7 Structural Analysis of Oil-Spill Booms
149
pressure and the gravity force. The potential energy of the external actions is defined
by the linear form l
l(v) =
ω
(p N · v) + (ρ ω g · v) dω,
(7.12)
where p N is the resultant normal pressure, ρ ω is the membrane surface density and
g is the gravity vector.
The non-linear equilibrium equation is given by
Find u admissible, such that
d
du
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 ∂ω A and ∂ω 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 ∂ω c in the vicinity of c.
u = 0
on ∂ω A and ∂ω B
(7.14)
u = 0 z
on ∂ω c
As a consequence, the boom hydrostatic pressure is defined as the reaction force
resulting from the Dirichlet boundary condition applied to ∂ω 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
N i (ξ 1 , ξ 2 ) =
1
4
(1 ± ξ 1 )(1 ± ξ 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
Find U admissible, such that F int (U) = F ext (U),
(7.16)
149
pressure and the gravity force. The potential energy of the external actions is defined
by the linear form l
l(v) =
ω
(p N · v) + (ρ ω g · v) dω,
(7.12)
where p N is the resultant normal pressure, ρ ω is the membrane surface density and
g is the gravity vector.
The non-linear equilibrium equation is given by
Find u admissible, such that
d
du
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 ∂ω A and ∂ω 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 ∂ω c in the vicinity of c.
u = 0
on ∂ω A and ∂ω B
(7.14)
u = 0 z
on ∂ω c
As a consequence, the boom hydrostatic pressure is defined as the reaction force
resulting from the Dirichlet boundary condition applied to ∂ω 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
N i (ξ 1 , ξ 2 ) =
1
4
(1 ± ξ 1 )(1 ± ξ 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
Find U admissible, such that F int (U) = F ext (U),
(7.16)