Research Article A Study on the Holding Capacity Safety Factors for Torpedo Anchors

19 Pages·2014·5.3 MB·English

HindawiPublishingCorporation
JournalofAppliedMathematics
Volume2012,ArticleID102618,18pages
doi:10.1155/2012/102618
Research Article
A Study on the Holding Capacity Safety Factors for
Torpedo Anchors
Luı´s V. S. Sagrilo,1 Jose´ Renato M. de Sousa,1 Edison C. P. Lima,1
Elisabeth C. Porto,2 and Jane V. V. Fernandes2
1LaboratoryofAnalysisandReliabilityofOffshoreStructures,CivilEngineeringDepartment,
CentrodeTecnologia,COPPE/UFRJ,CidadeUniversita´ria,BlocoI-2000,SalaI-116,IlhadoFunda˜o,
21945-970RiodeJaneiro,RJ,Brazil
2PETROBRASResearchandDevelopmentCenter(CENPES),CidadeUniversita´ria,Quadra7,
IlhadoFunda˜o,21949-900RiodeJaneiro,RJ,Brazil
CorrespondenceshouldbeaddressedtoJose´RenatoM.deSousa,[email protected]
Received19January2012;Accepted29March2012
AcademicEditor:IoannisK.Chatjigeorgiou
Copyrightq2012Lu´ısV.S.Sagriloetal. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproductioninanymedium,providedtheoriginalworkisproperlycited.
The use of powerful numerical tools based on the finite-element method has been improving
the prediction of the holding capacity of fixed anchors employed by the offshore oil industry.
One of the main achievements of these tools is the reduction of the uncertainty related to the
holding capacity calculation of these anchors. Therefore, it is also possible to reduce the values
of the associated design safety factors, which have been calibrated relying on models with
higheruncertainty,withoutimpairingtheoriginallevelofstructuralsafety.Thispaperpresents
a study on the calibration of reliability-based safety factors for the design of torpedo anchors
considering the statistical model uncertainty evaluated using results from experimental tests
andtheircorrespondentfinite-element-basednumericalpredictions.Bothworkingstressdesign
(cid:3)WSD(cid:4) and load and resistance factors design (cid:3)LRFD(cid:4) design methodologies are investigated.
ConsideringtheWSDdesignmethodology,thesinglesafetyisconsiderablylowerthanthevalue
typicallyemployedinthedesignoftorpedoanchors.Moreover,aLRFDdesigncodeformatfor
torpedoanchorsismoreappropriatesinceitleadstodesignshavingless-scatteredsafetylevels
aroundthetargetvalue.
1. Introduction
Torpedo anchors, as shown in Figure1, are a cost-effective and practical solution for
anchoringtautmooringlinesoffloatingunitsasshowninFigure2.Thisanchorhasa“rocket”
shape and is embedded into the seabed by free fall using its own weight as driven energy.
Thestructureofatypicaltorpedoanchorusuallycomprisesfourdifferentcomponents(cid:5)1,2(cid:6):
2 JournalofAppliedMathematics
(cid:3)a(cid:4) (cid:3)b(cid:4)
Figure1:Typicaltorpedoanchorwithfourflukes:(cid:3)a(cid:4)conicaltipand(cid:3)b(cid:4)topwithdetailofthepadeye.
Chain )D
W
Wire rope (h
Polyester pt
Chain rope de
Anchor Water
Chain Taut-leg radius=1×WD
Torpedo
Conventional mooring radius=3×WD
Figure2:Torpedoanchorformooringlinesoffloatingunits.
apadeyethatconnectsthefirstchainsegmentofthemooringlinetotheanchor;aballasted
shaftofcarbonsteel;fourflukes,whichareapproximatelyrectangular(cid:3)torpedoanchorswith
noflukesmayalsobeemployed(cid:4);aconicaltipthatisdesignedtohelptheembedmentofthe
anchor.Typicaltorpedoanchorsweightvariesfrom350kNupto980kN.
