# Probabilistic and Fuzzy Arithmetic Approaches for  the Treatment of Uncertainties in the Installation of  Torpedo Piles

Authors: Denise Margareth Kazue Nishimura Kunitaki; Beatriz Souza Leite Pires de  Lima; Alexandre Gonçalves Evsukoff; Breno Pinheiro Jacob
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

The “torpedo” pile is a foundation system that has been recently considered to anchor mooring lines and risers of floating production systems for offshore oil exploitation. The pile is installed in a free fall operation from a vessel. However, the soil parameters involved in the penetration model of the torpedo pile contain uncertainties that can affect the precision of analysis methods to evaluate its final penetration depth. Therefore, this paper deals with methodologies for the assessment of the sensitivity of the response to the variation of the uncertain parameters and mainly to incorporate into the analysis method techniques for the formal treatment of the uncertainties. Probabilistic and “possibilistic” approaches are considered, involving, respectively, the Monte Carlo method (MC) and concepts of fuzzy arithmetic (FA). The results and performance of both approaches are compared, stressing the ability of the latter approach to efficiently deal with the uncertainties of the model, with outstanding computational efficiency, and therefore, to comprise an effective design tool.


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## Body

## 1. Introduction

### 1.1. Context: Offshore Platforms, Mooring Systems, and Anchors

Petroleum
companies around the world have been faced with the challenge of developing
offshore oil production activities in deep and ultradeep waters. In
shallow water, the traditional solution consists in employing platforms
supported by fixed framed structures, such as thejackets where the foundation system consists of driven piles [1]. Presently, as oil fields have been
identified in deeper water such as in the Campos Basin (southeastern Brazil),
offshore platforms have included several types of floating units, such as the semisubmersible platforms, the tension-leg platforms (TLPs), and floating production, storage, and offloading (FPSOs) units based on ships.Floating platforms can be maintained in position by
different types of mooring systems, which in turn may employ anchors based on
different types of foundation elements. Semisubmersible platforms and FPSO
units, for instance, may be kept in position by mooring lines incatenary or taut-leg configurations. Mooring lines in a free-hanging catenary
configuration transmit essentially horizontal loads to the foundation system.
This fact introduces a greater flexibility in the selection of the appropriate
anchor type. However, the mooring radius (the horizontal distance, measured at
the sea bottom, from the center of the platform) is relatively large; typically,
about two to three times the water depth. Therefore, the application of
catenary configurations may not be feasible in deep or ultradeep waters, due to
the increased weight of the mooring lines, and also due to installation
problems that may arise in congested scenarios with several platforms close
together (as is the case of some oil fields in the Campos Basin).The taut-leg configuration has been proposed to tackle
these constraints. This configuration, where the lines are not slack, allows
the use of smaller line lengths. When associated with the use of new materials
(such as polyester fiber ropes) [2], this leads to
considerable reduction in the weight of the mooring system. Moreover, since at
the anchor point the lines are not in contact with the seabed, and may reach
inclinations around 45°, the mooring radius is typically equal to the water
depth, therefore, considerably shorter than in catenary configurations.However, taut-leg mooring systems transmit vertical
loads to the foundation system. This is also the case with the tension leg platforms,
which are moored by vertical tendons. Therefore, care should be taken in the
selection of anchor types that can withstand vertical loads.Amongst the foundation elements that have been applied
in deep water systems, two types of anchors can be mentioned: thesuction anchor and the vertically loaded anchor (VLA) [3]. However, some installation
difficulties have been reported for suction anchors, due to added mass effects
and the resonant period for the lifting system at the installation depth that
may approach the dominant wave period at the site [4].
Vertically loaded anchors are easier to install, but require drag procedures
that may hinder their correct positioning, mainly in congested areas with many others
nearby platforms.


### 1.2. The Torpedo Pile

Thetorpedo pile (illustrated in Figure 1) was proposed [5] as a solution to withstand
vertical loads while circumventing the problems associated with other types of
anchors. It consists simply in a metallic pipe, with closed tip, filled with
scrap chain, and concrete [6].Figure 1
Torpedo pile.The installation does not require drag procedures such
as employed in VLAs; the procedure is quite simple, and is illustrated in
Figure2.
First, the installation vessel hangs the pile (connected to the mooring line) at
a specified drop height, above the target point on the seabed. The design embedment
is then reached by simply releasing the pile, letting it accelerate, fall freely,
and then penetrate into the soil.Figure 2
Installation of torpedo pile [5].More than one hanging
configuration has been conceived, for instance, one of the alternatives (considered
for the installation of torpedo piles to anchor flexible risers ormobile drilling
units (MODUs)) has a chain loop at the top of the installation line, as
shown in Figure 3. As the pile falls, this loop
is pulled and unfolded. Therefore, the torpedo pile presents not only low cost
of manufacture, but also low cost of installation, since the same vessel can
transport and install the pile.Figure 3
Chain loop [6].There is another configuration, for permanent mooring
of production units, which does not present the chain loop, but requires two
vessels to hang, respectively, the installation line and the mooring line (to
which the torpedo pile is connected). In this configuration, the bottom end of
the installation line is connected to an intermediate point of the mooring line,
therefore, maintaining the pile suspended at the desired drop height. Above this
connection point, there is a trigger that allows the mooring line and the pile to
be released, causing the pile to fall (dragging with it the mooring line), and
penetrate the soil.Another advantage of the torpedo pile concept is that, since
it can withstand horizontal and vertical loads, it can be used with mooring
lines in a taut-leg configuration that, as mentioned before, is the preferred alternative
for semisubmersible platforms and FPSO units in deeper waters and congested
scenarios.


### 1.3. Objective of the Paper

The design of a torpedo pile should employ theoretical
models to predict pile penetration depth, such as the dynamic penetration model
proposed by True [7]. This model relies on soil
parameters whose values are assumed as known, fixed, and deterministic.However, it is well known that the soil properties
present a significant degree of variability that, associated to imprecisions in
the determination of their design values, can affect the accuracy of the
response given by the simulation method. The objective of this paper,
therefore, is to study techniques to deal with the uncertainty of the soil parameters,
and to associate these techniques to an analytical/numerical penetration model
for the torpedo pile.Two different approaches are considered for the
treatment of uncertainties of the penetration model. The first is a
probabilistic approach, based on the classical Monte Carlo method. The second is a “possibilistic” approach, derived using concepts fromfuzzy arithmetic and fuzzy sets.The following sections of the paper begin by describing
the theoretical model and solution procedure considered for the simulation of
the pile penetration. Firstly, the analytical formulation originally presented
by True [7] is described; then a numerical solution procedure in
the time domain is described, followed by an application where a pile dropped
from a height of 200 m above the seabed is analyzed for deterministic, fixed
values of the soil parameters.The paper then proceeds by describing the soil
parameters that are considered uncertain. Methodologies to assess the
sensitivity of the response to the variation of these uncertain parameters are
then presented, based on the Monte Carlo method
(MC) and fuzzy arithmetic (FA). More important, such methodologies allow the
designer to incorporate, into the analysis method, techniques for the formal
treatment of these uncertainties.Finally, results of applications of these concepts for
the treatment of uncertainties are presented for an actual case study,
beginning with results of deterministic parametric studies in order to assess
the sensitivity of the response to the variation of the uncertain parameters.
Results for the “probabilistic” MC approach are then presented, followed by the
novel implementation and application of the approach based on FA. The results
and performance of both approaches are compared, stressing the ability of the
latter approach to efficiently deal with the uncertainties of the model, with
outstanding computational efficiency, and therefore, to comprise an effective
design tool.


## 1.1. Context: Offshore Platforms, Mooring Systems, and Anchors

Petroleum
companies around the world have been faced with the challenge of developing
offshore oil production activities in deep and ultradeep waters. In
shallow water, the traditional solution consists in employing platforms
supported by fixed framed structures, such as thejackets where the foundation system consists of driven piles [1]. Presently, as oil fields have been
identified in deeper water such as in the Campos Basin (southeastern Brazil),
offshore platforms have included several types of floating units, such as the semisubmersible platforms, the tension-leg platforms (TLPs), and floating production, storage, and offloading (FPSOs) units based on ships.Floating platforms can be maintained in position by
different types of mooring systems, which in turn may employ anchors based on
different types of foundation elements. Semisubmersible platforms and FPSO
units, for instance, may be kept in position by mooring lines incatenary or taut-leg configurations. Mooring lines in a free-hanging catenary
configuration transmit essentially horizontal loads to the foundation system.
This fact introduces a greater flexibility in the selection of the appropriate
anchor type. However, the mooring radius (the horizontal distance, measured at
the sea bottom, from the center of the platform) is relatively large; typically,
about two to three times the water depth. Therefore, the application of
catenary configurations may not be feasible in deep or ultradeep waters, due to
the increased weight of the mooring lines, and also due to installation
problems that may arise in congested scenarios with several platforms close
together (as is the case of some oil fields in the Campos Basin).The taut-leg configuration has been proposed to tackle
these constraints. This configuration, where the lines are not slack, allows
the use of smaller line lengths. When associated with the use of new materials
(such as polyester fiber ropes) [2], this leads to
considerable reduction in the weight of the mooring system. Moreover, since at
the anchor point the lines are not in contact with the seabed, and may reach
inclinations around 45°, the mooring radius is typically equal to the water
depth, therefore, considerably shorter than in catenary configurations.However, taut-leg mooring systems transmit vertical
loads to the foundation system. This is also the case with the tension leg platforms,
which are moored by vertical tendons. Therefore, care should be taken in the
selection of anchor types that can withstand vertical loads.Amongst the foundation elements that have been applied
in deep water systems, two types of anchors can be mentioned: thesuction anchor and the vertically loaded anchor (VLA) [3]. However, some installation
difficulties have been reported for suction anchors, due to added mass effects
and the resonant period for the lifting system at the installation depth that
may approach the dominant wave period at the site [4].
Vertically loaded anchors are easier to install, but require drag procedures
that may hinder their correct positioning, mainly in congested areas with many others
nearby platforms.


