The second terms in (110) are
P ηj
Ni
η
η P
2
$
$
$
N i ds = ~ qr N i
= ∫~qr
∫
ds
-
2
2
P
a
I
a
s
Γ2 i
2
s i
s
s
P
P
~ 2
~qr 2 η - a qr η - η
(
)
$
=- a
(113)
$i $j
s i
LP
and
~ 2
~ qr 2 η - a qr η - η
(
)
$
P
(114)
I
=a
$ j $i
Γ2 j
s j
LP
where r is the radius of the semicircle. These terms can also be written in matrix form as
[ ]= [ ] η }+ [ ]
Γ {$
P
P
P
P
I
D
(115)
Γ2
2
where
~ qr 2
-
1
1
[]
a
Γ =-
P
(116)
- 1
1
2
LP
and
T
η
η
[]
$
$
= - ~ qr 2
, -
P
(117)
D
a
si
s j
The third terms in Equation 110 are
1
I Γ3 i = - ∫~ gN iPds = - ~ g∫N Pds = - ~ gLP = I P3 j
P
a
a
a
Γ
i
2
P
P
35
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