
s
i
g
ne
n
g
i
n
e
e
r
s
,probablythemostcompellingre品目onf
o
ru
s
i
n
g
ands
oat
h
o
r
o
u
g
hd
i
s
c
u
s
s
i
o
ni
si
no
r
d
e
r
.
f
r
a
c
t
u
r
em
e
c
h
a
n
i
c
s,
Thes
i
z
ee
f
f
e
c
ti
sd
e
f
i
n
e
dt
h
r
o
u
g
hacomp
釘 i
s
o
no
fg
e
o
m
e
t
r
i
c
a
l
l
ys
i
m
i
l
a
rs
t
r
u
c
t
u
r
e
so
fd
i
ι
andi
sc
o
n
v
e
n
i
e
n
t
l
yc
h
a
r
a
c
t
e
r
i
z
e
di
nt
e
r
m
so
ft
h
en
o
m
i
n
a
ls
t
r
e
s
s
σN a
tmaximum
f
e
r
e
n
ts
i
z
e
s,
entheσN v
a
l
u
e
sf
o
rg
e
o
m
e
t
r
i
c
a
l
l
ys
i
m
i
l
a
rs
t
r
u
c
t
u
r
田 o
fd
i
f
f
e
r
e
nt
(
u
l
t
i
m
a
t
e
)l
o
a
d,九. Wh
s
i
z
e
sa
r
et
h
es
晶me
,
wes
a
yt
h
a
.
tt
h
e
r
ei
snos
i
z
ee
f
f
e
c
t
.Ad
e
p
e
n
d
e
n
c
eo
fσN ont
h
es
t
r
u
c
t
u
r
e
s
i
z
e(
d
i
m
e
n
s
i
o
n
)i
sc
a
l
l
e
dt
h
es
i
問 e
f
f
e
c
t
.
o
tr
e
p
r
e
s
田 t姐 ya
c
t
ua
.
ls
t
r
e
s
si
nt
h
es
t
r
u
c
t
u
r
eb
u
tma
.
ybe
Then
o
m
i
n
a
ls
t
r
e
s
sn田 dn
u
/
b
dwhent
h
es
i
m
i
l
a
r
i
t
yi
stw
か d
i
m
e
n
s
i
o
n
a
l,
o
r出 凡/
c
Fwhen
d
e
f
i
n
e
ds
i
m
p
l
y回 σN =P
t
h
es
i
m
i
l
a
r
i
t
yj
st
h
r
e
e
-d
i
m
e
n
s
i
o
n
a
l
j b-t
h
i
c
k
n
e
s
so
ft
h
etw
令 d
i
m
e
n
s
i
o
n
a
ls
t
r
u
c
t
u
r
e, 組dd
a
.
ra
c
t
e
r
i
s
t
i
cd
i
m
e
n
s
i
o
no
ft
h
es
t
r
u
c
t
u
r
e,
w
h
i
c
hma
.
ybec
h
o
s
e
na
.
sa
.
nyd
i
m
e
n
s
i
o
n,
e
.
g
.パhe
c
h
,
o
ri
t
ss
p
a
n,
s
i
n
c
eo
n
l
yt
h
er
e
l
a
t
i
v
ev
a
l
u
e
so
fσN m
a
t
t
e
r
.
d
e
p
t
ho
ft
h
ebeam
A
c
c
o
r
d
i
n
gt
ot
h
ec
l
a
.
s
s
i
c
a
lt
h
e
o
r
i
e
s,
s
u
c
ha
se
l
a
s
t
i
ca
n
a
l
y
s
i
sw
i
t
ha
l
l
o
w
a
b
l
es
t
r
e
s
s,
p
l
a
.
si
i
c
I
im
i
tan
a
.
l
y
s
i
s,
a
.
sw
e
l
l却 姐yo
t
h
e
rt
h
田 r
i
e
swhichu
s
esomet
y
p
eo
fs
t
r
e
n
g
t
hl
i
m
i
to
rf
a
i
l
u
r
e
c
r
i
t
e
r
i
o
ni
nt
e
r
m
so
f耐 e
s
s
e
s(
e
.
g
.,v
i
s
c
o
e
l
a
.
st
i
c
i
t
y
,v
i
s
c
o
p
l
a
s
t
i
c
i
t
y
),
σN i
sc
o
n
s
t
a
n
t,出品 i
s,
i
n
d
e
p
e
n
d
e
n
to
ft
h
es
t
r
u
c
t
u
r
es
i
z
e
.T
h
i
sma
.
ybei
l
l
u
s
t
r
a
t
e
d,
e
.
g
.,
byc
o
n
s
i
d
e
r
i
n
gt
h
ee
l
a
.
s
t
i
c
組 dp
l
a
s
t
i
cf
o
r
m
u
l
a
.
sf
o
rt
h
e柑 e
n
g
t
ho
fbe
a
.
msi
nh
e
n
d
i
n
g
,
s
h
e
a
randt
o
r
s
i
o
n(
r
e
g
a
.
r
d
i
n
gt
h
e
d
e
f
i
n
i
t
i
o
n
σN =凡/
b
df
o
rt
o
r
s
i
o
n,
n
o
t
et
h
a
.
tonema
.
ys
e
t凡 =T
u
/
rwhereTu
=
=u
l
t
i
m
a
t
e
t
o
r
q
u
e,九==f
o
r
c
ea
c
t
i
n
gonanarm
,
r
,
s
u
c
ht
h
a
もr
/
Ho
rr
/
ai
sc
o
n
s
t
a
n
tf
o
rs
i
m
i
l
a
rs
t
r
u
c
t
u
r
e
s
o
fd
i
f
f
e
r
e
n
ts
i
z
e
s
jH =c
r
o
s
ss
e
c
t
i
o
nd
e
p
t
h,
a=c
r
a
c
kl
e
n
g
t
h
)
.I
ti
ss
e
e
nt
h
a
t油 田ef
o
r
m
u
l
a
s
釘 eo
ft
h
esameforme
x
c
e
p
tf
o
raf
a
.
c
t
o
r
.Thus,
i
fwep
l
o
もl
o
g
σN v
s
.l
o
gd,
t
h
ef
a
i
l
u
r
es
t
a
t
e
s
もr
e
n
g
t
ho
ry
i
e
l
dc
r
i
t
e
r
i
o
na
r
ea
l
w
a
.
y
sg
i
v
e
nbyah
o
r
i
z
o
n
t
a
l
l
i
n
e(
d
a
s
h
e
dl
i
n
e
a
c
c
o
r
d
i
n
gt
oas
i
nF
i
g
.1
.
4
)
.Sof
a
i
l
u
r
e
s乱c
c
o
r
d
i
n
gt
os
t
r
e
n
g
t
ho
ry
i
e
l
dc
r
i
t
e
r
i
ae
x
:
h
i
b
i
tnos
i
田 e
f
f
e
c
t
.
a
.
s
t
,
f
a
i
l
u
r
e
sg
o
v
e
r
n
吋 b
yl
i
n
e
a
re
l
a
s
t
i
cf
r
a
c
t
u
r
emech
a
.
n
i
c
se
x
h
i
b
i
tar
a
t
h
e
rs
t
r
o
n
g
Byc
o
n
t
r
s
i
z
ee
f
f
e
c
twhichi
nF
i
g
.1
.
4i
sd
回 c
r
i
b
e
dbyt
h
ei
n
c
l
i
n
e
dd
a
s
h
e
dl
i
n
eo
fs
l
o
p
e1
/
2
.Ther
e
a
l
i
t
y
o
ri
l
l
u
s
t
ra
.
t
e
dbyt
h
es
o
l
i
dc
u
r
v
ei
nF
i
g
.1.
4
.T
h
i
s
f
o
rc
o
n
c
r
e
t
es
t
r
u
c
t
u
r
e
si
sat
r
a
n
s
i
t
i
o
n
a
lbeh前 i
c
u
r
v
ea
p
p
r
o
a
c
h
e
st
h
eh
o
r
i
z
o
n
t
a
l
l
i
n
ef
o
rt
h
es
t
r
e
n
g
t
hc
r
i
t
e
r
i
o
ni
ft
h
es
t
r
u
c
t
u
r
ei
sv
e
r
ys
m
a
l
l,
組 dt
b
ei
n
c
l
i
n
e
ds
t
r
a
i
g
h
tl
i
n
ef
o
rl
i
n
e
a
re
l
a
s
t
i
cf
r
a
.
c
t
u
r
em
e
c
h
a
n
i
c
si
ft
h
es
t
r
u
c
t
u
r
ei
sv
e
r
yl
a
r
g
e
v
e
r
ys
m
a
l
l
"姐 d“
v
e
可 l
a
r
g
e
"w
i
l
lbec
l
a
r
i
f
i
e
dbyE
q
.5
.
1
1
)
.T
h
i
s
(
t
h
ep
r
e
c
i
s
emeaningo
f“
s
i
z
ee
f
f
e
c
t,
w
h
i
c
hi
sg
e
n
e
r
a
l
l
yi
g
n
o
r
e
dbyc
u
r
r
e
n
tc
o
d
e
s(
w
i
t
haf
e
we
x
c
e
p
t
i
o
n
s
),
i
so
b
v
i
o
u
s
l
y
i
m
p
o
r
t
a
n
ti
nd
e
s
i
g
n
.
i
1
i
t
y
.
Anothers
i
z
ee
f
f
e
c
twhichc
a
l
l
sf
o
rt
h
eu
s
eo
ff
r
a
c
t
u
r
em
e
c
h
a
n
i
c
si
se
f
f
e
c
to
fs
i
z
eond
u
c
t
larger
moststructures
・
s
;
z
e
)
log (
F
i
g
.
1
.
4
F
T
a
c
t
u
r
eM?chmesSizemeetf
o
rGeometriqllySimilustmct
u
r
e
so
fDi
J
f
e
r
e
n
tS
i
z
e
s
nom.
stress
O
N
re.
lde
f
l
.
F
i
g
.1
.5L
o
a
d
D
e
f
l
e
c
t
i
o
nD
i
a
g
r
a
m
so
fG
e
o
m
e
t
r
i
c
a
l
l
yS
i
m
i
l
a
rS
t
r
u
c
t
u
r
e
s
四
o
fD
i
f
f
e
r
e
n
tS
i
z
(b)
v
.
~~三
F
i
g
.
1
.
6
P
o
s
s
i
b
l
eF
i
n
a
lF
r
a
c
t
u
r
eL
o
c
a
t
i
o
n
sCausedbyRandomnesso
f
S
t
r
e
n
g
t
h
円・園田・ーー
幽
12
13
D
u
c
t
i
l
i
ho
fas
t
r
u
e
s
u
r
emaybec
h
a
r
a
C
4
e
r
i
z
e
dbyt
h
ed
e
f
o
r
m
a
t
i
o
na
twhich出 es
t
r
u
c
t
u
r
ef
a
i
l
s
i
ng
e
n
e
r
a
l,
f
a
i
l
u
r
eo
c
c
u
r
sr
i
g
h
ta
.
tt
h
ei
n
i
t
i
a
t
i
o
no
ft
h
em
a
c
r
o
s
c
o
pi
cc
r
a
c
kg
r
o
w
t
h
.F
o
rt
h
o
s
e
s
山 ctures,
f
u
n
c
t
i
o
n
σ(P
,
x)i
sk
n
o
w
n
;
σ(P
,
x
)=P
S
e
{
と
)where(=x/
d=relativeωrdinate
d= c
ha
.
r
a
.
c
t
e
r
i
s
t
i
cd
i
m
e
n
s
i
o
no
ft
h
es
t
r
u
c
t
u
r
e,
ands
e
(
{
)i
st
h
ee
l
a
.
s
t
i
cs
t
r
回 sd
i
s
t
r
i
v
e
c
t
o
r
s,
b
u
t
i
o
nduet
ou
n
i
t1
0
0
0(P= 1
)
.Thes
3
0
m
ea
s
s
u
m
p
t
i
o
n
sha
.
v
eo
f
t
e
nb
e
e
ni
m
p
l
i
e
di
nv
a
r
i
o
u
s
W
e
i
b
u
l
l
t
y
p
es
t
u
d
i
e
so
fc
o
n
c
r
e
t
es
t
r
u
c
t
u
r
e
s
.
C
o
n
c
r
e
t
es
t
r
u
c
t
u
r
e
s,
h
o
w
e
v
e
r
,
b
e
h
3
0
v
ed
i
f
f
e
r
e
n
t
l
y
. Duet
or
e
i
n
f
o
r
四m
enta
sw
e
l
la
st
h
e
3
o
i
n
s
o
f
もe
n
i
n
gi
naI
a
.
r
g
ez
o
n
eo
fm
i
c
r
o
c
r
a
c
k
i
n
ga
h
e
a
do
f出 et
i
po
fao
o
n
t
i
n
u
o
u
s
e
x
i
s
t
e
n
c
eo
fs
t
r
低k
,concretestruct町 田 donotf
a
i
la
tc
r
a
c
ki
n
i
t
i
a
t
i
o
n
.I
nf
a
c
t,
d
e
s
i
g
nc
o
d
e
sr
e
q
u
i
r
et
h
e
c
r
f
a
i
l
u
r
e10Mt
obes
i
g
n
i
f
i
c
叩 t
l
yh
i
g
h
e
rt
h
a
nt
h
ec
r
a
c
ki
n
i
t
i
a
t
i
o
nl
o
a
d
;f
o
rb
e
n
d
i
n
g,
.
a
tl
e
a
.
s
t1
.2
5
e
s
s
e
db
e
a
.
m
sand1
.2t
i
m
e
sh
i
g
h
e
rf
o
rp
r
e
s
t
r
e
s
s
e
dbe
a
.
m
s(
a
c
c
o
r
d
i
n
g
t
i
m
e
sh
i
g
h
e
rf
o
ru印 刷r
もoA
CI3
1
8
),
b
u
ti
np
r
a
.
c
t
i
c
et
h
i
sr
a
t
i
oi
su
s
u
a
l
l
ymuchh
i
g
h
e
r
.C
o
n
s
e
q
u
e
n
t
l
y
,
品
∞ncrete
a
c
r
o
s
c
o
p
i
cc
r
a
c
kgrow
t
hp
r
i
o
rt
o
s
t
r
u
c
t
u
r
eu
n
d
e
r
g
o
e
sp
r
o
n
o
u
n
c
e
di
n
e
l
a
s
t
i
cd
e
f
o
r
m
a
t
i
o
n岨 dm
s
u
c
ht
h
a
tt
h
es
t
r
e
s
sd
i
s
t
r
i
b
u
t
i
o
n
r
e
a
c
h
i
n
gt
h
ef
a
i
l
u
r
el
o
a
d
.T
h
i
sc
a
u
s
e
ss
t
r
e
s
sr
e
d
i
st
r
i
b
u
t
i
o
n
s,
σ(P
,
x)a
.
tf
a
i
l
u
r
ei
sv
e
r
yd
i
f
f
e
r
e
n
tf
r
o
mt
h
ee
l
a
s
t
i
cs
t
r
e
s
sd
i
s
t
r
i
b
u
t
i
o
ns
e(
C
).
T
h
i
sd
i
s
t
r
i
b
u
t
i
o
ni
sd
i
f
f
i
c
u
l
tt
od
e
t
e
r
m
i
n
e,
b
u
tt
h
en
e
a
r
t
i
pasympto
t
i
ce
l
a
s
t
i
cs
t
r
e
s
sf
i
e
l
d
,
PS
(
{
)
,
m
a
.
y
b
e
u
s
e
d
a
s
a
n
叩
p
r
似 i
m
a
t
i
o
n3
0
も
d
i
s
t
a
n
c
e
sn
o
t
t
o
o
f
a
.
r
f
r
o
m
a
n
d
n
o
t
t
o
o
c
1
0
s
e
e
a
c
r
o
c
r
a
c
ka
.
tt
h
emomento
ff
a
i
l
u
r
e
.Now
,
duet
os
i
n
g
u
l
a
r
i
t
yo
ft
h
i
sf
i
e
l
d,
t
ot
h
et
i
po
f出 em
t
h
es
t
r
e
s
sv
a
.
l
u
e
sf
,紅白e
rawayfromt
h
et
i
po
ft
h
em
a
c
r
o
c
r
a
c
ka
r
er
e
l
a
t
i
v
e
l
ys
m
a
l
la
.
ndmake
'
N (
E
q
.1
.2
)∞m
p
3
0
r
e
dt
o出e山 間 偲 i
nt
h
evolume竹 o
ft
h
e
an
e
g
l
i
g
i
b
l
ec
o
n
t
r
i
b
u
t
i
o
nt
oO
台a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
e砿 oundt
h
et
i
p
.T
h
i
so
fc
o
u
r
s
er
e
f
l
e
c
t
st
h
ef
a
.
c
tt
h
a
tt
h
evolumei
nw
h
i
c
h
t
h
em
a
c
r
o
c
r
a
c
kt
i
pa
tf
a
i
l
u
r
em
i
g
h
tbel
o
c
a
t
e
d(
a
sd
i
c
t
a
t
e
dbyt
h
el
a
w
so
fm
e
c
h
3
0
n
i
α
)i
sv
e
r
y
s
m
a
l
l(
e
.
g
.t
h
ed
i
a
g
o
n
a
ls
h
悶
c
r
a
c
ki
nF
i
g
.L6bc
a
.
n
notgrowt
o
w
a
r
dt
h
el
o
w
e
rmidspan
,
r
e
g
a
r
d
l
e
s
so
ft
h
es
t
r
e
n
g
t
hv
a
l
u
e
st
h
e
r
e
)
.
regiωort
o
w
a
r
dt
h
eu
p
p
e
rl
e
f
tc
o
r
n
e
ro
ft
h
ebeam
C
o
n
s
e
q
u
e
n
t
l
y
,thestatistica
.
ls
i
z
ee
f
f
e
c
tmustbes
m
a
l
l
e
rt
h
a
ni
nau
n
i
f
o
r
m
l
ys
t
r
e
s
s
e
d
wheref
a
.
il
u
r
ei
sp
r
e
c
i
p
i
t
a
t
e
dbyam3
o
c
r
o
c
r
a
c
k3
0
n
y
w
h
e
r
ew
i
thi
nt
h
evolumeo
ft
h
e
t
e
n
s
i
l
eb
a
r,
a
sp
o
i
n
t
e
do
u
tbyBa
.
zant(1986,1987a),
i
fo
n
ec
a
l
i
b
r
a
t
e
sWe
i
b
u
l
lpa
.
r
ame
t
e
r
目
b
a
r
. Thus,
m 姐 dσ ont
h
eb
a
s
i
so
fu
n
i
a
x
i
a
.
ls
t
r
e
n
g
t
ht
e
s
t
sa
.
n
dt
h
e
nu
s
e
st
h
esamep3
0
r
a
m
e
t
e
r
st
o
o
rdi
乱g
o
n
a
ls
h
e
a
rf
a
i
l
u
r
e
,
o
n
emustf
i
n
dt
h
es
t
a
t
i
s
t
i
c
3
0
ls
i
z
ee
f
f
e
c
tt
obe
p
r
e
d
i
c
tPJandO'N f
e
v
e
nt
h
o
u
g
hl
i
m
i
t
e
dt
e
s
tr
e
s
u
l
t
sf
o
rd
i
a
g
o
na
.
ls
h
e
a
rf
3
0
i
l
u
r
e
sha
.
veb
e
e
n
r
a
t
h
e
rs
m
a
l
l
. Thus,
s
u
c
c
e
s
s
f
u
l
l
yf
i
t
t
e
dbyf
o
r
m
u
l
a
sb
a
s
e
dont
h
eW
e
i
b
u
l
ld
i
s
t
r
i
b
u
t
i
o
n,
t
hes
i
z
ee
f
f
e
c
td
3
0
t
af
o
rb
o
t
h
t
h
ed
i
a
g
o
n
a
.
ls
h
e
a
.
rf
a
i
l
u
r
e
sa
.
ndt
h
eu
n
i
a
x
i
a
lf
a
i
l
u
r
e
sc
a
n
n
o
tbes
u
c
c
e
s
s
f
u
l
l
yf
i
t
t
e
du
s
i
n
gt
h
e
samem
a
t
e
r
i
a
.
lp
a
r
a
.
m
e
t
e
r
s(
u
n
l
e
s
s姐 i
且∞ r
r
e
c
もe
l
a
s
t
i
cs
t
r
e
s
sf
i
e
l
dP
s
.
(
(
)i
su
s
e
d
)
.
Thus
,
t
h
ep
r
i
n
c
i
pa
.
lf
a
u
l
to
ft
h
eW
e
i
b
u
l
l
t
y
p
es
t3
o
t
i
s
t
i
c
a
.
le
x
p
l
a
na
.
t
i
o
n
so
ft
h
es
i
z
ee
f
f
e
c
ti
n
民 e
f
f
e
c
tc
a
u
s
e
dbyt
her
e
d
i
s
t
r
i
b
u
t
i
o
no
fs
もr
e
s
s
c
o
n
c
r
e
t
es
t
r
u
c
t
u
r
e
si
sthωtheyi
g
n
o
r
et
h
es
i
σ(
P
,
x
)priort
ω
of
.
仏
悩
a
必i
l
u
r
e
.T
h
i
ss
i
z
ee
f
f
e
c
t,
w
h
i
c
hi
so
ft
h
ef
r
a
.
c
t
u
r
em配 ha
.
n
i
c
st
y
p
e
,
i
s踊 S
民 i
a
.
t
吋
w
i
t
ht
h
ee
n
e
r
g
yr
e
l
e
a
s
ei
n
t
ot
h
ef
r
o
n
to
fa
.l
a
r
g
ec
r
a
c
ka
.
n
dwo
ul
de
x
i
s
te
v
e
ni
ft
h
em
a
t
e
r
i
a
l
b
e
h
a
v
i
o
rw
e
r
ed
e
t
e
r
m
i
n
i
s
t
i
c
.
t
h
ep
r
o
p
e
ra
p
p
r
o
a
c
hi
st
of
i
tt
h
es
i
z
ee
f
f
e
c
td
a
t
a
.f
o
rac
O
D
e
r
e
t
es
t
r
u
c
t
ur
ef
i
r
s
t
T
h
e
r
e
f
o
r
e,
.f
r
a
c
t
u
r
em
e
c
h
a
n
i
c
st
h
e
o
r
y,
.
a
ndo
n
l
yi
fsomep
a
r
to
ft
h
eo
b
s
e
r
v
e
ds
i
z
ee
f
f
e
c
trem
a
.
i
n
s
bya
u
n
a
c
c
o
u
n
t
e
df
o
ri
tm晶ybea
t
t
r
i
b
u
t
e
dt
oW
e
i
b
u
l
l
t
y
p
es
ta
.
t
i
s
t
i
c
a
lphe
no
me
na(
Ba
.
z
a
n
t,
1
9
8
6,
1
9
8
7
a
)
.
I
lt
h
e
o
r
yt
h
e
nd
e
s
c
r
i
b
e
sas
i
z
ee
f
f
e
c
to
ft
h
evolumeo
ft
hes
t
r
u
c
t
u
r
e,
expr
e
s
si
n
gt
h
e
Weibu
m
お
む
ニ;
r
r
r
ニむ
と2
2
J
ぷ立
エ
7
1
J
i ozrJ;i;:::;
:eu;
A
i
s
pl
a
捌 l
e
n
tc
o
n
k
o
l(
ie
.i
叫
叫
m
n
叩
l
p
O
ω
s
吋 d
i
s
が
p
μ
I
a
.
蹴
c
回e
悶醐
m
1
吋
i
c
e
偲s
f
a
芯
i
li
泊
n色
出
h
隠
凶
e
i
rp
o
s
t
p
e
a
k,
s
t
r
3
o
i
n
s
o
f
t
e
n
i
n
gr
却炉・ I
na
.p
l
o
to
fσN v
e
r
s
u
sd
e
s
e
c
t
i
o
n,
t
h
e
d
e
v
3
o
r
a
.
c
t
e
r
i
z
e
dbyaもmgent(dashdl
i
n
ei
nFig.15)dac
e
r
t
a
i
neon
伽 n
も
f
a
i
l
u
r
ep
o
i
n
ti
sch
な丘公立政ぷh
;
2
2
e
;
出品治誌とぷねじ ;
Z
d
t
t
;
:
沈
t
y
p
es
t
1
O
L
ni
nFig.15.Asi
l
l
u
s
t
r晶,t
e
d,
f
a
i
l
u
r
eo
c
c
u
r
sc
l
o
s
e
rt
ot
h
epeak回 S
l
z
emcre
国 ω
.
T
h
i
se
f
T
e
di
s
a
g
a
i
ng
e
n
e
r
a
l
l
yp
r
e
d
i
c
t
e
dbyf
r
a
c
t
u
r
emechanichduet
ot
h
ef
a
c
tt
h
a
ti
na
l
3
0
r
g
e
rs
t
r
u
c
t
u
r
emo
陀 s
t
r
3
0
i
ne
n
e
r
g
yi
s肝 包l
3
0
b
l
eもod
r
i
v
et
h
ep
r
o
pa
.
g
a
t
i
o
no
ft
h
ef
a
il
u
r
ez
o
n
e
.
Ad
e
c
z
e
a
s
eo
fd
u
c
t
i
l
i
t
yo
fas
U
1
1
c
t
u
r
er
e
p
r
e
s
e
n
t
sani
n
c
r
e
a
s
ei
ni
t
sb
r
i
t
t
l
e
n
e
s
s
Thew
e
l
l
k
n
o
w
ne
f
f
e
c
to
fs
t
r
u
c
t
u
r
es
i
z
eo
rm四 l
b
e
rs
i
z
eonc
r
3
0
c
ks
p
3
0
c
i
n
ga
.
n
dc
r
a
c
kw
i
d
t
h
a
.
n
i
c
渇
・ T
hes
p
u
巾 u
se
f
f
e
c
to
fmeshs
i
田(Re剖on2
)c
組
i
sa
l
s
oe
x
p
l
i
C
3
0
b
l
ebyf
r
3
0
c
t
u
r
emech
ber
e
g
a
.
r
d
e
da
s0
3∞n
s
e
q
u
e
n
田 o
ft
h
es
t
r
u
c
t
u
凶 s
i
z
ee
f
f
e
c
t(
t
h
i
sc
姐 b
eshownby∞n
s
i
d
e
r
i
n
g
s
t
r
u
c
h
u
m
so
fd
i
f
e
r
e
n
ts
i
z
ebuwitht
h
esamemeshsim
, 組dt
h
e
ns
c
a
l
i
n
ga
l
ls
t
r
u
c
t
u
r
e
st
o
o
l
o
n
gw
i
t
ht
h
em
e
s
h
e
s
)
.
t
h
esames
i
z
e3
1
.2I
sWeibull'sS
t
a
t
i
s
t
i
c
a
lTheoryofS
i
z
eE
f
f
e
c
tApplicable?
T
r
a
d
i
t
i
o
n
a
l
l
y
,
t
h
es
i
z
ee
仔e
c
th
a
sb
e
e
ne
x
p
l
a
i
n
e
ds
t
a
t
i
s
t
i
c
a
l
l
y
,
byr
a
.
n
domnesso
f出 es
t
r
e
n
g
t
h
value.Thef
a
i
l
u
r
el
o
a
do
fac
h
&
i
ni
sdderminedbyt
h
eminimumv
a
l
u
eo
ft
h
es
t
r
e
n
g
t
ho
f
もt
h
el
o
n
g
e
rt
h
e
t
h
el
i
n
k
si
nt
h
echain-andt
h
esutMicalBime
t
r
e
e
ti
sduetobhef
a
c
ttha
c
h
a
i
n
,
t
h
esm
a
.
l
l
e
ri
sふ耐e
n
g
出 v
a
l
u
et
h
a
ti
sl
i
k
e
l
yt
obee
n
c
o
u
n
t
e
r
e
di
nt
h
ec
h
a
i
n
.T
h
i
s
e
x
p
l
a
n
a
t
i
o
n
,whichi
sc
e
r
t
a
i
n
l
yc
o
r
r
e
c
tf
o
rt
h
esimebdo
b
s
e
r
v
e
di
nt
h
ef
a
i
l
u
r
eo
fal
o
p
g
i
sd
e
s
c
r
i
b
e
dbyWeibd'sw
哨 e
北
a
k
悶
e
剖s
凶
も l
l
n
k
f
o
r
m
I
vs
h
r
e
s
s
e
dc
o
n
c
r
e
t
ebMi
nh
e
n
S
i
o
n(
F
i
g
.l
a
),
s
“
t
a
“
t
蜘
i
s
“
t
d
3
0eq
u
e
s
t
i
o
n
a
b
l
ef
o
rt
h
ef
o
l
l
o
w
i
n
gr
e
a
ω
s
o
n
.
凶
抗
胤
凶
紅
.
r
問
,
1
9
3
9
;
Z
a
i
b
e
v姐 dWitt
中ann,
1
9
7
3
;
A
c
c
o
r
d
i
n
gt
oW
e
i
b
u
l
l
w
p
et
h
e
o
r
yo
f
f
a
i
l
u
r
e
(
W
e
i
b
u
l
l
M
i
l
邸 h
ia
.
ndZ
a
i
t
s
e
v,
1
9
8
1
;組 dC
a
r
p
i
n
t
e
r
i,
1
9
8
6
),
t
h
ep
r
o
b
a
b
i
l
i
t
yo
ff
a
i
l
u
r
eo
fas
t
r
u
c
t
u
r
e
u
n
d
e
r1
0
0
0Pωdt
h
emeannomin
a
.
ls
t
r
e
s
s3
0
tf
a
i
l
u
r
ea
.
r
e
:
日
r
Lr但 豆 号斗
(
P
)=1-e
xp{-
I JVIσ'0
O
'
N
I
Vr
=主=土 f
∞[
1-P帥 (
P
)
]
d
P
J
(
1
.
1
)
(
1
.
2
)
b
d b
d1
0
WI
l
e
z
eP=meanload.x=coordinatev
e
c
t
o
r
s
,
V=volumeo
fs
b
mctuEe
,
E
4=r
e
p
r
e
s
e
n
も
3
0
t
i
v
e
volumeo
fm
a
h
e
r
i
a
l{
t
h
es
m
a
l
l
e
s
tvolumef
o
rwhicham
a
t
e
r
i
a
lw
i
t
hd
i
s
c
r
e
t
emicrostrmure
t
i
n
u
u
m
),
m =W
e
i
b
u
l
lmoduluso
ft
h
ema
も
e
r
i
a
.
l
,
σ。
=sca
.
l
ep
a
r
a
m
e
t
e
r
.
anbet
r
e
3
0
t
e
d回 a n
sf
u
n
c
t
i
o
nσ(P
,
x
),
r
e
p
r
e
s
e
n
t
i
n
gt
h
es
t
r
e
s
s
Thekeyt
oa
p
p
l
i
c
a
t
i
o
n
so
f
U
1
eW
e
i
b
u
l
l山 田 町 i
s
e
dbyl
o
3
o
dP 3
0
tp
o
i
n
tx(
f
o
r伽
t
h
並r
砕
e
e
吋d
im
叩
e
n
凶
s
B
叩
i
叩
1mp
戸
ふ
ふ
i
k
h
T
n
m
1
im
u
∞
,
。
1
4
1
5
f
a
c
も
もh
a
tt
h
el
a
r
g
e
rt
h
evolume
,
t
h
eg
r
帥 t
e
ri
st
h
ec
h
a
n
c
eo
fen
∞unteringacriticalmicroscopic
乱r加 t
h
el
o
a
.
da
ti
n
i
t
i
a
t
i
o
no
fc
r
a
ε
k
i
n
g(
o
rdamage)i
s
f
l
a
wt
h
a
tt
r
i
g
g
e
r
sf
a
i
l
u
r
e
. Thus, 回 f
∞n伺 med,
W
e
i
b
u
l
l
t
y
p
e油 田r
yi
s,
ofωurse,
a
p
p
l
i
c
a
b
l
e
.S
a
o
u
r
i
d
i
s(
1
9
8
9
)demons
t
I
叫edt
h
a
t
田 e
f
f
e
c
ti
nu
n
n
o
t
c
h
e
db
e
a
m
s
.I
n
d
田 d
,出 l
o
n
ga
.
st
h
e
r
e
bya
n
a
l
y
s
i
so
fL'
H
e
r
m
i
t
e
'
st
e
s
t
so
fs
i
a
r
enos
紅 白8 r
e
d
i
s
t
r
i
b
u
t
i
o
n
s,
i
.
e
.
,a
l
ls
t
r
側 部 a
r
ef
i
x
e
di
nt
h
e
i
rp
r
o
p
o
r
t
i
o
nt
ot
h
e1
0
0
0
,the
i
n
t
e
r
a
c
t
i
o
no
ft
h
ee
l
e
m
四 t
so
ft
h
es
t
r
u
c
t
u
r
ei
sm
a
t
h
e
m
a
t
i
c
a
l
l
ye
q
u
i
v
a
l
e
n
tt
oas
e
r
i
e
sc
o
u
p
l
i
n
g
o
fe
l
e
m
e
n
t
s
,
t
h
esamea
.
si
nau
n
i
a
x
i
a
l
l
ys
t
r
邸 s
e
db
a
ro
fv
a
r
i
a
b
l
ec
r
o
s
ss
e
c
t
i
o
n
.When出 e
r
ea
r
e
h
o
w
e
v
e
r,
t
h
es
t
r
u
c
t
u
r
eb
e
h
a
v
e
sa
sa∞m
b
i
n
a
t
i
o
no
fs
e
r
i
e
sandp
a
r
a
l
l
e
l
s
t
r
e
s
sr
e
d
i
s
t
r
i
b
u
t
i
o
n
s,
c
o
u
p
l
i
n
g
s
. ButW
e
i
b
u
l
l
'
st
h
e
o
r
yi
sva
.
l
i
do
n
l
yf
o
rs
e
r
i
e
sc
o
u
p
l
i
n
g, 舗 i
nac
h
a
i
no
fe
l
e
m
e
n
旬
(
h
e
n
c
et
h
eterm“
w
e
a
k
e
s
t
l
i
n
k山 t
i
s
t
i
c
s
")
.
吋
.
.
-
+-lz
1
b
(D)
hosuh︿ド・
H
・凶広
{
ロ)
whereB = (2G
b/h)
1/
2
/
f
;,
d
o= (h/2k)(d/α
)=r
e
f
e
r
e
n
c
es
i
z
ew
h
i
c
hd
e
p
e
n
d
sont
h
es
h
a
p
e
jE
o
ft
h
es
t
r
u
c
t
u
r
eb
u
ti
si
n
d
e
p
e
n
d
e
n
to
fs
t
r
u
c
t
u
r
es
i
z
ei
ft
h
es
t
r
u
c
t
u
r
e
sa
r
eg
e
o
m
e
t
r
i
c
a
l
l
ys
i
m
i
l
a
r
/
α=c
o
n
s
t
a
n
t
)
;刀 =t
e
n
s
i
l
es
t
問 n
g
t
h,
i
n
t
r
o
d
u
c
e
df
o
rc
o
n
v
e
n
i
e
n
c
e
jandh= w
i
d
t
h
(
b
e
c
a
u
s
ed
o
ft
h
ec
r
a
c
kbandf
r
o
n
t,
whichmaybet
r
e
a
t
e
da
p
p
r
o
x
i
m
a
t
e
l
y出 c
o
n
s
t
a
n
t,
i
n
d
e
p
e
n
d
e
n
t
C
t
u
r
es
i
z
e
. E
m
p
i
r
i
c
a
l
1
y,
do竺 nd
hereda = maximuma
g
g
r
e
g
a
t
es
i
z
eandn i
s
o
fs
t
r
u
aw
sv
a
r
i
e
d,
b
u
td
e
p
e
n
d
sons
t
r
u
c
t
u
r
eg
e
o
m
e
t
r
y
.E
q
.1
.4
a
p
p
r
o
x
i
m
a
t
e
l
yc
o
n
s
t
a
n
twhend01' d
ai
w
i
l
lbed
i
s
c
u
s
s
e
dmorei
nC
h
a
p
t
e
r5andd
e
r
i
v
e
di
nAppendix1
.
++zb
(
1
.4
)
︻
J
'i
s=d
/
ゐ
~
4ト
。d
,
.
、r、・
y
whereI
I=卯
po
t
総e
凶
n
t
i
凶
叫e
叩n
e
釘r
g
y
“
自
t
町
o
r
吋 i
泊
nt
出
hes
蜘
t
む
r
町
uc
t
旬u
r
凧
e
,
b=t
h
i
c
k
n
e
s
sand匂
Gf=f
r
a
c
t
u
r
ee
叩n
ぽ
e
r
g
y
2)
(
d
i
m
e
n
s
i
o
n刀
J
1
'
mη
=en
巴r
喝g
yn
e
e
d
e
dt
ω
o
c
α
r
白 t
旬e晶 f
仕
r
配
a
c
t
削u
r
児e
叫
o
rc
.
α
r
a
c
khando
fu山
n
1
t
h
et
h
i
r
dd
i
r
e
c
t
i
o
n
)u
n
i
tw
i
d
t
h
. Thev
a
l
u
eo
fGj i
sa
p
p
r
o
x
i
m
a
t
e
l
y∞n
s
t
a
n
tandr
e
p
r
e
s
e
n
t
s
.2f
o
rt
h
en
o
m
i
n
a
ls
t
r
e
s
s,
oneo
b
t
a
i
n
st
h
es
i
z
ee
f
f
e
c
七
am
a
t
e
r
i
a
lp
r
o
p
e
r
t
y
.S
o
l
v
i
n
gfromE
q
.1
lawp
r
o
p
o
s
e
dhyB
a
z
a
n
t(
1
9
8
4
a
)
:
σN= B
f:(I+β)ー1/2,
‘
-
ug UU仏∞ω切﹄咽阿古母音目的包ヨHEmEugu 刷局む回同
Ud
~
Thef
r
.
a
c
t
u
r
em
e
c
h
a
n
i
c
st
y
p
es
i
z
ee
f
f
e
c
t,
w
h
i
c
hi
sduet
oe
n
e
r
g
yr
e
l
e
a
s
e
,canbesimply
a
.
l
e
x
p
l
a
i
n
e
dbyc
o
n
s
i
d
e
r
i
n
gt
h
eu
n
i
f
o
r
m
l
ys
t
r
e
s
s
e
dp
l
a
n
e
lw
i
t
hac
r
a
c
ko
rc
r
a
c
kbando
fi
n
i
t
i
l
e
n
g
t
ha : 0 ao,
showni
nF
i
g
.1
.7
.I
tmaybeimaginedt
h
a
tt
h
ef
o
r
m
a
t
i
o
no
f.
ac
r
a
c
kband
o
ft
h
i
c
k
n
e
s
shr
e
d
u
c
e
st
h
es
t
r
a
i
ne
n
e
r
g
yd
e
n
s
i
t
y
σゐ
/2Ei
nt
h
ec
r
o
s
s
h
a
t
c
h
e
da
r
伺 t
oz
e
r
o(E
:0 e
l
a
s
t
i
cmoduluso
fc
o
n
c
r
e
t
e
)
. Whent
h
ec
ra
.
c
kba
.
nde
x
t
e
n
d
sbya,
t
h
ea
d
d
i
t
i
o
n
a
ls
t
r
a
i
n
e
n
e
r
g
yt
h
a
ti
sr
e
l
e
a
.
se
dcomesfromt
h
ed
e
n
s
e
l
yc
r
o
s
s
h
a
t
c
h
e
ds
t
r
i
po
fh
o
r
i
z
o
n
t
a
ld
i
m
e
n
s
i
o
n
.7
a
)
.I
ft
h
ef
a
i
l
u
r
emodesa
r
egωmetric
a
.
l
l
y5
i
凶 l
a
r,
拙i
su
s
u
a
l
l
y由 ec
剖 e
,
t
h
e
n
Aα(Fig.1
t
h
el
a
r
g
e
rt
h
ep
a
n
e
l,
t
h
el
o
n
g
e
ri
st
h
ec
r
a
c
kbanda
tf
a
i
l
u
r
e
.C
o
n
s
e
q
u
e
n
t
l
y
,
t
h
ea
r
e
ao
ft
h
e
b
e
i
n
gg
i
v
e
nbyh6a+
2ka6awherek=e
m
p
i
r
i
c
a
l
d
e
n
s
e
l
yc
r
o
s
s
h
a
t
c
h
e
ds
t
r
i
pi
sa
l
s
ol
a
r
g
e
r,
c
o
n
s
t
a
n
td
e
p
e
n
d
i
n
gont
h
es
ha
.
p
eo
ft
h
es
t
r
u
c
t
u
r
e
.T
h
i
si
l
l
u
s
t
r
a
t
e
st
h
a
.
t
,
i
na
.l
a
.
r
g
e
rs
t
r
u
c
t
u
r
e
,
moree
n
e
r
g
yi
sr
e
l
e
a
.
s
e
dbyt
h
esamee
x
t
e
n
s
i
o
no
ft
h
ec
r
a
c
kb
a
n
d
.Thee
n
e
r
g
yr
e
l
e
a
s
e
dfrom
t
h
es
t
r
i
pi
s6
I
T
/
6
α : 0 Gjb
,
i
.e
.,
土
(h6a+2ka6a)
生
,
b
6a、
'
2E=G
-J
回同
1
.3SimpleEnergyExplanationo
fS
i
z
eE
f
f
e
c
t
z
y
コ
17
16
n
em
i
g
h
tg
e
t出 ei
m
p
r
e
s
s
i
o
nt
h
a
t
h
h
i
se
x
p
l
a
n
a
t
i
o
no
ft
b
es
i
z
ee
f
e
c
tw
o
r
k
so
n
l
yf
o
r
L回 to
p d b
andb
u
tn
o
tforashrpl
i
n
ec
r
a
c
k,
c
o
n
s
i
d
e
rt
h
es
a
g
E
ep
a
n
e
lwithalinec
r
a
ι
ko
f
山 F
i
g
.1
.7bf
o
r5凶 叫a
n
e
l
so
f制 e
r
e
n
ts
問
Iωn
伺 いherei
sa
l
w
a
y
s
lengthωhow
h
e
a
dofthetipofacontinuouscmkof回 mef
i
n
i
t
el
e
n
g
t
h
i
z
e
a
b
l
e合acurepm-zonea
,
i
nt
h
ec
r
u
d
e
s
七a
p
p
r
o
x
i
m
a
t
i
o
n
,
bec
o
n
s
i
d
e
r
e
dcoastanb.Overb
h
el
e
n
g
t
ho
ft
h
h
i
c
hmay
. Becauseo
f出 i
sz
o
n
e,
t
h
e
zn t
h
et
r
a
n
s
v
e
r
s
enormalstmsg
r
a
d
u
a
l
l
yd
r
o
p
sfromれ oO
部 t
i
c
a
l
l
y叩 l
V
伽 tc
r
a
c
kl
e
n
g
t
hwhichc
a
u
附
t
h
e附 a
ω
s
s
鈴e
刊0
o
fs
t
r
日
a
叩
,
t
旬e
r
i
凶
a
l
i
抱
sl
o
n
g
e
r出
t
h釦
a
nt
h
ec
o
n
t
i
n
u
o
u
sc
α
r
a
配
.
c
k1
佃 g
t
出
h,
a,
bya
.d
i
s
t
組 問 c
whichi
sa
p
p
r
o
x
i
m
a
t
e
l
y
,cvazies,
b
u
tmuchl
e
s
sbhani
np
r
o
p
o
r
t
i
o
ntoUIe
am
a
t
e
r
i
a
lC
O
I
l
s
h
a
r
i
b
.
(
S
K
i
d
l
ys
p
e
a
k
i
n
g
,
ar
I
l
o
r
e
i
n
v
o
l
v
e
dd
e
r
i
v
a
t
i
o
ni
nwhichci
sv
a
r
i
a
b
l
ey
i
e
l
d
st
h
esuner
e
E
1
1
1
t,
md
size-Anyway
i
l
lbei
n
d
i
c
a
t
e
dj
u
s
もb
e
f
o
r
eE
q
.5
.
5
.
)
Whent
h
ec
r
a
c
ke
x
t
e
n
d
sbyl
e
n
g
t
hsa,
t
h
ef
r
邸 t
u
r
ep
r
o
c
e
s
sz
o
n
et
r
a
v
e
l
sw
i
t
ht
h
ec
r
a
c
k
t
i
p,
dt
h
ea
s
e
afromwhicha
d
d
i
t
i
o
n
a
ls
t
r
a
i
ne
n
e
r
g
yi
sr
e
l
e
a
s
e
dc
o
n
s
i
s
t
so
f
b
h
eM
r
i
p
so
fh
o
r
i
z
o
n
t
a
l
山a.
t
c
h
e
di
nF
i
g
.1
ル Theira
.
r
eai
sk
(a
o+c
)where
dimensionωwhicha
r
ed
e
n
s
e
l
yc
r
o
s
,
-6h/6a,must
ki
sa
p
p
r
o
x
i
m
a
t
e
l
yaconstant.Thee
n
e
r
g
yr
e
l
e
a
s
e
dP
e
ru
n
i
tc
r
a
c
ka
d
v
a
n
c
e
bwhereGJ~ t
h
e仕a.ct
u
r
ee
n
e
r
g
yo
ft
h
em
a
t
e
r
ia
.
l
,
and自O
e
q
u
a
lt
oGJ
,
l
_2
ヰ2k(ao+c)ßa;~ =GJb
•
(
1
.
5
)
S
o
l
v
i
n
gt
h
i
sequa
.
t
i
o
nf
o
rσN ,
onea
g
a
i
no
b
t
a
i
n
st
h
es
i
z
ee
f
f
e
c
tl
a
wi
nE
q
.1
.4i
nwhichnow
B = (EG
J
b
/k
C
)
1
/
2/
刀,
do=c
(
d
/α)=constani-
.
4,
w
h
i
c
hd
e
s
c
r
i
b
e
s出 et
r
a
n
s
i
t
i
o
n
畠1
s
i
民 e
f
f
e
c
tg
i
v
e
nby出 es
o
l
i
dc
u
r
v
ei
nF
i
g
.1
.
4,
E
q
.1
h山 oobdinedformiOIlsotherB
U
1
1
c
t
u
叫 g
e
o
m
e
t
r
i
e
s
.F
o
rl
a
.
r
g
es
i
z
民 t
h
ec
u
r
v
eo
fE
q
.1
.
4
i
nF
i
g
.1
.4a
p
p
r
o
a
c
h
e
sas
t
r
a
i
g
h
tl
i
n
eo
fs
l
o
p
e-1/2,
w
h
i
c
hr
e
p
r
白 e
n
t
st
h
es
i
z
ee
f
f
e
c
to
fl
i
n
e
a
r
l
a
s
t
i
cf
r
a
c
t
u
r
emech
回 i
c
s(
s
e
eS
e
c
.1
.2
)
.
Eq.1.4cana
l
s
obed
e
r
i
v
e
d
,
i
naC
O
I
n
p
l
e
b
e
1
yg
e
n
e
r
a
lway
,
byd
i
m
e
n
s
i
o
n
a
l姐 a
l
y
s
i
s姐 d
1
9
8
4
a
)・T
h
i
B
g
e
n
e
r
a
lddvMionmts?ntwoba
肌
h
y
p
o
t
h
e
s
e
s
:
dniliudeuguments(Batant,
s
t
a
n
te
n
e
r
g
y
(
1
)
t
h
ep
r
o
p
a
g
a
b
i
o
no
faf
r
a
c
t
u
r
eo
rc
r
a
c
kbandrequmsmapproxmately∞n
出ef
r
a
c
t
u
r
ee
n
e
r
g
y
,
GJ
)peru
n
i
ta
.
r
帥o
ff
r
a
c
t
u
r
ep
la
.
n
e,
and(
2
)t
h
ep
o
t
e
n
t
i
a
le
n
e
r
g
y
s
u
p
p
l
y(
l
e
a
s
e
dbyt
h
es
t
r
u
c
t
u
r
eduet
ot
h
ep
r
o
p
a
g
a
t
i
o
no
ft
h
ef
r
a
c
t
u
r
eo
rc
r
a
c
kbandi
saf
u
n
c
t
i
o
n
o
fbotht
h
ef
r
a
c
t
u
r
el
e
n
g
t
haadt
h
es
i
z
eo
ft
h
ef
r
a
c
t
u
r
ep
r
o
c
e
s
szonea
tt
h
ef
r
a
c
t
u
r
ef
r
o
n
t
.
I
tmustbekepti
nmindt
h
以 E
q
.1
.
4i
sa
p
p
r
o
泊m
ate
,
v
a
l
i
do
n
l
yw
i
t
h
i
nas
i
z
er
a
.
n
g
eo
f
t
h
el
a
r
g
e
s
ts
t
r
u
c
t
u
r
ef
o
rwhichEq.1.4canbea
p
p
l
i
e
di
sa
b
o
u
t20tiE
犯S
a
b
o
u
tl
:
2
0
(
i
.
e
.,
m
a
l
l
e
s
tshrMure)Forab
r
o
a
d
e
rsimrange
,
amorec
o
m
p
l
i
c
a
t
e
df
o
r
m
u
l
a
1
a
r
g
e
rぬmitles
由民 t
b
ea
f
o
r
e
m
e
n
t
i
o
n
e
ds
i
z
er
a
n
g
ei
ss
u
館c
i
e
n
tf
o
rmostp
r
a
c
t
i
c
a
l
wouldber
e
q
u
i
r
e
d
.N
e
v
e
r
t
h
e
l
p
u
r
p
o
s
e
s
句p
es
t
a
t
i
s
t
i
c
a
lt
h
e
o
r
yf
o
rt
h
es
i
z
ee
f
f
e
c
ti
st
h
抗
The-ainproblemw
i
t
ht
h
ew
e
i
b
u
l
l
t
h
ee
x
i
s
t
i
n
gw
o
r
k
si
g
n
o
r
es
t
r
e
s
sr
e
d
i
s
t
r
i
b
凶i
o
n
sandt
h
ec
o
n
s
e
q
u
e
n
te
n
e
r
g
yr
e
l
e
a
s
efromt
he
.
t
e
di
nF
i
g
.1
.7andm
a
n
i
f
,倒e
dbyh
h
es
i
z
eefectlaw).Thes
h
a
t
Mi?1
s
t
r
u
c
t
u
r
e(
a
si
l
l
u
s
t
ra
:
f
e
c
t、
s
h
o
u
l
dp
r
o
p
e
r
l
ya
p
p
e
a
ro
n
l
y出 回 a
d
d
i
t
i
o
nt
ot
h
ef
r
a
c
t
u
r
em
e
c
h
a
n
i
c
st
y
p
eS
l
z
e
e
:
f
e
c
t(
w
h
i
c
hi
sd
e
t
e
r
m
i
n
i
s
州, andwouldha
.
vet
od
e
s
c
r
i
b
eo
n
l
y山 tp
a
r
to
ft
h
es
i
z
ee
:
f
e
c
t
e
,
s
u
c
h∞m
p
a
r
i
s
o
n
sh
a
v
e
whichisn
o
te
x
p
l
M
r
i
e
dbyt
h
f
r
a
c
t
u
r
em
e
c
h
a
n
i
c
ss
i
z
eeRect.Sohr
n
o
ti
n
d
i
c
a
t
e
danyl
a
.
r
g
es
y
s
t
e
m
a
.
t
i
cd
e
v
i
a
t
i
o
n
sw
h
i
c
hwouldr
e
q
u
i
r
esomeo
t
h
e
rexp
l
.
組a
t
i
o
n,
s
u
c
ha
ss
t
a
t
i
s
t
i
c
a
l
.
A
p
p
l
i
c
a
t
i
o
n
so
fE
q
.1
.
4t
ob
r
i
t
t
l
ef
a
i
l
u
r
e
so
fc
o
n
c
r
e
t
es
t
r
u
c
t
u
r
e
sr
田 to
nt
woa
d
d
i
t
i
o
n
a
.
l
i
必l
u
r
emodeso
fg
ω
m
e
t
r
i
c
a
l
l
ys
i
m
i
l
a
.
r印 刷r
e
so
fd
i
f
f
e
r
e
n
ts
i
z
e
sa
.
r
e
h
y
p
o
t
h
e
s
e
s
:(
3
)もhef
回 m
e
t
r
i
c
a
l
l
ys
i
m
i
l
a
.
r(
e
.
g
.,
ad
ia
.
g
o
n
a
ls
h
e
a
rc
r
a
.
c
k1
同
a
tf
a
i
l
u
r
ea
.
boutt
h
es
a
.
mes
l
o
戸
d田 g
組dt
h
esamer
e
l
a
t
i
v
el
e
碍t
h
),
a
.
n
d(
4
)t
h
e抑 u
c
t
u
r
ed
附
n
o
tf
a
i
la
tc
r
a
c
ki
n
i
t
i
a
t
i
o
n(
w
h
i
c
h
田
S
I
沼
g
斗
n T
h
e
s
eh
y
p
o
t
h
e
s
e
sa
r
eu
s
u
a
l
l
ya
p
p
l
i
c
a
.
b
l
e,
b
u
tn
o
ta
l
ways
i
sar
e
q
u
i
r
e
m
e
n
to
fgoodd
e
組g
eo
fi
n
t
e
r
e
s
t
.A s
u
伍c
i
e
n
t
l
yl
a
.
r
g
ech
a
.
n
geo
fs
i
z
ema
.
ya
l
t
e
rt
h
ef
a
i
l
u
r
e
o
v
e
rt
h
ee
n
t
i
r
es
i
z
er
.
4i
n
a
p
p
l
i
ca
.
b
l
ebeyondt
h
a
ts
i
z
e
.(
T
h
i
sa
.
ppa
.
r
e
n
t
l
yi
st
hec
a
s
e
modeandt
h
u
sr
e
n
d
e
rE
q
.1
詔 i
l
l
i
a
ns
p
l
i
tc
y
l
i
n
d
e
rt
e
s
t
s
.
)
f
o
rt
h
eb
r
1
.4ExperimentalEvidenceforSizeEffectinStructures
E
x
t
e
n
s
i
v
et
e
s
t
sh
a
v
eb
e
e
nc
a
.
r
r
i
e
do
u
tt
ov
e
r
i
f
yE
q
.1
.
4f
o
rv
a
.
r
i
o
u
st
y
p
e
so
ff
a
i
l
u
r
eo
f
田 m
ento
fEq
.1
.
4w
i
t
ht
e
s
t
c
o
n
c
r
e
t
es
t
r
u
c
t
u
r
e
s(
u
s
i
n
gm
i
c
r
o
c
o
n
c
r
e
t
es
p
e
c
i
m
e
n
s
)
. Gooda
g
r
a
.
t
e
df
o
r
:
r
e
s
u
l
t
sh
a
sb
e
e
ndemonstr
1
.D
i
a
g
o
n
a
ls
h
e
a
rf
a
.
i
l
u
r
eo
fbeams(
Ba
.
Z
a
.
n
t組 dCω,
1
9
8
6
b
jBa
.
Z
組t組 dKazemi,
1
9
8
9
a
.
)
.
2
.Punchings
h
e
a
rf
a
.
i
l
u
r
eo
fs
l
a
b
s(Ba
.
Z
組tandCao
,1
9
8
7
)
.
3
.T
o
r
s
i
o
n
a
lf
心
,l
u
r
eo
fbeams(B
a
.
Z
a
n
t,
S
e
n
e
randP
r
a
t,
1
9
8
7)
.
4
.P
u
l
l
o
u
tf
a
i
l
u
r
eo
fr
e
i
n
f
o
r
c
i
n
gb
a
r
s(Ba
.
丞
姐b組 dS
e
n
e
r,
1
9
槌)
.
5
.D
o
u
b
l
e
p
u
n
c
ht
e
s
t
so
fc
y
l
i
n
d
e
r
s(M
a
.
r
t
i,
1
9
8
9
)
.
whichc
組 b
er
e
g
a
.
r
ded回 av
e
r
i
f
i
c
a
t
i
o
no
ft
hea
p
p
l
i
c
a
.
b
i
l
i
t
y
T
y
p
i
c
a
le
x
p
e
r
i
m
e
n
t
a
lr
e
s
u
l
t
s,
a
n
i
c
st
ot
h
eb
r
i
t
t
l
ef
a
i
l
u
r
e
so
f∞n
c
r
e
t
es
t
r
u
c
t
u
r
e
s,
a
.
r
eshowni
nF
i
g.1
ふ
o
ff
r
a
c
t
u
r
em配 h
1
.1
1(
t
e
s
t
sma
.
dea
tN
o
r
t
h
w
e
s
t
e
r
nUniv
e
r
s
i
t
y∞ m
i
c
r
o
c
o
恥 r
e
t
es
p
e
c
i
m
e
n
sw
i
tha
.
ggregωe
a
.
bi
l
i
t
yo
ff
:
日c
t
u
r
e
o
fmaximumd
i
a
m
e
t
e
r3/
8i
n
.o
r1
/
4i
n
.
)
. Asf
u
r
t
h
e
re
v
i
d
e
n
c
eo
fa
p
p
l
i
c
F
i
g
.1
.1
2s
h
o
w
s,
f
o
rt
h
ep
u
n
c
h
i
n
gs
h
e
a
rf
a
.
i
l
u
r
e
,
t
h
a
tt
h
ep
o
s
七
・p
e
a
.
k1
0add
r
o
p
m
e
c
h
a
n
i
c
s,
a
.
r
g
e
ra
st
h
es
i
z
ei
n
c
r
e
a
.
s
e
s
.T
h
i
si
sb
e
c
a
u
s
ei
nal
a
r
g
e
rs
p
e
c
i
m
e
nt
h
e
r
e
becomess
t
e
e
p
e
randl
)moreenぽ gyt
ober
e
l
e
a
s
e
di
n
t
oau凶 c
r
a
.
c
ke
x
t
e
n
s
i
o
n,
b
u
ts
i
n
c
et
h
e
i
s(
f
o
rt
h
esameσN
金a
c
t
u
r
ee
x
t
e
n
s
i
o
nd
i
s
s
i
p
a
同 t
h
e制 限 amounto
fe
n
e
r
g
y
,
t
h
ev
a
l
u
eo
fdN(
姐 d出 u
st
h
e1
0
吋)
mustb
er
e
d
u
c
e
d踊 t
h
es
t
r
u
c
t
u
r
es
i
z
ei
n
c
r
回 s
e
s
.
閃 (
F
i
g
.1
.8
・
1
.1
1
)d
o
e
sn
o
tr
e
a
.
ll
y
Notealωthatt
h
es
h
a
p
eo
ft
h
emeasureds
i
z
ee
f
f
e
c
tc
u
r
v
a
.
g
r
偲 W
l
もht
h
eW
e
i
b
u
l
l
t
y
p
es
t
a
t
i
s
t
i
c
a
lmodel
,
f
o
rw
h
i
c
ht
h
es
l
o
p
eo
ft
h
ec
u
r
v
ewouldha
.
ve
組
i
n
c
r
e
a
s
ew
i
t
hi
n
c
r
e
a
s
i
n
gs
i
z
eanda
p
p
r
oa
.
c
ha
.h
o
r
i
z
o
n
t
a
la
s
y
m
p
t
o
t
e
.
t
od
i
m
i
n
i
s
hr
a
t
h
e
rth
c
r
e
t
es
p
e
c
i
m
e
n
sw
i
t
hr
e
g
u
l
a
r
s
i
z
ea
g
g
r
e
g
a
t
er
e
p
o
r
t
e
di
n
Thee
x
i
s
t
i
n
gt
e
s
td
a
t
aon∞n
and山 en
e
e
do
fa
.f
r
a
c
t
u
r
em
e
c
h
a
n
i
c
sb
a
s
e
d
t
h
el
i
t
e
r
a
t
u
r
ea
l
s
oo
f
f
e
re
v
i
d
e
n
c
eo
fs
i
z
ee
f
f
e
c
t,
却l
a
.
na
t
i
o
nh
a
sbeenp
o
i
n
t
e
do
u
tbyv
a
.
r
i
o
u
sr
e
s
e
a
.
r
c
h
e
r
s,
e
.
g
.,
Re
i
n
ha
.
r
d
t(
1
9
8
1
a
,
1
9
8
1
b
)o
r
e
.
ndKim(
1
9
8
4
)
. Thed品同 fromt
h
el
i
t
e
ra
.
t
u
r
ea
r
eg
e
n
e
r
a
l
l
yf
o
u
n
dt
oa
g
r
関 w
it
h
B話 相ta
.4
,
bu
も
もh
i
se
v
i
d
e
n
c
ei
sn
o
tv
e
r
ys
t
r
o
n
gb
e
c
a
u
s
et
h
eda
.
t
ae
x
h
i
b
i
tv
e
r
yl
a
.
r
ges
t
a
.
t
i
s
t
i
c
a
l
F
i
g
.1
19
円
ー
18
¥
8Y BALANT & KAZEMI,
"
内
U
Elu=B
斤 1+s)-
λ 一 6.339
。
¥
¥
¥、ど
¥寸
ho-
do =0.35 i
礼
Bfi=741 psi
i
¥
九時
ヘ
Cl 一 0.830
吋
国
﹁
=Bj;s-
qd
12
/
1J
2
\で~l
0.0
f
、
-0.2
¥メ、
¥・
-0.
4
仰
,,~
¥
¥
と-0.5
¥
ヨ
。
0)
-0.5
s=d/do
¥
O¥
¥
¥
F
i
g
.1
.9T
e
s
tRe
s
u
l
t
s(Ba
.
Z
an
もa.
ndS
e
n
e
r,1
9
8
8
)o
nS
i
z
eE
f
f
e
c
ti
nP
u
l
l
a
rd
i
a
m
e
t
e
r
,
九=
=nomina
1p
u
l
l
o
u
ts
t
r
e
n
g
t
h,
o
u
tF
a
i
l
u
r
eo
fB
a
r
s(d~ =b
C
i,
K
i岨 d入。=empirica
1c
o
n
s
t
a
n
t
s
)
¥
0
X"log{db/da>
-0.4
ル
.
¥
0
.
2
¥
0
.
4
。
h
r
.、
、
。
¥
,4080p
s
i
0
.
2 k,
λ=0.328
822
C 一
ー 0.
λ 一 5.778
t
.
¥
〉
ー
¥
cl
¥O
。
句。日
:
:
1
-
〉
-mfaJ
泊Z ¥
2日刊
_
u0.
4
;
-
一れ、令、
.夫、ぷ、
一
、
r
ト
。
-mj 。
F
i
g
.1
.8T
e
s
tR
e
s
u
l
t
s(Ba
.
Z
a
n
tandKazemi,
1
9
8
9
乱)∞ S
i
z
eE
f
f
e
c
ti
nD
i
乱.
g
o
n
a
1ShearF
a
i
l
u
r
e
;vi
st
h
enominals
h
e
a
rs
t
r
e
n
g
t
h
0.6
¥
~;:S -0.3
,
ι
2
.
0
¥
1.5
,
、
、
0.5
1
.
0
log(s)
1
.
6
1
.
4
勺¥
2
0
.
0
、
、
¥
-1.
0
-0.5
-0.2
¥00
v=P/2bd
1
.
2
1
.
0
0
.
8
-0.1
¥
以二三
¥
(
b
)
-0.5
-0.6
0
.
8
1
.
4
1
.
0
1.
6
l
o
g (d/da
.
)
F
i
g
.1
.
1
0T
e
s
tR
e
s
u
l
t
s(Ba
.
Z
a
n
t,
S
e
n
e
randP
r
a
t,
1
9
8
9
)onS
i
z
eEf
I
e
c
ti
n
d=dept
ho
fbeam,
T
o
r
s
i
o
n
a
lF
a
i
l
u
r
eo
fP
l
a
i
nandR
e
i
n
{
o
r
c
e
dBeams(
りu
n
o
m
i
n
a
lt
o
r
s
i
o
na
1s
t
r
e
n
g
t
h,
,
i
CA
o e
m
p
i
r
i
c
a
lc
o
n
s
t
a
n
t
s)
=
=
2
1
20
4
qd
a4
}
凶
A
。
旬
2
.
0
0
.
0
1
0
n
o
n
l
i
n
e
a
r
i
>1
♂
0
.
0
1
5
d
e
f
l
e
c
:t
i
o
n(
in)
:
:
I
・
~
0
.
0
2
5
否五歪
百
1
2
柿
310
ー
.
2
•
c
=
.155f~(1 +
.
.
3
5t
>
)
¥
=
2
5
畠
'
.
8
1
.
0
1
.
2
‘
.
8
1
.
4
-A
山P﹄
J
-
086
RM4
匂(
d
/
d
.
)
ぷ 主 主、
L
-一一
ー
ミ
占
1
2
o
c
ムJ
t
i
f
4
2
x
v
ca
a
司
ザ
︼﹃
F
i
g
.l
.
llT
e
s
tR
e
s
u
l
t
s(
Ba
.
Z
叩ta
ndCao,
1
9
8
7
)onS
i
z
eE
f
f
e
c
ti
nD
i
a
g
O柄 d
ShearF
a
i
l
u
r
ei
nS
l
a
b
s(uu=nominals
h
e
a
rs
t
r
e
n
g
t
h,d=Blab
thickness, b=punchdame凧 v~ and向 =v
"andCf
o
r
"s
m
a
l
l
e
s
もs
l
a
b,
A
o=e
m
p
i
r
i
c
a
.
l
∞n
s
t
a
n
t
s
)
v t 111F l
ili-1Ill
ez
-! i1
.
0
&
d=2i
n
Group3
﹁
円
ー.
3
Group2
円
しV
ー
Group 1
寸u
c
=4In
0
.
1
否
;
三
0
.
3 0.
4
0.
5 0
.
6
d
e
f)
e
c
ti
o
ni
(凡)
F
i
g
.1
.1
2M
e
a
s
u
r
e
dL
o
a
d
D
e
s
e
c
t
i
o
nD
ia
g
r
a
m
so
fS
m
a
l
l,
Mediumand
品 川 姐 dCω ,
1
9
8
7
)
L
a
r
g
eS
l
a
b
s(Ba
22
s
c
a
t
t
e
r
. Dueto8
c
a
t
t
e
r,
a
b
o
u
te
q
ua
.
l
l
ygood五t
s臼.nbeob
句i
n
e
dw
i
出 o
t
h
e
rt
h
e
o
r
i
e
so
f
e
.
g
.,
W
e
i
b
u
l
l
'
ss
t
a
t
i
s
t
i
c
a
lt
h
e
o
r
y
.A
l
t
h
o
u
g
hp
a
r
to
ft
h
es
c
a
t
t
e
ri
si
n
e
v
i
t
a
b
l
eand
s
i
z
ee
f
f
e
c
t,
random
,
mosto
ft
h
eh
u
g
es
c
a
t
七e
ro
b
s
e
r
v
e
di
nb
r
i
t
t
l
e
df
a
i
l
u
r
e
ss
u
c
hωdiagonals
h
e
a
rp
r
o
b
a
b
l
y
s
t
e
m
sfromt
h
ef
a
c
tt
h
a
.
tt
h
et
e
s
ts
p
e
c
i
m
e
n
so
fva
.
r
i
o
u
ss
i
z
e
sw
e
r
eo
o
tg
e
o
m
e
t
r
i
c
a
l
l
ys
i
m
i
l
a
.
r
,
姐 ds
ot
h
e
o
r
e
t
i
c
a
la
d
j
u
s
t
m
e
n
t
smustbemadef
o
rt
h
ef
a
c
t
o
r
so
fs
h
a
p
eb
e
f
o
r
eac
o
m
p
a
r
i
s
o
n
.4c
a
nbemade.S
i
n
c
e出 ee
x
a
c
tt
h
e
o
r
yi
sn
o
tknown,
s
u
c
ha
d
j
u
s
t
m
e
n
旬 i
n
t
r
o
d
u
c
e
w
i
t
hE
q
.1
t
a
d
d
i
t
i
o
n
a
le
町'O
r
8
.1
na
d
d
ω
i
“
七
比1
叩
0
nt
句
ot
h
e“
自
t
r
u
c
t
町
u
1
げr
色佃
sl
i
s
t
e
dぬ
a
b仰o
v
久
,
ea
v
a
i
l
a
b
l
e∞mp紅
a
叩
r
i
s
o
n
sw
別i
t
山ht
旬
e
s
“
d
也a
t
句aa
l
s
o泊
i
n
c
山
l
l
u
d
e色
出
h
山
ebeamandr
i
n
gω
仏
.
f
'
♂
ω
a
i
リ
a
l
l
町
u
r
田
e
sofu町n
r
附
e
1
悶9
8
4
;B乱
a
.
Z
a
:
組
.
n
t組 dCa
ω
.
0,
1
9
8
筋6
吋
a
)
. Comparisonswiththebulko
ft
e
s
td
a
t
aa
.
v
a
i
l
a
bl
ei
n出 e
l
i
t
e
r
a
.
t
u
r
eweremadef
o
rd
i
a
g
o
n
a
ls
h
e
a
rf
a
i
l
u
r
e
so
fb
o
t
hu
n
p
r
e
s
t
r
e
s
s
e
dandp
r
e
s
t
r
e
s
s
e
dbeams,
姐 db
eamsw
i
t
h
o
u
tandw
i
t
hs
t
i
r
r
u
p
s(
B
a
z
a
n
tandKim,
1
9
8
4
;Ba
.
Z
叩ta
ndSun,
1
9
8
7
;Ba
.
Z
a
n
t
姐 dCω,
1
9
8
6
b
),
a
sw
e
l
la
st
o
r
s
i
o
n
a
lf
a
i
l
u
r
e
s(
B
a
z
a
:
ntandS
e
n
e
r1
9
8
7
)
.S
t
a
t
i
s
t
i
c
a
la
n
a
l
y
s
i
s
W剖 i
n
c
l
u
d
e
di
nt
h
e
s
es
t
u
d
i
e
s
.
23
p
P
Series Model
P
Uc
Uc
r
Uel
1
.5E
xplanationo
fS
i
z
eE
f
f
e
c
tonD
u
c
t
i
l
i
t
y
S
t
r
u
c
t
u
r
a
la
c
t
i
o
ni
sn
o
r
m
a
l
l
yac
o
m
b
i
n
a
t
i
o
no
fs
e
r
i
e
sandp
a
r
a
l
l
e
lc
o
u
p
l
i
n
g
so
ft
h
ec
r
a
.
c
k
i
n
g
却 l
a
i
n
e
dbyt
h
es
e
r
i
ω
z
o
n
e
sandt
h
eu
n
c
r
a
c
k
e
d(
e
l
a
s
t
i
c
)z
o
n
e
s
.Thes
i
z
ee
f
f
e
c
tond
u
d
i
l
i
t
yi
se
c
o
u
p
l
i
n
ga
s
p
e
c
t
.C
o
n
s
i
d
e
rac
r
a
c
k
i
n
ge
l
e
m
e
n
tc
o
u
p
l
e
dw
i
t
hane
l
a
s
t
i
ce
l
e
m
e
n
ta
sshowni
n
F
i
g
.1
.1
3
.Thel
o
a
d
d
i
s
p
l
a
c
e
m
e
n
td
ia
.
gramso
ft
h
e
s
ee
l
e
m
e
n
t
sa
r
ea
l
s
os
h
o
w
n
.S
i
n
白 山ef
o
r
c
e
t
h
er
e
s
p
o
n
s
eo
fb
o
t
h
i
nb
o
t
he
l
e
m
e
n
t
si
st
h
esameandt
h
ed
e
f
o
r
m
a
t
i
o
n
sa
r
es
u
p
e
r
i
m
p
o
s
e
d,
e
l
e
m
e
n
t
scombinedi
so
b
t
a
i
n
e
dbyp
a
s
s
i
n
gah
o
r
i
z
o
n
t
a
l
l
i
n
ea
te
a
c
hl
e
v
e
lP andsummingt
h
e
,
舗 s
howni
nF
i
g
.1
.1
3
.I
ft
h
ee
l
a
s
t
i
ce
l
e
m
e
n
ti
ss
u
f
f
i
c
i
e
n
t
l
y
c
o
r
r
e
s
p
o
n
d
i
n
gd
e
f
o
r
m
a
t
i
o
n
sa回 db
s
o
f
t,
t
h
i
sc
組
o
b
v
i
o
u
s
l
yp
r
o
d
u
c
eal
o
a
d
d
i自p
l
a
c
e
m
e
n
td
i
a
g
r
a
mw
h
i
c
he
x
h
i
b
i
t
st
h
es
o
c
a
且e
d
s
n
a
p
b
a
c
k,
i
nw
h
i
c
ht
h
ed
i
s
p
l
a
c
e
m
e
n
td
i
.
a
g
r
.
a
mt
u
r
n
sb
a
c
k.
a
tap
o
s
i
t
i
v
es
l
o
p
e
.Thes
n
.
a
pback
b
e
h
a
v
i
o
ri
su
n
s
t
a
b
l
ee
v
e
nu
n
d
e
rd
i
s
p
l
a
c
e
m
e
n
t∞n
t
r
o
l,
andt
h
es
t
r
u
c
t
u
r
ef
a
i
l
sa
tt
h
emaximum
d
i
s
p
l
a
.
c
ement,
l
a
.
b
e
l
e
da
sU cr • T
h
i
sd
i
s
p
l
a
c
e
m
e
n
tr
e
p
r
e
s
e
n
t
sa
.d
u
c
t
i
l
i
も
yl
i
m
i
tf
o
rt
h
es
y
s
t
e
m
.
S
i
n
c
et
h
ea
d
d
i
t
i
o
no
fa
:
ne
l
a
.
s
t
i
ce
l
e
m
e
n
ti
se
q
u
i
v
.
a
l
e
n
tt
oi
n
c
r
e
a
s
i
n
gt
h
es
i
z
eo
ft
h
es
t
r
u
c
t
u
r
e
,
i
ti
sc
l
e
a
rt
h
a
t姐 i
n
c
r
e
a
s
eo
fs
i
z
et
e
n
d
sもod
田 r
e
a
s
ed
u
c
t
i
l
i
t
y
.D
u
c
t
i
l
i
t
yi
sn
o
tama
.
t
e
r
i
a
l
a
n
i
c
sandd
e
p
e
n
d
s
p
r
o
p
e
r
t
yb
u
tas
t
r
u
d
u
r
ep
r
o
p
e
r
t
yw
h
i
c
hi
sg
o
v
e
r
n
e
dbyf
r
a
c
t
u
r
em配 h
ons
t
r
u
c
t
u
r
es
i
z
e(
a
sw
e
l
la
st
h
et
y
p
eo
f∞n
c
r
e
t
e
)
.I
ti
sworthn
o
t
i
碍 t
h
.
a
tHawkins(
1
9
8
4
)
i
d
e
n
t
i
f
i
e
dt
w
e
n
t
y
n
i
n
ep
r
o
v
i
s
i
o
n
si
nt
h
eACICodew
h
i
c
hseemt
obee
m
p
i
r
i
c
a
lb
u
tc
o
u
l
d
.
r
i
o
u
sd
u
c
t
i
l
i
t
yl
i
m
i泊 t
i
o
n
s,
p
r
o
b
a
b
l
ybee
x
p
l
a
i
n
e
dbyf
r
a
c
t
u
r
em
e
c
h
a
n
i
c
s
. Theyi
n
c
l
u
d
eva
c
o
n
d
i
t
i
o
n
sf
o
rminimumr
e
i
n
f
o
r
c
e
m
e
n
t,
c
r
a
c
ks
p
.
a
c
i
n
gandc
r
a
c
kw
i
d
t
h,
e
t
c
.
h
ee
x
p
e
r
i
m
e
n
t
a
l
l
yo
b
s
e
r
v
e
ds
t
r
u
d
u
r
a
ls
i
z
ee
宜e
c
t,
国 wella
.
st
h
er
e
l
a
t
e
d
Tosumup,t
,
p
r
e
s
e
n
t
st
h
emo
s
tp
o
t
e
nta
r
伊 m
entf
o
rt
h
ea
p
p
l
i
c
a
t
i
o
no
f
s
p
u
r
i
o
u
se
f
f
e
c
to
ft
h
emeshs
i
z
e
a
n
i
c
st
oc
o
n
c
r
e
t
es
t
r
u
c
t
u
r
e
s
.
f
r
a
c
t
u
r
em民 h
F
i
g
.1
.
13Ch
臼 g
ei
nP
o
s
t
P
e
a
kLoad-De
B
.
e
c
t
i
o
nDiagramDuet
oSe
r
i
e
s
C
o
u
p
l
i
n
gw
i
t
hE
l
a
s
t
i
cE
l
e
m
e
n
t
Mode I
江
E
F
i
g
2
.1ModesI
,
IImdI
I
I(
O
p
e
n
i
n
g
,
PMeshmAnti-PlaneSh
問
Fl
'
a
c
t
u
r
e
s
)
U
2
4
25
0,
C
h
a
p
t
e
r 2.ESSENTIA_L !!-~S!!_LTS FROMLINEAR
ELASTICFRACTUREMECHANICS
I
nl
i
n
e
a
re
l
a
s
t
i
cf
r
a
c
t
u
r
em
e
c
h
a
n
i
c
s(LEFM)i
ti
sassumedt
h
a
ta
l
lo
ft
h
ef
r
a
c
t
u
r
ep
r
o
c
倒
happensa
t出 ec
r
a
c
kt
i
pandt
h
a
ももh
ee
n
t
i
r
evolumeo
ft
h
ebodyr
e
m
a
i
n
se
l
a
s
t
i
c
. Under
t
h
eq
u
e
s
t
i
o
n
so
fc
r
拭 kp
r
o
p
a
g
a
t
i
o
nands
t
r
u
c
t
u
r
a
lf
a
i
l
u
r
ec
a
nbes
o
l
v
e
dby
t
h
i
sa
s
s
u
m
p
t
i
o
n,
methodso
fl
i
n
e
a
re
l
a
s
t
i
c
i
t
y
.
i
s
t
i
n
g
u
i
s
ht
h
r
e
ee
l
e
m
e
n
t
a
r
y{
r
a
c
t
u
r
emod
民 M
odes1
,
I
I姐 dI
I
I,
a
1s
o
I
ti
sc
o
n
v
e
n
i
e
n
ttod
t
h
ep
l
a
n
a
rs
h
e
a
rmodeandt
h
ea
n
t
i
p
l
a
n
es
h
e
a
rmode;s
e
eF
i
g
.2
.1
.
c
a
l
l
e
dt
h
eo
p
e
n
i
n
gmode,
Modes1andI
Ia
r
ep
l
a
n
a
rs
y
m
m
e
t
r
i
ca
.
nda
n
t
i
s
y
m
m
e
t
r
i
c,
w
h
i
l
eModeI
I
Ii
st
h
r
e
e
-d
i
m
e
n
s
i
o
n
a
l
.
.
a
durei
sal
i
n
e
a
rc
o
m
b
i
n
a
t
i
o
no
ft
h
e
s
et
h
r
e
em
o
d
e
s
.
G
e
n
e
r
a
lf
r
stress
r
e
l
i
e
f
zone
2
.
1S
t
r
e
s
sS
i
n
g
u
l
a
r
i
t
y
I
n
t
r
o
d
u
c
t
i
o
no
fa
.c
r
a
.
c
ki
n
も
oa
.l
i
n
e
a
re
l
a
s
t
i
cbodyp
r
o
d
u
c
e
ss
t
r
e
s
sc
o
n
c
e
n
t
r
a
.
t
i
o
n
sn
e
a
rt
h
e
a
.
t
e
dbyt
h
ep
e
r
t
u
r
b
a
t
i
o
no
ft
h
et
r
a
j
民 t
o
r
i
田 o
ft
h
emaximum
c
r
a
c
kt
i
p
s
.T
h
i
smaybei
l
l
u
s
t
r
wi
t
ha
1
1t
h
e
p
r
i
n
c
i
p
a
ls
t
r
e
s
sshowni
nF
i
g
.2
.
2
. Thes
t
r
e
s
sf
i
e
l
di
ss
i
n
g
u
l
a
ra
tt
h
ec
r
a
c
kt
i
p,
伺 S∞
mponentsa
p
p
r
o
a
c
h
i
n
gi
n
f
i
n
i
t
ya
st
h
er
a
.
di
a
ld
i
s
t
a
n
c
erfromt
h
ec
r
a
c
kt
i
p
D
o
n
z
e
r
os
t
r
t
h
es
紅 白B
t
e
n
d
st
oz
e
r
o(
F
i
g
.2
.
3
)
.l
nas
u
f
f
i
c
i
e
n
t
l
yc
l
o
s
en
e
i
g
h
b
o
r
h
o
o
do
ft
h
es
h
a
r
pc
r
a
c
kt
i
p,
components町 j a
r
et
h
esamer
e
g
a
r
d
l
e
s
so
ft
h
es
h
a
p
eo
ft
h
ebodyandt
h
emannero
fl
o
a
d
i
n
g
,
姐 dm
aybee
x
p
r
田s
e
d部:
σL=I
ufl(
l
}
)
(
2
付
t ,σu=J(llfi~[(())(2附)ー1/2 , σUI = kmLYI(6)(2πrt /2
1
12
/
F
i
g
.2.
2P
r
i
n
c
i
p
a
lSt
r
e
s
sT
r
a
j
e
c
t
o
r
i
e
si
naC
r
a
c
k
e
dSpecimen
(
2
.
1
)
H
e
r
et
h
es
u
b
s
c
r
i
p
t
sands
u
p
e
r
s
c
r
i
p
t
s1
,
1
1and1
1
1r
e
f
e
rt
ot
h
ee
l
e
m
e
n
t
a
r
ymodes,
8i
st
h
ep
o
l
a
r
J(
I,
/(IIa
nd1(111a
r
ep
a
r
a
m
e
t
e
r
sc
a
l
l
e
d七hes
t
r
e
s
si
n
t
e
n
s
i
t
yf
a
c
t
o
r
s,
andf
u
n
c
t
i
o
n
sん
a
n
g
l
e,
a
削 h
esamer
e
g
a
r
d
l
ω
so
ft
h
ebodygeom
向 a
ndt
h
emannero
fl
o
a
d
i
n
g
.F
o
rexample
,
f
[
l(
0
)
=0ωα(1-s
i
n
α
s
i
n
3
α
),
f(
0
)=∞sα(1+sinαsin3α),
f
l
2
(
0
)= Cοsαsin2αcos3α,
where
/
2
;s
e
ee
.
g
.,
Kn
o
t
t(
1
9
7
3
),
B
r
o
e
k(
1
9
7
4
),
OwenandFawkes(
1
9
8
3
),
H
e
l
l
a
n(
1
9
8
4
),
回d
α=0
o
p
e
l
a
r(
1
9
8
5
)
.
Kanninen姐 dP
ム
2
.
2EnergyC
r
i
t
e
r
i
o
n
Thef
a
c
tt
h
a
t,a
c
c
o
r
d
i
n
gt
ot
h
et
h
田 r
yo
fe
l
a
s
もi
c
i
t
y
,
もh
es
t
r
e
s
sn
e
a
rt
h
ec
r
晶c
kt
i
pa
p
p
r
o
a
ε
h
e
s
.
n
n
i
t
y
,
nom
a
t
t
e
rhows
m
a
l
lt
h
el
o
a
d
,
wasn
o
t
e
dbyG
r
i
f
f
i
t
h(
1
9
2
1,
1
9
2
4
)ont
h
eb
a
s
i
so
ft
h
e
i
n
i
fl
i
n
e
a
re
l
a
s
t
i
c
i
t
y
p
r
e
v
i
o
u
ss
o
l
u
t
i
o
nf
o
re
l
l
i
p
t
i
c
a
lh
o
l
e
sbyI
n
g
l
i
s(
1
9
1
3
).Hec
o
n
c
l
u
d
e
dt
h
a
t,
onec
a
n
n
o
ti
n
t
r
o
d
u
c
eas
t
r
e
n
g
t
hc
r
i
t
e
r
i
o
na
sac
o
n
d
i
t
i
o
no
ff
a
i
l
u
r
e
,
b凶 musti
n
s
t
e
a
d
i
su
s
e
d,
n
e
r
g
y
d
e
c
i
d
ef
a
i
l
u
r
eont
h
eb
a
s
i
so
fane
n
e
r
g
yc
r
i
t
e
r
i
o
n
. Ast
h
ec
r
a
c
kt
i
pp
r
o
p
a
g
a
t
e
s,e
n
e
r
g
yf
i
o
wi
s
f
l
o
w
si
n
t
ot
h
ec
r
a
c
kt
i
pw
h
e
r
ei
ti
sd
i
s
s
i
p
a
t
e
dbyt
h
ef
r
a
c
t
u
r
ep
r
o印 5S. Thee
c
h
a
r
a
c
t
e
r
i
z
e
dbyt
h
ee
n
e
r
g
yr
e
l
e
a
s
er
a
t
ew
h
i
c
hi
se
x
p
r
e
s
s
e
d出:
θ
l
l
(
α
)_
. 1r
T
T
I ,Lla, TTI .
d
.aJ
l
H
(
a
+す )-I
I
(
α ーτ )
J
b=ーすf 2E
(
2
.
2
)
a
.
rC
r
a
c
kT
i
p
F
i
g
.
2
.
3S
t
r
e
s
sD
i
s
t
r
i
b
u
t
i
o
nNe
26
27
i
nwhichI
I= U- W = p
o
t
e
n
t
i
a
len
釘g
yo
ft
h
es
t
r
u
c
t
u
r
e
,W = workofloads,
andU =
s
t
r
a
i
ne
n
e
r
g
yo
ft
h
es
t
r
u
c
t
u
r
ea
saf
u
n
c
t
i
o
no
ft
h
ec
r
a
c
kl
e
n
g
t
ha
.E
q
.2
.
2a
l
s
og
i
v
e
saf
i
n
i
t
e
d
i
f
f
e
r
e
n
c
ea
.
pproxim
叫i
o
nwhichma
.
ybeu
s
吋 t
oc
a
.
l
c
u
l
a
も
eG byt
h
ef
i
n
i
t
eelem
叩 t
method.
F
o
rt
h
a
tp
u
r
p
o
s
eonema
.
ymodelt
h
ec
r
a
c
k剖 al
i
n
egapb
e
t
w
.
四na
d
j
a
伺 n
te
l
e
m
e
n
t
sa
.
nd
c
a
l
c
u
l
a
.
t
et
h
es
t
r
a
i
ne
n
e
r
g
ys
t
o
r
e
di
nt
h
emeshf
o
rt
h
ec
r
a
c
kt
i
pd
i
s
p
l
a
c
e
dbye
i
もh
e
r-Aα/2
a
/
2
.I
n
s
t
e
a
do
fu
s
i
n
g姐 i
n
t
e
r
e
l
e
m
e
n
tl
i
n
ec
r
拭 k
,
onemay
,
f
o
rt
h
es
a
k
eo
fc
o
n
v
e
n
i
e
n
c
e
,
o
rA
modelt
h
ec
r
a
c
kbyωnsideringabando
fe
l
e
m
e
n
t
st
oh
a
v
ez
ぽ os
t
i
f
f
n
e
s
s(
i
foneu
s
e
samesh
姐a.
b
o
u
t0
.
1o
ft
h
ec
r
o
s
ss
e
c
t
i
o
nd
i
m
e
n
s
i
o
n,
t
h
e
o
fs
q
u
a
r
ee
l
e
m
e
n
t
swhoses
i
z
ei
sn
o
tmoret
h
閃 B
a
Z
a
.
n
t
e
r
r
o
ro
fc
o
n
s
i
d
e
r
i
n
gac
r
a
c
kbandi
n
s
t
e
a
do
fal
i
n
ec
r
a
c
ki
su
s
u
a
l
l
yl
e
s
s山 拍 1%;s
姐 dC
e
d
o
l
i
n,
1
9
7
9
)
.
伍t
h,
t
h
ec
o
n
d
i
t
i
o
no
fc
r
a
c
kp
r
o
p
a
g
a
t
i
o
n(
c
r
i
t
i
c
a
ls
t
a
t
e
)i
s
A
c
c
o
r
d
i
n
gt
oG
r
i
s
i
z
回, t
h
es
t
r
e
s
si
n
t
e
n
s
i
t
y{
a
c
t
o
ri
sp
r
o
p
o
r
t
i
o
n
a
lt
o出 es
q
u
a
r
er
o
o
to
fもhes
i
田, a
ndt
h
ee
n
e
r
g
y
r
e
l
e
a
s
er
a
t
ei
sp
r
o
p
o
r
t
i
o
n
a
lt
ot
h
es
i
z
eo
ft
h
es
t
r
u
c
t
u
r
e
.
t
h
ec
o
n
d
i
t
i
o
no
fmode1c
r
a
c
kp
r
o
p乱
.
g
a
t
i
o
n(
c
r
i
t
i
c
a
ls
t
a
t
e
)c
a
nbe
I
n
s
t
e
a
do
fE
q
.2
.
3,
e
x
p
r
e
s
s
e
di
nt
e
r
m
so
ft
h
es
t
r
e
s
si
n
t
e
n
s
i
t
yf
a
c
t
o
ra
s
:
,
Gi
st
h
ef
r
a
c
t
u
r
ee
n
e
r
g
y
,
whichh
a
sもhed
i
m
e
n
s
i
o
no
fJ/
が (
o
rN/m)andr
e
p
r
e
s
e
n
t
saba
.
si
c
m
a
t
e
r
i
a
lp
r
o
p
e
r
t
y
.F
o
rG く G ,
t
h
ec
r
a
c
kc
a
n
n
o
tp
r
o
p
a
g
a
t
e
,andforthecaseG >G
e
q
u
i
l
i
b
r
i
u
mi
si
m
p
o
s
s
i
b
l
e
.I
fG =G anddG
/
θa>0,
w
h
i
c
hi
sn
o
r
m
a
l
l
yt
h
ec
a
s
e
,
t
h
ec
r
a
c
k
吋 c
o
n
t
r
o
l(
i
.
e
.,
t
h
es
t
r
u
c
t
u
r
ef
a
i
l
s
)
.Theree
x
i
s
t,
'
howev
e
r
,
somec
a
s
e
si
n
i
su
n
s
t
a
b
l
eu
n
d
e
r1
0
andt
h
e
nt
h
ec
r
a
c
kc
a
ngrowu
n
d
e
rl
o
a
dc
o
n
t
r
o
li
nas
t
a
b
l
e
whichG =GfanddG/θα<0,
m
a
n
n
e
r
.
,
I
Ia
nd1
1
1maybee
x
p
r
e
s
s
e
dont
h
eb
a
s
i
so
ft
h
es
t
r
e
s
s
Thee
n
e
r
g
yr
e
l
e
a
s
er
a
t
ef
o
rmodes1
i
n
t
e
n
s
i
t
yf
a
c
t
o
r
sa
sf
o
I
I
o
w
s
:
,
,
,
/E
', GII=K
l
d
E
', Gm =
=K
l
μ
GI= Ki
I
I/
(
2
.
4
)
i
立w
h
i
c
hμ =e
l
a
s
t
i
cs
h
e
a
rmodulus,
f
o
rt
h
ec
a
s
eo
fp
l
a
n
es
t
r
e
s
sE'= E = Y
o
u
n
g
'
se
l
a
s
t
i
c
民 f
o
rt
h
ec
剖 eo
fp
l
a
n
es
t
r
a
i
nE'=E/(l-v2),
andv= P
o
i
s
s
o
n
'
sr
a
t
i
o
.F
o
rg
e
n
e
r
a
l
modul
l
o
a
d
i
n
g,
t
h
et
o
ta
.
le
n
e
r
g
yr
e
l
e
a
s
er
a
.
t
ei
s
:
G=GI+G
I
I+G m
(
2
.
5
)
a
.
lt
ot
h
ea
p
p
l
i
e
dl
o
a
d
,
andmayg
e
n
e
r
a
l
l
ybe
Thes
t
r
e
s
si
n
t
e
n
s
i
t
yf
a
c
t
o
r
sa
r
ep
r
o
p
o
r
t
i
o
n
e
x
p
r
e
s
s
e
di
nt
h
ef
o
r
m
:
α = α/d
(
2
.
6
)
i
nw
h
i
c
hfi
sac
e
r
ta
.
i
nn
o
n
d
i
m
e
n
s
i
o
n
a
lf
u
n
c
t
i
o
no
ft
h
er
e
l
.
a
t
i
v
ec
r
a
c
kl
e
n
g
t
hα(
d=c
h
a
r
a
c
t
e
r
and~(α) =f
(α)
.
j
石
:
:
:
:
:a
.
no
山e
rn
o
n
-d
i
m
e
n
s
i
o
n
a
lf
u
n
c
t
i
o
n
.F
o
r
i
s
t
i
cs
t
r
u
c
t
u
r
ed
i
m
e
n
s
i
o
n
),
v
a
r
i
o
u
ss
i
m
p
l
eg
e
o
m
e
t
r
i
e
so
fn
o
t
c
h
e
df
r
a
c
t
u
r
es
p
e
c
i
m
e
n
s
,accurateexpr田 sionsf
o
rf
u
n
c
t
i
o
n
fareavailablei
nもe
x
t
b
o
o
k
sandh
a
n
d
b
o
o
k
s(
e
.
g
.,
Tadae
ta
l
.,
1
9
8
5,
andMur
a
.
k
a
m
i
,
1
9
8
7
)
.
t
h
ef
u
n
c
t
i
o
nfc
a
na
l
w
a
y
sbec
a
l
c
u
l
a
t
e
dbyl
i
n
e
a
re
l
a
s
t
i
ca
n
a
l
y
s
i
s
je
.
g
.,
F
o
ro
t
h
e
rg
e
o
m
e
t
r
i
e
s,
t
h
r
o
u
g
haf
i
n
i
t
ee
l
e
m
e
n
tp
r
o
g
r
a
m
.F
o
rt
h
es
p
e
c
i
a
lc
a
s
eo
fas
i
n
g
l
el
i
n
ec
r
a
c
ko
fl
e
n
g
t
hai
nan
もi
n
f
i
n
i
t
yt
on
o
m
i
n
a
ls
t
r
回 SσN i
nt
h
ed
i
r
配 t
i
o
nnormalt
ot
h
ec
r
a
c
k
i
n
f
i
n
i
t
es
o
l
i
ds
u
b
j
e
c
t
e
da
on
eh硝 f
(
α)= 1
.E
q
.2
.
6s
h
o
w
s由民 f
o
rg
加 m
e
t
r
i
c
a
l
l
ys
i
m
i
l
a
.
rs
t
r
u
c
t
u
r
e
so
fd
i
f
f
e
r
e
n
t
p
l
a
n
e,
(
2
.
7
)
r
i
t
i
c
a
lv
a
l
u
eo
fK[,
w
h
i
c
hi
sa
l
s
oc
a
l
l
e
df
r
a
c
t
u
r
et
o
u
g
h
n
e
s
sandr
e
p
r
e
s
e
n
t
s
i
nwhichKlc=c
r
i
a
lp
r
o
p
e
r
t
y
;/(Ic= G
f
E
'
.HE
q
.2
.
7i
ss
u
b
s
t
i
t
u
t
e
di
n
t
oE
q
.2
.
6,
t
h
en
o
m
i
n
a
l抑 制
a回 以e
a
tf
a
i
l
u
r
e(
c
r
a
c
kp
r
o
p
a
g
a
t
i
o
n
)i
so
b
t
a
i
n
e
da
s
:
Kf
c
σN =百
(
2
.
3
)
G=G
石
f
(
α
)= b
:
:v'd~(α),
I
f
I=E
b
dVJ
.• -. ¥
,
d
KI=K1c
Kl
c
(
2
.
8
)
雨 =万 ね
I
tmayben
o
t
e
d出 a
t,
a
c
c
o
r
d
i
n
gt
oE
q
.2
.
8,
十ogd+∞M
(
2
.
9
)
l
o
g
σN =
・
3
T
h
i
sr
e
l
a
t
i
o
nshowst
h
a
.
tt
h
es
i
z
ee
f
f
e
c
tp
l
o
もa
c
c
o
r
d
i
n
gt
ol
i
n
e
a
.
re
l
a
s
t
i
cf
r
a
c
t
u
r
em
e
c
h
a
n
i
c
si
s
佃 i
吋 i
n
e
ds
t
r
a
i
g
h
tl
i
n
eo
fs
l
o
p
e-1/2(
F
i
g
.1
刈.
2
.
3LimitsofA
p
p
l
i
c
a
b
i
l
i
t
y
I
nr
e
a
l
i
t
y
,
t
h
ef
r
a
c
t
u
r
ep
r
o
c
e
s
sc
a
n
n
o
tt
a
k
ep
l
a
c
ea
tap
o
i
n
t
. Thef
旨a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
e
musth
a
v
esomef
i
n
i
t
es
i
民.A
c
c
o
r
d
i
n
gt
oI
r
w
i
n(
1
9
5
8
),
ac
r
u
d
ee
s
t
i
m
a
t
eo
ft
h
el
e
n
g
t
hr o
f
臼i
n
e
dbys
e
t
t
i
n
gt
h
et
r
a
n
s
v
e
r
s
en
o
r
m
a
ls
t
r
e
s
si
nE
q
.2.
1
t
h
ef
r
a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
emaybeob
.Thisy
i
e
l
d
s
ωbee
q
u
a
lt
ot
h
et
e
n
s
i
l
es
t
r
e
n
g
t
h刀
,
0 l
o
,1/(f1:(
.
=2
r =2
1
1
'
1
1
"
G
_
'
.
P
n
-=
-K~c _ E'
U
-
f
? -f
f
(
2
.
1
0
)
a
.
lp
r
o
p
e
r
t
i
e
s, 組dt
h
e
r
e
f
o
r
ei
s
Notet
h
a
tt
h
i
sl
e
n
g
t
hi
se
x
p
r
e
s
s
e
do
n
l
yi
nt
e
r
m
so
fm
a
t
e
r
i
am
a
t
e
r
i
a
.
lp
r
o
p
e
r
t
y
,
t
o
o
.Ana
l
t
e
r
n
a
t
i
v
ee
s
t
i
m
a
t
eo
ft
h
es
i
z
eo
ft
h
ef
r
a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
eo
f
∞ncretecanbeb出 edonthemaximum.
a
g
g
r
e
g
a
t
es
i
z
eda • B話相も andOh(
1
9
8
3
a
)c
o
n
c
l
u
d
e
d
もt
h
el
e
n
g
t
hande
f
f
e
c
t
i
v
ew
i
d
t
ho
ft
h
ef
r
a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
eo
fc
o
n
c
r
e
t
ei
nt
h
r
e
e
p
o
i
n
t
t
h
a
r
e
s
p
e
c
t
i
v
e
l
y
.
b
e
n
ds
p
e
c
i
m
e
n
sa
r
er
o
u
g
h
l
y1
2
d 組 d3da,
Ji
smuchs
m
a
l
l
e
rt
h
a
nt
h
ec
r
o
s
ss
e
c
L
i
n
e
a
re
l
a
s
t
i
cf
r
a
c
t
u
r
em
e
c
h
a
n
i
c
si
sa
p
p
l
i
c
a
b
l
ewhenr
a
.
t
i
s
f
I
edf
o
rmostc
o
n
c
r
e
t
es
t
r
u
c
t
u
r
e
s,
t
i
o
nd
i
m
e
n
s
i
o
no
ft
h
es
t
r
u
c
t
u
r
e
.T
h
i
sc
o
n
d
i
t
i
o
ni
sn
o
ts
w
i
t
ht
h
ep
o
s
s
i
b
l
ee
x
c
e
p
もi
o
no
fsomev
e
r
yl
a
.
r
g
es
t
r
u
c
t
u
r
e
ss
u
c
ha
sc
o
n
c
r
e
t
ed
a
m
s
. However,
amorep
r
e
c
i
s
ec
r
i
t
e
r
i
o
nf
o
rt
h
ea
p
p
l
i
c
a
b
i
l
i
t
yo
fl
i
n
e
a
re
l
a
s
t
i
c企配t
u
r
em
e
c
h
a
n
i
c
s,
whicha
l
s
o
t
a
k
e
si
n
t
oa
c
c
o
u
n
tt
h
es
t
r
u
c
t
u
r
es
h
a
.
peandt
h
em組 問ro
fl
o
a
d
i
n
g,
c
組
beg
i
v
e
ni
nt
e
r
m
so
f
t
h
e回 ・ 臼 且edb
r
i
t
t
l
e
n
e
s
snumber,
β,
w
h
i
c
hwi
I
Jb
ee
x
p
l
a
i
n
e
di
nS
e
c
.5
.
2
.
Thei
n
a
p
p
l
i
c
a
b
i
l
i
t
yo
fl
i
n
e
a
re
l
a
s
t
i
cf
r
a
c
t
u
r
em
e
c
h
a
n
i
c
st
ob
r
i
t
t
l
ef
a
i
l
u
r
e
so
ft
y
p
i
c
a
l∞E
c
r
e
t
es
t
r
u
c
t
u
r
e
si
sc
l
e
a
r
l
ya
p
p
a
r
e
n
tfromt
h
et
e
s
tr
e
s
u
l
t
sshowni
nF
i
g
.1
.
8
-1
.11
. Thed
a
t
a
.
p
o
i
n
t
si
n
d
i
c
a
t
eam
i
l
d
e
rs
i
z
ee
f
f
e
c
tt
h
回 t
h
es
t
r
a
i
g
h
tl
i
n
eo
fs
l
o
p
e1
/
2
.
Q
28
29
(0) Lineor Frocture
Chapter3
.NONLIN~AR_ ~~AC_TP"~ß MODELS
WITHSOFTENINGZONE
Ther
e
a
s
o
nf
o
rd
e
v
i
a
t
i
o
n
so
fc
o
n
c
r
e
t
eb
e
h
a
v
i
o
rf
r
o
ml
i
n
e
a
re
l
a
s
t
i
cf
r
a
c
t
u
r
em
e
c
h
a
n
i
c
s
i
st
h
ed
e
v
e
l
o
p
m
e
n
to
far
e
l
a
t
i
v
e
l
yl
a
r
g
e企a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
ewhichu
n
d
e
r
g
o
e
sp
r
o
g
r
e
s
s
i
v
e
s
o
f
t
e
n
i
n
gdamageduet
om
i
c
r
o
c
r
a
c
k
i
n
g
.Thee
f
f
e
c
to
ft
h
i
sm
i
c
r
o
c
r
a
c
k
i
n
gi
s
:(
1
)t
or
e
d
u
c
et
h
e
a
四 o
fe
n
e
曙 yt
h
a
tc
a
nber
e
l
e
a
s
e
di
n
t
o仕l
ec
r
a
c
kt
i
p
iand(
2
)a
tt
h
esameもimet
oi
n
c
r
e
a
s
e
t
h
ec
o
m
b
i
n
e
ds
u
r
f
a
c
ea
r
e
ao
fc
r
a
c
k
i
n
g,
andt
h
u
se
n
h
a
n
c
et
h
ee
n
e
r
g
ya
b
s
o
r
p
t
i
o
nc
a
p
a
.
b
i
l
i
t
yo
f
t
h
ef
r
拭 t
u
r
ep
r
o
c
e
s
sz
o
n
e
.
,
ar
e
l
a
t
i
o
nd
e
s
c
r
i
b
i
n
gt
h
es
o
f
t
e
n
i
n
gdamagen
e
e
d
st
obei
n
c
l
u
d
e
di
nt
h
ef
r
配 t
u
r
e
T
h
e
r
e
f
o
r
e
m
o
d
e
l
.T
h
i
sc
a
nbedoneb
a
s
i
c
a
l
l
yi
ntwoways
:(
1
)i
nt
h
eformo
fa
.s
t
r
e
s
s
d
i
s
p
l
a
c
e
m
e
n
t
,o
r(
2
)a
.s
t
r
e
s
s
s
t
r
a
i
nr
e
l
a
t
i
o
nf
o
rt
h
es
t
r
a
i
n
r
e
l
a
t
i
o
nf
o
rt
h
ef
r
o
n
t
a
lz
o
n
eo
fal
i
n
ec
r
a
c
k
s
o
f
t
e
n
i
n
g(
m
i
c
r
o
c
r
a
c
k
i
n
g
)r
e
g
i
o
ni
nf
r
o
n
to
ft
h
emainc
r
a
c
k
. Wew
i
l
lf
i
r
s
td
e
s
c
r
i
b
et
h
e
s
e
s
田 e
.
g
.P
l
a
n
舗
a
p
p
r
o
a
c
h
e
sandt
h
e
nd
i
s
c
u
s
st
h
e
i
rr
e
l
a
t
i
v
em
e
r
i
t
s(
F
o
rf
u
r
t
h
e
rd
i
s
c
u
s
s
i
o
n
s,
andE
l
i
c
e
s
,
1
9
8
8
a
;恥 t
s,
1
9
郎).
I
ng
e
n
e
r
a
l
,onem可 distinguishtwotypeso
fn
o
n
l
i
n
e
a
rf
r
a
c
t
u
r
em
e
c
h
a
n
i
c
s
:(
1
)d
u
c
t
i
l
e
(
m
e
t
a
l
s
),
and(
2
)n
o
n
d
u
c
t
i
l
e(
c
o
n
c
r
e
t
e,
a
sw
e
l
la
s田 r
a
m
i
c
s
)
;s
e
eF
i
g
.3.