## 1.2. The Torpedo Pile

Thetorpedo pile (illustrated in Figure 1) was proposed [5] as a solution to withstand
vertical loads while circumventing the problems associated with other types of
anchors. It consists simply in a metallic pipe, with closed tip, filled with
scrap chain, and concrete [6].Figure 1
Torpedo pile.The installation does not require drag procedures such
as employed in VLAs; the procedure is quite simple, and is illustrated in
Figure2.
First, the installation vessel hangs the pile (connected to the mooring line) at
a specified drop height, above the target point on the seabed. The design embedment
is then reached by simply releasing the pile, letting it accelerate, fall freely,
and then penetrate into the soil.Figure 2
Installation of torpedo pile [5].More than one hanging
configuration has been conceived, for instance, one of the alternatives (considered
for the installation of torpedo piles to anchor flexible risers ormobile drilling
units (MODUs)) has a chain loop at the top of the installation line, as
shown in Figure 3. As the pile falls, this loop
is pulled and unfolded. Therefore, the torpedo pile presents not only low cost
of manufacture, but also low cost of installation, since the same vessel can
transport and install the pile.Figure 3
Chain loop [6].There is another configuration, for permanent mooring
of production units, which does not present the chain loop, but requires two
vessels to hang, respectively, the installation line and the mooring line (to
which the torpedo pile is connected). In this configuration, the bottom end of
the installation line is connected to an intermediate point of the mooring line,
therefore, maintaining the pile suspended at the desired drop height. Above this
connection point, there is a trigger that allows the mooring line and the pile to
be released, causing the pile to fall (dragging with it the mooring line), and
penetrate the soil.Another advantage of the torpedo pile concept is that, since
it can withstand horizontal and vertical loads, it can be used with mooring
lines in a taut-leg configuration that, as mentioned before, is the preferred alternative
for semisubmersible platforms and FPSO units in deeper waters and congested
scenarios.


## 1.3. Objective of the Paper

The design of a torpedo pile should employ theoretical
models to predict pile penetration depth, such as the dynamic penetration model
proposed by True [7]. This model relies on soil
parameters whose values are assumed as known, fixed, and deterministic.However, it is well known that the soil properties
present a significant degree of variability that, associated to imprecisions in
the determination of their design values, can affect the accuracy of the
response given by the simulation method. The objective of this paper,
therefore, is to study techniques to deal with the uncertainty of the soil parameters,
and to associate these techniques to an analytical/numerical penetration model
for the torpedo pile.Two different approaches are considered for the
treatment of uncertainties of the penetration model. The first is a
probabilistic approach, based on the classical Monte Carlo method. The second is a “possibilistic” approach, derived using concepts fromfuzzy arithmetic and fuzzy sets.The following sections of the paper begin by describing
the theoretical model and solution procedure considered for the simulation of
the pile penetration. Firstly, the analytical formulation originally presented
by True [7] is described; then a numerical solution procedure in
the time domain is described, followed by an application where a pile dropped
from a height of 200 m above the seabed is analyzed for deterministic, fixed
values of the soil parameters.The paper then proceeds by describing the soil
parameters that are considered uncertain. Methodologies to assess the
sensitivity of the response to the variation of these uncertain parameters are
then presented, based on the Monte Carlo method
(MC) and fuzzy arithmetic (FA). More important, such methodologies allow the
designer to incorporate, into the analysis method, techniques for the formal
treatment of these uncertainties.Finally, results of applications of these concepts for
the treatment of uncertainties are presented for an actual case study,
beginning with results of deterministic parametric studies in order to assess
the sensitivity of the response to the variation of the uncertain parameters.
Results for the “probabilistic” MC approach are then presented, followed by the
novel implementation and application of the approach based on FA. The results
and performance of both approaches are compared, stressing the ability of the
latter approach to efficiently deal with the uncertainties of the model, with
outstanding computational efficiency, and therefore, to comprise an effective
design tool.


## 2. The Penetration Model

### 2.1. Original Formulation: Penetration of Projectiles

Studies on the
behavior of penetration of projectiles were initially intended for military
applications [8] and were followed by studies on
the prediction of final embedment depth of projectiles into soils [9, 10], and estimation of undrained shear strength [11, 12].The development of a dynamic penetration model by the US  navy
was required to represent the penetration of propellant-embedded plate anchors
into seafloor soils [13]. This kind of anchor is directly
positioned on the mud line and an explosion, caused by the propeller system,
pushes the anchor fluke down. In order to fulfill this objective, True [7]
took into account recommendations given by authors of empirical models (such as
Young [9]) and modified traditional bearing
capacity formulations (for deep foundations in cohesive soils) to consider
variations in penetration resistance with velocity and penetrator shape.The analytical model developed in [7] to simulate the dynamic penetration of plate anchors is
based on Newton's
second law. Considering that the penetrator velocity v can be expressed as v=dz/dt (where z stands for the soil depth), and
therefore, its acceleration dv/dt can be expressed as (dv/dz)(dz/dt),
the governing equation can be written as follows: (2.1)M′dvdzv=Ws−FD−FT−FS+FE, where M′ is the effective mass of the
penetrator, given by (2.2)M′=M+2ρ⋅V.In this latter
equation,M and V are, respectively, the structural mass and the volume of the
penetrator, and ρ is the mass density of the soil. It
can be seen that the term 2ρV is similar to the “added mass” term of
the Morison equation [14], which has
been traditionally employed to calculate hydrodynamic drag and inertia loads on
cylinders immersed in fluid. In the present case, when multiplied by the
acceleration at the left-hand side of (2.1), the term 2ρV introduces an additional inertia
force that corresponds to the contribution of the soil in which the penetrator
is immersed.The forces in the right-hand side of (2.1) are Ws (the submerged weight of the penetrator); FD, FT, and FS (which are, resp., the drag
force, the tip resistance, and the side resistance); and FE (the external driving force applied by the propeller system).The submerged weightWs is defined in terms of the weight in air W, volume V,
and the unit weight of soil γ by the following expression: (2.3)Ws=W−V⋅γ.The drag forceFD is similar to the longitudinal drag component given by Morison’s equation
[14], which is expressed as: (2.4)FD=12⋅v⋅|v|⋅Af⋅CD⋅ρ, where Af is the frontal or cross-sectional area of the penetrator
and CD is the empirical drag
coefficient, that can have the value
of 0.7 as proposed by True according to [13].The classic formulation forstatic bearing
capacity of deep pile foundation states that, for undrained conditions, the tip
resistance QT and side
resistance QS [15] are defined by (2.5)QT=Su⋅Nc⋅Af,QS=Su⋅α⋅As, where Su is the undrained soil shear strength; Nc   is the
bearing capacity factor (assumed equals to 9 for homogeneous clay); α is the dimensionless
side adhesion factor; and Af and As   are, respectively,
the frontal and lateral areas of the
pile.The dynamic tip and side resistanceFT and FS are now considered by the inclusion, in this classic static
formulation, of a side adhesion reduction
factorδ,
a soil strain rate factorSe˙, and the soil sensitivity value
Sti. The latter represents the
loss of shear strength that clays suffer when remolded, and is defined as the
ratio of undisturbed and remolded strengths [16].
Thus, the tip resistance FT and
side resistance FS are
defined by the following expressions:
(2.6)FT=Su⋅Nc⋅Af⋅Se˙,FS=Su⋅As⋅δSti⋅Se˙.Values for the side adhesion reduction factorδ were
determined in [17] based on results of model tests. An
expression for the strain rate factor  Se˙ was
also defined in [17], as a function of the velocity v and the diameter (or thickness) of the
penetrator d, the undrained soil
shear strength Su, and other
empirical parameters. This expression can be written as (2.7)Se˙=(Se1+(1/(Ce⋅v/Su⋅d)+C0)).Values for the empirical parametersSe  (maximum soil strain rate at high velocity values), Ce (strain
rate velocity factor), and C0   (strain rate
constant) were also determined in [17] based on results of model tests.In the penetration model considered for offshore
applications in the Campos Basin [5], the undrained shear strength Su of the soil is assumed to vary linearly with depth z,
according to the following expression: (2.8)Su(z)=Su0+Suk⋅z, where Su0 is the undrained shear strength at the mudline;
and Suk is the rate of
increase with depth.Original Solution Procedure
To solve (2.1), True [7] developed an incremental finite-difference algorithm
and considered that the penetrator is a point object at the ith depth increment, thus some
simplifications could be made:
(2.9)M'⋅vi⋅vi+1−vi−12Δz=Ws−FD−FT−FS.
Substituting the expressions for M′,FD,FT,FS, and Ws in (2.9), ignoring the external
driving force FE, and assuming C0=0.06
(according to [7]), the following expression is
obtained, which can be applied repeatedly to obtain the velocities of the
penetrator at each depth increment Δz: (2.10)vi+1=vi−1+2Δzvi(M+2ρiV)⋅((W−Vγi)−(12vi2AfCDρi)−Sui(AfNc+AsδSti))⋅(Se1+(1/(Cevi/Suid)+0.06)).
The final penetration depth can then be seen as the
product of the depth incrementΔz by the number of the increment for which
the penetrator velocity drops to zero. It should be recalled that, as in any
numerical solution procedure, the accuracy of the results also depends on the careful selection of the depth increment. This will depend on the
particular example that is analyzed, and may also involve the use of different
increment values to assess the convergence to an accurate solution.


