Guideline for Offshore Structural Reliability - DNVresearch.dnv.com/skj/OffGuide/Jacket_Application.pdf · A guideline for offshore structural reliability analysis of ... strength

TECHNICAL REPORT

DET NORSKE VERITAS

JOINT INDUSTRY PROJECT

GUIDELINE FOR OFFSHORE STRUCTURALRELIABILITY ANALYSIS:

APPLICATION TO JACKET PLATFORMS

REPORT NO. 95-3203

TECHNICAL REPORT

DET NORSKE VERITAS

JOINT INDUSTRY PROJECT

GUIDELINE FOR OFFSHORE STRUCTURALRELIABILITY ANALYSIS:

APPLICATION TO JACKET PLATFORMS

REPORT NO. 95-3203

DET NORSKE VERITAS

TECHNICAL REPORT

DET NORSKE VERITAS, Head Office: Veritasvn 1, N-1322 HØVIK, Norway Org. NO 945 748 931 MVA

Date of first issue:5 September 1996

Organisational unit:Struct. Reliability & Marine Techn.

DET NORSKE VERITAS ASDivision Nordic Countries

Approved by:Øistein HagenPrincipal Engineer

Veritasveien 1N-1322 HØVIK,NorwayTel. (+47) 67 57 99 00Fax. (+47) 67 57 74 74Org. No: NO 945 748 931 MVA

Client:Joint Industry Project

Client ref.:Rolf Skjong

Project No.:22210110

Summary:

A guideline for offshore structural reliability analysis of jacket structures is presented. Theguideline comprises experience and knowledge on application of probabilistic methods tostructural design, and provides advice on probabilistic modelling and structural reliabilityanalysis of jacket structures.

The characteristic features for jacket structures are outlined and a description of the analysissteps required for assessing the response in jacket structures exposed to environmental actionsis given.

Model uncertainties associated with the response analysis of jacket structures are discussedand recommendations are given for how to account for these uncertainties in the reliabilityanalysis.Important limit state functions that should be considered in a Level-III reliability analysis ofjacket structural components are defined and discussed.

The experience gained from two case studies involving probabilistic response analyses ofjacket structures, a fatigue failure limit state (FLS) and a total collapse limit state (ULS), aresummarised.

This report should be read in conjunction with the reports:

• Guideline for Offshore Structural Reliability Analysis - General, DNV Report no. 95-2018

• Guideline for Offshore Structural Reliability Analysis - Examples for Jacket Platforms, DNVReport no. 95-3204.

Report No.:95-3203

Subject Group:P12 Indexing terms

Report title:Guideline for Offshore Structural ReliabilityAnalysis:Application to Jacket Platforms

structural reliability

jacket platforms

environmental loads

capacity

Work carried out by:Gudfinnur Sigurdsson, Espen Cramer,Inge Lotsberg, Bent Berge

No distribution without permission from theClient or responsible organisational unit

Work verified by:Øistein Hagen Limited distribution within Det Norske Veritas

Date of this revision:05.09.96

Rev.No.:01

Number of pages:80 Unrestricted distribution

DET NORSKE VERITAS

TECHNICAL REPORT

DET NORSKE VERITAS, Head Office: Veritasvn 1, N-1322 HØVIK, Norway Org. NO 945 748 931 MVA

Guideline for Offshore Structural Reliability: Application to Jacket Plattforms Page No. 5

DNV Report No. 95-3203 Introduction

Sigurdsson,G; E. Cramer;I. Lotsberg, B.Berge “Guideline for Offshore Structural Reliability Analysis- Application to JacketPlatforms”, DNV Report 95-3203

Table of Contents

1. INTRODUCTION ...................................................................................................................................................7

1.1 OBJECTIVE ...............................................................................................................................................................71.2 DEFINITION OF A JACKET..........................................................................................................................................7

1.2.1 General ............................................................................................................................................................71.2.2 Types of Jackets ...............................................................................................................................................81.2.3 Structural Design Parameters .........................................................................................................................81.2.4 Jacket Design Analysis ....................................................................................................................................9

1.3 ARRANGEMENT OF THE REPORT...............................................................................................................................9

2. RESPONSE TO ENVIRONMENTAL ACTIONS .............................................................................................11

2.1 CLASSES OF RESPONSE...........................................................................................................................................112.2 ENVIRONMENTAL LOADS AND RESPONSE ..............................................................................................................12

2.2.1 Environmental Parameters............................................................................................................................122.2.2 Combination of Environmental Parameters ..................................................................................................122.2.3 Simulation of Wave Loads .............................................................................................................................132.2.4 Extreme Response Effects (ULS) ...................................................................................................................142.2.5 Fatigue (FLS) ................................................................................................................................................14

3. UNCERTAINTY MODELLING - TARGET RELIABILITY ..........................................................................16

3.1 GENERAL ...............................................................................................................................................................163.2 UNCERTAINTY MODELLING....................................................................................................................................16

3.2.1 Overview........................................................................................................................................................163.2.2 Types of Uncertainty......................................................................................................................................163.2.3 Uncertainty Implementation ..........................................................................................................................17

3.3 TARGET RELIABILITY .............................................................................................................................................173.3.1 General ..........................................................................................................................................................173.3.2 Selection of Target Reliability Level..............................................................................................................18

4. DISCUSSION OF LIMIT STATES.....................................................................................................................20

4.1 INTRODUCTION.......................................................................................................................................................204.2 BUCKLING FAILURE OF MEMBERS (ULS)...............................................................................................................22

4.2.1 Local Buckling of Members ...........................................................................................................................224.2.2 Global Buckling of Members .........................................................................................................................23

4.2.2.1 Background .............................................................................................................................................................. 234.2.2.2 Limit State Function................................................................................................................................................. 25

4.2.3 Buckling of Members Subjected to External Pressure...................................................................................264.2.3.1 Background .............................................................................................................................................................. 264.2.3.2 Limit State Function................................................................................................................................................. 28

4.3 JOINT FAILURE (ULS) ............................................................................................................................................284.3.1 Background....................................................................................................................................................284.3.2 Limit State Function ......................................................................................................................................32

4.4 FATIGUE FAILURE AT HOT-SPOT OF WELDED CONNECTIONS (FLS) .......................................................................334.4.1 General ..........................................................................................................................................................33

4.4.1.1 Overview.................................................................................................................................................................. 334.4.1.2 System Aspects ........................................................................................................................................................ 34

4.4.2 SN-Fatigue Approach ....................................................................................................................................354.4.2.1 General ..................................................................................................................................................................... 354.4.2.2 SN-Fatigue Modelling.............................................................................................................................................. 364.4.2.3 Uncertainty in SN-curves ......................................................................................................................................... 374.4.2.4 Fatigue Damage Model ............................................................................................................................................ 374.4.2.5 Limit State Formulation ........................................................................................................................................... 39

4.4.3 The FM-Approach for Fatigue Assessment ...................................................................................................394.4.3.1 General ..................................................................................................................................................................... 394.4.3.2 Crack Growth Rate................................................................................................................................................... 404.4.3.3 Crack Size over Time............................................................................................................................................... 414.4.3.4 Fatigue Quality......................................................................................................................................................... 434.4.3.5 Fatigue Crack Growth Material Parameters ............................................................................................................. 43




4.4.3.6 Limit State Formulation / Failure Criteria................................................................................................................ 444.4.4 Load and Response Modelling ......................................................................................................................45

4.4.4.1 General ..................................................................................................................................................................... 454.4.4.2 Sea State Description ............................................................................................................................................... 454.4.4.3 Global Structural Analysis ....................................................................................................................................... 504.4.4.4 Local Stress Calculation........................................................................................................................................... 52

4.4.5 Stress Range Distribution ..............................................................................................................................544.4.6 Formulation of Inspection Results.................................................................................................................574.4.7 Event Margins with Inspections Results:.......................................................................................................59

4.5 TOTAL STRUCTURAL COLLAPSE (ULS) ...................................................................................................................614.5.1 General ..........................................................................................................................................................614.5.2 Limit State Formulation.................................................................................................................................624.5.3 Distribution of the Annual Maximum Loading (Base-Shear) ........................................................................67

5. SUMMARY OF APPLICATION EXAMPLES .................................................................................................70

5.1 SUMMARY OF FATIGUE FAILURE LIMIT STATE - FLS EXAMPLE .............................................................................705.1.1 Modelling Approach ......................................................................................................................................705.1.2 Discussion of Results .....................................................................................................................................71

5.2 SUMMARY OF TOTAL COLLAPSE LIMIT STATE - ULS EXAMPLE.............................................................................725.2.1 Modelling Approach ......................................................................................................................................725.2.2 Discussion of Results .....................................................................................................................................73

6. REFERENCES ......................................................................................................................................................75




1. INTRODUCTION

1.1 ObjectiveThe objective of the application part of the project Guideline for Offshore Structural ReliabilityAnalysis for the structure types jacket, TLP and jack-up, is to give

• an overview of the characteristics of that structure's response to environmental actions,

• a detailed guidance on the reliability analysis of that structure with respect to severalimportant modes of failure,

• examples of reliability analyses applied to selected failure modes for that structure type.

The guidelines are intended for the application of Level III reliability analysis (DNV 1992b) tothe structure type; i.e. in which the joint probability distribution of the uncertain parameters isused to compute the probability of failure. This is usually a fairly demanding type of analysis,and is primarily expected to be applied in structural reassessment, in service inspection planning,code development/calibration and for detailed design verification of major load bearingcomponents of the structure. Hence, the guidelines prepared in this project concentrate on therequirements for these types of analyses, and do not make any attempt to embrace all aspects ofthe decision process. However, within these limitations, our aim is to cover significant aspects ofthe structural of reliability analysis.

1.2 Definition of a Jacket

1.2.1 GeneralFixed steel offshore structures are often called “jackets”. The name jacket originates from theearly days of the offshore industry when a trussed structure, jacket, was placed over the piles toprovide lateral stiffness to withstand wave, current and wind forces.

Jackets have been installed in water depths ranging from 0 to 400 metres, and in conceptualdesigns greater water depths have been considered. The steel weight and thus the cost increasesrapidly with water depth, therefore alternative platform solutions are often chosen for large waterdepths. Jackets have been designed to support topside weights of up to about 50000 tonnes, andit is feasible to design jackets for even larger topside weights.

The performance of jackets in hostile ocean environment has generally been good, although localfatigue damages have occurred in the earlier platforms. There have been very few total failures,and then only with the oldest platforms.




1.2.2 Types of JacketsA jacket may be used to support a large number of facilities, and depending on the purpose(drilling, production, utility, etc.) and ocean environment (water depth, waves, current, wind,earthquake, etc., it may be a simple or a very complex structure. Figure 1.2 shows a jacketdesigned to support drilling and production facilities.

Depending on the configuration, the jackets are classified (depending on the mode of installation)as:

- Self floater jacket

- Barge launched jacket

- Lift installed jacket

In early days the self floating jacket, which was floated out to the installation site and upended,was quite popular because it required a minimum of offshore installation equipment. The bargelaunched mode of installation has been most common as long as only “smaller” lifting vesselswere available. During the last ten years many platforms weighing less than 10.000 tonnes havebeen lift installed, thus minimising the need for temporary installation aids.

Most often jackets have piled foundations, but lately jackets have also been designed with platedfoundations, which reduce installation time. Among the piled jackets it is distinguished betweenthose with piles in the legs, template type jacket, and those with piles arranged as skirts andclusters, tower type jackets.

1.2.3 Structural Design ParametersThe jacket design is governed by the following:

- Functional requirements, i.e., support of topside, well conductors, risers, etc.

- Water depth

- Foundation soil conditions

- Environmental conditions, i.e., wave, current, wind, temperature, earthquake, etc.

Important items to be considered in an economical jacket design are:

- Jacket configuration

- Foundation (piled, plated, etc.)

- Type of installation (barge launch or lift installed)

- Use of high strength steel

- Use of cast nodes to improve fatigue performance.




1.2.4 Jacket Design AnalysisShallow water depth jackets are generally designed with adequate strength based on a staticanalysis where the wave loads are applied statistically on the structure. In addition a deterministicfatigue analysis and earthquake analysis (if required) are carried out. The jacket is in additiondesigned for the temporary installation phases. The natural period of the jacket is calculated toestablish the need for wave dynamic analysis.

Deep water jackets often exhibit dynamically amplified response when subjected to wave forces.The reason is that these platforms have a longer fundamental period of vibration (closer to thewave periods) than the shallow water platforms. These platforms need to be designed based onboth static and dynamic (stochastic) wave analyses. For the fatigue investigation a stochasticdynamic fatigue analysis may be more suited than a deterministic fatigue analysis. Earthquakeanalysis is carried out as required, and the jacket is designed for the temporary installationphases.

As mentioned above deep water platforms may be dynamically sensitive to wave forces. Thefrequency distribution of the random waves becomes a significant wave design parameter and theselection of wave spectra for design analyses is therefore extremely important. Due to the longfundamental period of vibration of the platform the fatigue behaviour may become one of thecritical design considerations.

1.3 Arrangement of the ReportThe response of Jacket structures to environmental loads are described in section 2, together withmethods for computation of the resulting load effects. The model uncertainties associated withthe computation of these load effects and the selection of target reliability are discussed insection 3. Important limit states are described in chapter 4, where also the stochastic modelling ofthese failure modes are discussed. Section 5 provides a summary of two reliability analyses,respectively for ultimate limit state and fatigue limit state for selected components in the Jacketstructure. The details of these analyses are presented in a separate report, Guideline for OffshoreStructural Reliability Analysis - Example for Jacket Platforms (DNV 1995b).

The present report is based on the general guidelines set out in the Guideline for ReliabilityAnalysis of Marine Structures - General, DNV (1995a). Companion applications are alsoavailable for jack-ups and TLP structures.




Figure 1.2 Jacket designed to support drilling and production facilities


DNV Report No. 95-3203 Response to Environmental Actions


2. RESPONSE TO ENVIRONMENTAL ACTIONS

2.1 Classes of ResponseAn important task in the reliability evaluation of an offshore structure is identification andmodelling of all significant loads and load combinations which the structure is exposed to duringthe service life.

The following Load Categories are defined for design and reassessment of jacket structures:

• Permanent Loads (P)

• Live Loads (variable functional loads) (L)

• Environmental Loads (E)

• Deformation Loads (D)

• Accidental Loads (A)

This section mainly considers environmental loads and load effects related to jacket structures.For structural engineering purposes, these environmental loads may be characterised mainly byover-water wind loads, by surface wave loads and by current loads that exist during severe stormconditions.

In the North Sea, the surface waves during storm conditions are of major importance in thedesign of Jacket structures for deep water environments, where the wind loads only represent acontribution of less that 5% of the total environmental loading. However, in the Gulf of Mexicothe wind loads are of major importance, having wind speeds during hurricane conditionsexceeding 50 m/s. Currents at a particular site can also contribute significantly to the total Jacketloading, where current generally refers to the motion of water that arises from sources other thansurface waves. E.g., tidal currents arise from the astronomical forces exerted on the water by themoon and sun, wind-drift currents arise from the drag of local wind on the water surface andocean currents arise from the drag of large-scale wind systems on the ocean.

During storm conditions, current velocities at the surface of more than 1 m/s are not uncommon,giving rise to more than 10% of the total induced environmental force.

The following sections give a more detailed description of environmental loads and responses onjacket structures. Regarding the other load categories, reference is made to DNV (1995a).




2.2 Environmental Loads and Response

2.2.1 Environmental ParametersThe parameters describing the environmental conditions shall be based on observations from, orin the vicinity of the actual location and on general knowledge about the environmentalconditions in the area. This is e.g. reflected in The Norwegian Petroleum Directorate, Guidelinesfor Loads and Load Effects (NPD (1996)) where it is stated that in designing, the recording ofwave data should have a duration of at least 10 years (if wave loads are of major importance).

The main environmental parameters governing jacket design are:

• wave height (H), wave period (T) and wave direction

• current velocity, current direction and current profile

• steady wind velocity, wind direction and wind profile

• water level variations (tidal, storm surge and potentially field subsidence)

Further details and descriptions of these parameters may be found in the General Guideline,Section 5 Loads. The above parameters are usually sufficient for jacket design in relativelyshallow waters with no structural dynamic effects present.

However, if the fundamental eigenperiods of the jacket system are at a level which may causeresonance phenomena, additional environmental parameters are needed in the design. For suchcircumstances the wave spectrum needs to be defined for different sea states, and the relativeoccurrence rate of significant wave height (Hs) and zero up crossing period (Tz) (or spectral peakperiod (Tp)) needs to be established. The wave spectra are usually of a single peak type (PM, orJONSWAP), however double peak spectra may also be applicable for some areas.

