Undrained Load Capacity of Torpedo Anchors in ... - laceo - UFRJ

Torpedo anchors were first employed to hold flexible risers where [ Δσ] = [ Δσ
] T
xx Δσ
yy Δσ
zz Δτ
xy Δτ
xz Δτ
yz is the
just after their touchdown points (TDP) in order to avoid that
the loads generated by the floating units movements were
incremental total stress vector;
transferred to subsea equipments such as wet Christmas trees [ Δε
] = [ Δε
] T
xx Δε
yy Δε
zz Δγ
xy Δγ
xz Δγ
yz is the incremental
(Medeiros Jr. [1]). In 2003, a high holding capacity torpedo total strain vector; and [ D
ep
] is the constitutive elasto-plastic
anchor was designed to sustain the loads imposed by the
matrix (Potts and Zdravkovic [4]):
mooring lines of Petrobras FPSO P-50 (Araújo et al. [2]). Since
E
[ Δσ] [ Dep ] [ Δε]
K s =
3⋅
1−
2⋅υ
(6)
then, other floating production systems have been anchored
T
using the torpedo concept (Brandão et al. [3]).
⎧∂P
[ ]
([ σ] ,[ m]
) ⎫ ⎧∂F( [ σ][ , k]
) ⎫
D ⋅ ⎨ ⎬ ⋅ ⎨ ⎬ ⋅ [ D]
Despite its large use in Brazilian offshore applications, the
[
calculus of the torpedo anchor load capacity and the prediction
] [ ]
⎩ ∂σ
⎭ ⎩ ∂σ
= D −
⎭
T
⎧∂F( [ σ][ , k]
) ⎫ ⎧∂P
of the stresses in its structure due to the loads imposed by the
[ ]
([ σ][ m]
) ⎫
⎨ ⎬ ⋅ D ⋅ ⎨
⎬
⎩ ∂σ
⎭ ⎩ ∂σ
⎭
attached mooring line remains as a great challenge. Thus,
(2)
aiming at accomplishing these tasks, a nonlinear threedimensional
finite element (FE) model is here proposed. This
where [ D ] is the elastic total stress constitutive matrix;
model employs isoparametric solid elements to represent both P ([ σ ],
[ m]
) is the plastic potential function; F ([ σ ],
[ k]
) is the
the soil and the anchor. These elements are capable of yield function; [ σ ] is the stress state in the element; [ m ] and
representing the physical nonlinear behavior of the soil and, [ k ] are state parameters related to the plastic potential and yield
eventually, of the anchor. Large deformations may also be
functions, respectively.
accounted for. Soil-anchor interaction is assured by surface to
The elastic total stress constitutive matrix can be split in
surface contact elements placed on the external surface of the
two parts, as follows:
anchor and the surrounding soil. A general overview of this
model is presented in Fig. 2.
[ D ] = [ D eff ] + [ D pore ]
(3)
where [ D
eff
] is the effective stress constitutive matrix and
[ D
pore
] is the pore fluid stiffness, which, considering an
isotropic material loaded under undrained conditions, are given
by (Zdravkovic and Potts [4]):
⎡ 4 2 2
⎤
⎢K
s + ⋅G
K s − ⋅G
K s − ⋅G
0 0 0
3 3 3
⎥
⎢
4 2
⎥
⎢
K
⎥
s + ⋅G
K s − ⋅G
0 0 0
⎢
3 3
⎥
[ D ] = ⎢
4
⎥
eff
⎢
K s + ⋅G
0 0 0
3
⎥
⎢
symmetric
G 0 0 ⎥
⎢
⎥
⎢
G 0 ⎥
Figure 2 – General view of the FE model.
⎢
⎥
⎣
G⎦
(4)
In this paper, a parametric study is conducted considering a
typical torpedo anchor embedded in a purely cohesive isotropic
⎡1
1 1 0 0 0⎤
soil. The effects on the holding capacity of the undrained soil
⎢
⎥
strength, the number of flukes in the anchor and the width of
⎢
1 1 1 0 0 0
⎥
⎢1
1 1 0 0 0⎥
these flukes are evaluated.
[ D pore ] = K pore ⋅ ⎢
⎥
(5)
⎢0
0 0 0 0 0⎥
⎢
⎥
FE MODEL
0 0 0 0 0 0
⎢
⎥
⎢⎣
0 0 0 0 0 0⎥⎦
Soil modeling
where K
s
is the bulk modulus of the soil, G is the effective
Constitutive matrix
The soil is assumed to be a perfectly elasto-plastic isotropic transverse modulus of the soil and K
pore
is the bulk modulus of
material with physical properties variable with depth. Hence, the fluid, which are given by:
the relation between stresses and strains is given in the form:
( )
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