### 2.2. Formulation for Free-Falling Pile

A free-falling cylindrical penetrometer, dropped
from a given height above the mudline, is studied in [12] for the prediction of penetration depth and undrained
shear strength. Equation (2.1),
with some modifications, is also applied to describe the movement of this
penetrometer. Firstly, since it is free falling and there is no external driving
force, the term FE is
omitted. Also, the added mass in (2.2)
is considered negligible for slender penetrometers moving along their long
axis, thus, the effective mass is equal to the structural mass (M′=M).
Therefore, also replacing the velocity v by dz/dt and considering that the acceleration a is equal to (dv/dz)(dz/dt), (2.1) becomes (2.11)Ma=Ws−FD−FT−FS.Obviously,
torpedo piles behave similarly to the free-falling penetrometers, therefore,
their motions can also be described by the True formulation [7], with the same considerations as employed in
[12], resulting in (2.11).
Moreover, the traditional Morison hydrodynamic formulation can also be applied
to describe the forces acting in the pile while it is still in the water,
before reaching the seabed, and the same submerged weight can be assumed for
both media.It should be
emphasized that such semiempirical formulations incorporate some assumptions,
which in turn leads to uncertainties in the model. However, as mentioned in the
introduction, this work is focused on the influence of the uncertainties in
selected soil parameters. Studies regarding uncertainties associated
with the penetration model itself will be dealt with in future works.Solution Procedure in the Time Domain
It can be observed that the procedure originally
proposed by True [7], as described in (2.9) and (2.10), involved the spatial integration of (2.1) to obtain velocities as functions of depth z. This indeed would be the more natural
solution procedure if one is concerned only with the representation of the
isolated pile and the natural random variability of the soil parameters with
depth. However, as will be commented later, there are other sources of
uncertainty to be concerned as well.
Moreover, as will be commented in the final section of
this paper, the final goal of the developments presented here is to incorporate
the penetration model (including the techniques that will be described later,
based on fuzzy arithmetic, for the formal treatment of uncertainties) in a finite
element (FE) spatial discretization scheme, associated to a time domain nonlinear
dynamic solver. The idea is to model not only the isolated pile, but also all
the other components of the system being installed (i.e., the mooring line
itself and the other lines and chains involved in the installation procedure),
in a complete 3D model submitted to other loadings such as marine current.
With this scenario in mind, it is more convenient to
integrate (2.11) in the time domain. At the current stage, where the focus is in
modeling an isolated pile and evaluating the uncertainties in the soil
parameters, the added mass can still be disregarded as assumed in [12]
and in (2.11). In a posterior implementation of
the penetration model in a time domain solver associated to the full FE model, the dynamic equations will also
incorporate the added mass effects of the complete configuration of the pile
with the installation and mooring lines.The solution in the time domain, in terms of the accelerationan+1,
velocity vn+1,
and displacement dn+1, at a given time tn+1,
can be accomplished by applying a time-integration algorithm such as the Chung and
Lee explicit method [18] that can be stated as: (2.12)M⋅an+1=fn;fn=Ws−FD−FT−FS,dn+1=dn+Δtvn+Δt2((1/2−β)an+βan+1),vn+1=vn+Δt((1−γ)an+γan+1), where β and γ are
algorithmic parameters defined as 1≤β≤28/27,γ=3/2; and Δt is the time step, which should not exceed a critical
time step (Δtc) in order to maintain the
stability of the numerical solution [18]. The full time domain solution procedure is shown in Algorithm 1. It should also be recalled that the displacements, velocities, and
accelerations are positive in the downward direction.Algorithm 1:Solution procedure in the time domain.
(a) Initial calculations1. Calculate the torpedo mass M.2. Initialize d0,v0,f0, and a0, where a0=f0/M.3. Select the appropriate algorithmic parameter β and
time step Δt,Δt≤Δtc and calculate the integration
constants,β1=Δt2(12−β),β2=Δt2β,γ1=−12Δt,γ2=32Δt.(b) For each time step(n=0,1,…,N−1).1. Calculate the forces,fn=Ws−FD−FT−FS.2. Calculate the acceleration at time tn+1=tn+Δt,an+1=fn/M.3. Calculate the displacement at time  tn+1=tn+Δt,dn+1=dn+Δtvn+β1an+β2an+1.4. Calculate the velocity at time tn+1=tn+Δt,vn+1=vn+γ1an+γ2an+1.5. n←n+1, go to (1) of step (b).Application
The application of the penetration model described
above is now illustrated for a problem also studied in [19], corresponding to a pile dropped from a
height of 200 m above the seabed. The pile and soil data are presented in Tables 1 and 2. It should be emphasized that the
soil parameters and the sensitivity value of 3 are related to a specific
deposit and may not necessarily be representative of general applications.Table 1
Problem definition: pile data.
W: weight in air396 kNWb: submerged weight340 kNd: diameter0.762 mTable 2
Problem definition: soil data.
Su0: undrained shear strength at the soil  surface5.0 kPaSuk: rate of increase with depth2.0 kPa/mSti: sensitivity3.0Nc: bearing capacity factor9.0δ: side adhesion reduction factor0.9Se: empirical maximum strain rate factor5.0Ce: empirical strain rate velocity factor0.02 kN⋅s/m2This application is not intended to represent an actual
installation procedure for a torpedo pile (such as the depicted in Figure2), since, as mentioned before, the current implementation
of the penetration model represents only the pile. Therefore, in order to take
into account the increase on drag effects due to the mooring line and chain
loop that are not explicitly represented, the model employs a value for the
drag coefficient CD equal to 2.7, larger than the
value of 0.7 as proposed by True according to [13]. In future
works, which will consider the implementation of a coupled finite element-based,
time domain simulation program, there will be no need to perform this “fudging”
of the drag coefficient CD,
since the coupled 3D model will explicitly include the complete installation
configuration (e.g., the mooring line and chain loop for the application in
MODUs described earlier).The time domain solution considered a total time of 15
seconds (enough, as will be seen, for the pile to fully penetrate the soil).
The analysis is performed with a time increment of 0.002 seconds. The value
considered for the  algorithmic parameterβ of the time-integration algorithm is β=28/27.The results are presented in Figure4, in terms of a graph relating the
vertical position of the pile to its velocity, and in Figure 5 in terms of time histories of depth
and velocity. The origin of the graph of Figure 4 corresponds
to the pile in its initial position, before being dropped (therefore, with
velocity and displacement equal to zero). It is seen that, as the pile drops in
the water, its velocity increases until it nearly reaches the so-called
“terminal velocity” due to the water drag (of course, this requires the pile to
be released from an appropriate height). As the pile reaches the seabed and
begins to penetrate in the soil, the velocity is reduced; when it returns to
zero the penetration is completed and the final depth of the pile tip is
reached.Figure 4
Deterministic pile penetration analysis: depthx velocity.Figure 5
Deterministic pile penetration analysis: time histories.


## 2.1. Original Formulation: Penetration of Projectiles

Studies on the
behavior of penetration of projectiles were initially intended for military
applications [8] and were followed by studies on
the prediction of final embedment depth of projectiles into soils [9, 10], and estimation of undrained shear strength [11, 12].The development of a dynamic penetration model by the US  navy
was required to represent the penetration of propellant-embedded plate anchors
into seafloor soils [13]. This kind of anchor is directly
positioned on the mud line and an explosion, caused by the propeller system,
pushes the anchor fluke down. In order to fulfill this objective, True [7]
took into account recommendations given by authors of empirical models (such as
Young [9]) and modified traditional bearing
capacity formulations (for deep foundations in cohesive soils) to consider
variations in penetration resistance with velocity and penetrator shape.The analytical model developed in [7] to simulate the dynamic penetration of plate anchors is
based on Newton's
second law. Considering that the penetrator velocity v can be expressed as v=dz/dt (where z stands for the soil depth), and
therefore, its acceleration dv/dt can be expressed as (dv/dz)(dz/dt),
the governing equation can be written as follows: (2.1)M′dvdzv=Ws−FD−FT−FS+FE, where M′ is the effective mass of the
penetrator, given by (2.2)M′=M+2ρ⋅V.In this latter
equation,M and V are, respectively, the structural mass and the volume of the
penetrator, and ρ is the mass density of the soil. It
can be seen that the term 2ρV is similar to the “added mass” term of
the Morison equation [14], which has
been traditionally employed to calculate hydrodynamic drag and inertia loads on
cylinders immersed in fluid. In the present case, when multiplied by the
acceleration at the left-hand side of (2.1), the term 2ρV introduces an additional inertia
force that corresponds to the contribution of the soil in which the penetrator
is immersed.The forces in the right-hand side of (2.1) are Ws (the submerged weight of the penetrator); FD, FT, and FS (which are, resp., the drag
force, the tip resistance, and the side resistance); and FE (the external driving force applied by the propeller system).The submerged weightWs is defined in terms of the weight in air W, volume V,
and the unit weight of soil γ by the following expression: (2.3)Ws=W−V⋅γ.The drag forceFD is similar to the longitudinal drag component given by Morison’s equation
[14], which is expressed as: (2.4)FD=12⋅v⋅|v|⋅Af⋅CD⋅ρ, where Af is the frontal or cross-sectional area of the penetrator
and CD is the empirical drag
coefficient, that can have the value
of 0.7 as proposed by True according to [13].The classic formulation forstatic bearing
capacity of deep pile foundation states that, for undrained conditions, the tip
resistance QT and side
resistance QS [15] are defined by (2.5)QT=Su⋅Nc⋅Af,QS=Su⋅α⋅As, where Su is the undrained soil shear strength; Nc   is the
bearing capacity factor (assumed equals to 9 for homogeneous clay); α is the dimensionless
side adhesion factor; and Af and As   are, respectively,
the frontal and lateral areas of the
pile.The dynamic tip and side resistanceFT and FS are now considered by the inclusion, in this classic static
formulation, of a side adhesion reduction
factorδ,
a soil strain rate factorSe˙, and the soil sensitivity value
Sti. The latter represents the
loss of shear strength that clays suffer when remolded, and is defined as the
ratio of undisturbed and remolded strengths [16].
Thus, the tip resistance FT and
side resistance FS are
defined by the following expressions:
(2.6)FT=Su⋅Nc⋅Af⋅Se˙,FS=Su⋅As⋅δSti⋅Se˙.Values for the side adhesion reduction factorδ were
determined in [17] based on results of model tests. An
expression for the strain rate factor  Se˙ was
also defined in [17], as a function of the velocity v and the diameter (or thickness) of the
penetrator d, the undrained soil
shear strength Su, and other
empirical parameters. This expression can be written as (2.7)Se˙=(Se1+(1/(Ce⋅v/Su⋅d)+C0)).Values for the empirical parametersSe  (maximum soil strain rate at high velocity values), Ce (strain
rate velocity factor), and C0   (strain rate
constant) were also determined in [17] based on results of model tests.In the penetration model considered for offshore
applications in the Campos Basin [5], the undrained shear strength Su of the soil is assumed to vary linearly with depth z,
according to the following expression: (2.8)Su(z)=Su0+Suk⋅z, where Su0 is the undrained shear strength at the mudline;
and Suk is the rate of
increase with depth.Original Solution Procedure
To solve (2.1), True [7] developed an incremental finite-difference algorithm
and considered that the penetrator is a point object at the ith depth increment, thus some
simplifications could be made:
(2.9)M'⋅vi⋅vi+1−vi−12Δz=Ws−FD−FT−FS.
Substituting the expressions for M′,FD,FT,FS, and Ws in (2.9), ignoring the external
driving force FE, and assuming C0=0.06
(according to [7]), the following expression is
obtained, which can be applied repeatedly to obtain the velocities of the
penetrator at each depth increment Δz: (2.10)vi+1=vi−1+2Δzvi(M+2ρiV)⋅((W−Vγi)−(12vi2AfCDρi)−Sui(AfNc+AsδSti))⋅(Se1+(1/(Cevi/Suid)+0.06)).
The final penetration depth can then be seen as the
product of the depth incrementΔz by the number of the increment for which
the penetrator velocity drops to zero. It should be recalled that, as in any
numerical solution procedure, the accuracy of the results also depends on the careful selection of the depth increment. This will depend on the
particular example that is analyzed, and may also involve the use of different
increment values to assess the convergence to an accurate solution.