Other environmental parameters which need to be evaluated in jacket design are:

• ice and snow

• marine growth (thickness, weight and variation with water depth)

• temperature (sea/air)

• earthquake

2.2.2 Combination of Environmental ParametersTraditionally jacket design is performed by assuming wind, waves and current acting in the samedirection. The assigned probability level for each of the environmental parameters whencombining them may vary depending on the applicable code.

The NPD Guideline for Load and Load Effects (NPD (1996)) presents a combination ofenvironmental loads which has been extensively used the last decade, see Table 2.1. Moredetailed procedures for assessing the combined environmental loading will normally be acceptedin design provided sufficient data and documentation are available. In this context one should beaware of that jacket design is usually governed by the wave loads.

If simultaneous time series of environmental parameters exists, long term joint environmentalmodels may be used. Alternatively, the environmental parameters may be approximated bymarginal distributions as reflected in Table 2.1. For further details, see DNV (1995a) Section 5.




Table 2.1 Combination of environmental loads with expected mean values (m) and annualprobability of exceedance 10-2 (ULS) and 10-4 (PLS), NPD (1996).

Limit State Wind Waves Current Ice Snow Earth-quake

Sealevel

UltimateLimit State

(ULS)

10-2

10-1

10-1

--

10-2

10-1

10-1

--

10-1

10-2

10-1

--

--

10-2

--

---

10-2

-

----

10-2

10-2

10-2

mmm

ProgressiveLimit State

(PLS)

10-4

10-2

10-1

-

10-2

10-4

10-1

-

10-1

10-1

10-4

-

----

----

---

10-4

m*m*m*m

2.2.3 Simulation of Wave LoadsThe Morris’s equation has been widely applied in the design of jacket structures in the lastdecades for assessing the wave induced loading. This is not a complete and consistentformulation which fully simulates the wave loads. The Morris’s equation has, however, proved togive reasonable reliable results by careful selection of the drag (Cd) and inertia (Cm) coefficientsin combination with an appropriate wave theory.

The jackets are usually made up of tubulars with outer diameters varying typically from 0.3m upto 6.0m (bottle legs). For deterministic static in-place analysis, a drag coefficient in the range 0.7-0.8 together with an inertia coefficient of 2.0 are often used in design. Anodes are usuallyincluded in the modelling by increasing the drag coefficient with 8-12% depending on theamount of anodes required.

Stokes’ 5th order wave theory is the most commonly applied wave theory in design of jackets.The higher Stokes theory has a good analytical validity in deep water, whereas the fit to theboundary conditions in shallow water is relatively poor. This theory is suitable as it describes thewave kinematics above the mean water level and give information about the crest height whichin turn is needed in e.g. air gap calculations.

First order wave theory may also be used when the procedure for extrapolating the wave profileabove (and below) the mean water level is carefully selected. The “Stream” function gives a goodanalytical validity over a wide range of wave conditions and is to be used in relatively shallowwaters. This theory also has a set of free parameters that can be adjusted to achieve the best fit tothe dynamic free boundary conditions. For very shallow waters Cnoidal & Solitary Wave may beapplicable. Other wave theories exists (e.g. New-Wave, Tromans et al. (1991)), however, theexperience with use is limited.

The energy distribution around the dominating wave direction is usually described by a cosinedistribution where the level of spreading is defined by the exponent in the cosine function,typically varying from 2 - 8. For extreme load conditions, it is usually not recommended toinclude wave spreading. This is e.g. reflected in NPD Guideline for Loads and Load Effects(NPD (1996)) where it is recommended not to include wave spreading for significant waveheights above 10 meters if it gives reduced load effect. This recommendation is based on actualmeasurements/recordings in the North Sea. It has also been proposed to set the cosine exponent




in the wave spreading function equal to the significant wave height (in meters). This impliesmore or less long-crested waves for significant wave heights above 10 meters.

2.2.4 Extreme Response Effects (ULS)For extreme load conditions a jacket is usually considered drag dominated. This is, however,dependent on the wave conditions and the dimensions of the tubulars. For relatively deep waterjackets the drag dominance is shifted towards the inertia regime due to large diameter tubulars atthe lower jacket levels. For fatigue calculations the inertia regime is also having a higherinfluence due to the importance of the intermediate wave heights in the fatigue damagecontribution.

In relatively shallow waters with low fundamental periods of the jacket, static deterministicanalyses will generally be sufficient. If the dynamic amplification is low (e.g. less than 5-10%),the dynamic effects can be simulated by dynamic amplification factors (DAF) in combinationwith static analyses. For extreme load analyses the dynamic amplification will be low, whereasfor fatigue analyses the degree of amplification will be higher and more important.

A spectral approach is required if the dynamic effects are dominant. For extreme load analysisthe level of dynamic amplification is limited due to the period spacing between jacketeigenperiods and extreme wave periods. These aspects are further commented below for fatigue.

Concerning linear vs. non-linear structural analyses, there are examples of jackets in water depthsof 150 -200m and fundamental eigenperiods beyond 3 seconds where non-linear effects related towave loads are found to be very important. These non-linear effects are typically surface effects,non-linear wave-current interaction and the non-linear drag forces. This implies that linearisedstochastic dynamic analyses may underestimate the response significantly if the dynamics aredominating.

Design wave analyses are usually considered conservative, but this depends, however, on theactual selection of design parameters in the analysis. For relatively deep water jackets it has beenfound that time domain simulations commonly give higher responses than what may bedetermined by single design wave analysis.

2.2.5 Fatigue (FLS)Depending on the level of dynamic amplification, either a long-term distribution of single waveheights (H) and associated wave periods (T), or a scatter diagram ( H Ts p− or H Ts z− ), isneeded in the fatigue assessment. As stated earlier, it is a requirement for the fatigue analysis thatthe long-term environmental distributions have been established based on relevant measurementsand subsequent statistical post-processing. Long-term single wave height distributions are usuallylimited to 10 - 20 H/T combinations whereas a scatter diagram may consist of up to 200 shortterm sea states.

The wave induced stress range response needs to be determined for the fatigue analysis. Differentapproaches may here be applied for assessing the stress range response. Usually different waves(H/T combinations) are stepped “through” the structure with a step interval of 10-15 degrees andfrom these curves the stress ranges are determined. Special considerations may be required forelements in the splash zone as these elements are intermittent in and out of the water as thewaves are passing.




The shape of the wave spectra has an influence on the response results. This is especially the casewhen the fundamental eigenperiod of the jacket system is high and there is little damping in thedynamic system such that resonance will occur. A dynamic system like this will e.g. givesignificantly higher responses at resonance with a Jonswap spectrum compared to a PMspectrum.

A linearisation of the drag forces is needed for dynamic analyses. Different methods exist forperforming this type of linearisation. One approach is to linearise with respect to a characteristicwave height for each wave period. Members with intermittent submergence need to be treatedseparately. The response results are strongly dependent on the chosen linearisation wave heights,and especial attention should be made in the linearisation evaluation in order not to achieve over-conservative results. Another and more consistent linearisation procedure is to apply the waveenergy spectrum, by assuming the ocean waves and the corresponding fluid kinematics to beGaussian processes.

Slamming on horizontal members in the splash zone needs to be taken into account in the FLSdesign. Different approaches may be applied to determine the dynamic response and the numberof oscillations due to wave slamming. However, usually this effect is minimised by carefullyplacing the horizontal levels of the jacket outside the splash zone.


DNV Report No. 95-3203 Uncertainty Modelling - Target Reliability


3. UNCERTAINTY MODELLING - TARGET RELIABILITY

3.1 GeneralThere is a close connection between the uncertainty modelling and the target reliability level, asthe obtained reliability against e.g. fatigue failure or ultimate collapse in a reliability analysis isdependent on the chosen uncertainty modelling, especially with respect to the implementation ofmodelling uncertainties.

3.2 Uncertainty Modelling

3.2.1 OverviewThis section provides general guidance in respect to uncertainty modelling as appropriate to theultimate and fatigue limit state modelling for jacket structures. In Section 4, the proposed modelsaccounting for the uncertainties related to the FLS and ULS analyses of jacket platforms aredescribed in detail.

For further guidance, see also Guideline for Offshore Structural Reliability Analysis - General(DNV 1995a), Section 5, and the applied uncertainty modelling in Guideline for OffshoreStructural Reliability Analysis - Examples for Jacket Platforms (DNV 1995b).

In DNV Classification Notes 30.6, Structural Reliability Analysis of Marine Structures (DNV1992b), a general description of the uncertainty modelling for marine structures is presented.

3.2.2 Types of UncertaintyUncertainties associated with an engineering problem and its physical representation in ananalysis have various sources which may be grouped as follows:

• physical uncertainty, also known as intrinsic or inherent uncertainty, is a natural randomnessof a quantity, such as the uncertainty in the yield stress of steel as caused by a productionvariability, or the variability in wave and wind loading.

• measurement uncertainty is uncertainty caused by imperfect instruments and sampledisturbance when observing a quantity by some equipment.

• statistical uncertainty is uncertainty due to limited information such as a limited number ofobservations of a quantity.

• model uncertainty is uncertainty due to imperfections and idealisations made in physicalmodel formulations for load and resistance as well as in choices of probability distributiontypes for representation of uncertainties.

This grouping of uncertainty sources is usually adequate. However, one shall be aware that othertypes of uncertainties may be present, such as uncertainties related to human errors. Transitionsbetween the quoted different uncertainty types may exist.




3.2.3 Uncertainty ImplementationUncertainties are represented in reliability analyses by modelling the governing variables asrandom variables. The corresponding probability distributions can be defined based on statisticalanalyses of available observations of the individual variables, yielding information on their meanvalues, standard deviations, correlation with other variables, and in some cases also theirdistribution types.

Variables for which uncertainties are judged to be important, e.g. by experience or by sensitivitystudy, shall be represented as random variables in a reliability analysis. Their respectiveprobability distributions shall be documented as far as possible, based on theoreticalconsiderations and statistical analysis of available background data.

Dependency among variables may be important appear and shall be assessed and accounted forwhen necessary. Correlation coefficients can be estimated by statistical analyses.

Model uncertainties in a physical model for representation of load and/or resistance quantitiescan be described by stochastic modelling factors, defined as the ratio between the true quantityand the quantity as predicted by the model for multiplicative correction factors. A mean value notequal to 1.0 for the stochastic modelling factor expresses a bias in the model, and the standarddeviation expresses the variability of the predictions by the model. An adequate assessment of amodel uncertainty factor may be available from sets of field measurements and predictions.Subjective choices of the distribution of a model uncertainty factor will, however, often benecessary. The importance of a model uncertainty may vary from case to case and should bestudied by interpretation of parameter sensitivities.

3.3 Target Reliability

3.3.1 GeneralTarget reliabilities have to be met in design in order to ensure that certain safety levels areachieved. A reliability analysis can be used to verify that such a target reliability is achieved for astructure or structural element. A difficulty in this context is that the uncertainties included in astructural reliability analysis will deviate from those encountered in real life. This is because;

• the reliability analysis does not include gross errors which may occur in real life

• the reliability analysis, due to lack of knowledge, includes statistical uncertainty and modeluncertainty in addition to the physical uncertainty (epistemic) which is present in real life

• the reliability analysis may include uncertainty in the probabilistic model due to distributiontail assumptions

This means that a reliability index calculated by a reliability analysis is an operational or nominalvalue, dependent on the analysis model and the distribution assumptions, rather than a truereliability value which may be given a frequency interpretation. Calculated reliabilities cantherefore usually not be directly compared with required target reliability values, unless the latterare based on similar assumptions with respect to analysis models and probability distributions.This is a limitation which implies that target reliability indices cannot, normally, be specified ona general basis, but only case by case for individual applications.

For a more detailed discussion of the subject of determining the target reliability level, referenceis given to the DNV (1995a).




3.3.2 Selection of Target Reliability LevelTarget reliabilities depend on the consequence and nature of failure, and to the extent possible,should be calibrated against well established cases that are known to have adequate safety. Incases where well established structures are not available for the calibration of target reliabilities,such target reliabilities may be derived by comparison of safety levels established for similarexisting structural design solutions or through decision analysis techniques.

By carrying out a reliability analysis of a structure satisfying a specified code using a givenprobabilistic model, the implicit required reliability level in this code will be obtained, whichmay be applied as the target reliability level. The advantage with this approach compared toapplying a predefined reliability level, is that the same probabilistic approach is applied in thedefinition of the inherent reliability of the code specified structure and the considered structure,reducing the influence of the applied uncertainty modelling in the determination of the targetreliability level.

The use of codes could with advantages be applied e.g. in the determination of the minimumacceptable reliability level, below which structural inspections for jacket structures exposed tofatigue degradation are required. In the NPD, Act, regulations and provisions for the petroleumactivities (NPD 1996), it is stated that for structural details with no access for inspection orrepair, the design factors specified in Table 3.1 are to be applied in the design, dependent on theconsequence of failure of the detail. This could be interpreted as that a structural detail does notneed to be inspected prior to one 10th, or one 3rd, of the fatigue design life for substantial and nosubstantial failure consequence, respectively. The reliability levels at these time periods (onetenth, or one third of the design life) then consequently also correspond to the target reliabilitylevel for which a structural inspection is required according to the code.

Table 3.1 Design fatigue factors when no access for inspection or repair exist

DamageConsequence

No access or inthe splash zone

Substantialconsequence

10

No substantialconsequence

3

In general, acceptable structural probabilities of failure, specified as minimum values of targetreliabilities, depend on the consequence and nature of failure. The evaluation of the consequenceof failure comprises an evaluation with regard to human injury, environmental impact andeconomical loss, whereas the nature or class of failure considers the type of structural failure.Required minimal reliability levels make sense only together with a specification of a referenceperiod. The reference period should reflect the nature of the failure and is generally equal to theanticipated lifetime of the structure, or simply one year. As a general statement it might beargued that an annual target failure probability should be used when human life is at stake whilelifetime target failure probabilities applies if the consequence is material cost only.

The economical aspect will mainly depend on the economical consequence of the failure due torepair cost, missing income and/or demand in the repair period. When the failure consequenceregards economic loss, minimum target structural failure probabilities may be specified by the




operator based on requirements from national authorities and company design philosophy and/orrisk attitude. The safety level may therefore in general vary between the individual structures.

The direct consequence of failure for the environment could be included in the target safety levelrelated to economical consequences.

A major accident is likely to have a negative influence on the reputation of the company, bothtowards the government and towards the society in general. The consequence of this effect isdifficult to quantify. It is probably related to the company philosophy or may simply beconsidered as a part of the economical consequence.

The consequence of failure related to human injury will in large extent depend on the type offailure and operational condition for the platform. E.g., in DNV CN 30.6 (DNV 1992b), thecriterion concerning human injury is to be formulated as the annual probability of failure (definedas total collapse of the platform) shall not exceed 10-6 for no warning and serious consequences.

The target reliability level may also be based upon the proposed values presented in Table 3.2,taken from DNV Classification Notes 30.6 (DNV 1992b). When predefined reliability levels areapplied as target values, care must, however, be made in the uncertainty modelling in order toaccount for the same level of uncertainty as is reflected in the predefined target reliability level.The target reliabilities, specified in Table 3.2., are therefore closely connected with the proposeduncertainty modelling described in the Classification Notes.

Table 3.2 Values of acceptable annual failure probability and target reliability index

Class of Failure Less SeriousConsequence

SeriousConsequence

I. Redundant structure PF = 10-3

β = 3.09

PF = 10-4

β = 3.71

II. Significant warning prior tooccurrence of failure in a non-redundant structure

PF = 10-4

β = 3.71

PF = 10-5

β = 4.26

III. No warning before theoccurrence of failure in a non-redundant structure

PF = 10-5

β = 4.26

PF = 10-6

β = 4.75


DNV Report No. 95-3203 Discussion of Limit States


4. DISCUSSION OF LIMIT STATES

4.1 IntroductionThe objective for all structural designs is to build structures which fulfil proper requirementswith respect to functionality, safety and economy. These three aspects are closely connected andan iterative process is necessary to achieve the most optimal design. Current design practice isbased on partial safety factors and control against several limit states. A structure, or a structuralcomponent, is considered not to satisfy the design requirements if one or more of the limit statesare exceeded. Four main categories of limit state are defined in the NPD regulations (NPD 1996):

• Ultimate Limit State (ULS) which is defined on the basis of danger of failure, large inelasticdisplacement or strains, comparable to failure, free drifting, capsizing and sinking.

• Fatigue Limit State (FLS) which is defined on the basis of danger of fatigue due to the effectof cyclic loading.

• Progressive Collapse Limit State (PLS) which is defined on the basis of danger of failure, freedrifting, capsizing or sinking of the structure when subjected to abnormal effects.

• Serviceability Limit State (SLS) which is defined on the basis of criteria applicable tofunctional capability, or durability properties under normal conditions.

Only the fatigue limit state and the ultimate limit state will be discussed further in this report.

