Environmental Engineering Reference
In-Depth Information
du
dr
u
r
Where
,
= the radial and
circumferential elastic strain increments at
r
j-
1
,
respectively, and
β
e
e
d
ε
d
ε
θ
ε
=−
...., ....
ε
θ
=−
(12)
rj
(
−
1
)
(
j
−
1
)
r
sin
sin
.
By selecting an arbitrary small value for dε
θ
,
the values of ε
θ
(
j
)
, ε
r
(
j
)
, ε
θ
(
j
)
,
r
j-
1
and
u
j
can be cal-
culated using eqs. (22), (23), (21) and (12) in that
order. The process is then repeated several times
to determine completely the strains and displace-
ments in the plastic region. The elastic strains at
rj
can be obtained by using the elastic stress-strain
relationship,
=
ψ
ψ
By dividing the plastic region with a number
of thin annular rings, one can obtain the radial
stress at
r
j
for the h-B yield criterion (Brown et al.,
1983).
1
2
(13)
σ
rj
bba
=− −
.
()
where
1
2
2
1
2
(14)
a
=
σ
−
4
k
m
σσ
+
s
σ
e
rj
(
−
1
)
1
crj
(
−
1
)
c
ε
=
[(
1
−
v
)(
σ
−−
p
)( )]
v
σ
θ
−
p
(25)
rj
()
rj
()
o
()
j
o
G
1
2
e
b
=
σ
r
(
j-1
)
+
k
1
mσ
c
(15)
ε
=
[(
1
−
v
)(
σ
−−
p
)( )]
v
σ
−
p
(26)
θ
()
j
θ
()
j
o
rj
()
o
G
2
r
−
−
r
where
G
= the shear modulus and
v
= the Poisson's
ratio of the rock.
The radial stress at the elastic-plastic interface,
in which
r
=
r
e
, is given for h-B criterion by
σ
j
−
1
j
(16)
k
=
1
r
r
j
−
1
j
a similar stepwise procedure was extracted in
this study for Mohr-coulomb criterion. The radial
stress at
r
j
for the M-c yield criterion is obtained
as following:
=−
pM
σ
(27)
re
o
c
2
M
mmp
m
1
24
p
p o
p
−
=
+
+
s
(28)
p
σ
8
1
c
σ
=
rj
()
1
+−
+
kK
k
(
1
ϕ
σσ σ
(
))
(17)
2
it is obtained for M-c criterion as following:
[(
−
)
+
σ
]
2
c
θ
(
j
−
1
)
rj
(
−
1
)
rj
(
−
1
)
2
1(
p
−
σ
ϕ
r
−
+
r
o
c
)
(29)
σ
=
j
−
1
j
k
=
(18)
re
+
K
2
r
r
j
−
1
j
1
1
+
−
sin
sin
ϕ
ϕ
Using eq. 27 and 29 as the starting point, suc-
cessive values of σ
r
(
j
)
can be calculated from eq. 13
and 17 for the radii determined from eq. 21.
critical deviatoric plastic strain at
rj
can be
approximately estimated as
k
()
ϕ
=
(19)
2
1
c
cos
sin
ϕ
ϕ
σ
=
(20)
c
−
1
2
{
}
p
using the radii given by
(30)
γ ε
=
−
ε
+
δ
−
δ
θ
(
j
−
1
)
r j
(
−
1
)
r j
(
−
1
)
θ
(
j
−
1
)
G
r
2
2
ε
−
ε
−
ε
j
θ
(
j
−
1
)
r j
(
−
1
)
r j
()
(21)
=
if
γ
p
exceeds
γ
p*
strength of rock mass reaches
residual condition.
r
ε ε
−
−
ε
j
−
1
θ
()
j
r j
(
−
1
)
r j
()
let the circumferential strain increment be:
3
aPPlicaTion
dε
θ
=
ε
θ
(
j
)
-
ε
θ
(
j
-1)
(22)
Data sets 1-1, 1-2 (related to rock masses with
elastic-strain-softening behavior) and 2 (related to
a rock mass with elastic-brittle-plastic behaviour)
are applied in this study.
in the first stage the sensitive analysis is imple-
mented for hoek-Brown criterion by changing
m
and
s
parameters.
Then, the radial strain increment at
rj
can be
obtained as
p p
ε β ε
θ
d
=−
d
(23)
r
e
e
d
ε
=
d
ε
−
βε ε
θ
(
d
−
d
)
(24)
rj
()
r
(
j
−
1
)
()
j
θ
(
j
−
1
)