1
.I
nc
o
n
t
r
却 t
t
ol
i
n
e
a
r
e
l
a
s
t
i
cf
r
a
c
t
u
r
em
e
c
h
a
n
i
c
s,
t
h
en
o
n
l
i
n
e
a
rz
o
n
ei
sl
a
r
g
ef
o
rb
o
t
ht
y
p
邸 o
fn
o
n
l
i
n
e
a
rf
r
a
c
t
u
r
e
m
e
c
h
a
n
i
c
s
.F
o
rd
u
c
t
i
l
ef
r
a
c
t
u
r
em
e
c
h
a
n
i
c
s,
mosto
ft
h
en
o
n
l
i
n
e
a
rz
o
n
eu
n
d
e
r
g
o
e
sp
l
a
.
s
t
i
c
,民t
p
l
制 t
i
c
i
t
y
,andt
h
ef
r
a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
ei
ss
t
i
l
1a
.v
e
r
ys
m
a
l
lp
a
r
to
f
h
a
r
d
e
n
i
n
go
rp
e
r
f
t
h
en
o
n
l
i
n
e
a
rz
o
n
e
. Byc
o
n
t
r
a
s
t,
f
o
rn
o
n
d
u
c
t
i
l
ef
r
邸 t
u
r
em
e
c
h
a
n
i
c
s,
w
h
i
c
hi
st
h
ec
a
s
ef
o
r
,
t
h
ef
r
a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
ei
sl
a
r
g
eando
c
c
u
p
i
e
sn
e
a
r
l
yt
h
ee
n
t
i
r
en
o
n
l
i
n
e
a
rz
o
n
e
.
c
o
n
c
r
e
t
e
Thus,
a
l
t
h
o
u
g
hmanyr
e
s
u
l
t
so
ft
h
ef
r
a
c
t
u
r
et
h
e
o
r
yo
fm
e
t
a
l
swhiche
v
o
l
v
e
de
a
r
l
i
e
ra
r
eu
s
e
f
u
l
,
mosto
fthemc
a
n
n
o
tbed
i
r
民 t
l
yt
r
a
.
n
s
p
l
a
n
t
e
d
.
Re
m
a
r
k
s
:Thef
r
a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
ei
sd
e
f
i
n
e
da
st
h
ez
o
n
ei
nwhicht
h
em
a
t
e
r
i
a
lu
n
d
e
r
,i
.
e
.,
t
h
es
t
r
e
s
sn
o
r
m
a
lt
ot
h
ec
r
a
c
k
・p
l
a
n
ed
e
c
r
e
a
s
e
sw
i
t
hi
n
c
r
e
a
s
i
n
g
g
o
e
ss
t
r
a
i
n
s
o
f
t
e
n
i
n
g
凶r
a
i
n
.Thes
t
r
e
s
sc
a
nbeu
n
d
e
r
s
t
o
o
da
st
h
em
a
c
r
o
s
t
r
e
s
so
ra
v
e
r
a
g
es
t
r
e
s
sσ
;
;t
h
a
ti
sc
a
l
c
u
l
a
t
e
d回 (
l
/
f
lt
)
t
i
m
e
st
h
ef
o
r
c
er
e
s
u
l
t
a
n
tt
r
a
.
n
s
m
i
t
t
e
da
c
r
o
s
sa
r
e
af
l
nw
h
i
c
ht
h
eh
e
t
e
r
o
g
e
1o
n岡 山 m
a
t
e
r
i
a
lw
i
t
ha
邸r
e
g
a
t
e
s回 dm
i
c
r
o
c
r
a
c
k
sc
a
nbea
p
p
r
o
x
i
m
a
t
e
d出 acontinuum(
o
n
t
h
em
a
c
r
o
s
c
a
l
e
)
.A
l
t
e
r
n
a
t
i
v
e
l
y
,
(
a
n
dc
u
s
t
o
m
a
r
i
l
y
),
σ匂=ん σ'
(
]
d
V
jv
.whereσ arethemi,
m
a
t
r
i
xandi
n
t
e
r
f
a
.
c
e
s,
whichshowh
i
g
hrandom
c
r
o
s
t
r
e
s
s
e
s(
a
c
t
u
a
ls
t
r
e
s
s
e
si
nt
h
ea
g
g
r
e
g
a
t
e
s
l
o
c
a
ls
c
a
t
t
e
r
),
a
.
ndv
,
.i
sf
o
rt
h
er
e
p
r
e
s
e
n
t
a
t
i
v
evolumeo
ft
h
em
a
t
e
r
i
a
l
,
d
e
f
i
n
e
db
e
l
o
wE
q
.1
.2
回 i
st
h
ec
h
a
r
a
c
t
e
r
i
s
t
i
cl
e
n
g
t
hi
)
. Thesec
o
n
c
e
p
t
s乱 開 d
e
v
e
l
o
p
e
dmorep
r
e
c
i
s
e
l
yi
nt
h
e
(
i
t
ss
i
s
t
a
t
i
s
t
i
c
a
lt
h
e
o
r
yo
fh
e
t
e
r
o
g
e
n
e
o
u
sm
a
t
e
r
i
a
l
s(
e
.
g
.K
r
'
るn
e
r
,
1
9
6
7
;Krumhansl,
1
9
6
8
)
.
(b) Metols
(c) Concrete
;
r
3
.
1SofteningStress-DisplacementRelations
T
h
i
sa
p
p
r
o
a
c
hd
e
v
e
l
o
p
e
d回 am
o
d
i
f
i
c
a
t
i
o
no
fas
i
m
i
l
a
ra
p
p
r
o
a
c
hp
r
e
v
i
o
u
s
l
yf
o
r
m
u
l
a
t
e
d
f
o
rm
e
t
a
l
s
. Dugdale(
1
9
6
0
)組 dB
a
r
e
n
b
l
a
t
t(
1
9
5
9,
1
9
6
2
)p
r
o
p
o
s
e
dt
h
a
tap
l
a
s
t
i
c(
y
i
e
l
d
i
n
g
c
o
h
e
s
i
v
e
)z
o
n
eo
fac
e
r
t
a
i
nf
i
n
i
t
el
e
n
g
t
hmuste
x
i
s
ta
tt
h
ef
r
o
n
to
fac
r
a
c
k(
F
i
g
.3
.
2
a
)
. The
もt
h
es
t
r
e
s
s
e
sfromt
h
ef
r
a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
ec
a
n
c
e
l
l
e
n
g
t
ho
ft
h
.
i
sz
o
n
emustbes
u
c
ht
h
a
t
h
es
t
r
e
s
ss
i
n
g
u
l
a
r
i
t
yc
a
u
s
e
da
tt
h
et
i
po
ft
h
ee
q
u
i
v
a
l
e
n
te
l
a
s
t
i
cc
r
a
c
kbyt
h
ea
p
p
l
i
e
dl
o
a
d
F
i
g
.
3
.
1
L
i
n
e
a
rZone(L),
N
o
n
L
i
n
e
a
rZone(
N
)andF
r
a
c
t
u
r
eP
r
o
c
e
s
s
Zo
ne(
S
)i
nF
r
a
c
t
u
r
eo
fD
i
f
f
e
r
e
n
tMa
.
t
e
r
i
a
l
s
31
30
(
i
.
e
.,
](1 =
0
)・ Thec
r
a
c
ko
p
e
n
i
n
ga
tt
h
eb
e
g
i
n
n
i
n
go
ft
h
ep
la
.
st
i
cz
o
n
e
,wherethes
も
r
e
s
s
回 l
yd
r
o
p
st
oz
e
r
o,
mayber
e
g
a
r
d
e
da
sam
a
t
e
r
i
a
lp
r
o
p
e
r
t
ywhichc
o
n
t
r
o
l
sp
r
o
p
a
g
a
t
i
o
n
s
u
d
d
(
F
i
g
.3
.
2
b
)
.
F
o
rsomem
e
t
a
l
sando
t
h
e
rm
a
t
e
r
i
a
l
si
twωlatern
o
t
e
dt
h
a
tt
h
ec
o
h
e
s
i
v
ez
o
n
es
h
o
u
l
d
邸 h
i
b
i
tag
r
a
d
u
a
lr
a
t
h
e
rt
h
阻 s
u
d
d
e
ns
t
r
e
s
sd
r
o
p,c
h
a
r
a
c
t
e
r
i
z
e
dbyas
o
f
t
e
n
i
n
gr
e
l
a
t
i
o
no
f
t
h
en
o
r
m
a
ls
t
r
e
s
sσa
.
c
r
o
s
st
h
ec
r
a
.
c
kv
s
.t
h
ec
r
a
c
k
o
p
e
n
i
n
gd
i
s
p
l
a
c
e
m
e
n
t
ι
(
F
i
g
.3
.
2
d
)
js
e
e
K
f
o
u
r
iandMi
I
l
e
r(
1
9
7
4
),
姐dWnuk(1974). Forconcrete,
t
h
i
st
y
p
eo
f
e
.
g
.Knauss(
1
9
7
4
),
modelwasp
r
o
p
o
s
e
dbyH
i
l
l
e
r
b
o
r
g,
Modeer姐 dP
悦 r
s
s
o
n(
1
9
7
6
)u
n
d
e
rt
h
enameo
ff
i
.c
t
i
t
i
o
u
s
c
r
8
心km
o
d
e
l
js
e
ea
l
s
oPet
位
e
,
r
s
“
旬
f
i
c
ω
叫
色
“i
t
“
1
叩
0
u
s
♂
"r
e
f
e
r
s色
ωothef.畠配
c
t
もt
出
ha
tt
h
e卯
po
r
尚
“
t
10
叩,
no
fac
r
a
c
kwhicht
r
a
n
s
m
i
t
st
e
n
s
i
l
es
t
r
e
s
sc
a
n
n
o
t
鈴e
p
鉱
a
r
叫
a
もt
i
o
no
ft
h
es
u
r
(
,
仏
a
α渇
町iも
吋
山
her
e
a
lc
r
a
c
ke
n
d
sa
tt
h
ep
o
ω
i
叫
n
beac
o
n
t
i
n
u
o
u
sc
r
a
c
kw
i
t
hf
u
l
ls
t
w
h
e
r
et
h
es
t
r
e
団 i
sr
e
d
u
c
e
dt
oz
e
r
O
jF
i
g
.3
.
2
d
.
)I
nt
h
i
smodel
,
whichha
日b
e
e
nw
i
d
e
l
ya
p
p
l
i
e
d
i
nf
i
n
i
もee
l
e
m
e
n
ta
n
a
l
y
s
i
so
fc
o
n
c
r
e
t
ef
r
a
c
t
u
r
e,
t
h
em
a
t
e
r
i
a
lf
r
a
c
t
u
r
ep
r
o
p
e
r
t
i
e
sa
r
ed
e
f
i
n
e
d
byt
h
es
o
f
t
e
n
i
n
gs
t
r
田s
-d
i
s
p
l
a
c
e
m
e
n
tr
e
l
a
.
t
i
o
n
:
“
(0)
E
仰M
C0
.d
(
3
.
2
)
叩
Oc
(e)
ε
F
i
g
.
3
.
2S
t
r
e
s
sD
i
s
t
r
i
b
u
t
i
o
n
s1
nandN
e
a
rt
h
eF
r
a
c
t
u
r
eP
r
o
c
e
s
sZ
o
nei
n
D
i
f
f
e
r
e
n
tModels
σ
J
九
∞
fIM
一
-G,
(
s
e
eF
i
g
.3
.
2
d
),
where円 i
st
h
e抗r
e
s
si
nt
h
ed
i
r
e
c
t
i
o
nn
o
r
m
a
lt
ot
h
ec
r
a
c
k
.The紅 e
au
n
d
e
r
a
.
c
t
u
r
ee
n
e
r
g
yo
ft
h
em
a
t
e
r
i
a
l,
i
.
e
.
:
t
h
ec
u
r
v
er
e
p
r
e
s
e
n
t
st
h
ef
r
(d)
(
c
)
噌
σ首 =f
(t
5
c)
qd
dc
、
‘
.
,
,
E--
担
,
,
、
・
1nnn~
Thec
r
a
c
kb
e
g
i
llS t
oopenwhent
h
es
t
r
e
s
sa
tt
h
et
i
pr
e
a
.
c
h
e
st
h
et
e
n
s
i
l
es
t
r
e
n
g
t
hl
i
m
i
tf
:
.
I
ft
h
es
h
a
p
eo
ft
h
es
o
f
t
e
n
i
n
gc
u
r
v
ei
sf
i
x
e
d,
t
h
e
nt
h
ef
r
a
c
t
u
r
epropertie~ a
r
ec
o
m
p
l
e
t
e
l
y
. T
hep
r
e
c
i
s
es
h
a
p
eo
ft
h
es
o
f
t
e
n
i
n
gs
t
r
e
s
s
c
h
a
r
a
c
t
e
r
i
z
e
dbytwop
a
r
a
m
e
t
e
r
s
: 刀 andGJ
d
i
s
p
l
a
.
c
ementdiagramh
a
sac
o
n
s
i
d
e
r
功l
ei
n
f
i
u
e
n
c
eont
h
ec
a
l
c
u
l
a
t
i
o
nr
e
s
u
l
比 I
nv
a
r
i
o
u
s
i
nwhichc
a
s
et
h
es
t
r
e
s
si
sr
e
d
u
c
e
d
w
o
r
k
st
h
i
ss
h
a
p
ei
sc
o
n
s
i
d
e
r
e
dt
r
i
a
n
g
u
l
a
ro
rb
i
l
i
n
e
a
r,
も
oz
e
r
oa
ta五n
i
t
ed
i
s
p
l
a
c
e
m
e
n
t6
x
p
o
n
e
n
t
i
a
ls
h
a
p
eh
a
sa
l
s
ob
e
e
nu
s
e
d
. Whent
h
e
0・ Ane
七h
eb
e
h
a
v
i
o
rI
s制 s
k
e
t
c
h
e
di
nF
i
g
.3
.
2
d
.
s
o
f
t
e
n
i
n
gz
o
n
ei
su
n
l
o
a
d
e
dandr
e
l
o
a
d
e
d(
F
i
g
.3
.
2
d
),
色i
t
i
o
u
sc
r
a
.
c
kmodelwasv
e
r
i
f
i
e
d岨 dc
a
l
i
b
r
a
七e
dbyv
a
r
i
o
u
sc
o
m
p
a
r
i
s
o
n
sw
i
t
h
H
i
l
l
e
r
b
o
r
g
'
sf
i
c
t
e
s
td
a
t
a
.
.However,
i
ts
e
e
m
st
h
a
.
ta
.
ne
x
h
a
u
s
t
i
v
ec
o
m
p
a
r
i
回 nw
i
t
ha
l
lt
h
ei
m
p
o
r
t
a
.
n
tc
o
n
c
r
e
t
e
f
r
a
c
t
u
r
eda
.
t
.
afromt
h
el
i
t
e
r
a
t
u
r
eh
a
sn
o
ty
e
tbe
四
p
r
e
s
e
n
t
e
d
.Butduet
oe
q
u
i
v
a
l
e
n
c
ew
i
t
h
,
もh
ee
x
t
e
n
s
i
v
ec
o
m
p
a
r
i
s
o
n
so
ft
h
el
a
t
t
e
rw
i
t
ht
e
s
tda
凶 (
B
a
z
a
n
tand
t
h
ec
r
a
c
kb叩 dmodel
Oh,
1
9
8
3a
.
)i
n
d
i
r
e
c
t
l
yv
a
l
i
d
a
t
et
h
ef
i
c
t
i
t
i
o
u
sc
r
a
c
km
o
d
e
l
.
Thes
h
a
p
eo
ft
h
es
t
r
e
s
s
d
i
s
p
l
a
c
e
m
e
n
tc
u
r
v
ewass
t
u
d
i
e
dont
h
eb
a
s
i
so
fm
i
c
r
o
m
e
c
h
a
n
i
c
so
f
.
a
c
k
sbyH
o
r
i
i(
1
9
8
8
),
H
o
r
i
ie
ta
l
.(
1
9
8
7,
1
9
8
8
),
a
.
n
dBa
.
i
a
n
t(
1
9
8
7
b
),
andonc
e
r
a
m
i
c
s
m
i
c
r
o
c
r
w
i
t
hs
i
m
i
l
.
a
rb
e
h
a
v
i
o
rbyO
r
t
i
z(
1
9
8
8
)
.Fromt
h
e
s
em
o
d
e
l
si
tt
r
a
n
s
p
i
r
e
dt
h
a
tt
h
em
i
c
r
o
c
r
a
c
k
a
.
b
i
l
i
t
yo
ft
h
es
t
r
e
s
s
d
i
s
p
l
a
c
e
m
e
n
tc
u
r
v
e
.
c
o
a
l
e
s
c
e
n
c
eg
e
n
e
r
a
l
l
yt
e
n
d
st
op
r
o
d
u
c
es
n
a
p
b
a
c
ki
n
s
t
t
h
ee
x
i
s
t
i
n
gme
a
.
s
u
r
e
m
e
n
t
s,
a
l
b
e
i
tl
i
m
i
t
e
di
ns
c
o
p
e
,
don
o
tshowa
.
nys
n
a
p
b
a
c
k
.1
t
However,
m.
a
y
w
e
l
lbet
h
a
.
tsomeo
t
h
e
rmechanism,
s
u
c
ha
.
sf
r
i
c
t
i
o
n
a
lp
u
l
l
o
u
to
fa
g
g
r
e
g
a
t
e.
a
ndf
r
a
g
m
e
n
t
s
32
成 8
,
e
l
i
m
i
n
a
t
e
st
h
es
n
a
p
b
a
c
k
.
fromt
h
ec
r
a
c
kf
a
3
.
2S
o
f
t
e
n
i
n
gS
t
r
e
s
s
S
t
r
a
i
nR
e
l
a
t
i
o
n
s
S
i
n
田 t
h
ec
r
a
c
k
si
nc
o
n
c
r
e
t
ea
r
en
o
ts
t
r
a
i
g
h
tb
u
tt
o
r
t
u
o
u
s,
andt
h
em
i
c
r
o
c
r
a
c
k
i
n
gz
o
n
ei
n
f
r
o
n
to
f出 ec
o
n
t
i
n
u
o
u
sf
r
a
c
t
u
r
ei
sn
o
tl
i
k
e
l
yt
od
e
v
e
l
o
pa
l
o
n
gas
t
r
a
i
g
h
tl
i
n
e,
t
h
eb
e
h
a
v
i
o
r
r
a
c
t
u
r
ep
r
o
c
伺 sz
o
n
ec
岨 e
q
u
a
l
l
yw
e
l
lbed
e
s
c
r
i
b
e
dbys
t
r
e
s
s
-s
t
r
a
i
nr
e
l
a
t
i
o
n
sw
i
t
h
o
f出 ef
品司s
o
f
t
e
n
i
n
g,
i
.
e
.,
d
e
c
l
i
n
i
n
gs
t
r
e
s
s抗 i
n
c
r
e
a
s
i
n
gs
t
r
a
i
n
.T
h
i
sa
p
p
r
o
a
c
hi
sq
u
i
t
ec
o
n
v
e
n
i
e
n
t
s
t
r
a
a
.
t
i
o
no
ft
h
en
o
d
e
so
ftwoa
d
j
a
c
e
n
te
l
e
m
e
n
t
sn
e
e
r
l
s
f
o
rc
o
m
p
u
t
e
rprogrammings
i
n
c
enos
e
p
a
r
t
ob
ei
n
t
r
o
d
u
c
e
dandf
r
a
c
t
u
r
ei
sh
a
n
d
l
e
dbya
d
j
u
s
t
m
e
n
t
so
ft
h
ei
n
c
r
e
m
e
n
t
a
ls
t
i
f
f
n
e
s
so
ff
i
n
i
t
e
b
a
s
i
c
a
l
l
yi
nt
h
esameway回 anyi
n
e
l
a
s
t
i
cb
e
h
a
v
i
o
r
.
e
l
e
m
e
n
t
s,
n
t
r
o
d
u
c
e
di
nf
i
n
i
t
ee
l
e
m
e
n
t
S
t
r
a
i
n
s
o
f
t
e
n
i
n
gi
nt
h
eformo
fas
u
d
d
e
nv
e
r
t
i
c
a
ld
r
o
pw踊 i
h
i
d(
1
9
6
8
)
.Then
e
e
dt
oc
o
n
s
i
d
e
rp
r
o
g
r
e
s
s
i
v
es
t
r
a
i
n
s
o
f
t
e
n
i
n
gi
n加 l
s
i
o
nwas
a
n
a
l
y
s
i
sbyRas
∞gni田 dbySc晶
.
n
l
a
n(
1
9
7
1
),
whoi
n
t
r
o
d
u
c
e
das
e
q
u
担問 o
fs
m
a
l
ls
t
r
e
s
sd
r
o
p
s
. Thef
a
c
t
r
e
x
p
e
r
i
m
e
n
t
a
l
l
yo
b
s
e
r
v
e
dbyL'H
e
r
m
i
t
e
thatωncretee
x
h
i
b
i
t
ss
t
r
a
i
n
s
o
f
t
e
n
i
n
gi
nt
e
n
s
i
o
nw槌 e
(
1
9
5
9
)
,RuschandH
i
l
s
d
o
r
f(
1
9
6
3
),Hughes釦 dCh
a
.
pman(
1
9
6
6
),andEvans乱
.
n
dMarathe
)
.Extensivea
.
ndc
a
r
e
f
u
l
l
yc
o
n
t
r
o
l
l
e
dm
e
a
s
u
r
e
m
e
n
t
sw
e
r
er
e
c
e
n
t
l
yr
e
p
o
r
t
e
dbyP
e
t
聞 s
o
n
(
1
9回
(
1
9
8
1
),Re
i
n
h
a
.
r
d
tandC
o
r
n
e
l
l
i
s
e
n(
1
9
8
4
),GopalaratnamandShah(
1
9
8
5
),ando
t
h
e
r
s
.
Fromt
h
econtinuumm
e
c
h
a
n
i
c
sv
i
e
w
p
o
i
n
t,
t
h
e∞n
c
e
p
to
fs
もr
a
i
n
s
o
f
t
e
n
i
n
gi
n
v
o
l
v
e
sc
e
r
t
a
i
n
回 m
a
t
h
e
m
a
t
i
c
a
ld
i
伍c
u
l
t
i
e
s,
s
u
c
ha
.
si
m
a
g
i
n
a
r
ywa
問 s
p
e
e
d(
o
rc
h
a
n
g
eo
ft
y
p
eo
ft
h
e
s
e
v
e
p
a
r
t
i
a
ld
i
f
f
e
r
e
n
t
i
a
le
q
u
a
t
i
o
no
fm
o
t
j
o
nf
r
o
mh
y
p
e
r
b
o
l
i
ct
oe
l
l
i
p
t
i
c
)andi
l
l
p
o
s
e
d
n
e
s
so
ft
h
e
,
w
h
i
c
hw
e
r
ep
o
i
n
t
e
do
u
tand叩 a
l
y
z
e
dbyHadamard(
1
9
0
3
),
Thom
剖
b
o
u
n
d
a
r
yv
a
l
u
ep
r
o
b
l
e
m
(
1
9
6
1
),B
a
z
a
n
t(
1
9
7
6
),S姐 d
l
e
r(
1
9
8
4
),
ReadandH号:
g
e
m
i
e
r(
1
9
8
4
),
WuandF
r
e
u
n
d(
1
9
8
4
),
Ba
.
i
a
n
t,
B
e
l
y
t
s
c
h
k
o組 dChang(
1
9
8
4
),
ando
t
h
e
r
s
. Thec
h
i
e
fproblemi
st
h
a
tt
h
ez
o
n
eo
f
e
n
e
r
g
yd
i
s
s
i
p
a
t
i
o
nt
e
n
d
st
ol
o
c
a
l
i
z
et
oaz
o
n
eo
fz
e
r
ovolume(
as
u
r
f
a
c
e
)s
ot
h
a
tt
h
et
o
t
a
l
ぽ g
yd
i
s
s
i
p
a
t
i
o
na
tf
a
i
l
u
r
ei
si
n
c
o
r
r
e
c
t
l
yi
n
d
i
c
a
t
e
dt
ob
ez
e
r
o(
c
f
.r
e
v
i
e
wbyBa
.
i
a
n
t,
1
9
8
6
)
.
en
侃c
u
l
t
i
e
sa
r
ec
i
r
c
u
m
v
e
n
t
e
di
fs
もr
a
i
n
s
o
f
t
e
n
i
n
gi
nf
i
n
i
t
ee
l
e
m
e
n
tm
o
d
e
l
si
si
n
t
r
o
d
u
c
e
d
T
h
e
s
ed
i
t
b
r
o
u
g
hf
r
a
c
t
u
r
ec
o
n
c
e
p
t
si
n0
問 。f
t
w
o
f
o
r
m
s
:(
1
)山 ec
r
a
c
kbandmodeland(
2
)t
h
en
o
n
l
o
c
a
l
mode
l
.Thel
a
t
t
e
rr
e
q
u
i
r
e
saf
i
n
e
rmeshb
u
ta
l
l
o
w
sb
e
t
t
e
rr
e
s
o
l
u
t
i
o
no
fl
o
c
a
l
i
z
e
ds
t
r
a
i
nf
i
e
l
d
s
.
33
o
fag
r
a
d
u
a
ls
o
f
t
e
n
i
n
g
.L
a
t
e
r∞mp
釘 i
s
o
n
sw
i
t
hnum
ぽ o
u
st
e
s
tr
e
s
u
l
t
s,howe
v
e
r
.i
n
d
i
c
a
t
e
d
h
u
sc
r
e
a
t
i
n
g
a
r
e
l
a
s
i
v
e
b
i
o
gf
r
a
e
b
u
r
e
t
h
a
tt
h
er
e
d
u
c
t
i
o
no
fs
t
r
e
s
st
oz
e
r
omustbegradud,t
聞
c
p
r
o
z
o
n
ea
h
e
a
do
ft
h
ef
r
o
n
to
f山 f
r
a
c
t
u
r
e(
F
i
g
.3
.
2
e
)
.T
h
i
s{
.
お
or
m
町m
叫晶机叫
4
l
似
o
∞
n,吋
wh
i
凶
c
hw剖
炉
g
i
討
v
佃,
e
n
.1
泊
nd
e
t
a
i
l匂
byB
加
a
.
i
a
副n
凶t(
仰
1
凶9
鈎
8
幻
2
)
,
制dBa
.
i
a
n
t凶 Oh(1983a
)h
a
sb
e
e
nshownhagmwith
cf
r
a
c
t
u
r
et
回t
d
a
t
aa
v
a
i
l
a
b
l
ei
nt
h
eH
t
e
r
畠t
u
r
e
,
p
a
r
t
i
c
u
l
a
r
J
yt
h
o
s
eont
h
e
e
s
s
e
n
t
i
a
l
l
ya
l
lt
h
eb担 i
,
t
h
eR
c
u
r
v
e(
s
e
eChap
附 4
),
andt
h
ed
i
f
f
e
r
e
n
c
e
s
e
f
f
e
c
to
fs
p
e
c
i
m
e
ns
i
z
eont
h
emaximuml
o
a
d
色w
eenv
a
r
i
o
u
ss
p
e
c
i
m
e
ng
印 m
e
t
r
i
e
s
.Att
h
esamet
i
m
ei
tw回 s
hownt
h
a
tt
h
ec
r
a
c
kband
be
exceP4when
modelaadm
i
l
e
r
b
O
I
r
g
'
sa
c
t
i
t
i
0
1
1
8c
r
a
c
kmodelg
i
v
ee
s
s
e
n
t
i
a
l
l
yt
h
es
a
m
e
r
e
s
u
l
t
s,
c
l
o
s
e
l
ys
p
a
c
e
dp
a
r
a
l
l
e
lc
r
a
c
k
soccur-(Thus,
t
h
ee
x
t
e
n
s
i
v
ee
x
p
e
r
i
m
e
n
t
a
lj
u
s
t
i
n
c
a
t
i
o
no
fb
h
e
t
i
t
i
o
u
sc
r
a
c
kmode
.
l)
c
r
a
c
kbandmodela
l
s
oi
n
d
i
r
e
c
t
l
yp
r
o
v
i
d
e
dj
u
s
t
i
f
i
c
a
t
i
o
nf
o
rt
h
e貧c
S
o
f
t
e
n
i
n
gi
sc
a
u
s
e
dbyf
r
a
c
t
u
r
i
n
g抗 日h,
t
f,
w
h
i
c
hi
ss
u
p
e
r
i
m
p
o
s
e
donS
h
ee
l
a
s
t
l
citmn.
h
o
o
s
i
n
.
l
!
:
Assuminga
1
1t
h
ec
r
a
c
k
st
ob
ep
a
r
a
l
l
e
lands
m
e
a
r
e
d(
c
o
n
t
i
n
u
o
u
s
l
yd
i
s
t
r
i
b
u
t
e
d
),andc
a
x
i
sX2t
oben
o
r
m
a
lもot
h
ec
r
a
c
k
s,
weh
a
v
ei
ntwod
i
m
e
n
s
i
o
n
s(
x
)-X2p
l
如 4
j
:
ロ
“
引
十
日
)
(i;)= [?;:?;;:i212/Fl(
The
,
∞lumnmatriCE泡 onthel
e
f
thand姐 dr
i
g
h
tbands
i
d
e
sc
o
n
s
i
s
to
ft
h
es
t
r
a
i
n阻 ds
t
r
側
c
o
m
p
o
n
e
n
t
s(γ=2
f
1
2= s
h
e
a
ra
n
g
l
e
),
C1111,
C1122 = C2211,
C2222 姐 dC1212 釘 et
h
ee
l
a
s
t
i
c
m
p
l
i
姐 C
民 組dβis姐 e
m
p
i
r
i
c
a
lf
a
c
t
o
r
,(
0く F壬 l
),
c
a
l
l
e
dt
h
es
}
m
r
!
?
e
t
e
n
t
i
o
nfadf
(
i
n
t
r
o
d
u
c
e
dbyS
u
i
d
a
nandS
c
h
n
o
b
r
i
c
h,
1
9
7
2,
叩dY
u
z
u
g
u
l
l
u組 dS
c
h
n
o
b
r
i
c
h,
1
9
7
3,
P
h
i
l
l
i
p
s,
lm
,岨dP
h
i
l
l
i
p
s組 dZ
i
e
n
k
i
e
w
i
e
z
,1976).Ift
h
em
a
t
e
r
i
a
li
si
s
o
k
o
p
i
c
,C1111=C2m=
22l
1=
りE',
Cll~12 :
:
:
:C
〆/
E',
C1212 = 2
{
i
nw
h
i
c
hE'= E,
v
':
:1f
o
rp
l
a
n
e
d
E
'=E/(l-1
1
),
〆=ν/(1-v)forplanestrain-ThefracturingMmincanbei
nc
o
r
p
o
r
a
t
d
i
n
t
ot
h
e∞m
p
l
i
a
n
c
e
,
i
nw
h
i
c
ht
h
ee
x
p
r
e
s
s
i
o
nf=,[ωα222/(1ω
)
J
σ
2
2h部 b田 ni
n
t
r
o
d
u
c
e
d
:
姐 b
er
e
g
a
r
d
e
da
.
sdamageandc
叩
b
ec
o
n
s
i
d
e
r
e
dt
obeaf
u
n
c
t
i
o
no
ft
hes
t
r
a
i
nnorm
i
.
l
ωC
t
ot
h
ec
r
a
c
k,
f
叫
o
rt
h
emaximump
r
i
n
c
i
p
a
ls
t
r
a
i
n
j
ω
/
(
1ー ω)=ゆ(
f
2
2
)
.I
n
i
t
i
a
l
l
yweha
.
v
e
ω
:
:
:
:0(
n
od叩 a
g
e
),
andc
o
m
p
l
e
t
edamage(
c
o
n
t
i
n
u
o
u
sf
r
a
c
t
u
吋 occursf
o
rω =1
. Always
r
a
c
t
u
r
ee
n
e
r
g
yi
so
b
t
a
i
n
e
da
s
0
:
:
;
ω 三1.Thef
∞
叫
べ
山口
,
3
.
2
.
1CrackBandModel
Theb
a
s
i
ci
d
e
ao
f出 ec
r
a
c
kbandmodel
,
w
h
i
c
hwasp
r
o
p
o
s
吋 b
yBa
.
ia
n
t(
1
9
7
6
),
i
s
:
(
1
)t
oc
h
a
r
a
c
t
e
r
i
配 t
h
em
a
t
e
r
i
a
lb
e
h
a
v
i
o
ri
nt
h
ef
r
a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
ei
nas
r
問 a
r
e
dmanner
.s
t
r
a
i
n
s
o
f
t
e
n
i
n
gc
o
n
s
t
i
t
u
t
i
v
er
e
l
a
.
t
i
o
n,
a
.
nd
t
h
r
o
u
g
ha
ft
h
ef
r
o
n
to
ft
h
es
t
r
a
i
n
s
o
f
t
e
n
i
n
gz
o
n
e(
c
r
a
c
kb
a
n
d
),
(
2
)t
oi
m
p
o
s
eaf
i
x
e
dw
i
d
t
hW c o
r
e
p
r
e
s
叩 t
i
n
gam
a
t
e
r
i
a
lp
r
o
p
e
r
t
y
.
Thei
m
p
o
s
i
t
i
o
no
fc
o
n
s
t
a
n
t叫 i
sr
e
q
u
i
r
e
di
no
r
d
e
rt
oa
v
o
i
ds
p
u
r
i
o
u
smeshs
e
n
s
i
t
i
v
i
t
y
姐 da
c
h
i
e
v
eo
b
j
e
c
t
i
v
i
t
y
,
a
s
s
u
r
i
n
gt
h
a
tt
h
ee
n
e
r
g
yd
i
s
s
i
p
a
t
i
o
nduet
of
r
a
c
t
u
r
ep
e
ru
n
i
tl
e
n
g
t
h
(
a
n
du
n
i
tw
i
d
t
h
)i
sac
o
n
s
t
a
n
t,
e
q
u
a
lt
ot
h
ef
r
a
c
t
u
r
ee
n
e
r
g
yo
ft
h
em
a
t
e
r
i
a
l,
GJ.I
nk
伺 p
i
n
g
w
i
t
ht
h
ec
l
a
.
s
s
i
c
a
la
p
p
r
o
a
c
ht
os
m
e
a
r
e
dc
r
a
c
k
i
n
g,
t
h
ed
e
t
a
i
l
e
df
o
r
m
u
l
a
t
i
o
no
ft
h
ec
r
a
c
kband
model(
Ba
.
i
a
n
tandC
e
d
o
l
i
n,1
9
7
9,1
9
8
0,1
9
8
3
)f
i
r
s
te
m
p
l
o
y
e
das
u
d
d
e
ns
t
r
e
s
sd
r
o
pi
n
s
旬a
d
(
3
.
3
)
J
GJ::::Wcf句 d
f
(
3.
4
)
Thef
u
l
l
yc
r
a
c
k
e
ds
t.
a
t
ei
so
b
t
a
i
n
e
df
o
rω →1. l
tw剖 shown(
B
a
z
a
n
tandOh,
1
9
8
3a
.
)
t
h
前 i
fE
q
.3.
4i
si
n
v
e
r
t
e
dandt
h
el
i
m
i
to
ft
h
ee
n
s
u
i
n
gs
t
i
汗n
e
s
sm
a
t
r
i
xf
o
rω → Oise
a
l
c
,
f
l
a
叫 ther田 ult∞incideswiththewel
1
k
r
附
ns
t
i
f
f
n
倒
m前 市 f
o
ramater
凶 t
h
a
tiscMEd
伽 n
a
l
l
y姐 dc
o
凶 削o
u
s
l
y
,
a
.
si
n
t
r
o
d
u
c
e
dbyR
a
.
s
h
i
d(
19
6
8)
・
u
n
i
d
i
r
e
E
q
.
3
.
4
r
e
g
e
c
t
so
n
l
yav
e
r
ys
p
e
c
i
a
lformo
fdamage.Ing
e
n
e
r
a
l,damagen
e
e
d
st
ob
e
i
n
t
r
o
d
u
c
e
dt
h
r
o
u
g
ht
h
ef
o
r
m
u
l
a
t
i
o
nO
ft
h
ei
n
t
e
r
n
a
lv
a
r
i
a
b
l
et
h
e
o
r
y(
f
o
rC
O
Z
I
e
r
e
b
e
, 間 e
.
g.
P
i
j
a
u
d
i
e
r
C
a
b
o
tandM
a
z
a
r
s,
1
9
8
9,
姐dPijaudier-Ca
.
b
o
tandBa
.
i
a
n
t
.1
9
8
8
)
蜘 a
i
nr
e
1
a
t
i
o
nu
n
d
e
r
l
y
i
n
gt
h
ec
α
r
舵
a
ckh n
dm
m
吋
O
d
由似
e
li
お
sd
Theu
n
i
a
x
i
a
ls
o
f
t
e
n
i
n
gs
t
r
e
s
s
凶凶i
砲刷
db
匂
'
y
f
加
叩
un
c
t
ω
t
1
叩
0
n
吋
ゆ
州
仲
仇
(
恥
仇
正
叫
匂
幻
2
叫
,
)
仏
2
)w
耐h
i
c
hd
e
t
白
i悶 d制晶略郁
g
酔e
ω The削 1
1
t
ω
t
s
a
r
陀'
e
, 却
a
ss
回叩
e
伽
n
邸
伽
凶
s
凶
s
討
i
i
川
t
も
u
伽
i
片
v
刊e
叫川
t 吋
o
ぱft
h
es
o
f
t
e
n
i
n
gs
t
r
色伺
s
目
仔
s
凶
古
仕
r
問a
吋
J
nr
,閃
e
l
晶
a
“
も
t
i
o
n卸 a
r
陀et
h
o
s
ef
o
rt
h
ef
貧
i
c
t
i
比
t
i
o
田
u
1
悶s
c
r
a
c
kmode
l
.1
ta
p
p
e
a
r
s
“ω
ω
“
λ
い
;
34
35
t
h
a
tt
h
es
i
m
p
l
ef
o
r
m
u
l
aゆ(
f
)=(Ej
刀)expa
(
f-f
w
i
t
he
m
p
i
r
i
c
a
lc
o
n
s
t
a
n
t
sα姐 dfp,
i
s
p),
i
.
e
.,
at
r
i
a
n
g
u
l
a
rs
t
r
e
s
s
s
t
r
a
i
nd
i
a
g
r
a
m
,h副
g
e
n
e
r
a
l
l
ya
d
e
q
u
a
t
e
. As
t
r
a
i
g
h
tl
i
n
es
o
f
t
e
n
i
n
g,
a
l
s
ob
e
e
nu
s
e
ds
u
c
c
e
s
s
f
u
l
l
y
.
Therea
r
etwo討mplev
a
r
i
a
n
t
st
ot
h
ec
阻止 b
andmode
.
lTheo
r
i
g
i
n
a
lone(Ba
.
Z
a
n
tandOh
,
1
9
8
3
a
)p
r
e
s
u
m
e
st
h
a
tt
h
es
m
e
a
r
e
dp
a
r
a
l
l
e
lc
r
a
c
k
s伽 r
tt
of
o
r
mi
nt
h
ed
i
r
e
c
t
i
o
nn
o
r
m
a
lt
o
t
h
emaximump
r
i
n
c
i
p
a
ls
t
r
e
s
sb
u
ts
u
b
s
e
q
u
e
n
t
l
yt
h
ec
r
a
c
ko
r
i
e
n
t
a
t
i
o
ni
sf
i
x
e
di
nt
h
em
a
t
e
r
i
a
l
白川 r
e
s
e
a
r
c
h(eιGuptaandAkbar
,
e
v
e
ni
ft
h
ep
r
i
n
c
i
p
a
ls
t
r
e
s
sd
i
r
e
c
t
i
o
nr
o
t
a
t
e
s
.Morer
e
1
9
8
4
;Cope
,1984)附 mstoindicatethati
ti
sb
e
t
t
e
rt
oassumet
h
a
tt
h
ec
r
a
c
ko
r
i
e
n
t
抗1
0
n
r
o
t
a
t
e
sw
i
t
ht
h
ed
i
r
e
c
t
i
o
no
ft
h
emaximump
r
i
n
c
i
p
a
ls
t
r
e
s
8,
w
h
i
c
hmeanst
h
a
ts
h
e
a
rs
もr
e
s
s
e
s
sv
a
r
i
a
n
t,
t
h
eg
e
n
e
r
a
lt
r
i
a
x
i
a
ls
t
r
e
s
s
s
t
r
a
i
nr
e
l
a
t
i
o
n
ont
h
ec
r
a
c
kp
l
a
n
ec
a
nn
e
v
e
ra
r
i
s
e
.F
o
r山 i
f
o
rt
h
em
i
c
r
o
c
r
a
c
k
e
dm
a
t
e
r
i
a
l叩 bew
r
i
t
t
e
ni
nt
h
ef
o
r
m(Ba.Z叫叫 L
i
n,
1
9
8
8
b
)
:
f;;
=l
Il
"
;;
;1
-
十ーーーーー一ー_'l7'
T
1
_.nr'rl__
l I'Tr
(
3
.
5
)
叶
i
nw
h
i
c
hni=
=d
i
r
e
c
t
i
o
nc
o
s
i
n
e
soft
h
ec
u
r
r
e
凶 ma
杭.
x
1
m
宜m
ω
もe
n
s
o
討叫 s
削u
b
s
c
α
n
中
p
t
匂si
m
p
l
ysummation)andC仰
,n i
st
h
ee
l
嗣 t
i
c(undamaged)c
o
m
p
l
i
a
n
c
e
t
e
n
s
o
r
.
Whent
h
ep
r
i
n
c
i
p
a
ls
t
r
e
s
sd
i
r
e
c
t
i
o
n
sr
o
同t
es
i
g
n
i
f
i
c
a
n
t
l
y
,
t
h
en
o
n
r
o
t
a
もi
n
gc
r
a
c
kmethod
mustbeg
e
n
e
r
a
l
i
z
e
dt
oa
l
l
o
wf
o
rt
h
ef
o
r
m
a
t
i
o
no
fs
e
c
o
n
d
a
r
yandt
e
r
t
i
a
r
yc
r
a
c
k
i
n
go
fd
i
f
f
e
r
e
n
t
o
r
i
e
n
t
a
t
i
o
n
s
.Suchm
u
l
t
i
d
i
r
e
c
t
i
o
n
a
ls
m
e
a
r
e
dc
r
a
c
k
i
n
gm
o
d
e
l
sw
e
r
ee
s
p
e
c
i
a
l
l
yp
e
r
f
e
c
t
e
dbyde
B
o
r
s
t(
1
9
8
4
)
.
1
nr
e
a
l
i
t
y
,
t
h
em
i
c
r
o
c
r
a
c
k
sp
r
i
o
rt
o出 .
ef
i
n
a
lc
o
n
t
i
n
u
o
u
sf
r
a
c
t
u
r
ea
r
ed
i
s
t
r
i
b
u
t
e
do
v
e
ra
l
l
o
r
i
e
n
t
a
t
i
o
n
s,
w
i
t
hd
i
f
f
e
r
e
n
tf
r
e
q
u
e
n
c
i
e
sf
o
rv
a
r
i
o
u
so
r
i
e
n
t
a
t
i
o
n
s
.T
h
i
sf
e
a
t
u
r
ei
sc
a
p
t
u
r
e
dby
.
Z
a
n
tandOh,
1
9
8
5
)w
h
i
c
hs
田 m
st
obep
h
y
s
i
c
a
l
l
yt
h
emostr
e
a
l
i
s
t
i
c
t
h
em
i
c
r
o
p
l
a
n
emodel(Ba
o
n
c
e
p
t
u
a
l
l
ys
i
m
p
l
e
s
もmod
e
lo
fdamageduet
oc
r
a
c
k
i
n
gb
u
ti
sv
e
r
ydemanding
a
sw
e
l
l邸 c
f
o
r∞mputerもime.1
nt
h
i
smodel(
w
h
i
c
hwi
l
Jb
ec
o
n
s
i
d
e
r
e
df
u
r
t
h
e
ri
nS
e
c
t
i
o
n3
.
2
.
5
),
t
h
e
h
y
p
o
t
h
e
s
i
st
h
a
tt
h
ec
r
a
c
k
i
n
gs
t
r
a
i
nt
e
n
s
o
ri
sa
d
d
i
t
i
v
et
ot
h
ee
l
a
s
t
i
cs
t
r
a
i
nt
e
n
s
o
ri
sabandon
吋
叫 c
r
a
c
k
i
n
gI
sm
o
d
e
l
l
e
da
ss
t
r
a
i
n
s
o
f
t
e
ni
ngs
e
p
a
r
a
t
均 onp
l
a
n
伺 o
fa
.
lo
r
i
e
n
t
a
t
i
o
n
s(
c
a
l
l
e
d
m
i
c
r
o
p
l
a
n
e
s
),
s
u
b
j
e
c
tt
o出 eh
y
p
o
t
h
e
s
i
st
h
a
tt
h
es
t
r
a
i
ncomponentsone
a
c
hm
i
c
r
o
p
l
a
n
ea
r
e
t
h
er
e
s
o
l
v
e
dcomponentso
ft
h
e(
m
a
c
r
o
s
c
o
p
i
c
)s
t
r
a
i
n
.Thes
t
r
e
蹴 s
a
.
f
t
e
rc
r
a
c
k
i
n
gont
h
ep
l
a
n
e
s
,
b
u
to
v
e
r
a
l
le
q
u
i
l
i
b
r
i
u
mi
se
n
f
o
r
c
e
dby
o
fv
a
r
i
o
u
so
r
i
e
n
t
a
t
i
o
n
sa
r
en
o
te
x
a
c
t
l
yi
ne
q
u
i
l
i
b
r
i
u
m
u
s
i
n
gt
h
ep
r
i
n
c
i
p
l
eo
fv
i
r
t
u
a
lw
o
r
k
.
I
tmaybep
o
i
n
t
e
do
u
tt
h
a
tl
i
n
e
a
.
re
l
a
s
t
i
cs
t
a
.
b
i
l
i
t
ya
.
n
a
l
y
s
i
si
nwhicht
h
em
i
c
r
o
c
r
a
c
k
sa
r
e
剖 s
umedt
ogrowi
nahomogen
∞uselasticcontinuumindicatesthatthesofteningstressd
i
s
p
l
a
c
即 l
e
n
to
rs
t
r
e
s
ss
t
r
a
i
nr
e
l
a
t
i
o
ns
h
o
u
l
de
x
h
i
b
i
tamaximumd
i
s
p
l
a
c
e
m
e
n
to
rs
t
r
a
i
n
a
f
t
e
rwhicht
h
es
t
r
e
s
ss
u
d
d
e
n
l
yd
r
o
p
st
oz
e
r
o(B
a
.
Z
姐t
,1987b). Buti
ti
sno
も
(
s
n
a
p
b
a
c
k
),
y
e
tknownw
h
e
t
h
e
rt
h
ep
r
e
d
i
c
t
i
o
no
fs
n
a
p
b
a
c
ka
l
s
or
e
s
u
l
t
sfrommodel
st
a
k
i
n
gi
n
t
oa
c
c
o
u
n
t
田,
i
n
e
l
a
s
t
i
cb
e
h
a
v
i
o
r,
andf
r
i
c
t
i
o
n
.
inhomog
e
n
e
i
t
i
.
p
p
r
o
x
i
m
a
t
e
l
yWc 竺 3
d
d
Thew
i
d
t
ho
ft
h
ec
r
a
c
kbandf
r
o
n
tc
a
nbeassumedt
obea
a(
a
= maximumaggregates
i
z
e
)
.T
h
i
sc
o
n
c
l
u
s
i
o
nw踊 drawnon出eb
a
s
i
so
foptimumf
i
t
t
i
n
g
.
Z
a
n
tandOh,
1
9
8
3a
.
)
.However
,
t
h
eoptimumwasn
o
ts
h
a
r
p
;叫
o
fnumeroust
e
s
td
a
t
a(Ba
v
a
l
u
e
sr
a
.
n
g
i
n
gfromda t
o6
d
"g
a
v
ealmo
s
te
q
ua
l
1
ygoodr
e
s
u
l
t
s,
p
r
o
v
i
d
e
d,
o
f∞u
r
s
e
,the
α
.
p
o
s
t
p
e
a
kf
u
n
c
t
i
o
nO
((22)w出 a
d
j
us
t
e
df
o
re
a
c
hd
i
f
f
e
r
e
n
tv
a
l
u
eo
f叫
Oncet
h
es
h
a
p
eo
ft
h
es
o
f
t
e
n
i
n
gs
t
r
e
s
s
s
もr
a
i
nr
e
l
a
t
i
o
ni
sf
i
x
e
d,
t
h
ec
r
a
c
kba
.
n
dmodeli
s
f
u
l
l
yc
h
a
r
a
.
c
t
e
r
i
z
e
dbyt
h
r
e
em
a
t
e
r
i
a
.
lp
a
r
a
m
e
t
e
r
s
: ,
Ga
.
n
dWc (
a
l
t
h
o
u
g
ht
h
ei
n
f
l
u
e
n
c
eo
f
a
l
u
ei
sr
a
t
h
e
rweakf
o
rs
i
t
u
a
-もi
o
n
sw
i
t
hi
s
o
l
a
t
e
dc
r
a
c
k
s
)
.Byc
o
n
t
r
a
s
t,
t
h
ef
i
c
t
i
t
i
o
u
s
t
h
e叫 -v
晶c
kmodelh
a
so
n
l
ytwob
a
s
i
cp
a
r
a
m
e
t
e
r
s,刀姐dGf ・ Whyt
h
ee
.
対r
a
.parameteri
nt
h
e
c
r
c
r
a
c
kb姐 dmodel?Thee
x
t
r
ap
a
r
a
m
e
t
e
r
,
ω
'
c,
i
si
m
p
o
r
t
a
n
to
n
l
yi
ns
i
t
u
a
t
i
o
n
swhent
h
e
r
ea
r
e
p叫 l
e
lc
r
a
c
k
s(
句
・
,i
nt
h
epresωoft
e
n
s
i
o
nr
e
i
n
f
o
r
c
e
m
e
n
t
)
;t
h
e
nw b
a
s
i
c
a
l
l
yd
e
t
e
r
m
i
n
e
s
a
sam
a
t
e
r
i
a
lp
r
o
p
e
r
t
y
.I
ts
h
o
u
l
dbep
o
i
n
t
e
do
u
t出品t
t
h
eminimump
o
s
s
i
b
l
ec
r
a
c
ks
p
a
c
i
n
g,
t
h
ef
i
c
t
i
t
i
o
u
sc
r
a
c
kmodel
,
b
e
c
a
u
s
eo
fi
t
sl
a
c
ko
ft
h
ee
x
t
r
aparameterωc,
c
組 g
i
v
er
田u
l
t
s
4hatuen
o
bo
b
j
e
c
t
i
v
e
i
ns
i
4
1
1
a
t
i
o
n
sw
i
t
hp
a
r
a
l
l
e
lc
l
o
s
e
l
ys
p
a
c
e
dc
r
a
c
k
s
-(Ac
e
r
t
a
i
nl
e
n
g
t
hh
a
s
d
d
i
t
i
o
n
a
lma
.
t
e
r
i
a
lp
a
r
a
m
e
t
e
ri
n出 ef
i
c
t
i
t
i
o
u
sc
r
a
ε
km
o
d
e
l
;h
o
w
e
v
e
r
,
a
l
s
ob
e
e
nd
e
f
i
n
e
d組 組 a
i
nc
o
n
t
r
a
s
ttoωc,
i
ti
sn
o
tani
n
d
e
p
e
n
d
e
n
tp
a
r
a
m
e
t
e
r
.
)
e
q
u
i
r
e
dbyt
h
ec
r
a
c
kb岨 dmodel,
maybet
∞smallin
Thef
i
n
i
t
ee
l
e
m
e
n
ts
i
z
eh=叫, r
t
h
ec
a
s
eo
fv
e
r
yl
a
r
g
es
t
r
u
c
t
u
r
e
s
.I
nt
h
a
t日 s
e,
i
ti
sp
o
s
s
i
b
l
et
oe
n
l
a
r
g
et
h
ee
l
e
m
e
n
ts
i
配 (
i
.
e
.
u
s
eh>叫 )p
r
o
v
i
d
e
dt
h
a
tt
h
es
o
f
も
匂
e
刷凶
n
1
自
帥
品mee
n
e
r
略
gyd
i
s
s
i
p
a
t
i
o
n
,GI- Thisi
si
l
l
u
s
t
r
a
t
e
di
nF
i
g
.3
.
3
.I
nv
i
e
wo
ft
h
es
e
r
i
e
sc
o
u
p
l
i
n
g
model
,
a
l
r
e
a
d
yd
i
s
c
u
s
s
e
di
nc
o
n
n
e
c
t
i
o
nw
i
t
hF
i
g
.1
.8
,thegivens
もr
e
s
s
s
t
r
a
i
ndiagramOPA
f
o
rt
h
es
t
r
a
i
n
s
o
f
t
e
n
i
n
gc
r
a
c
kbandn
田 d
st
ober
e
p
l
a
臼 d
,atincreasingelementsizeh,by
必a
g
r
a
m
sOPB,
OPC,
OPD,
e
t
c
.
,suchthatwhenthe紅 側 underanyofthesediagamh
esamef
r
a
山 r
ee
n
e
r
g
yv
a
l
u
eGJ i
so
b
t
a
i
n
e
d(
F
i
g
.3
.
3
c
)
.I
nt
e
r
m
so
f
i
sm
u
l
t
i
p
l
i
e
dbyht
t
h
es
t
r
e
s
s
d
i
s
p
l
a
c
e
m
e
n
tdiagramf
o
rlmghhho
f
S
n
i
t
ee
l
e
m
e
n
t
,
ac
h
a
a
g
ei
nhr
e
q
u
i
r
e
st
h
a
t
閃 m
entd
i
a
g
r
.
a
m0
1
2i
nF
i
g.3
.
3
bber
e
p
l
a
c
e
dbyd
i
a
g
r
a
m
s0
3
2,
0
4
2,
t
h
ea
c
t
u
a
ls
t
r
e
s
s
d
i
s
p
l
a
0
5
2,
e
t
c
.Theses
t
r
e
時 d
isplacemeMdiagramhavethepropedyt
h
a
tt
h
ea mu
n
d
e
rb
h
e
r
ni
c
o
n
s
t
a
n
t,
t
h
u
s回 s
u
r
i
n
gc
o
n
s
t
a
.
n
tf
r
a
c
t
u
r
ee
n
e
r
g
yG/
.
Onec
a
n,
o
fc
o
u
r
s
e,
a
l
s
ou
s
ee
l
e
m
e
n
t
sw
i
t
hhく ωc(
s
a
y0
.
1
ωふ p
r
o
v
i
d
e
dt
h
ep
o
s
t
p
e
a
k
i
a
g
r
a
mOPFi
nF
i
g
.3.3cωthatt
h
eσ-8dia~ram
s
l
o
p
ei
sd
e
c
r
e
a
s
e
da
sshownbyt
h
eσ-fd
0
8
2(
F
i
g
.3
品)woulda
g
a
i
nhav
et
h
esamea
r
e
a鎚 0
1
2
.I
nt
h
i
sc
槌 e
.therowd c
r
i
k
e
d
e
l
e
m
e
n
t
sc
a
nben
a
r
r
o
w
町七h
ant
h
ea
d
j
a
c
e
n
te
l
e
m
e
n
t
sandt
h
emeshmayl
o
o
ka
sshowni
n
r
e
p
l
a
c
e
dbyh
.O
b
v
i
o
u
s
l
y
,
whenh→ 0,
t
h
ec
r
a
c
kb姐 dmodeli
nt
h
e
F
i
g
.3.
4ab
u
tw
i
t
hω。
l
i
m
i
tbecomesi
判 icaltoHi
I
l
e
r
b
o
r
g'
smodel(
w
h
i
c
hi
nもh
iss
e
n
s
ei
sas
p
e
c
i
a
lc
蹴
o
ft
h
e
c
r
a
c
kbandmoden.
o
f
t
e
n
i
n
gs
l
o
p
eg
e
t
ss
t
e
e
p
e
r,u
n
t
i
lf
o
rac
e
r
t
a
i
n
Ast
h
ee
l
e
ments
i
z
ei
si
n
c
r
e
a
s
e
d,もhes
e
l
e
m
e
n
ts
i
z
ehav
e
r
t
i
c
a
ls
t
r
e
s
sd
r
o
pOPCor42iso
b
t
a
i
n
e
d
.
F
o
ras
t
i
l
ll
a
r
g
e
re
l
e
m
e
n
ts
i
z
e,
whichwouldcausenmericaldi
血 叫t
i
偲 J
thediagrmOPDor052woulde
x
h
i
b
i
ts
n
a
p
b
a
c
k
l
nt
h
a
.
tc
a
s
e
,
onema
.
yr
e
p
l
a
.
c
et
h
es
n
a
p
b
a
c
ksegment5
2o
ft
h
ee
l
e
ment鈴 阿 佐d
i
s
p
l
a
臼m
ent
67inFig.3.3b.ThepoiMofV
e
d
i
c
a
id
r
o
pi
sddermined
d
l
a
g
r
a
mbyaverbids
t
r
e
s
sd
r
o
p,
a
g
a
i
nfrom出 ec
o
n
d
i
t
i
o
nth4thea
r
e
au
n
d
e
rt
h
ediagram067mustbet
h
esame槌 t
h
e
a
r
e
au
n
d
e
rt
h
ed
i
a
g
r
a
m012o
r0
4
2.T
h
i
sc
o
n
s
i
d
e
r
a
t
i
o
ni
n
d
i
c
a
t
e
s出 a
tt
h
ee
q
u
i
v
a
l
e
n
tt
e
n
s
i
l
e
s
t
r
e
n
g
t
hj
んofthelargef
i
n
i
t
ee
l
e
m
e
n
to
fs
i
z
eh>ωci
sg
i
v
e
nby
:
1
,
:
1
、
吾
ん=
h
n=竺生 一紘
一
一
切
1
[
2 一 刀2
(
f
o
r
h三 h
o
)
(
3.
6
)
37
36
(
e
.
g
.,
BaZa
n
t阻 dC
e
d
o
l
i
n,
1
9
7
9,
1
9
8
0
)i
nw
h
i
c
hh
oi
st
h
ee
l
e
m
e
n
ts
i
z
ef
o
rw
h
i
c
hav
e
r
t
i
c
a
l
s
t
r
e
s
sd
r
o
p(
d
i
a
g
r
a
m0
4
2
)i
so
b
t
a
i
n
e
d
.N
o
t
et
h
a
tt
h
ee
xp
r
e
s
s
i
o
nf
o
rh
oi
ss
i
m
i
l
a
rt
oI
r
w
i
r
内
E
q
.2
.
1
0f
o
rt
h
es
i
z
eo
ft
h
ey
i
e
l
d
i
n
gz
o
n
e
.
も
ss
u
b
j
e
c
t
e
dt
o
Thef
o
r
e
g
o
i
n
ga
d
j
u
s
t
m
e
n
t
sa
r
ei
d
e
a
l
l
yd
e
f
i
n
e
df
o
rs
q
u
a
r
ef
i
n
i
t
ee
l
e
m
e
n
h
es
i
d
e
s
)
.However
,
e
x
t
e
n
s
i
o
n
st
on
o
n
s
q
u
a
r
ee
l
e
m
e
n
t
s
t
e
n
s
i
o
no
rs
h
e
a
r(
n
o
r
m
a
lo
rp
a
r
a
l
l
e
lもot
a
r
ep
o
s
s
i
b
l
e
.I
nt
h
i
sc
制 e
,
i
ft
h
ed
i
m
e
n
s
i
o
no
ft
h
ee
l
e
m
e
n
ti
ne
a
ε
hd
i
r
e
c
t
i
o
ni
sa
b
o
u
tt
h
esam
e
,
onemaya
g
a
i
nu
s
eE
q
.3
.
7i
nw
h
i
c
hh=ゾ
互
, whereA =elementarea(
i
n2
D
)
.
e
l
a
t
i
o
nhf;q/2E= hof?/2E= G/wherehr
e
p
r
e
s
e
n
t
se
i
t
h
e
r
E
q
.3
.
7f
o
l
l
o
w
sfromもher
t
h
ea
c
t
u
a
.
lw
i
d
t
ho
ft
h
ec
r
a
ε
k
i
n
ge
l
e
m
e
n
ti
nt
h
ed
i
r
配 t
i
o
nn
o
r
m
a
lt
ot
h
ec
r
a
c
k
s,
p
r
o
v
i
d
e
dt
h
e
elem
回 t
i
ss
q
u
a
r
eo
rr
e
c
t
a
n
g
u
l
a
r
,
o
rt
h
ee
f
f
e
c
t
i
v
ew
i
d
t
ho
ft
h
ee
l
e
m
e
n
td
e
f
i
n
e
da
.
sh=叫 A
.
,
f
A
6
w
h
e
r
eA
.
/,
Aba
r
e出 e紅 e
a
.
so
ft
h
ee
l
e
m
e
n
t姐 do
ft
h
ec
r
a
c
kbandw
i
t
h
i
nt
h
ee
l
e
m
e
n
t
.F
o
r
e
l
e
m
e
n
七w
i
d
t
bh<h
o,
f
o
rwhicht
h
eposιpeaks
l
o
p
ei
sr
e
d
u
c
e
db
u
tt
h
es
t
r
e
n
g
t
hl
i
m
i
もi
sk
e
p
t
,
出e向 e時蜘a.i
nd
i
a
g
r
a
mmaybee
x
p
r
e
s
s
e
d,
a
c
∞rdingtothes
e
r
i
e
smodel(
F
i
g
.1
.
8
),
a
8刀
a
.
sf
o
l
l
o
w
s
:
E
s
o
f
t+(1一生 )
f
u
n
h=
竺
f
o
rhく 1
1
0
)
(
3
.
7
)
f=竺Z
nl
L,
.
-ω
~c竺
Ab (
作
ト'
-+
V
-,~ð
!σ パゴ~
b/
ー
し/'
¥
工6
(
b
)
4
3
5
(c)σ
汗
f
f
;
eQ=
(
d
)
w~~国
TIN-上
f
.
o
/
t,
f
u
n
l= t
h
es
t
r
a
i
n
sc
o
r
r
田p
o
n
d
i
n
gt
ot
h
esames
t
r
e
s
sa
c
c
o
r
d
i
n
gt
ot
h
es
t
r
a
i
n
s
o
f
t
e
n
i
n
g
s
t
r
e
s
s
s
t
r
a
.
i
nd
i
a
g
r
a
mand叩 u
n
l
o
a
d
i
n
gs
t
r
e
s
s
s
t
ra
.
Ind
i
a
g
r
a
ms
t
a
r
t
i
n
gf
r
o
mt
h
epe
北 s
t
r
e
s
s
p
o
i
n
t
.
I
fl
a
r
g
e
re
l
e
m
e
n
t
sn
e
e
dt
obeu
s
e
d,
a
n
o
t
h
e
rp
o
s
s
i
b
i
l
i
t
yi
sもok
e
e
pt
h
es
i
z
eo
fもh
ee
l
e
m
e
n
t
s
ont
h
el
i
n
eo
fc
r
a
c
kbandc
o
n
s
t
a
n
t,
e
q
u
a
lt
os
i
z
ewc,
ande
n
l
a
r
g
ea
l
lt
h
er
e
m
a
i
n
i
n
ge
l
e
m
e
n
t
s
4
a.I
ft
h
ec
r
a
c
kbandi
smucht
h
i
n
n
e
rt
h
姐 t
h
ea
d
j
a
c
e
n
t血l
i
t
ee
l
e
m
e
n
t
s,
a
sshowni
nF
i
g
.3.
山 em
odel
,o
f∞u
r
s
e,
be
∞mespracticallyi
d
e
n
t
i
c
a
lt
ot
h
ef
i
c
t
i
t
i
o
u
sc
r
a
c
kmode
.
l
,thecrackbandmodela
sw
e
l
l
I
nt
h
ec
a
s
eo
fc
r
a
c
kp
r
o
p
a
g
a
t
i
o
ni
nana
r
b
i
t
r
a
r
yd
i
r
e
c
t
i
o
n
拙
t
h
ef
i
c
t
i
t
i
o
u
sc
r
a
c
kmodelr
e
q
u
i
r
e
sr
e
m
e
s
h
i
n
gs
o出 a
tt
h
eboundaryl
i
n
e
so
ft
h
ec
r
a
c
k
r
a
c
kl
i
n
ewouldc
o
n
f
o
r
mt
ot
h
ei
n
t
e
r
e
l
e
m
e
叫 b
o
u
n
d
a
r
i
e
s(
知 t
h
ef
i
c
t
i
t
i
o
u
s
b姐 d白 色hec
c
r
a
c
kmodel
,
t
h
e
s
et
e
c
h
n
i
q
t
瑚
w
e
r
et
op
e
r
f
e
c
t
e
dbyI
n
g
r
a
f
f
e
a(
1
9
8
5
),
I
n
g
r
a
f
f
e
a叩 dG
e
r
s
t
l
e
組dIngraffeaandSaouma(1985). I
ft
h
er
e
m
e
s
h
i
n
gi
sn
o
tdone
,i
ti
ss
t
i
l
Jp
o
回b
l
e
(
1
9
8
5
),
t
oa
p
p
r
o
x
i
m
a
t
e
l
yr
e
p
r
e
s
e
n
tf
r
a
c
t
u
r
er
u
n
n
i
n
gi
nana
r
b
i
t
r
a
r
yd
i
r
e
c
t
i
o
nbya
l
l
o
w
i
n
gt
h
ec
r
a
c
k
bandt
oh
a
v
eaz
i
g
z
a
gformωshowni
nF
i
g.3.
4
b
.Somea
d
j
u
s
t
m
e
n
t
so
ft
h
ef
r
a
c
t
u
r
ee
n
e
r
g
y
a
r
et
h
e
nne
偲 s
s
a
r
y七ot
a
k
ei
n
t
oa
ι
c
o
u
n
tt
h
e品 刊r
a
g
ew
i
d
t
ho
fs
u
c
haz
i
g
z
a
gband(
B
a
z
a
n
t,
1
9
部品)
Evenw
i
t
hs
u
c
ha
d
j
u
s
t
m
e
n
ω,
h
o
w
e
v
e
r,
t
h
e
r
ei
sac
e
r
t
a
i
nb
i
a
simposedbyt
h
emesho
r
i
i
fs
h
e
a
rs
t
r
e
s
s
e
sp
a
r
a
l
l
e
lt
ot
h
eo
v
e
r
a
l
lc
r
a
c
kb叫 dd
i
r
e
c
t
i
o
na
r
i
s
e
,the
e
n
t
a
t
i
o
n
.M
o
r
e
o
v
e
r,
z
i
g
z
a
gbandc
a
ni
n
t
r
o
d
u
ω
s
p
u
r
i
o
u
ss
h
e
a
.
rl
o
c
k
i
n
go
ft
h
eOPPωitef
a
c
e
s
.Thesep
r
o
b
l
emsc
a
n
∞mee
i
t
h
e
rbyi
n
もr
odudngan
o
n
l
o
c
a
lv
e
r
s
i
o
no
ft
h
emodel,
d
e
s
c
r
i
b
e
dl
a
句r
,
o
rby
beo
v
e
r
自∞n
t
i
n
u
o
u
sa
l
o
n
gt
h
eb
o
u
n
d
e
n
r
i
c
h
i
n
gt
b
ef
i
n
i
t
ee
l
e
m
e
n
te
i
t
h
e
rw
i
t
has
t
r
a
i
nf
i
e
l
dt
h
a
ti
sd
i
a
r
i
回 o
f姐 a
r
b
i
t
r
a
r
i
l
yo
r
i
e
n
t
e
db姐 d,
槌 proposedbyDroz(1987)f
o
rt
e
n
s
i
o
n,
(
s
i
m
i
l
a
rt
ot
h
e
m
o
d
e
l
sr
e
p
o
r
t
e
dbyO
r
t
i
ze
ta
l
.,
1
9
8
7
;LeroyandO
r
t
i
z,
1
9
8
9,
加dBelytschkoe
ta
l
.,
1
9
8
8f
o
r
s
h
e
a
rb姐 d
s
)o
rw
i
t
hd
i
s
p
l
a
c
e
m
e
n
tf
i
e
l
dd
i
s
c
o
n
t
i
n
u
o
u
sa
l
o
n
gal
i
n
e,
a
sp
r
o
p
o
s
e
dbyD
v
o
r
k
i
n
e
ta
1
.(
1
9
8
9
)
.
Thec
r
a
c
kbandmodelo
f
f
e
r
st
h
ep
o
s
s
i
b
i
l
i
t
yo
fi
n
t
r
o
d
u
c
i
n
gt
h
ei
n
s
u
e
n
c
eo
fno
n
s
i
n
g
u
l
a
r
q)
ε
4
σ
F
i
g.
3
.3E
f
f
e
c
to
fS
i
z
eont
h
eS
t
e
e
p
n
e
s
so
fPos
t
P
e
a
kS
o
f
t
e
n
i
n
g
(
α
)
(b)
F
i
g
.
3.
4M
eshest
oR
e
p
r
e
s
e
n
taVeryNarrowCrackBanda
n
daZ
i
g
Z
a
g
Band
39
38
t
h
r
e
e
d
i
m
e
n
s
i
o
n
a
ls
t
r
e
s
s
e
sonf
r
a
c
t
u
r
e
.F
o
rexample
,
i
ti
sknownt
h
a
tac
o
m
p
r
e
s
s
i
v
es
t
r
e
s
s
σ
z
p
a
r
a
l
l
e
lt
ot
h
ec
r
a
c
kp
l
a
n
ep
r
o
m
o
t
e
sc
r
a
c
k
i
n
g
.Based0 0t
h
eknowns
h
a
p
eo
fもheb
ia
x
:
i
a
lf
a
i
l
u
r
e
,
t
h
i
si
n
f
l
u
e
n
c
怠 (
f
o
rむ く 0
)maybet
a
k
e
ni
n
t
oa
c
c
o
u
o
tbyr
e
p
l
a
.
c
i
n
g
e
n
v
e
l
o
p
eo
fc
o
n
c
r
e
t
e
w
i
t
h
(
3
.
8
)
万=f
t
(
l+σ
,,/
f
D
:
f
(B
a
.
i
a
n
t,
1
9
8
5
a
)wheref~ = compr
白 s
i
o
ns
t
r
e
n
g
t
h
.F
o
rσ= 一f
疋~ (
c
o
m
p
r
e
s
s
i
o
nf
'
必
a
a
泊
i
l
l
町
u
l
開
r
e
),
.
on
whicha
g
r
問
w
i
t
hF
i
g
.3
.
9
. Note油 ot
h
a
t,
i
nc
o
n
t
r
a
.
s
t,
σ"h
剖 n
oe
f
f
e
c
t
musth
a
v
e斤 =0,
i
fl
i
n
e
a
re
l
a
s
t
i
cf
r
a
c
t
u
r
em民 h
a
n
i
c
si
su
s
e
d
;n
o
rwhenl
i
n
ec
r
a
c
km
o
d
e
l
sw
i
t
hs
o
f
t
e
n
i
n
gz
o
n
e
s
a
r
eu
s
e
d
.
a
r
g
eg
e
n
e
r
a
Jp
u
r
p
o
s
ef
i
n
i
t
ee
l
e
m
e
n
t
Thec
r
a
c
kbandmodelh
a
sb
e
e
ni
m
p
l
e
m
e
n
t
e
d泊 somel
∞des(
e
.
g
.DIANA,
TEMP-STRESS,
NONSAP).
3
.
2
.
2CrackLayerModel
Av
a
r
i
a
n
to
ft
h
ec
r
a
c
kbandmod
巴1
i
sr
e
p
r
e
s
e
n
t
e
dbyC
h
u
d
n
o
v
s
k
y
'
sc
r
a
c
kl
a
y
e
rmodel
(B
飽 s
e
n
d
o
r
f
fandChudnovsky
,
1
9
8
4
;Chudnovsky
,
1
9
8
6
)
.T
h
i
smodeli
smoreg
e
n
e
r
a
lt
h
叩 t
h
e
c
r
a
c
kba
.
ndmodeli
nt
h
a
tt
h
ewid
もho
f
t
h
ec
r
a
ε
kbandf
r
o
n
ti
sn
o
tc
o
n
s
i
d
e
r
e
dt
obeac
o
n
s
t
a
n
t
.
Ra
t
h
e
ri
ti
sa
l
l
o
w
e
dt
oexpando
rs
h
r
i
n
kb
a
s
e
donam
a
t
e
r
i
a
lp
r
o
p
e
r
t
y
.T
h
i
sp
r
o
p
e
r
t
yl
i
n
k
sa
c
e
r
t
a
i
np
a
t
h
i
n
d
e
p
e
n
d
e
n
ti
n
t
e
g
r
a
lt
h
a
.
tc
h
a
r
a
c
t
e
r
i
z
.
田 t
h
ee
n
e
r
g
yc
h
a
n
g
eduet
ot
h
ee
x
p
a
n
s
i
o
n
r
a
c
kf
r
o
n
twhichc
h
a
r
a
c
t
e
r
i
z
e
st
h
ee
n
e
r
g
y
t
oa
n
o
t
h
e
rp
a
t
h
i
n
d
e
p
e
n
d
e
n
ti
n
t
e
g
r
a
la
r
o
u
n
d出 ec
f
l
o
wi
n
t
ot
h
ec
r
a
c
kf
r
o
n
tduet
oc
r
a
c
kp
r
o
p
a
g
a
.
t
i
o
n(
町c
e
'
sJ
i
n
t
e
g
r
a
l
)
.
T
h
i
smodel,
how
e
v
e
r,
h
a
sn
o
tb
e
e
nd
e
v
e
l
o
p
e
df
o
ra
p
p
l
i
c
a
t
i
o
ni
n∞n
c
r
e
t
eandh
a
sn
o
tb伺 n
comp
a
r
e
dt
ot
y
p
i
c
a
lt
e
s
td
a
t
af
o
rc
o
n
c
r
e
t
e
. Thep
r
o
p
e
r
t
i
e
sg
o
v
e
r
n
i
n
gt
h
ebande
x
p
a
n
s
i
o
n
n
u
g
h
tbeu
n
i
d
e
n
t
i
f
i
a
b
l
efrom出 ee
x
p
e
r
i
m
e
n
t
a
ld
a
t
aa
tt
h
ec
u
r
r
e
n
td
e
g
r
e
eo
fs
o
p
h
i
s
t
i
c
a
t
i
o
n
o
ft
e
s
t
i
n
gr
o
e
t
h
o
d
s
.
3
.
2
.
3CompositeFr
actureModel
Thec
o
m
p
o
s
i
t
ef
r
a
c
t
u
r
emodeli
se
s
s
e
n
t
i
a
l
l
yana
d
a
p
t
a
t
i
o
no
ft
h
ec
r
a
c
kb
組 dm
odelt
o
f
i
n
i
t
ee
J
e
m
e
n
t
so
fal
a
r
g
e
rs
i
配 t
h
a
nt
h
ec
r
a
c
kbandw
i
d
t
hW c,
t
h
u
sp
e
r
m
i
t
t
i
n
gl
a
r
g
es
t
r
u
c
t
u
r
e
s
t
obea
n
a
l
y
z
e
dw
i
t
hac
r
u
d
e
rm
e
s
h
. Themodela
s
s
u
m
e
st
h
a
tac
r
a
c
kba
.
n
do
fw
i
d
t
hωc,
w
h
i
c
hi
sam
a
t
e
r
i
a
lp
r
o
p
e
r
t
y(
町 ane
q
u
i
v
a
l
e
n
tl
i
n
ec
r
a
c
k
)i
sembeddedw
i
t
h
i
na1
釘 g
e
rf
i
n
i
t
e
a
p
p
r
o
x
i
m
a
t
e
l
y酪 i
l
l
u
s
t
r
a
t
e
di
nF
i
g
.3
.
3
d
.T
h
i
si
d
伺
, whichi
ss
i
n
u
l
a
rt
oE
q
.3
.
8,
wω
e
l
e
m
e
n
t,
p
r
o
p
o
s
e
df
i
r
s
tf
o
rp
l
拙 t
i
cs
h
e
a
rba
.
nd
sbyP
i
e
t
r
u
s
z
c
z
a
kandMroz(
1
9
7
9
)andl
a
t
e
rd
e
v
e
l
o
p
e
d
f
o
rt
e
n
s
i
l
ec
r
a
c
k
i
n
go
fc
o
n
c
r
e
t
ebyWillame
ta
.
l(
1
9
8
5,1
9
8
6
)
.I
ft
h
ee
l
e
m
e
n
ts
i
z
es
h
r
i
n
k
s
t
h
i
smodelbecomese
s
s
e
n
t
i
a
l
l
yi
d
e
n
t
i
c
a
It
ot
h
ec
r
a
c
kbandmode
l
. Thec
r
a
c
kband
toωc,
w
i
t
h
i
nt
h
ef
i
n
i
t
ee
l
e
m
e
nもi
si
n
t
r
o
d
u
c
e
dbye
n
r
i
c
h
i
n
gt
h
esmoothd
i
s
t
r
i
b
u
t
i
o
nf
u
n
c
t
i
o
nw
i
t
ha
.
d
i
s
c
o
n
t
i
n
u
o
u
ss
t
r
a
i
nf
i
e
J
df
u
n
c
t
i
o
nw
h
i
c
hc
h
a
r
a
.
c
t
e
r
i
z
e
sc
r
a
ε
k
i
n
g
.The乱p
p
r
o
a
c
hi
ss
i
n
u
l
a
rt
o
t
h
o
s
ebyDroz(
1
9
8
7
),
O
r
t
i
ze
ta
J
.(
1
9
8
7
),
B
e
l
y
t
s
c
h
k
oe
ta
l
.(
1
9
8
8
),
L
e
r
o
yandO
r
t
i
z(
1
9
8
9
)
叫
dD
v
o
r
k
i
ne
ta
l
.(
1
9
8
9
)a
l
r
e
a
d
ym
e
n
t
i
o
n
e
da
tt
h
eendo
fS
e
c
.3
.
2
.1
.
3
.
2
.
4NoniocalContinuumwithLocalS
t
r
a
i
n(NonlocalDamage)
An
o
n
l
o
c
a
lcontinuumi
sac
o
n
t
i
n
u
u
mi
nw
h
i
c
hsomes
t
a
t
ev
a
r
i
a
b
l
ed
e
p
e
n
d
sn
o
to
n
l
yon
t
h
es
t
r
e
s
s
田 o
rs
t
r
a
i
n
sa
tt
h
esamep
o
i
n
tb
u
ta
l
s
oont
h
es
t
r
e
s
so
rs
七r
a
i
n五e
l
di
nt
h
en
e
i
g
h
b
o
r
hoodd t
h
epoint.Asshownbyk
r
i
s
n
e
r
(
1
9
6
7
),
kmmhand(1968),
kmin(
1
9
6
8
),
E
r
i
n
g
e
n
姐 dE
d
e
l
e
n
(
1
9
7
2
),
ando
t
h
e
r
s(cf.Bdan
t
,1986)
,thenonlocalconcepti
sa
p
p
r
o
p
r
i
a
t
ef
o
r
s
t
a
t
i
d
i
c
a
l
l
yh
e
t
e
r
o
g
e
n
e
o
u
sm
a
t
e
r
i
a
l
sandf
o
l
l
o
w
sl
o
g
i
c
a
l
l
yfroms
t
a
b
i
s
t
i
c
a
lm
i
c
r
o
m
e
c
h
a
n
i
c
a
l
∞nsidera
.
t
i
o
n
s
. R怠 切n
t
l
y
,thenonloca
.
la
p
p
r
o
a
c
hh
a
sb
e
e
nshownt
obev
e
r
ye
f
f
e
c
t
i
v
ef
o
r
企a
c
t
u
r
er
o
e
c
h
姐 i
c
so
fd
i
s
t
r
i
b
u
t
e
dc
r
a
c
k
i
n
g
.
Then
o
n
l
o
c
a
lc
o
n
c
e
p
tr
e
p
r
e
s
e
n
t
sag
e
n
e
r
a
la
p
p
r
o
a
c
hw
h
i
c
hmakesi
tp
o
s
s
i
b
l
et
ou
s
es
t
r
e
s
s
a
.
i
nr
e
l
a
t
i
o
n
sw
i
t
hs
t
r
a
i
n
s
o
f
t
e
n
i
n
g
. Asm
e
n
t
i
o
n
e
db
e
f
o
r
e
,i
ft
h
e5もr
a
i
n
s
o
f
t
e
n
i
n
g∞n
c
e
p
t
s
t
r
,
c
a
u
s
i
n
R
i
si
n
t
r
o
d
u
c
e
di
nt
h
el
o
c
a
lc
o
n
s
u
l
1
1
1
1
I
R
s
o
f
t
e
n
i
n
gz
o
n
e
sl
o
c
a
l
i
z
et
oav
a
n
i
s
h
i
n
gvolume
i
m
p
r
o
p
e
r∞n
v
e
r
g
e
n
明 o
nmeshr
e
f
i
r
悶 n
e
n
s,
s
p
u
r
i
o
u
smdEseMtivityandp
h
y
s
i
c
a
l
l
y
i
n
c
o
r
r
e
c
t
p
r
e
d
i
c
t
i
o
n
ss
u
c
ha
sf
a
i
l
u
r
ew
i
t
hv
a
n
i
s
h
i
n
ge
n
e
r
g
yd
i
s
s
i
p
a
t
i
o
n舗 もhemeshs
i
z
ei
sr
e
f
i
n
e
dt
o
z
e
r
o
. Toa
v
o
i
ds
u
c
hb
e
h
a
v
i
o
r,
t
h
ecomput
抗i
o
n
a
lmodelmusti
n
c
l
u
d
esomema
.
thema
t
i
c
a
l
d
e
v
i
田 t
h
a
.
tI
i
m
i
t
st
h
el
o
c
a
l
i
z
a
t
i
o
n(
Ba
.
z
a
n
t,
Be
l
y
t
s
c
h
k
oandC
1
加 1
9
,1984). Thesimplest,
a
l
b
e
i
tc
r
u
d
田 t
,
wayt
ol
i
n
u
tl
o
c
a
l
i
z
a
t
i
o
ni
st
oi
m
p
o
s
eaf
i
x
e
ds
i
z
e,
ωc,ont
hef
r
o
n
t
a
lf
i
n
i
t
e
e
l
e
m
e
n
t,
a
sdonei
nt
h
ec
r
a
c
kba
.
ndmodel(
u
n
l
e
s
st
h
ep
o
s
t
p
e
a
ks
t
r
e
s
s
s
t
r
a
i
nr
e
l
a
t
i
o
ni
s
宣c
i
a
l
l
ym
o
d
i
f
i
e
d
)
.However,
t
h
i
sa
p
p
r
o
a
c
hmakesi
ti
m
p
o
s
s
i
b
J
et
or
e
s
o
l
v
et
hef
i
e
l
di
nt
h
e
a
r
t
i
f
r
a
c
t
u
r
ep
r
o
α出 z
o
n
ei
t
s
e
l
f
,o
rd
e
t
e
r
n
u
n
ehowt
hef
r
a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
ew
i
d
t
hc
o
u
l
dv
a
r
y
c
t
u
r
eg
r
o
w
t
h(
n
om
o
d
i
f
i
c
a
t
i
o
no
fs
t
r
e
時 町a
i
nr
e
l
a
t
i
o
n
sc
o
u
l
dbep
e
r
m
i
も
tdf
O
}
d
u
r
i
n
g山 e企a
告
白
)
.
t
h
o
s
epurpo
A moreg
e
n
e
r
a
lwayt
ol
i
m
i
tl
o
c
a
l
i
z
a
t
i
o
n
,i
no
r
d
e
rt
oc
o
p
ew
i
s
hs
u
c
hp
r
o
b
l
e
m
s,
i
st
h
e
n
o
n
l
o
c
a
lc
o
n
t
i
n
u
u
m
.Thec
o
n
c
e
p
tmaybei
n
t
r
o
d
u
c
e
di
nv
a
.
r
i
o
u
sf
o
r
m
s,
s
u
c
h剖 a
v
e
r
a
g
i
n
go
f
s
t
r
a
i
n
so
rs
t
r
a
i
n
r
e
l
a
t
e
dqu
姐 t
i
t
i
田 o
v
e
ran
e
i
g
h
b
o
r
h
o
o
do
fap
o
i
n
t(
Bazant,
1
9
8
4
b
;Ba
.
i
a
n
t,
B
e
l
y
t
s
c
h
k
o叩 dChang
,
1
9
8
4
),
o
ri
山 吋u
c
t
i
o
no
f白 川 姐ds
e
c
o
n
d
s
p
a
t
i
a
lf
l
e
d
U
t
i
v
e
s
O
f町 ams
o
ro
t
h
e
rqu
岨 t
i
t
i
e
si
n
t
ot
h
ec
o
n
s
t
i
t
u
t
i
v
er
e
l
a
t
i
o
n(eιSchreyer岨 dChen,1
9
8
6;andf
o
r
机 m
ediumf
o
r
m
u
l
a
t
i
o
nf
o
rr
o
c
k
sbySuJem組 dVardo
由,
k
i
s,
1988)Thes
p
a
t
i
a
l
t
h
eC
o
s
s
e
r
も
i
o
nc
a
nbeo
b
t
a
i
n
e
dbye
x
p
a
n
d
i
n
g出 es
p
a
t
i
a
Ja
.
v
e
r
a
g
i
n
gi
nt
e
g
r
a
li
nT
a
y
l
o
r
d
e
r
i
v
a
t
i
v
ef
o
r
m
u
l
a
s
e
r
i
e
s
. Onlyt
h
es
p
a
t
i
a
l.
a
v
e
r
a
g
i
n
ga
p
p
r
o
a
c
hh倒 的 f
a
rb
四 nw
o
r
k
e
do
u
ti
nd
e
t
a
i
lf
o
r凶e
c
r
e
t
e
.
m
u
l
t
i
d
i
m
e
n
s
i
o
n
a
la
n
a
l
y
s
i
so
ff
r
a
c
t
u
r
eo
fm
a
t
e
r
i
a
l
ss
u
c
h出∞n
s
p
a
t
i
a
la
v
e
r
a
g
i
n
gwasa
p
p
l
i
e
dt
h
r
o
ught
h
et
o
t
a
ls
t
r
a
i
n
sus
e
di
n
1
nt
h
ee
a
r
l
yf
o
r
m
u
l
a
t
i
o
n,
t
h
a
もa
p
p
r
o
a
c
h,
w
h
i
c
hr
e
s
u
l
t
e
di
n
S
h
es
t
r
a
i
n
s
o
f
t
e
n
i
n
gc
o
n
s
t
i
t
u
t
i
v
er
e
l
a
t
i
o
n
.
H
o
w
e
v
e
r,
叫
a
l
1
凶
e
吋di
帥
m
b
伽
凶
r
i
c
臼a
t
旬e
c
∞
o
凶
n
凶
t
i
泊
n削
u
1
1
ha
刊
v
es
叩o
m
ロ
'
l
e∞
c
omp叫
u
l
比
凶
t
d
叫
a
品
t
i
o
n
a
l
l
yi
n
c
o
n
v
e
n
i
e
n
tp
戸r
叩
o
pe
釘r
凶
t
l
郎
e
8
.R
e
c
e
n
t
l
y(
Baian
tandP
i
j
a
u
d
i
e
r
-Cabot,
1
9
8
7,1
9
8
8
a
;P
i
j
a
u
d
i
e
r
C
a
b
o
t組 dBdmt,1987;BatantandL
i
n
,1988a, 邸
川
,i
tw
f
o
u
n
dt
h
a
tamoreeRectiveapproachi
st
oc
o
n
s
i
d
e
r
b
i
l
e
e
l
a
s
t
i
cs
t
r
a
i
n
sωlocal(
n
o r
a
g
i
n
g
),
a
干
da
p
p
l
yt
h
en
o
n
l
o
叫a.
v
e
r
叫ngonlytothesofteningpartoftheは 凶nswhichrepmejt
d
l
s
t
r
i
b
u
t
e
dm
i
c
r
o
c
r
a
z
k
i
n
g
T
h
I
Snewa
p
p
r
o
a
c
hh
a
st
h
eadvan
t
a
g
et
ha
tt
h
econtinuumhid
o
n
d
i
t
i
o
n
sr
e
m
a
i
nt
h
esame酪 f
o
rt
h
e
e
q
u
a
t
i
o
n
so
fe
q
u
i
l
i
b
r
i
u
ma
sw
e
l
la
s出 eboundaryc
c
“
“
h
…ム
40
41
c
l
a
s
s
i
c
a
ll
o
c
a
lcontinuum
,
andt
h
a
tt
h
e
r
ei
snop
o
s
s
i
b
i
l
i
t
yo
fz
e
r
o
e
n
e
r
g
yi
n
s
t
a
b
i
l
i
t
ymodes
均 shownt
oe
x
i
s
tf
o
rf
u
l
l
yn
o
n
l
o
c
a
lm
o
d
e
l
s
)
.I
th
a
sa
l
s
ob
e
e
nmathema
もi
c
a
l
l
yp
r
o
v
e
n
(
p
r
e
v
i
o
u
出品 t
h
ea
v
e
r
a
g
i
n
go
ft
h
ef
r
a
c
t
u
r
i
n
gp
a
r
to
ft
h
es
t
r
a
i
n
si
ss
u
伍c
i
e
n
tt
op
r
e
v
e
n
tl
o
c
a
l
i
z
a
t
i
o
n
o
fs
t
r
a
i
n
s
o
f
t
e
n
i
n
gdamagei
n
t
oaz
o
n
eo
fz
e
r
ovolume.Theu
n
l
o
a
d
i
n
g
r
e
l
o
a
d
i
n
gb
e
h
a
v
i
o
ri
s
a
l
w
a
y
sl
o
c
a
li
nt
h
i
sa
.
p
p
r
o
邸 h
.
.
a
t
i
a
.
la
v
e
r
a
g
i
n
go
p
e
r
.
a
t
o
r
,
d
e
n
o
t
e
dby姐 o
v
e
r
b
a
r,
maybed
e
f
i
n
e
di
nt
e
r
m
so
fs
t
r
a
.
i
n
Thes
p
e
l
la
sf
o
l
l
o
w
s
:
l
t
l
l
(
X
)= Y
T T~__\
(
s-x
)
{
f
l
l
(
s
)
}
dV
L
'
(
x-S
)
(
E
1
1
(
S
)
)
d
V
r(x)Jvα
¥
,
,
.
.
¥
,
,
.=J
vα
(
3
.
9
)
I
_
'
1
_
_
¥
.
.
I
t
I
"
_
I
_ _¥
_α
(
s-x
V
r(x)= l
_0
'
(
8-x)dV
,α
(x-s
)一一
一
一一)
V
r(
X
)
(
3
.
1
0
)
(
f
1
1
)=ε11i
fE
l
l>0, (
1
:
1
1
)=0i
ff
1
1壬O
(
3
.
1
1
)
i
nw
h
i
c
h
Jv-¥
-
--r"
,
-,
-
1
. 1
8-xl¥2
x
)= 1一 寸e
~I) 時- x
l<k
l
,otherwiseα(8-x)=0
t
z
』
s-x
!
0
x
1
i
1k2-'_kl_
ft
h
es
t
r
u
c
t
u
r
e
,
x=c
o
o
r
d
i
n
a
t
ev
e
c
t
o
r,
s=c
o
o
r
d
i
n
a
t
ev
e
c
t
o
r
so
fa
d
j
a
c
e
n
t
H
e
r
eV=volumeo
p
o
i
n
t
s,
α =g
i
v
e
nw
e
i
g
h
t
i
n
gf
u
n
c
t
i
o
no
ft
h
ed
i
s
t
組 c
eI
s-x
!,
c
o
n
s
i
d
e
r
e
dもob
eam
a
t
e
r
i
a
l
,
.hasapproximately,
b
u
tn
o
te
x
a
c
t
l
y
,
t
h
esamem田 n
i
n
g槌 t
h
er
e
p
r
争
p
r
o
p
e
r
t
y(
F
i
g
.3
.
5
)
.v
s
e
n
t
a
t
i
v
evolumei
nt
h
es
t
a
t
i
s
t
i
c
a
lt
h
e
o
r
yo
fh
e
t
e
r
o
g
e
n
e
o
u
sm
a
t
e
r
i
a
l
s
.
n
i
f
o
r
m(α=1
)o
v
e
rvolumev
,
.r
e
p
r
e
s
e
n
t
i
n
g
Thew
e
i
g
h
t
i
n
gf
u
n
c
t
i
o
nc
o
u
l
db
ed
e
f
i
n
e
d回 u
a
.c
i
r
c
l
ei
ntwod
i
m
e
n
s
i
o
n
s,
ab
a
rs
e
g
m
e
n
ti
noned
i
m
e
n
s
i
o
n
,
o
ras
p
h
e
r
ei
nt
h
r
e
ed
i
m
e
n
s
i
o
n
s,
w
i
t
haz
e
r
ov
a
l
u
eo
u
t
s
i
d
eVr • However,
i
th
a
sb
e
e
ne
x
p
e
r
i
e
n
c
e
dt
h
a
tt
h
ec
a
l
c
u
l
a
t
i
o
n
sc
o
n
v
e
r
g
e
自s
i
b
l
ec
h
o
i
c
e(
B
a
z
a
n
t組 dP
i
j
a
u
di
e
r
・C
abot,
b
e
t
t
e
ri
ft
h
ew
e
i
g
h
t
i
n
gf
u
n
c
t
i
o
ni
ss
m
o
o
t
h
.Onepo
1
9
8
7,
1
9
8
8
a
;Baz
組 t
andL
i
n
,
1
9
8
8
b
)i
saG
a
u
s
s
i
a
nd
i
s
t
r
i
b
u
t
i
o
nf
u
n
c
t
i
o
n,
which
,
h
o
w
e
v
e
r,
h
a
s
ts
e
e
m
sp
r
e
f
e
r
a
b
l
et
ou
s
eap
o
l
y
n
o
m
i
a
lb
e
l
l
s
h
a
p
e
d
n
o
n
z
e
r
ov
a
l
u
e
so
v
e
rt
h
ee
n
t
i
r
es
t
r
u
c
t
u
r
e
.I
e
.
g
.(
F
i
g
.3.
5
)
f
u
n
c
t
i
o
nw
h
i
c
hi
se
x
a
c
t
l
yz
e
r
obeyondac
e
r
t
a
i
nd
i
s
t
a
n
c
e,
α
(
ト
(0)
(
3
.
1
2
)
sanorm
a
.
l
i
z
i
n
gc
o
n
s
t
a
.
n
td
e
t
e
r
u
u
n
e
d50t
h
a
tt
h
ei
n
t
e
g
r
a
lo
fαover品 l
i
n
es
e
g
m
e
n
t,
H
e
r
eki
o
ra
.c
i
r
c
l
e,
o
ras
p
h
e
r
eo
fs
i
z
ek
lwouldbet
h
esame舗 もh
ei
n
t
e
g
r
a
lo
fα =1o
v
e
rt
h
esame
r
e
g
i
o
n
jk= 1
5
/
1
6= 0
.
9
3
7
5f
o
rl
D
,(
3
/
4
)
1
/
2= 0
.
9
0
8
6f
o
r2D,k= (
1
0
5
/
1
9
2
)
1
/
3= 0
.
8
1
7
8
.
r
a
.
m
e
t
e
rli
st
h
es
o
c
a
l
1
e
dc
h
a
.
r
a
c
t
e
r
i
s
t
i
cl
e
n
g
t
ho
ft
h
en
o
n
l
o
c
a
lc
o
n
t
i
n
u
町n
,
f
o
r3
D
.Thepa
w
h
i
c
hi
sam
a
t
e
r
i
a
lp
r
o
p
e
r
t
yt
h
a
td
e
f
i
n
e
st
h
es
i
z
eo
f出 ea
v
e
r
a
g
i
n
gvolumeandi
sd
e
t
e
r
m
i
n
e
d
byt
h
es
i
z
eo
ft
h
ei
n
h
o
m
o
g
e
n
e
i
t
i
e
si
nt
h
em
i
c
r
o
s
t
r
u
c
t
u
r
e
.F
o
ro
n
eq
u
i
t
et
y
p
i
c
a
lc
o
n
c
r
e
t
e,
m伺 s
u
r
e
m
e
n
t
s(
B
a
i
a
n
tandP
i
j
a
u
d
i
e
r
・C
abot,
1
9
8
8
b
)haveshownt
h
a
t{=2
.
7
d
.
.whered
.
.=
maximuma
g
g
r
e
g
.
a
t
es
i
z
e(
S
e
c
.6
.
9
)
.
叩 o
ft
h
ebody
,p紅色 o
ft
h
ea
.
v
e
r
a
g
i
n
gz
o
n
ep
r
o
t
r
u
d
e
s
I
fp
o
i
n
txi
st
o
oc
l
o
s
et
ot
h
es
u
r
f
a
beyondt
h
es
u
r
f
a
偲. T
h
i
si
sh
a
n
d
l
e
dbyd
e
l
e
t
i
n
gt
h
ep
r
o
t
r
u
d
i
n
gp
a
r
tfromt
h
ei
n
t
e
g
r
a
t
i
o
n
もi
o
nc
a
u
s
e
sv
,
.(
x
)t
od
e
p
e
n
donl
o
c
a
t
i
o
nx
.
r
e
g
i
o
n
.T
h
i
sd
e
l
e
t
h
ei
凶e
g
r
a
l
si
nE
q
.3
.
1
0阻 d3
.
1
1a
.a
r
ea
p
p
r
o
x
i
m
a
t
e
l
y
I
nf
i
n
i
t
ee
l
e
m
e
n
tc
a
l
c
u
l
a
t
i
o
n
s,
e
v
a
l
u
a
t
e
d舗 f
i
n
i
t
esumso
v
e
ra
l
lt
h
ei
n
t
e
g
r
a
.
t
i
o
np
o
i
n
も
so
fa
l
lもhee
l
e
m
e
n
t
si
nt
h
es
t
r
u
c
t
u
r
e
.
F
i
g
.
3
.
5W
e
i
g
h
t
i
n
gFunc
t
i
onf
o
rN
o
n
-Loc
a
lCo
ntinuumTbeoryandRe
p
r
e
s
e
n
t
a
t
i
v
eVolumeo
ft
h
eM
a
t
e
r
i
a
l
、‘、
、
、.
-、
0
.
3七
A
ー
。c
o
n
c
r
e
t
e
0-
m
o
r
t
ar
ト-ーご竺丸、
ユ
ー
0
.
5
、
、
、
E
E・0.7J
出
0
.
0
.
4
0
.
8
1
.2
1
.6
l
o
g
{
d
/
da)
F
i
g
.
3
.
6
S
i
z
eE
R
e
c
tC
a
l
c
u
l
a
t
e
dbyN
o
n
L
o
c
a
lContinuumT
h
e
o
r
y(
B
a
t
a
n
t
,1988b)forExponential(
d
a
s
h
e
dcurve)andL
i
n
e
a
r(
d
a
s
hd
o
t
andL
i
n
,
Comparedw
i
t
hS
i
z
eE
R
e
c
tLaw
v
e
)
T
e
n
s
i
l
eS
t
r
a
i
nS
o
f
t
e
n
i
n
gC
u
r
v
e
s
s
o
l
i
dcurve)andw
i
t
ht
e
s
tr
e
s
u
l
t
so
fBat
a
n
t組 d
o
fBasant(Eq.1.4,
a
.
t
ap
o
i
n
t
s
)
P
f
e
i
f
f
e
r(
1
9
8
7
)(
d
42
43
,
t
h
ei
n
t
e
g
r
a
t
i
o
np
o
i
n
t
swhosed
i
s
t
日 c
e
sf
r
o
mp
o
i
n
txe
x
c
e
e
dk
lmaybeo
m
i
t
t
e
d
However
fromt
h
esums
i
n
c
ef
o
rthemα=0.
I
d
e
a
l
l
y
,出ef
ormo
ff
u
n
c
t
i
o
n
α
(
s-x
)s
h
o
u
l
dbed
e
r
i
v
a
b
l
efrommicromecha
且i
c
a
.
l
m
o
d
e
l
i
n
g
o
ft
h
em
i
c
r
o
s
t
r
u
C
h
u
r
ew
i
t
ha
g
g
r
e
g
a
t
e
sandm
i
c
m
c
r
a
c
k
s,
samea
st
h
eformo
ff
u
n
c
t
i
o
n
O
(
ε
)i
n
,
t
h
i
sg
o
a
li
st
o
odimcuuatp
r
e
s
e
n
t,
w
h
i
l
ea
tt
h
esamet
i
m
et
h
e
E
q
.3.4姐d3.6.However
D
r
e
c
I
s
ef
o
r
mo
ff
u
n
c
t
i
o
n
α
(
s-x
)h
a
so
n
l
yl
i
t
t
l
ei
n
f
l
.
u
e
n
c
eonωmput
a
.
t
i
o
nr
e
s
u
l
t
s(
出i
sf
.
a
ι
b
即
tionαfromt
制 d
a
.
t
.
ap帥 lem
a
.
t
i
c
)・ Whati
si
m
p
o
r
t
a
n
ti
st
h
e
LakesideMiaωtiono
ff
a
l
u
eo
fi(
p
r
o
v
i
d
e
dt
h
ebodyi
sn
o
tも∞ l
a
r
g
ecomparedt
ol
)
.
ThesimpledEtonlocalmlSUMMErlodel,
appropmheforu
n
i
d
i
r
ω
i
o
n
a
lc
r
a
c
k
i
n
g,
i
so
b
t
a
i
n
e
dbyg
e
n
e
r
a
l
i
z
i
n
gt
h
eC
E
a
c
kbandmodel(B
話 相ta
ndL
i
n
,
1
9
8
8
b
)・ Onea
.
l
t
e
m
a
t
i
v
ei
st
o
,
t
h
e
M m et
h
ec
r
a
c
kd
i
r
e
c
t
i
o
nt
obeaxeda
tt
h
et
i
m
et
h
ec
r
a
c
k
ss
t
a
r
tt
oform.Int
h
i
scue
mnlodgeneralizationi
so
b
t
a
i
n
e
dbyreplacingωinE
q
.3.
4w
i
t
hn
o
n
l
o
叫 d
amageρ,
s
u
c
h
Ani
n
t
e
r
e
s
t
i
n
g
, 阻dp
o
t
e
n
t
i
a
l
l
yu
s
e
f
u
lp
r
o
p
e
r
t
yo
ft
h
en
o
n
l
o
c
a
lf
o
r
m
u
l
a
t
i
o
ni
st
h
a
tt
h
e
r
s
eo
fl
o
a
d
i
n
g
. The
w
i
d
t
ho
ft
h
ec
r
a
c
k
i
n
gz
o
n
ei
sn
o
tf
i
x
e
db
u
tv
a
r
i
e
sd
u
r
i
n
gt
h
e∞u
c
o
n
s
e
q
u
e
n
田 i
st
h
a
tt
h
en
o
n
l
o
c
a
lf
r
a
c
t
u
r
emodeld
o
e
sn
o
tc
o
r
r
e
s
p
o
n
dt
oau
n
i
q
u
es
t
r
e
s
s
d
i
s
p
l
a
c
e
m
e
n
td
i
a
g
r
a
mf
o
rt
h
el
i
n
ec
r
a
c
kmodel五
(c
t
i
t
i
o
u
sc
r
a
c
k
)
.T
h
i
sf
a
c
tmighte
却l
a
i
n
whyi
th
a
sp
r
o
v
e
dr
a
t
h
e
rd
i
f
f
i
c
u
l
tt
oo
b
t.
a
i
nu
n
i
q
u
ev
a
l
u
e
so
ft
h
ef
r
a
c
t
u
r
ee
n
e
r
g
yont
h
eb
8
0
s
i
s
o
ft
h
ef
i
c
t
i
t
i
o
u
sc
r
a
c
kmode
.
l
自o
ben
o
t
e
dt
h
a
tal
a
r
g
ef
i
n
i
t
ee
l
e
m
e
n
tp
r
o
g
r
a
.
mf
o
rt
h
en
o
n
l
o
c
a
lf
r
8
0
c
t
u
r
emodel
I
ts
h
o
u
l
da
l
I
l
yr
u
n
sf
a
s
t
e
rt
h
a
nt
h
ec
o
r
r
e
s
p
o
n
d
i
n
gl
o
c
a
lp
r
o
g
r
a
.
m
,d
e
s
p
i
t
et
b
ef
a
c
tt
h
8
0
t8od
d
i
t
i
o
na
.
l
u
s
u
a
∞mputertimei
sn
e
e
d
e
dもoc
a
l
c
u
l
a
t
et
h
es
p
a
t
i
a
l晶.
v
e
r
8
o
g
回 .T
her
同 s
ona
p
p
a
r
e
n
t
l
yi
st
h
a
も
もh
e
a
.
la
.
v
e
r
a
g
i
n
gs
t
a
.
b
i
l
i
z
e
st
h
er
e
s
p
o
n
s
e
,t
h
u
sc
晶u
s
i
n
gt
h
ei
t
e
r
a
.
t
i
o
n
si
nt
h
el
o
a
d
i
n
gs
t
e
p
s
n
o
n
l
o
c
t
oc
o
n
v
e
r
g
ef
a
s
t
e
r.
t
h
a
t
3.2.5 MultidirectionalCrackingandNontensileFr
acture
w
t1=cp(fI)
(
3
.