The Ultimate Limit State for a structure can be considered as the collapse of the structure. Thislimit state is difficult to describe through simple design equations, and therefore the design isnormally performed at a component level where the capacity of the single joints and membersbetween the joints are analysed/designed separately. Alternatively, the capacity for the UltimateLimit State can be assessed by non-linear analysis. At present non-linear analyses are performedfor reassessment and requalification purposes, but is not considered to be practical at a designstage. Also guidelines on how to perform such analyses are lacking. Therefore limit statefunctions for reliability analysis of jacket structures will in general also be based on designequations for single components.

The ULS limit state functions required for design of jacket structures are:

• Capacity of members between the joints with respect to yielding and buckling. This includesboth local buckling and global bending buckling of the member, section 4.2.1-2. The localcapacity is further affected by external pressure which may interact with global memberbuckling, section 4.2.3.

• Capacity of joints, section 4.3

Traditional ULS design are based on load effects determined by elastic frame analyses.

It should be noted that the design equations in the design standards are based on characteristicvalues which are defined at some fracture value or lower bound value. For reliability analyses thelimit states are based on the actual values, accounting for uncertainties, where the load andmaterial coefficients are not included in the equations for the limit state functions.




The Progressive Collapse Limit State is used to design the platforms for accidental events havinga probability of occurrence larger than 10-4. Accidental events such as explosion, fire and shipimpacts are considered. The accidental events are defined from Quantitative Risk Analyses, seeSection 2 of Guideline for Offshore Structural Reliability - General (1995a). Possible damage tothe structure is calculated based on an elasto-plastic analysis, and the structure is then analysedwith that damage for a given environmental loading. This analysis is similar to that of anUltimate Limit State analysis, but with different load and material coefficients in the designequation according to the NPD regulations, (NPD 1996).

The Serviceability Limit State is used for control of deflections and accelerations of the topsidestructures, but is hardly used for the design of jacket structures.

The potential application areas for structural reliability analyses of jacket structures are withindetailed design verification and for in-service inspection planning. For important components,the failure modes comprise;

• Jacket members (legs and braces) (ULS):

* Buckling of members:

- Local buckling of members

- Global buckling of members

- Buckling of members subjected to external pressure

* Total structural collapse due to environmental loading (e.g. wave and current loading onthe jacket and wind loading on the superstructure)

• Tubular joints (ULS):

* Joint failure

• Tubular joints and connections (FLS):

* Fatigue at hot-spots in welded connections

In the following sections the above component failure modes due to buckling failure of members,joint failure and fatigue failure are discussed, and examples for models which may be applied ina reliability analysis are given. Furthermore, a simplified limit state for system failure defined astotal structural collapse due to environmental loading is discussed, where the total structure isconsidered as a single component.




4.2 Buckling Failure of Members (ULS)

4.2.1 Local Buckling of MembersThe susceptibility to local buckling of tubular members is a function of member geometry andyield strength. The behaviour of a tubular subjected to a bending moment is shown in Figure 4.1.As the capacity behaviour is dependent on the geometry and material characteristics, it isconvenient to define the tubulars in section classes (Eurocode 3 (1993)) as illustrated in Table4.1

Table 4.1 Requirements to section classes in Eurocode 3

Section Class I Section Class II Section Class III Section Class IV

d/t ≤ 11750/fy 11750/fy ≤ d/t ≤ 16450/fy 16450/fy ≤ d/t ≤ 21150/fy 21150/fy ≤ d/t

fy = yield strength (MPa) d = diameter t = thickness

The section classes are defined as follows:

Class I : cross-sections are those which can form a plastic hinge with the rotation capacityrequired for plastic analysis.

Class II: cross-sections are those which can develop their plastic moment resistance, but havelimited rotation capacity.

Class III: cross-sections are those in which the calculated stress in the extreme compressionfibre of the steel member can reach its yield strength, but local buckling is liable toprevent development of the plastic moment resistance.

Class IV: cross-sections are those in which it is necessary to make explicit allowances for theeffects of local buckling when determining their moment resistance or compressionresistance. Tubulars belonging to this section class may also be defined as a shellstructure.

These section classes are not defined for conditions with external pressure, and tests or numericalanalyses must be carried out for documentation. This is controlled under section 4.2.3.

Θ

Figure 4.1 Tubular capacity in bending for different section class dependent on degree ofdeformation ΘΘΘΘ.




4.2.2 Global Buckling of Members

4.2.2.1 BackgroundThe procedure for design of tubulars subjected to a bending moment according to the NPDregulations is based on linear elastic analysis as if all tubulars were belonging to section class III.(Reference is made to Eurocode 3 (1993) with respect to requirements to section classes whichfor tubulars are shown in Table 4.1). This procedure is considered to be sufficient for tubularssubjected to external pressure.

The procedure for design of tubulars in air is considered conservative for section classes I and IIas yielding of the section is allowed (by definition of section class). A higher capacity accountingfor the plastic section modules is directly achieved through a non-linear analysis. The increase inthe bending capacity by going from elastic to plastic section modules is a factor of 4/π=1.27.

The effect of plastic section modulus is more directly incorporated in the API design equationsthan that of NPD although it is opened for plastic design also in the NPD regulations.

Other items related to buckling of tubular members are:

- effective buckling lengths

- buckling curves

- effect of external pressure.

In a design analysis it is common to assume a buckling length that is representative for typicalmember configurations as X-braces, K-frames, single braces, jacket legs and piles. The effectivebuckling length is dependent on the joint flexibilities and for X-braces also on the amount oftension force in the crossing element. It is also a matter of discussion whether the buckling lengthshould be measured from centreline to centreline of jacket legs which can be argued for in thecase of a combined collapse of the braces and the legs, or if the buckling length should beassociated with the face to face length between the legs which may be argued for consideringbuckling of a single brace. The effective buckling length may be derived from analyticalconsiderations. However, the effective buckling lengths derived from theoretical considerationsare longer than the buckling lengths obtained from tests of frame structures loaded until collapse.

It should be noted that the basis for the buckling curves in the different codes is different. TheAPI buckling curve is derived as a lower bound value for low slenderness while it is equal to theEuler stress for high slenderness values which may be considered as an upper bound value forthat region. Another definition of a buckling curve is used in the AISC (1986). The backgroundfor the buckling curves used in design of steel structures in European design standards is basedon work carried out within the European Convention for Constructional Steelwork which ispresented in Manual on Stability of Steel Structures (1976). The design curves are presented bytheir characteristic values which are defined as mean values minus two standard deviations alongthe slenderness axis. The test results are assumed normal distributed.

It is also noted that the requirements to allowable fabrication eccentricity are different associatedwith the various buckling curves. For the European buckling curves, a straightness deviation atthe middle of the column equal 0.0015 times the column length is allowed, while for API andAISC the corresponding numbers are 0.0010 and 0.00067, respectively.

Different buckling curves used for design of tubular members are shown in Figure 4.2.




The design equation for global member buckling in the NPD regulations (NPD 1996) reads

σ σ σγc b b

y

m

B Bf

where

+ + ≤*

σcNA

N

A

= =

=

=

design axial compressive stress

axial force

section area

B NN E

= =−

bending amplification factor1

1

N E = Euler buckling load

σ σγ

b cy

k

k

m E

EE

f

ff

ff

NA

* ( )( )= − −

=

1 1

f k = characteristic buckling strength derived from the buckling curve

Buckling stress

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3 3.5

Reduced slenderness

ECCS,NPD,DNVAPI LRFDAPI WSD/AISC

Euler

Figure 4.2 Different buckling curves used for design of tubular members




4.2.2.2 Limit State FunctionThe limit state function for global buckling of members can be formulated as

G f B B

f

NA

y c b b

y

c

= − + +

=

= =

( )*σ σ σ

σ

where

yield strength

design axial compressive stress

N

A

= axial force

= section area

B bending amplification factor NN E

= =−

1

1

Nf

fff

fNA

f

E

b cy

k

k

E

EE

k

=

= − −

=

=

Euler buckling load

buckling strength derived from the buckling curve

σ σ* ( )( )1 1




4.2.3 Buckling of Members Subjected to External Pressure

4.2.3.1 BackgroundThe capacity of tubulars subjected to axial force, bending and external pressure may be designedbased on the guidelines on design and analysis provided by the Norwegian Petroleum Directorate(NPD 1996) with additional guidance by Lotsberg (1993), or by a design procedure presented byLoh (1990). In the following the design procedure given by NPD and Lotsberg is given. It shouldbe noted that it is only the effective axial force that contributes to the axial stresses that enhancebuckling, see Figure 4.3. The axial stress resulting from the external pressure do contribute in theequation for the von Mises stress considering yielding, but does not contribute to the axial forcethat gives global buckling stress.

Figure 4.3 Illustration of effective axial force to be used for global buckling. (The totalstress is governing for the local structural behaviour in terms of yieldingand local buckling)

The equation for global buckling is modified to account for the effect of external pressure asfollows:

σacB B AC

A= + −2 4

2

where

Aff

y

ea= +1

2

2




Bf

f fy

ea epp= −( )

21

2

σ

Cf

ffp

y p

epy= + −σ

σ22 2

22

fea = elastic buckling stress with respect to axial force

f k E tlea

=−

��

πν

2

2

2

12 1( )

k l

r t rt

= + −

+1 0123 1

1150

4 2

2 2

. ( )

( )

ν

fep = elastic buckling stress in hoop direction with respect to external pressure

f trep

= ��

��0 25

2

.

σp = stress in hoop direction due to external pressure

The equation for global buckling is then modified as follows

σ σ σσ σ

γc b bac axp

m

B B+ + ≤−

*

where σaxp = axial stress in the tubular due to end cap pressure = σp/2.

For other notations see section 4.2.2. Note that σc now is derived as the effective axial stress(without including the end cap stress resulting from external pressure).

An example of the difference between the effective axial stress and the total stress in a tubularmember as function of the water depth is shown in Figure 4.4. It is noted that the difference issmall for water depths below say 100 metres, but that it becomes significant for deep-waterstructures.




4.2.3.2 Limit State FunctionThe limit state function for global buckling of members subjected to external pressure can beformulated as,

G B Bac axp c b b

= − − + +σ σ σ σ σ( )*

with notation as given in section 4.2.1.

0

20

40

60

80

100

120

140

160

180

200

0 200 400 600 800 1000

Waterdepth in m

Allo

wab

le st

ress

Effective stressTotal stress

Figure 4.4 Axial stress in the tubular as function of water depth and external pressureat global member buckling

4.3 Joint Failure (ULS)

4.3.1 BackgroundA number of design equations have been established for the static strength of tubular joints. Theequations in API (1991) and NPD (1996) show a similar shape although the coefficients aredifferent as also might be expected as the API RP2A is based on allowable stresses, while theNPD has based the design on the partial coefficient method since 1977. The following work isbased on the NPD regulations, but only small modifications would be required to revert toanother standard such as that of API or HSE.

It should be mentioned that work on joint capacities is being carried out within the developmentof a new ISO standard on design of steel offshore structures. This work should be considered asbasis for limit state functions when it is available.

The following symbols are used:




T = Chord wall thickness

t = Brace wall thickness

R = Outer radius of chord

r = Outer radius of brace

θ = Angle between chord and considered brace

D = Outer diameter of chord

d = Outer diameter of brace

a = gap (clear distance) between considered brace and nearest load-carrying brace measuredalong chord outer surface

β = r/R

γ = R/T

g = a/D

fy = Yield strength

Qf = See Table 4.3

Qg = See Table 4.2

Qu = See Table 4.2

Qβ = See Table 4.2

N = Axial force in brace

MIP = In-plane bending moment

MOP = Out-of-plane bending moment

Nk = Axial load capacity of brace(as governed by the chord strength)

MIPK = In-plane bending moment capacity of brace(as governed by the chord strength)

MOPK = Out-of-plane bending moment capacity of brace(as governed by the chord strength)

σax = Axial stress in chord

σIP = In-plane bending stress in chord

σax = Out-of-plane bending stress in chord




Table 4.2 Values for Qu (Characteristic values)

Type of joint and geometry Type of load in brace member

Axial In-plane bending Out-of-planebending

T & Y 2.5 +19β

X (2.7 +13β)Qβ 5.0γ0.5β 3.2/(1-0.81β)

K 0.90(2 +21β)Qg

Qβ

β ββ

β=

−>

≤

�

�

��

�

��

0 31 0 833

0 6

10 0 6

.( . )

.

. .

for

for

Qa T

gg =

− ≤

− >

�

��

��

18 01 20

18 4 20

. . /

.

for

for

γ

γ

but in no case shall Qg be taken less than 1.0.

Table 4.3 Values of Qf

Loading Qf

Axial 1.0-0.03γA2

In-plane bending 1.0-0.045γA2

Out-of-plane bending 1.0-0.021γA2

where

Af

ax IP OP

y

22 2 2

20 64=

+ +σ σ σ.

The characteristic capacity of the brace subjected to axial force is determined by

N Q Qf T

k u fy=

2

sinθ




The characteristic capacity of the brace subjected to in-plane moments is determined by

M Q Qdf T

IPk u fy=

2

sinθ

The characteristic capacity of the brace subjected to out-of-plane moments is determined by

M Q Qdf T

OPk u fy=

2

sinθ

NN

MM

MM

k

IP

IPk

OP

OPk m

+�

��

�

�� + ≤

21

γ

where γm is a material coefficient =1.15.

Figure 4.5 Simple Tubular Joint




Figure 4.6 Force displacement relationship for a tubular joint

4.3.2 Limit State FunctionThe limit state function for the static capacity of tubular joints can be formulated as

G NN

MM

MM

k

IP

IPk

OP

OPk

= − +�

��

�

�� +1

2

( ) or G NN

MM

MMk

IP

IPk

OP

OPk= − +

�

��

�

�� +log( )

2

where the equations given above are used to calculate Nk, MIPk and MOPk with Qu from Table 4.4and A as given below.

Table 4.4 Values for Qu based on 50 per cent fractiles (median values)


Axial In-plane bending Out-plane bend.

T & Y 2.8 +21β

X (3.0 +14.6β)Qβ 5.6γ0.5β 3.6/(1-0.81β)

K (2.6 +27β)Qg

The parameter A for calculation of Qf in Table 4.4 is obtained as:

Af

ax IP OP

y

22 2 2

2=+ +σ σ σ




The CoV values in Table 4.5 for Qu may be used for the reliability analysis based on thepresented limit state functions. Qu is normal distributed. For CoV for yield strength, see DNV(1995a).

Table 4.5 Values for CoV for Qu


Axial In-plane bending Out-plane bend.

T & Y 0.10

X 0.10 0.10 0.10

K 0.20

4.4 Fatigue Failure at Hot-spot of Welded Connections (FLS)

4.4.1 General

4.4.1.1 OverviewJacket structures of all types are generally subjected to cyclic loading from wind, current,earthquakes and waves, which cause time-varying stress effects in the structure. Theenvironmental quantities are of random nature and may be more or less correlated to each otherthrough the generating and driving mechanisms. Waves and earthquake loads are generallyconsidered to be the most important sources for structural excitations. However, earthquakeloads are only taken into account in the analysis of structures close to, or within tectonic areas,and will not be included here. Wind and current loads represent an insignificant contribution tothe fatigue loading and may be ignored in the fatigue analysis of jacket structures.

A fatigue analysis of offshore structures can in general terms be described as a calculationprocedure, starting with the environment (waves) creating stress ranges at the hot-spot regionsand ending with the fatigue damage estimation. The link between the waves and the fatiguedamage estimate is formed by mathematical models for the wave forces, the structural behaviourand the material behaviour. The probabilistic fatigue analysis may be divided into four mainsteps:

1) Probabilistic modelling of the environmental sea states (short- and long-term modelling)

2) Probabilistic modelling of the wave loading

3) Structural response analysis (global and local)

4) Stochastic modelling of fatigue damage accumulation.

The above steps are covered in DNV (1995a). In the following, it will be focused on theapplication to jacket structures.

In addition to the above steps, the analysis includes a stochastic modelling of the fatigue capacityand the probabilistic evaluation, i.e. the probabilistic derivation of the likelihood of the event thatthe accumulated fatigue damage exceeds the defined critical fatigue strength level.




In order to carry out a realistic fatigue evaluation of a jacket structure, it is necessary to introducesome simplifying assumptions in the modelling. These assumptions consist of:

• For a short term period (a few hours) the sea surface can be considered as a realisation of azero-mean stationary Gaussian process. The sea surface elevation is (completely)characterised by the frequency spectrum, which for a given direction of wave propagation, canbe described by two parameters, the significant wave height HS and some characteristicperiod like the spectral peak period TP or the zero-mean up-crossing period TZ .

• The long term probability distribution of the sea state parameters ( H TS P− or H TS Z−diagram) is known.