These anchors have been first employed to hold flexible risers just after their
touchdown points (cid:3)TDPs(cid:4) in order to avoid that the loads generated by the floating unit
movementsweretransferredtosubseaequipments,suchaswetChristmastrees(cid:5)1(cid:6).In2003
a high holding capacity torpedo anchor was designed to sustain the loads imposed by the
mooring lines of a floating production, storage, and offloading (cid:3)FPSO(cid:4) (cid:5)3(cid:6) and, since then,
various other floating production systems have been anchored using the torpedo concept
(cid:5)4(cid:6).Furthermore,recently,HenriquesJr.etal.(cid:5)4(cid:6)presentedanewtorpedoanchorwithatotal
weightof1200kNandlargerflukesthatwillbeemployedtomoorFPSOsinultradeepwaters.
Thesecharacteristics,altogether,significantlyincreaseitsholdcapacitywhencomparedtothe
traditionalanchorswith980kNinweight.
JournalofAppliedMathematics 3
Due to their “rocket” shape, the load position and inclination with respect to their
flukes, traditional methods, such as conventional beam-column analysis procedures (cid:3)other-
wiseknownastheP-Ymethod(cid:4)orthesimpleformulaeproposedbyAPIguidelines(cid:5)5(cid:6)for
predicting the axial capacity of slender piles, often cannot be directly employed to estimate
theholdingcapacityoftheseanchors.
However, numerical procedures, based on the finite element (cid:3)FE(cid:4) method, have
recently been proposed by Branda˜o et al. (cid:5)2(cid:6) and de Sousa et al. (cid:5)6(cid:6)(cid:4), and experimental
testsperformedbyPETROBRAShavedemonstratedtheirgoodaccuracy(cid:5)7(cid:6).Animportant
outcome of these FE-based models is the lower uncertainty in their predictions when
comparedtotraditionalmethods.Therefore,itispossibletoreducethedesignsafetyfactors
values that have been calibrated relying on prediction models with higher uncertainty,
withoutimpairingtheoriginallevelofstructuralsafety.
There is no specific guideline for the design of torpedo anchors. Therefore, it is
common practice to follow the indications from API RP 2A (cid:5)5(cid:6) for piles foundations to
support offshore structures. This guideline prescribes that the safety factor associated with
these piles, according to a WSD design methodology, under operating environmental
conditions and during producing operations is equal to 2.0. It seems that this relatively
large safety factor is mainly related to the high prediction model uncertainty due to the
large number of pile capacity prediction models available at the moment of the calibration
process.
This paper presents a study on the calibration of safety factors for the ultimate limit
state design of torpedo anchors. One important aspect of this study is the availability of
resultsfromholdingcapacitytestsofsixtorpedoanchorsinstalledinCamposBasin,offshore
Brazil(cid:5)7(cid:6).Theseresultsmadeitpossibletoassessthemodeluncertaintystatisticsassociated
withtheFE-basedmodelproposedbydeSousaetal.(cid:5)6(cid:6).
Both working stress design (cid:3)WSD(cid:4) and load and resistance factors design (cid:3)LRFD(cid:4)
methodologies are investigated. Concerning the traditional WSD methodology, it is shown
that,forthesameactualdesignsafetylevelofthetraditionaloffshorepiles,itssinglesafety
factor can be significantly lowered. However, the use of WSD design methodology results
in designs with very scattered safety levels depending on the ratio of the functional and
environmental load actions. Aiming at overcoming this drawback, a LRFD methodology
calibrationisalsoinvestigatedinthispaper.Theresultsshowthatthestructuralsafetylevels
ofLRFD-baseddesignsaremoreuniformthantheWSD-basedones.
Inthefollowingsections,firstly,abriefreviewoftheFEmodelproposedbydeSousa
et al. (cid:5)6(cid:6) is presented followed by a comparison between the theoretical and experimental
holdingcapacities.Afterthat,thereliabilitybased-designstudyispresented,and,finally,the
mainconclusionsofthisworkarestated.
2. Finite-Element (FE) Model
2.1. Overview
Recently, de Sousa et al. (cid:5)6(cid:6) have proposed a FE model devoted to predict the undrained
loadcapacityoftorpedoanchors.Thismodelemploysisoparametricsolidfiniteelementsto
modelboththesoilmediumandtheanchor.Theseelementsarecapableofrepresentingthe
physicalnonlinearbehaviorofthesoilmediumandoftheanchoritself.Largedeformations
mayalsobeconsidered.Thesoil-anchorinteractionisensuredbysurface-to-surfacecontact
elements placed on the external surface of the anchor and the surrounding soil medium. A
4 JournalofAppliedMathematics
general overview of this model is presented in Figure3. Next, its main aspects are briefly
reviewed.
