Civil Engineering Reference
In-Depth Information
For a three-dimensional problem, we take the zone of exclusion to be a sphere, as
shown in Figure 5.7.
S
H
HI
H\
n
I
S-S
H
\
H
Figure 5.7
Computation of integrals for the case that
P
=
Q,
three-dimensional case
In this case the first integral also approaches zero as Happroaches zero. The second
integral can be computed as
2
SS
cos
T
1
³
³³
u
Q
T
P
,
Q
dS
(
Q
)
u
(
P
)
H
d
I
d
\
u
(
P
)
(5.26)
2
2
4
SH
s
00
H
which for smooth surfaces gives the same result as before. Obviously, the same limiting
procedure can be made for elasticity problems. If
P=Q
the integral equation (5.16) can
be rewritten as
ª
º
1
³
³
«
»
(5.27)
u
P
lim
U
P Q
,
t
Q dS Q
(
)
ȉ
P Q
,
u
Q dS Q
(
)
2
«
»
H
o
0
¬
¼
SS
H
SS
H
If the boundary is not smooth but has a corner, as shown in Figure 5.8, then equation
(5.24) has to be modified. The integration limits are changed and now depend on the
angle J :
J
J
cos
T
1
J
³
³
³
(5.28)
u
Q
T
P
,
Q
dS
(
Q
)
u
(
P
)
H
d
I
u
(
P
)
d
I
u
(
P
)
2
SH
2
S
2
S
s
0
0
H
A more general integral equation can be written for potential problems
ª
º
(5.29)
«
³
³
»
¼
cu
P
lim
0
t
Q
U
P
,
Q
dS
Q
u
Q
T
P
,
Q
dS
Q
«
H
o
¬
S
S
S
S
H
H
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