• Applying frequency domain approach for assessing the structural response, the wave loadingon structural members must be linearised and the structural stress response must be assumedto be a linear function of the loading, i.e. the structural and material models are linear.

• The relationship between the sectional forces and the local hot-spot stresses (SCFs) is known,where an empirical parameter description is most common.

Fatigue is the process of damage accumulation in a material undergoing fluctuation stresses andstrains caused by time-varying loading. Fatigue failure occurs when the accumulated damage isexceeding a critical level. The fatigue process experienced by most offshore structures is high-cycle fatigue, i.e. the fluctuating nominal stress levels are below the yield strength and thenumber of cycles to failure is larger than 104 . Fatigue damage in welded structures is likely tooccur at the welded joints due to the stress concentration at areas of geometric discontinuity.Notches and initial defects caused by the welding processes may also occur in this area.

Traditional fatigue design of jackets is based on the SN-fatigue approach where fatigue failure isassumed to occur when the crack has propagated through the thickness of the member. However,at a design stage without any observed cracks in the structure, the estimated fatigue damagebased on fracture mechanics is normally less reliable than that derived from SN data due to thedifficulties involved in assessing the initial crack size. Applying the SN-approach, the fatiguedamage is measured in degree of damage, D , from an initial value 0 to ∆ , where ∆ is definedas the fatigue damage accumulation resulting in failure, depending on the detail considered andthe selected SN-curve.

When performing a reliability updating on the basis of structural inspections for cracks, theinspection outcome can not be used directly to update the degree of damage accumulation unlessthe fracture mechanics approach is applied. In order to also be able to perform reliabilityupdating when the SN approach is applied, a procedure for establishing a relationship betweenthese two fatigue approaches is proposed in the following.

4.4.1.2 System AspectsJacket structures are typically redundant with respect to brace failures and a total structuralcollapse will not occur before several members have failed. After a member has failed due to e.g.fatigue, the applied loading will be transferred by the remaining members, i.e. a redistribution ofthe load through the structure occurs. In the damaged structure, each remaining member hasalready some accumulated fatigue damage, and due to the redistribution of the stresses in thestructure the rate of damage accumulation will change. By accounting for the changes due tofailure in other members, the total damage at a section can by formulated mathematically. Once




the time to failure for each individual section in a sequence is defined, the sequence event isdefined as the intersection of a set of section failure events for which the time to failure for eachindividual section is less than the lifetime of the structure.

Usually, there will be many alternative sequences leading to collapse, and the total structuralfailure is the event that one of these collapse sequences occurs.

A system reliability approach is required when the probability of total structure failureaccounting for the progressive nature of collapse is to be estimated. One of the difficulties withsuch an approach is that for typical structures there are a very large number of sequences leadingto failure, and that it is not feasible to include all of these in the analysis. Usually, however, onlyfew of the failure sequences have significant contributions to the total failure probability.Therefore, in most structural reliability analyses, a search technique can be used to identifyimportant failure sequences and the system failure event is approximated as the union ofimportant sequences.

4.4.2 SN-Fatigue Approach

4.4.2.1 GeneralThe fatigue life of a joint may in general be characterised by three time intervals:

Tinitial The crack initiation period or first discernible surface cracking.

Tth The total time until the crack has propagated through the thickness.

Tsec The total time until gross loss of structural stiffness with extensive through thicknesscracking (defined as section failure).

Based on inspections for fatigue cracks in the joints, a fatigue reliability updating based on theoutcome of the inspections can be carried out applying Bayesian updating. The inspection resultscan for the SN-approach not be used directly to update the estimated accumulated fatiguedamage. However, if a relationship between the damage accumulator D in the SN-approach andthe crack size was available, it would be possible to utilise the inspection results for reliabilityupdating.

No guidelines or established procedures are available for establishing the relationship betweenthe accumulated fatigue damage from the SN-approach and the crack size. This relationship may,however, be obtained by calibrating the parameters describing the crack propagation in thefracture mechanics approach. In the following the parameters are calibrated by fitting theprobability of having a through thickness crack as a function of time obtained from the fracturemechanics approach to the results obtained from the SN-approach, applying e.g. least-squaresfitting.

It should be noted that calibrating the through thickness cracking to a SN-curve is in generalinconsistent, as the crack initiation period included in the SN-approach is not incorporated in thefracture mechanics formulation. This may lead to unconservative results in the reliabilityupdating based on the outcome of inspections.

More consistent results may be obtained by applying only the SN-curve for the crackpropagation period (if available) in the calibration of the fracture mechanics material parameters,i.e. a SN-curve describing the number of load cycles it takes for an already initialised crack to




propagate through the thickness. This approach assumes there exists a model available toestimate the crack initiation time and that the time period until inspection is greater than thecrack initiation time.

However, very limited information is available for describing the crack initiation time and theSN-curves for the crack propagation period. The calibration of the fracture mechanics parametersis therefore in the present study based on SN-curves where the crack initiation period is includedin the modelling of the fatigue capacity applying the SN approach. It should in this connectionalso be noted that for welded details, the crack initiation period is relatively small compared tothe whole fatigue life.

4.4.2.2 SN-Fatigue ModellingSN-data are experimental data giving the number of cycles N of stress range S resulting in fatiguefailure. These data are defined by SN-curves for different structural details.

The design SN-curves are based on a statistical analysis of experimental data. They are given aslinear or piece-wise linear relations between log10S and log10N. A design curve is defined as themean curve, minus two standard deviations of log10N obtained from the data fitting. The standarddeviation is computed based on the assumption of a fixed and known slope.

The design SN-curves are thus of the form

log log loglog10 10 10 102N a m SN= − −σ

or

N K S m= ⋅ − , S S> 0

where

N number of cycles to failure for stress range S

a a constant relating to the mean SN-curve

σ log10 N the standard deviation of log10N

m the inverse slope of the SN-curve

S0 stress range level for which change in slope occurs, i.e. for bilinear SN-curve or endurance limit for single slope SN-curve

log10 K log log10 102a N− σ

The bilinear SN-curve is defined as,

NK S S S

K S S S

m

m=

>

≤

�

��

��

−

−

;

;

0

2 02

K S K Sm m0 2 0

2− −=

where m2 is the inverse slope of the SN-curve ( ∞ for endurance limit at S0 ).

The numerical values for the relevant parameters are summarised in table 7.10 in DNV (1995a).For tubular joints, the T-curve (DNV 1984) is recommended for modelling the fatigue capacity.




In air, the T-curve has m=3, which changes to m2 =5 at N N= =0710 . For cathodically protected

structures in seawater the T-curve has m=3 and a cut-off value at N N= = ⋅082 10 . Knowing N0

and K, the stress range level S0 can be obtained by

S KN

m0

0

1

=�

��

�

��

The fatigue strength of welded joints is dependent on the plate thickness, t, with decreasingfatigue strength with increasing thickness. The design T-curve is used when the thickness t in atubular joint is less than 32 mm. For the thickness t ≥ 32 mm a modification of the T-curve isperformed, and the modified T-curve becomes,

log log log log10 10 10 104 32

N Km t

m S= − ⋅ �

��

�

�� − ⋅ S S> 0

or

N t K Sm m= ⋅ ⋅− −( / ) /32 4 S S> 0

The factor ( / ) /t m32 4− is denoted the thickness-effect factor.

4.4.2.3 Uncertainty in SN-curvesThe uncertainties associated with describing the fatigue capacity through empirical SN-curvesare accounted for by considering a stochastic SN-relation. This may be done by treating theparameters in the deterministic linear or bilinear SN-relation as random variables. I.e. bymodelling the inverse slope m as deterministic and fitting the log10N test data from the fatiguetests to the Normal distribution. The uncertainty modelling of the SN-curve can then be obtainedby modelling K as a Log-Normal distributed stochastic variable. E.g., for the T-curve withcathodic protection in seawater, where the inverse sloop m is modelled as deterministic and K ismodelled as Log-Normal distributed, the stochastic modelling of the SN-curve is defined by thefollowing properties:

[ ] [ ]E K Std Km m N Nm m N N

= ⋅ = ⋅= = ≤= = ∞ >

539 10 335 103

12 12

1 0

2 0

. .

The importance of modelling the cut-off level N0 as stochastic should also be evaluated. Forstochastic modelling of N0 the Normal distribution should be selected, e.g. with

[ ] [ ]E N CoV N08

02 10 010= ⋅ = .

4.4.2.4 Fatigue Damage ModelThe accumulated fatigue damage is computed from the representative stress distribution and theSN-capacity model. The accumulated damage depends on the number and magnitude of the




applied stress cycles. Assuming the accumulated fatigue damage independent of the sequence inwhich the stress cycles occur (no sequence effect), the damage accumulation D can be written as,

Dn

Ni

ii=

=�

1

where n n Si i= ( ) is the number of cycles of stress range Si in the stress history and N N Si i= ( )is the number of stress cycles of stress range Si necessary to cause failure. This formulation ofthe fatigue damage accumulation is usually denoted the Miner-Palmgren approach.

The failure criterion defines the degree of accumulated fatigue damage that results in failure. Fora constant amplitude stress variation, it follows directly from the damage definition above thatfailure occurs when D ≥ 1, as the SN-curves are originally derived from constant amplitudeloading.

For a variable amplitude loading, the value of the damage accumulation D at failure will typicallybe random due to the inherent randomness in the stress history and the potential influence ofsequence effects.

For offshore structures, the number of stress cycles resulting in fatigue failure will typically belarge and the inherent uncertainty in the damage accumulation will approach zero. The damageaccumulation, D, is then sufficiently represented by a summation of the expected value of m'thmoment of the local stress response process.

Modelling of the uncertainties associated with the fatigue capacity, ∆, is based on results fromrandom loading fatigue tests. Because the fatigue behaviour is influenced by many factors,among them the variability inherent in the material, it is difficult to interpret the test results.However, there seems to be some coherence to recent published results for welded details, wherea slight non-conservative bias is suggested implemented with uncertainties around 30-60%.

Experimental data suggests that the Miner-Palmgren rule predicts fatigue failure reasonable wellfor random loading on loaded components, and that the influence of sequence effects is usuallynegligible for random loading typical for offshore structures.

However, for welded joints it appears that the Miner-Palmgren rule is slightly non-conservative.Biases, in the ratio between the predicted damage and the measured damage, down to 0.7 to 0.8have been observed.

The response process in the estimation of the fatigue damage accumulation is usually assumed tobe a narrow banded Gaussian process. However, the fatigue stresses may typically be somewhatwide banded and a rainflow correction factor for wide banded response processes may thereforebe introduced.

The rainflow correction factor for wide banded processes indicates a compensating biascompared to the bias introduced due to the random loading on welded joints applying the Miner-Palmgren rule. Therefore, the Miner-Palmgren damage is in the following used unbiased and norainflow correction factor is included.

The uncertainty due to this phenomenon as well as model uncertainties are accounted for bymodelling fatigue failure to occur when the total damage D exceeds ∆, where ∆ is defined asstochastic, for which the Normal distribution is recommended with:

[ ] [ ]E CoV∆ ∆= =10 0 20. .




4.4.2.5 Limit State FormulationThe limit state function applied in the reliability analysis is expressed as,

g D D( , )∆ ∆= −

The random variable ∆ describes general uncertainty associated with the fatigue capacity and Dis the accumulated fatigue damage.

Defining the mean number of stress cycles per time unit to be ν 0,long term , the total accumulatedfatigue damage in a service period T can be expressed as

D T Dlong term cycle= ⋅ ⋅ν 0,

Dcycle is the expected damage per stress cycle, which depends on the distribution of the localstress range response process and the associated SN-curve.

Applying a bi-linear SN-curve (e.g. the T-curve) and assuming the stress range distributionwithin each sea state j , to be Rayleigh distributed, the expected damage per stress cycle in seastate j is taken as:

( ) ( )DK

Stm S

St K

m S

Stcycle j j

m

j

m

j

= ⋅ +�

�

��

�

�

��

�

�

��

�

�

��

+ ⋅ +�

�

��

�

�

��

�

�

��

�

�

��

12 2 1

2 2 2

12 2 1

2 2 22

2 2 0

2

0

2

( ) ;( )

;( )

σ γσ

σσ

Γ

where γ(; ) and Γ (; ) are the Incomplete and Complementary Incomplete Gamma functions,respectively, and St j ( )σ is the standard deviation of the stress process in sea state j .

The expected damage per stress cycle Dcycle is obtained by summing the weighted expecteddamage over all sea states, weighted by the relative number of stress cycles within each sea state.

For a Weibull distributed long term stress range distribution, the expected damage per stresscycle is calculated as:

DK

Am

B

S

A KA

m

B

S

Acycle

mB

mB

= +�

��

�

��

�

��

�

�� + +

�

��

�

��

�

��

�

��

11

11

2

2 2 0 0γ ; ;Γ

where A and B are distribution parameters in the Weibull distribution,

( ) ( )[ ]F s s AsB= − −1 exp /

4.4.3 The FM-Approach for Fatigue Assessment

4.4.3.1 GeneralThe damage D calculated by the SN fatigue approach and the Miner-Palmgren rule is a damagemeasure not related to any physically or measurable parameter. However, the size of thedeveloped fatigue crack may be applied as a measurable quantity to reflect the degree of fatiguedamage accumulation when the FM-fatigue approach is applied.

Applying the developed crack size as a measure for the fatigue damage accumulation, the extentof fatigue damage on the structure between the initial condition (design) and the failure condition




can be related to this physical measurable parameter. The degree of accumulated fatigue damagein a joint can then be assessed based on the outcome of inspections aiming at determining thesize of fatigue cracks in the joint.

4.4.3.2 Crack Growth RateThe basis for most fracture mechanics descriptions of crack growth is a relationship between theaverage increment in crack growth and the range ∆K of the stress intensity factor K during aload cycle. The factor is defined as the stress intensity because its magnitude determines theintensity of the stresses / strains in the crack tip region. The influence of external variables, i.e.the magnitude and type of loading and the geometry of the cracked body, is modelled in the cracktip region through the stress intensity factor.

The relationship between the crack growth rate and the stress intensity range ∆K has to bedetermined experimentally. Fatigue experiments are normally performed with simple standardspecimens with through-the-thickness cracks subjected to constant stress range.

The main motivation for applying a crack growth model where the stress distribution through thethickness is taken into account is that it has been observed that the propagation of fatigue cracksdepends highly on the stress distribution. The crack propagation depends further significantly onthe initial size and the initial aspect ratio of the crack.

In order to predict the fatigue crack growth of a surface crack, it is assumed that the crack growthper stress cycle at any point along the crack front follows the Paris and Erdogan equation. Thisequation states that, at a specific point along the crack front, the increment in crack size dr( )ϕduring a load cycle dN is related to the range of the stress intensity factor ∆Kr ( )ϕ for thatspecific load cycle through

dr

dNC K

r r

m( )( )( ( ))

ϕϕ ϕ= ∆

where Cr ( )ϕ and m are material parameters for that specific point along the crack front and ϕis the location angle.

To simplify the problem it is assumed that the fatigue crack initially has a semi-elliptical shapewith axes a and c, and that the shape remains semi-elliptical as the crack propagates. Thisimplies that the crack depth parameter a and the crack length 2c are sufficient parameters fordescribing the crack front. As a result of this simplification in the modelling of the crack frontcurvature, the general differential equation for the crack growth rate can be replaced by twocoupled differential equations,

da

dNC K m K K a N aA A A th= > =( ) ; ; ( )∆ ∆ ∆ 0 0

dc

dNC K K K c N cC C C th

m= > =( ) ; ( );∆ ∆ ∆0 0

The subscripts A and C refer to the deepest point and the end point of the crack at the surface,respectively. The material parameters CA and CC may differ due to the general triaxial stressfield. The material property m mainly depends on the fatigue crack propagation, assumed to beindependent of the crack size, both in the depth and surface directions. Normally the failure




criterion refers to a critical value of the crack depth a or the crack length 2c (for surface cracks);and the equations are conveniently rewritten as:

dc

da

C

C

K

Kc a cC

A

C

A

m

=�

�

��

�

�

��

=∆

∆; ( )

0 0

( )dN

da C KN a N

A A

m= =

10 0

∆; ( )

By either fixing the aspect ratio or by expressing the crack length as a function of crack depth,the first of the above equations is reduced to a constant, and the second equation can be solvedseparately. This is referred to as one dimensional crack growth model.

The general expression for the stress-intensity factor is K Y S atot= ⋅ π , where Stot is the appliedstress and Y is the geometry function accounting for the effect of the boundaries, i.e. the relevantdimensions of the structure (width, thickness, crack size, crack front curvature etc.).

4.4.3.3 Crack Size over TimeSince the stress intensity factors in the two-dimensional expression for the crack growth ratedepend on the crack size in a complicated manner, it is generally not possible to obtain a closedform analytical solution to the coupled differential equations, and numerical solution procedureshave to be applied to solve the coupled ordinary first order differential equations.