2.2. Soil Modeling
2.2.1.ConstitutiveMatrix
Thesoilisassumedtobeaperfectlyelastoplasticisotropicmaterialwithphysicalproperties
variablewithdepth.Hence,therelationbetweenstressesandstrainsisgiveninthefollowing
form:
(cid:2) (cid:3)
(cid:5)Δσ(cid:6)(cid:8) B ·(cid:5)Δε(cid:6). (cid:3)2.1(cid:4)
ep
Theconstitutiveelastoplasticmatrix,(cid:5)B (cid:6),isgivenby(cid:5)9(cid:6)asfollows:
ep
(cid:2) (cid:3) (cid:5)B (cid:6)·{∂P(cid:3)(cid:5)σ(cid:6),(cid:5)m(cid:6)(cid:4)/∂σ}·{∂F(cid:3)(cid:5)σ(cid:6),(cid:5)k(cid:6)(cid:4)/∂σ}T ·(cid:5)B (cid:6)
B (cid:8)(cid:5)B (cid:6)− el el . (cid:3)2.2(cid:4)
ep el {∂F(cid:3)(cid:5)σ(cid:6),(cid:5)k(cid:6)(cid:4)/∂σ}T ·(cid:5)B (cid:6)·{∂P(cid:3)(cid:5)σ(cid:6),(cid:5)m(cid:6)(cid:4)/∂σ}
el
Theelastictotalstressconstitutivematrixcanbesplitintotwoparts,asfollows(cid:5)9(cid:6):
(cid:4) (cid:5)
(cid:5)Bel(cid:6)(cid:8)(cid:5)Beff(cid:6)(cid:9) Bpore . (cid:3)2.3(cid:4)
In order to represent the nonlinear material behavior of the soil, the Drucker-Prager
model was chosen. This model approximates the irregular hexagon of the Mohr-Coulomb
failuresurfacebyacircle,and,consequently,theDrucker-Prageryieldfunctionisacylindrical
cone.Theyieldfunction,F(cid:3)·,·(cid:4),andplasticpotentialfunction,P(cid:3)·,·(cid:4),forthismodelaregiven
by(cid:5)10(cid:6)asfollows:
(cid:6)
F(cid:3)(cid:5)σ(cid:6),(cid:5)k(cid:6)(cid:4)(cid:8) J (cid:9)k(cid:3)1(cid:4)·I (cid:9)k(cid:3)2(cid:4),
(cid:6) 2 DP 1 DP (cid:3)2.4(cid:4)
P(cid:3)(cid:5)σ(cid:6),(cid:5)m(cid:6)(cid:4)(cid:8) J (cid:9)m(cid:3)1(cid:4) ·I (cid:9)m(cid:3)2(cid:4),
2 DP 1 DP
(cid:3)1(cid:4) (cid:3)2(cid:4) (cid:3)1(cid:4) (cid:3)2(cid:4)
wherek ,k ,m ,andm arestressstateparameters. Theseparameters dependonthe
DP DP DP DP
adopted approximation to the Mohr-Coulomb hexagon and are functions of the internal
friction angle, φ, the dilation angle, ψ, and the cohesion, c, of the soil. Figure4 presents
threepossibleapproximations.Intheproposedmodel,thecircumscribedDrucker-Pragerap-
proximationisused.ItisassumedthattheDrucker-PragercircletouchestheMohr-Coulomb
hexagonataLodeangleof(cid:9)30◦,whichleadstotheparameters(cid:5)9(cid:6)asfollows:
(cid:7) (cid:8) (cid:7) (cid:8) (cid:7) (cid:8)
k(cid:3)1(cid:4) (cid:8) √ 2(cid:2)·sen φ(cid:7) (cid:8)(cid:3), m(cid:3)1(cid:4) (cid:8) √ 2(cid:2)·sen ψ(cid:7) (cid:8)(cid:3), k(cid:3)2(cid:4) (cid:8)m(cid:3)2(cid:4) (cid:8) √6·(cid:2)c·cos φ(cid:7) (cid:8)(cid:3).
