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written as
v
iT
u
T
p
dS
N
T
p
dS
S
p
δ
=
δ
(6.35)
S
p
where
p
is the applied
su
rface loading. In terms of the noda
l d
isplacements
v
i
and the
equivalent nodal forces
p
i
0
(
the element loading vector due to
p
)
, the virtual work due to
v
iT
p
i
0
. This must be equal to the expression of Eq. (6.35), that is
the applied loading is
δ
v
iT
S
p
N
T
p
dS
v
iT
p
i
0
δ
=
δ
(6.36)
It would also be appropriate to include a loading vector due to the interelement forces.
Howe
v
er, this will not be considered here. Since
u
has
x
and
y
components, the applied
loads
p
can be separated into those acting in the
x
and those acting in the
y
direction. Thus,
p
ξ
(ξ
,
η)
p
(ξ
,
η)
=
(6.37)
p
η
(ξ
,
η)
For loading applied on the outer rim boundaries, the integration has to be p
er
formed around
the circumference (
S
p
) and not over the broad surface. The loading vector
p
i
0
of an element
is composed of the sum of the integrals for each of the loaded boundaries. For our elements
of Fig. 6.10, we can write
p
i
0
N
T
p
=
(ξ
,
η)
dS
S
p
a
1
0
a
1
0
N
T
N
T
=
(ξ
,
η
=
0
)
p
(ξ
,
η
=
0
)
d
ξ
+
(ξ
,
η
=
1
)
p
(ξ
,
η
=
1
)
d
ξ
b
1
0
b
1
0
N
T
N
T
+
(ξ
=
0
,
η)
p
(ξ
=
0
,
η)
d
η
+
(ξ
=
1
,
η)
p
(ξ
=
1
,
η)
d
η
(6.38)
To illustrate the application of Eq. (6.38), consider the element of Fig. 6.10 with a linearly
distributed load as shown in Fig. 6.12. For this element, all surface loads are zero except
h
p
0
x
2
p
0
2
4
3
b
1
2
x
FIGURE 6.12
The element of Fig. 6.10 with linearly varying loading
along one edge.
a
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