Environmental Engineering Reference
In-Depth Information
pressure and the gravity force. The potential energy of the external actions is defined
by the linear form
l
l
(
v
)
=
(
p
N
·
v
)
+
(ˁ
ˉ
g
·
v
)
d
ˉ,
(7.12)
ˉ
where
p
N
is the resultant normal pressure,
ˁ
ˉ
is the membrane surface density and
g
is the gravity vector.
The non-linear equilibrium equation is given by
d
du
Find
u
admissible, such that
e
(
u
)
·
v
=
l
(
v
)
∀
v
admissible
,
(7.13)
where the displacement field
u
respects the boundary conditions at the end-parts and
waterline of the boom domain.
Dirichlet boundary conditions are used at the two end-parts
∂ˉ
B
of the
boom domain boundary corresponding to the end-points
A
and
B
of the curvilinear
domain
c
. The same kind of boundary condition is applied to the displacement along
the vertical
z
of the membrane waterline
∂ˉ
A
and
∂ˉ
c
in the vicinity of
c
.
u
=
0n
∂ˉ
A
and
∂ˉ
B
(7.14)
u
=
0
z
on
∂ˉ
c
As a consequence, the boom hydrostatic pressure is defined as the reaction force
resulting from the Dirichlet boundary condition applied to
∂ˉ
c
.
A classical mooring device of a boom is composed of a transverse rigid beam
linked to two mooring cables, a buoy, a mooring chain and a dead-mass on the sea
floor. Such parts of a boom implementation are not detailed here. Adjacent barrier
or curtain sections can be connected by using standardized self-rigid ending beams
having a Z-shape. Boom inertial force can be neglected in the membrane equilibrium
as has been the case in Sect.
7.2.2
.
7.3.3 The Numerical Method
The Haug-Powell quadrilateral finite element is used to define the discrete problem
[
9
]. This element is based on the four bilinear shapes functions
1
4
(
N
i
(ʾ
1
,ʾ
2
)
=
1
±
ʾ
1
)(
1
±
ʾ
2
),
i
=
1
,
2
,
3
,
4
(7.15)
As for the curvilinear domain approach (
7.8
) the 3D non-linear equations of the
discrete equilibrium problem can be written
F
int
(
)
=
F
ext
(
),
Find
U
admissible, such that
U
U
(7.16)
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