DP 3· 3(cid:9)sen φ DP 3· 3(cid:9)sen ψ DP DP 3· 3(cid:9)sen φ
(cid:3)2.5(cid:4)
Itisworthmentioningthattherulesfordesigningdeepoffshorefoundations,suchas
theAPIRP2A(cid:5)5(cid:6),typicallyassumethatloadingoccursrapidlyenoughsothatnodrainage
JournalofAppliedMathematics 5
H
p
H
e
H
a
20D
(cid:3)a(cid:4) (cid:3)b(cid:4)
Figure3:GeneralviewofatypicalFEmesh.
σ
2
Circumscribed
Drucker-Prager
Mohr-Coulomb
(extension)
Circumscribed
Drucker-Prager
(compression)
Inscribed
Drucker-Prager
σ3 σ1
Figure4:Drucker-PragerandMohr-Coulombyieldsurfacesinthedeviatoricplane.
andhencenodissipationofexcessporepressuresoccur.Thisleadstoanundrainedcondition
and,asaconsequence,theundrainedshearstrengthofthesoil,s ,mayreplacethecohesion,
u
c,in(cid:3)2.5(cid:4).
On the other hand, if the rate of loading is slow enough such that no excess pore
pressures are developed and sufficient time has elapsed since any previous application of
stresses such that all excess pore pressures have been dissipated, a drained analysis would
havetobeperformed.Liengetal.(cid:5)11(cid:6),however,indicatethattheundrainedloadingcapacity
ofadeeppenetratinganchor,suchasatorpedoanchor,isquiteclosetothedrainedcapacity,
and this is also assumed in this work. Nevertheless, this aspect has yet to be validated for
torpedoanchors.
6 JournalofAppliedMathematics
2.2.2.FEMeshCharacteristics
TheundrainedresponseofatorpedoanchorembeddedinclayisaclassicalJ elastoplastic
2
problem that requires no locking for incompressible materials and reasonable bending
behaviorinordertoobtainacceptableanswers.Moreover,ashighplasticstrainsareexpected
in the soil, the elements may become highly distorted, and insensitivity to these distortions
is demanded. Finally, coarse meshes are often required for computational efficiency, and
elements capable of providing sufficient accuracy are also needed. Aiming at addressing
all these requirements, a possible choice, according to Wriggers and Korelc (cid:5)12(cid:6), is the
use of enhanced strain hexahedrical and prismatic isoparametric finite elements with three
degreesoffreedomateachnode:translationsindirectionsX,Y,andZ.Theseelementsavoid
both bend and volumetric locking by adding 13 internal degrees of freedom to the element
(cid:5)13,14(cid:6).
AnoutlineofthemaindimensionsofthesoilmeshisshowninFigure3.Theproposed
meshisacylinderwithabasediameterof20D,whereD isthepilediameter.Theheightof
the cylinder is given by the sum of the penetration of the torpedo anchor, H ,the length of
p
theanchor,H ,andthedistanceofthetipofthetorpedoanchortothebottomoftheFEmesh,
e
H .
a
Each“slice”ofthecylinderhasitsownphysicalproperties,and,consequently,variable
strength, density, longitudinal and transverse moduli, and Poisson coefficients may be
assumed.
Theverticaldisplacementofthecylinderisrestrainedatthenodesassociatedwithits
base.ThedisplacementsinXandZdirections(cid:3)seeFigure3(cid:4)ofnodesintheouterwallofthe
cylinder are restrained. Once a plane of symmetry is found, the out-of-plane displacements
are also restrained. In order to avoid any influence of these boundary conditions on the
response of the anchor, several mesh tests were performed. In these tests, the cylinder
diameter and the distance from the tip of the anchor to the bottom of the mesh (cid:3)H (cid:4) were
a
varied.Itwasobservedthatadiameterof20Dforthecylinderandadistanceof5.0mforH
a
wasenoughtosimulatean“infinite”media.