1
3
)
(
1-
.
l
山a
i
n臼 l
c
u
l
8
0
t
e
dfromt
h
en
o
n
l
o
c
a
l(
a
v
e
r
a
g
e
d
)
wheref
Ii
st
h
emaximumn
o
n
l
o
c
a
lp
r
i
n
c
i
pa
s
t
r
a
.
i
nt
e
n
s
o
rf
i
;
氏∞
c
∞
oI
n
J
1
Thes
e
s
叩u
medt
句or
o
t
a
.
t
ew
叩i
t
出
ht
h
emaximump
r
i
n
c
i
p
a
ls
t
r
e
s
s,
i
so
b
t
a
i
n
e
dbyr
e
p
l
a
c
i
n
gωwithwi
n
Eq.3.6
I
th
i
sb
e
e
nverihd畑 山n
tandL
i
n,1
9
8
8
b
)t
h
a
tt
h
i
sn
o
n
l
o
c
a
lg
e
n
e
叫 i
z
a
t
i
o
no
ft
h
e
c
r
a
c
kbandmodelロ閃e
r
m
首
i
i
t
日
s伽
t
h
eu
s
問
巴
eo
fh
脳
伝
f
i
n
泊
1
吋
i
向
s
叩
pu
巾 u
凶 sr
r
附
hs
鈴e
n
E
比
t
i
v
吋
i
i
t
句
yandi
m
p
r
o
p
e
rc
o
n
v
e
r
g
e
n
侃
F
o
re
l
e
m
e
nts
i
z
e
sno自m
a
l
l
e
rt
h
a
nl
,
t
h
en
o
n
l
o
c
a
l
m
o
d
e
lbecomesi
d
e
n
t
i
c
a
lt
ob
i
l
ec
r
a
£kb
andmodel
,
aadt
h
eskesE4rainE
e
l
a
t
i
o
n
f
o
rt
h
ef
i
n
i
t
ee
l
e
m
e
n
twhichu
n
d
e
r
g
o
田 c
r
a
c
k
i
n
gmustbea
d
j
u
s
t
e
da
c
∞rdingtotheelem
田も
s
i
z
e
ー
却
a
.
l
r
e
a
d
ye
x
p
l
a
i
n
e
d
.
i
th
a
sa
l
s
ob
e
e
nshownt
h
a
tt
h
enonlodmmedcmkingm
o
d
e
l
p
r
o
v
i
d
e
s祖 e
x
c
e
l
l
e
n
t
批 o
fv
a
r
i
o
u
sf
r
a
d
u
r
et
e
s
tdata-Seee
.
g
.,
Fig.3.6,
i
nwhichdIet
e
s
tr
e
s
u
l
t
sfromBdaab
andP
f
e
i
f
f
e
r(
1
9
8
7
)a
r
ecomparedw
i
出 n
o
n
l
o
ca
.
lf
i
n
i
t
ee
l
e
m
e
n
tr
e
s
u
l
t
so
fBaZa
n
tandL
i
n
回 h
ed組 dd
a
s
h
d
o
tc
u
r
v
e
唱
)
叫dw
i
t
ht
h
es
i
田 e
f
f
e
c
tlaw(
E
q
.1
ムsolidcurve)(the
(
1
9
8
8
b
)(
d
d
a
s
h
e
dc
u
r
v
ec
o
r
r
e
s
p
o
n
d
sもoas
o
f
t
e
n
i
n
gt
e
n
s
i
l
e山 倍 付 も 問nr
e
l
a
t
i
o
ni
n.
a
ne
x
p
o
n
e
n
t
i心 form
,
whichf
i
t
sもheda
.
t
ab
e
t
t
e
rt
h.
a
n出 eda
.
s
hd
o
tc
u
r
v
ew
h
i
c
h∞r
r
凶 p
o
n
d
st
oa
.l
i
n
e
a
rs
t
r
e
s
s
s
t
r
a
.
i
n
r
e
l
a
t
i
o
n
;
s
t
i
l
lb
e
t
t
e
ra
t
sc
o
u
l
d
p
r
o
b
a
b
l
y
b
e
o
b
t
a
i
n
e
dw
i
t
ho
b
h
e
rs
h
a
p
e
so
f
t
h
i
sr
e
l
a
t
i
o
n
)・Fr
om
Fig.35its
h
o
u
l
dbee
s
p
e
c
i
a
l
l
yn
o
t
e
dもh
8
0
tan
o
n
l
o
c
a
lf
i
n
i
t
ee
l
e
m
e
n
tc
o
d
ee
x
h
i
b
i
t
st
h
ec
o
r
r
e
c
t
剖 a
句p
戸
pr
似
o
X
l
凶
r
m
町】
胤
a
凶
t
.
e
l
yd
e
s
c
r
i
b
e
dbyt
h
e
t
r
a
n
s
i
t
i
o
n
a
.
ls
i
z
ee
f
f
e
c
to
fn
o
n
l
i
n
e
a
rf
r
a
c
t
u
r
em
e
c
h
a
n
i
c
s,
s
討i
z
ee
f
晶
T
島
'
e
c
tlawp
戸r
叩
o
p伺
o
s
e
吋
:
db
句yBa
おE
叫
も
1
北
ι
.By∞
c
o
n
t
r
a
泌
.
s
t,
も
山
h
el
o
c
a
lf
貧
i
凶
n
i
も
旬
ee
l
e
m
阻
e
nも乱
n
凶a
乱
.
々s
i
se
x
h
i
b
i
t
s
nos
討
i
z
民ee
釘e
c
d
t,
a
so
b
s
e
r
v
e
dong
m
m
e
t
r
i
c
a
l
l
ys
i
m
i
l
a
zs
p
e
c
i
m
e
n
so
fd
i
f
e
r
e
n
ts
i
z
e
s
,
w
i
t
hs
i
m
i
l
a
r
m
e
s
h
e
s
.
Onep
r
a
c
t
i
c
a
la
.
d
v
姐 t
a
g
eo
ft
h
en
o
n
l
o
c
a
lf
o
r
m
u
l
a
t
i
o
ni
st
h
a
ti
ft
h
ehitee
l
e
m
e
n
ts
i
z
e
s
u
e
l
e
s
st
h
a
na
b
o
u
tl/3oft
h
ec
h
a
r
a
c
t
e
r
i
s
t
i
cl
e
n
g
t
ht
,
t
h
e
nもh
e
r
ei
snod
i
r
e
c
t
i
o
n
a
lb
i
a
so
ft
h
e
mdlwithREardLoc
r
a
c
kpropagaM.ThisMbeen附 i
f
i
e
d(B
a
.
Z
a
n
tandL
i
n,
1
9
働 )by
i
t
h胡a.l
i
g
n
e
d叩 a
r
emesh組 da
.
mlyzingthefmureofbhesamer
e
c
t
a
n
g
u
l
a
rs
pぬ menw
訂 em
esh.Thus,
t
h
en
O
B
l
o
c
a
l
m
o
d
e
lcmbeu
s
e
di
ng
e
n
e
r
a
lf
E
a
z
u
r
es
i
t
u
a
b
i
o
n
s
s
l.
a
n
t
e
ds
q
u
ゐ
i
“
F
o
r∞n
c
r
e
t
es
t
r
u
c
t
u
r
e
ss
u
b
j
e
c
t
e
dt
o∞m
p
l
i
c
a
t
e
dl
o
a
.
d
i
n
gh
i
s
t
o
r
i
e
s,
i
tmaybene
伺 s
s
a
r
y
切組a.l
y
z
ef
r
a
c
t
u
r
et
a
.
k
i
n
gi
n
t
oa
c
∞unt出 eeXIstenceofmultidir庇 tionalcracking(BaZ祖t
1
9
8
3
;deB
o
r
s
t,1
9
8
4,1
9
8
7
,
.
a1
9
8
7
b
;deB
o
r
s
tandNa
.
u
t
a
,1
9
8
5
;O
t
t
o
s
e
n阻 dDahlblom
,1
9
8
6
;
Willame
t仏 1
9
8
7
;Ro
t
s,1
9
8
8
;Ro
t
s組 dB
l
品uwendr
a
.
a
d,1
9
8
9
;C
r
i
s
f
i
e
l
dandW
i
l
l
s,1
9
8
9
)
.
,whichwas
Ane
f
f
e
c
t
i
v
ea
p
p
r
o
a
c
ht
os
u
c
hb
e
h
a
v
i
o
ra
p
p
e
a
r
st
obet
h
em
i
c
r
o
p
l
a
n
emodel
山 由edam
a
.
g
el
a
w
.I
n
s
t
e
a
.
do
fam
a
c
r
c
ト
a
l
r
e
a
d
ym
e
n
t
i
o
n
e
db
e
l
o
wE
q
.3
.
6i
nc
o
n
n
e
c
t
i
o
nw
i
s
c
o
p
i
cs
t
r
e
s
s
s
t
r
a
i
nr
e
l
a
t
i
o
n,
t
h
em
a
t
e
r
i
a
lb
e
h
a
v
i
o
ri
nt
h
em
i
c
r
o
p
l
a
n
emodeli
sc
h
a
r
a
c
t
e
r
i
z
e
d
i
n
d
e
p
e
n
d
e
n
t
l
yonp
l
a
n
e
so
fv
a
r
i
o
u
so
r
i
e
n
t
a
t
i
o
n
s,
c
a
l
l
e
dt
h
em
i
c
r
o
p
l
a
n
e
s
.I
np
r
a
c
t
i
,
田 only0
8
吋
, .
a
c
c
o
r
d
i
n
gt
oan
u
m
e
r
i
ca
.
li
n
t
e
g
r
a
t
i
o
n
c
e
r
t
a
i
nnumbero
fd
i
s
c
r
e
t
em
i
c
r
o
p
l
a
n
e
si
sc
o
n
s
i
d
e
r
f
o
r
m
u
l
af
o
rs
p
8
0
t
i
a
ld
i
r
e
c
t
i
o
n
s
. Them
a
c
r
o
s
c
o
p
i
cr
e
s
p
o
n
s
ei
sd
e
t
e
r
m
i
n
e
da
sac
e
r
t
a
i
na
v
e
r
田 p
o
n
s
e
sfrom出 ei
n
d
i
v
i
d
u
a
lm
i
c
r
o
p
l
a
n
e
s,
o
b
t
a
i
n
e
da
c
c
o
r
d
i
n
gt
ot
h
ep
r
i
n
c
i
p
l
e
a
g
eo
ft
h
er
o
fv
i
r
t
u
a
lw
o
r
k
.T
h
i
smodel
,
whoseb
a
s
i
cc
o
n
c
e
p
ti
ss
i
m
i
l
a
rt
ot
h
es
l
i
pt
h
e
o
r
yo
fp
l
a
s
t
i
c
i
t
y
(
T
a
y
l
o
r,1
9
3
8
;B
a
t
d
o
r
f姐 dB
u
d
i
a
n
s
k
i,1
9
4
9
),wasf
i
r
s
td
e
v
e
l
o
p
e
di
na
.l
o
c
a
lf
o
r
m(Ba
加 l
t
a
n
dOh
,
1
9
8
5
)
.I
twasa
.
l
s
oshownt
h
a
tt
h
i
smodelc
a
nr
e
p
r
e
s
e
n
tw
e
l
ln
o
to
n
l
yt
h
ee
x
i
s
t
i
n
g
,
i
n
c
l
u
d
i
n
gt
h
ee
f
f
e
c
t
so
f出 et
r
a
n
s
t
e
s
td
a
t
aont
e
n
s
i
l
ef
r
a
c
t
u
r
eb
u
ta
.
l
s
ot
h
o
s
eonc
r
a
c
ks
h
e
a
r
v
e
r
s
en
o
r
m
a
ls
t
r
e
s
sands
h
e
a
r
i
n
d
u
c
e
de
x
p
a
n
s
i
o
n晶c
r
o
s
st
h
ec
r
a
c
k(Ba
おn
t叩 dGambarov
,
.
a
1
9
8
4
)
. Re
c
e
n
t
l
y(BaZa
叫 a
ndO
z
b
o
l
t,
1
9
8
9
),
t
h
em
i
c
r
o
p
l
a
n
emodelh
a
sbeeng
e
n
e
r
a
l
i
z
e
dt
o
an
o
n
l
o
c
a
lf
o
r
m
.
T
h
e
r
ea
r
eo
t
h
e
rt
y
p
e
so
ff
r
a
c
t
u
r
ew
h
i
c
hn
e
e
dt
obemodeledf
o
rc
o
n
c
r
e
t
es
t
r
u
c
t
u
r
e
s
.A
newt
h
即時 w
i
l
lhavet
obed
e
v
e
l
o
p
e
df
o
rB
r
a
z
i
l
i
a
nt
e
n
s
i
l
es
p
l
i
t
t
i
n
gf
r
a
c
t
u
r
eo
fv
e
r
yl
a
r
g
e
whichc
a
n
n
o
tb
ea
d
e
q
u
a
t
e
l
y阻 a
l
y
z
e
dw
i
t
ht
h
ee
x
i
s
t
i
n
gもe
n
s
i
l
ef
r
叫 t
u
r
em
o
d
e
l
s
.
c
y
l
i
n
d
e
r
s,
oa
n
a
l
y
s
i
sbyt
h
e
Ont
h
eo
t
h
e
rhand,s
h
e
a
rf
r
a
c
t
u
r
es
e
e
m
s,a
tl
e
a
s
tp
a
r
t
i
a
l
l
y
,am~nable t
e
x
i
s
t
i
n
gt
e
n
s
i
l
ec
r
a
c
k
i
n
gm
o
d
e
l
s
.I
nt
h
o
s
em
o
d
e
l
s,
t
h
es
h
e
a
rf
r
a
c
t
u
r
ei
sd
e
s
c
r
i
b
e
d祖 aband
o
fi
n
c
l
i
n
e
dm
i
c
r
o
c
r
a
ι
k
s,
g
o
v
e
r
n
e
dbyas
o
f
t
e
n
i
n
gs
t
r
e
s
s
s
t
r
a
.
i
nr
e
l8
o
t
i
o
n
.
3.3Stress-Displacementvs. Stress-StrainSofteningRelations
T
h
e
r
eh
a
sb
e
e
nani
n
c
e
s
s
a
.
n
td
e
b
a
t
eont
h
er
e
l
a
t
i
v
em
e
r
i
t
sandd
e
f
i
c
i
e
n
c
i
e
so
ft
h
el
i
n
e
44
4
5
c
r
a
c
kandc
r
a
.
c
kbandr
e
p
r
白 e
n
t
a
t
i
o
n
so
ff
r
a
c
t
u
r
e
.Therea
r
et
h
r
e
ev
i
e
w
p
o
i
n
t
sもom
e
n
t
i
o
n
:
3
.
3
.
1I
s
o
l
a
t
e
dCracks:MootPointComputationally
F町民 o
n
emustr
e
a
l
i
z
et
h
a
tt
h
el
i
n
ec
r
a
.
c
kmodel(
i
.
e
. t
h
ef
i
c
t
i
t
i
o
u
sc
ra
.
c
kmodelo
f
1
l
e
r
b
o
r
g
)andt
h
ec
ra
.
c
kb組 dmodely
i
e
l
da
b
o
u
tt
h
esamer
e
s
u
l
t
s(
w
i
t
hdぽe
r
e
n
側 o
fa
b
o
u
t
Hi
1
%)i
ft
h
es
t
r
e
s
sd
i
s
p
l
a
.
c
e
m
e
n
tr
e
l
a
t
i
o
ni
nt
h
ef
i
r
s
tmodeland出 es
もr
e
s
s
叫 r
a
i
nr
e
l
a
t
i
o
ni
nt
h
e
s
e
∞ndmodelareca
.
l
i
b
r
a
t
e
di
ns
u
c
hawayt
h
a
t
ι
=
ωc
.
f
'
(
3
.
1
4
)
i
.
e
.,
t
h
ec
r
a
.
c
ko
p
e
n
i
n
gd
i
s
p
l
a
.
c
ementi
stakenωthef
r
a
c
t
u
r
es
t
ra
.
i
n,
f
.
f,
a
.
c
c
u
m
u
l
a
t
e
do
v
e
rt
h
e
.T
h
i
se
q
u
i
v
a
l
e
n
c
ei
sa
l
r
e
a
d
ys
u
g
g
e
s
t
e
dbyt
h
ef
a
.
ctt
h
a
.
t
,
i
nl
i
n
e
a
r
w
i
d
t
hωco
ft
h
ec
r
a
c
kb岨 d
e
l
a
.
s
t
i
cf
ra
.
c
t
u
r
em
e
c
h
a
n
i
c
s,
ani
n
t
e
r
e
l
e
m
e
n
tl
i
n
ec
r
a
c
k組 das
i
n
g
l
e
-e
l
e
m
e
n
tw
i
d
ec
r
a
.
c
kband
w
i
t
hasymmetryl
i
n
e出 品 c
o
i
n
c
i
d
e
sw
i
t
ht
h
邑l
i
n
ec
r
a
c
kg
i
v
ee
s
s
e
n
t
i
a
l
l
yt
h
esamer
e
s
u
l
t
si
n
a
.
8
er
a
t
e,
w
i
t
hd
i
f
f
e
r
e
n
c
e
so
fo
n
l
ya
b
o
u
t1%,
p
r
o
v
i
d
e
dt
h
e
t
h
ec
a
l
c
u
l
a
t
i
o
no
ft
h
ee
n
e
r
g
yr
e
l
e
e
l
e
m
e
n
ts
i
z
e
sn
e
a
rt
h
ef
r
a
c
t
u
r
ef
r
o
n
tdon
o
te
x
c
e
e
da
b
o
u
t1
/
1
0o
ft
h
ec
r
o
s
ss
e
c
t
i
o
nd
i
m
e
n
s
i
o
n
(
B
a
i
姐 t
.
a
ndC
e
d
o
l
i
n,
1
9
7
9
)
.I
nf
a
.
c
t
,
t
h
ed
i
f
f
e
r
e
n
伺 b
e
t
w
e
e
nt
h
e
s
etwomethodso
fc
a
l
c
u
l
a
t
i
o
n
もl
a
r
g
e
rt
h
a
nt
h
ee
r
r
o
ro
f
o
ft
h
ee
n
e
r
g
yr
e
l
e
a
s
er
a
t
ef
o
rl
i
n
e
a
re
l
a
s
t
i
cf
r
a
c
t
u
r
em
e
c
h
a
n
i
c
si
sno
a
p
p
r
o
x
i
m
a
t
i
n
gt
h
econtinuumbyf
i
n
i
t
ee
l
e
m
e
n
t
si
nt
h
ef
i
r
s
tp
l
a
c
e
.
T
h
e
r
e
f
o
r
e,
t
h
eq
u
e
s
t
i
o
n“
l
i
n
ec
r
a
c
ko
rac
r
a
c
kb
a
n
d
?
"i
smootfromt
h
ev
i
e
w
p
o
i
n
to
f
a
p
p
r
o
x
i
m
a
t
i
n
gr
e
a
l
i
t
ybyc
o
m
p
u
t
a
t
i
o
n
a
lm
o
d
e
l
i
n
g
.Theo
n
l
yp
o
i
n
tworthyo
fd
e
b
a
t
ei
sc
o
m
t
h
etwom
o
d
e
l
sa
p
p
e
a
r
p
u
t
a
t
i
o
n
a
le
f
f
e
c
t
i
v
e
n
e
s
sandc
o
n
v
e
n
i
e
n
c
e
. Bute
v
e
ni
nt
h
a
tr
e
g
a
r
d,
t
obee
q
u
a
l
.
h
o
w
e
v
e
r,
d
e
s
e
r
v
ea
t
t
e
n
t
i
o
n
.L
e
i
b
e
n
g
o
o
d
,Darwin
V
a
r
i
o
u
sn
u
m
e
r
i
c
a
lm
o
d
e
l
i
n
ga
s
p
e
c
t
s,
叫 t
sf
o
rs
t
r
e
s
s
-d
i
s
p
la
.
c
ementands
t
r
e
s
s
s
t
r
a
i
nr
e
l
a
t
i
o
n
s
andDodds(
1
9
8
6
)showedt
h
a
tt
h
er
e
matchc
l
o
s
e
!
yi
ft
h
ec
r
a
c
k
i
n
gd
i
r
e
c
t
i
o
n
sa
tt
h
ei
n
t
e
g
r
a
t
i
o
np
o
i
n
t
sw
i
t
h
i
nt
h
ef
i
n
i
t
ee
l
e
m
e
n
t
s
a
r
ef
o
r
c
e
dt
obep
a
r
a
l
l
e
lt
o.
e
a
c
ho
t
h
e
r(
a
n
dt
ot
h
ea
c
t
u
a
lc
r
a
.
c
k
)
.I
ft
h
ec
r
a
c
k
i
n
gd
i
r
e
c
t
i
o
n
JS n
o
もk
nownap
r
i
o
r
i,
剖i
nt
h
eg
e
n
e
r
a
lc
a
s
e,
i
ti
sd
i
伍c
u
l
tt
oa
c
h
i
e
v
e出 i
sp
a
ra
.
l
l
e
l
n
e
s
s
. But
,i
fc
r
a
c
k
sa
r
ea
l
l
o
w
e
dt
oforma
td
i
t
f
e
r
e
n
to
r
i
e
n
t
a
t
i
o
n
sa
te
a
.
c
hi
n
t
e
g
r
a
t
i
o
np
o
i
n
t,
t
h
e
t
h
e
n
r
e
s
p
o
n
s
eo
ft
h
ef
i
n
i
t
ee
l
e
m
回 tm
odelw
i
t
hs
m
e
a
r
e
dc
r
a
ε
k
i
n
gi
ssomewhats
t
i
f
f
e
rt
h
叩 t
h
a
t
w
i
t
hd
i
s
c
r
e
t
ei
n
t
e
r
e
l
e
m
e
n
もc
r
a
c
k
i
n
g,
e
v
e
ni
ft
h
ee
l
e
m
e
n
ts
i
d
e
sa
r
epa
.
r
a
l
l
e
lt
ot
h
et
r
u
ec
r
配 k
.
3
.
3
.
2P
a
r
a
l
l
e
lCracks:ThirdParameter
Asa
l
r
e
a
d
ym
e
n
t
i
o
n
e
d,
i
ft
h
es
h
a
p
eo
ft
h
et
e
n
s
i
l
es
o
f
t
e
n
i
n
gc
u
r
v
ei
sf
i
x
e
d,
t
h
e
nt
h
el
i
n
e
c
r
a
.
c
k(
f
i
c
t
i
t
i
o
u
sc
r
a
ι
k
)modeli
sd
e
f
i
n
e
dbytwopa
陪
m
e
t
e
r
s,
Gf and,
刀 whilet
hec
r
a
c
k
貧n
edbyt
h
r
e
ep
a
ra
.
m
e
t
e
r
sG" andWc ・F
o
rt
h
e貧c
t
i
t
i
o
u
sc
r
a
c
kmodel
,
bandmodeli
sde
t
o
o,
a
.l
e
n
g
t
hp紅a.
m
e
t
e
r(
c
a
l
l
e
dt
h
e“
c
h
a
r
a
c
t
e
r
i
s
t
i
cl
e
n
g
t
h
"
)hil.9b
e
e
nd
e
f
i
n
e
d
;e
EG
,
/
刀
2
:
:
:
o
(
w
h
i
c
hi
ss
i
凶 l
a
rt
oE
q
.3
.
7a
sw
e
l
l回Ir
w
i
n
'
sE
q
.2
.
1
0
)
. However,出i
sp
a
r
a
m
e
t
e
ri
sa
.
d
e
r
i
v
e
dparam
e
t
e
r
,notanindependentone,
w
h
i
l
eωci
sani
n
d
e
p
e
n
d
e
n
tp訂 a
m
e
t
e
r
. Wh
y
四回?
t
h
i
sd
i
f
f
e
r
F
o
ri
s
o
l
a
t
e
dc
r
a
c
k
s,
i
tt
u
r
n
so
u
tt
h
a
tt
h
ee
t
f
e
c
to
f叫 ont
h
er
,回 u
l
も
si
sa
l
m
o
s
tne
g
l
i
g
i
b
l
e
,
:
f
,
p
r
o
v
i
d
e
dt
h
a
tt
h
es
o
f
t
e
n
i
n
gp
a
r
to
ft
h
e8
t
r
e
s
s
s
t
r
a
i
nd
i
a
g
r
a
mi
sad
j
u
s
t
ed8
0国 t
oa
l
w
a
y
sy
i
e
l
d
t
h
es
a
.
m
ef
r
a
c
t
u
r
ee
n
e
r
g
yG f
o
r組 Y叫・ However
,
t
h
ep
a
r
a
m
e
t
e
rWcdo
回 m
akead
i
t
f
e
r
e
n
c
ei
n
s
i
n
c
ei
tp
r
e
v
e
n
t
sa
d
j
a
c
e
ntc
r
a
c
k
sもobec
l
o
s
e
rt
h
姐
t
h
ec
a
s
eo
fd
e
n
s
e
l
ys
p
a
c
e
dp
a
r
a
l
l
e
lc
r
a
c
k
s,
t
h
ed
i
s
t
a
n
c
e叫 (
o
t
h
e
r
w
i
s
et
h
ea
d
ja
.
c
e
n
tc
r
a
ι
kb
a
n
d
swouldo
v
e
r
I
ap,
whichi
si
na
d
m
i
s
s
i
b
l
e)
.
l
i
a
c
a
n
c
eofU
1
ci
sn
o
tr
e
a
l
l
yul
ew
i
d
t
hofthea
c
t
u
a
lcmkingzo
nea
tth
$0t
h
ep
h
y
s
i
c
a
l勾 I
f
r
a
c
t
u
r
ef
r
o
n
t,
b
u
tt
h
eminimuma
d
m
i
s
s
i
b
l
es
p
a
c
i
n
go
fp
a
r
a
l
1
e
lc
r
民 k
s
.I
si
tne
偲 s
s
a
r
yt
h
抗
t
h
i
ss
p
a
c
i
n
gbeam
a
t
e
r
i
a
lf
r
a
c
t
u
r
ep
a
r
a
.
m
e
t
e
r
?
w
e
r
,c
o
n
s
i
d
e
rt
h
a
te
l
a
s
t
i
cl
o
n
g
i
t
u
d
i
n
a
l
f
i
b
e
r
sa
r
ebondedt
oal
ong,出 i
nc
o
n
c
r
e
旬
F
o
r組 組s
b
a
rs
u
b
j
e
c
t
e
dt
otension.Thec
r
o
s
ss
e
d
i
o
r
l
o
ftheabersi
Bs
olugeshbt
h
ec
o
m
p
o
s
i
s
e
四 e
x
h
i
b
i
t
snos
o
f
t
e
n
i
n
gu
n
d
e
rt
e
n
s
i
o
n,
a
l
t
h
o
u
g
h出 .
ec
o
n
c
r
e
t
ep
a
r
td
o
e
s
.Thent
h
e
s
p
e
c
i
m
s
t
a
t
eo
fu
n
i
f
o
r
ms
t
r
a
i
ni
sa
l
w
a
y
ss
t
a
b
l
e
. A
c
c
o
r
d
i
n
gt
ot
h
el
i
n
ec
r
a
.
c
kmodel,c
r
a
c
k
sc
姐
f
o
r
mω
a
r
b
i
k
a
r
i
l
yc
l
o
s
es
p
a
c
i
n
gi
ns
u
c
has
y
s
t
e
m
,a
ndd
e
p
e
n
d
i
n
gonもhes
p
a
c
i
n
g
,
もh
el
o
a
.
d
e
l
o
n
g
a
t
i
o
nd
i
a
.
gramP(u
)mayf
o
I
l
owa町 o
f
s
h
ed
i
a
g
r
a
m
sshownas0129,
0
1
3
9,
O149.orO159
i
nF
i
g
.3
.
7
. Thus
,t
h
e
r
ei
sana
m
b
i
g
u
i
t
y
,
町
0
r泊
i
加
n
ぬb
o
j
片
e
c
凶
t
t
i
吋
v
i
t
句
yo
fr
田
e
s
叩
'
p
∞
o
n
s
紙
Theproblemc
a
no
f
COusebee
l
i
m
i
n
a
t
e
dbye
n
r
i
c
h
i
n
gH
i
l
l
e
r
b
o
r
g
'
sl
i
n
ec
r
a
c
kmodelw
i
t
hat
h
i
r
di
n
d
e
pen
d
e
n
t
t
h
eminimumc
r
a
c
ks
p
a
c
i
n
g,
s(
i
n
d
e
p
e
n
d
e
n
to
fe
)
. Thi
si
s
,ofcourse,
m
a
t
e
r
i
a
lp
a
r
a
m
e
t
e
r,
e
q
u
i
v
a
l
e
n
tもoparameterωc・
“
3ふ 3RelationtoMicromechanismsofFr
acture
Then
o
r
m
a
lm
i
c
r
o
蜘 a
hsamsst
h
ef
r
a
c
t
u
r
ep
r
o
c
倒
z
o
n
emaybed
i
s
t
r
i
bu
t
e
dr
o
u
g
h
l
y舗
showni
nF
i
g
.3.
8
a.Thel
i
n
ec
r
a
c
kmodels
i
m
p
l
i
f
i
ω
t
h
i
ss
t
r
a
i
nd
i
s
t
r
i
b
u
t
i
o
nt
a
k
i
n
gi
ta
sa
∞nstaMplustheDiracdeltafunction,
Fig.3.8b.Thec
r
a
c
kbandmodels
i
u
p
l
i
f
t
e
si
t
.
t
o
o
t
a
k
i
n
gi
ta
sac
o
n
s
t
a
r
句 l
u
sa
n
o
t
h
e
rc
o
n
山 叫w
i
t
h
i
nttm
mckband(mta
n
g
u
l
a
zd
i
s
t
r
i
b
i
o
F
i
g
.3.
8
c
.Then
O
I
伽 a
lcontinuummodelg
i
v
e
sasmoothb
e
l
l
-s
hapedd
i
s
t
出 u
S
i
o
nacmsst
h
l
如 d
,
a
sshowni
nFig.3.8d(BahMandP
i
j
a
u
d
i
e
r
C
a
b
o
t,
1
9
8
8
a
),
andi
nthebiteelemmt
b
.
3
.8
e
.
f
o
r
ma
sFほ
M
e
a
s
u
r
e
e
n
t
so
ft
h
el
午前I
o
n
so
f乱c
o
u
s
t
i
ce
m
i
s
s
i
o
n
sd
u
r
i
n
gt
h
ef
r
a
c
t
ur
ep
r
o
閃 5
8(
Labuz
,Do?d時
, 1985;MaliandShah,1988)indi
c
a
t
e
,despi
t
et
h
e
i
ri
n
e
v
i
b
a
b
l
es
c
a
t
t
;
r
,that
S
h
a
h
t
h
ee
m
i
s
s
i
o
ns
o
u
r
ω
sa
r
el
o
c
a
t
e
do
v
e
rar
e
l
a
t
i
v
e
l
yw
i
d
ebandi
nt
h
ef
r
o
n
s
a
lr
e
g
i
o
no
ff
r
&
c
t
u
r
e
a
si
nt
h
ec
r
a
c
kbandm
o
d
e
l
.Ont
h
eo
t
h
e
rh組 d,
v
a
r
i
o
u
smeasurementso
fs
t
r
a
i
回
(
F
i
g
.3
.
8
f
),
バ
目
立
,Y
;
出官民;
:
?
;
;
z
J
2
2
足立己主;
J
:
;
:
4
2
2
'
z
r
;
r
よ
:
忠
良
z
o
n
ea
tt
h
ef
r
o
n
tf
r
a
c
t
u
r
e,
w
h
i
c
hm
i
g
h
tbeb
e
b
t
e
rmodeledbyal
i
n
ec
r
a
c
kmodel.I
もma
vbe
n
o
t
仙 h
a
tt
h
ef
r
a
.
c
t
u
r
e町 a
i
n
sm
i
g
h
tbemorel
o
c
a
!
i
z
e
da
tt
h
es
凶 c
eo
fas
p
e
c
i
m
e
n
m
,
duet
ot
h
ew
a
l
le
t
f
e
c
tando
t
h
e
re
f
f
e
c
t
s
.
t
h
ei
n
t
e
r
i
o
r
Asf
o
rt
h
ev
i
s
i
b
l
ec
o
n
t
i
n
u
o
u
sb
a
c
h
I
r
eb
e
h
i
n
db
h
ef
r
a
c
t
u
r
epr
o
c
e
s
sz
o
n
e
,
i
tmustbeE
m
b
e
d
ω i bi
sfmumslyh
i
g
h
l
yt04uoumm
e
a
n
d
e
r
i
n
gS
oeachddeo
ft
h
esymmetryl
i
n
ebya
姐 句o
fupt
oa
b
o
u
tt
h
emaxmuma
g
g
r
e
g
a
t
esim.Eveni
fdlm
i
c
r
o
c
r
a
c
k
i
n
gw
e
r
e∞n
c
e
n
d
i
s
t
h
a
t
e
dona
.
l
i
n
e
,
i
nv
i
e
wo
ft
h
et
o
r
t
u
o
s
i
t
yo
ft
h
i
sl
i
n
e,
t
hef
r
a
c
t
u
r
ei
sr
e
p
r
e
s
e
n
t
e
dnob
e
t
t
e
rby
?sMgMImecmkth
叩 b
yacmkbando
fwidUldoneorMageqatesi
z
e
si
however
p
u
t
a
t
i
o
n
a
lr
e
s
u
l
t
st
h
i
sw
i
d
t
h陀 a
l
l
yd
o
e
sn
o
tma
伽
i
nmost仙 討i
o
n
s, 前 4
e
n
t
i
o
n
e
J
I
o
r∞m
山
ふ
47
46
b
o
v
e
)
.
r
o
f
i
l
ea
c
r
o
s
st
h
ef
r
a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
e,
m
i
c
r
o
c
r
a
c
k
i
n
gi
sma
.
n
i
f
e
s
t
e
dbyani
n
A
l
o
n
g乱 p
品 s
ei
nt
h
ea
.
v
e
r
a
g
ema
.
g
n
i
t
u
d
eo
ft
h
et
r
姐 s
v
e
r
s
es
t
r
a
i
n
s
.T
h
i
si
n
c
r
e
a
s
ei
sp
r
o
b
a
b
l
yg
r
a
.
d
u
a
l,
c
r
.
a
sshowni
nF
i
g
.3
.
8
a,
o
ri
naf
i
n
i
t
ee
l
e
m
e
n
tformi
nF
i
g
.3
.8
e
.O
b
v
i
o
u
s
l
yt
h
ew
i
d
t
ho
f
t
h
ef
r
a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
eo
b
t
a
i
n
e
dbym
e
a
s
u
r
e
m
e
n
t
sd
e
p
e
n
d
sont
h
ec
h
o
i
c
eo
f出 .
ec
u
t
o
f
f
.
a
l
u
eo
ft
h
es
t
r
a
i
n
.T
h
i
sm
i
g
h
te
x
p
l
a
i
nt
h
ed
i
品 目n
c
e
si
ni
n
t
e
r
p
r
e
t
a
t
i
o
no
fv
a
r
i
o
u
st
y
p
e
so
f
v
o
b
s
e
r
v
a
t
i
o
n
s
.
t
h
e
r
es
田 m
st
obenoc
o
m
p
e
l
l
i
n
gr
e
a
s
o
n
sf
o
rr
e
j
e
c
t
i
n
ge
i
t
h
e
rt
h
ec
r
a
c
kband
Tosumup,
model,
c
h
a
r
a
c
t
e
r
i
z
e
dbys
o
f
t
e
n
i
n
gs
t
r
e
s
s
s
t
r
a
i
nr
e
l
a
t
i
o
n
s,
o
rt
h
el
i
n
ec
r
a
.
c
kmodel,
c
h
a
r
a
c
a
.
t
i
o
n
s
. Thec
h
o
i
c
ei
se
s
s
e
n
t
i
a
l
l
yam
a
t
t
e
ro
f
t
e
r
i
z
e
dbys
o
f
t
e
n
i
n
gs
t
r
e
s
s
d
i
s
p
l
a
c
e
m
e
n
tr
e
l
c
o
n
v
e
n
i
e
n
c
eo
fa
n
a
l
y
s
i
s
.
年一一一一平副ー P
3
品
+
ー
ム
Mesh1ト→ベ四空当出企
2H
It
ート:→ィ宍 =0.05
ア
/
3肝 隅 隅 密 画
/
σt
/
↑
f
'
.
司 Et=-Ec1
8
/
v 、
、 ε
/
フ
→
H
=
4
内
.
a-GOOF
CrackBandModef
,
2.3
Meshes1
C
〆
/
/
/
_
グ
く
uoo
a
.
3.
4Non!inearTr
i
a
x
i
a
lConstitutiveModelsf
o
rS
t
r
a
i
n
S
o
f
t
e:
n
ing
/
1川
t
i山
r
4 之1
f Mesh3 │ M。¢αω
剖
e
i
は
。n
d
d
/'¥Mesh2 commiteDGmqge
¥
/
___Mesh 1 ! odel
~~______,.,'-_," ,J M
¥μ-"
.
r 〆
「
¥ /
V
σA
ト¥
|
displacement ム (
j
n
.
)
~
8
0
.
0
1
F
i
g
.
3
.
7F
i
n
i
t
eE
l
e
m
e
n
tRe
s
u
l
t
sf
o
rD
i
f
f
e
r
e
n
tC
r
a
c
kS
p
a
c
i
n
,
gsi
naT
e
n
a
r
s
i
o
n
e
dR
β
i
n
f
o
r
c
e
db.
Thes
t
r
e
s
s
s
t
r
a
i
no
rs
t
r
e
s
s
d
i
s
p
l
a
c
e
m
e
n
tr
e
l
a
t
i
o
n
sf
o
rt
h
ef
r
a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
ewhich
h
a
v
ebe
回 d
e
s
c
r
i
b
e
ds
of
a
rt
a
k
ei
n
色oa
c
c
o
u
n
to
n
l
yt
e
n
s
i
l
ec
r
a
c
k
i
n
gi
noned
i
r
e
c
t
i
o
n
.However
,
t
h
ef
r
a
c
t
u
r
ep
r
o
c
問 sz
o
n
emayu
n
d
e
r
g
om
u
l
t
i
p
l
ec
r
a
c
k
i
n
gi
nmanyd
i
r
e
c
t
i
o
n
sandmaybe
回 s
i
v
es
t
r
e
s
s
e
sp
a
r
a
l
l
e
lt
ot
h
ec
r
a
c
kp
l
a
n
e,
a
sw
e
l
la
ss
h
e
a
rs
t
r
e
s
s
e
s
.
s
u
b
j
e
c
t
e
dt
oh
i
g
hcompr
Toh
a
n
d
l
es
u
c
hs
i
t
u
a
t
i
o
n
s,
f
i
n
i
t
ee
l
e
m
e
n
ta
n
a
l
y
s
i
sn
e
e
d
st
obeb
a
s
e
donag
e
n
e
r
a
ln
o
n
l
i
n
e
a
r
nt
h
ef
r
a
ε
t
u
r
ep
r
o
c
e
s
sz
o
n
e
,thematerialundergoesdegradationo
fi
t
s
t
r
i
a
x
i
a
lmode
.
l I
m
e
c
h
a
n
i
c
a
lp
r
o
p
e
r
t
i
e
s,
whichi
sb
a
s
i
c
a
l
l
yo
ftwok
i
n
d
s
:
a
.
d
a
t
i
o
no
fm
a
t
e
r
i
a
ls
t
i
f
f
n
e
s
sdueもodam
a
.
g
es
u
c
ha
sc
r
a
c
k
i
n
g,
whichi
sd
e
s
c
r
i
b
e
d
(
1
)d
e
g
r
byc
o
n
t
i
n
u
u
mdamagem
e
c
h
a
n
i
c
s(
o
rt
h
ef
r
a
c
t
u
r
i
n
gma
.
t
e
r
i
a
lt
h
e
o
r
yo
fD
o
u
g
i
l
l,
1
9
7
6
),
andi
s
c
h
a
r
a
c
t
e
r
i
z
e
dbyu
n
l
o
a
d
i
n
ga
c
c
o
r
d
i
n
gt
ot
h
es
e
c
a
n
tm
o
d
u
l
u
s
jand
(
2
)d
e
g
r
a
.
d
a
t
i
o
no
ft
h
es
t
r
e
n
g
t
ho
ry
i
e
l
dl
i
m
i
t,
w
h
i
c
hi
sd
e
s
c
r
i
b
e
dbymoderna
d
a
p
t
a
t
i
o
n
s
o
fp
l
a
s
t
i
c
i
t
y
.
e
s
c
r
i
b
e
d
Ther
e
a
lb
e
h
a
v
i
o
ri
sac
o
m
b
i
n
a
t
i
o
no
fb
o
t
h.Suchcombinedb
e
h
a
v
i
o
rh
a
sb田 nd
c
r
e
,
旬e
.
g
.,
byt
h
ee
n
d
o
c
h
r
o
n
i
ct
h
e
o
r
y
,
p
l
a
s
t
i
c
f
r
a
c
t
u
r
i
n
g出切r
yo
rt
h
edamageth
田 r
y
f
o
r∞n
o
fO
r
t
i
z
.F
o
rar
e
v
i
e
w, 附 Ba
.
i
a
n
t(
1
9
8
6
)
.
,t
r
i
a
x
i
a
.
lc
o
n
s
t
i
t
u
t
i
v
em
o
d
e
l
sc
a
n
n
o
tbea
.
c
commodatedi
nt
h
es
t
r
e
s
s
G
e
n
e
r
a
ln
o
n
l
i
n
e
a
r
自 f
o
rs
h
a
r
pl
i
n
ec
r
a
c
k
s,
whicha
r
eb
a
s
i
c
a
l
l
yc
o
u
n
t
e
r
p
a
r
旬 o
fu
n
i
a
x
i
a
l
d
i
s
p
l
a
c
e
m
e
n
tr
e
l
a
t
i
o
n
s
t
r
e
s
s
s
t
r
a
i
nr
e
l
a
t
i
o
n
s
.Thec
r
a
c
kbandmodelc
佃 b
ee
x
t
e
n
d
e
dt
oa
.
c
commodates
u
c
hm
o
d
e
l
s,
b
u
tf
o
rap
r
o
p
e
rもr
e
a
t
m
e
n
t,
t
h
en
o
n
l
o
c
a
la
p
p
r
o
叫 hs
田 m
st
obene
田 s
s
a
r
y
.
a
nh
a
n
d
l
eb
o
t
ht
h
ep
r
o
c
e
s
sz
o
n
eo
ft
e
n
s
i
l
e
Av
e
r
yg
e
n
e
r
a
lc
o
n
s
t
i
t
u
t
i
v
er
n
o
d
e
l,whichc
,
a
sw
e
l
l出 ∞m
p
r
e
s
s
i
o
nands
h
e
a
rdamage
,
andd
e
s
c
r
i
b
e
sw
e
l
lt
h
en
o
n
l
i
n
e
a
rt
r
i
a
x
i
a
l
f
r
a
c
t
u
r
e
b
e
h
.
a
v
i
o
ri
ng
e
n
e
r
a
l,
i
st
h
em
i
c
r
o
p
l
却 e
mode
l
.Thatt
h
i
smodelg
i
v
e
s出 ec
o
r
r
e
c
tr
田p
onsef
o
r
f
r
a
c
t
u
r
e
si
se
v
i
d
e
n
c
e
dbyt
h
ef
.
a
c
tt
h
a
ti
ty
i
e
l
d
s.
ac
o
r
r
e
c
tt
r
a
n
s
i
t
i
o
n
a
ls
i
z
ee
f
f
e
c
t
js
e
eF
i
g
.3
.
9
.
i
a
n
t組 dO
z
b
o
l
t(
1
9
8
9
)
.
t
a
k
e
nf
r
o
mBa
'
zp
a
r
a
l
l
e
lt
ot
h
ec
r
a
.
c
k,
a
tl
e
a
s
ts
i
m
p
l
ea
d
j
u
s
レ
Whent
h
e
r
ei
sal
a
r
g
e
rc
o
m
p
r
e
s
s
i
v
es
t
r
e
s
sσ
mento
f出 es
o
f
t
e
n
i
n
gs
t
r
e
s
s
d
i
s
p
l
a
c
e
m
e
n
tc
u
r
v
eo
f円 v
s
・d
rs
t
r
e
s
s
d
i
s
p
l
a
c
e
m
e
n
tc
u
r
v
eo
f
co
円 v
s
.fl n
白 d
st
obemade.B
i
a
x
i
a
lf
a
i
l
u
r
ee
n
v
e
l
o
p
eda
.
t
a
.i
n
d
i
c
a
t
et
h
a
tt
h
et
e
n
s
i
l
es
t
r
e
n
g
t
h
l
i
m
i
ts
h
o
u
l
dbes
c
a
.
l
e
ddownf
r
o
mf
:approximatelytothevaluef
igivena
l
r
回 .
d
yi
nE
q
.3
.
9
.
49
48
Whethert
h
e円 (
o
)c
u
r
v
eo
r円(ら)c
u
r
v
es
h
o
u
l
dbes
c
a
l
e
ddownp
r
o
p
o
r
t
i
o
n
a
l
I
y,
o
rs
h
i
f
t
e
d
c
,o
rs
u
b
j
e
c
t
e
dt
osomemorec
o
m
p
l
e
xt
r
a
n
s
f
o
r
m
a
t
i
o
nt
or
e
s
e
c
tt
h
ee
f
f
e
c
to
fむ i
sno
も
down
knowna
tp
r
田e
n
t
.Whenσ
zく o
andF
i
g
.3
.
9i
so
b
s
e
r
v
e
d,
t
h
e
n,
o
f∞u
r
s
e
,
t
h
ea
r
e
ou
n
d
e
r
t
h
e円 (
ι
)curvedoesnotr
e
p
r
e
s
四 t
t
h
ef
r
a
.
c
t
u
r
ee
n
e
r
g
yGJ,
b
u
tI
s1
ω
st
h
a
nG,
fi
.
e
.E
q
.3
.
2
I
si
n
v
a
l
i
d
.
.
。
3.5RandomParticleSimulationofMicrostructure
emmlsslons
F
i
g
.
3
.
8F
r
a
c
t
u
r
eP
r
o
c
e
s
sZo
ne,
ondS
t
r
e
s
sP
r
o
f
i
l
e
sA
c
r
o
s
s1
t
:(
a
)A
c
叫
, (
b
)ー(
c
)V
a
r
i
o
u
sR
e
p
r
e
s
e
n
t
乱t
i
o
n
t
u
、
、
、
ー0
.
3
¥
ー0
.
4
/
"
画
、
¥
切
ー
¥
t
5
、
"
、
て
二i
、
、
-0.5
¥
¥
¥
。 -0.6
¥
C
J
l
¥
¥
、
、
¥
。
-0.7
-0.8
0.4
0.6
¥
Size Effecl Law
FEM,Nonl.Microplane Model
Test results
0
.
8
¥
.
0
1
.2
1
.4
log(d/dg)
Fig.3.9MaximumL
o
a
d
so
fNotchedBeamS
p
e
c
i
m
e
n
sC
a
l
c
u
l
a
t
e
dby
,
Comparedt
oS
i
z
eE
f
f
e
c
tLawandt
oT
e
s
t
N
o
n
l
o
c
a
lM
i
c
r
o
p
l
a
n
eModel
.
z
a
n
t叩 dO
z
b
o
l
t,
1
9
8
9
)
Datι(Ba
,
y
e
tc
o
m
p
u
t
a
t
i
o
n
a
l
l
ye
x
t
r
e
m
e
l
ydem
組 d
i
n
g
,
modelf
o
rn
o
n
l
i
n
e
a
rf
r
a
c
Themostr
e
a
l
i
s
t
i
c
t
u
r
eb
e
h
a
v
i
o
ro
fc
o
n
c
r
e
t
ei
sp
r
o
v
i
d
e
dbyt
h
erandomp
a
r
t
i
c
l
es
i
m
u
l
o
t
i
o
n
.E
x
t
e
n
s
i
v
ei
n
v
e
s
t
i
st
y
p
eh
a
v
eb
e
e
nc
a
r
r
i
e
do
u
tbyWittmanne
ta
l
.,
(
1
9
8
4
)
;蹴 a
1
8
0R
o
e
l
f
s
t
r
ae
t
g
a
t
i
o
n
so
fぬi
,
.
1(1985),
andRo
e
l
f
前回 (
1
9
8
7
)
.C
r
a
c
kf
o
r
m
a
t
i
o
nandp
r
o
p
a
g
a
t
i
o
ni
ncomputerg
e
n
e
r
a
t
e
d
a
c
o
m
p
o
s
i
t
es
t
r
u
c
t
u
r
e
swas8
t
u
d
i
e
dn
u
m
e
r
i
c
a
l
l
y
.Thesei
n
v
e
s
t
i
g
a
t
o
r
su
s
e
ds
t
a
n
d
a
r
df
i
n
i
t
ee
l
e
mentt
e
c
h
n
i
q
u
e
s,
s
u
b
d
i
v
i
d
i
n
ge
a
.
c
ha
g
g
r
e
g
a
t
ep
i
e
c
eandt
h
em
o
r
t
a
rr
e
g
i
o
n
si
n
t
omanyf
i
n
i
t
e
anda
l
s
oc
o
n
s
i
d
e
r
i
n
gw
e
a
k
e
ra
g
g
r
e
g
a
te
-mortari
n
t
e
r
f
a
c
e
s.Thec
a
l
c
u
l
a
t
i
o
n
sp
r
o
e
l
e
m
e
n
t
s,
v
i
d
e
dv
a
l
ua
.
b
l
ei
n
s
i
g
h
t
sb
u
ta
r
ee
x
t
r
e
m
e
l
ydemandingonc
o
m
p
u
t
e
rt
i
m
eands
t
o
r
a
g
ewhich
madei
ti
m
p
o
s
s
i
b
l
et
omodelv
e
r
yl
a
r
g
es
t
r
u
c
t
u
r
e
s
.
a
l
s
oc
a
l
l
e
dt
h
ei
n
t
e
r
f
a
c
ee
l
ement
I
nt
h
es
i
m
p
l
e
s
tv
e
r
s
i
o
no
frandomp
a
r
t
i
c
l
es
i
m
u
l
a
t
i
o
n,
model(
Z
u
b
e
l
e
w
i
c
zandBa
.
z
a
n
t,
1
9
8
7
;Ba
.
z
a
n
t,
T
.
めb
a
r
a
,
KazemiandP
i
j
a
u
d
i
e
r
C
a
b
o
t,
1
9
8
9
),
t
h
ea
g
g
r
e
g
a
t
ep
i
e
c
回 a
r
ec
o
n
s
i
d
e
r
e
da
sr
ig
i
dd
i
s
c
so
rs
p
h
e
r
e
s
.Animport
a
n
t副 p
e
c
ti
st
h
a
tt
h
e
i
r
os
a
t
i
s
f
y
c
o
n
f
i
g
u
r
a
t
i
o
ni
sg
e
n
e
r
a
t
e
drandomlybyt
h
ec
o
m
p
u
t
e
r
.Amethodt
odot
h
i
ss
o回 t
もe
d
t
h
ep
r
e
s
c
r
i
b
e
dmixr
a
t
i
o
so
fa
g
g
r
e
g
ω
e
so
fv
a
r
i
o
u
s'
s
i
z
e
sh
a
sb
e
e
ns
u
c
c
e
s
s
f
u
l
l
yf
o
r
m
u
l
a
a
.
z
a
n
t,
Tabbara,
KazemiandP
i
j
a
u
d
i
e
r
C
a
b
o
t,
1
9
8
9
)
.
(B
T
h
e
s
em
o
d
e
l
sr
e
p
r
e
s
e
n
tana
d
a
p
t
a
t
i
o
no
fane
a
r
l
i
e
rmodelo
fC
u
n
d
a
l
l(
1
9
7
1)
,
S
e
r
r
a
n
oand
Rod
r
i
g
u
e
z
-O
r
t
i
z(
1
9
7
3
),
C
u
n
d
a
l
landS
t
r
a
c
k(
1
9
7
9
),
Kawai(
1
9
8
0)
,
andP
l
e
s
h
aandA
i
f
a
n
t
i
s
s
a
n
do
rg
r
a
v
e
l
.l
nc
o
n
t
r
a
s
tt
oC
u
n
d
a
l
l
'
smodel
,
(
1
9
8
3
)f
o
rg
r
a
n
u
l
a
rm
a
t
e
r
i
a
l
ss
u
c
ha
sr
o
c
k,
もh
es
i
m
p
l
ef
r
i
c
t
i
o
n
a
lt
e
n
s
i
o
n
r
e
s
i
s
t
i
n
gc
o
n
n
e
c
t
i
o
nb
e
t
w
e
e
np
a
r
もi
c
l
e
smustber
e
p
l
a
c
e
dbyan
I
n
e
l
a
s
t
i
ci
n
t
e
r
a
c
t
i
o
nw
i
t
hf
r
a
c
t
u
r
e
.T
h
i
sh
a
sb
e
e
ndon
ebya
s
s
u
m
i
n
ga
1
1t
h
ei
n
t
e
r
pa
r
t
i
c
l
e
七
七h
em
i
d
d
l
eo
ft
h
ei
n
旬r
p
a
r
t
i
c
l
ec
o
n
t
a
c
tl
a
y
e
r
d
e
f
o
r
m
a
t
i
o
n
st
obec
o
n
c
e
n
t
r
a
t
e
da
tap
o
i
n
ta
o
ft
h
em
a
t
r
i
x(mo
山 r
)
.I
nt
h
e0
柑n
a
lv
e
r
s
i
o
no
ft
h
i
smodel
,
bo
t
ht
h
en
o
r
m
a
lands
h
e
a
r
i
n
t
e
r
a
.
c
t
i
o
n
s(
b
u
tn
o
tmoments)w
e
r
et
a
.
k
eni
n
t
oa
c
c
o
u
爪 t
h
es
h
e
a
ri
n
t
e
r
a
c
t
i
o
nb
e
i
n
ge
l
副 i
c
,
組 dt
h
en
o
r
m
a
li
n
t
e
r
a
c
t
i
o
ne
x
h
i
b
i
t
i
n
gas
u
d
d
e
nl
o
s
so
fs
t
r
e
n
g
t
ha
f
t
e
r乱c
h
i
e
v
i
n
gt
h
ep
r
e
s
c
r
i
b
e
d
も
hl
i
m
i
tf
o
r出 ei
n
t
e
r
p
a
r
t
i
c
l
ef
o
r
c
e
. As
i
m
p
l
e
rv
e
r
s
i
o
n,
i
nw
h
i
c
ht
h
es
h
e
a
ri
n
t
e
r
a
c
t
i
o
n
s
t
r
e
n
g
b
e
t
w
e
e
np
a
r
t
i
c
l
e
si
sn
e
g
l
c
c
t
e
dandt
h
en
o
r
m
a
l(
a
x
i
a
l
)i
n
t
e
r
a
c
t
i
o
ni
sc
h
a
r
a
c
t
e
r
i
z
e
dbyaf
o
r
c
仔
d
i
s
p
l
a
c
e
m
e
n
tc
u
r
v
ew
i
t
hp
o
s
t
p
e
a
kg
r
a
d
u
a
l
ω
f
t
e
n
i
n
gi
nt
e
n
s
i
o
n,h
a
sb
e
e
nde
v
e
l
o
pedby
Ba
Za
n
t,
T,晶b
b
a
r
a
,Kazemi組 dPijaudier-Cabot(1989)andshownt
obes
u
f
f
i
c
i
e
n
tf
o
rmost
p
u
r
p
o
s
e
s,
e
x
伺 p
tもh
a
ti
tt
e
n
d
st
og
i
v
eaf
r
a
c
もu
r
ep
r
o
c
e
s
sz
o
n
et
h
a
ti
sn
a
r
r
o
w
e
rands
h
o
r
t
e
r
叩
t
h
a
to
b
t
a
i
n
e
dwheni
n
t
e
r
p
a
r
t
i
c
l
es
h
e
a
rs
t
i
f
f
n
e
s
si
st
a
k
e
ni
n
t
o乱c
c
o
u
n
t(
a
n
da
l
s
ot
h
組
t
h
t
h
a
te
x
p
e
c
t
e
dfroms
i
z
ee
f
f
e
c
td
a
t
af
r
o
m∞n
c
r
e
t
ef
r
a
c
t
u
r
et
e
s
t
s
)
.T
h
i
sv
e
r
s
i
o
no
ft
h
emode
l
i~ e
q
u
i
v
a
l
e
n
tt
oarandomt
r
u
s
sw
i
t
hs
o
f
t
e
n
i
n
gmembersandr
e
p
r
e
s
e
n
t
sag
e
n
e
r
a
l
i
z
a
t
i
o
no
f
o
u
g
i
l
l(
1
9
7
7
)
.Theprogrammingi
sq
u
i
t
es
i
凶 a
rt
on
o
n
l
i
n
e
a
rf
I
n
i
t
e
t
h
emodelo
fB
u
r
t副 D
e
l
e
m
e
n
tp
r
o
g
r
a
m
s
.
5
1
50
J
:
:
;
t
z
ぷ
♂
;
;
;
二
:
;
:
:
r
222:c:2c:21;:22:7iJZ
2222IZ
:
:
1
2
2
p
q
i
J
F
F
X
:
諮右
鵠話器
5
2
詰
芯
.
p
L
;::P;:;2:(;;;;)i1nipiJ1Ac
,fromwhich
pe
8
o
r
st
h
8
0
tt
h
ew
i
d
t
ho
ft
h
ebando
f
r
叫 t
u
r
et
e
n
d
st
obea
b
o
u
tt
h
r
田 m
aximuma
g
g
r
e
g
8
0
t
e
m
i
c
r
o
c
r
a
c
k
sa
h
e
8
0
do
ft
h
ec
o
n
t
i
n
u
o
u
sf
主ねじず:ぷ
ロ
ロ
エ
ヱ
:
:
川
222:;;t
p
r
e
p
o
s
t
e
r
o
凶 ydem
.
a
n
d
i
n
gf
o
r∞m
p
u
t
e
rt
im
J73Z22LTL
。
'
0
'
50
.
ミ
1
20
3A
、
s
.
、
c
l
.
回
。
w
(
0
)
吋拍
30
。
∞
O.
。‘。
0.02
0
06
Displacement U (
1
7
¥
1
7
¥
)
2C
'
:
l
了n
r
e
e Doi
n
t Be ~ di 刊 (Notc門 eョ)
D
i
r
e
c
tT
e
n
s
i
o
n(
U
n
n
o
t
c
h
e
d
)
.
a
句
﹁
♂
ぷ/イ
"
c
.0
電
今
、
町
¥AU
。
toV
。
判
ラ
ヘ
'
ミ
ミ
1口
、
、
、
-
.
4=0.378
C=2.143
s
A=1
.176
C=2.
005
仰h
u
4
-
(
b
)
・
10
,~
d/d
o
7
5
10
d/da
F
i
g
.
3
.
1
0Ra
.
n
domP
8
0
r
t
i
c
l
eS
i
m
u
l
a
t
i
o
no
fC
o
n
c
r
e
t
e(
8
0
),
Re
s
u
l
t
i
n
gL
o
a
d
D
e
f
l
e
c
t
i
o
nC
u
r
v
e
sandS
i
z
eE
f
f
e
c
t(
b
),
a
n
dC
r
a
c
k
i
n
gPa
t
t
e
r
n(
c
)
(
c
o
n
t
i
n
u
e
d
)
5
2
53
C
h
a
o
t
e
r4
.SPECIALNONLINEARFRACTURE
恥i
O
I
J
E
L
SBASEDONADAPTATIONSOFLEFM
S
p
e
c
i
a
ln
o
n
l
i
n
e
a
rf
r
a
.
c
t
u
r
emode
l
sdon
o
tωtemptt
omodel出 eb
e
h
a
v
i
o
ro
ft
hef
r
a
c
t
u
r
e
p
r
o
伺 8
Sz
o
n
et
h
r
o
u
g
hs
t
r
e
s
s
d
i
s
p
l
a
c
e
m
e
n
to
rs
t
r
e
s
s
s
t
r
a
i
nr
e
l
a
t
i
o
n
s
.I
n
s
t
e
a
d,
t
heyi
n
t
r
o
d
u
c
e
r
o
x
凶 a
t
e
l
yr
e
f
l
e
c
t
80mea
d
a
p
t
a
t
i
o
n
so
fl
i
n
e
a
re
l
a
s
t
i
cf
r
a
c
t
u
r
em
e
c
h
a
n
i
c
s(LEFM)which叩 p
a
.
c
t
u
r
eb
e
h
a
v
i
o
r
.
t
h
en
o
n
l
i
n
e
a
r
i
t
yo
ff
r
4
.
1E
f
f
e
c
t
i
v
eCrackModels
Thee
f
f
e
c
to
fal
a
r
g
e
s
i
z
ef
r
a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
ea
もt
h
et
i
po
f出 en
o
t
c
ho
rc
o
n
t
i
n
u
o
u
s
山田 t
h
e5もr
e
s
s
e
sn
e
a
rt
h
ec
r
a
.
c
kt
i
pandp
u
s
ht
h
epeaks
t
r
e
s
sf
a
r
t
h
e
ra
h
e
a
d
.
c
r
a
c
ki
st
or
e
Co
n
s
e
q
u
e
n
t
l
y
,
t
h
es
p
e
c
i
m
e
nb
e
h
a
v
e
sr
o
u
g
h
l
ya
sane
l
a
s
t
i
cs
p
e
c
i
m
e
nw
it
hal
o
n
g
e
r
,
e
f
f
e
c
t
i
v
e
c
r
叫 k1
叩 g
t
h,
ae • T
h
i
sl
e
n
g
t
hc
a
nbed
e
f
i
n
e
di
nv
a
r
i
o
u
sways
,givi
n
gsomewhatdi
f
f
e
r
e
n
t
r
e
s
u
l
t
s
.1
nD
u
g
d
a
l
e
B
a
r
e
n
b
l
a
もt
・t
y
p
em
o
d
e
l
s(
o
rd
u
c
t
i
l
em
e
t
a
l
s,
t
h
ee
f
f
e
c
t
i
v
ec
r
a
c
kl
e
n
g
t
hi
s
c
a
l
c
u
l
a
t
e
du
s
u
a
l
l
yfromt
h
ec
o
n
d
i
t
i
o
nt
h
a
tt
h
esumo
ft
h
es
t
r
ぉ si
n
t
e
n
s
i
t
yf
a
.
c
t
o
r
sa
tt
h
et
i
p
o
(t
h
ee
晶c
t
i
v
ec
r
a
c
kduet
ot
h
ef
a
rf
i
e
l
ds
t
r
e
s
s
e
sandt
o出 ey
i
e
l
ds
t
r
e
s
s
e
si
nt
h
ef
r
a
.
c
t
u
r
e
倒 s
z
o
n
ebez
e
r
o(
D
u
g
d
a
l
e
,
1
9
6
0
jB
a
r
e
n
b
l
a
t
t,
1
9
5
9
)
.
p
r
o
Butt
h
i
sd
e
f
i
n
i
t
i
o
ns
e
e
r
n
sn
e
i
t
h
e
rq
u
i
t
ea
p
p
r
o
p
r
i
a
t
en
o
rt
o
oc
o
n
v
e
n
i
e
n
tf
o
rb
r
i
t
t
l
emat
令
r
i
a
l
s
.F
o
rt
h
e
s
em
a
t
e
r
i
a
l
s,
ane
l
a
s
t
i
c
a
l
l
ye
q
u
i
v
a
l
e
n
t(Gr
i
伍
.
t
h
t
y
p
e
)e
f
f
e
c
t
i
v
ec
r
a
c
kh
a
sb
e
e
n
d
e
f
i
n
e
da
st
h
ec
r
a
c
kf
o
rw
h
i
c
hもhee
l
a
s
t
i
c
a
l
l
yc
a
l
c
u
l
a
t
e
dc
o
m
p
l
i
組 問 o
ft
h
es
p
e
c
i
m
e
ni
st
h
e
samea
st
h
emeasuredu
n
l
o
a
d
i
n
gc
o
m
p
l
i乱n
c
eCuf
o
ru
n
l
o
a
d
i
n
go
ft
h
es
p
e
c
i
m
e
nfromt
h
ep
e
a
k
i
v
ec
r
a
心km
o
d
e
l
sf
o
rc
o
n
c
r
e
t
eu
s
i
ngt
h
i
sd
e
f
i
n
i
t
i
o
nh
a
v
eb
e
e
n
l
o
a
dp
o
i
n
t(
c
r
i
t
i
c
a
ls
t
a
t
e
)
.E賀 町t
叩o
s
e
dbyNa
l
I
a
t
h
a
.
mbi加 dK
a
r
i
h
a
.
l
o
o(
1
9
8
6
a
)andS
w
a
r
t
zandR
e
f
a
i(
1
9
8
9
)
.Thee
f
f
e
c
t
i
v
e
p
r
c
r
a
c
kl
e
n
g
t
h
,a.,dependson出 emicrostructure,a
.
sw
e
l
l制 t
h
eg
曲 m
etryo
ft
h
es
p
e
c
i
m
e
n
.
WhenKlci
sc
a
l
c
u
l
a
t
e
du
s
i
n
gt
h
ee
f
f
e
c
t
i
v
ec
r
a
c
k,
t
h
er
e
s
u
l
t
ss
肥 mt
obea
pp
r
o
x
i
m
a
t
e
l
ys
i
z
e
a
ta
.andJ(Ic m
i
g
h
tbeu
s
e
da
stwof
r
a
.
c
t
u
r
ep訂 a
m
e
t
e
r
s
.
i
n
d
e
p
e
n
d
e
n
t
. Thusi
ta
p
p
e
a
r
sもh
Tot
h
i
send
,
e
m
p
i
r
i
c
a
lr
e
l
a
t
i
o
n
sf
o
rr
e
l
a
t
i
n
gt
h
ee
f
f
e
c
t
i
v
ec
r
a
c
kl
e
n
g
t
ht
ot
h
en
o
t
c
hl
e
n
g
t
hi
n
af
r
a
c
t
u
r
es
p
e
c
i
m
e
nhaveb
e
e
nd
e
v
e
l
o
p
e
dont
h
eba
.
s
i
so
fe
x
p
e
r
i
m
e
n
t
a
ld
a
t
abyN
a
l
l
a
山a
mbi
泳叫∞ (
1
9
8
6
a
)
.
andKar
Then
e
e
df
o
re
m
p
i
r
i
c
a
le
q
u
a
t
i
o
n
sr
e
l
a
t
i
n
gt
h
ee
f
f
e
c
t
i
v
ec
r
a
c
kl
e
n
g
t
hもot
h
en
o
t
c
hl
e
n
g
t
h
r
e
d
u
c
e
st
h
es
c
o
p
eo
fa
p
p
l
i
c
a
b
i
l
i
t
yo
ft
h
i
sm
e
t
h
o
d
.I
na
d
d
i
t
i
o
n,
onen
e
e
d
se
m
p
i
r
i
c
a
le
q
u
a
t
i
o
n
s
h
e
s
el
i
m
i
ta
t
i
o
n
sa
r
eo
v
e
r
∞mei
nt
h
et
w
o
p
訂 a
m
e
t
e
rmodel
f
o
re
a
c
hs
p
e
c
i
m
e
nge~~etry. T
p
r
o
p
o
s
e
dbyJ
e
n
qandS
h
a
h
.T
h
i
smodeli
sd
e
s
c
r
i
b
e
dn
e
x
t
.
4
.
2Two-ParameterModelofJenqandShah
F
i
g
.3
.
1
0c
o
n
t
i
n
u
a
t
i
o
n
T
h
i
smodel(
J
e
叫 a
ndShah,
1
9
8
5
a,
1
9
8
5
b
jShah
,1988),
w
h
i
c
ha
p
p
e
a
r
st
og
i
v
ear
a
th
e
r
r
e
引
i
抗i
cp
r
e
d
i
c
t
i
o
no
fc
o
n
c
r
e
t
ef
r
a
c
t
u
r
eb
e
h
a
v
i
o
r,
i
nv
o
l
v
e
stwof
r
a
.
c
t
u
r
eparame
t
e
r
s
:(
1
)t
h
e
cr~ticaI s
t
r
e
s
si
n
t
e
n
s
i
t
yf
a
c
t
o
rKlca
tt
h
et
ipo
fane
f
f
e
c
t
i
v
ec
r
a
.
c
ko
fl
e
n
g
もha
tP
u,
andt
h
e
ea
~~ítical v
a
l
u
eOCTODo
ft
h
ec
r
a
c
kt
i
po
p
e
n
i
n
gd
i
s
p
l
a
c
泡m
ent,
wh
i
chi
sc
a
l
c
u
1a
t
e
da
tt
h
et
i
po
f
e
e
x
i
s
t
i
n
gc~~ck o
rn
o
t
c
h,
wh~se iengthi
sd~noted 出 ao, anda
t凡.Thee
f
f
e
c
t
i
v
ec
r
a
c
k
t
h
ep
r
54
4
.
3Geometry-DependentR-CurveDeterminedfromS
i
z
eE
f
f
e
c
t
Law
Aq
u
a
s
i
e
l
a
s
t
i
ca
n
8
0
l
y
s
i
so
fn
o
n
l
i
n
e
a
rf
r
a
c
t
u
r
ec
姐
bea
c
c
o
m
p
l
i
s
h
e
dbyc
o
n
s
i
d
e
r
i
n
gt
h
e
e
n
e
r
g
yr
e
q
u
i
r
e
df
o
rc
r
a
c
kg
r
o
w
t
h
,R,tobevariabler
8
0
出e
rt
h
a
nc
o
n
s
t
a
n
t
. Thec
u
r
v
eo
fR
v
s
.t
h
ec
r
a
c
ke
x
t
e
n
s
i
o
nci
sc
a
U
e
dt
h
eR
c
u
r
v
e(
o
rr
e
s
i
s
t
組 問 c
u
r
v
e
)
.Thec
r
a
.
c
kp
r
o
p
a
g
a
t
i
o
n
c
o
n
d
i
t
i
o
ni
sG = R
(
c
)r
a
t
h
e
rt
h
a
nG= G" andi
ft
h
ev
a
l
u
eo
fR
(
c
)i
sknown,
t
h
er
e
s
p
o
n
s
e
i
sc
a
l
c
u
l
8
0
t
e
da
c
c
o
r
d
i
n
gt
ol
i
n
e
a
re
l
a
s
t
i
cf
r
a
c
t
u
r
em
e
c
h
a
n
i
c
s,
r
e
p
l
8
0
c
i
n
gG w
i
t
hR
(
c
)
.A
l
t
e
r
,
b
u
te
q
u
i
v
a
l
e
n
t
l
y
,
onec
a
nd
e
f
i
n
et
h
eRc
u
r
v
ei
nt
e
r
m
so
ft
h
ec
r
i
t
i
c
a
ls
t
r
e
s
si
n
句n
s
i
t
y
n
a
t
i
v
e
l
y
f
a
c
t
o
rKlt!w
i
も
h!
(
IR=
=(
E
'G )1/2,
a
.
ndu
s
et
h
ec
r
a
c
kp
r
o
p
a
g
a
t
i
o
nc
o
n
d
i
t
i
o
nK1=
=J(IRi
n
s
t
e
a
d
o
ff{1=J(lc・Thep
h
y
s
i
c
a
lr
e
邸 o
nt
h
a
tt
h
ee
n
e
r
g
yr
e
l
e
a
s
er
a
t
e,R,
r
e
q
u
i
r
e
df
o
rc
r
a
c
kg
r
o
w
t
h
臼 e
swi
山 ci
sもh
a
ti
ti
sd
e
t
e
r
m
i
n
e
dbyt
h
es
i
回 o
ft
h
ef
r
a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
e
,
whichg
r
o
w
s
.
inαe
Al
a
r
g
e
rp
r
o
c
回 sz
o
n
ed
i
s
s
i
p
a
t
e
smoreen
e
r
g
y
.Howt
h
ez
o
n
eg
r
o
w
s,
o
fc
o
u
r
s
e
,
depend
sa
l
s
o
oni
n
t
e
r
a
c
t
i
o
nw
i
t
ht
h
eb
o
u
n
d
a
r
i
e
s
.
fR
c
u
r
v
e
swass
u
g
g
e
s
t
e
dbyI
r
w
i
n(
1
9
5
8
)andf
o
r
m
u
l
a
t
e
df
o
rm
e
t
a
l
si
nd
e
t
a
i
l
The∞n切 pもo
色r
t
a
i
nt
y
p
e
so
f∞n
c
r
e
t
es
p
e
c
i
m
e
n
s,
R
c
u
r
v
e
sw
e
r
em
e
a
s
u
r
e
d
byK
r
a
f
f
te
ta
l
.(
1
9
6
1
)
.F
o
rc
byWecharatanaandShah(
1
9
8
0,
1
9
8
2
)andJ
e
n
qandShah(
1
9
8
5
a
)
.F
o
ral
o
n
gt
i
m
ei
twas
回
be
t
h
o
u
g
h
tt
h
a
tt
h
eR
c
u
r
v
emeasur
e
de
x
p
e
r
i
m
e
n
t
a
l
l
yonas
p
e
c
i
m
e
no
foneg
e
o
m
e
t
r
yc
u
s
e
du
n
i
v
e
r
s
a
l
l
y
, 副 am
a
t
e
r
i
a
lp
r
o
p
e
r
t
ya
p
p
l
i
c
a
b
l
ea
p
p
r
o
x
i
m
a
t
e
l
yt
oo
t
h
e
rg
e
o
m
e
t
r
i
e
s
. But
,
ε
f
f
ec
;
t
l
v
eC
;
,
. lti
t
hC
"
OC;K
pr&
・
c
n
l
l
c
a
tcrackQr
,
帥
1
1
t
h
C
r
i
t
i
c
a
lP
O
l
n
l i
nt
K,,,~
CTODcC
T
O
D
c
d
“
(
a
)
。
唱
l
e
n
g
t
h,
a
sm
e
n
t
i
o
n
e
db
e
f
o
r
e,
c
a
nbec
a
l
c
u
l
a
t
e
dfromt
h
ec
o
m
p
l
i
姐 c
eCu r
e
c
o
r
d
e
da
tu
n
l
o
a
d
i
n
g
四
i
sl
o
a
d
e
dt
ot
h
ec
r
i
t
i
c
a
ls
t
8
0
t
e(
F
i
g
.4
.
1
)
.J
e
n
qandShahe
x
p
e
r
i
m
e
n
t
a
l
l
y
whent
h
es
p
e
c
i
m
o
b
s
ぽv
edt
h
a
twhena
.wωcalculatedfromt
h
ec
o
m
p
l
i
a
n
c
em伺 s
u
r
e
m
e
n
t
sf
o
rn
o
t
c
h
e
dheam
t
h
ec
a
l
c
u
l
a
t
e
dv
a
l
u
e
so
fdCTODw
e
r
emoreo
rl
e
s
si
n
d
e
p
e
n
d
e
n
to
ft
h
es
i
z
e部 w
e
l
la
s
s
p
e
c
i
m
e
n
s,
g
e
o
m
e
t
r
yo
ft
h
es
p
e
c
i
m
e
n
s
.T
h
i
si
n
d
i
c
.
a
t
e
dt
h
a
tdCTODandKlcmaybef
u
n
d
a
m
e
n
t
a
lm
a
t
e
r
i
a
l
J
e
n
q組 dShaha
l
s
op
r
e
d
i
c
t
e
dt
h
eu
n
i
a
x
i
a
lt
e
n
s
i
l
e
p
a
r
a
m
e
t
e
r
s
.U
s
i
n
gt
h
e
s
etwop
a
r
a
m
e
t
e
r
s,
s
t
r
e
n
g
t
h,
t
h
es
p
l
i
tc
y
l
i
n
d
e
rs
t
r
e
n
g
t
h
,
t
h
es
i
z
ee
f
f
e
c
ton∞n
v
e
n
t
i
o
n
a
lK1e,
andt
h
es
i
z
ee
f
f
e
c
t
ont
h
emoduluso
fr
u
p
t
u
r
e
.
Thee
f
f
e
c
l
i
v
ec
r
a
c
kl
e
n
g
t
h
,
α
e,
c
a
l
c
u
l
a
t
e
dont
h
eb
a
s
i
so
ft
h
e
s
etwof
r
a
c
t
u
r
ep
a
r
a
m
e
t
e
r
si
s
ont
h
ec
o
m
p
r
e
s
s
i
v
es
t
r
e
n
g
t
ho
fc
o
n
c
r
e
t
e,
and
f
o
u
n
dt
odependont
h
es
i
z
eo
ft
h
es
p
e
c
i
m
e
n,
ont
h
es
t
r
a
i
nr
a
.
t
e
. Thev
a
l
u
eo
fa
.d
e
α
e
a
s
e
sw
i
出 i
n
c
r
e
a
s
i
n
gs
t
r
e
n
g
t
handw
i
t
hi
n
c
r
回 s
i
n
g
田
,t
出 e
o
fane
l
a
.
s
t
i
cp
e
r
f
l
yb
r
i
t
t
l
em
a
t
e
r
i
a
l,
dCTODa
p
p
r
o
品c
h
e
sz
e
r
oand
s
t
r
a
i
nr
a
t
e
.F
o
rt
h
ec
a
.a
p
p
r
o
a
c
h
e
sa
o
・
Thev
a
l
u
eo
fKlcfromt
h
i
smodelh
a
sb開 nshownt
obee
s
s
e
n
t
i
a
l
l
yi
n
d
e
p
e
n
d
e
n
to
ft
h
e
wedges
p
}
i
t
t
i
n
g
g
e
o
m
e
t
r
yo
ft
h
es
p
e
c
i
m
e
n
s
.R
β
s
u
l
t
so
b
t
a
i
n
e
du
s
i
n
gcompactt
e
n
s
i
o
nt
e
s
t
s,
c
u
b
et
e
s
t
sandl
a
r
g
e
,tapered,doublecantileverbeamsconductedundertheauspi田 so
f
RILEMCommitt
伺
FMT8
9showedt
h
a
tt
h
etwof
r
a
c
t
u
r
ep
a
r
a
m
e
t
e
r
s(Klc anddCTOD)
m
i
g
h
tbec
o
n
s
i
d
e
r
e
d剖 g
回 m
e
t
r
y
i
n
d
e
p
e
n
d
e
n
tm
a
t
e
r
i
a
lp
a
r
a
m
e
t
e
r
s(
J
e
n
qandSh
a
.
h,
1
9
8
8
8
0
),
a
l
t
h
o
u
g
hal
a
t
e
rs
t
u
d
y(
J
e
n
qandSha
.
h,
叩
1
9
8
部8
吋
b
)indicated8
0s
討
i
g
凶
n
1
1
f
包
i
c
a
.
組
n
叫
1
ti
n
f
f
丑
l
l
田
u
l
e
n
c
白e
o
fg
∞metげ
凶
1
I
恥t
s
h
o
u
l
dben
o
t
e
d出
t
ha
t出
t
h
i
ぬ
smodel臼 nbe叩
a
p凶
p
l
i
吋
e
d
t
ω
os
p
e
c
i
m
e
n
sand“
s
古
位
加
r
u
c
t
町
u
1
汀r
田
e
sw
叩i
t
出
h
o
ω
rw
i
比
t
h
o
叫
u
b
n
o
ω
t
c
h
e
ω
s
.Themodelh
a
sb
e
e
ne
x
t
e
n
d
e
dt
omixed-model
oa
.
d
i
n
gandt
oi
m
p
a
c
tl
o
a
.
d
i
n
g(
J
o
h
n
制 dS
hah,
1
9
8
5,1
9
8
6,
1
9
8
9
)
.
い¥
55
CMOD
T
y
pi
c
;o
IP
l
o
t0
1L
o
c
:V~ .
C
r
oC
:kM
outl
'
lO
o
e
n
t
n
gD
'
s
:
l
Ic
:
c
e
m
e
n
t
T
w
o
.
p
a
r
a
m
e
t
e
rf
r
a
c
t
u
r
:
cr
n
o
d
e
Jf
o
rc
o
n
c
r
:
c
t
e
(
b
)
F
i
g
.
4.
1TwoP
a
r
a
m
e
t
e
rF
r
a
c
t
u
r
eModelo
fJ
e
n
qa
n
dSh
ぬ(
1
9
8
5
)
t
L
へ
R(c)
,
P
1'
1
コ
コ
←
3向
コ
コ
』
エ
ーE
crackI
ength
C
c
F
i
g
.
4
.
2RC
u
r
v
e
sf
o
rD
i
f
f
e
r
e
n
tSpecimenG
e
o
m
e
t
r
i
e
sa
n
dT
l
w
I
rP
o
s
t
PeakA
d
j
u
s
t
m
e
n
t
s
C
56
57
a
s
e
.Tod
8
o
Yweknow(
e
.
g
.Wecharatana.
a
ndSh
油, 1
9
8
3
8
0
;
t
h
i
sh
a
sn
o
tt
u
r
n
e
do
u
tt
obeもhec
Z
姐
も
, KimandP
f
e
i
f
f
e
r,
1
9
8
6
)出 品 f
o
rc
o
n
c
r
e
t
e
,
R
c
u
r
v
e
sdepends
t
r
o
n
g
l
yont
h
es
h
叩 eo
f
Ba
もr
u
c
t
u
r
eo
rs
p
e
c
i
m
e
n
.T
h
e
r
e
f
o
r
e
,
t
oen
a
.
b
l
et
h
eu
s
eo
fR
c
u
r
v
e
swhichc
e
r
ta
.
i
n
l
yr
e
p
r
e
s
e
n
t
t
h
es
o
n
en
e
e
d
ssomemethodt
oo
b
t
a
.
i
nt
h
eR
c
u
r
v
ef
o
rag
i
v
e
ns
p
e
c
i
m
e
n
8
0v
e
r
ys
i
m
p
l
ea
p
p
r
o
a
c
h,
g
e
o
m
e
t
r
yfromsomeb.
a
s
i
cf
r
拭 t
u
r
ec
h
a
r
8
o
d
e
r
i
s
t
i
c
s
.
Asp
r
o
p
o
s
e
dbyBa
Zant.
a
ndC
e
d
o
l
i
n(
1
9
8
4
),
andB
a
z
a
n
t,
KimandP
f
e
i
f
f
e
r(
1
9
8
6
),
t
h
eR
.
nyg
i
v
e
ng
e
o
m
e
t
r
yma
.
ybed
e
t
e
r
m
i
n
e
dont
h
eba
.
si
so
ft
h
es
i
z
ee
f
f
e
c
tl
a
.
w(
S
e
c
.5
.
1
)
.
c
u
r
v
ef
o
ra
.
4
)app
回1"
Sa
d
e
q
u
a
t
ef
o
rt
h.
a
tp
u
r
p
o
s
es
i
n
c
e(
u
n
l
i
k
et
h
eR
c
u
r
v
e
)i
ti
sa
.
p
p
l
i
・
T
h
i
sl
a
w(
E
q
.1
c
a
b
l
e
,i
nt
h
es
8
0
m
ef
o
r
m,
t
omostg
e
o
m
e
t
r
i
回 w
i
t
h
i
nas
i
民 r
組 g
eupt
oa
.
b
o
u
t1
:
2
0,
which
印 鑑c
e
sf
o
rmosta
.
p
p
l
ic
a
.
t
i
o
n
s
.TheR
c
u
r
v
ei
st
h
e
nobtainedωthee
n
v
e
l
o
p
eo
ft
h
ef
r
a
c
t
u
r
e
四 f
o
rg
e
o
m
e
t
r
i
c
a
l
l
ys
i
m
i
l
a
rs
p
e
c
i
m
e
n
so
fva
.
r
i
o
u
ss
i
z
e
s
. Ane
回 yc
a
l
c
u
l
a
t
i
o
n
e
q
u
i
l
i
b
r
i
u
mc
u
r
v
o
ft
h
eR
c
u
r
v
ei
sp
o
s
s
i
b
l
eon
臼 t
h
ep
a
r
a
m
e
t
e
r
so
ft
h
es
i
z
ee
f
f
e
c
tlawh
a
v
eb
e
e
nd
e
t
e
r
m
i
n
e
d
.
8
o
r
,
t
h
eR
c
u
r
v
e,
g
i
v
i
n
gt
h
ef
r
8
0
c
t
u
r
ee
n
e
r
g
yr
e
q
u
i
r
e
dp
e
ru
n
i
tc
r
8
0
c
kgrowth
,
c
組
be
I
np
a
r
t
i
c
u
l
o
b
t
a
i
n
e
d(
低cordingt
oBa
.
z
a
n
tandKa
.
z
e
mi,1
9
8
8
)出 f
o
l
l
o
w
s(
f
o
rd
e
r
i
v
a
.
t
i
o
n,蹴 Appendix
1
)
:
l
ヱ (α=2
,
αo=22,
c=a-ao)
、 d'
d
R
(
c
)=Gfz:
色
J
g
(α。
)
cJ
--V
'
(
α
)(g
(
α
)
g
(α ¥
g
'
(
α
)
。
)
-, -
,
w
h
i
c
h ー =一一一 l一一一一 α+α。
I
Cf
.
・
U
)
(
4
.
1
)
(
4
.
2
)
whereGJ= f
r
a
c
t
u
r
ee
n
e
r
g
y(
o
b
t
a
i
n
e
df
r
o
mt
h
es
i
z
ee
f
f
e
c
t!
a
w
),
c=a- ao=c
r
a
c
ke
x
t
e
n
s
i
o
n
f
r
o
mt
h
en
o
t
c
ho
ri
n
i
t
i
a
lc
r
a
c
kt
i
p,
g
(
α
);
;n
o
n
d
i
m
e
n
s
i
o
n
a
le
n
e
r
g
yr
e
l
e
a
s
er
a
.
t
ed
e
f
i
n
e
dbyt
h
e
2
t
o
t
a
le
n
e
r
g
yr
e
l
e
a
.
ser
a
.
t
eG i
nt
h
eformG =g
(α)
P
2
/
{
E
b
d
)(
u
s
i
n
gG =K
}/E',
g
(
α)c
組
be
f
o
u
n
df
r
o
mt
h
eKl-v
a
l
u
e
sw
h
i
c
ha
.
r
ea
v
a
i
l
a
b
l
ei
nha
.
n
d
b
o
o
k
ss
u
c
h酎 Tadae
ta
1
.
,1
9
8
5,
and
Murakami,
1
9
8
7,
f
o
rt
h
eba
.
s
i
cs
p
e
c
i
m
e
ng
e
o
m
e
t
r
i
e
s,
andc
a
na
.
lw
a
.
y
sbeo
b
t
a
.
i
n
e
db
y
e
l
a
s
t
i
c
祖 a
l
y
s
i
s
)
;
α =a
o
/
d
,andCf = d
o
g
(
α)
/
g
'
(
α w
h
e
r
ed
sac
o
n
s
t
a
n
tfromt
h
es
i
z
ee
f
f
e
c
t
oi
l
a
w
;CJi
st
h
ee
恥c
t
i
v
el
e
n
g
t
ho
ft
h
ef
r
a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
ei
n叩 i
n
f
i
n
i
t
e
l
yl
a
r
g
es
p
e
c
i
m
e
n(
s
e
e
E
q
.5
.
5
),
d
e
f
i
n
e
da
st
h
ed
i
s
t
a
n
c
efromt
h
en
o
t
c
ht
i
pt
ot
h
et
i
po
ft
h
ee
q
u
i
v
a
l
e
n
te
l
a
凶i
cc
r
a
c
k
.
側
z
o
n
el
e
n
g
t
h(
d
e
f
i
n
e
da
st
h
ed
i
s
t
姐 c
ef
r
o
mt
h
en
o
t
c
ht
i
pt
ot
h
ep
o
i
n
to
f
Thea
c
t
u
a
lp
r
o
c
ma.x泊四もe
n
s
i
!
es
t
r
e
s
s
)i
sa
.
bou
も2
c
j
, a
.
si
n
d
i
c
a
.
t
e
dby∞m
p
a
r
i
s
o
n
sw
i
t
hv
a
r
i
o
u
sm
o
d
e
l
so
f
t
h
ep
r
o
c
e
s
sz
o
n
e(eιHorii,
1
9
8
8
;H
o
r
i
i,
e
ta
l
.,
1
9
8
7,
1
9
8
8
;P
l
姐剖 andElices,
1
9
8
8
a
;Ba
Z8
0
n
t
andKazemi,
1
9
8
8
)
.
.
2d
e
f
i
n
et
h
eRc
u
r
v
ep
a
r
a
m
e
t
r
ic
a
.
l
l
y
.KnowingGfandd
,
onemayc
h
o
o
s
e
E
q
s
.
4
.
1叩 d4
o
.
as
e
r
i
e
sofα-v
a
I
u
e
s,
a
.
ndf
o
re
a
c
ho
fthemc
a
l
c
u
l
a
.
t
e員 凶 cf
r
o
mE
q
.4
.
2組 dt
h
e
nR
(c
)from
E
q
.4
.1
. TheR
c
u
r
v
e
swhich.
a
r
eo
b
t
a
i
n
e
df
r
o
mEq
s
.4
.
14
.
2f
o
rv
e
r
yd
i
f
f
e
r
e
n
ts
p
e
c
i
m
e
n
回 m
e
t
r
i
e
sa
r
ev
e
r
yd
i
f
f
e
r
e
n
tf
r
o
me
a
c
ho
t
h
e
r, 部 i
l
l
u
s
t
r
a
t
e
di
nF
i
g
.4
.
2a
.
.
g
Re
c
e
n
t
l
yi
比
tw
晶制
sf
o
u
n
d但
(
Ba
量
z
.
a
n
帆 G
e
t
t
凶
u組 dK
拙 e
叫
m
1
l
o
a
d
ι
d
e
H
e
c
“
t
も
i
ぬ
∞
onc
u
r
v
ei
お
st
ω
obec
o
r
r
e
c
t
l
yp
r
e
d
i
c
t
e
d,
t
h
e泊
i
n
c
α
r
位
e
a
s
犯eo
fRmustbea
r
r
e
s
t
e
da
tt
h
e
戸e
a
k
l
o
a
dp
o
i
n
t(
p
o
i
n
tp i
nF
i
g
.4
.
2
), 回 i
n
d
i
c
a
t
e
dbyt
h
eh
o
r
i
z
o
n
t
a
Il
i
n
e1
3i
nF
i
g
.4
.
2
.
p
Ther
e
a
s
o
nt
h
ec
r
i
t
i
c
a
le
n
e
r
g
yr
e
l
e
a
.
s
er
a
t
eRc
a
n
n
o
te
x
c
田 dt
h
a
.
tf
o
rt
h
epeakI
o
a
ds
t
8
0
t
ei
s
t
h
a
t
:(
1
)t
h
ei
n
c
r
e
a
s
eo
fR i
sduet
oani
n
c
r
e
a
s
eo
ft
h
ef
r
a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
es
i
z
e(
al
a
r
g
e
z
o
n
ed
i
s
s
i
p
a
t
e
smoree
n
e
r
g
y
),
叩d(
2
)i
nt
h
ep
o
s
t
p
e
a
kr
e
s
p
o
n
s
e,
t
h
ef
r
a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
e
。
。 )
。
・
a
.
v
e
l
sf
o
r
w
8
o
r
dwi
山o
u
tg
r
o
w
i
n
g
d
e
t
a
c
h
e
si
t
s
e
l
ff
r
o
mt
h
et
i
po
ft
h
en
o
t
c
ho
ri
n
i
t
i
a
lc
r
a
c
k8
0
n
dt
r
戸
・ T
h
i
sbeh
a
.
v
i
o
rh剖 p
r
e
v
i
o
u
s
l
yb
e
e
ne
x
p
e
r
i
m
e
n
t
a
l
l
yi
d
e
n
t
i
f
i
e
d,
e
.
g
.,
f
r
o
mm伺 s
u
r
e
m
e
n
t
s
i
ns
i
a
Z
組t,LeeandP
f
e
i
f
f
e
r
,1987)飢 dwasv印 刷 especia
.
l
l
ybyt
倒 so
fr
o
c
k
onalumiDum(B
(B
話 相t
,
G
e
t
t
uandKazemi,
1
9
8
9
)andh
i
g
h
-s
t
r
e
n
g
t
h∞n
c
r
e
t
e(Ge
も
t
u,
B話 回t姐dK.
a
r
r,
1
9
8
9
)
.
Thes
t
u
d
yo
ft
h
eR
c
u
r
v
e
si
n
d
i
c
a
t
e
st
h
a
tt
h
eωn
田p
to
fe
l
a
.
s
t
i
c
a
l
l
ye
q
u
i
v
a
.
l
e
n
te
f
f
e
c
t
i
v
e
c
r
a
c
klmgthc
a
n
n
o
tg
i
v
ev
e
r
ygoodr
e
s
u
l
t
si
f
a
c
o
n
s
t
a
n
tf
r
a
d
u
E
ee
n
e
r
g
yv
a
l
u
ei
s回 目o
c
i
a
t
e
d
w
i
t
h出 ee
f
f
e
c
t
i
v
eロa
c
k
.C
a
l
c
u
l
a
t
i
o
na
c
c
o
r
d
i
n
gt
oE
q
s
.4
.
1
4
.
2i
m
p
l
i
e
st
h
a
tn
o
to
n
l
ymust
t
h
ec
r
a
c
kl
e
n
g
t
h,
a=ao+c
,d
i
t
r
e
rf
r
o
ma
o
,butalsotheenergyreleaseraterequiredf
o
r
.
s
ew
i
t
ht
h
ec
r
a
c
kl
e
n
g
t
hi
fa
.s
o
l
u
t
i
o
nt
h
a
.
ti
s
α
'
a
c
kg
r
o
w
t
hmustbec
o
n
s
i
d
e
r
e
dt
oi
n
α
'
ea
.
a
p
p
r
o
x
i
m
.
a
t
e
l
ye
q
u
i
v
a
l
e
n
tt
on
o
n
l
i
n
e
a
rf
r
a
c
t
u
r
em
e
c
h
a
n
i
c
ss
h
o
u
l
dbeo
b
t
a
i
n
e
d
.T
h
i
si
sn
o
t
白 川l
yp
r
o
p
o
s
e
dm
o
d
e
l
s(
e
.
g
.K
a
r
i
h
a
l
o
o
'
smodel
,S
e
c
.4
.
1and6
.
6
)・
d
o
n
ei
nsomer
e
59
58
。
=
C
h
a
p
t
e
r5_.__SI~E EFFECTANDBPJTTLENESSOF
STRUCTURES
s
e
q
u
e
n
c
eo
ff
ra
.
c
t
u
r
em舵 h叩 i
c
st
h
e
o
r
y
.T
h
e
r
e
f
o
r
e
,
Thes
i
z
ee
f
f
e
di
st
h
emosti
m
p
o
r
t
a
n
t∞n
i
ti
s1
0
g
i
c
a
.
It
od
e
t
e
r
m
i
n
ef
r
a
.
c
t
u
r
ep
r
o
p
e
r
も
i
e
sf
r
o
mi
t
.Knowledgeo
ft
h
es
i
田 e
f
f
e
c
ti
sn
e
e
d
e
d
ゆo
r
a
t
o
r
y
s
i
z
et
o
f
o
rt
h
edesimo
fs
t
r
u
c
M
r
e
hwhichg
e
n
e
r
a
l
l
yi
n
v
o
l
v
e
se
x
t
r
a
p
o
l
a
t
i
o
nfroml
r
叫
“
4
白
g
(
αJ
)~ g
(
α
0
)+g
'
(αo
)
(
c
J
/
d
)
5
.
1SizeEffect LawforMaximumNominalStress
凡
一M
en
N
σ
-
a
.
I
l
ys
i
m
i
l
a
rs
t
r
u
c
t
u
r
e
so
rs
p
e
c
i
m
e
n
sw
i
t
hg
e
o
m
e
t
r
i
α
l
l
ys
i
m
i
l
a
r
Wec
o
n
s
i
d
e
rg
e
o
m
e
t
r
ic
n
o
t
c
h
e
so
Ec
r
a
c
k
s
.Wew
i
l
lnowi
l
l
u
s
t
r
a
t
et
h
ea
n
a
l
y
s
i
sf
o
rs
i
m
i
l
a
r
i
b
yi
ntwod
i
m
e
n
s
i
o
n
s,
hthoughthemultsm t
h
esmeformihi
も
yinもh
r
e
e
-d
i
m
e
n
s
i
o
n
s(Ba
Z
a
n
t,
1
9
8
7
a
)・ The
no
凶n
a
.
l山 e
s
sa
tf
a
i
l
u
r
e(maximuro1
0
0
0
)ma
.
ybed
e
f
i
n
e
da
s
(
5:
1
)
ら
し
い
)
九
(
句
)
a
)
T
h
i
s四 p1'e
s
e
n
t
st
h
es
i
z
ee
f
f
e
c
tl
a
.
we
x
p
r
'
田 s
e
di
nt
e
1
'
mso
fm
a
t
e
r
i
a
lp
a
r
a
.m
e
t
e
r
s(B
話a.n
tand
Kazemi,
1
9
8
8
)
.E
q
.5
.
5i
se
q
u
i
v
a
l
e
n
tt
ot
h
es
i
z
ee
f
f
e
c
tl
a
wσ'
N= B刀(
1+s)-1/2,
β=d(
ゐ
.4
)i
fonemakest
h
en
o
t
a
t
i
o
n
s:
(
E
q
.1
。τり
一
q
α
i
j
、
u
fr
一
,
s
'
、
司
・
c
w
d
n3 n
(
5
.
3
)
内切
。
+(c/d)
1
a
B ~ ~(蒜7y/2
(
5
.
6
)
1
tm晶yben
o
t
e
dt
h
a
tamoreg
e
n
e
r
a
ls
i
z
ee
f
f
e
c
tl
a
.
wh
a
sb
e
e
nd
e
r
i
v
e
d
;
t
σN =B
I
:
(
1+
s
r 12
r
/
,
b
u
trd
e
p
e
n
d
songeomet
1
'
y,
andf
o
ra
l
lg
句 m
e
t
r
i
e
scombinedt
h
ev
a
l
u
er~ 1wasf
o
u
n
dt
o
bea
p
p
r
o
x
i
m
a
t
e
l
yoptimum(Ba
Z
a
n
tandP
f
e
i
f
f
e
r,
1
9
8
7
)
.
I
ti
sf
u
r
t
h
e
ru
s
e
f
u
lt
or
e
w
r
i
t
et
h
es
i
z
ee
f
f
e
c
tl
a
wi
nt
h
eform
/
叩=
(
部y
ムィ
i
nw
h
i
c
hg
(
α
)i
saknownf
u
n
c
t
i
o
no
ft
h
e凶 t
i
v
ec
1
'
a
c
kl
e
n
g
t
h
α =a
/
d
. Be
c
a
.
u
s
eG =
K]/
E,
f
u
n
c
t
i
o
ng
(
α)maybeo
b
t
a
i
n
e
df
r
o
mt
h
eexpmSionsf
o
rths
t
r
e
s
sintemtyf
a
c
t
o
r
s,
h
i
c
ha
r
ea
l
w
a
y
so
ft
h
eformKI= Pk(α)
/.
,
f
dwherek(α)=non-dimensionalfunction.For
t
y
p
i
c
a
ls
p
e
c
i
m
e
ng
e
o
m
e
t
r
i
e
s,
f
u
n
c
t
i
o
nk
(α
)c
叩
bed
e
t
e
r
m
i
n
e
dfromt
h
eKIva
.
l
u
e
sa
v
a
i
l
a
b
l
e
i
nhandbooksandb
e
x
t
b
o
o
k
s,
andf
o
ro
b
h
e
rg
E
o
m
d
r
i
e
si
tcmbeo
b
t
a
i
n
e
dbyl
i
n
e
a
re
l
a
s
U
c
a
n
a
l
y
s
i
s
N
o
n
l
i
n
e
a
rf
r
a
c
t
u
r
ew
i
七h a
.s
o
f
t
e
n
i
n
gc
o
h
e
s
i
v
e(
f
r
a
c
t
u
r
ep
r
o
c
e
s
s
)z
o
n
ei
sa
.
pproxim
a
.
t
e
l
y
e
q
u
i
v
a
.
I
e
n
tt
ol
i
n
e
a
.
re
l
a
s
t
i
cf
ra
.
c
t
u
r
eo
f回 i
n
c
r
e
a
s
e
dc
r
a
c
kl
e
n
g
t
h
(
5
.
5
)
σN=
一
一
i
nw
h
i
c
hP
.
. u
l
t
i
m
a
t
el
o
a
d,
b=t
h
i
c
k
n
e
s
so
ft
h
es
t
r
u
c
t
u
r
e
,
d=c
e
r
t
a
i
nc
h
o
s
e
nc
h
a
r
a
c
t
e
r
i
s
t
i
c
fもhes
t
r
u
c
t
u
r
e
,
andC
o
e
伍c
i
e
n
ti
n
t
r
o
d
u
c
e
df
o
r∞n
v
e
n
i
e
n
c
e
.(
F
o
rexample
,
i
n
n=c
蹴
o
fb
e
n
d
i
n
go
f乱 s
i
m
p
l
y叫 p
o
r
t
e
dbe
a
.
mo
fs
p
a
nla
.
n
dd
e
p
t
h,
d,
w
i
t
h∞ncentmtedlO
吋
t
h
ec
2
抗 出d
span,
wemayi
n
t
1
'
oduceσN品目 t
h
emaximumb
e
n
d
i
n
gs
t
r
e
,
回i
.
e
.
σN =3
凡l
/
2
b
d=
ιpjbdwhereら = 1
.5
l
/
d= c
o
n
s
t
a
n
t
;b
u
twec
a
ne
q
u
a
l
l
yw
e
l
lt
a
k
et
h
ec
h
a
r
a
.
c
t
e
1
'i
s
t
i
c
34hsidnω d=l=自pan
,anddenotingh=beamdepbhwethenhaveσN = c
.
.
Pu/b
I
.
wherecn=322/2M=consunt,
p
r
o
v
i
d
e
d
b
h
ebeamsa
r
es
i
m
i
l
a
r
)
.
斗 σゐ
/2Er
e
p
r
飴 e
n
t
st
h
enomina
.
Is
t
ra
.
i
ne
n
e
r
g
yd
e
n
s
i
t
y
,
もh
e町 a
i
ne
n
e
r
g
yi
5
N
o
t
i
n
gt
h
u= V(σゐ/2E)ψ(α),whe1'eV = 匂 bd
2= volumeo
ft
h
es
t
r
u
c
t
u
r
e(
e
o= c
o
n
s
t
a
n
t
)姐 d
=GId-Thee
n
e
r
g
yr
e
l
e
a
s
er
a
t
ei
sg
i
v
e
nbybG=-OU/θa=一δ
(U/
θa
)
/
d
.Nowwitht
h
e
n
o
t
a
t
i
(
α
)
C
!
e
o
/2=g
(
α),
t
h
ee
n
e
r
g
yr
e
l
e
蹴 凶eo
ft
h
es
t
r
u
c
t
u
r
ei
sf
o
u
n
dt
oh
a
v
et
h
e
form
P
2
g
(α)σμ
(
5
.
2
)
G=一 一 一 =--g(α)
2d c!E
Eb
α= 向 +C o
r α=α
(
5.
4
)
w
h
e
r
eαJ= c
J
/
d組 dg
'
(
α。
)= d
g
(
α。
)
/
d
α
.1
ti
sr
e
a
s
o
n
a
b
l
et
oassumet
h
抗 出eG
v
a
l
u
ea
t
.
lt
ot
h
eg
(
α
)
-va
.
I
uea
tt
h
ep
e
a
k1
0
0
0,
i
.
e
.
t
h
ep
e
a
k
l
o
a
di
sn
o
tc
o
n
s
t
a
n
tb
u
ti
sp
r
o
p
o
r
t
i
o
na
G= G
J
g
(
α)
/
g
(
αJ
)
.S
u
b
s
t
i
t
u
t
i
n
gt
h
i
sandEq.5.
4i
n
t
o5
.
2,
s
e
t
t
i
n
g凡 =σ
(N
b
d
/
c
2,
and
n)
s
o
l
v
i
n
gf
o
rσN,
o
n
eo
b
t
a
i
n
s
r
e
v
i
e
wi
t
sf
,
お
orm叫
u
1a
.
叫
b
i
∞
o
nandc
o
n
s
e
q
u
e
n
偲 5
i
nf
u
r
t
h
e
rd
e
t
a
i
l
.
。
w
h
e
r
eα t
h
ea
c
t
ua
.
l
l
e
n
g
t
ho
ft
h
ec
o
n
t
i
n
u
o
u
sc
r
a
.
c
ko
rn
o
t
c
h,
a
.
n
dc= d
i
s
t
a
.
n
c
eo
ft
h
et
i
p
o
ft
h
ee
q
u
i
v
a
l
e
n
te
l
a
.
s
t
i
cc
r
a
.
c
kfromt
h
et
i
po
ft
h
ei
n
i
t
i
a
lc
r
a
.
c
ko
rn
o
t
c
h(
F
i
g.4
.
2
)
.
.
t9
'
(
α
)>0,w
h
i
c
ha
p
p
l
i
e
st
ot
h
ema
.
j
o
r
i
t
yo
fs
p
e
c
i
m
佃
gωmetri
,
田 the
Assumingt
ha
f
r
a
d
u
r
ep
r
o
c
e
s
sz
o
n
ef
i
r
s
tg
r
o
w
si
ns
i
z
ew
h
i
l
er
e
m
a
i
n
i
n
ga
t
t
a
c
h
e
dt
ot
h
et
i
po
fn
o
t
c
ho
r
i
n
i
t
i
a
.
Ic
r
a
c
k
.S
i
n
c
et
h
ev
a
l
u
eo
fGmustdependont
h
el
e
n
g
t
ho
ft
h
ef
r
a
.
c
もu
r
ep
r
o
c
e
s
sz
o
n
e,
i
t
a
.
c
t
u
r
ee
n
e
r
g
yGJmaybe∞n
s
i
d
e
r
e
dt
obet
h
ev
a
l
u
eo
fGr
e
a
.
c
hed
r
o
u
s
tdependonc
.Thef
r
出 sC;
:
:c
Jwherec
Ji
sa
.r
o
a
t
e
r
i
a
lc
o
n
s
ta
.
n
t
. Thev
a
l
u
eo
fCa
tmaximuml
o
a
.
d
f
o
rc
r
a
c
kl
e
n
g
i
se
q
ua
.
It
oc
Jo
n
l
yf
o
rani
n
f
i
n
i
t
e
l
yl
a
r
g
es
p
e
c
i
m
e
n(
d→∞), buti
ss
m
a
l
l
e
rt
h
a
nc
Ji
nf
i
n
i
t
e
.
n
s
i
o
n
,
weh
a
v
e
s
i
z
es
p
e
c
i
m
e
n
s
.BasedonT
a
y
l
o
rs
e
r
i
e
se
x
pa
2
nw
h
i
c
h
(
5.