## 2.2. Formulation for Free-Falling Pile

A free-falling cylindrical penetrometer, dropped
from a given height above the mudline, is studied in [12] for the prediction of penetration depth and undrained
shear strength. Equation (2.1),
with some modifications, is also applied to describe the movement of this
penetrometer. Firstly, since it is free falling and there is no external driving
force, the term FE is
omitted. Also, the added mass in (2.2)
is considered negligible for slender penetrometers moving along their long
axis, thus, the effective mass is equal to the structural mass (M′=M).
Therefore, also replacing the velocity v by dz/dt and considering that the acceleration a is equal to (dv/dz)(dz/dt), (2.1) becomes (2.11)Ma=Ws−FD−FT−FS.Obviously,
torpedo piles behave similarly to the free-falling penetrometers, therefore,
their motions can also be described by the True formulation [7], with the same considerations as employed in
[12], resulting in (2.11).
Moreover, the traditional Morison hydrodynamic formulation can also be applied
to describe the forces acting in the pile while it is still in the water,
before reaching the seabed, and the same submerged weight can be assumed for
both media.It should be
emphasized that such semiempirical formulations incorporate some assumptions,
which in turn leads to uncertainties in the model. However, as mentioned in the
introduction, this work is focused on the influence of the uncertainties in
selected soil parameters. Studies regarding uncertainties associated
with the penetration model itself will be dealt with in future works.Solution Procedure in the Time Domain
It can be observed that the procedure originally
proposed by True [7], as described in (2.9) and (2.10), involved the spatial integration of (2.1) to obtain velocities as functions of depth z. This indeed would be the more natural
solution procedure if one is concerned only with the representation of the
isolated pile and the natural random variability of the soil parameters with
depth. However, as will be commented later, there are other sources of
uncertainty to be concerned as well.
Moreover, as will be commented in the final section of
this paper, the final goal of the developments presented here is to incorporate
the penetration model (including the techniques that will be described later,
based on fuzzy arithmetic, for the formal treatment of uncertainties) in a finite
element (FE) spatial discretization scheme, associated to a time domain nonlinear
dynamic solver. The idea is to model not only the isolated pile, but also all
the other components of the system being installed (i.e., the mooring line
itself and the other lines and chains involved in the installation procedure),
in a complete 3D model submitted to other loadings such as marine current.
With this scenario in mind, it is more convenient to
integrate (2.11) in the time domain. At the current stage, where the focus is in
modeling an isolated pile and evaluating the uncertainties in the soil
parameters, the added mass can still be disregarded as assumed in [12]
and in (2.11). In a posterior implementation of
the penetration model in a time domain solver associated to the full FE model, the dynamic equations will also
incorporate the added mass effects of the complete configuration of the pile
with the installation and mooring lines.The solution in the time domain, in terms of the accelerationan+1,
velocity vn+1,
and displacement dn+1, at a given time tn+1,
can be accomplished by applying a time-integration algorithm such as the Chung and
Lee explicit method [18] that can be stated as: (2.12)M⋅an+1=fn;fn=Ws−FD−FT−FS,dn+1=dn+Δtvn+Δt2((1/2−β)an+βan+1),vn+1=vn+Δt((1−γ)an+γan+1), where β and γ are
algorithmic parameters defined as 1≤β≤28/27,γ=3/2; and Δt is the time step, which should not exceed a critical
time step (Δtc) in order to maintain the
stability of the numerical solution [18]. The full time domain solution procedure is shown in Algorithm 1. It should also be recalled that the displacements, velocities, and
accelerations are positive in the downward direction.Algorithm 1:Solution procedure in the time domain.
(a) Initial calculations1. Calculate the torpedo mass M.2. Initialize d0,v0,f0, and a0, where a0=f0/M.3. Select the appropriate algorithmic parameter β and
time step Δt,Δt≤Δtc and calculate the integration
constants,β1=Δt2(12−β),β2=Δt2β,γ1=−12Δt,γ2=32Δt.(b) For each time step(n=0,1,…,N−1).1. Calculate the forces,fn=Ws−FD−FT−FS.2. Calculate the acceleration at time tn+1=tn+Δt,an+1=fn/M.3. Calculate the displacement at time  tn+1=tn+Δt,dn+1=dn+Δtvn+β1an+β2an+1.4. Calculate the velocity at time tn+1=tn+Δt,vn+1=vn+γ1an+γ2an+1.5. n←n+1, go to (1) of step (b).Application
The application of the penetration model described
above is now illustrated for a problem also studied in [19], corresponding to a pile dropped from a
height of 200 m above the seabed. The pile and soil data are presented in Tables 1 and 2. It should be emphasized that the
soil parameters and the sensitivity value of 3 are related to a specific
deposit and may not necessarily be representative of general applications.Table 1
Problem definition: pile data.
W: weight in air396 kNWb: submerged weight340 kNd: diameter0.762 mTable 2
Problem definition: soil data.
Su0: undrained shear strength at the soil  surface5.0 kPaSuk: rate of increase with depth2.0 kPa/mSti: sensitivity3.0Nc: bearing capacity factor9.0δ: side adhesion reduction factor0.9Se: empirical maximum strain rate factor5.0Ce: empirical strain rate velocity factor0.02 kN⋅s/m2This application is not intended to represent an actual
installation procedure for a torpedo pile (such as the depicted in Figure2), since, as mentioned before, the current implementation
of the penetration model represents only the pile. Therefore, in order to take
into account the increase on drag effects due to the mooring line and chain
loop that are not explicitly represented, the model employs a value for the
drag coefficient CD equal to 2.7, larger than the
value of 0.7 as proposed by True according to [13]. In future
works, which will consider the implementation of a coupled finite element-based,
time domain simulation program, there will be no need to perform this “fudging”
of the drag coefficient CD,
since the coupled 3D model will explicitly include the complete installation
configuration (e.g., the mooring line and chain loop for the application in
MODUs described earlier).The time domain solution considered a total time of 15
seconds (enough, as will be seen, for the pile to fully penetrate the soil).
The analysis is performed with a time increment of 0.002 seconds. The value
considered for the  algorithmic parameterβ of the time-integration algorithm is β=28/27.The results are presented in Figure4, in terms of a graph relating the
vertical position of the pile to its velocity, and in Figure 5 in terms of time histories of depth
and velocity. The origin of the graph of Figure 4 corresponds
to the pile in its initial position, before being dropped (therefore, with
velocity and displacement equal to zero). It is seen that, as the pile drops in
the water, its velocity increases until it nearly reaches the so-called
“terminal velocity” due to the water drag (of course, this requires the pile to
be released from an appropriate height). As the pile reaches the seabed and
begins to penetrate in the soil, the velocity is reduced; when it returns to
zero the penetration is completed and the final depth of the pile tip is
reached.Figure 4
Deterministic pile penetration analysis: depthx velocity.Figure 5
Deterministic pile penetration analysis: time histories.


## 3. Uncertainties of the Soil Parameters

Selected Parameters
As stated before in (2.8), for cohesive soils in offshore applications in Campos Basin [5], the undrained soil shear strength Su is assumed to vary linearly with
depth z, in terms of Su0  (undrained shear
strength at the mudline) and of Suk (the rate of increase with depth). For the normally consolidated clay
encountered offshore in the Campos Basin [5], typical values that may be considered for Su0 and Suk are, respectively, Su0=5 kPa
and Suk=2 kPa/m.
Therefore, (2.8) could be written as (3.1)Su(z)=5+2⋅z(kPa)
Actual values for these
parameters that affect the undrained soil shear strength Su for offshore sites may be obtained
fromin situ tests (such as CPT—cone
penetration tests [20], or vane tests based on a torsion procedure),
or from laboratory tests with undisturbed samples, such as triaxial and
minivane tests. It should also be recalled that the soil sensitivity Sti represents the loss of shear strength that clays suffer when remolded, and is
defined as the ratio of undisturbed and remolded strengths [16]. Remolded strength values can be
obtained from vane, triaxial, or minivane tests of disturbed soils.Thus, it can be seen that values for Su (undrained shear strength) and Sti (sensitivity) are obtained from testing. Traditionally,
a deterministic procedure is employed to obtain design values for these
parameters, by calculating the average of values obtained from several tests. However,
it is known that the results of both in-situ
or laboratory tests may be influenced by several factors. The latter tests can
be affected by factors such as mechanical disturbance in the soil samples, in
the process of extraction and remolding; by changes in the samples during
storage, and so forth. In-situ
tests can also be affected by mechanical interferences, inadequate execution,
and so on.Therefore, it can be intuitively understood that there
is a high degree of local soil variability, and imprecisions in the
determination of the design values of these soil parameters. Large variations
in the response of the torpedo pile, mainly in terms of the final penetration
depth reached by the pile, may be expected due to these uncertainties. The main
objective of this paper, then, is to present a methodology to take into account
uncertainties and imprecision in the values of input parameters that define the
physical and numerical models involved in the design and analysis of torpedo
piles.This work focuses on Su (specifically, the rate of increase with depth Suk) and Sti. Of course, other parameters (not
necessarily related only to the soil) could be considered; however, those can be
dealt with in future works.Sources of Uncertainty
Before proceeding further, it is
important to recall some basic concepts regarding sources of uncertainty. In soil
profile modeling, they may be grouped in two types [21–23]: (1) noncognitive, random natural variability, usually referred
as aleatory uncertainty; and (2) cognitive or epistemic uncertainties, that involve
abstraction or subjectivity.
The first group
comprises the inherent uncertainty type, due to natural heterogeneity or
in-situ variability of the soil, such as varying depths of strata during soil
formation, variation in mineral composition, and stress history [24]. This corresponds for instance
to the natural variability of the soil strength from point to point vertically
at the position where the pile is to be installed.
The second
group includes epistemic uncertainties due to lack of knowledge; in this case,
information about subsurface conditions is few, because soil profile
characteristics must be inferred from field or laboratory investigation of a
limited number of samples. It includes also uncertainties generated from sample
disturbance, test imperfections, human factors, and also, when engineering
properties are obtained through correlation with index properties, as in the case
of CPT tests where empirical models are used to calculate the undrained shear
resistance by applying correlation factors to the cone tip resistance [24].
According to
this classification, two major approaches, respectively, probabilistic or
“possibilistic” can be employed to deal with uncertainties [25, 26]. Therefore, the
remainder of this paper will deal with methodologies based on these approaches, to
assess the sensitivity of the response to the variation of the selected
parameters and mainly to incorporate, into the analysis method, techniques for
the formal treatment of uncertainties. Section 4 will describe a probabilistic
approach based on the Monte Carlo method and an approach based on fuzzy arithmetic (FA).
Before
proceeding, some additional comments should be presented regarding these
sources of uncertainty. Inherent or natural variability are random by nature and
cannot be reduced by increasing the number of tests [27]. The cognitive,
epistemic uncertainties are reducible; however, for offshore sites, they will
usually be present since the cost of performing in-situ tests at offshore sites
is very expensive. Such tests may not be performed for every installation site
and sometimes the values of the parameters are even estimated or extrapolated
from previous tests made at other locations. Moreover, disturbances in these
few samples are very common.
As strange as
it may seem to experienced geotechnical engineers, not involved in deepwater offshore
activities, this is precisely what has happened in soil investigations in the Campos Basin.
Those are the reasons whyepistemic uncertainties are always added to the
natural variability: the use of limited data, of data arising from disturbed
soil samples, and data from locations other than the one at which the torpedo
pile is to be installed. In summary, the fact that there may be no knowledge of
the exact site local soil variability is the very reason why (as presented in
the next section) probabilistic approaches may fail, and is the motivation of
the use of the approach based on FA.