In the following, the equivalent one dimensional crack growth model is applied for illustrationpurposes only. (For a fixed aspect ratio a c/ , or by expressing the crack length as a function ofcrack depth, an equivalent one dimensional crack growth model can be defined).

The crack growth rate, or the increment in the crack size per stress cycle, is for one-dimensionalcrack growth in the depth direction expressed as,

( )dadN

C K a c m= ⋅ ∆ ( , )

The variables in the differential equations may, for crack growth models not having a lowerthreshold, be separated and integrated to give,

( )( )da

Y aC S

m ma

a t

im

i

N t

π0 1

( ) ( )

� �= ⋅=

∆

where a t( ) is the crack depth at time t and N t( ) is the total number of stress cycles in the timeperiod [ ]0, t .

The number of stress cycles to fatigue failure until a critical crack size resulting in e.g. unstablefracture or plastic collapse is reached, is for offshore structures generally large, and the sum ofthe stress ranges can be expressed using the m’th moment of the stress range distribution.

The damage accumulation from the stress response process can then be expressed as,

[ ]ψ( ) ( )a C N t E SNm= ⋅ ⋅ ∆




where the term ψ( )a N is an indicator of the damage accumulated by the crack growth from aninitial crack size value a0 to a crack size aN after N stress cycles

( )ψ

π( )a

da

Y aN m m

a

aN= �

0

For variable amplitude loading, the sequential order of loads may have an influence on the crackgrowth rate, however, the sequence effect is typically of minor importance for offshorestructures.

The above model can be directly extrapolated to be valid also for crack growth models involvingthreshold levels on the stress intensity factor. Special attention then has to be made in thederivation of the m'th moment of the stress intensity range as the stress intensity is a function ofboth the crack size and the stress level. For crack growth models involving thresholds, thedamage indicator can be expressed as

( )ψ

π( )

( )a

da

G a Y aN m m

a

aN=

⋅�0

where G a( ) is a reduction factor in the range [ ]0 1− , depending on the threshold level ∆Kth andthe stress range process ∆S .

When the long-term distribution of stress ranges ∆S is defined through a Weibull distribution,with scale parameter A and shape parameter B, the m'th moment of the stress range is

[ ]E S AmB

m m∆ Γ= ⋅ +�

��

�

��1

and the reduction factor G a( ) can be shown to be,

G a

mB

KA Y a

mB

th

B

( )

;

=

+⋅

�

��

�

��

�

�

��

�

�

��

+�

��

�

��

Γ∆

Γ

1

1

π

where Γ( ) and Γ( ; ) are the gamma function and the complementary incomplete gammafunction, respectively.

For a stationary Gaussian stress range process, a good approximation in the accumulationanalysis is obtained by replacing the Gaussian process with an equivalent ideal narrow bandedprocess with the same spectral moments λ 0 and λ 2 . The m'th moment of the stress range thenbecomes,

[ ] ( )E Smm m

∆ Γ= +�

��

�

��2 2 1

20λ

and the mean number of stress cycles in a time period T is equal to the mean number of up-crossings of the mean stress level in that period




N T TT = = ⋅νπ

λ

λ0

2

0

1

2

4.4.3.4 Fatigue QualityOne element affecting the fatigue life of a component is the initial fatigue quality. The initialfatigue quality is a material and manufacturing property, thus representing material and processdefects such as inclusions, as well as damage caused during fabrication and installation which isnot detected by quality control.

For the purpose of design, the initial fatigue quality can be characterised by the initial crack sizeand/or the time to fatigue crack initiation. The initial crack size a0 and the time to crackinitiation T0 (or number of load cycles N 0 ) are often not well-known parameters and shouldtherefore be considered as random variables with a certain statistical distribution.

The time to crack initiation is being defined as the time from beginning of fatigue loading to thetime of possible crack detection. Data on N 0 for welded steel offshore structures, which arecoupled to a crack size a0 are sparse. However, for welded structures a common approach is toneglect the crack initiation time due to the presence of initial weld defects.

Cracks existing in the structure entering service include defects considered acceptable accordingto codes, as well as those undetected during fabrication and installation. It is a formidable taskcarrying out calculations allowing for the occurrence of defects in all shapes, locations andorientations which might arise, and it is common practice to simplify the modelling by assumingthe cracks to be of the same type, i.e. undercuts oriented normal to the principal stress at thelocation, or that they can be grouped. Since planar defects, like lack of penetration, undercuts,etc. are similar in nature to a crack, the number of cycles to initiate a fatigue crack from suchdefects is small compared to the overall life.

Surface defects are usually more dangerous than embedded defects as they are often located atstress concentrations and normal to the principal stress. Experience has shown that almost allfatigue cracks resulted from an initial surface defect.

4.4.3.5 Fatigue Crack Growth Material ParametersFatigue tests indicate a considerable amount of scatter in the obtained fatigue capacities, which isbelieved to be a result of inhomogeneous material properties. However, fatigue cracks associatedwith welded joints may propagate through different materials, i.e. the weld metal, the heat-affected-zone (HAZ) or the base (plate) material. For welded joints, cracks often initiate at theweld toe from undercuts, slag inclusions and/or initial cracks, and propagate through HAZ andinto the base material. Thus, the crack growth data used for fatigue life predictions must berepresentative, concerning inhomogeneous material and differences in material properties.

Relevant crack growth data for welded joints should be expressed through the materialparameters m and C in the Paris equation. Crack growth data are generated in the laboratoryunder constant cyclic loading on simple specimens with accepted characterising stress intensityfactors. The challenge is to define reasonable distributions for the material parameters and toestimate the distribution parameters based on available laboratory test results.




Several studies have indicated a high negative correlation between m and lnC. However, thechoice of randomisation should be based on a judgement of what is a reasonable representationof reality. The probability model implicit in the recommendations for Paris constants in BSI PD6493 (1991) and DNV (1984) is based on a deterministic m and randomised C. It is important forthis description that the typical crack growth rates (low and intermediate) are adequately fitted.

In Table 4.6, published values for lnC and m are given according to DNV (1984). The values arebased on collected data from various investigations and are recommended when other relevantinformation is not available.

Table 4.6 Modelling of lnC and m in Paris equation. Units [N, mm]

Environment m lnC (mean, std.dev.)

In air and non corrosive 3.1 Normal(-29.84, 0.55)

In sea water 3.5 Normal(-31.01, 0.77)

It should be noted that the values for lnC given above are only valid for the units [N, mm], andthat it is necessary to adjust the values for other units. The most typical conversion is from mm tom:

[ ] [ ]ln( ) ln( ) ( . ) ln( )C C mN,m N,mm= − ⋅ + ⋅15 1 1000

Offshore structures are subjected to numerous cycles in the low crack growth rate regime, and itis therefore of importance to establish the threshold values, ∆Kth , below which stress intensitythe crack is non-propagating. Considerable scatter has been reported for the modelling of thethreshold level, and for stress intensities close to the threshold level the crack growth rates arefound to be sensitive to the mean stress and environmental factors.

4.4.3.6 Limit State Formulation / Failure CriteriaMany uncertainties are related to the fatigue life predictions of offshore structures, both forapplication of SN-curves, the Miner type cumulative damage models, and the fracture mechanicsbased models. Uncertainties in the loading conditions, the material parameters, the initial fatiguequality and the stress intensity factor have to be considered. In probabilistic fracture mechanicsthese parameters are represented by random variables.

Reliability assessments for fatigue crack growth can be expressed as limit state formulations. Thefailure criteria may be defined as,

a aC N− ≤ 0

where aC (or cC ) is the critical crack size based on serviceability criteria, e.g. through thethickness crack or economic repair limits, or ultimate collapse criteria, i.e. unstable fracture andplastic collapse, or buckling that significantly reduces the static strength. aN is the size of thedeveloped crack after N stress cycles.




As the damage indicator Ψ( )a is a monotonically increasing function of the crack size, failureoccurs when the damage indicator for the critical crack size Ψ( )aC is exceeded by the

accumulated load effect C Sim

iN⋅ =� 1 . The failure criterion can then be written as

( )Ψ Ψ( ) ( )

( )a a

Y x xdx C SC N m m

a

aC

im

i

N− = − ⋅� �

=

1

0 1π

and the safety margin M is defined as

( )M

Y x xdx C S

m ma

aC

im

i

N= − ⋅� �

=

1

0 1( ) π

The failure probability, i.e., the probability that the size of the crack exceeds a critical limitwithin the time period T (or N) is then,

P P MF = ≤( )0

4.4.4 Load and Response Modelling

4.4.4.1 GeneralThe major time varying loads on jacket structures are generally wave induced loads. An adequatedescription of ocean waves is therefore necessary for assessing the fatigue accumulation in thestructure.

The long-term stress range response distribution is defined based on a weighted sum of Rayleighdistributed stress ranges within each short-term condition, i.e. the stress process for each short-term period is considered to be a narrow banded zero-mean stationary Gaussian process.

In the spectral fatigue analysis, only the load response caused by fluctuating wave loading isconsidered. The applied wave model assumptions do not give an exact description of the real seastate. However, from an engineering point of view they are very attractive due to thesimplifications they imply in the structural analysis.

This chapter focuses on the load and response modelling applied for fatigue assessment. First thesea environment model is considered. Then the load response model and the global structuralanalysis, defining the transfer functions for selected forces, are described. Finally the local stressanalysis is discussed. The sources of uncertainty and their treatment are also discussed.

4.4.4.2 Sea State DescriptionThe load model is based on a description of the wave conditions within a set of stationary shortterm sea states. Each sea state is characterised by

• Main wave direction θ0 , measured relative to a given reference direction

• Characteristic sea state parameters:

- Significant wave height, HS , defined as the average of the upper third of the wave heights

- Mean zero up-crossing period, TZ , defined as the time between successive up-crossing ofthe still water level, averaged over the number of waves.




• Wave spreading function

• Wave spectrum model e.g. PM or JONSWAP spectra

For each sea state, the long-term probabilities of the different main wave directions are givenalong with a wave scatter diagram for each direction. A wave scatter diagram defines theoccurrence probability for each set of HS and TZ values.

A unique wave spreading function may be assigned to all, or a subset, of the wave-statisticsdefined by each assigned scatter diagram.

Main wave directions:Sets of wave observations may be sorted with respect to the main wave directions ifdirectional buoy or hindcast data are available with statistics on the observations for differentsectors. Otherwise, the statistical properties for the waves may be assumed identical for allsectors. The main wave direction denotes the middle direction for each of the defined sectors,and the structural analysis is for simplicity only performed for waves at these discretedirections. Each main wave direction i is defined by the incoming wave direction angle θi ,measured relative to a given reference direction, defined as the structures global x-axis. Anexample of sector numbering and main wave directions is shown in Figure 4.7.

N

Reference direction(Global X-axis)

N

E

S

W

8 1

7

6

4

3

2

5

Main wave direction

no. 2

θ2

Wave spreadingfunction

a) b)

θ θ− 2

w( , )θ θ2

- 90

- 45

0

45

90

θ

Reference direction(Global X-axis)

Figure 4.7 a) Example of sector numbering.

b) Main wave direction in the structure co-ordinate system.

The main wave directions are given by a set of prescribed discrete directions. The probabilitydistribution of the main wave direction is given as a discrete distribution with

Piθ≡ probability that the main wave direction is θi , i=1,2,.., Nθ

where Nθ is the number of possible main wave directions, and




Pi

i

N

θ

θ

==� 1

1

Eight different main directions are considered i.e. Nθ =8 .

Wave scatter diagram:The scatter diagram gives the occurrence frequency of a discrete number of combinations of( HS ,TZ ), where the scatter diagram is commonly defined on a bi-variate discrete form. Thediscrete values of ( HS ,TZ ) data may be approximated by an analytical bi-variate distribution,e.g. a joint log-normal distribution,

( )F h th t

H T s zs z

S Z,

log,log

;=− −�

��

�

��Φ

µ

σ

µ

σρ1

1

2

2

where ( )Φ ; is the cumulative distribution function for a pair of standardised normallydistributed random variables with a correlation coefficient ρ . The marginal distribution forHS is

( )F h hH s

sS

= −�

��

�

��Φ log µ

σ1

1

and the conditional distribution of TZ given the value of HS is (see Figure 4.8)

( )( )

F t h

t h

T H z s

z s

Z S|

log log

,=

− + −�

��

�

��

−

�

�

��

�

�

��

Φ

µ ρσ

σµ

σ ρ

22

11

221




HS

TZ

hS jhSi

p t hT H Z SZ si i| ( | )p t hT H Z SZ sj j| ( | )

Figure 4.8 Marginal continuous probability density function for HS with continuousprobability density function for TZ given HS

The distribution function is thus specified by 5 parameters ( )µ σ µ σ ρ1 1 2 2, , , , and these areuniquely related to the moments of ( HS , TZ ) as

[ ]E HS = +�

��

�

��exp µ σ

112

2

[ ]E TZ = +�

��

�

��exp µ σ

222

2

[ ] [ ] ( )( )Var H E HS S= −212 1exp σ

[ ] [ ] ( )( )Var T E TZ Z= −222 1exp σ

[ ] [ ] [ ] ( )( )Cov H T E H E TS Z S Z, exp= −σ σ ρ1 2 1

From the available wave scatter diagram, the best estimates ( )� , � , � , � , �µ σ µ σ ρ1 1 2 2 for

( )µ σ µ σ ρ1 1 2 2, , , , are obtained. When different scatter diagrams are applied, i.e. separately foreach main wave direction, the fitting should be made separately for each wave direction.

Wave spreading function:The wave energy spreading function is introduced to account for the energy spreading amongdirections for a short crested sea. Real sea waves are not infinitely long crested and directionalspectra are required for a complete statistical description of the sea. The directional spectraaccounts for the spreading of wave energy by direction as well as frequency. A spectrum interms of direction θ is assumed of the form




( ) ( ) ( )S S wη ηω θ ω θ, =

where ( )w θ is the wave energy spreading function, which is herein assumed independent ofthe wave frequency.

It is commonly assumed that the wave energy is spread over a set of directions in a region ofπ / 2 on both sides of the main direction. The function is selected in such a way that it giveshigher weights to the directions closer to the main direction. For a long crested sea the waveenergy spreading is not introduced by definition. The wave energy spreading function for agiven main wave direction θi may in general depend on ( HS ,TZ ).

The common modelling of wave energy spreading function is a frequency independent cosinepower function of the form:

( ) ( )w

N

NiN

i iθ θπ

θ θ θ θ π, cos=+�

��

��

+��

��

− − <1 21

212

12

Γ

Γ

and zero otherwise. ( )Γ ⋅ is the gamma function, θi is the main wave direction no. i, and N is areal number. Figure 4.9 shows the directional function for different values of N . For largevalues of N , all the energy is concentrated around the main wave direction.

θ θ− i ( )degree

N = 20

N = 10

N = 4

N = 2

-90 -45 0 45 90

c = 0.5

Figure 4.9 The spreading function for different values of the cosine power N.

The spreading function weights are obtained by integration of the energy spreading functionover the proper ranges. The analytical spreading function is discretised. The analyticaldirectionality function is approximated by a histogram. The ordinate of each histogram boxcorresponds to the area of the analytical function over the width of the box.

Wave spectrum model:The wave spectrum defines the distribution of wave energy over different frequencies for aspecified sea state. A commonly applied wave spectrum is the one side Gamma spectrum,where the Gamma spectrum is uniquely defined in terms of the sea state parameters ( )H TS Z, ,

( ) ( )S A Bηξ ζω ω ω ω= − >− −exp ; 0




The gamma spectrum may have a variety of shapes depending on the values of the parameterξ giving the power of the high frequency tail and the parameter ζ describing the steepness ofthe low frequency part. The constants A and B are related to HS and TZ by,

A HTS

Z

=�

��

�

��

−�

��

�

��

−�

��

�

��

−

−

−

116

21

3

1

32

12

ζ πξ

ζ

ξζ

ξ

ξ

ξ

Γ

Γ

BTZ

=�

��

�

��

−�

��

�

��

−�

��

�

��

21

3

2

2

πξ

ζ

ξζ

ξ

ξ

ξ

Γ

Γ

The values ζ = 4 and ξ = 5 yields the PM spectrum (Pierson and Moskowitz 1964).

4.4.4.3 Global Structural AnalysisThe structural response to wave induced loading may be determined by the use of finite elementmethods (FEM). This includes modelling of the structural stiffness, the damping (only fordynamic analysis), the influence of marine growth, the stiffness from the foundation and thewave induced loading.

The finite element model is an idealised representation of the real structure, where the followingsimplifications are commonly introduced,

• Smaller eccentricities are not modelled.

• Eccentricities in the joints are often not modelled.