7
)
】
九
ロ7 てPu
n_ g
(
α。
)
=.
j
g
'
(
的)話 , D =訳可d
(
5.
8
)
~ere.7'N a
.
n
dD maybei
n
t
e
r
p
r
e
t
e
da
st
h
es
h
a
p
e
i
n
d
e
p
e
n
d
e
n
tn
o
m
i
n
a
ls
t
r
e
s
sa
tf
a
i
l
u
r
eand
a
.
pe
i
n
d
e
p
e
n
d
e
凶 c
h
a
r
a
c
t
e
r
i
s
t
i
cdimen
討o
no
ft
h
e山 u
c
t
u
r
e(Ba.孟姐も姐dK
a
z
e
r
r
u,
1
9
8
8
;
t
h
es
h
Z
a
.
n
t
,
G
e
t
t
u姐 dKazemi,
1
9
8
9
)・
Ba
60
61
1
nt
h
ec
回 eo
ft
h
r
e
&
d
i
m
e
n
s
i
o
n
晶l
自i
凶l
a
r
i
t
y
,
weh
a
v
eU = V(σ
ゐ/2E)ψ(α),V =匂♂ and
C
l
d
G= o
U
/
δa=一
θ
(U
/
δa
)
/
di
nw
h
i
c
he
o組 dC
la
r
esomec
o
n
s
t
a
n
t
s
.D
e
n
o
t
i
n
g
0
.
0
-
・
4
巳
.
宅 -u
E.M
z
b
a
・
1
thq
•.
点M
dnu
a‘,
a
{
,
,
・
,
,
.t
a
u
-
a
-
-
白
u'
•• .•
-
M
•
内
f
t
'
(
α
)
e
o
/2C
tc
!=g(α)weob凶 inagainEq.5.2,andalltheremainingderivationi
st
h
es
a
m
e
.
t
h
er
e
s
u
l
も8a
r
ea
g
a
i
nt
h
esameaBE
q
.5
.
5 5
.
8
.
T
h
e
r
e
f
o
r
e,
.
f
e
c
tl
a
w(
E
q
.1
ム5ム 5.7)国 na180bederivedverygenerallybymeansofd
i
Thes
i
z
ee
m
e
n
s
i
o
n
a
la
n
a
l
y
s
i
sands
i
凶 l
i
t
u
d
ea
r
g
u
m
e
n
t
s(
s
e
eBa
Z
a
n
t,
1
9
8
4
a
,
1
9
8
5
b,
f
o
rもwod
i
m
e
n
s
i
o
n
s
;
andB話 a
n
t,
1
9
8
7
a
,
f
o
rt
h
r
田
.d
i
m
e
n
s
i
o
n
s
)i
ft
h
ef
o
l
l
o
w
i
n
gh
y
p
o
t
h
e
s
i
si
sadop
もe
d
:
ぽ g
yr
e
l
e
a
s
ef
r
o
mt
h
es
t
r
u
c
t
u
r
eo
fany日i
z
edi
n
t
ot
h
ef
r
a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
e
Thet
o
t
a
len
d
e
p
e
n
d
sonb
o
t
h
:(
1
)t
h
el
e
n
g
t
h
,ao,oftheωntinuouscraιkornotch,and(
2
)al
e
n
g
t
h
∞n
s
t
a
n
to
ft
h
em
a
t
e
r
i
a
l.
1
ぽ e
i
omayrepresentthematerialconstantcombination
0
・H
l
o= E'G
J
!f
?(Irwin'scharacteristics
i
z
eo
fn
o
n
l
i
n
e
a
rn
e
a
r
t
i
pz
o
n
e
),
o
rt
h
ee
.
f
e
c
t
i
v
e1
田 g
t
h
C
Io
ft
h
ef
r
a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
ei
n姐 i
n
f
i
n
i
t
e
l
yl
a
r
g
es
p
e
c
i
m
e
n
,o
rt
h
ee
f
f
e
d
i
v
ew
i
d
t
hωco
f
fo
n
l
yp
a
r
t(
1
)o
ft
h
i
sh
y
p
o
t
h
e
s
i
si
sa
d
o
p
t
e
d
,
oneo
b
t
a
i
n
st
h
es
i
z
ee
.
f
e
c
to
fl
i
n
e
a
r
t
h
i
sz
o
n
e
.I
e
l
aB
t
i
cf
r
a
c
t
u
r
em
e
c
h
a
n
i
c
s(σNα d-1!
,
)
l
' 組 di
fo
n
l
yp
a
r
t(
2
)i
sa
d
o
p
t
e
d,
oneo
b
t
a
.
i
n
st
h
a
もo
f
p
l
a
s
t
i
cl
i
m
i
ta
n
a
l
y
s
i
s(
i
.
e
.nos
i
z
ee
.
f
e
c
t
)
.
舵 k
Fromt
h
i
sh
y
p
o
t
h
e
s
i
si
ta
p
p
e
a
r
st
h
a
tt
h
es
i
z
ee
f
f
e
c
tl
a
wi
sr
e
s
t
r
i
c
t
e
dn
e
i
t
h
e
rt
ot
h
ec
r
bandmodeln
o
rt
ot
h
el
i
n
ec
r
a
c
kmodelb
u
ti
sv
a
l
i
df
o
ranyn
o
n
l
i
n
e
a
.
rf
r
a
c
t
u
r
emodelw
i
t
h
al
a
r
g
ep
r
o
c
e
s
sz
o
n
e
.S
i
n
c
et
h
el
e
n
g
t
ho
rw
i
d
t
ho
ft
h
ef
r
a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
ei
sr
e
l
a
t
e
dt
o
t
h
ec
h
a
r
a
c
t
e
r
i
s
t
i
cl
e
n
g
t
hio
fa
.n
o
n
l
o
c
a
lc
o
n
t
i
n
u
u
m
,
t
h
es
i
z
ee
f
f
e
c
tlawa
l
s
oa
p
p
l
i
e
sf
o
rt
h
e
n
o
n
l
o
c
a
lcontinuuma
p
p
r
o
a
c
ht
of
r
a
c
t
u
r
e
.T
h
i
sh
a
sb
e
e
nd
e
m
o
n
s
t
r
a
t
e
dbyf
i
n
i
t
ee
l
e
m
e
n
t
s
f
o
ras
i
m
p
l
en
o
n
l
o
c
a
lsmearedc
r
a
c
k
i
n
gmodeli
nBa
Z
a
n
t祖 dL
i
n(
1
9
8
8
b
)(
s
e
eF
i
g
.3
.
6
)姐 d
f
o
rt
h
en
o
n
l
o
c
a
.
Jm
i
c
r
o
p
l
a
.
n
emodeJ(
w
h
i
c
ha
l
s
od
e
s
c
r
i
b
e
sw
e
l
ln
o
n
l
i
n
e
a
rt
r
i
a
x
i
a
lb
e
h
a
v
i
o
ri
n
c
o
m
p
r
e
s
s
i
v
es
t
前田)b
yB
a
z
a
n
tand0量b
o
l
t(
1
9
8
9
)(
F
i
g
. 3.
9
)
.P
a
r
t
i
c
l
es
i
m
u
l
a
t
i
o
no
fr
a
D
Tabbara
,
KazemiandP
i
j
a
u
d
i
e
r
C
a
b
o
t,
1
9
8
9
)山 os
h
o
w
sg
o
o
d
domm
i
c
r
o
s
t
r
u
c
t
u
r
e(
B
a
z
a
n
t,
a
g
r
e
e
m
e
n
tw
i
t
h出 es
i
z
ee
f
f
e
c
tl
a
w
.
cc
r
i
t
e
r
i
o
nf
o
ra
c
c
e
p
t
a
b
i
J
i
t
yo
fn
o
n
l
i
n
e
a
rf
i
n
i
t
ee
l
e
m
e
n
もc
o
d
e
sf
o
r∞n
c
r
e
t
es
t
r
u
c
t
u
r
e
s
AbaBi
i
st
h
a
tt
h
e
ymustd
e
s
c
r
i
b
et
h
et
r
a
n
s
i
t
i
o
n
a
ls
i
z
ee
.
f
e
c
t(
E
q
.1
.
4
)
.
f
e
c
t]
a
ww
i
t
hf
r
a
c
t
u
r
et
e
s
td
a
t
af
o
rd
i
.
f
e
r
e
n
ts
p
e
c
・
Ane
x
t
e
n
s
i
v
ec
o
m
p
a
r
i
s
o
no
ft
h
es
i
z
ee
imeng
e
o
m
e
t
r
i
e
sf
o
rc
o
n
c
r
e
t
eωwell剖 m
o
r
t
a
ri
sf
o
u
n
di
nB
a
z
a
n
t叩 dp
f
e
i
恥 r(
1
9
8
7
)
;s
e
e
o
/
d
,
,
)
.Theset
田 t
sh
a
v
eshownt
h
a
tv
e
r
yd
i
.
f
e
r
e
n
tt
y
p
e
so
fs
p
e
c
i
m
e
n
s
F
i
g
.5
.
1(
w
h
e
r
eA
o=d
,
c
回 t
r
i
ct
e
n
s
i
o
n
,
e
c
c
e
n
t
r
i
cc
o
m
p
r
e
s
s
i
o
n
)y
i
e
l
da
b
o
u
tt
h
esamef
r
a
c
t
u
r
ee
n
e
r
g
y(
e
v
e
n
(
b
e
n
d
i
n
g
t
l
町 p
a
r
a
m
e
t
e
rv
a
l
u
e
si
nF
i
g
.5
.
1d
i
.
f
e
rc
o
n
s
i
d
e
r
a
b
l
y
)
.
t
h
o
u
g
h出 eo
1
.
1
.
5
••
(
,
,
)
0
.
0
0
.
.
.
.
.
.
.
.
. __J
ι
.
。
J
e
c
tLaw
F
i
g
.
5.
1C
o
m
p
a
r
i
s
o
no
fS
i
z
eE1
.
・L
r
.
伺tR
.
e
a
u
l
t
son
E
q
.4
.
1
)w
I
t
h'
ミ・O.IO~IS叫
制
B・08
8
8
.
.
Y'.Y
‘
.
:
I
e
t
r
i
e
s,
S
p
e
c
i
m
e
n
so
fV
a
r
i
o
u
sG回 m
f
o
rh
o
t
hC
o
n
c
r
e
t
ea
n
dM
o
r
t
a
r(
a
.
f
t
e
r
~伺・ 18.80
8・O.15i .
o
・
'0
.
8
・
加.
1
_T)}})vJn:J
20
.
.
.
.
0
.
I .
.
.
.
.
・4凶ー~
・0.25・
0
.
. 0
.
8
Ba.Z姐t姐 .
dPf
e
i
1
J
e
r1
9
8
7
)
I
0-COnet.le
!
:
I
-¥
(orb.p
0
.
8
.
.
ー
・
1
.
0
1
.
2
l
o
C(
d
/
da)
・
.
J
1
.
8
,"
・
t1
d
l
v
l
c
l
.
.
.
1T
.
.
u
0.0
、
・
,
c
'・
o
.
z
2
、
、
0
.
'
:
Z
:
b
v
・
~ -0.
・
-0.
1
.
0
0.
'
・ ・u10&(d;a ) ・
0
O
.
U
1
.
a
1
.8
・
1
5
.
2BrittlenessNumber
(
b
)
E
q
.5
.
6e
x
p
r
e
s
s
e
s七hes
i
z
ee
.
f
e
c
tl
a
wi
nt
e
r
m
so
fm
a
t
e
r
i
a
lf
r
a
c
t
u
r
ep
a
r
a
m
e
t
e
r
s
,
GIandcJ
(
c
o
n
v
e
r
s
e
l
y
,
t
h
e
i
rd
e
f
i
n
i
t
i
o
nandmωurementmaybeb
a
s
e
dont
h
es
i
z
ee
.
f
e
c
tl
a
w
,
E
q
.1
.
4
)
.
T
h
i
sf
a
c
twase
x
p
l
o
i
t
e
dbyBa
Z
a
n
t(
1
9
8
7
乱
)t
od
e
f
i
n
et
h
陀e
砂 c
a
l
l
e
吋d凶
b
r
r
i
川
も
“
川
t
も
l
e
問
n
e
s
凶SD
l
1
即
a
p
p
r
o
x
i
m
a
t
e
l
yd
e
s
c
r
i
b
e
st
h
eb
r
i
t
ω
t
l
e
n
e
s
s0
ぱf
t
仕
出
hes
t
r
u
c
t
u
r
a
lr
e
s
p
o
n
s
er
e
g
a
r
d
l
e
s
so
ft
h
es
t
r
邸
u
1
κ
c
t
u
r
印e
s
h
a
p
e
.BasedonE
q
'5
.
7,
t
h
eb
r
i
t
t
l
e
n
e
s
snumberi
se
x
p
r
e
s
s
e
daB(Ba
Z
a
n
tandKazemi,
1
9
8
8
)
:
“
1
0
mall da'
'
'
'
'
. 90
'
N
~i__sma t1da
F
i
g
.
5
.
2C
u
r
v
e
so
fS
t
r
u
c
t
u
r
a
l
J
d 一勺
α
‘,
'
F
、ta、
a -JaE
何一
一
一
,
、
‘
,
g
コ
切
、
g
D 一勺
ρ
μ
一
一
官
S
i
z
eE
f
f
e
c
tf
o
rD
i
f
f
e
r
e
n
(
5
.
9
)
A邸 r
e
g
a
.
t
eS
i
z
e
s
logd
63
6
2
w
h
i
l
eont
h
eb
a
s
i
so
ft
h
es
i
z
ee
f
f
e
c
tl
a
.
we
x
p
r
e
s
s
i
o
ni
nE
q
.1
.
4i
ti
se
q
u
i
v
a
l
e
n
t
l
ye
x
p
r
e
s
s
e
d酪
(Ba
.
Z
叩t
,
1
9
8
7
a
;BazantandP
f
e
i
f
f
e
r
,
1
9
8
7
)
:
,
r
t
2
s=え=内州合;=AB2fd
(
5
.
1
0
)
)'i
vD
,
.
,,.‘、
a
sく 0
.
1
p
l
a
s
t
i
cl
i
m
i
ta
n
a
l
y
s
i
s
0
.
15
.s5
.10 nonlinearf
r
a
.
c
t
u
r
em
e
c
h
a
n
i
c
s
s>10 l
i
n
e
a
re
l
a
s
t
i
cf
r
a
c
t
u
r
em
e
c
h
a
n
i
c
s
F
o
rs<0
.
1,
t
h
ee
r
r
o
ro
fもhep
l
a
s
t
i
cl
i
m
i
ta
n
a
l
y
s
i
si
sl
e
s
st
h
a
.
n4.7%o
fσN・
andf
o
r
,
comparedt
ot
h
e
β>loth
ee
r
r
o
ro
fl
i
n
e
a
re
l
a
s
t
i
cf
r
a
c
t
u
r
em
e
c
h
a
n
i
c
si
sa
l
s
ol
e
s
sthan4.7%
n
o
n
l
i
n
e
a
rf
r
a
c
t
u
r
em
e
c
h
a
n
i
c
ss
o
l
u
t
i
o
n
.I
fane
r
r
o
ru
n
d
e
r2%i
sd
e
s
i
r
e
d,t
h
e
nt
h
en
o
n
l
i
n
e
a
r
ngemustbeexp
a
.
n
dedt
o1
/
2
5壬9三2
5
.
O
t
h
e
rd
e
a
n
i
t
i
o
n
so
ft
h
eb
r
i
t
t
l
e
E
l
e
s
snumbero
fa
s
b
r
u
c
t
u
r
ehaveb
e
e
np
r
o
p
o
s
e
db
e
f
o
r
e
.
r
e
s
e
a
r
c
h
e
r
si
nr
n
d
a
l
Eknewt
h
a
ts
t
r
u
c
t
u
r
a
lb
r
i
t
s
l
e
n
e
s
si
sb
a
s
i
B
e
g
i
n
n
i
n
gw
i
t
hI
r
w
i
n
(
1
9
5
8
),
c
a
l
l
yc
h
a
r
a
c
t
e
r
i
z
e
dbyt
h
er
a
t
i
oo
ft
h
es
t
r
u
c
t
u
r
es
i
z
et
oI
r
w
i
r
i
'
ss
i
z
eo
ft
h
en
o
n
l
i
n
e
a
rz
o
n
e
,
E
q
.2
.
1
0
.T
h
i
sd
e
f
i
n
i
t
i
o
nw回 c
o
o
p
t
e
df
o
rc
o
n
c
r
e
t
es
t
r
u
c
t
u
r
e
sbyH
i
l
l
e
r
b
o
r
g(
1
9
8
5
b
)(
c
f
.
a
l
s
oE1fgml,
1989)whop
r
o
p
o
s
e
dt
oc
h
a
r
a
c
t
e
r
i
z
et
h
es
t
r
u
c
t
u
r
a
lb
r
i
t
t
l
e
m
s
sbyt
h
er
a
t
i
od
/
ゐ
九一
何
日 C
Meme1hanicB
Eq59makesitpossiblehoedcuMepsolelyonthebuisofalinearelasticf
、f
o
rwhicht
h
es
h
a
p
eandt
h
el
e
n
g
t
ho
ft
h
eο
c凶 n
u
o
u
sc
r
a
.
c
ka
tu
l
t
i
m
a
t
el
o
a
.
d(
o
rt
h
e
s
o
l
u
t
i
o
n
n
o
t
c
hl
c
n
g
t
hi
naf
r
a
c
t
u
r
es
p
e
c
i
m
e
n
)mustbeknown.
伺 ?
witht
h
et
r
a
l
s
i
t
i
o
no
f
t
h
es
i
Z
E
Eq.510yieldst
h
eb
I
i
t
t
l
e
n
e
s
BE
1
1
I
n
b
e
r
w
h
i
c
h
e
x
a
c
t
l
ya
g
E
e
f
T
e
c
hl
a
wt
ot
h
ep
l
a
s
t
i
ca
n
a
l
y
s
i
ss
o
l
u
t
i
o
nwhent
h
es
t
r
u
c
t
u
r
e日i
z
ei
sv
e
r
ys
m
a
l
l
.Coe
伍c
i
e
n
もB
n
P
u
/
b
d
f
:whereP
u
.= u
l
t
i
m
a
.
t
el
o
a
d蜘 e
s
sa
.
c
c
o
r
d
i
n
gt
oap
l
越も i
c
c
a
nbed
e
t
e
r
m
i
n
e
da
sB=c
l
i
m
i
ta
n
a
l
y
s
i
sform
山 f
o
rt
h
es
t
r
u
c
t
u
r
a
.
lf
a
i
l
u
r
e
,s
叫 a
semploy
出 nt
h
ec
u
r
r
e
n
tωdesや
r
d
i
a
g
o
n
a
ls
h
e
a
zf
a
i
l
u
r
e
,
p
u
n
c
h
i
n
gs
h
e
a
rf
a
i
l
u
r
e
,
t
o
r
s
i
o
n
a
lf
a
i
l
u
r
e
,
etc.Eq.510ismoEea
c
c
u
r
a
巴
f
o
r自m
a
l
ls
,
w
h
i
l
eE
q
.5
.
9i
smorea
ι
c
u
r
a
t
ef
o
rl
a
.
r
g
es
.
Asat
h
i
r
dmethodf
O
i
:d
e
t
e
r
m
i
n
i
n
gb,
i
f出 emaximuml
o
a
d
sf
o
rag
i
v
e
ns
t
r
u
c
t
u
r
a
lgeome
t
r
y紅 eknownf
o
rf
:
li
g
n
i
五c
a
n
t
l
yd
i
f
f
e
r
e
n
ts
i
z
e
s,
e
.
g
.,
ont
h
eb
a
s
i
so
fl
a
.
b
o
r
a
t
o
r
yt
e
s
t
so
ront
h
e
b
a
s
i
so
fS
I
l
i
t
ee
l
e
m
e
n
ta
I
l
a
l
y
s
i
sbasdonsomemethodo
fn
o
n
l
i
n
e
a
rf
r
a
c
t
u
r
emιh
姐 i
c
s,
t
h
e
n
r
e
g
r
e
s
s
i
o
no
fもhet
e
s
tr
e
s
u
l
t
sbym位 n
so
ft
h
es
i
民 e
f
f
e
c
tl
a
wy
i
e
l
d
st
h
ep
a
r
a
m
e
t
e
rd,
from
whichb=d
/
ゐ
AItemauveiyi
も品I
s
os
u
伍C
国 t
oo
b
t
a
i
nt
h
ef
a
i
l
u
r
el
o
a
dP
uf
o
rav
e
r
ys
m
a
l
ls
i
z
e(
s→ 0
)by
plMicl
i
m
i
ta
n
a
i
y
s
i
s,
whichy
i
e
l
d
sB=du/
叫, a
.
n
dt
h
e
nd
e
t
釘 m
ineσN fromt
h
eLEFM
whichy
i
e
l
d
sd
o
/
d=σ
(N/B
f
D2.
o
l
u
t
i
o
n,
Thev
a
l
u
eF=1(ord=do)correspondsinU1esimefectp
l
o
ho
fl
o
g
σN v
s
.l
o
gdt
ot
h
e
r
i
z
o
n
t
a
l掛 Y
町 t
o
t
ef
o
rt
h
es
t
回 附hc
山 r
i
o
nandt
h
ei
n
d
i
n
e
d出 ymFo
七
e
p
o
i
n
twheret1叫o
f
o
rもhel
i
n
ea
.
re
l
a
s
t
i
cf
r
a
c
t
u
r
emech
姐 i
c
si
n
t
e
r
s
e
c
t
;F
i
g
.1
.4
.F
o
rs<1
,
t
h
eb
e
h
a
v
i
o
ri
sc
l
o
t
op
l
a
s
t
i
cl
i
m
i
t叩 乱l
y
s
i
s
,andf
o
rs> 1i
ti
sc
l
o
s
e
rt
ol
i
n
ea
.
re
l
a
s
t
i
cf
r
a
c
t
u
r
emech
岨 l
C
S・
I
f
s<
t
:
:1o
rs>
>1,
n
o
n
l
i
n
e
a
rf
r
a
c
t
u
r
em
e
c
h
a
n
i
c
si
sn
o
tn
e
印 s
s
a
r
y
.Themethodo
fa
n
a
l
y
s
i
smay
.
Z
a
n
t,
1
9
8
7
;Ba
.
Z
a
n
tandP
f
e
i
狂e
r,
1
9
8
7
)
:
bec
h
o
s
e
na
sf
o
l
l
o
w
s(Ba
w
h
e
r
ee
f
?(hecalleae
h
e“
c
h
a
r
a
c
t
e
r
i抗 i
c
l
叩 g
t
h
"
;b
u
tt
h
i
sc
o
n
s
i
c
t
sw
i
t
hp児吋・
o= EGJI
ot
o
u
st
e
r
m
i
n
o
l
o
g
yi
nn
o
n
l
o
c
a
lc
o
n
t
i
n
u
u
mt
h
e
o
η
)
.T
h
e
r
e
f
o
r
ewew
i
l
lc
a
l
l
l
ot
h
ec
h
a
r
a
c
t
e
r
i
s
t
i
c
.
r
p
i
n
t
e
r
i(
1
9
8
2
)p
r
o
p
o
s
e
dt
h
eb
r
i
t
t
l
e
n
e
s
snumbers= Gj/f
:
d
.Simi1a
.
r
p
r
ω
e
s
sz
o
n
es
i
z
e
. Ca
d
e
f
i
n
i
t
i
o
n
swぽ e回 r
l
i
e
rp
r
o
p
o
s
e
df
o
rc
e
r
a
m
i
c
sbyG
o
g
o
t
s
ie
ta
l
.(
1
9
7
8
)岨 dHomenye
ta
l
.
(
1
9
8
0
).Theb
r
i
t
t
1
e
n
e
s
snumberso
fm
削
凹l
l
的
e
E
出b
叫
o
r
喝
g,
Cω
むa
叫
r
叩
巾
戸
p
叩
阿
i
泊
n
t
凶
釘
凶i
,Gゆ
o
g
酔o
t
旬
凶
s
討i凶
a
n
吋d恥
H
Io附m
l
紅 en
o
ti
n
d
e
p
e
n
d
e
n
to
ft
出
hes
t
r
u
c
t
u
r
eg
印
e
ome
抗t
r
可
y却 dt
山
h凶u
sc
a
n
n
o
tbeu
s
e
da
su
n
i
v
e
r
s
a
l
,
a
.
b
so
l
u
t
ec
h
a
r
a
.
c
t
e
r
i
s
t
i
c
s(
e
.
g
.,
s= 3couldthenm品 naveryb
r
i
t
t
l
eb
e
h
a
v
i
o
rf
o
rones
t
r
u
c
t
u
r
e
gωmetry組 dav
e
r
yd
u
c
t
i
l
eb
e
h
a
v
i
o
rf
o
ra
n
o
t
h
e
rg
e
o
m
e
t
r
y
)
.
A
l
t
h
o
u
g
hE
q
s
.5
.
9and5
.
1
0d
e
f
i
n
et
h
eb
r
i
t
t
l
e
n
e
s
snumberi
nr
e
l
a
t
i
o
nt
ot
h
es
i
z
ee
f
f
e
c
t
もi
m
a
t
el
o
a
d,
もh
eb
r
i
t
t
l
e
n
e
s
snumber.
a
l
s
od
e
t
e
r
m
i
n
e
st
h
en
a
t
u
r
eo
ft
h
ep
o
s
t
p
e
a
.
k
ont
h
eu
l
r
e
s
p
o
n
s
e
,
p
a
r
t
i
c
u
l
a
r
l
yt
h
es
t
e
e
p
n
悶
o
ft
h
ep
o
s
t
p
e
a
.
ks
o
f
t
e
n
i
n
g(
1
0
0
0d
e
c
r
e
a
s
e叫 i
n
c
r
e
a
s
i
n
g
d
i
s
p
l
a
c
e
m
e
n
t
)
.Thatwasa
l
r
e
a
d
yi
l
l
u
s
t
r
a
t
e
dbyt
h
es
e
r
i
e
smodeli
nF
i
g
.1
.1
3
.
.
Z
叩t
,
i
tappe
町 st
h
a
t
I
nv
i
e
wo
ft
h
eu
n
i
v
e
r
s
a
l
i
t
yo
ft
h
eb
r
i
t
t
l
e
n
e
s
snumherp
r
o
p
o
s
e
dbyBa
as
i
m
p
l
ea
d
j
u
s
t
m
e
n
to
ft
h
ec
u
r
r
e
n
tl
i
m
i
t
a
n
a
l
y
s
i
s
-b
a
s
e
dc
o
d
ef
o
r
m
u
l
a
s,
t
a
k
i
n
gi
n
t
oa
c
c
o
u
n
t
a
.
c
t
u
r
em配 h
a
n
i
c
s
,
c
a
nb
emOOebyr
e
p
l
a
c
i
n
gt
h
en
o
m
i
n
a
ls
t
r
e
s
sa
tu
l
t
i
m
a
t
el
o
a
a,
n
o
n
l
i
n
e
a
rf
r
,
九a
sg
i
v
e
nw
i
t
h
o
u
tc
o
n
s
i
d
e
r
a
t
i
o
no
fs
i
z
ce
f
f
e
c
t,
byt
h
ee
x
p
r
四 S
l
o
n
(三 v~n)
(
5
.
1
2
)
However
,themethodtocalculated
o,
onw
h
i
c
hsd
e
p
e
n
d
s,
s
t
i
l
ln
e
e
d
st
ober
e
s
e
a
.
r
c
h
e
d
.
E
q
.5
.
1
2i
n
d
i
c
a
.
t
回 t
h
a
tt
h
e
r
em
i
g
h
tbeal
o
w
e
rl
i
m
i
tv:
:
a
mon出 enominalstrength,
duet
oa
心l
u
r
em配 h
a
n
i
s
m
.Anexampleo
fs
u
c
ha
p
o
s
s
i
b
l
et
r
a
n
s
i
t
i
o
nt
osomen
o
n
b
r
i
t
t
l
ef
r
i
c
t
i
o
n
a
lf
l
i
m
i
ti
ss
田 ni
nBr
招 i
l
i
a
ns
p
l
i
t
-c
y
l
i
n
d
e
rt
e
s
t
s(
s
e
ee
.
g
.B話 相t
,
1
9
8
7
a
)
.
5
.
3OtherSizeE:
f
f
ectsandLimitations
5
.
3
.
1EffectofResidualDuctileorFr
ictionaiLimit
I
nB
r
a
z
i
l
i
a
nt
e
s
t
so
fs
p
l
i
もc
y
l
i
n
d
e
rs
t
r
e
n
g
t
h,Hωegawae
ta
l
.
(
1
9
8
5
)ob
悶 v
edt
h
a
tt
h
e
,
乱
山r
e(
i
.
e
.,
t
h
es
p
l
i
tc
y
l
i
n
d
e
r抗r
e
n
g
t
h
)d
e
c
r
e
a
s
e
dw
i
t
hi
n
c
r
e
a
s
i
n
gd
i
a
m
e
t
e
r
n
o
m
i
n
a
ls
t
r
e
s
sa
tf
o
ft
h
ec
y
l
i
n
d
e
r,
h
o
w
e
v
e
r
,
beyondac
e
r
t
a
i
n1
a
r
g
es
i
z
eo
fc
y
l
i
n
d
e
r,
nof
u
r
t
h
e
rd
e
c
r
e
a
s
eo
ft
h
e
b
s
e
r
v
e
d(
t
h
i
sb
e
h
a
v
i
o
rh
a
snowb
e
e
nc
o
n
f
i
r
m
e
dbyt
e
s
t
so
fBazant,
Kazemi,
s
t
r
e
n
g
t
hw白 o
e
ta
l(
1
9
9
0
)
)
.T
h
i
ss
u
g
g
回 t
5t
h
a
tf
o
rac
e
r
t
a
i
ns
u
f
f
i
c
i
e
n
t
l
yl
a
r
g
es
i
z
e,
t
h
e
r
ei
sat
r
a
n
s
i
t
i
o
n
も
0叩 men
o
n
b
r
i
t
t
l
ef
a
i
l
u
r
em氏 h
a
n
i
s
ma
tt
h
emaximuml
o
a
d
.Ther
e
晶 o
nmightbet
h
a
tf
o
r
l
a
r
g
es
i
z
c
st
h
emaximuml
o
a
di
sd
e
c
i
d
e
dbyf
r
i
c
t
i
o
nonas
m
a
l
lwedg
e
-s
h
a
p
e
dr
e
g
i
o
nu
n
d
e
r
t
h
el
o
a
da
p
p
l
i
c
a
t
i
o
np
o
i
n
t
s
.F
o
rs
m
a
l
ls
p
e
c
i
m
e
nd
i
a
m
e
t
e
r
s,
t
h
el
o
a
dt
oc
a
u
s
e出 es
p
l
i
も
t
i
n
g
h
e1
0
0
0t
oc
a
u
s
ef
r
i
c
t
i
o
n
a
ls
l
i
po
ft
h
ew
e
d
g
e
s
h
a
p
e
dr
e
g
i
o
n
, 組d
c
r
a
c
ki
smuchh
i
g
h
e
r出 ant
ft
h
esi~ i
s~ery l
a
;
g
ea
b
o
u
t
t
h
e
r
e
f
o
r
et
h
eb
r
i
t
t
l
emechanismo
fc
r
a
c
k
i
n
gd
e
c
i
d
e
s
.How~ver, i
1
0
0d
",
t
h
en
o
m
i
n
a
l自t
r
回 S
机 w
h
i
c
ht
h
es
p
l
i
t
t
i
n
gf
r
a
c
t
u
r
eo
c
c
u
r
sb
e
c
o
m
e
sv
e
r
ys
m
a
l
l,
s
m
a
l
l
e
r
柏 t
h
en
o
m
i
n
a
l自t
r
e
s
s
σ
hw
h
i
c
hc
a
u
s
e
st
h
ef
r
i
c
t
i
o
n
a
ls
l
i
po
ft
h
ew
e
d
g
e
;s
e
eF
i
g
.5
.
2
.Thus,
t
h
~t 1eas~_ f
o
rt
h
i
st
y
p
eo
ff
a
i
l
u
r
e,
i
もs
e
e
m
sa
p
p
r
o
p
r
i乱t
et
op
u
tal
o
w
e
rl
i
m
i
tont
h
e5
i
z
ee
:
f
e
c
t
!
a
.
w,
σ
;
.
" beyondwhichσN C
姐 n
o
tbed
e
c
r
e
a
s
e
d
.T
h
i
smigh
もa
l
s
obet
r
u
ef
o
rsome0
仙e
r
65
64
f
♂l
u
r
emechanisms
,
e
s
p
e
c
i
a
l
l
yt
h
o
s
ei
n
v
o
l
v
i
n
gs
p
l
i
t
t
i
n
gc
r
a
c
k
si
nc
o
m
p
r
e
s
s
i
o
n
.
n
.
1
.
5
.
3
.
2S
t
a
t
i
s
t
i
c
a
lS
i
z
eE
f
f
e
c
tDuetoRandomStrength
S
t
a
t
i
s
t
i
c
a
lh
e
t
e
r
o
g
e
n
e
i
t
yo
ft
h
em
a
t
e
r
i
a
lnod
o
u
b
tp
l
a
y
sani
m
p
o
r
t
a
n
tr
o
l
ei
nt
h
em
i
c
r
o
m
e
c
h
a
n
i
s
m
sd
e
t
e
r
m
i
n
i
n
gt
h
em
a
t
e
r
i
a
ls
t
r
e
n
g
t
h
.I
td
e
s
c
r
i
b
e
st
h
ee
f
f
e
c
to
ft
h
es
i
z
ed
i
s
t
r
i
f
o
rr
e
a
s
o
n
s
b
u
t
i
o
no
ft
h
ef
l
a
w
si
nt
h
em
i
c
r
o
s
t
r
u
c
t
u
r
eont
h
em
a
t
e
r
i
a
ls
t
r
e
n
g
t
h
. However,
,山er
andomn
自由 o
fs
t
r
e
n
g
t
hduet
ot
h
eh
e
t
e
r
o
g
e
n
e
i
t
yo
ft
h
e
a
l
r
e
a
d
ye
x
p
l
a
i
n
e
di
nChapter1
m
a
t
e
r
i
a
l,
回 describedbyWeibull-typeprobabilisticmodels,
d
o
e
sn
o
tseemt
oh
a
v
eamajor
田 o
nt
h
es
i
z
ee
f
f
e
c
to
h
s
e
r
v
e
di
nb
r
il
ef
a
i
l
u
r
e
so
fmostc
o
n
c
r
e
t
es
t
r
u
c
t
u
r
e
s,
ex
田 p
七
i
n
f
l
u
e
n
t
h
o
s
ew
h
e
r
et
h
emaximums
色E伺 si
su
n
i
f
o
r
mo
v
e
ra
.l
a
r
g
epa
:
吋 o
ft
h
es
t
r
u
c
もu
r
e(
e
.
g
.,
al
o
n
g
u
n
i
f
o
r
m
l
y目t
r
e
s
s
e
ds
p
e
c
i
m
e
ni
nt
民
e
n
四
凶
s
s
討
山
i
o
n
小o
町rw
heret
h
es
t
r
u
c
t
u
r
ef
a
i
l
sa
t五r
s
tm
a
c
r
o
c
r
a
c
k
i
n
i
t
i
a
t
i
o
n
.
“
2
T
E
S
T
SB
Y8
A
L
A
N
T& K
A
Z
E
t
v
1l
, ‘9
89
5
.
3
.
3Diffusion-TypeS
i
z
eE
f
f
e
c
t
s
、
、
I
tmustbekepti
nmindt
h
a
ts
i
g
n
i
f
i
伺 n
ts
t
r
u
c
t
u
r
a
ls
i
z
ee
f
f
e
c
凶 C姐 b
eo
b
t
a
i
n
e
dduet
o
t
h
ed
i
f
f
u
s
i
o
np
r
o
c
e
s
so
fd
r
y
i
n
go
fc
o
n
c
r
e
t
ei
ns
t
r
u
c
t
u
r
e
s,
t
h
ec
o
n
d
u
c
t
i
o
no
fhe
a
.
tp
r
o
d
u
c
e
d
andt
h
en
o
n
u
n
i
f
o
r
m
i
t
i
e
so
fc
r
田 pp
r
o
d
u
c
e
dbyd
i
f
f
e
r
e
n
c
e
si
nt
e
m
p
e
r
a
t
u
r
eand
byh
y
d
r
a
t
i
o
n,
m
o
i
s
t
u
r
ec
o
n
t
e
n
tt
h
r
o
u
g
h
o
u
tt
h
es
t
r
u
c
t
u
r
e
.
.~
句
.
.
!
e
。
で
。
".
.
0
ま8
1
5
.
3
.
4W a
I
lE
f
f
e
c
ta
saSourceofS
i
z
eE
f
f
e
c
t
a
J
S
t
i
l
la
n
o
t
h
e
rt
y
p
eo
fs
i
z
ee
f
f
e
c
ti
sc
a
u
s
e
dbyt
h
ef
a
c
tt
h
a
taboundaryl
a
y
e
rn
e
a
rt
h
e
s
u
r
f
a
c
eo
fc
o
n
c
r
e
t
ei
n
e
v
i
t
a
b
l
yh
a
sad
i
f
f
e
r
e
n
t∞m
p
o
s
i
t
i
o
nands
t
r
e
n
g
t
ht
h
a
nt
h
ei
n
t
e
r
i
o
r
o
fc
o
n
c
r
e
t
es
t
r
u
c
t
u
r
e
.T
h
i
sl
a
y
e
r
,
whoset
h
i
c
k
n
e
s
si
sa
b
o
u
tonea
g
g
r
e
g
a
t
es
i
z
e
,containsa
,
邸r
e
g
a
t
e
sandah
i
g
h
e
rp
e
r
田 n
t
a
g
eo
fmorta
:
r
.T
h
i
sphenomenon
l
o
w
e
rperαntageo
fl
a
r
g
ea
i
sknowna
s“
w
a
l
le
f
f
e
c
t
"
.I
nas
m
a
l
ls
t
r
u
c
t
u
r
e,
t
h
ee
f
f
e
c
to
ft
h
i
sl
a
y
e
ri
sl
a
:
r
g
e
rもhani
na
l
a
r
g
es
t
r
u
c
t
u
r
eb
e
c
a
u
s
et
h
eboundaryl
a
y
e
rt
h
i
c
k
n
e
s
si
si
n
d
e
p
e
n
d
e
n
to
fs
t
r
u
c
t
u
r
es
i
z
e
.F
o
r
t
h
i
se
f
f
e
c
tbecomesn
e
g
l
i
g
i
b
l
e
.
v
e
r
yt
h
i
c
kc
r
o
s
ss
e
c
t
i
o
n
s,
R
J
ミ
=
b
2
4
6 丞
勺争
4
1
a•
6'
d (
i
γi
)
・
c
2
4
F
i
gふ 3D
e
v
i
a
t
i
o
nfromt
h
eS
i
z
eE
f
f
e
c
tLaw(
E
q
.4
.
1
)Ob
町 v
edi
nB
r
a
z
i
l
岨 C
o
m
p
r
e
s
s
i
o
nSp
1
i
t
t
i
n
gT
e
s
t
s
i
~.~.5 S
i
z
eE
f
f
e
c
tDuet
o3DS
i
n
g
u
l
a
r
i
t
ya
ttheEndsoftheCrackEdge
A
c
c
o
r
d
i
n
gt
ot
h
r
e
ed
i
m
e
n
s
i
o
n
a
ll
i
n
e
a
:
re
l
a
s
t
i
cs
o
l
u
t
i
o
n
s,
t
h
es
t
r
e
s
si
n
t
e
n
s
i
t
yf
a
c
t
o
ro
fa
出o
g
o
n
a
lf
r
o
n
te
d
g
ei
sn
o
tc
o
n
s
t
a
n
ta
l
o
n
gt
h
ee
d
g
e(
i
.
e
.
c
r
a
c
ki
nap
l
a
t
ew
i
t
has
t
r
a
i
g
h
to
r
a
c
r
o
s
st
h
et
h
i
c
k
n
e
s
so
ft
h
ep
l
a
t
e
)butd
r
o
p
st
oz
e
r
oa
tt
h
ei
n
t
e
r
s
e
c
t
i
o
n
sw
i
t
ht
h
es
u
r
f
a
c
e
so
f
a.Z組も叩dE
s
t
e
n
s
s
o
r
o,
1
9
7
9
)(
e
x
c
e
p
ti
ft
h
eP
o
i
s
s
o
n
'
sr
a
t
i
owerez
e
r
o
)
.Thec
o
n
t
h
ep
l
a
t
e(B
符 q
_
u
e
n
c
ei
st
h
a
tt
h
ef
r
o
n
te
d
g
eo
fap
r
o
p
a
g
a
t
i
n
gc
r
a
c
kmustbec
u
r
v
e
d,
s
u
c
ht
h
a
.
tt
h
es
u
r
f
a
c
e
吋 pointsofthecrackedgemustbetrailingbehindtheinteriorofapropagatingcrack.
f
f
e
c
to
fp
l
a
t
et
h
i
c
k
n
e
s
sont
h
ea
v
e
r
a
g
ev
a
l
u
eo
ft
h
es
t
r
e
s
si
n
もe
n
s
i
t
yf
a
c
t
o
r
T
h
i
se
n
g
e
n
d
e
r
s組 e
記 e
f
f
e
c
tmightn
o
tbel
a
r
g
eb
u
tc
a
nbec
o
m
p
l
e
t
e
l
y
o
v
e
rt
h
ee
n
t
i
r
ep
l
a
t
et
h
i
c
k
n
e
s
s
. Thi
ss
i
6
6
6
7
e
l
i
m
i
n
七
.
ae
di
ft
h
et
h
i
c
k
n
e
s
s
e
so
fs
p
e
c
i
m
e
n
so
fa
l
ls
i
z
e
sa
r
et
h
es
a
m
e
.
t
h
a
.
nt
h
a
tw
i
t
hau
n
i
f
o
r
ms
.
a
n
d
.
5
.
3
.
6EffectofAggi"egateSize
5
.
3
.
7Effect ofMatrixStrengthandHighSt:
r
engthConcretes
“
The合
fa
拭
蹴
訓£
f
o
ro
田
n
ea
組.n
dt
h
es
乱m
ec
∞
ο
n
c
α
r
e
t
,
e
鳥w
h
i
c
hi
m
p
l
i
e
sも
由
he拍s
ame喝
a
.
g
g
r
e
句:
,
g
a
t
e討
s
i
配 .I
ft
h
ea
g
g
r
e
g
a
.
t
es
i
z
ei
s
姐 g
叫 t
l
吋 凶u
r
ep
a
r
a
m
e
t
e
r
sa
.
ndもh
es
i
z
ee
f
f
e
c
tl
a
.
wpa
.
ram
山 sc
h
回 g
e.ABproposdv
c
h
Bdmt
,
t
h
esimefeetl
a
wn
e
e
d
st
obedjusted掴 f
o
l
l
o
w
s(
8
白 色1
8
0Baz
姐 t
阻 dK
im
,
1
9
8
4
)
:
品 ,:
f=
.
f
t
1
(
+
伺 , s=え
σN=
(
5.
1
3
)
廿
i
c
h伊
f
?i
拘
st
由
hed
i
r
e
氏c
七t
e
n
s
i
l
es
t
r
e
n
g
t
hf
o
rac
h
o
s
e
nr
e
f
e
r
e
n
c
e∞n
即c
r
陀e
も
旬
e,刀 i
st
出
h巴 d
i
r
伐
e
c
t
i
nw
h
i
i
匂
e
悶
n
s
討i
l
e蜘 附h
f
知
i
o
rma
x
対
泊
i
加
m
π
m
u
ぽ
包
s姐 a
.
l
o
伊
g
。u
凶sも
旬
ot
h
eP
e
t
c
hf
o
r
m
u
l
a
.f
o
rt
h
ee
宜
f
f
e
c
to
fg
r
釘
a
i
ns
討
i
問 ∞
ont
h
ey
i
e
l
ds
t
r
e
n
g
t
ho
fp
o
l
y
i
r
y
s
t
a
l
l
i
n
em
e
t
a
l
s
,
whichi
sd
e
r
i
v
e
dbyt
h
edishcauonthmEy.Eq.5.13hasbeenshownt
o
田 四 回o
n
ぬl
yw
e
l
lw
i
t
ht
h
ef
r
a
c
t
u
r
et
e
s
td
a
t
ao
fCh.
a
n
a
.(
1
9
8
1
),
T
a
y
l
o
r(
1
9
7
2
)姐 d1
9u
r
o
a
g
r
f
f
i
c
i
e
n
t向 h
a
sb
e
e
nc
a
l
i
b
r
a
.
t
e
d
.
e
ta
l
.(
1
9
8
5
), 組d∞e
F
i
g
.5
.
2(
B
a
i
a
.
n
t,
1
9
8
6
)∞mparest
h
es
i
z
eebdp
l
o
t
sa
c
c
o
r
d
i
n
gt
oEq
.
5
.
1
3
f
o
rB
p
e
c
i
m
?
n
s
叩 t
maximuma
g
g
r
e
g
a
.
t
es
i
z
e
sd
.
.
.The∞m
p
a
r
i
s
o
ni
smadeu
n
d
e
rt
h
ea
s
s
u
m
p
t
i
o
n
w
i
t
hd
i
f
f
e
r
thudo=dawheren=constant,
whichi
so
n
l
yapproxmate(andmightbei
n
v
a
l
i
di
fv
e
r
y
p
e
s,
e
.
g
. roundandv
e
r
ye
l
o
n
g
a
t
e
d.
a
gg
時耐久 o
rc
r
u
s
h
e
d組 d山 e
r
d
i
f
e
r
e
n
tagFegateもy
t
h
es
i
z
ee
t
r
e
c
tc
u
r
v
e
si
nt
h
ep
l
o
to
fl
o
gσN v
sl
o
g
d
a
g
g
r
e
g
a
t
e
sMeu
s
e
d
.
A
c
c
o
r
d
i
n
gt
oE
q
.
5
.
1
3,
.
ve出 esames
h
a
p
eforanyd,
i
.
e
.,
o
n
etransfOEmsMmottleEbybranSIMba-Ani
n
c
r
e
a
s
e
ha
o
fd
"n
o
to
n
l
ys
h
i
f
t
st
h
es
i
z
ee
宜e
c
tc
u
r
v
ev
e
r
t
i
c
a
l
l
ydownward(dueωtheterm匂 j
d
.
.)b
u
t
d
.
.i
sc
o
n
t
a
i
n
e
di
nt
h
eb
r
i
t
t
l
e
n
田 s
numbers
)・
a
l
s
ot
ot
h
er
i
g
h
t(
b
e
c
a
u
s
ed
o=n
C
o
n
s
e
q
u
e
n
t
l
y
,
t
h
es
i
z
ee
f
f
e
c
tc
u
r
v
e
sf
o
rd
i
π
e
r
e
n
td
.
.mayi
n
t
e
r
s
e
c
t,
a
sshowni
nF
i
g.5
.
2
(
t
h
i
smusthappeEu
n
l
e
s
st
h
em
a
t
e
r
i
a
lc
o
n
s
t
a
n
tc
u
i
slugeeno
略
的
・ Thus,
f
o
rasu
直c
i
e
n
t
l
y
s
m
a
l
ls
p
e
c
i
m
e
ns
i
z
e
,品 h
i
g
h
e
rn
o
m
i
n
a
ls
七r
e
n
g
t
hi
so
b
ta
.
i
n
e
dw
i
出 s
m
a
l
l
e
ra
g
g
r
e
g
a
t
es
i
z
e
sd
w
h
i
l
ef
o
ra
.s
u
伍d
e
n
t
l
yl
a
r
g
es
p
e
c
i
m
e
ns
i
z
e
,
ah
i
g
h
e
rn
o
m
i
n
a
l
s
t
r
e
n
g
t
hi
so
b
t
a
i
n
e
dw
i
U
1
l
a
r
g
e
r
a
.
t
es
i
z
e
s
.F
o
ri
n
t
e
r
m
e
d
i
a
t
es
t
r
u
c
t
u
r
es
i
z
e
s,
t
h
ea
g
g
r
e
g
a
t
es
i
z
em山 田 l
i
t
t
l
ed
i
f
f
e
r
叩 c
e
.
a
g
g
r
e
g
何回 i
nF
i
g.5
.
2i
sn
o
tknownv
e
r
ya
c
c
u
r
a
.
ωly
,
S
i
n
c
et
h
ei
n
t
e
r
s
e
c
t
i
o
np
o
i
n
to
ft
h
etwoc
u
t
h
e
r
emaybe乱 l
a
r
g
er
a
n
g
eo
fs
t
r
u
c
t
u
r
es
i
z
e
si
nw
h
i
c
ht
h
ee
f
f
e
c
to
ft
h
emaximuma
g
g
r
e
g
a
t
e
s
i
z
ei
su
n
c
e
r
t
a
i
n
.F
o
rs
m
a
l
lbeamsands
l
a
b
s,出i
sf
r.
a
c
t
u
r
e岨 a
l
y
s
i
swoulds
u
g
g
e
s
tu
s
i
n
ga
sm
a
.
l
la
.
g
g
r
e
g
a
.
t
e
.C
o
n
c
r
e
t
eda
.
msa
.
r
es
ol
a
.
r
g
et
h
a
tt
h
el
a
.
r
g
e
rt
h
ea
g
g
r
e
g
a
t
et
l
叫 e
t
t
e
r
,
ωfar
剖 t
h
edams
t
r
e
n
g
t
hi
sc
o
n
c
e
m
e
d
.However
,
r
e
c
e
n
tf
r
a
c
t
u
r
et
e
s
t
so
fdamc
o
n
c
r
e
t
田 o
fSaouma
a
t
e.
as
u
r
p
r
i
s
i
n
g
l
y
'
s
m
a
l
le
f
f
e
c
to
f.
a
g
g
r
e
g
a
t
es
i
z
ea
.
scompa
.
r
e
dt
ot
h
ee
f
f
e
c
t
e
ta
l
.(
1
9
8
9
)i
n
d
i
c.
o
fa
g
g
r
e
g
a
.
t
es
h
a
p
e
.
a
.
t
es
i
z
ea
.
n
dg
r.
a
d
a
t
i
o
ne
f
f
e
c
t
sh.
a
v
e叫s
ob
e
e
ns
t
u
d
i
e
df
o
rpolymer∞nCretes(ViP?
A
g
g
r
e
g
l
組and
岨 a
ndDharm
.
a
r
a
j.
a
n,
1
9
8
7,
1
9
8
8,
1
9
8
9
a
丸山 Dharmaraja
.
n
組dV
i
p
u
l
a
.
n
a
.
nd
a
.
n,
1
9
8
8
)
a
.
m
p
l
e
,
t
h
e
yf
o
u
n
dt
h
a
ta
.p
o
l
y
m
e
r∞n
c
r
e
t
ew
i
t
hwe
l
lg
r
a
d
e
ds
andh
a
s20%h
i
g
h
e
rKlc
F
o
rex
刷
,
Ther
e
目 n
tspec
同 四l
a
ra
d
v
a
n
c
e
si
nt
h
es
t
r
e
n
g
t
ho
f∞n
c
r
e
t
ehaveb
e
e
na
c
h
i
e
v
e
dm
a
i
n
l
y
byi
n
c
r
e
a
s
i
n
gt
h
es
t
r
e
n
g
t
ho
ft
h
em
a
t
r
i
xa
.
ndt
h
ea
.
g
g
r
e
g
a
te
-matrixbond. 1nhighs
t
r
e
n
g
出
c
o
n
c
r
e
t
e
s,
t
h
ed
i
f
f
e
r
e
n
c
e
sb
e
t
w
e
e
nt
h
es
t
r
e
n
g
t
hande
l
a
s
t
i
cmoduluso
ft
h
ea
g
g
r
e
g
a
.
t
ea
.
n
dt
h
e
m.
a
t
山 (
m
o
r
t
紅)a
r
emuchs
m
a
l
l
e
rt
h
a
.
nt
h
e
y訂 ef
o
rn
o
r
m
a
l抗 r
e
n
g
出 ∞n
c
r
e
t
e
.C
o
n
s
e
q
u
e
n
t
l
y
,
h
i
g
hs
t
r
e
n
g
t
hc
o
n
c
r
e
も
eb
e
h
a
v
e
sa
samorehomogeneousm
a
t
e
r
i
a
l
,a
.
n
dt
h
er
e
s
u
l
ti
st
h
a
tt
h
e
f
r
a
c
t
u
r
ep
r
侃
, 伺 sz
o
n
ebecomess
m
a
l
l
e
r(
J
o
h
nandSh
a
.
h,
1
9
8
9
b
;G
e
t
t
u,
BaZa
n
tandK
a
r
r,
1
9
8
9
).l
nv
i
e
wo
ft
h
ed
i
s
c
u
s
s
i
o
no
ft
h
es
i
z
ee
f
f
e
c
t(
伺p
e
c
i
a
l
l
yt
h
ef
a
c
tt
h
a
.
tc
o
e
f
f
i
.
c
i
e
n
td
o
i
nt
h
es
i
z
ee
f
f
e
c
tl
a
wi
sr
e
l
a
t
e
dt
o出 es
i
z
eo
ft
h
ef
r
a
c
t
u
r
ep
r
o
c
悶
z
o
n
e
),
i
ti
sc
l
e
a
rt
h
a
t,
f
o
rt
h
es
a
.
mes
t
r
u
c
もu
r
a
ls
i
z
e,
t
h
eb
e
h
a
v
i
o
ro
fh
i
g
hs
t
r
e
n
g
t
hc
o
n
c
r
e
t
ei
sc
l
o
s
e
rt
ol
i
n
e
a
re
l.
a
s
t
i
c
f
r
a
c
t
u
r
em
e
c
h
a
n
i
c
s,
i
.
e
.,
moreb
r
i
t
t
l
e
,
t
h
a
.
nt
h
ebeh
a
.
v
i
o
ro
ft
h
es
a
.
mes
t
r
u
c
t
u
r
emadeo
fn
o
r
m
a
l
∞n
c
r
e
t
e
.T
h
e
r
e
f
o
r
e
,f
r
a
c
t
u
r
em
e
c
h
a
n
i
c
s姐 a
l
y
s
i
sands
i
z
ee
f
f
e
c
t
sa
r
emuchmorei
m
p
o
r
t
a
n
t
f
o
rh
i
g
hs
t
r
e
n
g
t
hc
o
n
c
r
e
t
e
st
h
a
nf
o
rnorm
a
.
ls
t
r
e
n
g
t
hc
o
n
c
r
e
t
e
s(
e
.
g
.,
Shah,
1
9
8
8
)
.
5ふ 8SuppressionofSizeEffectbyYieldofReinforcement
.
r
to
ft
h
el
o
a
d,
A
s
i
d
efromp
r
o
v
i
d
i
n
gana
d
d
i
t
i
o
n
a
ldu
c
t
i
l
emechanismwhichc
a
r
r
i
e
spa
t
h
ee
f
f
e
c
もo
fr
e
i
n
f
o
r
c
e
m
e
n
ti
st
os
p
r
e
a
dt
h
ef
r
a
.
c
t
u
r
ep
r
o
c
e
s
sz
o
n
e
.T
h
e
r
e
f
o
r
e
,reinforced
a
ndt
h
ea
.
c
companyings
i
z
ee
f
f
e
c
t
s
c
o
n
c
r
e
t
ei
sg
en
e
r
a
l
l
yl
e
s
ss
u
s
c
e
p
t
i
b
l
et
of
r
a
c
t
u
r
ee
f
f
e
c
t
s.
t
h
a
nu
n
r
e
i
n
f
o
r
c
e
dc
o
n
c
r
e
t
e
.F
u
r
t
h
e
rs
t
u
d
i
e
sa
r
en
e
e
d
e
di
nt
h
i
sr
e
g
a
.
rd
.
I
七m
ustbek
e
p
ti
nmindt
h
o
u
g
h,
t
h
a
td
e
s
p
i
t
er
e
i
n
f
o
r
c
e
m
e
n
t,
m姐 yf
a
i
l
u
r
e
so
f∞n
c
r
e
t
e
u
r
r
e
n
tp
h
i
l
o
s
o
p
h
y
,
t
h
eu
l
t
i
m
a
.
t
el
o
a
di
sasumo
f
s
t
r
u
c
t
u
r
e
sa
r
eb
r
i
t
t
l
e
.A
c
c
o
r
d
i
n
gt
o出 ec
・
,they
i
e
l
d
i
n
go
fs
t
i
r
r
u
p
si
nd
i
a
g
o
n
a
ls
h
e
a
r
)a
.
n
dt
h
a
tdue
t
h
a
tduet
oy
i
e
l
dmechanisms(
e
.
g
t
oc
o
n
c
r
e
t
ea
l
o
n
e
,withoutreinforcingba
.
r
s
.T
h
i
smigh
もo
f
t
e
nbeac
o
n
s
e
r
v
a
t
i
v
ea
p
p
r
o
a
c
h,
w
h
i
c
hi
nf
a
c
ti
m
p
l
i
e
st
h
a
tt
h
ec
o
n
c
r
e
t
ec
o
n
t
r
i
b
u
t
i
o
nt
ot
h
eu
l
t
i
m
a
t
el
o
a
.
ds
h
o
u
l
dbea
.
n
a
.
l
y
z
e
d
a
c
c
o
r
d
i
n
gt
of
r
a
c
t
u
r
eme
c
h
a
n
i
c
s
.l
ti
sl
i
k
e
l
y
,however,tha
.
tt
h
ep
r
e
s
e
n
回 o
fs
t
e
e
landi
t
s
i
n
c
r
e
a
s
i
n
gt
h
ec
o
n
t
r
i
b
u
t
i
o
nt
of
a
i
l
u
r
el
o
a
d
y
i
e
l
d
i
n
ga
l
t
e
r
st
h
ef
r
a
c
t
u
r
eb
e
h
a
v
i
o
ro
fc
o
n
c
r
e
t
e,
d
u
et
oc
o
n
c
r
山 a
l
o
n
e(Ba
雲仙も a
ndSun,
1
9
8
7
).Th
悶
p
r
o
b
l
e
m
s{leedt
obes
t
u
d
i
e
df
u
r
t
h
e
r
.
6
8
69
(α)
b
Chapter6
.EXPERIMENTALO RA NALYTICALDETERMINATIONOFMATERIALFRACTUREPARAMETERS
Af
t
e
ras
l
o
wね 叫 madea
se
a
r
l
y部 1
9
6
1(
K
a
p
l
a
n,
1
9
6
1
),
f
r
a
c
t
u
r
et
e
s
t
i
n
go
fc
o
n
c
r
e
t
eh
a
s
d
e
v
e
l
o
p
e
dt
r
'
四 l
e
n
d
o
u
s
l
yd
u
r
i
n
gt
h
e1
9
8
0
'
s
. A numbero
fe
x
p
e
r
i
m
e
n
t
3
1t
e
s
t
i
n
gt
e
c
h
n
i
q
u
偲
姐 ds
p
e
c
i
m
四 t
y
p
伺 h
aveb
e
e
nt
r
i
e
d,
andt
h
ed
e
v
e
l
o
p
m
e
n
t
sc
r
y
s
t
a
l
i
z
e
di
n
t
osomee
f
f
e
c
t
i
v
e
method
目.
d
F
6
.
1NotchedBeamTests
.1NotchedBeamT
estsUsingLEFM
6
.1
Be
.
a
mt
e
s
t
shav
ebe
四 m
ostp
o
p
u
l
a
.
r
;p
r
o
b
a
b
l
yb
e
c
a
u
s
eo
fs
i
m
i
l
a
r
i
t
i
e
sw
i
t
ht
h
es
t
a
n
d
a
r
d
回 t組 db
e
c
a
u
s
et
h
ef
i
r
s
tLEFMt
e
s
ts
t
a
n
d
a
r
df
o
rmeta
.
l
su
s
e
dt
h
i
s
moduluso
fr
u
p
t
u
r
et
(ASTM,
1
9
8
3
)
.I
na
d
d
i
t
i
o
n
,testingproceduresむ esimplerth姐 f
o
ro
t
h
e
rg
回 m
e
t
r
i
田 a
nd
p
i
c
a
lbe
釘 ng
四 m
etryandt
h
r
骨 p
o
i
n
tl
o
a
d
i
n
g紅 ep
r
e
a
l
i
g
n
m
e
n
te
r
r
o
r
sa
r
em
i
n
i
m
a
l
. A もy
s
e
n
t
e
di
fF
i
g
.6
.
1
a
. Thes
p
a
nt
od
e
p
t
hr
a
t
i
of
r
e
q
u
e
n
t
l
yI
s4
:
1whichc
o
n
f
o
r
m
sw
i
t
ht
h
e
r
e
q
u
i
r
e
m
e
n
t
so
fもheASTMs
t
a
n
d
a
r
dE399(
1
9
8
3
)
.T
h
i
si
sb
e
c
a
u
s
ef
o
r
m
u
l
回
f
o
ro
p
e
n
i
n
g
mode耐 e
s
s
i
n
t
e
n
s
i
t
yf
a
c
t
o
r
sf
o
rbeamsa
r
eg
i
v
e
nf
o
rSjd=4o
r8-Tadae
t3
1
.,
(
1
9
8
4
)and
倒 a
l
t
h
o
u
g
hf
o
r
m
u
l
a
sc
a
.
na
1
s
obeo
b
t
a
.
inedf
o
ro
t
h
e
rSjd
Murakami(
1
9
8
7
)a
r
et
y
p
i
c
a
ls
o
u
r
Go,
S
w
a
r
t
zandHu,
1
9
8
6
)
.
r
a
t
i
o
s(
e
.
g
.,
Thet
e
s
tu
s
u
a
l
l
yc
o
n
s
i
s
t
so
ft
h
ef
o
l
l
o
w
i
n
gs
t
e
p
s
.
1
.N
otcht
h
ebeamt
od
e
p
t
h向 a
tm
i
d
s
p
a
n
.T
h
i
sc
o
u
l
dbedonebyas
a
w
c
u
to
rby姐
i
n
s
e
r
tc
a
s
ti
n
t
ot
h
ebeamandl
a
t
e
rr
e
m
o
v
e
d
.
(
b
l
P
d
Lood
p
9~" 0
'I
n
l
t
l
o
l
s
l
o
p
.
2
.U
s
i
n
gac
o
n
s
t
a
n
tr
a
.
t
e
,about1t
o1
0m
i
n
.t
of
a
.
il
u
r
e
,increa5etheload,desectiono
r
c
r
a
c
ko
p
e
n
i
n
gu
n
t
i
lf
a
i
l
u
r
e
. Re
∞rd七he1
0
吋 -d
e
f
l
e
c
t
i
o
n(OLPD)o
rI
o
a
d
c
r
a
c
kmouth
8
田p
o
n
s
ec
o
n
t
i
n
u
o
u
s
l
y
.
o
p
e
n
i
n
gd
i
s
p
l
a
c
e
m
e
n
t(
CMOD)r
3
.Thep
e
a
kl
o
a
d凡 o
rt
h
el
o
a
dPQ a
.
tt
h
ei
n
t
e
r
s
民 t
i
o
nw
i
t
ha
.s
e
c
姐も o
fs
l
o
p
e95%o
f
i
n
i
t
i
a
ls
l
o
p
e
,s
開 F
i
g
.6
.
1
h,
i
st
h
e
nu
s
e
dt
oc
a
l
c
u
l
.
a
t
et
h
ef
r
a
c
t
u
r
e七o
u
g
h
n
e
s
sfromt
h
e
a
.
t
i
o
n
s
h
i
p(ASTM,
1
9
8
3
;Ta
.
dae
ta
,
.
1 1985;Go,SwartzandHu,1986;Murakami,
r
e
l
1
9
8
7
)
K1c=F(S
,
d
,
b
,
αQ,
PuorP
Q)
(
6
.
1
)
0d
e
t
e
r
m
i
n
e
di
spresumedt
obet
h
ec
r
i
t
i
c
a
l,
o
p
e
n
i
n
gm
o
c
l
es
t
r
e
s
si
n
t
e
n
s
i
t
y
Thev
3
1
u
eo
fKlc5
o
c
i
a
t
e
dw
i
t
huns
t
a
b
1
ec
r
a
.
c
kg
r
o
w
t
h組 di
sa
.m
a
t
e
r
i
a
.
lp
r
o
p
e
r
t
y
.
f
a
c
t
o
r剖 s
I
fKlci
s,
i
nf
a
c
t,
ama
.
t
e
r
i
a
.
lp
r
o
p
e
r
t
yo
b
t
a
i
n
a
.
b
l
ef
r
o
mt
h
eu
n
m
o
d
i
f
i
e
du
s
eo
fLEFM,
t
h
e
f
o
l
l
o
w
i
n
gs
h
o
u
l
dbet
r
u
e
:(
1
)K1cs
h
o
u
l
db
ei
n
v
a
r
i
a
n
tw
i
t
hr
e
s
p
e
c
tt
obeams
i
z
ea
.
ndn
o
t
c
h
出. (
2
)Anyz
o
n
eo
f“p
l
回 t
i
c
i
句"o
r“凶α0
・c
r
a
c
k
i
n
g,
"i
.
e
.,t
h
ef
r
a
c
t
u
r
e
d
e
p
t
ho
rc
r
a
c
kd
e
p
間
z
o
n
e,
shou
l
dbev
e
r
ys
m
3
1
lcomparedt
ot
h
en
o
t
c
hd
e
p
t
h
.ASTME399(
1
9
8
3
)g
i
v
e
s
p
r
o
c
Displacement
凶 叫a
山
:2r(a)No
8
or 8
CMOD
LPO
T
h
r
e
e
P
o
i
n
tBend
b
)L叫
叩 nd(
D
i
s
p
l
舵 印 刷
7
0
7
1
af
o
r
m
u
l
at
od
e
t
e
r
m
i
n
ei
ft
h
i
sr
e
q
u
i
r
e
m
e
n
ti
ss
a
t
i
s
f
i
e
d
.T
h
i
sh
a
sb
e
e
na
d
a
p
t
e
dt
oc
o
n
c
r
e
t
e
(
S
e
c
.6
.
8
)
.(
3
)Thec
r
i
t
i
c
a
le
n
e
r
g
yr
e
l
e
a
s
er
a
t
ei
sr
e
l
a
t
e
dt
oK1cby
T
.
r2
す
c=
Gl
(
6
.
2
)
。
I
nwhicht
h
e(
s
l
i
g
h
t
)i
n
f
l
u
e
n
c
eo
fP
o
i
s
s
o
n
'
sr
a
t
i
oi
自n
e
g
l
e
c
t
e
dandEi
st
h
emoduluso
fe
l
a
s
t
i
c
i
t
y
.
I
n姐 a
t
t
e
m
p
tt
oe
s
t
i
m
a
t
et
h
ec
r
a
c
kl
e
n
g
t
hi
np
l
a
i
nc
o
n
c
r
e
t
ebeamss
u
b
j
e
c
t
e
dt
ot
h
r
e
e
n
d
i
r
e
c
tmethodb
a
s
e
dont
h
ec
o
m
p
l
i
a
n
c
eo
fn
o
t
c
h
e
dbeamswωproposed
p
o
i
n
tb
e
n
d
i
n
g, 姐 i
HuandJ
o
n
e
s(
1
9
7
8
)
.T
h
i
st
e
s
tp
r
o
c
e
d
u
r
ei
sa
sf
o
l
l
o
w
s
:
byS
w
a
r
t
z,
1
.Notchabeama
tm
i
d
s
p
a
n
.
2
. Mountad
i
s
p
l
a
ι
e
m
e
n
もg
a
g
et
omeasuret
h
ec
r
a
c
k
m
o
u
t
h
o
p
e
n
i
n
gd
i
s
p
l
a
c
e
m
e
n
tdCMOD
(
s
e
eF
i
g
.6
.
1
a
)andc
y
必 t
h
el
o
a
dont
h
ebeamt
h
r
e
et
i
m
e
sw
h
i
l
ep
l
o
t
t
i
n
gl
o
a
dP抑 制
dCMOD. T
hema
.
ximuml
o
a
di
ss
e
l
e
c
t
e
dt
obel
e
s
st
h
a
n1
/
3t
h
ee
x
p
e
c
t
e
df
a
i
l
u
r
e10M
f
o
rt
h
a
tn
o
t
c
hd
e
p
t
h
.
、
,i0・8
o-MA.一
a¥.c-υ.@υceこa
Eυ
。
6
.
1
.
2CcmplianceCalibrationMethod
60
@
・
C! o
rGlc,
w制 C
ο
n
s
t
a
n
tw
i
t
hr
e
s
p
e
c
tt
on
e
i
t
h
e
rbeam
Thee
a
r
l
yr
e
s
e
a
r
c
h
e
r
sf
o
u
n
dt
h
a
tKI
1
9
7
2,
1
9
7
6
)n
o
rn
o
t
c
hd
e
p
t
h(
S
w
a
r
t
z,
Hu,
F
a
r
t
卸 h姐 dH
uang,
1
9
8
2
)f
o
rc
o
n
c
r
e
t
e
s
i
z
e(
W
a
l
s
h,
抗i
o
nf
o
rt
h
i
swasp
r
o
v
i
d
e
dbyShah回 dMcGarry(
1
9
7
1
)who
b
e
a
m
s
. Onep
o
s
s
i
b
l
ee
x
p
l
a
n
n
o
t
e
dt
h
a
tc
r
a
c
k
sa
r
ea
r
r
e
s
t
e
dbyt
h
ea
g
g
r
e
g
a
t
ep
a
r
t
i
c
l
e
s
.A
d
d
i
t
i
o
n
a
le
n
e
r
g
yi
sr
e
q
u
i
r
e
dt
o
h
e
s
ep
a
r
t
i
c
l
e
s
. Anotherp
o
s
s
i
b
l
e阿 部onf
o
r
p
r
o
p
a
g
a
t
et
h
ec
r
a
c
k
st
h
r
o
u
g
ha
n
d
/
o
r訂 oundt
a
.
t
i
o
ni
nKlcme
a
.
s
u
r
e
dbyLEFMw凶 s
u
邸e
s
t
e
dbyS
w
a
r
t
zandc
か w
o
r
k
e
r
s(
S
w
a
r
t
z,
t
h
ev
a
r
i
Hu,
F
a
r
t
a
s
handHuang,
1
9
8
2
;S
w
a
r
t
z
,HuandJones,1978;SwartzandGo,1984). They
tt
h
ed
a
t
ae
v
a
l
u
a
t
i
o
n
si
nt
h
ee
a
r
l
yt
倒 s(
N
a
u
s姐 d1o
t
t,
1
9
6
9
;Walsh
,1972,
1
9
7
6
)
n
o
t
e
d出 a
α
Q
.T
h
i
sv
i
o
l
a
t
e
so
n
eo
ft
h
er
e
q
u
i
r
e
m
e
n
t
so
ft
h
e
werer
e
f
e
r
e
n
c
e
dt
ot
h
eo
r
i
g
i
n
a
ln
o
t
c
hd
e
p
t
h,
namelyt
h
a
tat
r
u
ec
r
a
c
k-n
o
tan
o
t
c
h-i
st
obeu
s
e
d
.Ofc
o
u
r
s
e
,
ASTMmethodf
o
rm
e
t
a
l
s,
me
a
.
s
u
r
i
n
gt
h
ea
c
t
u
a
lc
r
a
c
kl
e
n
g
t
hi
nc
o
n
c
r
e
t
ei
sq
u
i
t
ed
i
f
f
i
c
u
l
t'
and,
i
nfad,
ap
r
e
c
i
s
er
e
s
u
l
t
,
s
i
n
c
et
h
ec
r
a
c
kf
r
o
n
ti
ss
t
r
o
n
g
l
yn
o
n
u
n
i
f
o
r
mt
h
r
o
u
g
ht
h
es
p
e
c
i
m
e
nt
h
i
c
k
n
e
s
s
.
i
si
m
p
o
s
s
i
b
l
e
A
i
n
.
/
l
b
.=
0.57Ix
xTheoretlcal
40
20
e
r
s
u
sdCMODp
l
o
t
.T
h
i
si
st
h
ec
o
m
p
l
i
a
n
c
eCf
o
rt
h
e
3
. Me
a
.
s
u
r
et
h
ei
n
v
e
r
s
es
l
o
p
eo
ft
h
ePv
∞tchdepth.(Thisi
s,
o
fc
o
u
r
s
e
,
o
n
l
yan勾 p
a
r
e
n
ti
n
d
i
r
e
c
tmeasureo
fc
o
m
p
l
i
乱n
c
e
.The
t
r
u
ec
o
m
p
l
i
a
n
c
ewouldbeo
b
t
a
i
n
e
dbym
e
a
s
u
r
i
n
gt
h
ev
e
r
t
i
c
a
l
l
o
a
d
p
o
i
n
td
i
s
p
l
a
c
e
m
e
n
t
dLPD,
o
rbyl
o
a
d
i
n
gt
h
es
p
e
c
i
m
e
na
tt
h
ec
r
a
c
km
o
u
t
h
.
)
4
.Re
p
e
a
ts
t
e
p
sa
cf
o
rd
i
f
f
e
r
e
n
tn
o
t
c
hd
e
p
t
h
su
s
i
n
gt
h
esamebeam.
,theexperimenもa
l
l
yd
e
t
e
r
m
i
n
e
d∞m
p
l
i
a
n
c
ep
l
o
ti
su
s
e
dt
oe
s
t
i
m
a
t
et
h
e
S
u
b
s
e
q
u
e
n
t
l
y
HuandJ
o
n
e
s,
1
9
7
8
)・
i
n
i
t
i
a
lc
r
a
c
kl
e
n
g
t
ho
fp
r
e
c
r
a
c
k
e
dbeamsl
o
a
d
e
dt
of
a
i
l
u
r
e(
S
w
a
r
t
z,
Theu
s
eo
fp
r
e
c
r
a
c
k
e
dbe
a
.
mswasf
e
l
tt
obe町 田s
s
a
r
yi
no
r
d
e
rもoo
b
t
a
i
nv
a
l
i
df
r
a
c
t
u
r
e
。
0.2
0.4
o
d
。
5
.P
l
o
tC v
e
r
s
u
s
α I
d
.At
y
p
i
c
a
lp
l
o
ti
sshowni
nF
i
g
.6
.
2
.
0.6
.
_
,
.
.
.
喧
_
,
J
"
.
.
"
皿
塗
・
・
0.8
F
i
g
.
6
.
2ComplianceC
a
l
i
b
r
a
t
i
o
nC
u
r
v
ef
o
rNotchedB白 血
cm./N
72
73
da
.
t
ab
e
c
a
u
s
et
h
ephenomenono
fc
r
a
c
kc
l
o
s
i
n
gs
t
r
ω
s
e
smak
回 i
ti
m
p
o
s
s
i
b
l
et
omodela
r
e
a
lc
r
a
c
kw
i
t
han
o
t
c
h
.
Thep
r
o
c
e
d
u
r
eもop
r
e
c
r
a
c
ka(
d
i
f
f
e
r
e
n
t
)beamf
o
ras
u
b
叫 l
e
n
tl
o
a
dt
e
s
tw回 a
sf
o
l
l
o
w
s
:
6
.C凶 as
m
a
l
l
,
s
t
a
r
t
e
rn
o
t
c
ha
tm
i
d
s
p
a
n
.T
h
i
sw硝 t
y
p
i
c
a
l
l
y
α口/
d三0.
1
0
.
。岨由
oadt
h
ebeamt
oal
o
a
dbeyond九 t
ogrowac
r
a
c
k企omt
h
en
o
t
c
hu
n
t
i
lt
h
eP-OCMOD
7
.L
u
n
l
o
a
d
i
n
go
rr
e
l
o
a
d
i
n
gs
l
o
p
em
a
t
c
h
e
st
h
a
ta
s
s
o
c
i
a
t
e
dw
i
t
ht
h
ed
e
s
i
r
e
dc
r
a
c
kl
e
n
g
t
h
.
A
f
t
e
rt
h
ebeami
自p
r
e
c
r
a
.
c
k
e
d,
t
h
e
n
守
値
。
︻
.εE0.0ω-.三 o-.向 調 。 .auoυ
8
.L
o
adt
h
ebeamt
of
a
i
l
u
r
e姐 dme
回 u
r
e九 o
rPq
・C
a
l
c
u
l
a
t
eK1cfrom
(
6
.
3
)
。。ミ
同日
0・
。
・
悶
•
HE胃3
。
U句宮崎
前
N
惜
。
,
“
明
司
。制同
﹃
i
n
u
r
o
n
u
•
582u崎
-o-x ・
。柑両
,
。
Z
ω マ.守"・
a
﹄
。
6
.1
.3M
odisedComplianceC
a
l
i
b
r
a
t
i
o
nMethod
Them
o
d
i
f
i
c
a
t
i
o
nwast
ou
s
ep
r
e
c
r
a
c
k
e
dbeamsw
h
i
c
ha
r
ei
m
p
r
e
g
n
a
t
e
dbya
.d
yet
or
e
v
e
a
l
andt
h
e
nu
s
et
h
ee
x
t
e
n
d
e
dc
r
a
c
kl
e
n
g
t
ha
.t
od
e
t
e
r
m
i
n
e](1.
,
t
h
es
h
a
p
eo
ft
h
ec
r
a
c
kf
r
o
n
t,
Themethodwasf
i
r
s
tp
r
o
p
o
s
e
dbyGo(
S
w
a
r
t
zandGo,
1
9
8
4
jGoandSw
a
.
r
t
z,
1
9
8
6
)岨 d
S
w
a
r
t
zandR
e
f
a
i,
1
9
8
9
)
.P
r
e
c
r
a
c
k
e
dheamsa
r
eu
s
e
dbo
も
h
s
u
b
s
e
q
u
e
n
t
l
yr
e
f
i
n
e
dbyR品 i(
f
o
rt
h
ec
r
品もi
o
no
fac
o
m
p
l
i
a
n
c
ec
a
l
i
b
r
a
t
i
o
nc
u
r
v
eandf
o
rf
a
i
l
u
r
et
e
s
t
s
.Themethodh
a
st
h
e
d
i
s
a
d
v
a
n
t
a
g
et
h
a
.
ti
ti
sq
u
i
t
et
i
m
e
c
o
n
s
u
m
i
n
gandanumbero
fbeamsa
r
er
e
q
u
i
r
e
dt
oo
b
ta
.
i
o
.
sf
o
l
l
o
w
s
:
ac
a
l
i
b
r
a
t
i
o
nc
u
r
v
ei
n
s
t
e
a
do
fj
u
s
to
n
eb
e
a
m
.Thep
r
o
c
e
d
u
r
ef
o
re
a
c
hbeami
sa
c
Ee 守.ωω".
e-nwhωOON.寸 一 宇 。
-咽
N
o
t
i
c
et
h
a
tt
h
eo
n
l
yd
i
f
f
e
r
e
n
c
eb
e
t
w
e
e
nt
h
i
sandt
h
ee
a
r
l
i
e
rLEFMp
r
o
c
e
d
u
r
ei
st
h
eu
s
e
o
・
o
f
α
ii
n
s
t
e
a
do
fa
U
s
i
n
gt
h
er
e
l
a
t
i
o
no
fc
o
m
p
l
i
a
n
c
et
oc
r
a
c
kl
e
n
g
t
hc
a
l
c
u
l
a
t
e
dfromLEFM
,
onec
a
nde
同minet
h
ei
n
i
t
i
a
lc
r
a
c
kl
e
時出向 f
r
o
mt
h
es
l
o
p
eo
ft
h
eP -dCMOD c
u
r
v
e(
F
i
g
.6
.
3
)a
f
t
e
rt
h
e
i
n
i
もi
a
.
lc
r
a
c
kc
l
o
s
u
r
es
t
r
e
s
s
e
sa
r
eo
v
e
r
c
o
m
e
.
田 t
i
n
gr
e
q
u
i
r
e
st
h
eu
s
eo
fac
l
o
s
e
d
I
o
o
pe
l
e
c
t
r
o
-hydrodynamics
y
s
t
e
mi
n
T
h
i
smethodo
ft
w
h
i
c
ht
h
ec
o
n
t
r
o
l
l
e
d,
f
e
e
d
b
a
c
kv
a
.
r
i
a
b
l
ei
st
h
eo
u
t
p
u
to
ft
h
et
r
a
n
s
d
u
c
e
ru
s
e
dt
om伺.su
r
et
h
e
dCMOD・T
h
i
sh
a
st
r
a
d
i
t
i
o
n
a
l
l
yb
e
e
nc
a
l
l
e
d“
s
t
r
a
i
nc
o
n
t
r
o
l
.
"T
h
i
stermmaya
.
pproxim
a
.
t
e
l
y
a
p
p
l
yi
ft
h
ec
o
n
t
r
o
l
l
e
dv
a
r
i
a
b
l
ei
st
h
edCMODo
rsomeo
t
h
e
rbeamd
i
s
p
l
a
c
e
m
e
n
t,
b
u
tn
o
tt
h
e
s
t
r
o
k
ed
i
s
p
l
a
c
e
m
e
n
to
ft
h
et
e
s
t
i
n
gs
y
s
t
e
m
.
白も W 臼 m
adet
oshowad
i
r
e
c
tc
o
m
p
a
r
i
s
o
nb
e
t
w
e
e
nn
o
t
c
h
e
dbeamsandp
r
e
c
r
a
c
k
e
d
At
beamsf
o
l
l
o
w
i
n
gt
h
ep
r
o
c
e
d
u
r
e
sd
e
s
c
r
i
b
e
da
.
b
o
v
eandu
s
i
n
gE
q
.6
.
1f
o
rt
h
en
o
t
c
h
e
dbeamsa
n
d
twasf
o
u
n
dt
h
a
tt
h
ecomputedKlcf
o
rt
h
ep
r
e
c
r
a
c
k
e
d
E
q
.6
.
3f
o
rt
h
ep
r
e
c
r
a
c
k
e
db
e
a
m
s
.I
beamsw踊 a
l
w
a
y
sh
i
g
h
e
rt
h
組
f
o
rt
h
en
o
t
c
h
e
dbe
釦 1
5(
Swa
巾, Hu
,Fa
巾.s
h組 dHua
時,
1
9
8
2
).However
,
t
l
由 a
p
p
r
o
a
c
hw
ω
c
r
i
t
i
c
i
z
e
db
e
c
a
u
s
en
o
t
c
h
e
d-n
o
tp
r
e
c
r
a
c
k
e
d-beams
悶s
t
e
p
s日).Af
u
r
t
h
e
rω
i
t
i
c
i
s
m
w
e
r
eu
s
e
dt
oc
o
n
s
t
r
u
c
tt
h
ec
o
m
p
l
i
a
n
c
ec
a
l
i
b
r
a
t
i
o
nc
u
r
v
e
s(
wast
h
a
t,
duet
os
l
o
wc
r
a
c
kg
r
o
w
t
h,
t
h
ei
n
i
t
i
a
lc
r
a
c
kl
e
n
g
t
ha
im
i
g
h
tn
o
tbea
p
p
r
o
p
r
i
a
t
ef
o
r
h
e
r
e
f
o
r
et
h
i
smethodw出 modi
自e
d
.
d
e
t
e
r
m
i
n
i
n
gK1c'T
守
。
Klc= F(S
,
d,
b
,
αh九
, orP
Q)
7
4
75
1
. Cuta
.s
m
a
l
ls
t
a
r
t
e
rn
o
t
c
ha
tm
i
d
s
p
a
n
.
2
.U
s
i
n
garampingf
u
n
c
t
i
o
nands
t
r
a
i
nc
o
n
t
r
o
l,l
o
a
dt
h
ebeambeyond九, growac
r
a
c
k
,
p
l
o
tP v
e
r
s
u
sdCMODu
s
i
n
gamaximuml
o
a
d
andt
h
e
nremovet
h
el
o
a
d
.S
u
b
s
e
q
u
e
n
t
l
y
壬1
/
3t
h
emaximuml
o
.
a
da
s
s
o
c
i.
a
t
e
dw
i
t
ht
h
ec
r
a
c
k
.
3.I
n
t
r
o
d
u
c
et
h
edyeandc
y
c
l
et
h
el
o
a
dt
oworkt
h
edyei
n
t
o山 ec
r
a
c
k
e
ds
u
r
f
a
偲・ T
he
bea
.
mmustbel
o
a
d
e
dw
i
t
h出 ec
r
a
.
ckp
r
o
c
e
e
d
i
n
gf
r
o
mt
h
et
o
ps
u
r
f
a
c
edownw
a
.
r
d
.The
∞mecrackc
1
0
s
u
r
es
t
r
e
s
s
e
sandmusth
e
1
0
0
0mustbeg
r
e
a
t
e
rt
h
a
nt
h
a
tn
e
e
d
e
dt
oo
v
e
r
.
b
o
u
t1
/
3o
fP
u
.
l
e
s
st
h
a
na
4
. Dryt
h
edyeds
u
r
f
a
田 a
ndl
o
a
dt
h
ebe
a
.
mt
of
a
.
i
l
u
r
e
.P
l
o
tP .
a
ndsCMOD. Thei
n
i
t
i
a
l
.
.
s
l
o
p
ea
f
t
e
rc
r
a
c
kc
l
o
s
u
r
ei
sov
e
r
c
o
m
eg
i
v
e
st
h
ei
n
i
t
i
a
lc
o
m
p
l
i
a
n
c
eC
5
.A
f
t
e
rf
a
i
l
u
r
e,
m伺 s
u
r
et
h
edyeds
u
r
f
a
c
ea
r
e
a
.Thei
n
i
t
i
a
lc
r
a
c
kl
e
n
g
t
hi
s
向 =(
a
r
e
ao
fdyeds
u
r
f
a
田
)
/
b
(
6.
4
)
T
y
p
i
c
a
ldyeds
u
r
f
a
田 sa
.
r
eshowni
nF
i
g
.6.
4
.
α
i
6
. Re
p
e
a
ts
t
e
p
s15f
o
rd
i
f
f
e
r
e
n
tc
r
a
.
c
kd
e
p
t
h
sande
s
t
a
b
l
i
s
hc
a
.
l
i
b
r
a
t
i
o
nc
u
r
v
e
sr
e
l
a
.
t
i
n
gC
;
anda
i,凡回dα4・Thel
a
t
t
e
rr
e
l
a
t
i
o
n
s
h
i
pa
l
l
o
w
sonet
oo
b
t
a
i
nane
s
t
i
m
a
t
eo
ft
h
ec
r
a
.
c
k
l
e
n
g
t
ha
.
ss
o
c
i
a
.
t
e
dw
i
t
hanyl
o
a
.
dont
h
es
o
f
t
e
n
i
n
gp
a
r
to
ft
h
el
o
a
d
d
i
s
p
l
a
c
e
m
e
n
もp
l
o
t
.
・
Iu
c=F(S
,
d,
b
,
a
.,
0
.
9
5
P
)
u
i
す・ 0.511
SeriesB
。
i
0.580
す・ 0.761
SeriesC
(
6
.
5
)
Thev
a
l
i
d
i
t
yo
ft
h
ep
r
o
c
e
d
u
r
et
oe
s
t
i
m
a
t
et
h
ee
x
t
e
n
d
e
dc
r
a
c
kl
e
n
g
t
ha
.maybea
.
r
guedb
y
r
e
f
e
r
r
i
n
gt
oa
.l
o
a
d
u
n
l
o
a
d
r
e
l
o
a
dd
i
a
g
ram(
F
i
g
.6
.
5
)
.Theo
b
j
e
c
t
i
v
ei
st
od
e
t
e
r
m
i
n
et
h
ec
r
a
.
c
k
l
e
n
g
t
ha
tsomep
o
i
n
tont
h
es
o
f
t
e
n
i
n
gb
r
a
n
c
h-s
a
.
yp
o
i
n
tC(
w
h
i
c
hmaybeanyp
o
i
n
t
)
.I
f
もh
e
a
.
c
t
u
a
.
lu
n
l
o
.
ad
i
n
gt
r
a
.
c
ei
sa
.
va
.
i
l
.
a
b
l
e,
t
h
eu
n
l
o
.
ad
i
n
gc
o
m
p
l
i
a
n
c
eCu c
a
nr
e
a
d
i
l
ybeme
a
.
s
u
r
e
d
a
.
ndu
s
e
dw
i
t
hac
o
m
p
l
i
組 c
ec
a
l
i
b
r
a
t
i
o
nc
u
r
v
et
od
e
t
e
r
m
i
n
et
h
ee
x
t
e
n
d
e
dc
r
a
c
kl
e
n
g
t
ha
.
.
A
l
t
e
r
n
a
t
i
v
e
l
y
,
t
h
e九一 a;r
e
l
a
t
i
o
n
s
h
i
pma
.
ybeu
s
e
dwheret
h
el
o
a
d以 p
o
i
n
tCi
su
s
e
df
o
rP
u
.
I
nc
o
n
s
t
r
u
c
t
i
n
g出 e凡 -a;c
u
r
v
e
,
i
ti
sn
o
t
e
df
r
o
mF
i
g
.6
.
5t
h
a
tana
p
p
r
o
x
i
m
a
t
i
o
ne
x
i
s
t
si
n
uand implyt
h
eu
s
eo
ft
h
es
l
o
p
eo
fl
i
n
eOBi
n
s
t
e
a
do
ft
h
ea
c
t
u
a
l
l
i
n
eOA.Thee
r
r
o
r
t
h
a
tP
r
a
c
kl
e
n
g
t
hfromt
h
i
sa
p
p
r
o
x
i
m
a
t
i
o
nwasd
e
t
e
r
m
i
n
e
dt
obel
e
s
st
h回 6%
i
nd
e
t
e
r
m
i
n
i
n
g山 ec
w
i
t
hac
o
e
f
f
i
c
i
e
n
to
fv
a
r
i
a
t
i
o
no
f8.5%(
6
2s
a
m
p
l
e
s
)(
S
w
a
r
t
zandRe
f
a
i,
1
9
8
9
)
.
Ther
e
s
u
l
t
so
b
t
a
i
n
e
dbyu
s
i
n
gt
h
i
smethodon8I
n
.and1
2i
n
.d
e
e
pbe
a
.
m
sw
i
t
ha
.
/
d至
t
0
.
6
5showJ(Ict
obei
n
v
a
r
i
a
n
tw
i
t
hr
e
s
p
e
c
tt
ot
h
ec
r
a
c
kl
e
n
g
t
handbeams
i
z
e
,
w
i
t
hac
o
e
f
f
i
c
i
e
n
.
a
。・
τ
。
:
7
.Att
h
ep
o
i
n
tont
h
eu
n
l
o
a
.
d
i
n
gp
l
o
tc
o
r
r
e
s
p
o
n
d
i
n
gt
oもheo
n
s
e
to
fu
n
s
t
a
b
l
ec
r
a
c
kg
r
o
w
t
h
a
t0
.
9
5九 d
e
t
e
r
m
i
n
et
h
ee
x
t
e
n
d
e
dc
r
a
c
k1
叩 g
t
ha
.f
romt
h
e九一向 p
l
o
t
-t
a
k
e
nt
obe.
伺 t
e
rt
h
叩
a
.
/
d= 0.65. Using0.95凡
(
s
t
e
p6
)
. Thee
x
t
e
n
d
e
dl
e
n
g
t
hmustn
o
tbeg
r
組 da
.computeK1cfrom
F
i
g
.
6.
4DyedC
r
a
c
k
e
<
lS
u
r
f
a
c
e
s
p
m
A B
.
95凡
P
8CMOO
F
i
g
.
6
.
5Lo
adv
e
r
s
u
s6CMODw
i
t
hRe
l
o
a
da
t0
.
9
5凡
77
76
v
a
r
i
a
t
i
o
no
f5.5%(N= 1
9,
8i
n
.b
e
a
m
s
)and3
.
5
%(N= 1
9,
1
2i
n.beams)(
R
e
f
a
iand
~wartz, 1
9
8
8
;S
w
a
r
t
z姐 dR
e
f
a
i,
1
9
8
9
)・
6
.1
.4E
f
f
e
c
tofFr
i
c
t
i
o
n
.
d
v
a
n
t
a
g
eo
fn
o
t
c
h
e
dbeamt
e
s
t
si
st
h
a
tf
r
i
c
t
i
o
ne
f
f
e
c
ti
ss
m
a
l
l
. Toe
x
p
l
a
i
nt
h
i
s
Ana
d,
l
e
t
ηdbet
h
ed
i
s
t
a
n
c
ef
r
o
mt
h
ebottomf
a
c
e(
F
i
g
.6
.
1
)t
ot
h
ep
o
i
n
tont
h
ec
r
a
c
kp
l
a
n
e
a
.
t姐 a
x
i
a
lf
o
r
c
ep制 i
n
g出 r
o
u
g
ht
h
i
sp
o
i
n
twouldc
a
u
s
enode
f
l
.
e
仰 no
ft
h
ebeamj
u
c
ht
h
t
l
b
v
i
o
u
s
l
y向 /
dく ηく 1(typicallyη=0.
7
5
)
.I
ft
h
eh
o
r
i
z
o
n
t
a
.
lf
r
i
c
t
i
o
nf
o
r
c
回 配t
i
n
ga
.
tt
h
e
同e
a
.
ms
u
p
p
o
r
t
sa
r
ed
e
n
o
t
e
d剖 F,
t
h
eb
e
n
d
i
n
gmomenta
tmidspann~姐ed t
oc
a
u
s
ec
r
a
c
k
a
.
t
i
o
ni
sM =(
P
/
2
)
(
S
/
2
)-FηdwhereS=sp
a
.
n,
P= a
p
p
l
i
e
dl
o
a
.
d,
F = kP/2組 d
:
J
l
o
p
a
g
ki
s
-t
h
ec
o
e
f
f
i
c
i
e
n
to
ff
r
i
c
t
i
o
n
.D
e
n
o
t
i
n
g九 =4M/S,
w
h
i
c
hr
e
p
r
e
s
e
n
t
st
h
ef
o
r
c
en
e
e
d
e
dt
o
t
:
a
u
s
et
h
ec
r
a
c
kt
op
r
o
pa
.
ga
.
t
ei
ft
h
e
r
ew
e
r
enof
r
i
c
t
i
o
n(
k=0
)oneg
e
t
s九 =P-d
.
Pj w
here
P =m
e
a
s
u
r
e
da
p
p
l
i
e
df
o
r
c
e組 d(
a
c
∞r
d
i
n
gt
oB話相も):
u恥
,
圏
、
ロE
吋~y
、
ー
'
-EZ身S
。
"
司
、
、
.
.
〆
"
‘
、
孟且
,
個
、
旬
、
ー
'
,
圏
、
、
・
・
'
(d)
、
-
,E白 皆 宮 OA古
()
H
O
E
Sg
0
2
4
u ︿S
Uぷ
S3ht〆2
d骨同︻的安否ES司令ム)6azmg百万色白宮市三一身・品志津 句。・凶日
Q
,
目今
喧園
,
司
、
@
・
、
4
2ηd
thRf=3-kP
T
h
i
sr
e
p
r
e
s
e
n
t
st
h
ep
o
r
t
i
o
no
ft
h
ea
p
p
l
i
e
df
o
r
c
en
e
e
d
e
dt
oo
v
e
r
∞methef
r
i
c
t
i
o
n
. The
l
釘 g
e
rt
h
eS/
dr
a
.
t
i
o
,thesma
.
l
l
e
ri
sd
.
乃.Forη =0.
7
5a
.
ndk= 0
.
0
0
5f
o
rr
o
l
l
e
rbe
a
.
r
i
n
g
s
r
sg
i
v
eanupperv
a
.
l
u
eo
fk= 0
.
0
1
)a
.
ndf
o
rS/d= 2
.
5(
u
s
e
dbyBazant姐 d
!
m
a
n
u
i
a
c
t町 e
1
9
8
7
),
d
.Pj = 0.
0
0
3
P
. Thu8,
we5
田 t
h
a
tt
h
en
o
t
c
h
e
dbeamt
旬e
s
錦h
旬sa
.
閃 r
e
l
a
.
t
i
v
e
l
y
P
f
e
i
f
f
e
r,
i
凶
n
悶s
民
附
悶
e
釘叩
則
n
凶
l
h
旬yP
凹l
却
a
n
剖 a
ndE
凹l
i
c
e
剖5
,
1
9
8
8
b
)
6
.
2Wedge-SplittingTest
A
n
o
t
h
e
ru
5
e
f
u
lt
e
s
tf
o
rf
r
a
c
t
u
r
eo
fc
o
n
c
r
e
t
ei
st
h
ew
e
d
g
e
s
p
l
i
t
t
i
n
gt
e
s
t(
F
i
g
.6
.
6
)
.I
ti
s
a
.
c
tt
e
n
s
i
o
nt
e
s
tu
s
e
df
o
rm
e
t
a
l
s
. Wedges
p
l
i
t
t
i
n
gt
e
s
t
sweres
t
u
d
i
e
d
i
m
i
l
a
rt
ot
h
ecomp
o
rc
o
n
c
r
e
t
ebyH
i
l
l
e
m
i
e
r組 dH
i
l
s
d
o
r
i(
1
9
7
7
)andt
h
ep
r
e
s
e
n
ts
h
a
p
eo
ft
h
et
e
s
ts
p
e
c
i
m
e
n,
I
l
a
r
a
.
d
e
r
i
z
e
dbyaBもa
r
t
e
rn
o
t
c
handag
u
i
必ngg
r
o
o
v
ewh
i
chc
a
nbee
i
t
h
e
rmouldedo
rs
a
.
wn,
a
sp
r
o
p
o
s
e
dbyL
i
n
s
b
a
u
e
randTschegg(
1
9
8
6
)
. Thet
e
s
tw出 s
u
b
s
e
q
u
e
n
t
l
yr
e
f
i
n
e
dby
andB
r
u
h
w
i
l
e
randWittmann(
1
9
8
9
)whoc
o
n
d
u
c
t
e
d(
a
tt
h
eS
w
i
s
sF
e
d
e
r
a
l
H
r
u
h
w
i
l
e
r(
1
9
8
8
),
l
'3
0
0s
u
c
ht
e
s
t
sonn
o
r
m
a
lc
o
n
c
r
e
t
e,
damc
o
n
c
r
e
t
eand0七h
e
r
I
n
s
t
i
t
u
t
eo
iT
e
c
h
n
o
l
o
g
y
)O
V
e
.
m
e
n
t
i
t
i
o
u
sm
a
t
e
r
i
a
.
l
s
. Veryl
a
r
g
ewedges
p
l
i
t
t
i
碍 s
p
e
c
i
m
e
n
s,
o
fs
i
z
e
supt
o1
.
5m (
5f
t
.
),
附
r
e
c
e
n
t
l
yb
e
e
nt
e
s
t
e
dbyS
ω四 叫 B
r
o
z,
B
r
u
h
w
i
l
e
randBo
鰐 (
1
9
8
9
)a
tt
h
eU
n
i
v
e
r
s
i
t
y
C
A
l
o
r
a
d
o
,
t
os
t
u
d
yt
h
es
i
z
ee
f
f
e
c
ti
ndamc
o
n
c
r
e
t
e
.
F
i
g・6
.
6(
a
d
)s
h
o
w
sv
a
r
i
o
u
sp
o
s
s
i
b
l
ew
e
d
g
e
s
p
l
i
t
t
i
n
gs
p
e
c
i
m
e
ns
h
a
p
e
s
.S
p
e
c
i
m
e
n(
F
i
g
.
もh
e
ra
.d
e
e
pn
o
t
c
ho
ral
o
n
g
i
t
u
d
i
n
a
.
lg
r
o
o
v
eonb
o
t
hs
i
d
e
s,
i
no
r
d
e
rt
op
r
e
v
e
n
t
'
.
c
lr
e
q
u
i
r
e
se
i
訂 f
a
i
l
u
r
eo
fo
n
eo
ft
h
ec
a
n
t
i
l
e
v
e
r
s
.F
i
g
.6
.
6(
e
,
f
)i
l
1u
s
位以e
st
h
emethodo
ft
e
s
t
i
n
g.The
悶
twol
o
w
f
r
i
c
t
i
o
nr
o
l
l
e
ro
rn
e
e
d
l
ebωings(
o
n
!
n
b
l
yo
ftwow
e
d
g
e
si
sp
r
e
s
s
e
dbetw
副 e
)w
h
i
c
hd
e
v
e
l
o
pap
a
i
ro
ff
o
r
c
白 Nt
h
a
tt
e
n
dt
os
pl
i
tt
h
es
p
e
c
i
me
n(
F
i
g
.6
.
6
g
).The
l
I ,~" a
s
s
e
m
b
l
yi
sl
o
a
d
e
di
nas
t
a
t
i
c
a
l
l
yde
t
e
r
m
i
n
a
t
em姐 n
e
rs
ot
h
a
te
a
c
hwe
dger
e
田 i
v
郡 山e
78
79
samel
o
a
d
. Thed
i
m
e
n
s
i
o
n
so
ft
h
en
o
t
c
handt
h
eg
r
o
o
v
emustbec
h
o
s
e
n8
0t
h
a
tt
h
ec
r
a
c
k
a
.
ga
.
t
e
ss
y
m
m
e
t
r
i
c
a
l
l
y
.
prop
During出 et
e
s
t,
t
h
es
p
l
i
t
t
i
n
gf
o
r
c
eN (
F
i
g
.6
.
6
g
)mustbem回 s
u
r
e
dw
i
t
hs
u
f
f
i
c
i
e
n
ta
c
出 y
. Thec
r
a
.
c
kmoutho
p
e
n
i
n
gd
i
s
p
l
a
c
e
m
e
n
もOCMOD i
smeasuredbyat
r
a
n
s
d
u
c
e
ro
ra
c
u
r
もa
c
h
e
da
tt
h
el
e
v
e
lo
ft
h
es
p
l
i
t
t
i
n
gf
o
r
c
田, i
nw
h
i
c
h
dipgage(
F
i
g
.6.
6
f
)whichs
h
o
u
l
dbea
t
掛 eo
一CMODrepresentstheloOO-pointdispl民 町lentOLPD舗 回Cla胞 dwiththehorizontal
C
e
s
ti
sc
o
n
もr
o
l
l
e
dbyOCMODi
nac
l
o
s
e
d
I
o
o
ps
e
r
v
o
componento
ft
h
es
p
l
i
t
t
i
n
gf
o
r
c
eN.Thet
a8
t
a
b
l
et
e
s
tc
a
na
l
s
obep
e
r
f
o
r
m
e
dundera
c
t
u
a
t
o
r
h
y
d
r
a
u
l
i
ct
e
s
t
i
n
gm
a
ε
h
i
n
e
. However,
s
t
r
o
k
eC
O
I
l
s
m
lo
runderc
r
o
s
s
h
e
a
dd
i
s
p
l
a
c
m
I
e
n
tc
p
n
t
r
o
lu
s
i
n
gc
o
n
v
e
n
t
i
o
n
a
1
4
e
s
t
i
n
gm
a
c
h
i
n
e
s
.
I
nt
h
a
tc
a
s
e
,
t
h
ea
p
p
r
o
p
r
i
a
t
en
o
t
c
hl
e
n
g
t
hne
偲 s
s
a
r
yt
oe
n
s
u
r
es
t
a
b
i
l
i
t
ymustbei
d
e
n
t
i
f
i
e
dby
s
p
e
c
i
m
e
ns
t
i
f
f
n
e
s
sandm
a
t
e
r
i
.
a
l
c
o
n
s
i
d
e
r
i
n
gt
h
ei
n
t
e
r
a
c
t
i
o
nbetweent
e
s
t
i
n
gmachines
t
i
f
f
n
e
s
s,
p
r
o
p
e
r
t
i
e
s(
B
r
u
h
w
i
l
e
r,
1
9
8
8
;B
r
u
h
w
i
l
e
r副 Wittmann,
1
9
8
9
)
.
組
.
t
a
g
e
so
f出 .
ewedges
p
l
i
t
t
i
n
gt
e
s
ta
r
ea
sf
o
l
l
o
w
s
:
TheOOv
i
g
h
t,
s
i
n
c
et
h
er
a
t
i
oo
ff
r
a
c
t
u
r
ea
r
e
aもot
h
es
p
e
c
i
m
e
n
1
)Thes
p
e
c
i
m
e
n
sa
r
e∞mpactandl
山 (eι5
ユtim回 largerthanthatforthethree-poirル bend
volumei
sl
a
r
g
e
rthanf
o
ro
t
h
e
rt
e
1985)Thisi
se
s
p
e
c
i
a
l
l
yu
s
e
f
u
lf
o
rt
h
es
t
u
d
yo
fs
i
z
ee
f
e
c
b,
s
i
n
c
e
t
e
s
ta
c
c
o
r
d
i
n
gt
oRILEM,
出 C
担
beo
b
t
a
i
n
e
dw
i
t
hs
m
a
l
l
e
rs
p
e
c
i
m
e
nw
e
i
g
h
t
. Duet
ol
e
s
s
e
rw
e
i
g
h
t,
l
a
r
g
e
rf
r
a
c
t
u
r
ea
r
e
l
a
r
g
e
rs
p
e
c
i
m
e
n
sa
r
ee
a
s
i
e
rt
oh
a
n
d
l
e, 乱ndt
h
e
r
ei
sal
e
s
s
e
rr
i
s
ko
fb
r
e
a
k
i
n
gthemd
u
r
i
n
g
h
a
n
d
l
i
n
g
.
2
)Thec
u
b
i
c
a
lo
rc
y
l
i
n
d
r
i
c
a
ls
p
e
c
i
m
e
n
s(
F
i
g
.6
.
