Civil Engineering Reference
In-Depth Information
2
∫
⎫
φ
dx
1
2
⎡ ⎤
z
*
H
⎪
κ
⎡ ⎤
2
⎢ ⎥
4
ρ
B
m
ae
z
L
exp
⎪
⎢ ⎥
=
⋅
⋅
⎢ ⎥
2
1
4
1
2
⎪
ζ
∫
dx
φ
⎢ ⎥
⎢ ⎥
⎣ ⎦
ae
z
*
1
H
z
z
⎪
⎢
⎣ ⎦
⎬
⎡ ⎤
⎪
L
(6.31)
2
∫
dx
φ
*
θ
A
κ
⎡ ⎤
4
⎢ ⎥ ⎪
3
ρ
B
m
ae
L
θ
exp
⎢ ⎥
=
⋅
⋅
⎢ ⎥
⎪
2
1
4
ζ
∫
dx
φ
⎢ ⎥
⎢ ⎥ ⎪
⎣ ⎦
*
2
ae
θ
A
θ
θ
⎢ ⎥
⎪
⎣ ⎦
⎭
L
Example 6.1
The volume integral in the joint acceptance functions above, e.g. as first defined in Eq. 6.19 or as
normalised versions given in Eqs. 6.22, 6.28 and 6.29, may in general be expressed by
LL
exp
exp
m
⎫
(
)
2
J
=
∫∫
g
x
,
x
x x
y z
,,
=
θ
⎬
⎭
rr
rr
12
1 2
mn
mn
n
00
(
)
(
)
(
)
(
)
g
xx
,
G
x G x
x
k
uw
,
where:
=
⋅
⋅
ψ
Δ
,
=
. It will in most cases demand
rr
12
r
1
r
2
kk
mn
m
n
x
xx
a fine mesh, particularly in the region of small separation
Δ
=−
. The reason for this is
12
(
)
g
xx
will rapidly drop in the
,
that
ψ
, is usually rather steep close to zero, and thus,
kk
r
mn
12
x
x
region close to a diagonal plane through
. This difficulty may readily be overcome by
adopting Dyrbye & Hansen's [21] following procedure for turning a volume integral back into two
line integrals. The position coordinates
=
1
2
x
and
x
are interchangeable, and therefore
(
)
g
xx
will be symmetric about the plane through
,
x
x
=
. Thus,
r
mn
12
1
2
LL
⎡
⎤
exp
exp
2
⎢
(
)
⎥
J
2
∫ ∫
g
x
,
x
x x
dx
=
−
Δ
Δ
rr
rr
11
1
mn
mn
⎢
⎥
0
⎣
Δ
x
⎦
x
xx
Introducing the notation
=+
Δ
and
(
)
(
)
(
)
(
)
(
)
( )
(
)
g
xx xGxGx x
,
xGx xGx
x
−=
Δ
⋅
−⋅
Δ
ψ
Δ
=
+⋅
Δ
⋅
ψ
Δ
r
r
11
r
1
r
1
kk
r
r
kk
mn
m
n
m
n
then the following is obtained:
LL x
⎡
−
Δ
⎤
exp
exp
2
(
)
( )
(
)
∫ ∫
⎢
⎥
J
=
2
Gx xGx x
+
Δ
⋅
⋅
ψ
ΔΔ
xdx
rr
r
r
kk
mn
m
n
⎢
⎥
0
⎣
0
⎦
ˆ
It is usually convenient to introduce the normalised coordinate
xxL
=
/
and separation
exp
ˆ
ΔΔ
x
=
xL
/
. Thus, in a normalised format the joint acceptance function is given by
exp
11
⎡
−
Δ
x
ˆ
⎤
2
2
exp
(
)
( )
(
)
∫ ∫
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
J
=
2
L
G
x
+
Δ
x
⋅
G
x dx
⋅
ψ
ΔΔ
x d
x
⎢
⎥
rr
r
r
kk
mn
m
n
⎢
⎥
⎣
⎦
0
0
()
()
Let for instance
r
Gx xL
=
/
and
r
n
Gx xL
=
/
, then
1
1
exp
2
2
exp