• The marine growth is not included in the calculation of the natural frequencies.

• The jacket is modelled as a frame with members connected at rigid joints. In reality the jointsare flexible, and on the global level the joint flexibility is known to have some influence on thederived response, (Appendix C DNV (1977), Bouwkamp et al. (1980), Fessler and Spooner(1981), UEG (1984)). The joint flexibility affects the bending moments in braces, the axialforce distribution and the natural frequencies.

Wave load calculation:The linear Airy wave theory is adopted for fatigue analysis. In the Airy theory, the waterparticle velocity and accelerations are linear with wave amplitude. The linear wave theory isbased on the assumption that the wave height is much smaller than both the wave length andthe water depth.

Hydrodynamic loading on the jacket structure is calculated by Morison's equation, (Morisonet al. (1950)), not incorporating the structural motion. The in-line force p per unit length on avertical slender cylinder in unsteady flow is defined as,

p u u u= +CD

CD

d n n m nρ ρπ2 4

2

�

where ρ is the water density, D is the diameter, un and �u n are respectively the water particlevelocity and acceleration normal to the cylinder, and Cd and Cm are the drag and inertiacoefficients, respectively.




The uncertainty or bias introduced in the fatigue damage calculation using the linear Airywave theory is generally not significant for most structures, water depths and wave climatesconsidered.

Structural analysis:The major element of the frequency domain analysis is the determination of the response ofthe structure for a unit sinusoidal wave as function of the wave period, or angular frequency.This function is called the response transfer function, ( )H Fη ω .

The response transfer functions for section forces and moments in each beam end are derivedfor different wave directions, analysing the structure subjected to waves of different angularfrequencies.

The angular frequencies, or wave periods, should be selected in order to adequately define thetransfer function over the expected range of wave energy. Special care to be given in themodelling of the transfer function for wave periods close to the eigenperiods of the structure.

The relationship between the wave height and wave induced force is non-linear due to thedrag term in the Morison equation. To incorporate this non-linearity in a linear analysis, twobasic approaches exist.

One is to linearise the drag force and compute the response based on the linearised load, i.e.wave height linearisation.

The other approach is to compute the response using the non-linear force and then linearisethe response in one sea state, i.e. stochastic linearisation, (Borgman (1967)).

When a stochastic linearisation is applied, the influence of applying different sea states forthe linearisation should be considered.

The linearisation of the drag term introduces uncertainties in the response modelling formembers where the drag load is of importance. However, for the range of the waves mainlycontributing to the fatigue accumulation, the inertia forces are dominating for jacketstructures, and the relationship between the wave height and load response is approximatelylinear for the major part of the elements.

The linear wave theory does not account for the fluctuating water surface due to the passage ofwaves and is strictly applicable only up to the still water level (SWL). The use of a linearapproach can, therefore, not define realistic forces around the still water level. Variousmethods have been suggested to modify the linear wave theory to incorporate the variablesubmergence effect, e.g. (Chakrabarti (1971, 1976), Wheeler (1970), Hogben et al. (1977)). Itmust be expected that the establishment of transfer functions for these elements is associatedwith large uncertainties.

The uncertainty/bias introduced could be related to the significant wave height HS , e.g. bymultiplying the calculated transfer functions ( )Hcalc ω obtained in the structural analysis, by a2nd order polynomial function in HS , i.e. the applied ( )Happl ω transfer functions for a given

sea state ( )H TS z, is expressed as:

( ) ( ) ( )H H X X H X Happl calc a b S c Sω ω= ⋅ + ⋅ + ⋅ 2




where the parameters, Xa , Xb and Xc , define the uncertainty/bias in the transfer functionsdue to the applied wave theory. If no information is available for the uncertainty/bias in thecalculated transfer function, Xb and Xc should be set equal to zero (0.0) and the mean valueof Xa should be set equal to one (1.0).

4.4.4.4 Local Stress CalculationThe global FEM analysis discussed above yields the transfer functions H Fiη ω( ) for sectionforces and moments F ti ( ) in each beam end, e.g., for axial force, in-plane and out-of-planebending moments. These end reactions are used to calculate the nominal stresses in the braces.The nominal stresses from the global analysis are scaled with the Stress Concentration Factors(SCF) to account for local geometrical effects.

Existing design codes, e.g. (DNV (1984), AWS (1984), DoE (1984), API (1991, 1993),), usedifferent definitions of SCF. The hot-spot stress is here defined as: the greatest value around thebrace/chord intersection of the extrapolation to the weld toe of the geometric stress distributionnear the weld. This hot-spot stress definition incorporates the effects of the overall geometry butomits the stress concentrating influence of the weld itself which results in a local stressconcentration.

Parameteric formulas exist only for simple joints with members in one plane (e.g. Efthyminu(1985, 1988), Kuang et. al (1977)). In real structures one finds very few of these simple joints.No reference is made to sign, location, or orientation of the stress values representative of theSCFs. Little information is available on SCFs in overlapping and/or multiplanar and/or groutedand/or ring stiffened joints. An inherent shortcoming of the available SCF equations for K-jointsis that they were derived under balanced axial forces or self-equilibrated bending moments.Experimental work performed by (Dijkstra and de Back (1980)) shows that the SCFs are highlydependent on the type of loading on the individual member.

A comparison between various parameteric formulas available for an axially loaded T-joint atthe chord saddle (Lalani et al. (1986)), demonstrated that significant differences existed.

In general, there are six load cases for each free end. However, it is common approach in thefatigue assessment of jackets to neglect the effect of the torsional moments and the shear forcesin the analysis.

The hot-spot stress may be calculated as (DNV (1993b)):

σ hot ax ipbipb

local opbipb

localSCFN

ASCF

M

Iz SCF

M

Iy= ⋅ − ⋅ ⋅ − ⋅ ⋅' '

where

N the axial force in the brace

Mipb the bending moment in the brace about the IPB-axis

Mopb the bending moment in the brace about the OPB-axis

A the cross section area

I the moment of inertia for the pipe section




y zlocal local' , ' the co-ordinates of the stress point relative to the section centre of gravity, inthe in-plane/out-of-plan axis system

SCFax SCF for axial stress

SCFipb SCF for in-plan bending stress

SCFopb SCF for out-of-plan bending stress

It is a common practice to check the fatigue life of 8 points along the brace/chord intersection,i.e. the SCFs are calculated for eight locations around each brace/chord intersection.

Based on the transfer functions H Fiη ω( ) for all section forces (i.e. i=1: axial force, i=2: in-planbending moment and i=3: out-of-plan bending moment), the cross section properties and theSCFs, the spectral density of the hot-spot stress in a unidirectional sea state is defined from:

S I I H H Siji

j F Fi jσ η η ηω ω ω ω( ) ( ) ( ) ( )*= � ⋅� ⋅ ⋅ ⋅== 1

3

1

3

where the asterisk denotes the complex conjugate and

ISCF

Aax

1 =

ISCF

Izipb

local2 = '

ISCF

Iyiob

local3 = '

The parametric formulas for SCFs do not provide information about the variation of SCFsalong the intersection brace/chord. This lead to uncertainties in the estimation of the real hot-spot stress when the maximum resulting stress due to axial force, in-plane and out-of planebending moments is to be defined.

Because the position of the hot-spot is not known, a common procedure is to add themaximum stresses derived separately from the axial and bending loads in order to obtain thehot-spot stress. Such an approach will usually result in conservative estimates of the hot-spotstresses. The degree of conservatism depends on the actual geometry and the contribution ofbending stresses to the total hot-spot stress. In order to reduce the degree of conservatism, the

Uncertainty associated with the modelling of the SCFs may be defined in two levels:

• The first level is one single common uncertainty factor on all the stress concentrationfactors. The uncertainty in stress concentration is due to the fabrication inaccuracies andapproximations made in the stress calculation or joint classification.

• The second level is uncertainty on the SCFs for each degree of freedom, i.e. for axial load,in-plan bending moment and out-of-plan bending moment.




4.4.5 Stress Range DistributionThe fatigue strength is expressed in terms of the number of stress cycles N of stress range Sleading to failure. Statistics for the number of stress cycles and the stress range distribution mustconsequently be produced from the statistical description of the hot-spot stress process.

For a random stress process the definition of stress cycles is not unique and several differentmethods can be applied to define the number of stress cycles, including the peak countingmethod, the range counting method and the rain flow counting. A detailed description of themethods can be found in e.g. (Madsen et al. (1986)).

For a narrow band stress process there is only one maximum between two consecutive upcrossings of the mean level making the identification of stress cycles straightforward. The threementioned counting methods also give identical results for such a process. For a wide bandprocess or a process with a multi-mode spectral density function, there may be several localmaxima between one up crossing of the mean level and the following mean up crossing, and thethree counting methods will give different results.

The rain flow counting method gives generally results which are in better agreement withexperimental results than the peak counting method which tends to overestimate the fatiguedamage, and the range counting method which tends to underestimate the fatigue damage. Therain flow counting method is, however, used for time series of the stress process and has not yetbeen formulated for a response described by its spectral density function, even for a Gaussianstress process.

For offshore jacket structures with insignificant dynamic amplification, the hot-spot stressprocess tends to be narrow-banded as the wave loading is reasonably narrow banded and thestructure behaves in a quasi-static manner. However, cancellation effects may give rise to abimodal or even multi modal spectral density functions for the response.

Jacket structures which have a large resonant component in the response also have narrowbanded response as the damping ratio generally is small. It is therefore believed that there is notintroduced any error of importance by assuming the stress response process to be narrow bandedand applying the peak counting method.

The calculation of the fatigue life involves an estimation of the total number of stress cycles andthe "crack driving force", i.e. the m'th moment of the stress range distribution, E S m[ ] . Applyingthe peak counting method, the stress range is defined as two times the peak value and the numberof stress cycles is equal to the number of up crossings of the mean level. The mean number ofstress cycles for a stationary stress process in a time period T is then

N TT = ν 0

where ν0 is the mean up-crossing rate of the process. For a narrow banded Gaussian stressprocess the stress ranges are Rayleigh distributed with the distribution function

F ss

StdsS ( ) exp

( ( ));= − −

⋅

�

��

�

�� >1

80

2

2σ

where Std( )σ is the standard deviation of the stress process.

The m’th moment of the local stress range response process E S m[ ] is calculated as :




E S Stdmm m m[ ] ( ( )) ( ) ( )= ⋅ ⋅ +σ 2 2 12

Γ

Knowing the spectral density of the stress response, ν0 and Std( )σ are:

νπ

λ

λ0

2

0

1

2=

( ( ))Std σ λ20=

where λ i is the i'th spectral moment, defined as:

To estimate long term properties for E S m[ ] and ν0 , the long term distribution of sea states( H TS Z− ) must be taken into account.

The long term distribution of stress ranges is obtained as the weighted average of the short termdistributions, weighted with the relative number of stress cycles within each specific short termsea state. Each sea state is described by the significant wave height Hs , the mean zero crossingperiod Tz and the mean wave direction Θ . For each hot-spot, the short term stress rangedistribution function for the i'th sea state is defined as F s H TS s z i( , , , )Θ , where FS ( )⋅ is given by theequations above with standard deviation σ depending on ( , , )H Ts z Θ .

The fraction of sea states with the i'th combination of ( , , )H Ts z Θ is denoted by qi , and the meanzero crossing frequency for the stress process with this sea state parameter combination isdenoted as ν0,i . The long term mean zero crossing frequency is

ν ν0 0, , long term = ⋅�� qi iTzHs Θ

since the sum of the weights qi is unity. The expected number of stress cycles in a time period Tis obtained by multiplying ν 0, long term by T.

The long term distribution of stress ranges may now be determined as

F q F s H TS i iTzHs

S s z i,,

, ( , , , ) long term long term

= ⋅ ⋅�� ⋅1

00

νν

ΘΘ

This long term distribution function is of a somewhat complicated form and requiresconsiderable computation time.

When performing the updating of the estimated fatigue reliability based on the outcome ofinspections, there are usually no observations available of the environment, loads or response ofthe structure. It is therefore an unnecessary complication to apply a rather detailed load model inthe reliability updating based on inspections of cracks. A more computational advantageousprocedure is instead to model the load in terms of a long term stress distribution at each hot-spot,where the applied long term stress distribution is derived from the detailed analysis. A fit of thecomputed long-term distribution to a simpler distribution may be used.




The two parameter Weibull distribution has been confirmed in many studies to provide the bestfit for such a long term distribution of the load response. The load model with a large number ofrandom variables can be simplified to a model with only two uncertain distribution parametersapplying a long term Weibull stress range distribution:

The task is then to determine the values for the distribution parameters which result in a closeagreement between the original and the fitted distribution, where the goodness of the fit is judgedby the difference in the estimated fatigue damage by using the two approaches.

Experience shows that the value of the shape parameter B is typically in the area between 0.8 and1.2, with the lower values for drag loading dominated structures and the higher values for inertialoading dominated structures.

The estimated fatigue damage D within a time period T having a Weibull long term stress rangedistribution and a fatigue capacity defined through a SN-curve (not accounting for possiblethreshold or change in slope) is given as

[ ]DT

KE S

T

KA

mB

m m=⋅

⋅ =⋅

⋅ ⋅ +�

��

�

��

ν ν0 0 1, , long term long term Γ

The Weibull distribution parameters can be determined by fitting the Weibull distribution at twofractile levels. The fractiles corresponding to these two levels define three stress range intervals.An intuitively good choice for selecting the two fractile levels for which the fitting is to bedetermined is the fractile levels dividing the contribution to the fatigue damage into three equalintervals.

With the SN-curve slope parameter m = 3 and the Weibull shape parameter B = 1 , such adivision is obtained for the 95'th and 99'th percentile fractile levels. Defining the correspondingstress values for the original long term stress range distribution s95 and s99 , the Weibullparameters A and B for the fitted distribution are then

Ak s a

k=

−

−�

��

�

��exp

ln ln. .0 99 0 95

1

Bs A

=−

−

ln( ln . )

ln ln.

0 05

0 95

where

k =−

−=

ln( ln . )

ln( ln . ).

0 05

0 010 718

Experience shows that the fit is quite stable for varying choices of the fractile level, whichconfirms the goodness in the choice of the Weibull distribution. More elaborate fittingprocedures may involve fitting several fractile levels using with a least square, or other, fittingprocedure.

Uncertainties associated with the original long term distribution can be reflected through astochastic modelling of the fitted long term Weibull distribution parameters by assuming theparameters ln( )A and 1 / B to be e.g. bivariate normally distributed.

The stochastic Weibull distribution parameters can be fitted in an equivalent manner. Knowingthe variance and the second spectral moment of the hot-spot stress within each sea-state and the




long term distribution of sea states for the given structural location, the five quantities [ ]E Aln( ) ,Std A[ln( )] , [ ]E B1 / , Std B[ / ]1 and [ ]ρ ln( ), /A B1 can be estimated based on three cumulativeprobability levels approximately dividing the fatigue contribution into four areas of equalmagnitude, applying two stress levels at each probability level, (Skjong & Torhaug (1991)).

This procedure requires six FORM analyses in order to estimate the five parameters and thusrequires considerable resources, and for some cases an alternative procedure may be required dueto possible convergence problems in the FORM analyses. An alternative and much simplerstochastic fitting procedure is summarised in the following;

• calculate two (deterministic) fatigue lives Tlife−1 and Tlife−2 applying constant slope SN-curvefor two different, m1 and m2 , where the long term fatigue damage is calculated as a sum ofpartial damages within each short term sea state (e.g. a stochastic fatigue analysis using theSESAM software system).

• calculate the equivalent Weibull parameters A and B which give the same fatigue lives, bysolving (numerically) the following two equations with two unknown parameters (i.e. A andB):

TK

A m Blife

long termm− =

⋅ ⋅ +1

01

11ν , ( / )Γ

TK

A m Blife

long termm− =

⋅ ⋅ +2

02

21ν , ( / )Γ

The influence of different choices of m1 and m2 on the estimated value of the Weibull shapeand scale parameters A and B has been studied and is shown to be very limited.

• calculate the probability of failure as function of the service time, applying a probabilistic SN-fatigue approach, where the long term fatigue damage accumulation is calculated as a sum ofthe partial damages within each short term sea state.

• assume B to be deterministic and ln( )A to be Normal distributed with mean values equal tothe value obtained by solving for the two equations above. Calibrate the uncertainty in ln( )Asuch that the probability for fatigue failure over time, applying the Weibull distribution,approximates the results obtained when the long term fatigue damage accumulation iscalculated as the sum of the partial fatigue damages within each short term sea state.

This simple procedure is applied in the application example presented in DNV (1995b).

4.4.6 Formulation of Inspection ResultsThe objective of an in-service inspection plan is to keep the structure at an acceptable safety levelduring its service life with respect to human lives, pollution, operation (production) and costs ofstructure and equipment. The discussion of updating with respect to in-service inspectionspresented in this subsection is limited to fatigue related inspections for a single location.