## 4. Approaches for Treatment of Uncertainties

### 4.1. Probabilistic Approach: The Monte Carlo Method

As mentioned before, noncognitive sources of
uncertainty involve parameters that can be treated as random variables, and to
which a probabilistic distribution can be associated, based on statistical data. In
such cases, the probabilistic approach is traditionally recommended.Probabilistic approaches for treatment of uncertainties
can be divided in two main categories. The first one comprises statistical
methods that involve simulation, such as the classical Monte
Carlo simulation method and its variants. The second category
comprises nonstatistical methods such as those based on perturbation techniques. For instance,
the stochastic finite element method [26] falls in this latter category; it is based on expanding the random
parameters around their mean values via Taylor series, in order to formulate
linear relationships between some random characteristics of the response and
the random input parameters.In the implementation of the classical
Monte Carlo simulation,N samples of
the uncertain parameters are randomly generated using a given joint probability
density function. The deterministic analysis procedure is employed for each
sample of the simulation process [28], obtaining
then N responses that are statistically
treated to get the first two statistical moments of the response (mean and
standard deviation values).The MC method is completely
general, for linear or nonlinear analyses. However, in general, the accuracy of
the statistical response is only adequate when the number of sample dataN is sufficiently large; therefore, it
is usually considered too expensive in terms of computing time. This fact has
motivated studies on variants of the classical method, involving for instance
variance reduction techniques and the Neumann expansion [29].Due to its
robustness and ability to effectively treat the noncognitive, random
uncertainties, the classical MC simulation method has been used to calibrate
and validate all other probabilistic techniques. The studies presented in this
paper will also employ this method as a benchmark to compare the performance of
the approach based on
fuzzy arithmetic, which will be described in Section4.2, in the representation
of the random uncertainties.


### 4.2. Fuzzy Arithmetic (FA) Approach

It is important to recall that the cognitive sources of
uncertainty are related not to chance, but rather to imprecise or vague
information, involving subjectivity and/or dependent on expert judgment.
Moreover, the axioms of probability and statistics are not adequate to deal with
such types of uncertainties, which can be more effectively treated by
“possibilistic” approaches employing for instance the theory offuzzy sets.The theory of fuzzy sets was introduced by Zadeh in [30] to define classes of objects with continuous
membership graduations or associations in the interval [0, 1]. A fuzzy set has
vague limits, allowing graded changes from one class to another, instead of
exact limits characteristic of ordinary or crisp sets. In classical
Boolean algebra, the notion of false and true values is limited to 1 or zero.
In fuzzy logic, values that are “more or less” true or false can be treated,
defined by real numbers that vary continuously from 0 to 1.The treatment of uncertainties that derive from
imprecise information is then possible, avoiding the use of random information. Therefore, complex systems, that would be hard to
model with the theory of conventional sets, can be easily modeled by fuzzy
sets. The fuzzy set theory allows the representation of imprecise and uncertain
measures as fuzzy numbers, defined as:(4.1)A={(x,A(x)),x∈R,A(x)∈[0,1]}, where  x∈R is the numeric support of the fuzzy number A,
and A(x)∈[0,1]  is the membership function (MF).Fuzzy numbers are completely characterized by their MFs,
that are built based on knowledge of an expert, who can assign “low,” “probable,” or “high” values for
the desired parameters. Based on this subjective information, MFs can be
constructed presenting either linear or nonlinear shapes. The more usually
employed shapes for engineering problems are triangular, trapezoidal, and
sinusoidal; the choice will depend on the type of application, and will also
follow the assessment of the expert. In this work, triangular fuzzy MFs are
used, defined by estimating three values [31]:(i)
a more
reliable value,m, to which is
attributed a membership degree equals to1;(ii)
an
inferior value,a, that most
certainly will be exceeded by another value, and to which is attributed a
membership degree equals to 0;(iii)
a superior
value,b, that most certainly will
not be exceeded by another value, and to which is also attributed a membership
degree equals to 0.The membership function can then be defined as zero
outside the interval[a,b] of
possible values; taken as linear into this range, increasing from a to m,
and decreasing from m to b. This function is triangular, not
necessarily symmetric, and can be defined as parameterized piecewise linear
functions as: (4.2)A(x;a,m,b)={x−am−a,ifx∈[a,m]b−xb−m,ifx∈[m,b]0,otherwise, where a and b are, respectively, the lower and upper bounds, and m is the dominant value, as illustrated
in Figure 6.Figure 6
Triangular fuzzy number.Fuzzy numbers can also be defined byL (left) and R (right) MFs, resulting into the so-called L-R fuzzy numbers. In this context, a two-parameter modification of
an L-type MF applies to all x≤m,
whereas the R-MF defines A for x>m,
thus yielding (4.3)A(x)={L(m−xα),ifx≤m,α>0R(x−mβ),ifx>m,β>0.Therefore, the fuzzy number
can also be identified by the notationA=(m,α,β),
where α and β are the spreads of the number, which
represents its uncertainty [32].Fuzzy arithmetic (FA) operations, involving fuzzy numbers, can be used to propagate fuzziness from
inputs to outputs. General operations can be deduced from the extension
principle, which is used to transform fuzzy sets via functions [32]
, and plays a fundamental role in translating
set-based concepts into their fuzzy set counterparts. However, simplified formulae can be
obtained considering the L-R formulation of fuzzy numbers A=(m,α,β)LR and B=(n,γ,δ)LR.
The standard arithmetic operations are computed as follows.The addition of triangular fuzzy numbers results in
another triangular fuzzy number. Both addition and subtraction conserves the
linearity of the numbers. These operations are expressed as, respectively,(4.4)A+B=(m+n,α+γ,β+δ)LR,A−B=(m−n,α+δ,β+γ)LR.The multiplication of two
fuzzy numbers produces a quadratic number. However, a linear approximation can
be assumed when the spreadsα and β are small in comparison to the modal or
dominant values m. Therefore, this
operation can be approximated to (4.5)A⋅B=(mn,mγ+nα,mδ+nβ)LR(ifA>0,B>0),A⋅B=(mn,mα−nδ,mβ−nγ)RL(ifA>0,B<0),A⋅B=(mn,nα−mδ,nβ−mγ)RL(ifA<0,B>0),A⋅B=(mn,−nβ−mδ,−nα−mγ)RL(ifA<0,B<0).The
multiplication of a fuzzy number by a scalara is defined as: (4.6)aA=(am,aα,aβ)LR,(ifaispositive),aA=(am,−aβ,−aα)RL,(ifaisnegative).The division between two fuzzy
numbers is computed as(4.7)AB=(mn,δm+αnn2,ym+βnn2)LR.To apply the FA in a given
engineering problem, the uncertainty on each variable is modeled as a triangular
fuzzy number; moreover, all the operations related to them have their expressions
replaced by the corresponding FA expressions, as shown above on operations (4.4)
to (4.7)
.


## 4.1. Probabilistic Approach: The Monte Carlo Method

As mentioned before, noncognitive sources of
uncertainty involve parameters that can be treated as random variables, and to
which a probabilistic distribution can be associated, based on statistical data. In
such cases, the probabilistic approach is traditionally recommended.Probabilistic approaches for treatment of uncertainties
can be divided in two main categories. The first one comprises statistical
methods that involve simulation, such as the classical Monte
Carlo simulation method and its variants. The second category
comprises nonstatistical methods such as those based on perturbation techniques. For instance,
the stochastic finite element method [26] falls in this latter category; it is based on expanding the random
parameters around their mean values via Taylor series, in order to formulate
linear relationships between some random characteristics of the response and
the random input parameters.In the implementation of the classical
Monte Carlo simulation,N samples of
the uncertain parameters are randomly generated using a given joint probability
density function. The deterministic analysis procedure is employed for each
sample of the simulation process [28], obtaining
then N responses that are statistically
treated to get the first two statistical moments of the response (mean and
standard deviation values).The MC method is completely
general, for linear or nonlinear analyses. However, in general, the accuracy of
the statistical response is only adequate when the number of sample dataN is sufficiently large; therefore, it
is usually considered too expensive in terms of computing time. This fact has
motivated studies on variants of the classical method, involving for instance
variance reduction techniques and the Neumann expansion [29].Due to its
robustness and ability to effectively treat the noncognitive, random
uncertainties, the classical MC simulation method has been used to calibrate
and validate all other probabilistic techniques. The studies presented in this
paper will also employ this method as a benchmark to compare the performance of
the approach based on
fuzzy arithmetic, which will be described in Section4.2, in the representation
of the random uncertainties.


