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and
∂ψ/∂
z
, as well as
∂
δψ
and
∂
δψ
, in the triangle of Fig. 8.11a can be represented as
y
z
∂ψ
∂
y
=
ψ
−
ψ
∂ψ
∂
z
=
ψ
−
ψ
2
1
3
2
,
h
h
(3)
∂
∂
y
δψ
=
δψ
−
δψ
∂
∂
z
δψ
=
δψ
−
δψ
2
1
3
2
,
h
h
and those in Fig. 8.11b can be represented as
∂ψ
∂
y
=
ψ
−
ψ
∂ψ
∂
z
=
ψ
−
ψ
3
2
2
1
,
h
h
(4)
∂
∂
y
δψ
=
δψ
−
δψ
∂
∂
z
δψ
=
δψ
−
δψ
3
2
2
1
,
h
h
The total complementary virtual work is the sum of the virtual work for each of the triangles
k
. Thus,
∂ψ
∂
dy dz
∂
∂
z
∂ψ
+
∂ψ
∂
∂
W
∗
=
δ
y
δψ
−
2
δψ
=
0
(5)
z
∂
y
k
Replace the derivatives in (5) by their approximations, and replace
δψ
by the average of the
δψ
values of
at the three corners of the triangles. The total complementary virtual work of
the
k
th triangle then becomes
A
ψ
δψ
k
2
k
1
k
2
k
1
−
ψ
−
δψ
W
∗
=
δ
h
h
k
ψ
δψ
3
k
3
k
2
k
3
k
2
3
δψ
−
ψ
−
δψ
2
k
1
k
2
k
+
+
+
δψ
+
δψ
(6)
h
h
k
i
where
A
is the area of the
k
th triangle, and
ψ
,i
=
1
,
2
,
3 are the values of
ψ
at the corners
of the
k
th triangle. Rewrite Eq. (6) as
k
δ
W
∗
=
ψ
kT
k
k
ψ
k
p
k
δ
(
−
)
=
0
(7)
with
−
110
1
1
1
1
=
ψ
3
T
A
h
2
2
A
3
ψ
k
k
1
k
2
k
k
k
p
k
ψ
ψ
=
−
2
−
1
=
0
−
1
−
1
Equation (7) looks like the stiffness equations found in the finite element method, and in
fact, it can be treated in the same manner as the stiffness equations. Thus, the summation
process can be performed in the same way as assembling the element stiffness matrices
into the global equation as described in Chapter 5, and the boundary conditions can also
be imposed similarly. After the summation, Eq. (7) can be written as
δ
ψ
T
(
K
ψ
−
P
)
=
0
(8)
in which
ψ
2
,
2
ψ
2
,
3
ψ
3
,
2
ψ
3
,
3
]
T
and
K, P
are the same as those in Eq. (3) of Example 8.3. Also,
ψ
=
[
ψ
i, j
is the value of
ψ
at the
nodal point
i, j
.
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