In-service inspection is performed in order to assure that the existing cracks in the structure,which may have been present from the initial delivery or have arisen at a later stage during theservice time do not exceed maximum tolerable sizes. The reliability of a Non-Destructive




Examination (NDE) is described by the ability to detect an existing crack as a function of thecrack size and by the uncertainty associated with the sizing of an identified crack.

The detection ability as a function of a defect size is defined by the Probability Of Detection(POD) curve,

P x P x( ) ( )= detection of crack

where x is the size of the crack (usually crack length 2c ). It can be shown that the cumulativedistribution function for the smallest detectable crack size is expressed through the POD curve.

In the General Guideline (DNV 1995a) typical POD curves for different inspection scenarios arepresented. The curves are defined on the form,

( )P c

c x b( )/

2 11

1 2 0

= −+

where the values for the distribution parameters x0 and b depend on the inspection scenario.

In Table 4.7 typical values for x0 and b for different inspection scenarios are given. Thecorresponding POD curves are shown in Figure 4.10.

0 20 40 60 80 100 120Crack length (mm)

0.00

0.20

0.40

0.60

0.80

1.00

Pro

babi

lity

Of D

etec

tion

(PO

D)

MPI Under water

MPI Above water; ground test surface

MPI Above water; not ground test surface

Eddy current

Figure 4.10 POD curves for different inspection scenarios.




Table 4.7 POD distribution parameters for different inspection scenarios

Inspection Scenario x0 b

MPI under water 2.950 0.905

MPI above water;ground test surface

4.030 1.297

MPI above water;not ground test surface

8.325 0.785

Eddy Current 12.28 1.790

Regardless of whether or not cracks are detected, each inspection provides additional informationto that available at the design stage which can be used to update the reliability. This can lead tomodifications of future inspection plans, changes in the inspection method, or a decision onrepair or replacement.

When a repair of a detected crack is made it is important to account for the information that arepair was necessary. Often it is not possible to determine if the unexpected large crack size hasbeen caused by a large initial size, by material properties poorer than anticipated, or by a loadingof the crack area larger than anticipated.

The updating based on inspection results can be performed with the stress range distributionsresulting from detailed uncertainty modelling of the environmental conditions (sea scatterdiagram, wave energy spreading and wave spectrum), response transfer function and stressconcentrations.

It is, however, extremely time effective to calibrate a stress range distribution with a smallernumber of random variables. The distribution parameters, A and B, for the approximated long-term Weibull stress range distribution, are calibrated to include the uncertainties described above.

Inspection updating is based on the definition of conditional probability,

P F IP F I

P I( | )

( )

( )=

∩

P F I( | ) is the probability that event F occurs given that event I occurs. For example, if F is thefailure of a structural component and I is the inspection event, then P F I( | ) is the estimatedprobability of failure given the inspection outcome.

4.4.7 Event Margins with Inspections Results:The influence of the inspection results are in the reliability modelling formulated through eventmargins.

An inspection results in either no detection or the detection of a crack. This can be formulated asfollows,

i) 2 2c T ci pod( ) ≤




ii) 2 2c T cj obs( ) =

In the first case, no cracks were found in the inspection after the time Ti , implying that anycracks were smaller than the smallest detectable crack size 2cpod , where 2cpod is defined fromthe POD curve for the applied NDE method.

In the second case, a crack size 2cobs is observed after the time Tj , where 2cobs is random due touncertainties in the crack sizing.

For each inspection which does not result in a crack detection, an event margin Hi , i s= 1 2, ,� ,can be defined similar to the safety margin used to describe fatigue failure, where for a 1-D crackpropagation model may be formulated as:

( ) [ ]HY x x

dx C N E Si m ma

a c pod

im= − ⋅ ⋅�

1

0

2

( )

( )

π

This event margin is positive as the developed crack size is smaller than the detectable cracksize.

For each measurement resulting in the detection of a crack, an event margin H j can similarly bedefined as

( ) [ ]HY x x

dx C N E Sj m ma

a cobs

jm= − ⋅ ⋅�

1

0

2

( )

( )

π

This safety margin is zero as the developed crack size is equal to the observed crack size.

The situation is envisaged where no crack is detected in the first r inspections at a location, whilea crack is detected by the r+1'th inspection and its size is measured at this and the following s-1inspections. The updated failure probability is in this case

P P g H H H HFu

r r r s= ≤ > ∩ ∩ > ∩ = ∩ ∩ =+ +( ( ) | )X 0 0 0 0 01 1 � �

A more general situation involves simultaneous consideration of several locations withpotentially dangerous cracks for which inspections are carried out. The updating procedure stillapplies when due consideration is taken to the dependence between basic variables referring todifferent locations. A more detailed description concerning failure probability calculations ofparallel systems is given in the DNV PROBAN Theory Manual (DNV 1993a).

In addition to inspection, the knowledge that a repair has been performed can be used to updatethe failure probability. When a repair is made at time N rep stress cycles, the crack length a rep ismeasured. The event margin H rep is defined as

( ) [ ]HY x x

dx C N E Sm m

a

am

rep rep rep

= − ⋅ ⋅ =�1 0

0 ( ) π

The crack size present after repair and a possible inspection is a random variable a0,new and thematerial properties after repair are mnew and Cnew . These variables may or may not be the same




as the original depending on the type of repair, grinding or welding repair. The safety marginafter repair is Mnew ,

( ) [ ]MY x x

dx C N N E Sm m

a

am

C

newnew new rep

newnew

new

= − ⋅ − ⋅�1

0 ( )( )

, π

and the updated failure probability after repair is

P P M HFu

rep= ≤ =( | )new 0 0

4.5 Total structural collapse (ULS)

4.5.1 GeneralThe structural behaviour near a total collapse failure can be very complex and expensive toassess. This complexity can be due to the non-linear mechanical behaviour of the structure andthe applied loading and load distribution near failure. The structural behaviour beyond the firstmember-failure, depends not only on the degree of static indeterminacy, but also on the ability ofthe structure to redistribute the load and the post-failure behaviour, e.g. the ductility of theindividual members and joints.

In addition, the probability of system failure depends on the uncertainty of the load, uncertaintyof the member capacity and the correlation between the uncertain parameters. For a perfectlybalanced structure (i.e. the first member failure has the same probability of occurrence for allmembers in a linear analysis, or designed with the same utilisation ratio) the system effects foroverload capacity beyond the first member failure are strictly due to the randomness in themember capacities. In contrast, in a more realistic unbalanced structure the system effects areboth due to deterministic and probabilistic effects. In an unbalanced structure, the utilisation ratiois different for the different members, both in the intact and in the potentially damaged states ofthe structure. Hence, deterministic system effects are present because the remaining members inthe structure can still carry the load after one or several members have failed. In addition,randomness in the member capacities gives a probabilistic contribution to the collapse capacity.A major question is then how important the probabilistic effect is as compared to thedeterministic effect.

Based on results from reliability analyses, the following characteristic features of the ultimateload capacities for offshore structures are identified:

- The uncertainties in the structural capacity are much less than in the loading.

- Due to highly correlated load effects, the different failure sequences for the members arehighly correlated.

- Offshore structures are usually not balanced, which means that there is one or a few failuremodes which dominate.

A complete reliability analysis of a real multi-leg jacket structure with respect to structuralcollapse (overload failure) is very complicated and intractable, and simplifications(approximations) are required.

Experience based on simulation studies (Dalane (1993), Sigurdsson et al. (1994)), of bothbalanced and unbalanced designed jacket structures at water depth about 70 m, has shown that




the ultimate capacity, or collapse capacity, of the structures can be related directly to the totalbase-shear force on the structures. Also the load pattern (e.g. the wave height, the wave periodand the model describing the link between the wave kinematics and the wave forces, i.e. the dragcoefficient, the mass coefficient in Morison equation and marine growth) was shown to haveminor effect on the calculated collapse capacity obtained in the push-over analysis. It is,however, recommended that the sensitivity of the collapse capacity to the wave height and periodis analysed e.g. by performing push-over analyses applying different load patterns (differentwaves). For deeper water, the load pattern will have more effect and the collapse capacity should,if possible, be related directly to the total over-turning moment rather than to the total base-shearforce. However, this will dependent on the structural details being critical, for the legs it is theoverturning moment that is considered and for the braces it is the shear force.

In reliability assessment of jacket structures, the evaluation is generally load driven, i.e. theuncertainties in the load modelling are much greater than in the collapse capacity modelling. Theabove referred simulation studies of collapse capacity of jackets have shown that the coefficientof variation [ ]CoV L for the annual extreme base shear loading L on the structures are about 0.4,while the coefficient of variation [ ]CoV CC for the base-shear capacity of the structures are about0.05- 0.10, dependent on the applied CoV for the yield stress. Furthermore the simulationsshowed that the CoV for the collapse capacity was much smaller than the uncertainty in the yieldcapacity (approximately 50%) and it was shown that the Normal distribution gives an acceptablefit of the uncertainty distribution of the CC.

Reliability analysis of these structures have shown that the importance of modelling the collapsecapacity as a random variable is insignificant, which indicates that the median or the meancollapse capacity [ ]E CC can be used with good accuracy. In this analysis, data from the NorthSea were used for estimating the uncertainties in the sea state. The drag coefficient CD , the masscoefficient CM and the marine growth Mg were modelled as random variables, which reflects theuncertainties in prediction of extreme wave/current load condition for a given sea state. InSigurdsson et. al. (1994) and van de Graaf et. al (1994) it is shown that for [ ]CoV CC <01. , adeterministic description of the collapse capacity is suitable for quantification of the probabilityof collapse failure.

In the studies referred to above, only intact structures are considered. Similar simulation studieshave been performed for different damage-scenarios of the structures (results not published),indicating the same results as presented above.

The indication concerning the insignificance of modelling the collapse capacity CC as stochastic,is, however, based on study of a limited number of structures. More extensive studies on theimportance of modelling the randomness in the collapse capacity, including evaluation ofcorrelation between the load pattern and the collapse capacity, are needed for making a moregeneral conclusions in this subject.

4.5.2 Limit State FormulationBy relating the collapse capacity to the total base-shear (or the total over-turning moment) thecollapse capacity, for a given load direction θ , can be represented by a single random variableCC θ , and the annual extreme loading (i.e. the base-shear or the over-turning moment) can berepresented by a single random variable L θ . The limit state function for the reliability analysiscan then for load direction θ be expressed as,




( )g cc l cc lθ θ θ θ, = −

The collapse capacity CC θ , for a given load direction θ , can be expressed as

CC X X CCCC analysis CC model calcθ θ θ= ⋅ ⋅− −

where CCcalc θ is the calculated capacity, XCC analysis− is the model uncertainty, introduced inorder to account for the inconsistence in the analysis and XCC model− θ is the model uncertainty inthe calculated capacity, which is generally dependent on the geometry of the structure, materialproperties, the load pattern and the load direction, and other parameters.

The model uncertainty XCC model− θ can in general be assessed by comparing the calculation withresults obtained by experiments or advanced numerical analysis. Usually limited information isavailable for quantifying the model uncertainty and the uncertainty it is common to model thisuncertainty by a unconditioned Normal or Log-normal distributed stochastic variable.

The model uncertainty XCC analysis− can be assessed by considering the results obtained bydifferent engineers, considering the same structure and identical environmental criteria. It shouldbe noted that the analysis uncertainty include both the load and the response modelling, but isincluded here on the capacity model. The analysis uncertainty is function of the type of analysisundertaken. In DNV (1995a) Section 6, the analysis uncertainties for different type of analyses ofjack-ups and deep water floaters are outlined, and the CoV varies from about 10%-65%dependent on the complexity of the analysis. For collapse analysis of jackets, no data have beenfound in the literature and the results given in DNV (1995a) Section 6, can not be applieddirectly for jacket structures. As an indication, however, a CoV in the order of 10%~20% isreasonable.

The calculated capacity CCcalc θ is inherently stochastic due to uncertainties in material andgeometric properties. As discussed above, the calculated capacity may, however, for mostpractical purposes be assumed deterministic. The functional relation of the CCcalc θ and thewave/current profile applied for load calculation in the push-over analysis must be considered foreach case.

The annual extreme base-shear loading acting of the structure, L θ , can be expressed as;

L X L L X LL jack jack wind L deck deckθ θ θ θ= ⋅ + + ⋅− −

Ljack θ is the calculated hydrodynamic loading with associated model uncertainty X L jack− θ .Lwind θ is the wind loading. Ldeck θ is the calculated hydrodynamic loading on the deck structurewith associated model uncertainty X L deck− θ , only applicable when the wave hit the deck.

Ljack θ and Ldeck θ are stochastic due to randomness (aleatory uncertainties) in the sea state forgiven load direction θ , i.e. the wave height, the wave period, current speed and uncertainties(epistemic) in the hydrodynamic parameters applied in the load calculations.

The model uncertainties X L jack− θand X L deck− θ are introduced to account for uncertainties inthe applied models for hydrodynamic loading for a given environmental condition, which can ingeneral be assessed by comparison with results obtained by experiments or advanced numerical




analyses. Modelling of these uncertainties are highly dependent on the theoretical models and theuncertainty level included in Ljack θ and Ldeck θ .

A number of studies are available concerning the load prediction accuracy for given wavecondition, e.g. Haring et. al. (1979), Nerzic and Lebas (1988), Heideman and Weaver (1992),Elzinga and Tromans (1992). The results from these studies vary significantly, both in bias andscatter, e.g. the CoV varies in the range of 10%-35%. The predictions are based on differentwave theories with different choices of hydrodynamic coefficients, and some studies do notaccount for possible current loading. The main conclusion that can be drawn is, however, thatconsiderable uncertainty seems to be related to the load prediction.

In Nerzic and Lebas (1988), it has been demonstrated that the uncertainty in load prediction isnot independent of the calculated loading (the CoV decrease with increasing wave height).

Haver (1995) presents a review and a discussion of some available information aboutuncertainties in load and response estimates for jackets. The main conclusion is that uncertaintiesin the hydrodynamic forces are completely governed by the inherent randomness (aleatoryuncertainties) of the annual largest wave. It is therefore crucial that this quantity is properlymodelled. Concerning uncertainties being more of an epistemic nature, priority should be givento reduce the uncertainties related to the description of the wave conditions. Thereafteruncertainties related to the load calculation procedure could be attacked. In the study by Haver(1995), the model uncertainties are assumed to be unbiased and the impacts of possible bias isnot considered in the evaluation.

In Puskar et.al. (1994), a comparison of the calculated and observed platform damage during thehurricane Andrew in the Gulf of Mexico on August 24th-26th. 1992 is performed. The calculatedand the “true” ratio of capacity to load are compared, i.e. the model uncertainties in the capacityXCC model− and the jacket loading X L jack− are represented by a single bias factor. The conclusions

from this study, based on an evaluation of 13 platforms, indicated a bias factor with mean valuein the range of 1.1 - 1.2 and CoV of 0.1.

It should be noted that the model uncertainty depends on the procedures applied for calculatingthe wave force and the ultimate capacity, and the platform design. The platforms studied byPuskar et al. (1994) had failure mechanisms associated primarily with K-joints and thefoundation (pile hinging and pile plunging), and the results may differ for platforms with otherfailure modes.

The failure probability of the structure may be derived by considering a failure in each specifiedwave direction (θi diri N, =1� ) as a separate failure event with failure mode { }L CCi iθ θ≥ . Theannual system failure probability may be obtained by calculating the union of all the failureevents, i.e. a series system, given by

{ } { } { }P P L CC L CC L CCf annual N dir N dir≅ ≥ ≥ ≥�

��

��θ θ θ θ θ θ1 1 2 2� ��

As discussed above, the uncertainties in the environmental prediction usually dominating in thereliability analysis. When the difference between the considered environmental directions areconsiderable, say 45o or more, the uncertainties for the different directions can be assumed un-correlated. For small failure probabilities the total failure probability may be approximated as thesum of the individual failure probabilities for each wave directions,




P P Pf f i ii N

annual annualdir

≅ ⋅�=

( | ) ( ),

θ θ1

where ( )P θ is the probability of the load direction θ and Pf annualθ is the conditional annual

probability of collapse failure given the load direction θ , which is obtained by

{ } [ ] ( )

( )

P P L CC F f d

F f d

f L CC

CC L

annualθ θ θ= ≥ = −�

= �

1 Θ Θ

Θ Θ

( )

( )

x x x

x x x

X

X

where X is vector of the stochastic variables going into the reliability model, ( )FL Θ ⋅ and

( )FCC Θ ⋅ are the conditional cumulative annual probability distributions of the load and the

collapse capacity given the load direction Θ , respectively, and ( )f L Θ ⋅ and ( )fCC Θ ⋅ are the

corresponding probability density functions.

