Civil Engineering Reference
In-Depth Information
T can be computed by
§
1
·
1
T
cos
n
x
r
SIGN
(sin
T
)
©
r
¹
(5.44)
The first integral is evaluated as:
T
T
T
B
B
cos
TT
rd
1
T
B
1
³
³
e
'
T
d
T
T T
(5.45)
i
A
B
2
S
r
cos
T
2
S
2
S
2
S
T
T
T
A
A
A
B
n
T
rd
T
cos
T
L
e
r
A
d
T
y
r
A
r
B
x
T
B
T
T
A
h
P
i
Figure 5.12
Polar coordinate system used for the analytic evaluation of integral '
T
i
e
If
P
i
is at the centre of element
e
then we have to take the Cauchy principal value of
the integral. As shown in Figure 5.13, the integration is carried out over the region of
exclusion. The reader may verify that because of the anti-symmetry of T shown in
Figure 4.4 we obtain
i
T
'
The second integral is computed as
0
T
T
B
B
1
1
rd
T
1
§
h
·
hd
T
³
³
e
i
'
U
ln
ln
¨
¸
2
2
S
kr
cos
T
2
S
k
cos
T
©
¹
cos
T
T
T
A
A
(5.46)
T
ª
º
A
§
·
h
§
h
·
tan
T
ln
1
T
«
¨
¸
»
¨
¸
2
S
k
cos
T
«
©
¹
»
©
¹
¬
¼
T
B
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