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The normal slope
w
,a
is assumed to vary linearly
w
=
(
1
−
ξ)w
+
ξw
(13.153)
,a
,ai
,a j
Use the relationships of Eqs. (13.128), (13.129), (13.152), and (13.153) to find
w
w
,a
w
,s
=
L
ij
v
ij
(13.154)
where
yj
]
T
=
w
θ
θ
w
θ
θ
v
ij
[
i
xi
yi
j
xj
and
N
B
1
sN
B
3
−
cN
B
3
N
B
2
sN
B
4
−
cN
B
4
0
−
c
(
1
−
ξ)
−
s
(
1
−
ξ)
0
−
c
ξ
−
s
ξ
L
ij
=
(
N
B
1
,
ξ
)/
ij
(
sN
B
3
,
ξ
)/
ij
(
−
cN
B
3
,
ξ
)/
ij
(
N
B
2
,
ξ
)/
ij
(
sN
B
4
,
ξ
)/
ij
(
−
cN
B
4
,
ξ
)/
ij
Substitute Eqs. (13.151) and (13.154) into Eq. (13.150) to obtain
=
σ
T
C
ij
v
ij
U
Si j
(13.155)
where
ij
ij
1
0
R
ij
L
ij
dS
R
ij
(ξ)
C
ij
=
=
L
ij
(ξ)
d
ξ
0
The matrix
C
ij
is of order of 9
6 and the integration can be performed in closed form.
Summation of the integrals
U
Si j
for the three sides completes the integral of the second
term on the right hand side of Eq. (13.140)
×
σ
T
Cv
U
2
=
S
(w
q
a
−
w
,a
m
a
−
w
,s
m
t
)
dS
=
U
S
12
+
U
S
23
+
U
S
31
=
(13.156)
with
Cv
=
C
12
v
12
+
C
23
v
23
+
C
31
v
31
.
Note that
C
is a 9
×
9 matrix and
v
, which contains
all of the nodal variables, is a 9
1 vector. Substitution of Eqs. (13.145) and (13.156) into Eq.
(13.140) (without the last term) leads to
×
1
2
σ
T
B
∗
H
=−
+
σ
T
Cv
σ
(13.157)
Since, ultimately, the displacements are the desired variables, the expression of the stiffness
equation must be in terms of
v
, i.e.,
σ
should be expressed in terms of
v
. Note that the stress
σ
are independent of
v
. In order for
Π
∗
H
parameters
to be stationary, set the first variation
to be equal to zero [Chapter 2, Eq. (2.82)]
δ
Π
∗
H
=
∂
Π
∗
H
∂
σ
δ
σ
+
∂
Π
∗
H
0or
∂
Π
∗
H
∂
Π
∗
H
∂
δ
v
=
∂
σ
=
0
and
=
0
∂
v
v
Then
Π
∗
H
∂
∂
σ
=−
B
σ
+
Cv
=
0
This leads to
B
−
1
Cv
σ
=
(13.158)
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