Civil Engineering Reference
In-Depth Information
or in discretised form using line segments
8
8
¦
¦
e
e
e
e
(5.51)
u
P
'
T
(
P
)
'
U
(
P
)
a
a
a
e
1
e
1
where
e
³
e
³
(5.52)
'
T
T
(
P
,
Q
)
dS
(
Q
)
,
'
U
U
(
P
,
Q
)
dS
(
Q
)
a
e
a
e
S
S
e
e
The flows at
P
a
in x- and y-directions are computed by taking derivatives of (5.50)
§
·
w
u
w
U
w
T
¨
©
³
³
¸
¹
q
P
k
P
k
t
Q
P
,
dS
Q
u
Q
P
,
dS
Q
x
a
a
a
a
w
x
w
x
w
x
S
S
(5.53)
§
·
w
u
w
U
w
T
¨
©
¸
¹
³
³
q
P
k
P
k
t
Q
P
,
dS
Q
u
Q
P
,
dS
Q
y
a
a
a
a
w
y
w
y
w
y
S
S
where the derivatives of
U
have been presented previously and the derivatives of
T
are
given for two-dimensional problems as
ª
º
w w w w
« »
ww w w
T
U
U
n
n
x
y
x
x
x
y
¬
¼
(5.54)
ª
º
w w w w
« »
ww w w
T
U
U
n
n
x
y
y
y
x
y
¬
¼
For constant boundary elements, equation (5.53) can be replaced by
§
E
E
·
¦¦
¨
©
¸
¹
e
xa
e
e
xa
e
q
P
k
'
S
t
'
R
u
x
a
e
1
e
1
(5.55)
§
E
E
·
¦¦
¨
©
e
ya
e
e
ya
e
¸
¹
q
P
k
'
S
t
'
R
u
y
a
e
1
e
1
where the integrals
w
U
w
U
e
xa
³
e
ya
³
'
S
P
,
Q
dS
Q
;
'
S
P
,
Q
dS
Q
a
a
w
x
w
y
S
S
e
e
(5.56)
w
T
w
T
³
³
e
xa
e
ya
'
R
P
,
Q
dS
Q
;
'
R
P
,
Q
dS
Q
a
a
w
x
w
y
S
S
e
e
can be evaluated analytically over element
e
.
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