## 4.2. Fuzzy Arithmetic (FA) Approach

It is important to recall that the cognitive sources of
uncertainty are related not to chance, but rather to imprecise or vague
information, involving subjectivity and/or dependent on expert judgment.
Moreover, the axioms of probability and statistics are not adequate to deal with
such types of uncertainties, which can be more effectively treated by
“possibilistic” approaches employing for instance the theory offuzzy sets.The theory of fuzzy sets was introduced by Zadeh in [30] to define classes of objects with continuous
membership graduations or associations in the interval [0, 1]. A fuzzy set has
vague limits, allowing graded changes from one class to another, instead of
exact limits characteristic of ordinary or crisp sets. In classical
Boolean algebra, the notion of false and true values is limited to 1 or zero.
In fuzzy logic, values that are “more or less” true or false can be treated,
defined by real numbers that vary continuously from 0 to 1.The treatment of uncertainties that derive from
imprecise information is then possible, avoiding the use of random information. Therefore, complex systems, that would be hard to
model with the theory of conventional sets, can be easily modeled by fuzzy
sets. The fuzzy set theory allows the representation of imprecise and uncertain
measures as fuzzy numbers, defined as:(4.1)A={(x,A(x)),x∈R,A(x)∈[0,1]}, where  x∈R is the numeric support of the fuzzy number A,
and A(x)∈[0,1]  is the membership function (MF).Fuzzy numbers are completely characterized by their MFs,
that are built based on knowledge of an expert, who can assign “low,” “probable,” or “high” values for
the desired parameters. Based on this subjective information, MFs can be
constructed presenting either linear or nonlinear shapes. The more usually
employed shapes for engineering problems are triangular, trapezoidal, and
sinusoidal; the choice will depend on the type of application, and will also
follow the assessment of the expert. In this work, triangular fuzzy MFs are
used, defined by estimating three values [31]:(i)
a more
reliable value,m, to which is
attributed a membership degree equals to1;(ii)
an
inferior value,a, that most
certainly will be exceeded by another value, and to which is attributed a
membership degree equals to 0;(iii)
a superior
value,b, that most certainly will
not be exceeded by another value, and to which is also attributed a membership
degree equals to 0.The membership function can then be defined as zero
outside the interval[a,b] of
possible values; taken as linear into this range, increasing from a to m,
and decreasing from m to b. This function is triangular, not
necessarily symmetric, and can be defined as parameterized piecewise linear
functions as: (4.2)A(x;a,m,b)={x−am−a,ifx∈[a,m]b−xb−m,ifx∈[m,b]0,otherwise, where a and b are, respectively, the lower and upper bounds, and m is the dominant value, as illustrated
in Figure 6.Figure 6
Triangular fuzzy number.Fuzzy numbers can also be defined byL (left) and R (right) MFs, resulting into the so-called L-R fuzzy numbers. In this context, a two-parameter modification of
an L-type MF applies to all x≤m,
whereas the R-MF defines A for x>m,
thus yielding (4.3)A(x)={L(m−xα),ifx≤m,α>0R(x−mβ),ifx>m,β>0.Therefore, the fuzzy number
can also be identified by the notationA=(m,α,β),
where α and β are the spreads of the number, which
represents its uncertainty [32].Fuzzy arithmetic (FA) operations, involving fuzzy numbers, can be used to propagate fuzziness from
inputs to outputs. General operations can be deduced from the extension
principle, which is used to transform fuzzy sets via functions [32]
, and plays a fundamental role in translating
set-based concepts into their fuzzy set counterparts. However, simplified formulae can be
obtained considering the L-R formulation of fuzzy numbers A=(m,α,β)LR and B=(n,γ,δ)LR.
The standard arithmetic operations are computed as follows.The addition of triangular fuzzy numbers results in
another triangular fuzzy number. Both addition and subtraction conserves the
linearity of the numbers. These operations are expressed as, respectively,(4.4)A+B=(m+n,α+γ,β+δ)LR,A−B=(m−n,α+δ,β+γ)LR.The multiplication of two
fuzzy numbers produces a quadratic number. However, a linear approximation can
be assumed when the spreadsα and β are small in comparison to the modal or
dominant values m. Therefore, this
operation can be approximated to (4.5)A⋅B=(mn,mγ+nα,mδ+nβ)LR(ifA>0,B>0),A⋅B=(mn,mα−nδ,mβ−nγ)RL(ifA>0,B<0),A⋅B=(mn,nα−mδ,nβ−mγ)RL(ifA<0,B>0),A⋅B=(mn,−nβ−mδ,−nα−mγ)RL(ifA<0,B<0).The
multiplication of a fuzzy number by a scalara is defined as: (4.6)aA=(am,aα,aβ)LR,(ifaispositive),aA=(am,−aβ,−aα)RL,(ifaisnegative).The division between two fuzzy
numbers is computed as(4.7)AB=(mn,δm+αnn2,ym+βnn2)LR.To apply the FA in a given
engineering problem, the uncertainty on each variable is modeled as a triangular
fuzzy number; moreover, all the operations related to them have their expressions
replaced by the corresponding FA expressions, as shown above on operations (4.4)
to (4.7)
.


## 5. Implementation and Case Studies

Before proceeding with the study of the approaches
described above, this section will begin with deterministic studies to assess
the sensitivity of the response of the penetration model to the variation of
the selected soil parameters. Later, in order to reach the goal of incorporating
the formal treatment of uncertainties in the analysis of the penetration of
torpedo piles, this section will proceed by presenting the application of the
probabilistic approach based on the Monte Carlo method, followed by the implementation and application of the approach based on FA.Recalling that uncertainties related to the penetration
model involve a combination of both noncognitive (random, natural variability
of soil parameters) and cognitive (epistemic, due to incomplete or imprecise
information), it will be seen that, while the MC method can effectively deal
only with the random uncertainties, the implementation of the fuzzy approach presented
here can represent all sources of uncertainty.


### 5.1. Deterministic Sensitivity Studies

In order to perform an assessment of the sensitivity of
the torpedo pile penetration to the uncertainty of the selected soil
parameters, a parametric study is performed by considering the same problem
described in Tables1 and 2. The penetration model is applied to deterministic
and arbitrary variations on both uncertain parameters: the rate of increase with
depth of the undrained shear strength (Suk)
and the soil sensitivity (Sti).Recalling that according to Table2, the fixed, “deterministic” values are Suk=2.0 kPa/m
and Sti=3. Initially, Suk and Sti are individually increased by 10, 20, and 30%.  Then, their values are reduced, also by 10, 20
and 30%. The results of the analyses for the different values of the parameters
are presented in the graphs of Figure 7,
corresponding to analyses where they are increased and reduced, respectively.Deterministic sensitivity studies.
(a)(b)It should be noted that, since the drop height and the
characteristics of the pile have not been changed, the behavior of the pile
from the drop point until it reaches the seabed is the same as observed in
Figure4. Therefore, the graphs of Figure 7
represent only the behavior of the pile as it
penetrates the soil, beginning from the depth of 200 m (that corresponds to the
seabed) until it completes the penetration.A summary of the results of Figure7 is presented in Table 3 and Figure 8, in terms of penetration values (displacement
minus the drop height) for each variation of the parameters Suk and Sti. It can be verified that, as expected, reducing the undrained
shear strength (and therefore, the soil resistance) leads to the increase on
the final depth values. On the other hand, decreasing the sensitivity values
increases the soil resistance and, consequently, reduces the penetration of the
pile.Table 3
Deterministic sensitivity studies: summary of results.
Varied parameterPenetration (m)−30%−20%−10%0%10%20%30%Suk42.539.737.535.634.032.631.4Sti31.232.834.235.636.837.938.9Figure 8
Deterministic sensitivity studies: summary of results.


### 5.2. Probabilistic Analysis Using the Monte Carlo Method

Statistical Treatment of the Soil Input Parameters
In
the probabilistic analysis using the MC method, both uncertain parameters (the
undrained shear strength increase rate Suk and soil sensitivity Sti) are varied
simultaneously. Their values are randomly simulated, following a statistical
distribution and its associated values of mean and standard deviation, derived
from a given set of soil data from laboratory and/or in situ tests (in this case, the data were acquired from many
tests performed at different sites in a certain cluster of Campos Basin). This is
accomplished by performing a statistical treatment on the available data, representative
of offshore fields in Campos Basin. As a result, for
each uncertain parameter, the mean (which
in this case is the sample average) and standard deviation values were
estimated. These values are presented
in Table 4.Table 4
Statistical values from available soil data.
ParameterMeanStandard deviationSuk (kPa/m)1.90.9Sti (dimensionless)3.21.0In order to determine an appropriate probability
distribution function for the data, a
normality verification is performed for each parameter. Figure9 present the results, respectively, for Suk and Sti.
It can be observed that, despite Sti data fits better than Suk,
the normal pdf is not the ideal approximation for them. Hence, other functions
are fitted and verified, as presented in Figure 10. Observing this figure, it can be
verified that the lognormal pdf provides a better fit for both sets of
available data. Another advantage of this distribution is that it does not
generate negative values for the soil parameters, which does not have physical
meaning and can generate erroneous results.Normal probability verification of Suk and Sti data.
(a)(b)Probability function fitness for Suk and Sti data.
(a)(b)The number of simulations in an MC strategy is dictated
by the convergence of the mean value of the considered parameter to the
deterministic design value. In the present case, 1000 generations were needed
to obtain satisfactory convergence. Figure11 depicts the distribution of the
1000 randomly generated values of Suk and Sti, following the
lognormal distribution.Lognormal distribution of Suk and Sti.
(a)(b)Results
The probabilistic study then comprises a total number
of 1000 analyses with the penetration model, each taking a randomly generated
pair of values for the soil parameters Suk and Sti, following the lognormal
probability distribution with expected values and standard deviations given in
Table 4.
The results of the 1000 analyses are then gathered to
proceed to a statistical treatment, which will represent the penetration value
in terms of mean and standard deviation.
These results are presented in Table5.Table 5
Probabilistic MC approach: statistical values of the results.
MedianMeanStandard deviationPenetration (m)38.539.810.1These values will be compared
with the results obtained with the approach using FA, which will be presented in
Section5.3. We recall that the mean value of 39.8 m for the penetration cannot
be directly compared to the “deterministic” value of 35.6 m obtained in the
previous section, since the fixed, “deterministic” values for the soil
parameters were Suk=2.0 kPa/m and Sti=3.0, and the mean
probabilistic values gathered from the set of soil data considered are Suk=1.9 kPa/m and Sti=3.2. Anyway, the
results are consistent since, as could be observed in the results of the
deterministic sensitivity studies summarized in Figure 8, lower values of Suk and
higher values of Sti lead to
higher penetration values.