6a
・c
)c
a
nbee
a
s
i
l
yc
a
s
ta
tt
h
ec
o
n
s
t
r
u
c
t
i
o
n
h
esamemoldsasf
o
rs
t
r
e
碍t
ht
e
s
t
s,
andt
h
ec
y
l
i
n
d
r
i
c
a
ls
h
a
p
e
s(
F
i
g
.6
.
6b
d
)c
組
s
i
t
e戸時 t
a
l
s
obeo
b
t
a
i
n
e
dfromd
r
i
l
l
e
dc
o
r
回 f
rome
x
i
s
t
i
n
gs
t
r
u
c
t
u
r
田.
3
)Theu
s
eo
fwedgesf
o
ri
n
d
u
c
i
n
gt
h
el
o
a
di
n
c
r
e
a
s
e
st
h
es
t
i
f
f
n
e
s
so
ft
h
et
e
s
ts
e
t
叩 日d
makingi
tp
o
s
s
i
b
l
et
oc
o
n
d
u
c
tt
h
et
e
s
teveni
namachine
t
h
u
se
n
h
a
n
c
e
ss
t
a
b
i
l
i
t
yo
ft
h
et
e
s
t,
t
h
a
ti
sn
o
tv
e
r
ys
t
i
f
f
.
nc
ο
n
t
r
a
s
tt
on
o
t
c
h
e
dbeamt
e
s
t
s(
w
h
e
r
et
h
e
4
)t
h
ee
f
f
e
c
to
fs
e
l
f
w
e
i
g
h
ti
sn
e
g
l
i
g
i
b
l
ei
回 t
duet
oownw
e
i
g
h
tc
拍
beo
v
e
r50%o
ft
h
et
o
t
a
lb
e
n
d
i
o
gmoment)
b
e
n
d
i
n
gmom
tmustben
o
t
e
d七h
a
tt
h
ewedge1
0
晶d
i
n
gh組 乱1
5
0ad
i
s
a
d
v
a
n
t
a
g
e
Ont
h
eo
t
h
e
rhand,i
tP = a
p
p
l
i
e
dv
e
r
t
i
c
a
ll
o
a
d,
N=
=specimenr
e
a
c
t
i
o
n
s
a
si
ti
n
t
e
n
s
i
f
i
e
sf
r
i
c
t
i
o
n
a
le
f
f
e
c
t
s
. Le
償対e
dt
op
r
o
p
a
g
a
t
et
h
ec
r
a
c
k
swhicha
r
enormalt
ot
h
ewedges
u
r
f
a
c
ei
n
c
l
i
n
e
dbya
n
g
l
ea
n
=f
r
i
c
t
i
o
nc
o
e
f
f
i
c
i
叩 t
o
ft
h
eb
e
a
r
i
n
g
s
.Then,
t
h
ee
q
u
i
l
i
b
r
i
u
mc
o
n
d
i
t
i
o
no
f
(
F
i
g
.6
.
6
g
) andk=
田 a
c
t
i
n
g00t
h
ewedgey
i
e
l
d
sP =2(Ns
i
o
α +kN∞sα)=九 (1+kcotα
),
w
h
e
r
e
v
e
r
t
i
c
a
lf
o
r
c
凡 =2Nsinαist
h
;f
o
r
c
eneed;dt
op
r
o
pa
.
g
a
t
et
h
ec
r
a
c
ki
ft
h
e
r
ewerenof
r
i
c
t
i
o
n(
k= 0
)・
S
i
n
田 k
∞t
α<
t
:
:1
,
wehave九ど P
/(l+
kcotα)=P(l-kcotα)o
rP+
Q=
=P-I
:
:
.P where
(
a
c
c
o
r
d
i
n
gもoBa
Z
a
n
t
)
:
(
6
.
6
)
I
:
:
.
P = P/(c
o
tα
t
h
姐 f
o
rt
h
es
h
o
r
tn
o
t
c
h
e
dbeamso
fBa
Z
a
n
tandP
f
e
i
f
f
e
r(
1
9
8
7
),
andabout20
・t
i
m
e
sl
a
r
g
e
r
t
h
a
nf
o
rt
h
el
o
n
g
e
rn
o
t
c
h
e
dbeamsr
e
∞mmendedbyRILEM,for出 esamevalueofk(see
.
4
.
)
.T
h
i
sd
i
sa
.
d
v
a
n
ta
.
g
eo
ft
h
ewedges
p
l
i
t
t
i
n
gt
e
s
ti
ssurmounta
.
b
l
e, 岨df
r
i
c
t
i
o
n
a
l
S
e
c
.6
.1
姐
ber
e
d
u
c
e
dby(
1
)a
t
t
a
c
h
i
n
gh
a
r
d
e
n
e
ds
t
田 1i
n
s
e
r
t
sa
l
o
n
gt
h
ei
n
c
l
i
n
e
dwedge
e
f
f
e
c
t
sc
,
品
目
,(
2
)u叫 n
e
e
d
l
eb
e
a
r
i
n
g
s,and(
3
)c
a
r
e
f
u
i
l
yp
o
l
i
s
h
i
n
gt
h
ewedges
u
:
r
f
a
c
ea
sshown
s
u
r
f
i
l
l
e
m
e
i
ぽ a
ndH
i
l
s
d
o
r
f(
1
9
7
7
)whoe
x
p
e
r
i
m
e
n
t
a
l
l
yd
e
t
e
r
m
i
n
e
da/
(
v
a
l
u
e= 0
.
0
0
0
3
1f
o
r
byH
t
h
e
i
rwedgel
o
a
d
i
n
gs
e
t
u
pw
i
t
hn
e
e
d
l
eb
e
a
r
i
n
g
s
.
I
ft
h
ev
a
l
u
eo
fki
sn
e
a
r
l
yc
o
n
s
t
a
n
tandw
e
l
lr
e
p
r
o
d
u
c
i
b
l
e,
onemayi
n
t
r
o
d
u
c
et
h
ec
o
r
r
e
c
t
i
o
o
l
l
Pi
nt
h
e組 a
l
y
s
i
s
.However,
s
i
n
c
et
h
ev
a
l
u
eo
ff
r
i
c
t
i
o
nc
o
e
伍c
i
e
n
ti
so
f
t
e
nq
u
i
t
eu
n
c
e
r
t
a
i
n,
i
ti
sb
e
t
t
ぽ t
om'
品 s
u
r
et
h
es
p
l
i
t
t
i
n
gf
o
r
c
eN d
i
r
e
c
t
l
ybyi
n
s
t
r
u
m
e
n
t
i
n
gt
h
ewedgesandt
h
e
四.
s
h
a
f
t
st
h
a
tc
a
r
r
yt
h
eb
e
a
r
i
n
g
sw
i
t
hs
t
r
a
i
ngag
Thef
o
r
e
g
o
i
n
ga
n
a
l
y
s
i
sshowst
h
a
tav
e
r
ys
m
a
l
lwedgeangleαisu
n
f
a
v
o
r
a
b
l
efromt
h
e
t
h
es
m
a
l
l
e
rt
h
ea
n
g
l
e,
t
h
es
t
i
f
f
e
ri
st
h
es
p
e
c
i
m
e
n
v
i
e
w
p
o
i
n
to
ff
r
i
c
t
i
o
n
.Ont
h
eo
t
h
e
rh祖 d,
sa
.r
e
a
s
o
n
a
b
l
e∞m
p
r
o
m
i
s
e
.
m
a
c
h
i
n
ea
s
s
e
m
b
l
y
.Thea
n
g
J
eα =1
50 i
叫 al
a
r
g
ewedgea
n
g
l
e(α>3
00) i
su
n
d
e
s
i
r
a
b
l
eb
e
c
a
u
s
ei
tl
e
a
d
sもoas
i
g
n
i
f
i
c
a
n
tnormal
A
l
田 o
fs
u
c
hs
t
r
e
s
s
e
s
s
t
r
e
s
sp
a
r
a
l
l
e
lt
ot
h
ec
r
a
c
kp
l
a
n
ei
nt
h
ef
r
a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
e
.Thep
r
e
s
e
n
r
i
b
e
dbyE
q
.3
.
9
. The
maya
f
f
e
c
tt
h
es
o
f
t
e
n
i
n
gc
u
r
v
ef
o
rt
h
ef
r
a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
e, 制 d田 c
,
G jn
o
ri
st
h
e
a
r
e
au
n
d
e
rt
h
es
o
f
t
e
n
i
n
gc
u
r
v
ei
st
h
e
nn
o
tl
o
n
g
e
re
q
u
a
lt
ot
h
ef
r
a
c
t
u
r
ee
n
e
r
g
y
a
r
目 u
n
d
e
rt
h
el
o
a
d
d
i
s
p
l
a
c
e
m
e
n
tc
u
r
v
e
.
!
(
l
ci
so
b
t
a
i
n
e
dbyt
h
esamemethoda
sd
e
s
c
r
i
b
e
di
n
Thea
p
p
a
r
e
n
tf
r
a
c
t
u
r
et
o
u
g
h
n
e
s
s,
.
c
kl
e
n
g
t
h,
whicha
c
c
o
u
n
t
sf
o
rt
h
ef
r
a
c
t
u
r
e
S
e
c
t
i
o
n6
.
1f
o
rn
o
t
c
h
e
dbeamt
e
s
t
s
.Thee
f
f
e
c
t
i
v
ec
ra
恥 凶sz
o
n
e
,
i
sd
e
t
e
r
m
i
n
e
dbyt
h
ec
o
m
p
l
i
a
n
c
emethod,
b
a
s
e
don五n
i
t
ee
l
e
m
e
n
tc
a
l
i
b
r
a
t
i
o
n
.
p
r
,
u
n
l
o
a
d
r
e
l
o
a
dc
y
c
l
e
sa
r
ep
e
r
f
o
r
m
e
dd
u
r
i
n
gt
h
et
e
s
t
.Othermethodss
u
c
h
F
o
rt
h
a
tp
u
r
p
o
s
e
ee
v
a
l
u
a
t
i
o
no
ff
r
a
c
t
u
r
ee
n
e
r
g
yfromt
h
ea
r
e
a
.undert
h
el
o
a
.
d
d
i
s
p
l
a
c
e
m
e
n
tdiagramand
a
sもh
a
sd
e
s
c
r
i
b
e
di
nt
h
es
e
q
u
e
l
.
t
h
es
i
z
ee
f
f
e
c
tmethoda
r
ea
p
p
l
i
c
a
b
l
e,
唱
,
,
Pi
st
恥
heme
也a
邸
s
叩町
r
e
d1
0
0
0叩副
dI
:
:
.丹
Pf
0,
恒
討c
叫t
i
叩
o
l
l
.I
fα<45
I
:
:
.P i
slarg~r t
h
a
nkP,
whichmeanst
h
a
tf
r
i
c
t
i
o
n
a
le
f
f
e
c
t
sa
r
ee
n
h
a
n
c
e
d
f
byt
h
ewedgel
o
.
a
d
i
n
g
.
s
I
:
:
.PJ= 3
.7
3
k
P
.Themanuf
.
a
c
t
u
r
e
r
so
fr
o
l
l
e
rb
e
a
r
i
n
g
F
o
rt
h
et
y
p
i
c
a
lwedgea
n
g
l
eα =1
50,
組も田 0
.
0
1a
st
h
el
i
m
i
t
)・ Assuming
g
i
v
ek
v
a
l
u
e
srωgingfrom0
.
0
0
1t
o0
.
0
0
5(
a
n
dg
u
a
r
k= 0
.
0
0
5,
I
:
:
.P = O
.
0
1
9
P
.T
h
i
sf
r
i
c
t
i
o
n
a
le
f
f
e
c
ti
ss
i
g
n
i
f
i
c
a
n
tandi
sabout6
t
i
m
e
sl
a
.
r
g
e
r
,
,
,
,
6
.
3
.Work-of-Fr
actureMethod(悶LEM,
H
i
l
l
e
r
b
o
r
g
)
T
h
i
smethod,
whichw剖 o
r
i
g
i
n
a
l
l
yd
e
v
e
l
o
p
e
df
o
r田 r
a
.
凶ω(Nak
a
.
y
制 a
,
1
9
6
5
;Ta
.
t
t
e
r
s
a
l
l
dTappin,
1
9
6
6
),
i
st
h
ef
i
r
s
tmethodo
ft
田t
i
n
gf
o
rf
r
a
c
t
u
r
ep
r
o
p
e
r
t
i
回 o
fconc
同 et
obe
凶 a
sa
.s
t
a
n
d
a
r
d(RILEM,
1
9
8
5
)
. Theb
a
s
i
sf
o
ra
p
p
l
y
i
n
gt
h
i
smethodt
oc
o
n
c
r
e
t
e
p
r
o
p
o
1
9
8
5
b
)
.T
h
e
i
rmethodu
s
e
s
w
a
sd
e
v
e
l
o
p
e
dbyH
i
l
l
e
r
b
o
r
gandh
i
sc
o
w
o
r
k
e
r
s(
H
i
l
l
e
r
b
o
r
g,
f
i
c
t
i
t
i
o
u
sc
r
a
c
k
"c
o
n
c
e
p
t(
H
i
l
l
e
r
b
o
r
ge
ta
,
.
l 1976;H
i
l
l
e
r
b
o
r
g
,1980jPetersson,1981)
t
h
e“
sno
もa
nLEFMmethod.
(
F
i
g
.6
.
7
)i
m
p
l
i
c
i
t
l
yandもhusi
I
no
r
d
e
rもoc
o
n
t
r
a
s
tt
h
i
sw
i
t
hLEFM00t
h
eba
.
si
so
fe
n
e
r
g
yp
a
r
a
m
e
t
e
r
s,
r
e
c
乱It
h
a
tt
h
e
町g
yr
e
l
e
a
s
er
a
t
eGIci
st
h
ee
且e
r
g
yr
e
q
u
i
r
e
dp
e
ru
n
i
tc
r
a
<
ユke
x
t
e
n
s
i
o
ni
nam
a
t
e
r
i
a
l
c
r
i
t
i
c
a
le
n
~n ~hich t
h
e
r
ei
snop
r
o
c
e
s
szone
,thati
s,
a
l
lt
h
ee
n
e
r
g
yi
ss
u
r
f
a
田 e
n
e
r
g
y組 dnoe
n
e
r
g
y
h
ec
r
a
c
k'
t
i
p
.I
nf
a
c
t,
aproces~- z
o
n
ed
o
e
se
x
i
s
t~d t
h
e
r
e
f
o
r
eth~
i
sd
i
s
s
i
p
a
t
e
dawayfro~ t
t
?
t
a
le
n
e
r
g
yo
ff
r
a
c
t
u
r
ei
n
c
l
u
d
e
sa
1
1t
h
ee
n
e
r
g
yd
i
s
s
i
p
a
t
e
dp
e
ru
n
i
tp
r
o
p乱.
g
a
t
i
o
nd
i
s
t叩 c
eo
f
側
zone剖 aw
h
o
l
e
.T
h
i
si
sc
a
l
l
e
dt
h
ef
r
a
c
t
u
r
ee
n
e
r
g
yGJ(
F
i
g
.6
.
7
)
.
t
h
ef
r
a
c
t
u
r
ep
r
o
c
,themethodcanbeappliedtoavarietyoft
田 ts
p
e
c
i
m
e
ng
∞metriesbut
C
o
n
c
e
p
t
u
a
l
l
y
s
e
sabeamsp~cimen l
o
a
d
e
di
nt
h
r
e
ep
o
i
n
tbending
~withacent
r
a
l
t
h
ep
r
o
p
o
s
e
dst~ndard u
佃
8
1
e
d
l
!
:
en
o
t
c
h(
F
i
g
.6
.
1
a
)
. Co
m
p
l
e
t
ed
e
t
a
i
l
so
ft
h
ep
r
o
p
o
s
e
ds
t
a
n
d
a
r
da
r
eg
i
v
e
ni
nt
h
eRILEM
副 da
r
en
o
tr
e
p
e
a
t
e
dh
e
r
e
.B
r
i
e
f
l
y
,
t
h
et
e
s
tp
r
o
c
e
d
u
r
ec
o
凶s
t
so
f
t
h
ef
o
l
l
o
w
i
n
gs
t
e
p
s
.
b∞
mmendation(
1
9
8
5
)
1
.Theb
e
a
r
np
r
o
p
o
r
t
i
o
n
sa
r
es
e
l
e
c
t
e
di
nr
e
l
a
t
i
o
nt
omaximuma
g
g
r
e
g
a
.
t
es
i
z
e
.Them
i
n
i
mumd
e
p
t
hdi
sa
p
p
r
o
x
i
m
a
t
e
l
ys
i
xt
i
m
e
st
h
es
i
z
eo
ft
h
ea
g
g
r
e
g
a
t
e
. Ther
a
t
i
o
so
fS/d
.Thebeams
p
e
c
i
m
e
ni
sn
o
t
c
h
e
dt
oadep
出 向/
d=0ふ
v
a
r
yf
r
o
m8t
o4
.S
e
eT
a
b
l
e6
.1
A
l
2
.Thev
e
r
t
i
c
a
l
l
o
a
.
d
p
o
i
n
td
e
f
l
e
c
t
i
o
no
ft
h
ebeam(
c
a
l
l
e
ddLPDi
nF
i
g
.6
.
1
a
)i
st
obem
e
a
s
u
r
e
da
.
n
dp
l
o
t
t
e
dc
o
n
t
i
n
u
o
u
s
l
ya
l
o
n
gw
i
t
ht
h
ea
p
p
l
i
e
dJ
o
a
dP. Ther
e
s
u
l
t
i
n
gt
r
a
c
ei
s
showni
nF
i
g
.6
.
8
.
st
obec
o
n
d
u
c
t
e
di
nam組 問rもop
r
o
d
u
c
es
t
a
b
l
ec
r
a
c
kg
r
o
w
t
h
.I
fc
l
o
s
e
d
l
o
o
p
3
.The色白色 i
加 t
i
n
gi
su
s
e
dt
h
e
ns
t
r
a
i
nc
o
n
t
r
o
ls
h
o
u
l
dbes
e
l
e
c
t
e
d.I
fac
I
o
s
e
d
l
∞psystemi
sn
o
t
a
v
a
i
l
a
b
l
e,
t
h
e
nas
t
i
f
ft
e
s
t
i
n
gmachinei
sr
e
q
u
i
r
e
d(
s
t
i
f
f
n
e
s
sr
e
∞mmendationsa
r
eg
i
v
e
n
1
9
8
5
.
)
i
nRILEM,
4
.Thef
r
a
c
t
u
r
ee
n
e
r
g
yi
sc
a
l
c
u
l
a
t
e
da
s
Al
R _Wo+mgdo
一
一 日
J! 一
円
=
c
)
E
σ
σ
,'''
a
戸
‘
,s
﹃
4
E
(
6
.
8
)
一 回
' ~I・g
'
z
.
,
,
U
f
t
/
E
c
=
i
nw
h
i
c
hWo a
r
e
au
n
d
e
rP -dLPDc
u
r
v
eupt
odo;d
o d
i
s
p
l
a
c
e
m
e
n
twhenP r
e
t
u
r
n
st
o
m
l+2m2)g組 dm19=b
e
a
r
nw
e
i
g
h
tb
e
t
w
e
e
ns
u
p
p
o
r
t
s,
m29=w
e
i
g
h
to
ff
I
x
t
u
r
e
s
0
;mg=(
r
i
g
i
n
a
l,
u
n
c
r
a
c
k
e
dl
i
g
a
m
e
n
ta
r
e
a= b
(d-a
o
)
.
:
:
:
:o
w
h
i
c
hi
sc
a
r
r
i
e
dbyt
h
ebeam;andA1ig :
T
h
i
sf
o
r
m
u
l
ai
sv
a
l
i
di
ft
h
emovemento
fl
o
a
da
.
n
dhPDa
r
edownward. I
ft
h
ebeami
s
陥 t
e
d“
oni
t
ss
i
d
e
"白 山a
tt
h
ea
p
p
l
i
e
dl
o
a
.
dP i
sn
o
r
m
a
lt
o山 eb
e
a
m
'
sseHw
e
i
g
h
tv
e
c
t
o
r
,
t
h
e
nt
h
et
e
r
mmgdoi
sn
e
g
l
e
c
t
e
d
.A
l
s
o,
i
ft
h
ed
e
a
dw
e
i
g
h
ti
so
t
h
e
r
w
i
s
ec
o
m
p
e
n
s
a
t
e
d,
t
h
i
s
t
e
r
mi
自n
e
g
l
e
c
t
e
d
.
F
u
r
t
h
e
r
,
i
ft
h
emovementso
fl
oadanddLPDa
r
eupward-t
h
u
so
p
p
o
s
i
n
gt
h
es
e
l
fw
e
i
g
h
t
h
a
t(
S
w
a
r
t
zandYap,
1
9
8
8
)
v
e
c
t
o
r-t
h
e
ni
tお shownt
G
I
J= 尚一 ~mgðo
!- - - A -
f
{
f
{
d
J
w
(
6
.
7
)
w
F
i
g
.
6
.
7F
i
c
t
i
t
i
o
u
sC
r
a
c
kM
o
c
l
e
lD
e
s
c
r
i
p
t
i
o
no
fT
e
n
s
i
l
eF
ra
c
t
u
r
e
nw
h
i
c
hd
i
s
p
l
a
c
e
m
e
n
ta
t出ep
o
i
n
ton出eu
n
l
o
a
dp
o
r
t
i
o
no
ft
h
ep
l
o
t whenP =
=
o= d
屯t
/2+m2)g.
Eq~. 6
:
6a
.
n
d6
.
7w
e
r
ed
e
r
i
v
e
dbyS
w
a
r
t
zandYap(
1
9
8
8
).Thes
e
旺w
e
i
g
h
もt
ermmaybe
1
I
1
i
t
es
i
g
n
i
f
i
c
a
n
t,
e
s
p
e
c
i
a
l
l
yi
fy
o
u
r
i
g∞n
c
r
e
t
ei
sb
e
i
n
gt
e
s
t
e
do
rt
h
es
p
e
c
i
m
e
ni
sl
a
r
g
e
.
4laboratori~ i
n
∞r
p
o
r
a
t
i
n
ga
b
o
u
t7
0
0be
畠m
sw
e
r
e
E
x
t
e
n
s
i
v
er
o
u
n
d
r
o
b
i
nt
e
s
t
sfro~ 1
l
戸rtedbyHillerborg(1985c)Withregardtovariationo
fr
e
s
u
l
t
swidmag
i
v
e
nbmu
叶匂,~t~e c
o
e
f
f
i
c
i
e
n
to
fv
a
r
i
a
t
i
o
nr
a
n
g
e
df
;
o
ma
b
o
u
t2.5%t
o25%w
i
t
hmos
七r
e
s
u
l
t
sa
.
round
h
e悶 凶i
v
i
t
yo
ft
h
es
t
r
e
n
g
t
ho
fas
t
n
川u
r
ew
i
t
hr
e
g
a
r
d
ω
115%.Itwωnotedt
h
a
t仁 .t
略;
e
si
nG
7i
sn
o
r
m
a
l
l
yl
e
s
st
h
a
n1
/
3o
ft
h
es
e
n
s
i
t
i
v
i
t
yw
i
t
hr
e
g
a
r
dt
och
a
.
n
g
e
si
n~ormal
83
82
P(lood)
TABLE6
.
1
. SpecimenS
i
z
e
sRecommendedf
o
r
H
i
l
l
e
r
b
o
r
g
'
sMe
出 od(
RILEM. 1
9
8
5
)
明'
i
d
t
h
b
mm
Le
ngth
L
mm
1
・1
6
100土5
100ま5
840:
t10
800土5
1
6
.
1・32
200土5
100土5
1190土 10
1130士5
3
2
.
1・48
300土5
150土5
1450士官 O
1385土5
4
8
.
1・64
t5
400:
200:
t5
1640土 1
0
自
1.
0
,
m
TAB
日 6
.
2
. Re
g
r
e
s
s
i
o
nC
o
e
f
f
i
c
1
e
臨 ん .C
l
.司
.Dj
f
o
rKar1ha
l
o
oandN
a
l
l
a
血 ambl
・
sMe
血od.
(
i=
O
....
.
4
:j=
O
..• ..
5
)
1
/
j
。
L
凶 1
3I
l
d
2
.
5d
.
ー
4
s=d/d
o
6
"
F
i
g
.
6
.
9Th
即 時t
i
c
a
1S
i
z
eDependenceo
f悶 LEMFr
.
a
c
t
u
r
eEnergyG~
AI
C
I
同
D
j
3.6460
1
.5640
0.
4
607
1
.9560
l
-6.7890
-8.3200
0.0484
0.3982
2
39.2400
52.9500
0.
0063
-0.0553
3
-76.8200
-124.9000
-0.0003
0.0027
4
74.3300
122.9000
-0.0059
0.0202
0.0003
-0.0055
5
2
Span
S
mm
r
nu
a﹄
OAW
Depth
d
mm
一
一
F
i
g
.
6
.
8Lo
adV
e
r
s
u
sdLPD
Max1mum
Agg
r
.S
i
z
e
mm
.
84
85
“噸
s
t
児
r
e
n
l
紅 ea
b
o
u
t3t
i
m
e
s国 h
i
g
ha
si
nmosts
t
r
e
n
g
t
ht
e
山 (
H
i
1
l
e
r
b
o
r
g,
1
9
8
5
c
)"
・I
twa
.
sc
o
n
c
l
u
d
e
d
ont
h
i
sba
.
s
i
sandbye
x
a
m
i
n
i
n
gt
h
ec
h
a
n
g
e
si
nG
7叫 ues割 問atedwith仰 g田 inbeam
deptht
h
a
tt
h
es
p
u
r
i
o
u
si
n
f
l
u
e
n
c
eo
fs
p
e
c
i
m
e
ns
i
田 o
nme
a
.
s
u
r
e
dv
a
l
u
e
目o
f
,G
7i
sa
c
c
e
p
t
a
.
b
l
e
.
(However
,
t
h
es
i
z
ee
f
f
e
c
tmethodd
i
s
c
u
s
s
e
dl
a
t
e
re
l
i
m
i
n
a
同 t
h
i
si
n
f
l
u
e
n
団・)
Thus,
eventhought
h
e
r
ewωanu
n
d
e
s
i
r
a
b
l
es
i
z
edepend
e
ncyi
nt
h
et
e
s
tr
e
s
u
l
t
sf
o
rG
7
i
twωconsideredt
obeo
fnog
r
e
a
t
e
ri
m
p
o
r
t
a
n
c
eonbhc
a
l
c
u
l
a
t
e
ds
t
r
e
n
g
t
hd4hes
t
r
u
c
t
u
r
e
t
h
a
nt
h
es
i
m
i
l
a
rs
i
z
edependencyi
no
r
d
i
n
a
r
ys
t
r
e
n
g
t
ht
e
s
t
s
.
R怠 叫t
8from姐 o
t
h
e
rs
e
r
i
e
so
ft
e
s
t
s(
S
w
a
r
t
z組 d R
出, 1
9
8
9
;R
e
f
a
iandSw
叫
1
9
腿)
showas
i
m
i
l
a
rs
i
z
ee
f
f
e
c
ta
刈 a
l
s
oav
a
r
i
a
t
i
o
no
f w
i
t
hn
o
t
c
hd
e
p
t
h
.
l
nsummary
,
t
h
eRILEMw
o
r
k
o
f
f
r
a
.
c
t
u
r
emethodi
sa
.p
r
a
.
c
t
i
c
a
la
.
ppro
a
.
c
h,
whichi
ss
u
i
ι
a
b
l
ef
o
ru
s
ei
nl
a
b
o
r
a
t
o
r
i
e
swhichdon
o
thavev
e
r
ye
l
a
b
o
r
a
t
eequipmentandi
n
c
o
r
p
o
r
a
t
e
s
a
.numbero
fωmpromisesi
no
r
d
e
rt
os
i
m
p
l
i
f
yt
e
s
t
i
n
gw
h
i
l
es
t
i
l
lo
b
t
a
i
n
i
n
gu
s
e
f
u
lr
e
s
u
l
t
s
(H
i
l
1
e
r
b
o
r
g,
1
9
8
5
b
)
.Themethodi
sb
a
s
e
donat
h
e
o
r
ywhichh
a
sh品dw
i
d
e
'a
.
c
c
e
p
t
a
n
c
e
,
t
h
e
d
c
t
i
t
i
o
u
sc
r
a
ι
kmodel(
H
i
l
l
e
r
b
o
r
ge
ta
,
.
11976).Thef
r
a
.
c
t
u
r
ee
n
e
r
g
y∞n
c
e
p
ta
l
s
oh
a
sav
e
r
y
s
i
m
p
l
emeaning,
w
i
t
h
o
u
tu
s
i
n
gLEFM.
There釘 e,
however
,
c
e
r
t
a
i
na
.
s
p
e
c
t
si
nwhicht
h
eR1
LEMw
o
r
k
o
f
f
r
a
.
c
t
u
r
emethodn
e
e
d
s
t
obef
u
r
t
h
e
ri
m
p
r
o
v
e
d
.T
h
i
smethodd
o
e
sn
o
tg
i
v
er
ω
u
l
t
si
n
d
e
p
e
n
d
e
n
to
ft
h
es
i
z
eo
ft
h
e
fp
r
e
c
r
a
c
k
e
db白 血sa
r
eu
s
e
d,
andt
h
u
st
h
ev
a
l
u
e
c
r
a
c
k
e
da
r
e
af
o
rag
i
v
e
nbeamd
e
p
t
heveni
}mustbec
o
n
s
i
d
e
r
e
dt
obeo
n
l
yana
p
p
r
o
x
i
m
a
t
i
o
no
ft
h
et
r
u
ee
n
e
r
g
yp
a
r
a
m
e
t
e
rw
h
i
c
h
o
f01
char~cte出es t
h
es
u
r
f
a
田 e
n
e
r
g
yandt
h
ee
n
e
r
g
yo
fp
r
o
倒 s
z
o
n
ef
o
r
m
a
t
i
o
n.1
ti
ss
i
m
p
l
i
五edb
u
t
no
もu
n
r
e
a
.
s
o
n
a
b
l
et
oc
o
n
s
i
d
e
r出 i
ssuma
l
s
ot
ob
ec
o
n
s
t
a
.
n
tw
i
t
hr
e
s
p
e
c
tt
ot
h
ec
r
a
c
kl
e
n
g
t
h
.
T
h
i
si
m
p
l
i
e
st
h
a
tt
h
eshapeands
i
z
eo
ft
h
ep
r
o
c
e
s
sz
o
n
edon
o
tc
h
a
n
g
e
.F
u
r
t
h
e
r
m
o
r
e
,
r
e
f
i
n
e
m
e
n
もo
ft
h
eR1LEMmethodi
sn
e
e
d
e
d回 ar
e
s
u
l
to
ft
h
es
i
z
ee
f
f
e
c
t,whichw
i
l
lb
e
d
i
s
c
u
s
s
e
dn
e
x
t
.
0
7
6.
4SizeEffectinWork-of-Fr
actureMethod
0
7
o
b
ta
.
i
neda
.
c
c
o
凶時t
ot
h
ee
泊s
t
i
n
gRILEM
Thes
i
z
edepe
n
d
e
n
c
eo
ff
r
a
c
t
u
r
ee
n
e
r
g
y
r
e
c
o
n
u
n
e
n
d
a
t
i
o
nont
h
eworko
ff
r
乱c
t
u
r
emethodh
a
sb
e
e
ni
n
v
e
s
t
i
g
a
もe
dbyP
l
a
n
a
s姐 dE
l
i
c
e
s
(
1
9
8
8
b
)ont
h
eb
a
s
i
so
f日o
l
u
t
i
o
no
fani
n
t
e
g
r
a
lequa
.
t
i
o
nandbyBa
.
Z
a
.
nもan
dKazemi(
1
9
8
9
b
)
ont
h
eb
a
s
i
so
ft
h
es
i
z
ee
f
f
e
c
tl
a
w
. Thep
r
e
m
i
s
eo
ft
h
el
a
t
t
e
rana
.
l
y
s
i
swωtha
ももh
el
a
.
w
i
sa
p
p
l
i
c
a
b
l
et
os
i
田 r
a
n
g
回 u
pt
oa
b
o
u
t1
:
2
0andh
a
st
h
esameform(
f
o
rt
h
i
ss
i
z
er
a
.
n
g
e
)
fordls
p
e
c
i
m
e
ng
e
o
m
e
t
r
i
e
s
(
w
i
h
ho
n
l
yn
e
g
l
i
g
i
b
l
ee
r
r
o
r
s
)
.
T
h
ea
n
a
l
y
s
i
su
t
i
l
i
z
e
st
h
emethod
f
o
rc
a
l
c
u
l
a
t
i
n
gt
h
eRc
u
r
v
efroms
i
z
ee
f
f
e
c
tandl
o
a
d
d
e
f
l
e
c
t
i
o
nc
u
r
v
efromt
h
eR
c
u
r
v
ea
s
a
l
r
e
O
O
ye
x
p
l
a
i
n
e
d(
S
e
c
.5
.
1
)
.TheR叩 r
v
eo
b
t
a
i
n
e
di
nt
h
i
sm姐 n
e
ri
ss
t
r
o
n
g
l
yd
e
p
e
n
d
e
n
to
n
t
h
egeometryo
ft
h
es
p
e
c
i
m
e
n
.Thebωicr
e
l
a
t
i
o
n(
F
i
g
.6
.
9
b,
c
)
o~= 旦p , m=bfR(帆 Wz = bR(cm)(1-c
"
,
)
(
6
.
9
)
whereb= s
p
e
c
i
m
e
nt
h
i
c
k
n
e
s
s,
l:
:
;d-ao = l
i
g
a
.
mentl
e
n
g
t
h(
d= be
a
.
md
e
p
t
h
),
組dc",=
c
r
a
.
c
kl
e
n
g
t
hca
tpea
kl
o
a
d
.同 組dWzr
e
p
r
e
s
e
n
t出 eworkso
ff
r
a
ε
t
u
r
eb
e
f
o
r
eanda
I'岡山E
戸a
k1
0
0
0
. NotefromF
i
g6
.
9
b,
ct
h
以 Wzd
ep
e
ndss
t
r
o
n
g
l
yons
p
e
c
i
m
e
ns
i
z
e
,becausethe
v
a
l
u
eo
fc", d
e
p
e
n
d
sont
h
es
i
z
e
.
山 o
ft
h
i
si
n
v
e
s
t
i
g
a
.
t
i
o
nshowt
h
a
tG
7i
sn
o
ts
i
z
e
i
n
d
e
p
e
n
d
e
n
t, 出 onemight
Ther
e
s
b
u
td
e
p
e
n
d
ss
もr
o
n
g
l
yont
h
es
p
e
c
i
m
e
ns
i
z
ea
sshowni
nF
i
g6
.
9
a
.T
h
i
sdependen
田 i
s
d
e
s
i
r
e,
S
田 nt
obeev
e
ns
t
r
o
n
g
e
rt
h
a
.
nt
h
a
to
f山 eR-c
u
r
v
e
.Whent
h
es
p
e
c
i
m
e
ns
i
z
ei
se
x
t
r
a
p
o
l
a
t
e
d
,
G?c
o
i
n
c
i
d
e
自e
x
a
c
t
l
yw
i
t
hBa
.
Z
制'
sd
e
f
i
n
i
t
i
o
no
ff
r
a
c
t
u
r
ee
肘 r
g
yG/o
b
t
a
i
n
e
dby
t
oi
n
f
i
n
i
t
y
t
h
es
i
z
ee
f
f
e
c
tmethod.
T
h
i
si
n
v
e
s
t
i
g
a
t
i
o
na
l
s
oshowedt
h
a
.
tt
h
ep
r
e
p
e
a
kc
o
n
t
r
i
b
u
t
i
o
no
ft
h
eworko
ft
h
el
o
a
dt
o
t
h
ef
r
a
c
t
u
r
een
ぽ g
yGIi
sr
e
l
a
t
i
v
e
l
ys
m
a
l
l(
g
e
n
e
r
a
l
l
yunder6
%
)
.T
h
i
sc
o
n
c
l
u
s
i
o
nba
.
s
i
c
a
.
l
l
y
a
g
r
間
w
i
t
h出
a
.
to
b
t
a
i
n
e
dbyP
l
a
n
a
s叩 dE
l
i
c
e
s(
1
9
8
8
b
)i
nad
i
f
f
e
r
e
n
tm油 田r
.
6.5Two-ParameterF
r
actureModelofJenqandShah
U
n
l
i
k
et
h
ef
i
c
t
i
t
I
o
u
sc
r
a
c
kmodel
,thetwoparame
ほ m
odelo
fJenqandShah(
1
9
8
5
,
品
1
9
邸b
),
a
l
r
e
a
d
ye
x
p
l
a
i
n
e
di
nS
e
c
.4
.
2,
d
o
e
sn
o
tr
e
q
u
i
r
eap
o
s
t
p
e
a
.
k(
s
t
r
a
.
i
ns
o
f
t
e
n
i
n
g
)c
o
n
y
e
ti
もc
a
nd
e
s
c
r
i
b
et
h
en
o
n
l
i
n
e
a
rs
l
o
wc
r
a
c
kgrowthp
r
i
o
rt
opeakl
o
a
d
. The
s
t
i
t
u
t
i
v
elaw,
twopa
.
r
a
.
m
e
t
e
r
sa
r
et
h
ec
r
i
t
i
c
a
ls
t
r
e
s
si
山 悶t
yf
a
c
t
o
rKc
!andthecritica
.
lc
r
a
c
kt
i
po
p
e
n
i
n
g
d
i
s
p
l
a
c
ementdCMODr
e
l
a
t
i
o
n
s
h
i
pshowni
nF
i
g
.6
.
1
0.T
h
i
sr
e
l
a
t
i
o
n
s
h
i
pi
se
s
s
e
n
t
i
a
l
l
yl
i
n
e
a
r
s
c
e
n
d
i
n
gp
o
r
t
i
o
no
ft
h
ec
u
r
v
efromP= 0upt
oa
b
o
u
tt
h
el
o
a
dc
o
r
r
e
s
p
o
n
d
i
n
gt
oh
a
l
f
onぬea
t
h
emaximuml
o
a
.
dP
u・Att
h
i
ss
ta
.
g
e,
t
h
ec
r
a
c
kもi
po
p
e
n
i
n
gd
i
s
p
l
a
c
e
m
e
n
ti
sn
e
g
l
i
g
i
b
l
e姐 dKl
i
sl
e
s
s山 岨 0.5K
c
!(Fig.6.lOa).Astheloa
.
dP e
x
c
e
e
dt
h
ev
a
l
u
eo
f0
.
5凡
,i
n
e
l
a
s
t
i
cd
i
s
p
l
a
c
e
m阻 も 佃ds
l
o
wc
r
a
c
kgrowtho
c
c
u
rd
u
r
i
n
gt
h
en
o
n
l
i
n
e
a
rr
a
n
g
e(
F
i
g
.6
.
1
0
b
)
. Att
h
ec
r
i
t
i
c
a
l
p
ぬも(同.6
.
1
0
c
),
t
h
ec
r
a
c
kt
i
po
p
e
n
i
n
gd
i
s
p
l
a
.
cementr
e
叫 e
sa伺 t
i
c
a
1v
山 e叫 ん =K
f
c
.
F
o
rs
t
a
n
d
a
r
dp
l
a
i
nc
o
n
c
r
e
t
ebeamst
e
s
t
e
di
nt
h
r
e
φ
p
o
i
n
tb
e
n
d
i
n
g,
t
h
ec
r
i
t
i
c
a
.
lp
o
i
n
tc
組
be
a
p
p
r
o
x
i
m
a
t
e
db
e
t
w
e
e
nt
h
ep
o
i
n
to
fPua
.
ndt
h
ep
o
i
n
to
f0
.
9
5
凡 ont
h
ed
e
s
c
e
n
d
i
n
gb
r
a
n
c
ho
f
t
h
eP-dCMODp
l
o
t
.Thec
o
n
c
e
p
ti
sa
1
s
oshownf
o
rabeami
nb
e
n
d
i
n
gi
nF
i
g
.6.
1
1
.
Dependingupont
h
eg
回 me
t
r
yo
ft
h
es
p
e
c
i
m
e
n
,therateofloading,
andt
h
emethodo
f
l
o
a
d
i
n
g
,
f
u
r
t
h
e
rc
r
a
c
kg
r
o
w
t
hmayo
c
c
u
r.
a
tas
t
e
O
O
y
s
t
a
t
ev
a
l
u
eo
fK
f
c
.Denoting出 ec
r
a
c
k
t
h
el
o
a
dP w
i
l
lh
a
v
ea
.v
a
.
l
u
ee
q
u
a
lt
o九 whenK1 r
e
a
c
h
e
sI
<
!
ci
fd
I
<
I
/
d
α> O
.
l
e
n
g
t
ha,
I
fd
I
<
J
/d
aa
.
tt
h
i
sl
o
a
d
i
n
gs
t
a
g
ei
sn
e
g
a
t
i
v
e,
t
h
e
nbeyondt
h
ec
r
i
t
i
c
a
1p
o
i
n
tt
h
ec
r
a
c
kw
i
l
l
a
.
t
ea
tac
o
n
s
t
a
n
tv
a
l
u
eo
f[
(
!
candtheappliedloadwillincrea
.
s
eu
n
t
i
ldK
I
/
d
a=O
.
p
r
o
p
a
g
1
nt
h
em
a
j
o
r
i
t
yo
f
p
r
a
c
t
i
c
a
lc
a
s
e
s,
t
h
ev
a
l
u
eo
fdKl/dai
sp
o
s
i
t
i
v
e,
i
.
e.
,
I
<
li
sa
m
o
n
o
t
o
n
i
c
a
l
l
y
. Thr
e
e
-andf
o
u
r
p
o
i
n
tb
e
n
t,
ands
i
n
g
l
e
-edgenotchedandd
o
u
b
l
e
l
n
.
c
r
e
a
s
i
n
gf
u
n
c
t
i
o
no
fa
e
d
g
en
o
t
c
h
e
ds
p
e
c
i
m
e
n
ss
u
b
j
e
c
t
e
dt
ot
e
n
s
i
l
el
o
a
d
i
n
ga
r
ef
o
u
n
dt
osati~fy t
h
i
sc
o
n
d
i
t
i
o
n(
J
e
n
q
拍 dS
h
a
.
h,
1
9
8
5
b
)
.F
o
rt
h
i
sc
a
民 t
h
ev
a
l
u
e
so
fK
f
c拍 ddCTOD C組 beo
b
t乱i
n
e
dfrom出 e
a
.
kl
o
a
dandfromt
h
ek
n
o
w
l
e
d
g
eo
fana
s
s
o
c
i
a
t
e
de
f
f
e
c
t
i
v
ec
r
a
.
c
kl
e
n
g
t
ha
.・
m
e
a
s
u
r
e
dpe
Themethodh
a
sb
e
e
np
r
o
p
o
s
e
dt
oR1
LEM(
K
a
r
i
h
a
.
l
o
oandN
a
l
l
a
t
h
a
m
b
i,
1987~) 出a. s
t
a
n
d
a
r
dt
e
s
tmethodf
o
rd
e
t
e
r
m
i
n
a
t
i
o
no
f]{先制46cTODi
np
l
a
j
nc
o
n
c
r
e
t
eu
s
i
n
gbeams
p
民 1
m
e
n
s
. Thef
o
l
l
o
w
i
n
gd
i
m
e
n
s
i
o
n
sa
r
es
u
g
g
e
s
t
e
di
fa
.maximuma
g
g
r
e
g
a
t
es
i
z
en
o
te
x
c
e
e
d
i
n
g
on~ i
n
c
h(
2
5.
4mm)i
su
s
e
d
js
e
eF
i
g
.6
.
1,
i
nwh
i
c
hbxdxL= 3
i
n
.X6
i
n
.x2
8
i
n
.(
7
6
.
20
l
I
I
1
耳 1~2.4 m mx711.
2mm);S= 2
4in.e609.6mm)jao/d= 1/3;S/d=4.
‘
r
o
rla
.
r
g
e
rffiaXimuma
.
邸r
e
g
a
t
es
i
z
e
st
h
ed
i
m
e
n
s
i
o
n
ss
h
o
u
l
di
n
c
r
e
剖 ep
r
o
p
o
r
t
i
o
n
a
.
t
e
l
y
.
Ac
l
o
s
e
d
l
o
o
pt
e
s
t
i
n
gs
y
蜘 mw
i
t
hs
t
r.
a
i
nc
o
n
t
r
o
lu
s
i
n
gt
h
edCMOD gag
eo
re
l
s
e乱 s
t
i
f
f
m
a
c
h
i
n
e(
amachinet
h
a
もi
sa
.
tl
e
a
s
t1
0
t
i
m
e
ss
t
i
f
f
e
rt
h
a
nt
h
e凶 s
p
e
c
i
m
e
n
)i
sr
e
∞mmended
乱di
sr
e
a
c
h
e
di
n
t
oa
c
h
i
e
v
es
t
a
b
l
ef
a
i
l
u
r
e
.
T
h
er
a
t
eo
fl
o
a
d
i
n
gs
h
o
u
l
dbesuchshMpeakl
o
a
b
o
u
t5m
i
n
.
p
P
P
しV
-E .E
,・・
EE
E
'
a duoo
一
,
・
・
e-e-
kd
u
n
C
;
105KLdEぷ
1CTOD<CTODc
E
da--
Go
LVo
,
to
T・
0c
I
CMOD
αコ
σ
3
〆
ノ
,噌
P
m
ノ
J
r
.
u
o
o一
,
,,nuoo
n
_
n
_
ii
∞Tt
P
M
一
一
D
11
0
1
戸
し
M
oj,
ω。
O
Q
I
I~O!tl
9ヨ
(
a
)
/ kzd;C
I
CTOD=CTODc
CMOD
CMOD
(
b
t
-
(
c
)
,明 o
臼
F
i
g
.
6
.
1
0F
r
a
c
t
u
r
eResi
s
t
a
n
Stag
fP
l
a
i
nCo
n
c
r
et
e
_
,
,
.
_
CMODt
CMOD3
CMOD2
ド
4
~ド‘-
CTOO.O
(
a
J
t
b
J
I
c
'
α
3
P
h
t
. St
c
foloblωe!"Gc~ g
r
a
t
l
t
h
.
"
・
,
弓
・
2. p , oC&O~ Z ne p~, tt ・lI y dev.lopod.
'
1
5
・
f
d
o
y
o
l
o
p
e
c
l
.
・
!
I
t!!IP I・
・
川
'
"
a
.8tortO~ unotob!earackgr wtll•
戸円
pr I
J
d
胸1)
'
T
O
O
F
i
g.
6.
1
1dCMOD anddCTODBehavior&
tD
i
f
f
e
r
e
n
tLo
a
d
i
n
gS
t
A
g
e
s
司
89
88
Tod
e
t
e
r
m
i
n
et
h
etwof
r
a
c
t
u
r
ep
a
r
a
m
e
t
e
r
sK
f
canddCTODf
o
rp
l
司n∞n
c
r
e
も
e
も
,hem
a
x
i
mum凡 andt
h
ec
o
r
r
e
s
p
o
n
d
i
n
ge
l
a
s
t
i
c∞mponen
もo
ft
h
edCMOD,
d
e
n
o
t
e
ddbMODa
r
em品
s
u
r
,
巴d
.T
h
e
s
ev
a
l
u
e
s釘 e七henu
s
e
dt
oe
s
t
i
m
a
t
et
h
ee
f
f
e
c
t
i
v
ee
l
a
s
t
i
cc
即
kl
e
n
g
t
haes
u
c
hもh
a
t
t
h
ec
a
l
c
u
l
a
t
e
ddb
'
M O D(
bωedonLEFMe
q
u
a
t
i
o
n
s
)i
se
q
u
a
lt
ot
h
emeasuredv
a
l
u
e
.F
o
rag
i
v
e
n
i
n
i
t
i
a
ln
o
t
c
hl
e
n
g
もha
oanddependingupont
h
et
y
p
eo
ft
e
s
t,
t
h
etwof
r
a
c
t
u
r
ep
a
r
a
m
e
t
e
r
sa
r
e
田 F
i
g
.6
.
2組 d6
.
1
1
)
:
d
e
t
e
r
m
i
n
e
da
sf
o
l
l
o
w
s(
s
Themoduluso
fe
l
a
s
t
i
c
i
t
yE i
sd
e
t
e
r
m
i
n
e
du
s
i
n
gi
n
i
t
i
a
lcompJ
i
a
n
c
eC
jfrom
E-6SaoM(α0)
C
j
b
a
l
(
6
.
1
0
)
0
.
6
6
Y
i
.(α0)=0.76-2.28αo+3.870'~ -2・040'~ +万て0'0
)
2
(
6
.
1
1
)
内
u-e
α
α
一
同
一 vn
n
M
G
q
α
一
一
一ae
andα0
+(
a
o= h
o
)
/
(
d+h
)
,
h
o=c
l
i
pg
a
g
eh
o
l
d
e
rt
h
i
c
k
n
倒・
o
Thee
f
f
e
c
t
i
v
ec
r
a
c
kl
e
n
g
t
hi
sd
e
t
e
r
m
i
n
e
dfromE
q
.6
.
8byr
e
p
l
a
c
i
n
ga
owithαhαow
i
t
hα
e
組 dC w
i
t
h
C
"
w
h
e
r
e
的
=
(
a
.
+
h
o
)
/
(
W
+
h
o
)
岨
dC
i
s
t
h
e
u
n
l
o
a
d
i
n
g
c
o
m
p
l
i
a
n
c
e
a
t
0
.
9
5
u
i
P
U
'Thus,
(
6
.
1
2
)
w
h
i
c
hmaybe8
0
1
v
e
df
a
i
r
l
ye
a
s
i
l
ybyi
t
e
r
a
t
i
o
n
.
Then山.
ec
r
i
t
i
c
a
ls
t
r
e
s
si
n
何 回i
t
yf
a
c
t
o
ri
s
=E主
主
主Yi.(
α
)
[
(
1 ,
8)
2+(
ー1
.
149α+1
.0
8
1
)
(
,
82
)
]
1/
2
2E
b
d
。
(
6
.
1
5
)
a
.・
whereβ=α /
I
nu
s
i
n
gt
h
i
smethodf
o
rp
r
e
c
r
a
c
k
e
d-i
n
s
t
e
a
do
fn
o
t
c
h
e
d-beams,
r
e
p
l
a
c
ea
obya
j.
I
nJ
e
問 a
ndShah(
1
9
8
5
b
)i
ti
sshownt
h
a
tt
h
i
smethodg
i
v
e
s]
(
[
c
(
=]
(
!
c
)resultswhic~
a
r
ei
n
d
e
p
e
n
d
e
n
to
fs
p
e
c
i
m
e
ns
i
z
ewhenn
o
t
c
h
e
dbeamsa
r
eu
s
e
d
. Asshowni
nS
w
a
r
t
z姐 d
R
.
e
f
a
i(
1
9
8
9
)出 esamei
n
v
a
r
i
a
n
c
ei
so
b
t
a
.
i
n
e
dwhenp
r
e
c
r
a
.
c
k
e
dbeamsa
r
eu
s
e
danda
l
s
o
Kr
.i
si
n
d
e
p
e
n
d
e
n
to
fc
r
a
c
kl
e
n
g
t
hf
o
rα.
/
dく 0.
6
5
.I
ti
sa
l
s
oshownt
h
a
tt
h
em
o
d
i
f
i
e
d
-hliM-method(
s
e
c
t
i
o
n6
.
1
.
3
)
g
i
v
e
sv
i
r
t
u
a
l
l
ythsamemultf
o
rkICf
o
EpmMked
e
制 p
l
e,f
o
rd= 1
2i
n
.,anda
.
/
d三0
.
6
5,t
h
e乱.
v
e
r
a
g
beamsa
st
h
eJ
e
n
q
Shahm
e
t
h
o
d
.f
o
re
x
Kc
!= 1206kNm-3/2 (C.V.= 9.2%,N = 11)andthecompliancemethodaV釘 ageKlo:
;
3/2 (
1
l51kNmC
.
V
.=3.5%,
N= 1
9
)
.
I
ti
sf
u
r
t
h
e
rshownbyJ
e
n
q叩 dSh
a
.
h(
1
9
8
5
b
)t
h
a
.
tt
h
er
e
s
u
l
t
sf
o
rdCTOD a
r
er
e
l
a
t
i
v
e
l
y
c
o
n
s
t
a
n
tw
i
t
hr
e
s
p
e
c
tt
obeams
i
z
e
.T
h
i
sr
e
s
u
l
twa
.
sa
l
s
oo
b
t
a
i
n
e
df
o
rbeamst
e
s
t
e
dbyRe
f
a
i
岨 dS
w
a
r
t
z(
1
9
8
7
)b
u
tav
a
r
i
a
t
i
o
nw
i
t
ha
iw国 f
o
u
n
d
.
-3
w
i
t
h
α =α
.
/d
.Thec
r
i
t
i
c
a
lc
r
a
c
k
t
i
po
p
e
n
i
n
gd
i
s
p
l
a
四 m
enti
s
、A
(
6
.
1
4
)
何百一守
-.
,
f
i
+
一
ト
1 1.99α(1-α)(2.15-3.93α+2.
7
α
2)
(
1+2
α
)
(
1-α)
3
/
2
F(
α
)ー ー
M
s一
a
T
and
(
6
.
1
3
)
,
R一
A
一
r
h
i
smethod(
N
a
l
l
a
t
h
a
m
b
iandK
a
r
i
h
a
l
o
o
,1986a,1986b) hasa
1
8
0b
e
e
np
r
o
p
o
s
e
dt
o
1
9
8
7
)
.l
nt
h
i
sa
p
p
r
o
a
c
has
i
m
p
l
et
e
c
h
RILEMa
sas
t
a
n
d
a
r
d(
K
a
r
i
h
a
l
o
oandN
a
l
l
a
t
h
a
m
b
i,
,
l
Iq
u
ei
su
s
e
dwhereもhec
r
i
t
i
c
a
ls
t
r
e
s
si
n
t
e
n
s
i
t
yf
a
c
t
o
r]
(
[
c
(
=K
1
0
)andc
r
i
t
i
c
a
le
n
e
r
g
yr
e
l
e
a
s
e
c
(
=G
1
c
)a
r
ef
o
u
n
df
o
rb白 血 s
p
e
c
i
m
e
n
sbyLEFMmethodsa
f
t
e
rt
h
ec
蹴
kh
副 g
rown
r
d
t
eGl
f
r
o
ma
.n
o
t
c
ha
ot
oa
.
ne
f
f
e
c
t
i
v
el
e
n
g
t
hae
.
Thef
o
r
m
u
l
a
swhichf
o
l
l
o
wl
a
t
e
rw
e
r
ed
e
v
e
l
o
p
e
df
o
rc
e
r
t
a
i
nr
a
n
g
e
so
fs
p
e
c
i
m
e
nd
i
m
e
n
s
i
o
n
s
.
r
h
e
s
er
a
n
g
e
s,
w
h
i
c
ha
r
et
obeu
s
e
df
o
rma
.
x
imuma
.
g
g
r
eg
a
t
es
i
z
e
sda r
a
n
g
i
n
gfrom5t
o2
5
mm.w
i
t
hnoc
r
i
t
i
c
a
ld
i
m
e
n
s
i
o
nb
e
i
n
gl
e
s
st
h
a
nf
i
v
etim
回 d
a
r
e
:b=40・8
0mm(
三3d
jd
a,
a)
=5
0-3
0
0mm(
と3
d
00-1
8
0
0m m(
三3
d
iwi仕14壬S/d壬8
.A
l
s
oa
o
/
d=0.2-0.6
;S=2
a)
a)
0
.
3o
r0.
4i
sp
r
e
f
e
r
r
e
d
)
.
e
a
.
l
i
z
e
di
nF
i
g
.6.
1i
sshowni
nF
i
g
.6
.
1
2wheret
h
edLPD i
s
Thel
o
a
d
i
n
gs
e
t
u
pwhichi
si
d
m
e
a
s
u
r
e
dbyanLVDT.I
na
d
d
i
t
i
o
n,
t
h
edCMOD c
o
u
l
dbemeasuredbya
d
d
i
n
gac
l
i
pg
a
g
e
a
.
c
r
o
s
st
h
eno
もc
h
.P
r
e
f
e
r
a
b
l
yad
o
s
e
d
l
o
o
pt
e
s
t
i
n
gs
y
s
t
e
ms
h
o
u
l
dbeu
s
e
dw
i
t
hs
t
r
a
i
nc
o
n
t
r
o
l
f
P
.
e
db
a
c
kf
r
o
me
i
t
h
e
rt
h
edLPD o
fdCMODga
.
g
e
.O
t
h
e
r
w
i
s
e
,as
t
i
f
ft
e
日t
i
n
gmachines
h
o
u
l
dbe
v
e
n
t,
ac
o
n
t
i
n
u
o
u
sr
e
∞rdofPa
.
ndh P Di
smadeupもopeakl
o
a
dw
h
i
c
hs
h
o
u
l
d
n
s
e
d
.I
n組 ye
her
e
a
c
h
e
di
n1
・1
0m
i
n
u
t
e
s
.t
h
es
o
f
t
e
n
i
n
gr
田 p
o
n
s
ei
sno
もr
e
q
u
i
r
e
dw
i
も
ht
h
i
smethod
.
I
fi
ti
sn
o
tp
o
s
s
i
b
l
et
oo
b
t
a
i
nac
o
n
t
i
n
u
o
u
sl
o
a
d
d
i
s
p
l
a
田 m
entr
e
∞rd,
t
h
emethodc
a
ns
t
i
l
l
t
)
p u
s
e
db
u
tt
h
emoduluso
fe
l
a
s
t
i
c
i
t
y
,
E,
mustbed
e
t
e
r
m
i
n
e
dfromu
n
i
a
x
i
a
lt
e
s
t
sono
t
h
e
r
p
e
c
l
m
e
n
s
.
t
i
n
u
o
u
sr
e
∞rdo
fP v
s
.dLPDi
sa
.
va
.
il
a
b
l
e
,
d
e
t
e
r
m
i
n
eE f
r
o
m
I
fa∞n
E
'
d
;
3九 S.
K
l
c=五
J
i
i
7
i
'
e
F(
α
)
6.6E
f
f
e
c
t
i
v
eCrackModelo
fK
a
r
i
h
a
l
o
oandNallathambi
(
6
.
1
)
l
h
e
r
eP
i• d
ia
r
eme
a
.
s
u
r
e
da
.
も
姐yi
n
t
e
r
m
e
d
i
a
t
el
o
c
a
.
t
i
o
n00t
h
ei
n
i
t
i
a
la
s
c
e
n
d
i
n
gp
o
r
t
i
o
no
f
:
n
ec
u
r
v
eandwi
st
h
es
e
1
f
w
e
i
g
h
to
f出eb
e
a
m
.Thetermλisac
o
r
r
e
c
t
i
o
nf
a
c
t
o
rr
e
l
a
t
i
n
gde
市 t
i
o
n
so
f品 n
o
t
c
h
e
dt
oanu
n
n
o
t
c
h
e
dbeama
.
ndi
so
b
t
a
i
n
e
dfromf
i
n
i
t
ee
l
e
m
e
n
tc
a
l
c
u
l
a
t
i
o
n
s
入=ηlexp{
ゎ(
a
o
/
d
)
2+T
J
3
(
S
/
d
)
2+仇 (
a
o
/
d
)
(
S
/
d
)+η5
(
S
/
d
)
3
}
(
6
.
2
)
3,1
'
/
3=ー0
.
0
3
6,
り'
4= 0
.
2
0
1,and1
/
5 =0
.
0
0
4(
K
a
r
i
h
a
l
o
o
1
9
8
7
)
. Thent
h
ee
f
f
e
c
t
i
v
ec
r
a
c
kl
e
n
g
t
hαei
so
b
t
a
.
i
n
e
dfrom九 組dt
h
e
N
a
l
l
a
t
h
a
m
b
i,
.r
r
e
s
p
o
n
d
i
n
gd
i
s
p
l
a
c
e
m
e
n
td
"舗
A,
d'
1
1;
:
:1
.
3
7
9,
市
:
:
:
:
:
: -1必
a
c
/
d= 1- r
凡S
3
(
1+説
4
bd
36
c
uE
,
2
'
¥
1
'
"
(
6
.
3
)
u
.sdi
s“
s
m
a
l
l,
"t
h
ee
f
f
e
c
to
fs
h
er
.d
e
f
o
r
m
a
t
i
o
nc
a
nbe o
n
s
i
d
e
r
e
dbyu
s
i
n
g
乱
t
;
E =~i r~3(1 +義)↓ (1+ν)~L.
j l
6
4
b
♂(
1ー守
)
3
・ 2x
:b
d
(
1ー守 ,
.
J
)
(
6.
4
)
9
0
9
1
P
﹁ード﹂
b
e
r
e民 =s
h
e
a
rωnstant= 1
0
(
1+ν)/(12+ 1
1
1
1
)f
o
rar
e
c
t
a
n
g
u
l
a
rs
e
c
t
i
o
nand1i
sP
o
i
s
'
I
'
Sr
a
t
i
o
.T
h
i
sv
a
l
u
eo
fE i
st
h
e
ns
u
b
s
t
i
t
u
t
e
diI印刷.6.16ωobtainae
a
r
i
h
a
l
o
o回 d
・K
J
l
a
t
h
a
m
b
i(
1
9
8
7
)c
o
n
t
a
i
n
sr
e
g
r
e
s
s
i
o
nf
o
r
m
u
l
a
st
od
e
t
e
r
m
i
n
eaei
n
尚 a
do
fu
s
i
n
gE
q
.6
.
1
6
.
a
rhec
r
i
t
i
c
a
ls
t
r
e
s
si
n
t
e
n
s
i
t
yf
a
d
o
ri
st
h
e
ng
i
v
e
na
s
I
<
I
c=
σ
r
t
Y
ζF
(α)
¥
t
e
r
eF
(
α
)i
sg
i
v
e
nbyE
q
.6
.
1
6,
α
d
u
d
i
n
gs
e
l
f
w
e
i
g
h
te
f
f
e
c
t
si
s
=a./d
姐
(
6
.
5
)
dUn = 6M/b
♂. Them
idspanmomentM,
M=(
凡+
ω/
2
)S
/
4
(
6
.
6
)
h
ec
r
i
t
i
c
a
le
n
e
r
g
yr
e
l
e
a
s
er
a
t
ei
s
Gt= (
K
i
c
)
2
lC-E
一
(
6
.
7
)
I
m
p
r
o
v
e
d(
b
u
tmorec
o
m
p
l
i
c
a
t
e
de
x
p
r
e
s
s
i
o
n
sf
o
rK
I
c
{
=J
(
j
c
)andG1
c
(=G
i
c
)areobtained
i
凶ぬ e
l
e
m
e
n
tmethodsw
h
i
c
hc
o
n
s
i
d
e
rt
h
et
e
n
s
i
l
es
t
r
e
s
snormalもot
h
ec
r
a
c
kf
a
c
e
s,
vu
s
eo
ff
・
h
et
e
n
s
i
l
es
t
r
e
s
si
nt
h
ep
l
a
n
eo
ft
h
ec
r
a
c
kandt
h
es
h
e
紅 i
n
gs
t
r
e
s
s
.Thef
o
r
m
u
l
a
so
fJ
(
,cand
'
;
1
rare
J
(
I
C=
σn
{
α
.
)
1
/
2Y
j(
α
)日(
α,
s)
(
6
.
8
)
s
p
a
n
.S
ー
舎
(ι
)問
I
?
i
c
F
i
g
.
6
.
1
2L
o
a
d
i
n
gA
p
p
a
r
a
t
u
sa
.
n
dF
i
x
i
n
gArr
angemento
fLVTD
[
nt
h
e
s
eequa
もi
o
n
sa
g
a
i
n
(
6
.
9
)
=
α=a
.
/
d,
s S
/
d
;
ぺ
14
I
N
F
I
N
I
T
ES
I
Z
E
﹄
4
NORMALS
I
Z
E
γ,
(
α
)-Ao+Alα+A2
α2+A3α3+A4α〈九 α
(,
s
)=Bo+Bls+B2s2
+B3
β3+B4α
β +Bsαβ2,
Z
l
(α
)=Co+Cα
C
C
2
+
3
+
1 +C
α
α
α
2
3
4
Z
2
(
α,
β)=Do+Dlβ+D2β+D3β3+D4αs+Dsαs
2
.
1
)ELASTICFIELD
,,
・
・
・
,4
;j=0,
1,
.
•.
,
5
)a
r
eg
i
v
e
ni
nT
a
b
l
e6
.
2
.
Av
e
r
ye
x
t
e
n
s
i
v
et
e
s
t
i
n
gprogramwasc
a
r
r
i
e
do
u
tbyK
a
r
i
h
a
l
o
oandN
a
l
l
a
t
h
a
m
b
i(
N
a
l
;
;
th
a
m
b
ia
n
dK
a
r
i
h
a
l
o
o,
1
9
8
6
a
;K
a
r
i
h
乱
.
1
00組 dN
a
l
l
a
t
h
a
m
b
i,
1
9
8
7
)u
s
i
n
gt
h
i
smethod戸o
叫u
a
t
ed
a
t
af
r
o
mmoret
h
叩 9
5
0beamsi
nw
h
i
c
hmaximuma
g
g
r
e
g
a
t
es
i
z
e,beams
i
回 , 姐d
s
tr
e
自u
l
t
sw
e
r
eg
e
n
e
r
a
l
l
yv
e
r
yc
o
n
s
i
s
t
e
n
tw
i
t
h
'
d
a
t
i
v
en
o
t
c
hc
r
a
c
kd
e
p
t
hw
e
r
ev
a
r
i
e
d
.Theもe
pc
a
l
c
u
l
a
t
e
dv
a
l
u
e
s
;
t
h
ecoemdento
fv
a
r
i
a
b
i
o
ng
e
n
e
r
a
l
l
yr
a
n
g
e
dbebween6%and10%.T
田 t
w
i
t
hg
e
n
e
r
a
l
l
ye
x
c
e
l
l
e
M
a
g
r
e
e
m
e
n
tbdweent
h
e
a
L
af
r
o
mo
t
h
e
ri
n
v
e
s
t
i
g
a
t
o
r
sw
e
r
ea
l
s
ou
s
e
d,
,thedatafrom
J
t
l
i
t
so
ft
h
i
smethodandJenqmdS
h
a
h
'
smeq10df
o
rKIe-Forexample
釘 t
z(
1
9
8
7
)f
o
rd= 3
0
4m mg
a
v
eI
?
l
c:
:1074kNm-3 / 2 •, K~" :
:1206kNm-3/2
l
e
f
a
ia
n
dSw
m
t
la
n
dShah)andI
f
I
c=1151kNm3
/
2
(
c
o
m
p
l
i
i
n
c
em
e
t
h
o
d
)
.
'{崎悶s
i
o
nc
o
e
日c
i
e
n
t
sA;,
Bj
,
Ci,
D
j
(
i= 01
・
めASYMPTOTIC
ト
玉
E
L
A
S
T
I
CF
I
E
L
D
功NONL
l
NEAR
HARDENINGZONE
4
)FRACTURE
PROCESSZONE
ニ
(NONL
l
NEAR
SOFTENINGZONE1
H
.
iD
eterminationofMaterialParameters
byS
i
z
eE
f
f
e
c
tMethod
.
7
.
1AsymptoticD
e
f
i
n
i
t
i
o
nf
o
rI
n
f
i
n
i
t
eS
i
z
e
、
mcet
h
es
i
z
ee
f
f
e
c
t
F
i
g
.
6
.
1
3F
r
a
c
t
u
r
eP
r
o
c
e
s
sZo
n田 i
nD
i
f
f
e
陀 n
tSp
民 I
J
D
e
n
sandE
x
t
r
a
p
n
f
i
n
i
t
eS
i
z
e
o
l
a
t
i
o
nt
oI
i
s出 emosti
m
p
o
r
t
a
n
tp
r
a
c
t
i
c
a
lc
o
n
s
e
q
u
e
n
伺 o
ff
r
a
c
t
u
r
em
e
c
h
a
n
i
c
s,
i
t
92
93
b
.
ω
Z
2
d
d
s
4
ω ω
切gω‘
Jz
sz
a
sp
r
o
p
o
s
e
dby
i
sn
o
ti
l
l
o
g
i
c
a
lt
oe
x
p
l
o
i
ti
tf
o
rd
e
t
e
r
m
i
n
i
n
gt
h
em
a
t
e
r
i
a
lf
r
a
c
t
u
r
ep
r
o
p
e
r
t
i
e
s,
Ba
.
Za
n
t
.I
np
r
i
n
c
i
p
l
e,
t
h
es
i
z
ee
f
f
e
c
tp
r
o
v
i
d
e
st
h
eo
n
l
ym品 nsf
o
ranun
a
.
mbiguou5d
e
f
i
n
i
t
i
o
n
.
t
e
r
i
a
lf
r
a
c
t
u
r
ep
r
o
p
e
r
t
i
伺. T
hef
r
a
c
t
u
r
ee
n
e
r
g
ya
sw
e
l
la
.
so
t
h
e
rn
o
n
l
i
n
e
a
rf
r
a
.
c
t
u
r
e
o
fma
i
nw
h
i
c
hwehavee
x
p
r
e
s
s
e
dP
"a
c
c
o
r
d
i
n
gt
oE
q
.1
.2
.T
a
k
i
n
gt
h
el
i
m
i
td→∞,
f
o
rw
h
i
c
h
o
/
d,
weo
b
t
a
.
i
nt
h
ee
x
p
r
e
s
s
i
o
n
α→ αo= a
g
(α。
)
GJ= 一
一
!
:
2
d
o
g
(
α
0
)=一
ηE
'
"
A一
E
B2 .
<
/
2J
_f_ ¥
- V J¥-'V'
(
6
.
1
1
)
p
r
o
p
o
s
e
dbyB話 相t(
1
9
8
5
b
)i
nw
h
i
c
hAi
st
h
es
l
o
p
eo
fal
i
n
e
a
rr
e
g
r
田s
i
o
np
l
o
to
ft
e
s
tda
t
&
(
F
i
g
.6
.
1
4
)o
b
t
a
i
n
e
dbyalgebr
ぬc
a
l
l
yr
e
a
.
r
r
姐 g
i
n
gt
h
es
i
民 e
f
f
e
c
t}aw(
E
q
.1
.
4
)i
nt
h
ef
o
r
m
Y=AX+C,
inwhich
X =d, Y=(en/σ~) ,
, do=C/A
B= C-1/2
"。何ιIM~R
、
、
‘
、
2
¥
じ、
・・'.---~
FRACTUAE
舗ECH
且陶I
C
S
F
i
g
ふ1
4TheS
i
z
eE
f
f
e
c
tLaw(
E
q
.1
.
4
;B
ai
組 t
,
1
9
8
4
)
Y-AX+C
6l _ -1_
_
, -2
A-7.2~3 玄10. :
-i
n
.-:'psi
6
~-=.-2
c=22.34x.
10-'"psi
ωY
1
r
"0.1090 0
司
i
o
.
s
τ
一
(
6.
10
)
,
。
P
J
(α
) B2!
t
r
J
.1 ! _
d
GJ=J
i
mR=J
i
m
一
一
一
=一.
J
_
!
_ lim一一
一
→
∞ Ef
i
Jd
c~E .
.
_
-1+
d
/
ゐ limg(α)
~・
LOG(SIZE)
﹁︼︻圃的向田}
I
nE
q
.5
.
2weg
a
v
et
h
ee
x
p
r
e
s
s
i
o
nf
o
rt
h
een
ぽg
yr
e
l
e
a
s
er
a
t
ei
nt
e
r
m
so
ft
h
ea
p
p
l
i
e
dl
o
a
d
紅 白s
.S
e
tもingP=,
p川 wehaveG = R = specificenergyrequiredforcrack
o
rn
o
m
i
n
a
ls
,
a
.
n
dbye
x
t
r
a
p
o
l
a
t
i
o
nt
oi
n
f
i
n
i
t
es
i
z
eweha
.
v
e
growth
﹄
﹁
6
.
7
.
2Bas!
cRelations
/出足
、
、
/
;
,,
. f,
I
M
O
S
TE
X
l
s
r
l
附 T
ESTS
{C}
,
d T I N - - 、N(Zb¥吋)日ト
.
r
a
.
m
e
t
e
r
smaybeu
n
i
q
u
e
l
yd
e
f
i
n
e
dbyt
h
e
i
rv
a
.
l
u
e
se
x
t
r
a
.
p
o
l
a
.
t
e
dもoa
.s
p
e
c
i
m
田 o
fi
n
f
i
n
i
t
e
pa
回 s
o
ni
st
h
a
.
t
,
i
nani
n
f
i
n
i
t
e
l
yl
a
.
r
ges
p
e
c
i
m
e
n
,
t
h
ef
r
a
c
t
u
r
eprocωsz
o
n
eo
c
c
u
p
i
e
s
s
i
z
e.Ther
a
.
l
lf
r
a
c
t
i
o
no
ft
h
es
p
e
c
i
m
e
nvolume(
F
i
g
.6
.
1
3
)
. H佃 c
e
,nea
.
r
l
ya
l
lo
fもh
e
ani
n
f
i
n
i
t
e
l
ysm
l
a
s
t
i
cs
t
a
t
e
.Now,
froml
i
n
e
a
re
l
a
s
t
i
cf
r
a
c
t
u
r
em
e
c
h
a
n
i
c
si
ti
sknownt
h
a
t
s
p
e
c
i
m
e
ni
si
n組 e
.
s
y
m
p
t
o
t
i
cf
i
e
l
do
fd
i
s
p
l
a
.
c
e
m
e
n
t
s叫 抑 制e
s(
F
i
g
.6
.
1
3
)i
sもhesamer
e
g
a
.
r
d
l
e
s
s
t
h
en
e
a
r
t
i
pa
u
c
t
u
r
eg
e
o
m
e
t
r
y(
E
q
.2
.
1
)
.T
h
e
r
e
f
o
r
e,
t
h
ef
r
a
c
t
u
r
ep
r
o
畑 sz
o
n
ei
n
o
ft
h
es
p
e
c
i
m
e
no
r凶 r
a
.
ni
n
f
i
n
i
t
e
l
yl
a
.
r
g
es
p
e
c
i
m
田 i
se
x
p
o
s
e
doni
t
sbound
a
.
r
yt
ot
h
esames
t
r
e
s
sf
i
e
l
dr
e
g
紅 d
l
蜘
o
fs
t
r
u
c
t
u
r
egeometry
,
and5
0i
tmustbeha
.
v
ei
nt
h
e自amemanner. I
np
a
r
t
i
c
u
l
a
r
,i
tmus
七
d
i
s
s
i
p
a
.
t
et
h
esamee
n
e
r
g
yandha
.
v
et
h
es
a
.
mel
e
n
g
も
handw
i
d
t
h
.T
h
e
r
e
f
o
r
e
,剖 p
r
o
p
o
s
e
db
y
Ba
.
z
姐 t
(
1
9
8
7
a
), 組 unambiguousd
e
f
i
n
i
t
i
o
ni
sa
.
sf
o
l
l
o
w
s
:
.
ndt
h
ee
f
f
e
c
t
i
v
ef
r
拭 t
u
r
ep
r
o
c
e
s
sz
o
n
el
e
n
g
出 CJa
r
e出 ee
n
e
r
g
y
Thef
r
a
c
t
u
r
ee
n
e
r
g
yGJa
a
.
s
er
a
t
er
e
q
u
i
r
e
df
o
rc
r
a
c
kgrowtha
.
ndt
h
ed
i
s
t
a
n
c
efromt
h
en
o
t
c
ht
i
pt
ot
h
et
i
po
ft
h
e
r
e
l
e
a
.
l
e
n
tLEFMc
r
a
.
c
ki
na
.
ni
n
f
i
n
i
t
e
l
yl
a
.
r
ges
p
e
c
i
m
e
no
fa
.
n
ys
h
a
.
p
e
.
e
q
u
i
v
.
s
y
m
p
t
o
t
i
cd
e
f
i
n
i
t
i
o
n,
d
e
t
e
r
m
i
n
a
ι
i
o
no
ft
h
ef
r
a
c
t
u
r
ep
r
o
p
e
r
t
i
e
si
sr
e
d
u
c
e
dt
o
Witht
h
i
sa
a
.
w
.I
fweknewthes
i
z
ee
f
f
e
c
tl
a
we
x
a
c
t
l
y
,wewouldg
e
t
t
h
ec
a
l
i
b
r
a
t
i
o
no
ft
h
es
i
z
ee
f
f
e
c
tl
.
t
e
l
y
,theexacts
i
z
ee
f
f
e
c
tlaw
,a
p
p
l
ic
a
.
b
l
eupt
oi
n
f
i
n
i
t
es
i
z
e
,i
sn
o
t
e
x
a
c
tr
e
s
u
l
t
s
.U
n
f
o
r
t
u
na
t
h
i
smethod,
l
i
k
eo
t
h
e
r
s,
y
i
e
l
d
si
np
r
a
c
t
i
c
eo
n
l
ya
p
p
r
o
x
i
m
a
t
er
e
s
u
l
t
s
.
known. T
h
e
r
e
f
o
r
e,
a
l
i
d
i
もyo
ft
h
es
i
z
ee
f
f
e
c
tl
a
wp
r
o
p
o
s
e
dbyBa.i組も (
F
i
g
.6
.
1
4,E
q
.1
.4
)i
s
N
e
v
e
r
t
h
e
l
e
s
s,出ev
c
o
v
e
r
i
n
gar
姐 g
eo
fs
i
z
e
so
fp
e
r
h
a
.
p
s1
:
2
0,
w
h
i
c
hi
ssu
伍c
i
e
n
tf
o
rmos
もp
r
a
c
t
i
c
a
l
r
a
t
h
e
rb
r
o
a
d,
a
.
n
a
sandE
l
i
c
e
s(
1
9
8
8
a
)
.
p
u
r
p
o
s
e
s
js
e
eP
l
ELASTIC'TTOA
PI
.
AST'C'TY
。
。
。
O-TestData
x=d(
i
孔)
10
(
6.
1
2
)
g
r
e
s
s
i
o
nP
l
o
tf
o
rS
i
z
eE
f
f
e
c
tMethod
F
i
g
.
6
.
1
5Re
Thes
e
c
o
n
dn
o
n
l
i
n
e
a
rf
r
a
.
c
t
u
r
ep
a
r
a
m
e
t
e
r,
namelyt
h
ee
f
f
e
c
t
i
v
el
e
n
g
t
hcfo
ft
h
ef
r
a
c
t
u
r
e
四 s
sz
o
n
e
,
d
e
f
i
n
e
df
o
r姐 i
n
f
i
o
i
t
e
l
yl
a
r
g
es
p
e
c
i
m
e
n,
i
so
b
t
a
i
n
e
dfromE
q
.5
.
6a
sf
o
l
l
o
w
s
:
p
r
o
L
h
ep
r
o
p
o
s
a
lh
a
sb
e
e
nworkedo
u
ti
nde
同i
lf
o
r山i
st
y
p
eo
fs
p
e
c
i
m
e
n.(
8
伺 F
i
g
.6
.
1,
which
A
l
s
oe
x
p
l
a
.
in
st
h
en
o
t
a
t
i
o
n
s
.
)
AUt
h
ep
r
e
s
c
r
i
b
e
dd
e
t
a
i
l
sf
o
rs
p
e
c
i
m
e
np
r
e
p
a
r
M
i
o
n,
t
e
s
t
i
n
ganddataevdmuona
r
eg
i
v
i
nh
r
i
h
a
l
o
oandNdlathmbi(1987)andWHIn
o
tber
e
p
e
a
t
e
dhere.BNeay
,thet白 ting
p
r
o
g
r
a
mu
s
e
ss
p
e
c
i
m
e
n
so
fa
t1
助 成 仏r
e
e-b
u
tp
r
e
f
e
r
a
b
l
ymore・ 自i
z
e
sc
h
a
r
a
c
t
e
r
i
z
e
dbybeam
も
1
1
sd
3,
d
2,
etc.ands
p
a
n
sS
E,
S
2,
....Thes
m
a
l
l
e
s
td
e
p
t
hdImustn
o
tbel
a
r
g
e
rthan5
t
f
e
p
da
n
dt
h
el
a
r
g
e
s
td
e
p
t
hdn mustn
o
tb
es
m
a
l
l
e
rt
h
佃
1
5d~ ・ The r
a
t
i
odn/d
m
u
s
tbea
t
1
s
h
o
t
耐 b
e
k
e
p
μ
t
∞
c
∞
on
路
s
旬加n
t酎 s
h
o
u
l
dt
h
児.
e叩
spa
叩n/μdept出
hh
rriふt“
,
一ioS
めi/dj • A
l
e
a
s
t4
.Thebeamw
i
d
t
hb油
i
I
I
inimumo
ft
h
r
e
es
a
m
p
l
e
哩p
e
rg
i
v
e
ns
i
z
es
h
o
u
l
dbet
e
s
t
e
d
.
Oneo
ft
h
eadv
叩 t
ag-ofthemethodi
s
t
l
国 a
no
r
d
i
n
a
r
yu
n
i
a
x
i
a
lt
e
s
t
i
n
gmachinew
i
t
h
o
u
t
・
c
o
n
t
r
o
lo
rh
i
g
hs
t
i
f
f
n
e
s
sm品,
ybeu
s
e
d
. Theo
n
l
ymeasuredr
e
s
p
o
n
s
e
sn田 deda
r
et
h
e
arvo
p
e
a
kl
o
a
d
s乃 f
o
ra
l
ls
p
e
c
i
m
e
n
s,
j= 1
,
2,
.
.
.n.Theloadingrateshouldbe日electedsothat
t
h
emaximuml
o
a
dwouldber
e
a
c
h
e
di
no
n
et
ot
e
nminute~
4
g
(
α
0
)C
E
_
。
‘
内
g
(
α
0
)J
訳可7
ゐ=只石}五
、
,
CI -
)
95
n
o
・
唱
,
,
94
f=
A
g
a
i
n,
wes
e
et
h
a
tcf c
姐
bed
e
t
e
r
m
i
n
e
dfromt
h
el
i
n
e
a
rr
e
g
r
e
s
s
i
o
np
a
r
a
m
e
t
e
r
sA andC
(
B
a
z
姐 t
andKazemi,
1
9
8
8
)
.Ani
m
p
o
r
t
a
n
tp
o
i
n
ti
st
h
a
tt
h
et
e
s
td
a
t
amustωverasu
伍c
i
e
n
t
l
y
b
r
o
a
dr
a
n
g
eo
fs
i
z
e
s,自ot
h
a
tt
h
es
t
a
t
i
s
t
i
c
a
ls
c
a
t
t
e
rando
t
h
e
ri
n
f
l
u
e
n
c
e
swouldno
もo
v
e
r
s
h
a
d
o
w
t
h
es
i
z
ee
f
f
e
c
t.
E
q
s
.6
.
2
4and6
.
2
6f
o
r
mt
h
eb
a
s
i
sf
o
rt
h
ed
e
t
e
r
m
i
n
a
t
i
o
no
ft
h
ef
r
a
c
t
u
r
ee
n
e
r
g
y組 de
f
f
e
c
t
i
v
e
倒 s
z
o
n
el
e
n
g
t
hf
r
o
mt
h
es
i
z
ee
f
f
e
比 Ba
Z
a
n
t組 dP
f
e
i
f
f
e
r(
1
9
8
7
)v
e
r
i
f
i
e
dt
h
a
tv町 d
i
f
f
e
r
e
n
t
p
r
o
t
y
p
e
so
fs
p
回 i
m
e
n
s(
F
i
g
.5
.
1
),
i
n
c
l
u
d
i
n
gt
h
r
e
e
p
o
i
n
tbends
p
e
c
i
m
e
n
s,
e
d
g
e
n
o
t
c
h
e
dt
e
n
s
i
o
n
s
p
e
c
i
m
e
n
s回 de
c
c
e
n
t
r
i
cc
o
m
p
r
e
s
s
i
o
ns
p
e
c
i
m
e
n
s,
y
i
e
l
da
p
p
r
o
x
i
m
a
t
e
l
yt
h
esamev
a
l
u
e
so
ft
h
e
出a.I
ooandN
a
l
l
a
t
h
a
m
b
i(
1
9
8
7
),
S
w
a
r
t
z
f
r
a
c
t
u
r
ee
n
e
r
g
yf
o
rt
h
esamec
o
n
c
r
e
t
e
.S
t
u
d
i
e
so
fKa
andRe
f
a
i(
1
9
8
7
)andP
l
a
n
回 初dE
l
i
c
e
s(
1
9
8
8
b
)f
u
r
t
h
e
re
s
t
a
b
l
i
s
h
e
dt
h
a
tt
h
i
smethody
i
e
l
d
s
a
p
p
r
o
x
i
m
a
t
e
l
yt
h
esamer
e
s
u
l
t
sf
o
rt
h
ef
r
a
c
t
u
r
ee
n
e
r
g
y舗 JenqandS
h
a
h
'
stwo
・
p
a
r
a
m
e
t
e
r
m
e
t
h
o
d
. Ont
h
eo
t
h
e
rhand,
t
h
ev
a
l
u
eo
ff
r
a
c
t
u
r
ee
n
e
r
g
yo
b
t
a
i
n
e
dbyt
h
e
s
emethodsi
s
乱
AMwo
位d
巾
k
心o
i
f
企
r
拭
a
叫
.
t
cu
r
emethod(
i
.
e
.
g
e
n
e
r
a
l
l
yq
u
i
t
ed
i
f
f
e
r
e
n
tfromt
h
a
to
b
t
a
i
n
e
dfromt
h
eR
l
l
H
i
l
l
e
r
b
o
r
g
'
sm
e
t
h
o
d
)
.P
l
a
.
n
ωandE
l
i
c
e
s(
1
9
8
8
b
)andP
l
a
n
a
se
t札 (
1
9
8
9
)a
n
a
l
y
z
e
dぬe
s
e
d
i
f
f
e
r
e
n
c
e
se
x
h
a
u
s
t
i
v
e
l
yandc
o
n
c
l
u
d
e
dt
h
a
tt
h
er
e
a
s
o
nwhyHi
1
I
e
r
b
o
r
g'
.
smethody
i
e
l
d
sd
i
ι
乱l
u
e
si
sma
.
i
n
l
yt
h
eassumeds
h
a
p
eo
it
h
es
o
f
t
e
n
i
n
gs
t
r
e
s
s
d
i
s
p
l
a
田 m
entc
u
r
v
e
. By
i
e
r
e
n
tv
前e
e
p
e
ri
n
i
t
i
a
ld
e
c
l
i
n
e
,
e
x
t
e
n
d
e
dt
a
i
l
),
ab
e
t
t
e
ra
g
r
e
e
m
e
n
t
m
o
d
i
i
y
i
n
gt
h
es
h
a
p
eo
ft
h
i
sc
u
r
v
e(
i
nt
h
ev
a
l
u
e
so
iGJc
o
u
l
da
p
p
a
r
e
n
t
l
yb
eo
b
t
a
i
n
e
d
.
I
tmustben
o
t
e
dt
h
a
tE
q
.6.
2
6f
a
i
l
si
ig
'
(
α。
)a
p
p
r
o
a
c
h
e
sO.T
h
i
sha
.
ppens,
eιforac
e
r
t
a
i
n
回 ea
c
c
o
r
d
i
n
gt
o
c
r
a
c
kl
e
n
g
t
hi
nac
e
n
t
e
r
c
r
a
c
k
e
ds
p
e
c
i
m
e
nl
o
a
d
e
dont
h
ec
r
a
εk
.1
nt
h
a
tc
E
q
.6
.
2
6,
Cwouldhavet
ov
a
n
i
s
hi
fcJi
sc
o
n
s
t
a
n
t,
andt
h
e
na
l
s
od
o→ O
.Duet
os
t
a
t
i
s
t
i
c
a
l
,
t
h
er
a
t
i
oC
/
g
J
(
α。
)i
nE
q
.6
.
2
6becomesm
e
a
n
i
o
g
l
e
s
8when9
'(
α。
)i
st
∞ small.For
s
c
a
t
t
e
r
o
n
eo
b
t
a
i
n
sg
'(
a
o
)<0,
andi
nt
h
a
tc
a
s
ea
g
a
i
n
s
t
i
l
ls
h
o
r
t
e
rc
r
a
c
k
si
nt
h
i
sk
i
n
do
is
p
e
c
i
m
e
n,
E
q
.6
.
2
6c
a
n
n
o
tbeu
s
e
d
. However,
f
o
rt
y
p
i
c
a
lf
r
a
c
t
u
r
es
p
e
c
i
m
e
n
ss
u
c
hs
i
t
u
a
t
i
o
n
sdon
o
t
an
8
e
.
q
.1
.
4,t
h
es
i
z
ee
f
f
e
c
tl
a
wi
nt
h
eformo
iE
q
.5
.
6i
n
v
o
l
v
e
so
n
l
ym抗 e
r
i
a
l
I
nc
o
n
t
r
a
s
tt
oE
p
a
r
a
m
e
t
e
r
s,GJandc
J
.T
h
e
r
e
i
o
r
e,
もh
e
s
ep
a
r
a
m
e
t
e
r
8c
姐 b
eo
b
t
a
i
n
e
dd
i
r
e
c
t
l
ybyoptimum
f
i
t
t
i
n
go
fE
q
.5
.
6t
ot
h
emeasuredv
a
l
u
e
so
f1'N i
o
rv
a
r
i
o
u
sv
a
l
u
e
so
fD. Suchf
i
t
t
i
n
gc
a
nb
e
a
c
c
o
m
p
l
i
s
h
e
de
a
s
i
l
ybyn
o
n
l
i
n
e
a
rr
e
g
r
e
s
s
i
o
no
ranyo
p
t
i
m
i
z
a
.
t
i
o
ns
u
b
r
o
u
t
i
n
e
.
Th
es
p
e
c
i
m
e
ns
h
a
附
doo
o
tn
e
e
dt
obeg
e
o
m
e
t
r
i
c
a
l
l
ys
i
m
i
l
a
ri
iE
q
.5
.
6(
r
a
t
h
e
r伽 B
Eq
.1
.
4
)i
su
s
e
d
. However,
t
h
ep
a
r
a
m
e
t
e
r山 川 t
a
.
k
e
si
n
t
oa
c
c
o
u
n
tt
h
es
p
e
c
i
m
四
s
h
a
p
e
,
namelyt
h
er
a
t
i
og
'
(αo
)
/
g
(
α
0
),
i
so
n
l
ya
p
p
r
o
x
i
m
a
t
eandinvolvesωmee
r
r
o
r
. Toa
.
v
o
i
dt
h
i
s
,
i
ti
sp
r
e
f
e
r
a
b
l
et
ou
s
es
p
e
c
i
m
e
n
sw
h
i
c
ha
r
eg
'
ω
m
e
t
r
i
c
a
l
l
ys
i
m
i
l
a
r
.
e
r
r
o
r
Thes
i
z
ee
f
f
e
c
tmethodh
a
sa
1
8
0b
e
e
np
r
o
p
o
s
e
dbyRILEMCommitteeTC89i
o
raRILEM
∞mmendation.
Re
i
np
r
i
n
c
i
p
l
eanys
u
i
t
a
b
l
ef
r
a
c
t
u
r
es
p
e
c
i
m
e
nc
a
nbeu
s
e
d
.I
n
Asf
o
rt
h
es
p
e
c
i
m
e
ns
h
a
p
e,
v
i
e
wo
it
h
ep
o
p
u
l
a
r
i
t
yandc
e
r
t
a
i
np
r
a
c
t
i
c
a
la
d
v
a
n
t
a
g
e
so
it
h
et
h
r
e
e
-p
o
i
n
tbends
p
e
c
i
m
e
n
s
,
G
6
.
7
.
3C
a
l
c
u
l
a
t
i
o
nProcedureo
fS
i
z
eE
f
f
e
c
tMethod:
I
ft
h
el
e
n
g
t
hL
jofspecimenji
sa
1
mostt
h
esamea
s8ji
t
ssp
a
.
n,
t
h
e
ncompute
可=尽 +jmjg
(
6.
1
4
)
mw
h
i
c
hmjgi
st
h
ew
e
i
g
h
to
ft
h
es
p
e
c
i
m
e
nandt
h
es
p
e
c
i
m
e
ni
sl
o
a
d
e
ddownward.HLJis
muchJ
a
r
g
e
rt
h
加 S
jt
h
e
n
。
2-J
S
.
i- L
P
;
U
=
=P
;+ー
3m
・
0
2S
j
.
&
.
.
1
J
'''J3
•
(
6
.
1
5
)
[
ft
h
es
p
e
c
i
m
e
n
s訂 eg
'
田 m
e
t
r
i
c
a
l
l
ys
i
m
i
l
a
r制 P
;
*=1
1
・I
fn
o
t
.
.
.
d
_
S
:
P
:
*=P.・~
J
Jd 問
.
S
j
(
6
.
1
6
)
w
h
e
r
emr
e
f
e
r
st
ot
h
em
i
d
s
i
z
es
p
e
c
i
m
e
n
.Next,
p
l
o
t}
j(
o
r
d
i旧民)v
e
r
s
u
sXj (
a
b
s
C
I
s
s
a
)where
r
号
(
巧=
Xj=dj
(
6.
1
7
)
a
n
dddermineb
h
es
l
o
p
eo
ft
h
el
i
n
eAa
sshowni
nFig.6.i15lH
l b
也
heY
i
凶
n
t
町
e
'f
偲
C
e
p
μむC.The
f
任
r
a
総
叫
ぬ
c
ぺ
叫
t
h
h
1
u
?
;
;
斗
ζ
な
:
よ
:
t
r
A
r
m
o
m恥 6
.
却
2
6
. The叫 i
eo
fE f
o
rh m n
e
e
d
sb
obed
e
t
e
r
m
i
n
e
df mC04 i
rs
t
l
E
I
l
a
s
e
dfro-zu
s
i
n
ge
s
t
a
b
l
i
s
h
e
de
m
p
i
r
i
c
a
lf
o
r
m
u
l
a
s
.TheexpmSionf
o
y
g
(
α
0
)
24;;:
r
e
n
c
e
sf
o
rLEFMformulωThef
o
加 i
叩.r
ef
r
o
足
立
な沼野にで山長
F
(
α)=よ 1
.
9
9-α(1ー α
)
(
2均 一 3
.
9
3
α +2
.
7
(2)
、
f
i
(
1+2
α
)
(
1ー α)
3
/
2
(
6
.
1
8
)
9
6
も
"
.
‘
AWMV
'
・
&
)
=
f
h
,
αo
[
1
.
5F(
αoW
)
g
(
句
ρ
oa
、
F
(
α
)= 1
.
10
6-1
.552α+7
.
7
1
α
2-13.53α3+1
4
.
2
3
α
4
0
.F
i
n
a
l
l
y
,
L
i
n
e
a
ri
n
t
e
r
p
o
l
a
t
i
o
ni
sa
c
c
e
p
t
a
b
l
ep
r
o
v
i
d
e
d3~ S/d壬1
,
,
F
o
rS/d=8
:
97
•
ゐ
,;
:5.
4
d
and(
3
)f
o
rc
e
n
t
r
i
ct
e
l
凶o
ns
p
e
c
i
m
e
n
sd
p
.
8
d
"
.C
o
n
s
e
q
u
e
n
t
l
y
,theminimum
m,
o= 1
悶
s
p
e
c
i
m
e
n
sw
h
i
c
ha
r
en
e
e
d
e
df
o
ra
p
p
l
i
叫 i
l
i
t
yo
fLEFM(
w
i
出
"
r
o
S
ss
e
c
t
i
o
nd
e
p
t
h
so
ft
h
.
h
eもh
e
o
r
e
t
i
c
a
.
le
r
r
o
ro
f2%)a
.
r
e.
a
sf
o
l
l
o
w
s
:
(
6
.
2
0
)
I
nBa
.
Z
a
n
tandP
f
e
i
f
f
e
r(
1
9
8
7
)andK
a
r
i
h
a
l
o
o組 dN
a
l
l
a
.
thambi(
1
9
8
7
)t
h
e
r
ea
r
ed
i
s
c
u
s
s
i
o
n
s
o
fc
a
l
c
u
l
a
t
i
o
n
sn
e
e
d
e
dt
ov
e
r
i
f
yt
h
er
e
s
u
l
tf
o
rGJ岨 dt
oo
b
t.
a
i
nt
h
es
t
a
n
d
a
r
dd
e
v
i
a
t
i
o
n
.I
t
.
ts
c
a
t
t
e
r
e
dr
e
s
u
l
t
sc
組
a
r
i
s
ef
r
o
mo
t
h
e
rs
i
z
ee
f
f
e
c
t
自 (
S
e
c
.5
.
3
)
.T
h
i
s
i
sa
l
s
op
o
i
n
t
e
do
u
tもha
c
e
r
nf
o
rv
e
r
yl
a
r
g
eandt
h
i
c
ks
p
e
c
i
m
e
n
s,
s
i
n
c
eh
y
d
r
a
t
i
o
nh
e
a
tw
i
l
l
maybeo
fp
a
r
t
i
c
u
l
a
r∞n
p
r
o
d
u
c
ed
i
f
f
e
r
e
n
tt
e
m
p
e
r
a
t
u
r
e
si
nt
h
i
nandt
h
i
c
ks
p
e
c
i
m
e
n
sandt
h
u
sl
e
a
dt
o叩 a
d
d
i
t
i
o
n
a
l
8
1
田 e
f
f
e
c
tt
h
a
ti
ss
u
p
e
r
i
m
p
o
s
e
dont
h
a
td
e
s
c
r
i
h
e
dbyt
h
es
i
z
ee
f
f
e
c
tl
a
w
.I
nt
h
a
tc
描 et
h
i
s
method∞u
l
df
a
i.
lS
i
m
i
l
a
r
l
y
,
i
tc
o
u
l
df
a
i
lduet
od
i
f
f
e
r
e
n
c
ei
nm
o
i
s
t
u
r
ec
o
n
t
e
n
t,
s
i
n
c
et
h
i
n
c
ko
n
e
s.Thesee
f
f
e
c
t
s,
剖 wella
st
h
ew
a
l
le
f
f
e
c
t,
a
r
em
i
n
i
m
i
z
e
d
s
p
e
c
i
m
e
n
sd
r
yf
a
s
t
e
r出 組 出i
byu
s
i
n
gt
h
esames
p
e
c
i
m
e
nt
h
i
c
k
n
e
s
sf
o
ra
l
lt
h
es
i
z
e
s
.
6.7.
4ComparisonwithOtherMethods
K
a
r
i
h
a
l
o
o加 dNa
.
l
I
a
t
h
a
r
n
b
i(
1
9
8
7
)l
i
s
tmosto
ft
h
eknownf
r
a
c
t
u
r
et
e
s
tr
e
s
u
l
t
s,
i
nw
h
i
c
h
Ba
.
ia
n
t
'
ss
i
z
ee
f
f
e
c
tmethodi
sa
l
s
ocomparedw
i
t
hK
a
r
i
h
a
.
l
oo姐 dN
a
l
l.
a
t
h.
a
m
h
i
'
sm
e
t
h
o
d
.
s
eo
ft
h
eLEFMr
e
l
a
.
t
i
o
n
s
b
i
p
sp
u
r
p
o
s
e山 ev
a
l
u
田 o
fGJ w
e
r
ec
o
n
v
e
r
t
e
dt
oK1
F
o
r出 i
cbyu
/
)
1
/
2
.F
o
rt
h
ebeamr
e
s
u
l
t
sr
e
p
o
凶e
d,
t
h
eK1
.
l
u
e
sw
e
r
e0.847-0
.
8
9
2
k
N
;
;
.
3
2
K
l
c:
;
;(G
cva
JE
e=
a
.
n
dt
h
ev
a
l
u
e
sfromt
h
eK
a
r
i
h
a
l
o
oandNa
.
la
.
t
h
a
r
n
b
i
'
smethodw
e
r
eKic=
0
.
8
6
7andKI
c
n.Similara
1
.0
0
5
k
N
;
;
.
3
g
r
.
四 m
en
もw
asf
o
u
n
df
r
o
mt
h
eda
.
t
.
ao
fBa
.
Z
a
n
t姐 dP
f
e
i
f
f
e
r(
1
9
8
7
)
.
吋 b
yB
r
u
h
w
i
l
l
e
r(
1
9
8
8
)副 知umae
ta
1
.(
1
9
8
9
)t
o
Thes
i
z
ee
f
f
e
c
tmethodwasu
s
measuret
h
ef
r
a
c
t
u
r
ee
n
e
r
g
yo
fdamc
o
n
c
r
e
t
e
.
6
.
8S
i
z
eRequiredf
o
rA
p
p
l
i
c
a
b
i
l
i
t
yofLEFM
Ane
a
r
l
ya
.
t
t
e
m
p
tt
os
p
e
c
i
f
yt
h
eminimums
i
z
eo
fも回ts
p
e
c
i
m
e
ns
u
c
ht
h
a
tLEFMma
.
yb
e
h.
a
to
fWalsh(
1
9
7
2,
1
9
7
6
)
. Hea
r
g
u
e
dt
h
a
.
t
,
byc
o
m
p
a
r
i
n
gt
h
en
o
r
n
i
n
a
ls
t
r
e
s
sa
t
u
s
e
dw回 t
t
h
en
o
t
c
ht
i
pf
o
rabeams
p
e
c
i
r
悶
(
c
a
l
α
l
1
.
a
t
e
dbyu
s
i
n
ganLEFMform
由 f
o
r]
(
[
,
)w
i
t
ht
h
e
at
r
a
n
s
i
t
i
o
nd
i
m
e
n
s
i
o
nma
.
ybef
o
u
n
d
.Hes
t
a
t
e
dt
h.
a
tt
h
e
moduluso
fr
u
p
t
u
r
eo
ft
h
em
a
t
e
r
i
a
l,
c
h
a
r
a
c
t
e
r
i
s
t
i
cbeamd
i
m
e
n
s
i
o
ns
h
o
u
l
dbea
tl
e
a
.
s
td
o
u
b
l
et
h
i
si
no
r
d
e
rf
o
rLEFMt
obev
a
l
i
d
.
Ba
.
s
edonh
i
st
e
s
tda
.
t
.
ah
es
u
g
g
e
s
t
e
dt
h
en
o
t
c
h
e
dbeamt
obet
e
s
t
e
ds
h
o
u
l
dha
.
v
e.
ad
e
p
t
bn
o
t
l
e
s
st
h
a
.
n2
3
0m m(
9i
n
.
)
.
I
nt
h
el
i
g
h
to
fr
e
c
e
n
te
x
p
e
r
i
m
e
n
t
a
ls
t
u
d
i
e
sont
h
es
i
z
ee
f
f
,民t
i
nf
r
a
.
c
t
u
r
e
,
t
h
er
e
c
o
m
r
n
e
n
d
a
t
i
o
nf
o
rt
h
es
p
e
c
i
m
e
ns
i
z
ean.
a
l
y
z.
a
b
l
ebyLEFMnowa
p
p
e
a
r
st
obeg
r
o
s
s
l
yu
n
d
e
r
e
s
t
i
m
.
a
t
e
d
.
A
c
c
o
r
d
i
n
gt
ot
h
es
i
z
ee
f
f
e
c
tl
a
wo
fB話 a
n
t,
t
h
eb
r
i
t
t
l
e
n
e
s
snumbersmustbea
tl
e
a
s
も2
5i
f
抗i
o
nfromt
h
es
t
r
a
i
g
h
tl
i
n
ea
s
y
m
p
t
o
t
ef
o
rLEFMs
h
o
u
l
dbel
e
s
st
h
a
n2%. Ba
.
i
ωb
t
h
ed
e
v
i
却 dP
f
e
i
f
f
e
r
'
s(
1
9
8
7
)七e
s
t
so
fv
a
r
i
o
u
st
y
p
e
so
fs
p
e
c
i
m
e
n
sshowni
nF
i
g
.5
.
1i
n
d
i
c
a
t
et
h
a
t:(
1
)
f
o
re
c
c
e
n
t
r
i
cc
o
m
p
r
e
s
s
i
o
nf
r
a
c
t
u
r
es
p
e
c
i
m
e
n
sdo=1
.8
5d
(
2
)f
o
rt
h
r
e
e
p
o
i
n
tbends
p
e
c
i
r
r
削 S
m,
F
o
re
c
c
e
n
t
r
i
cc
o
m
p
r
e
s
s
i
o
ns
p
e
c
i
m
e
n
s
: dと4
6
d
"
F
o
rt
h
r
e
e
-p
o
i
n
tbends
p
e
c
i
m
e
n
s
: d三 1
3
5
d
"
F
o
rc
o
n
c
e
n
t
r
i
ct
e
n
s
i
o
ns
p
e
c
i
m
e
n
s
: dさ 4
2
0
d
"
(
6
.
2
1)
a
.
t
o
r
y七e
s
t
s,a
.
n
dt
h
e
i
ru
s
ewouldb
e
T
b
e
s
ed
i
m
e
n
s
i
o
n
sa
r
ei
m
p
r
a
c
t
i
c
a
b
l
yl
a
r
g
ef
o
rl
a
b
o
r
f
i
n
a
n
c
i
a
.
l
l
yw
a
s
t
e
f
u
ls
i
n
c
ek
n
o
w
l
e
d
g
eo
(t
h
es
i
z
ee
f
f
e
c
tc
a
np
r
o
v
i
d
ea
d
e
q
u
a
t
ei
n
f
o
r
m
a
t
i
o
non
a
t
e
r
i
a
lf
r
a
c
t
u
r
ep
a
r
a
.
m
e
t
e
r
su
s
i
n
gmuchs
m
a
l
l
e
rs
p
e
c
i
m
e
n
s
.
m.
6
.