2.3. Anchor Modeling
The torpedo anchor is modeled with eight node isoparametric solid elements analogous
to those used in soil representation. These elements, as pointed out before, are capable of
considering both material and geometrical nonlinearities. A typical FE mesh of a torpedo
anchorisshowninFigure5.
In order to properly represent the bending of the flukes and the shaft and,
consequently, avoid shear locking problems, the numerical integration of each element is
performedwiththeenhancedstrainmethod.
Itisworthmentioningthatneitherthepadeyeatthetopoftheanchornorthemooring
line is represented in the proposed model. The load from the mooring line is applied at the
gravitycenterofthepadeye,whereanodeisplacedandrigidlyconnectedtothetopofthe
anchorbyrigidbars,aspresentedinFigure6.
2.4. Anchor-Soil Interaction
ThemodelproposedbydeSousaetal.(cid:5)6(cid:6)employscontactelementsthatallowrelativelarge
displacement and separation between thesurfaces in contact. Theseelements are placed on
the external surface of the torpedo anchor and on the surrounding soil. A contact detection
JournalofAppliedMathematics 7
(cid:3)a(cid:4) (cid:3)b(cid:4)
Figure5:GeneralviewofaFEmeshforatorpedoanchor:(cid:3)a(cid:4)topand(cid:3)b(cid:4)bottompartoftheflukes.
R=L+E
Node where the load
is applied
Rigid bars
Figure6:Loadapplication.
algorithm based on the pinball technique and contact forces evaluated with the augmented
Lagrangianmethodareemployedintheformulationoftheseelements.
Relative sliding between the anchor and the soil is allowed when the shear stresses
along the outer wall of the anchor exceed the values estimated with the Mohr-Coulomb
frictionmodel,whichisgivenby(cid:5)5(cid:6)asfollows:
(cid:7) (cid:8)
τ (cid:3)z(cid:4)(cid:8)α(cid:3)z(cid:4)·s (cid:3)z(cid:4)(cid:9)K (cid:3)z(cid:4)·p (cid:3)z(cid:4)·tan χ . (cid:3)2.6(cid:4)
max u 0 o
Thefrictionanglebetweenthesoilandtheanchorwall,χ,isstatedas(cid:5)5(cid:6)
χ(cid:8)φ−5◦. (cid:3)2.7(cid:4)
8 JournalofAppliedMathematics
Theadhesionfactor,α,maybecalculatedbythefollowingformula(cid:5)5(cid:6):
⎧
⎨0.5·ψ(cid:3)z(cid:4)−0.5, ψ(cid:3)z(cid:4)≤1.0,
α(cid:3)z(cid:4)(cid:8) (cid:3)2.8(cid:4)
⎩0.5·ψ(cid:3)z(cid:4)−0.25, ψ(cid:3)z(cid:4)>1.0,
where
s (cid:3)z(cid:4)
ψ(cid:3)z(cid:4)(cid:8) u . (cid:3)2.9(cid:4)
p (cid:3)z(cid:4)
o
2.5. Solution Procedure and Implementation
The FE analysis is divided in three different steps. In the first one, the initial stress state,
that is, the stresses in the soil prior to the imposition of any kind of structural load to the
anchor,isgenerated.ThemodelproposedbydeSousaetal.(cid:5)6(cid:6)doesnotsimulatetheanchor
penetration in the soil, and it is assumed, by hypothesis, that the anchor is loaded with the
soilstressstateundisturbedbytheinstallationprocess.Theanchoristherebysupposedtobe
“wishedinplace,”andthisundisturbedstressstateisimposedtothesoilinthefirststepof
theanalysis.