### 5.3. Fuzzy Arithmetic: Implementation and Application

Implementation
In the computational implementation of the approach using
FA, the uncertain variables Suk and Sti are represented as triangular fuzzy numbers. Therefore, the
computational code is altered, and all operations performed with those
parameters in the solution procedure (as described in Algorithm 1 and (2.2)–(2.6)) have the traditional
arithmetic operators replaced by the fuzzy operators presented in (4.4) to (4.7).
As mentioned in Section4.2, the fuzzy operations of multiplication and division generate quadratic
numbers; however, when their spreads are small, they can be approximated by
linear ones. Therefore, although (4.5) to (4.7) are approximations for small
spreads, they are feasible for this specific work, since the values of final
penetration (dominant value, and the lower and upper bounds) are more important
than the shape of the membership function.
Once these fuzzy
operators are implemented in the computational code, it remains to determine the
values that define the triangular membership functions, which represent Suk and Sti as fuzzy numbers. As illustrated in Figure 6
, these values are the lower and upper bounds a and b, and the dominant value m;
they can be derived by investigating the statistical distribution of the soil
parameters, taking the lognormal distribution generated as described in the
previous item.
The lower and
upper boundsa and b can be assumed as defining an interval
of confidence of 75% corresponding to one standard deviation below and above
the mean. This criterion provides samples that have consistent values for the
uncertain parameters (positive values, Sti greater than 1.0, etc.), and is
illustrated in Figures 12 and 13, for
Suk and Sti distributions, respectively.
Regarding the
“dominant” valuem, the first choice
could be to take the mean value; however, since the most representative value
of a sample with large dispersion is the median, its value was chosen as m for each parameter. The values thus
obtained for a, b, and m that define the membership functions for Suk and Sti are presented in Table 6 and graphically represented in Figure 14.Table 6
Membership functions for the parameters Suk and Sti.
Lower boundaDominant valuemUpper boundbSuk (kPa/m)1.01.72.8Sti (dimensionless)2.23.04.3Figure 12
Definition of bounds for Suk.Figure 13
Definition of bounds for Sti.Figure 14
Triangular fuzzy representation for the soil uncertain parameters.Results
Finally, the evaluation of the uncertain response using
this FA approach consists simply in performing one analysis with the
penetration model. The uncertainties embedded in the fuzzy numbers that
represent the parameters Suk and Sti are incorporated in the
calculation of the terms FT and FS defined in (2.6), at the right-hand side of
step b.1 of Algorithm 1, and therefore are updated and propagated at each
time step of the solution procedure presented in that table. This fact points
to the remarkable computational advantage of this approach, compared to the
probabilistic MC method with conventional arithmetic that required a total
number of 1000 analyses.
The results of the fuzzy analyses, in terms of lower,
dominant, and upper bound for the final penetration of the torpedo pile are
presented in Table7.Table 7
FA approach: results of fuzzy analyses.
Lower boundDominant valueUpper boundPenetration (m)30.138.666.3


### 5.4. Comparison of Results

This section compares the results of analyses of the
torpedo pile with the penetration model, considering both MC and FA approaches.
Before comparing the final pile penetration, Figure15 presents the full behavior of the
pile as it penetrates the soil, in terms of penetration x velocity curves, beginning from
the depth of 200 m (that corresponds to the seabed) until it completes the
penetration.Figure 15
Comparison of results of MC and FA approaches.Three curves are presented for each approach,
corresponding to the “dominant” or “most probable” result, and a lower and
upper “bounds” of the response. For the MC simulation, the “most probable”
curve is represented by taking the median values of penetration and velocities at
each time step of the response; the lower and upper bounds are determined,
respectively, by taking the mean value and subtracting or adding one standard
deviation (similarly to the procedure applied to determine the bounds of the
fuzzy input parameters).For the FA approach, the “dominant” curve is
represented by taking the “crisp” result, that is, the values corresponding to
a degree of membership equal to one. The lower and upper bounds are determined
by the support of the fuzzy set defined by the values corresponding to a membership
degree greater than zero.Table8 summarizes and compares the results
presented in Tables 5 and 7 for the MC and fuzzy approaches.
Observing this table and also Figure 15, it can be observed that the
“dominant” or “most probable” results for the final penetration are practically
the same; the difference between the median value of the MC analyses and the
dominant value of FA analyses is insignificant.Table 8
Comparison of results.
MethodFinal penetration (m)MCMean – 1 std deviationMedian valueMean + 1 std deviation29.7 (–22.9%)38.549.9 (+29.6%)FALower boundDominant valueUpper bound30.1 (–22.0%)38.666.3 (+71.8%)Regarding dispersion of results, it should be recalled
that MC “lower” and “upper” results (defined by subtracting and adding one standard deviation
to the mean value) cannot be directly compared to the spreads of the fuzzy
results, where lower and upper bounds define the interval where a value can
possibly represent the calculated penetration. Anyway, it can be noted that the
uncertainties of the soil parameters are quite significant and can have a
decisive influence in the design of the torpedo pile.A better comparison for the final penetration can be
graphically assessed in terms of the probability distribution of the MC
simulation, and the membership function that characterizes the fuzzy number for
the FA approach. Therefore, Figure16 compares the
results obtained from the MC and FA approaches, in terms of the probability
distribution and membership function. For this example, while the assumed supports
of the fuzzy input parameters Suk and Sti corresponded to a certainty
interval of 75% of their lognormal distribution, the support of the fuzzy
number that represents the final penetration corresponds to a certainty
interval of around 80% of the MC distribution.Figure 16
Final penetration obtained from MC and FA approaches.Finally, the most remarkable comparison in the
performance of both methods can be stated in terms of the total CPU time
required. While the MC probabilistic approach required 1000 analyses with the
penetration model using the solution procedure of Algorithm1, only one analysis was required for the approach
employing FA.


## 5.1. Deterministic Sensitivity Studies

In order to perform an assessment of the sensitivity of
the torpedo pile penetration to the uncertainty of the selected soil
parameters, a parametric study is performed by considering the same problem
described in Tables1 and 2. The penetration model is applied to deterministic
and arbitrary variations on both uncertain parameters: the rate of increase with
depth of the undrained shear strength (Suk)
and the soil sensitivity (Sti).Recalling that according to Table2, the fixed, “deterministic” values are Suk=2.0 kPa/m
and Sti=3. Initially, Suk and Sti are individually increased by 10, 20, and 30%.  Then, their values are reduced, also by 10, 20
and 30%. The results of the analyses for the different values of the parameters
are presented in the graphs of Figure 7,
corresponding to analyses where they are increased and reduced, respectively.Deterministic sensitivity studies.
(a)(b)It should be noted that, since the drop height and the
characteristics of the pile have not been changed, the behavior of the pile
from the drop point until it reaches the seabed is the same as observed in
Figure4. Therefore, the graphs of Figure 7
represent only the behavior of the pile as it
penetrates the soil, beginning from the depth of 200 m (that corresponds to the
seabed) until it completes the penetration.A summary of the results of Figure7 is presented in Table 3 and Figure 8, in terms of penetration values (displacement
minus the drop height) for each variation of the parameters Suk and Sti. It can be verified that, as expected, reducing the undrained
shear strength (and therefore, the soil resistance) leads to the increase on
the final depth values. On the other hand, decreasing the sensitivity values
increases the soil resistance and, consequently, reduces the penetration of the
pile.Table 3
Deterministic sensitivity studies: summary of results.
Varied parameterPenetration (m)−30%−20%−10%0%10%20%30%Suk42.539.737.535.634.032.631.4Sti31.232.834.235.636.837.938.9Figure 8
Deterministic sensitivity studies: summary of results.


## 5.2. Probabilistic Analysis Using the Monte Carlo Method

Statistical Treatment of the Soil Input Parameters
In
the probabilistic analysis using the MC method, both uncertain parameters (the
undrained shear strength increase rate Suk and soil sensitivity Sti) are varied
simultaneously. Their values are randomly simulated, following a statistical
distribution and its associated values of mean and standard deviation, derived
from a given set of soil data from laboratory and/or in situ tests (in this case, the data were acquired from many
tests performed at different sites in a certain cluster of Campos Basin). This is
accomplished by performing a statistical treatment on the available data, representative
of offshore fields in Campos Basin. As a result, for
each uncertain parameter, the mean (which
in this case is the sample average) and standard deviation values were
estimated. These values are presented
in Table 4.Table 4
Statistical values from available soil data.
ParameterMeanStandard deviationSuk (kPa/m)1.90.9Sti (dimensionless)3.21.0In order to determine an appropriate probability
distribution function for the data, a
normality verification is performed for each parameter. Figure9 present the results, respectively, for Suk and Sti.
It can be observed that, despite Sti data fits better than Suk,
the normal pdf is not the ideal approximation for them. Hence, other functions
are fitted and verified, as presented in Figure 10. Observing this figure, it can be
verified that the lognormal pdf provides a better fit for both sets of
available data. Another advantage of this distribution is that it does not
generate negative values for the soil parameters, which does not have physical
meaning and can generate erroneous results.Normal probability verification of Suk and Sti data.
(a)(b)Probability function fitness for Suk and Sti data.
(a)(b)The number of simulations in an MC strategy is dictated
by the convergence of the mean value of the considered parameter to the
deterministic design value. In the present case, 1000 generations were needed
to obtain satisfactory convergence. Figure11 depicts the distribution of the
1000 randomly generated values of Suk and Sti, following the
lognormal distribution.Lognormal distribution of Suk and Sti.
(a)(b)Results
The probabilistic study then comprises a total number
of 1000 analyses with the penetration model, each taking a randomly generated
pair of values for the soil parameters Suk and Sti, following the lognormal
probability distribution with expected values and standard deviations given in
Table 4.
The results of the 1000 analyses are then gathered to
proceed to a statistical treatment, which will represent the penetration value
in terms of mean and standard deviation.
These results are presented in Table5.Table 5
Probabilistic MC approach: statistical values of the results.
MedianMeanStandard deviationPenetration (m)38.539.810.1These values will be compared
with the results obtained with the approach using FA, which will be presented in
Section5.3. We recall that the mean value of 39.8 m for the penetration cannot
be directly compared to the “deterministic” value of 35.6 m obtained in the
previous section, since the fixed, “deterministic” values for the soil
parameters were Suk=2.0 kPa/m and Sti=3.0, and the mean
probabilistic values gathered from the set of soil data considered are Suk=1.9 kPa/m and Sti=3.2. Anyway, the
results are consistent since, as could be observed in the results of the
deterministic sensitivity studies summarized in Figure 8, lower values of Suk and
higher values of Sti lead to
higher penetration values.