9I
d
e
n
t
i
f
i
c
a
t
i
o
nofNonlocalC
h
a
r
a
c
t
e
r
i
s
t
i
cLength
F
o
rb
h
em
n
l
o
c
a
lcon?inuummodelw
i
t
hl
o
c
a
I叫 r
a
i
n(
o
rn
o
n
l
o
c
a
ldamagec
o
n
t
i
n
u
u
m
)
.
.
h
ef
r
a
c
t
u
r
een
句 yGJi
sp
r
o
p
o
r
t
i
o
n
a
lt
ot
h
ea
r
e
aW.u
n
d
e
rt
h
e∞m
p
l
e
t
et
e
n
s
i
l
e町
制ー
ポr
a
i
nc
u
r
v
e
(
w
i
s
h
B
o
f
t
e
n
i
n
g
)
t
i
m
e
sb
h
ec
h
a
r
a
c
t
e
r
i
s
t
i
cl
e
n
g
t
h
,
t
,
andc
fi
sp
r
o
p
o
r
t
i
o
n
a
lt
ot
h
e
c
o
n
c
e
i
v
e
dbyBdant姐 d
c
h
u
a
c
t
e
r
i
s
t
i
cl
e
n
g
U
1
.
T
h
i
sf
a
c
ti
se
x
p
l
o
i
t
e
dbyas
i
m
p
l
emdhod,
P
i
j
a
u
必ぽ-C.
a
b
o
t(
1
9
8
8
b,
1
9
8
9
)
.I
tu
s
e
st
h
er
e
l
a
.
t
i
o
n
:
i
=皇位竺!
3]
W
.[
J
/
m
(
6
.
2
2
)
,
t
k
r
e
i
s
a
m
u
l
t
i
p
l
i
c
a
u
v
e
e
m
p
i
r
i
c
a
l
c
o
e
缶c
i
e
n
tont
h
er
i
g
h
もh
ands
i
d
e,
which
Z
M
o
r
e
p
r
e
c
i
s
e
l
y
1
8h
o
w
e
v
e
r,
刊r
yc
l
o
s
et
ol
(
i
bs
l
i
g
h
t
l
yd
e
p
e
n
d
sonb
h
et
y
p
eo
ft
h
en
O
B
l
o
c
a
lmodeladop
もe
d
,
,
.
u
df
o
r0
田 p
a
r
t
i
c
u
l
a
rmodelu
s
e
di
twas1
.0
2
)
.Sot
h
eo
n
l
yp
r
o
b
l
e
mi
showt
om倒 的 出
a 日e
r
g
yd
i
s
s
i
p
a
t
e
dp
e
ru
n
i
tvolume
,
W.・
Tot
h
i
send
,
onen
e
e
d
st
os
t
a
b
i
l
i
z
eS
h
es
p
e
c
i
m
e
nbybondedr
e
i
n
f
o
r
c
i
n
grodsmthatc
o
n
噌 E
eund
e
f
o
r
mh
o
m
o
g
e
n
e
o
u
s
l
yandr
e
m
a
i
ni
nam
a
c
r
o
s
c
o
p
i
c
a
l
l
yhomogeEmuss
b
a
b
ed
u
r
i
n
g
,
w
i
s
h
o
u
tanyl
o
c
a
l
i
z
a
t
i
o
no
fc
r
a
c
k
i
n
g
.
T
h
i
smaybea
c
c
o
m
p
l
i
s
h
e
du
s
m
s
o
f
t
e
n
i
n
gb
e
h
a
v
i
o
r
~t
h
es
p
e
c
i
m
佃 s
howni
nF
i
g
.
6
.
1
6
,
w
h
i
c
ha
l
s
os
h
o
w
sacompanionf
r
a
c
t
u
r
es
p
e
c
i
m
e
nu
s
e
d
n
f
.
d
e
t
e
r
m
i
n
a
b
i
0
1
1
o
fGI-Thel
o
n
g
e
rs
i
d
e
so
ft
h
er
e
c
t
a
n
g
u
l
a
rc
r
o
s
ss
e
c
t
i
o
na
r
er
e
s
t
r
a
i
n
e
dbv
.
u
n
gt
othem,
w
i
t
hepoxy
,
as
y
蜘 no
fpa
叫 e
lden
均 s
p
a
.
c
e
d仙 s
t
e
e
lrods-Thegaps
・
J
・
a
?
な
ど
な;
3
2
J
3
2
ぷ
:
:
:
:
:
:
ニ::22ztL222ぷ:
:
t
r
z
一
n?feetlnconcrete-Byc}loomgt11ecrossmtionsofumodstobemchmdlertharl
f 1 m m喝 r
e
仲間,t
h
er
仙 c
a
n
n
o
ta
f
f
e
c
tt
h
en
o
n
l
o
c
a
.
lbeha
.
v
i
o
ro
ft
h
em
a
t
e
r
i
a
li
n
a
n
s
v
e
r
s
ed
i
r
e
c
t
i
o
n
T
h
er
e
c
t
a
n
g
u
l
a
rc
r
o
s
ss
e
c
t
i
o
ni
se
l
o
n
g
a
t
e
d,
s
oaE4om
i
n
i
m
i
z
et
h
e
:
官会1
1
f
器Z
2
1
5
2
5
S
2
5
1
5
主主主忠誠i
;
k
n
州
99
98
Thec
o
m
b
i
n
e
dc
r
o
s
ss
e
c
t
i
o
no
f3
0
1
1t
hes
t
e
e
lr
o
d
si
ss
e
l
e
c
t
e
ds
oa
.
st
o出 s
u
r
et
h
et
a
n
g
e
n
t
i
a
l
,
・u
n
e
s
so
ft
h
ec
o
m
p
o
s
i
t
es
p
e
c
i
m
e
nωbea
l
w
a
y
sp
o
s
i
t
i
v
e
.T
h
i
sg
u
a
r
阻むe
e
st
h
es
p
e
c
i
m
四
h. ~ s
もa
b
l
e makingl
o
c
a
l
i
z
a
t
i
o
ni
n
s
t
a
b
i
l
i
t
yi
m
p
o
s
s
i
b
l
e
. Thef
o
r
c
ei
ns
o
f
t
e
n
i
n
g∞
c
o
町
n
c
r
問e
t
怜e
,
,
'
げ
1
:
1.
.
川
‘
1
.
t
e
舵e
l
r
o
也
d
sc
a
l
c
u
l
a
t
e
dfromt
h
emeasureds
t
r
a
i
ni
ss
u
b
t
r
a
.
c
t
e
dfromt
h
em他 s
叫u
r
,e
陀d
ω
も
。
叫
4
同a
lf
o
r
c
e
.
e
.
g
.,
bym
e
t
a
l
l
i
cp
l
a
t
e
sg
l
u
e
dbyepoxyt
ot
h
es
u
r
f
a
.
目
f
h
es
p
e
c
i
m
e
ni
sg
r
i
p
p
e
da
tt
h
ee
n
d
s,
,
r
¥~ hesteelrods.Thespecimeni
ss
u
b
j
e
c
t
e
dt
ot
e
n
s
i
o
ni
nac
l
o
s
e
d
l
o
o
pt
e
s
t
i
n
gma
.
c
h
i
n
e
,
a
.
nd
L
トサ l
o
to
ff
o
r
c
eVB. r
e
l
a
t
i
v
ed
i
s
p
l
a
c
e
m
e
n
to
v
e
rg
a
u
g
e1
田 g
t
hL i
nt
h
em
i
d
d
l
ep
a
r
to
ft
h
e
,
;ri
I
meni
sr
e
∞rded;seeFig.6.16c. Theindinedstraightlinei
n出 i
sp
l
o
tr
e
p
r
e
s
e
n
t
st
h
e
慢n
e
s
so
ft
h
er
o
d
sa
l
o
n
e
,
w
i
t
h
o
u
t回 yc
o
n
c
r
e
t
e
.F
o
rv
e
r
yl
a
r
g
ed
i
s
p
l
a
c
e
m
e
n
t
s,
t
h
er
田 p
o
n
s
e
仰r
叫 c
l
悶
t
h
i
sl
i
n
ea
s
y
m
p
t
o
t
i
c
a
l
l
y
. Work W~ , t
h
ed
e
s
i
r
e
dr
e
s
u
l
色
, i
sr
e
p
r
e
s
e
n
t
e
dbyt
h
e
吋 川・b
a
t
c
h
e
da
r
品 b
etweent
h
er
e
s
p
o
n
s
ec
u
r
v
ea
.
ndt
h
er
i
s
i
n
gs
t
r
a
i
g
h
tl
i
n
ef
o
rt
h
es
t
e
e
lr
o
d
s
山.川.‘
f
・
b 1
62
.
.
.圃
,
TI--T-14
(a)
⋮ u
l
F
o
rt
h
epa
.
r
t
i
c
u
l
a
.
rc
o
n
c
r
e
t
eu
s
e
dbyB話 回tandP
i
j
a
u
d
i
e
r
C
a
b
o
t(
am
i
c
r
o
c
o
n
c
r
e
t
ew
i
t
h
,
i
n
t
oE
q
.6
.
3
4
h
i
si
sr
a
t
h
e
rc
l
o
s
et
owhath
a
sb
e
e
n
)
'
t
l.
l
n
e
dt
h
en
o
n
l
o
c
a
lc
h
a
.
ra
c
t
e
r
i
s
t
i
cl
e
n
g
t
hf= 2.7da • T
け
J
!
ν
nedf
r
o
mf
i
n
i
t
ee
l
e
m
e
n
tf
i
t
t
i
n
go
ft
h
es
i
z
ee
l
f
e
c
td
a
t
afromf
r
a
c
t
u
r
et
e
s
t
s
.
a
.
n
so
ft
h
ef
i
c
t
i
t
i
o
u
s
[
tm
i
g
h
tben
o
t
e
dt
h
a
te
v
a
l
u
a
t
i
o
no
ft
h
ep
r
e
s
e
n
tt
y
p
eo
ft
e
s
tbyme
fonet
a
.
k
e
sc
a
r
et
op
r
o
d
u
c
ea
.n
e
a
r
l
yp
e
r
f
e
c
tr
e
s
t
r
a
i
n
t,
r
M
;
km
o
d
e
l.
a
p
p
e
a
r
sp
r
o
b
l
e
m
a
t
i
c
.I
fllI, ~tiple p
a
.
r
a
l
I
e
lf
i
c
t
i
もi
o
u
sc
r
a
c
k
sw
i
t
ha
r
b
i
t
ra
.
r
i
l
yc
l
o
s
es
p
a
c
i
n
gc
o
u
l
dbeo
b
t
a
i
n
e
di
ncompu~い 川s, ma
.
k
i
n
gt
h
er
e
s
u
l
t
sa
.
m
b
i
g
u
o
u
s
.Asa
.
l
r
e
a
d
yp
o
i
n
t
e
do
u
t,
t
h
i
sproblemc
a
nbea
v
o
i
d
e
d
I
川 崎r
i
c
h
i
n
gt
h
ef
i
c
t
i
t
i
o
u
sc
r
a
c
kmodelw
i
t
hat
h
i
r
di
n
d
e
p
e
n
d
e
n
tm
a
t
e
r
i
a
lp
a
r
a
m
e
t
e
r
,the
l
1
'
"
I
l
JI
I
I
U
ms
p.
a
c
i
n
go
ft
h
el
i
n
ec
r
a
c
k
s(
s
u
c
hs
p.
a
c
i
n
go
f∞u
r
s
ei
se
q
u
i
v
.
a
l
e
n
tt
ot
h
en∞
o
叫
n
1
1
o
ca
.
l
凶
Eh
a
瓜:
川.:
¥
'
l
In
叫
l
比
t
e
ぽr
e
飽s
“
叫
t
i
暗
n
gpoω
叫
m
もt
ω
o
n
ω
o
t
旬e
l
お
st
由
h抗
a
tt
h
ec
h
a
釘x
拭
a
c
t
ω
釘
e
n
叩s
“
叫
も
“i
比
cl
e
n
g
t
h,
,
.
fcannotbei
d
e
n
t
i
f
i
e
dfrom
倒 So
fu
n
r
e
s
t
r
a
i
n
e
ds
p
e
c
i
m
e
n
so
fd
i
f
f
e
r
e
n
t1
田 g
t
h
s(
e
.
g
.t
e
s
t
so
fv
a
nMier,
1
9
8
4,
l
'
o
ll
'
I
a
.
lt
I
P"
b
'u
rd
i
f
f
e
r
e
n
tg
a
g
el
e
n
g
t
h
s(
S
h
a
.
h.
a
ndS
a
n
k
a
r
,1987). Suchd.
a
t
a
.c
姐
bef
i
t
t
e
de
q
u
a
l
l
y
n
yv
a
l
u
eo
f
,
.
fi
n
c
l
u
d
i
n
gf
.=0(Ba
.
i
a
n
t,1
9
8
9a
.
)
.
I
!,I.~ing a
1
1
.
''
(
i
m
u
ma
g
g
r
e
g
a
t
es
i
z
e9.
5mm) s
u
b
s
t
i
t
u
t
i
o
no
ft
h
emeasuredva
.
l
ueo
fW~
1
1s
t
僧 1
r
噛
P
(c
.
)
i
堅
'
1nIdentificationofTensilePost-PeakSofteningStress.Strain
⋮臨
ーーム
1l il
l - U﹀翠
叫﹁
→い霊
rU
引コい引μlhui
Cpol}'
.て.I'~te.
t
A
¥
.
Ou
r
v
e
s
1
・・ i
r
l
e
n
t
i
f
yas
t
r
a
i
n
s
o
f
t
e
n
i
n
gt
r
i
a
x
i
a
lc
o
n
s
t
i
t
u
t
i
v
er
e
l
a
t
i
o
nfromt
e
s
td
a
ta
.
,onemusti
n
,.
i
n
v
e
r
s
eboundaryv
a
.
l
ueproblemt
a
k
i
n
gi
n
t
oa
c
c
o
u
n
ts
t
r
a
i
nl
o
c
a
l
i
z
a
t
i
o
n
,
'
:!
9
8
8
)
.T
h
i
si
sat
a
.
s
ko
ff
o
;
m
i
d
a
b
l
ec
o
m
p
l
e
x
i
t
子 H
owever,
i
d
e
n
t
i
f
i
c
a
t
i
o
no
fau
n
ia
.
x
i
a
l
(
O
r
l
'
,
u
a
i
l
' '.Jf
t
e
n
i
n
gs
t
r
e
s
s
s
t
r
a
i
nr
e
l
a
.
t
i
o
ni
sr
e
l
a
t
i
;
e
l
ysi~ple, p
r
o
v
i
d
e
dt
h
a
tt
h
ec
h
a
r
a
c
t
e
r
i
s
t
i
c
k-tl~! 1,
I、
d
e
t
e
r
m
i
n
e
di
na
d
v
a
n
c
e
卸
, alreadyexplained.AsshownbyBa.i組 内 (1989a)analysis
I
.
ni
a
!t
凶 d
a
t
a
.f
o
rs
p
e
c
i
m
e
n
sofVa
.
r
i
O
U
8l
e
n
g
も
h
s,
l
o
c
a
.
l
i
z
a
.
t
i
o
ni
nu
n
i
a
x
i
a
lt
e
s
ts
p
e
c
i
m
e
n
s
0
1u
l1
~:
I
,
r
.'
:
'!~u.a.t~ly d
e
s
c
r
i
b
e
dbyt
h
e
's
e
r
i
e
sc
o
u
p
l
i
n
gm~del, i
nwhichas
t
r
a
i
n
s
o
f
t
e
n
i
n
gl
o
a
d
i
n
g
I
C
4・
fI
.
'
n
g
t
hli
sc
o
u
p
l
e
di
n附 i
e
st
oanu
n
l
o
a
d
i
n
gz
o
n
e
.I
ft
h
ev
a
l
u
eo
f.
fi
sknown,
t
h
e間 i
a
l
es
t
r
a
i
n
s
o
f
t
e
n
i
n
gz
o
n
ec
o
r
r
e
s
p
o
n
d
i
n
gt
ou
n
i
a
x
i
a
ls
t
r
田 ssmayb
eapproxim
a
.
t
e
l
y
h車問・" ~ h
伊 1
1r 1 l
K
>
l
v
e姐
Figs.16Measuremento
fEBergD
i
s
s
i
p
a
t
i
o
nDuet
o
C
E
a
C
I
t
i
n
gPerUnis
阻 A
c
c
o
r
d
i
n
gt
oBaiantandP
i
ja
.
u
d
i
e
x
.
Volumei
naS
t
a
b
i
l
i
z
e
dSpecim
Ca
.
b
o
t(
1
9
8
8
)
1
0
1
1
0
0
_L
.E'
d、
r
‘
、
σ
,z
E
α
u
、
・
、‘‘ ,
,
,
,
f'h¥
L一
t
EE
σ
,
,
,
、
、
一
一
σ
‘
、
,
‘
(
﹄
,
F
‘
.
,
L一
t
c
a
l
c
u
l
a
t
e
da
s(
B話 回t1
9
8
9
a
)
:
t
uppersupporti
ngbeam
(
6
.
2
3
)
寸lid←
一
whereL ~ 1,
可
f
市
刷
(
司
σ
伊
例
)=me
叩&胡
n&
凶x
対i
叫
a
1柑
s
t
仕r
凶a
附
l
e
r
凶 o
肝v
e
白r
t
h
eg
a
g
el
e
n
g
ぬL
,
姐 df
u
(
σ
)= a
x
i
a
ls
t
r
a
i
ni
nt
h
eu
n
l
o
a
d
i
時 p
a
r
tc
o
r
r
e
s
p
o
n
d
i
n
g
m
.
組 c
he
m
a
n
a
t
i
n
gf
r
o
m出 ep
e
a
ks
t
r
e
s
sp
o
i
n
t.
toσforanu
n
l
o
a
d
i
n
gb
r
LD
6
.
1
1MaterialParametersf
o
rModeI
IandPlanarMixedMode
Fr
acture
F
o
rtwod
i
m
e
n
s
i
o
n
M
s
t
r
e
s
s
B
b
a
t
e
s
,
i
ng
e
n
e
r
a
li
tmaybepresumedt
h
a
tc
r
a
c
kpropagdon
c
t
i
o
no
fMode1(
o
p
e
n
i
n
g
)andModeI
I(
s
l
i
d
i
n
g
)
u
l
dt
a
k
ep
l
a
c
eu
n
d
e
rもhecombineda
d
e
f
o
r
m
a
t
i
o
n
.
I
nc
o
n
c
r
e
t
e
,however,
t
h
ee
f
r
e
e
b
o
fa
g
g
r
e
g
a
t
ei
n
t
e
r
l
o
c
k
,
wbichl
品 d
st
ov
o
l
u
m
e
d
i
l
a
t
i
o
n
.t
e
n
d
st
or
e
s
i
s
tc
r
a
c
kp
r
o
p
a
g
a
t
i
o
ni
nanYmode0
出e
rt
h
組
o
p
e
n
i
n
g(
B
a
z
a
n
t祖 d
j
o
v
a
-1
98
0}
.Ne
ve
r
t
h
e
l
e
s
s
,
t
h
e
r
eha
sbe
e
nane
x
対t
e
叩n
隠
s
抑 e晶m
釦o
G釧 ba
d
白o
且邸
e仰o
v
釘
e
r古
仏
ド
ey
戸e
一
…s, 組dcontinuingtod品y,i
nan叫 emptt
oc
a
u
s
ec
即 k
i
n
g姐 df
間 印 刷n
c
r
e
t
et
ooc
c
u
ri
nmode1
1o
ra
tl
e
a
s
ti
nmixedmode.
TheI
o
s
i
p
e
s
c
us
p
e
c
i
m
e
ngeomdry
,
showni
narnodigdformi
nF
i
g
.
6
.
1
7,
h
a
sb明 nu
詑d
byv
a
r
i
o
u
siwestigaws(Barre
ta
l
.,
1
9
8
7
iBahMandP
f
e
i
f
e
r
,1985h1985b,1986;Swartz
ndT
a
h
a
.1
9
8
7
)ands
ohωas
i
m
i
l
a
rgωmetryw
i
t
ho
n
l
yo
n
en
o
t
c
h(
A
r
r
e
a叩 dI
n
g
r
a
:
f
e
a
,
1981-SW4
巾,
TMlEMidR
e
f
a
i,1
9
8
8
)・ Beamsi
nt
h
r
e
e
p
o
i
n
tb
e
n
d
i
n
gw
i
t
ho
f
f
c
e
n
t
e
r
otcheshave--usedbyJ
e
叫 a
ndShah(
1
9
8
7
a
,1987b)andSwartz,
LuandTang(
1
9
8
8
)
(
s
e
eF
i
g
.
6
.
l
a
LB
i
a
x
i
a
lt
e
s
t
i
n
gs
y
s
t
e
m
shaveb
e
e
nd
e
v
e
l
o
p
e
di
nwhichc
o
m
b
i
n
a
t
i
o
n
so
fd
i
1989)ord
i
r
e
c
tt
e
n
s
i
o
nands
h
e
a
r
r
e
c
tt
e
n
s
i
o
nandcompmSiona
r
eu
s
e
d(binhardte44,
叩 l
z
a
d
e
he
ta
l
.,
1
9
8
7
)
(
H
a
s
N
u
m
e
r
i
c
a
lm
o
d
e
l
i
n
go
ft
h
eI
o
均e
sc
us
p
e
c
i
m
e
nwωdonebyA
r
r
e
a乱ndI
n
g
r
a
f
f
e
a(
1
9
8
1
)
,
I
n
g
r
a
f
f
e
a叩 dP
a
n
t
h
a
k
i(
1
9
8
5
),I
n
g
r
a
f
f
e
a
.
組dG
e
r
s
t
l
e(
1
9
8
5
),
BazantandP
f
e
i
鉦
'
e
r(
1
9
8
5
b
),
Ro
t
sanddeB
o
r
s
t(
1
9
8
7
), 組dSwen
則
(
1
9
8
6
)
・Ra
t
he
ri
n
t
e
r
偲 t
i
n
gwast
h
ef
i
n
d
i
n
gb
y
.
Z
姐tMldP
f
e
i
f
r
e
rt
h
a
ti
ft
h
es
h
e
a
rz
o
n
ei
nt
h
ebeami
sv
e
r
yn
a
r
r
o
w
,
t
h
ec
r
a
c
kp
r
o
p
a
g
a
4
6
Ba
.thec
r
o
s
s
s
e
d
i
o
np
l
a
n
er
a
t
h
e
r
S
h
a
ninani
n
c
l
i
n
e
ddiredion-Thisp
r
o
v
i
d
e
dac
h
a
l
l
e
n
g
e
a
.
lm
o
d
e
l
i
n
g
.A
c
c
e
p
t
a
b
l
ef
i
t
so
ft
e
叫 r
e
s
u
l
t
sw
e
r
eo
b
t
a
.
i
n
ed(
B話相も andP
f
e
i
:
f
e
r
,
i
c
1
9
8
5
b
)笛 sumingaModeI
Is
附
a
tt
h
ec
r
a
c
kt
恥 mod
e
l
山
aband山 伽dMod
拭c
k
s
目. O
t
h
e
r
so
b
ta
.
i
n
e
da
叫
.
c
c
白e
p
μ
t
功
a
b
l
e自
臥t
ω
sa
s
s
u
m
i
n
gも出】
hec
r
a
c
kp
戸r
叩
o
p喝
a
g
“
a
叫
t
i
ぬ
ont
ooc
c
a
n
l
yMode1d
e
f
o
r
m
a
t
i
o
natt
h
ec
叫
t
i
pa
lt
houghModeI
I
t
y
p
ed
e
f
o
r
m
a
t
i
o
n(
∞m
w
i
t
hModeI
le
x
i
s
t
si
m
m
e
d
i
a
t
e
l
yb
e
h
i
n
dt
h
ec
r
a
c
kfront.Thea
p
p
l
i
c
a
t
i
o
no
fS
w
e
E
l
g
o
ns
-d
e
lt
obe-st
e
s
t
e
dbySwutze
talinthreepoiMbending(Fig.618)mdfoumiM
紅 同 Lu
,
Ta
.
ngandR
.
e
f
a
i,
1
9
8
8
)(si出l~r t
oF
i
g
.6.U,
'
b
t
r
lw
i
t
h
'
o
o
l
yon
en
o
t
c
h
)
b
e
o
d
i
n
g(Sw
,
a
l
刷 l
a
.
t
i
v
e
s
h
o
w
sc
l
e
a
r
l
yt
h
a
tp
r
o
p
a
g
a
t
i
o
ni
nt
h
e
i
rt
y
p
e
o
f
t
e
s
to
c
c
u
r
si
nM04elHowever
e
x
p
l
a
n
a
s
i
o
n
s
o
ft
h
ef
r
a
c
t
u
r
ep
a
t
t
e
r
no
b
s
e
r
v
e
dbyBdMltandPfahr(1985b,
1
9
8
6
),
w
h
i
c
h
h
a
v
eb
e
e
no
f
e
r
e
dbyInudea姐 dP
a
n
t
h
a
k
i(
1
9
8
5
),
a
n
d
r
e
q
u
i
r
eo
n
l
yMod
eItypecra
ck
i
n
g,
・(
S
w
a
r
t
z回 dTah
,
.
a1987),
w
h
i
l
eR
o
も 5andd
eB
o
r
s
t(
1
9
8
7
)c
o
n
s
i
d
e
r
e
dt
h
ec
r
a
c
k
S
w
a
r
t
zeta
l
仔吋
ILPD
Test
Specimen
h
'owersupporti
ngbeom
L
u
r
e
a
c
ti
o
n
A,d 孟1
・
-
凶
ザ
F
i
g
.
6
.
1
7
1
ω
i
p
e
s
c
u
・
TypeS
h
e
a
rT,白tG
回 m
etry
。
'
行 x~
k
-x
S
F
i
g
_
6.
1
8B
e
a
r
nw
i
t
hO
f
f
-C
e
n
t
e
rN
o
t
c
h
e
s
103
102
nMode1ands
u
b
s
e
q
u
e
n
t
l
yt
or
e
s
p
o
n
dp
o
s
s
i
b
l
yi
nModeI
Ia
l
s
o
.I
n
g
r
a
.
f
f
e
a
o
p
e
n
i
n
gt
oi
n
i
t
i
a
t
ei
dPan
もh
a
k
ih
a
v
ea
r
g
u
e
dt
h
a
tt
h
eo
b
s
e
r
v
e
df
r
a
c
t
u
r
eo
r
i
g
i
n
a
もe
dfromt
e
n
s
i
l
es
p
l
i
t
t
i
n
g8
t
r
e
s
s
e
s
対s
t
e
n
c
eo
ft
h
e
s
e錦 町s
s
e
sw槌 叫 句
i
nt
h
er
e
g
i
o
nb
e
t
w
e
e
nt
h
en
o
t
c
ht
i
P
8(
F
i
g
.6
.
1
7
)
. Thee
showna
n
a
l
y
t
i
c
a
l
l
ybyR
o
t
s組 ddeB
o
r
8
t
.S
i
m
i
l
a
re
x
p
e
r
i
m
e
n
t
sbyS
w
a
r
t
ze
ta
l
.(
S
w
a
r
t
za
n
d
,1
9
8
7
)showedt
h
ep
r
i
n
c
i
p
a
lf
r
a
c
t
u
r
eo
r
i
g
i
n
a
.
t
i
n
gi
nt
h
i
sr
e
g
i
o
na
.
ndn
o
ta
tt
h
en
o
t
c
h
Taha
f
e
i
f
f
e
r(
1
9
8
5
b
)c
o
n
f
i
r
m
e
dt
h
ee
x
i
s
t
e
n
c
eo
f
t
i
p
s
. Thef
i
n
i
t
ee
l
e
m
e
n
ta
n
a
l
y
s
i
so
fBaza:叫 andP
s
i
g
n
i
f
i
c
a
n
tt
e
n
s
i
l
es
t
r
e
s
s
e
sb
e
t
w
e
e
nt
h
ec
r
a
c
kt
i
p
sb
u
tshowedt
h
ei
n
c
l
i
n
e
dp
r
i
n
c
i
p
a
lt
e
n
s
i
l
e
s
t
r
e
s
御成 t
h
ec
r
a
c
kt
i
p
st
obee
v
e
nh
i
g
h
e
r
.R民 e
n
tt
e
s
t
sa
tN
o
r
t
h
w
e
s
t
e
r
nU
n
i
v
e
r
s
i
t
y(K
回 e
凶
組 d
Bai
姐 t
,1
9
8
8
)i
n
d
i
c
a
t
e
dt
h
.
a
ti
ft
h
es
戸c
i
m
e
ni
nF
i
g6
.
1
7i
sl
o
a
.
deds
y
m
m
e
t
r
i
c
a
l
l
ybyP
/
2
a
te
a
c
hs
i
d
eo
fe
a
c
hc
r
a
c
k
m
o
u
t
hr
.
a
t
h
e
r出 叩 a
n
t
i
s
y
m
m
e
t
r
i
c
a
l
l
y
,t
h
e
ni
ts
p
l
i
t
sa
ta
.1
0
0
0n
o
t
muchh
i
g
h
e
rt
h
a
nt
h
emaximuml
o
a
.
df
o
rt
h
ea
n
t
i
s
y
m
m
e
t
r
i
cl
o
a
d
i
n
g自h
o
w
n
.T
h
i
ss
u
g
g
e
s
t
s
t
h
a
tt
h
es
p
l
i
t
t
i
n
gt
e
n
s
i
l
es
もr
e
s
sm
i
g
h
tbei
m
p
o
r
t
a
.
n
t.
Froms
i
z
ee
f
f
e
c
tt
由民 B
a
i
a
n
tandP
f
e
i
f
f
e
r(
1
9
8
6
)f
o
u
n
dt
h
eModeI
I台a
c
t
u
r
ee
n
e
r
g
yめ
bem
加 yt
i
m
e
sh
i
g
h
e
rt
h
a
nf
o
rMode1(
B
a
i
a
n
t組 dP
f
e
i
f
f
e
r,
1
9
8
5
a,
1
9
8
5
b
)
.
C
l
e
a
r
l
y
,aggregatei
n
t
e
r
1
0
c
kands
h
e
a
rf
r
i
c
t
i
o
np
l
a
yamajorr
o
l
ei
np
r
o
v
i
d
i
n
ge
n
e
r
g
y
h
et
i
pi
sduet
oMode1
t
or
e
s
i
s
tc
r
a
c
kp
r
o
p
a
g
a
t
i
o
ne
v
e
nt
h
o
u
g
ht
h
ed
r
i
v
i
n
ge
n
e
r
g
y8.tt
伺 db
emodeledp
r
o
p
e
r
l
ybya
p
p
r
o
p
r
i
a
t
eu
s
eo
fac
r
a
c
k
d
e
f
o
r
m
a
t
i
o
n
. Whethert
h
i
sc
a
ni
n
d
bandw
i
t
hi
n
c
l
i
n
e
dm
i
c
r
o
c
r
a
c
k
so
rf
r
i
c
t
i
o
ne
l
e
m
e
n
t
si
sn
o
tc
l
ea
.
ra
tp
r
e
s
e
n
t
.Att
h
esamet
i
m
e
,
t
h
e
r
e晶r
enogoodm
i
c
r
o
s
c
o
p
i
co
b
s
e
r
v
a
t
i
o
n
sa
v
a
i
l
a
b
l
et
oi
n
d
i
c
a
t
et
h
ep
r
e
s
e
n
c
e,
s
h
a
p
eands
i
z
e
田 o
ft
h
i
si
sc
l
e
a
rfromModeIb
e
n
d
i
n
gandt
e
n
s
i
o
nt
e
s
t
s
.
o
fap
r
o
c
e
s
sz
o
n
ea
l
t
h
o
u
g
he
v
i
d
e
n
a
.
r
dt
oModeno
rmixedmodec
r
a
ε
kp
r
o
p
a
g
a
t
i
o
n,
t
h
e
r
ei
st
h
u
sa
.g
e
n
e
r
a
l
l
a
c
ko
f
Withr
e
g
i
n
f
o
r
m
a
t
i
o
nandu
n
d
e
r
s
t
a
n
d
i
n
g
.T
h
e
r
ei
snog
e
n
e
r
a
la
g
r
田m
ent副 知 t
h
es
u
i
t
a
b
i
l
i
句o
ft
e
s
t
a
t
ae
v
a
l
u
.
a
t
i
o
no
rf
a
i
l
u
r
et
h
切 r
yt
op
r
e
d
i
c
tmixed-modec
r
a
c
kp
r
o
p
a
g
.
a
t
i
o
na
n
d
methods,d
f
r
a
c
t
u
r
ei
nc
o
n
c
r
e
t
e
.
組
12
1
10暗
唱
同
ヘ s
¥
ドミ
、
?
一
帽
6
4
2
6.12MaterialParametersforModeI
I
IFracture
I
ntheI
o
s
i
p
e
s
c
ut
y
p
emixed-modes
p
e
c
i
m
e
n
sa
l
r
e
a
.
dyd
i
s
c
u
s
s
e
d,
ap
e
r
{
.
舵 ta
n
t
i
s
y
n
u
n
e
t
r
y
i
sn
o
ta
c
h
i
e
v
e
d
.Thes
p
e
c
imenf
a
i
l
sw
i
t
hap
l
a
n
a
rc
r
a
ε
ki
nt
h
ec
r
o
s
ss
e
c
t
i
o
np
l
a
n
e
,
a
l
t
h
o
u
g
h
a
.
r
dt
ot
h
ec
r
o
s
ss
e
c
t
i
o
np
l姐 e
.H
o
w
e
v
e
r
,
t
h
ep
r
i
n
c
i
p
a
ln
o
r
m
a
ls
t
r
e
s
s
e
sw
e
r
ei
o
c
l
i
n
e
dw
i
t
hr
e
g
ap
e
r
f
e
c
ta
.
n
t
i
s
y
m
m
e
t
r
y
,r
e
q
u
i
r
e
df
o
rModeI
I
If
r
a
c
t
u
r
e,c
a
nbea
c
h
i
e
v
e
di
nac
y
l
i
n
d
r
i
c
a
l
.c
i
r
c
u
m
f
e
r
e
n
t
i
a
ln
o
t
c
hi
nt
h
em
i
d
d
l
e
,
s
u
b
j
e
c
t
e
dt
ot
o
r
s
i
o
n.Suchas
p
e
c
i
m
e
n
s
p
e
c
i
m
e
ow
i
t
ha
組dBaiant,
P
r
a
tandTabbara(
1
9
8
9
)
;F
i
g
.6
.
19
.
wasi
n
t
r
o
d
u
c
e
dbyB
a
i
a
n
tandP
r
a
t(
1
9
8
8
a
),
.
rs
p
e
c
i
m
e
n
so
fd
i
f
f
e
r
e
n
ts
i
z
e
s,
出
t
h
句
e
yf
o
u
n
dap
r
o
n
o
u
n
c
e
ds
釘
i
z
ee
f
.
“
ωt
T
e
s
t
i
n
gg
e
o
m
e
t
r
i
c
a
l
l
ys
i
m
i
la
姐 du
国s
e
吋di
比
t句
t
od
e
t
e
釘r
紅
n
I
凶
f
お
o
r
c
印e
,ぬe
yf
o
u
n
di
tt
obea
b
o
u
tt
h
r
e
e
t
i
m
e
sl
a
r
g
e
rt
h
a
nt
h
a
tf
o
rMode1
.However,t
h
e
y
o
b
s
e
r
v
edt
h
a
td
e
s
p
i
t
ep
e
r
f
e
c
t
l
ys
a
t
i
s
f
y
i
n
gt
h
ea
n
t
i
s
y
m
m
e
t
r
yo
fg
回 m
e
t
r
i
cc
o
n
d
i
t
i
o
n
s姐 d
I
Is
i
t
u
a
t
i
o
n,
aModeI
I
If
i
e
l
dm
i
g
h
tn
o
th
a
v
eb
e
e
n
l
o
a
d
i
n
ga
r
r
a
n
g
e
m
e
n
tr
e
q
u
i
r
e
df
o
r乱 ModeI
a
c
h
i
e
v
e
dI
o
c
a
l
l
yduet
ovolumee
x
p
a
n
s
i
o
no
ft
h
ef
r
a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
e
. Theya
l
s
oo
b
s
e
r
v
e
d
t
h
a
tt
h
eModeI
I
If
r
a
c
t
u
r
ee
n
e
r
g
yi
sa
p
p
a
r
e
o
t
l
yv
e
r
ys
e
n
s
i
t
i
v
et
ot
h
en
o
r
m
a
ls
t
r
e
s
sa
c
r
o
ss
t
h
ei
r
a
c
t
u
r
ep
r
o
c
邸 sz
o
n
e
.I
ta
p
p
e
a
r
st
h
a
tModeI
I
If
r
乱c
t
u
r
ee
n
e
r
g
yc
a
n
n
o
tbeama
も
'
e
r
i
a
l
c
o
n
s
t
a
n
t
jb
u
ti
tc
o
u
l
dbeam
a
t
e
r
i
a
lf
u
n
c
t
i
o
n,
d
e
p
e
n
d
i
n
g00t
h
et
r
a
n
s
v
e
r
s
en
o
r
m
a
ls
t
r
e
s
sa
o
d
p
o
s
s
i
b
l
yo
t
h
e
rva
.
r
i
a
.
b
l
e
s
.
y
,
s
i
m
i
l
a
.
rt
e
s
t
so
fXuandR
e
i
n
h
a
r
d
t(
1
9
8
9
)i
n
d
i
c
a
t
e
do
n
l
yan
e
g
l
i
g
i
b
l
es
I
z
ee
f
f
e
a
R怠 田 川l
I
Is
p
e
c
i
m
佃 s
.The.
r
モa
s
o
nf
o
rt
h
i
sd
i
f
f
e
r
e
n
c
ei
sn
o
tc
1
e
a
r
.
i
nt
o
r
s
i
o
n
a
lModeI
A,
. 0.312
C.
. 4.167
•
。
5
10
15
d/da
、
r
で¥ー0.3
¥
S
t
r
e
n
g
t
hG
r
i
t
e
r
i
o
n
。
¥
¥
"¥・ぞ
、;
;
/
\・~
トミー0.4
ー
、
0
'
)
¥
。
¥
、
~
-0
.
5
日 = 0.490
λa= 13.35
-0.6
0.4
0
.
0
1
.
0
1
.2
1
.
4
log(d/dQ. )
tR
.
e
g
r
e
s
討0
0P
l
o
tf
o
rM錨 u
r
e
m
e
n
to
fModeI
I
I(
a
f
t
e
r
F
i
g
.
6
.
1
9S
i
z
eE長 c
B
a
i
a
n
t,
P
r
a
t組 dTa
.
bb
訂
a
.
,1
9
8
9
)
104
105
s
l
r
e
n
a
l
h
c
r
i
l
e
r
i
o
n
-・恥白¥zb
C
h
a
p
t
e
r7
.FACTORSINFLUENCINGFRACTURE
PARAMETERS
7
.
1E
f
f
e
c
to
fLoadingRa
t
eandCreep
I
nnom
a
t
e
r
i
a
lc
a
nf
r
a
c
t
u
r
ehappeni
n
s
t
a
n
t
l
y
.C
r
a
c
k
st
a
k
eaf
i
n
i
t
et
i
m
et
of
o
r
mandt
o
.f
i
n
i
t
ed
i
s
t
佃 c
e
.Ther
e
a
s
o
ni
st
h
a
tt
h
ep
r
o
印 S
8o
fbondr
u
p
t
u
r
e
sp
r
o
c
田 d
s
p
r
o
p
a
g
a
t
eo
v
e
ra
a
ta
.c
位 t
a
i
nf
i
n
i
t
er
a
.
t
e
.Theba
.
s
i
cp
h
y
s
i
c
a
lthωryd
e
s
c
r
i
b
i
n
gt
h
ep
r
o
c
e
s
si
st
h
er
a
t
ep
r
o
c
e
s
s
切 r
y
,
i
nwhicht
h
er
a
t
eo
fbondr
u
p
t
u
r
白 i
sd
e
t
e
r
m
i
n
e
dbyt
h
ea
c
t
i
v
a
t
i
o
nen
町'
g
y
,
U
.The
t
h
p
r
o
ba
.
b
i
l
i
t
y(
o
rf
r
e
q
u
e
n
c
y
)o
fbondr
u
p
t
u
r
ed
e
p
e
n
d
sont
h
em
i
c
r
o
s
t
r
e
s
sl
e
v
e
la
.
tt
h
ec
r
a
.
c
kt
i
p
,
whichi
nt
u
r
ni
sd
e
t
e
r
m
i
n
e
dbye
i
t
h
e
rt
h
es
t
r
e
s
si
n
t
e
n
s
i
t
yf
a
c
t
o
r,
K],
o
rbyt
h
een
ぽ g
yr
e
l
e
ω
e
r
a
t
e
,G. Basedonthesephysicalobservations,
t
h
ef
o
l
1
owingapproxim
a
.
t
er
e
l
a
t
i
o
ne
n
s
u
e
s
叫
1
9
7
4
)
:
(
E
v
a= I
I
r
/
K
l
c
)
"
e
-U/R T
(
7
(K
.
1
)
c
!
!fost,peok i
n1
.
4s
+
+~suòl: peokIn"SOOs
~~ ~Iow. p
eak i
n1
4
0
0
0
s
*
キV
.
r
y$
I
o
w
.p
e
o
kI
n 253000$
0
.
1
0
.
1
B
r
i
t
f
l
e
n
e
s
s number,P=d/do
。,
Fig.7.1hmueofB
d
t
t
l
e
n
e
s
sNmber(Shia句 t
h
e町g
h
tont
h
eS
i
E
f
f
e
c
tC
u
r
v
e
)Duet
oI
n
c
r
e
a
.
s
i
n
gTimet
唱
。
,
(F
oFa
.
i
l
u
r
r
a
c
t
u
r
eS
p
e
c
i
m
e
n
s
I
.
Z
3
=
G
:的 .,,(I/T- へ
G~f
36.48 N/m
.
d
r
y=
Gf
f.O
70
J
・
20 c
=
・
﹁、、
恥
-h
4
E
u-U'Vw
¥HV
-ge
a
u
a
v
,
o
.
旬
-
、
、
、
(
b
)
0
.
0
,
.
.
﹄﹃
i
nwhicha= v
e
l
o
c
i
t
yo
fc
r
a
c
kt
i
pp
r
o
p
a
g
a
t
i
o
n,
T= a
b
s
o
l
u
t
etempera~ure, R = u
n
i
v
e
r
s
a
lg
a
s
c
o
n
s
t
a
n
t,
andI
I
,
n= e
m
p
i
r
i
c
a
lc
o
n
s
t
a
n
t
s
.Rοughly
,
n~ 3
0,
w
h
i
l
e110 c
a
nh
a
v
ev
e
r
yd
i
f
f
e
r
e
n
t
c
t
h
er
e
l
a
t
i
o
n
s
h
i
pb
e
t
w
e
e
nl
o
ga姐 dl
o
gK]s
h
o
u
l
dbel
i
n
e
a
r
.
v
a
l
u
e
s
.A
c
c
o
r
d
i
n
gt
oE
q
.7
.
1,
Butt
h
i
si
st
r
u
eo
n
l
ya
p
p
r
o
x
i
m
a
t
e
l
y
.E
x
p
e
r
i
m
e
n
t
si
n
d
i
c
a
t
eam
i
l
d
l
yn
o
n
l
i
n
e
a
rr
e
l
a
t
i
o
n
s
h
i
p
b
e
t
w
e
e
nlogKIandt
h
el
o
a
d
i
n
gr
a
t
e,
b
u
tnoc
l
e
a
ri
n
f
o
r
m
a
t
i
o
ne
x
i
s
t
sw
i
t
hregardωι
T
h
e
r
ee
x
i
s
to
t
h
e
ru
s
e
f
u
lr
e
l
a
t
i
o
n
8r
e
l
a
t
i
n
g七her
e
m
o
t
es
t
r
e
s
st
ot
h
er
e
m
o
t
es
t
r
e
s
sr
a
.
t
eo
r
r
e
m
o
t
es
t
r
a
i
nr
a
t
e,o
ru
n
l
o
a
d
i
n
gr
a
t
e(
e
.
g
.Re
i
n
h
a
r
d
,
七1
9
8
4
)
.Theya
r
ehoweverv
a
l
i
do
n
l
yf
o
r
a
.
b
l
e
.(
“
R
e
mote"a
削 h
el
o
c
a
.
t
i
o
D
S
c
e
r
t
a
i
np
a
r
t
i
c
u
l
a
rs
i
t
u
a
t
i
o
n
sanda
r
en
o
tg
e
n
e
r
a
l
l
yi
n
t
e
r
p
r
e
t
a
.
c
k8
0t
h
a
tt
h
es
も
r
e
s
si
sn
o
ts
i
g
n
i
f
i
c
a
n
t
l
y
a
f
f
e
c
t
e
dbyt
h
ec
r
a
c
k
)
.
5
0f
a
rfromt
h
ec
r
JohnandShah(
1
9
8
5,
1
9
8
6,
1
9
8
7,
1
9
8
9
a
ja
l
s
oJohn,
Sh
a
.
h組 dJenq,
1
9
8
7
)a
p
p
l
i
e
dt
h
et
w
φ
r
a
c
t
u
r
et
e
s
t
sc
o
n
d
u
c
t
e
da
.
td
i
f
f
e
r
e
n
t
p
a
r
a
m
e
t
e
rf
r
a
c
t
u
r
emodel七oMode1andMixedModef
5
t
r
a
i
nr
a
t
e
s
. Theyc
ο
n
c
l
u
d
e
d七h
a
tK
l
oi
sr
a
t
e
-i
n
d
e
p
e
n
d
e
n
tb
u
tt
h
a
t8
踊 w
e
l
lωthe
CTOD,
a
.
c
kl
e
n
g
t
ha
tp
e
a
kl
o
a
d,
d
e
c
r
e
a
s
e田 t
h
el
o
a
d
i
n
gr
a
t
ei
n
c
r
e
a
s
e
s
.
e
f
f
e
c
t
i
v
ec
r
I
ts
h
o
u
l
da
l
s
obeo
h
s
e
r
v
e
dt
h
a
tt
h
ee
f
f
e
c
t
so
fc
r
a
c
kg
r
o
w
t
hr
a
t
eandl
o
a
d
i
n
gr
a
t
ea
r
e,
i
n
t
h
e
i
rp
h
y
s
i
c
a
lmech
組 i
s
m,
r
e
l
a
t
e
dt
ot
h
es
t
r
e
s
sc
o
r
r
o
s
i
o
ne
f
f
e
c
ti
nf
r
a
c
t
u
r
e
.
I
na
n
a
l
y
z
i
n
gt
h
ef
r
a
c
t
u
r
eb
e
h
a
v
i
o
ro
fs
p
e
c
i
m
田 s
o
rs
t
r
u
c
t
u
r
e
su
n
d
e
rv
a
r
i
o
u
sl
o
a
d
i
n
gr
a
t
e
s,
t
h
ephenomenao
ff
r
a
.
c
t
u
r
e姐 dc
r
田 pa
r
ei
n
s
e
p
a
r
a
h
l
e
.Ther
e
s
u
l
t
sa
r
ei
n
f
l
u
e
n
c
e
dbyt
h
es
h
o
r
t
terml
i
n
e
a
rv
i
s
c
o
e
l
a
s
t
i
c
i
t
yo
f∞n
c
r
e
t
e,
whichi
sq
u
i
t
ep
r
o
n
o
u
n
c
e
d,
andi
nt
h
eh
i
g
hs
t
r
e
s
s
r
e
g
i
o
n
sn
e
a
rt
h
ef
r
a
c
t
u
r
ef
r
o
n
t,
byt
h
ea
d
d
i
t
i
o
n
a
ln
o
n
l
i
n
e
a
.
rc
r
田 po
f∞n
c
r
e
t
e
.F
u
r
t
h
e
r
m
o
r
e
,
t
h
es
t
r
a
i
n
-s
o
f
t
e
n
i
n
gbeha
.
v
i
o
ri
nt
h
ef
r
a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
ei
sl
i
k
e
l
yt
obee
v
e
nmoret
i
m
e
d
e
p
e
n
d
e
n
tt
h
組
t
h
en
o
n
l
i
n
e
a
rc
r
田 pi
nt
h
eh
a
r
d
e
n
i
n
gh
i
g
hs
t
r
e
s
sr
a
n
g
e
.R
e
c
e
n
ts
t
u
d
i
e
so
f
.
tv
a
r
i
o
u
sr
a
t
e
so
fl
o
a
d
i
n
g(
B
a
i
a
n
t飢 dGettu,1
9
8
9
)i
n
d
i
c
a
t
eav
e
r
ys
t
r
o
n
g
t
h
es
i
z
ee
f
f
e
c
ta
s
,田po
rs
t
r
e
s
sr
e
l
日 以i
o
ni
nt
h
es
t
r
a
i
n
s
o
f
t
e
n
i
n
gr
a
n
g
ewhichp
r
e
v
a
i
l
si
nt
h
ef
r
a
.
c
t
u
r
ep
r
o
c
e
s
c
r
z
o
n
e
.T
h
i
sphenomenons
e
e
m
st
or
e
d
u
c
et
h
es
i
z
eo
ft
h
ef
r配 t
u
r
ep
r
o
c
e
s
sz
o
n
e出 t
h
el
o
.
a
d
i
n
g
e
a
.
s
e
s
),
w
i
t
ht
h
e∞n
s
e
q
u
回目 t
h
a
.
tl
o
n
g
-七i
r
n
r
a
t
ed
i
m
i
n
i
s
h
e
s(
i
.
e
.t
h
et
i
m
et
of
a
i
l
u
r
ei
n
c
r
e
a
.
st
i
cf
r
a
c
t
u
r
em
e
c
h
a
n
i
c
st
h
a
ns
h
o
r
t
t
i
m
er
e
s
p
o
n
s
e
.T
h
i
s
r
e
s
p
o
n
s
emightbec
l
o
s
e
rt
ol
i
n
e
a
re
l
s
田 n
tda
.
t
ao
fBaza
瓜 叩 dG
e
t
t
u(
1
9
8
9
)i
nF
i
g
.・
7
.
1i
nwhicht
h
er
e
s
u
lt
i
si
l
l
u
s
t
r
a
t
e
dw
i
t
hr
e
2
.
5
d
QO
fZO
tlf
T
e
r
n
p
eγa
t
u
T
e(
・
0
ZDO
:U
f
~jg. ~.2 F
r
a
.
c
t
u
r
eEnergyDependenceonTemp
釘 a
t
u
r
eandM
o
i
s
t
u
r
e
Co
n
d
i
t
i
o
n
s(
a
f
t
e
rBa
.
Z
a
n
t姐 dPr
a
.
t
,
1
9
8
8
)
1
0
6
1
0
7
f
o
rd
i
f
f
e
r
e
n
tt
i
m
e
st
of
a
i
l
u
r
e
,
p
l
o
t
t
e
da
s
σN v
s
. もheb
r
i
t
t
l
e
n
e
s
snumbero
fB
a
z
a
n
t(
S
e
c
.5
.
1
and5
.
2
),
c
a
nbes
e
e
nt
os
r
u
f
tt
ot
h
er
i
g
h
ta
st
h
et
i
m
ei
n
c
r
e
a
s
e
s
.
a
.
rf
i
n
d
i
n
g
s,
i
n
d
i
c
a
t
i
n
gt
h
a
tt
h
ep
r
o
c
e
s
sz
o
n
es
i
z
e姐 df
r
a
c
t
u
r
ee
n
e
r
g
ydeαeaseおもh
e
S
i
m
i
l
t
i
m
et
of
a
i
l
u
r
ei
n
c
r
e
a
s
e
s,
ha
.
veb田 nr
e
c
e
n
t
l
yr
e
p
側 e
dbyWittm
.
a
nne
ta
l
.(
1
9
8
7
),
a
l
t
h
o
u
g
h
e
-i
n
d
e
p
e
n
d
e
n
tf
r
a
c
t
u
r
ec
h
a
.
r
a
c
t
e
r
i
s
t
i
c
s
.
t
h
i
ss
t
u
d
yd
i
dn
o
td
e
a
lw
i
t
hs
iz
L
e
sti
tbep
e
r
c
e
i
v
e
dt
h.
a
tt
h
ea
f
o
r
e
m
e
n
も
i
o
n
e
df
i
n
d
i
n
g
so
fBa
.
z
a
n
ta
.
n
dG
e
t
t
uc
o
n
t
r
a
d
i
c
t
も
h
o
s
eo
fJohna
.
n
dShah
,
ones
h
o
u
l
dr
e
a
l
i
z
et
h
a
tJohnandShah'
smethodd
e
t
e
r
m
i
r
悶
b
a
.
a
c
ho
ft
h
ef
r
a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
ewheret
h
e
r
ei
ss
i
g
n
i
f
i
c
姐 td
i
s
p
l
a
c
e
m
e
n
td
u
e
s
i
c
a
l
l
yt
h
er
e
t
om
i
c
r
o
c
r
a
c
k
i
n
g
.Buta
.
tt
h
el
o
c
a
t
i
o
n
sw
h
e
r
et
h
em
i
c
r
o
c
r
a
c
k
sa
r
eopenedmostw
i
d
e
l
y
,
t
h
e
.
vea
.
l
r
e
.
a
dyr
e
l
a
x
e
dt
oz
e
r
o
.T
h
i
sp
o
r
t
i
o
no
ft
h
em
i
c
r
o
c
r
a
c
k
i
n
gz
o
n
ewouldn
o
t
s
t
r
e
s
smayha
bema
.
n
i
f
e
s
t
e
di
nt
h
ep
r
o
c
e
s
sz
o
n
es
i
z
eo
b
t
a
i
n
e
dbyt
h
es
i
z
ee
f
f
e
c
tmethod,
whiche
v
e
n
t
u
a
l
l
y
回 s
m
i
t
もe
d
.I
ti
s
g
i
v
e
st
h
el
e
n
g
t
ho
ft
h
ez
o
n
eo
v
e
rw
h
i
c
hs
i
g
n
i
f
i
c
a
n
tb
r
i
d
g
i
n
gs
t
r
e
s
s
e
sa
r
et
r
duet
os
t
r
e
s
sr
e
l
a
x
a
t
i
o
n,
t
h
i
sl
e
n
g
t
hr
e
p
r
e
s
e
n
t
so
n
l
yasm
a
.
l
l
p紅 to
ft
h
e
q
u
i
t
ec
o
n
c
e
i
v
a
b
l
et
h
a
t,
l
e
n
g
出 o
ft
h
em
i
c
r
o
c
r
a
c
k
i
n
gz
o
n
e, 回 o
b
t
a
i
n
e
dbyJohnandS
h
a
h
'
smethod. Morer
i
g
o
r
oU8
s
t
u
d
i
e
sa
r
en伺 ded,
h
o
w
e
v
e
r
.
.
r
e
Thee
x
p
e
r
i
m
e
n
t
a
lt
e
c
h
n
i
q
u
e
sf
o
rm
e
a
s
u
r
i
n
gf
r
a
c
t
u
r
ep
r
o
p
e
r
t
i
e
sa
th
i
g
hl
o
a
d
i
n
gr
a
t
e
sa
d
i
s
c
u
s
s
e
dbyJohnandShah(
1
9
8
5
),Re
i
n
h
a
r
d
t(
1
9
8
4
),M
i
n
d
e
s
s(
1
9
8
4
),ando
t
h
e
r
s
.
7
.
2E
f
f
e
c
tofTemperatureandHumidityonFr
actureEnergy
,
と
W1
R
U一
n
2一
吋,
一
一
T
h
i
se
f
f
e
c
ti
sl
i
k
e
l
yt
oh
a
v
ea
tl
e
a
s
もp
a
r
t
l
yt
h
esamep
h
y
s
i
c
a
lmechanisma
st
h
ee
f
f
e
c
to
f
c
r
a
c
kv
e
l
o
c
i
t
yonl
o
a
d
i
n
gr
a
t
e
.Ther
a
t
eo
fbondr
u
p
t
u
r
e
sd
e
p
e
n
d
sn
o
to
n
l
yont
h
es
もr
e
s
s1
肝 e
l
b
u
ta
.
l
s
oont
e
m
p
e
r
a
t
u
r
e
,a
c
c
o
r
d
i
n
gt
oもhea
c
t
i
v
a
t
i
o
ne
n
e
r
g
yt
h
e
o
r
y
.U
s
i
n
gE
q
.7
.
1a
st
h
e
p
o
i
n
to
fd
e
p
a
r
t
u
r
e,
B
a
z
a
n
ta
.
n
dPr
a
.
t(
1
9
8
8
b
)ha
.
veshownt
h
a
t,
a
c
c
o
r
d
i
n
gt
ot
h
ea
c
t
i
v
a
.
t
i
o
n
e
n
e
r
g
yt
h
e
o
r
y
,
t
h
ef
r
a
c
t
u
r
ee
n
e
r
g
yo
fc
o
n
c
r
e
t
es
h
o
u
l
ddependont
h
ea
b
s
o
l
u
t
etempera
もu
r
e
a
sf
o
l
l
o
w
s
:
(
7
.
2
)
Gf= ~exp(ヱーユ)
, ~~nT
To
"
i
nwhich九 = r
e
f
e
r
e
nc
et
e
m
p
e
r
a
t
u
r
e,
G~ = f
r
a
c
t
u
r
ee
n
e
r
g
ya
.
tr
e
f
e
r
e
n
c
etemper
a
.
t
u
r
e,
a
n
d
γ =c
o
n
s
t.
a
n
t
.E
q
.7
.
2a
g
r
田 sw
e
l
lw
i
t
ht
田も r
e
s
u
l
t
so
ft
h
es
i
z
ee
f
f
e
c
tmethodo
b
t
a
i
n
e
df
o
r
v
a
r
i
o
u
stemper
a
.
t
u
r
e
sf
r
o
mr
∞m upt
o2
0
00C
js
田 F
i
g
.7
.
2
. Somer
e
s
u
l
t
sha
.
vea
l
s
ob
e
畑
o
b
t
a
i
n
e
df
o
rt
e
m
p
e
r
a
t
u
r
e
si
nt
h
ef
r
田 z
i
n
gr
a
n
g
e,
i
n
d
i
c
a
t
i
n
gt
h
et
r
e
n
di
nF
i
g
.7
.
2t
oc
o
n
t
i
n
u
e
o
downt
oa
b
o
u
t-20
C,叩dt
h
e
nr
e
v
e
r
s
i
n
g(
M
a
t
u
r
a
.
n
a
.e
ta
.
l
.
,1
9
8
8
)
.
As
i
m
i
l
a
.
rd
e
p
e
n
d
e
n
c
eo
fG onTwase
x
p
e
r
i
m
e
n
ta
.
l
l
yd
e
t
e
r
m
i
n
e
dbyBrameshuber(
1
9
8
8,
1
9
8
9
)who,
h
o
w
e
v
e
r,
u
s
e
dd
i
f
f
e
r
e
n
tf
o
r
m
u
l
a
.
s
もod
e
s
c
r
i
b
et
h
eo
b
s
e
r
v
e
dt
r
e
n
d
.
鉦
'
e
c
tt
e
s
t
sw
e
r
ec
o
n
d
u
c
t
e
da
.
tv
a
r
i
o
u
st
e
m
p
e
r
a
t
u
r
e
sons
.
a
t
u
・
Thea
f
o
r
e
m
e
n
t
i
o
n
e
ds
i
z
ee
r
a
t
e
d(
w
e
t
)∞n
c
r
e
t
e鎚 w
e
l
la
son∞n
c
r
e
t
ew
h
i
c
hwasd
r
i
e
db
e
f
o
r
et
h
et
e
st
.Thed
i
f
f
e
r
e
n
c
e
d
i
t
i
o
n
sappe
a
.
r
e
di
n
s
i
g
n
i
f
i
c
a
n
ta
troomtemper
a
.
t
u
r
e,
however
,
a
th
i
g
h旬 mi
nh
u
m
i
d
i
t
y∞n
p
e
r
a
t
u
r
e
st
h
ew
e
tc
o
n
c
r
e
t
eh.
a
damuchl
o
w
e
rf
r
a
c
t
u
r
ee
n
e
r
g
yt
h
a
nもhedryc
o
n
c
r
e
t
e(
F
i
g
.7
.
2
Z
必
ね
a
r
叫 a
ndPr
悶a
抗
,
ム
A1
ω
9
邸
8
8
吋
b
)
.Assumingl
i
n
e
a
rd
e
p
e
n
d
e
n
c
eOst
h
es
p
e
c
i
f
i
cw
a
t
e
rc
o
n
t
e
n
t
,
a
f
t
e
rBa
p
a
r
a
m
e
t
e
r"
1wa
.
se
x
p
r
e
s
s
e
da
s
(
7
.
3
)
"
I
1ー γ
0
)
γ=γ0+(
i
nw
h
i
c
hω =s
p
e
c
i
f
i
cw
a
t
e
rc
o
n
t
e
n
t,Wl i
t
sv
a
l
u
ea
ts
a
t
u
r
a
t
i
o
n
,andγ
0,γ
1= v
a
l
u
eo
fγ
0
i
ndryandwetc
o
n
d
i
t
i
o
n
s(
f
o
rもhe∞n
c
r
e
t
e同 t
e
d,
i
tw硝 f
o
u
n
dt
h
a
t"
1
0~ 6
0
0
[ (,
組d
1
1~ 1
9
0
0
0[
(
)
.
i
nwruchv
i
s
c
o
e
l
a
s
t
i
c
Thee
f
f
e
c
to
ft
e
m
p
e
r
a
t
u
r
ei
sq
u
i
t
ep
r
o
n
o
u
n
c
e
di
np
o
l
y
m
e
rc
o
n
c
r
e
t
e
s,
i
t
yo
ft
h
epolymerb
i
n
d
e
ri
snod
o
u
b
ta
.
ni
m
p
o
r
ta
.
n
tf
a
c
t
o
r(
V
i
p
u
l
姐a.n
danandDh
a
.
r
m
a
r
a
j
a
n,
1
9
8
7,
1
9
8
8,
1
9
8
9
a
丸c
)
.
7
.
3E
f
f
e
c
tofCyclicLoading
R
β
p
e
a
t
e
dl
o
a
d
i
n
gt
e
n
d
sもog
r
a
.
dua
.
l
l
yi
n
c
r
e
a
s
et
h
ec
r
a
c
kl
e
n
g
t
he
v
e
ni
ft
h
emaximum
s
t
r
e
s
si
n
t
e
n
s
i
t
yf
a
c
t
o
ro
ft
h
ec
y
c
l
e
si
sw
e
l
lb
e
l
o
wi
t
sc
r
i
t
i
c
a
lva
.
l
uef
o
rs
t
a
t
i
cl
o
a
d
i
n
g
. The
r
a
.
t
eo
fi
n
α
e
a
s
eo
ft
h
ec
r
a
c
kl
e
n
g
t
h
,a,
w
i
t
ht
h
enumbero
fcyd
,
四 N,
h
a
sb
e
e
nshownf
o
r
.
p
p
r
o
x
i
m
a
t
e
l
yd
e
s
c
r
i
b
e
dbyt
h
eP
a
r
i
slaw(
P
a
r
i
sa
.
ndErdogan,
1
9
6
3
)
:
manym
a
t
e
r
i
a
.
l
st
obea
da/dN= C(!
:
J
.Kd
[
(
I
c
)
r
ヘinwhichC andm areempirica
.
lc
o
n
s
t
a
n
t
s
. AlthoughE
q
.7
.
4i
s
.f
u
n
d
a
m
e
n
t
a
lp
h
y
s
i
c
a
.
lb
a
s
i
ss
u
c
ha
sE
q
.7
.
1,
i
ti
s
e
s
s
e
n
t
i
a
l
l
ye
m
p
i
r
i
c
a
landd
o
e
sn
o
thavea
n
e
v
e
r
もh
e
l
e
s
s組 e
q
u
a
t
i
o
no
fr
a
t
h
e
rg
e
n
e
r
a
la
p
p
l
i
c
a
.
b
i
l
i
t
y
.F
o
rc
o
n
c
r
e
t
e
,
h
o
w
e
v
e
r,
i
tw剖 f
o
u
n
d
(
Ba.Z叩t組 dXu,
1
9
8
9
)t
h
a
td
e
v
i
a
t
i
o
n
sf
r
o
mP
a
r
i
sl
a
wa
r
i
s
eduet
ot
h
es
i
z
ee
f
f
e
c
t
.Theyc
姐
b
ed
脱 出e
dbyt
h
ef
o
l
l
o
w
i
n
gg
e
n
e
r
a
l
i
叫 i
o
n(B
a.Z姐t
叩 dXu
,
1
9
8
9
)
:
表=
c
(
2
5
)
m,
出
i
t
h[
(
1
m= ι (
r
/
2
(
7
.
4
)
w
h
e
r
e(
1
0i
sa
.c
o
n
s
t
a
n
th
a
v
i
n
gas
i
m
i
l
a
rmeaningt
od
nt
h
es
i
z
ee
f
f
e
c
tl
a
.
w(
E
q
.1
.4
),
[
(
1
mi
st
h
e
oi
a
.
r
e
n
t(
s
i
z
e
d
e
p
e
n
d
e
n
t
)f
r
a
c
t
u
r
et
o
u
g
h
n
e
s
sc
a
.
l
c
u
l
a
t
e
dfromt
h
epeakl
o
a
d~f t
h
es
p
e
c
i
m
佃
a
p
p
組 dL
EFMr
e
l
叫i
o
n
s,叩ddi
s山 es
i
z
eo
ft
h
es
p
e
c
i
m
e
n
.T
e
s
tshowedt
h
a
td
0
d
hen
ow
o~ 1
t
h
e1
0
0
0i
sc
y
c
l
e
dbetwe
回 O阻 d8
0%o
f出 emaximuml
o
a
df
o
rm
o
n
o
t
o
n
i
cl
o
a
d
i
n
g
.E
q
.7
.
4
,
s
h
o
w
ni
nF
i
g
.7
.
3byt
h
et
h
r
e
es
もr
a
抱h
tl
i
n
e
s
,
i
ss
e
e
nt
oa
g
r
e
eq
u
i
t
ew
e
l
lw
i
t
ht
h
eda
.
t
.
af
o
r
トp
o
i
n
t
b
e
n
dn
o
t
c
h
e
dbeamso
fs
i
z
e
s1
:
2
:
4
. P.
a
r
i
sl
a
w,
byc
o
n
t
r
a
s
t,
y
i
e
l
d
si
nF
i
g
.7
.
3a
.
t
h
r
e
e
s
i
n
g
l
es
t
r
a
i
g
h
tl
i
n
e.
a
ndc
a
n
n
o
td
e
s
c
r
i
b
et
h
eo
b
s
e
r
v
e
ds
i
z
氾 e
f
f
e
c
t
.
a
b
o
u
tP
a
r
i
sl
a
wi
s出 a
tt
h
er
a
t
eo
fc
r
a
ι
ke
x
t
e
n
s
i
o
nd
o
e
sno
もd
epend
Ani
n
t
e
r
e
s
t
i
n
gp
o
i
n
t.
onぬ.
eu
p
p
e
ra
.
n
dl
o
w
e
rl
i
m
i
t
i
n
gv
a
l
u
e
so
ft
h
es
t
r
e
s
si
n
t
e
n
s
i
t
yf
a
c
t
o
r,
b
u
ta
.
pproxim
a
.
t
e
l
yo
n
l
y
四 回,
!
:
J
.
J
(
1・T
h
i
smigh
もb
ee
x
p
l
a
.
i
n
e
dbya
.8
0
r
to
fa
.shakedowni
nt
h
ef
r
a
c
t
u
r
e
O
Dt
h
e
i
rd
i
f
f
e
r
,
w
i
t
hd
e
v
e
l
o
p
m
e
n
to
fr
e
s
i
d
u
a
ls
t
r
e
s
s
e
sa
f
t
e
ranumbero
fc
y
c
l
e
s
.V
a
r
i
o
u
so
t
h
e
r
p
r
o
c
e
s
sz
o
n
e
i
n
t
e
r
e
s
t
i
n
gr
回 u
l
t
sonc
y
c
l
i
cf
r
a
c
t
u
r
e'
o
fc
o
n
c
r
e
t
e
,
w
r
u
c
hp
e
r
t
a
i
nt
op
a
r
t
i
c
u
l
a
rt
回 t
c
o
n
d
i
t
i
o
n
s,
h
a
.
v
eb
e
e
no
b
t
a
i
n
e
d
,f
o
rex
a
.
mple
,bySwartz,Hu相 dJ
O
l
悶
(
1
9
7
8
),S
w
a
r
t
z
,Hu組 ga
.
n
dHu
(
19
8
2
),
S
w
a
r
t
zandGo(1984)~ P
e
r
c
U
k
a
r
i
s組 dCalomino(
1
9
8
7
),
ando
t
h
e
r
s
.
Thei
n
t
e
r
p
r
e
t
a
t
i
o
no
ft
h
e企 Krvaluewhich8
h
o
u
l
dbes
u
b
s
t
i
t
u
t
e
di
nE
q
.7.
4i
ss
u
b
j
e
c
tt
o
ぬ~e q
u
e
s
t
i
o
nw
i
t
hr
e
g
a
r
dt
ot
h
es
i
z
ee
f
f
e
c
tont
h
ev
a
l
u
eo
fK
l
c
.Thev
a
l
u
eo
f!
:
J
.Kls
h
o
u
l
dbe
c
a
l
c
u
l
a
t
e
df
o
rt
h
ee
l
a
s
t
i
c
a
.
ll
ye
q
u
i
v
a
l
e
n
te
f
f
e
c
t
i
v
ec
r
a
c
kl
e
n
g
t
ha= ao+c,
whichmeans!
:
J
.[(l
s
h
o
u
l
dbem
o
d
i
f
i
e
dw
i
t
hr
e
s
p
e
c
もt
ot
h
el
e
n
g
t
ho
fc
r
a
c
ke
x
t
e
n
s
i
o
nf
r
o
Int
h
en
o
t
c
h(
F
i
g
.4
.
2
)・
108
1
0
9
ロ d
.1
.
!
5i
n
.
.
句 C_
1
6
.
7
・
1
.
0{・ d.
.
3i
n
.
,
同c
.
・
1
8
.
2
+“叫川
n-10.
6
-
Chapter8
.EFFECTOFREINFORCE
乱
1
:ENT
2
.
0
2
司
ち.
1E
f
f
e
c
tofReinforcingS
t
e
e
lBars
e
乏-3
.
0
。
明
向:
4
.
0
5
.
0
t
.
1
1
.3
1
.
7
1
.5
109(dKI
/
KI
f
)
F
i
g
.
7
.
3F
r
a
c
t
u
r
eo
fT
h
r
e
e
トP
o
i
n
tBendSpecim
租 s
unde
rC
y
c
l
i
cL
o
a
d
i
n
g
(
0
)
(
b
)
.0
II I f I I
(
c
)
:
1
L
.一
=一6一 1-P
a-U1-p+YEP
・
6
Whenc
o
o
c
r
e
t
ei
sr
e
i
n
f
o
r
伺 db
yar
e
g
u
l
a
rg
r
i
do
fr
e
i
n
f
o
r
c
i
n
gb
a
r
s,
i
ti
so
f
t
e
na
p
p
r
o
p
r
ia
.
t
e
.
rgridωoneh
e
t
e
r
o
g
e
n
o
u
sc
o
m
p
o
s
i
t
ema
.
t
e
r
i
a
.
lwhichi
s
ιc
o
n
s
i
d
e
rc
o
n
c
r
e
t
ew
i
t
ht
h
eba
辻Cf
c
r
h
o
m
o
g
回 目USo
nal
a
r
g
e
rs
c
a
l
e
.O
b
v
i
o
u
s
l
y
,
i
ns
u
c
hac
o
m
p
o
s
i
t
e
,
t
h
eb
a
r
si
n
h
i
b
i
tt
h
e
r
o
p
a
g
a
t
i
o
no
fc
r
a
c
k
s
.I
ft
h
e
r
ew
e
r
enobonds
l
i
p,
t
h
e
nt
h
eo
p
p
o
s
i
t
ef
a
c
e
so
f
r
m
a
t
i
o
n回 dp
,
:
r
a
c
kw
o
u
l
dn
o
tbea
b
l
eもoo
p
e
na
p
a
r
t,
ands
ot
h
e
r
ewouldbenof
r
a
c
t
u
r
eu
n
t
i
lt
h
eb
a
r
f
P
.
a
k
s
.T
h
i
ss
i
m
p
l
ec
o
n
s
i
d
e
r
a
t
i
o
ni
m
m
e
d
i
a
t
e
l
yc
l
a
r
i
f
i
e
st
ha
.
tbonds
l
i
pmustt
a
k
ep
l
a
c
eand
凶 t
b
et
a
k
e
ni
n
t
oa
c
c
o
u
n
t
. Thep
r
o
p
e
r
t
yw
h
i
c
hm抗 t
e
r
si
st
h
es
t
i
f
f
n
e
s
so
ft
h
ec
o
n
n
e
c
t
i
o
n
関 1
ba
.
r
sb
e
t
w
e
e
nt
h
eo
p
p
o
s
i
t
ef
a
c
e
so
ft
h
ec
r
a
c
ko
rc
r
a
.
c
kb
a
n
d
.
u
v
i
d
e
dbyt
h
es
t
A
s
s
u
m
i
n
gt
h
eu
l
t
i
m
a
t
ebonds
t
r
e
s
s
e
sUb t
obea
p
p
r
o
x
i
m
a
t
e
l
yc
o
n
s
t
a
n
t,
t
h
eb
a
rs
t
r
e
s
s,
.
'
1
i
c
hi
se
q
u
a
lt
o山 叫 t
h
ec
r
a
c
kc
r
o
s
s
i
n
g,
d
e
c
r
e
ωe
sl
i
n
e
a
r
l
yo
v
e
rt
h
ed
i
s
t
a
n
c
eo
fbonds
l
i
p
,
,
_o
ne
a
c
hs
i
d
eo
ft
h
ec
r
a
.
c
k
. Fromt
h
i
ss
t
r
e
s
sd
i
s
t
r
i
b
u
t
i
o
n,
o
n
ec
a
ne
s
t
i
m
a
t
et
h
er
e
l
a
t
i
v
e
.
ra
g
a
i
n
s
tc
o
n
c
r
e
t
eo
v
e
rl
e
n
g
t
h2
L
.
.F
u
r
t
h
e
r
m
o
r
e
,
i
na
.ma
.
c
r
o
s
∞p
i
c
a
l
l
y
p
l
a
c
e
m
e
n
to
ft
h
eba
'
I
l
f
o
r
mo
rq
u
a
s
i
u
n
i
f
o
r
mf
i
e
l
d,
t
h
ef
o
r
c
ec
a
r
r
i
e
dbyt
h
eba
.
ra
tt
h
ec
r
a
c
kc
r
o
s
s
i
n
gmust
o
r
c
ep
e
rb
a
rc
a
.
r
r
i
e
dj
o
i
n
t
l
ybyc
o
n
c
r
e
t
e組 ds
t
e
e
la
tl
o
c
a
.
t
i
o
n
sbeyondt
h
e
u
i
l
i
b
r
a
t
e七hef
,
n
ds
l
i
p1
田 g
t
h
.F
r
omt
h
e
s
ec
o
n
d
i
t
i
o
n
si
tf
o
l
l
o
w
st
h
a
t(
B
a
z
姐 t
andC
e
d
o
l
i
n,
1
9
8
0
;BaZa
n
t
;
:
l
i
c
)
:
l' I
F
i
g
.8
.
1E
f
f
e
c
to
fBondS
l
i
po
fRe
i
n
f
o
r
c
i
o
gB
a
r
sC
r
o
s
s
i
n
gC
r
a
c
kBands
o
rC
r
a
c
k
s
S
(
8
.
1
)
"
"
h
i
c
hp=s
t
e
e
lr
a
t
i
o
,
n士 s
t
e
e
l
t
o
c
o
n
c
r
e
t
er
a
t
i
oo
ft
h
ee
l
a
s
t
i
cm
o
d
u
l
i,
andA6=b
a
rc
r
o
s
s
もee
l
e
m
e
n
ta
n
a
l
y
s
i
s,
onemayf
u
r
t
h
e
ri
m
a
g
i
n
eane
q
u
i
v
a
l
e
n
t
'
o
i
o
n
.F
o
rt
h
ep
u
r
p
o
s
eo
ff
i
n
i
w
h
i
c
hi
s叩 c
h
o
r
e
dt
oc
o
n
c
r
e
t
ea
tt
h
en
o
d
e
so
ft
h
ef
i
n
i
t
ee
l
e
m
e
n
tmeshandh
a
snobond
"'",~es b
u
tas
h
o
r
t
e
re
q
u
i
v
a
l
e
n
tf
r
e
e
-s
l
i
pl
e
n
g
t
h
.Anothers
i
m
p
l
ef
o
r
m
u
l
af
o
rt
h
i
sl
e
n
g
t
hh剖
.四 d
e
v
i
s
e
dbyBaZa
n
tandC
e
d
o
l
i
n(
1
9
8
0
)
.
¥d
e
t
a
.
i
l
e
dd
e
s
c
r
i
p
t
i
o
no
fc
r
a
c
k
i
n
gi
nr
e
i
n
f
o
r
c
e
dc
o
n
c
r
e
t
er
e
q
u
i
r
田 am
ores
o
p
h
i
s
t
i
c
a
t
e
d
,
:
.
'y
;S白 e
.
g
.P
i
j
a
u
d
i
位ー C
abotandMa
.
z
a
r
s(
1
9
8
9
),
andB
r
e
y
s
s
eandMazars(
1
9
8
8
)
.These
回 t
i
cmodelf
o
rs
t
e
e
lb
a
r
s
.
l
i
e
su
s
e
ddamaget
h
e
o
r
yf
o
rc
o
n
c
r
e
t
ecombinedw
i
t
hane
l
a
s
t
o
p
l
r
e
s
u
l
t
sw
e
r
ei
n
t
e
r
e
s
t
i
n
g,
g
i
v
i
n
gi
n
f
o
r
m
a
t
i
o
nont
h
ee
v
o
l
u
t
i
o
no
fdamagez
o
n
回a.r
ound
C '
,
8
.d
e
v
e
l
o
p
m
e
n
to
fc
r
a
c
k
sandt
h
eg
l
o
b
a
lr
e
s
p
o
n
s
eo
ft
h
es
t
r
u
c
t
u
r
e
. Bonds
t
r
e
s
s
e
s姐 d
民dt
obea
l
s
oi
n
c
l
u
d
e
di
nad
e
t
a
i
l
e
da
n
a
l
y
s
i
s
;t
h
i
sh
a
sb
e
e
ndonebyP
i
j
a
u
d
i
e
r
Cabot
i
l
9
8
9
)i
nt
e
r
m
so
fan
o
n
l
o
c
a
ldamagemodelw
h
i
c
ha
l
s
oy
i
e
l
d
st
h
es
i
z
ee
f
f
e
c
t
.
~ F
r
a
c
t
u
r
ei
nFiber-ReinforcedConcrete
~ ~,nd ~lip o
!t
h
ef
i
b
e
r
si
saphenomenons
i
m
i
l
a
rt
ot
h
es
l
i
po
fr
e
i
n
f
o
r
c
i
n
gb
a
r
s,
anda
g
a
.
i
n
m
a
j
o
re
f
f
e
c
tonf
r
a
c
t
u
r
e,
a
l
o
n
gw
i
t
ht
h
ephenomenono
ff
i
b
e
rpull-~ut duet
os
h
o
r
t
,
.
n
t
i
a
l
l
yu
s
e
f
u
li
m
p
r
o
v
e
m
e
n
t
si
nt
h
em
e
c
h
a
n
i
c
a
lb
e
h
a
v
i
o
ro
fもe
n
s
i
o
n
-weakc
o
n
c
r
e
t
e
"c
a
nbee
f
f
e
c
t
e
dbyt
h
ei
n
c
o
r
p
o
r
a
t
i
o
no
ff
i
b
e
r
s
.S
i
m
i
l
a
rt
o七heb
e
h
a
v
i
o
ro
fp
l
a
i
n
1
1
1
110
c
o
n
c
r
e
t
e,
∞mpositef
,
品
.
il
u
r
eu
n
d
e
rmostt
y
p
e
so
fl
o
a
d
i
n
gi
si
n
i
t
i
a
t
e
dbyt
h
et
e
n
s
i
l
ec
r
a
c
k
i
n
go
f
.
t
r
i
xa
J
ongp
l
a
.
n
e
swheret
h
enorm
a
Jt
e
n
s
i
l
es
t
r
a
.
i
n
se
x
c
e
e
dt
h
ec
o
r
r
e
s
p
o
n
d
i
n
gp
e
r
m
i
s
s
i
b
l
e
t
h
ema
l
I
owedbym
u
l
t
i
p
l
ec
r
a
c
k
i
n
go
ft
h
em抗 r
i
xp
r
i
o
rt
oc
o
m
p
o
s
i
t
ef
r
a
.
c
t
u
r
e
.
v
a
l
u
e
s
.T
h
i
smaybef
o
i
ft
h
evolumef
r
a
c
t
i
o
no
ff
i
b
e
r
sa
同 副 館c
i
e
n
t
l
yh
i
g
h(
o
ri
ft
h
ef
i
b
e
r
sa
r
e∞n
t
i
n
u
o
u
s
)(A
v
e
s
t
O
D
e
ta
J
.
, 1
9
7
1
)
. However,whens
h
o
r
tf
i
b
e
r
sa
r
eu
s
e
d(
s
t
田1
,p
o
l
y
p
r
o
p
y
l
e
n
e,g
l
剖 5
,e
t
c
.
),o
n
句
t
h
emat
r
Ixh
a
.
sc
r
a
c
k
e
d
,
o
n
eo
ft
h
ef
o
l
l
o
w
i
n
gt
y
p
e
so
ff
a
.
il
u
r
ew
i
l
lo
c
c
u
r
:
。
。
。
"
“
岡
田
1
1
0
2
'
単
白
l
伽 同E
e
,
(
a
)Thec
o
m
p
o
s
i
t
ef
r
a
c
t
u
r
e
si
m
m
e
d
i
a
t
e
l
ya
f
t
e
rm
a
t
r
i
xc
r
a
.
c
k
i
n
g
.T
h
i
sr
e
s
u
l
t
sfromi
n
叫
q
u
a
t
ef
i
b
e
rc
o
n
t
e
n
ta
tt
h
ec
r
i
t
ic
a
.
ls
e
c
t
i
o
no
ri
n
s
u
f
f
i
c
i
e
n
tf
i
b
e
rl
e
n
g
t
h
st
ot
r
姐 s
f
i
釘 8
t
陶鞠
a
c
r
o
s
s仙 em
a
t
r
i
xc
r
a
c
k
.
(
b
)A
l
t
h
o
u
g
ht
h
emaximum1
0
0
0onもhe∞mposi
匂 i
sn
o
ts
i
g
n
i
f
i
c
a
n
t
l
yd
i
f
f
e
r
e
n
tfromt
h
a
to
f
t
h
em
a
t
r
i
xa
J
one
,
t
h
ec
o
m
p
o
s
i
t
e∞n
t
i
n
u
e
s
t
oc
紅 r
yd
e
c
r
田 8
i
n
gl
o
a
d
sa
f
t
e
rt
h
ep
e
a
k
.T
h
e
p
o
s
t
c
r
a
c
k
i
n
gr
e
s
i
s
t
a
n
c
ei
sp
r
i
m
a
r
i
l
ya
t
t
伽 t
e
dt
og
r
a
.
du
a
J(
r
a
t
h
e
r伽.nB
u
d
d佃)f
i
b
e
r
p
u
l
l
o
u
ta
.
ss
o
c
i
a
.
t
e
dw
i
t
hc
r
a
ε
kb
r
i
d
g
i
n
g
. Whilenos
i
g
n
i
f
i
c
a
n
ti
n
c
r
e
a
.
sei
nc
o
m
p
ω
i
t
e
,thecompositefractureenergyandtoughnessa
.
r
ec
o
n
s
i
d
e
r
a
b
l
y
s
t
r
e
n
g
t
hi
so
b
s
e
r
v
e
d
e
n
h
a
n
c
e
d(
F
i
g
.8
.
2
)(Gop
a
J
a
r
a
t
n.
a
m andSh.
a
h,
1
9
8
7
a
)
.
(
c
) Evena
.
!
t
e
rm
a
t
r
i
xc
r
a
c
k
i
n
g
,t
h
ec
o
m
p
o
s
i
t
ec
o
n
t
i
n
u
e
sもoc
a
.
r
r
yi
n
c
r
e
a
s
i
n
g1
0
0
0
5
.T
h
e
a
.
r
r
y
i
n
gc
a
.
p
a
c
i
t
yo
ft
h
ec
o
m
p
o
s
i
t
eandt
h
ec
o
r
r
e
s
p
o
n
d
i
n
gd
e
f
o
r
m
a
.
t
i
o
na
r
e
p
e
a
kl
o
a
d
-c
s
i
g
n
i
f
i
c
a
n
t
l
yg
r
e
a
t
e
rt
h
a
nt
h
a
to
ft
h
eu
n
r
e
i
n
f
o
r
c
e
dm
a
t
r
i
x
.D
u
r
i
n
gt
h
epr
e
-p
e
a
ki
o
p
r
o
g
r
飽 s
i
v
ed
e
b
o
n
d
i
n
g6
ft
h
ei
n
t
e
r
f
a
c
e組 d
e
l
a
s
t
i
cr
e
g
i
m
eo
ft
h
ec
o
m
p
o
s
i
t
er
e
s
p
o
n
s
e,
d
i
s
t
r
i
b
u
t
e
dm
i
c
r
o
c
r
a
c
k
si
nt
h
em
a
t
r
i
xm乱yber
e
s
p
o
n
s
i
b
l
ef
o
rt
h
ee
n
e
r
g
yd
i
s
s
i
p
a
t
i
o
D
側
(Mobasher,
C
a
s
t
r
o
-MonteroandShah,
1
9
8
9
)
.I
ti
sc
l
e
a
rt
h
a
t出 i
smodeo
f
p
r
o
c
∞mpositef
a
i
l
u
r
er
e
s
u
l
t
si
nt
h
ee
f
f
i
c
i
e
n
tu
s
eo
fb
o
t
ht
h
ec
o
n
s
t
i
t
u
e
n
t
s(Mob
a
.
s
h
e
r叩 d
Shah,1
9
8
9
;S
t
a
n
g
,M
obasherandShah,1
9
8
9
;Mobasher
,S
t
a
n
gandShah,1
9
8
9
)
.
1 f
6
.
u
sl'l倒閣
。
'
崎司自
圃圃圃目.占申司曙司
l
C
I
O
O
規制
1
0
1
>
>
。
10伺
DOPUCI
闇 調T
,
6 (悼制
F
i
g
.8
.
2T
y
p
i
c
a
.
lR怠s
u
l
t
ao
fS
t
r
e
s
s
D
i
s
p
l
a
c
e
m
e
n
tC
u
r
v
e
sO
b
t
a
i
o
e
dFrom
D
i
r
e
c
tT
e
n
s
i
o
oT
e
s
t
s00P
l
a
i
nMort
a
.
rM
a
t
r
i
xa
.
n
dS同 1F
i
b
e
rRe
i
o
f
o
r
c
e
dC
o
n
c
r
e
t
e
1
3
2
L_
'
t
t
,
.
.
丁lsll
一
,
AL J血 0 w向断固
ム
TI--ー
a
J
y
t
i
c
a
lm
o
d
e
l
su
s
e
df
o
rt
h
ep
r
e
d
i
c
t
i
o
no
f出 em
e
c
h
a
n
i
c
a
lb
e
h
a
v
i
o
ro
ff
i
b
r
o
u
scompωAn
i
t
e
sma
.
ybes
u
i
t
a
b
l
ef
o
ro
n
eo
rmoreo
ft
h
ea
b
o
v
et
y
p
e
so
ff
a
.
il
u
r
em
e
c
h
a
n
i
s
m
s
. Bas
e
<
ii
o
pa
.
r
tont
h
efundament
a
Ja
p
p
r
o
a
c
hi
nt
h
e
i
rf
o
r
m
u
l
a
t
i
o
n,
a
n
a
l
y
t
i
c
a
.
lm
o
d
e
l
sc
姐
be日 匂 伊
出 e
d(
G
o
p
a
l
a
r
a
t
n
制 副 S
hah,1
9
8
7
a
)剖: m
o
d
e
l
sb
a
s
e
dont
h
et
h
悶
yo
fm
u
l
t
i
p
l
efra.c~
t
u
r
e,c
o
m
p
o
s
i
t
em
o
d
e
l
s,s
t
r
a
i
n
r
e
l
i
e
fm
o
d
e
l
s,f
r
a
ι
t
u
r
em
e
c
h
a
n
i
c
Bm
o
d
e
l
s,i
n
t
e
r
f
.
a
c
emech
回 i
c
s
a
.
n
dm
i
c
r
o
m
e
c
h
a
n
i
c
sm
o
d
e
l
s
.Fa
.
ir
l
y
e
x
h
a
u
s
もi
v
er
e
v
i
e
w
so
f
t
h
e
s
em
o
d
e
l
sa
r
ea
v
a
.
i
l
.
a
b
l
e
m
o
d
e
l
s,
e
l
s
e
w
h
e
r
e(
G
o
p
a
l
a
.
r
atnamandShah,
1
9
8
7a
.
;M
i
n
d
e
s
s,
1
9
8
3
)
.B
r
i
e
fr
e
v
i
e
w
so
ft
h
ef
r
a
.
c
t
u
r
e
m配 h組 問 ba
.
s
edm
o
d
e
l
sandt
h
ei
n
t
e
r
f
a
c
em
e
c
h
a
n
i
c
sm
o
d
e
l
sa
r
eg
i
v
e
nh
e
r
e出 thωea
.
r
e
t
y
p
i
c
a
l
l
yt
h
emωts
u
i
t
a
b
l
ef
o
rm
o
d
e
l
i
n
gもhei
n
e
la
.
s
t
i
cp
r
o
c
e
s
5
田 i
ns
h
o
r
t
f
i
b
e
r∞m
p
o
s
i
t
e
s
.
Twob
r
o
a
dc
a
t
e
g
o
r
i
e
so
fm
o
d
e
l
sc
a
nbei
d
e
n
t
i
f
i
e
df
r
o
mamongt
h
ef
r
a
c
t
u
r
emech
a
.
n
i
c
s
ba
.
s
edm
o
d
e
l
s
. Themorefundamenta
.
lc
l
a
s
so
fm
o
d
e
l
su
s
et
h
ec
o
n
田p
t
so
fLEFMt
os
o
l
v
e
t
h
eproblemo
fc
r
a
c
ki
n
i
t
i.
a
t
i
o
n,
g
r
o
w
t
h
,
a
r
r
e
s
もa
nds
ta
.
b
i
l
i
t
yi
nt
h
ep
r
e
s
e
n
伺 o
ff
i
b
e
r
st
h
r
o
u
g
b
a
J
d
iandB
a
t
s
o
n,
1
9
6
3
;Ro
m叫 d
i
a
p
p
r
o
p
r
i
a
t
ec
h
a
n
g
e
si
nt
h
es
t
r
e
s
si
n
t
e
n
s
i
t
yf
a
c
t
o
r(Romu
andMandel,
1
9
6
4
)
.T
y
p
i
c
a
l
l
yt
h
偲 em
o
d
e
l
sa
.
s
sumep
e
r
f
e
c
tbondbetweent
h
ef
i
b
e
r組 dt
h
e
m
a
t
r
i
x,
anda
r
eon
e
p
a
r
a
m
e
t
e
rf
r
a
c
t
u
r
em
o
d
e
l
s
.U
n
l
i
k
et
h
ec
l
a
s
s
i
c
a
.
lLEFMm
o
d
e
l
s
自
,omeo!
t
h
el
a
t
e
rm
o
d
e
l
si
m
p
l
i
c
i
t
l
ya
c
c
o
u
n
tf
o
rt
h
ei
n
e
l
a
s
t
i
ci
n
t
e
r
f
a
c
er
e
s
p
o
n
s
ed
u
r
i
n
gc
r
a
c
kg
r
o
w
t
b
c
e
s
唱 ha
.n
o
n
l
i
n
e
a
r伽 e
s
s
-d
i
s
p
l
a
c
e
m
e
n
tr
e
l
a
t
i
o
n
s
h
i
pf
o
rt
h
ef
i
b
e
r
b
r
i
d
g
i
n
gz
o
n
e(
p
r
o
t
h
r
o
z
o
n
e
)
.T
h
i
sa
p
p
r
o
a
c
h
,
w
h
i
c
hh
a
s∞met
obeknown舗も hef
i
c
t
i
t
i
o
u
sc
r
a
c
kmodel(
H
i
l
l
e
r
b
叫
1
9
8
0
),
i
s∞n
c
e
p
t
u
a
l
l
ys
i
m
i
l
a
rt
oもh
a
td
e
s
c
r
i
b
e
di
nS
e
c
.3
.
1f
o
rt
h
ef
r
a
c
t
u
r
eo
fu
n
r
e
i
n
f
o
r
c
e
l
l
c
o
n
C
l
'
叫e
.Themajord
i
f
f
e
r
e
n
c
届 i
nt
h
ef
i
c
t
i
t
i
o
u
sc
r
a
c
km
o
d
e
l
s(
H
i
l
l
e
r
b
o
r
g,
1
9
8
0
;P
e
t
e
r
s
似
・
'
1
0
一一一
トー- 0 - " - - - - 0
-4
脚
.
.
.
.
.
.
.
.
.
.
. I 脚山山~-l
hF W
~ig. 9
.
1A
r
r
e
s
to
fP
a
r
a
l
l
e
1CrackDuet
oI
n
s
t
a
b
i
l
i
t
yandChangeo
f
Sp
a
.
c
i
n
gf
o
rL
e
a
d
i
n
gCr
a
.
c
k
s
1
1
2
1
1
3
1
9
8
0
;WecharatanaandShah,
1
9
8
3
b
jV
i
s
a
l
v
a
n
i
c
handNaaman,
1
9
8
2
)a
r
et
h
es
i
碍u
l
a
r
i
T
.
h
ec
即 k
t
i
p,
t
h
ec
r
i
t
e
r
i
au
s
e
df
o
rc
r
a
c
ki
n
i
t
i
a
t
i
o
nandgrow
もh
,
組dth~
a
s
s
u
m
p
t
i
o
n
s前 t
s
t
a
b
i
l
i
t
yo
ft
h
ec
r
a
c
kg
r
o
w
t
h
.Morer
e
c
e
n
t
l
y
,
JenqandShah(
1
9
8
6
)havep
r
o
p
o
s
e
daf
r
a
c
t
u
l
,
f
m
e
c
h
a
n
i
c
smodelt
op
r
e
d
i
c
tt
h
ec
r
a
c
kp
r
o
p
a
g
a
t
i
o
nr
e
s
i
s
t
a
n
c
eo
ff
i
b
e
rr
e
i
n
f
o
r
c
e
dc
o
n
c
r
e
t
et
h
a
r
p
p
r
o
配 h
e
s
.F
r
a
c
t
u
r
er
e
s
i
s
t
a
n
c
ei
nf
i
b
r
o
思
i
ssomewhatd
i
f
f
e
r
e
n
tfrome
i
t
h
e
ro
ft
h
e
s
eもwoa
o
o
m
p
o
s
i
t
e
sa
c
c
o
r
d
i
n
g七ot
h
i
smodeli
ss
e
p
a
r
a
t
e
di
n
t
of
o
u
rr
e
g
i
m
e
sw
h
i
c
hi
n
c
l
u
d
e
:l
i
n
e
a
l
e
g
i
n
n
i
n
g
e
l
a
s
t
i
cb
e
h
a
v
i
o
ro
ft
h
ec
o
m
p
o
s
i
t
e
js
u
b
c
r
i
t
i
c
a
lc
r
a
c
kg
r
o
w
t
hI
nt
h
em
a
t
r
i
x組 dもheb
e
rb
r
i
d
g
i
n
ge
f
f
e
c
t
jp
o
s
t
c
r
i
t
i
c
a
lc
r
a
c
kg
r
o
w
t
hi
nt
h
em
a
t
r
i
xs
u
c
ht
h
a
tt
h
en
e
t8
t
!
馳
o
ft
h
e五b
i
n
t
e
n
s
i
t
yf
a
c
t
o
rduet
ot
h
e乱p
p
l
i
e
dl
o
a
d叩 dt
h
ef
i
b
e
rb
r
i
d
g
i
n
gs
t
r
e
s
s
e
sr
e
m
a
i
nc
o
n
s
t
a
n
l
酔 w
h
e
r
et
h
er
e
s
i
s
t
a
n
c
et
oc
r
a
ι
ks
e
pa
.
r
a
t
i
o
ni
s
(
s
t
e
a
d
ys
t
a
t
ec
r
a
c
kg
r
o
w
t
h
)
;andt
h
ef
i
n
a
ls
t
a
p
r
o
v
i
d
e
de
x
d
u
s
i
v
e
l
ybyt
h
ef
i
b
e
r
s
. Themode1u
s
e
stwop
a
r
a
m
e
t
e
r
st
od
e
s
c
r
i
b
et
h
em
a
t
r
i
x
f
r
a
c
t
u
r
ep
r
o
p
e
r
t
i
e
s(
K
I
c
'm
o
d
i
f
i
e
dc
r
i
t
i
c
a
1s
t
r
e
s
si
n
t
e
n
s
i
t
yf
a
c
t
o
rb
a
s
e
donLEFMa
n
dt
h
e
anddCTOD,
t
h
ec
r
i
t
i
c
a
lc
r
a
c
kt
i
po
p
e
n
i
n
gd
i
s
p
1
a
c
e
m
e
n
t,
掛 described
e
f
f
e
c
t
i
v
ec
r
a
c
kl
e
n
g
t
h,
andaf
i
b
e
rp
u
l
l
o
u
ts
t
r
e
砕 c
r
a
c
k
w
i
d
七hr
e
1
a
t
i
o
n
s
1
中 ωthe
e
a
r
l
i
e
rf
o
ru
n
r
e
i
n
f
o
r
c
e
dc
o
n
c
r
e
t
e
),
品i
ci
n
p
u
ti
n
f
o
r
m
a
t
i
o
n
b
a
Ouyang叩 dShah(
1
9
8
9
)r
e
c
e
n
t
l
yd
e
v
e
l
o
p
e
danR
c
u
r
v
ea
p
p
r
o
a
c
hf
o
rf
r
a
c
t
u
r
e
Mobasher,
-curvecanbeuniquel
yd
e
t
e
r
.
o
fc
e
m
e
n
t
b
a
s
e
df
i
b
e
rc
o
m
p
o
s
i
t
e
s
. Thep
a
r
a
m
e
t
e
r
so
ft
h
eR
s
u
c
ha
sK
I
candOCTOD d
e
f
i
n
e
dbyJenqandS
h
a
h
mineda
c
c
o
r
d
i
n
gt
om
a
t
e
r
i
a
lp
r
o
p
e
r
t
i
e
s,
e
r
si
si
n
∞rpor叫 edintotheR-curvebyf
i
b
e
r
(
1
9
8
6
)
.T
o
u
g
h
e
n
i
n
go
ft
h
em
a
t
r
i
xbyt
h
e伍b
n
gp
r
e
s
s
u
r
ei
nt
h
ee
q
u
i
l
i
b
r
i
u
mc
o
n
d
i
t
i
o
n
sd
u
r
i
n
g
b
r
i
d
g
i
n
gp
r
e
s
s
u
r
e
_Byi
n
c
o
r
p
o
r
a
t
i
n
gt
h
ec
l
o日i
,
もh
eo
n
s
e
to
fi
n
s
もa
b
i
l
i
t
yo
fm
a
t
r
i
xi
sshownt
obed
e
p
e
n
d
e
n
to
n
t
h
es
t
a
b
l
ec
r
a
c
kp
r
o
p
a
g
a
t
i
o
n
t
h
ef
i
b
e
r
s
.
A
l
lo
ft
h
ef
i
c
t
i
t
i
o
u
sc
r
a
c
km
o
d
e
l
sr
e
l
yont
h
es
t
r
e
s
s
c
r
a
c
k
w
i
d
t
hr
e
l
a
t
i
o
n
so
b
t
a
i
n
e
de
x
p
e
r
.
i
m
e
n
t
a
l
l
y
.Thereh
a
v
eb
e
e
nsomea
t
t
e
m
p
t
sa
tp
r
e
d
i
c
t
i
n
gt
h
em
a
c
r
o
s
c
o
p
i
cs
t
r
e
s
s
c
r
a
c
k
w
i
d
t
h
e
r
m
a
t
r
i
xi
n
t
e
r
f
a
c
e
.T
h
e
y
r
e
l
a
t
i
o
n
so
ft
h
ec
o
m
p
o
s
i
t
efromas
t
u
d
yo
ft
h
em
e
c
h
a
n
i
c
so
ft
h
e五b
C叩 b
eg
r
o
u
p
e
d剖 m
o
d
e
l
sb
a
s
e
dont
h
es
h
ea
.
r
1
a
g山 田ryo
rm
o
d
i
f
i
c
a
t
i
o
n
st
1
悶 ωf
仏
(
La
刑w
r
e
凹E
町r
α
悶
s
止
;Lawse
ta
,
.
l1972jGopa1aratnamandShah,
1
9
8
7
b
;G
o
p
a
l
a
r
a
t
n
如 1a
ndCheng
,
1
9
8
8
)
,
1
9
7
2
f
r
a
c
t
u
r
em
e
c
h
a
n
i
c
sb
a
s
e
di
n
t
e
r
f
a
c
em
o
d
e
l
s(
S
t
a
n
gandShah,1
9
8
6
;M
o
r
r
i
s
o
ne
ta
l 1
9
8
8
)
1
9
8
8
;S
a
h
u
d
i
n,
1
9
8
7
)
. Manyo
ft
h
e
s
em
o
d
e
l
sh
a
v
e
andn
u
m
e
r
i
c
a
lmodels(
M
o
r
r
i
s
o
ne
ta
l
.,
be
ens
u
c
c
e
s
s
f
u
lt
ov
a
r
y
i
n
gd
e
g
r
e
e
si
np
r
e
d
i
c
t
i
n
gt
h
ep
e
a
kp
u
l
l
o
u
tl
o
a
d
sandt
h
el
o
a
d
s
i
l
p
叩 もr
e
m
e
n
d
o
u
s
1
y
r
e
s
p
o
n
s
eo
fi
d
e
a
1
i
z
e
da
l
i
g
n
e
ds
i
n
g
1
ef
i
b
e
rp
u
l
l
o
u
t
.T
h
e
s
em
o
d
e
l
sh
a
v
ebe
も
もh
ei
n
t
e
r
f
a
c
eands
h
o
w
i
n
g
u
s
e
f
u
li
nu
n
d
e
r
s
t
a
n
d
i
n
gt
h
eb
a
s
i
cm
e
c
h
a
n
i
c
so
fs
t
r
e
s
st
r
a
n
s
f
e
ra
もa
n
tr
o
1
ei
nt
h
ef
r
a
c
t
u
r
eo
fs
u
c
hc
o
m
p
o
s
i
t
e
s
.S
i
g
n
i
f
.
t
h
a
ti
n
t
e
r
f
a
c
i
a
ld
e
b
o
n
d
i
n
gp
l
a
y
sanimpor
how
巴v
e
r,
ben
e
e
d
e
dt
omodifyt
h
e
s
emodelst
op
r
e
d
i
c
t出 ep
u
l
l
o
u
t
i
c
a
n
tr
e
s
e
a
r
c
he
f
f
o
r
t
sw
i
l
l,
e
'
l
s
c
h
a
r
a
c
t
e
r
i
s
t
i
c
so
fi
n
c
l
i
n
e
df
i
b
e
r
st
h
a
ta
r
erandomlyo
r
i
e
n
t
e
da
tam
a
t
r
i
xc
r
a
c
k(random
n
町
吋
也
d
袖
lme
i
nb
o
t
ht
h
ea
n
g
u
l
a
ro
r
i
e
n
t
a
t
i
o
na
sw
e
l
la
st
h
e巴mbe
ね
a
釘
加時
I 乱姐
ndShah(
1
9
8
9
)p
r
o
p
o
s
e
dadamagemodelf
o
rt
h
es
t
u
d
yo
fd
i
批 i
b
u
t
e
dm
i
c
r
o
c
S
t
p
o
s
i
t
es
u
b
j
e
c
もe
dもou
n
i
a
x
i
a
lt
e
n
s
i
o
nw出 i
n
i
t
i
a
l
l
yd
e
r
i
v
e
d
r
a
c
k
i
n
g
.Thec
o
m
p
1
i
a
n
c
eo
ft
h
e∞m
a
c
c
o
r
d
i
n
gt
ot
h
es
h
e
a
rl
a
gt
h
e
o
r
y
. A damagev
a
r
i
a
b
1
e,
whichi
saf
u
n
c
t
i
o
no
fc
r
a
c
ks
p
a
c
r
a
c
ks
e
c
t
i
o
n,wasi
n
t
r
o
d
u
c
e
d
.A
f
t
e
rs
p
e
c
i
f
y
i
n
gt
h
e
i
n
gandf
i
b
e
rd
e
b
o
n
d
i
n
gl
e
n
g
t
ha
t山 ec
t
h
es
t
r
e
s
sands
t
r
a
i
nr
e
s
p
o
n
s
eo
ff
i
b
e
rr
e
i
n
f
o
r
c
e
d
damagee
v
o
l
u
t
i
o
nl
a
wbyadamages
u
r
f
a
c
e,
c
o
m
p
o
s
i
t
e
ss
u
b
j
e
c
t
e
dt
ou
n
i
a
x
i
a
lt
e
n
s
i
o
nc
a
nb
ec
a
l
c
u
1
a
t
e
d
.
