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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
,
η)
p
,
η) =
(6.37)
p
η
,
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
=
,
η)
dS
S p
a 1
0
a 1
0
N T
N T
=
,
η =
0
)
p
,
η =
0
)
d
ξ +
,
η =
1
)
p
,
η =
1
)
d
ξ
b 1
0
b 1
0
N T
N T
+
=
0 ,
η)
p
=
0 ,
η)
d
η +
=
1 ,
η)
p
=
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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