## 5.3. Fuzzy Arithmetic: Implementation and Application

Implementation
In the computational implementation of the approach using
FA, the uncertain variables Suk and Sti are represented as triangular fuzzy numbers. Therefore, the
computational code is altered, and all operations performed with those
parameters in the solution procedure (as described in Algorithm 1 and (2.2)–(2.6)) have the traditional
arithmetic operators replaced by the fuzzy operators presented in (4.4) to (4.7).
As mentioned in Section4.2, the fuzzy operations of multiplication and division generate quadratic
numbers; however, when their spreads are small, they can be approximated by
linear ones. Therefore, although (4.5) to (4.7) are approximations for small
spreads, they are feasible for this specific work, since the values of final
penetration (dominant value, and the lower and upper bounds) are more important
than the shape of the membership function.
Once these fuzzy
operators are implemented in the computational code, it remains to determine the
values that define the triangular membership functions, which represent Suk and Sti as fuzzy numbers. As illustrated in Figure 6
, these values are the lower and upper bounds a and b, and the dominant value m;
they can be derived by investigating the statistical distribution of the soil
parameters, taking the lognormal distribution generated as described in the
previous item.
The lower and
upper boundsa and b can be assumed as defining an interval
of confidence of 75% corresponding to one standard deviation below and above
the mean. This criterion provides samples that have consistent values for the
uncertain parameters (positive values, Sti greater than 1.0, etc.), and is
illustrated in Figures 12 and 13, for
Suk and Sti distributions, respectively.
Regarding the
“dominant” valuem, the first choice
could be to take the mean value; however, since the most representative value
of a sample with large dispersion is the median, its value was chosen as m for each parameter. The values thus
obtained for a, b, and m that define the membership functions for Suk and Sti are presented in Table 6 and graphically represented in Figure 14.Table 6
Membership functions for the parameters Suk and Sti.
Lower boundaDominant valuemUpper boundbSuk (kPa/m)1.01.72.8Sti (dimensionless)2.23.04.3Figure 12
Definition of bounds for Suk.Figure 13
Definition of bounds for Sti.Figure 14
Triangular fuzzy representation for the soil uncertain parameters.Results
Finally, the evaluation of the uncertain response using
this FA approach consists simply in performing one analysis with the
penetration model. The uncertainties embedded in the fuzzy numbers that
represent the parameters Suk and Sti are incorporated in the
calculation of the terms FT and FS defined in (2.6), at the right-hand side of
step b.1 of Algorithm 1, and therefore are updated and propagated at each
time step of the solution procedure presented in that table. This fact points
to the remarkable computational advantage of this approach, compared to the
probabilistic MC method with conventional arithmetic that required a total
number of 1000 analyses.
The results of the fuzzy analyses, in terms of lower,
dominant, and upper bound for the final penetration of the torpedo pile are
presented in Table7.Table 7
FA approach: results of fuzzy analyses.
Lower boundDominant valueUpper boundPenetration (m)30.138.666.3


## 5.4. Comparison of Results

This section compares the results of analyses of the
torpedo pile with the penetration model, considering both MC and FA approaches.
Before comparing the final pile penetration, Figure15 presents the full behavior of the
pile as it penetrates the soil, in terms of penetration x velocity curves, beginning from
the depth of 200 m (that corresponds to the seabed) until it completes the
penetration.Figure 15
Comparison of results of MC and FA approaches.Three curves are presented for each approach,
corresponding to the “dominant” or “most probable” result, and a lower and
upper “bounds” of the response. For the MC simulation, the “most probable”
curve is represented by taking the median values of penetration and velocities at
each time step of the response; the lower and upper bounds are determined,
respectively, by taking the mean value and subtracting or adding one standard
deviation (similarly to the procedure applied to determine the bounds of the
fuzzy input parameters).For the FA approach, the “dominant” curve is
represented by taking the “crisp” result, that is, the values corresponding to
a degree of membership equal to one. The lower and upper bounds are determined
by the support of the fuzzy set defined by the values corresponding to a membership
degree greater than zero.Table8 summarizes and compares the results
presented in Tables 5 and 7 for the MC and fuzzy approaches.
Observing this table and also Figure 15, it can be observed that the
“dominant” or “most probable” results for the final penetration are practically
the same; the difference between the median value of the MC analyses and the
dominant value of FA analyses is insignificant.Table 8
Comparison of results.
MethodFinal penetration (m)MCMean – 1 std deviationMedian valueMean + 1 std deviation29.7 (–22.9%)38.549.9 (+29.6%)FALower boundDominant valueUpper bound30.1 (–22.0%)38.666.3 (+71.8%)Regarding dispersion of results, it should be recalled
that MC “lower” and “upper” results (defined by subtracting and adding one standard deviation
to the mean value) cannot be directly compared to the spreads of the fuzzy
results, where lower and upper bounds define the interval where a value can
possibly represent the calculated penetration. Anyway, it can be noted that the
uncertainties of the soil parameters are quite significant and can have a
decisive influence in the design of the torpedo pile.A better comparison for the final penetration can be
graphically assessed in terms of the probability distribution of the MC
simulation, and the membership function that characterizes the fuzzy number for
the FA approach. Therefore, Figure16 compares the
results obtained from the MC and FA approaches, in terms of the probability
distribution and membership function. For this example, while the assumed supports
of the fuzzy input parameters Suk and Sti corresponded to a certainty
interval of 75% of their lognormal distribution, the support of the fuzzy
number that represents the final penetration corresponds to a certainty
interval of around 80% of the MC distribution.Figure 16
Final penetration obtained from MC and FA approaches.Finally, the most remarkable comparison in the
performance of both methods can be stated in terms of the total CPU time
required. While the MC probabilistic approach required 1000 analyses with the
penetration model using the solution procedure of Algorithm1, only one analysis was required for the approach
employing FA.


## 6. Final Remarks and Conclusions

The
torpedo pile has been acknowledged as a very promising alternative to anchor
mooring systems. It has recently been considered for use not only in mooring
lines of floating production systems, but also for mobile offshore drilling
units (MODUs) operating in deep and ultradeep waters. Therefore, oil
exploitation companies are devoting intense research and design activities in
order to deliver efficient mooring solutions using this concept.One of the main aspects concerning the design of
foundation systems are the uncertainties involved in the determination of
values for the soil parameters. For conventional onshore systems, this aspect
has been tackled by performing a large number of tests with soil samples.
However, on deepwater offshore sites the cost of performing such tests may be
very expensive, if not prohibitive; therefore, tests may not be performed for
every installation site, and sometimes results of tests made at other locations
are used to estimate or extrapolate the values of the soil parameters.This fact can severely affect the effectiveness of the
design and analysis of torpedo piles, leading to large discrepancies in the
response of the torpedo pile, mainly in terms of the final depth reached by the
pile. Therefore, it is very important to develop and employ methodologies to
properly assess the sensitivity of the response to the variation of these
parameters, and to incorporate, into the analysis method, techniques for the
formal treatment of the uncertainties.The classical probabilistic Monte Carlo simulation could be considered for this purpose, since it is a sound
methodology to estimate the effect of random uncertainties. Nevertheless, in
the problem described in this work, there are a great amount of epistemic
uncertainties in the model equations and parameters, and therefore, MC simulation
results provide only a rough estimation of the uncertainty. In addition, the
application of MC simulation requires excessive computational costs, as has
been confirmed in the case study considered in this work. More than 1000
simulations were needed to obtain the results.On the other hand, the computational efficiency of the fuzzy
arithmetic approach is outstanding—around three
orders of magnitude less. Therefore, the results of the application of the FA
approach demonstrated its ability to provide low-cost approximations of the
bounds of the uncertainties, and therefore, to comprise an effective design
tool for the practitioner.Future Developments
In this study, only two soil parameters, the undrained
shear strength and soil sensitivity, were considered as uncertain. Extensions
of the fuzzy methodology presented in this work could consider the treatment of
other uncertain parameters, such as for instance, the empirical maximum soil
strain rate factor (Se),
the empirical soil strain rate factor (Ce),
and the drag coefficient CD considered for the calculation of the soil drag force as the pile penetrates;
this latter parameter can vary for different anchor or pile shapes. Also, this
work did not consider the uncertainties associated with the penetration model
itself. These could also be considered, since it is a mainly empirical model
and involves imprecision in its formulation.
Finally, a promising approach for the design of
offshore systems would be to incorporate the pile penetration model, associated
with the fuzzy methodology, in the implementation of a coupled finite element-based,
time domain simulation program. In such implementation, not only the isolated torpedo
pile is considered, but also a full finite-element model of all components involved
in the installation of the pile (i.e., the mooring line itself and the other
lines and chains, illustrated in Figures2 and 3). The result is a complete 3D model, also
submitted to environmental loadings other than dead weight (such as marine
current). In such coupled model, there will be no need to “fudge” the drag
coefficient CD, to account
for the presence of the mooring line and chain loop.
Such computational tool would therefore comprise an
efficient tool for the design of mooring systems based on torpedo piles, and for the simulation of
the procedures needed for the installation of such complex offshore system.



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*Source: 512343-2008-06-23.xml*