For a small Pfannual, the total system failure probability Pf over a given time nlife (years) can be

estimated as,

P n Pf life fannual≈ ⋅

An overview over a procedure for probability analysis of structural collapse is shown in Figure4.11.




PROBAN

StructuralData

Joint EnvironmentalModel (HS ,TP ,Cur,...)

Uncertainties(H,T,Cd,Mg,Y,...)

Non-Linear Push-over Analysis

(E.G.: USFOS)

HydrodynamicLoading

(E.G.: WAJAC)

Results :• PF• Sensitivities

Limit StateCapacity - Load

Only one call for each wave direction, when theCollapse Capacity is modeled as deterministic

Figure 4.11 Probability analysis of structural collapse - Overview




4.5.3 Distribution of the Annual Maximum Loading (Base-Shear)In order to establish the probability distribution of the annual maximum base-shear load FL

acting on a given structure, a proper description of the environmental situation is needed.Quantities to be determined are the long term variations of sea state characteristics, the short termdescription of the environmental condition (wave, current, wind etc.) within a given sea state,and the load model for the base shear loading.

The joint environmental model described in DNV (1995a) Section 5 is recommended for thelong term modelling of the environmental conditions. In the following a brief description of thismodel is presented.

The following environmental parameters are included:

• 1-hour mean wind speed, Uw

• specified wind and main wave directions, Θ (assumed to be identical)

• current speed (collinear with wind and waves), Vc

• significant wave height, Hs

• characteristic spectral period, e.g. the spectral peak period Tp or the mean zero crossingperiod Tz

• sea water level, D

The long term description of the environmental conditions is thus given by the joint probabilitydensity function

( ) ( ) ( )f h t v u d f h t v u d fH T V U D s z c w H T V U D s z c ws z c w s z c wΘ Θ Θ, , , , , , , , ,θ θ θ= ⋅

The direction variable Θ is conveniently divided into a defined number Ndir of sectors and isdescribed by the corresponding probability mass function ( )p i Ni dirΘ θ , , ,=1� . Furthermore, thedescription for each sector is establish by factorising the joint distribution as follows,

( )f h t v u dH T V U D s z c ws z c w Θ , , , , θ =

( ) ( ) ( ) ( ) ( )f t h f v h f u h f d h f hT H z s V H c s U H w s D H s H sz s c s w s s sΘ Θ Θ Θ Θ, , , ,θ θ θ θ θ⋅ ⋅ ⋅ ⋅

i.e., given Hs and Θ , the random variables Tz , Vc , Uw and D are assumed to be mutuallyindependent.

There is no theoretical preference when it comes to deciding on probabilistic models for thevarious conditional density functions. The respective choices have therefore been based on anempirical basis. In DNV (1995a) a discussion of these probabilistic models is given.

The annual largest loading is assumed to occur when the largest wave (or wave crest) in thelargest annual storm, i.e. annual largest Hs , passes the structure. Under a Poissonian assumptionfor rare events, the distribution of the annual largest significant wave height, Hs max, θ , for a




given load direction θ , may be derived from the long term distribution of the arbitrary significantwave heights as,

( ) ( )[ ]F h F hH s H s

N

s s

storm

, max Θ Θθ θ=

where Nstorm is the number of annual storms.

However, in the load calculation the distribution of the largest wave height or crest height, andthe corresponding wave period are needed. Different models can be found in the literature, forpredicting the wave height distribution for a stationary sea state, see e.g. Haring (1976), Forristall(1978), Næss (1983), Krogstad (1985), Vinje (1988).

Knowing the conditioned wave height distribution, ( )F h hH H ss Θ ,θ , the distribution of the

largest wave height out of Nwave wave cycles, H hsmax ,θ , may be obtained as,

( ) ( )[ ]F h h F h hH H s H H s

N

s s

wave

max, ,Θ Θθ θ=

The number of waves Nwave in a sea state is in general a stochastic variable conditioned on thesignificant wave height and can be obtained as.

NT hwave

z s

=τ

where τ is the duration of the storm (e.g. 6 hours or 21600 sec.) and T hz s is the mean zerocrossing period conditioned on the significant wave height.

Forristall (1978) and Krogstad (1985) propose Weibull distribution for the wave heights of astationary sea state i.e.

( )F h hhhH H s

ss Θ , expθ δ

α

= − − ⋅�

��

�

��

�

�

�

�

1

where α = 213. and δ=2 26. (Krogstad (1985) propose δ=2 28. ). For this case the distribution ofthe largest wave height in the largest storm may be obtained as,

( )F h h Nh

hH H s waves

smax , max,max

,max

, exp expΘ

θ δ

α

= − ⋅ − ⋅�

�

��

�

�

��

�

�

�

��

��

��

�

��

��

Concerning wave-deck impact loads, the crest height of the waves becomes an importantparameter. In this case it appears that the crest height should be selected as the primary wavecharacteristic rather than the wave height. Haver (1995) discusses this problem and points outthat applicable models are available also for predicting non-Gaussian crest heights. However, atthe present some difficulties arise when the non-Gaussian crest height are associated with aproper wave profile. Fitting a Stokes 5th order wave to this crest leads to considerableoverestimation of the wave height, e.g. for North Sea location, the 100-years wave heightbecomes nearly 2m higher than the 100-years wave height obtained directly from the wave heightstatistics.




It is therefore recommended to apply the wave height as a primary wave characteristic andcombine this with a 5th order Stokian wave profile. The corresponding wave period may beobtained as,

TT

T

T

TH

p

z

p

z

max

.=

⋅

⋅

�

��

��

0 9or

1.2

when isapplied asa characteristic spectral period

when isapplied asa characteristic spectral period

Knowing the long term joint distribution of the environmental conditions, the conditional waveheight distribution and the uncertainties in the load calculation procedures (either global modeluncertainty parameters or uncertainties in the basic parameters as the hydrodynamic parametersetc.), the annual largest long term wave load distribution can be establish by combining a generalprobability analysis program (e.g. SESAM:PROBAN, DNV (1993a)) and a wave/current loadingprogram (e.g. SESAM:WAJAC, DNV (1992a)).

An alternative and much more efficient reliability procedure is to establish, once and for all, aresponse surface defining the total base-shear loading for each load direction θ as function ofcharacteristic parameters. The response surface gives a functional relationship between the totalbase-shear and a set of defined variables. E.g., a 6-dimensional response surfaces could bedefined as a function of hmax (5th order Stokes wave), tHmax

, vc , cD , cM and the thickness ofthe marine growth M g . The response surface is linked to the PROBAN application in order tocalculate the total base-shear loading for a given outcome of the stochastic variables.


DNV Report No. 95-3203 Summary of Application Examples


5. SUMMARY OF APPLICATION EXAMPLES

5.1 Summary of Fatigue Failure Limit State - FLS Example

5.1.1 Modelling ApproachAs discussed in Guideline for Offshore Structural Reliability Analysis - General (DNV 1995a), aprobabilistic fatigue analysis may be divided into the following four steps:

1. Probabilistic modelling of the environment (short-term and long-term modelling)

2. Probabilistic modelling of the wave loading

3. Stochastic assessment of the structural response (global and local)

4. Stochastic assessment of the fatigue damage accumulation.

In addition to the above steps, a stochastic modelling of the fatigue capacity is required. Theprobability against fatigue failure is obtained through a probabilistic evaluation of the likelihoodof the event that the accumulated fatigue damage exceeds the defined critical fatigue capacity.

In order to carry out a realistic fatigue evaluation of a jackets structure, it is necessary tointroduce some simplifying assumptions in the modelling. These assumptions consist of;

• For a short term period (a few hours) the sea surface can be considered as a realisation of azero-mean stationary Gaussian process. The sea surface elevation is (completely)characterised by the frequency spectrum, which for a given direction of wave propagation, canbe described by two parameters, the significant wave height HS and some characteristicperiod like the spectral peak period TP or the mean zero-mean up-crossing period TZ .

• The long term probability distribution of the sea state parameters ( H TS P− or H TS Z−diagram) is known.

• A frequency domain analysis is adequate. Applying a frequency domain approach forassessing the structural response, the wave loading on structural members must be linearisedand the structural stress response must be assumed to be a linear function of the loading, i.e.the structural and material models are assumed linear.

• The relationship between the sectional forces and the local hot-spot stresses (SCFs) is known.For this purpose, an empirical parametric description is most common.

The influence and consequence of the following modelling aspects are discussed in theapplication example;

• The effect of applying different wave spectra, i.e. PM and JONSWAP spectra.

• Influence of wave spreading on the estimated fatigue capacity.

• The effect of the linearisation of the wave loading. This can for some structures be ofsignificance. A study investigating the influence of performing the linearisation at differentsea-states is carried out. This study is based on a stochastic linearisation techniques for threedifferent sea states.

• Two different parametric equations for calculation the SCFs are compared.




• The long-term stress range distribution is defined through a calibrated Weibull distribution.

In order to account for the outcome of the structural inspections in the estimation of the fatiguereliability of the structure, the fatigue capacity needs to be evaluated applying the FractureMechanics (FM) fatigue approach. A procedure for calibrating the FM-fatigue model to the SN-fatigue model is further presented in the application example.

5.1.2 Discussion of ResultsThe selected North Sea jacket structure that is analysed for the FLS study is an existing NorthSea structure located at 107 m water depth. The choice of this particular structure is motivatedfrom the degree of structural redundancy, believed to be a typical characteristic for North Seajackets.

In order to identify the most fatigue sensitive structural elements, a frequency domain SN-fatigueanalysis (stochastic fatigue analysis) is performed.

The fatigue lives for the different members in one of the most critical joints in the jacketstructure, joint 589, is considered for the comparison analysis, see Table 5.2.

Table 5.2. Selected joint with associated members to be evaluated for the fatigue lifecomparison analysis of the North Sea jacket structure.

Joint Number 589

Member Number 123 152 372 373 401 402

Member diameter (m) Chord

Brace

3.50

0.90

3.50

1.00

3.50

1.40

3.50

1.40

3.50

1.10

3.50

1.30

Member thick. (mm) Chord

Brace

65.0

25.0

65.0

45.0

65.0

40.0

65.0

45.0

65.0

30.0

65.0

60.0

Joint type KTT YT KTK KTK KTK KTK

The base case for the fatigue analysis is the use of the transfer functions obtained applying adynamic analysis, the parametric equations proposed by Efthyminu for deriving the SCFs, thePM sea spectrum, and the assumption of long crested (uni-directional) sea.

The fatigue results obtained from the base case are compared with results from equivalent fatigueanalyses where different common modelling alternatives are considered. The followingvariations are compared; the use of a quasi-static approach for deriving the transfer functions, theuse of Kuang's model for deriving the SCFs, the use of a JONSWAP sea spectrum, and theinfluence of modelling different degrees of short crested sea.




The quasi-static and dynamic transfer functions as well as the SCF’s are calculated usingSESAM.

Based on the results from the analysis, the following conclusions are made:

• The study shows that insignificant differences in the calculated fatigue lives are obtained fordifferent selections of linearisation sea state, indicating that the loading on the consideredNorth Sea structure is dominated by linear inertia forces.

• The approximation of assuming narrow banded stress response is acceptable.

• A dynamic and a quasi-static approach for deriving the transfer functions are compared. It isobserved that applying a quasi-static approach, longer derived fatigue lives are obtained thanfor the dynamic approach (by a factor 1.3-4.0).

• The fatigue lives obtained applying the Efthyminu empirical model and the Kuang model forderiving the SCFs are compared. Longer fatigue lives are obtained applying the Kuang model(by a factor of 2.0-7.0).

• The fatigue lives obtained applying the Pierson-Moskowitz wave spectrum and theJONSWAP wave spectrum are compared. Only a minor increase in the estimated fatigue livesare observed using the JONSWAP wave spectrum compared to the PM spectrum.

• It is further observed that the estimated fatigue lives are increasing with the level of wavespreading, but only to a minor degree.

• Based on the obtained stress range distribution within each sea state, a long term stress rangedistribution is established and approximated to a Weibull distribution. The obtained fatigueresults applying the calibrated long term Weibull distribution for describing the stress rangesmatched the original obtained fatigue results over the service life.

• For the probabilistic fatigue analysis using the SN-approach, large influence of the uncertaintyassociated with the modelling of the SN-curve capacity and the modelling of the SCF factorsare identified.

• In order to carry out probabilistic inspection updating, it is necessary to express the fatiguereliability of the joints through a FM-approach. The example application study shows that it ispossible to give a good description of the fatigue reliability obtained from the SN-fatigueapproach applying the FM-fatigue approach when the crack initiation time is assumed to besmall.

• The probabilistic evaluation applying the FM-approach shows that the uncertainty associatedwith the modelling of the local SCFs, the geometry function and the material parameter C inthe FM model have a large influence on the uncertainty modelling.

• For probabilistic inspection updating, the inspection accuracy of the last inspection has a largeinfluence on the estimated updated reliability level.

5.2 Summary of Total Collapse Limit State - ULS Example

5.2.1 Modelling ApproachIn reliability analyses of jacket structures for structural collapse (ULS), the uncertaintiesassociated with the determination of the hydrodynamic loading are generally much greater than




for the estimation of the collapse capacity. As the problem formulation is load driven, a propermodelling of the hydrodynamic loading is therefore important.

The limit state function applied in the reliability analysis, is expressed as,

( )g cc l cc lθ θ θ θ, = −

where CC θ is the jacket collapse capacity measured as the total base-shear capacity of thestructure and L θ is the annual extreme base-shear loading, for a given load direction θ .

In the example application, the jacket collapse capacity is derived by performing a non-linearpush-over analysis with the computer program USFOS (USFOS (1996)). The hydrodynamicloads on the jacket are calculated by WAJAC (DNV (1992a)).

Modelling and physical uncertainties have been accounted for both in the capacity modelling andin the environmental description and hydrodynamic loading. See the Example Application report(DNV 1995b) for a more detailed description of the uncertainty modelling.

5.2.2 Discussion of ResultsThe selected North Sea jacket structure that is analysed for the ULS study is the same as for theFLS study, being an existing North Sea structure located at 107 m water depth.

The study shows that the uncertainty in the ultimate capacity model contributes to less than 15%of the total uncertainty, and that the major portion of the total uncertainty is associated with theenvironmental load description (~80%), where the inherent uncertainty in the wave height is thedominating parameter in the reliability analysis.

At design point, the wind loading contributes to less than 2% of the total loading and may forprobabilistic analyses of jacket structures be considered as deterministic. For the hydrodynamicloading, the wave loading contributed with about 75%, and the current loading with theremaining 25%.

A parameter study is carried out in order to investigate different modelling approaches on theestimated failure probability for total collapse. The results of the parameter study can besummarised as follows:

• CASE-1: Effect of uncertainties on Tz and Vc

Only the NW load direction is considered in the study. For given outcome of Hs ,the Tz and Vc are modelled as deterministic.

Results : For given outcome of Hs , the Tz and the Vc can be modelled asdeterministic.

• CASE-2: Effect of the model uncertainty in the calculated hydrodynamic loading X L jack− .

Variations in the mean value of X L jack− in the range of 0.9-1.1 and the CoV in therange of 0.0-0.4 are studied, considering both a Normal distribution and a Log-Normal distribution.




Result: The choice of distribution for X L jack− is not significant. It is also shownthat a 10% change in the mean value (bias) changes the failureprobability by approximately a factor of two.

• CASE-3: Effect of model uncertainty in the calculated collapse capacity XCC model− .

Variation in the mean value of XCC model− in range of 0.9-1.1 and the CoV in therange of 0.0-0.4 are studied, where XCC model− is modelled as Normal distributed.

Results: For CoV < 0.2, the effect of model uncertainty in the calculated collapsecapacity for the failure probability is similar as for X L jack− , but for largerCoV ( > 0.2) the estimated failure probability increases dramatically.

• CASE-4: Effect of ignoring the relationship between the capacity and the wave height andperiod.

The calculated collapse capacity is obtained by applying a 100 years wave conditionand the median value of the wave period.

Results : The 100 years wave condition can be applied in the push-over analysis inorder to obtain the calculated collapse capacity, and the relationshipbetween the capacity and the wave height and period can be omitted.

• CASE-5: Effect of ignoring the model uncertainties X L jack− and XCC model− in the reliability modelling.

Results : Ignoring the model uncertainties in the reliability model will lead tounderestimation of the failure probability by a factor of approximatelyseven.

It should be noted that the results presented in the case studies above are in general only valid forthe current example, but it is expected that the results will be similar also for other North Seajackets.


DNV Report No. 95-3203 References


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Documents

Guideline for Offshore Structural Reliability - DNVresearch.dnv.com/skj/OffGuide/Jacket_Application.pdf · A guideline for offshore structural reliability analysis of ... strength