Civil Engineering Reference
In-Depth Information
Evaluation of the matrices [
]and[
T
p
]
In order to calculate the elements of [
(h)
]
−
1
, the quantities
v
1
,v
3
,v
3
,σ
13
,σ
33
and
σ
33
must be evaluated from the potentials
ϕ
1
,ϕ
2
and
ψ
s
. The velocity component
v
1
will
also be calculated for the sake of completeness. The velocity components
v
1
and
v
3
are
obtained by using
ϕ
1
,ϕ
2
and
ψ
2
as given by Eqs. (11.22) and (11.23) in Equation (11.27).
i
v
1
A
i
)
cos
k
i
3
x
3
−
A
i
)
sin
k
i
3
x
3
}
=
jω
1
,
2
{−
jk
t
(A
i
+
k
t
(A
i
−
=
(11.35)
A
3
)
sin
k
33
x
3
+
A
3
)
cos
k
33
x
3
+
k
33
(A
3
+
jk
33
(A
3
−
i
v
3
A
i
)
sin
k
i
3
x
3
A
i
)
cos
k
i
3
x
3
=
jω
1
,
2
{−
k
i
3
(A
i
+
−
jk
i
3
(A
i
−
}
=
(11.36)
−
jk
t
(A
3
+
A
3
)
cos
k
33
x
3
−
k
t
(A
3
−
A
3
)
sin
k
33
x
3
In these equations the dependence on time and
x
1
has been removed. The velocities
v
1
and
v
3
can be evaluated from the displacement potentials
ϕ
1
,ϕ
2
and
ψ
f
, which are
related to
ϕ
1
,ϕ
2
and
ψ
s
by Equation (11.27)
i
v
1
1
,
2
{−
jk
t
µ
i
(A
i
+
A
i
)
cos
k
i
3
x
3
−
k
t
µ
i
(A
i
−
A
i
)
sin
k
i
3
x
3
}
=
jω
=
(11.37)
+
k
33
µ
3
(A
3
+
A
3
)
sin
k
33
x
3
+
jk
33
µ
3
(A
3
−
A
3
)
cos
k
33
x
3
i
v
3
A
i
)
sin
k
i
3
x
3
−
A
i
)
cos
k
i
3
x
3
}
=
jω
=
1
,
2
{−
k
i
3
µ
i
(A
i
+
jk
i
3
µ
i
(A
i
−
(11.38)
A
3
)
cos
k
33
x
3
A
3
)
sin
k
33
x
3
−
jk
t
µ
3
(A
3
+
−
k
t
µ
3
(A
3
−
The two components
σ
33
and
σ
13
of the stress tensor of the frame, and the component
σ
33
in the fluid, can be calculated by Equations (6.2) and (6.3)
+
2
N
∂u
s
3
∂x
3
σ
33
=
∇
·
u
s
∇
·
u
f
(P
−
2
N)
+
Q
(11.39)
N
∂u
s
1
∂u
s
3
∂x
1
σ
13
=
∂x
3
+
(11.40)
σ
33
=
u
f
u
s
R
∇
·
+
Q
∇
·
(11.41)