吋 a
sa
l
s
ob
e
e
ns
t
u
d
i
e
df
o
rg
l
a
s
s
f
i
b
e
rpolymer(
p
o
l
y
e
s
t
e
r
)c
o
n
c
r
e
t
e
.Suchm
a
t
e
r
i
ιl
s
F
r
a
c
t
U
I
a
.
k1
o
a
d, 組dもheR
c
u
r
v
e
s
showc
o
n
s
i
d
e
r
a
b
1
en
o
n
1
i
n
e
a
r
i
t
yands
t
a
b
l
ec
r
a
c
kgrowthp
r
i
o
rt
ope
,
1
9
8
7,
1
9
8
8
.
t
h
a
thaveb
e
e
nm
e
a
s
u
r
e
dr
e
f
l
e
c
tt
h
i
sb
e
h
a
v
i
o
r(
V
i
p
u
l
a
n
a
n
d
a
n組 dDharmarajan
ムc
)
.
1
9
8
9
a
吋
恥1
S
!
h
a
p
t
e
r9
. CRACKSYSTE
l
nc
o
n
t
r
a
s
もも om
e
t
a
l
l
i
cs
t
r
u
c
t
u
r
e
s
,
∞ncretestructure
so
o
n
t
a
i
nnumerousl
a
r
g
己cracks
l
o
s
eenought
oi
n
もe
r
a
c
t
.T
h
e
r
e
f
o
r
e,
u
n
d
e
r
s
t
a
n
d
i
n
go
fc
r
a
c
ki
n
t
e
r
a
c
t
i
o
ni
smore
h
i
c
h釘 ec
:
1
;
l
p
O
巾 n
tt
h
a
ni
ti
sf
o
rm
e
t
a
l
l
i
cs
t
r
u
c
t
u
r
e
s
.
{
'
.
lResponseofStructureswithI
n
t
e
r
a
c
t
i
n
gGrowingCracks
C
o
n
s
i
d
e
ra抗 r
u
c
t
u
r
ew
i
出 c
r
a
c
k
so
fl
e
n
g
t
h
sα
j
(
i= 1,
2
.
.
.n
)l
o
a
d
e
dbyas
ystemo
ff
o
r
c
e
s
b
o
u
n
d
紅 yd
i
s
p
l
a
c
e
m
e
n
t
s,
s
h
r
i
n
k
a
g
eo
rもherma
ld
i
l
a
t
i
o
np
r
o
p
o
r
t
i
o
n
a
lt
opa
.
r
ame
t
e
rA.The
目e
r
g
y(
u
n
d
e
rI
s
o
t
h
e
r
m
a
lc
o
n
d
i
t
i
o
n
s,
morep~ecÎs~ly _
t
h
_
eHel~oltz. fr~e e~ergy) o
ft
h
e山 u
c
a
.
ls
y
s
t
e
mi
st
h
esumo
f出 es
t
r
a
i
ne
n
e
r
g
yU,
whichd
e
p
e
n
d
sont
h
el
o
a
dp
a
r
a
m
e
t
e
r舗 w
e
l
l
I
l
r
、t
h
ec
r
a
c
kl
e
n
g
t
h
s,
andt
h
ee
n
e
r
g
yn
e
e
d
e
dt
op
r
o
d
u
c
et
h
ec
r
a
c
k
s
,
i
.
e
.,
!
"
F= U(
a
l, 川)+午
1
4
iR(αDぬ
;
(
9
.
1)
p
e
c
i
五ce
n
e
r
g
y(
p
e
ru
n
i
tc
r
a
c
kl
e
n
gt
h
)r
e
q
u
i
r
e
df
o
rc
r
a
c
kg
r
o
w
t
handα~ a
r
et
h
e
L
I
.w
h
i
c
hR=s
nt
h
ec
掛 e
o
fl
i
n
e
a
re
l
a
s
t
i
cf
r
a
c
t
u
r
em
e
c
h
a
n
i
c
s
,
R = G/
.
r
a
c
kl
e
n
g
t
hv
a
l
u
e
sbetween0andαt・I
•C
h
ee
q
u
i
l
i
b
r
i
u
mc
o
n
d
i
t
i
o
no
ft
h
es
y
s
t
e
mi
soF= E(
U,
i
+
R
(
a
i
)
)
o
a
i=0f
o
r組 yv
a
r
i
a
t
i
o
no
a
j,
h
i
c
hi
m
p
l
i
e
s出 a
t-U
刊 =R
(
a
i
)
'T
h
i
si
st
h
ew
e
l
l
k
n
o
w
nc
o
n
d
i
t
i
o
no
fc
r
a
c
kp
r
o
p
a
g
a
t
i
o
n
七t
h
eb
e
g
i
n
n
i
n
g
.Thes
t
a
b
i
1
i
t
yo
ft
h
es
t
r
u
c
t
u
r
ewit
ht
h
ec
r
a
c
k
sr
e
q
u
i
r
e
st
h
a
t
I
r
e
a
d
ys
t
a
t
e
da
2
a
.
r
i
a
t
i
o
no
F bep
o
s
i
t
i
v
ef
o
ranya
d
m
i
s
s
i
bl
eo
a
j(Ba
量姐ta
ndOhtsubo,
1
9
7
7
,
h
es
e
c
o
n
dv
l
f7
8
;lJoUIlUl
tandWahab,
1
9
7
9
)
.Fr
omE
q
.9
.
1,
wehave
o
R
(
a
i
)
2
2
;
_
U
,
i
j
O
a
i
O
α
j+~一一一
o
F= E
rE;
2
~ "J,
J • '
1 o
a
i (6d = H- AJMαj
r
(
9
.
2
)
,
1:
.w
h
i
c
hAi
q
u
a
r
em
a
t
r
i
x
jformas
Ai
j+2R
,
i
d
i
j
H
(
o
a
i
)
j= U,
(
9
.
3
)
I
,
j"
f
et
h
es
u
b
s
c
r
i
p
t
sf
o
l
l
o
w
i
n
ga∞mmar
e
p
r
e
s
e
n
tp
a
r
t
i
a
ld
e
r
i
v
a
t
i
v
e
sw
i
t
hr
e
s
p
e
c
もt
oa
i,
=lifi=j叩 d0i
fi=
Fj
;andH
(
o
a
i
)= 1i
fo
a
jと 0,
and0o
t
h
e
r
w
i
s
e
. No
t
et
h
a
t
,
r,
'j= Kl/E'whereKi= s
t
r
e
s
si
n
t
e
n
s
i
勿f
a
c
t
o
ro
fc
r
a
c
ka
i(
a
n
ymode
)
,
組d -E'UiJ=
.
日 /(i,
j=
2KjKj
,
i
'
.
¥
sもh
e1
0
a
d
i
n
gpa
.
r
a
m
e
t
e
rAi
n
c
r
e
回邸, t
h
ec
r
a
c
k
sg
r
o
w
. Ata民 r
t
a
i
nc
r
a
c
kl
e
n
gt
hi
tc
a
n
i・
岬 e
nt
h
a
tもh
e
r
ea
r
emoret
h
a
no
n
es
o
l
u
t
i
o
nf
o
rt
h
ec
r
a
c
ki
n
c
r
ements,
i
.
e
.,t
h
ep
a
t
ho
f
l
o
t
t
e
di
nt
h
es
p
a
田 w
i
t
hc
o
o
r
d
i
n
a
t
e
sa
ib
i
f
u
r
c
a
t
e
s
.I
tcanbeshownt
ha
tt
h
e
'
:
. ~ystem p
Ll
r
c
a
t
i
o
ncond
i
t
i
o
ni
s
detA
(
9.
4
)
i
j= 0
1
¥
i"
rb
i
f
u
r
c
a
t
i
o
n
,
t
h
epa
出 w
hichi
sf
o
l
l
o
w
e
di
st
h
a
tf
o
rw
h
i
c
h(Ba
.
Z
a叫
2F =m
F
o
rd
i
s
p
l
a
c
e
m
e
n
to
o
n
t
r
o
1
:o
in
2
F
o
rl
o
a
dc
o
n
t
r
o
l
: oF = max
(
9
.
5)
114
(o)
5
6 2
115
o
.
.
吋
,
.
.
.
.
.
-
T
h
i
sc
o
n
d
i
t
i
o
ni
sd
e
r
i
v
e
df
r
o
mt
h
ef
a
c
tt
h
a
tt
h
ep
a
t
hwhicho
c
c
u
r
smustbet
h
a
tw
i
t
h出e
ma
.
x
imumi
n
t
e
r
n
a
le
n
t
r
o
p
yi
n
c
r
e
m
e
n
t
.
田n
t
l
y
,Bazant(1989b)組 dBaz叩 tandTabbara
.(
1
9
8
9
)h
a
v
eshownもh
a
tt
h
e
Morer
e
t
a
n
g
e
n
t
i
a
l叫 i
f
f
n
e
s
sr
n
a
t
r
i
xc
∞t
a
i
n
i
n
gN i
n
t
e
r
a
c
t
i
n
gc
r
a
.
c
k
so
fl
e
n
g
t
h
sa
j
(
i=
=1,
.
.
.N)i
s
以
.
1
w
h
e
r
e
S
~14
1
.
匂
司H
α,
自国
=
s=
σ冴'X~
q = E'
/5x_
、
1
1
1o
,
,
!r-‘唱『
於
:11
・
'
"
3
5
・
ーーーー
O
.
S
.
.
・
・
・
.
.
-
.
.
I
I
;
"
'
t
)
-
'
.
o
u
Q
=
B
q
4
1,
0
F
i
g
.
9
.
3D
i
s
c
r
e
t
eC
r
a
c
k
i
n
gi
nF
r
a
c
t
u
r
eP
r
o
c
e
s
sZ
o
n
e岨 dT
h
ω
r
e
t
i
c
a
l
S
t
r
e
s
s
D
i
s
p
l
a
c
e
m
e
n
tDiagramf
o
rC
r
a
c
kC伺 J
e
s
伺 :
n
c
e
Thep
r
e
c
e
d
i
n
gf
o
r
m
u
l
a
t
i
o
nh
a
s姐 i
l
l
t
e
r
e
s
t
i
n
ga
p
p
l
i
c
a
t
i
o
nt
os
y
s
t
e
m
so
fp
a
r
a
l
l
e
lc
r
a
c
k
s
c
a
u
s
e
d,
e
.
g
.,
byd
r
y
i
n
gs
h
r
i
n
k
a
g
eo
rc
o
o
l
i
n
gs
t
r
回s
e
s,
o
rbyb
e
n
d
i
n
g
.I
nt
h
ei
d
e
a
l
i
z
e
dc
出 e
o
f
amω8ivew
a
l
lt
r
e
a
t
e
da
sah
a
l
f
s
p
a
c
e,
t
h
es
h
r
i
n
k
a
g
ec
r
a
c
k
s8
t
a
r
ta
tsomec
l
o
s
es
p
a
c
i
n
gsand
'
,
h
e
n
,
a
st
h
ed
r
y
i
n
gf
r
o
n
ta
d
v
a
n
c
e
si
n
t
ot
h
ew
a
l
l
,
t
h
e
yc
o
n
t
i
n
u
et
ogrowi
n
w
a
r
d
.C
a
l
c
u
l
a
t
i
o
n
s
も
embe
ω
, m
esu
n
s
t
a
b
l
ewhent
h
ec
r
a
c
kl
e
n
g
t
h
o
nt
h
eb
a
s
i
so
fE
q
.9
.
2h
a
v
eshownt
h
a
tt
h
es
y
s
r
e
a
c
h
e
st
h
ev
a
l
u
ea
p
p
r
o
x
i
m
a
t
e
l
y向 =1
.7
8,
a
twhichp
o
i
n
tt
h
ed
r
y
i
n
gf
r
o
n
ti
sa
p
p
r
o
x
i
m
a
t
e
l
y
a
tt
h
ed
e
p
t
h2
.
6
8(
Ba
.
Z
a
n
tandOhtsubo,
1
9
7
7
;B話 回t
,
Ohtsubo組 dAoh
,1979;BaZant姐 d
~ahab, 1
9
7
9
)
.AUhisp
o
i
n
tt
h
e
r
ei
sab
i
f
u
rt
i
o
no
ft
h
ee
q
u
i
l
i
b
r
i
u
mp
a
t
hi
nt
h
es
p
a
c
eo
fa
j
.
r
o
w
i
n
gandc
l
o
s
e
s
,
k
e
e
p
i
ngt
h
ei~ngth a
2,
~hile t
h
ei
n
t
e
r
m
e
d
i
a
t
e
E
v
e
r
yo
t
h
e
r~rack stop~ g
:
.
r
&c
k
so
fl
e
n
g
t
hα
jc
o
nt
in
u
et
og
r
o
w,
t
h
e
i
rs
p
a
c
i
n
gd
o
u
b
l
e
dt
o2
s
.Thea
r
r
e
s
to
ft
h
eg
r
o
w
t
h
si
sa
g
a
i
nr
e
p
e
a
t
e
dl
a
t
e
r
,andωon.This
I
md.c
l
o
s
i
n
go
fe
v
e
r
yo
t
h
e
ro
ft
h
el
e
a
d
i
n
gc
r&ck
&c
ks
y
s
t
e
m
sshowni
nF
i
g
.9
.1
.
'
)
r
o
d
u
c
e
st
h
ec
r
.
Thes
p
a
c
i
n
go
ft
h
ec
r
a
c
k
s,
8
,
i
simpo~tant f
o
rd
e
t
e
r
m
i
n
i
n
gt
h
e
i
rw
i
d
t
h
.Asi
sw
e
l
lknown,
ti
sd
e
s
i
r
a
b
l
et
ok田 pt
h
ec
r
a
c
kw
i
d
t
hl
e
s
st
h
a
na
b
o
u
tO.lmmt
oO
.
4
m
r
n
,mainlybecause
n
a
r
r
o
wc
r
a
c
k
sa
r
en
o
tr
e
a
l
l
yc
o
n
t
i
n
u
o
u
sanddo110も s
e
r
v
ea
sgoodc
o
n
d
u
i
t
sf
o
rm
o
i
s
t
u
r
eo
r
c
a
:
、、 (
,
.
,
r
7
.
U
"
.
'
0.0
0
.
0
n
=
9
.
2I
n
t
e
r
a
c
t
i
n
gP
a
r
a
l
l
e
lCracks
・'
.
'
1
T
.
Ir
;I
1
.
0
=
雷 同 =i
n
v
e
r
s
eo
ft
h
em
a
t
r
i
x
φk
m,
i
nwhich
m=伊 F/
θa
k
偽情 +
d
k
md凡n
/dαmiand
φk
九
‘(
a
g
i
v
e
nR
c
u
r
v
eo
ft
h
em-thc
r
a
c
k(
i
nLEFM,
凡
G" d
R
"
.
/
d
a
m
0
)
.
m)
a
.
6
q
.
伊 FJ
d
a
k
d
q
";
:
: 0f
o
ra
1
1m 壬 v and
E
q
.9
.
6i
sv
a
l
i
do
n
l
yi
f6
m = -EkWkmE
E
u
8
2
U
J
d
a
m
d
q
k;
:
:0f
o
ra
l
lm >ν. I
fn
o
t,
t
h
es
e
to
fp
r
o
p
a
g
a
t
i
n
gc
r
a
c
k
si
sd
i
f
f
e
r
e
n
tt
h
姐
a
s
s
u
m
e
d
.O
b
v
i
o
u
s
l
y
,variouss
e
t
so
fp
r
o
p
a
g
a
t
i
n
gc
r
a
c
k
smustbet
r
i
e
du
n
t
i
lt
h
ef
o
r
e
g
o
i
n
g
2
ム2
6
F =E
r
E
.
K
;
a
6
q
r
6
q
nwhichn
e
e
d
st
obe
c
o
n
d
i
t
i
o
n
sa
r
emetf
o
ra
l
lm. BasedonE
q
.9
もe
r
m
i
n
et
h
es
t
a
b
l
ep
a
t
ho
fe
v
o
l
u
t
i
o
no
ft
h
ec
r
拭 ks
y
s
t
e
m
.F
o
rf
u
r
t
h
e
rd
e
t
a
i
l
s,
s
e
e
u
s
e
dt
ode
t
h
et
e
x
t
b
o
o
kbyB
a
z
a
n
tandC
e
d
o
l
i
n(
1
9
9
0
)
.
1
.
2
.
0
旬
(
9
.
6
)
1
1=
numbero
fc
r
a
c
k
s七h
a
tp
r
o
p
a
g
a
t
ed
u
r
i
n
gl
o
a
d
i
n
g(
c
r
&c
k
sa川 , .
.
.
αN r
e
m
a
i
n
i
n
gs
ta
t
i
o
n
a
.
r
y
,
ν壬N
)
;
F
i
g
.9
.
2EnergyR
e
l
e
回 eZ
o
n
e
s00
時 四i
n
gC
r
a
c
kSp&ci
n
gi
nT
e
n
s
i
o
n
e
d
R
e
i
n
f
o
r
c
e
db
a
r
3
.
0
θ2
F
否a
,,8
q
.
J
(
:
.=伊 FJδqrdq.=secantstiffnessma
.
t
r
i
xo
fc
r
a
c
k
e
ds
t
r
u
c
t
u
r
e
j
三ζ+
s
d2
F
ゃ =d
i
s
p
l
配 e
m
e
n
t
s{
r 1
,
・
..
Nd,
Nd=
=numbero
fd
i
s
p
l
抑 m
ents)
j
j
S
_
=
k
f
a
25
Y呪 瓦
~ ~
.
‘
'
ー
ー
・
.
J
116
117
o
o
r
r
o
s
i
v
ea
g
e
n
t
s
.I
f
もh
ea
v
e
r
a
g
es
h
r
i
n
k
品,
ges
t
r
a
i
no
r∞o
l
i
n
gs
t
r
a
i
ni
sfO. 姐 dt
h
ebodyi
s
t
h
e
nt
h
eo
p
e
n
i
n
go
f出 ec
r
a
.
c
k
si
s
r
e
s
t
r
a
i
n
e
d,
i
nw
h
i
c
hd
.U =t
o
t
a
lr
e
l
e
a
s
eo
fs
t
r
a
i
ne
n
e
r
g
y
,
I
:
1Wf=GfAc=e
n
e
r
g
yn
e
e
d
e
dt
op
r
o
d
u
c
e
品c
t
u
r
e(
A
c=a
r
e
乱 o
fa
l
lt
h
ec
r
a
c
k
s
),
andI
:
1Wb :
n
e
r
g
yd
i
s
s
i
p
a
t
e
dbybonds
l
i
pw
h
i
c
h
;
:
;e
f
r
出 c
r
a
.
c
k
i
n
g
.
o
c
c
u
r
ss
i
m
u
l
t
a
n
e
o
u
s
l
yw
i
C
o
n
s
i
d
e
rnowc
o
n
c
r
e
t
et
h
a
ti
sr
e
s
t
r
a
i
n
e
dbys
t
e
e
lb
a
r
swhicha
r
es
u
伍c
i
e
n
t
l
ys
t
r
o
n
g旬
D
r
e
v
e
n
tt
h
ep
r
e
v
i
o
u
日l
yd
i
s
c
u
s
s
e
di
n
s
t
a
b
i
l
i
t
i
e
so
fc
r
a
c
ks
y
s
もem
自 i
nwhichsomec
r
a
c
k
sc
l
o
s
e
血dぬespacingo
ft
h
eopenc
r
a
.
c
k
sm
u
l
t
i
p
l
i
e
s
.F
o
rt
h
es
a
k
eo
fs
i
m
p
l
ei
l
l
u
s
t
r
a
t
i
o
n,
∞nsi
d
e
r
ar
o
u
n
dc
o
n
c
r
e
t
er
o
di
nt
e
n
s
i
o
n,
w
i
t
has
i
n
g
l
e批 e
lb
a
ri
nt
h
em
i
d
d
l
e(
F
i
g
.9
.
2
)
.T
h
i
sa
l
s
o
伺 t
h
eb
e
h
a
v
i
o
ro
fもhe∞n
c
r
e
t
ez
o
n
es
u
r
r
o
u
n
d
i
n
goneo
ft
h
eb
a
r
si
na
a
p
p
r
o
x
i
m
a
t
e
l
ys
i
m
u
l
a
t
.
,
i
nw
h
i
c
ht
h
ec
r
o
s
ss
e
c
t
i
o
n乱r
e
ai
ss
u
b
d
i
v
i
d
e
di
n
t
on
o
n
-o
v
e
r
l
a
p
p
i
n
gz
o
n
e
so
fc
o
n
c
r
e
t
e
,
beam
蹴 h
i
凶
n
も
旬
加
e
創r
a
a
n
dOh,
1
9
8
3
b,
1
9
8
4
)
.
,thes
t
r
a
品 a
tw
h
i
c
h
Case1
a
. StrengthLimitandNoBondS
l
i
p
. I
nt
h
i
sc
a
s
e
c
r
a
c
k
i
n
gb
e
g
i
n
si
s
:
(
9
.
1
0
)
f.'
2
:f
UE
ι
竺 S fO
(
9
.
7
)
Thed
r
y
i
n
gc
r
a
.
c
kst
h
a
th
a
v
es
t
o
p
p
e
dg
r
o
w
i
n
gw
i
l
lg
r
a
d
u
a
l
l
yc
l
o
s
e(
F
i
g
.9
.
1
)
.Th
民 o
n
e
w
h
i
c
he
v
o
l
v
e
sa
ss=s
o,
280
,
musts
u
b
s
t
i
t
u
t
ei
nE
q
.9
.
6t
h
es
p
a
c
i
n
go
ft
h
el
e
a
d
i
n
gc
r
a
c
k
s,
4s
,
8s
,
.
.
.
.Thespacingofthel
e
a
.
d
i
n
gc
r
a
.
c
k
sg
e
n
e
r
a
l
l
yr
e
m
a
i
n
sb
e
t
w
e
e
ns= aJ
/
1
.7姐 d
o
o
2ad1
.7,
i
.
e
.,
ont
h
ea
v
e
r
a
g
es~ 0
.
9
a
1・ Thus,
i
nv
i
e
wo
fE
q
.9
.
6,
t
h
eo
p
e
n
i
n
gw
i
d
t
ho
ft
h
e
s
h
r
i
n
k
a
.
g
ec
r
a
.
c
k
si
s,
ont
h
ea
v
e
r
a
g
e
,
o
d
.
9
a1f
(
9
.
8
)
c~ O
Theb
e
h
a
v
i
o
ri
sr
a
出.
e
rs
i
m
i
l
a
rf
o
rp
a
r
a
U
e
lc
r
a
.
c
k
sc
a
u
s
e
dbyc
o
o
l
i
n
g
,
a
sw
e
l
l回 p
a
r
a
l
l
e
l
,the
c
r
a
c
k
si
nbeamsc
a
u
s
e
dbyb
e
n
d
i
n
go
ra
x
i
a
lf
o
r
c
e
.I
nr
e
i
n
f
o
r
c
e
dc
o
n
c
r
e
t
es
t
r
u
c
t
u
r
e
s
.
c
k
sw
i
t
ht
h
ep
r
o
g
r
e
s
so
fd
r
y
i
n
go
rt
h
eb
e
n
d
i
n
gc
r
a
c
k
s
g
r
a
d
u
a
lo
p
e
n
i
n
go
ft
h
es
h
r
i
n
k
a
g
ec
ra
r
o
g
r
e
s
so
fl
o
a
d
i
n
g
,
i
sp
r
e
v
e
n
t
e
dbyr
e
i
n
f
o
r
c
e
m
e
n
t
.I
nf
a
c
t,
i
ti
sp
o
s
s
i
b
l
etoωIcul
a
.
t
e
w
i
t
h出 ep
(
p
r
o
v
i
d
e
dt
h
ebonds
l
i
pl
e
n
g
t
hi
sknown)t
h
eminimumr
e
i
n
f
o
r
c
e
m
e
n
tw
h
i
c
hi
sr
e
q
u
i
r
e
dt
o
a
.
c
k
sp
r
o
p
a
g
a
t
i
n
gw
i
t
he
q
u
a
ll
e
n
g
t
h
s抗 t
h
e
i
rs
m
a
l
li
n
i
t
i
a
ls
p
a
.
c
i
n
g
. Sucha
.r
e
k
e
e
pt
h
ec
r
i
n
f
o
r
c
e
m
e
n
tp
r
e
v
e
n
t
st
h
eo
p
e
n
i
n
gw
i
d
t
ho
ft
h
ec
r
a
c
k
sf
r
o
me
x
c
e
e
d
i
n
gac
e
r
t
a
i
nv
a
l
u
e
.The
ne
臼 s
s
a
r
yr
e
i
n
f
o
r
c
e
m
e
n
th
a
sb
e
e
nc
a
.
l
c
u
l
a
t
e
df
r
o
mt
h
es
t
a
b
出t
yc
o
n
r
u
t
i
o
nb
a
s
e
donE
q
.9
.
2
1
9
8
0
),
andi
tt
u
r
n
e
do
u
tt
h
a
t,
f
o
rat
y
p
i
c
a
ls
i
t
u
a
t
i
o
n,
t
h
eminimum
i
nB話 回t姐 dWahab(
r
e
i
n
f
o
r
c
e
m
e
n
twasa
b
o
u
t0.2%o
ft
h
ec
r
o
s
s
s
e
c
t
i
o
n
a
la
r
e
a
.o
fc
o
n
c
r
e
t
e
. Thuswes
e
et
h
a
も
もh
e
a
.
lr
u
l
e
sf
o
rminimumr
e
i
n
f
o
r
c
e
m
e
n
ta
r
ei
nf
a
c
te
x
p
l
i
c
a
b
l
ebymeanso
ff
r
a
.
ct
u
r
eme
e
m
p
i
r
i
c
c
h
a
n
i
c
s
.Ani
n
t
e
r
e
s
t
i
n
gp
o
i
n
t,
t
h
o
u
g
h,
i
st
h
a
.
tt
h
ep
r
e
c
i
s
ev
a
l
u
eo
ft
h
eminimumr
e
i
n
f
o
r
c
e
m
e
n
t
n
e
e
d
e
dt
op
r
e
v
e
n
tp
r
o
g
r
e
s
s
i
v
e
l
yw
i
d
e
ro
p
e
n
i
n
go
ft
h
ec
r
a
c
k
si
sn
o
tao
o
n
s
t
a
n
tb
u
td
e
p
e
n
d
s
もo
u
t
l
a
y
,
t
h
es
t
r
u
c
t
u
r
a
lg
e
o
m
e
t
r
y
,
t
h
ed
i
s
t
r
i
b
u
t
i
o
nandh
i
s
t
o
r
yo
fp
o
r
e
ont
h
er
e
i
n
f
o
r
c
e
m
e
n
h
u
m
i
d
i
t
yo
rt
e
m
p
e
r
a
t
u
r
e
,
e
t
c
.
Thephenomenonj
u
s
td
e
s
c
r
i
b
e
dmaya
l
s
ober
e
g
a
r
d
e
da
s晶 k
i
n
do
fl
o
c
a
l
i
z
a
t
i
o
no
ft
h
e
c
r
a
c
k
i
n
gs
t
r
a
i
ni
n
t
osomep
r
e
f
e
r
r
e
dc
r
a
c
k
s
.
9
.
3CrackSpacingand可Vidthi
nBeams
A
l
t
h
o
u
g
ht
h
ec
u
r
r
e
n
tc
o
d
ep
r
o
v
i
s
i
o
n
sr
e
g
u
l
a
.
t
i
n
gt
h
ew
i
d
t
ho
fc
r
a
c
k
sar
ee
m
p
i
r
i
c
a
l, 品
d配 peru
n
d
e
r
s
t
a
n
d
i
n
g伺 nbeg
a
i
n
e
dt
h
r
o
u
g
hf
r
a
c
t
u
r
em
e
c
h
a
n
i
c
s
.Onemus
もd
i
s
t
i
n
g
u
i
s
hbe
tw
悶
:(
1
)c
r
a
c
ki
n
i
t
i
a
t
i
o
n,
w
h
i
c
hc
o
r
r
e
s
p
o
n
d
st
ot
h
ep
e
a
kp
o
i
n
to
f山
t
h
児e
s
蜘t
r
e
悶s
悶s
s
t
回凶
r
a
i
n必
d1
噌
a
.
g
r
副
組刷
d叩 山 t
恥
he伽 蜘r
t
川o
fd
伽e
肝刊
V叫
e
l
ω
l
o
阿
p
m則
e
n
叫tぱ
0fm
削i
c
r
o
c
α
r
畠拭
c
叫
k
,
s
只 姐叫d(
ρ
2
)
μ
c
日 c
k∞m
p
l
e
t
i
o
n,
whichc
o
r
自p
o
n
d
st
oar
e
d
u
c
t
i
o
no
ft
h
et
e
n
s
i
l
es
t
r
e
s
st
oz
e
r
oandc
o
a
l
e
s
c
e
n
c
eo
fm
i
c
r
o
c
r
a
c
k
si
n
t
o
r
e
も
i
n
u
o
u
sc
r
a
c
k
s
.
c
o
n
Thec
r
a
c
ki
n
i
t
i
a
t
i
o
ni
sg
o
v
e
r
n
e
dbyt
h
es
t
r
e
n
g
t
hc
r
i
t
e
r
i
o
n
,whilecrackcompletioni
s
r
a
c
t
u
r
em
e
c
h
a
n
i
c
se
n
e
r
g
yc
r
i
t
e
r
i
o
n
.F
o
ra
p
p
r
o
x
i
m
a
t
ean
a
.
l
y
s
i
s,
o
n
en
e
e
d
g
o
v
e
r
n
e
dby出 ef
n
o
tw
r
i
t
et
h
ee
n
e
r
g
yb
a
l
a
n
c
ec
o
n
d
i
t
i
o
nf
o
ri
n
f
i
n
i
t
e
l
ys
m
a
l
lc
r
a
c
kl
e
n
g
t
hi
n
c
r
e
m
e
n
t
s,
b
u
to
n
e
伺.nw
r
i
t
ei
tf
o
rt
h
ee
n
t
i
r
ec
h
a
n
g
efromt
h
eu
n
c
r
a
c
k
e
dt
o.
af
u
l
l
yc
r
a
c
k
e
ds
t
a
t
e
:
I
:
1U =I
:
1Wf+I
:
1Wb
(
9
.
9
)
Casel
b
. StrengthLimitandBondS
l
i
p
.-Byf
o
r
c
ee
q
u
i
v
a
l
e
n
c
ec
o
n
d
i
t
i
o
n,
t
h
ef
o
r
c
e
a
α
o
s
st
h
ec
r
a
c
kp
l
a
n
e,Ac
,
刀 mustbeequalt
ot
h
ebondf
o
r
c
e叫ん accumula
t
edo
v
e
rt
h
e
四ゅん(巧=u
l
t
i
m
a
t
ebond抑制).S
i
n
α s'
2
:L
6.
a
ndAc:
;
:
;
付2
/
4(
b= ruame
t
e
r
b
o
n
ds
l
i
p1
ぱt
h
e∞n
c
r
e
t
ec
r
o
s
ss
e
c
t
i
o
n,
andt
h
ec
r
o
s
ss
e
c
t
i
o
no
ft
h
eb
a
ri
sn
e
g
l
e
c
t
e
d
),
t
h
ec
r
a
c
k
sc
a
n
b
e
g
i
nt
of
o
r
mi
nt
h
i
smιnneri
f
2
(
9
.
1
1)
s三π
(刀/4U
{
,
)b
Case2
a
.EnergyLimitandNoBondS
l
i
p
.-Asanapproxima
t
i
ono
n
ec
叩 i
m
a
.
g
i
n
e
ef
u
l
lf
o
r
m
a
t
i
o
no
fac
r
a
c
kr
e
l
i
e
v
e
st
h
es
t
r
e
s
sfromt
h
et
r
i
a
n
g
u
l
a
ra
r
e
制
c
r
o
s
s
占a
t
c
h
e
d
t
h
a
tもh
i
nF
i
g
.9
.
2
a
.Thevolumeo
ft
h
er
e
g
i
o
no
b
t
a
.
i
n
e
dbyr
o
ta
.
t
i
n
gt
h
i
sa
r
e
aa
.
bou
t凶 eb
a
ra
x
i
si
s
町=宵l
r
/
1
2
k
.T
h
e
r
e
f
o
r
e,
d
.U =v
i
.σ?/2E(
σ
1=E
f
.
)
.A
I
s
o叫ん =A
c
f
:
.N
e
g
l
e
c
t
i
n
gd
.Wし
web
a
v
e企 U'
2
:sW Sowef
i
n
dt
h
ec
o
n
d
i
t
i
o
nf
o
rc
o
m
p
J
e
t
ec
r
a
c
kf
o
r
m
a
t
i
o
n
:
,
,
2
らさ (
6kGI
E
)
1
/
2
b
-1
/
(
9
.
1
2
)
Case2
b
.EnergyLimitandBondS
l
i
p
.-I
nt
h
i
sc
a
s
e,
i
n
s
t
e
a
do
f町 =E
E
.weha
.
v
e,
2/
romI
I
I
でc
o
n
r
u
t
i
o
no
fe
q
u
i
l
i
b
r
i
u
mw
i
t
ht
h
ebond山 e
s
s
e
so
v
e
rl
e
n
g
t
hs,
町 宵b
4=【J{
.
s,
a
l
l
a
.
t
i
o
n
sb
e
i
n
gt
h
esamea
si
nc
a
s
e2
a
.I
tf
o
l
l
o
w
st
h
a
t~mplete cr~cks c
回 f
orm
t
h
eo
t
h
e
re
q
u
w
h
i
l
et
h
e
r
ei
sbonds
l
i
pi
f
32
/
s'
2
:(
3
π2
kEG /8U
t
,
)1
/
2
b
(
9
.
1
3
)
,
.
t
i
o
n
s(B
話a
n
tandOh
,1983b,1984)a
r
e凶 吋 ont
h
eassump
t
i
on
Theforegoingω!ωa
t
h
a
tt
h
es
もr
e
s
sr
e
l
e
a
s
ez
o
n
e
so
fa
d
j
a
c
e
n
tc
r
a
c
k
sdon
o
to
v
e
r
l
a
p,
ωshowni
nF
i
g
.9
.
2
a
.I
ft
h
e
y
d
o
.as
l
i
g
h
t
l
yd
i
f
f
e
r
e
n
tc
a
l
c
u
l
a
t
i
o
ni
sr
e
q
u
i
r
e
d
.
もi
o
n(B
aZa
n
tandOh,
1
9
8
3
b,
1
9
8
4
)y
i
e
l
d
sc
∞
ー
Asomewhatmores
o
p
h
i
s
t
i
c
a
t
e
dc
a
l
c
u
l
a
山内sr
e
l
a
.
t
i
o
n
sb
e
t
w
e
e
ns
p
a
c
i
n
gsands
t
r
a
i
nfc叩 宙e
n
t
i
n
gat
r
a
n
s
i
t
i
o
nfrommi
c
r
o
c
r配 k
~?ítíatjon t
oc
o
m
p
l
e
t
ec
r
a
c
kfo~ation. T
h
e
s
ecalc~lations h~ve b
e
e
nshownt
oa
g
r
偲 w
i
t
h
句 tr~s.ults o
fC
l
a
r
k(
1
9
5
6
),C
h
iandK
i
r
s
t
e
i
n(
1
9
5
8
),
MatheyandW
a
t
s
t
e
i
n(
1
9
6
0
),
t
h
et
H
o
g
n
e
s
t
a
d(
1
9
6
2
),
姐dkaarand
,
Mat
t
ock(1963).'
118
119
Asa
n
o
t
h
釘 i
l
l
u
s
t
r
a
t
i
o
no
ft
h
ef
r
a
c
t
u
r
ea
p
p
r
o
a
c
ht
ot
h
ef
o
r
m
a
t
i
o
no
fc
o
m
p
l
e
t
ec
r
a
.
c
k
s
‘
c
o
n
s
i
d
e
rt
h
ec
r
品c
k
i
n
go
ft
h
econcreteωvera
s
s
o
c
i
a
t
e
dwi
出 s
t
r
a
i
nf. i
nt
h
ec
o
n
c
r
e
句 b
紅
s
h
r
i
n
k
a
伊 a
ndt
e
m
p
e
r
a
t
u
r
ee
f
f
e
c
t
sd
i
s
r
e
g
a
r
d
e
d
;F
i
g
.9
.
2
b
.F
o
rt
h
es
a
k
eo
fs
i
r
叩l
i
c
i
t
y
,
c
o
n
s
i
d
e
r
o
-d
i
m
e
n
s
i
o
n
a
la
c
t
i
o
no
n
l
y
.F
o
r
m
a
t
i
o
no
f晶 c
o
m
p
l
e
句 c
r
a
c
kr
e
l
i
e
v
e
st
h
ei
n
i
t
i
a
ls
もr
a
i
neneuv
tw
d
e
n
s
i
t
yEf~/2 f
r
o
mvolume同:=kh2w
h
e
r
ey
i
.i
st
h
ec
r
o
s
sh
a
t
c
h
e
da
r
e
ashowni
nF
i
g
.9
.
2
b
,
姐 dk=回 mec
o
n
s
t
a
n
t
.N
e
g
l
e
c
t
i
n
gdWb,
wemusth
a
v
ed U主 d Wo
rG, bh 三 (Ef~/2) Yt.
H
e
r
ewemayB
u
b
s
t
i
t
u
t
eE.=d
.F
ur
t
h
e
r
m
o
r
e
,
a
s
s
u
m
i
n
gt
h
eb
a
rt
obes
u
缶c
i
e
n
t
l
ys
t
r
o
n
g
,
c/s
weknowfromt
h
ep
r
e
v
i
o
u
sa
.
na
l
y
s
i
so
fs
t
a
b
i
l
i
t
yo
fap
a
r
a
l
l
e
lc
r
a
c
ks
y
s
t
e
mt
h
a
tt
h
es
p
a
.
c
i
n
g
h
e
r
ek
ssomec
o
n
日t
a
n
t
.Thusweo
b
t
a
i
n
:
o
ft
h
eopenc
r
a
c
k
sw
i
l
lbes:= k
1hw
1i
r
e
l
a
t
i
o
n
sf
o
rt
h
ef
r
a
c
t
u
r
ep
r
o
c
e
8
Sz
o
n
ei
nn
o
n
l
i
n
e
紅 白n
i
t
ee
l
e
m
e
n
ta
n
a
l
y
s
i
ss
h
o
u
l
de
x
h
i
b
i
ta
a
.
i
n
)a
twhicht
h
es
t
r
e
s
ss
u
d
d
e
n
l
yd
r
o
p
st
oz
釘 o
.However
,
i
もI
S
maximumd
i
s
p
l
a
c
e
m
e
n
t(
o
rs
t
r
山e
rt
h
i
sc
o
n
d
u
s
i
o
ni
sa
l
s
ov
a
l
i
di
nt
h
ep
r
,伺e
n
田 o
ff
r
i
c
t
i
o
na
.
n
do
t
h
e
ri
n
e
l
a
s
t
i
c
n
o
t
ω
r
t
a
i
nwhe
m
e
n
a
p
h
e
n
o
.
a
.
t
h
e
rp
o
w
e
r
f
u
lapproxim
a
.
t
et
e
c
h
n
i
q
u
ef
o
r∞n
s
i
d
e
r
i
n
gi
n
t
e
r
民 t
i
o
n
sb
e
t
w
e
e
nrandomly
Ar
l
o
c
a
t
e
dc
r
a
c
k
si
nal
a
r
g
ebodyh却 肥 田n
t
l
yb
e
e
nd
e
v
e
l
o
p
e
dbyKachanov(
1
9
8
5,
1
9
8
7
)
;s
回
a
l
s
oB笹 川 口i
s
t
e,
Dvorak
,
Z
a
r
z
o
u
randWung(
1
9
8
8
)
.
・
ι
,
1
2
さん (2Gb
/
kE)1/2h
/
,
(
9
.
1
4
)
E
q
s
.9
.
9 9
.
1
3a
r
ei
n
t
e
r
e
s
t
i
n
gi
nt
h
a
t出 eyr
抗i
o
n
a
l
i
z
e出 ei
n
f
l
u
e
n
c
eo
ft
h
et
h
i
c
k
n
e
s
sb
o
fc
o
n
c
r
e
t
ea
r
o
u
n
dt
h
eb
a
r(
o
rb
a
rs
p
a
c
i
n
g
)andt
h
ei
n
f
l
u
e
n
c
eo
ft
h
et
h
i
c
k
n
邸
o
fc
o
n
c
r
e
t
e
c
o
v
e
r
.Withr
e
g
a
.
r
dt
oE
q
.9
.
1
3,
o
n
emayr
e
c
a
l
lt
h
ee
m
p
i
r
i
c
a
lf
o
r
m
u
l
ae
s
t
a
b
l
i
s
h
e
db
y
e
x
t
e
n
s
i
v
e
s
t
a
t
i
s
t
i
c
a
la
.
n
a
l
y
由 o
f
t
e
s
td
a
t
abyG
e
r
g
e
l
yandL
u
t
z(
1
9
6
8,
s
e
ea
l
s
oM
e
i
e
r岨 dG
e
r
g
e
l
y
,1
9
8
1
),
a
c
c
o
r
d
i
n
gt
ow
h
i
c
ht
h
ec
r
a
c
kw
i
d
t
hOci
sr
o
u
g
h
l
yp
r
o
p
o
r
t
i
o
n
a
lt
oA~/3. S
i
n
c
eAci
sr
o
u
g
h
l
y
t
h
i
si
so
n
l
ys
l
i
g
h
t
l
yd
i
f
f
e
r
四 t
fromE
q
.9
.
1
3
.UnderBomemores
o
p
h
i
s
t
i
c
a
t
e
d
p
r
o
p
o
r
t
i
o
n
a
lt
oh,
h
ep
r
o
p
o
r
t
i
o
n
a
l
i
t
yo
fd
oA~/3 C組 b
eo
b
t
a
i
n
e
dbyf
r
a
c
t
u
r
e田 町gy姐 a
l
y
s
i
s
a
s
s
u
m
p
t
i
o
n
s,t
ct
(
B
a
-z
a
n
tandOh
,1983b,
1
9
8
4
)
.
田 i
tc
l
e
a
.
rt
h
a
tv
a
r
i
o
u
se
x
i
s
t
i
n
gp
r
o
v
i
s
i
o
n
sr
e
g
a
r
d
i
n
gm
i
n
i
Thef
o
r
e
g
o
i
n
gd
i
s
c
u
s
s
i
o
nmak
田 m
entandc
r
a
c
kw
i
d
t
hc
o
u
l
dbea
tl
e
a
s
tp
a
r
t
i
a
l
l
yj
u
s
t
i
f
i
e
d叩 dp
r
o
b
a
b
l
ya
l
s
o
mumr
e
i
n
f
o
r
乱c
t
u
r
em
e
c
h
a
n
i
c
s
.
i
m
p
r
o
v
e
dbyt
h
eu
s
eo
ff
r
9
.
4I
n
t
e
r
a
c
t
i
n
gMicrocracks
M
o
d
e
l
i
n
go
ft
h
em
i
c
r
o
c
r
a
c
ks
y
s
t
e
mi
nt
h
ef
r
a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
eh
e
l
p
si
nd
e
v
e
l
o
p
i
n
g陪
t
i
o
n
a
ls
t
r
e
s
s
d
i
s
p
l
a
.
cemen
もo
rs
t
r
e
s
s
s
t
r
a
i
nr
e
l
a
t
i
o
n
sf
o
rt
h
ef
r
a
c
t
u
r
ep
r
o
c
e
s
sz
o
n
e
.I
ns
u
c
h
m
o
d
e
l
s,
i
ti
si
m
p
o
r
l
a
n
tもot
a
.
k
ei
n
t
oa
c
c
o
u
n
tt
h
ei
n
t
e
r
a
c
t
i
o
no
fi
n
d
i
v
i
d
u
a
lm
i
c
r
o
c
r
a
c
k
s
.T
h
i
s
i
n
t
e
r
a
c
t
i
o
nmayl
e
a
dt
oi
n
s
t
a
b
i
l
i
t
i
e
sandl
o
c
a
l
i
z
a
t
i
o
no
fc
r
a
c
k
i
n
g
.
n
山eもe
r
m
i
n
a
lp
h
a
s
eo
fm
i
c
r
o
c
C
o
n
s
i
d
e
rf
o
rexamplet
h
ei
d
e
a
l
i
z
e
ds
i
t
u
a
t
i
o
ni
nF
i
g
.9
.
3
.I
r
a
c
k
i
n
g,
twoa
d
j
a
c
e
n
tc
r
a
c
kt
i
p
sc
o
a
l
目 白 出 t
h
el
i
g
a
m
e
n
tb
e
t
w
e
e
nthemi
sg
e
t
t
i
n
gt
o
r
n
.T
h
i
s
∞ales田 n臼 mayberoughlymodeledastheterminalpropagationofacircumferentialcrack
i
nar
o
u
n
db
a
r(
F
i
g
.9.
3
)w
h
i
l
et
h
er
a
d
i
u
so
ft
h
ec
i
r
c
u
l
a
rl
i
g
a
m
e
n
ta
p
p
r
o
a
c
h
e
sz
e
r
o.B
a
s
e
d
,
onec
姐
c
a
l
c
u
l
a
t
et
h
e
ont
h
eknowns
o
l
u
t
i
o
n
so
ft
h
es
t
r
e
s
si
n
t
e
n
s
i
t
yf
a
c
t
o
rf
o
rt
h
i
sp
r
o
b
l
e
m
diagramo
ft
h
ea
p
p
l
i
e
dr
e
m
o
t
es
t
r
e
s
sv
s
.t
h
ed
i
s
p
l
乱ε
ementduet
oc
r
a
.
c
k
i
n
g
;F
i
g
.9
.
3
b
. An
偲 s
s
e
samaximumd
i
s
p
l
a
c
e
m
e
n
ta
f
t
e
rw
h
i
c
h
i
n
t
e
r
e
s
t
i
n
gp
r
o
p
e
r
t
yo
ft
h
i
sd
i
a
g
r
a
mi
st
h
a
ti
tp
o
s
t
h
e
r
ei
sas
n
a
p
b
a
c
ki
n
s
t
a
b
i
l
i
t
y
.
もC組 b
ep
r
o
v
e
nmoreg
e
n
e
r
a
l
l
yt
h
a
.
tamaximumd
i
s
p
l
a
ε
e
m
e
n
tw
i
t
hs
n
a
.
pbacki
n
s
t
a
b
i
l
i
t
y
I
もo
cωrf
o
re
v
e
r
yt
y
p
eo
fc
r
a
.
c
kl
i
g
a
.
mentt
e
a
r
i
n
g,
p
r
o
v
i
d
e
dt
h
el
i
g
a
m
e
n
tt
r
組 s
m
i
t
saf
o
r
c
e
.
mus
i
fo
n
l
yamomentb
u
tnof
o
r
c
ei
st
r
a
n
s
m
i
t
t
e
da
c
r
o
s
st
h
el
i
g
a
m
叩 t
,
t
h
e
nn
o
Ont
h
eo
t
h
e
rh日 d,
a
.
Z
副
,1
9
8
7
b
)
.S
i
n
回 t
h
elatterωei
su
n
l
i
k
e
l
yt
op
r
e
v
a
i
li
n伽 f
r
a
c
t
u
r
e
s
n
a
p
b
a
c
ko
c
c
u
r
s(B
倒
z
o
n
e,i
tf
o
l
l
o
w
sf
r
o
mt
h
i
sa
n
a
l
y
s
i
st
h
a
tt
h
e町 側 ー d
i
s
p
l
紅 白n
en
t(
o
rs
t
r
e
s
s
s
t
叩n
)
p
r
o
c
CONCLUDINGRE
乱1ARKS
Thep
r
e
s
e
n
te
x
p
o
s
i
t
i
o
no
ft
h
es
t
a
.
t
e
トo
f
t
h
e
-a
r
ti
nc
o
n
c
r
e
t
ef
r
a
c
t
u
r
ec
o
n
c
e
p
t
sandd
e
t
e
r
m
i
目e
a
r
c
ha
c
t
i
v
i
も
yi
nr
e
偲 n
tye
釘 S
n
a
t
i
o
no
fm
a
t
e
r
i
a
lp
r
o
p
e
r
t
i
e
sr
e
f
l
e
c
t
sat
r
e
m
e
n
d
o
u
ss
u
r
g
ei
nr
e
a
n
ddocument
8al
a
r
g
ed
e
g
r
田 o
fp
r
o
g
r
e
s
sw
h
i
c
hh
a
sb
e
e
na
c
h
i
e
v
e
do
v
e
rar
e
l
a
t
i
v
e
l
ys
h
o
r
t
s
p
a
no
ft
i
m
e
. Whileo
n
l
ya
b
o
u
tt
e
ny
e
a
r
sagot
h
ea
p
p
l
i
日 b
i
l
i
t
yo
ff
r乱c
t
u
r
emech
組 i
c
st
o
m組 ye
x
p
e
r
t
snowa
g
r
偲 t
h
a
tf
r
a
c
t
u
r
em
e
c
h
a
n
i
c
sc
a
nh晶v
e
c
o
n
c
r
e
t
es
t
r
u
c
t
u
r
e
sWasd
o
u
b
t
e
d,
ac
o
n
s
i
d
e
r
a
b
l
eimp
.
a
c
t,
i
m
p
r
o
v
i
n
gt
h
es
a
f
e
t
y叩 de
∞nomyof∞ncretestructuresandma
.
k
i
n
g
newd
e
s
i
g
n
sp
o
s
s
i
b
l
e
.
r
回 f
o
l
l
o
w
u
pr
e
p
o
r
t
s,
ACICommitt
田 4
4
6onF
r
a
c
t
u
r
e~ech組ics w
i
l
la
t
t
e
m
p
tもo
I
nもh
r
e
v
i
e
wt
h
e8
ta
.
t
e
-O
ιth
e
-a
r
ti
na
p
p
l
i
c
a
t
i
o
n
so
ff
r
a
c
t
u
r
em
e
c
h
a
n
i
c
st
os
t
r
u
c
t
u
r
a
lb
e
h
a
v
i
o
r,
w
i
t
h
.
av
i
e
wt
o
w
a
r
dp
o
t
e
n
t
i
a
lc
o
d
ei
m
p
r
o
v
e
m
e
n
t
s,
t
h
es
t
a
t
e
o
f
.
・
t
h
e
a
r
ti
nf
i
n
i
t
ee
l
e
m
e
n
t企a
c
t
u
r
e
c
r
e
t
es
t
r
u
c
t
u
r
e
s,
andt
h
es
t
a
t
e
o
f
t
h
e
-a
.
r
ti
nm
o
d
e
l
i
n
go
ft
h
er
a
t
ee
f
f
e
c
t
sand
a
n
a
l
y
s
i
so
f∞n
d
y
n
a
m
i
cf
r
a
c
t
u
r
eo
fc
o
n
c
r
e
t
e
.
Acknowledgment. Thanksf
o
rh
e
l
p
w
i
t
hc
o
m
p
i
l
a
t
i
o
no
ft
h
er
e
f
e
r
e
n
c
剖 a
ndv
a
r
i
o
u
s
m 泊i
o
n
sa
r
ed
u
et
oJ
.
-K
.Kim,
Y.Xi,
M.R.Tabba
叫 a
nd~.T. K回 e
凶 (
a
l
lgradua
もer
e
s
e
a
.
r
c
h
制
i
s
t
組 t
sa
tN
o
r
t
h
w
e
s
t
e
r
nU
n
i
v
e
r
s
i
t
y
)
.
1
2
0
REFERENCESANDBIBLIOGRAPHY
1
.ACI226R・8
0
:“
C
o
n
t
r
o
lo
fC
r
a
c
k
i
n
gi
nC
o
n
c
r
e
t
eS
t
r
u
c
t
u
r
e
s
,
"reportedbyACICommittee
2
2
6(
c
h
a
.
ir
e
dbyD
.D
a
r
w
i
n
)
.
2
.ACI224.1R
・8
4
:“
C
a
u
s
e
s,
E
v
a
l
u
a
t
i
o
nandR
.
e
pa
.
i
ro
fC
r
a
c
k
si
nC
o
n
c
r
e
t
eS
t
r
u
c
t
u
r
e
s,
"R勾 or凶
byACICommittee2
2
4(
c
ha
.
i
r
e
dbyD
.D
a
r
w
i
n
)
.
3
.A
r
r
e
a,
M.,
andI
n
g
r
a
飾
品
, A.R.,
1
9
8
1,
‘・Mixed-ModeCrackPropagationi
nMortara
n
dC
o
n
.
"Report31・13,
Corn
e
J
lU
n
i
v
e
r
s
i
t
y
,
I
t
h晶
c
a
.
c
r
e
t
e,
4
. ASTM,
1
9
8
3,
“
Sも岨d
a
r
dT
e
s
tMethodf
o
rP
l
a
n
e
.
S
t
r
a
i
nF
r
a
c
t
u
r
eT
o
u
g
h
n
e
s
so
fM
e
t
a
l
l
i
cMa
t
e
r
i
a
l
s
“
, StandardE399-83,
AnnualBooko
fASTMS
t
a
n
d
a
r
d
s,
v
.03.01,
p
p
.5
1
9
5
5
4
.
J
.,
Cooper
,G.A.,
andK
e
l
l
y
,
A
.,
1
9
7
1
“
,SingleandMultipleF
r
a
c
t
u
r
e,
"Proc.The
5
.A
v
e
s
t
o
n,
N
a
t
i
o
n
a
lP
h
y
s
i
c
a
lL
a
b
o
r
a
t
o
r
y
,
G
u
i
l
d
f
o
r
d,
S
u
r
r
e
y
,
U
.
K
.,
IPC
P
r
o
p
e
r
t
i
e
so
fF
i
b
e
rC
o
m
p
o
s
i
t
e
s,
S
c
i
e
n
c
eandT
e
c
h
n
o
l
o
g
yP
r
e
s
sL
t
d
.
,
B
.,
H
a
s
s
o
,A.,
andK
h
a
l
i
f
a
,
S
.,
1
9
8
7
“
,AStudyofModeI
IS
h
e
a
rF
r
a
c
t
u
r
eo
fN
o
t
c
h
e
d
6
.B
a
r
r
Beams,
C
y
l
i
n
d
e
r
sandC
o
r
e
s
,
"Proc.,
SEM-RILEMI
n
t
.C
o
n
f
.o
nF
r
a
c
t
u
r
eo
fC
o
n
c
r
e
t
ea
.
n
d
Rock,
e
d
s
.Sh
.
a
h,
S
.P
.,
.
a
ndS
w
a
r
t
z,
S
.E
.,
Houston,
p
p
.3
7
0
・3
8
2
.
7
.B
a
r
r
e
n
b
l
.
a
t
t,
G
.,
.
1 1959“
,TheFormationofEquilibriumCracksduringB
r
i
t
t
l
eF
r
a
c
t
u
r
e
,
G
e
n
e
r
a
lI
d
e
a
s.
a
ndHypoth
邸 i
s
,Axia
l
l
yS
y
m
m
e
t
r
i
cC
r
a
c
k
s,
"Pl'i
k
l
.M
a
t
. Mekh.,
V
.23,
N
o
.
p
p
.4
3
4
・4
44.
3,
8
.B
a
r
r
e
n
b
l
a
t
t,
G
.,
.
1 1962“
,TheMathematicalThωryofEquilibriumCraιksi
nB
r
i
t
t
l
eF
r
a
c
.
t
u
r
e
,
"Adv
.
a
n
c
e
dA
p
p
l
.M
e
c
h
.,
V
.7
,
p
p
.5
5
-1
2
9
.
,
S
.B
.,
andB
u
d
i
a
n
s
k
i,
B
.,
1
9
4
9
“
,AMathematicalTheoryo
fP
l
a
s
t
i
c
i
t
yBasedo
nt
h
e
9
.B
a
t
d
o
r
f
C
o
n
c
e
p
to
fS
l
i
p,
"NACATechNoteTN18
71
.
1
0
.B
a
i
a
n
t,
Z
.P
.,
1
9
7
6,
“
I
n
s
t
.
a
b
i
l
i
t
y
,
D
u
c
t
i
l
i
t
yandS
i
z
eE
t
f
e
c
ti
nS
t
r
a
i
n
-S
o
f
t
e
n
i
n
gC
o
n
c
r
e
t
e,
"J
.
i
v
.,
ASCE,
V
.1
0
2,
N
o
.EM2,
p
p
.3
3
1
・
3
4
4
.
o
ft
h
eE
n
g
n
g
.M
e
c
h
.D
1
.Bai.
a
n
t,
Z
.P
.,
1
9
8
2
“
,CrackBandModeJf
o
rF
r
a
c
t
u
r
eo
fG
e
o
m
a
t
e
r
i
a
l
s,
"Proc.,
4
t
hI
n
t
.C
o
n
f
.
1
o
fNumer.M
e
t
h
.i
nGeomech.,
e
d
.E
i
s
e
n
s
t
e
i
n,
Z
.,
Edmonton,
A
l
b
e
r
t
,
.
aV.3,
p
p
.1
1
3
7
1
1
5
2
.
1
2
1
ゅ.
Bai
胡t,Z
.P.,1985b,
“
F
r
a
c
t
u
r
ei
nC
o
n
c
r
e
t
eandRe
i
n
f
o
r
c
e
dC
o
n
c
r
e
t
e
,
"i
nM
e
c
h
a
n
i
c
8o
f
o
r
t
h
w
e
s
t
e
r
nU
n
i
v
.i
n1
9
8
3
),
0
0
.
G
ω
m
a
t
e
r
i
a
l
s(
P
r
o
c
.o
fIUTAMP
r
a
g
e
rSymp. heJd乱tN
Bai
胡t,
Z
.P
.,
JohnWiley&S
o
n
s,
C
h
i
c
h
e
s
t
e
randNewYork,
pp.259
・
3
0
3
.
げ. B
ai
a
.
n
t
,
Z
.P
.,
1
9
8
5
c
“
,Mecha
.
n
i
c
so
fF
r
a
c
t
u
r
eandP
r
o
g
r
e
s
s
i
v
eC
r
a
c
k
i
n
gi
nC
o
n
c
r
e
t
eS
t
r
u
c
-
,Chap1iEFTacureMechanicsofConcrete:Strudura』ApplicatioEmdNumerical
t
u
r
e
s
v
C
a
l
c
山.
t
i
o
n
,
e
d
.S
i
h,
G
.
C
.,
叩dDiTommaso,
A.
,
M
a
r
t
i
n
u
s
N
i
j
h
o
貧
"D
o
o
r
d
r
e
c
h
t
B
o
s
t
o
n,
p
p
.
1
9
.
I~. Bd
日 t
,
Z
.P
.,
1
9
8
6,
"Mech
a
.
l
I
i
c
so
fD
is
t
r
i
b
u
t
e
d白 紙 k
i
n
g,
"Appl
.Mech.R
e
v
.,
ASME
,
V
.3
9,
N
o
.5
,
p
p
.6
7
5
・
7
0
5
.
1
9
.B
a
z
a
.
n
t
,Z
.P
.,
1
9
8
7
晶
,
“
F
ractureEnergyofHeterogeneousMaterialsandSimilitude,
"Proc.
SEM-RILEMI
n
t
.Con
f
.onF
r
.
a
c
t
u
r
eo
fC
o
n
c
r
e
t
eandRo
c
k,
e
d
s
.Sh
油
,S
.P
.andS
w
a
r
t
z,
S
.
H
o
u
s
t
o
n,
p
p
.3
9
0
4
0
2
.
E
.,
品目t
,
Z
.P
.,
1
9
8
7
b
“
,Sn.
a
phackI
n
s
t
a
b
i
l
i
t
ya
tCraιkL
i
g
a
m
e
n
tT
e
a
r
i
n
ga
.
n
di
t
sI
m
p
l
i
c
a
t
i
o
n
2
0
. Ba
f
o
rF
r
a
.
c
t
u
r
唱 M
i
c
r
o
m
e
c
h
a
n
i
c
s,
"Cem.andConcr.Res.,
V
.1
7,
N
o
.6,
p
p
.9
5
1
・
9
6
7
.
.Bai胡t
,Z
.P
.,
1
9
8
8
“
,S
t
a
.
b
l
eS
t
a
t
e
sa
.
n
dP.
a
t
ho
fS
t
r
u
c
t
u
r
e
sw
i
t
hP
l
a
s
t
i
c
i
t
yo
rDam
.
a
g
e,
"J
.
21
E
n
g
.Mech.,
ASCE
,
V
.1
1
4,
N
o
.1
2
,
p
p
.2
0
1
3
2
0
3
4
.
2
2
.B
a
i
a
n
t,
Z
.P
.,
1
9
8
9
a
.
,
“I
d
e
n
t
i
f
i
c
土
.
aionofStrain-SofteningConstitutiveEqu.
a
t
i
o
nf
r
o
mS
p
e
c
i
menso
fD
i
f
f
e
r
e
n
tS
i
z
凶 fC
em.叩 dConα.Res.
,
V
.1
9,
p
p
.9
7
3・9
7
7
.
2
3
.Bai
.
a
n
t,
Z
.P
.,
1
9
8
9
b
“
,GeneralMethodf
o
rD
e
t
e
r
m
i
n
i
n
gT
a
n
g
e
n
t
i
a
lS
t
i
f
f
n
e
s
sMatrixf
o
rS
t
a
b
i
l
i
t
yA
n
a
l
y
s
i
so
fS
t
r
u
c
t
u
r
e
sw
i
t
hGrowingI
n
t
e
r
a
ι
t
i
n
gC
r
a
c
k
s,
"H
i
l
l
e
r
b
o
r
g
'
sA
n
n
i
v
e
r
sa
.
r
y
Volume(
P
r
o
c
.o
fC
o
n
f
. onF
r
a
c
t
u
r
eo
fC
o
n
c
r
e
t
eandRo
c
kh
e
l
di
nL
u
l
e
a,Sweden,June
1
9
8
9
)
.
2
4
.Bai
副
官
,Z
.P
.,
B
e
l
y
t
s
c
h
k
o,
T
.B
.,
andChang
,T.-P.,1984“
,ContinuumModelf
o
rS
t
r
a
i
n
S
o
f
t
e
n
i
n
g
,
"J
.E
n
g
n
g
.M
e
c
h
.,
ASCE,
V
.1
1
0,
N
o
.1
2,
p
p
.1
6
6
6
1
6
9
2
.
2
5
.Ba
.
Za
n
t,
Z
.P
.,
andC胡
,Z
.,
1
9
8
6
a
,
‘,S
i
z
eE晶 c
ti
nB
r
i
t
t
l
eF
a
i
l
u
r
eo
fU
n
r
e
i
n
f
o
r
c
e
dP
i
p
e
s
,
"ACI
J
.,
V
.8
3,
p
p
.3
6
5
3
7
3
.
ね,
n
t,
Z
.P
.,
.
a
ndCao
,Z 1986b“
,SizeEtf
e
c
ti
nS
h
e
a
rF
a
i
l
u
r
eo
fP
r
e
s
t
r
e
s
s
e
dC
o
n
c
r
e
t
e
2
6
.Ba
"ACIJ
.,
V
.8
3
,
p
p
.2
6
0
2
6
8
.
Beams,
叶
1
2
.B
a
i
a
n
t,
Z
.P
.,
1
9
8
3
“
,CommentonOrthotropicModelsf
o
rC
o
n
c
r
e
t
eandG
e
o
m
a
t
e
r
i
a
l
s
,
"J
.
ASCE
,
V
.1
0
9,
N
o
.3,
p
p
.8
4
9
8
6
5
.
E
n
g
n
g
.Mech.,
.
Z
a
n
t,
Z
.P
.,
回dCao,Z
.,
1
9
8
7“
,S
i民 E
t
f
e
c
ti
nP
u
n
c
h
i
n
gS
h
e
a
rF
a
i
l
u
r
eo
fS
l
a
b
s
,
"ACI
2
7
. Ba
S
t
r
u
c
t
.J
.,
V
.8
4,
p
p
.4
4
5
3
.
Z
.P.
,1984a“
,SizeE
f
f
e
c
ti
nB
l
u
n
tF
r
a
c
t
u
r
e
:C
o
n
c
r
e
t
e
,Rock,Metal,
"J
.E
n
g
n
g
.
1
3
.B
a
i
a
n
t,
,
ASCE
,
V
.1
1
0
,
N
o
.4,
p
p
.5
1
8
5
3
5
.
Mech.
.
Z
a
n
t,
Z
.P
.,
姐dCedolin,L
.,1
9
7
9
“
,BluntCrackBa
.
n
dP
r
o
p
a
g
a
t
i
o
ni
nF
i
n
i
t
eE
l
e
m
e
n
t
2
8
. Ba
"J
.E
n
g
n
g
.Mech.D
i
v
.,
ASCE,
V
.1
0
5,
N
o
.EM2,
P
r
o
c
.Paper1
4
5
2
9,
p
p
.2
9
7
・
315.
A
n
a
l
y
s
i
s,
Z
.P
.,1
9
8
4
b,
“ImbricateContinuum.
a
ndI
t
sV.
a
r
i
a
.
t
i
o
n
a
lD
e
r
i
v
a
t
i
o
n,
"J
.E
n
g
n
g
.
1
4.B
a
i
a
n
t,
Mech.,
ASCE,
V
.1
1
0,
N
o
.1
2,
p
p
.1
6
9
3
1
7
1
2
.
2
9
.Bai
拍 t
,
Z
.P
.,
.
a
ndC
e
d
o
l
i
n,
L
.,
1
9
8
0
“
,FractureMech.
a
I
U
c
so
fR怠i
n
f
o
r
c
e
dC
o
n
c
r
e
t
e,
"J
.E
n
g
n
g
.
i
v
.,
ASCE
,
V
.1
0
6,
N
o
. EM6,
P
r
o
c
. Pa
.
p
e
r1
5
9
1
7,
p
p
.1
2
8
7
1
3
0
6
;w
i
t
hD
i
s
c
u
s
s
i
o
n
Mech.D
andC
l
o
s
u
r
e,
V
.1
0
8,
EM2,
1
9
8
2,
p
p
.4
6
4
4
71
.
1
5
.B
a
i
a
n
t,
Z
.P
.,1
9
8
5
a,
“
F
r
a
c
t
u
r
eM
e
c
h
a
n
i
c
sandS
t
r
a
i
n
S
o
f
t
e
n
i
n
gi
nC
o
n
c
r
e
t
e,
"Preprints,
U.
S
.
J
a
p
a
nS
e
m
i
n
a
ronF
i
n
i
t
eE
l
e
m
e
n
tA
n
a
l
y
s
i
so
fR
e
i
n
f
o
r
c
e
dC
o
n
c
r
e
t
eS
t
u
c
t
u
r
e
s,
Tokyo
,
V
.
1,
p
p
.4
7
・
6
9
.
3
0
.B
a
i
a
n
t,
Z
.P
.,
組dCedolin,
L
.,
1
9
8
3
“
,FiniteElernentModelingo
fCraιkBandP
r
o
p
a
g
a
t
i
o
n
,
"
J
.S
t
r
u
c
t
.E
n
g
n
g
.,
ASCE,
V
.1
0
9,
N
o
.ST2,
p
p
.6
9
・9
2
.
1
2
2
1
2
3
31
.B
a
i
a
n
t,
Z
.P
.,
andC
e
d
o
l
i
n,
L
.、
1
9
8
4
“
,ApproximateLinearAnalysisofConcreteFractureby
R
c
u
r
v
e
s,
"J
.S
t
r
u
c
t
.E
n
g
n
g
.,
ASCE,
V
.1
1
0,
p
p
.1
3
3
6
・1
3
5
5
.
4
5
. Bai
叩t,
Z
.P
.,
andL
i
n
,F.-B.,1988a
“
,NonlocalYieldLimitDegradat
i
o
n,
"I
n
t
.J
.f
o
rNum.
,
V.2
6,
p
p
.1
8
0
5
・1
8
2
3
.
Meth.i
nEngng
.
3
2
. Ba
.
i
a
n
t,
Z.P
.,
姐dCedolin,
L
.,
1
9
9
0,
S
t
a
b
i
l
i
t
yo
fS
t
r
u
c
t
u
r
e
s
:P
r
i
n
c
i
p
l
e
so
fE
1
a
s
t
i
c,
I
n
e
l
a
s
t
i
c
andDamageT
h
e
o
r
i
e
s,
O
x
f
o
r
dU
n
i
v
.P
r
e
5
s,
NewYork,
i
np
r
e
s
s
.
4
6
.B話 相 t
,
Z
.P
.,
andL
i
n
,
F
.
B
.,
1
9
8
8
b
“
,NonlocalSmearedCrackingModelf
o
rC
o
n
c
r
e
t
eF
r
a
c
,
"J
.o
fS
t
r
u
c
t
.D
i
v
.,
ASCE
,
V
.1
1
4
,
N
o
.1
1,
p
p
.2
4
9
3
2
5
1
0
.
t
u
r
e
3
3
. Ba
.
i
姐t,Z
.P
.,
andEst
e
n
s
s
o
r
o
,1
.F
.,
1
9
7
9
“
,SurfaιeSingularityandCr;
u
:
kP
r
o
p
a
g
a
t
i
o
n
,
"
I
n
t
.J
.o
fS
o
l
i
d
sandS
t
r
u
c
t
.,
V
.1
5
,
p
p
.4
0
5
・
426(
a
l
5
0V
.1
6
,
p
p
.479
・
481
.
)
4
7
.Ba
.
i
回 t
,
Z
.P
.andOh,
B
.
H
.,
1
9
8
3
a
“
,CrackBandTheoryf
o
rF
r
a
c
t
u
r
eo
fCo
n
c
r
e
t
e
,
"Mater.
組dStruct.,
V
.1
6
,
N
o
.9
3
,
p
p
.1
5
5
・1
7
7
.
3
4
.B
a
i
a
n
t,
Z
.P
.,
叩dGambarova,
P
.,
1
9
8
0
“
,RoughCracksi
nR
e
i
n
f
o
r
c
e
dC
o
n
c
r
e
t
e
,
"J
.o
ft
h
e
S
t
r
u
c
t
.D
i
v.
,
ASCE,
V
.1
0
6
,No.ST4,
p
p
.8
1
9
8
4
2
.
却し Z
.P
.,
andOh,
B
.H
.,
1
9
8
3
b,
噌p.
a
c
i
n
go
fC
r
a
c
k
si
nR
e
i
l
l
f
o
r
c
e
dC
o
n
c
r
e
t
e,
"J
.o
f
4
8
. Bai
V
.1
0
9,
No
・ 9
,pp. 2066・2085;s
e
ea
l
s
o“
Cr
a
.
c
kS
p
a
c
i
n
gi
nR
e
i
n
f
o
r
c
e
d
S
t
r
u
c
t
.E
n
g
n
g
.,
C
o
n
c
r
e
t
e
:ApproximateS
o
l
u
t
i
o
n
,
"J
.o
fS
t
r
u
c
t
.E
n
g
n
g
.,
V
.1
0
9,
N
o
.
9p
p
.2
2
0
7
・
2
2
1
2
.
3
5
.Ba
.
i
叩t,Z
.P.
,andGambarova,R.G.,1984“
,CrackShea
.
ri
nC
o
n
c
r
e
t
e
: CrackB姐dM
i
c
r
o
p
l
a
n
eModel,
"J
.o
fE
l
l
g
n
g
.Mech.,
ASCE,
V
.1
1
0,
p
p
.2
0
1
5・2
0
3
5
.
3
6
. Ba
.
i
叩t,Z
.P
.,
叩dGettu,R.,1989“
,Determinationo
fN
o
n
l
i
n
e
a
rF
r
a
c
t
u問 C
h
a
r
a
c
t
e
r
i
s
t
i
c
s
"F
r
;
u
:
t
u
r
eo
fC
o
n
c
r
e
t
eandRock(
P
r
o
c
.of
I
n
t
.C
o
n
f
.
andTime-Dependencef
r
o
mS
i
z
e
E
f
f
e
c
t,
r
a
c
t
u
r
eo
fC
o
n
c
r
e
t
eh
e
l
da
tU
n
i
v
.o
fWales,
C
a
r
d
i
f
f
,UK,Sept. 1989),
onR
e
c
e
n
tD
e
v
.i
nF
e
d
s
.5h
油
,S
.P S
w
a
r
t
z,
S
.E
.,
andB
a
r
r,
B
.,
E
l
s
e
v
i
e
r
,
Londonp
p
.5
4
9
5
6
5
.
吋
3
7
. Ba
.
i
a
n
t,
Z
.P
.
;Gettu,
R
.
;andKazemi,
M.T
.,1
9
8
9,
"
I
d
e
n
t
i
f
i
c
a
t
i
o
no
fN
o
n
l
i
n
e
a
rF
r
a
c
t
u
r
e
e
t
r
y
D
e
p
e
n
d
e
n
tR
P
r
o
p
e
r
t
i
e
sf
r
o
mSi
z
eE
f
f
e
c
tT
e
s
t
sandS
t
r
u
c
t
u
r
a
lA
n
a
l
y
s
i
sB
a
s
e
do
nG回 m
c
u
r
v
e
s,
"ReportNo.89-3/498p,
Ce
n
t
e
r
f
o
rAdv
組c
e
dCemeI
ルB
asedM
a
t
e
r
i
a
l
s,
Northwe
御 m
U
n
i
v
.,
Ev
晶n
s
t
o
n
;s
u
b
m
i
t
t
e
df
o
rp
u
b
l
i
c
a
t
i
o
n,
I
n
t
.J
.Ro
c
kMech.M
i
n
.S
c
i
.,
1
9
9
0
.
3
8
. Baiant,
Z
.P
.,
andKazemi,
M.T.,
1
9
8
8
“
,DeterminationofFractureEnergy,ProcessZone
t,
w
i
t
hA
p
p
l
i
c
a
t
i
o
nt
oRockandC
o
n
c
r
e
t
e
,
"
LengthandB
r
i
t
t
l
e
n
e
s
sNumberf
r
o
mS
i
z
eE品 c
R
e
p
o
r
t8
8
-7
/
498dCent
e
rf
o
rC
o
n
c
r
e
t
ea日dG
e
o
m
a
t
e
r
i
a
l
s,
N
o
r
t
h
w
e
s
t
e
r
nU
n
i
v
.,
E
v
a
n
s
t
o
n
;a
l
s
o
V
.4
4,
1
9
9
0,
p
p
.1
1
1
・1
31
.
I
n
t
.J
.o
fF
r
a
c
t
.,
3
9
.B
a
z
a
n
t,
Z
.P
.,
andKazemi,
M.T
.,
1
9
8
9
a
,
‘'S
i
z
eE
f
f
e
c
tonD
i
a
g
o
n
a
lShe
a
.
rF
a
i
l
u
r
eo
fBe
白山
w
i
t
h
o
u
tS
t
i
r
r
u
p
s,
"ReportNo. 89-8/498S,Centerf
o
rAdvancedCement-BasedM
a
t
e
r
i
a
l
s
,
N
o
r
t
h
w
e
s
t
e
r
nU
n
i
v
.,
E
v
a
n
s
t
o
n
.
.
i
a
n
t,
Z
.P
.,
audKazemi,
M.T
.,
1
9
8
9
b
“
,SizeDependenceofConcreteFractureEnergy
4
0
. Ba
ra
c
t
u
r
eMethod,
"ReportNo. 89-12/B623s,Centerf
o
r
D
e
t
e
r
m
i
n
e
dbyRILEMW
o
r
k
o
f
-F
AdvancedCement-BasedM
a
t
e
r
i
a
l
s,
N
o
r
t
h
w
e
s
t
e
r
nU
n
i
v
.,
Ev
a
.
n
s
t
o
n
j5
u
b
m
i
t
t
e
dt
oI
n
t
.J
.
o
f
F
ra
c
t
.
41
.Ba
.
i
組t,
Z
.P.
,
Kazemi,
M.T
.,
Hasegaw
a
.
,
T
.,
Maza
.
r
s
,
J
.,
1
9
9
1
4
2
. Ba
.
i
却し Z.P.,
祖dKim,
J.K
.,
1
9
8
4
“
,SizeE保 c
ti
nSh
e
a
.
rF
a
i
l
u
r
eo
fL
o
n
g
i
t
u
d
i
n
a
.
l
l
yRe
i
n
f
o
r
c
e
d
Beams,
"J
.o
fACI,
V
.8
1,
N
o
.5,
p
p
.4
5
6
4
6
8
;D
i
s
c
u
s
s
i
o
nV
.82
,
No.4,
p
p
.5
7
9
-5
8
3
.
4
9
.B必
組
色
,Z
.P
.,
andOh,
B
.H
.,1
9
8
4
“
,DeformationsofProgressively白 紙kingReinforced
"ACIJ
.,
V
.8
1,
N
o
.3,
p
p
.2
6
8
2
7
8
.
C
o
n
c
r
e
t
eBeams,
5
0
. Ba
.
i
叩t
,
Z
.P
.,
andOh,
B
.H
.,
}
9
8
5
“
,Micropl組eModelf
o
rP
r
o
g
r
e
s
s
i
v
eF
r
a
c
tu
r
eo
fC
o
n
c
r
e
t
e
組dRock,
"J
.o
fE
n
g
n
g
.Mech.,
ASCE
,
V
.1
1
1,
p
p
.5
5
9
・ 5
8
2
.
51
.Ba
.
iant,
Z
.P
.,
andOhtsubo,
H
.,
1
9
7
7
“
,Stabil
i
t
yC
o
n
d
i
t
i
o
n
sf
o
rP
r
o
p
a
g
a
t
i
o
no
faSystemo
f
,
"Mech拙 i
c
sRe
s
e
a
r
c
hCommunications,
V
.4,
N
o
.5,
p
p
.3
5
3
3
6
6
.
C
r
a
c
k
si
naB
r
i
t
t
l
eS
o
l
i
d
.
i
組t,
Z
.P
.,
andOhtsubo
,
R
.
, 1978,
‘・GeothermalHeatExtr孔ctionbyWaterCirculation
5
2
.Ba
ThroughaL
a
r
g
eCracki
l
lDryHotRρckMass,
"I
n
t
.J
.f
o
rNum. andAna
l
. Meth. i
n
G伺 m
e
c
h
.,
V
.2,
p
p
.3
1
7
・
3
2
7
.
5
3
.Ba
.
i
a
n
t,
Z
.P
.,
Oh
.
t
subo
,R.,
組dAoh,K.,1979“
,S
t
a
b
i
l
i
t
yandP
o
s
t
C
r
i
t
i
c
a
lGrowtho
fa
S
y
s
t
e
mo
fC
o
o
l
i
n
gandS
h
r
i
n
k
a
g
eC
r
a
c
k
s,
"I
n
t
.J
.o
fF
r
a
.
c
t
.,
V
.1
5
,
pp.4
4
3
4
5
6
.
5
4
.Ba
.
i
叩t,
Z
.P
.,
andO
z
b
o
l
t,
J
.,
1
9
8
9
“
,NonlocalMicroplaneModelf
o
rF
r
a
c
t
u
r
e,
Damageand
S
i
z
eE
f
f
e
c
ti
nS
t
r
u
c
t
u
r
e
s,
"R怠port89・lO/498n,
C
e
n
t
e
rf
o
rAdvancedCe
ment
B
a
s
e
dM晶
,t
e
r
i
a
l
s,
N
o
r
t
h
w
e
s
t
e
r
nU
n
i
v
.,
E
v
a
n
s
t
o
n
.
5
5
.Ba
.
i
佃 t
,Z
.P
.,
andP
f
e
i
f
f
e
r,
P
.A
.,
}
9
8
5
a
“
,Testo
fS
h
e
a
rF
r
a
c
t
u
r
eandS
t
r
a
i
n
S
o
f
t
e
r
u
n
gi
n
C
o
n
c
r
e
t
e
,
"2ndSymposiumontheInteractiono
fN
o
n
-N
u
c
l
e
a
rM
u
n
i
t
i
o
n
sonS
t
r
u
c
tu
r
e
s,
PanamaC
i
t
yBe
.
a
c
h,
FL,
p
p
.2
5
4
・
2
6
4
.
5
6
.Ba
.
i
姐t
,
Z
.P andP
f
e
i
f
f
e
r,
P
.
A
.,
1
9
8
5
b
“
,CommentonIngraffeaandPanthaki'sAnalysisof
S
h
e
a
.
rF
r
a
c
t
u
r
eT
e
s
t
so
fC
o
n
c
r
e
t
e,
"F
i
n
i
t
eE
1
ementAn
a
.
l
y
s
i
so
fR息i
n
f
o
r
c
e
dC
o
n
c
r
e
t
eS
t
r
u
c
P
r
o
c
.,
US-Jap
a
.
nS
e
m
i
n
a
r,
e
d
. Meyer,
C
.,
andOkamura
,M.
,
h
e
l
di
nTokyo,
ASCE,
t
u
r
e
s,
NewYork,
p
p
.1
7
4
・1
8
3
.
吋
5
7
.Ba
.
i
加t,
Z
.P
.,
andP
f
e
i
f
f
e
r
,P
.A
.,
1
9
8
6
“
,ShearF
r
a
.
c
t
u
r
eT
e
s
t
so
fCo
n
-c
r
e
t
e
,
"Mater. and
.1
9
,
N
o
.1
1
0,
p
p
.1
1
1
・1
21
.
S
t
r
u
c
tV
咋
4
3
. Ba
.
i
a
n
t,
Z.P
.
;Kim.J
.K
.j.
a
l
l
dP
f
e
i
任
込r
,
P
.A
.,
1
9
8
6
“
,DeterminationofFractureProperties
,
"J
.o
f5
t
r
u
c
t
.E
l
l
g
n
g
.,
ASCE,
V
.1
1
2,
No
・2
,
p
p
.2
8
9
3
0
7
.
f
r
o
mS
i
z
eE
f
f
e
c
tT
e
s
t
s
叩t,
Z
.P
.,
Lee
,
S
.
G
.,
andP
f
e
i
仔e
r,
P
.A
.,
1
9
8
7,
"
S
i
z
eE
f
f
e
c
tT
e
s
t
s.
a
ndF
r
a
c
t
u
r
eCh
.
a
r
a
c
4
4
. Baz
t
e
r
i
s
t
i
c
so
fAluminum,
"Engng.Fra
c
t
.Mech.,
V
.2
6,
N
o
.1,
p
p
.4
5
・
5
7
.
5
8
.Ba
.
i
姐t
,
Z
.P
.,
andP
f
e
i
f
f
e
r
,P
.A
.,
1
9
8
7
“
,DeterminationofFractureEnergyPropertiesfrom
"ACrMater.J
.,
V
.8
4
,
N
o
.6,
pp.4
6
3
4
8
0
.
S
i
z
eE
f
f
e
c
tandB
r
i
t
t
l
e
n
e
s
sNumber,
1
2
4
5
9
. Ba
.
i
岨t
,
Z
.P
.,
叩dPijaudierCabot,
G
.,
1
9
8
7
“
,Modelingo
fD
i
s
t
r
i
b
u
t
e
dDamagebyN
o
n
l
o
c
a
1
Continnumw
i
t
hL
o
ca
JS
t
r
a
i
n,
"Preprints,
4
t
hI
n
t
.Co
n
I
.onNum.Meth.i
nF
r
a
c
t
.M.侭h
.
.
,D.R.J
.Owen,
andM.F
.Kanninen,
h
e
l
di
n8
a
nA
n
t
o
n
i
o,
T,
位
制
,P
p
.
e
d
s
.A
.R
.Luxmore
4
4
1
4
3
2
.
忌a
n
t,
Z
.P
.,
andP
i
j
a
u
d
i
e
r
C
a
b
o
t,
G.
,1988a
.
,
“NonlocalContinuumDamage,
L
o
c
a
l
i
z
a
t
i
o
n
.
6
0
. Ba
"J
.o
fApp
l
.M
e
c
h
.,
ASME
,
V
.5
5,
p
p
.2
8
7
2
9
3
.
I
n
s
t
a
b
i
l
i
t
yandC
o
n
v
e
r
g
e
n
c
e,
6
1
. Ba
.
i
a
n
t,
Z
.P
.,
叩dPijaudier-Cabot,
G
.,
1
9
8
8
b
“
,NonlocalContinuumDamagea
l
l
dMe
a
.
s
u
r
e
mento
fC
h
a
r
a
c
t
e
r
i
s
t
i
cL
e
n
g
t
h,
"i
nM
e
c
h
a
n
i
c
so
fC
o
m
p
o
s
i
t
eM
a
t
e
r
ia
]s-1
9
8
8,
AM.
D9
2
,
e
d
.
G
.J
.Dvor
a
.
k
組dN.Laws,Am. 80c. ofMech. Engrs.,N.Y.(JointASME/SESC
o
n
f
.
,
B
e
r
k
e
l
e
y
,
CA),
p
p
.7
9
・8
5
.
.
ia
n
t,Z
.P叶組dP
i
j
a
u
d
i
e
r
.
C
a
b
o
t
,G.,1989,
“Mea
.
suremento
fC
h
a
r
a
c
t
e
r
i
s
t
i
cL
e
n
g
t
ho
f
6
2
. Ba
n
1
o
c
a
JC
ontinuum,
"J
.o
fE
n
g
n
g
.Mech
.
,
ASCE
,
V.1
1
5,
N
o
.4,
pp.7
5
5
7
6
7
.
No
6
3
.B
a
z
a
n
t,
Z
.P
.,
a
n
dP
r
a
t,
P
.C
.,1
9
8
8
a
“
,Mea
.
suremento
fModeI
I
IF
r
a
c
t
u
r
eE
n
e
r
g
yo
fCon
"NuclearEngng.叩dDes.,
V
.1
0
6,
p
p
.1
・
8
.
c
r
e
t
e,
1
2
5
7
4
.B
e
l
y
t
s
c
h
k
o,
T
.,F
i
s
h,
J
.,
回dEng邑lmann,
B
.E
.,
1
9
8
8
“
,A FiniteElementwithEmbedded
L
oc
a
l
i
z
a
t
i
o
nZo
n
e
s
,
"Comp. Meth. i
nApp
l
.M
ecb_晶ndEngng
.
,V.70,
p
p
.5
9
・
8
9,
N
o
r
t
h
H
o
l
l
晶n
d
.
n
I
s
t
e,
Y
.,
Dvor
a
.
k
, G.J
.,
Z
a
r
z
o
u
r,
J
.,
andWung
,
E
.C
.J
.,
1
9
8
8
“
,OnInteractingCracks
7
5
.B
e
n
v
e
.
s
t
i
cMedia
.
,
"Report,Depa
.
r
t
m
e
n
to
fC
i
v
i
l
a
n
dComplexC
r
a
c
kC
o
n
f
i
g
u
r
a
t
i
o
n
si
nL
i
n
e
a
rE
la
E
n
g
i
n
e
e
r
i
n
g,
R
.
P
.
,
.
ITroy,
NY.
7
6
.B
e
s
s
e
n
d
o
r
f
f
,
M.,
andChudnovsky
,
A
.,
1
9
8
4
“
,TheCrackLayerApproacht
oT
o
u
g
h
n
e
s
sC
h
a
r
a
c
t
e
r
i
z
a
t
i
o
ni
nS
t
e
e
l
,
“Proc.,
I
n
t
.Conf
.onF
r
a
.
c
t
u
問 6
,
(
h
e
l
di
nNewD
e
l
h
i
,
I
n
d
i
a
)e
d
s
.V
a
l
l
u
r
i,
S
.
R
.e
ta
l
.,
V
.6
.
7
7
.B
r
a
m
e
s
h
u
b
e
r
,
W.
,1988
“
,F
r
a
c
t
.M
e
c
h
.P
r
o
p
e
r
t
i
e
so
fYoun
gC
o
n
c
r
e
t
e
,
"(
i
nGerm
加
)
, Doct
o
r
a
lD
i
!
!
s
e
r
t
a
t
i
o
n,
U
n
i
v
e
r
s
i
t
a
tK
a
r
l
s
r
u
h
e
,
Germany.
7
8
.B
r
a
m
e
s
h
u
b
e
r,
W.,
1
9
8
9
,
“
D
i
s
c
u
s
s
i
o
no
f“
E
f
f
e
c
to
fT
e
m
p
e
r
a
t
u
r
eandHUl
l
Ud
i
t
yonF
r
a
c
t
u
r
e
,
"ACIMater.J
.,
V
.8
6,
p
p
.3
3
0
・
3
31
.
E
n
e
r
g
yo
fC
o
n
c
r
e
t
e
Z
.P
.,
組dPrat,P
.C
.,1
9
8
8
b“
,E
f
f
e
c
to
fT
e
m
p
e
r
a
t
u
r
eandHumidityonF
r
a
c
t
u
r
e
6
4
_B
a
i
a
n
t,
"ACIMater.J
.,
V
.8
4,
p
p
.2
6
2
2
71
.
E
n
e
r
g
yo
fC
o
n
c
r
e
t
e,
7
9
.B
r
e
s
l
e
r,
B
.,
andW
o
l
l
.
a
c
k,
E
.(
1
9
5
2
)
“
,Sh品 rStrengthofConcrete,
"Report,Dept. ofC
i
v
i
l
U
n
i
v
e
r
s
i
t
yo
fCa
I
i
f
o
r
n
i
a
,Berkeley;presenteda
tAnnua
IC
o
n
v
e
n
t
i
o
n,
S
t
r
u
c
t
u
r
a
lE
n
E
n
g
n
g
.,
,
.
aheldi
nR
i
v
e
r
s
i
d
e,
CA
,
O
c
t
.1
7
,1952.
g
i
n
e
e
r
sA
s
s
o
c
i
a
t
i
o
no
fC
a
l
i
f
o
r
n
i
6
5
. Ba
.
i
a
n
t,Z
.P
.,Pr
.
a
t
,P
.C
.,
組dTabbara,M.R.,1988,
“AntiplaneShearF
r
a
c
t
u
r
eT
e
s
t
s
(Mode1
1
1
),
"ReportNo. 88-6/498a,Centerf
o
rC
o
n
c
r
e
t
eandG
e
o
m
a
t
e
r
i
a
l
s,
N
o
r
t
h
w
e
s
t
e
r
n
E
v
a
n
s
t
o
n
;a
.
l
s
oACIM
a
t
e
r
.J
.,
i
np
r
e
s
s
.
U
n
i
v
.,
,D.,
andMazar
s
,
J
.,
1
9
8
8
“
,SimplifiedApproachofNonlinearityi
nR-CBe
乱ms
,
"J
.
8
0
.B
r
e
y
s
s
e
V
.1
1
4,
N
o
.2,
p
p
.2
5
1
2
8
6
.
o
fS
t
r
u
c
t
.E
n
g
n
g
.,
.
i
組t
,
Z
.P
.,
a
n
dS
e
n
e
r,
S
.,
1
9
8
8
“
,S
i
z
eE
f
f
e
c
ti
nP
u
l
l
o
u
tT
e
s
t
s,
"ACIMater.J
.,
V
.8
5,
p
p
.
6
6
. Ba
・
3
51
.
3
4
7
6
7
.B
a
i
a
n
t,
Z
.P
.
;S
e
n
e
r,
S
.
;andP
r
a
t,
P
.C
.
,
1
9
8
8
“
,SizeE
f
f
e
c
tT
e
s
t
so
fT
o
r
s
i
o
na
.
lF
a
i
l
u
r
eo
fP
l
a
j
n
忠i
n
f
o
r
c
e
dC
o
n
c
r
e
t
eBeams,
"Mater.andS
t
r
u
c
t
.,
V
.2
1,
p
p
.4
2
5
・
4
3
0
.
andR
a
z
a
n
t,
Z
.P
.,
a
n
dSun,H
.
H
.,1
9
8
7
“
,S
i
z
eE
f
f
e
c
ti
nD
i
a
g
o
na
.
lS
h
e
a
.
rF
a
i
l
u
r
e
:I
n
f
l
u
e
n
c
eo
f
6
8
. B.
"ACJMater.J
.,
V
.8
4
,
N
o
.4,
p
p
.2
5
9
2
7
2
.
A
g
g
r
e
g
a
t
eS
i
z
eandS
t
i
r
r
u
p
s,
.
ia
n
t,
Z
.P
.,
a
l
l
dTabbara
,
M.R.,
1
9
8
9
“
,Bifurc.
a
t
i
o
na
n
dS
t
a
b
i
l
i
t
yi
nS
t
r
u
c
t
u
r
e
sw
i
t
hP
r
o
p
6
9
. Ba
a
g
a
t
i
n
gC
r
a
c
k
s
,
“Parts1
,
I
J
,
Re
p
o
r
t,
D
e
p
t
.o
fC
i
v
i
lE
n
g
n
g
.,
N
o
r
t
h
w
e
s
t
e
r
nU
n
i
v
.,
E
v
a
n
s
t
o
n
・
7
0
. Bai
皿 t
,Z
.P
.,
Tabbara
,M.R.,Kazemi,M.T.,andPijaudier-C.
a
b
o
t,G
.,1
9
8
9,
“
R
a
ndom
.
r
t
ic
1eModelf
o
rF
r
a
c
t
u
r
eo
fA
g
g
r
e
g
a
t
eo
rF
i
b
e
rC
o
m
p
o
s
i
t
e
s,
R疋p
o
r
tN
o
.89
-7
/
4
9
8
r
,
C
e
n
t
e
r
Pa
o
r
t
h
w
e
s
t
e
r
nU
n
i
v
.,E
v
a
z
l
s
t
o
n
;
a
l
s
o
io
fE
I
l
g
r
i
g
f
o
TAdvancedC
e
m
e
n
t
B
a
s
e
dM
a
t
e
r
i
a
l
s,N
ASCE,
V
.1
1
6,
1
9
9
0,
p
p
.1
6
8
6
1
7
0
5.
Mech.,
1
. Ba
.
iant,Z
.P
.,
a
n
dWahab,
A
.B
.,
1
9
7
8
“
,I
n
s
t
a
b
i
l
i
t
yandS
p
a
c
i
n
go
fC
o
o
l
i
n
go
rS
h
r
i
n
k
a
g
e
7
"J
.o
ft
h
eE
n
g
n
g
.M
e
c
l
t
.D
i
v
.,
ASCE
,
V
.1
0
5,
p
p
.8
7
3
・ 8
8
9
.
C
r
a
c
k
s,
Z
.P
.,
制dWahab,
A
.
B
.,
1
9
8
0
“
,S
t
a
b
i
l
i
t
yo
fP
a
r
a
l
l
e
lC
r
a
c
k
si
nS
o
l
i
d
sR
e
i
n
f
o
r
c
e
db
y
7
2
.B
a
i
a
n
t,
,
“I
nt
.J
.o
fS
o
l
i
d
sa
n
dS
t
r
u
c
t
.,
V
.1
6
,
p
p
.9
7
・1
0
5
.
B
a
r
s
Z
.
P
.,
andXu,
K
.,
1
9
8
9
“
,S
i
z
eE
f
f
e
c
ti
nF
a
t
i
g
u
eF
r
a
c
t
u
r
eo
fC
o
n
c
r
e
t
e,
"ReportNo・
7
3
.B
a
i
a
n
t,
8
9
・1
2/B627s,
C
e
n
t
e
rf
0
1
"A
dvancedC
e
m
e
n
t
B
a
s
e
dMa
.
t
e
r
ia
J
s
,
N
o
r
t
h
w
e
s
t
e
r
nU
n
i
v
.,
E
v
a
n
s
t
o
n
・
8
1
.B
r
o
e
k,
D
.,
1
9
8
2
“
,Elementa
巧 E
n
g
i
n
e
e
r
i
n
gF
r
a
c
t
.Mech.,
"3rde
d
.,
Ma
.
r
t
i
n
u
sN
i
j
h
o
f
f
,
Leyden,
N
e
t
h
e
r
l
a
.
n
d
s
,
(
山o4
t
he
d
.,
Ma
.
rt
i
n
u
sN
i
j
h
o
f
f
,
D
o
r
d
r
e
c
h
t,
1
9
8
6
.
)
8
2
.B
r
u
h
w
i
l
e
r,
E
.,
1
9
8
8
“
,Fr.
a
c
t
.Mech.o
fDamC
o
n
c
r
e
t
eS
u
b
j
e
c
t
e
dt
oQu
a
.
s
i
S
t
a
t
i
c細dS
e
i
s
m
i
c
"Laboratoryf
o
rB
u
i
l
d
i
n
gM
a
t
e
r
i
a
l
s(LMC),
S
w
i
s
sF
e
d
e
r
a
lI
n
s
t
.ofTech
.
,
L
o
a
d
i
n
gC
o
n
d
i
t
i
o
n
s,
L
a
u
s
a
n
n
e,
T
h
e
s
i
sN
o
.7
3
9(
i
nGerm
組).
8
3
.B
r
u
h
w
i
l
e
r,
E
.,
a
n
dWittmann,
F
.H
.,1
9
8
9
“
,TheWedgeS
p
l
i
t
t
i
n
gT
e
s
t,
A Methodo
fP
e
r
.
f
o
r
m
i
n
gS
t
a
b
l
eF
r
a
c
t
.Mech.Te
s
t,
"ContributiontotheI
n
t
.C
o
n
f
.onF
r
a
c
t
u
r
e岨dDamage
a
.1988;Engng.F
r
a
c
t
.Mech.(
i
np
r
e
s
s
)
.
o
fC
o
n
c
r
e
t
eandRocki
nV
i
e
n
n
8
4
.B
u
r
t,
W.J
.,
細dDougill,J
.W.,1
9
7
7
“
,ProgressiveFailurei
naModelo
fH
e
t
e
r
o
g
e
n
e
o
u
8
Medium,
"J
.o
ft
h
eE
n
g
n
g
.M
e
c
h
.D
i
v
.,
ASCE,
V
.1
0
3,
N
o
.EM3,
3
6
5
3
7
6p
p
.
8
5
.C
a
r
p
i
n
t
e
r
i,
A
.,
1
9
8
2
“
,Notch-SensitivityandF
r
a
c
t
u
陀 T
e
s
t
i
n
go
fAgg
民 g
a
t
eM
a
t
e
r
i
a
l
s,
"Enr
a
.
c
t
.Mech.,
V
.1
6,
N
o
.1
4,
p
p
.4
6
7
4
81
.
g
n
g
.F
8
6
.C
a
r
p
i
n
t
e
r
i,
A
.,
1
9
8
6
“
,Mech乱n
i
c
a
lDam
a
.
g
eandC
r
a
c
kGrowthi
nC
o
n
c
r
e
t
e
,
"Ma
.
rt
i
n
u
sN
i
j
h
o
f
f
1
i
s
h
e
r
s,
D
o
o
r
d
r
e
c
h
t
.
Pub
1
i
n,
L
.,D
e
iP
o
l
i,
S
.,
andI
o
r
i,
1
.,1
9
8
7
“
,TensileBehaviorofCo
n
c
r
e
t
e,
"J
.o
fE
n
g
n
g
.
8
7
. C吋 o
i
v
.,
ASCE,
V
.1
1
3,
N
o
.3,
p
.4
3
1
M
e
c
h
.D
,
P
.S
.,
1
9
8
1
“
,SomeAspectsofModelingtheBehaviorofReinforcedConcreωUnder
8
8
. Chana
~hea.r Loading,
"T
e
c
l
t
n
i
c
a
IRe
p
o
r
tN
o
.5
4
3,
Cementa
n
dC
o
n
c
r
e
t
eA
s
s
o
c
i
a
t
i
o
n,Wexham
,
pp.2
2
.
S
p
r
i
n
g
s
1
2
6
1
2
7
8
9
. Chi,
M.
,andKirstein,A.F.,1958“
,Flexura}Cra
.
c
k
si
nR
e
i
n
f
o
r
c
e
dC
o
n
c
r
e
t
eBe
乱
.
m
s
,
"J
.o
f
t
h
eACI
,
P
r
o
c
.,
V.5
4,
N
o
.1
0,
p
p
.8
6
5
8
7
8
.
1
0
5
.D
u
g
d
a
l
e,
D.S
.,
1
9
6
0
“
,YieldingofSteelSheetsConta
.
i
n
i
n
gS
l
i
t
s,
"J
.o
fMech.andP
h
y
s
.o
f
S
o
l
i
d
s,
V.8,
p
p
.1
0
0
1
0
8
.
9
0
. Chudnovsky
,A.,1986,
“Cr
乱
c
kL
a
y
e
rTheoryヘi
n1
0
t
hU
.
S
.N
a
t
i
o
na
JC
o
n
g
r
e
s
sonA
p
p
l
Mech.,
A
u
s
t
i
n
,
T
e
x
a
s
.
l
ω
.Dvo
出 n,
E
.N.,
C
u
i
t
i
n
o,
A
.M.,
組dGioia,G.,1989“
,FiniteElementswithDisplamm佃t
I
n
t
e
r
p
o
l
a
t
e
dEmbeddedL
o
c
a
l
i
z
a
t
i
o
nL
i
n
e
sI
ns
e
n
s
i
t
i
v
et
oMeshS
i
z
eandDi
s
t
o
r
t
i
o
n
s,
"sub
m
i
t
も
edt
oI
n
t
.J
.Num.Meth.i
nE
n
g
n
g
.
91
.C
l
a
r
k,
A.P
.,
1
9
5
6
“
,CrackinginReinforcedConcreteFlexuralMe.
m
b
e
r
,
"J
.o
fもheACI
,
P
r
o
c
.,
V
.5
2
,
N
o
.8,
p
p
.8
5
1
8
6
2
.
9
2
. Clough
,
R
.W.,
1
9
6
2
“
,StressDistributionofNorfolkDamn,
"DepartmentofCivilEngineer.
i
n
g,
S
e
r
i
e
s1
0
0,
I
S
5
u
e1
9,
U
n
i
v
e
r
s
i
t
yo
fC
a
l
i
f
o
r
n
i
a
,
B
e
r
k
e
l
e
y
.
9
3
. Cope,
R
.J
.,
1
9
8
4
“
,MateriaJModelingofRea
l,
R
e
i
n
f
o
r
c
e
dC
o
n
c
r
e
t
eS
l
a
b
s,
"Proc. oft
h
e
I
nt
.C
o
n
f
. onComputer-AidedA
n
a
l
y
s
i
sandD
e
s
i
g
no
fC
o
n
c
r
e
t
eS
t
r
u
c
t
u
r
e
s,
h
e
l
di
nS
p
l
i
t
,
Y
u
g
o
s
l
a
v
i
a,
Pa
.
r
t1
,
e
d
s
.D細 j
組i
e
,
F
.e
もa
l
.,
P
i
n
e
r
i
d
g
eP
r
e
s
s,
Sw
組sea,
U.
K
.,
8
5
1
1
8p
p
.
9
4
.C
r
i
s
f
i
e
l
d,
M.A.andW
i
l
l
s,
J
.,
1
9
8
9
“
,AnaJy
s
i
so
fR/CP
a
n
e
l
su
s
i
n
gDi
f
f
e
r
e
n
tC
o
n
c
r
e
t
eMode
l
s,
"J
.Engng.Mech.,
ASCE,
V.1
1
5,
N
o
.3,
p
p
.5
7
8
-5
9
7
.
9
5
.C
u
n
d
a
l
l,
P
.A.,
1
9
7
1
“
,A ComputerModelf
o
rS
i
m
u
l
a
t
i
n
gP
r
o
g
r
e
s
s
i
v
eLa
.
r
g
eS
c
a
l
eMovements
"Proc.ofI
n
t
.Symp.onRockF
r
a
c
t
u
r
e,
ISRM,
Nancy
,
F
r
a
n
c
e,
p
p
.
i
nB
l
o
c
k
yRockS
y
s
t
e
m
s,
2
8
.
9
6
.C
u
n
d
a
l
l,
P
.A
.,
andS
t
r
a
.
ck,
O.D
.L
.,
1
9
7
9,“
AD
i
s
c
r
e
t
eN
u
m
e
r
i
c
a
lModelf
o
rG
r
a
n
u
l
a
r
"Geotechnique,
V.29,
p
p
.4
7
6
5
.
S
u
b
a
s
s
e
m
b
l
i
e
s,
1
0
7
.到危r
e
n,
Z
.,
1
9
8
9,
e
d
.,
"
F
r
a
c
t
. Mech. o
fC
o
n
c
r
e
t
eS
t
r
u
c
t
u
r
e
s,
"(ReportofRILEMTechn.
Co~ittee TC90-FMA),
ChapmanandH
a
l
l,
1
o
n
d
o
n
.
加 制1,
andO
z
b
o
l
t,
J
.(
1
9
9
0
),
8
t
hEuropeanC
o
n
f
e
r
e
n
c
eonF
ra
c
t
u
r
e,
T
o
r
i
n
oーi
np
r
e
s
s
.
1
0
8
.E
l
i
g
e
h
1
0
9
.E
r
i
n
g
e
n,
A
.C
.,
回dEdelen,D.G.B.,
1
9
7
2
“
,OnNonlocalElasticity,
"I
n
t
.J
.Eng.S
c
i
.,
V.
,
p
p
.2
3
3
2
4
8
.
1
0
1
1
0
.E
v
a
n
s
,A.G.,1974“
,SlowCra
.
c
kGrowthi
nB
r
i
t
t
l
eM
a
t
e
r
i
a
l
su
n
d
e
rDyn
a
.
micC
o
n
d
i
t
i
o
n
s
,
"
I
n
t
.J
.o
fMech.,
V.1
0,
p
p
.2
5
1
2
5
9
.
1
1
1
.Ev
日 5
,
R
.H.,
a
l
l
dM
a
r
a
t
l
l
e
,M.S
.,
1
9
6
8
“
,MicrocrackingandStress-StrainCurvesf
o
rConc
r
e
t
ei
nT
e
n
s
i
o
n,
"Mater.乱
ndS
t
r
u
c
t
.,
V.1
,
p
p
.6
1
6
4
.