However, the undrained shear strength of the soil and its stress state are deeply
affected by the dynamic installation process and the setup time for reconsolidation or
recovery of shear strength after installation (cid:5)15(cid:6). The direct consideration of this modified
stress state and undrained shear strength is quite complex and, therefore, the hypothesis of
an undisturbed stress state, as employed in this work, is commonly assumed in the design
of different types of offshore anchors such as suction anchors or other types of dynamic
penetrating anchors (cid:5)16–18(cid:6). However, the validity of this hypothesis is subject of intense
debate. Lieng et al. (cid:5)11(cid:6), relying on FE simulations, postulate that reconsolidation after
installation of a typical dynamic anchor would occur in less than six months, and, as a
consequence, a holding capacity determined from undrained analyses with an adhesion
factor close to 1.0 is expected to be valid. Richardson et al. (cid:5)19(cid:6), relying on centrifuge tests
andtheoreticalmodels,indicatethat50%oftheoperationalcapacityofadynamicanchoris
attained 35 up to 350 days after its installation. According to the same authors, 90% of the
operational capacity is reached 2.4 up to 24 years after the installation of the anchor. These
valuesareextremelydependentonthehorizontalcoefficientofconsolidation,and,therefore,
authors recommend that dynamic anchors should be installed in soils that consolidation
proceeds quickly. Furthermore, Richardson et al. (cid:5)19(cid:6) also indicate that values higher than
1.0 for the adhesion factor are reached in the long term leading to the conclusion that the
undrainedholdingcapacityoftheanchorwouldincreasewithtime,andthiscapacitywould
evenbehigherthantheoneestimatedwithanundisturbedstressstate.Mirza(cid:5)20(cid:6)statethat
the time required by offshore piles to reach their full-operational capacity varies between
1 and 2 years, and Jeanjean (cid:5)21(cid:6) indicates that 90% of the operational capacity of suction
anchorsinstalledinsoftGulfofMexicoclaysisattainedin90daysorless.
A typical torpedo anchor is firstly loaded approximately three months after its
installation, and, therefore, a great part of its full operational capacity has already been
reached. Moreover, the initial imposed loads are normally much lower than its actual load
capacity,asitsdesignreliesonthemaximumloadobtainedinmooringanalysesthatconsider
JournalofAppliedMathematics 9
both intact and damaged (cid:3)one mooring line broken(cid:4) conditions in the mooring system and
also several environmental load cases. These load cases encompass, for instance, extreme
waves and currents. Hence, as the objective of the proposed FE model is to evaluate the
undrained operational holding capacity of the anchor, it seems to be reasonable to assume
thatfullreconsolidationisachievedbeforethemaximumoperationalloadisimposedtothe
anchor. Nevertheless, in order to improve the proposed design method, more experimental
andalsotheoreticalstudiesonthisaspecthavetobeperformed.
Inthesecondstepoftheanalysis,theselfweightoftheanchorisimposed.Theanalysis
isrestartedwiththestressstatefromthefirststep,andgravityactsontheanchor.Finally,in
thethirdstep,thetotalloadisappliedwithadaptableincrementssince,asthesurrounding
soilprogressivelyfails,thestabilityofthesolutionisaffected,andlowerloadincrementsare
required.Typically,thislastloadstepstartswithaninitialloadincrementof5%ofthetotal
loadapplied,andastheanalysisprogresses,itcanbereducedto1%ofthetotalload.
Ateachloadstep,convergenceisachievediftheEuclidiannorm(cid:3)L2(cid:4)oftheresidual
forces and moments is less than 0.1% of the absolute value of the total forces and moments
applied.
ThemodelwasimplementedinaprogramcalledESTACAS.Thissoftwaregenerates
FE meshes to be analyzed with ANSYS (cid:5)22(cid:6) program, where the following finite elements
areemployed:SOLID185inordertomodelthesoilandthetorpedoanchor;CONTA174and
TARGE170areusedtosimulatethecontactbetweenthesoilandtheanchor.
3. Comparison with Experimental Tests
Aiming at validating the torpedo anchor geotechnical and structural design methodology,
PETROBRAS conducted six full-scale tests in Campos Basin, offshore Brazil (cid:5)7(cid:6). These six
full-scaletestsweredividedintotwosets.
In the first set of tests, three torpedoes (cid:3)type 1(cid:4), each one weighing 350kN, were
installed at a location with a water depth of 500m. At this location, soil was characterized
as being pure clay after cone penetration tests (cid:3)CPTs(cid:4) had been performed. The torpedoes
embedment depths varied from 8m up to 11m. Besides, the torpedoes were pulled out
withloadinclinationsofabout40◦ (cid:5)7(cid:6).Thesecondsetoftestsconsistedofpullingoutthree
torpedoes(cid:3)type2(cid:4)weighting430kNeach.Thewaterdepthatthissecondlocationis150m,
andthesoilcomprisesbothclayandsandlayers.Thetorpedoesfinalembedmentdepths,in
thiscase,varied from6mto8m,andtheloadwasimposed withinclinations varying from
◦ ◦
50 to67 .
