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and the matrix of the components of the normal vector is [Eq. (1.57)]
a
x
0
a
y
A
T
=
(6.162)
0
a
y
a
x
General Solution
As the basis of the approximation procedure, choose an extended principle of virtual work
in the form of
CB
(Chapter 2). Equation
CB
is a combination of Eq. (B), the kinematics and
geometric boundary conditions, and Eq. (C), the principle of virtual work. From Chapter 2,
Table 2.4, form
CB
can be expressed as
V
i
δ
z
T
0
z
p
V
0
dV
z
T
dS
M
u
D
T
−
p
0
−
+
S
pi
δ
E
−
1
D
u
−
i
=
1
z
T
0
z
0
Au
dS
A
T
−
+
S
ui
δ
+
=
0
(6.163)
−
A0
where
i
is summed over all the elements (M), and the state vector
z
is given by
[
u
σ
]
T
z
=
Trial Functions
Follow the development of Section 6.9.1 for three dimensions. Assume that the displace-
ments and stresses can be represented by the trial functions.
N
u
u
=
u
and
σ
=
σ
p
+
N
σ
σ
where
σ
p
is the particular solution portion resulting from prescribed stresses. Place these
together as
u
σ
0
σ
p
N
u
0
0N
u
σ
N
z
=
z
=
z
p
+
z
=
+
(6.164)
σ
or
N
u
.
.
.
.
u
x
···
N
ux
···
·
···
·
···
·
···
·
···
.
.
.
.
u
y
···
σ
x
···
σ
y
···
σ
xy
N
uy
···
·
···
·
···
·
···
·
···
.
.
.
.
N
z
=
z
(6.165)
N
σ
x
···
·
···
·
···
·
···
·
···
.
.
.
.
N
σ
y
···
·
···
·
···
·
···
·
···
.
.
.
.
N
σ
xy
N
σ
We suppose that the transformation from generalized variables to the nodal response vari-
ables has been performed, e.g.,
N
ux
is replaced by
N
ux
G
u
.
We choose not to adjust the tilda
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