1
1
2
.G
e
r
g
e
l
y
,
P
.,
組dLutz,
L
.A.,
1
9
6
8,
"MaximumCrackWidthi
nRe
i
n
f
o
r
c
e
dC
o
n
c
r
e
t
eF
l
e
x
u
r
a
l
,
“Causes,
M
e
c
l
t
a
n
i
s
m
s,
andC
o
n
t
r
o
lo
fCr
a
.
c
k
i
n
gi
nC
o
n
c
r
e
t
e
,
SP-20,
ACI
,
D
e
t
r
o
i
t,
Member5
pp.87
・1
1
7
.
1
1
3
.G
e
t
t
u
‘R
.,
Baiant,
Z
.P
.,
andKarr,
M.E 1
9
8
9,
"
F
r
a
c
t
u
r
eP
r
o
p
e
r
t
i
e
sandB
r
i
t
t
l
e
n
e
s
so
fHigh
S
t
r
e
n
g
t
hConc
削 e
,
"Repor印 0・89-10jB627f,
Cent
ぽ 加A
dvancedCeme
叫 B
asedMat
ぽ i
a
l
s,
N
o
r
t
h
w
e
s
t
e
r
nU
n
i
v
.,
E
v
a
n
s
t
o
n
;a
l
s
oACIM
a
t
e
r
.J
.,
i
np
r
e
s
s,
1
9
9
0
.
叶
9
7
. Darwin,
D.,
1
9
8
5
“
,ConcreteCra
.
ckP
r
o
p
a
g
a
t
i
o
n-S
t
u
d
yo
fModelParame
七e
r
s,
"Proc.F
i
n
i
t
e
e
d
s
. Meyer,
C
.,andOkamura,
H
.,
E
l
e
m
e
n
tA
n
a
l
y
s
i
so
fR
e
i
n
f
o
r
c
e
dC
o
n
c
r
e
t
eS
t
r
u
c
t
u
r
e
s,
ASCE
,NewYork,
p
p
.1
8
4
2
0
3
.
9
8
. DeB
o
r
s
t
,
R
.,
1
9
8
4
“
,ApplicationofAdvancedSolutionTechniquestoConcreteCra
.
ck
i
n
g姐 d
七i
c
i
t
y
,
"NumericalMethodsf
o
rN
o
n
l
i
n
e
a
rProblems,
e
d
s
.T
a
y
l
o
r,
C
.e
t
N
o
n
A
s
s
o
c
i
a
t
e
dP
l
a
s
V
.2
,
P
i
n
e
r
i
d
g
巴 P
r
e
s
s,
Swansea
,
UK,
pp.3
1
4
3
2
5
.
a
l
.,
9
9
.DeB
o
r
s
t,
R
.,
1
9
87
a
“
,Smea
.
r
e
dc
r
a
c
k
i
n
g
,
P
l
a
s
t
i
c
i
t
y
,
C
r
e
e
pandThermal10ading-A U
n
i
f
i
e
d
Approach,
"Comp.Meth.Appl
.M
ech.E
n
g
n
g
.,
V.6
2,
p
p
.8
9
1
1
0
.
1
0
0
.DeB
o
r
s
t,
R
.,
1
9
8
7
b,
"Computationo
f
P
o
s
t
B
i
f
u
r
c
孔t
i
o
n乱
ndP
o
s
t
F
a
i
l
u
r
eBeh
品v
-i
o
ro
fS
t
r
a
i
n
.
,
"Comp.& Struct.,
V
.25,
N
o
.2
,
pp.2
1
1
2
2
4
.
S
o
f
t
e
n
i
n
gS
o
l
i
d
s
1
0
1
. DeB
o
r
s
t,
R
.andNauta
,P
.,
1
9
8
5
“
,Non-OrthogonalCra
.
c
k
si
naSm
e
a
r
e
dF
i
n
i
t
eE
l
e
m
e
n
t
Model,
"Engng.Computations,
V
.2
,
p
p
.3
5
4
6
.
1
0
2
. Dharmarajan,
N.
,andVipulanandan,
C叶 1
9
8
8
“
,C
r
i
t
i
c
a
lS
t
r
e
s
sI
n
te
n
s
i
t
yFa
c
t
o
rf
o
rEpoxy
Mortar,
"PolymerEngng.andS
c
i
.,
S
o
c
.o
fP
l
a
s
t
i
cE
n
g
r
s
.,
V.2
8
.p
p
.1
1
8
2
1
1
91
.
1
0
3
.D
o
u
g
i
l
l,
J
.W.,1
9
7
6
“
,OnStableProgressivelyFracturingSolids,
"J
.AppL Math. P
h
y
s
.
V.2
7,
p
p
.4
3
2
.
4
3
6
.
(ZAMP),
1
0
4
.Droz
,P.
,1987“
,Mod在l
enum
e
r
i
q
u
educomportem
巴n
tn
o
n
l
i
n
e
a
i
r
ed
'
o
u
v
r
a
.
g
e
smωsifse
n
b
e
t
o
nnonarme",
D
o
c
t
o
r
a
lT
h
e
s
i
s
,
E
c
o
l
eP
o
l
y
te
c
h
n
i
q
u
eFe
d
a
r
a
l
ed
eLausanne
,
S
w
i
t
z
e
r
l
a
n
d
.
,
C
.G
.,
andSwa
.
r
t
z,
S
.E
.,
1
9
8
6
“
,EnergyMethodsf
o
rF
r
a
c
t
u
r
e
Toughnes5D
e
t
e
r
m
i
n
a
t
i
o
n
1
1
4
.Go
i
nC
o
n
c
r
e
t
e,
"Exp.Mech.,
V.26,
N
o
.3,
p
p
.2
9
2
2
9
6
.
1
1
5
.Go,
C
.G S
w
a
r
t
z,
S
.E
.,
andHu
,K.K.,1984“
,StressIntensityFactorsf
o
rS
i
n
g
l
eEdge
"TechnicalNote,
J
.o
fE
n
g
n
g
.Mech.,
ASCE
,
V.1
1
0,
No.4,
p
p
.6
2
9
6
3
2
.
NotchBeam,
吋
1
1
6
.G
o
g
o
t
s
i
,
G.A G
r
o
u
s
h
e
v
s
k
y
,
Y.
,
.
1 andStrelov,
K.K.,
1
9
7
8
“
,TheSignificanceofNon-Ela
.
s
t
i
c
D
e
f
o
r
m
a
t
i
o
ni
nt
h
eF
r
a
c
t
u
r
eo
fH
e
t
e
r
o
g
e
n
e
o
u
sC
e
r
a
m
i
cM
a
t
e
r
i
a
l
s“
,
C
e
r
a
m
u
g
i
a
l
n
t
.,
U.K.,
V.
N
o
.3,
p
p
.1
1
3
1
1
8
.
4,
叫
1
1
7
.G
o
p
a
l
a
.
r
atnam,
V.S
.,
andS
h
a
l
l,
S
.P
.,
1
9
8
5
“
,SofteningResponseofPla
.
i
nC
o
n
c
r
e
t
ei
nD
i
r
e
c
t
Te
n
s
i
o
n
,
"J.ofACI,
V.8
2
.
1
18
.Gopalaratnam,
V
.S
.,
andCh
e
ng
,
J
.,
1
9
8
8,
"Ont
h
eMod
e
l
i
n
go
fI
n
e
l
a
.
s
t
i
cI
n
t
e
r
f
a
c
e
sI
nF
i
b
r
o
u
s
C
o
m
p
o
s
i
t
e
s,
"Bondingi
nC
e
m
e
n
t
i
t
i
o
u
sComposite
s,
e
d
s
.M
i
n
d
e
s
s,
S
.,
andShah,
S
.P
.,
M
a
t
e
r
.
R且s
.S
o
c
.,
Boston,
V
.1
1
4,
p
p
.2
2
5
2
31
.
1
1
9
.Gopalaratnam,
V.S
.,
andShah,
S
.P
.,
1
9
8
7
乱
,
“FailureMechanismsandFractureofFiber
R
e
i
n
f
o
r
c
e
d Concrete ,~ F
i
be
rR
e
i
n
f
o
r
c
e
dC
o
n
c
r
e
te
,ACISP-105,e
d
s
. Shah,S
.P
.,
and
Ba
.
t
s
o
n,
G.
,p
p
.1
2
5
.
1
2
9
1
2
8
1
2
0.Gopalaratnam,
V
.S
.,
andShah,
S
.P
.,1
9
8
7
b
,
“TensileFa
.
i
l
u
r
eo
fS
t
e
e
lF
i
b
e
r
R
e
i
n
f
o
r
c
e
d
Mortar
,
"J.ofEngng.Mech.,
ASCE
,
V
.1
1
3,
No.5,
pp.6
3
5
・6
5
2.
.G
r
i
f
f
i
t
h,
A.
,1921“
,ThePhenomenonofRuptureandFlowi
nS
o
l
i
d
s,
"Phil.Tran8. R
.
o
y
a
l
1
21
S
o
c.o
fLondon,
S
e
r
i
e
sA,
V
.2
2
1,
p
p
.1
6
3
1
9
8
.
侃t
h,
A
.,
1
9
2
4
“
,Theoryo
fRupt
u
r
e
,
"Proc.
,
1
s
tI
n
t
.C
o
n
g
r
e
s
sonApp
l
.Mech.
,
D
e
l
f
t,
p
p.
1
2
2
.G
r
i
・6
3.
5
5
1
2
3
.Gupta
,
A.K
.,
andAkbar,
H.
,
1
9
8
4
“
,Crackingi
nR
e
i
n
f
o
r
c
e
dC
o
n
c
r
e
t
eA
n
a
l
y
s
i
s,
"J
.o
fS
t
r
u
c
t
.
Engng.
,
ASCE,
V
.1
1
0
,
No.8
,
p
p
.1
7
3
5
.
1
2
4
.G
u
s
t
a
f
s
o
n,P
.J andH
i
l
l
e
r
b
o
r
g,
A
.,
1
9
8
4“
,Improvementsi
nC
o
n
c
r
e
t
eD
e
s
i
g
nA
c
h
l
e
v
e
d
,
"Preprints,
NATOAdvancedRes
e
a
r
c
hW
o
r
k
s
h
o
p
t
h
r
o
u
g
ht
h
eA
p
p
l
i
c
a
t
i
o
no
fF
r
a
c
t
. Mech.
r
a
c
t
.Mech.t
oC
e
m
e
n
t
i
t
i
o
u
sC
o
m
p
o
s
i
t
e
s,
N
o
r
t
h
w
e
s
t
e
r
nU
n
i
v
.,
E
v
a
n
s
t
o
n,
o
nA
p
p
l
i
c
a
t
i
o
no
fF
pp.4
8
7
・5
0
0
.
I'~. H
i
l
l
e
r
b
o
r
g
,
A
.,
1
9
8
5
b
“
,TheTheoreticalBω1sofMethodtoDeterminetheFrac-tureEnergy
G/o
fC
o
n
c
r
e
t
e
,
"Ma
.
t
e
r
.叫 dS
t
r
u
c
t.V
.1
8,
No.1
0
6,
p
p
.2
9
1
・
296
:
ω.
H
i
l
l
e
r
b
o
r
g
,
A.
,
1
9
8
5
c
,
“
Rs
u
l
t
so
fThreeC
o
m
p
a
r
a
t
i
v
eT
e
s
tS
e
r
i
e
sf
o
rD
e
t
e
r
m
i
n
i
n
gt
h
eF
r
a
c
t
u
r
e
怠
BEergGFo
fC
o
n
c
r
e
t
e
,
"Mater.aridS
t
r
u
c
t
.
,
v.18
,
No.107.
1
3
7.Hi11仰弘 A
.
.Modeer
,M.,andPetersson,
P.E.
,1976,“
A叫 s
i
so
fC凶 ? m : ? ? d
机
w
t
ii
凶
nCon t
ebyMe
組 so
fF
ra
c
t.Mech. andF
i
n
i
t
eE
l
師 団t
8,
"Cem.and
C凶 kGro
ふ
C
o
n
c
r
.R
e
s
.,
V.6,
p
p
.7
7
3
7
8
2
.
i
zt
;
2
:
叫
l
鈴
勺紅悶ごロ;お記
此
宅牛
ム
ぷ
l
込
ぷ
J
i
ぷ
;
:
;
4
ト
66
3.
吋
1
2
5
. Gus
t
a
f
s
on,
P
.J
.,
andH
i
l
l
e
r
b
o
r
g
,A.,1988“
,Sensitivityi
nS
h
e
a
rS
t
r
e
n
g
t
ho
fL
o
n
g
i
t
u
d
i
n
a
l
l
y
R且i
n
f
o
r
c
e
dCon
c
r
e
t
eBeamst
oF
r
a
c
t
u
r
eE
n
e
r
g
yo
fC
o
n
c
r
e
t
e
,
"ACIStruct.J
.,
V
.8
5
.
…
1
3
9
.Homeny
,
J.η 訂 r
o
u
d
i,
T
.,
a
.
ndB
r
a
d
t,
R.C.
,
1
9
8
0
“
,J-IntegralMeasurementso
ft
h
eF
r
a
c
t
u
r
e
of50%Alu;nblab- r
i
e
s,
"J
.Am.Ce
r
a
m
.S
o
c
.,
V
.6
3,
N
o
.5
・6
,
p
p
.3
2
6
・
3
3
1・
川
:
:
i
u
:
L
!
?
と
;
t
i
j
ば
誌
に
:
ロ
:
:
:
足
立
二
二
お
む(
2
2
i
J
Z
L
U
,
1
2
2
pp.325
・
3
3
7
.
1
2
6.Hadam釘 d,
J
.,
1
9
0
3,
"
L
e
c
o
n
ss
u
rl
ap
r
o
p
a
g
a
t
i
o
ndωondes,
"ChapVI,
HermannP
a
r
i
s
.
?
z
i
i
E
L
E
:
;
1
3
L
C
;
:
;
2
2
:
:
b
U
;
:
t
j
P
2
2
:
;
:
3
2
;
:
z
z
:
1
2
3
f
対話i
E
M
,
,
, ,
回
, ,
,
1
2
7.Hasegawa
,
T
.,
S
h
i
o
y
ムT
.,
乱
ndOk
a
.
da
,
T
.,
1
9
8
5,
'
‘
S
i
z
eE
f
f
e
c
tonS
p
l
i
t
t
i
n
gT
e
n
s
i
l
eS
t
r
e
n
g
t
ho
f
,
"Proc.JapanConcr.Inst.7thConf.
,
pp.3ω-3
1
2.
Concr
e
t
e
凶
1
2
8.H
a
s
s
a
n
z
a
.
deh,
M.
,
Hi
1
l
e
r
b
o
r
g
,A.,
andP
i
n
g
,
Z
.,
1
9
8
7,
“
T
e
s
t
so
fM
a
t
e
r
i
a
lP
r
o
p
e
r
t
i
e
si
nM
i
x
e
d
Mode1andI
!
,
"Proc.,SEM-RILEMI
n
t
. Conf
.onF
r
a
c
t
u
r
曹 o
fC
o
n
c
r
e
t
e叩 dRρck
,e
d
s.
Shah
,S.P.
,
andSw
剖・t
z,
S
.E.
,
H
o
u
s
t
o
n,
p
p
.3
5
3
・ 3
5
8
.
1
4
2.H
o
r
i
i,
H
.,
S
h
i
,Z
.,
andGong
,S.-X.,1988“
,Mode
1sofF
r
a
c
t
u
r
eP
r
o
c
e
s
sZo
n
ei
nC
o
n
c
r
e
t
e
,
R
.
o
c
k,
組dcerazdcsfi
ne
r
a
t
k
i
n
gandDamage
,proc.ofFrancdISWorkshoponStrain
L
o
c
a
l
i
z
a
t
i
o
nandS
i
z
eE
f
f
e
c
t,
Cach
a
.
n,
F
r
a
n
c
e,
e
d
s.Mazars,
J
.,
組dBaz拙 t,Z
.P.
,
E
l
s
e
v
i
e
r
A
p
p
l
i
e
dS
c
i
e
n
c
e
,
London,
1
9
8
9
.
N.M.
,
1
9
8
4,
“
TheR
o
l
ef
o
rF
r
a
c
t
u
r
eM
e
c
h
a
n
i
c
si
nC
o
n
v
e
n
t
i
on
a
lR
e
i
n
f
o
r
c
e
dC
o
n
c
r
e
t
e
1
2
9
.Hawkins,
D
e
s
i
g
n,
"NATOAdvancedRes
e
a
r
c
hWorkshoponApp
ic
1
a
t
i
o
no
fF
r
a
c
t
u
r
eM
e
c
l
l
a
n
i
c
st
o
S
e
p
t
e
m
b
e
r,
N
o
r
t
h
w
e
s
t
e
r
nU
n
i
v
.E
v
a
n
s
t
o
n,
I
L
.
C
e
m
e
n
t
i
t
i
o
u
sC
o
m
p
o
s
i
t
e
s,
1
3
0.H
e
l
l
a
n,
K.
,1984“
,Introductiont
oF
r
a
c
t
.Me
c
h
.
,
“McGrawHi
l
ICo
・
1
31
. Hi
1
l
e
m
e
i
e
r
,B
.,
andH
i
l
s
d
o
r
f
,H.K.,1977,
“
F
r
a
c
t.Mech.S
t
u
d
i
e
sonC
o
n
c
r
e
t
eCompounds,
"
Ce
m.a
ndConcr
.R
e
s
.,
V
.7
,
p
p
.5
2
3
5
3
6
.
l
1
e
r
b
o
r
g
,A.
,1980“
,AnalysisofF
r
a
c
t
u陀 byMeanso
ft
h
eF
i
c
t
i
t
i
o
u
sCrackMod
,
1
e Par1
3
2
.Hi
t
i
c
u
l
a
r
l
yf
o
rF
i
b
e
rR疋i
n
f
o
r
c
e
dC
o
n
c
r
e
t
e
ヘInt. J.ofCem.Compos-ites,V.2,No. 4,pp.
・1
8
5.
1
7
7
,A.
,1983“
,ExamplesofPracticalRes
u
l
t
sA
c
h
i
e
v
e
dbyMeanso
ft
h
eF
i
c
t
i
c
i
o
U
6
1
3
3
.H
i
l
l
e
r
b
o
r
g
CrackModel,
"i
nP
r
e
p
r
i
n
t
s,
P
r
a
g
e
rSymp.onMec
h
a
n
i
c
so
fGeom
at
e
r
i
a
l
s
:Roc
k
s,
C
o
n
c
r
e
回
,
S
o
i
l
s,
e
d.Ba
.
ま
a
n
t,
Z.P.
,
N
o
r
t
h
w
e
s
t
e
r
nU
n
i
v
.,
E
v
a
n
st
on,
pp.6
1
16
1
4.
司
1
3
4
.Hi
1
l
e
r
b
o
r
g,
A.
,
1
9
8
5
a
“
,NumericalMethodstoSimulateSoftening組dF
r
a
c
t
u
r
eo
fC
o
n
c
r
e
t
e
,
"
F
r
a
c
t.Me
c
h
.o
fC
o
n
c
r
e
t
e:S
t
r
u
c
t
UI
'
a
1A
p
p
l
i
c
a
t
i
o
nandNum
e
r
i
c
a
lC
a
l
c
u
l
a
t
i
o
n,
e
d
s
.S
i
h
,G.
andDiToma
.
s
s
o,
A
.,
M
a
r
t
i
n
u
sN
i
j
h
o
f
f
,
D
o
o
r
d
r
e
c
h
t
B
o
s
t
o
n,
p
p
.1
4
1
・1
7
0
.
C
.,
I
n
t.Conf
.Houston e
d
s
.Shah S
.P. dS
w
a
r
t
zS
.E.S
p
r
i
n
g
e
r
V
e
r
l
a
gB
e
r
l
i
n.
B
.P
.,
回dCh乱
pman,
B.P.
,
1
9
6
6,
“
TheC
o
m
p
l
e
t
eS
t
r
e
s
s
S
t
r
a
.
i
nCurvef
o
rCon
c
r
e
t
e
1
4
3
.H
u
g
h
e
s,
i
nD
i
r
e
c
tT
e
n
s
i
o
n,
"RILEM(Paris)Bul.
lNo.30,
p
p
.9
白5
9
肝7
.
““
…
1
4
4
.1
♂
酔
1
I
,
叫
0 M.
,Shi句
o
'
y
a,
T
.,
N吋
o
j
i
凶
r
i,
弘
Y
.
,
晶ndAki戸
y
am
s
t
r
児e
削 1o
d
f
L
a
u
r
g
e,
k
向
…
R
一
e
b
」
ぷ
仏
i
L
L
;
l
I
i
4
h
f
お
由
o
伽
ω
r
c
e
dCo
叩 悦B
e
帥晶
t
弘
ms川
u
n
附d
e
rU
J
凶
,
n
叫
,
i
i
江
f
お
0m
叫
口n
l
匂
yDi
凶
s
凶
附
凶
t
刷
r
r
i
出
bu
巾 dL
oad,
"Concrete
JapanSoc-ofC
i
v
i
lEREm-,
No
.5,
pp.1
3
7
1
5
4
.
L
i
b
r
a
r
yI
n
t
.,
n
g
l
i
s,
C
.E
.,
1
9
1
3
“
,Stressi
naP
l
乱
t
eDuet
ot
h
ePr
e
s
e
n
c
eo
fC
r
a
c
k
sandS
h
a
r
pCorn
e
r
s,
"
1
4
5.I
Tr加.I
ns
t
.N
a
v
a
lA
r
c
h
i
t
e
c
t
s,
p
p
.2
1
9・2
41
.
一
1
4
6
.I
n
g
r
a
.
f
t
'
e
,
a A.R.
,1985 “
F
r
a
c
t
u
r
ePropag
a
t
i
o
ni
nRρck,
"MechanicsofGeomaterials:Rβcks,
Co
n間隔, S
o
i
l
s,
e
d
.
a
n
t,
z.p,
JohnWil
e
yandS
o
n
s
,
I
n
c
.,
NewYork
,
N 1 J叫 258pp
1
47
.I
n
g
r
a
官e
,
aA
.R
.,
andP
a
n
t
h
a
k
i,
M.
J
.,
1
9
8
5
“
,ShearFractureTestsofConcreteBe乱
ms,
"Finite
E
l
;
m
e
n
tAnaly~is o
fRe
i
n
f
o
r
c
e
dC
o
n
c
r
e
t
eS
t
r
u
c
t
u
r
e
s,
Tokyo,
p
p
.1
5
1
1
7
3
.
1
4
8
.I
n
g
r
a
f
f
e
a
,A.R.
,and Gerstle,W.
,1985,
“Nonl
i
n
e
a
rF
r
a
c
t
u
r
eModelsf
o
rDi
s
c
r
e
t
eCr
a
.
ck
P~';;;~;;i~~:" 'Ap;ïi;ati;~-~fl'Ì'act. -Me~h. t
oCem
e
n
t
it
i
o
u
sC
o
m
p
o
s
i
t
e
s,
e
d
. Shah,
S.P.
,
M
a
r
t
i
n
u
sN
i
j
h
o
f
f
,
Pub
,
.
lDoordrechtandBoston,
p
p
.1
7
1
2
0
9
.
1
3
1
1
3
0
1
4
9
.I
n
g
r
a
f
f
e
a,
A
.R
.,
andSaoumaV
.,
1
9
8
5
“
,NumericalModelingofDiscreteCrackPropagatiou
l
a
.
inConcreもe,
"Fr晶c
t
.Mech.o
fCon
ロe
t
e
:S
t
r
u
c
t
u
r
a
lA
p
p
l
i
c
.
a
t
i
o
na
且d
i
nR
e
i
n
f
o
r
c
e
d姐dP
,e
d
s
.S
i
h,
G
.C
.,
andDiTomasso,
A
.,
Ma
.
r
t
i
n
u
sN
i
j
h
o
f
f
,
D
o
o
r
d
r
e
c
h
t
.
N
u
m
e
r
i
c
a
lC
a
l
c
u
l
a
t
i
o
n
p
p
.1
7
1
2
2
5
.
B
o
s
t
o
n,
1M
M
J
L
Z
Z
;
:
f
i
t
l
l
J
7
3
i
t
t
z
z
?
おお:
:
f
c
j
t
z
t
o
r
n
;
Z
1
'
.
)
.Kach阻 o
v,
M.,
1
9
8
5
“
,A SimpleTechniqueo
fS
t
r
e
s
sA
n
a
l
y
s
i
si
nE
l
a
s
t
i
cS
o
l
i
d
sw
i
t
hMany
C
r
a
c
k
s,
"I
n
t
.よ o
fF
r
ad.,
V.2
8,
R11
・
R
1
9
.
1
5
0
.I
r
w
i
n,
G.R.,1
9
5
8
“
,Fracture,
"HandbuchderPhysik,Vol
. VI,
e
d
.F
l
u
!
認e
,Springer,
P
p
.
5
5
1
5
9
0
.
I
I
,
!
.
.Kach回 ov,
M.
,1987
“
,EIωticSolidswithManyCracks:ASimpleMethodofAnalysis,
"I
n
t
・
J
.
o
fS
o
l
i
d
sandS
t
r
u
c
L
,
v.23,
No.l,
pp.23
・4
3
.
1
5
1
.Jenq
,
Y.S.
,
姐dSh油
,S
.P
.,
1
9
8
5
a
“
,AF
r
a
c
t
u
r
eT
o
u
g
h
n
e
s
sC
r
i
t
e
r
i
o
nf
o
rC
o
n
c
r
e
t
e
,
"Engug.
F
r
a
c
t
.Mech.,
V
.2
1,
N
o
.5,
p
p
.1
0
5
5
・1
0
6
9
.
:
, Kanninen,
M・F咋 畠ndP
o
p
e
l
a
r,
C
.H
.,
1
9
8
5
“
,AdvancedF
r
a
c
t
・Mech.,
"OxfordUniv.Press,
1
(
1
5
2
.J
e
且q
,Y
.S
.,
祖dShah,S
.P
.,
1
9
8
5
b
“
,TwoPa
.
r
a
m
e
t
e
rF
r
a
c
t
u
r
eModelf
o
rC
o
n
c
r
e
t
e,
"J
.o
f
E
n
g
n
g
.Mech.,
ASCE
,
V
.1
1
1,
N
o
.1
0
,
p
p
.1
2
2
7
・1
2
41
.
│
ドパ
・
[
n
c
.
K
a
p
l
a
n
,
M.F
.,
1
9
6
1
“
,CrackPropagation日 dtheF
r
a
c
t
u
r
唾 o
fC
o
n
c
r
e
t
e,
"ACIJ
.,
V
.回
, No.
1
5
3
.J
e
n
q
,
Y
.S
.,
andShah
,
S
.P
.,
1
9
8
6
“
,CrackPropag
.
a
t
i
o
ni
nF
i
b
e
rRe
i
n
f
o
r
c
e
dC
o
n
c
r
e
t
eヘJ
.
o
f
ASCE,
V
.1
1
2,
N
o
.1
,
p
p
.H
ι
3
4
.
S
t
r
u
c
.E
n
g
n
g
.,
1
1
.
a
l
∞
, B.L 組 dNallatha
.
m
b
i,
P
.,
1
9
8
7
“
,NotchedBeamTest:Mode1FractureTough!
o
9
.K ・h
"ωrep tt
o臨 聞 Committω89-FMT;F
r
a
c
t
.M民 h.ofC
ω:Tes
tM帥 吋
1
5
4
.Jenq,
Y
.S
.,
andShah,
S
.P
.,
1
9
8
7
,
.
a"Fract.Mech.andConstitutiveModelingo
fC
o
n
c
r
e
t
e
>
,
C
o
n
s
t
i
t
u
t
i
v
eL.
a
wsf
o
rE
n
g
i
n
e
e
r
i
n
gM
a
t
e
r
i
a
l
s
:TheoryandA
p
p
l
i
c
a
t
i
o
l
l
s
.
,
e
d
. Des
,
i
.
a C.S
.e
t
Vo
l
.1
1,
E
l
s
e
v
i
e
r
.
a
l
.,
l
7
0
.Kawai,
T
.,
1
9
8
0
“
,SomeConsiderationsontheFiniteElementMethodヘInt・J・Numer.Meth.
,
Y
.S
.,
andShah,
S
.P
.,
1
9
8
7
b,
‘,MixedModeF
r
a
c
t
u
r
eP
a
r
a
m
e
t
e
r
so
fC
o
n
c
r
e
t
e,
"P
r
o
c
.
,
1
5
5
.J
e
n
q
SEM-RILEMI
n
t
.Con
f
.onF
r
a
.
c
t
u
r
eo
fC
o
n
c
r
e
t
eandRo
c
k
,
e
d
s
.Sh
油
,S
.P
.andSwa
.
r
t
z,
S
.
Houston,
p
p
.3
5
9.
3
6
9
.
E
.,
e
s
l
,
ぽ C.E.;N乱.us
.D
.J
.
;andL
o
t
t,
J
.L
.,
1
9
7
1,
“
F
r
a
c
t
.Me
c
h
.
-I
tsAppli~a.bility t
oC
o
n
c
r
e
t
e,
"
1
7
1・K
p
r
o
c
.,
Int.conf.ohthehech
回 i
叫 B
eh
.
a
v
i
o
ro
fM
a
t
e
r
i
a
l
s,
Kyoω,
TheS
o
c
.o
fM.
a
t
低
S
c
i
.,
1
5
6
.Jenq,
Y.S
.,
andShah,
S
.
P
.,
1
9
8
8
a
“
,GeometricalEffectsonMode1FracturePa
.
r
a
m
e
t
e
r
s
"
,
R息p
o
r
tt
oRILEMCommittee8
9
-FMT.
1
7
2
.K
f
o
u
r
i,
A.
P
.,
a
.
ndM
i
l
l
e
r,
K
.J.
,
1
9
7
4
“
,StressDisplacement,
L
i
n
eI
n
t
e
g
r
a
l姐dC
l
o
s
u
r
eEnergy
1
5
7
. Jenq,
Y
.S
.,
andShah
,
S
.P 1
9
8
8
b
“
,OntheConcreteFractureT
e
s
t
i
l
l
gMethods,
"Preprints
I
nt
.WorkshoponF
r
.
a
c
t
u
r
eT
o
u
g
h
n
e
s
sandF
r
a
c
t
u
r
eE
n
e
r
g
y-T
e
s
tMethodsf
o
r
o
ft
h
eP
r
o
c
.,
Co
n
c
r
e
t
eandRoc
k,
Send
,
i
.
a J.
a
p
a
n
.
叶
R
.,andShah,S
.P
.,1
9
8
5“
,Stra
.
i
nR
a
.
t
eE
f
f
e
c
t
so
nMode1CrackPropag
.
a
t
i
o
ni
n
1
5
8
.John,
,
"Proc.,
I
n
t
.C
o
n
f
.onF
r
a
c
t
.Mech.o
fC
o
n
c
r
e
t
e,
L
a
u
s
a
n
n
e
.
C
o
n
c
r
e
t
e
よ
…
ふ
1
9
8
7
.
E
n
g
n
g
.,
V
.1
6,
p
p
.8
1
・1
2
0
.
Vo
l
.IV,
1
9
7
2,
p
p
.1
1
3
1
2
4
.
“
D蜘 m
凶
n
凶i
凶
n
a
抗叫
t
ω
ぬ
仰
o
i
悶
n
5o
fC悶 ii
T
叫
,
T
h
N
o
.3
,
p
p
.1
7
9
1
91
.
l
7
3
.Kn
.
a
u
s
s,
W.C
.,
1
9
7
4“
Ont
h
eS
t
e
a
d
yP
r
o
p
a
g
a
t
i
o
no
f.
aCraιk凶
a
.ViscoelasticSheet;Exp
e
r
i
m
e
脚
andAn
均i
i
s
fTheD
e
f
o
r
m
a
t
i
o
ni
nF
r
a
c
t
u
r
eHiOPolymers
,札 H
.H.Kau
叫,
PlenumP
r
e
s
s,
p
p
.5
0
1・5
41
.
l
7
4
.K
n
o
t
t,
J
.F
.,
1
9
7
3,
“
Fund
a
.
m
e
n
t
a
l
so
fF
r
a
c
t
.Mech.,
"BuUerworths,
Lo
ndon,
UK.
R
.,
andShah,
S
.P
.,
1
9
8
6
“
,Fractul'eofConcreteSubjectedtoImpaιtLoa
.
d
i
n
g
,
"Cem.,
1
5
9
. John,
C
o
n
c
r
.andA
g
g
r
e
g
a
t
e
s,
ASTM.
u
l
l
i
v
a
n,
A
.M.,
andB
o
y
l
e
,
R
.W.,
1
9
6
1,
"
E
f
f
e
c
to
fD
i
m
e
n
s
i
o
n
sonFa
s
tF
r
a
c
t
u
r
e
1
7
5
.K
r
a
f
f
t
.J
.M S
I
n
s
t
a
b
i
l
i
t
yo
fNotchedS
h
e
e
t
s,
CrackProp
.
a
g
a
t
i
o
nSymp.,
P
r
o
c
.,
U.K.
1
6
0
. John,
R
.,
andShah,
S
.P
.,
1
9
8
7
“
,E
f
f
e
c
to
fHighS
t
r
e
n
g
もha
ndRa
t
eo
fL
o
a
d
i
n
gonF
r
ω
t
u
r
e
P
a
r
a
m
e
t
e
r
so
fC
o
n
c
r
e
t
e,
"Proc.,
SEM-RILEMI
n
t
.Con
f
.onF
r
a
c
t
u
r
eo
fC
o
n
c
r
e
t
ea
.
ndR
ρck,
e
d
s
.Sh油
, S.P胡 dSwartz,
S
.E
.,
H
o
u
s
t
o
n,
p
p
.3
5
-5
2
.
1
7
6
.K
r
u
n
e
r
,
E.
,
1
9
6
7
“
,E
l
a
s
t
i
c
i
t
yTheoryo
fM
a
t
e
r
i
a
l
sw
i
t
hLong-R
甜酔 C
o
h
e
s
i
v
eF
o
r
c
e
s
ヘIn
t
.J
.
a
ndS
t
r
u
c
t
.,
N
o
.3,
p
p
.7
3
1
・
7
4
2
.
o
fS
o
l
i
d
s.
1
.John,
R
.,
andShah,
S
.P
.,
1
9
8
9
a
“
,MixedModeFractureofConcreteSubjectedtoImpacl
1
6
Lo
.
a
d
i
n
g
,
"J
.o
fS
t
r
u
c
t
.E
n
g
n
g
.,
ASCE,
Ma
.
r
.1
9
9
0
,
i
np
r
e
s
s
.
l,
J
.A
.,
1
9
6
8“
SomeC
o
n
s
i
d
e
r
.
a
t
i
o
n
so
ft
h
eR
e
l
.
a
t
i
o
nbetweenSo
1
i
dS
t
a
t
ePhy
8
I
c
s
1
7
7
.Krumh組 s
組n
叫dG
e
r
悶 a
d
山
叫l
且
i
包
z
m
e
d o
凶
n
t
“
i
出
n
ι
iumMe
吋c
h
閣
川
組
叩
a
凶
r
n
山
.
叩
l
i
同
c
α
5,
"M氏叫h
a
.
凶
削
a
.
凶
n
山
i
c
ω
s
o
f
G
e
r
附 a
山
l
i
i
包
z
e
dCon
山
ti
削m
u
叫
1
a
S
p
r
i
n
g
e
r,
B
e
r
l
i
n,
p
p
.3
3
0
3
4
0
.
1
6
2
.John,
R
.,
剖dShah,
S
.P
.,
1
9
8
9
b,
“
F
r
a
c
t
.Mech.A
n
a
l
y
s
i
so
fHi
ghS
t
r
e
n
g
t
hC
o
n
c
r
e
t
e
,
"J.of
ASCE,
V
.1
,
N
o
.4,
p
p
.1
8
5
1
9
8
.
M
a
t
e
r
.i
nC
i
v
i
lE
n
g
n
g
.,
1
6
3
. John,
R
.,
Shah,
S
.P
.,
andJ
e
n
q,
Y
.S
.,
1
9
8
7
“
,A F
r
a
.
c
t
. Mech. Modelt
oP
r
e
d
i
c
tt
h
eRωe
S
e
n
s
i
t
i
v
i
t
yo
fMode1F
r
a
c
t
u
r
eo
fC
o
n
c
r
e
t
e,
"Cem.andConcr.Res
.
叫
…
d
ふ ふ
凶n
,
I
.A 1968aTheTheoryofE
l
a
s
t
i
cMediaw
i
t
hM
i
c
r
o
s
t
r
u
c
t
u
r
eandt
h
eTheoryo
f
1
7
8
.Ku
Dido-ionsj'Mechanicso
fGe
悶 a
l
i
z
e
dC
o
n
t
i
n
u
a
,e
d
. Krn民 E
.,
S
p
r
i
n
g
e
r
,Berlin,pp.
3
3
0
3
4
0
.
叶
132
133
1
7
9
.Labuz,
J
.F
.,Shah
,S
.P
.,
andDowding,
C
.H
.,1
9
8
5,
“ExperimentalAnalysiso
fC
r
a
c
k
,
"I
n
t
.J
.o
fRρckMech.andM
i
n
.S
c
i
.,
V
.2
2
,
N
o
.2,
p
p
.8
5
9
8
.
P
r
o
p
a
g
a
t
i
o
ni
nG
r
a
n
i
t
e
S
.,
1
9
8
4“
,RateofLoadingE品 c
t
so
nt
h
eF
r
a
c
t
u
r
eofC
e
m
e
n
t
i
t
i
o
u
sM
a
t
e
r
i
a
l
s,
"
1
9
5.M
i
n
d
e
s
s,
P
r
e
p
r
i
n
t
s,
A
p
p
l
i
c
a
t
i
o
n
so
fF
r
a
c
t
. Mech. t
oC
e
m
e
n
t
i
t
i
o
u
sC
o
m
p
o
s
i
t
i
e
s,
NATO
ARW,
e
d
.
Shah,
S
.P
.,
N
o
r
t
h
w
e
s
t
e
r
nU
n
i
v
.
.
1
8
0
.Lawrence
,P
.,
1
9
7
2“
,SomeThωreticalConsiderationso
fF
i
b
r
eP
u
l
l
O
u
tfromanE
l
鎚 t
i
c
M
a
t
r
i
x
ヘJ
.o
fMat.S
c
i
.,
V.7
,
p
p
.1
6
.
1
8
1
.La
.
ws,
V
.,
Lawrence
,
P
.,
andN
u
r
s
e,
R
.W.,
1
9
7
2
,
‘'ReInforcementofB
r
i
t
t
l
eM
a
t
r
i
c
e
sbyG
l
a
s
s
ヘJ.ofPhys.D:AppJ
.P
h
y
s
.,
V
.6,
p
p
.5
2
3
5
3
7
.
F
i
b
e
r
s
1
8
2.L
e
i
b
e
n
g
o
o
d,
L
.D.
,
Darwin,
D
.,
andDodds
,
R
.H
.,
1
9
8
6
“
,ParametersAffectingFEAnalysis
"J
.S
t
r
u
c
t
.E
n
g
n
g
.,
V
.1
1
2,
N
o
.2,
p
p
.3
2
6
・
3
41
.
o
fC
o
n
c
r
e
t
eS
t
r
u
c
t
u
r
e
s,
1
8
3
.L
e
r
o
y
,
Y
.叩dO
r
t
i
z,
M.,
1
9
8
9
“
,FiniteElementAnalysisofStrainLocalizationi
nF
r
i
c
t
i
o
n
a
l
M
a
t
e
r
i
a
l
s,
"I
n
t
.J
.Numer.A
n
a
l
.M
e
t
h
.Geomech.,
V
.1
3,
p
p
.5
3
7
4
.
1
8
4
.L
i
n,
C
.S
.,
andS∞r
d
e
l
i
s
,
A
.C
.
,1975
“
,NonlinearAnalysisofRCS
h
e
l
l
so
fG
e
n
e
r
a
lForms
ヘ
J
.S
t
r
u
c
t
.Di
v
.,
ASCE,
V
.1
0
1,
p
p
.5
2
3
5
3
8
.
1
8
5
.L
i
n
s
b
a
u
e
r
,H.N.,andTschegg,E
.K
.
, 1986“
,FractnreEnergyDeterminationofConcrete
"ZementundBeton,
V
.3
2,
p
p
.3
8
・4
0(
i
nGerm
組).
w
i
t
hCube-ShapedS
p
e
c
i
m
e
n
s,
1
8
6
.L
'
H
e
r
m
i
t
e,
R
.,
1
9
5
9,
“Wh
a
tdoweKnowAboutt
h
eP
l
a
s
t
i
cD
e
f
o
r
m
a
t
i
o
nandC
r
e
e
po
fC
o
n
"RlLEMBul
,
.
lNo.1,
p
p
.2
2
5
1
.
c
r
e
t
e
?,
1
8
7.Maji,
A
.K
.,
andShah,
S
.P
.,
1
9
8
8
“
,ProcessZoneandAcousticEmissionMeasurementi
n
,
"Exp.Mech.,
V
.2
8
,
pp.2
7
3
3
.
C
o
n
c
r
e
t
e
1
8
8
.M
a
r
t
i,
P
.,
1
9
8
9,
‘,S
i
z
eE
f
f
e
c
ti
nD
o
u
b
l
e
P
u
n
c
hT
e
s
t
sonC
o
n
c
r
e
t
eC
y
l
i
n
d
e
r
s
,
"ACIMater.J
.,
V
.8
6
,
N
o
.6,
p
p
.5
9
7
・6
01
.
1
8
9
. Mathey,
R
.G
.,
加dWatstein,
D 1
9
6
0
“
,E
f
f
e
c
to
fT
e
n
s
i
l
eP
r
o
p
e
r
t
i
e
so
fR疋 i
n
f
o
r
c
e
m
e
n
tont
h
e
,
J
.o
ft
h
eACI
,
P
r
o
c
.,
V
.56
,
N
o
.1
2
,
p
p
.1
2
5
3
・7
3
.
F
l
e
x
u
r
a
lC
h
a
r
a
c
t
e
r
i
s
t
i
c
so
fBeams"
叶
1
9
0
. Maturana
,
P
.,
P
l
a
n
a
s,
J
.,
andE
l
i
c
e
s,
M.,
1
9
8
8
“
,EvaluationofF
r
a
c
t
u
r
eB
e
h
a
v
i
o
ro
fS
a
t
u
r
a
t
e
d
C
o
n
c
r
e
t
ei
nt
h
eLo
wTemper
a
.
t
u
r
eRan
g
e
,
"i
nF
r
a
c
t
u
r
e細 d D叩 l
a
g
eo
fC
o
n
c
r
e
t
eandRo
c
k
,
e
d
.Ros
s
m
a
n
i
t
h,
P
.,
I
n
t
.C
o
n
f
.,
V
i
e
n
n
a
.
.Meier,
S
..
W
.,
叩dGergely,
P.
,
1
9
8
1
“
,FlexuralCrackWidthi
nP
r
e
s
t
r
e
s
s
e
dC
o
n
c
r
e
t
eB品 ms"
,
1
91
J
.o
ft
h
eS
t
r
u
c
t
.D
iv
.,
ASCE
,
V
.1
0
7,
N
o
.ST2,
p
p
.4
2
9
・4
3
3
_
H
.,
andZ
a
i
t
s
e
v,
J
.W.,
1
9
8
1,
“
S
t
a
t
i
s
t
i
c
a
lN
a
t
u
r
eo
fC
r
a
c
kP
r
o
p
a
g
a
t
i
o
n,
"Section
1
9
2
.M
i
h
a
s
h
i,
4
・
2i
nR怠p
o
r
tt
oRILEMTC5
0・FMC,
e
d
.Wittmann,
F
.H
.
1
9
3
.M
i
l
l
e
r
,
R
.A
.
jSbah,
S
.P
.
j叩dB
j
e
l
k
h
a
g
e
n
,
H
.,
.
1 1988“
,CrackP
r
o
f
i
l
e
si
nMortarM
e
a
s
u
r
e
d
a
.
p
h
i
cI
n
t
e
r
f
e
r
o
m
e
t
r
y
,
"Exp.Mech.,
V
.2
8,
N
o
.4,
p
p
.3
3
8
3
9
4
.
byH
o
l
o
g
r
1
9
4
.M
i
n
d
e
s
s,
S
.,
1
9
8
3
“
,TheFractureofFibreR疋inforcedandPolymerImpregnatedConcretes:
A Re
v
i
e
w
,
“ Fract. Mech. ofConcrete,e
d
. Wittmann,
F
.
H
.,
E
I
s
e
v
i
e
rS
c
i
e
n
c
eP
u
b
l
i
s
h
e
r
9
,
Amsterdam
,TheNetherlands,
p
p
.4
8
1
・
5
01
.
1
9
6
.Mobasher
,
B
.,
Ouyang,
C
.
,
andShah,
S
.P
.,
1
9
8
9
“
,A R-CurveAppro品chtoPredictToughi
n
f
o
r
c
e
m
e
n
t,
"submittedf
o
rp
u
b
l
i
c
a
t
i
o
n
js
e
e
e
n
i
n
gofCement-BasedM
a
t
r
i
c
e
sduet
oF
i
b
e
rRe
,
B
.,
P
h
.
D
.T
h
e
s
i
s
,
N
o
r
t
h
w
e
s
t
e
r
nU
n
i
v
.,
1
9
8
9
.
a
l
s
oMobasher
1
9
7
.Mobasher
,
B
.,
加dShah,
S
.P
.,
1
9
8
9,
'
‘
I
n
t
e
r
a
c
t
i
o
nBetweenF
i
b
e
r
sandt
h
eCementM
a
t
r
i
xi
n
i
n
f
o
r
c
e
dC
o
n
c
r
e
t
e
,
"tobepublished,
ACIS
p
e
c
i
a
lV
o
l
u
m
e
.
G
l
a
s
sF
i
b
e
rRe
,B.,Stang,H.,andShah,S
.P 1
9
8
9
“
,Microcrackingi
nF
i
b
e
rR
e
i
n
f
o
r
c
e
dCon1
9
8
.Mobasher
c
r
e
t
e
,
"Acceptedforpublic
a
.
t
i
o
n,
Cem.andC
o
n
c
r
.R
e
s
.
.
円
1
9
9
.M
o
r
r
i
s
o
n,
J
.K
.,
Shah,
S
.P
.,
andJ
e
n
q,
Y
.S
.,
1
9
8
8
“
,AnalysisoftheDebonding組 dPull-Out
もe
sヘJ
.o
fE
n
g
n
g
.Mech.,
ASCE
,
V
.1
1
4,
N
o
.2,
p
p
.2
7
7
2
9
4
.
P
r
o
c
e
s
si
nF
i
b
e
rComposi
Y.,
1
9
8
7,
e
d
i
t
o
r
i
n
・c
h
i
e
f
“
,StressI
ntensityFactorsHandbook,
"Perga.monPres8,
2
0
0
.Murakami,
O
x
f
o
r
d
.
2
0
1
.Na
.
kayama
,J.
,1965,
“DirectMeasurementofF
r
a
c
t
u
r
eE
n
e
r
g
i
e
so
fB
r
i
t
t
l
eH
e
t
e
r
o
g
e
n
e
o
u
s
M
a
t
e
r
i
a
l,
"J.oftheAmer.Cera
.
m
.S
o
c
.,
V
.4
8,
N
o
.1
1
.
2
0
2
.N
a
l
l
a
t
h
a
m
b
i,
P
.,
andK
a
r
i
h
a
l
∞
, B.L
.,
1
9
8
6
a
“
,DetenninationofSpecimen-SizeIndependent
F
r
a
c
t
u
r
eToughn
邸 5o
fP
l
a
i
nC
o
n
c
r
e
t
e,
"Mag.o
fC
o
n
c
r
.Res
.,
V
.3
8,
N
o
.1
3
5,
p
p
.6
7
・7
6
.
回. N
a
l
l
a
t
h
a
m
b
i,
P
.,
andK
a
r
i
h
a
l
∞
, B.L
.,
1
9
8
6
b
“
,StressIntensityFa
.
c
t
o
randEnergyRe
l
e
a
s
e
2
Ra
t
ef
o
rT
h
r
e
e
P
o
i
n
tBendS
p
e
c
i
m
e
n
,
"Engng.F
r
a
c
t
.Mech.,
V.2
5,
N
o
.3
,
p
p
.3
1
5
・3
21
.
却4
.N叩5,
D
.J
.,
回dLot
t,
J
.L
.,
1
9
6
9
“
,FractureToughnωsofPortlandCementConcretes,
"ACI
J
.,
V
.6
6
,
p
p
.4
8
1
4
9
8
.
2
0
5
.Ngo,
D
.,
祖dSkordelis,
A
.C
.,
1
9
6
7
“
,FiniteElementAnalysisofReinforcedConcreteBe
a
.
r
n
s
,
"
ACIJ
o
u
r
n
a
l
,
V.6
4,
N
o
.3
,
p
p
.1
5
2
・
1
6
3
.
M.
,
1
9
8
8
“
,ExtractionofConstitutiveDatafromSpecimensUndergoingLoc
a
l
i
z
a
t
i
o
n,
"
2
0
6
.O
r
t
i
z,
s
u
b
m
i
t
t
e
dt
oJ
.o
fE
n
g
n
g
.Mech.,
ASCE.
M.,
L
e
r
o
y
,Y.,andNeedleman,A.,1987“
,A FiniteElementMethodf
o
rL
o
c
a
l
i
z
e
d
2
0
7
.O
r
t
i
z,
F
a
i
l
u
r
eA
n
a
l
y
s
i
s,
"Comp.Meth.App.
lMecl¥.Engr.,
V
.6
1,
p
p
.1
8
9
・2
2
4
.
N
.S
.,
andDahlblom,
0 1986
“
,SmearedCrackAr
はl
y
s
i
sU
s
i
n
gaN
o
n
l
i
n
e
a
rF
r
a
c
t
u
r
e
2
0
8
.O
t
t
o
s
e
n,
,
"NumeucalMethodsf
o
rN
o
n
l
i
n
e
出・ P
r
o
b
l
e
m
s3,
e
d
s
.T
a
y
l
o
r,
C
.e
ta
l
.,
Modelf
o
rC
o
n
c
r
e
t
e
P
i
n
e
r
i
d
g
骨 P
r
e
s
sL
t
d
.,
Sw乱,n
s
e
a,
U.K.,
p
p
.3
6
3
3
6
7
.
叶
2
0
9
.Owen,
D
.R
.J
.,
andFawkes,
A
.J
.,
1
9
8
3,
“
E
n
g
i
n
e
e
r
i
n
gF
r
a
c
t.M
e
c
h
.
:N
u
m
e
r
i
c
a
lMethods
andA
p
p
l
i
c
a
t
i
o
n,
"PineridgePressLtd.,
Swansea,
U.
K
.
2
1
0
.P
a
r
i
s,
P,
a
n
dErdog
a
.
n,
F.
,1967“
,A C
r
i
t
i
c
a
lA
n
a
l
y
s
i
so
fC
r
a
c
kPropag
a
.
t
i
o
nLa
.
ws,
"J.of
cE
n
g
n
g
.,
V
.87~ p
p
.5
2
8
5
3
4
.
B8si
1
3
4
21
1.P
e
r
d
i
k
a
.
r
i
s,
P
.C
.,
andCa
1
omino
,
A
.M.,
1
9
8
7
“
,KineticsofCrackGrowthinPla
.
i
nC
o
n
c
r
e
t
e
.
"
句r
i
n
t
s
,
S
犯EM川
/
R
則
川I
比
LEMI
n
加t
.C
ω
o
叫
n
f
.叩
0nF
r
悶
a
託c
凶
h
知
士u
r
e
Pr
S
.E.
S.P
.andSw
a
.
r
t
z,
2
1
2.P
e
t
e
r
s
ω,
D
>P
.E
.,1
9
8
0
a
“
,FTactureEnergyofCon
c
r
e
t
e
: Methodo
fDet
町m
i
n
a
t
i
o
n
.
"Cem
s
.,
V.1
0
,1980,
p
p
.河
7
肘
8
叫
7
町
9
;
‘
“
‘
F
r
削
a
c
伽
t
C
o
n
c
r
.Re
1Re
s
u
l
t
s,
"Cem.Concr.Res
.V
.1
0
,
p
p
.9
1
・1
0
1
.
andE
x
p
e
r
i
m
e
n
ta
2
1
3
.P
e
t
e
r
s
s
o
n,
P
.
E
.
,1980b,"FTactureMechanica
ICa
1
c
u
l
a
t
i
o
n
s姐dT
e
s
t
sf
o
rF
i
b
e
r
-R
e
i
n
f
o
r
c
e
d
,
"Proc.
,
A
d
v
a
n
c
e
si
nCementM
a
t
r
i
xC
o
m
p
o
s
i
t
e
s,
Mater.Rm.Soc-AnnualM
e
e
ι
C
o
n
c
r
e
t
e
i
n
g
,
B
o
s
t
o
n,
p
p
.9
5
・1
0
6
.
2
1
4
.P
e
t
e
r
s
s
o
n,
P
.E
.,
1
9
8
1
“
,CrackGrowthandDevelopmentofFTactureZonesinPla
.
inC
o
n
c
r
e
t
e
uldS
i
m
i
l
a
rM
a
t
e
r
i
a
I
s
fR
s
p
o
r
bTVBM-1006,
Div.ofB16langM
a
t
e
r
i
a
l
s
,
L
1
1
n
dI
n
s
t
.
o
f
T
e
c
h
.
,
Lund,
S
w
e
d
e
n
.
2
1
5
.P
h
i
l
l
i
p
s
,
D
.V
.,
andZ
i
e
n
k
i
e
w
i
c
z,
O
.C.
,
1
9
7
3
“
,Nonlinea
.
rAna
1
y
s
e
so
fS
t
r
u
c
t
u
r
a
lCo
n
c
r
e
t
ebv
"Ph.D.Thesis,
U
n
i
v
.o
fW
a
l
e
s,
Sw
叩 s
e
,
.
aUK.
.
F
i
n
i
t
eE
l
e
m
e
n
tMethods,
2
1
6
.P
h
i
l
l
l
p
s,
D
.V吋
制dZienkiewicz,
O
.C
.,
1
9
7
6
“
,FiniteElementNon-Linea
.
rA
n
a
l
y
s
i
so
fC
o
n
,
"Proc.1目前E・Civ.Engrs.,
Pa
.
r
t1
,
V
.6
1,
p
p
.5
9
8
8
.
c
r
e
t
eS
t
r
u
c
t
u
r
e
s
.
k
,S
.,
制dMr'
z
,Z
.,1
9
8
1“
,FiniteElementAna
1
y
s
i
so
fDeform
.
a
t
i
o
no
fS
t
r
a
i
n
・
2
1
7
.P
i
e
t
r
u
s
z
c
za
S
o
f
t
e
n
i
n
gM
a
t
e
r
i
a
l
s
,
"Int.J.ofNumer
-Meth-ofEnFg.,
v.17
,
pp.327
・
334
.
.
a
u
d
i
e
r
C
.
a
b
o
t,G.,a
n
dBa
.
i
a
n
t,
Z
.P
.,1
9
8
7,
“NonlocalDa
.
mageTh伺 ry,
"J.ofEngng.
2
1
8
.P
i
j
ASCE
,
V.1
1
3,
N
o
.1
1
0
,
p
p
.1
5
1
2
・
1
5
3
3.
M
e
c
h
.,
andB
a
z
a
n
t,
Z
.P
.,
1
9
8
8
“
,DynamicSt.
a
b
i
l
i
t
yA
n
a
l
y
s
i
sw
i
t
hN
o
n
l
o
c
a
l
2
1
9
.P
i
j
a
u
d
i
e
r
C
a
b
o
t,G.,
.
a
g
e
,
"Comp.& S
t
r
u
c
t
.,
V
.2
9
,
N
o
.
3
,
p
p
.5
0
3
5
0
7
.
Dam
.
a
u
d
i
e
r
C
a
b
o
t,G
.,.
a
ndMaz
a
.
r
s
,J
.,1
9
8
9
,
“ContinuumDa
.
m
a
.
g
eTheoryA
p
p
l
i
c
a
t
i
o
nt
o
2
2
0
.P
i
j
,
"J
.o
fEng
且g
.Mech.,
ASCE
,
V
.1
1
5,
N
o
.2
,
p
p
.3
4
5
-3
6
8
.
C
o
n
c
r
e
t
e
.P
i
j
.
a
u
d
i
e
r
C
a
b
o
t,G
.,M
a
z
a
r
s
,J
.,
祖dPolikowski,J
.,1
9
8
9
“
,Steel-ConcreteBondAnalysi目
2
21
w
i
t
hNo
n
Jo
c
a
lC
o
n
t
i
n
u
o
u
sDamage
,
"lnternalReportNo・96,
L
a
b
.o
fMech.& TechnoI句~,
E
c
o
l
eNormalS
u
p
e
r
i
e
u
r
,Cachan,
F
r
.
a
n
c
e
.
2
2
2.P
l
組
a
.
s
,J
.,
拍dEliωs,M.
,1988a“
,S
i
z
e
E
f
f
e
c
ti
nC
o
n
c
r
e
t
eS
t
r
u
c
t
u
r
e
s
: Mathematic
a
1Ap
p
r
o
x
i
m
a
t
i
o
n
s
a
a
dE
x
p
e
r
i
m
e
n
t
a
lv
a
l
i
d
a
t
i
o
n
,
"Cruking祖dDa
.
m
a
g
e
,
S
t
r
a
.
i
nL
o
c
a
l
i
z
a
t
i
o
n副 d
S
i
z
eEJ
f
e
c
t,
P
r
o
c.o
fF
rance
-Uふ Workshop,
Cachan,
F
T
組
問
,e
d
s
.Mazars
,
J・
姐dBai.
a
n
t,
Z
.
P
.,
p
p
.4
6
2
4
7
6
.
姐
a
.
s,
J
.,
andE
l
i
c
e
s,
M.,
1
9
8
8
b,
'
‘
C
O
l
l
c
e
p
t
u
a
landE
x
p
e
r
i
m
e
n
t
a
lP
r
o
b
l
e
m
si
nt
h
eD
e
t
e
r
m
i
2
2
3.P
l
,
"Proc.,
Int.WorkshoponqhcureToughne
Z
l
a
M
o
no
ft
h
eE
T
a
c
t
u
r
eEnergyo
fC
o
n
c
r
e
t
e
,
T
e
s
tMEthodsf
o
rC
o
n
c
r
e
t
eandRmk
,
"TohokuU
n
i
v
.,
S
e
n
d
a
i
,
J
a
p
a
n
,
andETacureB
E
e
r
g
y
p
p
.2
0
3
・2
1
2
.
1
3
5
2
2
4
.P
l
胡
舗
,J
.,
E
l
i
c
e
s,
M.,
a
.
n
dT
o
r
i
b
i
o
,
J
.,
1
9
8
9
“
,Approxim.
a
t
i
o
no
fCo
h
e
s
i
v
eCrackModelsby
"FTactureofConcreteandRoc
k
: Re
c
e
n
tD
e
v
e
l
o
p
m
e
n
t
s
,e
d
s
. Shah,S
.
R-CTODC
u
r
v
e
s,
S
w
a
r
t
z,
S
.E
.,
andBa
.
r
r
,B.,(
I
n
t
.C
o
n
f
.h
e
l
d.
a
tC
a
r
d
i
f
f
,U.K
.
)
, Elsevier,London,pp.
P
.,
2
0
3
・
2
1
2
.
,
M.E
.,
制dA
j
f
;
岨t
i
s,
E
.C
.,
1
9
8
3
“
,OntheModelingofRockswithMicrostructure,
"
2
2
5
.P
l
e
s
h
a
P
r
o
c
.,
2
4
t
hU
.
S.Symp.onRo
c
kM
e
c
h
a
n
i
c
s
,
Tex
a
.
sA&MUniv.,
C
o
l
l
e
g
eS
t
a
t
i
o
n,
p
p
.2
7
3
9
.
.
a
s
h
i
d,
Y.R
.
, 1船 8“
,AnalysisofPr飴 t
r
e
s
s
e
dCo
n
c
r
e
t
eP玲 s
s
u
r
eV
e
s
s
e
l
s,
"NuclearEngng.
2
2
6
.R
回dDes.,
V
.7,
N
o
.4,
pp.3
3
4
3
5
5
.
2
2
1
.R怠ad
,H.E.,
andHegemi色r
,G.P.
,1984
“
,S
t
r
a
.
i
n
S
o
f
t
e
n
i
n
go
fRρck
,S
o
i
landC
o
n
c
r
e
t
e-a
R
e
v
i
e
wA
r
t
i
c
l
e
,
"Mech.andMater.,
V
.3,
p
p
.2
7
1・2
9
4
.
f
a
i
,T.M.E
.,
拍dSwa
.
r
t
z,
S
.E
.,
1
9
8
7
,
“
F
r
a
c
t
u
r
eB
e
h
a
v
i
o
ro
fC
o
n
c
r
e
t
eBe
a
.
msi
nTh
f
.
骨
2
2
8
.Re
P
o
i
n
tB
e
n
d
i
n
gC
o
n
s
i
d
e
r
i
n
gt
h
eI
n
:
f
l
u
e
n
c
eo
fS
i
z
eE
f
f
e
c
t
s
,
"ReportNo. 190,Engineering
,
S
t
a
t
eU
n
i
v
.,
M訂 l
h
a
t
t
a
.
n
.
E
x
p
e
r
i
m
e
n
tS
t
a
t
i
o
n
f
a
i,
T
.M.E
.,
回dSwartz,
S
.E.
,1988
“
,Mode1FraιtureEnergyMeasurementsf
o
rCon2
2
9
.Re
,
"Exp.Mech.,
V
.2
8
.
c
r
e
t
e
i
n
h
a
r
d
t,H
.W.1
9
8
1
a
“
,S
i
z
eE
f
f
e
c
ti
nS
h
e
a
rT
e
s
t
si
nt
h
eL
i
g
h
to
fFr
a
.
c
t
. M
e
c
h
.“(
i
n
2
3
0
.Re
German)
,
B
e
t
o
n
u
n
dS
凶 l
b
e
t
o
n
b
a
u(
W
e
s
tBe
必n
)N
o
.1,
p
p
.1
9
2
1
.
2
3
1
.R
e
i
n
h
a
r
d
t
,H.W.1981b
“
,S
i
r
n
i
l
i
t
u
d
eo
fB
r
i
t
t
l
eF
r
a
.
c
t
u
r
eo
fS
t
r
u
c
t
u
r
a
lC
o
n
c
r
e
t
e,
"Advanced
i
n
f
o
r
c
e
dC
o
n
c
r
e
t
e
,
IABSEC
o
l
l
o
q
u
i
u
m,
D
e
l
f
t,
p
p
.6
9
9
2
.
M
e
c
h
.o
fRe
i
n
h
a
r
d
t,
H.W.,
1
9
8
4
“
,TensileFTactureofConcretea
tHi
ghR
a
t
e
so
fLo
a
d
i
n
g
,
"Preprints,
2
3
2
.Re
A
p
p
l
i
c
a
t
i
o
no
fF
T拭 t
.Mech. t
oC
e
m
e
n
t
i
t
i
o
u
sC
o
m
p
o
s
i
t
i
e
s,
NATO-ARW,
e
d
. Shah,
S
.P.
,
N
o
r
t
h
w
e
s
t
e
r
nU
n
i
v
.,
E
v
a
n
s
t
o
n
.
,
2
3
3
.R
e
i
nh
a
.
r
d
t,
H
.W.
,
岨dCornelissen,
H
.A
.W.,
1
9
8
4
“
,Post-peakCyclicBeh.
a
v
i
o
ro
fC
o
n
c
r
e
t
e
a
d
i
n
g
,
"Cem.Concr.Res.,
i
nU
n
i
a
x
i
a
lT
e
n
s
i
l
eandA
l
t
e
r
n
a
t
i
n
gT
e
n
s
i
l
ea
.
n
dC
o
m
p
r
e
s
s
i
v
eLo
V
.1
4,
N
o
.2,
p
p
.2
6
3
・
2
7
0
.
i
n
ha
.
r
d
t,H
.,C
o
r
n
e
l
i
s
s
e
n,H
.,
回dHordijk,D.,1987,
“MixedModeF
r
a
c
t
u
r
eT
e
s
t
son
2
3
4
.Re
C
o
n
c
r
e
t
e
,
"F
r
a
.
c
t
u
r
eo
fC
o
n
c
r
e
t
eandRρck,
e
d
s
.Sh
油
,S
.P.
,
andS
w
a
r
t
z,
S
.E
.,
(
I
n
t
.Co
n
(
.
S
p
r
i
n
g
e
r
V
e
r
l
a
.
g
,N.Y.,
1
9
8
9,
p
p
.1
1下 1
3
0
.
h
e
l
di
nH
o
u
s
t
o
n
),
1
9
8
5
・
TC50
・
FMC,
F
r
a
c
t
.Mech.o
(C
o
n
c
r
e
t
e
“
,DeterminationoftheFractureEnergy
2
3
5
.RILEM,
"RILEM
o
fM
o
r
t
a
randC
o
n
c
r
e
t
ebyMe
a
.
nso
fT
h
r
e
e
P
o
i
n
tBendT
e
s
t
sonN
o
t
c
h
e
dBeams,
Re
commendation,
Mate
r
.a
n
dS
t
r
u
c
t
.,
V
.1
8
,
N
o
.1
0
6
.
e
l
f
s
t
r
a
,
P.
E
.,
S
a
d
o
u
k
i,
H
.,
andWittmann,
F
.
H
.,
1
9
8
5,
"
L
eB
e
t
o
nNumerique
,
"M.
a
t
e
r
.and
2
3
6
.Ro
,
V
.1
0
7
.
S
t
r
u
c
t
.
e
l
f
s
t
r
a
,
P
.
E
.,
1
9
8
7
“
,NumericalAna
.
l
y
s
i
sandS
i
m
u
l
.
a
t
i
o
no
fC
r
a
c
kF
o
r
m
a
t
i
o
ni
nC
o
r
n
p
o
s
i
t
e
2
3
7
.Ro
M
a
t
e
r
i
a
l
ss
u
c
ha
.
sConcrete,
"i
nF
r
.
a
c
t
u
r
eo
fN
o
n
.
M
e
t
a
l
l
i
cM.
a
t
e
r
i
a
l
s,
e
d
s
.Herman,
K.P,
姐d
,
L
.H
.,
ECSC,
EEC,
EAEC,
B
r
u
s
8
邑1
8andLuxembourg,
p
p
.3
5
8
3
8
4
.
L
a
r
s
s
o
n
1
3
7
1
3
6
2
3
8
.Rρmualdi,
J
.P
.,
andB.
a
t
s
o
n,
G
.B
.,1
9
6
3
“
,Mech.
a
n
l
c
so
fC
r
a
ι
kA
r
r
e
s
ti
nC
o
n
c
r
e
t
e,
"J.o
f
ASCE
,
V
.8
9,
N
o
.EM3,
p
p
.1
4
7
1
6
8
.
Engng.M
e
c
h
.,
2
3
9
.Romualdi,
J
.P
.a
.
n
dMa
.
nd
,
J
eJ
.A
.,
1
前4
“
,TensileStrengthofConcreteAffectedbyUnifornuy
D
i
s
t
r
i
b
u
t
e
dC
l
o
s
e
l
yS
p
a
c
e
dS
h
o
r
tLen
g
t
h
sofWireRe
i
n
f
o
r
c
e
m
e
n
t,
"ACIJ.
,
V
.6
1,
N
o
.6,
p
p
.
6
5
7
6
7
0
.
2
4
0
.R
ρ
t
8,
J
.G.
,1988
“
,S
t
r
e
s
sRot
.
a
t
i
o
n.
a
ndS
t
r
e
s
sL
o
c
k
i
n
gi
nSme
a
.
r
e
dAn
a
.
l
y
s
i
so
fS
e
pa
.
r
a
.
t
i
o
n
"
,
P
r
o
c
.,
I
nt
.Workshopon“
F
r
a
c
t
u
r
eT
o
u
g
h
n
e
s
8a
.
n
dF
r
a
.
c
t
u
r
eE
n
e
r
g
y
,
T
e
s
tMethodsf
o
rC
o
n
c
k,
"TohokuUniv.,
Send
,
i
.
a J.
a
p.
a
n
.
c
r
e
t
eandRo
1
.Rρts,
J
.G
.,
組dBla
.
auwendr
品 d
,
J
.,
1
9
8
9
,
“
Cr
.
a
c
kModelsf
o
rC
o
n
c
r
e
t
e
:D
i
s
c
r
e
t
eo
rS
m
e
a
r
e
d
7
2
4
F
i
x
e
d,
M
u
l
t
i
-D
i
r
e
c
t
i
o
na
.
lo
rRo
t
.
a
t
i
n
g
,
"HERON,
V
.3
4,
N
o
.1
.
2
4
2
.R
ρ
t
s,
J
.G.,
andd
eB
o
r
s
t,
R
.,1
9
8
7
“
,AnalysisofMixed-ModeF
r
a
.
c
t
u
r
ei
nC
o
n
c
r
e
t
e,
"J
.o
f
ASCE,
V
.1
1
3,
N
o
.1
1,
p
p
.1
7
3
9
・
1
7
5
8
.
Engng
.Mech.,
t
s,
J
.G
.,
N.
a
u
t
,
.
aP
.,
K
u
s
t
e
r
s,
G
.M.A
.,
釘l
dB
l
a
.
a
.
uwendr
a
.
a
d,
J
.
,1985“
,Sme
.
a
r
e
dCr
a
.
c
k
2
4
3
. Ro
ApproachandF
r
a
.
c
t
u
r
eLoc
a
.
l
i
z
a
t
i
o
ni
nC
o
n
c
r
e
t
e,
"HERON,
V
.3
0,
N
o
.1
.
2
4
4
.Rusch
,H.,.
a
ndH
i
l
s
d
o
r
f
,H.,1963“
,Deform乱
.
t
i
o
nCha
.
r
a
c
t
e
r
i
s
t
i
c
so
fC
o
n
c
r
e
t
eu
n
d
e
rAxi
a
l
T
e
n
s
i
o
n,
"Vorunterschungen,
Munich
,
B
e
r
i
c
h
t4
4
.
2
4
5
. Sa
.
h
u
d
i
n,
A.
H
.,1
9
8
7,
“Nonlinea
.
rFiniteElementStudyofAxisymmetricFiberPull-Out,
f
l
M
.
S
.T
h
e
s
i
s,
U
n
i
v
.o
fM
i
s
s
o
u
r
i
C
o
l
u
m
b
i
,
.
app.1
1
0
.
,
.
aV.E.,
I
ng
r
a
.
f
f
e
a
,
A
.R
.,
andC
a
t
a
l
a
n
o
,
D
.M.,
1
9
8
2,
“
F
r
a
c
t
u
r
eT
o
u
g
h
n
e
s
so
fC
o
n
c
r
e
t
e
2
4
6
. Saoum
:K R
e
v
i
s
i
t
e
d,
"J
.o
fE
n
g
n
g
.M
e
c
h
.D
i
v
.,
V
.1
0
8,
N
o
.EM6
,
p
p
.1
1
5
2
・1
1
6
6
.
2
4
7
.Sand
1
e
r,
1
.S
.,
1
9
8
6
“
,Stra
.
i
且-Softeningf
o
rS
t
.
a
t
i
c.
a
ndDyna
.
m
i
cP
r
o
b
l
e
m
s,
"i
nP
r
o
c
.o
fS
y
m
p
.
a
.
t
i
o
n
s
:M
i
c
r
o,
Ma
.
c
r
oandComput
.
a
t
i
o
n
a
lA
s
p
e
c
t
s,
ASMEW
i
n
t
e
rA
n
n
u
a
l
onC
o
n
s
t
i
t
u
t
i
v
eEqu
M
e
e
t
.,
NewO
r
l
e
a
.
n
s
,
e
d
.W
i
l
l
i
a
.
m
, K.,
ASME
,
N
.
Y
.,
p
p
.2
1
7
2
31
.
2
4
8
.S
a
o
u
r
i
d
i
s,
C
.,
1
9
8
9,
D
o
c
t
o
r
a
lT
h
e
s
i
so
nDamageA
n
a
l
y
s
i
si
nC
o
n
c
r
e
t
e
,
Labo
fM
e
c
h
a
n
i
c
sa
n
d
,
E
c
o
l
eNormalSupr
i
e
u
r
,
Cachan,
F
r
a
n
c
e
.
T
e
c
h
n
o
l
o
g
y
,
V
.E
.
,B
r
.
但
,J
.J.
,
B品 w
i
l
e
r,
E
.,
a
n
dB
o
g
g
s,
H.
,1989,"Fra
.
c
t
u
r
eP
r
o
p
e
r
t
i
e
so
fD叩1
2
4
9
.Saouma
C
o
n
c
r
e
t
e
.P
a
r
t1
:L
a
b
o
r
a
t
o
r
yE
x
p
e
r
i
m
e
n
t
s,
"submittedt
oJ
.o
fM
a
t
e
r
.i
nC
i
v
i
lE
n
g
n
g
.,
ASCE.
A
.,1
9
7
1“
,TimeDependentDef
i
e
c
t
i
o
n
so
fRe
i
n
f
o
r
c
e
dC
o
n
c
r
e
t
eS
l
.
a
b
s,
"DoctoraJ
2
5
0
.S
c
a
n
l
o
n,
T
h
e
s
i
s,
U
n
i
v
.o
fA
l
b
e
r
t
,
.
aEdmonton,
Can
a
.
d
a
.
.Schreyer,
H
.L
.,
叩dChen,Z.
,1986
“
,OneDimensionalSofteningwithLocalization,
"J
.o
f
2
51
.M
ech.,
V
.5
3,
p
p
.7
9
1
・7
9
7
.
Appl
A
.
A
.,
andR
ρ
d
r
i
g
u
e
z
O
r
t
i
z,
J
.
M
.,
1
9
7
3
“
,A Contributiont
ot
h
eM
e
c
h
a
n
i
c
so
fH
e
t
e
r
o
2
5
2
.S
e
r
r
a
n
o,
g
e
n
回 u
sGr
姐
叫
a
.
rMedia,
"Proc.,
Symp.o
nP
l
a
.
s
t
i
c
i
t
y.
a
ndS
o
i
lMech.,
C.
a
m
b
r
i
d
g
e
,
U.K.
a
.
h
,S
.P
.,
1
9
8
8
“
,FraεtureToughnessofCementBa
.
s
e
dM.
a
t
e
r
i
a
l
s,
"M.
a
t
e
r
.a
.
n
dS
t
r
u
c
t
.,
V
.
2
5
3
. Sh
p
p
.1
4
5
・1
5
0
.
2
1,
2
5
4
.S
h
a
h,
S
.P
.,
姐dMcGa
.
r
r
y
,
F・J
.,
1
9
7
1
“
,G
r
i
f
f
i
t
hF
r
a
c
t
u
陀 C
r
i
t
e
r
i
o
nandC
o
n
c
r
e
t
e,
"J
.o
ft
h
e
BIIEng.Mech.Div.,
ASCE,
V091.97
,
No.EM6,
pp.1663
・ 1
6
7
6
.
2
5
5
.S
h
a
h,
S
.P
.,
andS
姐k,
r
.
a R,
1
9
8
7
“
,InternalCra
.
c
k
i
n
gandS
t
r
a
.
in-SofteningResponseofConc
r
e
t
eu
n
d
e
rU
n
i
a
x
i
a
lCompressionJ'ACIM
a
t
e
r
.
J
.,
v.84
,
pP20$212.
2
5
6・S
t
a
n
g,
H
.,
MobasherB andShah
,
s.p.,
1
9
8
9
“
,QuantitativeDama
.
g
eCh
a
.
r
a
.
ct
e
r
i
z
a
t
i
o
ni
n
,
"Amptedforp
耐 iωion
,
Cem.乱ndConιRes
PolypropyleneFiber-K4fomdC
o
n
c
r
e
t
e
姐g,H.,.
a
ndShah,S
.P
.,1
9
8
6,
“F泊lureofFiberReinforcedCompositesbyPull-Out
2
5
1
.S
t
ETutunfJ.ofMas.sd.
,
v.21
,
No.3
,
p
p
.
9
3
5
9
5
7
.
2
5
8
.S
t
叩g,H.,.
a
ndSh
a
l
1
, S
.P
.,1
9
8
9,
“Da
.
mageEvolutioni
nFRCMa
.
t
e
r
i
a
l
sM
o
d
e
l
l
i
n
gand
民 R
e
c
e
n
tDevelopmω5
,
ExperimiωobmvjtioL"FiberReinfonedementsandConmt
eds.swamy
,
R.N.,
andB
a
r
r,
B
.,
E
l
s
e
v
i
e
rA
p
p
l
i
e
dS
c
i
e
n
c
e,
pp.37$387.
組
, M.,
組dSchnobrich,
W.C
.,
1
9
7
3
“
,FiniteElementAnalysisofReinforcedConcrete,
"
2
5
9
.S
u
i
d
J
.
o
fS
t
r
u
e
t
.
E
n
E
n
g
.
,
ASCEV.99
,
No.10
,
pp.2109-2122
.
2
6
0
.S
u
l
e
m,J
.,
組dVardoulakis,
,
.
1 1989“
,BifurcationAnaJy
s
i
so
ft
h
e百 i
a
.
x
i
a
lT
e
s
tonRock
"CrazkingandDamage
,
P
r
o
c
.,
F
r
a
n
c
e
U
SWorkshoponS
t
r
a
i
nL
o
c
a
l
i
z
a
t
i
o
nand
s
p
e
c
i
m
e
n
s,
S
i
z
eE
f
e
e
tduet
oCraEkingmdDmage
,
(hdda
teachan,
F
r
a
n
c
e,
s
e
p
1
9
8
8
),
eds.Mazazs,
J
.,
副dBaZa.nt,
Z
.
P
.,
E
l
s
e
v
i
e
rA
p
p
l
i
e
dS
c
i
e
n
c
e,
1o
ndon,
pp.3
0
8
・3
2
2
.
2
6
1
.Swa
.
r
t
z,
S.E
.andGo,
C
.G
.,
1
9
8
4
“
,ValidityofComplianceCalibrationtoCra
.
c
k
e
dCo
n
c
r
e
t
e
Be
a
.
m
si
nBending"
,Exp.Mech.,
V
.2
4,
No
・2
,
p
p
.1
2
9
1
3
4
.
2
6
2
.Sw
a
.
r
t
z,
S
.E
.,
Hu
,
K
.K.,
andJ
o
n
e
s,
G
.L
.,
1
9
7
8
“
,Comp1
ia
n
c
eM
o
n
i
t
o
r
i
n
go
fCr
a
.
c
kGrowth
ヤ oftheEngng.Mech.Div.,ASCE,V.104,No・EM4,pp.789・800.
i
nCon-teJ
a
.
r
t
z,
S
.E
.,
Hu,
K
.K.,
Fa
.
r
tωh,
M.,
a
.
ndHu
組g,
C
.M.J.
,
1
9
8
2
“
,S
t
r
e
s
sI
n
t
e
n
s
i
t
yF
a
c
t
o
r
s
2
6
3
.Sw
山 COBmt;iih
;
1
d
i
昨 P
r
e
n
o
t
c
h
e
dV
e
r
s
u
sP
r
e
c
r
a
c
k
e
dBeams,
"Exp.Mech.,
V
.2
2,
f
o
rP
l
N
o
.1
1,
p
p
.4
1
2
・4
1
7
.
S.
E
.,
Hu
叩g,
J
.,
andHu,
K
.K
.,
1
9
8
2
“
,Cr.
a
c
kGrowthandF
r
a
.
c
t
u
r
ei
nP
l
a
i
nC
o
n
c
r
e
t
e
2
6
4
.S
w
a
r
t
z,
,
"inFatiguedConmustructumed.sh
油
,S
.P
.,
ACISP-75
,
S
t
a
t
i
c
v
s FatigueioAing
p
p
.
7
2
0
.
a
.
n
dTang,
L
.D
.,
1
9
8
8
“
,Mixed-ModeFr.
a
c
t
u
r
eT
o
u
g
h
n
e
s
sT
e
a
t
i
n
g
2
6
5
.S
w
a
r
t
z
.S
.E
.
.1
u
.L
.W.,
o
fC
o
n
c
r
e
t
eBe
乱
間 inThre令 PointBending,
"M.
a
t
e
r
.andS
t
r
u
c
t
.,
V
.2
1,
p
p
.3
3
・4
0
.
a
.
r
t
z,
S
.E
.,
1u,
L
.W.,
T制 g,
L
.D
.,
andRe
f
,
i
.
a T.M.E 1988
“
,ModeI
IF
r
a
.
c
t
u
r
ePar
a
.
m
e
t
e
r
2
6
6
.Sw
p
e
c
i
m
e
n
s,
"Exp.Mech.
,
V.28
,
N
o
.2,
p
p
.1
4
6
-1
5
3
.
E
s
t
i
m
a
t
e
sf
o
iC
o
n
c
r
e
t
efromBe乱m S
吋
S
.E
.,
叩dRefai,
T
.M.E
.,
1
9
8
9
“
,InfluenceofS
i
z
eE
f
f
e
c
t
sonOpeningModeF
ra
.
c
t
u
r
e
2
6
7
.S
w
a
r
t
z,
.r祖伽sf
o
rP
r
e
c
ra
.
ムdConcreteBea
.
m
ai
nB
e
n
d
i
n
g
,
"F
ractureo
fC
o
n
c
r
e
t
eandRo
c
k,
吋S.
Pa
Shah
,s.p.,aBdSwartz,s.E,SEM-RILEMInt.Conf.,HouEton,1987,springer-verlag,
p
p
.2
4
2
2
5
4
.
:
油
a
.
.N.M 1987
“
,Prelimina
.
r
yI
n
v
e
s
t
i
g
.
a
t
i
o
no
ft
h
eS
u
i
t
a
b
i
1
i
t
yo
ft
h
e
2
6
8
.S
w
a
r
t
z
.S
.E
.
.組dT
1
0
吻e
1
1T
e
s
is
p
e
c
i
m
e
n
'
f
o
rD
E
t
e
r
m
i
n
i
n
gMixedModeF
r
a
c
t
u
r
ep
r
o
p
e
r
t
i
e
so
fConeMeJ
Re
p
o
r
tNo~ 1
9
1
:.
E
n
g
i
n
e
e
r
i
n
gE
x
p
e
r
i
m
e
n
t
S
t
.
a
t
i
o
n,
Kans
a
.
sS
t
a
t
eU
n
i
v
.,
Manhattan.
叫
139
138
、
.
'
'
e
c
h
a
r
以a.n
a
,
M.,
andShah,
S
.P
.,
1
9
8
2
“
,SlowCra
.
c
kGrowthi
nCementCompo
s
i
t
e
s,
"J・o
f
2
6
9
.S
w
a
r
t
z,S
.E
.,
祖dY.
a
p,S
.T
.,1
9
8
8,
“TheI
nf
i
u
e
n
c
eo
fDe
a
.
dLoadonF
r
a
c
t
u
r
eE
o"t酌
M
e
a
s
u
r
e
m
e
n
t
sU
s
i
n
gt
h
eRILEMMethod
,
"Mater.andS
t
r
u
c
t
.,
V.2
1,
p
p
.4
1
0
4
1
5
. ,
.
2
7
0
.S
w
e
n
s
o
n
,D.V.,1986“
,ModelingM
.
i
xed-ModeDyn
a
.
m
i
cC
r
a
c
kP
r
o
p
a
g
a
t
i
o
nU
s
i
n
gF
i
E
l
e
m
e
n
t
s,
"Depa
.
r
t
m
e
n
to
fS
t
r
u
c
t
.E
n
g
n
g
.,
Re
p
o
r
tN
o
.81
)
.1
0,
C
o
r
n
e
l
lU
n
i
v
.,
I
t
h
a
c
a
.
2
7
1
.Ta
.
d
a
.
, H.,
P
a
r
i
s
,
P
.C
.,
組dI
r
win
,1985
“
,τ
'
h
eS
t
r
e
s
sA
n
a
l
y
s
i
so
fC
r
a
c
k
sHandbook2
n
d
Pa
.
r
i
sP
r
o
d
u
c
t
i
o
n
s,
S
t
.Lo
凶8
.
2
7
2
.T
a
t
t
e
r
s
a
l
l
,H
.G.,
回dTappin,G.,1966“
,TheWorko
fF
r
a
c
t
u
r
eandi
t
sM
e
a
s
u
r
e
m
e
U
1
M
e
t
a
l
s
,
C
e
r
a
m
i
c
s姐 dO
t
h
e
rM
a
t
e
r
i
a
l
s
,
"J
.o
fM
a
t
e
r
.S
c
i.
,
V
.1
,
No.3
,
p
p
.2
9
6・3
01
.
2
7
3
.T
a
y
l
o
r
,
G
.
,
.
1 1938
“
,P
l
a
s
t
i
cS
t
r
a
i
ni
nM
e
t
a
l
s,
"J
.o
ft
h
eI
n
s
t
.o
fM
e
t
a
l
s
,
Lo
ndon,
UK
,
V
.
I
.
1
pp.307
・
3
2
4
.
ASCE,
v.108,
No.6,
pp.1400
・1
41
3
.
t
q
eS
t
r
E
l
d
D
i
v
.,
則
、¥echara
M.a
n
dShah
,S
.P
.,
1
9
8
3
品
'
“Predictionso
fN
o
n
l
i
n
e
a
rF
r
a
c
t
u
r
ePr
o
c
e
s
sZo
n
e
副
乱,
t
'
mC
o
n
c
r
e
t
e
,
"J・Engng.Mech.,
V
.1
0
9,
N
o
.5,
p
p
.1
‘
2
3
1
1
2
4
6
.
‘
附a
;
3
:
立
:
T
V
.1
5
3
,
p
p
.1
・5
5.
i
P:
ロよ
f
2
7
8
.V
i
p
u
l
a
n
a
n
d
a
n,
C
.,
叩dDharmara伊n,
N
.,
1
9
8
7
“
,FrωturePropertieso
fEpoxyP
o
l
y
m
e
rC
O
D
,
"e
d
s
.Shah
,S
.P
.,
叩dSwa
.
rt
z,
S
.E
.,
SEM-RILEMI
n
t
.C
o
n
f
.onF
r
a
c
t
u
r
eo
fC
o
n
c
r
e
f
.
c
r
e
t
e
叩 dR
oc
k,
H
o
u
s
t
o
n,
p
p
.6
6
8
6
7
8
.
2
7
9
.V
i
p
u
l
a
n
a
且d
an
,C.,
組dDha
.
r
m訂
a
.
j
如
,N
.,1
9
8
8,
“E
f
f
e
c
to
fT
e
m
p
e
r
a
t
u
r
eont
h
eF
r
a
.
c
t
u
r
.
,
P
r
o
p
e
r
t
i
e
so
fEpoxyP
o
l
y
m
e
rCo
n
c
r
e
t
e
,
"Cem.andConcr.Res.,
V.1
8
,
N
o
.2
,
p
p
.2
6
5
・2
7
6
.
2
8
0
.V
i
p
u
l
a
n
a
n
d
a
n
,
C
.,
andDh
訂ma
.
r
a
j
a
n,
N
.,
1
9
8
9
a
“
,C
r
i
t
i
c
a
lC
r
a
c
kT
i
pOpeningD
i
s
p
l
a
c
e
m
凹
f
o
rP
o
l
y
m
e
rC
o
m
p
o
s
i
t
e
s
“
, Engng.F
r
a
c
t
.M
e
c
h
.,
V
.3
3
,No.3,
p
p
.4
0
9
4
2
0
.
1
.Vip叫副知d日
,C
.,
andD
h
a
r
m
a
r
a
j
a
n,
N.
,
1
9
8
9
b
“
,Analysiso
fF
r
a
ι
t
u陀 Pa
.
r
a
.
m
e
t
e
r
so
fE
p
o
x
y
2
8
,
"ACIMateria
.
lJ
.,
V
.8
6
,
N
o
.4,
p
p
.4
0
9
・4
2
0
.
P
o
l
y
m
e
rC
o
n
c
r
e
t
e
2
8
2
.V
i
p
u
l
孤 島d
回
, C
.,andDharmara
品n,
N
.,1
9
8
9
c
,
“Infiuenceo
fA
g
g
r
e
g
a
t
eont
h
eF
r
a
.
c
t
n
r
t
'
P
r
o
p
e
r
t
i
e
so
fP
o
l
y
e
s
t
e
rP
o
l
y
m
e
rC
o
n
c
r
e
t
e,
"ACIS
p
e
c
i
a
lP
u
b
l
i
c
a
t
i
o
n(
S
P
1
1
6
)
.
2
8
3
.V
i
s
a
l
v
a
n
i
c
h
,
K
.
,a
.
ndNaaman,
A
.E
.,
1
9
8
2
,
“
F
r
a
c
t
u
r
eModelf
o
rF
i
b
e
rR息i
n
f
o
r
c
e
dC
o
n
c
r
e
t
e,
ACIJ
.,
V
.8
0
,
N
o
.2,
p
p
.1
2
8
1
3
8
.
2
8
4
.Walsh,
P
.F
.,
1
9
7
2
,
“
F
ractureo
fP
l
a
i
nC
o
n
c
r
e
t
e
,
"I
nd
i
a
nC
o
n
c
r
.J
.,
V
.46
,
N
o
.1
1
.
2
8
5
. Walsh,
P
.F
.,
1
9
7
6
“
,CrackI
n
i
t
i
a
t
i
o
ni
nP
l
a
i
nC
o
n
c
r
e
t
e
,
"Mag. o
fC
o
n
c
r
. Re
s
.V
.2
8,
p
p
.
.
3
7
41
2
8
6
.Wechar
.
a
t
a
n
a
,M.,
組dSh油
,S
.P
.,
1
9
8
0
“
,DoubleTor8ionTestsf
o
rS
t
u
d
y
i
n
gS
l
o
wC
r
a
l
h
Growtho
fP
o
r
t
l
a
n
dCementM
o
r
t
a
r
,
"Cem.andConcr.Res.,
V.1
0,
p
p
.8
3
3
8
4
4
.
点むななす点目::;:fZZf:i
tbt;2Jt:1itz:-
R
e
i
n
f
o
r
c
e
dC
o
n
c
r
e
t
e,
p
p
.1
4
9
・1
7
2
.
W
i
l
凧 K.J
九
,到
Bm
i
比
同
c
臼a
伽
凧
n
i
,
"i
凶
n“
F
r
乱dur
符eT
o
u
g
h
n
e
s
sandF
、
r
a
c
疋
q.l-SofteningC
o
m
p
u
t
a
t
i
o
n
so
fC
o
n
c
r
e
t
e
:E
悶α
c
r
e
悦t
総
凧
鳥
e,
川 Wi伽 拍n,
F
.H
.,
E
到l
1
児剛巾肌 N卵 川
Y叫 pp 附 6
2
0
J
;
:
芯
2
7
5
.Thomas,
T
.Y
.,
1
9
6
1
“
,P
l
a
s
t
i
cFlowandF
r
a
c
t
u
r
ei
nS
o
l
i
d
s,
"ACa
.
d
e
m
i
c,
NewY
o
r
k
.
2
7
7
.v
a
nM
i
e
r
,
J
.G.M.,
1
9
8
6
“
,Multiaxia
.
lS
t
r
a
i
n
S
o
f
t
e
n
i
n
go
fC
o
n
c
r
e
t
e
jP
a
r
t1
:F
r
a
c
t
u
r
e,
P
a
r
t1
Lo
adH
i
s
t
o
r
i
e
s,
"Mater.andS
t
r
u
c
t
.,
V
.1
1
1,
N
o
.1
9
,
p
p
.1
7
9
2
0
0
.
誌
W
e
i
b
u
l
l,
W.,
1
9
3
9,
μPhenomenono
fR
u
p
t
u
r
ei
nS
o
l
i
d
s,
"Ingeni伺 rsvetenskap・saka
.
dHandl,
2
7
4
.T
a
y
l
o
r,
H
.P
.J
.,
1
9
7
2,
"TheS
h
e
a
rS
t
r
e
n
g
t
ho
fLa
.
r
g
eBeams,
"J
.o
fS
t
r
u
c
t
.E
n
g
n
g
.,
A
S
C
?
pp.2473
・2
4
9
0
.
2
7
6
.vanM
i
e
r
,J
.G.M.,1
9
8
4
“
,Strain-Softeningo
fC
o
n
c
r
e
t
eu
n
d
e
rM
u
l
t
i
a
x
i
a
lL
o
a
d
i
n
gC
o
n
d
t
i
o
n
s,
"DoctoralThesis,
D
e
l
f
tU
n
i
v
.o
fT
e
c
h
n
o
l
o
g
y
,
TheN
e
t
h
e
r
l
a
n
d
s
.
m
主γ
怜
吋
立
立
討
士
r
討
ぷ
T
は
:
官
枕
ゴ口
れ
ネ
立
詑
出
出
;
♂
必
ど
立J出ご古出出叩
%
?
Y
三
i
:
f
1
:
z
:
:
:
と
t
: ど
烈
マ
寸j
?
l
r
1
:
討
詑
E
t
士
;
吠
之
ど
;
ご
ji7
つ
t
立
:
:
r
官
;
知
r
知
?
悩
:
t
お
お
部
:2
悶
;
お
;
訟
お
ナ
z
z
?
;
ω
ω
b
恥
附
十
:
f
山
脳
忌
…
i
M
泌
抑
s
抗
由t
5
伽
捌
畠
a
叫…
E
n
e
r
g
y
ニ
ト
aJ21tfszz:K23121j;:; ob::23tz:::fJ;1:12:21:rcy;
S
w
a
r
t
z,
S
.E
.,
SEM,
B
e
t
h
e
l,
p
p
.1
9
2
2
0
7
.
2
'
11
.Wittmann,
F
.H
.,
e
d
.,
1
9
8
3,
"
F
r
a
.
ct
.M
e
c
h
.o
fC
o
n
c
r
e
t
e,
"E
l
s
e
v
i
e
r
,
Am
s
t
e
r
d
a
r
o
.
寸
Wittmann
,
F
.
H
.,
R
o
e
l
f
s
t
r
a
,
P
.
E
.,
andS
.
a
d
o
u
k
i,
H
.,
1
9
8
4,
“
S
i
m
u
l
.
a
t
i
o
na
ndAnal
y
s
i
so
fCom刈 E
n
g
n
g
.,
V
.6
8,
p
p
.2
3
9・ 2
4
8
.
p
o
s
i
t
esujctn-J'Mater.sci a
th. Wittm
組n,
F
.
H
.,
R
o
e
l
f
s
t
r
a,
P・E
.,
M
i
h
a
s
h
i,
H
.,
Hu
.
a
n
g,
Y
.
Y
.,
Zh胡 g,
X
.
H
.,
回dNomura,
N
.,
1
9
8
7“
・Inhenceo
fAgeo
fL
o
a
d
i
n
g
,
Water-CementR
A
t
i
oZ
L
E
M
IRAteo
fL
o
a
d
i
n
gonbatture
fConc
問 t
e,
"Mater.andS
t
r
u
c
t
.,
V
.2
0
,
p
p
.1
0
3
1
1
0
En o
叩
・
1
.W
nuk.M.P← 1
9
74
““
, QuasiS
t
a
t
i
cE
x
t
e
n
s
i
ono
faT
e
n
s
i
l
eCr
a
.
c
kC
o
n
t
a
i
n
e
di
nV
i
s
c
o
e
l
ω
t
i
c
P
l
s
o
l
i
d
J
'
J
. fApii.Mech.,
ASME
,
v.4l,
N0
・1
,
p
p
.2
3
4
・2
4
8
.
槌
…
川.
。
Wu
,
F
.H
.,
andF
r
eund,
L
.B
.,
1
9
8
4,
“
D
e
f
o
r
m
a
t
i
o
nT
r
.
a
p
p
i
n
gduet
oThermoplastkI
n
s
t
a
b
i
l
i
t
y
,
V32
,
No2Jp.
i116ne-DimeIMEalwavePropaptionfJ ofMech.andPhys o
f
S
o
l
i
d
s
1
1
9
.
1
3
2
.
2
,
.~ Xu
,D.,andRe
i
n
h
a
r
d
t,H
.W.,1
9
8
9“
,S
o
f
t
e
n
i
且g
ofC
o
n
c
r
e
t
e1
1
I
L
d
e
rTor
s
i
o
n
dLoadingf
F
r
a
c
t
u
r
eo
fC
o
n
c
r
e
t
ea
.
山 Rock:Recen
tDev
e
l
o
p
m
e
n
t
s,
e
d
s
.Sh
油
,S
.P
.,
Swar
民 S
.E
.,
and
B-;;:B~, Ï~t~C:~f..h:ïd-a~~C~~diff: K
.
, Els
e
v
i
e
rA
p
p
l
i
e
dSdence
,
Lo
ndon,
U.
K
,p
p
.3
9
い5
0
U
.
任
Y
u
z
u
g
u
l
l
u,
0
.,
andS
c
h
n
o
b
r
i
c
h,
W.C
.,
1
9
7
3
“
,ANumericalProceduref
o
rt
h
eDe
t
e
r
r
n
i
n
.
a
t
i
on
"ACIJ
.,
v.7o
,
No.7,
pp.47
4・4
7
9
.
o
ft
h
eBehwioro
faS
h
e
a
rWanETameS
y
s
t
em,
二
ヱ ヰ
よ
。
ii23JZ:::::;::;;;:;2JF;::?:::::
Z1 2 2 J4LZ: ;tz:::
a
t
e
r
ia
J
s
,
Kyoto,
J
.
a
pan,
7
0
5p
.
1
9
7
3Symp.onM
e
c
h
a
n
i
c
a
lB
e
h
a
v
i
o
ronM.
,,
, , “
,I
n
t
e
r
f
a
c
eE
l
e
m
e
n
tM
o
d
e
l
i
n
go
fF
r
a
c
t
u
r
ei
nA
g
g
r
e
山 s
,
"J
.o
fEng
略 M
e
c
h
.D
i
v
.,
ASCE
,
V
.日3,
No.1
1,
p
p
.1
6
時 1
6
3
0
.
g
a
t
eCompL
司2
・Z
u
b
e
l
e
w
i
c
zA
. andB
a
i
a
n
tZ
.P
.1
9
8
7
140
Appendix1
. DerivationsofSomeFormulas
圃
D
e
r
i
v
a
t
i
o
no
fE
q
.1
.4
: Thes
i
z
ee
f
f
e
c
tl
a
wi
nE
q
.4
.
1c
a
nbed
e
r
i
v
e
dmostg
e
n
e
r
a
l
l
yb
¥
l
y
s
i
sands
i
m
.
i
l
i
t
u
d
ea
r
g
u
m
e
n
t
s(
B
a
.
Z
a
n
t,
1
9
8
4
a
)ont
h
eb
a
s
i
so
ft
h
efollowin~
d
i
m
e
n
s
i
o
n
a
l姐 a
h
y
p
o
t
h
e
s
e
s
:(
I
.
)Thee
n
e
r
g
yr
e
l
e
a
s
eo
ft
h
e山 u
c
t
u
r
ei
saf
u
n
c
t
i
o
no
f
b
o
t
h
:(
a
)t
h
el
e
n
g
t
h
a,
叩 d(
b
)t
h
ec
h
a
r
a
c
t
e
r
i
s
t
i
cf
r
a
c
t
u
r
ep
r
o
慣 sz
o
n
e,
Cj・ (
1
1
.
)Thel
e
n
g
t
ha
,
a
t
o
ff
r
a
c
t
u
r
e,
sp
r
o
p
o
r
t
i
o
n
a
lt
os
t
r
u
c
t
u
r
es
i
配 d
,
wh
i
1
ecJi
sas
t
r
u
c
t
u
r
a
J
maximuml
o
a
di
sn
o
tn
e
g
l
i
g
i
b
l
e姐 di
p
r
o
p
e
r
t
yi
n
d
e
p
e
n
d
e
n
to
fd.
Thet
o
t
a
lamounto
fe
n
e
r
g
yr
e
l
e
a
s
e
dfromt
h
es
t
r
u
c
t
u
r
ei
n
t
ot
h
ef
r
a
c
t
u
r
emustb
ee
x
.
p
r
e
s
s
i
b
l
ei
n出 eform
2
W =.
.
.
.
!
σ
τ
t
b
d
F
("
川),。
8
2
2
E
'
UN
"
.
.
.
.
.
1
.
'¥
1
."2)> V1=E
=d
'
"
2=2
=d
(
A.
1
)
J
t
,8
r
ei
n
d
e
p
e
n
d
e
n
tn
o
n
d
i
m
e
n
s
i
o
n
a
lp
a
r
a
m
e
t
e
r
sも
(h
e
i
rnumberf
o
U
o
Ws
whereσN = P/bdj(
2a
fromBuckingham
込 山 田remo
fd
i
m
e
n
s
i
o
n
a
l組 a
l
y
s
i
s
),
andF i
sac
e
r
t
a
i
nf
u
n
c
t
i
o
nw
h
i
c
hma
.
y
r
omt
h
ec
r
a
c
kp
r
o
p
a
g
a
t
i
o
nc
o
n
d
i
t
i
o
n8W/
δ
α =Gf
b
,
o
n
eg
e
t
s
bee
x
p
e
c
t
e
dt
obes
m
o
o
t
h
.F
σL= 竿 問ヲ~rl
(
A
.
2
)
WL--T
・
1
1/
コa
、
、'f
A
一
一
F
q&
c
、
.
/'tin-
σ
一
一N
E
-
Wenowc
h
o
o
s
et
h
es
t
a
t
e(
J2 =0(
w
h
i
c
hc
o
r
r
e
s
p
o
n
d
st
od→∞)ωther
e
f
e
r
e
n
四 s
t
a
t
e
, 組d
exp
組 d8
F/
δ8
i
n
t
o
T
a
y
l
o
r
s
e
r
i
e
s
a
b
o
u
t
t
h
i
s
s
t
a
t
e
.
i
e
.
,
aF
/
δ
0
=
凡
+
F
,
)
'
8
,
)
'
+
九時+九時+
1
1
…whereFl,
凡 一a
r
ec
o
n
s
t
a
n
t
si
fg
e
o
m
e
t
r
i
c
a
1
1
ys
i
m
.
il
a
rs
h
a
p
e
s(
s
叩 l
e(
1
)a
r
ec
o
n
s
i
d
e
r
e
d
.
q
.(
A
.
2
)姐 dt
r
u
n
c
a
t
i
o
no
ft
h
es
e
r
i
e
sa
f
t
e
rt
h
el
i
n
e
a
rtermy
i
e
l
d
s
:
S
u
b
s
t
i
t
u
t
i
o
ni
n
t
oE
(
A
.
3
)
,
/
j
.
T
h
i
sy
i
帥 E
q
.4
.
1i
fo
n
eden
o
t
e
sB= (
2
E
'
G
andn
o
t
e
告側
o=凡C F1,
J)/(九叫)佃dd
B anddoa
r
ec
o
n
s
t
a
n
t
s
.
田 e
f
f
e
c
tl
a
w(
E
q
.1
.4
),
weh
a
v
ep2=σ
(N
b
d
/c
.
,
.)
2= (
B
!
u
b
d
/c
.
,
.)
2
d
J
(
d+
d
.
)
Fromt
h
es
i
o
o
S
u
b
s
t
i
t
u
t
i
n
gf
o
rp2i
nE
q
.5
.
2andn
o
t
i
n
gE
q
.6
.
2
4f
o
rG weg
e
t
,
(
α
)
G(α)=G/
一
一
一
Jg
(一一
α )d
+
d
。
o
(
A.
4
)
Thec
r
i
t
i
c
a
ls
t
a
t
eo
c
c
u
r
si
ff
r
a
c
t
u
r
ep
r
o
p
a
g
a
t
i
o
ni
sp
o
s
s
i
b
l
ea
l
s
of
o
rt
h
en
e
x
ta
d
j
a
c
e
n
t
t
a
t
e
,
i
.
e
. ifθ(G-R
)
J
8
d= O
.B
e
c
a
u
s
eθR
(
c
)
/
8
d= 0,
wemusth
a
v
eθ
G
/
θd= O
.S
i
n
c
e
α=a
J
d=(
a
o+c
)
/
d
,
wef
i
n
d8
α
/
δd=c
J
♂ =(αoー α)
/
d
,
組 d80s
u
b
s
t
i
t
u
t
i
o
no
fE
q
.A
.
4
i
n
t
oθGJ
θd=[
o
G
(
α)
J
θ
α
]
θ
α
/
δd+θGJ
δd=0p
r
o
v
i
d
e
s
日
d+d
o=dog(α)J !9'(α)(α 一 αo~
(~司
S
u
b
s
t
i
t
u
t
i
碍 t
h
i
sf
o
r(
d+d
)i
nE
q
.Aム wef
u
r
t
h
e
ro
b
t
a
i
nG(α)=G,
旬'
(
α)
J
g
(
α'
o
)
]
c
f
d
o
・
o
F
i
n
a
U
y
,notingthatdo= cg
(α)
/
g
'
(
α
0
)(
f
r
o
mE
q
.5
.
6
),weo
b
t
a
i
nE
q
.4
.
1 (Bai
組 主 回d
凶, 1
9
8
8
)
.F
u
r
t
h
e
r
m
o
r
e
,s
u
b
s
t
i
t
u
t
i
n
gE
q
.4
.
1i
n
t
oE
q
.A.4i
nw
h
i
c
h(R=G)回 d
Kaze
e
l
i
m
.
in
a
t
i
n
gdw
i
t
ht
h
eh
e
l
po
fE
q
.A.5,
E
q
.4
.
2e
n
s
u
e
s
.
,。
PartI
I
ConferencePapers
Fly UP