Table1 presents the relation between the experimental measured pull-out loads and
thoseestimatedwiththemodelproposedbydeSousaetal.(cid:5)6(cid:6).
Despite all uncertainties related to the model and the geotechnical parameters
involved, Table1 indicates that the pull-out loads estimated with the FE model agree quite
welltothemeasuredloads.Hence,theseresultswillserveasabasistothereliability-based
designthatispresentednext.
4. Reliability-Based Design
Modern probabilistic design codes aim at designing new structures with a specified target
probabilityoffailure.Thiscanbeachievedbyfullyprobabilisticordeterministiccodes(cid:5)23(cid:6).
The latter is more common in engineering design. In these codes the target probability of
failureisnotstatedbutitisimplicitlyachievedbymeansofsafetyfactors.Thesesafetyfactors
10 JournalofAppliedMathematics
Table1:Relationbetweenmeasuredandnumericalpull-outloads.
Experimental/numericalratio
Torpedo
T35 T43
1 1.04 1.11
2 0.84 1.08
3 0.85 1.04
arecalibratedbyusingstandardstructuralreliabilitymethods(cid:5)23,24(cid:6),inwhichallrandom
variables(cid:3)andtheiruncertainties(cid:4)arerepresentedbymeansofprobabilitydistributions.
The two main deterministic design methodologies for ultimate limit state (cid:3)ULS(cid:4) are
the working stress design (cid:3)WSD(cid:4) and load and resistance factor design (cid:3)LRFD(cid:4). Focusing
specificallyonthecaseoftorpedoanchorsformooringsystems(cid:3)seeFigure6(cid:4)undertheWSD
methodology,thedesignequationcanbewrittenas
R
k ≥L (cid:9)E , (cid:3)4.1(cid:4)
k k
S
F
where the characteristic functional load effect, L , is associated with the initial mooring
k
line pretension applied to the anchor, and the characteristic environmental load effect, E ,
k
correspondstothetotaltensionactingontheanchor,whenenvironmentalloadsareapplied,
minusthefunctionalone.Accordingtothisclassification,onecanobservethatthefunctional
loadeffecthassmalleruncertaintiesthantheenvironmentalone.
InthecaseoftheLRFDmethodology,thedesignequationcanbewrittenas
R ≥γ L (cid:9)γ E , (cid:3)4.2(cid:4)
k L k E k
where γ and γ are the partial safety factors associated with functional and environmental
L E
loadeffects,respectively.
Thesafetyfactorsof(cid:3)4.1(cid:4)and(cid:3)4.2(cid:4)areobtainedbymeansofacalibrationprocess.The
mainideaofthecalibrationprocessistoobtainsafetyfactorsthatguaranteeacertaintarget
failureprobability,pf ,forthedesign.Thetargetprobabilityoffailuremaybestatedas
T
(cid:7) (cid:8)
pf (cid:8)Φ −β . (cid:3)4.3(cid:4)
T T
The calibration process is undertaken by means of reliability analyses considering
various design cases with different E /L ratios and also different statistical descriptions
k k
ofloadingsandanchorresistancesthatareexpectedtobefoundinpractice.Then,thesafety
factorscanbedefined,forinstance,forLRFDmethodology(cid:3)similarprocedureappliestothe
WSDmethodology(cid:4)throughanoptimizationprocesstosolvethefollowingproblem:
(cid:7) (cid:8) 1 (cid:12)N (cid:7) (cid:7) (cid:8)(cid:8)
ming γ ,γ (cid:8) β −β γ ,γ 2, (cid:3)4.4(cid:4)
L E T i L E
N
i(cid:8)1
where β is the reliability index of a given torpedo anchor design case that can be obtained
i
byareliabilityanalysismethod,forexample,first-orderreliabilitymethod(cid:3)FORM(cid:4),second-
orderreliabilitymethod(cid:3)SORM(cid:4),orMonteCarlosimulation(cid:5)23,24(cid:6).Inordertopursuethis